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See discussions, stats, and author profiles for this publication at: http://www.researchgate.net/publication/11652190 Fluid Mechanics of Vascular Systems, Diseases, and Thrombosis ARTICLE in ANNUAL REVIEW OF BIOMEDICAL ENGINEERING · FEBRUARY 1999 Impact Factor: 12.45 · DOI: 10.1146/annurev.bioeng.1.1.299 · Source: PubMed CITATIONS 253 DOWNLOADS 1,800 VIEWS 196 2 AUTHORS: David M Wootton The Cooper Union for the Advancement of Sc… 48 PUBLICATIONS 676 CITATIONS SEE PROFILE David N Ku Georgia Institute of Technology 150 PUBLICATIONS 6,255 CITATIONS SEE PROFILE Available from: David N Ku Retrieved on: 16 June 2015

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  • Seediscussions,stats,andauthorprofilesforthispublicationat:http://www.researchgate.net/publication/11652190

    FluidMechanicsofVascularSystems,Diseases,andThrombosisARTICLEinANNUALREVIEWOFBIOMEDICALENGINEERINGFEBRUARY1999ImpactFactor:12.45DOI:10.1146/annurev.bioeng.1.1.299Source:PubMed

    CITATIONS253

    DOWNLOADS1,800

    VIEWS196

    2AUTHORS:

    DavidMWoottonTheCooperUnionfortheAdvancementofSc48PUBLICATIONS676CITATIONS

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    DavidNKuGeorgiaInstituteofTechnology150PUBLICATIONS6,255CITATIONS

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    Availablefrom:DavidNKuRetrievedon:16June2015

  • Annu. Rev. Biomed. Eng. 1999. 01:299329Copyright q 1999 by Annual Reviews. All rights reserved

    15239829/99/08200299$08.00 299

    Fluid Mechanics of VascularSystems, Diseases, and Thrombosis

    David M. Wootton1 and David N. KuG.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology,Atlanta, Georgia 303320405; e-mail: [email protected],[email protected]

    Key Words platelets, shear, arteriosclerosis, stenosis, intimal hyperplasiaAbstract The cardiovascular system is an internal flow loop with multiple

    branches circulating a complex liquid. The hallmarks of blood flow in arteries arepulsatility and branches, which cause wall stresses to be cyclical and nonuniform.Normal arterial flow is laminar, with secondary flows generated at curves andbranches. Arteries can adapt to and modify hemodynamic conditions, and unusualhemodynamic conditions may cause an abnormal biological response. Velocity profileskewing can create pockets in which the wall shear stress is low and oscillates indirection. Atherosclerosis tends to localize to these sites and creates a narrowing ofthe artery lumena stenosis. Plaque rupture or endothelial injury can stimulate throm-bosis, which can block blood flow to heart or brain tissues, causing a heart attack orstroke. The small lumen and elevated shear rate in a stenosis create conditions thataccelerate platelet accumulation and occlusion. The relationship between thrombosisand fluid mechanics is complex, especially in the post-stenotic flow field. New con-vection models have been developed to predict clinical occlusion from platelet throm-bosis in diseased arteries. Future hemodynamic studies should address the complexmechanics of flow-induced, large-scale wall motion and convection of semisolid par-ticles and cells in flowing blood.

    CONTENTS

    Introduction ..................................................................................... 300Physiologic Environment.................................................................... 300Flows in Specific Arteries................................................................... 302

    The Carotid Arteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302The Aorta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303Flow at the Left Coronary Artery Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304Flows in the Heart and Great Vessels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

    Biological Responses to Hemodynamics ............................................... 305Hemodynamics of Stenoses ................................................................ 308

    1Department of Biomedical Engineering, Johns Hopkins University School of Medicine,Baltimore, Maryland 21205

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    Diagnosis of Disease. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309Shear-Dependent Thrombosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

    Arterial Thrombosis .......................................................................... 310Cellular and Molecular Mechanisms of Thrombosis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

    Hemodynamics and Thrombosis .......................................................... 312Hemodynamics in Advanced Atherosclerosis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313Hemodynamics and Thrombus Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313Shear and Platelet Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313Shear-Linked Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

    Modeling Clinical Thrombosis ............................................................ 317Model Based on Ex Vivo Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318A Model of Occlusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

    Conclusions ..................................................................................... 322

    INTRODUCTION

    Nutrient and waste transport throughout the body is the primary function of thecardiovascular system. The heart serves to pump blood through a sophisticatednetwork of branching tubes. The flow is not steady but pulsatile. The blood vesselsdistribute blood to different organs while maintaining vessel integrity. The arteriesare not inert tubes but adapt to varying flow and pressure conditions by growingor shrinking to meet changing hemodynamic demands.

    It is important to study blood flows during disease as well as under normalphysiologic conditions. The majority of deaths in developed countries are fromcardiovascular diseases. Most cardiovascular diseases are associated with someform of abnormal blood flow in arteries. This review focuses on some selectedareas of importance to cardiology.

    PHYSIOLOGIC ENVIRONMENT

    The fluid blood is a complex mixture of semisolid and liquid material. Blood iscomposed of cells, proteins, lipoproteins, and ions by which nutrients and wastesare transported. Red blood cells (RBCs) typically comprise ;40% of blood byvolume. In most arteries, blood behaves in a Newtonian fashion, and the viscositycan be taken as a constant 4 centipoise (cP) for a normal hematocrit. The non-Newtonian viscosity is extensively studied in the field of biorheology and hasbeen reviewed by others (e.g. 21, 89).

    Blood flow and pressure are unsteady. The cyclic nature of the heart pumpcreates pulsatile conditions in all arteries. The aorta serves as a compliance cham-ber that provides a reservoir of high pressure during diastole as well as systole.Flow is zero or even reversed during diastole in some arteries such as the external

  • FLUID MECHANICS AND THROMBOSIS 301

    carotid, brachial, and femoral arteries. These arteries have a high distal resistanceduring rest, and flow is on/off with each cycle. Flow during diastole can also behigh if the downstream resistance is low, as in the internal carotid or the renalarteries.

    Pulsatile flows dominate many of the problems in the cardiovascular system.The existence of unsteady flow forces the inclusion of a local acceleration termin most analyses. In contrast to unsteadiness, several features of biological flowsmay often be neglected as being of secondary importance for particular situations.These include vessel wall elasticity, non-Newtonian viscosity, slurry particles inthe fluid, body forces, and temperature. Although each of these factors is presentin physiology, the analysis is greatly simplified if they can be justifiably neglected,which is the case in most arterial flows.

    Biologists are often concerned with the local hemodynamic conditions in aparticular artery or branch. The important fluid mechanic parameter is often a de-tailed local description of the fluid-wall shear stress in a blood vessel for a givenpulsatile flow situation. The three-dimensional nature of many of these unsteadyflows has provided an important challenge to computational methods, becausethe computational time required is enormous.

    The arterial system is tortuous and must branch many times to reach an endorgan. The cross-sectional area along the axis may enlarge at branch points,sinuses, and aneurysms. However, if the area diverges, the flow must decelerate,and an adverse pressure gradient can exist. In this situation, flow separation ispossible and typically occurs along the walls of the sinus.

    As blood flows across the endothelium, a shear stress is generated to retardthe flow. The wall shear stress is proportional to the shear rate c (velocity gradient)at the wall, and the fluid dynamic viscosity l: s 4 lc . Shear stress for laminarsteady flow in a straight tube is

    11 13s 4 32lqp D ,

    where q is volume flow rate, and D is tube diameter. This approximation is areasonable estimate of the mean wall shear stress in arteries. For situations inwhich the lumen is not circular or the blood flow is highly skewed, as it is atbranch points, shear stress must be determined by detailed measurements of veloc-ity near the wall. Shear stress is not easily measured for pulsatile flows. Thevelocity and velocity gradient must be measured very close to a wall, which istechnically difficult. The gradient will depend highly on the shape of the velocityprofile and the accurate measurement of distance from the wall. For blood flow,the viscosity very near a wall is not precisely known because the red cell con-centration is reduced. Thus, arterial wall shear stress measurements are estimatesand may have errors of 20%50%.

    At the lumenal surface, shear stress can be sensed directly as a force on anendothelial cell. In contrast, cells cannot sense flow rates directly. Determinationof the flow rate would require knowledge of blood velocities far away from cellsin the artery wall, as well as some way to integrate the velocities to give the

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    volume flow rate. Thus, it is natural for endothelial cells to sense and respond toshear stress.

    Arteries will typically adapt to maintain a wall shear stress of ;15 dyn/cm2(41). This appears to be true for different arteries within an animal, betweenanimal species, as well as after large changes within a single artery. The blood-wall shear stress modulates diameter adaptive responses, intimal thickening, andplatelet thrombosis. The wall shear stress is thus central to the vascular responseto hemodynamics.

    The other major hemodynamic force on an artery is the transmural pressureacross the thickness of the wall. Arteries have a mean pressure of ;100 mmHg,whereas veins have pressures of ;10 mmHg. The hoop stress can be estimatedby Laplaces Law as

    11r 4 0.5PDt ,

    where t is wall thickness, D is vessel inner diameter, and P is transmural pressure,for vessels with circular lumens that are not too thick (38). It is possible that theprimary determinant of smooth muscle cell response is the local strain of thesecells. The arterial wall may remodel in response to both static and cyclical loadingconditions by secretion and organization of collagen and elastin, respectively (88).

    FLOWS IN SPECIFIC ARTERIES

    There are four major arteries that are subject to the most clinical disease. Theseinclude the carotid bifurcation, the abdominal aorta, the left coronary artery, andthe heart and proximal aorta.

