In this talk, I will present the basics of the three main electromagnetic simulation approaches for assessment of MRI RF coils: finite difference time domain, finite element modeling and integral equations. The strengths and weaknesses of these techniques will be compared. I will introduce the co-simulation approach to fast coil analysis. Finally, I will talk about the availability of body models for each technique as well as major applications such as RF coil safety assessment, design optimization and calculation of ultimate metric bounds.
With increasing B0 field strength, the EM fields created by transmit and receive RF coil become more and more dependent on the load (patient). For receive coils, the SNR is proportional to the ratio of B1- sensitivity profiles and the square root of thermal losses in the coil and load (depends on the electric field). For transmit coils, the metric of interest in the strength of the B1+ field divided by the square root of the specific absorption rate (SAR). For multi-channel coils, both the SNR and B1+/sqrt(SAR) metrics depend on the coil combination. These metrics can be computed using EM simulation, which allows assessment of RF coils without having to actual build prototypes.
FDTD solves Maxwell equations by discretization of the spatial domain using a rectangular grid pattern (1). The grid is not necessarily regular (it can be smaller in regions of small details and greater in air regions), however the minimum time step required for stable convergence is inversely related to the size of smallest discretization element. Thus, highly detailed discretization schemes are associated with prohibitively long computation times with FDTD, although this can be mitigated somewhat by the use of graphics processing units (2). Since FDTD solves Maxwell’s equations in time, it can quickly assess the broadband response of the coil in a large frequency range. Another advantage of FDTD is that it is compatible with voxel models of the human anatomy, which is by far the most convenient discretization scheme for creating human body models (3). A major weakness of FDTD is the difficulty of modeling multi-channel coils, since each port needs to be simulated independently with FDTD resulting in long computation times.
FEM usually uses a tetrahedral discretization of the simulation domain, although this is not a requirement. FEM, unlike FDTD, is highly flexible with respect to the discretization scheme, which is very practical as this allows efficient simulation of both small and large geometrical scales in the simulation domain without yielding excessively long computation times (4). FEM is a frequency domain method, therefore it is better suited to the analysis of narrow-band RF systems (this is the case in MRI since we tune our coils to the Larmor frequency) with multiple channels since these can be solved simultaneously in a single FEM solve (this is in contrast with FDTD which requires N solves to simulate a N-port system). A disadvantage of FEM with tetrahedral discretization is that it requires surface mesh body models, which are difficult to obtain since the complexity of the human body is not easily modeled by surface meshes (5).
The last main EM simulation method for MRI applications is the integral equation approach. Historically, the surface integral equations (SIE) method has received an outsize amount of focus in the past 30 years. Recently however, volume integral equation (VIE) was revived for MRI (6). VIE is an ideal technique for MRI applications since it discretizes the spatial domain in regular voxels and, like FDTD, is compatible with body models generated using MR and CT data. VIE/SIE are frequency domain methods that are very efficient for solving Maxwell’s equation at discrete frequencies (narrow-band) and for multiple ports. VIE can be interfaced with SIE to provide a combined method that is extremely efficient as modeling both flat surfaces (coil, shield) and dielectric volumes (patient) (7). Moreover, the SIE/VIE coupling is analytical, which means that the empty air space between the coil and the patient does not have to be discretized thus minimizing the number of triangle and voxel elements as well as simulation time. Another advantage of SIE/VIE over FDTD and FEM is that it does not require the imposition of boundary conditions: The domain is placed analytically in an infinite medium of arbitrary conductivity and permittivity (usually cond=0 S/m and rel_perm=1 for air/vacuum) which precludes the need for absorbing boundary layers such as ABC and PML.
FDTD, FEM and VIE/SIE are compatible with the so-called co-simulation strategy (8). In this approach, tuning and matching is performed using a circuit simulator as opposed to within the EM solve step. This is much more efficient since circuit simulation is fast (on the order of seconds) whereas EM simulation is slow (on the order to tens of minutes or even hours). In the co-simulation approach, lumped elements (capacitors and inductors) with unknown value are replaced by ports and the fields and S-parameters for each of these ports are computed in the EM solve step. This allows modeling any possible capacitance and inductance value in the circuit simulator S-matrix representation of the coil electrical response very quickly using a single EM solve.
EM simulation can be very useful for MRI. Transmit and receive coil performance can be predicted (9), which is helpful to compare possible designs and weigh tradeoff metrics such as complexity, cost, SNR, SAR, efficiency etc… (10-12) For transmit coil, EM simulation is a crucial tool that is required for safety evaluation as there are no possible direct measurement of SAR. Finally, EM simulation can be used to assess the maximum performance of transmit and receive coils allowed by Maxwell’s equation (13-17). These “ultimate metrics” can be useful to assess the performance of coil design and to assess the room for improvement of coil performance as a function of number of channels.
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