Fast electric field solvers for coil placement optimization: the auxiliary dipole method and its extensions

Background. During transcranial magnetic stimulation (TMS), a coil placed on the scalp induces an electric field in the brain, and the dose delivered to a target depends strongly on where the coil is placed. Choosing the placement that maximizes the electric field at a target, or that is robust across a population, requires the electric field for thousands to millions of candidate placements. Standard solvers compute the electric field for one coil placement per run, which makes such optimization impractical.

Objective. Compute the TMS-induced electric field for essentially all admissible coil placements from a handful of simulations, so that coil placement optimization and uncertainty quantification become routine.

Approach. The auxiliary dipole method uses electromagnetic reciprocity: the average electric field in a region of interest for any coil placement is obtained from a single simulation with an auxiliary dipole source placed in that region. The adaptive cross approximation auxiliary dipole method treats the electric field as a bivariate function of coil position and brain observation point and exploits its low-rank structure to recover the field over medium-sized regions of interest (up to about 10 cm in diameter). The probabilistic matrix decomposition auxiliary dipole method extends the same idea to whole-head models.

Main results. The auxiliary dipole method delivers the average electric field in a small (under 2 cm) region of interest for about one million coil placements from a single set of auxiliary simulations. The adaptive cross approximation extension recovers the electric field over a 4 cm region for about one million placements in under two hours. After an initial set-up stage, the probabilistic matrix decomposition extension predicts the whole-brain electric field for any coil placement in 2–3 ms with about 2% error, and computes over one million placements in 9.5 hours instead of the more than five years a brute-force approach would require.

Significance. These methods turn coil placement optimization from a computational bottleneck into a routine step and underpin our group-level dosimetry, real-time solver, and uncertainty quantification work. The auxiliary dipole method code is on GitHub, and the method is also included in the SimNIBS platform; our other solvers are listed on the Software section of the Research page.

Publications.

L. J. Gomez, M. Dannhauer, and A. V. Peterchev, "Fast computational optimization of TMS coil placement for individualized electric field targeting," NeuroImage, vol. 228, pp. 117696, 2021. link

D. Wang (PD), N. I. Hasan (G), M. Dannhauer, A. C. Yucel, and L. J. Gomez, "Fast computational E-field dosimetry for transcranial magnetic stimulation using adaptive cross approximation and auxiliary dipole method (ACA-ADM)," NeuroImage, vol. 267, pp. 119850, 2023. link

N. I. Hasan (G), D. Wang (PD), and L. J. Gomez, "Fast and accurate computational E-field dosimetry for group-level transcranial magnetic stimulation targeting," Computers in Biology and Medicine, vol. 167, pp. 107614, 2023. link

V. Sabino (G), A. Murugesan (PD), N. I. Hasan (G), S. S. Vaezi (G), A. Walenciak (G), P. Jayatissa (UG), and L. J. Gomez, "Advances in Computational Electromagnetics for Enhanced Noninvasive Brain Stimulation: E-field dosimetry, uncertainty quantification, optimization, and neural response modeling," IEEE Antennas and Propagation Magazine, vol. 68, no. 2, pp. 22-34, 2026 (invited). link

D. Wang (PD), M. Dannhauer, A. C. Yucel, and L. J. Gomez, "Adaptive Cross Approximation for E-field-Guided Noninvasive Magnetic Brain Stimulation," International Applied Computational Electromagnetics Society Symposium, August 2021.

N. I. Hasan (G), D. Wang (PD), and L. J. Gomez, "Optimal Population Level Transcranial Magnetic Stimulation via Probabilistic Matrix Decomposition," IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, July 2022.