The Geometrical Multiscale Modelling of the Cardiovascular System: Coupling 3D FSI and 1D Models

Bugalho de Moura, Alexandra

The Geometrical Multiscale Modelling of the Cardiovascular System: Coupling 3D FSI and 1D Models

Friday 25th May 2007

Formaggia, L.

The subject of this thesis is the geometrical multiscale approach for the modelling and simulation of the cardiovascular system, which was introduced to deal with the complexity and diversity of the human circulatory system. In particular, in this thesis we have focused on the coupling between three-dimensional (3D) and one-dimensional (1D) fluid-structure interaction (FSI) models describing blood flow inside compliant vessels. The 1D model is a hyperbolic system of partial differential equations. The 3D model consists of the Navier-Stokes equations for incompressible Newtonian fluids coupled with a model for the vessel wall dynamics. We have studied the stability of the global 3D FSI - 1D coupling. We show that the stability at the continuous level can be achieved by means of a convenient reformulation of the Navier-Stokes equations. Moreover, several models for the structure vessel wall on the 3D FSI problem were considered and analyzed, showing how the 3D FSI - 1D coupling depends on them. From the numerical view point, we have studied each subproblem, 3D FSI and 1D, separately, namely we have demonstrated the stability of the discretization of the proposed non standard formulation of the Navier-Stokes equations, and we have analyzed the global 3D FSI - 1D coupling problem. We propose both a staggered explicit algorithm and an implicit coupling, as well as different coupling strategies. Intensive numerical tests were carried out, including the coupling at both inflow and outflow artificial sections, as well as a realistic physiological case of interest consisting of the coupling between a 3D anatomically geometric carotid bifurcation with a realistic 1D network, representing the circle of Willis.