JP7639447B2 - Peak separation method and apparatus - Google Patents

Description

The present invention relates to a method and device for separating a signal containing multiple overlapping peaks into individual peaks and a baseline using a nonlinear function and a baseline function, and for accurately calculating the quantity of each peak.

Chromatography is a method used to separate mixtures of organic and inorganic substances. With the mobile phase flowing, a sample is introduced into a column filled with gels (separation materials) with various functional groups, and the mixture is separated based on differences in interactions, and the output is monitored using various detectors. This method performs qualitative analysis based on the elution time of the separated peaks, and quantitative analysis based on the degree of detector output. Ideally, accurate qualitative and quantitative analysis is possible by completely separating the mixture, but there are restrictions and limitations to separation performance, and there are many cases where complete separation is not possible (see Figure 1). If the separation of each component is insufficient, generally vertical slicing as shown in Figure 1b, base processing as shown in Figure 1c, and skimming as shown in Figure 1d are performed to calculate the area of each component, etc., and provide for quantification.

However, when the separation performance is extremely poor or when there is a large difference in the quantitative relationship between the components, it becomes difficult to accurately calculate the area, etc., regardless of the processing method. In such cases, the peak separation (peak fitting) method may be used. This is a method in which a part of the chromatogram where separation is insufficient is divided into individual peaks through mathematical processing (see Figure 2). This method assumes that the shape of the peak obtained in the chromatogram is theoretically normally distributed, and determines the coefficients by changing the coefficients of the normal distribution coefficient and repeating calculations so that the difference between the sum and the original chromatogram is minimized, that is, by the least squares method. This repeated calculation using the least squares method is an effective means of quantification for unseparated peaks, but there are many cases where accurate quantification is not possible.

First of all, the above-mentioned method is based on the assumption that the peaks are normally distributed. In actual chromatograms, peaks often have tailing or leading states due to non-specific adsorption of the sample to the gel or an inappropriate sample loading amount. In other words, the peak shape is no longer symmetrical and is not normally distributed, so accurate fitting calculations may not be possible (see Figure 3).

In addition, in chromatography, it is desirable to perform measurements with as stable a baseline as possible; an unstable baseline makes it impossible to perform accurate quantitative calculations. In reality, various factors, such as changes in the environment or the composition of the eluent used, can cause the baseline to tilt in a certain direction (drift) (Figure 4a) or undulate periodically (Figure 4b). When the peak intensity is strong, even slight fluctuations in the baseline do not have a significant effect on quantification, but when the peak intensity is weak, accurate quantification is not possible. The same is true when quantification is performed using peak separation (peak fitting). Furthermore, when the peak shape is not a symmetrical normal distribution and base fluctuations occur, it becomes difficult to perform accurate quantification using peak separation (peak fitting) (see Figure 5).

In addition, when performing peak separation using the least squares method, the initial value of the peak function is important. If an inappropriate initial value is used, the calculation results may not converge. Also, if the baseline in the region where the peak elutes is not horizontal, large errors will occur in the calculation.

To solve these problems, for the former, a distribution function other than the normal distribution function that produces an asymmetric peak shape is sometimes used. For the latter, measures are sometimes taken to offset the baseline before peak separation using the least squares method, bringing the baseline fluctuation in the calculation interval closer to zero. Even when such measures are taken, accurate peak separation is still not possible, and there are many cases in which the quantitative results are not reliable.

The object of the present invention is to provide a method for separating peaks in an unseparated chromatogram, which has been made to solve the above-mentioned problems, and which appropriately sets the initial value for peak separation and enables accurate peak separation even in the presence of baseline fluctuations, as well as a liquid chromatography device equipped with the above-mentioned peak separation method.

