Impact of aortic dynamic modes on heart and brain hemodynamics for advanced diagnostics and therapeutics

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Content Impact of aortic dynamic modes on heart and brain hemodynamics
for advanced diagnostics and therapeutics

By

Arian Aghilinejad

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

MAY 2023

Copyright 2023 Arian Aghilinejad
ii

Dedication
To my mother and my father who love me unconditionally.

iii

Acknowledgements
I would like to express my sincere gratitude to my PhD advisor, Dr. Niema Pahlevan, for
his full support, expert guidance and encouragement throughout my study and research. His
guidance and knowledge have been a major factor in my ability to excel at the University of
Southern California (USC). I appreciate all his contributions of time, ideas, and experiences to
make my PhD studies productive and stimulus. I am also grateful to my clinical mentors, Dr.
Gregory Magee from division of vascular surgery and endovascular therapy at USC and Dr.
Kevin King, neuroradiologist and a professor in the department of neuroradiology at Barrow
Neurological Institute, whose great expertise was a valuable asset for conducting my research. I
would like to thank Dr. Paul Newton, Dr. Satwindar Sadhal and Dr. John Wood for being my
committee members and giving suggestions on my thesis. I am also thankful for the guidance I
received from Dr. Faisal Amlani during my PhD.
This thesis was supported by the American Heart Association predoctoral fellowship award
(grant number: 915728) during the last year of my PhD, and the Alfred E Mann innovation in
engineering doctoral fellowship award during the third and fourth years of my PhD. I am also
grateful to the Viterbi school of engineering for their entering Ph.D. fellowship award supported
me in my first year of PhD studies.
I highly appreciate the support of all my dear friends: Rashid Alavi, Heng Wei, Coskun
Bilgi, Mohammad Aghaamoo, Darius Saadat-Moghaddam, Navid Niknafs, and Mahdi
Aghapour. I am also thankful to Kimia Noorbakhsh for her support during my PhD. Lastly, I
would like to express my deepest gratitude and warmest appreciation to my family including my
parents (Mashaallah Aghilinejad and Mojgan Ashrafi), my aunt (Marjan Ashrafi), and my
grandmother (Parvaneh Mehrabi) for their continuous love and encouragement.
iv

TABLE OF CONTENTS
Dedication ....................................................................................................................................... ii
Acknowledgements............................................................................................................... iii
List of Tables ..................................................................................................................... xiii
List of Figures ..................................................................................................................... xiv
Abstract ............................................................................................................................. xvii
CHAPTER 1 : Introduction ...................................................................................................... 1
1.1 Motivation ............................................................................................................................. 1
1.2 Thesis objectives .................................................................................................................... 3
1.3 Thesis outline ......................................................................................................................... 4
CHAPTER 2 : Dynamic effects of aortic arch stiffening on pulsatile energy transmission to
cerebral vasculature: a core determinant of healthy brain-heart coupling........................................ 7
2.1 Chapter abstract ..................................................................................................................... 7
2.2 Introduction ........................................................................................................................... 8
2.3 Theoretical indicators of an optimum wave condition ............................................................. 9
2.4 Methods ............................................................................................................................... 12
2.4.1 Physical model .............................................................................................................. 12
2.4.2 Mathematical and computational model ........................................................................ 14
2.4.3 Hemodynamic analysis ................................................................................................. 19
2.5 Results ................................................................................................................................. 20
2.5.1 Physiological accuracy of the model.............................................................................. 20
2.5.2 Effect of heart rate on transmitted pulsatility to the brain ............................................... 22
v

2.5.3 Flow and pressure pulsatility indices versus pulsatile power percentage......................... 24
2.6 Discussion ........................................................................................................................... 25
2.7 Conclusion ........................................................................................................................... 29
CHAPTER 3 : Mechanistic insights on age-related changes in heart-aorta-brain hemodynamic
couplings using in-silico model of the entire circulation ............................................................ 30
3.1 Chapter abstract ................................................................................................................... 30
3.2 Introduction ......................................................................................................................... 31
3.3 Methods ............................................................................................................................... 33
3.3.1 Physical model of the entire human circulation .............................................................. 33
3.3.2 Computational model and numerical solver ................................................................... 34
3.3.3 Time-varying elastance heart model .............................................................................. 38
3.3.4 Hemodynamic analysis ................................................................................................. 38
3.4 Results ................................................................................................................................. 41
3.4.1 Physiological accuracy of the model.............................................................................. 41
3.4.2 Effect of LV contractility on transmitted pulsatility to the brain ..................................... 43
3.4.3 Effect of heart rate on transmitted pulsatility to the brain ............................................... 46
3.4.4 Effect of LV-aorta dynamics on wave intensity ............................................................. 48
3.4.5 Effect of LV-aorta dynamics on Brain Perfusion ........................................................... 51
3.5 Discussion ........................................................................................................................... 52
3.5.1 Impact of LV contractility on brain hemodynamics ....................................................... 53
3.5.2 Presence of the optimum heart rate ................................................................................ 55
3.5.3 Impact of aortic stiffness on brain hemodynamics ......................................................... 56
vi

3.5.4 Wave intensity analysis ................................................................................................. 58
3.5.5 Limitations.................................................................................................................... 59
3.6 Conclusion ........................................................................................................................... 59
CHAPTER 4 : Accuracy and applicability of non-invasive evaluation of aortic wave intensity
using only pressure waveforms in humans ............................................................................... 60
4.1 Chapter abstract ................................................................................................................... 60
4.2 Introduction ......................................................................................................................... 61
4.3 Materials and methods ......................................................................................................... 63
4.3.1 Participants and data ..................................................................................................... 63
4.3.2 Wave intensity analysis ................................................................................................. 66
4.3.3 Estimating Wave Intensity from Only Pressure Measurements ...................................... 67
4.3.4 Measurement sites and excess pressure calibration ........................................................ 68
4.3.5 Statistical analysis ......................................................................................................... 71
4.4 Results ................................................................................................................................. 72
4.4.1 Accuracy of pressure-only WI from radial-based measurements .................................... 73
4.4.2 Accuracy of pressure-only WI from carotid-based measurements .................................. 76
4.4.3 Accuracy of pressure-only WI under healthy and diseased conditions ............................ 79
4.5 Discussion ........................................................................................................................... 82
4.5.1 Carotid versus radial-based estimates of WI .................................................................. 82
4.5.2 Pressure-only WI cannot capture backward wave contributions ..................................... 83
4.5.3 Performance of pressure-only WI in healthy and diseased individuals............................ 83
4.5.4 Failure rates of the reservoir pressure algorithm ............................................................ 84
vii

4.5.5 On the accuracy of peak flow velocity calibration.......................................................... 85
4.5.6 Critique of method ........................................................................................................ 85
4.6 Conclusion ........................................................................................................................... 86
CHAPTER 5 : Hybrid Fourier decomposition-machine learning approach for pressure-only
aortic wave intensity estimation in Framingham heart study ....................................................... 87
5.1 Chapter abstract ................................................................................................................... 87
5.2 Introduction ......................................................................................................................... 88
5.3 Materials and methods ......................................................................................................... 90
5.3.1 Participants and data ..................................................................................................... 90
5.3.2 Wave intensity analysis ................................................................................................. 91
5.3.3 Fourier representation and physics-based feature selection ............................................ 93
5.3.4 Machine learning models .............................................................................................. 95
5.3.5 Statistical analysis ......................................................................................................... 96
5.4 Results ................................................................................................................................. 97
5.4.1 Accuracy of pressure-only WI amplitudes ..................................................................... 97
5.4.2 Accuracy of pressure-only WI timings .......................................................................... 99
5.4.3 Difference comparison between ML algorithms .......................................................... 101
5.4.4 Analytical versus Fourier-based ML pressure-only WI ................................................ 103
5.4.5 Sensitivity analysis for the input Fourier modes........................................................... 104
5.4.6 Sensitivity analysis for the training size ....................................................................... 104
5.5 Discussion ......................................................................................................................... 105
5.5.1 Towards non-invasive pulse wave analysis .................................................................. 105
viii

5.5.2 Principal findings ........................................................................................................ 107
5.5.3 Applicability of machine learning in cardiovascular engineering ................................. 108
5.5.4 Study limitations ......................................................................................................... 109
5.6 Conclusion ......................................................................................................................... 110
CHAPTER 6 : Model-based fluid-structure interaction approach for evaluation of thoracic
endovascular aortic repair endograft length in type B aortic dissection ...................................... 111
6.1 Chapter abstract ................................................................................................................. 111
6.2 Introduction ....................................................................................................................... 112
6.3 Materials and methods ....................................................................................................... 114
6.3.1 Physical problem ........................................................................................................ 114
6.3.2 Mathematical formulation ........................................................................................... 116
6.3.3 Implementations of the boundary conditions ............................................................... 119
6.3.4 Numerical method ....................................................................................................... 121
6.3.5 Hemodynamic analysis ............................................................................................... 125
6.3.6 Patient description and invasive clinical measurement ................................................. 127
6.4 RESULTS .......................................................................................................................... 128
6.4.1 Physiological accuracy of the model............................................................................ 128
6.4.2 Effect of endograft length on left ventricular workload ................................................ 130
6.4.3 Effect of endograft length on FL flow reversal ............................................................ 130
6.4.4 Effect of LV contractile state on FL flow reversal ....................................................... 133
6.5 Discussion ......................................................................................................................... 134
6.5.1 Model validation against invasive clinical measurements ............................................ 134
ix

6.5.2 Impact of endograft length on LV workload ................................................................ 135
6.5.3 Impact of Endograft Length on FL Thrombosis ........................................................... 136
6.5.4 Effect of medical therapy on FL thrombosis ................................................................ 137
6.5.5 Study limitations ......................................................................................................... 137
6.6 Conclusion ......................................................................................................................... 138
CHAPTER 7 : Framework development for patient-specific compliant aortic disease phantom
model fabrication: magnetic resonance imaging validation and deep-learning segmentation ........ 140
7.1 Chapter abstract ................................................................................................................. 140
7.2 Introduction ....................................................................................................................... 141
7.3 Materials and methods ....................................................................................................... 142
7.3.1 Patients description ..................................................................................................... 142
7.3.2 Prototyping the artificial patient-specific dissection models ......................................... 143
7.3.3 Segmentation and deep learning .................................................................................. 146
7.3.4 Magnetic resonance imaging of the artificial phantoms................................................ 148
7.3.5 In-vitro hemodynamic measurements .......................................................................... 149
7.3.6 Invasive clinical data collection ................................................................................... 150
7.4 Results ............................................................................................................................... 151
7.4.1 Accuracy of the deep-learning for dissection segmentation .......................................... 151
7.4.2 Accuracy of the phantom structure via MRI ................................................................ 153
7.4.3 Accuracy of the In-vitro pressure measurements inside the phantom model ................. 156
7.5 Discussion ......................................................................................................................... 157
7.6 Conclusion ......................................................................................................................... 158
x

CHAPTER 8 : Longitudinal stretching-based wave pumping in compliant tubes: a
bio-inspired approach .......................................................................................................... 159
8.1 Chapter abstract ................................................................................................................. 159
8.2 Introduction ....................................................................................................................... 160
8.3 Materials and methods ....................................................................................................... 163
8.3.1 Physical problem ........................................................................................................ 163
8.3.2 Governing equations ................................................................................................... 165
8.3.3 Implementation of the boundary conditions ................................................................. 166
8.3.4 Computational model and numerical method ............................................................... 167
8.3.5 Analysis method ......................................................................................................... 168
8.4 Results and discussion........................................................................................................ 170
8.4.1 Wave speed and natural frequency .............................................................................. 170
8.4.2 Effect of frequency ..................................................................................................... 171
8.4.3 Pumping mechanism ................................................................................................... 174
8.4.4 Effect of wave speed ................................................................................................... 179
8.4.5 Effect of tube length .................................................................................................... 181
8.4.6 Theoretical analysis of the net flow ............................................................................. 181
8.5 Conclusion ......................................................................................................................... 187
CHAPTER 9 : Aortic stretch and recoil create pumping in the systemic circulation: an assist
mechanism for left ventricular function ................................................................................. 189
9.1 Chapter abstract ................................................................................................................. 189
9.2 Introduction ....................................................................................................................... 189
xi

9.3 Methods ............................................................................................................................. 191
9.3.1 Hydraulic circuit ......................................................................................................... 191
9.3.2 Aortic phantom fabrication.......................................................................................... 192
9.3.3 Longitudinal stretching mechanism ............................................................................. 193
9.3.4 Measurement devices .................................................................................................. 193
9.3.5 Experimental procedure .............................................................................................. 194
9.3.6 Flow visualization study.............................................................................................. 195
9.3.7 Pulse wave velocity analysis ....................................................................................... 195
9.3.8 Aortic compliance measurements ................................................................................ 195
9.4 Results ............................................................................................................................... 196
9.4.1 Physiological accuracy of the fabricated phantom models............................................ 198
9.4.2 Sample hemodynamic waveforms ............................................................................... 199
9.4.3 Effect of frequency on the longitudinal wave pumping ................................................ 202
9.4.4 Effect of aortic stiffness on the longitudinal wave pumping ......................................... 204
9.4.5 Effect of aortic stretch on the longitudinal wave pumping............................................ 204
9.4.6 Wave intensity analysis ............................................................................................... 205
9.5 Discussion ......................................................................................................................... 208
9.6 Conclusion ......................................................................................................................... 213
CHAPTER 10 : Role of aortic longitudinal wave pumping on heart-brain hemodynamic
coupling ............................................................................................................................ 214
10.1 Chapter abstract ............................................................................................................... 214
10.2 Introduction ..................................................................................................................... 214
xii

10.3 Methods ........................................................................................................................... 215
10.4 Results and discussion ...................................................................................................... 216
10.5 Conclusion ....................................................................................................................... 220
References ......................................................................................................................... 221

xiii

List of Tables
Table 2.1 The baseline values of the physical characteristics for the relevant arterial segments............................... 14
Table 3.1 Physical parameters used in this study. .................................................................................................. 38
Table 3.2 Impact of LV contractility at two levels of aortic stiffness on the transmitted pulsatility to the brain. ...... 46
Table 3.3 Impact of LV contractility on carotid WI indices at different aortic stiffness (PWV). .............................. 50
Table 3.4 Impact of heart rate on carotid WI indices at different aortic stiffness (PWV). ........................................ 51
Table 4.1 Baseline Characteristics of Patient Data (N = 1617). .............................................................................. 65
Table 4.2 Accuracy of radial-based estimated wave intensity (WI) analysis for peak amplitudes, timings, and
reflection coefficients (N = 1617). ........................................................................................................................ 74
Table 4.3 Accuracy of carotid-based estimated wave intensity (WI) analysis for peak amplitudes, timings, and
reflection coefficients (N = 1617). ........................................................................................................................ 79
Table 4.4 Accuracy of carotid-based estimated wave intensity (WI) analysis for peak amplitudes and reflection
coefficients for healthy and diseased participants. ................................................................................................. 81
Table 5.1 Baseline Characteristics of Patient Data (N = 2640). .............................................................................. 91
Table 5.2 Regression statistics between predicted and exact WI peak amplitudes. .................................................. 98
Table 5.3 Regression statistics between predicted and exact WI peak times. ........................................................ 100
Table 5.4 Comparison between the proposed Fourier-based ML and the previous ODE-based analytical models. . 103
Table 6.1 Geometric and material parameters used in the computational models. ................................................. 116
Table 6.2 Comparison between invasive clinical measurements and the results from our computational model..... 129
Table 7.1 Baseline Characteristics of Patient Data (N = 19). ................................................................................ 143
Table 7.2 Anatomical and Geometrical Parameters of the Type B Dissection Patients Utilized for Phantom
Fabrication. ........................................................................................................................................................ 144
Table 7.3 Dice similarity index between the predicted and the ground truth. ........................................................ 153
Table 8.1 Physical parameters used in the model. ................................................................................................ 165
Table 8.2 Parameter settings for different case studies. ........................................................................................ 169
Table 9.1 Dynamical and physiological properties of the fabricated artificial aorta............................................... 199
xiv

List of Figures
Fig. 2.1 Physical model of the 1D vascular network and inlet flow. ....................................................................... 13
Fig. 2.2 Sample pressure and flow data in caortid artery. ....................................................................................... 21
Fig. 2.3 The carotid flow augmentation index versus the aortic pressure augmentation index. ................................ 22
Fig. 2.4 Average transmitted pulatile power to the brain per cardiac cycle at different wave dynamics conditiosns. 23
Fig. 2.5 Pulsatile power transmission to the brain as a function of the Heart Rate and Aortic Arch PWV. ............... 24
Fig. 2.6 Sensitivity of different pulsatility indices to various wave dynamics states. ............................................... 25
Fig. 3.1 Schematic of the closed-loop cardiovascular system model. ...................................................................... 34
Fig. 3.2 Components of the interest and study design for varying relevant parameters in Chapter 3. ....................... 41
Fig. 3.3 Effects of LV contractility on central hemodynamics. ............................................................................... 43
Fig. 3.4 Compounded and isolated impacts of contractity on transmited pulsatile energy to the brain. .................... 44
Fig. 3.5 Impact of contractility on the transmitted energy to the brain at different wave states. ............................... 45
Fig. 3.6 Impact of heart rate on the transmitted energy to the brain at different wave states and LV contractility..... 47
Fig. 3.7 Impact of heart rate on the brain flow pulsatility index at different wave states and LV contractility. ......... 48
Fig. 3.8 Sample carotid Wave Intensity (WI) patterns at different heart rates and contractilites. ............................. 49
Fig. 3.9 Impact of aortic stiffness on brain perfusion at different cardiac conditions. .............................................. 52
Fig. 4.1 Schematic of the computation of exact WI (d𝐼 ) and pressure-only estimate of WI (d𝐼 ). ............................. 70
Fig. 4.2 Visualization of the reservoir pressure approach and corresponding wave intensity patterns. ..................... 73
Fig. 4.3 Bland-Altman plots for Wf1, Wf2 and Wb1. Agreement of peak amplitude values between exact wave
intensities and those estimated by only radial pressure measurements. ................................................................... 76
Fig. 4.4 Bland-Altman plots for Wf1, Wf2 and Wb1. Agreement of peak amplitude values between exact wave
intensities and those estimated by only carotid pressure measurements. . ............................................................... 77
Fig. 5.1 Schematic of the computation of exact WI and pressure-only estimate of WI. ........................................... 93
Fig. 5.2 Associated error between the reconstructed pressure waveform based on different number of Fourier
modes and the measured pressure waveform.. ....................................................................................................... 95
Fig. 5.3 Scatter and Bland-Altman plots for Wf1, Wf2 and Wb1 amplitudes. ......................................................... 99
Fig. 5.4 Scatter and Bland-Altman plots for Wf1, Wf2 and Wb1 times. ............................................................... 101
Fig. 5.5 Boxplots for WI parameters and the comparison between different ML models. ...................................... 102
xv

Fig. 5.6 Impact of number of Fourier modes on the proposed method accuracy. ................................................... 104
Fig. 5.7 Sensitivity of precision in terms of normalized root mean square (NRMSE) to the number of the
training data. ...................................................................................................................................................... 105
Fig. 6.1 Sample CT image for type B aortic dissection patient and the employed model-based approach in this
chapter. .............................................................................................................................................................. 115
Fig. 6.2 Sensitivity of the wall dispalcement to different leveles of aortic stiffness. .............................................. 119
Fig. 6.3 Time-varying end systolic elastance heart model used in this chapter. ..................................................... 120
Fig. 6.4 Algorithm for the LV-dissection system model. ...................................................................................... 125
Fig. 6.5 In-vivo pressure measurement data during the operation of type B aortic dissection patient. .................... 127
Fig. 6.6 Sample pressure and flow data in the developed dissection model. .......................................................... 129
Fig. 6.7 Impact of endograft length and heart rate on LV workload. ..................................................................... 130
Fig. 6.8 Spatial distribution of fluid and solid behavior in the FSI type-B dissection model at various times
during the cardiac cycle. ..................................................................................................................................... 131
Fig. 6.9 Simulated septum wall displacement waveform for different graft lengths during one cardiac cycle. ....... 131
Fig. 6.10 Simulated flow velocity waveform inside the false lumen for different graft lengths during one
cardiac cycle....................................................................................................................................................... 132
Fig. 6.11 Impact of endograft length and heart rate on thrombose formation in FL.. ............................................. 133
Fig. 6.12 Impact of LV contractile state on FL thrombosis. ................................................................................. 133
Fig. 7.1 Fabrication overview for patient-specific 3D printed aortic dissection phantom. ...................................... 146
Fig. 7.2 The deep learning pipeline for dissection auto-segmentation. .................................................................. 148
Fig. 7.3 Full hydraulic circuit for in-vitro flow modeling of the patient-specific phantoms with the
corresponding physical components. ................................................................................................................... 150
Fig. 7.4 Sample segmented for the aortic dissection. ............................................................................................ 152
Fig. 7.5 Schematic and axial MR images captured at different section along phantoms 1 and 2. ........................... 154
Fig. 7.6 Schematic, coronal and axial MR images captured at different section along phantoms 3 and 4. .............. 155
Fig. 7.7 Comparison between measured clinical and in-vitro data.. ...................................................................... 156
Fig. 8.1 Schematic representation of the longitudinal impedance pump. ............................................................... 164
Fig. 8.2 Implemented boundary condition at the tube root based on the reported physiological measurement. ...... 167
xvi

Fig. 8.3 Implementation of FSI algorithm to solve the longitudinal wave pumping system model. ........................ 168
Fig. 8.4 System’s response to impulse stretching. ................................................................................................ 171
Fig. 8.5 Sample dipslacement waveforms at the root and flow wavefroms at the outlet. ....................................... 172
Fig. 8.6 Flow-frequency analysis at the baseline model parameters. ..................................................................... 174
Fig. 8.7 Flow-frequency analysis at different levels of the root displacements. ..................................................... 175
Fig. 8.8 Spatial distributions of flow behavior in the longitudinal impedance pump model at different
snapshots of time during cycle 𝑻 and for different frequencies............................................................................. 176
Fig. 8.9 Tube wall displacement as a function of frequency. ................................................................................ 177
Fig. 8.10 Pressure-flow (P-Q) loops at six locations along the tube for stretching frequency of 1.6 Hz. ................ 179
Fig. 8.11 Impact of tube stiffness on the generated flow in the longitudinal impedance pump. .............................. 180
Fig. 8.12 Impact of tube length on the generated flow in the longitudinal impedance pump. ................................. 181
Fig. 8.13 Comparison between the computed Normalized flow rate from the theory and numerical simulations.... 187
Fig. 9.1 Clinical motivation of investigating aortic stretch and recoil wave pumping and the in-vitro setup........... 198
Fig. 9.2 Measured hemodynamic waveforms at the aortic root for different values of the stretching frequency. .... 201
Fig. 9.3 Flow visualization due to the aortic stretch and recoil as well as flow-frequency behavior. ...................... 203
Fig. 9.4 Flow-frequency behavior at different levels of aortic stiffness measured by wave speed. ......................... 204
Fig. 9.5 The effect of stretching amplitude on flow-frequency pattern. ................................................................. 205
Fig. 9.6 Wave analysis on the propagated waves in the vasculature due to the aortic stretch and recoil. ................ 207
Fig 10.1 Schematic representation of the in-vitro hydraulic circuit to conduct the experiments. ............................ 216
Fig 10.2 Measured pressure and flow waveforms at the carotid artery for different values of the stretching
frequency. .......................................................................................................................................................... 217
Fig 10.3 Flow-frequency relation at the carotid artery at different levels of aortic stiffness. .................................. 218
Fig 10.4 Flow-frequency relation at the carotid artery.......................................................................................... 219
Fig 10.5 Snapshots of the flow over time at the carotid artery due to the aortic stretch and recoil. ........................ 220

xvii

Abstract
This dissertation focuses on the role of aortic biomechanics in the heart-aorta-brain system.
Combination of in-vitro experimentations, numerical simulations, and machine learning
approaches are employed to achieve the aims of this thesis. We uncovered a new pumping
mechanism in the arterial system based on the ascending aortic longitudinal stretch and recoil
due to the left ventricle systolic long-axis shortening. Our findings indicated that stretching-
based aortic wave pumping generates significant flow which can assist left ventricle.
Furthermore, results show that this mechanism has a major impact on the cerebral blood flow.
This complex pumping effect is a function of the wave dynamic conditions that are mainly
dictated by the heart rate and the wave speed inside the aorta. We also demonstrated that wave
dynamics in the aorta dominate the pulsatile hemodynamics of the brain. Findings indicated the
existence of an optimum wave state, near normal human heart rate, where destructive pulsatile
energy transmission to the brain is minimized. Based on the impact of aortic biomechanics on
cardiovascular and cerebrovascular system, we developed a hybrid Fourier decomposition-
machine learning algorithm that facilitates wave energy calculation via single non-invasive
pressure measurement. Our method was tested in a large clinical cohort and could successfully
capture the wave energy features of arterial system. The findings of this thesis are expected to
provide valuable insights regarding the impaired heart-aorta-brain system and can be an initial
step towards the development of diagnostic tools and assist devices for patients suffering from
heart diseases and vascular brain damage.
1

CHAPTER 1 : Introduction
1.1 Motivation
Neurodegenerative diseases such as Alzheimer’s and other related dementias have reached
an epidemic proportion with a significant impact on public health. In 2017, it is estimated that
there were 5.7 million Americans with Alzheimer’s disease (AD) which was associated with a
cost of 232 billion dollars for care and lost productivity for patients and their caregivers [1].
There is no cure for AD and there are different major risk factors recognized for Alzheimer
development such as increasing age, family history, hypertension, hypotension, and high
cholesterol levels [2]. Although the cause or cure for AD is not fully understood, the
identification of the risk factors that cause brain injury may offer important ways to mitigate the
development of this disease [3]. Vascular risk factors such as hypertension increase the risk for
AD and serve as potential targets for prevention [4]. Recent studies have demonstrated that
identifying arterial stiffening can improve the prediction of hypertension-related risk for both the
cerebral microvascular ischemic disease and neurodegenerative changes associated with AD [5].
In cerebral circulation, a common manifestation of hypertensive damage is white matter
hyperintensities (WMH) [6]. WMH predict risk for significant morbidity with aging including
risk of death, functional impairment and dementia [7]. In the population-based Dallas heart study
[8, 9], it was shown that aortic arch pulse wave velocity (PWV) measured by magnetic resonance
is a more a robust predictor of WMH volume than clinical assessments of blood pressure or
hypertension. Furthermore, the presence of aortic stiffness was independent of—but additive
to—the presence of hypertension in predicting WMH [8]. These results indicate the existence of
a potential link between arterial stiffening and AD. While there are several biomarkers for
2

quantification of arterial stiffness such as total arterial compliance and carotid-femoral PWV,
recent clinical data reveals a stronger association of aortic PWV with cerebrovascular and other
extra-cardiac events [9].
Furthermore, population-based clinical studies have suggested that Heart Failure (HF)
patients who suffer from impaired LV function have worse degrees of cognitive impairment than
age-matched individuals without HF [2, 10]. In general, HF is defined as a progressive condition
that leads to inadequate cardiac output for meeting metabolic demands. HF has been proposed as
a risk factor for Alzheimer’s disease (AD), where the current clinical hypothesis is that the
decreased cerebral blood flow due to HF may contribute to the dysfunction of the neurovascular
unit and hence may lead to impaired clearance of amyloid beta [2, 11-13]. In addition to the
consequences of HF, age-associated changes in ventricular wall thickening and stiffening may
trigger heart remodeling that can also affect cerebral hemodynamics. Previous studies have not
adequately addressed the effect of interactions between the aorta and the left ventricle (LV) on
destructive pulsatile energy transmission to the cerebral circulation nor on brain perfusion. This
is due to the fact that there are inherent difficulties in studying the isolated effects of aortic wave
dynamics and cardiac function on brain hemodynamics [14, 15].
Another dynamic mode of the aorta in the heart-aorta-brain system is related to its
longitudinal stretch and recoil due to the LV systolic long-axis shortening. Recent clinical studies
showed that in an optimal heart-aorta coupling, LV systolic contraction displaces the aortic
annulus and produces a considerable longitudinal stretch in the ascending aorta (AA). The force
associated with this mechanical coupling increases the systolic load on the LV but also stores
energy in the elastic elements of the proximal aorta. While the importance of the stretch-related
aortic work has shown in relation to diastolic filling and HF [16], the full aspects of this dynamic
3

mode on LV workload is not known. Specifically, the connection between the longitudinal aortic
stretching and cardiac pumping efficiency is not fully understood. In addition, the fluid dynamic
mechanisms relating stretch-related dynamic mode of the aorta to the cerebrovascular perfusion
have not been investigated.
All in all, although previous studies have attempted to elucidate the underlying mechanisms
involved in vascular brain damage, these studies do not adequately address the hemodynamic
couplings at heart-aorta-brain interfaces. Importantly, the dynamic behavior of the aorta in
transferring blood to the cerebral vasculature in healthy individuals as well as the diseased one
(including the heart failure population or patients suffering from aortic diseases) have not been
investigated. Much remains to be understood about the impact of major aging mechanisms in the
arterial system and cardiac function on brain hemodynamics. In order to identify potential
therapeutic targets for intervention in patients suffering from heart and brain disease, there is an
essential need for a well-designed quantitative study to investigate the association between aortic
wave dynamic modes (such as the impact of its stiffness or stretch-related work) and the pulsatile
load on the heart as well as the energy transmission to the cerebrovascular network.
Understanding this physics can further contribute to the development of better diagnostic tools
that are able to quantify cardiovascular and cerebrovascular function.
1.2 Thesis objectives
The specific aims that we are going to follow in this dissertation are:
- To investigate the effect of age-related changes in the aortic stiffness on energy and
blood flow transmission to the brain.
4

- To investigate the effect of left ventricular contractility and cardiac dynamics on energy
and blood flow transmission to the cerebral circulation.
- To develop more efficient and affordable methods to compute energy-based indices that
quantify cardiovascular and cerebrovascular function.
- To investigate the fluid dynamics in the aortic dissection, as an example of life-
threatening complex aortic disease, and its impact on heart-aorta coupling.
- To investigate the effects of ascending aortic longitudinal stretch and recoil on heart
workload as well as the volume blood flow transmission to the brain.
1.3 Thesis outline
In chapter 2, we use an energy-based analysis of hemodynamic waves to quantify the effect
of aortic stiffening on transmitted pulsatility to cerebral vasculature, employing a computational
approach. We show the existence of an optimum wave condition—occurring near normal human
heart rates—that minimizes pulsatile energy transmission to the brain. We further demonstrate
the major role of aortic biomechanics on heart-brain coupling. The importance of the energy-
based indices for capturing the transmitted pulsatility to the brain is also discussed in this
chapter.
In chapter 3, employing 1D model of the entire human circulation, we investigate how age-
related changes in the cardiac function and vasculature affect underlying mechanisms involved in
LV-aorta-brain system. We show that LV contractility alone affects the pulsatile energy
transmission to the brain even at the preserved cardiac output. We further show that at low levels
of LV contractility which can happen in HF, the reduction in cerebral blood flow due to age-
related changes in aortic stiffness becomes more pronounced. Our findings in this chapter
5

demonstrate the level of coupling in the LV-aorta-brain system and show how such coupling is
affected by age-related changes in the cardiovascular system.
In chapter 4, we investigate an applicability of a new method for computing wave intensity
which a well-established tool for quantifying the energy carried in arterial waves. The primary
drawback of the conventional methods to quantify wave intensity is the need for concurrent
measurements of both pressure and flow waveforms. We provide a comprehensive analysis on
the accuracy of pressure-only estimates of wave intensity derived from radial and carotid
pressure waveforms in large population of healthy and diseased individuals. The strengths and
weaknesses of this method, which is based on the mathematical decomposition of the pressure
waveform, is discussed in this chapter.
Inspired by the findings in chapter 4, we propose a new approach based on Fourier-
decomposition Machine learning to compute wave intensity using single pressure measurement
in chapter 5. It is shown that this new method is able to fully capture the wave intensity patterns
in large clinical database.
In chapter 6, we investigate the fluid dynamics in a life-threatening aortic disease, called
aortic dissection. We develop a model-based approach based on the fluid-structure interaction
computational fluid dynamics to study the hemodynamic variables in dissection. Using energy-
based indices, the impact of this disease and the current treatment strategies on the heart-aorta
coupling was quantified.
Motivated by the aims defined in chapter 6, we extend our analysis on fluid dynamics in
dissection to patient-specific models through chapter 7. We propose a new approach toward fully
automated patient-specific type B aortic dissection model fabrication. The developed framework
is helpful for in-vitro flow modeling in dissection.
6

In chapter 8, we introduce the concept of a longitudinal stretching-based impedance pump, a
novel pumping mechanism inspired by the human aorta. The pump’s behavior is quantified as a
function of stretching and material characteristics. It is shown that like conventional impedance
pumps, both the direction and magnitude of the net flow depend on wave characteristics in
compliant tubes.
In chapter 9, we investigate the isolated effects of stretch-related aortic dynamic mode on
wave pumping in the cardiovascular system. We show that stretching-based longitudinal wave
pumping in a compliant aorta creates wave propagation and reflections, which generate
significant flow. This complex pumping effect is a function of the wave dynamic parameters,
dominated by the stretching frequency and the pulse wave speed inside the aorta.
In chapter 10, we further investigate the impact of longitudinal dynamic mode of the aorta
on the brain circulation. Two modes are identified based on the wave dynamics condition: a
pumping mode and the suction mode. We show that aortic stretch and recoil due to the left
ventricle long-axis shortening has a major impact on the cerebral blood flow.

7

CHAPTER 2 : Dynamic effects of aortic arch stiffening on pulsatile energy
transmission to cerebral vasculature: a core determinant of healthy brain-
heart coupling

This chapter is based on the following published manuscript: Aghilinejad, A., Amlani, F., King,
K.S. and Pahlevan, N.M., 2020. Dynamic effects of aortic arch stiffening on pulsatile energy
transmission to cerebral vasculature as a determinant of brain-heart coupling. Scientific reports,
10(1), p.8784.

2.1 Chapter abstract
Aortic stiffness increases with age and is a robust predictor of brain pathology including
Alzheimer’s and other dementias . Aging causes disproportionate stiffening of the aorta compared
with the carotid arteries, reducing protective impedance mismatches at their interface and
affecting transmission of destructive pulsatile energy to the cerebral circulation. Recent clinical
studies have measured regional stiffness within the aortic arch using pulse wave velocity (PWV)
and have found a stronger association with cerebrovascular events than global stiffness
measures. However, effects of aortic arch PWV on the transmission of harmful excessive
pulsatile energy to the brain is not well-understood. In this chapter, we use an energy-based
analysis of hemodynamic waves to quantify the effect of aortic arch stiffening on transmitted
pulsatility to cerebral vasculature, employing a computational approach using a one-dimensional
model of the human vascular network. Results show there exists an optimum wave condition—
occurring near normal human heart rates—that minimizes pulsatile energy transmission to the
brain. This indicates the important role of aortic arch biomechanics on heart-brain coupling. Our
8

results also suggest that energy-based indices of pulsatility combining pressure and flow data are
more sensitive to increased stiffness than using flow or pressure pulsatility indices in isolation.
2.2 Introduction
The proximal aorta acts as a coupling device between the heart and the brain, regulating the
amount of pressure and flow pulsatility transmitted to the cerebral vasculature [17]. In young
healthy adults, the conduit arteries arising from the aorta, such as the carotid artery, have higher
stiffnesses than the aortic arch. This distinct difference in compliance between the highly elastic
aorta and the more muscular branch vessels results in a high impedance mismatch. This
mismatch causes a large pulse wave reflection at the aorta-carotid interface which protects the
brain microvasculature from high pulsatile energy. Disproportionate age-related stiffening of the
aorta (relative to the carotid arteries) is theorized to reduce this protective impedance mismatch
at the interface and thereby affect the wave reflection [18]. These age-related changes in the
biomechanics of the aorta may therefore have a significant impact on the transmission of
potentially deleterious pulsatile energy into the microcirculation resulting in impaired regulation
of local blood flow and in tissue damage. The brain is particularly susceptible to pulsatile
damage resulting from low vascular resistance related to its high resting rate of blood flow and
low impedance [19].
There is an essential need for a well-designed quantitative study to investigate the
association between aortic arch stiffness and the pulsatile energy transmission to the
cerebrovascular network (that causes brain insult) in order to identify potential therapeutic
targets for intervention. Hence, in this chapter, we are interested in studying how changes in
aortic stiffness and their corresponding wave dynamics lead to excessive pulsatile energy
transmission to the brain. To this end, we have employed a one-dimensional (1D) computational
9

model of the arterial network that has been validated and used extensively for studying the
pressure and flow wave propagation along the arterial network [20, 21].
In clinical studies, flow and pressure pulsatility indices in the carotid arteries are commonly-
used parameters for determining pulsatility transmission to the brain [22]. In this chapter, we
have performed an energy-based analysis for quantification of the hemodynamic pulsatility.
Specifically, we have focused on the pulsatile portion of the net power transmitted to the brain.
In the Dallas Heart Study [8], it has been shown that aortic arch PWV (which is a contributor to
the pulsatile portion of the net power) and increased blood pressure (which is a contributor to the
steady portion of the power) have independent associations with brain vascular insult. Due to this
independence, we put the focus of the current study only on the pulsatile portion of the
transmitted power in order to capture the dynamic effects of arterial stiffening on pulsatility
transmitted to the cerebral vasculature. This analysis considers the combined effects of flow and
pressure propagation into the brain and demonstrates greater sensitivity to increased stiffness
than using a pressure or flow pulsatility index alone. Simulations have been performed at
different HRs to account for different wave conditions [23, 24].
2.3 Theoretical indicators of an optimum wave condition
As an introduction to the energy-based analysis used in this chapter, it is helpful to consider
the theoretical relation for the transmitted power through waves. Following the analysis of
Alastruey et al. [25] and Passerini [26], the governing equations for pressure and flow
propagation that are employed later in this work can be linearized about a reference state in the
space-time domain. Under the assumption of periodicity for the propagated pressure and flow
waves inside the vasculature, the solutions for pressure 𝑝 (𝑥 ,𝑡 ) and flow 𝑞 (𝑥 ,𝑡 ) at a time t and at
a position x can be sought as harmonic waves of the form
10

𝑝 (𝑥 ,𝑡 )=𝑝 0
𝑒 𝑖 (𝜔𝑡 −𝑘𝑥 )
, (2.1)
𝑞 (𝑥 ,𝑡 )=𝑞 0
𝑒 𝑖 (𝜔𝑡 −𝑘𝑥 )
, (2.2)
where 𝑝 0
and 𝑞 0
are the (possibly complex) amplitudes of the pressure and flow at (𝑥 ,𝑡 )=
(0,0) , 𝜔 is the frequency of the oscillation, and k is the wavenumber. It can be shown [25] that
for a complex wavenumber k, the corresponding dispersion relation for temporal frequency 𝜔 is
given by
𝜔 =
𝑖 (𝐶 2
𝑘 2
+𝐶 3
)±√−(𝐶 2
𝑘 2
+𝐶 3
)
2
+4𝐶 1
𝑘 2
2
(2.3)
and the subsequent phase velocity 𝑐 𝑤 = 𝜔 /𝑘 is given by
𝑐 𝑤 =
𝑖 (𝐶 2
𝑘 +
𝐶 3
𝑘 )±
√
−(𝐶 2
𝑘 +
𝐶 3
𝑘 )
2
+4𝐶 1
2
, (2.4)
where 𝑖 =√−1 and where the constants 𝐶 1
, 𝐶 2
, and 𝐶 3
are combinations of physical
parameters that ultimately account for wall compliance, flow inertia and the resistance to the
flow.
Assuming that pressure amplitude 𝑝 0
is real-valued, a mass conversation argument yields a
resulting complex amplitude for flow as a function of phase velocity [25], i.e., 𝑞 0
=𝑞 0
(𝑐 𝑤 ) .
Hence the corresponding physical solutions, which are represented by the real parts of Eq. (2.1)
and Eq. (2.2), are given by
ℜ(𝑝 (𝑥 ,𝑡 ))=𝑝 0
𝑒 ℑ(𝑘 )𝑥 cos (𝜔𝑡 −ℜ(𝑘 )𝑥 ) , (2.5)
ℜ(𝑞 (𝑥 ,𝑡 ))=𝑒 ℑ(𝑘 )𝑥 (ℜ(𝑞 0
(𝑐 𝑤 ))cos (𝜔𝑡 −ℜ(𝑘 )𝑥 )−ℑ(𝑞 0
(𝑐 𝑤 ))sin(𝜔𝑡 −ℜ(𝑘 )𝑥 )) , (2.6)
11

where ℜ and ℑ denote the real and imaginary parts of a complex number, respectively. The
complex-valued amplitude of the flow given by Eq. (2.2) indicates a (frequency-dependent)
phase-shift between the pressure p and the flow q that is expressed in Eq. (2.6). The
corresponding instantaneous transmitted power, derived by multiplying the propagated pressure
and flow at a point x, is hence given by
𝑊 ̇ =ℜ(𝑝 (𝑥 ,𝑡 ))∙ℜ(𝑞 (𝑥 ,𝑡 ))=
1
2
𝑝 0
ℜ(𝑞 0
(𝑐 𝑤 ))𝑒 2ℑ(𝑘 )𝑥 +
1
2
𝑝 0
|𝑞 0
(𝑐 𝑤 )|𝑒 2ℑ(𝑘 )𝑥 sin(2𝜔𝑡 −2ℜ(𝑘 )𝑥 +𝜑 (𝑐 𝑤 )) , (2.7)
where |𝑞 0
(𝑐 𝑤 )| denotes the complex amplitude of the flow rate and 𝜑 (𝑐 𝑤 ) is a (frequency-
dependent) phase difference.
Even for the linearized system of governing equations considered in the above analysis, one
can note the complex (nonlinear) dependencies on the wavenumber for oscillation frequency 𝜔
(Eq. (2.3)) and for wave speed 𝑐 𝑤 (Eq. (2.4)). In particular, these dependencies are prominent in
the corresponding transmitted power given by Eq. (2.6); they may be reasonably assumed to be
further nonlinear when incorporating more elastic/viscoelastic effects as well as wave reflections
that result from interfaces in vasculature (see Methods). This very simplified analysis suggests
that extrema may be found in pulsatility power curves as a function of both frequency (i.e., HR)
and wave speed (i.e., PWV). This seems to be the case even in the absence of wave reflections
(i.e., branching of the vasculature) and the fully-coupled nonlinear formulation considered later
in this paper. Thus, the primary motivation of this chapter is to investigate these dynamic effects
through a computational approach that treats a more physiologically relevant and physically
accurate viscoelastic hemodynamics model for understanding the behavior of these physical
variables on energy transmission to the brain.
12

All in all, wave dynamics in a compliant tube depend on three parameters: 1) fundamental
frequency of the propagating waves, 2) wave speed as a function of material properties, and 3)
reflection sites [24]. In what follows, we have focused primarily on investigating the effect of the
first two parameters on the transmitted pulsatile power to the brain. However, the relative change
of aortic arch stiffness with respect to the branches will affect the wave reflections as well, and
hence the third is implicitly studied through the physiological relevance of the physical and
mathematical models.
2.4 Methods
2.4.1 Physical model
A validated 1D model of the vascular network based on space-time variables has been
employed in this study [27, 28]. 1D arterial models have been shown to be a powerful tool for
studying hemodynamics and wave dynamics in both large systemic arteries [29] as well as the
entire adult circulation [30]. The physical model used in this study consists of 55 larger systemic
arteries, where each artery is modeled as a visco-elastic tube characterized by its diameter,
length, Young’s mo dulus, viscosity and wall thickness. To consider the effect of visco-elasticity,
the Voigt-type model (a combination of a linear spring and a linear viscous dashpot connected in
parallel [31]) has been employed. The arterial wall is assumed to be thin, incompressible,
homogenous and isotropic. In this chapter, our focus was to investigate the effect of proximal
aorta stiffness on pulsatility transmission into the brain. For this purpose, the first three portions
of the aorta (1-ascending aorta, 2-proximal aortic arch feeding the brachiocephalic and left
common carotid arteries, and 3-distal aortic arch feeding the left vertebral and left subclavian
arteries) were altered while all the other segments of the aorta and systemic vasculature were
kept constant (Fig. 2.1(a)).
13

Fig. 2.1 Physical model of the 1D vascular network and inlet flow. (a) Schematic of the systemic vasculature with a zoom on
the aortic arch (dashed box). (b) The physiological inflow waveform prescribed at the aortic root of the baseline model where the
corresponding cardiac output is 5.2 l/min.

