Molin Zhang^{1}, Nicolas Arango^{1}, Jason Stockmann^{2}, Jacob White^{1}, and Elfar Adalsteinsson^{1,3,4}

^{1}EECS, MIT, Cambridge, MA, United States, ^{2}Athinoula A. Martinos Center for Biomedical Imaging, Charlestown, MA, United States, ^{3}Harvard-MIT Health Sciences and Technology, MIT, Cambridge, MA, United States, ^{4}Institute for Medical Engineering and Science, MIT, Cambridge, MA, United States

Shim arrays provide nonlinear, time-varying fields that can extend the possibility of linear gradient field for RF excitations. In this work, we explore the possibility of designing a selective refocusing 180° pulse for a restricted slice pattern of 1.2cm thickness, aimed at application for the fetal brain. We use an auto-differentiable Bloch simulator framework for the design, but without the presentence of crusher gradients. Decomposition property is employed to overcome the undetermined initial magnetization. The selective refocusing profile achieves selective refocusing for 85% of voxels in the ROI (My< -0.9).

In this work, we present numerical simulations of selective refocusing on a restricted slice pattern of 1.2cm thickness in the fetal brain by a joint optimization of a set of time-varying, spatially-nonlinear shim array fields

While the optimization-based design requires exact initial and desired magnetization for each voxel, the distribution of the initial magnetization is unknown for the refocusing pulse design due to several factors, e.g, B0 inhomogeneity and various TEs. To mitigate this problem, we investigate the linear/decomposition property of the Bloch equation on three components, Mx, My and Mz as Bloch equation describes a series of rotations, R = Rn…R1. To clarify this,

$$\left[\begin{array}{l}M x^{\prime} \\M y^{\prime} \\M z^{\prime}\end{array}\right]=R\left[\begin{array}{l}M x \\M y \\M z\end{array}\right]=R\left[\begin{array}{c}M x \\0 \\0\end{array}\right]+R\left[\begin{array}{c}0 \\M y \\0\end{array}\right]+R\left[\begin{array}{c}0 \\0 \\M z\end{array}\right] = \left[\begin{array}{l}M x 1^{\prime} \\M y 1^{\prime} \\M z 1^{\prime}\end{array}\right] + \left[\begin{array}{l}M x 2^{\prime} \\M y 2^{\prime} \\M z 2^{\prime}\end{array}\right]+ \left[\begin{array}{l}M x 3^{\prime} \\M y 3^{\prime} \\M z 3^{\prime}\end{array}\right]$$

Specifically, for a selective refocusing pulse design with x axis as the rotation axis, the following requirements should be satisfied as shown in figure 1.

2.

3.

The optimization could be formulated as below for any initial magnetization distributions,

$$ \underset{I \in \mathbb{R}^{n_{T} \times n_c}, b \in \mathbb{C}^{n_{T}}}{\arg \min }\mathcal{L}=||W*Mx-W*Mx_d|| + ||W*My-W*My_d|| + 0.5*||W*Mz_{xy}|| +\mathcal{R}(b)$$

where $$$Mx$$$, $$$My$$$ are the magnetization distribution actual profile generated by the RF and time-varying nonlinear B field using time-discrete Bloch simulator with the initial magnetization $$$Mx = 1$$$ and $$$My = 1$$$, respectively. $$$Mx_d$$$, $$$My_d$$$ are the desired magnetization distribution. $$$Mz_{xy}$$$ is the transverse magnetization distribution with the initial magnetization $$$Mz = 1$$$. $$$W$$$ is the loss weight of ROI and ROI interferers.$$$I$$$ is the shim coil current. $$$b$$$ is the RF pulse. $$$n_T$$$ is the number of time points and $$$n_c$$$ is the number of shim coils. $$$\mathcal{R}(b)$$$ is the regularizer.

We first show that decomposition theory works by designing a slice-selective refocusing pulse via optimization only on a time-varying RF pulse. Figure 3 shows that all the grids are perfectly refocused even without the presentence of crusher gradient. However, as the resolution is 1.2cm, the pulse may not successfully refocus all grids at a finer resolution.

Further we show the design of selective refocusing pulse for the restricted slice of fetal brain. Both the RF pulse and shim coil current are optimized. Figure 4 shows the results of the slice profile in the ROI based on decomposition theory with different crusher gradient conditions. Figure 5 shows the results of the slice profile in the ROI interferers (not refocusing regions).

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2. Stockmann, Jason P., et al. "A 32‐channel combined RF and B0 shim array for 3T brain imaging." Magnetic resonance in medicine 75.1 (2016): 441-451.

3. Luo, Tianrui, et al. "Joint Design of RF and gradient waveforms via auto-differentiation for 3D tailored excitation in MRI." arXiv preprintarXiv:2008.10594 (2020).

4. Pauly, John, et al. "Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm (NMR imaging)." IEEE transactions on medical imaging 10.1 (1991): 53-65.

5. Zhang M, Arango NS, Stockmann JP, Jacob W, Adalsteinsson E. Inner Volume Excitation via Joint Design of Time-varying Nonlinear Shim-array Fields and RF Pulse. In Proceedings of the 29th Annual Meeting of ISMRM, 2021. p. 0912.

An illustration of designing refocusing pulse with decomposition property. As Bloch equation is linear for Mx, My and Mz, the refocusing effect can be expressed in a decomposition way as shown in figure 1 which solves the undermined initial magnetization issues.

The details of the model and 64 shim coils. A. The contour of the mother's body and fetal brain slice (red slice) to be refocused. B The distribution of 64 shim coils along the mother's body. C. Representative field of one coil in the shim array. The red dashed line represents the contour of the fetal brain.

The results of optimization only on RF pulse for a slice selective refocusing task. A. The optimized RF pulse. B-C. Resulting slice profile on grids. Note that applying crusher gradient helps eliminate the magnetization in the stop band.

The results of optimization on both RF pulse and shim coil current for the fetal brain slice selective refocusing task. A. Optimized RF pulse. B. Optimized shim current. C-E. Statistics of refocusing profile in the ROI in terms of Mx, My and Mz. Note that the optimization is operated without crusher gradients.

The results of optimization on both RF pulse and shim coil current for the fetal brain slice selective refocusing task. A. Optimized RF pulse. B. Optimized shim current. C-E. Statistics of refocusing profile in the ROI interferers in terms of Mx, My and Mz.

DOI: https://doi.org/10.58530/2022/4062