Gabriel Ramos-Llordén^{1}, Santiago Aja-Fernández^{2}, Congyu Liao^{3}, Kawin Setsompop^{3}, and Yogesh Rathi^{1}

In this work, we generalize conventional GRAPPA-based dMRI reconstruction by exploiting joint information from the

Our novel method, joint-diffusion GRAPPA, is validated with in-vivo multi-slice dMRI data, where we show it always outperforms conventional GRAPPA in terms of image

**Joint-diffusion GRAPPA**

Let $$$S_{c}^{q}(k_y - bR_{in-plane}\Delta_{k_y}) $$$ denote the set of k-space data points that are defined at phase-encoding line $$${k_y - bR_{in-plane}\Delta_{k_y}}$$$ ($$$b=-B,...,B$$$), ( $$$R_{in-plane}$$$ and $$$\Delta_{k_y}$$$ are the acceleration and sampling rate, respectively), and that are acquired at the $$$c$$$-th coil-channel and probed with a given q-space point $$$q$$$.

The non-acquired k-space data points $$$S_{c'}^{q'}(k_y - m\Delta_{k_y})$$$ are recovered as (joint-diffusion GRAPPA, see Fig.1):

$$S_{c'}^{q'}(k_y - m\Delta_{k_y}) = \sum_{q \in N_{q'}}\sum_{c=1}^C\sum_{b=-B}^B W(q',c',m,q,c,b)S_{c}^{q}(k_y - bR_{in-plane}\Delta_{k_y}),$$

where $$$N_{q'}$$$ is the set of q-space points that are ‘neighbors’ of the target q-space point, $$$q'$$$, and $$$W$$$, the GRAPPA kernel. Observe that for a particular point $$$q’$$$, the missing phase-encoding lines are not learned from the whole dataset but only from a subset of the k-space dataset that contains k-space data from DW images which bear structural similarity with the DW image to be recovered at point $$$q’$$$. In this work, we partition the set of q-space points into $$$k$$$ clusters. Then, for a given point $$$q$$$, $$$N_q$$$ are the q-space points of the cluster where $$$q$$$ belongs. Kernel $$$W$$$ is trained with 21 ACS lines^{1} acquired at each point $$$q$$$ , and estimated with standard least-squares+Tikhonov regularization^{1}.

**Experiments**

Joint diffusion GRAPPA was compared to zero-filled reconstruction and conventional GRAPPA^{2}. To that end, single-shell, fully sampled k-space data of an axial slice (3T, single-shot EPI, 2 $$$mm^3$$$ isotropic resolution, $$$C=8$$$ coil-channels, 20 repetitions) was acquired with one $$$b=0$$$ plus 15 gradient directions ($$$b=1200s/mm^2$$$). Fully sampled k-space data were retrospectively undersampled with acceleration factors of $$$R_{in-plane}= [2, 3, 4, 5, 6]$$$, after which data were reconstructed with joint-diffusion and conventional GRAPPA^{2}. The 15 diffusion directions were clustered into $$$k=3$$$ partitions with the k-means algorithm. The kernel in conv. GRAPPA was learned from the baseline dataset (21 ACS lines), and each of the k-space datasets for a given gradient direction was reconstructed individually. Reconstructed images where coil-combined with the Sum of Squares (SoS) method to get magnitude DW images only. For each gradient direction and repetition, the Normalized Root Mean-Squared Error (NRMSE) was computed (fully sampled data is the ground-truth). See caption of Fig. 3 for details.

We also performed diffusion tensor estimation^{3}, after which we computed the fractional anisotropy (Fig.4). The NRMSE was also used to assess the reconstruction quality in terms of FA estimation (Fig.5).

We have shown that, by extending conv. GRAPPA along diffusion directions, it is possible to reconstruct undersampled k-space data with reasonably good image quality at substantially high acceleration rates, where conv. GRAPPA often fails. We envisage promising extensions of joint-diffusion GRAPPA, which can further improve the preliminary results shown here. Joint-diffusion GRAPPA can accommodate virtual-coil k-space data information as well as complementary undersampling along diffusion directions^{4}. Moreover, to select ‘similar’ k-space datasets along diffusion direction, more sophisticated mechanisms than clustering in the q-space can be incorporated, e.g., machine/deep learning. Finally, an iterative process may be devised to reduce the number of packages of ACS lines, in the same spirit as in^{4,5,6}

^{1}Bilgic, B. et al., “Improving parallel imaging by jointly reconstructing multi-contrast data”. Magn. Reson. Med., 80: 619-632

^{2}Griswold, M. A. et al., “Generalized autocalibrating partially parallel acquisitions (GRAPPA)”. Magn. Reson. Med., 47: 1202-1210

^{3}Tristán-Vega, A., et al., “Least squares for diffusion tensor estimation revisited: Propagation of uncertainty with Rician and non-Rician signals”., Neuroimage, 59(4):4032-4043

^{4}Liao, C., et al., “Joint Virtual Coil Reconstruction with Background Phase Matching for Highly Accelerated Diffusion Echo-Planar Imaging”., Proc. Intl. Soc. Mag. Reson. Med. 26 (2018): 0465

^{5}Huang, F. et al., “ k‐t GRAPPA: A k‐space implementation for dynamic MRI with high reduction factor”. Magn. Reson. Med., 54: 1172-1184.

^{6}Breuer, F.A., et al., “Dynamic autocalibrated parallel imaging using temporal GRAPPA (TGRAPPA)”., Magn. Reson. Med., 53: 981-985.

With joint-diffusion GRAPPA, the missing
k-space lines from a k-space dataset probed at a given q-value, *q*, (e.g., a green
point on the sphere) are recovered from information
1) along coil channels dimension and from 2) ‘neighbor’ k-space datasets (e.g., rest of the
green points on the sphere).

Example of coil-combined (SoS) reconstructed DW images from 1) fully sampled k-space data, and from undersampled k-space data ($$$R_{in-plane}$$$ = 4) with 2) conventional GRAPPA and 3 ) the proposed joint-diffusion GRAPPA method. Top: magnitude images from a given diffusion direction and repetition. Bottom: Relative absolute error maps. (Fully sampled reconstructed image is considered the ground-truth)

Quantitative results to assess the performance of joint-diffusion GRAPPA in terms of image quality reconstruction. NRMSE is calculated for each diffusion-weighted image and for each repetition (20 in total), and then averaged to provide a single value for each $$$R_{in-plane}$$$ value and method.

Color-encoded Fractional Anisotropy maps obtained from DW images
reconstructed from 1) fully sampled k-space, and undersampled k-space data ($$$R_{in-plane}$$$ = 4 and $$$R_{in-plane}$$$ = 5 ) with 2) conventional and 3) joint-diffusion GRAPPA.
Sample-mean (20 realizations) of the estimated FA maps are displayed
here.

Quantitative results to assess the performance of joint-diffusion GRAPPA in terms of diffusion-metrics reconstruction, e.g., FA. NRMSE is calculated for each diffusion-weighted image and for each repetition (20 in total), and then averaged to provide a single value for each $$$R_{in-plane}$$$ value and method. Observe that joint-diffusion GRAPPA always outperforms conventional GRAPPA.