Simulation of Intrathrombus Fluid and Solute Transport Using In VivoClot Structures with Single Platelet Resolution

ROMAN S. VORONOV,1 TIMOTHY J. STALKER,2 LAWRENCE F. BRASS,2 and SCOTT L. DIAMOND1

1Department of Chemical and Biomolecular Engineering, Institute for Medicine and Engineering, University of Pennsylvania,Philadelphia, PA 19104, USA; and 2Department of Medicine and Pharmacology, University of Pennsylvania, Philadelphia, PA,

USA

(Received 4 December 2012; accepted 11 February 2013; published online 20 February 2013)

Associate Editor Konstantinos Konstantopoulos oversaw the review of this article.

Abstract—The mouse laser injury thrombosis model providesup to 0.22 lm-resolved voxel information about the porearchitecture of the dense inner core and loose outer shellregions of an in vivo arterial thrombus. Computationalstudies were conducted on this 3D structure to quantifytransport within and around the clot: Lattice Boltzmannmethod defined vessel hemodynamics, while passiveLagrangian Scalar Tracking with Brownian motion contri-bution simulated diffusive-convective transport of variousinert solutes (released from lumen or the injured wall). For aninput average lumen blood velocity of 0.478 cm/s (measuredby Doppler velocimetry), a 0.2 mm/s mean flow rate wasobtained within the thrombus structure, most of whichoccurred in the 100-fold more permeable outer shell region(calculated permeability of the inner core was 10211 cm2).Average wall shear stresses were 80–100 dyne/cm2 (peakvalues >200 dyne/cm2) on the outer rough surface of thethrombus. Within the thrombus, small molecule tracers(0.1 kDa) experienced ~70,000 collisions/s and penetrated/exited it in about 1 s, whereas proteins (~50 kDa) had~9000 collisions/s and required about 10 s (tortuosity ~2–2.5). These simulations help define physical processes duringthrombosis and constraints for drug delivery to the throm-bus.

Keywords—Blood, Lattice Boltzmann, Confocal microscopy,

Brownian, Diffusion, Permeability, Stresses, Drug delivery,

Computation, Modeling.

INTRODUCTION

Inadequate thrombus formation can result inbleeding, while excessive platelet aggregation or coag-ulation can lead to pathogenic thrombosis. The

thrombus growth rate, mechanical stability, and risk ofembolism are all emergent properties that depend onreaction processes within the porous structure as itforms under flow conditions. As platelets are recruitedfrom the blood stream to the injury site, a complexinterplay between their activation state, transport ofcoagulation factors and hemodynamic forces isresponsible for determining the thrombus structureand function.

Despite the underlying complexity of the coupledprocesses involved in hemostasis, the resulting throm-botic structures do have some organization. Recentmicroscopy studies have observed the thrombusstructure to be hierarchical, with two distinct regions: adensely packed ‘‘core’’ nearest the injury site (com-posed of highly activated platelets that have undergonea-granule exocytosis and phosphatidylserine exposure)and a loose ‘‘shell’’ overlaying the core composed ofless activated platelets that have retained a discoidshape.3 Moreover, from own microscopy studies, thethrombus shell consists of loosely associated plateletsthat are subject to continual removal by flow forces.34

Within the thrombus core, platelet agonists maybecome enriched since molecular transit may bereduced by the core having restricted pores.7

In this work, we investigated the coupling betweenblood flow, transport of inert solutes, and the innerarchitecture of the porous thrombus. Utilizing thecapability of intravital confocal microscopy to provide3D thrombus structure with a single platelet resolu-tion, we implemented a combination of in vivo exper-iments, biomedical imaging, and computer simulationsin order to quantify the intrathrombic microenviron-ment (Fig. 1). This approach offers several advantagesover the conventional blood systems biology models(for example platelet adhesive dynamics). Specifically,

Address correspondence to Scott L. Diamond, Department of

Chemical and Biomolecular Engineering, Institute for Medicine and

Engineering, University of Pennsylvania, Philadelphia, PA 19104,

USA. Electronic mail: sld@seas.upenn.edu

Annals of Biomedical Engineering, Vol. 41, No. 6, June 2013 (� 2013) pp. 1297–1307

