A method for quantitatively measuring the content and concentration of a particulate component in a mixture

Abstract

The application provides a method for quantitatively measuring the content and concentration of a particle component in a mixture, comprising: (1) performing spICP-MS measurement on the mixture and each potential component; (2) performing probability density estimation on the measured signal to obtain a probability density function of each potential component; (3) establishing a finite mixture model by using the probability density function and estimating model parameters to obtain the content of each potential component; (4) based on the obtained content, combining the content threshold value, and reducing the component number of the potential component to obtain the component number, composition and content of the real component in the mixture; (5) based on the information of the real component, repeating step (3) to obtain the content of each real component after the potential component is limited; (6) based on the obtained content, combining the transmission efficiency and sampling speed data of spICP-MS to obtain the concentration of each real component. The method improves the measurement resolution and accuracy of the particle size, content and concentration.

Description

Technical Field

[0001] This invention belongs to the field of analytical testing technology, and relates to a method for detecting mixtures, and more particularly to a method for quantitatively measuring the content and concentration of particulate components in a mixture. Background Technology

[0002] The physical and chemical properties of nanoparticles are highly dependent on their size range; therefore, accurately determining the particle size distribution and content of nanoparticles is a crucial step in understanding their properties and technological applications. Currently, various methods exist for determining the particle size distribution and content of nanoparticles. For example, dynamic light scattering (DLS) is one of the most commonly used size characterization techniques; however, this technique requires a size difference of more than three times to distinguish different particle components. As a counting method, nanoparticle tracking analysis (NTA) has better size resolution than DLS, but still requires approximately 50% size difference to distinguish different nanoparticle components and tends to overestimate the number of larger particles in a mixture.

[0003] To improve the size resolution of hybrid systems, researchers have developed a promising alternative—a combined technique that integrates separation and detection methods. Numerous size-related separation techniques can be used to achieve this, such as field flow separation (FFF), chromatography, density gradient centrifugation, centrifugal liquid sedimentation (CLS), electrophoresis, and membrane filtration; detection techniques include DLS, UV-Vis spectroscopy, multi-angle light scattering (MALS), inductively coupled plasma optical emission (ICP-OES), and mass spectrometry (ICP-MS).

[0004] Sánchez-Cachero et al. (A. Sanchez-Cachero, S. Lopez-Sanz, N.R. Farinas, A. Rios, R. Martin-Doimeadios, Talanta 222(2021)121513.) investigated the effect of organic matter on the size of platinum nanoparticles using asymmetric flow field separation (AF4-ICP-MS). The results showed that under optimal elution conditions, 5 nm and 30 nm platinum nanoparticles could be well separated, but the elution peaks of 30 nm and 50 nm platinum nanoparticles partially overlapped under the same elution conditions. Techarang et al. (T. Techarang, Anal Chim Acta 1144(2021)102-110.) achieved partial separation of a mixture of 5 nm and 15 nm gold nanoparticles (AuNPs) using electric field flow separation. Johnson et al. (ME. Johnson, AR. Montoro Bustos, MR. Winchester, Anal Bioanal Chem408(27)(2016)7629-7640.) achieved good separation of mixtures of 30 nm, 80 nm, and 150 nm AuNPs using sucrose density gradient centrifugation. However, due to the decrease in particle size difference, the separation effect of mixtures of 30 nm, 60 nm, and 100 nm AuNPs from mixtures of 20 nm, 50 nm, and 100 nm AuNPs was poor.

[0005] Despite significant advancements in coupled techniques, the complete separation and precise quantification of particulate components with small size differences remains a formidable challenge, particularly when the diameter of the particles to be separated in the mixture is greater than 20 nm. Separation of mixtures of particles with size differences of less than 10 nm is difficult. Furthermore, separation techniques may introduce additional interferences and biases, such as adsorption and segregation effects, causing the physical properties measured by downstream detection techniques to deviate from the original properties of the sample.

[0006] Single-particle inductively coupled plasma mass spectrometry (spICP-MS) is a newly emerging mass spectrometry technique used in recent years to measure particle size distribution and particle number concentration. Mitrano et al. (DM Mitrano, A. Barber, A. Bednar, P. Westerhoff, C.P. Higgins, J.F. Ranville, Journal of Analytical Atomic Spectrometry 27(7)(2012)1131.) found that although the integrated intensity histograms of particles in a mixture of 40 nm and 80 nm silver nanoparticles (AgNPs) were well separated, the integrated intensity histograms of particles in a mixture of 60 nm and 80 nm AgNPs showed a large overlap. Westerhoff et al. (X.Bi, S.Lee, J.F.R.F.R., P.S.Sattigeri, A.Spanias, P.Herckes, P.Westerhoff, Journal of Analytical Atomic Spectrometry 29(9)(2014)1630.) used the K-means clustering algorithm to find that the minimum distinguishable size difference between primary and secondary nanoparticles was approximately 20 nm. However, due to further overlap in particle distribution, mixtures composed of particles with similar sizes still could not be correctly distinguished.

[0007] Therefore, how to provide a method for quantitatively measuring the content and concentration of particulate components in a mixture, and improve the measurement resolution and accuracy of the particle size, content and concentration of the measured particles, has become an urgent problem that needs to be solved by those skilled in the art. Summary of the Invention

[0008] The purpose of this invention is to provide a method for quantitatively measuring the content and concentration of particulate components in a mixture. The method is based on kernel density estimation and finite mixing model analysis to process spICP-MS data, thereby improving the measurement resolution and accuracy of the particle size, content and concentration of the measured particles.

