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Design of transmit array coils by minimizing the modal reflected power values and increasing B1+ efficiency
Ehsan Kazemivalipour1,2, Giorgio Bonmassar3, Laleh Golestanirad4,5, and Ergin Atalar1,2
1Electrical and Electronics Engineering Department, Bilkent University, Ankara, Turkey, 2National Magnetic Resonance Research Center (UMRAM), Bilkent University, Ankara, Turkey, 3AA. Martinos Center, Massachusetts General Hospital (MGH), Harvard Medical School, Boston, MA, United States, 4Department of Radiology, Feinberg School of Medicine, Northwestern University, Chicago, IL, United States, 5Department of Biomedical Engineering, McCormick School of Engineering, Northwestern University, Evanston, IL, United States

Synopsis

The vast majority of RF coil design techniques are based on reducing the coupling level to increase the coils' power efficiency. The assessment of scattering (S) parameters plays a crucial role in these techniques. However, just by checking S-parameters, we cannot be sure that the coil's accepted power generates a field with useful characteristics. Here, we propose to perform the co-simulation on transmit array coils and replace unknown capacitors with power sources. Extraction of the field profiles of the coil's lumped ports/elements could provide the opportunity to involve the field-dependent parameters in the minimization procedure of finding the coil's capacitors.

Introduction

Transmit array (TxArray) coils with multiple transmit elements provide the additional degrees of freedom that can be used to enhance field uniformity1, accelerate acquisition time2, enable RF shimming while intending to mitigate specific absorption rate (SAR) hotspots3, and increase power efficiency4. How a TxArray coil is designed can have a significant impact on its gain from parallel transmission technology. Magnetic and electric interactions between the individual channels cause extra losses leading to an increase in the power consumption by TxArray coils. Hence the decreasing coupling ratio and consequently increasing coil's power efficiency are considered a significant challenge on the design process of the TxArray coils5.
The most popular approach for designing TxArray coils relies on minimizing the magnitude of the scattering (S) parameters without taking excitation signals into account6,7. However, the excitation signals significantly impact the level of total accepted power by TxArray coils8. In our earlier studies, we proposed that the eigenmode analysis of the S-matrix9,10 can provide a quantitative and compact representation of TxArray coils' power transmission capabilities according to their excitation signals. Then, we proposed using this approach in the design process of the TxArray coils. We demonstrated that by minimizing the modal reflected power values (eigenvalues of SHS), the dimension of the excitation space with high power efficiency could be enlarged. Increasing the coil's total power does not necessarily increase the power delivered to the load. The total power absorbed by the load and the power loss due to conductors and radiation cannot be distinguished by merely assessing the S-matrix.
Furthermore, maximizing the power delivered to the load is not directly linked to generating a field with useful characteristics. Here we propose a simulation approach by combining the co-simulation strategy11 and the eigenmode analysis to perform field analysis during the design process of the TxArray coils. We show that the proposed approach could increase the coil's B1+ efficiency for some critical modes of operation, such as circularly-polarized (CP) mode.

