Alessandro Sbrizzi^{1}, Arian Beqiri^{2}, Hans Hoogduin^{1}, Joseph Hajnal^{2}, and Shaihan J Malik^{2}

Turbo Spin-Echo sequences (TSE) are frequently characterized by high local specific absorption rate (SAR), a limiting factor for their application. Here we show that the direct signal control (DSC) framework can drastically reduce the local SAR response of a TSE sequence by expanding the search-space for the amplitude and phase RF weights. A solution is found which enforces optimal contrast behavior and local SAR limits across different shim settings. A cardiac exam at 3 Tesla MRI is used as simulated test case.

The Optimal control DSC problem is outlined as:

$$ \begin{array}{rl}\mathbf{P}^{\text{opt}}=\arg\min&\|\mathbf{f}(\mathbf{P})-\mathbf{t}\|^2\\ \text{such that}&\sum_{j = 1}^J|p_{j,\ell}|^2\leq \pi_{\max},\quad\ell=1,\dots,L\\& |p_{j,\ell}|^2\leq \text{peak}^2_{\max},\quad \ell=1,\dots,L\quad\text{and }j=1,\dots J\end{array}$$

where $$$\mathbf{P}=(p_{j,\ell})$$$ contains all dynamic (complex) weighting terms. $$$\ell$$$ and $$$j$$$ denote, respectively, the channel and time indexes. $$$\mathbf{t}$$$ is the vector of target signal responses. $$$\mathbf{f}$$$ represents the signal responses of the SR-EPG model (i.e. the $$$F^+_0$$$ configuration state). Constraints on the maximum average power per channel, $$$\pi_{\max}$$$, and peak power are included.

To include the local SAR constraint, we assume that virtual-observation-points matrices $$$\mathbf{Q}_n$$$ are available^{[3]}.

Denoting by $$$b_j(t)$$$ the RF pulse waveform at the $$$j$$$-th excitation (these could be different for each excitation), the SAR for the $$$n$$$-th VOP over the whole TSE sequence is: SAR$$$_n=\sum_{j=1}^J\pi_j\mathbf{p}_j^H\mathbf{Q}_n\mathbf{p}_j$$$ where $$$\pi_j$$$ is the power of the $$$j$$$-th RF waveform (a fixed parameter) and $$$\mathbf{p}_j$$$ the corresponding RF weights. The local SAR can be calculated directly from the dynamic complex weights $$$\mathbf{P}$$$.

Expanding the original problem, we obtain the local-SAR optimized version:

$$ \begin{array}{rl}\mathbf{P}^{\text{opt}} =\arg\min&\|\mathbf{f}(\mathbf{P})-\mathbf{t}\|^2\\ \text{such that} & \sum_{j=1}^J|p_{j,\ell}|^2\leq\pi_{\max},\quad\ell=1,\dots,L\\& |p_{j,\ell}|^2\leq \text{peak}^2_{\max},\quad \ell=1,\dots,L\quad\text{and }j=1,\dots J\\ &\sum_{j=1}^J\pi_j\mathbf{p}_j^H\mathbf{Q}_n\mathbf{p}_j\leq \text{SAR}_{\max},\quad n=1,\dots,N.\end{array}$$

The derivatives of the SAR constraint are quickly calculated by the compact expressions:

$$ \begin{array}{ccc}\frac{\partial}{\partial\mathbf{p}_{m}^{R}}\sum_{j=1}^J\pi_j\mathbf{p}_j^H\mathbf{Q}_n\mathbf{p}_j&=&2\pi_j\left[(\mathbf{p}^R_m)^T\mathbf{Q}_n^R+(\mathbf{p}^I_m)^T\mathbf{Q}_n^I\right]\\ \frac{\partial}{\partial\mathbf{p}_{m}^{I}}\sum_{j=1}^J\pi_j\mathbf{p}_j^H\mathbf{Q}_n\mathbf{p}_j&=&2\pi_j\left[(\mathbf{p}^I_m)^T\mathbf{Q}_n^R-(\mathbf{p}^R_m)^T\mathbf{Q}_n^I\right] \end{array}$$

where $$$R$$$ and $$$I$$$ denote, respectively, the Real and Imaginary part of the variable: $$$\mathbf{p}_m=\mathbf{p}^R_m+\imath\mathbf{p}^I_m$$$ and $$$\mathbf{Q}_n=\mathbf{Q}^R_n+\imath\mathbf{Q}^I_n$$$.

We consider a 2D cardiac TSE sequence at 3T with 16 echoes in the train. The target response corresponds to a standard 90$$$^o$$$-180$$$^o$$$ scheme. $$$(T_E,T_R)=(5.9,1060)$$$ms. An 8 Tx channel coil^{[4] }is employed. The measured transmit fields^{[5] }are shown in Fig.1. The NORMAN model^{[6]} was employed for SAR calculations, which is compressed to 419 VOPs.

The DSC design algorithm is run for three configurations:

1) 1W/Kg maximum local SAR,

2) 2W/Kg maximum local SAR,

3) without local SAR constraints.

These configurations are all constrained by average and peak power per channel.In addition, we report also the obtained SAR and signal response for the quadrature-mode excitation.

The DSC problem is implemented in Matlab on a desktop PC. Exact derivatives are calculated as in [2] and as outlined in the previous paragraph.

The present research was funded by the following grants:

Dutch Technology Foundation (STW) Grant 14125

EPSRC (UK research councils) grant number EP/L00531X/1

[1] Malik, S. J., Beqiri, A., Padormo, F. and Hajnal, J. V. (2015), Direct signal control of the steady-state response of 3D-FSE sequences. Magn Reson Med, 73: 951-963. doi: 10.1002/mrm.25192

[2] Sbrizzi A, Hoogduin H, Hajnal JV, van den Berg CA, Luijten PR, Malik SJ. Optimal control design of turbo spin-echo sequences with applications to parallel-transmit systems. Magnetic resonance in medicine. 2016 Jan 1.

[3] Eichfelder G, Gebhardt M. Local specific absorption rate control for parallel transmission by virtual observation points. Magnetic resonance in medicine. 2011 Nov 1;66(5):1468-76.

[4] Vernickel P, Roeschmann P, Findeklee C, Luedeke K-M, Leussler C, Overweg J, Katscher U, Graesslin I, Schuenemann K. Eight-channel transmit/receive body MRI coil at 3T. Magn. Reson. Med. 2007;58:381-9.

[5] Nehrke K, Boernert P. DREAM-a novel approach for robust, ultrafast, multislice B1 mapping. Magnetic resonance in medicine. 2012 Nov 1;68(5):1517-26.

[6] Dimbylow PJ. FDTD calculations of the whole-body averaged SAR in an anatomically realistic voxel model of the human body from 1 MHz to 1 GHz. Phys. Med. Biol. 1997;42:479-490.