Anal. Chem. 1998, 70, 218-225
Articles
Standardization of Second-Order Chromatographic/ Spectroscopic Data for Optimum Chemical Analysis Bryan J. Prazen, Robert E. Synovec,* and Bruce R. Kowalski
Center for Process Analytical Chemistry, Department of Chemistry, Box 351700, University of Washington, Seattle, Washington 98195
Chemical analysis using second-order data collected on hyphenated instruments has proven advantages over firstorder or zero-order techniques due to what is known as the second-order advantage. The primary second-order advantage is the ability to perform analysis in the presence of unknown interferences. This work demonstrates another key advantage of second-order chemical analysis, that is, the ability to standardize data sets of a secondorder chromatographic analyzer under conditions which result in retention time variations along the chromatographic axis. An objective technique to standardize second-order chromatographic-spectral data is both theoretically and experimentally developed and tested. This method corrects for retention time shifts that occur between the analysis of the calibration sample and “unknown” samples. When this technique is combined with bilinear data analysis techniques like generalized rank annihilation method (GRAM), standardization and quantitation can be performed in the presence of unknown interferences with a single calibration sample. Most signal inconsistencies in second-order chromatographic data are confined to shifts of the time axis in the chromatographic profile. This retention time shift correction method is objective because it relies upon spectral signal shape and an understanding of the instrumentation. Retention time correction of this type would not be objective for first-order chromatographic analysis because retention time is the only qualitative information present. In one example of experimental evaluation, quantitation of a single analyte in a sample of five chemical components is performed using liquid chromatography with absorbance detection (LC/UV-vis). Both the chromatographic and spectral signals of these five chemical components are highly overlapped. In this example, a retention time shift between the calibration and “unknown” data sets of 0.2 s resulted in a 20% quantitation error prior to standardization. After alignment of the data sets using secondorder chromatographic standardization, quantitative error was reduced to nearly 1%. Theoretical simulations which evaluate the performance of this technique as a second-
order chromatographic retention time correction method were performed for a wide range of resolution and signalto-noise values. In simulations where chromatographic resolution was 0.3 or below, quantitative precision improvements resulting from second-order chromatographic standardization ranged from 3-fold to 10-fold. The standardization method presented should be generally applicable to chromatography hyphenated with all forms of spectroscopic detection, such as gas chromatography/ mass spectrometry (GC/MS).
218 Analytical Chemistry, Vol. 70, No. 2, January 15, 1998
S0003-2700(97)00633-1 CCC: $15.00
Chromatographers are very aware of retention time variation. In practice, this problem is traditionally overcome by fully separating the component of interest from interferences. Quantitation in the presence of retention time variation can then be performed by integration of a time window larger than the peak of interest. The implementation of this traditional method requires considerable analysis time, since increasing chromatographic resolution often requires a longer analysis time. As the importance of shortening analysis time continues to gain interest, the requirement of baseline resolution becomes less favorable. The power of hyphenated chromatographic methods that create second-order, bilinear data sets has been appreciated for some time.1,2 The use of a second-order tensor of data per sample for data analysis can lead to the second-order advantage. The second-order advantage allows for the accurate quantitation of multiple analytes using a calibration sample containing multiple chemical components without knowledge of the interfering chemical components that may be