Computational Analysis of the Effects of Exercise on Hemodynamicsin the Carotid Bifurcation
H. F. YOUNIS,1 M. R. KAAZEMPUR-MOFRAD,1 C. CHUNG,1 R. C. CHAN,2 and R. D. KAMM 1
1Department of Mechanical Engineering and the Division of Biological Engineering, Massachusetts Institute of TechnoloCambridge, MA and2Boston Heart Foundation, Harvard/MIT Division of Health Sciences and Technology, Cambridge, MA
Due to the strong correlations known to exist betwesites of atherosclerosis and variations in local hemonamics, considerable research has been performed toon arterial blood flow in bifurcations, junctions, angrafts. These studies include numerical simulations tallow detailed and direct examination of the flow ashear stress patterns in regions of interest. In a conting attempt to increase the realism of these simulationew factors have been introduced including no
Address correspondence to Roger D. Kamm, 77 Massachusettsenue, Room 3-260, Cambridge, MA 02139. Electronic [email protected]
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Newtonian blood properties, arterial wall compliancand at times, treating the blood as a two-phase mediEach added factor represents another step toward aistic prediction of wall shear stress~and other relatedparameters!, so that these might be correlated with mesures of disease or the various biological factors knoto cause it.
For example, a study by Gijsenet al.14 on the influ-ence of non-Newtonian properties of steady blood flin the carotid bifurcation demonstrated that notable dferences exist between Newtonian and non-Newtonmodels. The axial velocity field of the non-Newtoniafluid was flattened, and it had higher velocity gradientsthe outside wall and lower gradients at the flow dividthan the Newtonian model. On the other hand, Perktet al.28 found that in the case of pulsatile flow, the geeral flow characteristics between Newtonian and nNewtonian models were largely unchanged in the carobifurcation. An experimental investigation by Friedmaet al.11 supported this conclusion by noting that the effeof fluid rheology in the aortic bifurcation is minimal inquantifying the correlation between intimal thickeninand shear rate.
Another aspect of arterial blood flow that has bemuch investigated is the effect of arterial wall compance on the flow characteristics. Friedmanet al.11 dem-onstrated that the effect of compliance is also insigncant. Steinman and Ethier30 have observed that onlyminor changes in overall wall shear stress patternscurred when the effect of wall distensibility was includein an end-to-side anastomosis. In separate simulatiPerktold and colleagues27 demonstrated that flow separation and wall shear stress are reduced in a distenswall model of the carotid bifurcation. Consistent witPerktoldet al.27 and Zhaoet al.,35 wall distensibility wasfound to have little effect on the wall shear stress pterns at the outer wall of the carotid bulb, generabelieved to be the site most prone to disease. Significdifferences in wall shear stress, of up to 25%, only apeared at the flow divider.27
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996 YOUNIS et al.
Despite the large number of previous studies, onlfew have rigorously examined the effect of normal, davariations in local hemodynamic conditions.19,23–25 Thisis rather surprising since one might expect these vations, due to large changes in cardiac output, to farceed the small effects associated with non-Newtonrheology or even wall compliance. Depending on tlevel of activity and the variations that an individuexperiences during the course of a day, changes in mblood flow rate might expectedly cause more significvariations in the above-mentioned parameters. In adtion, due to the effect of pulse wave reflections on shing the local pressure waveform, even the patternpressure and velocity variations will change. These vations are not only due to intense exercise, but can ebe seen in mild exertion such as brisk walk. Here,consider these effects in the context of blood flothrough a healthy carotid bifurcation.
METHODS
Pulsatile blood flow in a model carotid bifurcationsimulated using fully coupled fluid–structure interactifinite element methods at two different heart rates:bpm representing a person at rest and the other atbpm ~50% increase!. Two cases are considered at 1bpm heart rate:~a! peripheral resistance is uniformlreduced resulting in an increase in mean flow, and~b!peripheral resistance reduced, except in the brain whit is held constant and, hence, mean flow to carotid artis nearly unchanged. For each case, wall shear stvalues, relevant integrated parameters of wall shear stover the cycle as well as temporal gradients of wall shstress are compared to identify significant differencamong them.