DOI: 10.1007/s10439-013-0764-z

0090-6964/13/0600-1297/0 � 2013 Biomedical Engineering Society

1297

while the typical simulations rely on generating thethrombus structure mathematically, in image-basedmodeling it is obtained from experiment. Therefore,instead of having to couple four different components(fluid mechanics, coagulation cascade, cell mechanics,and receptor-ligand binding),40 the model is reduced tojust two: hydrodynamics and transport of plateletagonists. This is a significant gain, since the bypassedparts of the simulations are the most complex ones andoften rely on assumptions due to lack of experimentalparameters. At the same time, the disadvantages of theimage-based modeling methodology are relativelyinsignificant: the thrombus structure is static and themesh resolution is limited by the quality of themicroscope. Hence, the trade-off involved is consid-ered to be a lucrative opportunity to obtain a top-down insight into the intrathrombic microenvironmentthat would otherwise be inaccessible.

In order to build this model, a thrombus wasinduced via laser injury in a mouse cremaster musclearteriole and imaged using fluorescent confocalmicroscopy. The 3D thrombus reconstruction recoversthe pore space and its connectivity with up to 0.22 lmresolution (voxel-to-voxel distance) and this complexobject was used in silico to model the fluid dynamicenvironment experienced by the thrombus due toblood flow in microvasculature as follows: (1) LatticeBoltzmann method (LBM) was implemented in orderto obtain bulk fluid velocities around the thrombusand surface stresses; (2) permeability of the thrombus

to flow of blood plasma was calculated in order tocharacterize the convective component of innerthrombus transport; and (3) convective/diffusivetransport of inert molecular species, with prescribeddiffusivity, was modeled within the thrombus porespace using passive Lagrangian Scalar Tracking (LST)with a Brownian motion component (Fig. 1). Becausethe Brownian contribution to the motion of moleculesis largely a function of their size, molecular weights ofcalcium ion (Ca2+), adenosine diphosphate (ADP), orFactor X (FX) were chosen as the representative low,medium and high MW tracers, respectively. Twomodes of tracer release and transport were investi-gated: (1) release from the blood stream into thethrombus toward the injury site, and (2) release fromthe injury site across the thrombus and into the blood.Improved analysis of molecular and fluid transportwithin the thrombus pore space facilitates deeperunderstanding of thrombogenesis, therapeutic deliveryof drugs, and in vivo diagnostic imaging.

METHODS

In Vivo Laser Injury Model and Intravital Microscopy

Procedures and protocols in this study wereapproved by the Institutional Animal Care and UseCommittee (IACUC) of the University of Pennsylva-nia. Thrombus formation was visualized in the cre-master muscle microcirculation of mice according to

FIGURE 1. Schematic of the multidisciplinary investigation approach implemented in this manuscript. Red color marks activatedthrombus core; blue marks the thrombus shell; green marks the lumen and interstitial space within the thrombus.

VORONOV et al.1298

procedures originally developed by Falati et al.16

Briefly, in order to obtain the thrombus structure,intravital microscopy experiments were performed inmale C57Bl/6J mice 8–12 weeks of age (Jackson Lab-oratories, Bar Harbor, ME). The structure of thethrombus formed due to the laser injury was imagedusing confocal microscopy. Alexa-fluor monoclonalantibody labeling kits from Invitrogen (Carlsbad, CA)were used to label antibodies. Anti-CD41 F(ab)2fragments (clone MWReg30, BD Biosciences, SanDiego, CA) were used to visualize platelet surfaces,anti-P-selectin (clone RB40.34, BD Bioscience) wasused to visualize degranulated platelets, and Alexa-fluor 488 tagged dextran (3 kD, Invitrogen) was usedto image the lumen and the interstitial space betweenthe thrombus cells. The center line maximum velocitythrough the mouse blood vessels was measured usingoptical Doppler velocimetry, and divided by a factor of1.6 (to account for an artifact of the measuring tech-nique that causes velocity profiles to appear slightlyblunted and non-parabolic due to out-of-focus cellsmodulating the light intensity signals) in order toobtain the average blood vessel velocity of 4.78mm/s.5,13,18 The velocimetry measurement was takenin a region away from the thrombus where the velocityfield was not affected by its presence.

