## 5 Discussion
The reconstructed conductivity value shows overestimation compared to the real value when violating the assumption, such as region 2 of the phantom simulation. This assumption is close to the assumption used in the conventional approach, which is \(\sigma>>\omega\varepsilon\)[25]. Third, the spatial derivatives of \(B_{z}\) are negligible compare to \(B_{1}^{+}\) and \(B_{1}^{-}\), which appear rational in the central region of the birdcage coil from the simulation. From the perspective of clinical application, the \(B_{z}\) information cannot be directly acquired. The optimization problem of the proposed method includes a second-order TGV regularization term that can suppress the noise while omitting the staircase effect. ADMM was applied to address the problem and then obtain accurate conductivity images. Compared with DRM, the proposed method eliminates the staircase effect and shows better results in terms of noise suppression. The implementation of the proposed method requires four parameters, which are determined by using RMSE as a criterion. The optimal parameters for different regions vary. Thus, selecting spatially dependent parameters provides the potential to obtain a more accurate result than before, as observed in other applications [51]. During the calculation, there are a few iterations to acquire moderate conductivity results, indicating that the time cost is short. The phantom and brain reconstructions take approximately 2 s and 10 s, respectively, on a standard PC with an Intel i5 (2.9 GHz) and 16 Gb of RAM. The proposed method reformulates the central equation to remove the Laplacian operator through divergence properties. In the discretization for the imaging region, different points are used to construct an inverse problem compared with DRM. Inspired by the recent research [52], we can also use the divergence theorem to transform the equation into an integral formula that can suppress noise, further improving the reconstruction. The distribution of conductivity is reconstructed using only transceive phase information. Thus, any sequences can be used to obtain the transceive phase, theoretically. However, other factors, such as the homogeneity of the main field strength, eddy currents, and the phase contribution from flow and motion contaminate the phase data [16]. In this study, SE was chosen to acquire the phase. In other studies, the steady-state free precession (SSFP) sequence shows advantages for scanning because of its short time [53]. During the present experiment, a volume coil was used for transmitting and receiving, which is a typical configuration in clinical application. The multi-channel coil as the receive coil improves the SNR of the MR image and saves time [54]. It would be exploring in future research. In addition, when the phase image is achieved, the magnitude image is also obtained. The magnitude image includes the structural information of the object, the incorporation of which should be explored to get better results. Conceptualization, X.S. ; methodology, X.S. and L.L. ; validation, X.S. ; investigation, X.S. and Y.M. ; resources, Y.M. and X.L. ; writing--original draft preparation, X.S. ; writing--review and editing, L.L., L.Q. and W.C.; supervision, X.L. and W.C.; project administration, X.L. and W.C.; funding acquisition, W.C. All authors have read and agreed to the published version of the manuscript. This work was supported by the Key-Area Research and Development Program of Guangdong Province under Grant 2018B030333001, the National key research and development program under grant 2016 YFC0104003, the Natural Science Foundation of Guangdong Province under grants 2016A030313577, and the Program of Pearl River Young Talents of Science and Technology in Guangzhou under grant 201610010011. The authors declare no conflict of interest.