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Joint-diffusion GRAPPA: enabling higher acceleration rates in dMRI by exploiting joint information from the k- and q-space
Gabriel Ramos-Llordén1, Santiago Aja-Fernández2, Congyu Liao3, Kawin Setsompop3, and Yogesh Rathi1

1Brigham and Women's Hospital, Harvard Medical School, Boston, MA, United States, 2Laboratorio de Procesado de Imagen, Universidad de Valladolid, Valladolid, Spain, 3Massachusetts General Hospital, Harvard Medical School, Charlestown, MA, United States

### Synopsis

In this work, we generalize conventional GRAPPA-based dMRI reconstruction by exploiting joint information from the k-and q-space simultaneously. Higher acceleration in-plane rates than those commonly reported may be achieved when the missing k-space lines are learned using all information available in the whole k-space data set, that is, considering multi-coil channel information as well as the k-space data probed at different q-space points.

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 quality, and works reasonably well for regimes where conventional GRAPPA results in significant noise penalty ($R_{in-plane}$ > 3 ).

### Introduction

Conventionally, in-plane GRAPPA-based reconstruction in diffusion MRI (dMRI) is limited to acceleration factors of 2 or 3, to maintain a compromise between image quality and reduction in acquisition time. Recent work has shown that higher acceleration rates may be obtained in parametric MRI, by exploiting information from multiple coil-channels and redundancies across multiple images with different settings/contrasts1. Driven by these findings and targeting highly accelerated diffusion MRI, in this work, we extend conventional GRAPPA2 dMRI reconstruction by learning the missing k-space lines not only from different coil channels but also from the k-space data that correspond to diffusion-weighted (DW) images acquired at different q-space points.

### Methods

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 lines1 acquired at each point $q$ , and estimated with standard least-squares+Tikhonov regularization1.

Experiments

Joint diffusion GRAPPA was compared to zero-filled reconstruction and conventional GRAPPA2. 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 GRAPPA2. 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 estimation3, 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).

### Results

NRMSE values reported in Fig. 3 confirmed the superiority of joint-diffusion GRAPPA over conventional GRAPPA, for all values of $R_{in-plane}$. Substantial gains in reconstruction quality become notorious for high acceleration factors ($R_{in-plane}> 2$). Note that joint-diffusion GRAPPA at $R_{in-plane} = 5$ performs equally well as conventional GRAPPA for $R_{in-plane} = 3$. Visual results in Fig. 2 agree with quantitative results. FA estimation also benefits from joint-diffusion GRAPPA reconstruction, in comparison to conventional GRAPPA. NRMSE results show that joint-diffusion GRAPPA outperforms conv. GRAPPA in all the cases (Fig.5). Furthermore, for substantially high $R_{in-plane}$ (4 and 5), color-encoded FA obtained from the data reconstructed with joint-diffusion GRAPPA yet presents a reasonably good quality, whereas conv. GRAPPA clearly fails (Fig.4).

### Discussion

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 directions4. 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 in4,5,6

### Conclusions

This preliminary work showcases that high in-plane acceleration rates in GRAPPA-based dMRI reconstruction ( >3x ) can be reached if conventional GRAPPA is generalized to recover missing k-space lines not only with information derived from different coil channels but also from different DW images. We expect to get even higher acceleration rates by combining joint-diffusion GRAPPA with multi-band techniques and virtual-coil information.

### Acknowledgements

NIH grant R01 MH116173 (PIs: Setsompop, Rathi)

### References

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

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

3Tristán-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

4Liao, 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

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

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

### Figures

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.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)
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