This is the abstract of a talk prepared for the Oeiras Mathematical and Computational Biology Workshop. June 20, 2003, Instituto Gulbenkian de Ciência
Abstract: Mathematical and numerical models together with computer simulations are playing an increasingly relevant role in biology and medicine. Applications to blood flow in the human circulatory system (and to its inherent pathologies) are certainly one of the major mathematical challenges of the next years. Relevant features have already been addressed but many fundamental issues have still to be fully understood. Blood is a multi-component mixture with complex rheological characteristics. It interacts both mechanically and chemically with vessel walls producing complex fluid-structure interactions whose mathematical analysis is still incomplete and which are practically impossible to simulate in its entirety.
In this talk we address some mathematical issues arising from the modelling of the cardiovascular system through problems of different complexity. In particular we will specifically consider the fluid-structure interaction problem of an incompressible generalized Newtonian shear-thinning fluid flowing inside a thin compliant vessel whose walls undergo small deformations under the action of the fluid. The numerical approach is based on a finite element approach for the coupling of the fluid equations in a moving domain, described in an Arbitrary Lagrangian Eulerian (ALE) frame, with a simple structural model for the vessel wall. The system is completed with suitable matching conditions which play the role of boundary conditions for the submodels. Some numerical results are already available and will be presented.
This is a highly interdisciplinary research work which demands a close connection between mathematical modelling, numerical simulation and clinical experiments. It is built on a strong international collaboration with internationally renowned mathematicians, bioengineers, computational fluid-dynamicists and medical researchers, in Europe and USA, in the framework of some ongoing national and international projects.