At the inlet, we impose a physiological flow wave at the aortic root as shown in Fig. 2.1(b).
The blood is assumed to be an incompressible Newtonian fluid with density of ρ=1050 kg/m
3

and viscosity of µ=4 mPa∙s. Different levels of aortic arch rigidity are considered by employing
multiplicative factors of a minimum rigidity level 𝐸 1
(𝑥 ) that corresponds to the baseline PWV of
c1 initially prescribed to the model. The baseline properties for aortic segments that were altered
in this study are presented in Table 2.1. In this table, 𝒄 𝒊𝒏
and 𝒄 𝒐𝒖𝒕 refer to the wave speed at the
inlet and outlet of the segment, respectively.

(a)
(b)
14

Table 2.1 The baseline values of the physical characteristics for the relevant arterial segments.
Name Length (cm)
𝒄 𝒊𝒏
→𝒄 𝒐 𝒖 𝒕 (
m
s
)
Ascending Aorta 5.8 3.95→3.96
Aortic Arch1 2.3 4.15→4.2
Aortic Arch2 4.5 4.35→4.39
Right Common Carotid 10.8 5.32→6.47
Right Vertebral 17.1 8.03→8.73
Left Common Carotid 16 5.51→6.78
Left Vertebral 17 8.03→8.73
Values are adopted from Alastruey [32].
2.4.2 Mathematical and computational model
Conservation of mass and momentum applied to a 1D impermeable and deformable tubular
control volume of an incompressible Newtonian fluid, flowing with a constant axisymmetric
velocity profile, yields the system of equations
{

𝜕𝐴
𝜕𝑡
+
𝜕 (𝐴𝑈 )
𝜕𝑥
=0, (2.8)
𝜕𝑈
𝜕𝑡
+𝑈 𝜕𝑈
𝜕𝑥
+
1
𝜌 𝜕𝑝
𝜕𝑥
=
𝑓 𝜌𝐴
, (2.9)

where x is the axial coordinate along the vessel, t is the time, A(x,t) is the cross-sectional
area of the lumen, U(x,t) is the average axial velocity, p(x,t) is the average internal pressure over
the cross-section, and f is the friction force per unit length. For mathematical simplification it is
assumed that the Coriolis coefficient (velocity shape factor) is unity, resulting in a flat velocity
profile for the 1D model and hence a corresponding friction force per unit length of 𝑓 =
15

−22𝜇𝜋𝑈 [33]. In order to close the system of Eq. (2.8) and (2.9) for the three unknowns A(x,t),
U(x,t) and p(x,t), a constitutive relation between the sectional pressure p and area A can be
implemented by a Voigt-type viscoelastic tube law. This relationship accounts for the fluid-
structure interaction of the problem and can be derived as [34]
𝑝 =𝑝 ext
+
𝐺 (𝑥 )
𝐴 0
(√𝐴 −√𝐴 0
)+
𝜆 (𝑥 )
𝐴 0
√𝐴 𝜕𝐴
𝜕𝑡
, (2.10)
where 𝑝 ext
is the constant external pressure and 𝐴 0
is the constant cross-sectional area at
equilibrium state (p, U)=(𝑝 ext
,0). The spatially-varying functions 𝐺 (𝑥 ) and 𝜆 (𝑥 ) are related to
the elastic and visco-elastic properties of the arterial wall, respectively, and can be given in terms
of material properties as
𝐺 (𝑥 )=
√𝜋 𝐸 (𝑥 )ℎ(𝑥 )
(1−𝜗 2
)
, (2.11)
𝜆 (𝑥 )=
√𝜋 𝜑 (𝑥 )ℎ(𝑥 )
2(1−𝜗 2
)
, (2.12)
where E(x) is the Young’s modulus, 𝜑 (𝑥 ) is the vessel wall viscosity, h(x) is the wall
thickness, and 𝜗 is the Poisson’s ratio of the wall (taken to be 𝜗 =1/2 assuming the wall is
incompressible). Note that following the definition of the local PWV in terms of area and
pressure, the wave speed can be written in terms of the elasticity factor 𝐺 (𝑥 ) as
𝑐 =√
𝐴 𝜌 𝜕𝑝
𝜕𝐴
=√
𝐺 (𝑥 )
2𝜌 𝐴 0
𝐴 1/4
. (2.13)
The system of partial differential equations (PDEs) in Eq. (2.8) and (2.9) can be represented
in matrix form as
16

{

∂U
𝜕𝑡
+
𝜕 F
𝜕𝑥
=H
𝑈 , (2.14)
U=[
𝐴 𝑈 ] , H
𝑈 =[
0
𝑓 𝜌𝐴
] , (2.15)
F=F
𝑒 +F
𝑣 , [
𝐴𝑈
𝑈 2
2
+
𝑝 𝑒𝑥𝑡 +
𝐺 (𝑥 )
𝐴 0
(√𝐴 −√𝐴 𝑑 )
𝜌 ]+ [
0
−
𝜆 (𝑥 )
𝐴 0
√𝐴 𝜕 (𝐴𝑈 )
𝜕𝑥
] , (2.16)

where Eq. (2.8) has been used to replace 𝜕𝐴 /𝜕𝑡 with 𝜕 (𝐴𝑈 )/𝜕𝑥 in Eq. (2.16). In order to
solve this system of hyperbolic PDEs, a discontinuous Galerkin scheme can be employed for
simplicity and fast convergence without causing excessive dispersion or diffusion errors [35].
Consider a spatial domain Ω=(a,b) discretized into a mesh of 𝑁 𝑒𝑙
elemental non-overlapping
regions Ω
𝑒 =(𝑥 𝑒 𝑙 ,𝑥 𝑒 𝑢 ) , e=1,…, 𝑁 𝑒𝑙
, where 𝑥 𝑒 𝑢 =𝑥 𝑒 +1
𝑙 and ⋃ Ω
𝑒 𝑁 𝑒𝑙
𝑒 =1
=Ω. The weak form of Eq.
(2.14) is given by
(
𝜕 U
𝜕𝑡
,𝜑 )
Ω
+(
𝜕 F
𝜕𝑥
,𝜑 )
Ω
=(H
𝑈 ,𝜑 )
Ω
, (2.17)
where 𝜑 (𝑥 ) is an arbitrary function on the domain Ω and (𝑣 ,𝑢 )
Ω
=∫ 𝑢𝑣𝑑𝑥 Ω
is the standard
𝐿 2
(Ω) inner product. The discrete form of the conservative representation in Eq. (2.17) can be
given by [32]
∑ [(
𝜕 U
𝛿 𝜕𝑡
,𝜑 𝛿 )
Ω
𝑒 +(
𝜕 F(U
𝛿 )
𝜕𝑥
,𝜑 𝛿 )
Ω
𝑒 +[𝜑 𝛿 ∙{F
𝑢 −F(U
𝛿 )}]
𝑥 𝑒 𝑙 𝑥 𝑒 𝑢 ]=∑ (H
𝑈 𝛿 ,𝜑 𝛿 )
Ω
𝑒 𝑁 𝑒𝑙
𝑒 =1
𝑁 𝑒𝑙
𝑒 =1
, (2.18)
where, following a traditional Galerkin approach, the superscript 𝛿 indicates that the
variable is approximated in the finite space of piecewise polynomial vector functions (the trial
space) and F
𝑢 is the approximation of the flux at an interface. In the trial space, the expansion
basis is chosen to be Legendre polynomials due to their orthogonality with respect to the 𝐿 2
(Ω
𝑒 )
17

inner product. Hence, the approximated solution on each elemental region U
𝑒 𝛿 can be expanded to
order M as
U
𝑒 𝛿 (𝜒 𝑒 (𝜉 ),𝑡 )=∑ 𝐿 𝑗 (𝜉 )U
̂
𝑒 𝑗 (𝑡 )
𝑀 𝑗 =0
, (2.19)
where 𝐿 𝑗 (𝜉 ) is the Legendre polynomial of order j with corresponding time-varying
coefficient U
̂
𝑒 𝑗 (𝑡 ) , and 𝜒 𝑒 (𝜉 )=𝑥 𝑒 𝐿 (1−𝜉 )/2+𝑥 𝑒 𝑅 (1+𝜉 )/2 is the elemental mapping.
Substituting Eq. (2.19) into Eq. (2.18) and letting φ
𝑒 𝛿 =U
𝑒 𝛿 yields a system of M+1 ordinary
differential equations (ODEs) for each U
̂
𝑒 𝑗 (𝑡 ) , j=0,…,M as
𝑑 U
̂
𝑒 𝑗 (𝑡 )
𝑑𝑡
=𝜓 (U
𝑒 𝛿 )=−(
𝜕 F(U
𝛿 )
𝜕𝑥
,𝐿 𝑗 )
Ω
𝑒 −
2
𝑥 𝑒 𝑅 −𝑥 𝑒 𝐿 [𝐿 𝑗 ∙{F
𝑢 −F(U
𝛿 )}]
𝑥 𝑒 𝐿 𝑥 𝑒 𝑅 +(H
𝑈 𝛿 ,𝐿 𝑗 )
Ω
𝑒 . (2.20)
Solving the ODE in Eq. (2.20) for each U
̂
𝑒 𝑗 (𝑡 ) yields the coefficients required to reconstruct
the physical solution given by Eq. (2.19). To calculate the fluxes at each interface between
elements, F
𝑢 is decomposed into the elastic term F
𝑒 𝑢 and the viscous term F
𝑣 𝑢 . The elastic term is
determined by solving the Riemann problem and the viscous term can be treated by the average
of the lower and upper limits in an elemental region. In order to numerically resolve U
̂
𝑒 𝑗 (𝑡 ) at a
discrete time 𝑡 =𝑡 𝑛 +1
, a second-order Adams-Bashforth time integration scheme is applied to Eq.
(2.20) for each j=0, …, M and e=1, …, 𝑁 𝑒𝑙
. This yields the iterative sequence with a time-step
Δ𝑡 [32]
(U
̂
𝑒 𝑗 (𝑡 𝑛 +1
))
𝑛 +1
=(U
̂
𝑒 𝑗 (𝑡 𝑛 ))+
3Δ𝑡 2
𝜓 (U
𝑒 𝛿 (𝑡 𝑛 ))−
Δ𝑡 2
𝜓 (U
𝑒 𝛿 (𝑡 𝑛 −1
)) , (2.21)
where we have taken the notational license U
𝑒 𝛿 (𝑡 )=U
𝑒 𝛿 (𝜒 𝑒 (𝜉 ),𝑡 ) .
18

A physiological flow wave (Fig. 2.1(b)) has been applied as an inlet flow to the aortic root
and scaled to give a cardiac output of 5.2 L/min for any given HR. At arterial segment junctions
and bifurcations (Fig. 2.1(a)), the boundary conditions are prescribed by enforcing conservation
of mass and continuity of the total pressure 𝑝 +.5𝜌 𝑈 2
. By decomposing the governing system of
equations (Eqs. (2.8) and (2.9)) into the characteristic variables, the system can be interpreted in
terms of forward and backward traveling waves. At any bifurcations and junctions, we have six
unknowns: (𝐴 𝑝 ,𝑈 𝑝 ) in the parent vessel, (𝐴 𝑑 1
,𝑈 𝑑 1
) in its first daughter vessel and (𝐴 𝑑 2
,𝑈 𝑑 2
) in
its second daughter vessel. This a set of six independent equations within the parent vessel, the
information can only reach the bifurcation by the forward traveling wave, while within the
daughter vessels the information can only reach the bifurcation by the backward traveling wave.
Therefore, the first three equations can be obtained by imposing that the characteristic variables
in each vessel (parent and daughters) remain constant (𝑑𝑊 /𝑑𝑡 =0, where 𝑊 is the
characteristic variable) [32]. The other three independent equations can be obtained by requiring
the conservation of mass and continuity of the momentum balance. The latter condition leads to
continuity of the total pressure at the boundary. It has been shown that energy losses at arterial
segment junctions only change the mean pressure and flow by less than 0.5% [36], hence they
are disregarded in the model. Finally, at terminal boundaries, three-element RCR Windkessel
models are employed as 0D lumped parameters that act as the outflow boundary condition on
pressure p(t) and flow q(t)=AU at each peripheral branch. This is given by the ODE
𝑑𝑝
𝑑𝑡
=𝑅 1
𝑑𝑞
𝑑𝑡
+
1
𝑅 2
𝐶 ((𝑅 1
+𝑅 2
)𝑞 −𝑝 ) , (2.22)
19

for inflow resistance R1 (that matches the characteristic impedance of the terminal vessel),
peripheral compliance C and outflow resistance R2—all of which are chosen within the average
physiological range [37].
A validated code called “Nektar” was used to solve the d iscretized equation [20, 34, 38-40].
This code has been developed for solving the nonlinear 1D equations of blood flow in a given
network of compliant vessels subjected to boundary and initial conditions. Importantly, the code
has been validated against in vitro [34, 36] and in vivo [29, 39, 41] experiments. A Linux
operating system has been used to compile the code on a standalone workstation equipped with
an Intel Core i7 CPU (6 cores and 3201 MHz) with 32GB memory. Each simulation is run at a
time-step of Δ𝑡 =10µs. At least 10 cardiac cycles are simulated in order to ensure that a
periodic steady state is reached. The results are then processed and analyzed using MATLAB
(The MathWorks, Inc., MA, USA).
2.4.3 Hemodynamic analysis
The total power 𝑃̅
𝑡𝑜𝑡𝑎𝑙 transmitted to the brain over a cardiac cycle of length T is calculated
as the average of the product of the pressure p(t) and the flow q(t) in each brain segment. The
steady power 𝑃̅
𝑠 is computed as the product of mean pressure 𝑝 𝑚𝑒𝑎𝑛 and mean flow 𝑞 𝑚𝑒𝑎𝑛 in
each segment. The pulsatile transmitted power 𝑃̅
𝑝𝑢𝑙𝑠𝑒 is the difference between the total power
and the steady power. Each of these power quantities are respectively given by
𝑃̅
𝑡𝑜𝑡𝑎𝑙 =
1
𝑇 ∫ 𝑝 (𝑡 )𝑞 (𝑡 )𝑑𝑡 𝑇 0
, (2.23)
𝑃̅
𝑠 =𝑝 𝑚𝑒𝑎𝑛 𝑞 𝑚𝑒𝑎𝑛 , (2.24)
𝑃̅
𝑝𝑢𝑙𝑠𝑒 =𝑃̅
𝑡𝑜𝑡𝑎𝑙 −𝑃̅
𝑠 . (2.25)
20

Based on the above equations, the pulsatile power percentage PPP is defined as the ratio
between the pulsatile transmitted power and the total power, i.e.,
𝑃𝑃𝑃 =
𝑃̅
𝑝𝑢𝑙𝑠𝑒 𝑃̅
𝑡𝑜𝑡𝑎𝑙 . (2.26)
Common clinical parameters such as flow pulsatility index FPI and pressure pulsatility
index PPI [22] are defined as
𝐹𝑃𝐼 =
𝑞 𝑚𝑎𝑥 −𝑞 𝑚𝑖𝑛 1
𝑇 ∫ 𝑞 (𝑡 )𝑑𝑡
𝑇 0
, (2.27)
𝑃𝑃𝐼 =
𝑝 𝑚𝑎𝑥 −𝑝 𝑚𝑖𝑛 1
𝑇 ∫ 𝑝 (𝑡 )𝑑𝑡
𝑇 0
, (2.28)
where 𝑞 𝑚𝑖𝑛 and 𝑞 𝑚𝑎𝑥 (resp. 𝑝 𝑚𝑖𝑛 and 𝑝 𝑚𝑎𝑥 ) are the minimum and maximum flow (resp.
pressure) transmitted to the brain during a cardiac cycle.
2.5 Results
Simulations are run for five different levels of aortic arch PWV, starting from the baseline
PWV of a healthy individual (c1) and moving towards different levels by multiplicative factors of
c1 given by c2=1.25c1, c3=1.5c1, c 4=2c1, c5=3c1 (see values in Table 2.1). Each case has been run
for eight HRs (30, 47, 63, 75, 94, 126, 150, and 189 beats per minute (bpm)). In all simulations,
the Cardiac Output (CO), the peripheral resistance (PR), the terminal compliance, the shape of
the inflow wave and all the outflow boundary conditions are kept constant.
2.5.1 Physiological accuracy of the model
A sample of flow and pressure in the left common carotid artery is shown in Fig. 2.3. The
expected fiducial features of pressure and flow waveforms, including the pressure inflection
21

point, the pressure dicrotic notch, and the peaks of the flow (Q 1 and Q2), can be seen in Fig.
2.2(a) and 2.2(b).

Fig. 2.2 Sample pressure and flow data in caortid artery. The simulated flow (a) and pressure (b) in the left common carotid
artery at the baseline aortic arch PWV (see Table1) and HR of 75 bpm.
Fig. 2.3 demonstrates the comparison between our simulated data with clinical data
(reproduced from data by Hashimoto et al [22]). The clinical data consists of recorded Doppler
waveforms in 286 patients with hypertension in order to measure the carotid flow augmentation
index defined as the ratio of late systolic flow height (Q2-Qmin) over early systolic wave height
(Q1-Qmin). Pressure augmentation indices (augmented pressure over the maximum height of the
pressure waveform) are computed from Tonometric pressure waveforms. The red curve in Fig.
2.3 is the exponential fitted curve (r=0.71) on clinical data [22]. The dashed upper and lower
curves are based on the error bars reported by Hashimoto et al. [22].
(a) (b)

22

Fig. 2.3 The carotid flow augmentation index versus the aortic pressure augmentation index. Red is the exponential fitted
curve (r=0.71) on the clinical data produced by Hashimoto et al.[22]. Blue diamonds represent simulation results for the baseline
HR. The dashed upper and lower curves are based on the error bars reported by Hashimoto et al. [22].
2.5.2 Effect of heart rate on transmitted pulsatility to the brain
Fig. 2.4(a) gives 𝑃̅
𝑝𝑢𝑙𝑠𝑒 computed by Eq. (2.25) as a function of HR for different levels of
the aortic arch PWV. As mentioned previously, the CO is kept constant for all cases. Results are
computed from the left common carotid artery hemodynamic waveforms (the only cerebral
branch that is directly connected to the aortic arch). As HR increases, the value of 𝑃̅
𝑝𝑢𝑙𝑠𝑒
decreases until the HR reaches an optimum point where 𝑃̅
𝑝𝑢𝑙𝑠𝑒 is minimized. 𝑃̅
𝑝𝑢𝑙𝑠𝑒 increases
with HR beyond this optimum point (Fig. 2.4(a)). Note that this phenomenon is present for all
different multiplicative factors of aortic arch PWV. Fig. 2.4(b) demonstrates the pulsatile power
as a function of the aortic arch PWV at different HRs. As expected, pulsatile power in the carotid
artery increases at all HRs when the aortic arch PWV increases.
23

Fig. 2.4 Average transmitted pulatile power to the brain per cardiac cycle at different wave dynamics conditiosns. Impact
of (a) the HR at different levels of aortic arch stiffness (PWV) and (b) the PWV at different HRs.
Fig. 2.5 depicts a 3D interpolation mapping of the pulsatile power with respect to the aortic
arch stiffness (as measured by PWV) and the HR. There is an optimum wave condition region in
which pulsatile power transmission is minimized (red arrow in Fig. 2.5). This optimal region
occurs at the baseline aortic arch PWV (for a normal adult) around a value of HR=75 bpm.
(a) (b)
24

Fig. 2.5 Pulsatile power transmission to the brain as a function of the Heart Rate and Aortic Arch PWV. Red arrow
indicates the optimum region in which pulsatile power is minimized.
2.5.3 Flow and pressure pulsatility indices versus pulsatile power percentage
Fig. 2.6 compares pulsatility indices and PPP at different aortic arch PWVs for three cases
of the HR: 1) a normal heart rate, 2) the highest simulated heart rate (HR=189 bpm), and 3) the
lowest simulated heart rate (HR=30 bpm). This range of HRs covers a large domain of different
wave conditions.

25

Fig. 2.6 Sensitivity of different pulsatility indices to various wave dynamics states. Flow Pulsatility Index (a), Pressure
Pulsatility Index (b), and Pulsatile Power Percentage (c) within the left common carotid artery at diiferent aortic arch PWVs for a
normal HR (red), the lowest investigated HR (green) and the highest investigated HR (blue).
2.6 Discussion
In this chapter, we have employed a physiologically accurate computational model of the
systemic vasculature to investigate the effect of aortic arch stiffening on the transmission of
excessive wave pulsatility to the cerebral circulation. Our results suggest that: (1) there exists an
optimum wave condition in the aorta that minimizes the harmful pulsatile energy (power)
transmission to the brain, (2) at different wave conditions (i.e. different HR and aortic arch
PWV), this optimum wave condition occurs around a value near the normal human HR (75bpm),
and (3) an index based on pulsatile power (i.e., a percentage of it) is a more sensitive measure for
excessive pulsatility transmission to the brain compared to conventional measures such as
pressure and flow pulsatility indices alone.
Transmission of arterial pulsatility to the brain has long been known to promote vascular
events such as ischemic and hemorrhagic stroke. More recently, hypertension has been shown to
promote dementia, accounting for up to 30% of cases [42]. The hemodynamic mechanisms
underlying these associations have not been well characterized. Aortic stiffening is the primary
cause of systolic hypertension with aging [43]. In the prior work, it has been shown that the
aortic arch stiffening that occurs with aging [44] is a much more powerful predictor of insult to
(a) (b) (c)
26

the microvasculature in the brain than blood pressure or the presence of hypertension treatment
[8]. Our results affirm that aortic stiffening does indeed increase transmission of harmful
pulsatility to the brain. More importantly we also saw that this increase was several folds more
severe when other parameters such as HR also become suboptimal. Results from simulations
have been compared to published clinical data in order to verify the clinical relevancy of the
computational model (Fig. 2.3). As it has been demonstrated, the calculated carotid flow
augmentation index and pressure augmentation index from simulation data are well within the
range of clinical data and follow similar trends. This confirms the physiological accuracy of our
study for purposes of investigating pulsatility transmission to the cerebral vasculature. This may
help us identify with much greater accuracy those persons at risk of cerebrovascular events and
accelerated brain aging due to harmful effects of excessive arterial pulsatility.
We studied the effect of aortic arch stiffening on the pulsatile energy transmission to the
brain across a physiological range of HRs while keeping CO(=5.2 L/min) and other vascular
parameters constant. Our results show that there is an optimum HR at which the transmitted
pulsatile energy to the brain is minimized (Fig. 2.4(a)). The pulsatile energy decreases with
increasing HR until it reaches this minimum value. Beyond the value, waves transmitted to the
brain start acting destructively and, as a result, the pulsatile power starts elevating as HR
increases. The existence of an optimum wave reflection has been shown in different contexts
related to ventricular workload in animals [23, 45], in-vitro experimental data , and
computational data [24]. Results are consistent with previous studies that have suggested the idea
that aortic wave optimization is one of the design characteristics found in the mammalian
cardiovascular system. However, the connection of the wave optimization in a heart-brain
coupling framework has been thus far unknown. In this chapter, we have demonstrated (Fig. 2.5)
27

the presence of an optimum wave condition —found near the normal human heartbeat—for the
transmitted energy to the brain across different aortic arch rigidities and HRs.
Unfortunately, as people age and become frail, the resting HR increases [46] even when it is
impaired from appropriately increasing in response to physical exertion [47]. Effects of harmful
pulsatile energy transmission to the brain that result from aortic stiffening with aging is likely
compounded by a harmful interaction with increased HR among frail elderly at rest and by a
decreased HR during exertion. This finding has not been reported before from prior in-vivo
experiments that involve evaluating arterial stiffness and blood flow for a person at rest. In vivo-
work fails to capture the impact on cerebral hemodynamics of the changes in HR that occur
throughout the day due to physical exertion.
Pressure and flow pulsatility indices (PPI and FPI) are the conventional dimensionless
parameters for monitoring hemodynamic pulsatility transmission to the brain. Fig. 2.6 displays
results based on the PPI and FPI. As expected, these indices capture the effect of the aortic arch
stiffness on pulsatility, and they additionally support the presence of an optimum wave condition
when passing from a low HR to a high HR. In other words, increasing or decreasing the HR (the
green and blue curves in Fig. 2.6) will render the transmitted wave suboptimal. Since aortic
aging affects the transmitted flow and pressure waves to the brain simultaneously, there is a need
to employ a parameter which considers the combined effects of pressure and flow pulsatilities.
Hence, we have utilized an energy-based index defined as the ratio of the pulsatile power over
the total power transmitted to the brain per cardiac cycle. We have compared the performance of
this index with PPI and FPI. The results show that the increase in PPP is much more significant
than the other two indices. This suggests that the PPP is more sensitive to changes in wave
dynamics and can provide better insight into the effect of aortic arch stiffness and its subsequent
28

behavior on excessive pulsatile power transmission to the brain. Additionally, on the effect of the
HR, it has been found that flow pulsatility is more consistent with PPP. Both the FPI as well as
the PPP have their highest values at higher HRs (blue curves in Fig. 2.6), while the PPI has its
highest value at lower HRs (green curve in Fig. 2.6). This work shows that pressure and flow
pulsatility may not be considered as interchangeable measures of cerebral pulsatility. A
combined consideration of pressure and flow are needed to properly understand the power
transmitted to the brain, and future clinical studies should include both assessments.
A limitation of this study lies in the 1D vasculature model formulation. The model used here
may not necessarily reveal all aspects of flow distribution in the arterial network, especially that
of the cerebral arteries. In addition, a detailed exploration of how pulsatility is conveyed in the
brain and the effect of the circle of Willis on the transmitted pulsatility to the brain are beyond
the scope of the current work. In this work, we have focused on the interaction between the aorta
and the cerebral circulation; the model has demonstrated capability of capturing the main
features of flow and pressure wave pulsatility in larger arteries (Fig. 2.2). Therefore, it is a good
starting point for investigating the effects of arch stiffness on transmitted pulsatility to the brain.
Future works involves studying the widespread variability in the circle of Willis and its potential
effects on the transmission of arterial pulsatility into different cerebrovascular beds.
Additionally, employing a linear model to describe the dynamics of vessel walls may
introduce errors relating to the wall stress relaxation. However, it has been shown previously
that, under normal physiological conditions, this error is not very significant [34]. A further
limitation of the current model is the approximation of the dynamics of the heart as a flow source
imposed as a flow wave at the inlet. Although in general the heart is neither a flow nor pressure
source, the behavior of the normal heart is closer to a flow source, and this interpretation has
29

been employed in literature to validate the 1D model and has been shown to be a reasonable
approximation [23, 24].
2.7 Conclusion
We have demonstrated that at different aortic arch rigidities, there is an optimum wave
condition that minimizes the pulsatile energy transmitted to the brain. This optimum condition
occurs near the normal HR and remains constant across a wide range of aortic arch stiffnesses.
Based on an energy-based analysis of the waves at the carotid artery, pulsatile power percentage
was used as an index to consider the combined effects of pressure and flow changes as the aortic
arch become stiffer. This non-dimensional parameter was compared across different wave
conditions with pulsatility indices that are based on pressure and flow. Results demonstrate that
pulsatile power percentage can capture the transmitted pulsatility to the brain more clearly than
the other pulsatility indices due to a higher sensitivity to different wave conditions. Previous
work has discussed pathological waves—defined as abnormalities in aortic and coronary wave
dynamics—as a potential trigger towards cardiac death in the presence of the cardiovascular
disease [48]. The pathology in the wave dynamics and the detection of wave condition signatures
for different aortic arch rigidities may provide further insight into the underlying wave dynamics
of the arterial system, particularly for the brain-heart coupling portion of the vasculature.
Understanding the physics can potentially be a first step towards contributing in the development
of new therapeutic strategies for neurodegenerative diseases like Alzheimer’s dementia.

30

CHAPTER 3 : Mechanistic insights on age-related changes in heart-aorta-
brain hemodynamic couplings using in-silico model of the entire
circulation

3.1 Chapter abstract
Age-related changes in aortic biomechanics can impact brain through reducing blood flow
and increasing pulsatile energy transmission. Furthermore, clinical studies have shown that heart
failure patients who suffer from impaired cardiac function have worse degrees of cognitive
impairment. Although previous studies have attempted to elucidate the complex relationship
between age-associated aortic stiffening and pulsatility transmission to the cerebral network,
these studies have not adequately addressed the effect of interactions between aortic stiffness and
left ventricle (LV) contractility on such energy transmission nor on brain perfusion. In this
chapter, we utilized a 1D model of the entire human circulation to investigate how age-related
changes in the cardiac and vasculature affect underlying mechanisms involved in LV-aorta-brain
system. Results suggest that LV contractility alone affects the pulsatile energy transmission to
the brain even at the preserved cardiac output. Results show the presence of the optimum heart
rate (near normal human heart rate) that minimizes energy transmission to the brain at different
levels of contractility. Our findings further suggest that at low levels of LV contractility, the
reduction in cerebral blood flow due to age-related changes in aortic stiffness becomes more
pronounced. In order to regulate the blood flow to the brain at the limit of vasodilatory response,
the body needs to either increase the contractility or heart rate. The former consistently leads to
higher pulsatile power transmission and the latter can either increase (for values less than normal
31

heart rate) or decrease (for values beyond normal heart rate) subsequent pulsatile power
transmission to the brain.
3.2 Introduction
The circulatory system operates based on a delicate hemodynamic balance between the
heart, the aorta, and major target organs such as the brain [18]. In healthy young adults,
interactions between the left ventricle (LV) and the aorta are optimized to guarantee the delivery
of cardiac output with modest pulsatile hemodynamic load on the LV [5]. In such a cohort, the
low impedance of a compliant aorta interacts with usually stiff conduit arteries such as the
carotid artery. This creates impedance mismatches and wave reflections at the aorta-brain
boundaries that limit the transmission of excessive pulsatile energy into the cerebral
microcirculation and that protect the brain tissue [17, 49]. This optimum hemodynamic coupling
between the LV, the aorta, and the brain can be impaired due to age-related changes in aortic
stiffness [18, 19]. Indeed, the stiffness increase with age is one of the earliest pathological
changes within the arterial wall, affecting the wave dynamics in the vasculature. For heart-aorta
coupling, previous studies have shown that elevated aortic stiffness increases the LV pulsatile
load, leading to an increase in LV mass which, in turn, contributes to heart failure (HF) [50]. At
the aorta-brain interface, it has been shown that aortic stiffening increases aortic impedance,
reduces impedance mismatches, and results in an increased transmission of harmful pulsatile
energy into the cerebrovascular network—ultimately leading to cognitive impairments such as
Alzheimer’s and other related vascular dementia [6, 8] .
Furthermore, population-based clinical studies have suggested that HF patients who suffer
from impaired LV function have worse degrees of cognitive impairment than age-matched
individuals without HF [2, 10]. HF has been proposed as a risk factor for Alzheimer’s disease
32

(AD), where the current clinical hypothesis is that the decreased cerebral blood flow due to HF
may contribute to the dysfunction of the neurovascular unit and hence may lead to impaired
clearance of amyloid beta [2, 11-13]. In addition to the consequences of HF, age-associated
changes in ventricular wall thickening and stiffening may trigger heart remodeling that can also
affect cerebral hemodynamics. Although previous studies have attempted to elucidate the
complex relationship between aortic stiffness and pulsatile energy transmission to the brain,
these studies have not adequately addressed the effect of interactions between the aorta and the
LV on such energy transmission (nor on brain perfusion). Indeed, most recent work has focused
only on aorta-brain coupling and has neglected the impact of cardiac dynamics on cerebral
perfusion. This is due to the fact that there are inherent difficulties in studying the isolated effects
of aortic wave dynamics and cardiac function on brain hemodynamics [14, 15].
The aim of the current chapter is to investigate the impact of major aging mechanisms in the
arterial system and cardiac function on brain hemodynamics. The state of LV-aorta-brain
coupling is mainly dominated by: LV contractility (a major determinant of LV function), heart
rate (the determinant of the fundamental frequency of propagated arterial waves), and aortic
stiffness (a determinant of the buffering function of the aorta for transferring pulsatile flow from
the LV to the brain). The optimal state of LV-aorta-brain coupling is achieved via the interplay
of these three determinants [51]. In order to gain a better understanding of the age-associated
impacts of these determinants on blood transfer and associated pulsatility to the brain, we
employ in this chapter a physiologically-validated one-dimensional (1D) computational model of
the entire human circulation using a high-order numerical methodology [52]. Since the
relationship between HF and AD becomes increasingly important with aging, it is essential to
understand the underlying mechanisms associated between cardiovascular hemodynamics and
33

brain perfusion. In what follows, we model and discuss age-related changes in the arterial system
and its associated cardiac dynamics. We also propose an explanation as to the possible
underlying mechanisms that may be involved in heart-aorta-brain coupling.
3.3 Methods
3.3.1 Physical model of the entire human circulation
A validated 1D model of the complete systemic and venous vascular network, based on
space-time variables, has been employed in this chapter [27, 28] . The physical model includes
122 larger systemic arteries and 162 veins, where each artery is characterized by its diameter,
length, Young’s modulus, and wall thickness. Fig. 3.1 illustrates the closed-loop cardiovascular
model that consists of such 1D segments for modeling wave propagation in larger arteries/veins,
and 0D compartments for modeling all four heart chambers (including the left ventricle) as well
as the microvasculature. The arterial wall is assumed to be thin, incompressible, homogenous
and isotropic. In this chapter, our focus is to investigate the effect of LV dynamics and aortic
stiffness on pulsatility transmission to the brain. Different levels of aortic rigidity are considered
by employing multiplicative factors of a minimum rigidity level 𝐸 1
(𝑥 ) that corresponds to the
baseline PWV (𝑐 0
) initially prescribed in the model (values are presented in Table 1). To
simulate different states of LV contractility, end-systolic elastance (𝐸 es
) is varied since it is
considered a common measure of contractility [53-55]. In this study, a value of 𝐸 es
= 2.5
mmHg/mL is taken to be the control and normotensive case, while values below 1.5 mmHg/mL
and larger than 3.5 mmHg/mL are considered to be low and high contractility, respecitvely [56].
34

Fig. 3.1 Schematic of the closed-loop cardiovascular system model. Our model consists of 1D segments coupled to 0D
lumped-parameter models of the heart and microvasculature.
3.3.2 Computational model and numerical solver
In order to simulate the complete circulation with, it is essential to adopt a nonlinear and
physiologically-relevant fluid-structure model. For cross-sectional area 𝐴 =𝐴 (𝑥 ,𝑡 ) and flow
velocity averaged over the cross-section 𝑈 = 𝑈 (𝑥 ,𝑡 ) (yielding the flow rate as 𝑄 =𝐴𝑈 ), such a
model can be expressed as a reduced-order nonlinear system for each segment as
(
𝜕𝐴
𝜕 t
(𝑥 ,𝑡 )
𝜕𝑈
𝜕 𝑡 (𝑥 ,𝑡 )
)= − (
𝜕 (𝐴𝑈 )
𝜕𝑥
(𝑥 ,𝑡 )
𝑈 𝜕𝑈
𝜕𝑥
(𝑥 ,𝑡 )+
1
𝜌 𝜕𝑃
𝜕𝑥
(𝑥 ,𝑡 )+
2(𝜉 +2)𝜋𝜇𝑈 (𝑥 ,𝑡 )
𝜌𝐴 (𝑥 ,𝑡 )
) (3.1)
where 𝜌 is a (constant) blood density, 𝜇 is a (constant) blood viscosity and 𝜉 is a given
constant of an assumed axisymmetric velocity profile. The system is closed by an assumed
Systemic enous
1D Segments
eft Heart
ime varying lastance 0D odel
Systemic ascular ed
0D odel
ight Heart

Pulmonary irculation
n silico odel of
ntire irculation
Systemic Arterial
1D Segments
35

elastic (tube law) that accounts for the fluid-structure interaction and is given by the constitutive
law
𝑃 =𝑃 ext
+
𝛽 (𝑥 )
𝐴 d
(√𝐴 −√𝐴 d
), 𝛽 (𝑥 )=
4
3
√𝜋 𝐸 (𝑥 )ℎ(𝑥 ) (3.2)
where 𝑃 ext
is the external and reference pressure, 𝐴 d
is the diastolic area, and 𝛽 (𝑥 ) is an
expression of the arterial wall material properties in terms of elastic modulus 𝐸 (𝑥 ) (a measure of
stiffness) and wall thickness ℎ(𝑥 ) . 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
𝑃 𝑝 +
𝜌 2
𝑈 𝑝 =𝑃 𝑑 ,i
+
𝜌 2
𝑈 𝑑 ,i
, i=1,2, (3.3)
A
𝑝 U
𝑝 +A
𝑑 ,1
U
𝑑 ,1
+A
𝑑 ,2
U
𝑑 ,2
=0. (3.4)
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. The complete non-linear system given by (3.1) is solved using
a numerical scheme based on an accelerated Fourier continuation (FC) methodology for accurate
Fourier expansions of non-periodic functions. Considering an equispaced Cartesian spatial grid
on, for example, the unit interval [0, 1] (given by the discrete points 𝑦 =
𝑖 𝑁 −1
,𝑖 =1,2,…,𝑁 −
1), Fourier continuation algorithms append a small number of points to the discretized function
36

values 𝑈 (𝑥 𝑖 ) and 𝐴 (𝑥 𝑖 ) in order to form (1+𝑑 ) -periodic trigonometric polynomials 𝑈 𝑐𝑜𝑛𝑠𝑡 (𝑥 𝑖 )
and 𝐴 𝑐𝑜𝑛𝑠𝑡 (𝑥 𝑖 ) that are of the form of
𝑈 cont
(𝑥 )=∑ 𝑢 𝑘 𝑒 2𝜋𝑖𝑘𝑥 𝑑 +1
𝑀 𝑘 =−𝑀 , (3.5)
𝐴 cont
(𝑥 )=∑ 𝑎 𝑘 𝑒 2𝜋𝑖𝑘𝑥 𝑑 +1
𝑀 𝑘 =−𝑀 , (3.6)
And that the match of discrete values of 𝑈 (𝑥 𝑖 ) and 𝐴 (𝑥 𝑖 ) for 𝑖 =1,2,…,𝑁 −1. Spatial
derivatives of the governing system are then computed by exact-termwise differentiation of (3.5)
and (3.6) as
𝜕𝑈
𝜕𝑥
(𝑥 𝑖 )=
𝜕 𝑈 cont
𝜕𝑥
(𝑥 𝑖 )=∑ (
2𝜋𝑖𝑘𝑥 𝑑 +1
)𝑢 𝑘 𝑒 2𝜋𝑖𝑘𝑥 𝑑 +1
𝑀 𝑘 =−𝑀 , (3.7)
𝜕𝐴
𝜕𝑥
(𝑥 𝑖 )=
𝜕 𝐴 cont
𝜕𝑥
(𝑥 𝑖 )=∑ (
2𝜋𝑖𝑘𝑥 𝑑 +1
)𝑎 𝑘 𝑒 2𝜋𝑖𝑘𝑥 𝑑 +1
𝑀 𝑘 =−𝑀 . (3.8)
From practical point of view, FC algorithms add a (fixed) handful of additional values to the
original discretized function in order to form a periodic extension in [0,1+𝑑 ] that transitions
smoothly from 𝑈 (1) back to 𝑈 (0) (similarly for 𝐴 ). The resulting continued functions can be
viewed as sets of discrete values of periodic and smooth functions that can be approximated to
high-order on slightly larger intervals by a trigonometric polynomial. Once these discrete
periodic continuation functions have been constructed, corresponding Fourier coefficients 𝑢 𝑘 , 𝑎 𝑘
in Eq. (3.5) and (3.6) can be obtained rapidly from applications of the Fast Fourier Transform
(FFT). Employing these discrete continuations in order to evaluate spatial function values and
derivatives on the discretized physical domain modeled by the wave equations, the algorithm is
completed by employing the explicit fourth-order Adams-Bashforth scheme to integrate the
corresponding ordinary differential equations in time from the given initial conditions 𝐴 (𝑥 ,𝑡 ) =
37

