# Magnetic Resonance-Electrical Properties Tomography by Directly Solving Maxwell's Curl Equations
## 1 Introduction  
Magnetic Resonance-Electrical Properties Tomography (MR-EPT) is an imaging technique which maps the electrical properties (permittivity and conductivity) of a subject by analyzing the radiofrequency (RF) magnetic field measured during an MR scan. This technique has shown promise for clinical diagnosis and interests in ultra-high field Magnetic Resonance Imaging (MRI), because of its potential in the evaluation of patient-specific absorption rate (SAR) [1]. MR-EPT has alsoshown promise for clinical diagnosis and guiding the treatment of malignant tissues [2; 3] and improving the patient-specific electromagnetic (EM) modelling of hyperthermia [4]. The concept of MR-EPT was introduced by [PERSON] et al. in 1991 [5], and [PERSON] et al. [6] verified this concept through experiments in 2003. The initial development of MR-EPT was based on the utilization of the Helmholtz equation, in which the permittivity and conductivity were assumed constant (also known as the \"local homogeneity assumption\" (LHA)) [7; 8]. The methods based on this LHA have been found to be sensitive to noise, and severe distortions at the boundaries of different tissues may occur [9; 10]. To reduce the noise amplification in the utilization of second-order derivatives, several methods have been proposed. Additional regularization terms were added to solve the EPT inverse problem [11; 12], the Laplacian operator was replaced with a surface integral [8; 13], the k-space data were weighted properly [14], and the region of interest (ROI) was expanded with more voxels [15]. Numerical and experimental results in References [11; 12; 13; 14; 15] demonstrated that these methods improved the noise robustness but failed to correct the edge artefacts caused by the LHA. To eliminate these edge artefacts, a new set of partial differential equations (PDE) was introduced [16; 17; 18; 19; 20; 21]. In References [16; 17; 18], the special gradients of EPs were considered as unknown variables in the PDE, which were resolved by using multi-channel parallel transmit coils. In this scheme, the condition number of the constructed PDE was reduced, and the inverse solution was thus improved. An alternative method to resolve the unknown EP variables is to transform the PDE into a linearly regularized system and search for its global solution [19; 20; 21]. Although the methods in References [16; 17; 18; 19; 20; 21] have been tested in experiments, it is noteworthy that the EP reconstructions were implemented in an inverse process, in which the solution robustness could be weakened by the ill-conditioned property of the linear system. For these reasons, the development of new EPT approaches without the involvement of second-order PDE (i.e., Laplacian-operator free) is of interest. Recently, EP reconstruction in MRI has been alternatively investigated using the integral equations (IE). The IE-based EPT methods rely on a global convolution between the Green function and the contrast source [22; 23; 24; 25; 26] With the integral of global convolution, the noise robustness is theoretically improved in those IE-based methods. However, the development of IE-EPT is still at an initial stage and facing several practical problems. The technical problems include the requirement of the estimation of the incident electromagnetic fields, low spatial resolution, the accuracy issue associated with the use of the free-space Green function to construct the contrast source and the necessity of using global convolution. In this paper, a novel MR-EPT method is proposed. In the presented method, the EP reconstruction was achieved by directly solving [PERSON]'s equations [27; 28]. That is, the first-order PDE was numerically solved by using an in-house finite-difference-time-domain (FDTD) method with the graphics processing unit (GPU)-based acceleration.