A method for peak separation comprising the steps of: separating a group of unseparated peaks consisting of two or more component peaks; and quantifying each of the peaks, comprising the steps of:
a first step of specifying at least three times, i.e., a start time, an end time, and a virtual time for peak separation, acquiring corresponding chromatogram output values, calculating a second- or higher-order polynomial from the three sets of times and outputs, calculating each coefficient of the polynomial, and acquiring a virtual baseline;
The peak function used in the peak separation is a non-normal distribution function,
Obtaining an initial value of a coefficient related to a peak top time among the coefficients of the non-normal distribution function for each peak from the group of peaks in the unseparated state;
a second step of applying an initial value of a coefficient relating to a peak width, among the coefficients of the non-normal distribution function for each peak, based on a value calculated from the peak top time and a predetermined number of columns;
Varying each coefficient of the virtual baseline obtained in the first step and a coefficient of a non-normal distribution function for each peak obtained in the second step;
The present invention provides a peak separation method, characterized in that processing is performed by the least squares method so that the difference between the sum of the non-normal distribution functions of each peak and the sum of the baseline function and the data to be processed is minimized, and the baseline function and each peak function are identified and quantified, and a liquid chromatography device equipped with the peak separation method is provided.

This is a method for separating peaks from an unseparated chromatogram, and it is possible to provide a method that can properly set the initial value for peak separation and can perform accurate peak separation even when there is baseline fluctuation, as well as a liquid chromatography device equipped with the peak separation method.

The process of the present invention will now be described in detail.
FIG. 6 is a flow chart showing the flow of the process of the present invention. FIG. 7 is a diagram showing a chromatogram in an unseparated state. The thick solid line in the figure shows the chromatogram. First, in the first step, at least three times are specified, namely, the start time TS (●), the end time TE (▲), and the virtual time TC (◯), at which peak separation is performed, and the corresponding output values (YS, YE, YC) of the chromatogram are obtained. Next, each coefficient of a polynomial of degree 2 or higher is calculated from the three sets of times and outputs, and a virtual baseline is obtained. Here, an example is described in which the virtual point is one point and is approximated by a quadratic expression. The virtual point may be a time between the start time and the end time, a time before the start time, or a time after the end time, and it is desirable to specify a time at which no peak is estimated to exist. In the case of a quadratic expression, the coefficient can be mathematically calculated by the following formula 1.

When there are two or more virtual points, the coefficients a, b, and c can be calculated using the least squares method. Here, the start point TS is set to 4.000 minutes, the virtual point TC to 6.000 minutes, and the end point TE to 14.000 minutes, and the coefficients of the quadratic equation are calculated as shown in Table 1.

In the second step, the number of columns N used for separation, the number of peaks n to be separated, and the apex time T of each peak are specified, and the corresponding output value y is obtained. The time can be specified manually while looking at the chromatogram, or it can be obtained automatically from the inflection point of the differential curve, and there is no limit to the method of specification.

Next, the theoretical peak width w is calculated from the number of plates N and the time t of the peak top. The formula for calculating the number of plates varies depending on the standard, but any formula can be used since the peak width is used as the initial value of the coefficient related to the peak width of the function that performs peak fitting. Here, the commonly used formula 2 is used for the calculation.

As shown in Table 2, four peaks were specified here: Peak_1: 7,500 minutes (output 104.249), Peak_2: 9,700 minutes (output 85.454), Peak_3: 10,600 minutes (output 68.381), and Peak_4: 11,900 minutes (output 28.473). In addition, the number of stages N was calculated as 4000 stages. The peak width is calculated as shown in the following formula 3.

Next, specify the peak function to be used for peak separation. In chromatography, theoretically, peaks are said to have a symmetric normal distribution, but in reality, due to various factors such as non-specific adsorption and sample overload, leading and tailing states occur, and the distribution is not symmetrical. In order to perform accurate peak separation, a non-normal function is specified as the peak function. Here, an asymmetric sigmoid function, which is one of the non-normal distribution functions, is applied as the peak function (see equation 4).