present in the sample being analyzed. In spectroscopy, standardization3,4 is used to adjust the calibration model to make it applicable to another instrument producing different responses, or to the same instrument under different conditions. Standardization is important for all types of instrumentation, but the common variation in chromatographic retention times makes chromatographic standardization especially important. Unfortunately, standardization methods designed for spec(1) Hirschfeld, T. Anal. Chem. 1980, 52, 297A-308A. (2) Booksh, K. S.; Kowalski, B. R. Anal. Chem. 1994, 66, 782A-91A. (3) Wang, Y.; Kowalski, B. R. Anal. Chem. 1993, 65, 1174-80. (4) Wang, Y.; Lysaght, M. J.; Kowalski, B. R. Anal. Chem. 1992, 64, 562-4. © 1998 American Chemical Society Published on Web 01/15/1998
trometers are not applicable to chromatography, because the variations found in chromatography are of a different kind. When the run-to-run instrument response for an analyte of interest is not sufficiently reproducible, the analytical chemist looks for a standardization method which solves the problemwithout sacrificing the integrity of the analysis. The most objective standardization methods limit corrections in the instrument response function to explainable variations of the instrument. Often, injection timing imprecision, flow rate changes, and temperature fluctuations cause shifts in the chromatographic profiles. The standardization method presented herein corrects for retention time shifts found in bilinear chromatographic data. Linear corrections are made to short time windows in the chromatographic profiles. Second-order calibration algorithms call for a consistent instrumental response function. Responses of satisfactory precision can be obtained with many spectroscopic techniques. It is more of a challenge to create chromatographic data with such a high level of instrumental response precision.5,6 Rank annihilation methods demand that the signal for the analyte of interest be nearly identical for both the sample and calibration data and that the signal of the analyte be independent of the interfering components. Many publications, which used the generalized rank annihilation method (GRAM) and other second-order calibration methods, note that, although these techniques are successful in controlled laboratory settings, the effects of retention time variation between the calibration and unknown samples were observed and are problematic.7-9 Curve resolution methods10 appear to overcome strict response function precision requirements by decomposing each matrix individually. Even so, unless a theoretical relationship exists between the measured response and the concentration, calibration samples must also be used for curve resolution techniques. Thus, a good match of the signals is necessary for objective calibration, regardless of the method. Recently, the importance of retention time precision on quantitation of first-order chromatographic data was theoretically and experimentally examined in the context of chromatographic resolution.6 A peak width-based retention time precision was defined, δ, as δ ) st/4σt, where st is the run-to-run standard deviation of retention times, and 4σt is the baseline peak width. The study concluded that chromatographic resolution, Rs, alone is insufficient and ambiguous for judging the quantitative analysis precision for deconvoluting overlapped peaks. Quantitative optimization of ill-resolved peaks should be based on both retention time precision and resolution. The standardization method presented here uses the precision of the spectral information to increase the precision of the chromatographic retention time and quantitation. Other authors have offered solutions to the problem of retention time precision. Most of these addressed first-order (5) Goldberg, B. J. Chromatogr. Sci. 1971, 9, 287-92. (6) Bahowick, T. J.; Synovec, R. E. Anal. Chem. 1995, 67, 631-40. (7) Poe, R. B.; Rutan, S. C. Anal. Chim. Acta 1993, 283, 845-53. (8) Ramos, L. S.; Sanchez, E.; Kowalski, B. R. J. Chromatogr. 1987, 385, 16580. (9) Kim, R. R. Ph.D. Thesis, Massachusetts Institute of Technology, 1985. (10) Tauler, R.; Barcelo, D. Trends Anal. Chem. 1993, 12, 319-27.
