Sihui Luo

Nuclear magnetic resonance (NMR) spectroscopy presents an important analytical tool for composition analysis, molecular structure elucidation, and dynamic study in the fields of chemistry, biomedicine, food science, energy and more. As a basic function, exponential functions can be applied to model NMR signals of free induction decay, relaxation, and diffusion. In this paper, we will review Fourier and Laplace NMR exponential signals separately, as well as the performance of state-of-the-art machine learning on NMR applications.

As a non-invasive technology, nuclear magnetic resonance (NMR) spectroscopy can perform qualitative and quantitative analyses on chemical components, molecular structures, and even dynamic studies based on the detection of molecular level information of chemical shifts, 

 couplings, relaxation rates, and diffusion coefficients (1,2). Benefiting from the multitudinous information provided by NMR, it has been widely used in various fields such as chemistry, biomedicine, food science, and energy.

Exponentials are the basic functions for acquired signals in NMR spectroscopy. Signals can be divided into Fourier NMR signals (3) and Laplace NMR signals (4) according to different application scenarios. For Fourier NMR signals, the free induction decay (FID) signals contain the basic information of chemical shifts and 

couplings, which can be readily extracted and understood for elucidating chemical components and molecular structures after suitable spectrum reconstructions (5,6). For Laplace NMR signals, relaxation measurements in low-field NMR are an important analytical tool for recording molecular dynamic changes (7). Diffusion measurements are adopted to explore the transportation properties of molecules in solution (8).

Machine learning (ML) aims to derive a corresponding model by learning (modeling) known data to predict new objects (9). For NMR applications, conventional ML algorithms include principal component analysis (PCA) (10), singular value decomposition (SVD) (11), partial least squares (PLS) (12), support vector machine (SVM) (13) and compressed sensing (CS) (14). These algorithms work well on small group datasets. However, with the recent explosion of datasets, deep learning (DL) has become a hot topic in current ML, training a neural network with multiple layers and parameters to discover fruitful features (15,16). ML has extensively promoted the development of NMR spectroscopy.

In this paper, we will review how the state-of-the-art ML methods process exponential NMR signals. The structure of this article is as follows. First, the NMR theory is introduced. Then, the mathematical modeling of the Fourier and Laplace NMR signals are presented separately, and some examples for data processing by ML are previewed. Finally, the challenges and trends of ML in the NMR research are discussed.

Relaxation is widely used in physics to indicate the re-establishment of thermal equilibrium after some perturbation is applied (4). In NMR experiments, two main relaxation effects are usually considered: longitudinal relaxation and transverse relaxation.

As Figure 1 shows, the axis parallel to the magnetic field is called the 

-axis is called longitudinal magnetization. The magnetization in the 

 plane is called transverse magnetization. This radio frequency (RF) field corresponds to a 90° pulse. After the pulse, the spins resume their precessional motion. The precession frequency of the transverse magnetization is named Larmor frequency (17). The transverse magnetization induces a decaying current in the receive coil, which is called the FID signal (4).

FIGURE 1: The net magnetization vector. For a NMR spectrometer, an RF is applied perpendicular to the 

-axis. The spins experience two magnetic fields: a static field (B

) and RF field. When the frequency of the RF field is equal to the precession frequency of nuclei, the resonance is matched. This allows the RF field to rotate the magnetization from 

The Bloch equation (18) describes the relaxation process of the magnetization vector, and the solution of this equation can be written in the form of an exponential function (19). The process by which the longitudinal magnetization is returned to its equilibrium value is called longitudinal relaxation, or spin-lattice relaxation. The relaxation rate is proportional to the disequilibrium value of longitudinal magnetization, which can be expressed as

 represent the longitudinal magnetization and equilibrium magnetization, respectively, while 

 is the longitudinal relaxation rate constant. Equation 1 can be integrated to get:

Similar to longitudinal relaxation, the process by which transverse magnetization decays away to its equilibrium value of zero is called transverse relaxation, or dipole-dipole relaxation. The expression of transverse relaxation rate constant 

 represents the transverse magnetization. Equation 3 may be integrated to get:

Therefore, the exponential modeling fits the relaxation effect of NMR signals.

Diffusion ordered spectroscopy (DOSY) is an effective tool for identifying chemical substances in mixtures and detecting intermolecular interactions by differentiating NMR signals of a compound mixture according to their diffusion coefficients (20,21).

