Mathematical modeling and simulation of blood circulation

In this talk we address some mathematical issues arising from the modelling of blood circulation.

The interest in the use of mathematical modelling and numerical simulation in the study of the cardiovascular system (and its inherent pathologies) has greatly increased in the past few years. Blood flow interacts mechanically and chemically with vessel walls producing a complex fluid-structure interaction problem, which is practically impossible to simulate in its entirety.

Several reduced models have been developed which may give a reasonable approximation of averaged quantities, such as mean flow rate and pressure, in different sections of the cardiovascular system. They are, however, unable to provide the details often needed for understanding a local behavior, such as, e.g., the effect on the shear stress distribution due a modification in the blood flow consequent to a partial vessel stenosis.

We will consider models with different complexity. The most complex model is based on the coupling of the Navier-Stokes equations with structural models for vessel walls. An intermediate model is derived from integrating these equations on a section of a vessel geometry, and it is formed by a system of hyperbolic equations for the evolution of mean pressure and flow rate. An even simpler model that will be considered is based on the solution of a system of ordinary differential equations which describe electrical networks.

The derivation of these models will be presented together with schemes for their numerical solution. Furthermore, we will specifically address the coupling problem.

This heterogeneous approach looks a viable solution to obtain detailed numerical simulation of sections of the cardiovascular apparatus, while properly accounting for the presence of the global system. It is expected that this could open new possibilities for the use of numerical modelling for medical research.