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Pulsatile non-Newtonian blood flow in three-dimensional carotid bifurcation models: anumerical study of flow phenomena underdifferent bifurcation anglesK. Perktold, R.O. Peter, M. Resch and G. LangInstitute of Mathematics, Technical University Graz, Graz, AustriaReceived May IWI, accepted May I!WIABSTRACTFlow and stressatterns in human carotid artery brfircation models, which differ in the bifurcation angle, are analysednumerically under physiologically relevant flow conditions. The governing Navier-Stokes equations describingpulsatile.three-dimensionaljow of an incompressible non-Newtonian fluid are approximated using a pressure correction finiteelement method, which has been developed recently. The non-Newtonian behaviour of blood is modelled using C&onsrelation, based on measured dynamic viscosity. l%e study roncentrates on jlow and slresscharacteristics in the carotidsinus. The results show that the complex flow in the sinus is affected by the angle variation. The magnitude of reversedflozv, the extension of the recirculation zone in the outer sinus region and the duration of jlow separation during the pulsecycle as well as the resulting wall shear stress are clearly different in the small angle and in the large angle bifurcation.The haemodynamic phenomena, which are important in atherogenesis, are more pronounced in the large anglebifurcation.Keywords: carotid artery, bifurcation models, Navier-Stokes equations, numerical analysis

INTRODUCTIONAtherosclerotic alterations occur preferentially in thecarotid artery bifurcation and especially in the carotidsinus. From the fluid dynamic point of view the flowis strongly disturbed in this region. The influence ofdisturbed flow on atherogenesis has been studiedintensively over the last twenty years. Correlationsbetween fluid dynamics and the localized genesis andthe development of atherosclerosis are well known.The findings indicate that flow separation and recir-culation are important factors in the deposition ofplatelet thrombi and in the occurrence of earlyatherosclerotic lesions.To investigate the flow field and the stresses in thecarotid bifurcation considerable experimental- andtheoretical- research work has been performed.The fluid dynamics studies show that flow separationphenomena occur in the carotid sinus. This investiga-tion analyses the bifurcation angle dependency of theflow characteristics in the sinus during the pulse cycle.In the study four carotid bifurcation models, whichdiffer in the bifurcation angle, are analysed numeri-cally. The angle between common carotid axis andinternal axis was chosen from the range 1.5 to 50.The basic shape of the model, except the bifurcationCorrespondence and reprint requests to: Karl Perktold, Institute ofMathematics, Technical University Graz, Steyrergasse 30/3, A-8010Graz, Austria0 19!)1 Butterworth-Heinemann for BES011 I-5425/91/060507-O!)

angle, agrees with data from the literature-. Thecalculations have been carried out under thephysiological pulse waveform and flow division ratiointernal to external carotid published by Ku et al..The mean flow rate in the common carotid artery is5.1 mls-.The calculations are based on the pulsatile three-dimensional (SD) N avier-Stokes equations wherenon-Newtonian blood viscosity is assumed. The non-Newtonian behaviour of blood has been describedusing the Casson relation, to which experimentaldynamic viscosity data are fitted. The computersimulation of the pulsatile flow field and of the wallshear stress has been carried out using a recentlydeveloped pressure correction finite element proce-dure. The numerical method is described only brieflyhere. Our study concentrates on the presentation anddiscussion of the numerical results for flow velocity(axial and secondary velocity, flow separation, stasisand transient flow reversals) and wall shear stressdistribution during the pulse cycle.

GEOMETRICAL MODELS AND FLOWCONDITIONSThe four geometrical models considered in this studyare based on the model used by Ku et aL7 and byReneman et al.. The diameter of the common

