CN117388127B - Method for quantitatively measuring content and concentration of particle components in mixture - Google Patents
Method for quantitatively measuring content and concentration of particle components in mixtureInfo
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Abstract
The invention provides a method for quantitatively measuring the content and concentration of particle components in a mixture, which comprises the steps of (1) carrying out spICP-MS measurement on the mixture and each potential component, (2) carrying out probability density estimation on a measured signal to obtain a probability density function of each potential component, (3) establishing a limited mixing model by using the probability density function and estimating model parameters to obtain the content of each potential component, (4) deleting the component number of the potential component based on the obtained content and combining a content threshold value to obtain the component number, composition and content of the real component in the mixture, (5) repeating the step (3) based on the information of the real component to obtain the content of each real component after the potential component is limited, and (6) based on the obtained content, combining the transmission efficiency and sample injection speed data of spICP-MS, and converting to obtain the concentration of each real component. The method improves the measurement resolution and accuracy of the particle size, content and concentration of the particles.
Description
Technical Field
The invention belongs to the technical field of analysis and detection, relates to a method for detecting a mixture, and particularly relates to a method for quantitatively measuring the content and concentration of particle components in the mixture.
Background
The physical and chemical properties of nanoparticles are highly dependent on their size ranges, so accurate determination of the particle size distribution and content of nanoparticles is an important step in understanding their properties and technical applications. Currently, there are a variety of measurement methods that can determine 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 more than three times the size difference to distinguish between the different particle components, nanoparticle Tracking Analysis (NTA) has better size resolution than DLS as a counting method, but still requires about 50% of the size difference to distinguish between the different nanoparticle components, and tends to overestimate the number of larger particles in the mixture.
In order to increase the dimensional resolution of the mixed system, researchers have developed a promising alternative, a combination of separation and detection techniques. A number of size-dependent 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 including DLS, ultraviolet visible spectroscopy (UV-vis), multi-angle light scattering (MALS), inductively coupled plasma-optical emission (ICP-OES) and mass spectrometry (ICP-MS).
S nchez-Cachero et al (A. Sanchez-Cachero, S. Lopez-Sanz, N.R.Farinas, A.Rios, R.Martin-Doimeadios, talanta 222 (2021) 121513) studied the effect of organic on platinum nanoparticle size using asymmetric flow field flow separation (AF 4-ICP-MS) techniques. The results show that under optimal elution conditions, 5nm and 30nm platinum nanoparticles can achieve good separation, but that the elution peaks of 30nm and 50nm platinum nanoparticles overlap partially under the same elution conditions. TECHARANG et al (T.Techarang, anal Chim Acta 1144 (2021) 102-110.) achieve partial separation of 5nm and 15nm gold nanoparticle (AuNPs) mixtures by electric field flow separation. Johnson et al (M.E.Johnson, A.R.Montoro Buston, M.R. Winchester, anal Bioanal Chem 408 (27) (2016) 7629-7640.) performed better separation of 30nm, 80nm, 150nm AuNPs mixtures using sucrose density gradient centrifugation. However, the separation of 30nm, 60nm, 100nm AuNPs mixtures from 20nm, 50nm, 100nm AuNPs mixtures was poor due to the reduction in particle size difference.
Despite significant advances in combinatorial techniques, complete separation and accurate quantification of small size differential particle components remains a serious challenge, particularly when the diameter of the separated particles in the mixture is greater than 20nm, and separation of particle mixtures having a size differential of less than 10nm is difficult, and the separation technique may introduce additional disturbances and deviations, such as adsorption, segregation, etc. that cause the physical parameters measured by the back-end detection technique to deviate from the original properties of the sample.
Single particle inductively coupled plasma mass spectrometry (spICP-MS) is an emerging mass spectrometry technique for particle size distribution and particle number concentration measurement in recent years. Mitrano et al (D.M.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 40nm and 80nm silver nanoparticle (AgNPs) mixture separated well, there was a substantial overlap in the integrated intensity histograms of particles in a 60nm and 80nm AgNPs mixture. Westerhoff et al (X.Bi,S.Lee,J.F.Ranville,P.Sattigeri,A.Spanias,P.Herckes,P.Westerhoff,Journal of Analytical Atomic Spectrometry 29(9)(2014)1630.) used the K-means clustering algorithm to obtain a minimum distinguishable size difference of about 20nm for the primary and secondary nanoparticles. However, due to the further overlapping of the particle distributions, the mixture of closely sized particle components still cannot be correctly resolved.
Therefore, how to provide a method for quantitatively measuring the content and concentration of the particle components in the mixture, and improving the measurement resolution and accuracy of the particle size, content and concentration of the measured particles, is an urgent problem to be solved by those skilled in the art.
Disclosure of Invention
The invention aims to provide a method for quantitatively measuring the content and concentration of particle components in a mixture, which is based on a nuclear density estimation method and a finite mixing model analysis processing spICP-MS data, and improves the measurement resolution and accuracy of the particle size, the content and the concentration of the measured particles.
In order to achieve the aim of the invention, the invention adopts the following technical scheme:
The present invention provides a method for quantitatively measuring the content and concentration of particulate components in a mixture, which is directed to mixtures of known potential constituent components.