chromatographic data.11-14 Others have chosen to use only the spectral data from second-order chromatographic data to perform calibration in the presence of retention time shifts,15,16 thus performing first-order analysis on second-order data. Standardization of second-order chromatographic analyzers has been addressed using a standard sample which is run on the instrument under each instrumental response condition.3 In this method, a comparison of data from the standard under different instrumental response conditions allows transformation from one instrument response to the other to be built. The method we present does not require the analysis of standards under each operational condition, thus allowing for correction for small retention time variations which occur within short periods of time. The problem of chromatographic retention time shifts in second-order analysis has previously been addressed using Bessel’s inequality.17 This method was successful for standardization of the simulated data presented. Standardized data were then calibrated by rank annihilation factor analysis (RAFA). Bessel’s inequality method and RAFA require the chromatographic profiles of the pure analytes of interest (resolved standards). Often in second-order analysis, the pure chromatographic profiles are not obvious due to a rotational ambiguity. The method presented here is advantageous because pure chromatographic profiles are not a requirement, making it appropriate for standardization prior to GRAM calibration when the calibration samples contain multiple chemical components.18 The method we present, second-order chromatographic standardization, allows for the reduction of run-to-run retention time variation to a degree which is determined by chromatographic data acquisition rate. This can be seen as yet another secondorder advantage. Both the Bessel’s inequality method and the method presented here demonstrate effective ways to use the spectral selectivity and the precision found in second-order chromatographic data to increase retention time precision. The information necessary to apply second-order chromatographic standardization is very similar to that necessary to perform GRAM calibration. The method presented calls for an estimation of the number of chemical components in the time window of the sample being analyzed. This is not a handicap of this method, because numerous papers have covered the estimation of the number of components, or pseudorank, of a bilinear data matrix.19-21 Our estimates of the pseudorank are based on principal component analysis using singular value decomposition (SVD). It is also necessary that each of the components in the time window of the calibration sample be present to some extent in the sample being analyzed. This method also requires that the data be bilinear, (11) Webster, G. H.; Cecil, T. L.; Rutan, S. C. J. Chemom. 1988, 3, 21-32. (12) Andersson, R.; Ha¨ma¨la¨inen, M. D. Chemom. Intell. Lab. Syst. 1994, 22, 4961. (13) Cecil, T. L.; Rutan, S. C. Anal. Chem. 1990, 62, 1998-2004. (14) Malmquist, G.; Danielsson, R. J. Chromatogr. A 1994, 687, 71-88. (15) Gampp, H.; Maeder, M.; Meyer, C. J.; Zuberuehler, A. D. Anal. Chim. Acta 1987, 193, 287-93. (16) Burns, D. H.; Callis, J. B.; Christian, G. D. Anal. Chem. 1986, 58, 141520. (17) Grung, B.; Kvalheim, O. M. Anal. Chim. Acta 1995, 305, 57-66. (18) Sanchez, E.; Kowalski, B. R. Anal. Chem. 1986, 58, 496-9. (19) Faber, N. M.; Buydens, L. M. C.; Kateman, G. Anal. Chim. Acta 1994, 296, 1-20. (20) Faber, N. M.; Buydens, L. M. C.; Kateman, G. Chemom. Intell. Lab. Syst. 1994, 203-26. (21) Malinowski, E. R. J. Chemom. 1988, 3, 49-60.
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Figure 1. Mesh plot of two LC/UV-vis data sets augmented on the time axes to perform second-order chromatographic standardization.
with the exception of retention time imprecision. Chromatography combined with many spectrometric techniques fits this requirement. We demonstrate this standardization method on data obtained with an instrument combining reversed-phase liquid chromatography and a CCD array UV-visible absorbance spectrometer (LC/UV-vis). THEORY The following discussion will use these conventions: italic lowercase letters for scalars; bold lowercase letters for vectors; bold uppercase letters for matrices; and superscript T for transpose matrices and vectors. We begin with the sample data which may be represented as a matrix, M, each element of which, mij, represents the intensity measured at the ith retention value and the jth spectral value (e.g., wavelength). It is also necessary to run a calibration sample of known concentrations which results in a similar type of data matrix, N. This standardization technique, along with GRAM and many other second-order methods designed for calibration of analytes in complex mixtures, calls for bilinear data. c
Z ) XYT )
∑x y
T n n
profile. The rank of the augmented matrix is the total number of different chemical components in the sample and calibration sample. Augmenting a truly bilinear matrix does not change the mathematical rank; if all of the components in the calibration sample are also in the sample being calibrated, the rank of the augmented matrix is equal to that of the sample matrix. If, on the other hand, the chromatographic retention time changed between the time the calibration data and the sample data were collected, the augmented matrix will not be bilinear, and thus the mathematical rank will be greater than the number of chemical components. This is because each chemical component is not the outer product of two vectors. The demonstrated technique uses this indicator of a retention time shift to standardize the calibration data to the sample data. In order to standardize the calibration sample and the sample being analyzed, the time window of the calibration sample is shifted until a minimum pseudorank is found. The term “pseudorank” describes the estimation of the rank of a real data set. The true rank must be estimated for real data because all data contain noise. The steps necessary to perform second-order chromatographic standardization are shown below. Step 1. The pseudorank of the sample matrix is determined by inspection of the singular values.19 There are many statistical methods for determining pseudorank.19-22 Step 2. The two data sets are augmented so that the spectral axis is twice as long, and principal component analysis is preformed using singular value decomposition (SVD). SVD of [M|N] leads to a list of the positive square roots of the eigenvalues, s, for the cross product or covariance of the augmented matrices (s2 ) uT([M|N][M|N]T)u). Step 3. The sum of the secondary eigenvalues, λ0, or those beyond the pseudorank of the sample is divided by the sum of all the singular values and multiplied by 100 and a term related to the degrees of freedom to calculate the residual percent variance.22 This can be explained by the fact that the variance due to error is the sum of the eigenvalues associated with the error divided by the degrees of freedom of the error components.
