Asymptotic Models of Anisotropic Heterogeneous Elastic Walls of Blood Vessels

Using the dimension reduction procedure in the three-dimensional elasticity system, we derive a two-dimensional model for elastic laminate walls of a blood vessel. The wall of arbitrary cross-section consists of several (actually three) elastic, anisotropic layers. Assuming that the wall’s thickness is small compared with the vessel’s diameter and length, we derive a system of the limit equations. In these equations, the wall’s displacements are unknown given on the two-dimensional boundary of a cylinder, whereas the equations themselves constitute a second order hyperbolic system. This system is coupled with the Navier–Stokes equations through the stress and velocity, i.e. dynamic and kinematic conditions at the interior surface of the wall. Explicit formulas are deduced for the effective rigidity tensor of the wall in two natural cases. The first of them concerns the homogeneous anisotropic laminate layer of constant thickness like that in the wall of a peripheral vein, whereas the second case is related to enforcing of the media and adventitia layers of the artery wall by bundles of collagen fibers. It is also shown that if the blood flow stays laminar, then the describing cross-section of the orthotropic homogeneous blood vessel becomes circular.