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Sci Rep
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Scientific Reports
2045-2322
Nature Publishing Group UK London

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37731
10.1038/s41598-023-37731-8
Article
A Monte Carlo algorithm to improve the measurement efficiency of low-field nuclear magnetic resonance
Guo Pan guopan@cqnu.edu.cn

1
Zhang Ruoshuang 1
Zhang Jiawen 1
Shi Junhao 1
Li Bing 2
1 grid.411575.3 0000 0001 0345 927X College of Physics and Electronic Engineering, Chongqing Normal University, Chongqing, China
2 grid.433158.8 0000 0000 8891 7315 Urumqi Power Supply Company, State Grid Xinjiang Electric Power Co, LTD, Urumqi, China
29 6 2023
29 6 2023
2023
13 1053325 1 2023
27 6 2023
© The Author(s) 2023
https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Nuclear magnetic resonance (NMR) has shown good applications in engineering fields such as well logging and rubber material ageing assessment. However, due to the low magnetic field strength of NMR sensors and the complex working conditions of engineering sites, the signal-to-noise ratio (SNR) of NMR signals is low, and it is usually necessary to increase the number of repeated measurements to improve the SNR, which means a longer measurement time. Therefore, it is especially important to set the measurement parameters appropriately for onsite NMR. In this paper, we propose a stochastic simulation using Monte Carlo methods to predict the measurement curves of T1 and D0 and correct the measurement parameters of the next step according to the previous measurement results. The method can update the measurement parameters in real time and perform automatic measurements. At the same time, this method greatly reduces the measurement time. The experimental results show that the method is suitable for the measurement of the self-diffusion coefficient D0 and longitudinal relaxation time T1, which are frequently used in NMR measurements.

Subject terms

Energy science and technology
Engineering
Physics
http://dx.doi.org/10.13039/100014717 National Outstanding Youth Science Fund Project of National Natural Science Foundation of China No. 51707028 Guo Pan http://dx.doi.org/10.13039/501100005230 Natural Science Foundation of Chongqing No. cstc2021jcyj-msxmX0470 Guo Pan Science and Technology Funds of Chongqing Municipal Education CommissionKJQN202100533 Guo Pan Innovation and entrepreneurship training program for college studentsS202210637053 Guo Pan issue-copyright-statement© Springer Nature Limited 2023
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pmcIntroduction

Nuclear magnetic resonance (NMR) can be used to explore the structure and properties of substances from the microscopic level through non-destructive methods and is now widely used in the fields of medicine, chemistry, materials, biology, petroleum, geology1, etc. The application of portable NMR sensors and unilateral magnetic resonance devices has attracted the attention of many researchers2–5. However, the low magnetic field strength and uniformity and complex working conditions result in a low signal-to-noise ratio (SNR), long measurement time and low measurement efficiency. These are currently urgent problems that need to be solved. A more effective way is to optimize the measurement process and set the measurement parameters appropriately to improve the efficiency.

There are some reports on the parameter optimization algorithm in NMR measurements. Some scholars have noticed the influence of the magnetic resonance measurement parameters on the measurement results, but their research mainly focuses on the frequency domain6–9, that is, exploring the data acquisition method that can reduce the measurement time and ensure that the frequency domain inversion spectrum has sufficient precision10–14. In 2017, Xing D. et al.15 presented an adaptive method for determining an acquisition parameter t0 in a modified CPMG sequence for measuring the internal magnetic field gradient distribution of samples. This method can reduce the difficulties of operating T2-G experiments. In 2018, A. Reci16 proposed an optimization method for the measurement parameters of the NMR liquid self-diffusion coefficient based on the Cramér-Rao lower bound (CRLB) theory, which assumes that the self-diffusion coefficient of the sample satisfies the log-normal distribution. In 2019, A. Reci 17 further improved the sampling method based on CRLB theory, making it also applicable to signal acquisition with double exponential decay. In 2021, Guest et al. studied the relationship between the experimental parameters and SNR for diffusion coefficient measurement18.

