1Worcester Polytechnic Institute, Worcester, Massachusetts, U. S. A.
2Beijing Normal University, China
3Shinshu University, Japan
4Georgia Institute of Technology, Atlanta, Georgia,\forcenl U. S. A.
Artery collapse is believed to be related to cyclic bending, stretching and compression of the artery which can lead to plaque cap rupture, stroke and heart attack. The exact mechanism behind this collapse-rupture process is not well understood.
In this paper, a nonlinear three-dimensional model with fluid-wall interaction is introduced to simulate blood flow in stenotic carotid arteries. The incompressible Navier-Stokes equations are used as the governing equations for the fluid. The tube wall is assumed to be elastic, homogeneous and isotropic. Nonlinear stress-strain relationship for the tube wall made of PVA gel was measured experimentally and used in the model. The radial strain was 62\% when the tube was subjected to a 36\% axial strain and 100 mmHg inner pressure. The axial and circumferential bending moments are assumed to be proportional to the deviations of the tube axial and circumferential curvatures from its resting shape. The moment equilibrium equations involving the stresses and strains of the tube wall and the pressure and shear stresses from the fluid are used to determine the wall deformations. No-slip conditions are specified at the tube wall. Pressure is specified at the inlet and outlet of the tube and the external pressure is set to be zero. The geometry and the ranges of the parameters in the model were chosen to match the experimental set-up so that the computational results can remain physiologically relevant.
A numerical method using the Generalized Finite Difference Method (GFDM) and boundary iteration techniques is developed to solve the computational model. The GFDM derives finite difference schemes from non-uniform, non-rectangular and unstructured grids and is especially suitable for irregular geometries. The Navier-Stokes equations for the fluid and the moment equilibrium equations for the tube wall were solved iteratively to determine the wall deformations and the flow and pressure fields.
Using the three-dimensional model, we were able to simulate wall collapse under physiological conditions consistent with experimental observations. Three-dimensinal wall deformation and stress-strain distributions, pressure, velocity and shear stress fields provided information for many further investigations. With an 80\% axisymmetric stenosis, the tube wall starts to collapse when pressure drop reaches 80 mmHg with inlet pressure set at 100 mmHg. Once the collapse condition is reached, a small change in the pressure condition can lead to large change in the wall deformation which changes all the other flow parameters. The pressure field and wall deformation obtained from the 3-D model improves the axisymmetric results considerably with the 3-D results give lower minimum pressure and much larger nonsymmetric wall deformation under the same pressure conditions. This does not come as a surprise because the axisymmetric pressure can be viewed as an average around the axis of symmetry and the wall cannot buckle under axisymmetric assumption. Wall compression varies along the circumferential direction as the curvature changes. The behavior of shear stresses in the buckled region distal to the stenosis is complicated. Both high and low shear stresses were observed and both can have important clinic implications. Experiments have demonstrated that cyclic bending and compression of arteries can lead to plaque cap rupture. Further research simulating unsteady flow in a collapsible tube with asymmetric stenosis will lead to a better understanding of the collapse-rupture process. {\bf Acknowledgment.} This research was supported in part by a grant from the Whitaker Foundation and NSF grant DMS-9505685.