Material Models and Solution Process
The simulations employ finite element analys~ADINA, version 7.3, Automatic Dynamic IncrementaNonlinear Analysis, Watertown, MA! and incorporate anonlinear, isotropic, hyperelastic model for the arterwall. Blood is treated as an incompressible, Newtonfluid ~an assumption that has repeatedly been showhold well for large arteries,4,28 where shear rates geneally exceed 100 s21! and the flow is assumed laminar.
The arterial solid response is modeled using the sdard Lagrangian formulation for large displacements alarge strains.2 An isotropic form of the strain energdensity function for the~nearly! incompressible arterywall is specified.9
W5a
b@e@b/2~ I 123!#21#, ~1!
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wherea andb (a544.25 kPa,b516.73) are elastic constants that reflect the elastic properties and tissue cposition; their values are obtained using a nonlineargression fit to pressure–diameter inflation tests ofcarotid artery.8,31 I 1 is the first invariant of the straintensor. In the Taylor series expansion,a has the signifi-cance of the elastic modulus. This exponential formappropriate for arterial mechanics since it portrayswell-known strain-stiffening behavior of collagen.
Continuity and the full Navier–Stokes equations asolved for the fluid. For fluid domains with movinboundaries, we utilize the arbitrary Lagrangian Euleri~ALE! formulation of the momentum equation.2 Fullcoupling between the fluid and solid domains requithat displacement continuity and force equilibrium asatisfied at the fluid–structure interface at each step. Tis done iteratively between the fluid and structure solvat each time step until sufficient convergence is reach
Fluid–structure interaction analysis is generally important when large scale motions of the arteriespresent and impact the flow field considerably, asstudies of arterial collapse when largely obstructistenoses are present3,10 or where physiologic movemenof blood vessels~e.g., coronary arteries! during the car-diac cycle affects blood flow patterns significantly.22 Inthe carotid bifurcation, even though the motions duearterial wall motion are relatively small, wall complianccan have a locally significant effect~up to 25% differ-ence in wall shear stress! at the divider wall27 and wasthus included in the simulations herein. The results psented focus on the fluid domain; the arterial wall stradistributions are presented elsewhere.34
Model Description and Boundary Conditions
An idealized but realistic three-dimensional~3D! solidmodel of the carotid bifurcation~extending from 3 cmdistal to 3 cm proximal to the bifurcation! based on thework of Bharadvajet al.6,7 and Delfinoet al.9 was cre-ated in SolidWorks~SolidWorks Corporation, ConcordMA ! and imported into the finite element software fmeshing and analysis. Only half of the geometry wmodeled and symmetry boundary conditions were eployed. The~solid! model used and boundary conditionimposed are illustrated in Fig. 1. The arterial wall wmeshed with an unstructured grid consisting of 43,3eleven-noded 3D tetrahedral elements and 118,nodes. The fluid domain was filled with an unstructurgrid consisting of 56,209 four-noded tetrahedral elemewith 11,597 nodes. The meshes used for the fluid asolid domains are shown in Fig. 2. Computations weperformed using a SGI Origin 2000 computer equippwith 4 processors and 6 GB of RAM.