    The Carotid Arteries

    The carotid arteries are located along the sides of the neck. These arteries supplythe brain and face with blood. Atherosclerosis, which develops right at the bifur-cation, causes the majority of strokes in patients. The branch is unique in thatthere is an anatomic sinus or expansion at the origin of the internal carotid. Themean Reynolds number is ;300, and the Womersley parameter is ;4. The daugh-ter branches are ;258 off-axis of the parent artery, on average.

    Measurements of velocity have been made in machined plastic models of thisbifurcation by laser Doppler anemometry (65). Secondary flows are produceddownstream of the bifurcation (Figure 1). Velocity profiles obtained by laserDoppler anemometry and computational fluid dynamics quantify the extent ofreverse velocities at the outer wall of the internal carotid sinus (Figure 2). Aregion of transient flow separation is created along the posterior wall of the carotidsinus, which is prominent during the downstroke of systole. The artery wall inthe sinus region would experience oscillations in near-wall velocity and a lowmean wall shear stress. Atherosclerotic plaque is highly localized to a small area

  • FLUID MECHANICS AND THROMBOSIS 303

    FIGURE 1 Hydrogen bubble visualiza-tion of flow through a model carotid bifur-cation illustrating the laminar flow at theflow divider and separation of flow at theposterior wall of the internal carotid sinus.The separation region of transient reversevelocities is also the site of secondary vor-tex patterns. (Reprinted with permission ofthe American Heart Association, Inc.)

    within this sinus region and correlates with low wall shear stress with coefficientsgreater than 0.9, p , 0.001. Comparison of the unsteady, three-dimensional invitro results against in vivo measurements with Doppler ultrasound confirms thatthe assumptions of the modeling are valid (66). Several groups have recently usedcomputational fluid dynamics to study the effects of wall elasticity and non-Newtonian viscosity (4, 86). These effects are small in comparison with the ana-tomic and flow variations between patients (79, 83).

    The Aorta

    The aorta is the large vessel from the heart that traverses the middle of the abdo-men and bifurcates into two arteries supplying the legs with blood. The renalarteries have a low resistance so that two-thirds of the entering flow leaves theabdominal aorta through these branches at the diaphragm. Curiously, atheroscle-rotic disease extends along the posterior wall of the relatively straight abdominalaorta downstream of the renal arteries in all people. Little disease is ever presentin the upstream thoracic aorta.

    In vitro measurements in a glass-blown aorta model show that outflow con-ditions combine with curvature to create an oscillation in velocity direction at theposterior wall of the aorta, with a corresponding low average wall shear stress(77). The area of low wall shear stress correlates very well with the location of

  • 304 WOOTTON n KU

    FIGURE 2 a. Axial velocityprofiles in the sinus region of athree-dimensional model of thecarotid bifurcation, using laserDoppler anemometry (LDA) andcomputational fluid dynamics(CFD). b. Flow in the carotidsinus is unsteady with a transientreverse flow at the outer wallshown in this three-dimensionalplot of velocity vs diameter posi-tion and time. (Reprinted from 65with permission from ElsevierScience, Ltd.)

    atherosclerotic plaque measured in autopsy specimens, p , 0.001 (35, 77). Asverification, measurements of in vivo flow in humans exhibit the same skewingand time-varying velocity profiles as are produced in the model (76).

    Flow at the Left Coronary Artery Bifurcation

    Flow at the left coronary artery bifurcation is complicated by several importantfeatures (10). First, the left main coronary artery is quite short, leading to anentrance type flow at a small Womersley parameter of 3. Second, the flow wave-form in the left coronary artery is reversed in comparison with that in most arter-

  • FLUID MECHANICS AND THROMBOSIS 305

    ies, having more flow during diastole. Flow can be reversed during systole. Thehigh pressure in the myocardium during systolic contraction causes the bloodflow to reverse direction in the coronary arteries. Third, the bifurcation does notlie in a single plane but curves around the heart while branching. The curvatureslikely set up secondary flows during part of the cardiac cycle. The actual fluiddynamics have been characterized with large-scale experimental models (103)and spectral-element computational modeling (50, 51). Comparison of the flowfield with maps of atherosclerotic disease locations yields a strong correlationbetween oscillations in shear stress and probability of plaque (r . 0.85, p ,0.001) (51). Surprisingly, variations in branch angle do not alter the overall flowfield regimes in a dramatic way (50). However, changes in the coronary flowwaveform affect the magnitudes of oscillation significantly (50).

    Flows in the Heart and Great Vessels

    Flows in the heart and great vessels are dominated by inertial forces over viscousforces. Reynolds numbers at peak systole are ;4000. The flow in the aorta andpulmonary trunk is similar to an entrance-type flow, which is not developed.Consequently, the core of the flow can be considered as an inviscid region awayfrom a developing boundary layer at the wall. The pressure and velocity patternsin a complex chamber of the heart can be modeled in three dimensions, evenincluding a moving boundary condition that develops tension (80, 113).

    The analysis of hemodynamics in this representative set of arteries enablesone to develop a general understanding of the fluid mechanics in the normalcardiovascular system. It should be remembered that arteries are not fixed tubes.They are biological organs, which remodel over time.

    BIOLOGICAL RESPONSES TO HEMODYNAMICS

    The artery reacts to the dynamic changes in mechanical stress. Several physiologicresponses are essential to the maintenance of normal functioning of the circulatorysystem. The responses of arteries to the hemodynamic environment may createnormal adaptation or pathological disease.

    Hemostasis is the arrest of bleeding. Trauma is a common occurrence, and thebody must be able to deal with this possibility. In this hemodynamic environmentof high shear stresses, hemostasis is maintained primarily by platelet adherenceand activation. Platelets pass quickly over the injury site, and adherence mustoccur in milliseconds.

    On a longer time scale, an artery can respond to minute-to-minute changes inhemodynamics. The blood vessels must adapt to differing physiologic demandsand conditions from changes in blood pressure and flow. This response is typicallygoverned by the need to control systemic vascular resistance, venous pooling,and intravascular blood volume.

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    Arteries adapt to long-term increases or decreases in wall shear stress. Theresponse to increased wall shear stress is to vasodilate and then remodel to alarger diameter with the same arterial structure. This situation is commonly seenwith the creation of an arteriovenous fistula for hemodialysis access. Decreasedflow rates will cause a thickening of the intimal layer to reestablish a normal wallshear stress (41). Eventually, the artery may maintain a thickened intima orremodel to a normal artery of smaller diameter.

    On an even longer time scale of weeks to months, arteries will remodel theirintima and media layers. The medial thickness is influenced by the local amountof hoop stress and nutrition. As described above, as the blood pressure increases,the hoop stress will proportionally increase (22). Because the formation of alamellar unit requires the proliferation of smooth muscle cells and the creation ofa highly organized extracellular structure, the process may take several days.Alterations in the pulsatile pressure lead to changes in organization of the elastinand collagen structure within the media (41, 88).

    The effects of flow, shear stress, and stretch on arteries in vivo have beenstudied by several groups. Flow can be augmented through an artery by the crea-tion of an arteriovenous fistula. Such increased flow causes a dilation of the arteryuntil the wall shear stress reaches the baseline level of the artery (60, 116). Thisbaseline appears to be ;1520 dyn/cm2 for most arteries in a wide range ofspecies (41). Conversely, restricted flow through an artery produces a smaller-diameter vessel (68).

    Several pathological states may arise from an excessive or uncontrolledresponse to a hemodynamic stimulus. Long-term hypertension produces a gen-eralized medial thickening of blood vessels. Studies of intimal hyperplasia in acanine model clearly indicate that low shear stresses can accelerate intimal thick-ening. Shear stress can also be varied in a single artery by using tapered vasculargrafts with differing diameters. In this case, intimal thickening follows from lowshear stresses even for a constant flow rate as depicted in Figure 3a (94).

    Atherosclerotic disease forms over decades. Atherosclerosis is highly localizedto only a few places in the systemic vasculature. The primary locations of ath-eroma are at the carotid artery sinus, the coronary arteries, the abdominal aorta,and the superficial femoral arteries. In each of these arteries, there are localizedsites where the mean wall shear stress is very low and oscillates between positiveand negative directions during the cardiac cycle. Comparisons of the sites ofdisease with the local hemodynamic conditions reveal a consistent curve wherelow wall shear stress is strongly correlated with atherosclerotic intimal thickening(Figure 3b) (51, 67, 77). Typically, most intimal thickening is found where theaverage wall shear stress is , 10 dyn/cm2 and follows the curve shape shown forintimal hyperplasia and arterial adaptation. Thus the biological pattern of arterialresponse to shear stress appears to be consistent and preprogrammed.

    Currently, a field of cellular and tissue engineering is developing that attemptsto subject cultured cells and tissues to well-defined stresses in an in vitro envi-

  • FLUID MECHANICS AND THROMBOSIS 307

    FIGURE 3 a. Neointi-mal hyperplasia thickeningvs wall shear stress in adog arterial graft. Theinverse relationship indi-cates more thickening atlow shear stresses. b. Ath-erosclerotic intimal thick-ening vs wall shear stressin human carotid arteries.The reciprocal relationshipholds for mean and maxi-mum wall shear stressesand correlates directlywith oscillatory shearstress.

    ronment. The creation of flow chambers that recreate physiologically realistic invivo stresses is an important area of research (53, 75).

    The effects of hemodynamics on convective mass transfer should not beneglected. Most biologically active molecules are convected from one site toanother. These molecules may be nutrients, wastes, growth factors, or vasoactivecompounds. Systemic hormones reach an artery by convection and then maydiffuse through the wall, with the intima as a major barrier. However, convectivemass transport may be a limiting factor for small molecules such as nitrous oxideand oxygen, which diffuse rapidly through the wall. Such convection would beimpaired in areas of flow separation or reversing wall shear at sites prone toatherosclerosis (70, 71). Alternatively, biologically active molecules released byendothelial cells may have an effect downstream if the molecules are trapped ina boundary layer near the wall.