[0009] To achieve this objective, the present invention adopts the following technical solution:

[0010] The present invention provides a method for quantitatively measuring the content and concentration of particulate components in a mixture, the method being applicable to mixtures with known potential constituent components.

[0011] The method includes the following steps:

[0012] (1) Perform spICP-MS measurements on the mixture and each potential component of the mixture individually to obtain the measured signals of the mixture and each potential component;

[0013] (2) The probability density distribution of the measured signal of each potential component obtained in step (1) is estimated by using the kernel density estimation method to obtain the probability density function of each potential component.

[0014] (3) Use the probability density function of each potential component obtained in step (2) to establish a finite mixing model, and use the measured signal of the mixture obtained in step (1) to estimate the model parameters of the finite mixing model to obtain the content of each potential component in the mixture.

[0015] (4) Based on the content obtained in step (3), and combined with the specified content threshold, the component number of potential components in the mixture is reduced to obtain the component number, composition and content of the real components in the mixture.

[0016] (5) Based on the information of the real components in the mixture obtained in step (4), repeat step (3) to obtain the content of each real component in the mixture after the potential components are defined;

[0017] (6) Based on the content obtained in any of the steps (3)-(5), combined with the transmission efficiency and injection speed data of spICP-MS, the concentration of each real component in the mixture is obtained after conversion.

[0018] This invention uses a finite mixture model (FMKDE) composed of kernel estimates of probability density functions to describe mixtures with known potential constituents. The measurement signals obtained by spICP-MS are analyzed to obtain the content and concentration of each actual component in the mixture with known potential constituents. Because this method fully utilizes the statistical characteristics of the measured signals of each potential component, the particle size resolution and the accuracy of content and concentration measurements are significantly improved when spICP-MS is applied to the measurement of mixtures with known potential constituents.

[0019] Specifically, the method is based on kernel density estimation and finite mixing model analysis to process spICP-MS data, enabling quantitative measurement of the content and concentration of particulate components in mixtures. It can reduce the particle size resolution limit of particulate mixtures to 5 nm, and in multi-component mixtures, the absolute content deviation is reduced to below 3%. This significantly improves the particle size resolution and content accuracy of spICP-MS multi-component testing, and eliminates the need for complex separation processing, which helps to promote the application of spICP-MS methods in nanotechnology-related fields.

[0020] Furthermore, this invention first uses spICP-MS to measure each potential component individually, and then uses kernel density estimation to estimate the probability density distribution of each potential component, thereby avoiding pre-setting constraints on the functional form of the probability density function of each potential component.

[0021] In this invention, the content mentioned in steps (3)-(5) includes the percentage of the number of particles contained in each component (potential component or real component) of the mixture in the total number of particles; the concentration mentioned in step (6) includes the number concentration or mass concentration of particles or particles of a specific size in the mixture.

[0022] Preferably, the mixture of known potential constituent components is specifically a mixture of at least two known types of potential components in a known or unknown proportion, for example, it may be 2, 5, 10, 15, 20, 25 or 30 types, but is not limited to the listed values, and other unlisted values ​​within this range are also applicable.

[0023] The method provided by this invention targets a mixture in which the specific types of potential components are known, without specifying the exact proportions of each potential component. For example, the mixture can be an initial mixture or a mixture that has undergone storage, use, or other treatment; the content of each potential component in the mixture may not have changed or may have changed, or the content of some potential components may have decreased to 0, while the content of the remaining potential components is undetermined. All of the above situations apply to the method provided by this invention, and therefore are not specifically limited herein.

[0024] In this invention, the potential components in steps (1)-(5) correspond to the constituent components in the initial mixture or to the potential constituent components in the subsequent mixture; the actual components in steps (4)-(6) correspond to the actual constituent components in the subsequent mixture inferred from the measurement results; if the actual constituent components are known, there is no need to speculate, and it is only necessary to set both the potential components and the actual components as the actual constituent components.

[0025] Preferably, the measured signal of the mixture is represented by a finite mixture model to express the probability density function, as shown by the formula:

[0026]

[0027] Where g(x; ψ) is the probability density function of the finite mixture model; x is the combined intensity of a single particle measured by spICP-MS; ψ = (π1,...,π) g ,a1,...a g ,h1,...h g ,f1,...,f g ) represents the model parameters; g is the total number of potential components; π j f represents the content of the j-th potential component; j (x) is the probability density function of the j-th potential component; a j This is a drift correction parameter set to address potential instrument signal drift during the measurement of the j-th potential component and the mixture; h j For bandwidth.

[0028] In this invention, the comprehensive intensity x refers to the integrated intensity of the continuous pulse count signal corresponding to the particle during spICP-MS measurement, or its converted physical quantity, including particle mass and equivalent particle size. Because the integrated intensity corresponding to the particle can be converted into particle mass or equivalent particle size using standard particles or other calibration methods, the converted physical quantity is still applicable to this invention. The integrated intensity refers to the sum or integral value of the continuous pulse count signals associated with the measurement event of a single particle obtained by spICP-MS measurement, or it can be the total value after deducting background. The data of the integrated intensity can be read from the data provided by the ICP-MS instrument, or it can be obtained through more basic data processing of the continuous pulse count signal.