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997Effects of Exercise on Hemodynamics in the Carotid Bifurcation
The fluid–structure simulations were performedtwo stages: An ‘‘inflation’’ stage~from 0 to 1 s! in whichthe fluid velocity vectors where ramped up from zerodiastolic values and within which simultaneouslyaxial stretch of 10%~as measuredin situ by Delfino8!was applied incrementally to the solid domain~and sub-sequently stretched the fluid domain!. This was followedby the ‘‘transient’’ stage, in which the results from th‘‘inflation’’ stage were used as initial conditions; the aterial wall stretch remained fixed at 10% and the fluvelocity pulsatile boundary conditions were applied.achieve a periodic solution, the transient stage flow coputation was allowed to run over two heart cycles, sing only the second one. The boundary conditions~flowrate and blood pressure profiles—see Fig. 3! to the tran-sient stage were obtained using the one-dimensional~1D!arterial model of Ozawaet al.26 This is a 32 segmentlinked piecewise 1D model of the entire arterial netwowhich has been shown to reproduce the flow and psure waveforms observed at various locations inbody.35 Using this model, two situations were simulatethought to bracket the behavior in the carotid bifurcatioIn the first, the heart rate was increased while reducsystemic vascular resistance to keep the mean artpressure nearly constant~13 kPa in the 72 bpm vs. 12kPa in the 108 bpm case!, thus causing cardiac output t
FIGURE 1. Idealized model of the carotid bifurcation used forfluid–structure interaction „FSI… simulations showing bound-ary conditions imposed on the arterial wall „see the text fordescription of fluid boundary conditions …. Due to symmetry,a half model of the carotid bifurcation was utilized. ‘‘P’’ rep-resents the inner surfaces, where the FSI boundary condi-tion was applied. The solid arrows represent an applied axialstrain of 10%. Comparisons in wall shear stress are con-ducted along ‘‘line 1’’ labeled above which spans the entirelength „along the normalized coordinate variable s… of thecommon-internal wall, including the carotid bulb.
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increase. Mean flow rate in the common carotid arteincreased by 31% from 12.5 ml/s~72 bpm! to 16.4 ml/s~108 bpm!. A second, somewhat more realistic represetation of exercise is achieved by following this samprocedure, but holding peripheral vascular resistancethe brain fixed on the assumption that during exercithe largest changes in peripheral resistance are confito the exercising muscle. Consequently, the primaryfects influencing flow conditions are the increase in herate and the change in flow and pressure waveform wthe mean flow is maintained constant.
The transient stage was subsequently solved inmore steps, similar to the commonly used procedure fiintroduced by Perktoldet al.28 To achieve arterial pulsatility, the pressure profiles shown in Fig. 3 were supimposed on the overall pressure drops across the domas computed in the inflation/stretch procedure. FollowPerktold et al.,28 a Womersley-type flow correspondinto the flow rate profile in the common carotid artery15
was first imposed node by node at the inlet to the comon carotid and a plug flow was specified at the exitthe internal carotid, while the external carotid was spefied as having zero normal traction. Flow profiles at toutput of the external carotid artery at the end of thstep were then used as a boundary condition to theond computational step of this ‘‘transient’’ stage, whiretaining Womersley flow at the common carotid aimposing a traction-free boundary condition at the intnal carotid. This second step was then consideredfinal computed solution.28 The entire procedure was repeated three times; first for 72 bpm and then for the tcases at 108 bpm.
FIGURE 2. Finite element meshes used for the solid „a… andfluid „b… domains.
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998 YOUNIS et al.
Wall shear stress, oscillatory shear index and mamum wall shear stress temporal gradients are then cputed for comparison between the two cases. ‘‘Size’’the recirculation region is defined as the normalizlength along the common-internal carotid adjoining w~line 1 of Fig. 1!, which exhibits reversed shear stressis normalized to the entire length of the common-intercarotid adjoining wall as simulated in the model, withe carotid bulb spanning the nondimensional lengths50.40– 0.73, wheres is measured along line 1. Thparameters denotes the normalized length along the li1, with s50 at the inlet to the common carotid artery
The wall shear stress components are calculated~using the Einstein summation convention!:
FIGURE 3. Pressure „top panel … and flow rate boundary con-ditions on the fluid domain representing two states: 72 bpm„middle panel … and 108 bpm „lower panel … as obtained fromthe 1D distributed arterial tree model of Ozawa et al. „Ref.26…. Labeled points for comparison are: accelerative „a…,peak flow „b… and decelerative „c… phase of systole as well asthe point of lowest flow rate „d….