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tD=62mm I _c

a b

CFigure 1 Four human carotid artery bifurcation models. a, CAR 1; b, CAR 2; C,CAR 3; d, CAR 4. The essential geometric data are indicated; A,B, C, D; E, F indicate flow cross-section levels where numerical results are displayed

carotid artery (D = 6.2mm), of the internal (Oi =0.70) and of the external (DeX = 0.590) as well asthe sinus shape (maximum sinus diameter D, =1.06D) correspond to their assumptions. The anglebetween common carotid axis and internal carotidaxis is 15 (CAR I), 25 (CAR 2), 40 (CAR 3) and 50(CAR 4); between common carotid and externalcarotid it is 25 (CAR 1, 2, 3) and 50 (CAR 4). In allcases the branching plane is the plane of symmetry.The four bifurcation models, including the essentialgeometrical data, are shown in Figure 7. The capitalletters, A, B, C, D, E, F, G, H, I indicate differentcross-section locations where numerical results arepresented. Cross-section A belongs to the commoncarotid; B, C, D, E, F, are located in the internalcarotid where level B is proximal to the carotid sinus,level D agrees with the maximum sinus diameter, andlevel F is distal to the internal carotid. G, H, I belongto the external carotid. The flow divider is slightlyrounded; the vessel wall is assumed to be rigid. Figure2 shows a three-dimensional representation of CAR 1where the inner, outer and side walls are defined.The calculations are carried out under unchangednon-Newtonian pulsatile flow conditions. Thecomplex rheological behaviour of blood has beenapproximated using Cassons relation on the basis of

experimental viscosity data corresponding to anoscillation of 2 Hz (ref. 17). The Casson relationexpresses the viscosity as a function of the shear strainInternalcarotidartery

Outer wall

Carotid 1 \lnner wall-Side wall

Figure 2 Three-dimensional representation of the small anglecarotid CAR I; definition of the inner, outer and side wal1

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Carot id low phenomena: K . Perktol d et al.

1O1 r Systole Oiastole

I I I I I10-l l oo 10' l o2 l o3+a

Figure 3 Dynamic viscosity of human blood, hemaocrit c = 43%;the experimental data measured in oscillatory flow (2H) (ref. 17) arefitted using Cassons relation

rate (and of hematocrit; here 43%). The shear strainrate dependent blood viscosity is shown in Figure 3.The pulse waveform in the common carotid and inthe internal carotid used in this investigation has beenpublished by Ku et al7 Figure 4 shows the flow ratepulse waveform in the common and in the internalcarotid. The mean flow rate in the common carotid isassumed to be 5.1 ml s-, and the corresponding meanflow velocity is 0;) = 16.9 cm s-. Using the commondiameter D = 6.2 mm and a representative kinematicreference viscosity Y = 0.035 cm2 s- in the defini-tion, the mean reference Reynolds number is Re =X)0 where Re = UJl/v. The assumed pulsefrequency is HO strokes min-. The correspondingreference Womersley number CY 4.8 where(Y = D/:!v%& w is angular frequency.EQUATIONSThe mathematical analysis was carried out using thetime-dependent non-linear Navier-Stokes equations.In this study blood is assumed to be an incompres-sible Casson fluid. The equation system can bewritten as

i l-.L = ()ilx, i,j= 1,2,3where ui, i = 1, 2, 3, are velocity components in theCartesian coordinate system Xi, i = I, 2, 3; p is pres-sure, p is constant fluid density and p is shear ratedependent dynamic viscosity.As shown by Perktold et al. Ix the viscosity ingeneral Casson flow can be expressed as,i = I(2 I.&) (k,,(c)+ k , (c) w

The experimental data are fitted using thehematocrit-dependent paramaters k,) = 0.6125 andk I = 0.174 whereby hematocrit c = 43%. D is thesecond invariant of the strain rate tensor. Equation 3describing the apparent viscosity fits blood data quitewell at shear rates higher than 1 s-l, From previouscarotid bifurcation flow studies it is known that theshear rates to be expected are higher than 1 s-.

O I I I I I / I I , , I0.20 0.40 0.60 0.80 1 ootitp --cFigure 4 Pulsatile velocity waveform in the common and internal

carotid artery according to Ku ef al.; tp denotes the time for one pulsecycle; mean common carotid flow 5.1 mls . -. common carotidflow; ---, internal carotid flow