[0029] In this invention, formula (1) uses a negative sign correction for the observed values ​​of the mixture, i.e., xa j Those skilled in the art should understand that the negative sign here only relates to the relativity of the drift, and x+a can also be used. j In the form of, and a j Taking the opposite number; or not correcting the observations of the mixture, but only correcting the observations of the j-th potential component, both have the same correction effect; or if there is no instrument signal drift, then a j Take zero.

[0030] Preferably, the measured signal in step (1) is the comprehensive intensity data corresponding to each particle in the mixture sample. Furthermore, the expression for the finite mixing model, i.e., formula (1), includes the comprehensive strength data corresponding to each particle in the potential components.

[0031] In this invention, the A set of independent and identically distributed samples, sampled from the probability density function g(x; ψ) of the overall intensity of the mixture; Let f be a set of independent and identically distributed samples, and the probability density function f of the combined intensity of each potential component be given. j (x).

[0032] Preferably, the kernel density estimation method in step (2) is expressed by the formula:

[0033]

[0034] in, K is the kernel estimate of the probability density function of the j-th potential component; K() is the kernel function; m is the sample of observed values ​​for the potential component. The number of observations in h; j For bandwidth.

[0035] Preferably, the kernel function is in the form of any one of the following: Gaussian function, rectangular function, trigonometric function, cosine function, or Epanechnikov function; more preferably, it is a Gaussian function, expressed by the formula:

[0036]

[0037] Right now This is the Gaussian kernel function.

[0038] Preferably, the method for determining the bandwidth includes manual assignment or calculation assignment, and more preferably calculation assignment.

[0039] In this invention, the manual assignment specifically refers to: manually setting different bandwidths h. j It is worth drawing a picture. The graph of the function, and then select a reasonable h based on the graph. j value.

[0040] Preferably, the criterion or equation upon which the calculation assignment is based includes any one of the following: Silverman's rule, Scott's rule, unbiased cross-validation, biased cross-validation, Sheather-Jones method, or bootstrap method. More preferably, it is Silverman's rule, expressed as a formula:

[0041] l O =min(sd(x) i ),IQR(x i (4) / 1.34)

[0042] h j =0.9×l O ×m -0.2 (5)

[0043] Among them, sd(x i () is a sample of potential component observations The sample standard deviation; IQR(x i ) represents the interquartile range of the observed sample, i.e., the difference between the third quartile and the first quartile; min() is the minimum value function; m is the number of observed values ​​for the j-th potential component.

[0044] Preferably, the estimation method for the model parameters in step (3) includes any one or a combination of at least two of the following: maximum likelihood estimation, Bayesian discriminant method, or least squares estimation. Typical but non-restrictive combinations include the combination of maximum likelihood estimation and Bayesian discriminant method, or the combination of least squares estimation and Bayesian discriminant method.

[0045] Preferably, the estimation method includes the following steps:

[0046] (3.1) The content of each potential component in the mixture is initialized, expressed by the formula:

[0047]

[0048] Where j = 1, 2, ..., g, and g is the total number of potential components.

[0049] The present invention can also use other values ​​within the range (0,1) to initialize the content of each potential component in the mixture.

[0050] (3.2) The drift correction parameters for the measured signal of the mixture are initialized, expressed by the formula:

[0051]

[0052] Where j = 1, 2, ..., g, and g is the total number of potential components.

[0053] This invention can also use other values ​​to initialize the drift correction parameters of the measured signal of the mixture, typically selecting values ​​close to 0. For example, -4 to 4 can be selected for the integral intensity; -1 nm to 1 nm can be selected for the particle diameter. When the absolute value of the initial value is too large, it may lead to convergence failure or convergence to outliers, and a j An excessively large absolute value of the convergence value indicates severe instrument drift, requiring the instrument to be adjusted to a stable state before further measurement. When instrument signal drift is nonexistent or negligible, then a... j Always take zero.

[0054] (3.3) The estimation methods for model parameters include any one or a combination of at least two of the following methods:

[0055] (3.3A) The maximum likelihood estimation includes estimating the model parameters π under the maximum likelihood probability conditions using numerical optimization methods. j It can also include a j and h j The value is as follows:

[0056] The log-likelihood probability (the logarithmic value of the likelihood probability) is used as the object of numerical optimization, expressed by the formula:

[0057]

[0058] Where logL is the log-likelihood probability; n is the sample of observed values ​​for the mixture. The number of observations in the data; f j Estimate (x).

[0059] In this invention, the meanings of the remaining parameters in formula (8) are the same as those in formula (1), and therefore will not be repeated here; parameter π j and a j The initial values ​​are given in formulas (6)-(7); parameter a j It is possible to exclude a from optimization. j The parameter h is always 0. j It can be excluded from optimization and fixed as the bandwidth parameters of each potential component obtained from the kernel density estimation in step (2), as shown in formulas (2)-(5). Alternatively, it can be included in optimization and used as the initial values.

[0060] In this invention, the log-likelihood probability is used as the object of numerical optimization, which can effectively prevent numerical overflow.

[0061] Preferably, the numerical optimization method includes any one of the following: Brent algorithm, Nelder-Mead algorithm, BFGS quasi-Newton algorithm, L-BFGS-B quasi-Newton algorithm, conjugate gradient algorithm, or SANN simulated annealing algorithm.

[0062] Preferably, the numerical optimization process also includes optimizing a. j Between and / or h j Assumptions and constraints are made regarding the quantitative relationships between parameters to reduce the number of parameters being fitted, thereby improving convergence characteristics.