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    FIGURE 4 X-ray contrast angiogram of a dis-eased carotid bifurcation illustrating the focalnature of a stenosis. The stenosis (arrow) willreduce blood flow and pressure to the brain. (FromStrandness DE and van Breda A, 1994. VascularDiseases: Surgical and Interventional Therapy.Reprinted with permission of Churchill Living-stone Inc.)

    HEMODYNAMICS OF STENOSES

    When arteries become severely diseased, the arterial lumen becomes restrictedover a short distance of about 1 cm. This constriction is commonly referred to asa stenosis. An example of an atherosclerotic carotid artery stenosis is depicted inFigure 4.

    In clinical medicine, percent stenosis is commonly defined as percent occlusionby diameter, as follows:

    % stenosis 4 (D11D2)/D1 2 100%,

    where D1 is upstream diameter and D2 is the minimum diameter in the stenosis.As disease advances, the percent stenosis increases.

    Stenotic flows have been well characterized by a number of investigators.Some important summary features are that flow separation (Figure 5b) occurs inthe expansion region at Reynolds numbers of .10 for a 70% stenosis, a strongshear layer develops between the central jet and the recirculation region, thecritical upstream Reynolds number for turbulence is ;300 (114), turbulenceintensity levels reach up to 100% of the upstream velocity values, and the tur-bulence is high for ;1.56 diameters downstream (69).

    For stenoses .75%, flow is limited severely by two mechanisms. Intense tur-bulence downstream of the stenosis creates large pressure losses. In addition, lowpressure at the stenosis throat, owing to a Bernoulli-type pressure drop, can causelocal collapse in severe stenoses.

  • FLUID MECHANICS AND THROMBOSIS 309

    FIGURE 5 Steady flowthrough a moderate steno-sis (50% diameter reduc-tion, 1.2 cm long, 4-mmdiameter, Re 4 160) (98).a. Stenosis configuration.b. Streamlines show sepa-ration distal to the throat.c. Wall shear rate. Shearrate increases sharply inthe converging stenosis,reaching a peak justupstream of the throat.Shear rate is negative andlow in the post-stenoticrecirculation region.

    Two important clinical consequences arise from the collapse of stenoses. Oneis that the flow rate can be limited by choking, beyond that of purely turbulentlosses. This flow limitation or critical flow rate has long been observed by phys-iologists and described as the coronary flow reserve that is limited even withdecreases in distal resistance. Estimates of coronary flow reserve should includethis choking flow limitation as well as other forms of viscous losses (46, 95). Asecond consequence is that of the imposed loading conditions on an atheroscle-rotic plaque. Stenotic flow collapse creates a compressive stress that may bucklethe structure. The oscillations in compressive loading may induce a fracturefatigue in the surface of the atheroma, causing a rupture of the plaque cap.Because plaque cap rupture is the precipitating event in most heart attacks andstrokes, the fluid-solid mechanical interactions present in high-grade stenosis maycontribute to the catastrophic material failure (74).

    Diagnosis of Disease

    There are a wide variety of clinical applications for hemodynamic studies ofstenoses. One area of investigation revolves around the diagnosis of severe ste-nosis. The most accepted clinical predictors of impending heart attack, stroke,and lower-limb ischemia are based on the presence of hemodynamically signifi-cant stenoses. Currently, the best indicator for surgical treatment of arterioscle-rosis is the degree of stenosis. Although X-ray angiography is currently thestandard, cost and morbidity are distinct disadvantages.

    Doppler ultrasound can be used to measure the increased velocities in thestenotic jet and back out a percent stenosis. This technique is widely used todetermine levels of stenosis in carotid artery disease, with an accuracy of 90%.Doppler ultrasound can also be used to measure the flow waveform in the leg

  • 310 WOOTTON n KU

    arteries. Normal arteries have a characteristic triphasic pattern, whereas diseasedarteries with a stenosis exhibit a blunted monophasic pattern.

    Recently, magnetic resonance imaging (MRI) has been proposed as a lessexpensive, less morbid alternative to X-ray angiography (115). In contrast toDoppler techniques, which require an acoustic or optical window, MRI uses anelectromagnetic window that does not contact the flow. Thus, much more of thebody can be studied.

    Shear-Dependent Thrombosis

    Stenotic flows become critical to clinical medicine in the acute symptoms ofatherosclerosis. After the plaque cap ruptures, the revealed contents of the ath-eroma stimulate a blood-clotting reaction called thrombosis. For the arterial sys-tem, thrombosis is initiated by the adherence of platelets at the surface with rapidaccumulation of additional platelets. Although a number of confusing in vitroexperiments are described in the literature, studies with nonanticoagulated bloodthrough stenoses indicate that platelets stick at the throat of the stenosis. Theadherence and accumulation of these platelets are shear dependentwith moreaccumulation at higher shear rates. The time scale of adhesion is on the order ofmilliseconds. Likewise, the adhesion strength must be enormous because the shearstresses on the platelet are large and increasing as the throat fills with clot. Thefollowing sections explore some of the relationships between thrombosis andhemodynamics and how these relationships may be used to understand the riskof clinical thrombosis.

    ARTERIAL THROMBOSIS

    Thrombosis is the formation of a blood clot, called a thrombus, inside a livingblood vessel. The mechanisms of thrombosis are identical to the mechanisms ofhemostasis, the clotting system that protects the body from excessive blood loss.A thrombus is composed primarily of two blood cell types, platelets and RBCs.The cells are bound together by molecules in the cell membrane of the platelets,called membrane glycoproteins (GPs), by a variety of plasma proteins, and by anetwork of polymerized plasma protein called fibrin.

    Arterial thrombosis is an extremely significant health problem because it islinked to the onset of acute clinical symptoms in atherosclerosis. Thrombus super-imposed on ruptured atherosclerotic plaque is commonly found in autopsy studiesof heart disease (2427). Thrombosis is also associated with carotid artery plaquerupture in stroke and transient ischemic attack (24, 85). Platelets and fibrin emboliare frequently found in the myocardium (heart muscle) of victims of heart disease(28, 37). Clinical studies confirm the link between thrombosis and atherosclero-sisantithrombotic drugs significantly reduce the risk of clinical ischemia (40,81, 108).

  • FLUID MECHANICS AND THROMBOSIS 311

    FIGURE 6 Thrombosis in late-stage ath-erosclerosis. a. Plaque rupture exposes sub-endothelium to the blood, causing plateletadhesion. b. Platelet aggregation forms aplatelet plug. c. Coagulation and plateletaggregation may cause occlusion.

    Cellular and Molecular Mechanisms of Thrombosis

    Thrombosis is a complicated interaction of platelets and plasma proteins. At afunctional level acute thrombosis is described by three platelet functions (adhe-sion, activation, and aggregation) and the coagulation cascade (Figure 6). Thesemechanisms can occur simultaneously and have multiple interactions, with theenzyme thrombin playing a central role.

    Adhesion Thrombosis is triggered when a thrombogenic surface is exposed toblood (Figure 6a). Thrombogenic surfaces include most artificial surfaces, thesubendothelial and medial layers of blood vessels, and subendothelial componentsof atherosclerotic lesion such as fibrous plaque cap and atheromatous core (30).

    Platelets adhere to proteins in the surface via platelet membrane GPs (62).Subendothelial tissue and atheroma contain collagen, to which platelets bind viaglycoprotein GPIa/IIa (91), and von Willebrand factor (vWF), to which plateletsbind via two GPs. GPIb mediates a rapid but transient binding to vWF, whereasGPIIb/IIIa mediates more permanent binding (96). GPIIb/IIIa can bind to manyother plasma and vessel wall proteins, including fibrinogen, fibrin, fibronectin,thrombospondin, and vitronectin (62).

    Activation Activation refers to platelet functions triggered by chemical or physi-cal agonists (stimuli). Chemical agonists include ADP, thrombin, thromboxaneA2 (TxA2), fibrillar collagen, platelet-activating factor, and serotonin (23). Shearstress (in the presence of vWF, ADP, and Ca2`) can activate platelets (52). Plate-lets may also be activated by biomaterials via the complement system (39, 59).

    Perhaps the most important activation function is a conformational change inGPIIb/IIIa that allows it to bind to plasma proteins. GPIIb/IIIa activation has beenestimated to occur within 0.1 s (84) and is required for aggregation and permanentadhesion to vWF (96).

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    Activation causes shape change with pseudopod extension, which increasesthe strength of adhesion and may decrease the resistance of platelets to aggre-gation. Activation triggers upregulation and local clustering of GPIIb/IIIa, whichmay also strengthen adhesion. Activated platelets contract, consolidating loosecells and fibrin into compacted thrombus, and release granular contents (23).

    Several activation functions are positive feedback mechanisms for activationof other platelets (23). Activated platelets synthesize platelet agonist TxA2 andrelease ADP from dense granules; inhibition of either the ADP (87) or TxA2 (64)activation pathway significantly reduces thrombus growth. Activated plateletsalso catalyze thrombin production (23).

    Aggregation Aggregation is essential to formation of a platelet plug (Figure6b). Aggregation can proceed via several mechanisms. At low to moderate shearrates, activated platelets can bind to other activated platelets via fibrinogen orfibrin and GPIIb/IIIa. At higher shear rates, platelets aggregate primarily via vWF(3, 58, 73). It is not clear whether activation occurs before or after initial vWFbinding. Incoming platelets may be activated by passing through an agonistcloud of thrombin, TxA2, or ADP before interacting with an adherent throm-bus (57, 84). Alternatively, vWF on the surface of aggregated and activated plate-lets could support binding of unactivated platelets via GPIb, followed byactivation and permanent binding via GPIIb/IIIa.

