Combined effects of pulsatile flow and dynamic curvature on wall shear stress in a coronary artery bifurcation model
Introduction
It is widely believed that the development and progression of atherosclerosis is related to the complex flow field occurring in curvatures and bifurcations of large and medium arteries (Caro et al., 1971, Ku et al., 1985, Friedman et al., 1987). Numerical studies based on rigid, idealized (Steinman et al., 2000, Bertolotti and Deplano, 2000) as well as realistic geometries (Myers et al., 2001, Steinman et al., 2002, Zeng et al., 2003) have been useful in understanding the unique complex features of these flows. Most of these investigations focus on the effects of the geometry, pulsatile flow and non-Newtonian behavior of blood in non-moving vessels. Perktold and Rappitsch (1995) and Zhao et al. (2000) analyzed the effect of distensible wall on the local flow field. Some authors have suggested that the flexibility and motion of coronary artery during each contraction and expansion of the heart is important in the study of the flow dynamics. Several investigations used models in which the vessel was represented by a single flexible tube (Schilt et al., 1996, Santamarina et al., 1998, Zeng et al., 2003).
Comparatively less is known about the effects of vessel movement on the blood flow patterns in coronary arteries at bifurcations. Zeng et al. (2003) simulated a realistic arterial motion based on biplane cineangiograms in human right coronary arteries but did not consider any branches. Weydahl and Moore (2001) were among the first to consider the model of coronary artery at bifurcation with time-varying geometry. They showed that curvature variation is important in determining temporal wall shear stress variations. However, they considered only steady inflow.
The objective of the present work is to analyze the combined effect of dynamic geometry and pulsatile inflow on the flow dynamics and wall shear rate in a simple model of a coronary artery at bifurcation. This will allow us to take into account any phase difference between the two unsteady phenomena that may arise in cardiac disease.
Normal coronary flow velocities are characterized by a small forward flow during systole and a large forward flow during diastole (Guyton, 1986). In cardiac disease other patterns can arise. For example, if there is an infarcted region of myocardium, the cardiac muscle in that region does not contract as healthy muscle. Consequently, the epicardial blood flow variation and the deformations of the ventricular wall masses may both change. If the infarct is due to current thrombotic obstruction, outflow resistance in the corresponding intramural segments is not relieved in diastole. Matsuo et al. (1988) have connected the change in phasing information to aortic regurgitation. Using a bidirectional Doppler flowmeter catheter to examine coronary flow velocity in patients who had aortic valve disease, they found a decreased diastolic and increased systolic coronary flow. In some cases, the diastolic flow velocity was less than the systolic. These flow velocity patterns became notable with severe aortic regurgitation. Such phenomena have a direct effect on the phase difference between the arterial motion studied here and the pulsatile inflow imposed in our models.
While approximate values for epicardial artery curvature and its variation through a heartbeat are known, precise values are problematic to obtain. Biplane or multiplane angiograms taken in vivo are useful for estimating the axes of vessel segments through the cardiac cycle but do not register lumenal diameters or details sufficiently accurately, and do not show artery wall tissue characteristics. Intravascular ultrasound can supply lumenal geometry and wall tissue information but does not of itself find vessel curvature, takes some time to generate in vivo, and is limited to larger-diameter coronary vessels. MRI, while potentially providing geometric information and wall structure details for coronary arteries, is still not at a state-of-art where spatial resolution is as good as desired for the present study. Thus, to explore fluid dynamic features of curvature variation as well as curvature and flow pulsatility we have chosen to examine flow in a branched vessel model with prescribed but representative parameter values.
Section snippets
Basic assumptions
A three-dimensional bifurcating model is defined by an analytical intersection of two cylindrical tubes lying on a sphere that represents an idealized heart surface (Fig. 1). Due to the sharp edge at the junction of arterial branches the solution has a numerical singularity that is localized in a small region close to the junction and does not seem to affect the results in other parts of the domain. The heart motion is simulated by changing the sphere radius, ; the center of the sphere is
Methods and verification
The three-dimensional, unsteady, incompressible Navier–Stokes equations are cast in an arbitrary Lagrangian Eulerian (ALE) frame and solved using the parallel solver NEKTAR that employs the new generation of spectral/hp element methods (Karniadakis and Sherwin, 1999). The mesh, constructed using the commercially available mesh generator Gridgen (Pointwise, Inc.), has 6649 tetrahedral elements (Fig. 2). The initial domain corresponds to the model geometry with sphere radius . The advantage
Results
We have summarized the cases we have simulated in Table 1. In the two cases marked with , the geometry was fixed at the minimum and maximum radii for the dynamic case with , which correspond to and 61.875 mm, respectively. Next, we present the main results for representative cases only.
Summary and conclusions
We have presented new results on the effects of unsteady flow in an unsteady (flexing) geometry—a side-branched, curved tube flexing rhythmically in curvature—as an approximation of the flows in and motions of epicardial coronary arteries. These results go further than the recent study of Zeng et al. (2003), which lacked a side branch, and the earlier study of Weydahl and Moore (2001), which included a side branch but lacked flow pulsatility. The tube geometry, the flexing motion and the flow
Acknowledgements
This work was supported by an NSF-ITR Grant (CCR-0086065) and by BSF (IBSF 2001150). We would like to thank S.J. Sherwin of Imperial College and Z. Yosibash and A. Yakhot of Ben Gurion University for useful discussions. Computations were performed on the AHPCC Los Lobos Linux cluster.
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