University of Southern California Dissertations and Theses

Modeling and simulation of circulating tumor cells in flow. Part I: Low-dimensional deformation models for circulating tumor cells in flow; Part II: Procoagulant circulating tumor cells in flow

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Modeling and simulation of circulating tumor cells in flow. Part I: Low-dimensional deformation models for circulating tumor cells in flow; Part II: Procoagulant circulating tumor cells in flow

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Content MODELING AND SIMULATION OF CIRCULATING TUMOR CELLS IN FLOW Part I - Low-dimensional deformation models for circulating tumor cells in ow Part II - Procoagulant circulating tumor cells in ow by Angela Meeyoun Lee A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements of the Degree DOCTOR OF PHILOSOPHY (AEROSPACE ENGINEERING) May 16, 2014 Copyright 2014 Angela Meeyoun Lee Dedication To my sons Elliott and Brandon, for being my greatest sources of inspiration. ii Acknowledgements The accomplishment of this dissertation would not have been possible without a whole host of people who have contributed to its completion in many ways. First, I would like to thank my advisor, Paul Newton, for his guidance and support throughout the entire thesis process. His wisdom and warmth have inspired me to achieve my academic and personal goals, and his humbleness is a truly a testament to his intelligence. I would also like to acknowledge my thesis committee for their meaningful discussions and valuable feedback: Eva Kanso, for her wit and bright personality and for being a role model in how to enjoy academic success while raising two young boys; Paul Macklin, for his infectious enthusiasm for science and for giving me motivation to accomplish big things every time we talk; and a special thank you to Owen McCarty, for ying out from Portland to be a member of my committee. His energetic passion for the important things in life is inspiring, and I am grateful for our fruitful collaborations and his endless support of my work. There are many whom I befriended during my time as a doctoral student at USC. I would like to thank Sam Graves for oering words of encouragement (and chocolate) whenever I visit her oce, and for her constant support, both logistically and emotionally, through two pregnancies. I would like to thank my research group for sharing this experience of growth with me (in order of whom I met): Roxana Tiron, Fangxu Jing, Babak Oskouei, iii Stephen Liao, Jeremy Mason, Andrew Tchieu, Prabu Sellappan, Kevin Phillips (OHSU), Garth Tormoen (OHSU), Brian Hurt, Jerey West, and Zaki Hasnain. I would also like to thank my few but valuable girlfriends in the AME department, Shanling Yang, Gauri Khanolkar, Okjoo Park, and Sydnie Lieb for our many enjoyable lunch dates. And I would like to acknowledge the ARCS Foundation for their generous nancial support of my studies as an ARCS Scholar for the past three years. My sincerest gratitude goes to my family for the constant love they poured upon me, especially my husband, Grant, for unconditionally loving me as I am; my two sons, Elliott and Brandon, for being a wonderful blessing and constantly giving me great joy; my parents, Young and Stella, for being there for me every single step of the way; my sister Diane and my brother Sam, for putting up with my idiosyncracies; my parents-in-law, Dong Ha and Kyung Ja, for their valuable academic advice throughout my graduate studies; my late grandfather, who showed me that the pursuit of knowledge and a career in education are fullling and rewarding; my best friends, Stella and Jane, for being my lifelong sisters in Christ; and my church family for their prayers for wisdom and strength that extend beyond my own abilities. Finally, all glory and honor and praise goes to the living God, who gives value to my work and a higher purpose to my life. iv Table of Contents Dedication ii Acknowledgements iv List of Figures vii List of Tables x Abstract xi Part I Low-dimensional deformation models for circulating cancer cells in ow 1 Chapter 1 Introduction 2 Chapter 2 The cell deformation and release experiment 7 2.1 MDA-MB-231 cell deformation and release experiment . . . . . . . . . . . . 7 2.2 Active shape model and principal components of deformation . . . . . . . . . 8 Chapter 3 The low-dimensional deformation model and simulation results 12 3.1 The mathematical modeling assumptions . . . . . . . . . . . . . . . . . . . . 12 3.1.1 The cell surface model and constitutive assumption . . . . . . . . . . 12 3.1.2 Fluid ow computation . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1.3 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 The cell deformation simulation . . . . . . . . . . . . . . . . . . . . . . . . . 21 Part II Procoagulant circulating tumor cells in ow 31 Chapter 4 Introduction 32 4.1 Blood coagulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 Previous models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3 Description of the biology of thrombosis . . . . . . . . . . . . . . . . . . . . 36 v Chapter 5 The clot formation model 43 5.1 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.3 Mathematical infrastructure and Green's functions approach . . . . . . . . . 47 5.3.1 Fluid dynamics and concentration eld equations . . . . . . . . . . . 47 5.3.2 The Green's function approach . . . . . . . . . . . . . . . . . . . . . 48 5.3.3 Concentration eld gradient tracking diagnostics . . . . . . . . . . . . 51 5.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.4.1 Two-dimensional concentration elds . . . . . . . . . . . . . . . . . . 52 5.4.2 Three-dimensional concentration elds . . . . . . . . . . . . . . . . . 54 5.4.3 Gradient tracking results . . . . . . . . . . . . . . . . . . . . . . . . . 57 Chapter 6 The brin formation model 63 6.1 Experimental methods and results . . . . . . . . . . . . . . . . . . . . . . . . 63 6.2 Model in a closed circular domain . . . . . . . . . . . . . . . . . . . . . . . . 64 Chapter 7 The computational uid dynamics models 70 7.1 Development of the model from patient samples to computational simulation 71 7.2 Description of the mathematics and multiphysics used in COMSOL models . 72 7.2.1 Laminar two-phase ow with moving mesh . . . . . . . . . . . . . . . 74 7.2.2 Transport of diluted species . . . . . . . . . . . . . . . . . . . . . . . 76 7.2.3 Particle tracking for uid ow . . . . . . . . . . . . . . . . . . . . . . 78 7.2.4 The implicit time-dependent solver algorithm . . . . . . . . . . . . . 79 7.3 Modeling building process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 7.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.4.1 Breast cancer single cell in a channel venule . . . . . . . . . . . . . . 84 7.4.2 Colon cancer single cell and cluster of two cells in a branching venule 87 7.4.3 Lung cancer cluster in a branching venule . . . . . . . . . . . . . . . 94 Chapter 8 Discussion and future directions 99 8.1 Future work for deformation of cancer cells in ow (Part I) . . . . . . . . . . 99 8.2 Future work for procoagulant circulating tumor cells in ow (Part II) . . . . 101 Bibliography 106 vi List of Figures 1.1 Schematic of intravasation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Images of cell-release experiment . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Adhesion by ligands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Principal components of deformation . . . . . . . . . . . . . . . . . . . . . 11 3.1 Gaussian cell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 The training experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Cell surface parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4 x-z plane of the computational domain . . . . . . . . . . . . . . . . . . . . 26 3.5 Normal force on cell surface . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.6 Simulated forces acting on the cell surface . . . . . . . . . . . . . . . . . . 28 3.7 Pressure gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.8 Cell surface parameters for simulated run . . . . . . . . . . . . . . . . . . . 30 4.1 Schematic of coagulation reactions . . . . . . . . . . . . . . . . . . . . . . . 34 4.2 Circulating tissue factor in blood vessel . . . . . . . . . . . . . . . . . . . . 35 4.3 Schematic of deep vein thrombosis . . . . . . . . . . . . . . . . . . . . . . . 38 4.4 Schematic of artery vs. vein . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.1 Experimental setup for blood occlusion . . . . . . . . . . . . . . . . . . . . 44 5.2 Experimental results comparing CTC burden with occlusion time . . . . . 46 5.3 Schematic for atmospheric dispersion modeling . . . . . . . . . . . . . . . . 50 5.4 Schematic of gradient tracker . . . . . . . . . . . . . . . . . . . . . . . . . . 52 vii 5.5 2D simulation results for n = 100 cells . . . . . . . . . . . . . . . . . . . . . 53 5.6 Long-time 2D simulation results for n = 100 cells . . . . . . . . . . . . . . 55 5.7 Computational domain for 3D model of diusing cells . . . . . . . . . . . . 56 5.8 3D simulation results for n = 4 cells . . . . . . . . . . . . . . . . . . . . . . 59 5.9 Initial positions and concentration levels for 3D simulation . . . . . . . . . 60 5.10 3D simulation results for n = 100 cells . . . . . . . . . . . . . . . . . . . . . 61 5.11 Chemical eld gradient tracking simulation . . . . . . . . . . . . . . . . . . 62 6.1 Single cell clotting assay experiment . . . . . . . . . . . . . . . . . . . . . . 64 6.2 2D simulation of CTCs in a circular blood vessel with no blood ow . . . . 66 6.3 Fibrinogen bridge simulation with two cells near center of vessel . . . . . . 68 6.4 Fibrinogen bridge simulation with two cells near circular wall of vessel . . . 69 7.1 Development from patient sample to computational simulation . . . . . . . 73 7.2 Model building: Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 7.3 Model building: Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.4 Model building: Step 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.5 Model building: Step 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.6 Model building: Step 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.7 Model building: Step 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.8 Simulation results for the breast cancer model . . . . . . . . . . . . . . . . 86 7.9 Colon cancer model development from image to mesh . . . . . . . . . . . . 88 7.10 Simulation results for the colon cancer model . . . . . . . . . . . . . . . . . 89 7.11 Particle tracking simulation for the colon cancer model . . . . . . . . . . . 91 7.12 Wake structure formation of the chemical thrombin eld in the colon cancer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.13 Lung cancer model development from image to mesh . . . . . . . . . . . . 95 viii 7.14 Simulation results for the lung cancer model . . . . . . . . . . . . . . . . . 97 7.15 Particle tracking simulation for the lung cancer model . . . . . . . . . . . . 98 8.1 Flow diagram of modeling procedure for cell deformation model . . . . . . 100 8.2 Experimental setup of cancer cell passing through micro uidic channel . . . 105 ix List of Tables 7.1 Components in the bloodstream with their densities. . . . . . . . . . . . . . 71 7.2 Velocities of vessels of dierent sizes in the venous system. . . . . . . . . . . 75 7.3 Coagulation factors with their respective diusion coecients, D. . . . . . . 77 x Abstract In this thesis, we mathematically model and computationally simulate several aspects asso- ciated with the dynamics of circulating tumor cells in the bloodstream. We focus on physical processes that initiate cancer metastasis, such as intravasation and the subsequent diusion of thrombin by the expression of tissue factor (TF) on the surface of the circulating tumor cells that are of epithelial origin. In Part I, we develop a low-dimensional parametric deformation model of a cancer cell under shear ow. The surface deformation of MDA-MB-213 cells is imaged using DIC microscopy imaging techniques until the cell releases into the ow. We post-process the time sequence of images using an Active Shape Model (ASM) to obtain the principal components of deformation, which are then used as parameters in an empirical constitutive equation to model the cell deformations as a function of the uid normal and shear forces imparted. The cell surface is modeled as a 2D Gaussian interface with three active parameters: H (height), x (x-width), and y (y-width). Fluid forces are calculated on the cell surface by discretizing the surface with regularized Stokeslets, and the ow is driven by a stochastically uctuating pressure gradient. The Stokeslet strengths are obtained so that viscous boundary conditions are enforced on the surface of the cell and the surrounding plate. We show that the low- dimensional model is able to capture the principal deformations of the cell reasonably well and argue that Active Shape Models can be exploited further as a useful tool to bridge the gap between experiments, models, and numerical simulations in this biological setting. xi In Part II, we describe a mathematical and computational model for diusion-limited procoagulant circulating tumor cells (CTCs) in ow. We rst build a model based on an exact formulation of Green's function solutions for domains with a blood vessel wall and for closed domains. Time-dependent gradient trackers are used to highlight the result that concentration elds build up near boundaries (vessel walls), in regions surrounding the dif- fusing particles, and in complex time-dependent regions of the ow where elds associated with dierent particles overlap. Then, as a next step to deal with more complex blood vessel geometries and actual CTC shapes obtained from DIC images of uid biopsy samples, we use COMSOL Multiphysics and CellProler software to simulate initial congurations and various geometries relevant to venule sizes in the body. We develop CFD models based on nite element meshes of the imaged single cells and circulating tumor cell clusters. Our results indicate that the thrombin chemical elds diuse to and collect at the blood vessel walls, and that the domain geometry combined with the spatial distribution of CTCs in ow determine local thrombin concentrations. The nal chapter outlines what would be the next steps in the model developments of both Part I and Part II of the thesis. xii Part I Low-dimensional deformation models for circulating cancer cells in ow 1 Chapter 1 Introduction Portions of the work in Chapters 1, 2, and 3 were originally published by the American Institute of Physics in Physics of Fluids 2012; Volume 24, Article 081903. Reprinted with kind permission The ` uid phase' of cancer begins when cancer cells making up the primary tumor mass are released into the bloodstream [48]. Some of these circulating tumor cells (CTCs) will become the future `seeds' when they implant themselves at distant sites, and the deadly extravasation step in the metastatic cascade will commence [79]. The schematic associated with the main physical mechanisms involved in this early stage event is shown in Figure 1.1. Mechanical models for this intravasation step [41, 69] are important to develop, primarily for the purposes of gaining a fundamental understanding of the initiation of metastatic spread. The reality of developing a high-delity model of circulating tumor cell and cell cluster deformation and release requires knowledge of (i) the competing forces acting on the cell surface from the incoming blood ow which act to potentiate the separation of the CTC 2 from the primary tumor, and the ligand-receptor bonds which hold the cell in place [10, 73]; (ii) knowledge of the constitutive equation associated with the cell membrane [73, 78] and a ne scale discretization of the deformable cell surface for a high delity numerical simulation; and (iii) modeling assumptions on the (blood) ow eld, which is near the vessel wall, hence in the Stokes regime [22, 29, 46]. An example of free-standing computational models of cell deformation under ow are those of [42, 65] which focus on the `rolling' regime of 3D deformable cells near vessel walls. A very comprehensive overview of the biomechanical properties of cancer cells is given in [73] along with references therein. The primary goal of this thesis is not to develop and describe a free standing physics based model of circulating tumor cell deformation and release, but to show how a software tool called the `Active Shape Model' (ASM) [19] can be exploited for use in bridging the gap that often exists between experiment and model development in the context of cancer cell deformation. In general terms, active shape models are statistical models that can track the shape of an object as it dynamically deforms. The models need to be trained on a sequence of images (the `training' experiment) from which they learn the principal components of deformation of an object. In our context, we then use those trained principal components of deformation as inputs for a low-dimensional parametric deformation model of the cancer cell under ow. This allows us to compute the uid forces on a model cell surface and calibrate/tune the parameters in an empirical constitutive equation relating forces to response, so that the response matches the trained principal components obtained from the experiment. Although `active shape models', `active appearance models' [18], and `active 3 Figure 1.1: Schematic of intravasation Schematic diagram showing the early stage of cancer progression, called the `intravasation' stage. Primary tumor cells in contact with the bloodstream experience shear forces that can be strong enough to exceed the adhesion forces keeping them attached to the primary tumor. Upon release into the incoming ow, they become circulating tumor cells capable of spreading the disease to remote sites. contour models' [43] have been used in medical applications [20, 67], to our knowledge, their use in the context described here, i.e. to calibrate/tune parameters for use in an empirical constitutive equation, is new and potentially useful in situations where a rst- principle constitutive equation is not known. The main goals and steps of the procedures described in this thesis are laid out as follows: 4 1. In the physical experiment described, we cannot measure forces on the cell surface directly. We can only visualize the surface deformation/shape as a function of the external ow parameters and time (as shown in Figure 2.3). 