University of Southern California Dissertations and Theses

Pulsatile and steady flow in central veins and its impact on right heart-brain hemodynamic coupling in health and disease

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Pulsatile and steady flow in central veins and its impact on right heart-brain hemodynamic coupling in health and disease

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Content Pulsatile and Steady Flow in Central Veins and Its Impact on Right
Heart-Brain Hemodynamic Coupling in Health and Disease
By
Heng Wei
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfilment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
AUGUST 2024
Copyright 2024 Heng Wei

ii
Dedication
To my mother, my wife and to my father,
who selflessly love me without expecting anything in return.

iii
Acknowledgements
First and foremost, I express my deepest gratitude to my PhD advisor, Professor Niema
Pahlevan. His continuous support, wisdom, encouragement, sage advice, and expert guidance were
invaluable to my PhD training and achievements.
I am particularly grateful to my clinical collaborators, Dr. Andrew Cheng and Dr. Cynthia
Herrington from CHLA, whose clinical insights and guidance were essential to this thesis. I also
appreciate the help and guidance of my committee members: Professor John Wood, Professor Paul
Newton, Professor Satwindar Sadhal, and Professor Michael Khoo. My deepest appreciation goes
to my collaborators: Professor Faisal Amlani (Université Paris-Saclay), Dr. Hossein Gorji (Swiss
Federal Laboratories), Professor Paul Ronney (USC), Professor David Hutchins (USC), Professor
Matthew Gilpin (USC), Dr. Jing Liu (USC), and Dr. Seth Wieman (USC Viterbi Machine Shop).
I also thank my friends and lab-mates in Professor Pahlevan’s lab: Arian Aghilinejad and
Rashid Avali, with whom I closely worked as the first generation of Pahlevan’s lab members;
Coskun Bilgi, Soha Niroumandi, Deniz Rafieianzab, Elena Lottich, Mahmood Alfayoumi, Haojie
Geng, Jianjun Li, and Haonan Meng. I deeply appreciate their support and friendship.
Lastly, words cannot express my gratitude to my family, especially my father Yongtie Wei,
my mother Qi Zang, and my wife Jianing Yu, for their patience, steadfast support, and constant
inspiration. I owe all my accomplishments to their unwavering support and deep love.

iv
Table of Contents
Dedication ...................................................................................................................................... ii
Acknowledgements ......................................................................................................................iii
List of Tables................................................................................................................................ xi
List of Figures............................................................................................................................. xiii
Nomenclature ............................................................................................................................ xxx
Abstract.................................................................................................................................... xxxiv
Chapter 1: Introduction and Motivation ................................................................................... 1
1.1 Background ........................................................................................................................... 1
1.2 Thesis objectives ................................................................................................................... 6
1.3 Thesis outline ........................................................................................................................ 8
Chapter 2: Direct 0D-3D coupling of a lattice Boltzmann methodology for fluid–structure 11
hemodynamic flow simulations.................................................................................................... 11
2.1 Chapter Introduction ........................................................................................................... 12
2.2 Materials and Methods........................................................................................................ 14
2.2.1 Governing formulations................................................................................................ 14
2.2.2 3D lattice Boltzmann equations.................................................................................... 15
2.2.3 2D elastic wall equations.............................................................................................. 17
2.2.4 3D elastic wall equations.............................................................................................. 17
2.2.5 Implementations of the 0D Boundary Conditions: Varying-elastance and lumped 20
parameter models....................................................................................................... 20
2.2.6 A direct 0D-3D coupling for ODE-based boundary equations and lattice Boltzmann 22
solvers........................................................................................................................ 22
2.2.7 On the particularities of the specific hybrid ODE-Dirichlet LV model ....................... 24
2.2.8 Algorithmic details....................................................................................................... 26
2.2.9 Fluid–structure interactions (FSI)................................................................................. 27
2.2.10 Efficient boundary condition-enforced immersed boundary method......................... 29
2.3 Validation of the 0D-3D framework on Left-heart ............................................................. 32

v
2.3.1 Introduction of the validation ....................................................................................... 32
2.3.2 The complete FSI solver coupled with ODE-based heart model (time varying 33
elastance model)........................................................................................................ 33
2.3.3 An example physiological case: wall shear stress in the aorta..................................... 35
Chapter 3: 3D Numerical investigation of the significance of wall stiffness and non- 41
Newtonian effects on jugular vein pulsatile hemodynamics ........................................................ 41
3.1 Chapter introduction............................................................................................................ 42
3.2 Methodology and material................................................................................................... 44
3.2.1 Boundary switching procedure (tricuspid valve modeling): ........................................ 45
3.3 Computational Modeling..................................................................................................... 46
3.3.1 Right Heart-Internal Jugular Vein setup (IJV) model .................................................. 46
3.4 Application and Computational Results.............................................................................. 47
3.4.1 0D-3D model of IJV-Right Heart model...................................................................... 47
3.4.2 Non-Newtonian effects in IJV flow dynamics............................................................. 53
3.4.3 3D IJV modeling using the 0D-3D right-heart-brain coupled FSI Framework ........... 55
3.5 Discussion ........................................................................................................................... 65
3.6 Conclusion........................................................................................................................... 68
3.7 Clinical and Research Implications..................................................................................... 68
Chapter 4: Methodological development for 3D simulations of solid fluid wave interactions 70
for jugular vein right heart coupling ............................................................................................. 70
4.1 Chapter introduction............................................................................................................ 70
4.2 Fourier continuation methods.............................................................................................. 72
4.2.1 Spatial discretization of Fourier continuation .............................................................. 72
4.2.2 Accelerated Fourier continuation: FC(Gram) .............................................................. 73
4.3 Some new FC operators for high-order PDEs..................................................................... 75
4.3.1 Temporal discretization................................................................................................ 79
4.4 Applications to flexible filaments ....................................................................................... 80
4.4.1 Hanging filament under gravity.................................................................................... 80

vi
4.4.2 Convergence study ....................................................................................................... 82
4.4.3 Performance study ........................................................................................................ 84
4.5 Conclusions......................................................................................................................... 86
Chapter 5: Experimental investigation of Pulsatile Flow in Physiologically Accurate 87
Collapsible Jugular Vein-right heart Model using Three-Dimensional particle tracking 87
velocimetry ............................................................................................................................... 87
5.1 Chapter introduction............................................................................................................ 87
5.2 In-vitro Hemodynamic Simulator for the Cerebral venous Circulation.............................. 90
5.2.1 Description of the in-vitro experimental setup............................................................. 91
5.2.2 Sketch of Experimental setup....................................................................................... 93
5.2.3 Physiologically accurate artificial organ fabrication.................................................... 96
5.2.4 Procedures and Measurements..................................................................................... 97
5.2.5 Compliance measurement ............................................................................................ 98
5.3 Particle Image Velocimetry............................................................................................... 100
5.3.1 Indexed matching ....................................................................................................... 102
5.3.2 Volumetric 3D PIV Calibration.................................................................................. 104
5.4 Experimental conditions and ‘Reynolds Number’ Calculation:........................................ 106
5.4.1 Reynolds Number:...................................................................................................... 106
5.5 Hemodynamic quantification and analysis: ...................................................................... 106
5.6 IJV Waveform Pressure measurement .............................................................................. 108
5.7 2D experimental results..................................................................................................... 109
5.7.1 2D-PIV particle + vector: In and out of plane motion for vector field....................... 109
5.7.2 3D-PIV particle identification .................................................................................... 110
5.7.3 Comparison of 3D-PIV and 3D-CFD......................................................................... 110
5.8 3D experimental results..................................................................................................... 112
5.8.1 Vector plot:................................................................................................................. 113
5.8.2 Velocity & Vorticity plot: .......................................................................................... 116

vii
5.8.3 Sample IJV pressure waveform:................................................................................. 120
5.8.4 Reversed Flow Index & pressure waveform: ............................................................. 121
5.8.5 Reversed Flow Index & Volume change.................................................................... 121
5.8.6 Master plot 1-RFI Q V.S. Frequency ......................................................................... 123
5.8.7 Master plot 2-RFI Stiffness........................................................................................ 124
5.8.8 Reversed Flow Index & Vortex wave ........................................................................ 125
5.8.9 Master plot 3-Vorticity............................................................................................... 126
5.9 Discussion ......................................................................................................................... 128
5.10 Clinical importance ......................................................................................................... 133
5.11 Conclusion....................................................................................................................... 134
Chapter 6: The global effect of the right heart dynamics on jugular vein wave dynamics 136
using 1D hemodynamic modeling of the entire circulatory system ........................................... 136
6.1 Chapter introduction.......................................................................................................... 136
6.2 Method and Governing equations ..................................................................................... 140
6.2.1 1D (2D-axisymmetric) fluid-structure dynamics in a vessel segment ....................... 140
6.2.2 0D lumped parameter modeling ................................................................................. 142
6.2.3 Numerical methodology and computational approach............................................... 146
6.2.4 System parameters/coefficients.................................................................................. 149
6.3 Results and discussion....................................................................................................... 151
6.3.1 Baseline characteristics .............................................................................................. 152
6.3.2 Isolated effects of vessel stiffness .............................................................................. 153
6.3.3 Effects of heart rate .................................................................................................... 154
6.3.4 Effects of heart contractility ....................................................................................... 156
6.3.5 Effects of tricuspid valve conditions.......................................................................... 158
6.3.6 Effects of Fontan circulation ...................................................................................... 160
6.4 Conclusion......................................................................................................................... 164

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Chapter 7: The significance of steady blood flow shear-rate-dependency in modeling of 166
Fontan hemodynamics................................................................................................................ 166
7.1 Introduction of non-Newtonian effect in human venous system ...................................... 167
7.2. Method ............................................................................................................................. 170
7.2.1 Flow solver................................................................................................................. 170
7.2.2 Fluid viscosity models: non-Newtonian vs. Newtonian............................................. 172
7.2.3. LBM Algorithm......................................................................................................... 174
7.2.4 Fontan hydraulic circuit model................................................................................... 175
7.2.5 Boundary conditions................................................................................................... 176
7.2.6 Numerical simulations................................................................................................ 176
7.3 Hemodynamic analysis ..................................................................................................... 176
7.3.1 Shear stress:................................................................................................................ 176
7.3.2 Power Loss: ................................................................................................................ 177
7.3.3 Viscous Dissipation:................................................................................................... 177
7.3.4 Non-Newtonian Importance Factor:........................................................................... 178
7.3.5 The pulmonary blood flow distribution:..................................................................... 179
7.4 Results............................................................................................................................... 179
7.4.1 Qualitative flow patterns ............................................................................................ 181
7.4.2 Quantitative analysis .................................................................................................. 184
7.5 Discussion ......................................................................................................................... 194
7.5.1 Limitations.................................................................................................................. 197
7.5.2 Clinical implication .................................................................................................... 198
7.6 Conclusions....................................................................................................................... 199
Chapter 8: The Impact of Blood Viscosity Modeling on Computational Fluid Dynamic 201
Simulations of Pediatric Patients with Fontan Circulation......................................................... 201
8.1 Chapter introduction.......................................................................................................... 202
8.2 Methods............................................................................................................................. 204

ix
8.2.1 Patient-specific Geometry .......................................................................................... 204
8.3 Numerical Methods........................................................................................................... 205
8.3.1 3D Lattice Boltzmann Equations................................................................................ 205
8.3.2 Non-Newtonian Fluid Modeling ................................................................................ 206
8.3.3 Fluid–Structure Interactions (FSI).............................................................................. 208
8.3.4 Concentration Field .................................................................................................... 209
8.3.5 Simulation conditions................................................................................................. 210
8.4 Hemodynamic Analysis and Metrics ................................................................................ 211
8.4.1 Power Loss ................................................................................................................. 211
8.4.2 Viscous Dissipation Rate............................................................................................ 211
8.4.3 Non-Newtonian Importance Factor............................................................................ 212
8.4.4 Hepatic Blood Flow Distribution ............................................................................... 212
8.5 Statistical Analysis............................................................................................................ 213
8.6 Results............................................................................................................................... 213
8.6.1 Demographics of Fontan Patients............................................................................... 213
8.6.2 Non-Newtonian Effects on Flow Patterns and Energy Loss...................................... 214
8.7 Discussion ......................................................................................................................... 218
8.8 Clinical Significance ......................................................................................................... 221
8.9 Limitations ........................................................................................................................ 222
8.10 Conclusions..................................................................................................................... 223
Chapter 9: Hemodynamically efficient artificial right atrium design for univentricular 225
heart patients ............................................................................................................................. 225
9.1 Chapter introduction.......................................................................................................... 227
9.2 Methods............................................................................................................................. 229
9.2.1 Flow solver................................................................................................................. 229
9.2.2 Solid solver................................................................................................................. 233
9.2.3 Hydraulic circuit configuration and dimensions for an artificial right atrium ........... 237

x
9.2.4 Shape-selection procedure for hemodynamic optimization ....................................... 237
9.2.5 Boundary Conditions.................................................................................................. 239
9.2.6 Numerical Simulations............................................................................................... 239
9.2.7 Hemodynamic analysis............................................................................................... 241
9.3 Results............................................................................................................................... 243
9.3.1 Hemodynamically optimized long-axis plane (LAP)................................................. 243
9.3.2 3D compliant artificial right atrium (ARA)................................................................ 243
9.3.3 Qualitative flow analysis............................................................................................ 246
9.3.4 Evaluation of the total volume compliance ................................................................ 250
9.4 Discussion ......................................................................................................................... 250
9.5 Limitations ........................................................................................................................ 255
9.6 Clinical implications ......................................................................................................... 256
9.7 Conclusions....................................................................................................................... 257
Appendix A: An immersed boundary-lattice Boltzmann method for porous media .................. 258
Appendix B: Supplementary Materials for Fontan..................................................................... 262
References................................................................................................................................... 268

xi
List of Tables
Table 4-1 Maximum L∞ errors (over all space and all time) and orders-of-convergence for
FC solver applied to problem derived from the manufactured solution of Eq. (4.20). The
timestep is chosen small enough so that errors are dominated by the spatial discretization. All
solutions are advanced to the same final time using 409,600 timesteps....................................... 84
Table 4-2 Comparison of maximum L∞ errors (over all space and time) as well as
computational times between FC and a second-order (central) finite difference (FD) method
[145] for resolving the solution given by Eq. (4.23). Discretizations are chosen for each so
that they achieve a similar error.................................................................................................... 86
Table 5-1. Dynamic and physiological properties of the fabricated IJV (C=0.1-0.2). ................. 99
Table 6-1 Heart model baseline parameters for the left atrium (LA), left ventricle (LV), right
atrium (RA), and right ventricle (RV), where Tper = HR/60 is the length of a cardiac cycle.
Together with the baseline heart rate of HR = 75 bpm, the corresponding nominal cardiac
output is CO = 5.9 L/min. ........................................................................................................... 150
Table 6-2 Baseline parameters for the venous segments of interest, adopted from [125].
Initial area A0 and initial pulse wave velocity c0 are the same throughout the vessel (no
tapering in each subsegment). Changes in stiffness for the studies of this chapter are affected
by modifying the pulse wave speed c0. All other segments of the entire circulation (394 in
total) are identical to those used in [125, 166]............................................................................ 150
Table 6-3 Normal and diseased parameters for the tricuspid valve employed in the
simulations of this chapter (Ko and Kc are in units of cm2
·s2
/g)................................................ 151
Table 7-1 Comparison of percent power loss between Newtonian and Non-Newtonian........... 185

xii
Table 7-2 Comparison of indexed power loss between Newtonian and Non-Newtonian.......... 186
Table 7-3 Comparison of Total Viscous Dissipation between Newtonian and NonNewtonian................................................................................................................................... 186
Table 7-4 Global Non-Newtonian Importance Factor (Zero for Newtonian cases). .................. 186
Table 7-5 Global Non-Newtonian Importance Factor without the central area. ........................ 187
Table 7-6 Comparison of pulmonary blood flow (difference in % flow to LPA) between
Newtonian and non-Newtonian models...................................................................................... 189
Table 8-1 Characteristics of patient cohort. SD: standard deviation, IQR: interquartile range.. 213
Table 9-1 The averaged PRT determined from 2D simulations of LAP1–LAP4. LAP1–
LAP4 have similar LAP areas. The T-junction configuration is the reference case for
minimum PRT. PRT: particle residence time (PRT); LAP: long-axis plane.............................. 243
Table 9-2 The average particle residence time (PRTave) for 3D compliant models. Case 1-3
have the same total Volume........................................................................................................ 246
Table 9-3 Compliance (C) (m3/Pa ∙ 10−10) and pressure (P) rise values for 3D compliant
models. Cases 1-3 have the same volume as the sphere. % C reduction is calculated as
CSphere Analytical−C
CSphere Analytical % P difference is calculated asP−PSphere Numerical
PSphere Numerical ....................................... 250

xiii
List of Figures
Figure 2.1 A representative illustration of the complete 3D computational domain defined by
Ω+∂Ω1+∂Ω2, where Ω denotes the fluid interior (governed by lattice Boltzmann equations),
∂Ω1 denotes the compliant solid boundary (governed by elastodynamics PDEs and
incorporated by any appropriate fluid–structure interaction algorithm), and ∂Ω2 denotes the
coupled 0D-3D boundary (governed by time dependent ODEs). Figure from Wei H et al.
International Journal for Numerical Methods in Biomedical Engineering (2023): e3683. .......... 15
Figure 2.2 An illustrative example diagram of the direct (explicit) coupling between a 0D
model (e.g., the ODEs corresponding to the LV-heart model and the Windkessel model) and
the 3D lattice Boltzmann (LB) model. The flow in ∂Ω2 is computed in terms of the LB
distribution functions at a timestep n which is fed into ODEs governing the 0D model. The
corresponding pressure produced by the 0D model is then re-translated into distribution
functions on the boundary via the non-equilibrium extrapolation................................................ 25
Figure 2.3 An illustration of the exponential-based C∞-function St,td, W which is employed
to smoothly reduce lattice Boltzmann velocity amplitudes as the valve closes in the 0D LVelastance model (i.e., the ODE-Dirichlet switch). ........................................................................ 26
Figure 2.4 A flowchart describing the implementation of the particular hybrid ODE-Dirichlet
0D LV-elastance model that is of interest..................................................................................... 27
Figure 2.5 A fluid–structure interaction (FSI) procedure facilitated by the immersed boundary
(IB) method................................................................................................................................... 31
Figure 2.6 Diagram of the simplified straight aorta test case coupled to the LV-elastance
model............................................................................................................................................. 33

xiv
Figure 2.7 (Left) Physiological pressure profiles at the inlet for successively-refined
discretizations of a straight aorta as produced by the LV-elastance model. (Right) The
corresponding L∞ errors (relative to the finest solution) .............................................................. 34
Figure 2.8 (Left) Flow profiles at the inlet for successively-refined discretizations of a
straight aorta as produced by the LV-elastance model. (Right) The corresponding L∞ errors
(relative to the finest solution)...................................................................................................... 34
Figure 2.9 (Left) Pressure profiles at the midpoint for successively-refined discretizations of
a straight aorta as produced by the LV elastance model. (Right) The corresponding L∞ errors
(relative to the finest solution)...................................................................................................... 35
Figure 2.10 (Left) Diagram of a physiologically relevant 3D aortic domain (with carotid and
renal branches) coupled to an LV-elastance model at the aortic inlet and a lumped-parameter
Windkessel model at the aortic outlet. (Right) A temporal snapshot of the normalized flow
velocity magnitude produced by the solver. Figure from Wei H et al. International Journal
for Numerical Methods in Biomedical Engineering (2023): e3683. ............................................ 37
Figure 2.11 Aortic pressure at the inlet (blue) and the corresponding ventricular pressure
(dashed red) for healthy patient parameters, simulated by the 3D FSI solver. As expected,
aortic inlet pressure is equal to LV pressure during the systolic phase (when the valve is
opened).......................................................................................................................................... 38
Figure 2.12 (left) Pressure profiles at the inlet, midpoint and outlet of the 3D aorta model,
demonstrating the expected amplification as flow propagates downstream. (Right)
Corresponding flow profiles simulated at the inlet, midpoint and outlet, demonstrating the
expected decrease in flow amplitude as the wave propagates downstream.................................. 39

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Figure 2.13 (Left) Experimental data from an in vitro LV-aortic simulator. (Right)
Experimental flow data (of a different run) from the same setup. The overall morphologies
and physiological ranges are in agreement with those of the simulations presented in this work.
....................................................................................................................................................... 39
Figure 2.14 Simulated wall shear stress (WSS), for both normal and high contractility cases,
at a location between the midpoint and outlet of the 3D aorta model. ......................................... 40
Figure 3.1. Diagram of the simplified internal jugular vein system coupled to the RA-RVelastance model............................................................................................................................. 46
Figure 3.2. Sample waveform of RAP and RVP. Figure from [127, 128]. .................................. 47
Figure 3.3. IJV waveform and its physiological meaning behind it [129, 130]. .......................... 48
Figure 3.4. IJV waveform trace [131] (Source: Gray’s Anatomy, Fig. 558.), [132, 133]. ........... 49
Figure 3.5. Pressure waveform in RA and IJV for different conditions from the 0D-3D model. 50
Figure 3.6. Wave forms from previous 0D-heart model [134]..................................................... 51
Figure 3.7. Flow (Ux/Uref, Uref = 0.1 m/s) and pressure (mmHg) distribution for Eh =
1.0, 1.5, 3.0 Eh0. ........................................................................................................................... 52
Figure 3.8. Flow (Ux/Uref, Uref = 0.1 m/s) and pressure (mmHg) distribution for Eh =
1.0Eh0 HR = 120, and Low Ea case with HR = 60. .................................................................. 52
Figure 3.9. Pressure propagation for both the Newtonian and non-Newtonian scenarios and
Non-Newtonian importance factor in IJV flow field.................................................................... 54
Figure 3.10 Inlet (heart-side) Pressure waveform and Internal Jugular Venous Pressure (JVP)
waveform for stiff IJV with Newtonian blood model................................................................... 57
Figure 3.11 Inlet (heart-side) Pressure waveform and Internal Jugular Venous Pressure (JVP)
waveform for stiff IJV with non-Newtonian blood model. .......................................................... 57

xvi
Figure 3.12 Snapshot of Flow in Internal Jugular Vein (IJV) for stiff IJV model with
Newtonian (left) and non-Newtonian (right) blood models. Time at T=0.3, 0.5 and 0.8
Cardiac Cycle................................................................................................................................ 59
Figure 3.13 Inlet (heart-side) Pressure waveform and Internal Jugular Venous Pressure (JVP)
waveform for soft IJV (half of the stiffness Eh) with Newtonian blood model. .......................... 60
Figure 3.14 Inlet (heart-side) Pressure waveform and Internal Jugular Venous Pressure (JVP)
waveform for soft IJV (half of the stiffness Eh) with non-Newtonian blood model.................... 61
Figure 3.15 Snapshost of Flow in Internal Jugular Vein (IJV) for soft IJV model with
Newtonian (left) and non-Newtonian (right) blood models. Time at T=0.3, 0.5 and 0.8 Cardiac
Cycle. ............................................................................................................................................ 62
Figure 3.16 Inlet (heart-side) Pressure waveform and Internal Jugular Venous Pressure (JVP)
waveform for stiff IJV (left), soft IJV (right) with Newtonian blood model under elevated
RAP Condition.............................................................................................................................. 63
Figure 3.17 Snapshots of Flow in Internal Jugular Vein (IJV) for high RAP condition with
stiff IJV model (left) and soft IJV model (right). Time at T=0.5 Cardiac Cycle.......................... 64
Figure 4.1 An example Fourier continuation of a non-periodic function [64]. The original
function on 0,1 is translated by a distance of length NcontΔx whose values are filled-in by the
sum of “blend-to-zero" continuations (dashed lines) in order to render the function periodic.
Triangles and circles represent the discrete dl, dr = 5 matching points, and squares represent
the discrete Ncont = 25 continuation points that comprise the extension. .................................. 75
Figure 4.2. An example Fourier continuation of the non-periodic function Xs =
expsin5.4πs − 2.7π − cos2πs, x ∈ 0,1. The operators Pl, Pr perform the blending-to-zero

xvii
via projection onto a precomputed FC basis. The new function values Xcont are periodic
(in a discrete sense) on the interval 0, b........................................................................................ 79
Figure 4.3 (Left) Superposition of filament positions over time (plotted every 0.02 sec),
where the blue dot represents a simply-supported end and the red dot represents a free end.
(Right) The X-component displacement of the free end as a function of time. ............................ 82
Figure 4.4 Snapshots at two different times (increasing time from left to right) of the solution
to the problem derived from Eq. (4.20). The blue dot represents a clamped-like (non-zero)
boundary condition and the red dot represents a simply-supported-like (non-zero) boundary
condition. ...................................................................................................................................... 84
Figure 4.5 : Snapshots at two different times (increasing time from left to right) of the
solution to the problem derived from Eq. (4.23). The blue dot represents a standard clamped
boundary condition. ...................................................................................................................... 85
Figure 5.1. Schematic representation of the heart-head axis. ....................................................... 91
Figure 5.2. Schematic of IJV experimental setup design corresponding to the right
atrioventricular-IJV hemodynamic simulator setup...................................................................... 92
Figure 5.3 Sketch of the experimental setup with PIV system..................................................... 93
Figure 5.4 Sketch of the experimental setup with measurement device....................................... 93
Figure 5.5. Setup includes the complete in-vitro circulation system consisting of the RV
(pump), the IJV (inside close tank), the RV (compliant chamber) and the venours return tank. . 94
Figure 5.6. Overall view of the entire Setup includes the complete in-vitro circulation
system and the PIV system. .......................................................................................................... 95
Figure 5.7. Photo of the IJV mold and fabricated IJVs/RAs. ....................................................... 96
Figure 5.8. Compliance measurement setup of IJV...................................................................... 98

xviii
Figure 5.9. PIV system setting and a zoomed-in view. .............................................................. 101
Figure 5.10. Index matching result and reflective index measurement (zoom in at bottom). .... 104
Figure 5.11. PIV calibration procedure result and PIV laser sheet............................................. 105
Figure 5.12. Pressure waveform in internal jugular vein with HR=60 Q=0.5 and 0.75 ml/min. 108
Figure 5.13. Pressure waveform in internal jugular vein with HR=60 Q=0.5 and 0.75 ml/min. 108
Figure 5.14. 2D-PIV results with particle and vector field......................................................... 109
Figure 5.15. 3D-PIV particle identification results..................................................................... 110
Figure 5.16. 3D-PIV vector filed for HR=60 Q=0.5 and 0.75 ml/min and a sample 3D-CFD
demonstration. For the 3D-PIV images the tube boundary demonstrates the IJV wall at the
initial neutral position. ................................................................................................................ 110
Figure 5.17 Snapshots of the particle velocity vector (colored by the velocity magnitude) in
the jugular vein vessel (tube). (a)Front view and (b) Side view for model one (M1), flowrate
Q=0.75 L/min and heartrate HR=1.0 Hz; (c)Front view and (d) Side view for material one
(M1), flowrate Q=0.75 L/min and heartrate HR=2.0 Hz. T1 to T2 represents the time before,
during (T2 and T3) and after (T4) the collapse of the tube. Red and blue arrows show the
direction of the venous flow (top to bottom) and the direction of the heart wave (bottom to
top).............................................................................................................................................. 115
Figure 5.18 Snapshots of the velocity magnitude field in the jugular vein vessel (tube).
(a)Front view and (b) Side view for material one (M1), flowrate Q=0.75 L/min and heartrate
HR=1.0 Hz; (c)Front view and (d) Side view for material one (M1), flowrate Q=0.75 L/min
and heartrate HR=2.0 Hz. T1 to T2 represents the time before, during (T2 and T3) and after
(T4) the collapse of the tube. All the timings match the ones in Figure 3. The flow direction

xix
is represented by the red arrow (top to bottom), and the wave direction is represented by the
blue arrow (bottom to top). ......................................................................................................... 117
Figure 5.19 Snapshots of the vorticity magnitude field in the jugular vein vessel (tube).
(a)Front view and (b) Side view for material one (M1), flowrate Q=0.75 L/min and heartrate
HR=1.0 Hz; (c)Front view and (d) Side view for material one (M1), flowrate Q=0.75 L/min
and heartrate HR=2.0 Hz. T1 to T2 represents the time before, during (T2 and T3) and after
(T4) the collapse of the tube. All the timings match the ones in Figure 5.17............................. 119
Figure 5.20. Sample Pressure waveform in internal jugular vein with HR=60 Q=0.5 and
0.75 ml/min................................................................................................................................. 120
Figure 5.21 Reverse flow index (RFI) changes profile within one cardiac cycle (top row) and
the IJV pressure waveform profile in one cardiac cycle (bottom row) under different flow
conditions with material one (M1). (a) Heartrate HR=1.0 Hz flowrate Q=0.50 L/min (noncollapse mode); (b) Heartrate HR=1.0 Hz flowrate Q=0.75 L/min (collapse mode); (c)
Heartrate HR=1.0 Hz flowrate Q=0.50 L/min (collapse mode with oscillatory); (d) Heartrate
HR=2.0 Hz flowrate Q=0.75 L/min (collapse mode with higher frequency). Note that (b) and
(d) are corresponding to the cases (flow field) in Figure 5.17 to 5.19 (with T1 to T4 marked
in the figure)................................................................................................................................ 121
Figure 5.22 Jugular vein tube volume (V(t) normalized be the mean volume) changes profile
within one cardiac cycle (top row) and the corresponding Reverse flow index (RFI) changes
profile within one cardiac cycle (bottom row). (a) to (d) represent the cases with different
flow conditions (all cases here are with collapsed mode) with material one (M1). (a)
Heartrate HR=1.0 Hz flowrate Q=0.75 L/min; (b) Heartrate HR=1.0 Hz flowrate Q=1.0
L/min; (c) Heartrate HR=2.0 Hz flowrate Q=0.75 L/min; (d) Heartrate HR=2.0 Hz flowrate

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Q=1.0 L/min. The starting and ending of the RFI wave are marked by the dashed lines and
matched with the volume change profile. Note that (a) and (c) are corresponding to the
cases (flow field) in Figure 5.17 to 5.19 (with T1 to T4 marked in the figure).......................... 121
Figure 5.23 calculation of the mean Reverse flow index (RFI) with respective to time in one
cardiac cycle. All cases here are for material one (M1) with same stiffness. Different colors
represent different hear rate: Blue 1.0Hz, Red 2.0 Hz, Yellow 1.66 Hz. (The measurement
error from PIV system is around 1.25 cm/s calculate from calibration error 50 micron/pixel
Camera to World Error and 1 pixel World to Camara Error with dt=4ms)................................ 123
Figure 5.24 calculation of the mean Reverse flow index (RFI) with respective to time in one
cardiac cycle for all cases here with different stiffness (material one to four (M1 to M4). M1
to M4 are marked by different marker styles and the colors represent different hear rate: Blue
1.0Hz, Red 2.0 Hz, Yellow 1.66 Hz. .......................................................................................... 124
Figure 5.25 Reverse flow index (RFI) changes profile within one cardiac cycle (top row) and
the corresponding average vorticity (AV) profile in one cardiac cycle (bottom row) under
different flow conditions with material one (M1). (a) Heartrate HR=1.0 Hz flowrate Q=0.50
L/min (non-collapse mode); (b) Heartrate HR=1.0 Hz flowrate Q=0.75 L/min (collapse
mode); (c) Heartrate HR=1.0 Hz flowrate Q=0.50 L/min (collapse mode with oscillatory); (d)
Heartrate HR=2.0 Hz flowrate Q=0.75 L/min (collapse mode with higher frequency). Note
that (b) and (d) are corresponding to the cases (flow field) in Figure 5.17 to 5.19 (with T1 to
T4 marked in the figure). ............................................................................................................ 125
Figure 5.26 calculation of the mean of the volume Average vorticity (AV) with respective to
time in one cardiac cycle. All cases here are for material one (M1) with same stiffness.
Different colors represent different hear rate: Blue 1.0Hz, Red 2.0 Hz, Yellow 1.66 Hz.......... 126

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Figure 5.27 calculation of the mean of the volume Average vorticity (AV) with respective to
time in one cardiac cycle for all cases here with different stiffness (material one to four (M1
to M4). M1 to M4 are marked by different marker styles and the colors represent different
hear rate: Blue 1.0Hz, Red 2.0 Hz, Yellow 1.66 Hz................................................................... 127
Figure 6.1 A schematic illustration of the multiscale closed-loop model employed in this
chapter, where all heart chambers interact with each other via a septal elastance (dashed grey
arrows). The complete system is composed of 394 individual vascular segments, and
vascular beds are modeled with any arbitrary number of connecting arteries and veins. (LA:
left atrium; LV: left ventricle; RA: right atrium; RV: right ventricle.) ...................................... 139
Figure 6.2 Two consecutive cycles of pressure waveforms at baseline for the right atrium
(RA, solid blue line) and the right ventricle (RV, dashed red line). ........................................... 152
Figure 6.3 Baseline pressure waveforms (left) and flow waveforms (right) of the right
internal jugular vein I.................................................................................................................. 153
Figure 6.4 Pressure waveforms (left) and flow waveforms (right) of the right internal jugular
vein I at baseline stiffnesses (solid blue line) and at three times the baseline stiffnesses
(dashed red line) of the segments in Table 6.2. .......................................................................... 154
Figure 6.5 Two consecutive cycles of pressure waveforms of the right atrium (RA) and the
right ventricle (RV) at heart rates of HR = 40 bpm (left) and HR = 160 bpm (right). ............ 155
Figure 6.6 Pressure waveforms (left) and flow waveforms (right) at HR = 40 bpm of the
right internal jugular vein I at baseline stiffness (solid blue line) and at three times the
baseline stiffness (dashed red line) of the segments in Table 6.2............................................... 155

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Figure 6.7 Pressure waveforms (left) and flow waveforms (right) at HR = 160 bpm of the
right internal jugular vein I at baseline stiffness (solid blue line) and at three times the
baseline stiffness (dashed red line) of the segments in Table 6.2............................................... 156
Figure 6.8 Two consecutive cycles of pressure waveforms of the right atrium (RA) at a heart
rate of HR = 75 bpm at baseline heart contractility (solid blue line), at low contractility
(1/2 of baseline contractility, red dashed line), and at high contractility (2.5 times baseline,
dash-dotted yellow line).............................................................................................................. 157
Figure 6.9 Pressure waveforms (left) and flow waveforms (right) at HR = 75 bpm and at
baseline stiffness (c0) for the right internal jugular vein I at baseline heart contractility (solid
blue line), at low contractility (1/2 of baseline contractility, red dashed line), and at high
contractility (2.5 times baseline, dash-dotted yellow line)......................................................... 157
Figure 6.10 Pressure waveforms (left) and flow waveforms (right) at HR = 75 bpm and at
high stiffness (3c0) for the right internal jugular vein I at baseline heart contractility (solid
blue line), at low contractility (1/2 of baseline contractility, red dashed line), and at high
contractility (2.5 times baseline, dash-dotted yellow line)......................................................... 158
Figure 6.11 Two consecutive cycles of pressure waveforms of the right atrium (RA) at a
heart rate of HR = 75 bpm for a normal tricuspid valve (solid blue line), one undergoing
stenosis (dashed red line), and one inducing flow regurgitation (dash-dotted yellow line). ...... 159
Figure 6.12 Pressure waveforms (left) and flow waveforms (right) at HR = 75 bpm and at
baseline stiffness (c0) of the right internal jugular vein I for a normal tricuspid valve (solid
blue line), one undergoing stenosis (dashed red line), and one inducing flow regurgitation
(dash-dotted yellow line). ........................................................................................................... 159

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Figure 6.13 Pressure waveforms (left) and flow waveforms (right) at HR = 75 bpm and at
high stiffness (3c0) of the right internal jugular vein I for a normal tricuspid valve (solid blue
line), one undergoing stenosis (dashed red line), and one inducing flow regurgitation (dashdotted yellow line). ..................................................................................................................... 160
Figure 6.14 Pressure waveforms (left) and flow waveforms (right) at HR = 75 bpm at
baseline with the complete heart (solid blue line) and for Fontan circulation (dashed red) of
the superior vena cava................................................................................................................. 161
Figure 6.15 Pressure waveforms (left) and flow waveforms (right) at HR = 75 bpm at
baseline with the complete heart (solid blue line) and for Fontan circulation (dashed red) of
the right internal jugular vein I. .................................................................................................. 162
Figure 7.1. Viscosity of the shear-thinning Carreau-Yasuda model matching with Fontan
patient-specific data. Figure from Wei H et al. European Journal of Mechanics-B/Fluids 84
(2020): 1-14. ............................................................................................................................... 172
Figure 7.2 Schematic of the Fontan circuit. The superior and inferior cava (SVC and IVC)
are the inlets with diameter D = 1.2cm. The right and left pulmonary arteries (RPA and LPA)
are the outlets with diameter D = 0.9cm. Bold arrows indicate the direction of fluid flow. ...... 175
Figure 7.3 Velocity magnitude (nondimensionalized by v/Uref) for Newtonian (left column)
and Non-Newtonian (right column) cases under an equal SVC/IVC distribution (50/50). The
velocity profiles of Non-Newtonian cases are more blunted. There is a large area of
stagnation in the center. The top row is for cardiac output (CO=2.5L/min), the middle row is
for cardiac output (CO=1.5L/min) and the bottom row is for cardiac output (CO=0.5L/min). . 180
Figure 7.4 Velocity magnitude (nondimensionalized by v/Uref) for Newtonian (left column)
and Non-Newtonian (right column) cases under a SVC/IVC distribution of 30/70. Similar to

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50/50 cases, the velocity profiles of Non-Newtonian cases were more blunted. The top row is
for cardiac output (CO=2.5L/min), the middle row is for cardiac output (CO=1.5L/min) and
the bottom row is for cardiac output (CO=0.5L/min)................................................................. 181
Figure 7.5 Shear stress (nondimensionalized by τγ/ρUref 2
) distributions in Newtonian (left
column) and Non- Newtonian cases (right column) under a SVC/IVC distribution of 50/50.
At any given CO, the magnitudes of shear stress are higher in non-Newtonian cases, and the
high-stress regions cover larger area in non-Newtonian cases than in Newtonian cases. The
top row is for cardiac output (CO=2.5L/min), the middle row is for cardiac output
(CO=1.5L/min) and the bottom row is for cardiac output (CO=0.5L/min)................................ 183
Figure 7.6 Shear stress (nondimensionalized by τγ/ρUref 2
) distributions in Newtonian and
non-Newtonian cases under a SVC/IVC distribution of 30/70. At any given CO, the
magnitudes of shear stress are higher in non-Newtonian cases, and the high-stress regions
cover larger area in non-Newtonian cases than in Newtonian cases. The top row is for
cardiac output (CO=2.5L/min), the middle row is for cardiac output (CO=1.5L/min) and the
bottom row is for cardiac output (CO=0.5L/min)....................................................................... 184
Figure 7.7 Box plots for absolute (left) and percentage (right) difference of indexed Power
Loss (iPL) between Newtonian and non-Newtonian across a wide range of cardiac outputs
(0.5 to 2.5 L/min) under various SVC/IVC distributions. .......................................................... 185
Figure 7.8 Viscous dissipation (nondimensionalized by Φ/ρUref 2
/Tref) of Newtonian (left
column) and Non-Newtonian (right column) cases for the SVC/IVC distribution of 50/50.
The top row is for cardiac output (CO=2.5L/min), the middle row is for cardiac output
(CO=1.5L/min) and the bottom row is for cardiac output (CO=0.5L/min)................................ 189

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Figure 7.9 Viscous dissipation (nondimensionalized by Φ/ρUref 2
/Tref) of Newtonian (left
column) and Non-Newtonian (right column) cases for the SVC/IVC distribution of 30/70.
The top row is for cardiac output (CO=2.5L/min), the middle row is for cardiac output
(CO=1.5L/min) and the bottom row is for cardiac output (CO=0.5L/min)................................ 190
Figure 7.10 Box plots for the absolute (left) and the percentage (right) difference of total
viscous dissipation (nondimensionalized by Φ/ρUref 2
/Tref) between Newtonian and nonNewtonian at different cardiac output (0.5 to 2.5 L/min) under various SVC/IVC. .................. 191
Figure 7.11 Local non-Newtonian importance factor (IL = μ/μ∞) distributions in NonNewtonian cases. All cases in the left column are for the SVC/IVC distribution of 50/50, and
all cases in the right column are for the SVC/IVC distribution of 40/60. The top row is for
cardiac output (CO=2.5L/min), the middle row is for cardiac output (CO=1.5L/min) and the
bottom row is for cardiac output (CO=0.5L/min)....................................................................... 192
Figure 7.12 Local non-Newtonian importance factor (IL = μ/μ∞) distributions in NonNewtonian cases. All cases in the left column are for the SVC/IVC distribution of 30/70, and
all cases in the right column are for the SVC/IVC distribution of 20/80. The top row is for
cardiac output (CO=2.5L/min), the middle row is for cardiac output (CO=1.5L/min) and the
bottom row is for cardiac output (CO=0.5L/min)....................................................................... 193
Figure 7.13 Box plots for Global Non-Newtonian Importance Factor with (left) and without
(right) the central stagnation area for different cardiac outputs (0.5 to 2.5 L/min) under
various SVC/IVC distributions. (Red lines denote the median of the data. Top and bottom
borders of the boxes denote first and third quartiles. Whiskers denote minimum and
maximum values.). (For interpretation of the references to color in this figure legend, the
reader is referred to the web version of this article.) .................................................................. 194

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Figure 8.1 Workflow for creating CFD simulations from MRI images. .................................... 205
Figure 8.2 Viscosity of shear-thinning Casson model based on patient-specific data. Red
circles are mean viscosity measurements from 61 Fontan patients. ........................................... 207
Figure 8.3 Streamlines and velocity magnitudes in four sample Fontan patients. NonNewtonian models demonstrate lower velocity values, decreased flow rotation, and more
stagnant flow in the TCPC region (dashed red boxes). SVC: superior vena cava, IVC:
inferior vena cava, RPA: right pulmonary artery, LPA: left pulmonary artery.......................... 214
Figure 8.4 Patient-specific variation in local non-Newtonian importance factor (note Patients
4 and 15 have a different color scale since the values are larger). SVC: superior vena cava,
IVC: inferior vena cava, RPA: right pulmonary artery, LPA: left pulmonary artery................. 215
Figure 8.5 (a) Comparison of indexed power loss between Newtonian and non-Newtonian
viscosity models. (b) Comparison of indexed viscous dissipation rate between Newtonian
and non-Newtonian viscosity models. Patient-level differences are denoted with a black line. 216
Figure 8.6 (a) Comparison of mean wall shear stress (WSS between Newtonian and nonNewtonian viscosity models. (b) Comparison of low WSS area (as a percentage of total
Fontan area) between Newtonian and non-Newtonian viscosity models. Patient-level
differences are denoted with a black line.................................................................................... 216
Figure 8.7 Dye simulations for calculation of hepatic blood flow distribution (HFD). Dye
was tracked from IVC to RPA and LPA. SVC: superior vena cava, IVC: inferior vena cava,
RPA: right pulmonary artery, LPA: left pulmonary artery......................................................... 218
Figure 8.8 (a) Comparison of hepatic flow distribution between Newtonian and nonNewtonian fluid models for each patient. (b) Difference in hepatic flow distribution between
Newtonian and non-Newtonian fluid models. ............................................................................ 218

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Figure 9.1. Schematic of the artificial right atrium (ARA) circuit. The superior and inferior
vena cava (SVC and IVC) are the inlets, and the pulmonary artery (PA) is the outlet. Arrows
indicate the blood flow direction. Figure from Wei H et al. Physical Review Fluids 6.12
(2021): 123103............................................................................................................................ 226
Figure 9.2 Viscosity of the shear-thinning Carreau-Yasuda model matching with Fontan
patient specific data [52]. ............................................................................................................ 231
Figure 9.3 A summary of the overall algorithm for the complete solver. FSI: fluid-solid
interaction; IBM: immersed boundary method; LBM: Lattice-Boltzmann method; FEM:
finite element method. ................................................................................................................ 236
Figure 9.4 Schematic of the artificial right atrium (ARA) circuit. The superior and inferior
vena cava (SVC and IVC) are the inlets, and the pulmonary artery (PA) is the outlet. Bold
arrows indicate the net direction of blood flow. ......................................................................... 237
Figure 9.5 Flow chart for the overall procedure of identifying hemodynamically optimum
ARA shapes with minimum PRT (relative to the cases considered in this study). ARA:
artificial right atrium; LAP: long-axis plane; PRT: particle residence time............................... 239
Figure 9.6 Schematics of 2D rigid cases used for identifying and evaluating LAP-opt. The
superior and inferior cava (SVC and IVC) are the inlets with diameter Di = D = 1.2 cm.
The pulmonary artery (outlet) are the outlets with diameter Do = 0.75D = 0.9 cm. The
curvature for LAP1 is D at both concave and convex side, the curvature for LAP2 is 2.0D at
the convex side, and the curvature for LAP3 is 2.5D at the convex and 0.5D at the concave
side etc. Bold arrows indicate the direction of fluid flow. .......................................................... 240

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Figure 9.7 Schematics of the corresponding 3D compliant configurations. Cases 1, 2, and 3
have the same total volume. Case 4 is the reference case for the minimum particle residence
time. ............................................................................................................................................ 241
Figure 9.8 Qualitative visualization of Particle Resident Time (PRT) for 2D simulations of
LAP1-4 All cases have similar LAP area. .................................................................................. 244
Figure 9.9 Particle Resident Time (PRT) distribution in long axis plane (LAP) of compliant
ARA models (3D FSI simulations). Cases 1-3 have the same total volume. Case 4 has the
same LAP as case 3 but without out of plane bulge (see Fig. 9.7). ............................................ 245
Figure 9.10 Velocity magnitude (nondimensionalized by v/Uref) and streamlines in the long
axis planes (LAPs) of compliant models. Cases 1-3 have the same overall volume. Case 4
has the same LAP as case 3, but it has lower volume compared to others (see Fig. 9.7 d)........ 247
Figure 9.11 Velocity magnitude (nondimensionalized by v/Uref) and streamlines in the long
axis planes (LAPs) of compliant models. Cases 1-3 have the same overall volume. Note that
Case 4 has the same LAP as case 3............................................................................................. 248
Figure 9.12 Pressure distributions (nondimensionalized by P/ρUref 2
) in long axis planes of
compliant 3D models. As expected, case 4 has the highest overall pressure due to its low
volume (low compliance). .......................................................................................................... 249
Figure 9.13 Uniform flat velocity profile at the inlet of the extension tube boundary model
(left) and the corresponding fully developed velocity profile at the entrance of the ARA
(right). ......................................................................................................................................... 253
Figure 9.14 Velocity magnitudes (nondimensionalized by 𝑣/𝑈𝑟𝑒𝑓) for the 2D LAP cases. ..... 254
Figure 9.15 PRT on short-axis planes at centerline (Cases 1 to 4 from left to right). ................ 255

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Figure 9.16 PRT on short-axis planes at slightly higher (0.2D) than the centerline (Cases 1 to
4 from left to right). .................................................................................................................... 257

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Nomenclature
AD Alzheimer's disease
AV Aortic valve
ARA Artificial right atrium
BGK Bhatnagar-Gross-Krook
BSA Body surface area
CO Cardiac output
CNS Central nervous system
CVP Central venous pressure
CFD Computational fluid dynamics
CHD Congenital heart diseases
CHF Congestive heart failure
FFT Fast Fourier Transform
FDM Finite Difference Method
FEM Finite element method
FSI Fluid–structure interaction
FC Fourier Continuation
HR Heart rate
HBF Hepatic blood flow
IB Immersed boundary
IB-LBM Immersed boundary-lattice Boltzmann method
iPL Indexed Power Loss
IJV Internal jugular vein
JVP Jugular venous pulse
JVR Jugular venous reflux
LB Lattice Boltzmann
LA Left atrium
LPA Left pulmonary arteries
LV Left ventricle
LAP Long axis plane
MRI Magnetic resonance imaging
MSR Mean shear rate

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MMS Method of manufactured solutions
MCI Mild cognitive impairment
MV Mitral valve
MS Multiple sclerosis
ODE Ordinary differential equation
PDE Partial differential equation
PIV Particle Image Velocimetry
PRT Particle resident time
PVA Polyvinyl alcohol
PL Power loss
PA Pulmonary artery
PV Pulmonary valve
PP Pulse pressures
RI Refractive index
RFI Reverse flow index
Re Reynolds number
RAP Right atrial pressure
RA Right atrium
RPA Right pulmonary arteries
SGFD Self-implemented staggered grid finite difference
SRT Single-relaxation-time
SVD Singular Value Decomposition
TAV Time-average velocity
TARFI Time-averaged reverse flow index
TCPC Total Cavopulmonary Connection
TV Tricuspid valve
VAD Ventricular assist device
WSS Wall shear stress
WENO Weighted essentially non-oscillatory
Wo Womersley number
XG Xanthan gum
𝜔̅ Average vorticity

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𝐸𝐼 Bending stiffness
𝑍𝑤 Characteristic impedance
𝑓𝑖 Distribution function
𝐶𝑤 Effective chamber compliance
𝜇𝑒𝑓𝑓 Effective viscosity
𝐸𝑒𝑠 End-systolic elastance
𝑓cont Fast-converging interpolating trigonometric polynomial
𝜌 Fluid density
𝒇 Force density
𝐼𝐺
̅ Global non-Newtonian importance factor
𝜗 Kinematic viscosity
𝑐𝑠 Kinematic viscosity
𝛿𝑖𝑗 Kronecker delta
𝒔 Lagrangian coordinates
𝑭𝐿 Lagrangian force
∆𝑥 Lattice space
𝐼𝐿 Non-Newtonian importance factor
𝑅𝑤 Peripheral resistance
𝜈̂ Poisson’s ratio
𝒙 Position
𝐷𝐼𝐼 Second invariant of the rate-of-strain tensor
𝛾̇ Shear rate
𝑐 Sound speed
𝑆𝛼𝛽 Strain tensor
𝐸ℎ Stretching stiffness
𝑇 Tension force
ℎ Thickness
t Time
∆𝑡 Time step
𝐶𝑣 Time varying compliance
𝒆𝑖 Velocity set
𝑝𝑣 Ventricular pressure
𝑉𝑣 Ventricular volume

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𝜇 Viscosity
Φ𝑉𝐷 Viscous dissipation
𝜌𝑠 Wall density
𝑿 Wall position
ωi Weighting factor
𝐸 Young's modulus

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Abstract
The study of hemodynamics in cerebral venous circulation is crucial for understanding
neurodegenerative diseases and cerebrovascular function. The physics of fluids in cerebral veins
are more complex than in arteries, and research in this area is relatively sparse. In healthy
conditions, pulsatile fluid dynamics play a critical role, with the internal jugular vein (IJV) being
a significant contributor. However, in diseased conditions like Fontan patients, non-Newtonian
effects become significant. For better understanding the underlying mechanisms, it is necessary to
develop accurate models that can improve prediction and treatment strategies. To achieve such
aims, we implemented both computational and experimental approaches to investigate the fluid
dynamics of the central venous system and its impact on brain-heart hemodynamic coupling, under
both healthy and diseased conditions. We developed a 0D-3D coupling computational framework
to capture wave dynamics of IJV in a right-heart-brain coupled system towards investigating nonNewtonian effects. Additionally, we designed an experimental setup with a 3D particle image
velocimetry measurement system for capturing IJV wave dynamics in realistic scenarios and for
accurately measuring the 3D flow. The global effects on IJV and how they affect right-heart-brain
coupling were examined using a 1D closed-loop model of the circulatory system. Following this,
we demonstrated the applications of our simulations for Fontan patients. In such conditions, the
flow dynamics shift non-negligibly, highlighting the importance of non-Newtonian blood
rheology. Findings from this thesis may lead to novel diagnostic and therapeutic strategies for
various cardiovascular and neurological disorders, potentially offering valuable insights into their
pathophysiology.

1
Chapter 1: Introduction and Motivation
1.1 Background
The physics of fluids inside cerebral venous system is in general more complex than on the arterial
side and poorly understood, especially regarding its clinically relevant anatomy and physiology.
While the arterial system has been extensively studied, the hemodynamics of cerebral venous
circulation remain underexplored [1]. Recent studies have shown that the brain's vascular system
is closely linked to heart's function, and alterations in the hemodynamic coupling between these
two systems can contribute to the development of various cardiovascular and cerebrovascular
diseases [2]. Understanding the cerebrovascular system in both health and disease is crucial for
identifying the system-level changes in the circulation that contribute to neurological and
neurodegenerative diseases.
There is a growing recognition of the significance of investigating cerebral venous circulation.
Studies have demonstrated that abnormalities in this system can significantly impact cerebral
blood flow and cognitive function. Advances in imaging techniques have increased interest in the
biophysical and hemodynamic parameters of the intracranial venous system. For example, higher
aortic stiffness has been associated with reduced cerebral blood flow and cognitive performance
in older adults [2]. Arterial stiffness contributes to microvascular brain disease and cognitive
impairment, while abnormalities in cerebral endothelial cells have been linked to multiple sclerosis
pathogenesis (MS) [3-6]. Pulsatility in venous system and global cerebral blood flow have been
associated with MS, and jugular venous flow abnormalities correlate with MS clinical and imaging
variables [5, 7, 8]. Impaired right ventricular function has been found to reduce cerebral blood

2
flow in obstructive sleep apnea patients [9], highlighting the role of the jugular vein in right heartbrain hemodynamic coupling.
Despite these associations, the role of the venous system in the pathogenesis of inflammatory
neurological and neurodegenerative diseases remains underexplored [10-12]. Given the
importance of the cerebral venous system in maintaining normal brain function, a deeper
understanding of its physiology and pathophysiology is necessary to develop effective treatments
for neurodegenerative diseases. Neurodegenerative diseases, such as mild cognitive impairment
(MCI) and Alzheimer's disease (AD), are increasingly common with aging populations and pose
significant public health concerns. While the exact causes of these conditions remain unclear,
recent research suggests that abnormalities in the extracranial venous system, particularly jugular
venous reflux (JVR), may play a significant role in their pathophysiology [13, 14]. The internal
jugular vein (IJV) is one of the primary tracts for cerebral venous drainage and is easily observed
using color-coded Doppler sonography. Study has identified internal jugular venous reflux as a
potential cause of cerebral venous outflow impairment and neurologic dysfunction [13, 14]. JVR,
characterized by increased venous pressure and impaired cerebral venous outflow, has been linked
to structural brain changes in MCI and AD patients (it suggests that JVR may transmit venous
hypertension into the brain and result in vasogenic edema), potentially leading to brain tissue
swelling and neurologic dysfunction [13-15].
Understanding the extracranial venous system's role in Central nervous system (CNS) disorders is
crucial for advancing knowledge of these conditions' underlying mechanisms. The complex nature
of the extracranial venous system makes it challenging to assess without invasive measurements.
However, bedside analysis of the jugular venous pulse can provide critical information regarding

3
right heart hemodynamics, aiding in diagnosing certain diseases [16]. Further research is needed
to explore the links between JVR and neurodegenerative diseases, as well as to develop
noninvasive methods for assessing venous abnormalities.
In the context of heart function, assessing the jugular venous pulse (JVP) offers a window (crucial
perspective) into the right heart. This assessment provides valuable insights into the right heart's
function and its hemodynamics, helping to understand its overall performance and potential
abnormalities. However, analyzing the JVP is often disregarded during routine physical
examinations due to the belief that invasive measurements are necessary [17]. Current methods of
estimating right atrial pressure (RAP) are imprecise, and diagnosing cerebral venous outflow
impairment relies on imaging modalities [18]. Recent advances suggest that interventions like
intravenous angioplasty and stenting may restore normal blood circulation and alleviate symptoms
[19]. Determining mean central venous pressure (CVP) through ultrasound assessment of the JVP
could provide a safer, noninvasive method for acquiring important clinical parameters [20] [21].
Research on the impact of space travel on the cardiovascular system has revealed that astronauts
are at risk for venous thrombosis, emphasizing the need for further investigation and preventive
measures [22, 23]. Space travel exposes astronauts to unique physiological stressors such as
microgravity, radiation, and isolation from Earth's environment, which can significantly affect
their cardiovascular system, leading to venous thrombosis [24-26]. The reduced internal jugular
vein flow in space, likely due to negative transmural pressure and increased resistance, underscores
the importance of understanding the underlying mechanisms [27] [28]. NASA's surveillance
program for evaluating astronauts' venous thrombosis highlights the necessity for continued

4
research and the development of effective prevention and treatment strategies for long-duration
space missions [29].
From a hemodynamic perspective, the JVP is a useful tool for evaluating cardiac function and
diagnosing heart diseases. Studies have shown that the JVP can be obtained by measuring the
internal jugular vein's cross-sectional area using ultrasound, with the pressure gradient driving the
hemodynamics [30]. The fluid dynamics within the jugular vein are distinct from those in arteries,
with pressure waves propagating from the heart to the head at a wave speed of C, while the
direction of internal jugular vein blood flow (U) is from the head towards the heart. The JVP
regulates IJV blood flow and pressure waves are transmitted from the heart toward the brain
through the IJV wall. Extensive research on the pressure-flow relationship of collapsible tubes and
veins has provided insight into how these systems behave under various conditions [31-34].
However, most studies have focused on non-pulsatile flow in collapsible flexible tubes, which are
commonly known as Starling resistors [35-40]. These studies have investigated the behavior of
collapsible tubes under steady flow conditions, examining their role in regulating blood flow and
the factors influencing the pressure-flow relationship, such as tube geometry, material properties,
and fluid dynamics [41-44]. Additionally, research has explored self-excited and flow-induced
oscillations, along with the onset of self-oscillation, collapsing, and instability in various
conditions, including elliptical tube and channel flow [41-44] [45, 46]. Further investigations have
focused on mass transport in flexible tubes, particularly steady flow in flexible tubes with selfexcited oscillations, simplified oscillatory waves, or pulsatile flow in rigid tubes [33, 47-51].
Despite considerable research on the self-oscillatory and collapsing behavior of flexible tubes
induced by steady flow or pressure conditions, there has been no exploration of the pulsatile

5
dynamics of collapsible tubes and the interaction between pressure waves and pulsatile flow.
Investigating the pulsatile flow dynamics in the venous system is crucial, as it remains an
unexplored area. A comprehensive investigation of this fundamental fluid dynamic problem,
specifically pulsatile fluid dynamics in a collapsible tube, has yet to be carried out. This area of
study is essential as it has the potential to uncover important insights into the behavior of fluids in
collapsible tubes under pulsatile flow conditions. Thorough exploration could provide a better
understanding of how these systems work, informing the development of novel medical devices
and industrial applications. Therefore, further research is required to comprehensively investigate
this important fluid dynamic problem.
In diseased conditions, such as in Fontan patients, the physics of flow dynamics changes
significantly. In Fontan circulation, the flow is typically steady with a low shear rate, making the
non-Newtonian effects of blood more pronounced [52-54]. Studying the impact of non-Newtonian
blood rheology and its effects on important clinical metrics and risk factors is critical in such
disease cases. Developing a methodology to accurately model and investigate these blood flow
conditions is necessary for better prediction and simulation, leading to improved understanding
and treatment of specific cardiovascular diseases.
Overall, these recent studies provide evidence of the importance of right heart-brain hemodynamic
coupling and the potential role of the veins in this process, where understanding these factors is
crucial for developing effective treatments and interventions for a variety of health and diseased
conditions, highlighting the need for further investigation into the pulsatile fluid dynamics of the
vein and its impact on the cerebrovascular system. Therefore, to better understand the

6
hemodynamics of cerebral veins, specific analytical tools and conceptual frameworks need to be
developed.
1.2 Thesis objectives
This thesis aims to examine and provide a necessary foundation on the impact of pulsatile and
steady fluid dynamics in the central venous system on the hemodynamic coupling between the
right heart and the brain under both healthy and diseased conditions, particularly focusing on cases
where cerebral veins are obstructed (e.g., collapse of the internal jugular vein) and in Fontan
patients (patient with unfunctional heart). The research seeks to provide novel insight into this
critical aspect of the cardiovascular system, emphasizing the mechanisms behind internal jugular
venous reflux and the effects of central venous pressure (CVP) and non-Newtonian dynamics on
Fontan circulation. To achieve these goals, the study will utilize both experimental and
computational models. The specific objectives of this research are:
1. Develop a 0D-3D Coupled Fluid-Solid Interaction Computational Framework: Create
a robust computational model to simulate the interactions between fluid dynamics and
vessel wall mechanics in the central venous system.
2. Numerically Investigate Right Heart Function Effects on the Jugular Vein: Explore
how the right heart function affects pulsatile hemodynamics (pressure and flow) in large
cerebral veins (e.g., internal jugular vein) and study the characteristics of backward wave
propagation from the right atrium (RA) to internal jugular veins (IJVs), including the nonNewtonian effects on this pulsatile system.

