Chen Qian^{1}, Zi Wang^{1}, Xinlin Zhang^{1}, Boxuan Shi^{1}, Di Guo^{2}, Boyu Jiang^{3}, Ran Tao^{3}, and Xiaobo Qu^{1}

^{1}Department of Electronic Science, Biomedical Intelligent Cloud R&D Center, Fujian Provincial Key Laboratory of Plasma and Magnetic Resonance, National Institute for Data Science in Health and Medicine, Xiamen University, Xiamen, China, ^{2}School of Computer and Information Engineering, Xiamen University of Technology, Xiamen, China, ^{3}United Imaging Healthcare, Shanghai, China

Diffusion-weighted imaging (DWI) is widely employed in clinical diagnosis and white matter connections mapping. Multi-shot interleaved echo planer imaging (ms-iEPI) is used for acquiring images with higher spatial resolution and fewer distortion but suffers from motion-induced shot-phase variations. In this work, a joint reconstruction model is proposed to obtain the shot-phase and composite magnitude image simultaneously. Specifically, the smoothness of the shot-phase is exploited to construct low-rank constraints. Experiment results show that the proposed method has better performance than state-of-the-art methods on shot-phase estimation and image reconstruction.

$$

\begin{equation}

\mathop{\min}_{\boldsymbol{\rm S_{\textit{j}},E}} {\frac{\lambda}{2}}{\sum_{j=1}^{J}}{\sum_{i=1}^{I}}{\lVert \boldsymbol {\rm Y_{\textit{ij}}} - {\mathcal{UF}\boldsymbol{\rm C_{\textit{i}}\odot S_{\textit j}\odot E}} \rVert}_{\rm F}^{2}+{\sum_{j=1}^{J} \lVert {\mathcal{PF} \boldsymbol{\rm S_{\textit j}\odot E}} \rVert_{*}},

\tag{1}

\end{equation}

$$

where the $$$\boldsymbol{\rm{Y}_{\textit{ij}}} \in {\mathbb{C}^{N_x \times N_y}}$$$ denotes i-th coil and j-th shot sampled k-space data, $$$\boldsymbol{\rm C_{\textit{i}}}$$$ is the i-th coil sensitivity map, $$$\boldsymbol{\rm S_{\textit{j}}}$$$ is the j-th shot-phase, which is updated within the iterative solving steps. $$$\boldsymbol{\rm{E}} \in {\mathbb{C}^{N_x \times N_y}}$$$ is the target composite magnitude image, $$$I$$$ is the number of coils and $$$J$$$ is the number of shots, $$$\mathcal{U}$$$ is an under-sampling operator, $$$\mathcal{F}$$$ is the Fourier transform operator, $$$\mathcal{P}$$$ is an operator to build structured matrix[5], $$$\lambda$$$ is a regularization parameter, $$$\lVert {\cdot}\rVert _{\rm F}$$$ represents Frobenius norm, $$$\lVert {\cdot}\rVert _{*}$$$ denotes nuclear norm, $$$\odot$$$ means Hadamard product.

we adopt the Projections Onto Convex Sets algorithm to solve the proposed model[10]. The solution $$$\boldsymbol{\rm{E}}$$$ can be considered to lie in a closed convex subset in a Hilbert space. Two convex sets defined by data consistency and regularizer items have a non-empty intersection [10]. Each point in this intersection is a potential solution for model (1). The image reconstruction problem can be solved by building a projection operator for each convex set [11]. In the iterative solving process, the shot-phase is estimated from intermediate images with smoothness constraints.

Figure 2 compares estimated shot-phases in the iterative solving process. POCS-ICE fails to get smooth shot-phase and motion-induced ghosting artefacts still exists in the final magnitude image. Conversely, the estimated shot-phase of the proposed method becomes increasingly smooth and finally leads to a reconstructed image without ghosting artefacts. The result demonstrates that the low-rank constraint could facilitate the shot-phase estimation and magnitude image reconstruction.

Figure 3 shows the averaged DWI image of 15 diffusion directions. Obvious ghosting artefacts have been marked with yellow arrows in Figure 3 (b)-(d). The 8-coil 8-shot data is challenging for the separated shots reconstruction methods, which try to obtain each shot image following a parallel imaging procedure. The proposed method outperforms other state-of-the-art methods visually.

