Merry Mani^{1}, Mathews Jacob^{1}, Graeme McKinnon^{2}, Baolian Yang^{2}, Brian Rutt^{3}, Adam Kerr^{3}, and Vincent Magnotta^{1}

Multi-shot diffusion-weighted (

EPI-based diffusion imaging using single-shot readout methods are prone to geometric distortions and blurring, which preclude its utility for high-resolution imaging. Multi-shot diffusion-weighted (msDW) imaging overcomes the above problems by splitting the single long readout into multiple shorter readouts that are acquired over multiple TRs. The shorter k-space readout duration limits the geometric distortions and blurring and achieves a shorter echo-time suitable for high-resolution diffusion imaging. However, the acquisition spanning multiple TRs to cover a single k-space reduce the time efficiency of multi-shot methods. Hence the practical utility of multi-shot methods for high angular resolution studies currently remains limited.

Simultaneous multi-slice^{1,2} (SMS) imaging acceleration techniques can improve the time efficiency of msDW imaging without sacrificing SNR. A blipped-CAIPI^{3} EPI trajectory modified for multi-shot acquisitions coupled with a multi-band RF pulse can simultaneously encode and acquire k-space data from multiple slice locations. However, the reconstruction of this data is extremely challenging. The images from the above acquisition will be not only slice aliased, but will exhibit severe artifacts due to inter-shot phase inconsistencies arising from the msDW encoding. Reconstruction of artifact-free DWIs from the above acquisition requires simultaneous phase compensation and slice unfolding. Existing methods depend on explicit^{4} or implicit^{5,6} navigators to calibrate the phase associated with each shot which is then used in a simultaneous slice-unfolding and phase-compensation reconstruction. We present a novel navigator-free reconstruction method, SMS MUSSELS, to achieve the above.

The proposed method exploits the low-rank property that exists between the multi-shot diffusion data. The MUSSELS^{7} formulation for the navigator-free reconstruction of msDW data proved the existence of annihilation relationships between multi-shot data of the form:

\begin{equation} \label{eq_8} {\bf{\cal H}}(\widehat{m_{i}})\cdot \widehat{\phi_j}[{\bf{k}}]- {\bf{\cal H}}(\widehat{m_{j}})\cdot\widehat{\phi_i}[{\bf{k}}]=0, \end{equation}

where $$$m_i(x), m_j(x)$$$ and $$$\phi_i(x), \phi_j(x)$$$ denote the DWI from shots $$$i, j$$$ and their phases respectively, $$$\widehat{m_i}[k], \widehat{m_j}[k],\widehat{\phi_i[k]}, \widehat{\phi_j[k]}$$$ denote the corresponding Fourier samples, and $$${\bf{\cal H}}$$$ represents the construction of the Hankel-structured convolution matrix. The existence of such linear relations assert that the block-Hankel matrix

\begin{equation}{\bf{H_1}}=[{\bf{\cal H}}(\widehat{m_{1}})\hspace{5mm}~{\bf{\cal H}}(\widehat{m_{2}}) \hspace{5mm}\cdots~\hspace{5mm}{\bf{\cal H}}(\widehat{m_{N_s}})],\end{equation}

formed out of the k-space data of each shots is rank-deficient. Here, $$$N_s$$$ denote the number of shots. In the SMS setting where $$$L$$$ slices are simultaneously excited, the above relation holds true independently for the shot data corresponding to each slice. Thus, we can state that

\begin{equation}{\bf{H}}_l=[{\bf{\cal H}}(\widehat{m_{1,l}})\hspace{5mm}~{\bf{\cal H}}(\widehat{m_{2,l}}) \hspace{5mm}\cdots\hspace{5mm}{\bf{\cal H}}(\widehat{m_{N_s,l}})];~~ l=1:L\end{equation}

is rank-deficient. We propose to make use of the above property to reconstruct the slice unaliased DWIs from the SMS-accelerated acquisition that are unaffected by phase artifacts. We pose the reconstruction of the slice unalised k-space data corresponding to each shot and each slice, $$${\bf{\widehat{m}}}$$$, as

\begin{equation}\label{use Hankel}\widehat{\tilde {\bf m}}= argmin_{{\bf{\widehat{m}}}} \underbrace{||{\mathcal{A}}\left({\bf{\widehat{m}}}\right)-{\bf{\widehat{y}}}||^2_{\ell_2}}_{\mbox{slice unfolding}} + \lambda\underbrace{\sum_{l=1}^{L}||{\bf H}_l({\bf{\widehat{{m}}}})||_*.}_{\mbox{low rank penalty}}\end{equation}

