Accelerated Magnetic Resonance Spectroscopy with Model-inspired Deep Learning
Zi Wang1, Yihui Huang1, Zhangren Tu2, Di Guo2, Vladislav Orekhov3, and Xiaobo Qu1
1Department of Electronic Science, National Institute for Data Science in Health and Medicine, Xiamen University, Xiamen, China, 2School of Computer and Information Engineering, Xiamen University of Technology, Xiamen, China, 3Department of Chemistry and Molecular Biology, University of Gothenburg, Gothenburg, Sweden


Multi-dimensional nuclear magnetic resonance (NMR) spectroscopy is an invaluable biophysical tool but often suffers from long measurement. Several methods have been established for spectra reconstruction from undersampled data, two of which are model-based optimization and data-driven deep learning. Combining the main merits of them, we present a model-inspired flexible deep learning framework, for reliable, robust, and ultra-fast spectra reconstruction. Besides, we demonstrate that the model-inspired network needs very few parameters and is not sensitive to training datasets, which greatly reduces the demand for memory footprints and can work effectively in a wide range of scenarios without re-training.


NMR spectroscopy serves as an indispensable biophysical tool in modern chemistry and life science. Since the duration of NMR experiments increases rapidly with dimensionality, the non-uniform sampling (NUS) approach1, 2 is commonly used for accelerating the measurement. Over the past two decades, many remarkable methods have been established in the NMR field to reconstruct high-quality spectra from NUS data. Two main strategies of them are model-based optimization1-14 and data-driven deep learning15-19. The former has the explicit structure with insights from NMR spectroscopy but needs lengthy computational time, while the latter greatly reduces the reconstruction time but is trained as a black-box17 and is almost over-parameterized. In this work, we demonstrate the effectiveness of merging optimization and deep learning. The proposed method enables 10 times faster spectra reconstruction than optimization methods, and has 10 times fewer parameters than the existing deep learning method. Besides, it is not sensitive to the training dataset and can work effectively in a wide range of scenarios without re-training.


The proposed Model-inspired Deep learning framework, called MoDern, its design of architecture starts from the sparse prior and adopts the main idea from the compressed sensing (CS) algorithm: iterative soft-thresholding (IST)6, 19. Let $$$\bf r$$$ be the complete NMR time-domain signal, and the forward Fourier transform $$$\bf F$$$ converts it into a NMR spectrum $$$\bf x=Fr$$$ . In NMR spectroscopy, the CS model states that the sparsest solution can be always found by the $$$l\scriptsize 1$$$ norm optimization6, 7, 19. This task can be efficiently solved by the IST algorithm, and its $$$k^{th} (k=1,...,K)$$$ iteration process can be written as follows6, 19:$${({\bf Data\ Consistency}):\ {\bf d}_{k}={\bf x}_{k-1}+{\bf FU}^{T}({\bf y}-{\bf UF}^{H}{\bf x}_{k-1}),\ (1a)}$$$${({\bf Soft-thresholding}):\ {\bf x}_{k}=S({\bf d}_{k},\theta),\ (1b)}$$where $$$\bf y$$$ is the FID signal undersampled by operator $$$\bf U$$$, $$${\bf F}^{H}$$$ is the inverse Fourier transform, $$$\bf d$$$ is the spectrum after data consistency, the superscript $$$T$$$ is the transpose operator, $$$\theta$$$ is the threshold, and $$$S(\cdot,\theta)=sgn(\cdot)\times max(0,\mid \cdot\mid-\theta)$$$ is the soft-thresholding operator. Initialized with $$${\bf x}_{0}={\bf FU}^{T}{\bf y}$$$ , CS reconstructs the spectrum by alternating the data consistency and soft-thresholding.
Once the overall number of iterations is fixed, the data flow can be viewed as an unfolded deep learning network, as shown in Figure 1. Same to Eq. (1a), the spectrum is forced to maintain the data consistency to the sampled signal. Since the proper choice of thresholds is still of great demand and challenge, thus, instead of Eq. (1b), we use a learnable network $$$LS$$$ which can change thresholds with the characteristics of the input data, for adaptive soft-thresholding. The single network $$$LS$$$ is composed of convolutional layers, fully-connected layers, and a soft-thresholding. The overall number of iterations in our implementation is 10. With the increase of iterations, artifacts are gradually removed, and finally a high-quality reconstructed spectrum can be obtained.
Given a proof-of-concept of training neural networks using solely synthetic data with the exponential functions has been presented in paper15, we also employ this scheme to train our network to learn the best internal parameters and an optimal mapping $$$f$$$ by minimizing the mean square error between outputs of the learnable adaptive soft-thresholding and fully sampled spectra. For a well-trained network, the spectrum can be reconstructed reliably and fastly from an undersampled signal via $$$f$$$ .