In this function, x: time, y0: coefficient related to output offset, A: coefficient related to intensity, xc: coefficient related to peak apex time, w1: coefficient related to peak width (whole), w2: coefficient related to peak width (forward), w3: coefficient related to peak width (backward). There are three coefficients related to peak width, w1 to w3, in this function formula, and the peak width calculated from the number of stages is set as the initial value of w1. For w2 and w3, the value of 1/2 of w1 is set as the initial value. The values calculated in this manner are shown in Table 2 and Figure 7a. As can be seen from this, differences remain between the peak curves obtained from each peak function and the output of the raw chromatogram. Therefore, the coefficient A of each peak function is adjusted so that the height of the peak curve obtained from each peak function and the output of the raw chromatogram are the same (Figure 7b, Table 2A correction).

The peak function coefficients Xc, A, w1, w2, w3 of each peak, and the baseline function coefficients a, b, and c are sequentially changed, and calculations are repeated to minimize the difference between their sum and the raw chromatogram, deriving the best fitting coefficients. Figures 8 and 9 show the degree of fitting during the repeated calculation process. The left diagram in Figure 8 shows the chromatogram, peak function, and sum of the baseline function, while the right diagram in Figure 8 shows the error. Figure 9 shows the relationship between the number of calculations and the error. Table 3 also shows the changes in the baseline function coefficients a, b, and c during the repeated calculation process. In this way, it can be seen that the peak function and baseline function coefficients also change during the calculation process to minimize the error.

FIG. 1 is a schematic diagram showing a peak detection method in an insufficiently separated state. FIG. 1 is a schematic diagram showing general peak separation (peak fitting) for a group of peaks in an insufficiently separated state. FIG. 1 is a schematic diagram showing a peak shape in chromatography. FIG. 1 is a schematic diagram showing baseline fluctuation in chromatography. FIG. 1 is a diagram showing a schematic diagram of a peak shape when there is baseline fluctuation. 1 is a flowchart showing the flow of peak separation (peak fitting) processing according to the present invention. FIG. 1 is a diagram showing initial values in peak separation (peak fitting) of the present invention. FIG. 2 is a diagram showing the calculation process in peak separation (peak fitting) of the present invention. FIG. 2 is a diagram showing the calculation process in peak separation (peak fitting) of the present invention. 1 shows a system configuration used to demonstrate the effects of the present invention. FIG. 2 is a diagram showing the structure of a sample (polymer) used for verification in the examples. 10 is a flowchart showing a process flow in an embodiment, in which a quadratic expression is applied as a baseline function and the number of virtual points is one. 1 shows a chromatogram (sample: polystyrene) used for verification in Example 1. FIG. 1 shows sections where peak separation was performed in Example 1 and the positions of the peaks (six peak points). FIG. 1 shows the results of waveform separation/peak fitting by analysis method 1 (conventional method) in Example 1. FIG. 1 shows the results of waveform separation/peak fitting by analysis method 2 (conventional method) in Example 1. FIG. 13 shows the results of waveform separation/peak fitting by analysis method 3 (conventional method) in Example 1. FIG. 13 shows the results of waveform separation/peak fitting by analysis method 4 (the present invention) in Example 1. FIG. 13 is a diagram showing the process of waveform separation/peak fitting by analysis method 4 (the present invention) in Example 1. FIG. 1 shows a baseline function from the final result of peak fitting in Example 1. FIG. 1 shows sections where peak separation was performed in Example 1 and the positions of the peaks (8 peak points). FIG. 13 shows the results of waveform separation/peak fitting by analysis method 4 (the present invention) in Example 1. 1 is a chromatogram (sample: epoxy resin) used for verification in Example 2. FIG. 13 shows the results of waveform separation/peak fitting by analysis method 5 (conventional method) in Example 2. FIG. 13 shows the results of waveform separation/peak fitting by analysis method 6 (the present invention) in Example 2. FIG. 13 is a diagram showing the process of waveform separation/peak fitting by analysis method 6 (the present invention) in Example 2. 1 is a diagram comparing the conventional method (analysis method 5) and the present invention (analysis method 6), in which the horizontal axis indicates the peak and the vertical axis indicates the area of each peak function finally obtained.

The following describes examples of the present invention, but the present invention is not limited to these examples. For the verification, a chromatogram was obtained using a molecular exclusion chromatograph, a high-speed GPC device HLC-8320GPC manufactured by Tosoh Corporation. Figure 10 shows the system configuration used in this verification.

