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A. ComerfordCentre for Bioengineering,

University of Canterbury,Private Bag 4800,

Christchurch, New Zealand

M. J. PlankCentre for Bioengineering,

University of Canterbury,Christchurch, New Zealand;Department of Mathematics,

University of Canterbury,Christchurch, New Zealand

T. DavidCentre for Bioengineering,

University of Canterbury,Christchurch, New Zealand

e-mail: [email protected]

Endothelial Nitric Oxide Synthaseand Calcium Production inArterial Geometries: AnIntegrated Fluid Mechanics/CellModelIt is well known that atherosclerosis occurs at very specific locations throughout thehuman vasculature, such as arterial bifurcations and bends, all of which are subjected tolow wall shear stress. A key player in the pathology of atherosclerosis is the endothelium,controlling the passage of material to and from the artery wall. Endothelial dysfunctionrefers to the condition where the normal regulation of processes by the endothelium isdiminished. In this paper, the blood flow and transport of the low diffusion coefficientspecies adenosine triphosphate (ATP) are investigated in a variety of arterial geometries:a bifurcation with varying inner angle, and an artery bend. A mathematical model ofendothelial calcium and endothelial nitric oxide synthase cellular dynamics is used toinvestigate spatial variations in the physiology of the endothelium. This model allowsassessment of regions of the artery wall deficient in nitric oxide (NO). The models hereaim to determine whether 3D flow fields are important in determining ATP concentrationand endothelial function. For ATP transport, the effects of a coronary and carotid waveform on mass transport is investigated for low Womersley number. For the carotid, theWomersley number is then increased to determine whether this is an important factor. Theresults show that regions of low wall shear stress correspond with regions of impairedendothetial nitric oxide synthase signaling, therefore reduced availability of NO. How-ever, experimental work is required to determine if this level is significant. The resultsalso suggest that bifurcation angle is an important factor and acute angle bifurcationsare more susceptible to disease than large angle bifurcations. It has been evidenced thatcomplex 3D flow fields play an important role in determining signaling within endothelialcells. Furthermore, the distribution of ATP in blood is highly dependent on secondaryflow features. The models here use ATP concentration simulated under steady conditions.This has been evidenced to reproduce essential features of time-averaged ATP concen-tration over a cardiac cycle for small Womersley numbers. However, when the Womersleynumber is increased, some differences are observed. Transient variations are overallinsignificant, suggesting that spatial variation is more important than temporal. It hasbeen determined that acute angle bifurcations are potentially more susceptible to athero-genesis and steady-state ATP transport reproduces essential features of time-averagedpulsatile transport for small Womersley number. Larger Womersley numbers appear to bean important factor in time-dependent mass transfer. �DOI: 10.1115/1.2838026�

Keywords: CFD, mass transfer, nitric oxide

Introduction

A key player in the pathobiology of atherosclerosis is the en-othelium. Originally, this monolayer of cells was thought to be aassive barrier between the blood and the artery wall, but now it isrmly established that the endothelium participates actively inaintaining vascular homeostasis and the regulation of vascular

one �by maintaining the balance between vasodilators and vaso-onstrictors�, prevention of platelet aggregation, and anticoagulantffects, all of which are essentially acting to prevent atherosclero-is �1�. During the early pathology of atherosclerosis �which in-olves an inflammatory response�, the regulation of these pro-esses by the endothelium is diminished �2�. For example, a

Contributed by the Bioengineering Division of ASME for publication in the JOUR-

AL OF BIOMECHANICAL ENGINEERING. Manuscript received April 10, 2006; final manu-cript received May 14, 2007; published online February 11, 2008. Review con-
ucted by David A. Steinman.
ournal of Biomechanical Engineering Copyright © 20

om: https://biomechanical.asmedigitalcollection.asme.org/ on 01/06/2016 T

reduction in the bioavailability and activity of vasodilators, suchas nitric oxide �NO� and prostacyclin, has been observed �3,2�.This altered regulation leads to upregulation of the vasoconstrictorendothelin and increased smooth muscle cell �SMC� proliferation.The release of vasodilators, and regulation of other endothelialprocesses, is linked to agonists �4,5� and wall shear stress �4,6,1�,although the precise mechanisms are not well understood at thisstage. It is therefore of the utmost importance to understand theimplications that various geometries have on the concentrationand transport of agonists and the distribution of wall shear stress�WSS�. In particular, lesions are generally localized in regions ofthe vasculature subjected to flow disturbances such as flow de-tachment, flow recirculation, and low time-averaged WSS, asfound near arterial bifurcations and bends �7,6�.

Calcium �Ca2+� is an intracellular second messenger that playsa key role in many of the endothelial regulatory processes.Changes in calcium concentration are known to elicit a number of
responses. Blood-borne agonists, such as adenosine nucleotides,
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ind to G-protein coupled P2Y receptors on the surface of thendothelium; this in turn causes the formation of inositol trispho-phate �IP3� in the cytosol. Once formed, IP3 binds to specificites on the surface of the endoplasmic reticulum �ER�, openingon channels which release internally stored calcium from the ERnto the cytosol. Depletion of internal stores then leads capacitivealcium entry via the plasma membrane. Further to agonist-egulated calcium release, there is evidence that shear stress di-ectly regulates cellular dynamics via mechanosensitive cationhannels �8�.

The activity of endothelial nitric oxide synthase �eNOS�, whichs responsible for the synthesis of NO, is modulated by cytosolica2+ and also by a Ca2+-independent pathway activated by shear

tress �9�. Endothelial NOS in endothelial cells �ECs� resides inaveolae, which are specific microdomains high in the proteinaveolin; this protein binds to eNOS, inhibiting its enzymatic ac-ivity �3�. NO is essential for vascular homeostasis, counters theffects of vasoconstrictors, and has key atheroprotective charac-eristics �10,2,11�. Free Ca2+ in the cytosol, resulting from WSS/gonist-activated pathways, binds with calmodulin; this causesNOS to disassociate from calveolin, activating the production ofO. Hence, a reduction in endothelial calcium concentration can

ead to reduced bioavailability of NO at the artery wall �12�. Lackf NO is associated with endothelial dysfunction, which is becom-ng increasingly identifiable as an early marker of vascular regionsusceptible to intimal wall thickening �3�.