The generalized Navier-Stokes system equations(l)-(3) can be solved for given boundary andinitial conditions. Here at the inflow boundary theWomersley velocity profiles are prescribed. Thesefully developed profiles were calculated as longstraight tube profiles correspondin8

to the velocitywaveform in the common carotid Figure 4). At therigid vessel wall the no slip condition is applied. Atthe outlet of the two branches time-dependent flowdivision ratio is prescribed.At the outflow boundary of the external carotid,which is assumed at a distance of eight times theexternal diameter from the divider, outflow velocityprofiles are calculated in a first step. During this stepfully developed flow is applied at the internal outletlocated downstream at a distance of 3.5 times theinternal diameter from the sinus end. The externaloutflow velocity profiles resulting from this step areused as boundary conditions in a second step wheretraction-free boundary conditions are assumed at theinternal outlet. The second step is the actual calcula-tion step. Studies to estimate the appropriate length ofthe two branches have been carried out.The numerical procedure recently developed isbased on the pressure correction technique and usesthe Galerkin finite element method and implicit finitedifferences for the time derivatives (1 ). Lo.The non-linear viscosity and the convection terms are linea-rized using Picard iteration. The finite elementdiscretization employs eight-node isoparametric brickelements with tri-linear velocity approximation andconstant pressure. The validation of the method in theNewtonian case has been carried out by Hilber? andby Perktold et al. where a comparison of curvedtube results with results by van de Vosse et al_hasbeen done. A comparison of non-Newtonian results isperformed in Perktold and Resch.The finite element subdivision of the computa-tional domain (symmetry is taken into account)creates 7228 elements. The total node number (threevelocity components and pressure) is 34498.

NUMERICAL RESULTSThe presentation of the results concentrates on theaxial and secondary velocity field, the recirculation

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t/to = 0 80

Figure 5 Axial velocity profiles in the branching plane duringsystolic deceleration l/lp = 0.14 and during diastolic phase l/tp = 0.X.a, CAR 1; b, CAR 2; c, CAR 3; d, CAR 4

zones and the wall shear stress distribution during thepulse cthe in flycle. The comparison of the results illustratesuence of the bifurcation angle on the flowparameters.Comparison of the flow velocityThe axial velocity profiles in the branching plane ofthe four models at different locations and at differentfractions of the pulse period t/ tp = 0.14 (systolicdeceleration phase), t/ tp = 0.8 (diastolic phase) areshown in Figure 5. In the common carotid, upstreamof the branching, the symmetric inflow profilesremain relatively unchanged. In the two branches,extremely skewed profiles with high velocity near theinner (divider) wall occur especially for high flowrates. At the outer wall the velocity decreases.During decelerated flow in the outer sinus regionflow separation and recirculation appear. Themagnitude of the reversed velocity and the extensionof the reversed flow area increase with increasingbifurcation angle. During the diastolic phase low axialvelocity in the outer sinus region can be observed; inthe case of CAR 4 the plot demonstrates diastolic flowstagnation near the outer sinus wall. Fi gure 6 shows athree-dimensional representation of the axial velocityprofiles in CAR 1 and CAR 4 during the systolicdeceleration phase at different flow cross-sections.The location of the cross-sections is indicated in

a

b D EFigure 6 Three-dimensional representation of axial velocity in thecommon carotid (level A) and in the internal carotid (levels B, D, E) ofthe small angle (CAR 1) and the large angle bifurcation (CAR 4)during systolic deceleration l/tp = 0.14. The location of the cross-sections is indicated in Figure 7. a, CAR I; b, CAR 4

Fi gure 1. The comparison illustrates a more complexflow field in the sinus (locations B, D, E) in the case ofCAR 4.Some results on secondary motion are displayed inFigure 7 and Table 1. The comparison of thesecondary velocity profiles in CAR 1 and CAR 4during systolic deceleration (t/ tp = 0.14) and duringthe diastolic phase ( t / p = 0.8) demonstrates anincrease of the secondary velocity with increasingbifurcation angle (Figure 7). The secondary motion


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Carortdjlow phenomena: K. Perktold et al.t/tp = 0.14

j Outerwallr/tr, = 0.80

r/tp = 0.80

t-ii>@@

a b 50cms- - ,Figure 7 Secondary flow velocity at the specified flow cross-sections (A. common carotid; B, C. D, E, E, internal carotid) in CAR I and CAR 4during systolic deceleration f/tp = 0.14 and during diastolic phase f/t/~ = 0.8. a, CAR 1; b, CAR 2

results from the branching and the curvature effect.At the location proximal sinus (B) the branchingeffect is dominant. Table 1 shows the magnitude ofmaximum secondary velocity at the location ofmaximum sinus diameter D during the pulse cycle.During systolic acceleration, at maximum flow rateand during diastolic phase the maximum secondaryvelocity approximately doubles from the small angle(CAR 1) to the large an le bifurcation (CAR 4).During decelerated flow t/ tp = 0.14) only minordifferences occur. This behaviour can also be seen inFigure 7.Table 1 Magnitude of maximum secondary velocity at the locationof maximum sinus diameter D

