Global Maxwell Tomography: a novel technique for electrical properties mapping without symmetry assumptions or edge artifacts

Jose E.C. Serralles^{1}, Athanasios Polimeridis^{2}, Manushka V. Vaidya^{3,4}, Gillian Haemer^{3,4}, Jacob K. White^{1}, Daniel K. Sodickson^{3,4}, Luca Daniel^{1}, and Riccardo Lattanzi^{3,4}

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 B_{1}+ 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 B_{1}+
map that is compared to a measured B_{1}+ 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 |B_{1}+| 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 |B_{1}+|, 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 |B_{1}+| 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.

1. Stuchly MA, Athey TW, Samaras GM, Taylor GE. Measurement of Radio Frequency Permittivity of Biological Tissues with an Open-Ended Coaxial Line. IEEE Transactions on Microwave Theory and Techniques 1982;30(1):87-92.

2. Brown BH, Barber DC. Electrical impedance tomography; the construction and application to physiological measurement of electrical impedance images. Med Prog Technol 1987;13(2):69-75.

3. Gencer NG, Ider YZ, Kuzuoglu M. Electrical impedance tomography using induced and injected currents. Clin Phys Physiol Meas 1992;13(Suppl A):95-99.

4. Katscher U, Voigt T, Findeklee C, Vernickel P, Nehrke K, Dössel O. Determination of electric conductivity and local SAR via B1 mapping. IEEE Trans Med Imaging. 2009 Sep;28(9):1365-74.

5. Sodickson DK, Alon L, Deniz CM, Brown R, Zhang B, Wiggins GC, Cho GY, Ben Eliezer N, Novikov DS, Lattanzi R, Duan Q, Sodickson LA, Zhu Y. Local Maxwell Tomography Using Transmit-Receive Coil Arrays for Contact-Free Mapping of Tissue Electrical Properties and Determination of Absolute RF Phase. 20th Scientific Meeting of the International Society of Magnetic Resonance in Medicine (ISMRM), 2012; Melbourne, Australia. p 387.

6. Balidemaj E, van den Berg CA, Trinks J, van Lier AL, Nederveen AJ, Stalpers LJ, Crezee H, Remis RF. CSI-EPT: A Contrast Source Inversion Approach for Improved MRI-Based Electric Properties Tomography. IEEE Trans Med Imaging. 2015 Sep;34(9):1788-96.

7. A.G. Polimeridis, J.F. Villena, L. Daniel, J.K. White, Stable FFT-JVIE solvers for fast analysis of highly inhomogeneous dielectric objects, Journal of Computational Physics, Vol. 269, 15 July 2014, p. 280-296, ISSN 0021-9991.

8. Fernandez Villena, J.; Polimeridis, A.G.; Hochman, A.; White, J.K.; Daniel, L., "Magnetic resonance specific integral equation solver based on precomputed numerical Green functions," in Electromagnetics in Advanced Applications (ICEAA), 2013 International Conference on , vol., no., pp.724-727, 9-13 Sept. 2013.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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