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Hemodynamics in the carotid artery bifurcation:
Journal of Biomechanics 33 (2000) 137}144
Hemodynamics in the carotid artery bifurcation: a comparison between numerical simulations and in vitro MRI measurements ReneH Botnar!, Gerhard Rappitsch", Markus Beat Scheidegger!, Dieter Liepsch#, Karl Perktold", Peter Boesiger!,* !Institute of Biomedical Engineering, University of Zurich and Swiss Federal Institute of Technology, Gloriastrasse 35, CH-8092 Zurich, Switzerland "Institute of Mathematics, Technical University Graz, A-8010 Graz, Austria #Fachhochschule Mu( nchen, D-80335 Mu( nchen, Germany Received 1 June 1998; accepted 9 August 1999
Abstract The presence of atherosclerotic plaques has been shown to be closely related to the vessel geometry. Studies on postmortem human arteries and on the experimental animal show positive correlation between the presence of plaque thickness and low shear stress, departure of unidirectional #ow and regions of #ow separation and recirculation. Numerical simulations of arterial blood #ow and direct blood #ow velocity measurements by magnetic resonance imaging (MRI) are two approaches for the assessment of arterial blood #ow patterns. In order to verify that both approaches give equivalent results magnetic resonance velocity data measured in a compliant anatomical carotid bifurcation model were compared to the results of numerical simulations performed for a corresponding computational vessel model. Cross sectional axial velocity pro"les were calculated and measured for the midsinus and endsinus internal carotid artery. At both locations a skewed velocity pro"le with slow velocities at the outer vessel wall, medium velocities at the side walls and high velocities at the #ow divider (inner) wall were observed. Qualitative comparison of the axial velocity patterns revealed no signi"cant di!erences between simulations and in vitro measurements. Even quantitative di!erences such as for axial peak #ow velocities were less than 10%. Secondary #ow patterns revealed some minor di!erences concerning the form of the vortices but maximum circumferential velocities were in the same range for both methods. ( 2000 Elsevier Science Ltd. All rights reserved. Keywords: Atherosclerosis; MRI #ow measurement; Numerical #ow simulations; Carotid bifurcation; Secondary #ow pattern
1. Introduction The presence of atherosclerotic plaques has been shown to be closely related to the vessel geometry (Nerem et al., 1980; Friedman et al., 1983; Caro, 1994). The carotid bifurcation, the coronary arteries, the infrarenal aorta and the vessels supplying the lower extremities may be markedly diseased while other vessels are rather spared. Due to these "ndings local hemodynamics is considered to play an important role in the initiation and development of atherosclerosis. Studies
* Corresponding author. Tel.: #41-1-632-4581; fax: #41-1-6321193. E-mail address: [email protected] (P. Boesiger)
on postmortem human arteries (Zarins et al., 1983; Ku et al., 1985) and on the experimental animal (Sawchuk et al., 1994) showed positive correlation between plaque thickness and the presence of low shear stress, departure of unidirectional #ow and regions of #ow separation and recirculation. Comparison between #ow velocity measurements and plaque distributions in human carotid artery specimens revealed that at the outer wall of the carotid sinus, where wall shear stress and #ow velocities are low, plaque thickness is maximal whereas at the #ow divider wall, where wall shear stress and velocities are high, plaque thickness is minimal (Zarins et al., 1983). At the side walls, where circumferential #ow velocities (secondary #ow) are present plaque thickness is medium. Various #ow quanti"cation techniques such as laser Doppler anemometry (LDA) (Liepsch et al., 1984; Ku
0021-9290/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 9 9 ) 0 0 1 6 4 - 5
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R. Botnar et al. / Journal of Biomechanics 33 (2000) 137}144
et al., 1987; Steiger, 1990), ultrasound Doppler (US) (Hatle et al., 1985; Steinke et al., 1989; Sugawara et al., 1989) or magnetic resonance imaging (MRI) (Nayler et al., 1986; Boesiger et al., 1992; Kilner et al., 1993) are commonly used to study arterial hemodynamics. The choice of one of these techniques depends on the speci"c application and the demand on spatial and temporal resolution. An alternative and complementary approach to study arterial hemodynamics uses numerical simulations (Perktold et al., 1995a,b,c,1998; Reuderink, 1991; Rindt et al., 1990). The development of e$cient numerical methods and new computer generations allows the calculation of local #ow behavior under physiologic and anatomically realistic conditions. This approach may be very useful for detailed quantitative studies of the in#uence of various local vessel geometries on speci"c vessel pathologies. The aim of this study was to verify that numerical simulations and #ow velocity measurements such as MRI lead to same #ow patterns if the computational geometric vessel model is an exact copy of the experimental bifurcation and if the same #ow conditions are used.