[0063] (3.3B) The model parameter π in step (3.3A) j Alternatively, Bayesian discriminant analysis can be used for estimation, including the following steps:

[0064] (3.3B.1) Calculate the posterior probability that the comprehensive intensity observation of the mixture belongs to a certain potential component, expressed by the formula:

[0065]

[0066] Where i = 1, 2, ..., n, and n is the number of comprehensive strength observations of the mixture; j = 1, 2, ..., g, and g is the total number of potential components.

[0067] (3.3B.2) Update the content of each potential component in the mixture, expressed by the formula:

[0068]

[0069] (3.3B.3) Using formulas (9) and (10) to calculate π j and λ j The iteration is performed, with superscripts t and t+1 indicating the iteration steps. The termination condition of the iteration is based on π. j The change in can be expressed by the formula:

[0070]

[0071] Where ε is the specified tolerance, preferably ε = 10. -5 .

[0072] (3.3C) The least squares estimation includes the following steps:

[0073] (3.3C.1) The probability density distribution of each observation of the mixture is estimated using the kernel density estimation method, expressed by the formula:

[0074]

[0075] Where K() is the kernel function; n is the sample of mixture observations. The number of observations in the data; h is the bandwidth.

[0076] In this step, the estimation method for the kernel density is similar to formulas (2)-(5), so it will not be elaborated here.

[0077] (3.3C.2) The probability density of each observation in the mixture is estimated using a finite mixture model, expressed by the formula:

[0078]

[0079] (3.3C.3) Using the probability density data obtained in step (3.3C.1) as experimental values ​​and the probability density data obtained in step (3.3C.2) as theoretical values, the model parameter π under the least squares condition is estimated using numerical optimization methods. j It can also include a j and h j The value of .

[0080] Preferably, the numerical optimization method includes any one of the Levenberg-Marquardt algorithm, the trust region reflection algorithm, or the trust region dogleg algorithm.

[0081] Preferably, the numerical optimization process also includes optimizing a. j Between and / or h j Assumptions and constraints are made regarding the quantitative relationships between parameters to reduce the number of parameters being fitted, thereby improving convergence characteristics.

[0082] In this invention, the model parameter π in step (3.3C) j Alternatively, Bayesian discriminant analysis (3.3B) can be used for estimation, which includes steps (3.3B.1)-(3.3B.3), and will not be elaborated here.

[0083] In this invention, when the instrument's performance is stable, the instrument's drift correction parameter a can be adjusted.j The above estimation algorithm still applies when the parameter π is always set to 0; however, when only the parameter π needs to be estimated... j Alternatively, instead of using maximum likelihood estimation or least squares estimation, the improved Bayesian discriminant estimation method can be used, i.e., steps (3.3B.1)-(3.3B.3) can be used to directly obtain the parameter π. j The estimated value.

[0084] Compared with the prior art, the present invention has the following beneficial effects:

[0085] The method provided by this invention is based on kernel density estimation and finite mixing model analysis to process spICP-MS data, realizing the quantitative measurement of the content and concentration of particulate components in mixtures. It can reduce the particle size resolution limit of particulate mixtures to 5 nm, and in multi-component mixtures, the absolute content deviation is reduced to less than 3%. It significantly improves the particle size resolution and content accuracy of spICP-MS multi-component testing, and does not require complex separation processing, which helps to promote the application of spICP-MS methods in nanotechnology-related fields. Attached Figure Description

[0086] Figure 1 These are probability density function images of the four potential components in the method provided in Example 1;

[0087] Figure 2 These are the probability density function images of the mixture samples in the method provided in Example 1 and the probability density function images corresponding to each potential component normalized by content;

[0088] Figure 3 These are the probability density function images of the mixture samples in the method provided in Example 2, and the probability density function images corresponding to each real component normalized according to its content. Detailed Implementation

[0089] The technical solution of the present invention will be further illustrated below through specific embodiments. Those skilled in the art should understand that the embodiments described are merely illustrative of the present invention and should not be construed as limiting the invention in any way.

[0090] This invention provides a method for quantitatively measuring the content and concentration of particulate components in a mixture. The method is for mixtures with known potential constituent components, specifically: mixtures composed of at least two known potential components mixed in known or unknown proportions. The measured signal of the mixture is represented by a finite mixing model as a probability density function, expressed by the formula:

[0091]

[0092] Where g(x; ψ) is the probability density function of the finite mixture model; x is the combined intensity of a single particle measured by spICP-MS; ψ = (π1,...,π) g ,a1,...a g ,h1,...h g ,f1,...,f g ) represents the model parameters; g is the total number of potential components; π j f represents the content of the j-th potential component; j (x) is the probability density function of the j-th potential component; a j This is a drift correction parameter set to address potential instrument signal drift during the measurement of the j-th potential component and the mixture; h j For bandwidth.

[0093] The method includes the following steps:

[0094] (1) The mixture and each potential component in the mixture are individually measured by spICP-MS to obtain the measured signal of the mixture and each potential component, and the measured signal is the comprehensive intensity data corresponding to each particle in the mixture sample. Combined intensity data for each particle in the potential component sample

[0095] (2) The probability density distribution of the measured signal of each potential component obtained in step (1) is estimated using the kernel density estimation method to obtain the probability density function of each potential component; the kernel density estimation method is expressed by the formula:

[0096]

[0097] in, K is the kernel estimate of the probability density function of the j-th potential component; K() is the kernel function; m is the sample of observed values ​​for the potential component. The number of observations in h; j For bandwidth.