Previously, a mathematical model that links the prevailing hae-odynamics and biochemical stimulus with the underlying cellu-

ar dynamics has been studied in the idealized two-dimensionalackward facing step geometry by Plank et al. �13�. This model isow applied to various three-dimensional geometries representingrtery bifurcations and bends in order to elucidate the importancef three-dimensional flow structures on the endothelium. Althoughtill idealized, these are much more representative of human arte-ial geometries in which atherosclerosis frequently occurs, en-bling predictions to be made about the susceptibility of differenteometric configurations and regions to cardiovascular disease.

The main aim of the model is to investigate spatial variation inC function and the consequences for the initiation of atheroscle-

otic plaque formation. ECs are sensitive to a number of bio-hemical species transported in the bloodstream, such as adenos-ne nucleotides, and to mechanical forces exerted on the cell

embrane. In particular, we model the EC response to adenosineriphosphate �ATP� and adenosine diphosphate �ADP� and to thehear stress on the artery wall due to viscous blood flow. All thesetimuli are spatially varying and this requires a model of three-imensional fluid flow and convective-diffusive mass transport.hese stimuli are linked to EC function via a cell signaling model.novel aspect of this model is that it includes activation of eNOS

by a Ca2+-dependent and a Ca2+-independent pathway�, which

Fig. 1 Example geometries with mes„c… mesh slice

irectly affects the pathophysiology of the artery wall. This en-

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ables the susceptibility of the endothelium to dysfunction to beevaluated in different parts of the artery wall and in differentarterial geometries. Hence, factors associated with heightened lo-calized risk of atherosclerosis may be identified. Furthermore, un-derstanding whether 2D simulations are sufficient to infer howECs respond in complex 3D environments, therefore the transportof agonist and WSS variations, is of utmost importance as cou-pling can be very computationally intensive in complex 3D envi-ronments. Furthermore, experiments are predominantly under-taken in 2D environments.

The transport of ATP has been studied extensively in 2D geom-etries �14–17�. In Refs. �14� and �16�, an analytical solution wasderived that fully described the transport of low diffusion coeffi-cient species in terms of the applied WSS. Computational fluiddynamical �CFD� studies have also looked at the transport whichis convection dominated, modeling species such as low-densitylipoprotein and oxygen in 3D arterial geometries �18–20�. Theresults obtained for the mass transfer field in this study couldapply to any blood-borne species of low diffusivity, for example,low density lipoprotein �LDL�. Time-dependent mass transfer hashad little attention; so, an in-depth look into transient variations isan important step to determine whether the time averaged andsteady state exhibit similar characteristics.

2 Model Development

2.1 Geometries. Previous work �21� had investigated, using asimple endothelial model, ATP concentrations for flow over a two-dimensional bifurcation. In order to investigate whether there wasany significant difference in the higher-dimensional case, fullthree-dimensional geometries were utilized. The arterial geom-etries were developed in SolidWorks �SolidWorks, Concord, MA�.Three arterial bifurcation angles were investigated �=37.5 deg,75 deg, and 135 deg; this was in order to provide insight into howdiffering bifurcation angles affect the transport of nucleotides, thedistribution of WSS, and the subsequent intracellular dynamics.Figure 1�a� shows an example bifurcation geometry with meshoverlain. The dimensions of the bifurcation were a 3 mm �D1�inlet diameter that bifurcates into two 2.38 mm �D2� daughterarteries, in accordance with “Murray’s law” �22�. These dimen-sions are representative of medium-sized arteries, in which athero-sclerosis is known to occur. Arterial bends are another region ofthe human vasculature known to be prone to atherogenesis. Onearterial bend geometry was developed. The bend had a constantdiameter of 3 mm �D3� throughout; the geometry with mesh over-lain is shown in Fig. 1�b�. This is similar to the bend Wada andKarino �20� use in their computational study of LDL transport.The bend consists of a mild curve in the entrance region, followedby a substantial curve in the opposite direction, and a mild curve

verlain: „a… bifurcation, „b… bend, and
h o
on exit aligned the same as the inlet curve.

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2.2 Numerical Formulation. In order to solve the flow andass transfer field, the commercial finite volume CFD package

LUENT �Fluent Inc., Lebanon, NH� was utilized. The flow field isolved iteratively via the continuity and momentum equations �1�nd �2�, on the assumptions that blood is incompressible and ofewtonian rheology:

� · v = 0 �1�

��v · ��v = − �p + ��2v �2�

here v represents the velocity vector �v= �u ,v ,w��, p the bloodressure, � the density, and � the dynamic viscosity.

The species mass transport was solved simultaneously via theonservation of species equation:

v · �� − D�2� = 0 �3�

here � is the species concentration and D the diffusion coeffi-ient. The walls of the model were assumed to be rigid and sta-ionary. For the majority of simulations, the flow field was as-umed to be steady; this is in order to obtain the underlyingharacteristics of the flow field, and previous studies have high-ighted that, during pulsatile flow, the time-averaged nucleotideoncentration does not differ significantly from the steady-stateoncentration �21,15�. The governing equations were solved itera-ively via the SIMPLE pressure-velocity coupling, and second-rder upwinding for the convective terms.

Boundary conditions used were inlet velocity condition coupledith the outlet condition of FLUENT on the domain exits; this

ondition assumes zero diffusive flux for all flow variables. Theutlet condition was located sufficiently downstream �25D2 arte-ial diameters for the bifurcation and 20D3 for the bend� to ensurehat fully developed flow had formed, as required when using thisoundary condition. A fully developed Poiseuille flow was speci-ed at the inlet via a compiled user-defined function �UDF�. Forass transfer, the inlet boundary condition used was a uniformTP concentration of 0.1 �M across the lumen, based on the pa-

ameters used in the simulations of John and Barakat �15�. Toake this condition more physiological, it is important to allow

he development of a mass transfer boundary layer. Hence, thenlet to the domains extended approximately 7D1 upstream of thepex of the bifurcation, and 4D3 of the bend inlet. On the outlets,zero species gradient condition was imposed.To model surface hydrolysis of ATP ��1� to ADP ��2� at the

rtery wall, a reactive boundary condition was used:

�DATP��1

��

�=0= KATP�1 − SATP �4�

his boundary condition represents a conservation of mass state-ent, where the flux of ATP by diffusion �� representing the

urface normal coordinate� to the endothelium is balanced by ATPydrolysis and release. The hydrolysis of ATP at the endothelialurface takes place via Michaelis–Menten kinetics, but previoustudies �14,15� have shown that rate saturation effects only occurt high ATP concentration ��100 �M�; so, the first-order termsed in Eq. �4�, where KATP represents the rate of uptake of ATP,s a very accurate approximation. The release of ATP, which haseen observed by a number of groups experimentally �23�, isodeled in the same manner to previous work �15� given by the

ollowing sigmoidal function for the rate SATP of ATP release:

SATP�x� = Smax�1 − exp�− �w�x��0

�3

�5�

ere, �0 represents the characteristic WSS and governs how theaximum ATP release rate is obtained. Two different release ratesere considered, in order to simulate the conditions of both slow

nd rapid releases. Smax represents the maximum ATP release ratend has a value of 10−6 �M m s−1.