    Coagulation Exposure of a thrombogenic surface is also likely to trigger thecoagulation cascade, leading to fibrin coagulation (Figure 6c). In normal hemo-stasis, injury exposes tissue factor, which rapidly leads to thrombin generation.Tissue factor is found at high concentrations in the necrotic core of the atheroma(110) and may be exposed by plaque rupture. Coagulation may also be triggeredby exposure of collagen or an artificial surface (23).

    The ultimate reaction in the coagulation cascade converts prothrombin tothrombin. Thrombin cleaves fibrinogen so that it can polymerize to form fibrin,which traps red cells in the clot and supports platelet adhesion. Thrombin is alsoone of the most potent platelet agonists, causing activation, release of granularcontents, and irreversible aggregation (23). This interaction between platelets andthrombin is important in thrombosis lasting longer than 10 or 15 min (48, 49,61).

    HEMODYNAMICS AND THROMBOSIS

    Thrombosis is fundamentally linked to hemodynamics because blood transportscells and proteins to the thrombus and applies stresses that may disrupt the throm-bus. In this section, we review how blood flow conditions affect the rate andlocalization of platelet accumulation, platelet activation, and fibrin coagulation.

  • FLUID MECHANICS AND THROMBOSIS 313

    TABLE 1 Mean blood flow parameters for human arteries commonly subject to occlusivethrombosis in atherosclerosis

    Vessel (reference)Diameter

    (mm)Average flow

    rate (ml/s)

    MeanReynoldsnumber

    Mean wallshear

    ratea (s!1)

    Mean wallshear stressa

    (dyne/cm2)Femoral artery (44) 5.0 3.7 280 300 11Common carotid (65) 5.9 5.1 330 250 8.9Internal carotid (65) 6.1 5.0 220 220 8Left main coronary (51) 4.0 2.9 240 460 16Right coronary (50) 3.4 1.7 150 440 15aMean wall shear rate and shear stress are estimated from the Poiseuille profile.

    Hemodynamics in Advanced Atherosclerosis

    Most thrombosis experiments with controlled flow report the wall shear rate,c 4 s/l. In major arteries subject to occlusive clinical thrombosis (Table 1), meanshear rate normally ranges from 200 to 500 s11 and mean flow Reynolds numbersrange from 100 to 400. Where atherosclerosis creates a stenosis, shear rateincreases to a peak just upstream of the stenosis throat (Figure 5c). The wall shearrate may be estimated by using scaling based on the Reynolds number and geom-etry (99). Peak shear rate increases with Reynolds number and stenosis severity,to ;10,000 s11 for moderate stenoses and ;100,000 s11 for severe stenoses.Distal to the stenosis, a recirculation region may develop, with unusually highresidence time and low shear rate.

    Hemodynamics and Thrombus Composition

    The composition of a thrombus depends on local flow conditions. In static andlow-shear recirculating flow, the bulk of a thrombus consists of RBCs trapped infibrin. But in unidirectional flow at shear rates of 100 s11 and higher, the bulkof an acute thrombus consists of platelets (18, 92, 101). At arterial and stenoticshear rates, mechanisms of platelet adhesion, activation, and aggregation domi-nate, and the thrombus size can be estimated by counting the number of plateletsthat accumulate.

    Shear and Platelet Accumulation

    Platelet Accumulation Rate Increases with Shear Rate An increase in plateletaccumulation is directly related to the shear rate. This has been observed in vitrofor platelet deposition on subendothelium (106) and collagen-coated surfaces (3,8, 93, 100). The effect of shear rate has also been demonstrated in human (11,92), baboon (72), and porcine (8) ex vivo experiments, which are dominated bythe aggregation phase of thrombosis. The rate of platelet accumulation on fibrillar

  • 314 WOOTTON n KU

    FIGURE 7 Average platelet accumulation rate in ex vivo baboon (72), pig (8), andhuman (11, 92) experiments, as a function of peak wall shear rate. Platelet accumulationrate on collagen I is averaged over 15 min, measured on tubes () and stenoses (2) (72),and in U-channels (n) (8). Platelet accumulation rate on collagen III over 5 min (m,n)(11, 92), estimated from thrombus volume by a linear correlation of data published for thesame system (93), platelets/thrombus volume 4 9 2 1010 platelets/ml.

    collagen increases for shear rates from 100 to at least 10,000 s11 (Figure 7). Atlow shear rates, accumulation is roughly proportional to shear rate. At highershear rates, there may be a divergence from this trend. One experiment shows adecrease in deposition rate between 10,000 and 32,000 s11 (11), whereas anotherexperiment shows an increase in deposition rate between 4,300 and 20,000 s11(72).

    Platelets Adhere Preferentially in High-Shear Regions Shear also appears toaffect where platelets are deposited. Platelet accumulation on collagen-containingstenotic surfaces is highest at the stenosis throat, where shear rate is highest (8,72), for peak shear rates ranging from 1,300 to .20,000 s11 (72). On smoothartificial surfaces by contrast, platelet accumulation may be depressed in highshear regions (15, 97).

    Shear-Linked Mechanisms

    The correlations between shear and platelet accumulation may be explained interms of several shear-linked mechanisms: platelet transport, platelet activation,and embolization.

    Transport Platelet transport is important in acute thrombosis because plateletaccumulation on highly thrombogenic surfaces in vivo may be transport limitedor transport modulated for shear rates up to at least 20,000 s11 (111). Transport

  • FLUID MECHANICS AND THROMBOSIS 315

    in blood is still not completely understood, partly because there is no fundamentaltheory to predict dispersion in a concentrated suspension like blood. But exper-imental studies have identified two mechanisms that influence the rate of plateletinteraction with a thrombogenic surface: (a) RBC motion, the dominant mecha-nism, increases small-scale transport by several orders of magnitude (44); (b)nonuniform platelet concentration may increase platelet transport by a factor of110. Both of these mechanisms increase platelet transport as shear rate increases.

    RBC motion RBCs exhibit randomlike transverse motion and rotation in shearflow (43), which displaces plasma and platelets and increases lateral transport.The rate of platelet transport has often been quantified by an effective diffusivity,derived by fitting experimental data to a species transport model of platelet adhe-sion (e.g. 106). Power law correlations in the form De 4 D1 (c/c1)n, where c isthe shear rate and c 4 1 s11) give power n and coefficient D1 that are functionsof hematocrit and the stiffness and size of the RBCs. For platelets or chemicalsolutes in anticoagulated human blood at 40% hematocrit, n ranges from 0.49 to0.89, and D1 ranges from 1019 to 3 2 1018 cm2/s (1, 5, 107, 109). From thesecorrelations, De ranges from 2 2 1018 to 3 2 1017 cm2s11 at shear rate 4 100s11 and from 5 2 1017 to 1 2 1015 cm2s11 at shear rate 4 10,000 s11, 1 to4 orders of magnitude above the thermal diffusivity for platelets in plasma (1.62 1019 cm2s11).

    Estimates of De vary by up to an order of magnitude between experiments.One source of variation may be the variability of platelet adhesion rates withdifferences in anticoagulation and platelet handling; the adhesion rate begins toaffect the deposition rate in vitro for shear rates . 300 s11 (107). This difficultycan be avoided by using a global model of enhanced diffusivity in sheared con-centrated suspensions (117). Assuming that RBC rotation is relatively unimpor-tant, the effective diffusivity De is the sum of the RBC dispersion and the thermaldiffusivity:

    D 4 D ` D (1)e R swhere DR is the RBC dispersion coefficient and Ds is the thermal diffusivity ofthe solute or platelets in the stationary blood. The dispersion coefficient for RBCsis correlated to experiments by

    (2)DR 4 a2ccfp(11fp)m

    with c 4 0.15 5 0.03 and m 4 0.8 5 0.3, where a is the RBC major radius, cis the shear rate, and up is the hematocrit. For platelets, De is essentially propor-tional to c for c . 10 s11. The model is consistent with transport rates for avariety of solutes in whole blood and was later confirmed for macromoleculetransport (63).

    Nonuniform Concentration RBCs are concentrated in the center of a bloodvessel, and appear to force increased platelet concentration toward the vessel wall.

  • 316 WOOTTON n KU

    This effect has been studied most heavily in narrow vessels (e.g. 43, 104) but hasalso been observed in 3-mm-diameter tubes at arterial and higher shear rates (2).Platelet concentration at the wall increases with increasing hematocrit, shear rate,and platelet concentration. For example, in a 3-mm tube with a 40% hematocritand a 0.25 billion/ml average platelet count, near-wall concentration is a factorof 2 to 4 higher than the average platelet count as the shear rate increases from240 s11 to 1200 s11 (2).

    Both RBC motion and enhanced platelet concentration link high shear toincreased platelet deposition. As long as molecular mechanisms of adhesion andaggregation are rapid enough to permit platelet incorporation into a thrombus,increasing shear will drive more platelets into the thrombus, resulting in morerapid thrombus growth.

    Activation The role of shear stress activation in clinical thrombosis is not clear.The threshold shear exposure required for platelet activation in vitro has beenmeasured for shear rate (in whole blood) ranging from ;103 to 107 s11 (52) andfit to a platelet stimulation function, PSF (16), such that PSF 4 s t 0.452, wheres is shear stress in dynes per square centimeter and t is exposure time in seconds.The threshold for shear-induced activation is PSF $ 1000 (16). High shear stressactivates platelets with short exposure, whereas lower shear stress activates plate-lets over longer durations.

    A platelet flowing through a stenosis in vivo is exposed to high shear stress,but the exposure time is at least one order of magnitude lower than the thresholdfor shear-induced platelet activation (16). Shear stress exposure may not bedirectly responsible for platelet activation in most cases of relatively severe ath-erosclerosis, if activation is required for the initial interaction between a circu-lating platelet and growing thrombus. Shear stress exposure time will exceed theactivation threshold only if a platelet adheres to a stenosis.