The operator $$${\mathcal{A}}\left({\bf{\widehat{m}}}\right)$$$ incorporates SMS slice folding and enforces data consistency with the measured k-space data, $$$\bf{\widehat y}$$$, of dimension $$$N_1\times N_2/N_s \times N_c\times N_{s}$$$. Here, $$$N_1\times N_2$$$ represents the size of the DWIs, and $$$N_c$$$ represents the number of channels. The different steps involved in the computation of $$${\mathcal A}\left({\bf{\widehat{m}}}\right)$$$ are illustrated in figure 1. The second term in eq. [4] enforces the low-rank property of the block-Hankel matrix for each slice by minimizing its nuclear norm. $$$ \lambda$$$ is a regularization parameter. The above cost function can be minimized using alternating minimization^{7} schemes that alternate between the data consistency term and the rank penalty term.

To test the proposed reconstruction, we collected diffusion data using a multi-shot blipped-CAIPI EPI sequence at various multi-shot and multi-band (MB) factors. A Stajeskal-Tanner sequence with b-value=700 s/mm^{2}, 25 diffusion directions and one non-DWI was used for data acquisition using a 32 channel coil on 3T MRI. MB=2,3 and Ns=2,4 were tested. Other imaging parameters were: FOV= 210mm; slice thickness=2mm, NEX=1 with matrix sizes of 128x88 and 192x120. The proposed reconstructions were compared with SMS-MUSE^{6}, another navigator-free reconstruction method.

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Figure 1: Forward model for SMS MUSSELS involves taking (i) the inverse Fourier transform ($$${\cal {F}}^{-1}$$$) of the k-space data of L slices, (ii) multiplying the resulting image data by the coil sensitivities ($$${\cal {S}}$$$), (iii) taking the Fourier transform ($$${\cal {F}}$$$) to get the k-space channel-by-channel data of each slice, (iv) introducing slice shift by multiplying each slice data by phase factors ($$$\theta_l$$$) and adding the data from all slices in a channel-by-channel manner and finally (v) multiplying each channel data by a sampling mask ($$${\cal {M}}$$$) corresponding to the multi-shot acquisition.

Figure 2: In-vivo multi-band multi-shot data for MB=2,3 and Ns=2 for a matrix size of 128. (a) shows the images from different slice locations from the multi-band acquisition before slice unfolding and phase correction. (b) shows the SMS SENSE reconstruction which performs slice unfolding but does not involve phase compensation. Thus residual phase artifacts are seen in the slice unfolded images. (c) shows the SMS MUSE reconstruction which performs both slice unfolding and phase compensation. (d) shows the SMS MUSSELS reconstruction which recovers the slice unfolded DWIs via matrix completion of the k-space shots. SMS MUSSELS show superior recovery of DWIs compared to SMS MUSE.

Figure 3: In-vivo multi-band multi-shot data for MB=3 and Ns=4 for a matrix size of 128. (a) shows the images from different slice location from the multi-band acquisition before slice unfolding and phase correction. (b) shows the SMS SENSE reconstruction which performs slice unfolding but does not involve phase compensation. (c) shows the SMS MUSE reconstruction which performs both slice unfolding and phase compensation and (d) shows the SMS MUSSELS reconstruction. SMS MUSSELS show superior recovery of DWIs compared to SMS MUSE at the challenging case of MB=3 and Ns=4.

Figure 4: In-vivo multi-band multi-shot data for MB=3 and Ns=4 for a matrix size of 192. (a) shows the SMS MUSE reconstruction and (b) shows the SMS MUSSELS reconstruction from various slice locations for a given diffusion direction.

Figure 5: The msDW acquisition (b) drastically improves the image quality of DWIs and the fiber direction estimation as observed from the color-coded FA maps. Coupled with SMS acceleration, high-quality DWIs can be acquired in a time comparable to that of a traditional single-shot acquisition (a).