3D Image Acquisition and Reconstruction

SlidebookTM 5 software was used in order toacquire images from the confocal microscope at0.22 lm pixel-to-pixel distance and 0.5 lm inter-sliceseparation (interpolated to 0.22 lm to be consistentwith the pixel size). ‘‘Constrained Iterative’’ 3Ddeconvolution algorithm (developed by David Agardat UCSF) was used to deblur the images.1 A customMatlab� code was then used to enhance the signal-to-noise ratio of the 2D image stack (See SupplementalMethods for a further discussion of image processingand accuracy). Platelets in the thrombus were detectedand segmented from the rest of the image based ontheir characteristic intensities and shapes. The imageswere then assembled into a 3D stack in order toreconstruct a cubic lattice representative of theimpermeable, solid phase geometry (an example shownin Fig. 1). Also, the Qhull algorithm was used to definethe total thrombus volume.2 The obtained Qhull maskwas eroded with a cubic structuring element (size 93

voxels) using Matlab in order to avoid the outerboundary of the thrombus, where exceptionally highshear rates occur. The final thrombus geometry wasused for flow dynamics calculations as described in thenext section, and the Qhull volume was used to averageany intrathrombic results. Since the imaged thrombusstructure is fixed, the underlying conjecture of this

methodology is that the thrombus deformation occursslower than does the transport within it. Thus, theresults presented herein provide a glimpse of the in-trathrombic microenvironment during a snapshot intime. Furthermore, the accuracy of the study is limitedby the image resolution, which limits the capture ofany physics or details of the thrombus structureoccurring at length scales smaller than 0.22 lm in theplane of the confocal images and 0.5 lm in the per-pendicular direction. Though, not many features belowthis resolution are expected to be present in thethrombus, since most of the fibrin in arterial clots istypically located in the extravascular space.34

Lattice Boltzmann Method (LBM) Simulationof Fluid Flow

Convection within the blood vessel and within theporous thrombus was modeled via LBM. LBM is anumerical technique for simulating fluid flow thatconsists of solving the discrete Boltzmann equa-tion.9,35,36 In addition to computational advantages,e.g., LBM is inherently parallelizable,20,39 LBM tech-niques have been used in a wide spectrum of applica-tions (turbulence,12 non-Newtonian flows,6,17,43 andmultiphase flows37). LBM is especially appropriate formodeling pore-scale flow within porous media due tothe simplicity with which it handles complicatedboundaries. A parallelized, in-house LBM code wasused for this work employing a D3Q15 lattice31 inconjunction with the single-relaxation time approxi-mation of the collision term given by Bhatnagar et al.4

The no-slip boundary condition was applied at the wallfaces using the ‘‘bounce-back’’ technique.36 The LBMresults were validated for several flow cases havinganalytical solutions: flow in a slit, pipe, or around anarray of spheres, which is particularly relevant forporous media flow.38

The average lumen blood velocity in LBM wasmatched to the in vivo value obtained by opticalDoppler velocimetry for the 65 lm diameter vessel.Since the thrombus was imaged 5 min post injury, thelaser injury underneath the thrombus was completelysealed (no blood flow was detected out of the arteri-ole). Also, by 5 min post injury, the thrombus struc-ture does not change shape significantly. Maximumthrombus dimensions (corresponding to the dimen-sions of a smallest bounding box) were: 108.4 lmlength (along the direction of the arteriole), 26.0 lmwidth (of projection shown in Fig. 4-right), and34.1 lm height (of projection shown in Fig. 4-right).While the fluid dynamic viscosity of blood is known tobe shear-dependent,26,41 we assume Newtonian bloodviscosity of l � 0.03 g/cm/s.21,23 This is consideredto be a good assumption due to the low hematocrit

Intrathrombus Transport Simulation Based on In Vivo Images 1299

typically observed in microvasculature.14 The simula-tions were considered converged when smallest, aver-age, and highest flow velocities in the whole simulationdomain varied by less than 0.001% per 1000 steps.Approximately 80,000 steps were needed for full con-version. The image-based LBM simulation results werevalidated against an idealized homogeneous porousmedia model of a thrombus using a commercial fluiddynamics package COMSOL�.