2. For the numerical simulation, we do not have a good model for the constitutive equation for how the cell surface responds to forces, hence we cannot do a `stand-alone' numerical simulation of the uid-cell surface interaction and deformation. 3. Our goal is to obtain an `empirical' constitutive law, trained from an experimental run. Once this is obtained, we can use the `trained' model for Stokes ow numerical simulations in a more general setting. 4. To obtain the empirical constitutive force-response law, we make a `Hooke's law' (lin- ear) constitutive assumption. Then, we use the sequence of shape changes obtained from the training experiment as inputs to our Gaussian surface model (some generic shapes are shown in Figure 3.1). This allows us to compute the uid forces on the `low-dimensional' deforming Gaussian surface using a Stokes ow simulation with dis- tributed Stokeslets on the Gaussian surface. The outputs from the training run are shown in Figures 3.2 and 3.3. 5. We then use the `trained' empirical constitutive equation to carry out a new ow simulation, both deterministic and stochastic. The ow simulation is driven by a deterministic and stochastic pressure gradient. The outputs from the ow simulation using the trained model is shown in Figures 3.7 and 3.6. 5 These highlight the main steps which we describe in more detail. We start by describing the experiment which we use to train the model. 6 Chapter 2 The cell deformation and release experiment 2.1 MDA-MB-231 cell deformation and release exper- iment The \training" experiment proceeds as follows. MDA-MB-231 cells (donated from Dr. Tlsty at the University of California, San Francisco, CA) were plated onto glass coverslips and incubated at 37 o C and 5 CO2 for 24 hours. Glass coverlips with plated MDA-MB-231 cells were assembled onto a ow chamber tted with a silicon rubber gasket with a ow width of 0:25 cm and thickness of 0:005 in (Glycotech). See [7] for a general discussion of velocity elds for dierent ow chamber designs. The ow chamber was mounted on the stage of a Zeiss Axiovert 200M inverted microscope (Carl Zeiss). Dulbecco's Modied Eagle Medium (DMEM, Invitrogen) with 10% fetal bovine serum (FBS, Invitrogen) was perfused over MDA-MB-231 cells at a uid shear stress of 1 dyne=cm 2 . The ow rate was doubled every 10 minutes to achieve shear stresses of 1, 2, 4, 8, 16, 32dynes=cm 2 . Real-time dierential interference contrast (DIC) microscopy images were captured every 15 seconds 7 during perfusion with a 40; 0:75 NA lens using Slidebook software (Intelligent Imaging Innovation). A representative sequence of cell images under shear ow (obtained from 12 runs of the experiment) are shown in Figure 2.1. Flow is from left to right. At some point during each run, the cell releases and leaves behind the \ligand footprints" which kept it adhered to the slide. The ligand footprints are shown in Figure 2.2 in this case, 54 minutes into the run, 2 minutes after the cell releases. Figure 2.1: Images of cell-release experiment DIC images of a MDA-MB-231 adherent cell subjected to increasing shear ow from 0 to 32 dynes/cm 2 . The contact ring around the outer region of the cell `rues' during the run. Flow is from left to right. 2.2 Active shape model and principal components of deformation We post-process the sequence of images (roughly 200 images per run) using the open-source software tool `Active Shape Model' (ASM). The ASM algorithm [19] is trained from a time sequence of manually drawn contours used as training images. We use the DIC images from 8 Figure 2.2: Adhesion by ligands Ligand `footprints' immediately after cell release into the ow. Adhesion force of cell ligands to slide must be overcome by shear force from the uid-cell-surface contact for transition from tumor cell to circulating tumor cell to occur. the experiment. With each image in the time sequence, we manually mouse-click the outline of the outer contour of the cell. This process could easily be automated for use in a larger setting, using an `active contour' type of model [43], for example. The ASM algorithm then nds the main variations in the training data using Principal Component Analysis (PCA), which enables the model to automatically recognize if a contour is a possible or good object contour. For our purposes, we only use the two largest singular values that the model produces (in each frame) to obtain the elliptical shape of the outer contour as a function of time. The experiment was run 12 times and sequences of images for each run were retained. 9 We show an example of one of our post-processed sequences in Figure 2.3. In each case, the semi-major and semi-minor axes ( x , y ), which the ASM produces, are shown. Also shown is the average cell shape (in white) produced by the ASM algorithm (averaged over 10 frames spanning 2 minutes) during each of the 10 minute windows in which the pressure gradient is held xed. The black elliptical curves are the contours of the cell surface model described next. These sequences give rise to experimentally produced values for ( x , y ) in a time sequence. The cell height is not directly imaged/measured in these runs, hence, we make the assumption that cell volume is held constant which enables us to calculate height once we have the other two deformation parameters in the model. 10 1 dyne/cm 2 λ x λ y 2 dyne/cm 2 λ y λ x 4 dyne/cm 2 λ y λ x 8 dyne/cm 2 λ x λ y λ x λ y 16 dyne/cm 2 32 dyne/cm 2 λ y λ x Figure 2.3: Principal components of deformation Principal components of deformation ( x , y ) for 6 representative frames in a 52 min run. Time sequence shown: 5 min, 14 min, 25 min, 39 min, 43 min, 50 min. Cell releases into ow at 52 min. The white curve is the average cell shape obtained from the ASM algorithm during the 10 minute window in which the ow rate is xed. The black curves are the contours of the Gaussian cell surface model. Thick black curve is the closest elliptical contour to the white curve. Cell width is roughly 20 m across. Flow is from left to right. 11 Chapter 3 The low-dimensional deformation model and simulation results 3.1 The mathematical modeling assumptions We next describe how this experiment is used to tune the parameters in our low-dimensional model. 3.1.1 The cell surface model and constitutive assumption In light of the fact that the cell deformations, as shown in Figure 2.1, are relatively small (compared with the cell size), we use a parametrically deformable surface model for the cell which is of Gaussian shape [81]: h(x;y) =H exp[(a(xx 0 ) 2 + 2b(xx 0 )(yy 0 ) +c(yy 0 ) 2 )]; (3.1) 12 with parameters: a = cos 2 2 2 x + sin 2 2 2 y ; (3.2) b = sin 2 4 2 x + sin 2 4 2 y ; (3.3) c = sin 2 2 2 x + cos 2 2 2 y : (3.4) Here, the cell surface height is given by h(x;y), with peak cell height H, x-width x , y- width y , and the center of the cell marked by the coordinates (x 0 ;y 0 ). The cell orientation parameter is held xed during the simulation, although for more complex cell tracking experiments, both (x 0 ;y 0 ) and could vary. Contours of the cell surface are shown in Figure 2.3. Since the cell height is not directly available, we use the (approximately correct) assumption that the volume enclosed by the cell (above the plate) is constant as the cell deforms, which provides a constraint on the three parameters (H; x ; y ). Examples of the cell surface model in `z-stack' form (contour slices of the surface at various heights), with two representative parameter values are shown in Figure 3.1. In general terms, the 2D Gaussian surface we generate this way closely contours the 2D shape of the cell surface, but we do not yet know how to vary the parameters of the model (H; x ; y ) as a function of the uid forces produced on the cell surface. To link the parametric cell deformation model with the uid ow forces produced, we start with the constitutive assumptions x (f x ;f y ), y (f x ;f y ) relating cell deformation (response) to forces imparted in each of thex (streamwise) andy (spanwise) directions. In the experiment, 13 ï 10 0 10 ï 10 0 10 0 5 h (µm) x (µm) y (µm) 1 2 3 4 5 6 7 ï 10 0 10 ï 10 0 10 0 5 h (µm) x (µm) y (µm) 1 2 3 4 5 6 7 (a) = 0, x = 5, y = 5, H = 8; (b) = 0, x = 4, y = 6:25, H = 8; Figure 3.1: Gaussian cell model Cell model with two dierent parameterizations shown in `z-stack' form. The parameters x , y , andH vary, but are constrained so that the total volume above the glass slide inside the cell is held constant. the ow is nearly unidirectional (along thex-direction) so the deformation response is much stronger in this direction than in the cross- ow (y) direction. For this reason, @ x =@f y 0, and@ y =@f x 0, so we can use a simpler assumption x (f x ), y (f y ). The pressure gradient driving the ow is increased in steps, hence the corresponding forces imparted increase in steps as well. We therefore Taylor expand the deformation functions: x (f x ) = x (0) + 0 x (0)(f x ) +:::; (3.5) y (f y ) = y (0) + 0 y (0)(f y ) +:::; (3.6) 14 truncating the expansions after the linear terms in (f x ;f y ). This is what we call our `linear constitutive' assumption. Based on these equations, we set up an iterative scheme of the form: (n+1) x = (n) x + x f x (3.7) (n+1) y = (n) y + y f y ; (3.8) where x 0 x (0), y 0 y (0), and (n) x x (0), (n) y y (0). We call these equations the `linear force-response' equations. We then iterate these equations (from frame to frame) throughout the ow run. The variables (n+1) x and (n+1) y are the width variables (in frame n + 1) that we use in our Gaussian surface model, while (n) x and (n) y are the corresponding values before the iteration step (in framen). We obtain these from the principal components of deformation ( x , y ) from the ow experiment. The variables f x and f y are the x- component and y-component of the surface normal force on the cell that we numerically compute. From these equations, we obtain the linear parameters x and y at each iteration (the data points in Figure 3.2), which we then average over each 10 minute window (shown in Figure 3.2) while the deterministic component of the pressure dierence driving the ow is held at a xed nominal value. Since the ow is coming entirely from thex direction, and not the y direction, the magnitudes of the forces in these two directions are of dierent orders. Hence, the coecients x and y are of dierent orders of magnitude, which is what we expect. In addition, due to the uni-directional nature of the incoming ow (approximately 15 one-dimensional), our simple form for the constitutive equation captures the deformations reasonably well. The deformations of the cell surface parameters are shown in Figure 3.3, which depict how the cell height H varies throughout the experimental run. We have divided the time regime into a region to the left of the vertical dashed line (i.e. T < 15), and a region to the right (i.e. T > 15). The horizontal lines in these two regions depict the average cell height. Note the change in this height in these two regimes, the rst being fairly constant, followed by an abrupt attening out of the cell. Roughly speaking, this change in height follows from the fact that both x and y both increase, and the volume under the cell is held constant. In the rst regime (T < 15), the cell seems to be responding in a mechanically passive way to the ow forces. In the second regime (T > 15), the cell response seems to become active, as it is actively attempting to atten itself to the incoming ow. Various types of active responses have been discussed, documented, and modeled in the literature. For example, [38] discuss a cell's propensity to orient in such a way as to reduce the total force on its nucleus. Active nucleus movement in response to shear has also been reported in [49], while active cytoplasmic response to shear is reported in [50]. Whether or not the transition we see atT 15 is due to any of these eects is hard to pin down from these particular sets of experiments without more detailed nucleus and cytoplasmic visualization. 16 3.1.2 Fluid ow computation To obtain the forces on the cell surface (which are not measured directly in the experiment), we perform a stochastic uid ow simulation. Since the ow relevant to the cell release experiment is very near the glass plate, on which we assume viscous boundary conditions, the Reynolds number is small (Re =UL= << 1), whereU is a representative ow velocity, L is taken to be the ow chamber length, and is the uid kinematic viscosity of blood plasma. Therefore, as in [29], we assume the ow is governed by the Stokes equations and the continuity equation: ~ u = r p (3.9) r ~ u = 0; (3.10) wherer p is the pressure gradient driving the ow in the chamber, is the viscosity of the surrounding uid, whose value we take as that for blood plasma, and ~ u = (u ;v ) is the convective velocity of the uid. The denotes quantities that are dimensional, and unstarred quantities will denote dimensionless variables. Our rst task is to scale the dimensional variables in a way that is consistent with the experiment. We use the following scalings: x = lx;y = ly, p = Pp, and u = Uu = Ph 2 8L u. l = 10m is a representative length of the cell, h = 0:0127 cm andL = 5 cm are the height and length of the ow chamber, P = 10dyne=cm 2 is a representative pressure drop which drives the ow down the chamber across the cell surface, and = 0:9cP , which is the 17 value for the HBSS media used in the experiment (viscosity of blood is roughly = 3:2cP ). If we assume a parabolic ow prole [7]: u(y) = P 2 y(hy); (3.11) then the maximum velocity occurs at the centerliney =h=2 from which we take our velocity scale to be U = Ph 2 =8L. The dimensionless equations then become: 2 8 ~ u = rp (3.12) r~ u = 0: (3.13) The pressure conditions are then normalized to 1 (incoming) and 0 (outgoing), and the uid velocity at the plate y = 0 is ~ u(x;y = 0) = 0. = h=l is the dimensionless parameter relating the chamber height to the length of the cell, which in our experiment is roughly = 1:27. To compute the forces on the cell surface, we follow the approach described in [21], namely, we consider the dimensionless system (3.12), (3.13) augmented with an external force ~ F which accounts for the interaction between the uid ow and the cell membrane. Hence we use: 2 8 ~ u = rp ~ F (3.14) r~ u = 0: (3.15) 18 3.1.3 Numerical method We use the method of regularized Stokeslets described in [21] to simulate the ow in a chamber with one cell attached to the plate. A Stokeslet is a fundamental solution to the steady Stokes equations, and it represents the velocity due to a concentrated external force at a point in the uid. Smoothing the force over a small ball, instead of concentrated at a point as a Dirac delta function, produces regularized Stokeslets, and this resolves many of the numerical issues that arise from singularities in the Stokes equations. We discretize the force ~ F on the cell surface with a distribution of regularized Stokeslets [21, 36] located at the grid points ~ x i , i = 1;:::;N, where: ~ F (~ x) = N X i=1 ~ f i (i) (~ x~ x i ); (3.16) where ~ f i is the strength of the ith Stokeslet, ~ x i is its location, and (i) is a regularizing function which has the property: lim !0 (~ x) =(~ x); (3.17) where (~ x) is the usual Dirac delta function. We use the function: (~ x) = 2r 2 + 5 2 (r 2 + 2 ) 5=2 ; (3.18) 19 r 2 =k~ xk 2 , which is a standard regularization discussed in [21]. The Stokeslet `blob' velocity eld in R 3 is then given by: 2 8 ~ u(~ x) = ( ~ f i r)rB (~ x~ x i ) ~ f i G (~ x~ x i ); (3.19) where G (~ x) is the solution to: G = (~ x); (3.20) and B (~ x) is the solution to: B =G (~ x) (3.21) in innite space. G can be thought of as the regularized Green's function for the problem. More specically, we want to determine the forces along the plate and the surface of the cell. A two-dimensional slice of the computational domain is shown in Figure 3.4. The ow (left to right) is driven by a constant pressure gradient that has reached a steady state. The in ow and out ow velocity conditions are set to be the parabolic Poiseuille ow prole, and viscous no-slip boundary conditions are enforced at the walls of the chamber and along the bottom plate, including the cell surface. In the 3-D numerical model, the computational domain is set to be:L<x<L;L< y < L; 0 < z < W , where L = 40 is the length of the domain in the x and y directions. 