For ADP, a similar condition to Eq. �4� was used:

ournal of Biomechanical Engineering


�DADP��2

��

�=0= KADP�2 − KATP�1 �6�

where the first term on the right-hand side represents hydrolysis ofADP to AMP, and the second represents production of ADP fromATP. The above conditions were incorporated into FLUENT througha compiled UDF, and applied to all walls of the artery geometry.

The mesh for the geometries contained 1.5�106 volumes forthe bifurcations, and 1.2�106 for the bend. A hybrid mesh with amajority of hexahedral elements was used in both cases; this is inpart due to the high aspect ratio cells required to keep the mesh inpractical limits, which, with tetrahedral meshing, will lead to amesh of poor quality. Due to the extremely low diffusion coeffi-cient of ATP �large Peclet number�, the transport is convectiondominated. Therefore, it was important to divide the geometry inthe immediate vicinity of the wall into sufficient elements to fullycapture steep species gradients, and prevent nonphysical solutionoscillations. This region contained a structured boundary layercomprising of hexahedral elements that progressed from approxi-mately 0.0021D2 �5 �m� to a total thickness of 0.064D2�152 �m� over 14 layers. In the streamwise direction, the mini-mum element size was approximately 0.025D2 �in the region offlow divider�, progressing to a maximum of 0.34D2 at the flowexits. In the circumferential direction, the daughter artery con-tained 90 elements for all geometries, while the inlet contained120 elements. Similarly, for the bend, the circumferential direc-tion was divided into 100 elements, and in the streamwise direc-tion, the minimum element was 0.013D3. Figure 1�c� shows aslice through the bend’s computational mesh, demonstrating thefine resolution elements near the wall. Mesh sensitivity tests werecarried out to determine whether the solution of ATP concentra-tion was sufficiently converged at two different Peclet numbers�5�106 and 8.3�106�. An off-wall mesh spacing of 5 �m wasfound to be sufficient for convergence. Global mass balance ofATP was also used as a convergence criterion. The validity andaccuracy of the numerical code were checked by comparing thenumerical calculation of the Sherwood number �Sh=2�� /�� tothat calculated from the analytical Graetz–Nusselt solution foraxisymmetric developing mass transfer in a pipe �24�. The com-parison in Fig. 2 shows that the numerical code is fully capable ofcapturing steep species gradients that arise when modeling lowdiffusion coefficient species.

Simulations were carried out at Reynolds numbers of 200 and

Fig. 2 Comparison of the numerical solution of low diffusioncoefficient species against the classic Graetz–Nusselt analyti-cal solution „developing mass transfer boundary layer in apipe…

500 �based on the inlet diameter� for the arterial bifurcations, and

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00 and 500 for the arterial bend, following Ref. �20�. This was torovide an understanding on how different flows, such as thearger recirculations expected at higher Reynolds numbers, affecthe transport of ATP to the surface and the resultant endothelialignaling. The elevated Reynolds number is expected to be repre-entative of peak diastole conditions.

It has been reported that the time-averaged nucleotide concen-ration during pulsatile flow differs little from the concentrationuring steady flow �15�, but this has yet to be validated in regionshere the flow has characteristics such as flow detachment, andeneral pulsatile mass transport in 3D geometries has had littlettention. Therefore, pulsatile simulations were performed. Theolution methodology is as for the steady flow case, but with andditional term ��v /�t and �c /�t� in Eq. �2� and �3�, respectively.wo wave forms are considered, as shown in Fig. 3. The carotidave form is based on the work of Ku et al. �25�, and the coronaryave form is from Matsuo et al. �26�. The mean Reynolds num-ers were 200 and 300 for the bifurcation and bend, respectively.ach cycle was divided into 100 time steps. Most simulationsere carried out with a Womersely number �Wo� of 2.1. This was

o capture the effects of different wave forms on the transport ofTP at the surface. However, simulations were also carried out atWo of 4 for the carotid wave form in the bifurcation. This was inrder to determine the effects of Wo on the transport of ATP. Anmportant part of time-dependent species computation is the eradi-ation of start up transients, which can persist for multiple cycles.or the low Wo, this required 8 cycles and for the higher Wo0 cycles; at this point, the agreement in ATP concentration be-ween successive cycles differed by less than 1%, and no variationas observed in WSS. The solution method used was the FLUENT

mplicit time-stepping algorithm, with the pressure-implicit withplitting of operators �PISO� scheme for pressure-velocity cou-ling.

2.3 Mathematical Model of Endothelial Cell Signaling.he intracellular calcium and eNOS dynamics of ECs were mod-led using a set of ordinary differential equations outlined below.he model has been adapted from the original models of Wiesnert al. �27,28� and Plank et al. �13�. The model outlined below �forull details, see Ref. �13�� describes the rate of change of fourependent variables: IP3 �i�, cytosolic free Ca2+ �Cc�, stored Ca2+

Cs�, and activated eNOS �n�. The dynamics of these quantities

ig. 3 Pulsatile inlet wave forms, carotid wave form given byu et al. †25‡, and coronary wave form given by Matsuo et al.

