Estimation and Correction of Systematic Bias Inherent in Sparsely Undersampled Sodium Imaging of the Human Brain at High-Field
Yasmin Blunck1, Sonal Josan2, Brad A. Moffat3, Shawna Farquharson4, Roger J. Ordidge3, and Leigh A. Johnston1

1Electrical & Electronic Engineering, University of Melbourne, Melbourne, Australia, 2Siemens Healthcare, Melbourne, Australia, 3Anatomy & Neuroscience, University of Melbourne, Melbourne, Australia, 4Florey Institute of Neuroscience and Mental Health, Melbourne, Australia

### Synopsis

Sodium plays a vital role in various physiological processes and functions due to the delicate balance between intra- and extracellular sodium concentration. Disturbance to this electrochemical gradient is considered to be a sensitive, early indicator of cell breakdown and provide an insight into cellular integrity. Iterative reconstruction of sparsely undersampled data has been shown to cause inaccuracies in the estimation of tissue sodium content. This work investigates the bias in sodium concentration estimates and presents a method for correction that recovers the quantitative property of sodium imaging for high undersampling factors.

### Purpose

Sodium plays a vital physiological role due to the delicate balance between intra- and extra-cellular sodium concentration. Given the important role of sodium, it is of interest to develop imaging techniques that provide quantitative sodium concentration estimates.

Iterative reconstruction of sparsely undersampled data can recover a signal from fewer samples than the Nyquist theorem predicts, and has been shown to be of great potential benefit in accelerating MRI acquisition2. A preliminary study has demonstrated the feasibility of this form of reconstruction in Sodium Imaging, however, inaccuracies in the estimation of tissue sodium concentrations were reported3. In this work, we investigate the regularisation bias introduced by sparse undersampling and iterative reconstruction, and present a correction that restores quantitative sodium concentration estimates across undersampling factors.

### Methods

Acquisition was performed on a research 7T MRI scanner (Siemens Healthcare, Erlangen, Germany) with a transmit/receive dual-tuned 1H-23Na head coil (QED, USA). Data was collected with a prototype sequence (Siemens Healthcare) for radial 3D UTE image acquisition with TE=0.1ms, TR=100ms, a flip angle of 20°, 10 averages.

A sodium concentration phantom consisted of four 2.5cm-diameter vials filled with different amounts of sodium (150, 110, 70, and 30 mMol) and placed into a water-filled 14cm-diameter cylinder. The sodium-filled vials contained 3% agar to closely match in-vivo tissue T2 values4. The phantom data set was acquired with 12000 projections, with FOV=16cm and isotropic resolution of 2.5mm.

Human in-vivo data was acquired on a female healthy volunteer. Data was collected over 1000 projections to reduce the total acquisition time of all averages to 16:40min. The in-vivo FOV was 20cm to allow full brain coverage resulting in an isotropic resolution of 3.1mm.

Datasets were corrected for B1 field inhomogeneities.

The fully-sampled datasets were randomly undersampled and iteratively reconstructed by minimising

$$\|NUFFT(x)-y\|_2^2+\lambda_1\|x\|_1+\lambda_2TV(x)$$

Data consistency was based on a non-uniform FFT5 of the randomly undersampled projections, sparsity was minimised in the image domain3. The investigation was performed over a range of undersampling factors. The sampling ratio ranged from 0.1 to fully-sampled, with an increment of 0.025. Sparsity and total variation (TV) weightings $\lambda_1$ and $\lambda_2$ were examined over a range of values (0 0.0001 0.0005, 0.001, 0.0025, 0.005, 0.0075, 0.01)3. Optimisation was performed in Matlab using a nonlinear conjugate gradient method6.

### Results

The iterative reconstruction results for the phantom data were used to determine the dependence of intensity bias on undersampling factor, TV and sparsity weighting. It was found that intensity bias was highly dependent on the undersampling factor, while the influence of $\lambda_1$ and $\lambda_2$ was less variable across the parameter ranges tested. The mean intensity within the phantom vials converged to a biased fully-sampled mean intensity for higher percentages of sampled projections (Fig. 1). Hence for further analysis, $\lambda_1$ and $\lambda_2$ were fixed to 0.0001. The effect on standard deviation within each vial shows the regularisation property of the iterative reconstruction, which increases with higher undersampling, thus decreasing standard deviation.

