Jose E.C. Serralles1, Athanasios Polimeridis2, Manushka V. Vaidya3,4, Gillian Haemer3,4, Jacob K. White1, Daniel K. Sodickson3,4, Luca Daniel1, and Riccardo Lattanzi3,4
1Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, United States, 2Center for Computational and Data-Intensive Science and Engineering, Skolkovo Institute of Science and Technology, Moscow, Russian Federation, 3Center for Advanced Imaging Innovation and Research (CAI2R) and Center for Biomedical Imaging, Department of Radiology, New York University School of Medicine, New York, NY, United States, 4The Sackler Institute of Graduate Biomedical Science, New York University School of Medicine, New York, NY, United States
Synopsis
We
introduce Global Maxwell Tomography (GMT), a novel volume integral
equation-based technique for the extraction of electric properties
from MR data. GMT is framed as an unconstrained optimization problem
in which the error between measured and simulated B1+
magnitude maps is minimized. Due to its global nature, GMT is not
subjected to edge artifacts. By using exclusively B1+
magnitude, GMT does not rely on symmetry assumptions to estimate B1+
phase. In one numerical example, three tumors were inserted into a
head model, and starting from a tumorless initial guess, GMT
accurately inferred the electrical properties and locations of these
tumors.Introduction
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.
Theory
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.
Methods
and Results
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.
Discussion
and Conclusions
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.
Acknowledgements
No acknowledgement found.References
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