• RF modelling plays an important role in RF coil design and patient safety evaluation.

• Popular full-wave numerical methods for RF modelling include method of moments (MoM) and finite difference time domain (FDTD). These two methods will be discussed in the context of MRI applications.

Target Audience- MR researchers and clinicians who want to learn about RF modelling principles and methods for MRI applications

Outcome/Objectives- Attendees will learn the basics of RF modelling methods, be able to choose one suitable method according to their needs, and also learn how to set up conditions for accurate RF modelling.

The need for full-wave analysis of RF problems in MRI

The MR signal arises from the interaction between a radiofrequency (RF) wave and the nuclear spins of hydrogen atoms inside the human body, such as water molecules. A knowledge of RF field propagation and distribution in the imaged subject is important in terms of image quality and patient safety. However, RF field quantities are difficult to measure in living tissues; therefore, RF modelling has become a necessary approach used in RF coil designs and safety evaluations.

RF modelling involves the numerical approximation of Maxwell’s equations to characterize the electromagnetic (EM) behavior of RF coils. For relatively low field strengths (≤1.5T), RF coil modelling is often based on circuit analysis that invokes quasi-static field approximations (using Biot-savart law). These quasi-static calculations are valid at lower field strengths, because the RF coils are small compared with the wavelength of the RF field inside the biological load and thus the coil–tissue interactions are considered negligible. In high-field MRI (≥3T), the dimensions of the RF coils are comparable to, or less than that of, the operational wavelength, causing the RF field inside the biological load to exhibit prominent wave behavior. In addition, the RF field and source currents in the RF coil are strongly perturbed by the biological load. For the high frequency RF coil designs, it is necessary to resolve these complex coil-tissue interactions and also their concomitant RF heating problems. Quasi-static approximations are no longer appropriate and the accurate prediction of the EM behavior requires a full-wave solution of the involved Maxwell's equations.

There is a range of full-wave numerical methods that are suitable for the MRI applications, with the most popular techniques including the finite difference time domain (FDTD), the finite element method (FEM) and the method of moments (MoM). Each of these modelling techniques has its own strengths and limitations. More recently, hybrid numerical methods, such as hybrid MoM/FDTD and MoM/FEM algorithms, have been used to design RF coils and to accurately evaluate the EM field distributions inside the biological samples.

Method of moments (MoM)

In RF coil modelling, MoM typically incorporates piecewise triangular functions to discretize the coil surface currents on metallic wires and patches. It solves unbounded problems without the discretization of the complete 3D space, thus it is very well suited for modeling complex coil structures and shields. Compared with other modelling methods, MoM offers narrowband EM solutions in a more efficient and accurate manner, particularly for the RF coil structures that are unloaded or loaded with simple dielectric volumes. On the other hand, as a frequency domain approach, MoM does not easily handle the complex and arbitrary-shaped dielectric volumes. This is because it needs to use a complicated Green function to represent the lossy, dielectric loads, and the formed dense system matrix equations are not easy to solve. This makes it relatively unpopular in modelling large-scale RF problems involving biological loads.

Finite Difference Time Domain (FDTD)

The FDTD method is a grid-based time-domain numerical technique that directly solves Maxwell's curl equations in the partial differential form. It outputs broadband EM solutions from a single execution of the program. This helps us to understand the fields within patients and general temporal field behavior during an MRI scan, thus offering insights into the fundamental EM problems related to MRI. The FDTD method has been widely used for RF modelling because of its simplicity and efficiency in handling large scale, near-field problems in MRI.

A comprehensive consideration of the loading effects is essential in the design/analysis of high-frequency RF coils. The FDTD method has proven to be the most efficient approach in the characterization of field–tissue interaction problems. It allows the modelling of dielectric bodies with arbitrary shapes and heterogeneous dielectric properties. Various anatomically detailed digital models and the dielectric data of human and animals are available in the literature. These body models generally have tens of tissue types and the spatial resolution is in millimeters. For FDTD computations, the models can be mapped onto a defined grid with volume-averaged dielectric properties. Such realistic patient models play a central role in the RF hardware design at high field MRI.

In FDTD simulation, the entire 3D space must be discretized, and the perfectly matched layers (PMLs) are usually needed as an EM field absorbing boundary condition. To model complex RF field–tissue interactions, conventional CPU-based FDTD calculations offer limited computing performance in a standard PC environment. To boost the FDTD computing efficiency, hardware-based parallel computing frameworks have recently been used in RF modelling. With the help of the latest graphics processing unit (GPU) technology, the FDTD simulation of RF system designs can achieve up to three orders of acceleration in the solution time. The GPU-enabled acceleration paves the way for FDTD to be applied for both detailed forward modelling and iterative, inverse design of MRI coils that were previously challenging or infeasible. In this talk, computational strategies and specific details of implementing the FDTD method in a GPU architecture will be discussed.

