Synopsis

Keywords: Electromagnetic Tissue Properties, Electromagnetic Tissue Properties

We propose a novel approach to electrical property (EP) estimation that also estimates the incident fields generated by a coil, by solving a basis pursuit problem. We also propose a novel Global Maxwell Tomography (GMT) formulation that uses the MR signal instead of $$$B_1^+$$$. We tested our approach experimentally by reconstructing the average EP of a homogeneous cylinder. We obtained < 5% estimation error using only $$$B_1^+$$$ data obtained from MR Fingerprinting. We then estimated the receive sensitivity maps $$$B_1^-$$$ by using the new signal-based GMT.

Introduction

The distribution of electrical properties (EP) of an object, namely relative permittivity $$$\epsilon_{R}$$$ and conductivity $$$\sigma$$$, dictates how the object interacts with electromagnetic fields. The ability to infer these EP from measured data would improve a myriad of applications, such as radiofrequency (RF) ablation, RF hyperthermia, and cancer detection¹.
The process of EP estimation is commonly referred to as inverse scattering or electrical property tomography (EPT). Most existing magnetic resonance (MR)-based EPT techniques rely on simplifying assumptions, such as the transceive phase approximation, limiting their applicability^2,3. Global Maxwell Tomography (GMT) is a global formulation of EPT that requires RF coil incident fields and assumes that the coupling between the coil and body can be neglected, which is its only simplifying assumption¹. In this work, we propose the use of basis pursuit with an electromagnetic basis to replace the coil model entirely. In addition, we propose a new cost function for GMT that uses MR signal data, rather than $$$B_1^+$$$. We demonstrate the effectiveness of our approach by estimating both the EP of a homogeneous cylinder and the coil's incident fields with a 7T experiment.