Furthermore, we test algorithm performance on under-sampled reconstruction. Figure 4 shows the proposed method has better robustness on under-sampled data. It demonstrates that the joint shots strategy reduces the scale of unknowns which decrease the difficulty of solving this underdetermined equation.

The proposed method has been extended with a weighted total variation to increase the signal to noise ratio. See more details and results in the full-length paper: https://csrc.xmu.edu.cn/.

This work was supported in part by the National Natural Science Foundation of China under grants 62122064, 61971361, 61871341, and 61811530021, the National Key R&D Program of China under grant 2017YFC0108703, and the Xiamen University Nanqiang Outstanding Talents Program. The authors thank Hua Guo for assisting in data acquisition and very helpful advice for the abstract. The authors thank Lijun Bao, Weibin Zhou and Zunquan Chen for the discussion on fractional anisotropy maps.

The correspondence should be sent to Prof. Xiaobo Qu (Email: quxiaobo@xmu.edu.cn)

[1] D. K. Jones et al., ''Diffusion MRI,'' Oxford University Press, 2010.

[2] M. A. Bernstein et al., ''Handbook of MRI pulse sequences,'' Elsevier, 2004.

[3] A. W. Anderson et al., ''Analysis and correction of motion artifacts in diffusion weighted imaging,'' Magnetic Resonance in Medicine, vol. 32, pp. 379-87, 1994.

[4] M. Mani et al., ''Multi‐shot sensitivity‐encoded diffusion data recovery using structured low‐rank matrix completion (MUSSELS),'' Magnetic Resonance in Medicine, vol. 78, pp. 494-507, 2017.

[5] Y. Huang et al., ''Phase-constrained reconstruction of high-resolution multi-shot diffusion weighted image,'' Journal of Magnetic Resonance, vol. 312, pp. 106690, 2020.

[6] N.-k. Chen et al., ''A robust multi-shot scan strategy for high-resolution diffusion weighted MRI enabled by multiplexed sensitivity-encoding (MUSE),'' Neuroimage, vol. 72, pp. 41-47, 2013.

[7] H. Guo et al., ''POCS‐enhanced inherent correction of motion‐induced phase errors (POCS‐ICE) for high‐resolution multishot diffusion MRI,'' Magnetic Resonance in Medicine, vol. 75, pp. 169-180, 2016. [8] X. Zhang et al., ''Image reconstruction with low-rankness and self-consistency of k-space data in parallel MRI,'' Medical Image Analysis, vol. 63, pp. 101687, 2020.

[9] J. P. Haldar, ''Low-rank modeling of local k -space neighborhoods (LORAKS) for constrained MRI,'' IEEE Transactions on Medical Imaging, vol. 33, pp. 668-681, 2013.

[10] A. A. Samsonov et al., ''POCSENSE: POCS-based reconstruction for sensitivity encoded magnetic resonance imaging,'' Magnetic Resonance in Medicine, vol. 52, pp. 1397-1406, 2004.

[11] M. L. Chu et al., ''POCS-based reconstruction of multiplexed sensitivity encoded MRI (POCSMUSE): A general algorithm for reducing motion-related artifacts,'' Magnetic Resonance in Medicine, vol. 74, pp. 1336-48, 2015.

[12] M. Uecker et al., ''ESPIRiT—an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets GRAPPA,'' Magnetic Resonance in Medicine, vol. 71, pp. 990-1001, 2014.

[13] H.-K. Jeong et al., ''High-resolution human diffusion tensor imaging using 2-D navigated multishot SENSE EPI at 7 T,'' Magnetic Resonance in Medicine, vol. 69, pp. 793-802, 2013.

Figure 4. Under-sampled reconstruction results of one direction. (a) is reconstructed by IRIS with a navigator as a reference and all 8 shots data are employed. (b)-(e) are under-sampled reconstruction results of POCS-ICE, MUSSELS, PLRHM, and the proposed method, respectively and only shot 1,3,5,7 is employed for reconstruction.