Quantifying Specific Absorption Rate (SAR) for Parallel Transmission (PTx) MRI systems remains a challenge¹. A straightforward method to allow for SAR for any weighting applied to the PTx system to be computed quickly from electromagnetic (EM) field simulations is to compute Q-matrices². These are Hermitian matrices containing information on SAR from each channel and coupling between channels.

Generating these Q-matrices from simulation can be time consuming in itself as the matrices must be 10g averaged to compute local SAR according to international standards³. EM simulation software packages usually provide optimised methods for performing this 10g averaging for specific channel weights but generally do not have a way of utilising this optimised averaging to generate SAR Q-matrices directly. Here we describe a simple method for generating complex valued Q-matrices from a minimal set of real valued SAR maps that can be computed directly in EM field solver software.

After running an EM simulation for a multichannel array, it is often possible to generate local SAR distributions for arbitrary combinations of these channels directly in the EM simulation software. These SAR distributions are real valued. However if sufficient SAR maps are created using linearly independent weightings on the different channels, it is possible to reproduce complex valued Q-matrices. Zhu proposed a method for doing exactly this for measurement of the global Q-matrix for an N channel PTx MRI system using a minimal set of power measurements (N² in total) with an optimal set of weights⁴. We use the same method for obtaining Q-matrices from EM simulations as a post processing step. We produce local SAR maps using linear combinations of array elements as defined by Zhu’s algorithm⁴, in each case allowing the EM software to perform the 10g averaging.

As an example, for a two-channel system the SAR at a location $$$\mathbf{r}$$$ from a 2x2 Q-matrix is quantified as:

$$\mathrm{SAR}_{[w_1,w_2]}(\mathbf{r})=[w_1^*\, w_2^*]\begin{bmatrix}Q_{11}(\mathbf{r}) & Q_{12}(\mathbf{r})\\Q_{21}(\mathbf{r}) & Q_{22}(\mathbf{r})\end{bmatrix}\begin{bmatrix}w_1\\w_2\end{bmatrix}$$

The local SAR distribution from channel 1 is simply $$$Q_{11}$$$, likewise channel 2 and $$$Q_{22}$$$ – i.e. $$$\mathbf{w}$$$ = [1 0] and $$$\mathbf{w}$$$ = [0 1] respectively. $$$Q_{12}$$$ (= $$$Q^*_{21}$$$ since $$$\mathbf{Q}$$$ is Hermitian) can be found by using two other channel combinations: $$$\mathbf{w}$$$ = [1 1] and $$$\mathbf{w}$$$ = [1 i] (the latter corresponds to 90 degree phase shift on channel 2). These real numbers are combined to make $$$Q_{12}$$$ as follows:

$$\mathrm{SAR}_{[1,1]}(\mathbf{r})=Q_{11}(\mathbf{r}) + Q_{12}(\mathbf{r})+Q_{12}^*(\mathbf{r}) + Q_{22}(\mathbf{r}) = Q_{11}(\mathbf{r}) + Q_{22}(\mathbf{r})+2\mathrm{Re}(Q_{12}(\mathbf{r}))\\\mathrm{SAR}_{[1,i]}(\mathbf{r})=Q_{11}(\mathbf{r}) + iQ_{12}(\mathbf{r})-iQ_{12}^*(\mathbf{r}) + Q_{22}(\mathbf{r}) = Q_{11}(\mathbf{r}) + Q_{22}(\mathbf{r})-2\mathrm{Im}(Q_{12}(\mathbf{r}))$$

therefore

$$\mathrm{Re}(Q_{12}(\mathbf{r})) = \dfrac{\mathrm{SAR}_{[1,1]}(\mathbf{r})-Q_{11}(\mathbf{r}) - Q_{22}(\mathbf{r})}{2}\\\mathrm{Im}(Q_{12}(\mathbf{r})) = \dfrac{\mathrm{SAR}_{[1,i]}(\mathbf{r})-Q_{11}(\mathbf{r}) - Q_{22}(\mathbf{r})}{-2}\\$$

This methodology was tested using a simulation of the NORMAN male voxel model⁵ located heart centred in a 2-channel birdcage system run in CST Microwave Studio 2013. Local SAR was calculated directly in the software and from the Q-matrices calculated as above for a set of three random complex RF shims.

Figure 1 shows the local SAR distribution in a central, coronal slice through the voxel model for the SAR directly calculated in the simulation software (top row), the SAR calculated using the proposed method (centre row), and the percentage deviation between the two (bottom row). Each column is the data for a different complex, random RF shim set with constant norm. Within the body, the results are completely identical – some errors only occur in the averaging regions that extend outside the body and the maximum residual error over the entire volume for the different shims is <3%.We have demonstrated a simple way to generate SAR Q-matrices using combinations of PTx channels for which the 10g averaged SAR maps can be calculated directly by EM simulation software. This process can easily be scripted using the internal scripting functions of most EM simulation software to ensure accurate, automated Q-matrix computation.