L H. Back Group Supervisor, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109 Fellow ASME

D. W. Crawford Professor, Department of Medicine, Cardiology Division, University of Southern California, Los Angeles, CA 90033

Wall Shear Stress Estimates in Coronary Artery Constrictions1 Wall shear stress estimates from laminar boundary layer theory were found to agree fairly well with the magnitude of shear stress levels along coronary artery constrictions obtained from solutions of the Navier Stokes equations for both steady and pulsatile flow. The relatively simple method can be used for in vivo estimates of wall shear stress in constrictions by using a vessel shape function determined from a coronary angiogram, along with a knowledge of the flow rate.

Introduction Blood flowing through vessels exerts a shear stress on the wall by virtue of the difference between its motion and the relatively stationary wall. The velocity difference between blood and vessel wall increases where the lumen is constricted because flow velocities increase to satisfy conservation of mass flow. Consequently, wall shear stress increases along a constriction where the flow is accelerating (e.g., see the earlier investigations by Fry, 1968; Lee and Fung, 1970; Yu and Goldsmith, 1973; Young and Tsai, 1973, although there have been more recent studies as well). The purpose of this paper is to present some results on an approximate method for estimating wall shear stresses in the contraction region of coronary artery constrictions. The method is evaluated by comparison to more exact numerical calculations involving the time dependent Navier Stokes equations for axisymmetric flow and inferences from measured wall pressure distributions in coronary artery models. Direct measurements of coronary wall shear stress are not possible in man, and are very difficult to make in vessel models. Consequently, there is interest in wall shear stress prediction methods, particularly those simple enough to be used in clinical practice.

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(2)

In this expression/,,, is the slope of the nondimensional velocity profile at the wall in the transformed plane in the solution method, and depends upon a free-stream velocity gradient parameter /3 i.e.,/„,(/?), which in turn depends on the amount of constriction

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(3)

di

Thus the function F depends upon the percent constriction (l-d/d,) x 100 percent where d/dt is the ratio of local to upstream diameter, and the half angle X of the constriction as shown in Fig. 1. Numerical values are given in Table 1. As is evident in Fig. 1, increases in the percent constriction and steepness of constriction i.e., larger X, lead to larger values of the shape function F, and larger wall shear stresses. Approximate Method In the present applications, this method is used for steady The wall shear stress TW estimate derived earlier from laminar flow, and also for pulsatile flow by using the quasi-steady boundary layer theory for steady flow through a conical con- boundary layer approximation. striction by using a local similarity method (Back, 1975), is given by Results Back et al, (1986) have measured the steady flow pressure (1) ,3/2 dl' distributions along a hollow vascular axisymemtric replica of T„ depends upon the 3/2 power of flow rate Q (typical of a segment of the left circumflex coronary artery of man with laminar boundary layer flow), upstream diameter dh fluid mildly atherosclerotic diffuse disease. In particular, the measproperties, density p and viscosity /x, and a shape function F ured pressure distribution normalized by the dynamic pressure at the inlet is shown in Fig. 2 by the open circles for an elevated given by coronary flow at a Reynolds number, Re, = 353. The fluid used was a 33 percent sugar-water solution with kinematic The research described in this paper was carried out at the Jet Propulsion viscosity, v = 0.035 cm2/s, simulating that for blood, and the Laboratory, California Institute ofTechnology, under contract with the National Aeronautics and Space Administration. flow rate, Q = 180 ml/min. The enlarged lumen radius norContributed by the Bioengineering Division for publication in the JOURNAL malized by the inlet radius, r„//> is shown above the pressure OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering distribution to accentuate the contraction. The inlet diameter Division September 8, 1991; revised manuscript received December 16, 1991. dj = 3.08 mm. Note that the axial distance x is normalized by Associate Technical Editor: L. Talbot.

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Journal of Biomechanical Engineering

NOVEMBER 1992, Vol. 114 / 515

Copyright © 1992 by ASME

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Table 1 Values of shape function for wall shear stress d_ d, 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 . 0.2

/3

/„

X=5°

0.21 0.44 1.0. 1.71 2.67 4.0 6.0 9.3 16.0

0.695 0.88

0.76 0.80 1.12 1.73 2.94 5.55 12.0 32.8 135.6

1.23

1.56 1.92 2.35 2.86 3.55 4.65

SHAPE 10° 1.07 1.13 1.58 2.45 4.15 7.84 17.0 46.3 191.7

FUNCTION F 20°

30°

1.50 1.58 2.22 3.43 5.82 11.0 23.9 65.0 268.9

1.81 1.91 2.69 4.15 7.03 13.3 28.8 78.5 325.0

45° 2.15 2.27 3.20 4.94 8.37 15.8 34.3 93.4 386.6

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Fig. 1

(1-—)x100%

Shape function tor wall shear stress

an increment Ax that was used in the numerical calculations involving the Navier-Stokes equations. The relationship between the inlet radius and Ax is 4Ax = /•,-. Between axial location number, xlAx, of #17 and #45 there is an approximate conical contraction (shown by the dashed line) with a small constriction half angle X ~ 2.4 deg, and amount of constriction of about 30 percent. Local lesion variations are evident along the constriction so that the wall was not smooth. The basis for determining the relevance of the approximate analysis rests on the correspondence between the measured pressure distributions (open circles) and the values obtained from a numerical solution of the Navier-Stokes equations (Back et al., 1977) that is shown in Fig. 2 by the solid circles. At the narrowest location #45, the calculated pressures were identical to the measured values. There were some deviations along the inlet of the constriction that may be due to inertial effects associated with turning of the flow into the constriction. As noted by Back et al, (1986) the calculated pressures are average values across the flow, so that the wall pressure measurements may differ somewhat from the average values in the inlet region.

d = lumen diameter (cm) fw = slope of nondimensional velocity profile at wall F = shape function, Eq. (2), dimensionless P = experimentally measured pressure (dyne/cm 2 ) Q = volume flow rate (cmVmin); cmVs in evaluating Eq. (1) rw = lumen radius (cm) 516 / Vol. 114, NOVEMBER 1992

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Fig. 2 Lumen contour, dimensionless pressure and wall shear stress versus axial location along the lumen; Re; = 353; o experimental data; • prediction Navier-Stokes equations; — prediction, Eq. (1)

The distribution of wall shear stress (normalized by the dynamic pressure upstream) obtained from the solution of the Navier-Stokes equations is shown in Fig. 2 by the solid circles. The general trend indicates increases in shear stresses in the constriction region to peak values at the narrowest location #45. The sharp variations in shear stress are associated with local wall irregularities associated with diffuse disease i.e., local peak values occur at the tops of the local plaques, and lower values occur in the trough regions between plaques. These

Re, = Reynolds number based on inlet lumen diameter (4/7r) (Q/W/) t = time (s) u = axial velocity (cm/s) X = axial distance (cm) P = free-stream velocity gradient parameter X = half angle of constriction (deg) /* = viscosity (poise)

v = kinematic viscosity (= /i/p) (cm 2 /s) p = density (gm/cm 3 ) TW = wall shear stress (dyne/cm2) Subscripts B = blood /' = inlet condition Superscript (~) = time average Transactions of the ASME

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Wall shear stress estimates in coronary artery constrictions.

Wall shear stress estimates from laminar boundary layer theory were found to agree fairly well with the magnitude of shear stress levels along coronar...
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