Conforming Discretizations of Boundary Element Solutions of the Electroencephalography Forward Problem

In this paper we present a new discretization strategy for the boundary element formulation of the Electroencephalography (EEG) forward problem. Boundary integral formulations, classically solved with the Boundary Element Method (BEM), are widely used in high resolution EEG imaging because of their recognized advantages in several real case scenarios. Unfortunately however, it is widely reported that the accuracy of standard BEM schemes is limited, especially when the current source density is dipolar and its location approaches one of the brain boundary surfaces. This is a particularly limiting problem given that during an high-resolution EEG imaging procedure, several EEG forward problem solutions are required for which the source currents are near or on top of a boundary surface. This work will first present an analysis of standardly discretized EEG forward problems, reporting on a theoretical issue of some of the formulations that have been used so far in the community. We report on the fact that several standardly used discretizations requires the expansion term to be a square integrable function. Instead, those techniques are not consistent when a more appropriate mapping is considered. Such a mapping allows the expansion function term to be a less regular function, thus sensibly reducing the need for mesh refinements and low-precisions handling strategies that are currently required. These mappings, however, require a different and conforming discretization which must be suitably adapted to them. To do this we adopt a mixed discretization based on dual boundary elements residing on a suitably defined dual mesh. Finally we show how the resulting EEG problems has favorable properties with respect to previously proposed schemes and we show their applicability to real case modeling scenarios obtained from MRI data.