Large scale PDE constrained optimization of cardiac defibrillation

Abstract : The bidomain model consist of a system of elliptic partial differential equations coupled with a non-linear parabolic equation of reaction-diffusion type, where the reaction term, modeling ionic transport is described by a set of ordinary differential equations. An extra elliptic equation for the solution of an extracellular potential needs to be solved on the torso domain. The optimal control approach is based on minimizing a properly chosen cost functional depending on the extracellular current as input at the boundary of torso domain, which must be determined in such a way that wavefronts of transmembrane voltage in cardiac tissue are smoothed in an optimal manner. We establish the existence of the finite element scheme, and convergence of the unique weak solution of the bidomain-torso model. The convergence proof is based on deriving a series of a priori estimates and using a general compactness criterion. The optimal control framework for the cardiac defibrillation is provided and proof of the first order optimality conditions is shown. Anatomically realistic such multiscale models of torso embedded whole heart electrical activity are computationally expensive endeavour on its own right and solving optimal control of such models in an optimal manner is the most challenging issue. A primal-dual active set strategy is employed for treating inequality control constraints. In this talk, a parallel finite element based algorithm is devised to solve an optimal control problem on such complex geometries and its efficiency is demonstrated not only for the direct problem but also for the optimal control problem.