The Discontinuous Galerkin Finite Element Method for Solving the MEG and the Combined MEG/EEG Forward Problem

Abstract

In Electro- (EEG) and Magnetoencephalography (MEG), one important requirement of source reconstruction is the forward model. The continuous Galerkin finite element method (CG-FEM) has become one of the dominant approaches for solving the forward problem over the last decades. Recently, a discontinuous Galerkin FEM (DG-FEM) EEG forward approach has been proposed as an alternative to CG-FEM (Engwer et al., 2017). It was shown that DG-FEM preserves the property of conservation of charge and that it can, in certain situations such as the so-called skull leakages, be superior to the standard CG-FEM approach. In this paper, we developed, implemented, and evaluated two DG-FEM approaches for the MEG forward problem, namely a conservative and a non-conservative one. The subtraction approach was used as source model. The validation and evaluation work was done in statistical investigations in multi-layer homogeneous sphere models, where an analytic solution exists, and in a six-compartment realistically shaped head volume conductor model. In agreement with the theory, the conservative DG-FEM approach was found to be superior to the non-conservative DG-FEM implementation. This approach also showed convergence with increasing resolution of the hexahedral meshes. While in the EEG case, in presence of skull leakages, DG-FEM outperformed CG-FEM, in MEG, DG-FEM achieved similar numerical errors as the CG-FEM approach, i.e., skull leakages do not play a role for the MEG modality. In particular, for the finest mesh resolution of 1 mm sources with a distance of 1.59 mm from the brain-CSF surface, DG-FEM yielded mean topographical errors (relative difference measure, RDM%) of 1.5% and mean magnitude errors (MAG%) of 0.1% for the magnetic field. However, if the goal is a combined source analysis of EEG and MEG data, then it is highly desirable to employ the same forward model for both EEG and MEG data. Based on these results, we conclude that the newly presented conservative DG-FEM can at least complement and in some scenarios even outperform the established CG-FEM approaches in EEG or combined MEG/EEG source analysis scenarios, which motivates a further evaluation of DG-FEM for applications in bioelectromagnetism.

Solving High-Resolution Forward Problems for Extra- and Intracranial Neurophysiological Recordings Using Boundary Element Fast Multipole Method

“…Fig. 4d indicates that the BEM-FMM generates a smaller error for all considered mesh resolutions and that its accuracy exceeds the FEM field accuracy reported in (Piastra et al, 2018) by a factor of 1.5-4.…”

“…We test the accuracy and speed of the proposed BEM-FMM numerical solver against previously published first-order FEM solutions (continuous Galerkin or CG and discontinuous Galerkin or DG) for EEG and MEG problems (Engwer, Vorwerk, Ludewig, and Wolters, 2017; Piastra et al, 2018). We replicate the benchmark tests using our approach and use the published results as a reference.…”

“…Fig. 3a shows the problem geometry and conductivity values previously employed (Engwer, Vorwerk, Ludewig, and Wolters, 2017; Piastra et al, 2018).…”

“…4a and separated by 0.99 mm from the CSF boundary (with the same eccentricity factor of 0.987). Following (Piastra et al, 2018), the magnetic field is evaluated on a sphere surface with the radius of 110 mm also shown in Fig. 4a.…”

“…One main challenge in applying the finite-element methods to solve the EEG/MEG forward problem is to deal with the strong singularity at the current dipole source. Different singularity extraction approaches have been proposed; among them the subtraction approach developed in (Wolters et al, 2007;Engwer, Vorwerk, Ludewig, and Wolters, 2017;Piastra et al, 2018) and reported in Table 1. This approach very significantly increases the overall FEM run time as Table 1 shows.…”