Analysis of Transcranial Magnetic Stimulation Based on the Surface Integral Equation Formulation

Goal: The aim of this paper is to provide a rigorous model and, hence, a more accurate description of the transcranial magnetic stimulation (TMS) induced fields and currents, respectively, by taking into account the inductive and capacitive effects, as well as the propagation effects, often being neglected when using quasi-static approximation. Methods: The formulation is based on the surface integral equation (SIE) approach. The model of a lossy homogeneous brain has been derived from the equivalence theorem and using the appropriate boundary conditions for the electric field. The numerical solution of the SIE has been carried out using the method of moments. Results: Numerical results for the induced electric field, electric current density, and the magnetic flux density distribution inside the human brain, presented for three typical TMS coils, are in a good agreement with some previous analysis as well as to the results obtained by analytical approach. Conclusion: The future work should be related to the development of a more detailed geometrical model of the human brain that will take into account complex cortical columnar structures, as well as some additional brain tissues. Significance: To the best of authors knowledge, similar approach in modeling TMS has not been previously reported, albeit integral equation methods are seeing a revival in computational electromagnetics community.