Reconstructing the electrical properties (EP) in vivo using MRI has drawn significant attention due to its great potential of high specificity for tumor diagnosis. However, the assumptions of equal transmit and receive phase distribution in a quadrature volume coil and locally homogenous tissue EP values in the Helmholtz equation may substantially degrade reconstruction fidelity, especially in the breast with a highly complex structure. These hurdles could be overcome by employing multiple transmit B₁ maps obtained from a coil array, a potent tool which could significantly push EPT further towards assumption-free reconstruction¹.

The central equation of EPT can be written as a partial differential equation concerning each specific transmit B₁ ($$$B_1^+$$$) distribution as$$\tag{1}{\begin{bmatrix}\color{red}{\nabla^2B_1^+}\\{\color{red}{\left(\frac{\partial{B_1^+}}{\partial{x}}-i\frac{\partial{B_1^+}}{\partial{y}}\right)}+\color{blue}{\frac{1}{2}\frac{\partial{B_z}}{\partial{z}}}}\\{\color{red}{\frac{\partial{B_1^+}}{\partial{z}}}-\color{blue}{\frac{1}{2}\left(\frac{\partial{B_z}}{\partial{x}}+i\frac{\partial{B_z}}{\partial{y}}\right)}}\end{bmatrix}^T\begin{bmatrix}\gamma_c\\\frac{\partial{\gamma_c}}{\partial{x}}+i\frac{\partial{\gamma_c}}{\partial{y}}\\\frac{\partial{\gamma_c}}{\partial{z}}\end{bmatrix}=\color{red}{-\omega^2\mu_0B_1^+}}$$ which relates conductivity $$$\sigma$$$ and relative permittivity $$$\epsilon_r$$$ in $$$\gamma_c=\frac{1}{\epsilon_r\epsilon_0-i\sigma/\omega}$$$ to time-harmonic $$$B_1^+$$$ and $$$B_z$$$ field at angular frequency $$$\omega$$$.^2,3 The $$$\color{red}{red}$$$ terms in the equation are $$$B_1^+$$$ dependent of which the magnitude and relative phase can be mapped using a multi-transmit coil array^4,5. However, the $$$B_z$$$-related terms in $$$\color{blue}{blue}$$$ are not directly tractable from MRI signal. 1) For reconstruction strategies ignoring EP gradient, the central equation reduces to Helmholtz equations⁶$$\tag{2}{\sigma=\color{red}{\frac{1}{\omega\mu_0}Im\left(\frac{\nabla^2B_1^+}{B_1^+}\right)}}$$$$\tag{3}{\epsilon_r=\color{red}{-\frac{1}{\omega^2\mu_0\epsilon_0}Re\left(\frac{\nabla^2B_1^+}{B_1^+}\right)}}$$It is desirable to combine multiple transmit channels for elevated reconstruction performance as a result of improved $$$B_1^+$$$ coverage and SNR. 2) For strategies considering EP gradient when the full equation (1) is employed, a carefully designed coil, which has enhanced $$$\color{red}{red}$$$ terms dominating over the $$$\color{blue}{blue}$$$ terms, should be used. Long rectangular microstrips⁷ and loops⁸ aligning in z direction are good candidates for serving this purpose due to their relatively constant $$$B_1^+$$$ along z direction and relatively insignificant $$$B_z$$$ variation in the middle of the coil. However, for the loop element, due to the opposite polarity of currents flowing in the two long sides, the $$$E_z$$$-related term $$$\color{red}{\left(\frac{\partial{B_1^+}}{\partial{x}}-i\frac{\partial{B_1^+}}{\partial{y}}\right)}$$$ destructively interferes while the $$$B_z$$$-related term constructively interferes with each other, causing difficulty to retrieve EP gradient without $$$B_z$$$ information. This interference is not prominent for microstrips as the field is generated by current flowing in a single-sided conductor. Therefore, it is expected that EPT reconstruction with EP gradient consideration prefers microstrips to loops.

Eight-channel microstrip and loop array coils for a breast model were constructed in SEMCAD and loaded with either a homogenous phantom mimicking saline water (conductivity 0.56S/m, relative permittivity 76) or a realistic female chest model (breast part from Phantom Repository, UMCEM; chest part from Ella, Virtual Population). All microstrips are 150mm in length and 30mm in width, and distributed azimuthally following the circumference of a half-cylinder 180mm in diameter. All loops are 140mm in length and 50mm in width in plane, decoupled by proper overlaps, and arched to the same half-cylinder circumference (Figure 1). Input power to each coil element at 128MHz was normalized to 1 Watt. The coil elements were driven sequentially, and the resultant steady-state time-harmonic $$$B_1^+$$$ maps for each element were extracted. Same level of complex white noise was added separately to $$$B_1^+$$$ maps from the two arrays loaded with the homogeneous phantom. The final reconstructed conductivity of the phantom was a $$$|B_1^+|$$$-weighted sum of the reconstructed conductivity using all eight individual complex B₁ map based on Helmholtz equation. The conductivity of the breast model was reconstructed by solving the concatenated linear system of equations associated with all eight channels regularized by Total Variation⁹. Helmholtz reconstruction of the conductivity in the phantom is shown in figure 2. Compared to the microstrips, reconstruction is more accurate and precise with the loop coil, especially at locations further away from the coils because of deeper $$$B_1^+$$$ penetration. At SNR=50, reconstructed mean ± standard deviation in the box is 0.52±0.11S/m for microstrips and 0.54±0.05S/m for the loops. The reconstructed conductivity distributions in the breast model are shown in figure 3. Result from the loop array coil manifests substantially lower contrast between tissues due to its more dominant $$$B_z$$$-related term which results in inaccurate preservation of the EP gradient information as discussed above. The reconstruction error pervades to homogeneous tissue as well owing to the nature of spatial partial differential equation which links together all pixels in the region of interest. With theoretical analysis and numerical simulation results, this study has demonstrated that loop array is preferable for local EPT reconstruction within homogeneous tissue while microstrip array is superior when considering EP gradient in the reconstruction process. It is expected that other specially designed coil hardware could further improve the reconstruction fidelity of the EP values for various body parts.