(1)
Z is bilinear in the sense that, if one set of unknown variables (X or Y) is held constant, the resulting system is linear in the other (Y or X). Thus, a bilinear estimation can be applied whenever the signal of each chemical component can be expressed as the outer product of two vectors, x and y. Each component in the data of this report can be described by the outer product of its chromatographic profile, y, and its spectral profile, x. Z contains n chemical components and can be represented as the spectra of each of the components multiplied by the chromatographic profile of each. In the ideal situation, no noise is present, and each chemical component has a spectral and chromatographic profile which is unique and not a linear combination of the other components. In this case, the rank of the data matrix is equal to the number of chemical components. If the unknown sample data matrix, M, and calibration matrix, N, are augmented so that the spectral axis is twice as long, as shown in Figure 1, the spectral axis of each component common to N and M becomes its spectral profile adjoined to the same 220
Analytical Chemistry, Vol. 70, No. 2, January 15, 1998
( )( ) c
∑λ
n)1
% residual variance ) 100 ×
0
j
j)n+1
cr
c
(c - n)(r - n)
∑λ
(2)
j
j)1
where n is the number of chemical components, and c and r are the dimensions of the augmented data matrix. Step 4. In order to find the alignment with the minimum variance in the insignificant principal components or minimum rank, the time window of the standard is shifted, and steps one and two are repeated over a range determined by the expected retention time variance. If the degrees of freedom due to the chemical component-related principal components are constant, the minimum pseudorank of the augmented matrix will occur at the minimum percent variance in the principal components describing error. The percentage of the residual variance plotted for each shift position will be referred to as the standardization (22) Malinowski, E. R.; Howery, D. G. Factor Analysis in Chemistry, 2nd ed.; Wiley: New York, 1991.
profile. The minimum of the standardization profile indicates the time shift correction to be made. At this point, the spectral profiles of the analytes of interest can be compared to the spectral profiles of the calibration matrix N. The correlation of these profiles can be evaluated by a vector comparison method such as least squares to verify that the chromatographic shift standardized the calibration matrix to the correct chromatographic profiles. EXPERIMENTAL SECTION Isocratic, isothermal HPLC separations were performed with a 30 mm long × 2.1 mm i.d. Hyperprep ODS column containing 8 µm particles and 120 Å pores (Alltech, Deerfield, IL). The mobile phase was a mixture of 70% methanol and 30% water by volume, delivered at 1.5 mL/min by a pump (Model 114M, Beckman, Fullerton, CA). Samples of 1.27 µL were injected with an automated 10-port valve (Vici Valco, EC10U, Houston, TX). A single switch injected the sample and began data acquisition. The effluent of the column was monitored through a 1 µL flow cell made in-house. The flow cell was a Z-configuration with a 400 µm optical fiber (Model FP-4-UHT, 3M, West Haven, CT) at each end so that the effluent flowed directly past the end of the fiber. The optical fibers were held in the cell by inserting the fiber into a short length of PEEK HPLC tubing (Upchurch Scientific, Oak Harbor, WA) held by stainless steel ferrules. This provided a highpressure seal around the optical fibers. A xenon lamp was used as a light source (Model C4264, Hamamatsu Photonics, Hamamatsu City, Japan). The transmitted light was carried via the previously described fiber optic to a spectrometer (Model PC1000, Ocean Optics, Dunedin, FL) mounted on a card in a P5-90 personal computer (Gateway 2000, North Sioux City, SD). In the spectrometer, a 1200 line holographic grating distributed the wavelength region from 200 to 450 over a 1024 CCD array. A total of 127 absorbance spectra were collected every 3 s. Ten of these spectra were averaged for each spectrum recorded. Five hundred spectra were recorded for each run. The spectrometer was run and transmittance signals were converted to absorbance signals and recorded in ASCII format by the SpectraScope software distributed with the Ocean Optics