[Paper Review] The Unfitted Discontinuous Galerkin Method for Solving the EEG Forward Problem

Andreas Nüßing, Carsten H. Wolters+2 more|arXiv (Cornell University)|Jan 28, 2016

Advanced MRI Techniques and Applications34 references3 citations

TL;DR

This paper introduces the unfitted discontinuous Galerkin finite element method (UDG-FEM) for solving the EEG forward problem using a structured, non-conforming hexahedral mesh that implicitly incorporates complex head geometry via level set functions. The method achieves higher accuracy than conventional DG-FEM on hexahedral and conforming tetrahedral meshes, especially in modeling smooth tissue interfaces and reducing skull leakage errors, while simplifying the simulation pipeline without requiring body-fitted meshing.

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.

Motivation & Objective

• To develop a robust, geometry-implicit method for solving the EEG forward problem that avoids complex, patient-specific mesh generation.
• To address skull leakage artifacts common in regular hexahedral FEM by leveraging discontinuous Galerkin formulations.
• To improve accuracy in modeling smooth tissue interfaces without requiring conforming tetrahedral meshes.
• To evaluate the method’s convergence, accuracy, and computational efficiency in spherical and realistic head models.
• To demonstrate that UDG-FEM can achieve performance comparable or superior to conforming tetrahedral DG-FEM while simplifying the simulation pipeline.

Proposed method

• The method employs a structured hexahedral background mesh that does not conform to tissue boundaries, eliminating the need for complex volume mesh generation.
• Geometry is implicitly represented using level set functions, which define tissue interfaces without requiring mesh alignment.
• The discontinuous Galerkin formulation allows for discontinuous basis functions across element boundaries, enabling local conservation and better handling of material discontinuities.
• Integration over cut cells is performed using sub-triangulation of the level set interface, enabling accurate weak formulation on non-conforming elements.
• The method maintains discrete conservation laws and supports anisotropic conductivity modeling, crucial for realistic EEG simulations.
• A transfer matrix approach is used to efficiently solve for multiple electrode configurations after a single system assembly.

Experimental results

Research questions

• RQ1 Can UDG-FEM achieve accurate EEG forward solutions without requiring body-fitted tetrahedral meshes?
• RQ2 How does UDG-FEM perform in terms of convergence and accuracy compared to DG-FEM on conforming tetrahedral and hexahedral meshes?
• RQ3 To what extent does UDG-FEM reduce skull leakage artifacts compared to standard CG-FEM and DG-FEM on hexahedral meshes?
• RQ4 Can UDG-FEM maintain high accuracy on coarser meshes due to improved geometric representation?
• RQ5 How does the method perform in a realistic head model with complex, smooth tissue boundaries?

Key findings

• UDG-FEM demonstrates proper convergence in multi-layer sphere models as mesh resolution increases, confirming theoretical consistency.
• For radial sources, UDG-FEM achieves a MAG% error of approximately -1% at eccentricity 0.9686, significantly outperforming geometry-adapted CG-FEM, which reports ~3.5–4% error.
• UDG-FEM on structured hexahedral meshes achieves higher accuracy than DG-FEM on the same mesh type, indicating improved handling of interface discontinuities.
• UDG-FEM outperforms DG-FEM on conforming tetrahedral meshes in terms of accuracy, particularly for radial dipole sources.
• In a realistic head model simulation, UDG-FEM produced a smooth, physiologically plausible potential distribution over the scalp, consistent with auditory evoked potential patterns.
• The method reduces the need for high-resolution meshes due to better geometric approximation, leading to potential reductions in overall computational cost despite increased DOFs in cut cells.