Maximum magnitude of secondary velocity in cm s location D

t/t/l CAR 1 CAR 2 CAR 3 CAR 4o.oF, 2.7 3.6 5.0 5.90.10 !?.!I 12.8 16.6 18.30.14 16.1 17.3 17.0 15.70.2kI.27 6.2.1 6.3.2 7.3.5 7.17.50.36 4.2 5..5 6.8 7.40.46 3.5 4.8 6.2 7.00.80 2.4 3.3 4.4 5.1

Additional information on the secondary motionis gained from the mean axial vorticity2,, which isdefined as [,= l /A$siidY where A = nd2/8 isthe area surrounded by the path S along the tubediameter d in the symmetry plane and the semicircle.Figure 8 shows the mean axial vorticity in the internal

C E c E C E C Ea b c dFigure 8 Mean axial vorticity in the internal carotid from thelocation proximal sinus (B) to the distal internal level F at l/lp = 0.14(top) and at t/tp = 0.8 (bottom) for CAR l-4 (a-d, respt)

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Carotidflow phenomena: K. Perktold et al.carotid from location B (proximal sinus) to location F(distal internal) during systolic deceleration t/ tp =0.14 and during the diastolic phase t/tp = 0.8. At theproximal sinus the branching effect is dominant, andhigh axial vorticity occurs in all models considered.Along the sinus the vorticity decreases and reaches aminimum at location E. This minimum is caused bythe cross-section reduction at the downstream end ofthe sinus together with the high axial velocity in theregion of the inner sinus wall. The resulting effect iscontrary to the secondary motion effect from thecurvature and the branching. For increasing bifurca-tion angle an increase of the axial vorticity isapparent.Comparison of reversed flow zones in the carotidsinusThe analysis of the bifurcation angle-dependentlocation and extension of flow separation zones, andof the duration of reversal flow during the pulse cycleare crucial points in the present study. In these zones,relatively low flow velocity and velocity gradientsand consequently low wall shear stresses occur. These

0.32 0.1 0.28 0.26 0.24

0.24 \

0.46-l .O 3

b VFigure 9 Zonesof reversed flow at the branching plane during thepulse cycle at discrete pulse phases (A(Ulp) = 0.02). The lines are zeroaxial velocity levels. a, CAR 1; b, CAR 4

facts are of essential importance in the deposition ofplatelet thrombi and in the occurrence of earlyatherosclerotic lesions.The zones of reversed axial flow in the plane ofsymmetry of the models CAR 1 and CAR 4 duringthe pulse cycle are plotted in Figure9. The lines in theflow field are zero axial velocity contour linescorresponding to different pulse phases. Major flowseparation and recirculation occur during systolicdeceleration. The plots show that flow separation atthe outer sinus wall first appears at the locationproximal sinus (level B) during late systolic accelera-tion. In the case of CAR 1 the beginning of the flowseparation is detected at t/tp = 0.1; with an increas-ing angle, separation can be seen somewhat earlier.In CAR 4 the beginning is at t/tp = 0.06. Duringthe systolic phase the zone of reversed axial velocitygrows and reaches its maximum extension in thesecond half of the systolic deceleration phase,approximately at t/ tp = 0.16, 0.18. Subsequently therecirculation zone decreases and disappears approxi-mately at t/ tp = 0.28 in CAR 1 and t/ tp = 0.26 inCAR 4. The location of the disappearance is shifteddownstream up to the maximum sinus diameter level.In all cases at the proximal sinus location, minorseparation reoccurs during the end of the systolicphase. In the figure it can be seen that there are onlyminor differences in the duration of the systolic flowseparation, but in CAR 4 additional separation is

a

bFigure 10 Zones of reversed flow (hatched areas) at specified flowcross-sections (A, common carotid, B, C, D, E, F, internal carotid) inCarotid 1 and Carotid 4 during systolic deceleration t/tp = 0.14. a,CARl;b,CAR4


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Carotidj l ow phenomena: K . Perkt old et al.