7
3. Develop a Fourier-Based Solid Mechanics Solver: Implement a solid mechanics solver
to accurately capture the motion of collapsible vessel walls, enhancing the computational
model's accuracy.
4. Experimentally Investigate Pulsatile Dynamics on the Jugular Vein: Examine the
pulsatile behavior of the collapsible internal jugular vein in a physiologically accurate
hemodynamic setup and quantify the 3D flow structure using a 3D Particle Image
Velocimetry (PIV) system.
5. Study Global Circulatory Effects Using 1D Simulations: Investigate the global impact
of the entire circulatory system, including the effects of diseased conditions like stenosis,
regurgitation, heart failure, on wave dynamics and pulsatile hemodynamics of large
cerebral veins (such as IJV). And assess the right heart function using the central venous
pressure (CVP).
6. Investigate Non-Newtonian Effects in Fontan Circulation: Examine how nonNewtonian blood rheology affects the venous system in Fontan patients, considering the
low shear flow conditions typical of this patient group.
7. Assess Patient-Specific Non-Newtonian Rheology Effects: Study the individualized
effects of non-Newtonian blood properties on Fontan patients, aiming to understand
patient-specific variations.
8. Design an Artificial Right Atrium for Fontan Patients: Develop an artificial right atrium
optimized for Fontan patients to enhance the performance of ventricular assist devices.
This research aims to deepen our understanding of cerebral venous hemodynamics and the
hemodynamic coupling between the right heart and brain through innovative experimental and
computational methodologies. The ultimate objective is to uncover new insight into these

8
mechanisms and their associations with cardiovascular and neurodegenerative diseases, potentially
leading to more effective diagnostic and therapeutic strategies.
1.3 Thesis outline
Chapter 1 provides the background information that led to this thesis, summarizing relevant
research findings and identifying gaps this study aims to fill. It outlines the objectives of previous
research related to the current study and details the goals this research aims to accomplish.
Chapter 2 describes our in-house direct 0D-3D coupling fluid-structure hemodynamic flow
computational framework. It enables the coupling of zero-dimensional (0D) boundary conditions,
governed by complex ordinary differential equations (ODEs), to three-dimensional (3D) lattice
Boltzmann (LB)-based fluid-structure systems. The chapter focuses on the numerical approach
and implementation of this coupling method, essential for simulating complex fluid-structure
interactions in hemodynamics. The framework, incorporating a Lattice Boltzmann method solver,
can handle fluid flow and account for non-Newtonian behavior. We evaluate its performance using
the coupled LV-arterial system to generate physiological pressure and flow waveforms.
Chapter 3 applies the 0D-3D framework to a right-heart brain coupled internal jugular vein (IJV)
fluid dynamics system. It explores the pulsatile dynamics of the 3D IJV under varying stiffness,
heart function, and non-Newtonian blood effects, aiming to identify conditions where nonNewtonian effects are significant in the wave dynamics of the RA-IJV-Brain coupling.
In chapter 4, We introduce a new solid dynamics procedure based on the Fourier Continuation
(FC) method to capture wave dynamics details in a collapsible tube. This chapter focuses on the
methodological advancements necessary for accurate simulation of these dynamics.

9
Chapter 5 details the novel experimental setup designed for the right heart-internal jugular vein
coupled system, including setup configuration, critical component preparation (such as artificial
organ fabrication), and advanced measurement procedures. Specifically, it describes the
implementation of a 3D particle image velocimetry (PIV) system to analyze complex flow patterns
and dynamics of the collapsible vessel within the coupled system. The chapter outlines the PIV
setup, index matching, and calibration procedures, and presents results of IJV pressure
measurement and 3D PIV demonstrating IJV collapse and backflow propagation under different
wave dynamics.
Chapter 6 studies the global effects of the entire human circulatory system and the impact of
diseased conditions, such as stenosis or heart failure conditions, on the pulsatile hemodynamics of
the IJV. Using our 1D numerical framework, we explore these systemic interactions.
Chapter 7 elucidates the details of the methodology of the lattice Boltzmann (LB) approach for
incorporating non-Newtonian fluid effects. It systematically investigates non-Newtonian effects
in Fontan simulations, comparing the flow behavior of non-Newtonian and Newtonian fluids in a
simplified Fontan model and using clinical metrics to evaluate the findings.
Chapter 8 demonstrates the following study on patient-specific variance of non-Newtonian
rheology effects. It emphasizes the necessity of accurately capturing fluid dynamic behavior using
non-Newtonian models to enhance our understanding of patient-specific differences in Fontan
circulation.
In chapter 9, we present our novel design of an artificial right atrium for Fontan patients, aimed
at optimizing ventricular assist devices. This design focuses on avoiding blood clotting and over-

10
pressurizing the cerebral and hepatic veins, addressing critical challenges in Fontan circulation
management.

11
Chapter 2: Direct 0D-3D coupling of a lattice Boltzmann
methodology for fluid–structure hemodynamic flow simulations
This chapter is based on the following published manuscript: Wei, Heng, Faisal Amlani, and
Niema M. Pahlevan. "Direct 0D‐3D coupling of a lattice Boltzmann methodology for fluid‐
structure aortic flow simulations." International Journal for Numerical Methods in Biomedical
Engineering (2023): e3683.
This work introduces a numerical approach and implementation for the direct coupling of arbitrary
complex ordinary differential equation- (ODE-)governed zero-dimensional (0D) boundary
conditions to three-dimensional (3D) lattice Boltzmann-based fluid–structure systems for
hemodynamics studies. In particular, a most complex configuration is treated by considering a
dynamic left ventricle- (LV-)elastance heart model which is governed by (and applied as) a
nonlinear, non-stationary hybrid ODE-Dirichlet system. Other ODE-based boundary conditions,
such as lumped parameter Windkessel models for truncated vasculature, are also considered.
Performance studies of the complete 0D-3D solver, including its treatment of the lattice Boltzmann
fluid equations and elastodynamics equations as well as their interactions, is conducted through a
variety of benchmark and convergence studies that demonstrate the ability of the coupled 0D-3D
methodology in generating physiological pressure and flow waveforms—ultimately enabling the
exploration of various physical and physiological parameters for hemodynamics studies of the
coupled LV-arterial system. The methods proposed in this paper can be easily applied to other
ODE-based boundary conditions as well as to other fluid problems that are modeled by 3D lattice
Boltzmann equations and that require direct coupling of dynamic 0D boundary conditions.

12
2.1 Chapter Introduction
Cardiovascular modeling is a challenging fluid–structure interaction problem that involves
treatment of complex geometries and boundary conditions in order to effectively capture the
physiological dynamics [55]. Computational fluid dynamics (CFD) is a widely-used approach for
simulating blood flow in the circulatory system, [56-65] which includes applications to 1D [64-
66], 2D or 3D [59-63, 67, 68] formulations. The lattice Boltzmann (LB) method, [69-72]
originating from classical statistical physics, is a powerful alternative to conventional continuumbased CFD methods that use Navier–Stokes equations. The LB method uses simplified kinetic
equations combined with a modified molecular-dynamics approach to model both Newtonian and
non-Newtonian fluid flow in any complex geometry (the fluid is modeled as particles that stream
and collide over a discrete lattice mesh). Indeed, a particular advantage of LB-based
hemodynamics solvers is their ability to easily model non-Newtonian effects via its right-handside; capturing such effects may be important for small vessels or vessels where the shear rate is
low [73, 74]. The accuracy and usefulness of the LB method have been demonstrated in a variety
of fluid dynamics problems including turbulence [75] and multiphase flow [76]. As highlighted in
previous studies, [72, 75-78] LB methods have been shown to be particularly suitable for
hemodynamics simulations since many flow features of clinical interest may require efficient
numerical treatment of fully 3D computational domains (and local nature of LBM collision
calculations enables highly-parallelizable implementations).
In order to extend the clinical applicability of fluid–structure blood flow solvers based on LB
equations applied to large vessels, this work introduces a direct 0D-3D coupling for the treatment
of physiological boundary conditions that are governed by ordinary differential equations (ODEs)

13
such as lumped parameter Windkessel models [79, 80] or more complex hybrid ODE-Dirichlet
systems such as time-varying elastance organ models (both known as 0D models due to the
absence of spatial dependence). Previous contributions on the 0D-3D coupling for finite element
methods [81, 82] have been implicit and iterative, and for lattice Boltzmann [83, 84] blood flow
models usually only a Dirichlet or Neumann pressure or flow is prescribed during the entirety of
a cardiac cycle (precluding the use of more sophisticated and nonstationary, i.e., switching,
boundary conditions [64]).
Additionally, recent work on LB-based hemodynamics solvers have assumed only rigid walls, and
have applied 0D lumped parameter models externally through an iterative procedure where the
heart model is evolved and precomputed entirely independently [84] (such that the resultant
pressure profile is applied on a 3D LB domain simply as a Dirichlet condition, i.e., not a true
mathematical coupling).
This work, on the other hand, presents a first direct 0D-3D coupling for fully fluid–structure 3D
pulsatile blood flow solvers based on LB and elastodynamics equations. In particular, the 0D
equations considered in this work govern a highly-complex and non-stationary dynamic left
ventricle (LV-)elastance heart model [64] (that switches between an ODE and a Dirichlet boundary
condition in a “non-stationary” fashion [64]) in order to generate physiologically-accurate
hemodynamic conditions (instead of simply assigning a given inlet flow or pressure, as is
commonly done [84-86]). Such a coupled model represents the most complicated boundary
configuration found in the circulatory system [64]: a hybrid ODE-Dirichlet boundary condition
representing the left ventricle, where the time at which the ODE-governed condition transitions to
a Dirichlet condition is itself determined by the corresponding solution of the governing fluid–

14
structure LB system. Hence the methodology introduced in this work can be trivially extended to
the application of non-switching ODE-based boundary conditions such as lumped parameter
models based on Windkessels [64, 79, 80] (also treated in this contribution).
This work presents a numerical approach for directly coupling these 0D LV-elastance and
Windkessel boundary conditions (or any simpler ODE-based boundary condition) to a 3D LBbased fluid–structure interaction solver for hemodynamics (where the solid is governed by elastic
equations). The methodology introduces, for both ODE-based as well as non-stationary
boundaries, a discrete explicit-in-time modification to an LB non-equilibrium extrapolation
method [69, 87, 88] that has been previously proposed for fluid-only problems (i.e., no solid
interaction) and only for given (often analytical) Dirichlet-based pressure or velocity boundary
conditions (a particular novelty here is in the non-stationary switching between pressure and
velocity). The ultimate aim is to enable accurate physiological 0D-3D coupling conditions for
cardiovascular studies of, for example, the effects of left ventricle contractility on pulsatile
hemodynamics in the aorta.
2.2 Materials and Methods
2.2.1 Governing formulations
This section presents the governing equations employed in the numerical solver described in
Section 2.3: those for the fluid domain of a vessel (governed by lattice Boltzmann equations); those
for the LV-elastance model for the fluid inlet (governed by hybrid ODE-Dirichlet equations); and
those for the solid vessel walls (governed by elastodynamics equations). An illustration of the

15
complete coupled fluid–structure computational domain is presented in Figure 2.1 for the (interior)
fluid domain Ω, the solid wall ∂Ω1 and the 0D-3D coupled domain ∂Ω2.
Figure 2.1 A representative illustration of the complete 3D computational domain defined by Ω+∂Ω1+∂Ω2, where Ω denotes
the fluid interior (governed by lattice Boltzmann equations), ∂Ω1 denotes the compliant solid boundary (governed by
elastodynamics PDEs and incorporated by any appropriate fluid–structure interaction algorithm), and ∂Ω2 denotes the
coupled 0D-3D boundary (governed by time dependent ODEs). Figure from Wei H et al. International Journal for
Numerical Methods in Biomedical Engineering (2023): e3683.
2.2.2 3D lattice Boltzmann equations
The Immersed boundary-lattice Boltzmann method (IB-LBM) was used for the analysis of fluid
flow with moving boundaries. To solve the pressure and flow fields in the fluid domain, a singlerelaxation-time (SRT) incompressible LBM was used as an efficient solver of Navier-Stokes
equations [89, 90]. In such a method, the synchronous motions of the particles on a regular lattice
are enforced through a particle distribution function. This distribution function enforces mass and
momentum conservation. It also ensures that the fluid is Galilean invariant and isotropic. The
evolution of the distribution functions on the lattice is governed by the discrete Boltzmann
equation with the BGK (Bhatnagar-Gross-Krook) collision model and the forcing term to couple
the fluid and solid domains as,
𝑓𝑖
(𝒙 + 𝒆𝑖∆𝑡,𝑡 + ∆𝑡) − 𝑓𝑖
(𝒙,𝑡) = −
1
𝜏
[𝑓𝑖
(𝒙,𝑡) − 𝑓𝑖
𝑒𝑞(𝒙,𝑡)] + ∆𝑡𝐹𝑖
, 𝑖 = 0, … , 𝑁0 − 1, (2.1)

16
where 𝑓𝑖
(𝒙,𝑡) is the distribution function for particles with velocity 𝒆𝑖 at position 𝒙 and time t. ∆𝑡
and ∆𝑥 are the time step and lattice space, respectively. The sound speed is 𝑐 =
∆𝑥
∆𝑡
= 1. τ is a
dimensionless relaxation time constant which is associated with fluid viscosity in the form 𝜇 =
𝜌𝜗 = 𝜌𝑐𝑠
2
(𝜏 −
1
2
)∆𝑡, where 𝜗 is the kinematic viscosity and 𝑐𝑠 =
1
√3
𝑐 is the lattice sound speed.
Here, 𝑁0 = 19 since a D3Q19 (19 discrete velocity vectors) stencil is applied (and a D2Q9 stencil
is employed for the 3D-axisymmetric cases, i.e., 𝑁0 = 9.). The equilibrium distribution function
for incompressible LBM and the forcing term are defined as
𝑓𝑖
𝑒𝑞 = 𝜔𝑖𝜌0 + 𝜔𝑖𝜌 [
𝒆𝑖
∙ 𝒗
𝑐𝑠
2 +
(𝒆𝑖
∙ 𝒗)
2
2𝑐𝑠
4 −
𝒗
2
2𝑐𝑠
2
], (2.2)
𝐹𝑖 = (1 −
1
2𝜏
) 𝜔𝑖 (
𝒆𝑖 − 𝒗
𝑐𝑠
2 +
𝒆𝑖
∙ 𝒗
𝑐𝑠
4
𝒆𝑖) ∙ 𝒇, (2.3)
where ωi
is the weighting factor, ρ0 is related to the pressure by 𝜌0 =
𝑝
𝑐𝑠
2
, 𝒇 is the force density at
the Eulerian point, and velocity 𝐯 can be calculated by
𝜌0 = ∑𝑓𝑖
, (2.4)
𝜌𝒗 = ∑𝒆𝑖𝑓𝑖 +
1
2
𝒇∆𝑡. (2.5)
At the interface of the aortic and septal wall with the fluid the IB algorithm was used, and the
bounce-back boundary condition was used for modeling the fluid flow at the interface of the rigid
boundary (endograft). A source term was considered to satisfy the axisymmetric condition at the
centerline [91].

17
2.2.3 2D elastic wall equations
To compute the deformation of the elastic aortic and septum wall, the dynamic motion of these
two in the Lagrangian form is solved using:
𝜌𝑠ℎ
𝜕
2𝑿
𝜕𝑡
2 =
𝜕
𝜕𝑠 [𝐸ℎ ((
𝜕𝑿
𝜕𝑠 ∙
𝜕𝑿
𝜕𝑠 )
1/2
− 1)
𝜕𝑿
𝜕𝑠 −
𝜕
𝜕𝑠 (𝐸𝐼
𝜕
2𝑿
𝜕𝑠
2
)] + 𝑭𝐿
. (2.6)
where 𝑠 is the arclength of the wall, ℎ is the thickness, 𝑿 = (𝑋(𝑠,𝑡), 𝑌(𝑠,𝑡)) is the position of the
wall, 𝜌𝑠
is the density of the wall, 𝐸ℎ is the stretching stiffness, 𝐸𝐼 is the bending stiffness, and 𝑭𝐿
is the Lagrangian force exerted on the wall by the surrounding fluid. The simple support boundary
condition applied at the fixed points of the two sides of the septum wall [92], which is given by,
𝑿 = 𝑿0,
𝜕
2𝑿
𝜕𝑡
2 = (0,0). (2.7)
For the same geometrical configuration, the material parameter which affects the deformation of
the vessel wall governed by Eq. (2.6) is only the material elasticity (E). Since there is a range for
reported physiological values for vessel wall elasticity and also there are uncertainties in
determining the septum properties, it is essential to investigate the impact of selected material
parameter on the solution of the dynamical model Eq. (2.6).
2.2.4 3D elastic wall equations
In order to account for fluid–structure interactions in a vessel, the solid boundary ∂Ω1 is assumed
to be a thin wall that can be described by deformation of a compliant (elastic) wall in a Lagrangian
coordinate system, [93] that is,

18
𝜌𝑠ℎ
𝜕
2𝑿
𝜕𝑡
2 = ∑
𝜕
𝜕𝑠 [𝐸ℎ𝜑𝑖𝑗 (√
𝜕𝑿
𝜕𝑠𝑖
∙
𝜕𝑿
𝜕𝑠𝑗
− 𝛿𝑖𝑗)
𝜕𝑿
𝜕𝑠𝑗
−
𝜕
𝜕𝑠𝑗
(𝐸𝐼𝛾𝑖𝑗
𝜕
2𝑿
𝜕𝑠𝑖𝜕𝑠𝑗
)]
2
𝑖,𝑗=1
+ 𝑭𝐿
, (2.8)
where 𝜌𝑠
is the density of the solid wall; ℎ is the wall thickness; 𝑿(𝒔,𝑡) is the position of the solid
wall in Lagrangian coordinates 𝒔 = (𝑠1, 𝑠2); 𝛿𝑖𝑗 is the Kronecker delta; and 𝑭𝐿
is the Lagrangian
force exerted on the wall by the fluid. The product 𝐸ℎ and coefficient 𝐸𝐼
represent the stretching
and bending stiffnesses, respectively. The matrices 𝜑 and 𝛾 represent in- and out-of-plane effects
as a function of Poisson’s ratio 𝜈̂, respectively defined as
𝜑 =
(

1
1
2(1 + 𝜈̂)
1
2(1 + 𝜈̂)
1
)
, 𝛾 = (
1 1
1 1
) . (2.9)
For all 3D simulations of this work (including the physiological example study), the solid
deformation given by compliant wall equations (Eq. 2.8) is numerically simulated by the nonlinear
finite element method (FEM) solver of previous study [94] where the large-displacement and
small-strain deformation problems are handled by co-rotational schemes. The numerical strategy
has been successfully implemented in previous works for resolving a wide range of fluid–structure
interaction (FSI) problems incorporating elastic structures [93, 95-97]. Briefly, such a method uses
three-node triangular elements to describe the deformation using six degrees of freedom (three
displacement components and three angles of rotation) [98]. An iterative strategy is then used for
the time integration of the subsequent nonlinear systems of algebraic equations in order to ensure
second-order accuracy. A further detailed description of the particular finite element method
employed in this work can be found elsewhere [99]. In this work, the solid computational domain
consists of ~10,000 triangular elements and ~5,000 nodes, where the size of each solid element

19
is comparable to the lattice spacing (𝛥𝑥) in order to ensure the stability of the fluid-solid coupling
(described in what follows).
For the 3D-axisymmetric performance studies included in above section, a self-implemented
staggered grid finite difference (SGFD) methodology [92] is employed in the Lagrangian
coordinate system (where 𝑠 = 𝑠 ∈ 𝑅 is the arc length), where only the tension force given by
𝑇 = 𝐸ℎ (√
∂𝑋
∂𝑠
⋅
∂𝑋
∂𝑠
− 1), (2.10)
is defined on the interface (the displacement variable 𝑋(𝑠,𝑡), for example, is defined on all the
nodes). The solid deformation governed by Eq. (2.6) is subsequently solved by such a finite
difference methodology in a strong form [92, 100, 101]. That is, for an arbitrary variable, the
central, downwind and upwind difference approximations to the first-order derivatives, are given
by:
𝐷𝑠
0𝑋 = (𝑋(𝑠 + Δ𝑠/2) − 𝑋(𝑠 − Δ𝑠/2))/Δ𝑠,
𝐷𝑠
+𝑋 = (𝑋(𝑠 + Δ𝑠) − 𝑋(𝑠))/Δ𝑠, (2.11)
𝐷𝑠
−𝑋 = (𝑋(𝑠) − 𝑋(𝑠 − Δ𝑠))/Δ𝑠,
such that the corresponding second-order central difference approximation can be defined as
𝐷𝑠
+𝐷𝑠
−𝑋 = (𝑋(𝑠 + Δ𝑠) − 2𝑋(𝑠) + 𝑋(𝑠 − Δ𝑠))/Δ𝑠
2
, (2.12)
where the same difference approximation is applied for the time derivative. The tension force term
is hence approximated as

20
𝐷𝑠
(𝑇𝐷𝑠𝑋) = 𝐷𝑠
0
(𝑇𝐷𝑠
0𝑋) =
𝑇𝐷𝑠
0𝑋𝑠+Δ𝑠/2 − 𝑇𝐷𝑠
0𝑋𝑠−Δ𝑠/2
Δ𝑠
. (2.13)
Similarly. the bending force term can be approximated as
−𝐷𝑠𝑠(𝐸𝐼𝐷𝑠𝑠𝑋) = −𝐷𝑠
+𝐷𝑠
−(𝐸𝐼𝐷𝑠
+𝐷𝑠
−𝑋) = −𝐸𝐼
𝐷𝑠
+𝐷𝑠
−𝑋𝑠+Δ𝑠 − 2𝐷𝑠
+𝐷𝑠
−𝑋𝑠 + 𝐷𝑠
+𝐷𝑠
−𝑋𝑠−Δ𝑠
Δ𝑠
2
. (2.14)
2.2.5 Implementations of the 0D Boundary Conditions: Varying-elastance and
lumped parameter models
The LV/RV was modeled as time-varying coupled with the aorta. The extension tube outflow
boundary model was used for the truncated vasculature at the outlet of our 3-D FSI solver. At the
inlet, the pressure 𝑝𝑣
(𝑡) inside the LV/RV and the corresponding volume 𝑉𝑣
(𝑡) in the LV/RV are
connected via time varying compliance 𝐶𝑣
(𝑡) given by,
𝑉𝑣
(𝑡) − 𝑉𝑑𝑒𝑎𝑑 = 𝐶𝑣
(𝑡)𝑝𝑣
(𝑡). (2.15)
In Eq. (2.15), the constant 𝑉𝑑𝑒𝑎𝑑 known as the dead volume is the limit for pressure generation.
Substituting the relation between the flow into the aorta with the 𝑉𝑣
(𝑡) and differentiating Eq.
(2.15) with respect to t, we can get the following ordinary differential equation (ODE) for the
pressure inside the LV and RV:
𝜕𝑝𝑣
(𝑡)
𝜕𝑡 = −
1
𝐶𝑣
(𝑡)
[
𝜕𝐶𝑣
(𝑡)
𝜕𝑡 𝑝𝑣
(𝑡) + 𝑄(𝑥 = 0,𝑡)], (2.16)
Clinically, 𝐶𝑣
(𝑡) stands for inverse of LV/RV end-systolic elastance (𝐸𝑒𝑠) which is the measure
of LV/RV contractility [64, 102] . Once 𝑃𝑣
(𝑡) is greater than the pressure at the interface of the
aorta and the LV/RV, the valve opens and 𝑝(𝑥 = 0, 𝑡) = 𝑝𝑣
(𝑡) with the flow condition given by

21
the fluid solver (the ODE condition). Once the inflow reaches zero (or, numerically, the time at
which 𝑄(𝑥 = 0, 𝑡) ≤ 0), the valve closes, and the left boundary condition remains 𝑄(𝑥 = 0,
𝑡) = 0 (a Dirichlet-type condition). The empirically given time-varying compliance (𝐶𝑣
(𝑡) )
reported from clinical data for normal contractile state of LV/RV [102].
At the terminal boundary 𝑥 = 𝐿, the physical outflow boundary model approximates the effect of
the truncated vasculature and peripheral vessels. This extension tube boundary model is a simple
outflow boundary condition for three-dimensional fluid-structure interaction (FSI) simulation of
pulsatile blood flow in compliant vessels. In this structural model, the computational domain is
extended with an elastic tube connected to a rigid contraction to account for the compliance,
resistance, and the wave reflection of the truncated vascular network. Previous studies showed that
this model is suitable for cardiac transient (non-periodic) events [103, 104]. The parameters of the
outflow boundary condition model are given in previous study, where the contraction ratio κ is the
ratio of the radius of the rigid boundary tube (after the contraction) to the original radius (before
the contraction).
At the outlet boundary ∂Ωoutlet ⊂ ∂Ω2 of the 3D physiological aorta considered in this study, a
conventional 0D lumped parameter model, a so-called circuit-like Windkessel model, is employed
to represent the effects of truncated vasculature. The outlet of the fluid domain is coupled to such
a model through a matching characteristic impedance 𝑍𝑤 that is related to the fluid inductance and
overall outgoing aortic compliance. Together with an effective chamber compliance 𝐶𝑤 and a total
peripheral resistance 𝑅𝑤, the pressure 𝑃𝑤 in the terminal compliance chamber is related to the
aortic pressure 𝑃(𝑥,𝑡) at the outlet boundary 𝒙 ∈ ∂Ωoutlet through an ODE given by

22
∂𝑃𝑤
∂𝑡
(𝑡) =
1
𝐶𝑤𝑍𝑤
𝑃(𝑥,𝑡) −
𝑅𝑤 + 𝑍𝑤
𝐶𝑤𝑅𝑤𝑍𝑤
𝑃𝑤(𝑡). (2.17)
The corresponding 0D outflow 𝑄(𝒙,𝑡) at the outlet 𝒙 ∈ ∂Ωoutlet is given by
𝑄(𝒙,𝑡) =
1
𝑍𝑤
(𝑃(𝒙,𝑡) − 𝑃𝑤(𝑡)). (2.18)
Further details on the parameters, usage and implementation of both this outgoing Windkessel
model, as well as the LV-elastance heart model above, can be found elsewhere [64].
2.2.6 A direct 0D-3D coupling for ODE-based boundary equations and lattice
Boltzmann solvers
For the 3D-axisymmetric case, the D2Q9 velocity model is applied in the LBM with the sound
speed c where the velocity set is given by:
𝒆𝑖 =
{

0 𝑖 = 0
(𝑐𝑜𝑠 [(𝑖 − 1)
𝜋
2
], 𝑠𝑖𝑛 [(𝑖 − 1)
𝜋
2
]) 𝑐 𝑖 = 1, 2, 3, 4
√2 (𝑐𝑜𝑠 [(𝑖 − 5)
𝜋
2
+
𝜋
4
], 𝑠𝑖𝑛 [(𝑖 − 5)
𝜋
2
] +
𝜋
4
) 𝑐 𝑖 = 5, 6, 7, 8
. (2.19)
Axisymmetric LBM is implemented in this study using an incompressible D2Q9 BGK model. In
pseudo-Cartesian coordinates (𝑥, 𝑟) for describing 3D axisymmetric flow, Eq. (3.1) can be
transformed into
𝑓𝑖
(𝑥 + 𝑒𝑖𝛥𝑡,𝑡 + 𝛥𝑡) − 𝑓𝑖
(𝑥,𝑡) = −
1
𝜏
[𝑓𝑖
(𝑥,𝑡) − 𝑓𝑖
𝑒𝑞(𝑥,𝑡)] + 𝛥𝑡𝐹𝑖
(𝑥,𝑡) + 𝐻𝑖
(𝑥,𝑡), (2.20)
where a source term 𝐻𝑖
(𝑥,𝑡) is given by
𝐻𝑖
(𝑥,𝑡) = 𝛥𝑡ℎ𝑖
(1)
(𝑥,𝑡) + 𝛥𝑡
2ℎ𝑖
(2)
(𝑥,𝑡), (2.21)

23
ℎ𝑖
(1) = −
𝜔𝑖𝜌𝑣𝑟
𝑟
, (2.22)
ℎ𝑖
(2) = −𝜔𝑖
3𝜈
𝑟
[𝜕𝑦𝑃 + 𝜌𝜕𝑥𝑣𝑥𝑣𝑟 + 𝜌𝜕𝑟𝑣𝑟𝑣𝑟 + 𝜌(𝜕𝑟𝑣𝑥 − 𝜕𝑥𝑣𝑟
)𝑒𝑖𝑥]. (2.23)
𝐻𝑖
(𝑥,𝑡) is the added source term into the collision step defined based on ℎ𝑖
(1)
and ℎ𝑖
(2) with 𝑃 =
𝑐𝑠
2
∙ 𝜌𝑜. The source term is added to recover the extra terms caused by the curvature from the
continuity equation and Navier–Stokes equation in cylindrical coordinates [90, 105]. For
calculating the derivatives of the velocity vector along the radial and axial directions, the terms
𝜕𝑟𝑣𝑥 + 𝜕𝑥𝑣𝑟
, 𝜕𝑥𝑣𝑥 and 𝜕𝑟𝑣𝑟 can be obtained by the following equation [90]
𝜌𝜈(𝜕𝛽𝑣𝛼 + 𝜕𝛼𝑣𝛽) = − (1 −
1
2𝜏
)∑(𝑓𝑖 − 𝑓𝑖
𝑒𝑞)
8
𝑖=0
𝑒𝑖𝛼𝑒𝑖𝛽 + 𝑜(𝜀
2
), (2.24)
where substituting 𝛼 = 𝑥 and 𝛽 = 𝑟 gives us a relation for 𝜕𝑟𝑣𝑥 + 𝜕𝑥𝑣𝑟
;substituting 𝛼 = 𝛽 = 𝑥
gives us a relation for 𝜕𝑥𝑣𝑥 ; and substituting 𝛼 = 𝛽 = 𝑟 gives us a relation for 𝜕𝑟𝑣𝑟
. For
calculating 𝜕𝑟𝑣𝑥 − 𝜕𝑥𝑣𝑟
in Eq. (2.23) the only the value left unknown is 𝜕𝑥𝑣𝑟
. Below is a finite
difference method employed to obtain 𝜕𝑥𝑣𝑟 at lattice node (𝑖,𝑗) with the following expression:
(𝜕𝑥𝑣𝑟
)(𝑖,𝑗) =
(𝑣𝑟
)(𝑖+1,𝑗) − (𝑣𝑟
)(𝑖−1,𝑗)
2∆𝑥
. (2.25)
This section proposes a discrete, direct methodology for the coupling of 3D lattice Boltzmann
equations with dynamic (ODE-based) 0D models which, as described before, are often found in
inflow and outflow conditions for cardiovascular configurations. The hybrid ODE-Dirichlet
system governed by Eq. (2.16) represents a most complex form of the myriad such formulations
found in cardiovascular modeling, and its particular treatment is discussed in above Section

24
(although the method proposed in what follows is straightforwardly applicable to other ODE-based
0D boundary equations such as Windkessel models). The general strategy for such
multidimensional coupling of the solver presented in this work is based on the non-equilibrium
extrapolation method, [69, 106] which has been introduced as an alternative to typical “bounceback” methods40 used to implement (given) pressure and velocity boundary conditions for LB
methods in order to preserve a consistent order-of-accuracy between the boundaries and the orderof-accuracy inherent to LB formulations [106]. In particular, such a method is ideally suited for
curved fluid boundaries [88, 107] (such as those provided by fluid–structure interfaces of interest
in this work), where the physical boundary need not coincide with the regular fluid lattice.
2.2.7 On the particularities of the specific hybrid ODE-Dirichlet LV model
The non-stationary switching condition of the specific 0D LV-elastance equations presented in
above section leads to a loss in regularity of the solution near boundaries governed by the hybridODE Dirichlet system. Indeed, the corresponding velocity at the time 𝑡 = 𝑇𝑑 of valve closure (also
known as a dicrotic notch on the corresponding pressure waveform) is generally non-zero, and
hence the switch to a Dirichlet condition for diastole leads to a discontinuity in the velocity solution
close to the 0D-3D boundary ∂Ω2. That is, for a point 𝑥𝑓 ∈ Ω neighboring a boundary node 𝑥 ∈
∂Ω2 with a velocity solution during the systolic phase (𝑡 ∈ [0, 𝑇𝑑
]) defined as 𝑣0
(𝑡) = 𝑣(𝑥𝑓,𝑡),
the velocity over a complete cardiac cycle of length T can be expressed as
𝑣(𝑡) = 𝑣0
(𝑡),𝑡 ∈ [0, 𝑇𝑑
] (𝑣𝑎𝑙𝑣𝑒𝑜𝑝𝑒𝑛), 𝑎𝑛𝑑 𝑣(𝑡) = 0,𝑡 ∈ (𝑇𝑑, 𝑇] (𝑣𝑎𝑙𝑣𝑒𝑐𝑙𝑜𝑠𝑒𝑑), (2.26)

25
Figure 2.2 An illustrative example diagram of the direct (explicit) coupling between a 0D model (e.g., the ODEs
corresponding to the LV-heart model and the Windkessel model) and the 3D lattice Boltzmann (LB) model. The flow in
∂Ω2 is computed in terms of the LB distribution functions at a timestep n which is fed into ODEs governing the 0D model.
The corresponding pressure produced by the 0D model is then re-translated into distribution functions on the boundary
via the non-equilibrium extrapolation.
which, again, can be discontinuous. Such a loss in the smoothness of the velocity derived from the
switching solutions to the LV-elastance boundary equations may lead to spurious reflections
(including in the form of artificial backflow) at the point of closure of the valve, i.e., at 𝑡 = 𝑇𝑑. In
order to ensure there is no spurious backflow or artificial oscillations resulting from immediately
setting a zero velocity, we introduce a smooth transition function to the velocity profile upon valve
closure. Such a function can be defined as a continuously differentiable 𝐶
∞ smoothing-to-zero
function 𝑆(𝑡) and can be derived from an exponential or erf-based partition-of-unity as
𝑆(𝑡) = 𝑆(𝑡,𝑡0, 𝐿𝑆
) = exp (
2𝑒
−1/𝑢(𝑡)
𝑢(𝑡)−1
) , 𝑢(𝑡) =
𝑡−𝑡0
𝐿𝑆
,𝑡0 ≤ 𝑡 ≤ 𝑡0 + 𝐿𝑆
, (2.27)
where 𝑡0 = 𝑇𝑑 is the location of the start of the smoothing-to-zero (i.e., the time of valve closure)
and 𝐿𝑆
is the interval over which to smoothly transition to zero (indeed, there is a noninstantaneous valve closure in physical reality).

26
An illustrative example of the smoothing function 𝑆 for closure time 𝑡0 = 𝑇𝑑 = 0.5𝑠 , Δ𝑡 =
0.005 𝑠, and 𝑁𝑆 = 15 is shown in Figure 3.3.
Figure 2.3 An illustration of the exponential-based 𝑪
∞-function 𝑺(𝒕, 𝒕𝒅, 𝑾) which is employed to smoothly reduce lattice
Boltzmann velocity amplitudes as the valve closes in the 0D LV-elastance model (i.e., the ODE-Dirichlet switch).
A flowchart summarizing the complete implementation and smooth switching of the 0D LVelastance model with the 𝐶
∞ smoothing employed in this work is presented in Figure 2.4.
2.2.8 Algorithmic details
The following details the numerical algorithms/methods employed for the LB fluid solver and the
solid (elastic) solver (which utilizes finite differences for 3D-axisymmetric and finite elements for
3D). Their interactions, described in previous section, can be facilitated by any appropriate fluid–
structure algorithm: in this work, an immersed boundary method is used.

27
Figure 2.4 A flowchart describing the implementation of the particular hybrid ODE-Dirichlet 0D LV-elastance model that
is of interest.
2.2.9 Fluid–structure interactions (FSI)
The solid deformation equation (Eq. (2.6)) was solved by Finite Difference Method (FDM) and
Finite Element Method (FEM) [99]. For coupling the fluid and solid systems, any suitable FSI
coupling strategy can be employed. For this particular work, the immersed boundary (IB) method
is used to couple the LB method of the fluid with the 3D FEM (or 3Daxisymmetric FDM) of the
solid [92, 93]. This method has been extensively used to simulate FSI problems in cardiovascular
biomechanics [108-111]. The IB method was used to couple the fluid and solid solvers.
Particularly, an explicit velocity correction-based IB approach was used in this study which has
been extensively used to simulate the FSI problems in cardiovascular biomechanics [109, 110]. In
this method, the body force term 𝒇 is used as an interaction force between the fluid and the
boundary to enforce the no-slip velocity boundary condition by introducing an intermediate
velocity 𝒗
∗ by

28
𝒗(𝒙,𝑡) = 𝒗
∗
(𝒙,𝑡) + 𝛿𝒗(𝒙,𝑡). (2.28)
The relation between the velocity correction 𝛿𝒗 and the body force term 𝒇 is,
𝜌𝛿𝒗(𝒙,𝑡) =
1
2
𝒇(𝒙,𝑡)𝛿𝒕. (2.29)
While in the conventional IBM, 𝒇 is computed in advance and then the velocity correction 𝛿𝒗 and
corrected velocity 𝒗(𝒙,𝑡) are explicitly computed, there is no guarantee the velocity at the
boundary satisfies the no-slip boundary condition [112]. In the revised implicit velocity correctionbased immersed boundary approach, the velocity correction 𝛿𝒗 term at the Eulerian point (fluid
domain) can be first obtained by the following Dirac delta function interpolation as
𝛿𝒗(𝒙,𝑡) = ∫𝛿𝑽(𝑠,𝑡)𝛿(𝒙 − 𝑿(𝑠,𝑡))𝑑𝑠
𝛤
, (2.30)
Where 𝛿(𝒙 − 𝑿(𝑠,𝑡)) is smoothly approximated by a continuous kernel distribution and 𝛿𝑽(𝑠,𝑡)
is the unknown velocity correction vector at every Lagrangian point at the FSI boundary 𝛤 as
proposed by previous works [112]. Note that in the notation above, 𝒙 is the Eulerian coordinates
related to the fluid phase while 𝑿 stand for Lagrangian coordinates related to the solid phase. In
order to meet the non-slip boundary condition, the fluid velocity at the boundary point 𝛺 obtained
by the smooth 𝛿 function interpolation must be equal to the wall velocity 𝑽 at the same position.
Its mathematical expression is
𝑽(𝑠,𝑡) = ∫ 𝒗(𝒙,𝑡)𝛿(𝒙 − 𝑿(𝑠,𝑡))𝑑𝒙
𝛺
. (2.31)
Substituting Eq. (2.29) and (2.30) into Eq. (2.31), we can get the following equation:

29
𝑽(𝑠,𝑡) = ∫ 𝒗
∗
(𝒙,𝑡)𝛿(𝒙 − 𝑿(𝑠,𝑡))𝑑𝒙
𝛺
+ ∫ [∫𝛿𝑽(𝑠,𝑡)𝛿(𝒙 − 𝑿(𝑠,𝑡))𝑑𝑠
𝛤
] 𝛿(𝒙 − 𝑿(𝑠,𝑡))𝑑𝒙
𝛺
, (2.32)
where the only unknown velocity correction 𝛿𝑽(𝑠,𝑡) can be obtained by solving this equation. In
the utilized IB approach, after determining the velocity correction terms via Eq. (2.30), the force
density acting on the fluid phase 𝒇 can be calculated using Eq. (2.29). Lastly, the boundary force
density at Lagrangian points 𝑭𝐿 can be explicitly found by
𝑭𝐿
(𝑠,𝑡) = − ∫ 𝒇(𝒙,𝑡)𝛿(𝒙 − 𝑿(𝑠,𝑡))𝑑𝒙
𝛺
. (2.33)
2.2.10 Efficient boundary condition-enforced immersed boundary method
The following well-defined equation system is obtained in discrete form,
𝑉(𝑋𝑗) = ∑𝐷
𝑖
(𝑥𝑖 − 𝑋𝑗)𝑣
∗ℎ
𝑛 + ∑𝐷(𝑥𝑖 − 𝑋𝑗)∑𝐷(𝑥𝑖 − 𝑋𝑗)δ𝑉(𝑋𝑗)δ𝑠𝑗
𝑖 𝑗
ℎ
𝑛
, (2.34)
where δ𝑉(𝑋𝑗)δ𝑠𝑗
is unknown and all the other terms are pre-determined. The well-defined Eq.
(2.34) can be cast into matrix form as,
𝑨𝒀 = 𝒃, (2.35)
Where we have:
𝒀 = [δ𝑉(𝑋𝑗)δ𝑠𝑗
], 𝒃 = [𝑉(𝑋𝑗)/ℎ
𝑛
] − 𝐷[𝑣
∗
(𝑥𝑖
)],𝑨 = 𝑫𝑫𝑇
, 𝑎𝑛𝑑 𝑫 = [𝐷(𝑥𝑖 − 𝑋𝑗)]. (2.36)
Eq. (2.36) implicitly implements the boundary condition and thus removes unphysical penetration
of streamlines. By plugging the solution:
𝒀 = 𝑨
−1𝒃. (2.37)

30
This IBM that enforces boundary conditions initially accurately interprets the no-slip boundary
condition. However, it becomes computationally burdensome when simulating moving boundary
problems because a large matrix must be assembled at every time step, and there is an implicit
resolving process. The computational complexity of O(𝑁
2
) grows significantly as the number of
Lagrangian points (𝑁) distributed on the immersed boundary increases. To overcome these
limitations, and to enhance the efficiency of the boundary condition-enforced IBM: the explicit
technique is employed. The IBM using the explicit technique is a non-iterative approach based on
error analysis with a computational complexity of O(𝑁). The explicit technique presented in this
section aims to resolve velocity correction in IBM explicitly in one step by:
𝑌𝑗 = 𝑑𝑗
−1
⋅ 𝑏𝑗
, 𝑤𝑖𝑡ℎ 𝑑𝑗 = ∑𝐴𝑗𝑖
𝑖
. (2.38)
Such explicit technique-based IBM presents very simple procedures. Overall, the explicit
technique requires no iteration. The explicit technique-based IBM has a computational complexity
of O(𝑁), which implies its high efficiency in applications [113].
Such a formulation of the IB numerical strategy has been successfully applied to a wide range of
FSI problems, [93, 96, 114, 115] including those governed by the dynamics of fluid flow over a
circular flexible plate and an inverted flexible plate [93, 96]. The overall numerical algorithm for
solving the complete FSI system is summarized in the block-diagram of Figure 2.5, and its
corresponding pseudocode implementation is provided in Algorithm 1. At the start of each
numerical simulation, the 0D and 3D domains can be initialized with 𝑈0 = 0, 𝑄0 = 0 and an
arbitrary 𝑃0 ≠ 0, respectively.

31
Figure 2.5 A fluid–structure interaction (FSI) procedure facilitated by the immersed boundary (IB) method.

32
2.3 Validation of the 0D-3D framework on Left-heart
2.3.1 Introduction of the validation
This section introduces a series of performance studies validating the effectiveness and precision
of the fluid and solid solvers outlined here. Initially, we examine a simplified model of the aorta,
followed by an exploration of a physiological study involving oscillatory wall shear stress within
a 3D aorta, incorporating carotid and renal branches.
Advanced congestive heart failure (CHF) often leads to decreased blood flow in the aorta due to
reduced cardiac output (CO) and a low ejection fraction, affecting nearly half of the patients.
Various factors, including the direct coupling between the left ventricle (LV) and the arterial
system, can influence the heart's pumping capacity. [102, 116, 117] In this chapter, we employ a
hybrid ODE-Dirichlet boundary condition known for its effectiveness in capturing the dynamic
and nonlinear aspects of this complex coupling. This boundary condition alternates between
systole—an ordinary differential equation (ODE)—and diastole—a Dirichlet condition. Utilizing
a 0D elastance-(compliance-) based LV model, we generate physiological pressure and flow
waveforms capable of accommodating diverse contractility levels and cardiac outputs. By
integrating this model with 3D lattice Boltzmann equations, we investigate how the heart's
dynamics impact 3D fluid–structure interactions, particularly those related to near-wall shear stress
(WSS). Notably, both mechanical experiments [118] and in vivo/in vitro studies [119-122] have
revealed the presence of negative WSS, indicative of retrograde flow during specific intervals
within the cardiac cycle. These effects have demonstrated strong correlations with the state of
CHF.

33
2.3.2 The complete FSI solver coupled with ODE-based heart model (time varying
elastance model)
In order to verify the complete FSI solver coupled to a 0D hybrid ODE-Dirichlet heart model, one
can consider the axisymmetric straight aorta configuration (of length 25D) presented in Figure 2.6.
Adopting parameters of the LV-elastance hybrid ODE-Dirichlet model from previous work
corresponding to an end-systolic LV elastance 𝐸𝑒𝑠 2.2 𝑚𝑚ℎ𝑔/𝑚𝑙 mmHg/ml and a 𝐶𝑂 =
4.3 𝐿/𝑚𝑖𝑛.
Figure 2.6 Diagram of the simplified straight aorta test case coupled to the LV-elastance model.
Figure 2.7 (left) presents the expected physiologically-accurate pressure profiles at the 0D-3D
interface as simulated by the complete solver for discretizations corresponding to Δx =
1/20,1/32,1/64,1/100 and 1/128, where physical parameters of Wo = 16, a non-dimensionalized D
= 1 (corresponding to 24 mm), μ = 3.5cP and ρ = 1000kg/m3
are employed. The timestep is fixed
again and is taken small enough so that errors are dominated by the spatial discretization. Figure
2.7 (right) presents the corresponding L
∞
errors (relative to the finest solution), where a
convergence between first and second order can be observed (and is expected from the secondorder nature of the lattice Boltzmann solver and the first-order discretizations of the LV-elastance
ODEs). Figure 2.8 (left) and Figure 2.9 (left) additionally present the simulated physiological flow
profiles at the inlet and the pressure profiles at the midpoint of the vessel, respectively. The
corresponding L∞
errors (relative to the finest discretization of Δx = 1/128) are presented in Figure

34
2.8 (right) and Figure 2.9 (right), respectively. As before, one can appreciate the convergence and
accuracy as expected from the second-order fluid discretization and the first order LV-elastance
ODE time integration.
Figure 2.7 (Left) Physiological pressure profiles at the inlet for successively-refined discretizations of a straight aorta as
produced by the LV-elastance model. (Right) The corresponding L∞ errors (relative to the finest solution)
Figure 2.8 (Left) Flow profiles at the inlet for successively-refined discretizations of a straight aorta as produced by the LVelastance model. (Right) The corresponding L∞ errors (relative to the finest solution)

35
Figure 2.9 (Left) Pressure profiles at the midpoint for successively-refined discretizations of a straight aorta as produced
by the LV elastance model. (Right) The corresponding L∞ errors (relative to the finest solution)
2.3.3 An example physiological case: wall shear stress in the aorta
As described above. A predominant effect of advanced congestive heart failure (CHF) is reduced
blood flow in the aorta that results from a reduction in cardiac output (CO) and a low ejection
fraction (in almost half of the patients). Many factors can influence the heart's pumping ability,
including those related to the direct coupling between the LV and the arterial system [102, 116,
117]. The hybrid ODE-Dirichlet boundary condition considered in this paper has been chosen for
its ability to model the non-stationary and nonlinear effects of such complex coupling (which is
expressed as an alternating boundary condition between systole—an ODE—and diastole—a
Dirichlet condition). The 0D elastance-(compliance-) based LV model enables the generation of
physiological pressure and flow waveforms that can account for different contractility and cardiac
outputs, and its corresponding coupling to 3D lattice Boltzmann equations can enable investigation
of the heart's influence on corresponding 3D fluid–structure effects such as those related to near
wall shear stress (WSS). Indeed, mechanical experiments [118] and both in vivo/in vitro studies
[119-122] have shown there can exist negative WSS corresponding to a retrograde flow during a

36
substantial interval within a cardiac cycle, and such effects have been strongly correlated with the
state of CHF [118].
As a demonstration of the applicability of the proposed solver toward exploring these parameters
for studying pathophysiological conditions in the cardiovascular system (e.g., CHF), a
computational model of a simplified 3D aorta that includes carotid and renal branches is
considered and illustrated in Figure 2.7 (left), where the elastic wall is discretized by finite
elements and the fluid by lattice Boltzmann as described in previous section. For an effective aortic
diameter 𝐷 (taken to be unity in the non-dimensionalized configuration), such a domain
corresponds to Cartesian coordinates for a non-dimensionalized 𝐷 (corresponding to 24mm). The
fluid is considered Newtonian (although the LBM is well-known to also handle non-Newtonian
flow) with a Reynolds number of 𝑅𝑒 = 884. For the compliant wall, a linear elastic material with
Young's modulus of 𝐸 = 0.5 MPa and a wall thickness corresponding to h=1 mm is considered.
The complete fluid–structure (immersed boundary) solver, where a no-slip condition is imposed
at the fluid–structure interface, is coupled to the 0D LV-elastance heart model at the inlet and a
Windkessel ODE at the outlet. At all peripheral branch outlets, extension tube boundary models
are employed. Figure 2.10 (right) presents the corresponding normalized velocity magnitudes
produced by a simulation that employs discretizations of Δx = 1/32D, Δt = 1/50000 s and is
advanced up to a time T = 5 s (where 1 s corresponds to the period of a cardiac cycle). The LV
parameters (including the compliance function) and the Windkessel lumped parameters are
adopted from previous work and correspond to a healthy case with normal contractility.
Additionally, Figure 2.11 illustrates the expected physiological characteristics of the LV and aortic
pressures, particularly the equality during systole (in the absence of a diseased valve condition)

37
between ventricular pressure and the aortic pressure at the coupled boundary. Figure 2.12 (left)
further demonstrates that the simulations capture the expected increase in pressure amplitude as
the LV-sourced waves propagate downstream. Figure 2.12 (right) provides the corresponding flow
profiles as simulated at the inlet, midpoint and outlet of the 3D aorta, demonstrating the
physiologically-expected decrease in amplitude as flow propagates downstream. For an
experimental reference, Figure 2.13. additionally presents experimental, demonstrating similar
morphology characteristics and physiological ranges for both pressure and flow waveforms.
Differences between the simulated results of Figures 2.11 and 2.12 and the experiments of Figure
2.13 can be attributed primarily to differences in the physiological parameters employed (HR = 60
BPM, CO = 3.5 L/min for the simulations; HR = 75 BPM, CO = 5.0 L/min for the pressure data;
and HR = 75 BPM, CO = 3.0 L/min for the flow data).
Figure 2.10 (Left) Diagram of a physiologically relevant 3D aortic domain (with carotid and renal branches) coupled to an
LV-elastance model at the aortic inlet and a lumped-parameter Windkessel model at the aortic outlet. (Right) A temporal
snapshot of the normalized flow velocity magnitude produced by the solver. Figure from Wei H et al. International Journal
for Numerical Methods in Biomedical Engineering (2023): e3683.
In investigating WSS (as a relevant hemodynamic biomarker in CHF [118]), two contractility
cases (representing a low flow rate and a high flow rate) can be considered employing the same
Womersely number Wo = 16. For normal contractility, the end-systolic LV elastance Ees is set to
1.76 mmHg/ml, which corresponds to a cardiac output of CO ≈3.5 l/min and is in accordance with

38
values employed in experimental studies. [118] For the high contractility scenario, Ees = 2.75
mmHg/ml. Again, for both cases, the Womersley number Wo = 16 (corresponding to a heart rate
of HR = 60 BPM[118]) is fixed. Figure 2.14 presents the corresponding wall shear stress (WSS)
for both normal and high contractility cases, as calculated through the fluid points next to the solid
wall [118] via the expression
𝑊𝑆𝑆(𝒙,𝑡) = 𝜇
𝜕𝑣1
𝑥3
(𝒙,𝑡). (2.39)
where 𝜇 is the fluid viscosity (corresponding to 3.5 centipoise), 𝑣1 is the simulated axial velocity,
and 𝑥3 is the dimension normal to the wall. As expected, [118] a negative WSS (corresponding to
a retrograde flow) is evident for the normal contractility case parameters, and such effects
disappear in the high contractility case since the corresponding flow rate is very high. These results
are in agreement with the experimental results presented in Gharib and Beizaie. [118]
Figure 2.11 Aortic pressure at the inlet (blue) and the corresponding ventricular pressure (dashed red) for healthy patient
parameters, simulated by the 3D FSI solver. As expected, aortic inlet pressure is equal to LV pressure during the systolic
phase (when the valve is opened).

39
Figure 2.12 (left) Pressure profiles at the inlet, midpoint and outlet of the 3D aorta model, demonstrating the expected
amplification as flow propagates downstream. (Right) Corresponding flow profiles simulated at the inlet, midpoint and
outlet, demonstrating the expected decrease in flow amplitude as the wave propagates downstream.
Figure 2.13 (Left) Experimental data from an in vitro LV-aortic simulator. (Right) Experimental flow data (of a different
run) from the same setup. The overall morphologies and physiological ranges are in agreement with those of the simulations
presented in this work.

40
Figure 2.14 Simulated wall shear stress (WSS), for both normal and high contractility cases, at a location between the
midpoint and outlet of the 3D aorta model.
Overall, this work presents a direct 0D-3D coupling for dynamic (ODE-based) boundary
conditions applied to lattice Boltzmann solvers for hemodynamic flow. Benchmark performance
studies and a physiological case of wall shear stress in a simplified 3D aorta are treated in order to
validate the proposed methodology and its implementation. In particular, this work treats a most
complicated configuration of such coupling conditions: a hybrid non-stationary ODE-Dirichlet
boundary condition. Such a methodology produces a physiologically accurate hemodynamics
solver (with a heart model) for studying wave propagation and pulsatile blood flow in arterial and
venous vessels. The methodology introduced in this paper can be easily extended to non-switching
ODE conditions such as 0D lumped parameter models (e.g., Windkessels for truncating
vasculature at vessel outlets), as well as to other methods for treating fluid–structure interactions
(facilitated here by an immersed boundary method). Such a direct 0D-3D coupling with the
proposed regularization can also be applied to any other fluid problem that is governed by lattice
Boltzmann equations and that requires direct time-dependent ODEs as boundary conditions.

