Magnetic Resonance Imaging (MRI) technology, especially ultra-high-field (UHF) systems, has attracted extensive attention in clinical practice due to its high resolution, no ionizing radiation, and capability to provide various imaging parameters [1]. The imaging parameters include proton density, T1, and T2; the versatility makes MRI an indispensable tool in medical applications. As a critical component of an MRI scanner, the radio frequency (RF) coil is used to transmit and receive MRI signals [2]. The performance of the RF coil directly determines the imaging quality. In UHF MRI applications, the resonance frequency of the MRI signal increases, inducing short-wavelength effects. When the wavelength is close to the size of the imaging load, a strong wave behavior occurs [3,4]. This will reduce the uniformity of the RF (B1) field, leading to poor imaging results and bringing RF safety issues associated with the concentration of specific absorption rate (SAR) values in biological tissues [5]. To address these issues, a novel design and analysis of high-performance RF coils is required for UHF MRI applications.

The methods for designing RF coils mainly include the method of moments [6,7] (MoM), the finite difference time domain method (FDTD) [8,9], and the finite element method [10] (FEM). These numerical algorithms can deal with electromagnetic field distribution problems with complex structural parameters, each with advantages and weaknesses. MoM solves integral equations in the frequency domain and is suitable for modeling the current distribution on metal conductors. It requires no boundary condition setup and meshing of the intermediate air domain. However, MoM is not suitable for solving electromagnetic problems involving complex biological tissue. This is because using the complex Green's Eq. [11] will significantly reduce the computational efficiency. FDTD is a numerical method for solving differential equations in the time domain. It directly discretizes Maxwell's differential equations in time and space and solves the electromagnetic problems in a time-step manner. It can efficiently solve the electromagnetic field distribution of objects with complex permittivity and conductivity [11]. However, it is difficult to accurately model a metal conductor with a curved structure because its mesh grid is usually a cubic cell, which can introduce staircasing errors. It also needs to apply additional absorbing boundary conditions, which is unsuitable for accurately modeling RF coils. FEM is also a method for solving differential equations in the frequency domain, for which a matrix equation must be constructed. It can accurately model objects of any shape. Like MoM, the solution efficiency of FEM is low for media with complex electromagnetic parameters, such as human tissue [11]. In addition, the FEM solution also needs to discretize the intermediate air domain, and a boundary condition is also required. Hence, it is not efficient in modeling MRI RF coil-tissue interactions.

Considering the merits and disadvantages of these numerical methods, we develop a hybrid FDTD/MoM algorithm [11,12] to model the MRI RF coil. The MoM is used to solve the RF coil's current distributions, while the FDTD is used to solve the electromagnetic field distribution inside the imaging load. The application of this hybrid algorithm in the field of MRI was first proposed by Chen et al. [13] in 1998. In their article, the formulas of FDTD and MoM were deduced in detail, but the interaction between the two algorithms was not considered. Later, Li et al. [12] further developed the method and introduced the Huygens' equivalent surface to connect the two algorithms. In this case, the effect of biological load on the coil current at high frequencies can be considered. Furthermore, the staircasing errors that arise in FDTD's current mapping process can also be eliminated using Huygens' equivalent surface. In their works, MoM is implemented by commercial software (FEKO), and the FDTD calculation is implemented by an in-house program [14], so it needs a specially designed data connection program, which indicates the implementation of the hybrid algorithm is not flexible. In 2011, Xin et al. [15] used a hybrid FDTD/MoM algorithm based on Huygens' equivalent surface to study a multi-channel RF coil. Their study adopted two commercial software tools (SEMCAD and FEKO) and required a complex interface program to connect the two software tools. This extra interface design brings concerns about the numerical accuracy and efficiency of modeling complex RF coil-tissue interactions.

Since the implementation of existing hybrid FDTD/MoM algorithms is not flexible, the computing environment is not unified, and the calculation time and accuracy cannot be well balanced. In this paper, the two algorithms are integrated and tuned to achieve optimal numerical performance. In the FDTD region, a non-uniform meshing technique [16,17] is used to improve the computational efficiency of the algorithm. In addition, the MoM region adopts the Rao-Wilton-Glisson (RWG) basis function [18] to model the RF coil structure accurately. Unlike commercial software-involved hybrid implementation, our FDTD-MoM algorithm is used explicitly for MRI RF coil design with biological load, in which various parameters can be flexibly controlled, and numerical efficiency and accuracy can be optimally balanced. The article mainly consists of three parts: the first part describes the MoM algorithm and its validation; the second part presents the calculation and verification of the FDTD algorithm; the third part is on the construction and validation of the hybrid algorithm. All calculations in this paper are implemented in MATLAB (MathWorks, Natick, Massachusetts, USA).