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High-Resolution Sodium Imaging Using Anatomical and Sparsity Constraints for Denoising and Recovery of Novel Features

Yibo Zhao^1,2, Rong Guo^1,2, Yudu Li^1,2, Keith R. Thulborn³, and Zhi-Pei Liang^1,2
¹Department of Electrical and Computer Engineering, University of Illinois, Urbana-Champaign, Urbana, IL, United States, ²Beckman Institute for Advanced Science and Technology, University of Illinois, Urbana-Champaign, Urbana, IL, United States, ³Center for Magnetic Resonance Research, University of Illinois at Chicago, Chicago, IL, United States

Synopsis

Quantitative sodium MRI is a unique tool for assessing tissue viability noninvasively. A major obstacle to widespread clinical applications of sodium MRI is low sensitivity, which leads to low spatial resolution even with long scan times. We present a novel method for reconstruction of high-resolution sodium maps from noisy, limited k-space data. The proposed method has been validated using simulated and experimental data, producing high-SNR and high-resolution tissue sodium concentration maps. These high-resolution maps were also shown to improve the detection of tumor responses to radiation therapy.

Introduction

Tissue sodium concentration (TSC) has long been recognized as a quantitative indicator of ion homeostasis that reflects tissue viability.^1-3 Quantitative sodium MRI techniques have been developed and used to measure TSC in the human brain non-invasively, successfully detecting tumor responses to therapy^4,5 and neural cell death in Alzheimer’s disease⁶.
However, clinical applications of sodium MRI remain impeded by several longstanding technical challenges, including low sensitivity, fast biexponential T₂ relaxation, long data acquisition time and low spatial resolution.^3,7 Significant advances have been made in the past decades to address these challenges, from hardware development^8,9 to novel pulse sequences^10-15. However, the capabilities of sodium MRI at 3T are still rather limited, with nominal spatial resolution around $$$5\times5\times5$$$ mm³ and data acquisition time on the order of 10 minutes.¹³
Constrained reconstruction methods, such as compressed sensing^16,17 and anatomically constrained reconstruction^18,19, have recently emerged to enhance SNR and resolution and reduce artifacts in sodium MRI. Although compressed sensing has been used to remove or reduce noise-like artifacts, its practical performance has been limited due to low SNR and limited k-space coverage of the acquired data. Constrained reconstruction incorporating anatomical prior information from high-quality proton image can also enhance SNR and spatial resolution for image features evident in the companion proton image, but with a limited capability to reconstruct sodium-dependent novel features (e.g., lesions). In addition, anatomically constrained reconstruction is known to be sensitive to inter-scan subject motion, which affects their robustness needed for practical applications.
We have developed a new method for reconstruction of high-quality sodium images from noisy, limited k-space data in the presence of inter-scan subject motion. The new method has been tested using simulated and experimental data from human subjects, producing significantly better results than the existing methods. The high-resolution sodium concentration maps obtained using our new method were also shown to improve the detection of tumor responses to radiation therapy.

Methods

Our proposed method, illustrated in Fig. 1, synergistically integrates motion-compensated constrained reconstruction with compressed sensing for reconstruction of high-quality sodium images from noisy, limited k-space data. The key features of our method include: (a) use of anatomical prior information for denoising and resolution enhancement, (b) built-in capability for detection and correction of inter-scan subject motion, and (c) use of sparsity constraints for effective recovery of sodium-dependent novel features.
More specifically, our proposed method represents the desired sodium image $$$\rho(\boldsymbol{x})$$$ using the following model:
$$\rho(\boldsymbol{x})=\sum_{m=-M}^{M} \alpha_m \rho_{\mathrm{ref}}(\boldsymbol{x};\boldsymbol{\theta})e^{-i2\pi m\Delta \boldsymbol{k\cdot x}}+\rho_{\mathrm{s}}(\boldsymbol{x}),\\
\mathrm{subject~to~}\|\Phi(\rho_{\mathrm{s}}(\boldsymbol{x}))\|_1\le\varepsilon,$$
where $$$\rho_{\mathrm{ref}}(\boldsymbol{x};\boldsymbol{\theta})$$$ represents a motion-compensated reference, $$$M$$$ the order of the generalized series (GS) model²⁰, $$$\alpha_m$$$ the GS coefficients, $$$\rho_{\mathrm{s}}(\boldsymbol{x})$$$ an image component which is forced to be sparse under sparsifying transform $$$\Phi$$$. As compared with the conventional Fourier model, the proposed model enables the effective use of a priori information for denoising and resolution enhancement. As compared with the modern compressed sensing model, the proposed model enables better use of the measured data and the sparsity constraint for recovery of novel image features by modeling the “prior” and “novel” components separately.
To generate the reference $$$\rho_{\mathrm{ref}}(\boldsymbol{x};\boldsymbol{\theta})$$$, we first segment the proton image into different tissue compartments, then estimate the average sodium concentration of each tissue compartment
$$\{\hat{c}_r\}=\arg\min_{\{c_r\}}\left\|d(\boldsymbol{k}) - \Omega\mathcal{F}\left\{\sum_{r=1}^{R} c_rM_r(\boldsymbol{x})\right\}\right\|_2^2.$$
Then compartmental concentration variations are reconstructed using the GS model
$$\{\hat{\alpha}_m\}=\arg\min_{\{\alpha_m\}}\left\|d(\boldsymbol{k}) - \Omega\mathcal{F}\left\{\sum_{m=-M}^{M} \alpha_m\left(\sum_{r=1}^{R} \hat{c}_rM_r(\boldsymbol{x})\right)e^{i2\pi m\Delta \boldsymbol{k\cdot x}}\right\}\right\|_2^2.$$
To avoid potential image reconstruction artifacts due to inter-scan subject motion, our model also includes a built-in capability for motion estimation and correction.
We finally use sparsity constraint to recover the novel features by solving the following problem:
$$\hat{\rho}_{\mathrm{s}}(\boldsymbol{x})=\arg\min_{\rho_{\mathrm{s}}(\boldsymbol{x})}\|d(\boldsymbol{k})-\Omega\mathcal{F}\left\{\hat{\rho}_{\mathrm{ref}}(\boldsymbol{x};\boldsymbol{\theta})+\rho_{\mathrm{s}}(\boldsymbol{x})\right\}\|_2^2 + \lambda\|\Phi(\rho_{\mathrm{s}}(\boldsymbol{x}))\|_1.$$
Our method has a distinct advantage over the conventional sparsity-constrained reconstruction methods for solving the sodium imaging problem because of its ability to “remove” the “prior” component so that the sparsity constraint is applied to $$$\rho_{\mathrm{s}}(\boldsymbol{x})$$$ only. Since $$$\rho_{\mathrm{s}}(\boldsymbol{x})$$$ is much sparser than the overall image $$$\hat{\rho}_{\mathrm{ref}}(\boldsymbol{x};\boldsymbol{\theta})+\rho_{\mathrm{s}}(\boldsymbol{x})$$$, the sparsity constraint is more effective for recovery of the sodium-dependent novel features that are not present in the companion proton image.

