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- Permanent Link:
- https://ufdc.ufl.edu/UFE0000580/00001
## Material Information- Title:
- Multi-Scale computation in hemodynamics
- Creator:
- N'Dri, Narcisse Abou (
*Author, Primary*) - Publication Date:
- 2002
- Copyright Date:
- 2002
- Language:
- English
## Subjects- Subjects / Keywords:
- Adhesion ( jstor )
Cell adhesion ( jstor ) Inlets ( jstor ) Interfacial tension ( jstor ) Kinetics ( jstor ) Leukocytes ( jstor ) Ligands ( jstor ) Receptors ( jstor ) Velocity ( jstor ) Viscosity ( jstor )
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
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- Copyright N'Dri, Narcisse Abou. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 12/27/2005
- Resource Identifier:
- 53335816 ( OCLC )
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MULTI- SCALE COMPUTATION IN HEMODYNAMICS By NARCISSE ABOU N'DRI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2002 Coppight 2002 by Narcisse Abou N'Dri To Jean-Claude Lelievre, ACKNOWLEDGMENTS I would like to thank Dr. Tran- Son- Tay not only for giving me the opportunity to finalize my education in USA but also for helping me since my arrival in Gainesville. To Dr. Shyy Wei, I would like to extend my appreciation and respect for gving me the possibility to conduct my research in computational fluid dynamics with moving boundaries. As stated by Hobbes, "All thought is a kind of Computation." I thank him for allowing me to discover this challenging field and for all the help and support to achieve this work. For Dr. Tran-Son-Tay and Dr. Shyy, I would like to quote one of my favorite thoughts, which is "When you follow an elephant, the dew cannot touch you." I thank them again for everything. I sincerely thank Professor Urich Kurzweg, Professor Nicolae Cristescu and Professor Richard Dickinson for serving as members of my supervisory committee. While we were looking for a post-doctoral position, my former advisor, JeanClaude Lelievre, decided that it would be better for me to pursue my study in the USA and go for a PhD degree instead. I told him that this would be difficult because of the language barrier. His answer was that when I arrived in France, I had to go through difficulties, but I managed to overcome them, so I can succeed anywhere if I put my mind on it. I thank professor Lelievre for believing in me. I would like to thank all the colleagues of the computational thermo-fluid group, in particular Marianne Francois, Inane Senocak, and the colleagues of the cellular mechanics and biorheology group. I would like to thank Patricia and Mark for all the years spent together. I have a lot to write about Kathleen, but I can summarize it through Boorstin's thought, "So long as man marked his life only by the cycles of nature--the changing seasons, the waxing or waning moon--he remained a prisoner of nature. If he was to go his own way and fill his world with human novelties, he would have to make his own measures of time." I would like also to extend my thanks to her parents, Mr. and Mrs. Anderson. I would like to thank my Mom, Ipou Madeleine, and my sister, N'Dri Marie, for all the prayers said and moral support given throughout the years I spent in the USA and in France. I thank them also for being there when I needed them he most. I thank my dad, N'Dri Bernard, and my uncle, Ipou Jean, for always being supportive of all decisions I have made through my years of education. TABLE OF CONTENTS Page A CKN OW LED GM EN TS . iv LIST OF TABLES . ix LIST OF FIGU RE S . x N OM EN CLA TURE . xiv A B S T R A C T . xv i CHAPTER I M O TIV A TION AN D IN TROD U CTION . 1 2 REVIEW ON CELL AD HE SION . 7 3 COMPUTATIONAL METHOD BACKGROUND . 21 Proj ecti on M ethod . 21 Continuity Equation . 21 M om entum Equation . 21 Boundary Conditions . 22 Im m ersed Boundary Technique (IBT) . 24 Interface Inform ation . 24 M aterial Property A ssignm ent . 25 Source Term Com putation . 26 Interfacial V elocity Com putation . 27 4 DEVELOPMENT OF THE MULTI- SCALE METHOD . 29 M odel Overview . 29 M acro- Scale M odel . 30 M icro- Scale M odel . 31 M icro-M acro Coupling . 32 M acro and M icro Test Cases . 33 M icro-Scale Test Cases . 33 M acro-Scale Test Cases . 36 M icro-M acro Coupling . 37 5 R E S U L T S . 4 1 Liquid D rop M odel . 41 Effect of Bond M olecule . 44 Effect of the Reverse Constant kr . . 44 Effect of the Spring Constant . 44 Com parison to Existing Results . 45 Com parison to N um erical Results . 46 Com parison to Experim ental Results . 47 Com pound D rop M odel . 48 Cell M ovem ent and D eform ation . 49 Effect of the N ucleus . 49 Effect of the V iscosity . 50 Effect of the Surface Tension . 51 Effect of the Inlet Boundary Condition . 52 Effect of the Capillary N um ber . 53 Peeling Tim e and Rolling V elocity Calculation . 55 Effect of the Inlet V elocity . 55 Effect of the R eceptor D ensity . 57 Effect of the Ligand D ensity . 58 Effect of the Surface Tension . 60 Effect of the Param eter P . 61 Effect of the Param eter u . . 62 Effect of the Contact Length . 63 Com parison to Experim ental Results . 65 Effect of Cell V iscosity on Bond Lifetim e . 65 Effect of Cell Surface Tension on Bond Lifetim e . 65 Effect of Applied Hydrodynamic Force on the Bond Lifetime . 67 Bond Rupture Force Fbr . 68 Effect of V iscosity on Bond Rupture Force . 69 Effect of Interfacial Tension on Bond Rupture Force . 70 M ultiple Bond Case . 70 Effect of the Cell D iam eter . 72 Effect of the V essel D iam eter . 72 Effect of the Pulsatile Inlet Boundary Condition . 73 Comparison of the Peeling Time for the U niform and Pulsatile Conditions . 74 Effect of the Surface Tension on the Pulse N um ber . 75 Effect of the Param eter u. on the Pulse N um ber . 76 Effect of the Initial Bond Force . 77 6 CON CLU SION AN D D ISCU SION . 79 L IST O F R E F E R E N C E S . 84 B IO G R A PH IC A L SK E T C H . 92 LIST OF TABLES Table Pg 2-1 Overview of some fundamental adhesion kinetic models. 11 2-2 Adhesion parameters computed using Bell and Hookean spring models. 15 5-1 Average macro-model parameters . 42 5-2 Average micro-model parameters . 42 5-3 Reference values used for non-dimensionalization. 42 LIST OF FIGURES Figure Page 1-1 M ulti-Scale phenom ena in hem odynam ics . 4 1-2 Illustration of the multi-scale nature of hemodynamics . 5 2-1 Force balance on an adhered neutrophil mediated by a single attachment . . 12 2-2 Effects of shear stress and Bond force on the reverse reaction rate . 15 2-3 Membrane tether formation and tear- drop- shaped deformation . 17 2-4 Effect of wall shear rate on the bond lifetime . 18 2-5 State diagram for adhesion. Four different states are shown . 19 3-1 velocity and pressure location on a given grid cell . 23 3-2 Interface representation and num bering . 25 3-3 H eaviside function definition illustration . . 26 3-4 Source term computation at the grid point P and marker points contributing to the computation is also shown . . 27 3-5 Interfacial point velocity computation and grids contributing to the computation is also shown . 28 4 -1 M acro -M icro m o d el . 3 0 4-2 Flow chart showing the interaction between the micro-model an d th e m ac ro -m o d el . . 3 3 4-3 G eom etry of the test case problem . 34 4-4 Maximum length of a bond before breaking as a function of the reverse reaction rate, for different values of forward reaction rate . 34 4-5 Effect of the ligand density on the de-bonding time . 35 4-6 Effect of the slippage constant f, on the de bonding time for cy (circle) and c7t, (square) held constant respectively . 36 4-7 Instantaneous droplet shapes for cy = 0.01 N/m (left) and cy = 0.1 N/m (right) . . 37 4-8 Instantaneous droplet shapes for two different values of the droplet viscosity . 37 4-9 Illustration of the instantaneous membrane shape resulting from the pressure field 3 8 4- 10 Forward reaction rate as a function of tim e . 39 4-11 R everse reaction rate as a function of tim e . 40 5-1 Schematic of the problem statement for the simple liquid drop . 42 5-2 Instantaneous position of an interfacial point on the cell . 43 5-3 Effect of the bond molecules on the rolling of the cell along the wall . 44 5-4 Effect of the reverse reaction rate kr,) on the rolling of the cell . 45 5-5 Effect of the spring constant on the rolling of the cell . 45 5-6 C om parison of the cell rolling velocity . 47 5-7 Comparison of the bond lifetimes in the simple drop case . 48 5-8 Schematic of the problem statement for the compound drop model . . 49 5-9 Effect of the nucleus on the rolling of the cell along the vessel wall . . 50 5- 10 Effect of the viscosity on the rolling of the cell along the vessel wall . 50 5-11 Effect of Viscosity on the instantaneous cell Shape . 51 5-12 Effect of the interfacial tension on the rolling of the cell along the vessel wall . 51 5-13 Instantaneous cell shape for different values of the interfacial tension . 52 5-14 Effect of the inlet velocity on the rolling of the cell along the vessel wall . 53 5-15 Instantaneous cell shape for different values of inlet velocities . 53 5-16 Effects of v, t, and y for Ca 10 on the cell behavior, kr,, = 1.0 . 54 5-17 Effect of the inlet velocity U on the peeling time T . 56 5-18 Effect of the inlet velocity on the rolling velocity of the leukocyte . 56 5-19 Effect of the receptor density N, on the peeling time T . 57 5-20 Effect of the receptor density N, on the rolling velocity F . . 58 5-21 Effect of the ligand density F, on the peeling time T . 59 5-22 Effect of the ligand density F, on the rolling velocity F . . 59 5-23 Effect of the surface tension on the peeling time T . 60 5-24 Effect of the surface tension on the rolling velocity F . . 61 5-25 E ffect of P on the peeling tim e . 6 1 5-26 E ffect of P on the rolling velocity . 62 5-27 E ffect of u. on peeling tim e . 62 5-28 E ffect of u. on the rolling velocity . 63 5-29 A ngle and contact length definitions . 63 5-30 Effect of the contact length on the peeling time T . 64 5-31 Effect of the surface tension on the rolling velocity F . . 64 5-32 Comparison of the bond lifetimes for the compound drop case . 66 5-33 Effects of surface tension and shear rate on the bond lifetime . 66 5-34 Bond lifetimes as a function of applied force . 67 5-35 Bond lifetime t as a function of bond force Fbr . 69 5-36 Effect of the viscosity on the bond rupture force . 69 5-37 Effect of the surface tension on the bond rupture force . 70 5 - 3 8 Compari son of the computed bond lifetimes in the case of compound drop and multiple bonds . 71 5-39 Effect of cell diameter on the peeling time of the cell . 72 5-40 Effect of vessel diameter on the peeling time of the cell . 73 5-41 Pulsatile and uniform inlet boundary conditions used in the present work . . 74 5-42 Comparison of peeling time for the pulsatile and uniform inlet boundary condition for different values of the surface tension . 75 5-43 Effect of the cell surface tension on the number of pulses rip . 76 5-44 Effect of the parameter u. on the number of pulses, rip . 77 5-45 Comparison of peeling time for the zero and non-zero initial bond force for different values of the surface tension . 78 NOMENCLATURE acel Area of the cell fSlippage constant fb Force of a bond Fb Total force of bonds kfo Equilibrium forward reaction rate kf Forward reaction rate kro Equilibrium reverse reaction rate kr Reverse reaction rate KbT Thermal energy (kb = Boltzmann constant, T temperature) Nb Density of a bond Nbo Initial bond density N1 Ligand density Nr Receptor density Re Cell radius Rt Number of receptors U Inlet Velocity Xm Current position of a bond on the membrane Cy Spring constant Gyts Transition spring constant c~ts Transition spring constant k Equilibrium length of a bond y Cell surface tension Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MULTI- SCALE COMPUTATION TN T{EMODYNAMICS By Narcisse Abou N'Dri December 2002 Chair: Dr. Tran-Son- Tay Roger Cochair: Dr. Shyy Wei Major Department: Mechanical and Aerospace Engineering Blood flow dynamics involves length scales across vastly different ranges. For example, adhesion of leukocytes to substrate entails dimensions ranging from nm to tm in length. Understanding the adhesive and rheological behavior of these cells is essential not only for describing microcirculatory flow dynamics, but also for understanding their function and behavior in health and disease. Models of cell adhesion do not combine molecular and full cellular information. In this study, a multi-scale computational approach for studying the adhesion kinetics and the deformation and movement of a cell on a substrate is presented. The cellular level model consists of a continuum representation of the field equations and a moving boundary tracking capability to allow the cell to change its shape continuously. At the receptor-ligand level, a bond molecule is mechanically represented by a spring and a reversible kinetics model is used to describe the association and dissociation of bonds. Communication between the macro- and micro-scale models is facilitated interactively with iterations between models until both levels approach compatible solution in each time step. The computational model is assessed using a cell adhering and deforming along the vessel wall under imposed shear flows. It confirms existing numerical and experimental results. In previous studies, the cell was modeled as solid body or a liquid drop. In the case of adhesion, only a small portion of the contact area was allowed to peel away from the wall. In this study, modeling the cell as a compound drop, we show that the presence of the nucleus increases the bond lifetime. We also show that increasing cell surface tension decreases the bond lifetime and that vessel diameter affects the cell rolling velocity. Furthermore, we find that the peeling time for a uniform inlet boundary condition is higher than that of a pulsatile one. In addition, increasing cell viscosity for a fixed hydrodynamic force increases linearly the critical bond force, whereas increasing cell surface tension inversely decreases it. Finally, we have shown that a non-zero initial bond force increases the peeling time. The present work shows that cell deformability and hydrodynamic flow affect cell adhesion. CHAPTER 1 MOTIVATION AND) INTRODUCTION Understanding the adhesion and rowing of a white blood cell on the endothelium wall and its impact on the whole cardiovascular system flow and the blood flow through the aorta and its bifurcations motivate this dissertation. Fundamentally, the blood flow through the cardiovascular system is a multi- scale fluid mechanics problem. Blood dynamics involves length scales across vastly different ranges. An example is provided by leukocyte adhesion to a substrate, which is an important biomedical problem that has been the subject of extensive research. Analyzing the adhesive and rheological behavior of leukocytes is essential not only for describing microcirculatory flow dynamics, but also for understanding cell function and behavior in health and disease. Computational fluid dynamics (CFD) techniques provide us with powerful tools for tacling a variety of problems in engineering involving fluid flow accompanied by mass, momentum and energy transport. These methods have recently been applied to a variety of problems related to hemodynamics, such as cardiovascular and capillary fluid flow, cellular physics, and adhesion dynamics. Computational fluid dynamics techniques are amongst the most powerful tools available to the engineering branches dealing with the motion of fluids and exchange of mass, momentum and energy. In particular, fluid dynamics studies of the normal circulation and pathogenesis of cardiovascular diseases such as atherosclerosis, thrombosis and aneurysms have been performed by numerous authors. In large vessels, dynamics of an individual cell and its impact on fluid properties are negligible but it is not the case for small vessels, where the effect is important. The study of the effect of leukocyte adhesion on blood flow in microvessels is of significant interest for understanding the resistance changes in microcirculation. Several computational fluid dynamic studies have been done to understand flow resistance and drag forces due to adhering leukocytes. Cell adhesion involves receptors, cells, and vessels with dimensions on the order of nm, irn, and cm, respectively. The study of the effect of leukocyte adhesion on blood flow in microvessels is important for understanding the resistance changes in microcirculation (Pedley, 1980; Skalak, 1972; Weiss, 1990). Knowledge of the effects of the adhesion parameters, rheological properties (viscosity and surface tension), and geometry (cell and tube diameters) of the leukocyte on the adhesion process is important for establishing guidelines for the design of drug particle size and receptor density with reference to the residence time required for the drug to be effective. Models proposed previously in the literature (Bell, 1978; Dembo et al., 1988) do not take into account the rheological properties of the leukocyte. Today, these properties have been shown to have substantial effect on the adhesion process (Dong et al., 1999; Dong and Lei, 2000; Hodges and Jensen, 2002). Multi-scale modeling of blood flow through arteries is important for understanding the physiological functions of the circulatory system and the pathophysiology of disease processes such as atherosclerosis, hypertension and diabetes, as well as for the diagnosis and medical cure of certain cardiovascular diseases. For example, in large arteries with diameters ranging from 3 cm down to 1 mm, a change in the dicrotic wave due to reflection indicates changes in the vessel wall and peripheral elasticity, while the absence of the dicrotic wave indicates that the patient may suffer from diabetes or hypertension. Thus, severe artery stenoses (constrictions) can be diagnosed by measuring the pulse wave propagation and reflection in terms of blood pressure or heart sounds. In addition, the location of artery disorders (atherogenesis) correlates with the spatial or temporal distributions of shear stresses exerted by the macroscopic blood flow on the artery vessel wall (Fry,1987; Giddens et al., 1993; Jin et al., 2001; Kilner et al., 1993), as well as with the microscopic mass transport across the arterial endothelium and within the vessel wall. Issues spanning from cell surface receptors to blood flow in arteries involve disparate scales, as shown in Figure 1-1. A direct computation encompassing all relevant scales, from large vessels to receptor-ligand interaction is beyond he reach of current and foreseeable computational resources (N'Dri et al., 2002). The multi-scale technique allows us to assess the effect of a reflected wave on the distribution of the flow through the aorta for example. An effective overall model can be devised globally using the lumped (ID) model to estimate the pressure and flow rate at a given location, and locally using the 3D field equations to analyze the detailed fluid flow in a given artery. Figure 1-2 illustrates the broad range of dimensions involved in hemodynamics. The adhesion of a leukocyte to the endothelium wall involves the deformation of a cell whose length is on the order of micrometers, whereas the adhesion equilibrium bond length involves scales on the order of the nanometers. Blood flows from large arteries whose diameters are on the order of centimeters to capillaries whose diameters are on the order of a few micrometers. To simulate the adhesion process and blood flow in arteries, a multi-scale modeling approach must be developed. Multi-Scale modeling of cell adhesion (gim to nm) I e I Blood vessel (po st-capillary venule) Receptors Site of inflammation Figure 1-1: Multi- Scale phenomena in level to the human cardiovascular system. hemodynamics. From cellular and/or molecular (a) (b) A B> Bonds Lumped 3 model model Figure 1-2: Illustration of the multi-scale nature of hemodynamics. (a) Cell adhesion modeling, and (b) 3D flow model and lumped model of the human circulation. Blood flow through the cardiovascular system is a multi-scale fluid mechanics problem involving flow in arteries, through the capillary network, and in veins. Localized abnormal physiological conditions due to atherogenosis or stenosis can influence the whole network. Change in flow resistance in one vessel will affect the flow in the upstream and downstream branches (Kilner et al., 1993; Liu, 2000). To describe such interactions, the interface between the lumped model and the detailed flow model must be mediated through effective boundary conditions. The multi-scale computation in hemodynamics can be achieved by combining the models at different levels through appropriate interfaces (N'Dri et al., 2002). The long-term goal of this project is to develop a multi-scale modeling framework spanning form cell surface receptors to blood flow in arteries. However, this is beyond the scope of this dissertation. This study will focus only on the multi-scale modeling of cell deformation and adhesion to provide a starting point for the more challenging multi- scale problems encountered in the human body. The major computational challenge for the present type of studies is in the treatment of the disparate length scales that exist between cell deformation, whose length scale is on the order of tm, and cell adhesion kinetics, whose length scale is on the order of nm. However, the developed framework will provide a scheme for treating general disparate length scale encounter in engineering problems. In this dissertation, we will first review the modeling and experimental studies for the leukocyte adhesion in the microvessels, with an overview of the numerical methods used. Then, we will describe the computation technique developed for this work and present the results. Finally, we will provide a discussion of the findings. CHAPTER 2 REVIEW ON CELL ADHESION The study of the effect of leukocyte adhesion on blood flow in micro-vessels is of significant interest for understanding the resistance changes in microcirculation. Adhesion to vascular endothelium (layer of cells lining a blood vessel) is a prerequisite for the circulating leukocytes in order to migrate into tissues. These adhesion events are mediated by coordinated regulation of adhesion molecules expressed by both leukocyte and endothelium. This three-step process is mediated by a series of different endothelial cell-leukocyte adhesion molecules. There are three basic classes of leukocyte-endothelium adhesion molecules: selectins, integrins and immunoglobulins (Long 1995). All the adhesion molecules are membrane proteins. Significant progress has been made toward understanding the receptor-mediated cell adhesion that is involved in the white blood cell and the endothelium cells interactions (Lauffenburger and Linderman 1993). Detailed experimental studies of the adhesive bonds have suggested that adhesion molecules of the selectin family are involved in maintaining the initial rolling of the leukocytes on the endothelium, whereas the integrin bonds are responsible for firm and prolonged white blood cells attachment to the endothelium cells. The reader is referred to Hammer and Tirrell (1996) for a review of the fundamental parameters that characterize biomolecule function in cell adhesion. Receptor-mediated cell adhesion plays an important role in many physiological and biotechnological processes. For example, Springer (1990) shows that leukocyte and tumor cell homing to particular tissues is accomplished by specific receptor interaction with endothelium ligand. The ability to design biomaterial substrate with known and quantifiable densities of adhesion molecules makes it possible to manipulate cell behavior by the correct preparation of the substrate. Kitano et al. (1996) show that bacterial adhesion to biomaterial was prevented by specific blockage of cell adhesion molecule receptors. It is also known that receptor-ligand binding and the resulting adhesion influence cell growth, differentiation, and motility. Analyzing the effect of leukocyte adhesion on blood flow in micro-vessels is important for understanding flow resistance changes in microcirculation. Adhesion of leukocytes to the blood vessel wall is critical in inflammation (Kaplanski et al., 1993; Lawrence and Springer, 1991) and plays an important role in the function of the immune system (Long, 1995). Rolling of leukocytes under physiological flow is a first step in the migration of cells toward an infection site. Adhesion of a cell to a surface often slows down but does not prevent the cell motion (Alon et al., 1997; Alon et al., 1998). It is facilitated by a complex interaction of receptor-ligand bond formation and dissociation. This initial adhesion is mediated by the P- and E-selectin found on the endothelium surface and by the L-selectin found at the tip of leukocyte microvilli (Hammer and Apte, 1992; Springer, 1990). Blood flow exerts a pulling hydrodynamic force on the adhesion bonds. It has been shown that this pulling force can shorten the adhesion bond lifetime or even extract receptor molecules from the cell surface. Several models have been proposed for describing the interaction of the leukocyte with the endothelium cells. Mathematical models of cell adhesion can be classified into two classes based on equilibrium (Evans, 1983; Evans, 1985a) and kinetics concept (Dembo et al., 1988; Hammer and Lauffenburger, 1987). The kinetics approach is more capable of handling the dynamics of cell adhesion and rolling. In this approach, formation and dissociation of bonds occur according to the reverse and forward rate constants. Using this concept, Hammer and Lauffenburger (1987) studied the effect of external flow on cell Adhesion. The cell is modeled as a solid sphere, and the receptors at the surface of the sphere are assumed to diffuse and to convect into the contact area, as reviewed by Shyy et al. (2001). The main finding is that the adhesion parameters, such as the reverse and forward reaction rates and the receptor number, have a strong influence on the peeling of the cell from the substrate. Dembo et al. (1988) developed a model based on the ideas of Evans (1985a,b) and Bell (1978). In this model, a piece of membrane is attached to the wall, and a pulling force is exerted on one end while the other end is held fixed. The cell membrane is modeled as a thin inextensible membrane. The model of Dembo et al. (1988) was subsequently extended via a probabilistic approach for the formation of bonds by Cozens-Roberts et al. (1990). Other authors used the probabilistic approach and Monte Carlo simulation to study the adhesion process as reviewed by Zhu (2000). Dembo's model has also been extended to account for the distribution of microvilli on the surface of the cell and to simulate the rolling and the adhesion of a cell on a surface under shear flow. Hammer and Apte (1992) modeled the cell as a microvilli- coated hard sphere covered with adhesive springs. The binding and breakage of bonds and the distribution of the receptors on the tips of the microvilli are computed using a probabilistic approach. Hammer and Apte (1992) studied the effects of the number of receptors, density of ligands, rates of reaction between receptor and ligand, and stiffness of the receptor-ligand spring on the adhesion of the cell. They also identified their effects on the peeling process. Several minor modifications of these models have been presented, but none takes into account the cell deformability, which has been shown to be necessary for calculating the magnitude of the adhesion force. Table 2-1 summarizes some of the efforts reported in the literature that address the modeling of cell adhesion dynamics. It is clear that much further studies including the role of cell deformation in adhesion are needed. Such knowledge would be particularly helpful in understanding tumor cell adhesion, because metastasizing tumor cells have been reported to deform differently than their non metastasizing counterparts (Weiss, 1990). Dong et al. (1999) and Dong and Lei (2000) have modeled the cell as a liquid drop encapsulated into an elastic ring. They show how the deformability and the adhesion parameters affect the leukocyte and Endothelium cell adhesion process in shear flow. In their study, only a small portion of the adhesion length is allowed to peel away from the vessel wall. This means that their results cannot truly represent what happens in vivo. So a more sophisticated model has to be developed. Shao and Hochmuth (1996) used the micropipette technique to measure the adhesion bond force and tether formation from neutrophil membrane. They found that a minimum force of 45pN is required for tether formation and the adhesion force increases linearly with the tether velocity. Shao et al. (1998) used the methodology developed by Shao and Hochmuth (1996) to measure the static and dynamic length of neutrophil microvilli. In all their experiments, they found a free motion rebound length, which is independent of the applied suction pressure. An explanation of this behavior is