# Magnetic Resonance-Electrical Properties Tomography by Directly Solving Maxwell's Curl Equations
## 3 Results
### Experimental Results  
The new EPT method was further verified by using experimental data obtained from the 9.4T MRI system. The measured \(B_{1}^{+}\) field data were compared with that from the GPU-FDTD solver, as shown in Figure 6, indicating that an accurate \(B_{1}^{+}\) field can be calculated for both the magnitude and phase. These results confirmed the accuracy of the forward solver in the new EPT approach, thus showing that stable and fast convergence can be expected. The measured \(B_{1}^{+}\) data (including the magnitude and phase information) was firstly used to generate the EP profile by using the conventional method with the LHA. As shown in Figure 7b,f, large reconstruction artefacts can be clearly observed in the two solutions. In the inner and outer regions of the acrylic container, more distortions are observed compared with the  
Figure 5: The root-mean-square-errors (RMSE) concerning the reconstructed \(\varepsilon_{\text{r}}\) and \(\sigma\) by using the traditional LHA-based method (blue line), the recently proposed modified Finite-Difference (mFD) method (purple line) and the proposed method (red line) under different SNR levels (from 55 dB to 15 dB). (**a,d**) retrieve EP’s values in the cerebrospinal fluid (CSF) CSF region, (**b,e**) retrieve EP’s values in the white matter’s region, (**c,f**) the retrieved EP’s values in the gray matter’s region. Figure 6: (**a,c**) indicate the magnitude and phase of the measured \(B_{1}^{+}\) data. (**b,d**) indicate the magnitude and phase of the calculated \(B_{1}^{+}\) data from the GPU-FDTD solver. results from mFD and the proposed method. This is because the LHA-based method is susceptible to measurement noise; hence EP distortions still appear in those homogenous regions. Figure 7c,g showed the reconstruction results by using the mFD and Figure 7d,h are the solutions of the new EPT method. It can be observed that although the artifacts were reduced when using the mFD method, and the specific EPs were still deviated from the measured values. With the proposed method, an accurate estimation of the EPs in the two solutions can be achieved. The proposed method attempted to seek the optimum EP values of the inner and outer regions by fitting the measured and calculated \(B_{1}^{+}\) field. This process mainly considers the global \(B_{1}^{+}\) field distribution, thereby mitigating the local measurement noise effect and significantly improving the robustness of the algorithm. Owing to this feature, accurate EP values of the inner and outer regions can be calculated. Table 2 shows that permittivity and conductivity values can be calculated with less than 0.6% errors, which indicated the superiority of the proposed EPT in a practical utilization. The excellent results shown in Figure 7 are obtained with the available exact acrylic boundaries derived from the acquired MR image, and the EPs in segmented regions are taken as unknown constants to reconstruct through optimization. The precise position of the acrylic pixels is also known from segmentation and excluded in the EP reconstruction. In addition to EP reconstruction, the new EPT can also provide information about the \(B_{z}\) field, which is usually neglected in current differential and integral-based MR-EPT methods. Figure 8 shows the exported \(B_{z}\) components from the proposed solution procedure. As expected, in the central plane of the birdcage coil, the \(|B_{z}|\) was quite small. Close to the coil ends, the \(|B_{z}|\) value was elevated. The availability of these magnetic field components should be helpful to improve the EPT solution and its applications in MRI. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Compartment**} & \multicolumn{1}{c}{**Probe Measurement**} & \multicolumn{1}{c}{**Reconstructed**} \\ \hline \multirow{2}{*}{Inner} & \(\epsilon_{\mathbf{r}}\) & 78.5 & 78 \\  & \(\sigma\) & 0.78 & 0.7768 \\ \hline \multirow{2}{*}{Outer} & \(\epsilon_{\mathbf{r}}\) & 78.5 & 78 \\  & \(\sigma\) & 0.43 & 0.4324 \\ \hline \hline \end{tabular}
\end{table}