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

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.