that this length is in fact the natural or static tether length. After the microvillus reaches its natural length, it will extend under a small pulling force or form a tether under a high pulling force. Table 2- 1: Overview of some fundamental adhesion kinetic models. Kinetic Main Major Findings Model Assumptions/Features - Bonds are equally stressed in the Adhesion occurrence depends on Point contact area (flat) values of dimensionless quantities Attachment - Binding & dissociation occur that characterize the interaction according to characteristic rate between the cell and the surface (Hammer and constants Lauffenberger, - Receptors diffuse and convect 1987) into the binding area of contact Clamped elastic membrane Critical tension to overcome the Bond stress and chemical rate tendency of the membrane to constants are related to bond spread over the surface can be Peeling strain calculated - Bonds are linear springs fixed in Predictions of model depend (Dembo et al., the plane of the membrane whether the bonds are catch1988) - Chemical reaction of bond bonds or slip-bonds formation and breakage is If adhesion is mediated by catchreversible bonds, then no matter how much -Diffusion of adhesion molecules tension is applied, it is impossible is negligible to separate membrane & surface. Microvilli- -Combined point attachment model Model can describe rolling, coated hard with peeling model transit attachment, firm adhesion sphere Binding determined by a random A critical adhesion modulator is covered w/ statistical sampling of a the spring slippage (it relates the adhesive probability distribution that strain of a bond to its rate of springs describes the binding (or breakage; the higher the slippage, unbinding) the faster the breakage for the (Harmner and same strain Apte, 1992) 2D elastic Bond density related to the Shear forces acting on the entire ring kinetics of bond formation cortical shell of the cell are Bonds are elastic springs transmitted on a relatively small (Dong et al. Interaction between moving fluid "peeling zone" at the cell's trailing 1999) & adherent cell edge Both microvilli extension and tether formation are found to be important in the rolling of the cell on the endothelium. Extension of the microvillus and formation of a tether can affect the magnitude of the bond force. AL- T, y x R F Figure 2-1: Force balance on an adhered neutrophil mediated by a single attachment. L is the total length of the microvillus, 1 the moment arm length, R the cell radius, b the bond force, F, the force and T, the shear stress imposed by the shear flow on the cell. For a microvillus extension, the spring constant derived by Shao et al. (1998) under the assumption that no threshold force exists is 43pN/tm. At low Reynolds number typical of blood flow through the capillaries, inertial effects on the fluid motion are negligible and a solution derived by Goldman et al. (1967b) may be used to compute the force of a single bond (Shao et al., 1998). Referring to the geometry in Figure 2-1, the angle 0 determining the direction of the bond force can be computed as follows: 0 = arctan- + arccos , (1) I 2Li271 where L is the total length of the microvillus, 1 the moment arm length, and R the cell radius. A force balance in the x-direction gives Fbcos0 = F = 32.05rR2, (2) where Fb is the bond force, F, the force, and -c the imposed shear stress. The bond force is defined as the force on an adhesive bond. A torque balance with respect to point A gives Fblsin0 = T + RF = 43.91R3, (3) where T, is the torque imposed by the shear flow on the cell. For the extension of a microvillus, the bond force Fb is given by Fb = k,(L- Lo), (4) where k, is the bond spring constant and L0 is the initial length of the microvillus. For the formation of a tether formation, Fb is computed as F =F0 k2A (5) dt where Fo is the initial bond force, dL/dt is the velocity at which the tether is pulling away from the wall and k2 is the change of the spring constant with time. For a neutrophil, lq = 43pN/ tm, k2 =1 lpN.s/[tm and F0 = 45pN (Shao and Hochmuth, 1996). Their results show that the bond force decreases in time, and drops 50% in 0.2 s before reaching a plateau. The moment arm and tether length quickly increase within the first 0.2 s, and then slowly increase. Shao and Hochmuth (1999) studied the strength of anchorage of the transmembrane receptors to cytoskeleton, which is believed to be important in cell adhesion and migration. In this experiment, micropipette suction was used to apply force to human neutrophils adhering to latex beads coated with antibodies of the CD626L (L-selectin), CD18 (32 integrins), or CD45. The adhesion bond lifetime is defined as the time period between the moment the cell is pulled and the moment the cell resumes its free motion. They computed the pulling force on the cell based on the theory derived by Shao et al. (1998) as a function of bond lifetime. They found an exponential relationship between the pulling force and the bond lifetime. c = 18e 0085F (6) Where F (pN) is the pulling force and -c (s) the bond lifetime. To explain the rolling of the cells over the endothelium, two mechanisms have been proposed: intrinsic kinetics of bond dissociation (kinetics in the absence of force) and reactive compliance (the susceptibility of the dissociation reaction to an applied force). Leukocytes have been observed to roll faster on bselectin than on E-selectin and P-selectin. To determine which of these two mechanisms can better describe this difference, Alon et al. (1997) studied the kinetics of tethers and the mechanics of selectinmediated rolling. Figure 2-2 shows the best fit of their experimental points to the spring model (Dembo et al., 1988), the Bell (1978) equation, and the linear model (Alon et al., 1997). They show that the Bell equation and the spring model both fit the data better than the linear relationship, suggesting an exponential dependence of kr on Fb. Because the leukocyte is modeled as a rigid body in their force computation, the error in the force computation is estimated to be about + 21%, so the computed model constants have to be viewed as an approximation. Alon et al. (1998) also studied the kinetics of transient and rolling interactions of leukocytes with L-selectin immobilized on a substrate, and measured the rolling velocity for different values of ligand density (L-selectin and P-selectin). For the L-selectin, increasing the ligand density reduces the rolling velocity. The rolling velocity corresponding to the P-selectin is smaller than that of the L-selectin, and there is a threshold shear stress of 0.4 dyn/cm2 above which rolling can occur. A frame-by-frame observation performed by Alon et al. (1998) show that leukocytes do not move smoothly on selectin substrates but rather in a jerky fashion. The same behavior was also observed by Chen and Springer (1999). - , E.4S A rr2ESL 1 30 2 .0 4 2 - : ;III I. . o0.0 0.2 &4 0.6 0.1 1.0 12 1.4 I- 1.8 wal .hear flrms (dypfmz D dO 0 120 16W 260 240 force Ft {pM) Figure 2-2: Effects of shear stress and Bond force (Fb) on the reverse reaction rate kÃ¯Â¿Â½f also called l in text. The solid and dashed lines correspond to the fit to existing models, as displayed in the legend. The squares correspond to ULselectin, the circles to E-selectin, and the rhombus to P-selectin. (Copyright 1998 National academy of Sciences, USA) Table 2-2: Adhesion parameters computed using Bell and Hookean spring models Bell model Hookean spring model Substrate kro (s-1) ro (A) kro (s-1) cy/ (N/m) Smith et and 2.4 + 0.47 0.39 + 0.08 3.7 + 0.51 2.5 + 0.61 P-selectin Lawrence (1999) Alon et al. (1995) 0.93 0.4 + 0.08 ND ND Smith and 2.6Ã¯Â¿Â½ 0.45 0.18 + 0.03 3.35 + 0.47 9.12 + 1.9 E-selectin Lawrence (1999) Alon et al. (1995) 0.5-0.7 0.31 + 0.02 ND ND Smith et and Smith et and 2.8+ 0.72 1.11 + 0.12 8.2 0.17 0.9 + 0.1 L-selectin Lawrence (1999) Alon et al. (1997) 7-9.7 0.24 + 0.02 9.7 + 0.66 6.31 + 0.96 The predicted values of the reactive compliance r and reverse reaction rate lko for L- P- and E-selectins using the Bell equation and cy/f, and again 1% using the Hookean model are shown in Table 2-2. A comparison between Smith and Lawrence (1999) and Alon et al. (1997) is also shown. The differences observed in Table 2-2 are due to the type of experimental analysis. In the study of Smith and Lawrence (1999), a high temporal and spatial resolution microscopy is used to allow the capture of features that previous studies could not capture (Alon et al., 1997, 1998). The effect of the ligand density on the rolling velocity was investigated by Greenberg et al. (2000) who reported an increase of rolling velocity with shear stress and a decrease of the rolling velocity with increasing ligand density. The mean rolling velocity found by Greenberg et al. study lies between 25 to 225 [tm/sec. Smith et al. (1999) found that the rolling velocity of a neutrophil rolling over immobilized L-selectin ranges from 50 to 125 [tm/sec, while the rolling velocities for P-selectin and E-selectin are 8 [tm/sec and 6 tm/sec respectively. The dissociation rate constant 14 is determined from the distribution of pause times observed during leukocyte adhesion, and has been used to quantify the effect of force on bond lifetimes (Kaplanski et al., 1993). By comparing the pause time for neutrophils tethering on P-, E-, L-selectin for estimated bond force ranging from 37 to 250 pN, they found that the pause times for neutrophils interacting with E-selectin or P-selectin are significantly longer than those mediated by L- selectin. Smith et al. (1999) found that the measured dissociation constants for neutrophil tethering events at 250 pN/bond are lower than the values predicted by the Bell and Hookean spring models. The plateau observed in their graph of the shear stress versus the reaction rate kr suggests that there is a force value above which the Bell and spring models are not valid. Since model proposed so far considers the cell as a rigid body, whether the plateau is due to molecular, mechanical or cell deformation is not clear at this time. Schmidtke and Diamond (2000) studied the interaction of neutrophils with platelets and P-selectin in physiological flow using high-speed, high-resolution videomicroscopy. Under wall shear rates ranging from 50 to 300 s-1, an elongated tether is pulled from the neutrophil causing a slight tear-drop-shaped deformation, as seen in Figure 2-3. Figure 2-3: Membrane tether formation and tear-drop-shaped deformation. (Reproduced from the journal of cell biology, 2000, vol. 149, pp. 719-729 by copyright permission of the Rockefeller University Press) The average length of the tether formed by a neutrophil interacting with a spread platelet is estimated to be 5.9 + 4.1 tm. The average tether lifetime is found to be between 630 and 133 ms for shear rates ranging from 100 s-I (Fb=86pN) to 250 Ã¯Â¿Â½1 (Fb=172pN), consistent with previous results (Alon et al., 1997; Bell, 1978). Figure 2-4 illustrates the dependence of the tether lifetime on the shear rate. Schmidtke and Diamond (2000) also studied the rolling of neutrophils over Pselectin-coated surfaces, and observed that when a neutrophil rolls on P-selectin, a tether develops similar to that observed for the spread platelets. The measured tether length and lifetime are 8.9 Ã¯Â¿Â½ 8.8 tm and 3.79 Ã¯Â¿Â½ 3.32 s, respectively. Chang et al. (2000) used computer simulation of cell adhesion to study the initial tethering and rolling process based on Bell's model, and constructed a state diagram for cell adhesion under an imposed shear rate of 100 s- using the Bell equation shown below kr kroexp(r~~) (7) kbT where kro) is the unstressed dissociation rate constant, kbT the thermal energy, ro the reactive compliance, and f the bond force. 1.2 1.0 o0.8 S0.8 .(4-0 50 1 10 -2t0 25::00 shear rate (s-1) Figure 2-4: Effect of wall shear rate (s1) on the bond lifetime, P < 0.001 means significant difference for the tether duration. (Reproduced from the journal of cell biology, 2000, vol. 149, pp. 719-729 by copyright permission of the Rockefeller University Press) Figure 2-5 shows the computed values of 4. as a function of ro (y in the Figure). The symbols correspond to the experimental data obtained for different receptor-ligand pair using the Bell equation. Adhesion occurs for high values of 4,) and low values of mo, as indicated by the wide area between the no-and fimn-adhesion zones in Figure 2-5. As ro increases, kro) has to decrease in order for adhesion to take place. In the simulation, both association rate kf and wall shear rate are kept constant. Varying kf does not change the shape of the state diagram but shifts the location of the rolling envelope in the 4,0-ro plane. They also found that as the shear rate increases, there is an abrupt change from firm adhesion to no adhesion without rolling motion. For values of ro less than 0.1 A and high reverse rate constant values kro, the rolling velocity is independent of the spring constant. A detail on how the computation is performed can be found in N'Dri et al. (2002). 10 001 5 Figure 2-5: State diagram for adhesion. Four different states are shown. The dotted curve represents velocity of 0.3 VH and the dashed curve represents velocity of 0.1 Mfi. The dots correspond to the data listed in Table 2-5 for h-selectin (squares, [Alon et al., 1997, Smith and Lawrence, 1999]), P-selectin (triangles, [Ramachandra et al., 1999, Smith and Lawrence, 1999]), L-selectin (diamonds, [Alon et al., 1998; Chen and Springer, 1999; Ramachandra et al., 1999]). (Copyright 2000 National academy of Sciences, USA) Tees et al. (2002) used the approach described in Chang et al. (2000) to study the effect of particle size on the adhesion. Three different spherical particles with radii 5.0, 3.75, and 2.5 tm and a shear rate G =100 Ã¯Â¿Â½s1 are used in their study. The spring constant and the association rate constant were kept fixed. Tees et al. (2002) observed that an increase of the particle size raises the rolling velocity. This is consistent with the experimental finding of Shinde Patil et al. (2001). In both studies (Chang et al., 2000; Tees et al., 2002), the Bell model is used to construct the state diagram but same results can be achieved using the spring model (Dembo et al., 1988). Dong et al. (1999) and Dong and Lei (2000) modeled the cell as a liquid drop encapsulated by an elastic ring and 20 illustrated the effect of the deformability and adhesion parameters on the interaction between the leukocyte and endothelium in shear flow. Only a small portion of the adhesion length is allowed to peel away from the vessel wall. This constraint is not physically sound, and a more sophisticated model is required. CHAPTER 3 COMPUTATIONAL METHOD BACKGROUND The cell adhesion-modeling problem is solved using a fixed Cartesian grid while the interface moves through the mesh, the so-called Eulerian-Lagrangian approach. The advantage of the fixed grid approach is that grid topology remains simple while large distortions cf the interface take place. Two approaches can be used to solve this kind of problem. The first one is the Immersed Boundary Technique (IBT) originally used by Peskin (1977) to simulate the heart flow and by Fauci and Peskin (1988) to model the swimming organism, and later extended and applied by, amongst others, (Juric and Tryggvason, 1996; Kan et al., 1998; Udaykumar et al., 1997; Unverdi and Tryggvason, 1992). The second approach is called the Interface Cut-Cell Method (Kwak and Pozrikidis, 1998; Shyy et al., 1996; Ye et al., 1999, 2001). In this dissertation, we will discuss the first approach. Projection Method The problem of leukocyte deformation, adhesion and movement is solved using the Navier- Stokes equations. For incompressible flow, these equations are given Continuity Equation Vi70 (8) Momentum Equation paii p(- + (.V)ii) VP + pV2ii +F (9) at where ii is the velocity vector, t is the time, p is the pressure, F is the source term, t is the fluid viscosity and p is the fluid density. Boundary Conditions Continuity condition (V)mterface = (u.n)l = (u.n)2, (10) Normal stress balance, the dynamic Young-Laplace equation P2 p1A )4 '+/12 (O )2 i-'( )I 11 On an where K is the curvature for two dimensional flows and twice the mean curvature for three dimensional flows, y is the interracial tension and n is the normal vector at the interface. Equations (8-11) are solved using the projection method on a fixed Cartesian collocated grid. The projection method or fractional steps is divided into three steps: Step 1: Momentum equations solved without pressure aii a- (iiV)ii = =vV2#i +r (12) a~t where v is the kinematic viscosity. Step 2: Solve the pressure equation (Poisson type) VP ( p - .1 (13) {1> V (-V)p =-I V.w op At Step 3: Correction step Numerical Implementation The convection terms are explicitly treated using the Adams-Bashforth scheme, while the diffusion terms are treated implicitly using the Crank-Nicholson schemes. Both schemes are 2d order accurate. Figure 3-1 shows the location of the velocity component and the pressure on a grid cell. Evaluation of the intermediate velocity u* and * for 2D problem jjii tin Viq)_ (ii.,Viiy-i 1 VV2iin _VV2ii 14 ( )+ (3(u.Vu - U +S (14) At 2 2 where n is the time step level and S the source term The pressure equation is derived by assuming that the velocity satisfies the continuity equation at n+1 time step level. U Y4 Figure 3-1: velocity and pressure location on a given grid cell V.W1~~ 0z .-p ) -V.UCj (15) p At And finally, the correction step is done as follows: At the cell center: j+l= W*- At(Ã¯Â¿Â½ 1 n+l. (16) P At the cell face: n1 At(Vp IcP~lf (17) Where cc stands for cell center and fc stands for face cell Immersed Boundary Technique (TBT) The TBT approach incorporates the interfacial condition into the field equation without explicitly tracking the interface. As detailed in Udaykumar et al. (1997), the interface can be handled by using markers points. The issues in this approach are the following: How the interface is represented into the grid domain? How to treat the material properties jump? How to communicate the force stored at the interface to the flow field? How to compute the interfacial velocity once the flow field velocity is known? How to move the marker point? Interface Information The immersed interface is denoted by C(t), the interface is either a curve for 2D problem or a surface for 3D problem. The interface is represented by K marker points of coordinates ik (s) with k 1.K and s is the arclength. Figure 3-2 shows the interface numbering and representation. The markers points are regularly separated 0. 5h K! ds K! 1. 5h where h is the grid size. The interface is parameterized as a function of the arclength s by fitting quadratic polynomials Xk (s) 54) +ks+e( through three consecutive marker points of coordinates k1' < .Once the position of the interface is known, the normal and the curvature are evaluated. The convention adopted is that the unit normal point form Fluid 2 to Fluid 1. In 2D the normal is given by: Ys and n = Xs Where the subscript s denote d/ds. The curvature is then obtained by taking the divergence of the normal vector IC = ui Figure 3-2: Interface representation and numbering Material Property Assignment With the known interface position, the material properties are assigned using a Heaviside step function. (20) where 3 is any material property such as density p or dynamic viscosity [t. The subscript 1 and 2 denote Fluid 1 and Fluid 2 respectively as shown in Figure 3-2. And H(- -k) is the discrete Heaviside step function defined as follows: (18) (19) /P /A1+ (8 2 -P I )H(j" - jOk dim 1 (x -(" x)k) n(5m )k)) m- (1+ +- sin ' if 2 -xk 0 ifx -xk >+d I 1 otherwise (21) Where dim is the spatial dimension, d=2h with h the grid size, Y is the grid coordinate and Xk is the interfacial point coordinates. Figure 3-3 shows the definition of the Heaviside function. Transition area 0 4h fluid2 (132) H= 1.0 Figure 3-3: Heaviside function definition illustration. The next step is to communicate the force stored at the interface to the near by grid points. Source Term Comoutation Let recall here the momentum equation p(-- + (i.V)ii) at Vp + pV2ii + F The force F exerted by the interface on the flow is transmitted to the momentum equation by means of integral source terms. Here we show this force only on the discretized form: (22) = - fk k( k )Ak k H(f- k)= Interface fluidly (P1) H= 0.0 The force at the grip point P is computed based on the sum of the interfacial force fk of the marker point located inside a circle of radius 2h weighted by the Delta function as shown in Figure 3-4. The delta function is computed as follows and the delta function spreads over 4h and is the derivative of the Heaviside step function. Ldim 1 (I~ (X +(osifImi - k)! 10 otherwise 0 o 0 0 0 0 0 o 0 0 0 0 0 0 0 0 0 0 Circle of radius 2h -~around P Markers points Contributing to Force F Figure 3-4: Source term computation at the grid point P and marker points contributing to the computation is also shown. Once the interfacial position is known, the material properties can be assigned properly then the interfacial force is spread to near by grid points and finally the flow field equation can be solved. The next step is to compute the interfacial point velocity. Interfacial Velocity Computation The velocity at the marker point is denoted by M and should satisfy the continuity condition, so in discretized form the interfacial velocity is: 1F 3 xk5(.i-:,)h2(z Where h is the grid size, i and j are the grid location indices and ii is the fluid velocity. The computation of the interfacial velocity is illustrated in Figure 3-5. The 1) 28 velocity is computed by taking the sum of all the grid points located inside a circle of radius 2h. Circle of radius 2h 0 0 0 0 0 around Marker Point 0 00 0 0 0 o 0 0 0 Grid Points Contributing to Interface Velocity TA'~ 0 o 0 0 0 0 Figure 3-5: Interfacial point velocity computation and grids contributing to the computation is also shown The last step is the advection of the interface and this is done using the following equation: This method is used to study the cell adhesion and movement along the endothelium wall. CHAPTER DEVELOPMENT OF THE MULTI- SCALE METHOD Although in large vessels, blood cells influence minimally blood flow, they play an important role in flows through small vessels with a lumen diameter comparable to cell size. The ability of a blood cell to deform and flow through capillaries or to migrate in tissues is governed by its mechanical properties. This aptitude to deform is critical to our immune system. The goal here is to develop novel numerical techniques to better understand and predict the effects of a deformable cell in contact with the endothelium, and the force of adhesion between them under quasi- steady sate flow. Most of the studies of the adhesion were done by assuming a flow over a plate and by using a shear flow at the inlet. But most of the existing experiments are performed using a parallel flow chamber. In addition, in the post capillaries, the diameter of cell and that of the vessel are in the same order; this can affect the rolling of a cell along the vessel wall. Finally, in-vivo the cell is exposed to a pulsatile flow. These flow and geometric characteristics have not been taken into account in previous studies. Furthermore, it has been shown in our laboratory (Kan et al., 1998, 1999ab) that a compound drop model for the leukocyte reconciles different experiments performed in different laboratories. So in the present a compound drop will be used to study the adhesion of a cell on a substrate. Model Overview In order to form a comprehensive modeling framework to treat the disparate scales between cell deformation ([[in) and cell adhesion (tim), a multi- scale model to account for both macroscopic (continuum) and microscopic (ligand- receptor) levels of phenomena is needed. Recently, N'Dri et al. (2000) have implemented such an approach. In their multiscale model, the macroscopic component deals with the deformation of the cell and the microscopic part takes care of the adhesion aspect. The continuum scale problem is the one of a fluid flowing over a cell attached to one wall of a tube by an area of contact defined by the membrane segment A-B (Figure 4-la). The ligand-receptor bond scale problem involves the interaction between the membrane segment A-B and the substrate surface (Figure 4- lb). (a) (b) A B Bonds A b Figure 4- 1: Macro-Micro model. The cellular model is shown on the left, whereas the enlarged area on the right shows the micro domain for the receptor model. Macro- Scale Model The macroscopic problem consists of a fluid flowing over a cell attached to a flat surface of a tube. Solution to the problem will provide information about the cell shape, velocity, and pressure. That information will be passed to the micro-scale model. Computation of the source term F k Where D is the delta function, y is the interfacial tension, K is the curvature, As is the arclength, n is the normal direction vector and hb is the bond stress which is defined by the micro- scale model. Mvicro- Scale Model The interaction between the cell and surface at the microscopic level is analyzed with the model proposed by Dembo et al. (1988). This model treats a bond as a spring, and the force of a bond,fb, is given by: fb = (X (x ) and ]7 = Nbfb (27) Where cy is the spring constant, xm and k are the current and the equilibrium lengths of a bond, Fb is the total bond force, and Nb is the bond density. In Dembo's model, major assumptions were made: 1) the bonding stresses are the only distributed stresses acting on the membrane; 2) the bonds are fixed in the plane of the membrane; and 3) the chemical reaction of bond formation and breakage is reversible. Those assumptions have some drawbacks since membrane experience forces due to surrounding forces such as the non specific potential forces, and it is also known that adhesion molecules are able to diffuse laterally in the plane of the membrane. Nonetheless, a significant understanding can be gained from the present study. Calculation of the bond density, Nb The balance equation of the formation and dissociation of bonds is given ty a simple kinetic relationship: at = kf (NI Nb)(Nr Nb) -krNb (28) Where Nb is the bond density, k, and kf are the reverse and forward reaction rate coefficients, respectively, N, is the initial ligand density on the surface, and N, is the initial density of receptors on the cell membrane. The reverse and forward reaction rate coefficients are, respectively, given by Dembo et al. (1988): k, k,,oexp( (29) 2kbT and k f kfo exp( -or, (Xm A)2) (30) 2kbT The initial bond density, Nb0, is found by solving the following equation: kf0 (NI Nbo)(N, Nbo) -kNbO 0 (31) Where the subscript o refers to the initial equilibrium state. The kinetic equation is solved using a 4h order Runge-Kutta method. The micro model has been used to analyze the case of a membrane being pulled away from a surface at a constant velocity (N'Dri et al., 2000; Shyy et al., 2001). The effects of reaction rates, ligand density, and other parameters on the adhesion process have been reported. Ivlcro-Macro Coupling The macro-model is first solved to determine the position of the interface while freezing the micro-model. Knowing the position of the interface allows the computation of the actual length of the adhesion molecule. Then, this information is transmitted to the micro-model while freezing the macro-model. The adhesion bond stress is computed in the micro-model and is transmitted to the macro-model via the source term. Information between he two models is continuously exchanged during the course of the computation. The computational flow chart of their approach is shown in Figure 4-2. Figure 4-2: Flow chart showing the interaction between the micro-model (receptor scale) and the macro-model (cellular scale). Macro and Micro Test Cases In the first part of this study, the macroscopic and the microscopic model were solved separated in order to study the effect of the Reynolds number and the capillary number on the deformation of a cell attached to the wall of a tube and the adhesion parameters such as ligand number, bond stiffness respectively. The main founding of this part of their study can be found in Shyy et al. (2001). Micro-Scale Test Cases A cell attached to a wall of a tube is pulled away from the substratum with a specified velocity, v as indicated in Figure 4-3. The bond is considered disrupted based on the criterion offered by equation NbC = 10 4Nbo. This computation is conducted to see the effect of the different parameters on the adhesion process. A v B >Bonds Figure 4-3: Geometry of the test case problem Figure 4-4 shows the effects of the reverse reaction rate, kro, on the maximum length of a bond before breaking, for different values of the forward reaction rate kf0. We see that when k0 increases, the maximum length reached by a bond before breaking is greater. For a given value of k, the maximum length of a bond decreases as the reverse reaction rate increases. The range of the values of kro used in this study is 0. 