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Inverse problem in electrocardiography via factorization method of boundary problems : how reconstruct epicardial potential maps from measurements of the torso ?

Julien Bouyssier 1, * Nejib Zemzemi 1 Jacques Henry 1
* Corresponding author
1 CARMEN - Modélisation et calculs pour l'électrophysiologie cardiaque
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest, IHU-LIRYC
Abstract : In this work, we present a new approach solving the inverse problem of electrocardiography. This approach is based on an invariant embedding method: the factorization method of boundary values problems. 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 define) from the torso skin to the epicardium surface. For the direct problem this method provides an equivalent formulation with two Cauchy problems evolving on this moving boundary and which have to be solved successively in opposite directions. This method calculates Neumman-Dirichlet and Dirichlet-Neumann operators on this moving boundary that satisfy Riccati equations. 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. One of the advantages of this method is the computation of the potential at different times during cardial cycle : it is not necessary to repeat the resolution of all the equations at every time.
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Julien Bouyssier, Nejib Zemzemi, Jacques Henry. Inverse problem in electrocardiography via factorization method of boundary problems : how reconstruct epicardial potential maps from measurements of the torso ?. 2013. ⟨hal-00926344⟩

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