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t i5Ti2~Tjnj !ni , ~2!
where Tj52meji ni is the net traction vector at a poinon the wall with the local surface normalnW and thesymmetric rate of strain tensorei j . Both the instanta-neous,tw and the temporal mean (tw)ave values of wallshear stress are used in the results.
Maximum tw temporal gradient (]tw /]t)max is themaximum value of the gradient in wall shear stress cculated between every two consecutive time points ding the cycle: (]tw /]t)max is defined as
maxU]itwi]t U,
where itwi is the magnitude of the local shear strevector.
The oscillatory shear index~OSI! is a nondimensionameasure that quantifies the fractional time a particuwall region in the cycle experiences cross or reveflow. The mean flow direction is defined by the tempomean of the shear stress vector. For purely oscillatflow, the OSI approaches 0.5.17,25 OSI is defined as
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wheretw* is the component of wall shear stress actinga direction opposite to the temporal mean value. Nthat the OSI does not take into account the magnitudethe shear stress vectors, only the directions.
RESULTS
Wall Shear Stress
Previous studies suggest that the hemodynamicrameter of greatest relevance to atherogenesis is the mnitude of the wall shear stress vector.18 This is presentedin the form of band plots of the wall shear stress avaged over the cycle (tw)ave for the 72 bpm~case 1! and108 bpm~cases 2 and 3 for the high and low flow raterespectively! solutions~Fig. 4!. Comparing first cases 1and 2, (tw)ave increases due to the higher flow ratranging up to 50% but generally in the range of 20%30% at the higher heart rate depending on location. Aof note is that the area of low (tw)ave(,1.0 Pa) shrinksconsiderably with increasing heart rate~108 bpm vs. 72bpm!. Overall, shear stress levels in the common caroare uniform and the increase in (tw)ave in the 108 bpmcase 2 over the 72 bpm case 1~;2.3 Pa vs. 1.8 Parespectively! almost directly reflects the 30% increaseaverage flow rate. When cranial resistance is held cstant, however, these differences largely disappear.
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999Effects of Exercise on Hemodynamics in the Carotid Bifurcation
band plots of (tw)ave for these two cases are virtuallindistinguishable@Figs. 4~a! and 4~c!#, despite the higherfrequency, and the corresponding increase in Womersnumber.
It is of particular interest to consider the outer wallthe carotid bifurcation from the common trunk to thinternal carotid~line 1 on Fig. 1!, which spans the carotid bulb, a commonly diseased site. The time poichosen for comparison are at peak acceleration~A!, peakflow ~B!, and peak deceleration~C!, as well as the timeof lowest flow rate~D!, which occurs at the flow ratethrough following the second peak@Fig. ~3b!#. Instanta-neous values oftw along line 1 at these several timpoints exhibit a general similarity despite the considable difference in frequency and, for case 2, flow ra~Fig. 5!. Flow separation is observed in the carotid bustarting approximately at time~B! and continuing until atime during flow acceleration prior to time~A!. Point ~D!is the time at which there exists the largest recirculatzone for all three cases, although at the higher frequethe region of reverse flow extends all the way to tentrance of the model. Values oftw in the commoncarotid generally reflect the relative flow rates, wicases 1 and 3 being most similar and case 2 somewgreater in magnitude. Values in the carotid bulb tend ato be similar, except that the time of maximum negatishear stress shifts from time~C! to time ~D! at the higherfrequency, cases 2 and 3. Times~A! and~C! were chosenso that they correspond to approximately the same flrate, but shear stress is markedly lower at all locatioand for all three cases during flow deceleration. Tablesummarizes the wall shear stress values and normallengths of the recirculation regions along this line.