Example 1
As a sample, a standard polystyrene kit (PStQuickMP-M F-80, F-10, A-5000, A-500 manufactured by Tosoh Corporation) used for preparing calibration curves was used. Table 4 shows the measurement conditions.

Figure 13 shows the chromatogram obtained under the above conditions. When this standard polystyrene kit is separated, monodisperse peaks are seen around 6.2 minutes (MW: 37,900) and 7.9 minutes (MW: 5,970), and unseparated peaks (MW: 682-266) are seen around 9.5-10.6 minutes. Here, peak separation was performed on this group of unseparated peaks using the following four analysis methods (methods 1-3 are conventional methods, and method 4 is the method of the present invention), and the effectiveness was verified. The processing range was start point: 8,750 minutes, end point: 11,000 minutes, virtual point: 9,000 minutes, and number of peaks: 6 points.

Analysis method 1 (conventional method)
After performing offset processing to set the start point of processing to zero, peak fitting was performed using a normal distribution function (Equation 5). The coefficients that were changed were the coefficients of the normal function. (The output of 30.063 at the start point of 8.750 minutes was subtracted from the output of the processing region.)

Analysis method 2 (conventional method)
After performing offset processing to set the start point of processing to zero, peak fitting was performed using a non-normal distribution function (Equation 4). The coefficients that were changed were the coefficients of the non-normal function. (The output of 30.063 at the start point of 8.750 minutes was subtracted from the output of the processing region.)

Analysis method 3 (conventional method)
A quadratic equation calculated from three points, the start point, end point, and virtual point of the treatment, was used as the baseline (see Table 5), and peak fitting was performed using a non-normal distribution function (Equation 4). The coefficients that were changed were the coefficients of the non-normal function.

Analysis method 4 (present invention)
The quadratic equation calculated from the three points of the start point, end point, and virtual point of the treatment was used as the initial value of the baseline function (see Table 6), and peak fitting was performed in combination with the non-normal distribution function (Equation 4). The coefficients that were changed were the coefficients a, b, and c of the quadratic equation and each coefficient of the non-normal function.

The initial values of the coefficients of the functions (normal/irregular) used for fitting were determined as follows (see Table 7). The coefficients related to time were calculated back from the peak apex time of each peak in the chromatogram to be processed, and the coefficients related to peak width were calculated back from the number of column stages and the peak time set in advance.

FIG. 18 shows the results of fitting by the method of the present invention (analysis method 4 (present invention) 6 peaks), and FIG. 19 shows the process of fitting by the method of the present invention (analysis method 4 (present invention) 6 peaks). FIG. 19a shows each peak function (solid line) and baseline function (dashed line), FIG. 19b shows the sum of each peak function and baseline function (solid line) and raw chromatogram (dashed line), and FIG. 19c shows the sum of each peak function and baseline function and the sum of the differences between the raw chromatogram (ΣΔ ² ). The left diagram of FIG. 19 shows the fitting state with the initial value, and the number of repeated calculations increases toward the right. As can be seen from this, each peak function gradually changes with each calculation, and as shown in Table 8, each coefficient of the baseline function also changes, and the sum also becomes similar to the original chromatogram. In this case, the error was minimized by 140 repeated calculations.

Figure 20 shows the raw chromatogram and the baseline function. The solid line shows the raw chromatogram, the long-dashed line shows the final baseline function, and the short-dashed line shows the baseline function of the initial fitting value (same as the coefficients in analysis method 3). As can be seen from this, not only do the coefficients of the peak function change, but it can also be seen that the baseline function changes significantly (see Table 8). In other words, this suggests that more accurate fitting can be achieved by treating the baseline as one of the variables rather than using a fixed value, as in the present invention.

