Kazem Hashemizadeh^{1}, Rong-Rong Chen^{1}, Edward V. R. DiBella^{1,2}, Leslie Ying^{3}, and Ganesh Adluru^{2}

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.

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.

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.

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.

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[2] Setsompop K, Gagoski BA, Polimeni JR, Witzel T, Wedeen VJ, Wald LL. Blipped-controlled aliasing in parallel imaging for simultaneous multislice echo planar imaging with reduced g-factor penalty. Magnetic Resonance in Medicine 2012;67(5):1210–1224.

[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.

Figure 1: The ground truth, reconstructed single slices images, and corresponding error maps for an SMS diffusion-weighted image with $$$N_s=3$$$. The images are SOS coil-combined. Error maps for the optimal and sub-optimal CC-SSG are darker, and more uniform than those of SG and SSG. These agree with the numerical results presented in Table 1.

Table 1: The normalized root-mean-square-error (RMSE) between reconstructed and ground truth slices for different methods and slice accelerations. Each entry in the table is obtained by averaging over the results from three different sets of equally distanced reconstructed slices. The normalized RMSE increases with slice acceleration factor. While SG and SSG have smaller RMSE for individual coil images, optimal and sub-optimal CC-SSG have much smaller RMSE for the SOS coil-combined images.

Figure 2: Reference and reconstructed DTI maps and corresponding error images for the middle slice in Figure 1. MD error images for optimal and sub-optimal CC-SSG are clearly darker than those for SG and SSG. FA error images are also darker for optimal and sub-optimal CC-SSG compared to SG and SSG. The FA-weighted color-coded (RGB) images are similar for all methods. These agree with the average error values presented in Table 2.

Table 2: The errors between reconstructed and reference DTI maps for the four methods and different slice accelerations. For mean diffusion (MD) and fractional anisotropy (FA), the errors are RMSE and percentages are RMSE normalized by average MD or FA values. Angle errors (errors between primary eigen-vectors) are calculated for voxels with FA greater than 0.3. MD errors for optimal and sub-optimal CC-SSG are significantly smaller than those for SG and SSG. The FA errors also show improvement for optimal and sub-optimal CC-SSG. The angle errors are similar for all four methods.

Figure 3: Normalized RMSE between reconstructed and ground truth coil and coil-combined images as functions of a global tuning parameter $$$\alpha$$$ for different slice accelerations. The curves are computed using training data (baseline $$$b=0$$$). While SSG ($$$\alpha_{i,z}=1$$$) minimizes the MSE of coil images, the proposed sub-optimal CC-SSG ($$$\alpha_{i,z}=\alpha^{\star}$$$) minimizes the MSE for coil-combined images. A larger $$$\alpha$$$ results in more suppression of the intra-slice leakage than that of the inter-slice leakage.