Inverse Problem in Electrocardiography via Factorization Method of Boundary Value Problems Julien Bouyssier, Nejib Zemzemi, Jacques Henry, CARMEN team, Inria Bordeaux Sud-Ouest 200 avenue de la vieille tour, 33405 Talence Cedex Electrocardiographic Imaging (ECGI) is a new imaging technique that noninvasively images cardiac electrical activity on the heart surface. In ECGI, a multi-electrode vest records body-surface potential maps (BSPMs); then, using geometrical information from CT-scans and a mathematical al- gorithm, electrical potentials, electrograms and isochrones are reconstructed on the heart surface. The reconstruction of cardiac activity from BSPMs is an ill-posed inverse problem. In this work, we present an approach based on an invariant embedding method: the factorization method of boundary values problems [1, 2]. The idea is to embed the initial problem into a family of similar problems on subdomains bounded by a moving boundary from the torso skin to the epicardium surface. For the direct problem this method pro- Inverse Problem in Electrocardiography via Factorization Method of Boundary Value Problems : How reconstruct epicardial potential maps from measurements of the torso ? Julien Bouyssier, Nejib Zemzemi, Jacques Henry julien.bouyssier@inria.fr, nejib.zemzemi@inria.fr, jacques.henry@inria.fr Motivation and goal Solve the inverse problem in electrocardiography from measurements of the torso. factorizationmethod to compute epicardial potential maps. simplified presentation of the method by considering a cylindrical geometry of our problem. Conclusions and perspectives Conclusions : Direct optimal estimation of t and ! before using any discretisation : =# Analyse ill-posedness and propose a better regularization and discretization Equations for P and Q depend only of the geometry : =# Not necessary to repeat resolution at every time step of cardiac cycle Perspectives : Apply the method to 3D case where the moving boundary S will be a deformed surface : =# First : model of spheres =# Then : realistic geometries : how compute 3D surfaces ? + numerical cost ? . This method calculates Neumman-Dirichlet and Dirichlet-Neumann operators on the moving boundary using Riccati equations. Mathematical analysis allows to write an optimal estimation of the epicardial potential based on a quadratic criterion. The analysis of the of the inverse problem ill-posedness allows to compare different regularisation terms and choose a better one. For numerical simulations we first construct a synthetical data based on the ECG solver [3]. The electrical potential on the torso boundary is then extracts from the forward solution to be used as an input of the in- verse problem. The first obtained results using this method in 3D show that we can capture the wave front. Whereas the amplitude of the inverse problem solution is too low compared to the forward solution. References [1] Jacques Henry and Angel Manuel Ramos , La methode de factorisation des problemes aux limites, book (in preparation), (2013) [2] Fadhel Jday, Completion de donnees frontieres : la methode de plonge- ment invariant, PHD thesis, (2012) [3] Nejib Zemzemi,  Etude theorique et numerique de l'activite electrique du coeur: Applications aux electrocardiogrammes, PHD thesis, (2009) 2