To verify the superiority of the present invention, fitting was also performed in the conventional analysis methods 1, 2, and 3. Figure 15 shows analysis method 1 (conventional method): the result of peak fitting with a normal distribution function after offset processing with the start point of processing set to zero, Figure 16 shows analysis method 2 (conventional method): the result of peak fitting with a non-normal distribution function after offset processing with the start point of processing set to zero, and Figure 17 shows analysis method 3 (conventional method): the result of peak fitting with a non-normal distribution function using a quadratic equation calculated from the three points of the start point, end point, and virtual point of processing as the baseline. Figure 17a shows each peak function (solid line) and baseline function (dashed line), and Figure 17b shows the sum of each peak function and baseline function (solid line) and raw chromatogram (dashed line).

Table 9 lists the final results of each analysis method and the fitting according to the present invention. The sample used for the verification is a polymer in which styrene groups are polymerized, as shown in Figure 11a. In molecular exclusion chromatography, in the low molecular weight range, separation is performed according to the degree of polymerization (n), but as n increases, separation becomes worse.

As can be seen from Figure 15, the peak fitting method using the normal distribution function (analysis method 1) provides a reasonable degree of fitting in the low molecular weight region, but does not provide accurate fitting in the high molecular weight region, resulting in a large final error. This is mainly due to the fact that the actual peak shape is not symmetrical, and also because the baseline is not completely horizontal.

Analysis method 2 fits the baseline as a straight line, while analysis methods 3 and 4 fit the baseline as a quadratic equation, but when comparing the two, analysis methods 3 and 4 have extremely small final errors, about 1/3. This shows that it is preferable to use a quadratic equation for the baseline. Furthermore, in the present invention, a quadratic equation is used as the baseline, but it can be seen that further accuracy improvements can be achieved by performing repeated calculations with the coefficient of the quadratic equation also as one of the fitting functions. Table 9 lists the conditions for each analysis method and the error at final convergence.

In order to verify the superiority of the present invention, fitting was also performed when the number of peaks was set to 8 (two peaks were added on the polymer side). Figure 21 shows the start point, end point, virtual point, and each peak point. The initial parameters were determined using the same method as described above (see Table 10).

The fitting results are shown in Figure 22. Figure 22a shows each peak function (solid line) and baseline function (dashed line), Figure 22b shows the sum of each peak function and baseline function (solid line) and raw chromatogram (dashed line), and Figure 22c shows the sum of the differences between each peak function and baseline function and raw chromatogram ( ^ΣΔ2 ). The final error was 0.3818, which is 1/10 of the error when the number of peaks was 6, and this also shows that the fitting method of the present invention has high accuracy (rightmost column of Table 9).

Example 2
The epoxy resin E1001 (manufactured by Mitsubishi Chemical Corporation) was used as the sample. Table 11 shows the measurement conditions.

FIG. 23 shows the chromatogram obtained under the above conditions. When epoxy resin (E1001) is separated, it is separated from 8 minutes to 14 minutes due to the difference in polymerization degree (n). After 12.5 minutes, it is almost completely separated, but before that (high molecular weight side), the separation is insufficient. Peak separation was performed on such complex unseparated peaks, and the effect was verified. The processing range was start point: 7.5 minutes, end point: 14.2 minutes, virtual point: 12.49 minutes, and number of peaks: 15 points. For comparison, fitting was also performed using the conventional method (corresponding to analysis method 1 in Example 1). Analysis method 5 (conventional method) (corresponding to analysis method 1 in Example 1)
After performing offset processing to set the start point of processing to zero, peak fitting was performed using a normal distribution function. The coefficients changed were the coefficients of the normal function. (The output of 30.032 at the start point of 7.50 minutes was subtracted from the output of the processing region.)

Analysis method 6 (present invention) (corresponding to analysis method 4 in Example 1)
The quadratic equation calculated from the three points of the start point, end point, and hypothetical point of the process was used as the initial value, and peak fitting was performed in combination with the non-normal distribution function. The coefficients that were changed were the coefficients a, b, and c of the quadratic equation and each coefficient of the non-normal function.