20 W = 10 is the length of the domain inz direction. The grid spacing is determined bydx = 1. N = 81 81 11 = 72; 171 are the total number of Stokeslets used in the simulation. The regularization parameter,, that we use is one-fourth of the grid spacing. For the simulation, Stokeslets are placed at every grid point in the computational domain, and the grid spacing is uniform in all three dimensions. Figure 3.4 shows the xz slice of the computational domain. The parameters of the cell surface are: H, the maximum height of the cell; x , the width in the x-direction. We set the plate, which cuts o the Gaussian at the base, at a = 0:005H. The Stokeslets above the plane z =a represent the cell surface while those below represent the plate. The rest of the Stokeslets (not pictured in the schematic) lying above the Gaussian surface, represent the uid. To simulate the ow induced by the constant pressure gradient, a constant force (P; 0; 0) is imposed on the grid covering the interior of the chamber. The velocity due to this force is computed on the channel walls using the regularized Stokeslets solution for velocity. Then, the forces along the plate and along the cell surface are computed so that the boundary conditions are enforced. The forces on the cell surface, which we are more interested in, are plotted in Figure 3.5 and Figure 3.6. Details of how to use the regularized Stokeslets to compute forces on surfaces can be found in [1, 21]. 3.2 The cell deformation simulation The cell deformation simulation using the trained model proceeds as follows. We carry out both a deterministic and a stochastic simulation. For the stochastic simulation, we obtain 21 a nominal value of the pressure gradient chosen to match the experiment, then we add a random uctuation with amplitude taken to be 10% of the base value for the deterministic pressure. The random uctuation is taken to be a uniformly distributed random variable. This is shown in Figure 3.7. The forces on the model cell surface are computed throughout a ow simulation in which we double the pressure gradient in each 10 minute interval of time. The forces on the cell surface are computed using the regularized Stokeslet method, and the parameters x and y in the empirical constitutive equations (3.7), (3.8) are averaged across 10 minute intervals, taking a moving average in which a window of 20 minutes is used to perform the averages. The forces on the cell surface and the parametric deformations are shown in Figures 3.6 and 3.8, respectively. Figure 3.8 is particularly instructive and interesting with respect to how the cell height H varies throughout the run. We have divided the time regime into a region to the left of the vertical dashed line (i.e. T < 15), and a region to the right (i.e. T > 15). The horizontal lines in these two regions depict the average cell height. Note the change in this height in these two regimes, the rst being fairly constant, followed by an abrupt attening out of the cell. Roughly speaking, this change in height follows from the fact that both x and y both increase, and the volume under the cell is held constant. In the rst regime (T < 15), the cell seems to be responding in a mechanically passive way to the ow forces. In the second regime (T > 15), the cell response seems to become active, as it is actively attempting to atten itself to the incoming ow. Various types of active responses have been discussed, documented, and modeled in the literature. For example, [38] discuss a cell's propensity to 22 orient in such a way as to reduce the total force on its nucleus. Active nucleus movement in response to shear has also been reported in [49], while active cytoplasmic response to shear is reported in [50]. Whether or not the transition we see at T 15 minutes is due to any of these eects is hard to pin down from these particular sets of experiments without more detailed nucleus and cytoplasmic visualization. Figure 3.6 shows the simulated forces acting on the cell surface from both deterministic and stochastic runs. This gure should be compared with Figure 3.2(b), which are the forces from the training experiment. The force plots are qualitatively similar. Figure 3.8 shows the simulated parameter deformations through the ow simulation. In this gure, we show a deterministic simulation and a stochastic simulation, where the stochastic uctuations are taken to be 10% of the deterministic base ow. Comparisons of Figures 3.3 and 3.8 show that the parametric deformation of x , y , and H between the experiment and the low- dimensional model are trending in the same direction and are qualitatively similar, although quantitative details dier, due presumably to the stochastic component to our ow eld and the low-dimensionality of our model. 23 (a) 0 10 20 30 40 50 60 ï 0.2 ï 0.1 0 0.1 0.2 Time (min) _ x _ x mean _ x 0 10 20 30 40 50 60 ï 2 0 2 4 x 10 9 Time (min) _ y _ y mean _ y (b) 0 10 20 30 40 50 60 0 10 20 Time (min) f x 0 10 20 30 40 50 60 ï 5 0 5 x 10 ï 9 Time (min) f y 0 10 20 30 40 50 60 0 10 20 Time (min) f p Figure 3.2: The training experiment (a) Force-response parameters x , y as a function of time. The orders of magnitude are dierent because the ow is incoming in thex-direction, hence forces are much larger in this direction compared to the cross- owy-direction. (b) Forces acting on the cell surface. Force in x-direction: f x ; Force in y-direction: f y ; Peak normal force: f p . Note the dierence in magnitude of the forces in the x and y directions, due to the fact that the ow is incoming and outgoing along the x-direction. Force units are in dynes. 24 0 10 20 30 40 50 60 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 Time (min) Gaussian parameters m x m y H Figure 3.3: Cell surface parameters Deformation of the cell surface parameters x , y , H, as a function of time. To the left of the dashed line (T < 15),H remains relatively constant (horizontal lines denote the average height in the two regimes). To the right of the dashed line (T > 15), H abruptly attens out. Since there is no corresponding abrupt change in the ow, we view the regime T < 15 as a `passive mechanical response' regime, whereas for T > 15, the cell seems to be actively responding to the ow. 25 x z L -L H dx z = a W Stokeslets σ x Figure 3.4: x-z plane of the computational domain The 3D computational domain for our cell deformation model isL<x<L,L<y<L, and 0 < z < W , where L = 40, W = 10, dx = 1, and a = 0:005H. H is the maximum height of the cell surface, and x is the x-width parameter in the Gaussian cell surface shape. Regularized Stokeslets are placed at every grid point on the cell surface and in the uid domain. The Stokeslets above the plane z = a represent the cell surface, while those below the plane represent the plate. 26 (a) ï 20 ï 10 0 10 20 ï 20 ï 10 0 10 20 0 1 2 3 4 x y z ï 20 ï 15 ï 10 ï 5 0 5 10 15 20 f p (b) ï 20 ï 10 0 10 20 ï 20 ï 10 0 10 20 0 1 2 3 4 y x z ï 20 ï 15 ï 10 ï 5 0 5 10 15 20 Figure 3.5: Normal force on cell surface Normal force on cell surface computed by discretizing the surface with regularized Stokeslets whose strengths are chosen to enforce the viscous boundary conditions. (a) Front of cell (with respect to incoming ow). f p is the maximum normal force on the cell surface.; (b) Back of cell (with respect to incoming ow). Force units are in dynes. 27 0 10 20 30 40 50 60 0 10 20 Time (min) f x deterministic stochastic 0 10 20 30 40 50 60 ï 1 0 1 x 10 ï 8 Time (min) f y deterministic stochastic 0 10 20 30 40 50 60 0 10 20 Time (min) f p deterministic stochastic Figure 3.6: Simulated forces acting on the cell surface Force in x-direction: f x ; Force in y-direction: f y ; Peak normal force: f p . Note the dierence in magnitude of the forces in thex andy directions, due to the fact that the ow is incoming and outgoing along thex-direction. Force units are indynes. Plot should be compared with Figure 3.2(b). 28 0 10 20 30 40 50 60 0 1 2 3 4 5 6 7 8 9 Time (min) Pressure difference deterministic stochastic Figure 3.7: Pressure gradient Pressure dierence (in ow minus out ow) driving the ow as a function of time in the nu- merical simulation. Pressure dierence doubles every 10 minute interval of time. Stochastic uctuation is taken to be 10% of the base value. Units are in dynes=cm 2 . 29 0 10 20 30 40 50 60 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 Time (min) Gaussian parameters m x (det) m y (det) H (det) m x (stoch) m y (stoch) H (stoch) Figure 3.8: Cell surface parameters for simulated run Deformation of the cell surface parameters x , y ,H, as a function of time as obtained from the numerical simulation. Stochastic uctuations are taken to be 10% of the deterministic base ow. The plot should be compared with the experimental run shown in Figure 3.3. 30 Part II Procoagulant circulating tumor cells in ow 31 Chapter 4 Introduction Portions of the work in Chapters 4 and 5 were originally published by Frontiers in Oncology 2012; Volume 2, Article 108. Reprinted with kind permission Blood clotting, or thromboembolism, is the second highest cause of death in cancer pa- tients [45]. Unregulated coagulation is dangerous because blood clots that occur in the brain can lead to stroke, and clotting in the lungs can lead to pulmonary embolisms. Circulating tumor cells express tissue factor, which circulates in ow and initiates a coagulation cascade often seen in metastatic cancer patients [57]. Metastatic cancer is associated with increased levels of intravascular tissue factor and a high risk of venous thromboembolism [30]. Concen- tration elds of serine proteases, enzymes generated following the exposure of tissue factor, diuse to and from CTCs as they propagate through the bloodstream [16]. Our goal is to model and computationally simulate this process in simple settings. At this stage, chemical 32 reactions are not included in the model, and only the diusion of dierent species under ow is examined. 4.1 Blood coagulation Blood coagulation is a process of the hemostatic system, which also includes platelet aggre- gation and brinolysis. The main functions of the hemostatic system are to heal injured blood vessels and to prevent bleeding. The clotting interactions are extremely complex and require a delicate balance of hemostasis. If this balance is disrupted, the results can be very dangerous and even fatal, with one side of the imbalance resulting in persistent bleeding and hemorrhage and the other side giving rise to thrombosis (intravascular blood clotting) and possible vessel occlusion due to the thrombi (intravascular clots). Cancer metastasis and the circulating tumor cells that are the vehicles of metastasis can be a cause of thrombosis [62, 76]. Coagulation involves a network of tightly-regulated reactions of enzymes in the blood (Figure 4.1) [28, 32]. The initiation of coagulation by circulating tissue factor occurs on the surfaces of procoagulant cells (Figure 4.2) [34]. The kinetics of tissue-factor initiated enzyme activation are dependent on the physiological and chemical properties of TF-bearing surface and its interface with blood. Coagulation occurs in the presence of owing blood, and therefore it is strongly in uenced by the uid dynamics [56]. We aim to model and study the eects of ow on the diusion of tissue factor expressed by CTCs. 33 ATIII ATIII TFPI ATIII ADP Xa X IXa IX Xa Va Prothrombin THROMBIN Va VIIa Unactivated Platelet Fibrin TF TF VII VIIIa VIII V Va Va VII Fibrinogen Va:Xa TF:VIIa VIIIa VIII V Activated Platelet Subendothelium X VIIIa:IXa Figure 4.1: Schematic of coagulation reactions Schematic of coagulation reactions. Dashed magenta arrows show cellular or chemical acti- vation processes. Blue arrows indicate chemical transport in the uid or on a surface. Green segments with two arrowheads depict binding and unbinding from a surface. Rectangular boxes indicate surface-bound species. Solid black lines with open arrows show enzyme action in a forward direction, while dashed black lines with open arrows show feedback action of enzymes. Red disks indicate chemical inhibitors. 4.2 Previous models There have been numerous models that have examined parts of the hemostatic system. The previous models loosely fall under two categories: hydrodynamic modeling of the mechanics 34 Figure 4.2: Circulating tissue factor in blood vessel Circulating tissue factor (CTF) is expressed from the circulating tumor cells (CTCs). En- dothelial cells (EC) form the lining of the blood vessel. Blood ow is shown as a shear Poiseuille prole. Extracellular matrix and other TF-factor expressing cells lie outside of the blood vessel. The inset shows the spatial separation between the surface of two CTCs, and TF is one of the initial enzymes that starts the cascade of biochemical reactions. The end product, brin, forms the bers of the blood clot. of blood ow during coagulation, and the kinetics of coagulation biochemisty. In the former category, Dzwinel et al. used a discrete-particle model of blood dynamics in capillary blood vessels and concluded that aggregation of red blood cells in capillary vessels can be stimulated by depletion forces and hydrodynamic interactions [25]. Anand et al. incorporated the mechanics of ow with a growing blood clot by modeling both blood and clots as shear- thinning viscoelastic uids [3]. The model included reaction equations of prothrombinase and tenase, but they did not represent all of the reactions in the coagulation system. Bodnar and Sequira used a simplied version of Anand's model, in which the physical domain is one-dimensional, to simulate blood clotting in three dimensions [11]. 35 In the latter category, the Hockin-Mann reaction network model, which is a homogenous ordinary dierential equation (ODE) reaction scheme, accurately predicts the initiation times of blood coagulation at various levels of tissue factor [39]. Building upon the Hockin-Mann model, Chatterjee et al. modeled the kinetics of coagulation initiation, specically platelet activators [15]. Surprisingly, human blood was found to clot in vitro due to the activity of Factors XIIa and XIa, which proceeds in the absence of tissue factor. Fogelson et al. has developed an extensive model with both platelet aggregation under ow with coagulation biochemistry [26, 27]. His comprehensive model is the rst to couple the interactions of both phenomenons, and it shows the importance of platelet interaction with surface-bound chemicals and competition for binding sites during the maintenance and regulation of the coagulation cascade. Gregg further improved upon this model by accounting for the signicant growth of a thrombus, or blood clot, and the physical eects of this growth on local ow and transport of chemical enzymes [33, 51]. The previous models examine coagulation at a site of injury or a cut in the blood vessel, not the coagulation that is activated by circulating tumor cells in metastatic cancer patients. The model we propose focuses on the chemical concentration elds of tissue factor expressed from procoagulant cancer cells in blood ow, but it does not yet include chemical reaction equations. 4.3 Description of the biology of thrombosis Metastatic cancer accounts for the majority of deaths caused by cancer. Metastasis is be- lieved to result from tumor cells from a primary site, migrating towards and intravasating 36 into a blood vessel, navigating the blood circulation to arrive at a distant site whereby it arrests from the blood ow, extravasates and establishes a metastatic tumor site. The process of metastasis thereby exposes a tumor cell to a variety of new environments, and poses signicant physical challenges the tumor cell must overcome if it is to successfully metastasize. Deep vein thrombosis, or deep venous thrombosis (DVT), is the formation of a blood clot in a deep vein, predominantly in the legs (refer to Figure 4.3). The detachment of a large clot that travels to the lungs can result in a pulmonary embolism, which is a life-threatening med- ical condition. Metastatic cancer patients are especially at risk of developing this potential complication, since circulating tumor cells are known to increase the occurrence of clotting in blood vessels. Large blood clots in the lower extremities caused by circulating tumor cells typically occur in the venous system of the circulatory system (shown in Figure 4.4), as op- posed to the arterial system, because veins are low-pressure vessels that carry deoxygenated blood back to the heart, creating conditions that favor the formation of DVTs. Therefore, in the following chapters, we create models based on physiological blood parameters found in the venous system, such as blood vessel diameter and ow rate. The interactions between circulating tumor cells and blood coagulation proteins have not been fully characterized. Activation of the bloods coagulation system has been associated with cancer, particularly metastatic cancer, for centuries [30]. The exact mechanism(s) underlying the activation of blood coagulation in cancer remain ill-dened [30, 45]. Tumor cell expression of tissue factor (TF) has been associated with advancing stages of cancer 37 Figure 4.3: Schematic of deep vein thrombosis Deep vein thrombosis is the medical condition where a blood clot has been formed in a large vein, typically in the legs. Symptoms include pain and swelling as blood builds up behind the clot. progression, and has been shown to correlate with metastatic potential in vivo [2, 58, 59]. TF is a transmembrane glycoprotein that is normally expressed by cells outside of the blood vasculature. The exposure of blood to TF, as occurs in the event of a blood vessel injury, is a physiological initiator of coagulation [34, 61]. TF serves as the cell membrane receptor for and enzyme cofactor of coagulation factor VIIa (FVIIa). In complex, TF-FVIIa activates the extrinsic pathway of coagulation leading to the formation of thrombin which can then convert brinogen to brin in order to form a plug that stops bleeding at the injury site in order to maintain blood ow and volume. 