The highest values oftw in the carotid bulb occurduring peak flow, but only at the ends of the bulb regias separation has already started near the upstreamIn the 72 bpm case, the maximumtw in the bulb overthe cycle occurs at the time of peak systolic flow rate~B!and is 2.9 Pa. This compares favorably with the 108 bcase, where it also occurs at time~B! and has a value o3.0 Pa. Comparing the values at similar points in tcycle between the two heart rates, we also see sodifference intw during the accelerative phase of systo~A! ~2.0, 2.5 and 1.8 Pa for cases 1, 2, and 3, resptively! that scale approximately with flow rate. Howeveboth at peak systole~B! and during the decelerativphase of systole~C!, virtually no difference is observedin tw as heart rate is increased along with flow rate~case2!, and roughly a 20%decreaseis observed as the hearate is increased while flow rate is maintained const~case 3!. The most significant difference is experiencat the point of lowest flow rate~D! wheretw increasesby 225% to 2.6 Pa in case 2 or by 137% to 1.8 Pacase 3.
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1000 YOUNIS et al.
FIGURE 5. Wall shear stress „WSSÄtw… along the wall of thecommon-internal carotid „along line 1 of Fig. 1 … at differenttime points „A,B,C, and D as identified in Fig. 3 …: 72 bpm „a…;108 bpm †„b… and „c…‡.
Even though the overall size of the recirculation rgion over the entire cycle is relatively unaffected bheart rate~see the OSI plots in Fig. 6!, there are somevariations during the cardiac cycle. In general, increasthe heart rate has mixed effects on the length ofrecirculation zone~see Table 1!. At the time points ofhighest ~B! and lowest~D! flow rate, the length of therecirculation zone increases by increasing heart rate fcase 1 to case 2. In contrast, at the point of peak fldeceleration~C!, increasing heart rate leads to a sligdecrease in the length of recirculation region. Increasheart rate at constant flow rate~i.e., from case 1 to case3!, decreases the length of recirculation zone excepthe lowest flow rate. At peak acceleration, the flow rmains attached to the walls in all three cases studiedtime point D of the 108 bpm runs, the flow profilcontains a region of reverse flow even at the entrancethe common carotid artery due to the fully developWomersley flow boundary condition associated with tsharp deceleration there.
In contrast, no flow separation was seen alongoutside wall of the external carotid~results not shown!.Curvature of the external carotid artery failed to produan adverse pressure gradient of sufficient strengthseparate the flow from the outside wall of the artery~thecommon-external adjoining wall!. Comparison of themaximum wall shear stress values in that region~at peaksystole! reveals an increase in peaktw levels from 14.9to 19.5 Pa with the increase in heart rate from case 1case 2, but a much smaller increase to 15.2 Pa in casThe same comparison during peak systole at the carapex reveals an increase from 15.8 Pa at 72 bpm to 2Pa at 108 bpm in case 2, but a slight decrease to 15.3in case 3. All of these changes correspond most closto changes in mean flow rate with the frequency exerta secondary influence.
Although not explicitly computed, it is possible tinfer approximate values for the spatial gradient intw
from the band plots of Fig. 4 and the instantaneodistributions of Fig. 5. Because there are no significdifferences in the overall pattern oftw , the spatial gra-dients would follow the same overall tendencies dcussed above fortw .
OSI and Wall Shear Stress Maximum Temporal Gradi
Two other parameters that have been investigatedthe oscillatory shear index~Fig. 6! as defined above18
and the maximum wall shear stress temporal grad(]tw /]t)max over the cycle~Fig. 7!. Nonzero values ofthe OSI in all three cases studied are confined toregion of the carotid bulb, with magnitudes and distribtion patterns that are notable for their similarity.
The magnitudes of (]tw /]t)max are considerablyhigher in the 108 bpm case with high flow rate~case 2!