400 300 200 0 2 4 6 8 10 kro(S-1) Figure 4-4: Maximum length of a bond before breaking as a function of the reverse reaction rate, for different values of forward reaction rate., (n) for k- = 10-7cm2/sec, (? ) for k1-= 10-10cm2/sec, (,) for k. = 1O-12cm2/sec, (.) for kfo = 1O-14cm2/sec. The velocity v = 0. 1 jtm/sec. In all the following figures, v = 0.1[tm/sec, 4o = 10-14cm2/sec, ko = .1s-1, rc = 4x10-4 cm, dt= 0.001s, k 5x 10-6 cm, kbT 3.8x10-7 dyne.cm, Nr 5x10 cm-2. 6.2 E 5.8 0. 5.4 5 1 3 5 7 9 number of ligand Figure 4-5: Effect of the ligand density (X 0'Ã¯Â¿Â½cf2) on the de-bonding time (s). t= 0.1. Gts 4.5dynlcm and c = 5dyn/cm. In Figure 4-5, we observe that the de-bonding time increases as the number of the ligand increases. The sharp slope observed at the beginning is due to the fact that the number of ligands is less than the number of receptors, so there are a lot of free receptors at the contact area. Therefore, by increasing the ligand density, the number of bonds increases until the number of ligands outnumbers the number of receptors. At that instant, the bond formation is decreased due to the unavailable receptors at the contact area. The peeling time decreases as the spring constant increases. constant is the less elastic is the spring and therefore it breaks faster. Figure 4-6 shows the effect of the slippage constant f, bonding time. The slippage constant appears in the computation reaction (equation 30). The higher the spring ('-o'ta)/o" on the deof the reverse rate 7 6.5 W6 E 5 4.5 0 0.1 0.2 0.3 0.4 f,, (slippage constant) Figure 4-6: Effect of the slippage constant f on the de bonding time (s) for CY (circle) and Gts (square) held constant respectively. The circle dots correspond to cy 5dynlcm, and Gts is allowed to vary with t. The square dots correspond to cGts =4.5dynlcm, and (3 is allowed to vary with fNr =N, =5 x10' cmf' In this computation, (7t is held constant while aT is varying, then aT is held constant while (7t, is varying. Figure 4-6 shows that f, and the peeling time are inversely proportional. When f, 0.]1, the peeling time for (7t, is smaller than that of cy constant while forfa 0.]1 the peeling time for fixed value of (7t, and aT are similar. Macro- Scale Test Cases In this section, a droplet adhering to a wall under flow conditions is studied. This computation is done by using equations (8-11). No adhesion molecule is present in this test. In the following computation, the size of the tube is 1. 0 x2. 0, the diameter of the droplet is 0. 6, the inlet velocity is taken equal to 1. The properties of the droplet and the surrounding flow are p -0. 0] Pa. s, p -].O0kgm anda- .O]Nmanda- .]NIm, the capillary number Ca -p Ul - 1. 0 and 0.]1 respectively and the Reynolds number Re -(pUdlp)- 100. Figure 4-7 shows the instantaneous droplet shapes for two values of the surface tension for a fixed value of the inlet velocity. It is observed that the surface tension has an effect on the shape of the droplet. The droplet shape is smoother for high surface tension. 1 1 0.5 n A ~ 0 0.5 1 1.5 2 0N 0.5 1 1.5 1 Figure 4- 7: Instantaneous droplet shapes for cy 0.0 1 N/rn (left) and cy 0. 1 N/rn (right). 5- 0 05 1 1.5 2 0 . 5 1 1.5 2 Figure 4-8: Instantaneous droplet shapes for two different values of the droplet viscosity [L 0.01 Pa.sec (left) and [L 1.0 Pa.sec (right) and cy 0.01 N/rn for both computations. The viscosity has a significant effect on the droplet shapes and the flow field velocity as shown by Figure 4- 8. Ivijero-Macro Coupling In this paragraph, the interaction of both models was also studied. We will review some of the results for this aspect. An initial membrane shape given by the macro-model is used as the input for the micro-model. The membrane is modeled as a thin inextensible body (Dembo et al., 1988). A difference of pressure across the membrane, based on the 1 0. macroscopic solution is applied to find the new position of the membrane. During the first step of this computation, the bond model is not activated. In the next and subsequent time steps, the bond density and force are computed based on the receptor-ligand model. Using this information the new location of the interface is found at every time instant. In the computation, the following values are used: N, Nr 5.O0x 1010 cm2 , (T 5. 0 dyne/cm, (7, 4.5dyne/cm, kbT= 3.8x 10-7 dyne.cm. Instantaneous membrane shapes between points A-B shapes after various time steps are illustrated in Figure 4-9. Again, the initial curve represents the original membrane location segment before a flow is imposed. Initially, due to the action of the hydrodynamic forces, the membrane is pulled away from the substrate. This membrane movement activates the receptor-ligand model, and, consequently, the deformation of the membrane becomes more limited as a higher force is required to deform the bonds. The jump between the initial and first membrane locations in Figure 4-9 is because we didn't start the macroscopic and microscopic models simultaneously. This is done mainly to illustrate the interaction between the two models. 1.6 Y Initial 0.8 0.4 0 0 .1 0.2 x Figure 4-9: Illustration of the instantaneous membrane shape resulting from the pressure field, based on the macroscopic model, and the bond force, based on the receptor-ligand model. The vertical axis is normalized by the bond length, and the horizontal axis by the cell radius. To understand the behavior observed in Figure 4-9, we plot in Figure 4-10 and Figure 4-11 the variation of the reverse and the forward reaction rates as a function of time. 0.999960.99986-he 0.99976 - a X =6.130E M M 0 0.99966 X = 0.062 0.99956 1 1 0 0.002 0.004 0.006 0.008 0.01 time Figure 4-10: Forward reaction rate as a function of time for two different locations of fie bonds. X is the abscise of the bond. The distance between the membrane and the wall has a major impact on the rate constants, as evidenced by Eqs. (29) and (30). Hence, the initial upward movement of the membrane during the first time step, shown in Figure 4-9, is responsible for the initial decrease in the forward rate constant, kf, in Figure 4-10. The subsequent downward movement of the membrane reverses the trend accordingly. The two curves correspond to the quarter- and half-point of the membrane segments. The above argument is applicable to both forward and reverse rate coefficients, except that the reverse reaction rate coefficient, k, has an opposite behavior as shown in Figure 4-11. The main findings of N'Dri et al. (2000) for the kinetic model are: " The maximum bond length decreases as the reverse reaction rate increases, and increases with an increasing forward reaction rate. " As the cell velocity increases during the de-bonding process, the maximum bond length increases while the total peeling time decreases. * The rate of de-bonding decreases as the number of ligands increases. * The bonds are strong enough to resist the hydrodynamic force generated by a moderate flow and to affect the local shape of the cell membrane. * The peeling time decreases with increasing spring or slippage constants. * The peeling time reaches a maximum when the number of ligands equals its receptor counterpart. 1.0006 1.0004- * # O062* X = 0.130 1.0002 1 /"' 0 0.002 0.004 0.006 0.008 0.01 time Figure 4-11: Reverse reaction rate as a function of time for two different locations of the bonds. X is the abscise of the bond. CHAPTER-5 RESULTS in this chapter, the cell is allowed to deform and to roll along the vessel wall contrary to the precedent computation where the cell remains still during the computation. A 2D dimensional representation of a cell modeled as either simple or compound drop is studied in a planar channel under imposed flow. The cell is attached to one channel wall with adhesive bonds governed by kinetics model proposed by Dembo et al. (1988). A uniform flow is used a the inlet because in vitro experiment uses a parallel flow chamber and in vivo the cell is exposed to a pulsatile flow with a uniform description at a given time. Most of the studies use a shear flow over a plate with cell attached to it. It is also believed that changing the tube or vessel diameter can have an impact on cell adhesion. Furthermore, it has been shown in our laboratory (Kan et al., 1998, 1999ab) that a compound drop model for the leukocyte reconciles different experiments performed in different laboratories. Liquid Drop Model In this paragraph, a two- dimensional cell is attached to a vessel wall, and a uniform flow is imposed at the inlet of the vessel tube as shown in Figure 5-1 in order to study its peeling from the vessel wall using the immersed boundary technique described in Chapter 3. Table 5-1 and Table 52 show the adhesion parameters used in the study. All the results presented below will be based on the non-dimensional parametric variations normalized by the reference scales shown in Table 5-3. Figure 5-1: Schematic of the problem statement for the simple liquid drop Table 5-1: Average macro-model parameters. Tube diameter ([tm) 30 Cell viscosity (dyne sec/cm2) 10-1000 Cell diameter ([tm) 6 - 8 Plasma viscosity (dyne sec/cm2) 1.0 Tube length ([tm) 120 Plasma density (g/cm3) 1.0 U ([tm/sec) 50-3200 Interfacial tension (dyne/cm) (Evans, 1983) 103 - 8.0 Table 5-2: Average micro-model parameters. Parameter Value Parameter Value Nr (cm-2) 2.0 - 5.0 10'0 kb(dyne.cm/K) 1.38 10-16 (Bell et al., 1984) (Boltzman Constant) kfo (cm2/s) (Hammer and 10-14 (cm) 5.0 10-6 (Bell et al., 1984) Lauffenburger, 1987) kro (s-1) 10-10 r (cm) 4.0 10-4 (Bell, 1978) (Schmid- Shonbein et al., 1980) a (dyne/cm) (dyne/cm) 0.5 - 10 F, = (yo-ats)/Y 0.1 (Dembo et al.,1988) Gts (dyne/cm) 048 95 N (cm-2) 20 51010 (Dembo et al.,1988) (Lawrence and Springer, 1991) Table 5-3: Reference values used for non-dimensionalization. Length 30 tm Viscosity 1 dyne sec/cm2 Velocity 600 [tm/sec Density 1.0 g/cm3 Spring constant 5.0dyne/cm Bond density 5.1010cm-2 Reverse reaction rate 0.1 sec-1 Interfacial tension 0.1 dyne/cm In all the computations, unless specified otherwise, the values used for 1%, 0, and ots are 0.1 sl, 5 dyne/cm, and 4.5 dyne/cm, respectively. Using these parameters, the effects of the reverse reaction rate, the wall, and the spring constant c on the rolling/displacement of the cell along the endothelium are evaluated. The cell is first modeled as a liquid drop with a constant viscosity and surface tension. This is the simplest model one can use to describe qualitatively certain rheological behaviors of leukocytes (Evans and Yeung, 1989). Figure 5-2a shows the instantaneous position of an interfacial point on the cell surface for a given 4,. It is seen that the cell translates along the wall and rotates at the same time. The combination of these two movements leads to the rolling of the cell along the wall. The cell initially rolls along the wall and starts to slide along the vessel wall when the bonds offer no more resistance as shown by the plateau observed in Figure 5-2b. The curve trend can be qualitatively compared to the one obtained by Shao et al. (1998) in their study where they found a fast increase of the bond length follow by a slow increase and a plateau. In the subsequent computations, a position of the instantaneous interfacial point will be plotted as indicated in Figure 5-2a. Fa]yIRAt t5 t40 0.14 0.12 0.1 0.08 0 2 4 6 8 10 t(s) Figure 5-2: Instantaneous position of an interfacial point on the cell for kro) 1.0 and 17=0.1, t= 100,y= 1.0. Effect of Bond Molecule The effect of bond molecules on the behavior of the cell is shown in Figure 5-3. The effect of the wall on the cell displacement is obtained by not activating the micro model (no bond force). It is found that the bond force is strong enough to delay the rolling of the cell and the peeling of the cell from the vessel wall. 0.125rf Without activating micro model 0.105- ------- With micro model (k. - 1.0) 0.085 1.81 1.85 1.89 1.93 x/R Figure 5-3: Effect of the bond molecules on the rolling of the cell along the wall, U=0.1, t= 100,y= 1.0. Effect of the Reverse Constant kro Figure 5-4 shows the effect of the reverse reaction rate kro on the rolling of the cell. It is observed that an increase in the value of ko decreases the resistance force from the bond molecules. On the other hand, a lower value of 40 makes the cell rolls for a longer distance, and tends to increase the bond length of the adhesion molecule before breakage as demonstrated by the upper asymptote obtained. Effect of the Spring Constant The effect of the spring constant cy on the rolling of the cell is provided in Figure 5-5. Changing the spring constant value by a factor of two has little effect cn the rolling of the cell along the vessel wall. The work by Chang et al. (2000) shows also the little dependence of the spring constant on the adhesion process for some adhesion parameters. 0.12 0.08 1.81 1.85 1.89 1.93 xIR Figure -5-4: Effect of the reverse reaction rate 1. on the rolling of the cell, U 0. 1, [t l00, Y= 1.0 0.08 1.81 1.85 1.89 1.93 xIR Figure 5-5: Effect of the spring constant on the rolling of the cell. =0.1, k') 1.0, aU 0. 1 , i 100, Y= 1.0. Comparison to Existing Results In the following computations, a bond is considered peeled away from the surface if the cell travels 2 radii. This assumption is based on the shape taken by the cell as a function of time. Kuo et al. (1997) made similar assumptions in their study into which the leukocyte is modeled as a solid body. They assumed that bonds are broken if the cell travels 10 radii. The best assumption should be based on the adhesion molecule bond strength. The work by Evans (1999) gives a iange of adhesion bond strength but not an exact bond strength for a given molecule bond. Comparison to Numerical Results Dong et al. (1999) and Dong and Lei 2000) have modeled the cell as a drop enclosed into an elastic ring. In their approach, the initial shape of the elastic ring is the one taken from the picture of an experiment of a cell adhering on a surface subjected to a known shear rate. They assume that only a small portion of the adhesion contact can be peeled away from the wall, but that length is not specified in their work. This assumption allows them to use an energy approach to calculate the cell rolling velocity. In the present model, we do not make that assumption. We solve the full flow field and fluidinterface interactions, and let the flow dictate the contact area. We use the same parameter as in Dong et al. for the adhesion parameters and cell viscosity. The only unknown parameters in their study are the cell surface tension, the contact length and the number of bonds. In the Figure 56, a comparison to Dong results is shown for j7= 4.0, 5.0 and 10. The top line corresponds to the higher surface tension, and is in agreement with the results of Dong and Lei (2000). This is expected, since as the surface tension increases, the liquid drop model should provide results similar to the 2D elastic model. However, because our approach is more general, additional information can be found. For example, we do not constraint the cell to a peeling motion only, we allow the cell to be lifted away from the surface. 20 15 E 10 0 500 1000 1500 2000 i(dyn sec/cm2) Figure 5-6: Comparison of the cell rolling velocity. The dashed line corresponds to Dong et al. results and the lines to the present study. U7 1.0, Nr 0.4, N1 =1.06, U 0.1, f, 0. 04, )7 =4.0, 5. 0, and 10 from bottom to top respectively. Comparison to Experimental Results A comparison of the bond lifetimes computed in this study and the experimental results by Schmidtke and Diamond (2000) for one bond is shown in Figure -5-7. For this comparison, no information about the reaction rates and spring constant exist. But, we have shown that, for the adhesion parameters considered in this study, these values do not significantly affect the rolling and displacement of the cell along the vessel wall. The key parameters are the cell rheological properties, the receptor density and the ligand density. In the present study, we have supposed an excess of ligand number to the number of receptors, the dimensionless cell surface tension is taken to be 1.2 and the dimensionless viscosity ranges from 50 to 300. The top line corresponds to the higher viscosity. Good agreement is seen for the parameters used. An increase in the cell surface tension for a fixed value of cell viscosity will move all the simulated results upward. A surface tension of 1.2 and a viscosity of 200 provide the best fit. 1.2 0.8 0.4 0 50 100 150 200 250 300 shear rate (s-) = 300 100 t = 200 50 Figure 5-7: Comparison of the bond lifetimes in the simple drop case. The dot corresponds to Schmidtke and Diamond (2000) results and the line to the present study. N, = 0.02, N, = 1. 0, ff = 0. 1, 0. 04, )7 = 1.2. Compound Drop Model Figure 5-8 illustrates the problem considered for the compound drop case. Specifically, a nucleated cell is attached to a vessel wall, and a uniform flow is imposed at the inlet of the vessel tube. In the model, the nucleus occupies 44% of the volume of the cell (Schmid- Schonbein et al., 1980), and the theological properties (viscosity and surface tension) of the nucleus are taken to be 10 times those of the cytoplasm and cellular membrane. The kinetics parameters used are the same as in the simple drop model. The problem to be solved is the same as before: a cell is attached to a vessel wall, with a uniform flow imposed at the inlet of the vessel tube. It should be noted that the nucleus of a neutrophil is small and segmented. Its contribution may not be as large as the one predicted by the present model but it is difficult to assess this at the present time. A neutrophil nucleus is asymmetric and it has been shown by Kan et al. (1999) that nucleus eccentricity affects the instantaneous shapes of the cell during recovery. In addition, Kan et al. (1998, 1999) have shown that the presence of a nucleus, although small in size, is needed in order to reconcile the various leukocyte theological data published in the literature. The compound drop model describes better the structure of a lymphocyte, but it is still a good model for evaluating the effect of key parameters on the theology and adhesion of leukocytes in general. Plasma Cytoplasm Nucleus Figure 5-8: Schematic of the problem statement for the compound drop model. Cell Movement and Deformation In this paragraph, the displacement and deformation of the cell are studied for different values of cell viscosity, cell surface tension and the inlet boundary condition. Effect of the Nucleus Figure 59 shows the effect of the nucleus. The nucleus tends to delay the rolling of the cell along the vessel wall but does not appear to affect the bond length before peeling. We also found that the reverse reaction constant within the range studied, 10-2 <- k,, :! 10-1 , has no significant effect on the position and displacement of the cell or nucleus. The curves of y/R plotted against t for the compound drop are identical to the one for the liquid drop in Figure 5-2b. 0.14 0.12 WitWith nucleus 0.08 I 1.31 1.35 1.39 1.43 xIR Figure -5-9: Effect of the nucleus on the rolling of the cell along the vessel wall. ljo 1.0, U~oi 01t= 10,)'= 1.0. Effect of the Viscosity The effect of viscosity value on the rolling of the cell is investigated for jt 10 and 100 and is shown in Figure 5-10. A cell with a lower cytoplasmic viscosity deforms more and remains closer to the surface as shown in Figure 5-11. 0.14 0.12- 00 S100 0.1 - 10 1.31 1.36 1.41 1.46 1.51 XIR Figure 5-10: Effect of the viscosity on the rolling of the cell along the vessel wall, 1% 1. 0,U =0. 1 , )/ 1. 0. Time (t) [t10 [t =100 0.50 1.0 1.5 Figure 5-11: Effect of Viscosity on the corresponds to the wall. instantaneous cell Shape. The horizontal line Effect of the Surface Tension Surface tension is also found to have a major effect on cell adhesion and rolling as can be seen in Figure 5-12. A lower surface tension allows the cell to maintain a larger curvature, which enable the cell to remain attached with a larger contact area (Figure 513). Consequently, a lower surface tension delays the liffing of the cell from the wall. 0.14 0.12 0.08 -I 1 1 1.31 1.35 1.39 1.43 xIR Figure 5-12: Effect of the interfacial tension on the rolling of the cell along the vessel wall, kr10 1. 0,U =0.lj t 100. Time (t) y = 5 Y = I Y = 0.1 1.0 2.0 3.0 5.0 Figure 5-13: Instantaneous cell shape horizontal line corresponds to the wall. Effect of the Wet Boundarv Condition for different values of the interfacial tension. The The effect of the inlet velocity is shown in Figure 5-14. As the inlet velocity is increased, the movement of the cell along the vessel wall accelerates and the bond breaks at a shorter distance from the wall. This is due to the fact that when the inlet velocity is increased, the cell is more deformed and stays closer to the wall as shown in Figure 5-15. 0.14 0.12 0.1 0.08 1.5 1.7 Figure 5-14: Effect of the inlet velocity on = 1.0, 4t = 100,)y= 1.0. the rolling of the cell along the vessel wall, 1% Time (t) U=0.1 U=0.5 U=1.0 1.0 0 0 Ã¯Â¿Â½ 2.0 Q 0 0 3.0 Q0 Q 0 Figure 5-15: Instantaneous cell shape for different values of inlet velocities. horizontal line corresponds to the wall. The Effect of the CaDillarv Number Figure 5-16 shows the relative effects of U, [t, and y on the behavior of the cell for a given capillary number (Ca=U~t/y). - - - - - - - - - . 'I - 0.298 0.288 - " - -- U =0.1, I= 10, Y 1.0 0.278 U =0.2, I= 10, y= 2.0 , U =0.1, L=20, y= 2.0 0.268 1.23 1.33 1.43 1.53 1.63 x/R Figure 5-16: Effects of v, [L, and y for Ca 10 on the cell behavior, kro = 1.0 We should point out that when we vary, say, viscosity, other parameters like the viscosity ratio between media, as well as the contribution of the bond force also change. On the other hand, if we hold all other dimensionless parameters the same, and vary Ca, then we will see a unique correlation. This is better understood by looking at the dimensionless form of the equation of motion in the x-direction, for example. By rewriting the equation of motion: pu = - + FÃ¯Â¿Â½F (32) at y) a2ab In dimensionless form using the following reference values, the subscript i refers to the medium of interest: Uf: = 0U, Pref = Yo RUI to = 7O R b Nf= ref-, -, U ~,Fref 2, -- efb (33) U-' R2 R Where U =inlet velocity, R =tube radius, [to =viscosity of the plasma, y =interfacial tension, cy spring constant, kX equilibrium bond length, Nr =receptor density, we obtain: Ni (T 02iT 02iT 1- R2LaNb(4 Re =- -Ã¯Â¿Â½+a (- + ~ r)+- +F (4 at ay N2 62 Ca Y0U0 Where ai [ti/[to is the viscosity ratio. Peeling Time and Rolling Velocity Calculation In what follow, we compute the peeling time, the rolling velocity of a cell adhering to a wall and the force of a bond molecule for different cell properties and adhesion parameters. The following defined parameters will be used for the subsequent computations. a=-' and PA"'l (35) PO10 lI Where [to is the plasma viscosity, [t, is the cytoplasm viscosity and [tn is the nucleus viscosity. The reference parameters used are given in Table 5-3 previously listed in the liquid drop model paragraph. Effect of the Inet Velocity Figure 5-17 shows the effect of the inlet velocity flow on the peeling time of the cell from the substratum. An increase of the inlet velocity decreases the peeling time. An inverse relationship can be used to fit the curve as shown in Figure -5-17, and is of the form: T =26.84t7047 (36) where T is the dimensionless peeling time and CU is the inlet velocity flow. E 60 a40 0 0 0.2 0.4 0.6 0.8 1 U Figure 5-17: Effect of the inlet velocity U on the peeling time a.c 100, 13 10, Nr 1. 0, N, 1. 0, ff 1. 0, f~0.2, 7 1. 0. The dot points correspond to the numerical results and the line is the curve fitting. Figure 5-18 shows the effect of the inlet velocity flow on the rolling velocity of the leukocy-te. The rolling velocity is defined as the ratio of the distance traveled by the cell to the peeling time. Here only the displacement of the cell along the xaxis is shown. An increase of the inlet flow increases the rolling velocity of the cell. The relationship between inlet velocity flow and cell rolling velocity, 17, is given by V, = 0.0055C3 -0.014U2 + 0.017UÃ¯Â¿Â½+0.002. (37) 0.011 0.009 >0.007 0.005 0 0.2 0.4 0.6 0.8 1 U Figure -5-18: Effect of the inlet velocity on the rolling velocity of the leukocyte. a~ 100, P3 = 0 N' 1. 0, N1 =1. 0, ff 1. 0, f, 0.2, 7 1. 0. The dot points correspond to the numerical results and the line is the curve fitting. Effect of the Receptor Density Figure 5-19 and Figure 5-20 show the effects of the receptor density on the peeling time and rolling velocity. An increase of the receptor density increases the peeling time and decreases the rolling velocity. Dong et al. (1999) also found an increase of the peeling time with receptor density. The evolution of the peeling time as a function of the receptor density can be described by the followAing relationship. T =3.495Ln(N,)Ã¯Â¿Â½+76.90 (38) V,~ 0. 003 7N 046 (39) 79 77 E cm 7571 1 0.2 0.4 0.6 0.8 1 1.2 1.4 Nr Figure -5-19: Effect of the receptor density N.on the peeling time t . a~ 100, J3 10, U =0. 1, N, 1.0, ff 1.0, f~0.2, y 1.0. The dot points correspond to the numerical results and the line is the curve fitting. 0.00395 0.00365 0.00355IIIII 0.2 0.4 0.6 0.8 1 1.2 1.4 4 Figure -5-20: Effect of the receptor density K, on the rolling velocity ,. a =100, J 10, U=0.1, N, =1.0, f =1.0, 0.2, y=1.0. The dot points correspond to the numerical results and the line is the curve fitting. Effect of the Ligand Density In Figure 5-21 and Figure 5-22, the effects of ligand density on the peeling time and the rolling are shown. An increase of the ligand density increases the peeling time and decreases the rolling velocity for N, <1 .0. For N, > 1.0, the peeling time and the rolling velocity remain the same. The following relations are used to describe the dependence: T 2.63Ln(N,) + 76.91 (40) V, 0.00 003 52 (41) The plateau is reached because no receptor is available for binding as the ligand density increases. Dong at al. (1999) and Alon et al. (1998) made the same observation in their simulation and experiments respectively. 78 S76 4 72 0.2 Figure 5-21: Effect of the ligand density N1 on the peeling time T. 100, P3 10, U=0.1, N r1.0, Y=1.0, f,=0.2, Y-=1.0. The dot points correspond to the numerical results and the line is the curve fitting. 0.0039 0.0038 0.0037 0.0036 0.4 0.6 0.8 1 1.2 Figure 5-22: Effect of the ligand density N1 on the rolling velocity VÃ¯Â¿Â½. U=0.1, Nr=1.0, 6;=I.0, f,=0.2, y-=1.0. The dot points numerical results and the line is the curve fitting. c = 100, 3 = 10, correspond to the 0.4 0.6 0.8 1 1.2 1.4 Effect of the Surface Tension The effect of the surface tension on the peeling time shows an inverse dependence as observed in Figure 5-23. 180 S130 ~80 30 0 2 4 6 8 Figure 5-23: Effect of the Nr 1.0, N, 1.0, f =I surface tension on the peeling [.0, f,=0.2, /3=10. The time T. u. 100, U =0. 1, dot points correspond to the numerical results and the line is the curve fitting. The following relation can be used to describe this dependence T =82.78y-0.3483 (42) In addition, an increase of the surface tension increases the rolling velocity as shown in Figure 5-24. For 7 V' =0.0018)72Ã¯Â¿Â½ 0.0043)7+ 0.001, (43) while for )T >!1. 0, the relation is given by -7 .10-6y3 _10-4y2Ã¯Â¿Â½o13oo2 (4 (44) 1 0.3 0.5 0.7 0.9 0.007 >0.005 0.003 3 5 7 9 Figure 5-24: Effect of the surface tension on the rolling velocity 17,. a = 100, U=0.1, Nr 1.0, N, =1.0, U=1.0, f, =0.2, 3 =10, 7<1! .0 (a) and 7>1A.0 (b). The dot points correspond to the numerical results and the line is the curve fitting. Effect of the Parameter 13 Figure -5-25 and Figure 5-26 show the effect of the parameter f3, which is the ratio of the nucleus viscosity to the cytoplasm viscosity, on the peeling time and rolling velocity. An increase of j3 increases the peeling time and decreases the rolling velocity. A third order polynomial can be used to fit the computational points. 0 E 0 60 50 2 4 6 8 Figure 5 -25: Effect of P3 on the peeling time. a = 100, U7 0. 1, Nr 1. 0, N1 =1. 0, ff 1.0, f, 0.2. The dot points correspond to the numerical results and the line is the curve fitting. 0.004 0.003 0.002 0.001 0.005 0.004 0.003 2 4 6 8 10 0 Figure -5-26: Effect of P on the rolling velocity. a = 100, 17 = 0. 1, IV, = 1.0, 91 = 1.0, ff = 1.0, f, = 0.2. The dot points correspond to the numerical results and the line is the curve fitting. Effect of the Parameter a The effects of the parameter u., defined as the ratio of the cytoplasm viscosity to the plasma viscosity, on the peeling time and rolling velocity are illustrated in Figures 527 and 528 respectively. An increase of u. slows the breakage of the formed bonds and the cell rolling velocity. A third order polynomial and an inverse power are found to best fit the responses of the peeling time and the velocity, respectively. 200 150 E 100 CL 5 0 0 0 50 100 150 200 250 300 a Figure 5-27: Effect of u. on peeling time. 10, 17 = 0. 1 , 1. 0, 1. 0, ff 1. 0 , f, = 0.2. The dot points correspond to the numerical results and the line is the curve fitting. 