Michael Beck^{1}, Dennis Parker^{1}, and Rock Hadley^{1}

Full wave simulations are known for their high accuracy, but simulation optimization is not feasible with FDTD and FEM for vast numbers of MRI coil applications. Optimization strategies do have feasible runtimes for quasi-static solutions, but the system being simulated must be small compared to the electromagnetic wavelength since they do not account for boundary conditions. This work uses multiple single loop coils of different diameters and three phantoms with a simple geometry to compare the accuracy and usefulness of full wave and quasi-static solutions of RF coils at 3T. Full wave simulations proved to be significantly more accurate.

Three
cylindrically shaped phantoms, Fig 1, were used with solutions of 1.955g CuSO_{4}
x H_{2}0 combined with 1.094g NaCl, 3.020g NaCl, and 4.915g NaCl to
achieve 0.3, 0.6, and 0.9S/m conductivities (at 20°Celsius), respectively. The phantom height and radius were 10.1 and
13.9cm.

Eight circular 16 AWG copper wire coil loops ranging from 3-10cm in diameter in 1cm increments were constructed for each phantom (24 total). The coils had 1, 2, 3, and 4 tune capacitors for the 3,4cm, 5,6cm, 7,8cm, and 9,10cm diameter loops, respectively. Coils were tuned and matched to the same impedance. The preamplifier socket connected directly to the match circuit, eliminating a cable. The same preamplifier was used for all measurements.

SNR measurements were acquired using 2D axial GRE sequences positioned through the center of each coil with noise calculated as the standard deviation of noise-only image voxels. Images were acquired on a 3T MRI scanner (MAGNETOM Tim Trio, Siemens Healthcare, Erlangen, DE).

Full
wave simulations were performed using Computer Simulation Technology Microwave
Studio (CST MWS). The FDTD method was used with mesh cell counts ranging from
2-11 million mesh cells, simulation accuracy of -80dB, and an “Enhanced
(Preview)” AR-filter with a max error for steady state of 0.001. CST Design
Studio (DS) co-simulation was used to tune the coils and calculate the
composite magnetic field (B1^{-}) and total noise (R_{t})^{2-4}.
Total noise was calculated by removing the match capacitor and measuring the
real part of the Z_{1,1} parameter. The active decoupling diode on the
match circuit between the signal and ground was included in the CST simulations
as a 0.8pF capacitor. Total MWS run times ranged from 24m to 4h22m per coil.

Quasi-static
simulations used Biot-Savart for magnetic field calculations and vector
potential for noise calculations^{5}. Calculations of sample resistance
(R_{s}) and coil resistance (R_{c}) were done separately and
total noise was calculated by R_{t} = R_{s}+R_{c}.
Run time was less than 4 seconds per coil.

Total
noise bench measurements were made using the same method used for the full wave
measurements^{3}.

Preamplifier noise was not included in any of the noise calculations or measurements. Full wave and quasi-static SNR results were scaled (one scaler for each method) to make the SNR calculations of the 3 cm loop on the 0.3S/m phantom match that of the SNR measurement at 8 cm depth from the loop center. We assumed a uniform body coil excitation.

The full wave
simulations correctly represented the signal and noise while the quasi-static
simulations did not. Superficial B1^{-} discrepancies in the
full wave simulation were mainly due to flip angles not being 90° in the
phantom. Discrepancies in the quasi-static signal and noise are largely due to
the assumption that boundary conditions are correctly accounted for at infinity^{1}.
The quasi-static method did not accurately calculate B1^{-} nor
did it account for the increase in noise with coil diameter for the 0.3S/m
phantom. This anomalous increase was due to a larger electric field strength for
coils larger than 7 cm when compared to that in the 0.6 and 0.9S/m phantoms.

We have demonstrated
that the frequency at 3T is high enough for a large system to have destructive
wave effects, requiring a full wave simulation for accurate results. Further
work would include whether a system that is small compared to the
electromagnetic wave could be accurately characterized with a quasi-static
solution or if a software package like MARIE using integral equations is
necessary and/or practical for all optimizations at 3T^{6}.

1. Larsson, Jonas. Electromagnetics from a quasistatic perspective. Am. J. Phys. 2007;75(3):230-239.

2. Hoult, D.I. The principle of reciprocity in signal strength calculation-A mathematical guide. Concepts Magn Reson. 2000;12(4):173-187.

3. Lemdiasov et al. A numerical post processing procedure for analyzing radio frequency MRI coils. Concepts Magn Reson Part A. 2011;38A(4):133-147.

4. Horneff et al. An EM simulation based design flow for custom-build MR coils incorporating signal and noise. IEEE Trans Med Imaging. 2018;37(2):527-535.

5. Roemer et al. The NMR phased array. Magn Reson Med. 1990;16(2):192-225.

6. Villena et al. Fast Electromagnetic Analysis of MRI Transmit RF Coils Based on Accelerated Integral Equation Methods. IEEE Trans Biomed Eng. 2016;63(11):2250-2261.

Fig 1: Top, cylindrical shaped phantom, 10.1 (height) and 13.9cm (radius),
used for experiments. There were three
total phantoms with 0.3, 0.6, and 0.9S/m. Bottom, total noise (R_{t}) for the quasi-static, full
wave, and measurements. The full wave and measurement noise values included an
active decoupling diode on the match circuit, between the signal and ground, which
increases the total noise by approximately 15-30% compared to the match circuit
without the diode.

Fig 2: SNR images of quasi-static, full wave, and
measurement data for the 6cm loops. The
bottom plots are the logarithmic transformation (dB) of the SNR images. The dB plots easily show destructive wave
effects in the full wave and measurement images that are not present in the
quasi-static images. All coils and phantoms showed these destructive wave effects
in full wave simulations and measurements. Images were acquired using a 2D
axial GRE sequence positioned through the axis of each coil (TE/TR=4.02/500ms,
flip angle=90°, matrix size=192x192, FOV=200x200mm, bandwidth=260Hz/pixel, slice
thickness=5cm)

Fig 3: Comparison of quasi-static, full wave, and
measurement data with a 0.3 S/m phantom. Coil is separated from the solution by 6mm polycarbonate.

Fig 4: Comparison of quasi-static, full wave, and
measurement data with a 0.6S/m phantom. Coil is separated from the solution by 6mm polycarbonate.

Fig 5: Comparison of quasi-static, full wave, and
measurement data with a 0.9S/m phantom. Coil is separated from the solution by 6mm polycarbonate.