[0098] The kernel function used is the Gaussian kernel function, expressed by the formula:

[0099]

[0100] The method for determining the bandwidth is based on Silverman's rule of thumb, expressed by the formula:

[0101] l O =min(sd(x) i ),IQR(x i (4) / 1.34)

[0102] hj =0.9×l O ×m -0.2 (5)

[0103] Among them, sd(x i () is a sample of potential component observations The sample standard deviation; IQR(x i ) represents the interquartile range of the observed sample, i.e., the difference between the third quartile and the first quartile; min() is the minimum value function; m is the number of observed values ​​for the j-th potential component.

[0104] (3) Establish a finite mixture model using the probability density function of each potential component obtained in step (2), and estimate the model parameters to obtain the content of each potential component in the mixture; the estimation method of the model parameters includes any one or a combination of at least two of the following: maximum likelihood estimation, Bayesian discriminant method, or least squares estimation; the estimation method includes the following steps:

[0105] (3.1) The content of each potential component in the mixture is initialized, expressed by the formula:

[0106]

[0107] Where j = 1, 2, ..., g, and g is the total number of potential components.

[0108] (3.2) The drift correction parameters for the measured signal of the mixture are initialized, expressed by the formula:

[0109]

[0110] Where j = 1, 2, ..., g, and g is the total number of potential components.

[0111] (3.3) The estimation methods for model parameters include any one or a combination of at least two of the following methods:

[0112] (3.3A) The maximum likelihood estimation includes estimating the model parameters π under the maximum likelihood probability conditions using numerical optimization methods. j It can also include parameter a j and h j The value is as follows:

[0113] The log-likelihood probability is used as the object of numerical optimization, expressed by the formula:

[0114]

[0115] Where logL is the log-likelihood probability; n is the sample of observed values ​​for the mixture. The number of observations in the data; f j Estimate (x).

[0116] The numerical optimization method includes any one of the following: Brent algorithm, Nelder-Mead algorithm, BFGS quasi-Newton algorithm, L-BFGS-B quasi-Newton algorithm, conjugate gradient algorithm, or SANN simulated annealing algorithm.

[0117] The numerical optimization process also includes adjusting a. j between and / or h j Assumptions and constraints are made regarding the quantitative relationships between parameters to reduce the number of parameters being fitted, thereby improving convergence characteristics.

[0118] (3.3B) The model parameter π in step (3.3A) j Alternatively, Bayesian discriminant analysis can be used for estimation, including the following steps:

[0119] (3.3B.1) Calculate the posterior probability that the comprehensive intensity observation of the mixture belongs to a certain potential component, expressed by the formula:

[0120]

[0121] Where i = 1, 2, ..., n, and n is the comprehensive intensity observation value of the mixture; j = 1, 2, ..., g, and g is the total number of potential components.

[0122] (3.3B.2) Update the content of each potential component in the mixture, expressed by the formula:

[0123]

[0124] (3.3B.3) Using formulas (9) and (10) to calculate π j and λ j The iteration is performed, with superscripts t and t+1 indicating the iteration steps. The termination condition of the iteration is based on π. j The change in can be expressed by the formula:

[0125]

[0126] Where ε is the specified tolerance, and ε = 10 -5 .

[0127] (3.3C) The least squares estimation includes the following steps:

[0128] (3.3C.1) The probability density distribution of each observation of the mixture is estimated using the kernel density estimation method, expressed by the formula:

[0129]

[0130] Where K() is the kernel function; n is the sample of mixture observations. The number of observations in the data; h is the bandwidth.

[0131] (3.3C.2) The probability density of each observation in the mixture is estimated using a finite mixture model, expressed by the formula:

[0132]

[0133] (3.3C.3) Using the probability density data obtained in step (3.3C.1) as experimental values ​​and the probability density data obtained in step (3.3C.2) as theoretical values, the model parameter π under the least squares condition is estimated using numerical optimization methods. j It can also include parameter a j and h j The value of; the numerical optimization method includes any one of the Levberg–Marquardt algorithm, the trust region reflection algorithm, or the trust region dogleg algorithm; the numerical optimization process also includes adjusting a. j Between and / or h j Assumptions and constraints are made regarding the quantitative relationships between parameters to reduce the number of parameters being fitted, thereby improving convergence characteristics.

[0134] The model parameter π in step (3.3C) j Alternatively, Bayesian discriminant analysis (3.3B) can be used for estimation, which includes steps (3.3B.1)-(3.3B.3), and will not be elaborated here.

[0135] (4) Based on the content obtained in step (3), and combined with the specified content threshold, the number of potential components in the mixture is reduced to obtain the number of components, composition and content of the real components in the mixture.

[0136] (5) Based on the information of the real components in the mixture obtained in step (4), repeat step (3) to obtain the content of each real component in the mixture after the potential components are defined.

[0137] (6) Based on the content obtained in any of the steps (3)-(5), combined with the transmission efficiency and injection speed data of spICP-MS, the concentration of each real component in the mixture is obtained after conversion.

[0138] Example 1

[0139] This embodiment provides a method for quantitatively measuring the content and concentration of particulate components in a mixture. Four types of gold nanoparticles with different particle sizes, A, B, C, and D, are used as potential components. Two to three of these potential components are selected and mixed in different proportions to obtain four mixture model systems whose actual component numbers, composition, and content are yet to be determined. These are used as mixture samples of known potential components to be measured and are numbered 11-14.