Methods

We simulated a 3T 4-channel TxArray coil loaded with a detailed human head model12 using the co-simulation strategy11. Figure 1 depicts the coil's numerical model, along with the geometrical details of the head model. Figure 2 demonstrates the schematic model of the RF coil without considering the load and shield effects. In this model, cd denotes decoupling capacitor, ct denotes tuning capacitor, cm denotes matching capacitor, and cs denotes the series matching capacitor. In this study, we aim to minimize the normalized reflected power for all excitation eigenmodes (eigenvalues of SHS, i.e., λn values)9 and B1+ efficiency for the CP mode ($$$Tx_{eff}^{CP}(\text{ }\mu \text{T/}\sqrt{\text{W}})$$$). We defined the B1+ efficiency as the mean of B1+ in the central axial plane for a unit total incident power. The optimization problem has, therefore, been formulated as follows to find the optimum capacitor values (λ-opt):
\[\begin{matrix}\lambda\text{-opt:}&\begin{matrix}\underset{{\mathbf{\bar{c}}}}
{\mathop{\min}}\,&\left(\frac{1}{4}\sum\limits_{n=1}^{4}{{{\lambda}_{n}}^{2}}\right)+\\\end{matrix}{{\left(Tx_{eff}^{\text{target}}-Tx_{eff}^{CP}\right)}^{2}}\\\end{matrix}\]
For each capacitor set, the corresponding S-matrix can be attained using the co-simulation strategy; then λ values can be determined. However, to perform the optimization, B1+ distribution on the central axial plane also needs to be determined. To find B1+ profile, all capacitors can be replaced by lumped ports in the circuit simulator of the co-simulation method (Figure 3). Then, for ports individually feeding with a unit voltage, B1+ distributions can be extracted. For a specific voltage input of the 4-port coil, i.e., $$$\mathbf{\bar{V}}_{in}^{\text{4-port}}$$$, and a set of capacitors, the input voltage of the 20-port coil, i.e., $$$\mathbf{\bar{V}}_{in}^{\text{20-port}}$$$, can be derived as follows
\[\mathbf{\bar{V}}_{in}^{\text{20-port}}=\left({{Z}_{0}}{{{\mathbf{\bar{\bar{Z}}}}}_{\text{20-port}}}^{-1}+{{{\mathbf{\bar{\bar{U}}}}}_{20\times20}}\right){{\left(\left[\begin{matrix}{{Z}_{0}}{{{\mathbf{\bar{\bar{U}}}}}_{4\times4}}&\mathbf{0}\\\mathbf{0}&{{{\mathbf{\bar{\bar{Z}}}}}_{caps}}\\\end{matrix}\right]{{{\mathbf{\bar{\bar{Z}}}}}_{\text{20-port}}}^{-1}+{{{\mathbf{\bar{\bar{U}}}}}_{20\times20}}\right)}^{-1}}\left[\begin{matrix}\mathbf{\bar{V}}_{in}^{\text{4-port}}\\\mathbf{0}\\\end{matrix}\right]\]
where Z0 denotes the reference impedance, $$${{\mathbf{\bar{\bar{Z}}}}_{\text{20-port}}}$$$ denotes the impedance matrix of the 20-port coil, U denotes the identity matrix, and $$${{\mathbf{\bar{\bar{Z}}}}_{caps}}=\text{diag}({{Z}_{{{c}_{d1}}}},\,{{Z}_{{{c}_{d2}}}},\,...,\,{{Z}_{{{c}_{t4}}}})$$$. As a result,
\[\mathbf{B}_{1}^{+}(\mathbf{\vec{r}})=\sum\limits_{i=1}^{20}{\mathbf{\bar{V}}_{in}^{\text{20-port}}(i)\mathbf{B}_{1i}^{+}(\mathbf{\vec{r}})}\]
where $$$\mathbf{B}_{1i}^{+}(\mathbf{\vec{r}})$$$ is the B1+ distribution when the ith port is excited. Therefore, the field profiles for each set of capacitors can be determined. We have also designed the coil based on one of the conventional minimization approaches (S-opt), which formulated as
\[\begin{matrix}\mathbf{S}\text{-opt:}&\begin{matrix}\begin{matrix}\underset{{\mathbf{\bar{c}}}}{\mathop{\min}}\,&\frac{1}{N}\sum\limits_{n=1}^{N}{\sum\limits_{m=1}^{N}{|{{s}_{mn}}{{|}^{2}}}}\\\end{matrix}\\\text{s}\text{.t}\text{.}\,\,\,\,|{{s}_{nn}}|\,\le\,\,-15\,\text{dB}\\\end{matrix}\\\end{matrix}\]

Results

Figure 4 illustrates the magnitude and phases for B1+ distributions when each port is individually fed by 1 volt. After performing the λ-opt and S-opt minimization strategies, the B1+ distributions in the CP mode within the head model at the central axial plane was achieved (Figure 5A). Figure 5B summarized the coil's performance for two different sets of capacitors acquired by the λ-opt and S-opt minimization approaches. In the λ-opt approach, B1+ efficiency is increased by 11%. As it can be observed, one excitation eigenmode has a high total reflection for both cases. However, the average of the λn values (λav) in the S-opt approach is 2.2-fold greater than λav in the λ-opt approach.