The method comprises the following steps:
(1) Performing spICP-MS measurements on the mixture and each potential component of the mixture separately to obtain a measured signal for the mixture and each potential component;
(2) Carrying out probability density distribution estimation on the detected signals of each potential component obtained in the step (1) by using a nuclear density estimation method to obtain probability density functions of each potential component;
(3) Establishing a limited mixing model by using the probability density function of each potential component obtained in the step (2), and estimating model parameters of the limited mixing model by using the measured signals of the mixture obtained in the step (1) to obtain the content of each potential component in the mixture;
(4) Deleting the component number of potential components in the mixture based on the content obtained in the step (3) and combining with a specified content threshold value to obtain the component number, composition and content of real components in the mixture;
(5) Repeating step (3) based on the information of the real components in the mixture obtained in step (4), to obtain the content of each real component in the mixture after the potential components are limited;
(6) And (3) based on the content obtained in any one of the steps (3) - (5), combining spICP-MS transmission efficiency and sample injection speed data, and obtaining the concentration of each real component in the mixture after conversion.
The invention uses a finite mixture model (FMKDE) consisting of a kernel estimation of probability density functions to describe a mixture of known potential constituents, and analyzes the measured signals obtained by spICP-MS to obtain the content and concentration of each real component in the mixture of known potential constituents. The method fully utilizes the statistical characteristics of the measured signals of each potential component, so that the resolution, content and concentration measurement accuracy of the particle size are obviously improved when spICP-MS is applied to the measurement of the mixture of known potential components.
Specifically, the method is based on a nuclear density estimation method and a limited mixing model for analyzing and processing spICP-MS data, realizes quantitative measurement of the content and concentration of the particle components in the mixture, can reduce the grain size resolution limit of the particle mixture to 5nm, reduces the absolute content deviation to below 3% in the multicomponent mixture, remarkably improves the grain size resolution and content accuracy of spICP-MS multicomponent test, does not need complex separation treatment, and is beneficial to promoting popularization and application of spICP-MS in the related field of nano technology.
In addition, the invention adopts spICP-MS to measure each potential component separately, and then uses a nuclear density estimation method to estimate the probability density distribution of each potential component, thereby avoiding the preset constraint on the function form of the probability density function of each potential component.
In the present invention, the contents of steps (3) to (5) include the percentage of the total particle count of the number of particles contained in each component (potential component or true component) of the mixture, and the concentration of step (6) includes the number concentration or mass concentration of the particles or particles of a specific particle size in the mixture.
Preferably, the mixture of known potential constituent components is specifically a mixture of at least 2 potential components of known type mixed in known or unknown proportions, for example 2, 5,10,15,20, 25 or 30, but not limited to the values recited, other non-recited values within the range of values being equally applicable.
The method provided by the invention is directed to mixtures in which the specific types of potential components need to be known, without the specific proportions of each potential component having to be specified. For example, the mixture may be an initial mixture, or may be a mixture after being placed, used or otherwise treated, wherein the content of each potential component in the mixture is unchanged or has been changed, or wherein the content of a part of the potential components is reduced to 0, and the content of the remaining potential components is left undetermined. The above-mentioned cases are applicable to the method provided by the present invention, and are not particularly limited herein.
In the invention, the potential components in the steps (1) - (5) correspond to the constituent components in the initial mixture or the potential constituent components in the subsequent mixture, the actual components in the steps (4) - (6) correspond to the actual constituent components of the subsequent mixture deduced from the measurement results, and if the actual constituent components are known, no presumption is needed, and only the potential components and the actual components are required to be set as the actual constituent components.
Preferably, the measured signals of the mixture are expressed as probability density functions in a finite mixture model, expressed as:
Where g (x; ψ) is the probability density function of the finite mixture model, x is the integrated intensity ;ψ＝(π1,...,πg,a1,...ag,h1,...hg,f1,...,fg) of the individual particles measured by spICP-MS, g is the total number of potential component types, pi j is the content of the j-th potential component, f j (x) is the probability density function of the j-th potential component, a j is the drift correction parameter set for the possible instrument signal drift phenomenon when the j-th potential component and the mixture are measured respectively, and h j is the bandwidth.
In the invention, the comprehensive strength x refers to the integral strength of a continuous pulse count signal corresponding to particles or the physical quantity converted from the integral strength, including the mass of the particles and the equivalent particle diameter, during spICP-MS measurement. Because the integral intensity corresponding to the particles can be converted into the mass of the particles or equivalent particle size by matching with standard particles or other calibration methods, the converted physical quantity is still applicable to the invention. The integrated intensity refers to the sum or integral value of the continuous pulse count signal associated with the measurement event of the single particle measured by spICP-MS, and can also be the total count value after background subtraction, and the data of the integrated intensity can be read from the data provided by the ICP-MS instrument or can be obtained by the data processing of the continuous pulse count signal based on the more foundation.
In the invention, the formula (1) corrects the observed value of the mixture by negative sign, namely x-a j, and it should be clear to a person skilled in the art that the negative sign only relates to the relativity of drift, the form of x+a j can also be adopted, and a j takes the opposite number, or the observed value of the mixture is not corrected, but only the observed value of the j potential component is corrected, the correction effects of the two are the same, or when the instrument signal drift phenomenon does not exist, a j takes zero.