In recent years, NMR has been widely used in industrial automation19,20, but few studies have provided a more general time-domain sampling strategy for T1 and D0 measurement experiments in on-site NMR. At the same time, some users of NMR instruments may not have background knowledge of the NMR principles, so it is difficult for them to adjust the measurement parameters. Therefore, automatic and intelligent measurement algorithms are wanted by users. In this paper, we introduce a Monte Carlo algorithm-based intelligent search method for NMR measurement parameters. This method can automatically set the key parameters in the T1 and D0 measurements. The experimental results demonstrate that the method could achieve a 3–4 times acceleration effect in the T1 and D0 measurement experiments, and the systematic error between the results and the accurate value was less than 5%.

Algorithm introduction

Introduction to the Monte Carlo method

The Monte Carlo method, also known as the statistical simulation method, is a numerical calculation method guided by probabilistic statistical theory21–29. The characteristic of this method is the random sampling of the measurement variable, and an approximate result is calculated in real time for the NMR measurement constrained with the available dataset. In this paper, an intelligent search method for NMR measurement parameters based on the Monte Carlo algorithm is proposed to achieve the measurements of T1 and D0.

Measurement sequence

Figure 1 shows the static gradient spin echo (SGSE) sequence 1 for D0 measurement in low-field NMR when a static gradient magnetic field is present30. where the time interval between the 90° and 180° pulses (Td) is variable. When measuring the diffusion properties, Td is increased at each Td point, and the spin echoes are recorded. Therefore, the diffusion decay curve of a liquid molecule under a gradient magnetic field can be fitted. To improve the SNR, the sequence shown in Fig. 1 needs to be repeated several times and superimposed, and it is therefore quite time-consuming.Figure 1 SESG-CPMG measurement of D0 sequences and their parameters.

Currently, the common way to increase the Td value is uniform stepping, if N signals need to be measured to fit the curve of D0. Td is set at a uniformly increasing step rate.

The diffusion curve is estimated by using these measured values, which consumes considerable time. At the same time, some users of NMR instruments may not have background knowledge of the NMR principles, so it is difficult for them to adjust the measurement parameters. Therefore, automatic and intelligent measurement algorithms are wanted by users.

The inversion recovery (IR) sequence used to measure T1 is shown in Fig. 2. Similar to the SGSE sequence and variable Td, Ti is varied to measure the T1 recovery of the sample.Figure 2 IR-CPMG measurement of T1 sequences and their parameters.

Method

In T1 and D0 measurements, adjusting Td and Ti during the measurement needs to be performed. In our method, many diffusion curves and T1 decay curves are randomly generated by Monte Carlo simulations, and then the unreasonable curves are excluded according to the data already measured, and the remaining curves can be used to estimate the measurement parameters for the next step. The detailed steps of the algorithm are as follows.

First, the initial range of B and D0 are determined in Eq. (1), and the two-dimensional space determined by the initial ranges of these two parameters is called the initial parameter space. The initial parameter space is set to B ∈ [0, 2], D0∈[10–6 mm2/s, 10–2 mm2/s]. The D0 range can cover most of the self-diffusion coefficients of the tested samples. The variation range is [2 ms, 180 ms], which can reflect the attenuation curve of all samples in the above D0 range. A Monte Carlo sampling method is used to randomly select a specified number of data points in this initial parameter space and thus draw a cluster of t-f(t) curves to calculate the uncertainty of the model at this point (expressed as the product of the variance of B and D0 in this space). f(t) is the diffusion decay curve, and t is the abscissa of the measured data points. The most divergent time point t1 in this curve cluster is selected as the first sampling point, and the actual corresponding measurement value f(t1) is obtained by measurement. From this measurement dataset, (t1, f(t1)) constitutes the first data point of the dataset, and the initial parameter space of B and D0 is constrained under the constraints of this data point (i.e. only the parameter space that matches the (t1, f(t1)) data point is retained), which in turn yields the B and D0 parameter subspaces. Similarly, the t-f(t) curve cluster is plotted in this parameter subspace, and the model uncertainty is calculated, while the point t2, which makes the curve cluster most divergent, is still selected as the next sampled data point, and the measurement is performed. At this time, the data space is expanded to ((t1, f(t1), (t2, f(t2))), and the parameter space of B and D0 will be further reduced under the constraint of these two measurement data points. The above steps are repeated until the model uncertainty no longer significantly decreases; then, the parameter space of B and D0 under the data space constraint consisting of all measured data will be close to the true B and D0 values.

