Abstract of the scientific program
There are two topics on which we will be working during the three years of the project. The first topic is conductivity parameters identification in the electrocardiography imaging inverse problem. The second topic concerns the ionic model parameters identification problem.
In the first topic, we will address the electrocardiography imaging inverse problem using new approaches. The novelty is that we will be optimizing the conductivities of the organs in the thoracic cage while constructing the electrical potential on the heart surface. This question has not been addressed before in the literature, although, we know that the electrical information is affected by the torso conductivity heterogeneity [16, 17]. The aim is to see if combining conductivities optimization with the electrical reconstruction allows significantly improving the quality of the reconstructed signals. In the second topic we will be using the dynamic model of the electrical activity of the heart which is generally based on the stateoftheart monodomain, bidomain [14] or Eikonal model [15], the pathological information that we will introduce to the numerical model are either structural information in this case they will be linked to the geometry (hypertrophy, hypotrophy, malformation,…) or they could be related to the functional aspect of electrical activity of the heart. In the latter case, the parameters of reaction diffusion model like the conductivities (diffusion parameters) or the ionic model parameters (reaction parameters) have to be fitted to the patient condition. This topic is challenging in terms of mathematical analysis and numerical implementation. On the theoretical side, we will rely to the two following questions: 1) For a given electrical measurement taken from a patient (either in the heart or on the body surface), is there a unique set of parameters which allows the model to fit these measurements. 2) In case the identification problem has a unique solution, how sensitive is this solution to the errors in the measurements. On the numerical side, we will be using gradient descent methods, we will consider the adjoint problem corresponding to a given misfit functional representing the measure of the gap between the observations and the model.
Scientific progress program
Parameters identification and estimation problems
In this year, we have been working on the problem of ionic parameter identification using the strongly coupled bidomain model. We succeed to prove the identifiability under certain conditions on the conductivities on the heart domain and a very important condition on the ionic current of the cell membrane model that could be summarized as follows: “The condition is that there exist a time t0 at which all the ionic current are linearly independent almost everywhere in space”. We also started implementing the numerical schemes to numerically solve the parameter estimation problem. First, we wrote the adjoint problem of the bidomain equation combined with the Luo and Rudy model. We are now testing the convergence of gradient descent method to solve the parameter estimation problem for a set of ionic conductance parameters.
Electrocardiography imaging inverse problem
We also have been working this year on the assessment of fifteen algorithms for solving the electrocardiography imaging inverse problem. We assessed the performance of each algorithm in the localization of pacing sites in the left and right ventricular walls. We used both in silico and exvivo data that has been collected from the Torso tank experiment performed at the LIRYC institute. Unfortunately, there are many issues with the experimental data due to the fact that pig hearts have been explanted. In fact, many ischemic regions appear in the signals influencing the accuracy of the gold standard measured data. There are also many contact electrodes that are measuring a high level of noise du to the movement of the heart during the experiment.
In this year a strong effort has been made in writing and finalizing papers and working on the reviewers requests for the different papers that have been published/accepted this year (see publication section). In particular for the papers that have been accepted recently in the special issue “Mathematical modeling in Cardiology” that would appear soon in the MMNP journal.
Joint publications
Journal papers that have been published/accepted in 2018

 Abidi, Yassine, et al. “On the identification of multiple space dependent ionic parameters in cardiac electrophysiology modelling.” Inverse Problems 34.3 (2018): 035005..
 Amel Karoui, Laura Bear, Pauline Migerditichan, Nejib Zemzemi. Evaluation of fifteen algorithms for the resolution of the electrocardiography imaging inverse problem using exvivo and insilico data Frontiers in Physiology, Frontiers, In press .
 Saloua Aouadi, Wajih Mbarki, Nejib Zemzemi. Towards the modelling of the Purkinje/ myocardium coupled problem: A wellposedness analysis Journal of Computational and Applied Mathematics, Elsevier, In press .
We considerably participated in a special issue « Mathematical Modelling in cardiology » that would be published soon in the Mathematical Modelling and Natural Phenomena journal :

 Najib Fikal, Rajae Aboulaich, Emahdi Guarmah, Nejib Zemzemi. Propagation of two independent sources of uncertainty in the electrocardiography imaging inverse solution Mathematical Modelling of Natural Phenomena, EDP Sciences, In press .
 Mohammed Addouche, Nadra Bouarroudj, Fadhel Jday, Jacques Henry, Nejib Zemzemi. Analysis of the ECGI inverse problem solution with respect to the measurement boundary size and the distribution of noise Mathematical Modelling of Natural Phenomena, EDP Sciences, In press .
 Yassine Abidi, Mourad Bellassoued, Moncef Mahjoub, Nejib Zemzemi. Ionic parameters identification of an inverse problem of strongly coupled PDE’s system in cardiac electrophysiology using Carleman estimates Mathematical Modelling of Natural Phenomena, EDP Sciences, In press .
 Rabeb Chamekh, Abderrahmane Habbal, Moez Kallel, Nejib Zemzemi A nash game algorithm for the solution of coupled conductivity identification and data completion in cardiac electrophysiology. Mathematical Modelling of Natural Phenomena, EDP Sciences, In press .
Previous years journal papers
 Y Abidi, M Bellassoued, M Moncef, N Zemzemi. On the identification of multiple space dependent ionic parameters in cardiac electrophysiology modelling. Journal of Inverse Problems (2018). .
 S. Aouadi, W. Mbarki and N. Zemzemi. Stability analysis of decoupled time stepping schemes for the specialized conduction system/myocardium coupled prob lem in cardiology. (2017).
 J. Lassoued, M. Mahjoub, N. Zemzemi, Stability results for the parameter identi fication inverse problem in cardiac electrophysiology, Inverse Problems 32, 2016, p. 131. [hal:hal01399373] .
 R. Aboulaich, N. Fikal, E. M. EL Guarmah, N. Zemzemi(2016). Stochastic Finite Element Method for torso conductivity uncertainties quantification in electrocardiography inverse problem. Mathematical Modelling of Natural Phenomena, 11(2), 119.
 C. Corrado, J. Lassoued, M. Mahjoub, N. Zemzemi, Stability analysis of the POD reduced order method for solving the bidomain model in cardiac electrophysiology, Mathematical Biosciences, December (2015). [hal:hal01245685]
Software
We have an ADT (Pauline Migerditichan) that contributes to the developpement of the MUSIC platform in particular all the methods that have been developed partially in the Master thesis of Amel Karoui and this year in her PHD studies are now implemented in the MUSIC software:
Carmen Music Plugins
the password is: musicPassword