Results

Our method has been validated using simulated and experimental data, producing high-SNR and high-resolution TSC maps. Figure 2 shows reduction in noise level and improvement in spatial resolution provided by our method in the presence of novel features in a Monte-Carlo simulation study. Figure 3 shows some representative high-quality reconstructions from phantoms and healthy subjects. Linear calibration curves with small variances from the phantom data and reproducible TSC values across different brain regions and different subjects were obtained. Figure 4 shows the capability of our method to resolve small anatomical structures and to reduce partial volume effects from surrounding CSF. Figure 5 shows high-resolution TSC changes in response to treatment in a tumor patient. The TSC decline detected by the proposed method indicated the return of infiltrated brain tissue to normal cell packing.

Conclusions

A novel method for high-resolution sodium imaging has been proposed to provide an effective solution to the low resolution and low SNR problems that have limited clinical applications of sodium imaging. Simulated and experimental results have successfully demonstrated the feasibility of achieving high-quality TSC maps using the proposed method. The reduction in partial volume effects was shown to improve the detection of tumor responses to radiation treatment.

Acknowledgements

The authors acknowledge Dr. Ian C. Atkinson and Dr. Aiming Lu for useful discussions about image reconstruction from twisted projection data and for handling p-files from GE 3T scanner. The work reported in this paper was supported, in part, by the National Institutes of Health (NIH); contract grant number: KRT RO1 CA1295531-01A1.

References

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Figures

Figure 1. Illustration of the proposed reconstruction method. Tissue-based structural information from high-quality proton images is incorporated into the reconstruction of normal image features using a motion-compensated GS model. Sodium-dependent novel features (e.g., lesions) are reconstructed from the residues using a sparse model.

Figure 2. Simulation results without and with novel features. As can be seen, the proposed method produced much better reconstructions in terms of SNR and resolution than the existing methods, which include Fourier reconstruction, conventional anatomically constrained reconstruction, and compressed sensing. Also note that the lesion was not well reconstructed using the existing methods but was well recovered by the proposed method.

Figure 3. Phantom and healthy subject results. (A) Phantom ¹H reference image, ²³Na FFT and proposed reconstruction, with corresponding calibration curves. Note that both curves are linear, but the FFT reconstruction had much larger standard deviation (5.3%) than the proposed method (2.8%). (B) Images from two healthy subjects. Note the marked improvement in image quality by the proposed method. (C) Regional TSC values obtained using the proposed method, which were consistent across different brain regions and subjects, demonstrating the reproducibility of the proposed method.

Figure 4. Healthy subject results focusing on hippocampal structures. The proposed method produced better delineation of CSF-tissue boundaries. The high hippocampal TSC values in the FFT reconstructions (45.2 mM) were due to partial volume effects from the surrounding CSF, which were significantly reduced by the proposed method (38.1 mM) to values closer to the rest of the brain as indicated by the 30 mM line on the TSC line profiles. This result matches our expectation that the proposed method should reduce the partial volume effects from CSF due to its high-resolution capability.

Figure 5. Experimental results across radiation therapy for a patient following incomplete resection with residual tumor infiltrating the posterior margin of the resection site in the right frontal lobe. Proton image at day 0 (start of radiation treatment) and TSC maps are shown with corresponding TSC line profiles at day 10 (red squares), 24 (blue circles) and 45 (green diamonds). The tumor regions (horizontal black lines) show systematic decreases in TSC across treatment time. The proposed method improves the resolution of the tumor from the CSF in the interhemispheric fissure.

Proc. Intl. Soc. Mag. Reson. Med. 29 (2021)

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