A High-Order Discretization Scheme for Surface Integral Equations for Analyzing the Electroencephalography Forward Problem

TL;DR

The paper tackles the accurate solution of the EEG forward problem using surface integral equations in multilayer head models. It introduces a Nyström-based high-order discretization where interpolation points on surface patches are distinct from mesh nodes, enabling higher-order basis function flexibility without re-meshing. The method is extended to four SIE formulations (DL, ADL, ISA, IADL) and incorporates specialized numerical integration strategies for far, near, and self-interaction terms. Numerical experiments on spherical head phantoms demonstrate improved accuracy and robustness, particularly for large skull-conductivity contrasts, underscoring the approach's potential for EEG forward modeling.

Abstract

A Nystrom-based high-order (HO) discretization scheme for surface integral equations (SIEs) for analyzing the electroencephalography (EEG) forward problem is proposed in this work. We use HO surface elements and interpolation functions for the discretization of the interfaces of the head volume and the unknowns on the elements, respectively. The advantage of this work over existing isoparametric HO discretization schemes resides in the fact that the interpolation points are different from the mesh nodes, allowing for the flexible manipulation of the order of the basis functions without regenerating the mesh of the interfaces. Moreover, the interpolation points are chosen from the quadrature rules with the same number of points on the elements simplifying the numerical computation of the surface integrals for the far-interaction case. In this contribution, we extend the implementation of the HO discretization scheme to the double-layer and the adjoint double-layer formulations, as well as to the isolated-skull-approach for the double-layer formulation and to the indirect adjoint double-layer formulation, employed to improve the solution accuracy in case of high conductivity contrast models, which requires the development of different techniques for the singularity treatment. Numerical experiments are presented to demonstrate the accuracy, flexibility, and efficiency of the proposed scheme for the four SIEs for analyzing the EEG forward problem.

Figures (7)

• Figure 1: The head volume model.
• Figure 2: An example for the discretization of $S_{N_{\mathrm{i}}}$ using the quadratically curved triangular patches with six mesh nodes along the boundary of each patch and using the $2$nd order interpolation function with six interpolation points on each patch.
• Figure 3: The space mapping between the $(x,y,z)$ Cartesian and $(\alpha, \beta)$ parametric coordinate systems for the $p^{\mathrm{th}}$ patch on $S_i$.
• Figure 4: The relative error of the solution after using the proposed scheme versus the reciprocal of the average length of the edges for the DL, ADL, ISADL, and IADL approaches.
• Figure 5: The relative error of the solution after using the proposed scheme versus the order of the interpolation function for the DL, ADL, ISADL, and IADL approaches.