38 Figure 4.4: Schematic of artery vs. vein Arteries and veins have dierent functions and dierent structures. Arteries carry oxygenated blood to various organs in the body, whereas veins carry deoxygenated blood back to the heart. Deep vein thrombosis occurs in the venous system, as opposed to the arterial system. In the context of a metastasizing tumor cell, a TF-expressing circulating tumor cell (CTC) may expose blood within an uninjured blood vessel to TF [8, 45, 62, 74, 76]. Levels of intravascular TF correlate with cancer progression and to some extent with the formation of pathological clots or thrombi in the veins of patients with cancer. Thrombosis, the formation of pathological thrombi, accounts for the second leading cause of death for patients with cancer and constitutes a signicant source of morbidity in these patients [76]. Anticoagulant measures taken after a thrombotic event are eective at reducing the formation of subsequent thrombi, but no current laboratory assay is capable of predicting which patients are at risk to develop thrombosis. The incidence of thrombosis is known to correlate with cancer type and 39 tissue of origin, suggesting that the cancerous cells themselves have a role in the formation of pathological thrombi [9]. In vitro, cancer cells are capable of independently initiating coagulation and clotting blood plasma. Similarly, functionally blocking TF on cancer cells prevents the cells ability to clot blood plasma. Therefore, mounting evidence suggests that cancer cell expressed TF is a likely culprit for the initiation of blood coagulation associated with cancer [8, 53, 54, 70, 80, 83]. The ability of TF to activate blood coagulation is dependent upon the presence of phos- pholipids, suggesting that only cell surface-expressed TF or cell membrane-derived TF bear- ing microvesicles are capable of activating coagulation [60]. This also indicates that the activation of coagulation by TF is essentially a surface phenomenon, requiring coagulation factors to transport from the blood to the surface-expressed TF in order to participate in co- agulation. In vitro, the tracking of soluble coagulation factors from bulk to a TF-expressing phospholipid surface is rate-limiting with respect to enzyme activation [31, 35, 56]. Further, TF activity is augmented in the presence of blood ow where convective transport sup- plants diusive transport as the dominant mode of transport for coagulation factors to TF. Taken together, the ability for a TF-expressing CTC to activate blood coagulation is likely dependent upon its spatio-temporal relationship with the blood. In vitro, the coagulation kinetics for cancer cells in suspension is dependent upon the number of cells added to plasma [8, 74, 80, 83]. Further work has suggested that TF-expressing cells in suspension show syn- ergistic eects on their ability to initiate and propagate coagulation, with the time to initiate coagulation enzymes and the rate at which these enzymes are generated correlating with the 40 average separation distance between cells rather than the overall cell count [74]. The eects of spatial separation on coagulation kinetics are consistent in assays that utilize closed sys- tems under well-mixed conditions as well as open systems under laminar ow. However, a CTC would experience dierent ow regimes if it were circulating on the arterial side versus the venous side or if it were free- owing or adherent to the cell wall, and the eects that these dierent conditions have on coagulation kinetics have not been established. We model the concentration of thrombin generated by dispersed CTCs under laminar ow. Our model is based upon exact solutions used in the atmospheric dispersion community [72] whereby a source of pollution near the ground (i.e. a smokestack) emits a pollutant which enters the atmosphere and is dispersed and diused downstream as a `Gaussian plume or a `Gaussian pu [72]. We adapt and use the solutions, which are based on a Greens function formulation for the concentration eld equations [44], to model the dispersing and diusing thrombin concentration eld entering the blood. Since the concentration eld equations are linear, we can superpose as many elds from each of the CTCs as needed. We assume that the transport of coagulation factors is diusion-limited as viscous forces dominate inertial forces of the cells. Our results suggest that thrombin generated by a CTC collects at the blood vessel wall and correlates with the number and spatial distribution of CTCs in the blood, supporting a role for the CTC count in predicting risk for developing thrombosis. Modeling and simulation of coagulation processes has been performed [11, 15, 26, 27, 33, 51], but to 41 our knowledge, our work is the rst to model the physiologically relevant scenario of a TF- expressing cell entering into and circulating within the bloodstream and simulate its eects on coagulation processes. 42 Chapter 5 The clot formation model 5.1 Experimental methods The set of experiments performed in the McCarty lab at Oregon Health Sciences University, Portland, were aimed at characterizing how the phenotype of tissue-factor carriers is related to procoagulant and prothrombotic activity, and specically, how it is dependent on the car- rier number, or density, of circulating tissue factor [8, 74]. The experimental setup is shown in Figure 5.1. Blood is placed in the upper chamber, and then it is allowed to ow through a collagen-lined capillary tube into the bottom chamber. When various concentrations of cells coated with tissue factor are introduced into the upper chamber, a blood clot forms within the order of minutes, and ow into the bottom chamber is occluded. The time for the blood solutions to occlude ow was recorded as the time to occlusion. The experiment was performed with three dierent types of cells, otherwise known as TF carriers: TF-coated microspheres, a model cell line for lymphoma called U937 cells, and a metastatic breast cancer cell line called MDA-MB-231 cells. The cancer cells naturally 43 Figure 5.1: Experimental setup for blood occlusion Blood in the upper chamber is allowed to ow into the bottom chamber. Once TF-carriers are introduced, a blood clot will form that occludes the ow of blood. h b is the height of the blood in the upper chamber, h c is the height of the capillary tube, and h pbs is the height of the capillary tube submerged in the bottom chamber lled with phosphate buered saline (PBS). express tissue factor. The microspheres, however, were dipped into a tissue factor coating solution to allow for coating for 60 minutes at room temperature. The TF carriers were then analyzed on a FACSCaliber ow cytometer for carrier density before being placed into the experimental chamber. Human monocytic U937 cells were purchased from ATCC (Manassas, VA). The metastatic adenocarcinoma cells, MDA-MB-231, were obtained from Dr. Tlsty (University of Califor- nia, San Francisco, CA). Polymeric microspheres with a diameter of 9.86 micrometers (the size of human monocytes is approximately 7-10 micrometers in diameter) were purchased 44 from Bangs Laboratories (Fishers, IN). All blood donations from healthy human subjects were obtained in accordance with Oregon Health Sciences University's IRB approval. The spatial separation, or the average distance between TF carriers in suspension, was obtained by calculating the cubic root of the volume of liquid divided by the number of TF carriers added. This approach assumes that TF carriers are uniformly distributed in suspen- sion. The reported values represent the average value for a minimum of three experiments, and the data are reported as mean standard error. 5.2 Experimental results The three types of TF carriers: synthetic TF microspheres, U937 cells, and MDA-MB-231 cells, were found to be procoagulant in a carrier number- and TF-dependent manner. The experiments determined that the time to occlusion is linearly correlated with the spatial separation of TF carriers, as in Figure 5.2, and therefore, the addition of TF-expressing cells to blood plasma results in shortened clotting times in a cell concentration manner. The spatial separation of TF surfaces strongly correlates with procoagulant and prothrombotic activity. These results suggest that coagulation initiated by circulating tissue factor is ki- netically limited by the mass transport of TF to the surfaces of cells, where the biochemical enzymes reactions occur. Also, the experimental results suggest the potential for circulating tissue factor to increase prothrombotic activity based solely on changes in spatial separation. 45 Figure 5.2: Experimental results comparing CTC burden with occlusion time The three plots show a linear relationship between clotting time and the spatial separation of cells. R 2 is the coecient of determination that provides a measure of the goodness of the linear t. As R 2 approaches 1, the regression line perfectly ts the data. These plots show the results for experiments done with microspheres coated in tissue factor, a cell line for lymphoma, and a cell line for breast cancer. 46 5.3 Mathematical infrastructure and Green's functions approach 5.3.1 Fluid dynamics and concentration eld equations The computational model is based on the partial dierential equations for the diusing concentration eld, coupled with the equations for incompressible uid ow [44, 66]: @ ~ C (i) @t +~ ur ~ C (i) = i ~ C (i) +(~ x ~ x i ) (5.1) _ ~ x i = ~ u (5.2) r~ u = 0 (5.3) ( @~ u @t +~ ur~ u) = rp +~ u (5.4) ~ u(~ x; 0) = ~ f(~ x) (5.5) Here, the concentration eld associated with the ith species is denoted ~ C (i) (x;y;z; t), with diusion coecient i . The uid (blood) velocity eld is denoted by ~ u (~ x;t), blood pressure denoted p, density , and each of the `i' CTCs (i = 1;:::;n) are located at ~ x i (t), their time derivatives are denoted by the `dot' superscript. in Equation 5.1 is the Dirac- delta function which is zero everywhere but where the argument is zero, which in this case are the locations of each of the CTCs. The initial locations of the CTCs are given by the functionf(x). The diusion coecient in Equation 5.4 is denoted by. Equation 5.4 are the Navier-Stokes equations representing the background plasma, which at this level of model 47 approximation we treat as an incompressible Newtonian uid. The ow takes place in the upper-half space, above a solid wall which models the vessel wall, hence boundary conditions for the concentration eld and blood velocity at the wall are: @ ~ C @n wall = 0 (5.6) ~ uj wall = 0 (5.7) The rst is a no penetration condition for the concentration eld, while the second is the viscous no slip boundary condition at the wall. 5.3.2 The Green's function approach In our models, the concentration elds are diusing away from the CTCs, which are modeled as point particles propagating with the ow. The initial concentration that is emitted from the CTCs at inital time t = 0 is described by a Delta function. Therefore, we use the Green's functions solution to the advection-diusion equation, (5.1). The Green's function, also called the fundamental solution, is the solution of a dierential equation corresponding to the initial condition of an initial point source at a known position. More details on the derivation of Green's functions can be found in book by Duy [24]. In this chapter, we focus on the simple geometry associated with the upper-half plane in 2D and upper-half space in 3D, making it possible to use an analytical Green's function 48 formulation to form solutions that satisfy exact boundary conditions. In 2D, with no ow (u = 0), we use the standard Green's function associated with the 2D diusion equation [44]: C (i) (x;y; t) = p 4t exp (xx i ) 2 (yy i ) 2 4t (5.8) C (i) image (x;y; t) = p 4t exp (xx i ) 2 (y +y i ) 2 4t (5.9) Here, each particle is placed in the upper-half plane (y > 0), at positions (x i ;y i ) for i = 1;:::;n, and image particles are placed at (x i ;y i ). The no penetration condition Equation 5.6 for the concentration eld at the wall is enforced exactly, with no other explicit boundary conditions needed. For our three-dimensional model, we used an adapted form of the Gaussian plume model developed by Stockie [72]. Although the application he used is dierent from cancer cells in ow, atmospheric dispersion modeling is very useful for our purposes. Figure 5.3 shows a schematic of the Gaussian plume model, which models a contaminant plume emitted from a continuous point source. In our model, the CTCs are not continuously emitting tissue factor and are not in a stationary position like the chimney stack shown in Figure 5.3. Therefore, we use the Gaussian pu model, which takes into account these dierences and more closely follows our model conditions, as opposed to the Gaussian plume model. 49 Figure 5.3: Schematic for atmospheric dispersion modeling A contaminant plume emitted from a continuous point source, with the wind direction aligned with the x-axis. The Gaussian shapes of the plume cross-sections are shown. In 3D, with ow u = constant, the corresponding Green's function is given by [44, 72]: C(r;x i ;y i ;z i ; t) = Q T 8(r) 3 2 exp (x i ut) 2 +y 2 i 4r exp (z i H) 2 4r + exp (z i +H) 2 4r (5.10) Here, r = x i =u is a new scaled independent variable. The symbol Q T expresses the total amount of thrombin expressed by the CTC. Using these solutions as the basic building blocks for ow simulations based on Equations 5.1 through 5.7, we are able to perform highly resolved concentration- ow simulations described in Section `Results.' 50 5.3.3 Concentration eld gradient tracking diagnostics Since the concentration eld is fairly complex, we need a diagnostic tool to help with visu- alization during a simulated run. It is useful to use what we call `passive gradient trackers,' which are diagnostic particles placed in the ow. The gradient trackers do not disturb the ow, but move toward regions of high TF concentration and low TF concentration in time, as the simulation proceeds. A schematic of one of these trackers is shown in Figure 5.4. If the tracker is placed at position (x;y;z) at time `t,' the concentration eld at that point is given by ~ C(x;y;z; t). The tracker then measures the concentration eld at six neighbor- ing points in the eld: ~ C(x;y;z); 0 < << 1, and measures the dierences in concentration at these six points compared to the concentration at ~ C(x;y;z; t). Thus, it measures the quantities ( ~ C(x +;y;z; t) ~ C(x;y;z; t)), ( ~ C(x;y;z; t) ~ C(x;y;z; t)), ( ~ C(x;y+;z; t) ~ C(x;y;z; t)), ( ~ C(x;y;z; t) ~ C(x;y;z; t)), ( ~ C(x;y;z+; t) ~ C(x;y;z; t)), ( ~ C(x;y;z; t) ~ C(x;y;z; t)), and if it is seeking high concentration regions, it moves to the point yielding the largest increase in concentration. If it seeks low concentrations, it moves to the point yielding the largest decrease. Thus, it tracks `gradients' in the concentra- tion eld at each time step, and as time evolves, the particles will gather in high or low TF concentration regions giving a useful visual diagnostic tool. For our simulations, we use `red' trackers to follow increases in gradient, and `blue' trackers to follow decreases. We note that there is an inherent timescale associated with the tracking, which is essentially governed by the size of . In the limit as this parameter goes to zero, the discrete trackers approximate derivatives in concentrations, hence gradients. 51 up down right left back front gradient tracker Figure 5.4: Schematic of gradient tracker The gradient tracker `polls' the chemical eld concentrations in six directions, and then updates its position according to the greatest increase or decrease in concentration. 5.4 Simulation results 5.4.1 Two-dimensional concentration elds A two-dimensional simulation of developing concentration gradients for 100 diusing CTCs with no ow (u = 0) is shown in Figure 5.5. The top row in Figure 5.5 shows the concen- tration eld at times T = 1; 5; 15 with u = 0 in Equation 5.1. The CTCs are randomly placed in the upper-half plane (y> 0), with the vessel wall at y = 0. On the vessel wall, we use the no penetration condition Equation 5.6 for the concentration eld. No other explicit boundary conditions are needed when using the Green's functions formulas. The middle row of Figure 5.5 shows the concentration proles aty = 0; 150; 300, while the bottow row shows the 3D surface plots of the concentration elds in the (x;y) plane. The CTCs are placed in the region y > 0, while their images are placed appropriately at y < 0 (see Equations 5.8 and 5.9) so that boundary conditions are enforced. The (dimensionless) diusion coecient 52 for each particle is taken to be i = 1:5. We note that here, and in all of the following sim- ulations, equations, and parameters are to be interpreted non-dimensionally since explicit comparisons with in vivo experiments are not described in this dissertation. n = 100 particles in upper−half plane T = 1 x y −500 0 500 0 100 200 300 400 500 10 20 30 40 50 60 70 80 90 n = 100 particles in upper−half plane T = 5 x y −500 0 500 0 100 200 300 400 500 0 20 40 60 80 n = 100 particles in upper−half plane T = 15 x y −500 0 500 0 100 200 300 400 500 0 20 40 60 80 −500 −400 −300 −200 −100 0 100 200 300 400 500 0 10 20 30 40 50 60 70 80 90 x C Concentration profile T = 1 y = 300 y = 150 y = 0 −500 −400 −300 −200 −100 0 100 200 300 400 500 0 10 20 30 40 50 60 70 80 90 x C Concentration profile T = 5 y = 300 y = 150 y = 0 −500 −400 −300 −200 −100 0 100 200 300 400 500 0 10 20 30 40 50 60 70 80 90 x C Concentration profile T = 15 y = 300 y = 150 y = 0 (a) T = 1 (b) T = 5 (c) T = 15 Figure 5.5: 2D simulation results for n = 100 cells 2D results for 100 cells with no ow, ~ u = 0, at three times in the simulation. The rst row shows the evolving concentration elds at timesT = 1; 5, and 15. The second row shows the concentration prole at dierent slices along the y-direction, at y = 0; 150, and 300. Note that the blood vessel wall is at y = 0. The third row shows the 3D contour plot of the concentration elds and gives a sense of how much the concentration diuses away as time progresses. 