Resolving and Quantifying Overlapped Chromatographic Bands by

Department of Chemistry and Chemical Biology, Stevens Institute of Technology, Castle Point, Hoboken, New Jersey 07030. Anal. Chem. , 2000, 72 (18), ...
0 downloads 0 Views 314KB Size
Download PDF
Anal. Chem. 2000, 72, 4363-4371

Resolving and Quantifying Overlapped Chromatographic Bands by Transmutation Edmund R. Malinowski*

Department of Chemistry and Chemical Biology, Stevens Institute of Technology, Castle Point, Hoboken, New Jersey 07030

A new chemometric technique called “transmutation” is developed for the purpose of sharpening overlapped chromatographic bands in order to quantify the components. The “transmutation function” is created from the chromatogram of the pure component of interest, obtained from the same instrument, operating under the same experimental conditions used to record the unresolved chromatogram of the sample mixture. The method is used to quantify mixtures containing toluene, ethylbenzene, m-xylene, naphthalene, and biphenyl from unresolved chromatograms previously reported. The results are compared to those obtained using window factor analysis, rank annihilation factor analysis, and matrix regression analysis. Unlike the latter methods, the transmutation method is not restricted to two-dimensional arrays of data, such as those obtained from HPLC/DAD, but is also applicable to chromatograms obtained from single detector experiments. Limitations of the method are discussed. Finding experimental conditions to effect complete resolution of chromatographic bands poses a serious problem, often requiring considerable time and effort. A number of chemometric techniques, such as iterative target transformation factor analysis (ITTFA),1 evolving factor analysis (EFA),2,3 window factor analysis (WFA),4,5 subwindow factor analysis (SFA),6,7 rank annihilation factor analysis (RAFA),5 generalized rank annihilation method (GRAM),8 matrix regression analysis (MATRA),5 and heuristic evolving projections (HELP)9,10 have been developed and applied to quantify the components of overlapping chromatographic bands. These methods require two-dimensional arrays of data, such as that obtained from HPLC/DAD. Some methods, such as alternat* Direct correspondence to E. R. Malinowski, 5042 SE Devenwood Way, Stuart, FL 34997. (1) Gemperline, P. J. J. Chem. Inf. Comput. Sci. 1984, 24, 206-212. (2) Maeder, M. Anal. Chem. 1987, 59, 527-530. (3) Keller, H. R.; Massart, D. L. Anal. Chim. Acta 1991, 246, 379-390. (4) Malinowski, E. R. J. Chemom. 1993, 6, 29-40. (5) Schostack, K. J.; Malinowski, E. R. Chemom. Intell. Lab. Syst. 1993, 20, 173-182. (6) Manne, R.; Shen, H.; Liang, Y. Chemom. Intell. Lab. Syst. 1999, 45, 171176. (7) Shen, H.; Manne, R.; Xu, Q.; Chen, D.; Liang, Y. Chemom. Intell. Lab Syst. 1999, 45, 323-328. (8) Wilson, B.; Sanchez, E.; Kowalski, B. R. J. Chemom. 1989, 3, 493. (9) Kvalheim, O. M.; Liang, Y. Z. Anal. Chem. 1992, 64, 936-946. (10) Liang, Y. Z.; Kvalheim, O. M.; Keller, H. R.; Massart, D. L.; Kiechle, P.; Erni, F. Anal. Chem. 1992, 64, 946-953. 10.1021/ac000407m CCC: $19.00 Published on Web 08/04/2000

