Mathematical and Numerical Models for Focal Cerebral Ischemia and Electrocardiology
Monday 11th January 2010
In this work we deal with the mathematical and numerical modeling of two challenging applications coming from biomedical subjects: focal cerebral ischemia and electrocardiology. These topics have a great medical importance since heart and cerebrovascular diseases are major causes of death in industrialized countries. The mathematical modeling, at earlier stages, can help the comprehension of these complex biophysical phenomena and, at later stages, can possibly become a predictive tool. Depending on the complexity of the problem and on the mathematical models available in literature the development of a research field can take years, decades or even more. Mathematical modeling of cerebral ischemia is still an open issue. Several models have been presented in last years, each of them designed to capture particular mechanisms or aspects of the whole complex phenomenon. Difficulties in this context arise also in the identification and evaluations of parameters. Moreover, the validation of the proposed models relies mostly on qualitative comparisons with experimental data. Recently, a coordinated proposal for studying cerebral ischemia has been introduced by Dronne et al., and several aspects pointed out in the initial proposal have been developed in subsequent papers. To our knowledge, numerical simulations presented in the literature in this field are so far limited to 2D simple geometries. In this work we focus the attention on the ischemic dynamics related to blood perfusion of the brain region at hand. Consequently our model considers tissue metabolism, ion dynamics and hemodynamics, together. This choice allows us to describe both the tissue metabolism based on the concentration of oxygen and glucose in blood and tissues and the hemorrhage that can occur after the occlusion. To our knowledge it is the first time that hemodynamics is explicitly modeled in combination with tissue metabolism; in [95, 89] only an extremely simple lumped model for the blood flux was presented. In order to model the ionic currents across the membrane, we use the Tuckwell and Miura model , able to capture the so called Spreading Depression (SD) phenomenon. The strong nonlinearities of the model and the different time scales at hand make the problem hard to be managed by numerical methods. Mathematical modeling in Electrocardiology is nowadays an almost mature field. Models for action potential propagation at a cell level and in the whole tissue are, by now, consolidated. Also, well-posedness results have been obtained for a large variety of propagation and ionic models. Now, part of the research efforts are addressed to find efficient numerical methods to solve the set of equations. Strategies to reduce computing time without loosing accuracy resort to time and/or space adaptivity and to set up efficient preconditioners to solve the linear systems coming from the discretization of the equations governing the phenomena, possibly based on parallel architectures. In this work we introduce a second order scheme for advancing in time. The proposed scheme is a generalization of the Rush-Larsen first order method, which maintains some favourable properties of the original scheme. Then, we devise a time-adaptive algorithm that results in a great reduction of computing time. We also devise a block-triangular model based preconditioner for the solution of the well known Bidomain system for the potential propagation (see , and, for more details, ). With respect to the ILU-CG preconditioner, the proposed preconditioner reduces significantly both the computational costs and the memory storage on serial architectures.