B, D). The maximum negative shear stress is2.6 Nm-2 and a?

pears at the location of maximumsinus diameter D) in CAR 4. Negative wall shearstress indicates reversed flow at the specified walllocations B, D. From the shear stress distribution itcan be detected that the duration of systolic flowseparation at the outer wall locations B and D of CAR1 and CAR 4 are only slightly different. However, inCAR 4 the flow separation begins somewhat earlier

detected during the diastolic phase from t/p = 0.46to the end of the pulse cycle. A further difference canbe observed in the location of the separation zones. Inmodel CAR 1 flow separation occurs already at theterminal end of the common carotid, while in modelCAR 4 the separation zone develops at the beginningof the internal carotid artery. At the outer wall of theexternal carotid in CAR 4 flow separation is alsoobserved where the maximum extension appearsagain at the systolic deceleration phase t/ t$t = 0.16.Figure 70 illustrates the cross-sectional extension offlow separation in the sinus of CAR 1 and CAR 4 atsystolic deceleration t/ tp = 0.14. The areas ofreversed axial velocity are hatched. The plots showthe different shape of the cross-sectional separationareas. With increasing bifurcation angle the shape ofthe zero axial velocity contour lines indicates thedevelopment of so-called C-shaped axial velocitydistribution, where the area of reversed flow isrelatively concentrated in the vessel centre. This C-shaped axial velocity distribution occurs because ofthe high axial velocity near the inner sinus wall anddue to the movement of the high velocity particlestowards the outer wall by circumferential secondarymotion.

Comparison of wall shear stressThe wall shear stress at different locations at the innerand the outer wall (A, B, D, F) in the cases of CAR 1and CAR 4, as well as at the side wall (B,D) in CAR 1,2, 3, 4 during the pulse cycle is displayed in Figures7 l- 73. The sign corresponds to the tangentialvelocity direction near the wall.

Figure 7 7 shows the difference in the shear stressdistribution at the inner and at the outer wall. Shearstress is significantly higher at the inner wall; thesystolic maximum is 14 N rnd2 and can be seen nearthe flow divider tip (level B) in CAR 1. While theshear stress at the inner wall of the sinus is maximum,negative shear stress occurs at the outer wall (location,

6 CA

-0 ok-0 t//p/tpFigure 11 Shear stress at the outer and at the inner wall in the small angle bifurcation Carotid 1and in the large angle bifurcation Carotid 4; a,level A: common carotid; b, level B: proximal carotid sinus; C, level D: maximum carotid sinus diameter; d, level F: distal internal carotid (levellocations shown in F&W 7). II I , CAR I; A, CAR 4 (open and solid symbols represent inner and outer wall respectively)

- 4YEz: 2

al , Ia

_ 47EZ2 2

-I I I I 1

b t/tp 1 .oFigure 12 Wall shear stress magmtude at the side wall of the carotidsmus (levels R and D) during the pulse cycle. n, CAR I; 13, CAK 2;+. CAR 3; A. CAR 4. a, B (side); b, D (side)

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Caroti djow phenomena: K. Perkt old et al.in the pulse cycle. The diastolic outer sinus wall shearstress is low and nearly constant.In contrast to the behaviour of the other threemodels used, in CAR 4 negative diastolic wall shearstress occurs at the location proximal sinus (B). Thisimplies flow separation during the diastolic phase.The wall shear stress results indicating the durationof flow separation agree with the results shown inFi gure 9.The magnitude of the shear stress vector at the sidewall locations B and D is displayed in Figure 72.During the entire pulse cycle the highest side-wallshear stress occurs in the case of CAR 4. At location Bthe systolic peak shear stress is about four timeshigher, and during the diastolic phase about twice ashigh, when compared with CAR 1. Figure 73 illus-trates the angle of the shear stress vector at the sidewall (locations B, D) with respect to the internal axis.The maximum angle is about 9.5 and occurs in CAR1 during the systolic peak flow. During the diastolicphase the maximum angle of the side wall shear stressvector appears in CAR 4.

