On some inverse conductivity recovery problem in a sphere: - Uniqueness and reconstruction results with applications to EEG

Electroencephalography (EEG) is a non invasive imaging technique that measures the effect of the electric activity of active brain regions, called sources, through values of the electric potential furnished by a set of electrodes placed at the surface of the scalp. A fundamental problem there is the inverse problem of source localization which aims at locating the sources of the electric activity using the acquired EEG measurements [2]. The quality of the source estimation depends on the accuracy of the conductivity model used to solve the problem. Among the head tissues, the skull conductivity is the one that influences most the accuracy of EEG source localization [3]. Indeed, modelling the electrical conductivity values for the scalp and the brain are relatively well-known and do not vary much across subjects, but this is not the case for the skull. We examine the inverse skull conductivity estimation problem, which aims at recovering the electrical conductivity properties of the skull from measurements given at the surface of the head by EEG measurements. Our goal is to show uniqueness and a constructive scheme for the inverse skull conductivity estimation problem using partial boundary EEG data from a single experiment, in the preliminary case of an homogeneous skull conductivity. This is a version of the many inverse conductivity issues still under study nowadays [1]. The head is assumed to be an isotropic piecewise homogeneous medium and we examine a layered spherical head model made of three concentric nested spheres, each of them modelling scalp, skull and brain tissues (from the outermost to the innermost layer). Each of the three layers is supposed to have a constant conductivity. We also assume that the conductivities of the brain and the scalp are known, while the conductivity to be recovered is the one of the intermediate spherical layer (skull). We solve the above conductivity estimation problem from the available EEG partial boundary data, expanded on the spherical harmonics basis, and transmitted over the spherical interfaces by transfer functions, while we consider that the source term is already estimated (through a number of coefficients of its spherical harmonics expansion). Linear algebra computations then allow us to find polynomials that possess a root which should coincide with the unknown skull conductivity, thus solving the estimation problem. We derive uniqueness properties and a reconstruction algorithm for the skull conductivity. It uses a non-linear least squares minimization scheme applied to the computed spherical harmonics coefficients of the solution in the three layers. A numerical study shows that the algorithm is able to accurately estimate the skull conductivity, with good robustness properties with respect to various levels of noise. This work was supported by the Région Provence-Alpes-Côte d’Azur, France, and BESA GmbH, Germany.