2. Material and methods 2.1. Mathematical model The local #uid dynamics was described using the three-dimensional time-dependent Navier}Stokes equations for incompressible Newtonian #uids: Lu 1 #(u ' +)u!l*u# +p"0, Lt o
(1)
+ ' u"0
(2)
with the velocity vector "eld u"(u, v, w)T and the pressure p.l"k/o represents the kinematic viscosity, o and k stand for the constant #uid density and the dynamic viscosity, respectively. For the numerical solution of the #ow problem the "nite element method was applied: The approximation applies eight-node isoparametric brick elements with trilinear interpolation functions for the velocity vector "eld and constant pressure in each element. The equations are solved using the "nite element Galerkin method with implicit Euler backward di!erences for the time derivatives and Picard iteration for the non-linear convection terms. The solution of the time-dependent Navier}Stokes equations is performed applying a recently developed velocity}pressure correction method, in which the occurring variables (velocity components and pressure) are uncoupled (Perktold et al., 1994). In each time (or iteration) step of the solution procedure this method takes the following steps:
1. Calculation of an auxiliary velocity "eld from the equations of motion using the known velocity and pressure values from the previous time (or iteration) step. 2. Calculation of the pressure correction from a pressure Poisson equation using the concentrated mass matrix. 3. Calculation of the divergence-free velocity "eld. 4. Updating the pressure. Flow velocity patterns were calculated for an anatomically realistic carotid artery bifurcation model. The computational geometric model was generated from an optically digitized anatomically correct cast of a postmortem carotid bifurcation (Liepsch et al., 1984,1992) from which the bifurcation replica was produced. For generation of the model for the mathematical simulation a recently developed "nite element grid generator (Perktold et al., 1995b) was applied. The procedure consists of two steps: the generation of the model surface using weighted least-squares splines and the discretization of the inner grid by a local optimization algorithm of geometric grid properties (Fig. 1). For an appropriate computational representation the vessel surface was described by 12.712 "nite elements. The #ow domain was described by 50.568 "nite elements. The total node number of the three velocity components and the pressure was 223.333. The kinematic viscosity l"0.037 cm2/s of the #uid was chosen the same as in the experimental study which is in the physiological range of blood. In#ow velocity pro"les were taken from MRI measurements. 2.2. Bifurcation model Measurements of pulsatile #ow velocity were performed in an anatomically realistic replica of a human carotid artery bifurcation (true-to-scale elastic model) made from silicon rubber. It is casted in a box "lled with 1.5% agarose gel (Fig. 2). By this casting procedure a model with nearly rigid walls arises. To mimic the rheologic properties of blood a water}glycerin mixture with a kinematic viscosity of l"0.037 cm2/s was used.
Fig. 1. Computational geometry of the bifurcation model.
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139
Fig. 2. MRI compatible #ow model circuit: The input wave form is produced by a computer controlled step motor driven pump. Long-term stability was controlled by an electromagnetic #ow meter. For synchronization of he MRI measurement with the #ow pulsatility every pump cycle a trigger signal was transmitted to the MR acquisition unit via an optical "ber.
The time-averaged mean #ow rate was 9.4 ml/s corresponding to a Reynolds number of Re"488. The mean vessel diameters of the examined cross sections of the common, midsinus and endsinus internal and external carotid artery were (A) 6.6 (B) 6.2 (C) 5.3 and (D) 4.8 mm (Fig. 3) with a #ow ratio of 0.5 for both daughter tubes (Fig. 4). The length of the sinus was 7 mm and the bifurcation angle approximately 453. Fig. 3. Schematic drawing of the compliant and anatomically realistic bifurcation model. The location of interest is the carotid sinus (B, C), whose #ow patterns are highly complex. The temporal and spatial #ow behavior in the common carotid artery segment (A) and the #ow ratio between common and the two daughter tubes (internal (B,C), external (D)) were used as input data for the numerical simulations.
The experiment was carried out under non-stationary #ow conditions with a mid-systolic #ow rate of 28 ml/s (the corresponding maximum Reynolds number Re"1454) in the common carotid artery segment. A programmable step motor driven pump allowed to reproduce physiologic #ow conditions. The #ow direction of the #uid was controlled by two mechanical valves. During the dead time of the pulse cycle the #uid was sucked from a water reservoir into the piston and pumped towards the bifurcation during the working cycle. To guarantee fully developed #ow pro"les the pump was connected to the bifurcation with a 4.5 m long rigid Plexiglas tube (/"10 mm). Fluid leaving the daughter tubes #owed into a normal pressure over#ow tank, which was connected with a 5 m long Plexiglas tube to the water reservoir of the pump. The #ow pulse rate was 60 cycles per minute. Long-term stability of the produced pulse wave was monitored during the experiment with an electromagnetic #ow meter (Tecmag II, Endress#Hausser, Reinach, Switzerland) which was installed close to the pump. The sampling rate was 25 Hz.
2.3. MR-yow velocity measurements Cross sectional velocity measurements of the axial and circumferential velocity component were performed at the common, midsinus and endsinus internal and at the external carotid artery on a 1.5 T Philips Gyroscan ACS II (Philips Medical Systems, Best, Netherlands) MRI-scanner using a prospectively triggered phase contrast sequence. Synchronization of the velocity measurement with respect to the periodic #ow cycle was achieved by sending a periodic 5 V TTL trigger signal to the MR acquisition unit. To obtain a high signal-to-noise (SNR) level a 7 cm diameter surface coil was placed directly above the bifurcation. With a FOV of 140*140 mm2 and an image matrix of 256*256 the spatial resolution was 0.5*0.5 mm2. To minimize intra voxel velocity gradients the slice thickness was chosen to be 4 mm. Numerically calculated velocity maps presented in this paper therefore are always averaged over a 4 mm thick slice. Flow voids, geometric image artifacts and #ow quanti"cation errors which may occur under complex #ow conditions were reduced by shortening the echo time using a partial echo acquisition scheme (62.5% of the full echo) (Botnar et al., 1996). Minimum echo time was 8.9 ms, time resolution 25 ms and #ip angle 153. The missing samples were zero padded. Velocity encoding was adapted to the expected velocity range and was
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Fig. 4. Flow rates measured in the common carotid artery, at the mid- and endsinus position of the internal and in the external carotid artery. Four systolic time points (acceleration: t/tp"0.075, peak #ow: t/tp"0.15, deceleration: t/tp"0.25, and minimum #ow: t/tp"0.325) are marked for which the axial and circumferential velocity patterns were measured and calculated, respectively.