Preferably, the measured signal of step (1) is integrated intensity data corresponding to each particle in the mixture sampleAnd the expression of the finite mixture model, equation (1), includes the integrated intensity data corresponding to each particle in the latent component
In the invention, theIs a set of independent samples distributed at the same time and is sampled by a probability density function g (x; psi) of the comprehensive strength of the mixtureIs a set of independent co-distributed samples and is sampled at a probability density function f j (x) of the integrated intensity of each potential component.
Preferably, the nuclear density estimation method of step (2) is expressed as:
wherein, the Kernel estimation of probability density function for jth potential component, K () kernel function, m potential component observation sampleH j is the bandwidth.
Preferably, the functional form of the kernel function includes any one of a gaussian function, a rectangular function, a trigonometric function, a cosine function or an Epanechnikov function, and further preferably, a gaussian function, and is expressed as a formula:
I.e. Is a gaussian kernel function.
Preferably, the bandwidth value method comprises manual assignment or calculation assignment, and further preferably calculation assignment.
The manual assignment is specifically that different bandwidth h j values are manually set to be plotted and drawnAnd then selecting a reasonable h j value according to the image.
Preferably, the criterion or equation based on which the calculation assignment is based includes any one of Silverman rule of thumb, scott rule of thumb, unbiased cross-validation, biased cross-validation, sheather-Jones method or bootstrap method, and further preferably Silverman rule of thumb, expressed as:
lO＝min(sd(xi),IQR(xi)/1.34) (4)
hj＝0.9×lO×m-0.2 (5)
Wherein sd (x i) is a sample of the observations of potential components IQR (x i) is the difference between the quarter-bit distance, i.e., three-quarters-bit, and one-quarter-bit of the observation sample, min () is the minimum function, and m is the number of observations of the j-th potential component.
Preferably, the estimation method of the model parameter in the step (3) includes any one or a combination of at least two of maximum likelihood estimation, bayesian discrimination, and least square estimation, and typically, but not limited to, a combination of maximum likelihood estimation and bayesian discrimination, or a combination of least square estimation and bayesian discrimination.
Preferably, the estimation method comprises the steps of:
(3.1) initializing the content of each potential component in the mixture, expressed by the formula:
Where j=1, 2, once again, g, and g is the total number of types of potential components.
Other values within interval (0, 1) may also be employed in the present invention to initialize the level of each potential component in the mixture.
(3.2) Initializing drift correction parameters of the measured signals of the mixture, and expressing the drift correction parameters as the formula:
Where j=1, 2, once again, g, and g is the total number of types of potential components.
Other values may be used to initialize the drift correction parameters of the measured signal of the mixture, typically a value close to 0. For example, -4 to 4 for integrated intensity and-1 nm to 1nm for particle diameter. When the absolute value of the initial value is too large, convergence failure or convergence to an abnormal value may occur, and an excessively large absolute value of the convergence value a j means that the instrument drifts seriously, and the instrument needs to be adjusted to a stable state for measurement. When the instrument signal drift phenomenon is not present or negligible, then a j is constant at zero.
(3.3) The method of estimating model parameters includes any one or a combination of at least two of the following methods:
(3.3A) the maximum likelihood estimation includes estimating a model parameter pi j under the condition of maximum likelihood probability by adopting a numerical optimization method, and may further include values of a j and h j, specifically:
The log likelihood probability (the logarithmic value of likelihood probability) is adopted as the object of numerical optimization, and expressed as the following formula:
wherein logL is log likelihood probability, n is mixture observation sample The number of observations in (a); Is an estimate of f j (x).
In the invention, the meanings of the rest parameters of the formula (8) are the same as those of the formula (1), so that description is omitted herein, the initial values of the parameters pi j and a j are shown in formulas (6) - (7), the parameter a j does not participate in optimization, namely a j is constant to 0, the parameter h j does not participate in optimization and is fixed as the bandwidth parameters of each potential component obtained by the kernel density estimation in the step (2), namely formulas (2) - (5), and the bandwidth parameters are also used as initial values.
In the invention, the log likelihood probability is adopted as the object of numerical optimization, so that the overflow of the calculated numerical value can be effectively prevented.
Preferably, the numerical optimization method comprises any one of a Brent algorithm, a Nelder-Mead algorithm, a BFGS quasi-Newton algorithm, a L-BFGS-B quasi-Newton algorithm, a conjugate gradient algorithm or a SANN simulated annealing algorithm.
Preferably, the numerical optimization process further includes the step of assuming and defining the quantitative relationship between a j and/or h j to reduce the number of fitted parameters, thereby improving convergence characteristics.
(3.3B) the model parameter pi j in step (3.3A) may also be estimated using Bayesian discriminant, comprising the steps of:
(3.3B.1) calculating a posterior probability that the composite intensity observations of the mixture belong to a certain potential component, expressed as:
Where i=1, 2,..n, and n is the number of integrated intensity observations of the mixture, j=1, 2,..g, and g is the total number of classes of potential components.