26‡

re affected by the local surface values of the shear stress �w and

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ATP+ADP concentration �.The meaning of symbols and their values are given in Table 1.

di

dt= ki

�

Kc + �

Cc

K1 + Cc− �1i �7�

dCc

dt= qrel − qres + qin − qout �8�

dCs

dt= − Vr�qrel − qres� �9�

dn

dt=

kdisCc

K6 + Cc− �2n + �1 − �gmaxF��w� �10�

Equation �7� represents IP3 production and decay. IP3 produc-tion is activated by external nucleotide ��=�1+�2 is the concen-tration of ATP+ADP� at a maximum rate ki, and is accelerated bythe presence of Cc via a positive feedback mechanism. The qiterms in Eqs. �8� and �9� represent Ca2+ release from internalstores into the cytosol �qrel�, Ca2+ resequestration by the internalstores �qres�, Ca2+ influx �qin�, and Ca2+ efflux �qout�. These aregiven by the following:

qrel = krel� i

K2 + i�3

Cs �11�

qres = kres� Cc �2

�12�

Table 1 Parameter values for simulations, taken from Plank etal. †13‡

Parameter Value and units Denotation

KATP 1.68�10−6 m s−1 ATP uptake rateKADP 6.45�10−7 m s−1 ADP uptake rateSmax 1�10−6 �M m s−1 Maximum ATP release�0 1.75, 3.16 Pa Characteristic WSSDATP 2.4�10−10 m2 s−1 Diffusion coefficient for ATPDADP 2.4�10−10 m2 s−1 Diffusion coefficient for ADP� 1000 kg m−3 Density of blood� 4�10−3 N s m−2 Dynamic viscosity of blood�0 0.1 �M Inlet ATP concentrationki 5.46�10−3 �M s−1 IP3 production rate�1 0.2 s−1 IP3 Decay ratekrel 6.64 s−1 Ca2+ release ratekret 5 �M s−1 Ca2+ resequestration ratekout 24.7 �M s−1 Ca2+ efflux ratekdis 0.09 �M s−1 eNOS-caveolin disassociation rate�2 0.0167 s−1 eNOS-caveolin association rateqmax 17.6 �M s−1 Max. WSS-induced Ca2+ influx ratekCCE 5.7�10−6 s−1 CCE rateCs,0 2830 �M Resting stored Ca2+ concentrationCex 1500 �M External Ca2+

Kc 0.026 �M Michaelis–Menten constantsK1 0 �MK2 0.2 �MK3 0.15 �MK4 5 �MK5 0.32 �MK6 0.45 �MVr 3.5 Ratio of cytosol to ER volume 2 Zero shear open channel constant 0.1 Relative strength of the Ca2+-dependent

pathway for eNOS activationW0 1.4 Pa−1 Shear gating constantgmax 0.06 �M s−1 Max. WSS-induced eNOS activation� 2.86 Pa Membrane shear modulus

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qin =kCCE

K4 + n�Cs,0 − Cs��Cex − Cc� + qmaxF��w� �13�

qout =koutCc

K5 + Cc�14�

here

F��w� =1

1 + exp�− W��w��

W��w� = W0

��w + 16�2 + �w2 − 4��2

�w + 16�2 + �w2

qrel is an increasing function of IP3 concentration, with maxi-um rate krel. qin includes a term representing capacitive calcium

ntry �CCE� where influx is stimulated by depleted internal stores29�, and a term qmaxF��w� that stimulates calcium entry via WSS8�. CCE is reduced in the presence of eNOS �30�. qres and qoutepresent the action on cytosolic calcium of Ca2+-ATPases locatedn the ER and plasma membrane, respectively.

Equation �10� is a new addition to the model, representing ac-ivation and deactivation of eNOS. Endothelial NOS is activatedy Cc at maximum rate kdis and deactivated at constant rate �2.lso, this equation includes a source term, gmaxF��w�, for activa-

ion of eNOS directly by WSS, as there is considerable evidenceuggesting that eNOS is activated in a calcium-independent man-er via Akt-dependent phosphorylation of eNOS �9�. Few dataxist regarding the manner of this response, but the results of an inivo experiment by Cheng et al. �31� suggest a sigmoidal relation-hip between WSS and eNOS activation, which has been adoptedor the present model. The relative contribution of thea2+-dependent �Eq. �13�� and -independent �Eq. �10�� eNOS ac-

ivation pathways is governed by the parameter . This is taken toe 0.1 for the simulations since evidence suggests that thea2+-independent mechanism is dominant �9�. Simulations withifferent values of yield the same qualitative results, but furtherxperimental data are required to determine the relative impor-ance of the Ca2+-dependent and -independent pathways for eNOSctivation.

Equations �7�–�10� were solved via an adaptive step Runge–utta routine, until steady state; previously, single cell resultsave shown that there is an initial transient response that plateaus13�. This allows for the steady-state concentrations of calciumnd eNOS, at any point on the endothelial surface, to be exam-

Fig. 4 WSS plotted along a cut plane through the outer wall odecreases as � decreases.

ned. The above equations were incorporated into the FLUENT



model via a compiled UDF. The code takes as inputs WSS andsurface ATP concentration, calculated as described in Sec. 2.2.These environmental stimuli are then substituted into the set ofdifferential equations: WSS �w in Eqs. �10� and �13� and surfaceATP+ADP concentration � in Eq. �7�. The program loops overall faces of the surface mesh solving equations �7�–�10� at everysolution variable storage point �cell center in this code�, thus re-turning the spatial variation of the steady-state concentrations ofIP3, cytosolic Ca2+, stored Ca2+, and eNOS.

3 Results

3.1 Arterial Bifurcation

3.1.1 Flow Characteristics. The flow features of the bifurca-tion are similar to previous studies, with the curvature and branch-ing between the parent and daughter artery causing the flow toassume a helicoidal trajectory rotating about the streamwise axis.This secondary flow is highly dependent on the bifurcation angle,with the symmetric counter-rotating helicoidal flows each side ofthe midplane strengthening with increasing bifurcation angle �.The stronger secondary flow leads to larger velocity gradients atthe artery wall �due to high inertia fluid near the inner wall�,which maintains the WSS in a higher range with increasing �.

As of result of counter-rotating flows, following the flow di-vide, the velocity profile is skewed to the inner wall, leading tohigh WSS in this zone. On the outer wall, a large region of lowvelocity flow exists, with the central part of the low velocity re-gion extending into the vessel and forming a horseshoe profile;this profile results from the characteristic Dean vortices, broughtabout by curvature. For smaller angle bifurcations, the sameskewing of the profile occurs, but the low velocity region spans awider portion of the outer wall, due to substantially reduced sec-ondary flows and reduced circumferential fluid velocity.

