P 211. Efficient method for computing the TMS-induced electric field in spherically symmetric head models, the triangle Construction

Abstract

In transcranial magnetic stimulation (TMS), we are interested in calculating the TMS-induced electric field (E-field), which defines the given stimulus. This is known as the forward problem, and there exist no general analytic solution for it. This is because the electric field is dictated by Maxwell’s equations, which are partial differential equations. Thus, for non-trivial boundary conditions, numerical methods are needed. General methods for solving these equations are the finite element method (FEM), where we split the volume into small polyhedrons, and the boundary element method (BEM), where we divide the system into uniform-conductance regions and split the formed boundaries into small polygons. Both of these methods can be applied for nearly arbitrary geometries but are hard to parallelize. However, often one can simplify the analysis by using a spherically symmetric head model. For example, if one is to assess the general properties of a stimulator coil, using a more complicated head model will just increase the computational cost and complicate the analysis of the results. Also when the head can be approximated being locally spherical, the spherical symmetric head model allows a significant simplification in the computations. This is because the spherical symmetry of the conducting medium allows us to compute an analytic solution for the forward problem. We will present an efficient method for computing the induced E-field. The model works for spherically symmetric head models and is based on the triangle construction, (Ilmoniemi RJ, Hamalainen MS, Knuutila J. The forward and inverse problems in the spherical model. In: Biomagnetism: Applications & Theory, H. Weinberg, G. Stroink, T. Katila (eds.), Pergamon Press, Amsterdam 1985, pp. 278–282); see Fig. 1 . Devices based on the triangle construction have been used to calibrate magnetoencephalography (MEG) and TMS systems. Now, to obtain the two tangential components (the radial component is always zero) of the induced E-field in a point, one simply has to calculate the mutual inductance between the TMS coil and two perpendicular triangular loops with one corner at the sphere center and the opposite edge tangential to the spherical surface. For the mutual-inductance calculations, the TMS coil can be modeled as a set of current-carrying filaments. Then, the mutual inductance between the coil and the triangle loops can be calculated using analytic formulas. Using the triangle construction, we developed an O (n) time and memory algorithm for the TMS forward problem, where n is the number of points where the induced E-field is computed. This is a considerable improvement over O (M2) for the BEM or O (N) for the FEM, where M is the number of surface elements, and N is the number of volume elements. (For FEM, we need much more 3D volume elements than points in our construction because we have to sample also the volume surrounding the system. For BEM, we need to sample all the 2D surfaces, also the uninteresting ones. Thus, for a small region of interest, N ≫ M ≫ n.) The developed algorithm is very simple compared to those for BEM or FEM as it does not require any tessellation. Also, because in our model the computation for the field in a point does not depend on the field in the other points, the algorithm is embarrassingly parallel, i.e., trivial to parallelize with minimal overhead.