(3.3B.2) updating the content of each potential component in the mixture, expressed by the formula:
(3.3B.3) iterating pi j and lambda j using equation (9) and equation (10), the upper corner marks t and t+1 representing the iteration steps, the termination condition of the iteration based on the variation of pi j expressed as:
Where epsilon is a specified tolerance, preferably epsilon=10 -5.
(3.3C) the least squares estimation comprising the steps of:
(3.3C.1) estimating probability density distribution of each observed value of the mixture by using a nuclear density estimation method, wherein the probability density distribution is expressed as the formula:
wherein K () is a kernel function and n is a sample of the observed values of the mixture H is the bandwidth.
In this step, the estimation method of the kernel density estimation is similar to the formulas (2) - (5), so that the description thereof will be omitted here.
(3.3C.2) estimating probability densities of observations of the mixture using a finite mixture model, expressed as:
(3.3C.3) taking the probability density data obtained in the step (3.3C.1) as an experimental value, taking the probability density data obtained in the step (3.3C.2) as a theoretical value, estimating a model parameter pi j under the condition of least square by adopting a numerical optimization method, and further comprising values of a j and h j.
Preferably, the numerical optimization method comprises any one of a Levenberg-Marquardt algorithm, a trusted region reflection algorithm or a trusted region dogleg algorithm.
Preferably, the numerical optimization process further includes the step of assuming and defining the quantitative relationship between a j and/or h j to reduce the number of fitted parameters, thereby improving convergence characteristics.
In the present invention, the model parameter pi j in the step (3.3C) may also be estimated by using a bayesian discriminant method (3.3B), specifically including steps (3.3 b.1) - (3.3 b.3), which will not be described herein.
In the invention, when the performance of the instrument is stable, the drift correction parameter a j of the instrument can be constantly 0, the estimation algorithm is still applicable, and when only the parameter pi j is needed to be estimated, the maximum likelihood estimation or the least square estimation can be not used, and the improved Bayesian judgment estimation method, namely the steps (3.3 B.1) - (3.3 B.3), can be used for directly obtaining the estimation value of the parameter pi j.
Compared with the prior art, the invention has the following beneficial effects:
the method provided by the invention is based on a nuclear density estimation method and a finite mixing model analysis processing spICP-MS data, realizes quantitative measurement of the content and concentration of the particle components in the mixture, can reduce the grain size resolution limit of the particle mixture to 5nm, reduces the absolute content deviation to below 3% in the multicomponent mixture, obviously improves the grain size resolution and content accuracy of spICP-MS multicomponent test, does not need complex separation treatment, and is beneficial to promoting popularization and application of spICP-MS in the related field of nano technology.
Drawings
FIG. 1 is a probability density function image of four potential components of the method provided in example 1;
FIG. 2 is a probability density function image of a sample mixture for the method provided in example 1, and corresponding to normalized probability density function image for each potential component by content;
FIG. 3 is an image of the probability density function of a sample mixture in the method provided in example 2, corresponding to the normalized probability density function of each real component by content.
Detailed Description
The technical scheme of the invention is further described by the following specific embodiments. It will be apparent to those skilled in the art that the examples are merely to aid in understanding the invention and are not to be construed as a specific limitation thereof.
The present invention provides a method for quantitatively measuring the content and concentration of particulate components in a mixture, which is directed to mixtures of known potential constituent components, in particular mixtures of at least 2 potential components of known kind mixed in known or unknown proportions. The measured signals of the mixture are expressed as probability density functions by a finite mixture model, and expressed as:
Where g (x; ψ) is the probability density function of the finite mixture model, x is the integrated intensity ;ψ＝(π1,...,πg,a1,...ag,h1,...hg,f1,...,fg) of the individual particles measured by spICP-MS, g is the total number of potential component types, pi j is the content of the j-th potential component, f j (x) is the probability density function of the j-th potential component, a j is the drift correction parameter set for the possible instrument signal drift phenomenon when the j-th potential component and the mixture are measured respectively, and h j is the bandwidth.
The method comprises the following steps:
(1) Performing spICP-MS measurement on the mixture and each potential component in the mixture separately to obtain a measured signal of the mixture and each potential component, wherein the measured signal is the comprehensive intensity data corresponding to each particle in the mixture sample Integrated intensity data corresponding to each particle in a potential component sample
(2) And (3) estimating probability density distribution of the detected signals of each potential component obtained in the step (1) by using a kernel density estimation method to obtain probability density functions of each potential component, wherein the kernel density estimation method is expressed as follows:
wherein, the Kernel estimation of probability density function for j-th potential component, K () is kernel function, m is potential component observation sampleH j is the bandwidth.
The kernel function adopts a Gaussian kernel function and is expressed as a formula:
The bandwidth value method is based on Silverman rule of thumb and expressed as the following formula:
lO＝min(sd(xi),IQR(xi)/1.34) (4)
hj＝0.9×lO×m-0.2 (5)
Wherein sd (x i) is a sample of the observations of potential components IQR (x i) is the difference between the quarter-bit distance, i.e., three-quarters-bit, and one-quarter-bit of the observation sample, min () is the minimum function, and m is the number of observations of the j-th potential component.