From the numerical simulations, it was found that the regionsof low WSS are localized on the outer wall of the bifurcation; thisresult is irrespective of bifurcation angle and Reynolds number. Atthe inner wall of the bifurcation, the WSSs, observed are high,regardless of the bifurcation angle; so, this area is unlikely to beof significance for the onset of cardiovascular disease. ExaminingWSS in detail along a cut plane through the outer wall �Fig. 4�reveals a relationship between low WSS and smaller bifurcationangle. Clearly, the Reynolds number plays a central role: At Re=500, lower overall WSSs are observed on the outer wall; this isattributed to the stronger secondary flows that lead to a larger lowvelocity fluid region in the vicinity of the outer wall. However, it

e bifurcation; „a… Re=200 and „b… Re=500. The minimum WSS
f th
is the spatial variation in WSS that is of primary interest here, and

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he key observation is that the outer wall of the bifurcation isubjected to relatively low WSS irrespective of the Reynoldsumber, and the region of low WSS is larger for smaller bifurca-ion angles.

3.1.2 Nucleotide Concentration. ATP concentration undergoesn overall decline in the streamwise direction, but increases/ecreases in response to various flow conditions. Previous studiesave looked at ATP concentration at the inner walls of 2D bifur-ations �21�, and it was found that ATP followed the WSS distri-ution very closely. Additionally, another study investigated ATPoncentration over a backward facing step, and again WSS wasignificant �16�. While WSS magnitude is reasonably related inD, the most important controller is the WSS vector. This is ef-ected by complex flow structures such as secondary vortices,esponsible for the growth of the mass transport boundary layer inhe outer wall regions, which results in substantial depletion ofTP at the surfaces. This occurs due to the convection of ATP

rom the outer to the inner wall, hence reduced convection of ATPnto the boundary layer. In simplified 2D geometries, secondaryow features do not exist; so, essential features are lost in theimplification that cannot be inferred by simply extrapolating theelocity field. Figure 5 shows surface ATP concentration contoursunder no-release conditions� on the outer wall of a 75 deg bifur-

ig. 5 ATP contours with limiting streamlines overlain, �75 deg, Re=500. This is a perspective view looking at theuter wall where the streamlines converge on the outer wallorresponds with minimum ATP concentration „hence massransfer…; as the bifurcation angle is reduced, this region ex-ands undergoing further depletion „data not shown….

Fig. 6 Nucleotide concentration along the outer wall of the 7tration for �0=3.16. „b… ATP+ADP concentration for different A
is subject to reduced combined nucleotide concentration relativ
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cation with limiting streamlines �vector valued WSS trajectoriesresolved over the surface �32��. Evidently, the surface depletionfollows these flow patterns very closely and the location of re-duced ATP concentration corresponds to the meeting of two Deanvortices; additionally, the variation being strongly dependent onthese patterns shows the dependence on secondary flows. Thedepletion at the surface is greater for smaller bifurcation anglesand this results from weakened secondary vortices. Although thereis evidence that ADP is also capable of eliciting responses by theendothelium, it is also important to consider ATP alone to under-stand the mass transfer processes occurring and also because ATPis the sole agonist that activates the P2X receptor �ion channel�,which has recently been identified as a vasoactive signaling recep-tor �33�.

The present simulations have also included the effects of flow-induced ATP release. Figure 6�a� shows the results for a Reynoldsnumber of 500 and �0=3.16. Although the surface ATP is elevatedwith respect to the bulk concentration �0.1 �M�, there is a definitereduction of nucleotide concentration in the vicinity of the outerwall, and unpublished data have shown that under the right flowconditions �torsional�, the ATP is depleted with respect to thebulk. Similar results were obtained throughout the investigatedbifurcation angles with lower nucleotide concentration on theouter wall as � decreased. Reducing the value of the referenceWSS �0 results in a similar profile but with higher nucleotideconcentrations �Fig. 6�b��.

The current model of ATP release contains a number of limita-tions. For example, there would be uneven release over the sur-face of the endothelium, and the stores of intracellular ATP woulddeplete �if the supply of glucose and oxygen to the cells could notkeep pace� after a certain period of activation, eventually leadingto no-release conditions. Therefore, the response would possiblyconsist of an initial transient release that decays. Future modelswill attempt to address these issues and include the cell dynamicsthat lead to ATP release. The present analysis including both norelease and varying degrees of release covers the different situa-tions that may be encountered in vivo.

3.1.3 Endothelial Cell Signaling. The surface distributions ofshear stress, ATP concentration �under slow release conditions�,and ADP described above act as the external factors driving theintracellular signaling of the ECs. At each grid point on the arterywall, the local WSS �w and surface concentration �=�1+�2 ofATP+ADP are used as the “inputs” to the model presented in Sec.2.3 via Eqs. �7�, �10�, and �13�. This enables the steady-state dis-tribution of important intracellular chemicals such as calcium and

eg bifurcation, Re=500: „a… ATP, ADP, and ATP+ADP concen-release rates. It is clear that the outer wall of the bifurcation

5 dTP

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NOS to be evaluated. Again, following the WSS and ATPADP concentration patterns, regions of reduced cytosolic cal-ium are found predominantly on the outer wall �data not shown�.his, and also low WSS directly, leads to reduced activation ofNOS. Figure 7 shows the resulting eNOS concentration contours.ll bifurcation angles are subjected to a region of low eNOS

oncentration. Observing a cut plane �Fig. 8�, it is evident that,lthough the level of depletion does not change considerably forhe investigated range, smaller angle bifurcations contribute toustained low eNOS concentration over a larger portion of thendothelial surface. The extended region of depleted eNOS con-entration at a bifurcation angle of �=37.5 deg results from flowecirculation. There is a clear relationship between the eNOS con-entration and the WSS, but ATP does not show a strong relation-

Fig. 7 Contours of eNOS concentration, Re=500: „a… �=37.5signaling increase with decreasing �

ig. 8 Endothelial NOS concentration plotted along a cutlane through the outer wall of the bifurcation, Re=500. As theifurcation angle reduces, the region of depleted eNOS con-entration is more sustained. The small rise in the reducedNOS region at �=37.5 deg results from the flow recirculation
ccurring in this area.


ship with the resulting distribution.The results suggest that the low WSS region found on the outer

wall will be more susceptible to endothelial dysfunction at allbifurcation angles, but it is hypothesized that this is more likely tooccur on the outer wall of acute angle bifurcations. In addition tothe outer wall depletion, there is a region of low eNOS concen-tration located on the top and bottom of the bifurcation shown inFig. 7. However, the severity and extent of this depletion havebeen observed to change significantly depending on the curvatureof the geometry in this region.

