Surface enthalpy and elastic properties of blood vessels

Abstract

1. Blood vessels form one of the most complicated and important functional systems of humans and other mammals subjected to many risks and poorly amenable to medicinal treatment. The mathematical simulation of blood flow through the aorta, arteries, veins, peripheral vessels, and capillaries both sepa rately and as a whole still remains an important problem. At the same time, the published bloodflow models give no possibility to take into account the complex (composite and anisotropic) structure of walls of vessels. Moreover, the formulation itself of experiments on determining elastic characteristics of walls of ves sels is at the stage of development (see [1, 2] and the review of literature in monograph [3, sect. 8]). This study is devoted to derivation of certain relations for deformable bloodvessel walls on the basis of the conventional representation of their laminar structure (see, for example, [1, 4, 3]). By means of the dimensionreduction procedure and on the basis of the concept of surface enthalpy, we derived a simple formula for the effective elasticmodulus tensor. Contrary to the habitual formulation of the mathematical problem, the blood vessel is not beforehand assumed as a circular cylinder, and the occurrence of a variable curvature, which does not complicate principally the asymptotic analysis, enables us to investigate the deformation of walls of arteries due to damage, such as inhomogeneous calcification (hyalinosis or calcinosis), oblong atherosclerotic deposits (cholesterolplaques), or various surgical treatments (clipping, stenting, and stitching).