[0140] In this embodiment, because the instrument is stable and the instrument signal drift is negligible, the drift correction parameter a of the finite mixture model is used. j Set to 0, bandwidth parameter h j Without optimization, a modified Bayesian discriminant analysis (BD) method was used to estimate the model parameters. The preparation content, concentration, and spICP-MS measurement results of the above four groups of mixture samples are shown in Table 1. The specific implementation process is demonstrated below:

[0141] (1) Conventional spICP-MS measurements were performed on four potential components (A, B, C, and D) in the single-particle time-resolved analysis (TRA) mode of the ICP-MS instrument to obtain the comprehensive intensity data corresponding to each nanoparticle in each potential component, i.e., four sets of data. The data, where m is the number of observed nanoparticles in each potential component; the comprehensive intensity data specifically selects the integrated intensity. The obtained comprehensive intensity data is calibrated using standard nanoparticles to obtain the average particle size of the nanoparticles in each of the four potential components A, B, C, and D, corresponding to 33nm, 44nm, 49nm, and 61nm respectively (particle size data is not essential in component content measurement; this data is provided in this embodiment to facilitate a better understanding of the improved size resolution of the present invention by those skilled in the art).

[0142] (2) The probability density distribution of the comprehensive intensity of each potential component obtained in step (1) is estimated using the kernel density estimation method. The results are obtained by calculating the probability density distribution using the Gaussian kernel function shown in formula (3) and the Silverman empirical rules shown in formulas (4)-(5). Where j = A, B, C, D, and x is the overall intensity; the probability density function images of the four potential components obtained by the kernel density estimation method are shown below. Figure 1 ,and Figure 1 Histograms are used for comparison. Although the combined intensity distributions of the four potential components all exhibit tailing, the kernel density estimation method still has good descriptive power.

[0143] (3) Conventional spICP-MS measurements were performed on the four mixture samples (numbered 11-14) containing known potential constituent components to be tested, and the comprehensive intensity observation values ​​corresponding to each nanoparticle in each mixture sample were obtained, i.e., the four groups Data, where n is the number of nanoparticles observed in each mixture sample.

[0144] (4) The comprehensive intensity observation data obtained in step (3) are processed according to formulas (6)-(7) and formula (9).

[0145] -(11) Perform π j and λ j Iterative calculation and let a j The constant is 0, where j = A, B, C, D. In formula (9) This is what was obtained in step (2). And j = A, B, C, D. Based on the iteration termination condition shown in formula (11), π is obtained. j The convergence value is the content of each of the four potential components A, B, C, and D in each mixture sample. The content measurement results of the four mixture samples numbered 11-14 in this embodiment are shown in Table 1; using π... j The convergence value yields the probability density function of the mixture sample and the corresponding probability density function of each potential component normalized to its content. The graphs of these correlation functions are shown below. Figure 2 .

[0146] (5) In this embodiment, the content of each potential component obtained in step (4) is further processed as needed to determine the composition, structure, and content of the true components in each mixture sample. Specifically, potential components with a content of less than 1% are removed. Taking mixture sample No. 11 as an example, after removing potential components C and D, only potential components A and B are retained as the true components of mixture sample No. 11. The content of the true components can be directly obtained from the content measurement value obtained in step (4), or the potential components can be limited to A and B, and step (4) can be repeated to obtain the content measurement value after limiting the components. The content of the true components in each mixture sample is shown in Table 1. As can be seen from Table 1, the difference between the two measurement values ​​of each mixture sample in this embodiment is less than 1%, and the composition, structure, and content of the true components are consistent with the actual situation, with a content difference of less than 3%.

[0147] (6) In this embodiment, based on the content data obtained in step (4) and / or step (5), combined with the transmission efficiency data of spICP-MS, the content data is multiplied by the total number of observations per minute of the mixture sample to obtain the number of observations per minute of each real component. Then, it is divided by the product of the transmission efficiency and the injection speed to obtain the particle number concentration of each real component in each mixture sample, as shown in Table 1.

[0148] Table 1

[0149]

[0150]

[0151] As shown in Table 1, the spICP-MS measurement results obtained by the method provided in this embodiment are basically consistent with the configuration values ​​of the mixture. Among them, the particle size difference of the B+C two-component particles in the No. 12 mixture sample is only 5 nm, and the correct content measurement results are also obtained. The absolute deviation of the content measurement values ​​of each group of mixture samples in Table 1 is less than 3%, which further proves the accuracy and applicability of this method.

[0152] Example 2

[0153] This embodiment provides a method for quantitatively measuring the content and concentration of particulate components in a mixture. Two types of gold nanoparticles, B and C, are used as potential components and are mixed in different proportions to form three sets of mixture model systems with undetermined contents. These are used as mixture samples of known potential components to be measured and are numbered 21-23.

[0154] This embodiment performs spICP-MS testing under conditions of significant instrument signal drift. The test results are then corrected for drift-free performance (i.e., a value is always a for any j). j =0) and includes drift correction (i.e., there exists j such that a = 0) and includes drift correction (i.e j The analysis was performed using a finite mixture model (≠0). The composition, concentration, and spICP-MS measurement results of the three mixture samples are shown in Table 2. The specific implementation process is demonstrated below:

[0155] (1) Conventional spICP-MS measurements were performed on potential components B and C in the single-particle time-resolved analysis (TRA) mode of the ICP-MS instrument to obtain the comprehensive intensity data corresponding to each nanoparticle in each potential component, i.e., two sets of data. The data, where m is the number of observed nanoparticles in each potential component; the comprehensive intensity data specifically selects the integral intensity.