Methods

The traditional GMT optimization problem is described as:
$$\begin{aligned}\min_{\epsilon\in\mathbb{C}^{N}}\quad & \sum_k \sum_l\left\| w_{kl}\odot \left[b_k^+\!\left(\epsilon\right)\odot\overline{b_l^+\!\left(\epsilon\right)}-\hat{b}_k^+\odot \overline{\hat{b}_l^+} \right] \right\|_2^2 \\ \textrm{s.t.}\quad & \text{Re}\!\left\{\epsilon\right\}\geq 1 \\ &\text{Im}\!\left\{\epsilon\right\}\leq 0\end{aligned}\tag{1}$$
The decision variables are complex relative permittivities, defined as $$$\epsilon=\epsilon_{\rm R}+\frac{\sigma}{j2\pi f\epsilon_0}$$$, where $$$f$$$ is the Larmor frequency. The cost function compares measured products of $$$\hat{b}_k^+$$$ with corresponding products of simulated $$$b_k^+$$$, computed using the (known) coil's incident fields with the EP estimate at the current iteration.
We surrounded the cylinder mask with volumetric current sources and approximated the mappings from sources to electric field $$$U_e$$$ and magnetic field $$$U_h$$$ using randomized SVD⁴. We then pose the task of estimating the coil's incident fields as a basis pursuit problem, using the same cost function as in Problem 1:
$$\begin{aligned}\min_{\alpha\in\mathbb{C}^{M\times N_{\rm tx}}}\quad &\sum_k\sum_l\left\|w_{kl}\odot \left[b_k^+(\alpha_k)\odot\overline{b_l^+(\alpha_l)}-\hat{b}_k^+\odot\overline{\hat{b}_l^+}\right]\right\|_2^2 \\ \textrm{s.t.}\quad &\text{Re}\!\left\{\alpha_{1,1}\right\}\geq 0 \\ &\text{Im}\!\left\{\alpha_{1,1}\right\}=0 \\\end{aligned}\tag{2}$$
The decision variables are the complex-valued basis coefficients $$$\alpha$$$, and the constraints help to reduce the size of the set of possible solutions. We perform this joint optimization over EP $$$\epsilon$$$ and basis coefficients $$$\alpha$$$ by first solving the basis pursuit problem given an initial guess of EP, and then solving the EP problem. We repeat this alternation, or "leap-frog", approach until we achieve the desired accuracy.
Given the challenges associated with the experimental measurement of $$$B_1^+$$$, we also propose a new MR signal-based GMT formulation:
$$\begin{aligned}\min_{\epsilon\in\mathbb{C}^{N}}\quad &\sum_k\sum_l\left\|w_{kl}\odot\left[s_{kl}(\epsilon)-\hat{s}_{kl}\right]\right\|_2^2 \\ \textrm{s.t.}\quad &\text{Re}\!\left\{\epsilon\right\}\geq 1 \\ &\text{Im}\!\left\{\epsilon\right\}\leq 0\\ \end{aligned}\tag{3}$$
where $$$s_{kl}$$$ denotes the MR signal associated with the transmit field $$$b_k^+$$$ and receive sensitivity map $$$b_l^-$$$. We also employ a corresponding basis pursuit problem:
$$\begin{aligned}\min_{\alpha\in\mathbb{C}^{M\times N_{\rm tx}}}\quad &\sum_k\sum_l\left\|w_{kl}\odot \left[s_{kl}(\alpha_k,\alpha_l)-\hat{b}_k^+\odot\overline{\hat{b}_l^+}\right]\right\|_2^2 \\ \textrm{s.t.}\quad &\text{Re}\!\left\{\alpha_{1,1}\right\}\geq 0 \\ &\text{Im}\!\left\{\alpha_{1,1}\right\}=0\\ \end{aligned}\tag{4}$$
To test our approach, we filled a cylinder with a solution of water and table salt (NaCl). We used a 7T ($$$f$$$ = 297.2MHz) 8-channel transmit/receive coil array⁵. We used MR Fingerprinting⁶ to obtain the 8 reference $$$|\hat{b}_k^+|$$$ maps. We then used the raw signals $$$s_{kl}$$$ to extract the relative phase information. We generated a numerical basis for the given geometry and solved the $$$b_1^+$$$-based optimization problems in an alternating fashion, starting from initial guesses of $$$\epsilon_{\rm R}=51$$$, $$$\sigma=0.13$$$S/m, $$$\alpha_{1,1}=1$$$, and 0 for the rest of the basis coefficients. We used $$$w_{kl} = |\hat{b}_k^+| \odot |\hat{b}_l^+|$$$ as the weights for the above equations and ran the leapfrog approach over the mean value of EP. Once we finished optimizing using $$$B_1^+$$$ data, we used the EP values to attempt to estimate the transmit and receive incident fields using the signal data.

Results & Discussion

Due to artifacts in the MR data, for this preliminary experiment we constrained all the EP to one value, from knowing that the phantom was homogeneous. We will remove this constraint in future experiments. Using a dielectric probe with expected error of 5%-10%, we measured the permittivity and conductivity to be 79.9 and 0.73S/m, respectively. Fig. 2 shows the evolution of permittivity and conductivity for various iterations. Fig. 3 shows the evolution of $$$|b_1^+|$$$ and relative phase between channels 1 and 2 along the same central slice. After 37 iterations, $$$\epsilon_{\rm R}$$$ converged to 81.2 and $$$\sigma$$$ converged to 0.71S/m, corresponding to 1.6% and 2.7% deviation, respectively, from measured probe values. Using the estimated EP, we inferred the receive coil sensitivities (Fig. 4).

Conclusions

In this work, we proposed a new formulation of $$$B_1^+$$$-based GMT that completely abandons the coil model in favor a numerical basis. The reconstructed average EP values matched the values measured by the probe almost exactly. Removing the need of an accurate coil model simplifies GMT experiments and does not require worrying about the coupling between the body and the coil, which was neglected by the original GMT. This comes at the expense of computational cost. In addition, we also demonstrated that is possible to infer the transmit coil sensitivities using the new MR signal-based optimization approach. Future work will involve determining the root cause of artifacts in the experimental data and also speeding up the convergence of the alternating approach. Once we figure out those details, we will reconstruct objects that are not homogeneous.

Acknowledgements

This work was supported by NIH R01 EB024536.