##### Assessement of torso conductivities effect on the ECGI inverse problem solution

During the first year, we have been working on the effect of torso conductivity inhomogeneity on the ECGI inverse problem solution. We proceeded following two different strategies. The first idea is simple: We simulate the forward problem using a heterogeneous 3D real life human torso that takes into account different organs conductivities (lungs, bones and the rest of the tissue). Then, we solve the inverse problem using a homogeneous torso. We assume that the mathematical and geometrical models are perfect. Only the conductivities are supposed to be unknown. This test is very important because it isolates the issue of torso inhomogeneity and its effect on the ECGI inverse solution. The first results show that taking into account the heterogeneities allows a gain of 20% in terms of accuracy when we don’t consider errors in the electrical potential measurements. It also shows that this accuracy is altered when high level of noise is introduced in the electrical measurements [CINC1]. The synthetic data produced in this work would be used later to test the different methods developed in the EPICARD framework. In the second approach, we used a stochastic finite element method in order to quantify the effect of torso conductivity uncertainties on the ECGI-inverse solution. The problem is formulated using a cost function that we minimize using a conjugate gradient method. We minimize the mean value of a stochastic energy functional where the stochastic part comes from the conductivity uncertainties. The results show that the error in the lungs conductivities has more effects than the other organs [FIMH1].

##### Nash game approach for solving ECGI inverse problem

During the visit of M. Kallel to Inria in May, we have been working on using the Nash-game approach for solving the ECGI inverse problem. This method has been used in 3D but for spherical geometries where the solution is an analytical function. We have adapted the FreeFem++ code to the 3D anatomical model and we conducted some numerical simulations. The main reason for using FreeFem++ is that M. Kallel and A. Habbel used this type of finite element to approximate some fluxes [11]. We rose up some technical issues related to the computational time. The use of Raviart Thomas finite elements method in order to minimize one of the two players cost function makes the optimization process very time consuming. We started looking for a new strategy to solve this issue.

##### Extended domain approch for the ECGI inverse problem regularization

F. Ben Belgacem and F. Jelassi have introduced the method in the field of the data completion problem for the Laplace equation and have proved its robustness in 2D [17]. N. Zemzemi has implemented the extended domain approach used as a regularization method for solving the ECGI inverse problem. But the first results did not show any improvement compared to the non-extended Steklov-Poincaré variational approach. These results have been discussed with F. Ben Belgacem. He started with F. Jelassi looking at the problem using a 2D section of the 3D anatomical geometry.

##### Ischemia dection based on a variational approach combined with a levelset method

C. E. Chavez, N. Zemzemi, Y. Coudière, F. Alonzo and A. Alvarez have been working on introducing a level set approach for detecting ischemia regions in the heart. This method has been used for localizing cancer regions. We used the state-of-the-art monodomain model coupled to two state variables ionic model. We characterize the ischemic region using two parameters. The results obtained in [FIMH2] show that this method allows finding the ischemic region with a very good accuracy. In [CINC4] we compared the accuracy of level set approach combined with the adjoint problem is more accurate than two other state-of-the-art method. We still need to validate this method against physiological recordings.

##### New formulation for the missing electrodes problem in ECGI inverse problem

M. Addouche, Nadra Belaiib, J. Henry, Fadhel Jday and Nejib Zemzemi have been working on the mathematical formulation of how to introduce missing electrodes issue in the ECGI inverse problem using the factorization of boundary values method. This method has been introduced to the ECGI inverse problem by J. Bouyssier, J. Henry and N. Zemzemi in [ISBI2015]. We started by formulating the problem using Robin boundary conditions by introducing a Robin to Dirichlet operator. This approach didn’t provide good results due to a singularity at the interface between the known and unknown potential boundaries. We found a better way to solve this problem by introducing the torso potential missing data in the control variables. We use Dirichlet to Neumann and Neumann to Dirichlet operators as introduced. Using this formulation we obtain three equations: Two equations give the potential and its flux on the heart boundary and the other gives the value of the potential on the torso surface where the data is missing.