Modelling and B1 Shim Analysis of 16-Element Transceiver Array at 7 T

Abstract

The work examines the workflow of using commercially available software for electromagnetic modelling and validation of a transceiver array coil operating at 298 MHz for magnetic resonance imaging and spectroscopy at 7 T. The coneshaped, tight-fit parallel transmit head array consists of two rows with eight loop coils per row and transmits two distinct spatial distributions by means of B₁ shimming. Considerations for finite-difference time-domain simulation setup and post-processing with circuit-domain co-simulation are examined, as is the generation of virtual observation points suitable for online safety monitoring.

I. Introduction

The ability to generate differing radiofrequency (RF) distributions is an important aspect of multi-element parallel transmit (pTx) coil arrays. At ultra-high fields, such distributions have been developed for RF-based selective spatial excitation, which is particularly useful for suppression of extracranial signal in magnetic resonance spectroscopic imaging [1]. However, the B₁ wavelength is shortened significantly at 7 T such that in order to generate varying RF distributions, spatial interference between the B₁ fields of array elements must be considered. In this work, we consider how simulations of multi-coil arrays can perform RF shimming in terms of resulting B₁ and E distributions and the consequent specific absorption rate (SAR).

II. METHODS

A. Electromagnetic Modelling

Analysis is performed using full-wave electromagnetic (EM) simulation. A 16-channel (2 × 8) split transceiver array [2] loaded with various anatomical body models is modelled at 298 MHz using the finite-difference time-domain (FDTD) method. We apply the self-organising migrating algorithm (SOMA) for minimising a cost function to optimise a 16 × 16 S-parameter matrix of the driven ports for the transceiver array. Improving on the scheme discussed in [3], our procedure optimised both capacitive and inductive reactance components for each lumped element. RF shimming of the model is validated by experimental data acquired on nine equally-spaced axial slices through the brain. The optimisation outputs a finalised scattering matrix for all ports, and the resulting B₁ and E fields for each array element are imported into MATLAB (The MathWorks Inc., Natick, MA, USA) for post-processing. Our study investigates practical aspects of modelling transmit arrays and shows the potential of using combined methods of EM simulation and SOMA to discern B₁ homogeneity and SAR distribution.

1). FDTD Simulation Setup:

The coil models were meshed with a nominal 1-mm grid in XFdtd v7.7.0.5 (Remcom Inc., State College, PA, USA) and 2-mm grid in Sim4Life v4.4.2 (Zurich MedTech AG, Zurich, Switzerland). Our initial investigations using Sim4Life indicated the 1-mm and 2-mm grids yielded similar results; thus, we use the 2-mm grid for Sim4Life to reduce computational requirements. Our investigation on the influence of different meshing sizes on simulation results will be published in a separate paper. As shown in Fig. 1, the RF coil is loaded with the Duke model [4] (male, 34 years old, 1.77 m, 70.3 kg, 305 tissue types) from the Virtual Population (ViP) v3.0 developed by IT'IS Foundation. For the sake of computation power, only the head of the human model is meshed in a high-resolution grid, while the body parts are meshed with a coarse grid. However, this meshing setup is able to preserve necessary features of the body.

Fig. 1.

The Duke human body model and 16-channel transmit array coil meshed in FDTD. The green rectangular box illustrates the simulation boundaries, where FDTD applies an absorbing boundary condition.

2). Actual Lumped Elements Simulation:

In simulation, solder resistances and equivalent series resistance (ESR) are added into the lumped element definitions; the actual coil components are high-Q multilayer ceramic capacitors and wire-wound inductors. We employ bench measurements to confirm the simulated coils and in vivo coils have similar input impedances in the unloaded condition. During measurement of the input impedances, we did not include a matching circuit at the simulation or physical coil inputs as doing so would prevent validation of the standalone coil structures. Using a vector network analyzer (E5071C, Keysight Technologies, Santa Rosa, CA, U.S.), the measured single coil (the first channel bottom row coil) input impedance is 4.2 + j147.0 Ω. In XFdtd the input impedance is 4.5 + j193.6 Ω, and 3.3 + j164.2 Ω in Sim4Life. Matching circuits are embedded into the FDTD simulation for producing results of RF B₁ shimming and associated 10-g SAR and virtual observation point (VOP) distributions.

