Global Maxwell Tomography Using the Volume-Surface Integral Equation for Improved Estimation of Electrical Properties

TL;DR

This paper tackles the problem of accurately mapping tissue electrical properties from MR measurements by improving Global Maxwell Tomography (GMT). It replaces the traditional volume integral equation (VIE) forward problem with a volume-surface integral equation (VSIE) formulation that re-estimates coil currents at every optimization iteration, thereby capturing coil loading effects as EP updates. The VSIE-based GMT demonstrates superior accuracy over VIE-based GMT in both simulations (head models and phantoms) and a 7 T phantom experiment, reducing relative errors in permittivity and conductivity and achieving competitive SSIM scores. The work shows that accounting for EP-induced changes in coil currents yields more reliable reconstructions, advancing EP mapping toward potential in vivo applications, while noting practical considerations such as calibration, shielding, and computational demand.

Abstract

Objective: Global Maxwell Tomography (GMT) is a noninvasive inverse optimization method for the estimation of electrical properties (EP) from magnetic resonance (MR) measurements. GMT uses the volume integral equation (VIE) in the forward problem and assumes that the sample has negligible effect on the coil currents. Consequently, GMT calculates the coil's incident fields with an initial EP distribution and keeps them constant for all optimization iterations. This can lead to erroneous reconstructions. This work introduces a novel version of GMT that replaces VIE with the volume-surface integral equation (VSIE), which recalculates the coil currents at every iteration based on updated EP estimates before computing the associated fields. Methods: We simulated an 8-channel transceiver coil array for 7 T brain imaging and reconstructed the EP of a realistic head model using VSIE-based GMT. We built the coil, collected experimental MR measurements, and reconstructed EP of a two-compartment phantom. Results: In simulations, VSIE-based GMT outperformed VIE-based GMT by at least 12% for both EP. In experiments, the relative difference with respect to probe-measured EP values in the inner (outer) compartment was 13% (26%) and 17% (33%) for the permittivity and conductivity, respectively. Conclusion: The use of VSIE over VIE enhances GMT's performance by accounting for the effect of the EP on the coil currents. Significance: VSIE-based GMT does not rely on an initial EP estimate, rendering it more suitable for experimental reconstructions compared to the VIE-based GMT.

Figures (8)

• Figure 1: The homogeneous phantom used for calibration (left) and two- compartment phantom cylindrical phantom (right). The homogeneous phantom was filled with a solution of distilled water and sodium chloride, while for the two- compartment phantom, we also added polyvinylpyrrolidone and manganese(II) chloride to control the permittivity and T1, respectively.
• Figure 2: (top) The model of the triangular RF coil array loaded with the Billie head model. (bottom left) Top view of the coil geometry. The annotations indicate the width of the rectangular section, the radius of the semicircles, and the overall length of the coil. (bottom right) Circuit diagram of three representative elements of the triangular coil array. The coil array consisted of eight triangular elements that were capacitively decoupled from their nearest neighbors and inductively decoupled from their next nearest neighbors.
• Figure 3: (left) Interior of the $8$- channel triangular transceiver RF coil array. Yellow skin markers were placed $6$ cm away from the end ring to help matching the location of the sample relative to the coil in the forward problem simulation. (right) The coil loaded with a cylindrical phantom.
• Figure 4: Relative error of the $L_2$ norm of the coil currents updated in each VSIE-GMT iteration with respect to the coil currents associated with ground- truth EP values. The results are presented for different cylinders to show the effect of the distance of the cylinder from the coil conductors.
• Figure 5: PNAE distribution of the relative permittivity (top) and conductivity (bottom) with respect to the ground truth for all voxels in the head model and each GMT reconstruction.