Lagrangian Scalar Tracking (LST) of Solute Transport

Macroscopic solute transport was modeled usingLST.28,29 The fundamental hypothesis of this methodis that solute transport behavior of passive, nonbind-ing tracers is a combination of convection (obtainedusing the velocity field from the LBM simulations) anddiffusion (obtained from a mesoscopic Monte-Carloapproach that simulates Brownian motion). The LSTmethod follows tracers that are passive (i.e., they donot affect the flow field, bind the solid phase, orexperience molecularly hindered diffusion). The newposition of a tracer at time t + 1 is calculated from itsprevious position ~X at time t based on convection anddiffusion, as given by:

~Xtþ1 ¼ ~Xt þ Dt � ~Ut þ D~X; ð1Þ

where U!

t is the fluid velocity at the particle location alsoat time t. The velocity is interpolated from the steady-state solution obtained from the LBM simulation viatrilinear interpolation using the eight nearest cubic lat-tice points. In order to stay consistent with the LBMbounce-back boundary condition, the LST particles arereflected to their previous position upon collision with asolid wall. Although the nature of the particle reflectioncould range anywhere between bounce-back (e.g., mostfluid–solid no slip interfaces) to adsorption/adhesion/desorption kinetics (e.g., particles with strong interac-tions) to bounce forward (e.g., gas–fluid full slip inter-face), depending on the solute force interactionswith thethrombus surface, the current study is limited to theformer no slip case for simplicity.

This method allows simulation of various Schmidtnumber solutes (Sc = m/D where m is kinematic vis-cosity and D is unhindered Brownian diffusivity inwater at 37 �C). The rate of molecular dispersion inone direction, say the X direction, is proportional to itsdiffusivity, as described by Einstein’s theory forBrownian motion15:

@ �X2

@t¼ 2D: ð2Þ

The random diffusion of tracers follows a normaldistribution with a zero mean and standard deviation

r, as denoted by Nð0; rÞ. The ‘Mersenne Twister’random number generator with a cycle of length(219937 2 1) is used for all random number generationin the LST code.25 The molecular dispersion in eachone direction has a standard deviation that is given byr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi

2D0Dtp

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2mDt=Scp

, where D0 is the nominalmolecular diffusivity (i.e., diffusivity that the tracerswould have if their motion was purely Brownian). Theproperties of the fluid, i.e., the Schmidt number of thefluid, are thus taken into account. Equation (2) cannow be written as follows:

~Xtþ1 ¼ ~Xt þ ~UðLBMÞt Dtþ D~XðrandomÞt : ð3Þ

In one mode of release, the LST tracers are releasedfrom the site of the injury and tracked as they movethrough the rest of the thrombus structure (where theinjury is defined as the surface of the blood vesselunder the upstream half of the thrombus). In the sec-ond mode of release, the tracers are released at theblood vessel inlet and their motion is tracked to theinjury site. Since tracer numbers greater than 10,000were found to be unnecessary in order to reproduce theresults reported in this manuscript, this was the num-ber used for the LST simulations.

Continuum Modeling of Thrombus Transport DynamicsUsing Finite Element Method

To validate the image-based LBM/LST results, ahomogeneous porous media model of a thrombus in ablood vessel was simulated in COMSOL� version 4.2(a commercial finite element fluid dynamics solver). Inthe model the idealized thrombus structure is repre-sented by two distinct porous media regions: ahomogeneous core with porosity of 0.3 and a homo-geneous shell with a porosity of 0.7 (porositiesobtained from the 3D reconstruction of the confocalmicroscopy image). The shell and core permeabilities,which were needed to characterize the porous media inCOMSOL, were estimated using the Blake–Kozeny–Carman equation:

k ¼ e3

1� eð Þ21

S2K; ð4Þ

where e is the porous medium porosity, S is the specificsurface area, and K is the Kozeny ‘‘constant’’ (exper-imentally observed to be �5 for a wide variety ofporous media with a range of porosities between 0.4and 0.7).33 The specific surface area, similarly toporosity, was calculated from the 3D reconstruction ofthe fluorescent confocal microscopy images. Theobtained value for S was found to be ~24,400 cm2/cm3.The final estimates for the core and shell permeabilitieswere 1.85 9 10211 and 1.28 9 1029 cm2, respectively.