60BE8Yca 30

I I I I Ia 01

011.0b t/tp

Figure 13 Variation of the direction of the wall shear stress vector atthe carotid sinus side wall (levels B and D) during the pulse cycle; thedirection 0 = 0 agrees with the direction of the internal axis. 0,CAR 1; 0, CAR 2; 0, CAR 3; L?, CAR 4. a, B (side); b, D (side)

CONCLUSIONDetailed analysis of the flow dynamic characteristicswere the basis for the investigation of the role ofhaemodynamic phenomena in atherogenesis. In ourstudy four human carotid bifurcation models, whichdiffer in the bifurcation angle, were analysed numeri-cally under physiologically relevant pulsatile flowconditions. In the investigation the axial andsecondary velocity, the extension and location of theflow recirculation zones, and the duration of flowseparation in the carotid sinus, and the shear stressdistribution at specified locations at the inner, outerand side wall are calculated. In a previous investiga-tion concerning the geometric factor in atherogenesisflow studies have been carried out for carotidbifurcation models with different sinus shape where-by idealized constant flow division ratio was assumed.The parameter study showed that the fluid dyna-mic variables in the sinus are significantly affected bythe angle between common carotid axis and internalcarotid axis. With increasing bifurcation angle fromCAR 1 to CAR 4 increased reversed flow andenhanced flow recirculation in the outer sinus regionoccur. In the case of the large angle bifurcation CAR4 at location proximal sinus flow separation accom-panied with low negative wall shear stress was foundover the major part of the pulse cycle, while in thesmall angle bifurcation CAR 1 flow separation occursonly during the systolic phase. In CAR 1 flowseparation was found at the terminal end of thecommon carotid; in CAR 4 separation develops atthe beginning of the internal carotid.From the clinical point of view it is known thatlarge angle bifurcations have unfavourable character-istics in the development of atherosclerosis?. In theliterature it is documented that flow separation andrecirculation and the resulting low wall shear stressfavour the development of atheroscleroticlesions 4,7.27-2.haemodynamic The present study supports thetheory of atherogenesis, that thespecific occurrence of this disease may reflect theindividual variations in the vascular geometry.One aspect, which confirms the crucial importanceof flow separation and recirculation and of low wallshear stresses in atherogenesis, is described byDeSyo. It is reported that in higher level carotidbifurcations (larger angles) atherosclerotic plaques aretypically located a short distance downstream fromthe starting point of the internal carotid artery, whilein lower level bifurcations (smaller angles) typicalatherosclerotic processes can be observed at theterminal end of the common carotid artery and at thevery beginning of the internal carotid artery. Thisbifurcation angle-dependent location of deposits andstenosis, reflects the variation in the position of thezones of flow separation as analysed in this study.

ACKNOWLEDGEMENTThis investigation is supported by the AustrianResearch Fund (Fonds zur. Forderung der wissens-chaftlichen Forschung in Osterreich), Projekt-Nr. P7726 PHY, Vienna, Austria.


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IX. Perktold K, Peter R and Resch M. Pulsatile non-Newtonian blood flow simulation through a bifurcationwith an aneurysm. Biorheology I!%!); 26: 101 l-30.

l!). Perktold K. On numerical simulation of three-dimensional physiological flow problems. Ber Math StatSektion, ForschungsgesJoanneurn, Graz, 1!)87; 280.

20. Hilbert D. An efficient Navier-Stokes solver and itsapplication to fluid flow in elastic tubes. In: Greenspan D,Rozsa P, eds. Numerical Methods, Coil. Sot. J Bolyai 50,Amsterdam: North Holland, l!*X7; -4X1-~11.21. Hilbert D. Ein Finite Elementr Aufspaltungsve~ahren ournumerischen LDsung der Navier-Stokes Gleichungen und seineAnwendung auf die Stramung in Rohren mit elastischen Wanden.(Thesis]. Technical University Graz, Austria, 1087.

22. Perktold K, Nerem RM, Peter RO. A numerical calcula-tion of flow in a curved tube model of the left maincoronary artery. J Biomechanics l!l!l I ; 24: 175-X9.Xi. Van de Voss FN, van Steenhoven AA, Segal A,JanssenJD. A finite element analysis of the steady laminarentrance flow in a !I() curved tube. Int J Num Meth Fluia!sl!lXl; 9: L7.5-x7.

