The paper is devoted to the analysis of electroencephalography (EEG) in neonates. The goal is to investigate the impact of fontanels on EEG measurements, i.e. on the values of the electric potential on the scalp. In order to answer this clinical issue, a complete mathematical study (modeling, existence and uniqueness result, realistic simulations) is carried out. A model for the forward problem in EEG source localization is proposed. The model is able to take into account the presence and ossification process of fontanels which are characterized by a variable conductivity. From a mathematical point of view, the model consists in solving an elliptic problem with a singular source term in an inhomogeneous medium. A subtraction approach is used to deal with the singularity in the source term, and existence and uniqueness results are proved for the continuous problem. Discretization is performed with 3D Finite Elements of type P1 and error estimates are proved in the energy norm (H 1-norm). Numerical simulations for a three-layer spherical model as well as for a realistic neonatal head model including or not the fontanels have been

In this paper we present some aspects of an inverse source problem in an elliptic equation with varying coefficients, using partial Dirichlet boundary measurements. A uniqueness result is established for dipolar sources including their number. Additionally, assuming the number of dipoles known, a stability result is obtained and an efficient numerical identification method is developed. Finally, numerical experiments illustrate the effectiveness of the approach and a discussion is given on electroencephalography (EEG) in neonates.

Electrocardiographic Imaging (ECGI) is a non-invasive procedure that allows to reconstruct the electrical activity of the heart from body surface potential map (BSPM). In this paper, we present a volume model to solve the electrocardiography inverse problem capable to take into account structural informations obtained by imaging techniques. The direct problem maps a volume current in the cardiac muscle (ventricles) to the body surface electrical measures. The model is based on coupling bidomain heart model with torso conduction. The corresponding inverse problem is solved with the Tikhonov regularization. Simulated database are used for the evaluation of this method and we compared them to standard method of fundamental solutions (MFS). The sensitivity to noise is also assessed. The correlation coefficients (CC) and the relative error (RE) of activation times were computed. Results show that the CC (respectively RE) median is respectively 0.75 for the volume model and 0.4 for the MFS (respectively 0.31 vs 0.35) on the epicardium. On the endocardium, the CC and the RE median are 0.65 and 0.33 for the volume method. In conclusion, the volume method performs better than the method of fundamental solutions (MFS) for any noise level, and reconstruct in addition endocardial information.

We propose a methodology to take into account the location of scars in ECGI problem. The method is to consider the whole body, including blood, heart and remaining volume as a conductor with an electric current source field localized in the heart. We identify the source best matching a given body surface potential map, by solving the classical quadratic optimization problem with a Tikhonov regularization term. The method behaves better than the MFS method in presence of a scar. The correlation coefficients of the activation times around the scar are improved up to 10 % on the epicardium, and 7 % on the endocardium, by adapting the Tikhonov regularization parameter and conductivity coefficient in the scar.