Transcranial magnetic stimulation (TMS) is often targeted using a model of TMS-induced electric field (E). In such navigated TMS, the E-field models have been based on spherical approximation of the head. Such models omit the effects of cerebrospinal fluid (CSF) and gyral folding, leading to potentially large errors in the computed E-field. So far, realistic models have been too slow for interactive TMS navigation. We present computational methods that enable real-time solving of the E-field in a realistic five-compartment (5-C) head model that contains isotropic white matter, gray matter, CSF, skull and scalp.Using reciprocity and Geselowitz integral equation, we separate the computations to coil-dependent and -independent parts. For the Geselowitz integrals, we present a fast numerical quadrature. Further, we present a moment-matching approach for optimizing dipole-based coil models.We verified and benchmarked the new methods using simulations with over 100 coil locations. The new quadrature introduced a relative error (RE) of 0.3-0.6%. For a coil model with 42 dipoles, the total RE of the quadrature and coil model was 0.44-0.72%. Taking also other model errors into account, the contribution of the new approximations to the RE was 0.1%. For comparison, the RE due to omitting the separation of white and gray matter was >11%, and the RE due to omitting also the CSF was >23%.Using a standard PC and basic GPU, our solver computed the full E-field in a 5-C model in 9000 points on the cortex in 27 coil positions per second (cps). When the separation of white and gray matter was omitted, the speed was 43-65 cps. Solving only one component of the E-field tripled the speed.The presented methods enable real-time solving of the TMS-induced E-field in a realistic head model that contains the CSF and gyral folding. The new methodology allows more accurate targeting and precise adjustment of stimulation intensity during experimental or clinical TMS mapping. (Matti Stenroos) 2. Theory
Reciprocity relationLet there be a system of conductors. Let primary current distribution J p 1 give rise to electric field E 1 . Correspondingly, J p 2 gives rise to E 2 . Then, assuming the head electrically resistive and small compared to the wavelength of electromagnetic waves, we get the reciprocity relation [10]Letting J p 1 be a dipolar current element Q at r Q , i.e. J p 1 ( r ) = Qδ( r − r Q ), and J p 2 a current loop (coil) with current I flowing in a thin wire c, we get [11]