1001Effects of Exercise on Hemodynamics in the Carotid Bifurcation
TABLE 1. Maximum wall shear stress „tw… in the carotid bulb and size of the recirculation region at different times during thecardiac cycle: peak acceleration „A…, peak flow „B…, and peak deceleration „C… as well as at the time of lowest flow rate „D… †see Fig.3„b…‡ for the case of a person at rest „72 bpm … and a person exercising moderately „108 bpm …. „À… signifies that the wall shearstress vector reversed direction. Length of the recirculation region is defined as the normalized length along the common-internalcarotid adjoining wall „line 1 of Fig. 1 … with reversed shear stress. The carotid bulb spans the region sÄ0.40– 0.73, where s is
normalized distance along line 1. Values in parentheses represent the range of s over which flow reversal occurs.
Timepoint incardiaccycle
72 bpm(case 1)
108 bpm(case 2)
108 bpm(case 3)
Max tw inbulb (Pa)
Recirculationlength
Max tw inbulb (Pa)
Recirculationlength
Max tw inbulb (Pa)
Recirculationlength
A 2.0 None 2.5 None 1.8 None
B 2.9 0.09 (0.4 to 0.49) 3.1 0.12 (0.38 to 0.52) 2. 0.06 (0.41 to 0.47)
C 1.8(2) 0.46 (0.32 to 0.78) 1.8(2) 0.44 (0.33 to 0.77) 1.4(2) 0.43 (0.33 to 0.76)
D 0.8(2) 0.45 (0.32 to 0.77) 2.6(2) 0.87 (0 to 0.87) 1.9(2) 0.87 (0 to 0.87)
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but contrary to the comparisons oftw these differencescannot be explained by the higher flow rates alone.the common carotid, (]tw /]t)max averages 140 Pa/s~72bpm! and increases to 215 and 160 Pa/s at 108 bcases 2 and 3, respectively. In the carotid bu(]tw /]t)max doubles in value from 35 to 70 and 60 Pawith the increase in heart rate at increased flow rate~case2! and fixed flow rate~case 3!, respectively. In contrast(]tw /]t)max at the region of curvature in the wall adjoining the common-external carotid reaches 285 P~72 bpm! and 430 and 410 Pa/s at 108 bpm for caseand 3, respectively. At the carotid apex, (]tw /]t)max isan order of magnitude greater than at the bulb areaches values of 600 Pa/s~72 bpm! and 800 Pa/s at 108bpm for both cases 2 and 3. Within the external carofurther down from the region of curvature, (]tw /]t)max
is seen to vary between 350 and 500 Pa/s~72 bpm!,550–700 Pa/s~108 bpm! in case 2, and 400–550 Pa/scase 3.
Pressure Drop
Pressure drops were compared at point~C! during thedecelerative phase of systole in all cases; this time sexperiences some of the largest adverse pressure gents throughout the cycle~results not shown!. A pressuredrop ~which can be explained largely on the basisconvective rather than temporal acceleration! of 1.0, 1.3,and 0.7 kPa was recorded at this time step in cases3, respectively, which is at most only 8% of the applisystolic pressure.
DISCUSSION
Most of the recent literature on the relationship btween hemodynamics and atherosclerosis point to wshear stress as the most important fluid mechanic pareter. It is now widely accepted16,18,29that low tw regions
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are more prone to atherosclerosis. Wall shear stressels greater than 1.5 Pa are generally believed to indendothelial quiescence and an atheroprotective genepression profile, whereas low shear stress levels~lessthan 1.0 Pa! stimulate an atherogenic phenotype andgenerally observed at atherosclerosis-prone sites.21 Thisis evident in ourtw results ~Figs. 4 and 5!. With theincrease in heart rate, our results differ, dependingwhether or not we allow flow rate to increase. Althougthe current study only examines the carotid bifurcatiothe two cases at elevated heart rate could be viewedrepresenting two physiologic limits, the first corresponing to arteries in regions that experience an increaseflow with exercise, and the second corresponding toregions that do not. In the former case,tw values aregenerally higher, nearly 50% higher in some locatiobut only in proportion to the increase in flow rate elswhere, and the lowtw region in the carotid bulb reducein size. This observation may provide some supportthe hypothesis that exercise promotes an atheroprotecphenotype.12,13
Our results for case 3, however, when flow rateheld constant, suggest a very different conclusion. Shstress magnitudes and distribution are virtually uchanged, and are even slightly higher in the carotid bat the lower heart rate, at least at times A and C. Tsimilarity between these two cases is strong and sowhat counterintuitive. Because of the high Womerslvalue at higher heart rates, one would expect tothinner boundary layers and, consequently, higher pshear stress values. Even these small differences dipear, however, if we only consider (tw)ave, which is asexpected because OSI maps~Fig. 6! show very little signof significant negative shear stress, and hence, Womley number should have no effect on mean wall shstress (tw)ave if the mean flow is unchanged.@It must benoted that mean wall shear stress (tw)ave is calculated
