Seyed Amir Hossein Hosseini^{1,2}, Steen Moeller^{2}, Sebastian Weingärtner^{1,2,3}, Kâmil Uğurbil^{2}, and Mehmet Akçakaya^{1,2}

Long scan duration remains a challenge in coronary MRI. A scan-specific machine learning technique, called Robust Artificial-neural-network for k-space Interpolation (RAKI) has recently shown promising results in accelerating MRI. However, RAKI was originally designed for uniform undersampling patterns. In this study, we propose a technique, called SPIRiT-RAKI that enables RAKI with arbitrary undersampling using scan-specific convolutional neural networks to enforce self-consistency among coils. Regularization terms are also incorporated in the new formulation. Our results indicate that SPIRiT-RAKI can successfully accelerate 3D targeted coronary MRI.

Coronary MRI: Targeted right
coronary MRI was acquired on two healthy subjects at 3T with a 30-channel
body-coil using a T_{2}-prepared GRE sequence. Imaging parameters were FOV=300x300x60mm^{3} and resolution=1x1x3mm^{3}. The data were retrospectively
undersampled, both uniformly (acceleration rate of 2x2 along k_{y} and
k_{z} directions) and randomly (acceleration rates of 4 and 5). For
uniform sampling, no regularization term was used for fair comparison to
conventional parallel imaging methods. ACS region was selected as the central
45x10 k_{y}-k_{z} lines.

SPIRiT-RAKI: A 4-layer CNN architecture with 3-dimensional kernels was employed to find a nonlinear mapping function from acquired data points to missing data (Fig. 1). In contrast to RAKI, convolutional kernels do not use dilation and the output included the k-space across all coils, which significantly reduced the required time for both calibration and reconstruction. Therefore, the network consisted of input and output channels, where represents the number of coils. The factor of is due to complex k-space being mapped to a real field. All layers except the output layer were followed by rectifier linear units (ReLU). Tikhonov regularization was applied to the weights at each layer to avoid overfitting. The network was trained on the ACS region, with a mean square error objective function and an ADAM optimizer. Following the training of the CNN, reconstruction was performed using the following objective function:

||**y **- **DX**||_{2}^{2} + β||**x **- **G(x)**||_{2}^{2} + γ||**WEx**||_{1 }

_{ }

where **x** is the desired k-space data across all coils, **y**
is the noisy acquired data, **D** is the under-sampling operator, **G(.)** represents the CNN nonlinear operations to
enforce self-consistency, **E** is the operator combing coil images into a
SENSE Rate-1 image, and **W** transforms this image into the wavelet domain. β and γ were empirically chosen.

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The 4-layer CNN architecture to enforce self-consistency among all coils.

A representative slice from
the right coronary MRI of a healthy subject using SPIRiT and SPIRiT-RAKI with
uniform k_{y} - k_{z} undersampling
rate of 4.

A representative slice from
the right coronary MRI of another healthy subject using SPIRiT and SPIRiT-RAKI
with uniform k_{y} - k_{z} undersampling rate of 4.

The same slice from the dataset in Fig. 3 using L1-SPIRiT
and L1-SPIRiT-RAKI when random k_{y} – k_{z} undersampling
rates of 4 and 5 are utilized. Arrows indicate smoothing artifacts.