Speaker: Sunsica Canic.
Title: Blood Flow Through Axi-Symmetric Arterial Sections Before and After Endovascular Repair: Modeling, Analysis and Numerical Simulation.
Affiliation: University of Houston.
Date: Monday, 3 February 2003.
Place and Time: Room 102 - Love Building, 3:35-4:30 pm.
Refreshments: Room 204 - Love Building, 3:00 pm.

Abstract. The complexity of the cardiovascular system features a tremendous variety of districts like large arteries, vessels of medium caliber as well as capillaries. Except for the tiny capillaries, the blood flow can be assumed to behave like a continuum, as well as incompressible, except for some severe pathological situations. The incompressible Navier-Stokes equations can be used to model the flow in large, or incompressible Stokes equations in small arteries. To analyze some relevant properties of blood flow in specific arterial districts for specific medical problems, two important effects often need to be taken into account: pulsatile nature of the flow and the compliant nature of the vessel walls.
   Despite the incredible power of supercomputing now-a-days, it is still impossible to take all these effects into account in order to simulate large sections of the human cardiovascular system in a realistic time frame. This is why simplified models describing fluid-structure interaction between the pulsatile blood flow and the compliant vessel wall are crucial in fast, real-time calculations, often needed by medical specialists.
   This talk will address a rigorous mathematical approach in the derivation of simplified equations using the axi-symmetric nature of compliant arterial sections (either treated with axy-symmetric prostheses or not). The compliant arterial sections are modeled using the Navier equations for a linearly elastic membrane. Depending on the size of the vessel, the resulting simplified (effective) equations are either hyperbolic (derived from the coupling between the Navier-Stokes equations for the flow and the Navier equations for the vessel wall), or parabolic (derived from the coupling between the Stokes equations for the flow and the Navier equations for the vessel wall). The effective equations are rigorously justified through a weak convergence result and through the error estimates. The error estimates are new in the literature on incompressible Stokes flow in compliant channels.
   Movies showing numerical solutions of two models will be presented. One is a model of creeping flow in small (coronary) arteries, and the other is model of blood flow through the abdominal aorta treated with stents in nonsurgical treatment of aortic abdominal aneurysm.
Collaborators:
1. Prof. Andro Mikelic, Universite Claude Bernard Lyon 1, France.
2. Medical Doctors: Z. Krajcer, Texas Heart Institute, (St. Luke's Hospital), G. Dorros, Arizona Heart Institute.
3. Prof. Ravi-Chandar, Aerospace Engineering, University of Texas in Austin.