𝑈 (𝑥 ,𝑡 )=0 up to a final given time. The final full solver enables high-order accuracy and nearly
dispersionless resolution of propagating waves with mild, linear Courant-Friedrichs-Lewy
constraints on the temporal discretisation---properties that are important for adequate resolution
of the different spatial and temporal scales. Both implicit and explicit FC-based partial
differential equation solvers have been successfully constructed and utilized for a variety of
physical problems including those governed by radiative transfer equations, Navier-Cauchy
elasto-dynamics equations, Navier-Stokes fluid equations, and fluid-structure hemodynamics
equations [52, 54, 57, 58].
Following the works of Mynard and Smolich [59], three types of vascular beds are
considered in our computational model of the entire circulation: generic vascular beds (shown in
Fig. 3.1), a hepatic vascular bed and coronary vascular beds. The generic vascular bed model is
used for all microvasculature beds except the liver and myocardium. It is based on commonly
used three-element Windkessel model and consists of the characteristic impedances 𝑍 𝑎𝑟𝑡 and
𝑍 𝑣𝑒𝑛 (to couple the connecting 1D arteries to the vascular bed), lumped compliances for the
arterial and venous microvasculature (𝐶 𝑎𝑟𝑡 and 𝐶 𝑣𝑒𝑛 ) and the vascular bed resistance (𝑅 𝑝 ). The
resistance is assumed to be pressure dependent to account for the atrio-venous pressure
difference. The hepatic vascular bed is a modification of the above to account for both arterial
and venous inlets in liver. It includes a compartment for the flow from hepatic artery (𝑅 𝑎𝑟𝑡 , 𝐶 𝑎𝑟𝑡 )
which connects to another compartment (𝐶 𝑝 𝑎 ) with common portal/arterial pressure. The coronary
vascular bed model represents blood flow through intramyocardial. The coronary vessels
experience a large time-varying myocardial pressures 𝑃 𝑖 𝑚 caused by the contracting heart muscle.
38

3.3.3 Time-varying elastance heart model
The relationship between the pressure and the volume of a heart chamber is given by
𝑃 =𝑃 𝑝𝑐
+
𝐸 nat
𝐸 sep
𝑃 ∗
+𝐸 nat
(𝑉 − 𝑉 𝑃 =0
)−𝑅 𝑠 𝑞 , (3.9)
where 𝑃 𝑝𝑐
is the pericardiac pressure (assumed to depend exponentially on the total chamber
volumes, 𝐸 nat
is the native elastance of the chamber, 𝐸 sep
is the septal elastance, 𝑉 𝑃 =0
is the
volume of the chamber in zero pressure, 𝑅 𝑠 is the source resistance, and 𝑃 ∗
is the pressure in the
contralateral chamber. Parameters varied in this study and their corresponding range are listed in
Table 3.1.
Table 3.1 Physical parameters used in this study.
Physical Parameter Unit Baseline Range
Aortic PWV m/s 4.66 [4.66, 13.98]
Heart Rate bpm 75 [30, 180]
LV End Systolic Elastance
mmHg
mL
2.5 [0.6, 5.0]
LV End Diastolic Volume mL 136 [65, 465]
Ejection Fraction % 55 [14, 70]
Stroke Volume mL 74 [19, 186]
Cardiac Output L/min 5.56 [1.4, 7.1]

3.3.4 Hemodynamic analysis
The total power 𝑃̅
𝑡𝑜𝑡𝑎𝑙 transmitted to the brain over a cardiac cycle of length T is calculated
as the average of the product of the pressure P(t) and the flow Q(t). For computing energy, we
employ pressure and flow data from the left common carotid artery, since it is the only cerebral
branch that is directly connected to the aortic arch. The steady power 𝑃̅
s
is computed as the
product of mean pressure 𝑃 mean
and mean flow 𝑄 mean
in each segment. The pulsatile transmitted
39

power 𝑃̅
pulse
is the difference between the total power and the steady power. Each of these
power quantities are respectively given by
𝑃̅
total
=
1
𝑇 ∫ 𝑃 (𝑡 )𝑄 (𝑡 )d𝑡 𝑇 0
, (3.10)
𝑃̅
𝑠 =𝑃 mean
𝑄 mean
, (3.11)
𝑃̅
pulse
=𝑃̅
total
−𝑃̅
s
. (3.12)
Total fluid flow transmitted to the cerebral network is computed by a summation of the
average flow over one cardiac cycle (i.e., integrating over time) for all four arteries that are
connected to the brain (two carotid and two vertebral). An additional well-established clinical
metric considered here for similarly quantifying energy transmission to the brain is wave
intensity (WI). WI is defined as the power per unit cross-sectional area of an artery due to blood
pressure 𝑃 =𝑃 (𝑡 ) and average cross-sectional blood flow velocity 𝑈 =𝑈 (𝑡 ) . Mathematically
speaking, WI is computed as the product of the change in pressure (d𝑃 ) times the change in
velocity (d𝑈 ) during a small interval, i.e.,
d𝐼 =d𝑃 .d𝑈 . (3.13)
To remove the dependency of d𝐼 on sampling time, the derivative of pressure and velocity
are divided by the time interval (denoted as
d𝑃 d𝑡 and
d𝑈 d𝑡 , respectively), yielding units of power per
unit area per unit time (𝑊 .𝑠 −2
.𝑚 −2
) [60]. WI patterns determine both the direction and intensity
of arterial wave propagation at any time instance during a cardiac cycle [61]. For example, a
d𝐼 >0 at a fixed time during the cycle indicates that forward waves (which largely originate from
the left ventricle) are dominant at that moment. Conversely, if d𝐼 <0, backward waves, which are
40

mostly related to wave reflections [62], are dominant. As a third and final measure employed in
this work, we also consider the Carotid (Flow) Pulsatility Index (CPI), a clinical parameter based
on single flow waveform measurements that is defined as
𝐶𝑃𝐼 =
𝑞 max
−𝑞 min
1
𝑇 ∫ 𝑞 (𝑡 )d𝑡 𝑇 0
, (3.14)
where 𝑞 min
and 𝑞 max
are, respectively, the minimum and maximum flow transmitted to the
brain through the carotid artery during a cardiac cycle.
Fig 3.2 (b) illustrates the impact of 𝐸 es
on the LV pressure-volume loop. Varying the
elastance 𝐸 es
, while fixing the preload and LV end-diastolic volume (LVEDV), leads to different
cardiac outputs (COs). In order to keep the CO constant at different levels of contractility, we
adjust the LV end-diastolic volume (LVEDV). The underlying physiological mechanism for LV
remodeling to fix a constant cardiac output is summarized in Fig. 3.2 (c).
41

Fig. 3.2 Components of the interest and study design for varying relevant parameters in Chapter 3. (a) Schematic
representation of the human circulatory system; (b) interventricular pressure-volume loop for different cases of contractility
(demarcated in different line styles and colors); and (c) the compounded impact (a fixed LVEDV) and the isolated impact (a fixed
CO) of contractility.
3.4 Results
3.4.1 Physiological accuracy of the model
Fig. 3.3 presents various pressure and flow waveforms, simulated via the numerical
methodology described above, for cases of decreased (𝐸 es
=1.2
mHg
mL
) and increased (𝐸 es
=
5.0
mHg
mL
) contractility, where LVEDV is adjusted to have the same cardiac output for both
(5.60 L/min ). These cases are computed at a baseline heart rate (75 bpm) and aortic PWV
(4.66 m/s), which are within physiological ranges. The presented pressure and flow waveforms
demonstrate the expected dynamics of the LV and the aorta during systole, including the
(b)
(c)

es, high

es, norm

es, low

es, low
(ad usted )
olume
Pressure
nd diastolic olume
ardiac utput
ncreased ontractility
Normal ontractility
Decreased ontractility
(a)
solated mpact
ompound mpact
S
norm
D
norm
42

presence of the pressure dicrotic notch as well as the physiological point-to-point consistency of
the pressure with flow. For the case of increased contractility, all waveforms have steeper
upstrokes at the onset of ejection and reach their respective peaks earlier in systole. Even though
the cardiac output is preserved by varying LVEDV, the maximal flow value is significantly
higher for the increased contractility case. Additionally, while the pulse pressure is almost
conserved at the carotid artery, the shape of the pressure waveform is also affected by
contractility.
Fig. 3.3 further presents the computed carotid WI for the decreased and increased
contractility cases. The curves fully capture the typical pattern of WI [62]: a large-amplitude
forward (positive) peak corresponding to the initial compression caused by LV contraction
(Forward Compression Wave Intensity, FCWI); a subsequent small-amplitude backward
(negative) peak corresponding to the reflection of the initial contraction (Backward Compression
Wave Intensity, BCWI); and a final moderate-amplitude forward decompression wave in
protodiastole (Forward Expansion Wave Intensity, FEWI). The overall results of Fig. 3.3
demonstrate the general ability of our in-silico computational model to reproduce physiological
characteristics of the LV, the aorta, and the carotid artery.
43

Fig. 3.3 Effects of LV contractility on central hemodynamics. The Cardiac output is the same for both set of the figures on the
right and left panels. The figures on the left panel demonstrate the impact of reduced contractility and the Fig.s on the right panel
demonstrate the impact of increased contractility.

3.4.2 Effect of LV contractility on transmitted pulsatility to the brain
Fig. 3.4 (a) presents the carotid pulsatile power (CPP) transmitted to the brain as a function
of contractility (measured by) for different levels of aortic PWV at fixed LVEDV. The data are
computed at the baseline heart rate (75 bpm). Since LVEDV is fixed, changes in contractility
lead to corresponding changes in CO (Fig. 3.2), further compounding the overall effect of
varying contractility by 𝐸 es
. Fig. 3.4(b) presents the isolated impact of contractility at a fixed CO
(achieved by adjusting LVEDV) on the transmitted pulsatile power to the brain, where it can be
observed that, at all values of 𝐸 es
, pulsatile power in the carotid artery increases as function of
aortic PWV.

es
= 1. (mmHg m )

es
= 5.0 (mmHg m )

44

Fig. 3.4 Compounded and isolated impacts of contractity on transmited pulsatile energy to the brain. Carotid pulsatile
power (CPP) per cardiac cycle versus the contractility (measured by 𝑬 𝒆𝒔
) at different levels of aortic stiffness at (a) fixed LVEDV
(changing CO) and (b) fixed CO.
Fig. 3.5 presents CPP as a function of contractility (𝐸 es
) at different levels of both aortic
PWV and heart rate (HR). As before, to achieve a constant CO (5.6 L/min) at baseline aortic
PWV (c
0
), values of LVEDV are accordingly adjusted; hence, at each PWV, the changes in CPP
are a consequence of the isolated changes in contractility. Results demonstrate a trend toward
increased transmitted pulsatile power to the brain as contractility increases. However, the rate of
this increase depends on the heart rate. Table 3.2 additionally presents a comparison between
baseline and increased aortic PWV (which can result from aging) on CPP transmitted to the
brain, and further presents corresponding values of carotid pulsatility index (CPI) computed
using Eq. (3.14).

ompound mpact
(a) (b)
solated mpact
Fi ed
45

Fig. 3.5 Impact of contractility on the transmitted energy to the brain at different wave states. Carotid pulsatile power
(CPP) per cardiac cycle versus the contractility (measured by 𝑬 𝒆𝒔
) at different levels of aortic stiffness at (a) heart rate of 30bpm,
(b) heart rate of 50bpm, (c) heart rate of 100bpm, and (d) heart rate of 125bpm.

(a)
(c) (d)
(b) = 0 (bpm) = 50 (bpm)
= 100 (bpm) =1 5 (bpm)
46

Table 3.2 Impact of LV contractility at two levels of aortic stiffness on the transmitted pulsatility to the brain.
Contractility (mmHg/ml) 0.6 1.2 1.8 2.5 3.5 5.0
Baseline aortic PWV, 𝐜 𝟎
Carotid Pulsatile Power, CPP (mWatt) 2.19 2.20 2.25 2.44 2.83 3.49
Carotid Pulsatility Index, CPI 5.00 4.96 4.96 5.04 5.90 7.55
Increased aortic PWV, 𝟑 𝐜 𝟎
Carotid Pulsatile Power, CPP (mWatt) 5.11 5.24 5.42 5.88 6.78 8.24
Carotid Pulsatility Index, CPI 8.50 8.52 8.61 8.86 9.43 11.16
* All values are reported at heart rate of 75 bpm.
3.4.3 Effect of heart rate on transmitted pulsatility to the brain
Fig. 3.6 presents values of CPP as a function of heart rate for different levels of aortic PWV.
The data in each plot is obtained at different levels of contractility (as measured by 𝐸 es
). CO is
fixed at each level of aortic PWV in a manner as has been described before. As heart rate
increases, the value of CPP decreases until the heart rate reaches an optimum point
corresponding to where CPP is minimized. CPP increases with heart rate beyond this optimum
point. Note that this phenomenon is present for all the different multiplicative factors of aortic
PWV considered here, as well as all the different levels of contractility. In all cases, the optimum
point is located near the normal human heart rate (75 bpm).
47

Fig. 3.6 Impact of heart rate on the transmitted energy to the brain at different wave states and LV contractility. Carotid
pulsatile power (CPP) per cardiac cycle versus the heart rate at different levels of aortic stiffness at (a) 𝑬 𝒆𝒔
of 0.6 mmHg/mL, (b)
𝑬 𝒆𝒔
of 1.2 mmHg/mL, (c) 𝑬 𝒆𝒔
of 1.8 mmHg/mL, (d) 𝑬 𝒆𝒔
of 2.5 mmHg/mL, (e) 𝑬 𝒆𝒔
of 3.5 mmHg/mL, and (f) 𝑬 𝒆𝒔
of 5.0
mmHg/mL.
Fig. 3.7 presents carotid pulsatility index (CPI) as a function of heart rate for different levels
of aortic PWV. The data in each plot is obtained at different levels of contractility (as measured
by 𝐸 es
). As before, CO is fixed at each level of aortic PWV. The results suggest a trend toward
increased CPI as heart rate increases.
(a) (b) (c)
(d) (e) (f)

es
= 0. (mmHg m )
es
= 1. (mmHg m )
es
= 1.8 (mmHg m )

es
= .5 (mmHg m )
es
= .5 (mmHg m )
es
= 5.0 (mmHg m )
48

Fig. 3.7 Impact of heart rate on the brain flow pulsatility index at different wave states and LV contractility. Carotid
pulsatility index (CPI) per cardiac cycle versus the heart rate at different levels of aortic stiffness at (a) 𝑬 𝒆𝒔
of 0.6 mmHg/mL, (b)
𝑬 𝒆𝒔
of 1.2 mmHg/mL, (c) 𝑬 𝒆𝒔
of 1.8 mmHg/mL, (d) 𝑬 𝒆𝒔
of 2.5 mmHg/mL, (e) 𝑬 𝒆𝒔
of 3.5 mmHg/mL, and (f) 𝑬 𝒆𝒔
of 5.0
mmHg/mL.
3.4.4 Effect of LV-aorta dynamics on wave intensity
Fig. 3.8 presents calculated carotid WI patterns from simulations at different levels of aortic
PWV for different heart rates and contractility. Similarly to Fig. 3.3, these patterns capture all the
well-known fiducial features [62], including the large-amplitude forward (positive) peak FCWI
that is followed in sequence by both the small-amplitude backward (negative) peak BCWI and
the moderate-amplitude forward decompression wave FEWI.
(a) (b) (c)
(d) (e) (f)

es
= 0. (mmHg m )
es
= 1. (mmHg m )
es
= 1.8 (mmHg m )

es
= .5 (mmHg m )
es
= .5 (mmHg m )
es
= 5.0 (mmHg m )
49

Fig. 3.8 Sample carotid Wave Intensity (WI) patterns at different heart rates and contractilites (measured by 𝑬 𝒆𝒔
). Each
plot contains data obtained at different levels of aortic stiffness (quantified by PWV).
Table 3.3 presents peak amplitudes of the major features of WI (FCWI, BCWI, and FEWI)
at different levels of contractility. The data are presented at normal heart rate for both baseline
aortic PWV and an increased PWV.

es
= 1. (mmHg m )
es
= .5 (mmHg m )
= 75 (bpm)

es
= 5.0 (mmHg m )
= 50 (bpm) = 0 (bpm)
50

Table 3.3 Impact of LV contractility on carotid WI indices at different aortic stiffness (PWV).
Contractility (mmHg/mL) 0.6 1.2 1.8 2.5 3.5 5.0
Baseline PWV 𝐜 𝟎
FCWI (W.𝑚 −2
.𝑠 −2
10
5
) 9.5 14.2 19.4 26.7 39.0 60.2
BCWI (W.𝑚 −2
.𝑠 −2
10
5
) 1.7 2.4 2.9 3.6 4.4 5.2
FEWI (W.𝑚 −2
.𝑠 −2
10
5
) 8.8 6.1 3.8 2.5 2.9 5.5
Increased PWV 𝟑 𝐜 𝟎
FCWI (W.𝑚 −2
.𝑠 −2
10
5
) 18.7 27.6 37.1 50.7 74.0 113.0
BCWI (W.𝑚 −2
.𝑠 −2
10
5
) 3.4 4..6 5.8 7.1 8.9 11.4
FEWI (W.𝑚 −2
.𝑠 −2
10
5
) 8.9 19.0 15.6 13.1 11.6 10.6
* All values are reported at heart rate of 75 bpm. Effect of contractility is isolated by fixing the CO using LVEDV.
Table 3.4 presents peak amplitudes of the major features of WI (FCWI, BCWI, and FEWI)
at different heart rates. The data are presented for both baseline aortic PWV and an increased
PWV at a state of normal contractility (𝐸 es
= 2.5 mmHg/mL).

51

Table 3.4 Impact of heart rate on carotid WI indices at different aortic stiffness (PWV).
Heart Rate (bpm) 30 50 75 100 125
Baseline PWV 𝐜 𝟎
FCWI (W.𝑚 −2
.𝑠 −2
) 27.1 27.3 26.7 28.3 28.9
BCWI (W.𝑚 −2
.𝑠 −2
) 4.9 4.4 3.5 3.1 2.5
FEWI (W.𝑚 −2
.𝑠 −2
) 2.5 2.9 2.5 6.8 19.9
Increased PWV 𝟑 𝐜 𝟎
FCWI (W.𝑚 −2
.𝑠 −2
) 40.4 47.7 50.7 55.2 58.2
BCWI (W.𝑚 −2
.𝑠 −2
) 8.1 8.2 7.1 6.4 5.5
FEWI (W.𝑚 −2
.𝑠 −2
) 10.9 14.3 13.2 18.6 46.3
* All values are reported at the contractility of 2.5 mmHg/mL.
3.4.5 Effect of LV-aorta dynamics on Brain Perfusion
Fig. 3.9 demonstrates how changes in aortic stiffness (as measured by PWV) at different
levels of contractility affect transmitted cerebral blood flow (CBF). The data in each plot are
obtained at different heart rates. At each wave state, the percentage change is computed by the
change in flow relative to the baseline PWV. Note that at each contractility, LVEDV remains
fixed. Results suggest a trend toward decreased cerebral flow as aortic stiffness increases. The
rate of this change depends both on the contractility and on the heart rate.
52

Fig. 3.9 Impact of aortic stiffness on brain perfusion at different cardiac conditions. Change in the transmitted flow to the
brain versus the aortic stiffness (as measured by PWV) at different levels of contractility at (a) heart rate of 30bpm, (b) heart rate
of 75bpm, (c) heart rate of 125bpm, and (d) heart rate of 180bpm.
3.5 Discussion
In this chapter, we have investigated the effect of LV-aortic dynamics on brain perfusion
and on the transmission of excessive wave pulsatility to the cerebral circulation. We have
modeled age-related changes on the arterial system and on cardiac dynamics. Our results suggest
that: (1) LV contractility alone affects the pulsatile energy transmission to the brain (even at the
preserved cardiac output); (2) at different levels of LV contractility and aortic stiffness, there
exists an optimum wave condition, occurring near the normal human heart rate (75bpm), in
which excessive pulsatile energy (power) transmission to the brain is minimized; and (3) at a
given heart rate and LV contractility, greater aortic stiffness leads to lower cerebral blood flow.
(a)
(c) (d)
(b) = 0 (bpm) = 75 (bpm)
= 1 5 (bpm) =180 (bpm)
53

At the limit of brain autoregulation, the compensatory mechanism for adjusting the cerebral flow
is achieved either by increasing the LV contractility or increasing the heart rate. Our results
suggest that the former consistently leads to higher pulsatile power transmission and the latter
can either increase (for values less than normal heart rate) or decrease (for values beyond normal
heart rate) subsequent pulsatile power transmission to the brain.
3.5.1 Impact of LV contractility on brain hemodynamics
We have utilized a reduced-order 1D model of the entire human circulation in order to
study and elucidate the underlying mechanisms involved in LV-aorta-brain hemodynamic
coupling. The numerical solver employed in this chapter incorporates the nonlinear and
nonstationary coupling of various cardiovascular system components (including a hybrid-ODE,
four-chamber heart model with valves), where the validation results that have been presented
support its suitability for the objectives of this chapter. Indeed, in order to assess the
physiological relevancy of the model, we have computed pressure and flow waveforms at
different anatomical sites of the closed-loop system including inside the LV, the ascending aorta,
and the carotid artery (Fig. 3.3). At baseline, the simulations employed a heart rate of 75 bpm at
two different levels of contractility defined by 𝐸 es
=1.2 mmHg/mL and 𝐸 es
=5.0 mmHg/mL at
fixed cardiac output. Fig. 3.3 demonstrates that our model is able to generate the main
physiological features of the pressure and flow waveforms as found in the human cardiovascular
system. These include: 1) the physiological development of the pressure inside both the LV and
the aorta during systole (corresponding to LV-arterial coupling); 2) the presence of the dicrotic
notch due to the aortic valve closure (corresponding to LV-arterial decoupling); 3) the increase in
pulse pressure as the wave propagates downstream towards side branches including the carotid
artery; and 4) the physiological point-to-point consistency of the pressure and flow in the
54

ascending aorta [63-66]. Results further demonstrate that our model is adequately able to capture
the effects of contractility on central and peripheral pressure waveforms. The expectedly steeper
upstrokes in both pressure and aortic flow waveforms for increased contractility are well-
captured in this model and are consistent with previous studies [14]. Our findings suggest that an
increase in LV contractility alone can directly alter central and peripheral hemodynamics, even
for unchanged arterial loads and cardiac outputs. These observations are consistent with previous
experimental and clinical studies [67]---verifying the physiological accuracy of our model for the
purposes of investigating the coupling mechanisms behind the LV-aorta-brain hemodynamic
system.
A first principal finding in this chapter is related to examining the impact of LV
contractility on transmitted energy and pulsatility to the cerebral network, where we have used
end-systolic elastance (𝐸 es
) as a measure to quantify the state of LV contractility. Fig. 4 presents
a comparison between both the compounded and isolated impact of LV contractility on
pulsatility transmission to the brain at a normal human heart rate (75 bpm). Here, isolated refers
to conditions where the LV stroke volume is compensated in order to fix the cardiac output. In a
physiological setting, the underlying mechanism to restore such stroke volumes is governed by
the Frank-Starling law or by the LV remodeling (e.g., LV dilation) [14] that can be observed in
Fig. 3.2. Indeed, Fig. 4 demonstrates the true effect of contractility on pulsatile energy
transmission to the brain, where our results suggest that even at fixed cardiac output, an increase
in contractility alone can lead to elevated levels of harmful pulsatile energy transmission to the
brain. This behavior can be observed at different levels of aortic stiffness (Fig. 3.4). However,
the rate of increase in pulsatile energy transmission as a function of contractility is smaller when
the CO is compensated for than when the CO is affected by changes in 𝐸 es
(corresponding to a
55

fixed LVEDV). Since CO and the total arterial resistance of the system is the same for different
levels of contractility (at the same aortic PWV), the steady portion of the transmitted power does
not change. However, since the shape of the pressure also changes due to contractility, the total
power increases (Eq. (3.10)) lead to an increase in the transmitted pulsatile power (Eq. (3.12)).
Results also suggest that the impact of contractility on brain perfusion depends on heart rate (Fig.
3.5 and Table 3.2). At lower heart rates, changes in carotid pulsatile power are more pronounced
than at higher heart rates. Indeed, the heart rate can be interpreted as a fundamental frequency of
the cardiovascular system and hence is an important parameter in determining the overall wave
state in arteries. At a fixed travel time (i.e., keeping segment lengths and wave speeds constant),
changing the heart rate affects the interaction between the compression waves generated by the
LV and the reflected waves due to vessel branching [53]. The sample net effect of these two
types of waves is illustrated in the WI patterns of Fig. 3.3 and Fig. 3.8. These interactions
become less sensitive to contractility at higher heart rates, and hence the pulsatile portion of the
power varies less. This pattern can be observed at all levels of aortic stiffness considered in this
work.
3.5.2 Presence of the optimum heart rate
Our results indicate there is an optimum heart rate at which the transmitted carotid pulsatile
energy is minimized (Fig. 3.6). Pulsatile energy decreases with increasing heart rate until it
reaches this minimum value. Beyond this value, waves transmitted to the brain begin to act
destructively, and, as a result, pulsatile power starts increasing as heart rate increases. This has
implications on the aging population, where resting HR increases in general [46, 47].
Additionally, aging leads to the stiffening of the aorta, further increasing the pulsatile energy
transmission to the brain. Indeed, our findings are consistent with previous studies [18, 23]
56

which have suggested that aortic wave optimization is one of the key design characteristics found
in the mammalian cardiovascular system. To the best of our knowledge, the presence of the
optimum heart rate at different levels of contractility in the LV-aorta-brain system has not been
reported in prior studies (including from in-vivo experiments).
In contrast to pulsatile energy which requires both flow and pressure waveforms to
calculate, the carotid flow pulsatility index (CPI) is a conventional dimensionless parameter,
based on only flow measurements, for also quantifying hemodynamic pulsatility transmission to
cerebrovasculature [18, 50]. Fig. 3.7 presents CPI values corresponding to the same values of
heart rate, contractility and aortic PWV considered for our pulsatile energy analysis. A trend of
increasing CPI with increasing heart rate can be observed in all cases. However, the CPI curves
do not capture the non-linearity and the presence of a minimum that can be found in the CPP
curves of Fig. 3.6. This suggests that considering the flow waveform alone may not be adequate
in properly quantifying the pulsatility transmitted to the brain, and that consideration of both
pressure and flow are needed, and hence we suggest that future clinical studies should include
both indices in their assessments. This is particularly prudent since aortic aging affects both the
transmitted flow and pressure waves to the brain simultaneously [51].
3.5.3 Impact of aortic stiffness on brain hemodynamics
Aortic stiffening is the primary cause of systolic hypertension with aging [43]. In prior
work, we have shown such age-related stiffening [44, 51] is a powerful predictor of insult to the
microvasculature in the brain, more so than blood pressure [8]. Our results confirm that aortic
stiffening does indeed increase transmission of harmful pulsatility to the brain. This excessive
pulsatility can be observed at all contractility states in both CPI (Fig. 3.7) and CPP (Fig. 3.6).
More importantly, results from Fig. 3.9 suggest that at a fixed heart condition (i.e., a fixed
57

contractility), greater aortic stiffness leads to lower cerebral blood flow. These findings are
consistent with a recent population-based clinical study by Jefferson et al. [68], where it was
reported that greater aortic stiffening relates to lower cerebral blood flow, especially among
individuals with increased genetic predisposition for Alzheimer’s disease. he authors have
hypothesized that this mechanism is due to microcirculatory remodeling in response to higher
pulsatility in the cerebrovascular network. In our present investigation, the increased aortic
impedance due to stiffening leads to an extra workload for the LV which, under fixed
contractility, in turn leads to decreased blood flow transmission to the brain [68]. Therefore, the
effects of harmful carotid pulsatile energy transmission are likely compounded with the
decreased flow transmission to the brain as a result of such age-related aortic stiffening.
A decreased cerebral blood flow, which can result from systemic diseases such as heart
failure, can contribute to the dysfunction of the neurovascular unit [2]. This is the current
prevailing view of the underlying mechanism of heart failure-induced Alzheimer’s disease [10].
Our results suggest that at low levels of LV contractility, the reduction in cerebral blood flow
due to age-related changes in aortic stiffness becomes even more pronounced (Fig. 3.9). In order
to compensate and regulate the blood flow to the brain, the body needs to either increase the
heart rate or increase the contractility. The latter leads to higher pulsatile energy transmission to
the brain (Figs. 3.4 and 3.5). On the other hand, increasing the heart rate can both decrease or
increase this energy transmission (Fig. 3.6), depending on whether the increasing heart rate is
approaching or diverging from the minimum, respectively. Since the resting heart rate in the
elderly is usually higher than the normal human heart rate (i.e., the latter case), an increase in
pulsatile energy transmission will be observed (which, as explained before, can have detrimental
effects on brain structure). Overall, our results demonstrate that age-relating aortic stiffening can
58

lead to a cascade of detrimental effects, due to both changes in contractility as well as heart rate,
on both cerebral perfusion and pulsatile energy transmission to the brain.
3.5.4 Wave intensity analysis
Wave intensity (WI) analysis is a well-established method for quantifying the energy carried
in arterial waves, providing valuable information about cardiovascular and cerebrovascular
function [69]. While WI has been traditionally computed using both pressure and flow
measurements, there has been recent efforts to expand the applicability of this method by
employing only a single waveform measurement [70, 71]. In a recent population-based clinical
study, Chiesa et al. [72] showed that elevated carotid WI, captured in FCWI amplitudes (e.g., in
Fig. 3.3), predicts faster cognitive decline in long-term follow-ups independently of other
cardiovascular risk factors. Their findings suggest that exposure to increased WI in mid- to late-
life may contribute to the observed association between arterial stiffness in mid-life and the risk
of dementia in the following decades. This cannot be detected using common carotid phenotypes
[72]. Our results are consistent with such observations, where one can observe that elevated
aortic stiffness leads to higher WI (Fig. 3.8 and Table 3.4). Our results also demonstrate that
elevated FCWI not only depends on aortic stiffness, but also strongly upon LV contractility (Fig.
3.8 and Table 3.3). This can be mostly attributed to the larger d𝑃 /𝑑𝑡 that results from higher
contractility, presenting as sharper pressure slopes in systole and stronger forward compression
waves (Table 3.3). This is notable since abnormal ventricular-arterial interaction in HF patients
with preserved ejection fraction can lead to higher end-systolic elastance compared to age- and
blood pressure-matched controls [56]. Our results suggest that these patients may also suffer
from excessive FCWI which can have detrimental effects on the brain structure [5, 8, 18, 72].

59

3.5.5 Limitations
The primary limitation of this study is in the vessel wall assumptions of the 1D vasculature
model formulation, i.e., neglecting the viscoelasticity (which may be important to consider in
certain vessels [73, 74]). However, our model still employs an effective nonlinear/hyperelastic
wall model that has been shown previously to be appropriate under normal physiological
conditions, and does not lead to considerable differences with viscoelastic considerations [75]. In
addition, we have not included any auto-regulatory models for brain circulation. However, since
the objective of this study is to investigate the impact of LV-aorta dynamics on general brain
perfusion, the feedback response of the brain on such dynamics is beyond our scope in this
chapter. Incorporating auto-regulatory models, together with our closed-loop circulatory model,
will be a subject of future work.
3.6 Conclusion
We have demonstrated that LV contractility alone can affect pulsatile energy transmission to
the brain. We have additionally demonstrated an optimum wave condition, existing at different
levels of contractility, heart rate, and aortic stiffness, that minimizing this energy. We have
shown that this optimum condition occurs near the normal human heart rate and remains constant
across a wide range of aortic arch stiffnesses and LV contractility. Our findings also suggest that
at a given contractile state of the LV, greater aortic stiffness leads to higher pulsatile energy
transmission to the brain and to decreased cerebral blood flow. These principal findings not only
demonstrate the level of coupling in the LV-aorta-brain system, but also how such coupling is
affected by age-related changes to cardiovasculature. Understanding the underlying physical
mechanisms involved here can potentially be a first step towards contributing to the development
of new therapeutic strategies for vascular-related neurodegenerative diseases.
60

CHAPTER 4 : Accuracy and applicability of non-invasive evaluation of
aortic wave intensity using only pressure waveforms in humans

This chapter is based on the following published manuscript: Aghilinejad, A., Amlani, F., Liu, J.
and Pahlevan, N.M., 2021. Accuracy and applicability of non-invasive evaluation of aortic wave
intensity using only pressure waveforms in humans. Physiological Measurement, 42(10),
p.105003.

4.1 Chapter abstract
Wave intensity (WI) analysis is a well-established method for quantifying the energy carried
in arterial waves, providing valuable clinical information about cardiovascular function. The
primary drawback of this method is the need for concurrent measurements of both pressure and
flow waveforms. We have for the first time investigated the accuracy of a novel methodology for
estimating wave intensity employing only single pressure waveform measurements; we studied
both carotid- and radial-based estimations in a large heterogeneous cohort. Tonometry was
performed alongside Doppler ultrasound to acquire measurements of both carotid and radial
pressure waveforms as well as aortic flow waveforms in 2640 healthy and diseased participants
(1439 female) in the Framingham Heart Study. Patterns consisting of two forward waves (Wf1,
Wf2) and one backward wave (Wb1) along with reflection metrics were compared with those
obtained from exact WI analysis. Carotid-based estimates correlated well for forward peak
amplitudes (Wf1, 𝑟 =0.85, 𝑝 <0.05; Wf2, 𝑟 =0.72, 𝑝 <0.05) and peak time (e.g., Wf1, 𝑟 =0.94,
𝑝 <0.05; Wf2, 𝑟 =0.98, 𝑝 <0.05), and radial-based estimates correlated fairly to poorly for
amplitudes (Wf1, 𝑟 =0.62, 𝑝 <0.05; Wf2, 𝑟 =0.42, 𝑝 <0.05) and peak time (Wf1, 𝑟 =0.04, 𝑝 =0.10;
Wf2, 𝑟 =0.75, 𝑝 <0.05). In all cases, estimated Wb1 measures were not correlated. Reflection
61

metrics were well correlated for healthy patients (𝑟 =0.67, 𝑝 <0.05), moderately correlated for
valvular disease (𝑟 =0.59, 𝑝 <0.05) and fairly correlated for CVD (𝑟 =0.46, 𝑝 <0.05) and heart
failure (𝑟 =0.49, 𝑝 <0.05). These findings indicate that pressure-only WI produces accurate results
only when forward contributions are of primary interest and only for carotid pressure waveforms.
The pressure-only WI estimations of this chapter provide an important opportunity to further the
goal of uncovering clinical insights through wave analysis affordably and non-invasively.
4.2 Introduction
Information that is contained in blood pressure and flow waves provides valuable insight
into ventriculo-arterial function and is important for the development of non-invasive diagnostic
methods. The effective condition of major biomechanical properties of the heart and circulatory
system may be altered due to cardiovascular conditions such as heart failure, hypertension, or
valvular disease, and such alterations lead to changes in waveform shapes. Numerous wave
analysis techniques have been proposed to study the patterns found in these waves in order to
uncover clinically valuable information, such as pulse wave analysis [76] as well as separation
into backwards and forwards components [77]. Many of these methods have demonstrated
efficacy in predicting or capturing relevant clinical metrics through analysis of both pressure and
flow measurements [78], although pressure wave-only analysis methods (such as reservoir-
excess pressure [79, 80], wave condition number [81] and intrinsic frequency [82, 83]) have
recently enjoyed increased traction due to the immense interest in obtaining blood pressure
measurements cheaply, easily and non-invasively [84, 85].
Wave intensity (WI) is one such method for characterizing hemodynamic coupling,
accomplished by quantifying the power per area transported by forward and backward-traveling
waves (as well as quantifying their respective timings) in the arterial system [86]. This enables
62

calculation of reflection measures that have been experimentally and clinically shown to provide
valuable prognostic information above standard cardiovascular risk factors [87], including for
myocardial infarction [88] and ischemic heart disease [89]. So far, WI analysis has provided
insights on arterial physiology and pathophysiology in the systemic circulation [90], on cerebral
vasometer tone [91], and on pulmonary circulation [92]. In addition, a recent population-based
clinical study has reported that, for mid- to late-life patients, elevated WI in the carotid artery
predicts faster cognitive decline in long-term follow-ups independently of other cardiovascular
risk factors [72].
WI is determined by incremental changes in pressure and flow velocity, and hence requires
measurements of both. The typical pattern of WI consists of a large amplitude forward (positive)
peak Wf1 (corresponding to the initial compression caused by a left ventricular contraction),
followed by a small amplitude backward (negative) peak Wb1 (corresponding to the reflection of
the initial contraction), and finally followed by a moderate amplitude forward decompression
wave Wf2 (in protodiastole). In order to enable WI analysis using only single (possibly non-
invasive) pressure measurements in the absence of the corresponding flow waveforms, Hughes et
al. [70] recently proposed an estimated WI calculation by combining it with a reservoir pressure
approach [79, 80] used for estimating the flow increments. Such an approach assumes that the
aortic pressure waveform can be decomposed into a sum of a reservoir pressure (accounting for
the net compliance of the arteries) and an excess pressure (determined by local wave behavior)
[93]. Leveraging the morphological similarity between the waveforms, the latter can be
considered approximately proportional to the flow in the aortic root [94] and hence can act as a
surrogate for estimating the flow wave increments required for WI calculations [70].
63

Previous studies on this promising pressure-only WI method have been based only on
transferred radial measurements and have focused only on adequate feasibility and
reproducibility employing very limited healthy individuals. The actual accuracy of such
pressure-only analysis in capturing WI measures such as peak amplitudes, timing (arrival) and
reflection metrics has not been studied, and hence its potential applicability is unknown. It is
imperative to study and validate the performance of such a method since its use in clinical
studies is already underway (e.g., Michail et al. [95]). Thus our objective was to conduct a
rigorous analysis on the accuracy and practical applicability of the pressure-only WI method
using a large heterogeneous population of healthy and diseased women and men in the
Framingham Offspring Cohort. To the best of our knowledge, this is the first time that such a
quantitative comparison has been made in any clinical cohort. Using non-invasive tonometry
measurements of pulse pressure, we examined the ability of pressure-only WI analysis to capture
features and corresponding metrics from an “e act” analysis (which required both pressure and
flow wave measurements), including as a function of measurement site, calibration choice and
cardiovascular health.
4.3 Materials and methods
4.3.1 Participants and data
We used a subgroup of Framingham Heart Study (FHS) data, a population-based
epidemiological cohort analysis, in the present work. The sample was drawn from the eight-
examination cycle of an Offspring cohort which has been previously described [96, 97]. The
sample participants all underwent comprehensive and non-invasive assessment of central
hemodynamics, providing an extensive collection of tonometry recordings of both carotid and
radial pressure waveforms as well as aortic flow waveforms [96]. The initial population
64

represented a heterogenous cohort of N=2640 participants including 1201 male and 1439 female.
The age of the participants ranged from 40 to 91 years old, including at least 279 patients who
suffered from cardiovascular diseases (CVD) (defined as myocardial infarction, coronary
insufficiency, stroke, heart failure, or cardiovascular-related death [98]), 177 who suffered from
valvular diseases and 32 with congestive heart failure (CHF, defined based on FHS criteria [99]).
Of the initial 2640 participants, reservoir analysis failed for radial pressure measurements in
1014 (38%) and failed for carotid pressure measurements in only 35 (1.3%), excluding these
participants from the analysis. Failure was defined as poor exponential fitting in the methodology
that is described later, ultimately yielding non-physical reservoir pressure components. All
participants provided written informed consent, and the protocols were approved by the Boston
University Medical Campus and Boston Medical Center Institutional Review Board.

65

Table 4.1 Baseline Characteristics of Patient Data (N = 1617).
Variable Value
Clinical measures
Age, y 66 ± 9
Women, n (%) 803(49.7)
Height, cm 168 ± 10
Weight, kg 80 ± 17
Body mass index, kg/m
2
28.0 ± 5.1
Heart rate, bpm 61 ± 10
Brachial blood pressure, mmHg
Systolic 142 ± 19
Diastolic 69 ± 9
Pulse 73 ± 18
Hypertension, n (%) 610 (38)
Diabetes mellitus, n (%) 157(10)
Valve Disease, n (%) 101 (6)
Arrythmia, n (%) 102 (6)
Hemodynamics measures
𝑞 𝑚𝑎𝑥 , ml/s 314 ± 69
𝑈 𝑚𝑎𝑥 , m/s 0.94 ± 0.15
Time (𝑈 𝑚𝑎𝑥 ), ms 193 ± 24
Time (maximum pressure), ms 224 ± 52
All values are (mean ± SD) except as noted. 𝑞 𝑚𝑎𝑥 is the maximum aortic flow during a
cardiac cycle. 𝑈 𝑚𝑎𝑥 is the maximum aortic velocity. Time is measured from the beginning of
systole.