[0156] (2) The probability density distribution of the integral intensity of each potential component obtained in step (1) is estimated by using the kernel density estimation method. The Gaussian kernel function shown in formula (3) and the Silverman empirical rules shown in formulas (4)-(5) are used to calculate the probability density distribution. Where j = B, C, and x is the integral intensity.

[0157] (3) Conventional spICP-MS measurements were performed on the three groups of mixture samples (numbered 21-23) containing known potential constituent components to be tested, and the integrated intensity observation values ​​corresponding to each nanoparticle in each mixture sample were obtained, i.e., the three groups Data, where n is the number of nanoparticles observed in each mixture sample.

[0158] (4) Perform drift-free correction on the integral intensity observation data obtained in step (3) (i.e., for any j, a).j The finite mixture model description is given by (=0). This case is similar to the data processing in Example 1, using an improved Bayesian discriminant estimation method (BD) for estimation, and assuming a... j The constant is 0, where j = B, C, and π is calculated according to formulas (6)-(7) and (9)-(10). j and λ j The iterative calculation, the probability density distribution function in formula (9) This is what was obtained in step (2). And j = B, C. Based on the iteration termination condition shown in formula (11), π is obtained. j The convergence value is the content of each potential component B and C in each mixture sample. The content measurement results of the three mixture samples numbered 21-23 in this embodiment are shown in Table 2; using π... j The convergence value yields the probability density function of the mixture sample and the corresponding probability density function of each potential component normalized to its content. The graphs of these correlation functions are shown below. Figure 3 .

[0159] (5) Perform drift correction on the integral intensity observation data obtained in step (3) (i.e., there exists j such that a) j A finite mixture model description (≠0) is used. This embodiment employs the maximum likelihood estimation (MLE) method to estimate the content parameter π. j Drift parameter a j and bandwidth parameter h j Estimate the parameter π using the L-BFGS-B quasi-Newton algorithm. j a j and h j To maximize the value of the log-likelihood function defined by formula (8), we obtain the parameter π. j a j and h j The optimized value; where the parameter π is... j j = B,C represents the content of the two potential components B and C in the mixture sample. The content measurement results of mixture samples numbered 21-23 in this embodiment are shown in Table 2. Using the parameter π... j a j and h j The optimized value yields the probability density function of the mixture sample and the corresponding probability density function of each potential component normalized to its content. The graphs of these correlation functions are shown below. Figure 3 .

[0160] This embodiment also employs the least squares estimation method (NLS) to estimate the content parameter π. j Drift parameter a j and bandwidth parameter h j The estimation; optimization of parameter π using the trust region reflection algorithm.j a j and h j To minimize the difference in probability density estimates of the mixture calculated by formulas (12) and (13) under the least squares condition, the parameter π is obtained. j a j and h j The optimized value; where the parameter π is... j j = B,C represents the content of the two potential components B and C in the mixture sample. The content measurement results of mixture samples numbered 21-23 in this embodiment are shown in Table 2. Using the parameter π... j a j and h j The optimized value yields the probability density function of the mixture sample and the corresponding probability density function of each potential component normalized to its content. The graphs of these correlation functions are shown below. Figure 3 .

[0161] Table 2

[0162]

[0163]

[0164] As shown in Table 2, the spICP-MS measurement results with drift correction given by the finite mixing model and related algorithms described in this embodiment are basically consistent with the configuration values ​​of the mixture. In the case of significant instrument drift, the content values ​​given by the finite mixing model with drift correction are more reliable, with an absolute content deviation of less than 3%, further proving the applicability and accuracy of the finite mixing model and related algorithms.

[0165] Therefore, the method provided by this invention, based on kernel density estimation and finite mixing model analysis of spICP-MS data, achieves quantitative measurement of the content and concentration of particulate components in mixtures. It can reduce the particle size resolution limit of nanoparticle mixtures to 5 nm, and in multi-component mixtures, the absolute content deviation is reduced to below 3%. This significantly improves the particle size resolution and content accuracy of spICP-MS multi-component testing, and eliminates the need for complex separation processing, which helps to promote the application of spICP-MS methods in the field of nanotechnology.

[0166] The applicant declares that the above description is only a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Those skilled in the art should understand that any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention fall within the scope of protection and disclosure of the present invention.

Claims

1. A method for quantitatively measuring the content and concentration of particulate components in a mixture, characterized in that, The method is for mixtures with known potential constituent components; The method includes the following steps: (1) Perform spICP-MS measurements on the mixture and each potential component of the mixture separately to obtain the measured signals of the mixture and each potential component; (2) The probability density distribution of the measured signal of each potential component obtained in step (1) is estimated by using the kernel density estimation method to obtain the probability density function of each potential component; (3) Establish a finite mixing model using the probability density function of each potential component obtained in step (2), and estimate the model parameters of the finite mixing model using the measured signal of the mixture obtained in step (1) to obtain the content of each potential component in the mixture. (4) Based on the content obtained in step (3), and combined with the specified content threshold, the component number of potential components in the mixture is reduced to obtain the component number, composition and content of the real components in the mixture; (5) Based on the information of the real components in the mixture obtained in step (4), repeat step (3) to obtain the content of each real component in the mixture after the potential components are defined; (6) Based on the content obtained in any of the steps (3)-(5), combined with the transmission efficiency and injection speed data of spICP-MS, the concentration of each real component in the mixture is obtained after conversion.

