Nikolai J Mickevicius^{1}, Eric S Paulson^{1}, and Andrew S Nencka^{2}

This work aims to extend the RAKI method for artificial intelligence-based k-space interpolation to non-Cartesian acquisitions. It was tested in radial acquisitions up to acceleration factors of 7. This method performs similarly well, or better than total-variation regularized sensitivity encoding.

The
Pseudo-Cartesian Robust Artificial-neural-network for K-space Interpolation
(PC-RAKI) method was implemented as a regression in Keras with a TensorFlow
backend. For each non-acquired Cartesian grid location, a kernel selecting two
nearby source points was determined. A separate neural network was trained for
each kernel shape to reconstruct missing target points. Each network consisted
of three layers. The input to the network was the concatenated real/imaginary
k-space values for each coil and source point. The output size for each layer
was as follows: #1: 128, #2: 128, #3: 2*N_{coils}. The sizes for layers
1 and 2 were chosen heuristically. The multiplicative factor of two in the
final layer accounts for the real/imaginary components of the target point. Rectified
linear unit (ReLU) activations were applied in the 1st and 2nd
layers. The PC-RAKI method is summarized graphically in Figure 1. The networks
were trained using 2000 randomly selected target points in the NUFFT-gridded[6]
Cartesian k-space from the fully sampled radial acquisition. Note: like standard
RAKI, this is a database-free, scan-specific deep learning approach.

A parallel imaging-based, nearest-neighbor gridding (via GROG) of non-Cartesian k-space samples was performed with golden-angle radial data acquired on an Elekta 1.5T MR-Linac. Eight receive coils were used. The abdomen of a free-breathing, consenting healthy volunteer was scanned using a spoiled gradient echo 3D stack-of-stars acquisition. An in-plane matrix size of 256x256 with 40 slice partitions was prescribed. The data were retrospectively undersampled to 144, 89, and 55 spokes corresponding to acceleration factors of R=2.8, R=4.5, and R=7.3, respectively. To interpolate the skipped points for each acceleration factor, 25, 69, and 156 kernels (i.e. trained networks) were required, respectively. For a central slice of the acquisition, PC-RAKI was compared with a reference image (610 spokes), NUFFT, GROG-gridding, and total variation (TV)-regularized CG-SENSE[7].

[1] Pruessmann KP, Weiger M, Börnert P, Boesiger P. Advances in sensitivity encoding with arbitrary k-space trajectories. Magn Reson Med 2001;46:638–51. doi:10.1002/mrm.1241.

[2] Lustig M, Donoho D, Pauly JM. Sparse MRI: The application of compressed sensing for rapid MR imaging. Magn Reson Med 2007;58:1182–95. doi:10.1002/mrm.21391.

[3] Uecker M, Zhang S, Frahm J. Nonlinear inverse reconstruction for real-time MRI of the human heart using undersampled radial FLASH. Magn Reson Med 2010;63:1456–62. doi:10.1002/mrm.22453.

[4] Akçakaya M, Moeller S, Weingärtner S, Uğurbil K. Scan-specific robust artificial-neural-networks for k-space interpolation (RAKI) reconstruction: Database-free deep learning for fast imaging. Magn Reson Med 2018. doi:10.1002/mrm.27420.

[5] Seiberlich N, Breuer FA, Blaimer M, Barkauskas K, Jakob PM, Griswold MA. Non-Cartesian data reconstruction using GRAPPA operator gridding (GROG). Magn Reson Med 2007;58:1257–65. doi:10.1002/mrm.21435.

[6] Fessler JA, Sutton BP. Nonuniform fast fourier transforms using min-max interpolation. IEEE Trans Signal Process 2003;51:560–74. doi:10.1109/TSP.2002.807005.

[7] Block KT, Uecker M, Frahm J. Undersampled radial MRI with multiple coils. Iterative image reconstruction using a total variation constraint. Magn Reson Med 2007;57:1086–98. doi:10.1002/mrm.21236.

[8] Seiberlich N, Breuer F, Heidemann R, Blaimer M, Griswold M, Jakob P. Reconstruction of undersampled non-Cartesian data sets using pseudo-Cartesian GRAPPA in conjunction with GROG. Magn Reson Med 2008;59:1127–37. doi:10.1002/mrm.21602.

The radial data are interpolated onto a
Cartesian grid via GROG. The kernel shapes necessary to interpolate missing
data points are determined. For each kernel shape, a 3-layer Neural network is
trained using Cartesian calibration data. The weights are then applied to the
undersampled data to create the interpolated data output.

Results for R=2.8 (144 spokes)

Results for R=4.5 (89 spokes)

Results for R=7.3 (55 spokes)