An Efficient 3-D Eddy-Current Solver Using an Independent Impedance Method for Transcranial Magnetic Stimulation

Abstract

In many important bioelectromagnetic problem settings, eddy-current simulations are required. Examples are the reduction of eddy-current artifacts in magnetic resonance imaging and techniques, whereby the eddy currents interact with the biological system, like the alteration of the neurophysiology due to transcranial magnetic stimulation (TMS). TMS has become an important tool for the diagnosis and treatment of neurological diseases and psychiatric disorders. A widely applied method for simulating the eddy currents is the impedance method (IM). However, this method has to contend with an ill conditioned problem and consequently a long convergence time. When dealing with optimal design problems and sensitivity control, the convergence rate becomes even more crucial since the eddy-current solver needs to be evaluated in an iterative loop. Therefore, we introduce an independent IM (IIM), which improves the conditionality and speeds up the numerical convergence. This paper shows how IIM is based on IM and what are the advantages. Moreover, the method is applied to the efficient simulation of TMS. The proposed IIM achieves superior convergence properties with high time efficiency, compared to the traditional IM and is therefore a useful tool for accurate and fast TMS simulations.

Comparison of the induced fields using different coil configurations during deep transcranial magnetic stimulation

“…Assuming that the electric conductivities are isotropic and constant in all direction in each voxel, the head model is represented as a 3D network of impedances. The magnetic fields and the induced electric fields were calculated using the impedance method [37–39]. we have successfully employed the impedance method in the modelling of brain stimulation and electromagnetic dosimetry [40, 41].…”

Low-parametric Induced Current - Magnetic Resonance Electrical Impedance Tomography for quantitative conductivity estimation of brain tissues using a priori information: A simulation study

“…(3) time-efficient Independent Impedance Method (IIM) as forward eddycurrent solver: Since a forward eddy-current solver needs to be evaluated many times in an iterative loop for solving the inverse problem, an efficient forward model is needed so to avoid prohibitive computational times. We use a recently developed IIM [11] that is based on the impedance method where acceleration of computations is carried out by solving a linear system of independent equations. (4) iterative scheme for cost function minimization associated to the IC-MREIT inverse problem: The simulated MR phase differences needs to approximate the measured MR phase differences.…”

“…However, the linear system of equations Z · J eddy = V that have to be solved in IM, can be significantly ill-conditioned, leading to a poor numerical convergence or even in some situation no solution at all [15]. In this paper, we use a recently developed IIM [11], based on IM, whereby a set of independent equations is identified by defining independent loops in the 3D circuit. This results in a reduction of the number of equations that have to be solved, to the benefit of memory burden and computational time.…”

How much detail is needed in modeling a transcranial magnetic stimulation figure-8 coil: Measurements and brain simulations

“…A variety of computational models of TMS induced currents in the brain exists, such as Finite Element Modeling (FEM), Boundary Element Modeling (BEM) [4,5] and Impedance Methods (IM) [6]. The aforementioned studies focus on how different brain tissues, anisotropy and shape influence the induced electric fields.…”