Conditions for numerically accurate TMS electric field simulation

Abstract

HIGHLIGHTS FDM and 1st order FEM with 1.5 mm average mesh edge length have numerical errors above 7%. BEM or 2nd order FEM are most efficient for achieving numerical errors < 2%.  Coil wire cross-section must be accounted to achieve E-field errors below < 2%. Coil eddy currents can account for > 2% of E-field when very brief pulses are used. ABSTRACTBackground: Computational simulations of the E-field induced by transcranial magnetic stimulation (TMS) are increasingly used to understand its mechanisms and to inform its administration. However, characterization of the accuracy of the simulation methods and the factors that affect it is lacking.Objective: To ensure the accuracy of TMS E-field simulations, we systematically quantify their numerical error and provide guidelines for their setup. Method:We benchmark the accuracy of computational approaches that are commonly used for TMS E-field simulations, including the finite element method (FEM), boundary element method (BEM), finite difference method (FDM), and coil modeling methods. Results:To achieve cortical E-field error levels below 2%, the commonly used FDM and 1 st order FEM require meshes with an average edge length below 0.4 mm, whereas BEM and 2 nd (or higher) order FEM require edge lengths below 1.5 mm, which is more practical. Coil models employing magnetic and current dipoles require at least 200 and 3,000 dipoles, respectively. For thick solid-conductor coils and frequencies above 3 kHz, winding eddy currents may have to be modeled.Conclusion: BEM, FDM, and FEM methods converge to the same solution. However, FDM and 1 st order FEM converge slowly with increasing mesh resolution; therefore, the use of BEM or 2 nd (or higher) order FEM is recommended. In some cases, coil eddy currents must be modeled. Both electric current dipole and magnetic dipole models of the coil current can be accurate with sufficiently fine discretization.

Electric field simulations for transcranial brain stimulation using FEM: an efficient implementation and error analysis

“…8, reveal a clear improvement in numerical accuracy when employing finer meshes, with larger gains when the density is low. At a density of 1.0 node/mm , we obtain errors of 2.6 % for TMS and 2.9 % for TES, close to the 2% threshold proposed by [21]. Figure 9 shows the time used in running an entire simulation (denoted as "All" in the figure legend), including calculating TMS coil fields and placing TES electrodes, for assembling and solving the FEM systems (denoted as "FEM" in the figure legend) in SimNIBS 2.1 and SimNIBS 3.0, and for running an additional simulation re-using the stiffness matrix and preconditioner (denoted as "Additional FEM" in the figure legend) in SimNIBS 3.0.…”

“…In a related manner, this study is limited to FEM with first order tetrahedral elements. Employing numerical methods other than first order FEM and SPR, such as higher order FEM [21], DG-FEM [36,47], BEM [48,49], and BEM-FMM [20,21,50] will likely lead to better numerical accuracy. However, for all these methods, the results can still only be accurate as far as the head model is accurate, meaning that we still expect the errors, when comparing to a more accurate head model, to quickly reach a lower bound.…”

“…Equally important, this also slows down the development and broad adoption of more advanced applications of realistic and individualized field calculations that rely on the evaluation of many simulations, such as the optimization of multichannel TES montages [14][15][16] or the systematic uncertainty quantification of the simulation outcome [17][18][19]. There is also a lack of studies trying to quantify how much the error in modelling tissue boundaries impacts the accuracy in the electric field calculations, as previous studies addressing simulation accuracy have been focused on comparing different methods for TMS [20,21] or TES [22] simulations rather than changes in the anatomical fidelity of the underlying head model.…”

Conditions for numerically accurate TMS electric field simulation

“…All solvers where developed in-house, with implementation details provided in the Supplement. All of the solvers are freely available online at [46]. The FMM acceleration of the BEM is achieved using FMM libraries [47].…”

Solving High-Resolution Forward Problems for Extra- and Intracranial Neurophysiological Recordings Using Boundary Element Fast Multipole Method

“…It has been applied to modeling the transcranial magnetic stimulation (TMS) fields and has demonstrated a fast computational speed and superior accuracy for high-resolution head models as compared to both the standard boundary element method and the finite element method of first order (Makarov, Noetscher, Raij, and Nummenmaa, 2018;Htet et al, 2019). These results have been further confirmed in (Gomez, Dannhauer, Koponen, and Peterchev, 2018).…”