66

4.3.2 Wave intensity analysis
The principles and derivation of WI analysis, which enables a decomposition of arterial
waves into forward and backward components, have been previously described [74, 100, 101].
WI is defined as the total rate of working, or the power per unit cross-sectional area 𝐴 , of an
artery due to blood pressure 𝑃 =𝑃 (𝑡 ) and average cross-sectional blood flow velocity 𝑈 =
𝑈 (𝑡 ) . Mathematically speaking, WI analysis is based on the method of characteristics and
follows the propagation of infinitesimal waves in one-dimensional space. A hyperbolic system of
equations in space and time is transformed into a system of ordinary differential equations
(ODEs) along the two families of characteristics defined as 𝑑𝑧 /𝑑𝑡 =𝑈 ±𝑐 , where 𝑈 is the
average velocity over the cross section and 𝑐 =√(𝐴 /(𝜌 𝜕𝐴 /𝜕𝑃 )) is the speed of propagation for
fluid density 𝜌 [102]. The Riemann invariants can then be defined on the characteristic lines as
d𝑅 ±
=d𝑈 ±
1
𝜌𝑐
d𝑃 . (4.1)
Solving for d𝑃 and d𝑈 as increments (changes) of the pressure and velocity over a time
instance, the instantaneous WI is defined by their corresponding product d𝐼 , given in units of
power per unit area (W/m
2
) as
d𝐼 ≔d𝑃 d𝑈 =
𝜌𝑐
4
(d𝑅 +
2
−d𝑅 −
2
) . (4.2)
Here, forward running waves (d𝑅 +
) add to the WI, while backward running waves (d𝑅 −
)
subtract from it. Hence if d𝐼 >0 at a fixed time during the cardiac cycle, forward waves will
dominate at that moment. Conversely, if d𝐼 <0, backward (reflected) waves will dominate.
Hence WI patterns determine the direction and intensity of arterial wave propagation at any time
during a cardiac cycle, ultimately quantifying the respective contributions of forward waves (that
67

largely originate from the left ventricle) and backward waves (that are mostly related to wave
reflections [62]). As can be noted from Eq. (4.2), simultaneous measurements of both pressure
and flow (velocity) are needed to determine values of WI. Previous clinical studies have
established that carotid pressure waveforms can be used as accurate surrogates for aortic pressure
waveforms [103-105]. Hence, carotid pressure measurements were employed as surrogates for
central pressure waveforms for all exact WI calculations in this work.
4.3.3 Estimating Wave Intensity from Only Pressure Measurements
A pressure-only estimate of WI can then be obtained based on an observationally-founded
assumption that the flow velocity 𝑈 (𝑡 ) is directly proportional to the excess pressure 𝑃 xs
(𝑡 ) [70,
76, 94, 106, 107], i.e.,
𝑈 (𝑡 )=
𝑃 xs
(𝑡 )
𝜌 ∙𝑐 =
𝑃 (𝑡 )−𝑃 res
(𝑡 )
𝜌 ∙𝑐 (4.3)
for given blood density 𝜌 (kg/m
3
) and pulse wave velocity 𝑐 (m/s) [70], where the reservoir
pressure 𝑃 𝑟𝑒𝑠 (𝑡 ) is extracted from a given pressure waveform 𝑃 (𝑡 ) . A substitution of this
expression into that for exact wave intensity d𝐼 in Eq. (4.2) yields a pressure-only WI estimate,
which we denote d𝐼 ̃
, given in terms of this excess pressure as
d𝐼 ̃
≔
1
𝜌 ∙𝑐 d𝑃 d(
𝑃 −𝑃 res
𝜌 ∙𝑐 ). (4.4)
The corresponding reservoir pressure can be extracted from pressure measurements by an
overall conservation of mass for the circulation, described via a first-order ODE given as [108]
𝑑 𝑃 res
𝑑𝑡
= 𝑘 𝑠 (𝑃 (𝑡 )−𝑃 res
(𝑡 ))− 𝑘 𝑑 (𝑃 res
(𝑡 )− 𝑃 𝑣 ) (4.5)
68

for a systolic rate constant 𝑘 𝑠 , diastolic rate constant 𝑘 𝑑 , and a ceasing pressure 𝑃 𝑣 [108].
Since aortic inflow (velocity) is normally zero during the diastolic phase, Eq. (4.3) implies that
the first term in Eq. (4.5) may be neglected over 𝑇 𝑠 ≤𝑡 ≤𝑇 (where 𝑇 𝑠 is the time corresponding
to the end of systole). Hence the corresponding solution to Eq. (4.5) over the entire cardiac cycle
is given by
𝑃 res
(𝑡 )={
𝑘 𝑑 𝑘 𝑑 +𝑘 𝑠 𝑃 𝑣 + 𝑒 − (𝑘 𝑑 +𝑘 𝑠 )𝑡 [∫ 𝑘 𝑠 𝑃 (𝜏 )𝑒 (𝑘 𝑑 +𝑘 𝑠 )𝜏 𝑑 𝜏 +𝑃 res
(0)−
𝑘 𝑑 𝑘 𝑑 +𝑘 𝑠 𝑃 𝑣 𝑡 0
],
(𝑃 (𝑇 𝑠 )−𝑃 𝑣 )𝑒 − 𝑘 𝑑 (𝑡 −𝑇 𝑠 )
+ 𝑃 𝑣 ,
0≤𝑡 <𝑇 𝑠 (systole )
𝑇 𝑠 ≤𝑡 ≤𝑇 (diastole )
(4.6)
where 0≤𝑡 <𝑇 𝑠 and 𝑇 𝑠 ≤𝑡 ≤𝑇 corresponds to the systolic and diastolic time intervals,
respectively, and 𝑃 res
(0) is assumed equal to the measured pressure at the beginning of systole
[108]. The unknown constants 𝑘 𝑠 ,𝑘 𝑑 and 𝑃 𝑣 in Eq. (4.6) were calculated by first determining 𝑘 𝑑
and 𝑃 𝑣 through exponential fitting of the solution in the diastolic phase 𝑇 𝑠 ≤𝑡 ≤𝑇 to the
measured pressure data using the Matlab fminsearch routine; as illustrated in Fig. 4.2(a), the
reservoir effect is the main driver of the exponential pressure fall in diastole [70]. The
corresponding value of 𝑘 𝑠 was determined by enforcing continuity at 𝑡 =𝑇 𝑠 between the systolic
and diastolic solutions of 𝑃 res
(𝑡 ) in Eq. (4.6) via non-linear optimization in Matlab. Fig. 4.2(a)
illustrates the overall relationships between a measured pressure 𝑃 and the correspondingly
extracted reservoir pressure 𝑃 res
, as well as between a measured flow 𝑄 =𝐴𝑈 and the
corresponding excess pressure 𝑃 xs
. All WI calculations (exact and estimated) employed
differences (e.g., d𝑃 ,d𝑈 ) calculated via Savitsky-Golay filters [108].
4.3.4 Measurement sites and excess pressure calibration
Since the direct central blood pressure is not usually accessible in practice, we obtained
exact (pressure- and flow-based) and estimated (pressure-only) WI patterns by employing a
central pulse pressure extracted from the available (carotid and radial) tonometry through two
69

common methods. That is, for each WI calculation technique, we performed (1) a radial-based
WI analysis, where the central pulse pressure was obtained from the radial pressure waveform
via a generalized transfer function, and (2) a carotid-based WI analysis, where the carotid
pressure pulse, due to the adjacency of the carotid artery to the aorta, was employed as a direct
surrogate of the central blood pressure (which, as aforementioned, is a well-established and
accurate surrogate [103, 105]). Given the Fourier representations 𝑃̃
rad
(𝜔 ) and 𝑃̃
cen
(𝜔 ) of the
radial and central pressure waveforms (calculated via a Fast Fourier Transform of the time-
signals), the radial-based transfer function was defined as
𝐻 cen−rad
(𝜔 )=
𝑃̃
rad
(𝜔 )
𝑃̃
cen
(𝜔 )
=
|𝑃̃
rad
(𝜔 )| 𝑒 𝑖 𝜑 rad
|𝑃̃
cen
(𝜔 )|𝑒 𝑖 𝜑 cen
, (4.7)
or
𝐻 cen−rad
(𝜔 )=
|𝑃̃
rad
(𝜔 )|
|𝑃̃
cen
(𝜔 )|
𝑒 𝑖 𝜑 rad−cen
, (4.8)
for angular frequency 𝜔 and phase 𝜑 rad
(𝜔 ) (resp. 𝜑 cen
(𝜔 ) ) of the radial (resp. central)
pressure waveforms. Here, |∙| denotes the complex magnitude of the respective Fourier
component. Phases were adjusted by adding or subtracting integral values of 2𝜋 such that all
angles consistently fell within (−𝜋 ,𝜋 ]. The corresponding generalized transfer function was
obtained by averaging all such individual transfer functions (computed from each pair of radial
and central pressure waveform patient data) over bins of length 1Hz. Use of this transfer function
for radial-based measurements has been well-established and validated clinically [70, 80, 95,
109].
All velocity data was measured by Doppler ultrasound applied to the aortic input [96]. These
flow waveforms were combined with carotid pressure waveforms in order to determine exact
70

(conventional) WI for subsequent comparisons with (either carotid or transferred radial)
pressure-only estimated WI. Fig. 4.1 illustrates the overall procedure for computing the exact and
estimated WI for both measurement sites.

Fig. 4.1 Schematic of the computation of exact WI (𝐝 𝑰 ) and pressure-only estimate of WI (𝐝 𝑰 ̃
). The pressure wave
(blue) and the flow wave (red) are required to compute exact WI d𝐼 . Only the pressure waveform (blue) is needed to compute d𝐼 ̃
.
Here, ∁ is determined based on calibration of excess pressure.
For facilitating comparisons between exact and estimated WI, we used two methods for
calibrating the excess pressure generated in the reservoir analysis of the pressure waveform for
the latter. In order to enable a truly flow-independent analysis, we considered a population
method denoted as Calibration 1 that employed a scaling of the excess pressure by an assumed
peak aortic flow velocity of 1m/s, a value suggested by previous population-based clinical
studies [110, 111] as well as employed in the proof-of-concept study of pressure-only WI
analysis [70]. In order to investigate the reasonability of this choice using our extensive FHS
71

data, a second method denoted as Calibration 2 employed a (patient-specific) scaling of the
excess pressure with the measured peak aortic flow in each individual.
4.3.5 Statistical analysis
Baseline characteristics for the study sample were tabulated, and continuous variables
derived from the sample data summarized as mean ± standard deviation (SD). Bland-Altman
analysis was performed to demonstrate the agreement between exact and estimated WI peaks and
troughs found in typical WI patterns (see Results), presented as mean differences with limits of
agreement (mean bias ± 1.96 SD of the differences). These patterns include two forward waves
and a backward-running wave, from which we investigated their peak amplitudes and timings.
We additionally investigated corresponding reflection measures from this pressure decomposed
analysis, such as reflection index (RI, defined as the ratio of the peak backward pressure with
total pressure) as well as the ratio of peak backward to peak forward pressure (Pb/Pf) [77]. We
assessed the statistical accuracy of these measures for carotid-based versus radial-based WI
measurements, peak flow calibration of 1m/s for all patients versus patient-specific measured
peak flow, as well as correlations for healthy participants, those with CVD, those with only
valvular disease, and those with CHF. Exact WI values for all cases were determined using
central flow measurements together with carotid pressure measurements that, again, served as a
surrogate for central pressure [103, 105]. In order to assess the error of the estimated values with
those of exact WI analysis, we evaluated both Pearson correlation coefficients 𝑟 , where 𝑟 <0.05 is
considered as not correlated (denoted NC in the corresponding tabular fields), and root mean
square errors (RMSE) of peak values in the patterns (normalized to the corresponding exact
mean WI values). Statistical significance was defined as a correlation p-value<0.05. All
72

mathematical analysis on the clinical data was performed using custom written codes and
algorithms implemented in Matlab (R2020b, The Mathworks Inc).
4.4 Results
Study exclusions based on the success of reservoir analysis on both radial and carotid
measurements yielded a final sample size of 1617 participants. The characteristics of these
remaining participants are presented in Table 4.1. Fig. 4.2(b) and 4.2(c) illustrate typical
pressure-only patterns of estimated WI along with exact WI in two patients: a 50-year-old female
with a heart rate of 54 bpm (Fig. 4.2B) and a 74-year-old male with a heart rate of 50 bpm (Fig.
4.2(c)). Evident in these example curves is the ability of pressure-only WI to adequately capture
the typical patterns of exact WI. These major wave peaks are consistent with what is expected
for conventional exact WI [62] and were chosen for comparison in this study.
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Fig. 4.2 Visualization of the reservoir pressure approach and corresponding wave intensity patterns. (a) A reservoir
pressure 𝑷 𝒓𝒆𝒔 (𝒕 ) (solid grey) is extracted from a given pressure waveform 𝑷 (𝒕 ) (solid black) by Eq. (4.6), with the corresponding
difference defined as excess pressure 𝑷 𝒙𝒔
(𝒕 ) (dashed gray). 𝑸 =𝑨𝑼 (dashed black) is the corresponding flow for velocity 𝑼 and
cross-sectional area 𝑨 (rescaled by the characteristic impedance in order to align the upstrokes of pressure and flow). (b,c) Exact
WI calculations that employ both pressure and flow (solid gray) are overlaid on Estimated WI calculations (from two different
excess pressure calibrations) that employ only measured carotid pressure waveforms. Examples are of two Framingham Heart
Study (FHS) patients: (b) a 50-year-old female with a heart rate of 54bpm, and (c) a 74-year-old male with a heart rate of 50bpm.
Peaks of the forward wave-dominated contributions (Wf1 and Wf2) and the backward-dominated contribution (Wb1) are
identified accordingly.
4.4.1 Accuracy of pressure-only WI from radial-based measurements
Table 4.2 presents correlation and errors between radial-based pressure-only estimates of WI
with exact WI for major features commonly found in WI patterns, including the peak amplitudes
of Wf1, Wf2, and Wb1 (and their corresponding timings) as well as measures of RI and pressure
(a)
(b) (c)
74

ratio (Pb/Pf) which quantify the contributions of reflected and backward pulse pressure waves.
The absolute Mean ± SD (both in W m
2
s
2
⁄ ×10
4
) of error between the estimated WI and exact
WI are as follows: Calibration 1, Wf1 Peak Amplitude: 36 ± 60, Wf2 Peak Amplitude: 31 ± 20,
Wb1 Peak Amplitude: 17 ± 13, and Calibration 2, Wf1 Peak Amplitude: 44 ± 48, Wf2 Peak
Amplitude: 33 ± 18, Wb1 Peak Amplitude: 16 ± 13.
Table 4.2 Accuracy of radial-based estimated wave intensity (WI) analysis for peak amplitudes, timings, and reflection
coefficients (N = 1617).
Hemodynamics Variable
Correlation
Coefficient* (𝒓 )
Mean ± SD
Normalized
RMSE
Limit of
agreement
(Mean
Difference)
Calibration 1 (Umax = 1m/s)
Wf1 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.61

147 ± 59 0.440 235 (35.7)
Wf2 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.42 30 ± 11 0.569 77 (30.9)
Wb1 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.20 23 ± 13 0.815 52 (17.2)
Calibration 2 (patient-specific Umax)
Wf1 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.77 139 ± 63 0.423 188 (43.4)
Wf2 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.55 28 ± 10 0.583 71 (32.9)
Wb1 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.26 22 ± 14 0.806 52 (15.9)
Wf1 Peak Time (ms) NS (𝑝 =.10) 59 ± 11 0.302 55 (-11)
Wf2 Peak Time (ms) 0.75 321 ± 30 0.066 82 (5)
Wb1 Peak Time (ms) NS (𝑝 =.07) 118 ± 35 0.595 205 (51)
Reflection Index (RI) 0.68 0.21 ± 0.04 0.169 0.13 (-0.01)
Pressure ratio (Pb/Pf) 0.69 0.26 ± 0.06 0.205 0.20(-0.02)
* Correlations are calculated between the exact and estimated WI. Correlations have 𝑝 -values<0.05 unless otherwise
indicated. Timing, reflection index and pressure ratio calculations are independent of calibration and separated by the dashed
line. 95% limits of agreement are used. RMSE indicates Root Mean Square Error (normalized to mean exact wave intensity);
NS, Not Significant.

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Average peak values for exact WI calculations were found to be 183 ± 74 (Wf1), 61 ± 22
(Wf2) and 6 ± 4 (Wb1). For Calibration 1 (Umax = 1m/s) of estimated wave intensity, the peak
amplitudes of Wf1 were well correlated (.60 to .70) and the peak amplitudes of Wf2 were only
moderately correlated (.40 to .50). For comparison, Calibration 2 (patient-specific measured
Umax) was strongly correlated for Wf1 (above .75) and moderately correlated for Wf2 (.50 to
.60). For reference, normalized RMSE values are also provided; however, we are interested
mostly in the ability of estimated WI analysis in capturing (i.e., correlating with) generalized
waveform features of exact WI. The lower correlations (and errors) observed in the patient-
specific calibration was expected through use of a more physiologically accurate calibration
assumption (using velocity measurements). In all cases, the peak amplitude of the backward
wave Wb1 was either poorly correlated or not correlated at all. Additionally, the timings of these
peaks (which indicate physiological forward and reflection arrival times) is only well correlated
for Wf2, but not correlated for the timings of the peaks of Wf1 and Wb1. Exact WI values for
these timings were 47 ± 9 ms (Wf1), 326 ± 30 ms (Wf2) and 169 ± 41 ms (Wb1). For reference,
Bland-Altman plots of the corresponding differences with exact WI peak amplitudes for both
calibrations is presented in Fig. 4.3.
76

Fig. 4.3 Bland-Altman plots for Wf1, Wf2 and Wb1. Agreement of peak amplitude values between exact wave intensities
and those estimated by only radial pressure measurements. (a, c, e) represents estimations for Wf1, Wf2 and Wb1
(respectively) based an assumed peak velocity of 1m/s. (b, d, f) represent respective estimations based on the measured peak
velocity in each individual.
4.4.2 Accuracy of pressure-only WI from carotid-based measurements
Bland-Altman plots indicating the agreement between all intra-individual differences in the
three major peak and trough amplitudes (Wf1, Wf2, and Wb1) between exact WI (whose average
(a) (b)
(c) (d)
(e) (f)
77

values have been described above) and carotid-based pressure-only WI (using both calibrations)
are shown in Fig. 4.4.

Fig. 4.4 Bland-Altman plots for Wf1, Wf2 and Wb1. Agreement of peak amplitude values between exact wave intensities
and those estimated by only carotid pressure measurements. (a, c, e) represents estimations for Wf1, Wf2 and Wb1
(respectively) based an assumed peak velocity of 1m/s. (b, d, f) represent respective estimations based on the measured peak
velocity in each individual.
(a) (b)
(c) (d)
(e) (f)
78

Table 4.3 presents correlation and errors between these carotid-based pressure-only
estimates of WI with exact WI for peak amplitudes and timings of Wf1, Wf2 and Wb1, as well
as RI and the pressure ratio. It should be noted that a majority of excluded patients from the
radial case were still successful in carotid-based measurement analysis. In both calibrations, the
peak amplitudes of both Wf1 and Wf2 were strongly and significantly correlated, respectively,
with Wf1 and Wf2 calculated via exact WI. As with radial-based analysis, the peak amplitudes
of the backwards wave Wb1 were poorly to not at all correlated in all cases. Additionally, the
timings of these peaks were strongly correlated for Wf1 (in contrast to radial-based
measurements) and Wf2, but poorly correlated for the timings of the peaks of Wb1 (similarly to
radial-based measurements). The absolute Mean ± SD (both in W m
2
s
2
⁄ ×10
4
) of error between
the estimated WI and exact WI are as follows: Calibration 1, Wf1 Peak Amplitude: 41 ± 42, Wf2
Peak Amplitude: 1 ± 17, Wb1 Peak Amplitude: 7 ± 11, and Calibration 2, Wf1 Peak Amplitude:
30 ± 27, Wf2 Peak Amplitude: 3 ± 14, Wb1 Peak Amplitude: 6 ± 11.

79

Table 4.3 Accuracy of carotid-based estimated wave intensity (WI) analysis for peak amplitudes, timings, and reflection
coefficients (N = 1617).
Hemodynamics Variable
Correlation
Coefficient* (𝒓 )
Mean ± SD
Normalized
RMSE
Limit of
agreement
(Mean
Difference)
Calibration 1 (U max = 1m/s)
Wf1 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.85 224 ± 76 0.247 163 (-41.1)
Wf2 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.72 61 ± 24 0.264 68 (-0.7)
Wb1 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) NC 13 ± 9 0.806 42 (7.4)
Calibration 2 (patient-specific U max)
Wf1 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.96 212 ± 88 0.176 107 (-29.8)
Wf2 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.82 58 ± 24 0.224 54 (3.1)
Wb1 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) NC 12 ± 10 0.801 42 (6.6)
Wf1 Peak Time (ms) 0.94 45 ± 9 0.087 13 (2)
Wf2 Peak Time (ms) 0.98 320 ± 31 0.026 23 (6)
Wb1 Peak Time (ms) NS (𝑝 =.06) 117 ± 47 0.650 250 (52)
Reflection Index (RI) 0.64 0.19 ± 0.03 0.320 0.10 (-0.04)
Pressure ratio (Pb/Pf) 0.63 0.24 ± 0.05 0.398 0.16 (-0.06)
*Correlations are calculated between the exact and estimated WI. Correlations have 𝑝 -values<0.05 unless otherwise
indicated. Timing, reflection index and pressure ratio calculations are independent of calibration and separated by the dashed
line. 95% limits of agreement are used. RMSE indicates Root Mean Square Error; NC, Not Correlated; NS, Not Significant.
4.4.3 Accuracy of pressure-only WI under healthy and diseased conditions
Table 4.4 presents the corresponding correlations and errors of estimated WI amplitudes,
timings and reflection measures for different categories of patient health. These categories were
defined as healthy participants (N=1344), participants with CVD (N=192), participants with only
valvular disease (N=81), and participants with heart failure (N=18). The analysis of Table 4.4
employed carotid-based WI measurements only, consistent with the higher accuracy of such
measurements as presented in previous tables. Additionally, since the aim of this chapter is to
analyze a completely flow-independent pressure-only WI method, the peak flow velocity was
80

assumed 1m/s in all cases (i.e., Calibration 1), consistent with previous results. The resulting
peak amplitudes of both Wf1 and Wf2 were strongly and significantly correlated, respectively,
with Wf1 and Wf2 peaks produced by exact WI for all healthy and diseased cases. The peak
amplitudes of the backward wave Wb1 were poorly to not at all correlated in all cases. RI and
pressure-ratios were well correlated for healthy individuals (as expected from Table 4.3 since
they are the majority of patients), moderately correlated for valvular disease, and fairly
correlated for CHF and CVD.

81

Table 4.4 Accuracy of carotid-based estimated wave intensity (WI) analysis for peak amplitudes and reflection
coefficients for healthy and diseased participants.
Hemodynamics Variable
Correlation
Coefficient*
(𝐫 )
Mean ± SD
Normalized
RMSE
Limit of agreement
(Mean Difference)
Healthy Individuals (N = 1344)
Wf1 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.85 222 ± 73 0.249 158 (-42.0)
Wf2 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.73 61 ± 25 0.259 67 (-1.4)
Wb1 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) NC 13 ± 9 0.789 40 (7.7)
Reflection Index (RI) 0.67 0.19 ± 0.03 0.320 0.10 (-0.04)
Pressure ratio (Pb/Pf) 0.67 0.24 ± 0.05 0.397 0.15 (-0.06)
Individuals with CVD (N = 192)
Wf1 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.85 232 ± 83 0.243 178 (-39.0)
Wf2 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.65 62 ± 24 0.297 78 (1.7)
Wb1 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) NC 12 ± 13 0.898 57 (6.0)
Reflection Index (RI) 0.46 0.19 ± 0.04 0.320 0.14 (-0.04)
Pressure ratio (Pb/Pf) 0.43 0.24 ± 0.06 0.406 0.22 (-0.06)
Individuals with only valvular disease (N = 81)
Wf1 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.85 242 ± 103 0.244 221 (-31.0)
Wf2 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.81 61 ± 25 0.259 63 (5.5)
Wb1 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) NS (𝑝 =.07) 11 ± 8 0.793 37 (4.9)
Reflection Index (RI) 0.59 0.20 ± 0.03 0.320 0.12 (-0.04)
Pressure ratio (Pb/Pf) 0.58 0.25 ± 0.05 0.396 0.18 (-0.06)
Individuals with CHF (N = 18)
Wf1 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.87 236 ± 91 0.255 180 (-46.0)
Wf2 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) 0.70 65 ± 21 0.220 61 (0.57)
Wb1 Peak Amplitude (W m
2
s
2
⁄ ×10
4
) NS (𝑝 =.30) 9 ± 8 0.922 41 (3.5)
Reflection Index (RI) 0.49 0.18 ± 0.03 0.274 0.13 (-0.03)
Pressure ratio (Pb/Pf) NS (𝑝 =.06) 0.22 ± 0.04 0.342 0.19 (-0.04)
* Correlations are calculated between the exact and estimated WI. Correlations have 𝑝 -values<0.05 unless otherwise
indicated. 95% limits of agreement are used. Peak velocity in each individual is assumed to be 1m/s. RMSE indicates Root
Mean Square Error; NC, Not Correlated; NS, Not Significant.
82

4.5 Discussion
In the population-based Framingham Heart Study, we investigated the accuracy of pressure-
only estimates of WI (in terms of peak wave amplitudes, timings and reflection measures)
derived from radial and carotid pressure waveforms in an extensive general cohort which
included both healthy and diseased individuals. Wave intensities provide an energy-based index
that is particularly of interest due to the emerging popularity of energy-based cardiovascular
analysis [18, 51, 95, 112]. The most significant drawback of conventional WI, similarly to a
number of other wave analysis methods [113], is the need for measurements of both pressure and
flow waveforms, which is often very difficult to obtain outside a research setting. Calculation of
the reservoir and excess pressure can be obtained from a single measured pressure waveform,
and hence previous studies have tried to incorporate such a pressure decomposition for
measuring wave intensities and corresponding wave reflections [70, 95].
4.5.1 Carotid versus radial-based estimates of WI
A first major novel finding of our study is that aortic WI estimated based on carotid pressure
measurements is substantially more accurate (in terms of both correlations and errors for peak
amplitudes and timings) than estimations based on radial pressure. Radial-based estimates were
unable to capture the timing of Wf1, which is clinically significant since it measures the behavior
of the incident waves generated by the left ventricle. However, this is to be expected: a linear
transfer function cannot fully capture the central dynamics using radial measurements. Such
dynamics are determined by the nonlinear characteristics and relationships of pulsatile blood
flow between the two sites as defined by the governing fluid-structure (Navier-Stokes coupled
with solid wall motion) systems. This, combined with the nonlinear dynamics of wave
reflections, implies that use of a linearized transfer function for the radial site (as similarly used
83

in previous contributions on pressure-only WI [70, 95]) will be naturally unable to fully capture
the forward wave components and completely unable to capture the highly non-linear behavior
of the backward wave components (such dynamics are already physiologically compromised due
to the larger distance of the radial site from central [114]]). Only a solution to the true inverse
operator of the governing partial differential equations can potentially transfer such information
between the two measurement sites.
4.5.2 Pressure-only WI cannot capture backward wave contributions
Pressure-only WI analysis was unable in general to capture the peak backwards amplitude of
Wb1 for both carotid- and radial-based estimates. Analytically speaking, this is consistent with
the known flaws of the reservoir method in modeling wave dynamics (introducing errors in
arterial wave analysis [115] as well as the known shortcomings of Windkessel-based models (on
which the reservoir approach is based) in capturing wave reflections in general [115-117]. Such
drawbacks do not affect the ability (in both carotid and radial cases) of capturing forward
contributions of WI, since such components are dominated by the primary shape of left
ventricular flow whose morphology is well captured by the excess (reservoir) pressure.
4.5.3 Performance of pressure-only WI in healthy and diseased individuals
The assessment of wave reflection has been consistently associated with measures of
cardiovascular events, independent of conventional risk factors in people with treated
hypertension [77, 87, 100]. The importance of reflection measures such as RI (Pb/[Pb + Pf]) and
pressure ratio (Pb/Pf) have been demonstrated clinically and experimentally to capture changes
in hemodynamic coupling as well as intricacies of wave propagation [77]. Since, as discussed
above, Table 4.2 and Table 4.3 suggest that carotid-based estimated WI is well-correlated with
exact WI (even for flow-independent constant peak velocity of 1m/s), we employed such data
84

and assumptions for comparing diseased subgroups in Table 4.4. We found that, although
backward waves were not well-captured (as aforementioned), the corresponding reflection
measures were fairly to strongly correlated among the general population, independent of
cardiovascular disease. This implies that pressure-only WI performs more-or-less equally well in
the 1,617 healthy and diseased individuals (it was previously only shown in a small healthy
population of 34 participants [70]), and is able to capture important reflection coefficients for
each employing only carotid pressure measurements. This is promising for reproducing the same
clinical predictions and wave reflections that are provided by exact WI analysis [87]. For
example, valvular disease specifically affects the dynamics of the incident waves generated by
the left ventricle---the forward peaks by estimated WI in Table 4.4 are very strongly correlated
with exact WI.
4.5.4 Failure rates of the reservoir pressure algorithm
We also found that reservoir-based analysis has a substantially higher rate of failure for
radial-based pressure waveforms when compared to carotid measurements. Reservoir pressure
analysis finds exponential fitting parameters in a non-linear formulation, and we found that it
failed for radial measurements in 38% of the initial N=2640 heterogeneous population
(preliminary studies reported around 15% failure of radial in a small healthy N=34 [70]).
Reservoir analysis of carotid measurements failed in only 1.3% of the initial N=2640 population.
The high failure rate observed in radial-based reservoir analysis may be largely attributed to a
combination of possibly poor-quality tonometry traces and the utilization of a transfer function to
derive central pressure waveforms from radial measurements. This results in a possible lack of
identifiable fiducial points that are required for the determination of reservoir pressure and the
exponential fitting (these findings are consistent with those reported previously [70, 95]).
85

4.5.5 On the accuracy of peak flow velocity calibration
We additionally investigated the assumption of peak flow velocity equal to 1m/s, as
suggested by previous population-based studies [110, 111] and employed in preliminary studies
of pressure-only wave intensity [70, 95]. This peak velocity is necessary to calibrate excess
pressure to reproduce the appropriate scale of conventional (exact) WI. We found that such a
value was more than acceptable by comparing it to the ideal scenario: patient-specific peak
velocities determined from the corresponding flow measurements in FHS (see Table 4.1 for the
range). As expected, patient-specific values performed better---however, the differences were
only marginal (consistent with smaller population studies [70]), and hence assumed peak flow
velocities may be appropriate for enabling truly flow-independent WI estimation in a large
population. However, further studies are necessary in order to determine whether other assumed
values may be more accurate, particularly as a function of patient characteristics such as age and
health.
4.5.6 Critique of method
ur study has certain limitations that should be considered. ne limitation is that we don’t
have invasively-measured aortic pressure waveforms for determining exact WI. However, our
choice of using the carotid pressure waveform as a surrogate for aortic is well-established [103,
105]. Even by comparing estimates with exact wave intensities that were determined using the
transferred radial pressure (instead of the carotid as a surrogate) together with measured central
flow, the poor correlations of radial-based pressure-only estimates persist. Additionally, our
findings may also not be generalizable to other racial and ethnic groups since data was composed
primarily of white participants of Western European descent.
86

4.6 Conclusion
In the present chapter, we have provided a comprehensive analysis on the accuracy of
pressure-only estimates of WI (in terms of peak wave amplitudes, timings and reflection
measures) derived from radial and carotid pressure waveforms in 1,617 healthy and diseased
women and men. We showed that non-invasive carotid pressure-only estimates of wave intensity
adequately captured important features found in forward components of exact wave intensity
(determined by pressure and flow) ---albeit with varying degrees of success. However, the ability
of pressure-only WI to reproduce reflection metrics demonstrates promise in future clinical use,
where entirely flow-independent analysis can be achieved. The pressure-only WI estimations of
this work provide an important opportunity to further the goal of uncovering clinical insights
through wave analysis affordably and non-invasively. However, further studies are needed to
independently assess the ability of pressure-only WI in capturing clinical measures by itself
(without regard to exact WI) as well as cardiovascular risk factors in experimental or clinical
settings.

87

CHAPTER 5 : Hybrid Fourier decomposition-machine learning approach for
pressure-only aortic wave intensity estimation in Framingham heart study

This chapter is based on the following published manuscript: Aghilinejad, A., Wei, H. and
Pahlevan, N.M., 2023. Non-Invasive Pressure-Only Aortic Wave Intensity Evaluation Using
Hybrid Fourier Decomposition-Machine Learning Approach. IEEE Transactions on Biomedical
Engineering.

5.1 Chapter abstract
Recent clinical studies have demonstrated the clinical significance of wave intensity (WI)
method for diagnosis of cardiovascular and cerebrovascular diseases. The primary drawback of
this method is the need for concurrent measurements of pressure and flow waveforms. In this
chapter, we aimed to investigate the accuracy of pressure-only WI based on Fourier-based
machine learning (ML) approach. Collection of tonometry recordings of carotid pressure and
ultrasound measurements for aortic flow waveforms is used from the Framingham Heart
database (2640 individuals; 55% women). Feature selection for physics-based ML training is
performed by Fourier decomposition of the pressure waveforms. WI parameters consisting of
two forward waves (Wf1, Wf2) and one backward wave (Wb1) are employed for comparison.
Method-derived estimates are significantly correlated for forward peak amplitudes (Wf1, 𝑟 =0.88,
p<0.05; Wf2, 𝑟 =0.84, p<0.05) and peak time (Wf1, 𝑟 =0.80, p<0.05; Wf2, 𝑟 =0.97, p<0.05). For
backward component (Wb1), method-derived estimates correlated strongly for amplitude
(𝑟 =0.71, p<0.05) and fairly strongly for peak time (r=0.60, p<0.05). In all cases, the Bland-
Altman analysis shows negligible bias in the estimations and maximum error is bounded to
14.7%. The results suggest the accuracy of this approach can be further improved by tuning ML
algorithms and input Fourier modes as the determinant of input size. The proposed pressure-only
88

Fourier-based ML approach provides accurate estimates for WI parameters, which are
traditionally difficult to obtain. The WI estimations of this study expand the usage of WI into
more affordable and non-invasive clinical settings such as wearable healthcare electronics.
5.2 Introduction
There has been an emerging interest in employing energy-based approaches for better
understanding the cardiovascular function in healthy and diseased conditions. The potential
clinical application can range from identifying the therapeutic targets for vascular-related
neurodegenerative diseases [51] to hemodynamic analysis of new assist devices for
cardiovascular disease patients [84] and optimizing the design for vascular stents and grafts . WI
is a well-established energy-based index representing the amount of energy carried by arterial
waves generated by the left ventricle. Wave Intensity Analysis (WIA) was introduced by Parker
and Jones [60] as a time domain method for separating forward and backward waves. In this
method, WI is computed as the product of the blood pressure and the velocity changes during
short time intervals. The typical pattern of WI consists of a large amplitude forward (positive)
peak Wf1 (corresponding to the initial compression caused by a left ventricular contraction),
followed by a small amplitude backward (negative) peak Wb1 (corresponding to the reflection of
the initial contraction), and finally followed by a moderate amplitude forward decompression
wave Wf2 (in protodiastole) [62]. Previous studies have shown the applicability of WIA in
arterial physiology and pathophysiology in the systemic circulation [90], on cerebral vasomotor
tone [91], and on pulmonary circulation [92]. In addition, recent population-based clinical study
done by Chiesa et al. [72] has demonstrated that elevated carotid WI, captured in Wf1
amplitudes, predicts faster cognitive decline in long-term follow-ups independently of other
cardiovascular risk factors.
89

While WIA has proved an increasingly valuable approach to understanding the
hemodynamics of normal and pathophysiological state, the current requirement of having
concurrent measurements of pressure and flow is a limitation for the application of WIA in
routine clinical practice [62, 101]. In particular, measuring the aortic flow in the clinical setting
can be relatively expensive due to the requirement of aortic flow measurement systems (e.g.,
echocardiogram) and also time consuming with the need for the trained personnel. Recently,
Hughes et al. [70] proposed an estimation method for WIA utilizing pressure measurements
alone, rather than together with measurements of flow. This method is based on the reservoir-
model of hemodynamic waves in the cardiovascular system [116]. In this model, the aortic blood
pressure is separated into components representing reservoir and excess pressures, where the
excess pressure has been shown clinically to be a surrogate of left ventricular outflow tract flow
velocity [113]. The pressure-only estimate demonstrated adequate feasibility and reproducibility
in a small dataset [70]. In addition, previous population-based study by our group [69] showed
that implementing the reservoir-based method on carotid waveform with the assumption of
constant peak velocity (1m/s to calibrate excess pressure) can outperform the typical use of radial
waveform. However, it was shown that the pressure-only WI estimate based on reservoir
analysis is not able to capture the backward components of WI, neither the amplitude nor the
timing of Wb1 [69]. In addition, the general agreement of the estimated forward components of
WI (i.e., Wf1 and Wf2) with the exact one is not significant and there is a bias in this estimation
method. Therefore, there is still an essential need for introducing a reliable pressure-only
estimate of WI to expand the clinical applicability of this method.
The objective of this chapter is to employ hybrid Fourier decomposition and machine
learning (ML) approach, to estimate WI metrics using only pressure waveform measurement. We
90

used the large heterogeneous population of the Framingham Offspring Cohort to conduct a
rigorous analysis on the accuracy of the proposed Fourier-based ML model to estimate pressure-
only WI. While the common feature selection techniques available in the literature solely relies
on the training data without considering the underlying physiology, our physics-based approach
in this study is based on the information of the hemodynamic waveforms enabling us to extract
the physiologically-relevant data. We also tested the generalizability of this approach in different
ML algorithms. The systemic work conducted here lays the groundwork for wider application of
Fourier-based ML models to estimate energy-based indices such as WI that can provide
clinically valuable insights of the physiology of the cardiovascular system.
5.3 Materials and methods
5.3.1 Participants and data
We used a subgroup of Framingham Heart Study (FHS) data, a population-based
epidemiological cohort analysis in this study. The sample was drawn from the eight-examination
cycle of an Offspring cohort which has been previously described [96]. The characteristics of
these participants are presented in Table 5.1. The initial population represented a heterogeneous
cohort of N=2640 participants including 1201 male and 1439 female with the age range from 40
to 91 years old. This population included 279 patients who suffered from cardiovascular-related
diseases (defined as myocardial infarction, coronary insufficiency, stroke, heart failure [98]), 177
who suffered from valvular diseases and 32 with congestive heart failure (defined based on FHS
criteria [99]). All participants provided written informed consent, and the protocols were
approved by the Boston University Medical Campus and Boston Medical Center Institutional
Review Board. The sample participants all underwent comprehensive and non-invasive
assessment of central hemodynamics, providing an extensive collection of tonometry recordings
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of carotid pressure waveforms [96]. The aortic flow waveforms were obtained via Two-
dimensional echocardiography of the left ventricular outflow tract (LVOT) followed by a pulsed
Doppler from an apical 5-chamber view to acquire the aortic flow waveform [96].
Table 5.1 Baseline Characteristics of Patient Data (N = 2640).
Variable Value
Clinical measures
Age, y 66 ± 9
Women, n (%) 1439 (55)
Height, cm 167 ± 10
Weight, kg 78 ± 17
Body mass index, kg/m
2
27.9 ± 5.1
Heart rate, bpm 62 ± 10
Brachial blood pressure, mmHg
Systolic 141 ± 20
Diastolic 69 ± 9
Pulse 72 ± 19
Hypertension, n (%) 978 (37)
Diabetes mellitus, n (%) 229 (9)
Valve Disease, n (%) 177 (5)
Arrythmia, n (%) 182 (5)
All values are (mean ± SD) except as noted.
5.3.2 Wave intensity analysis
WI is defined as the power per unit cross-sectional area 𝐴 , of an artery due to blood pressure
𝑃 =𝑃 (𝑡 ) and average cross-sectional blood flow velocity 𝑈 =𝑈 (𝑡 ) . The principles and
derivation of WI analysis, which enables a decomposition of arterial waves into forward and
backward components, have been previously described in detail [60, 62, 74]. Mathematically
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speaking, WI is computed as the product of the change in pressure (𝑑𝑃 ) times the change in
velocity (𝑑𝑈 ) during a small interval, given by
𝑑𝐼 =𝑑𝑃𝑑𝑈 . (5.1)
To remove the dependency of the WI to the sampling time, the derivative of pressure and
velocity are divided by the time interval (denoted as
𝑑𝑃
𝑑𝑡
and
𝑑𝑈
𝑑𝑡
, respectively), hence the WI in the
units of power per unit area per unit time (W.s
-2
.m
-2
). WI patterns determine both the direction
and intensity of arterial wave propagation at any time instance during a cardiac cycle. For
example, if 𝑑𝐼 >0 at a fixed time during the cardiac cycle, forward waves that largely originate
from the left ventricle will dominate at that moment. Conversely, if 𝑑𝐼 <0, backward waves that
are mostly related to wave reflections [62]) will dominate. As can be noted from Eq. (5.1),
simultaneous measurements of both pressure and flow (velocity) are needed to determine values
of WI which is the typical requirement for the energy-based indices. Fig. 5.1 illustrates a typical
pattern of WI where the three major wave peaks (chosen for comparison in this study) are
labeled.
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Fig. 5.1 Schematic of the computation of exact WI and pressure-only estimate of WI. The pressure wave (blue) and the flow
wave (red) computed from cross-sectional averaged velocity are required to compute exact WI. Only the pressure waveform
(blue) is needed to compute. Figure also illustrates the three major peaks of WI and the ML pipeline.
5.3.3 Fourier representation and physics-based feature selection
In this study, we utilized Fourier series decomposition for input feature selection (Fig. 5.1).
Originally, the carotid pressure waveforms from tonometry measurements are sampled at the rate
of 1000Hz, which for the typical heart rate (HR) of 60 beats per minute in human beings
(corresponding to cycle size of 1 second) leading to 1000 data points per single pressure
measurement. This high dimensionality of the input signal renders naive ML constructs that are
significantly limited for practical data-driven applications [118]. In this chapter, to transform
data from this high-dimensional space to a low-dimensional one, we utilized Fourier-based
analysis to retain meaningful properties of the original data. The Fourier series represents a
synthesis of a periodic function by summing harmonically related sinusoids and cosinusoides.
An arbitrary periodic pressure function P(t) can be represented as a Fourier series with N
entral Pressure
Data Processing
f 1 Amplitude
f ime
b 1 Amplitude
f Amplitude
b 1 ime
f 1 ime

entral Pressure
Aortic Flow

Parameters
Derivative
Product

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oscillatory components. A common form of the Fourier series decomposition, in the Sine-Cosine
form is defined as:
𝑃 (𝑡 )=
𝑎 0
2
+∑ (𝑎 𝑛 𝑐𝑜𝑠 (
2𝜋 𝑇 𝑛𝑡 )+𝑏 𝑛 𝑠𝑖𝑛 (
2𝜋 𝑇 𝑛𝑡 ) )
𝑁 𝑛 =1
, (5.2)
where T is the period of the pressure function P(t) (i.e., the cardiac cycle or inverse of HR
for blood pressure waveform). In conducting the Fourier decomposition here, we used the Sine-
Cosine form rather than an Amplitude-Phase form to make the inputs independent from each
other. This is due to the fact that the system of all sine and cosine functions at different
frequencies builds a complete orthonormal set. Additionally, including the phase into the inputs
may introduce non-linear relationships between each of the inputs. Coefficients 𝑎 𝑛 and 𝑏 𝑛 are
associated with each individual harmonics (cosine and sine) corresponding to different
frequencies 𝑓 𝑛 =
𝑛 𝑇 , and can be calculated by the Fourier transform given by:
𝑎 𝑛 =
2
𝑇 ∫ 𝑃 (𝑡 )𝑐𝑜𝑠 (
2𝜋 𝑇 𝑛𝑡 )𝑑𝑡 𝑇 0
,𝑏 𝑛 =
2
𝑇 ∫ 𝑃 (𝑡 )𝑠𝑖𝑛 (
2𝜋 𝑇 𝑛𝑡 )𝑑𝑡 𝑇 0
,𝑛𝜖 (0, ∞). (5.3)
Obtaining the coefficients using Eq. (5.3), the represented pressure waveform 𝑃̃
(𝑡 ) with
finite selected frequency (i.e., n = 0 to N) is achieved by adding up each individual frequency
component (Eq. (5.2)). In this study, the features of the pressure wave were extracted as the first
𝑁 𝑡 ℎ
low frequency components of the waveforms using the Fast Fourier Transform (FFT). This
is the advantage of using FFT-based input reduction for clinical pressure waveform as high
frequency components do not provide any significant and additional physiological information in
computing the WI, since it is not sensitive to high frequency data [84]. Fig. 5.2 represents the
associated error between the clinically-measured pressure waveforms and the reconstructed
pressure waveform based on the number of Fourier coefficients. The outliers of the data are
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identified with four standard deviation convention, leading to 263 outliers. As it can be noticed
from this figure, the error between the measured pressure and the reconstructed one is marginal
after N=10. The selected input features in this study (depending on the chosen number of Fourier
modes; FN) is consists of FN cosine coefficients, (FN - 1) sine coefficients (sine since 𝑏 0
is
always equal to zero), cardiac time and the notch time, lead to the input size of (2FN + 1).