1002 YOUNIS et al.
FIGURE 6. Oscillatory shear index „OSI… plots: 72 bpm „a…;108 bpm †„b… and „c…‡.
FIGURE 7. Maximum wall shear stress temporal gradient„WSSTGmaxÄ„tw Õt…max… „in Pa Õs…: 72 bpm „a…; 108 bpm †„b…and „c…‡.
1003Effects of Exercise on Hemodynamics in the Carotid Bifurcation
FIGURE 8. Contour plots of velocity magnitude „in m Õs… at four time points in the heart cycle. Letters A–D refer to the timepoints indicated in Fig. 3. Left panel: 72 bpm „case 1 …; middle panel: 108 bpm „case 2 …; and right panel: 108 bpm „case 3 ….
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1004 YOUNIS et al.
using the absolute valueof the instantaneous sheastress.#
To further address the little variation in peak washear stress as the Womersly number changes, conplots of velocity magnitude are shown in Fig. 8. As tfigures demonstrate, the flow features are qualitativsimilar at the time points of interest. The most prnounced difference quantitatively is found at time D, i.the point of lowest flow rate, where case 3 showsconsiderably slower velocities. Interestingly, cases 13 are quite similar at time points A–C, despite the dferent waveform and different Womersley number.
This can be explained by the importance of mometum convective mixing and the resulting tendency of tflow to behave in a nearly quasi-steady manner desvalues of the Womersley number in the range of 5.5–based on the fundamental heart frequency. The mixdue to secondary flows, essentially increases the effecdiffusivity for momentum, and thereby reduces the effetive Womersly number. This is supported by the fact ththe only pronounced differences in velocity magnituare experienced at time point D, i.e., where the convtive mixing is less dominant. This may be also attributto the limited effect of Womersley number between vues of 5.5 and 7.5 based on the fundamental heartquency. That is, considering that the Stokes layer isversely proportional to Womersely number, a changeWomersley number from 5.5 to 7.5 would roughly mea 25% reduction in the Stokes layer in the unsepararegions. At these Womersely numbers, this reductdoes not exert appreciable changes in the velocity mnitude pattern between cases 1 and 3, particularlyhigher instantaneous flow rates, i.e., in systole~see Fig.8, panels A–C as compared to panel D!.
This explains the indistinguishable mean wall shepatterns between cases 1 and 3~see Fig. 4!. Of the twohigh heart rate cases considered~cases 2 and 3!, thelatter case is probably more relevant. It is believed tduring exercise the peripheral resistance in the brainmains fairly constant and instead higher flow rates tarthe oxygen needing muscles.5
The maximum wall shear stress temporal gradie(]tw /]t)max, has been related to the expressionatherogenesis-related genes in endothelial cells~ECs!.Baoet al.1 have found that the temporal gradient in shebut not steady shear stimulates the expression of mocyte chemoattractant protein-1~MCP-1!, a potent chemo-tactic agent for monocytes, and platelet-derived growfactor A ~PDGF-A!, a potent mitogen and chemotactagent for smooth muscle cells. The higher values(]tw /]t)max experienced during the 108 bpm simulatiosuggest, therefore, that exercise may lead to anopathway that works toincreaserather than decrease thatherogenicity of ECs. This, of course, must be consered in the context of the hypothesis that hemodyna
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forces generated by complex flow patterns can actboth positive and negative stimuli in atherogenesiseffects on endothelial cell gene expression.12 It is wellknown that endothelial cells in different regions of tharterial tree acquire both functional and dysfunctionphenotypes due to regional hemodynamics.13 This com-plex character of endothelial cells may contribute to tnotion that atherosclerosis is a multifactorial phenomeFinally, it must be remarked that the flows in the externcarotid are overestimated due to the lack of a tributthere, leading to high temporal gradients of wall shestress on the external carotid artery during exercise@seeFigs. 7~b! and 7~c!#.