As with Example 1, the peak function used was the normal distribution function in analysis method 5, and the asymmetric sigmoid function in equation 4 was used as the nonlinear function in analysis method 6. The initial values of the coefficients of the functions (normal/non-normal) used for fitting were determined as follows. The coefficients related to time were calculated backwards from the peak apex time of each peak in the chromatogram to be processed, and the coefficients related to the peak width were calculated backwards from the number of columns set in advance and the peak time.

Fig. 26 is a diagram showing the fitting process by the method of the present invention (analysis method 6). Fig. 26a shows each peak function (solid line) and the baseline function (dashed line), Fig. 26b shows the sum of each peak function and the baseline function (solid line) and the raw chromatogram (dashed line), and Fig. 26c shows the sum of the differences between each peak function and the sum of the baseline function and the raw chromatogram ( ^ΣΔ2 ).

The left image shows the fitting state with the initial values, and moving to the right shows the state where the number of repeated calculations increases. As can be seen from this, with each repeated calculation, the peak functions gradually change, and as shown in Table 14, the coefficients of the baseline function also change, with their sum becoming similar to the original chromatogram. In this case, the error was minimized after 463 repeated calculations.

To verify the superiority of the present invention, peak fitting was also performed using analysis method 5 (corresponding to analysis method 1 in Example 1), which is a conventional method. Figure 24 shows the results of analysis method 5 (conventional method): peak fitting using a normal distribution function after performing offset processing with the starting point of processing set to zero. Figure 24a shows each peak function (solid line) and baseline function (dashed line), and Figure 24b shows the sum of each peak function and baseline function (solid line) and the raw chromatogram (dashed line). Table 15 also lists the final fitting results for each analysis method and the present invention.

As shown in Figure 11b, the sample used for the verification is epoxy resin, a thermosetting synthetic resin with reactive epoxy groups at the terminals, synthesized by a condensation reaction between bisphenol A and epichlorohydrin. In molecular exclusion chromatography, in the low molecular weight range, separation is performed according to the degree of polymerization (n), but as n increases, separation becomes worse.

As can be seen from FIG. 24, the peak fitting method using a normal distribution function (analysis method 5: conventional method) has a final error three times larger than that of the present invention, and the calculation error is large in the region from 11.0 to 12.5 minutes, where the peak apex can be confirmed despite insufficient separation. Furthermore, the very small peak (Peak_13) at 12.7 minutes is not recognized as a peak. On the other hand, in the case of the present invention, the very small peak (Peak_13) at 12.7 minutes is also recognized as a peak, and accurate peak fitting is performed. This is because the original baseline is not a straight line, but is wavy. In the present invention, the baseline is also treated as a fit function, so it can handle baseline fluctuations and enable accurate peak fitting, resulting in more accurate quantitative analysis.

1. Eluent 2. Degasser 3. Liquid delivery pump (sample side)
4. Liquid delivery pump (reference side)
5. Sample injection mechanism 6. Analytical column 7. Reference column 8. Thermostatic bath 9. Refractometer

Claims (2)

A method for peak separation, comprising: performing peak separation on a group of unseparated peaks consisting of two or more component peaks; and quantifying each of the peaks,
a first step of specifying at least three times, i.e., a start time, an end time, and a virtual time for peak separation, acquiring corresponding chromatogram output values, calculating a second- or higher-order polynomial from the three sets of times and outputs, calculating each coefficient of the polynomial, and acquiring a virtual baseline;
The peak function used in the peak separation is a non-normal distribution function,
Obtaining an initial value of a coefficient related to a peak top time among the coefficients of the non-normal distribution function for each peak from the group of peaks in the unseparated state;
a second step of applying an initial value of a coefficient relating to a peak width, among the coefficients of the non-normal distribution function for each peak, based on a value calculated from the peak top time and a predetermined number of columns;
Varying each coefficient of the virtual baseline obtained in the first step and a coefficient of a non-normal distribution function for each peak obtained in the second step;
A peak separation method for chromatography, characterized in that processing is performed using the least squares method so that the difference between the sum of the non-normal distribution functions of each peak and the sum of the baseline function and the data to be processed is minimized, and the baseline function and each peak function are identified and quantified. A liquid chromatography device equipped with the peak separation method described in claim 1.