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.

The worst-case local SAR can be derived from a set of VOPs A_j by selecting the maximum eigenvalue λ_max 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.

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

The SAR_{estimated 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 SAR_{estimated bound}, SAR_{worst case} and the ratio of both.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.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.