Introduction

Introduction For parallel transmit systems, control of local SAR is very important to ensure safety. Since calculating SAR over the whole set of SAR matrices obtained from simulations comes with a very high computational cost, compression by clustering around virtual observation points (VOPs) has been proposed by Eichfelder et al. (1). Later, Lee et al. (2) proposed an even more efficient compression method without clustering that can outperform Eichfelder’s method by an order of magnitude in terms of compression efficiency. Yet, Lee’s method still leaves room for improvement since it produces redundant VOPs. We present an enhanced method that improves compression efficiency and speed at the same time (3).

Methods

We propose an enhanced algorithm that can reduce the redundant VOPs by iteratively reducing the overestimation. An overview of the algorithm is shown in Figure 1. In its core, the algorithm uses Lee’s algorithm together with a speed enhancement proposed by Kuehne et al. (4). It starts by sorting all matrices by their respective highest eigenvalue in decreasing order. The matrix with the highest eigenvalue is used as the first matrix in the subset Vsub. A matrix for the overestimation term is chosen, which can for example be derived from global SAR (2), local SAR (5) or simply be a diagonal matrix. Then Lee’s algorithm is run with Kuehne’s expansion. Instead of finishing the VOP calculation after completing a run of Lee’s algorithm, the resulting subset Vsub containing the matrices identified as VOPs (but without overestimation) is used at the start of a rerun of the algorithm with the full set of matrices, but with an overestimation lowered by multiplying with a factor R compared to the previous run. After each run of Lee’s algorithm, the resulting set of VOPs can be saved and represents a valid solution for the compression with its respective overestimation.

To compare the enhanced algorithm to Lee’s original algorithm, the SAR matrices of three different arrays made from micro strip lines with meanders (6) and operating at the proton resonance frequency of 7 Tesla were used. The first array is a local 8-channel array placed directly on the body (7), while the second and the third array are remotely positioned behind the bore liner in a 2x4-channel and 1x16-channel configuration, respectively (Fig. 2). All simulations were performed in CST Microwave Studio 2017 (CST AG, Darmstadt, Germany).

The algorithms were implemented in Matlab (The Mathworks Inc., Natick, MA, USA) with a high degree of vectorization and other optimizations to speed up the calculations.

The enhanced algorithm is compared to the exact same implementation of Lee’s algorithm as used in the core loop of the enhanced algorithm to minimize the influence of the implementation on timing differences between the algorithms. In both cases, the overestimation is defined by a diagonal matrix with all diagonal elements equal to a fraction of the worst-case local SAR.

Results

Figure 3 shows the comparison between Lee’s algorithm and the enhanced algorithm for all three arrays. The left column shows the resulting number of VOPs for different overestimations given in percent of the worst-case local SAR, while the right column shows the time necessary to calculate the results. Two different reduction factors R were used.

The results show in all cases that after very few iterations the enhanced algorithm approximately halves the number of VOPs compared to the original algorithm. The smaller reduction factor R performs slightly better.

The effect on calculation time in the right column of Figure 3 shows a strong dependence on the array model. While the higher R value (smaller reduction step) takes approximately twice the calculation time as Lee’s algorithm for the 8-channel local array, it is faster for the 2x4-channel remote array. The larger reduction step (lower R value) proved to be faster in all cases. It should be noted that the duration given for the calculation time of the enhanced algorithm includes the calculation of all previous intermediate results.

Figure 4 shows scatter plots for local SAR results of the compressed data versus the results of the uncompressed data for the 8-channel local array for one million random excitation vectors.

No underestimation occurred in any scenario.

Discussion

The results show that the proposed algorithm is capable of approximately halving the number of VOPs for a certain overestimation. When the reduction factor R is chosen carefully, the enhanced algorithm is also faster than the original algorithm. The reason for this is twofold; firstly, a lower number of VOPs makes the optimization in Lee’s algorithm faster when trying to find the coefficients to dominate a certain matrix; secondly, the search for other dominated matrices in the step introduced by Kuehne finds more matrices that are dominated during the first runs of Lee’s algorithm, making successive runs faster.

Conclusion

The presented algorithm is an expansion of the algorithm presented by Lee et al. It significantly reduces the number of VOPs for a given overestimation while at the same time being potentially faster. Furthermore, the algorithm has more flexibility because it provides intermediate results that are valid VOP sets with an overestimation decreasing with each iteration step.

Acknowledgements

The authors would like to thank Emil Orzada, whose constant search for Lego® blocks with intermediate sorting during home office inspired the idea for the presented algorithm.