# Magnetic Resonance-Electrical Properties Tomography by Directly Solving Maxwell's Curl Equations
## 3 Results  
### Simulation Results  
The simple phantom described in Section 2.3 was first used to evaluate the performance of the proposed EPT method. To implement the FDTD calculation, the Yee-cell size was 1 mm, and the entire computational domain was divided into \(D_{x}\times D_{y}\times D_{z}=112\times 112\times 112\approx\) 1,404,928 cells. Each GPU-FDTD iteration time was approximately 18 seconds, and the total time for the optimization was around 30 min. In the evaluation, six different noise levels were added to the simulated \(B_{1}^{+}\) field. As shown in Table 1, the EPs can be retrieved with high accuracy using the proposed approach, even under an extreme scenario in which the SNR level declined to 15 dB (the lowest SNR level used in current MR-EPT numerical evaluations was 35 dB). The maximum errors were 3% and 0.63% for the permittivity and conductivity reconstruction, respectively. These results show the strong robustness of the method to noise.  
Figure 3: (**a**) The photo of the phantom used in the experiment and (**b**) the MR image obtained at 9.4 Tesla (T).  
Following the simple phantom evaluation, the numerical head model, as described in Section 2.3, was used to assess the performance of the proposed EPT method. The target and reconstructed EP profiles are shown in Figure 4. It can be seen that accurate EPs can be retrieved by using the proposed EPT method.  
To further illustrate the effectiveness of the new method, the traditional algorithm based on LHA and a recently developed EPT solution using a modified Finite-Difference (mFD) scheme [18] were both used for comparison. As a recently developed second-order differential equation-based approach, the mFD approach is a good candidate for comparison between the conventional EPT method with the proposed one. The root-mean-square-error (RMSE) [16] was used to evaluate the performance quantitatively, and the corresponding results are shown in Figure 5. It can be seen from this figure that the new EPT method achieved significantly better results than those from compared methods across SNR levels from 15 to 55 dB. In the extreme case, in which the SNR level was 15 dB, both comparison methods obtained RMSE values larger than 1.5 for the permittivity reconstruction and 1.2 for the conductivity reconstruction. However, the proposed method was capable of retrieving more accurate EP's values with an RMSE less than 0.2 for both the permittivity and conductivity for all SNR levels.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**SNR**} & \multicolumn{2}{c}{**Inner**} & \multicolumn{2}{c}{**Outer**} \\ \cline{2-5}  & \(\varepsilon_{\mathbf{r}}\) & \(\sigma\)**(S/m)** & \(\varepsilon_{\mathbf{r}}\) & \(\sigma\) \\ \hline Target & 78.5 & 0.43 & 78.5 & 0.78 \\ Noise-free & 78 & 0.4309 & 78 & 0.7784 \\ SNR = 15 & 77 & 0.4428 & 79 & 0.7644 \\ SNR = 25 & 77 & 0.4212 & 78 & 0.7918 \\ SNR = 35 & 78 & 0.4278 & 78 & 0.7843 \\ SNR = 45 & 78 & 0.4326 & 78 & 0.7762 \\ SNR = 55 & 78 & 0.4322 & 78 & 0.7824 \\ \hline \hline \end{tabular} SNR: signal-to-noise ratio. \(\sigma\) in S/m.  
\end{table}
Table 1: Reconstructed EPs of the phantom (based on simulated data).  
Figure 4: The target and reconstructed EP profiles under different SNR levels (15\(\sim\)35 dB). (**a**,**e**) are the target EPs. (**b**,**f**) are the results under SNR level is 15 dB. (**c**,**g**) are the results under SNR level is 25 dB. (**d**,**h**) are the results under SNR level is 35 dB.