120 cm/s for through plane and 50 cm/s for in-plane velocities. 2.4. Velocity data analysis Data analysis was performed on a DEC-Alpha (Digital Equipment Corporation, USA) workstation with a #ow analysis software package written in PV-Wave (Visual Numerics, Boulder CO, USA) which has been developed at the Institute of Biomedical Engineering in Zurich. Vessel contours were drawn using movable and deformable ellipses. The contour identi"cation could either be done on the modulus or the velocity (phase) images of each heart phase. The obtained contours were copied to the corresponding modulus or velocity (phase) images. After vessel wall segmentation axial and circumferential velocity maps were either visualized as 3D-mesh plots or as vector plots to allow a proper comparison with the results of the numerical simulations.
3. Results As expected #ow in the carotid sinus proofed to be highly complex. Axial and circumferential (secondary) #ow velocities were measured and calculated at the midsinus (C) and the endsinus (D) level of the internal carotid bifurcation (Fig. 3) during early systole (t/tp"0.075), mid-systole (t/tp"0.15), late systole (t/tp"0.25) and
end-systole (t/tp"0.325). High spatial and temporal #ow variations were observed for both approaches leading to high mainly axial directed #ow velocities at the #ow divider wall and small but still antegrade velocities at the opposite outer wall. In early systole (acceleration) the axial #ow pro"le is mainly axial symmetric but rapidly changes to a highly skewed velocity pro"le during midsystole which can be seen until early diastole. The same temporal behavior was observed for the secondary #ow pattern. It starts to develop at mid-systole and stays visible until early diastole. Fig. 4 shows the #ow rates measured in the common, the midsinus, the endsinus and the external carotid artery. Early systole, mid-systole, late systole and end-systole are marked with respect two the #ow cycle. 3.1. Axial yow A 3D representation of the axial velocity patterns reveals for both methods a skewed velocity pro"le with slow velocities at the outer and high velocities at the #ow divider (inner) wall (Figs. 5 and 6). The overall correspondence of both approaches is quite convincing for the axial #ow pattern. Despite slight variations the peak velocity curves do not di!er more than 10%. Even though the branching of the bifurcation is not exactly planar a high symmetry of the axial velocity "eld with respect to the both side walls can be observed.
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Fig. 5. Axial #ow pro"les at the midsinus position B in the internal carotid artery during early systolic acceleration (t/tp"0.075), mid systolic peak #ow (t/tp"0.15), late systolic deceleration (t/tp"0.25) and end systolic #ow reversal (t/tp"0.325). a) Numerical simulations. (b) MRI measurements. The #ow pro"les are highly skewed. With both approaches small velocities were observed at the outer, medium velocities at the side walls and high velocities at the #ow divider (inner) wall.
Fig. 6. Axial #ow pro"les at the endsinus position C in the internal carotid artery during early systolic acceleration (t/tp"0.075), mid-systolic peak #ow (t/tp"0.15), late-systolic deceleration (t/tp"0.25) and end-systolic #ow reversal (t/tp"0.325). (a) Numerical simulations. (b) MRI measurements. The #ow pro"les are highly skewed. With both approaches small velocities were observed at the outer, medium velocities at the side walls and high velocities at the #ow divider (inner) wall.
3.2. Secondary yow Contrary to these "ndings the secondary #ow pattern di!ers slightly between in vitro measurements and numerical simulations. However, in both cases secondary #ow is of the same order of magnitude. The simulations show a highly pronounced #ow from the outer wall throughout the center line of the vessel towards the #ow divider wall where it separates into two streams resulting in two asymmetric vortices (Figs. 7a and 8a). The main stream back to the outer wall is along the lower side wall whereas only a small back stream can be observed at the
upper side wall. This asymmetry can be observed for the midsinus as well as the endsinus position of the internal carotid artery. The in vitro measurements revealed during mid-systole only a very poor #ow throughout the center of the vessel towards the #ow divider wall which increased slightly when the #uid started to decelerate in late systole (Figs. 7b and 8b). Flow along the side walls towards the outer wall was balanced resulting in two more or less symmetric vortices. No great di!erences could be observed in the #ow patterns between the midsinus and the endsinus location except of a somewhat more pronounced #ow
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Fig. 7. Vector #ow patterns (secondary #ow) at the midsinus position B in the internal carotid artery during early systolic acceleration (t/tp"0.075), mid-systolic peak #ow (t/tp"0.15), late systolic deceleration (t/tp"0.25) and end systolic #ow reversal (t/tp"0.325). (a) Numerical simulations. (b) MRI measurements. Two vortices can be observed. In the center line of the vessel #ow is directed from the outer towards the #ow divider wall (i.e. from left to right in the "gure) whereas at the side walls it is directed in the opposite direction. Center line #ow is more pronounced in the simulations.
Fig. 8. Vector #ow patterns (secondary #ow) at the endsinus position C in the internal carotid artery during early systolic acceleration (t/tp"0.075), mid-systolic peak #ow (t/tp"0.15), late systolic deceleration (t/tp"0.25) and end-systolic #ow reversal (t/tp"0.325). (a) Numerical simulations. (b) MRI measurements. Two vortices can be observed. In the center line of the vessel #ow is directed from the outer towards the #ow divider wall (i.e. from left to right in the "gure) whereas at the side walls it is directed in the opposite direction. Center line #ow is more pronounced in the simulations.
along the side walls for the endsinus position. The main di!erence between the in vitro model study and the simulations was the less pronounced center line #ow from the outer wall to the #ow divider wall. The maximum circumferential velocity values are for both approaches about one-third of the axial main stream velocities. From early systole to end-diastole the ratio between the peak circumferential and the peak axial velocity varies between 0.15 to 0.35. For both methods the lower of the two vortices is more pronounced and the velocity pattern di!ers slightly from a double helical
secondary pattern which would be expected for an ideal planar bifurcation. Compared to the axial velocity component di!erences of the secondary #ow "eld between the in vitro measurements and the numerical simulations are more evident but still small. 3.3. Accuracy of the MRI results The spatial resolution for the MRI measurements has been selected from in vivo protocols. Thus, similar resolution may be achieved for in vivo experiments. Careful
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examinations and numerical simulations show that for secondary #ow due to #ow induced image artifacts inaccuracies of about 5% are to be expected for vessel diameter of 10 mm and up to 25% for 4 mm, respectively. So for the current experiments with diameters from 5.3 to 6.2 mm inaccuracies of 15}20% are expected. For axial components the errors are considerably lower.