The diffusion decay curve ft satisfies the following form30:1 ft=B∗e-tD03,

This equation is correct for the constant gradient case, where the parameter vector A =D0,B is to be determined. t is the abscissa of the measured data points. Considering the form of Eq. (1), if there is no measurement error, B should be equal to the peak of the measured echo, and the maximum Maxy˙i, y˙i is the real measurement data. Considering the measurement error, the parameter search range can be set to Maxy˙i±5σ; and the D0 of common samples is between 0 and 10-8cm2·s-1. Therefore, D0 and B are set to obey the uniform distribution in their respective intervals:2 D0∼U0,10-8B∼UMaxy˙i-5σ,Maxy˙i+5σ,

Multiple random sampling according to the above distribution is performed.3 ∑i=1Ny˙i-B∗e-tD032<Nσ2,

When there are sufficient curves that satisfy the condition of Eq. (3), DYt can be calculated, and then, it can be substituted into t˙i+1=t|DYt=MaxDYt to obtain t˙i+1. Td is set to ti+1 to obtain the new measured value of t˙i+1, and it is fit by least squares to obtain a^i+1. The iterations are stopped when |a^0i-a^0i-1|<ε is satisfied. The measurement of T1 was performed similarly.

The entire algorithm flow is shown in Fig. 3.Figure 3 Algorithm flow chart.

Experimental verifications

The equipment used in the experiment was a spectrometer (KEA2, Magritek Inc. New Zealand), RF power amplifier (BT00500 ALPHA-SA, Tomoco Inc., Australia) and unilateral nuclear magnetic resonance sensor (130 mT). To verify the effectiveness of the above algorithm, the diffusion coefficient of pure water was measured at points taken with the algorithms D0 and T1. Moreover, ti obtained by the algorithm is inputted into the programmable spectrometer, y˙i is obtained from the actual measurement, and then, y˙i is returned to the algorithm to calculate ti+1. This process is shown in Fig. 4.Figure 4 Measurement flow chart.

The upper limit of error ε=10-10 is set, and the algorithm stops iterating after 12 points are measured. With the increase in measurement data, the prediction dataset continues decreasing, giving the prediction dataset when the known data are 1, 5, 8, and 12, respectively, as shown in Fig. 5, and the diffusion curve plotted by the actual measurement data as shown in Fig. 6.Figure 5 D0 Convergence of the predicted dataset in the measurement with increasing measurement data.

Figure 6 Pure water measured diffusion curve.

The same algorithm is used to measure T1 of the pure water, the upper error limit is set to ε=10-10, and the algorithm stops iterating after measuring 13 points. The predicted dataset with the known number of Data Points 1, 5, 8, and 13 is shown in Fig. 7, and the actual measured T1 curve is shown in Fig. 8.Figure 7 Convergence of the prediction dataset with an increasing measurement data in the T1 measurement.

Figure 8 Pure water actual measurement T1.

Table 1 shows the data of D0 and T1 obtained by the Monte Carlo intelligent algorithm and manual measurements using the same experimental equipment with pure water and glycerol as samples at 20 °C room temperature, respectively.Table 1 D0 and T1 of deionized water and glycerine by the Monte Carlo method and manual, where M-C replaced Monte Carlo, and points represents the measurement points in the D0 and T1 experiments.

	D0 (mm2/s)	Points	T1 (ms)	Points	
M-C	Manual	M-C	Manual	M-C	Manual	M-C	Manual	
Water	2.49 × 10–3	2.35 × 10–3	4	15	2653	2650	4	15	
Glycerine	0.124 × 10–3	0.12 × 10–3	3	15	227	230	3	15	

To evaluate the accuracy and stability of the Monte Carlo optimization-based algorithm in the D0 and T1 measurement experiments, 10 repeated D0 and T1 measurements were performed on pure water and glycerol samples, respectively. The deviation of the mean value of the 10 replicate measurements relative to the manual measurements was used to measure the systematic error of the algorithm, and the standard deviation of the 10 replicate measurements was used to measure the accuracy of the algorithm30, as shown in Table 2.Table 2 Bias and standard deviation of D0 and T1 by repeating the measurement 10 times.

