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Reconstructing High-Quality Sodium MR Images from Limited Noisy k-Space Data with Model-Assisted Deep Learning1Beckman Institute for Advanced Science and Technology, University of Illinois, Urbana-Champaign, Urbana, IL, United States, 2Department of Electrical and Computer Engineering, University of Illinois, Urbana-Champaign, Urbana, IL, United States, 3Center for Magnetic Resonance Research, University of Illinois at Chicago, Chicago, IL, United States
Synopsis
Sodium MRI can acquire important biological information about cell integrity and tissue viability, but its clinical application has been limited by low SNR and poor spatial resolution. We propose a novel method to reconstruct high-quality sodium images from limited and noisy k-space data. The new method synergistically integrates model-based reconstruction with deep learning. Simulation and experimental results show that the proposed method can reconstruct high-SNR and high-resolution sodium images, which clearly delineate lesions such as brain tumors.
Introduction
Sodium MRI, complementary to proton MRI, has long been recognized as a powerful tool for noninvasive measurement of tissue sodium concentration (TSC), which is a direct, quantitative bioscale for cell integrity and tissue viability1-6. However, its clinical applications remain limited by low sensitivity due to its low gyromagnetic ratio and biological concentrations. With state-of-the-art data acquisition technology, sodium MRI scans at 3T typically acquire 44×44×44 encodings in 10 minutes with marginal SNR7-9. Larger k-space coverage (e.g., 64×64×64 encodings) can be achieved but at the expense of longer scan time and further reduced SNR10-14. The low spatial resolution resulting from limited k-space coverage and fast transverse relaxation leads to large partial volume effects that are especially problematic when the region of interest is close to CSF spaces.Constrained image reconstruction has shown great potential for overcoming the low SNR and poor resolution problems with sodium MRI. Conventional constrained reconstruction methods exploit the anatomical information from a companion proton scan15,16 or transformed sparsity in sodium images17-20 to enhance the SNR and spatial resolution. However, these methods have several known limitations: (a) minimal improvement in sodium-dependent novel features (e.g., lesion), (b) potential reconstruction error due to inaccurate anatomical boundaries, and (c) ineffectiveness of the sparsity constraint for low-SNR data. Deep learning (DL) methods have recently been proposed for reconstructing sodium images from sparsely sampled data21 or noisy data22. These methods achieved encouraging results, but they also require large amounts of experimental data with ground truth for training and may suffer from the well-known instability problem23,24.
To address these problems, we propose a model-assisted DL method to reconstruct high-quality sodium images from limited noisy k-space data. The new method has been validated using simulation and experimental data, producing very encouraging results.
Methods
The proposed method decomposes the desired sodium image $$$\rho(\boldsymbol{x})$$$ as:$$\rho(\boldsymbol{x})=\sum_{m=-M}^{M}\alpha_{m}\rho_{\mathrm{ref}}(\boldsymbol{x})e^{-i2\pi m\Delta \boldsymbol{k}\cdot \boldsymbol{x}}+\rho_{\mathrm{s}}(\boldsymbol{x}),$$
where $$$\rho_{\mathrm{ref}}(\boldsymbol{x})$$$ represents a reference image for the generalized series (GS) model25-28 used to incorporate anatomical constraints, and $$$\rho_{\mathrm{s}}(\boldsymbol{x})$$$ a sparse component used to capture novel features. This image model was introduced in a recent work29, in which $$$\rho_{\mathrm{ref}}(\boldsymbol{x})$$$ was obtained using a tissue-based compartmental model and $$$\rho_{\mathrm{s}}(\boldsymbol{x})$$$ was obtained using a conventional L1-regularized reconstruction.
In this work, we further enhance the capability and robustness of this model by integrating the GS model with a DL network, as illustrated in Fig. 1. The proposed method has the following key features: (a) utilizing an unrolled neural network to learn a sparsity-promoting regularization to compensate the GS model, (b) stabilizing the DL module with the output of the GS model as a conditional prior, and (c) updating the basis functions of the GS model progressively using the output of the DL module. Because the DL module is used to compensate the GS model, it can be trained using simulated data. This feature is very desirable since it eliminates the need of large amounts of experimental data with ground truth as in the case of most DL methods.
More specifically, in the first iteration, the GS component $$$\hat{\rho}_{\mathrm{gs}}^{(1)}(\boldsymbol{x})$$$ was estimated using a tissue-based compartmental model29. Then a network that unrolled the alternating direction method of multipliers (ADMM) algorithm was used30-32, as described in Fig. 2. Mathematically, it is equivalent to finding an optimal regularization functional $$$R(\cdot)$$$ for $$$\rho_{\mathrm{s}}(\boldsymbol{x})$$$ to compensate the GS model the best:
$$\hat{\rho}_{\mathrm{s}}^{(1)}(\boldsymbol{x})=\arg\min_{\rho_{\mathrm{s}}(\boldsymbol{x})}\left\|d(\boldsymbol{k})-\Omega\mathcal{F}\left(\hat{\rho}_{\mathrm{gs}}^{(1)}(\boldsymbol{x})+\rho_{\mathrm{s}}(\boldsymbol{x})\right)\right\|_2^2+R\left(\rho_{\mathrm{s}}(\boldsymbol{x});\hat{\rho}_{\mathrm{gs}}^{(1)}(\boldsymbol{x})\right).$$
The DL network was trained using synthetic data generated from 160 in vivo datasets with more than 16,000 2D images. After training, the network effectively reconstructed sodium-dependent novel features and compensated any boundary mismatch artifacts that the GS model failed to capture. It is worth noting that both $$$\rho_{\mathrm{s}}(\boldsymbol{x})$$$ and $$$\hat{\rho}_{\mathrm{gs}}^{(1)}(\boldsymbol{x})$$$ were included as the network input so that $$$\hat{\rho}_{\mathrm{gs}}^{(1)}(\boldsymbol{x})$$$ can be effectively used as a conditioner to bring additional prior information to the DL network. The additional information helped stabilize the network for better recovery of $$$\rho_{\mathrm{s}}(\boldsymbol{x})$$$.
The network, in turn, helped improve the GS model basis functions by updating the reference with $$$\rho_{\mathrm{ref}}^{(2)}(\boldsymbol{x})=\hat{\rho}_{\mathrm{gs}}^{(1)}(\boldsymbol{x})+\hat{\rho}_{\mathrm{s}}^{(1)}(\boldsymbol{x})$$$. With the updated reference (shown in Fig. 3), the GS model provided better reconstruction results. The progressively improved GS reconstructions made $$$\rho_{\mathrm{s}}(\boldsymbol{x})$$$ sparser and easier to recover from limited and noisy data. The alternative update of the GS model and neural network was performed several times until convergence.
Results
Our method has been validated using simulated and experimental data, producing high-SNR and high-resolution sodium images. Figure 4 shows simulation results using Fourier reconstruction, anatomically constrained reconstruction33, end-to-end neural network21 and our proposed method compared with the ground truth. As can be seen, the proposed method showed significant improvement of SNR and spatial resolution in both anatomical features and the lesion, which matched well with the ground truth. Figure 5 shows TSC maps of brain tumor patients. Again, the proposed method produced high-quality reconstructions with reduced noise-introduced variations and smaller partial volume effects from CSF, leading to much more clear delineation of the lesion.Conclusions
A novel model-assisted DL method has been proposed to enable robust reconstruction of high-quality sodium images from limited noisy k-space data. The proposed method provides an effective solution to the low-SNR and low-resolution problems associated with sodium MRI, which may significantly enhance its clinical use at 3T.Acknowledgements
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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