Introduction

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 researchers2,3,4,\({\mathrm{T}}_{\mathrm{d}}\) 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, \({T}_{i}\) is varied to measure the T1 recovery of the sample.

Figure 2
figure 2

IR-CPMG measurement of T1 sequences and their parameters.

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Method

In \({\text{T}}_{1}\) and \({\text{D}}_{0}\) measurements, adjusting \({\text{T}}_{\text{d}}\) and \({\mathrm{T}}_{\mathrm{i}}\) during the measurement needs to be performed. In our method, many diffusion curves and \({\text{T}}_{1}\) 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 \({\text{D}}_{0}\) 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], \({D}_{0}\)∈[10–6 mm2/s, 10–2 mm2/s]. The \({D}_{0}\) 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 \({\text{D}}_{0}\) 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-\(\mathrm{f}(\mathrm{t})\) 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 \({\mathrm{t}}_{1}\) in this curve cluster is selected as the first sampling point, and the actual corresponding measurement value \(\mathrm{f}{(\mathrm{t}}_{1})\) is obtained by measurement. From this measurement dataset, (\({\mathrm{t}}_{1}\), \(\mathrm{f}{(\mathrm{t}}_{1})\)) 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 (\({\mathrm{t}}_{1}\), \(\mathrm{f}{(\mathrm{t}}_{1})\)) data point is retained), which in turn yields the B and D0 parameter subspaces. Similarly, the t-\(\mathrm{f}(\mathrm{t})\) curve cluster is plotted in this parameter subspace, and the model uncertainty is calculated, while the point \({\mathrm{t}}_{2}\), 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 ((\({\mathrm{t}}_{1}\), f(\({\mathrm{t}}_{1}\)), (\({\mathrm{t}}_{2}\), f(\({\mathrm{t}}_{2}\)))), 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 \(\mathrm{f}\left(\mathrm{t}\right)\) satisfies the following form30:

$$\begin{array}{c}f\left(\mathrm{t}\right)=B*{\mathrm{e}}^{-{\left(\frac{\mathrm{t}}{{\mathrm{D}}_{0}}\right)}^{3}},\end{array}$$
(1)

This equation is correct for the constant gradient case, where the parameter vector A =\(\left[{\mathrm{D}}_{0},\mathrm{B}\right]\) 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, \(\mathrm{B}\) should be equal to the peak of the measured echo, and the maximum \(\mathrm{Max}\left({\dot{\mathrm{y}}}_{\mathrm{i}}\right)\), \({\dot{\mathrm{y}}}_{\mathrm{i}}\) is the real measurement data. Considering the measurement error, the parameter search range can be set to \(\left[\mathrm{Max}\left({\dot{\mathrm{y}}}_{\mathrm{i}}\right)\pm 5\upsigma \right]\); and the \({\mathrm{D}}_{0}\) of common samples is between 0 and \({10}^{-8}{\mathrm{cm}}^{2}\cdot {\mathrm{s}}^{-1}\). Therefore, \({\mathrm{D}}_{0}\) and B are set to obey the uniform distribution in their respective intervals:

$$\begin{array}{c}\left\{\begin{array}{c}{\mathrm{D}}_{0}\sim U\left[0,{10}^{-8}\right]\\ B\sim U\left[\mathrm{Max}\left({\dot{\mathrm{y}}}_{\mathrm{i}}\right)-5\upsigma ,\mathrm{Max}\left({\dot{\mathrm{y}}}_{\mathrm{i}}\right)+5\upsigma \right]\end{array}\right.,\end{array}$$
(2)

Multiple random sampling according to the above distribution is performed.

$$\begin{array}{c}{{\sum }_{\mathrm{i}=1}^{\mathrm{N}}\left[{\dot{\mathrm{y}}}_{\mathrm{i}}-\mathrm{B}*{\mathrm{e}}^{-{\left(\frac{\mathrm{t}}{{\mathrm{D}}_{0}}\right)}^{3}}\right]}^{2}<N{\upsigma }^{2},\end{array}$$
(3)

When there are sufficient curves that satisfy the condition of Eq. (3), \({\mathrm{DY}}_{\mathrm{t}}\) can be calculated, and then, it can be substituted into \({\dot{\mathrm{t}}}_{\mathrm{i}+1}=\mathrm{t}|{\mathrm{DY}}_{\mathrm{t}}=\mathrm{Max}\left({\mathrm{DY}}_{\mathrm{t}}\right)\) to obtain \({\dot{\mathrm{t}}}_{\mathrm{i}+1}\). \({\mathrm{T}}_{\mathrm{d}}\) is set to \({\mathrm{t}}_{\mathrm{i}+1}\) to obtain the new measured value of \({\dot{\mathrm{t}}}_{\mathrm{i}+1}\), and it is fit by least squares to obtain \({\widehat{\mathrm{a}}}_{\mathrm{i}+1}\). The iterations are stopped when \(|{\widehat{\mathrm{a}}}_{0\left(\mathrm{i}\right)}-{\widehat{\mathrm{a}}}_{0\left(\mathrm{i}-1\right)}|<\upvarepsilon\) is satisfied. The measurement of \({\mathrm{T}}_{1}\) was performed similarly.

The entire algorithm flow is shown in Fig. 3.

Figure 3
figure 3

Algorithm flow chart.

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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 \({\mathrm{D}}_{0}\) and \({\mathrm{T}}_{1}\). Moreover, \({\mathrm{t}}_{\mathrm{i}}\) obtained by the algorithm is inputted into the programmable spectrometer, \({\dot{\mathrm{y}}}_{\mathrm{i}}\) is obtained from the actual measurement, and then, \({\dot{\mathrm{y}}}_{\mathrm{i}}\) is returned to the algorithm to calculate \({\mathrm{t}}_{\mathrm{i}+1}\). This process is shown in Fig. 4.

Figure 4
figure 4

Measurement flow chart.

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The upper limit of error \(\upvarepsilon ={10}^{-10}\) 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
figure 5

\({\mathrm{D}}_{0}\) Convergence of the predicted dataset in the measurement with increasing measurement data.

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Figure 6
figure 6

Pure water measured diffusion curve.

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The same algorithm is used to measure \({\mathrm{T}}_{1}\) of the pure water, the upper error limit is set to \(\upvarepsilon ={10}^{-10}\), 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 \({\mathrm{T}}_{1}\) curve is shown in Fig. 8.

Figure 7
figure 7

Convergence of the prediction dataset with an increasing measurement data in the T1 measurement.

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Figure 8
figure 8

Pure water actual measurement \({\mathrm{T}}_{1}\).

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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.
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To evaluate the accuracy and stability of the Monte Carlo optimization-based algorithm in the D0 and \({\mathrm{T}}_{1}\) measurement experiments, 10 repeated D0 and \({\mathrm{T}}_{1}\) 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.
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The Monte Carlo-based optimization algorithm was experimentally demonstrated to be well adapted to two D0, pure water and glycerol, in \({\mathrm{T}}_{1}\) 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 \({\mathrm{D}}_{0}\) and \({\mathrm{T}}_{1}\) parameters of the measured pure water under this algorithm are basically accurate.

In addition, with the algorithm optimization, the D0 and \({\mathrm{T}}_{1}\) 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 \({\mathrm{D}}_{0}\) and \({\mathrm{T}}_{1}\). 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 \(\mathrm{NMR}\). 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.