53 The 2D simulations with no ow clearly show the diusing elds from each particle merging and smoothing over time, with concentration persisting at the vessel wall because of the no penetration boundary condition. This is seen most clearly in Figure 5.6 which shows the concentration prole for T = 500 at y = 0; 150; 300. Figure 5.6(a) shows the persistence of the highest concentration at the wall (y = 0). Figure 5.6(b) shows the peak concentration at y = 0; 150; 300 as time progresses. The vertical line in this gure separates two distinct temporal regimes: (i) 0 < T < 3; (ii) T > 3. The rst early regime represents a `rapid mixing' regime where the concentration elds quickly merge to form a complex combined overlap domain of elds associated with the dierent sources merging together. The `long-time' regime (T > 3) shows that the peak combined concentration eld continues to decay, but rather slowly, with the peak wall concentration (y = 0) dominating. 5.4.2 Three-dimensional concentration elds We next performed a high resolution (exact, since we are using the Green's function formu- lation) simulation of CTCs in three- dimensions with a constant ow velocity prole (u = constant). Figure 5.7 shows the general schematic diagram with ow only in the direction of x, with diusing CTCs initially placed in the domain at random heights z =H 1 ;H 2 ;H 3 . The vessel wall in these simulations is located atz = 0. Because the concentration elds are spatially complex and time-dependent, we build in particle gradient tracking capability in our code, also shown schematically in Figure 5.7. 54 −500 −400 −300 −200 −100 0 100 200 300 400 500 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 x C Concentration profile T = 500 y = 300 y = 150 y = 0 0 5 10 15 0 100 200 300 400 500 600 T C Peak concentration along 1D slices y = 300 y = 150 y = 0 (a) Concentration proles for long time (b) Peak concentration vs. time Figure 5.6: Long-time 2D simulation results for n = 100 cells (a) Concentration proles at various y-slices. After a long time, the concentration at the wall is greater than anywhere else in the semi-innite plane. (b) Peak concentration vs. time. There are two time regimes involved in this simulation. First, a short time region up to aboutT = 3 represents the fast mixing period of the concentration elds. Then, after the vertical dashed line, the concentration elds decrease much slower and decay very slowly, showing that in the long time region, the elds of TF have mixed and are slowly diusing away. Next, we performed a 3D concentration eld simulation (without gradient trackers) using four CTCs, where we show a top down z-projection view of the (x;y) plane for values z = 0 (wall), z = 45, and z = 90, progressively in time T = 1; 10; 40; 75 in Figure 5.8. In order to compare with the 2D simulations, we have chosen the same dimensionless diusion coecient values i = 1:5. For these simulations, the initial locations of the four CTCs are (x i ;y i ;z i ) = (300; 300; 45); (180; 400; 30); (300; 100; 30); (275; 200; 60), as shown in Figure 5.9. Careful examination of the coloring of the elds indicates (i) the persistence of the strongest concentration region near the wall, (ii) strong concentrations in overlap domains from dierent CTCs, and (iii) strong concentrations near each of the CTCs which express 55 H 1 H 2 H 3 wall up down right left back front gradient tracker x z y u Figure 5.7: Computational domain for 3D model of diusing cells 3D computation domain with a bottom blood vessel wall. CTCs are placed at random initial positions and travel with the uid ow, which is constant in the x-direction. The concentration elds diuse away as `pus', not plumes, according to the experiment with synthetic TF microspheres. In the upper right-hand corner of the domain, we indicate the six directions that the gradient is `polled' for the gradient tracker calculations. TF. These numbers and results are consistent with the experiment described in [74] in which small numbers of TF-coated micro-spheres were placed in blood solution and clotting time was carefully measured. Typically, in metastatic patients, measured numbers of CTCs would be in the range of 1 100 CTCs/ml. For a direct comparison with the 2D results, we show in Figure 5.10 the 3D concentration eld simulations using 100 CTCs placed randomly (initially). The overall concentration eld again persists near the wall, but the overall concentration eld level is higher, roughly increasing linearly with the number of CTCs in the ow, also in agreement with results 56 from [74]. As the ow progresses in time, ow visualization tools become crucial to help understand the eld patterns that develop. 5.4.3 Gradient tracking results Circulating tumor cells and other biological cells in the body have the capability to follow chemical gradients in the blood or other uid environment. This phenomenon, called chemo- taxis, is an especially important ability that allows cancer cells to detect certain biochemical agents or enzymes in the blood and direct their movements toward or away from them [68]. To track developing TF concentration gradient patterns, we include gradient tracking capa- bility to our simulation. Figure 5.11 illustrates dierent time points (T = 8 220 s) as the ow progresses. The red particles move toward regions of high CTC concentration, whereas the blue move to regions of low CTC concentration. The patterns that develop with the red and blue particles depend on the comparison of relative timescales as determined by the concentration eld diusion rates, i , as well as the timescales on which the gradient trackers move. The rst three panels in the gure clearly show the red gradient trackers gathering in highly concentrated regions near each of the CTCs, lining up in elongated columnar strands. The timescale in which these trackers locate the diusing CTCs is short compared with the timescales on which the diusion elds spread. The last two gures in the panel show the particles then moving toward the vessel wall where concentration elds persist. This move- ment of the trackers to the vessel walls takes place on a longer timescale, well after the CTCs have been located in the ow. 57 Gradient tracking conrms our simulation result of the diusion of tissue factor collecting at the vessel wall, which is also supported by experiments and numerical models performed by Haynes et al. [37]. Prothrombin species were generated at increasing shear rates, and they examined the dilutional eects of variations with shear rate. Their model results validate that there is little diusion away from the wall region. The hypothesis that clot formation can only occur in a relatively small region close to the vessel walls has been previously suggested by Fogelson's numerical modeling studies of coagulation under ow, and it is also supported by our computational model. 58 y T = 1 Concentration slice at z = 90 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 5 10 15 y Concentration slice at z = 45 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 5 10 15 x y Concentration slice at z = 0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 5 10 15 y T = 10 Concentration slice at z = 90 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 5 10 15 y Concentration slice at z = 45 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 5 10 15 x y Concentration slice at z = 0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 5 10 15 y T = 40 Concentration slice at z = 90 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 5 10 15 y Concentration slice at z = 45 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 5 10 15 x y Concentration slice at z = 0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 5 10 15 y T = 75 Concentration slice at z = 90 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 5 10 15 y Concentration slice at z = 45 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 5 10 15 x y Concentration slice at z = 0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 5 10 15 Figure 5.8: 3D simulation results for n = 4 cells Top-down view of the x-y plane, with four CTCs randomly placed and in ow with the blood. There are three dierent z-slices at z = 0 (the wall), z = 45 (the source height), and z = 90 (above the source). The concentration builds up near the wall and is highest near the wall and in the complex over-lap regions where the concentration elds have mixed together. 59 0 100 200 300 400 500 0 100 200 300 400 500 x y CTC 1 (Q T = 80, H = 30) CTC 2 (Q T = 80, H = 45) CTC 3 (Q T = 40, H = 30) CTC 4 (Q T = 40, H = 30) Figure 5.9: Initial positions and concentration levels for 3D simulation Top-down view of the x-y plane, with four CTCs randomly placed. The initial positions, (x i ;y i ;z i =H), and the total concentration emitted, Q T , are as follows: CTC 1: (180; 400; 30), Q T = 80; CTC 2: (300; 300; 45), Q T = 80; CTC 3: (275; 200; 30), Q T = 40; CTC 4: (300; 100; 30), Q T = 40. 60 y T = 1 Concentration slice at z = 90 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 20 40 60 80 100 y Concentration slice at z = 45 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 20 40 60 80 100 x y Concentration slice at z = 0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 20 40 60 80 100 y T = 10 Concentration slice at z = 90 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 20 40 60 80 100 y Concentration slice at z = 45 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 20 40 60 80 100 x y Concentration slice at z = 0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 20 40 60 80 100 y T = 40 Concentration slice at z = 90 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 20 40 60 80 100 y Concentration slice at z = 45 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 20 40 60 80 100 x y Concentration slice at z = 0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 20 40 60 80 100 y T = 75 Concentration slice at z = 90 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 20 40 60 80 100 y Concentration slice at z = 45 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 20 40 60 80 100 x y Concentration slice at z = 0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 0 20 40 60 80 100 Figure 5.10: 3D simulation results for n = 100 cells Top-down view of thex-y plane, with 100 CTCs randomly placed and in ow with the blood. There are three dierent z-slices at z = 0 (the wall), z = 45 (the source height), and z = 90 (above the source). The concentration builds up near the wall and is highest near the wall and in the complex over-lap regions where the concentration elds have mixed together. 61 100 200 300 400 500 600 0 200 400 600 0 50 100 x y z 50 100 150 200 250 T = 8 100 200 300 400 500 600 0 200 400 600 0 50 100 x y z 50 100 150 200 250 T = 46 100 200 300 400 500 600 0 200 400 600 0 50 100 x y z 20 40 60 80 100 120 140 T = 99 (a) T = 8 (b) T = 46 (c) T = 99 100 200 300 400 500 600 0 200 400 600 0 50 100 x y z 10 20 30 40 50 60 70 80 90 100 T = 167 100 200 300 400 500 600 0 200 400 600 0 50 100 x y z 10 20 30 40 50 60 70 T = 220 (d) T = 167 (e) T = 220 Figure 5.11: Chemical eld gradient tracking simulation This simulation takes the 3D results from Figure 5.8 and displays a 3D view with red and blue gradient trackers that track increasing and decreasing concentration elds, respectively. The ve panels show the simulation at times T = 8; 46; 99; 167, and 220 s. 62 Chapter 6 The brin formation model 6.1 Experimental methods and results A novel experiment was developed to observe brin formation by a colon cancer cell line, SW480. The cancer cell(s) were allowed to attach to a substrate in a glass chamber, and then coagulation factors were introduced in the chamber. As the coagulation reactions occurred, the end product of clot bers, or brin, are visible. A brin front can be observed, as the brin strands propagate radially outwards from the cancer cell, and the bers get denser as the experiment progresses. It is important to note that the experiment is performed under no ow conditions. Figure 6.1 shows an experimental run of one colon cancer cell at several time points. This single cell clotting assay reveals that cancer cells nucleate blood clots. The cell is approximately 12 microns in diameter, and the experimental images are produced with DIC microscopy imaging techniques. Both the experiments and microscopy are performed by the McCarty Lab at Oregon Health and Science University. 63 Figure 6.1: Single cell clotting assay experiment Time lapse microscopy of brin formation by SW480 colon adenocarcinoma cells immobilized onto a silanized glass surface, immersed in plasma under conditions of coagulation for (top row) pooled plasma and (bottom row) coagulation factor VII-depleted plasma. 6.2 Model in a closed circular domain This series of experiments set the stage for the development of a model of brin forma- tion in a closed circular domain, in which there is no ow. Our model was based on the experiments described above. Again, the cancer cell is modeled as a point source, with a concentration eld of thrombin diusing out from the initial position of the cell. This time, the concentration eld builds up more quickly at the circular vessel wall and persists for a longer period of time than in the semi-innite domain. Our mathematical model is based on 64 the exact solutions found using the Green's function formulation, this time however, in polar coordinates. The advection-diusion equation in polar coordinates with no ow in written as the two dimensional heat equation: @g @t @ 2 g @r 2 + 1 r @g @r + 1 r 2 @ 2 g @ 2 = (r)( 0 )(t) 2r (6.1) withg representing the Green's function, is the diusion constant, is the initial position along the radius r, 0 is the initial angle where = 0 is along the positive x axis, and t is time for the region inside the circle 0r <a. The boundary conditions are the Neumann boundary conditions at the circular wall (no penetration condition), and the Green's function solution is a series of Fourier-Bessel functions, also known as Dini series: g = 1 a 2 m=1 X m=1 n=1 X n=1 J m ( n r)J m ( n ) (1 m 2 na 2 )J 2 m ( n a) +J 02 m ( n a) cos[m( 0 )]e 2 n t (6.2) where n is the nth positive root of J 0 m ( n a) = 0. Some identities which are useful in simplifying the Green's function solution are as follows: J 0 m (z) = 1 2 [J m1 (z)J m+1 (z)] (6.3) J 0 0 (z) = J 1 (z) (6.4) Figure 6.2 shows a 2D simulation with ve randomly placed CTCs in a circular domain in no ow conditions, showing the surface plots and contour plots at 3 dierent times (T = 65 0:4; 1:5; 3 s). The ve CTCs have dierent diusion constants, and one can see that the concentration eld of thrombin is highest at the source locations, in complicated overlap regions, and persists for longer times at the wall. This simulation is signicant in showing what the concentration of thrombin elds may look like in parts of the circulatory system where thrombin elds may build up due to minimized blood ow, for example, in situations where blood ow is restricted by a growing clot or a cluster of cells. Figure 6.2: 2D simulation of CTCs in a circular blood vessel with no blood ow Five cancer cells that are represented as point particles are placed randomly in a closed circular domain. The concentration eld is shown as a contour plot in the top row and also as a surface plot in the bottom row. The frames shown are at T = 0:4; 1:5, and 3 s. The next two sets of gures (Figure 6.3 and Figure 6.4) show the evolving thrombin elds with two CTCs having the same diusion constant. Figure 6.3 has two CTCs placed near the 66 center of the circular domain, and Figure 6.4 has two CTCs placed near the wall of the circular domain. We included a propagating radial diusion front based on a constant concentration assumption inside the growing ring, which leads to the approximation of Equation 6.5 r(t) p t (6.5) where r is the radius of the growing ring, is the diusion constant, and t is time. We also placed white gradient trackers that follow high concentration elds of thrombin, as in Chapter 5, using the nearest neighbor polling technique. The white trackers are randomly placed initially near the location of the two CTCs. For the CTCs in the center, they are placed in a smaller co-centric circle with a radius that is half of that of the circular domain, and for the CTCs near the wall, the trackers are placed in a wedge-like manner of the circular domain that spans an angle of =2. As the white gradient trackers move to high concentration zones, they eectively build brinogen bridges, which are composed of the brins displayed in the experiments of brin formation described earlier in this chapter. Upon performing these simulations based on exact analytical Green's functions solutions for the concentration eld of point particles in a closed domain, we then developed nite element models that employ nonlinear time dierence schemes to solve more complex ow conditions and blood vessel geometries, even including a deformable boundary for the CTCs. We use the exact thrombin eld results obtained in this chapter as test models, as a tool for comparison, for the nite element CFD models discussed in the following chapter. 