Fig. 5.2 Associated error between the reconstructed pressure waveform based on different number of Fourier modes and
the measured pressure waveform. The error is reported as the normalized root mean square error (NRMSE) averaged over the
whole population.
5.3.4 Machine learning models
In the present chapter, we employed six well-established ML algorithms to evaluate the
accuracy of the proposed Fourier-based method on WI estimation. These models include Lasso
regressor, kernel ridge regressor (KRR), support vector regressor (SVR), random forest
regressor, gradient boosted decision-tree, and neural network. Each one of these algorithms are
trained on features derived from the Fourier decomposition of the pressure waveform, notch time
and cardiac time. The training and testing data split for all machine learning analyses was 70%
and 30% respectively. Models are strictly trained on the training population and the test data are
only used at the last step for model evaluation. Python’s sklearn and ensorFlow packages are
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used for data pre-processing (e.g., normalization), training, and testing the algorithms. The
pandas and numpy packages were also used for data processing. The hyperparameters in the
models were found by a ten-fold cross validation (CV) scheme using the GridSearchCV library.
Particularly for neural networks, the “Adam” optimizer was chosen to optimize the weights of
neurons in the hidden and output layer, and the number of epochs was set to be equal to 1500.
For all models, the combination of hyperparameters attributed to the highest accuracy for
training are chosen.
5.3.5 Statistical analysis
Baseline characteristics for the study sample were demonstrated in Table 5.1, and
continuous variables derived from the sample data summarized as mean ± standard deviation
(SD). Two forward waves (Wf1 and Wf2) and a backward-running wave (Wb1) from the pattern
of WI during one cardiac cycle are chosen to evaluate the performance of the proposed Fourier-
based ML methodology. We investigated both the peaks and troughs amplitudes with their
corresponding timings. Exact WI values for all cases were determined using central flow
measurements together with carotid pressure measurements that served as a surrogate for central
pressure [103, 105]. In order to assess the error of the estimated values with those of exact WI
analysis, we evaluated both Pearson correlation coefficients 𝑟 , and root mean square errors
(RMSE) of peak values in the patterns. Statistical significance was defined as 𝑝 -value<0.05. We
also reported the Normalized RMSE (NRMSE) which is computed based on the range of the
dependent variable (difference between the maximum and the minimum). The agreement and the
bias between the exact WI variables and estimated ones are evaluated by Bland-Altman analysis
presented as mean differences with limits of agreement (mean bias ± 1.96 SD of the differences).
Levene’s test for homogeneity of variance is conducted to e amine the inter -algorithm
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differences between F-ML models for evaluating the WI parameters. Lastly, Kruskal- allis’
rank-sum test and the Dunn’s test with onferroni ad ustment are employed for overa ll and two-
by-two comparisons for WI parameters considering non-normal distribution. Mathematical
analysis on the clinical data was performed using custom written codes and algorithms
implemented in Python ((Python Software Foundation, Python Language Reference, version
3.9).
5.4 Results
5.4.1 Accuracy of pressure-only WI amplitudes
Table 5.2 presents the correlation, errors, and agreement between pressure-only estimates of
WI with exact WI for the peak amplitudes of Wf1, Wf2, and Wb1. The analysis is conducted for
different ML models to evaluate the applicability of the Fourier-based reduction of the pressure
waveforms. In this analysis, we utilized the first 20 Fourier modes of the pressure waves, which
guarantees the capture of all features of the pressure waveform (see Fig. 5.2). In all six
algorithms, the peak amplitudes are well-correlated, where both forward peaks were significantly
correlated, and backward peak was fairly strongly correlated. As demonstrated by the mean
difference reported in Table 5.2, the systemic bias between the estimated values and exact WI
measurements is negligible.

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Table 5.2 Regression statistics between predicted and exact WI peak amplitudes.
ML Algorithm Lasso KRR SVR Random Forrest Gradient Boosting Neural Network
Wf1 Peak Amplitude
Correlation Coefficient (r) 0.83 0.88 0.88 0.83 0.85 0.88
RMSE 36.6 30.8 31.2 38.1 34.5 32.5
Normalized RMSE (%) 8.8 7.4 7.5 9.1 8.3 7.8
Limit of Agreement 143.2 120.7 122.5 150.0 135.6 125.0
Mean Difference 1.6 0.7 -1.1 -0.4 -0.4 1.2
Wf2 Peak Amplitude
Correlation Coefficient (r) 0.75 0.83 0.84 0.77 0.80 0.82
RMSE 13.6 11.3 11.1 13.0 12.2 11.8
Normalized RMSE (%) 10.6 8.8 8.7 10.1 9.5 9.2
Limit of Agreement 53.2 44.2 43.8 51.3 48.6 45.2
Mean Difference -0.5 -0.2 -0.6 0.4 0.3 0.5
Wb1 Peak Amplitude
Correlation Coefficient (r) 0.57 0.69 0.71 0.63 0.62 0.69
RMSE 3.1 2.8 2.7 3.0 2.9 2.9
Normalized RMSE (%) 13.9 12.4 12.0 13.6 13.3 13.0
Limit of Agreement 12.3 10.8 10.5 11.9 11.6 11.3
Mean Difference 0.1 0.1 0.3 -0.2 -0.3 0.2
*Correlations have 𝑝 -values<0.05 unless otherwise indicated. All wave intensity amplitudes are reported in units of
𝑊 𝑚 2
𝑠 2
×10
4
. SVR indicates the support vector regressor. KRR indicates the kernel ridge regressor. Underline values
show the highest correlations.
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Fig. 5.3 demonstrates the Bland-Altman plots indicating the agreement between all intra-
individual differences in the three major peak and trough amplitudes (Wf1, Wf2, and Wb1) as
well as the corresponding scatter plots between exact and method-derived pressure-only WI.
These results are reported for the neural network.

Fig. 5.3 Scatter and Bland-Altman plots for Wf1, Wf2 and Wb1 amplitudes. The plots are demonstrated for the test
data (N=714). Correlation (top row) and Agreement (bottom row) of peak amplitude values between exact wave intensities and
those estimated by Fourier-based neural network model.
5.4.2 Accuracy of pressure-only WI timings
Table 5.3 presents the corresponding correlations, errors, and agreements of estimated WI
peak timings for different ML algorithms. Similar to the peak amplitudes, it can be noted that the
systemic bias in the estimation of WI timings is negligible. Among the six ML models, neural
network outperforms the rest in terms of correlation coefficient and errors. In this model, the
peak timings for forward WI components (Wf1 and Wf2) are significantly correlated, and the
timing for the backward peak is fairly strongly correlated.
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Table 5.3 Regression statistics between predicted and exact WI peak times.
ML Algorithm Lasso KRR SVR Random Forrest Gradient Boosting Neural Network
Wf1 Peak Time
Correlation Coefficient (r) 0.55 0.75 0.75 0.51 0.68 0.80
RMSE 7.3 5.9 6.1 7.7 6.5 5.4
Normalized RMSE (%) 13.0 10.6 10.8 13.7 11.6 9.8
Limit of Agreement 28.5 23.2 23.8 30.1 25.1 20.5
Mean Difference -0.3 0.1 0.2 0.0 0.0 0.7
Wf2 Peak Time
Correlation Coefficient (r) 0.92 0.97 0.97 0.84 0.92 0.97
RMSE 11.3 7.1 8.1 16.7 11.7 6.8
Normalized RMSE (%) 5.6 3.5 4.0 8.2 5.7 3.3
Limit of Agreement 44.1 27.9 31.9 65.5 43.9 27.0
Mean Difference 1.1 0.4 0.6 -1.1 -0.8 -2.3
Wb1 Peak Time
Correlation Coefficient (r) 0.49 0.59 0.59 0.45 0.55 0.60
RMSE 24.0 22.0 21.9 24.8 23.0 22.2
Normalized RMSE (%) 14.2 13.0 13.0 14.7 13.6 13.1
Limit of Agreement 94 86.2 85.9 97.4 93.1 88.5
Mean Difference -0.4 0.3 0.5 -0.3 -0.2 -0.5
*Correlations have 𝑝 -values<0.05 unless otherwise indicated. All wave intensity timings are reported in units of 𝑚𝑠 . SVR
indicates the support vector regressor. KRR indicates the kernel ridge regressor. Underline values show the highest correlations.
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For reference, scatter plots for pressure-only estimates of WI peak timing versus the exact
ones as well as the Bland-Altman plots of the corresponding differences for the neural network
model is presented in Fig. 5.4.

Fig. 5.4 Scatter and Bland-Altman plots for Wf1, Wf2 and Wb1 times. The plots are demonstrated for the test data (𝑵 =714).
Correlation (top row) and Agreement (bottom row) of peak timings between exact wave intensities and those estimated by
Fourier-based neural network model.
5.4.3 Difference comparison between ML algorithms
Fig. 5.5 demonstrates the distribution of the testing data among different F-ML algorithms.
Levene's test suggests that for all WI variables including the peak amplitudes and timings, the
variances for each group are different. p-values from Kruskal- allis’ test for overall comparison
of the algorithms is also reported in this figure. We also conducted the pairwise comparison
using Dunn's test between the models for Wf1 and Wb1 amplitudes and timings. The results of
Dunn's test suggest that for Wf1 and Wb1 timing there is a significant difference between the
neural network model and the other algorithms. For Wf1 and Wb1 amplitudes, there is no
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significant difference between F-ML models except between neural networks and support vector
regressor (p=0.0236).

Fig. 5.5 Boxplots for WI parameters and the comparison between different ML models. The plots are demonstrated for the
test data (𝑵 =714). All wave intensity amplitudes are reported in units of
𝑾 𝒎 𝟐 𝒔 𝟐 ×𝟏𝟎
𝟒 and timings are reported in 𝒎𝒔 . The
reported p-values are from the Kruskal- allis’ test for overall comparison of the algorithms .
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5.4.4 Analytical versus Fourier-based ML pressure-only WI
Table 5.4 presents the correlation and agreement between the pressure-only estimates of WI,
the one which is proposed in this study (Fourier-based ML) and the previous approach based on
the reservoir model of the hemodynamic waves (ODE-based analytical model). The comparison
is conducted between the peak amplitudes and times of the WI parameters including Wf1, Wf2,
and Wb1. The output data from neural network model is used for the comparison. 𝑁𝐶 in this
table indicates no correlation.
Table 5.4 Comparison between the proposed Fourier-based ML and the previous ODE-based analytical models.
Pressure-only WI Method
Wf1
Amplitude
Wf1
Time
Wf2
Amplitude
Wf2
Time
Wb1
Amplitude
Wb1
Time
Fourier-based ML
Correlation Coefficient (r) 0.88 0.80 0.82 0.97 0.69 0.60
Limit of Agreement 125 20.5 45.2 27.2 11.3 88.5
Mean Difference 1.2 0.7 0.5 -2.3 0.2 -0.5
Analytical Method
Correlation Coefficient (r) 0.85 0.94 0.72 0.98 𝑁𝐶 𝑁𝐶
Limit of Agreement 107 13 54 23 42 250
Mean Difference -29.8 2 3.1 6 6.6 52
*Correlations have 𝑝 -values<0.05 unless otherwise indicated. All wave intensity amplitudes are reported in units of
𝑊 𝑚 2
𝑠 2
×10
4
and timings are reported in 𝑚𝑠 . 𝑁𝐶 indicates no correlation. The data for ODE-based analytical model are adopted
from [69].

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5.4.5 Sensitivity analysis for the input Fourier modes
Fig. 5.6 presents the correlation between the estimated WI and the exact one for three major
peak amplitudes and timings as a function of different Fourier modes to represent the input
pressure waveform. Neural network and support vector regressor are chosen as the best
performing prediction models. As it can been seen from this figure, there is an optimum range
for choosing the number of Fourier modes (10 to 20 modes for our database) that can result in
the highest correlation. The trend is consistent between both models.

Fig. 5.6 Impact of number of Fourier modes on the proposed method accuracy. Correlations between the estimated WI and
the exact one for three major peak amplitudes and timings as a function of different Fourier modes.
5.4.6 Sensitivity analysis for the training size
Fig. 5.7 presents the RMSE of estimated WI peak amplitudes and their corresponding timing
as a function of the training size. For both timing and the amplitude, it can be observed that
including more data instances to the training sample after reaching the 40% of the entire dataset
only affect the accuracy of the estimation marginally.
Neural Network
Support ector egressor
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Fig. 5.7 Sensitivity of precision in terms of normalized root mean square (NRMSE) to the number of the training data.
The 100% of the training size corresponds to the whole clinical dataset.
5.5 Discussion
In this chapter, we investigated the accuracy of pressure-only estimates of WI (in terms of
peak wave amplitudes and timings) derived from a novel Fourier-based ML approach in an
extensive general cohort of the Framingham heart database. This clinical database included both
healthy and diseased individuals. Our results show that this approach works great for capturing
all the main features of WI using only pressure measurements and significantly outperforms the
previous models for pressure-only WI [69]. In addition, our results suggest that the Fourier-based
representation of the pressure wave can be tuned to reduce the input feature size without the
major influence on the accuracy of the model.
5.5.1 Towards non-invasive pulse wave analysis
In recent years, a variety of new medical technologies have been developed to assess cardiac
function and health. These technologies utilize wave analysis to uncover clinical insights
affordably and non-invasively [119]. There is therefore great merit in developing new strategies
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for quantifying pulse wave measures [120]. Aortic wave intensity is among such measures that is
particularly of interest due to the emerging popularity of energy-based cardiovascular analysis
[62, 69, 70]. The significance of the WI analysis has recently been pronounced in clinical studies
by showing its capability in quantifying the excessive arterial pulsatility that may contribute to
cognitive decline during the mid- to late-life [72]. The most notable drawback of conventional
WI, similarly to a number of other wave analysis methods, is the need for concurrent
measurements of both pressure and flow waveforms. This can either happen by invasive
simultaneous measurement of flow and pressure (which makes it clinically challenging), or
non-invasive measurement of flow using imaging modalities (e.g., echocardiography) and
tonometry for pressure. Both these approaches are clinically challenging and time-consuming.
These factors limit the usage of WI as it is often very difficult to obtain outside a research
setting. The recent study of Hughes et al. [70] demonstrated the feasibility of a single pressure
waveform analysis of WI on a very limited population size (e.g., N=34 healthy participants)
based on the reservoir wave method. The accuracy of this method is further examined by our
group on population-based clinical database (Framingham) and was shown that reservoir-based
pressure-only WI produces accurate results only when the amplitude of forward contributions is
of primary interest [69]. This method is not able to capture backward contributions. In addition, a
significant bias was observed in this estimation method which reduces the generalizability of this
technique. In the present study, we have comprehensively investigated, for the first time, the
accuracy of wave intensities calculated through use of only single pressure waveform
measurements using a novel Fourier-based ML approach.
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5.5.2 Principal findings
A first major finding of our study is that aortic WI estimated based on the proposed Fourier-
based ML approach significantly outperforms estimations based on reservoir analysis (as
employed in previous studies) in terms of both correlations and errors for peak amplitudes and
timings. Table 5.2 suggests that forward contributions of WI (amplitudes of Wf1 and Wf2) are
strongly to significantly correlated with exact WI (maximum correlation of 0.88 for Wf1 and
0.84 for Wf2). The maximum error associated for different ML algorithms is 10.6%
corresponding to the Lasso regression. As demonstrated in Table 5.3, there is a significant
correlation between the peak timings of predicted Wf1 and Wf2 with the exact one (correlations
of 0.80 and 0.97 respectively corresponding to the neural network). Regarding the Backward
component of WI, in contrast to the previous reservoir model which was not able to capture this
feature (correlation coefficient below 0.2) [69], the proposed method here can capture the
amplitude of Wb1 with strong correlation (0.71 corresponding to SVR in Table 5.3) and its
timing with fairly strong correlation (0.61 corresponding to neural network in Table 5.3). Table
5.2 and Table 5.3 also suggest that there is a high agreement between the method-derived
estimations of WI (both amplitude and timing) with the exact one with negligible systemic bias.
High agreement can be also observed from the sample Bland-Altman plots demonstrated for
three peak amplitudes demonstrated in Fig. 5.3 and three peak times demonstrated in Fig. 5.4.
We also found that choosing the number of Fourier modes to decompose the input pressure
waveform to the machine has a substantial impact on the correlation and the error of predicted
WI (Fig. 5.6). Depending on the chosen number of Fourier modes, some information about the
pressure waveform may be missed as shown in Fig. 5.2 (e.g., the exact shape of the notch).
However, to capture the WI peaks, these higher frequency features may not be necessarily
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required and hence it is essential to find the optimum number Fourier modes. This is particularly
important for reduced-order approaches for ML modeling [118]. Results suggest that there is an
optimum number of Fourier modes to include as feature dimension for training purposes. Results
show that 10 to 20 modes of Fourier results in highest correlation and minimal error based on the
utilized population in this study. A large input size with uncorrelated variables (features) could
reduce the overall accuracy of the ML model. On the other hand, employing a very small number
of Fourier modes can also lead to inability of the input features to sufficiently represent the
pressure waveform for the purpose of capturing WI. All in all, these results suggest that in order
to take the most advantage of the proposed Fourier-based ML approach, it is essential that
different numbers of Fourier modes be examined for the purpose of enhancing the accuracy of
the model with minimal input feature size.
5.5.3 Applicability of machine learning in cardiovascular engineering
In the present study, we investigated the performance of different ML algorithms on Fourier
decomposed pressure waveforms with the goal of predicting the WI. Recent advances in AI and
ML along with the availability of large clinical datasets bring new research possibilities and
approaches in cardiovascular engineering. As examples, Bikia et al. [121, 122] demonstrated the
potential applicability of ML-based methodology for predicting aortic hemodynamics and
cardiac contractility for the goal of non-invasive monitoring. Tavallali et al. [123] showed the
applicability of the regression analysis in estimating carotid-femoral pulse wave velocity which
is the gold standard measurement of vascular aging. In another study, Jin et al. [124] showed
possibility of assessing the vascular ageing based on a single peripheral pulse wave instead of
carotid-femoral pulse wave velocity using ML models. Following above, this study is in line
with the direction of introducing AI and ML into the field of cardiovascular engineering.
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The inter-algorithm comparison between different F-ML models, conducted by Kruskal-
allis’ and pairwise Dunn's tests, demonstrated that there is no statistical difference in capturing
the output of WI peak amplitudes. This suggests that utilizing Fourier representation of the
pressure waveforms, linear models such as Lasso regression predictor led to strong correlations
in capturing WI amplitudes and may not be necessary to utilize complex algorithms which
require significant hyperparameter tuning with expensive computational costs. In addition, the
linear models take advantage of better interpretability compared to more complex ones such as
the neural network. Results from pairwise Dunn's tests demonstrated that the only model which
is statistically different from the rest in capturing WI peak timings is the neural network as
shown in Fig. 5.5. Overall, a more delicate dependence of WI timing with the shape of the
waveform necessitates utilizing more complex models for achieving accurate estimations. Lastly,
we e amined the model’s sensitivity to the relative training size of the utilized dataset in this
study. The training size was modified from 90% to 10% of the total number of cases (Fig. 5.7).
Results suggest that NRMSEs decreased gradually with increasing training size. The
improvement in the accuracy however for neural network and SVR models are negligible after
the training size reaches 40% of the dataset. This sensitivity analysis suggests that the employed
training size for this study is sufficient.
5.5.4 Study limitations
One limitation is that we do not have invasively-measured aortic pressure waveforms for
determining exact WI. However, our choice of using the carotid pressure waveform as a
surrogate for aortic is well-established [103, 105]. Each modeling strategy is limited by
assumptions and data collection is dependent on several factors, including clinical context,
physician preferences, or other clinical decisions which influence the model development [122].
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With regards to the future work, further studies are needed to independently assess the ability of
the proposed pressure-only WI in capturing clinical measures by itself as well as cardiovascular
risk factors in experimental or clinical settings.
5.6 Conclusion
This chapter included a large heterogeneous sample of healthy subjects and patients with
cardiovascular diseases in order to examine the accuracy of the pressure-only estimation method
of WI in the Framingham Heart Study population. This study employed a novel Fourier-based
ML approach to estimate different WI parameters. The strength of our study includes the use of a
single-pressure waveform measurement for approximating wave intensity which potentially
enables analysis by only non-invasive pressure measurements. Contrary to conventional WI
methodologies which require measurements of both flow and pressure waveforms, the novel
proposed technique requires only the central arterial pressure waveform to perform the analysis.
Our results suggest that the proposed pressure-only estimate of WI has straightforward
application for large sample sizes and this technique can ultimately increase the clinical
usefulness of WI. The results of this model show excellent agreement with conventional WI
(which required both pressure and flow wave measurements), significantly outperform the
previous approach based on the reservoir model and can be used as a reliable model to compute
WI based on single pressure measurement. The pressure-only WI estimations of this work
provide an important opportunity to further the goal of uncovering clinical insights through wave
analysis affordably and non-invasively.

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CHAPTER 6 : Model-based fluid-structure interaction approach for
evaluation of thoracic endovascular aortic repair endograft length in type
B aortic dissection

This chapter is based on the following published manuscript: Aghilinejad, A., Wei, H., Magee,
G.A. and Pahlevan, N.M., 2022. 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, 10.

6.1 Chapter abstract
This chapter and the next one is on the fluid dynamic modeling of the complicated aortic
disease, called aortic dissection, and its impact on heart-aorta coupling. Aortic dissection is a
catastrophic life-threatening aortic emergency that can result in aortic rupture, myocardial
infarction, pericardial tamponade, stroke, acute kidney failure, bowel ischemia, lower extremity
ischemia, and in the long-term congestive heart failure and aortic aneurysms [125, 126].
Thoracic endovascular aortic repair (TEVAR) is a commonly performed operation for patients
with type B aortic dissection (TBAD). The goal of TEVAR is to cover the proximal entry tear
between the true lumen (TL) and the false lumen (FL) with an endograft to induce FL
thrombosis, allow for aortic healing, and decrease the risk of aortic aneurysm and rupture. While
TEVAR has shown promising outcomes, it can also result in devastating complications including
stroke, spinal cord ischemia resulting in paralysis, as well as long-term heart failure, so treatment
remains controversial. Similarly, the biomechanical impact of aortic endograft implantation and
the hemodynamic impact of endograft design parameters such as length are not well-understood.
In this chapter, a fluid-structure Interaction (FSI) computational fluid dynamics (CFD) approach
112

was used based on the immersed boundary and Lattice-Boltzmann method to investigate the
association between the endograft length and hemodynamic variables inside the TL and FL. The
physiological accuracy of the model was evaluated by comparing simulation results with the true
pressure waveform measurements taken during a live TEVAR operation for TBAD. The results
demonstrate a non-linear trend towards increased FL flow reversal as the endograft length
increases but also increased left ventricular pulsatile workload. These findings suggest a medium
length endograft may be optimal by achieving FL flow reversal and thus FL thrombosis, while
minimizing the extra load on the left ventricle. These results also verify that a reduction in heart
rate with medical therapy contributes favorably to FL flow reversal.
6.2 Introduction
Acute aortic dissection is a tear in the aortic wall, resulting in high-pressure blood flow
through a false passage within the smooth muscle layer of the aorta, creating a false lumen (FL)
channel. This FL may flow back into the original aortic flow channel (the true lumen; TL)
distally or proximally from the original tear. Anatomically, aortic dissections are categorized
into Stanford type A involving the ascending aorta and Stanford type B aortic dissection (TBAD)
which occurs in the aortic arch or distally, and usually extend down to the thoracoabdominal
aorta [127]. While type A dissections typically undergo immediate open repair of the ascending
aorta, the thoracoabdominal segment of aorta cannot be repaired at the same time, so patients are
typically left with a residual dissection, which is anatomically similarly to a de novo type B
dissection [128, 129]. The first line treatment for TBAD is medical treatment to decrease systolic
blood pressure and heart rate which decreases the risk of rupture and progression of disease, but
there is growing evidence that early thoracic endovascular aortic repair (TEVAR) may result in
improved outcomes to medical management alone [127, 129]. TEVAR for TBAD occludes the
113

flow of blood across the proximal aortic tear and shunts it back into the TL. This decompresses
the FL, causes thrombosis within the FL, and thereby allows it to heal [130].
By decreasing FL flow, TEVAR thus allows for aortic healing and decreases the risk of
subsequent aortic aneurysm and rupture [128, 131, 132]. Clinical data have found that TBAD
patients with complete FL thrombosis have improved outcomes, whereas failure of FL
thrombosis, and persistent FL flow is a predictor of adverse outcomes [133-135]. However, flow
patterns in TBAD are poorly understood due to the complexity of patient-specific anatomy and
physiology as well as the limitations of imaging modalities [136]. While TEVAR has shown
promising results in the treatment of TBAD patients, the permanent implantation of a prosthetic
endograft can cause its own set of problems including spinal cord ischemia with resulting
paralysis, stroke, and long-term heart failure. Current endografts have significantly great
stiffness and anisotropy compared to the native aorta [137]. The compliance mismatch between
the endograft and the native aorta can lead to a cascade of hemodynamic alterations which affect
the aortic wave dynamic and may contribute to subsequent cardiovascular complications such as
congestive heart failure [138, 139]. Deleterious effects of compliance mismatch can even occur
proximal to the endograft by affecting delicate hemodynamic balance between the left ventricle
(LV) and vascular network which exists in normal physiological condition [140]. Therefore,
much remains to be understood about the biomechanical consequences of TEVAR for TBAD
and what length of endograft is optimal for treatment.
The objective of this chapter is to evaluate the impact of endograft implantation in TEVAR
on the unique fluid dynamics behavior of the pulsatile blood flow in the TL and FL. We used an
idealized geometry to focus on the overall behavior of the hemodynamics independent of
individual patient anatomy. Due to the extensive endograft-related variability in TEVAR, this
114

study focused primarily on the impact of endograft length. We examined the impact of endograft
length on the LV pulsatile workload (as an indicator of global cardiovascular state [141]) and the
FL flow reversal (as a predictor for FL thrombosis [142, 143]). While the optimal treatment
modality for type B dissection is currently the subject of considerable debate, this study provides
insight on the impact of TEVAR on aortic fluid dynamics.
6.3 Materials and methods
6.3.1 Physical problem
A schematic representation of the 3D axisymmetric model of the dissected aorta along with
the illustrative images from TBAD patient is shown in Fig. 6.1. In our idealized model, it is
assumed that the TL is located concentric within the aorta and the FL is formed uniformly
around the TL, connected with the flexible and compliant septum in the middle [144, 145]. For
modeling the dynamics of the LV, the time-varying elastance model is used as an inlet condition
of the dissection model [54]. The importance of outflow boundary conditions to capture
physiologically accurate hemodynamic waveforms is highlighted in the previous works [146,
147]. In this study, the extension tube boundary model was used as the outflow boundary
condition to capture the compliance, resistance, and wave reflections of the downstream
vasculature [148]. The dimensions of the model are chosen within the average physiological
range; the length of the TL is chosen from descending aorta to the bifurcation and the length of
the septum is chosen from descending aorta to renal arteries [128]. The length of the endograft is
varied in the range of 4cm to 20cm to cover the whole range of currently utilized endografts
[134]. To investigate the effect of endograft length in this study, endograft-septum length ratio
(λ) is defined as λ=
Graft Length
Septum Length
. Based on the utilized parameters in this study, λ∈
115

(0.13, 0.26, 0.40, 0.53, 0.66) , where λ= 0.13 is considered to be short endograft, λ= 0.40
is considered to be medium endograft, and λ= 0.66 is considered to be long endograft. To
account for the compliance mismatch between the replaced endograft and native aorta, the aortic
wall and septum are considered to be compliant with stretching coefficient of the human aorta
while the endograft is assumed to have a rigid wall. The physical parameters of this study are
summarized in Table 6.1.

Fig. 6.1 Sample CT image for type B aortic dissection patient and the employed model-based approach in this chapter. CT
image (a) axial and (b) sagittal planes of the type B dissection patient. (c) Idealized model of type B aortic dissection with arrows
indicating different segments of the model.

(a)
(b)
(c)
116

Table 6.1 Geometric and material parameters used in the computational models.
Name Variable Value Reference
Length of the aortic model (cm) L 40

Length of the septum (cm) L
septum
29 [135]
Radius of the aorta (cm) r
aorta
1 [149]
Bending coefficient of wall (Pa∙m
3
) EI 2×10
−7
[150]
Length of the outflow boundary model (cm) L
boundary
15 [37]
Contraction ratio of the rigid boundary model κ 0.4

Volume compliance of the boundary model (m
3
/Pa) C
outflow
3.14×10
−11

LV compliance (ml/mmHg ) C
v
(t) Fig. 6.3 [55]
LV dead volume (ml) V
dead
4 [54]

6.3.2 Mathematical formulation
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
single-relaxation-time (SRT) incompressible LBM was used as an efficient solver of Navier-
Stokes equations [151, 152]. 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,
f
i
(𝐱 +𝐞 i
∆t,t+∆t)−f
i
(𝐱 ,t)= −
1
τ
[f
i
(𝐱 ,t)−f
i
eq
(𝐱 ,t)]+∆tF
i
, (6.1)
117

where f
i
(𝐱 ,t) is the distribution function for particles with velocity 𝐞 i
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= ρc
s
2
(τ−
1
2
)∆t, where ϑ is the kinematic viscosity and c
s
=
1
√3
c is the lattice sound speed.
The equilibrium distribution function for incompressible LBM and the forcing term are defined
as
f
i
eq
= ω
i
ρ
0
+ω
i
ρ[
𝐞 i
∙𝐯 c
s
2
+
(𝐞 i
∙𝐯 )
2
2c
s
4
−
𝐯 2
2c
s
2
], (6.2)
F
i
=(1−
1
2τ
)ω
i
(
𝐞 i
−𝐯 c
s
2
+
𝐞 i
∙𝐯 c
s
4
𝐞 i
)∙𝐟 , (6.3)
where ω
i
is the weighting factor, ρ
0
is related to the pressure by ρ
0
=
p
c
s
2
, 𝐟 is the force
density at the Eulerian point, and velocity 𝐯 can be calculated by
ρ
0
=∑f
i
, (6.4)
ρ𝐯 =∑𝐞 i
f
i
+
1
2
𝐟 ∆t. (6.5)
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 [152] to satisfy the axisymmetric
condition at the centerline [153]. To compute the deformation of the elastic aortic and septum
wall, the dynamic motion of these two in the Lagrangian form is solved using
ρ
s
h
∂
2
𝐗 ∂t
2
=
∂
∂s
[Eh(1−(
∂𝐗 ∂s
∙
∂𝐗 ∂s
)
−1/2
)
∂𝐗 ∂s
−
∂
∂s
(EI
∂
2
𝐗 ∂s
2
)]+𝐅 L
, (6.6)
118

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, ρ
s
is the density of the aortic and septum wall, Eh is the stretching stiffness, EI is the
bending stiffness, and 𝐅 L
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 [154], which is given by,
𝐗 =𝐗 0
,
∂
2
𝐗 ∂t
2
=(0,0) . (6.7)
For the same geometrical configuration, the material parameter which affects the
deformation of the vessel wall governed by Eq. (6.6) 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. (6.6). Fig. 6.2 shows the
sensitivity analysis of the radial displacement of both the intimal septum and aortic wall
computed at the center of the model during one cardiac cycle with different material elasticities.
While the results show our model is able to capture the effect of elasticity on dynamic motion of
the wall, the overall shape of the displacement waveform for different elasticities is preserved. In
this study, we used the baseline parameters reported in Table 6.1.
119

Fig. 6.2 Sensitivity of the wall dispalcement to different leveles of aortic stiffness. Radial vessel wall displacement waveform
at the center of the model for (a) intimal septum, and (b) the aortic wall for various elasticities.
6.3.3 Implementations of the boundary conditions
The LV 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 p
v
(t) inside the LV and the corresponding volume V
v
(t) in the LV are
connected via time varying compliance C
v
(t) given by,
V
v
(t)− V
dead
= C
v
(t)p
v
(t) . (6.8)
In Eq. (6.8), the constant V
dead
known as the dead volume is the limit for pressure
generation. Substituting the relation between the flow into the aorta with the V
v
(t) and
differentiating Eq. (6.8) with respect to t, we can get the following ordinary differential equation
(ODE) for the pressure inside the LV
∂p
v
(t)
∂t
= −
1
C
v
(t)
[
∂C
v
(t)
∂t
p
v
(t)+Q(x=0, t)], (6.9)
Clinically, C
v
(t) stands for inverse of LV end-systolic elastance (E
es
) which is the measure
of LV contractility [54, 55] (Fig. 6.3(a)). Once P
v
(t) is greater than the pressure at the interface
(a) (b)
120

of the aorta and the LV, the valve opens and p(x=0, t)= p
v
(t) with the flow condition given
by 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). Fig. 6.3(a) shows the empirically given time-varying
compliance (C
v
(t) ) reported from clinical data for normal contractile state of LV [55]. Fig. 6.3(b)
demonstrates the computed sample flow response to the LV model at the aortic root in our model
resulting in 5.7 L/min for average cardiac output (CO) over the cycle 𝑇 .

Fig. 6.3 Time-varying end systolic elastance heart model used in this chapter. (a) The time-varying left ventricular
compliance 𝑪 𝒗 (𝒕 ) , and (b) computed flow waveform at the aortic root using our computational model.
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. The parameters of the outflow boundary condition model are given in Table 6.1, where
(a) (b)
121

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). The presence of the rigid contraction is more
attributed to the required resistance for the system, while the elastic portion accounts for the
compliance of the eliminated vasculature.
6.3.4 Numerical method
The D2Q9 velocity model is applied in the LBM with the sound speed c where the velocity
set is given by
𝐞 i
= {
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
. (6.10)
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. (6.1) can
be transformed into
f
i
(x+e
i
Δt,t+Δt)−f
i
(x,t)=−
1
τ
[f
i
(x,t)−f
i
eq
(x,t)]+ΔtF
i
(x,t)+H
i
(x,t), (6.11)
where a source term H
i
(x,t) is given by
H
i
(x,t)=Δth
i
(1)
(x,t)+Δt
2
h
i
(2)
(x,t) , (6.12)
h
i
(1)
=−
ω
i
ρv
r
r
, (6.13)
h
i
(2)
=−ω
i
3ν
r
[∂
y
P+ρ∂
x
v
x
v
r
+ρ∂
r
v
r
v
r
+ρ(∂
r
v
x
−∂
x
v
r
)e
ix
]. (6.14)
H
i
(x,t) is the added source term into the collision step defined based on h
i
(1)
and h
i
(2)
with
P=c
s
2
∙ρ
o
. The source term is added to recover the extra terms caused by the curvature from the
122

continuity equation and Navier–Stokes equation in cylindrical coordinates [152, 155]. For
calculating the derivatives of the velocity vector along the radial and axial directions, the terms
∂
r
v
x
+∂
x
v
r
, ∂
x
v
x
and ∂
r
v
r
can be obtained by the following equation [152]
ρν(∂
β
v
α
+∂
α
v
β
)=−(1−
1
2τ
)∑ (f
i
−f
i
eq
)
8
i=0
e
iα
e
iβ
+o(ε
2
), (6.15)
where substituting α=x and β=r gives us a relation for ∂
r
v
x
+∂
x
v
r
;substituting α=β=
x gives us a relation for ∂
x
v
x
; and substituting α=β=r gives us a relation for ∂
r
v
r
. For
calculating ∂
r
v
x
−∂
x
v
r
in Eq. (6.14) the only the value left unknown is ∂
x
v
r
. Below is a finite
difference method employed to obtain ∂
x
v
r
at lattice node (i,j) with the following expression
(∂
x
v
r
)
(i,j)
=
(v
r
)
(i+1,j)
−(v
r
)
(i−1,j)
2∆x
. (6.16)
The solid deformation equation (Eq. (6)) was solved by Finite Element Method (FEM)
[156]. The IB method was used to couple the fluid and solid solvers. Particularly, implicit
velocity correction-based IB approach was used in this study which has been extensively used to
simulate the FSI problems in cardiovascular biomechanics [157, 158]. 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
𝐯 (𝐱 ,t)=𝐯 ∗
(𝐱 ,t)+δ𝐯 (𝐱 ,t) . (6.17)
The relation between the velocity correction δ𝐯 and the body force term 𝐟 is
ρ δ𝐯 (𝐱 ,t)=
1
2
𝐟 (𝐱 ,t)δ𝐭 . (6.18)
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
123

boundary satisfies the no-slip boundary condition [159]. In the revised implicit velocity
correction-based 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
, (6.19)
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 [159]. 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
. (6.20)
Substituting Eq. (6.17) and (6.19) into Eq. (6.20), we can get 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
, (6.21)
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.
(6.17), the force density acting on the fluid phase 𝐟 can be calculated using Eq. (6.18). Lastly,
the boundary force density at Lagrangian points 𝐅 L
can be explicitly found by
𝐅 L
(s,t)= −∫ 𝐟 (𝐱 ,t)δ(𝐱 −𝐗 (s,t))d𝐱 Ω
. (6.22)
124

The clinical and physical quantities were connected to the numerical quantities using
dimensionless parameters including the Womersley number Wo=r
aorta√
αρ
μ
where r
aorta
is the
reference length (radius of the aorta) and α is the pulsation frequency (i.e., heart rate) [156, 160].
For spatial and temporal discretization, each simulation was run at
D
∆x
=32 with a time step of
T
Δt
=50,000 (T=2∙π/α) . Mesh independence studies are done on the pressure profiles at
different cross sections of the model to ensure that this mesh density and time step are sufficient
for the accurate calculations. Simulations were run on US ’s center for advanced research
computing cluster nodes, each node equipped with 20 cores (2600 MHz) with 64GB memory. At
least 10 cardiac cycles were simulated to ensure a periodic steady state was reached. The
complete FSI solver for the LV-dissection model is summarized in the pseudo-code of the
algorithm shown in Fig. 6.4.
125

Fig. 6.4 Algorithm for the LV-dissection system model. Steps for implementation of IB-LBM-FEM algorithm to numerically
solve the LV-Dissection system model with time-varying elastance LV input.

6.3.5 Hemodynamic analysis
The pulsatile power (P
̅
pulse
) was used in this chapter to quantify the LV power requirement.
P
̅
pulse
is the difference between the total power P
̅
total
and the steady power P
̅
s
. The total power
was calculated based on the average product of the pressure p(t) and flow q(t) during one
cardiac cycle T, while steady power was calculated based on the product of the average pressure
and average flow during a cardiac cycle. Each of these power quantities are respectively given by
P
̅
total
=
1
T
∫ p(t)q(t)dt
T
0
, (6.23)
P
̅
s
=p
mean
q
mean
, (6.24)

0
=0
<

0

, +
, +

+
+

, , [0,

]
126

P
̅
pulse
=P
̅
total
−P
̅
s
. (6.25)
Reverse Flow Index (RFI) is calculated to quantify the flow reversal as a measure to predict
thrombose formation, following the works done by Birjiniuk et al. [136, 142, 143]. RFI is
defined as the ratio of the retrograde flow Q
reverse
(which is in the opposite direction of the
systemic circulation) over the absolute summation of the antegrade flow Q
forward
(which is in
the same direction of the systemic circulation) and retrograde flow, given by
RFI=
|∫ Q
reverse
dt
T
0
|
|∫ Q
reverse
dt
T
0
|+ |∫ Q
forward
dt
T
0
|
×100 . (6.26)
To quantify Q
reverse
and Q
forward
, velocity profiles in each lumen were integrated across
luminal cross-sections at different zones (Fig.s 6.5(a) and 6.5(b)) at each cardiac phase.
127

Fig. 6.5 In-vivo pressure measurement data during the operation of type B aortic dissection patient. Schematic of type B
aortic dissection (a) pre- and (b) post-TEVAR. Different zones are classified for computing hemodynamic quantities. (c) Invasive
pressure data measured post-TEVAR via ComboMap system from the patient.

