Melanie Schellenberg^{1}, Armin M. Nagel^{1,2}, Peter Bachert^{1}, Mark E. Ladd^{1}, and Nicolas G. R. Behl^{1}

^{23}Na MRI provides important information for many pathologies. However, its low SNR entails low spatial resolutions and long acquisition times. The
proposed work reconstructs 3D radially undersampled *in vivo* 30-channel ^{23}Na head data at B_{0}=7T with a sensitivity encoding using a nonlinear
conjugate gradient method (CG-SENSE) including a total variation and a discrete
cosine transform. With CG-SENSE using iteratively generated Lagrangian coil sensitivities, image quality and contrast within the object are improved compared to
sum of squares (SOS) and adaptive combination (ADC) reconstructions.

CG-SENSE iteratively minimizes the following cost function using a nonlinear
conjugate gradient algorithm^{6}:

$$\bf{ \arg\min_m||FCm -
y||_2^2 +\sum_{i=1}^N{\lambda_i || \Psi_i m ||_1},}$$

where $$$\bf{F}$$$ is a non-uniform fast
Fourier transform operator (NUFFT)^{7}, $$$\bf{C}$$$ are the coil
sensitivities, $$$\bf{m}$$$ is the reconstructed
image, $$$\bf{y}$$$ are the k-space data, and $$$\bf{\Psi_i}$$$ is one of $$$\bf{N}$$$ sparsifying transform operators with $$$\bf{\lambda_i}$$$ as its corresponding regularization tuning constant.
In this work, three differently
generated coil sensitivities $$$\bf{C}$$$ are used. The
conventional sum of squares (SOS) and birdcage (BC) coil sensitivities are created
by dividing each gaussian filtered and phase-corrected individual coil image by the
SOS or BC image, respectively. The Lagrangian coil sensitivities $$$\bf{C_{opt}}$$$ are generated by minimizing another cost
function using an iterative Augmented Lagrangian Method^{8}:

$$\bf{C_{opt}\stackrel{\wedge}{=} \arg\min_C \frac{1}{2} ||Z-DC||_W^2 + \frac{\lambda}{2}||RC||_2^2 ,}$$

where $$$\bf{Z}$$$ are the single-coil images, $$$\bf{D}$$$ is the BC image, $$$\bf{C}$$$ are the coil
sensitivities, $$$\bf{W}$$$ is a weighting matrix, $$$\bf{\lambda}$$$ is a regularization tuning constant, and $$$\bf{R}$$$ is a finite difference matrix. The weighting matrix acts
as a threshold and is subjectively adjusted to the noise level of the BC image.
The ^{23}Na data were acquired
using a density-adapted 3D radial projection pulse sequence^{2} on a 7T whole body MR
system (MAGNETOM 7T, Siemens Healthcare GmbH, Erlangen, Germany) and a
double-resonant ^{1}H/^{23}Na Tx/Rx quadrature volume head coil
integrating a 30-channel ^{23}Na Rx phased array (Rapid Biomedical
GmbH, Rimpar, Germany). The data were acquired with the following parameters: (Δx)^{3}=(3mm)^{3},
15000 projections, TE/TR=0.35ms/30ms, α=53°, TA=30min, USF= 1, and N_{avg}=4.
The CG-SENSE was
regularized with a 3D total variaton^{3} (TV) and discrete cosine
transform (DCT)^{9}. The regularization tuning constants were empirically
determined by visual inspection. The reconstruction was initialized with a (72x72x72)
matrix of zeros. Convergence was assumed after a maximum of 50 iterations (Figure
1).
The performance of CG-SENSE using Lagrangian coil sensitivities is compared to SOS
and adaptive combination (ADC) reconstructions. Blurring
artifacts in the outer parts of the CG-SENSE images are suppressed by applying
a binary mask obtained with the BC data. Furthermore, the influence of the
differently generated coil sensitivities on CG-SENSE is analysed.

1. Madelin G et al., Prog Nucl Magn Reson Spectrosc (2014) 79:14-47.

2. Nagel AM et al., Magn Reson Med (2009) 62:1565-73.

3. Behl NG et al., Magn Reson Med (2016) 75:1605-16.

4. Lustig M et al., Magn Reson Med (2007) 58:1182-95.

5. Wright KL et al., J Magn Reson Imaging (2014) 40:1022-1040.

6. Feng L et al., J Magn Reson Med (2014) 72:707-717. 5].

7. Fessler J, University of Michigan, https://web.eecs.umich.edu/~fessler/.

8. Allison M J et. Al., IEEE Trans Med Imag (2013) 32:556-564.

9. Myronenko A, NVIDIA, Santa Clara, California, https://sites.google.com/site/myronenko/.