Acknowledgements
4. Discussion
References
Velocity measurements performed in a anatomically realistic carotid bifurcation model with nearly rigid walls and simulations performed on the corresponding computational geometric model with rigid walls (for di!erences in the #ow patterns for rigid and compliant walls see Perktold et al., 1995c) show that numerically calculated axial #ow velocity patterns lead to similar results as MRI #ow velocity measurements provided that both are performed for exactly the same vessel geometry and under the same #uid dynamic conditions. Secondary #ow "elds prove to be very sensitive to di!erences in the vessel geometry. Such di!erences between the experimental situation and the mathematical model never can be removed completely. Even in our case where such slight di!erences cannot be excluded the #ow patterns show very similar global structures. The results are important to show the signi"cance of numerical simulations to predict correctly the #ow behavior in vessel segments with complicated #ow patterns where atherosclerotic plaques most often occur. The high correlation between in vitro velocity measurements in vessel models and in vivo examinations in human bifurcations was shown by Ku and Giddens (1985). In an earlier study we found that the global #ow pattern in the human carotid artery bifurcation in vivo could be well predicted by in vitro studies but that there are some di!erences in the secondary #ow pattern showing in most of the cases only single vortex #ow (Botnar et al., 1996). These di!erences were hypothesized to be due to the more bended course of the bifurcation plane. Based on the good correspondence between MRI velocity measurements and numerical simulations these two approaches both can be used to study in detail the role of di!erent #ow patterns for the initiation and ampli"cation of atherosclerotic plaque sedimentation. Accurate estimation of wall shear stress in vivo can be performed from simulations using in#ow pro"les and vessel geometry measured by MRI. The resulting pro"les from simulations then may be compared for various sites with MRI results to guarantee the agreement of the simulations with the in vivo conditions. The results also may be of great importance for vessel surgery as well as for the design of new implants such as grafts and arti"cial heart valves whose long term performance still su!ers from blood clot formation and thromboembolic events.
Boesiger, P., Maier, S.E., Kecheng, L., Scheidegger, M.B., Meier, D., 1992. Visualization and quanti"cation of the human blood #ow by magnetic resonance imaging. Journal of Biomechanics 25, 55}67. Botnar, R., Scheidegger, M.B., Boesiger, P., 1996. Quanti"cation of blood #ow patterns in human vessels by magnetic resonance imaging. Technology and Health Care 4, 97}112. Caro, C.G., 1994. Alterations of arterial hemodynamics associated with risk factors for artherosclerosis induced by pharmacological means: implications for the development/management of atherosclerosis. In: Hosoda, S., Yaginuma, T., Sugawara, M., Taylor, M.G., Caro, C.G. (Eds.), Recent Progress in Cardiovascular Mechanics. Harwood Academic Publishers, Chur, Switzerland, pp. 197}213. Friedman, M.H., Deters, O.J., Mark, F.F., Bargeron, C.B., Hutchins, G.M., 1983. Arterial geometry e!ects hemodynamics * a potential risk factor for atherosclerosis. Atherosclerosis 46, 225}231. Hatle, L., Angelsen, B.P., 1985. Doppler Ultrasound in Cardiology. Lea and Felbiger, Philadelphia, PA. Kilner, P.J., Yang, Z.Y., Mohiaddin, R.H., Firmin, D.N., Longmore, D.B., 1993. Helical and retrograde secondary #ow patterns in the aortic arch studied by three-dimensional magnetic resonance velocity mapping. Circulation 88, 2235}2247. Ku, D.N., Giddens, D.P., 1985. Hemodynamics of the normal carotid bifurcation. Ultrasound in Medicine and Biology 11 (1), 13}26. Ku, D.N., Giddens, D.P., 1987. Laser Doppler anemometer measurements of pulsatile #ow in a model carotid bifurcation. Journal of Biomechanics 20, 407}421. Ku, D.N., Giddens, D.P., Zarins, C.K., Glagov, S., 1985. Pulsatile #ow and atherosclerosis in the human carotid bifurcation. Arteriosclerosis 5 (3), 293}302. Liepsch, D., Moravec, S., 1984. Pulsatile #ow of non-Newtonian #uid in distensible models of human arteries. Biorheology 21, 571}586. Liepsch, D., Moravec, S., Baumgart, R., 1992. Some #ow visualization and Laser-Doppler measurements in a true-to-scale elastic model of a human aortic arch * a new model technique. Biorheology 29, 563}580. Nayler, G.L., Firmin, D.N., Longmore, D.B., 1986. Blood #ow imaging by cine magnetic resonance imaging. Journal of Computer Assisted Tomography 12, 715}722. Nerem, R.M., Cornshill, J.F., 1980. The role of #uid mechanics in atherogenesis. ASME Journal of Biomechanical Engineering 102, 181}189. Perktold, K., Hofer, M., Rappitsch, G., Loew, M., Kuban, B.D., Friedman, M.H., 1998. Validated computation of physiologic #ow in a realistic coronary artery branch. Journal of Biomechanics 31, 217}228. Perktold, K., Rappitsch, G., 1994. Mathematical modeling of local arterial #ow and vessel mechanics. In: Crolet, J.M., Ohayon, R. (Eds.), Computational Methods for Fluid-Structure Interactions. Pitman Research Notes in Mathematics Series 306. Longman, Essex, UK, pp. 230}245. Perktold, K., Rappitsch, G., 1995a. Computer simulation of arterial blood #ow: vessel diseases under the aspect of local haemodynamics. In: Ja!rin, M.Y., Caro, C.G. (Eds.), Biological Flows. Plenum Press, New York, pp. 83}114.