spectrometer. Preprocessing included removal of the first 13 and last 23 spectral units, which contained a low signal-to-noise ratio. The spectral axis was then boxcar averaged in groups of 10. This reduced the size of the matrices by a factor of 10 making data analysis easier. The dinitrophenylhydrazine (DNPH) derivative preparation was explained in a previous publication.23 Standard solutions of these derivatives in methanol and volumetric dilutions of these standards were made. Simulation data sets were also constructed by combining Gaussian profiles. Each of the simulated samples contained three components in the chromatographic time window being analyzed. The calibration sample contained only the single component being analyzed. Retention times of the calibration sample were randomly generated in a normal distribution so that 96% of the retention times were (10 time units, or 1 s, from the correct retention time. Spectral acquisition rate, spectral selectivity, and retention time variation parameters were programmed to simulate realistic high-speed gas chromatographic situations. A spectral (23) Renn, C. N.; Synovec, R. E. Anal. Chem. 1990, 62, 558-64.
Table 1. Least-Squares Comparison of Spectral Profiles before and after Second-Order Chromatographic Standardization least-squares spectral difference
experiment two-component model five-component mixture calibrating with MVK-DNPH 1 five-component mixture calibrating with MVK-DNPH 2 three-component strong mobile phase
before standardization (% rel dev)
after standardization (% rel dev)
0.139 0.313
0.138 0.265
0.340
0.286
0.824
0.775
acquisition rate of 10 spectra/s, retention time of 40 s for the analyte of interest, peak widths of 4 s, and retention time variation resulted from a 1% mobile phase or carrier gas flow rate fluctuation. Data sets were 80 units on the chromatographic axis and 60 units on the spectral axis. All data processing and simulations were done on a Sun Sparc using Matlab (Mathworks Inc., Sherborn, MA). The implementations used to perform singular value decomposition and the QZ algorithms for GRAM were those included with Matlab. RESULTS AND DISCUSSION The first application is the analysis of one component which is highly overlapped in time with an “unknown” component which gives a very similar spectral shape. Data from a sample of only the analyte of interest, methyl vinyl ketone DNPH (MVK-DNPH), were added to data from methyl ethyl ketone-DNPH (MEKDNPH), which is considered an “unknown” interference. This combination of data allows the standardization results to be verified. These sample data were calibrated with data of a sample of MVK-DNPH of twice the concentration according to a volumetric dilution. Visual inspection of the two MVK-DNPH data sets and standardization of the data sets without interferences confirmed that the MVK-DNPH data used for the sample were retained just less than 2 data acquisition time units more than the MVK-DNPH data used to make the calibrated mixture. This translates to approximately a 0.2 s deviation in retention time. Spectra of MVK-DNPH and MEK-DNPH and the analyzed sample chromatographic profiles can be found in Figure 2A and B, respectively. Note that the chromatographic resolution of MVKDNPH and MEK-DNPH is only 0.15, or a time difference of 0.96 s. The standardization profile obtained following the procedure outlined in the Theory section is shown in Figure 2C. The shift necessary to standardize the calibration sample is identified as an unambiguous minimum at shift of -2 units or -0.2 s, which is on the same order of magnitude as the retention time difference between the analyte and interference. Thus, the standardization is effective on a challenging case. The accuracy of this standardization is confirmed by the difference in retention times of the analyte of interest before the sample was complicated by a second component. Standardization was also confirmed by the leastsquares comparison of the SVD spectra from the augmented matrix and the spectra of the standard. The results of the leastsquares confirmation can be found in Table 1 (two-component model). A small improvement in the spectral fit is seen in the Analytical Chemistry, Vol. 70, No. 2, January 15, 1998
221
A
B
C
