Matrix gradient coils have recently been introduced for shimming as well as spatial encoding.^1-4 The coil design used in this work is a cylindrical head-insert with a length of 70cm and an inner diameter of 39cm. It consists of 84 coil elements, distributed on 7 rings with 12 elements each (Fig. 1).⁵ Due to the high current requirements for imaging, amplifiers are expensive and therefore not every element can be driven individually. Fewer amplifiers can be used if coil elements are grouped and connected in series within a group. These configurations are optimized to allow for the creation of a desired target field shape.⁶ However, different target field shapes require different configurations, therefore the possibility to switch between configurations within an MR sequence (Fig. 2) is required. This can, in principle, be achieved with a switching circuit that allows the currents to take any path through this network of coil elements.^7,8 For the given coil, this requires >18000 switches, which is again technically unfeasible. In this work, we propose an algorithm to reduce the number of switches necessary to switch between a limited number of configurations.

Minimizing the number of switches can be split into two sequential optimization routines. The first minimizes the number of switches between coil element pins, the second minimizes the number of switches for connecting the amplifiers to the matrix coil. Both problems are NP-hard combinatorial optimization problems, therefore we use Simulated Annealing⁹ to find an approximate solution fast. The problem is similar to the Traveling Salesman Problem (TSP); however, here we have $$$n$$$ salesmen (configurations $$$C_k$$$) and the aim is to reduce the number of necessary streets (connections) between cities (coil elements) and junctions (switches).

1. Minimizing the number of switches between coil elements

This part is modelled with graph theory. Every configuration $$$C_k$$$ of $$$k=1...n$$$ target fields $$$T_k$$$ has an associated adjacency matrix $$$A_k$$$, dependig on the specific connections between pins. The non-zero entries of $$$A_c=\sum_{k=1}^{n}A_k$$$ indicate necessary connections. Every entry where $$$a_c^{i,j}=n$$$ indicates a hardwired connection because this connection exists for every $$$C_k$$$, hence no switch is required. Therefore, the number of necessary switches is$$s=\sum\limits_{i,j;\,\,j<i}\begin{cases}0&\mathrm{if}\quad\,a_c^{i,j}=0\\0&\mathrm{if}\quad\,a_c^{i,j}=n\\1&\mathrm{else}\end{cases}$$which serves as the cost function. For optimization, the following degrees of freedom can be exploited:

a) The order in which the elements are connected within a group can be arbitrary.

b) Each element is equipped with a bridge consisting of 4 switches that allows current to be routed through the element in both directions (Fig. 3). Hence each element can be connected in both orientations.

Therefore the annealing step in the Simulated Annealing algorithm either changes the ordering of elements within a group or flips the direction in which the elements are connected. Both of these actions change the individual adjacency matrices $$$A_k$$$. This is illustrated in Fig. 4.

2. Minimizing the number of switches from the amplifiers to the coil element network

It is irrelevant which of the available amplifiers drives which group. Furthermore, the positive and negative pin of the amplifier can be connected to either the entry- or exit-point of a group because the sign of the current can be changed. This is exploited to minimize the number of switches.

To evaluate the algorithm, we minimize the number of switches for 3, 7 and 15 different configurations optimized for different spherical harmonics and 12 amplifiers.

The coil elements within the every group of every configurations are given, however, the initial ordering is random. The number of necessary switches for this inital configuration is used as reference. Note that another random ordering will require a different number of switches. The results are shown in Table 1. After optimization, the number of switches are reduced by 60, 211 and 511 for $$$n=3$$$, $$$n=7$$$ and $$$n=15$$$, respectively.

We show that the number of switches can be substantially reduced, which reduces the complexity of the circuit and the costs associated with it. The proposed framework, in combination with the work introduced in ⁶, allows retaining most of the matrix gradient coil's flexibility although fewer amplifiers than coil elements are available. The optimization problems can be merged into a single optimization, which in our experience complicated the optimization without improving the result. The resulting circuit is inherently much more flexible and allows for more paths than optimized for. Analysing the actual flexibility of the circuit is part of future research. Furthermore, one may define a measure for flexibility and maximize it for a fixed number of switches, which will also be explored in the future.