(3) Establishing a finite mixing model by using the probability density function of each potential component obtained in the step (2), and estimating model parameters to obtain the content of each potential component in the mixture, wherein the estimating method of the model parameters comprises any one or a combination of at least two of maximum likelihood estimation, bayesian discrimination or least square estimation, and the estimating method comprises the following steps:
(3.1) initializing the content of each potential component in the mixture, expressed by the formula:
Where j=1, 2, once again, g, and g is the total number of types of potential components.
(3.2) Initializing drift correction parameters of the measured signals of the mixture, and expressing the drift correction parameters as the formula:
Where j=1, 2, once again, g, and g is the total number of types of potential components.
(3.3) The method of estimating model parameters includes any one or a combination of at least two of the following methods:
(3.3A) the maximum likelihood estimation includes estimating a model parameter pi j under the condition of maximum likelihood probability by adopting a numerical optimization method, and may further include values of parameters a j and h j, specifically:
The log likelihood probability is adopted as an object of numerical optimization, and expressed as a formula:
wherein logL is log likelihood probability, n is mixture observation sample The number of observations in (a); Is an estimate of f j (x).
The numerical optimization method comprises any one of a Brent algorithm, a Nelder-Mead algorithm, a BFGS quasi-Newton algorithm, a L-BFGS-B quasi-Newton algorithm, a conjugate gradient algorithm or a SANN simulated annealing algorithm.
The numerical optimization process also comprises the steps of supposing and limiting the number relation between a j and/or h j so as to reduce the number of fitted parameters and improve convergence characteristics.
(3.3B) the model parameter pi j in step (3.3A) may also be estimated using Bayesian discriminant, comprising the steps of:
(3.3B.1) calculating a posterior probability that the composite intensity observations of the mixture belong to a certain potential component, expressed as:
Where i=1, 2,..n, and n is the combined strength observation of the mixture, j=1, 2,..g, and g is the total number of classes of potential components.
(3.3B.2) updating the content of each potential component in the mixture, expressed by the formula:
(3.3B.3) iterating pi j and lambda j using equation (9) and equation (10), the upper corner marks t and t+1 representing the iteration steps, the termination condition of the iteration based on the variation of pi j expressed as:
Where ε is a specified tolerance and ε=10 -5.
(3.3C) the least squares estimation comprising the steps of:
(3.3C.1) estimating probability density distribution of each observed value of the mixture by using a nuclear density estimation method, wherein the probability density distribution is expressed as the formula:
wherein K () is a kernel function and n is a sample of the observed values of the mixture H is the bandwidth.
(3.3C.2) estimating probability densities of observations of the mixture using a finite mixture model, expressed as:
And (3.3C.3) taking the probability density data obtained in the step (3.3C.1) as an experimental value, taking the probability density data obtained in the step (3.3C.2) as a theoretical value, estimating a model parameter pi j under the least square condition by adopting a numerical optimization method, and further comprising the values of parameters a j and h j, wherein the numerical optimization method comprises any one of a Levenberg-Marquardt algorithm, a trusted region reflection algorithm or a trusted region dog leg algorithm, and the numerical optimization process further comprises the steps of carrying out assumption and limitation on the quantitative relation between a j and/or between h j so as to reduce the number of fitted parameters, thereby improving convergence characteristics.
The model parameter pi j in the step (3.3C) may also be estimated by using a bayesian discriminant method (3.3B), specifically including steps (3.3 b.1) - (3.3 b.3), which will not be described herein.
(4) And (3) deleting the component number of the potential components in the mixture based on the content obtained in the step (3) and combining the specified content threshold value to obtain the component number, composition and content of the real components in the mixture.
(5) Repeating the step (3) based on the information of the real components in the mixture obtained in the step (4), so as to obtain the content of each real component in the mixture after the potential components are limited.
(6) And (3) based on the content obtained in any one of the steps (3) - (5), combining spICP-MS transmission efficiency and sample injection speed data, and obtaining the concentration of each real component in the mixture after conversion.
Example 1
The embodiment provides a method for quantitatively measuring the content and concentration of particle components in a mixture, which takes A, B, C, D gold nanoparticles with different particle diameters as potential components, selects 2-3 of the potential components to be mixed according to different proportions, obtains a four-group mixture model system with undetermined component number, composition and content of real components, takes the four-group mixture model system as a mixture sample of known potential component to be measured, and is numbered 11-14.
In this embodiment, since the instrument state is stable and the instrument signal drift phenomenon is negligible, the drift correction parameter a j of the finite-hybrid model is set to 0, the bandwidth parameter h j is not optimized, and the improved Bayesian Discriminant (BD) is used to estimate the model parameters. The preparation content and concentration of the four mixture samples and spICP-MS measurement results are shown in Table 1, and the following demonstrates the specific implementation process:
(1) Conventional spICP-MS measurement is carried out on A, B, C, D potential components under a single particle Time Resolution Analysis (TRA) mode of an ICP-MS instrument to obtain comprehensive intensity data corresponding to each nanoparticle in each potential component, namely four groups And data, wherein m is the observed number of the nano particles in each potential component, and the integrated intensity data specifically selects integrated intensity. The resulting integrated intensity data was calibrated with standard nanoparticle size to give A, B, C, D individual nanoparticle average sizes of the four potential components, 33nm, 44nm, 49nm, 61nm, respectively (particle size data is not necessary in component content measurements, and is provided in this example to facilitate a better understanding of the dimensional resolution improvement of the present invention by those skilled in the art).