67 Figure 6.3: Fibrinogen bridge simulation with two cells near center of vessel Two cancer cells that are represented as point particles are placed near the center of a closed circular domain. The concentration eld is shown with the diusion front demarcated by a thick black line. White gradient trackers highlight the brinogen bridge formation as time progresses. The frames shown are at T = 0:2; 0:4; 1; 1:5; 2, and 3 s. 68 Figure 6.4: Fibrinogen bridge simulation with two cells near circular wall of vessel Two cancer cells that are represented as point particles are placed near the vessel wall of a closed circular domain. The concentration eld is shown with the diusion front demarcated by a thick black line. White gradient trackers highlight the brinogen bridge formation as time progresses. The frames shown are at T = 0:2; 0:4; 1; 1:5; 2, and 3 s. 69 Chapter 7 The computational uid dynamics models Our mathematical models of CTCs and their respective thrombin elds involving both non- constant blood ow and more complicated geometries than the simple cases of a semi-innite plane and circular domain, as we examined earlier, require computational tools such as nite element analysis and computational uid dynamics techniques. This is due to the fact that analytical solutions do not exist for the advection-diusion equation in these complex problems. Therefore, to create ow models where the dynamics of the CTCs and the chemical gradient elds are acting in response to the ow in addition to one another in the ow, we build computational models using COMSOL Multiphysics (Version 4.3b), which is a nite element analysis (FEA)software package. The CFD (computational uid dynamics) model is developed using nite element analysis to solve the partial dierential equations (Equation 7.6) and to maintain boundary conditions on the cell surface and the blood vessel walls. 70 Components in blood Density Platelets 150 400 million /mL Red blood cells 4 5 million /mL White blood cells 4; 500 10; 000 /mL CTCs 2 100 /mL Table 7.1: Components in the bloodstream with their densities. 7.1 Development of the model from patient samples to computational simulation Fluid biopsy samples are drawn from patients diagnosed with cancers of the breast, lung and colon at the Scripps Medical Hospital in La Jolla, California. Dr. Peter Kuhn's Lab at The Scripps Research Institute processes these blood samples, and the HD-CTCs (high denition circulating tumor cells) are run through cell assays, stained and cultured, so that they can be processed through the system identication modules in the Kuhn Lab [17, 47, 55]. CTCs are deemed an extremely rare event, even in highly metastatic patients, as the number of CTCs found in the blood is in the range of 2 200/mL. As a point of reference, the number of platelets is 150 400 million/mL. Table 7.1 gives the average counts of the various particles found in the bloodstream of a cancer patient. Once the CTCs are identied, the blood samples are then sent to be imaged and char- acterized at OHSU by Dr. Owen McCarty's Lab. These samples are imaged using DIC 71 microscopy as shown in Figure 7.1(a), and various cell parameters were determined, which include cell density, mass, volume, and area. We then take the DIC images and the cell metrics as inputs into our computational models. In order to accurately demarcate the boundary of the cancer cells and clusters of cells, we use a program called CellProler, which obtains the outline of the cell in white and also identies the center of the outline as a white dot, both shown in Figure 7.1(b). CellProler is widely used as an image processing tool to identify and characterize images of biological cells and organisms. The shape of the cell or cluster from CellProler is then generated in our computational domain created in COMSOL Multiphysics, shown in Figure 7.1(c). The inside of the cell and the surrounding uid outside the cell are discretized into triangular mesh elements, where the uid physics are solved at the grid points. Lastly, the cell or cluster is placed in a blood vessel, either a channel ow blood vessel or a branching blood vessel, as in Figure 7.1(d). 7.2 Description of the mathematics and multiphysics used in COMSOL models COMSOL Multiphysics is a nite element analysis solver and simulation package for various engineering and physics applications, particularly for coupled phenomena, otherwise known as multiphysics. The specic modules within COMSOL that we utilized for our models include the Micro uidics Module and the Particle Tracing Module. The physics that are 72 o (a) DIC image of cancer cell (b) Outline of cell from Cell Proler (c) Shape of cell generated in mesh (d) Cell placed in a channel venule Figure 7.1: Development from patient sample to computational simulation The uid biopsy is sampled from patient, in this case a breast cancer patient, and then (a) imaged using DIC microscopy. CellProler (b) outlines the shape of the cell, which is (c) generated as a mesh using COMSOL. Lastly, (d) the cell is placed in the computational domain, which in this simulation, is a channel venule. coupled in our model include uid ow, chemical species transport, and particle tracing, which will be described in detail below. 73 7.2.1 Laminar two-phase ow with moving mesh The blood ow velocity eld is described by the full Navier-Stokes equations ( @~ u @t +~ ur~ u) = rp +~ u (7.1) r~ u = 0 (7.2) ~ u(~ x; 0) = ~ f(~ x); (7.3) whereu is the blood velocity, p is the pressure, is the blood density, and is the dynamic viscosity. We use the values of density and dynamic viscosity found in literature of whole blood at body temperature of 37 C = 310K, which are = 1060 kg/m 3 and = 310 3 Pa s). We also assume the blood is Newtonian and incompressible, which is theoretically valid at the ow conditions we are modeling in the venous system of the circulatory system. The blood ow assumes a parabolic ow prole (Poiseuille ow) at the inlet. The ow prole is set up by a pressure gradient between the in ow and the out ow. The boundary conditions at the blood vessel walls are the viscous no-slip conditions, and at the cell wall, the velocity of the blood is equal to the velocity of the cell boundary. ~ uj wall = 0 (7.4) ~ uj cell = velocity of cell boundary (7.5) 74 Vessel Diameter Velocity (cm s 1 ) Vena cava 1 cm 10 1 Veins 2 9 mm 10 0 Venules 7 50 m 10 1 Capillaries 1 m 10 2 Table 7.2: Velocities of vessels of dierent sizes in the venous system. As stated in the Introduction to Part II, we are studying the incidence of blood clotting in cancer patients that occur in the venous system, as opposed to the arterial system. Table 7.2 gives a list of the dierent types of blood vessels in the venous system, with the vena cava having the largest diameter size and capillaries having the smallest. The typical velocities (listed as an order of magnitude in cm/s) are also given, with the vena cava having the fastest velocity and capillaries the slowest. In our models and simulations, we use diameters that are typically found in venules, and therefore, the blood velocity is also on the order of 10 1 cm/s, in order to be as physiologically relevant as possible. The uid ow physics we specify in our model is a laminar two-phase ow, which models the laminar ow of two immiscible uids. The velocity eld, pressure, and mesh deformation are solved for, and a moving triangular mesh is used to track the position of the uid- uid interface. Therefore, we are modeling the interaction of two uids: the uid outside the cell represents blood, and the uid inside the cell represents the internal mechanics of the cell, which also has a specied density and dynamic viscosity. The density of the 75 cell is characterized by the McCarty lab as the DIC images are produced by calculating the density as a function of the mass and volume of the cells that are being imaged. The dynamic viscosity is set as cell = 1:36 Pa s, which is three orders of magnitude greater than that of blood. This is to simulate the cancer cells as having a much greater viscosity than blood, in order to exhibit behavior related to a thicker uid, as deformable cells do have an internal viscosity. The deformability of the cell is enabled by the surface tension criteria, which creates an interface between the two uids. The surface tension is inversely proportional to the deformability of the cell, meaning that a lower surface tension results in a more deformable cell interface. The surface tension values in our models range from 7 10 2 to 5 10 7 N/m. Also the uid ow prole is determined by a pressure gradient between the inlet and the outlet, with pressure dierences in our model ranging from 0:1 to 1:5 Pa. 7.2.2 Transport of diluted species The concentration elds of thrombin associated with each CTC is given by the advection- diusion equation @C (i) @t +~ urC (i) = D i C (i) (7.6) @C @n wall,cell = 0; (7.7) 76 Coagulation Factor D (10 7 cm 2 s 1 ) VII 5.1 IX 4.4 VIII 2 X 5 V 1.9 II 4.4 Table 7.3: Coagulation factors with their respective diusion coecients, D. where C is the concentration eld, and D i is the diusion constant associated with coagu- lation factors. In our simulations, we currently model one coagulation factor, but in theory, we can include more than one factor in the coagulation cascade. We use the diusion coe- cients for coagulation factors in the blood, tabulated in Table 7.3. All of our simulations use diusion coecients in the range from approximately D = 2 5 10 7 cm 2 /s. The bound- ary conditions of the concentration eld at the vessel walls and at the cell walls are the no penetration conditions (Neumann boundary conditions). Equation 7.6 is slightly dierent from the advection-diusion equation given in Equation 5.1 because of the omission of the delta term. This is because the CTCs are no longer represented as point particles, since they now have a distinct shape and a deformable cell boundary, as seen in Figure 7.1(c). 77 To model the concentration elds of thrombin, we use the transport of diluted species physics, which models the transport of diluted species by solving for the species concentra- tion. The transport mechanisms include diusion and convection in the uid representing the blood only, not inside the cell, since the thrombin elds diuse out from the surface of the cancer cell into the surrounding blood environment. As stated earlier, the diusion coecients used in our model are in the range specied by Table 7.3. 7.2.3 Particle tracking for uid ow We incorporate a particle tracking mechanism, where thousands of particles follow increasing gradients of thrombin concentrations. Instead of the nearest neighbor polling technique used in Chapters 5 and 6, we let the particles follow the dynamics of the concentration eld according to Equation 7.8 @~ q @t = 1 c 0 (C +) rC jjrC +jj (7.8) where q is the location of the particle, c o is a velocity scaling coecient, and is a small number that prevents issues arising from singularities whenC is equal to zero. The particles are massless and only in the blood ow, not inside the cell. When they make contact with either the boundary of the cell or the blood vessel boundaries, they are set to disappear. 78 7.2.4 The implicit time-dependent solver algorithm The nite element discretization of our time-dependent PDE problem is solved by the generalized- time stepping method, which is one of the nonlinear solver options provided within COMSOL. The generalized- method is an implicit, second-order accurate method with a parameter to control the damping of solutions with sharp gradients. We set the error tolerance used for each time step to be 1 10 5 . The other nonlinear solver option is the BDF (backwards dierentiation formula) solver, which is also implicit but uses a variable-order, variable step-size implementation. The BDF solver is known for its stability, but it can have severe damping eects, which results in a loss of accuracy due to the smoothing of solutions with sharp gradients. Therefore, we chose to use the generalized- solver, which causes much less damping and is more accurate. For the same reason, it is also less stable, but for our models, this solver has sucient stability properties. COMSOL also builds adaptive nite element meshes, with continual remeshing respecting error tolerance inputs. The automatic remeshing, together with the moving mesh in our two- phase uid ow, assures a satisfactory mesh quality throughout the simulation. After each remeshing, the time integration is restarted using the generalized- nonlinear solver, again with an error tolerance of 1 10 5 . 79 7.3 Modeling building process In this section, we describe the step-by-step modeling building/development we used in COMSOL. In order to ascertain that the models were accurately solving for the results we desired, we started with the simplest settings and added complexities one step at a time. 1. Begin with channel blood vessel with no cells, only blood ow Figure 7.2 gives the blood velocity eld for a simple channel geometry, with a parabolic prole set up by a pressure gradient between the inlet and outlet. The boundary conditions at the top and bottom walls are the viscous boundary conditions. Figure 7.2: Model building: Step 1 The velocity eld for simple channel ow. 2. Add a circularly-shaped cancer cell in the ow Figure 7.3 gives the blood velocity eld in the same channel with one deformable cell placed vertically in the middle of the channel. The ow eld has been changed in the presence of the cancer cell. 3. Add a chemical thrombin eld diusing outwards from the surface of the cell Figure 7.4 gives the concentration eld of thrombin diusing from the cells boundary and 80 Figure 7.3: Model building: Step 2 The velocity eld with one deformable circular cell. The two frames shown are at T = 0:8 and 3:7 s. also advecting with the blood ow. The thrombin eld leaves the channel at the out ow boundary, and we enforce the no penetration condition at the two vessel walls. Figure 7.4: Model building: Step 3 The concentration eld of thrombin for one deformable circular cell. The two frames shown are at T = 0:8 and 3:7 s. 4. Create a branching blood vessel geometry by adding two branches to the channel Figure 7.5 shows the same simulations as above for a branching geometry. The cancer cell is placed higher in the vertical direction to allow it to travel down the upper branch. 81 Figure 7.5: Model building: Step 4 The concentration eld of thrombin for one deformable circular cell in a branching vessel. The two frames shown are at T = 1:5 and 12:2 s. 5. Add particle trackers that follow the velocity ow eld Figure 7.6 shows the passive trackers following the blood velocity eld. Figure 7.6: Model building: Step 5 Particle trackers following the blood velocity eld for one deformable circular cell in a branch- ing vessel. The two frames shown are at T = 0:9 and 2:15 s. 6. Add another cell and particle trackers that follow the chemical thrombin eld Figure 7.7 shows the gradient trackers following the thrombin eld interaction between two 82 CTCs. The particles gather on ridges of high concentration between the two cells before the cells split o and travel into their respective branching vessels. Figure 7.7: Model building: Step 6 Particle trackers following high concentration levels for thrombin eld for two deformable circular cell in a branching vessel. The four frames shown are atT = 0:3; 0:8; 1:4; and 2:3 s. At each step in the model building process, we were condent that the simulation results were physically accurate before moving onto the next step in the development of the nal model. We then added more cells, including clusters of cells, scaled the dimensions to match the physiologically relevant parameter space, and included blood and measured cell 83 properties for the density and dynamic viscosity values to obtain the nal models that we describe in the next section. 7.4 Simulation results We describe below the models and simulation results for cancer cells of three dierent types: breast, colon, and lung. Various shapes of cells (both single cells and clusters of cells) and various blood vessel shapes are modeled. 7.4.1 Breast cancer single cell in a channel venule The rst model is based on a breast cancer cell, and the model was shown earlier in Figure 7.1, when we described how the computational model was built based on patient uid biopsy samples. A single breast cancer cell is placed in a channel venule that is 100 m in length and 75m in height. The size of the breast cell is approximately 10m in length and 20m in height. It is initially placed closer to the upper boundary layer region as opposed to in the middle of the channel. This initial location causes the cell to rotate in a counter-clockwise motion, due to the eects from the boundary layer region because the ow is slower closer to the venule wall and fastest in the middle of the venule. Figure 7.8 shows the simulation results for the blood velocity eld, the thrombin concen- tration eld, and the adaptive mesh generation. According to Figure 7.8(a), the maximum velocity is 0:026 cm/s, created by a pressure dierence of 0:1 Pa between the inlet and the outlet. Figure 7.8(b) shows the concentration eld diusing outwards and advecting with 84 the blood ow. The diusion coecient in this simulation is D = 1 10 7 cm 2 /s. The thrombin eld persists at the upper vessel wall, and is highest near the boundary of the cell. Figure 7.8(c) shows how the adaptive mesh deforms and regenerates at the three time points. The initial complete mesh consists of 2552 domain elements and 183 boundary elements for this simulation. One can see that the mesh is very ne near the cell boundary, in order to accurately solve for the ow physics near the cell's relatively complicated shape, and the mesh is coarser in the channel away from the cell where the physics is simpler to solve. 