Recent studies32,33 suggest that spatial shear stregradients may play a key role in the morphological rmodeling of the vascular endothelium. Tardyet al.32 ob-served that endothelial cell division increased in thecinity of flow separation whereas cell loss was elevaboth upstream and downstream in the regions whereshear gradient diminished. The current study suggethat exercise results in no significant impact on the ovall patterns oftw and hence the spatial gradients therein the carotid bulb, believed to be the early site athesclerosis.
These solutions also provide insight into the impotance of performing a coupled fluid and structure intaction ~FSI! solution. Since many codes lack this capbility, it is important to assess the need for this additioncomplexity. The maximum pressure drop along therotid bifurcation ~due to inertial effects! is ;8% of thesystolic pressure. This small change in the wall pressdistribution supports the notion that FSI analyses mnot be necessary in assessing the wall mechanical strwhen examining arteries with nonsevere stenoses. Wrespect to the fluid mechanical parameters, PerktoldRappitsch27 show that the significant differences in washear stress, of up to 25%, only appeared at the divwall between the distensible and nondistensible modThey observed no difference in wall shear stress betwthe two models at the primary region of atherogeninterest, the carotid bulb. Thus, in numerical studaimed at understanding atherogenesis in the carotid bFSI analyses may not be necessary. They may be nesary, however, in cases of severe disease~90% stenosesand above! because blood pressure will be higher, andwill the pressure drop across the stenosis, resultinglarger scale wall motion, even with the presence ostiffer diseased arterial wall.3
We believe the present model is a fair representatof the common characteristics observed in a normhealthy carotid bifurcation. It incorporates a fairly reaistic, though idealized, 3D geometry that replicates mgeometrical features in typical healthy carotid bifurctions. Further, the arterial wall mechanics involvesnonlinear model using Lagrangian formulation for lar
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1005Effects of Exercise on Hemodynamics in the Carotid Bifurcation
displacements and large strains. It incorporates a stenergy density function that portrays the well knowstrain-stiffening behavior of healthy arterial wall. Furthethe FSI formulation implemented in the present motakes into account the artery wall compliance, which chave locally significant hemodynamics effects. Therefothe model has the essential features that govern therotid bifurcation hemodynamics in normal healthy hman subjects. To investigate the impact of detailed atomical and morphological features that are specificdifferent human subjects would, however, requsubject-specific models based onin vivo images of thecarotid bifurcation.
Finally, it is important to note that exercise affects tcardiovascular system via a complex mechanism, givrise to a wide range of changes in the dynamics ofsystem. Here, we only focused on the carotid bifurcatand studied how a relatively narrow range of physiolocally conceivable changes in carotid artery blood flowould impact the atherogenically relevant hemodynampatterns experienced therein. A detailed analysis ofeffect of exercise requires a comprehensive model ofwhole cardiovascular network, incorporating various dnamical and neural pathways that play a role in tcomplex mechanism.20
ACKNOWLEDGMENTS
Support from the National Heart, Lung, and BlooInstitute ~Grant No. HL61794! is gratefully acknowl-edged. The authors gratefully acknowledge fruitful dcussions with Thomas Heldt. The first two autho~H.F.Y. and M.R.K.! equally contributed to this work.
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