0.012 0.008 0.004 0 0 50 100 150 200 250 300 Oc Figure 5 - 28: Effect of a on the rolling velocity. 10, U = 0. 1 1. 0, N, = 1. 0, ff = 1.0, f, = 0.2. The dot points correspond to the numerical results and the line is the curve fitting. Effect of the Contact Length The effect of the initial contact length was also investigated in this study. An increase of the contact length augments the initial bond number as indicated by Equation (45): nb = 1.40 + 7 (45) where rib is the number of bonds and 0 is defined in Figure 5-29. 0 Contact length Figure 5-29: Angle and contact length definitions In addition, an increase of the contact length rises the peeling time and reduces the cell rolling velocity as shown in Figures 5-30 and 5-31, respectively. The 64 relationships between the initial contact length, the peeling time and the velocity are given by: 7- = 0.0125 02 + 0.8250 + 67.813 (46) (47) V, = -4. 10-'0 + 0. 0041 20 30 angle Figure 5-30: Effect of the contact length on the peeling time T. a = 100, U=0.1, N, = 1.0, 91 = 1.0, ff = 1.0, f, = 0.2, 0 = 10, 1.0. The dot points correspond to the numerical results and the line is the curve fitting. 0.00350.0030.00250.002 10 20 30 angle Figure 5-31: Effect of the surface tension on the rolling velocity 17. a = 100, U=0.1, N, = 1.0, 91 = 1.0, ff = 1.0, f, = 0.2, 0 = 10. The dot points correspond to the numerical results and the line is the curve fitting. Comparison to Experimental Results A comparison of our results to the experimental results obtained by Schmidtke and Diamond (2000) is conducted by first fixing the cell surface tension and varying the cell viscosity and then the cell viscosity is held fixed while varying cell surface tension. In this computation, the number of bonds is taken to be small. Effect of Cell Viscosity on Bond Lifetime Comparisons between the experimental data by Schmidtke and Diamond (2000) and the compound drop model results of N'Dri et al. (2002) are shown in Figure 5-32, for viscosity values of 50, 100 200 and cytoplasmic viscosity to nucleus viscosity ratios of 5 and 10. The interfacial tension is fixed at 1.2. The lifetime of the bonds decreases with the amount of applied shear rate, and if the cell cytoplasm is less viscous. Within the standard deviations of the experimental data, the compound drop model with a cytoplasmic viscosity equals to 50 fits the data. However, the model with a cytoplasmic viscosity of 100 describes better the experimental data for shear rates up to 250 ! '. it is also seen that if the nucleus is more viscous, it will increase the lifetime of the bonds. Effect of Cell Surface Tension on Bond Lifetime As shown in Figure 5-33, fixing the cell viscosity and varying the surface tension allows us to assess the surface tension effect on the bond lifetime. Higher surface tension shows agreement with the experimental results obtained by Schmidtke and Diamond (2000). Again an increase of surface tension decreases the bond lifetime. ---------- shear rate s1 a=200,3 10 ------- a =200,3 5 . . . 100, = 10 a50, P 10 Figure 5-32: Comparison of the bond lifetimes for the compound drop case. The corresponds to Schmidtke and Diamond (2000) results and the lines to N'Dri et (2002). Nr 0.02, N1 =1.0, ff 0.1, f, 0.04, )7=1.2. 1.2 0.8 0.4 0 100 150 200 7 =O.8 )7 1.0 7-3.0 250 300 shear rate (s-1) Figure 5-33: Effects of surface tension and shear rate on the bond lifetime: compound drop model. The dot corresponds to Schmidtke and Diamond (2000) results and the lines to the present study. NVr 1. 0, N1 =1. 0, ff 0. 1, f, 0. 04,c a l=100and P3 10. 0.4 0 50 Effect of Applied Hydrodynamic Force on the Bond Lifetime The bond lifetime is shown as a function of the applied hydrodynamic force for a cytoplasmic viscosity equal to 200 in Figure 5-34. An exponential function, as proposed by Shao and Hochmuth (1999), can be used to describe the results as indicated in Figure 4-34a. The hydrodynamic force is computed as follows F =pUd (48) where t is the plasma viscosity, U the inlet velocity and d the tube diameter. 0.8 0.8 0.7- b 0.7 06-)~ 0.60.6 0.5- 0.50.4 0.4 600 1000 1400 1800 600 1000 1400 1800 F (g~dy ne) F(Jidyne) Figure -5-34: Bond lifetimes as a function of applied force. N.0.02, N1 =1. 0, ff 0. 1, f, 0.04, )7 1.2, /3 1I0, u. 200. The dot points correspond to the numerical results and the line is the curve fitting, (a) exponential fit; (b) inverse power fit. The exponential relationship, t =0.95e-0004 (49) provides a good fit to the data, however, an inverse correlation between the applied force or pulling force and the bond lifetime, gives the best fit to the data (Figure 5-34b). An increase of the pulling force decreases the bond lifetime as observed by many authors (Evans, 1999; Smith and Lawrence, 1999). t19.1 (50) Tp, Bond Rupture Force Fbr The force of rupture of a single bond, hur, was also determined. This force is the force reaches by the bond before it breaks. The critical bond forces computed by Goldman et al. (1967b) and Dong et al. (1999) are reported to be between 100 and 400 pN and 100 and 200 pN, respectively. In the literature, experimental measurements give forces of rupture between 37 and 250 pN (Shao and Hochmuth, 1996; Smith and Lawrence, 1999). Evans (1999) found bond strength of 200-300 pN for biotinstreptavidin and a value of bond strength of order 160pN for biotin-avidin. In the work of N'Dri et al. (2002), the values of the bond molecule force vary from 250 to 400 pN for a range of shear flows between 2 and 20 dyne/cm2 and a spring constant ff = 0.1. These results are shown in Figure 5-35 for c = 100. The value of ff= 0.1 corresponds to a dimensional value of 0.5 dyne/cm as used in the computation of Dong and Lei (2000). Lower values of Fbr can be obtained by decreasing the value of the spring constant. A lower force of rupture means a shorter bond lifetime. Figure 5-35 demonstrates a linear relationship between bond lifetime and force of rupture, t = 0.0013Fb, - 0.206 (51) N'Dri et al. (2002) found that using a bond stiffness - _< 0.06 (cy = 0.3 dyn/cm) gives the range of bond forces reported in the literature. It is important to note that Shao and Hochmuth (1996) have reported a spring constant value of 43pN/[tm (0.043 dyne/cm), which is much smaller than the values cited above. Using the range of force computed in this study, the length reached by a bond before breakage was computed and is found to lie between 0.25 and 0.85 tm. The measured bond length found in the literature is between 0.1 and 0.7 tm (Shao et al., 1998). Effect of Viscosity on Bond Rupture Force The bond lifetime as a function of the shear for different values of the viscosity was shown in Figure 5-33. Using equation (51), one can assess the effect of viscosity on rupture force of a bond for a fixed value of the inlet boundary condition. The result is shown in Figure 5-36. An increase of the viscosity for a fixed applied hydrodynamic force increases linearly with the rupture force. 0.4 0.3 0.2 0.1 011 200 300 400 Fbr(PN) Figure 5-35: Bond lifetime t as a function of bond force Fbr. Nr 0.02, N1 =1.0, ff 0. 1, f, 0. 04, j7 1. 2, /3 10. The dot points correspond to the numerical results and the line is a curve fitting. 700 z U 300II 50 100 150 Figure 5-36: Effect of the viscosity on the bond rupture force. N, = 0.02, N1 =1.0, ff 0. 1, f, 0.04, )7 1.2, /3= 10. Effect of Interfacial Tension on Bond Rupture Force Using Figure 5-34 and Equation (51), we can assess the effect of the surface tension on the bond rupture force. An increase of the surface tension inversely decreases with the bond rupture force for a fixed applied hydrodynamic force as shown in Figure 5 37. 1550 -6 1050550 0.1 1.1 2.1 Y Figure 5-37: Effect of the surface tension on the bond rupture force. 0.02, N, = 1.0, ff = 0.1, 0.04, 10, u=100. Multiple Bond Case Finally, the case of multiple bonds is shown in Figure 5-38. In Schmidtke and Diamond (2000), only a shear rate of 150 1 was studied. In the present computation, as in the case of a single bond, shear rates vary between 100 and 250 91, the cytoplasmic viscosity is 100, and the interfacial tension ranges from 0.1 to 3.0. The number of bonds is not known in the work of Schmidtke and Diamond (2000), but in the simulation of N'Dri et al. (2002) the bond number is 7. The bond lifetime is higher for a cell with a lower surface tension. The top curve corresponds to the cell with the lowest surface tension. In addition, an increase of bonds or cell viscosity will extend the bond lifetime as suggested in Figures 5-19, 5-25, 5-27. 8 Y0. 1 Y= 0.4 .I7. y =1. 0 . Y = 3.0 (4 2 0 50 100 150 200 250 300 shear rate (s-1) Figure 5-38: Comparison of the computed bond lifetimes in the case of compound drop and multiple bonds. The dot corresponds to Schmidtke and Diamond (2000) results and the line to N'Dri et al. (2002). a -100, N, = 1.0, N1 =1.0, U 1.0, f, 0.04, /3= 10. In summary, from the work of N'Dri et al. (2002) using the multi-scale approach outlined in chapter 4, it is shown that cell deformability affects the rolling of a cell along a vessel wall. A cell with a higher surface tension rolls faster. These results are in agreement with those of Dong and Lei (2000) where only a small section of the membrane is allowed to deform. The results also show that as the shear rate increases, the bond lifetime decreases as observed by Schmidtke and Diamond (2000). The bond lifetime is also found to increase as the cell or nucleus viscosities increase. Effect of the Cell Diameter There are three types of leukocytes with different diameters, the lymphocytes with 7[Lm, the neutrophils with 8[Lm, and the monocytes with 10[Lm in diameter in average. The effect of different cell diameters for a fixed tube diameter on the peeling time is shown in Figure 5-39. A larger cell diameter decreases the peeling time and increases the cell rolling velocity as shown in Tran- Son-Tay et al. (2002). This result is consistent with the simulation and experimental results found by Tees et al. (2002) and by Patil et al. (2001), respectively. A larger cell causes a higher blockage in the tube, which results in larger hydrodynamic forces. Considering that the velocity scales inversely and linearly with the tube opening (and hence scales linearly with the cell size) to satisfy mass continuity, and the pressure force scales quadratically with the velocity, the hydrodynamic force on the cell increases more than linearly with the cell diameter. 3.6 3.3 3 2.7 7 8 9 10 cell diameter Wli) Figure 5-39: Effect of cell diameter on the peeling time of the cell. a = 100, U 0. 1, N, 1.0, N, 1.0, U 1.0, f0 0.2, 5 5. Effect of the Vessel Diameter The blood vessel diameter varies from centimeter to micrometer in the circulatory system, in order to study the effect of the vessel diameter on cell adhesion, we compute the peeling time of an adherent cell for three different diameters. Figure 5-40 shows the result of such a computation, where we found that an increase of the vessel diameter decreases the peeling time and increases the cell rolling velocity. Increasing the vessel diameter increases the applied hydrodynamic force. 2.95 2.85 2.75 2.65 0.6 0.7 0.8 0.9 1 Figure 540: Effect of vessel diameter on the peeling time of the cell. U = 100, U = 0. 1, N, = 1.0, N, = 1. 0, ff = 1.0, 0.2, 5 . Effect of the Pulsatile Inlet Boundary Condition The effect of the inlet pulsatile boundary condition on the peeling of the cell is also assessed. Equation (52) shows the boundary condition used and Figure 5-41 shows the number of pulses studies in the present work. 1 4 U, (t) = - Y sin(nir t) (52) Where Up is the inlet velocity and t the time. 0.3 0.2 0.1 0 0 5 10 15 20 -0.1 t Figure 5-41: Pulsatile (line) and uniform (dashed line) inlet boundary conditions used in the present work. Comparison of the Peeling Time for the Uniform and Pulsatile Conditions Figure 5-42 shows a comparison of the pulsatile and uniform inlet boundary conditions for different values of the surface tension. The uniform boundary condition is computed by taking the average of the pulsatile velocity over one period. U=- f Up Q) dt (53) T 0 where T is the time period. We observed that the peeling time is higher for the uniform inlet velocity than that of the pulsatile boundary condition. 70 .E "" 55 CL 40 25 1 2 3 4 5 6 Figure 5-42: Comparison of peeling time for the pulsatile (line) and uniform (dashed line) inlet boundary condition for different values of the surface tension. U = 100, N, = 1.0, 91 = 1. 0, ff = 1.0, f, = 0.2, 0 = 10. Effect of the Surface Tension on the Pulse Number Figure 5-43 shows the effect of the cell surface tension on the number of pulses needed to peel the cell away from the wall. An increase of the surface tension decreases the number of pulses. The approximate curve that can be used to fit the experimental points is given below: np = -0. 821n07)+ 2.08 (54) where np is the number of pulses 4 3 CL 4- 2 0 E M z 0 0 1 2 3 4 7 Figure 5-43: Effect of the cell surface tension on the number of pulses np. a 100, N, = 1.0, NI = 1.0, ff = 1.0, 0.2, 0 =I 0. The dot corresponds to the numerical results and the line is the curve fitting. Effect of the Parameter a on the Pulse Number The study of the effect of the parameter P, ratio of the nucleus viscosity to the cytoplasmic viscosity, has shown no difference for the range (1:! P:! 10) used in the present study. On the other hand, the effect of the parameter U, ratio of the cytoplasmic viscosity to the plasma viscosity, shows a linear dependence of the pulse number 4 with u. as shown in Figure 5-44. np = 0.01a (55) In summary, the pulsatile inlet boundary condition has shown that surface tension and the parameter u., the ratio of the cytoplasm viscosity to the plasma viscosity, have major impact on the number of pulses needed to peel the cell away from the vessel wall while the parameter P, the ratio of the nucleus viscosity to the cytoplasm viscosity, has shown no impact on the number of the pulses needed to detach the cell for the inlet boundary conditions used in the present study. 3 CL CL 4- 2 0 E M Z 100 200 300 a Figure 544: Effect of the parameter a on the number of pulses, r . 1.0, 91 = 1.0, U = 1.0, f, = 0.2, 0 =I 0. The dot corresponds to the numerical results and the line is the curve fitting. Effect of the Initial Bond Force In all the previous computations, the initial bond force was taken to be zero, in the following computation; we assume an initial bond force, which varies with surface tension. The initial bond force is computed as follows: Fb, =ry (56) where r is the cell radius and y is the cell surface tension Figure 5-45 shows a comparison of the peeling time for both non-zero and zero initial bond force for different values of surface tension. The peeling time is higher in the case of non-zero initial bond force. In most of the studies conducted over the past years, no initial bond force was assumed in the computation of the bond force. 78 79 69 E P 0)59 0-49 39 1 2 3 4 5 6 Figure 5-45: Comparison of peeling time for the zero (line) and non-zero (dashed line) initial bond force for different values of the surface tension. u. = 100, 10, 1. 0, 91 = 1.0, f, = 0.2 CHAPTER 6 CONCLUSION AND) DISCUSION In this work, a multi-scale modeling for biofluid dynamics with an emphasis on cell adhesion was presented. This method breaks the computational work into two separate but interrelated domains. At the cellular level, a continuum model satisfying the field equations for momentum transfer and mass continuity was adopted. At the receptorligand or molecular level, a bond molecule was mechanically represented by a spring. A reversible two-body kinetic model characterized the association and dissociation of a bond. Communication between the macroscopic and microscopic scale models was facilitated interactively in the course of the computation. The computational model was assessed using an adherent cell allowed to roll along the vessel wall under imposed shear flows. The cell was first modeled as a liquid drop with constant surface tension to illustrate the computational approach, and then as a compound drop in order to evaluate the effect of the nucleus. The results compare very well with those obtained computationally (Chang and Hammer, 2000;Dong et al., 1999; Dong and Lei, 2000; Tees et al., 2002) and experimentally (Patil et al., 2001; Schmidtke and Diamond, 2000). We have shown that cell deformability affects the rolling of a cell along a vessel wall. A cell with a higher surface tension rolls faster. These results are in agreement with those of Dong and Lei (2000), where only a small section of the membrane is allowed to deform. The results also showed that as the shear rate increases, the bond lifetime decreases as observed by Alon et al. (1997, 1998) and Evans (1999), Schmidtke and Diamond (2000), Smith et al. (1999). The bond lifetime was also found to increase as the cell or nucleus viscosities increase. A decrease in cell diameter increases the peeling time and decreases the rolling velocity as observed by Tees et al. (2002) and Patil et al. (2001). We also studied the effect of vessel diameter on the adhesion process. We found that increasing the vessel diameter decreases the peeling time and increases the rolling velocity. In addition, we have shown that the peeling time is smaller for a pulsatile inlet condition than that of a uniform one and a non-zero initial bond force increases the peeling time. The bond lifetime was also shown to increase with the critical bond force, which is the force reaches by a bond molecule before breaking. An increase of the cell viscosity increases the critical bond force while increasing cell surface tension decreases the bond rupture force. In this work, we use the kinetic model proposed by Dembo et al. (1988), a drawback with this model is the assumption that the bonding stresses are the only distributed stresses acting on the membrane and bonds molecules are fixed on cell surface while Bell's model takes into account the role of the non specific potential forces in biological chemistry and the diffusion of the bond molecules to the contact area. Nonetheless, the results obtained in this study won't be affected by the Bell's equation for the reverse reaction rate. Many computational studies have considered the cell as a solid body. Some state that the deformation of the cell would not have a significant impact on rolling of the cell, but microvillus deformation is more important. However, the microvilli are connected to the cell surface; hence, a deformed microvillus also means cell deformation. The study by N'Dri et al. (2002) has shown that a tether was pulled away from the cell membrane at high values of shear stress, but no membrane pulling was observed at low values of shear stress (Figure 5-15). These observations are consistent with what have been seen in the literature (Alon et al., 1997; Schmidtke and Diamond, 2000; Shao et al., 1998). Shao et al. (1998) have found that after the microvillus reaches its natural length, it will extend under a small pulling force or form a tether under a high pulling force. In addition, we have also observed a liffing of the cell from the vessel wall leading to its peeling as shown in Figure 5-2b. Similar observations were made by Sukumaran and Seifert (2001), who studied the influence of the shear flow on vesicles near a wall, and by Hodges and Jensen (2002) in their numerical study. The graph of the height of center of the cell above the plane increases rapidly with time, then reaches a plateau as observed in Figure -5-2b of the present study. In addition, in the study by Hodges and Jensen (2002), they found that changing the suspending fluid viscosity does not affect the rolling velocity of the cell along the wall. In their study, the viscosity of the cell was taken to be very small, so the fluid inside the cell was considered as invinscid. While in this study we have shown that fixing the suspending viscosity and varying either the cytoplasmic or/and the nucleus viscosity affect the rolling velocity of the cell showing that hydrodynamics do affect cell adhesion. It is also known that a higher shear stress can enhance bond formation. How this phenomenon occurs is not clear. Does an increase of the shear stress increase first the diffusion and convection of bond molecules toward the contact area, or does an increase of the shear stress lead to an increase of the contact area due to cell deformation? Although, in our study, bond foundation is not considered per se due to the numerical resolution (we consider that a bond is formed when the distance between a receptor and ligand is less than a bond length, taken as 5xI 0-6CM), we can nevertheless comment from Figure 5-14 that an increase of the shear stress keeps the cell closer to the wall, and increases the contact area, as shown in Figure 5-15. The finding of the present work is also consistent with the cell shape observed experimentally by Schmidtke and Diamond (2000) and numerically by Dong et al. (1999) and Dong and Lei (2000), who have shown that the cell goes from a spherical form to a tear-like drop shape. Although the proposed approach was able to describe key features of cell adhesion, several issues still need to be addressed. For example, experimental results have shown that there is a threshold stress above which cell rolling occurs. However, the models proposed in the literature cannot capture this feature, indicating that the straightforward analogy between a bond molecule and a spring model is incomplete. A bond model with a yield force can be used to help resolve this deficiency. Furthermore, in the present study we have neglected membrane roughness and the effect of nonspecific forces in the bond force computation. With respect to cell theology, although the compound liquid drop model can capture most of the features of cell recovery and explain the reasons for the different published values for leukocyte viscosity, it does not include elastic effects which may be needed to fully describe leukocyte theology (TranSon-Tay et al., 1994, 1998; Drury and Dembo, 1999, 2001; Kan et al., 1998, 1999). In addition, we assumed that the bond molecules are fixed on the membrane surface, which is not the case since bond molecules have been shown to diffuse laterally to the contact area. The last issue is the assumption of a 2D cell model, which has some limitations such as force estimation with respect to 3D assumption. Another issue is the error related to the multi-scale model. It is noted that the adhesion model is kinetics-based and is statistical. One approach to address the error associated with the micro-macro interactions is to consider the uncertainty of the microscopic prediction and assign a distribution function. This error distribution will then be incorporated into the adhesion model, based on which a statistical solution will be obtained, with the mean transmitted to the macroscopic model. In order to compute the bond lifetime, we have assumed that bonds are broken if the cell travels 2 cell-radii based on non-physical cell shape beyond this distance. This assumption is solely based on cell mechanic and not on a single bond mechanic. To overcome this difficulty, we propose to use bond molecule strength as a condition for bond breakage. Therefore, the bond lifetime will be the unknown parameter that will depend on the hydrodynamic pulling force and the cell theological properties. Nevertheless, the present effort has offered a comprehensive framework to couple the cellular and the receptor-ligand dynamics and significant knowledge has been gained from this work. In the future, we will exploit the fict that the time scale for the formation of a new interface is fairly short, of the order on I nanosecond (ns) or less, and can therefore be tracked explicitly by large-scale parallel molecular dynamics simulations. Similar to the present model, the boundary conditions on the molecular simulation will be obtained from the continuum simulation and vice versa. It is simply a matter of time that a more comprehensive framework will be developed to offer new insights into the very complicated processes that occur in the dynamics of blood cells and blood flow. |

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PAGE 1 MULTI SCALE COMPUTATION IN HEMODYNAMICS By NARCISSE ABOU NDRI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2002 PAGE 2 Copyright 2002 by Narcisse Abou NDri PAGE 3 To Jean Claude Lelievre, PAGE 4 iv ACKNOWLEDGMENTS I would like to thank Dr. Tran Son Tay not only for giving me the opportunity to finalize my education in USA but also for helping me since my arrival in Gainesville. To Dr. Shyy Wei, I would like to extend my appreciation and respect for g iving me the possibility to conduct my research in computational fluid dynamics with moving boundaries. As stated by Hobbes, All thought is a kind of Computation. I thank him for allowing me to discover this challenging field and for all the help and s upport to achieve this work. For Dr. Tran Son Tay and Dr. Shyy, I would like to quote one of my favorite thoughts, which is When you follow an elephant, the dew cannot touch you. I thank them again for everything. I sincerely thank Professor Ulrich Kurz weg, Professor Nicolae Cristescu and Professor Richard Dickinson for serving as members of my supervisory committee. While we were looking for a post doctoral position, my former advisor, Jean Claude Lelievre, decided that it would be better for me to purs ue my study in the USA and go for a PhD degree instead. I told him that this would be difficult because of the language barrier. His answer was that when I arrived in France, I had to go through difficulties, but I managed to overcome them, so I can succ eed anywhere if I put my mind on it. I thank professor Lelievre for believing in me. PAGE 5 v I would like to thank all the colleagues of the computational thermo fluid group, in particular Marianne Francois, Inanc Senocak, and the colleagues of the cellular mech anics and biorheology group. I would like to thank Patricia and Mark for all the years spent together. I have a lot to write about Kathleen, but I can summarize it through Boorstins thought, So long as man marked his life only by the cycles of nature -t he changing seasons, the waxing or waning moon -he remained a prisoner of nature. If he was to go his own way and fill his world with human novelties, he would have to make his own measures of time. I would like also to extend my thanks to her parents, Mr. and Mrs. Anderson. I would like to thank my Mom, Ipou Madeleine, and my sister, NDri Marie, for all the prayers said and moral support given throughout the years I spent in the USA and in France. I thank them also for being there when I needed them t he most. I thank my dad, NDri Bernard, and my uncle, Ipou Jean, for always being supportive of all decisions I have made through my years of education. PAGE 6 vi TAB LE OF CONTENTS Page ACKNOWLEDGMENTS ................................ ................................ ................................ .. iv LIST OF TABLES ................................ ................................ ................................ .............. ix LIST OF FIGURES ................................ ................................ ................................ ............. x NOMENCLATURE ................................ ................................ ................................ ......... xiv ABSTRACT ................................ ................................ ................................ ...................... xvi CHAPTER 1 MOTIVATION AND INTRODUCTION ................................ ................................ ...... 1 2 REVIEW ON CELL ADHESION ................................ ................................ .................. 7 3 COMPUTATIONAL METHOD BACKGROUND ................................ ..................... 21 Projection Method ................................ ................................ ................................ ......... 21 Continuity Equation ................................ ................................ ...................... 21 Momentum Equation ................................ ................................ ..................... 21 Boundary Conditions ................................ ................................ .................... 22 Immersed Boundary Technique (IBT) ................................ ................................ .......... 24 Interface Information ................................ ................................ .................... 24 Material Property Assignment ................................ ................................ ...... 25 Source Term Computation ................................ ................................ ............ 26 Interfacial Velocity Computation ................................ ................................ .. 27 4 DEVELOPMENT OF THE MULTI SCALE METHOD ................................ ............ 29 Model Over view ................................ ................................ ................................ ........... 29 Macro Scale Model ................................ ................................ ................................ ....... 30 Micro Scale Model ................................ ................................ ................................ ....... 31 PAGE 7 vii Micro Macro Couplin g ................................ ................................ ................................ 32 Macro and Micro Test Cases ................................ ................................ ........................ 33 Micro Scale Test Cases ................................ ................................ ................. 33 Macro Sca le Test Cases ................................ ................................ ................ 36 Micro Macro Coupling ................................ ................................ ................. 37 5 RESULTS ................................ ................................ ................................ ..................... 41 Liquid Drop Mod el ................................ ................................ ................................ ....... 41 Effect of Bond Molecule ................................ ................................ ............... 44 Effect of the Reverse Constant k ro ................................ ................................ 44 Effect of the Spring Constant ................................ ................................ ........ 44 Comparison to Existing Results ................................ ................................ .... 45 Comparison to Numerical Results ................................ ............................ 46 Comparison to Experimental Results ................................ ........................ 47 Compound Drop Model ................................ ................................ ................................ 48 Cell Movement and Deformation ................................ ................................ 49 Effect of the Nucleus ................................ ................................ ................. 49 Effect of the Viscosity ................................ ................................ ............... 50 Effect of the Surface T ension ................................ ................................ .... 51 Effect of the Inlet Boundary Condition ................................ .................... 52 Effect of the Capillary Number ................................ ................................ 53 Peeling Time and Rolling Velocity Calculation ................................ ........... 55 Effect of the Inlet Velocity ................................ ................................ ........ 55 Effect of the Receptor Density ................................ ................................ .. 57 Effect of the Ligand Density ................................ ................................ ..... 58 Effect of the Surface Tension ................................ ................................ .... 60 Effect of the Parameter b ................................ ................................ .......... 61 Effect of the Parameter a ................................ ................................ .......... 62 Effect of the Contact Length ................................ ................................ ..... 63 Comparison to Experimental Results ................................ ............................ 65 Effect of Cell Viscosity on Bond Lifetime ................................ ............... 65 Effect of Cell Surface Tension on B ond Lifetime ................................ .... 65 Effect of Applied Hydrodynamic Force on the Bond Lifetime .................... 67 Bond Rupture Force F br ................................ ................................ ................ 68 Effect of Viscosity on Bond Rupture Force ................................ .............. 69 Effect of Interfacial Tension on Bond Rupture Force ............................... 70 Multiple Bond Case ................................ ................................ ...................... 70 Effect of the Cell Diameter ................................ ................................ ........... 72 Effect of the Vessel Diameter ................................ ................................ ....... 72 Effect of the Pulsatile Inlet Boundary Condition ................................ .......... 73 Comparison of the Peeling Time for the Uniform and Pulsatile Conditions ................................ ................... 74 Effect of the Surface Tension on the Pulse Number ................................ 75 Effect of the Parameter a on the Pulse Number ................................ ....... 76 Effect o f the Initial Bond Force ................................ ................................ .... 77 PAGE 8 viii 6 CONCLUSION AND DISCUSION ................................ ................................ ............. 79 LIST OF REFERENCES ................................ ................................ ................................ .. 84 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 92 PAGE 9 ix LIST OF TABLES Table Page 2 1 Overview of some fundamental adhesion kinetic models. ................................ ........ 11 2 2 Adhesion parameters computed using Bell and Hookean spring models ................. 15 5 1 Average macro model parameters. ................................ ................................ ............ 42 5 2 Average micro model parameters. ................................ ................................ ............. 42 5 3 Reference values used for non dimensionalization. ................................ .................. 42 PAGE 10 x LIST OF FIGURES Figure Page 1 1 Multi Scale phenomena in hemodynamics ................................ ................................ .. 4 1 2 Illustration of the multi scale nature of hemodynamics ................................ .............. 5 2 1 Force balance on an adhered neutrophil mediated by a single attachment. .............. 12 2 2 Effects of shear stress and Bond force on the reverse reaction rate .......................... 15 2 3 Membrane tether formation and tear drop shaped deformation ................................ 17 2 4 Effect of wall shear rate on the bond lifetime ................................ ............................ 18 2 5 State diagram for adhesion. Four different states are shown ................................ .... 19 3 1 velocity and pressure location on a given grid cell ................................ ................... 23 3 2 Interface representation and numbering ................................ ................................ .... 25 3 3 Heaviside function definition illustration. ................................ ................................ 26 3 4 Source term computat ion at the grid point P and marker points contributing to the computation is also shown. ......................... 27 3 5 Interfacial point velocity computation and grids contributing to the computat ion is also shown ................................ ........ 28 4 1 Macro Micro model ................................ ................................ ................................ ... 30 4 2 Flow chart showing the interaction between the micro model and the mac ro model. ................................ ................................ .............................. 33 4 3 Geometry of the test case problem ................................ ................................ ............ 34 PAGE 11 xi 4 4 Maximum length of a bond before breaking as a function of the reverse reaction rate, for different values of forward reaction rate. ........................ 34 4 5 Effect of the ligand density on the de bonding time ................................ ................. 35 4 6 Effect of the slippage constant f s on the de bonding time for s (circle) and s ts (square) held constant respectively ................................ ....... 36 4 7 Instantaneous droplet shapes for s = 0.01 N/m (left) and s = 0.1 N/m (right). ........ 37 4 8 Instantaneous droplet shapes for two different values of the droplet viscosity ......... 37 4 9 Illustration of the instantaneous membrane shape resulting from the pressure field 38 4 10 Forward reaction rate as a function of time ................................ ............................. 39 4 11 Reverse reaction rate as a function of time ................................ .............................. 40 5 1 Schematic of the problem statement for the simple liquid drop ................................ 42 5 2 Instantan eous position of an interfacial point on the cell ................................ .......... 43 5 3 Effect of the bond molecules on the rolling of the cell along the wall ...................... 44 5 4 Effect of the reverse reaction rate k ro on the rolling of the cell ................................ 45 5 5 Effect of the spring constant on the rolling of the cell ................................ .............. 45 5 6 Comparison of the cell rolling velocity ................................ ................................ ..... 47 5 7 Comparison of the bond lifetimes in the simple drop case ................................ ........ 48 5 8 Schema tic of the problem statement for the compound drop model. ........................ 49 5 9 Effect of the nucleus on the rolling of the cell along the vessel wall. ....................... 50 5 10 Effect of the viscosity on the rolling of the cell along the vessel wall .................... 50 5 11 Effect of Viscosity on the instantaneous cell Shape ................................ ................ 51 5 12 Effect of the interfacial tension on the rolling of the cell along the vessel wall ..... 51 5 13 Instantaneous cell shape for different values of the interfacial tension ................... 52 5 14 Effect of the inlet velocity on the rolling of the cell along the vessel wall ............. 53 PAGE 12 xii 5 15 Instantaneous cell shape for different values of inlet velocities .............................. 53 5 16 Effects of v, m and g for Ca = 10 on the cell behavior, k ro = 1.0 ............................ 54 5 17 Effect of the inlet velocity U on the peeling time t ................................ .............. 56 5 18 Effect of the inlet velocity on the rolling velocity of the leukocyte ........................ 56 5 19 Effect of the receptor density r N on the peeling time t ................................ ........ 57 5 20 Effect of the receptor density r N on the rolling velocity r V ................................ ... 58 5 21 Effect of the ligand density l N on the peeling time t ................................ ............ 59 5 22 Effect of the ligand density l N on the rolling velocity r V ................................ ...... 59 5 23 Effect of t he surface tension on the peeling time t ................................ ................ 60 5 24 Effect of the surface tension on the rolling velocity r V ................................ .......... 61 5 25 Effect of b on the peeling time ................................ ................................ ................ 61 5 26 Effect of b on the rolling velocity ................................ ................................ ............ 62 5 27 Effect of a on peeling time ................................ ................................ ...................... 62 5 28 Effect of a on the rolling velocity ................................ ................................ ........... 63 5 29 Angle and contact length definitions ................................ ................................ ....... 63 5 30 Effect of the contact length on the peeling time t ................................ .................. 64 5 31 Effect of the surface tension on the rolling velocity r V ................................ .......... 64 5 32 Comparison of the bond lifetimes for the compound drop case .............................. 66 5 33 Effects of surface tension and sh ear rate on the bond lifetime ................................ 66 5 34 Bond lifetimes as a function of applied force ................................ .......................... 67 5 35 Bond lifetime t as a function of bond force F br ................................ ........................ 69 5 36 Effect of the viscosity on the bond rupture force ................................ .................... 69 PAGE 13 xiii 5 37 Effect of the surface tension on the bond rupture forc e ................................ ........... 70 5 38 Comparison of the computed bond lifetimes in the case of compound drop and multiple bonds ................................ ................. 71 5 39 Effect of ce ll diameter on the peeling time of the cell ................................ ............. 72 5 40 Effect of vessel diameter on the peeling time of the cell ................................ ......... 73 5 41 Pulsatile an d uniform inlet boundary conditions used in the present work. ............ 74 5 42 Comparison of peeling time for the pulsatile and uniform inlet boundary condition for different values of the sur face tension ...................... 75 5 43 Effect of the cell surface tension on the number of pulses n p ................................ .. 76 5 44 Effect of the parameter a on the number of pulses, n p ................................ ............ 77 5 45 Comparison of peeling time for the zero and non zero initial bond force for different values of the surface tension ................................ .. 78 PAGE 14 xiv NOMENCLAT URE a cell Area of the cell f s Slippage constant f b Force of a bond F b Total force of bonds k fo Equilibrium forward reaction rate k f Forward reaction rate k ro Equilibrium reverse reaction rate k r Reverse reaction rate K b T Thermal energy (k b = Boltz mann constant, T = temperature) N b Density of a bond N bo Initial bond density N l Ligand density N r Receptor density R c Cell radius R t Number of receptors U Inlet Velocity X m Current position of a bond on the membrane s Spring constant s ts Transit ion spring constant PAGE 15 xv s ts Transition spring constant l Equilibrium length of a bond g Cell surface tension PAGE 16 xvi Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MULTI SCALE COMPUTATION IN HEMODYNAMICS By Narcisse Abou NDri December 2002 Chair: Dr. Tran Son Tay Roger Cochair: Dr. Shyy Wei Major Department: Mechanical and Aerospace Engineering Blood flow dynamics involves length scales ac ross vastly different ranges. For example, adhesion of leukocytes to substrate entails dimensions ranging from nm to m m in length. Understanding the adhesive and rheological behavior of these cells is essential not only for describing microcirculatory fl ow dynamics, but also for understanding their function and behavior in health and disease. Models of cell adhesion do not combine molecular and full cellular information. In this study, a multi scale computational approach for studying the adhesion kineti cs and the deformation and movement of a cell on a substrate is presented. The cellular level model consists of a continuum representation of the field equations and a moving boundary tracking capability to allow the cell to change its shape continuously. PAGE 17 xvii At the receptor ligand level, a bond molecule is mechanically represented by a spring and a reversible kinetics model is used to describe the association and dissociation of bonds. Communication between the macro and micro scale models is facilitated interactively with iterations between models until both levels approach compatible solution in each time step. The computational model is assessed using a cell adhering and deforming along the vessel wall under imposed shear flows. It confirms existing n umerical and experimental results. In previous studies, the cell was modeled as solid body or a liquid drop. In the case of adhesion, only a small portion of the contact area was allowed to peel away from the wall. In this study, modeling the cell as a c ompound drop, we show that the presence of the nucleus increases the bond lifetime. We also show that increasing cell surface tension decreases the bond lifetime and that vessel diameter affects the cell rolling velocity. Furthermore, we find that the pe eling time for a uniform inlet boundary condition is higher than that of a pulsatile one. In addition, increasing cell viscosity for a fixed hydrodynamic force increases linearly the critical bond force, whereas increasing cell surface tension inversely d ecreases it. Finally, we have shown that a non zero initial bond force increases the peeling time. The present work shows that cell deformability and hydrodynamic flow affect cell adhesion. PAGE 18 1 CHAPTER 1 MOTIVATION AND INTRODUCTION Understanding the adhesion and rol ling of a white blood cell on the endothelium wall and its impact on the whole cardiovascular system flow and the blood flow through the aorta and its bifurcations motivate this dissertation. Fundamentally, t he blood flow through the cardiovascular system is a multi scale fluid mechanics problem. Blood dynamics involves length scales across vastly different ranges. An example is provided by leukocyte adhesion to a substrate, which is an important biomedical problem that has been the subject of extensive r esearch. Analyzing the adhesive and rheological behavior of leukocytes is essential not only for describing microcirculatory flow dynamics, but also for understanding cell function and behavior in health and disease. Computational fluid dynamics (CFD) te chniques provide us with powerful tools for tackling a variety of problems in engineering involving fluid flow accompanied by mass, momentum and energy transport. These methods have recently been applied to a variety of problems related to hemodynamics, s uch as cardiovascular and capillary fluid flow, cellular physics, and adhesion dynamics. Computational fluid dynamics techniques are amongst the most powerful tools available to the engineering branches dealing with the motion of fluids and exchange of ma ss, momentum and energy. In particular, fluid dynamics studies of the normal circulation and pathogenesis of cardiovascular diseases such as atherosclerosis, thrombosis and aneurysms have been performed by numerous authors. PAGE 19 2 In large vessels, dynamics of a n individual cell and its impact on fluid properties are negligible but it is not the case for small vessels, where the effect is important. The study of the effect of leukocyte adhesion on blood flow in microvessels is of significant interest for underst anding the resistance changes in microcirculation. Several computational fluid dynamic studies have been done to understand flow resistance and drag forces due to adhering leukocytes. Cell adhesion involves receptors, cells, and vessels with dimensions on the order of nm, m m, and cm, respectively. The study of the effect of leukocyte adhesion on blood flow in microvessels is important for understanding the resistance changes in microcirculation (Pedley, 1980; Skalak, 1972; Weiss, 1990). Knowledge of the effects of the adhesion parameters, rheological properties (viscosity and surface tension), and geometry (cell and tube diameters) of the leukocyte on the adhesion process is important for establishing guidelines for the design of drug particle size and re ceptor density with reference to the residence time required for the drug to be effective. Models proposed previously in the literature (Bell, 1978; Dembo et al., 1988) do not take into account the rheological properties of the leukocyte. Today, these pr operties have been shown to have substantial effect on the adhesion process (Dong et al., 1999; Dong and Lei, 2000; Hodges and Jensen, 2002). Multi scale modeling of blood flow through arteries is importan t for understanding the physiological functions of the circulatory system and the patho physiology of disease processes such as atherosclerosis, hypertension and diabetes, as well as for the diagnosis and medical cure of certain cardiovascular diseases. For example, in large arteries with diameters rangin g from 3 cm down to 1 mm a change in PAGE 20 3 the dicrotic wave due to reflection indicates changes in the vessel wall and peripheral elasticity, while the absence of the dicrotic wave indicates that the patient may suffer from diabetes or hypertension. Thus, sev ere artery stenoses (constrictions) can be diagnosed by measuring the pulse wave propagation and reflection in terms of blood pressure or heart sounds. In addition, the location of artery disorders (atherogenesis) correlates with the spatial or temporal d istributions of shear stresses exerted by the macroscopic blood flow on the artery vessel wall (Fry,1987; Giddens et al., 1993; Jin et al., 2001; Kilner et al., 1993), as well as with the microscopic mass transport across the arterial endothelium and withi n the vessel wall. Issues spanning from cell surface receptors to blood flow in arteries involve disparate scales, as shown in Figure 1 1. A direct computation encompassing all relevant scales, from large vessels to receptor ligand interaction is beyond t he reach of current and foreseeable computational resources (NDri et al., 2002). The multi scale technique allows us to assess the effect of a reflected wave on the distribution of the flow through the aorta for example. An effective overall model can b e devised globally using the lumped (1D) model to estimate the pressure and flow rate at a given location, and locally using the 3D field equations to analyze the detailed fluid flow in a given artery. Figure 1 2 illustrates the broad range of dimensions i nvolved in hemodynamics. The adhesion of a leukocyte to the endothelium wall involves the deformation of a cell whose length is on the order of micrometers, whereas the adhesion equilibrium bond length involves scales on the order of the nanometers. Bloo d flows from large arteries whose diameters are on the order of centimeters to capillaries whose diameters are on the PAGE 21 4 order of a few micrometers. To simulate the adhesion process and blood flow in arteries, a multi scale modeling approach must be develope d. Migration Firm adhesion Leukocyte rolling Site of inflammation Endothelial lining Blood vessel (post capillary venule) Blood flow Receptors Ligands Lumped model (cm to m m) 3D modeling of Fluid/Structure (cm) Multi Scale modeling of cell adhesion ( m m to nm) Figure 1 1: Multi Scale phenomena in hemodynamics. From cellular and/or molecular level to the human cardiovascular system. PAGE 22 5 (a) A B Bonds A B (b) Lumped model 3D model Figure 1 2: Illustration of the mul ti scale nature of hemodynamics. (a) Cell adhesion modeling, and (b) 3D flow model and lumped model of the human circulation. Blood flow through the cardiovascular system is a multi scale fluid mechanics problem involving flow in arteries, through the capi llary network, and in veins. Localized abnormal physiological conditions due to atherogenosis or stenosis can influence the whole network. Change in flow resistance in one vessel will affect the flow in the upstream and downstream branches (Kilner et al. 1993; Liu, 2000). To describe such interactions, the interface between the lumped model and the detailed flow model must be mediated through effective boundary conditions. The multi scale computation in hemodynamics can be achieved by combining the mod els at different levels through appropriate interfaces (NDri et al., 2002). The long term goal of this project is to develop a multi scale modeling framework spanning form cell surface receptors to blood flow in arteries. However, this is beyond the scop e of this dissertation. This study will focus only on the multi scale modeling of cell deformation and adhesion to provide a starting point for the more challenging multi PAGE 23 6 scale problems encountered in the human body. The major computational challenge fo r the present type of studies is in the treatment of the disparate length scales that exist between cell deformation, whose length scale is on the order of m m, and cell adhesion kinetics, whose length scale is on the order of nm. However, the developed fr amework will provide a scheme for treating general disparate length scale encounter in engineering problems. In this dissertation, we will first review the modeling and experimental studies for the leukocyte adhesion in the microvessels, with an overview o f the numerical methods used. Then, we will describe the computation technique developed for this work and present the results. Finally, we will provide a discussion of the findings. PAGE 24 7 CHAPTER 2 REVIEW ON CELL ADHESION The study of the effect of leukocyte adhesion on blood flow in micro vessels is of significant interest for understanding the resistance changes in microcirculation. Adhesion to vascular endothelium (layer of cells lining a blood vessel) is a prerequisite for the circulating leukocytes in order to mi grate into tissues. These adhesion events are mediated by coordinated regulation of adhesion molecules expressed by both leukocyte and endothelium. This three step process is mediated by a series of different endothelial cell leukocyte adhesion molecules. There are three basic classes of leukocyte endothelium adhesion molecules: selectins, integrins and immunoglobulins (Long 1995). All the adhesion molecules are membrane proteins. Significant progress has been made toward understanding the receptor media ted cell adhesion that is involved in the white blood cell and the endothelium cells interactions (Lauffenburger and Linderman 1993). Detailed experimental studies of the adhesive bonds have suggested that adhesion molecules of the selectin family are inv olved in maintaining the initial rolling of the leukocytes on the endothelium, whereas the integrin bonds are responsible for firm and prolonged white blood cells attachment to the endothelium cells. The reader is referred to Hammer and Tirrell (1996) for a review of the fundamental parameters that characterize biomolecule function in cell adhesion. Receptor mediated cell adhesion plays an important role in many physiological and biotechnological processes. PAGE 25 8 For example, Springer (1990) shows that leukocyt e and tumor cell homing to particular tissues is accomplished by specific receptor interaction with endothelium ligand. The ability to design biomaterial substrate with known and quantifiable densities of adhesion molecules makes it possible to manipulate cell behavior by the correct preparation of the substrate. Kitano et al. (1996) show that bacterial adhesion to biomaterial was prevented by specific blockage of cell adhesion molecule receptors. It is also known that receptor ligand binding and the res ulting adhesion influence cell growth, differentiation, and motility. Analyzing the effect of leukocyte adhesion on blood flow in micro vessels is important for understanding flow resistance changes in microcirculation. Adhesion of leukocytes to the blo od vessel wall is critical in inflammation (Kaplanski et al., 1993; Lawrence and Springer, 1991) and plays an important role in the function of the immune system (Long, 1995). Rolling of leukocytes under physiological flow is a first step in the migration of cells toward an infection site. Adhesion of a cell to a surface often slows down but does not prevent the cell motion (Alon et al., 1997; Alon et al., 1998). It is facilitated by a complex interaction of receptor ligand bond formation and dissociatio n. This initial adhesion is mediated by the P and E selectin found on the endothelium surface and by the L selectin found at the tip of leukocyte microvilli (Hammer and Apte, 1992; Springer, 1990). Blood flow exerts a pulling hydrodynamic force on the a dhesion bonds. It has been shown that this pulling force can shorten the adhesion bond lifetime or even extract receptor molecules from the cell surface. Several models have been proposed for describing the interaction of the leukocyte with the endotheliu m cells. Mathematical models of cell adhesion can be classified into PAGE 26 9 two classes based on equilibrium (Evans, 1983; Evans, 1985a) and kinetics concept (Dembo et al., 1988; Hammer and Lauffenburger, 1987). The kinetics approach is more capable of handling the dynamics of cell adhesion and rolling. In this approach, formation and dissociation of bonds occur according to the reverse and forward rate constants. Using this concept, Hammer and Lauffenburger (1987) studied the effect of external flow on cell a dhesion. The cell is modeled as a solid sphere, and the receptors at the surface of the sphere are assumed to diffuse and to convect into the contact area, as reviewed by Shyy et al. (2001). The main finding is that the adhesion parameters, such as the r everse and forward reaction rates and the receptor number, have a strong influence on the peeling of the cell from the substrate. Dembo et al. (1988) developed a model based on the ideas of Evans (1985a,b) and Bell (1978). In this model, a piece of membr ane is attached to the wall, and a pulling force is exerted on one end while the other end is held fixed. The cell membrane is modeled as a thin inextensible membrane. The model of Dembo et al. (1988) was subsequently extended via a probabilistic approac h for the