VORONOV et al.1300

The porous thrombus was placed at the bottom of a65 lm diameter cylinder, which represented the bloodvessel. The fluid viscosity in the COMSOL model wassame as in LBM: l = 0.03 g/cm/s. The inlet conditionto the blood vessel was set to match a parabolic profileexpected in pressure driven flow at the average bloodvessel velocity that was measured from velocimetry.The outlet condition was set to be zero pressure. Inorder to reduce the computational intensity of thesimulation, a symmetry condition was applied alongthe plane that cuts the cylinder and thrombus vertically,in half, along the direction of flow. Solute transport inCOMSOL was modeled using an Eulerian approach,rather than Lagrangian, and the symmetry conditionwas imposed by applying a no mass flux requirement inthe normal direction across the boundary.

RESULTS AND DISCUSSION

Intrathrombus Velocity Field

To simulate conditions in a mouse cremaster musclearteriole laser injury model, LBM simulations wereperformed using a geometry reconstructed from con-focal microscopy. From the LBM results, the averageintrathrombus velocity in the stream-wise direction is0.2 mm/s, which is about an order of magnitude lessthan average blood velocity in the arteriole. Clearly,there is a net flow in the stream-wise direction thattakes place within the thrombus, while the convectionin the other directions through the thrombus is negli-gible.

Since the thrombus structure is non-homogeneous,it is interesting to compare convection within differentlocations in its pore space. For this purpose, the localintrathrombus velocity field visualized in 3D is shown

in Fig. 2, where the velocity vectors (left) and stream-lines (right) are plotted through a thrombus length-wise cross-section. Here, most of the flow occurs in theshell, while the compact core experiences significantlyless permeation.

Fluid Stresses Experienced by the Thrombus

Since the thrombus partially obstructs blood flowwithin the vasculature, it experiences forces exerted bythe passing fluid. These forces can drive thrombusbreak-up and embolism. Additionally, platelet adhe-sion and aggregation27 and activation19,32 are influ-enced by local shear rate. Therefore, it is interesting tocalculate the stresses experienced by the plateletsmaking up the blood clot structure. It is important tonote that only stresses due to fluid shear are reported inthis study, while events due to blood cell collisions withthe thrombus are considered minimal due to the lowhematocrit in microvasculature and are, thus, nottaken into account.

The thrombus examined in this study obstructsabout half way into the blood vessel (50% thrombusheight to blood vessel diameter ratio), which is typ-ical of thrombi observed during in vivo laser injury.In order to calculate the surface stresses due to fluidflow experienced by the thrombus within our model,the 3D velocity field obtained from LBM was dif-ferentiated following the scheme suggested by Porteret al.30

r � l1

2

� �

rU!þrU

!T� �

; ð5Þ

where r is the shear stress tensor, and U!

is the localvelocity vector. The maximum fluid stresses in any

FIGURE 2. Intrathrombus velocity field (left) and convective stream lines (right). Red color marks the blood vessel; gray scale isthe porous thrombus structure. In both cases the simulation domain is cut in half along the flow direction in order to show theintrathrombus environment. One half of the thrombus is shown, with the cut plane facing the reader. The other half is omitted.

Intrathrombus Transport Simulation Based on In Vivo Images 1301

direction are calculated at each surface node as thelargest eigenvalues of r. The final result is compared toCOMSOL results in Fig. 3.