The study was supported by the Austrian Science Foundation, Project No. P 11 982-TEC, Vienna, by EUREKA Project No.EU 1353, and by the Swiss Commission for Technology and Innovation KTI, Project No. 3030.2.
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Perktold, K., Rappitsch, G., 1995b. Computer simulation of local blood #ow and vessel mechanics in a compliant carotid artery bifurcation model. Journal of Biomechanics 28, 845}856. Perktold, K., Rappitsch, G., 1995c. Mathematical modeling of arterial blood #ow and correlation to atherosclerosis. Technology and Health Care 3, 139}151. Reuderink, P., 1991. Analysis of the #ow in a 3D distensible model of the carotid artery bifurcation. Thesis, Eindhoven ISBN 90-9004413-2. Rindt, C.C.M., Steenhoven, A.A., van Janssen, J.D., Reneman, R.S., Segal, A., 1990. A numerical analysis of steady #ow in a threedimensional model of the carotid artery bifurcation. Journal of Biomechanics 23, 461}473. Sawchuk, A.P., Unthank, J.L., Davis, T.E., Dalsing, M.C., 1994. A prospective, in vivo study of the relationship between blood #ow
hemodynamics and atherosclerosis in a hyperlipidemic swine model. Journal Vascular Surgery 19, 58}64. Steiger, H.J., 1990. Pathophysiology of development and rupture of cerebral aneurysms. Acta Neurochirurgica, Suppl. 48. Springer, Wien, New York. Steinke, W., Hennerici, M., 1989. Three dimensional ultrasound imaging of carotid artery plaques. Journal of Cardiovascology Technology 8, 15}22. Sugawara, M., Kajiya, F., Kitabatake, A., Matsuo, H., 1989. Blood Flow in the Heart and Large Vessels. Springer, Tokyo. Zarins, C.K., Giddens, D.P., Bharadavaj, B.K., et al., 1983. Carotid bifurcation atherosclerosis: quantitation of plaque localization with #ow velocity pro"les and wall shear stress. Circulation Research 53, 502}514.
Hemodynamics in the carotid artery bifurcation: a comparison between numerical simulations and in vitro MRI measurements ReneH Botnar!, Gerhard Rappitsch", Markus Beat Scheidegger!, Dieter Liepsch#, Karl Perktold", Peter Boesiger!,* !Institute of Biomedical Engineering, University of Zurich and Swiss Federal Institute of Technology, Gloriastrasse 35, CH-8092 Zurich, Switzerland "Institute of Mathematics, Technical University Graz, A-8010 Graz, Austria #Fachhochschule Mu( nchen, D-80335 Mu( nchen, Germany Received 1 June 1998; accepted 9 August 1999
Abstract The presence of atherosclerotic plaques has been shown to be closely related to the vessel geometry. Studies on postmortem human arteries and on the experimental animal show positive correlation between the presence of plaque thickness and low shear stress, departure of unidirectional #ow and regions of #ow separation and recirculation. Numerical simulations of arterial blood #ow and direct blood #ow velocity measurements by magnetic resonance imaging (MRI) are two approaches for the assessment of arterial blood #ow patterns. In order to verify that both approaches give equivalent results magnetic resonance velocity data measured in a compliant anatomical carotid bifurcation model were compared to the results of numerical simulations performed for a corresponding computational vessel model. Cross sectional axial velocity pro"les were calculated and measured for the midsinus and endsinus internal carotid artery. At both locations a skewed velocity pro"le with slow velocities at the outer vessel wall, medium velocities at the side walls and high velocities at the #ow divider (inner) wall were observed. Qualitative comparison of the axial velocity patterns revealed no signi"cant di!erences between simulations and in vitro measurements. Even quantitative di!erences such as for axial peak #ow velocities were less than 10%. Secondary #ow patterns revealed some minor di!erences concerning the form of the vortices but maximum circumferential velocities were in the same range for both methods. ( 2000 Elsevier Science Ltd. All rights reserved. Keywords: Atherosclerosis; MRI #ow measurement; Numerical #ow simulations; Carotid bifurcation; Secondary #ow pattern
1. Introduction The presence of atherosclerotic plaques has been shown to be closely related to the vessel geometry (Nerem et al., 1980; Friedman et al., 1983; Caro, 1994). The carotid bifurcation, the coronary arteries, the infrarenal aorta and the vessels supplying the lower extremities may be markedly diseased while other vessels are rather spared. Due to these "ndings local hemodynamics is considered to play an important role in the initiation and development of atherosclerosis. Studies
* Corresponding author. Tel.: #41-1-632-4581; fax: #41-1-6321193. E-mail address: [email protected] (P. Boesiger)
on postmortem human arteries (Zarins et al., 1983; Ku et al., 1985) and on the experimental animal (Sawchuk et al., 1994) showed positive correlation between plaque thickness and the presence of low shear stress, departure of unidirectional #ow and regions of #ow separation and recirculation. Comparison between #ow velocity measurements and plaque distributions in human carotid artery specimens revealed that at the outer wall of the carotid sinus, where wall shear stress and #ow velocities are low, plaque thickness is maximal whereas at the #ow divider wall, where wall shear stress and velocities are high, plaque thickness is minimal (Zarins et al., 1983). At the side walls, where circumferential #ow velocities (secondary #ow) are present plaque thickness is medium. Various #ow quanti"cation techniques such as laser Doppler anemometry (LDA) (Liepsch et al., 1984; Ku
0021-9290/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 9 9 ) 0 0 1 6 4 - 5