	Bias	Standard deviation	
D0	T1	D0	T1	
Water	 − 0.34%	 − 0.48%	1.75%	2.68%	
Glycerine	 − 0.83%	0.96%	1.66%	2.23%	

The Monte Carlo-based optimization algorithm was experimentally demonstrated to be well adapted to two D0, pure water and glycerol, in T1 samples with values tens of times different, and the systematic error relative to the manually measured values is less than 1%. Considering the effect of temperature, it can be considered that the D0 and T1 parameters of the measured pure water under this algorithm are basically accurate.

In addition, with the algorithm optimization, the D0 and T1 measurement experiments can be completed with only 3 to 4 data points and achieves a speed increase of approximately 6 times compared to manual measurements of 15 to 25 data points, even without taking into account the manual process of repeatedly adjusting the measurement parameters, greatly improving the measurement efficiency.

In the application of low-field unilateral magnetic resonance, due to its poor magnetic field uniformity and low signal-to-noise ratio, the parameters of T2 generally do not need to be selected, and the echo time needs to be set as small as possible. Therefore, this study does not involve the adaptive adjustment of the measurement parameters in the T2 experiment30.

Conclusion

In this paper, the diffusion and relaxation curves of the samples are estimated by Monte Carlo simulation, and the algorithm selects the optimal measurement parameters according to the estimated values to achieve D0 and T1. The algorithm selects the optimal measurement parameters based on the estimated values and automates the parameter setting during the measurement, saves the measurement time, significantly reduces the measurement threshold, and facilitates NMR. The algorithm has been experimentally verified to be able to obtain more accurate results than the previous algorithm. The algorithm has been experimentally verified to obtain more accurate measurement results. Notably, the proposed intelligent search algorithm is based on the premise of single-component samples, and it needs to be extended for multi-component samples, which is also the future research direction of our team.

Author contributions

All authors contributed to the study conception and design. The experimental preparation, data collection and analysis were carried out by P.G. and R.Z., the analysis and manuscript preparation were carried out by J.Z. and J.S. and B.L. put forward constructive suggestions for discussion and analysis. The first draft of the manuscript was written by P.G., and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Funding

This research was funded by the National Natural Science Foundation of China (No. 51707028), Chongqing Natural Science Foundation (No. cstc2021jcyj-msxmX0470), Science and Technology Funds of Chongqing Municipal Education Commission (KJQN202100533), and Innovation and Entrepreneurship Training Program for College Students (S202210637053).

Data availability

The datasets generated and/or analysed during the current study are not publicly available due private reasons but are available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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References