(2) Estimating probability density distribution of the comprehensive strength of each potential component obtained in the step (1) by using a kernel density estimation method, and calculating by using a Gaussian kernel function shown in the formula (3) and Silverman rule of thumb shown in the formulas (4) - (5)Wherein j=A, B, C, D and x are the comprehensive intensity, the probability density function image of the four potential components obtained by the kernel density estimation method is shown in figure 1, and the histogram is used as a comparison in figure 1. The nuclear density estimation method still has good descriptive ability despite the tailing of the integrated intensity distribution of all four potential components.
(3) Conventional spICP-MS measurements were performed on each of four sets of samples (numbered 11-14) of mixtures of known potential constituent components to be tested to obtain corresponding integrated intensity observations for each nanoparticle in each set of samples, i.e., four setsData, where n is the observed number of nanoparticles in each mixture sample.
(4) The data of the comprehensive intensity observation value obtained in the step (3) are calculated according to formulas (6) - (7) and (9)
- (11) An iterative calculation of pi j and lambda j is performed and a j is set constant to 0, where j=a, B, C, D. In formula (9)Is obtained in the step (2)And j=a, B, C, D. And obtaining a convergence value of pi j according to the iteration termination condition shown in the formula (11), namely the respective contents of A, B, C, D potential components in each group of mixture samples. The content measurement results of the four groups of mixture samples numbered 11-14 in this example are shown in Table 1, the probability density function of the mixture samples and the probability density function corresponding to the normalization of the contents of the potential components can be obtained by using the convergence value of pi j, and the image of the correlation function is shown in FIG. 2.
(5) In this embodiment, the content of each potential component obtained in step (4) is further processed as needed to determine the component number, composition and content of the real component in each mixture sample, specifically, to remove the potential component with a content of less than 1%, taking the 11 # mixture sample as an example, and after removing the potential components C and D, only the potential components a and B remain as the real components of the 11 # mixture sample. The content of the real component can be directly measured by using the content measurement value obtained in the step (4), or the potential component can be limited to be the content measurement value obtained by repeating the step (4) for the potential component A and the potential component B, and the content of the real component in each mixture sample is shown in the table 1. As can be seen from Table 1, the difference between the two measured values of each mixture sample in this example was less than 1%, and the composition, constitution and content of the actual components were identical to those of the actual conditions, and the difference in content was less than 3%.
(6) In this embodiment, the content data is multiplied by the total number of observations per minute of the mixture samples based on the content data obtained in step (4) and/or step (5), in combination with the transmission efficiency data of spICP-MS, to obtain the number of observations per minute of each real component, and then divided by the product of the transmission efficiency and the sample injection speed to obtain the particle number concentration of each real component in each mixture sample, as shown in table 1.
TABLE 1
As can be seen from Table 1, the result of spICP-MS measurement obtained by the method provided by this example is substantially identical to the configuration value of the mixture, wherein the difference in particle size of the B+C bi-component particles of the No. 12 mixture sample is only 5nm, and the correct content measurement result is also obtained, and the absolute deviation of the content measurement values of each group of mixture samples in Table 1 is less than 3%, thus further proving the accuracy and applicability of the method.
Example 2
The embodiment provides a method for quantitatively measuring the content and concentration of a particle component in a mixture, wherein two gold nanoparticles B and C are used as potential components, mixed according to different proportions to form a three-group mixture model system with undetermined content, and the three-group mixture model system is used as a mixture sample of known potential constituent components to be measured and is numbered 21-23.
In this embodiment, spICP-MS test is performed under the condition that the instrument signal has a significant drift phenomenon, and the test results are analyzed by adopting a finite mixture model with no drift correction (i.e. a j =0 is given for j) and with drift correction (i.e. a j +.0 is given for j). The configuration content and concentration of the three mixture samples and spICP-MS measurement results are shown in Table 2, and the following demonstrates the specific implementation process:
(1) Conventional spICP-MS measurement is carried out on the two potential components B and C respectively under the single particle Time Resolution Analysis (TRA) mode of an ICP-MS instrument to obtain comprehensive intensity data corresponding to each nanoparticle in the potential components, namely two groups And data, wherein m is the observed number of the nano particles in each potential component, and the integrated intensity data specifically selects integrated intensity.
(2) Estimating probability density distribution of the integral intensity of each potential component obtained in the step (1) by adopting a kernel density estimation method, and calculating by using a Gaussian kernel function shown in the formula (3) and Silverman rule of thumb shown in the formulas (4) - (5)Where j=b, C, x is the integrated intensity.
(3) Conventional spICP-MS measurements were performed on three sets of mixture samples (numbered 21-23) of known potential constituent components to be tested, respectively, to obtain integrated intensity observations corresponding to the nanoparticles in each set of mixture samples, i.e., three setsData, where n is the observed number of nanoparticles in each mixture sample.
