A software toolkit for TMS electric-field modeling with boundary element fast multipole method: an efficient MATLAB implementation

Abstract

Background: Transcranial magnetic stimulation (TMS) is currently the only non-invasive neurostimulation modality that enables painless and safe supra-threshold stimulation by employing electromagnetic induction to efficiently penetrate the skull. Accurate, fast, and high resolution modeling of the electric fields (E-fields) may significantly improve individualized targeting and dosing of TMS and therefore enhance the efficiency of existing clinical protocols as well as help establish new application domains.Objective: To present and disseminate our TMS modeling software toolkit, including several new algorithmic developments, and to apply this software to realistic TMS modeling scenarios given a high-resolution model of the human head including cortical geometry and an accurate coil model. Method:The recently developed charge-based boundary element fast multipole method (BEM-FMM) is employed as an alternative to the 1st order finite element method (FEM) most commonly used today. The BEM-FMM approach provides high accuracy and unconstrained field resolution close to and across cortical interfaces. Here, the previously proposed BEM-FMM algorithm has been improved in several novel ways. Results and Conclusions:The improvements resulted in a threefold increase in computational speed while maintaining the same solution accuracy. The computational code based on the MATLAB ® platform is made available to all interested researchers, along with a coil model repository and examples to create custom coils, head model repository, and supporting documentation. The presented software toolkit may be useful for post-hoc analyses of navigated TMS data using high-resolution subject-specific head models as well as accurate and fast modeling for the purposes of TMS coil/hardware development.

Degree of improving TMS focality through a geometrically stable solution of an inverse TMS problem

“…For the forward-problem solution, we use the boundary element fast multipole method formulated in terms of induced surface charge density ρ ( r ) residing at the conductivity interfaces or BEM-FMM ( Makarov et al, 2018 , Makarov et al, 2020 , GitHub Repository 2021 ). The method possesses high numerical accuracy, which was shown to exceed that of the comparable finite element method of the first order ( Gomez et al, 2020 ).…”

“…While the BEM-FMM forward solver itself was tested previously ( Makarov et al, 2018 , Makarov et al, 2020 , Gomez et al, 2020 ), the inverse-problem solution was not. To do so, we have rerun…”

“…This table simultaneously presents the results for the normal field optimization, for the CoolB35 coil. The normal fields just inside and outside correlate with each other ( Makarov et al, 2020 ) and are optimized identically. These results appear to be quite different from the total-field results (a better optimization and a higher instability).…”

“… Laakso and Hirata, 2012 , Saturnino et al, 2019 , Gomez et al, 2020 ) or, more recently, also the boundary element method (cf. Stenroos and Koponen, 2019 , Makarov et al, 2018 , Makarov et al, 2020 ).…”

“…The numerical implementation of the TMS-IP solution uses the boundary element fast multipole method (BEM-FMM) ( Makarov et al, 2018 , Makarov et al, 2020 ) as the forward solver and a simple gradient descent search method with a small step size in a full six-dimensional search space as the inverse solver. This search space includes three independent coil coordinates and three independent coil rotation angles as suggested in the recent TMS protocol ( Balderston et al, 2020 ).…”