Mathematical analysis of aortic deformation in aneurysm and wall dissection

Aortic dissection is an extremely severe pathology. From the viewpoint of mechanics, the aorta is a multilayered anisotropic reinforced shell, which is subjected to periodic loading under the action of pulsed blood pressure. Various issues of mathematical modeling of dissection of the aorta and large arteries are considered in the present study. Modern mathematical models of the aortic and arterial wall structures obtained by processing experimental data on biaxial stretching of samples are reviewed. These mathematical models can be conventionally divided into two classes: 1) effective models, where the internal structure of the walls is ignored, but mechanical parameters of the material “averaged” over the wall thickness are introduced; 2) structured models, which take into account the multilayered (up to three layers) structure of the artery with addition of one to four families of reinforcing fibers. One of the most popular models (Holzapfel–Gasser–Ogden model) is considered in detail. This model describes a two- or three-layered artery with two families of reinforcing fibers. For this model, tables of design parameters are provided, and numerical simulations of arterial rupture and dissection are performed. The blood vessel is subjected to pulse pressure of blood flowing through it. It is shown that rupture of the inner layer of the vessel leads to an increase in the stress at the outer wall of the vessel. As the rupture thickness and length increase, the stress at the outer wall of the vessel is also increased. If there is an aneurism of the vessel, the stress is twice that in the vessel without the aneurism. Dissection of the inner wall of the vessel leads to an increase in the stress at the wall: the stress decreases with increasing rupture width for a straight vessel and increases for a vessel with an aneurism. The stress calculations on the “tip” of delamination show that the maximum stress is reached at the outer wall of the rupture.