6.3.6 Patient description and invasive clinical measurement
Data from a TBAD patient undergoing TEVAR was studied and utilized to examine the
physiological accuracy of our model. The participant was provided with written informed
consent and all protocols were approved by the Keck Medical Center of the University of
Southern California (USC) Institutional Review Board. The dissection started distal to the origin
of the left subclavian artery and extended to the infrarenal aorta and the TEVAR endograft
extended from proximal to the left subclavian to the mid-descending thoracic aorta. The entire
patient’s aorta was imaged before an d after the TEVAR with computed tomography angiography
(CTA) with 1mm slices, and illustrative images in the axial and sagittal planes are shown in Fig.
6.1(a) and 6.1(b). The ComboMap system with a ComboWire guide wire (Philips Volcano
Corporation) was used to acquire pressure and flow data inside the TL and FL. The guide wire
(a) (b) (c)
128

was 0.36 mm in diameter and 185 cm in length. The sensor contained a pressure transducer and
an ultrasound transducer, both mounted in a single housing at the tip of the guide wire. Data
collected during invasive assessment were extracted directly from the ComboMap system at
200Hz sampling rate. The measurements were done at all different aortic zones as demonstrated
in Fig.s 6.5(a) and 6.5(b). Samples of the invasive measured pressure waveforms are shown in
Fig. 6.5(c) at different zones inside the FL post-TEVAR.
6.4 RESULTS
6.4.1 Physiological accuracy of the model
A sample pressure inside the TL and FL at zone 4 is shown in Fig. 6.6(a). The expected
fiducial features of the pressure wave inside the TL including the pressure dicrotic notch can be
seen in this figure. The shape of the FL pressure waveform match well with the measured data
shown in Fig. 6.3(c). Fig. 6.6(b) demonstrate the computed flow waveform inside the FL at the
place where the endograft is implanted. The flow pattern consists of systolic biphasic flow which
is similar to the findings of Rudenik et al. [145] who reported the phase-contrast magnetic
resonance imaging of 31 patients with AD.
129

Fig. 6.6 Sample pressure and flow data in the developed dissection model. The simulated (a) pressure inside the TL and FL at
zone 4 and (b) flow inside the FL at zone 4 for 𝝀 =0.66 and HR=60 bpm.
Table 6.2 presents the comparison between the results of our computational model with our
measured invasive clinical data. Note that the Womersley number and the endograft-septum
length ratio are matched in accordance with the clinical values based on patient’s characteristics
(Wo ≈ 11.2) and TEVAR procedure (λ ≈ 0.66). Relative pulse pressure (RPP) inside the FL is
used to compare the computational and clinical data, defined as RPP(i, j) =
pp
zone
i
−pp
zone
j
pp
zone
j
for
i=5,6,7 and j=4. This hemodynamic parameter is related to the overall fluid motion inside
the FL, and it is controlled more by the underlying physics rather than the patient-specific
geometry. Therefore, it is suitable to be utilized for the comparison in this study.
Table 6.2 Comparison between invasive clinical measurements and the results from our computational model.
Hemodynamic Variable RPP(4,5) RPP(4,6) RPP(4,7)
Measurement Type
Invasive Clinical Data 0.059

0.088 0.294
FSI Computational Model 0.044 0.073 0.327
* RPP(i, j) =
pp
zone
i
−pp
zone
j
pp
zone
j
are calculated for comparing the clinical and computational data. Zones’ classification is
illustrated in Fig. 6.5a.
(a) (b)
130

6.4.2 Effect of endograft length on left ventricular workload
Fig. 6.7(a) gives the left ventricular pulsatile power requirement P
̅
pulse
as a function of the
endograft-septum length ratio (λ) for different heart rates (HRs). In these cases, the CO of the LV
is kept constant at the value of 5.7 L/min . The calculated pulsatile power is based on the
pressure and flow data at Zone 1 in the TL. As expected, LV pulsatile power increases at all HRs
when the endograft length increases. Fig. 6.7(b) demonstrate the left ventricular pulsatile power
requirement as a function of HRs for different endograft-septum length ratios.

Fig. 6.7 Impact of endograft length and heart rate on LV workload. Average LV pulsatile power requirement per cardiac
cycle versus (a) the 𝝀 (endograft-septum length ratio) at different HRs and versus (b) the HR at different 𝝀 .
6.4.3 Effect of endograft length on FL flow reversal
Fig. 6.8 presents the fluid velocity amplitudes in the fluid domain as well as the septum and
aortic wall displacements at various snapshots in time during a cardiac cycle of length T for short
and long grafts. The displacement waveform of the intimal septum 5cm proximal to the distal
tear in the presence of short, medium and long endografts is shown in Fig. 6.9.
(a) (b)
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Fig. 6.8 Spatial distribution of fluid and solid behavior in the FSI type-B dissection model at various times during the
cardiac cycle. The zig-zag boundary shows the graft (rigid) schematically and the dashed wall represent the axis of the
symmetry. The flow direction is from left to the right.

Fig. 6.9 Simulated septum wall displacement waveform for different graft lengths during one cardiac cycle. The data is
collected 5cm proximal to the distal tear.

132

Fig. 6.10 presents the sample of velocity profile inside the FL for short, medium and long
endografts. The velocity is computed at the center of the of the FL 5 cm proximal to the distal
tear. Fig. 6.11(a) demonstrates RFI (to quantify FL flow reversal) as a function of λ for different
HRs. RFI is reported based on the average of the values computed at Zones 4, 5 and 6 inside the
FL (Fig. 6.3(b)). Similar to the previous section, the CO of the LV is kept constant at the value of
5.7 L/min . Fig. 6.11(b) shows RFI as a function of HRs for different endograft-septum length
ratios.

Fig. 6.10 Simulated flow velocity waveform inside the false lumen for different graft lengths during one cardiac cycle. The
data is collected 5cm proximal to the distal tear.

133

Fig. 6.11 Impact of endograft length and heart rate on thrombose formation in FL. Average Reverse Flow Index inside the
FL per cardiac cycle versus (a) the 𝝀 (endograft-septum length ratio) at different HRs and versus (b) the HR at different 𝝀 .
6.4.4 Effect of LV contractile state on FL flow reversal
Fig. 6.12(a) demonstrates the pressure inside the TL at zone 4 for three different LV
contractility demonstrated by E
es
. Fig. 6.12(b) presents RFI as a function of endograft-septum
length ratio (λ) for these three different contractile states of the left ventricle (E
es
=
2.05mmHg /ml corresponds to CO=5.7L/min ). These simulations run at fixed HR of 60 bpm .

Fig. 6.12 Impact of LV contractile state on FL thrombosis. (a) Simulated pressure inside the TL at zone 4 for different levels
of LV contractility, and (b) Average Reverse Flow Index inside the FL per cardiac cycle versus the 𝝀 (endograft-septum length
ratio) at different levels of LV contractility. 𝑬 𝒆𝒔
=𝟐 .𝟎𝟓 𝒎𝒎𝑯𝒈 /𝒎𝒍 corresponds to 𝑪𝑶 =𝟓 .𝟕 𝒍 /𝒎𝒊𝒏 , 𝑬 𝒆𝒔
=𝟐 .𝟎𝟎 𝒎𝒎𝑯𝒈 /
𝒎𝒍 corresponds to 𝑪𝑶 =𝟓 .𝟐 𝒍 /𝒎𝒊𝒏 , and 𝑬 𝒆𝒔
=𝟏 .𝟗𝟓 𝒎𝒎𝑯𝒈 /𝒎𝒍 corresponds to 𝑪𝑶 =𝟒 .𝟕 𝒍 /𝒎𝒊𝒏 .

(a) (b)
(a) (b)
134

6.5 Discussion
In this chapter, we investigated clinically relevant hemodynamic patterns inside the TL and
FL after endovascular repair using a physiologically accurate idealized model of TBAD. Our
results suggest that: (1) There is a non-linear trend towards increased FL flow reversal as the
endograft length increases but with an increased LV workload, (2) at a given heart cardiac
output, lower HR enhances FL flow reversal and recirculation independent of the endograft
length, and (3) at a given HR, a reduced LV contractility enhances FL flow reversal and reduces
the systolic blood pressure.
6.5.1 Model validation against invasive clinical measurements
We utilized FSI computational model of the coupled LV-aorta system to gain insight on the
biomechanical behavior of blood flow in type B dissection following TEVAR. Numerous
computational models, both patient-specific and lumped parameter [51, 161, 162] are available
in the literature and provide additional information on flow patterns in aortic dissection which
are not possible by imaging alone [163]. While there are significant data supporting the impact of
intimal septal motion on disease progression [142, 161], past studies on dissection modeling
assumed rigid vessel wall. This assumption leads to neglecting the septum dynamics and wall
compliance which has been shown to play critical role in understanding hemodynamics [142,
161]. In addition, due to inability of such models in capturing wave dynamics, they are unable to
describe detailed pulsatile flow and wave reflection [144, 161]. Our model is among the first
which is able to capture the septal motion in TBAD. Results from simulations have been
compared to invasively measured clinical data acquired during a TEVAR operation (Fig. 6.5(c))
to verify the clinical relevancy of the computational model (Fig. 6.6 and Table 6.2). The
dimensionless pressure index inside the FL (RPP) was utilized to compare the in-vivo results
135

with our simulation. Table 6.2 shows that the calculated RPP from simulation data is within the
range of clinical data and follow a similar trend. The computed flow waveform inside the FL
(Fig. 6.6(b)) shows the similar characteristics with the reported clinical MRI data in the literature
[145]. These confirm the physiological accuracy of our study for the purposes of investigating
hemodynamics of TBAD.
6.5.2 Impact of endograft length on LV workload
The first novel finding in this chapter is related to examining the impact of endograft-aortic
compliance mismatch on LV power requirement which is a global hemodynamic metric of
cardiovascular system. The replacement of highly elastic native aorta with non-compliant
endograft reduces compliance and alters the aortic wave dynamics. This alteration has been
shown to translate into additional workload on the LV, eventually inducing adaptive hypertrophy
[164]. However, to the best of our knowledge, the effect of this compliance mismatch between
the aorta and the endograft on hemodynamic variables has not been quantitatively studied. In this
study, we investigated the effect of this compliance mismatch via changing the endograft length
on LV pulsatile load. LV pulsatile load is the result of complex wave dynamics and LV-aorta
coupling and has been used as a global hemodynamic index to monitor different wave conditions
in the vasculature [141]. Indeed, previous clinical studies suggested that reducing LV pulsatile
load is an important therapeutic target in HF [50]. Our results suggest a trend towards increased
LV workload as endograft length increases at different heart rates (Fig. 6.7). This finding is in
line with previous observations in terms of increase in pulsatile load due to the overall decrease
in aortic compliance [141]. While longer endografts have the advantage of covering more tears
in AD, this undesirable effect can be a limiting factor for clinicians when choosing endograft
length.
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6.5.3 Impact of Endograft Length on FL Thrombosis
Current understanding indicates that increased FL flow reversal enhances thrombosis and
patients exhibiting reversed flows within the FL may be more likely to develop complete FL
thrombosis [142, 165]; this is considered as a positive prognostic indicator [133, 134]. Fig. 6.8
presents the spatial distributions of the flow velocity and wall displacement in the presence of
short and long endografts. As expected, there is significant difference in the septum wall
displacement during the cardiac cycle between these two models; the presence of a longer
endograft leads to the decrease in the overall compliance of the system and smaller radial
displacement of the intimal septum which is quantified in Fig. 6.9. The velocity profile for short,
medium and long endografts is presented in Fig. 6.10. Regarding the overall dynamics of the
septum and the flow, lower compliance of the repaired aorta with longer endografts leads to the
earlier development of the antegrade flow inside the false lumen. To be mentioned that RFI
which is the measure for thrombose prediction is the ratio of the retrograde flow over the total
flow. Therefore, although the amplitude of both the antegrade and retrograde component of the
flow data is smaller inside in the model with longer endografts (Fig. 6.10), the averaged RFI of
different sites in these models is higher (Fig. 6.11). In other words, our results suggest that
increase in endograft length enhances FL flow reversal. This may be attributed to a reduction in
the overall compliance of the septal wall as the native aorta is replaced with a rigid endograft,
leading to less unidirectional flow into the FL and an increase in FL flow reversal. However,
while the FL flow reversal enhances significantly as the endograft length increases from the
short-size to medium-size (e.g., at HR=60 bpm , 65% increase in RFI from λ=0.13 to λ=
0.40), there is a minor enhancement in RFI as the endograft length increases beyond λ=0.40
(e.g., at HR=60 bpm , a 12% increase in RFI from λ=0.40 to λ=0.66). This finding
137

suggests that medium-size endograft replacement (λ=0.40) may achieve high FL flow reversal
(predictor of FL thrombosis) with minimal extra pulsatile load on LV.
6.5.4 Effect of medical therapy on FL thrombosis
Although many TBAD patients undergo surgical aortic repair, medical therapy remains an
essential part of their treatment. The primary objective of this pharmacological therapy is the
reduction of the rate of rise of systolic aortic pressure [127, 166, 167]. Beta-blocking agents are
the mainstay of pharmacologic therapy for TBAD as they reduce the HR and decrease the
intrinsic contractile state of the heart. This chapter evaluated the effect of both these parameters
(HR and LV contractility) on FL flow reversal. The results demonstrated that decreasing HR at a
fixed CO enhances FL flow reversal (Fig. 6.12(b)). Furthermore, lower HR led to increased flow
reversal index (Fig. 6.11(b)) after endograft deployment. This implies lower HRs have favorable
outcomes in terms of FL thrombus formation. To investigate the impact of different contractile
states of LV on FL thrombosis, end systolic elastance was decreased in our LV model to
simulate the physiological response to beta blockers (reduced contractility). The results indicated
that reduced contractility (at a fixed HR) enhances FL flow reversal. Ultimately, results suggest
that medical therapy in TBAD patients not only achieves the therapeutic goal of reducing the
systolic blood pressure (Fig. 6.12(a)), but also contributes favorably to FL flow reversal (Fig.
6.12(b)).
6.5.5 Study limitations
This study has certain limitations that should be considered. The dissection model used in
this study is constructed based on average physiological values in TBAD patients and is a based
on a simplified (idealized) model of TBAD. This model is limited by the number of tears
considered in the septum model as well as the exclusion of aortic branches and the aortic arch.
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While the geometry of TBAD can be very complex due to tortuosity, irregularities of luminal
diameter along the dissection, multiple fenestrations in the septum wall and partial FL
thrombosis, our model is intended to contribute to the understanding of the hemodynamics in
TBAD independent of each individual. This generic model is ideal to provide insights on the
impact of one parameter at a time (e.g., endograft length) while controlling all other parameters.
We also utilized Newtonian flow assumption for the fluid in this study. This assumption is still
conventionally used in both experimental and CFD studies in large arteries [168]. However,
future studies are needed to investigate the significance of non-Newtonian flow behavior in
TBAD modeling in terms of FL flow reversal after TEVAR. Another major assumption in this
study is to model the endograft as a rigid material. While current commercially available
endografts are not fully rigid, previous studies reported the measured elasticity of endografts are
up to sixteen times larger than that of the aorta [169]. For this reason, the assumption of rigid
endograft in this study is reasonable.
6.6 Conclusion
The present chapter provides a comprehensive analysis of the role of endograft length on
both global and local hemodynamic variables in TBAD anatomy. The computational model used
here illustrates the amplitude and the form of the septum displacement in TBAD (Fig.s 6.8 and
6.9). The significance of the septum displacement necessitates the FSI modeling for capturing
the wave dynamics in this disease. Trends towards increased FL flow reversal (Fig. 6.11) and
increased pulsatile workload with increasing the endograft lengths were observed (Fig. 6.7). This
trade-off between desirable impact on FL flow reversal via longer endografts and their
undesirable impact on LV workload suggest that there may exist an optimal endograft length that
can lead to improved long-term clinical outcomes. Based on the non-linear increase in FL flow
139

reversal with increased endograft length (Fig. 6.11), our results suggest medium-length
endografts can lead to relatively high FL flow reversal (and consequent FL thrombosis) with
minimal extra load on the LV. Another major finding of this study is related to the role of
medical therapy on the hemodynamic state in TBAD. Our results indicate that medical therapy
can achieve the therapeutic goal of reducing the systolic blood pressure and contribute favorably
to FL flow reversal and FL thrombosis. Further clinical studies are needed to assess the role of
endograft length on hemodynamic variables following TEVAR. Further patient-specific
modeling can also be conducted utilizing the FSI approach to provide additional information on
flow patterns and the comparison among different TBAD patients in the presence of the patient-
specific septum dynamics. Employing such an approach is also helpful in identifying the possible
factors involved in the formation of distal aneurysm and distal re-entry [144, 170].
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CHAPTER 7 : Framework development for patient-specific compliant aortic
disease phantom model fabrication: magnetic resonance imaging
validation and deep-learning segmentation
7.1 Chapter abstract
Due to the complexity of patient-specific characteristics, only limited information on flow
patterns in dissected aortas has been reported in the literature. Leveraging the medical imaging
data for patient-specific in-vitro modeling can complement the hemodynamic understanding of
aortic dissections. In this chapter, we propose a new approach toward fully automated patient-
specific type B aortic dissection model fabrication. Our framework uses a novel deep-learning-
based segmentation for negative mold manufacturing. Deep-learning architectures were trained
on a dataset of 15 unique computed tomography scans of dissection subjects and were blind-
tested on 4 sets of scans, which were targeted for fabrication. Following segmentation, the 3D
models were created and printed using polyvinyl alcohol. These models were then coated with
latex to create compliant patient-specific phantom models. The Magnetic resonance imaging
(MRI) structural images demonstrate the ability of the introduced manufacturing technique for
creating intimal septum wall and tears based on patient-specific anatomy. The in-vitro
experiments show the fabricated phantoms generate physiologically-accurate pressure results.
The deep-learning models also show high similarity metrics between manual segmentation and
auto-segmentation where Dice metric is as high as 0.86. The proposed deep-learning-based
negative mold manufacturing method facilitates an inexpensive, reproducible, and
physiologically-accurate patient-specific phantom model fabrication suitable for aortic dissection
flow modeling.
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7.2 Introduction
Type B aortic dissection (TBAD) is a catastrophic cardiovascular event that is associated
with considerable mortality and morbidity [125, 126, 167]. Current standard of care treatment for
TBAD is medical anti-impulse therapy with thoracic endovascular aortic repair (TEVAR) in
select patients to cover the proximal entry tear [127, 129, 134]. When properly performed,
TEVAR maintains blood flow in the true lumen (TL) and obstructs antegrade flow into the
proximal aspect of the false lumen (FL) [130]. Although TEVAR has shown promising results in
the treatment of TBAD, the long-term prognosis for patients with TBAD remains poor with a
near 50% mortality at 10 years [53, 171]. In addition, only limited information on the flow
patterns in dissected aortas has been reported in the literature due to the complexity of patient-
specific characteristics.
In-vitro hemodynamic modeling can supplement the current fluid mechanic and wave
dynamic understanding of aortic dissections. These experiments may provide data that could
reduce the high expense and risks associated with clinical trials [51, 172]. Recent advances in
additive manufacturing and the development of robust flow measurement techniques make in-
vitro models a practical tool to test vascular implants and improve therapeutic approaches [173].
Previous studies have introduced various in-vitro hemodynamic simulators and have
experimentally investigated the hemodynamics of aortic dissections [174-177]. However, most
of the fabricated phantoms had rigid walls that cannot capture wave dynamics of the
cardiovascular system [173, 178]. In addition, previous studies have used simplified and generic
geometry that cannot mimic patient-specific characteristics [174-177].
In this chapter, we propose a systematic approach toward fabricating a physiologically-
accurate, compliant, patient-specific phantom model for TBAD. The proposed phantom
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manufacturing pipeline starts with auto-segmentation of the aorta followed by mold generation
and coating. Instead of conventional manual delineation for segmentation that requires
significant time and effort in the model generation process, we used an automated deep-learning
model-based approach. This minimizes operator variability and its related error as well as the
cost, by streamlining the segmentation process, and improves the reproducibility of the phantom
fabrication. The inner geometry and structure of the fabricated patient-specific phantoms are
validated using Magnetic Resonance Imaging (MRI). Lastly, a hydraulic model of the human
systemic circulation (aortic simulator [141, 176]) is used to collect hemodynamic measurements
from a patient-specific phantom model and these measurements are compared with the invasive
clinical data collected during the in vivo operation on that patient.
7.3 Materials and methods
7.3.1 Patients description
Data from 19 TBAD patients were utilized to develop our deep-learning-based negative
mold manufacturing framework. The baseline characteristics of these patients are presented in
Table 7.1. This study was approved by the University of Southern California Institutional
Review Board. Each patient’s entire aorta was imaged with computed tomography (CT)
angiography with 1 mm slices. Only pre-TEVAR images were used to train the deep-learning
model and fabricate phantom models.

143

Table 7.1 Baseline Characteristics of Patient Data (N = 19).
Variable Value
Clinical measures
Age, y 55 ± 12
Women, n (%) 7 (37)
Hypertension, n (%) 12 (63)
Height, cm 170 ± 15
Weight, kg 79 ± 22
Body Mass Index, kg/𝑚 2
27 ± 5
* All values are (mean ± SD) except as noted.

7.3.2 Prototyping the artificial patient-specific dissection models
Four patients were chosen for prototyping based on anatomical variations such as helical
angle, FL length, and renal artery connection to the false lumen. Table 7.2 demonstrates the
anatomical characteristics of these selected patients. These parameters include the number of
segments of complete FL thrombosis (no flow inside the FL) and the overall geometry of the
dissected aorta. The helical angle, the average helical twist, and how the renal arteries originate
from the dissected aorta are parameters that help to characterize the overall geometry of the
dissected aorta [179]. The length of the false lumen segmentation was also reported in Table 7.2.
This measurement was done on sagittal planes and have errors of ±1 mm based on resolution.

144

Table 7.2 Anatomical and Geometrical Parameters of the Type B Dissection Patients Utilized for Phantom Fabrication.
Phantom 1 Phantom 2 Phantom 3 Phantom 4
FL Length (mm) 230 293 233 345
Major Thrombosis Site in FL present absent absent present
Helical Angle (Deg) -105 -157 103 -167
Average Helical Twist (Deg/cm) 1.60 0.58 0.70 1.00
Renal Arteries Connecting to FL None One Both One
* The method for computing the helical angle and average helical twist is adopted from [179].
The 3D geometry of the TBAD was acquired by an open-source software package used for
medical imaging research (3D Slicer) [180]. It was then exported as two STL (Standard Triangle
Language) files. One STL file contains the combined segmentations of patent false and true
lumens, and the other file for the segments of the aorta that were thrombosed. The detail for the
segmentation is described in the next section. Renal and carotid arteries were manually added to
the STL files using SolidWorks (SolidWorks Corp.), in accordance with the CT images. The
aortic root and femoral artery diameters were modified to install the fabricated phantoms in a
flow simulator [67, 141]. After the modifications, the patient-specific geometry is hollowed to
accelerate the 3D printing process and separated into pieces to satisfy the maximum print size of
our 3D printer.
A dual injection 3D printer (S5, Ultimaker B.V.) was used to print these mold pieces of
artificial phantom models. The main printing material was chosen as water-soluble polyvinyl
alcohol (PVA), except for the complete thrombosed section. The supports for these pieces were
printed with non-soluble polylactic acid (PLA). The completely thrombosed regions were printed
using PLA as the main material and PVA as the support. Use of different main and support
145

materials was preferred to easily separate the printed pieces from their supports and to have a
smooth mold surface. After the printing process, these pieces were brought together with a
water-soluble glue to assemble a complete patient-specific mold. The molds were then dipped
into natural latex (Chemionics Corp.) and subsequently left to dry for two hours at standard room
temperature (25 ̊ C). The dipping process was repeated for fifteen times for every mold. The
number of times dipped can control the thickness (and wave speed) for mimicking the age-
related changes in the aortic wall. The used natural latex conforms with the properties of the
human aorta to fabricate physiologically-accurate vascular models [67, 141, 176]. After the
dipping process, the prepared aortic models were removed from their PVA molds. The removal
process starts by injecting water in between the coated latex and the mold. The aortic models
were then immersed to a water bath for 72 hours at 25 ̊ C to ensure the inner mold was dissolved.
Lastly, to enhance surface quality, the latex aortas are submerged in regular bleach (Clorox
Company) for 12 hours, followed by another 12 hours of submerging in water for the final curing
process. The overall manufacturing process is shown in Fig. 7.1.
146

Fig. 7.1 Fabrication overview for patient-specific 3D printed aortic dissection phantom.

7.3.3 Segmentation and deep learning
To capture patient-specific geometries, DICOM (Digital Imaging and Communications in
Medicine) images obtained by CT scans were used. Ground truth masks for the deep learning
algorithm were created by manual segmentation using 3D Slicer. The true and false lumens were
identified in the axial, sagittal, and coronal views. After the identification, the false and true
lumens were segmented using every other slice in the axial view. Contouring (to define mask)
was performed to create an outline based on the pixel intensity. For the thrombosed regions, the
Draw module of the 3D Slicer was used to outline the false lumen. The intimal flap as well as the
aortic wall were not considered during segmentation, resulting in space between the false lumen
and true lumen segmentations.
In order to overcome the shortcomings of manual segmentation, a deep-learning-based
automatic segmentation framework is proposed and represented in Fig. 7.2. To demonstrate the
generalizability of our approach, three deep-learning architectures were independently used for
linical Data Acquisition eometry econstruction Phantom Design P A old Preparation
old Post processing ate oating Dissolving the old esting the Phantom

147

segmentation: U-Net [181], DeepLab [182], and mask region-based convolutional neural
network (Mask-RCNN) [183]. The details of each architecture can be found elsewhere [181-
185]. This framework can be extended to any deep neural network architecture that is suitable
for image segmentation. Our primary interest in this study was the binary segmentation mask to
create a model for mold fabrication. All the tested deep-learning networks were trained on the
CT images of 15 TBAD patients. Each model was then examined on 4 unique patients isolated
from the training dataset for a blind-test. A high-pass threshold of 500 pixels was applied to the
raw input patient image (256x256 pixel) in the data preparation stage. 4405 training image pairs
(the raw image and mask) were selected from the 15 patients. To mitigate the need for large
datasets of CT segmentation data, a synthetically augmented dataset was generated from the
training data. The synthetic augmentation was performed by shifting, rotating, and zooming the
initial images. Finally, a total of 26,430 images from 16 patients and their corresponding masks
were generated for our training database.
The training of our deep learning models was supervised by the manual ground truth
segmentation (see Fig. 7.2). Cross-entropy loss function (𝐿 ) was used to compute the
summarized pixel-wise probability loss between the predicted probabilistic output 𝑃 for image
𝑖 and the corresponding target ground truth segmentation mask 𝐺 given by,
𝐿 (𝑃 𝑖 ,𝐺 𝑖 )=−
1
𝑛 ∑ 𝐺 𝑖 log(𝑃 𝑖 )
𝑛 𝑖 =1
(7.1)
where 𝑛 is the number of data samples at each iteration [186]. The segmentation predictions
made by deep-learning networks were evaluated using the Dice similarity index between the
ground truth (manual segmentation) and auto-segmentation predictions [184].
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Fig. 7.2 The deep learning pipeline for dissection auto-segmentation. Data preparation consists of slicing and augmentation.
Three architectures are independently used for training the deep learning models.
7.3.4 Magnetic resonance imaging of the artificial phantoms
o evaluate the phantom models’ structure, scans were performed using a 0.55 Tesla
scanner (prototype MAGNETOM Aera, Siemens Healthineers). The scanner is equipped with
high-performance gradients (45 m m amplitude, 00 m s slew rate) [187]. The fabricated
patient-specific phantom models were filled with water and placed between a 6-channel body
coil (anterior) and the integrated spine arrays (posterior). These phantoms were imaged using a
T1 weighted gradient recalled echo volume interpolated breath-hold examination sequence with
the following parameters: repetition time, 8.86 ms; echo-time, 1.75 ms; field of view 350 mm x
205 mm; voxel size, 1.1 mm x 1.1 mm x 3.0 mm; and the number of slices, 88. The subsequent
DICOM images were analyzed and converted to portable network format using MicroDicom
DICOM Viewer (MicroDicom Ltd).

Data Preparation raining
UN
Deep ab
NN
oss
round ruth
abel
Predicted
abel
Update Parameters
lind est Data
Similarity nde
rained odel
Slicing and Augmentation
onvergence
149

7.3.5 In-vitro hemodynamic measurements
An in-vitro systemic circulation system, called aortic simulator (similar to the one employed
by Pahlevan and Gharib [141, 176]) was used for the experiments to obtain pressure
measurements in the fabricated patient-specific phantom models (Fig. 7.3). Briefly, the aortic
simulator is a hydraulic model that has physical and dynamical properties similar to the human
circulatory system [141, 176]. This hydraulic model consists of a piston-in-cylinder pump
(ViVitro Labs Inc.) that generates a physiologically accurate pulsatile flow (using a
programmable waveform generator ViVigen) and a pump head, including a silicone ventricle
membrane, mitral valve, and aortic valve. The patient-specific phantom models were installed to
this setup, and the outlets (arteries) were connected to the end organ units that include a half-
filled air syringe (to simulate compliance) and the resistance clamp. The aortic simulator also
includes two compliance chambers with a hydraulic resistance between them. These compliance
chambers are placed at the end of the aortic circuit. The last component of the aortic simulator is
a reservoir tank that is located between the second compliance chamber and the inlet of the
pump. The further details of this experimental setup (the aortic simulator) have been explained
in detail in previous publications [141, 176]. The pressure data are collected by high-fidelity
pressure sensors (Millar MIKRO-TIP® Catheter Transducer). These sensors are connected to a
data acquisition system (PowerLab 4/35, ADInstruments) and their respective data are collected
with the LabChart v7 software (ADInstruments). The utilized in-vitro hydraulic circuit and its
corresponding components are presented in Fig. 7.3.
150

Fig. 7.3 Full hydraulic circuit for in-vitro flow modeling of the patient-specific phantoms with the corresponding physical
components.
7.3.6 Invasive clinical data collection
The ComboMap system (Philips Volcano Corporation) was used to acquire pressure data at
the aortic root, in the TL and the FL, prior to TEVAR endograft implantation in the patient. The
measurements were performed using a ComboWire guide wire (Volcano Corp.). The guide wire
was 0.36 mm in diameter and 185 cm in length. The sensor contained a pressure transducer and
an ultrasound transducer, both mounted in a single housing at the tip of the guide wire. Data
collected during invasive assessment were extracted directly from the ComboMap system at 200
Hz sampling rate. Invasive data collected from one of the TBAD patients undergoing TEVAR

151

were utilized to examine the hemodynamic accuracy of the fabricated patient-specific phantom
model. This patient’s images were used for the fabrication of Phantom 4 in Table 7.2.
7.4 Results
7.4.1 Accuracy of the deep-learning for dissection segmentation
Fig. 7.4 demonstrates the batch of manually segmented dissection masks (the blue mask)
compared with a corresponding sample of automatically segmented masks from the deep-
learning (the red mask). The sample auto-segmented batch in this figure is generated by the
Mask-RCNN architecture for Phantom 2.
152

Fig. 7.4 Sample segmented for the aortic dissection. The blue segmentation is the manual ground truth and the red
segmentation is the deep learning auto-segmentation based on the Mask-RCNN for Phantom 2.

153

The Dice metrics of the testing data (fabricated phantoms) for three deep-learning
architectures are summarized in Table 7.3. This metric is a quantitative measure of similarity
between the automatically and manually segmented images (Eq. 7.1).
Table 7.3 Dice similarity index between the predicted and the ground truth.
Phantom 1 Phantom 2 Phantom 3 Phantom 4 Average
U-Net Architecture 0.88 0.84 0.89 0.84 0.86
DeepLab Architecture 0.75 0.71 0.84 0.71 0.75
Mask-RCNN Architecture 0.87 0.85 0.88 0.82 0.86
* Mask-RCNN stands for mask region-based convolutional neural network.
7.4.2 Accuracy of the phantom structure via MRI
Fig. 7.5 presents the structural images obtained using MRI for Phantoms 1 and 2 in the axial
slices. Four slices (annotated on Fig. 7.5) are chosen for each phantom to show the intimal
septum layer and the tears on it. Slice 1 is located near the root, slice 2 is at the top of the
ascending aorta, slice 3 is at the descending aorta, and slice 4 is located at the abdominal aorta.
The red zones demonstrate the areas of thrombosis.
154

Fig. 7.5 Schematic and axial MR images captured at different section along phantoms 1 and 2. Red zones represent
thrombosis.
Fig. 7.6 demonstrates the obtained MRI structural images for Phantoms 3 and 4 in the
coronal and axial slices. Note that the axial MRI images were acquired for all four phantoms and
then two of the phantoms (one with and one without thrombosis) were chosen to acquire coronal
images. The first three slice locations are chosen similar to Phantoms 1 and 2. The last slice for
phantom 3 is chosen to show the renal artery connection to the false and true lumens. For the
phantom 4, this slice shows the simulated area of thrombosis (shown with red on the schematic
and blue arrow on the MRI image).
Phantom 1 Phantom Phantom

Phantom
155

Fig. 7.6 Schematic, coronal and axial MR images captured at different section along phantoms 3 and 4. Red zones
represent thrombosis. The dashed red line demarcates the root of the fabricated phantom model.

llustration A ial oronal

Phantom Phantom 4

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7.4.3 Accuracy of the In-vitro pressure measurements inside the phantom model
Fig. 7.7 demonstrates the in-vitro pressure waveforms measured inside the fabricated
patient-specific phantom model (bottom row) and the invasive data collected during the live
operation of that patient (top row). True and false lumen pressures are obtained at the abdominal
aorta. Note that the simulated heart rate in in-vitro experiments is matched with the clinical one
(55 beats per minute). The expected fiducial features of the pressure waveforms at the aortic root
and in the true lumen, including the dicrotic notch, can be seen in Fig. 7.7. The FL pressure
collected during the in-vitro experiments is slightly higher (≈4 mmHg ) than the TL, which
shows good agreement with the clinical measurements.

Fig. 7.7 Comparison between measured clinical and in-vitro data. Top panel demonstrated the invasive pressure data
measured during the live operation of the dissection patient. Bottom panel indicate the in-vitro pressure measurement on the
patient-specific phantom fabricated based on the images of that individual.

ime (s)

Pressure (mmHg)

ime (s)

Pressure (mmHg)

ime (s)

Pressure (mmHg)

ime (s)

Pressure (mmHg)

ime (s)

Pressure (mmHg)

ime (s)

Pressure (mmHg)
linical nvasive easurement of a Patient
n vitro easurement from Patient specific Phantom odel
Aortic oot
rue umen False umen
Aortic oot
rue umen False umen
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7.5 Discussion
This chapter presents a novel fabrication method for compliant patient-specific aortic
dissection phantoms. We propose a deep-learning based model for segmenting the patient’s CT
images and a negative mold manufacturing with 3D printing. Results demonstrate that the deep-
learning model can accurately capture the dissection geometry from patients’ medical images
(Table 7.3 and Fig. 7.4). The MRI images of the fabricated phantoms (Fig.s 7.5 and 7.6)
demonstrate the intimal septal wall and tears in agreement with the patients’ anatomy. In
addition, the hemodynamic measurements reveal that the expected fiducial features from the live
invasive measurement can be accurately captured using the fabricated phantoms in the in-vitro
flow simulator (Fig. 7.7).
Our findings showed that our proposed approach has the ability to capture wave dynamics
due to the physiologically accurate compliance of the fabricated phantom models [69, 73, 74].
Results from Fig. 7.7 demonstrate the ability of the fabricated phantom to mimic the main
physiological features of the pressure waveforms measured invasively during a TEVAR
operation. The MRI images of these fabricated phantoms (Fig.s 7.5 and 7.6) also validate the
presence of the septal wall and expected tears connecting the true and false lumens.
Our proposed approach in this study consists of the deep-learning segmentation aiming to
fabricate aortic dissection phantom models accurately and reproducibly. To test the
generalizability of our approach, three popular architectures are selected including U-Net,
DeepLab, and Mask-RCNN. Our results demonstrated that U-Net and Mask-RCNN yield great
similarity (Dice of 0.86) and DeepLab yields good similarity (Dice of 0.75) between the auto-
segmented images and ground truth (manual segmentation). This observation is in-line with
previous work comparing these algorithms [188, 189]. Although both U-Net and Mask-RCNN
158

performed well on auto-segmentation, UNET has much lower computational cost (U-Net used
2,140,065 trainable parameters while Mask-RCNN used 44,401,393).
The time required to fabricate these patient-specific phantoms consists mostly of the
segmentation, printing, and coating time. The manual segmentation time of a patient’s medical
images can take several hours of human effort. Our approach reduces the segmentation time to
less than a minute by taking advantage of deep-learning-based auto-segmentation. The printing
time of a complete dissection mold is around 24 hours. The coating process takes approximately
120 hours including dipping, drying, and dissolving the mold. Therefore, the complete
fabrication process with our method is around 144 hours (~ 6 days). The average mass of the
four fabricated phantoms is 199 grams. The overall cost for complete fabrication is due to the
3D-printed PVA mold (~ $133/kg) and the amount of coated latex (~ $32/kg). Hence, the total
material cost for a patient-specific phantom fabricated with our approach is in the range of $37 to
$46.
7.6 Conclusion
Our proposed deep-learning-based negative mold manufacturing method can create
physiologically accurate aortic dissection phantom models. This method is fast, affordable, and
reproducible. In addition, the fabricated phantoms are compliant which makes them suitable for
capturing aortic wave dynamic features and conducting fluid-structure interaction during in-vitro
studies. Our method can greatly facilitate aortic dissection experimental flow studies and has the
potential to ultimately improve treatment of TBAD and pre-clinical testing of vascular
endografts.

159

CHAPTER 8 : Longitudinal stretching-based wave pumping in compliant
tubes: a bio-inspired approach

8.1 Chapter abstract
This chapter investigates the physics of a longitudinal stretching-based impedance pump, a
novel pumping mechanism inspired by the human aorta. An impedance pump is a valveless
pump that operates based on the principles of wave propagations and reflections. In its simplest
form, an impedance pump consists of a fluid-filled elastic tube connected to rigid tubes at its
ends with a wave generator (e.g., pincher) located off-center relative to the two ends. Previous
studies have shown that the aorta acts as an impedance pump where the left ventricle pulsatile
dynamic creates waves. Another dynamic mode in human cardiovascular system is due to the
substantial systolic displacement of the aorta. Inspired by this aortic stretching mechanism, we
conducted a comprehensive analysis of a longitudinal impedance pump where waves are
generated by stretching of the elastic wall and its passive elastic recoil. We developed a
computational finite element model consisting of a fluid-filled elastic tube with fully coupled
fluid-structure interaction. he pump’s beh avior is quantified as a function of stretching and
material characteristics. Our results indicate that stretch-related wave propagation and reflection
can induce frequency-dependent pumping. We found a non-linear relation in the flow-frequency
pattern, and quarter wave theory is discussed in the context of impedance pump behavior to
explain the underlying physical mechanism of this pattern. It was shown that like other
impedance pumps, both the direction and magnitude of the net flow depend on wave
characteristics in compliant tubes.