To demonstrate the reliability of the proposed MoDern on experimental data, we reconstruct several spectra under NUS. Herein, we show two of them: 2D HSQC spectrum from cytosolic CD79b and 3D HNCACB of GB1-HttNTQ7 protein. Pearson correlation coefficient R2 is calculated as a measure of the peak intensity correlations between the fully sampled spectra and the reconstructed spectra.
The reconstruction of 2D spectra in Figure 2 shows that, (a) MoDern can faithfully reconstruct it using 20% NUS data, and its peak intensity correlation reached 0.9998 with high fidelity of the lineshape reconstruction. (b) MoDern is comparable with, or may even surpass the state-of-the-art reconstruction method DLNMR15 in spectra quality, while being robustness and can maintain excellent performance at low NUS densities. For the reconstruction of 3D spectra in Figure 3 shows that, similar to DLNMR, MoDern also has great potential in fastly high-quality reconstruction, the peak intensity correlation can reach 0.99 even at 20 times acceleration.
The most important advantage is that, MoDern abandons a large number of redundant convolution layers, which is often used in the data-driven deep learning15, 16, to achieve the dramatic reduction of trainable parameters and computational complexity. Figure 4 shows that, (a) The parameters of MoDern is ca. 9% of that needed for DLNMR, resulting in a significant reduction in network training time and reconstruction time without loss of spectral reconstruction quality. (b) MoDern is very flexible and robust, which means it is not sensitive to spectra sizes, types, and NUS densities. So that, re-training of MoDern is not necessary and it may alleviate the common “mismatch” problem in practice.


In summary, we propose a model-inspired deep learning framework, called MoDern, as a reliable, robust, and ultra-fast technique for obtaining high-quality spectra from NUS data. This work is a powerful demonstration of the effectiveness of merging optimization and deep learning in biological NMR.


This work was supported in part by the National Natural Science Foundation of China (NSFC) under grants 61971361, 61871341, and U1632274, the Joint NSFC-Swedish Foundation for International Cooperation in Research and Higher Education (STINT) under grant 61811530021, the National Key R&D Program of China under grant 2017YFC0108703, the Natural Science Foundation of Fujian Province of China under grant 2018J06018, the Fundamental Research Funds for the Central Universities under grant 20720180056, the Xiamen University Nanqiang Outstanding Talents Program, the Science and Technology Program of Xiamen under grant 3502Z20183053, the Swedish Research Council under grant 2015–04614, and the Swedish Foundation for Strategic Research under grant ITM17-0218.

The correspondence should be sent to Prof. Xiaobo Qu (Email: quxiaobo@xmu.edu.cn)


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Figure 1. MoDern: The proposed Model-inspired Deep Learning framework for NMR spectra reconstruction. The recursive MoDern framework that alternates between the data consistency (DC), which is same to Eq. (1a), and the learnable adaptive soft-thresholding (LS) inspired by Eq. (1b). With the increase of iterations, artifacts are gradually removed, and finally a high-quality reconstructed spectrum can be obtained. Note: “FT” is the Fourier transform. A data consistency followed by a learnable adaptive soft-thresholding constitutes an iteration.

Figure 2. 2D 1H-15N HSQC spectrum of the cytosolic domain of CD79b protein. (a) The fully sampled spectrum. (b)-(c) are reconstructed spectra using DLNMR and MoDern from 20% data, respectively. (d) is zoomed out 1D 15N traces. (e) is the peak intensity correlation obtained with two methods under different NUS levels. The insets of (b)-(c) show the corresponding peak intensity correlation between fully sampled spectrum and reconstructed spectrum. Note: The average and standard deviations of correlations in (e) are computed over 100 NUS trials.

Figure 3. The sub-region of the projections on 1H-15N and 1H-13C planes of the 3D 1H-15N HNCACB spectrum of GB1-HttNTQ7. (a) and (d) are projections of the fully sampled spectrum. (b) and (e) are projections of the DLNMR reconstructed spectrum. (c) and (f) are projections of the MoDern reconstructed spectrum. (g) and (h) are the peak intensity correlations obtained by DLNMR and MoDern, respectively. Note: 5% NUS data were acquired for reconstruction. The contours of all projections are at the same level.

Figure 4. Comparison of DLNMR and MoDern on the number of trainable parameters and the sensitivity to training datasets. (a) is the number of trainable parameters of two networks. (b) Peak intensity correlations of DLNMR and MoDern trained using 15% NUS density dataset respectively, to reconstruct 2D 1H-15N HSQC spectrum of the cytosolic domain of CD79b protein, under a series of NUS densities ranging from 10% to 25%. Note: The peak intensity correlations of DLNMR (orange line) is shifted horizontally for clear display but the values unchanged.

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