41
Chapter 3: 3D Numerical investigation of the significance of wall
stiffness and non-Newtonian effects on jugular vein pulsatile
hemodynamics
The primary objective of designing and developing the 0D-3D coupled Fluid-Structure Interaction
(FSI) solver within a Lattice Boltzmann Method (LBM) framework is to thoroughly investigate
the complex physics and fluid/pulsatile wave dynamics within the right heart-venous system. The
Lattice Boltzmann Method is particularly well-suited for capturing intricate fluid dynamics due to
its ability to handle complex boundary conditions, multiphase flows, and turbulent behaviors with
high accuracy. By integrating this method into our 0D-3D coupled FSI solver, we aim to achieve
a comprehensive understanding of the interactions between the right heart and venous system
under various physiological and pathological conditions. This advanced solver is intended to
bridge the gap between simplified 0D models, which provide a broad overview of circulatory
dynamics of the cardiovascular system, capturing essential aspects of hemodynamics, and detailed
3D models, which offer a more precise representation of the local fluid and structural interactions
occurring within the cardiovascular system. We can simulate the entire right heart-venous system
with greater fidelity. This combination allows us to study pulsatile flow dynamics, venous pressure
variations, and the mechanical interactions between the heart and veins in a more holistic manner.
By integrating the strengths of both 0D and 3D Lattice Boltzmann Method within this 0D-3D
coupled FSI framework, we can simulate and analyze the detailed behavior of blood flow and
pressure wave propagation through the right atrium and venous system under various physiological
and pathological conditions (including the study of pulsatile flow dynamics, venous pressure
fluctuations, and the mechanical interactions between the heart and veins) and explore how

42
changes in heart function, blood rheology, and vessel elasticity impact overall circulatory health.
This research has significant implications for improving diagnostic techniques, enhancing the
design of medical devices, and developing more effective treatments for cardiovascular diseases,
ultimately leading to better patient outcomes.
3.1 Chapter introduction
The fluid dynamics within the cerebral venous system are complex and poorly understood
compared to the arterial system, despite its significant clinical relevance. The cerebral venous
circulation has not been studied as extensively, though recent research highlights its close
connection to heart function and its role in various cardiovascular and cerebrovascular diseases
[1].
Understanding cerebral venous circulation is crucial for identifying changes contributing to
neurological and neurodegenerative diseases. Abnormalities in cerebral venous circulation can
impact cerebral blood flow and cognitive function. Advances in imaging techniques have increased
interest in the biophysical and hemodynamic parameters of the intracranial venous system. For
example, aortic stiffness has been linked to reduced cerebral blood flow and cognitive
performance, and arterial stiffness to microvascular brain disease. Venous pulsatility and jugular
venous flow abnormalities are associated with conditions like multiple sclerosis (MS) [2]. The role
of the venous system in neurological and neurodegenerative diseases remains underexplored,
despite its association with several central nervous system diseases [3-6]. A deeper understanding
of cerebral venous physiology and pathophysiology is essential for developing treatments for
neurodegenerative diseases. Conditions like mild cognitive impairment (MCI) and Alzheimer's
disease (AD) are increasingly common and linked to abnormalities in the extracranial venous

43
system, such as jugular venous reflux (JVR) [13, 14]. JVR can impair cerebral venous outflow,
potentially causing neurologic dysfunction and structural changes in the brain. Understanding the
extracranial venous system's role in CNS disorders is crucial for advancing knowledge of these
conditions' underlying mechanisms. Bedside analysis of the jugular venous pulse can provide
critical information about right heart hemodynamics, aiding in the diagnosis of certain diseases.
Assessing jugular venous pulse (JVP) provides insights into right heart function, though invasive
measurements are often deemed necessary [17]. Improved non-invasive methods, such as
ultrasound assessment of the jugular venous pulse, offer promising alternatives for determining
central venous pressure (CVP) [21]. Research on internal jugular vein dynamics is also crucial for
space exploration, where astronauts face unique physiological stressors that can affect the
cardiovascular system and lead to conditions like venous thrombosis [22, 23].
Understanding fluid dynamics in the jugular vein and its behavior under pulsatile flow conditions
remains a significant area of study with implications for both medical and spaceflight applications.
Further research in these areas is needed to uncover important insights and develop effective
interventions. In this chapter, we applied our 0D-3D computational framework to study the rightheart system, focusing on how variations in heart model parameters and vessel stiffness impact the
dynamic coupling between the right heart and the internal jugular vein (IJV). By leveraging our
0D-3D Computational Fluid Dynamics (CFD) and Fluid-Structure Interaction (FSI) framework,
we conducted a comprehensive analysis to understand these interactions in greater detail.
Our advanced computational approach allows us to simulate the right-heart-IJV system under
different physiological conditions. Specifically, we examined how changes in heart model
parameters, such as right atrial elastance, and variations in IJV wall stiffness influence the fluid

44
dynamics and mechanical responses within the system. These simulations help us understand the
propagation of pressure waves, the impact of vessel compliance on flow distribution, and the
resulting venous pressure dynamics. Additionally, our framework enables the investigation of the
non-Newtonian effects of blood flow. Unlike traditional Newtonian fluid models that assume
constant viscosity, non-Newtonian models account for the shear-thinning properties of blood,
providing a more accurate representation of its behavior under varying flow conditions. This aspect
of our study is crucial for capturing the realistic dynamics of blood flow in the right-heart-IJV
system, as the non-Newtonian properties of blood significantly affect its flow characteristics and
interaction with vessel walls.
Overall, the application of our 0D-3D CFD FSI framework in this chapter provides valuable
insights into the complex interplay between the right heart and the IJV. The comprehensive
insights gained from this solver can lead to a deeper understanding of how changes in heart
function, blood properties, and vessel elasticity impact overall circulatory health. These findings
contribute to a better understanding of cardiovascular mechanics, with potential implications for
improving clinical diagnostics, designing medical devices, and developing effective treatments for
cardiovascular conditions.
3.2 Methodology and material
Inspired by the LV-aortic model, we further design the IJV-RA/RV based on our 0D-3D
hemodynamic framework. Figure 3.1. illustrate the simplified IJV configuration (straight
compliant tube). The complete fluid–structure (immersed boundary) solver, where a no-slip
condition is imposed at the fluid–structure interface, is coupled to the 0D RA/RV-elastance heart
model (Figure 3.1) at the inlet and Windkessel ODE at the outlet and other components (including

45
SVC, IVC and Pulmonary artery (PA)). A flow boundary condition coupled with Windkessel
module (constant flow rate 𝑄0 and transient flow 𝑄𝑉𝑅(𝑡)) is implemented at the upper stream of
the IJV. The implementation is the same as the one described above.
For the right atrium modeling, same as the one for LV and RV, the following ODE for the pressure
inside the RA:
𝜕𝑝𝑎
(𝑡)
𝜕𝑡 = −
1
𝐶𝑎
(𝑡)
[
𝜕𝐶𝑎
(𝑡)
𝜕𝑡 𝑝𝑣
(𝑡) + 𝑄(𝑥 = 0,𝑡)]. (3.1)
Clinically, 𝐶𝑎
(𝑡) stands for inverse of RA elastance which represent the measure of RA elasticity
and contractility. The corresponding parameter values of the present 0D right heart model can be
found in the previous 0D or 1D studies [123-126].
3.2.1 Boundary switching procedure (tricuspid valve modeling):
Similarly, to the mitral valve switching technique in LV-aortic model, in the IJV-RA/RV solver a
non-stationary switching condition 0D boundary switching procedure (for tricuspid valve opening
and closure) is implemented. When the tricuspid valve is open, the boundary pressure is achieved
as 𝑃 = 𝑃𝐴 = 𝑃𝑉 where the right atrial pressure is equal to right ventricular pressure. Here both RA
and RV are connected to the 3D solver and have contributions to the boundary pressure 𝑃. When
the flowrate from RA to RV reaches to zero (𝑄𝐴𝑉 = 0) the tricuspid valve closes and the 0D-3D
boundary switches to a RA-only component (The RV is not connected to the 3D domain side in
this case). After the valve closure, the right atrial pressure and right ventricular pressure are
calculated separately, and the boundary pressure is achieved as 𝑃 = 𝑃𝐴 < 𝑃𝑉. When 𝑃𝐴 > 𝑃𝑉, the
tricuspid valve will open and the 0D boundary switches back where both RA and RV connected
to the 3D solver. A diagram summarizing the complete implementation and tricuspid valve

46
switching of the 0D RA/RV-elastance model with the in this work is presented in Figure 3.1. In
this model a right heart 0D component is coupled with 3D internal jugular vein solver. Other
components (SVC, IVC, Venous return) are modeled with Windkessel modules.
3.3 Computational Modeling
3.3.1 Right Heart-Internal Jugular Vein setup (IJV) model
Right-heart model with RA and RV coupled with the tricuspid wave is shown above
Figure 3.1. Diagram of the simplified internal jugular vein system coupled to the RA-RV-elastance model.

47
3.4 Application and Computational Results
3.4.1 0D-3D model of IJV-Right Heart model
The parameter of the IJV wall property and the right heart component model can be found in the
following references [123-126]. Readers can find the details about the RA/RV model in those
literature.
Waveforms - Sample waveforms of the right heart:
Here are some sample waveforms of the right atrium and ventricle from the literature [127, 128].
Figure 3.2. Sample waveform of RAP and RVP. Figure from [127, 128].
Physiological IJV waveform (IJV) trace:
Distinguishing the JVP from the carotid pulse can be achieved through several methods. Firstly,
the JVP "beats" twice (in quick succession) in the cardiac cycle. In other words, the JVP exhibits
a multiphasic pattern, characterized by two quick successive beats during each contractionrelaxation cycle by the heart. The first wave corresponds to the atrial contraction (a), while the
second represents venous filling of the right atrium against a closed tricuspid valve (v), not
“ventricular contraction” as commonly mistaken. However, some medical conditions may alter
these waveforms, rendering this method less reliable. Conversely, the carotid artery produces only

48
one beat per cardiac cycle. Secondly, the JVP is not palpable, and any pulse felt in the neck area
typically corresponds to the common carotid artery. Lastly, the JVP can be temporarily stopped by
occluding the internal jugular vein by lightly pressing against the neck, and it fills from above.
Figure 3.3. IJV waveform and its physiological meaning behind it [129, 130].
The jugular venous pulsation exhibits a biphasic waveform with several distinguishable
components. The first wave, known as the 'a' wave, corresponds to right atrial contraction and
ends synchronously with the carotid artery pulse. The 'a' wave peak demarcates the end of atrial
systole. Following the 'a' wave is the x descent, representing atrial relaxation and rapid atrial filling
due to low pressure. The 'c' wave corresponds to right ventricular contraction, causing the closed
tricuspid valve to bulge towards the right atrium during RV isovolumetric contraction.
Subsequently, the x' descent follows the 'c' wave and occurs because of the right ventricle pulling
the tricuspid valve downward during ventricular systole, indicating ventricular ejection/atrial
relaxation (end of right ventricular contraction creases extra space within pericardium). The x'
descent's magnitude can be used as a measure of right ventricle contractility. The 'v' wave
represents venous filling against a closed tricuspid valve, and venous pressure increases from
venous return, occurring during and following the carotid pulse. Lastly, the 'y' descent corresponds
to the rapid emptying of the atrium into the ventricle following the opening of the tricuspid valve.

49
In summary, we can state that the 'a' wave corresponds to atrial contraction, causing atrial pressure
to rise. The 'x' descent represents the relaxation of the right atrium, allowing the ventricles to shrink
and the atrium to fill. The 'c' wave signifies the closure of the tricuspid valve at the start of systolic
contraction. The 'v' wave corresponds to the venous filling of the right atrium as the atrium relaxes,
resulting in a slight rise in JVP. Lastly, the 'y' wave represents the opening of the tricuspid valve,
causing the rapid emptying of the right atrium.
Figure 3.4. IJV waveform trace [131] (Source: Gray’s Anatomy, Fig. 558.), [132, 133].
Demonstrated below are examples of the results obtained from the 0D-3D right heart coupling
model. Firstly, we showcase the right atrium pressure (RAP) under healthy conditions.
Subsequently, we present the jugular vein pressure (IJV) under different circumstances, including
varying the IJV wall stiffness (Eh) and heart rate. Lastly, we depict a scenario where the right
atrium elasticity is reduced by half, resulting in a thick atrium wall.

50
Figure 3.5. Pressure waveform in RA and IJV for different conditions from the 0D-3D model.
According to the findings presented in a previous study with 0D-heart model [134]. It can be seen
that, our current right atrium heart model is capable of generating right atrium pressure (RAP)
values that closely align with those obtained from a 0D heart model. These results suggest that our
model is able to replicate the performance of the previous model with a high degree of accuracy.

51
Figure 3.6. Wave forms from previous 0D-heart model [134].
Below are the flow and pressure distributions for different scenarios that correspond to variations
in IJV wall stiffness (Eh) and heart rate, as well as a case where right atrium elasticity is reduced.
These results provide insight into how changes in these physiological parameters can affect the
flow and pressure dynamics of the RA-IJV-Brain system.

52
Figure 3.7. Flow (𝑼𝒙/𝑼𝒓𝒆𝒇, 𝑼𝒓𝒆𝒇 = 𝟎. 𝟏 𝒎/𝒔) and pressure (𝒎𝒎𝑯𝒈) distribution for 𝑬𝒉 = 𝟏. 𝟎, 𝟏. 𝟓, 𝟑. 𝟎 𝑬𝒉𝟎.
Figure 3.8. Flow (𝑼𝒙/𝑼𝒓𝒆𝒇, 𝑼𝒓𝒆𝒇 = 𝟎. 𝟏 𝒎/𝒔) and pressure (𝒎𝒎𝑯𝒈) distribution for 𝑬𝒉 = 𝟏. 𝟎𝑬𝒉𝟎 𝑯𝑹 = 𝟏𝟐𝟎, and Low
Ea case with 𝑯𝑹 = 𝟔𝟎.
Both heart rate (HR) and the stiffness of the internal jugular vein (IJV) wall (Eh) significantly
influence the dynamics of the IJV, altering the distribution of flow and pressure. These changes
can, in turn, impact the collapsing behavior of the vein. Additionally, a change in right atrium

53
elasticity leads to an increase in right atrial pressure (RAP), which further affects the IJV dynamics
resulting in internal jugular vein Distention.
The main goal of this study is to use a 0D-3D coupled solver to address real-world problems in a
comprehensive 3D framework. By doing so, we aim to achieve a more accurate representation of
the IJV's behavior under various physiological conditions. Additionally, we plan to integrate the
FC-wall solver with our model to capture high-order wave modes, thereby enhancing our
understanding of the complex dynamics within the IJV.
This approach allows for a more detailed and accurate simulation of the interactions between HR,
IJV wall stiffness, and RAP, providing valuable insights into the factors that influence IJV
behavior. The findings from this study could have important implications for medical research and
clinical practice, particularly in the development of treatments and devices that rely on accurate
modeling of venous dynamics.
3.4.2 Non-Newtonian effects in IJV flow dynamics
The primary objective of this study is to enhance the precision of understanding fluid dynamic
behavior. To do this, next, we consider a more comprehensive fluid model by investigating nonNewtonian fluid effects. Specifically, we aim to assess the impact of non-Newtonian properties on
the flow dynamics within the internal jugular vein (IJV) and identify the conditions under which
these effects are most significant.
To achieve this goal, we conducted detailed simulations of fluid flow in a 3D-axisymmetric model,
comparing pressure propagation from the right atrium (RA) to the IJV under both Newtonian and

54
non-Newtonian conditions. We evaluated the non-Newtonian importance factor, which quantifies
the influence of non-Newtonian characteristics on the flow dynamics (refer to the figure below).
Through comprehensive data analysis, we aim to uncover insights into the fluid behavior under
varying scenarios, thereby deepening our understanding of the system's underlying physics. This
study contributes to the expanding research on non-Newtonian fluids and their effects on biological
systems. The findings could have significant implications for the design and optimization of
medical devices and treatments involving fluid flow, potentially leading to improved patient
outcomes and more effective therapeutic strategies.
Figure 3.9. Pressure propagation for both the Newtonian and non-Newtonian scenarios and Non-Newtonian importance
factor in IJV flow field.
In Figure 3.9, the pressure propagation behavior for both Newtonian and non-Newtonian cases is
illustrated within the 3D-axisymmetric setup. The comparison indicates that the non-Newtonian
effects do not significantly impact pressure propagation in this specific configuration. However,
further analysis reveals a noteworthy non-Newtonian importance factor, represented by the ratio
μ/μ∞, on the brain side. This ratio compares the fluid's viscosity under non-Newtonian conditions
to its viscosity in the Newtonian case.

55
This finding is crucial as it suggests that, although the non-Newtonian effect might not be
immediately evident in terms of pressure propagation, it can still substantially influence fluid
behavior in particular regions. Specifically, the high non-Newtonian importance factor on the brain
side indicates that the fluid is more likely to exhibit non-Newtonian behavior in this area.
Understanding this localized influence is important because it highlights the complex nature of
fluid dynamics in biological systems, where non-Newtonian properties can play a critical role in
certain conditions. These insights could be vital for designing medical interventions and devices
that need to account for such specific non-Newtonian effects to improve their efficacy and
reliability.
3.4.3 3D IJV modeling using the 0D-3D right-heart-brain coupled FSI Framework
To gain deeper insights into the pulsatile fluid dynamics and understand the complex mechanisms
behind the collapsing and pulsatile wave behavior of the internal jugular vein (IJV) under realistic
and physiologically accurate conditions, we utilized our advanced 3D solver to perform
comprehensive simulations across various scenarios.
These scenarios included conditions with high elasticity (stiff) and low elasticity (soft, half of Eh)
for the IJV wall, under both Newtonian and non-Newtonian fluid assumptions. Additionally, we
modeled a scenario with increased right atrium elastance (a stiffer atrium) to simulate elevated
Right Atrium Pressure (RAP), which can potentially cause internal jugular vein distention.
To ensure physiological relevance, the venous return flow from the brain was set at 0.6 L/min,
corresponding to an average velocity of 10 cm/s, fitting within the typical physiological range. The
external pressure was maintained at 10 mmHg throughout the simulations.

56
By conducting these detailed simulations, we aimed to capture the nuances of the IJV's response
to different elasticity conditions and fluid dynamics properties. This approach allows us to better
understand the behavior of the IJV under varying physiological states, providing valuable insights
that could inform medical research and the development of clinical interventions. Our findings
have the potential to improve the accuracy of models used in designing treatments and devices that
interact with the venous system, ultimately enhancing patient outcomes.
Figures 3.10 and 3.11 provide a comparative analysis of pressure wave dynamics at the inlet
(originating from the heart) and the resulting jugular venous pressure (JVP) under different blood
models for a stiff internal jugular vein (IJV) condition.
Figure 3.10 illustrates the pressure wave propagation and the corresponding JVP when assuming
a Newtonian blood model. This figure highlights how the pressure wave from the heart influences
the venous pressure in the IJV under the assumption that blood behaves as a Newtonian fluid,
which is characterized by a constant viscosity regardless of the flow conditions. In contrast, Figure
3.11 presents the pressure wave dynamics and the resultant JVP under the assumption of a nonNewtonian blood model. In this scenario, the blood exhibits varying viscosity depending on the
shear rate, more accurately reflecting the complex rheological behavior of real blood.
By comparing these figures, we can observe the differences in pressure wave propagation and JVP
under Newtonian versus non-Newtonian conditions for a stiff IJV. This comparison is crucial for
understanding the impact of blood's non-Newtonian properties on venous pressure dynamics,
especially in pathological states where the IJV stiffness is altered. These insights can inform more
accurate modeling and prediction of venous behavior in clinical settings, potentially leading to
better diagnostic and therapeutic strategies for conditions affecting venous circulation.

57
Figure 3.10 Inlet (heart-side) Pressure waveform and Internal Jugular Venous Pressure (JVP) waveform for stiff IJV with
Newtonian blood model.
Figure 3.11 Inlet (heart-side) Pressure waveform and Internal Jugular Venous Pressure (JVP) waveform for stiff IJV with
non-Newtonian blood model.
Figure 3.12 provides a snapshot of the flow distribution and the motion of the IJV for the stiff vein
model under both Newtonian (left) and non-Newtonian (right) blood conditions. This comparison
allows us to visualize the differences in flow patterns and vein deformation between the two blood
models. Similarly, Figure 3.15 shows a snapshot of the flow distribution and IJV motion for a soft

58
vein model under both Newtonian (left) and non-Newtonian (right) blood conditions. By
comparing these figures, we can observe how the elasticity of the IJV wall, and the rheological
properties of blood interact to influence flow dynamics and vein motion. The velocity is
normalized by (𝑊𝑟𝑒𝑓 = 20 𝑐𝑚/𝑠).
These visual comparisons are critical for understanding the combined effects of blood viscosity
and vein wall elasticity on venous flow and pressure distribution. They provide valuable insights
into how different physiological and pathological conditions can affect the behavior of the IJV,
informing more accurate models for clinical applications and the development of targeted
treatments.

59
Figure 3.12 Snapshots of Flow in Internal Jugular Vein (IJV) for stiff IJV model with Newtonian (left) and non-Newtonian
(right) blood models. Time at T=0.3, 0.5 and 0.8 Cardiac Cycle.
Figure 3.13 illustrates the pressure waveform at the inlet (originating from the heart) and the
corresponding jugular venous pressure (JVP) waveform for a soft internal jugular vein (IJV)
condition, characterized by half the stiffness (Eh) of the stiff condition, using a Newtonian blood
model. This figure highlights how reduced vein stiffness affects the transmission of pressure waves
from the heart through the venous system when the blood is assumed to have a constant viscosity.
On the other hand, Figure 3.14 depicts the pressure waveform at the inlet and the corresponding

60
JVP waveform for the same soft IJV condition (half the stiffness Eh) but with a non-Newtonian
blood model. This scenario assumes the blood exhibits varying viscosity depending on the shear
rate, providing a more realistic representation of its behavior.
These figures are essential for understanding how the elasticity of the IJV wall influences pressure
wave propagation and venous pressure dynamics under different blood rheology assumptions. By
comparing the pressure waveforms and JVP for Newtonian and non-Newtonian blood models in
both stiff and soft IJV conditions, we can gain deeper insights into the complex interactions
between blood flow and venous wall mechanics, which are crucial for developing accurate
physiological models and effective clinical interventions.
Figure 3.13 Inlet (heart-side) Pressure waveform and Internal Jugular Venous Pressure (JVP) waveform for soft IJV (half
of the stiffness Eh) with Newtonian blood model.

61
Figure 3.14 Inlet (heart-side) Pressure waveform and Internal Jugular Venous Pressure (JVP) waveform for soft IJV (half
of the stiffness Eh) with non-Newtonian blood model.

62
Figure 3.15 Snapshots of Flow in Internal Jugular Vein (IJV) for soft IJV model with Newtonian (left) and non-Newtonian
(right) blood models. Time at T=0.3, 0.5 and 0.8 Cardiac Cycle.
After altering the stiffness of the internal jugular vein (IJV) and varying the blood model, the next
phase of our study involved modifying the right heart model. We examined a scenario with
increased right atrial pressure (RAP) by increasing the elastance of the right atrium (stiffer RA).
Figure 3.16 (left) illustrates the pressure wave at the inlet (originating from the heart) for the stiff
IJV condition under the High RAP model. This figure shows how increased right atrial pressure
influences the pressure wave propagation in a stiff IJV scenario. Correspondingly, Figure 3.16

63
(right) presents the pressure wave at the inlet and the jugular venous pressure (JVP) for the soft
IJV condition (half the stiffness Eh) under the High RAP model. This figure demonstrates the
effects of increased RAP on pressure wave dynamics and venous pressure in a more compliant IJV
scenario.
By comparing these figures, we can understand how changes in right atrial elastance, and increased
RAP affect the pressure dynamics in both stiff and soft IJV conditions. This comparison is crucial
for comprehending the interaction between heart function and venous pressure, providing insights
that could inform the development of more accurate models for clinical diagnostics and therapeutic
strategies.
Figure 3.16 Inlet (heart-side) Pressure waveform and Internal Jugular Venous Pressure (JVP) waveform for stiff IJV (left),
soft IJV (right) with Newtonian blood model under elevated RAP Condition.
Additionally, Figure 3.17 presents snapshots of the flow distribution and internal jugular venous
(IJV) motion for cases with increased right atrium elastance (stiffer atrium with elevated
contractility). These snapshots compare the scenarios using a Newtonian blood model under
different IJV stiffness conditions.

64
Figure 3.17 Snapshots of Flow in Internal Jugular Vein (IJV) for high RAP condition with stiff IJV model (left) and soft
IJV model (right). Time at T=0.5 Cardiac Cycle.
On the left, the figure illustrates the flow distribution and IJV motion for the stiff IJV model. This
snapshot provides insights into how a stiffer IJV, combined with a stiffer right atrium, affects the
flow dynamics and venous motion. On the right, the figure shows the flow distribution and IJV
motion for the soft IJV model. This visualization helps to understand the impact of a more
compliant IJV under the same high RAP conditions.
By examining these snapshots, we can observe the differences in flow patterns and vein motion
between stiff and soft IJV models in the presence of elevated right atrial contractility. This
comparison is vital for understanding the interplay between heart and vein mechanics, offering
valuable information for the development of better diagnostic tools and treatment options for
conditions involving venous and cardiac interactions.

65
3.5 Discussion
From our 3D results, it is evident that non-Newtonian blood has some effect on the pressure
waveform and wall motion within the internal jugular vein (IJV). However, its impact, particularly
on the wave dynamics, is not as significant in pulsatile cases compared to conditions like steady
flow, where shear rates are very low. This finding aligns with our 2D simulation results, which
also showed that non-Newtonian effects are more pronounced in 3D simulations, where they affect
flow and solid motion to a greater extent than in 2D scenarios. Notably, non-Newtonian effects
are significant during collapsing events but are nearly negligible during non-collapsing periods
(before and after T~0.5). (During collapsing periods, non-Newtonian effects play a crucial role in
altering the pressure, and solid motion dynamics within the IJV. However, during non-collapsing
periods (before and after T~0.5), these effects are minimal.)
Understanding the differences in flow behavior between Newtonian and non-Newtonian fluids is
crucial, especially when investigating the internal jugular vein (IJV). The pressure waveform can
effectively quantify the impact of these fluid properties on wave dynamics within the vein. The
primary distinction lies in the additional viscous stress component present in non-Newtonian
fluids, represented by the elastic stress tensor 𝜎𝑖𝑗 = −𝑃𝛿𝑖𝑗 + 𝜀𝑖𝑗 . This tensor accounts for the
pressure and fluid's viscous properties (shear stress) and affects the IJV wall dynamics (motion).
Pressure is a key parameter in analyzing the effects of fluid properties (viscosity) on wave
reflection and propagation within the IJV. Our findings indicate that pressure measurements show
no significant difference in wave reflection and propagation between Newtonian and nonNewtonian fluids under most conditions. However, notable differences emerge during periods of
low pressure, particularly during vein collapse.

66
The main difference in the dynamic’s behavior of the IJV motion between Newtonian and nonNewtonian conditions occurs primarily during the collapsing (low pressure) period, as quantified
by the pressure waveform. During this low-pressure phase, the viscous effects of the fluid become
more significant in the total elastic stress adding to the wall (the effects are usually negligible with
pressure is high). Specifically, the viscous effects alter the viscous stress portion of the elastic
stress tensor (non-Newtonian model may cause extra viscous stress) impacting the overall stress
(with low pressure) and the IJV wall motion, leading to changes in pressure waveform and slight
variations in the shape and geometry of the collapsing IJV. These shape changes subsequently alter
the flow structure due to fluid-structure interaction (FSI). When the pressure is high, the impact of
viscous stress is negligible. However, during the collapse of the IJV, the lower pressure conditions
amplify the significance of viscous stress on wave dynamics. This effect slightly modifies the IJV's
motion and geometry, resulting in changes to the overall flow structure.
Our experimental studies, detailed in later chapters, focus on investigating the overall wave
dynamics and reflection within the IJV system. While our current research primarily addresses
these broader dynamics, a more detailed analysis of flow and shear structures may require the use
of non-Newtonian fluids to accurately replicate blood behavior. Future research will need to
overcome challenges such as achieving refractive index matching with non-Newtonian fluids. For
example, adding xanthan gum (XG) to simulate blood can cause the liquid to appear blurry,
complicating precise measurements. Addressing these challenges is essential for advancing our
understanding of IJV fluid dynamics and their implications for cardiovascular and cerebrovascular
health.

67
On the other hand, the stiffness of the IJV has a more substantial impact on the propagation of
pressure waves through the vein and on its motion, particularly during collapsing. In softer IJV
cases, we observed less 'V' wave after collapsing events, for both Newtonian and non-Newtonian
fluids. This alteration in pressure waveform post-collapse leads to different IJV motion behaviors,
characterized by increased collapsing in soft IJV, where the vein exhibits more pronounced
deformations and changes in flow patterns. Moreover, the fluid-solid interaction within the IJV
results in significant changes in flow distributions. This interaction between the blood flow and
the compliant vessel wall is critical for understanding venous dynamics under varying
physiological conditions, providing insights into how vessel compliance affects blood flow and
pressure dynamics.
Another aspect of our study involved varying the elastance of the right heart model, focusing on
conditions of increased right atrial pressure (RAP) with both stiff and soft IJV models. In both
cases, we observed distension of the IJV. The stiff IJV condition does not collapse due to increased
pressure, resulting in distension. Conversely, the soft IJV condition exhibits more pronounced
behavior, including larger distension and deformation during the collapsing period. Therefore, we
found that right heart pressure significantly affects overall IJV motion. Meanwhile, the stiffness
of the IJV influences how the IJV responds to the input pressure waveform (from right heart),
leading to different flow structures.
These findings underscore the complex interplay between heart function, IJV stiffness, and fluid
dynamics in the venous system. They provide critical insights into how physiological changes and
pathologies can influence venous behavior, which is essential for improving clinical

68
understanding, developing accurate diagnostic tools, and designing effective treatments for
cardiovascular diseases.
3.6 Conclusion
In conclusion, our study has provided comprehensive insights into the dynamics of the right heartvenous system under various physiological conditions, focusing on the effects of non-Newtonian
blood behavior, internal jugular vein (IJV) stiffness, and right atrial pressure (RAP) on fluid
dynamics and vessel mechanics.
From our 3D simulations, we found that non-Newtonian blood affects the pressure waveform and
wall motion within the IJV, with more pronounced effects in steady flow conditions compared to
pulsatile flow, where shear rates are lower. These dynamics significantly influence flow and solid
motion but have minimal effects during non-collapsing phases. IJV stiffness critically influences
pressure wave propagation and motion characteristics, particularly during collapsing. Softer IJV
conditions changes flow distributions due to fluid-solid interactions. Increasing RAP by increasing
right atrium elastance caused IJV distension in both stiff and soft IJV models, while soft IJV
exhibited more extensive distension and deformation during collapsing phases. These results
underscore the significant influence of right heart pressure on IJV motion, with IJV stiffness
impacting vessel response to pressure waveforms and flow structure variations.
3.7 Clinical and Research Implications
Understanding these complex interactions between heart function, venous stiffness, and fluid
dynamics is crucial for improving diagnostic methods and developing effective treatments for
cardiovascular diseases. Our findings suggest that incorporating non-Newtonian blood (not very

69
important here) behavior and IJV stiffness into computational models can enhance their accuracy
in simulating cardiovascular dynamics. This knowledge could potentially lead to advancements in
medical device design and therapeutic strategies aimed at improving cardiovascular health
outcomes.
These insights contribute to advancing our understanding of cardiovascular mechanics and have
significant implications for clinical practice and biomedical research. Future studies should
continue to explore these interactions to further refine our models and improve patient care in
cardiovascular medicine.

70
Chapter 4: Methodological development for 3D simulations of solid
fluid wave interactions for jugular vein right heart coupling
The methods to be employed here are based on the recently introduced concept of Fourier
continuation (FC), which broadens the applicability of Fourier-based partial differential equation
(PDE) computational approaches by resolving the well-known Gibbs "ringing" phenomenon to
enable high-order convergence of Fourier series approximations to non-periodic functions. FCbased solvers maintain the well-established qualities of other spectral solvers for time-domain
equations including accuracy by means of coarse discretization and a faithful preservation of the
dispersion characteristics of the underlying continuous problems (i.e., errors do not compound
over space and time, which may not be true for many standard finite difference or finite elementbased methods). The ultimate goal is to integrate the high-fidelity FC-based method for elasticity
into a lattice Boltzmann-based fluid-structure solver.
4.1 Chapter introduction
The Fourier continuation (FC) methodology [64, 135-137] enables Fourier series representations
of non-periodic functions, ultimately facilitating the construction of fast, high-order solvers with
limited numerical dispersion (or “pollution errors”).
Fourier continuation (FC) methods produce highly-accurate Fourier series representations of nonperiodic functions while avoiding the well-known Gibb’s “ringing effect" [138]. Such techniques
expand the applicability of Fourier-based partial differential equation (PDE) solvers towards
general (physical) boundary conditions and computational domains [64] [136, 139]. Corresponding
solutions provide Fast Fourier Transform- (FFT-)speed high-order accuracy and faithfully capture

71
the dispersion or diffusion characteristics of the underlying continuous problems. A number of
time-domain FC-based solvers have been constructed for a variety of physical equations including
those governing wave propagation [64, 136] [140-142] and diffusion [139] [143]. The resulting highorder solvers enjoy a number of desirable properties for scientific computation: accuracy by means
of relatively coarse discretizations; little-to-no numerical dispersion or diffusion errors; mild
(linear) CFL constraints on time integration; and efficient parallelization for distributed-memory
high-performance computing.
FC algorithms have been successfully coupled to other methods including discrete ordinates
methods for radiative transfer [143] and weighted essentially non-oscillatory (WENO) shockcapturing schemes for conservation laws [140]. This work is ultimately interested in developing an
FC-based elastodynamics solver to couple with a lattice Boltzmann fluid solver [144] for capturing
high-frequency wall effects in fluid-structure hemodynamics simulations. Such coupling can be
facilitated by an immersed boundary method [145], where the wall is modeled as a flexible filament
or thin plate/shell in Lagrangian coordinates. The corresponding PDEs contain fourth-order
derivatives in space, requiring multiple conditions at a domain boundary (e.g., a simply-supported
or clamped end).
Hence this contribution proposes various new differentiation operators for treating these multiple
conditions at boundaries, extending the class of FC methods to high-order PDEs in general. A
preliminary application to flexible filaments in two-dimensions is also presented with
corresponding verification and convergence studies, including comparisons with a wellestablished method employed in fluid-structure configurations.

72
4.2 Fourier continuation methods
4.2.1 Spatial discretization of Fourier continuation
For the numerical treatment of the spatial variables and directional derivatives, one can consider
discrete point values 𝑓(𝑥𝑖
) of a given smooth function 𝑓(𝑥):[0,1] → ℝ defined on a structured
uniform discretization 𝑥𝑖 = 𝑖𝛥𝑥, 𝑖 = 0, … , 𝑁 − 1, 𝛥𝑥 = 1/(𝑁 − 1), the FC method constructs a
fast-converging interpolating trigonometric polynomial (Fourier series representation)
𝑓cont:[0, 𝑏] → ℝ on a region [0, 𝑏] that is slightly larger than the original physical domain od
definition [0,1]:
𝑓cont = ∑ 𝑎𝑘
𝑀
𝑘=−𝑀
𝑒
2𝜋𝑖𝑘𝑥
𝑏 s.t. 𝑓cont(𝑥𝑖
) = 𝑓(𝑥𝑖
), 𝑖 = 0, … , 𝑁 − 1, (4.1)
where 𝑀 = (𝑁 + 𝑁cont)/2 is a full bandwidth parameter for a number of points 𝑁cont added to the
original domain (such that 𝑏 = (𝑁 + 𝑁cont)𝛥𝑥). The FC function 𝑓cont renders the original function
𝑓 discretely periodic, i.e., 𝑓cont approximates 𝑓 to high-order in the original domain [0,1] and is
approximately periodic on the slightly larger domain [0, 𝑏], 𝑏 > 1. Spatial derivatives of a PDE
can then be produced by exact termwise differentiation of this series as
∂𝑓cont
∂𝑥
(𝑥) = ∑ (
2𝜋𝑖𝑘
𝑏
)
𝑀
𝑘=−𝑀
𝑎𝑘𝑒
2𝜋𝑖𝑘𝑥
𝑏 . (4.2)
This ultimately provides the numerical derivatives of 𝑓 to high-order by restricting the domain of
∂𝑓cont/ ∂𝑥 to the original unit interval. Hence the approximation rests in the construction of Eq. (1)

73
from which the computation of the derivative in Eq. (2) can be facilitated by the Fast Fourier
Transform (FFT). Note here that 𝑓 has been defined on [0,1] without loss of generality.
The overall errors are found only in the production of Eq. (4.1), from where the calculation of the
derivative (Eq. (4.2)) is easily facilitated by a Fast Fourier Transform (FFT). This overall idea can
also be easily extended to any general interval [𝑠0, 𝑠𝑁] via affine transformations [138].
4.2.2 Accelerated Fourier continuation: FC(Gram)
The coefficients 𝑎𝑘 of Eq. (4.1) are found in the most intuitive treatment [146] via the solution to the
least squares problem given by
min
𝑎𝑘
∑|𝑓cont(𝑥𝑖
) − 𝑓(𝑥𝑖
)|
2
𝑁−1
𝑖=0
, (4.3)
by the Singular Value Decomposition (SVD). This can become rather costly for 3D problems as
well as time-dependent solutions of complex boundary-valued PDEs (where each spatial
dimension requires application of SVDs at each timestep). An accelerated method [136, 139],
known as FC(Gram) [64, 135-137], can circumvent such expense by employing small vectors of
only a handful of function values near the left and right endpoints at 𝑥 = 0 and 𝑥 = 1 which can
then be projected onto a Gram polynomial basis (whose continuations are precomputed through
solving the corresponding least squares problem of Eq. (3) by high-precision SVD). That is, one
utilizes a subset of the given function values on small numbers 𝑑ℓ and 𝑑𝑟 of matching points
{𝑥0, . . . , 𝑥𝑑𝑙−1} and {𝑥𝑁−𝑑𝑟
, . . . , 𝑥𝑁−1} contained in small subintervals on the left and right ends of
the interval [0,1] to produce a discrete periodic extension of size 𝑁cont. This is accomplished by
projecting these end values onto a Gram basis up to degree 𝑑ℓ − 1 (or 𝑑𝑟 − 1) of polynomials

74
(producing a polynomial interpolant) whose FC extensions are precomputed and whose
orthogonality is enforced by the natural discrete scalar product defined by the discretization points.
This effectively forms a “basis" of continuation functions with which to quickly and accurately
extend the given function 𝑓 to provide a smooth transition from 𝑓(𝑥 = 0) back to 𝑓(𝑥 = 1) over
the interval [0, 𝑏].
Defining the vectors of matching points for the left and right as
𝒇ℓ = (𝑓(𝑥0
), 𝑓(𝑥1
), . . . , 𝑓(𝑥𝑑ℓ−1))
𝑇
, 𝒇𝑟 = (𝑓(𝑥𝑁−𝑑𝑟
), 𝑓(𝑥𝑁−𝑑𝑟+1), . . . , 𝑓(𝑥𝑁−1
))
𝑇
, (4.4)
the continuation operation can be expressed in a block matrix form as
𝒇cont = [
𝒇
𝐴ℓ𝑄ℓ
𝑇𝒇ℓ + 𝐴𝑟𝑄𝑟
𝑇𝒇𝑟
], (4.5)
where 𝒇 = (𝑓(𝑥0
), … , 𝑓(𝑥𝑁−1
))
𝑇
is a column vector containing the discrete point values of 𝑓;
𝒇cont is a vector of the 𝑁 + 𝑁cont continued function values; 𝐼 is the 𝑁 × 𝑁 identity matrix; and
𝐴ℓ
, 𝐴𝑟 contain the corresponding 𝑁cont values that perform the continuation from the left and the
right (such that the sum of leftward and rightward continuations provides the necessary smooth
transition). The columns of 𝑄 = 𝑄ℓ
, 𝑄𝑟 contain the 𝑑 = 𝑑ℓ
, 𝑑𝑟 point values of each element of the
corresponding Gram polynomial basis, produced from a 𝑄𝑅 decomposition of a Vandermonde
matrix 𝑉 of monomials, i.e.,
𝑉 =
(

1 𝑥0
(𝑥0
)
2
. . . (𝑥0
)
𝑑−1
1 𝑥1
(𝑥1
)
2
. . . (𝑥1
)
𝑑−1
⋮ ⋮ ⋮ ⋮ ⋮
1 𝑥𝑑−1
(𝑥𝑑−1
)
2
. . . (𝑥𝑑−1
)
𝑑−1)
= 𝑄𝑅. (4.6)

75
Figure. 4.2 illustrates an example Fourier continuation of a non-periodic function. The resulting
continued vector 𝐟cont can be interpreted as a set of discrete values of a smooth and periodic
function that can be approximated to high-order via FFT on an interval of size (𝑁 + 𝑁cont)𝛥𝑥.
Following others [64, 136, 139] [141], this work employs 𝑁cont = 25 and 𝑑ℓ
, 𝑑𝑟 = 5. Further
technical details on particular FC(Gram) algorithm have been provided previously [64, 135-137].
It should be noted that the FC procedure relies on polynomial interpolation (not discussed here);
hence choices of 𝑑𝑙 = 5, 𝑑𝑟 = 5 will lead to fifth-order convergence in space [64, 135-137].
Figure 4.1 An example Fourier continuation of a non-periodic function [64]. The original function on [𝟎, 𝟏] is translated
by a distance of length 𝑵cont𝜟𝒙 whose values are filled-in by the sum of “blend-to-zero" continuations (dashed lines) in
order to render the function periodic. Triangles and circles represent the discrete 𝒅𝓵, 𝒅𝒓 = 𝟓 matching points, and
squares represent the discrete 𝑵cont = 𝟐𝟓 continuation points that comprise the extension.
4.3 Some new FC operators for high-order PDEs
The Fourier continuation method is known to handle Dirichlet boundary conditions [135] (where
displacement values are known at the boundary) and, more recently, Neumann-like boundary
conditions [136, 137] (where normal first derivative values are known at the boundary).

76
For high-order PDEs, such as the Euler-Bernoulli beam equations (with a fourth-order derivative
in space), multiple conditions need to be satisfied at a boundary 𝑥0 , e.g., for clamped
(𝑓(𝑥0
), ∂𝑓(𝑥0
)/ ∂𝑥) , simply-supported (𝑓(𝑥0
), ∂
2𝑓(𝑥0
)/ ∂𝑥
2
) or free ends (∂
2𝑓(𝑥0
)/
∂𝑥
2
, ∂
3𝑓(𝑥0
)/ ∂𝑥
3
). The methodology outlined above applies only to known function values at
the end points, which is sufficient for a Dirichlet problem of a lower-order PDE, but insufficient
for high-order PDEs. However, modified polynomial interpolants can be introduced to match
multiple combinations of function values and derivative values at endpoints.
The example of new matching vectors (matching both value and the second derivative at the
absolute boundary) are defined (similarly to Eq. (4.4)) as
𝑿𝑚𝑖𝑥𝑒𝑑,𝑙 = (𝑋
′′(𝑠0
), 𝑋(𝑠0
), 𝑋(𝑠1
), … , 𝑋(𝑠𝑑𝑙−2))
𝑇
, 𝑋𝑚𝑖𝑥𝑒𝑑,𝑟 = 𝑿(𝑠𝑁−𝑑𝑟+1), … , 𝑋(𝑠𝑁−1), 𝑋′′(𝑠𝑁−1)))^𝑇, (4.7)
where values at 𝑋(𝑠𝑑𝑙−1) and 𝑋(𝑠𝑁−𝑑𝑟
) are no longer necessary to provide sufficient conditions
for a unique 𝑑𝑙 − 1(𝑟𝑒𝑠𝑝. 𝑑𝑟 − 1) polynomial interpolant.
Such interpolants can be obtained by orthonormalizing, instead of the columns of Eq. (4.6), the
corresponding columns of a Vandermonde-like matrix that is appropriately modified for the
conditions of interest. For example, such modified matrices can be given for a clamped end as
𝑉m =
(

0 1 2𝑥0 . . . (𝑑 − 1)(𝑥0
)
𝑑−2
1 𝑥0
(𝑥0
)
2
. . . (𝑥0
)
𝑑−1
1 𝑥1
(𝑥1
)
2
. . . (𝑥1
)
𝑑−1
⋮ ⋮ ⋮ ⋮ ⋮
1 𝑥𝑑−2
(𝑥𝑑−2
)
2
. . . (𝑥𝑑−2
)
𝑑−1 )

(4.8)
A reconstruction of the coefficients from the clamped case to the original Gram polynomial basis
can be simply obtained by replacing the operator 𝑄ℓ
(resp. 𝑄𝑟
) in Eq. (5) with a modified operator

77
𝑄‾
ℓ
(resp. 𝑄‾
𝑟
) that directly performs such a (re-)projection. The new operator can be found by first
solving for new coefficients 𝐚 = (𝑎1, 𝑎2, . . . , 𝑎𝑑−1
)
𝑇 via the expression
𝑉m𝐚 = (∂𝑓(𝑥0
)/ ∂𝑥, 𝑓(𝑥0
), 𝑓(𝑥1
), … , 𝑓(𝑥𝑑−2
))
𝑇
, (4.9)
which, from a QR-decomposition 𝑉m = 𝑄m𝑅m, can be solved as
𝐚 = 𝑅m
−1𝑄m
𝑇
(∂𝑓(𝑥0
)/ ∂𝑥, 𝑓(𝑥0
), 𝑓(𝑥1
), … , 𝑓(𝑥𝑑−2
))
𝑇
. (4.10)
From the original decomposition of the original Vandermonde matrix 𝑉 = 𝑄𝑅 of Eq. (3.6) [136],
the corresponding coefficients are similarly given by
𝑉𝐚 = 𝑄𝑅𝐚(𝑓(𝑥0
), 𝑓(𝑥1
), . . . , 𝑓(𝑥𝑑−1
))
𝑇
. (4.11)
Substitution of Eq. (4.10) into Eq. (4.11) yields the expression
𝑄𝑅𝑅m
−1𝑄m
𝑇
(∂𝑓(𝑥0
)/ ∂𝑥, 𝑓(𝑥0
), 𝑓(𝑥1
), . . . , 𝑓(𝑥𝑑−2
))
𝑇
= (∂𝑓(𝑥0
)/ ∂𝑥, 𝑓(𝑥0
), 𝑓(𝑥1
), . . . , 𝑓(𝑥𝑑−2
))
𝑇
.
Defining 𝑄‾ = (𝑅𝑅m
−1𝑄m
𝑇
)
𝑇
further gives
𝑄‾ 𝑇
(∂𝑓(𝑥0
)/ ∂𝑥, 𝑓(𝑥0
), 𝑓(𝑥1
), … , 𝑓(𝑥𝑑−2
))
𝑇
= 𝑄
𝑇
(∂𝑓(𝑥0
)/ ∂𝑥, 𝑓(𝑥0
), 𝑓(𝑥1
), … , 𝑓(𝑥𝑑−2
))
𝑇
. (4.12)
Hence, for a domain with two-clamped boundaries, the new block matrix form of the continuation
with the modified operators are given by
𝒇cont = [
𝒇
𝐴ℓ𝑄‾
ℓ
𝑇𝒇‾
ℓ + 𝐴𝑟𝑄‾
𝑟
𝑇𝒇‾
𝑟
] (4.13)

78
where 𝐟
‾
ℓ = (∂𝑓(𝑥0
)/ ∂𝑥, 𝑓(𝑥0
), 𝑓(𝑥1
), . . . , 𝑓(𝑥𝑑ℓ−2))
𝑇
, etc., and where 𝐴ℓ
, 𝐴𝑟 are the same
original continuation operators. Such a formulation enables mixing and matching of left and right
boundary conditions (e.g., clamped on one end, free on the other). The corresponding projections
for simply-supported or free ends can be found similarly using the modified Vandermonde
matrices respectively given by
𝑉m =
(

0 0 2 . . . (𝑑 − 2)(𝑑 − 1)(𝑥0
)
𝑑−3
1 𝑥0
(𝑥0
)
2
. . . (𝑥0
)
𝑑−1
1 𝑥1
(𝑥1
)
2
. . . (𝑥1
)
𝑑−1
⋮ ⋮ ⋮ ⋮ ⋮
1 𝑥𝑑−2
(𝑥𝑑−2
)
2
. . . (𝑥𝑑−2
)
𝑑−1 )

or
(

0 0 0 . . . (𝑑 − 3)(𝑑 − 2)(𝑑 − 1)(𝑥0
)
𝑑−4
0 0 2 . . . (𝑑 − 2)(𝑑 − 1)(𝑥0
)
𝑑−3
1 𝑥1
(𝑥1
)
2
. . . (𝑥1
)
𝑑−1
⋮ ⋮ ⋮ ⋮ ⋮
1 𝑥𝑑−2
(𝑥𝑑−2
)
2
. . . (𝑥𝑑−2
)
𝑑−1 )

. (4.14)
In summary, the FC(Gram) algorithm appends 𝑁cont values to a given discretized function in order
to form a periodic extension in [1, 𝑏] those transitions smoothly from 𝑋(𝑠𝑁−1
) back to 𝑋(𝑥0
). An
illustrative example is provided in Figure 4.2, where a non-periodic function is originally
discretized on [0,1] and then translated by a distance 𝑁contΔ𝑥 with the subsequent interval filledin by the sum of leftward and rightward continuations. The resulting continued vector 𝑋cont can be
interpreted as a set of discrete values of a smooth and periodic function that can be approximated
to high order via FFT on an interval of size (𝑁 + 𝑁cont)Δ𝑠 (and hence accurate termwise). As has
been suggested previously [64, 135-137], a small number of 𝑑𝑙
, 𝑑𝑟 = 5 matching points with a
periodic extension comprised of 𝑁cont = 25 points are often used. The matrices
𝐴𝑙
, 𝐴𝑟
,𝑄𝑙
, 𝑄𝑟
,𝑄mixed,𝑙 and 𝑄mixed,𝑟 are computed only once (before numerically solving the
governing system) and stored in file for use each time the FC procedure is invoked.

79
Figure 4.2. An example Fourier continuation of the non-periodic function 𝑿(𝒔) = 𝐞𝐱𝐩(𝐬𝐢𝐧(𝟓. 𝟒𝛑𝒔 − 𝟐. 𝟕𝛑) −
𝐜𝐨𝐬(𝟐𝛑𝒔)) , 𝒙 ∈ [𝟎, 𝟏]. The operators 𝑷𝒍
,𝑷𝒓 perform the blending-to-zero via projection onto a precomputed FC basis. The
new function values 𝑿cont are periodic (in a discrete sense) on the interval [𝟎, 𝒃].
4.3.1 Temporal discretization
To complete the solver, any suitable 𝑠-step explicit time integration scheme may be employed,
e.g., for ∂𝑋/ ∂𝑡, the expression given by
𝑋𝑖
(𝑡
𝑛+1
) = 𝑋𝑖
(𝑡
𝑛) + Δ𝑡∑α𝑗𝐹𝑖 (𝑡
𝑛−𝑗
, 𝑋𝑖(𝑡
𝑛−𝑗
))
𝑠
𝑗=0
, (4.15)
for a right-hand-side term 𝐹𝑖
(derived from a first-order temporal representation of Eq. (4.2)) and
for corresponding integration weights α𝑗
. In order to be consistent with the high-order spatial
scheme described above, a fourth-order Adams-Bashforth method is employed, similarly to other
successful FC-based solvers [64, 135-137, 141].
Upon completion of the methodology development, the implementation of the 2D and 3D FC-wall
solver is expected to be a relatively straightforward process. This is because the methodology
serves as a comprehensive guide and set of instructions for how to develop and implement the
solver effectively. Additionally, any potential issues or roadblocks that may arise during the
implementation process can be addressed using the methodology as a reference. As a result, the

80
development and implementation of the 2D and 3D FC-wall solver can be expected to proceed
efficiently and with minimal complications.
4.4 Applications to flexible filaments
The governing equations in Lagrangian form for the position 𝐗(𝐬,𝑡) = (𝑋(𝑠,𝑡), 𝑌(𝑠,𝑡))
𝑇
∈ ℝ2 of
a massive and inextensible flexible filament in two dimensions can be given by [144, 145]
𝜌𝑠ℎ
∂
2𝐗
∂𝑡
2 =
∂
∂𝑠
[𝐸ℎ (√
∂𝐗
∂𝑠
⋅
∂𝐗
∂𝑠
− 1)
∂𝐗
∂𝑠
−
∂
2
∂𝑠
2
(𝐸𝐼
∂
2𝐗
∂𝑠
2
)] + 𝐅(𝐬,𝑡), (4.17)
where 𝜌𝑠 ∈ ℝ is the solid density; ℎ ∈ ℝ is a constant filament thickness; 𝑠 ∈ ℝ is the Lagrangian
coordinate along the filament; 𝐅(𝐬,𝑡) is the Lagrangian force exerted on the filament (e.g, gravity
or a surrounding fluid [145]); and 𝐸ℎ, 𝐸𝐼 are stretching and bending stiffnesses (respectively). For
all the numerical experiments of this section, the FC method is applied to Eq. (4.17) for spatial
discretization and spatial differentiation (𝑁cont = 25, 𝑑ℓ = 𝑑𝑟 = 5), and a fourth-order AdamsBashforth scheme is employed for temporal integration (similarly to other FC-based solvers [64,
136]).
4.4.1 Hanging filament under gravity
As a first verification exercise, the hanging filament case of [145] is considered, where a filament
of length 𝐿 = 1 m is initially held stationary at an angle 𝑘 = 0.1𝜋 radians from the vertical and
subjected to a gravitational force 𝐅(𝑠,𝑡) = (10,0)
𝑇 m/s2
. At 𝑠 = 0, the filament has a free end
given by

81
𝑇(0,𝑡) = 0,
∂
2𝐗
∂𝑠
2
(0,𝑡) = 𝟎,
∂
3𝐗
∂𝑠
3
(0,𝑡) = 𝟎, (4.18)
and at 𝑠 = 1, a simply-supported condition is considered, i.e,
𝐗(1,𝑡) = 𝟎,
∂
2𝐗
∂𝑠
2
(1,𝑡) = 𝟎. (4.19)
Fig. 4.3 presents a superposition of filament positions (left) and the time evolution of the 𝑌
displacement of the free end produced by an FC-based simulation using 𝑁 = 21 spatial
discretization points and a timestep of 𝛥𝑡 = 10−3
. These results are in excellent agreement with
those produced by other (finite difference-based) solvers [144, 145].

82
Figure 4.3 (Left) Superposition of filament positions over time (plotted every 𝟎. 𝟎𝟐 sec), where the blue dot represents a
simply-supported end and the red dot represents a free end. (Right) The 𝑿-component displacement of the free end as a
function of time.
4.4.2 Convergence study
For the proposed FC-based methodology, the method of manufactured solutions (MMS) can be
employed in order to verify both its numerical accuracy as well as its code implementation. Such
a method postulates a (sufficiently smooth) solution of Eq. (4.17) and, upon substitution,
incorporates the corresponding right-hand side and boundary conditions as non-trivial forcing
terms. The resulting system enables the proposed function to be an exact (analytical) solution of a
forced filament system. For example, one can postulate a solution 𝐗(𝑠,𝑡) as
𝐗ex(𝑠,𝑡) = (sin(𝑘1𝑠 − 𝜔1𝑡)cos(𝑘2𝑠 − 𝜔2𝑡), sin(𝑘3𝑠 − 𝜔3𝑡)cos(𝑘4𝑠 − 𝜔4𝑡))
𝑇
, 𝑠 ∈ [0,2], 𝑡 ≥ 0, (4.20)
for {𝑘1, 𝜔1, 𝑘2, 𝜔2, 𝑘3, 𝜔3, 𝑘4, 𝜔4} = {2.08,1.05,5.00,0.30,8.33,4.80,3.75,2.40}.

83
Substituting Eq. (13) into Eq. (12) yields the right-hand-side 𝐅(𝑠,𝑡) for the corresponding problem,
with the initial condition given by 𝐗(𝑠, 0) = 𝐗ex(𝑠, 0) . To complete the problem, boundary
conditions must be chosen similarly, e.g., for a simply-supported condition at 𝑠 = 0,
𝐗(0,𝑡) = 𝐗ex(0,𝑡),
∂
2𝐗
∂𝑠
2
(0,𝑡) =
∂
2𝐗ex
∂𝑠
2
(0,𝑡), (4.21)
and for a clamped condition at 𝑠 = 2,
𝐗(2,𝑡) = 𝐗ex(2,𝑡),
∂𝐗
∂𝑠
(2,𝑡) =
∂𝐗ex
∂𝑠
(2,𝑡). (4.22)
Fig. 4.4 presents snapshots of an FC-based solution at two different times. Applying the FC solver
on a series of discretizations corresponding to integer multiples of 𝑁 = 40 (where all timesteps
are chosen small enough so that errors are dominated by spatial discretizations), Table 4.1 presents
the maximum absolute 𝐿
∞ errors, over all space and all time (for 409 600 timesteps), for both
components of 𝐗(𝑠,𝑡). Orders-of-convergence are also presented and illustrate the expected fifthorder accuracy [64] [139] of the operators for the FC parameters employed in this work.

84

Figure 4.4 Snapshots at two different times (increasing time from left to right) of the solution to the problem derived
from Eq. (4.20). The blue dot represents a clamped-like (non-zero) boundary condition and the red dot represents a
simply-supported-like (non-zero) boundary condition.
Table 4-1 Maximum 𝑳
∞ errors (over all space and all time) and orders-of-convergence for FC solver applied to problem
derived from the manufactured solution of Eq. (4.20). The timestep is chosen small enough so that errors are dominated by
the spatial discretization. All solutions are advanced to the same final time using 409,600 timesteps
𝑋(𝑠,𝑡) 𝑌(𝑠,𝑡)
𝑁 𝐿
∞ error 𝒪(𝐿
∞) 𝑁 𝐿
∞ error 𝒪(𝐿
∞)
40 1.85 × 10−2 — 40 1.29 × 10−1 —
80 1.54 × 10−4 6.91 80 2.51 × 10−3 5.69
160 6.09 × 10−6 4.66 160 6.33 × 10−5 5.31
320 1.53 × 10−7 5.32 320 1.40 × 10−6 5.50
640 3.18 × 10−9 5.59 640 3.02 × 10−8 5.53
4.4.3 Performance study
Fig. 4 presents snapshots at two different times of a problem derived from the manufactured exact
solution given by
𝐗(𝑠,𝑡) = (𝑠, 𝐴sin(𝑛𝜋𝑠)cos(𝜋𝑡)), 𝑠 ∈ [0,2], 𝑡 ≥ 0, (4.23)

85
with 𝐴 = 0.05 and 𝑛 = 5. Employing clamped boundary conditions on both ends (whose timedependent values are determined accordingly by Eq. (4.23), as before), Table 4.2 presents a comparison
of numerical 𝐿
∞ errors over all space and time of the FC method and a commonly-used secondorder staggered-grid finite difference (FD) method [144, 145] (which solves implicitly for the
stretching term). The solutions are advanced to a final time of 𝑡 = 1 s (corresponding a full period
of oscillation), and spatial discretizations for both methods are chosen such that the corresponding
solvers achieve similar accuracy (three, four and five digits). As can be observed, the FC-based
solver enables significantly faster computations and significantly less memory requirements than
the well-established FD method. Equivalent findings attesting to the advantages of FC-based
simulations, in relation to cost or memory demands, have been showcased for various FC-based
solvers in prior works [64, 136, 139].

Figure 4.5 : Snapshots at two different times (increasing time from left to right) of the solution to the problem derived
from Eq. (4.23). The blue dot represents a standard clamped boundary condition.

86
Table 4-2 Comparison of maximum 𝑳
∞ errors (over all space and time) as well as computational times between FC and a
second-order (central) finite difference (FD) method [145] for resolving the solution given by Eq. (4.23). Discretizations are
chosen for each so that they achieve a similar error.
FC FD
𝑁 𝐿
∞ error Time (s) 𝑁 𝐿
∞ error Time (s)
𝟒𝟓 𝟏. 𝟖𝟎 × 𝟏𝟎−𝟑 0.54 s 𝟕𝟑 𝟏. 𝟖𝟔 × 𝟏𝟎−𝟑 0.93 s
𝟔𝟕 𝟏. 𝟖𝟖 × 𝟏𝟎−𝟒 1.32 s 𝟐𝟐𝟖 𝟏. 𝟖𝟕 × 𝟏𝟎−𝟒 16.22 s
𝟏𝟎𝟎 𝟏. 𝟖𝟕 × 𝟏𝟎−𝟓 2.84 s 𝟕𝟐𝟎 𝟏. 𝟖𝟖 × 𝟏𝟎−𝟓 689.25 s
4.5 Conclusions
This contribution introduces new Fourier continuation operators for treating boundary conditions
that are required for high-order PDEs, demonstrating their efficacy through an example solver for
flexible filaments in Lagrangian coordinates. Current ongoing efforts involve extending the
implementation to 3D thin wall/shell dynamics, with the ultimate goal of integrating an FC-based
method for elasticity into a lattice Boltzmann-based fluid-structure solver for cardiovascular
hemodynamics simulations [144]. Preliminary results presented in this work suggest that the FC
method is a promising approach for capturing high-frequency wall oscillations with relatively
coarse discretizations and with relatively low computational costs versus existing methods.

