Diffusion MRI Consistent Wire Models for Efficient Solutions of the Anisotropic Forward Problem in Electroencephalography

The surface Boundary Element Method (BEM) is one of the most commonly employed formulations to solve the forward problem in electroencephalography, but the applicability of its classical incarnations is lamentably limited to piecewise homogeneous media. Several head tissues, however, are strongly anisotropic due to their complex underlying microstructure. This implies that standard boundary integral formulations oversimplify the electrical properties of the head and produce unrealistic solutions, something that drastically limits the suitability and impact of BEM technologies to brain imaging. This contribution addresses this issue by observing that the brain anisotropy in the white matter is due to the presence of neuronal wire-like structures. We then extend the well known wire integral equations used for high frequency problems to the imperfectly conducting quasi-static case and we propose a new hybrid wire/surface/volume integral equation. When applied on multimodal magnetic resonance images combined with tractography, this new approach can flexibly and realistically handle the conductivity anisotropy in any head compartment providing high level of accuracy and efficiency. The beneficial properties of the new formulation together with its impact on brain imaging is demonstrated via numerical results on both canonical and realistic case scenarios.