Modeling transcranial magnetic stimulation coil with magnetic cores

Objective. Accurate modeling of transcranial magnetic stimulation (TMS) coils with the magnetic core is largely an open problem since commercial (quasi) magnetostatic solvers do not output specific field characteristics (e.g. induced electric field) and have difficulties when incorporating realistic head models. Many open-source TMS softwares do not include magnetic cores into consideration. This present study reports an algorithm for modeling TMS coils with a (nonlinear) magnetic core and validates the algorithm through comparison with finite-element method simulations and experiments. Approach. The algorithm uses the boundary element fast multipole method applied to all facets of a tetrahedral core mesh for a single-state solution and the successive substitution method for nonlinear convergence of the subsequent core states. The algorithm also outputs coil inductances, with or without magnetic cores. The coil–core combination is solved only once i.e. before incorporating the head model. The resulting primary TMS electric field is proportional to the total vector potential in the quasistatic approximation; it therefore also employs the precomputed core magnetization. Main results. The solver demonstrates excellent convergence for typical TMS field strengths and for analytical B–H approximations of experimental magnetization curves such as Froelich’s equation or an arctangent equation. Typical execution times are 1–3 min on a common multicore workstation. For a simple test case of a cylindrical core within a one-turn coil, our solver computed the small-signal inductance nearly identical to that from ANSYS Maxwell. For a multiturn rodent TMS coil with a core, the modeled inductance matched the experimental measured value to within 5%. Significance. Incorporating magnetic core in TMS coil design has advantages of field shaping and energy efficiency. Our software package can facilitate model-informed design of more efficiency TMS systems and guide selection of core material. These models can also inform dosing with existing clinical TMS systems that use magnetic cores.