Section: New Results
Electrophysiology
Participants : Muriel Boulakia, Miguel Ángel Fernández, Jean-Frédéric Gerbeau, Céline Grandmont, Nejib Zemzemi.
Decoupled time-marching schemes
M.A. Fernández and N. Zemzemi have investigated the approximation of the cardiac bidomain equations, either isolated or coupled with the torso, via first order semi-implicit time-marching schemes involving a fully decoupled computation of the unknown fields (ionic state, transmembrane potential, extracellular and torso potentials). For the isolated bidomain system, M.A. Fernández and N. Zemzemi show that the Gauss-Seidel and Jacobi like splittings do not compromise energy stability; they simply alter the energy norm. Time-step constraints are only due to the semi-implicit treatment of the non-linear reaction terms. Within the framework of the numerical simulation of electrocardiograms (ECG), these bidomain splittings are combined with an explicit Robin-Robin treatment of the heart-torso coupling conditions. They show that the resulting schemes allow a fully decoupled (energy) stable computation of the heart and torso fields, under an additional mild CFL like condition. Numerical simulations, based on anatomical heart and torso geometries, illustrate the stability and accuracy of the proposed schemes. These results have been reported in [40] .
Inverse problems
M. Boulakia and C. Grandmont have been working with A. Osses (University of Chili, Santiago) on the parameter identification problem for the Allen-Cahn or bistable equation which can be viewed as a simplified model in cardiac electrophysiology. In [18] , through suitable Carleman estimates, they recover parameters of the ionic model from volume or surface measurements and they obtain Lipschitz stability results.
Work in progress
M. Boulakia, M.A. Fernández, J.-F. Gerbeau and N. Zemzemi are working on the numerical identification of parameters of the cardiac model from measurements and in particular from electrocardiograms. Since the computation of the direct problem is quite costly, we first consider reduced order models obtained using for instance the POD method.
