## Section: New Results

### Homogeneous Neumann condition on the torso for solving inverse problems

The electrical activity of the heart creates an electrical field in the body. This phenomenon is classically modelled in a quasistatic manner by Laplace's equation. The non-invasive electrocardiographic imaging (ECGI) problem consists in retrieving the best electrical map on the heart from given torso measurements. Classically, the solution is found as the best fit between data generated by a forward problem and the actual torso measurements, and it needs a regularization. Hence the inverse solution depends on the matrix of the forward problem, called the transfer matrix, and the choice of the regularization procedure. In 2006, a meshless method based on the method of fundamental solutions (MFS) was adapted by Y. Wang and Y. Rudy [54] to directly solve the inverse problem, combined with a 0-th order Tikhonov regularization. The MFS method is notably more robust than previous methods (e.g. BEM) to the uncertainties introduced by the segmentation of the geometries. In the MFS, the potential is expressed as summation of the fundamental solution of the Laplace equation over a discrete set of virtual point sources placed outside of the domain of interest. The inverse solution is searched as the set of sources that best fit the boundary conditions on the torso, up to the regularization term. This formulation yields a linear system, which matrix depends on the torso and heart geometries, and the boundary conditions at the torso surface. The regularization parameter also heavily depends on the properties of the transfer matrix. The boundary conditions considered in [54] are: i) the Dirichlet conditions, meaning that the potentials at the torso surface are fitted to the recorded ones, ii) homogeneous Neumann conditions (HNC) meaning that the normal flux of current is minimized.

Numerically, the HNC requires to build accurate directions at each measurement location of the body surface, which is a first difficulty. In addition, the body is cut at the top and the bottom where no-flux conditions are probably not relevant. Lastly, the matrix coefficients related to the HNC appears to be much smaller than the ones from the Dirichlet condition, due to the distance between the torso and the actual electrical source (the heart).

J. Chamorro-Servent, Y. Coudière and R. Dubois studied the effect of the HNC on the matrix. They showed that enforcing the Neumann condition has a negligible effect on the solution of the inverse problem. Reconstructed potentials and activation time maps were built for in-silico data. No major differences were found between the standard MFS and the MFS removing the HNC in terms of potentials and activation times. In addition, removing the HNC reduces the ill-conditioning of the problem and the computational burden: the normal at the torso surface is not required anymore, and the problem size is divided by 2. The results of this work were presented as a poster in CinC 2016, and collected in a proceeding for the same conference by J. Chamorro-Servent et al. [18].