160

8.2 Introduction
Valveless impedance-based pumping is a promising technique for producing net flow with
simple structure. In its conventional form, pumping is achieved via periodic radial compression
of an elastic tube at an asymmetric location from rigid ends [190]. The impedance mismatch,
which is due to the connection of tube from different compliances, create wave reflection sites in
this system [51, 191]. By compressing the elastic section periodically, traveling waves are
emitted from the compression that can reflect at the impedance mismatches, generating a net
flow [192]. While this pumping mechanism is first observed by Liebau in 1954 [193], the first
direct experimental evidence that this mechanism is involved in the nature is reported by
Forouhar et al.. Due to its complexity and dependence to many variables, it has been subjected to
numerous analytical, computational, and experimental studies that modeled possible underlying
mechanisms responsible for this phenomenon [194]. The conventional models mostly consist of
fluid-filled elastic tubes in open and closed loops excited by a periodic total or partial
compression of the tube [195]. Depending on wave parameters such as the frequency, an
impedance pump may assist circulation of a fluid in a compliant tubing system in either
direction, which means impeding flow in the opposite direction [48]. The first direct
experimental
Jung and Peskin [196] were among the first who reported a strong non-linear dependence of
the net flow on pinching frequency in the impedance pump. Later in a comprehensive
experimental study, Hickerson et al. [197] showed that the direction of flow is also dependent on
the frequency of compression. Their results also showed that the net pressure head and net flow
as a function of the compression frequency had distinct peaks at selected frequencies [197].
Based on these findings, a one-dimensional wave model was proposed to predict the
161

characteristics exhibited by the experiments [198]. Using this model, they showed the major role
of wave reflection and interaction in the behavior of the impedance pump. More recently,
Avrahami and Gharib [199] investigated the interplay of pressure, flow and elasticity in the
impedance pump using a computational approach. They showed that pumping is the result of
constructive wave interaction located at the extremity of the elastic tube distant to the pincher
[199]. They also showed that the interaction location is very sensitive to the timing, and therefore
to the frequency of excitation [199]. In another work, Loumes et al. [200] showed that at
resonance frequency, maximum energy transmission between the elastic tube and the fluid
occurs and therefore, the elastic tube itself works as a pump which generates frequency
dependent flow.
Besides the abovementioned studies, there has been efforts to extend the applicability of
impedance pumping and introduce non-conventional impedance pumps. Carmigniani et al. [201]
has recently developed a free-surface wave pumping mechanism where resonance pumping can
be used for hydraulic energy harvesting. Similar to the conventional impedance pump,
considerable bidirectional flow near certain frequencies was observed [201]. It was discussed
that this type of impedance pump has strong application for renewable energy extraction. In
2006, Forouhar et al. [202] provided the first experimental evidence that the impedance-based
pumping mechanism is involved in driving blood flow. They showed that in the embryonic heart
tube, the role of the pincher is played by a band of active contractile cells near the heart entrance
[202]. Several other studies have also shown the recurrence of the impedance pump mechanism
in biological systems. Pahlevan and Gharib [141] showed the similar pumping behavior to
impedance pump in the human aorta. The difference between an aortic pump and an impedance
pump is that there is no external pincher in the aortic wave pumping mechanism; rather, the flow
162

waves are generated by the heart and the aorta works as a fluid-filled elastic tube with multiple
reflection sites. This aortic wave pumping effect can generate a bidirectional flow depending on
the state of the wave dynamics [141]. While these impedance pumps can serve as an energy-
efficient and simple flow generators with interest in biomedical research, there are technical
challenges that limits the applicability of such devices [190, 195]. The low mean flow rates seen
in the previous studies are troublesome since they offer little use for any real flow improvement
in physiologically relevant settings [203]. In addition, there are disagreements on the optimal
performing condition of conventional impedance pumps due to the complexity in their
characteristic behavior [197, 204].
In this chapter, we present a novel extension of impedance wave pumping based on the
longitudinal stretching of a compliant tube. This wave pumping mechanism is inspired by a
pumping mechanism inside the cardiovascular system. In an optimal left ventricle-aorta
coupling, the left ventricular systolic contraction displaces the aortic annulus and produces a
considerable longitudinal stretch of the ascending aorta [205]. This longitudinal (axial)
displacement results in energy storage in the vessel’s spring-like elements that enhances early
diastolic left ventricle recoil and creates suction in the heart that facilitates diastolic filling [16,
206]. Longitudinal stretching of the ascending aorta during the contraction of the left ventricle
has been recognized by the medical community as an important determinant of aortic
biodynamics [16, 207]. The longitudinal wave pumping, based on the aortic characteristics, and
the conventional impedance pumping differ in their wave generating system, but both pumping
mechanisms rely on the principles of wave propagations and reflections.
Our central hypothesis is that stretch-related wave propagation and reflection in a fluid-filled
compliant tube, inspired by the aortic natural mechanism, can create a wave pumping
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mechanism. To understand the underlying mechanism in the longitudinal stretching-based
impedance pump, we developed a finite element model of a fluid-filled elastic tube with fluid-
structure interaction. he pump’s behavior was characterized as a function of various stretching
and compliant tube parameters. Employing principles of wave dynamics, the behavior of the net
generated flow at different frequencies was discussed. Lastly, a simplified theoretical model is
developed and utilize to investigate the relation between the parameters involved in the
longitudinal wave pumping.
8.3 Materials and methods
8.3.1 Physical problem
The physical problem of interest consisted of a straight cylindrical tube filled with water,
with one end fixed and the other stretched cyclically at frequency 𝑓 (Fig. 8.1). Each period of
the cyclic extension consisted of two phases: stretching and recoil. The elastic wall of the pump
is first stretched to a distance then released during the recoil phase. The recoil is passive,
meaning it is only due to the elasticity of the tube wall. The elastic recoil results in wave
propagation along the wall with initial amplitude 𝐷 𝑤 which is reflected upon reaching the fixed
outlet.

164

Fig. 8.1 Schematic representation of the longitudinal impedance pump. Different phases during one cycle are shown. An
illustrative sketch of the axisymmetric model and the computational boundaries are also shown.

Since our investigation is inspired by the human aorta, baseline parameters were selected
within the average physiological ranges [51, 74, 208]. The parameters used in this study are
summarized in Table 8.1.

w
Solid (
s
,
s
)
Fluid (
f
,
f
)
Fluid solid nterface

Stress free oundary
No slip oundary

f

s
A is of Symmetry
utlet
Fi ed nd

Stretch
Stretching
ecoiling
165

Table 8.1 Physical parameters used in the model.
Physical Parameter Symbol Value
Solid wall thickness (cm)
t
s

0.1
Solid wall density (g/cm
3
) 𝜌 s
1.05
Poisson ratio of the solid wall 𝜗 0.45
Fluid density (g/cm
3
) 𝜌 𝑓 1.05
Fluid viscosity (Pa. s) 𝜇 𝑓 0.005
Fluid domain radius (cm)
r
f

1.5
8.3.2 Governing equations
Fluid motion was calculated by using the continuity and conservation of momentum
equations of an incompressible fluid,
∇∙𝒗 =0, (8.1)
𝜌 𝑓 𝒗 ̇ +𝜌 𝑓 (𝒗 ∙∇)𝒗 −∇∙𝑻 𝐟 −𝒇 =0, (8.2)
𝑻 𝐟 =−𝑝 𝑰 +𝜇 (∇𝒗 +∇𝒗 T
) , (8.3)
where 𝒗 is fluid velocity, 𝜌 f
is fluid density, 𝑻 𝐟 is the fluid stress tensor (for the Newtonian
and incompressible flow), 𝒇 is body force, 𝑝 is pressure, 𝑰 is the identity tensor, and 𝜇 is dynamic
viscosity. Motion of the linearly elastic vessel wall was calculated using the Lagrangian form of
the momentum balance equations given by
𝜕 𝑻 𝐬 𝜕 𝑿 +𝑭 =𝜌 s
𝒖 ̈ 𝐬 , (8.4)
where 𝑿 is the position of a material point, 𝑻 𝒔 is the solid stress tensor, 𝑭 is the vector of
external force, 𝜌 𝑠 is vessel wall density, and 𝒖 ̈ 𝒔 is wall acceleration. At the fluid-structure
interface the fluid was fully coupled to the solid. The fundamental conditions applied to the
fluid-structure interface were displacement compatibility and traction equilibrium between the
166

two surfaces. Applying a no slip boundary condition at this interface, the fluid-structure coupling
conditions are given by
r=𝑟 f
: 𝒗 = 𝒖 ̇, (8.5)
r=𝑟 f
: 𝒏 ∙𝑻 𝐟 =𝒏 ∙𝑻 𝐬 , (8.6)
where 𝒖 ̇ is the solid velocity, and 𝒏 is the normal vector of the fluid-solid interface. To
ensure total wave reflection, outlet of the vessel was fixed, such that
𝒖 =0, at z=𝐿 (8.7)

8.3.3 Implementation of the boundary conditions
In the fluid domain, zero pressure was applied to the inlet and outlet while an axisymmetric
boundary condition was applied to the inner edge (𝑟 =0). At the interface of the fluid and solid
domains, no slip boundary condition was applied (Eq. (2.5) and (2.6)). In the solid domain, zero
traction (𝑻 𝐬 =𝟎 ) was applied to the outer edge (𝑟 s
=𝑟 f
+𝑡 s
), and the inlet edge was extended
cyclically using a custom hybrid boundary condition. One cycle of the boundary condition
consists of stretching and recoil. During stretching, a prescribed velocity profile is applied to the
wall to elongate the tube. The shape of the velocity waveform was taken from root displacement
measurements using cardiovascular magnetic resonance by Codreanu et al. [207]. Fig. 8.2
demonstrates their reported data overlaid by the prescribed velocity profile used in this study
during the stretching phase (𝑇 stretch
). The maximum amount of stretch is determined by
computing the area under the velocity-time curve and normalized by the length of the tube
(stretching coefficient, 𝑆𝑅 ). his value is controlled by changing the ma im um velocity ( 𝑉 max
).
Recoil begins after peak displacement is reached, when the prescribed velocity is replaced by a
167

zero-traction boundary condition and the tube relaxes due to its own elasticity. Once the cycle
period has elapsed, the cycle repeats.

Fig. 8.2 Implemented boundary condition at the tube root based on the reported physiological measurement. The
prescribed velocity profile applied to the root wall during the stretching phase overlaid on top of the reported magnetic resonance
measurements done by Codreanu et al [207].
8.3.4 Computational model and numerical method
The 2D axisymmetric model consisted of a fluid domain and a solid domain coupled at the
inner edge of the tube wall. For solving the fluid and solid domains numerically, we used the
finite element method. The fluid-structure interaction model employs the Arbitrary Lagrangian-
Eulerian method. The system was initialized at rest, with zero pressure and velocity in the fluid,
and zero stress and strain in the solid. The fluid and solid domains consisted of 1040 and 260
elements, respectively, and the time step was 0.001s. Note that these numbers were obtained for
the largest tube length used in this study (i.e., L=50cm). A Newton-Raphson iteration scheme
was used for time integration. The period of the cycle was determined by 𝑓 and the relative
durations of the stretching and recoil were determined by duty cycle (𝐷𝐶 ). A computational
168

model of the physical problem was generated and solved using the finite element solver ADINA
version 9.7 (ADINA R&D Inc., MA). Simulations were run on a standalone workstation
equipped with an Intel Core i7 CPU (6 cores and 3201 MHz) with 32GB memory. At least 35
cardiac cycles were simulated for each case to ensure a periodic steady state was reached. The
inputs, algorithm, and outputs of the computational model are summarized as pseudo-code in
Fig. 8.3. The output of the numerical solver is the fluid velocity (𝒗 (𝒙 ,𝑡 ) ), fluid pressure
(𝑝 (𝒙 ,𝑡 ) ), and solid displacement (𝒅 (𝑿 ,𝑡 ) ).

Fig. 8.3 Implementation of FSI algorithm to solve the longitudinal wave pumping system model.
8.3.5 Analysis method
Wave dynamics in an elastic tube depends on three major parameters: i) frequency of the
wave generator, ii) wave speed, and iii) reflection sites [51]. To validate our simulations, the

fluid filled elastic tube model , , , , ,
boundary parameters , ,duty cycle ( )
discretization ( ) and number of cycles to simulate ( final time
f
)
solid and fluid domains with zero velocity and stress
1

prescribe velocity profile to the tube root active stretching (Figure 8. )
4
5 zero traction to the tube root passive recoil

7 solve for fluid pressure and velocity fields at via qs. (8.1) and (8. )
8 prescribe the fluid forces at the fluid structure interface via q. (8. )
solve for solid displacement and stress fields at via q. (8.5)
10 apply no slip condition at the fluid solid interface , via qs. (8.5) and (8. )
11
Numerical solutions for and on
169

natural frequency of the system, as the inherent property of the model, was computed. The
relation between the natural frequency and the wave speed in a fluid-filled elastic tube can be
calculated from the empirically-derived formula [195]
𝑓 𝑛 =𝑛 𝑐 2𝐿 , (8.8)
where 𝑓 𝑛 is the natural frequency of the system corresponding to the 𝑛 𝑡 ℎ
harmonic, 𝑐 is the
speed at which the waves propagate in the tube (wave speed), and 𝐿 is tube length. To compute
the wave speed, we utilized the Moens-Korteweg equation given by,
𝑐 =√
𝐸 ℎ
2𝑟𝜌
, (8.9)
Where 𝐸 is the Young modulus of the wall, ℎ is the wall thickness, 𝑟 is the internal radius,
and 𝜌 is the fluid density. Simulations were run for different levels of wave speed inside the tube
(modified by changing the elasticity). At each wave speed, the simulations were done for 9
different frequencies (0.5, 0.8, 1, 1.25, 1.6, 2, 2.5, 3.2, and 4 Hz). The simulations were also run
for different tube lengths in order to investigate the effect of change in the location of the
reflection site. Ranges of the parameters are listed in Table 8.2. In this table, 𝑆𝑅 is defined as the
stretch ratio which is amount of stretch divided by the tube length.
Table 8.2 Parameter settings for different case studies.
Physical Parameter Symbol Range
Length of the tube (cm) L [12.5, 50]
Solid wall elasticity (kPa) 𝐸 s
[100, 900]
Duty Cycle 𝐷𝐶 [0.25, 0.5]
Stretching coefficient 𝑆𝑅 [0.01, 1.1]
Stretching frequency (Hz) f [0.5, 4]
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8.4 Results and discussion
8.4.1 Wave speed and natural frequency
Given the baseline characteristic in our model (𝐿 =50cm, 𝑟 =1.5cm, 𝐸 s
=100KPa, density of
1.05 g/cm
3
, duty cycle of 0.33, and thickness of 0.1cm), Eq. (8.9) yields 𝑐 =1.78m/s.
Therefore, the first three expected natural frequencies for this model, computed by Eq. (2.8), are
𝑓 1
=1.78Hz , 𝑓 2
=3.56Hz , and 𝑓 3
=5.34Hz . Fig. 8.4 shows the computed displacement of the
root (inlet) for a free-vibration test with stretching coefficient of 𝑆𝑅 =0.1, which corresponds to
5cm stretch for 50cm tube length. The displacement of the root is sampled for 10s, beyond which
the amplitude of the displacement is nearly zero. A Fast Fourier Transform (FFT) was applied to
identify the model’s natural frequencies , shown in Fig. 8.4(b). The computed natural frequencies
match well with the predicted values. This comparison shows that the employed fluid-structure
interactive model based on the Arbitrary Lagrangian-Eulerian algorithm (Fig. 8.3) is capable of
capturing the wave dynamics accurately.
171

Fig. 8.4 ’ . (a) The transient response of wall displacement, and (b) the power spectrum
density of the outlet flow signal using FFT. The peaks correspond to the natural frequencies of the system.
8.4.2 Effect of frequency
Fig.s 8.5(a) and 8.5(b) depict the root displacement and outlet flow profile during one
oscillatory steady-state period (𝑇 ) at different frequencies. The displacement profile during the
stretching phase is due to the applied velocity profile to the wall (Fig. 8.2) and therefore, for
different frequencies it has a similar shape. As the recoil starts, the displacement-time variation
exhibits the characteristics of a mass-spring system reacting to an initial force with an overshoot
and a subsequent oscillation. The exact response of the tube and the oscillation at the root varies
across different frequencies. Like conventional impedance pump, as the excitation happens, the
waves are allowed to travel along the tube and are then partially reflected at the outlet interface.
As the frequency changes, even for the same wave speed and tube length (fixed travel time), the
interactions between the forward running waves and the reflected ones got affected. In Fig.
8.5(b) outlet flow was computed by integrating fluid velocity across the outlet elements. Similar
to the displacement profile, the flow profile varies dramatically across different frequencies in
terms of both amplitude and direction.

= 1. Hz

= . Hz

= 4. 8 Hz
(a) (b)
172

Fig. 8.5 Sample dipslacement waveforms at the root and flow wavefroms at the outlet. (a) Root (inlet) displacement and (b)
outlet flow at given different excitation frequencies.
Fig. 8.6(a) presents mean outlet flow 𝑄̅
averaged over one period 𝑇 as a function of
frequency given the baseline parameters for the model (𝐿 =50cm, 𝑟 =1.5cm, 𝐸 s
=100KPa, density
of 1.05 g/cm
3
, duty cycle of 0.33, and thickness of 0.1cm). The mean outlet flow 𝑄̅
is an
indicator of the bulk flow motion for the specific wave condition. The data presented in Fig.
8.6(a) suggest a non-linear flow-frequency relationship similar to the previously-observed trends
in the experimental and computational studies of the conventional impedance pumps [197-199].
The dominant negative flow (toward stretched and excited inlet) reported in Fig. 8.6 suggests
that overall, the impact of the suction created due to the stretching is larger than the compression
wave at the outlet (due to the reflection). However, there are certain frequencies which affect this
balance and cause positive net flow generation (toward the fixed outlet). The frequency spectra
of oscillatory part of the outlet flow obtained by Fourier transform of three sample frequencies
(the flow pattern changes) are plotted in Fig.s 8.6(b), 8.6(c), and 8.6(d). Results suggest that for
the case with negative net flow (suction mode), the first harmonic is dominant while for the
positive cases (pumping mode) the second mode is dominant. As a result, there is an obvious
difference in the wave interactions between the two modes. From wave reflection spectral
(a) (b)
173

analysis, it can be shown that there are specific frequencies that minimize the impedance of the
elastic tube (called ‘impedance frequencies’ in this study). At these frequencies, the
superposition of the incident and reflected waves is minimized and therefore the modulus of
impedance becomes small [209]. Around the impedance frequencies, the reflected pressure
harmonics subtract from the incident pressure harmonic and cancels out, whereas the reflected
flow harmonic adds to the incident flow harmonic, thus increasing its amplitude. These
frequencies happen to occur at frequencies close to the one-quarter wavelength frequency (𝑓 ̂
1
=
𝑐 𝜆 1
given 𝜆 1
=4𝐿 ) and three-quarter wavelength frequency (𝑓 ̂
2
=
𝑐 𝜆 2
given 𝜆 2
=4𝐿 /3). In our
system, the theoretical one-quarter wavelength frequency is 𝑓 ̂
1
=0.9Hz and 𝑓 ̂
2
=2.6Hz (Note
that 𝑐 above is computed based on Eq. (8.9).). Results from Fig. 8.6 suggests that at similar
frequencies, the generated positive flow due to the interference of the traveling and reflected
waves at the outlet dominate the negative flow generated by the suction (due to the stretch) and
hence, the net flow is positive. In other words, these frequencies alter the flow-frequency trend
exists for the longitudinal impedance pump, which is due to the net effect of the suction and the
compression, leading to the non-linear flow-frequency pattern. This difference shows itself in the
frequency domain by altering the dominancy of either first or second harmonics.
174

Fig. 8.6 Flow-frequency analysis at the baseline model parameters. (a) Mean outlet flow of the longitudinal impedance pump
as a function of excitation frequency. The power spectral density of the outlet flow at the frequencies of (b) 1Hz, (c) 1.6Hz, and
(d) 2.5Hz.
8.4.3 Pumping mechanism
Based on the currently prevailing view, conventional impedance pumps operate based on the
pressure wave generation by periodic tube wall excitations [195, 198, 199]. It was shown that for
achieving the maximum pressure difference to generate flow, the system has to be excited near
its natural resonant frequencies [199]. However, there are some studies that reported significant
generated flow at frequencies far below the natural frequency [195]. Either way, the unique
frequency dependence of the net flow rate implies an impedance-driven flow. Our findings
indicate that there are two modes in the longitudinal impedance pump, the suction mode at which
= 1.0Hz
= .5Hz = 1. Hz
(a) (b)
(c) (d)
= 1. Hz
= 1.0 Hz
= .5 Hz
175

the first harmonic in the power spectra is dominant and the pumping modes where the second
mode in the power spectra become dominant. Fig. 8.7(a) and 8.7(b) presents the flow-frequency
pattern computed in the longitudinal impedance pump model for stretching of 2cm and 2.5cm
respectively. The power spectra for the outlet flow at the local positive and negative peaks for
both cases are also demonstrated in this plot. Similar to the findings from Fig. 8.6, results suggest
that the second harmonic is dominant in the pumping mode in the longitudinal impedance pump,
while the first harmonic is dominant in the suction mode.

Fig. 8.7 Flow-frequency analysis at different levels of the root displacements. Mean outlet flow of the longitudinal impedance
pump as a function of excitation frequency for stretching of (a) 2 cm corresponding to SR=0.04, and (b) 2.5cm corresponding to
SR=0.05. The power spectral density of the outlet flow at the frequencies corresponding to the peak flows are also demonstrated.

Fig. 8.8 presents distributions of the fluid pressure at various snapshots in time during one
cycle. The longitudinal impedance pump demonstrates both positive flow generation (pumping)
and negative flow generation (suction) based on the stretching frequency. In the negative flow
cases, as the pressure drops due to the stretching, a flow begins to fill the low-pressure region
(stretched portion of the tube) from a higher-pressure region (resting portion of the tube). By
modifying the stretching frequency and fixing the travel time (constant tube length and wave
(a) (b)
176

speed), a coordination between the forward running and reflected waves can be achieved such
that net unidirectional flow is sustained. The stretching of the aortic root results in a suction
wave that travels downstream in the tube as demonstrated in Fig. 8.8. During recoil, the flow
generation is determined primarily by the compression reflected pressure waves. The radial
displacement profiles of the vessel wall given various frequencies are also presented in Fig. 8.8.
Because the tube material is linearly elastic, there is a strong association between fluid pressure
and wall displacement [73, 74]. The wave propagation and reflection, which their interaction
lead to wall displacement (Fig. 8.9), create the compression pressure wave. The summation of
suction and the pressure wave ultimately determines the direction of the net flow.

Fig. 8.8 Spatial distributions of flow behavior in the longitudinal impedance pump model at different snapshots of time
during cycle 𝑻 and for different frequencies. The plots in each row present pressure distributions. For the visualization
purpose, the deflection of the wall is neglected.
From Fig. 8.9, it can be noted that radial wall displacement is relatively small at frequencies
such as 𝑓 =1𝐻𝑧 and 𝑓 =2.5𝐻𝑧 . These frequencies happen to be close to the impedance
frequencies (𝑓 ̂
1
and 𝑓 ̂
2
). It can be shown for a single harmonic excited at a frequency
corresponding to ¼ wavelength (considering the wave speed), the reflected wave is delayed half
= 0.8Hz
= 1.0Hz
= 1. Hz
= .5Hz
Pa
177

a period relative to the forward harmonic, and the reflected and forward pressure harmonics thus
cancel it out [209]. While in our longitudinal pump model we have more complex wave
propagation than simple sinusoids, the findings can be explained based on the same principle.
For these cases, the main determinant of flow is the compression waves generated due to the
reflection. This explains the net positive outlet flow at these frequencies. Note that the amplitude
of the radial wall displacement reaches its maximum at frequencies near natural resonant
frequency. For example, at 𝑓 =1.6Hz which is near system’s natural frequency ( Fig. 8.4), the
amplitude of the radial displacement is larger than the rest.

Fig. 8.9 Tube wall displacement as a function of frequency. Computed radial wall displacements along the tube length 𝑳 at
different snapshots in time for (a) 𝒇 =𝟎 .𝟖𝑯𝒛 , (b) 𝒇 =𝟏 .𝟎𝑯𝒛 , (c) 𝒇 =𝟏 .𝟔𝑯𝒛 , and (d) 𝒇 =𝟐 .𝟓𝑯𝒛 .
Fig. 8.10 presents the sampled pressure and flow during an oscillatory steady-state cycles at
different locations along the tube. The data presented in this figure show that loop direction tends
= 0.8Hz = 1.0Hz
= .5Hz
= 1. Hz
(a) (b)
(c) (d)
178

to flip once along the length of the tube. Near the inlet where the stretching occurs, the elastic
wall does work on the fluid, and the 𝑃 -𝑄 loop is thus counterclockwise. Closer to the outlet, the
interaction of the suction and compression waves impedes flow, resulting in work done by the
fluid on the walls and clockwise 𝑃 -𝑄 loop. These sampled data are collected for frequency of
1.6Hz . Avrahami and Gharib [199] reported a nonlinear trend for wall pumping work in the
pinching impedance pump, which reached its maximum around the resonant frequency. It was
suggested that at these resonant frequencies, the elastic tube contributes to pumping by
transmitting energy to the fluid. This energy exchange contributes to the vessel wall
displacement profile (demonstrated in Fig. 8.9) which affects the significance of the compression
waves and the net flow generation.
179

Fig. 8.10 Pressure-flow (P-Q) loops at six locations along the tube for stretching frequency of 1.6 Hz. Arrows indicate the
direction of the loop. The axial distance from the inlet (stretching site) are (a) 𝒛 = 𝟐𝒄𝒎 , (b) 𝒛 = 𝟖𝒄𝒎 , (c) 𝒛 = 𝟏𝟒𝒄𝒎 , (d)
𝒛 = 𝟑𝟎𝒄𝒎 , (e) 𝒛 = 𝟑𝟔𝒄𝒎 , and (f) 𝒛 = 𝟒𝟐𝒄𝒎 .
8.4.4 Effect of wave speed
Fig. 8.11(a) presents 𝑄̅
as a function of frequency for different wave speeds inside the tube.
Wave speed is varied by changing the tube elasticity [51]. Simulations were run for four different
wave speeds, starting from 𝐶 1
, corresponding to the baseline parameters of the model (𝑟 =1.5cm,
𝐸 =100KPa, and thickness of 0.1cm). Results suggest that changing the tube characteristics affect
the net flow generation in longitudinal impedance pump. Alteration of the wave speed affects the
wave travel time and consequently changes the interactions of the forward-running and reflected
waves (which creates the compression waves). As described earlier, the flow generation in the
longitudinal impedance pump is the result of the net effect of the stretching suction and the
compression waves due to the wave propagation and reflection. The alteration in the material
characteristics and wave speed affect the intensity of the compression waves (due to the changes
in the input energy) and hence, results in a different flow-frequency pattern as shown in Fig.
(a) (b) (c)
(d) (e) (f)
180

8.9(a). These results suggest that at some frequencies (e.g., 3.2Hz), for the same stretching
amplitude, the flow can be either positive or negative depending on the wave speed on the wall
(tube wall elasticity or thickness).
Flow-frequency relation as a function of wave speed were further investigated using
dimensionless wave condition number (𝑊𝐶𝑁 ) demonstrated in Fig. 8.10(b). This normalization
helps investigating the behavior of longitudinal impedance pump across different range of
parameters. 𝑊𝐶𝑁 is a quantity that determines the state of wave dynamics in an elastic tube
(computed by 𝑊𝐶𝑁 =
𝑓𝐿 𝑐 [54]). In this figure, the flow is also normalized with respect to its
corresponding range. These results are in agreement with our hypothesis that the net effect of the
suction and compression waves changes at different wave states in the longitudinal impedance-
driven flow. As demonstrated by this figure, the computed results at different wave speeds
collapse on top of each other and the non-linearity in flow-frequency pattern is well-captured
using 𝑊𝐶𝑁 . Hence, this scaling facilitates understanding the non-linear flow-frequency pattern
which exists in the longitudinal impedance pump across wide range of parameters.

Fig. 8.11 Impact of tube stiffness on the generated flow in the longitudinal impedance pump. (a) Mean outlet flow of the
longitudinal impedance pump against the excitation frequency for different levels of pulse wave velocity 𝒄 . (b) The normalized
outlet flow of the longitudinal impedance pump against the wave condition number (𝑾𝑪𝑵 ) for different levels of pulse wave
velocity 𝒄 .

(a) (b)
181

8.4.5 Effect of tube length
Fig. 8.12(a) presents mean outlet flow 𝑄̅
for 𝑆𝑅 =0.1 as a function of frequency for different
tube lengths. Changing tube length changes the location of the only reflection site in our model –
the end of the tube. This affects the wave interaction. Simulations were run for four lengths,
starting from 𝐿 1
=50𝑐𝑚 . As the wave state changes due to the alteration in the tube length, the
net effect of the suction and compression waves got affected and therefore, the peaks in the flow-
frequency pattern shift. Fig. 8.12(b) depicts the normalized outlet flow at each tube length as a
function of 𝑊𝐶𝑁 .

Fig. 8.12 Impact of tube length on the generated flow in the longitudinal impedance pump. (a) Mean outlet flow as a
function of excitation frequency for different tube lengths. (b) Normalized outlet flow as a function of wave condition number for
different tube length.
8.4.6 Theoretical analysis of the net flow
As a complement to the computational analysis used in this chapter, analytical relations
between the longitudinal impedance pump parameters are presented here. To find an analytical
relation for the outlet flow, the following assumptions were made: (i) the propagated wave inside
the system have reached an oscillatory steady-state condition ,and (ii) the tube radius (𝑟 𝑓 ) is
much smaller than the tube length which is the case in our computational model as well, and (iii)
(a) (b)
182

we considered the long wave solution (k<<1 where k is the wavenumber) [210]. Fluid motion
can be written in dimensionless form as
𝜕𝑉
𝜕𝑇
+𝑉 𝜕𝑉
𝜕𝑅
+𝑈 𝜕𝑉
𝜕𝑍
=
𝜕 𝜕𝑅
(
1
𝑅 𝜕 𝜕𝑅
(𝑅𝑉 ))+
𝜕 2
𝑉 𝜕 𝑍 2
−
𝜕𝑃
𝜕𝑅
, (8.10)
𝜕𝑈
𝜕𝑇
+𝑉 𝜕𝑈
𝜕𝑅
+𝑈 𝜕𝑈
𝜕𝑍
=
1
𝑅 𝜕 𝜕𝑅
(𝑅 𝜕𝑈
𝜕𝑅
)+
𝜕 2
𝑈 𝜕 𝑍 2
−
𝜕𝑃
𝜕𝑍
, (8.11)
1
𝑅 𝜕 𝜕𝑅
(𝑅𝑉 )+
𝜕𝑈
𝜕𝑍
=0, (8.12)
where 𝑈 and 𝑉 are the dimensionless components of the velocity vector in (Z, R) directions
(axial and radial velocities, respectively) and scaled by (
𝜗 𝑟 𝑓 ) where 𝜗 is the kinematic viscosity
and 𝑟 𝑓 is the tube radius, 𝑇 is time scaled by
𝑟 𝑓 2
𝜗 , 𝑃 is pressure scaled by 𝜌 𝑓 (
𝜗 𝑟 𝑓 )
2
where 𝜌 𝑓 is the
fluid density. Considering the displacement of the wall (𝑌 𝑤 ) is represented by the propagated
sinusoidal waves, the boundary conditions to solve the above system of equation can be given by
𝑌 =0: 𝑉 =0,
𝜕𝑈
𝜕𝑅
=0,𝑌 =𝑌 𝑤 (𝑍 ,𝑇 ): 𝑈 =0,𝑉 =
𝜕 𝑌 𝑤 (𝑍 ,𝑇 )
𝜕𝑇
(8.13)
𝑌 𝑤 (𝑍 ,𝑇 )=𝐴𝑐𝑜𝑠 (𝐾 (𝑍 −𝐶𝑇 ))+1 (8.14)
Where 𝐴 is the dimensionless amplitude of the wave (scaled by 𝑟 𝑓 ), 𝐾 =2π/Λ is
dimensionless wavenumber (𝛬 is the wavelength scaled by 𝑟 𝑓 ), and 𝐶 is the dimensionless wave
speed (normalized by (
𝜗 𝑟 𝑓 )). We also define a constraint, based on the assumption that any fluid
motion in the Z-direction is induced by wave propagation along the wall and hence, there is no
external imposed pressure gradient, translating to
183

𝜕𝑃
𝜕𝑍
|
𝑚 =0, (8.15)
Where 𝑚 denotes the mean value. The analysis is simplified by introducing frame of
reference moving with the wave phase speed. The relevant transformation is
𝑧 =𝑍 −𝐶𝑇 , (8.16)
𝑟 =𝑅 . (8.17)
The governing equation in the transformed direction can then be written as
𝜕𝑣
𝜕𝑟
+(𝑢 −𝑐 )
𝜕𝑣
𝜕𝑧
=
𝜕 𝜕𝑟
(
1
𝑟 𝜕 𝜕𝑟
(𝑟𝑣 ))+
𝜕 2
𝑣 𝜕 𝑧 2
−
𝜕𝑃
𝜕𝑟
, (8.18)
𝑣 𝜕𝑢
𝜕𝑟
+(𝑢 −𝑐 )
𝜕𝑢
𝜕𝑧
=
1
𝑟 𝜕 𝜕𝑟
(𝑟 𝜕𝑢
𝜕𝑟
)+
𝜕 2
𝑢 𝜕 𝑧 2
−
𝜕𝑃
𝜕𝑧
, (8.19)
1
𝑟 𝜕 𝜕𝑟
(𝑟𝑣 )+
𝜕 𝑢 𝜕𝑧
=0, (8.20)
Subject to the following boundary conditions
𝑣 (0)=0,
𝜕𝑢
𝜕𝑟
(0)=0,𝑢 [𝑦 𝑤 (𝑧 )]=0,𝑣 [𝑦 𝑤 (𝑧 )]=−𝑐 d𝑦 𝑤 d𝑧 , (8.21)
where
𝑦 𝑤 (𝑧 )=𝐴𝑐𝑜𝑠 (𝐾𝑧 )+1. (8.22)
Considering the long wave assumption (K →0) [210-212], the unknowns of the interest can
be expanded as
(𝑢 ,𝑣 )=(𝑢 0
,𝑣 0
)+𝐾 (𝑢 1
,𝑣 1
)+𝑂 (𝐾 2
) , (8.23)
𝑃 =𝐾 −1
𝑝 −1
+𝑝 0
+𝑂 (𝐾 ) , (8.24)
184

𝑄 =𝑄 0
+𝐾 𝑄 1
+𝑂 (𝐾 2
) . (8.25)
The solution domain is regularized using the transformation in the form of
𝜉 =𝐾𝑧 , (8.26)
𝜂 =𝑟 (𝐴 cos ( 𝐾𝑧 )+1)
−1
. (8.27)
Inserting the expansions in (8.23) and (8.24) into the governing equations in the regularized
domain and keeping only the leading-order terms result in
𝑣 0
=0, (8.28)
𝜕 𝑝 −1
𝜕𝜂
=0, (8.29)
𝜂 𝜕 2
𝑢 0
𝜂 2
+
𝜕 𝑢 0
𝜕𝜂
−𝜂 (1+𝐴𝑐𝑜𝑠 (𝜉 ))
2
𝜕𝑝
−1
𝜕𝜉
=0. (8.30)
General form for the solution to the (8.30) is given by
𝑢 0
=
(1+𝐴𝑐𝑜𝑠 (𝜉 ))
2
4
𝑑𝑝
−1
𝑑𝜉
(𝜂 2
−1)+𝐶 1
log(𝜂 )+𝐶 2
, (8.31)
Subject to
𝑢 0
(𝜂 =1,𝜉 )=0,
𝜕 𝑢 0
(𝜂 =0,𝜉 )
𝜕𝜂
=0. (8.32)
Considering the boundary condition and the constraint that 𝑢 0
(𝜂 =0,𝜉 ) is finite yields
𝑢 0
=
(1+𝐴𝑐𝑜𝑠 (𝜉 ))
2
4
𝑑𝑝
−1
𝑑𝜉
(𝜂 2
−1) , (8.33)
Hence, the flow can be computed by
185

𝑄 0
=2𝜋 ∫ 𝑢 0
𝑟𝑑𝑟 =
𝑦 (𝑧 )
0
−
𝜋 4
(1+𝐴𝑐𝑜𝑠 (𝜉 ))
4
𝑑 𝑝 −1
𝑑𝜉
. (8.34)
To complete the derivation for axial velocity and outlet flow, we need to compute
𝑑𝑃
−1
𝑑𝜉
. We
use the stream-function formulation as 𝑢 =
1
𝑟 𝜕𝜓
𝜕𝑟
and 𝑣 =−
1
𝑟 𝜕𝜓
𝜕𝑧
. Stream-function at the tube wall
can be written as
𝑑 𝜓 𝑤 =𝑟𝑈𝑑𝑟 −𝑟𝑉𝑑𝑧 . (8.35)
Utilizing the chain rule and integrating Eq. (8.35), the stream-function at the wall can be
written as
𝜓 𝑤 =𝐶𝐴𝑐𝑜𝑠 (𝜉 )+
1
4
𝐶 𝐴 2
cos(2𝜉 )+𝐶 𝑤 . (8.36)
To compute outlet flow, we also need to compute the stream-function at the center (so
subtracting it from the stream-function at the wall [211]). Similar to the Eq. (8.25), the centerline
stream function gives a constant value (𝐶 center
). Hence, the flow can be computed as
𝑄 =2𝜋 (𝜑 𝑤 −𝜑 center
) , (8.37)
𝑄 =2𝜋𝐶𝐴𝑐𝑜𝑠 (𝜉 )+
𝜋 2
𝑐 𝐴 2
cos(2𝜉 )+𝑄 mean
. (8.38)
Using Eq. (3.25) and (3.29),
𝑑𝑃
−1
𝑑𝜉
is given by
𝑑 𝑝 −1
𝑑𝜉
=
−8𝑐𝐴𝑐𝑜𝑠 (𝜉 )
(1+𝐴𝑐𝑜𝑠 (𝜉 ))
4
−2
𝑐 𝐴 2
cos(2𝜉 )
(1+𝐴𝑐𝑜𝑠 (𝜉 ))
4
−
4𝑄 𝑚𝑒𝑎𝑛 𝜋 (1+𝐴𝑐𝑜𝑠 (𝜉 ))
4
. (8.39)
Lastly, eliminating for mean pressure gradient [211, 212] results in the following relation for
mean flow
186

𝑄 mean
=−
𝐴 2
𝐶 (16𝜋 −𝜋𝐴
2
)
2(3𝐴 2
+2)
. (8.40)
Eq. (8.40) states that for one simple harmonic, the outlet flow has a linear relation with wave
amplitude (𝐴 ) as well as characteristic wave speed (𝐶 ). Fig. 8.13 compares the theoretical model
with the computational results on the log-log scale. Note that this equation is only derived for
one harmonic under the assumption of long wave amplitudes. Therefore, the numerical points
that are chosen for comparison are among those who have only one dominant harmonic in their
flow spectrum (not close to the impedance frequencies). The parameters of the theoretical model
are chosen based on the values of the computational simulations with baseline wave speed of
𝑐 0
=1.78 m/s . To consider the complete range for wave amplitudes, a region is defined in the
figure (gray area). This comparison suggests that the observed non-linearity in the flow-
frequency pattern (Fig.s 8.6, 8.11 and 8.12) is mostly due to the flow behavior in the impedance
frequencies. These results suggest that while interactions of the suction and compression waves
result in a complex pumping behavior, we can expect mostly linear relation between the flow,
stretching parameters and material characteristics at frequencies other than the impedance
frequencies.
187

Fig. 8.13 Comparison between the computed Normalized flow rate from the theory and numerical simulations.
Computational data at different wave condition numbers are demonstrated.
8.5 Conclusion
The present chapter provides a comprehensive analysis on a novel type of the impedance
pump with longitudinal stretching as an excitation mechanism. This impedance pump model was
inspired by the stretch-related biodynamics of the human aorta. In this type of impedance pump,
waves are created by longitudinal stretching of the elastic wall. Our results indicate that stretch-
related wave propagation and reflection in a fluid-filled compliant tube can create a wave
pumping mechanism. y e amining a wide range of parameters, the pump’s nature is describe d
and the concept of quarter-wave theory and its role in flow-frequency pattern in the longitudinal
wave pumping mechanism is discussed. It was shown that the pumping characteristics in such
188

systems is the result of the net impact of the suction waves, created due to the stretching, and the
compression waves, created as a result of net interaction of the propagating and reflected waves.
Like other impedance pumps, the unique frequency-dependence of the net flow rate implies an
impedance-driven flow. It was shown that both the direction and magnitude of the net flow
depend strongly on wave dynamic characteristics including the stretching frequency, tube wall
elasticity and the tube length. Results suggest that at the frequencies near the impedance
frequencies (𝑓 ̂
=
𝑐 𝜆 , where 𝜆 =4𝐿 𝑜𝑟 4𝐿 /3), the compression waves compensate (partially or
fully) the impact of suction waves generated due to the stretch. This effect show itself in the
spectra of the flow by switching the dominancy of the first harmonic with the second harmonic.
For other frequencies, it was demonstrated that the suction waves dominate, and the net outlet
flow is negative. Our results also suggest that wave condition number can be used to capture the
non-linearity in flow-frequency relation across different scales. All in all, the results presented in
this study provide fundamental understanding the underlying physics of stretch-related wave
pumping and can serve as a design guideline for future use of longitudinal impedance pumps.