2. The method according to claim 1, characterized in that, The mixture of known potential constituent components is specifically defined as a mixture of at least two known types of potential components in known or unknown proportions.

3. The method according to claim 2, characterized in that, The measured signal of the mixture is represented by a probability density function using a finite mixture model, expressed by the formula: （1） in, is the probability density function of the finite mixture model; x is the combined intensity of a single particle as measured by spICP-MS. =( ,..., , ,... , ,... , ,..., ) represents the parameters of the model; g is the total number of potential components; Let be the content of the j-th potential component; Let be the probability density function of the j-th potential component; It is a drift correction parameter set to address the potential instrument signal drift that may occur when measuring the j-th potential component and the mixture. For bandwidth.

4. The method according to claim 3, characterized in that, The measured signal in step (1) is the comprehensive intensity data corresponding to each particle in the mixture sample. Combined intensity data for each particle in the potential component sample .

5. The method according to claim 4, characterized in that, The kernel density estimation method described in step (2) is expressed by the following formula: （2） in, K is the kernel estimate of the probability density function of the j-th potential component; K() is the kernel function; m is the sample of observed values ​​for the potential component. The number of observations in the data; For bandwidth.

6. The method according to claim 5, characterized in that, The kernel function can be any one of the following: Gaussian function, rectangular function, trigonometric function, cosine function, or Epanechnikov function.

7. The method according to claim 6, characterized in that, The kernel function is a Gaussian function, expressed by the formula: （3） Right now This is the Gaussian kernel function.

8. The method according to claim 5, characterized in that, The bandwidth is determined by either manual assignment or calculation.

9. The method according to claim 8, characterized in that, The bandwidth is determined by calculation and assignment.

10. The method according to claim 8, characterized in that, The criteria or equations upon which the computational assignment is based include any one of the following: Silverman's rule of thumb, Scott's rule of thumb, unbiased cross-validation, biased cross-validation, Sheather-Jones method, or bootstrap method.

11. The method according to claim 10, characterized in that, The criterion for the calculation and assignment is the Silverman rule of thumb, which can be expressed as a formula: （4） （5） in, Sample of potential component observations The sample standard deviation; is the interquartile range of the observed sample, i.e., the difference between the third quartile and the first quartile; min() is the minimum value function; m is the number of observed values ​​for the j-th potential component.

12. The method according to claim 3, characterized in that, The estimation method for the model parameters in step (3) includes any one or a combination of at least two of the following: maximum likelihood estimation, Bayesian discriminant method, or least squares estimation.

13. The method according to claim 12, characterized in that, The estimation method includes the following steps: (3.1) The content of each potential component in the mixture is initialized, expressed by the formula: （6） Where j = 1, 2, ..., g, and g is the total number of potential components; (3.2) The drift correction parameters of the measured signal of the mixture are initialized, as expressed by the formula: （7） Where j = 1, 2, ..., g, and g is the total number of potential components.

14. The method according to claim 12, characterized in that, The maximum likelihood estimation includes estimating the model parameters under the maximum likelihood probability conditions using numerical optimization methods. , and The value is as follows: (3.3A) The log-likelihood probability is used as the object of numerical optimization, expressed by the formula: （8） in, The log-likelihood probability is given by n, where n is the sample of observed values ​​in the mixture. The number of observations in the data; for The estimate.

15. The method according to claim 14, characterized in that, The numerical optimization method includes any one of the following: Brent algorithm, Nelder-Mead algorithm, BFGS quasi-Newton algorithm, L-BFGS-B quasi-Newton algorithm, conjugate gradient algorithm, or SANN simulated annealing algorithm.

16. The method according to claim 15, characterized in that, The numerical optimization process also includes... Between and / or Assumptions and limitations are made regarding the quantitative relationships between them.

17. The method according to claim 12, characterized in that, The model parameters estimated by the Bayesian discriminant method include This includes the following steps: (3.3B.1) Calculate the posterior probability that the overall intensity observation of the mixture belongs to a certain potential component, expressed by the formula: （9） Where i = 1, 2, ..., n, and n is the number of comprehensive intensity observations of the mixture; j = 1, 2, ..., g, and g is the total number of potential components; (3.3B.2) Update the content of each potential component in the mixture, expressed by the formula: （10） (3.3B.3) Using formulas (9) and (10) to... and The iteration is performed, with superscripts t and t+1 representing the iteration steps. The termination condition of the iteration is based on... The change in can be expressed by the formula: （11） in, The specified tolerance.

18. The method according to claim 17, characterized in that, The specified tolerance =10 -5 .

19. The method according to claim 12, characterized in that, The least squares estimation includes the following steps: (3.3C.1) The probability density distribution of each observation of the mixture is estimated using the kernel density estimation method, expressed by the formula: （12） Where K() is the kernel function; n is the sample of mixture observations. The number of observations in the data; h is the bandwidth; (3.3C.2) The probability density of each observation in the mixture is estimated using a finite mixture model, expressed by the formula: （13） (3.3C.3) Using the probability density data obtained in step (3.3C.1) as experimental values ​​and the probability density data obtained in step (3.3C.2) as theoretical values, the model parameters under the least squares condition are estimated using numerical optimization methods. , and The value of .

20. The method according to claim 19, characterized in that, The numerical optimization method includes any one of the Levenberg–Marquardt algorithm, the trust region reflection algorithm, or the trust region dogleg algorithm.

21. The method according to claim 20, characterized in that, The numerical optimization process also includes... Between and / or Assumptions and limitations are made regarding the quantitative relationships between them.

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