87
Chapter 5: Experimental investigation of Pulsatile Flow in
Physiologically Accurate Collapsible Jugular Vein-right heart
Model using Three-Dimensional particle tracking velocimetry
5.1 Chapter introduction
The complexities of fluid dynamics within the cerebral venous system remain relatively
understudied, particularly in comparison to the arterial side [1]. Recent findings have underscored
the close relationship between the brain's vascular system and cardiac function, indicating that
alterations in their hemodynamic coupling may contribute to various cardiovascular and
cerebrovascular diseases [2]. Understanding the cerebrovascular system's intricacies is vital for
identifying circulatory changes associated with neurological and neurodegenerative diseases.
Emerging research has highlighted the significant impact of cerebral venous circulation
abnormalities on cerebral blood flow and cognitive function. Advances in imaging techniques have
facilitated the exploration of the intracranial venous system, shedding light on its biophysical and
hemodynamic parameters [5, 7, 8]. Abnormalities in venous circulation, such as jugular venous
reflux [13, 14], have been implicated in conditions like multiple sclerosis and Alzheimer's disease,
suggesting a critical role for the venous system in the pathogenesis of inflammatory neurological
and neurodegenerative diseases [15].
Furthermore, the implications of venous abnormalities extend beyond neurological disorders, with
implications for space medicine. Astronauts face unique physiological challenges, including
alterations in cardiovascular function and the risk of venous thrombosis [22, 23]. Recent incidents
of internal jugular venous thrombosis in astronauts highlight the necessity for further research into

88
the mechanisms underlying this condition and the development of preventive measures [28].
Thrombotic disease might be underdiagnosed in those exposed to prolonged microgravity, with
underlying factors still poorly understood. Recently, an internal jugular vein thrombosis was
diagnosed in a low-risk female astronaut after a 7-week space mission. Additionally, six out of ten
crew members showed risk factors for jugular venous flow issues, such as suspicious stagnation
or retroversion, though they remained asymptomatic. Observations in space, along with clinical
and in vitro microgravity studies on Earth designed to mimic space conditions, suggest impacts on
blood flow stasis, coagulation, and vascular function [24-26].
From a hemodynamic standpoint, the jugular venous pulse serves as a valuable indicator of cardiac
function and filling pressures. Recent studies have demonstrated the potential of ultrasound
measurements to assess the internal jugular vein and derive important hemodynamic parameters.
However, despite extensive research on the behavior of collapsible tubes under steady flow
conditions [31-34], the pulsatile dynamics of collapsible tubes, particularly within the internal
jugular vein, remain largely unexplored [33, 47-51].
Investigating pulsatile fluid dynamics in collapsible tubes could yield crucial insights into the
behavior of fluids under pulsatile flow conditions, informing the development of medical devices
and industrial applications. Further research in this area is essential for a comprehensive
understanding of cerebral venous circulation and its implications for neurological health and space
exploration.
Moreover, previous studies on the dynamics of collapsed tubes have predominantly utilized 2D
PIV systems to capture flow behavior. While these studies have provided valuable insights, the 2D
PIV method is inherently limited as it only captures flow information within a single plane, failing

89
to account for the complex three-dimensional motion and interactions that occur in such systems.
Consequently, the full spectrum of fluid dynamics in collapsible tubes has remained inadequately
explored. In this research, we break new ground by employing a 3D PIV experimental system to
study the flow structure in a collapsible tube. This marks the first time that 3D flow dynamics in
such a context have been visualized and analyzed with this level of detail and accuracy. Our 3D
PIV system allows us to capture volumetric flow fields, providing a comprehensive view of the
in-plane and out-of-plane motions that are critical for understanding the true nature of the fluidstructure interactions in the collapsible tube. Our findings from the 3D PIV analysis reveal
complex flow structures and interactions that were not detectable in earlier studies using 2D
methods. This comprehensive approach not only enhances our understanding of the fluid dynamics
in collapsible tubes but also sets a new standard for future research in this field. The
implementation of 3D PIV thus represents a significant advancement, allowing for a deeper and
more nuanced exploration of the behavior of collapsible tubes under various flow conditions.
This chapter focuses on conducting in-vitro experimental studies of pulsatile flow within a
collapsible tube. The complexity of this system stems from the pulsatile wave generated by the
heart, which opposes the direction of the mean flow. Due to the complicated interaction between
these pulsatile and mean flow components, numerical simulations may not fully capture all
dynamic features accurately (Numerical simulations, while powerful, may struggle to fully
replicate certain features of the system). Therefore, experimental methodology is employed as it
offers the capability to comprehensively capture these dynamic features. Experimental setups
allow for a detailed examination of the complex interactions between the pulsatile wave and mean
flow that occur in the cardiovascular system, which are crucial for a thorough understanding of the
flow dynamics in collapsible tubes in venous system. By conducting these experiments, we aim to

90
provide insights that are challenging to obtain through numerical simulations alone, thereby
advancing our understanding of the biomechanics and fluid dynamics in such systems.
Role of Experimental Methodology: To overcome these challenges, we employ experimental
methods that offer several advantages:
• Realistic Dynamic Features: Experimental setups can replicate the exact conditions of
pulsatile flow with high fidelity, including the complex interaction between the heart's
pulsations and the tube's collapsibility.
• Direct Observation: Experimental setups allow for direct observation and measurement of
dynamic features that may not be easily quantifiable in simulations.
• Validation of Numerical Models: Experimental data serves as a crucial validation tool for
numerical models, helping to refine and improve their accuracy.
Significance of the Study: This study is significant as it provides a bridge between theoretical
numerical simulations and real-world physiological conditions. By focusing on experimental
study, we enhance our understanding of how pulsatile flow dynamics influence collapsible tube
behavior. This knowledge can potentially lead to advancements in medical diagnostics and
treatment strategies, particularly in cardiovascular health.
Our experimental study investigates the dynamics of pulsatile flow within a collapsible tube. This
setup mimics the physiological conditions found in the cardiovascular system, where the artificial
heart's pulsations cause varying flow rates and pressures.
5.2 In-vitro Hemodynamic Simulator for the Cerebral venous Circulation
We developed a hydro-mechanical setup for Cerebral venous loop that remarks most of the
functionality contributors of the internal jugular vein and right heart such as 1) venous flow and
right heart pressure wave propagate again each other; 2) the contractile state of the RV; 3) mean
central venous pressure and external venous pressure; 4) The RV-RA coupling; 5) Geometry and

91
elastic properties of the internal jugular vein. The setup is designed in such a way that the basal
RV is first coupled with RA then connected to IJV. Our proposed system generates physiologically
accurate waveforms for pressure and flow in cerebral venous segments including the IJV and RA.
This setup would be a very useful tool for simulating and studying various cerebrovascular and
neurodegenerative diseases, thereby assessing hemodynamic change in such diseases.
Figure 5.1. Schematic representation of the heart-head axis.
Figure 5.1 illustrated the special homodynamic system in IJV, where a pressure wave propagating
from the heart toward the head with velocity 𝑐 is represented together with the direction of the
internal jugular vein blood velocity 𝑢 that propagates from the head toward the heart (Opposite to
each other).
5.2.1 Description of the in-vitro experimental setup
Hydraulic Circuit Components. The main components of our in-vitro experimental setup are i)
the atrioventricular simulator; ii) the internal jugular vein simulator. The atrioventricular simulator

92
consists of a compliant RV sac in the pump head that is connected to the open tank reservior (as
pulmonary artery (PA)) on one side via a pulmonary valve and that is connected to the artificial
RA on the other side via a tricuspid valve. The RV sac employed in this simulator is installed
inside pump heard of a computer-controlled piston-in-cylinder pulsatile pump (ViVitro Labs Inc,
SuperPump, AR SERIES) programmed by the ViVigen software (using a physiological input
waveform for the pump displacement; the piston faces the apical RV). The internal jugular vein
simulator includes the artificial IJV and immersed in a closed chamber, the latter of which consists
of fluid-air filled syringes in order to adjust for the compliance and external vessel pressure in the
neck. The venous return from brain consists of an open tank that is connected to the IJV from the
top. The experimental setup design is shown in the following figure.
Figure 5.2. Schematic of IJV experimental setup design corresponding to the right atrioventricular-IJV hemodynamic
simulator setup.

93
5.2.2 Sketch of Experimental setup
Figure 5.3 Sketch of the experimental setup with PIV system
Figure 5.3. demonstrate a sketch of the 3DPIV right heart IJV coupled setup. The main components
are the IJV chamber, laser generating a rectangular beam, 3D camara, and right heart pumping
system (piston pump with right ventricle chamber connected to a right atrium).
Figure 5.4 Sketch of the experimental setup with measurement device

94
Figure 5.4. demonstrates a sketch of the right heart IJV setup with Pressure and flow measurement
device. The main components are the IJV chamber and right heart pumping system as well as the
devices that have been used for the process of measurement, include: i) pressure (via a Millar
MIKRO-TIP® Catheter Transducer (Mikro-Cath) using a PowerLab 4/35 from ADInstruments);
ii) flow (via a Transonic Flowmeter (TS410)).
The actual experimental setup is demonstrated below, showcasing the apparatus that replicates
pulsatile flow in a collapsible tube, including detailed schematics and descriptions of the
equipment used for measuring flow rates, pressures, and tube deformation.
Figure 5.5. Setup includes the complete in-vitro circulation system consisting of the RV (pump), the IJV (inside close tank),
the RV (compliant chamber) and the venours return tank.

95
Figure 5.6. Overall view of the entire Setup includes the complete in-vitro circulation system and the PIV system.
Hydraulic Circuit Functioning. The compliant RV sac is contracted (squeezed) inside the pump
head using a pulsatile pump programmed by the ViVigen interface (ViVitro Labs Inc.). Using
physiological input waveforms for the piston pump displacement. After the pressure starts to
reduce inside the RV sac during the diastole, the tricuspid valve opens, and sucks the fluid back
from the RA and IJV while generating hemodynamic waves that propagate up to the IJV. The
testing fluid then flows into the reservoir tank on the pulmonary side, as the LV/RV starts to
contract, the mechanical pulmonary valve opens, and the fluid subsequently pumps back to the
venous return from the pulmonary side. This completes the in-vitro venous circulation. A
schematic of the full hydralutic circuit corresponding to the right atrioventricular-IJV
hemodynamic simulator setup and its corresponding circulation path is presented above.

96
5.2.3 Physiologically accurate artificial organ fabrication
The artificial organs installed in the setup (e.g., IJV and RA) are fabricated using silicone rubber
(RTV-3040, Freeman Manufacturing & Supply Company) and Solaris (Smooth-On, Macungie,
PA) which is an alternative for high compliance cases (natural latex rubber (Chemionics Corp.)
can be used for RA fabrication for high compliance cases). These materials are chosen based on
their characteristic ability to replicate the stiffness of a physiological IJV and RA [147, 148]. The
artificial IJV are fabricated in-house based on the one-to-one human-scale stainless-steel metal
mold considering the tapering of IJV with length of 150mm, inlet diameter 8mm and outlet
diameter 12mm. The artificial RA are fabricated using polyvinyl alcohol (PVA) molds via a 3D
printer (Ultimaker S5 Dual Extrusion). Figure 5.7 shows molds that are employed in this work.
Figure 5.7. Photo of the IJV mold and fabricated IJVs/RAs.
Dipping and coating casting methods are used for the natural latex and silicone-based fabrications,
respectively. For the silicone-based fabrications, the following steps are taken: i) mixing the
catalyst and the base of the silicon rubber (RTV-3040, Freeman Manufacturing & Supply
Company, Solaris (Smooth-On, Macungie, PA) is used an alternative for high compliance cases)

97
with a mass ratio of 10 (base) to 1 (catalyst); ii) removing the mixture bubbles using a vacuum
pump chamber; iii) coating a light silicon sheet using 10 g of the mixed solution using a soft-tip
acrylic brush; iv) curing the coated material for 16 hours at standard room temperature (25℃); v)
repeating the coating for more layers (as necessary) so as to achieve the desired IJV compliance
(steps i to iv). For ensuring surface uniformity, the mold is turned upside down at each drying step.
Table 5.1 presents the measured IJV compliances of the final fabricated IJV employing either
silicone or latex.
5.2.4 Procedures and Measurements
Different input parameters to the system are systematically changed/adjusted in a way that such
physiological conditions and variability is thoroughly covered. The in-vitro atrioventricular-IJV
simulator are able to be tested across a wide range of physiological hemodynamic conditions (e.g.,
heart rate (HR)=60 to 120 bpm, cardiac output (CO)=0.5 to 1.0 L/min. For assigning cardiac
parameters to the setup, user-defined input waveforms are imported into the pulsatile pump
controller (ViViGen interface). The frequency of the operation for the pump (which determines
the HR) is also modified using the Vivigen interface on the computer unit. The input profile of the
pulsatile pump system is adjusted to simulate the impact of RV contractility on system
hemodynamics. Different physiological stiffnesses are achieved by changing the number of
applications of dipping (for latex aortas) or coating (for silicone IJV or RA).
Different mechanical parameters are directly measured during each run of the system at different
measurement stations. A list of the parameters that are measured in this study, as well as the
devices that have been used for the process of measurement, include: i) pressure (via a Millar
MIKRO-TIP® Catheter Transducer (Mikro-Cath) using a PowerLab 4/35 from ADInstruments);

98
ii) flow (via a Transonic Flowmeter (TS410)). Two measurement sites are selected for the
hemodynamic data collection: the central IJV, and outside IJV with same height in the closed
chamber. I the experiments, Water/Glycerin (42%/58%) mixture is used as the circulating fluid in
all experiments, and any visible air bubbles are removed prior to running experiments. Where the
viscosity is around 9.6 𝑚2
/𝑠 (4.6 for blood with Water/Glycerin (45%/55%)).
5.2.5 Compliance measurement
IJV Compliance is measured by adding incremental volumes of fluid and measuring the
corresponding incremental change in pressure. Details on acquiring IJV Compliance
measurements are shown in the below figure [147, 148].
Figure 5.8. Compliance measurement setup of IJV.
After casting the models using either latex or silicone, the prepared IJV or RA are removed from
their respective molds. For metal molds, removal is achieved through gentle injection of water at
the material/mold interface. For PVA molds, removal is achieved by immersing the models inside

99
water at standard room temperature. The PVA mold is then dissolved in the water, and the PVA
residue is manually detached from the latex or silicon at the end of the process.
The materials and molds described above are chosen based on their ability to provide physiological
results as well as adding flexibility for further studies on cardiovascular fluid dynamics. Both
materials (silicone and latex) provide characteristics with the ability to replicate compliance and
PWVs of physiological organs. Silicone enables the production of transparent IJV which are
preferable for optical methods for flow visualization such as particle image velocimetry (PIV). On
the other hand, latex provides more control over the compliance and PWV of fabricated IJV and
RA. There is no difference in the final product as one uses either metal or PVA molds. Metal mold
has previously shown to provide reproducible and reliable models for producing smooth IJV tube,
while PVA molds can be potentially used for arbitrary geometry fabrications of the RA.
Fabrication procedures for the RA are similar to the fabrication procedure of the silicone based
IJV. The designed molds are printed for each RA via a 3D printer using PVA material, and the
same steps as the aforementioned fabrication steps for the silicone based IJV are taken. For
fabricating artificial organs in this work, elastomer-based materials, i.e., silicone or natural latex,
have been used. Elastomers are rubber-like materials which are categorized as soft materials with
a low Young's modulus, so the fabricated artificial organs are able to return to their initial condition
when the external stress is diminished [149-152]. The control volume of our fabricated RA is
120ml.
Table 5-1. Dynamic and physiological properties of the fabricated IJV (C=0.1-0.2).
IJV No. AC (mL/mmHg) Thickness(mm) Material

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IJV-1 0.25 ~0.4 Silicone (Solaris)
IJV-2 0.077 ~0.4 Silicone
IJV-3 0.129 ~0.3 Silicone
IJV-4 0.167 ~0.2 Silicone
5.3 Particle Image Velocimetry
PIV, or Particle Image Velocimetry, is an experimental technique that is used to optically measure
fluid velocity fields by tracking particle displacements with high spatial and temporal resolution.
To achieve accurate results, this technique requires clear models and high-speed cameras that can
track illuminated particles using a laser. In this study, the three-dimensional velocity fields were
captured using DDPIV, as originally developed by Pereira and Gharib [153] was used to determine
velocity fields within volumes downstream of the cylinder exit.
Figure 5.9 provides an illustration of the experimental setup, where the flow was illuminated by a
dual-head Nd:YAG laser with 100 mJ/pulse. To create a triangular column of laser light, two
negative 50-mm cylindrical lenses were positioned in front of the beam exit with perpendicular
orientations. In order to limit the horizontal thickness of the illumination volume to around 20 mm,
beam blocks were utilized. The tracers used were white Polycrystalline particles with an average
diameter of approximately 70 μm. (Fluorescent red particle can also be an alternative to white
particle) The volumetric camera system consisted of three apertures placed in an equilateral
triangle configuration, with a side length of 170 mm. Each aperture included a CCD array featuring
12 million pixels (2,048 x 2,048) of size 7.4 microns and depth 12 bits. Three 50-mm Nikon camera
lenses were utilized with aperture settings of f#16. The camera was mounted at an optical distance

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of 700 mm from the back plane of the measurement region at a 90-degree angle to the illuminating
light. The camera's field of view allowed for a measurement region of 140 mm in the horizontal
and vertical directions. The V3V camera frames and laser pulses were synchronized by a TSI
610035 synchronizer with 1 ns time resolution. Each of the three apertures captured pairs of
images. The two lasers emitted pulses timed to straddle neighboring camera frames in order to
produce images suitable for 3D particle tracking. The time between frame-straddled laser pulses
(dt) was 2,000 μs.
Figure 5.9. PIV system setting and a zoomed-in view.
In DDPIV, particle depth information is determined by quantifying the natural blurring of the
particle as it moves out of the focal plane and captured by three cameras. The defocusing PIV
technique uses the image shift produced by the apertures to measure the depth of the particle from
the camera. The DDPIV system in this study utilizes a three-aperture mask arranged in the form
of a triangle to eliminate the ambiguity in determining the depth. The DDPIV system is able to
capture a volume of 100*100*100 mm with 40,000 particles and 0.034 particles/pixel.

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The IJV flow was captured up to 10 cm downstream of the IJV outlet, which is able to capture the
main flow structure. A total of 180 frames at the rate of 60 Hz were acquired for each experiment.
This set of experiments was conducted at the Pahlevan research lab at USC. To determine
instantaneous velocity fields, the following four steps were employed: identification of 2D particle
locations from each of the three apertures, determination of 3D particle locations in space, tracking
of individual particles within the volume, and interpolation of the resulting randomly spaced
vectors onto a Cartesian grid. Here we only briefly describe the process, for more details account
of each of these steps please refer to previous publications of this system [154, 155].
5.3.1 Indexed matching
In the field of biomechanics, PIV is a useful tool for capturing complex physiological flow patterns
in various vascular territories and has been compared to flows obtained from medical imaging
techniques such as magnetic resonance imaging (MRI). However, to achieve this, anatomically
realistic in vitro models are necessary and clear models are required so that high-speed cameras
can track particles illuminated by a laser, which can be challenging and prohibitively expensive to
manufacture through commercial vendors. In addition, it is also necessary for the models to match
the refractive index (RI) of the working fluid to avoid image distortion between the solid-liquid
interface, which enables the accurate measurement of entire velocity fields. PIV analysis has been
conducted using rigid, patient-specific models that were 3D-printed in the past. However, these
materials often have a higher refractive index (RI) than water (1.33), which can make matching
the refractive index of the working fluid and the material challenging without the use of hazardous
chemicals such as sodium iodide [156].

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In this study, particle illumination was tested in the real-size IJV models made with silicon (RTV3040, Freeman Manufacturing & Supply Company, similar to the Sylgard 184). The models were
successfully index matched with a 58:42 glycerol/water mixture (RI = 1.412 nm) and the particles
were imaged without distortion or light scattering at the background wall (mesh box) of both
models. (Solaris (Smooth-On, Macungie, PA) can be an alternative for high compliance cases.)
[156] (sodium iodide can be an alternative for adjusting the viscosity and index matching)

104
Figure 5.10. Index matching result and reflective index measurement (zoom in at bottom).
5.3.2 Volumetric 3D PIV Calibration

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Figure 5.11. PIV calibration procedure result and PIV laser sheet.
As depicted in the above image, the results of the PIV calibration indicate that the system was
successfully calibrated, with a low error rate of 0.5 pixels in the world-to-camera transformation.
This calibration is a critical step in ensuring accurate and reliable measurements of blood flow
dynamics using the 3DPIV system. For further details, please refer to our previous publication on
the 3DPIV system [153] [154, 155].

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5.4 Experimental conditions and ‘Reynolds Number’ Calculation:
5.4.1 Reynolds Number:
The realistic Reynolds number is 55~337 using a generalized Reynolds number definition
(effective Reynolds with effective viscosity with Carreau parameters of whole blood) [157], the
advantage of this is that we can consider the non-Newtonian blood rheology effect by using a
Newtonian fluid.
𝑅𝑒𝑒𝑓𝑓 =
8𝜌𝑢
2
𝜇𝑒𝑓𝑓𝛾̇𝑤𝑎𝑙𝑙
=
𝜌𝑢𝐷
𝜇𝑒𝑓𝑓
, 𝜇𝑒𝑓𝑓 = 𝜇(𝛾̇𝑤𝑎𝑙𝑙), (5.1)
With 𝑄𝑟𝑒𝑓 = (0.7 ± 0.27)/2 𝐿/𝑚𝑖𝑛 [158], (0.74 ± 0.21)/2 𝐿/𝑚𝑖𝑛 [159], or (0.36 ±
0.23) & (0.31 ± 0.23) 𝐿/𝑚𝑖𝑛 [160]. And diameter 𝐷 = 0.87~1.25 𝑐𝑚 [161].
In our study, we selected flowrate 𝑄 from 0.5 𝑡𝑜 1.0 𝐿/𝑚𝑖𝑛, viscosity 8.56 𝑚𝑃𝑎. 𝑠 measured by
viscometer, a tapered internal jugular vein with diameter 𝐷 from 0.8 𝑡𝑜 1.25 𝑐𝑚, and heart rate
from 60 𝑡𝑜 120 𝑏𝑝𝑚 (1 − 2𝐻𝑧). And the corresponding Reynolds number is 118 to 335, which
fits in with the physiological range.
5.5 Hemodynamic quantification and analysis:
Reverse flow index (RFI): Quantifies the flow going towards the brain.
RFI is defined as the ratio of the retrograde flow 𝑄𝑟𝑒𝑣𝑒𝑟𝑠𝑒 (which is in the opposite direction of the
systemic circulation) over the absolute summation of the antegrade flow 𝑄𝑓𝑜𝑟𝑤𝑎𝑟𝑑 (which is in the
same direction of the systemic circulation) and retrograde flow, given by

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𝑅𝐹𝐼 =
|𝑄𝑟𝑒𝑣𝑒𝑟𝑠𝑒|
|∫ 𝑄𝑟𝑒𝑣𝑒𝑟𝑠𝑒𝑑𝑡 𝑇
0
| + |∫ 𝑄𝑓𝑜𝑟𝑤𝑎𝑟𝑑𝑑𝑡 𝑇
0
|
. (5.2)
To quantify 𝑄𝑟𝑒𝑣𝑒𝑟𝑠𝑒and 𝑄𝑓𝑜𝑟𝑤𝑎𝑟𝑑, velocity profiles in IJV were integrated and averaged across
all the IJV cross-sections at each cardiac phase.
Average vorticity: Quantifies the flow rotations.
The average vorticity magnitude inside the IJV is used as a metric to quantify the overall fluid
blending in the ventricle using,
𝜔̅ =
1
𝑉∭ ‖𝛻 × 𝒗‖2 𝑑𝑉
V
𝑉
, (5.3)
where V denotes the initial volume of the IJV prior to pumping (𝑉 = 13.55 𝑚𝑙).

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5.6 IJV Waveform Pressure measurement
The figures below show the sample waveforms of IJV traces (red line) under varying conditions.
Notably, a physiological IJV waveform can be observed prior to the onset of collapsing (left).
Figure 5.12. Pressure waveform in internal jugular vein with HR=60 Q=0.5 and 0.75 ml/min.
Figure 5.13. Pressure waveform in internal jugular vein with HR=60 Q=0.5 and 0.75 ml/min.

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5.7 2D experimental results
Our pressure waveform exhibits physiological behaviors, making it a crucial aspect of our study.
To begin, we utilize our 2D Particle Image Velocimetry (PIV) system to capture two-dimensional
flow patterns. This initial step allows us to determine if a 2D representation can sufficiently capture
the complex three-dimensional collapsing motion of the tube. We aim to establish a baseline
understanding of the flow dynamics within the collapsible tube. If the 2D data proves sufficient, it
simplifies the experimental process and reduces computational demands. However, if
discrepancies arise, it underscores the need for a more comprehensive 3D analysis to fully capture
the intricate behaviors of the system.
5.7.1 2D-PIV particle + vector: In and out of plane motion for vector field
Figure 5.14. 2D-PIV results with particle and vector field.

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5.7.2 3D-PIV particle identification
Figure 5.15. 3D-PIV particle identification results.
5.7.3 Comparison of 3D-PIV and 3D-CFD
Figure 5.16. 3D-PIV vector filed for HR=60 Q=0.5 and 0.75 ml/min and a sample 3D-CFD demonstration. For the 3D-PIV
images the tube boundary demonstrates the IJV wall at the initial neutral position.
The figures above present the results from both 2D and 3D Particle Image Velocimetry (PIV)
analyses of the internal jugular vein's (IJV) collapsing motion. While both methods successfully
captured the flow field within the IJV, it is important to note that 2D PIV provides data from only

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a single plane of the flow field. Consequently, the 2D vector field may not fully capture the
complex physics of the flow, including critical in-plane and out-of-plane motions necessary for a
complete understanding of the flow behavior. Given that this problem inherently involves threedimensional motion and flow structures, as evidenced by our 3D CFD sample results, it is crucial
to use a 3D PIV system to achieve a comprehensive understanding of the flow behavior. The results
obtained thus far underscore the need for and justify the implementation of 3D PIV in this study.
Limitations of 2D PIV: The primary limitation of the 2D PIV technique lies in its inability to
capture the full complexity of the flow, especially in cases involving significant out-of-plane
movements. The flow within the IJV, particularly during collapsing events, involves intricate
interactions between the fluid and the vessel walls, with flow patterns that extend beyond a single
plane. As such, relying solely on 2D PIV data may lead to an incomplete or inaccurate
understanding of these interactions, as it cannot account for the three-dimensional nature of the
flow structures.
Need for 3D PIV Analysis: The necessity for a comprehensive analysis using 3D PIV becomes
apparent when considering the complex nature of the problem. 3D PIV offers a more holistic
approach by capturing volumetric flow fields and providing a complete visualization of both inplane and out-of-plane motions. This method involves using multiple cameras and advanced image
reconstruction techniques to create a detailed map of the flow within the entire volume of the IJV.
The 3D PIV system can accurately measure velocity vectors in all three spatial dimensions, thereby
offering a thorough representation of the fluid dynamics and interactions within the vessel.
The initial results obtained from our 3D PIV system demonstrate its capability to capture the threedimensional collapsing motion of the internal jugular vein (IJV). This capability is crucial for

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investigating the flow structure within the IJV, especially in the context of a Right Heart-IJV
coupled system. Unlike traditional 2D PIV methods, the 3D PIV system enables a more
comprehensive analysis of the flow dynamics, encompassing all aspects of 3D flow motions. This
thorough examination is essential for accurately understanding the physics of flow in our study.
The 3D PIV system offers an unprecedented view into the flow patterns and dynamic behaviors
within the Right Heart-IJV coupled system. By capturing volumetric flow fields, it provides a
detailed and accurate representation of how the fluid interacts with the vessel walls and responds
to various physiological conditions. This depth of analysis is critical for identifying and
understanding the complex flow phenomena that occur within the IJV, particularly during
collapsing events.
Overall, the implementation of the 3D PIV system represents a significant advancement in the
study of fluid dynamics within the Right Heart-IJV coupled system. It not only enhances our ability
to observe and analyze the three-dimensional aspects of flow but also serves as an invaluable tool
for investigating the complex interplay between the heart and the IJV. This comprehensive
approach will undoubtedly contribute to a deeper understanding of the flow patterns and dynamics
within this physiological system, paving the way for future research and potential medical
applications.
5.8 3D experimental results
In previous studies on the internal jugular vein (IJV), researchers have predominantly relied on 2D
PIV. However, as illustrated in the figures above, 2D PIV is insufficient for capturing the in-plane
and out-of-plane motions involved in the collapsing of the IJV. The asymmetric motion created
during the collapse of the IJV cannot be fully captured by 2D PIV, highlighting its limitations.

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Therefore, employing 3D PIV is necessary to accurately study the complex and asymmetric
dynamics of IJV collapse.
5.8.1 Vector plot:
(a)

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(b)
(c)

115
(d)
Figure 5.17 Snapshots of the particle velocity vector (colored by the velocity magnitude) in the jugular vein vessel (tube).
(a)Front view and (b) Side view for model one (M1), flowrate Q=0.75 L/min and heartrate HR=1.0 Hz; (c)Front view and
(d) Side view for material one (M1), flowrate Q=0.75 L/min and heartrate HR=2.0 Hz. T1 to T2 represents the time before,
during (T2 and T3) and after (T4) the collapse of the tube. Red and blue arrows show the direction of the venous flow (top
to bottom) and the direction of the heart wave (bottom to top)

116
5.8.2 Velocity & Vorticity plot:
(a)
(b)

117

(c)
(d)
Figure 5.18 Snapshots of the velocity magnitude field in the jugular vein vessel (tube). (a)Front view and (b) Side view for
material one (M1), flowrate Q=0.75 L/min and heartrate HR=1.0 Hz; (c)Front view and (d) Side view for material one (M1),

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flowrate Q=0.75 L/min and heartrate HR=2.0 Hz. T1 to T2 represents the time before, during (T2 and T3) and after (T4)
the collapse of the tube. All the timings match the ones in Figure 3. The flow direction is represented by the red arrow (top
to bottom), and the wave direction is represented by the blue arrow (bottom to top).

(a)
(b)

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(c)
(d)
Figure 5.19 Snapshots of the vorticity magnitude field in the jugular vein vessel (tube). (a)Front view and (b) Side view for
material one (M1), flowrate Q=0.75 L/min and heartrate HR=1.0 Hz; (c)Front view and (d) Side view for material one (M1),

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flowrate Q=0.75 L/min and heartrate HR=2.0 Hz. T1 to T2 represents the time before, during (T2 and T3) and after (T4)
the collapse of the tube. All the timings match the ones in Figure 5.17.
Figure 5.17 shows the snapshots of the particle velocity vector in the jugular vein tube from the
front view ((a) and (c)) and the side view ((b) and (d)), under different heart rate (HR=1.0 Hz for
(a) and (b); HR=2.0 Hz for (c) and (C)) with same stiffness (M1) and same flowrate Q=0.75 L/min
(under collapsing mode). T1 and T4 represent the time before and after the collapsing happening;
T2 and T3 represent the time during the collapse of tube. Note that the flow direction is from top
to bottom (brain to heart) represented by the red arrow, and the heart wave direction is from bottom
to top (heart to brain) represented by the blue arrow. The conditions chosen and shown here are
the ones with collapsed behavior (at T3 and T4). The particle velocity vectors are colored by the
by the velocity magnitude of the flow. The corresponding flow velocity and vorticity of the field
are demonstrated in Figure 5.18 and 5.19 respectively. The views and timing in Figure 5.18 and
Figure 5.19 are matched with the ones in Figure 5.17.
5.8.3 Sample IJV pressure waveform:
Figure 5.20. Sample Pressure waveform in internal jugular vein with HR=60 Q=0.5 and 0.75 ml/min.

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5.8.4 Reversed Flow Index & pressure waveform:
Figure 5.21 Reverse flow index (RFI) changes profile within one cardiac cycle (top row) and the IJV pressure waveform
profile in one cardiac cycle (bottom row) under different flow conditions with material one (M1). (a) Heartrate HR=1.0 Hz
flowrate Q=0.50 L/min (non-collapse mode); (b) Heartrate HR=1.0 Hz flowrate Q=0.75 L/min (collapse mode); (c)
Heartrate HR=1.0 Hz flowrate Q=0.50 L/min (collapse mode with oscillatory); (d) Heartrate HR=2.0 Hz flowrate Q=0.75
L/min (collapse mode with higher frequency). Note that (b) and (d) are corresponding to the cases (flow field) in Figure 5.17
to 5.19 (with T1 to T4 marked in the figure).
5.8.5 Reversed Flow Index & Volume change
Figure 5.22 Jugular vein tube volume (V(t) normalized be the mean volume) changes profile within one cardiac cycle (top
row) and the corresponding Reverse flow index (RFI) changes profile within one cardiac cycle (bottom row). (a) to (d)
represent the cases with different flow conditions (all cases here are with collapsed mode) with material one (M1). (a)
Heartrate HR=1.0 Hz flowrate Q=0.75 L/min; (b) Heartrate HR=1.0 Hz flowrate Q=1.0 L/min; (c) Heartrate HR=2.0 Hz
flowrate Q=0.75 L/min; (d) Heartrate HR=2.0 Hz flowrate Q=1.0 L/min. The starting and ending of the RFI wave are
marked by the dashed lines and matched with the volume change profile. Note that (a) and (c) are corresponding to the
cases (flow field) in Figure 5.17 to 5.19 (with T1 to T4 marked in the figure).

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Figure 5.20 shows the sample pressure for HR=60 Q=0.5 and 0.75 ml/min. Figure 5.21 shows the
reverse flow index (RFI) profile coupled with the IJV pressure waveform profile under different
flow condition with M1, heartrate HR=1.0 Hz and flowrate Q from 0.5 to 1.0 L/min ((a), (b), (c)),
and M1, heartrate HR=2.0 Hz, flowrate Q=0.75 L/min. Large RFI appears when flow reach to 0.75
L/min and more with low IJV pressure (lower than external). Two cases (b) and (d) are
corresponding to the flow field in Figure 5.17 to 5.19, where T1 to T4 represents the time before,
during and after the collapse of the tube. Accordingly, Figure 5.22 demonstrates the jugular tube
volume change (normalized by its mean) coupled with reverse flow index (RFI) change under 4
conditions all with collapse behavior of the tube ((a)(b): HR=1.0 Hz, Q=0.75 and 1.0 L/min; (c)(d):
HR=2.0 Hz, Q=0.75 and 1.0 L/min, respectively). The dashed lines marked the start and end of
reverse flow happens which is matched with the increase of the tube volume. T2 and T3 for in
Figure 5.22 (a) and (c) (corresponding to the cases in Figure 5.17 to 5.19) represent the period
during the collapse of the tube which is within the start and end of the reverse flow.

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5.8.6 Master plot 1-RFI Q V.S. Frequency
Figure 5.23 calculation of the mean Reverse flow index (RFI) with respective to time in one cardiac cycle. All cases here are
for material one (M1) with same stiffness. Different colors represent different hear rate: Blue 1.0Hz, Red 2.0 Hz, Yellow
1.66 Hz. (The measurement error from PIV system is around 0.16 mm/s calculate from calibration error 25 micron/pixel
Camera to World Error and 0.025 pixel World to Camara Error (Pereira and Gharib (2002)) with dt=4ms)

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5.8.7 Master plot 2-RFI Stiffness
Figure 5.24 calculation of the mean Reverse flow index (RFI) with respective to time in one cardiac cycle for all cases here
with different stiffness (material one to four (M1 to M4). M1 to M4 are marked by different marker styles and the colors
represent different hear rate: Blue 1.0Hz, Red 2.0 Hz, Yellow 1.66 Hz.

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5.8.8 Reversed Flow Index & Vortex wave
Figure 5.25 Reverse flow index (RFI) changes profile within one cardiac cycle (top row) and the corresponding average
vorticity (AV) profile in one cardiac cycle (bottom row) under different flow conditions with material one (M1). (a)
Heartrate HR=1.0 Hz flowrate Q=0.50 L/min (non-collapse mode); (b) Heartrate HR=1.0 Hz flowrate Q=0.75 L/min
(collapse mode); (c) Heartrate HR=1.0 Hz flowrate Q=0.50 L/min (collapse mode with oscillatory); (d) Heartrate HR=2.0
Hz flowrate Q=0.75 L/min (collapse mode with higher frequency). Note that (b) and (d) are corresponding to the cases (flow
field) in Figure 5.17 to 5.19 (with T1 to T4 marked in the figure).
Figure 5.25 shows the reverse flow index (RFI) profile coupled with the average vorticity profile
under different flow condition with M1, heartrate HR=1.0 Hz and flowrate Q from 0.5 to 1.0 L/min
((a), (b), (c)), and M1, heartrate HR=2.0 HZ, flowrate Q=0.75 L/min. Large RFI appears when
flow reach to 0.75 L/min and more with larger average vorticity. Two cases (b) and (d) are
corresponding to the flow field in Figure 5.17 to 5.19, where T1 to T4 represents the time before,
during and after the collapse of the tube. Accordingly, Figure 5.22 demonstrates the jugular tube
volume change (normalized by its mean) coupled with reverse flow index (RFI) change under 4
conditions all with collapse behavior of the tube ((a)(b): HR=1.0 Hz, Q=0.75 and 1.0 L/min; (c)(d):
HR=2.0 Hz, Q=0.75 and 1.0 L/min, respectively). The dashed lines marked the start and end of
reverse flow happens which is matched with the increase of the tube volume. T2 and T3 for in
Figure 5.22 (a) and (c) (corresponding to the cases in Figure 5.17 to 5.19) represent the period
during the collapse of the tube which is within the start and end of the reverse flow.

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5.8.9 Master plot 3-Vorticity
Figure 5.26 calculation of the mean of the volume Average vorticity (AV) with respective to time in one cardiac cycle. All
cases here are for material one (M1) with same stiffness. Different colors represent different hear rate: Blue 1.0Hz, Red 2.0
Hz, Yellow 1.66 Hz.

127
Figure 5.27 calculation of the mean of the volume Average vorticity (AV) with respective to time in one cardiac cycle for all
cases here with different stiffness (material one to four (M1 to M4). M1 to M4 are marked by different marker styles and
the colors represent different hear rate: Blue 1.0Hz, Red 2.0 Hz, Yellow 1.66 Hz
Figure 5.23 shows the average reverse flow index (RFI) profile under different flow conditions
(heartrate and flowrate) with M1. The color of the plot indicates the heartrate/frequency of the
flow. It can be seen that the RFI increased the Q in all three heartrate conditions and the increase
rate is larger for HR=1.0 Hz. Moreover, the average reverse flow index (RFI) profile for all the
conditions (adding stiffness different) in our experiments are shown in Figure 5.24. Note that the
marker style represents different stiffness of the tube (M1 to M4). The RFI increase significantly
from Q=0.5 and 0.75 L/min in all the conditions with different HR and stiffness (which is

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correlated to the non-collapse and collapse mode of the tube). Interestingly, RFI is larger for
medium stiffness (M2 and M3) and smaller for low and high stiffness (M1 and M4) when Q=0.75
L/min (collapse mode), showing a non-linear effect of the stiffness to the collapse motion of the
tube. The mean of the volume Average vorticity (AV) over time is shown in Figure 5.26 and Figure
5.27 with same marker and color style as for Figure 5.23 and 5.24. The total vorticity increases as
the Q increases among all the cases and the frequency and stiffness of the tube has less impact
comparing with RFI.
5.9 Discussion
In our study, we experimentally investigated the fluid dynamics in a collapsible tube representing
the internal jugular vein (IJV). Our aim was to explore the pulsatile dynamics of the IJV and its
potential effects on the heart-brain coupling system. We utilized a physiological right-heart
system, including the right atrium and ventricle, to generate venous flow in the IJV tube. The flow
within the jugular vein was measured using a 3D Particle Image Velocimetry (PIV) system. We
conducted in-vitro experiments under various conditions, including different flow rates, heart
rates, and vessel stiffness, to determine the dynamic modes of the IJV, such as non-collapse and
collapse modes. The details of the experimental setup are described in the methodology section
above. Our results indicate that the jugular vein exhibits collapse behavior under certain
conditions, particularly at high flow rates. The 3D PIV system successfully captured the flow
motion inside the collapsing tube (see Figure 5.17), revealing a three-dimensional structure
observable from both front and side views. This structure includes a collapse on one side and a
bulge on the other side (see Figures 5.17 to 5.19), underscoring the necessity of 3D PIV over the
traditionally used 2D PIV in previous studies. A unique aspect of our findings is that, unlike in

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arteries where the flow and heart pressure wave travel in the same direction, in the jugular vein,
which connects the brain and the right heart, the flow and pressure wave move in opposite
directions. This dynamic is clearly visible in our PIV results (Figure 5.17), showing the flow vector
direction and the heart wave-induced collapse wave direction.
In our study, the collapse and oscillatory behavior occurred at higher flow rates, given the fixed
external and inlet pressure differences of the tube. As the flow rate increased, a collapse motion
appeared. At even higher flow rates (1.0 L/min), self-excited oscillations also emerged in the tube
(refer to Figures 5.17 and 5.21). This collapse motion significantly affects the fluid dynamics of
the tube, particularly the self-pumping phenomenon and vortex formation. Interestingly, our
results revealed the presence of reverse flow phenomena within the jugular tube flow, correlated
with the tube's collapse. From T2 and T3 in Figures 5.17 to 5.19, we observed that the collapse of
the tube propagated upwards following the direction of the heart pressure wave, which is opposite
to the mean flow direction. This unique collapsing and propagating behavior subsequently caused
the reverse flow towards the top.
Figure 5.22 illustrates that the reverse flow index (RFI) increases significantly during the collapse
of the internal jugular vein (IJV). The collapse wave propagates from the bottom of the tube (heart
side) to the top (brain side), opposite to the mean venous flow, as observed in moments T2 and T3
in Figures 5.17 to 5.19. This indicates a direct link between RFI and the collapse of the IJV. The
reverse flow and collapse wave are also detectable through changes in the tube's total volume
(V(t)). When the total volume recoils (increases from a lower peak) after the collapse starts (T2 in
Figure 5.22(a) and (c)), the RFI begins to rise. This increase continues until the collapse wave
reaches the top, and the tube returns to its normal volume (as shown by the dashed lines in Figure

130
5.22). This demonstrates that the collapse of the tube creates a unique pumping phenomenon,
generating a backflow as the collapse wave propagates "back" from the heart to the brain, opposite
to the mean flow direction from the brain to the heart. Note that, this backflow pumping is unique
to the jugular venous system due to the opposite direction of the heart wave generating the collapse
compared to the mean flow direction. Our study confirms that this backflow is directly correlated
with the collapse, as evidenced by the RFI, total volume changes, and the collapse period of the
tube shown in Figures 5.17, 5.18, 5.21, and 5.22. The reverse flow occurs precisely when the
collapse propagates upward during moments T2 and T3, accompanied by a reduction and recoil in
total volume V(t). These findings indicate that the abnormal flow phenomena result from the
unique pulsatile dynamics of the jugular vein, including collapse and self-excitation. More
importantly, the collapse-induced back-pumping behavior can generate a backflow to the brain
from the venous return, potentially leading to neurological and neurodegenerative diseases.
To investigate the effect of dynamic parameters such as flow rate, heart rate, and stiffness on
reverse flow, the time-averaged reverse flow index (TARFI) over a cardiac cycle was calculated
as a metric representing the overall backflow ratio and plotted against these parameters. Figure
5.23 shows the effect of flow rate and heart rate on TARFI for the jugular vein with model one
(M1). The data indicate that TARFI increases significantly with the flow rate. This is because
reverse flow is induced by the back-pumping from the upward collapse propagation. The collapse
behavior of the tube is highly correlated with the flow rate in our experimental configuration.
When the flow rate (Q) is low (0.5 L/min), the jugular tube is not in a collapse mode, resulting in
a low TARFI (as seen in Figure 6(a)). When Q reaches 0.75 L/min, the jugular tube transitions to
a collapse mode and, under certain conditions, to a highly oscillatory collapse mode. As shown in
Figure 8, TARFI increases significantly with Q across all heart rates (frequencies). Four cases

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marked as “A” to “D” in Figure 5.23 correspond to the four cases in Figure 5.22, illustrating the
effect of heart rate on reverse flow. For a low heart rate (1 Hz), TARFI increases more significantly
(from A to B) compared to a higher heart rate (2 Hz) (from C to D). This difference is caused by
the dynamic changes in the tube under different frequencies. In Figures 5.22A and 5.22B, the
jugular tube exhibits a highly oscillatory mode when the flow rate changes from 0.75 L/min to 1.0
L/min for 1 Hz cases. For 2 Hz cases (C and D), the IJV tube maintains a single collapse motion
mode with more significant collapse (more change in total volume V(t)) for higher flow rates. The
latter cases (C and D) show an increase in TARFI due to enhanced collapse motion, while the first
cases (A and B) represent a mode change from collapse to a highly oscillatory mode where multiple
RFI peaks can be found in one cardiac cycle. This dynamic mode change creates more collapse
and oscillation, resulting in a more significant reverse flow increase with the flow rate. These
results indicate that the backflow phenomenon is highly correlated to the collapsing and selfexcitation of the jugular vein, characterized by the mean flow rate (Q) and heart rate (frequency).
To explore the material characteristics of the IJV tube and their impact on dynamics and backflow
behavior, TARFI for cases with all four stiffness values chosen in this study are summarized in
Figure 5.24. Similar to the M1 case, all configurations show an increase in TARFI as the IJV
transitions from a non-collapse to a collapse mode with increasing flow rate from 0.5 to 0.75
L/min. However, unlike a linear increase with flow rate, stiffness demonstrates a non-linear effect
(Figure 5.24 M1 to M4). During the collapse mode at Q = 0.75 L/min, both low stiffness (M1) and
high stiffness (M4) configurations exhibit similar total backflow (represented by TARFI) across
all heart rates, whereas the medium stiffness cases (M2 and M3) show significantly higher TARFI
under the same flow rate and heart rates. This non-linear effect of tube wall stiffness on reverse
flow indicates that softer and much stiffer materials with thicker walls (M1 and M4) induce less

132
significant collapse-induced reverse flow compared to stiffer materials with thinner walls (medium
stiffness). The non-linear effect of stiffness on the back pumping flow ratio can be attributed to
the non-linear behavior of heart wave propagation and reflection as stiffness increases (changes in
Pulse Wave Velocity and Wave Condition Number). Our experiments demonstrate that an IJV
tube with medium stiffness may cause larger backflow when collapse occurs. These findings
suggest that the non-linear effects of stiffness on wave dynamics can significantly alter back
pumping phenomena, further highlighting the correlation between collapse and backflow with the
unique wave dynamics in the IJV, where the heart wave propagates against the mean venous return
flow from the brain.
Unlike the Reverse Flow Index (RFI), which is influenced by dynamic parameters of the IJV tube
(flow rate coupled with frequency and stiffness), the average vorticity over time is primarily
associated with flow rate and shows a weaker correlation with tube stiffness or heart rate. This
suggests that vortex formation is primarily dependent on the flow rate, as it significantly impacts
the overall fluid dynamics of the tube due to a fixed viscosity considerations. As depicted in Figure
5.18, vortices typically form along the walls of the IJV tube. Therefore, in our experiment with
fixed dimensions and viscosity, the time-averaged vorticity is controlled by the flow rate Q, which
is proportional to the Reynolds number (Re) and governs vortex shedding. Under constant
viscosity conditions, high vorticity near the wall can result in elevated wall shear stress (WSS).
This indicates that, in the context of the experiment, the flow rate is the dominant factor influencing
the occurrence and strength of vortices, while tube stiffness and heart rate play a lesser role.

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5.10 Clinical importance
The internal jugular veins (IJV), as the primary vessels connecting the brain and right heart, carry
venous return flow from the brain to the heart. The potential backflow to the brain induced by
collapsing and back-propagated heart waves can have significant impacts on the hemodynamic
balance between the brain and heart, potentially leading to cardiovascular and cerebrovascular
diseases. Abnormalities in cerebral venous blood flow have been linked to neurological disorders
such as normal pressure hydrocephalus (NPH). Additionally, issues in the extracranial venous
system, particularly jugular venous reflux (JVR), may play a role in the pathophysiology of
neurodegenerative diseases like Alzheimer's disease (AD). Studies have suggested that JVR may
also be associated with neurologic symptoms such as transient global amnesia or transient
monocular blindness. More recently, studies on retrograde internal jugular vein blood flow during
spaceflight have indicated that reverse flow in the jugular vein can be linked to IJV thrombosis.
Vortex formation in venous vessels, changes in shear stress, increased turbulence and recirculation,
and elevated wall shear stress (WSS) due to high vorticity near the wall (under constant viscosity)
are significant factors in these phenomena [162]. Numerous studies have implicated altered local
hemodynamic metrics, such as vorticity and WSS, in the initiation and progression of thrombosis
and atherosclerosis [163-165]. In particular, vortex formation and increased WSS can lead to
higher power loss in the blood flow between the brain and heart. These factors highlight the critical
role of IJV dynamics in the development and exacerbation of cardiovascular and cerebrovascular
diseases, as well as neurological disorders associated with altered venous flow patterns and
hemodynamics.

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5.11 Conclusion
The study of internal jugular veins (IJV) and their dynamics in the context of right heart-brain
coupling has revealed significant insights into the complex interplay between venous flow,
hemodynamics, and potential health implications. Internal jugular veins serve as crucial conduits
for cerebral venous return, linking the brain and heart. Our experimental investigation, employing
3D-PIV and other methodologies, has elucidated several key findings.
Firstly, the collapse and back propagation of heart waves induce reverse flows in the IJV, a
phenomenon that can potentially disrupt the normal hemodynamic balance between the brain and
heart. This backflow, observed under conditions of increased flow rate and certain material
stiffnesses, may contribute to cardiovascular and cerebrovascular diseases. Specifically,
abnormalities such as jugular venous reflux (JVR) have been linked to neurodegenerative diseases
like Alzheimer's disease and neurological symptoms such as transient global amnesia. Secondly,
our study underscores the role of altered local hemodynamic metrics, including vorticity and wall
shear stress (WSS), in the initiation and progression of thrombosis and atherosclerosis. Vortex
formation, increased turbulence, and high WSS near the vessel wall due to high vorticity are
critical factors in these pathophysiological processes. The findings highlight the necessity for
further research into the mechanisms underlying IJV dynamics and their implications for
cardiovascular and neurological health. Future studies should continue to explore how factors such
as flow rate, heart rate, and vessel stiffness influence venous flow patterns and hemodynamics.
Additionally, investigations into therapeutic interventions targeting abnormal venous flow could
potentially mitigate the risk of associated diseases.

135
In conclusion, our study contributes to a deeper understanding of the complex dynamics of internal
jugular veins and their role in cardiovascular and cerebrovascular health. By shedding light on
these mechanisms, we aim to pave the way for improved diagnostic and therapeutic strategies in
the treatment and prevention of venous-related diseases.

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Chapter 6: The global effect of the right heart dynamics on jugular
vein wave dynamics using 1D hemodynamic modeling of the
entire circulatory system
6.1 Chapter introduction
On the heart side, assessing the jugular venous pulse (JVP) offers a window into the right heart,
providing valuable insights into the functioning of the right heart and its hemodynamics. However,
there is a widespread belief that analyzing the jugular venous pulse waveform at the bedside is
only a theoretical exercise and that invasive measurements are necessary in real-world scenarios.
As a result, the analysis of the jugular venous pulse is frequently disregarded during routine
physical examinations [17]. Current methods of estimating right atrial pressure (RAP), such as
bedside jugular venous pressure exams or inferior vena cava measurement during a comprehensive
echocardiogram, offer imprecise estimates of actual RAP. However, the diagnosis of cerebral
venous outflow impairment largely depends on a combinatory use of imaging modalities, as
standard diagnostic criteria are not yet available [18]. Cerebral venous outflow impairment is a
relatively new and underexplored area of research, but the implications of such impairment are
significant. Impaired cerebral venous outflow has been linked to various neurologic disorders,
suggesting that these disorders are related to cerebral venous outflow impairment [13].
Extracranial venous abnormalities, especially jugular venous outflow disturbance, were initially
viewed as nonpathological phenomena due to a lack of understanding of their features and clinical
significance. Recent advances suggest that intravenous angioplasty with stenting can safely restore
normal blood circulation, alleviate symptoms, and improve quality of life. When endovascular

137
intervention is not feasible, surgical removal of structures constraining the internal jugular vein
may be an alternative or complementary option [19]. Efforts to understand, diagnose, and treat
cerebral venous outflow impairment are still in their early stages. In a proof-of-concept study, the
impact of central venous pressure (CVP) on internal jugular vein cross-sectional area (CSA) and
blood flow time-average velocity (TAV) was evaluated. Results showed that TAV and CSA lagged
behind CVP, with an inverse correlation between CSA and TAV [20]. Ultrasound assessment of
the jugular venous pulse to determine mean CVP offers a noninvasive alternative to current
invasive procedures. The autocorrelation signals of CVP and IJV-CSA pulses were similar,
allowing for accurate estimation of mean CVP using IJV-CSA. This method shows promise as a
safer and more efficient approach for measuring mean CVP [21].
To investigate the global effects of the entire human circulatory system on the internal jugular vein
(IJV) dynamics, we employ a validated 1D model of the complete systemic and venous vascular
network, based on space-time variables. This 1D simulation allows us to explore the interactions
and influences of the overall circulation on IJV wave dynamics. By performing these simulations,
we aim to uncover the relationship between the jugular venous pressure (JVP) and the right atrial
pressure (RAP). This involves detailed analysis of the hemodynamic interactions within the entire
circulatory system, which can reveal how changes in one part of the system affect pressures and
flows in another. Our 1D model allows us to simulate various physiological and pathological
conditions, providing a comprehensive understanding of how RAP influences JVP. Specifically,
by adjusting parameters such as the rigidity of the internal jugular vein and the contractility of the
right ventricle and right atrium, we can observe how these changes impact the pressure dynamics.
This includes understanding the transmission of pulsatility and the resulting waveforms in the IJV
and right atrium. Furthermore, by analyzing the data from these simulations, we can identify

138
patterns and correlations that elucidate the mechanistic link between JVP and RAP. This
relationship is crucial for diagnosing and managing conditions like heart failure, where elevated
RAP can lead to increased JVP, reflecting the compromised hemodynamic state of the patient. In
addition to providing insights into normal physiological conditions, our model can simulate disease
states such as tricuspid valve stenosis or regurgitation, which may alter the JVP-RAP relationship.
Ultimately, this study aims to enhance our ability to predict clinical outcomes based on JVP
measurements and provide a more robust framework for assessing the right heart function through
non-invasive methods. The insights gained from this research could lead to better diagnostic and
therapeutic strategies for managing cardiovascular disorders.
Our study focuses on examining the impact of IJV dynamics on the pulsatility transmission from
the right heart to the brain. By using our 1D circulatory solver, we can simulate these global effects
with high precision. We consider different levels of IJV rigidity by applying multiplicative factors
of a minimum rigidity level 𝐸1
(𝑥) that corresponds to the baseline PWV (𝑐0) initially prescribed
in the model. To simulate different states of RV and RA contractility, we vary the end-systolic
elastance (𝐸es), a common measure of contractility. All components in the model are assigned
physiological ranges of parameters based on the natural cardiovascular system, ensuring realistic
and relevant simulations. This 1D model offers the advantage of capturing the global effects on
IJV hemodynamics comprehensively. By simulating the entire circulatory system, we can quantify
the hemodynamic parameters of the IJV under various conditions, providing valuable insights into
the pulsatility transmission and its implications for brain-heart coupling and overall circulatory
dynamics.

139
Figure 6.1 A schematic illustration of the multiscale closed-loop model employed in this chapter, where all heart chambers
interact with each other via a septal elastance (dashed grey arrows). The complete system is composed of 394 individual
vascular segments, and vascular beds are modeled with any arbitrary number of connecting arteries and veins. (LA: left
atrium; LV: left ventricle; RA: right atrium; RV: right ventricle.)