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R. Botnar et al. / Journal of Biomechanics 33 (2000) 137}144
et al., 1987; Steiger, 1990), ultrasound Doppler (US) (Hatle et al., 1985; Steinke et al., 1989; Sugawara et al., 1989) or magnetic resonance imaging (MRI) (Nayler et al., 1986; Boesiger et al., 1992; Kilner et al., 1993) are commonly used to study arterial hemodynamics. The choice of one of these techniques depends on the speci"c application and the demand on spatial and temporal resolution. An alternative and complementary approach to study arterial hemodynamics uses numerical simulations (Perktold et al., 1995a,b,c,1998; Reuderink, 1991; Rindt et al., 1990). The development of e$cient numerical methods and new computer generations allows the calculation of local #ow behavior under physiologic and anatomically realistic conditions. This approach may be very useful for detailed quantitative studies of the in#uence of various local vessel geometries on speci"c vessel pathologies. The aim of this study was to verify that numerical simulations and #ow velocity measurements such as MRI lead to same #ow patterns if the computational geometric vessel model is an exact copy of the experimental bifurcation and if the same #ow conditions are used.
2. Material and methods 2.1. Mathematical model The local #uid dynamics was described using the three-dimensional time-dependent Navier}Stokes equations for incompressible Newtonian #uids: Lu 1 #(u ' +)u!l*u# +p"0, Lt o
(1)
+ ' u"0
(2)
with the velocity vector "eld u"(u, v, w)T and the pressure p.l"k/o represents the kinematic viscosity, o and k stand for the constant #uid density and the dynamic viscosity, respectively. For the numerical solution of the #ow problem the "nite element method was applied: The approximation applies eight-node isoparametric brick elements with trilinear interpolation functions for the velocity vector "eld and constant pressure in each element. The equations are solved using the "nite element Galerkin method with implicit Euler backward di!erences for the time derivatives and Picard iteration for the non-linear convection terms. The solution of the time-dependent Navier}Stokes equations is performed applying a recently developed velocity}pressure correction method, in which the occurring variables (velocity components and pressure) are uncoupled (Perktold et al., 1994). In each time (or iteration) step of the solution procedure this method takes the following steps:
1. Calculation of an auxiliary velocity "eld from the equations of motion using the known velocity and pressure values from the previous time (or iteration) step. 2. Calculation of the pressure correction from a pressure Poisson equation using the concentrated mass matrix. 3. Calculation of the divergence-free velocity "eld. 4. Updating the pressure. Flow velocity patterns were calculated for an anatomically realistic carotid artery bifurcation model. The computational geometric model was generated from an optically digitized anatomically correct cast of a postmortem carotid bifurcation (Liepsch et al., 1984,1992) from which the bifurcation replica was produced. For generation of the model for the mathematical simulation a recently developed "nite element grid generator (Perktold et al., 1995b) was applied. The procedure consists of two steps: the generation of the model surface using weighted least-squares splines and the discretization of the inner grid by a local optimization algorithm of geometric grid properties (Fig. 1). For an appropriate computational representation the vessel surface was described by 12.712 "nite elements. The #ow domain was described by 50.568 "nite elements. The total node number of the three velocity components and the pressure was 223.333. The kinematic viscosity l"0.037 cm2/s of the #uid was chosen the same as in the experimental study which is in the physiological range of blood. In#ow velocity pro"les were taken from MRI measurements. 2.2. Bifurcation model Measurements of pulsatile #ow velocity were performed in an anatomically realistic replica of a human carotid artery bifurcation (true-to-scale elastic model) made from silicon rubber. It is casted in a box "lled with 1.5% agarose gel (Fig. 2). By this casting procedure a model with nearly rigid walls arises. To mimic the rheologic properties of blood a water}glycerin mixture with a kinematic viscosity of l"0.037 cm2/s was used.
Fig. 1. Computational geometry of the bifurcation model.
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Fig. 2. MRI compatible #ow model circuit: The input wave form is produced by a computer controlled step motor driven pump. Long-term stability was controlled by an electromagnetic #ow meter. For synchronization of he MRI measurement with the #ow pulsatility every pump cycle a trigger signal was transmitted to the MR acquisition unit via an optical "ber.
The time-averaged mean #ow rate was 9.4 ml/s corresponding to a Reynolds number of Re"488. The mean vessel diameters of the examined cross sections of the common, midsinus and endsinus internal and external carotid artery were (A) 6.6 (B) 6.2 (C) 5.3 and (D) 4.8 mm (Fig. 3) with a #ow ratio of 0.5 for both daughter tubes (Fig. 4). The length of the sinus was 7 mm and the bifurcation angle approximately 453. Fig. 3. Schematic drawing of the compliant and anatomically realistic bifurcation model. The location of interest is the carotid sinus (B, C), whose #ow patterns are highly complex. The temporal and spatial #ow behavior in the common carotid artery segment (A) and the #ow ratio between common and the two daughter tubes (internal (B,C), external (D)) were used as input data for the numerical simulations.