(4) And (3) carrying out drift-free correction on the integral intensity observed value data obtained in the step (3) (i.e. taking j to have a j =0). This case is similar to the data processing procedure of example1, estimated using the modified Bayesian discriminant estimation (BD) with j=B, C constant a j, and performing iterative calculations of pi j and lambda j according to equations (6) - (7) and equations (9) - (10), the probability density distribution function in equation (9)Is obtained in the step (2)And j=b, C. And obtaining a convergence value of pi j according to the iteration termination condition shown in the formula (11), namely the respective contents of the two potential components B and C in each group of mixture samples. The results of measuring the contents of three groups of mixture samples numbered 21-23 in this example are shown in Table 2, and the probability density function of the mixture samples and the probability density function corresponding to normalization of the contents of the potential components can be obtained by using the convergence value of pi j, and the image of the correlation function is shown in FIG. 3.
(5) And (3) carrying out limited mixed model description containing drift correction on the integral intensity observed value data obtained in the step (3) (i.e. j exists so that a j is not equal to 0). In the embodiment, a maximum likelihood estimation Method (MLE) is adopted to estimate a content parameter pi j, a drift parameter a j and a bandwidth parameter h j, an L-BFGS-B quasi-Newton algorithm is used to optimize parameters pi j、aj and h j, the value of a log likelihood function defined by a formula (8) is maximized, and optimized values of parameters pi j、aj and h j are obtained, wherein the parameters pi j, j=B and C are the contents of two potential components B and C in a mixture sample. The content measurement results of the mixture samples numbered 21 to 23 in this example are shown in Table 2. The probability density function of the mixture sample can be obtained by using the optimized values of the parameters pi j、aj and h j, the probability density function corresponding to the normalization of the potential components by content, and the related function image is shown in figure 3.
In the embodiment, a least squares estimation method (NLS) is adopted to estimate the content parameter pi j, the drift parameter a j and the bandwidth parameter h j, a trust zone reflection algorithm is used to optimize the parameters pi j、aj and h j, the probability density estimated difference of each observed value in the mixture obtained by calculation in the formula (12) and the formula (13) is minimized under the least square condition, and the optimized values of the parameters pi j、aj and h j are obtained, wherein the parameters pi j, j=B and C are the contents of two potential components B and C in a mixture sample. The content measurement results of the mixture samples numbered 21 to 23 in this example are shown in Table 2. The probability density function of the mixture sample can be obtained by using the optimized values of the parameters pi j、aj and h j, the probability density function corresponding to the normalization of the potential components by content, and the related function image is shown in figure 3.
TABLE 2
As can be seen from Table 2, the limited mixing model and the related algorithm described in this example give spICP-MS measurement results with drift correction which are substantially consistent with the configuration values of the mixture, and in the case of significant drift of the apparatus, the limited mixing model with drift correction gives more reliable content values, the absolute deviation of the content of which is less than 3%, which further proves the applicability and accuracy of the limited mixing model and the related algorithm.
Therefore, the method provided by the invention is based on a nuclear density estimation method and a finite mixing model analysis processing spICP-MS data, realizes quantitative measurement of the content and concentration of the particle components in the mixture, can reduce the particle size resolution limit of the nanoparticle mixture to 5nm, and obviously improves the particle size resolution and content accuracy of spICP-MS multicomponent test without complex separation treatment in the multicomponent mixture, thereby being beneficial to promoting popularization and application of spICP-MS in the nanotechnology field.
The applicant declares that the above is only a specific embodiment of the present invention, but the scope of the present invention is not limited thereto, and it should be apparent to those skilled in the art that any changes or substitutions that are easily conceivable within the technical scope of the present invention disclosed by the present invention fall within the scope of the present invention and the disclosure.
Claims (21)
1. A method for quantitatively measuring the content and concentration of particulate components in a mixture, characterized in that the method is directed to mixtures of known potential constituent components;
The method comprises the following steps:
(1) Performing spICP-MS measurements on the mixture and each potential component of the mixture separately to obtain a measured signal for the mixture and each potential component;
(2) Carrying out probability density distribution estimation on the detected signals of each potential component obtained in the step (1) by using a nuclear density estimation method to obtain probability density functions of each potential component;
(3) Establishing a limited mixing model by using the probability density function of each potential component obtained in the step (2), and estimating model parameters of the limited mixing model by using the measured signals of the mixture obtained in the step (1) to obtain the content of each potential component in the mixture;
(4) Deleting the component number of potential components in the mixture based on the content obtained in the step (3) and combining with a specified content threshold value to obtain the component number, composition and content of real components in the mixture;
(5) Repeating step (3) based on the information of the real components in the mixture obtained in step (4), to obtain the content of each real component in the mixture after the potential components are limited;
(6) And (3) based on the content obtained in any one of the steps (3) - (5), combining spICP-MS transmission efficiency and sample injection speed data, and obtaining the concentration of each real component in the mixture after conversion.
2. The method according to claim 1, wherein the mixture of known potential constituent components is in particular a mixture of at least 2 potential components of known type mixed in known or unknown proportions.
