Steen Moeller^{1}, Sudhir Ramanna^{1}, Essa Yacoub^{1}, Kamil Ugurbil^{1}, and Mehmet Akcakaya^{1,2}

A k_{z}-dependent shift-variant 3D GRAPPA approach for
reconstructing 2D under-sampled k-space is proposed. The method results in equal
or lower g-factors compared to a conventional shift-invariant 3D kernel. In
turn, this permits higher 2D k_{y}-k_{z} accelerations, and promises significant advantages for functional and diffusion imaging. The method is
demonstrated with anatomical and diffusion imaging using thin slabs.

Several methods for better k-space interpolation
have been proposed for GRAPPA, leading to improved performance. These include
various regularization approaches^{3-5}, different calibration
approaches^{6-9}, and extension of the initial GRAPPA system^{10-12}.
Recently, analysis of GRAPPA via Reproducing
Kernel Hilbert Spaces suggests the conventional linear shift-invariant GRAPPA
kernels may be sub-optimal^{13}. While GRAPPA kernels have been
extensively studied and analyzed for 2D imaging^{14,15}, their design
and optimization have not been thoroughly studied for the 3D case.

In this work, we propose an estimation procedure
that depends on the slice encoding (kz) location for generating
multiple 3D GRAPPA kernels for reconstructing different parts of volumetric
k-space. We compare its performance to the standard 3D GRAPPA kernel approach^{15}
in terms of image quality, noise amplification, and stability across dynamic
images and under challenging conditions including thin slabs with limited coil
sensitivity variations. We use anatomical imaging with undersampling along both
the phase and slice-encoding and we use dMRI with 2X undersampling along the
slice-encoding direction. For
dMRI we use SQUASHER encoding with SE-EPI for generating a quadratic phase
across the slab^{16}.

k_{z} -GRAPPA:
The
proposed reconstruction utilizes region-specific 3D kernels in the slice
encoding direction. In order to estimate the missing k-space points ($$$k_{z}^{miss} $$$) from the
measured data ($$$k_{z'}^{meas} $$$),
at the slice encoding location $$$z \neq z'$$$ , a unique kernel with
elements, $$$w_{nj}^{k_z} $$$, of size 3×4×2 (n_{RO}×n_{PE}×n_{PE2})
across all coils is utilized to
reconstruct the data in coil *j*. The
kernels are calibrated using k_{x}
(readout) and k_{y} (phase-encoding)
points and with the choice of kernel the closest $$$k_{z'}^{meas} $$$ to $$$k_{z}^{miss} $$$ in
the matched ACS data. The undersampling in *k _{y}* is treated in a shift-invariant
manner and for a given k

3D
GRAPPA: For comparison,
a shift-invariant 3D GRAPPA algorithm was implemented^{14,15}, with a 3×4×4
kernel size. The kernel was calibrated across all available ACS data and all
coils. The corresponding reconstruction equation is depicted in Figure 1, where R_{y} and R_{z} are the reduction factors in k_{y} (phase-encoding)
and k_{z} (partition encoding), respectively, following the original
convolution-based notation^{2}.

Imaging:
Data was acquired on a 3T Siemens Prisma scanner with a 32-channel head coil. **In-vivo and Phantom GRE** : Fully-sampled k-space using a 5120µs sinc RF pulse with a time bandwidth
product of 4 and sequence parameters: TE/TR/FA= 30/35ms/15°, resolution=1.3×1.3×1.3
mm^{3}. In-vivo: FOV=330×225×50
mm^{3}, 46.7%OS . phantom:
FOV=332×332×20.8 mm^{3} 25%OS.

**DWI
Multislab SE-EPI with SQUASHER ^{17} Encoding:**
Excitation/Refocusing = HS2R12/HS2R14, duration 7680µs,
1mm

[1] SENSE: sensitivity encoding for fast MRI. Pruessmann KP1, Weiger M, Scheidegger MB, Boesiger P. Magn Reson Med. 1999 Nov;42(5):952-62.