3.2 Arterial Bend. Arterial bends are also subject to dis-turbed flow conditions, such as detachment and recirculation, andare prone to atherosclerosis �7�. Simulations were carried out atReynolds numbers of 300 and 500. Additionally, unsteady simu-lations were carried out to investigate the variation of nucleotidesunder this flow condition.

3.2.1 Flow Characteristics. The flow characteristics in the ar-terial bend are as expected, exhibiting secondary flows �Dean vor-tices�, and skewed velocity profiles toward the outer walls of thebends, due to the radial pressure gradient �or centripetal accelera-tion� induced by the curvature. A standing recirculation zoneforms at the inner wall of the second bend, where peripheral flowcomes down from the outer wall surrounding the main flow, slow-ing rapidly before traveling in the upstream direction, and finallybeen pulled rapidly back into the main flow. The extremities ofthis range define a point of flow separation and flow reattachment.The flow characteristics for the two Reynolds numbers are verysimilar, just amplified more at Re=500.

Figure 9 shows that, for both Reynolds numbers investigated,the inner and outer walls of the bend both exhibit a region of lowWSS. For Re=500, there is a small region of higher WSS withinthe recirculation zone, but this region is still predominantly a lowwall shear region. The recirculation zone is similar for both Rey-nolds numbers, but occupies a larger region of the inner wall atRe=500.

3.2.2 Nucleotide Concentration. The flow characteristics suchas flow separation �nodal point� and reattachment �saddle point�lead to impaired mass transfer, with both locations subject to re-duced ATP concentration, again following the flow patterns

, „b… �=75 deg, and „c… �=135 deg regions of impaired eNOS
deg
closely. Due to the flow detachment, there is a significant growth

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f the concentration boundary layer observed at the inner wall.ithin the recirculation zone, convective transport results in a

light increase in the ATP concentration. At Re=500, the low con-entration region moves to either side of the central plane, leavingt the point of flow detachment, extending along each side of theentral plane and rejoining at the point of flow reattachment; thisistribution follows the WSS gradient very closely. At Re=300,he lower flow velocities and reduced secondary flow allow fur-her depletion of ATP at the artery wall.

Following the addition of ATP release �Fig. 10�, again there islevated concentration with respect to the bulk, but as has previ-usly been demonstrated in 2D backward facing step geometries13�, the concentration at the stagnation point is minimum. In theresent case, there is no stagnation streamline; rather in the limit,here is a streamline that defines the location of reattachment; thisocation corresponds to the minimum in ATP concentration in Fig.0. At the location of flow detachment, the ATP concentrationrops rapidly, but due to the peak in ADP concentration resultingrom this, there is no rapid drop exhibited by the combined stimu-us. Interestingly, the first curvature of the bend results in a deple-ion of both ATP and ADP. This is due to the combined effects of

ig. 9 WSS plotted along a cut plane through the walls of theend, Re=300 and 500: „a… inner wall and „b… outer wall. Essen-

ially, the elevated Reynolds number provides for amplifiedesponses.

eripheral flow convecting ATP away and general decrease in wall

11010-8 / Vol. 130, FEBRUARY 2008


shear just prior to the first mild bend; hence, hydrolysis of ATPcounters the effects of ATP release, and the strong peripheralflows mean that ADP is also depleted.

3.2.3 Endothelial Cell Signalling. Figure 11 shows the spatialvariation of eNOS. The inner and outer walls are both subjected toreduced eNOS levels �due to the combined effect of impairedcalcium signaling and low WSS�. Again, these correspond to re-gions of flow recirculation and, in particular, low WSS. For thehigher Reynolds number, the region of impaired signaling is re-duced, but the inner and outer walls are still areas of relativelylow eNOS activity. The inner wall contains a small portion of higheNOS concentration, as a consequence of the WSS in this regionresulting from a strong recirculating flow. Plotting along the cen-terline of the bend’s walls �Fig. 12� highlights the key aspects:eNOS concentration is low in regions of low WSS, particularlywhere the flow detaches or reattaches. The region of constanteNOS concentration, of around 4 �M, prior to the flow detach-ment on the inner wall is due to the very high WSS in this area,which results in saturation of eNOS activation.

Fig. 10 Nucleotide concentration under slow release condi-tions plotted along the inner wall of the arterial bend, Re=500.The location of minimum nucleotide concentration corre-sponds to the flow reattachment point.

Fig. 11 Contours of eNOS concentration: „a… Re=300 and „b…Re=500. Reduced eNOS signaling is observed at the inner andouter walls, which will result in reduced production of NO. Theregions of reduced signaling are more significant at the lower
Reynolds number.


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3.3 Pulsatile Flow. In order to validate the use of steady-stateimulations, pulsatile simulations were also performed using thenlet wave form given in Fig. 3. The results presented here arepplicable to small arteries for the low Wo and large arteries forigher Wo. For low Wo, Fig. 13 compares the steady ATP concen-ration along a central cut plane of the bends inner and bifurca-ions outer walls with the time-averaged concentration over a car-iac cycle of period T:

�̄ =1

T�0

T

�dt

This was plotted along the cut plane of the bifurcations outerall and the inner and outer walls of the bend, so comparisons

ould be made to the steady-state concentration. For the bifurca-ion, the time varying concentration is reduced compared with theteady state �Fig. 13�a��. To understand this further depletion,ariation of the instantaneous wall shear vector was investigatedy calculating the oscillatory shear index �OSI�. This was showno spike up in the region where the time-averaged ATP concentra-ion drops. The primary contribution to this OSI was a brief periodf flow separation induced by the adverse pressure gradient lateiastole �coronary� and late systole �carotid�. This period of sepa-ation results in attenuated ATP delivery, hence reduced surfaceoncentration compared with steady state. The inclusion of ATPelease results in significantly closer tracking between the steadynd unsteady �Fig. 13�b��.

