Synopsis

Simultaneous multi-slice (SMS) acquisition accelerates diffusion MR imaging by acquiring multiple slices simultaneously. In this work, we propose a new method, termed coil-combined split slice-GRAPPA (CC-SSG), to improve the quality of SMS diffusion imaging reconstruction. By optimizing split-slice-GRAPPA (SSG) kernels specifically for coil combining, our approach allows for a better trade-off for suppressing inter-slice and intra-slice leakages and minimizes the mean-square-error (MSE) of coil-combined images. The proposed CC-SSG method improves the estimation of diffusion tensor imaging (DTI) parameters over existing methods.

Introduction

Simultaneous multi-slice (SMS) acquisition accelerates diffusion MR imaging by acquiring multiple slices simultaneously [1]. When combined with blipped controlled aliasing in parallel imaging (CAIPI) [2], it can reduce scan time for EPI acquisitions. We propose a new method, termed coil-combined split slice-GRAPPA (CC-SSG), to improve the quality of SMS diffusion imaging reconstruction. By optimizing split-slice-GRAPPA (SSG) [3] kernels specifically for coil combining, our approach allows for a better trade-off for suppressing inter-slice and intra-slice leakages and minimizes the mean-square-error (MSE) of coil-combined images. When applied to in-vivo multi-shot RESOLVE datasets, the proposed CC-SSG method improves the estimation of diffusion tensor imaging (DTI) parameters over existing methods.

Theory

Consider a set of diffusion-weighted images (DWIs) with $N_c$ coils, $N_s$ slices, $N_d$ diffusion-encoded directions, denoted by $m_{i,z,n}(x,y)$, where $i=1,\cdots, N_c$, $z=1,\cdots, N_s$, and $n=1,\cdots, N_d$. Let $M_{i,z,n}(k_x,k_y)$ denote the k-space representation of $m_{i,z,n}(x,y)$. Each slice $z$ is phase modulated by $\phi_z$ and the resulting phase modulated slice is given by $$m_{i,z,n}^{(\phi_z)}(x,y) = \mathscr{F}^{-1}\{M_{i,z,n}^{(\phi_z)}(k_x,k_y)\} = \mathscr{F}^{-1}\{e^{j\phi_z(k_x,k_y)}M_{i,z,n}(k_x,k_y)\},\quad (1)$$ where $\mathscr{F}^{-1}\{.\}$ is the 2D inverse Fourier transform. The k-space SMS data $R_{i,n}(k_x,k_y)$ is given by $$R_{i,n}(k_x,k_y) =\sum_{z=1}^{N_s}M_{i,z,n}^{(\phi_z)}(k_x,k_y) =\sum_{z=1}^{N_s}e^{j\phi_z(k_x,k_y)}M_{i,z,n}(k_x,k_y).\quad (2)$$

k-space kernels are applied to SMS data to estimate phase modulated single slices $\hat{M}_{i,z,n}^{(\phi_z)}(k_x,k_y)$ by $$\hat{M}_{i,z,n}^{(\phi_z)}(k_x,k_y) =\sum_{j=1}^{N_c}\sum_{b_x=-B_x}^{B_x}\sum_{b_y=-B_y}^{B_y} a_{i,z,j}^{b_x,b_y}R_{j,n}(k_x-b_x,k_y-b_y), \quad (3)$$ where $a_{i,z,j}^{b_x,b_y}$ is the kernel coefficient at position $(b_x,b_y)$, used to weight the SMS data acquired by the $j$-th coil to reconstruct data for $i$-th coil and slice $z$.

Non-SMS baseline data ($b=0$) is applied to (3) for kernel training. We rewrite (3) in a compact form $$\mathcal{M}_{i,z} = \left(\sum_{z=1}^{N_s}\mathcal{P}_{z}\right)\mathcal{K}_{i,z},\quad (4)$$ where $\mathcal{K}_{i,z}$ is the k-space kernel in a vectorized form of $\{a_{i,z,j}^{b_x,b_y}|_{j=1}^{N_c}|_{b_x=-B_x}^{B_x}|_{b_y=-B_y}^{B_y}\}$. Each row in matrix $\mathcal{P}_{z}$ represents a cubic patch extracted from $z$-th slice, in a vectorized form of $\{M^{(\phi_z)}_{j,z,0}(k_x-b_x,k_y-b_y)|_{j=1}^{N_c}|_{b_x=-B_x}^{B_x}|_{b_y=-B_y}^{B_y}\}$. Similarly, $\mathcal{M}_{i,z}$ represents single slice baseline data in the vectorized form of $\{M_{i,z,0}^{(\phi_z)}(k_x,k_y)|\forall(k_x,k_y)\}$. The least square (LS) solution to (4) is the slice-GRAPPA (SG) [2] kernel. The split-slice-GRAPPA (SSG) kernel generalizes the SG kernel to control both intra-slice leakage, by enforcing $\mathcal{M}_{i,z} =\mathcal{P}_{z} \mathcal{K}_{i,z}$ for the slice of interest $z$, and inter-slice leakage, by enforcing $0= \mathcal{P}_{z'} \mathcal{K}_{i,z},$ for every $z' \ne z$. In [3], SSG is generalized to allow tuning parameters to weigh intra-slice and inter-slice leakages differently. However, there is no detailed study on how to tune such parameters or the effect of such tuning on image reconstruction quality.

