A layer potential approach to inverse problems in brain imaging

Paul Asensio (1) , Jean-Michel Badier , Juliette Leblond (1) , Jean-Paul Marmorat (2) , Masimba Nemaire (1, 3)

1 FACTAS - Analyse fonctionnelle pour la conception et l'analyse de systèmes
2 CMA - Centre de Mathématiques Appliquées
3 IMB - Institut de Mathématiques de Bordeaux

We study the inverse source (primary current) localisation problem using the electrical potential measured point-wise inside the head with stereo-ElectroEncephaloGraphy (sEEG), the electrical potential measured point-wise on the scalp with ElectroEncephaloGraphy (EEG) or the magnetic flux density measured point-wise outside the head with MagnetoEncephaloGraphy (MEG). We present a method that works on a wide range of models of primary currents, in particular we give details for primary currents that are assumed to be smooth vector-fields that are supported on and normally oriented to the grey/white matter interface. Irrespective of the data used we need to understand the transmission of the electric potential associated with a recovered source through the head hence we solve the cortical mapping problem. To ensure that the electric potential and normal currents are continuous in the head, the electric potential is expressed as a linear combination of double layer potentials and the magnetic flux density is expressed as a linear combination of single layer potentials. Numerically, we solve the problems on meshed surfaces of the grey/white matter interface, cortical surface, skull and scalp. A main feature of the numerical approach we take is that on the meshed surfaces we can compute the double and single layer potentials exactly and at arbitrary points. Because we study the transmission of the electric potential in head irrespective of the modality used, this enables the coupling of electric and magnetic data in the source recovery problem. We provide numerical examples of the source recovery and inverse cortical mapping using synthetic data.

Mots clés: inverse problems; layer potentials; EEG; MEG; sEEG

Domaines: Mathématiques [math]

Paul Asensio, Jean-Michel Badier, Juliette Leblond, Jean-Paul Marmorat, Masimba Nemaire. A layer potential approach to inverse problems in brain imaging. Journal of Inverse and Ill-posed Problems, 2023, ⟨10.1515/jiip-2023-0041⟩. ⟨hal-03977724v2⟩