189

CHAPTER 9 : Aortic stretch and recoil create pumping in the systemic
circulation: an assist mechanism for left ventricular function

9.1 Chapter abstract
Understanding the hemodynamic interactions between the heart and vasculature is crucial in
the development of new therapeutic strategies and assist devices. One such interaction is the
physical connection between ascending aorta and left ventricle. Clinical studies have visualized
downward aortic root motion due to the left ventricle systolic long-axis shortening. While this
displacement results in energy storage in aorta’s spring -like elements, it is unknown whether the
stretch-related aortic dynamic mode can result in a wave pumping effect. To investigate the
isolated effects of this phenomenon, we employ an in-vitro experimentations using artificial
phantom models that mimic the characteristics of the human aorta and tested them under
physiological stretching. Results suggest that stretching-based longitudinal wave pumping in a
compliant aorta creates wave propagation and reflections, which generate significant flow. This
complex pumping effect is a function of the wave dynamic parameters, dominated by the
stretching frequency and the pulse wave speed inside the aorta.
9.2 Introduction
Interactions between the left ventricle and the aorta are optimized in normal conditions to
guarantee the delivery of cardiac output with modest pulsatile hemodynamic load on the left
ventricle. Recent clinical studies showed that in an optimal left ventricle-aorta coupling, left
ventricle systolic contraction displaces the aortic annulus and produces a considerable
longitudinal stretch in the ascending aorta [205]. The force associated with this mechanical
190

coupling increases the systolic load on the left ventricle but also stores energy in the elastic
elements of the proximal aorta. While the importance of the stretch-related aortic work has
shown in relation to the diastolic filling and heart failure, the full aspects of this dynamic mode
on heart workload is not known. Specifically, the connection between the longitudinal aortic
stretching and cardiac pumping efficiency is not fully understood. Longitudinal pumping may
provide the heart a supplementary pumping mechanism that helps reduce its workload. This can
have significant impact in heart failure patients who have impaired left ventricular function or
heart-aortic coupling. However, there is no study on the association between stretch-related wave
pumping in the aorta and heart pumping ability.
In a recent study, Pahlevan and Gharib [141] showed that the wave reflection in the aorta
creates a pumping effect similar to an impedance pump. Impedance pump is a simple valveless
pump that works based on wave propagation and reflections and can generally generate a
bidirectional flow depending on the state of the wave dynamics [141]. Previous studies showed
that aortic wave dynamics plays an important role in terms of adding favorable or unfavorable
pumping effect to the mean heart output [141]. In a zebrafish embryo model, Forouhar et al.
provided the first experimental evidence that the impedance-based pumping mechanism is
involved in driving blood flow at early stage of heart development. Our hypothesis is that the
stretch-related aortic dynamic mode can also result in a wave propagation and reflection in the
cardiovascular system which ultimately lead to the wave pumping mechanism in the aorta. This
longitudinal wave pumping, and impedance pumping differ in their wave generating system, but
both pumping mechanisms rely on the principles of wave propagations and reflections.
We will test our hypothesis through investigating the effects of ascending aortic
longitudinal stretch on flow generation and pumping. To this end, we employ an in-vitro
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hydraulic model that has hemodynamic properties similar to the human systemic circulation. A
driving component consisting of a cam-follower mechanism mounted on the stepper motor is
employed to simulate stretch-related dynamic mode. This model mimics the active stretching of
the aortic root during systole and passive recoil during the diastole in-line with the physiological
evidence. Our choice of in-vitro modeling makes it possible to focus on the isolated effect of
longitudinal wave pumping mechanism, which is inherently difficult (or even impossible)
through clinical or pre-clinical studies. Different stretching conditions (amplitude and frequency)
and aortic characteristics are modeled to gain a comprehensive insight over this mechanism.
9.3 Methods
The experimental setup consists of a hydraulic circuit, a longitudinal stretching mechanism,
and an artificial human aorta. The details of the experimental system and the fabrication method
of the aortic phantom are presented in the following subsections.
9.3.1 Hydraulic circuit
The hydraulic circuit provides a closed-loop flow circuit between the aortic phantom and the
reservoir (See Fig. 9.1). In this system, the aortic phantom is installed into the system by placing
luer lock-enabled barbed connectors to its major branches and connecting them with soft Tygon
tubes that can be fitted to the ports on the side walls of this container. The ports of the carotid
and femoral arteries are lumped with their respective branches and connected to the reservoir
tank. The remaining branches are isolated from the circulation by placing stopcock valves to
their downstream. As the final installation step, the aortic root is directly connected to the
longitudinal stretching mechanism, and it is connected to the reservoir tank with a Tygon tube to
complete the closed-loop circulation. Four resistance clamps are placed downstream of the
arteries to mimic the resistances of the eliminated vasculature.
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9.3.2 Aortic phantom fabrication
The artificial aortas are fabricated using an anatomically accurate one-to-one human-scale
stainless steel mold that contains major arterial branches and aortic tapering. This mold is either
coated with natural latex (Chemionics Corp.) or silicone (RTV-3040, Freeman Manufacturing &
Supply Company). Both of these materials conform with the mechanical properties of the human
aorta to fabricate physiologically accurate vascular phantoms. The natural latex is coated on the
mold by dipping, whereas the silicone is applied by brushing the metal mold. The dipping
procedure can be explained in three steps: i) metal mold is fully dipped into a container filled
with natural latex for ten seconds; ii) the coated mold is placed on a hanger to cure for 2 hours at
room temperature (25
∘
𝐶 ); iii) repeating the previous steps (as many times as desired) to control
the wall thickness, thus the aortic compliance and pulse wave velocity. The brushing method is
summarized in five steps: i) 25 grams of silicone mixture is prepared by mixing the silicone base
and its catalyst with a mass ratio of 10:1; ii) the silicone mixture is placed in a vacuum chamber
that removes the bubbles to improve the surface uniformity; iii) coating the mold with a layer of
silicone using a soft-tip acrylic brush; iv) the mold is hanged to dry for 16 hours at room
temperature (25
∘
𝐶 ); v) repeating the previous steps (as many times as desired) to control the
thickness. The coated mold is hung from opposite ends at each drying step for both fabrication
techniques to achieve a uniform surface. After the desired material thickness develops on the
mold, the phantom model is removed by injecting water between the mold and the material.
Natural latex phantoms require final surface treatment to increase their durability. Such treatment
is achieved by submerging them in regular bleach (Clorox) for 12 hours.
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9.3.3 Longitudinal stretching mechanism
The phantom is connected to the longitudinal stretching mechanism from its aortic root and
the carotid arteries. The latter is placed in a fixture that constrains the movement of the aortic
arch. The aortic root is connected to the stretching apparatus, which can move longitudinally to
generate waves in the phantom model. This apparatus is actively stretched and recoiled by a
cam-follower mechanism. Cam is made of an aluminum mounted on a stepper motor. The
actuation frequency is regulated by the software of the motor controller, whereas the maximum
cam radius determines the aortic stretching amplitude. These cams are designed with a spiral
shape (see Fig. 9.1(b)) that brings the stretcher to the maximum radius and holds it for 1/3 of the
working period for the active stretching phase. Then, the cam suddenly releases the stretching
apparatus to its initial position to create the passive recoil. Such cam design is inspired by the
motion of the aortic root during the systolic and diastolic phases of the left heart (Fig. 9.1).
9.3.4 Measurement devices
A high-fidelity piezoelectric pressure catheter (Mikro-Cath, Millar Inc., Houston, TX) is
inserted into the Tygon tube that connects the longitudinal stretching mechanism to the reservoir.
A clamp-on ultrasonic flowmeter (ME-16PXN, Transonic Systems Inc.) is placed on top of the
pressure sensor. These two sensors are connected to a PowerLab data acquisition device
(ADInstruments, Colorado Springs, CO), and their data are sampled at 1000Hz by LabChart Pro
7 software. Flowmeter calibration session is performed for this tube. During this experiment, a
range of flow rate values (5-400 mL/min) in both forward and backward directions is introduced
by a syringe pump (PHD Ultra, Harvard Apparatus, MA), and the voltage values the flowmeter
generated are collected. This procedure allows us to create a linear regression line for the
measured voltage values against the known flow rates and obtain the calibration coefficients for
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this tube. These coefficients are used to convert the sampled voltage values to flow rate data that
are presented in the Results section. The pressure catheter is calibrated to ≈0 𝑚𝑚𝐻𝑔 at the
atmospheric pressure at the tube’s elevation before it is inserted into the data collection position.
9.3.5 Experimental procedure
The hydraulic circuit is filled with water, and the visible air bubbles are removed from the
system. Before the experiments, the cam is attached to the stepper motor, and the longitudinal
stretcher is actuated at 3 Hz to inspect for possible leakages and systematical problems. As the
first step of each experiment, the data for the pulse wave velocity (PWV) analysis (See section
Pulse wave velocity analysis) are collected for all the phantom models. For this procedure, one
pressure catheter is placed near the aortic root, and a second catheter is inserted into the
abdominal aorta from one of the renal arteries. Consecutively, the distance between the sensor
locations of these catheters is measured. Then, the stretching mechanism is actuated at the
frequency of 1 Hz.
After the PWV data is collected, the pressure catheter is relocated to the Tygon tube that
connects the aortic stretcher to the reservoir, and the flow meter is placed on top of this sensor.
Upon this, the stepper motor starts to actuate the stretching mechanism at 0.5 Hz until the system
reaches the oscillatory steady state (≈ ten cycles). Following this, the operating speed of the
stretcher is increased by 0.5 Hz, and the system is brought to the steady oscillatory state (by
waiting for ≈ ten cycles). The previous step is repeated until the stretching mechanism is at 3.0
Hz. After the frequency range (0.5-3.0 Hz) is covered, the cam is changed to test the effect of the
maximum stretching amplitude.
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9.3.6 Flow visualization study
For the flow visualization, a dye solution is prepared by diluting a food coloring with water
with a volume ratio of 1:50. A small amount of this dye solution (≈5−10 𝑚𝐿 ) is injected into
the aortic root of a silicone phantom from the coronary arteries. The choice of silicon models is
due to their transparency. When the dye sufficiently covers the aortic root region, the
longitudinal stretching mechanism is activated to generate the flow field that eventually carries
the dye into the reservoir tank. Visualizing different parameters (stretching amplitude or
frequency) necessitates additional injections of the dye solution to the aortic root. This process
can be repeated until the color of the fluid in the reservoir tank starts impairing the visualization.
9.3.7 Pulse wave velocity analysis
The pulse wave velocity (PWV) analysis data is collected at the beginning of each
experiment by inserting two pressure catheters and measuring the distance between their sensors.
The foot-to-foot method is utilized to compute the PWV of the phantoms. This method relies on
measuring the time delay (Δ𝑡 ) between the foot of the first propagating wave at the aortic root
and the foot of the second propagating wave at the abdominal aorta, together with the distance
(Δ𝐿 ) between the two measurement sites. Then the PWV can be calculated by the following
formula, 𝑃𝑊𝑉 =Δ𝐿 /Δ𝑡 .
9.3.8 Aortic compliance measurements
The procedure for the aortic compliance measurements can be summarized in the following
steps: i) The branches of the aortic phantoms are connected to barbed-ended connectors with luer
locks (Qosina, Long Island, NY); ii) One-way stopcock valves are attached to the luer ports; iii)
the aortic root is sealed with a connector and a plastic stopper; iv) the phantom is filled with
water, and all the air bubbles are removed through the valves; v) a pressure catheter is inserted to
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the aortic phantom, and the excess water is drained until the catheter reports the atmospheric
pressure (≈0𝑚𝑚𝐻𝑔 ); vi) a known amount of water (Δ𝑉 ) is injected from one of the valves; vii)
the pressure value is recorded after the catheter reports a steady value; viii) The water injection
and recording of pressure steps (vi and vii) are repeated until the phantom pressure reaches
150𝑚𝑚𝐻𝑔 . ix) the aortic compliance (AC) can be computed by the following formulation 𝐴𝐶 =
Δ𝑉 Δ𝑃 ; x) The reported AC values correspond to the mean operating pressure of the experiments.
9.4 Results
Schematic representation of the aortic stretch and recoil during the cardiac cycle is shown in
Fig. 9.1(a). At the end-diastole, the heart muscle relaxes, and the aorta is in its unstretched
length. As the systole starts, left ventricle starts to contract and the base of the heart start moving
downward toward the stationary apex. Because of minimal displacement of the aortic arch during
systole, movement of the aortic annulus towards the apex produces longitudinal stretch of the
spring-like elements of the aorta. At end-systole, the left ventricle has imposed maximal
downward force on the aortic root, which achieves maximal longitudinal stretch. In early
diastole, relaxation of the left ventricle allows the preloaded elastic elements in the walls of the
aortic root to recoil. This recoil is passive and is due the presence of the elasticity in the aortic
wall. To simulate this longitudinal stretch, a mechanism consists of a cam-follower design
mounted on the stepper motor, drives aorta, and will be installed in the hydraulic simulator (Fig.
9.1(b) and 9.1(c)). To control the frequency of the stretching, speed profile of the stepper motor
will be modified (Fig. 9.1(d)). The duty cycle (the ratio of the active stretching over the whole
cycle) is chose as one third which is the typical systole to cardiac ratio in the physiological
setting. Fig. 9.1(e) demonstrates the stretching and recoil of the artificial aortic phantom made
from silicon in the simulator. The snapshots at the left side of this figure show the unstretched
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position and the ones on the right-hand side show the aorta in the displaced (stretched) position.
Lastly, the compliance profile of this artificial phantom is measured and reported in Fig. 9.1(f).
The procedure to compute the compliance is described in the methods section. Experiments are
run for six different levels of aortic stiffness (measured by aortic pulse wave velocity) covering
the range for healthy individuals to pathophysiologically aged ones. Cases have been run for
different frequencies, ranging from 0.5H till 3Hz which is translated to the heart rates in the
range of 30bpm to 180bpm. Each case has been run for four different stretching amplitudes
ranging from 0.5cm to 2cm. In all cases, the terminal resistance and the height of the reservoir
kept constant.
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Fig. 9.1 Clinical motivation of investigating aortic stretch and recoil wave pumping and the in-vitro setup. (a) Schematic
representation of longitudinal stretch and recoil of the ascending aorta in heart-aorta coupling. At the end-diastole, the aorta is in
its unstretched length. As the systole starts, left ventricle starts to contract and the base of the heart start moving downward
toward the stationary apex. Because of minimal displacement of the aortic arch during systole, movement of the aortic root
towards the apex produces longitudinal stretch of the spring-like elements of the aorta. At end-systole, the left ventricle has
imposed maximal downward force on the aortic root, which achieves maximal longitudinal stretch. In early diastole, relaxation of
the left ventricle allows the preloaded elastic elements in the walls of the aortic root to recoil. (b) Design of the cam in the cam-
follower mechanism for mimicking the longitudinal stretch of the aorta. The ratio of the active stretching (engagement of the cam
and follower) over the whole cycle defines the duty cycle. (c) The in-vitro hydraulic circuit to conduct the experiments. The
pressure and flow measurements are conducted at the location of the ‘sensor’. (d) The close-shot of the physical setup consisting
of the cam-follower design mounted on a stepper motor which is connected to the actuator. (e) the unstretched and stretched
states of the artificial aorta, made from silicon, in the hydraulic circuit. (f) The sample compliance curve for the artificial
phantom demonstrated in (e).

9.4.1 Physiological accuracy of the fabricated phantom models
Table 9.1 summarizes the measured hemodynamic parameters for the 6 fabricated artificial
aortic models. The dimension of the aorta is chosen based on the average physiological range
release
stretch
(a)
(b)
(c)
(d)
(e) (f)
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and generic geometry. The foot-to-foot method is used to compute the pulse wave velocity of
each phantom model. The characteristics impedance was calculated using 𝑍 𝑐 =
𝜌𝑐
𝐴 where 𝑐 is the
measured pulse wave velocity, 𝜌 is the fluid density, and 𝐴 is the cross-section area of the aorta
at the root. Aortic compliance is measured by adding the incremental volumes of the fluid and
measuring the corresponding incremental change in pressure. The process and procedure to
conduct these measurements are described in detail in the method section. The physiological
range for the measured parameters is also reported in Table 9.1 [213, 214] . Overall, results
demonstrate the ability of our in-house fabricated phantom models to reproduce wave dynamic
characteristics of the human aorta.
Table 9.1 Dynamical and physiological properties of the fabricated artificial aorta.
Artificial Aortas
Wave Speed, c
(m/s)
Aortic Compliance
(mL/mmHg)
Char. Impedance
(MPa.s/m
3
)
Material

Phantom Model 1 7.2 1.5 10.1 Latex
Phantom Model 2 10.3 1.2 14.5 Latex
Phantom Model 3 14.8 0.8 20.8 Latex
Phantom Model 4 22.7 0.3 31.9 Latex
Phantom Model 5 10.0 0.5 14.0 Silicon
Phantom Model 6 15.5 0.4 21.8 Silicon
* Physiological range for aortic wave speed is in the range of 5 to 24 m/s, for aortic compliance is in the range of 0.34 to
2.35 mL/mmHg, and the physiological range for characteristic impedance is in the range of 2.5 to 13 MPa.s/m
3
.
9.4.2 Sample hemodynamic waveforms
Fig. 9.2 presents various pressure and flow waveforms measured at the aortic root for
different values of the stretching frequency of 0.5, 1.0, 1.25, 1.5, and 2.0Hz (demonstrated in
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blue). These cases are measured for the aorta with pulse wave velocity of 10.3 m/s which is
within the normal physiological range (phantom model 2). The time-averaged values for the
measured pressure and flow are also reported on top of each plot. The baseline pressure of the
setup in the absence of the stretching is 21 mmHg. Results show that the pressure gradient during
one cycle is almost similar at different frequencies. However, the flow changes significantly in a
non-linear fashion as frequency increases. Data also suggest that the peak of the pressure
waveform happens at the constant time from the beginning of the oscillatory cycle (around 0.4 s
from the beginning of the cycle) independent of the stretching frequency. The dashed line in
these plots demonstrate the end of the stretching phase and start of the recoil. The duty cycle is
kept fixed to 1/3 in this study.
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Fig. 9.2 Measured hemodynamic waveforms at the aortic root for different values of the stretching frequency. The first
row presents the pressure waveforms measured with high fidelity Millar system. The average pressure during one cycle is
reported on top of each plot. The second row presents the flow waveforms measured with ultrasound Transonic flow system. The
average generated flow during one cycle is reported on top of each plot where positive flow represents the pumping mode and
negative flow represents the suction mode. The third and fourth row presents the derivative of the pressure and flow waveforms
respectively. The fifth row presents the wave intensity defined as the product of the time derivative of the pressure with the time
derivative of the velocity. Velocity data are computed by dividing the measured flow waveforms by the cross-section area of the
aorta.
Fig. 9.2 further presents the derivative of the measured pressure and flow waveforms
(demonstrated in red). Results suggest that the generated compression and expansion waves,
based on the dP/dt, follow the same pattern. Results also suggest that there is a shift in the
minimums and maximums of the flow waveform. The combined effect of these two derivatives
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can be observed in the wave intensity dI pattern of Fig. 9.2 as well (demonstrated in black).
Wave intensity is defined as the power per unit cross-sectional area of fluid-filled tube due to
fluid pressure 𝑃 and average cross-sectional flow velocity 𝑈 . Mathematically speaking, Wave
intensity (dI) is computed as the product of the change in pressure (d𝑃 ) times the change in
velocity (d𝑈 ) during a small interval. Wave intensity patterns determine both the direction and
intensity of arterial wave propagation at any time instance during an oscillatory cycle. For
example, a d𝐼 >0 at a fixed time during the cycle indicates that forward waves are dominant at
that moment. Conversely, if d𝐼 <0, backward waves are dominant.
9.4.3 Effect of frequency on the longitudinal wave pumping
Fig. 9.3(a) presents the flow direction inside the mounted silicon aorta (phantom model 5)
with pulse wave velocity of 10m/s at two different frequencies; frequency of 1Hz which leads to
the suction (negative flow measured by the flow meter), and frequency of 1.5Hz which leads to
the pumping (positive flow measured by the flow meter). Results show that the net flow in the
aorta due to the wave pumping can be created in both directions depending on the stretching
frequency. Fig. 9.3(b) and 9.3(c) present mean measured flow 𝑄̅
averaged over one period 𝑇 as a
function of frequency at different values of aortic stiffness (measured by pulse wave speeds).
Fig. 9.3(b) depicts the results for three aortas with smaller pulse wave speed (phantom models 1,
2, 5) and Fig. 9.3(c). demonstrates the results for the three aortas with larger wave speed
(phantom models 3, 4, 6). Wave speed in the aorta depends on its material characteristics as well
as thickness and is one of the major determinants of the wave dynamics in the compliant aortas.
Results of Fig. 9.3 shows that for the aorta with the same pulse wave speed, the flow-frequency
relation is almost identical.

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Fig. 9.3 Flow visualization due to the aortic stretch and recoil as well as flow-frequency behavior. (a) The figures on the top
panel demonstrate the generated flow direction due to the aortic stretch and recoil, excited at the frequency of 1.5 Hz at various
snapshots in time. The installed silicon aorta is the phantom model 6. The figures on the bottom panel present the flow direction
for the same aorta which is generated due to the aortic stretch and recoil, excited at the frequency of 1 Hz. Results suggest the
bidirectionality of the generated flow depending on the excitation frequency. (b) Measured mean generated flow of the aortic
wave pumping against the excitation frequency for different levels of pulse wave velocity of 7.2, 10.0, and 10.3 m/s. (c)
Measured mean generated flow of the aortic wave pumping against the excitation frequency for different levels of pulse wave
velocity of 14.8, 15.5, and 22.7 m/s.

Pumping
Suction
(a)
(b) (c)
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9.4.4 Effect of aortic stiffness on the longitudinal wave pumping
Fig. 9.4 presents 𝑄̅
as a function of frequency for small, moderate, and large aortic wave
speed (a measure of stiffness). Results clearly show that both the pattern and the amplitude of the
generated flow due to the longitudinal stretching of the aorta depends on the measured wave
speed.

Fig. 9.4 Flow-frequency behavior at different levels of aortic stiffness measured by wave speed. This figure shows that both
direction and magnitude of the net flow can change as tube stiffness changes. Results suggest that at some stretching frequencies
(e.g., 1Hz) different tube stiffness can produce net flow in the opposite direction.

9.4.5 Effect of aortic stretch on the longitudinal wave pumping
Fig. 9.5(a) and 9.5(b) present 𝑄̅
as a function of frequency for different values of the
longitudinal stretch 𝑙 max
at two levels of aortic pulse wave velocity of 10.3 and 22.7 m/s,
respectively. Fig. 9.5(c) and 9.5(d) demonstrate 𝑄̅
as a function of stretching amplitude for the
same aortas. Stretch amplitude are modified by installing cams of different sizes as shown in Fig.
9.1. Results demonstrate a non-linear trend toward increased 𝑄̅
as the stretching amplitude
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increases. Results further show insignificant generated flow at lower values of the stretching
amplitude.

Fig. 9.5 The effect of stretching amplitude on flow-frequency pattern. (a) and (b) measured mean generated flow of the aortic
wave pumping against the excitation frequency for different levels of stretching amplitude of 0.5, 1.0, 1.5, and 2.0 cm for two
phantom models with aortic wave speed of 10.3 and 22.7m/s respectively. (c) and (d) the flow-stretching relation for two
phantom models with aortic wave speed of 10.3 and 22.7m/s respectively. For both models, it follows a parabolic-like curve.

9.4.6 Wave intensity analysis
To detect the flow driving mechanism, we performed the wave intensity analysis. Fig. 9.6(a)
and 9.6(b) demonstrate the pattern of the wave intensity derived from the pressure and flow
measurements for the aortas with different pulse wave velocities at the frequency of the 2Hz.
This frequency is chosen since the direction of the generated flow for all the aorta is the same
(a) (b)
(c) (d)
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(positive, pumping mode). Fig. 9.6(e) and 9.6(f) demonstrate the pattern of the wave intensity
derived from the pressure and flow measurements for the aortas with different pulse wave
velocities at the frequency of the 1Hz. Note that at this frequency, the direction of the generated
flow depends on the pulse wave speed, where in some of the phantoms the pumping exists and,
in the others, suction happens. Based on the wave intensity analysis and the derivative of the
measured pressure, driving forces can be divided into four types based on their characteristics;
forward running compression waves where d𝐼 >0 and d𝑃 /d𝑡 >0; forward running expansion
waves where d𝐼 >0 and d𝑃 /d𝑡 <0; backward running compression waves where d𝐼 <0 and d𝑃 /
d𝑡 >0; and backward running expansion waves where d𝐼 <0 and d𝑃 /d𝑡 <0. Fig. 9.6 further
presents the rate of the change in the measured pressure waveform (d𝑃 /d𝑡 ) for all the six
phantom models at stretching frequencies of 1 and 2Hz. The compression wavefronts in arteries
(d𝑃 /d𝑡 >0) are associated with an expansion in arterial diameter, while expansion wavefronts
(d𝑃 /d𝑡 <0) are associated with a reduction in arterial diameter.
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Fig. 9.6 Wave analysis on the propagated waves in the vasculature due to the aortic stretch and recoil. Wave intensity
pattern at the stretching frequency of 1Hz in the phantom models with wave speed of (a) 7.2, 10.0 and 10.3 m/s, and (e) 14.8,
15.5, and 22.7 m/s. Wave intensity pattern at the stretching frequency of 2Hz in the phantom models with wave speed of (b) 7.2,
10.0 and 10.3 m/s, and (f) 14.8, 15.5, and 22.7 m/s. Corresponding to each wave intensity plot, the time derivative of the pressure
is plotted in (c), (d), (g), and (h). The wave intensity pattern along with the pressure derivative one can help understanding the
direction as well as the type of the propagated waves.
(a) (b)
(c) (d)
(e) (f)
(g) (h)
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9.5 Discussion
Our aim in this chapter was to examine the effect of ascending aortic longitudinal stretch on
wave pumping in the cardiovascular system. This aim was achieved by utilizing the unique
hydraulic aortic simulator. This in-vitro setup has physical, physiological, and dynamical
properties similar to the human aorta and it is suitable for isolating the effect of longitudinal
stretching [176]. A driving component was used to mimic the stretch-related dynamic mode of
the ascending aorta. Our results suggest that aortic longitudinal wave pumping alone can create a
significant flow even in the absence of any other pumping mechanism. The unique frequency
dependence of the net flow rate implies an impedance-driven flow. Results suggest that both the
direction and magnitude of the net flow depend strongly on wave dynamic characteristics
including the stretching frequency and tube wall characteristics. Depending on the wave
characteristic parameters, longitudinal wave pumping can create both positive (pumping) and
negative (suction) flow.
The presence of ascending aortic longitudinal stretch is supported by previous clinical
observations of substantial axial sinotubular junction displacement. Aortic stretch requires
significant force, and this force and the resulting displacement represent energy that is stored in
the elastic elements of the ascending aorta. During early diastole, the energy stored as aortic
stretch leads to aortic recoil, which pulls upward on the base of the heart. Since the LV apex is
fixed in location by pericardial attachments, aortic recoil facilitates lengthening of the LV in
early diastole and aids in early filling. In other words, the aortic longitudinal (axial) displacement
results in energy storage in its spring-like elements, that enhances early diastolic LV recoil and
provides a suction mechanism in the heart that facilitates diastolic filling. However, the impact of
the aortic longitudinal stretch and recoil on wave generation and pumping has not been
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investigated. This is due to the fact that there are inherent difficulties in studying the isolated
effects of aortic wave dynamics in in-vivo and preclinical settings. Our results in this study
showed for the first time that aortic stretch and recoil can create a wave pumping mechanism
(Fig. 9.2 and 9.3). This wave pumping mechanism has similar characteristics to an impedance
pump. The net flow generated by the wave pumping mechanism in our aortic system depends on
the stretching frequency, the tube′s compliance, and the stretching amplitude (Fig. 9.4 and 9.5).
Aorta is a compliant tube that acts as a conduit for propagation and reflection of the waves.
The wave dynamics in a compliant tube is a complex nonlinear phenomenon that includes wave
interactions and resonance [7–9]. Waves in compliant tubes can create a pumping effect as
observed in impedance pumps (Liebau pump) [7,8,10–13]. In the stretching-based longitudinal
aortic wave pumping, as the waves propagate along the tube wall and interact with each other,
the dynamic pressure difference between the two ends of the compliant tube is created (Fig. 9.2)
which in turns generate the net flow. The mean measured flow over a cardiac cycle 𝑄̅
is an
indicator of the bulk flow motion for the specific wave condition. The data presented in Fig. 9.3
suggest a non-linear flow-frequency relationship similar to the previously-observed trends in the
experimental and computational studies of the conventional impedance pumps [197-199]. Based
on the currently prevailing view, conventional impedance pumps operate based on the pressure
wave generation by periodic tube wall excitations [195, 198, 199]. To reach the maximum
pressure difference to generate flow, the system has to be excited near its natural resonant
frequencies [199]. However, there are some studies that reported significant generated flow at
frequencies far below the natural frequency [195]. Either way, the unique frequency dependence
of the net flow rate implies an impedance-driven flow.
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In-line with the results of the previous experiments which showed the direction of the flow
depends on the excitation frequency, the longitudinal aortic wave pumping based on the
ascending aorta stretch and recoil demonstrates both positive flow (pumping) and negative flow
(suction) depending on the stretching frequency. By modifying the stretching frequency and
fixing the travel time (constant tube length and wave speed), a net effect of the propagating and
reflected wave to create flow pumping changes (Fig. 9.3). The alteration in the material
characteristics and wave speed affect the intensity of the compression waves as well as the
interactions between the propagated and reflected waves. This results in a different flow-
frequency pattern as shown in Fig. 9.4. Data measured in three phantom models with pulse wave
speeds of 10.3, 15.5, and 22.7m/s are used to demonstrate different flow-frequency relationships.
This figure shows that both direction and magnitude of the net flow can change as tube stiffness
changes. Results suggest that at some stretching frequencies (e.g., 1Hz) different tube stiffness
can produce net flow in the opposite direction. Overall, any small changes in the wave
characteristics may significantly affect the wave dynamic in an elastic tube.
Fig. 9.5 has shown that the stretching amplitude significantly influences the net generated
flow due to the aortic stretch and recoil. For example, at the frequency of 2Hz, an increase in the
stretching amplitude from 1cm to 1.5cm results in the net flow increase from no flow to 0.4
L/min for the phantom model with pulse wave speed of 10.3m/s and increase from almost no
flow to 0.5 L/min for the phantom model with pulse wave speed of 22.7m/s, respectively. Under
the assumption of periodicity for the propagated pressure and flow waves inside the vasculature,
simple wave energy equation can be derived from the harmonics of the flow. A single harmonic
flow wave in an elastic tube can be written as 𝑞 (𝑥 ,𝑡 )=𝑞 0
𝑒 𝑖 (𝜔𝑡 −𝑘𝑥 )
, where 𝑞 0
is the (possibly
complex) amplitude of the flow, 𝜔 is the frequency of the oscillation, and k is the wavenumber.
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In the absence of the reflections in the system, 𝑞 0
can be approximated with the real value which
depends on the initial amplitude of the excitations and the fluid-filled elastic tube characteristics.
One can write 𝑞 0
=𝑍 .𝑠 where 𝑠 is the wave amplitudes and 𝑍 depends on the system and is a
function of the tube’s material properties, fluid density, fluid viscosity, and tubes diameter.
Therefore, the flow wave harmonic can be written as
𝑞 (𝑥 ,𝑡 )=𝑍 .𝑠 𝑒 𝑖 (𝜔𝑡 −𝑘𝑥 )
(9.1)
For any wave with a mechanical nature, the transmitted power via waves in an elastic tube is
proportional to the product of the spatial derivative and time derivative of the wave [215]. Using
Eq. (9.1), the time averaged power generated via aortic stretch and recoil is proportional to 𝑊 ̇ ∝

1
𝑇 ∫
𝜕𝑞
𝜕𝑡
.
𝜕𝑞
𝜕𝑥
d𝑡 𝑇 0
where T is the period of one cycle. Computing the integral based on (9.1) at a
fixed stretching frequency for the same aortic model and circulating fluid yields
𝑊 ̇ ∝ 𝑠 2
. (9.2)
It can be concluded from Eq. (9.2) that the energy carried by the wave is proportional to the
square of the wave amplitude. Similar non-linear parabolic relation for the flow-stretching curves
can be observed in Fig. 9.6. Results suggest that for maximizing the energy harvesting for the
flow pumping from the aortic stretch and recoil mechanism, after tuning the frequency, the
stretching amplitude can significantly enhance the generated flow.
Wave intensity analysis is a well-established method for quantifying the energy carried in
arterial waves, providing valuable information about cardiovascular function [69-71]. Results
presented in Fig. 9.6 suggests that for positive pumping (2Hz), the pattern of the wave intensity
starts with the forward running waves (d𝐼 >0). Since for these cases d𝑃 /d𝑡 <0 (Fig. 9.6), these
212

waves are expansion waves which are created due to the stretching of the aortic root. The waves
propagate forward through the compliant aorta where they are reflected by the mismatch in
impedance in the bifurcations with small arteries. This reflection lead to the backward running
compression waves or reflected waves. After these two dominant waves, the sequence of re-
reflections happens in the vasculature where their superposition determines the wave intensity.
While the pattern of wave intensity for the first two dominant peaks at the beginning of the cycle
are similar between different phantom models, as time passes the interaction of the forward
running and reflected waves got affected by the pulse speed (even at the same stretching
frequency) and therefore, the pattern changes among different phantom models. Note that the
amplitude of the first two peaks is more significant than the others in all the phantom models. At
frequency of the 1Hz, the first two dominant forward running and reflected waves are also
present. However, since the cardiac cycle is longer, wave trapping, which is a result of
reflections a re-reflections, become more apparent and show itself as low-amplitude oscillations
in the pattern of the wave intensity. Another important observation from the data in Fig. 9.6 at
frequency of 1Hz is related to the pattern of wave intensity for the phantom model with pulse
wave speed of 15.5m/s. Flow measurements for this phantom model suggested suction mode
with net negative flow at frequency of 1 Hz (Fig. 9.4). The wave intensity pattern suggests that
the amplitude of the first two peaks is not significant for this phantom. This means that the
interplay of the wave state conditions at frequency of 1Hz could not generate strong forward
running expansion wave and flow direction may get dominated by backward running
compression waves which ultimately generates the negative flow (suction mode).
213

9.6 Conclusion
Overall, our findings in this chapter indicates that aortic stretch and recoil creates a pumping
effect in the cardiovascular. We conducted the wave analysis to understand the underlying
mechanisms of this effect. The aortic stretch and recoil can generate net flow in both positive
(pumping mode) and negative (suction mode) directions depending on the state of the wave
dynamics. In the pumping mode, the generated flow due to this aortic wave pumping can assist
the left ventricle to reduce the workload. Taking advantage of this mechanism can be of
particular interest in systemic diseases such as heart failure, where the pumping ability of the
heart is impaired.

214

CHAPTER 10 : Role of aortic longitudinal wave pumping on heart-brain
hemodynamic coupling
10.1 Chapter abstract
In this chapter, we investigate the impact of longitudinal dynamic mode of the aorta on the
brain circulation. We use the same setup as the previous chapter and measure the pressure and
flow in the left common carotid. While the importance of the aortic hemodynamics in regulating
the amount of pressure and flow pulsatility transmitted to the cerebral vasculature from the heart
is well-accepted, there is an unmet need to study how changes in stretch-related aortic dynamics
affect the pulsatile flow transmission to the brain The findings of this aim are helpful to identify
therapeutic strategies for intervention in heart-failure induced brain injured population based on
aortic biomechanics optimization.
10.2 Introduction
It has been recently proposed that heart failure (HF) is a risk factor for Alzheimer’s
disease (AD) [2]. The current clinical hypothesis is that the decreased cerebral blood flow due to
the HF may contribute to the dysfunction of the neurovascular unit and leads to the impaired
clearance of amyloid beta [2]. AD and HF often occur together and therefore increase the cost of
care and health resource utilization [13]. In 2017 alone, it is estimated that there were 5.7 million
Americans with Alzheimer’s disease (AD) which was associated with a cost of billion
dollars for care and lost productivity for patients and their caregivers [1]. Although previous
studies have attempted to elucidate the complex relationship between HF and AD, these studies
do not adequately address the hemodynamic couplings at LV-aortic-brain interfaces.
Importantly, the dynamic behavior of the stretch-related wave pumping mechanism on
215

transferring blood to the cerebral vasculature in the HF population have not been investigated.
There is an essential need for a well-designed basic quantitative study to investigate the
association between longitudinal stretch of the aorta in HF and the transmission of pulsatile
energy into the cerebral circulation.
The focus of this chapter is on understanding the effect of stretch-related wave pumping
mechanisms on volume blood flow to the brain. We will employ a physiologically accurate
experimental model to simulate the flow behavior inside the aorta and quantify the flow
transmission to the brain. There are significant data supporting the presence and importance of
brain injury in the HF population due to the reduced cardiac output (CO) [10]. While under
normal condition, the level of cerebral blood flow (CBF) maintained almost constant due to the
central nervous autoregulatory system, contemporary data suggested that in HF population the
level of CBF is jeopardized due to the reduced CO [11, 216]. Clinical studies have shown that
decreased flow to the brain affects lacunes, microinfarcts, white matter hyperintensities (WMH),
(sub)cortical atrophy, and white matter/gray matter integrity [8, 17]. In addition, recent studies
showed that the changes in aortic dynamic modes have a significant impact on the transmission
of potentially deleterious pulsatile energy into the cerebral microcirculation [51]. Therefore, it is
of a great clinical importance to understand the mechanism associating aortic stretch and recoil
(as a major dynamic mode of the aorta) with the flow transmission to the brain.
10.3 Methods
The employed in-vitro setup to collect hemodynamic data in this chapter is similar to the
one used in chapter 9. The cerebral pressure and flow data are collected at one of the cerebral
branches connected to the aorta. The details to conduct the measurements as well as the
216

fabrication process for the artificial aortas are described in Chapter 9. Fig. 10.1 presents the
schematic of the setup and the location of the sensors to do the measurements in this chapter.

Fig 10.1 Schematic representation of the in-vitro hydraulic circuit to conduct the experiments. The pressure and flow
measurements are conducted at the location of the ‘sensor’. For the cererbral flow, data are collected at the carotid artery.
10.4 Results and discussion
Fig. 10.2 presents the sample pressure and flow data collected in the left common carotid at
three different sample frequencies. Results suggest that while the average pressure during a
cardiac cycle does not change significantly, the alterations in the average flow amplitudes are
noticeable. These data are collected for the stretching of the 2 cm at the aortic root.
217

Fig 10.2 Measured pressure and flow waveforms at the carotid artery for different values of the stretching frequency. The
first row presents the flow waveforms and the average flow during one cycle is reported on top of each plot. The second row
presents the pressure waveforms and the average pressure during one cycle is reported on top of each plot.
Fig. 10.3 presents the mean measured flow 𝑄̅
during one cycle as a function of frequency at
different values of aortic pulse wave speeds. Findings suggest that the transmitted flow to the
brain due to the aortic stretching depends on the stretching frequency, which is translated into the
heart rate in the physiological settings. This frequency dependence is the common feature of
wave pumping mechanisms. Results suggest that the amplitude of the generated flow can be
significant (as high as 300 mL/min for stiff aorta). Fig. 10.3(a) depicts the results for three aortas
with smaller pulse wave speeds (7.2, 10.0 and 10.3 m/s) and Fig. 10.3(b) demonstrates the results
for the three aortas with larger wave speeds (14.8, 15.5 and 22.7m/s). As discusses previously in
Chapters 8 and 9, wave speed is one of the major determinants of the wave condition and
218

changing it, even at the fixed frequency would affect the net impact of propagating and reflected
waves which ultimately determines the flow amplitude and direction. Our results here suggest
that the transmitted flow to the brain is dominated by the aortic wave speed.

Fig 10.3 Flow-frequency relation at the carotid artery at different levels of aortic stiffness. Measured mean generated flow
at the carotid artery due to the aortic wave pumping against the excitation frequency for different levels of pulse wave velocity of
(a) 7.2, 10.0, and 10.3 m/s, and (b) 14.8, 15.5, and 22.7 m/s
Fig. 10.4 demonstrates the mean measured flow 𝑄̅
at the carotid artery during one cycle as a
function of stretching amplitude of the aortic root. Results suggest a non-linear trend toward the
increased flow at the carotid artery as stretching increases. This flow-stretching pattern is in
consistency with the measured data at the aortic root.
(a) (b)
219

Fig 10.4 Flow-frequency relation at the carotid artery. (a) Flow-stretching relation for a phantom model with aortic wave
speed of 22.7m/s. (b) measured mean generated flow inside the carotid artery due to the aortic wave pumping against the
excitation frequency for different levels of stretching amplitude of 0.5, 1.0, 1.5, and 2.0 cm for a phantom models with aortic
wave speed of 22.7m/s

Fig. 10.5 presents the flow direction inside the mounted silicon aorta with pulse wave
velocity of 10m/s at two different frequencies; frequency of 1Hz which leads to the suction
(negative flow measured by the flow meter) of the flow from the brain, and frequency of 1.5Hz
which leads to the pumping (positive flow measured by the flow meter) flow to the brain. Results
show that the net flow in the carotid due to the aortic longitudinal wave pumping can be in both
directions depending on the stretching frequency. The dependency of both the amplitude and the
direction of the generated flow in the carotid artery due to the longitudinal wave pumping of the
aorta is the result of the wave propagation and reflection which has been extensively discussed in
Chapters 8 and 9.
(a) (b)
220

Fig 10.5 Snapshots of the flow over time at the carotid artery due to the aortic stretch and recoil. Figures on the top panel
demonstrate the generated flow direction inside the carotid artery due to the aortic stretch and recoil at the frequency of 1.5 Hz at
various snapshots in time (pumping mode). The figures on the bottom panel present the flow direction for the same aorta inside
the carotid artery at the frequency of 1 Hz (suction mode).

10.5 Conclusion
In this chapter, we showed that the wave pumping inside the vasculature due to the aortic
stretch and recoil can also create pumping and suction in the carotid artery. The amplitude of the
generated flow is significant, and it has a strong frequency-dependence. It also depends on the
wave characteristics of the aorta including aortic stiffness (measured by pulse wave speed).
These findings suggest longitudinal aortic wave pumping has a major impact on the cerebral
blood flow and can be used as mechanism in designing new assist devices for patients suffering
from vascular brain damage.

Pumping ode (1.5Hz)
Suction ode (1.0Hz)
221

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Asset Metadata

Creator Aghilinejad, Arian (author)

Core Title Impact of aortic dynamic modes on heart and brain hemodynamics for advanced diagnostics and therapeutics

Contributor Electronically uploaded by the author (provenance)

School Andrew and Erna Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Degree Conferral Date 2023-05

Publication Date 03/23/2025

Defense Date 03/09/2023

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

Tag biomechanics,brain disorder,cardiovascular engineering,fluid dynamics,heart failure,hemodynamics,non-invasive diagnostics,OAI-PMH Harvest,vascular dementia

Format theses (aat)

Language English

Advisor Pahlevan, Niema (committee chair), Newton, Paul (committee member), Sadhal, Satwindar (committee member), Wood, John (committee member)

Creator Email aghiline@usc.edu,arian.aghili1298@gmail.com

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Legacy Identifier etd-Aghilineja-11516

Document Type Dissertation

Format theses (aat)

Rights Aghilinejad, Arian

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Type texts

Source 20230324-usctheses-batch-1011 (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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Abstract (if available)

Abstract This dissertation focuses on the role of aortic biomechanics in the heart-aorta-brain system. Combination of in-vitro experimentations, numerical simulations, and machine learning approaches are employed to achieve the aims of this thesis. We uncovered a new pumping mechanism in the arterial system based on the ascending aortic longitudinal stretch and recoil due to the left ventricle systolic long-axis shortening. Our findings indicated that stretching-based aortic wave pumping generates significant flow which can assist left ventricle. Furthermore, results show that this mechanism has a major impact on the cerebral blood flow. This complex pumping effect is a function of the wave dynamic conditions that are mainly dictated by the heart rate and the wave speed inside the aorta. We also demonstrated that wave dynamics in the aorta dominate the pulsatile hemodynamics of the brain. Findings indicated the existence of an optimum wave state, near normal human heart rate, where destructive pulsatile energy transmission to the brain is minimized. Based on the impact of aortic biomechanics on cardiovascular and cerebrovascular system, we developed a hybrid Fourier decomposition-machine learning algorithm that facilitates wave energy calculation via single non-invasive pressure measurement. Our method was tested in a large clinical cohort and could successfully capture the wave energy features of arterial system. The findings of this thesis are expected to provide valuable insights regarding the impaired heart-aorta-brain system and can be an initial step towards the development of diagnostic tools and assist devices for patients suffering from heart diseases and vascular brain damage.