[2] Griswold MA, Jakob PM, Heidemann RM, Nittka M, Jellus V, Wang J, Kiefer B, Haase A. Generalized autocalibrating partially parallel acquisitions (GRAPPA). Magn Reson Med 2002;47(6):1202-1210.

[3] High-pass GRAPPA: an image support reduction technique for improved partially parallel imaging. Huang F, Li Y, Vijayakumar S, Hertel S, Duensing GR. Magn Reson Med. 2008 Mar;59(3):642-9

[4] Discrepancy-based adaptive regularization for GRAPPAreconstruction. Qu P, Wang C, Shen GX. J Magn Reson Imaging. 2006 Jul;24(1):248-55

[5] Paradoxical Effect of the Signal-to-Noise Ratio of GRAPPA Calibration Lines: A Quantitative Study Yu Ding, Hui Xue, Rizwan Ahmad, Ti-chiun Chang, Samuel T. Ting, and Orlando P. Simonetti Magn Reson Med. 2015 July ; 74(1): 231–239.

[6 ]Improvement of temporal signal-to-noise ratio of GRAPPA accelerated echo planar imaging using a FLASH based calibration scan. Talagala SL, Sarlls JE, Liu S, Inati SJ. Magn Reson Med. 2016 Jun;75(6):2362-71

[7] Reducing sensitivity losses due to respiration and motion in accelerated echo planar imaging by reordering the autocalibration data acquisition. Polimeni JR, Bhat H, Witzel T, Benner T, Feiweier T, Inati SJ, Renvall V, Heberlein K, Wald LL. Magn Reson Med. 2016 Feb;75(2):665-79

[8] Brau AC, Beatty PJ, Skare S, Bammer R. Comparison of reconstruction accuracy and efficiency among autocalibrating data-driven parallel imaging methods. Magn Reson Med 2008;59(2):382-395.

[9] Samsonov AA. On optimality of parallel MRI reconstruction in k-space. Magn Reson Med 2008;59(1):156-164

[10 ]Improved radial GRAPPA calibration for real-time free-breathing cardiac imaging. Seiberlich N, Ehses P, Duerk J, Gilkeson R, Griswold M. Magn Reson Med. 2011 Feb;65(2):492-505

[11] Nonlinear GRAPPA: a kernel approach to parallel MRI reconstruction. Chang Y, Liang D, Ying L. Magn Reson Med. 2012 Sep;68(3):730-40

[12] Scan-specific robust artificial-neural-networks for k-space interpolation (RAKI) reconstruction: Database-free deep learning for fast imaging. Akçakaya M, Moeller S, Weingärtner S, Uğurbil K. Magn Reson Med. 2018 Sep 18

[13] Parallel Magnetic Resonance Imaging as Approximation in a Reproducing Kernel Hilbert Space. Athalye V, Lustig M, Uecker M. Inverse Probl. 2015 Apr 1;31(4):045008.

[14] Breuer FA, Blaimer M, Mueller MF, Seiberlich N, Heidemann RM, Griswold MA, Jakob PM. Controlled aliasing in volumetric parallel imaging (2D CAIPIRINHA). Magn Reson Med 2006;55(3):549-556.

[15] Breuer FA, Blaimer M, Mueller MF, Seiberlich N, Heidemann RM, Griswold MA, Jakob PM. Controlled aliasing in volumetric parallel imaging (2D CAIPIRINHA). Magn Reson Med 2006;55(3):549-556.

[16] SQUASHER: Slice Quadratic Phase with HSn Encoding and Reconstruction. Moeller, Steen Wu, Xiaoping Harel, Noam Garwood, Mike Akcakaya, Mehmet . ISMRM 2017, pp 1522

[17] Progress in the use of SQUASHER for Diffusion weighted imaging. Moeller Steen , Ramanna Sudhir, Yacoub Essa , Ugurbil Kamil , Akcakaya Mehmet. ISMRM, 2018, pp 1624