High-Resolution EEG Source Reconstruction with Boundary Element Fast Multipole Method Using Reciprocity Principle and TES Forward Model Matrix

Abstract

Background: Accurate high-resolution EEG source reconstruction (localization) is important for several tasks, including rigorous and rapid mental health screening. Objective: The present study has developed, validated, and applied a new source localization algorithm utilizing a charge-based boundary element fast multipole method (BEM-FMM) coupled with the Helmholtz reciprocity principle and the transcranial electrical stimulation (TES) forward solution. Methods: The unknown cortical dipole density is reconstructed over the entire cortical surface by expanding into global basis functions in the form of cortical fields of active TES electrode pairs. These pairs are constructed from the reading electrodes. An analog of the minimum norm estimation (MNE) equation is obtained after substituting this expansion into the reciprocity principle written in terms of measured electrode voltages. Delaunay (geometrically balanced) triangulation of the electrode cap is introduced first. Basis functions for all electrode pairs connected by the edges of a triangular mesh are precomputed and stored in memory. A smaller set of independent basis functions is then selected and employed at every time instant. This set is based on the highest voltage differences measured. Results: The method is validated against the classic, yet challenging problem of median nerve stimulation and the tangential cortical sources located at the posterior wall of the central sulcus for an N20/P20 peak (2 scanned subjects). The method is further applied to perform source reconstruction of synthesized tangential cortical sources located at the posterior wall of the central sulcus (12 different subjects). In the second case, an average source reconstruction error of 7 mm is reported for the best possible noiseless scenario. Conclusions: Once static preprocessing with TES electrodes has been done (the basis functions have been computed), our method requires fractions of a second to complete the accurate high-resolution source localization.

Fast and Accurate EEG/MEG BEM-Based Forward Problem Solution for High-Resolution Head Models

“…(by a full GMRES solution) and "refine" until a convergence criterion for relative inter-iteration change in skin potential was reached. While the method produced quite accurate results [16],…”

“…In a previous approach [15], [16], an h-refinement method was applied wherein facets of the model were selected for 4:1 barycentric subdivision according to the absolute value of total charge 𝑞 𝑚 = |𝐴 𝑚 𝜌 𝑚 | upon them, where 𝐴 𝑚 and 𝜌 𝑚 denote the area and charge density, respectively, on the facet with index 𝑚. The solution was carried out in alternating steps of "solve"…”

“…It has long been known [14] that the BEM cannot accurately model the response of singular sources at such short distances, no matter how many neighbor surface integrals are precomputed analytically. Therefore, an adaptive mesh refinement (AMR) is necessary close to the sources [16]. While possessing an unconstrained numerical accuracy, the corresponding solution suggested in [15], [16] is still relatively slow for practical purposes.…”

“…Therefore, an adaptive mesh refinement (AMR) is necessary close to the sources [16]. While possessing an unconstrained numerical accuracy, the corresponding solution suggested in [15], [16] is still relatively slow for practical purposes.…”

“…Section 2 describes the idea of the method and its realization. It also describes three test cases: comparison with an analytical solution, comparison with the precise AMR algorithm [15], [16] (which is considered as the ground truth) for realistic head models with three challenging dipole locations, and finally, application to practical source reconstruction for experimental data on median nerve stimulation. Section 3 reports the obtained results in all three cases.…”