3. The method of claim 2, wherein the measured signal of the mixture is formulated as a finite mixture model representing a probability density function expressed as:
(1)
wherein, the X is the integrated intensity of single particles measured by spICP-MS;=(,...,,,...,,...,,..., ) G is the total number of types of potential components; is the content of the j potential components; probability density function for the j-th potential component; is a drift correction parameter set for the possible instrument signal drift phenomenon during the respective measurement of the jth potential component and the mixture; Is the bandwidth.
4. A method according to claim 3, wherein the measured signal of step (1) is integrated intensity data corresponding to each particle in the mixture sampleIntegrated intensity data corresponding to each particle in a potential component sample。
5. The method of claim 4, wherein the kernel density estimation method of step (2) is formulated as:
(2)
wherein, the Kernel estimation of probability density function for j-th potential component, K () is kernel function, m is potential component observation sampleThe number of observations in (a); Is the bandwidth.
6. The method of claim 5, wherein the functional form of the kernel function comprises any one of a gaussian function, a rectangular function, a trigonometric function, a cosine function, or an Epanechnikov function.
7. The method of claim 6, wherein the kernel function is in the form of a gaussian function formulated as:
(3)
I.e. Is a gaussian kernel function.
8. The method of claim 5, wherein the bandwidth valuation method comprises manually assigning or calculating a value.
9. The method of claim 8, wherein the bandwidth valuation method is a calculation valuation.
10. The method of claim 8, wherein the criteria or equations upon which the calculated assignment is based include any of Silverman rule of thumb, scott rule of thumb, unbiased cross-validation, biased cross-validation, sheather-Jones method, or boottrap method.
11. The method of claim 10, wherein the criterion upon which the calculation is based is Silverman rule of thumb, formulated as:
(4)
(5)
wherein, the For potential component observations samplesIs a sample standard deviation of (2); is the difference between the quarter bit distance, i.e. the three-quarter bit number, and the quarter bit number of the observed value sample, min () is the minimum function, and m is the observed value number of the j-th potential component.
12. A method according to claim 3, wherein the method of estimating model parameters in step (3) comprises any one or a combination of at least two of maximum likelihood estimation, bayesian discriminant method or least square estimation.
13. The method according to claim 12, wherein the estimation method comprises the steps of:
(3.1) initializing the content of each potential component in the mixture, expressed by the formula:
(6)
Where j=1, 2, once again, g, and g is the total number of types of potential components;
(3.2) initializing drift correction parameters of the measured signals of the mixture, and expressing the drift correction parameters as the formula:
(7)
Where j=1, 2, once again, g, and g is the total number of types of potential components.
14. The method of claim 12 wherein the maximum likelihood estimation comprises estimating model parameters under maximum likelihood probability conditions using a numerical optimization method、AndSpecifically the values of (2):
(3.3A) using log likelihood probability as an object of numerical optimization, expressed as:
(8)
wherein, the Is log likelihood probability, n is mixture observation value sampleThe number of observations in (a); Is that Is a function of the estimate of (2).
15. The method of claim 14, wherein the numerical optimization method comprises any one of a Brent algorithm, a Nelder-Mead algorithm, a BFGS quasi-Newton algorithm, a L-BFGS-B quasi-Newton algorithm, a conjugate gradient algorithm, or a SANN simulated annealing algorithm.
16. The method of claim 15, wherein the process of numerical optimization further comprises the steps ofBetween and/orThe quantitative relationship between them is assumed and defined.
17. The method according to claim 12, wherein the model parameters estimated by the bayesian discriminant method compriseComprising the following steps:
(3.3B.1) calculating a posterior probability that the composite intensity observations of the mixture belong to a certain potential component, expressed as:
(9)
Wherein i=1, 2,..n, and n is the number of integrated intensity observations of the mixture; j=1, 2,..g, and g is the total number of classes of potential components;
(3.3B.2) updating the content of each potential component in the mixture, expressed by the formula:
(10)
(3.3B.3) pairs using equation (9) and equation (10) AndPerforming iteration, wherein upper corner marks t and t+1 represent iteration steps, and the termination condition of the iteration is based onIs expressed as:
(11)
wherein, the Is a specified tolerance.
18. The method of claim 17, wherein the specified tolerance=10-5。
19. The method of claim 12, wherein the least squares estimation comprises the steps of:
(3.3C.1) estimating probability density distribution of each observed value of the mixture by using a nuclear density estimation method, wherein the probability density distribution is expressed as the formula:
(12)
wherein K () is a kernel function and n is a sample of the observed values of the mixture H is the bandwidth;
(3.3c.2) estimating probability densities of observations of the mixture using a finite mixture model, expressed as:
(13)
(3.3C.3) estimating model parameters under the least squares condition by using probability density data obtained in the step (3.3C.1) as experimental values and probability density data obtained in the step (3.3C.2) as theoretical values through a numerical optimization method 、AndIs a value of (2).
20. The method of claim 19, wherein the method of numerical optimization comprises any one of a Levenberg-Marquardt algorithm, a trusted zone reflection algorithm, or a trusted zone dog leg algorithm.
21. The method of claim 20, wherein the process of numerical optimization further comprises the steps ofBetween and/orThe quantitative relationship between them is assumed and defined.
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