Fast FEM-based electric field calculations for transcranial magnetic stimulation

Objective. To provide a finite-element method (FEM) for rapid, repeated evaluations of the electric field induced by transcranial magnetic stimulation (TMS) in the brain for changing coil positions.Approach. Previously, we introduced a first-order tetrahedral FEM enhanced by super-convergent patch recovery (SPR), striking a good balance between computational efficiency and accuracy (Saturninoet al2019J. Neural Eng.16066032). In this study, we refined the method to accommodate repeated simulations with varying TMS coil position. Starting from a fast direct solver, streamlining the pre- and SPR-based post-calculation steps by implementing these steps as parallel sparse matrix multiplications strongly improved the computational efficiency. Additional speedups were achieved through efficient multi-core and GPU acceleration, alongside the optimization of the volume conductor model of the head for TMS.Main Results. For an anatomically detailed head model with ∼4.4 million tetrahedra, the optimized implementation achieves update rates above 1 Hz for electric field calculations in bilateral gray matter, resulting in a 60-fold speedup over the previous method with identical accuracy. An optimized model without neck and with adaptive spatial resolution scaled in dependence to the distance to brain grey matter, resulting in ∼1.9 million tetrahedra, increased update rates up to 10 Hz, with ∼3% numerical error and ∼4% deviation from the standard model. Region-of-interest (ROI) optimized models focused on the left motor, premotor and dorsolateral prefrontal cortices reached update rates over 20 Hz, maintaining a difference of <4% from standard results. Our approach allows efficient switching between coil types and ROI during runtime which greatly enhances the flexibility.Significance. The optimized FEM enhances speed, accuracy and flexibility and benefits various applications. This includes the planning and optimization of coil positions, pre-calculation and training procedures for real-time electric field simulations based on surrogate models as well as targeting and dose control during neuronavigated TMS.