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.