    Even if shear stress is not the sole activating agonist in vivo, the history ofshear stress exposure may change the threshold of platelet activation by chemicalagonists (42, 45). Compared with flow that has a gradually changing shear rate,stenotic flow with a rapid increase in shear stress may significantly increase plate-let activation and platelet deposition (54, 111, 112).

    Embolization Another feature of thrombosis is embolization, the removal ofparts of the thrombus owing to fluid mechanical stress. A theoretical model hasbeen developed for embolization in steady and pulsatile flows (14), based onmodels of drag on a protrusion into steady (12) and pulsatile (13) flow. The stresson the thrombus depends on the particle Reynolds number, Rep 4 cHp2/t, whereHp is the thrombus height and t is the kinematic viscosity. For small thrombi(Rep , 1), stress on an isolated thrombus is four- to fivefold the wall shear stressof the approaching flow. For larger thrombi, stress becomes a function of throm-bus height, and stress increases rapidly as the thrombus grows.

  • FLUID MECHANICS AND THROMBOSIS 317

    The missing part of the model is quantitative data on the stress required forplatelet removal from a surface. Mechanical properties of platelets are the subjectof ongoing study (47), so the critical stress for embolization may soon be withinreach, using a combination of modeling and experiments.

    Differences in platelet embolization stress may explain the difference betweenplatelet accumulation patterns on collagen (72) or damaged artery (7) and accu-mulation on Lexan (97). Platelets probably adhere more strongly to collagen inthe natural surfaces than to the smooth Lexan surface and can support largerthrombi without embolization. Ultrasound measurements of embolization fromknitted Dacron or collagen surfaces in ex vivo experiments show low emboliza-tion rates (111).

    Recirculation and Residence Time The effect of hemodynamics on thrombosisis well documented in uniform or unidirectional shear flow. But separated flowoccurs at bifurcations, and downstream of stenoses that occur in atherosclerosis.In regions of complex flow, the relationships between flow and thrombosis arenot very clear.

    Convection patterns and high residence times may modulate thrombosis inseparated flows. In vitro experiments show increased platelet accumulation nearthe reattachment point in Lexan stenoses, presumably caused by increased con-vection (15). Platelets may recirculate in the separated region long enough tobecome activated and form small aggregates. Residence time and convectionpatterns have also been related to fibrin polymerization in shear flow (36, 82).

    Based on steady-flow experiments, residence time on the order of at least 10s is required for significant shear-induced aggregation (56) or fibrin polymeraccumulation (82). In physiologic pulsatile flow, 10-s residence is quite long; forexample, .99% of particles are washed out of the recirculation zone of a 75%or 95% area reduction stenosis within 10 s (19). One potential location for phys-iologic high residence time would be along the trailing edge of a sharp geometricflow separator, which could be created by a tear or flap following plaque rupture,or by a poorly designed prosthetic valve. A sudden expansion, which has a geo-metric flow separator, creates an environment favoring a fibrin-rich red thrombus(18). High residence time could also occur distal to a flow-limiting acute plateletplug, in which case occlusion becomes the primary cause of high residence timeand fibrin coagulation.

    MODELING CLINICAL THROMBOSIS

    Despite the well-documented role of thrombosis in clinical ischemia, thrombosisrisk is not used as a surgical indicator in atherosclerosis. Stenosis severity is usedto identify patients who are good candidates for surgical treatment because itcorrelates with risk of ischemia (6, 20, 78). The relationships between shear rate,stenotic flow, and thrombus growth rate are probably reflected in the clinical

  • 318 WOOTTON n KU

    statistics. However, using only stenosis severity misses patients with moderate ormild stenoses (,50% diameter reduction), who have a significant risk of ischemicattack and death (17, 25, 105).

    Clinically it is important to know the likelihood that thrombosis will lead toocclusion following plaque rupture or ulceration. A model of occlusion risk couldbe combined with a model of plaque rupture risk to decide which patients aregood candidates for surgical treatment and which patients can be managed med-ically. A clinical thrombosis model has not been developed yet, owing to thecomplexity of thrombosis and the wide range of data produced by different throm-bosis experiments. But there is enough experimental data available to begin build-ing a model, based on a theoretical mechanistic thrombosis model. The modelcan estimate the time required for a thrombus to occlude a vessel, based onhemodynamics and geometry, and the occlusion time can be used as a measureof risk of occlusion in the event of plaque rupture.

    Model Based on Ex Vivo Experiments

    Only a few experiments are representative of clinical occlusive thrombosis. Invivo and ex vivo experiments, which use nonanticoagulated blood or mildly hep-arinized blood, include all of the major mechanisms of thrombosis and can occurover time periods long enough for occlusion of a major artery. In contrast, invitro experiments require anticoagulation and often involve extensive blood sam-ple manipulation, and platelet deposition rates are much lower in in vitro than inex vivo experiments (9, 30, 92). For model development, ex vivo experimentshave an advantage over in vivo experiments; precise control of flow rate andthrombogenic surface geometry allows the shear rate to be calculated.

    Relevant thrombosis experiments need to match the hemodynamic environ-ment of stenotic arteries subject to clinical occlusive thrombosis. The controllinghemodynamic variable is the shear rate on the thrombogenic surface, influencedby the shear rate history of platelets flowing over the thrombogenic surface instenotic flow. The Reynolds number does not appear to be significant for forwardflow, apart from its direct effect on the shear rate.

    Several experiments approximate the hemodynamic and hematologic environ-ment expected in atherosclerosis in vivo. Baboon (72) and porcine (7, 8, 30) exvivo experiments span physiologic shear rates and approach a physiologic Rey-nolds number, using little or no anticoagulation; the baboon experiments alsohave well-characterized stenotic flow (72). Human ex vivo experiments (11, 92)match physiologic shear rates without anticoagulation, although the duration ofthe experiments is limited. These experiments can be used to guide and test amodel of occlusive thrombosis. Well-characterized in vivo canine experimentscan also be used to test an occlusion model (101).

    Model Development

    Several experiments provide insight into occlusion when platelets may adhere tothe entire lumen surface, a relatively severe injury. In stenotic geometry, the

  • FLUID MECHANICS AND THROMBOSIS 319

    FIGURE 8 The characteristic time course of platelet accumulation on collagen-coatedtubes, in ex vivo baboon experiments (m) (72). The experiment can be divided into threephases: (I) an accelerating phase, lasting for about 5 min, (II) an acute phase, lasting ;50min, and (III) a slow phase, which extends to the end of the experiment, characterized bya lower accumulation rate than the acute phase. The total platelet accumulation can beapproximated by a 5-min delay, constant accumulation at the acute rate, and no accumu-lation during the slow phase (solid line).

    stenosis throat is the location of most rapid platelet accumulation and of occlusion(72, 101). For 4-mminside-diameter stenoses at a 100-ml/min flow rate, occlu-sion occurs for smaller lumen sizes (,2.7 to 3 mm) and for higher shear rates(.600 s11) (72). Occlusion occurs consistently and rapidly for narrower lumensand higher shear rates (33, 101).

    A theoretical model can help scale experimental data to other flow conditions.Occlusion can be estimated by predicting the size of the thrombus, which isproportional to the number of accumulated platelets, because platelets comprisethe bulk of the thrombus. The time course of platelet accumulation in ex vivoexperiments (72) is dominated by an acute phase, which eventually deceleratesto a slow phase (Figure 8). In some experiments, a platelet plug occludes thelumen, slowing flow and platelet accumulation, but in other experiments, the rateof accumulation is limited by a drop in the aggregation rate. To first order, thefinal size of a thrombus is proportional to the acute rate of platelet accumulationand the duration of the acute phase of platelet deposition.

    The first objective of the model is to estimate the acute rate of platelet accu-mulation, as a function of hematologic and hemodynamic variables. Several goodtheoretical thrombosis models have been developed to understand thrombosisexperiments. Some models treat platelets as discrete particles (31, 55, 84, 90,102); this approach has the potential to be more accurate as molecular models ofadhesion are developed, but can become complicated. Current particle modelsidealize or ignore the particle-fluid interactions and thrombus shape or are explor-

  • 320 WOOTTON n KU

    atory tools. Other models treat platelets as a continuous chemical species (29, 32,34, 106, 112) reacting with a reactive surface. Species transport models are quan-titative and relatively simple but have not been applied to clinical thrombosis andocclusion.

    A modified species transport model has been developed to compute plateletaccumulation rates based on hemodynamics, geometry, platelet count, and aggre-gation rate (111, 112). Unactivated platelets in the blood are treated as a chemicalspecies, which is transported by convection and shear-enhanced diffusion (117).Near-wall platelet concentration is enhanced by a factor of two above averageplatelet concentration, consistent with experiments in similar-sized tubes (2).Platelets at the surface are incorporated into the thrombus by a first-order reactionstep that includes both aggregation and activation. Flow and transport equationscan be solved analytically in tubular flow. For a stenosis, a commercial compu-tational fluid dynamics package is used to compute the flow field and plateletaccumulation rate. This approach predicts the acute platelet accumulation rate oncollagen-coated tubes and in the upstream, converging, and throat sections ofcollagen-coated stenoses (111, 112) of differing stenosis severity (Figure 9c). Theplatelet accumulation rate is highest at the stenosis throat and increases withincreasing percent stenosis (Figure 9b). The model is less successful in recircu-lating post-stenotic flow, but in experiments the maximum platelet deposition rateis located in the throat section (7, 72), where occlusion occurs (101), so the modelis applicable to predicting occlusion in stenotic flow.

    A Model of Occlusion

    A model of acute platelet accumulation rate can be used to estimate thrombussize and occlusion risk if the duration of the acute phase can be predicted. Unfor-tunately, the mechanisms that are responsible for reducing the accumulation rateare not well studied. Embolization has been assumed an important limiting mech-anism, but embolization loss is difficult to measure, and large emboli appear tobe infrequent in ex vivo experiments (111). A model of embolization has beenderived (14), but the embolization stress is unknown. In addition, systemicchanges may reduce the rate of platelet activation, or the concentration of plateletactivation agonists may decrease locally as the thrombus size increases.