1. Brown RW Cheng Y-CN Haacke EM Thompson MR Venkatesan R Magnetic Resonance Imaging: Physical Principles and Sequence Design 1999 Wiley
2. Blümich B Haber-Pohlmeier S Zia W Compact NMR 2014 De Gruyter
3. Blümich B Perlo J Casanova F Mobile single-sided NMR Prog. Nucl. Magn. Reson. Spectrosc. 2008 52 197 269 10.1016/j.pnmrs.2007.10.002
4. Perlo J Casanova F Blümich B Single-sided sensor for high-resolution NMR spectroscopy J. Magn. Reson. 2006 180 274 279 10.1016/j.jmr.2006.03.004 16580238
5. Zheng X Xianjun C Kaikai M Yunfeng X Novel unilateral NMR sensor for assessing the aging status of silicone rubber insulator IEEE Sens. J. 2016 16 1168 1175 10.1109/JSEN.2015.2499765
6. Eghbalnia HR Bahrami A Tonelli M Hallenga K Markley JL High-resolution iterative frequency identification for NMR as a general strategy for multidimensional data collection J. Am. Chem. Soc. 2005 127 12528 12536 10.1021/ja052120i 16144400
7. Wang HN Wang Y The research progress of De-Noising Methods in low-field NMR signal Chin. J. Med. Phys. 2013 34 4261 4265
8. Wu F Fan Y Deng S Xing D Liu X Wu Z Automatic matching technology for determining acquisition parameters of formation rock T2-G experiment Zhongguo Shiyou Daxue Xuebao (Ziran Kexue Ban) J. China Univ. Pet. 2014 38 50 56
9. Zeng Qunying, C. S., Zhu Xiaoqin Chen Zhong. Optimized parameters in intermolecular multiple-quantum coherence NMR with CRAZED sequence. J. Xiamen Univ. Nat. Sci. 02, 176–180 (2005).
10. Hansen AL Brüschweiler R Absolute minimal sampling in high-dimensional NMR spectroscopy Angew. Chem. Int. Ed. Engl. 2016 55 14169 14172 10.1002/anie.201608048 27723193
11. Jaravine VA Orekhov VY Targeted acquisition for real-time NMR spectroscopy J. Am. Chem. Soc. 2006 128 13421 13426 10.1021/ja062146p 17031954
12 Masuda R Gupta A Stait-Gardner T Zheng G Torres A Price W Shortening NMR experimental times Magn. Reson. Chem. 2018 56 847 851 10.1002/mrc.4749 29777626
13. Q SY. A machine learning based adaptive method for multi-parametric NMR experiments. Halifax: 14th ICMRM (2017).
14 Tang Y Song YQ Realtime optimization of multidimensional NMR spectroscopy on embedded sensing devices Sci. Rep. 2019 10.1038/s41598-019-53929-1 31882748
15 Xing D An adaptive method for determining an acquisition parameter t0 in a modified CPMG sequence J. Magn. Reson. 2017 10.1016/j.jmr.2017.01.006 28923777
16. Reci A Sederman AJ Gladden LF Optimising magnetic resonance sampling patterns for parametric characterisation J. Magn. Reson. 2018 294 35 43 10.1016/j.jmr.2018.06.010 30005192
17. Reci A Ainte MI Sederman AJ Mantle MD Gladden LF Optimising sampling patterns for bi-exponentially decaying signals Magn. Reson. Imaging 2019 56 14 18 10.1016/j.mri.2018.09.029 30413334
18. Guest J Kiraly P Nilsson M Morris GA Signal-to-noise ratio in diffusion-ordered spectroscopy: How good is good enough? Magn. Reson. 2021 2 733 739 10.5194/mr-2-733-2021
19. Fan Yiren NZ Shaogui Deng Wenjie Zhao Bingkai Liu Experimental study on NMR measurement parameters and reservoir properties Pet. Geol. Exp. 2005 27 6 624 626
20. Fu Yonghong, S., Kaichen, Z., Liang, W., Qian, D. Preliminary study and comparison of porosity measurement parameters of shale core gas measurement. Spec. Oil Gas Reserv., 25(3), (2018).
21 Abou JA The Monte Carlo Methods—Recent Advances, New Perspectives and Applications 2022 IntechOpen
22. Bidokhti PS Theory, Application, and Implementation of Monte Carlo Method in Science and Technology 2019 Books on Demand
23. Gabrié M Rotskoff GM Vanden-Eijnden E Adaptive Monte Carlo augmented with normalizing flows Proc. Natl. Acad. Sci. U. S. A. 2022 119 e2109420119 10.1073/pnas.2109420119 35235453
24. Guth PA Kaarnioja V Kuo FY Schillings C Sloan IH A Quasi-Monte Carlo method for optimal control under uncertainty SIAM/ASA J. Uncertain. Quantif. 2021 9 354 383 10.1137/19M1294952
25. Heng J, Bishop AN, Deligiannidis G, Doucet A. Controlled sequential Monte Carlo. arXiv: Computation (2017).
26 Hübler C Analysing binding stoichiometries in NMR titration experiments using Monte Carlo simulation and resampling techniques PeerJ Anal. Chem. 2022 4 e23 10.7717/peerj-achem.23
27 Jamroz BF Williams DF Consistency in Monte Carlo uncertainty analyses Metrologia 2020 57 065008 10.1088/1681-7575/aba5aa
28 Leal JES da Silva JA Arencibia RV Contributions to the adaptive Monte Carlo method J. Braz. Soc. Mech. Sci. Eng. 2020 10.1007/s40430-020-02548-3
29. Rahbari A Hens R Ramdin M Moultos OA Dubbeldam D Vlugt TJH Recent advances in the continuous fractional component Monte Carlo methodology Mol. Simul. 2020 47 804 823 10.1080/08927022.2020.1828585
30. Rata DG Casanova F Perlo J Demco DE Blümich B Self-diffusion measurements by a mobile single-sided NMR sensor with improved magnetic field gradient J. Magn. Reson. 2006 180 229 235 10.1016/j.jmr.2006.02.015 16545589