Analyses théoriques et numériques de la méthode de factorisation pour les problèmes inverses en électrocardiographie.

Abstract

In this work, we provide a new mathematical formulation for the inverse problem in electrocardiographic imaging. The novelty in this ap- proach is that we account for missing measurements on the body surface. The electrocardiography imaging inverse problem is formulated as a data completion problem for the Laplace equation. The Neumann boundary condition is given at the whole body surface. The di¢ culty comes from the fact that the Dirichlet boun- dary condition is only given on a part of the body surface and thus an incomplete boundary is adjacent to a complete boundary. In order to construct the electri- cal potential on the heart surface, we use an optimal control approach where the unknown potential at the external boundary is also part of the control variables. We theoretically compare this case to the case where the a Dirichlet boundary condition is given on the full accessible surface. We then compare both cases and based on the distribution of noise in the measurements, we conclude whether or not it is worth to use all the data. We use the method of factorization of ellip- tic boundary value problems combined with the nite element method. The idea is to embed the initial problem into a family of similar problems on subdomains bounded by a moving boundary (along a axis of evolution that we de ne) from the torso skin to the epicardium surface. Regarding the inverse problem, mathematical analysis allows to write an optimal estimation of the epicardial potential based on a quadratic criterion. Then, we can analyse the ill-posed behaviour of the inverse problem and propose a better regularization and discretization of the problem. We illustrate the theoretical results by some numerical simulations in a cylindrical domain. We numerically study the e¤ect of the size of the missing data zone and the noise distribution on the accuracy of the inverse solution.