Figure 2. (A) Normalized spectra of methyl ethyl ketone dinitrophenylhydrazine (MEK-DNPH) and methyl vinyl ketone dinitrophenylhydrazine (MVK-DNPH) over the wavelength region monitored. (B) Chromatographic profile of two-component sample made by combining individual MVK-DNPH and MEK-DNPH data sets. (C) Secondorder chromatographic standardization profile for the two-component mixture of MVK-DNPH and MEK-DNPH using MVK-DNPH as the analyte and MEK-DNPH as an interference. A sharp minimum is shown at the correct retention time shift of -0.2 s.
standardized data. The improvement in spectral fit is small because the spectra of the interferences in the sample used are very similar to that of the analyte of interest. The similarity of the spectra makes this an exceptionally challenging case. A substantial improvement in the quantitation can be seen in Table 2 (two-component model), where the calibration performed with the generalized rank annihilation method (GRAM) before and after 222
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standardization is compared to quantitation determined by comparing dilutions used to prepare samples from a stock solution. The second and third examples are the analyses of MVKDNPH in a mixture of five DNPH derivatives using two different calibration standards. The first calibration sample contained MVKDNPH at 1.44 times its concentration in the sample. The second calibration sample contained MVK-DNPH at 0.722 times its concentration in the sample. Unlike the previous example (Figure 2), the data set being analyzed is an actual mixture and is not made by the addition of single-component data sets. Thus, an external validation of the standardization results is more ambiguous, but conditions are more realistic. One source of validation is that the first of the two calibration samples is known to have a retention of nearly 0.2 s earlier than the second calibration sample. Thus, the retention time correction determined for the fivecomponent mixture with the two calibration samples should differ by approximately 0.2 s. The sample contains the two DNPH derivatives from the previous example, along with propionaldehyde dinitrophenylhydrazine (PRO-DNPH), isovaleraldehyde dinitrophenylhydrazine (Iso-DNPH), and valeraldehyde dinitrophenylhydrazine (Val-DNPH). The differences of the normalized spectra of each of the interferences from the analyte of interest are shown in Figure 3A. The spectral differences were plotted because spectra of all components are so similar. The chromatographic profile of the five-component sample mixture is shown in Figure 3B. The results of the two standardizations performed on the five-component mixture are shown in Figure 3C and D. The first calibration sample calls for a shift of -0.30 s, and the second calls for a shift of -0.20 s. The unexpected difference of 0.10 s, or 1 unit, is probably due to the fact that the shifts are quantitized instead of being continuous. This standardization technique could be extended with a two-dimensional interpolation in one variable to make standardization continuous, but this would be computationally intensive and would result in a relatively small improvement. A comparison of the results of these two examples with the first example demonstrates the effect of crowding of the chromatographic time window on this technique. Although the three standardization results depicted in Figures 2C, 3C, and 3D are accurate indicators of the retention time correction, the standardization profile in Figure 2C has a much sharper minimum than those of Figure 3C and D. This is attributed to each interference adding some amount of nonlinear response, which acts similarly to increasing background noise. Also, each interference masks some of the increase in pseudorank created by the retention time shift in their principal components. Again, the leastsquares fit of the spectra can be found in Table 1, and the calibration results can be found in Table 2 (five-component mixtures 1 and 2). The improvements in spectral fits validate the standardizations. Both standardizations considerably improved the calibration results. The difference between the errors found in the two five-component calibrations can be attributed to the fact that these are individual cases and not statistically averaged results. In the fourth example, MVK-DNPH in a sample containing ProDNPH and Val-DNPH is standardized and calibrated. This sample is very challenging in that the signal height of the analyte of interest is approximately equal to each of the interferences, and the chromatographic resolution of the analyte, MVK-DNPH,