    Absent a clear mechanism to limit thrombus growth, the occlusion time canbe estimated from the acute accumulation rate and lumen diameter, assuming thatthe acute phase does not end. This extrapolated occlusion time can be used as arisk indicator; a short occlusion time indicates a higher risk of occlusion whenthere is plaque rupture.

    For a fully reactive surface, occlusion occurs when the thrombus height reachesthe lumen radius. The occlusion time is

    D Clumen thT 4 ` t , (3)occlusion d2fjlumen

  • FLUID MECHANICS AND THROMBOSIS 321

    FIGURE 9 Model of platelet accumulation on 4-mm-inner-diameter collagen-coatedstenoses (111). a. Collagen-coated surface consists of straight segment from x 4 11.2cm to 10.6 cm, a cosine-shaped stenosis from 10.6 cm to 0.6 cm, and a straight segmentfrom 0.6 cm to 1.2 cm. b. Acute platelet accumulation rate (j*) vs axial location (x) in50%, 75%, and 90% area reduction stenoses. j* peaks just upstream of the throat. Peakaccumulation rate increases with increasing stenosis severity. c. Average platelet accu-mulation in the stenosis throat section (x 4 10.48 cm to x 4 0.48 cm), for 50% (m),75% (n), and 90% (l) area reduction (72), compared with model (lines).

  • 322 WOOTTON n KU

    where Dlumen is the diameter of the vessel lumen (throat diameter in a stenosis),Cth is the concentration of platelets in arterial thrombus [estimated to be 75 billion/ml from ex vivo experiments (111)], and f is the ratio of thrombus height tothrombus cross-section area, which accounts for the roughness of the thrombus(estimated to be ;2 based on experimental occlusion times in stenoses). A 5-mindelay (td) is used to model the effect of the accelerating phase of thrombosis. Theacute rate of platelet accumulation jlumen is computed by using the species trans-port model. Equation 3 estimates occlusion times of 16, 28, and 66 min for 90%,75%, and 50% area reduction stenoses, respectively. In experiments, occlusiontimes were 1825 min for the 90% stenosis and 2535 min for the 75% stenoses(72), relatively close to the model. The 50% stenosis does not occlude, so anocclusion time somewhere between 30 and 60 min indicates a low risk of occlu-sive thrombosis.

    The model predicts increased risk of occlusion (decreasing occlusion time)with increasing shear rate, decreasing lumen diameter, and increasing plateletcount. Because shear rate increases and lumen diameter decreases with increasingpercent stenosis, the correlation is consistent with clinical studies linking risk ofischemia and benefit of surgery with percent stenosis. Platelet count is a hema-tologic parameter that should also have a strong influence on risk of occlusion,based on this model.

    Future Directions

    The knowledge that shear affects platelets is already being applied to the designof cardiovascular devices, to minimize shear stress and residence time in bloodpumps, cardiopulmonary bypass devices, and prosthetic valves.

    Clinical application of an occlusive thrombosis model depends on a betterunderstanding of mechanisms that limit thrombus growth after the acute aggre-gation phase that is typically observed. Embolization and systemic negative feed-back may contribute to subocclusive thrombus under some flow conditions. Asecond requirement for an occlusive thrombosis model is a risk model for plaquerupture. Combined understanding of plaque rupture and thrombosis, along withmeasurements of degree of stenosis, could increase the accuracy of screeningpatients for surgical treatment of atherosclerosis.

    CONCLUSIONS

    The study of hemodynamics is a rich field that allows one to characterize thebiological responses to mechanical forces. Specific arteries exhibit flow charac-teristics that are three-dimensional and developing. Diseased arteries can createhigh levels of turbulence, head loss, and a choked flow condition in tubes thatcan collapse. The pulsatile nature of the flow creates a dynamic environment withmany interesting fundamental fluid mechanics questions. The fundamental knowl-edge can be used to predict and change blood flow to alter the course of disease.

  • FLUID MECHANICS AND THROMBOSIS 323

    Shear stress and shear rate have emerged as important parameters that modulateboth chronic and acute biological responses.

    The relationships between thrombosis and fluid mechanics are complicated. Aspecies transport model can be used to estimate clinical thrombosis risk based onthe hemodynamic environment. Future studies will be driven by the need tounderstand the complex effect of hemodynamics on cells and the design of newdevices to modulate this effect.

    Visit the Annual Reviews home page at http://www.AnnualReviews.org.

    LITERATURE CITED

    1. Aarts PAMM, Steendijk P, Sixma JJ,Heethaar RM. 1986. Fluid shear as apossible mechanism for platelet diffusiv-ity in flowing blood. J. Biomech.19(10):799805

    2. Aarts PAMM, van den Broek SA, PrinsGW, Kuiken GDC, Sixma JJ, HeethaarRM. 1988. Blood platelets are concen-trated near the wall and red blood cells,in the center in flowing blood. Arterio-sclerosis 8(6):81924

    3. Alevriadou BR, Moake JL, Turner NA,Ruggeri ZM, Folie BJ, et al. 1993. Real-time analysis of shear-dependent throm-bus formation and its blockade byinhibitors of von Willebrand factor bind-ing to platelets. Blood 81(5):126376

    4. Anayiotos AS, Jones SA, Giddens DP,Glagov S, Zarins CK. 1994. Shear stressat a compliant model of the humancarotid bifurcation. J. Biomech. Eng.116:98106

    5. Antonini G, Guiffant G, Quemada D,Dosne AM. 1978. Estimation of plateletdiffusivity in flowing blood. Biorheology15:11117

    6. Asymptomatic Carotid AtherosclerosisStudy (ACAS). 1995. Endarterectomyfor asymptomatic carotid artery stenosis.JAMA 273(18):142128

    7. Badimon L, Badimon JJ. 1989. Mecha-nisms of arterial thrombosis in nonpar-allel streamlines: Platelet thrombi growon the apex of stenotic severely injured

    vessel wall. Experimental study in thepig model. J. Clin. Invest. 84(4):113444

    8. Badimon L, Badimon JJ, Galvez A,Chesebro JH, Fuster V. 1986. Influenceof arterial damage and wall shear rate onplatelet deposition. Ex vivo study in aswine model. Arteriosclerosis 6:31220

    9. Badimon L, Badimon JJ, Lassila R,Heras M, Chesebro JH, et al. 1991.Thrombin regulation of platelet interac-tion with damaged vessel wall and iso-lated collagen type I at arterial flowconditions in a porcine model: effects ofhirudins, heparin, and calcium chelation.Blood 78(2):42334

    10. Bargeron CB, Deters OJ, Mark FF,Friedman MH. 1988. Effect of flowpartition on wall shear in a cast of ahuman coronary artery. Cardiovasc. Res.22(5):34044

    11. Barstad RM, Kierulf P, Sakariassen KS.1996. Collagen induced thrombus for-mation at the apex of eccentric sten-osesa time course study withnon-anticoagulated human blood.Thromb. Haemost. 75(4):68592

    12. Basmadjian D. 1984. The hemodynamicand embolizing forces acting on thrombi,from incipient attachment of single cellsto maturity and embolization. J. Bio-mech. 17(4):28798

    13. Basmadjian D. 1986. The hemodynamicand embolizing forces acting on

  • 324 WOOTTON n KU

    thrombiII. The effect of pulsatileblood flow. J. Biomech. 19(10):83745

    14. Basmadjian D. 1989. Embolization: criti-cal thrombus height, shear rates, and pul-satility. Patency of blood vessels. J.Biomed. Mater. Res. 23(11):131526

    15. Bluestein D, Niu L, Schoephoerster RT,Dewanjee MK. 1997. Fluid mechanics ofarterial stenosis: relationship to thedevelopment of mural thrombus. Ann.Biomed. Eng. 25:34456

    16. Boreda R, Fatemi RS, Rittgers SE. 1995.Potential for platelet stimulation in criti-cally stenosed carotid and coronary arter-ies. J. Vasc. Invest. 1(1):2637

    17. Brown BG, Gallery CA, Badger RS,Kennedy JW, Mathey D, et al. 1986.Incomplete lysis of thrombus in the mod-erate underlying atherosclerotic lesionduring intracoronary infusion of strepto-kinase for acute myocardial infarction:quantitative angiographic observations.Circulation 73(4):65361

    18. Cadroy Y, Horbett TA, Hanson SR.1989. Discrimination between platelet-mediated and coagulation-mediatedmechanisms in a model of complexthrombus formation in vivo. J. Lab. Clin.Med. 113(4):43648

    19. Cao J, Rittgers SE. 1998. Particle motionwithin in vitro models of stenosed inter-nal carotid and left anterior descendingcoronary arteries. Ann. Biomed. Eng.26(2):19099

    20. Chaitman BR, Fisher LD, Bourassa MG,Davis K, Rogers WJ, et al. 1981. Effectof coronary bypass surgery on survivalpatterns in subsets of patients with leftmain coronary artery disease. Reportof Collaborative Study in CoronaryArtery Surgery (CASS). Am. J. Cardiol.48(4):76577

    21. Chien S. 1970. Shear dependence ofeffective cell volume as a determinant ofblood viscosity. Science 168:977

    22. Clark JM, Glagov S. 1985. Transmuralorganization of the arterial wall: the

    lamellar unit revisited. Arteriosclerosis5:1924

    23. Colman RW. 1993. Mechanisms ofthrombus formation and dissolution.Cardiovasc. Pathol. 2(3):S2332

    24. Constantinides P. 1990. Cause of throm-bosis in human atherosclerotic arteries.Am. J. Cardiol. 66(16):G37G40

    25. Davies MJ. 1990. A macro and microview of coronary vascular insult in ische-mic heart disease. Circulation 82(Suppl.3):II3846