From both LBM and COMSOL simulations, thethrombus base (nearest the blood vessel wall) expe-riences stress values that are comparable to, or lowerthan, the upstream wall shear stress of 17.65 dyn/cm2

expected for an empty blood vessel at this flow rate.For comparison, experimentally measured wallstresses for similar size microvasculature of catmesentery were found to be on the order of 40dyn/cm2.24 The top part of the thrombus, on theother hand, protrudes into the center of the bloodvessel and experiences stresses that are several foldhigher than the wall stresses for the same shear rate:the ratio of stress experienced by the top of thethrombus top is roughly twice in the COMSOLmodel and up to four-to-five higher in the LBMmodel than the wall stress. The reason for this dif-ference between the two models is that in COMSOLthe thrombus structure is idealized and smooth,while in the LBM model it is composed of individualplatelets where surface roughness causes heterogene-ity of the surface shear stress. Therefore, theCOMSOL model is representative of the lowerbound of the expected stresses for the given flowconditions, while the LBM model is expected to bemore accurate, since it takes into account the chan-ges in the flow field due platelet-scale roughness ofthe actual thrombus surface. Nonetheless, the stressespredicted by both of the models are likely well belowthose needed for shear-induced platelet activation(although it is not well studied for mouse platelets).Since the stresses are dependent on viscosity, theerror due to the Newtonian fluid assumption wastested against the Carreau blood viscosity model andwas found to be approximately 6.3% for the peakstresses experienced by the thrombus.

Thrombus Darcy Permeability

Permeability is a measure of the ability of a porousmaterial to allow fluid to pass through it. It is thecentral characteristic of the porous media that isimportant to understanding convective transportthrough the thrombus. In order to calculate perme-ability, k, we use Brinkman’s empirical modification ofDarcy’s law8:

@P

@x¼ � l

kuþ lr2u; ð6Þ

where the pressure drop across the thrombus is cal-culated by applying the Navier–Stokes equations tothe converged velocity field from LBM. Figure 4 showsresulting Darcy permeability map averaged in the flowdirection over the length of the thrombus for theCOMSOL model (left) and for the LBM model (right).The resulting Darcy permeabilities ranged over twodegrees of magnitude: from ~1 9 10211 cm2 in thethrombus core to ~1 9 1029 cm2 in the thrombusshell. From Fig. 4, the two models compare well, withthe LBM model results showing more detail of per-meability variations between the different regions inthe thrombus. The fact that there is a two orders ofmagnitude difference between the permeability of thecore and the shell reaffirms the finding that most of theconvection is expected to take place in the shell regionof the thrombus structure.

Intrathrombus Diffusive Transport of Proteins:Injury-to-Lumen Release

In order to measure how various molecular speciesmove within a blood clot, tracers with different diffu-sive properties were released from the injury site andtracked throughout the thrombus geometry. Sincemolecular diffusivity is a function of molecular size, a

FIGURE 3. Maximum viscous fluid surface stresses experiences by the thrombus within microvasculature calculated fromCOMSOL (left) and LBM (right).

VORONOV et al.1302

set of small, medium, and high molecular weights(corresponding to that of Ca2+, ADP and Factor X,respectively) were used as the representative species inthe simulation. Literature diffusivity values were usedfor ADP and FX, while the Ca2+ diffusivity wasestimated using the Stokes–Einstein correlation fordiffusion of spherical particles through liquid underlow Reynolds number conditions (see Table 1):

D ¼ kBT

6plR; ð7Þ

where kB is the Boltzmann constant, T is the absolutetemperature (assumed to be body temperature of310.15 K), l is the viscosity of the medium andR = 1.14 A is the radius of a Ca2+ ion.

Figures 5 and 6 are time plots of the percentremaining tracers in the thrombus and the averagecollision rates in the thrombus, respectively. FromFig. 5, a much longer time is required for the tracersrepresenting the large molecules to exit the thrombusthan it does for the smaller-sized ones: 0.7 s for Ca2+,1.7 s for ADP and 7.3 s for FX. This is because thesmaller more dynamic molecules can make their way tothe periphery of the thrombus faster and become en-trained by the lumen flow there. The collision rates inFig. 6 are consistent with the observation that thesmaller molecules are more dynamic, and, therefore,have higher collision rates than the larger molecules:Ca2+ undergoes almost 10 times more collisions withthe thrombus surface per second than does FX.Interestingly, but expectedly, the average collisionrates decrease with time for all three molecule types,because more of tracers begin to populate the porousshell where collisions with the thrombus surface areless probable.