Table 2. Quantitation Results before and after Second-Order Chromatographic Standardization GRAM ratio of concentrations experiment two-component model five-component mixture calibrating with MVK-DNPH 1 five-component mixture calibrating with MVK-DNPH 2 three-component strong mobile phase
% error
volumetric ratio of concentrations
without standardization
with standardization
without standardization
with standardization
2.0 1.44
1.46 2.14
2.00 1.67
27 48
0.32 15
0.722
0.871
0.713
21
1.3
0.750
0.606
0.704
19
6.2
A
C
B
D
Figure 3. (A) Differences of the normalized absorbance spectra of (a) propionaldehyde dinitrophenylhydrazine (PRO-DNPH), (b) isovaleraldehyde dinitrophenylhydrazine (Iso-DNPH), (c) valeraldehyde dinitrophenylhydrazine (Val-DNPH), and (d) methyl ethyl ketone dinitrophenylhydrozine (MEK-DNPH) from the normalized spectra of the analyte methyl vinyl ketone dinitrophenylhydrazine (MVK-DNPH). (B) Chromatographic profile of five-component sample mixture containing Iso-DNPH, MEK-DNPH, MVK-DNPH, Pro-DNPH, and Val-DNPH. (C) Second-order chromatographic standardization profile for the five-component mixture of Iso-DNPH, MEK-DNPH, MVK-DNPH, Pro-DNPH, and Val-DNPH. A minimum in % residual variance at the retention time shift of -0.30 s indicates the retention time correction. (D) Second-order chromatographic standardization profile for the five-component mixture of Iso-DNPH, MEK-DNPH, MVK-DNPH, Pro-DNPH, and Val-DNPH with a less concentrated calibration standard. A minimum in % residual variance at the retention time shift of -0.20 s indicates the retention time correction.
withboth interferences was purposely reduced by decreasing the polarity of the mobile phase. The resolution of MVK-DNPH with Pro-DNPH and Val-DNPH was 0.47 and 0.73, respectively, in this example. The chromatograms and standardization results are among those shown in Figure 4. The spectral differences for these components are shown on Figure 3A. Table 2 shows the improvement in quantitation resulting from second-order standardization (see three-component strong mobile phase). The broad minimum of the standardization profile in Figure 4B indicates that this was the most challenging standardization of those shown. Errors in concentration prediction before standard-
ization shown in Table 2 demonstrate the importance of retention time precision for second-order chromatographic calibration. Simulations that examine the effects of chromatographic resolution and noise on quantitative precision before and after retention time standardization were performed in order to make generalizations regarding the method. Table 3 lists the results of this study. The spectral profiles are shown in Figure 5A. An example chromatographic profile in which the analytes were separated by a resolution of 0.3 is shown in Figure 5B. Chromatographic resolutions of 0.2, 0.3, 0.4, and 0.5 and noise values of 0.0001, 0.005, and 0.001 were studied. The noise values listed Analytical Chemistry, Vol. 70, No. 2, January 15, 1998
223
A
A
B
B
Figure 4. (A) Chromatographic profile of a separation of MVKDNPH, Pro-DNPH, and MEK-DNPH. In this example, the signals of interfering components, Pro-DNPH and MEK-DNPH, are approximately equal to the height of MVK-DNPH. (B) Second-order chromatographic standardization profile for data from the separation of MVK-DNPH, Pro-DNPH, and MEK-DNPH, indicating the correct retention time shift of 1.1 s. Table 3. Effects of Noise and Resolution on Quantitative Precision for Analysis Using Second-Order Chromatographic Standardization Method for Analysis of Simulated Data Shown in Figure 5a resolution noise standardization
0.2
0.3
0.4
0.5
0.0001
no yes
39 ( 2 4(1
18.1 ( 0.8 3.5 ( 0.4
9.8 ( 0.4 5 ( 3 3.3 ( 0.5 3.4 ( 0.8
0.0005
no yes
39 ( 1 18.4 ( 0.8 9.4 ( 0.4 4.8 ( 0.9
9.6 ( 0.7 4.0 ( 0.1 3.3 ( 0.8 3.3 ( 0.2
0.001
no yes
57 ( 25 14.6 ( 0.7
19 ( 1 11.7 ( 0.4 7.7 ( 0.5 6.3 ( 0.8 3.8 ( 0.4 3.9 ( 0.9
a Quantitative precision results are listed as percent relative deviation of predicted concentrations from actual concentrations. Each percent relative deviation result listed is the average of 1000 simulations.