Conclusion and Discussion

The analysis of field-dependent parameters such as transmit efficiency, specific absorption rate (SAR), B1+ uniformity, etc., are widespread during the RF shimming process of predesigned TxArray coils13,14. However, considering these parameters, which are not directly linked to the coil's S-matrix, is an issue. In this study, we proposed a method to involve the coil's field profiles in the minimization procedure of finding the coil's capacitors. We demonstrated that TxArray coils could be designed to have good performance for some critical modes of operation.

Acknowledgements

No acknowledgement found.

References

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[2] K. Setsompop et al., "Parallel RF transmission with eight channels at 3 Tesla," Magnetic Resonance in Medicine, vol. 56, no. 5, pp. 1163-1171, Nov 2006.

[3] U. Katscher and P. Bornert, "Parallel RF transmission in MRI," NMR in Biomedicine, vol. 19, no. 3, pp. 393-400, May 2006.

[4] A. S. Childs, S. J. Malik, D. P. O'Regan, and J. V. Hajnal, "Impact of number of channels on RF shimming at 3T," Magnetic Resonance Materials in Physics Biology and Medicine, vol. 26, no. 4, pp. 401-410, Aug 2013.

[5] Guérin B, Gebhardt M, Serano P, et al. Comparison of simulated parallel transmit body arrays at 3 T using excitation uniformity, global SAR, local SAR, and power efficiency metrics. Magn Reson Med. 2015;73:1137‐1150.

[6] Avdievich NI, Giapitzakis IA, Pfrommer A, et al. Decoupling of a double‐row 16‐element tight‐fit transceiver phased array for human whole‐brain imaging at 9.4 T. NMR Biomed. 2018;31:e3964.

[7] Yan X, Ole Pedersen J, Wei L, et al. Multichannel double‐row transmission line array for human MR imaging at ultrahigh fields. IEEE Trans Biomed Eng. 2015;62:1652‐1659.

[8] Kozlov M, Turner R. Effects of tuning condition, head size and position on the SAR of a MRI dual‐row transmit array at 400 MHz. In Proceedings Progress in Electromagnetics Research Symposium (PIERS), Taipei, Taiwan. 2013. p. 422‐426.

[9] Kazemivalipour E, Sadeghi-Tarakameh A, Atalar E. Eigenmode analysis of the scattering matrix for the design of MRI transmit array coils. Magn Reson Med. 2020.

[10] Kazemivalipour E, Sadeghi‐Tarakameh A, Atalar E. Design of transmit array coils for MRI by minimizing the modal reflection coefficients. In Proceedings of the ISMRM & SMRT Virtual Conference & Exhibition, 2020. p. 0762.

[11] Kozlov M, Turner R. Fast MRI coil analysis based on 3-D electromagnetic and RF circuit co-simulation. J Magn Reson. 2009;200(1):147-152.

[12] A. Massire et al., "Thermal simulations in the human head for high field MRI using parallel transmission," J. Magn. Reson. Imaging, 2012, doi: 10.1002/jmri.23542.

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Figures

Figure 1 – EM simulation model of a shielded 3T 4-channel TxArray coil loaded with a detailed human head model12. The shield is slit into four segments evenly distributed along the axial direction, where the adjacent slits are connected via two 3 nF capacitors at positions facing the coil's end-rings.

Figure 2 – Schematic of the 4-channel RF TxArray coil. Vq for q = 1, 2, 3, and 4 represents the voltage across the port of the qth loop.

Figure 3 – (A) Circuit schematic of the 4-channel TxArray coil. According to the co-simulation strategy, initially, all capacitors were replaced by the lumped ports, and then EM simulation on a 20-channel coil was performed. Later in the circuit simulator, 4 lumped ports were connected to the power sources, and 16 lumped ports were connected to the capacitors. (B) All lumped ports/elements connected to the power sources.

Figure 4 - Magnitude and phase profiles of the B1+ field for all lumped ports/elements in the central axial plane. For each B1+ profile, the corresponding port was fed by a 1 volt, and other ports were terminated.

Figure 5 – (B) |B1+|-profiles of the 4-channel TxArray coil in the central axial plane designed based on the S-opt the λ-opt approaches. (B) Summary of performance of the TxArray coil designed based on the S-opt the λ-opt strategies.

Proc. Intl. Soc. Mag. Reson. Med. 29 (2021)
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