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

Objective. Transcranial magnetic stimulation (TMS) and transcranial electric stimulation (TES) modulate brain activity non-invasively by generating electric fields either by electromagnetic induction or by injecting currents via skin electrodes. Numerical simulations based on anatomically detailed head models of the TMS and TES electric fields can help us to understand and optimize the spatial stimulation pattern in the brain. However, most realistic simulations are still slow, and the role of anatomical fidelity on simulation accuracy has not been evaluated in detail so far. Approach. We present and validate a new implementation of the finite element method (FEM) for TMS and TES that is based on modern algorithms and libraries. We also evaluate the convergence of the simulations and estimate errors stemming from numerical and modelling aspects. Main results. Comparisons with analytical solutions for spherical phantoms validate our new FEM implementation, which is three to six times faster than previous implementations. The convergence results suggest that accurately capturing the tissue geometry in addition to choosing a sufficiently accurate numerical method is of fundamental importance for accurate simulations. Significance. The new implementation allows for a substantial increase in computational efficiency of FEM TMS and TES simulations. This is especially relevant for applications such as the systematic assessment of model uncertainty and the optimization of multi-electrode TES montages. The results of our systematic error analysis allow the user to select the best tradeoff between model resolution and simulation speed for a specific application. The new FEM code is openly available as a part of our open-source software SimNIBS 3.0.