When actual lumped elements are placed in coil gaps for simulation, the results do not yield optimum S-parameters. The 16-channel S-parameter matrix is shown in Fig. 2a; the S_nn is not optimally tuned, however we scaled up the simulated complex field maps with maximum input power. In the real scanning scenario, pTx coil elements are often not decoupled and matched to the ideal case.

Fig. 2.

Simulated S-parameter matrices of the 16 loop coil elements (a) using actual lumped elements and (b) optimised by SOMA. The scale bar is in decibels.

3). Circuit Domain Co-Simulation:

Using the circuit domain co-simulation in combination with FDTD, we can tune and match the coil with very fine resolution. Most commercial FDTD software packages include routines or interfaces for post-processing with circuit domain co-simulation. Our post-processing code is implemented in MATLAB and freely available to the public.

In a two-port system, with port 1 connected to the drive port and port 2 connected to a lumped element, the following relationships describe the incident (a_n) and the reflected (b_n) waves and the S-parameters

$$b_1=S_{11}{a}_1+S_{12}{a}_2$$ (1)

$$b_2=S_{21}{a}_1+S_{22}{a}_2$$ (2)

and

$$a_2=\frac{Z_L-50}{Z_L+50}{b}_2$$ (3)

where Z_L is the impedance seen, when looking toward the lumped elements from the lumped element port. Combining (1)-(3) yields

$$\frac{b_1}{a_1}=S_{11}+S_{12}\frac{Z_L-50}{Z_L+50}{\left(1-S_{22}\frac{Z_L-50}{Z_L+50}\right)}^{-1}{S}_{21}.$$ (4)

By minimising the result in (4), we can find (Z_L − 50)/(Z_L + 50). In addition, if we assume the incident wave a₁ (in V) at of the feed port is known, we can calculate the incident wave at the lumped element using

$$a_2=\frac{Z_L-50}{Z_L+50}{\left(1-S_{22}\frac{Z_L-50}{Z_L+50}\right)}^{-1}{S}_{21}{a}_1.$$ (5)

Since the B₁ fields are proportional to the square root of input power of each port, which is proportional to the incident waves presented at each port, we can approximate this relationship as

$$B_1=a_1{B}_{1,1}+a_2{B}_{1,2}$$ (6)

where B_1,i is the B₁ field created by the incident wave in the port i. Similar to (4), for the np + nl port network, where np ports are connected with the drive ports and nl ports are connected with the lumped elements ports, we can derive a similar equation

$$\frac{b_{np}}{a_{n{p}^{\prime }}}=S_{np\times n{p}^{\prime }}+S_{np\times nl}\Sigma (1-S_{nl\times nl}\Sigma {)}^{-1}{S}_{nl\times n{p}^{\prime }}.$$ (7)

Note np and np′ both denote drive port numbers, but they are not always equal. Here ∑ is a scattering matrix described in [3] containing information on the impedance of the lumped elements. SOMA, including consideration for reactive components in each coil element, is used to minimise the diagonal elements of (7) and weight the off-diagonal elements below the worst-case coupling between coils if the coils are tuned to 298 MHz. Once we optimise the results in (7), we can calculate the combined B₁ fields using equations similar to (5) and (6).

B. B₁ RF Shimming

In our Siemens 7T whole body human scanner environment, eight transmit channels with independent RF amplifiers are split into 2 × 8 channels to drive the 16 loop elements, with 30° phase shift between the top row of eight elements and the bottom row of eight elements.

The peak voltage presented at the coil plug is 110 V. Therefore, we need to find the simulated B₁ maps associated with the voltage values at the coil plug. Given the Siemens 7T setup, the actual time-averaged input power (in W) is calculated by

$$P_{\text{in}}=\frac{V_{\text{peak}}^2}{8Z_0}$$ (8)

which in this case equals 30.25 W. To scale the simulated B₁ amplitude to the actual B₁ amplitude, we use the transmission efficiency equivalency relationship

$$\frac{B_{1,\text{sim}}}{\sqrt{P_{\text{in,sim}}}}=\frac{B_{1,\text{act}}}{\sqrt{P_{\text{in,act}}}}.$$ (9)

The phase map of received images P_k,j, where k denotes coil number for transmission and j denotes the coil number for reception, is

$$P_{k,j}={\varphi }_{B_{1,k}^{+}}+{\varphi }_{B_{1,j}^{-{}^{\ast }}}+{\varphi }_{0,j,k}+{\varphi }_{\text{ind}}$$ (10)

where ${\varphi }_{B_{1,k}^{+}}$ is transmit B₁ field relative phase, ${\varphi }_{B_{1,j}^{-{}^{\ast }}}$ is receive B₁ field relative phase, ϕ_0,j,k is the coil zero-order phase, and ϕ_ind comprises coil-independent factors including common magnetisation and receiver phase, B₀ inhomogeneities, and eddy currents, as described by Van de Moortele et al. [5]. Within our console B₁ scout software routine, 45° incremental phases are added from channel 1 to 8 during transmission to account for the relative polar position of the elements, thus eliminating the zero-order phase ϕ_0,j,k.

