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 account^6,7. 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-matrix^9,10 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 S^HS), 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 S^HS, i.e., λ_n 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 λ_n values (λ_av) in the S-opt approach is 2.2-fold greater than λ_av 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 coils^13,14. 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.