Conditions for numerically accurate TMS electric field simulation

Background 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 1st order FEM require meshes with an average edge length below 0.4 mm, whereas BEM and 2nd (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 1st order FEM converge slowly with increasing mesh resolution; therefore, the use of BEM or 2nd (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. Funding Research reported in this publication was supported by the National Institute of Mental Health and the National Institute of Neurological Disorders and Stroke of the National Institutes of Health under Award Numbers RF1MH114268 and R01NS088674-S1. The content of current research is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. 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.