formation of bonds by Cozens Roberts et al. (1990). Other authors used the probabilistic approach and Monte Carlo simulation to study the adhesion process as reviewed by Zhu (2000). Dembos model has also been extended to account for the distri bution of microvilli on the surface of the cell and to simulate the rolling and the adhesion of a cell on a surface under shear flow. Hammer and Apte (1992) modeled the cell as a microvilli coated hard sphere covered with adhesive springs. The binding an d breakage of bonds and the distribution of the receptors on the tips of the microvilli are computed using a probabilistic approach. Hammer and Apte (1992) studied the effects of the number of receptors, density of ligands, rates of reaction between recep tor and ligand, PAGE 27 10 and stiffness of the receptor ligand spring on the adhesion of the cell. They also identified their effects on the peeling process. Several minor modifications of these models have been presented, but none takes into account the cell defo rmability, which has been shown to be necessary for calculating the magnitude of the adhesion force. Table 2 1 summarizes some of the efforts reported in the literature that address the modeling of cell adhesion dynamics. It is clear that much further st udies including the role of cell deformation in adhesion are needed. Such knowledge would be particularly helpful in understanding tumor cell adhesion, because metastasizing tumor cells have been reported to deform differently than their non metastasizing counterparts (Weiss, 1990). Dong et al. (1999) and Dong and Lei (2000) have modeled the cell as a liquid drop encapsulated into an elastic ring. They show how the deformability and the adhesion parameters affect the leukocyte and Endothelium cell adhesi on process in shear flow. In their study, only a small portion of the adhesion length is allowed to peel away from the vessel wall. This means that their results cannot truly represent what happens in vivo. So a more sophisticated model has to be develo ped. Shao and Hochmuth (1996) used the micropipette technique to measure the adhesion bond force and tether formation from neutrophil membrane. They found that a minimum force of 45pN is required for tether formation and the adhesion force increases linea rly with the tether velocity. Shao et al. (1998) used the methodology developed by Shao and Hochmuth (1996) to measure the static and dynamic length of neutrophil microvilli. In all their experiments, they found a free motion rebound length, which is ind ependent of the applied suction pressure. An explanation of this behavior is that this length is in fact the natural or static tether length. After the microvillus reaches its PAGE 28 11 natural length, it will extend under a small pulling force or form a tether un der a high pulling force. Table 2 1: Overview of some fundamental adhesion kinetic models. Kinetic Model Main Assumptions/Features Major Findings Point Attachment (Hammer and Lauffenberger, 1987) Bonds are equally stressed in the contact area (flat) Bi nding & dissociation occur according to characteristic rate constants Receptors diffuse and convect into the binding area of contact Adhesion occurrence depends on values of dimensionless quantities that characterize the interaction between the cell and t he surface Peeling (Dembo et al., 1988) Clamped elastic membrane Bond stress and chemical rate constants are related to bond strain Bonds are linear springs fixed in the plane of the membrane Chemical reaction of bond formation and breakage is reve rsible Diffusion of adhesion molecules is negligible Critical tension to overcome the tendency of the membrane to spread over the surface can be calculated Predictions of model depend whether the bonds are catch bonds or slip bonds If adhesion is mediate d by catch bonds, then no matter how much tension is applied, it is impossible to separate membrane & surface. Microvilli coated hard sphere covered w/ adhesive springs (Hammer and Apte, 1992) Combined point attachment model with peeling model Bindin g determined by a random statistical sampling of a probability distribution that describes the binding (or unbinding) Model can describe rolling, transit attachment, firm adhesion A critical adhesion modulator is the spring slippage (it relates the strain of a bond to its rate of breakage; the higher the slippage, the faster the breakage for the same strain 2D elastic ring (Dong et al. 1999) Bond density related to the kinetics of bond formation Bonds are elastic springs Interaction between moving fluid & adherent cell Shear forces acting on the entire cortical shell of the cell are transmitted on a relatively small peeling zone at the cells trailing edge PAGE 29 12 Both microvilli extension and tether formation are found to be important in the rolling of the c ell on the endothelium. Extension of the microvillus and formation of a tether can affect the magnitude of the bond force. x y A Figure 2 1: Force balance on an adhered neutrophil mediated by a single attachment. L is the total leng th of the microvillus, l the moment arm length, R the cell radius, F b the bond force, F s the force and T s the shear stress imposed by the shear flow on the cell. For a microvillus extension, the spring constant derived by Shao et al. (1998) under the assum ption that no threshold force exists is 43pN/ m m. At low Reynolds number typical of blood flow through the capillaries, inertial effects on the fluid motion are negligible and a solution derived by Goldman et al. (1967b) may be used to compute the force of a single bond (Shao et al., 1998). Referring to the geometry in Figure 2 1, the angle q determining the direction of the bond force can be computed as follows: 22 22 arcta n arcco s, 2 R Ll l L Rl q + =+ + ( 1 ) where L is the total length of the microvillus, l the moment arm length, and R the cell radius. A force balance in the x direction gives 2 co s 32.05 bs F FR qt == ( 2 ) PAGE 30 13 where F b is the bond force, F s the force, and t the imposed shear stress. The bond force is defined as the force on an adhesive bond. A torque balance with respect to point A gives 3 si n 43.91 b ss F l T R FR qt = += ( 3 ) where T s is the torque imposed by the shear flow on the cell. For the extension of a microvillus, the bon d force F b is given by 10 () b F k LL =, ( 4 ) where k 1 is the bond spring constant and L 0 is the initial length of the microvillus. For the formation of a tether formation, F b is computed as 02 b dL FFk dt =+ ( 5 ) where F o is the initial bond force, dL/dt is the velocity at which the tether is pulling away from the wall and k 2 is the change of the spring constant with time. For a neutrophil, k 1 = 43pN/ m m, k 2 = 11pN.s/ m m and F 0 = 45 pN (Shao and Hochmuth, 1996). Their results show that the bond force decreases in time, and drops 50% in 0.2 s before reaching a plateau. The moment arm and tether length quickly increase within the first 0.2 s, and then slowly increase. Shao and Hochmut h (1999) studied the strength of anchorage of the transmembrane receptors to cytoskeleton, which is believed to be important in cell adhesion and migration. In this experiment, micropipette suction was used to apply force to human neutrophils adhering to latex beads coated with antibodies of the CD626L (L selectin), CD18 ( b 2 integrins), or CD45. The adhesion bond lifetime is defined as the time period between the moment the cell is pulled and the moment the cell resumes its free motion. They computed the pulling force on the cell based on the PAGE 31 14 theory derived by Shao et al. (1998) as a function of bond lifetime. They found an exponential relationship between the pulling force and the bond lifetime. 0.085 18 F e t = ( 6 ) Where F (pN) is the pulling force and t (s) the bond lifetime. To explain the rolling of the cells over the endothelium, two mechanisms have been proposed: intrinsic kinetics of bond dissociation (kinetics in the absence of force) and reactive complianc e (the susceptibility of the dissociation reaction to an applied force). Leukocytes have been observed to roll faster on L selectin than on E selectin and P selectin. To determine which of these two mechanisms can better describe this difference, Alon et al. (1997) studied the kinetics of tethers and the mechanics of selectin mediated rolling. Figure 2 2 shows the best fit of their experimental points to the spring model (Dembo et al., 1988), the Bell (1978) equation, and the linear model (Alon et al., 1 997). They show that the Bell equation and the spring model both fit the data better than the linear relationship, suggesting an exponential dependence of k r on F b Because the leukocyte is modeled as a rigid body in their force computation, the error in the force computation is estimated to be about 21%, so the computed model constants have to be viewed as an approximation. Alon et al. (1998) also studied the kinetics of transient and rolling interactions of leukocytes with L selectin immobilized on a substrate, and measured the rolling velocity for different values of ligand density (L selectin and P selectin). For the L selectin, increasing the ligand density reduces the rolling velocity. The rolling velocity corresponding to the P selectin is small er than that of the L selectin, and there is a threshold shear stress of 0.4 dyn/cm 2 above which rolling can occur. PAGE 32 15 A frame by frame observation performed by Alon et al. (1998) show that leukocytes do not move smoothly on selectin substrates but rather in a jerky fashion. The same behavior was also observed by Chen and Springer (1999). Figure 2 2: Effects of shear stress and Bond force (F b ) on the reverse reaction rate o off k also called k r in text. The solid and dashed lines correspon d to the fit to existing models, as displayed in the legend. The squares correspond to L selectin, the circles to E selectin, and the rhombus to P selectin. (Copyright 1998 National academy of Sciences, USA) Table 2 2: Adhesion parameters computed using B ell and Hookean spring models Bell model Hookean spring model Substrate k ro (s 1 ) r 0 ( o A ) k ro (s 1 ) s /f s (N/m) Smith et and Lawrence (1999) 2.4 0.47 0.39 0.08 3.7 0.51 2.5 0.61 P selectin Alon et al. (1995) 0.93 0.4 0.08 ND ND Smith and Lawrence (1999) 2.6 0.45 0.18 0.03 3.35 0.47 9.12 1.9 E selectin Alon et al. (1995) 0.5 0.7 0.31 0.02 ND ND Smith et and Lawrence (1999) 2.8 0.72 1.11 0.12 8.2 0.17 0.9 0.1 L selectin Alon et al. (1997) 7 9.7 0. 24 0.02 9.7 0.66 6.31 0.96 PAGE 33 16 The predicted values of the reactive compliance r 0 and reverse reaction rate k ro for L P and E selectins using the Bell equation and s /f s and again k ro using the Hookean model are shown in Table 2 2. A comparison betwee n Smith and Lawrence (1999) and Alon et al. (1997) is also shown. The differences observed in Table 2 2 are due to the type of experimental analysis. In the study of Smith and Lawrence (1999), a high temporal and spatial resolution microscopy is used to allow the capture of features that previous studies could not capture (Alon et al., 1997, 1998). The effect of the ligand density on the rolling velocity was investigated by Greenberg et al. (2000) who reported an increase of rolling velocity with shear st ress and a decrease of the rolling velocity with increasing ligand density. The mean rolling velocity found by Greenberg et al. study lies between 25 to 225 m m/sec. Smith et al. (1999) found that the rolling velocity of a neutrophil rolling over immobili zed L selectin ranges from 50 to 125 m m/sec, while the rolling velocities for P selectin and E selectin are 8 m m/sec and 6 m m/sec respectively. The dissociation rate constant k r is determined from the distribution of pause times observed during leukocyte adhesion, and has been used to quantify the effect of force on bond lifetimes (Kaplanski et al., 1993). By comparing the pause time for neutrophils tethering on P E L selectin for estimated bond force ranging from 37 to 250 pN, they found that the pau se times for neutrophils interacting with E selectin or P selectin are significantly longer than those mediated by L selectin. Smith et al. (1999) found that the measured dissociation constants for neutrophil tethering events at 250 pN/bond are lower than the values predicted by the Bell and Hookean spring models. The plateau observed in their graph of the shear stress versus PAGE 34 17 the reaction rate k r suggests that there is a force value above which the Bell and spring models are not valid. Since model propose d so far considers the cell as a rigid body, whether the plateau is due to molecular, mechanical or cell deformation is not clear at this time. Schmidtke and Diamond (2000) studied the interaction of neutrophils with platelets and P selectin in physiologic al flow using high speed, high resolution video microscopy. Under wall shear rates ranging from 50 to 300 s 1 an elongated tether is pulled from the neutrophil causing a slight tear drop shaped deformation, as seen in Figure 2 3. Figure 2 3: Membrane t ether formation and tear drop shaped deformation. (Reproduced from the journal of cell biology, 2000, vol. 149, pp. 719 729 by copyright permission of the Rockefeller University Press) The average length of the tether formed by a neutrophil interacting wit h a spread platelet is estimated to be 5.9 4.1 m m. The average tether lifetime is found to be between 630 and 133 ms for shear rates ranging from 100 s 1 (F b =86pN) to 250 s 1 (F b =172pN), consistent with previous results (Alon et al., 1997; Bell, 1978). Figure 2 4 illustrates the dependence of the tether lifetime on the shear rate. Schmidtke and Diamond (2000) also studied the rolling of neutrophils over P selectin coated surfaces, and observed that when a neutrophil rolls on P selectin, a tether develop s similar to that observed for the spread platelets. The measured tether length and PAGE 35 18 lifetime are 8.9 8.8 m m and 3.79 3.32 s, respectively. Chang et al. (2000) used computer simulation of cell adhesion to study the initial tethering and rolling proces s based on Bells model, and constructed a state diagram for cell adhesion under an imposed shear rate of 100 s 1 using the Bell equation shown below exp () o r ro b rf kk kT = ( 7 ) where k ro is the unstressed dissociation rate constant, k b T the thermal energy, r 0 the reactive compliance, and f the bond force. shear rate (s 1 ) Figure 2 4: Effect of wall shear rate (s 1 ) on the bond lifetime, P < 0.001 means significant difference for the tether duration. (Reproduced from the j ournal of cell biology, 2000, vol. 149, pp. 719 729 by copyright permission of the Rockefeller University Press) Figure 2 5 shows the computed values of k ro as a function of r 0 ( g in the Figure). The symbols correspond to the experimental data obtained fo r different receptor ligand pair using the Bell equation. Adhesion occurs for high values of k ro and low values of r 0 as indicated by the wide area between the no and firm adhesion zones in Figure 2 5. As r 0 increases, k ro has to decrease in order for a dhesion to take place. In the simulation, both association rate k f and wall shear rate are kept constant. Varying k f does not change the shape of the state diagram but shifts the location of the rolling envelope in the k ro r 0 PAGE 36 19 plane. They also found that as the shear rate increases, there is an abrupt change from firm adhesion to no adhesion without rolling motion. For values of r 0 less than 0.1 0 A and high reverse rate constant values k ro the rolling velocity is independent of the spring constant. A detail on how the computation is performed can be found in NDri et al. (2002). Figure 2 5: State diagram for adhesion. Four different states are shown. The dotted curve represents velocity of 0.3 V H and the dashed curve represe nts velocity of 0.1 V H The dots correspond to the data listed in Table 2 5 for E selectin (squares, [Alon et al., 1997, Smith and Lawrence, 1999]), P selectin (triangles, [Ramachandra et al., 1999, Smith and Lawrence, 1999]), L selectin (diamonds, [Alon et al., 1998; Chen and Springer, 1999; Ramachandra et al., 1999]). (Copyright 2000 National academy of Sciences, USA) Tees et al. (2002) used the approach described in Chang et al. (2000) to study the effect of particle size on the adhesion. Three differe nt spherical particles with radii 5.0, 3.75, and 2.5 m m and a shear rate G = 100 s 1 are used in their study. The spring constant and the association rate constant were kept fixed. Tees et al. (2002) observed that an increase of the particle size raises the rolling velocity. This is consistent with the experimental finding of Shinde Patil et al. (2001). In both studies (Chang et al., 2000; Tees et al., 2002), the Bell model is used to construct the state diagram but same results can be achieved using th e spring model (Dembo et al., 1988). Dong et al. (1999) and Dong and Lei (2000) modeled the cell as a liquid drop encapsulated by an elastic ring and PAGE 37 20 illustrated the effect of the deformability and adhesion parameters on the interaction between the leukoc yte and endothelium in shear flow. Only a small portion of the adhesion length is allowed to peel away from the vessel wall. This constraint is not physically sound, and a more sophisticated model is required. PAGE 38 21 CHAPTER 3 COMPUTATIONAL METHOD BACKGROUND The cell a dhesion modeling problem is solved using a fixed Cartesian grid while the interface moves through the mesh, the so called Eulerian Lagrangian approach. The advantage of the fixed grid approach is that grid topology remains simple while large distortions o f the interface take place. Two approaches can be used to solve this kind of problem. The first one is the Immersed Boundary Technique (IBT) originally used by Peskin (1977) to simulate the heart flow and by Fauci and Peskin (1988) to model the swimming organism, and later extended and applied by, amongst others, (Juric and Tryggvason, 1996; Kan et al., 1998; Udaykumar et al., 1997; Unverdi and Tryggvason, 1992). The second approach is called the Interface Cut Cell Method (Kwak and Pozrikidis, 1998; Shyy et al., 1996; Ye et al., 1999, 2001). In this dissertation, we will discuss the first approach. Projection Method The problem of leukocyte deformation, adhesion and movement is solved using the Navier Stokes equations. For incompressible flow, these equ ations are given Continuity Equation .0 u = r r ( 8 ) Momentum Equation 2 ( (.)) u u up uF t rm + =+ + r r rr r rr ( 9 ) PAGE 39 22 where u r is the velocity vector, t is the time, p is th e pressure, F is the source term, m is the fluid viscosity and r is the fluid density. Boundary Conditions Continuity condition in t 12 ( ) (. ) (.) erface V u n un == ( 10 ) Normal stress balance; the dynamic Young Laplace equation 2 1 2 2 11 ( ) () nn uu pp nn g k mm -= +, ( 11 ) where k is the curvature for two dimensional flows and twice the mean curvature for three dimensional flows, g is the interfacial tension and n is the normal vector at the interface. Equations (8 11) ar e solved using the projection method on a fixed Cartesian collocated grid. The projection method or fractional steps is divided into three steps: Step 1: Momentum equations solved without pressure 2 (.) u u u uF t n + = + r rr r rr ( 12 ) where n is the kinematic viscosity. Step 2: Solve the pressure equation (Poisson type) 1 11 ( ). .0 u p t pu t u r r = = D r r r rr r r r ( 13 ) Step 3: Correction step Numerical Implementation PAGE 40 23 The convection terms are explicitly treated using the Ad ams Bashforth scheme, while the diffusion terms are treated implicitly using the Crank Nicholson schemes. Both schemes are 2 nd order accurate. Figure 3 1 shows the location of the velocity component and the pressure on a grid cell. Evaluation of the inte rmediate velocity x u and y u for 2D problem 1 2 2* 3 ( .) ( .) ( ) () 22 n n nn u u u u u u uu S t nn + =+ D rr r r r r r r rr ( 14 ) where n is the time step level and S the source term The pressure equation is derived by assuming that the velocity satisfies the continuity equation at n+1 time step level. x u y u y U x U p Figure 3 1: velocity and pressure location on a given grid cell 1 1* 11 0 (). nn U pU t r ++ = = D r r r r rr ( 15 ) And finally, the correction ste p is done as follows: At the cell center: 1 *1 1 () nn cc uutp r ++ = D r rr ( 16 ) At the cell face: PAGE 41 24 1 *1 1 () nn fc UUtp r ++ = D r rr ( 17 ) Where cc stands for cell center and fc stands for face cell Immersed Bound ary Technique (IBT) The IBT approach incorporates the interfacial condition into the field equation without explicitly tracking the interface. As detailed in Udaykumar et al. (1997), the interface can be handled by using markers points. The issues in thi s approach are the following: How the interface is represented into the grid domain? How to treat the material properties jump? How to communicate the force stored at the interface to the flow field? How to compute the interfacial velocity once the flow fi eld velocity is known? How to move the marker point? Interface Information The immersed interface is denoted by C(t), the interface is either a curve for 2D problem or a surface for 3D problem. The interface is represented by K marker points of coordinate s () k xs r with k=1,.,K and s is the arclength. Figure 3 2 shows the interface numbering and representation. The markers points are regularly separated 0. 5 1.5 h d sh where h is the grid size. The interface is parameteri zed as a function of the arclength s by fitting quadratic polynomials 2 () k k kk x s a s b sc = ++ r r rr through three consecutive marker points of coordinates 11 ,, k kk x xx -+ r rr Once the position of the interface is known, the normal and the curvature are evaluated. The convention adopted is that the unit normal point form Fluid 2 to Fluid 1. In 2D the normal is given by: PAGE 42 25 2 2 22 ss xy s s ss yx n an dn x y xy = -= ++ ( 18 ) Where the subscript s denote d/ds. The curvature is then obtained by taking the divergence of the normal vector n k = r r ( 19 ) Fluid 2 Fluid 1 n r ds 1 2 K h Figure 3 2: Interface representation and numbering Material Property Assignment With the known interface position, the mate rial properties are assigned using a Heaviside step function. 1 21 ( ) () k H xx b bbb = + -rr ( 20 ) where b is any material property such as density r or dynamic viscosity m The subscript 1 and 2 denote Fluid 1 and Fluid 2 respecti vely as shown in Figure 3 2. And () k H xx rr is the discrete Heaviside step function defined as follows: PAGE 43 26 dim 1 ( ( ) ) ( ( )) 11 ( 1 si n) 2 ( )0 1 m m k m mk k m kk x x xx i f x xd dd H x x i f x xd otherwise p p = -+ + = >+ r r rr rr r r rr ( 21 ) Where dim is the spatial dimension, d=2h with h the grid size, x r is the grid coordinate and k x r is the interfacial point coordinates. Figure 3 3 shows the definition of the Heaviside function. Transition area 0 PAGE 44 27 The force at the grip point P is computed based on the sum of th e interfacial force f k of the marker point located inside a circle of radius 2h weighted by the Delta function as shown in Figure 3 4. The delta function is computed as follows and the delta function spreads over 4h and is the derivative of the Heaviside step function. dim 1 ( () 1 ( 1 co s ) || () 2 0 m mk k m k xx i fx xd xx dd otherwise p d = + -= rr rr rr ( 23 ) Markers points Contributing to Force F Circle of radius 2 h around P P h Figure 3 4: Source term computation at the grid point P and marker points contributing to the computation is also shown. Once the interfacial position i s known, the material properties can be assigned properly then the interfacial force is spread to near by grid points and finally the flow field equation can be solved. The next step is to compute the interfacial point velocity. Interfacial Velocity Compu tation The velocity at the marker point is denoted by V k and should satisfy the continuity condition, so in discretized form the interfacial velocity is: 2 () k i jk ij V u xxh d =r r rr ( 24 ) Where h is the grid size, i and j are the grid location indices and u r is the fluid velocity. The computation of the interfacial velocity is illustrated in Figure 3 5. The PAGE 45 28 velocity is computed by taking the sum of all the grid points located inside a circle of radius 2h Grid Points Contributing to Interface Velocity V ( k ) Circle of radius 2 h around Marker Point Figure 3 5: Interfacial point velocity computation and grids contributing to the computation is also shown The last step is the advection of the interface and this is done using the following equation: 1 () k n nn kk xx tV + = +D r rr ( 25 ) This method is used to study the cell adhesion and movement along the endothelium wall. PAGE 46 29 CHAPTER 4 DEVELOPMENT OF THE MULTI SCALE METHOD Although in large vessels, blood cells influence minimally blood flow, they play an import ant role in flows through small vessels with a lumen diameter comparable to cell size. The ability of a blood cell to deform and flow through capillaries or to migrate in tissues is governed by its mechanical properties. This aptitude to deform is critic al to our immune system. The goal here is to develop novel numerical techniques to better understand and predict the effects of a deformable cell in contact with the endothelium, and the force of adhesion between them under quasi steady sate flow. Most of the studies of the adhesion were done by assuming a flow over a plate and by using a shear flow at the inlet. But most of the existing experiments are performed using a parallel flow chamber. In addition, in the post capillaries, the diameter of cell an d that of the vessel are in the same order; this can affect the rolling of a cell along the vessel wall. Finally, in vivo the cell is exposed to a pulsatile flow. These flow and geometric characteristics have not been taken into account in previous studi es. Furthermore, it has been shown in our laboratory (Kan et al., 1998, 1999a,b) that a compound drop model for the leukocyte reconciles different experiments performed in different laboratories. So in the present a compound drop will be used to study th e adhesion of a cell on a substrate. Model Overview In order to form a comprehensive modeling framework to treat the disparate PAGE 47 30 scales between cell deformation ( m m) and cell adhesion (nm), a multi scale model to account for both macroscopic (continuum) and microscopic (ligand receptor) levels of phenomena is needed. Recently, NDri et al. (2000) have implemented such an approach. In their multi scale model, the m acroscopic component deals with the deformation of the cell and the microscopic part takes care of the adhesion aspect. The continuum scale problem is the one of a fluid flowing over a cell attached to one wall of a tube by an area of contact defined by t he membrane segment A B (Figure 4 1a). The ligand receptor bond scale problem involves the interaction between the membrane segment A B and the substrate surface (Figure 4 1b). (a) (b) A B Bonds A B Figure 4 1: Macro Micro model. The cellular model is shown on the left, whereas the enlarged area on the right shows the micro domain for the receptor model. Macro Scale Model The macroscopic problem consists of a fluid flowing over a cell attached to a flat surface of a tube. Solution to the problem will provide information about the cell shape, velocity, and pressure. That information will be passed to the micro scale model. Computation of the source term F PAGE 48 31 ( ) ( ) ( ) ( ) () ( ) () kk kkk b k F D x X Fns gk = +D ( 26 ) Where D is the delta function, g is the interfacial tens ion, k is the curvature, D s is the arc length, n is the normal direction vector and F b is the bond stress which is defined by the micro scale model. Micro Scale Model The interaction between the cell and surface at the microscopic level is analyzed with th e model proposed by Dembo et al. (1988). This model treats a bond as a spring, and the force of a bond, f b is given by: () b m b bb f x an d F Nf sl = -= ( 27 ) Where s is the spring constant, x m and l are the current and the equilibrium lengths of a bond, F b is the total bond force, and N b is the bond density. In Dembos model, major assumptions were made: 1) the bonding stresses are the only distributed stresses acting on the membrane; 2) the bonds are fixed in the plane of the membrane; and 3) the chemical reaction of bond formation and breakage is reversible. Those assumptions have some drawbacks since membrane experience forces due to surrounding forces such as the non specific potential forces, and it is also known that adhesion mole cules are able to diffuse laterally in the plane of the membrane. Nonetheless, a significant understanding can be gained from the present study. Calculation of the bond density, N b The balance equation of the formation and dissociation of bonds is given b y a simple kinetic relationship: b r b r b l f b N k N N N N k t N = ) )( ( ( 28 ) PAGE 49 32 Where N b is the bond density, k r and k f are the reverse and forward reaction rate coefficients, respectively, N l is the initial ligand density on the surface, and N r is