BIORHEOLOGY, 29; 337351,1992 0006355X/92 $5.00 + .00 Printed in the USA. Copyright (c) 1992 Pergamon Press Ltd. All rights reserved.
A NUMERICAL SIMULATION OF INTIMAL THICKENING UNDER SHEAR IN ARTERIES D. Lee and J. J. Chiu Institute of Aeronautics and Astronautics National Cheng Kung University Tainan, Taiwan, ROC 700 (Received 27.1.1992; accepted in revised form 29.6.1992 by Editor H. Niimi)
ABSTRACT
A model of intima thickening proposed by Friedman and his coworkers (1,2) is incorporated in our computer code to simulate the growth of intima under shear. The computer code is based on a finite volume method in a boundaryfitted coordinate system. It is found that the model yields an evenlydistributed thickening over a straight, smooth vessel wall. However, in a stenosed or a curved artery, thicker intima can be formed in preferential regions due to unevenlydistributed wall shear stresses. The results clearly demonstrate the correlations among the geometry, wall shear rate and the plaque localization in arteries. The model is applied to a straight artery with a stenosis or sinus, a smooth curved artery and a stenosed curved artery. The effects of stenosis/sinus and lumen curvature on the flows and the intimal thickening are studied. The simulation provides a numerical visualization of the intimal thickening in a dynamic way.
INTRODUCTION It is well known that hemodynamics plays an important role in atherogenesis and thrombosis. Numerous theories have been proposed. Among them, the controversy continues between the high shear theory (3) and the low shear theory (4). Fry suggested that elevated shear rate favors intimal disease (3). The finding of Corn hill and Roach (5) offered the evidence that plaques occur in regions of elevated shear stress. They found that sudanophilic lesions about the intercostal ostia of cholesterolfed rabbits tend to occur distal to the flow divider edge. Similar findings by Sinzinger et aI. (6) have been reported about the ostia of aortic branches in human infants. On the other hand, Caro et al. (4) suggested that low wall shear rates retard the transport of circulating particles away from the wall,
KEY WORDS:
intimal thickening, arterial flows, wall shear rate, hemodynamics, finite volume method
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resulting in increased intimal accumulation of lipids. Evidences on this theory seem more frequently founded in the literature. Among them, Zarins et al. (7) and Ku et al. (8) concluded that intimal thickening and atherosclerosis develop largely in regions of relatively low wall shear stress, flow separation. Analyses of Friedman et al. (1) and Friedman (2) suggested that the intimal thickness at sites subjected to elevated shear rates increases quickly to a modest value and grows slowly thereafter. While for those low shear rate regions, intima thickens more slowly in a "young" vessel, but, reaches higher values as vessel ages. This analysis includes both the high and low shear theories and provides a more complete insight of the intimal thickening. The high shearlow shear behavior is explained consistently by the model proposed by Friedman and his coworkers. In the model, the intimal permeability and removal rate both increase as the shear stress is raised. Intimal thickness is related to the amount of the substance in the wall. Although the model is simple, it shows that the results can be the consequence of competing sheardependent processes. This model unifies the high and low shear theories and provides a possible solution to the high shearlow shear controversy. The expression derived has been written in terms of the wall shear rate and the age of the artery. The physiological processes involved are included implicitly in the rate coefficients as will be discussed in the next section. In this article, we incorporate the above model in our computer code to simulate the intimal thickening in arteries. The intimal thickness along the wall of an artery at different vessel "age" is computed. Growth of intima can be visualized in a dynamic way. Simulations on plaque formation in a twodimensional aortic bifurcation by using a critical shear stress can be found in the works of Nazemi et al. (9) and Kleinsteuer et al. (10). In these studies, a finite element method is employed. An ad hoc, thin sinusoidal mass layer is incorporated at the sites which are identified as areas for the onset of plaque formation. In the present work, the intimal thickness is computed by using an expression proposed in the model which is derived from the experimental data. Subsequent change of the wall shear rate due to the intimal thickening can be accounted for in the present method. Most arteries are at least slightly curved. Patterns of femoral artery lesion distribution have indicated early and moderately advanced atheromas clustered mainly on the inner curvature of a curved vessel (11). Calculation of a curved tube flow can provide the hemodynamic information such as the shear rates, pressure distribution and the location of recirculation zones if exist. The calculated shear rate can then be used in the above model to compute the intima thickness at the site. To take into account the changes of flow field due to the growth of intima, geometry of the artery is updated during the course of computation. The grid generation technique is also addressed. INTIMAL THICKENING MODELS The model is first proposed by Friedman and his coworkers (1) and later remodeled by Friedman (2). In the earlier model, to fit the experimental data, the following formula is used (1): (5 (s, T)
= p[ 1 + n(sI)] x exprpC1 s)/s] x [ 1 exp {T exp[p(sl)/s]} ].
(1)
where (5 and s are the intimal thickness and the corresponding local shear rate. The quantity T measures the extent to which the thickening process have progressed and can be regarded as the nondimensional "biological age" of the vessel. In the present calculation, T represents only the "age" of the vessel, and has only implicit relation with the real time.
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The P has the units of length and is the steady state thickness at a site exposed to the average shear. The parameter a measures the sensitivity of the uptake process to shear rate. The best fit value is close to unity. The parameter ~ is regarded as a nondimensional activation energy for a rate process that retards thickening. These parameters in reality, are not constants but vary among individuals and over time. Some features of the above expression can be observed from the thickening rate of intima as the vessel ages. Take the derivative of intima thickness with respect to the "age" of vessel, one obtains
ao [ ( )]  T x exp'aT = P 1 + a s  1 x exp
~(sl )
(2)
The intimal thickness as a function of normalized maximum shear rate is plotted together with the thickening rate in Fig. 1. It is seen that for young vessel (e.g. T6 in the figure). This behavior clearly shows that the low shear theory (4) dominates in the "old" vessels in this model. The increment of the intimal thickness L10 in an arbitrary time interval L1T can be approximated by the following equation:
(3)
0.10
0.3~
....,
0.30
.,.,'"
. ....::
0'1"
0.08
>::
>::
.:.:: ~ ..:: ....
