William Allyn Grissom^{1}, Xinqiang Yan^{1}, and Zhipeng Cao^{1}

A large number of coils is desirable in parallel transmission to achieve uniform excitation while controlling SAR in a subject-adaptive manner, but only a small number of transmit channels are available on most ultra-high field scanners. We describe an algorithm that optimally groups a large number of coils into a small number of channels, based on the fact that if two coils are in the same channel, the matrix formed by collecting their slice-by-slice RF shims will have rank one. The algorithm was used to optimize the coil-to-channel mappings of a 30-coil array at 7T, based on 8 representative heads.

**Algorithm**

Our algorithm is based on the observation that two coils in the same channel receive scaled copies of the same vector of RF shim values, so the two-column matrix formed by their shims will have rank one. That is, if the matrix has a large second singular value σ_{22} the coils should be in different channels, and vice-versa. We can determine a set of RF shims and σ_{22} values for making coil-to-channel assignments by solving:

$$\hat{\mathbf{B}}=\mathop{argmin}_{\mathbf{B}}\frac{1}{2}\sum_{i=1}^{N_{slices}}\left\{\left\Vert\left\vert\mathbf{A}_{i}\mathbf{b}_{i}\right\vert-\mathbf{1}\right\Vert^2+\beta\mathbf{b}^H_i\mathbf{S}\mathbf{b}_i\right\}+\sum_{l=1}^{N_{coils}}\sum_{m=1}^{l}\lambda_{l,m}\sigma_{22}\left(\mathbf{b}_l,\mathbf{b}_m\right)$$

where $$$\mathbf{A}_i$$$ contains B_{1}^{+} maps for slice $$$i$$$, $$$\mathbf{b}_i$$$ contains the shim values for slice $$$i$$$, $$$\mathbf{S}$$$ is a global SAR matrix, and $$$\sigma_{22}\left(\mathbf{b}_l,\mathbf{b}_m\right)$$$ is the paired σ_{22 }for coils $$$l$$$ and $$$m$$$.

A variable splitting with continuation strategy [5] is used to repeatedly solve Equation 1. After the first solution, the two coils with the largest σ_{22} are assigned to channels 1 and 2. Thereafter, the coil with the largest minimum σ_{22} with assigned coils is assigned to a new channel, until eight channels are initialized. After that, coils are assigned to the channel with which they have the smallest maximum σ_{22}. Each time a coil is assigned, its $$$\lambda$$$ with coils in other channels is set to zero, and its $$$\lambda$$$ with coils in its own channel is set to a large value. The final coil amplitudes and phases for each channel are determined by SVD of its RF shims [1].

**Electromagnetic Simulations**

8- and 30-loop (Figure 1b) coil arrays were modeled at 298 MHz in 10 human head models (Figure 1c). The models represented average male and female dimensions in five countries. The arrays had 28 cm diameters to accommodate a receive coil, and height 16 cm (8 loops) and 17.8 cm (30 loops). Each loop in the 30-element array was 7 cm^{2}.^{ }Loops were overlapped 2 cm in different rows, while loops in the same row were gapped 5.5 mm for self-decoupling [6].

**Multisubject Design and Comparisons**

The algorithm was used to co-optimize slice-by-slice RF shims and coil-to-channel assignments across subjects 1-4 and 7-10, in three slice orientations. The coil design corresponding to the knee of the flip angle coefficient of variation (CoV; standard deviation divided by mean) versus global SAR curve was then used to design 8-channel RF shims for subjects 5 and 6, while varying SAR regularization. This optimized design was compared to a conventional array-compressed coil designed with geometric coil-to-channel assignments, and to the 8-coil array with one and two-spoke pulses [7].

Figure 3 shows that RF shimming with 8 coils achieved the worst flip angle CoV versus SAR tradeoff among coil and pulse configurations, and 30 coils with optimized coil-to-channel assignments achieved the best tradeoff. The 8-coil/2-spoke excitations had similar minimum CoV as the optimized 30 coil shims, but with twice the SAR and RF pulse duration. Figure 4 shows slice-by-slice normalized flip angle maps in subject 5, for a matched global SAR of 0.4 W/kg/slice. At this SAR level, the 30 coil RF shims with optimized coil-to-channel assignments were the only ones with average CoV better than 0.05.

Figure 5 shows the amplitudes and phases of the geometric and optimized coil weights for each of the eight channels. The optimized channels contained coils distributed around the circumference of the array, often in clusters. The presented consecutive geometric coil-to-channel assignments performed better than any other geometric assignments (data not shown).

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