140
6.2 Method and Governing equations
The problem of propagation of pressure and flow waves in the cardiovascular system can be
modeled by decomposing the domain into segments of elastic tubes of given lengths whose
properties can be described by single axial coordinates [166-170]. For the jugular vein studies of
this chapter, we employ a complex multiscale model of the entire closed-loop circulation, replete
with a four-chamber heart model, that is adopted from a validated physiologically-relevant
formulation [171] [125] [172]. Figure 6.1 presents a schematic illustration summarizing the
complete closed-loop system (composed of 394 segments) that is detailed in what follows.
6.2.1 1D (2D-axisymmetric) fluid-structure dynamics in a vessel segment
For a mean pressure 𝑃 = 𝑃(𝑥,𝑡) and a mean velocity 𝑈 = 𝑈(𝑥,𝑡) over a cross-sectional area 𝐴 =
𝐴(𝑥,𝑡), propogation in each vessel segment of length ℓ can be expressed [125, 171-173] as a
reduced-order nonlinear system given by
(
∂𝐴
∂𝑡
(𝑥,𝑡)
∂𝑈
∂𝑡
(𝑥,𝑡)
) = −
(

∂(𝐴𝑈)
∂𝑥
(𝑥,𝑡)
𝑈
∂𝑈
∂𝑥
(𝑥,𝑡) +
1
𝜌
∂𝑃
∂𝑥
(𝑥,𝑡) +
2(𝜁 + 2)𝜇𝜋𝑈(𝑥,𝑡)
𝜌𝐴(𝑥,𝑡) )
, (6.1)
where 𝜌 ∈ ℝ is a (constant) blood density, 𝜇 ∈ ℝ is a (constant) blood viscosity and 𝜁 ∈ ℝ is a
given constant of an assumed axisymmetric velocity profile. The internal pressure 𝑃 = 𝑃(𝑥,𝑡),
which accounts for the fluid-structure interaction of the problem, can be given by a number of
constitutive tube laws [125, 173]. We adopt a non-linear elastic relation [173] given by the
expression

141
𝑃(𝑥,𝑡) − 𝑃ext = 2𝜌𝑐0
(𝑥)
2
[(
𝐴(𝑥,𝑡)
𝐴0
(𝑥)
)
1/2
− 1], (6.2)
where 𝑃ext(𝑥) is the external pressure, 𝐴0
(𝑥) is the (initial) diastolic area, and 𝑐0
(𝑥) is the initial
pulse wave velocity (which, in some formulations [173], can be written in terms of elastic modulus
𝐸(𝑥) and wall thickness ℎ(𝑥), i.e., 𝑐0 is a measure of the stiffness as well). At any given spacetime location (𝑥,𝑡) for a vessel, the corresponding pulse wave speed can be expressed in terms of
the cross-sectional area 𝐴(𝑥,𝑡) as
𝑐(𝑥,𝑡) = 𝑐0
(𝑥) (
𝐴(𝑥,𝑡)
𝐴0
(𝑥)
)
1/2
. (6.3)
Vessel branching
In order to simulate multiple vessels, including vascular bifurcations or trifurcations, it is necessary
to treat the fractal structure of the circulation network and, namely, branching points. These
junctions effectively act as mathematical discontinuities in cross-sectional area and material
properties. Physically, one must enforce a continuity of total pressure and a conservation of mass
at each junction point. For example, given a parent vessel 𝑝 that splits into two daughter vessels
𝑑, 𝑖 = 1,2, the corresponding mathematical conditions are given by
𝑃𝑝 +
𝜌
2
𝑈𝑝
2 = 𝑃𝑑,𝑖 +
𝜌
2
𝑈𝑑,𝑖
, 𝑖 = 1,2 and 𝐴𝑝𝑈𝑝 + 𝐴𝑑,1𝑈𝑑,1 + 𝐴𝑑,2𝑈𝑑,2 = 0. (6.4)
Numerically, these equations are implemented through the solution of a corresponding Riemann
invariant problem that enforces compatibility of propagating characteristics and provides the final
three equations given by

142
𝑤1𝑝 = 𝑈𝑝 + 4(𝑐𝑝 − 𝑐0) and 𝑤2𝑑,𝑖 = 𝑈𝑑,𝑖 − 4(𝑐𝑑,𝑖 − 𝑐0,𝑑,𝑖), 𝑖 = 1,2, (6.5)
where 𝑤1𝑝 represents the outgoing characteristic from the parent vessel and 𝑤2𝑑,𝑖
the outgoing
characteristic from the daughter vessels (𝐴 and 𝑈 can both be represented at the boundary in terms
of both incoming and outgoing characteristics).
6.2.2 0D lumped parameter modeling
Four-chamber heart model
We employ the same heart model proposed in [125], which includes interaction of all four
chambers with each other via the septum. Pressure 𝑃c
in each chamber is given by
𝑃c
(𝑡) = 𝑃pc + 𝐸nat(𝑉(𝑡) − 𝑉0
) − 𝑅s𝑄c
(𝑡) +
𝐸nat
𝐸sep
𝑃CL(𝑡), (6.6)
where 𝑅s = 𝐾𝐸nat(𝑉 − 𝑉0
) is a source resistance coefficient, 𝑉 is the volume ( 𝑉0 the
initial/unstressed volume), and 𝑃𝐶𝐿 is the pressure in the contra-lateral chamber. The external
pressure from the pericardium, 𝑃pc, is given for a pericardial volume 𝑉pc by the expression
𝑃pc = 𝐾pc𝑒𝑥𝑝 (
𝑉pc − 𝑉0,pc
𝛷
) , (6.7)
where 𝛷, 𝐾pc, 𝑉0,pc are constants. The native elastance 𝐸nat in a chamber is given in terms of free
wall elastance 𝐸fw and septal elastance 𝐸sep = 𝜅𝐿𝐸fw,L + 𝜅𝑅𝐸fw,R (for left and right constants 𝜅𝐿
and 𝜅𝑅) by the expression
𝐸nat =
𝐸fw𝐸sep
𝐸fw + 𝐸sep
. (6.8)

143
Similarly to others [125, 172], the free wall elastance is given by a closed-form function in terms
of maximum and minimum elastances 𝐸max and 𝐸min by the expression
𝐸fw(𝑡) = (
𝐸max − 𝐸min
max [(
𝑔1
1 + 𝑔1
) (
1
1 + 𝑔2
)]
) (
𝑔1
1 + 𝑔1
) (
1
1 + 𝑔2
) + 𝐸min, (6.9)
where
𝑔1 = (
𝑡 − 𝑡0
𝜏1
)
𝑚1
and 𝑔2 = (
𝑡 − 𝑡0
𝜏2
)
𝑚2
, (6.10)
for the time of onset of contraction 𝑡0, contraction and relation time offsets 𝜏1, 𝜏2, and contraction
and relation rate constants 𝑚1, 𝑚2.
Valve model
The valve model employed in our solver is based on Bernoulli’s equation (with a dependence on
transvalvular pressure) and is similar to the formulation of [124, 125]. For a pressure difference
𝛥𝑃 and flow 𝑄v across the valve, the governing model is given by
𝛥𝑃 − 𝐵𝑄v
|𝑄v
| − 𝐿
∂𝑄v
∂𝑡
= 0, (6.11)
where 𝐿 is the (Bernoulli) inertia and 𝐵 the (Bernoulli) resistance, both given respectively by
𝐵 =
𝜌
2𝐴eff
2
(6.12)
and
𝐿 =
𝜌ℓeff
𝐴eff
. (6.13)

144
Here, ℓeff and 𝐴eff are effective valve length and orifice area, where the latter is determined by the
expression given by
𝐴eff(𝑡) = (𝐴eff, max − 𝐴eff,min)𝜉(𝑡) + 𝐴eff,min (6.14)
for valve state 𝜉(𝑡), minimum orifice area 𝐴eff,min, and maximum orifice area 𝐴eff, max. The valve
state varies from 0 (closed) to 1 (open), i.e. 𝜉(𝑡) ∈ [0,1], and is updated during valve opening by
the ODE given by
𝑑𝜉
𝑑𝑡 = 𝐾o
(1 − 𝜉)𝛥𝑃, (6.15)
and during closure by
𝑑𝜉
𝑑𝑡 = 𝐾c
(𝜉)𝛥𝑃, (6.16)
where 𝐾o
,𝐾v are coefficients of the rate of opening and closing, respectively. In this work, four
valves are modeled: the aortic valve (AV), the pulmonary valve (PV), the mitral valve (MV), and
the tricuspid valve (TV).
Vascular bed models
All lumped-parameter vascular bed models connecting arterial to venous sides of the circulation
are adopted from those proposed by Mynard and Smolich [125]. These beds are connected to all
terminal arteries and veins, employing circuit-like relations incorporating resistive and compliant
elements (similarly to standard three-element Windkessels used for open-loop models [173]). For
effective arterial and venous compliances 𝐶art and 𝐶ven, generic vascular beds can be modeled by
the ordinary differential equation given by

145
𝑄art =
𝑃art − 𝑃ven
𝑅
+ 𝐶art
𝑑𝑃art
𝑑𝑡 = ∑𝑄1𝐷,𝑖
𝑛art
𝑖=1
(6.17)
on the arterial side, and
𝑄ven =
𝑃art − 𝑃ven
𝑅
− 𝐶ven
𝑑𝑃ven
𝑑𝑡 = ∑𝑄1𝐷,𝑖
𝑛ven
𝑖=1
(6.18)
on the venous side. Here, any arbitrary number 𝑛art (resp. 𝑛ven) of arterial (resp. venous) segments
can be connected to each vascular bed. The resistances 𝑅 are pressure-dependent variables that are
given by
𝑅 = {
𝑅0 (
𝑃tm0 − 𝑃𝑧𝑓
𝑃tm − 𝑃𝑧𝑓 ) , 𝑃tm > 𝑃zf
∞, 𝑃tm ≤ 𝑃zf
(6.19)
for a reference resistance 𝑅0 at a reference transmural pressure 𝑃tm0 and a zero-flow pressure 𝑃zf
(taken to be 5 mmHg following others [125, 171]). The hepatic vascular bed is similarly modeled
but with an additional resistance term to match the incoming penetrating hepatic artery and portal
vein, as described in [125]. Coronary vascular beds use volume-dependent resistances and external
pressures from the heart ventricles, i.e.,
𝑅𝑖
(𝑡) = 𝑅0,𝑖
𝑉0,𝑖
2
𝑉𝑖
2
, 𝑅𝑚(𝑡) = 𝑅0,𝑚 (0.75
𝑉0,1
2
𝑉1
2 + 0.25
𝑉0,2
2
𝑉2
2
) , (6.20)
where
𝑉𝑖
(𝑡) = 𝑉0,𝑖 + ∫ 𝐶𝑖
𝑡
0
𝑑𝑃tm,i
𝑑𝜏 𝑑𝜏. (6.21)

146
Here, 𝑉 is the volume in the compliant compartment, 𝑉0,𝑖
is a reference volume and 𝐶𝑖
is an
intramyocardial compliance. Further details on such formulations are provided in [125, 171].
6.2.3 Numerical methodology and computational approach
The Fourier continuation (FC) method [64, 139, 171] is employed to solve the governing PDEs of
Equation 6.1. Such a method enables Fourier series representations of non-periodic functions,
ultimately facilitating the construction of fast, high-order solvers with limited numerical dispersion
(or “pollution errors"). Indeed, for the numerical treatment of the spatial variables and derivatives
of Equation 6.1, one can consider discrete point values of velocity (for example) 𝑈(𝑥𝑖
) of a given
smooth function 𝑈(𝑥):[0,1] → ℝ (defined on the unit interval without loss of generality) through
a uniform discretization of size 𝑁, i.e., 𝑥 = 𝑖𝛥𝑥, 𝑖 = 0, … , 𝑁 − 1, 𝛥𝑥 = 1/(𝑁 − 1) . The FC
method constructs a fast-convergent Fourier series (or trigonometric interpolant) 𝑈cont:[0, 𝑏] → ℝ
(where 𝑏 is only slightly larger than 1) that is given by
𝑈cont = ∑ 𝑎𝑘
𝑊
𝑘=−𝑊
𝑒
2𝜋𝑖𝑘𝑥
𝑏 s.t. 𝑈cont(𝑥𝑖
) = 𝑈(𝑥𝑖
), 𝑖 = 0, … , 𝑁 − 1, (6.22)
where 𝑊 = (𝑁 + 𝑁cont)/2 is the corresponding bandwidth for an 𝑁cont-sized truncated Fourier
series (here, 𝑁cont is defined as the number of discrete points added in the interval [1, 𝑏], i.e., 𝑏 =
(𝑁 + 𝑁cont)𝛥𝑥). The continued (or “extended") function 𝑈cont renders the original 𝑈(𝑥) periodic,
discretely approximating 𝑈 to very high-order in [0,1] but demonstrating periodicity on the
slightly larger [0, 𝑏] . Directional spatial derivatives on a dimension 𝑥 = 𝑥𝑑 of 𝑈(𝐱,𝑡) =
𝑈(𝑥1, 𝑥2,𝑡) in Equation 6.1 can then be calculated by exact termwise differentiation of the
trigonometric series given by Equation 6.22 as

147
∂𝑈cont
∂𝑥
(𝑥) = ∑ (
2𝜋𝑖𝑘
𝑏
)
𝑊
𝑘=−𝑊
𝑎𝑘𝑒
2𝜋𝑖𝑘𝑠
𝑏 . (6.23)
By restricting the domain of ∂𝑈cont/ ∂𝑠 to the original unit interval, the numerical derivatives of
𝑈 are hence approximated to high-order. The overall errors are found only in the production of
Equation 6.22, from where the calculation of the derivative (Equation 6.23) is easily facilitated by
a Fast Fourier Transform (FFT). This overall idea can also be easily extended to any general
interval [𝑥0, 𝑥𝑁] via affine transformations.
The function 𝑈cont, of size 𝑁 + 𝑁cont, can be constructed discretely through a technique known as
FC(Gram) [64, 139], which relies on a small number of points at the left and right boundaries of
𝑈 and projects them onto an interpolating polynomial basis whose Fourier continuations have been
preconstructed. That is, defining the discretized function as a vector U = (𝑈(𝑥0
), . . . ,𝑈(𝑥𝑛
))
𝑇
,
together with its first 𝑑ℓ and final 𝑑𝑟 points defined as
𝐔ℓ = (𝑈(𝑥0
),𝑈(𝑥1
), . . . ,𝑈(𝑥𝑑ℓ−1))
𝑇
, 𝐔𝑟 = (𝑈(𝑥𝑁−𝑑𝑟
),𝑈(𝑥𝑁−𝑑𝑟+1), . . . , 𝑈(𝑥𝑁−1
))
𝑇
, (6.24)
then the discrete form of 𝑈cont can be found by the overall FC operation summarized by the
expression
𝐔cont = [
𝐔
𝐴ℓ𝑄ℓ
𝑇𝐔ℓ + 𝐴𝑟𝑄𝑟
𝑇𝐔𝑟
] = [
𝐔
𝑃ℓ𝐔ℓ + 𝑃𝑟𝐔𝑟
] (6.25)
where 𝐔 = (𝑈(𝑥0
), … ,𝑈(𝑥𝑁−1
))
𝑇
contains the discrete point values of 𝑈; 𝐔cont is a vector of the
𝑁 + 𝑁cont discretely-periodic function values; 𝐼 is the identity matrix; and 𝑄ℓ
, 𝑄𝑟
(of size 𝑑ℓ × 𝑑ℓ
,
𝑑𝑟 × 𝑑𝑟
) are projections onto precomputed Fourier continuation blendings contained in 𝐴ℓ
, 𝐴𝑟
,

148
respectively. Such projections are produced from a 𝑄𝑅 decomposition of a Vandermonde matrix.
Further technical details on particular FC(Gram) algorithm have been provided previously [64,
139]. It should be noted that the FC procedure relies on polynomial interpolation (not discussed
here); hence choices of 𝑑ℓ = 5, 𝑑𝑟 = 5 will lead to fifth-order convergence in space [64, 139].
In summary, the FC(Gram) algorithm appends 𝑁cont values to a given discretized function in order
to form a periodic extension in [1, 𝑏] that transitions smoothly from 𝑈(𝑥𝑁−1
) back to 𝑈(𝑥0
). The
resulting continued vector 𝐔cont can be interpreted as a set of discrete values of a smooth and
periodic function that can be approximated to high-order via FFT on an interval of size
(𝑁 + 𝑁cont)𝛥𝑥 (and hence accurate termwise derivative representations for Equation 6.1). As has
been suggested previously [64, 139], a small number of 𝑑ℓ
, 𝑑𝑟 = 5 matching points with a periodic
extension comprised of 𝑁cont = 25 points are often used. The matrices 𝐴ℓ
, 𝐴𝑟
,𝑄ℓ
, and 𝑄𝑟 are
computed only once (before numerically solving the governing system) and stored in file for use
each time the FC procedure is invoked.
Implementation
To complete the solver, any suitable 𝑠-step explicit time integration scheme may be employed to
resolve the corresponding ODEs in time, e.g., for ∂𝑈/ ∂𝑡, via the expression given by
𝑈𝑖
(𝑡
𝑛+1
) = 𝑈𝑖
(𝑡
𝑛) + 𝛥𝑡∑𝛼𝑗
𝑠
𝑗=0
𝐹𝑖 (𝑡
𝑛−𝑗
,𝑈𝑖(𝑡
𝑛−𝑗
)) , (6.26)
for a right-hand-side term 𝐹𝑖
(derived from a first-order temporal representation of Equation 6.1)
and for corresponding integration weights 𝛼𝑗
. In order to be consistent with the high-order spatial
scheme described above, a fourth-order Adams-Bashforth method is employed (𝑠 = 3 above),

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similarly to other successful FC-based solvers [64, 139]. The ODEs for each of the lumped
parameter models (heart, valves, vasculature) similarly employ a fourth-order Adams-Bashforth
method (in contrast to the low order extrapolations/integrations commonly used [125, 173]).
All codes are implemented in MATLAB together with the use of C++ MEX files (obtained via the
MATLAB Coder). The solver has been extensively validated numerically and
physiologically [171] [172], including with all cases proposed in the open-loop community
benchmark paper of Boileau, et al. [173].
6.2.4 System parameters/coefficients
Almost all parameters we employ are adopted from [125], which are validated against
physiological data. For Equation 6.1, Fluid (blood) viscosity is taken to be 𝜇 = 0.0035 P⋅s, density
is taken to be 𝜌 = 1060 kg/m 3
,, the viscous friction coefficient is taken to be 𝜁 = −22
(corresponding to a relatively flat velocity profile with a small boundary layer), and the external
pressure is taken to be 𝑃ext = 0 (due to lack of data). For all segments (394 in total), initial
reference pulse wave speeds (corresponding to stiffness characteristics) are calculated from initial
diameters 𝐷0 by the expression
𝑐0
2 =
2
3𝜌
(𝑘1exp(𝑘2𝐷0/2) + 𝑘3
) (6.27)
for constants 𝑘1, 𝑘2, 𝑘3 that vary between systemic arteries, systemic veins, hepatic portal veins,
pulmonary arteries, and pulmonary veins. All the above vessel values and constants that are
invoked in this work are identical to and can be found in [125].

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The baseline heart parameters employed in this work are given in Table 6.1. For the compliance
studies presented later, max𝐸fw is accordingly adjusted, and a baseline heart rate (HR) of 75 bpm
is considered (corresponding to a cardiac output of CO = 5.9 L/min). The baseline parameters for
the juglar veins and vena cava (the segements of interest for the hypothesis of this work) are
reported in Table 6.2. For the stiffness studies presented in the next section, initial 𝑐0 is adjusted
accordingly. Finally, tricuspid valve (TV) parameters are presented in Table 6.3 for the normal
case, for severe stenosis, and for regurgitation, all of which are studied in the next section. All
other (unreported) parameters are adopted exactly from [125].
Table 6-1 Heart model baseline parameters for the left atrium (LA), left ventricle (LV), right atrium (RA), and right
ventricle (RV), where Tper = HR/60 is the length of a cardiac cycle. Together with the baseline heart rate of HR = 75 bpm,
the corresponding nominal cardiac output is CO = 5.9 L/min.
Heart model baseline parameters
LA LV RA RV
𝑚𝑎𝑥 𝐸fw (mmHg/mL) 0.150 3.200 0.150 0.450
𝑚𝑖𝑛 𝐸fw (mmHg/mL) 0.090 0.060 0.045 0.035
𝑉𝑝=0
(mL) 3 10 7 40
𝑉𝑡=0
(mL) 71 136 67 172
𝐾 (10−3
s/mL) 0.25 0.50 0.50 1.00
𝜏1
(–) 0.07500𝑇per 0.26875𝑇per 0.07500𝑇per 0.26875𝑇per
𝜏2
(–) 0.3125𝑇per 0.4525𝑇per 0.3125𝑇per 0.4525𝑇per
𝑚1
(–) 11.2 21.9 11.2 21.9
𝑚2
(–) 1.99 1.32 1.99 1.32
𝜅 (–) 2 6 2 6
𝑡0
(s) 0 0 0.8125𝑇per 0.8125𝑇per
Table 6-2 Baseline parameters for the venous segments of interest, adopted from [125]. Initial area A0 and initial pulse
wave velocity c0 are the same throughout the vessel (no tapering in each subsegment). Changes in stiffness for the studies
of this chapter are affected by modifying the pulse wave speed c0. All other segments of the entire circulation (394 in total)
are identical to those used in [125, 171].
Baseline parameters for the venous segments of interest
Length ℓ (cm) 𝐴0 (cm2
) 𝑐0 (m/s)
Left Internal Jugular Vein I 17.8 0.255 3.292

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Baseline parameters for the venous segments of interest
Left Internal Jugular Vein II 5.2 0.212 3.471
Right Internal Jugular Vein I 17.8 0.950 2.043
Right Internal Jugular Vein II 5.2 0.882 2.104
Superior Vena Cava I 2.0 1.767 1.627
Superior Vena Cava II 4.0 2.545 1.477
Table 6-3 Normal and diseased parameters for the tricuspid valve employed in the simulations of this chapter (Ko and Kc
are in units of cm2
·s2
/g).
Tricuspid valve parameters
𝐾o 𝐾c 𝑀st 𝑀rg
Normal 0.03 0.04 1.00 0.00
Severe stenosis 0.03 0.04 0.15 0.00
Regurgitation 0.03 0.04 1.00 0.5
6.3 Results and discussion
We investigated the internal jugular vein (IJV) wave dynamics under various conditions to
understand their effects on hemodynamics. The conditions tested included different heart rates,
varying levels of IJV stiffness indicated by pulse wave velocity (PWV), and varying contractility
of both the right heart (RV/RA) and left ventricle (LV), encompassing low, normal, and high
contractility scenarios. Additionally, we examined the effects of and pathological conditions such
as tricuspid valve regurgitation and stenosis.
Our analysis included measuring the right atrial pressure (RAP) and jugular venous pressure (JVP)
waveforms, as well as determining the mean pressure in the brain. This comprehensive assessment
allowed us to capture the intricate interactions and resulting hemodynamic changes in response to
different physiological and pathological states.

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Through these investigations, we aim to provide a detailed understanding of how various factors
influence IJV wave dynamics, offering insights that could inform clinical approaches to managing
cardiovascular conditions.
6.3.1 Baseline characteristics
Figure 6.2 presents pressure waveforms of the right atrium and the right ventricle at baseline for a
normal valve at heart rate HR = 75 bpm. The resulting cardiac output (from the left ventricle) at
such parameters is CO = 5.95 L/min, and the resulting average pressure in the brain (i.e., the
venous side of the cerebral vascular bed) is 𝑃brain = 9.45 mmHg. Figure 6.3 presents the
corresponding baseline pressure and flow waveforms of the right internal jugular vein.
Figure 6.2 Two consecutive cycles of pressure waveforms at baseline for the right atrium (RA, solid blue line) and the
right ventricle (RV, dashed red line).

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Figure 6.3 Baseline pressure waveforms (left) and flow waveforms (right) of the right internal jugular vein I.
6.3.2 Isolated effects of vessel stiffness
Figure 6.4 presents pressure and flow waveforms at a heart rate HR = 75 bpm in the right internal
jugular vein I for baseline stiffnesses and three times the baseline stiffnesses of all the segments in
Table 6.2. The resulting shapes, maximums, and minimums are noticeably altered between the
baseline case and the higher pulse wave speed (stiffness) case. Both these stiffness configurations
are considered in all subsequent results in this work. The mean brain pressure remains relatively
the same in both cases (9.46 mmHg at baseline, 9.43 mmHg at three times baseline stiffness).

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Figure 6.4 Pressure waveforms (left) and flow waveforms (right) of the right internal jugular vein I at baseline stiffnesses
(solid blue line) and at three times the baseline stiffnesses (dashed red line) of the segments in Table 6.2.
6.3.3 Effects of heart rate
In addition to the normal (baseline) heart rate of HR = 75 bpm, this section considers a low heart
rate (HR = 40 bpm) and high heart rate (HR = 160 bpm), at both baseline and high stiffnesses
(as defined in the previous subsection). Figure 6.4 presents the corresponding pressure waveforms
of the right atrium and the right ventricle at these heart rates. As expected, there are noticeable
differences with the baseline case of Figure 6.2. The associated cardiac outputs for the low and
high heart rates are determined to be 4.70 L/min and 6.42 L/min, respectively.
Figure 6.6 (resp. Figure 6.7) presents pressure and flow waveforms of the right internal jugular
vein I at HR = 40 bpm (resp. HR = 160 bpm) where, in all cases, the waveform shapes, minima,
and maxima are noticeably different than the normal heart rate cases of Figure 6.3 and Figure 6.4.
For the low heart rate, mean brain pressure is found to be 8.70 mmHg in both the baseline and
high stiffness cases. For the high heart rate, both stiffness configurations result in a mean brain
pressure of 9.71 mmHg (cf. 9.46 mmHg for the normal heart rate).

155
Figure 6.5 Two consecutive cycles of pressure waveforms of the right atrium (RA) and the right ventricle (RV) at heart
rates of HR = 𝟒𝟎 bpm (left) and HR = 𝟏𝟔𝟎 bpm (right).
Figure 6.6 Pressure waveforms (left) and flow waveforms (right) at HR = 𝟒𝟎 bpm of the right internal jugular vein I at
baseline stiffness (solid blue line) and at three times the baseline stiffness (dashed red line) of the segments in Table 6.2.

156
Figure 6.7 Pressure waveforms (left) and flow waveforms (right) at HR = 𝟏𝟔𝟎 bpm of the right internal jugular vein I
at baseline stiffness (solid blue line) and at three times the baseline stiffness (dashed red line) of the segments in Table
6.2.
6.3.4 Effects of heart contractility
In order to study the effects of heart contractility, the values of max𝐸fw for the left ventricle, right
atrium, and right ventricle are simultaneously decreased by one-half (low contractility) or
increased by 2.5 (high contractility). Heart rate is fixed to the normal baseline of HR = 75 bpm.
Figure 6.8 presents the corresponding right atrial pressure for these cases and the baseline, where
noticeable changes in shape and amplitude can be observed. The corresponding cardiac outputs
for the low and high contractility are found to be 4.44 L/min and 6.93 L/min, respectively (cf.
5.95 L/min at baseline). Figures 6.9 and 6.10 additionally present the associated pressure and flow
waveforms in the right internal jugular vein I for the baseline stiffness and high stiffness cases,
respectively. Mean brain pressure at both baseline and high stiffness is determined to be 8.57
L/min for the low contractility case and 9.71 L/min in the high contractility case.

157
Figure 6.8 Two consecutive cycles of pressure waveforms of the right atrium (RA) at a heart rate of HR = 𝟕𝟓 bpm at
baseline heart contractility (solid blue line), at low contractility (𝟏/𝟐 of baseline contractility, red dashed line), and at high
contractility (𝟐. 𝟓 times baseline, dash-dotted yellow line).
Figure 6.9 Pressure waveforms (left) and flow waveforms (right) at HR = 𝟕𝟓 bpm and at baseline stiffness (𝒄𝟎) for the
right internal jugular vein I at baseline heart contractility (solid blue line), at low contractility (𝟏/𝟐 of baseline
contractility, red dashed line), and at high contractility (𝟐. 𝟓 times baseline, dash-dotted yellow line).

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Figure 6.10 Pressure waveforms (left) and flow waveforms (right) at HR = 𝟕𝟓 bpm and at high stiffness (𝟑𝒄𝟎) for the
right internal jugular vein I at baseline heart contractility (solid blue line), at low contractility (𝟏/𝟐 of baseline
contractility, red dashed line), and at high contractility (𝟐. 𝟓 times baseline, dash-dotted yellow line).
6.3.5 Effects of tricuspid valve conditions
The effects of various tricuspid valve (TV) diseases/conditions can be studied by considering the
parameters provided in Table 6.3 for a normal TV, for stenosis, and for regurgitation. At a normal
heart rate of HR = 75 bpm, Figure 6.11 presents the right atrial pressure for each of these three
cases, where a distinct increase in pressure can be observed in the diseased conditions. The
corresponding cardiac outputs are determined to be 3.49 L/min and 3.52 L/min for stenosis and
regurgitation, respectively (cf. 5.95 L/min for baseline). Figures 6.11 and 6.12 present the pressure
and flow waveforms in the right internal jugular vein I at baseline stiffness (𝑐0) and high stiffness
(3𝑐0), respectively (the corresponding mean brain pressures are found to be 9.40 mmHg and 8.90
mmHg). One observes an expected increase in pressure (from the valve not opening or closing
adequately) and a large decrease in flow. Significant alterations in shapes can also be seen,
implying that the jugular vein can be used as a proxy to potentially observe such conditions in a
realistic setting.

159
Figure 6.11 Two consecutive cycles of pressure waveforms of the right atrium (RA) at a heart rate of HR = 𝟕𝟓 bpm for a
normal tricuspid valve (solid blue line), one undergoing stenosis (dashed red line), and one inducing flow regurgitation
(dash-dotted yellow line).
Figure 6.12 Pressure waveforms (left) and flow waveforms (right) at HR = 𝟕𝟓 bpm and at baseline stiffness (𝒄𝟎) of the
right internal jugular vein I for a normal tricuspid valve (solid blue line), one undergoing stenosis (dashed red line), and
one inducing flow regurgitation (dash-dotted yellow line).

160
Figure 6.13 Pressure waveforms (left) and flow waveforms (right) at HR = 𝟕𝟓 bpm and at high stiffness (𝟑𝒄𝟎) of the
right internal jugular vein I for a normal tricuspid valve (solid blue line), one undergoing stenosis (dashed red line), and
one inducing flow regurgitation (dash-dotted yellow line).
6.3.6 Effects of Fontan circulation
Furthermore, we examined a specific case of Fontan Total Cavopulmonary Connection (TCPC)
circulation. The Fontan procedure is the final stage of surgical treatment for patients with a
univentricular heart defect [174]. This operation establishes a total cavopulmonary connection
(TCPC), where all systemic venous blood bypasses the heart and flows directly into the lungs
[174]. This creates a non-physiologic circulation marked by low-shear, non-pulsatile pulmonary
blood flow, central venous hypertension, and reduced cardiac output [175]. In this scenario, the
systemic venous return bypasses the right heart entirely and is directly connected to the pulmonary
arteries. This unique condition allowed us to study the hemodynamic consequences of altered
circulatory pathways, particularly focusing on the impact of the bypass on IJV dynamics and
overall venous return.

161
The effects of Fontan circulation (i.e., no right heart) can be studied by bypassing the right atrium
and right ventricle and forcing the superior vena cava, the coronary sinus, and the inferior vena
cava merge directly into the pulmonary artery. At a normal heart rate of HR = 75 bpm,
Figure 6.14 and 6.15 presents the pressure and flow waves for the superior vena cava and the right
internal jugular vein I, respectively. The corresponding cardiac output is 1.71 L/min. As expected,
pressure significantly increases in both segments as a result of not having the right heart (no
pumping from the right). Indeed, this lack of right heart pumping leads to relatively flat pressure
and flow profiles in both cases.
Figure 6.14 Pressure waveforms (left) and flow waveforms (right) at HR = 𝟕𝟓 bpm at baseline with the complete heart
(solid blue line) and for Fontan circulation (dashed red) of the superior vena cava.

162
Figure 6.15 Pressure waveforms (left) and flow waveforms (right) at HR = 𝟕𝟓 bpm at baseline with the complete heart
(solid blue line) and for Fontan circulation (dashed red) of the right internal jugular vein I.
In our study, we began with a baseline analysis to ensure our results accurately represented
physiological conditions. As shown in Figures 6.2 and 6.3, we successfully captured physiological
right ventricular (RV) and right atrial (RA) pressures, particularly the characteristic a, c, and v
peaks in right atrial pressure (RAP). Our measurements of jugular venous pressure (JVP) and
jugular venous flow rate (JVQ) also exhibited physiological features.
Next, we investigated the effect of internal jugular vein (IJV) stiffness, represented by pulse wave
velocity (PWV), on IJV waveform with same heart function. Our results indicated minimal
differences between low and high stiffness cases. We concluded that in our 1D simulation, JVP is
primarily influenced by the heart model, with IJV stiffness affecting only the pause pressure, while
the mean brain pressure remained consistent. Thus, JVP proves to be a stable and robust indicator
for assessing RAP and right heart function.
We then examined the impact of heart rate on hemodynamics and wave dynamics of RA and IJV.
With preserved contractility, changes in heart rate significantly affected cardiac output (CO) and

163
RAP. A low heart rate resulted in a flatter a-c wave in RAP, while a high heart rate produced a
very low v wave. Also the IJV pressure wave difference between low stiffness (PWV) and high
stiffness (PWV) of IJV cases become more significant when heart rate is higher, although with
same pattern. Interestingly, these variations were mirrored in JVP, which showed a flat peak in the
a-c wave at low heart rates and no distinct v peak at high heart rates, corresponding with our
experimental measurements and confirming the physiological accuracy of our 1D framework. We
also studied the effect of heart contractility by altering the E max parameter. Changes in E max
resulted in varying mean pressures and pulse pressures (PP) in RAP, with low E producing high
mean, low amplitude PP, and high E yielding low mean, high amplitude PP (since the contractility
of the LV and RV is also changing in the same way with RA). This trend was consistently reflected
in JVP across all contractility conditions (low, normal, and high), demonstrating that JVP can
reliably capture RAP features and trends.
To explore more complex conditions, we simulated tricuspid valve disease, including stenosis and
regurgitation. Our results accurately represented the physiological RAP features for both scenarios.
In stenosis, RAP increased significantly before tricuspid valve closure, indicating resistance from
a dysfunctional valve. In regurgitation, RAP increased markedly after valve closure, signifying
improper valve closure and flow leakage from RA to RV (result in increased preload pressure).
Correspondingly, JVP precisely reflected these complex right heart pressure waveforms,
particularly demonstrating the elevated JVP after the a-c wave in regurgitation, which signifies
valve leakage. In this disease scenario, our findings suggest that JVP can effectively evaluate heart
valve disease, offering an alternative to directly assessing RAP. The ability of JVP to reflect
complex variations indicates its potential to represent the status of the right atrium in both healthy
and diseased conditions.

164
Another diseased condition we investigate is Fontan circulation, under which the venous
circulation completely changes, with the SVC bypassing the right heart and connecting directly to
the lung. As part of this most complicated case, we compared the SVC pressure and IJV pressure
waveforms under healthy and Fontan conditions (firstly, with preserved LV contractility). Due to
the absence of the right heart, our results clearly show a flatter pressure and flow waveform in both
SVC and IJV. As expected, the mean pressures in SVC and IJV increased in Fontan's condition
(with a very low pause), despite low CO, which can lead to over-pressurization of the cerebral
veins. To simulate a condition where the left heart works harder, we simulated a Fontan case with
high LV contractility. Overall, all the results agree with Fontan's physiological findings. Hence,
IJV can still reflect heart function in a disease like Fontan circulation, which shows that our model
is capable of reflecting even the most complex disease conditions.
Our study tested common heart function variations (heart rate and contractility) and more
uncommon conditions (valve disease). Across all scenarios, JVP consistently reflected RAP
waveforms, reinforcing its potential as a window into right heart function. We systematically
examined the relationship between JVP and RAP under various heart conditions, confirming JVP's
ability to assess right heart status across a wide range of conditions. This finding holds significant
clinical implications for noninvasively diagnosing right heart diseases.
6.4 Conclusion
Our study demonstrates that the function of the right heart, as represented by right atrial pressure
(RAP), has a significant impact on jugular venous pressure (JVP) waveform. This correlation
indicates that JVP can effectively reflect changes in the right atrial function, suggesting that the
internal jugular vein (IJV) can serve as a window into the right heart's performance. Moreover, we

165
found that the stiffness of the IJV, measured by pulse wave velocity (PWV), also affects JVP. The
stiffness can alter the right-heart pressure waveform, especially in terms of pause pressure
differences. This finding highlights the importance of considering venous stiffness in the
assessment of right-heart hemodynamics.
In conclusion, our study suggests that JVP measurements could be a valuable noninvasive tool for
examining or assessing RAP. By monitoring JVP, clinicians can gain crucial insights into the
function of the right heart. This method has the potential to enhance the diagnosis and management
of right heart diseases, providing a safer and more accessible alternative to traditional invasive
techniques. By further refining this approach, JVP measurement could become an integral part of
clinical practice for evaluating right-heart function and diagnosing associated cardiovascular
conditions.

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Chapter 7: The significance of steady blood flow shear-ratedependency in modeling of Fontan hemodynamics
This chapter is based on the following published manuscript: Wei, Heng, Andrew L. Cheng, and
Niema M. Pahlevan. "On the significance of blood flow shear-rate-dependency in modeling of
Fontan hemodynamics." European Journal of Mechanics-B/Fluids 84 (2020): 1-14.
This chapter delves into the pathological conditions of the central venous system, specifically
examining scenarios where blood flow is steady and characterized by low shear rates. In these
diseased states, such as those seen in conditions like Fontan circulation, the dynamics of blood
flow differ significantly from the pulsatile flow typically observed in a healthy venous system.
Under these circumstances, the non-Newtonian properties of blood, which include its variable
viscosity and shear-thinning behavior, become more pronounced and have a greater impact on
hemodynamics.
In healthy individuals, blood flow in the central venous system is generally pulsatile, influenced
by the rhythmic contractions of the heart. This pulsatile nature means that the shear rates within
the vessels vary significantly, causing the blood to exhibit Newtonian fluid characteristics where
its viscosity remains relatively constant. However, in diseased states where the flow becomes more
steady and less pulsatile, the shear rates decrease, and the blood’s non-Newtonian properties
become more evident. These properties include the blood’s tendency to exhibit shear thinning, a
behavior critical to understanding the flow dynamics under pathological conditions. For instance,
in patients with Fontan circulation, a surgical condition often performed to treat complex
congenital heart defects, the flow is typically steady and exhibits low shear rates. In such cases,
the non-Newtonian nature of blood significantly affects the viscous stress exerted on the vessel
walls. This chapter emphasizes the critical differences in flow dynamics between healthy and

167
diseased states of the central venous system. By focusing on the significant role of non-Newtonian
effects in steady, low shear rate conditions, it provides a comprehensive understanding of how
these factors influence hemodynamics and the potential clinical implications for managing and
treating venous system diseases.
7.1 Introduction of non-Newtonian effect in human venous system
For patients born with a univentricular heart defect, the Fontan procedure is the final stage of
palliative surgery. The surgery separates the pulmonary and systemic circulations by completing
a non-physiologic total cavopulmonary connection (TCPC), where blood from the central systemic
veins (superior vena cava (SVC) and inferior vena cava (IVC)) bypasses the heart and drains
directly into the pulmonary arteries [176]. This procedure has markedly improved early survival
and resulted in beneficial short-term outcomes for these patients. However, a univentricular heart
is still less efficiently than a normal biventricular heart.
Thus, after the Fontan procedure, these patients are still at risk for early mortality and are prone to
developing severe long-term complications [177-180]. Even in the modern surgical era, patients
who survive the Fontan surgery still have a nearly 20% mortality rate at age 20 years, and half of
Fontan patients develop severe comorbidities [181-183].
The Fontan procedure results in low shear non-pulsatile pulmonary blood flow and central venous
hypertension in all patients [184]. This has been shown to result in endothelial dysfunction by
altering the responsiveness of endothelial nitric oxide synthase [185, 186]. Even though all patients
have the same fundamentally abnormal circulation, the variability in timing and presentation of
Fontan failure is currently unexplained [187]. It has been shown that many patient-specific factors,
such as the size of the blood vessels, the shape and location of surgical anastomoses, and the

168
distribution of pulmonary blood flow all affect the energy efficiency of the circulation [188].
However, the effect of shear rate- dependent blood viscosity in the Fontan circulation has been
ignored in many studies.
Patient-specific computational fluid dynamic (CFD) studies have provided valuable insights into
the pathophysiology of Fontan failure and have helped physicians understand the clinical
importance of hemodynamic metrics such as power loss (PL), pulmonary blood flow distribution,
and wall shear stress (WSS) [189] [190]. Previous works indicate that PL in the Fontan circuit is
a substantial reason for decline in exercise capacity, and that PL increases with cardiac output and
somatic growth [190] [191]. WSS plays a vital role in maintaining normal vascular function and
growth of the cardiovascular system [192]; recent studies show that abnormal WSS is associated
with pulmonary hypoplasia and blood clot formation [193, 194].
As a non-Newtonian fluid with shear-thinning behavior, blood has an apparent viscosity that
increases exponentially at shear rates less than 100 s−1 [195]. In large arteries (e.g., aorta) where
shear rates are typically above this threshold, it is generally accepted that blood flow characteristics
follow Newtonian fluid behavior [196]. Therefore, the assumption of blood as a Newtonian fluid
is very common in both in-vitro experiments and computational hemodynamics studies [197].
However, the non-Newtonian behavior of blood flow has been shown to be substantial in
microvasculature [73] and in hemodynamics conditions where low shear flow is more common
[198-200]. Since the Fontan circuit is a low-shear-rate system by design, non-Newtonian changes
in blood viscosity should be readily apparent in this environment. However, previous
computational and experimental studies of the Fontan circuit generally assumed blood to be a

169
Newtonian fluid (i.e., having a constant viscosity that is independent of shear rate) without
evaluating the error introduced by this assumption [201, 202].
Since shear rates in large veins in the Fontan circulation are typically around 10 s−1 [203], nonNewtonian behavior should be critically important in this circulation. Prior investigations of nonNewtonian behavior in blood vessels have suggested that treating blood as Newtonian fluid may
lead to incorrect conclusions about clinically important hemodynamic metrics such as PL,
pulmonary blood flow distribution, and WSS distribution [198-200, 204, 205]. Clinically, nonNewtonian behavior has also been shown to affect pulmonary blood flow in patients with Glenn
and Fontan circulations [206]. In a recent experimental study, Cheng et al. [181] used 4D flow
magnetic resonance imaging (MRI) to demonstrate how realistic non-Newtonian viscosity affects
flow in 3D patient-specific experimental models of the Fontan circulation. They showed that a
non-Newtonian blood-mimicking fluid creates different flow profiles, PL, and shear stress that
also varied with model geometry and hemodynamic conditions [181].
To the best of our knowledge there is no study that has used CFD to systemically investigate the
impact of shear-rate-dependent viscosity (non-Newtonian effect) in the Fontan-circulation. The
main goal of this study is to evaluate the non-Newtonian viscosity effect on Fontan hemodynamics
under different cardiac outputs and various caval flow ratios using CFD. We employed a
computational approach using a Carreau–Yasuda model combined with Lattice Boltzmann method
(LBM) to accurately capture the non-Newtonian flow characteristics [207]. In our study, we
systematically compared the hemodynamics of flow resulting from a Newtonian viscosity
assumption with those resulting from a non-Newtonian viscosity assumption in a simplified Fontan
circulation with different cardiac outputs and different SVC/IVC flow distributions. Our

170
systematic approach quantifies the discrepancies stemming from the results of the non-Newtonian
model and Newtonian model by comparing the clinically relevant Fontan hemodynamic metrics
— PL, viscous dissipation, WSS, and pulmonary flow distribution.
7.2. Method
7.2.1 Flow solver
Lattice Boltzmann Method (LBM):
LBM uses simplified kinetic equations combined with a modified molecular-dynamics approach
to simulate fluid flows as an alternative method for conventional computational fluid dynamics
(CFD) methods that use Navier-Stokes equations. The accuracy and usefulness of LBM were
demonstrated in various fluid dynamics problems including turbulence [208], multiphase flow
[76], and hemodynamics [72]. As highlighted in previous studies, LBM methods have been shown
to be particularly suitable for hemodynamics simulations [72, 76, 208, 209].
In LBM method, the synchronous motions of the particles on a regular lattice are enforced through
a particle distribution function. This distribution function enforces mass and momentum
conservation. It also ensures that the fluid is Galilean invariant and isotropic [210]. In this study,
a single-relaxation-time (SRT) incompressible lattice Boltzmann method was used to solve the
incompressible fluid flow [211]. The evolution of the distribution functions on the lattice was
governed by the discrete Boltzmann equation with the BGK (Bhatnagar-Gross-Krook) collision
model:
𝑓𝑖
(𝒙 + 𝒆𝑖∆𝑡,𝑡 + ∆𝑡) − 𝑓𝑖
(𝒙,𝑡) = −
1
𝜏
[𝑓𝑖
(𝒙,𝑡) − 𝑓𝑖
𝑒𝑞(𝒙,𝑡)], (7,1)

171
where 𝑓𝑖(𝒙,𝑡) is the distribution function of the particles in phase space, 𝒆𝒊
is the discrete velocity
at position 𝒙 and time 𝑡, 𝜏 is a non-dimensional relaxation time, and 𝑓𝑖
𝑒𝑞 is the equilibrium
distribution function. Here, i=0, 1, …, 18 since a D3Q19 (19 discrete velocity vectors) stencil was
applied.
The non-dimensional relaxation time, 𝜏, is related to fluid viscosity 𝜇 as
𝜇 = 𝜌𝜗 = 𝜌𝑐𝑠
2 (𝜏 −
1
2
) ∆𝑡, (7,2)
where 𝜈 is the kinematic viscosity, 𝜌 is the incompressible fluid density (e.g. blood density), and
𝑐𝑠 =
∆𝑥
∆𝑡√3
is the lattice sound speed. Δ𝑡 and Δ𝑥 are discrete time step and lattice space,
respectively. In this study, we used uniform discretization and set ∆𝑥
∆𝑡
= 1.
The equilibrium distribution function, 𝑓𝑖
𝑒𝑞, for incompressible Lattice Boltzmann model is defined
as [211]
𝑓𝑖
𝑒𝑞 = 𝜔𝑖
𝑝
𝑐𝑠
2 + 𝜔𝑖𝜌 [
𝑒𝑖
∙ 𝑣
𝑐𝑠
2 +
(𝑒𝑖
∙ 𝑣)
2
2𝑐𝑠
4 −
𝑣
2
2𝑐𝑠
2
], (7,3)
where 𝜔𝑖
is the weighting factor, 𝜌𝑜 is related to the pressure by 𝜌o =
𝑝
cs
2
and velocity 𝒗 can be
calculated by
𝜌o = ∑ 𝑓𝑖
𝑖
, (7,4)
𝜌𝒗 = ∑ 𝒆𝑖𝑓𝑖
𝑖
, (7,5)

172
7.2.2 Fluid viscosity models: non-Newtonian vs. Newtonian
Fluid viscosity models: non-Newtonian vs. Newtonian in our non-Newtonian simulations, blood
was modeled as a non-Newtonian fluid using the Carreau–Yasuda model.
In our non-Newtonian simulations, blood was modeled as a non-Newtonian fluid using the
Carreau–Yasuda model. This model has been widely used for the shear-thinning behavior of blood
in hemodynamical simulations [212, 213]. In this study, the model was curve-fitted with Fontan
patient specific data from a prior study by Cheng et al. [213].
Figure 7.1. Viscosity of the shear-thinning Carreau-Yasuda model matching with Fontan patient-specific data. Figure from
Wei H et al. European Journal of Mechanics-B/Fluids 84 (2020): 1-14.
The apparent viscosity of the Carreau–Yasuda model is given by
μ(γ̇) = μ∞ + (μ𝑜 − μ∞)[1 + (λγ̇)
𝑎]
𝑛−1
𝑎 , (7.6)
where 𝛾̇ is the shear rate, 𝜇𝑜 is the zero-shear viscosity, and 𝜇∞ is the Newtonian viscosity (when
the shear rate goes to infinity).

173
In this model, 𝜆 (time constant), a, and n (power-law index) are empirically determined constant
parameters. The main advantage of the Carreau–Yasuda model is that it is continuous for all 𝛾̇ ≥
0.
The shear rate (𝛾̇) in the non-Newtonian models is computed from the second invariant of the rateof-strain tensor (𝐷𝐼𝐼):
γ̇ = 2√𝐷𝐼𝐼. (7.7)
where the DII is defined as
𝐷𝐼𝐼 = ∑ 𝑆αβ
𝑙
α,β=1
𝑆αβ, (7.8)
where l = 3 for a three-dimensional model.
In the LBM, the strain tensor 𝑆𝛼𝛽 can be calculated locally at each node using below equation (Eq.
(7.9)) [212]:
𝑆αβ = −
3
2τ∑(𝑓𝑖 − 𝑓𝑖
𝑒𝑞)𝑒𝑖α
𝑖
𝑒𝑖β. (7.9)
Thus, the shear strain can be calculated directly and locally without calculating the derivatives of
the velocity; this is particularly advantageous for parallel simulation [214].
The stress tensor for an incompressible fluid can be then computed as
𝜎𝛼𝛽 = −𝑝𝛿𝛼𝛽 + 2𝜇𝑆𝛼𝛽, (7.10)
where 𝑝 is the pressure, 𝛿𝛼𝛽 is the Kronecker delta.

174
In our simulations 𝜇𝑜 = 56cP (1cP = 0.001Pa s), 𝜇∞ = 4.5cP, 𝜆 = 3.133s, 𝑎 = 2, and 𝑛 = 0.3568
were obtained from patient specific data as described in the previous work [215].
μ = ρ𝑐𝑠
2 (τ −
1
2
) Δ𝑡, (7.11)
Indicate the relationship between the viscosity and local relaxation time. For our non-Newtonian
cases, the shear-rate-dependent effect of non-Newtonian blood flow was implemented into the
LBM using Eq. (7.11) where the microscopic relaxation time and the macroscopic fluid viscosity
are coupled [212]. For our Newtonian cases, the viscosity was set to be the infinite shear viscosity
𝜇∞ = 4.5cP. This means that the relaxation time of Eq. (7.11) is constant in Newtonian LBM
simulations.
7.2.3. LBM Algorithm
The core non-Newtonian LBM algorithm consists of a cyclic sequence of sub steps, with each
cycle corresponding to one timestep:
1. Compute the macroscopic moments ρ0 and v from fi via Eq. (7.4) and (7.5).
2. If desired, write the macroscopic fields ρ0, v and/or p to the hard disk for visualization or postprocessing.
3. Obtain the equilibrium distribution f eq
i from Eq. (7.3).
4. Compute strain tensor Sαβ and shear rate 𝛾˙ from fi and f eq
i via Eq. (7.9) and Eq. (7.7).
5. Obtain the apparent viscosity µ at each point by Eq. (7.6) and calculate relaxation time τ locally
from Eq. (7.2).

175
6. Perform collision (relaxation) and streaming (propagation) to update fi via Eq. (7.1).
7. Increase the time step, setting t to t + ∆t, and go back to step 1 until the last time step or
convergence has been reached. Further details about the LBM algorithm can be found in previous
publications [209] [211] [216].
Figure 7.2 Schematic of the Fontan circuit. The superior and inferior cava (SVC and IVC) are the inlets with diameter D =
1.2cm. The right and left pulmonary arteries (RPA and LPA) are the outlets with diameter D = 0.9cm. Bold arrows indicate
the direction of fluid flow.
7.2.4 Fontan hydraulic circuit model
A simplified model of the TCPC (Glenn, Fontan conduit, and pulmonary arteries) was used in our
simulation. The dimensions of the model were based on prior similar in vitro studies [53] [52]

176
where the diameters of the superior and inferior cava (SVC and IVC) were set to D = 1.2 cm, and
the diameters of the right and left pulmonary arteries (RPA and LPA) were set to D = 0.9 cm. A
schematic of the simplified Fontan circuit is shown in Figure. 7.2.
7.2.5 Boundary conditions
The TCPC tube walls were considered rigid with a no-slip boundary condition on all solid walls.
For each hemodynamics case, steady flow rate 𝑄 (the extension tube boundary model [217] was
used at the inlet to make the velocity profile get developed by itself) was applied at the inlet (e.g.
SVC and IVC) and constant pressure (p0 = 10 mmhg) was applied at the outlets (e.g. RPA and
LPA).
7.2.6 Numerical simulations
Simulations were performed for twenty hemodynamic cases at five different flow rates (𝑄 = 0.5,
1, 1.5, 2, 2.5 L/min) and four different flow distributions of SVC and IVC (50/50, 40/60, 30/70,
and 20/80). In our study, we varied the SVC/IVC flow distribution, because the ratio changes from
childhood to adulthood, and varies slightly person to person [218]. Total mesh element number
(N) of 6527241 with 𝐷/∆𝑥 = 40 was used in the simulations. Mesh independence studies were
done to ensure that this mesh density is sufficient to ensure the accuracy of the calculations.
7.3 Hemodynamic analysis
7.3.1 Shear stress:
The shear stress 𝜏𝑠 was calculated from
τ𝑠 = μγ̇, (7.12)

177
where 𝜇 is the dynamic viscosity calculated from Eq. 7.6, 𝛾̇ is the shear rate calculated from Eq.
7.7.
7.3.2 Power Loss:
Power loss was calculated as
∑ 𝑄 (𝑝 +
1
2
ρ𝑣
2)
𝑖𝑛𝑙𝑒𝑡𝑠
− ∑ 𝑄 (𝑝 +
1
2
ρ𝑣
2)
𝑂𝑢𝑡𝑙𝑒𝑡𝑠
. (7.13)
Since power loss is dependent on flow rate and fluid density, the flow-independent indexed power
loss (iPL) [53] [190] was used as:
𝐼𝑛𝑑𝑒𝑥𝑒𝑑 𝑃𝑜𝑤𝑒𝑟 𝐿𝑜𝑠𝑠 𝑖𝑃𝐿 =
∑ 𝑄 (𝑝 +
1
2
ρ𝑣
2
𝑖𝑛𝑙𝑒𝑡𝑠 ) − ∑ 𝑄 (𝑝 +
1
2
ρ𝑣
2
𝑂𝑢𝑡𝑙𝑒𝑡𝑠 )
ρ𝑄3/𝐴2
, (7.14)
where 𝜌 is the fluid density, 𝑄 is the total volume blood flow, and 𝐴 is the cross section of the
RPA/LPA tube (pulmonary artery cross sectional area used as a surrogate for body surface area).
7.3.3 Viscous Dissipation:
Viscous dissipation is clinically important since it is associated with the exercise intolerance of
Fontan patients. The main advantage of viscous dissipation as a hemodynamic biomarker is that it
only relies on the velocity field; therefore, it can be computed noninvasively using imaging
modalities such as magnetic resonance imaging or (e.g. 4D flow MRI). The local viscous
dissipation per unit volume (Φ𝑉𝐷) was computed using below equation [219]:
Φ𝑉𝐷 =
1
2
μ∑∑[(
∂𝑣𝑗
∂𝑥𝑖
+
∂𝑣𝑖
∂𝑥𝑗
) −
2
3
(∇ ⋅ 𝑣)δ𝑖𝑗]
2
𝑖 𝑗
. (7.15)

178
The total viscous dissipation was calculated by the integral of the unit viscous dissipation as:
∫ Φ𝑉𝐷𝑑𝑉 = ∑Φ𝑉𝐷𝑉𝑖
𝑁
𝑖=1
, (7.16)
where 𝑁 is the total element number in the computational domain and 𝑉𝑖
is the volume for each
element.
7.3.4 Non-Newtonian Importance Factor:
The non-Newtonian effect was quantified using the non-Newtonian importance factor [220]
defined as:
𝐼𝐿 =
μ
μ∞
. (7.17)
This ratio gives an indication of the overall importance of non-Newtonian effects in the flow [221].
The global non-Newtonian importance factor was computed using:
𝐼𝐺
̅ =
1
𝑁
[∑ (μ − μ∞)
2
𝑁 ]
1
2
μ∞
× 100, (7.18)
where 𝑁 is the total element number in the entire computational domain.
𝐼𝐿 highlights the areas within the fluid domain where the non-Newtonian effects is more
significant. 𝐼𝐺 shows the percentage of the averaged relative difference of each viscosity value
from the Newtonian viscosity value in the full domain.

179
7.3.5 The pulmonary blood flow distribution:
The pulmonary blood flow (% flow to LPA) is calculated from the fraction of the flow rate at the
LPA over the total CO.
7.4 Results
As described in the method section, the presented results are from twenty hemodynamic cases at
various volume flow rates ranging from 0.5 l/min to 2.5 l/min and different IVC/SVC flow
distributions ranging from 50/50 to 20/80.
In our study, Reynolds number (Re) range was from 126 to 632, where Re defined as
𝑅𝑒 =
ρ𝑈𝑟𝑒𝑓𝐿𝑟𝑒𝑓
μ∞
. (7.19)
with 𝑈𝑟𝑒𝑓 = 𝑄/𝐴 and 𝐿𝑟𝑒𝑓 = 𝐷 (D is the diameter of SVC), µ∞ is the Newtonian viscosity.

180
Figure 7.3 Velocity magnitude (nondimensionalized by ‖𝒗‖
𝑼𝒓𝒆𝒇
) for Newtonian (left column) and Non-Newtonian (right column)
cases under an equal SVC/IVC distribution (50/50). The velocity profiles of Non-Newtonian cases are more blunted. There
is a large area of stagnation in the center. The top row is for cardiac output (CO=2.5L/min), the middle row is for cardiac
output (CO=1.5L/min) and the bottom row is for cardiac output (CO=0.5L/min).

181
Figure 7.4 Velocity magnitude (nondimensionalized by ‖𝒗‖
𝑼𝒓𝒆𝒇
) for Newtonian (left column) and Non-Newtonian (right column)
cases under a SVC/IVC distribution of 30/70. Similar to 50/50 cases, the velocity profiles of Non-Newtonian cases were more
blunted. The top row is for cardiac output (CO=2.5L/min), the middle row is for cardiac output (CO=1.5L/min) and the
bottom row is for cardiac output (CO=0.5L/min).
7.4.1 Qualitative flow patterns
Velocity Distribution (Velocity Magnitude & streamlines):
Velocity distributions are shown in Fig. 7.3 and Fig. 7.4. The velocity magnitude was
nondimensionalized by ‖𝒗‖
𝑈𝑟𝑒𝑓
for Newtonian (left column) and Non-Newtonian (right column) cases
under a SVC/IVC distribution of 50/50 and 30/70. The qualitative differences in flow patterns

182
between the non-Newtonian flow and control Newtonian flow are easily recognizable. The overall
velocity profiles were more blunted for the non-Newtonian flow. The qualitative differences
become more significant with lower flow rate (See Fig. 7.3 and Fig. 7.4). All other cases with
different CO and SVC/IVC distributions showed similar behavior and are provided in the
supplementary material.
Shear stress:
Fig. 7.5 and Fig. 7.6 illustrate the shear stress distribution under the same hemodynamic conditions
described above, the general distributions of the areas with high shear stress (nondimensionalized
by 𝜏𝛾̇
1
2
𝜌𝑈𝑟𝑒𝑓
2
) were considerably larger in non-Newtonian flow (right column) than Newtonian (left
column) cases. For all cases, the average magnitude of shear stress was also higher in nonNewtonian flow models. As expected, there was a large area of stagnation in the center of the
circuit for 50/50 cases. All other cases with different CO and SVC/IVC distributions showed
similar behavior and are provided in the supplementary material.

183
Figure 7.5 Shear stress (nondimensionalized by 𝝉𝜸̇
𝟏
𝟐
𝝆𝑼𝒓𝒆𝒇
𝟐
) distributions in Newtonian (left column) and Non- Newtonian cases
(right column) under a SVC/IVC distribution of 50/50. At any given CO, the magnitudes of shear stress are higher in nonNewtonian cases, and the high-stress regions cover larger area in non-Newtonian cases than in Newtonian cases. The top
row is for cardiac output (CO=2.5L/min), the middle row is for cardiac output (CO=1.5L/min) and the bottom row is for
cardiac output (CO=0.5L/min).