the initial density of receptors on the cell membrane. The reverse and forward reaction rate coefficients are, respectively, given by Dembo et al. (1988): ) 2 ) )( ( ( exp 2 T k x k k b m ts ro r l s s = ( 29 ) and ) 2 ) ( exp( 2 T k x k k b m ts fo f l s = ( 30 ) The initial bond density, N bo is found by solving the following equation: 0 ) )( ( = bo ro bo r bo l fo N k N N N N k ( 31 ) Where the subscript o refers to the initial equilibrium state. The kinetic equation is solved using a 4 t h order Runge Kutta method. The micro model has been used to analyze the case of a membrane being pulled away from a surface at a constant velocity (NDri et al., 2000; Shyy et al., 2001). The effects of reaction rates, ligand density, and other paramete rs on the adhesion process have been reported. Micro Macro Coupling The macro model is first solved to determine the position of the interface while freezing the micro model. Knowing the position of the interface allows the computation of the actual lengt h of the adhesion molecule. Then, this information is transmitted to the micro model while freezing the macro model. The adhesion bond stress is computed in the micro model and is transmitted to the macro model via the source term. Information between t he two models is continuously exchanged during the course of the computation. The computational flow chart of their approach is shown in Figure 4 2. PAGE 50 33 Input parameters from the macro model (Freeze the macro model) Compute the bond density N b Co mpute the bond force F b Transfer the bond force to the macro model (Freeze the micro model) Apply interfacial conditions Solve the momentum equation Solve the pressure equation Interfacial condition Identify and Track interface movement No Yes Figure 4 2: Flow chart showing the interaction between the micro model (recept or scale) and the macro model (cellular scale). Macro and Micro Test Cases In the first part of this study, the macroscopic and the microscopic model were solved separated in order to study the effect of the Reynolds number and the capillary number on the deformation of a cell attached to the wall of a tube and the adhesion parameters such as ligand number, bond stiffness respectively. The main founding of this part of their study can be found in Shyy et al. (2001). Micro Scale Test Cases A cell attached t o a wall of a tube is pulled away from the substratum with a PAGE 51 34 specified velocity, v as indicated in Figure 4 3. The bond is considered disrupted based on the criterion offered by equation bo bc N N 4 10 = This computation is conducted to see the e ffect of the different parameters on the adhesion process. A B Bonds v Figure 4 3: Geometry of the test case problem Figure 4 4 shows the effects of the reverse reaction rate, k ro on the maximum length of a bond before breaking, for differ ent values of the forward reaction rate k fo We see that when k fo increases, the maximum length reached by a bond before breaking is greater. For a given value of k fo the maximum length of a bond decreases as the reverse reaction rate increases. The rang e of the values of k ro used in this study is 0.1 PAGE 52 35 In all the following figures, v = 0.1 m m/sec, k fo = 10 14 cm 2 /sec, k ro = 0.1s 1 rc = 4 10 4 cm, dt = 0.001s, l = 5 10 6 cm, k b T = 3.8 10 7 dy ne.cm, N r = 5 10 10 cm 2 5 5.4 5.8 6.2 1 3 5 7 9 number of ligand peeling time Figure 4 5: Effect of the ligand density ( 10 10 cm 2 ) on the de bonding time (s). f s =0.1. s ts = 4.5dyn/cm and s = 5dyn/cm. In Figure 4 5, we observe that the de bonding time increases as the number of the ligand increases. The sharp slope observed at the beginning is due to the fact that the number of ligands is less than the number of receptors, so there are a lot of free receptors at the contact area. Therefore, by increasing the ligand density, the num ber of bonds increases until the number of ligands outnumbers the number of receptors. At that instant, the bond formation is decreased due to the unavailable receptors at the contact area. The peeling time decreases as the spring constant increases. The higher the spring constant is the less elastic is the spring and therefore it breaks faster. Figure 4 6 shows the effect of the slippage constant f s = ( s s ts )/ s on the de bonding time. The slippage constant appears in the computation of the reverse rate r eaction (equation 30). PAGE 53 36 4.5 5 5.5 6 6.5 7 0 0.1 0.2 0.3 0.4 f s (slippage constant) peeling time Figure 4 6: Effect of the slippage constant f s on the de bonding time (s) for s (circle) and s ts (square) held constant respectively. The circle dots correspond to s = 5dyn/cm, and s ts is allowed to vary wi th f s The square dots correspond to s ts = 4.5dyn/cm, and s is allowed to vary with f s N r = N l =5 10 10 cm 2 In this computation, s ts is held constant while s is varying, then s is held constant while s ts is varying. Figure 4 6 shows that f s and the peeli ng time are inversely proportional. When f s > 0.1 the peeling time for s ts is smaller than that of s constant while for f s = 0.1 the peeling time for fixed value of s ts and s are similar. Macro Scale Test Cases In this section, a droplet adhering to a wal l under flow conditions is studied. This computation is done by using equations (8 11). No adhesion molecule is present in this test. In the following computation, the size of the tube is 1.0 2.0 the diameter of the droplet is 0.6 the inlet velocity is taken equal to 1 The properties of the droplet and the surrounding flow are m = 0.01 Pa.s, r = 1.0 kg/m 3 and s = 0.01 N/m and s = 0.1 N/m, PAGE 54 37 the capillary number Ca = m U/ s =1.0 and 0.1 respectively and the Reynolds number Re =( r Ud/ m )= 100 Figure 4 7 s hows the instantaneous droplet shapes for two values of the surface tension for a fixed value of the inlet velocity. It is observed that the surface tension has an effect on the shape of the droplet. The droplet shape is smoother for high surface tension. 0 0 5 1 1 5 2 0 0 5 1 0 0 5 1 1 5 2 0 0 5 1 Figure 4 7: Instantaneous droplet shapes for s = 0.01 N/m (left) and s = 0.1 N/m (right). 0 0 5 1 1 5 2 0 0 5 1 0 0 5 1 1 5 2 0 0 5 1 Figure 4 8: Instantaneous droplet shapes for two different values of the droplet viscosity m = 0.01 Pa.sec (left) and m = 1.0 Pa.sec (right) and s = 0.01 N/m for both computations. The viscosity has a significant effect on the droplet shapes and the flow field velocity as shown by Figure 4 8. Micro Macro Coupling In this paragraph, the interaction of both models was also studied. We will review some of the resul ts for this aspect. An initial membrane shape given by the macro model is used as the input for the micro model. The membrane is modeled as a thin inextensible body (Dembo et al., 1988). A difference of pressure across the membrane, based on the PAGE 55 38 macrosc opic solution is applied to find the new position of the membrane. During the first step of this computation, the bond model is not activated. In the next and subsequent time steps, the bond density and force are computed based on the receptor ligand mod el. Using this information the new location of the interface is found at every time instant. In the computation, the following values are used: N l = N r = 5.0 10 10 cm 2 s = 5.0 dyne/cm, s ts = 4.5dyne/cm, k b T = 3.8 10 7 dyne.cm. Instantaneous membrane shap es between points A B shapes after various time steps are illustrated in Figure 4 9. Again, the initial curve represents the original membrane location segment before a flow is imposed. Initially, due to the action of the hydrodynamic forces, the membran e is pulled away from the substrate. This membrane movement activates the receptor ligand model, and, consequently, the deformation of the membrane becomes more limited as a higher force is required to deform the bonds. The jump between the initial and f irst membrane locations in Figure 4 9 is because we didn't start the macroscopic and microscopic models simultaneously. This is done mainly to illustrate the interaction between the two models. 0 0.4 0.8 1.2 1.6 0 0.1 0.2 X Y Initial First Second Third Fourth Figure 4 9: Illustration of the instantaneous membrane shap e resulting from the pressure field, based on the macroscopic model, and the bond force, based on the receptor ligand model. The vertical axis is normalized by the bond length, and the horizontal axis by the cell radius. PAGE 56 39 To understand the behavior observed in Figure 4 9, we plot in Figure 4 10 and Figure 4 11 the variation of the reverse and the forward reaction rates as a function of time. 0.99956 0.99966 0.99976 0.99986 0.99996 0 0.002 0.004 0.006 0.008 0.01 time k f X = 0.062 X =0.130 Figure 4 10: Forward reaction rate as a function of time for two different locations of t he bonds. X is the abscise of the bond. The distance between the membrane and the wall has a major impact on the rate constants, as evidenced by Eqs. (29) and (30). Hence, the initial upward movement of the membrane during the first time step, shown in Fi gure 4 9, is responsible for the initial decrease in the forward rate constant, k f in Figure 4 10. The subsequent downward movement of the membrane reverses the trend accordingly. The two curves correspond to the quarter and half point of the membrane segments. The above argument is applicable to both forward and reverse rate coefficients, except that the reverse reaction rate coefficient, k r, has an opposite behavior as shown in Figure 4 11. The main findings of NDri et al. (2000) for the kinetic mo del are: The maximum bond length decreases as the reverse reaction rate increases, and increases with an increasing forward reaction rate. As the cell velocity increases during the de bonding process, the maximum bond length increases while the total peel ing time decreases. PAGE 57 40 The rate of de bonding decreases as the number of ligands increases. The bonds are strong enough to resist the hydrodynamic force generated by a moderate flow and to affect the local shape of the cell membrane. The peeling time decreas es with increasing spring or slippage constants. The peeling time reaches a maximum when the number of ligands equals its receptor counterpart. 1 1.0002 1.0004 1.0006 0 0.002 0.004 0.006 0.008 0.01 time k r X=0.062 X = 0.130 Figure 4 11: Reverse reaction rate as a function of time for two different location s of the bonds. X is the abscise of the bond. PAGE 58 41 CHAPTER 5 RESULTS In this chapter, the cell is allowed to deform and to roll along the vessel wall contrary to the precedent computation where the cell remains still during the computation. A 2D dimensional represent ation of a cell modeled as either simple or compound drop is studied in a planar channel under imposed flow. The cell is attached to one channel wall with adhesive bonds governed by kinetics model proposed by Dembo et al. (1988). A uniform flow is used a t the inlet because in vitro experiment uses a parallel flow chamber and in vivo the cell is exposed to a pulsatile flow with a uniform description at a given time. Most of the studies use a shear flow over a plate with cell attached to it. It is also be lieved that changing the tube or vessel diameter can have an impact on cell adhesion. Furthermore, it has been shown in our laboratory (Kan et al., 1998, 1999a,b) that a compound drop model for the leukocyte reconciles different experiments performed in d ifferent laboratories. Liquid Drop Model In this paragraph, a two dimensional cell is attached to a vessel wall, and a uniform flow is imposed at the inlet of the vessel tube as shown in Figure 5 1 in order to study its peeling from the vessel wall using the immersed boundary technique described in Chapter 3. Table 5 1 and Table 5 2 show the adhesion parameters used in the study. All the results presented below will be based on the non dimensional parametric variations normalized by the reference scales shown in Table 5 3. PAGE 59 42 Cytoplasm Plasma Figure 5 1: Schematic of the problem statement for the simple liquid drop Table 5 1: Average macro model parameters. Tube diameter ( m m) 30 Cell viscosity (dyne sec/cm 2 ) 10 1000 Cell diameter ( m m) 6 8 Plas ma viscosity (dyne sec/cm 2 ) 1.0 Tube length ( m m) 120 Plasma density (g/cm 3 ) 1.0 U ( m m/sec) 50 3200 Interfacial tension (dyne/cm) (Evans, 1983) 10 3 8.0 Table 5 2: Average micro model parameters. Parameter Value Parameter Value N r (cm 2 ) (Bell et al ., 1984) 2.0 5.0 10 10 k b (dyne.cm/K) (Boltzman Constant) 1.38 10 16 k fo (cm 2 /s) (Hammer and Lauffenburger, 1987) 10 14 l (cm) (Bell et al., 1984) 5.0 10 6 k ro (s 1 ) (Bell, 1978) 10 11 10 r c (cm) (Schmid Shonbein et al., 1980) 4.0 10 4 s (dyne/cm) (Demb o et al.,1988) 0.5 10 F s = ( s s ts )/ s 0.1 s ts (dyne/cm) (Dembo et al.,1988) 0.48 9.5 N l (cm 2 ) (Lawrence and Springer, 1991) 2.0 5. 10 10 Table 5 3: Reference values used for non dimensionalization. Length 30 m m Viscosity 1 dyne sec/cm 2 Velocity 600 m m/sec Density 1.0 g/cm 3 Spring constant 5.0dyne/cm Bond density 5.10 10 cm 2 Reverse reaction rate 0.1 sec 1 Interfacial tension 0.1 dyne/cm In all the computations, unless specified otherwise, the values used for k ro s and s ts are 0.1 s 1 5 dyne/cm, and 4.5 dyne/cm, respectively. Using these parameters, the effects of the reverse reaction rate, the wall, and the spring constant s on the PAGE 60 43 rolling/displacement of the cell along the endothelium are evaluated. The cell is first modeled as a liquid drop with a constant viscosity and surface tension. This is the simplest model one can use to describe qualitatively certain rheological behaviors of leukocytes (Evans and Yeung, 1989). Figure 5 2a shows the instantaneous pos ition of an interfacial point on the cell surface for a given k ro It is seen that the cell translates along the wall and rotates at the same time. The combination of these two movements leads to the rolling of the cell along the wall. The cell initiall y rolls along the wall and starts to slide along the vessel wall when the bonds offer no more resistance as shown by the plateau observed in Figure 5 2b. The curve trend can be qualitatively compared to the one obtained by Shao et al. (1998) in their stud y where they found a fast increase of the bond length follow by a slow increase and a plateau. In the subsequent computations, a position of the instantaneous interfacial point will be plotted as indicated in Figure 5 2a. 0.08 0.1 0.12 0.14 0 2 4 6 8 10 t(s) y/R x/R y/R t 1 t 40 t 5 a b Figur e 5 2: Instantaneous position of an interfacial point on the cell for k ro = 1.0 and 0.1 U = m = 100, g = 1.0. PAGE 61 44 Effect of Bond Molecule The effect of bond molecules on the behavior of the cell is shown in Figure 5 3. The effect of the w all on the cell displacement is obtained by not activating the micro model (no bond force). It is found that the bond force is strong enough to delay the rolling of the cell and the peeling of the cell from the vessel wall. 0.085 0.105 0.125 1.81 1.85 1.89 1.93 x/R y/R Without activating micro model With micro model (k ro = 1.0) Fig ure 5 3: Effect of the bond molecules on the rolling of the cell along the wall, 0.1 U = m = 100, g = 1.0. Effect of the Reverse Constant k ro Figure 5 4 shows the effect of the reverse reaction rate k ro on the rolling of the cell. It is observed that an increase in the value of k ro decreases the resistance force from the bond molecules. On the other hand, a lower value of k ro makes the cell rolls for a longer distance, and tends to increase the bond length of the adhesion molecule bef ore breakage as demonstrated by the upper asymptote obtained. Effect of the Spring Constant The effect of the spring constant s on the rolling of the cell is provided in Figure 5 5. Changing the spring constant value by a factor of two has little effect o n the rolling PAGE 62 45 of the cell along the vessel wall. The work by Chang et al. (2000) shows also the little dependence of the spring constant on the adhesion process for some adhesion parameters. 0.08 0.12 1.81 1.85 1.89 1.93 x/R y/R k ro = 1.0 k ro = 0.1 k ro = 0.01 Figure 5 4: Effect of the reverse re action rate k ro on the rolling of the cell, 0.1 U = m = 100, g = 1.0 0.08 0.1 0.12 0.14 1.81 1.85 1.89 1.93 x/R y/R s = 1.0 s = 2.0 Figure 5 5: Effect of the spring constant on the rolling of the cell. f s = 0.1, k ro = 1.0, 0.1 U = m = 100, g = 1.0. Comparison to Existing Results In the following computations, a bond is considered peeled away from the surface if the cell travels 2 radii. This assumption is based on the shape taken by the cell as a function of time. Kuo et al. (1997) made similar as sumptions in their study into which PAGE 63 46 the leukocyte is modeled as a solid body. They assumed that bonds are broken if the cell travels 10 radii. The best assumption should be based on the adhesion molecule bond strength. The work by Evans (1999) gives a r ange of adhesion bond strength but not an exact bond strength for a given molecule bond. Comparison to Numerical Results Dong et al. (1999) and Dong and Lei 2000) have modeled the cell as a drop enclosed into an elastic ring. In their approach, the initia l shape of the elastic ring is the one taken from the picture of an experiment of a cell adhering on a surface subjected to a known shear rate. They assume that only a small portion of the adhesion contact can be peeled away from the wall, but that length is not specified in their work. This assumption allows them to use an energy approach to calculate the cell rolling velocity. In the present model, we do not make that assumption. We solve the full flow field and fluid interface interactions, and let t he flow dictate the contact area. We use the same parameter as in Dong et al. for the adhesion parameters and cell viscosity. The only unknown parameters in their study are the cell surface tension, the contact length and the number of bonds. In the Fig ure 5 6, a comparison to Dong results is shown for g = 4.0, 5.0 and 10. The top line corresponds to the higher surface tension, and is in agreement with the results of Dong and Lei (2000). This is expected, since as the surface t ension increases, the liquid drop model should provide results similar to the 2D elastic model. However, because our approach is more general, additional information can be found. For example, we do not constraint the cell to a peeling motion only, we al low the cell to be lifted away from the surface. PAGE 64 47 0 5 10 15 20 0 500 1000 1500 2000 m ( dyn sec/cm 2 ) V r ( m m/sec) Figure 5 6: Comparison of the cell rolling velocity. The dashed line corresponds to Dong et al. results and the lines to the present study. 1.0 U = 0.4 r N = 1.06 l N = 0.1 s = 0.04 f s = g =4.0, 5.0, and 10 from bottom to top respectively. Comparison to Experimental Results A comparison of the bond lifetimes computed in this study and the experimental results by Schmidtke and Diamond (2000) for one bond is shown in Figure 5 7. For this comparison, no information about the reaction rates and spring constant exist. But, we have shown that, for the adhesion par ameters considered in this study, these values do not significantly affect the rolling and displacement of the cell along the vessel wall. The key parameters are the cell rheological properties, the receptor density and the ligand density. In the present study, we have supposed an excess of ligand number to the number of receptors, the dimensionless cell surface tension is taken to be 1.2 and the dimensionless viscosity ranges from 50 to 300. The top line corresponds to the higher viscosity. Good agreem ent is seen for the parameters used. An increase in the cell surface tension for a fixed value of cell viscosity will move all the simulated results upward. A surface tension of 1.2 and a viscosity of 200 provide the best fit. PAGE 65 48 0 0.4 0.8 1.2 50 100 150 200 250 300 shear rate (s -1 ) t(s) m = 300 m = 200 m = 100 m = 50 Figure 5 7: Comparison of the bond lifetimes in the simple drop case. The dot corresponds to Schmidtke and Diamond (2000) results and the line to the present study. 0.02 r N = 1.0 l N = 0.1 s = 0.04 f s = g =1.2. Compound Drop Model Figure 5 8 illustrates the problem considered for the compound drop case. Specifically, a nucleated cell is attached to a vessel wall, and a uniform flow is imposed at the in let of the vessel tube. In the model, the nucleus occupies 44% of the volume of the cell (Schmid Schonbein et al., 1980), and the rheological properties (viscosity and surface tension) of the nucleus are taken to be 10 times those of the cytoplasm and cel lular membrane. The kinetics parameters used are the same as in the simple drop model. The problem to be solved is the same as before: a cell is attached to a vessel wall, with a uniform flow imposed at the inlet of the vessel tube. PAGE 66 49 It should be noted th at the nucleus of a neutrophil is small and segmented. Its contribution may not be as large as the one predicted by the present model but it is difficult to assess this at the present time. A neutrophil nucleus is asymmetric and it has been shown by Kan et al. (1999) that nucleus eccentricity affects the instantaneous shapes of the cell during recovery. In addition, Kan et al. (1998, 1999) have shown that the presence of a nucleus, although small in size, is needed in order to reconcile the various leuko cyte rheological data published in the literature. The compound drop model describes better the structure of a lymphocyte, but it is still a good model for evaluating the effect of key parameters on the rheology and adhesion of leukocytes in general. Cytoplasm Plasma Nucleus Figure 5 8: Schematic of the problem statement for the compound drop model. Cell Movement and Deformation In this paragraph, the displacement and deformation of the cell are studied for different values of cell viscosity, cell surfa ce tension and the inlet boundary condition. Effect of the Nucleus Figure 5 9 shows the effect of the nucleus. The nucleus tends to delay the rolling of the cell along the vessel wall but does not appear to affect the bond length before peeling. We also found that the reverse reaction constant within the range studied, 21 1 0 10 ro k -, has no significant effect on the position and displacement of the cell or nucleus. The curves of y/R plotted against t for the compound drop are identical to the o ne for the liquid drop in Figure 5 2b. PAGE 67 50 0.08 0.1 0.12 0.14 1.31 1.35 1.39 1.43 x/R y/R Without nucleus With nucleus Figure 5 9: Effect of the nucleus on the rolling of the cell along the vessel wall. k ro = 1.0, 0.1 U = m = 100, g = 1.0. Effect of the Viscosity The effect of viscosit y value on the rolling of the cell is investigated for m = 10 and 100 and is shown in Figure 5 10. A cell with a lower cytoplasmic viscosity deforms more and remains closer to the surface as shown in Figure 5 11. 0.08 0.1 0.12 0.14 1.31 1.36 1.41 1.46 1.51 x/R y/R m = 100 m = 10 Figure 5 10: E ffect of the viscosity on the rolling of the cell along the vessel wall, k ro = 1.0, 0.1 U = g = 1.0. PAGE 68 51 Time (t) m = 10 m = 100 0.5 1.0 1.5 Figure 5 11: Effect of Viscosity on the instantaneous cell Shape. The horizontal line corresponds to the wall. Effect of the Surface Tension Surface tension is also found to have a major effect on cell adhesion and rolling as can be seen in Figure 5 12. A lower surface tension allows the cell to maintain a larger curvature, which enab le the cell to remain attached with a larger contact area (Figure 5 13). Consequently, a lower surface tension delays the lifting of the cell from the wall. 0.08 0.1 0.12 0.14 1.31 1.35 1.39 1.43 x/R y/R g = 5.0 g = 1.0 g = 0.1 Figure 5 12: Effect of the interfacial tension on the rolling of the c ell along the vessel wall, k ro = 1.0, 0.1 U = m = 100. PAGE 69 52 Time (t) g = 5 g = 1 g = 0.1 1.0 2.0 3.0 5.0 Figure 5 13: Instantaneous cell shape for differen t values of the interfacial tension. The horizontal line corresponds to the wall. Effect of the Inlet Boundary Condition The effect of the inlet velocity is shown in Figure 5 14. As the inlet velocity is increased, the movement of the cell along the vess el wall accelerates and the bond breaks at a shorter distance from the wall. This is due to the fact that when the inlet velocity is increased, the cell is more deformed and stays closer to the wall as shown in Figure 5 15. PAGE 70 53 0.08 0.1 0.12 0.14 1.3 1.5 1.7 1.9 2.1 x/R y/R U = 0.1 U = 0.5 U = 1.0 Fig ure 5 14: Effect of the inlet velocity on the rolling of the cell along the vessel wall, k ro = 1.0, m = 100, g = 1.0. Time (t) U=0.1 U=0.5 U=1.0 1.0 2.0 3.0 Figure 5 15: Instantaneous cell shape for different values of inlet velocities. The horizontal line correspon ds to the wall. Effect of the Capillary Number Figure 5 16 shows the relative effects of U, m and g on the behavior of the cell for a given capillary number (Ca=U m / g ). PAGE 71 54 0.268 0.278 0.288 0.298 1.23 1.33 1.43 1.53 1.63 x/R y/R U = 0.1, m = 10, g = 1.0 U = 0.2, m = 10, g = 2.0 U = 0.1, m = 20, g = 2.0 Figure 5 16: Effects of v, m and g for Ca = 10 on the cell behavior, k ro = 1.0 We should point out that when we vary, say, viscosity, other parameters like the viscosity ratio between media, as well as the contribution of the bond force also change. On the other hand, if we hold all other dimensionless parameter s the same, and vary Ca, then we will see a unique correlation. This is better understood by looking at the dimensionless form of the equation of motion in the x direction, for example. By rewriting the equation of motion: 22 22 () ib u p uu FF t x xy g rm = + + ++ ( 32 ) In dimensionless form using the following reference values, the subscript i refers to the medium of interest: 0 0 00 2 0 , ,, r re f re f re f bref U RN UU p t FF R U RR g g sl m = ==== ( 33 ) PAGE 72 55 Where U = inlet velocity, R = tube radius, m 0 = viscosi ty of the plasma, g = interfacial tension, s = spring constant, l = equilibrium bond length, N r = receptor density, we obtain: 22 22 00 1 R e () r ib u p u u RN FF t x xy C aU g ls a m = + + ++ ( 34 ) Where a i = m i / m 0 is the viscosity ratio. Peeling Time and Rolling Ve locity Calculation In what follow, we compute the peeling time, the rolling velocity of a cell adhering to a wall and the force of a bond molecule for different cell properties and adhesion parameters. The following defined parameters will be used for the subsequent computations. 0 cn c and mm ab mm == ( 35 ) Where m 0 is the plasma viscosity, m c is the cytoplasm viscosity and m n is the nucleus viscosity. T he reference parameters used are given in Table 5 3 previously listed in the liquid drop model paragraph. Effect of the Inlet Velocity Figure 5 17 shows the effect of the inlet velocity flow on the peeling time of the cell from the substratum. An increase of the inlet velocity decreases the peeling time. An inverse relatio nship can be used to fit the curve as shown in Figure 5 17, and is of the form: 0.47 26.84 tU = ( 36 ) where t is the dimensionless peeling time and U is the inlet ve locity flow. PAGE 73 56 20 40 60 80 0 0.2 0.4 0.6 0.8 1 U peeling time Figure 5 17: Effect of the inlet velocity U on the peeling time t a = 100, b = 10, 1.0 r N = 1.0 l N = 1.0 s = 0.2 f s = 1.0 g = The dot points correspond to the numerical results and the line is the curve fitting. Figure 5 18 shows the effect of the inlet velocity flow on the rolling velocity of the leukocyte. Th e rolling velocity is defined as the ratio of the distance traveled by the cell to the peeling time. Here only the displacement of the cell along the x axis is shown. An increase of the inlet flow increases the rolling velocity of the cell. The relation ship between inlet velocity flow and cell rolling velocity, r V is given by 32 0.005 5 0.01 4 0.01 7 0.002 r V UUU = ++ ( 37 ) 0.003 0.005 0.007 0.009 0.011 0 0.2 0.4 0.6 0.8 1 U V r Figure 5 18: Effect of the inlet velocity on the rolling velocit y of the leukocyte. a = 100, b = 10, 1.0 r N = 1.0 l N = 1.0 s = 0.2 f s = 1.0 g = The dot points correspond to the numerical results and the line is the c urve fitting. PAGE 74 57 Effect of the Receptor Density Figure 5 19 and Figure 5 20 show the effects of the receptor density on the peeling time and rolling velocity. An increase of the receptor density increases the peeling time and decreases the rolling velocity. Dong et al. (1999) also found an increase of the peeling time with receptor density. The evolution of the peeling time as a function of the receptor density can be described by the following relationship. 