In solution NMR experiments, the information about translational motion can be encoded in NMR data sets through the use of pulsed field gradient NMR (PFG-NMR) experiments (22). Then, the diffusion spectra can be obtained by incrementing the areas of the gradient pulses (G) in PFG-NMR and transforming the NMR signals’ amplitudes with respect to 

 (23). According to the Stejskal-Tanner equation (22), the echo signal of single free diffusion components at 

 is the signal intensity without diffusion, 

 is the diffusion time between two gradient pulses and γ is the gyromagnetic ratio (24). The γ, G, δ, and Δ are expressed as (25):

Thus, in most research, one can see that the diffusion signal is expressed as:

After the initial labeling, the molecules are allowed to diffuse for a time 

, and then the initial positions are compared with the final positions. The bigger the difference between the two positions is, the higher the diffusion coefficient 

 is. It is worth noting that, in common experiments, 

) are measured, and the echo signal is acquired by changing 

FIGURE 2: Pulsed field gradient spin-echo sequence for diffusion measurements (8,25).

The general model of an exponential signal is

), it represents the function of parameter 

 represents the transverse relaxation attenuation performance suffering from magnetic field inhomogeneities, which is given as (26):

 is the relaxation time in the magnetic field inhomogeneity across a sample. The magnetic field inhomogeneity can be eliminated or alleviated by using a 180° pulse, such as spin-echo pulse sequence (26).

The basic signal modeling of the FID without parameter p is given as (3):

 represent phase and frequency, respectively, and 

 is the imaginary unit. By performing the Fourier transform on 

 relaxation signals can be obtained by a Carr-Pursell-Meiboon-Gill (CPMG) pulse sequence (27,28), or a 90° pulse followed by a series of 180° pulses (Figure 3). In the exponential signal model in Equation 8, if the parameter or kernel function 

CPMG pulse sequence will prolong the lifetime of NMR signals compared to FID signals when samples are placed in an inhomogeneous field. In this situation, spin echoes can be inverted with inverse Laplace transform (ILT) to obtain 

The typical 2D relaxation correlation signals include 

 signals can be measured with Inversion Recovery-CPMG pulse sequence (31) (IR-CPMG), which consists of a 180° pulse followed by a CPMG pulse sequence (Figure 4a). The parameter function 

 signals can be measured with Diffusion-Editing sequence (32) (Figure 4b). The parameter function for 

 signals can be measured with the pulse sequence that consists of two CPMG pulse sequences separated by a mixing time (Figure 4c) (33). During the mixing time, diffusion coupling is evolved. The parameter functions are:

FIGURE 4: Pulse sequences for 2D relaxation measurements (27,28,31–33). (a) 

In general, the 2D relaxation correlation signals, 

 are expressed as the Fredholm integral equation of the first kind:

) is the correlation distribution to solve. The matrix form of 2D relaxation correlation signals is (11,34):

, the 1D representation of equation 18 is (11,34):

 are vectors obtained by lexicographically ordering matrices 

 relaxation time reflects the pore size distribution. The larger the pore size is, the longer the relaxation time 

 distributions (Figure 5) combined with conventional ML methods has been proven to be effective in evaluating water content, porosity and permeability (37).

FIGURE 5: The inversion Laplace transform of 

PCA is a common supervised analysis tool to interpret the internal characteristics among numerous variables (37,38). The PCA plots provide a comprehensive overview of relaxation data and variables (Figure 6). However, some multi-components have very close relaxation properties, leading to overlapping in the PCA plot.

FIGURE 6: Illustration of a PCA score plot.

PLS regression is another unsupervised analysis method to build multivariate regression models (12). Model performance is evaluated by the determination coefficient and root mean square error (12). This method was used to determine predicted values (dependent variables) based on transverse relaxation data (independent variables) by linear regression (39–41).

 distributions, the valuable information in 

 decay curves can also be extracted, like what SVM does (13). The SVM method projects data in its original dimension into a higher dimension to find the best hyperplane (Figure 7). If taking SVM is the only means for decay curves analysis, model classification will be affected by the same relaxation features, leading to relatively long relaxation times and biased samples (13).

FIGURE 7: Illustration of a linear SVM model.

The above algorithms can be analyzed by software, such as Pirouette (40), Unscrambler (41), and SPSS (12).

As an emerging research technology of ML, DL has received widespread attention in NMR. Convolutional neural networks (CNN) is a basic structure of DL (16), which aims at automatic feature extraction and classification. For example, both original 

 distributions were involved in the analysis with CNN (42). Image and signal amplitude are the inputs of 2D CNN and 1D CNN (Figure 8), respectively. In the data set, 468 samples were divided into training, test, and validation sets according to the ratio 7:1:1. The model performance of 2D CNN was affected by the same relaxometry mechanisms and spatial position of the data, with an accuracy of 92.31%. Notably, the 1D CNN for original decay curves obtained a remarkable classification accuracy of 100% since this neural network avoids inversion instability.

