Compensation of domain modelling errors in the inverse source problem of the Poisson equation: Application in electroencephalographic imaging

In the inverse source problem of the Poisson equation, measurements on the domain boundaries are used to reconstruct sources inside the domain. The problem is an ill-posed inverse problem and it is sensitive to modelling errors of the domain. These errors can be boundary, structure and material property errors, for example. In this paper, we investigate whether the recently proposed Bayesian approximation error (BAE) approach could be used to alleviate the source estimation errors when an approximate model for the domain is employed. The BAE is based on postulating a probabilistic model for the uncertainties, in this case the geometry and structure of the domain, and to carry out approximate marginalization over these nuisance parameters. We particularly consider electroencephalography (EEG) source imaging as an application. EEG is a diagnostic brain imaging modality, and it can be used to reconstruct neural sources in the brain from electric potential measurements along the scalp. In the feasibility study, we assess to which degree one can recover from the modelling errors that are induced by the use of the three concentric circle head model instead of an anatomically accurate head model. The studied domain modelling errors include errors in the geometry of the exterior boundary and the structure of the interior. We show that, in particular with superficial dipole sources, the BAE yields estimates that can in some cases be considered adequately accurate. This would avoid the need for the extraction of the accurate head features which is conventionally carried out via expensive and time consuming auxiliary imaging modalities such as magnetic resonance imaging.