Another interesting observation of tracer movementwithin the thrombus is that the dominant mode of exit

for the tracers was out the sides of the clot, despite thefact that most of the intrathrombic convection takesplace in the stream-wise direction according to the fluiddynamics solvers. This is because the thrombus ispreferentially elongated in the stream-wise dimension(a typical thrombus structure thickness is approxi-mately a third of its length in the stream wise direction,possibly due to the shape of the injured endothelial cellthat serves as its base), and the tracers can traverse thethrombus width faster than its length.

In order to further explore how convection affectstracer transport within the thrombus, we artificiallyturned off the flow in the simulation and modeledtracer release from the injury site using purely diffusivetransport. Both Figs. 5 and 6 show that in the no-flowcase the tracers remain in the thrombus considerablylonger than in the case where the flow was present.Moreover, the trends in Fig. 6 indicate that the tracershave more opportunity to migrate to the shell of thethrombus, because the collision rate for the no-flowcase becomes significantly lower with time. This con-clusion is also supported by Fig. 7, where 3D collisionprobability (upper) and time (lower) ‘‘heat’’ maps are

FIGURE 4. Comparison of thrombus permeability (left) obtained from COMSOL (right) calculated based on LBM results. Both areaveraged over the length of the thrombus in the flow direction (perpendicular to the image).

FIGURE 5. Percent of tracers remaining in the thrombusafter the initial release at the injury site, as a function of timeand molecule weight. Flow and no flow cases are presentedfor each tracer type.

Intrathrombus Transport Simulation Based on In Vivo Images 1303

shown for the case of ADP transport in the presence offlow (left) and without it (right). From Fig. 7, thethrombus core has a significantly higher probability ofcollisions relative to the thrombus shell in the presenceof flow. In the ‘‘no flow’’ case, however, the particleshad more opportunity to reach the thrombus shell dueto a longer residence time. Similar trends wereobserved for Ca2+ and FX. The fact that tracers areremoved from the thrombus porosity by the lumenflow before they can ever reach the shell, also explainswhy there is not a significant decrease with time in therate of collisions for the flow cases in Fig. 6.

Intrathrombus Diffusive Transport of Proteins:Lumen-to-Injury Release

In order to explore transport of inert solutes fromthe blood lumen to the injury site, the simulations wererepeated with the tracers being released at the inlet ofthe blood vessel instead of at the injury site. Periodicboundary conditions were applied to the tracer motionin the stream-wise dimension with the aim of mim-icking a constant influx of the coagulation factors.Figure 8 is a plot of thrombus tracer concentrationratio relative to that in the bulk lumen after the initialrelease. All three tracers showed an increase in thethrombus concentration as the released tracers maketheir initial pass by the blood clot. After the tracersbegin to loop around the simulation domain the

increase in concentration becomes slower, which wasmore representative of continuous flow of coagulationfactors past the thrombus. In agreement with previousobservations, the smaller, more dynamic tracers madetheir way into the thrombus faster and took less timeto reach the final value of 1 for the clot-to-lumenconcentration ratio: ~0.1 s for Ca2+, ~1 s for ADP and~10 s for FX.

Intrathrombus Diffusive Transport of Diffusible Species:Effect of Pore Constrictions

Since the thrombus pore walls present barriers tothe tracers (even when the pore diameter is muchgreater than the size of the solute), the effectivemolecular diffusivity within the clot is reduced. On theother hand, the presence of convective flows within thethrombus can propel the motion of the tracers beyondtheir diffusive abilities. To compare the interplaybetween these two opposing effects, the intrathrombictransport results from the LST simulations are sum-marized in Table 1. The effective diffusivity due to thepore constrictions was measured by modeling tracertransport in the thrombus without flow. From theresults in Table 1, the thrombus pores constricted theparticle diffusion in the direction of flow, but notdramatically: the effective diffusivity was approxi-mately 50% lower than in a free fluid. This corre-sponds to a tracer path tortuosity of approximately 2,if the tortuosity is defined as follows:

s ¼ limt!1

D0

D tð Þ �D0

Deff: ð8Þ

The tortuosity of the tracer path in the stream-wisedirection did not seem to depend on the tracer typesignificantly, as expected for solutes that are dramati-cally smaller than the pore diameter. The addition offlow, as expected, enhanced transport within thethrombus. In order to quantify the effect of flow rela-tive to diffusion, we calculated the Peclet number,which is defined as the product of the Reynolds andSchmidt numbers. A characteristic length of 1 lm(from mouse platelet size) was used for the calculationof the intrathrombus Reynolds number. The Pecletnumber results shown in Table 1 indicate thatthe molecular transport within the thrombus spans

FIGURE 6. Average rate of collisions with the thrombussurface that each particle experiences after being released atthe injury site, as a function of time and molecular weight.Data represented as moving average trend line with thenumber of periods equal to 10. Flow and no flow cases arepresented for each tracer type.