are standard deviations of normally distributed noise which was added to both the sample and calibration data. If retention time variation was not present, a noise value of 0.001 would translate to a signal-to-noise ratio of 1000 when working with the singlechannel chromatographic data at the most sensitive spectral channel for the analyte of interest. Standard signal-to-noise measurements of this type do not evaluate the data well because 224 Analytical Chemistry, Vol. 70, No. 2, January 15, 1998
Figure 5. (A) Simulated spectral profiles used to evaluate the performance of second-order chromatographic standardization. (B) Simulated chromatographic profiles used to evaluate the performance of second-order chromatographic standardization. Chromatographic resolution of 0.3 is shown. Resolutions of 0.2, 0.3, 0.4, and 0.5 were also evaluated. Results are in Table 3.
retention time variation adds a large heteroscedastic noise to the data. Retention time variation for all cases mimicked a 1% random mobile phase flow rate fluctuation, as described in the Experimental Section. One thousand simulations were performed at each of the noise and resolution values studied. The averages and standard deviations of the percent relative deviations of predicted concentrations are listed in Table 3. All concentrations were predicted with GRAM calibration. A comparison of the results for calibration with and without standardization demonstrates the importance of second-order chromatographic standardization under typical chromatographic conditions. As would be expected, at low signal-to-noise ratio and resolution, the quantitative precision is poorer. In the most beneficial case studied with regard to applying standardization, a resolution of 0.2 and a noise of 0.0001 led to an average quantitative deviation of 39% before standardization. After the application of secondorder chromatographic standardization, the average quantitative deviation was reduced to 4%, or a 10-fold improvement. Table 3 shows a trend of improving calibration results both before and after standardization as signal-to-noise ratio and chromatographic resolution increase. In all cases, the average quantitative results improve substantially with the application of second-order chromatographic standardization.
CONCLUSION In all cases examined, an improvement in the calibration accuracy was observed when second-order chromatographic standardization was applied. Second-order chromatographic standardization does not eliminate the importance of retention time precision, and all possible instrumental or experimental improvements should be made to increase retention time precision. Highspeed second-order chromatographic analysis methods such as LC/UV-vis, GC/MS,24 and comprehensive two-dimensional GC25 should all benefit from second-order chromatographic standardization. (24) Synovec, R. E.; Prazen, B. J.; Kowalski, B. R. Proc. 19th Int. Symp. Cap. Chromatogr. Electrophoresis, Wintergreen, VA, 1997; pp 366-7. (25) Synovec, R. E.; Bruckner, C. A.; Prazen, B. J. Proc. 19th Int. Symp. Cap. Chromatogr. Electroporesis, Wintergreen, VA, 1997; pp 148-9.
ACKNOWLEDGMENT This work was supported by the Center for Process Analytical Chemistry (CPAC), a National Science Foundation, University/ Industrial Cooperative Research Center at the University of Washington. Received for review June 18, 1997. Accepted October 22, 1997.X AC9706335
X
Abstract published in Advance ACS Abstracts, December 15, 1997.
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