    26. Davies MJ, Thomas A. 1984. Thrombo-sis and acute coronary-artery lesions insudden cardiac ischemic death. N. Engl.J. Med. 310(18):113740

    27. Davies MJ, Thomas AC. 1985. Plaquefissuringthe cause of acute myocar-dial infarction, sudden ischaemic death,and crescendo angina. Br. Heart J.53(4):36373

    28. Davies MJ, Thomas AC, Knapman PA,Hangartner JR. 1986. Intramyocardialplatelet aggregation in patients withunstable angina suffering sudden ische-mic cardiac death. Circulation 73(3):41827

    29. Eckstein EC, Belgacem F. 1991. Modelof platelet transport in flowing bloodwith drift and diffusion terms. Biophys.J. 60(1):5369

    30. Fernandez-Ortiz A, Badimon JJ, Falk E,Fuster V, Meyer B, et al. 1994. Charac-terization of the relative thrombogenic-ity of atherosclerotic plaque compo-nents: implications for consequences ofplaque rupture. J. Am. Coll. Cardiol.23(7):156269

    31. Fiechter J, Ku DN. 1998. Numericalstudy of platelet transport in flowingblood. Proc. ASME Biomed. Eng. Div.,NYC. Adv. Bioeng. 39:1112, Anaheim,Ca

    32. Fogelson AL. 1992. Continuum modelsof platelet aggregation: formulation andmechanical properties. SIAM J. Appl.Math. 52(4):1089110

    33. Folts JD, Crowell EB Jr, Rowe GG.

  • FLUID MECHANICS AND THROMBOSIS 325

    1976. Platelet aggregation in partiallyobstructed vessels and its eliminationwith aspirin. Circulation 54(3):36570

    34. Friedman LI, Liem H, Grabowski EF,Leonard EF, McCord CW. 1970. Incon-sequentiality of surface properties for ini-tial platelet adhesion. Trans. Am. Soc.Artif. Intern. Organs 16:6373

    35. Friedman MH, Bargeron CB, Hutchin-son GM, Mark FF, Deters OJ. 1980.Hemodynamic measurements in humanarterial casts and their correlation withhistology and luminal area. J. Biomech.Eng. 102:247

    36. Friedrich P, Reininger AJ. 1995. Occlu-sive thrombus formation on indwellingcatheters: in vitro investigation and com-putational analysis. Thromb. Haemost.73(1):6672

    37. Frink RJ, Rooney PA Jr, Trowbridge JO,Rose JP. 1988. Coronary thrombosis andplatelet/fibrin microemboli in death asso-ciated with acute myocardial infarction.Br. Heart J. 59(2):196200

    38. Fung YC. 1984. Biodynamics: Circula-tion. New York: Springer-Verlag. 404pp.

    39. Gemmell CH, Black JP, Yeo EL, SeftonMV. 1996. Material-induced up-regula-tion of leukocyte CD11b during wholeblood contact: material differences and arole for complement. J. Biomed. Mater.Res. 32(1):2935

    40. Gent M, Blakely JA, Easton JD, EllisDJ, Hachinski VC, et al. 1989. TheCanadian American Ticlopidine Study(CATS) in thromboembolic stroke. Lan-cet 1(8649):121520

    41. Glagov S, Zarins C, Giddens DP, KuDN. 1988. Hemodynamics and athero-sclerosis: insights and perspectivesgained from studies of human arteries.Arch. Pathol. Lab. Med. 112:101831

    42. Goldsmith HL, Frojmovic MM, BraovacS, McIntosh F, Wong T. 1994. Adeno-sine diphosphate-induced aggregation ofhuman platelets in flow through tubes.III. Shear and extrinsic fibrinogen-

    dependent effects. Thromb. Haemost.71(1):7890

    43. Goldsmith HL, Marlow JC. 1979. Flowbehavior of erythrocytes. II. Particlemotions in concentrated suspensions ofghost cells. J. Colloid Interface Sci.71(2):383407

    44. Goldsmith HL, Turitto VT. 1986. Rheo-logical aspects of thrombosis and hae-mostasis: basic principles andapplications. ICTH-ReportSubcom-mittee on Rheology of the InternationalCommittee on Thrombosis and Haemos-tasis. Thromb. Haemost. 55(3):41535

    45. Goldsmith HL, Yu SS, Marlow J. 1975.Fluid mechanical stress and the platelet.Thromb. Diath. Haemorrh. 34(1):3241

    46. Gould KL. 1978. Pressure-flow charac-teristics of coronary stenoses in unse-dated dogs at rest and during coronaryvasodilation, Circ. Res. 43(2):24253

    47. Haga JH, Beaudoin AJ, White JG, StronyJ. 1998. Quantification of the passivemechanical properties of the restingplatelet. Ann. Biomed. Eng. 26(2):26877

    48. Hanson SR, Harker LA. 1988. Interrup-tion of acute platelet-dependent throm-bosis by the synthetic antithrombinD-phenylalanyl-L-prolyl-L-arginyl chlo-romethyl ketone. Proc. Natl. Acad. Sci.USA 85(9):318488

    49. Hanson SR, Markou C, Lindahl AK, KuDN, Scarborough RM, et al. 1993. Inhi-bition of in vivo thrombus formation inan arterial stenosis model. Thromb. Hae-most. 69:540 (Abstr.)

    50. He X. 1993. Numerical simulations ofblood flow in human coronary arteries.PhD thesis. Georgia Inst. Technol.,Atlanta. 249 pp.

    51. He X, Ku DN. 1996. Pulsatile flow in thehuman left coronary artery bifurcation:average conditions. J. Biomech. Eng.118:7482

    52. Hellums JD. 1994. 1993 Whitaker lec-ture: biorheology in thrombosis research.Ann. Biomed. Eng. 22:44555

  • 326 WOOTTON n KU

    53. Helmlinger G, Geiger RV, Schreck S,Nerem RM. 1991. Effects of pulsatileflow on cultured vascular endothelial cellmorphology. J. Biomech. Eng. 113:12331

    54. Holme PA, Orvim U, Hamers MJ,Solum, NO, Brosstad FR, et al. 1997.Shear-induced platelet activation andplatelet microparticle formation at bloodflow conditions as in arteries with asevere stenosis. Arterioscler. Thromb.Vasc. Biol. 17(4):64653

    55. Huang PY, Hellums JD. 1993. Aggre-gation and disaggregation kinetics ofhuman blood platelets: Part I. Develop-ment and validation of a population bal-ance model. Biophys. J. 65(1):33443

    56. Huang PY, Hellums JD. 1993. Aggre-gation and disaggregation kinetics ofhuman blood platelets: Part II. Shear-induced platelet aggregation. Biophys. J.65(1):34453

    57. Hubbell JA, McIntire LV. 1986. Plateletactive concentration profiles near grow-ing thrombia mathematical considera-tion. Biophys. J. 50:93745

    58. Ikeda T, Handa M, Kawano K, KamataT, Murata M, et al. 1991. The role of vonWillebrand factor and fibrinogen inplatelet aggregation under varying shearstress. J. Clin. Invest. 87:123440

    59. Kalman PG, McCullough DA, Ward CA.1990. Evacuation of microscopic air bub-bles from Dacron reduces complementactivation and platelet aggregation. J.Vasc. Surg. 11(4):59198

    60. Kamiya A, Togawa T. 1980. Adaptiveregulation of wall shear stress to flowchange in the canine carotid artery. Am.J. Physiol. 239:H1421

    61. Kelly AB, Marzec UM, Krupski W, BassA, Cadroy Y, et al. 1991. Hirudin inter-ruption of heparin-resistant arterialthrombus formation in baboons. Blood77(5):100612

    62. Kieffer N, Phillips DR. 1990. Plateletmembrane glycoproteins: functions in

    cellular interactions. Annu. Rev. CellBiol. 6:32957

    63. Kim D, Beissinger RL. 1993. Aug-mented mass transport of macromole-cules in sheared suspensions to surfaces.J. Colloid Interface Sci. 159:920

    64. Kotze HF, Lamprecht S, van BadenhorstPN, vanWyk V, Roodt JP, Alexander K.1993. In vivo inhibition of acute platelet-dependent thrombosis in a baboon modelby Bay U3405, a thromboxane A2-receptor antagonist. Thromb. Haemost.70(4):67275

    65. Ku DN, Giddens DP. 1987. Laser Dopp-ler anemometer measurements of pulsa-tile flow in a model carotid bifurcation.J. Biomech. Eng. 20:407

    66. Ku DN, Phillips DJ, Giddens DP, Strand-ness DE. 1985. Hemodynamics of thenormal human carotid bifurcation: invitro and in vivo studies. UltrasoundMed. Biol. 11:1326

    67. Ku DN, Zarins CK, Giddens DP, GlagovS. 1985. Pulsatile flow and atheroscle-rosis in the human carotid bifurcation:positive correlation between plaquelocalization and low and oscillating shearstress. Arteriosclerosis 5:292302

    68. Langille BL, ODonnell F. 1986. Reduc-tion in arterial diameter produced bychronic decreases in blood flow are endo-thelium-dependent. Science 231:4057

    69. Lieber BB, Giddens DP. 1990. Post-ste-notic core flow and its effects on wallshear stress. J Biomech. 23(6):597605

    70. Lutostansky EM. 1996. The role of con-vective mass transfer in atherosclerosis.PhD thesis. Georgia Inst. Technol.Atlanta. 195 pp.

    71. Ma P, Li X, Ku DN. 1994. Heat and masstransfer in a separated flow region forhigh Prandtl and Schmidt numbers underpulsatile conditions. Int. J. Heat MassTransf. 37(17):272336

    72. Markou CP, Hanson SR, Siegel JM, KuDN. 1993. The role of high wall shearrate of thrombus formation in stenoses.Proc. ASME Biomed. Eng. Div., NYC.