184
Figure 7.6 Shear stress (nondimensionalized by 𝝉𝜸̇
𝟏
𝟐
𝝆𝑼𝒓𝒆𝒇
𝟐
) distributions in Newtonian and non-Newtonian cases under a
SVC/IVC distribution of 30/70. At any given CO, the magnitudes of shear stress are higher in non-Newtonian cases, and
the high-stress regions cover larger area in non-Newtonian cases than in Newtonian cases. The top row is for cardiac output
(CO=2.5L/min), the middle row is for cardiac output (CO=1.5L/min) and the bottom row is for cardiac output
(CO=0.5L/min).
7.4.2 Quantitative analysis
Power loss
Table 7.1 compares the percentage of power loss between the two fluid models (Newtonian Vs.
non-Newtonian). Under all hemodynamics conditions, the percentage of power loss for non-

185
Newtonian cases was significantly higher than Newtonian cases. The difference in percent power
loss increased with lower CO and uneven SVC/IVC flow distributions. Since power loss is
dependent on flow rate and fluid density, the flow-independent indexed power losses (iPL) were
also computed, as shown in Table 7.2. The overall pattern in iPL results were similar to percentage
power loss with iPL for non-Newtonian flow cases being significantly higher than Newtonian
cases). Fig. 7.7 shows the absolute difference and the percentage difference of iPL between the
Newtonian and non-Newtonian results. The difference between Newtonian and non-Newtonian
flow became substantially larger at low cardiac output (more than 50% for CO = 0.5 L/min). The
overall differences between the two models were substantial (greater than 20%) even at high
cardiac output.
Figure 7.7 Box plots for absolute (left) and percentage (right) difference of indexed Power Loss (iPL) between Newtonian
and non-Newtonian across a wide range of cardiac outputs (0.5 to 2.5 L/min) under various SVC/IVC distributions.
Table 7-1 Comparison of percent power loss between Newtonian and Non-Newtonian.

186
Table 7-2 Comparison of indexed power loss between Newtonian and Non-Newtonian.
Table 7-3 Comparison of Total Viscous Dissipation between Newtonian and Non-Newtonian.
Table 7-4 Global Non-Newtonian Importance Factor (Zero for Newtonian cases).

187
Table 7-5 Global Non-Newtonian Importance Factor without the central area.
Viscous dissipation
Figs. 7.8 and 7.9 show the Viscous dissipation (nondimensionalized by Φ
1
2
𝜌𝑈𝑟𝑒𝑓
2
/𝑇𝑟𝑒𝑓
) in Newtonian
(left column) and non-Newtonian (right column) cases for the SVC/IVC distribution of 50/50 and
30/70. Qualitatively, the area of high viscous dissipation is larger (near the wall) in non-Newtonian
cases than Newtonian cases at any given CO. This effect is more substantial at lower COs. All
other cases with different CO and SVC/IVC distributions showed similar behavior and are
provided in the supplementary material. Then the total viscous dissipation values (computed using
Eq. (7.16)) over the whole volume of all twenty cases for both non-Newtonian and Newtonian
cases are summarized in Table 7.3 (nondimensionalized by Φ
1
2
𝜌𝑈𝑟𝑒𝑓
2
/𝑇𝑟𝑒𝑓
). The values for nonNewtonian cases are higher than Newtonian cases at any given CO across all SVC/IVC flow
distributions.
Fig. 7.10 shows the box plots of the absolute difference (left) and the percentage difference (right)
of total viscous dissipation (nondimensionalized by Φ
1
2
𝜌𝑈𝑟𝑒𝑓
2
/𝑇𝑟𝑒𝑓
) between the Newtonian and nonNewtonian cases. The difference between Newtonian and non-Newtonian cases increases
exponentially as cardiac output decreases. The difference is more than 50% for Q = 0.5 L/min: and
the difference between the two fluid models is not negligible even at high cardiac output (average
difference is greater than 10%).

188
Non-Newtonian importance factor
Fig. 7.11 shows the local non-Newtonian importance factors (IL) for SVC/IVC distributions of
50/50 (left column) and 40/60 (right column) over a range of cardiac outputs. Fig. 7.12, shows the
(IL) for SVC/IVC distribution of 30/70 (left column) and 20/80 (right column). The area with a
high non-Newtonian importance factor becomes larger as CO decreases. Table 7.4 summarizes the
values of global non-Newtonian Importance Factor (IG; Eq. (7.17)) over the whole volume for all
twenty cases. A large area with high IL was observed in 50/50 SVC/IVC distribution in the central
area of stagnation. This was due to a very low shear rate and consequently very high viscosity in
the stagnation area. Therefore, the IG calculations were repeated after excluding this central area
(Table 7.5). Fig. 7.13 shows the box plots of IG for all cases with (left plot) and without (right plot)
the central stagnation area. As shown in this figure, IG decreases with increasing CO.

189
Figure 7.8 Viscous dissipation (nondimensionalized by 𝚽
𝟏
𝟐
𝝆𝑼𝒓𝒆𝒇
𝟐
/𝑻𝒓𝒆𝒇
) of Newtonian (left column) and Non-Newtonian (right
column) cases for the SVC/IVC distribution of 50/50. The top row is for cardiac output (CO=2.5L/min), the middle row is
for cardiac output (CO=1.5L/min) and the bottom row is for cardiac output (CO=0.5L/min).
Table 7-6 Comparison of pulmonary blood flow (difference in % flow to LPA) between Newtonian and non-Newtonian
models.

190
Figure 7.9 Viscous dissipation (nondimensionalized by 𝚽
𝟏
𝟐
𝝆𝑼𝒓𝒆𝒇
𝟐
/𝑻𝒓𝒆𝒇
) of Newtonian (left column) and Non-Newtonian (right
column) cases for the SVC/IVC distribution of 30/70. The top row is for cardiac output (CO=2.5L/min), the middle row is
for cardiac output (CO=1.5L/min) and the bottom row is for cardiac output (CO=0.5L/min).

191
Figure 7.10 Box plots for the absolute (left) and the percentage (right) difference of total viscous dissipation
(nondimensionalized by 𝚽
𝟏
𝟐
𝝆𝑼𝒓𝒆𝒇
𝟐
/𝑻𝒓𝒆𝒇
) between Newtonian and non-Newtonian at different cardiac output (0.5 to 2.5 L/min)
under various SVC/IVC.
Pulmonary flow distribution
Table 7.6 summarizes the percentage difference of the outlet flow distributions between
Newtonian and non-Newtonian cases. These differences were not substantial with the largest value
occurring at the highest CO (2.5 L/min) and 20/80 SVC/IVC flow distribution (3.54%). In
symmetric SVC/IVC distribution cases (50/50), no difference was observed between Newtonian
and non-Newtonian cases

192
Figure 7.11 Local non-Newtonian importance factor (𝑰𝑳 =
𝛍
𝛍∞
) distributions in Non-Newtonian cases. All cases in the left
column are for the SVC/IVC distribution of 50/50, and all cases in the right column are for the SVC/IVC distribution of
40/60. The top row is for cardiac output (CO=2.5L/min), the middle row is for cardiac output (CO=1.5L/min) and the
bottom row is for cardiac output (CO=0.5L/min).

193
Figure 7.12 Local non-Newtonian importance factor (𝑰𝑳 =
𝛍
𝛍∞
) distributions in Non-Newtonian cases. All cases in the left
column are for the SVC/IVC distribution of 30/70, and all cases in the right column are for the SVC/IVC distribution of
20/80. The top row is for cardiac output (CO=2.5L/min), the middle row is for cardiac output (CO=1.5L/min) and the
bottom row is for cardiac output (CO=0.5L/min).

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Figure 7.13 Box plots for Global Non-Newtonian Importance Factor with (left) and without (right) the central stagnation
area for different cardiac outputs (0.5 to 2.5 L/min) under various SVC/IVC distributions. (Red lines denote the median of
the data. Top and bottom borders of the boxes denote first and third quartiles. Whiskers denote minimum and maximum
values.). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this
article.)
7.5 Discussion
In this study, we systematically studied the effect of the shear rate dependence (non-Newtonian
effect) of blood on Fontan total cavopulmonary connection (TCPC) hemodynamics in a simplified
geometry with different cardiac outputs (CO) and SVC/IVC distributions. We employed a
computational approach using LBM with the Carreau–Yasuda viscosity model to complete this
study. Both qualitative and quantitative results indicate that the Newtonian fluid model introduces
substantial errors (up to 55% for the iPL) in simulation of Fontan hemodynamics. The qualitative
differences in velocity distribution between the two viscosity models were substantial. The
velocity magnitude profiles were more blunted for the non-Newtonian flow (Fig. 7.3, 7.4) at any
given CO and SVC/IVC flow distribution.
The qualitative differences became more substantial when CO decreased, the maximum difference
occurring at the lowest CO. For cases with 50/50 SVC/IVC distributions, a large area of nearly
stagnant flow was observed in the central area between inlet branches. This stagnation area was

195
larger in non-Newtonian cases at any given CO. These results are consistent with previous in-vitro
experimental studies [53] [52]. The area of stagnation is clinically important since it promotes
thromboemboli formation, concentric remodeling, and vessel stenosis due to its low wall shear
stress [222]. Therefore, simulations with Newtonian assumption can underestimate the risk of
these clinical complications in Fontan patients.
The shear stress distribution was significantly different between the Newtonian flow and nonNewtonian cases, which is consistent with previous experimental studies [53] [52] [221] [223]. In
particular, the shear stress near the vessel wall was higher in all non-Newtonian cases at any given
CO or inlet flow distribution. The high shear region had also a larger extension near the center of
the TCPC circuit and the outlets (LPA and RPA) in non-Newtonian cases. These regions correlated
with regions of higher viscous dissipation, and differences between the two fluids were
significantly larger at low CO (e.g. Q=0.5 L/min). These observations support that the mechanism
of increased power loss with a non-Newtonian fluid is due to increased frictional energy loss along
the vessel walls. While high shear stress can lead to increased frictional energy loss in the Fontan
circulation, low shear stress is also clinically important since it plays a role in thrombus formation,
adverse pulmonary vascular remodeling, and endothelial dysfunction [224] [225] [226]. However,
since the Fontan circulation is globally a low-shear-rate system, it is unclear how shear stress
differences of the magnitude that we observed would manifest clinically. Rather than global
differences in non-Newtonian vs. Newtonian shear stress, local differences in patient-specific
regions at risk for particularly low shear (such as the central region of the model used in this study)
are more likely to be clinically relevant. As demonstrated in Tables 7.1–7.2 and Fig. 7.7, both
percentage of power loss and flow-independent indexed power losses (iPL) were significantly
higher for non-Newtonian cases than Newtonian cases. The Newtonian model underestimated

196
these hemodynamic power metrics in all cases and the underestimation became significantly larger
at lower CO (the error increased almost exponentially with decreasing CO). This is particularly
interesting since chronic low cardiac output is common in Fontan patients [50]. Our results suggest
that the iPL can be underestimated up to 55% when Newtonian fluid model is used. As discussed
above, this difference was most likely due to larger shear stress along the vessel walls with the
non-Newtonian fluid which caused a larger frictional energy loss. The power loss across the Fontan
circuit is a primary contributor to decreased exercise capacity [190] [191], neglecting the nonNewtonian effect of blood viscosity in patient specific hemodynamic simulations may result in
erroneous assessment of a patient’s potential exercise performance.
While both power loss and viscous dissipation can be used to quantify flow efficiency, the clinical
utility of viscous dissipation is higher since it can be measured noninvasively (i.e. using cardiac
MRI) and has fewer limiting assumptions [227]. Qualitatively, the areas of high viscous dissipation
were larger in non-Newtonian cases than Newtonian cases at any given CO. This was more
substantial at lower CO. Similar to other hemodynamic metrics, the viscous dissipation increased
with decreasing CO. The flow distribution did not affect the overall viscous dissipation (Table
7.3), but it was slightly larger in the 20/80 SVC/IVC flow distribution. These results indicate that
Newtonian assumption introduces a very large error (up to 55%) in estimation of viscose
dissipation in hemodynamic modeling of Fontan circulation. The non-Newtonian importance
factors IL (local) and IG (global) were used to evaluate the importance of non-Newtonian behavior
qualitatively (using IL) and quantitatively (using IG). A large area of nearly stagnant flow in the
center of the circuit caused a very large viscosity for non-Newtonian cases in even (50/50)
SVC/IVC distribution cases. In order to better investigate the overall impact of non-Newtonian in
areas where the flow behavior is non-trivial, the IG evaluations were repeated after excluding the

197
central stagnation are (the area between the inlet branches). Overall, areas with high IL grew larger
as CO decreased. The IG also increased exponentially (similar to the growth of viscosity with
decreased shear rate in non-Newtonian fluids) with decreased CO. In contrast, the SVC/IVC flow
distribution had negligible effect on IG. Based on the cut-off value of IG (0.25) suggested by
Johnston et al. [220], the non-Newtonian effect was substantial across all COs and SVC/IVC
distributions that were investigated in this manuscript (See box plots of Fig. 7.13). It can be
concluded based on these results that the effects of non-Newtonian behavior are substantial and
must not be ignored in the Fontan circulation modeling.
The Newtonian assumption did not affect the pulmonary flow distribution (%LPA flow) in our
simplified models. There were negligible differences (< 4%) between Newtonian and nonNewtonian cases. No difference observed between Newtonian and non-Newtonian in symmetric
SVC/IVC distribution cases. It seems that a Newtonian assumption does not introduce error in
prediction of pulmonary flow distribution. However, this is only true if one assumes that
downstream vascular resistance remains constant. We also used a simplified outflow model in our
study cases and did not consider downstream pulmonary branches. Since the non-Newtonian
behavior affect viscous dissipation in the downstream vessels, it will affect the total resistance,
and the changes in the resistance will affect the flow distribution if it is uneven. Therefore, a
Newtonian assumption would likely introduce substantial error in estimation of flow distribution
in patient-specific models.
7.5.1 Limitations
One notable limitation of our study is that the Fontan TCPC circuit was relatively simple hydraulic
circuit, and the effects of other physiologic variables such as vessel elasticity, downstream vascular

198
resistance, and lung compliance were not included. Vessel elasticity can modulate non-Newtonian
behavior by altering local shear rate [228], while downstream vascular resistance and lung
compliance can affect the distribution of pulmonary blood flow. However, even with this
simplified model the substantial effect of the shear-rate-dependence of blood viscosity in the
Fontan circulation was revealed. Furthermore, we considered asymmetric geometry for IVC and
SVC. This limited us ability to explore the effect of asymmetry combined with non-Newtonian
behavior of pulmonary flow distribution. Also there is some pulsatility in Fontan flow due to
changes in intrathoracic pressure with respiration [229, 230]. Further studies are warranted to
evaluate how non-Newtonian behavior changes with respiratory-induced variations in blood flow.
7.5.2 Clinical implication
The Fontan circulation is characterized by abnormally low shear-rate pulmonary blood flow,
central venous hypertension, and chronic low CO. Flow inefficiency, quantified by power loss or
viscous dissipation, through this abnormal circulation has been linked to poor functional outcomes
including decreased exercise capacity. Our study demonstrates that using a Newtonian blood
viscosity assumption for these calculations results in substantial underestimation compared to
when a realistic non-Newtonian viscosity model is used. Notably, the degree of error increases at
lower CO, which is commonly observed in Fontan patients. Thus, using a Newtonian fluid
assumption results in overestimation of circulation efficiency in these patients, potentially leading
to delayed recognition of patients who are at risk for poor functional outcomes. Further patientspecific studies are warranted to investigate how non-Newtonian behavior is affected by patientspecific blood viscosity, blood vessel geometry, and blood vessel compliance.

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7.6 Conclusions
In our study of the fluid mechanics inside a simplified rigid wall model of the Fontan circulation
we systematically explored several circulation efficiency metrics by numerical simulation. shearrate-dependent non-Newtonian viscosity was found to have a substantial influence on overall
qualitative flow patterns, velocity distributions, and shear stress distributions, as well as the
quantitative metrics, power loss, non-Newtonian importance factor, and viscous dissipation. NonNewtonian behavior was substantial and varied with cardiac output (CO) and superior vena cava
(SVC)/inferior vena cava (IVC) flow distributions. Clinically impactful Fontan optimization
parameters (power loss, viscous dissipation, and wall shear stress) were found to be markedly
larger for a non-Newtonian fluid. Differences between non-Newtonian and Newtonian fluid
models were more marked at lower CO, which is notable since Fontan patients commonly have
decreased CO. Thus, we have demonstrated that a Newtonian fluid assumption for blood viscosity
introduces considerable error into the fluid dynamics of the low-shear-rate Fontan circulation,
which may lead to overestimation of circulation efficiency in patients at risk of Fontan failure.
Shear-rate-dependent non-Newtonian viscosity was found to have a substantial influence on
overall qualitative flow patterns, velocity distributions, and shear stress distributions, as well as
the quantitative metrics, power loss, non-Newtonian importance factor, and viscous dissipation.
Non-Newtonian behavior was substantial and varied with cardiac output (CO) and superior vena
cava (SVC)/inferior vena cava (IVC) flow distributions. Clinically impactful Fontan optimization
parameters (power loss, viscous dissipation, shear stress and non-Newtonian importance factor)
were found to be markedly larger for a non-Newtonian fluid. Differences between non-Newtonian
and Newtonian fluid models were more marked at lower CO, which is notable since Fontan

200
patients commonly have decreased CO. Thus, we have demonstrated that a Newtonian fluid
assumption for blood viscosity introduces considerable error into the fluid dynamics of the lowshear-rate Fontan circulation, which may lead to overestimation of circulation efficiency in
patients at risk of Fontan failure.

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Chapter 8: The Impact of Blood Viscosity Modeling on
Computational Fluid Dynamic Simulations of Pediatric Patients
with Fontan Circulation
This chapter is based on the following published manuscript: Wei, H., Bilgi, C., Cao, K., Detterich,
J, Pahlevan, N, & Cheng, A. (2024). Examining the Influence of Non-Newtonian Blood Viscosity
in Computational Fluid Dynamic Modeling of the Fontan Circulation: A Case Series of 20
Pediatric Patients. Scientific Reports (Submitted to Revision)
The next phase of this study involved creating patient-specific models for a cohort of twenty
individuals. These models were meticulously developed to reflect the unique anatomical and
physiological characteristics of each patient. By simulating low cardiac output conditions, the
study aimed to gain deeper insights into the hemodynamic variations and potential complications
associated with reduced cardiac efficiency. This patient-specific approach allows for a more
accurate and individualized understanding of the cardiovascular dynamics under pathological
conditions, paving the way for more personalized and effective medical interventions.
For univentricular heart patients, the Fontan circulation presents a unique pathophysiology due to
chronic non-pulsatile low-shear-rate pulmonary blood flow, where non-Newtonian effects are
likely substantial. This study evaluates the influence of non-Newtonian behavior of blood on fluid
dynamics and energetic efficiency in pediatric patient-specific models of the Fontan circulation.
We used immersed boundary-lattice Boltzmann method simulations to compare Newtonian and
non-Newtonian viscosity models. The study included models from twenty patients exhibiting a
low cardiac output state. We quantified metrics of energy loss (indexed power loss and viscous
dissipation), non-Newtonian importance factors, and hepatic flow distribution. We observed
significant differences in flow structure between Newtonian and non-Newtonian models.

202
Specifically, the non-Newtonian simulations demonstrated significantly higher local and average
viscosity, corresponding to a higher non-Newtonian importance factor and larger energy loss.
Hepatic flow distribution was also significantly different in a subset of patients. These findings
suggest that non-Newtonian behavior contributes to flow structure and energetic inefficiency in
the low cardiac output state of the Fontan circulation.
8.1 Chapter introduction
The Fontan procedure is the final stage of surgical palliation for patients born with a univentricular
heart defect [174]. This procedure completes a total cavopulmonary connection (TCPC), where all
systemic venous blood bypasses the heart and drains directly into the lungs [174]. The result is a
non-physiologic circulation characterized by low-shear non-pulsatile pulmonary blood flow,
central venous hypertension, and decreased cardiac output [175].
Due to the abnormal hemodynamics of the Fontan circulation, patients commonly develop
multiorgan dysfunction and a progressively increasing risk of Fontan circulation failure,
requirement for heart transplant, or premature death [231]. In an effort to decrease these
complications, many researchers have used computational fluid dynamic (CFD) modeling to better
understand the pathophysiology of the Fontan circulation [232-234], test new surgical approaches
[194, 235-237], and design novel devices to augment blood flow through the circulation [238,
239].
The majority of prior CFD studies of the Fontan circulation have assumed blood to be a Newtonian
fluid with constant viscosity [194, 232, 235, 236, 238, 239]. While a Newtonian viscosity model
may be a suitable assumption in large systemic arteries where shear rates are sufficiently high
(above 100 s-1) [91, 240], blood is most accurately modeled as a shear-thinning non-Newtonian

203
fluid in which viscosity increases exponentially at shear rates less than 100 s-1 [241-244]. Since
shear rates in the Fontan pulmonary vasculature fall in this range, it is likely that non-Newtonian
effects on blood viscosity significantly influence fluid dynamics. We previously demonstrated that
non-Newtonian behavior is an important determinant of circulatory efficiency in simplified in vitro
and in silico models of the Fontan circulation [52-54]. In these studies, we observed significant
differences in qualitative flow patterns, particularly areas of flow stagnation and recirculation,
between Newtonian and non-Newtonian viscosity models. When we evaluated a simplified
vascular geometry consisting of straight cylinders both in vitro and in silico, measures of
circulation efficiency (viscous dissipation and indexed power loss) were worse with nonNewtonian viscosity models. However, when we evaluated two patient-specific vascular
geometries (in vitro only), we observed a complex relationship between efficiency metrics,
viscosity model, vascular geometry, and cardiac output. Given these complex interactions and the
significant heterogeneity in vascular geometry among patients with Fontan circulation, we sought
to evaluate the impact of non-Newtonian behavior on Fontan circulation performance more
extensively in a larger number of diverse patient-specific models.
The objective of this study was to evaluate the extent to which non-Newtonian behavior
contributes to Fontan circulation performance in pediatric patient-specific models under a low
cardiac output state. In this study, we used immersed boundary-Lattice Boltzmann (IB-LBM)
based CFD coupled with a physiologically accurate non-Newtonian blood model to simulate the
flow in patient-specific Fontan circulations. We hypothesized that non-Newtonian increases in
blood viscosity at low shear rates would increase power loss and affect hepatic blood flow
distribution to the lungs.

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8.2 Methods
8.2.1 Patient-specific Geometry
The study was approved by the Institutional Review Board (IRB) of Children’s Hospital Los
Angeles (CHLA-19-00385). The IRB determined that the study was exempt from the requirement
for informed consent per federal regulations (45 CFR 46). Research was conducted in accordance
with the Declaration of Helsinki.
Cardiac magnetic resonance imaging (MRI) scans were performed on a 1.5T Philips Achieva
system (Philips Healthcare, Best, The Netherlands) for clinical indications. Images of the thoracic
vasculature and heart were obtained using a 3-dimensional steady state free precession sequence
with respiratory navigator gating and electrocardiogram triggering. We used 3D Slicer 5.6
(https://www.slicer.org/) to segment the Fontan pulmonary vasculature from the MRI images and
create a patient-specific vasculature geometry of the vascular geometry. We then used ADINA
9.8.2 (ADINA R&D, Inc.) to refine the mesh. Figure 1 shows the workflow from MRI images to
patient specific CFD models.

205
Figure 8.1 Workflow for creating CFD simulations from MRI images.
8.3 Numerical Methods
We employed a computational approach using a Casson viscosity model combined with IB-LBM
to accurately capture the non-Newtonian characteristics of blood flow [53, 245]. LBM uses
simplified kinetic equations combined with a modified molecular-dynamics approach to simulate
fluid flows as an alternative method to conventional CFD methods that use Navier–Stokes
equations. The accuracy and usefulness of LBM were demonstrated in various fluid dynamics
problems including bio-inspired fluid-structure interactions [246, 247], porous media [248],
thermal flow [249], non-Newtonian flow [54, 250] and hemodynamics [144, 251].
8.3.1 3D Lattice Boltzmann Equations
We used a single-relaxation-time incompressible LBM formulation to obtain the pressure and flow
fields [89, 90]. In such a method, the synchronous motions of the particles on a regular lattice are
enforced through a particle distribution function that inherently satisfies mass and momentum
conservations. This distribution function also ensures that the fluid is Galilean invariant and

206
isotropic [89, 90]. The evolution of the distribution functions on the lattice is governed by the
discrete Boltzmann equation with Bhatnagar-Gross-Krook (BGK) collision model as,
𝑓𝑖
(𝒙 + 𝒆𝑖∆𝑡,𝑡 + ∆𝑡) − 𝑓𝑖
(𝒙,𝑡) = −
1
𝜏
[𝑓𝑖
(𝒙,𝑡) − 𝑓𝑖
𝑒𝑞(𝒙,𝑡)] + ∆𝑡𝑭𝒊
, 𝑖 = 0, … , 𝑁0 − 1, (8.1)
where 𝑓𝑖
(𝑥,𝑡) is the distribution function for particles with velocity 𝑒𝑖 at position 𝑥 and time t and
𝐹𝑖
is the forcing term to couple the fluid and solid domains. ∆𝑡 and ∆𝑥 are the time step and lattice
space, respectively that are assigned as equal to each other in the simulations. 𝜏 is a dimensionless
relaxation time constant which is associated with fluid viscosity in the form of 𝜇 = 𝜌𝜗 =
𝜌𝑐𝑠
2
(𝜏 −
1
2
)∆𝑡, where 𝜗 is the kinematic viscosity, 𝜌 is the fluid density and 𝑐𝑠 =
1
√3
is the lattice
sound speed. In this model, a 19 discrete velocity vector stencil (D3Q19) is used, hence, 𝑁0 = 19
[101, 250]. The local equilibrium distributions for the incompressible LBM and the forcing term
are defined as 𝑓𝑖
𝑒𝑞 = 𝜔𝑖
𝑝
𝑐𝑠
2 + 𝜔𝑖𝜌 [
𝑒𝑖
∙𝑣
𝑐𝑠
2 +
(𝑒𝑖
∙𝑣)
2
2𝑐𝑠
4 −
𝑣
2
2𝑐𝑠
2
], and 𝑭𝑖 = (1 −
1
2𝜏
) 𝜔𝑖 (
𝑒𝑖−𝑣
𝑐𝑠
2 +
𝑒𝑖
∙𝑣
𝑐𝑠
4
𝑒𝑖) 𝒃𝒇
,
where 𝜔𝑖
is the weighting factor, 𝒃𝒇
is the force density at Eulerian grid; 𝑝 is the pressure and 𝑣
is the velocity vectors. The macroscopic variables are calculated by, 𝑝 = 𝑐𝑠
2 ∑𝑖 𝑓𝑖 and 𝜌𝒗 =
1
2
𝒃𝒇∆𝑡 + ∑𝑖 𝒆𝑖𝑓𝑖
. We used an immersed boundary algorithm at the interface of the fluid and solid
domains to obtain the force density, 𝒃𝒇
, and bounce-back conditions at the rigid inlets and outlets.
8.3.2 Non-Newtonian Fluid Modeling
Non-Newtonian properties of blood were modeled by applying the modified Casson model to the
fluid domain [52, 209, 252, 253]. This model has been widely used to mimic the shear-thinning
behavior in hemodynamic simulations [209, 213]. In this study, the Casson parameters were
obtained by curve-fitting to in vitro viscosity data from a historical cohort of Fontan patients

207
(Figure 8.2) [53, 213]. The historical cohort was a mean age of 15.2 years (SD 5.3 years) and was
41% female.
Figure 8.2 Viscosity of shear-thinning Casson model based on patient-specific data. Red circles are mean viscosity
measurements from 61 Fontan patients.
The apparent viscosity of the modified Casson model is given by
𝜇(𝛾̇) = (√𝜏0
(1 − 𝑒
−𝑚𝛾̇)
𝛾̇
+ √𝜇∞)
2
, (8.2)
where 𝛾̇ is the shear rate, 𝜏0 is the yield stress, and 𝜇∞ is the infinite-shear plateau viscosity. The
model parameters were empirically determined from patient data as 𝜇∞ = 3.149 𝑚𝑃𝑎 ∙ 𝑠, 𝜏0 =
12.97 𝑚𝑃𝑎, 𝑚 = 3.913 𝑠.
The shear rate (𝛾̇) is computed from the second invariant of the rate-of-strain tensor (𝐷𝐼𝐼): 𝛾̇ =
√2𝐷𝐼𝐼, where the 𝐷𝐼𝐼 is defined as 𝐷𝐼𝐼 = ∑ 𝑆αβ
3
α,β=1 𝑆αβ. In LBM, the strain tensor 𝑆𝛼𝛽 can be
calculated locally at each node using [209]: 𝑆αβ = −
3
2τ
∑ (𝑓𝑖 − 𝑓𝑖
𝑒𝑞
𝑖 )𝑒𝑖α 𝑒𝑖β. Thus, the shear strain

208
is obtained locally without using the derivatives of the velocity [214]. For the baseline Newtonian
cases, the dynamic viscosity is set to be the infinite shear viscosity of the Casson model: 𝜇 = 𝜇∞ =
3.149 𝑚𝑃𝑎 ∙ 𝑠.
8.3.3 Fluid–Structure Interactions (FSI)
For coupling the fluid and patient-specific structure systems, an explicit velocity correction-based
IB method is used [92, 93]. This method has been extensively applied to cardiovascular
biomechanics problems to capture the fluid-solid interactions [109-111, 250]. In this method, the
body force, 𝑓, enforces the no-slip velocity boundary condition at the interface of the fluid and
solid domains by introducing velocity correction, 𝛿𝒗:
𝑣(𝑥,𝑡) = 𝑣
∗
(𝑥,𝑡) + 𝛿𝑣(𝑥,𝑡), (8.3)
where the uncorrected velocity is 𝒗
∗ =
1
𝜌
∑𝑖 𝒆𝑖𝑓𝑖
.
In the velocity correction-based immersed boundary approach, 𝛿𝑣 term at the Eulerian point (fluid
domain) is obtained by the following Dirac delta interpolation function [112] as
𝛿𝒗(𝒙,𝑡) = ∫ 𝛿𝑽(𝑠,𝑡)𝛿(𝒙 − 𝑿(𝑠,𝑡))𝑑𝑠
𝛤
, (8.4)
where 𝛿(𝑥 − 𝑋(𝑠,𝑡)) is smoothly approximated by a continuous kernel distribution, and 𝛿𝑉(𝑠,𝑡)
is the unknown velocity correction vector at every Lagrangian point s of the solid boundary [112].
Note that 𝑥 denotes the Eulerian coordinates related to the fluid field, while 𝑋 stands for
Lagrangian coordinates of the solid domain. In order to meet the non-slip boundary condition, the
fluid velocity must be equal to the wall velocity 𝑉 at the same position that can be described as

209
𝑽(𝑠,𝑡) = ∫ 𝒗(𝒙,𝑡)𝛿(𝒙 − 𝑿(𝑠,𝑡))𝑑𝒙 𝛺
. (8.5)
Substituting Eq. (8.3) and (8.4) into Eq. (8.5), we can have the following equation:
𝑽(𝑠,𝑡) = ∫ 𝒗
∗
(𝒙,𝑡)𝛿(𝒙 − 𝑿(𝑠,𝑡))𝑑𝒙 𝛺
+ ∫ [∫ 𝛿𝑽(𝑠,𝑡)𝛿(𝒙 − 𝑿(𝑠,𝑡))𝑑𝑠
𝛤
] 𝛿(𝒙 − 𝑿(𝑠,𝑡))𝑑𝒙 𝛺
. (8.6)
After obtaining 𝛿𝑽(𝑠,𝑡) via solving Eq. (8.6), the body force density, 𝑓, is obtained by using the
following relations: 𝛿𝒗(𝑥,𝑡) =
1
2𝜌
𝑓(𝑥,𝑡)𝛿𝑡. These IB equations are solved explicitly with the
efficient boundary condition enforced method, whose details can be found elsewhere [113].
8.3.4 Concentration Field
Dye concentration evolution 𝐶(𝑥,𝑡) subjected to a velocity field 𝑣(𝑥,𝑡) is governed by the
homogeneous advection-diffusion equation [139], i.e.,
𝜕𝐶
𝜕𝑡 + 𝑣 ⋅ 𝛻𝐶 − 𝐷𝛻
2𝐶 = 0, (8.7)
In this study, we used an additional distribution function, 𝑔𝑖
(𝑥,𝑡), in our LBM model to solve the
advection-diffusion equation [249]. This distribution function is also governed by BGK collision
model [254] with 19 discrete velocity vectors. The Boltzmann equation for the concentration field
can be written as
𝑔𝑖
(𝑥 + 𝑒𝑖∆𝑡,𝑡 + ∆𝑡) − 𝑔𝑖
(𝑥,𝑡) = −
1
𝜏𝑐
[𝑔𝑖
(𝑥,𝑡) − 𝑔𝑖
𝑒𝑞(𝑥,𝑡)] + ∆𝑡𝐺𝑖
, 𝑖 = 1, … ,19. (8)
Here, 𝜏𝐶 is dimensionless relaxation time constant associated with diffusion coefficient, 𝐷 =
𝑐𝑠
2
(𝜏𝐶 − 1/2)∆𝑡; the local equilibrium distribution (𝑔𝑖
𝑒𝑞) and immersed boundary force term for
the concentration field (𝐺𝑖
) can be written as [255] 𝑔𝑖
𝑒𝑞 = 𝜔𝑖𝐶 [1 +
𝑒𝑖
∙𝑣
𝑐𝑠
2 +
(𝑒𝑖
∙𝑣)
2
2𝑐𝑠
2 −
𝑣
2
2𝑐𝑠
2
], and 𝐺𝑖 =

210
(1 −
1
2𝜏𝑐
) 𝜔𝑖𝑏𝑐
, where 𝑏𝑐
is the immersed boundary force for concentration field in Eulerian
coordinates. Macroscopic dye concentration can be calculated by 𝜌𝐶(𝒙,𝑡) =
∆𝑡
2
𝑏𝑐 +
∑ 𝑔𝑖
(𝒙,𝑡)
19
𝑖=1
.
8.3.5 Simulation conditions
The mean cardiac index measured by MRI was 2.5 L/min/m2 (interquartile range (IQR) 2-2.8
L/min/m2). In all simulations, we apply the patient-specific flow rates by multiplying the patient
specific body surface area (BSA) with a constant cardiac index ( 𝑐𝑎𝑟𝑑𝑖𝑎𝑐 𝑜𝑢𝑡𝑝𝑢𝑡 =
𝑐𝑎𝑟𝑑𝑖𝑎𝑐 𝑖𝑛𝑑𝑒𝑥 × 𝑏𝑜𝑑𝑦 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎) of 2 L/min/m2 to model a low cardiac output state. We
assigned the flow distribution of the inlets as 30% from superior vena cava (upper body), and 70%
from inferior vena cava (lower body). The inlet flow profile is assigned by applying a uniform
velocity profile (plug flow) with an inlet flow extension (plug flow conditions at the inflow of the
extension). We set pulmonary vascular resistances at the outlets as a constant of 2 Wood Units ∙
m2 without assigning any outlet flow conditions. These simulation parameters were kept
consistent for both Newtonian and non-Newtonian fluid models in order to observe the isolated
effect of non-Newtonian behavior on Fontan hemodynamics. A fluid mesh cardinality of 𝑁𝑚𝑒𝑠ℎ =
2000/𝑚𝑙 is used for all the simulations (corresponding to a total number of fluid mesh 𝑁𝑡𝑜𝑡𝑎𝑙 =
3.4 × 106
for patient 1). The solid mesh size (Δ𝑠) and fluid element size (Δx) employed here
satisfy the comparability requirement (Δ𝑠 < 2Δx) to ensure the stability of the fluid-solid coupling
by immersed boundary method. [113] The patient specific Fontan meshes are generated by an
open-source software MeshLab [256].

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8.4 Hemodynamic Analysis and Metrics
8.4.1 Power Loss
We quantified power loss in the overall flow field by calculating the indexed power difference
between the inlets and outlets [53, 227, 257, 258],
𝑖𝑃𝐿 =
∑ 𝑄 (𝑃 +
1
2
𝜌𝑢
2
𝐼𝑛𝑙𝑒𝑡𝑠 ) − ∑ 𝑄 (𝑃 +
1
2
𝜌𝑢
2
𝑂𝑢𝑡𝑙𝑒𝑡𝑠 )
𝜌𝑄3/𝐴2
(8.9)
where 𝜌 is the fluid density, Q is total inlet volume flow rate, and A is the average cross-sectional
area of the inlets.
8.4.2 Viscous Dissipation Rate
We computed total viscous dissipation (Φ𝑉𝐷) using:
Φ𝑉𝐷 = ∭2𝜇 [(
𝜕𝑢𝑥
𝜕𝑥 )
2
+ (
𝜕𝑢𝑦
𝜕𝑦 )
2
+ (
𝜕𝑢𝑧
𝜕𝑧 )
2
]
⬚
𝑉
+ 𝜇 [(
𝜕𝑢𝑥
𝜕𝑦 +
𝜕𝑢𝑦
𝜕𝑥 )
2
+𝜇 (
𝜕𝑢𝑦
𝜕𝑧 +
𝜕𝑢𝑧
𝜕𝑦 )
2
+ 𝜇 (
𝜕𝑢𝑧
𝜕𝑥 +
𝜕𝑢𝑥
𝜕𝑧 )
2
] 𝑑𝑉(8.10)
where 𝑉 is the volume for the fluid domain. Accordingly, the indexed viscous dissipation was
calculated as:
𝑖Φ𝑉𝐷 =
Φ𝑉𝐷
𝜌𝑄3/𝐴2
(8.11)
where Q is the total inlet volume flow rate and A is the average cross-sectional area of the outlets.

212
8.4.3 Non-Newtonian Importance Factor
We quantified the importance of shear-thinning effects using the non-Newtonian importance factor
[54, 215] to have an overall indication of non-Newtonian effects in the flow field [259]. This metric
is defined as:
𝐼𝐿 =
𝜇
𝜇𝑁𝑒𝑤𝑡𝑜𝑛𝑖𝑎𝑛
=
𝜇
𝜇∞
. (8.12)
Global and average non-Newtonian importance factors were computed as:
𝐼𝐴
̅ =
∑ (
𝜇
𝜇∞
𝑁 )
𝑁
, 𝑎𝑛𝑑 𝐼𝐺
̅ =
1
𝑁
√∑ (𝜇 − 𝜇∞)
2
𝑁
𝜇∞
× 100, (8.13)
where 𝑁 is the total element number in the fluid domain. Overall, 𝐼𝐿 highlights the areas where the
non-Newtonian effects are more substantial. 𝐼𝐴
̅ and 𝐼𝐺
̅ shows the average and deviation from the
Newtonian viscosity in the fluid domain.
8.4.4 Hepatic Blood Flow Distribution
We calculated the hepatic blood flow (HBF) at the outlets (left and right pulmonary arteries) by
tracking the concentration field with a threshold (𝐶0 = 0.001) to avoid numerical error,
𝐻𝐵𝐹𝑖 = ∫ {
𝑢, 𝐶 ≥ 0.001
0, otherwise 𝑑𝐴
⬚
𝛺∈ 𝑜𝑢𝑡𝑙𝑒𝑡 𝑖
, 𝑖 = 𝐿𝑒𝑓𝑡, 𝑅𝑖𝑔ℎ𝑡 (8.14)
We calculated hepatic blood distribution (HFD) as the ratio of HBF at the left pulmonary artery
(LPA) to the total HBF at the outlets [233, 234]:
𝐻𝐹𝐷 =
𝐻𝐵𝐹𝐿𝑒𝑓𝑡
𝐻𝐵𝐹𝐿𝑒𝑓𝑡 + 𝐻𝐵𝐹𝑅𝑖𝑔ℎ𝑡
(8.15)

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8.5 Statistical Analysis
Categorical variables were summarized as counts and percentages. Continuous variables were
summarized as mean with standard deviation for normally distributed variables or median with
interquartile range for non-normally distributed variables. Normality was assessed by the ShapiroWilk test. We assessed differences in outcome measures between viscosity models using paired
Student’s t-test for normally distributed variables and Wilcoxon signed-rank test for non-normally
distributed variables.
8.6 Results
8.6.1 Demographics of Fontan Patients
Twenty patients (7 females) with Fontan circulation were included. MRI was performed at a mean
age of 10.8 years (SD 4.4 years), which corresponded to a mean time of 6.1 years from the date of
Fontan surgery (all non-fenestrated extracardiac Fontan). Patient demographics are shown in Table
8.1.
Table 8-1 Characteristics of patient cohort. SD: standard deviation, IQR: interquartile range.
Number of patients 20
Age, years (SD) 10.8 (4.4)
Female, n (%) 7 (35%)
Body surface area, m2 (IQR) 1.12 (0.88, 1.54)
Race, n (%)
Hispanic
White
Asian
Unknown
15 (75%)
3 (15%)
1 (5%)
1 (5%)
Cardiac Diagnosis, n (%)
Hypoplastic left heart syndrome
Double outlet right ventricle
7 (35%)
3 (15%)

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Atrioventricular septal defect
Double inlet left ventricle
Other
3 (15%)
2 (10%)
4 (25%)
Extracardiac Fontan, n (%) 20 (100%)
8.6.2 Non-Newtonian Effects on Flow Patterns and Energy Loss
Flow structures
Flow structures of four patients with diverse vascular geometries are presented in Figure 8.3. The
main differences between viscosity models on the flow structures are observed in the TCPC region
(denoted by red dashed boxes). In this low shear region, non-Newtonian viscosity leads to lower
velocity values, decreased flow rotation, and more stagnant flow.
Figure 8.3 Streamlines and velocity magnitudes in four sample Fontan patients. Non-Newtonian models demonstrate lower
velocity values, decreased flow rotation, and more stagnant flow in the TCPC region (dashed red boxes). SVC: superior
vena cava, IVC: inferior vena cava, RPA: right pulmonary artery, LPA: left pulmonary artery.

215
Non-Newtonian Importance Factor
The local non-Newtonian importance factors for the four patients are shown in Figure 8.4 only for
the non-Newtonian cases (this value is equal to 1 for Newtonian simulations, cf. Eq (8.13).). The
mean 𝐼𝐴 was 9.06 (standard deviation 2.88, range 4.18-13.18) and mean 𝐼𝐺 was 2.95 (standard
deviation 1.79, range 1.13- 5.22). Thus, the mean effective viscosity was nearly ten-fold higher for
the non-Newtonian simulations compared to Newtonian simulations.
Figure 8.4 Patient-specific variation in local non-Newtonian importance factor (note Patients 4 and 15 have a different color
scale since the values are larger). SVC: superior vena cava, IVC: inferior vena cava, RPA: right pulmonary artery, LPA:
left pulmonary artery.
Power Loss and Viscous Dissipation Rate
Figure 8.5 (a) shows the indexed power loss for both Newtonian and non-Newtonian viscosity
models. We observed a statistically significant increase in power loss with non-Newtonian models
vs. Newtonian models (median 1.96 [IQR 0.87, 3.98] vs. 1.30 [IQR 0.47, 2.84], p<0.0001).
Indexed viscous dissipation rate for each case is shown in Figure 8.5 (b). Non-Newtonian models
yielded significantly higher viscous dissipation rate vs. Newtonian models (median 0.72 [IQR
0.38, 1.45] vs. 0.38 [IQR 0.25. 0.81], p<0.0001).

216
Figure 8.5 (a) Comparison of indexed power loss between Newtonian and non-Newtonian viscosity models. (b) Comparison
of indexed viscous dissipation rate between Newtonian and non-Newtonian viscosity models. Patient-level differences are
denoted with a black line.
Figure 8.6 (a) Comparison of mean wall shear stress (WSS between Newtonian and non-Newtonian viscosity models. (b)
Comparison of low WSS area (as a percentage of total Fontan area) between Newtonian and non-Newtonian viscosity
models. Patient-level differences are denoted with a black line.
Non-Newtonian Effects on Wall Shear Stress (WSS) and Low WSS Area
We computed the wall shear stress (WSS, 𝜏𝑤 = 𝜇(𝛾̇𝑊𝑎𝑙𝑙) ∙ 𝛾̇𝑊𝑎𝑙𝑙) distribution for all the patients
with Newtonian and non-Newtonian models (see supplementary Figure 8.3). To quantify the
changes in WSS with the viscosity model, we calculated the mean WSS among the total Fontan
geometry [260, 261] and evaluated the low WSS area defined by the percentage ratio of the area
experiencing WSS less than 0.4 Pa, to the total Fontan wall area (𝐴𝑊𝑆𝑆<0.4𝑃𝑎/𝐴𝑇𝑜𝑡𝑎𝑙) [261].

217
Figure 8.6 (a) shows the mean WSS for both Newtonian and non-Newtonian models. A significant
increase in mean WSS with non-Newtonian vs. Newtonian (median 3.19 [IQR 2.48 3.57] vs. 1.26
[IQR 1.04 1.54], p<0.0001) can be observed. Figure 8.6 (b) shows the low WSS area for each case,
where the non-Newtonian models represent significant small low WSS area (%) vs. Newtonian
models. (median 1.89 [IQR 0.44 7.22] vs. 15.52 [IQR 7.86 24.29], p<0.0001). The mean shear rate
(MSR) of the whole Fontan fluid domain (reported in supplementary Table 2) are also found to be
between 2.28 to 14.26 s-1 (median MSR = 5.13 Std = 3.01, for Newtonian model), and from 1.59
to 11.22 s-1 (median MSR = 3.81 Std = 2.35, for non-Newtonian model).
Non-Newtonian Effects on Quantification of Hepatic Blood Flow Distribution
Figure 8.7 shows dye simulations and HFD in four diverse patients. When considering all cases
together, there was not an overall statistically significant difference in HFD between Newtonian
and non-Newtonian models (54.2% ± 17.3% vs. 55.1% ± 18.4%, p=0.36). However, we observed
that patient geometry has non-negligible effects on HFD. As shown in Figure 8, both the magnitude
of difference between Newtonian and non-Newtonian models and which viscosity model yielded
higher HFD was highly variable between patients.

218
Figure 8.7 Dye simulations for calculation of hepatic blood flow distribution (HFD). Dye was tracked from IVC to RPA and
LPA. SVC: superior vena cava, IVC: inferior vena cava, RPA: right pulmonary artery, LPA: left pulmonary artery.
Figure 8.8 (a) Comparison of hepatic flow distribution between Newtonian and non-Newtonian fluid models for each
patient. (b) Difference in hepatic flow distribution between Newtonian and non-Newtonian fluid models.
8.7 Discussion
Our study delves into the critical area of Fontan failure risk assessment through LBM based CFD,
specifically focusing on the impact of non-Newtonian effects on hemodynamics. The results imply
that there is a significant impact of blood viscosity models on Fontan hemodynamics, indicating
that the Newtonian assumption might introduce considerably large errors into performance

219
metrics. Importantly, we observed substantial variation between different vascular geometries,
emphasizing the patient-specific nature of non-Newtonian effects.
Across all cases, substantial differences in blood flow structure, particularly near the center TCPC
region, were clearly present and varied significantly between patients. For example, in patient 3
(Figure 8.3), the streamlines became flatter for the non-Newtonian model and impacted a circular
helical structure in the TCPC region. In contrast, non-Newtonian simulations for patient 4 revealed
a larger stagnation area, corresponding to high local viscosity/non-Newtonian importance factors.
The patient-specific effects imply that differences are not easily predictable from a Newtonian
model with any simple corrections.
The non-Newtonian simulations consistently exhibited significantly higher viscosity compared to
Newtonian models. In particular, the center TCPC region exhibited higher viscosity (because of
low shear), corresponding to an area of stagnation. High global non-Newtonian importance factors
were observed with a substantial average and global deviation, indicating that the Newtonian
assumption significantly underestimates blood viscosity. These differences varied widely between
different patient geometries.
Both indexed power loss, representing the energy difference between the inlets and outlets, and
the indexed viscous dissipation rate, representing global energy loss, showed a substantial increase
when shear-thinning effects were considered. This increase can be attributed to differences in flow
and pressure distribution with the non-Newtonian viscosity model, as well as viscosity changes in
non-Newtonian conditions.

220
In our simulations, the median value of the mean shear rate is ~5 s-1 for Newtonian models (and
~4 s-1 for Newtonian models). This shear rate level of our simulations falls within the critical
viscosity range (𝜇 > 𝜇𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 = 3𝜇∞, 𝑤ℎ𝑒𝑟𝑒 𝑠ℎ𝑒𝑎𝑟 𝑟𝑎𝑡𝑒 𝛾̇ < 7.69 𝑠
−1
) as defined in previous
study [162, 262, 263]. Therefore, the mean shear rate serves as an additional evidence indicating
that the non-Newtonian blood model has considerable effects in Fontan circulation hemodynamics.
This finding aligns well with prior conclusions asserting the importance of the non-Newtonian
effect in venous system, particularly those characterized by lower flow pulsatility and velocities
[162].
We investigated the impact of non-Newtonian viscosity model by analyzing WSS, which serves
as a clinically relevant hemodynamic parameter [260, 261, 264, 265]. Furthermore, we examined
the low WSS area, a metric commonly reported in previous on Fontan studies that is associated
with atherosclerosis and thrombosis risk [261, 265]. Given that the Fontan circulation exhibited
lower flow pulsatility and velocity compared to the arterial model, our analysis again revealed
larger WSS values (as shown in supplementary Figure 8.2), together with the greater mean WSS
(as illustrated in Figure 8.7(a)) in the non-Newtonian cases. The difference in mean WSS between
Newtonian and non-Newtonian cases is well correlated with the global non-Newtonian importance
factor 𝐼𝐺 (2.95 times). This observation is consistent with previous research suggesting that the
increased viscosity in non-Newtonian cases is significantly greater, consequently leading to
elevated wall shear stress, despite lower velocity gradients [261]. Moreover, the non-Newtonian
case exhibited a significant decrease in the low WSS area due to the substantial increase in WSS.
Interestingly, both mean WSS and low WSS area calculations demonstrated notable patientspecific variations (Figure 8.7(a) and (b)), highlighting the inherent error associated with the

221
Newtonian blood assumption and the importance of utilizing a physiologically accurate nonNewtonian model.
Differences in HFD between Newtonian and non-Newtonian simulations also varied significantly
between patients. Furthermore, neither viscosity model uniformly predicted higher HFD, again
highlighting that differences are not easily predictable from a Newtonian model with simple
corrections.
8.8 Clinical Significance
Our findings demonstrate that non-Newtonian blood viscosity influences multiple facets of Fontan
circulation fluid dynamics that may have clinical significance -- hemodynamic efficiency, flow
stagnation, and HFD. Prior CFD studies have evaluated Fontan circulation performance using
several different measures of hemodynamic efficiency, including indexed power loss and viscous
dissipation [227]. Indexed power loss has most often been used as a global marker of circulation
performance since it correlates moderately well with exercise capacity [232]. We observed that
both indexed power loss and viscous dissipation were significantly lower when using a Newtonian
blood model vs. a non-Newtonian model, and that the discrepancy varied significantly between
patients. Therefore, when used for prognostic purposes these metrics, if calculated using a
Newtonian assumption, may underestimate the risk of Fontan circulation failure in a subset of
patients. Similarly, when used to evaluate different surgical approaches to the Fontan procedure
these metrics may lead to erroneous conclusions about the most optimal approach.
Largely due to chronic low-shear passive pulmonary blood flow, Fontan patients have a high risk
of thrombosis and thromboembolism [266]. Areas in the Fontan conduit or vasculature with flow
stagnation pose an increased risk of thrombus formation [267]. In our study we noted significant

222
differences in areas of flow stagnation between Newtonian and non-Newtonian simulations. These
differences varied widely depending on patient-specific vascular geometry. Thus, neglecting nonNewtonian behavior in CFD modeling may underestimate the risk of thrombus formation.
Symmetric distribution of hepatic blood flow to all lung segments is necessary to suppress the
formation of pulmonary arteriovenous malformations in Fontan patients [268]. While the overall
difference in HFD between our Newtonian and non-Newtonian simulations was not statistically
significant, there was a subset of patients in whom the difference was as large as ~14%. A
difference of this magnitude would likely translate to a clinically meaningful difference in risk of
developing pulmonary arteriovenous malformations. Also, since we did not include the peripheral
pulmonary vasculature in our models, we cannot conclude whether small differences in hepatic
blood flow distribution to the right vs. left lung could translate to larger differences to individual
lung segments.
In summary, this study highlights the limitations of the Newtonian blood model in CFD
simulations of the Fontan circulation and argues that non-Newtonian viscosity must be considered
to accurately assess hemodynamics. Further investigation is needed to determine the extent of
correlation between these and clinical outcomes.
8.9 Limitations
The scope of flow conditions analyzed in this study does not encompass the entirety of potential
scenarios. We also did not account for the effects of respiration on pulmonary blood flow or the
effects of vascular compliance. Further limitations of our study are simplifications of the
simulation conditions such as a constant cardiac index, rigid solid wall and a simplified inlet profile
(plug flow with extension tubes). Although the plug profile with extension is an established

223
assumption in patient specific simulations [257, 265] [261], the impact of such a simplification on
the hemodynamic quantities may be significant in some scenarios [269]. Future investigations
should aim to incorporate more dynamic and varying flow conditions by incorporating patient
specific flow rate waveform and profiles, consider non-rigid body dynamics, and thoroughly
explore the influences of respiration to provide a more comprehensive understanding of the subject
matter. Since the patients included in this study did not have blood viscosity measurements, we
instead used average viscosity measurements from a historical cohort of Fontan patients who were
on average 4 years older than the patients in this study. Among the historical cohort there was a
moderate correlation between viscosity and age. Therefore, the differences we detected in this
study between the Newtonian and non-Newtonian simulations might have been slightly larger than
what would have been measured had we been able to use patient-specific viscosity measurements.
8.10 Conclusions
Our investigation illuminated significant variances in flow structures when comparing Newtonian
and non-Newtonian models. The study demonstrates that non-Newtonian simulations consistently
exhibit significantly higher viscosity compared to Newtonian models with potential implications
for thrombosis in certain patients. Non-Newtonian effects also manifested in elevated power loss,
indicating that using the conventional Newtonian blood assumption might lead to a significant
underestimation of both power loss and viscous dissipation and, consequently, overestimation of
exercise capacity. Furthermore, non-Newtonian behavior influenced hepatic blood flow
distribution in a patient-specific manner. This comprehensive exploration underscores the
imperative consideration of non-Newtonian effects in the optimization of the Fontan circulation.

224
In essence, our research substantiates the crucial need to transcend the confines of Newtonian
assumptions, emphasizing the necessity for a more accurate approach that integrates nonNewtonian effects. Incorporating these factors into the optimization strategies for the Fontan
procedure stands to significantly improve patient outcomes and refine our understanding of
hemodynamics in complex circulatory systems.

225
Chapter 9: Hemodynamically efficient artificial right atrium design
for univentricular heart patients
This chapter is based on the following published manuscript: Wei, H., Herrington, C. S.,
Cleveland, J. D., Starnes, V. A., & Pahlevan, N. M. (2021). Hemodynamically efficient artificial
right atrium design for univentricular heart patients. Physical Review Fluids, 6(12), 123103. DOI:
10.1103/PhysRevFluids.6.123103
Utilizing such non-Newtonian fluid solver, we also investigated the existence of a
hemodynamically optimized geometry for an artificial right atrium (ARA) that can act as a
reservoir for circulatory support and may potentially lessen the risk of implementing biventricular
supports in Fontan patients. We theorized that an implantable artificial right atrium (ARA), if
appropriately designed and implanted in place of the Fontan graft, provides a reservoir for blood
and allow for full circulatory support. An ideal artificial atrium should be clinical feasible,
adjustable in volume (to account for changes in patient body surface area), compatible with all
commercially available VADs, and possess efficient intracavitary flow (to prevent blood
stagnation and clotting). Also, it should decompress (depressurized) congested cerebral veins and
hepatic veins. This decompression is clinically important since it can alleviate hepatic congestion,
protein loosing enteropathy, and liver cirrhosis. Our goal in this study was to identify the
hemodynamically optimized geometry for an artificial right atrium suitable for Fontan Circulation.
Our target hemodynamics was based on 1- minimizing particle resident time (PRT) and 2-
preserving total volume compliance. In this work, we treated blood as a non-Newtonian fluid
(using Carreau-Yasuda model), and employed a computational approach using Lattice Boltzmann
method (LBM; for the fluid domain) and Immersed Boundary Method (IBM; for the solid domain).
Our results indicate that a hemodynamically optimum shape for an ARA is convex at the outlet
and concave at the inlets that resembles the healthy anatomical right atrium [108].

226
Figure 9.1. Schematic of the artificial right atrium (ARA) circuit. The superior and inferior vena cava (SVC and IVC) are
the inlets, and the pulmonary artery (PA) is the outlet. Arrows indicate the blood flow direction. Figure from Wei H et al.
Physical Review Fluids 6.12 (2021): 123103.
Infants born with single ventricle physiology pose a significant challenge for mechanical
circulatory support. The Fontan circulation consists of passive blood flow returning to the lungs
via direct connections to the pulmonary artery without a pumping chamber to generate flow. As
these patients age they tend to develop heart failure and may require heart transplant and
mechanical assist. Mechanical assist devices have revolutionized cardiac transplantation by
enabling patients to safely wait for a heart to become available and as they wait, recover from their
heart failure sequela to be healthier candidates for transplantation. These patients frequently need
bi ventricular support; support for both the systemic and pulmonary circulation. Mechanical
support requires a reservoir to attach to and there is no blood reservoir in the closed Fontan
circulation. We propose the design for an artificial right atrium (ARA) that would assist in
supporting single ventricle patients who have reached the Fontan stage of repair and have
developed subsequent heart failure. At the time of surgery for mechanical support the Fontan
gortex conduit is removed and the ARA is implanted in its place, thus providing a reservoir for
blood and allow for full circulatory support. Non-Newtonian Fluid-Structure Interaction (FSI)

227
simulations employing Lattice-Boltzmann and immersed boundary method were utilized to
evaluate the optimum geometrical design of ARA. Our results indicate that the optimized design
should resemble the atrium of a healthy human (with a convex outlet and a concave inlet). With a
minimized particle residence time the ARA is able to reduce the reduce the chance of blood
stagnation and clotting and also provide suitable volume compliance to prevent cerebral/hepatic
over pressurization.
9.1 Chapter introduction
Within the spectrum of surgical management for congenital heart diseases (CHD; occurrence
varies between 4 and 50 patients per 1000 live births [270]), infants born with single-ventricle
physiologies experience significant challenges throughout their lives [271]. Burdened with a
univentricular heart (either a left or a right ventricle only), these patients are palliated through a
series of three-staged surgical operations with the goal of improving systemic and pulmonary
blood circulation. These operations are called Norwood [272], Glenn [273], and Fontan [274],
which, together, are designed to dedicate the only ventricle to the systemic circulation (Norwood)
and to recreate the pulmonary circulation through direct communication with the pulmonary
arteries (Glenn and Fontan).
Due to recent advances in surgical techniques and patient care, more patients survive through all
three stages [275] than in the past. However, since univentricular hearts are hemodynamically less
efficient, these patients develop diastolic and/or systolic dysfunction. Eventually, they become
heart transplant candidates as their level of heart failure progresses. Severe comorbidities occur in
half of Fontan patients [53, 276, 277], and Fontan patients who survive the Fontan surgery have
around a 20% mortality rate by the age of 20. Given the national shortage of available donor organs

228
and correspondingly extended wait-list times, heart transplantation remains a limited treatment
option (the number of donor hearts available for children is fixed at around 500 per year) [278].
Many institutions have attempted to support these patients with standard ventricular assist devices
(VADs) until they can be given a transplant [279-282]. The primary difficulty in establishing
mechanical support for Fontan concerns the lack of a blood reservoir for the pulmonary circulation
in such diseased conditions. Several attempts have been made to recreate a tissue atrium but,
unfortunately, they have uniformly failed [283, 284].
We theorized that an artificial right atrium (ARA), if appropriately designed and implanted in place
of the Fontan graft, provides an adequate reservoir for blood that subsequently improves
circulatory support. An ideal ARA should be clinically feasible; adjustable in volume (to account
for changes in patient body surface area); compatible with all commercially available VADs; and
possess efficient intracavitary flow (to prevent blood stagnation and clotting). Furthermore, it
should decompress (depressurize) congested cerebral and hepatic veins. This decompression is
clinically important since it can alleviate hepatic congestion, protein-losing enteropathy, and liver
cirrhosis. Our goal in this study was to identify a hemodynamically optimized geometry for an
artificial right atrium to be potentially employed for Fontan circulation. Our target hemodynamic
characteristics were based on (1) minimizing particle residence time (PRT) and (2) preserving total
volume compliance. In the present work, we treated blood as a non-Newtonian fluid (using the
Carreau-Yasuda model) and employed a computational approach employing the lattice-Boltzmann
method (LBM; for the fluid domain [207]) coupled with an immersed boundary method (IBM; for
the solid domain [285]).