85 (a) Blood velocity eld (b) Thrombin concentration eld (c) Adaptive mesh generation Figure 7.8: Simulation results for the breast cancer model We show (a) the velocity eld of the blood, (b) the concentration eld of thrombin, and (c) the adaptive mesh generation, at three dierent time points in the simulation. The rst row is at time T = 0:044 s, the second row is at time T = 0:1475 s, and the third row is at time T = 0:2185 s. 86 7.4.2 Colon cancer single cell and cluster of two cells in a branch- ing venule The second model is based on colon adenocarcinoma cancer cells, and the model development is shown in Figure 7.9. These cells are relatively circular in shape, as opposed to the non- circular shape of the breast cancer cell. A single colon cancer cell and a cluster of two cells are placed in a branching venule that is 100 m in length and 50 m in height before it branches o into two branches that are each 25 m in height, with an angle of 30 between them. The single cell and the cluster are placed such that the single cell will travel into the upper branch, and the cluster will travel into the lower branch. Vertically, the cells are placed so that the cluster is very close to the lower boundary layer region. This results in the cluster rotating in a clockwise motion, due to the boundary layer eects. Figure 7.10 shows the simulation results for the blood velocity eld, the thrombin concen- tration eld, and the adaptive mesh generation. According to Figure 7.10(a), the maximum velocity is 0:08 cm/s, created by a pressure dierence of 1:5 Pa between the inlet and the outlet. Figure 7.10(b) shows the concentration eld diusing outwards and advecting with the blood ow into the two branching venules. The diusion coecient in this simulation is D = 3 10 7 cm 2 /s. The thrombin eld collects in the region between the single cell and the cluster before they travel down their respective branches, and then the thrombin elds persist at the vessel walls. Note that the upper branch shows that the thrombin persists at a higher concentration level near the upper boundary of that branch. Figure 7.10(c) shows how the adaptive mesh deforms and regenerates at the three time points. The initial complete 87 o o (a) DIC image of cancer cells (b) Outline of cells from Cell Proler (c) Shape of cluster generated in mesh (d) Cells placed in a branching venule Figure 7.9: Colon cancer model development from image to mesh The uid biopsy is sampled from a colon cancer patient and then (a) imaged using DIC microscopy. CellProler (b) outlines the shape of the cells, which is (c) generated as a mesh using COMSOL. Lastly, (d) the single cell and cluster is placed in the computational domain, which in this simulation, is a branching venule. mesh consists of 5696 domain elements and 334 boundary elements for this simulation. One 88 can see that the mesh for this model is ner than for the breast cancer model, and again, the mesh is discretized extremely nely near the cell boundary, especially in the cluster. (a) Blood velocity eld (b) Thrombin concentration eld (c) Adaptive mesh generation Figure 7.10: Simulation results for the colon cancer model We show (a) the velocity eld of the blood, (b) the concentration eld of thrombin, and (c) the adaptive mesh generation, at three dierent time points in the simulation. The rst row is at time T = 0:027 s, the second row is at time T = 0:060 s, and the third row is at time T = 0:087 s. Particle trackers, or passive particles in the ow, are placed in the simulation to track the increasing gradient in the thrombin concentration eld, shown in Figure 7.11. The 89 particles are released three times at T = 0, 0:01, and 0:02 s, with 5; 000 particles released each time. One can see in Figure 7.11(a) that there is a front of particles traveling towards the cells (in the region where the concentration eld is colored yellow), and there is also a line created by the particles between the single cell and the cluster (in the region where the concentration eld is orange). This line of particles represents the highest concentration zone at this time point, and eectively demonstrates where a brinogen bridge may occur in these circumstances. Figure 7.11(b) shows the line and the front moving behind the cells as the cells ow past the particle trackers, and Figure 7.11(c) shows that the front has now become another line of particles in between the single cell and the cluster, right as they ow into their respective branching venules. Lastly, we performed one more simulation with the colon cancer model that shows the wake structure of the thrombin concentration eld, as shown in Figure 7.12. In this simula- tion, we changed two parameters: the pressure gradient and the diusion coecient, while all the other parameters are held the same. The pressure gradient was changed from 1:5 to 0:15 Pa, which is one order of magnitude smaller, and the diusion coecient changed from D = 3 10 7 cm 2 /s to D = 5 10 9 cm 2 /s, which is two orders of magnitude smaller than in all of our other models. These two parameter changes result in two distinct dierences from the previous simulation. The rst is that the cluster travels much faster than the single cell, as it ows farther in the lower branch than the single cell has traveled in the upper branch. This is because at this ow rate, the single cell seems to travel closer to the upper boundary layer region before it reaches the branching point, which results in smaller velocity 90 (a) (b) (c) Figure 7.11: Particle tracking simulation for the colon cancer model The three time points shown in the gure are at time T = 0:037, 0:047, and 0:054 s. 91 closer to the vessel wall. The cluster, although initially placed close to the lower boundary layer region, is positioned in the center of the branching venule as it enters the lower branch. The second dierence between this simulation and the previous one is that the concentration eld clearly shows an interesting wake structure. Although this diusion coecient is not physically relevant, it is still useful in determining how the circularly shaped cells rotate as they ow in the blood vessel. The single cell seems to rotate in a counter-clockwise motion, and combined with the tendency of the thrombin to persist at the walls, results in a wake structure of the chemical thrombin eld, as shown in Figure 7.12(b) and (c). For the cluster, it seems to rotate in a clockwise motion, resulting in a wake structure in which the concen- tration eld is trailing the cluster at the upper boundary layer and leading the cluster in the lower boundary layer in the branch, as shown in Figure 7.12(b) and (c). 92 (a) (b) (c) Figure 7.12: Wake structure formation of the chemical thrombin eld in the colon cancer model The three time points shown in the gure are at time T = 0:100, 0:200, and 0:300 s. 93 7.4.3 Lung cancer cluster in a branching venule The third model is based on lung cancer cells, and the model development is shown in Figure 7.13. Here we have a cluster of many cells, specically 17 cells. Some of these cells may be white blood cells that were captured in the cluster outline in Cell Proler. Nevertheless, the cluster is a highly non-uniform shape that is approximately 65 m in length and 45 m in height. The blood vessel is another branching venule, as in the colon cancer model. However, since the cluster is relatively larger than in the previous model, the size dimensions of the venule are also larger, with the length being 125 m and the height being 80 m. The two branches are now each 40m in height, with an angle of 30 between them. One can see that the angle between the branches, which previously formed a sharp point, has been smoothed to have a radius of 1:5 m. This smoothing was done in order to prevent the simulation from breaking down when the cluster reaches the branching point. The initial position of the center of mass of the lung cluster is placed vertically in the middle of the venule. The center of mass is determined in Cell Proler and shown in Figure 7.13(b). Figure 7.14 shows the simulation results for the blood velocity eld, the thrombin concen- tration eld, and the adaptive mesh generation. According to Figure 7.14(a), the maximum velocity is 0:088 cm/s, created by a pressure dierence of 0:1 Pa between the inlet and the outlet. Upon reaching the branching point, the cluster signicantly obstructs the blood ow in the computational domain. The ow into the upper branch is much slower than in the lower branch. The cluster also begins to deform, with the upper part of the cluster deform- ing slightly in a clockwise motion and the lower part of the cluster deforming slightly in a 94 o (a) DIC image of cancer cells (b) Outline of cluster from Cell Proler (c) Shape of cluster generated in mesh (d) Cluster placed in a branching venule Figure 7.13: Lung cancer model development from image to mesh The uid biopsy is sampled from a lung cancer patient and then (a) imaged using DIC microscopy. CellProler (b) outlines the shape of the cluster, which is (c) generated as a mesh using COMSOL. Lastly, (d) the cluster is placed in the computational domain, which in this simulation, is a branching venule. counter-clockwise motion. Figure 7.14(b) shows the concentration eld diusing outwards and advecting with the blood ow into the two branching venules. The diusion coecient in this simulation is D = 1 10 7 cm 2 /s. The thrombin eld persists at the vessel walls, and one can note that the thrombin concentration eld is greater in the upper branch than 95 in the lower branch because the ow being greater in the lower branch causes more of the thrombin to travel away from the cluster at a higher velocity. Figure 7.14(c) shows how the adaptive mesh deforms and regenerates at the three time points. The initial complete mesh consists of 4872 domain elements and 346 boundary elements for this simulation. One can see that the mesh for this model is nely discretized near the cluster boundary, as well as in the smoothed branching point. Particle trackers are placed in this simulation as well to track the increasing gradient in the thrombin concentration eld, shown in Figure 7.15. The particles are released four times at T = 0, 0:15, 0:3, and 0:45 s, with 5; 000 particles released each time. Since there is one cluster in the simulation, there is no brinogen bridge formation as shown in the previous colon cancer model, but the front of particles is exhibited again. The particles show that early in the simulation, the thrombin eld closely resembles the shape of the cluster as shown in Figure 7.15(a) and (b), whereas later in the simulation, the thrombin eld is dictated by the blood vessel geometry as shown in Figure 7.15(c) and (d). 96 (a) Blood velocity eld (b) Thrombin concentration eld (c) Adaptive mesh generation Figure 7.14: Simulation results for the lung cancer model We show (a) the velocity eld of the blood, (b) the concentration eld of thrombin, and (c) the adaptive mesh generation, at three dierent time points in the simulation. The rst row is at time T = 0:121 s, the second row is at time T = 0:281 s, and the third row is at time T = 0:837 s. 97 (a) (b) (c) (d) Figure 7.15: Particle tracking simulation for the lung cancer model The four time points shown in the gure are at time T = 0:070, 0:210, 0:400, and 0:600 s. 98 Chapter 8 Discussion and future directions 8.1 Future work for deformation of cancer cells in ow (Part I) The cell deformation model we developed, in which the cell surface is allowed to deform parametrically as a family of Gaussian surfaces, does a reasonably good job of capturing the principal deformations of a cell under realistic ow conditions when the parameters in the empirical constitutive equations are chosen to match the experiment. The Active Shape Model was used to extract deformation parameters from the experiment and tune the cor- responding parameters in an empirical constitutive equation for the cell surface response to the ow, and it provided the necessary link between our experiment and a numerical simulation. For this particular experiment, a linear constitutive assumption suced due to the relatively mild deformations of the cell, but one could easily imagine extending the tech- nique to situations requiring nonlinear constitutive assumptions in which case higher order terms in (3.5), (3.6) would be retained and more complicated assumptions, which include 99 cross- ow terms f x f y , would need to be used. In that case, a series of dierent experiments with ow coming from dierent directions would be used to estimate the parameters. Figure 8.1 shows a ow diagram of the steps we used in our model to obtain ow simulations using our trained force-response law. An important future step that closes the loop is indicated by the red dashed arrow, which is to re-run experiments under dierent ow conditions in order to get better and better information about the constitutive equations. Figure 8.1: Flow diagram of modeling procedure for cell deformation model Schematic ow diagram of the steps we used to develop the cell deformation model in Part I. Note that the red dashed arrow, which closes the loop, has not been completed yet, but it is an important future step to strengthen the model's constitutive equations. In this thesis, the full capability of the ASM algorithm is yet not being realized in the sense that once a model is trained by a series of experiments, the algorithm can then be used to simulate the ways in which the trained model acts under a much wider set of conditions than those from which it was originally trained. In principle, for example, one 100 can use it to generate computer animations of a deformable cell which deforms in ways that are consistent with the training sets, but do not necessarily identically match any of the individual experimental runs. As a potentially useful way the algorithm could be used to understand cancer cell deformation, one could imagine setting up two separate ow experiments for training models, one using a normal white blood cell, the other using a circulating tumor cell. The ASM algorithm could be used in this setting to (i) uncover the dierences in the modes of deformation of these two kinds of cells, and (ii) run these two dierently trained cell models through new environments, such as simulated capillary beds, to quantify dierences in their respective responses. We believe ASM and related algorithms, such as the Active Appearance Models [18] and Active Contour Models [43], are ripe for further exploitation in these cellular level biological settings. 8.2 Future work for procoagulant circulating tumor cells in ow (Part II) In this thesis, we develop novel computational tools to model and characterize the movement of diusing thrombin elds emitted from CTCs in ow, with the aim of scaling up the techniques to more complex arterial environments and more complex, time-dependent ow assumptions [66]. A main nding from our model is the build-up and persistence of thrombin concentrations near vessel walls and in complex time-dependent overlap regions of the ow. The build-up near walls occurs on a relatively long timescale compared to the timescale in 101 which the concentration elds diuse (regulated by the diusion constants i ). We expect this main nding to persist under more complex and realistic ow geometries, as locally, near a vessel wall, the boundary curvature should not play a big role. Furthermore, very near the vessel wall where viscous ow boundary conditions dominate, blood velocity magnitudes are small, so should have minimal eects on disturbing the concentration elds that build-up there, therefore we do not expect the inviscid uid boundary conditions used in this study to qualitatively alter this main nding. In complex capillary beds where branching of the ow into several dierent regions is the reality, we do expect the tracking of the diusive overlap regions of many CTCs to be more computationally challenging, yet the main nding that concentration levels are relatively high in overlap regions should remain valid. Thrombin has several well characterized eects on endothelial cells and platelets which are also located at blood vessel walls under ow. In addition, the proximity of CTCs to each other determined the persistence of thrombin concentrations in the ow, which may help explain the eect of cell count on coagulation kinetics [74], suggesting that CTC counts may hold signicance to understanding the role for CTCs in activating blood coagulation. The gradient tracking capability described here holds strong potential to aid in the understanding of the fate of CTC-generated thrombin in more complex settings that arise within the circulation. We build models for diusing CTCs in ow with vessel wall boundaries, and exact an- alytical formulations allow us to clearly see the concentration buildup near walls, diusing particles, and complex overlap ow regions. We also develop nite element computational 102 models that are used under a wider range of boundary conditions and ow conditions. In order to best simulate the ow of circulating tumor cells and their developing chemical con- centration gradients, we model the blood ow as a parabolic Poiseuille ow prole and use initial congurations of actual patient uid biopsy samples in channel and branching vessel geometries. The future work we propose would more closely model the physiological conditions in the human body, by further developing our COMSOL models to include more realistic blood vessel geometries of the venous system, where blood clots are most susceptible to occur. As our COMSOL models are currently in 2D, further developing 3D computational models (and COMSOL does have the capability to build 3D models) and performing high resolution simulations would also enable us to better understand and interpret the ow physics and concentration gradient eld evolution in our models. In order to more accurately develop these complex models in 3D, we propose to further develop the DIC imaging tools used in the McCarty group at OHSU to obtain 3D spatial information about circulating tumor cells identied in patient uid biopsy samples. Perhaps individualized data about each patient's vasculature and circulatory system can be obtained using non-invasive imaging techniques, as in [4] using MR and CT angiography. If we can even characterize specic areas in the venous system where blood clotting may be more at risk for thrombosis, specic cancer clotting models can be developed, enabling physicians to practice improved therapeutic care using a quantitative risk analysis tool that is tailored to individual patients. 