The experiment was carried out under non-stationary #ow conditions with a mid-systolic #ow rate of 28 ml/s (the corresponding maximum Reynolds number Re"1454) in the common carotid artery segment. A programmable step motor driven pump allowed to reproduce physiologic #ow conditions. The #ow direction of the #uid was controlled by two mechanical valves. During the dead time of the pulse cycle the #uid was sucked from a water reservoir into the piston and pumped towards the bifurcation during the working cycle. To guarantee fully developed #ow pro"les the pump was connected to the bifurcation with a 4.5 m long rigid Plexiglas tube (/"10 mm). Fluid leaving the daughter tubes #owed into a normal pressure over#ow tank, which was connected with a 5 m long Plexiglas tube to the water reservoir of the pump. The #ow pulse rate was 60 cycles per minute. Long-term stability of the produced pulse wave was monitored during the experiment with an electromagnetic #ow meter (Tecmag II, Endress#Hausser, Reinach, Switzerland) which was installed close to the pump. The sampling rate was 25 Hz.
2.3. MR-yow velocity measurements Cross sectional velocity measurements of the axial and circumferential velocity component were performed at the common, midsinus and endsinus internal and at the external carotid artery on a 1.5 T Philips Gyroscan ACS II (Philips Medical Systems, Best, Netherlands) MRI-scanner using a prospectively triggered phase contrast sequence. Synchronization of the velocity measurement with respect to the periodic #ow cycle was achieved by sending a periodic 5 V TTL trigger signal to the MR acquisition unit. To obtain a high signal-to-noise (SNR) level a 7 cm diameter surface coil was placed directly above the bifurcation. With a FOV of 140*140 mm2 and an image matrix of 256*256 the spatial resolution was 0.5*0.5 mm2. To minimize intra voxel velocity gradients the slice thickness was chosen to be 4 mm. Numerically calculated velocity maps presented in this paper therefore are always averaged over a 4 mm thick slice. Flow voids, geometric image artifacts and #ow quanti"cation errors which may occur under complex #ow conditions were reduced by shortening the echo time using a partial echo acquisition scheme (62.5% of the full echo) (Botnar et al., 1996). Minimum echo time was 8.9 ms, time resolution 25 ms and #ip angle 153. The missing samples were zero padded. Velocity encoding was adapted to the expected velocity range and was
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Fig. 4. Flow rates measured in the common carotid artery, at the mid- and endsinus position of the internal and in the external carotid artery. Four systolic time points (acceleration: t/tp"0.075, peak #ow: t/tp"0.15, deceleration: t/tp"0.25, and minimum #ow: t/tp"0.325) are marked for which the axial and circumferential velocity patterns were measured and calculated, respectively.
120 cm/s for through plane and 50 cm/s for in-plane velocities. 2.4. Velocity data analysis Data analysis was performed on a DEC-Alpha (Digital Equipment Corporation, USA) workstation with a #ow analysis software package written in PV-Wave (Visual Numerics, Boulder CO, USA) which has been developed at the Institute of Biomedical Engineering in Zurich. Vessel contours were drawn using movable and deformable ellipses. The contour identi"cation could either be done on the modulus or the velocity (phase) images of each heart phase. The obtained contours were copied to the corresponding modulus or velocity (phase) images. After vessel wall segmentation axial and circumferential velocity maps were either visualized as 3D-mesh plots or as vector plots to allow a proper comparison with the results of the numerical simulations.
3. Results As expected #ow in the carotid sinus proofed to be highly complex. Axial and circumferential (secondary) #ow velocities were measured and calculated at the midsinus (C) and the endsinus (D) level of the internal carotid bifurcation (Fig. 3) during early systole (t/tp"0.075), mid-systole (t/tp"0.15), late systole (t/tp"0.25) and
end-systole (t/tp"0.325). High spatial and temporal #ow variations were observed for both approaches leading to high mainly axial directed #ow velocities at the #ow divider wall and small but still antegrade velocities at the opposite outer wall. In early systole (acceleration) the axial #ow pro"le is mainly axial symmetric but rapidly changes to a highly skewed velocity pro"le during midsystole which can be seen until early diastole. The same temporal behavior was observed for the secondary #ow pattern. It starts to develop at mid-systole and stays visible until early diastole. Fig. 4 shows the #ow rates measured in the common, the midsinus, the endsinus and the external carotid artery. Early systole, mid-systole, late systole and end-systole are marked with respect two the #ow cycle. 3.1. Axial yow A 3D representation of the axial velocity patterns reveals for both methods a skewed velocity pro"le with slow velocities at the outer and high velocities at the #ow divider (inner) wall (Figs. 5 and 6). The overall correspondence of both approaches is quite convincing for the axial #ow pattern. Despite slight variations the peak velocity curves do not di!er more than 10%. Even though the branching of the bifurcation is not exactly planar a high symmetry of the axial velocity "eld with respect to the both side walls can be observed.
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Fig. 5. Axial #ow pro"les at the midsinus position B in the internal carotid artery during early systolic acceleration (t/tp"0.075), mid systolic peak #ow (t/tp"0.15), late systolic deceleration (t/tp"0.25) and end systolic #ow reversal (t/tp"0.325). a) Numerical simulations. (b) MRI measurements. The #ow pro"les are highly skewed. With both approaches small velocities were observed at the outer, medium velocities at the side walls and high velocities at the #ow divider (inner) wall.
Fig. 6. Axial #ow pro"les at the endsinus position C in the internal carotid artery during early systolic acceleration (t/tp"0.075), mid-systolic peak #ow (t/tp"0.15), late-systolic deceleration (t/tp"0.25) and end-systolic #ow reversal (t/tp"0.325). (a) Numerical simulations. (b) MRI measurements. The #ow pro"les are highly skewed. With both approaches small velocities were observed at the outer, medium velocities at the side walls and high velocities at the #ow divider (inner) wall.
