To introduce a fast regularized reconstruction scheme for under-sampled multi-shot diffusion weighted (MS-DW) data recovery that does not rely on motion-induced phase estimates.

Multi-shot diffusion imaging holds great potential for enabling high spatial resolution diffusion imaging as well as short echo time imaging to enhance studies at higher field strengths. However, the imaging throughput of multi-shot diffusion scheme is low. To increase the efficiency, under-sampled multi-shot acquisitions can be employed. However the conventional multi-shot diffusion-weighted imaging (DWI) reconstructions that rely on motion-induced phase estimates are not appropriate for such acquisitions. This is because a good estimate of the motion-induced phase cannot be obtained from such under-sampled acquisitions unless a navigator is used which further decreases the throughput of the imaging. The aim of this work is to develop a regularized method to reconstruct high-throughput MS-DW data that does not depend on the motion-induced phase estimates. It directly recovers the missing samples of the individual k-space shots using a nuclear norm minimization of the Hankel matrix of the multi-shot data based on the annihilating filter bank formulation in k-space^1-5.

Denoting the multi-shot k-space data of an $$$N_s$$$-shot DWI acquisition as $$$\hat {\bf{ m}}(k)$$$, we can pose the recovery of $$$\hat {\bf{ m}}(k)$$$ using the following nuclear norm minimization problem: \begin{equation}\tilde {\bf m}= \text{argmin}_{{\bf \hat{m}}} {||\bf{{\cal{A}}\hat{m}-\hat y}}||^2_{\ell_2} + \lambda||{\bf{\cal{H}}_1(\hat{m})}||_* , \hspace{30mm}[1]\end{equation} where $$$\bf{y}$$$ is the matrix of measured multi-channel multi-shot k-space data of dimension $$$N_1\times N_2\times N_c\times N_{s}$$$. The operator $$$\bf{{\cal{A}}}$$$ concatenates $$$\bf{{\cal{F}}}\circ\bf{{\cal{S}}}\circ\bf{{\cal{F}}^{-1}}$$$ where $$$\bf{{\cal{F}}}$$$ and $$$\bf{{\cal{F}}^{-1}}$$$ represent the Fourier transform and the inverse Fourier transform operation and $$$\bf{{\cal{S}}}$$$ represents multiplication by coil sensitivities. $$$\lambda$$$ is the regularization parameter and $$${\bf{\cal{H}}_1}$$$ is the block Hankel matrix consisting of Hankel matrices of data combined from all channels of each shot. In the case of under-sampled acquisitions, the above reconstruction produces highly noisy images. Additional constraints such as total-variation (TV) penalty can be imposed on the images to improve the conditioning of the reconstruction as given by: $$$\tilde {\bf m}= \text{argmin}_{{\bf \hat{m}}} {||\bf{{\cal{A}}\hat{m}-\hat y}}||^2_{\ell_2} + \lambda_1||{\bf{\cal{H}}_1(\hat{m})}||_*+ \lambda_2||{\bf{m}}||_{TV}.$$$ The advantage of the annihilating filter formulation is that one can derive new annihilating filters and the corresponding low rank Hankel structured matrices in a weighted k-space domain for the sparsifying constraints such as TV^3,5. This leads us to a consolidated nuclear-norm minimization problem \begin{equation}\tilde {\bf m}= \text{argmin}_{{\bf \hat{m}}} {||\bf{{\cal{A}}\hat{m}-\hat y}}||^2_{\ell_2} + \lambda||{\bf{\cal{H}}}_1({\bf{\cal{R}}(\hat{m}))}||_*,\hspace{30mm}[2]\end{equation} for the total-variation regularized recovery of under-sampled multi-shot diffusion acquisitions with comparable computational complexity as of the un-regularized version in Eq [1]. The operator $$${\bf{\cal{R}}}$$$ represents multiplication by $$$-j\omega$$$ in the Fourier domain to generate the weighted k-space data, which is equivalent to doing a finite difference operation in the image domain to get the edge information. The above nuclear norm minimization problem can be solved using singular value thresholding by employing a variable splitting strategy in an augmented Lagrangian setting.

Diffusion data was collected on GE MR950 7T scanner at the University of Iowa equipped with a 32-channel Tx/Rx coil in quadrature mode using a 4-shot Stajeskal-Tanner sequence (b-value=1000s/mm2; FOV was 220mmx220mm; matrix size=128x128, slice thickness=1.7mm, TE=69.6ms). The data was retrospectively under-sampled by a factor of 2 for showing the results of regularized recovery. We compare the reconstruction of the multi-channel MS-DW data to the standard MUSE method⁶, that relies on motion-induced phase estimates for the recovery of the multi-shot data. To fairly compare the techniques, a similar TV-regularization has been added to the MUSE reconstruction (TV-MUSE) also.

Figure (1) shows three DWIs and the color-coded fractional anisotropy maps recovered from 2x-under-sampled 4-shot data using MUSE and TV-MUSE. Figure (2) shows the corresponding images recovered using the proposed method with no TV regularization and with TV regularization corresponding to the reconstruction given in Eq [1] and [2] respectively. Since the motion-induced phase estimates are highly corrupted for the cases of under-sampled acquisition, the MUSE-based reconstructions show poor performance, even after adding TV regularization. On the other hand, the proposed method does not depend on motion-induced phase estimates. As is observed from fig (2), the unregularized reconstruction using the proposed method gives highly noisy images which is stabilized by using TV-regularized reconstruction.We proposed a fast and robust reconstruction scheme for under-sampled multi-shot diffusion data recovery that does not rely on motion induced phase estimates. To the best of our knowledge, this is the first work that have been proposed for the recovery of under-sampled multi-shot diffusion acquisition where navigator data is not available.