Systematic bias in both the phantom and in-vivo datasets was observed (Fig. 2). Iterative reconstruction induced a large linear bias in higher intensities, particularly evident in the in-vivo dataset. In order to correct for the regularisation-induced bias, the relationship between undersampling factor and bias in the linear higher intensity region was investigated, revealing clear functional dependencies for both the slope and offset (intercept) of the linear bias across undersampling factors (Fig. 3).

The empirically-derived relationship between bias and undersampling factor was used to restore the image reconstructions (Fig. 4). Correction was employed over the linear, higher intensity region, and led to an increase in image contrast (Fig. 4 top row), along with a more accurate estimation of tissue sodium content (Fig. 4 bottom row).

### Discussion

The analysis of sodium image intensities derived from iterative reconstruction of sparsely undersampled data demonstrated a systematic bias dependent primarily on the degree of undersampling. Models of the parameters of the linear bias-undersampling relationship were derived. It was demonstrated that the bias can be removed by inverse mapping of these models, thus restoring the quantitative property of sodium imaging, with a concomitant increase in sodium image contrast.

### Conclusion

It has been shown that the bias in sodium image intensities caused by iterative reconstruction with higher undersampling factors ($\ge$3) can be modelled and corrected, permitting faster sodium image acquisition while restoring accuracy in tissue sodium content quantitation. We are currently investigating inter-subject bias variability and the effect of iterative reconstruction on bias using other sampling techniques such as Twisted Projection Imaging4,7.

### Acknowledgements

No acknowledgement found.

### References

1. Madelin G, Lee J-S, Regatte R, et al. Sodium MRI: Methods and applications. Prog Nucl Mag Res Sp. 2014; 79:14-47.

2. Geethanath S, Reddy R, Konar A S, et al. Compressed Sensing MRI: A Review. Crit Rev Biomed Eng. 2013;41(3):213-235.

3. Madelin G, Chang G, Otazo R, et al. Compressed sensing sodium MRI of cartilage at 7T: Preliminary study. J Magn Reson. 2012;214:360-365.

4. Lu A, Atkinson I, Claiborne T, et al. Quantitative Sodium Imaging With a Flexible Twisted Projection Pulse Sequence. Magn Reson Med. 2010;63:1583-1593.

5. Fessler J, Sutton B. Nonuniform Fast Fourier Transforms Using Min-Max Interpolation. IEEE Trans Signal Process. 2003;51:560-574.

6. Lustig M, Donoho D, Pauly J. Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging. Magn Reson Med. 2007;58:1182-1195.

7. Boada F, Gillen J, Shan G, et al. Fast Three Dimensional Sodium Imaging. Magn Reson Med. 1997;37(3):470–47.

### Figures

Figure 1: Mean and standard deviation of reconstruction results for the concentration phantom. Data points represent results for different sparsity and TV weightings dependent on the sampling ratio. Solid line indicates mean and standard deviation of the respective vial in the original image. Phantom cross section shows investigated ROIs.

Figure 2: Reconstruction of intensities over a range of undersampling ratios. Each plot depicts fully-sampled intensity vs reconstructed intensity from sparse sampling. In-vivo data reconstruction (left column), concentration phantom reconstruction (right column). Solid black line indicates an ideal, unbiased reconstruction.

Figure 3: Reconstruction of intensities show a linear bias for high intensities. The slope and offset of the bias linear relationship demonstrate logarithmic growth and exponential decay, respectively, across undersampling ratios.

Figure 4: Correction of undersampling-induced bias for in-vivo data. Fully sampled image shown on left. Iterative reconstruction results and correction for different undersampling ratios in the upper row. Lower row illustrates intensity reconstruction for uncorrected and corrected images. Solid line indicates ideal reconstruction.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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