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kz-GRAPPA for 3D parallel imaging with localized estimation of interpolation kernels
Steen Moeller1, Sudhir Ramanna1, Essa Yacoub1, Kamil Ugurbil1, and Mehmet Akcakaya1,2

1University of Minnesota, Minneapolis, MN, United States, 2Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN, United States

Synopsis

A kz-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 ky-kz accelerations, and promises significant advantages for functional and diffusion imaging. The method is demonstrated with anatomical and diffusion imaging using thin slabs.

Introduction

Several methods for better k-space interpolation have been proposed for GRAPPA, leading to improved performance. These include various regularization approaches3-5, different calibration approaches6-9, and extension of the initial GRAPPA system10-12. Recently, analysis of GRAPPA via Reproducing Kernel Hilbert Spaces suggests the conventional linear shift-invariant GRAPPA kernels may be sub-optimal13. While GRAPPA kernels have been extensively studied and analyzed for 2D imaging14,15, 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 approach15 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 slab16.

Methods

kz -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 (nRO×nPE×nPE2) across all coils is utilized to reconstruct the data in coil j. The kernels are calibrated using kx (readout) and ky (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 ky is treated in a shift-invariant manner and for a given kz location only one kernel is generated for the kx – ky directions. The corresponding reconstruction equations is shown in Figure 1A, and a schematic illustration of kz-GRAPPA is in Figure 1C.

3D GRAPPA: For comparison, a shift-invariant 3D GRAPPA algorithm was implemented14,15, 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 Ry and Rz are the reduction factors in ky (phase-encoding) and kz (partition encoding), respectively, following the original convolution-based notation2.

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 mm3. In-vivo: FOV=330×225×50 mm3, 46.7%OS . phantom: FOV=332×332×20.8 mm3 25%OS.

DWI Multislab SE-EPI with SQUASHER17 Encoding: Excitation/Refocusing = HS2R12/HS2R14, duration 7680µs, 1mm3, TE/TR of 92.2/1610ms with 12slices/slab, 10 slabs, FOV 210x210x120 mm3, iPAT=2, Volume acquisition time (VAT)=26s. Diffusion data were acquired across 2shells (b=1500 and b=3000 s/mm2) using different diffusion encoding vectors $$$\vec{b}$$$. Each kz plane (from a single-shot acquisition) is corrected first for in-plane undersampling and for physiological induced phase-variation from the diffusion sensitizing gradient. 3D GRAPPA and kz-GRAPPA kernels are calculated from a b=0 ACS acquisition and applied to undersampling along kz, where no data has been acquired for alternating kz planes, ie all even kz planes.

Results

For a head shaped resolution phantom, reconstructions are shown in Figure 2 for both kz-GRAPPA and GRAPPA with a 2x2 synthesized undersampling. The top row shows a central axial slice and the bottom rows have both sagittal and coronal reformats. The differences to a fully sampled reference are shown in column 2 and 4, respectively. For an in-vivo acquisition, the reconstruction with a 3x2 undersampling is shown in Figure 3, with GRAPPA and kz-GRAPPA along with the corresponding g-factors for a central slice in the slab. The reconstruction of the diffusion volumes are shown in Figure 4 and 5. The 3D GRAPPA and kz-GRAPPA are applied to 4 different $$$\vec{b}$$$. The top row is a fully kz sampled acquisition. The middle and bottom rows show 3D GRAPPA and kz GRAPPA, respectively, with the right column showing a sagittal reformat prior to removal of the slab oversampled signal.

Discussion

For the resolution phantom both algorithms produce artifact free images without loss of resolution from the slab-varying kernels. For the in-vivo GRE acquisition, an increase in SNR is seen with kz-GRAPPA and also supported by the g-factor map. For the diffusion data, the stability under different diffusion contrast and SNR achieved by different b values is shown. For the diffusion application, kz-GRAPPA requires no additional ACS data acquisition in the slice encoding direction compared to 3D GRAPPA. Thus, the gain in scan time is exactly two-fold. In both in-vivo cases kz-GRAPPA yields significantly better performance (less noise) compared with regular GRAPPA

Acknowledgements

Grant support: NIH U01 EB025144, NIH P41 EB015894, NIH R00 HL111410, and NSF CAREER CCF-1651825.

References

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[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, U─čurbil 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

Figures

Figure 1 A: The reconstruction equation for kz-GRAPPA algorithm, for missing data along both ky and kz where bx is the kernel size along the readout, Rx and Ry the undersampling along ky and kz, and w a shift invariant interpolation kernel. For kz-GRAPPA the interpolation weigths wkz, have a kz dependence, captured with a Kronecker Delta function in the condensed formulation with kspj, the full k-space with zeros inserted for missing data. B: The reconstruction equation for 3D-GRAPPA algorithm. C: A schematic of how the kz-GRAPPA kernels are recovering missing data between planes in kspace, and how different kernels are used.

Figure 2 Reconstructions of a resolution phantom with a 2x2 undersampling pattern. Top row: Axial images, bottom row, coronal and sagittal views. From left to right, 3D GRAPPA, difference with 3D GRAPPA to fully sampled data, kz-GRAPPA, and difference with kz-GRAPPA to fully sampled data.

Figure 3 Reconstruction of in-vivo GRE data. Left are the reconstructions with 3D-GRAPPA and kz-GRAPPA respectively, and right are the corresponding g-factors, which shows the difference in noise to be from the reconstruction kernels.

Figure 4 Reconstruction of diffusion data with b=1500, for different diffusion direction. Top row is the reconstructed data with a 2D self-navigator applied to each kz-plane, and with standard GRAPPA applied to each kx-ky plane. The middle row is the 3D-GRAPPA applied across kz from full data in kx-ky, and bottom is with kz-GRAPPA. The first four columns are for different diffusion direction, listed above. The right-most column is a 3D reformat of the multi-slab data, prior to removing of oversampled data. Each slab is processed independently.

Figure 5 Reconstruction of diffusion data with b=3000, for different diffusion direction. Top row is the reconstructed data with a 2D self-navigator applied to each kz-plane, and with standard GRAPPA applied to each kx-ky plane. The middle row is the 3D-GRAPPA applied across kz from full data in kx-ky, and bottom is with kz-GRAPPA. The first four columns are for different diffusion direction, listed above. The right-most column is a 3D reformat of the multi-slab data, prior to removing of oversampled data. Each slab is processed independently.

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