3.49 5( ) 76.90 r t L nN =+ ( 38 ) 0.0466 0.0037 r r VN = ( 39 ) 71 73 75 77 79 0.2 0.4 0.6 0.8 1 1.2 1.4 N r peeling time Figure 5 19: Effect of the receptor density r N on the peeling time t a = 100, b = 10, 0.1 U = 1.0 l N = 1.0 s = 0.2 f s = 1.0 g = The dot points correspond to the numerical results and the line is the curve fitting. PAGE 75 58 0.00355 0.00365 0.00375 0.00385 0.00395 0.2 0.4 0.6 0.8 1 1.2 1.4 N r V r Figure 5 20: Effect of the receptor density r N on the rolling velocity r V a = 100, b = 10, 0.1 U = 1.0 l N = 1.0 s = 0.2 f s = 1.0 g = The dot points correspond to the numerical results and the line is the curve fitting. Effect of the Ligand Density In Figure 5 21 and Figure 5 22, the effects of ligand density on the peeling time and the rolling are shown. An incr ease of the ligand density increases the peeling time and decreases the rolling velocity for 1.0 l N For l N > 1.0, the peeling time and the rolling velocity remain the same. The following relations are used to describe the dependence: 2.6 3( ) 76.91 l t L nN =+ ( 40 ) 0.0352 0.0037 l r VN = ( 41 ) The plateau is reached because no receptor is available for binding as the ligand density increases. Dong e t al. (1999) and Alon et al. (1998) made the same observation in their simulation and experiments respectively. PAGE 76 59 72 74 76 78 0.2 0.4 0.6 0.8 1 1.2 1.4 N l peeling time Figure 5 21: Effect of the ligand density l N on the peeling time t a = 100, b = 10, 0.1 U = 1.0 r N = 1.0 s = 0.2 f s = 1.0 g = The dot points correspond to the numerical results and the line is the curve fitting. 0.0036 0.0037 0.0038 0.0039 0.2 0.4 0.6 0.8 1 1.2 1.4 N l V r Figure 5 22: Effect of the ligand density l N on the rolling velocity r V a = 100, b = 10, 0.1 U = 1.0 r N = 1.0 s = 0.2 f s = 1.0 g = The dot points correspond to the numerical results and the line is the curve fitting. PAGE 77 60 Effect of the Surface Tension The effect of the surface tension on the peeling time shows an inverse dependence as observed in Figure 5 23. 30 80 130 180 0 2 4 6 8 10 g peeling time Figure 5 23: Effect of the surface tension on the peeling time t a = 100, 0.1 U = 1.0 r N = 1.0 l N = 1.0 s = 0.2 f s = 10 b = The dot points correspond to the numerical results and the line is the curve fitting. The following relation can be used to describe this dependence 0.3483 82.78 t g = ( 42 ) In addition, an increase of the surface tension increases the rolling velocity as shown in Figure 5 24. For 1.0 g the relationship can be described by 2 0.001 8 0.004 3 0.001 r V gg = -++ ( 43 ) while for 1.0 g the relation is given by 6 3 42 5.1 0 1 0 0.001 3 0.0023 r V g gg -= ++ ( 44 ) PAGE 78 61 0.001 0.002 0.003 0.004 0.1 0.3 0.5 0.7 0.9 g V r a 0.003 0.005 0.007 1 3 5 7 9 g V r b Figure 5 24: Effect of the surface tension on the rolling v elocity r V a = 100, 0.1 U = 1.0 r N = 1.0 l N = 1.0 s = 0.2 f s = 10 b = 1.0 g (a) and 1.0 g (b). The dot points correspond to the numerical results and the line is the curve fitting. Effect of the Parameter b Figure 5 25 and Figure 5 26 show the effect of the parameter b which is the ratio of the nucleus viscosity t o the cytoplasm viscosity, on the peeling time and rolling velocity. An increase of b increases the peeling time and decreases the rolling velocity. A third order polynomial can be used to fit the computational points. 50 60 70 80 90 2 4 6 8 10 b peeling time Figure 5 25: Effect of b on the peeling time. a = 100, 0.1 U = 1.0 r N = 1.0 l N = 1.0 s = 0.2 f s = The dot points correspond to the numerical results and the li ne is the curve fitting. PAGE 79 62 0.003 0.004 0.005 2 4 6 8 10 b V r Figure 5 26: Effect of b on the rolling velocity. a = 100, 0.1 U = 1.0 r N = 1.0 l N = 1.0 s = 0.2 f s = The dot points correspond to the numerical results and the line is the curve fitting. Effect of the Parameter a The effects of the parameter a defined as the ratio of the cytoplasm viscosity to the plasma viscosity, on the peeling time and rolling velo city are illustrated in Figures 5 27 and 5 28 respectively. An increase of a slows the breakage of the formed bonds and the cell rolling velocity. A third order polynomial and an inverse power are found to best fit the responses of the peeling time and t he velocity, respectively. 0 50 100 150 200 0 50 100 150 200 250 300 a peeling time Figure 5 27: Effect of a on peeling time. b = 10, 0.1 U = 1.0 r N = 1.0 l N = 1.0 s = 0.2 f s = The dot points correspond to the numerical results and the line is the curve fitting. PAGE 80 63 0 0.004 0.008 0.012 0 50 100 150 200 250 300 a V r Figure 5 28: Effect of a on the rolling velocity. b = 10, 0.1 U = 1.0 r N = 1.0 l N = 1.0 s = 0.2 f s = The dot points correspond to the numerical results and the line is the curve fitting. Effect of the Contact Length The effect of the initial contact length was also investigated in this study. An increase of the contact length augments the initial bond number as indicated by Equation (45): 1. 47 b n q =+ ( 45 ) where n b is the number of bonds and q is defined in Figure 5 29. q Contact length Figure 5 29: Angle and contact length definitions In addition, an increase of the contact length rises the peeling time and reduces the cell rolling velocity as shown in Figures 5 30 and 5 31, respectively. The PAGE 81 64 relationships between the initial contact length, t he peeling time and the velocity are given by: 2 0.012 5 0.82 5 67.813 t qq = ++ ( 46 ) 5 4.1 0 0.0041 r V q = -+ ( 47 ) 75 95 115 10 20 30 40 angle peeling time Figure 5 30: Effect of the contact length on the peeling t ime t a = 100, 0.1 U = 1.0 r N = 1.0 l N = 1.0 s = 0.2 f s = 10 b = 1.0 g = The dot p oints correspond to the numerical results and the line is the curve fitting. 0.002 0.0025 0.003 0.0035 10 20 30 40 angle V r Figure 5 31: Effect of the surface tension on the rolling velocity r V a = 100, 0.1 U = 1.0 r N = 1.0 l N = 1.0 s = 0.2 f s = 10 b = The dot points correspond to the numerical results and the line is the curve fitting. PAGE 82 65 Comparison to Experimental Results A co mparison of our results to the experimental results obtained by Schmidtke and Diamond (2000) is conducted by first fixing the cell surface tension and varying the cell viscosity and then the cell viscosity is held fixed while varying cell surface tension. In this computation, the number of bonds is taken to be small. Effect of Cell Viscosity on Bond Lifetime Comparisons between the experimental data by Schmidtke and Diamond (2000) and the compound drop model results of NDri et al. (2002) are shown in Figu re 5 32, for viscosity values of 50, 100 200 and cytoplasmic viscosity to nucleus viscosity ratios of 5 and 10. The interfacial tension is fixed at 1.2. The lifetime of the bonds decreases with the amount of applied shear rate, and if the cell cytoplasm is less viscous. Within the standard deviations of the experimental data, the compound drop model with a cytoplasmic viscosity equals to 50 fits the data. However, the model with a cytoplasmic viscosity of 100 describes better the experimental data for s hear rates up to 250 s 1 It is also seen that if the nucleus is more viscous, it will increase the lifetime of the bonds. Effect of Cell Surface Tension on Bond Lifetime As shown in Figure 5 33, fixing the cell viscosity and varying the surface tension a llows us to assess the surface tension effect on the bond lifetime. Higher surface tension shows agreement with the experimental results obtained by Schmidtke and Diamond (2000). Again an increase of surface tension decreases the bond lifetime. PAGE 83 66 0 0.4 0.8 1.2 50 100 150 200 250 300 shear rate (s -1 ) t(s) a = 200, b = 10 a = 100, b = 10 a = 50, b = 10 a = 200, b = 5 Figure 5 32: Comparison of the bond lifetimes for the compound drop case. The dot corresponds to Schmidtke and Diamond (2000) results and the lines to NDri et al. (2002). 0.02 r N = 1.0 l N = 0.1 s = 0.04 f s = g =1.2. 0 0.4 0.8 1.2 50 100 150 200 250 300 shear rate (s -1 ) t(s) 0.8 g = 1.0 g = 3.0 g = Figure 5 33: Effects of surface tension and shear rate on the bond lifetime: compound drop model. The dot corresponds to Schmidtke and Diamo nd (2000) results and the lines to the present study. 1.0 r N = 1.0 l N = 0.1 s = 0.04 f s = a = 100 and b = 10. PAGE 84 67 Effect of Applied Hydrodynamic Force on the Bond Lifetime The bon d lifetime is shown as a function of the applied hydrodynamic force for a cytoplasmic viscosity equal to 200 in Figure 5 34. An exponential function, as proposed by Shao and Hochmuth (1999), can be used to describe the results as indicated in Figure 4 34a The hydrodynamic force is computed as follows F Ud m = ( 48 ) where m is the plasma viscosity, U the inlet velocity and d the tube diameter. 0.4 0.5 0.6 0.7 0.8 600 1000 1400 1800 F( m dyne) t(s) a 0.4 0.5 0.6 0.7 0.8 600 1000 1400 1800 F( m dyne) t(s) b Figure 5 34: Bond lifetimes as a function of applied force. 0.02 r N = 1.0 l N = 0.1 s = 0.04 f s = g =1.2, 10 b = a = 200. The dot points correspond to the numerical results and the line is the curve fitting, (a) exponential fit; (b) inverse power fit. The exponential relationship, 0.0004 0.95e F t = ( 49 ) provides a good fit to the data, however, an inverse correlation between the applied force or pulling force and the bond lifetime, gives the best fit to the data (Figur e 5 34b). An increase of the pulling force decreases the bond lifetime as observed by many authors (Evans, 1999; Smith and Lawrence, 1999). 19.1 t F = ( 50 ) PAGE 85 68 Bond Rupture Force F br The force of rupture of a sing le bond, F br was also determined. This force is the force reaches by the bond before it breaks. The critical bond forces computed by Goldman et al. (1967b) and Dong et al. (1999) are reported to be between 100 and 400 pN and 100 and 200 pN, respectively In the literature, experimental measurements give forces of rupture between 37 and 250 pN (Shao and Hochmuth, 1996; Smith and Lawrence, 1999). Evans (1999) found bond strength of 200 300 pN for biotin streptavidin and a value of bond strength of order 160pN for biotin avidin. In the work of NDri et al. (2002), the values of the bond molecule force vary from 250 to 400 pN for a range of shear flows between 2 and 20 dyne/cm 2 and a spring constant 0.1 s = These results are shown in Figure 5 35 for a = 100. The value of 0.1 s = corresponds to a dimensional value of 0.5 dyne/cm as used in the computation of Dong and Lei (2000). Lower values of F br can be obtained by decreasing the value of the spring constant. A lower force of rupture means a shorter bond lifetime. Figure 5 35 demonstrates a linear relationship between bond lifetime and force of rupture, 0.001 3 0.206 b r tF =( 51 ) NDri et al. (2002) found that using a bond stiff ness 06 0 s ( s = 0.3 dyn/cm) gives the range of bond forces reported in the literature. It is important to note that Shao and Hochmuth (1996) have reported a spring constant value of 43pN/ m m (0.043 dyne/cm), which is much smaller than the values cited above. Using the range of force computed in this study, the length reached by a bond before breakage was computed and is found to lie between 0.25 and 0.85 m m. The measured bond length found in the literature is between 0.1 and 0.7 m m (Shao et al., 1998). PAGE 86 69 Effect of Viscosity on Bond Rupture Force The bond lifetime as a function of the shear for different values of the viscosity was shown in Figure 5 33. Using equation (51), one can assess the effect of viscosity on rupture force of a bond f or a fixed value of the inlet boundary condition. The result is shown in Figure 5 36. An increase of the viscosity for a fixed applied hydrodynamic force increases linearly with the rupture force. 0 0.1 0.2 0.3 0.4 200 300 400 F br (pN) t(sec) Figure 5 35: Bond lifetime t as a function of bond force F br 0.02 r N = 1.0 l N = 0.1 s = 0.04 f s = g =1.2, 10 b = The dot points correspond to the numerical results and the line is a curve fitting. 300 500 700 50 100 150 200 a F br (pN) Figure 5 36: Effect of the viscosity on the bond rupture force. 0.02 r N = 1.0 l N = 0.1 s = 0.04 f s = g =1.2, 10 b = PAGE 87 70 Effect of Interfacial Tension on Bond Rupture Force Using Figure 5 34 and Equation (51), we can assess the effect of the surface tension on the bond rupture force. An increase of the surface tensio n inversely decreases with the bond rupture force for a fixed applied hydrodynamic force as shown in Figure 5 37. 550 1050 1550 0.1 1.1 2.1 g F br (pN) Figure 5 37: Effect of the surface tension on the bond rupture force. 0.02 r N = 1.0 l N = 0.1 s = 0.04 f s = 10 b = a =100. Multiple Bond Case Finally, the case of multiple bonds is shown in Figure 5 38. In Schmidtke and Diamond (2000), only a shear rate of 150 s 1 was studied. In the present computation, as in the case of a single bond, shear rates vary between 100 and 250 s 1 the cytoplasmic viscosity is 100, and the interfacial tension ranges from 0.1 to 3.0. The number of bonds is not known in the work of Schmidt ke and Diamond (2000), but in the simulation of NDri et al. (2002) the bond number is 7. The bond lifetime is higher for a cell with a lower surface tension. The top curve corresponds to the cell with the lowest surface PAGE 88 71 tension. In addition, an increas e of bonds or cell viscosity will extend the bond lifetime as suggested in Figures 5 19, 5 25, 5 27. 0 2 4 6 8 50 100 150 200 250 300 shear rate (s -1 ) t (s) 0.1 g = 0.4 g = 1.0 g = 3.0 g = Figure 5 38: Comparison of the computed bond lifetimes in the case of compound drop and multiple bonds. The dot corresponds to Schmidtke and Diamond (2000) results and the line to NDri et al. (2002). a = 100, 1.0 r N = 1.0 l N = 1.0 s = 0.04 f s = 10 b = In summary, from the work of NDri et al. (2002) using the multi scale approach outlined in chapter 4, it is shown that cell deformability affects the rolling of a cell along a vessel wall. A cell with a higher surface tension rolls faster. These results are in agreement with tho se of Dong and Lei (2000) where only a small section of the membrane is allowed to deform. The results also show that as the shear rate increases, the bond lifetime decreases as observed by Schmidtke and Diamond (2000). The bond lifetime is also found to increase as the cell or nucleus viscosities increase. PAGE 89 72 Effect of the Cell Diameter There are three types of leukocytes with different diameters, the lymphocytes with 7 m m, the neutrophils with 8 m m, and the monocytes with 10 m m in diameter in average. The effect of different cell diameters for a fixed tube diameter on the peeling time is shown in Figure 5 39. A larger cell diameter decreases the peeling time and increases the cell rolling velocity as shown in Tran Son Tay et al. (2002). This result is con sistent with the simulation and experimental results found by Tees et al. (2002) and by Patil et al. (2001), respectively. A larger cell causes a higher blockage in the tube, which results in larger hydrodynamic forces. Considering that the velocity scal es inversely and linearly with the tube opening (and hence scales linearly with the cell size) to satisfy mass continuity, and the pressure force scales quadratically with the velocity, the hydrodynamic force on the cell increases more than linearly with t he cell diameter. 2.7 3 3.3 3.6 7 8 9 10 cell diameter ( m m) t(s) Figure 5 39: Effect of cell diameter on the peeling time of the cell. a = 100, 0.1 U = 1.0 r N = 1.0 l N = 1.0 s = 0.2 f s = 5 b = Effect of the Vessel Diameter The blood vessel diameter varies from centimeter to micrometer in the circulatory system, in order to study the effect of the vessel diameter on cell adhesion, we compute PAGE 90 73 the peelin g time of an adherent cell for three different diameters. Figure 5 40 shows the result of such a computation, where we found that an increase of the vessel diameter decreases the peeling time and increases the cell rolling velocity. Increasing the vessel diameter increases the applied hydrodynamic force. 2.65 2.75 2.85 2.95 0.5 0.6 0.7 0.8 0.9 1 d t (s) Figure 5 40: Effect of vessel diameter on the peeling time of the cell. a = 100, 0.1 U = 1.0 r N = 1.0 l N = 1.0 s = 0.2 f s = 5 b = Effect of the Pulsatile Inlet Boundary Condition The effect of the inlet pulsatile boundary condition on the peeling of the cell is also assessed. Equation (52) shows the bou ndary condition used and Figure 5 41 shows the number of pulses studies in the present work. 4 1 1 ( ) sin () 10 P n U t nt p = = ( 52 ) Where U P is the inlet velocity and t the time. PAGE 91 74 -0.1 0 0.1 0.2 0.3 0 5 10 15 20 t U Figure 5 41: Pulsatile (li ne) and uniform (dashed line) inlet boundary conditions used in the present work. Comparison of the Peeling Time for the Uniform and Pulsatile Conditions Figure 5 42 shows a comparison of the pulsatile and uniform inlet boundary conditions for different va lues of the surface tension. The uniform boundary condition is computed by taking the average of the pulsatile velocity over one period. 0 1 () T P U U t dt T = ( 53 ) where T is the time period. We observed that the peeling time is higher for the uniform inlet velocity than that of the pulsatile boundary condition. PAGE 92 75 25 40 55 70 1 2 3 4 5 6 g peeling time Figure 5 42: Comparison of peeling time for the pulsatile (line) and uniform (dashed line) inlet boundary condition for different val ues of the surface tension. a = 100, 1.0 r N = 1.0 l N = 1.0 s = 0.2 f s = 10 b = Effect of the Surface Tension on the Pulse Number Figure 5 43 shows the e ffect of the cell surface tension on the number of pulses needed to peel the cell away from the wall. An increase of the surface tension decreases the number of pulses. The approximate curve that can be used to fit the experimental points is given below: 0.82ln () 2.08 p n g = -+ ( 54 ) where n p is the number of pulses PAGE 93 76 0 1 2 3 4 0 1 2 3 4 g Number of pulses, n p Figure 5 43: Effect of the cell surface tension on the number of pulses n p a = 100, 1.0 r N = 1.0 l N = 1.0 s = 0.2 f s = 10 b = The dot corresponds to the numerical results and the line is the curve fitting. Effect of the Parameter a on the Pulse Number The study of the effect of the parameter b ratio of the nucleus viscosity to the cytoplasmic viscosity, has shown no difference for the range ( 1 10 b ) used in the present study. On the other hand, the effect of the parameter a ratio of the cytoplasmic viscosity to the plasma viscosity, shows a linear dependence of the pulse number n p with a as shown in Figure 5 44. 0.01 p n a = ( 55 ) In summary, the pulsatile inlet boundary condition has shown that surface tension and the parameter a the ratio of the cytoplasm viscosity to the plasma viscosity, have major impact on the number of pulses needed to peel the cell away from the vessel wall while the parameter b the ratio of the nucleus viscosity to the cytoplasm viscosity, ha s shown no impact on the number of the pulses needed to detach the cell for the inlet boundary conditions used in the present study. PAGE 94 77 1 2 3 100 200 300 a Number of pulse, n p Figure 5 44: Effect of the parameter a on the number of pulses, n p 1.0 r N = 1.0 l N = 1.0 s = 0.2 f s = 10 b = The dot corresponds to the numerical results and the line is the curve fitting. Effect of the Initial Bond Force In all the previous co mputations, the initial bond force was taken to be zero, in the following computation; we assume an initial bond force, which varies with surface tension. The initial bond force is computed as follows: bo Fr g = ( 56 ) where r is the cell radius and g is the cell surface tension Figure 5 45 shows a comparison of the peeling time for both non zero and zero initial bond force for different values of surface tension. The peeling time is higher in the case of non ze ro initial bond force. In most of the studies conducted over the past years, no initial bond force was assumed in the computation of the bond force. PAGE 95 78 39 49 59 69 79 1 2 3 4 5 6 g Peeling Time Figure 5 45: Comparison of peeling time for the zero (line) and non zero (das hed line) initial bond force for different values of the surface tension. a = 100, 10 b = 1.0 r N = 1.0 l N = 0.2 f s = PAGE 96 79 CHAPTER 6 CONCLUSION AND DISCUSION In this work, a multi scale modeling for biofluid dynamics with an emphasis on cell adhesion was present ed. This method breaks the computational work into two separate but interrelated domains. At the cellular level, a continuum model satisfying the field equations for momentum transfer and mass continuity was adopted. At the receptor ligand or molecular l evel, a bond molecule was mechanically represented by a spring. A reversible two body kinetic model characterized the association and dissociation of a bond. Communication between the macroscopic and microscopic scale models was facilitated interactively in the course of the computation. The computational model was assessed using an adherent cell allowed to roll along the vessel wall under imposed shear flows. The cell was first modeled as a liquid drop with constant surface tension to illustrate the com putational approach, and then as a compound drop in order to evaluate the effect of the nucleus. The results compare very well with those obtained computationally (Chang and Hammer, 2000;Dong et al., 1999; Dong and Lei, 2000; Tees et al., 2002) and experi mentally (Patil et al., 2001; Schmidtke and Diamond, 2000). We have shown that cell deformability affects the rolling of a cell along a vessel wall. A cell with a higher surface tension rolls faster. These results are in agreement with those of Dong and Lei (2000), where only a small section of the membrane is allowed to deform. PAGE 97 80 The results also showed that as the shear rate increases, the bond lifetime decreases as observed by Alon et al. (1997, 1998) and Evans (1999), Schmidtke and Diamond (2000), Smi th et al. (1999). The bond lifetime was also found to increase as the cell or nucleus viscosities increase. A decrease in cell diameter increases the peeling time and decreases the rolling velocity as observed by Tees et al. (2002) and Patil et al. (2001 ). We also studied the effect of vessel diameter on the adhesion process. We found that increasing the vessel diameter decreases the peeling time and increases the rolling velocity. In addition, we have shown that the peeling time is smaller for a pulsa tile inlet condition than that of a uniform one and a non zero initial bond force increases the peeling time. The bond lifetime was also shown to increase with the critical bond force, which is the force reaches by a bond molecule before breaking. An inc rease of the cell viscosity increases the critical bond force while increasing cell surface tension decreases the bond rupture force. In this work, we use the kinetic model proposed by Dembo et al. (1988), a drawback with this model is the assumption that the bonding stresses are the only distributed stresses acting on the membrane and bonds molecules are fixed on cell surface while Bells model takes into account the role of the non specific potential forces in biological chemistry and the diffusion of th e bond molecules to the contact area. Nontheless, the results obtained in this study wont be affected by the Bells equation for the reverse reaction rate. Many computational studies have considered the cell as a solid body. Some state that the deformat ion of the cell would not have a significant impact on rolling of the cell, but microvillus deformation is more important. However, the microvilli are connected to the cell surface; hence, a deformed microvillus also means cell deformation. The study PAGE 98 81 by NDri et al. (2002) has shown that a tether was pulled away from the cell membrane at high values of shear stress, but no membrane pulling was observed at low values of shear stress (Figure 5 15). These observations are consistent with what have been seen in the literature (Alon et al., 1997; Schmidtke and Diamond, 2000; Shao et al., 1998). Shao et al. (1998) have found that after the microvillus reaches its natural length, it will extend under a small pulling force or form a tether under a high pulling f orce. In addition, we have also observed a lifting of the cell from the vessel wall leading to its peeling as shown in Figure 5 2b. Similar observations were made by Sukumaran and Seifert (2001), who studied the influence of the shear flow on vesicles ne ar a wall, and by Hodges and Jensen (2002) in their numerical study. The graph of the height of center of the cell above the plane increases rapidly with time, then reaches a plateau as observed in Figure 5 2b of the present study. In addition, in the st udy by Hodges and Jensen (2002), they found that changing the suspending fluid viscosity does not affect the rolling velocity of the cell along the wall. In their study, the viscosity of the cell was taken to be very small, so the fluid inside the cell wa s considered as invinscid. While in this study we have shown that fixing the suspending viscosity and varying either the cytoplasmic or/and the nucleus viscosity affect the rolling velocity of the cell showing that hydrodynamics do affect cell adhesion. I t is also known that a higher shear stress can enhance bond formation. How this phenomenon occurs is not clear. Does an increase of the shear stress increase first the diffusion and convection of bond molecules toward the contact area, or does an increas e of the shear stress lead to an increase of the contact area due to cell deformation? PAGE 99 82 Although, in our study, bond formation is not considered per se due to the numerical resolution (we consider that a bond is formed when the distance between a receptor a nd ligand is less than a bond length, taken as 5 x 10 6 cm), we can nevertheless comment from Figure 5 14 that an increase of the shear stress keeps the cell closer to the wall, and increases the contact area, as shown in Figure 5 15. The finding of the prese nt work is also consistent with the cell shape observed experimentally by Schmidtke and Diamond (2000) and numerically by Dong et al. (1999) and Dong and Lei (2000), who have shown that the cell goes from a spherical form to a tear like drop shape. Althoug h the proposed approach was able to describe key features of cell adhesion, several issues still need to be addressed. For example, experimental results have shown that there is a threshold stress above which cell rolling occurs. However, the models prop osed in the literature cannot capture this feature, indicating that the straightforward analogy between a bond molecule and a spring model is incomplete. A bond model with a yield force can be used to help resolve this deficiency. Furthermore, in the pre sent study we have neglected membrane roughness and the effect of non specific forces in the bond force computation. With respect to cell rheology, although the compound liquid drop model can capture most of the features of cell recovery and explain the r easons for the different published values for leukocyte viscosity, it does not include elastic effects which may be needed to fully describe leukocyte rheology (Tran Son Tay et al., 1994, 1998; Drury and Dembo, 1999, 2001; Kan et al., 1998, 1999). In addi tion, we assumed that the bond molecules are fixed on the membrane surface, which is not the case since bond molecules have been shown to diffuse laterally to the contact PAGE 100 83 area. The last issue is the assumption of a 2D cell model, which has some limitation s such as force estimation with respect to 3D assumption. Another issue is the error related to the multi scale model. It is noted that the adhesion model is kinetics based and is statistical. One approach to address the error associated with the micro m acro interactions is to consider the uncertainty of the microscopic prediction and assign a distribution function. This error distribution will then be incorporated into the adhesion model, based on which a statistical solution will be obtained, with the mean transmitted to the macroscopic model. In order to compute the bond lifetime, we have assumed that bonds are broken if the cell travels 2 cell radii based on non physical cell shape beyond this distance. This assumption is solely based on cell mechani c and not on a single bond mechanic. To overcome this difficulty, we propose to use bond molecule strength as a condition for bond breakage. Therefore, the bond lifetime will be the unknown parameter that will depend on the hydrodynamic pulling force and the cell rheological properties. Nevertheless, the present effort has offered a comprehensive framework to couple the cellular and the receptor ligand dynamics and significant knowledge has been gained from this work. In the future, we will exploit the f act that the time scale for the formation of a new interface is fairly short, of the order on 1 nanosecond (ns) or less, and can therefore be tracked explicitly by large scale parallel molecular dynamics simulations. Similar to the present model, the boun dary conditions on the molecular simulation will be obtained from the continuum simulation and vice versa. It is simply a matter of time that a more comprehensive framework will be developed to offer new insights into the very complicated processes that o ccur in the dynamics of blood cells and blood flow. 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PAGE 109 92 BIOGRAPHICAL SKETCH Narcisse Abou NDri was born in Ivory Coast (Cote dIvoire) whe re he received his bachelors degree in applied mathematics at the University of Cocody. Then, he earned a fellowship from his government to pursue his study in France. After graduating with a doctoral degree in biorheology at the University of Paris 6, he decided to complete his academic training by expanding his knowledge in computational fluid dynamics. He entered the PhD program at the University of Florida in 1999. |