Numerical issues of modelling blood flow in networks of vessels with pathologies

Abstract

The synthesis of the blood circulation model and the elastic fiber model of the vessel wall allows us to take into account the influence of possible vessel pathologies on the global blood flow. The interaction is based on the state equation representing the dependence of the transmural pressure on the cross-section of the vessel. Numerical properties of both models are considered in the paper. The mathematical modelling of blood circulation is a fundamental problem lying at the junction of several disciplines, such as differential equations, numerical analysis, elasticity theory, and physiology. Several numerical implementations of blood circu- lation models taking into account elastic properties of blood vessels were created in the last decade (6,9,10,14,21,22). Previously we proposed an approach to synthesis of the blood circulation model and the elastic model of the vessel wall (24) taking into account the influence of possible vessel pathologies on the global blood flow. The distinctive feature of the approach is the use of merely one-dimensional dif- ferential operators, which provided us with an efficient numerical simulation tech- nology. The mathematical blood flow model is a system of differential equations for each vessel linked by boundary conditions at the points of vessel junctions (22). The mathematical model of the elastic vessel wall is based on the fiber approach (17,18) to the calculation of the reaction force as a response to the deformation of a fiber. The representation of an elastic body by sets of fibers of different configurations was successfully used for simulation of cardiac work (13) and collapsed veins (18). In our model we used the same types of fibers as in (18). The synthesis of both models is based on the state equation representing the dependence of the transmural pressure on the cross-section area of the vessel. This