103 With these tools in place, we can begin to develop diusing particle simulations to include more complex and realistic geometries and ow conditions. We can also include chemical enzyme reactions to model the coagulation cascade, as in the previous models presented by Fogelson and Gregg [51, 12]. The overall goal is to develop a spatial-temporal mathematical model of blood clotting from circulating tumor cells in ow, which couples the physical process of chemical concentration gradients under blood ow with the biochemical enzyme reactions that regulate the hemostatic system in the body. Capturing the delicate and complex interplay between these physical and chemical processes is the key to understanding blood coagulation in metastatic cancer. We have developed models and simulations in COMSOL Multiphysics, giving us the tools to create computational and mathematical models for new and other potentially useful applications. One such new application is the experimental setup created by Parag Mallick and his group at Stanford University [14], with whom we have had extensive conversations on their experiments investigating the deformability and surface friction of cancer cells. They characterize cancer cells passing through a constriction at the end of a suspended micro uidic channel (Figure 8.2), and we are interested in modeling how the cell deforms as a result of the surface friction and internal mechanics of the cell as it ows through the constriction. This is one example of many potential collaborations that may result from our development of computational tools using COMSOL Multiphysics. We believe that the research areas of mathematical and computational modeling and simulation of various biophysical phenomena 104 surrounding metastatic cancer cells are currently ripe with opportunities for growth and progress. Figure 8.2: Experimental setup of cancer cell passing through micro uidic channel Schematic diagram of the suspended microchannel resonator (SMR) with a constriction located between Points 3 and 4. Numbers 1-5 indicate the dierent positions in the mi- crochannel that signify the ow trajectory of a cell. 105 Bibliography [1] Ainley, J., et al., The method of images for regularized stokeslets. J. Comp. Phys, 2008. 227: p. 4600-4616. [2] Amirkhosravi, A., et al., Tissue factor pathway inhibitor reduces experimental lung metas- tasis of B16 melanoma. Thromb. Haemost., 2002. 8: p. 930-936. [3] Anand, M., et al., A model incorporating some of the mechanical and biochemical factors underlying clot formation and dissolution in owing blood. J. Theoret. Med., 2003. 5: p. 183-218. [4] Antiga, L., et al., Computational geometry for patient-specic reconstruction and meshing of blood vessels from MR and CT angiography. IEEE T. Med. Imaging, 2003. 22(5): p. 674-684. [5] Ataullakhanov, F. I., et al., Spatiotemporal dynamics of clotting and pattern formation in human blood. Biochim. Biophys. Acta, 1998. 1425: p. 453-468. [6] Ataullakhanov, F. I., et al., Spatio-temporal dynamics of blood coagulation and pattern formation. A theoretical approach. Int. J. Bifurcat. Chaos, 2002. 12(9): p. 1985-2002. [7] Bakker, D.P., et al., Comparison of velocity elds for dierent ow chamber designs used in studies of microbial adhesion to surfaces. Applied Environ. Microbio., 2003. 69(10): p. 6280-6287. [8] Berny-Lang, M.A., et al., Promotion of experimental thrombus formation by the proco- agulant activity of breast cancer cells. Phys. Biol., 2011. 8: p. 015014. [9] Blom, J. W., et al., Malignancies, prothrombotic mutations, and the risk of venous throm- bosis. JAMA, 2005. 293: 715-722. [10] Boal, D., Mechanics of the Cell. Cambridge: Cambridge University Press, 2002. [11] Bodnar, T. and A. Sequeira, Numerical simulation of the coagulation dynamics of blood. Comp. and Math. Meth. in Medicine, 2008. 9: p. 83-104. 106 [12] Brummel-Ziedins, K. E., et al., Modeling thrombin generation: plasma composition based approach. J. Thromb. Thrombolysis, 2013. 0929: p. 5305. [13] Butler, T.P. and P.M. Gullino, Quantitative cell shedding into eerent blood of mammary adenocarcenoma. Cancer Res., 1975. 35: p. 512-516. [14] Byun, S., et al., Characterizing deformability and surface friction of cancer cells. PNAS, 2013. 100(19): p. 7580-7585. [15] Chatterjee, M.S., et al., Systems biology of coagulation initiation: kinetics of thrombin generation in resting and activated human blood. PLoS Comput. Biol., 2010. 6: e1000950. [16] Che, S., et al., Role of tissue factor and tissue factor pathway inhibitor complex in cancer cell adhesion. Third Annual NCI Physical Sciences-Oncology Centers (PS-OCs) Network Investigators Meeting, poster session, 2012. [17] Cho, E. H., Circulating tumor cells as emerging tumor biomarkers in lung cancer. J. Thorac. Dis., 2012. 4(5): p. 444-445. [18] Cootes, T.F., et al., Active appearance models. IEEE TPAMI, 2001. 23(6): p. 681-685. [19] Cootes, T.F., et al., Active shape models | their training and application. Comput. Vision Image Understanding, 1995. 61(1) p. 38-59. [20] Cootes, T. F., et al., The use of active shape models for locating structures in medical images. Image Vision Comput., 1994. 12(6): p. 355-366. [21] Cortez, R., The method of regularized stokeslets. SIAM J. Sci. Comp, 2001. 23(4): p. 1204-1225. [22] Couzin, C., et al., Critical stresses for cancer cell detachment in microchannels. Eur. Biophysics J., 2009. 38: p. 1035-1047. [23] Dashkevich, N. M., et al., Thrombin activity propagates in space during blood coagulation as an excitation wave. Biophys. J., 2012. 103: p. 2233-2240. [24] Duy, D.G., Green's Functions with Applications. Boca Raton: Chapman & Hall/CRC, 2001. [25] Dzwinel, W., et al., A discrete-particle model of blood dynamics in capillary vessels. J. Colloid Interface Sci., 2003. 258: p. 163-173. [26] Fogelson, A.L., Cell-based models of blood clotting. Mathematics and Biosciences in Interaction, Basel: Birkhauser Verlag, 2007. p. 243-269. [27] Fogelson, A.L. and N. Tania, Coagulation under ow: the in uence of ow-mediated transport on the initiation and inhibition of coagulation. Pathophysiol. Haemost. Throm., 2005. 34: p. 91-108. 107 [28] Forman, S.D. and Y. Nemerson, Membrane-dependent coagulation reaction is indepen- dent of the concentration of phospholipid-bound substrate: uid phase factor X regulates the extrinsic system. Proc. Natl. Acad. Sci., 1986. 83: p. 4675-4679. [29] Gaver III, D.P. and S.M. Kute, A theoretical model study of the in uence of uid stress on a cell adhering to a microchannel wall. Biophys. J., 1998. 75: p. 721-733. [30] Gay, F.J. and B. Felding-Habermann, Contribution of platelets to tumour metastasis. Nat. Rev. Cancer, 2011. 11: p. 123-134. [31] Gemmell, C. H., et al., Flow as a regulator of the activation of factor X by tissue factor. Blood, 1988. 72: p. 1404-1406. [32] Giesen, P.L.A., et al. Production of thrombin by the prothrombinase complex is regulated by membrane-mediated transport of prothrombin. J. Biol. Chem., 1991. 266: p. 1379-1382. [33] Gregg, K.L., A mathematical model of blood coagulation and platelet deposition under ow. 2010. [34] Gomez, K. and J.H. McVey, Tissue factor initiated blood coagulation. Front. Biosci., 2006. 11: p. 1349-1359. [35] Hall, C. L., et al., Factor Xa generation at the surface of cultured rat vascular smooth muscle cells in an in vitro ow system. J. Biomech. Eng., 1998. 120: p. 484-490. [36] Hasimoto, H. and O. Sano, Stokeslets and eddies in creeping ow. Ann. Rev. Fluid Mech., 1980. 12: p. 335-363. [37] Haynes, L.M., et al., Dilutional control of prothrombin activation at physiologically rel- evant shear rates. Biophys. J., 2011. 100: p. 765-773. [38] Hazel, A.L. and T.J. Pedley, Vascular endothelial cells minimize the total force on their nuclei. Biophys. J., 2008. 78: p. 47-54. [39] Hockin M.F., et al., A model for the stoichiometric regulation of blood coagulation. J. Biol. Chem., 2002. 277: p. 18322-18333. [40] Hundsdorfer, W. and J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diusion Reaction Equations. Berlin: Springer-Verlag, 2003. [41] Hunter, K.K., et al., Mechanisms of metastasis. Breast Cancer Res., 2008. 10(1): S2. [42] Jadhav, S., et al., A 3-D computational model predicts that cell deformation aects selectin-mediated leukocyte rolling. Biophys. J., 2005. 88: p. 96-104. [43] Kass, M., et al., Snakes: Active Contour Models. Int. J. of Computer Vision, 1988. p. 321-333. 108 [44] Kevorkian, J., Partial Dierential Equations: Analytical Solution Techniques. Pacic Grove, CA: Wadsworth & Brooks/Cole Mathematics Series, 1989. [45] Khorana, A.A., et al., Thromboembolism is a leading cause of death in cancer patients receiving outpatient chemotherapy. J. Thromb. Haemost., 2007. 5(3): p. 632-4. [46] Kim, S. and S.J. Karrila, Microhydrodynamics: Principles and Selected Appli- cations. Mineola: Dover Pubs, 2005. [47] Kuhn, P. and K. Bethel, A uid biopsy as investigating technology for the uid phase of solid tumors. Phys. Biol., 2012. 9: p. 010301. [48] Ledford, H., Cancer theory faces doubts. Nature, 2011. 472: p. 273. [49] Lee, J.S.H., et al., Cdc42 mediates nucleus movement and MTOC polarization in Swiss 3T3 broblasts under mechanical shear stress. Molecular Bio. of the Cell, 2005. 16: p. 871-880. [50] Lee, J.S.H., et al., Ballistic intracellular nanorheology reveals ROCK-hard cytoplasmic stiening response to uid ow. J. Cell Sci., 2006. 119: p. 1760-1768. [51] Leiderman, K. and A.L. Fogelson Grow with the ow: a spatial-temporal model of platelet deposition and blood coagulation under ow. Math. Med. Biol., 2011. 28: p. 47-84. [52] Levchenko, A. and P. A. Iglesias, Models of eukaryotic gradient sensing: application to chemotaxis of amoebae and neutrophils. Biophys. J., 2002. 82: p. 50-63. [53] Marchetti, M., et al., All transretinoic acid modulates the procoagulant activity of human breast cancer cells. Thomb. Res., 2011. 128: p. 368-374. [54] Marchetti, M., et al., Characterization of thrombin generation potential of leukemic and solid tumor cells by the calibrated automated thrombography. Haematologica, 2012. 97: p. 1173-1180. [55] Marrinucci, D., et al., Fluid biopsy in patients with metastatic prostate, pancreatic, and breast cancers. Phys. Biol., 2012. 9(1): p. 016003. [56] McGee, M.P., et al., Diusion control in blood coagulation. J. Biol. Chem., 1992. 267: p. 24333-24339. [57] Molnar, B., et al., Circulating tumor cell clusters in the peripheral blood of colorectal cancer patients. Clin. Cancer Res., 2001. 7: p. 4080-4085. [58] Mueller, B. M., et al., Expression of tissue factor by melanoma cells promotes ecient hematogenous metastasis. Proc. Natl. Acad. Sci. U.S.A., 1992. 89: p. 11832-11836. [59] Mueller, B. M. and W. Ruf, Requirement for binding of catalytically active factor VIIa in tissue factor dependent experimental metastasis. J. Clin. Invest., 1998. 101: p. 1372-1378. 109 [60] Nemerson, Y., The phospholipid requirement of tissue factor in blood coagulation. J. Clin. Invest., 1968. 47: p. 72-80. [61] Okorie, U. M., et al., Determination of surface tissue factor thresholds that trigger coagulation at venous and arterial shear rates: amplication of 100 fM circulating tissue factor requires ow. Blood, 2008. 111: p. 3507-3513. [62] Otero, L.L., et al., Tissue factor as a novel marker for detection of circulating cancer cells. Biomarkers, 2011. 16: p. 58-64. [63] Panteleev, M. A., et al., Spatial propagation and localization of blood coagulation are regulated by intrinsic and protein C pathways, respectively. Biophys. J., 2006. 90: p. 1489- 1500. [64] Panteleev, M. A., et al., Task-oriented modular decomposition of biological networks: trigger mechanism in blood coagulation. Biophys. J., 2010. 99: p. 1751-1761. [65] Pawar, P., et al., Roles of cell and microvillus deformation and receptor-ligand binding kinetics in cell rolling. Am. J. Physiol. Heart Circ. Physiol., 2008. 295: p. H1439-H1450. [66] Pedley, T. J., The Fluid Mechanics of Large Blood Vessels. Cambridge Mono- graphs on Mechanics and Applied Mathematics. Cambridge: Cambridge University Press, 1980. [67] Roberts, M.G., et al., Automatic segmentation of lumbar vertebrae on digitized radio- graphs using linked active appearance models. Proc. Medical Image Understanding and Analysis, 2006. 2: p. 120-124. [68] Roussos, E.T., et al., Chemotaxis in cancer. Nat. Rev. Cancer, 2011. 11: p. 573-587. [69] Sahai, E., Mechanisms of cancer cell invasion. Curr. Opin. Genetics and Development, 2005. 15: p. 87-96. [70] Saito, Y., et al., The inhibition of pancreatic cancer invasion-metastasis cascade in both cellular signal and blood coagulation cascade of tissue factor by its neutralization anitbody. Eur. J. Cancer, 2011. 47: p. 2230-2239. [71] Stern, M. D., Coagulation makes waves. Biophys. J., 2012. 103: p. 2049. [72] Stockie, J.M., The mathematics of atmospheric dispersion modeling. SIAM Rev., 2011. 53: p. 349-372. [73] Suresh, S., Biomechanics and biophysics of cancer cells. Acta Materialia, 2007. 55: p. 3989-4014. [74] Tormoen, G.W., et al., The role of carrier number on the procoagulant activity of tissue factor in blood and plasma. Phys. Biol., 2011. 8: p. 066005. 110 [75] Tosenberger, A., et al., Modelling of thrombus growth in ow with a DPD-PDE method. J. Theor. Biol., 2013. 337C: p. 30-41. [76] Versteeg, H.H., et al., Tissue factor and cancer metastasis: the role of intracellular and extracellular signaling pathways. Mol. Med., 2004. 10: p. 6-11. [77] Wajima, T., et al., A comprehensive model for the humoral coagulation network in humans. Nature, 2009. 86(3): p. 290-298. [78] Wang, Y. and P. Dimitrakopoulos, Nature of the hemodynamic forces exerted on vascular endothelial cells or leukocytes adhering to the surface of blood vessels. Phys. Fluids, 2006. 18: p. 087107. [79] Weinberg, R.A., The Biology of Cancer. New York: Garland Science, 2006. [80] Welsh, J., et al., Tissue factor expression determines tumor cell coagulation kinetics. Int. J. Lab. Hematol., 2012. 34: p. 396-402. [81] Yamaguchi, T., et al., Computational mechanical model studies on the spontaneous emergent morphogenesis of the cultured endothelial cells. J. Biomech., 2000. 33: p. 115-126. [82] Yanenko, N.N., The Method of Fractional Steps. Berlin: Springer-Verlag, 1971. [83] Yates, K. R., et al., Pancreatic cancer cell and microparticle procoagulant surface char- acterization: involvement of membrane-expressed tissue factor, phosphatidylserine and phosphatidylethanolamine. Blood Coagul. Fibrinolysis, 2011. 8: p. 680-687. [84] Yu, A., et al., Spatiotemporal dynamics of contact activation factors of blood coagulation. Biochim. Biophys. Acta, 2002. 1569: p. 86-104. [85] Zarnitsina, V. I., et al., A mathematical model for the spatio-temporal dynamics of intrinsic pathway of blood coagulation. The model description. Thromb. Res., 1996. 84(4): p. 225-236. [86] Zarnitsina, V. I., et al., A mathematical model for the spatio-temporal dynamics of intrinsic pathway of blood coagulation. Results. Thromb. Res., 1996. 84(5): p. 333-344. 111

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Creator Lee, Angela Meeyoun (author)

Core Title Modeling and simulation of circulating tumor cells in flow. Part I: Low-dimensional deformation models for circulating tumor cells in flow; Part II: Procoagulant circulating tumor cells in flow

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Aerospace Engineering

Publication Date 05/09/2014

Defense Date 12/09/2013

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

Tag chemical gradient tracking,circulating tumor cell induced hypercoagulation,intravastion models,low-dimensional deformation,OAI-PMH Harvest,procoagulant circulating tumor cells,prothombin and thrombin fields,tissue factor and coagulation

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Language English

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Advisor Newton, Paul K. (committee chair), Kanso, Eva (committee member), Macklin, Paul (committee member), McCarty, Owen J. T. (committee member)

Creator Email angelamc@usc.edu,angiemcho@gmail.com

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