Section: New Results

Inverse problems

• A Steklov-Poincaré approch to solve the inverse problem in electrocardiography [23]

  In the cardiac electrophysiology imaging commu- nity the most widely used approach to solve the inverse prob- lem is the least square formulation with different Thikhonov regularizations. Clinicians are not yet fully satisfied by the technology that solves the inverse problem. Reformulating the inverse problem could bring new techniques to solve it. In this paper we use the Steklov-Poincare ́ formulation of the Cauchy problem in order to solve the inverse problem in electrocardiography imaging. We present in this work the technique and an algorithm of gradient descent. We also show numerical results based on simulated synthetical data.

• A machine learning regularization of the inverse problem in electrocardiography imaging [22]

  Radio-frequency ablation is one of the most ef- ficient treatments of atrial fibrillation. The idea behind it is to stop the propagation of ectopic beats coming from the pulmonary vein and the abnormal conduction pathways. Medical doctors need to use invasive catheters to localize the position of the triggers and they have to decide where to ablate during the intervention. ElectroCardioGraphy Imaging (ECGI) provides the opportunity to reconstruct the electrical potential and activation maps on the heart surface and analyze data prior to the intervention. The mathematical problem behind the reconstruction of heart potential is known to be ill posed. In this study we propose to regularize the inverse problem with a statistically reconstructed heart potential, and we test the method on synthetically data produced using an ECG simulator.

• Inverse problem in electrocardiography via factorization method of boundary value problems : How to reconstruct epicardial potential maps from measurements on the torso ? [26]

  We are working on a new approach for solving the inverse problem of electrocardiography. This approach is based on an invariant embedding method: the factorization method of boundary values problems [35] . 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 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, the ill-posed behaviour of the inverse problem can be analyzed and a better regularization and discretization of the problem can be proposed. 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. In a first time the simplar case of a cylinder is considered. In a second time the method is applied to the 3D model of concentric spheres. The next step will be to use 3D deformed surfaces.

• Reconstruction of 3D depolarization wavefronts from surface optical mapping images [33]

  Starting from the diffusion-absorption equation of light in a tissue we solved the forward problem for excitation light using the FreeFem++ software (www.freefem.org/ff++ ). We first considered a spherical wave front expanding in time: the tissue is depolarized inside the sphere. This choice allowed us to locate the position of the excitation. Using this representation of the wavefront, we obtained in silico data. We defined a functional to minimize and implemented the BFGS method to solve the inverse problem. We tested our method on in silico data and obtained good results. We next compared our results with an approach developed by Khait [36] and found that our method is more accurate and that we have less restrictions for the convergence of the method. We modified the wave front into ellipsoid in order to start working on experimental data.