We propose CC-SSG to find optimal SSG kernels that minimize the MSE of coil-combined DWIs. Similar to [3], the kernels are defined as LS solution to $$\begin{pmatrix}0 \\\vdots \\\alpha_{i,z}\mathcal{M}_{i,z} \\\vdots \\0\end{pmatrix} = \begin{pmatrix}\mathcal{P}_{1} \\\vdots \\\alpha_{i,z}\mathcal{P}_{z} \\\vdots \\\mathcal{P}_{N_s}\end{pmatrix}\mathcal{K}_{i,z},\quad (5),$$ where $\{\alpha_{i,z}\}$ are weighting parameters to balance intra-slice and inter-slice leakages.The LS solution to (5) is $$\mathcal{K}_{i,z} = \alpha_{i,z}^2 \left(\sum_{\substack{z'=1 \\ z'\neq z}}^{N_s}\mathcal{P}^{\dagger}_{z'}\mathcal{P}_{z'} + \alpha_{i,z}^2\mathcal{P}^{\dagger}_{z}\mathcal{P}_{z}\right)^{-1}\mathcal{P}^{\dagger}_{z}\mathcal{M}_{i,z},\quad (6)$$ where $(\cdot)^{\dagger}$ represents the Hermitian operator.

The main novelty of CC-SSG is to optimize the kernels in (6) specifically for coil-combining. When using root-sum-of-squares (SOS) coil combining, the optimized CC-SSG kernels are found by setting the optimal weights $\{\alpha_{i,z}^{\star}\}$ to be $$\{\alpha_{i,z}^{\star}\}= \arg\min_{\{\alpha_{i,z}\}}\left\|\sqrt{\sum_{i=1}^{N_c}\left|\mathscr{F}^{-1}\left\{\bigg(\sum_{z=1}^{N_s}\mathcal{P}_{z}\bigg)\mathcal{K}_{i,z}\right\}\right|^2}-\sqrt{\sum_{i=1}^{N_c}\left|\mathscr{F}^{-1}\left\{\mathcal{M}_{i,z}\right\}\right|^2}\right\|^2 \;\; \text{(CC-SSG)} \quad (7).$$ Note that $\mathcal{K}_{i,z}$ in (7) depends on $\{\alpha_{i,z}\}$ through (6). A conjugate gradient algorithm is applied to find the optimal solution of (7). As shown in (7), CC-SSG optimizes $\{\alpha_{i,z}\}$ to minimize the MSE between coil-combined de-alised images and ground truth baseline images. This is in contrast to existing work that train SSG kernels to minimize the MSE of coil images only. Equation (7) reduces to a single-variable optimization by letting $\{\alpha_{i,z}=\alpha|\forall i,z\}$. This is referred to as sub-optimal CC-SSG.

Acquisition

Results

Reconstructed coil and coil-combined images are compared in Figure 1 and Table 1. In Figure 2 and Table 2, retrospective reconstructed DTI maps are compared. These results demonstrate that the proposed methods minimize the MSE of coil-combined images and improve the estimation of DTI maps. Figure 3 shows normalized RMSE variations for coil and coil-combined images as functions of a global tuning parameter.

Conclusion

Acknowledgements

References

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[3] Cauley SF, Polimeni JR, Bhat H,Wald LL, Setsompop K. Interslice leakage artifact reduction technique for simultaneous multislice acquisitions. Magnetic Resonance in Medicine 2014;72(1):93–102.

[4] Porter DA, Heidemann RM. High resolution diffusion-weighted imaging using readout-segmented echo-planar imaging, parallel imaging and a two-dimensional navigator-based reacquisition. Magnetic Resonance in Medicine 2009;62(2):468–475.