Mathematical modeling of flows in blood vessels of complex shape

Abstract

A study of the structure and topography of the cerebral arterial bed of 156 virtually healthy individuals verified 17 variants of the arterial circle of the great brain, most of which are shown in actual photographs. An analysis of various approaches to modeling of cerebral vessels was carried out and it is shown that all of them are based on the representation of a blood vessel in the form of a linear cylinder. In reality, in the blood system of the brain, such sections are only rare short elements of vessels that vary in diameter and direction. Cylindrical coordinates have been most commonly used to construct mathematical models of cerebral vessels, but they are not applicable to bending vessels. The aim of this work: to build a mathematical apparatus allowing to adequately describe various types of vessels containing sections with complex geometry (bends, especially U-shaped ones, spirals, etc.). Four typical sections of the vascular system were identified, in correspondence with which mathematical surfaces were introduced, and the corresponding mathematical description was selected for each of them. The mathematical apparatus introduced into consideration allowed us to build a description of hydrodynamic processes occurring in vessels of complex shape. The results of geometrical modeling in the Mathlab system and the corresponding fragments of the script are presented. The developed mathematical software made it possible to construct a system of transfer equations for vessels of complex shape.