FIGURE 8: A classification model based on CNN structure. The inputs of CNN are 

 distribution, respectively. The input of 1D CNN is signal amplitude, and the input of 2D CNN is image (42).

DL is also a great way to improve inversion stability. The aforementioned 

 distributions are obtained by software (43,44), but the ILT is an ill-posed problem (45). It is a great challenge to properly obtain 

 distributions which best represent the relaxation mechanisms of samples. In 2021, Parasram and associates (46) proposed a fully connected neural network (Figure 9) for 

 analysis. The input data is simulated by 1D relaxation signals, and the output is the 

 distribution. By comparing with non-linear least squares and Frobenius norm (11) methods, 

 values obtained by Parasram are more accurate. However, this method is unsuitable for narrow peaks and low-intensity peaks.

FIGURE 9: The structure of an artificial neural network with full connections (46).

The measured data of 2D relaxation correlation signals consists of hundreds of thousands of echos (47), which slow down acquisition and consume a lot of computational memory. Therefore, it is necessary to compress echos before the inversion process (Figure 10). Many methods like window averaging (48), SVD (34) and PCA (10) were proposed.

FIGURE 10: The processing scheme of 2D relaxation correlation data (47,49).

The window averaging method (48) divides echo data into different window lengths to compress, but its compression ratio is low.

The SVD method projects the kernel matrix 

 into a lower-dimensional submatrix (11). The kernel matrix 

 is a diagonal matrix made up of singular values in descending order. The original kernel matrix is compressed by choosing the front singular values. This method has a high compression ratio but takes a long computing time, especially for multidimensional experiments (50).

The PCA calculates the covariance matrix 

 is the zero mean normalization result of kernel matrix 

 (10). Then, the eigenvalues ξ and eigenvectors λ of 

Third, the eigenvectors that correspond to the front 

 eigenvalues are chosen to form a transform matrix:

and then apply the transformation matrix to 

 at the same time to achieve compression.

Unlike SVD, which directly decomposes the kernel matrix 

, fast SVD (49) creates a submatrix based on a Hadamard matrix, then applies SVD on the submatrix, which largely decreases sampling time compared to SVD. Random SVD (50) seeks two lower-dimensional submatrices by random sampling to reduce computing time and memory.

After inversion, the relaxation correlation distributions need interpretation. Different types of relaxation distributions have their own parameters for analysis. In 

 ratio is an important value to identify fluid components and determine the properties of complex fluids, like viscosity (51). However, when 

 mapping is an alternative way to tell the difference by exploring diffusion properties (52). In 

 maps, symmetric nature provides diffusion coupling of porous media, which detects water exchange between different pores (53–55). The number of spectral peaks on the main diagonal represents the number of relaxation components, and the exchange between relaxation components is represented by a pair of off-diagonal peaks.

Gaussian mixture models (51) for relaxation correlation distributions exhibit a good performance for fluid characterization according to the 

 ratio range and angle difference. This method overcomes the subjectivity of manual labeling. Peng et al (56) used two ML models for analysis (Figure 11). The unsupervised model reduces the dimensions of the relaxation joint data to form interpretable data for user-friendly quantitative analysis. The supervised models are used to compare the analysis efficiency of machines and humans, showing that the machine model only needs 30 sec to complete the binary decision.

FIGURE 11: ML for decisive making. Adapted from reference (56).

As the resolution and sampling dimensions of the NMR spectrum increase, the data acquisition time increases rapidly (16). For example, when comparing the sampling time of a 2D spectrum and 4D spectrum, it rises from several minutes to dozens of days (57). The excessive sampling time can hinder the application of NMR spectroscopy.

To reduce data acquisition time, nonuniform sampling (NUS) is applied to obtain partial data (58). However, NUS will introduce spectrum artifacts, and thus, reconstructing high-fidelity spectra is very important.

In 2006, Donoho (59) and Candès and associates (60) proposed the CS, which requires the signal be sparse under a certain transform domain, incoherent sampling, and a nonlinear optimization algorithm to solve the sparse recovery problem.

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A review of exponential signal models with machine learning in nuclear 
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Sihui Luo

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Review and Prospect: Applications of Exponential Signals with Machine Learning in Nuclear Magnetic Resonance