A method to approximate maximum local SAR in multi-channel transmit MR systems without transmit phase information
Stephan Orzada1, Mark E. Ladd1,2, and Andreas K. Bitz2

1Erwin L. Hahn Institute, Essen, Germany, 2Medical Physics in Radiology, German Cancer Research Center (DKFZ), Heidelberg, Germany

Synopsis

The capability of multi-channel transmit systems to drive different waveforms in the individual transmit channels results in an increased complexity for the SAR supervision In this work we propose a method based on virtual observation points (VOPs) to derive a conservative upper bound for the local SAR with a reasonable safety margin without knowledge of the transmit phases of the channels. In six different scenarios we demonstrate that the proposed method can be superior to the simple worst case method often used when only amplitude and no phase information is available.

Purpose

The capability of multi-channel transmit systems to drive different waveforms in the individual transmit channels results in an increased complexity for the SAR supervision since the instantaneous power loss density inside the human body depends on the complex weighting of the transmit fields of the multi-channel RF coil1. Hence, to calculate local SAR correctly, knowledge of amplitude and phase of the signal in each transmit channel has to be known. In this work we propose a method based on virtual observation points (VOPs)2 to derive a conservative upper bound for the local SAR with a reasonable safety margin without knowledge of the transmit phases of the channels, which can be used in systems without the capability to supervise phases or as a secondary monitoring system.

Methods

The worst-case local SAR can be derived from a set of VOPs Aj by selecting the maximum eigenvalue λmax over all VOPs and multiplying this value with the square of the norm of the N-dimensional excitation vector U:

$$$SAR_{worst\ case}=\lambda_{max} \parallel \mathbf U\parallel_2^2$$$

This poses a large overestimation. This overestimation can be reduced if information on the actual power distribution is taken into account. With the absolute elements of the excitation vector U and a fixed random set of phases P one can find a correction factor ξ to satisfy:

$$$ ξ\cdot {\max\limits_{j=1,\dots ,J} \left(\left(\left|u_1\right|\overline{p_1},\dots ,\left|u_N\right|\overline{p_N}\right){{\mathbf A}}_j\left( \begin{array}{c} \left|u_1\right|p_1 \\ \vdots \\ \left|u_N\right|p_N \end{array} \right)\right)\ }\ge {\max\limits_{υ=1,\dots ,J} \left({{\rm U}}^H{{\mathbf A}}_υ{\rm U}\right)\ \ ={SAR}_{max\ local}\ \forall \ \ {\rm U}\in {{\mathbb C}}^N\ }$$$

This can be enhanced to cover K sets of phases:

$$${SAR}_{estimated\ local}={\min\limits_{k=1,\dots ,K} {\left(ξ_k\cdot {\max\limits_{j=1,\dots ,J} \left(\left(\left|u_1\right|\overline{p_{1,k}},\dots ,\left|u_N\right|\overline{p_{N,k}}\right){{\mathbf A}}_j\left( \begin{array}{c}\left|u_1\right|p_{1,k} \\ \vdots \\ \left|u_N\right|p_{N,k} \end{array}\right)\right)\ }\right)}_k\ }$$$

$$$\ge {\max\limits_{ν=1,\dots ,J} \left({{\rm U}}^H{{\mathbf A}}_ν{\rm U}\right)\ \ \forall \ \ {\rm U}\in {{\mathbb C}}^N\ }$$$

This estimation can still be larger than the worst-case approximation. Therefore, the worst case-SAR is used as an upper bound for the estimation:

$$$SAR_{estimated\ bound}=\min \left\{{SAR}_{estimated\ local},{SAR}_{worst\ case}\right\}$$$

This leads to:

$$$SAR_{worst\ case}\geq SAR_{estimated\ bound}\geq SAR_{max\ local}$$$

This scheme of approximating local SAR was tested in 6 different scenarios. (a) an integrated (rigid) body coil of meander elements, (b) a flexible body coil of meander elements in the liver kidney region, (c) the same coil in the pelvic region, (d) a head coil of meander elements, (e) a c-shaped shoulder coil, and (f) a spine array consisting of loops. All of the arrays used have 8 transmit channels. The coil arrays were loaded with the DUKE model from the virtual family. Simulation models are shown in Fig. 1.

Full-wave EM simulations were performed in CST Microwave Studio (CST AG, Darmstadt, Germany).

The correction factors for each scenario were calculated using nested optimizations.

For each scenario, eight phase vectors P were chosen complying to the CP modes of a cylindrical 8 channel coil.

To compare the overestimation of the proposed method with the worst case method, 1,000,000 random excitation vectors were used for SAR calculations with each of the coil arrays.

Results

During 1,000,000 tests for each setup, no underestimation of the SAR occurred. Figure 2 shows scatter plots of SARestimated local over SARestimated local for 1,000,000 points each. Note that none of the points lie below the line of equality. For the calculation of SARestimated bound every result above the worst case line is set to SARworst case.

The SARestimated bound value showed a mean improvement of roughly 3% to 50% over the worst-case SAR, depending on the scenario. Figure 3 shows a table of the mean overestimation for SARestimated bound, SARworst case and the ratio of both.

Discussion

The best results were obtained for the scenarios a)-e), while proposed method shows low effectiveness for scenario f). This might be due to the used phase sets. While the arrays in the first 5 scenarios are more or less circular, the array in scenario f) is not. The effectiveness of the method for scenario f) could possibly be increased by choosing optimal sets of phases.

Conclusion

The proposed method provides a large improvement of maximum local SAR estimation compared to the worst case SAR when knowledge of the phases in a multichannel transmit system is not available. This can be useful in systems that cannot provide phase information, and in systems that need a real time SAR-supervision as a backup for during the dead-time of a non-real time local SAR supervision system.

Acknowledgements

The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 291903 MRexcite.

References

[1] Graesslin I, Vernickel P, Bornert P, Nehrke K, Mens G, Harvey P, Katscher U. Comprehensive RF safety concept for parallel transmission MR. Magn Reson Med 2014.

[2] Eichfelder G, Gebhardt M. Local specific absorption rate control for parallel transmission by virtual observation points. Magn Reson Med 2011;66(5):1468-1476.

Figures

Array models used in this work: a) 8-channel integrated body coil, b) flexible body coil around liver-kidney region, c) flexible body coil around the pelvic region, d) head coil, e) shoulder coil at right shoulder, f) spine array. All models include magnet, bore liner and patient table.

Scatter plots of SARworst case over SARestimated local for 1,000,000 points each. Note that none of the points lie below the line of equality. For SARestimated bound, every result above the worst-case line is set to SARworst case. SAR was normalized to an overall input power of 20 mW. For this calculation 8 phase vectors (K = 8) were used for each case.

Factor of mean overestimation for all coil arrays and positions, for the optimized solution using 8 phase vectors (SARestimated bound) and for the worst- case method (SARworst case). The last column shows the reduction in the SAR estimation compared to the worst-case method when using the optimized proposed method.



Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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