1Worcester Polytechnic Institute, Worcester, Massachusetts, U. S. A.
2Beijing Normal University, China
3Shinshu University, Japan
4Georgia Institute of Tech., Atlanta, Georgia, U. S. A.
There has been increasing interest in the research for blood flow in collapsible arteries. Arteries with high grade stenoses may collapse under physiologic conditions. The collapse may cause mechanical conditions and hemodynamic conditions which may lead to artery fatigue and plaque cap rupture. These again may lead to stroke and heart attack. The mechanism related to the collapse process is not well understood.
An axisymmetric computational model with fluid-wall interactions and large wall deformation is introduced to study viscous flow in collapsible stenotic tubes. The flow is assumed to be Newtonian, viscous, laminar and incompressible. The tube wall is assumed to be elastic, homogeneous and isotropic. The bending moments are assumed to be proportional to the deviations of the tube axial and circumferential curvatures from its resting shape. {\it In vitro} experiments were performed to determine the nonlinear stress-strain relationship (tube law) for both bovine arteries and elastic tubes made of PVA gel whose mechanical properties are close to bovine arteries. Dimensions and physical parameters were chosen to match blood flow in human carotid arteries so that the results would be physiological relevant. The radial expansion of the PVA gel tube was 62\% under 100 mmHg internal pressure and a 36.2\% axial stretch.
A numerical method using the Generalized Finite Difference Method (GFDM) and boundary iteration techniques is developed to solve the computational model. The Navier-Stokes equations for the fluid and the moment equilibrium equations for the tube wall were solved iteratively to determine the wall deformation and the flow and pressure fields. Staggered meshes and upwind techniques were used to prevent the wiggling of the pressure field.
Extensive computations were conducted to quantify the effects of the stenosis severity, wall stiffness and inlet and outlet pressures on the collapse conditions and location of the collapse. Our results indicate that the pressure field near a stenosis has a complex pattern. Negative pressures ranging from -3 mmHg to -45 mmHg occurred for stenoses with severities from 40\%-90\% by diameter under inlet pressure 100 mmHg and outlet pressure less than 30 mmHg. While the minimum negative pressure was found at the throat of the stenoses, tube wall collapse was observed about one diameter distal to the stenosis experimentally. To explain this, a new collapse criterion for stenotic tubes is proposed which involves not only the transmural pressure, but also longitudinal tension, tube axial curvature, and tube stiffness. Predictions made by the new criterion is consistent with experiments. Maximum shear stresses in the order of 2,000 dyn/${\rm cm^2}$ were observed at the throat of the stenoses. The stiffness of the tube wall has considerable effect on wall deformation and the change of dynamic severity of the stenoses which in term have considerable effects on the pressure and flow fields. Good agreements were found between computational and experimental results on flow rates, velocities, wall deformation and dynamic severity changes.
Negative transmural pressure, conditions and locations of artery collapse, peak shear stress and plaque cap rupture have important clinic implications. Further studies of the pathology identifying an exact location of plaque cap rupture may lead to a better understanding of the mechanism behind rupture. Acknowledgment. This research was supported in part by a grant from the Whitaker Foundation and NSF grant DMS-9505685.