The FDTD method is suitable for use in MRI RF modelling, however, it has one major technical drawback. Using the stair-casing approach, the FDTD method discretizes the 3D space into cubic cells to model dielectric samples and RF coils and this can sometimes lead to unacceptable numerical errors for the cases with complex EM structures. A variety of numerical strategies (such as sub-gridding, sub-cell techniques) have been devised to improve the modelling accuracy of the FDTD algorithm.

FDTD-MoM

Recently, hybrid numerical methods that integrate the desirable features of two or more distinctive full-wave EM modelling techniques have been developed for RF coil modelling and EM field calculations. In the case of the hybrid FDTD/MoM method, the RF coil is accurately modelled using the MoM algorithm, and the biological load is handled with the FDTD algorithm. Effective communication between these two algorithms is realized through a Huygence equivalent surface. With the appropriate numerical interfacing, this approach can provide more efficient and accurate results in modelling loaded coils at high frequencies.

Full-wave EM software package

Most of the existing commercial EM packages are now capable of producing simulation results close to those obtained experimentally, provided that they are used appropriately. Most of these packages can provide comparable outputs for coil design and Specific Absorption Rate (SAR) estimations. In these packages, an accurate setup of the driving source is important in terms of bridging the gap between the simulation and experiment. Various EM sources, including voltage or current sources, can be used to drive the coil. To achieve satisfactory simulation results, it is important to consider a favorite driving scheme for a specific application in MRI. The software selection depends on the user interface, software stability, affordability and personal preference. In practice, a balance needs to be achieved between the numerical accuracy, storage space and simulation time. In general, the software with the MoM kernel excels at simulations of RF coils that are unloaded or loaded with simple dielectric subject, and those FDTD packages perform better when handling strong field-tissue interactions at high fields.

RF coil design and analysis based on EM-modelling

In MRI, the design of the RF coil plays a major role in optimizing the signal-to-noise-ratio (SNR) of MR images and ensuring the RF safety inside the biological sample. In the early days, prototypes were constructed and tested to investigate the performance of the RF coils. Often, the design was improved with several iterations to achieve an acceptable result. However, with the advancements in computational electromagnetic techniques, RF coil simulation allows for the accurate evaluation of coil performance on a large variety of parameters, which include field penetration, excitation homogeneity, coil efficiency and array decoupling. It has now become the modus operandi of coil design, which also reduces the financial cost, time and effort spent in designing and prototyping RF coils.

RF simulation assists in the development of different kinds of RF coils such as surface coils, volume coils and array coils. For surface coils, coil shape and size can be modelled and optimized for specific applications, and the capacitors are adjusted to enable the coil to resonate at the desired frequency. For the volume coils, such as birdcage coils, circularly polarized transverse magnetic fields can be obtained by driving the coil at two ports in a quadrature mode. For array coils, RF coupling between coil elements will be numerically minimised using decoupling networks or an overlapping scheme between adjacent coils. In these RF simulations, coil performance is typically investigated under realistic loading conditions.

RF modelling can also be valuable in developing imaging methods such as parallel imaging (PI). Investigating the SENSE technique for each element of an array coil involves computing the B1-receive field (coil sensitivity) profile, as well as the electric fields in the volume of interest. Using these field components, the optimum SNR and g-factor (spatial interaction between coils) can then be calculated. The array coil geometry can be optimized to achieve the best parallel imaging performance. Taking a subject-specific approach, RF modelling can also help design parallel transmission at high fields.

RF safety is a major concern in high field MRI and must be strictly monitored to limit tissue heating in patients. RF modelling permits us to investigate a wide range of design parameters to arrive at optimal solutions for RF coil designs and reliable results for RF safety involving local/global SAR. With further development, the GPU-accelerated FDTD computing may lead to the ability to perform real-time, subject-specific SAR assessment and reduction during the RF transmission at high fields. High performance RF modelling also enables accurate simulation of the EM environment with interventional devices such as pacemaker leads and can be used for the safety evaluation of different implantable devices.

RF modelling is now routinely performed in the design and analysis of MRI RF systems. It can be effectively employed for the validation or fine-tuning of the engineering designs to reduce expensive hardware prototyping, thus significantly easing the research and development in RF coil development.

Recently, the precise evaluation of field distributions inside the biological samples has become a critical design requirement in high field MRI scenarios. RF modelling can achieve precise knowledge of field-tissue interactions and provide guidance for the deign of novel EM hardware and imaging methodologies for safe, high-quality human imaging at 7 T, thus facilitating the transition of ultra-high field MRI into clinical practice.

This talk shares insights into technical details of implementing the most popular numerical EM methods. In particular, hybrid full-wave EM methods and GPU computing are highlighted, which creates a powerful theoretical prototyping platform for the design of novel RF systems. It is hoped that this talk can aid those who intend to implement demanding computational experiments for the research and development of RF coil designs for high-field MRI applications.