229
9.2 Methods
9.2.1 Flow solver
The lattice Boltzmann Method (LBM):
LBM uses simplified kinetic equations combined with a modified molecular-dynamics approach
in order to simulate fluid flows. In essence, it is a methodology that is an alternative to the
conventional continuum-based computational fluid dynamics (CFD) methods that use NavierStokes equations. The accuracy and usefulness of the LBM approach are well established for
various fluid dynamics problems including turbulence [208], multiphase flow [76], and
hemodynamics [72]. As highlighted in previous studies, LBM methods have been shown to be
particularly valuable for simulating hemodynamics [72, 76, 208, 209].
In such a method, the synchronous motions of the particles on a regular lattice are enforced through
a particle distribution function. This distribution function enforces mass and momentum
conservation. It also ensures that the fluid is Galilean invariant and isotropic [210]. In the present
work, a single-relaxation-time (SRT) incompressible lattice-Boltzmann method was used to solve
the incompressible blood flow [211]. The evolution of the distribution functions on the lattice was
governed by the discrete Boltzmann equation employing the BGK (Bhatnagar-Gross-Krook)
collision model:
𝑓𝑖
(𝑥 + 𝑒𝑖Δ𝑡,𝑡 + Δ𝑡) − 𝑓𝑖
(𝑥,𝑡) = −
1
τ
[𝑓𝑖
(𝑥,𝑡) − 𝑓𝑖
𝑒𝑞(𝑥,𝑡)] + Δ𝑡𝐹𝑖
, (9.1)
where 𝑓𝑖(𝑥,𝑡) is the distribution function of the particles in phase space, 𝑒𝑖
is the discrete velocity
at position 𝑥 and time 𝑡, 𝜏 is a non-dimensional relaxation time and 𝑓𝑖
𝑒𝑞 is the equilibrium

230
distribution function. Here, i=0, 1, …, 18 since a D3Q19 (19 discrete velocity vectors) stencil was
applied.
The non-dimensional relaxation time, 𝜏, is related to fluid viscosity 𝜇 as
μ = ρν = ρ𝑐𝑠
2 (τ −
1
2
) Δ𝑡, (9.2)
where 𝜈 is the kinematic viscosity, 𝜌 is the incompressible fluid density (e.g. blood density), and
𝑐𝑠 =
∆𝑥
∆𝑡√3
is the lattice sound speed. ∆𝑡 and ∆𝑥 are time step and lattice space, respectively. In this
study, we used uniform discretization and set ∆𝑥
∆𝑡
= 1.
The equilibrium distribution function, 𝑓𝑖
𝑒𝑞, for incompressible Lattice Boltzmann model [211] and
the forcing term (𝐹𝑖
) [286] are defined as
𝑓𝑖
𝑒𝑞 = ω𝑖ρ0 + ω𝑖ρ [
𝑒𝑖
⋅ 𝑣
𝑐𝑠
2 +
(𝑒𝑖
⋅ 𝑣)
2
2𝑐𝑠
4 −
𝑣
2
2𝑐𝑠
2
], (9.3)
𝐹𝑖 = (1 −
1
2τ
) ω𝑖 (
𝑒𝑖 − 𝑣
𝑐𝑠
2 +
𝑒𝑖
⋅ 𝑣
𝑐𝑠
4
𝑒𝑖) ⋅ 𝑓, (9.4)
where 𝜔𝑖
is the weighting factor (1/3 for i=1, 1/18 for i=2 to 7, and 1/36 for the rest [211]), 𝜌0 is
related to the microscopic pressure by 𝜌0 =
𝑃
𝑐𝑠
2
and velocity 𝑣 can be calculated by
ρ𝑜 = ∑𝑓𝑖
𝑖
, (9.5)
ρ𝑣 = ∑m𝑒𝑖
𝑖
𝑓𝑖 +
1
2
𝑓Δ𝑡. (9.6)

231
Figure 9.2 Viscosity of the shear-thinning Carreau-Yasuda model matching with Fontan
patient specific data [52].
Fluid viscosity model
The significance of non-Newtonian effects in Fontan circulation has been shown in previous
studies [54, 287]. In our simulations, blood was thusly modeled as a non-Newtonian fluid using
the Carreau-Yasuda model, which has been widely used to account for the shear-thinning behavior
of blood in hemodynamics simulations [209] [52]. The model was curve fitted with Fontan
patientspecific data provided by a prior study from Cheng et al. [52], as depicted in Fig. 9.2.
The apparent viscosity of the Carreau-Yasuda model is given by
μ(γ̇) = μ∞ + (μ𝑜 − μ∞)[1 + (λγ̇)
𝑎]
𝑛−1
𝑎 , (9.7)

232
where 𝛾̇ is the shear rate, 𝜇0 is the zero shear viscosity, and 𝜇∞ in the Newtonian viscosity (when
the shear rate goes to infinity). In this model, 𝜆 (time constant), 𝑎, and 𝑛 (power-law index) are
empirically determined constant parameters. The main advantage of the Carreau-Yasuda model is
that it is continuous for all 𝛾̇ ≥ 0.
In our simulations 𝜇0 = 56 𝑐𝑃 (1 𝑐𝑃 = 0.001 𝑃𝑎 𝑠), 𝜇∞ = 3.5 𝑐𝑃 , 𝜆 = 3.133 𝑠 , 𝑎 = 2, and
𝑛 = 0.3568 were obtained from patient specific data as Provided by Cheng et al [220].
In our study, the shear-rate-dependent effect of non-Newtonian blood flow was implemented into
the LBM using Eq.9.7 with Eq.9.2 where the microscopic relaxation time and the macroscopic
fluid viscosity are coupled [209].
LBM Algorithm
The core non-Newtonian LBM algorithm consists of a cyclic sequence of sub steps, with each
cycle corresponding to one timestep:
1. Compute the macroscopic moments 𝜌0 and 𝒗 from 𝑓𝑖 via Eq. 9.5 and 9.6.
2. Obtain the equilibrium distribution 𝑓𝑖
𝑒𝑞 from Eq.9.3.
3. Obtain the apparent viscosity 𝜇 at each point by Eq.9.7 and calculate relaxation time 𝜏 locally
from Eq. 9.2.
4. Perform collision (relaxation) and streaming (propagation) to update 𝑓𝑖 via Eq. 9.1.
Further details about the LBM algorithm can be found in previous publications [209] [211] [216].

233
9.2.2 Solid solver
Compliant wall modeling
To describe the deformation of the compliant wall in a Lagrangian coordinate system, the structure
equation was considered as,
ρ𝑠ℎ
∂
2𝑋
∂𝑡
2 =
∂
∂𝑠
[𝐸ℎ (1 − (
∂𝑋
∂𝑠
⋅
∂𝑋
∂𝑠
)
−1/2
)
∂𝑋
∂𝑠
−
∂
∂𝑠
(𝐸𝐼
∂
2𝑋
∂𝑠
2
)] + 𝐹𝐿
, (9.8)
where 𝜌𝑠
is the density of the solid wall, ℎ is the thickness, 𝑠 is the Lagrangian coordinate along
the solid wall, 𝑋 is the position of the solid wall, and 𝐹𝐿
is the Lagrangian force exerted on the
solid wall by the fluid. 𝐸ℎ and 𝐸𝐼 are the stretching and bending stiffness, respectively.
The boundary condition of the solid wall at a free head as
1 − (
𝜕𝑿
𝜕𝑠 ⋅
𝜕𝑿
𝜕𝑠 )
−1/2
= 0,
𝜕
2𝑿
𝜕𝑠
2 = (0,0),
𝜕
3𝑿
𝜕𝑠
3 = (0,0), (9.9)
and at a fixed end is
𝑋 = 𝑋𝑜,
∂𝑋
∂𝑠
= (−1,0). (9.10)
The reference quantities such as fluid density 𝜌, velocity 𝑈∞ = 𝑄/𝐴 (A is the area of SVC) and
length 𝐿 = 𝐷 (D is the diameter of SVC) were chosen to non-dimensionalize the above
formulations. The non-dimensional parameters considered in our simulations are the Reynolds
number 𝑅𝑒 = 𝜌𝑈∞𝐿/𝜇 , the bending coefficient 𝐾 = 𝐸𝐼/𝜌𝑈∞
2 𝐿
3
, the tension coefficient 𝑆 =
𝐸ℎ/𝜌𝑈∞
2 𝐿, and the mass ratio of the solid wall to the fluid 𝑀 = 𝜌𝑠ℎ/𝜌𝐿.

234
The solid deformation equation (8) was solved by nonlinear Finite Element Method (FEM) solver
as described by Doyle [94] where the large-displacement deformation problem was handled by corotational scheme. The detailed description of the finite-element method can be found in [94].
Briefly, the three-node triangular element describes the deformations where each node has six
degrees of freedom (three displacement and three angles of rotation) [288]. The large-displacement
and small-strain deformation in the structural solver was handled using co-rotational scheme [289].
Iterative strategy was used for the time stepping of the nonlinear system of algebraic equations to
ensure a second-order accuracy.
Fluid Solid Coupling
The immersed boundary (IB) method [285] was used to couple the LBM and nonlinear FEM [145,
290]. This method has been extensively used to simulate the fluid structure interaction (FSI)
problems in cardiovascular biomechanics [54]. The body force term 𝑓 in Eq.4 was used as an
interaction force between the fluid and the boundary to enforce the no-slip velocity boundary
condition at the FSI boundary. The Lagrangian force between the fluid and structure 𝐹𝐿 was
calculated by the penalty scheme [285],
𝐹𝐿 = α [∫ [𝑉𝑓(𝑠,𝑡
′) − 𝑉𝑠(𝑠,𝑡
′)]d
𝑡
0
𝑡
′] + β[𝑉𝑓
(𝑠,𝑡) − 𝑉𝑠
(𝑠,𝑡)], (9.11)
where 𝛼 and 𝛽 are negative large penalty parameters selected based on previous studies [290]
[291]. 𝑉𝑠 = 𝜕𝑋/𝜕𝑡 is the velocity of Lagrangian material point of the solid wall and 𝑉𝑓 is the fluid
velocity at the position 𝑋 obtained by
𝑉𝑓
(𝑠,𝑡) = ∫ m𝑣(𝑥,𝑡)δ(𝑥 − 𝑋(𝑠,𝑡))d𝑥 . (9.12)

235
Here, v(x, t) is the fluid velocity at the fluid domain x. The body force 𝑓 on the Eulerian points can
be calculated using:
𝑓(𝑥,𝑡) = − ∫ m𝐹𝐿
(𝑠,𝑡)δ(𝑥 − 𝑋(𝑠,𝑡))d𝑠 , (9.13)
where 𝛿(𝑥 − 𝑋(𝑠,𝑡)) is a Dirac delta function.
The above Eulerian body force 𝑓 and Lagrangian interaction force 𝐹𝐿 were explicitly obtained by
penalty IB strategy and were included in Eqs.9.4 and 9.8, respectively [285].
The overall numerical strategy employed in this study has been successfully applied to a wide
range of FSI problems [290] [291] [246] [247], including those concerning the dynamics of fluid
flow over a circular flexible plate [290] and the dynamics of an inverted flexible plate [291]. A
summary of the complete algorithm, including the FSI coupling procedure, is provided in Fig. 9.3.

236
Figure 9.3 A summary of the overall algorithm for the complete solver. FSI: fluid-solid interaction; IBM: immersed
boundary method; LBM: Lattice-Boltzmann method; FEM: finite element method.

237
Figure 9.4 Schematic of the artificial right atrium (ARA) circuit. The superior and inferior vena cava (SVC and IVC) are
the inlets, and the pulmonary artery (PA) is the outlet. Bold arrows indicate the net direction of blood flow.
9.2.3 Hydraulic circuit configuration and dimensions for an artificial right atrium
The overall dimensions of the artificial right atrium (ARA) including inlet and outlet were taken
within physiological range [53] [52]. Specifically, the diameters of the superior and inferior cava
(SVC and IVC) were set to 𝐷𝑖 = 𝐷 = 1.2 cm, and the diameters of the pulmonary artery outlet
were set to 𝐷𝑜 = 0.75𝐷 = 0.9 cm (Fig. 9.4).
9.2.4 Shape-selection procedure for hemodynamic optimization
The overall procedure for identifying a hemodynamically optimum shape for an ARA with
minimum PRT (relative to the cases considered in this study) is summarized in Fig. 9.5. We
assumed that the geometry of the long axis plane (LAP: the center plane with inlets and outlet; see
Fig. 9.4) contributes the most to a particle’s residence time. Therefore, our first step was to study
the hemodynamics of flow in LAPs (of various shapes) using 2D models in which the cross-

238
sectional area was preserved. We studied four relevant configurations of LAPs: sphere-like convex
on both sides (LAP1); flat at the outlet but convex on the opposite side (LAP2); convex at the
outlet and moderately concave on the opposite side (LAP3); and convex at the outlet and a concave
curvature with a large radius (compared to LAP3) at the opposite side (LAP4). We additionally
studied a T junction without stagnation as a reference for the minimum PRT. We did not consider
cases where the outlet curvature is concave since the resulting outflow would be behind a
stagnation point (and trivially produce the largest PRT). The schematics of all the considered 2D
cases are summarized and illustrated in Fig. 9.6. With such a 2D-LAP approach, we were able to
identify the hemodynamically optimized LAP (LAP-opt) in which PRT was minimized.
Such 2D configurations were then extruded for 3D compliant FSI models of the ARA to further
validate and study the significance of LAP geometry as well as to ensure that the geometry
associated with LAP-opt does indeed yield a minimum PRT while preserving a high compliance
value. The schematics of all the corresponding 3D compliant cases are summarized and illustrated
in Fig. 9.7. Note that Case 1 and Case 2 are derived from the shapes of LAP1 and LAP2,
respectively, whereas Case 3 and Case 4 employ the shape of LAP3 in their long-axis planes.

239
Figure 9.5 Flow chart for the overall procedure of identifying hemodynamically optimum ARA shapes with minimum PRT
(relative to the cases considered in this study). ARA: artificial right atrium; LAP: long-axis plane; PRT: particle residence
time.
9.2.5 Boundary Conditions
The extension tube boundary models were used at inlets and outlets [217] with specified steady
flow 𝑄 = 1.5 L/min with flat velocity profile at the inlet of the boundary models and constant
pressure (𝑝0 = 10 mmHg) for the outlet of the boundary model. The non-slip condition was
imposed at the solid-fluid interface. For the compliant wall we considered linear elastic material
with Young's modulus of E=0.5 MPa and thickness of h=1mm).
9.2.6 Numerical Simulations
Simulations were performed for 6 hemodynamic cases with different LAP shapes in 2D (to identify
the optimum LAP) and 5 ARA with different shapes in 3D. In our study, we fixed the SVC/IVC
flow distribution to be 50/50. Mesh size 𝐷/∆ 𝑥 = 40 was used in the simulations. Mesh

240
independence studies were performed to ensure the convergence of the calculations. In our study,
Reynolds number (Re) was 378 at outlet and 189 at each inlet, where Re defined as
𝑅𝑒 =
ρ𝑈ref𝐿ref
μ
, (9.14)
where 𝑈𝑟𝑒𝑓 = 𝑄/𝐴, 𝐿𝑟𝑒𝑓 = 𝐷.
Figure 9.6 Schematics of 2D rigid cases used for identifying and evaluating LAP-opt. The superior and inferior cava (SVC
and IVC) are the inlets with diameter 𝑫𝒊 = 𝑫 = 𝟏. 𝟐 𝒄𝒎. The pulmonary artery (outlet) are the outlets with diameter 𝑫𝒐 =
𝟎. 𝟕𝟓𝑫 = 𝟎. 𝟗 𝒄𝒎. The curvature for LAP1 is D at both concave and convex side, the curvature for LAP2 is 2.0D at the
convex side, and the curvature for LAP3 is 2.5D at the convex and 0.5D at the concave side etc. Bold arrows indicate the
direction of fluid flow.

241
Figure 9.7 Schematics of the corresponding 3D compliant configurations. Cases 1, 2, and 3 have the same total volume. Case
4 is the reference case for the minimum particle residence time.
9.2.7 Hemodynamic analysis
Two clinically relevant fluid dynamics criteria were considered to identify the optimum ARA
geometry: 1-particle residence time (PRT) and 2- compliance. Minimum PRT is desirable since it
reduces the change of blood stagnation and clot formation, and maximum compliance is beneficial
since it reduces the chance of over pressurizing hepatic and/or cerebral veins.
Particle residence time (PRT)
The PRT were computed using below equations [210]:
∂τ𝑝
∂𝑡
∣𝑥+ 𝑢 ⋅ τ𝑝 = 𝐻(𝑥), (9.15)
Here, H(x) is the Heaviside function, 𝜏𝑝 is the total time that a particle, occupying a spatial position
𝑥 at time 𝑡, spent in the subdomain Ω. The source term given by the Heaviside function ensures
that the time is accumulated only when the particle is inside the subdomain.

242
The space average 𝑃𝑅𝑇𝑎𝑣𝑒 within target area was calculated using below equation (9.16) [210]:
𝑃𝑇𝑅𝑎𝑣𝑒 =
1
Ω
∫ 𝐻(𝑥)τ𝑝
(𝑥,𝑡)𝑑Ω . (9.16)
Based on the definition above, the PRT quantifies how much time fluid particles have spent at a
certain location after entering from the inlet. The value of the PRT is zero at the inlet since flow is
downstream and the fluid particles at the inlet have just entered and could not have spent any time
there. PRT is larger in stagnation regions since these particles have remained in the ARA chamber
for a longer period of time since entering the ARA domain.
Volume compliance evaluation
We compared cases at the same total volume to ensure they all had similar base-level total volume
compliance. Since geometry can affect the volume compliance of fluid flow, we evaluated the
compliance for each chamber using the differentiation principle 𝐶 =
∆𝑉
∆𝑃
, where we added a known
volume (∆V) of the blood into the chamber models and computed corresponding pressure changes
(∆P). The injected volume ∆V was chosen to be the same as the volume change in the steady flow
simulation of the ARA (around 10% of the total ARA volume). This ensures that the computed
compliance is close to the operating compliance of the ARA during flow simulations. The
compliance values were compared against a spherical chamber with the same volume as the control
case and whose compliance can be determined analytically as of 𝐶 =
∆𝑉
∆𝑃
=
2𝜋𝑅
4
𝐸ℎ
, where R is the
radius of the sphere.

243
9.3 Results
9.3.1 Hemodynamically optimized long-axis plane (LAP)
Table 9.1 presents the average particle residence time (PRTave) for each of the LAP cases (and Tjunction reference case) as summarized by Fig. 9.6. The results of Fig. 9.8 illustrate PRT
distributions for cases LAP1–LAP4. LAP3 demonstrates the lowest PRT whereas LAP2
demonstrates the highest (corresponding velocity distributions of each case are provided in the
online supplementary materials).
9.3.2 3D compliant artificial right atrium (ARA)
Averaged particle residence time (PRTave)
Figure 9.9 presents the PRT distributions for the four cases of 3D compliant ARAs in their longaxis planes. Table 9.2 provides the corresponding values of PRTave, including those of the spherical
control case. Percentage improvements from the sphere were defined relative to the baseline
improvement of Case 4 (the model without an out-of-plane bulge, i.e., the lowest compliance) by
the expression (PRTSphere − PRT)/(PRTSphere − PRTCase 4 ) × 100. Our results indicate that Case 3
(the model corresponding to LAP-opt) has the lowest PRT among ARAs with similar total volume,
and that the sphere has (expectedly) the highest PRT.
Table 9-1 The averaged PRT determined from 2D simulations of LAP1–LAP4. LAP1–LAP4 have similar LAP areas. The
T-junction configuration is the reference case for minimum PRT. PRT: particle residence time (PRT); LAP: long-axis
plane.

244
Figure 9.8 Qualitative visualization of Particle Resident Time (PRT) for 2D simulations of LAP1-4 All cases have similar
LAP area.

245
Figure 9.9 Particle Resident Time (PRT) distribution in long axis plane (LAP) of compliant ARA models (3D FSI
simulations). Cases 1-3 have the same total volume. Case 4 has the same LAP as case 3 but without out of plane bulge (see
Fig. 9.7).

246
Table 9-2 The average particle residence time (𝑷𝑹𝑻𝒂𝒗𝒆) for 3D compliant models. Case 1-3 have the same total Volume.
9.3.3 Qualitative flow analysis
Fig. 9.10 and 9.11 shows the velocity magnitude (nondimensionalized by ‖𝑣‖
𝑈𝑟𝑒𝑓
) and streamlines for
3D compliant models in long axis and short axis planes respectively. The pressure distributions
(nondimensionalized by 𝑃
𝜌𝑈𝑟𝑒𝑓
2
) in long axis planes are shown in Fig. 9.12. As expected, case 4 has
the highest overall pressure due to its low volume (low compliance).

247
Figure 9.10 Velocity magnitude (nondimensionalized by ‖𝒗‖
𝑼𝒓𝒆𝒇
) and streamlines in the long axis planes (LAPs) of compliant
models. Cases 1-3 have the same overall volume. Case 4 has the same LAP as case 3, but it has lower volume compared to
others (see Fig. 9.7 d).

248
Figure 9.11 Velocity magnitude (nondimensionalized by ‖𝒗‖
𝑼𝒓𝒆𝒇
) and streamlines in the long axis planes (LAPs) of compliant
models. Cases 1-3 have the same overall volume. Note that Case 4 has the same LAP as case 3.

249
Figure 9.12 Pressure distributions (nondimensionalized by 𝑷
𝝆𝑼𝒓𝒆𝒇
𝟐
) in long axis planes of compliant 3D models. As expected,
case 4 has the highest overall pressure due to its low volume (low compliance).

250
9.3.4 Evaluation of the total volume compliance
Table 9.3 presents compliance values and net pressure rises for the 3D compliant ARA models. It
also provides both analytically and numerically computed values corresponding to the trivial case
of a sphere. The small differences observed for the sphere between analytical and numerical
compliances is due to the added volume and fluid flow in the numerical model. The analytical
solution is an ideal condition that assumes no flow motion exists. The numerical simulation is
based on the added-volume method, which produces a small error in the compliance computation
due to volume increase. As expected, Case 4 produces the lowest compliance (since it does not
contain an out-of-plane bulge) and hence the highest pressure rise. Cases 1–3 have the same
volume as the sphere. Case 3 has the highest compliance and lowest pressure drop among all ARA
cases.
Table 9-3 Compliance (C) (𝒎𝟑/𝑷𝒂 ∙ 𝟏𝟎−𝟏𝟎) and pressure (P) rise values for 3D compliant models. Cases 1-3 have the same
volume as the sphere. % C reduction is calculated as 𝑪𝑺𝒑𝒉𝒆𝒓𝒆 𝑨𝒏𝒂𝒍𝒚𝒕𝒊𝒄𝒂𝒍−𝑪
𝑪𝑺𝒑𝒉𝒆𝒓𝒆 𝑨𝒏𝒂𝒍𝒚𝒕𝒊𝒄𝒂𝒍
. % P difference is calculated as𝑷−𝑷𝑺𝒑𝒉𝒆𝒓𝒆 𝑵𝒖𝒎𝒆𝒓𝒊𝒄𝒂𝒍
𝑷𝑺𝒑𝒉𝒆𝒓𝒆 𝑵𝒖𝒎𝒆𝒓𝒊𝒄𝒂𝒍
.
9.4 Discussion
In this study, we investigated the existence of a hemodynamically optimized geometry for an
artificial right atrium (ARA) design. Our hemodynamics criteria consisted of the minimum particle
residence time (PRT) and maximum volume compliance at a fixed total ARA volume. From a

251
physiological point of view, low PRTs are important since they reduce the probability of blood
clot formation. Volume compliance is also clinically important since low compliance can result in
pressurization of cerebral and hepatic veins. The pressurization (compression) of the liver is of
great clinical concern since it can promote hepatic congestion and liver cirrhosis.
We employed a computational approach where non-Newtonian (Carreau-Yasuda viscosity modelbased) FSI simulations were produced by a lattice-Boltzmann method (LBM) coupled with an
immersed boundary method (IBM). We first studied the shape of the long-axis plane (LAP) in an
ARA by using 2D models to identify the optimum cross-sectional shape of an ARA in which PRT
is minimized (LAP-opt, i.e., LAP3). We then extruded these LAP shapes to 3D compliance models
of ARAs in order to confirm that the ARAs with LAP-opt Cases 3 and 4 have the lowest PRT
when the total volume is preserved. Our results ultimately suggest that a configuration with a
concave curvature at the outlet and a convex curvature at the inlet is associated with minimized
PRT (as demonstrated in Table 9.1). This can be attributed to a reduction in the concentration of
particles behind the stagnation point (see Fig. 9.8). Although this configuration comes with the
expense of a slight increase in PRT at the outlet, it still provides the minimum PRT among all
other relevant LAP configurations. Additionally, our results demonstrate that an LAP with
moderately concave curvature (moderate radius) (0.5D) on the opposite side produces the
minimum PRT in the long-axis plane (i.e., LAP-opt = LAP3).
The results summarized in Table 9.2 suggest that Cases 3 and 4 (ARAs corresponding to LAPopt) do indeed produce the smallest PRTave over the entire ARA volume, even though the
visualizations of Fig. 9.10 and 9.11 indicate in these cases that the concave shapes on the opposite
sides create large vortices near the outlet. These vortices and the stagnation areas are the main

252
causes of PRT increase (Fig. 9.9). However, they occupy significantly less space than those
associated with the nonconcave Cases 1 and 2. Overall, two flow regimes with high PRT exist in
each ARA: (1) the stagnation area and vortices close to the outlet, and (2) the stagnation region
and vortices at the opposite side of the outlet. Based on our results, the contribution of outlet
stagnation regions and vortices on the elevated PRT values is significantly less than those of the
opposite side of the outlet. This is mainly due to the fact that particles trapped around the outlet
have a higher chance of exiting the ARA domain than the fluid particles trapped on the opposite
side of the outlet. It can be observed in Fig. 9.10 that the primary exit pathway for the oppositeside particles (those trapped in stagnation regions or vortices of the opposite side of the outlet) is
through the narrow centerline “strip.” This is the main reason that PRT is observed to be large
around the centerline. In other words, the PRT is high in the narrow centerline stripe since it is
occupied by “old” particles that have been trapped on the opposite side for a long time. This means
that when the stagnation area and vortices are not close to the outlet, blood clotting is more likely
to occur. Therefore, in order to obtain an ARA shape with minimized PRTs, the stagnation region
at the opposite side of the outlet should be minimized (similarly to Case 3). Comparing the
corresponding pressure distributions between Cases 3 and 4 (corresponding to minimal PRTs) in
Fig. 9.12 demonstrates that Case 3 carries the lowest pressure compared to all the other cases. This
suggests that Case 3 (which contains an out-of-plane bulge) could be a suitable configuration for
preventing overpressurization of the IVC (liver) and SVC (brain), in contrast to the large SVC/IVC
pressures observed in Case 4 (due to its low total volume compliance resulting from lack of such
a bulge). Further analysis of the differences in pressure rises produced by various LAP geometries
(Table 9.3) additionally demonstrates that Case 3 provides the largest compliance with the lowest

253
resultant pressure (although the reference spherical case provides better ratios, it is not
hemodynamically efficient since it has the highest PRTs).
Overall, both qualitative and quantitative comparisons suggest that LAP3, with a convex curvature
at the outlet and a concave curvature at the inlet, produces the minimum PRT (LAP-opt), and that
the 3D compliant ARA associated with LAP3 that contains a bulge (Case 3) is the preferable design
configuration with the highest volume compliance. Interestingly, the most hemodynamically
efficient design suggested by our results is similar in shape to a healthy anatomical right atrium.
Figure 9.13 Uniform flat velocity profile at the inlet of the extension tube boundary model (left) and the corresponding fully
developed velocity profile at the entrance of the ARA (right).

254
Figure 9.14 Velocity magnitudes (nondimensionalized by v/Uref) for the 2D LAP cases.

255
Figure 9.15 PRT on short-axis planes at centerline (Cases 1 to 4 from left to right).
9.5 Limitations
One notable limitation of this study is that only steady flow was considered. There is no right
ventricle in the Fontan circulation. In the absence of the hydraulic pulsatile force of the right
ventricle, Fontan circulation is a single “pump” circulatory system with passive, nonpulsatile flow
to the lungs (pulmonary arteries) [292]. Thus, we only consider steady cases since there is no flow
pulsatility except for a negligible pressure pulsation crosstalk that might exist in some patients
between arterial vessels (i.e., aorta and carotids) and large veins (i.e., SVC, IVC, jugular vein).
However, small amplitude flow pulsations may propagate back into the ARA if a pulsatile-type
assist device is connected to the ARA [293] [294]. The effect of the corresponding pulsatile flow
on the performance of our proposed ARA is beyond the scope of this paper and is a subject of
future studies. Although pulsatile flow may not affect the overall findings of this study (i.e., the
best shape for minimization of particle residence time and preservation of compliance), it may
induce different dynamic behaviors including collapse and self-excitation. Future experimental
studies are needed to confirm that our proposed optimal ARA geometry is not suboptimal when it
comes to self-excitation and collapse. We also did not consider the complete material mechanics

256
of the wall (e.g., viscoelasticity, isotropy, and nonlinearity). Future studies are needed to identify
what type of material is best suited for an artificial right atrium.
9.6 Clinical implications
The series of operations used to palliate infants with single-ventricle diseases have improved
outcomes over the last thirty years. Instead of dying in infancy, they now survive until adolescence
or young adulthood, although eventually dying of heart failure with a failing Fontan circulation.
Single-ventricle cardiac diseases encompass hypoplastic left heart syndrome, severe Shone’s
syndrome, hypoplastic right heart syndrome, and severely unbalanced AV canal defects. The
sequence of procedures for repair of all these lesions ends with the Fontan. Ultimately, patients
may become cardiac transplant candidates. Unfortunately, the use of mechanical support for these
patients (while they sit on such transplant waiting lists) has proved to be difficult [281, 283, 284].
The major issue is that Fontan patients lack a right-sided compliance that can be attached to
ventricular-assist devices. In circumstances where both right and left heart support is needed, there
is no standardized way to successfully perform such support, particularly for the right heart. By
removing the Fontan graft and replacing it with a hemodynamically optimized ARA, the required
reservoir can hopefully be provided.

257
Figure 9.16 PRT on short-axis planes at slightly higher (0.2D) than the centerline (Cases 1 to 4 from left to right).
9.7 Conclusions
In this study, we proposed a hemodynamically optimized geometry for an artificial right atrium
(ARA) suitable for Fontan circulation. The implantation of the ARA can potentially lessen the risk
of biventricular support in Fontan patients. Such an ARA can also be connected to currently
available FDA-approved ventricular assist devices. Our results indicate that the overall optimum
shape of an ARA with minimum PRT (relative to the cases considered in this study) is convex at
the outlet (towards the lungs) and concave at the opposite side of the outlet, which not
coincidentally also resembles the actual healthy human right atrium [295-297]. The proposed ARA
geometry can also reduce the risk of thrombosis and blood clotting (via the minimization of the
PRT) and also prevent the pressurization of cerebral and/or hepatic veins in Fontan circulation (via
the preservation of volume compliance).

258
Appendix A: An immersed boundary-lattice Boltzmann method for
porous media
Immersed boundary-lattice Boltzmann method (IB-LBM) is used for the analysis of fluid flow
with moving boundaries in a porous media. This method employs simplified kinetic equations in
conjunction with a modified molecular-dynamics approach, where the motions of particles on a
regular lattice are enforced by distribution functions that enforce mass and momentum
conservation. It also ensures that the fluid is Galilean invariant and isotropic. For particles with
velocity 𝒆𝑖 at position 𝒙 and time t, the distribution functions of then are governed by the
Bhatnagar-Gross-Krook (BGK) collision model through the expressions:
𝑓𝑖
(𝒙 + 𝒆𝑖∆𝑡,𝑡 + ∆𝑡) − 𝑓𝑖
(𝒙,𝑡) = −
1
𝜏
[𝑓𝑖
(𝒙,𝑡) − 𝑓𝑖
𝑒𝑞(𝒙,𝑡)] + ∆𝑡𝐹𝑖
, (𝐴1.1)
where 𝑓𝑖
(𝒙,𝑡) is the distribution function (𝑓𝑖
𝑒𝑞 is the corresponding equilibrium distribution). ∆𝑡
and ∆𝑥 are the time step and lattice space, respectively. 𝐹𝑖
is the forcing term. The lattice speed is
𝑐 =
∆𝑥
∆𝑡
= 1. 𝜏 is a dimensionless relaxation time constant which is associated with fluid viscosity
in the form 𝜇 = 𝜌𝜗 = 𝜌𝑐𝑠
2
(𝜏 −
1
2
)∆𝑡, where 𝜗 is the kinematic viscosity and 𝑐s =
1
√3
𝑐 is the
lattice sound speed. To model incompressible fluid flow in a porous medium governed by the
above equations in the LBM framework. The generalized equilibrium distribution function the
forcing term that include the effect of the porous medium are now defined as
𝑓𝑖
𝑒𝑞 = 𝜔𝑖𝜌0 + 𝜔𝑖𝜌 [
𝒆𝑖
∙ 𝒗
𝑐s
2 +
(𝒆𝑖
∙ 𝒗)
2
2𝜖𝑐s
2 −
𝒗
2
2𝜖𝑐s
2
], (𝐴1.2)

259
𝐹𝑖 = (1 −
1
2𝜏
) 𝜔𝑖 (
𝒆𝑖
∙ 𝑮
𝑐s
2 +
𝒗𝑮: (𝒆𝑖𝒆𝑖 − 𝑐s
2
𝑰)
𝜖𝑐s
4
) , (A1.3)
where 𝜔𝑖
is the weighting factor, 𝜌0 is related to the microscope pressure by 𝜌0 =
𝑝
𝑐𝑠
2
, 𝑮 represents
the total body force due to the presence of a porous medium and other external force fields, and is
given by 𝑮 = −𝐴
𝜖𝜌𝒗
𝐾
+ 𝜖𝒇 + 𝜖𝑱. 𝐴 denote a binary constant discriminating the fluid phase (𝐴 =
0) from the porousmatrix (𝐴 = 1), 𝐾 and 𝜖 are the permeability and the porosity of the porous
medium, 𝒇 is the fluid-solid interaction force density at the Eulerian point, 𝑱 is the additional
buoyant force density cause by the temperature difference at the Eulerian point, and velocity 𝒗 can
be calculated by
𝜌0 = ∑𝑓𝑖
, (𝐴1.4)
𝜌𝒗 = ∑𝒆𝑖𝑓𝑖 +
1
2
𝑮∆𝑡, (𝐴1.5)
In this study, we implemented a mechanical mathematical model that aims to perform small-scale
flux modeling and assess the physical, chemical, and biological constraints of processes that cooccur in Trichodesmium colonies and accurately reproduce empirical measurements of Oxygen,
Ammonium or Nitrate concentrations and fluxes. In this model, the solute concentration within
the fluid filed is governed by an advection-diffusion reaction equation which the biological
activities will be accounting by adding a reaction term:
𝜀
𝜕𝐶
𝜕𝑡 + 𝒖 ∙ (𝛁𝐶) = 𝛁 ∙ (𝐷𝜀𝛁𝐶) + 𝑅𝑟
, (𝐴1.6)

260
where 𝜀, 𝐶,𝐷 represent the porosity, concentration, and diffusion coefficient. 𝑅𝑟
is the reaction
term associated with the consumption of the solutes. The governing equation was also solved by a
Lattice Boltzmann method scheme.
Here the effects of the temperature differences on the flow motion are considered as well. The
buoyancy force term 𝑱 can be formulated by the Boussinesq approximation to depend linearly on
the temperature 𝑇(𝒙,𝑡), i.e
𝑱 = 𝜌𝛽𝑔0
(𝑇(𝒙,𝑡) − 𝑇𝑚)𝒋, (𝐴1.7)
where 𝛽 is the thermal expansion coefficient, 𝑔0 is the acceleration due to gravity, 𝑇𝑚 is the
average temperature, and 𝒋 is the vertical direction opposite to that of gravity. A convection
diffusion LB is coupled with the momentum porous LB where the evolution equation for
temperature is given as follows:
𝑔𝑖
(𝒙 + 𝒆𝑖∆𝑡,𝑡 + ∆𝑡) − 𝑔𝑖
(𝒙,𝑡) = −
1
𝜏𝑇
[𝑔𝑖
(𝒙,𝑡) − 𝑔𝑖
𝑒𝑞(𝒙,𝑡)], (𝐴1.8)
where 𝑔𝑖
(𝒙,𝑡) is the distribution function for the temperature field, 𝜏𝑇 is the relaxation time
constant which is associated with thermal diffusivity in the form 𝜅 = 𝜌𝑐𝑠
2
(𝜏𝑇 −
1
2
)∆𝑡. The
corresponding equilibrium distribution can be defined as:
𝑔𝑖
𝑒𝑞 = 𝜔𝑖𝑇 [1 +
𝒆𝑖
∙ 𝒗
𝑐s
2
]. (𝐴1.9)
Here, the temperature 𝑇(𝒙,𝑡) at each lattice node is then computed by:
𝑇 = ∑𝑔𝑖
. (𝐴1.10)

261
Further details on the lattice Boltzmann method (LBM) can be found in the literature below.
We used our own in-house code to reproduce the results from previous studies which have shown
that Trichodesmium colonies and aggregates are characterized by steep concentration gradients of
internal gases and nutrients. Pressure field and Concentration field for sinking marine aggregates:
𝑅𝑒 = 10, 𝐷𝑎 = 10−3 & 10−6
for literature(left) and present(right).

262
Appendix B: Supplementary Materials for Fontan
Supplementary Table 1. Summary of hemodynamic metrics for all patients. NN: non-Newtonian.
Mode
l Viscosity Phi
Indexed
Phi
Power Loss
(%)
Indexed
Power
Loss
Average
NN
Importanc
e Factor
Global
deviatio
n of
NNIF
LPAQ(H
FD)
1
Newtonia
n 0.000157 0.290636 3.112497 0.49114 1 0
0.43597
1 NN 0.00036 0.657935 6.723742 1.066043 9.702 3.453 0.484107
2
Newtonia
n 1.33E-04 0.3302 2.736 0.6729 1 0
0.806019
2 NN 1.94E-04 0.3814 3.6676 0.9022 6.17 1.735 0.835736
3
Newtonia
n 0.000122 0.350121 1.623179 0.424641 1 0
0.41928
3 NN 0.000179 0.515385 2.315457 0.606339 5.105 1.2 0. 378463
4
Newtonia
n 7.38E-05 0.3821 12.0746 4.2578 1 0
0.593975
4 NN 1.47E-04 0.757 20.7704 7.3241 11.035 3.434 0.577596
5
Newtonia
n 3.42E-04 0.8179 11.1155 2.2305 1 0
0.272144
5 NN 7.23E-04 1.6779 15.2235 3.0642 12.646 4.652 0.291237
6
Newtonia
n 1.87E-04 0.172 3.8 0.4685 1 0
0.510953
6 NN 3.55E-04 0.3241 6.4815 0.799 6.607 2.181 0.648323
7
Newtonia
n 2.47E-04 0.6163 6.9418 1.3115 1 0
0.668852
7 NN 4.59E-04 1.1279 11.6931 2.208 11.394 3.856 0.666785
8
Newtonia
n 2.32E-04 0.3325 9.1918 1.1103 1 0
0.444651
8 NN 3.03E-04 0.4331 11.0991 1.3407 7.361 2.809 0.353018
9
Newtonia
n 5.62E-05 0.2419 8.2834 2.525 1 0
0.671566
9 NN 8.74E-05 0.3739 13.2126 4.0188 7.409 2.606 0.716167
10
Newtonia
n 2.46E-04 1.0235 4.0087 1.3587 1 0
0.621634
10 NN 3.50E-04 1.4589 4.9257 1.6693 4.181 1.133 0.644051
11
Newtonia
n 6.16E-05 0.22955 11.87002 3.38623 1 0
0.27808
11 NN 9.82E-05 0.362341 17.9618 5.120711 8.847 2.853 0.285119

263
12
Newtonia
n 6.48E-05 0.080618 3.391323 0.345565 1 0
0.875159
12 NN 0.00013 0.161214 5.804588 0.591243 10.192 3.011 0.878673
13
Newtonia
n 4.62E-04 1.4745 9.5865 2.9385 1 0
0.467586
13 NN 7.74E-04 2.4582 12.6459 3.8758 6.927 2.185 0.482065
14
Newtonia
n 5.79E-05 0.4508 12.10872 4.252563 1 0
0.601744
14 NN 1.30E-04 0.9711 22.93744 8.084705 13.18 5.216 0.598385
15
Newtonia
n 3.54E-04 0.38 12.7611 1.9947 1 0
0.840487
15 NN 5.60E-04 0.679 16.8481 2.6214 11.819 4.428 0.854448
16
Newtonia
n 2.43E-04 0.7846 6.7654 1.7793 1 0
0.363634
16 NN 4.47E-04 1.431 10.5063 2.7621 12.407 3.704 0.374494
17
Newtonia
n 0.000386 1.089292 9.127237 2.433842 1 0
0.521946
17 NN 0.000889 2.468192 14.13107 3.768091 12.092 3.498 0.521529
18
Newtonia
n 0.000369 1.166623 12.60949 3.13363 1 0
0.43336
18 NN 0.000753 2.34444 17.08899 4.254152 11.711 3.9 0.40883
19
Newtonia
n 4.81E-04 0.1459 5.4937 0.74 1 0
0.61528
19 NN 8.02E-04 0.301 8.559 1.1529 6.893 1.805 0.627113
20
Newtonia
n 2.37E-04 0.4766 3.8344 1.0427 1 0
0.389905
20 NN 3.58E-04 0.7608 5.4497 1.4817 5.434 1.339 0.387669

264
Supplementary Figure 1. Flow structures (streamlines and velocity magnitudes) for patients not shown in Manuscript

265
Supplementary Figure 2. Contours showing wall shear stress for Newtonian and non-Newtonian viscosity models and the
stress difference between two models.

266
Supplementary Table 2. Summary of mean wall shear stress (WSS), mean shear rate, and low WSS area for all patients.
NN: non-Newtonian.
Patient Viscosity Mean WSS (Pa) Mean Shear Rate (s-1
) Low WSS Area (%)
1 Newtonian 1.2 4.15 12.2
1 NN 3.51 3.15 0.44
2 Newtonian 1.05 6.15 18.25
2 NN 2.13 4.43 3.14
3 Newtonian 1.31 10.2 9.66
3 NN 2.29 7.56 2.92
4 Newtonian 1.83 3.3 15.91
4 NN 5.64 2.49 1.21
5 Newtonian 0.93 3.16 35.72
5 NN 2.47 2.48 17.68
6 Newtonian 1.55 7.36 7.78
6 NN 3.61 5.86 1.65
7 Newtonian 1.13 5.28 32.26
7 NN 2.72 4.09 4.25
8 Newtonian 1.2 8.28 22.65
8 NN 2.49 6.02 2.17
9 Newtonian 2.31 6.85 2.25
9 NN 5.29 5.03 0.92
10 Newtonian 1.84 14.36 5.64
10 NN 3.11 11.22 2.91
11 Newtonian 1.97 4.97 3.71
11 NN 5.18 3.53 1.03
12 Newtonian 0.88 4.47 20.6
12 NN 2.3 2.96 7.61
13 Newtonian 1.47 6.67 13.74
13 NN 3.53 5.32 1.48
14 Newtonian 1.52 2.28 7.93
14 NN 5.3 1.59 1.19
15 Newtonian 1.02 4.11 30.15
15 NN 2.73 3.08 8.89
16 Newtonian 0.92 3.95 22.47
16 NN 2.42 2.71 7.22
17 Newtonian 0.99 2.98 26.84
17 NN 3.36 2.34 0.9
18 Newtonian 1.05 3.26 25.93
18 NN 3.5 2.56 2.13
19 Newtonian 1.35 6.13 15.12

267
19 NN 3.27 4.82 0.67
20 Newtonian 1.36 9.93 6.21
20 NN 2.7 7.63 1.22
Supplementary Figure 3. Fluid mesh and the Fontan geometry (patient 1) used in the simulation shown from (a) an isometric
view, and (b) a front-view. The velocity magnitude profile of the centerline of the cross-section in the middle of the Fontan
with different total number of mesh 𝑵𝒕𝒐𝒕𝒂𝒍 .
In our simulations, the fluid domain is a rectangular prism that covers and extends beyond the
region of Fontan (see Figure S3. a). A Mesh sensitivity analysis is done by using the geometry of
Patient 1 (Figure S3. b). The velocity magnitude profiles of the centerline of mid-Fontan crosssection are compared in Figure S3. c. The error (Normalized RMSE) between the finer cases
𝑁𝑡𝑜𝑡𝑎𝑙 = 6.65 × 106
and 𝑁𝑡𝑜𝑡𝑎𝑙 = 3.4 × 106
is negligible ( < 1.2% ). Thus, the mesh
cardinality of 𝑁𝑚𝑒𝑠ℎ = 2000/𝑚𝑙 (corresponding to 𝑁𝑡𝑜𝑡𝑎𝑙 = 3.4 × 106
for Patient 1) is chosen
for the all the simulations considering the accuracy of the simulations and computational
efficiency.

268
References
[1] B. Schaller, "Physiology of cerebral venous blood flow: from experimental data in animals
to normal function in humans," Brain research reviews, vol. 46, no. 3, pp. 243-260, 2004.
[2] A. L. Jefferson et al., "Higher aortic stiffness is related to lower cerebral blood flow and
preserved cerebrovascular reactivity in older adults," Circulation, vol. 138, no. 18, pp.
1951-1962, 2018.
[3] T. T. van Sloten, A. D. Protogerou, R. M. Henry, M. T. Schram, L. J. Launer, and C. D.
Stehouwer, "Association between arterial stiffness, cerebral small vessel disease and
cognitive impairment: a systematic review and meta-analysis," Neuroscience &
Biobehavioral Reviews, vol. 53, pp. 121-130, 2015.
[4] A. Minagar, W. Jy, J. J. Jimenez, and J. S. Alexander, "Multiple sclerosis as a vascular
disease," Neurological research, vol. 28, no. 3, pp. 230-235, 2006.
[5] C. B. Beggs et al., "Aqueductal cerebrospinal fluid pulsatility in healthy individuals is
affected by impaired cerebral venous outflow," Journal of Magnetic Resonance Imaging,
vol. 40, no. 5, pp. 1215-1222, 2014.
[6] C. B. Beggs et al., "Internal jugular vein cross-sectional area and cerebrospinal fluid
pulsatility in the aqueduct of sylvius: a comparative study between healthy subjects and
multiple sclerosis patients," PLoS One, vol. 11, no. 5, p. e0153960, 2016.
[7] M. Mancini et al., "Internal jugular vein blood flow in multiple sclerosis patients and
matched controls," PLoS One, vol. 9, no. 3, p. e92730, 2014.
[8] M. Farina, E. Novelli, and R. Pagani, "Cross-sectional area variations of internal jugular
veins during supine head rotation in multiple sclerosis patients with chronic cerebrospinal
venous insufficiency: a prospective diagnostic controlled study with duplex ultrasound
investigation," BMC neurology, vol. 13, no. 1, pp. 1-12, 2013.
[9] E. Y. Joo, W. S. Tae, S. J. Han, J.-W. Cho, and S. B. Hong, "Reduced cerebral blood flow
during wakefulness in obstructive sleep apnea-hypopnea syndrome," Sleep, vol. 30, no. 11,
pp. 1515-1520, 2007.

269
[10] B. V. Zlokovic, "Neurodegeneration and the neurovascular unit," Nature medicine, vol. 16,
no. 12, pp. 1370-1371, 2010.
[11] M. D'haeseleer, M. Cambron, L. Vanopdenbosch, and J. De Keyser, "Vascular aspects of
multiple sclerosis," The Lancet Neurology, vol. 10, no. 7, pp. 657-666, 2011.
[12] J. S. Alexander, L. Prouty, I. Tsunoda, C. V. Ganta, and A. Minagar, "Venous endothelial
injury in central nervous system diseases," BMC medicine, vol. 11, no. 1, pp. 1-15, 2013.
[13] C. Beggs et al., "Jugular venous reflux and brain parenchyma volumes in elderly patients
with mild cognitive impairment and Alzheimer’s disease," BMC neurology, vol. 13, pp. 1-
8, 2013.
[14] C.-P. Chung and H.-H. Hu, "Jugular venous reflux," Journal of Medical Ultrasound, vol.
16, no. 3, pp. 210-222, 2008.
[15] C. B. Beggs, "Venous hemodynamics in neurological disorders: an analytical review with
hydrodynamic analysis," BMC medicine, vol. 11, pp. 1-17, 2013.
[16] R. Zivadinov, "Is there a link between the extracranial venous system and central nervous
system pathology?," vol. 11, ed: BioMed Central, 2013, pp. 1-4.
[17] P. J. Devine, L. E. Sullenberger, D. A. Bellin, and J. E. Atwood, "Jugular venous pulse:
window into the right heart," Southern medical journal, vol. 100, no. 10, pp. 1022-1027,
2007.
[18] L. Istrail, J. Kiernan, and M. Stepanova, "A novel method for estimating right atrial
pressure with point-of-care ultrasound," Journal of the American Society of
Echocardiography, vol. 36, no. 3, pp. 278-283, 2023.
[19] D. Zhou et al., "Understanding jugular venous outflow disturbance," CNS neuroscience &
therapeutics, vol. 24, no. 6, pp. 473-482, 2018.
[20] C. Beggs et al., "Spectral characteristics of the internal jugular vein and central venous
pressure pulses: a proof of concept study," Veins and Lymphatics, vol. 10, no. 1, 2021.

270
[21] P. Zamboni et al., "Central venous pressure estimation from ultrasound assessment of the
jugular venous pulse," PLoS One, vol. 15, no. 10, p. e0240057, 2020.
[22] S. M. Auñón-Chancellor, J. M. Pattarini, S. Moll, and A. Sargsyan, "Venous thrombosis
during spaceflight," New England Journal of Medicine, vol. 382, no. 1, pp. 89-90, 2020.
[23] K. Marshall-Goebel et al., "Assessment of jugular venous blood flow stasis and thrombosis
during spaceflight," JAMA network open, vol. 2, no. 11, pp. e1915011-e1915011, 2019.
[24] M. M. Elahi, A. N. Witt, E. L. Pryzdial, and P. B. McBeth, "Thrombotic triad in
microgravity," Thrombosis Research, 2023.
[25] U. Limper et al., "The thrombotic risk of spaceflight: has a serious problem been
overlooked for more than half of a century?," European Heart Journal, vol. 42, no. 1, pp.
97-100, 2021.
[26] D. S. Kim et al., "The effect of microgravity on the human venous system and blood
coagulation: a systematic review," Experimental Physiology, vol. 106, no. 5, pp. 1149-
1158, 2021.
[27] M. Simka, P. Latacz, and W. Redelbach, "Blood flow in the internal jugular veins during
the spaceflight–Is it actually bidirectional?," Life Sciences in Space Research, vol. 25, pp.
103-106, 2020.
[28] M. Lan et al., "Proposed mechanism for reduced jugular vein flow in microgravity,"
Physiological Reports, vol. 9, no. 8, p. e14782, 2021.
[29] J. Pavela et al., "Surveillance for jugular venous thrombosis in astronauts," Vascular
Medicine, vol. 27, no. 4, pp. 365-372, 2022.
[30] F. Sisini, E. Toro, M. Gambaccini, and P. Zamboni, "The oscillating component of the
internal jugular vein flow: the overlooked element of cerebral circulation," Behavioural
Neurology, vol. 2015, 2015.
[31] R. W. Brower and A. Noordergraaf, "Pressure-flow characteristics of collapsible tubes: A
reconciliation of seemingly contradictory results," Annals of biomedical engineering, vol.
1, no. 3, pp. 333-355, 1973.

271
[32] S. Rodbard, "Flow through collapsible tubes: augmented flow produced by resistance at
the outlet," Circulation, vol. 11, no. 2, pp. 280-287, 1955.
[33] C. Caro, J. Fitz-Gerald, and R. Schroter, "Atheroma and arterial wall shear-observation,
correlation and proposal of a shear dependent mass transfer mechanism for atherogenesis,"
Proceedings of the Royal Society of London. Series B. Biological Sciences, vol. 177, no.
1046, pp. 109-133, 1971.
[34] J. B. Grotberg and O. E. Jensen, "Biofluid mechanics in flexible tubes," Annu. Rev. Fluid
Mech., vol. 36, pp. 121-147, 2004.
[35] J. Wang, Y. Chew, and H. Low, "Effects of downstream system on self‐excited oscillations
in collapsible tubes," Communications in numerical methods in engineering, vol. 25, no.
5, pp. 429-445, 2009.
[36] R. Kamm and T. Pedley, "Flow in collapsible tubes: a brief review," 1989.
[37] A. I. Katz, Y. Chen, and A. H. Moreno, "Flow through a collapsible tube: experimental
analysis and mathematical model," Biophysical Journal, vol. 9, no. 10, pp. 1261-1279,
1969.
[38] J. Holt, "Flow through collapsible tubes and through in situ veins," IEEE Transactions on
Biomedical Engineering, no. 4, pp. 274-283, 1969.
[39] W. A. Conrad, "Pressure-flow relationships in collapsible tubes," IEEE Transactions on
Biomedical Engineering, no. 4, pp. 284-295, 1969.
[40] H. Low and Y. Chew, "Pressure/flow relationships in collapsible tubes: effects of upstream
pressure fluctuations," Medical and Biological Engineering and Computing, vol. 29, pp.
217-221, 1991.
[41] P. Flaud, C. Oddou, and D. Geiger, "High amplitude wave propagation in collapsible tube.
I.—Relation between rheological properties and wave propagation," Journal de Physique,
vol. 46, no. 5, pp. 691-698, 1985.
[42] O. E. Jensen and M. Heil, "High-frequency self-excited oscillations in a collapsiblechannel flow," Journal of Fluid Mechanics, vol. 481, pp. 235-268, 2003.

272
[43] C. Bertram and M. Sheppeard, "Interactions of pulsatile upstream forcing with flowinduced oscillations of a collapsed tube: mode-locking," Medical Engineering & Physics,
vol. 22, no. 1, pp. 29-37, 2000.
[44] B. Brook, S. Falle, and T. Pedley, "Numerical solutions for unsteady gravity-driven flows
in collapsible tubes: evolution and roll-wave instability of a steady state," Journal of Fluid
Mechanics, vol. 396, pp. 223-256, 1999.
[45] R. J. Whittaker, M. Heil, J. Boyle, O. E. Jensen, and S. L. Waters, "The energetics of flow
through a rapidly oscillating tube. Part 2. Application to an elliptical tube," Journal of fluid
mechanics, vol. 648, pp. 123-153, 2010.
[46] X. Luo, Z. Cai, W. Li, and T. Pedley, "The cascade structure of linear instability in
collapsible channel flows," Journal of Fluid Mechanics, vol. 600, pp. 45-76, 2008.
[47] E. Watson, "Diffusion in oscillatory pipe flow," Journal of Fluid Mechanics, vol. 133, pp.
233-244, 1983.
[48] T. Mizushina, T. Maruyama, S. IDE, and Y. MIZUKAMI, "Dynamic behaviour of transfer
coefficient in pulsating laminar tube flow," Journal of Chemical Engineering of Japan,
vol. 6, no. 2, pp. 152-159, 1973.
[49] G. I. Taylor, "Dispersion of soluble matter in solvent flowing slowly through a tube,"
Proceedings of the Royal Society of London. Series A. Mathematical and Physical
Sciences, vol. 219, no. 1137, pp. 186-203, 1953.
[50] Z. Huang and J. Tarbell, "Numerical simulation of mass transfer in porous media of blood
vessel walls," American Journal of Physiology-Heart and Circulatory Physiology, vol.
273, no. 1, pp. H464-H477, 1997.
[51] C. A. Dragon and J. B. Grotberg, "Oscillatory flow and mass transport in a flexible tube,"
Journal of Fluid Mechanics, vol. 231, pp. 135-155, 1991.
[52] A. L. Cheng, N. M. Pahlevan, D. G. Rinderknecht, J. C. Wood, and M. Gharib,
"Experimental investigation of the effect of non-Newtonian behavior of blood flow in the
Fontan circulation," European Journal of Mechanics-B/Fluids, vol. 68, pp. 184-192, 2018.