  • FLUID MECHANICS AND THROMBOSIS 327

    Adv. Bioeng. 26:55558, New Orleans,La

    73. Markou CP, Lindahl AK, Siegel JM, KuDN, Hanson SR. 1993. Effect of block-ing the platelet GPIb interaction with vonWillebrand factor under a range of shear-ing forces. Ann. Biomed. Eng. 21:220(Abstr.)

    74. McKinsey J, McCord BN, Aoki T, KuDN. 1991. Can mechanical stress causefatigue of the atherosclerotic plaque?Surg. Forum 42:319

    75. Moore JE Jr, Burki E, Suciu A, ZhaoSM, Burnier M, et al. 1994. Device forsubjecting vascular endothelial cells toboth fluid shear stress and circumferen-tial cyclic stretch. Ann. Biomed. Eng.22:41622

    76. Moore JE Jr, Maier SE, Ku DN, BoesigerP. 1994. Hemodynamics in the abdomi-nal aorta: a comparison of in vitro andin vivo measurements. J. Appl. Physiol.76:152027

    77. Moore JE Jr, Xu C, Glagov S, Zarins CK,Ku DN. 1994. Fluid wall shear stressmeasurements in a model of the humanabdominal aorta: oscillatory behaviorand relationship to atherosclerosis. Ath-erosclerosis 110(2):22540

    78. North American Symptomatic CarotidEndarterectomy Trial Collaborators(NASCET). 1991. Beneficial effect ofcarotid endarterectomy in symptomaticpatients with high-grade carotid stenosis.N. Engl. J. Med. 325:44553

    79. Perktold K Rappitch G. 1995. Computersimulation of local blood flow and vesselmechanics in a compliant carotid arterybifurcation model. J. Biomech. 28:84556

    80. Peskin CS McQueen DM. 1989. A three-dimensional computational method forblood flow in the heart. I. Immersed elas-tic fibers in a viscous incompressiblefluid. J. Comput. Phys. 81:372405

    81. Physicians Health Study ResearchGroup (PHSRG). 1989. Final report onthe aspirin component of the ongoing

    Physicians Health Study. Steering Com-mittee of the Physicians Health StudyResearch Group. N. Engl. J. Med.321(3):12935

    82. Reininger AJ, Reininger CB, HeinzmannU, Wurzinger LJ. 1995. Residence timein niches of stagnant flow determinesfibrin clot formation in an arterialbranching modeldetailed flow analysisand experimental results. Thromb. Hae-most. 74(3):91622

    83. Reuderink PJ. 1991. Analysis of the flowin a 3D distensible model of the carotidartery bifurcation. PhD thesis. Tech-nische Univ. Eindhoven, Netherlands

    84. Richardson PD. 1973. Effect of bloodflow velocity on growth rate of plateletthrombi. Nature 245(5420):1034

    85. Ricotta JJ, Schenk EA, Ekholm SE,DeWeese JA. 1986. Angiographic andpathologic correlates in carotid arterydisease. Surgery 99(3):28492

    86. Rindt CC, van Steenhoven AA, JanssenJD, Reneman RS, Segal A. 1990.A numerical analysis of steady flowin a three-dimensional model of thecarotid artery bifurcation. J. Biomech.23(5):46173

    87. Roald HE, Barstad RM, Kierulf P, Skjor-ten F, Dickinson JP, et al. 1994. Clopi-dogrela platelet inhibitor whichinhibits thrombogenesis in non-anticoa-gulated human blood independently ofthe blood flow conditions. Thromb. Hae-most. 71(5):65562

    88. Rodbard S. 1970. Negative feedbackmechanisms in the architecture and func-tion of the connective and cardiovasculartissues. Perspect. Biol. Med. 13:50727

    89. Rodkiewicz CM, Sinha P, Kennedy JS.1990. On the application of a constitutiveequation for whole blood. J. Biomech.Eng. 112:198206

    90. Ruckenstein E, Marmur A, Gill WN.1977. Growth kinetics of plateletthrombi. J. Theor. Biol. 66(1):14768

    91. Saelman EU, Nieuwenhuis HK, HeseKM, de Groot PG, Heijnen HFG, et al.

  • 328 WOOTTON n KU

    1994. Platelet adhesion to collagen typesI through VIII under conditions of stasisand flow is mediated by GPIa/IIa (alpha2 beta 1-integrin). Blood 83(5):124450

    92. Sakariassen KS, Joss R, Muggli R, KuhnH, Tschopp TB, Sage H, BaumgartnerHR. 1990. Collagen type III induced exvivo thrombogenesis in humans: role ofplatelets and leukocytes in deposition offibrin. Arteriosclerosis 10(2):27684

    93. Sakariassen KS, Kuhn H, Muggli R,Baumgartner HR. 1988. Growth and sta-bility of thrombi in flowing citratedblood: assessment of platelet-surfaceinteractions with computer-assisted mor-phometry. Thromb. Haemost. 60(3):39298

    94. Salam TA, Lumsden AB, Suggs WD, KuDN. 1996. Low shear stress promotesintimal hyperplasia thickening. J. Vasc.Invest. 2:1222

    95. Santamore WP, Bove AA, Carey RA.1982. Tachycardia induced reduction incoronary blood flow distal to a stenosis.Int. J. Cardiol. 2:2327

    96. Savage B, Saldivar E, Ruggeri ZM.1996. Initiation of platelet adhesion byarrest onto fibrinogen or translocation onvon Willebrand factor. Cell 84(2):28997

    97. Schoephoerster RT, Oynes F, Nunez G,Kapadvanjwala M, Dewanjee MK. 1993.Effects of local geometry and fluiddynamics on regional platelet depositionon artificial surfaces. Arterioscler.Thromb. 13(12):180613

    98. Siegel JM. 1992. Wall shear stressthrough an arterial stenosis and its impli-cations to thrombosis. MS thesis. Geor-gia Inst. Technol., Atlanta. 88 pp.

    99. Siegel JM, Markou CP, Ku DN, HansonSR. 1994. A scaling law for wall shearstress through an arterial stenosis. J. Bio-mech. Eng. 116:44651

    100. Sixma JJ, de Groot PG. 1994. Regulationof platelet adhesion to the vessel wall.Ann. NY Acad. Sci. 714:19099

    101. Strony J, Beaudoin A, Brands D, Adel-

    man B. 1993. Analysis of shear stressand hemodynamic factors in a model ofcoronary artery stenosis and thrombosis.Am. J. Physiol. 265(5):H178796

    102. Tandon P, Diamond SL. 1997. Hydro-dynamic effects and receptor interactionsof platelets and their aggregates in linearshear flow. Biophys. J. 73(5):281935

    103. Tang TD. 1990. Periodic flow in a bifur-cating tube at moderate Reynolds num-ber. PhD thesis. Georgia Inst. Technol.,Atlanta. 211 pp.

    104. Tilles AW, Eckstein EC. 1987. The near-wall excess of platelet-sized particles inblood flow: its dependence on hematocritand wall shear rate. Microvasc. Res.33(2):21123

    105. Torvik A, Svindland A, Lindboe CF.1989. Pathogenesis of carotid thrombo-sis. Stroke 20(11):147783

    106. Turitto VT, Baumgartner HR. 1975.Platelet deposition on subendotheliumexposed to flowing blood: mathematicalanalysis of physical parameters. Trans.Am. Soc. Artif. Intern. Organs 21:593601

    107. Turitto VT, Weiss HJ, Baumgartner HR.1980. The effect of shear rate on plateletinteraction with subendothelium exposedto citrated human blood. Microvasc. Res.19(3):35265

    108. United Kingdom Transient IschaemicAttack Study Group (UK-TIA). 1988.United Kingdom transient ischaemicattack (UK-TIA) aspirin trial: interimresults. UK-TIA Study Group. Br. Med.J. Clin. Res. Ed. 296(6618):31620

    109. Wang NHL, Keller KH. 1985. Aug-mented transport of extracellular solutesin concentrated erythrocyte suspensionsin Couette flow. J. Colloid Interface Sci.103(1):21025

    110. Wilcox JN, Smith KM, Schwartz SM,Gordon D. 1989. Localization of tissuefactor in the normal vessel wall and inthe atherosclerotic plaque. Proc. Natl.Acad. Sci. USA 86(8):283943

    111. Wootton DM. 1998. Mechanistic mod-

  • FLUID MECHANICS AND THROMBOSIS 329

    eling of occlusive arterial thrombosis.PhD thesis. Georgia Inst. Technol.,Atlantic. 421 pp.

    112. Wootton DM, Markou CP, Hanson SR,Ku DN. 1998. Mechanistic model ofarterial thrombosis in collagen-coatedstenoses. Proc. ASME Biomed. Eng.Div., NYC. Adv. Bioeng. 39:11314,Anaheim, Ca

    113. Yoganathan AP, Lemmon JD Jr, KimYH, Walker PG, Levine RA, Vesier CC.1994. A computational study of a thin-walled three-dimensional left ventricleduring early systole. J. Biomech. Eng.116:30717

    114. Young DF. 1979. Fluid mechanics of

    arterial stenoses. J. Biomech. Eng. 101:15775

    115. Yucel EK. 1994. Magnetic ResonanceAngiography in Vascular Diseases: Sur-gical and Interventional Therapy, ed DEStrandness Jr, pp. 289302. New York:Churchill Livingstone

    116. Zarins CK, Zatina MA, Giddens DP, KuDN, Glagov S. 1987. Shear stress regu-lation of artery lumen diameter in ex-perimental atherogenesis. J. Vasc. Surg.5:41320

    117. Zydney AL, Colton CK. 1988. Aug-mented solute transport in the shear flowof a concentrated suspension. Physico-Chem. Hydrodyn. 10:7796