Electrical property estimation has long been pursued as a possible contrast mechanism for biological tissue.^[2][3][4] However, despite attempts over several decades, there is yet no practical technique to carry out this procedure non-invasively. Recently, magnetic resonance has been explored as a way of obtaining internal measurements that can dramatically improve the conditioning of the estimation procedure.^[4][5][6] However, techniques that have been proposed to date rely on some form of symmetry assumptions, which significantly constrain the application of such methods and which result in numerical edge artifacts and low effective resolution due to reliance on local derivatives.^[4][5][6]

We present a new approach, dubbed Global Maxwell Tomography or GMT, that relies solely on B₁+ magnitude maps for electrical properties estimation from MR data. Our technique is formulated as an inverse problem: Given a guess at the material properties of the scanned object, a full electromagnetic simulation is used to obtain a simulated B₁+ map that is compared to a measured B₁+ map, and relative permittivity and conductivity are then adjusted accordingly to reduce the discrepancy between simulation and measurement. The optimization procedure is repeated until the error in |B₁+| is sufficiently small.

Behind this formulation is a fast volume integral equation solver, MARIE, which was specifically designed for highly inhomogeneous, lossy dielectric media, and which enables rapid full-domain electromagnetic simulation.^[7][8] The integral equation formulation obtains an equivalent current distribution that matches the incident field data via an iterative solving procedure for the resulting second-kind integral equation system.^[7][8] Similarly, the gradient of the cost function that is minimized is derived analytically in terms of the elementary integro-differential operators that constitute the integral equation formulation, enabling rapid optimization.^[7]

In order to solve the systems of equations of the formulation efficiently, a biconjugate gradient stabilized method is used. The optimization algorithm that is used is a quasi-Newton–type algorithm for large-scale unconstrained optimization that is known commonly as Limited-Memory BFGS or L-BFGS. Solving for the equivalent currents and for the gradient, which involves solving an adjoint formulation that uses the same operators, constitute the inner loop of the algorithm, whereas the quasi-Newton stepping constitutes the outer loop of our implementation.

The cost function that is used seeks to minimize the error in estimates of |B₁+|, or more specifically, the error in the square of the magnitude at each location within the sample. In order to condense this three-dimensional tensor of error to a scalar value, we sum the square of each entry in this error tensor. Multiple scans are handled by weighting and summing the measures of error across all scans.

In our first example, we attempted to reconstruct electrical properties for a tissue-mimicking numerical phantom from a homogeneous guess, in the absence of noise, at a resolution of 10 mm, at an operating frequency of 297.2 MHz. As shown in Fig. 1, the properties of all of the structures in the phantom were reconstructed correctly solely from |B₁+| data, although the edges are blurred, as is the case with local methods.

In our second example, we attempted to detect and characterize tumors that have been artificially inserted into the open-source Duke head model, at a resolution of 3 mm, a frequency of 297.2 MHz, and an SNR of 80 in the magnitude maps. The relative permittivity and the conductivity of the tumors were set to values in the ranges (50, 60) and (1.1, 1.2) S/m, respectively.^[1] The initial guess is precisely the Duke model without the three tumors. As shown in Fig. 2, the tumors were correctly identified and the reconstructed electrical properties of these tumors were mostly correctly inferred, albeit with some blurring. The method converged after 124 iterations.

The proposed GMT technique is promising: In our first example, when the initial guess was completely incorrect and no assumptions were made about the electrical properties of the materials within the phantom, GMT successfully converged to a close approximation of the true property distribution. In our second example, when the initial guess was close to the actual values but missed some important unknown elements, i.e. the tumors, these were successfully reconstructed, even in the presence of noise. No edge artifacts were found in either case, and the blurring probably occurs due ill-conditioning of the outer loop. Further work on GMT would include preconditioning the outer optimization loop to achieve faster convergence, and the inclusion of proper regularization, such as additive total variation, to deal with the impact of noise.