TABLE 1. Intrathrombus transport simulation results for coagulation factors of different molecular weights.

FX (Sc = 60,000) ADP (Sc = 12,500) Ca (Sc = 4516)

D0 (cm2/s) 5 9 1027 (Ref. 22) 2.4 9 1026 (Refs. 11, 42) 6.64 9 1026 (Einstein–Stokes prediction)

Deff (cm2/s) 2.5 9 1027 1 9 1026 3 9 1026

s 2 2.4 2.21

PeCLOT 8.04 2.00 0.67

VORONOV et al.1304

different regimes depending on the tracer type, rangingfrom roughly equal convective-diffusive contributionsfor Ca2+ to predominately convective for FX-sizedproteins.

CONCLUSIONS

The intrathrombus fluid dynamics simulationresults showed that the shell experiences higher flows,stresses and is significantly more permeable than thecore. At the same time, the Brownian transport resultssuggest that under normal physiological flow condi-tions molecules originating from the damaged endo-thelium would not be able to reach the shell prior toexiting the thrombus. Thus, it can be hypothesized that

FIGURE 7. Comparison of intrathrombus diffusive transport of ADP-sized tracers for flow conditions (left) and no flow conditions(right). Upper panel is collision probability (i.e., number of collisions at each solid surface node relative to the total number ofcollisions that occurred in the thrombus during the simulation time), and lower panel is average collision time shown at eachsurface node locally (grayscale means that no collisions occurred at that particular surface node). In all cases the simulationdomain is cut in half along the flow direction in order to show the intrathrombus environment. One half of the thrombus is shown,with the cut plane facing the reader. The other half is omitted.

FIGURE 8. Ratio of tracer concentration in the thrombus totracer concentration in the lumen, as a function of time andmolecular weight.

Intrathrombus Transport Simulation Based on In Vivo Images 1305

the contributing reasons for the shell-and-core struc-ture of the thrombus could be: (a) mechanical—the cellbonds in the shell are not sufficient for platelet activa-tion, and/or (b) chemical—platelet agonists from theinjury do not reach the shell. Furthermore, theobserved differences in the intrathrombus transportenvironment could have implications for design ofdrugs that target either the core or the shell preferen-tially. The latter would be especially relevant for anti-clotting drug applications, since the shell is responsiblefor most of the embolism and, yet, does not seem toserve a biological purpose. Thus, the ability to dissolveit selectively, while keeping the core intact, would paveway for anti-clotting therapies that do not increase therisk of bleeding. Finally, we hope that this work willhelp to advocate the use of experiments and simulationscoupled via imaging for future studies, as it guaranteesa realistic thrombus structure and could be easilyextended to other systems (e.g., aggregate formation inin vitro microfluidics devices using human blood10).

ELECTRONIC SUPPLEMENTARY MATERIAL

The online version of this article (doi:10.1007/s10439-013-0764-z) contains supplementarymaterial, which is available to authorized users.

ACKNOWLEDGMENTS

We acknowledge financial support from NIHR01-HL103419 and AHA 11POST6890012 grants.Computation was carried under TERAGRID super-computing allocation TG-IBN110004 on the Lonestarlinux cluster (Texas Advanced Computing Center). Wewould also like to acknowledge Dr. Papavassiliou’sComputational Transport Processes Laboratory at theUniversity of Oklahoma, since a considerable portionof the code used in this study was written as a part of aPhD dissertation there. Finally, we would like to thankDr. Henry J. Neeman, Dr. Raghu Reddy, and thewhole OU Supercomputing Center for Education &Research (OSCER) team for useful discussions andsuggestions.

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