For the bend, similar characteristics were observed with theime variance exhibiting very similar characteristics to the steady;gain, the rapid drop in ATP concentration corresponds with a lowean WSS. However, in addition, WSS is highly oscillatory in

his location, nearly purely �0.48 where 0.5 corresponds to purelyscillatory�. The depletion at the surface being lower under steadyonditions �Fig. 13�c�� is due to the standing recirculation zonehat forms, while in unsteady flow, this is nonexistent during theystolic phase, with the flow being unidirectional. In Fig. 13�c�,he location of low ATP concentration is due to attenuated trans-ort within the recirculation zone and the more sustained loweroncentration over the cardiac cycle is involved with time periodt the location of low mean WSS. The second drop in ATP con-entration occurs where the mean WSS is low �due to stagnantow�. In this location, the OSI is at an intermediate to low level.dditionally, at flow separation, there is also a very low meanSS.If the Wo is increased to 4 �representative of Wo in the carotid

rtery�, the story is a little different. Figure 14 details the time-

Fig. 12 Endothelial NOS concentration plotted along the centand „b… outer wall. Endothelial NOS concentration is low in th

veraged ATP concentration along the outer wall of the bifurcation



for a carotid wave form at this elevated Wo number. Evidently,there are some clear differences between the low and higher Wo;particularly, the ATP concentration undergoes a greater depletionof ATP concentration and then “plateaus.” Further downstream,the concentration increases; this is in contrast to the lower Woflow where ATP decreases in the streamwise direction. There are anumber of possible reasons for these different phenomena. Firstly,for higher Wo, the underlying characteristics of the flow field areconsiderably different. The mean WSS is fairly similar, but theOSI is considerably larger due to larger periods of recirculationinduced by the higher slew rates. This recirculation leads to thegreater depletion observed at the outer wall. Secondly, the con-centration plateau suggests that convection to the surface is bal-anced by surface degradation. This is potentially due to the effectsof secondary flow-induced convective mixing resulting in an in-crease in the effective diffusivity of ATP; hence, to a certain de-gree, delivery overcomes hydrolysis.

The results suggest that there is a difference in concentrationbetween pulsatile and steady flows, but for low Wo, this differenceis quite small and was within 4% for the arterial bend and was 7%for the arterial bifurcation. Whereas, the concentration spatialvariation over the entire surface is in the range of 0.075–0.098. Incontrast, the distribution along the outer wall for increased Wowas quite different and suggests that the steady state results aremore applicable to small arteries. Importantly, for our model, thedistribution of depleted ATP concentration regions is not signifi-cantly altered. Therefore, the simplification of the model to steadyflow �for small arteries� is justified as the spatial location of re-gions of low eNOS is largely unaffected.

Previously, little information has been reported on transient ef-fects of low diffusion coefficient species. Figure 15 shows thetransient effects at specific spatial locations for low Wo. Figure15�a� shows the sampling locations for the bifurcation and bend�looking at the inner wall�. Locations A, B, and C in the bifurca-tion are progressing along the outer wall in a low shear region,while D is in a region where the WSS follows the inlet wave formpattern. Transient variations for higher Wo in the bifurcation werealso negligible. Bend Location 1 corresponds with a higher shearregion �again WSS following the inlet wave form�, while Loca-tions 2, 3, and 4 progress through the recirculation zone. Samplingin other locations around the artery wall provided for very similarcharacteristics �data not shown�.

For a carotid wave form �Fig. 15�b�� during the systolic phase,the Dean vortices strengthen and move toward the outer wall,resulting in enhanced convective delivery of ATP in the circum-

axis of the arterial bend walls, Re=300 and 500: „a… inner wallegions corresponding to low WSS.

ral

ferential direction, which essentially disperses the low concentra-

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ion region, leading to elevated surface levels. During the systoliceceleration, the vortices move closer together and away from theuter wall. This “drags” the boundary layer out resulting in alight reduction in endothelial ATP concentration. This boundaryayer growth is due to the formation of a small recirculation zonebecause of an adverse pressure gradient� at the end of systole.he vortices slowly track toward the outer wall during the dias-

olic phase until there is relatively little movement. With a coro-ary wave form �Fig. 15�c��, the rise in ATP concentration is ob-erved during the diastolic phase. This is because secondary flowortices move toward the outer wall enhancing convective deliv-ry. During the deceleration of this phase, the vortices track backoward the inner wall, thus reducing the concentration. The for-

ation of a recirculation zone also occurs during this deceleration.For the bend �Fig. 15�d��, similar characteristics are observed

xcept in the location of flow separation where transient variationsre significant �sampling Location 2�. Secondary flows are againesponsible for ATP transients with two counter-rotating vorticesach side of the vertical plane �of the lumen cross section�. Theseortices rotate away from the vertical plane �and the inner wall�uring the systolic phase, leading to a decrease in concentration athe surface. During the large forward flow of the coronary dias-olic phase, the predominant mechanism for ATP delivery is aarge recirculation zone, which enhances convective transport tohe inner wall, providing the rise in ATP concentration for sam-ling Locations 3 and 4. In the high shear region, such as sam-

Fig. 13 ATP concentration in steady flow and time-averaged cinner wall of the bend and the outer wall of the bifurcation: „a…and „c… bend. Evidently, the steady state exhibits essential fe

ling Location 1, the skewing of the velocity profile toward the

11010-10 / Vol. 130, FEBRUARY 2008


lower wall is the predominant mechanism for ATP delivery. Figure16 demonstrates that the boundary between the high shear region

centration in pulsatile flow plotted along medial planes of theer wall bifurcation, „b… outer wall bifurcation with ATP release,res of the time-averaged profile.

Fig. 14 Comparison of Wo for carotid wave form. For thelarger Wo, the ATP concentration plateaus in the daughter ar-

onoutatu

tery and then increases in the streamwise direction.



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nd flow separation is associated with the largest transient varia-ions of ATP concentration �sampling Location 2 of Fig. 15�d�� isn this region�.