273
[53] A. L. Cheng, C. P. Wee, N. M. Pahlevan, and J. C. Wood, "A 4D flow MRI evaluation of
the impact of shear-dependent fluid viscosity on in vitro Fontan circulation flow,"
American Journal of Physiology-Heart and Circulatory Physiology, vol. 317, no. 6, pp.
H1243-H1253, 2019.
[54] H. Wei, A. L. Cheng, and N. M. Pahlevan, "On the significance of blood flow shear-ratedependency in modeling of Fontan hemodynamics," European Journal of MechanicsB/Fluids, vol. 84, pp. 1-14, 2020.
[55] C. A. Taylor and C. A. Figueroa, "Patient-specific modeling of cardiovascular mechanics,"
Annual Review of Biomedical Engineering, vol. 11, pp. 109--134, 2009.
[56] C. A. Taylor and D. A. Steinman, "Image-based modeling of blood flow and vessel wall
dynamics: applications, methods and future directions," Annals of Biomedical
Engineering, vol. 38, no. 3, pp. 1188--1203, 2010.
[57] G. Pennati, F. Migliavacca, G. Dubini, and E. L. Bove, "Modeling of systemic-topulmonary shunts in newborns with a univentricular circulation: state of the art and future
directions," Progress in Pediatric Cardiology, vol. 30, no. 1-2, pp. 23--29, 2010.
[58] I. E. Vignon-Clementel, A. L. Marsden, and J. A. Feinstein, "A primer on computational
simulation in congenital heart disease for the clinician," Progress in Pediatric Cardiology,
vol. 30, no. 1-2, pp. 3--13, 2010.
[59] R. Mittal, S. P. Simmons, and H. S. Udaykumar, "Application of large-eddy simulation to
the study of pulsatile flow in a modeled arterial stenosis," Journal of Biomechanical
Engineering, vol. 123, no. 4, pp. 325--332, 2001.
[60] H. Asgharzadeh and I. Borazjani, "A non-dimensional parameter for classification of the
flow in intracranial aneurysms. I. Simplified geometries," Physics of Fluids, vol. 31, no. 3,
p. 031904, 2019.
[61] H. Asgharzadeh, H. Asadi, H. Meng, and I. Borazjani, "A non-dimensional parameter for
classification of the flow in intracranial aneurysms. II. Patient-specific geometries,"
Physics of Fluids, vol. 31, no. 3, p. 031905, 2019.
[62] J. H. Seo et al., "A method for the computational modeling of the physics of heart
murmurs," Journal of Computational Physics, vol. 336, pp. 546--568, 2017.

274
[63] J. Wu and S. C. Shadden, "Coupled simulation of hemodynamics and vascular growth and
remodeling in a subject-specific geometry," Annals of Biomedical Engineering, vol. 43,
no. 7, pp. 1543--1554, 2015.
[64] F. Amlani and N. M. Pahlevan, "A stable high-order FC-based methodology for
hemodynamic wave propagation," Journal of Computational Physics, vol. 405, p. 109130,
2020.
[65] A. Aghilinejad, F. Amlani, K. S. King, and N. M. Pahlevan, "Dynamic effects of aortic
arch stiffening on pulsatile energy transmission to cerebral vasculature as a determinant of
brain-heart coupling," Scientific Reports, vol. 10, no. 1, pp. 1--12, 2020.
[66] Y. Shi, P. Lawford, and R. Hose, "Review of zero-D and 1-D models of blood flow in the
cardiovascular system," Biomedical Engineering online, vol. 10, no. 1, pp. 1--38, 2011.
[67] D. A. Steinman and C. A. Taylor, "Flow imaging and computing: large artery
hemodynamics," Annals of Biomedical Engineering, vol. 33, no. 12, pp. 1704--1709, 2005.
[68] D. A. Steinman, "Image-based computational fluid dynamics modeling in realistic arterial
geometries," Annals of Biomedical Engineering, vol. 30, no. 4, pp. 483--497, 2002.
[69] K. Timm, K. Halim, K. Alexandr, S. Orest, S. Goncalo, and E. M. Viggen, The Lattice
Boltzmann Method: Principles and Practice (Graduate Texts in Physics). Springer, 2016.
[70] R. Benzi, S. Succi, and M. Vergassola, "The lattice Boltzmann equation: theory and
applications," Physics Reports, vol. 222, no. 3, pp. 145--197, 1992.
[71] X. Shan and H. Chen, "Lattice B oltzmann model for simulating flows with multiple phases
and components," Physical Review E, vol. 47, no. 3, p. 1815, 1993.
[72] H. Fang, Z. Wang, Z. Lin, and M. Liu, "Lattice Boltzmann method for simulating the
viscous flow in large distensible blood vessels," Physical Review E, vol. 65, no. 5, p.
051925, 2002.
[73] K. Sriram, M. Intaglietta, and D. M. Tartakovsky, "Non-Newtonian flow of blood in
arterioles: consequences for wall shear stress measurements," Microcirculation, vol. 21,
no. 7, pp. 628--639, 2014.

275
[74] D. S. Sankar and K. Hemalatha, "A non-Newtonian fluid flow model for blood flow
through a catheterized artery—steady flow," Applied Mathematical Modelling, vol. 31, no.
9, pp. 1847--1864, 2007.
[75] J. A. Cosgrove, J. M. Buick, S. J. Tonge, C. G. Munro, C. A. Greated, and D. M. Campbell,
"Application of the lattice Boltzmann method to transition in oscillatory channel flow,"
Journal of Physics A: Mathematical and General, vol. 36, no. 10, p. 2609, 2003.
[76] X. Shan and H. Chen, "Lattice Boltzmann model for simulating flows with multiple phases
and components," Physical Review E, vol. 47, no. 3, p. 1815, 1993.
[77] J. Boyd, J. M. Buick, and S. Green, "Analysis of the Casson and Carreau-Yasuda nonNewtonian blood models in steady and oscillatory flows using the lattice Boltzmann
method," Physics of Fluids, vol. 19, no. 9, p. 093103, 2007.
[78] H. Wei, A. L. Cheng, and N. M. Pahlevan, "On the significance of blood flow shear-ratedependency in modeling of Fontan hemodynamics," European Journal of MechanicsB/Fluids, 2020.
[79] N. Westerhof, F. Bosman, C. J. De Vries, and A. Noordergraaf, "Analog studies of the
human systemic arterial tree," Journal of Biomechanics, vol. 2, no. 2, pp. 121--143, 1969.
[80] I. E. Vignon and C. A. Taylor, "Outflow boundary conditions for one-dimensional finite
element modeling of blood flow and pressure waves in arteries," Wave Motion, vol. 39, no.
4, pp. 361--374, 2004.
[81] M. E. Moghadam, I. E. Vignon-Clementel, R. Figliola, A. L. Marsden, and et al., "A
modular numerical method for implicit 0D/3D coupling in cardiovascular finite element
simulations," Journal of Computational Physics, vol. 244, pp. 63--79, 2013.
[82] H. J. Kim et al., "On coupling a lumped parameter heart model and a three-dimensional
finite element aorta model," Annals of Biomedical Engineering, vol. 37, no. 11, pp. 2153-
-2169, 2009.
[83] S. Moore, K. Halupka, and S. Zhuk, "Towards RealTime 3D Coronary Hemodynamics
Simulations During Cardiac Catheterisation," 2018.

276
[84] R. Sadeghi, S. Khodaei, J. Ganame, and Z. Keshavarz-Motamed, "Towards non-invasive
computational-mechanics and imaging-based diagnostic framework for personalized
cardiology for coarctation," Scientific Reports, vol. 10, no. 1, pp. 1--19, 2020.
[85] I. E. Vignon-Clementel, C. A. Figueroa, K. E. Jansen, and C. A. Taylor, "Outflow boundary
conditions for three-dimensional finite element modeling of blood flow and pressure in
arteries," Computer methods in applied mechanics and engineering, vol. 195, no. 29-32,
pp. 3776--3796, 2006.
[86] I. E. Vignon-Clementel, C. A. Figueroa, A. L. Marsden, J. A. Feinstein, K. E. Jansen, and
C. A. Taylor, "Outflow boundary conditions for three-dimensional simulations of nonperiodic blood flow and pressure fields in deformable arteries," Journal of Biomechanics,
no. 39, p. S431, 2006.
[87] G. Zhao-Li, Z. Chu-Guang, and S. Bao-Chang, "Non-equilibrium extrapolation method for
velocity and pressure boundary conditions in the lattice B oltzmann method," Chinese
Physics, vol. 11, no. 4, p. 366, 2002.
[88] X. Kang, Q. Liao, X. Zhu, and Y. Yang, "Non-equilibrium extrapolation method in the
lattice Boltzmann simulations of flows with curved boundaries (non-equilibrium
extrapolation of LBM)," Applied Thermal Engineering, vol. 30, no. 13, pp. 1790--1796,
2010.
[89] Y. Wang, C. Shu, C. Teo, and J. Wu, "An immersed boundary-lattice Boltzmann flux
solver and its applications to fluid–structure interaction problems," Journal of Fluids and
Structures, vol. 54, pp. 440-465, 2015.
[90] T. Lee, H. Huang, and C. Shu, "An axisymmetric incompressible lattice BGK model for
simulation of the pulsatile flow in a circular pipe," International journal for numerical
methods in fluids, vol. 49, no. 1, pp. 99-116, 2005.
[91] C. Bilgi and K. Atalık, "Effects of blood viscoelasticity on pulsatile hemodynamics in
arterial aneurysms," Journal of Non-Newtonian Fluid Mechanics, vol. 279, p. 104263,
2020.
[92] W.-X. Huang, S. J. Shin, and H. J. Sung, "Simulation of flexible filaments in a uniform
flow by the immersed boundary method," Journal of Computational Physics, vol. 226, no.
2, pp. 2206--2228, 2007.

277
[93] R.-N. Hua, L. Zhu, and X.-Y. Lu, "Dynamics of fluid flow over a circular flexible plate,"
Journal of Fluid Mechanics, vol. 759, pp. 56--72, 2014.
[94] J. F. Doyle, Nonlinear analysis of thin-walled structures: statics, dynamics, and stability.
Springer Science & Business Media, 2013.
[95] H. Dai, H. Luo, and J. F. Doyle, "Dynamic pitching of an elastic rectangular wing in
hovering motion," Journal of Fluid Mechanics, vol. 693, pp. 473--499, 2012.
[96] C. Tang, N.-S. Liu, and X.-Y. Lu, "Dynamics of an inverted flexible plate in a uniform
flow," Physics of Fluids, vol. 27, no. 7, p. 073601, 2015.
[97] H. Huang, H. Wei, and X.-Y. Lu, "Coupling performance of tandem flexible inverted flags
in a uniform flow," Journal of Fluid Mechanics, vol. 837, pp. 461--476, 2018.
[98] J.-L. Batoz, K.-J. Bathe, and L.-W. Ho, "A study of three-node triangular plate bending
elements," International Journal for Numerical Methods in Engineering, vol. 15, no. 12,
pp. 1771--1812, 1980.
[99] J. F. Doyle, Nonlinear analysis of thin-walled structures: statics, dynamics, and stability.
Springer Science & Business Media, 2001.
[100] B. S. H. Connell and D. K. P. Yue, "Flapping dynamics of a flag in a uniform stream,"
Journal of fluid mechanics, vol. 581, pp. 33--67, 2007.
[101] W.-X. Huang and H. J. Sung, "Three-dimensional simulation of a flapping flag in a uniform
flow," Journal of Fluid Mechanics, vol. 653, pp. 301--336, 2010.
[102] D. S. Berger, J. Li, and A. Noordergraaf, "Differential effects of wave reflections and
peripheral resistance on aortic blood pressure: a model-based study," American Journal of
Physiology-Heart and Circulatory Physiology, vol. 266, no. 4, pp. H1626-H1642, 1994.
[103] N. M. Pahlevan, F. Amlani, M. H. Gorji, F. Hussain, and M. Gharib, "A physiologically
relevant, simple outflow boundary model for truncated vasculature," Annals of biomedical
engineering, vol. 39, no. 5, pp. 1470-1481, 2011.

278
[104] J. Kang, A. Aghilinejad, and N. M. Pahlevan, "On the accuracy of displacement-based
wave intensity analysis: Effect of vessel wall viscoelasticity and nonlinearity," PloS one,
vol. 14, no. 11, p. e0224390, 2019.
[105] H. Huang and X.-Y. Lu, "Theoretical and numerical study of axisymmetric lattice
Boltzmann models," Physical Review E, vol. 80, no. 1, p. 016701, 2009.
[106] G. Zhao-Li, Z. Chu-Guang, and S. Bao-Chang, "Non-equilibrium extrapolation method for
velocity and pressure boundary conditions in the lattice Boltzmann method," Chinese
Physics, vol. 11, no. 4, p. 366, 2002.
[107] Z. Guo, C. Zheng, and B. Shi, "An extrapolation method for boundary conditions in lattice
Boltzmann method," Physics of Fluids, vol. 14, no. 6, pp. 2007--2010, 2002.
[108] H. Wei, C. S. Herrington, J. Cleveland, S. D, Vaughn A, and N. M. Pahlevan,
"Hemodynamically efficient artificial right atrium design for univentricular heart patients,"
Physical Review Fluids , , volume = 6 , issue = 12 , pages = 123103 , numpages = 22,
2021.
[109] C. S. Peskin, "The immersed boundary method," Acta numerica, vol. 11, pp. 479-517,
2002.
[110] R. Mittal and G. Iaccarino, "Immersed boundary methods," Annu. Rev. Fluid Mech., vol.
37, pp. 239-261, 2005.
[111] J. H. Seo, V. Vedula, T. Abraham, and R. Mittal, "Multiphysics computational models for
cardiac flow and virtual cardiography," International journal for numerical methods in
Biomedical Engineering, vol. 29, no. 8, pp. 850--869, 2013.
[112] J. Wu and C. Shu, "Implicit velocity correction-based immersed boundary-lattice
Boltzmann method and its applications," Journal of Computational Physics, vol. 228, no.
6, pp. 1963-1979, 2009.
[113] X. Zhao, Z. Chen, L. Yang, N. Liu, and C. Shu, "Efficient boundary condition-enforced
immersed boundary method for incompressible flows with moving boundaries," Journal
of Computational Physics, vol. 441, p. 110425, 2021.

279
[114] H. Ye, H. Wei, H. Huang, and X.-y. Lu, "Two tandem flexible loops in a viscous flow,"
Physics of Fluids, vol. 29, no. 2, p. 021902, 2017.
[115] E. Guilmineau and P. Queutey, "A numerical simulation of vortex shedding from an
oscillating circular cylinder," Journal of Fluids and Structures, vol. 16, no. 6, pp. 773--
794, 2002.
[116] S. B. David, A. R. Kimberly, and G. S. Sanjeev, "Wave Propagation in Coupled Left
Ventricle-Arterial System," Hypertension, vol. 27, no. 5, pp. 1079 - 1089, 1996.
[117] K. B. Campbell, L. C. Lee, H. F. Frasch, and A. Noordergraaf, "Pulse reflection sites and
effective length of the arterial system," American Journal of Physiology-Heart and
Circulatory Physiology, vol. 256, no. 6, pp. H1684 - H1689, 1989.
[118] M. Gharib and M. Beizaie, "Correlation between negative near-wall shear stress in human
aorta and various stages of congestive heart failure," Annals of Biomedical Engineering,
vol. 31, no. 6, pp. 678--685, 2003.
[119] J. E. Moore Jr, C. Xu, S. Glagov, C. K. Zarins, and D. N. Ku, "Fluid wall shear stress
measurements in a model of the human abdominal aorta: oscillatory behavior and
relationship to atherosclerosis," Atherosclerosis, vol. 110, no. 2, pp. 225--240, 1994.
[120] E. M. Pedersen, H.-W. Sung, A. C. Burlson, and A. P. Yoganathan, "Two-dimensional
velocity measurements in a pulsatile flow model of the normal abdominal aorta simulating
different hemodynamic conditions," Journal of Biomechanics, vol. 26, no. 10, pp. 1237--
1247, 1993.
[121] S. Oyre, E. M. Pedersen, S. Ringgaard, P. Boesiger, and W. P. Paaske, "In vivo wall shear
stress measured by magnetic resonance velocity mapping in the normal human abdominal
aorta," European Journal of Vascular and Endovascular Surgery, vol. 13, no. 3, pp. 263--
271, 1997.
[122] E. M. Pedersen, S. Kozerke, S. Ringgaard, M. B. Scheidegger, and P. Boesiger,
"Quantitative abdominal aortic flow measurements at controlled levels of ergometer
exercise," Magnetic Resonance Imaging, vol. 17, no. 4, pp. 489--494, 1999.
[123] A. Siviglia and M. Toffolon, "Steady analysis of transcritical flows in collapsible tubes
with discontinuous mechanical properties: implications for arteries and veins," Journal of
fluid mechanics, vol. 736, pp. 195-215, 2013.

280
[124] J. Mynard, M. Davidson, D. Penny, and J. Smolich, "A simple, versatile valve model for
use in lumped parameter and one‐dimensional cardiovascular models," International
Journal for Numerical Methods in Biomedical Engineering, vol. 28, no. 6-7, pp. 626-641,
2012.
[125] J. P. Mynard and J. J. Smolich, "One-dimensional haemodynamic modeling and wave
dynamics in the entire adult circulation," Annals of biomedical engineering, vol. 43, pp.
1443-1460, 2015.
[126] J. M. Huttunen, L. Kärkkäinen, and H. Lindholm, "Pulse transit time estimation of aortic
pulse wave velocity and blood pressure using machine learning and simulated training
data," PLoS computational biology, vol. 15, no. 8, p. e1007259, 2019.
[127] J. A. Chirinos, Textbook of Arterial Stiffness and Pulsatile Hemodynamics in Health and
Disease. Academic Press, 2022.
[128] D. Timms, M. Hayne, K. McNeil, and A. Galbraith, "A complete mock circulation loop
for the evaluation of left, right, and biventricular assist devices," Artificial organs, vol. 29,
no. 7, pp. 564-572, 2005.
[129] I. Wilkinson et al., Oxford handbook of clinical medicine. Oxford university press, 2017.
[130] P. Kumar and M. L. Clark, Kumar and Clark's clinical medicine E-Book. Elsevier health
sciences, 2012.
[131] R. D. Conn and J. H. O’Keefe, "Simplified evaluation of the jugular venous pressure:
significance of inspiratory collapse of jugular veins," Missouri medicine, vol. 109, no. 2,
p. 150, 2012.
[132] E. A. Ashley and J. Niebauer, "Cardiology explained," 2004.
[133] A. Noordergraaf, Circulatory system dynamics. Elsevier, 2012.
[134] K. Hemalatha and M. Manivannan, "A study of Cardiopulmonary interaction
haemodynamics with detailed lumped parameter model," International Journal of
Biomedical Engineering and Technology, vol. 6, no. 3, pp. 251-271, 2011.

281
[135] N. Albin and O. P. Bruno, "A spectral FC solver for the compressible Navier–Stokes
equations in general domains I: Explicit time-stepping," Journal of Computational Physics,
vol. 230, no. 16, pp. 6248-6270, 2011.
[136] F. Amlani and O. P. Bruno, "An FC-based spectral solver for elastodynamic problems in
general three-dimensional domains," Journal of Computational Physics, vol. 307, pp. 333-
354, 2016.
[137] F. Amlani, H. Wei, and N. M. Pahlevan, "A new pseudo-spectral methodology without
numerical diffusion for conducting dye simulations and particle residence time
calculations," arXiv preprint arXiv:2112.05257, 2021.
[138] O. P. Bruno and M. Lyon, "High-order unconditionally stable FC-AD solvers for general
smooth domains I. Basic elements," Journal of Computational Physics, vol. 229, no. 6, pp.
2009-2033, 2010.
[139] F. Amlani, H. Wei, and N. M. Pahlevan, "A Fourier-based methodology without numerical
diffusion for conducting dye simulations and particle residence time calculations," Journal
of Computational Physics, vol. 493, p. 112472, 2023.
[140] K. Shahbazi, J. S. Hesthaven, and X. Zhu, "Multi-dimensional hybrid Fourier
continuation–WENO solvers for conservation laws," Journal of computational physics,
vol. 253, pp. 209-225, 2013.
[141] F. Amlani et al., "Supershear shock front contribution to the tsunami from the 2018 M w
7.5 Palu, Indonesia earthquake," Geophysical Journal International, vol. 230, no. 3, pp.
2089-2097, 2022.
[142] F. Amlani, "A new high-order Fourier continuation-based elasticity solver for complex
three-dimensional geometries," California Institute of Technology, 2014.
[143] E. L. Gaggioli, O. P. Bruno, and D. M. Mitnik, "Light transport with the equation of
radiative transfer: the Fourier Continuation–Discrete Ordinates (FC–DOM) Method,"
Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 236, p. 106589, 2019.
[144] H. Wei, F. Amlani, and N. M. Pahlevan, "Direct 0D‐3D coupling of a lattice Boltzmann
methodology for fluid–structure aortic flow simulations," International Journal for
Numerical Methods in Biomedical Engineering, vol. 39, no. 5, p. e3683, 2023.

282
[145] W.-X. Huang, S. J. Shin, and H. J. Sung, "Simulation of flexible filaments in a uniform
flow by the immersed boundary method," Journal of computational physics, vol. 226, no.
2, pp. 2206-2228, 2007.
[146] J. P. Boyd and J. R. Ong, "Exponentially-convergent strategies for defeating the Runge
phenomenon for the approximation of non-periodic functions, part I: single-interval
schemes," Comput. Phys, vol. 5, no. 2-4, pp. 484-497, 2009.
[147] R. Amelard, N. Flannigan, C. A. Patterson, H. Heigold, R. L. Hughson, and A. D.
Robertson, "Assessing jugular venous compliance with optical hemodynamic imaging by
modulating intrathoracic pressure," Journal of Biomedical Optics, vol. 27, no. 11, p.
116005, 2022.
[148] R. N. Baird and W. M. Abbott, "Elasticity and compliance of canine femoral and jugular
vein segments," American Journal of Physiology-Heart and Circulatory Physiology, vol.
233, no. 1, pp. H15-H21, 1977.
[149] R. W. Ogden, Non-linear elastic deformations. Courier Corporation, 1997.
[150] M. Shojaeifard, S. Niroumandi, and M. Baghani, "Programming shape-shifting of flat
bilayers composed of tough hydrogels under transient swelling," Acta Mechanica, vol. 233,
no. 1, pp. 213-232, 2022.
[151] J. Cassels and N. Hitchin, "Nonlinear elasticity: theory and applications," 2001.
[152] A. Aghilinejad, R. Alavi, B. Rogers, F. Amlani, and N. M. Pahlevan, "Effects of vessel
wall mechanics on non-invasive evaluation of cardiovascular intrinsic frequencies,"
Journal of Biomechanics, p. 110852, 2021.
[153] F. Pereira and M. Gharib, "Defocusing digital particle image velocimetry and the threedimensional characterization of two-phase flows," Measurement Science and Technology,
vol. 13, no. 5, p. 683, 2002.
[154] A. Falahatpisheh, N. M. Pahlevan, and A. Kheradvar, "Effect of the mitral valve’s anterior
leaflet on axisymmetry of transmitral vortex ring," Annals of biomedical engineering, vol.
43, pp. 2349-2360, 2015.

283
[155] D. R. Troolin and E. K. Longmire, "Volumetric velocity measurements of vortex rings
from inclined exits," Experiments in fluids, vol. 48, pp. 409-420, 2010.
[156] K. L. Ruedinger, R. Medero, and A. Roldán-Alzate, "Fabrication of low-cost patientspecific vascular models for particle image velocimetry," Cardiovascular engineering and
technology, vol. 10, pp. 500-507, 2019.
[157] C. Bilgi and N. M. Pahlevan, "Viscosity-model-independent generalized Reynolds number
for laminar flow of shear-thinning and viscoplastic fluids," arXiv preprint
arXiv:2308.05319, 2023.
[158] P. Zamboni, "Why current Doppler ultrasound methodology is inaccurate in assessing
cerebral venous return: the alternative of the ultrasonic jugular venous pulse," Behavioural
Neurology, vol. 2016, no. 1, p. 7082856, 2016.
[159] H. Müller, G. Hinn, and M. Buser, "Internal jugular venous flow measurement by means
of a duplex scanner," Journal of ultrasound in medicine, vol. 9, no. 5, pp. 261-265, 1990.
[160] G. Ciuti, D. Righi, L. Forzoni, A. Fabbri, and A. M. Pignone, "Differences between internal
jugular vein and vertebral vein flow examined in real time with the use of multigate
ultrasound color Doppler," American Journal of Neuroradiology, vol. 34, no. 10, pp. 2000-
2004, 2013.
[161] C. R. Giordano, K. R. Murtagh, J. Mills, L. A. Deitte, M. J. Rice, and P. J. Tighe, "Locating
the optimal internal jugular target site for central venous line placement," Journal of
Clinical Anesthesia, vol. 33, pp. 198-202, 2016.
[162] S. Lynch, N. Nama, and C. A. Figueroa, "Effects of non-Newtonian viscosity on arterial
and venous flow and transport," Scientific Reports, vol. 12, no. 1, p. 20568, 2022.
[163] P. Di Achille, G. Tellides, C. Figueroa, and J. Humphrey, "A haemodynamic predictor of
intraluminal thrombus formation in abdominal aortic aneurysms," Proceedings of the
Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 470, no. 2172, p.
20140163, 2014.
[164] J. Biasetti, F. Hussain, and T. C. Gasser, "Blood flow and coherent vortices in the normal
and aneurysmatic aortas: a fluid dynamical approach to intra-luminal thrombus formation,"
Journal of The Royal Society Interface, vol. 8, no. 63, pp. 1449-1461, 2011.

284
[165] S.-W. Lee, L. Antiga, and D. A. Steinman, "Correlations among indicators of disturbed
flow at the normal carotid bifurcation," 2009.
[166] D. S. Berger, J. K. Li, and A. Noordergraaf, "Differential effects of wave reflections and
peripheral resistance on aortic blood pressure: a model-based study," American Journal of
Physiology-Heart and Circulatory Physiology, vol. 266, no. 4, pp. H1626 - H1642, 1994.
[167] K. Azer and C. S. Peskin, "A one-dimensional model of blood flow in arteries with friction
and convection based on the Womersley velocity profile," Cardiovascular Engineering,
vol. 7, pp. 51-73, 2007.
[168] F. N. Van de Vosse and N. Stergiopulos, "Pulse wave propagation in the arterial tree,"
Annual Review of Fluid Mechanics, vol. 43, pp. 467-499, 2011.
[169] T. J. Hughes and J. Lubliner, "On the one-dimensional theory of blood flow in the larger
vessels," Mathematical Biosciences, vol. 18, no. 1-2, pp. 161-170, 1973.
[170] D. S. Berger, K. A. Robinson, and S. G. Shroff, "Wave Propagation in Coupled Left
Ventricle–Arterial System: Implications for Aortic Pressure," Hypertension, vol. 27, no. 5,
pp. 1079-1089, 1996.
[171] F. Amlani and N. M. Pahlevan, "A novel Fourier-based (pseudo)spectral approach to 1D
Navier-Stokes equations with fluid-structure interaction for the comprehensive simulation
of hemodynamics and wave propagation in the entire cardiovascular system," ed. to be
submitted, 2024.
[172] A. Aghilinejad, F. Amlani, S. P. Mazandarani, K. S. King, and N. M. Pahlevan,
"Mechanistic insights on age-related changes in heart-aorta-brain hemodynamic coupling
using a pulse wave model of the entire circulatory system," American Journal of
Physiology-Heart and Circulatory Physiology, vol. 325, no. 5, pp. H1193-H1209, 2023.
[173] E. Boileau et al., "A benchmark study of numerical schemes for one‐dimensional arterial
blood flow modelling," International journal for numerical methods in biomedical
engineering, vol. 31, no. 10, p. e02732, 2015.
[174] P. Khairy, N. Poirier, and L.-A. Mercier, "Univentricular heart," Circulation, vol. 115, no.
6, pp. 800-812, 2007.

285
[175] J. Rychik et al., "Evaluation and management of the child and adult with Fontan
circulation: a scientific statement from the American Heart Association," Circulation, vol.
140, no. 6, pp. e234-e284, 2019.
[176] P. Khairy, N. Poirier, and L.-A. e. Mercier, "Univentricular heart," Circulation, vol. 115,
no. 6, pp. 800--812, 2007.
[177] J. A. Feinstein et al., "Hypoplastic left heart syndrome: current considerations and
expectations," Journal of the American College of Cardiology, vol. 59, no. 1 Supplement,
pp. S1--S42, 2012.
[178] R. W. Elder et al., "Risk Factors for Major Adverse Events Late after F ontan Palliation,"
Congenital heart disease, vol. 10, no. 2, pp. 159--168, 2015.
[179] C. C. Long, M. C. Hsu, Y. Bazilevs, J. A. Feinstein, and A. L. Marsden, "Fluid--structure
interaction simulations of the Fontan procedure using variable wall properties,"
International journal for numerical methods in Biomedical Engineering, vol. 28, no. 5, pp.
513--527, 2012.
[180] E. Tang et al., "Effect of Fontan geometry on exercise haemodynamics and its potential
implications," Heart, vol. 103, no. 22, pp. 1806--1812, 2017.
[181] A. L. Cheng, C. P. Wee, N. M. Pahlevan, and J. C. Wood, "A 4D flow MRI evaluation of
the impact of shear-dependent fluid viscosity on in vitro Fontan circulation flow,"
American Journal of Physiology-Heart and Circulatory Physiology, vol. 317, no. 6, pp.
H1243--H1253, 2019.
[182] C. L. Poh and Y. d’Udekem, "Life after surviving Fontan surgery: a meta-analysis of the
incidence and predictors of late death," Heart, Lung and Circulation, vol. 27, no. 5, pp.
552--559, 2018.
[183] C. Schilling et al., "The Fontan epidemic: population projections from the Australia and
New Zealand Fontan registry," International journal of cardiology, vol. 219, pp. 14--19,
2016.
[184] M. Gewillig and S. C. Brown, "The Fontan circulation after 45 years: update in
physiology," Heart, vol. 102, no. 14, pp. 1081--1086, 2016.

286
[185] R. Henaine et al., "Effects of lack of pulsatility on pulmonary endothelial function in the
Fontan circulation," The Journal of thoracic and cardiovascular surgery, vol. 146, no. 3,
pp. 522--529, 2013.
[186] S. Khambadkone, L. J., D. L. M. R., C. S., D. J. E., and R. A. N., "Basal pulmonary vascular
resistance and nitric oxide responsiveness late after Fontan-type operation," Circulation,
vol. 107, no. 25, pp. 3204--3208, 2003.
[187] J. M. Chen et al., "Current Topics and Controversies in Pediatric Heart Transplantation:
Proceedings of the Pediatric Heart Transplantation Summit 2017," World Journal for
Pediatric and Congenital Heart Surgery, vol. 9, no. 5, pp. 575--581, 2018.
[188] F. M. Rijnberg et al., "Energetics of blood flow in cardiovascular disease: concept and
clinical implications of adverse energetics in patients with a Fontan circulation,"
Circulation, vol. 137, no. 22, pp. 2393--2407, 2018.
[189] S. Pant, B. Fabrèges, J. F. Gerbeau, and I. Vignon‐Clementel, "A methodological paradigm
for patient‐specific multi‐scale CFD simulations: from clinical measurements to parameter
estimates for individual analysis," International journal for numerical methods in
biomedical engineering, vol. 30, no. 12, pp. 1614-1648, 2014.
[190] R. H. Khiabani et al., "Exercise capacity in single-ventricle patients after Fontan correlates
with haemodynamic energy loss in TCPC," Heart, vol. 101, no. 2, pp. 139-143, 2015.
[191] M. Restrepo et al., "Energetic implications of vessel growth and flow changes over time in
Fontan patients," The annals of thoracic surgery, vol. 99, no. 1, pp. 163-170, 2015.
[192] P. F. Davies, "Hemodynamic shear stress and the endothelium in cardiovascular
pathophysiology," Nature clinical practice Cardiovascular medicine, vol. 6, no. 1, pp. 16-
26, 2009.
[193] A. M. Malek, S. L. Alper, and S. Izumo, "Hemodynamic shear stress and its role in
atherosclerosis," Jama, vol. 282, no. 21, pp. 2035-2042, 1999.
[194] W. Yang, F. P. Chan, V. M. Reddy, A. L. Marsden, and J. A. Feinstein, "Flow simulations
and validation for the first cohort of patients undergoing the Y-graft Fontan procedure,"
The Journal of thoracic and cardiovascular surgery, vol. 149, no. 1, pp. 247-255, 2015.

287
[195] O. K. Baskurt and H. J. Meiselman, "Blood rheology and hemodynamics in Seminars in
thrombosis and hemostasis," ed, 2003.
[196] P. F. Davies, "Hemodynamic shear stress and the endothelium in cardiovascular
pathophysiology," Nature Reviews Cardiology, vol. 6, no. 1, p. 16, 2009.
[197] J. Kang, A. Aghilinejad, and N. M. Pahlevan, "On the accuracy of displacement-based
wave intensity analysis: Effect of vessel wall viscoelasticity and nonlinearity," PloS one,
vol. 14, no. 11, 2019.
[198] C. Bachmann, G. Hugo, G. Rosenberg, S. Deutsch, A. Fontaine, and J. M. Tarbell, "Fluid
dynamics of a pediatric ventricular assist device," Artificial Organs, vol. 24, no. 5, pp. 362-
-372, 2000.
[199] M. G. Al-Azawy, A. Turan, and A. Revell, "Investigating the impact of non-Newtonian
blood models within a heart pump," International journal for numerical methods in
Biomedical Engineering, vol. 33, no. 1, p. e02780, 2017.
[200] C. Tu and M. Deville, "Pulsatile flow of non-Newtonian fluids through arterial stenoses,"
Journal of biomechanics, vol. 29, no. 7, pp. 899--908, 1996.
[201] W. Yang, F. P. Chan, V. M. Reddy, A. L. Marsden, and J. A. Feinstein, "Flow simulations
and validation for the first cohort of patients undergoing the Y-graft Fontan procedure,"
The Journal of thoracic and cardiovascular surgery, vol. 149, no. 1, pp. 247--255, 2015.
[202] P. M. Trusty, M. Restrepo, K. R. Kanter, A. P. Yoganathan, M. A. Fogel, and T. C.
Slesnick, "A pulsatile hemodynamic evaluation of the commercially available bifurcated
Y-graft Fontan modification and comparison with the lateral tunnel and extracardiac
conduits," The Journal of thoracic and cardiovascular surgery, vol. 151, no. 6, pp. 1529--
1536, 2016.
[203] L. D. C. Casa, D. H. Deaton, and D. N. Ku, "Role of high shear rate in thrombosis," Journal
of vascular surgery, vol. 61, no. 4, pp. 1068--1080, 2015.
[204] B. Liu and D. Tang, "Influence of non-Newtonian properties of blood on the wall shear
stress in human atherosclerotic right coronary arteries," Molecular \& cellular
biomechanics: MCB, vol. 8, no. 1, p. 73, 2011.

288
[205] S. P. Shupti, M. G. Rabby, M. Molla, and et al., "Rheological behavior of physiological
pulsatile flow through a model arterial stenosis with moving wall," Journal of Fluids, vol.
2015, 2015.
[206] A. L. Cheng, C. M. Takao, R. B. Wenby, H. J. Meiselman, J. C. Wood, and J. A. Detterich,
"Elevated low-shear blood viscosity is associated with decreased pulmonary blood flow in
children with univentricular heart defects," Pediatric cardiology, vol. 37, no. 4, pp. 789--
801, 2016.
[207] M. Ashrafizaadeh and H. Bakhshaei, "A comparison of non-Newtonian models for lattice
Boltzmann blood flow simulations," Computers & Mathematics with Applications, vol. 58,
no. 5, pp. 1045-1054, 2009.
[208] J. Cosgrove, J. Buick, S. Tonge, C. Munro, C. Greated, and D. Campbell, "Application of
the lattice Boltzmann method to transition in oscillatory channel flow," Journal of Physics
A: Mathematical and General, vol. 36, no. 10, p. 2609, 2003.
[209] J. Boyd, J. M. Buick, and S. Green, "Analysis of the Casson and Carreau-Yasuda nonNewtonian blood models in steady and oscillatory flows using the lattice Boltzmann
method," Physics of Fluids, vol. 19, no. 9, 2007.
[210] D. A. Wolf-Gladrow, Lattice-gas cellular automata and lattice Boltzmann models: an
introduction. Springer, 2004.
[211] X. He and L.-S. Luo, "Lattice Boltzmann model for the incompressible Navier–Stokes
equation," Journal of statistical Physics, vol. 88, pp. 927-944, 1997.
[212] J. Boyd, J. M. Buick, and S. Green, "Analysis of the Casson and Carreau-Yasuda nonNewtonian blood models in steady and oscillatory flows using the lattice B oltzmann
method," Physics of Fluids, vol. 19, no. 9, p. 093103, 2007.
[213] A. L. Cheng, N. M. Pahlevan, D. G. Rinderknecht, J. C. Wood, and M. Gharib,
"Experimental investigation of the effect of non-Newtonian behavior of blood flow in the
Fontan circulation," European Journal of Mechanics-B/Fluids, vol. 68, pp. 184--192, 2018.
[214] A. Artoli, "Mesoscopic computational haemodynamics," Ponsen \& Looijen, Wageningen,
2003.

289
[215] B. M. Johnston, P. R. Johnston, S. Corney, and D. Kilpatrick, "Non-Newtonian blood flow
in human right coronary arteries: steady state simulations," Journal of biomechanics, vol.
37, no. 5, pp. 709--720, 2004.
[216] K. Timm, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Silva, and E. Viggen, "The lattice
Boltzmann method: principles and practice," Cham, Switzerland: Springer International
Publishing AG, 2016.
[217] N. M. Pahlevan, F. Amlani, M. Hossein Gorji, F. Hussain, and M. Gharib, "A
physiologically relevant, simple outflow boundary model for truncated vasculature,"
Annals of biomedical engineering, vol. 39, pp. 1470-1481, 2011.
[218] M. A. Salim, T. G. DiSessa, K. L. Arheart, and B. S. Alpert, "Contribution of superior vena
caval flow to total cardiac output in children: a Doppler echocardiographic study,"
Circulation, vol. 92, no. 7, pp. 1860-1865, 1995.
[219] M. Cibis et al., "The effect of resolution on viscous dissipation measured with 4D flow
MRI in patients with Fontan circulation: Evaluation using computational fluid dynamics,"
Journal of biomechanics, vol. 48, no. 12, pp. 2984-2989, 2015.
[220] B. M. Johnston, P. R. Johnston, S. Corney, and D. Kilpatrick, "Non-Newtonian blood flow
in human right coronary arteries: steady state simulations," Journal of biomechanics, vol.
37, no. 5, pp. 709-720, 2004.
[221] P. Ballyk, D. Steinman, and C. Ethier, "Simulation of non-Newtonian blood flow in an
end-to-side anastomosis," Biorheology, vol. 31, no. 5, pp. 565-586, 1994.
[222] P. D. Coon, J. Rychik, R. T. Novello, P. S. Ro, J. W. Gaynor, and T. L. Spray, "Thrombus
formation after the Fontan operation," The Annals of thoracic surgery, vol. 71, no. 6, pp.
1990-1994, 2001.
[223] J. Chen, X.-Y. Lu, and W. Wang, "Non-Newtonian effects of blood flow on hemodynamics
in distal vascular graft anastomoses," Journal of Biomechanics, vol. 39, no. 11, pp. 1983-
1995, 2006.
[224] R. Henaine et al., "Effects of lack of pulsatility on pulmonary endothelial function in the
Fontan circulation," The Journal of thoracic and cardiovascular surgery, vol. 146, no. 3,
pp. 522-529, 2013.

290
[225] S. Ovroutski et al., "Absence of pulmonary artery growth after Fontan operation and its
possible impact on late outcome," The Annals of thoracic surgery, vol. 87, no. 3, pp. 826-
831, 2009.
[226] F.-J. S. Ridderbos et al., "Adverse pulmonary vascular remodeling in the Fontan
circulation," The Journal of Heart and Lung Transplantation, vol. 34, no. 3, pp. 404-413,
2015.
[227] Z. A. Wei, M. Tree, P. M. Trusty, W. Wu, S. Singh-Gryzbon, and A. Yoganathan, "The
advantages of viscous dissipation rate over simplified power loss as a Fontan hemodynamic
metric," Annals of biomedical engineering, vol. 46, pp. 404-416, 2018.
[228] F. Kabinejadian and D. N. Ghista, "Compliant model of a coupled sequential coronary
arterial bypass graft: Effects of vessel wall elasticity and non-Newtonian rheology on blood
flow regime and hemodynamic parameters distribution," Medical Engineering & Physics,
vol. 34, no. 7, pp. 860-872, 2012.
[229] T.-Y. Hsia, S. Khambadkone, A. N. Redington, F. Migliavacca, J. E. Deanfield, and M. R.
de Leval, "Effects of respiration and gravity on infradiaphragmatic venous flow in normal
and Fontan patients," Circulation, vol. 102, no. suppl_3, pp. Iii-148-Iii-153, 2000.
[230] A. L. Marsden, I. E. Vignon-Clementel, F. P. Chan, J. A. Feinstein, and C. A. Taylor,
"Effects of exercise and respiration on hemodynamic efficiency in CFD simulations of the
total cavopulmonary connection," Annals of biomedical engineering, vol. 35, pp. 250-263,
2007.
[231] T. Alsaied et al., "Predicting long‐term mortality after Fontan procedures: a risk score
based on 6707 patients from 28 studies," Congenital heart disease, vol. 12, no. 4, pp. 393-
398, 2017.
[232] R. H. Khiabani et al., "Exercise capacity in single-ventricle patients after Fontan correlates
with haemodynamic energy loss in TCPC," Heart, 2014.
[233] P. Frieberg, P. Sjöberg, E. Hedström, M. Carlsson, and P. Liuba, "In vivo hepatic flow
distribution by computational fluid dynamics can predict pulmonary flow distribution in
patients with Fontan circulation," Scientific Reports, vol. 13, no. 1, p. 18206, 2023.

291
[234] X. Liu et al., "Surgical planning and optimization of patient-specific Fontan grafts with
Uncertain post-operative boundary conditions and anastomosis displacement," IEEE
Transactions on Biomedical Engineering, vol. 69, no. 11, pp. 3472-3483, 2022.
[235] M. Restrepo et al., "Surgical planning of the total cavopulmonary connection: robustness
analysis," Annals of biomedical engineering, vol. 43, pp. 1321-1334, 2015.
[236] F. Zhu et al., "Hemodynamic effects of a simplified Venturi Conduit for Fontan circulation:
A pilot, in silico analysis," Scientific Reports, vol. 10, no. 1, p. 817, 2020.
[237] R. Prather, A. Das, M. Farias, E. Divo, A. Kassab, and W. DeCampli, "Parametric
investigation of an injection-jet self-powered Fontan circulation," Scientific Reports, vol.
12, no. 1, p. 2161, 2022.
[238] W. P. Lin, M. G. Doyle, S. L. Roche, O. Honjo, T. L. Forbes, and C. H. Amon,
"Computational fluid dynamic simulations of a cavopulmonary assist device for failing
Fontan circulation," The Journal of thoracic and cardiovascular surgery, vol. 158, no. 5,
pp. 1424-1433. e5, 2019.
[239] W. Yang, T. A. Conover, R. S. Figliola, G. A. Giridharan, A. L. Marsden, and M. D.
Rodefeld, "Passive performance evaluation and validation of a viscous impeller pump for
subpulmonary fontan circulatory support," Scientific Reports, vol. 13, no. 1, p. 12668,
2023.
[240] P. D. Morris et al., "Computational fluid dynamics modelling in cardiovascular medicine,"
Heart, 2015.
[241] O. K. Baskurt and H. J. Meiselman, "Blood rheology and hemodynamics," in Seminars in
thrombosis and hemostasis, 2003, vol. 29, no. 05: Copyright© 2003 by Thieme Medical
Publishers, Inc., 333 Seventh Avenue, New …, pp. 435-450.
[242] O. K. Baskurt, M. R. Hardeman, and M. W. Rampling, Handbook of hemorheology and
hemodynamics. IOS press, 2007.
[243] X. Luo and Z. Kuang, "A study on the constitutive equation of blood," Journal of
biomechanics, vol. 25, no. 8, pp. 929-934, 1992.

292
[244] T. Sochi, "Non-Newtonian rheology in blood circulation," arXiv preprint
arXiv:1306.2067, 2013.
[245] H. Wei, K. Cao, N. M. Pahlevan, and A. L. Cheng, "Shear-Dependent Changes in Blood
Viscosity Negatively Affect Energetic Efficiency in Patient-Specific Models of the Fontan
Circulation," Circulation, vol. 148, no. Suppl_1, pp. A12955-A12955, 2023.
[246] H. Ye, H. Wei, H. Huang, and X.-y. Lu, "Two tandem flexible loops in a viscous flow,"
Physics of Fluids, vol. 29, no. 2, 2017.
[247] H. Huang, H. Wei, and X.-Y. Lu, "Coupling performance of tandem flexible inverted flags
in a uniform flow," Journal of Fluid Mechanics, vol. 837, pp. 461-476, 2018.
[248] H. Wei, D. A. Hutchins, P. D. Ronney, and N. M. Pahlevan, "Fluid-based microbial
processes modeling in Trichodesmium colony formation," Physics of Fluids, vol. 35, no.
10, 2023.
[249] C. Bilgi, F. Amlani, H. Wei, N. Rizzi, and N. M. Pahlevan, "Thermal and Postural Effects
on Fluid Mixing and Irrigation Patterns for Intraventricular Hemorrhage Treatment,"
Annals of Biomedical Engineering, pp. 1-14, 2023.
[250] H. Wei, C. S. Herrington, J. D. Cleveland, V. A. Starnes, and N. M. Pahlevan,
"Hemodynamically efficient artificial right atrium design for univentricular heart patients,"
Physical Review Fluids, vol. 6, no. 12, p. 123103, 2021.
[251] A. Aghilinejad, H. Wei, G. A. Magee, and N. M. Pahlevan, "Model-based fluid-structure
interaction approach for evaluation of thoracic endovascular aortic repair endograft length
in type B aortic dissection," Frontiers in Bioengineering and Biotechnology, vol. 10, p.
825015, 2022.
[252] A. Razavi, E. Shirani, and M. Sadeghi, "Numerical simulation of blood pulsatile flow in a
stenosed carotid artery using different rheological models," Journal of biomechanics, vol.
44, no. 11, pp. 2021-2030, 2011.
[253] P. R. Vijayaratnam, C. C. O’Brien, J. A. Reizes, T. J. Barber, and E. R. Edelman, "The
impact of blood rheology on drug transport in stented arteries: steady simulations," PloS
one, vol. 10, no. 6, p. e0128178, 2015.

293
[254] P. L. Bhatnagar, E. P. Gross, and M. Krook, "A model for collision processes in gases. I.
Small amplitude processes in charged and neutral one-component systems," Phys. Rev.,
vol. 94, no. 3, p. 511, 1954.
[255] S. K. Kang and Y. A. Hassan, "A direct-forcing immersed boundary method for the thermal
lattice Boltzmann method," Computers & Fluids, vol. 49, no. 1, pp. 36-45, 2011.
[256] P. Cignoni, M. Callieri, M. Corsini, M. Dellepiane, F. Ganovelli, and G. Ranzuglia,
"Meshlab: an open-source mesh processing tool," in Eurographics Italian chapter
conference, 2008, vol. 2008: Salerno, Italy, pp. 129-136.
[257] C. M. Haggerty et al., "Fontan hemodynamics from 100 patient-specific cardiac magnetic
resonance studies: a computational fluid dynamics analysis," The Journal of thoracic and
cardiovascular surgery, vol. 148, no. 4, pp. 1481-1489, 2014.
[258] K. K. Whitehead, K. Pekkan, H. D. Kitajima, S. M. Paridon, A. P. Yoganathan, and M. A.
Fogel, "Nonlinear power loss during exercise in single-ventricle patients after the Fontan:
insights from computational fluid dynamics," Circulation, vol. 116, no. 11_supplement,
pp. I-165-I-171, 2007.
[259] P. D. Ballyk, D. A. Steinman, and C. R. Ethier, "Simulation of non-Newtonian blood flow
in an end-to-side anastomosis," Biorheology, vol. 31, no. 5, pp. 565--586, 1994.
[260] S. S. Bossers et al., "Long-term serial follow-up of pulmonary artery size and wall shear
stress in Fontan patients," Pediatric Cardiology, vol. 37, pp. 637-645, 2016.
[261] P. Frieberg, N. Aristokleous, P. Sjöberg, J. Töger, P. Liuba, and M. Carlsson,
"Computational fluid dynamics support for fontan planning in minutes, not hours: the next
step in clinical pre-interventional simulations," Journal of cardiovascular translational
research, vol. 15, no. 4, pp. 708-720, 2022.
[262] X. Liu, Y. Fan, X. Deng, and F. Zhan, "Effect of non-Newtonian and pulsatile blood flow
on mass transport in the human aorta," Journal of biomechanics, vol. 44, no. 6, pp. 1123-
1131, 2011.
[263] G. De Nisco et al., "Modelling blood flow in coronary arteries: Newtonian or shearthinning non-Newtonian rheology?," Computer Methods and Programs in Biomedicine,
vol. 242, p. 107823, 2023.

294
[264] R. Gerrah and S. J. Haller, "Computational fluid dynamics: a primer for congenital heart
disease clinicians," Asian Cardiovascular and Thoracic Annals, vol. 28, no. 8, pp. 520-
532, 2020.
[265] Z. Wei et al., "Non-Newtonian Effects on Patient-Specific Modeling of Fontan
Hemodynamics," Annals of Biomedical Engineering, vol. 48, no. 8, pp. 2204--2217, 2020.
[266] M. Firdouse, A. Agarwal, A. K. Chan, and T. Mondal, "Thrombosis and thromboembolic
complications in fontan patients: a literature review," Clinical and Applied
Thrombosis/Hemostasis, vol. 20, no. 5, pp. 484-492, 2014.
[267] K. Itatani et al., "Optimal conduit size of the extracardiac Fontan operation based on energy
loss and flow stagnation," The annals of thoracic surgery, vol. 88, no. 2, pp. 565-573, 2009.
[268] B. W. Duncan and S. Desai, "Pulmonary arteriovenous malformations after cavopulmonary
anastomosis," The Annals of thoracic surgery, vol. 76, no. 5, pp. 1759-1766, 2003.
[269] U. Morbiducci, R. Ponzini, D. Gallo, C. Bignardi, and G. Rizzo, "Inflow boundary
conditions for image-based computational hemodynamics: impact of idealized versus
measured velocity profiles in the human aorta," Journal of biomechanics, vol. 46, no. 1,
pp. 102-109, 2013.
[270] J. I. Hoffman and S. Kaplan, "The incidence of congenital heart disease," Journal of the
American college of cardiology, vol. 39, no. 12, pp. 1890-1900, 2002.
[271] E. R. Griffiths et al., "Evaluating failing Fontans for heart transplantation: predictors of
death," The Annals of thoracic surgery, vol. 88, no. 2, pp. 558-564, 2009.
[272] W. I. Norwood Jr, "Hypoplastic left heart syndrome," The Annals of thoracic surgery, vol.
52, no. 3, pp. 688-695, 1991.
[273] G. J. Bittle et al., "Exosomes isolated from human cardiosphere–derived cells attenuate
pressure overload–induced right ventricular dysfunction," The Journal of thoracic and
cardiovascular surgery, vol. 162, no. 3, pp. 975-986. e6, 2021.
[274] W. I. Norwood Jr, M. L. Jacobs, and J. D. Murphy, "Fontan procedure for hypoplastic left
heart syndrome," The Annals of thoracic surgery, vol. 54, no. 6, pp. 1025-1030, 1992.

295
[275] A. D. McCormick and K. R. Schumacher, "Transplantation of the failing Fontan,"
Translational Pediatrics, vol. 8, no. 4, p. 290, 2019.
[276] C. Poh and Y. d’Udekem, "Life after surviving Fontan surgery: a meta-analysis of the
incidence and predictors of late death," Heart, Lung and Circulation, vol. 27, no. 5, pp.
552-559, 2018.
[277] C. Schilling et al., "The Fontan epidemic: population projections from the Australia and
New Zealand Fontan registry," International journal of cardiology, vol. 219, pp. 14-19,
2016.
[278] W. G. Members et al., "Heart disease and stroke statistics—2010 update: a report from the
American Heart Association," Circulation, vol. 121, no. 7, pp. e46-e215, 2010.
[279] J. B Clark, L. B Pauliks, J. L Myers, and A. Undar, "Mechanical circulatory support for
end-stage heart failure in repaired and palliated congenital heart disease," Current
cardiology reviews, vol. 7, no. 2, pp. 102-109, 2011.
[280] L. W. Miller, "Left ventricular assist devices are underutilized," Circulation, vol. 123, no.
14, pp. 1552-1558, 2011.
[281] M. Griselli, R. Sinha, S. Jang, G. Perri, and I. Adachi, "Mechanical circulatory support for
single ventricle failure," Frontiers in Cardiovascular Medicine, vol. 5, p. 115, 2018.
[282] A. L. Marsden, Y. Bazilevs, C. C. Long, and M. Behr, "Recent advances in computational
methodology for simulation of mechanical circulatory assist devices," Wiley
interdisciplinary reviews: Systems biology and medicine, vol. 6, no. 2, pp. 169-188, 2014.
[283] C. L. Poh et al., "Ventricular assist device support in patients with single ventricles: the
Melbourne experience," Interactive CardioVascular and Thoracic Surgery, vol. 25, no. 2,
pp. 310-316, 2017.
[284] J. R. Miller, T. S. Lancaster, C. Callahan, A. M. Abarbanell, and P. Eghtesady, "An
overview of mechanical circulatory support in single-ventricle patients," Translational
Pediatrics, vol. 7, no. 2, p. 151, 2018.
[285] D. Goldstein, R. Handler, and L. Sirovich, "Modeling a no-slip flow boundary with an
external force field," Journal of computational physics, vol. 105, no. 2, pp. 354-366, 1993.

296
[286] X. He, X. Shan, and G. D. Doolen, "Discrete Boltzmann equation model for nonideal
gases," Physical Review E, vol. 57, no. 1, p. R13, 1998.
[287] Z. Wei et al., "Non-Newtonian effects on patient-specific modeling of Fontan
hemodynamics," Annals of biomedical engineering, vol. 48, pp. 2204-2217, 2020.
[288] J. L. Batoz, K. J. Bathe, and L. W. Ho, "A study of three‐node triangular plate bending
elements," International journal for numerical methods in engineering, vol. 15, no. 12, pp.
1771-1812, 1980.
[289] H. Dai, H. Luo, and J. F. Doyle, "Dynamic pitching of an elastic rectangular wing in
hovering motion," Journal of Fluid Mechanics, vol. 693, pp. 473-499, 2012.
[290] R.-N. Hua, L. Zhu, and X.-Y. Lu, "Dynamics of fluid flow over a circular flexible plate,"
Journal of fluid mechanics, vol. 759, pp. 56-72, 2014.
[291] C. Tang, N.-S. Liu, and X.-Y. Lu, "Dynamics of an inverted flexible plate in a uniform
flow," Physics of Fluids, vol. 27, no. 7, 2015.
[292] T. B. Fredenburg, T. R. Johnson, and M. D. Cohen, "The Fontan procedure: anatomy,
complications, and manifestations of failure," Radiographics, vol. 31, no. 2, pp. 453-463,
2011.
[293] P. M. Trusty et al., "An in vitro analysis of the PediMag and CentriMag for right-sided
failing Fontan support," The Journal of Thoracic and Cardiovascular Surgery, vol. 158,
no. 5, pp. 1413-1421, 2019.
[294] P. M. Trusty et al., "In vitro examination of the ventriflo true pulse pump for failing fontan
support," Artificial organs, vol. 43, no. 2, pp. 181-188, 2019.
[295] F. F. Faletra, S. Y. Ho, and A. Auricchio, "Anatomy of right atrial structures by real-time
3D transesophageal echocardiography," JACC: Cardiovascular Imaging, vol. 3, no. 9, pp.
966-975, 2010.
[296] R. Shereen, S. Lee, S. Salandy, W. Roberts, and M. Loukas, "A comprehensive review of
the anatomical variations in the right atrium and their clinical significance," Translational
Research in Anatomy, vol. 17, p. 100046, 2019.

297
[297] M. Tadic, "The right atrium, a forgotten cardiac chamber: An updated review of
multimodality imaging," Journal of Clinical Ultrasound, vol. 43, no. 6, pp. 335-345, 2015.

Abstract (if available)

Abstract The study of hemodynamics in cerebral venous circulation is crucial for understanding neurodegenerative diseases and cerebrovascular function. The physics of fluids in cerebral veins are more complex than in arteries, and research in this area is relatively sparse. In healthy conditions, pulsatile fluid dynamics play a critical role, with the internal jugular vein (IJV) being a significant contributor. However, in diseased conditions like Fontan patients, non-Newtonian effects become significant. For better understanding the underlying mechanisms, it is necessary to develop accurate models that can improve prediction and treatment strategies. To achieve such aims, we implemented both computational and experimental approaches to investigate the fluid dynamics of the central venous system and its impact on brain-heart hemodynamic coupling, under both healthy and diseased conditions. We developed a 0D-3D coupling computational framework to capture wave dynamics of IJV in a right-heart-brain coupled system towards investigating non-Newtonian effects. Additionally, we designed an experimental setup with a 3D particle image velocimetry measurement system for capturing IJV wave dynamics in realistic scenarios and for accurately measuring the 3D flow. The global effects on IJV and how they affect right-heart-brain coupling were examined using a 1D closed-loop model of the circulatory system. Following this, we demonstrated the applications of our simulations for Fontan patients. In such conditions, the flow dynamics shift non-negligibly, highlighting the importance of non-Newtonian blood rheology. Findings from this thesis may lead to novel diagnostic and therapeutic strategies for various cardiovascular and neurological disorders, potentially offering valuable insights into their pathophysiology.

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Creator Wei, Heng (author)

Core Title Pulsatile and steady flow in central veins and its impact on right heart-brain hemodynamic coupling in health and disease

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Degree Conferral Date 2024-08

Publication Date 07/11/2024

Defense Date 06/24/2024

Publisher Los Angeles, California (original), University of Southern California (original), University of Southern California. Libraries (digital)

Tag computational fluid dynamics,Fontan,hemodynamic,internal jugular vein,Lattice Boltzmann methods,OAI-PMH Harvest,particle image velocimetry,pulsatile flow,steady flow

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Language English

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Advisor Pahlevan, Niema (committee chair), Khoo, Michael (committee member), Sadhal, Satwindar (committee member), Wood, John (committee member)

Creator Email hengwei421@outlook.com,weiheng@usc.edu

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