2-C. Perktold K, Resch M. Numerische Losung der Navier-Stokes Gleichungen fur verallgemeinerte NewtonscheFluide. ZAMX4 - Z angew Math Mech 1!)91; 71: T-llti-18.

2.5. Olson DE, Snyder B. The upstream scale of flowdevelopment in curved circular pipes. ./ Fluid Mech 1985;150: 1X-58.

26. DeSyo D. Radiogammetric analysis of carotid bifur-cation: Hemodynamic-atherogenetic repercussionson surgical practice. Proc. 2nd Int. Symp. BiofluidMechanics and Biorheology, Munich l!l#); 71-2.

27. Caro CG, Fitz-Gerald JM, Schroter KC. Atheroma andarterial wall shear: Observation, correlation and proposalof a shear dependent mass transfer mechanism of ather-ogenesis. Pro< Roy Sttc Land 1117 ; 177: IO!)-59

28. Friedman MH, Deters OJ, Mark FF, Bargeson CB,Hutchins GM. Arterial geometry affects hemodynamics -a potential risk factor for atherosclerosis. Atherosclerosis1083; 46: 22.5-31.

Y!). Mark FF, Bargeson CB, Deters OJ and Friedman MH.Variations in geometry and shear distribution in castsof human aortic bifurcations. J Biomech 1989; 22: 577-x2.

REFERENCESI. Bharadvaj BK, Mabon RF, Giddens DP. Steady flow in a

model of the human carotid bifurcation. Part I - Flowvisualization. JBiomechanics 1982; 15: 34!)-62.

L. Bharadvaj BK, Mabon RF, Giddens DP. Steady flow in amodel of the human carotid bifurcation. Part II - LaserDoppler anemometer measurements. J Biomechanics 19X2;15: 363-78.

3. Ku DN, Giddens DP. Pulsatile flow in a model carotidbifurcation. Arteriosclerosis 1983; 3: 3 I-9.

.4. Zarins CK, Giddens DP, Bharadvaj BK. Sottiura VS,Mabon RF, Glagov S. Carotid bifurcation atherosclerosis.Circ Res 1983; 53: 502-14.

5. Motomaiya M, Karino T. Flow patterns in the humancarotid artery bifurcation. Stroke 19X-l; 15: 50-X.

6. Reneman RS, van Merode T, Hide P, Hoeks APG. Flowvelocity patterns in and distensibility of the carotid arterybulb in subjects of various ages. Circulation l!lH5; 71: .500-!I.

7. Ku DN. Giddens DP, Zarins CK, Glagov S. Pulsatile flowand atherosclerosis in the human carotid bifurcation.Arteriosclerosis 198.5;5: 293-302.

8. Ku DN, Giddens DP. Laser Doppler anemometermeasurements of pulsatile flow in a model carotid bifurca-tion. J Biomerhanics l!)X7; 20: 407-2 1.!I. Perktold K, Hilbert D. Numerical simulation of pulsatileflow in a carotid bifurcation model. J Biomed Eng l!lX6; 8:I!&!).

10. Rindt CCM. Analysis of the three-dimensionaljowfield i n thecarotid artery bifurcation. IPhD Thesis]. Eindhoven Univer-sity of Technoiogv 19X!).I I. Rindt CCM, van.Steenhoven AA, Janssen JD, RenemanKS, Segal A. A numerical analysis of steady flow in a 3D-model of the carotid artery bifurcation. J Biomechanirs1lOO. 23. 461-73.\ , .12. Keuderink PJ, Willems PJB, Schreurs PJG and vanSteenhoven AA. Fluid flow through distensible models ofthe carotid artery bifurcation. Proc 2nd Int Symp BiofluidMechanics and Biorheology, Munich l!jH!l; 455-62.

1:-i. Van de Vosse FN, van Steenhoven AA, Janssen JD andReneman KS. A two-dimensional numerical analysis ofunsteady flow in the carotid artery bifurcation. Biorheologyl!l!)O; 27: lti%H!).

14. Perktold K, Resch M. Numerical flow studies in humancarotid bifurcations: Basic discussion of the geometricfactor in atherogenesis. J Biomed Eng l!l!lO; 12: 11 l-23.

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