The Unfitted Discontinuous Galerkin Method for Solving the EEG Forward Problem

TL;DR

This study shows the first application of the UDG-FEM approach to the EEG forward problem, and shows the results show convergence to the quasi-analytical solution and indicate a good accuracy of UDG -FEM.

Abstract

Objective: The purpose of this study is to introduce and evaluate the unfitted discontinuous Galerkin finite element method (UDG-FEM) for solving the electroencephalography (EEG) forward problem. Methods: This new approach for source analysis does not use a geometry conforming volume triangulation, but instead uses a structured mesh that does not resolve the geometry. The geometry is described using level set functions and is incorporated implicitly in its mathematical formulation. As no triangulation is necessary, the complexity of a simulation pipeline and the need for manual interaction for patient specific simulations can be reduced and is comparable with that of the FEM for hexahedral meshes. In addition, it maintains conservation laws on a discrete level. Here, we present the theory for UDG-FEM forward modeling, its verification using quasi-analytical solutions in multi-layer sphere models and an evaluation in a comparison with a discontinuous Galerkin (DG-FEM) method on hexahedral and on conforming tetrahedral meshes. We furthermore apply the UDG-FEM forward approach in a realistic head model simulation study. Results: The given results show convergence and indicate a good overall accuracy of the UDG-FEM approach. UDG-FEM performs comparable or even better than DG-FEM on a conforming tetrahedral mesh while providing a less complex simulation pipeline. When compared to DG-FEM on hexahedral meshes, an overall better accuracy is achieved. Conclusion: The UDG-FEM approach is an accurate, flexible and promising method to solve the EEG forward problem. Significance: This study shows the first application of the UDG-FEM approach to the EEG forward problem.

Figures (7)

• Figure 1: Construction of the subtriangulation for a single bilinear level set function. In the left image, two domains are delimited by a bilinear level set function and in the right image, the resulting discrete domains are shown. The dashed lines delimit the elements of the subtriangulation which are only used for integration.
• Figure 2: Sections of the different meshes used in the multi-layer sphere verification. From left to right, the images show the DG tet 1447k, DG hex 3057k and UDG 1335k models. The different colors represent the different conductivity values.
• Figure 3: A section of the cut cell mesh for the realistic head model: the thick black lines show the cut cells on the fundamental mesh on which the basis functions are defined, while the dashed black lines depict the subtriangulation used for integration in the UDG-FEM approach.
• Figure 4: Verification and convergence study for the UDG-FEM approach: $\operatorname{RDM\%}$ (upper row) and $\operatorname{MAG\%}$ (lower row) errors for radial (left column) and tangential (right column) sources for the three different resolutions 39k (green), 218k (red) and 1335k (blue). Note that the x-axis is logarithmically scaled.
• Figure 5: Comparison between the UDG-FEM approach using model 1,335k (blue) and the DG-FEM approach on a conforming mesh with tetrahedral elements using model 1,447k (green): $\operatorname{RDM\%}$ (upper row) and $\operatorname{MAG\%}$ (lower row) errors for radial (left column) and tangential (right column) sources. Note that the x-axis is logarithmically scaled. With vertical yellow lines, the maximal, mean and minimal (from left to right) element diameter is indicated of those elements in the inner compartment touching the compartment boundary. This diameter is given relative to the radius of the inner sphere. Note that this information is only relevant for the DG-FEM approach.