Overall, the transients in ATP concentration are negligible; but,otentially in certain regions of the vasculature where substantialow recirculation occurs, transients may be significant. AlthoughTP concentrations in this particular flow separation zone are sig-ificant, the Ca2+ and thence eNOS may not necessarily followhis, as the time scales are much longer than the cardiac cycle fora2+ dynamics. Further work is required to determine the re-

ponse in large arteries subject to substantial recirculation.The overall lack of temporal response of ATP concentration is

ue to the three orders of magnitude difference between momen-um diffusivity and mass transfer diffusivity. Considering theransport equations, for momentum, an increase in Wo results inonsiderable different flow phenomena; however, for mass trans-er, the diffusion coefficient is very small and hence increases in

o that does not impart temporal changes due to the largechmidt number ��17,000�. The low diffusivity of ATP effec-

ively means that the ATP concentration has limited variation inesponse to the prevailing temporal changes of the flow field.

DiscussionThere is a significant amount of literature that identifies the

Fig. 15 Time variations in ATP concentration for different flobend „looking at the inner wall…, „b… bifurcation with carotid wwith coronary wave form

uter walls of bifurcations and recirculation zones as regions that



are more susceptible to the formation of atherosclerotic plaque�34,6�. The current model, which incorporates the prevailing hae-modynamics, transport of blood-borne agonists in bifurcation andbend geometries, and underlying endothelial cellular dynamics,shows strong correlation between regions of low WSS and lowintracellular eNOS activity. Regions of reduced Ca2+ and eNOSsignaling imply reduced bioavailability of NO, which has beenlinked with endothelial dysfunction and the subsequent onset ofatherogenesis �3,2�.

Simulations in an idealized arterial bifurcation and bend haveidentified the areas that are subjected to low WSS and reducedeNOS activity, and hence most susceptible �by the above hypoth-esis� to endothelial dysfunction and disease. These are the outerwalls of the bifurcation and the proximal outer wall and distalinner wall of the bend. The extent and severity of the pathologicalconditions vary with the Reynolds number of the flow, but theseregions always suffer from low WSS and eNOS relative to otherparts of the endothelial surface. Hence, these regions are identifiedas “hot spots” for the initiation of atherosclerotic plaque forma-tion, due to endothelial dysfunction and impaired NO-mediatedvasodilation. Furthermore, the size of this distribution is depen-dent on the 3D nature of the haemodynamics, with reduced sec-ondary flows resulting in a larger region of low eNOS activity.There is a possibility that the upper and lower walls of the bifur-

ave forms: „a… sampling locations on the bifurcation and theform, „c… bifurcation with coronary wave form, and „d… bend

w wave

cation may also be prone to disease, as regions of low WSS and

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ence reduced eNOS exist. However, this phenomenon is highlyependent on the curvature in this region, which will vary mark-dly in human arterial bifurcations.

In the present study, the concentration of ATP and ADP at therterial wall appears to be of less significance for eNOS activationhan the WSS. This is particularly true when the effects of flow-nduced ATP release are included. This observation suggests that

SS is the dominant factor determining the spatial variation inndothelial function, although this statement should be accompa-ied by the following caveats. It has been assumed in this modelhat the ECs are sensitive only to the combined surface concen-ration of ATP+ADP, which does not vary greatly over the regionf endothelium studied. There are a number of other signalingathways that have not been explicitly included in the presentodel. For example, the P2X ion channel pathway is only sensi-

ive to ATP. Future work will include the addition of this pathway,hich may increase the sensitivity of eNOS activity to ATP con-

entration. Furthermore, certain rate constants in the cell modelre not known; so, the sensitivity to ATP may be greater thanredicted with the current parameter set.

Although there is always a spatial relationship between theuter wall of the bifurcation and a pathological environment, theifurcation angle has been found to play an important role in theesponse. Regions of low WSS are more significant at smallerifurcation angles, due to weakened secondary flow. There haveeen some reports �35,36� that acute angle bifurcations are corre-ated with a larger lateral distribution of low WSS �as observed inhe present study� and atherosclerotic plaque. In contrast, othertudies have reported that larger angle bifurcations are more sus-eptible �37�. However, the evidence to date is inconclusive andhis remains a controversial issue. Other factors, such as parent-aughter artery area ratios and out-of-plane bifurcation effects38�, may be important, and a definitive relationship betweenranching angle, fluid shear stress, and disease has yet to be firmlystablished, and may very well depend on the location within theasculature. The theoretical predictions of this model thereforeall for further experimental study in this area.

The results of this model agree well with in vitro research thatas identified regions of low time-averaged WSS/disturbed flow,uch as those found on the outer wall of bifurcations, with thenset of cardiovascular disease �7,39�. Further experimental re-earch needs to be focused on in vitro cellular function �in par-icular, endothelial calcium dynamics and NO production� in ordero validate model predictions regarding eNOS activity and themplications for cardiovascular disease. Studies of this natureave been undertaken in parallel plate flow chambers �40�. How-

Fig. 16 Time variation of over „a… ATP and „b… WSS plotted alsteps.

ver, the cell culture in such a setup is exposed to a uniform flow

11010-12 / Vol. 130, FEBRUARY 2008


field. In contrast, there has been relatively little experimental re-search on cell function in a spatially varying mechanical and bio-chemical environment, which is key to understanding the local-ization of atherosclerotic lesions in vivo.

The assumption that there are no major differences betweenATP concentration in steady flow and the time-averaged concen-tration over a pulsatile cardiac cycle has been validated by per-forming pulsatile flow simulations for small Wo. The time-averaged concentration was found to differ from the steady stateby no more than 7%. However, when an increased Wo was inves-tigated �for the carotid wave form�, there were significant differ-ences between the steady and time averaged. This was attributedto the effects of convective mixing. Temporal variation in nucle-otide concentration generally did not vary significantly throughoutthe course of the cardiac cycle; however, in the arterial bend, theflow detachment was subjected to significant transients, meaningin regions of flow recirculation �and generally low diffusion coef-ficient species� may undergo significant variation, but this may nothappen with eNOS as noted above.

Future work will involve the addition of NO release to themodel, and the effects of NO-dependent arterial wall vasomotion.Endothelium-derived NO induces relaxation of subendothelialSMC and consequent vasodilation �3�. The altered vessel diameterwill in turn modify the fluid flow field, acting as a feedback con-troller. In addition, this will allow realistic blood wave forms to becoupled to the underlying cellular dynamics. Although agonisttransport is not greatly affected by pulsatile flow, the potentiallynonlinear interaction between spatially and temporally varyingWSS and vasomotion may be significant.

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