IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

Inverse Boundary Value Problems in Vascular Mechanics

  • April 14, 2014
  • 1 p.m.
  • LeConte 312

Abstract

Theoretical studies of vascular growth and remodeling are based on fundamental solid mechanics, the concept of a preferred mechanical state for the arterial wall, and understood mechanisms by which vascular tissue sense and respond to their local mechanical environment. Predictive mathematical models which include inverse boundary problems (BVPs) formulated from these principals are realized in a host of commercially available software, ultimately providing powerful tools to explain vascular function, development, and the origins and progression of some vascular disorders. The inverse approach yields the final outcomes of arterial remodeling without postulation of growth laws, the analytical form of which remain an open question. In general, the formulation of an inverse BVP requires specification of a system of “n” transcendental equations with “n” unknown model parameters. Each equation represents a relation between the analytical expression for a certain mechanical quantity and its prescribed numerical value. The present seminar will provide results from the following two studies to illustrate the benefits of inverse methods in vascular mechanics. I. Understanding the effects of tissue structure on the geometrical outputs of arterial remodeling. Currently, there is debate about the relative importance of various factors in determining the final outcome of adaptive vascular remodeling. Inverse BVPs are formulated and solved to show that accounting for changes in collagen fiber configuration significantly impacts theoretical predictions of pressure-induced arterial remodeling outcomes, while accounting for changes in the mass fractions of structural constituents has comparatively minimal influence. Conversely, predicted flow-induced remodeling outcomes were relatively insensitive to both of these factors. II. Examining the concept of “optimal mechanical operation” along the length of the aorta. It is welldocumented that the geometrical dimensions, the longitudinal stretch ratio in situ, certain structural mechanical descriptors such as compliance and pressure-diameter moduli, as well as the mass fractions of structural constituents vary along the length of the descending aorta. A mathematical model is proposed and inverse BVPs are solved for the equations that follow from finite elasticity, structure-based constitutive modeling within constrained mixture theory, and stress-induced control of aortic homeostasis. Published experimental data are used to illustrate the predictive power of the proposed model. The results obtained are in agreement with published experimental data and support the proposed principle of optimal mechanical operation for the descending aorta.

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