A Fast Volume Integral Equation Solver with Linear Basis Functions for the Accurate Computation of Electromagnetic Fields in MRI

Abstract

This paper proposes a stable volume integral equation (VIE) solver based on polarization/magnetization currents, for the accurate and efficient computation of the electromagnetic scattering from highly inhomogeneous and high contrast objects. Methods: We employ the Galerkin Method of Moments to discretize the formulation with discontinuous piecewise linear basis functions on uniform voxelized grids, allowing for the acceleration of the associated matrix-vector products in an iterative solver, with the help of FFT. Results: Numerical experiments are conducted to study the accuracy and convergence properties of the proposed framework. Their results are compared against standard low order (piecewise constant) discretization schemes, a more conventional VIE formulation based on electric flux densities, and a commercial software package that employs the finite difference time domain method. Conclusion:The results illustrate the superior accuracy and wellconditioned properties of the proposed scheme. Significance: The developed solver can be applied to accurately analyze complex geometries, including realistic human body models, typically used in modeling the interactions between electromagnetic waves and biological tissue, that arise in magnetic resonance scanners.Index terms-Electromagnetic scattering, high contrast/inhomogeneity, high-order basis functions, magnetic resonance modeling, method of moments, volume integral equations.

Magnetic-resonance-based electrical property mapping using Global Maxwell Tomography with an 8-channel head coil at 7 Tesla: a simulation study

“…One possible solution that will be explored to make that feasible is to compress the VIE operators using the Tucker decomposition [47], [48], which should allow us to perform the necessary matrix-vector products in GPUs even at high spatial resolutions [49]. Moreover, GMT could be further accelerated to facilitate in-vivo applications by applying a recently proposed preconditioner for the solution of the forward problem [25].…”

“…The estimation of the B + 1 (the forward problem) is performed using the method of moments technique, where the robust current-based VIE (JVIE) [28] is discretized over a uniform grid, in order to significantly accelerate the matrix-vector products with the help of the fast Fourier transform (FFT) [29]. In this work we employed piecewise linear basis functions, which have recently been shown to provide accurate EM fields calculations in MRI applications [25]. The solution of the forward problem starts by computing the polarisation currents in the scatter (j b ) from the incident electric fields (e inc ) and an estimated EP distribution (ǫ) :…”

“…GMT uses an iterative non-linear least squares optimization to estimate the EP from multiple measurements of the absolute value and relative phase of the transmit field B + 1 . It is accurate because it relies on higher-order volume integral equations (VIE) [25] for rapid and reliable solutions of the forward problem. GMT was first assessed in simulation with realistic human body models (RHBM), using the first eight eigenvectors of a numerical EM basis as hypothetical MR transmitters, to investigate the technique independently from particular coil designs [20].…”

Memory Footprint Reduction for the FFT-Based Volume Integral Equation Method via Tensor Decompositions

“…The unknown volumetric currents should be expanded in some discrete set of appropriate basis functions e.g., PWC [10] or PWL [12] as follows:…”

“…Towards that direction, a magnetic-resonance integral equation-based suite (MARIE) [9]- [11] has been developed, where polynomial basis functions are used for the fast and precise EM modeling of the interactions between human tissue and MR coils. Specifically, by expanding the unknowns with higher order polynomials [12], superior numerical accuracy is in place, contrary to standard low-order approximations, even for the challenging dielectric shimming technique [13], [14]. When VIEs are discretized over a uniform grid with polynomial basis functions, the arising Galerkin Method of Moments (MoM) system matrix has block-Toeplitz with Toeplitz-blocks (BTTB) structure, owing to the translation invariance property of the Green's function.…”