Seng-Wei Chieh^{1}, Mostafa Kaveh^{1}, Mehmet Akcakaya^{1,2}, and Steen Moeller^{2}

Conventional non-Cartesian parallel imaging reconstruction in k-space necessitates large amounts of calibration data for successful estimation of region-specific interpolation kernels. In this work, we propose a self-calibration strategy for obtaining region-specific non-Cartesian interpolation kernels from a single calibration dataset. This enables simple and efficient high-quality reconstruction of non-Cartesian parallel imaging.

**Introduction **

Non-Cartesian undersampled acquisition for parallel imaging
is commonly reconstructed using CG-SENSE^{1} type methods. k-space
interpolation methods can also be used, but it is challenging to acquire
sufficient number of identical patches of an undersampled patch for determining
interpolation kernels and applying it to recover missing data^{2}. Region-specific
kernels can be calibrated from repeatedly acquired fully sampled calibration
scans, and provide improved reconstruction because of better match with the
theoretical model, but with the cost of large number of calibration scans^{3,4}.

In this work, we propose a scalable
non-Cartesian region-specific GRAPPA method called SING (**S**calable self-calibrated **i**nterpolation
of **n**on-Cartesian data with **G**RAPPA) which uses a single calibration
dataset. The proposed method resamples ACS data
for calibration of region-specific kernels. Subsequently, an explicit noise-based
regularized solution is utilized to estimate region-specific kernels, similar
to TT-GRAPPA^{3}. The efficacy of the method is demonstrated in 2D
radial cardiac imaging.

**Method **

Imaging: Cine cardiac MRI was acquired at 3T (Siemens
Prisma) using a 30-channel body array with a GRE sequence. Imaging parameters
were: TE/TR/α=2.3ms/3.9ms/12°, bandwidth=440Hz/pixel,
resolution 2×2 mm^{2}, temporal resolution=45ms. Data was fully sampled
with 216 views using a linear view order in a single breath hold. For TT-GRAPPA
calibration, additional 20 free-breathing fully-sampled acquisitions were
obtained shortly after the breath hold. Data was retrospectively undersampled
uniformly for each cardiac phase, which was rotationally shifted between
successive phases by the angle of a single view (180/216=0.83°).

SING: The proposed self-calibration strategy is depicted in **Figure 1**. First, the central k-space
data is resampled onto a Cartesian grid, then the region-specific kernel is
calibrated on this data. Subsequently, the system is regularized in step 4 by
adding i.i.d. Gaussian noise to synthetic points on each equation in order to
match the SNR of undersampled patch, which is lower on outer k-space. The least-squares
solution of the new system

$$\hspace{9.1cm} (b+Δb)=(A+ΔA)x,\hspace{9.1cm} [1]$$

is the calibrated kernel for the undersampled patch. $$$b$$$ and $$$A$$$ are synthetic target and source points. The corresponding rows in $$$Δb$$$ and $$$ΔA$$$ have the same noise level, and different rows have different noise levels. After reconstructing the whole k-space by iterating through step 1-5 for each region-specific kernel, gridding reconstruction is applied to generate an image with root-sum-of-squares coil combination.

TT-GRAPPA: TT-GRAPPA was implemented for comparison, with each region-specific kernel calibrated from local measurements spanning ±1 readouts and ±2 views in separate free-breathing calibration data.

Effect of regularization: The SNR-matched regularization strategy of Eq. [1] was compared with Tikhonov regularization. The Tikhonov regularization parameter was chosen in a readout-dependent manner and modelled by the theoretical SNR decay along the readout (1/r^{2}).

Non-Cartesian GRAPPA reconstructions are shown in **Figure 2**. At R=6, the reconstruction quality of TT-GRAPPA and SING are visually similar to the reference, outperforming CG-SENSE. At R=16, both TT-GRAPPA and SING suffer from reconstruction artifacts, although their quality is visually comparable. RMSE errors further show that SING
performs similar to TT-GRAPPA.