3.2. Secondary yow Contrary to these "ndings the secondary #ow pattern di!ers slightly between in vitro measurements and numerical simulations. However, in both cases secondary #ow is of the same order of magnitude. The simulations show a highly pronounced #ow from the outer wall throughout the center line of the vessel towards the #ow divider wall where it separates into two streams resulting in two asymmetric vortices (Figs. 7a and 8a). The main stream back to the outer wall is along the lower side wall whereas only a small back stream can be observed at the
upper side wall. This asymmetry can be observed for the midsinus as well as the endsinus position of the internal carotid artery. The in vitro measurements revealed during mid-systole only a very poor #ow throughout the center of the vessel towards the #ow divider wall which increased slightly when the #uid started to decelerate in late systole (Figs. 7b and 8b). Flow along the side walls towards the outer wall was balanced resulting in two more or less symmetric vortices. No great di!erences could be observed in the #ow patterns between the midsinus and the endsinus location except of a somewhat more pronounced #ow
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Fig. 7. Vector #ow patterns (secondary #ow) at the midsinus position B in the internal carotid artery during early systolic acceleration (t/tp"0.075), mid-systolic peak #ow (t/tp"0.15), late systolic deceleration (t/tp"0.25) and end systolic #ow reversal (t/tp"0.325). (a) Numerical simulations. (b) MRI measurements. Two vortices can be observed. In the center line of the vessel #ow is directed from the outer towards the #ow divider wall (i.e. from left to right in the "gure) whereas at the side walls it is directed in the opposite direction. Center line #ow is more pronounced in the simulations.
Fig. 8. Vector #ow patterns (secondary #ow) at the endsinus position C in the internal carotid artery during early systolic acceleration (t/tp"0.075), mid-systolic peak #ow (t/tp"0.15), late systolic deceleration (t/tp"0.25) and end-systolic #ow reversal (t/tp"0.325). (a) Numerical simulations. (b) MRI measurements. Two vortices can be observed. In the center line of the vessel #ow is directed from the outer towards the #ow divider wall (i.e. from left to right in the "gure) whereas at the side walls it is directed in the opposite direction. Center line #ow is more pronounced in the simulations.
along the side walls for the endsinus position. The main di!erence between the in vitro model study and the simulations was the less pronounced center line #ow from the outer wall to the #ow divider wall. The maximum circumferential velocity values are for both approaches about one-third of the axial main stream velocities. From early systole to end-diastole the ratio between the peak circumferential and the peak axial velocity varies between 0.15 to 0.35. For both methods the lower of the two vortices is more pronounced and the velocity pattern di!ers slightly from a double helical
secondary pattern which would be expected for an ideal planar bifurcation. Compared to the axial velocity component di!erences of the secondary #ow "eld between the in vitro measurements and the numerical simulations are more evident but still small. 3.3. Accuracy of the MRI results The spatial resolution for the MRI measurements has been selected from in vivo protocols. Thus, similar resolution may be achieved for in vivo experiments. Careful
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examinations and numerical simulations show that for secondary #ow due to #ow induced image artifacts inaccuracies of about 5% are to be expected for vessel diameter of 10 mm and up to 25% for 4 mm, respectively. So for the current experiments with diameters from 5.3 to 6.2 mm inaccuracies of 15}20% are expected. For axial components the errors are considerably lower.
Acknowledgements
4. Discussion
References
Velocity measurements performed in a anatomically realistic carotid bifurcation model with nearly rigid walls and simulations performed on the corresponding computational geometric model with rigid walls (for di!erences in the #ow patterns for rigid and compliant walls see Perktold et al., 1995c) show that numerically calculated axial #ow velocity patterns lead to similar results as MRI #ow velocity measurements provided that both are performed for exactly the same vessel geometry and under the same #uid dynamic conditions. Secondary #ow "elds prove to be very sensitive to di!erences in the vessel geometry. Such di!erences between the experimental situation and the mathematical model never can be removed completely. Even in our case where such slight di!erences cannot be excluded the #ow patterns show very similar global structures. The results are important to show the signi"cance of numerical simulations to predict correctly the #ow behavior in vessel segments with complicated #ow patterns where atherosclerotic plaques most often occur. The high correlation between in vitro velocity measurements in vessel models and in vivo examinations in human bifurcations was shown by Ku and Giddens (1985). In an earlier study we found that the global #ow pattern in the human carotid artery bifurcation in vivo could be well predicted by in vitro studies but that there are some di!erences in the secondary #ow pattern showing in most of the cases only single vortex #ow (Botnar et al., 1996). These di!erences were hypothesized to be due to the more bended course of the bifurcation plane. Based on the good correspondence between MRI velocity measurements and numerical simulations these two approaches both can be used to study in detail the role of di!erent #ow patterns for the initiation and ampli"cation of atherosclerotic plaque sedimentation. Accurate estimation of wall shear stress in vivo can be performed from simulations using in#ow pro"les and vessel geometry measured by MRI. The resulting pro"les from simulations then may be compared for various sites with MRI results to guarantee the agreement of the simulations with the in vivo conditions. The results also may be of great importance for vessel surgery as well as for the design of new implants such as grafts and arti"cial heart valves whose long term performance still su!ers from blood clot formation and thromboembolic events.
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The study was supported by the Austrian Science Foundation, Project No. P 11 982-TEC, Vienna, by EUREKA Project No.EU 1353, and by the Swiss Commission for Technology and Innovation KTI, Project No. 3030.2.
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