Yasmin Blunck^{1,2}, Scott C. Kolbe^{2,3}, Bradford A. Moffat^{2,3}, Roger J. Ordidge^{2,3}, Jon O. Cleary^{2,3}, and Leigh A. Johnston^{1,2}

The clinical application of Sodium MRI is hampered due to relatively low image quality and associated long acquisition times (TA). Compressed Sensing (CS) aims at a reduction of TA, but has been found to encompass quantitative estimation bias when used in low SNR x-Nuclei imaging. This work analyses CS in human brain Sodium MRI from both angles, acquisition speed-up and quantification, and presents an optimized setup allowing an up to four-fold TA reduction with recommendations for quantitative assessments. The demonstrated global optima of CS weighting parameters and achievable reduction in TA greatly support the transition of Sodium MRI into clinical routine.

Imaging was performed on a research 7T MRI scanner (Siemens Healthcare, Erlangen, Germany) with a transmit/receive dual-tuned ^{1}H-^{23}Na head
coil (QED, USA). Data was collected with a custom-built 3D-radial sequence. Five healthy volunteers were imaged with the following parameters: TR=160ms, TE=0.4ms, FA=90°, 10,000 projections, FOV=(20cm)^{3}, TA=26min40s.

Image reconstruction was performed offline in Matlab onto an isotropic 3.1mm-grid. A standard NUFFT^{5}-reconstruction of the full raw-data (I_{ref}) was compared with randomly undersampled reconstructions (I_{u}) from NUFFT and CS. The employed CS optimization was

$$\hat{x}=\arg\min_x\|\mathcal{F}_u(x)-y\|_2^2+\lambda_1\|\Psi x\|_1+\lambda_2 \text{TV}(x)$$

where $$$\|\cdot\|_1$$$ and $$$\|\cdot\|_2$$$ denote the l_{1} and l_{2}-norm, respectively, $$$\mathcal{F}_u$$$ is the undersampled NUFFT-operator, x is the reconstructed image, y the undersampled raw-data, Ψ the applied sparsity transform and λ_{1} and λ_{2} the weights on the transform sparsity and total variation (TV), respectively. Reconstructions were investigated for undersampling factors (USF) ranging from 2 to 10 with increments of 1 and across regularization weights, λ_{1} (sparsity domain) = [0,1.0] step size 0.1; λ_{2} (TV) = [0,0.0001,0.0005,0.001,0.005,0.01,0.05,0.1,0.5]. The employed sparsity domains Ψ were identity, i.e. the inherent image sparsity, Discrete CosineTransform (DCT) and Wavelet transform (WT).

Image reconstruction performance of each I_{u} was assessed by calculating the MSE, pSNR and sSim with respect to I_{ref}.

_{}Figure 1 illustrates reconstruction performance of CS for all investigated regularization weights and USF; red line surrounds area where CS reconstruction achieves higher scores than NUFFT-based reconstruction at the same USF. DCT and WT clearly outperform image-sparsity based and standard NUFFT-reconstructions in all investigated quality measures. WT shows a robust performance over a wide range of parameters (moderate to high λ_{1}, low to moderate λ_{2}).

Figure 2 shows mean and standard errors of quality scores with regards to λ_{1} (Figure 2a) and λ_{2} (Figure 2b) across subjects for all USF. The curves re-emphasize the overall better performance of WT reconstructions. Vertical red lines indicate the joint optimum across all image quality measures at λ_{1}=0.2 and λ_{2}=0.01. sSim scores were found to show good agreement with visual assessments; MSE and pSNR show a bias towards higher TV weights where images were visually smoothed.

Figure 3 illustrates cross sections of I_{ref} and I_{u}. CS reconstructions with highest sSim score are displayed. Up to an USF of 4, WT reconstructions show good structural detail.

Figure 4 plots the correlation between I_{ref} and I_{u} for all USF. Point clouds show some non-linear behavior, particularly for DCT and Identity-based CS reconstructions. Linear fits through point clouds (Figure 4b) show a decreasing slope manifested in an overall increase in noise and decrease in high image intensities. WT reconstruction fit residuals are distributed around zero highlighting the good linear correlation (Figure 4c).

In Figure 5 mean intensities of 200 evenly spaced intensity intervals in I_{ref} and I_{u} are plotted for all undersampled reconstruction schemes. The non-linear intensity reconstruction behavior is re-accentuated, dynamic ranges decrease with higher USF. Iterative CS reconstruction show better noise suppression. Wavelet-based reconstructions follow a more linear trend and lead to smaller drop in high intensities.

This work investigated the optimization of sparsity and TV-regularized CS reconstructions with regards to TA reduction, image quality and intensity accuracy. A clear advantage towards Wavelet-based CS results was found, enabling USF=4 with great image quality and structural detail while maintaining good signal intensity reconstruction performance (mean linear bias slope: 0.96±0.02, Figure 4). Nevertheless, the use of a range of Sodium concentration phantoms across the whole spectrum of brain Sodium concentrations (20-150mM) is advised if quantitative information of Sodium content are to be extracted from CS reconstructed images; the use of background signal as a 0mM reference should be avoided due to the non-linear denoising properties of iterative reconstructions.

Apart from TA reduction, CS reconstructions can be further exploited for SNR enhancements. Additional regularization terms, e.g. ^{1}H-constraints^{6} or learned dictionaries^{7}, have shown great SNR results and could be included in Eq[1]. The radial acquisition scheme can furthermore be used for flexible time-resolved imaging^{8,9,10}.

In summary, the demonstrated global optima of CS weighting parameters and achievable reduction in TA greatly support the transition of Sodium MRI into clinical routine.

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CS Reconstruction Analysis. (a) Structural similarity, (b) peak SNR and (c) MSE of in vivo CSreconstruction across undersampling factors, TV and sparsity weighting in investigated sparsity domains. Red dottedline surrounds area with a higher quality score than NUFFT-based reconstructions at the respective undersamplingfactor. USF, undersampling factor; DCT, Discrete Cosine Transform; TV, total variation

Optimal weighting parameters. Mean and standard error of (a) sparsity and (b) TV weighting across subjects and undersampling factors for all investigated quality measures. Dashed red line indicates a joint optimum across all investigated quality parameters. USF, undersampling factor; TV, total variation; DCT, Discrete CosineTransform

In vivo reconstruction. Axial and sagittal cross section of in vivo reconstructions. Top row: Fully sampled (10,000 projections) reconstruction (I_{ref}); lower rows: NUFFT and CS reconstructions across different undersampling factors. CS reconstructions with highest sSim score are displayed. USF, undersampling factor; DCT, Discrete CosineTransform, I_{ref}, reference image

CS Intensity Reconstruction Performance. Correlation between reference image intensity I_{ref} and undersampled reconstruction intensities I_{u} for all undersampled reconstruction schemes. (a) Point cloud overlay of reconstruction accuracy for all USF; black dashed line highlights a perfect correlation. (b) Linear least-square fits through point clouds at each USF. (c) Residuals of linear fit results. USF, undersampling factor; I_{ref}, reference image; I_{u}, undersampled image; DCT, Discrete Cosine Transform

Investigation of non-linearity in intensity reconstruction. Mean intensity of voxel groups resulting from subdivision of I_{ref} into 200 equally-spaced intensity groups. For each intensity voxel group, mean intensity values in I_{u} over the mean intensity values in I_{ref} are plotted across all USF. Red solid box and blue dashed box show zoom-in view for lower and upper 15% of intensity groups, respectively. USF, undersampling factor; I_{ref}, reference image; I_{u}, undersampled image; DCT, Discrete Cosine Transform