# Magnetic Resonance-Electrical Properties Tomography by Directly Solving Maxwell's Curl Equations
## 2 Theory and Method
### GPU-Accelerated FDTD Solver  
The in-house built FDTD solver [29] was constructed based on the differential form of the Maxwell equations (ME). As the first-order central differential scheme was used to implement the partial derivatives in the ME numerically, the noise effects were only amplified in the first-order rate, which was more robust than conventional EPT methods based on second-order wave equation. In the FDTD computational domain, a uniform mesh was used to discretize the subject-coil space, and specific cell sizes were problem-dependent. The calculation volume was truncated with perfectly matched  
Figure 1: Frameworks of different types of Electrical Properties Tomography (EPT) methods, (**a**) the local homogeneity assumption (LHA)-based solution [7, 8], (**b**) the second-order differential wave equation-based solution [16, 17, 18, 19, 20], (**c**) integral equation-based solution [23, 24, 25, 26, 11], and (**d**) the proposed EPT approach. It is noted that the proposed method finds the EPT solution without accessing the phase information of the radiofrequency (RF) field.  
layers (PML). The PML was composed of 8 to 10 layers, and the conductivities of each layer were calculated based on the method presented in Reference [30].  
In the FDTD solver, the vector forms of the electric and magnetic fields were calculated for the evaluation of \(\left|B_{1}^{+}\right|=\left|B_{x}+iB_{y}\right|/2\)[31]. Here, as [PERSON]'s curl equations state, the calculated \(\left|B_{1}^{+}\right|\) fields were a function of EPs of the imaged subject. For more details about the in-house developed FDTD solver, please refer to our previous work [29].  
In the constructed optimization problem, the forward process was implemented in each iteration step; thus the FDTD solver was required to run efficiently for the evaluation of the \(\left|B_{1}^{+}\right|\) field. To this end, a GPU-accelerated FDTD solver [29] was applied to the proposed EPT solution procedure. In this study, a 12-rung birdcage coil was modelled as an EM source and loaded with a subject (phantom or head), as shown in Figure 2. A perfect electric conductor (PEC) was placed around the birdcage coil to simulate the RF shield, and PML boundary conditions were placed at the top and bottom of the computation domain. In the FDTD simulation, [PERSON]'s curl equations are approximated with [PERSON]'s 'leap-frog' finite-difference algorithm, and the entire computational space is translated from standard Cartesian space to Yee-cell space, where the FDTD equations exhibit inherent strong parallelism. At each Yee-cell, this parallelism permits the simultaneous calculation of magnetic and electric field components in both space and time domains. The explicit field updating procedure was easily implemented in multiple, synchronized threads within the parallel-computing framework of a GPU. In this work, all computations have been executed on an Intel Xeon CPU E5-2680 V2 @ 2.80 GHz (128 GB RAM) equipped with an NVIDIA TITAN V GPU (12 GB RAM, 5120 CPUs, and 1455 MHz clock frequency) and more details about the GPU-FDTD computing can be found in Reference [29].