The transmit phase map of the k^th channel relative to the 1^st channel is calculated as

$$P_{k,j}-P_{1,j}={\varphi }_{B_{1,k}^{+}}-{\varphi }_{B_{1,1}^{+}}=\text{angle}\left(\frac{P_k}{\mid P_k\mid }\cdot \frac{P_1^{\ast }}{\mid P_1\mid }\right)$$ (11)

where the complex B₁ for the k^th transmit channel is P_k, * denotes the complex conjugate, and angle() is a MATLAB function that returns the phase of a complex number from −π to π. As described in [1], for the spectroscopy mode (i.e., homogeneous mode) shim, the phase shift for the eight channels are

$${\Phi }_k=-\overline{{\varphi }_{B_{1,k}^{+}}-{\varphi }_{B_{1,1}^{+}}}k=1,2,\dots ,8$$ (12)

while for the ring mode shim, the phase shift terms are

$${\Phi }_k=-\overline{{\varphi }_{B_{1,k}^{+}}-{\varphi }_{B_{1,1}^{+}}}+\frac{3}{4}\pi (k-1)k=1,2,\dots ,8$$ (13)

where the overline indicates the mean phase calculated in the region of interest (ROI) at the centre of the human head. The two excitation profiles, i.e., spectroscopy and ring modes, are illustrated in Fig. 3.

Fig. 3.

∣B₁∣ in an axial slice for (a) spectroscopy and (b) ring excitation modes, from 0–1000 Hz (42.58 Hz = 1 μT). The cyan line outlines the head.

C. 10-g SAR and Virtual Observation Points

An in-house 10-g averaging function, conforming to the IEEE C59.3 standard, is used to calculate the 10-g averaged Q matrix. The Q matrix, as described by [6], is based on the combinations of individual channel complex field maps of all pTx channels. The 10-g averaged Q matrix is fed into an in-house VOP calculation program. The VOPs are calculated based on the method proposed by [7], and we define the overestimation as within 5% of worst-case SAR when the norm of channel weights is equal to one. The time-averaged SAR values assume an RF pulse duty cycle of 3%.

III. RESULTS

A comparison of simulated S-parameter maps using actual lumped elements versus SOMA optimisation is shown in Fig. 2. Fig. 4 presents a comparison of both phase and amplitude maps between actual lumped element FDTD simulation, in vivo scan data, and SOMA co-simulation. A high degree of B₁ uniformity is observable in the centre of the brain tissue in the spectroscopy mode (i.e., homogeneous mode) as shown in Fig. 3a, while B₁ is suppressed in the centre of the head in the ring mode as shown in Fig. 3b, both in agreement with previously-reported in vivo data. Meanwhile, it is apparent there is less agreement between the SOMA results and the in vivo data shown in Fig. 4.

Fig. 4.

Individual channel phase (in degrees) and amplitude (42.58 Hz = 1 μT) maps comparing the simulation model with actual lumped element components (top), in vivo scan data (middle), and SOMA optimised model (bottom). The modelled field maps at top are masked by head tissue, and the dark circles in the center of the simulated phase maps are the ROIs over which the mean phases are calculated.

In the spectroscopy mode, the 10-g averaged SAR local maximum and VOPs are localised at the periphery of the head, as shown in Fig. 5. The expected estimation accuracy of the VOPs is confirmed in Fig. 6.

Fig. 5.

VOP locations superimposed on 10-g averaged SAR maps from the spectroscopy excitation mode. The 10-g averaged SAR are plotted in decibels (0 dB = 10.57 W/kg).

Fig. 6.

VOP magnitude vs. worst-case local 10-g averaged SAR (overestimation = 5% of worst-case SAR).

IV. CONCLUSION

The work shows the actual lumped elements simulation have great value to be used for validating pTx coil complex B₁ maps. Additionally, the combined complex field maps after B₁ shim of each channel and associated SAR distributions can be derived based on the simulation data.