For the effect of regularization, images reconstructed from SING using no regularization, Tikhonov regularization and the SNR-matched regularization in Eq. [1] are shown in **Figure 3**. The proposed regularization in Eq. [1] reduces noise and blurring artifacts compared to the unregularized
and Tikhonov regularized approaches, respectively.

A temporal view through a cross-section of the heart is depicted
in **Figure 4**, showing similar temporal
fidelity between SING, TT-GRAPPA and fully sampled acquisitions. This is consistent with the fact that none of
these approaches utilize temporal regularization.

In 2D non-Cartesian imaging, SING is able to reconstruct images with similar quality to TT-GRAPPA without separate calibration acquisitions. This self-calibration strategy has the potential to scale up to 3D non-Cartesian imaging, since it does not require additional calibration volumes, which is especially prohibitive in 3D acquisitions.

The proposed regularization in Eq. [1] is established upon the inherent noise regularization effect in TT-GRAPPA. The superior reconstruction of TT-GRAPPA over other parallel imaging reconstruction methods may result from the implicit inclusion of noise on the target points.

Previous
work has considered a SENSE based composite method^{5 }for obtaining region specific kernel and^{ }is outperformed by this newer strategy. We also note a related method to SING
was proposed^{6}, with the main difference being that shifted ACS data is
utilized for calibration of kernels without a noise-aware of calibration.

The effect of correlation in the composite data generated from Kaiser-Bessel convolution was not considered, but warrants further investigation.

- BTRC P41 EB015894
- NIH R00HL111410
- NSF CCF-1651825

- Pruessmann KP, Weiger M, Bornert P, Boesiger P. Advances in sensitivity encoding with arbitrary k-space trajectories. Magn Reson Med. 2001;46(4):638-651.
- Griswold M, Heidemann R, Jacob P. Direct parallel imaging reconstruction of radially sampled data using GRAPPA with relative shifts. In Proceedings of the 11th Annual Meeting of the ISMRM, Toronto, Ontario, Canada, 2003. p. 2349.
- Seiberlich N, Ehses P, Duerk J, et al. Improved radial GRAPPA calibration for real-time free-breathing cardiac imaging. Magn Reson Med. 2011;65:492-505.
- Sayin O, Saybasili H, Zviman M et al. Real-time
free-breathing cardiac imaging with self-calibrated through-time radial GRAPPA.
Magn Reson Med. 2017;77(1):250-264.
- Chieh S, Akcakaya M, and Moeller S. A scalable composite through-time radial GRAPPA method. ISMRM 2017; p. 1045.
- Luo T, Noll D, Fessler JA, and Nielsen JF. A fast and general non-Cartesian GRAPPA reconstruction method. ISMRM 2018; p. 2821.

Figure 1: The flowchart of calibrating a
region-specific kernel and applying it for data interpolation. The undersampled
patch is firstly determined. Next, the dynamic undersampled data is combined
into a single fully-sampled composite data. Subsequently, this data is gridded
onto Cartesian grid. The gridded data is then used to resample non-overlapping
patches which match a given undersampled patch. These resampled patches
establish an overdetermined linear system. Then, kernel weights are obtained
through solving for the regularized system. Missing data is recovered by
applying the kernel onto undersampled data.

Figure 2: Image reconstructions of undersampled data
using CG-SENSE (top row), TT-GRAPPA (middle row) and SING (bottom row) for low
and high acceleration rates. CG-SENSE with Tikhonov regularization (λ=0.01) is
used. We varied
from 0.001 to 0.1 and selected λ=0.01
based on best reconstruction quality. The difference images between the
reconstructed images and the reference image are shown in right half. RMSE
errors are plotted in bottom-left.

Figure 3: Different regularizations are
applied for SING calibration at an acceleration rate R=16. From left to right:
No regularization, Tikhonov regularization and the proposed noise based
regularization.

Figure 4:
Temporal changes of a line through the heart (yellow dash line) in
reconstructed images with TT-GRAPPA (top row) and SING (bottom row) for low and
high acceleration rates. The reference is the temporal changes of the same line
in a fully sampled image. For R=16, a dark band (red arrows) is observed in
TT-GRAPPA reconstructed temporal changes, whereas not observed in SING
reconstructed temporal changes and the reference.