© 2000 American Chemical Society

ing trilinear decomposition (ATLD),11 alternating coupled vectors resolution (ACOVER),12 and parallel factor analysis (PARAFAC2),13 require a three-dimensional array of data, not easily achievable. Recent papers14,15 have focused on comparing various chemometric techniques. The present investigation concerns the development of a new method, called “transmutation”, that can be used to resolve overlapping chromatographs obtained from single as well as multiple detector instruments. THEORY The “transmutation technique” is based on the assumption that the shapes of the overlapping chromatographic bands are proportionately congruent (i.e., exact same shapes but different intensities). Although, it is well known that the bands broaden as a function of their retention times, if the band overlap is confined to a relatively narrow time window, these differences will be small and will not perturb the results to any significant extent. The transmutation method is not a “black box” approach. It is based on sound mathematical principles. Consider c (a row vector) to be a digitized chromatogram of a pure standard solution of the component of interest. The elements of c consist of detector responses as a function of digitized time intervals. A typical chromatogram, as shown in Figure 1, is broad and exhibits tailing. The transmutation function, T, is a matrix designed to convert c into an ideal chromatogram. The most ideal chromatogram is a Dirac delta (δ) function, infinitely thin with a height directly proportional to the concentration of the standard. Our studies, described in the following section, using experimental as well as model data, indicate that transmutation into a Dirac δ function requires ultimate precision along the time axis and nearly perfect chromatographic congruency, a task not achievable by present-day instrumentation. A more realistic ideal chromatogram is a triangular function, which, because it has a finite bandwidth, can tolerate a degree of error along the time axis as well as chromatographic incongruency. The Dirac δ function can be looked upon as a triangular function with an infinitely small bandwidth. Because bandwidth is directly related to the standard deviation in the measurement process, the δ function cannot account for experimental error manifested in band broadening. Nevertheless, it is pedagogically (11) Wu, H.; Shibukawa, M.; Oguma, K. J. Chemom. 1998, 12, 1-26. (12) Jiang, J.; Wu, H.; Li, Y.; Yu, R. J. Chemom. 1999, 13, 557-578. (13) Bro, R.; Anderson, C. A.; Kiers, H. A. L. J. Chemom. 1999, 13, 295-309. (14) Dunkerley, S.; Brereton, R. G.; Crosby, J. Chemom. Intell. Lab. Syst. 1999, 48, 99-119. (15) Cuesta Sanchez, F.; Rutan, S. C.; Gil Garcia, M. D. D.; Massart, L. Chemom. Intell. Lab. Syst. 1997, 36, 153-164.

Analytical Chemistry, Vol. 72, No. 18, September 15, 2000 4363

Figure 1. Chromatogram of m-xylene standard solution recorded at 295 nm.

important in our development of the transmutation method to investigate the potential use of this important function since it represents the Holy Grail of chromatography. The transmutation matrix is determined from two matrices, a real matrix and an ideal matrix, as explained below. The real matrix, C(real), is constructed from c, the chromatogram of the pure component of interest. Before constructing C(real), c is expanded 3-fold by adding an equivalent number of time points with zero intensity to the front of the experimental time window as well as to the end of the time window. This is done to prevent aberrations inside the experimental time window, which would result if the chromatographic profiles were truncated at the edges of the experimental time window. The first row of C(real) contains the expanded vector c, after shifting the vector so its retention time (peak maximum) coincides with the first time point of the expanded time scale, truncating all responses that fall outside the expanded time scale. The second row of C(real) contains c after shifting the expanded c so its peak maximum coincides to the second time point. Similarly, the third, fourth, fifth, etc., rows are created by shifting c so their maximums correspond to their appropriate time points. Responses that fall outside the expanded time window are deleted. A three-dimensional plot of a typical C(real) matrix is presented in Figure 2, which, for clarity, only shows every ninth row and ninth column of a 219 × 219 matrix obtained by expanding the chromatogram of m-xylene (Figure 1), which consists of 73 equally spaced time points. The ideal matrix, C(ideal), is constructed in a similar fashion, using an ideal chromatographic vector in place of c. If the Dirac δ function is selected to be the ideal shape function then

Figure 2. Real matrix, C(real), constructed from the chromatogram of m-xylene shown in Figure 1. (For clarity, the figure displays every ninth row and ninth column of the 219 × 219 matrix.)

The transmutation function, T, is defined to be the matrix that converts C(real) into C(ideal) by postmultiplication as shown in eq 2. If C(ideal) were constructed from a Dirac δ function, then

C(real) T ) C(ideal)

(2)

eq 2 would represent an orthogonal transformation. There is nothing strange or mysterious here. Such transformations are well known. Because C(real) and C(ideal) are both known, T can be obtained by rearranging eq 2 to eq 3. It is important to note that

T ) C(real)-1C(ideal)

(3)

the transmutation matrix not only converts C(real) into C(ideal) but also converts the detector response units into concentration units, relative to the response of the standard solution. As shown in Appendix I, the transmutation matrix can be used to convert a mixed chromatographic vector exhibiting overlapping bands, c(real, mix), into c(ideal, mix) as shown in eq 4. Of course,

c(real, mix)T ) c(ideal, mix)

(4)

(1)

c(real, mix) must be recorded with the same instrument operating under the same experimental conditions used to generate T. Prior to transmutation, c(real,mix) must be expanded 3-fold along the time axis as described earlier. After transmutation, the expanded c(ideal,mix) vector is reduced to its original time scale by deleting the augmented time points.

where co (a scalar quantity) represents the concentration of the pure component standard solution and I is the identity matrix, containing ones as the diagonal elements and zeros as the offdiagonal elements. Multiplying the ideal shape function by the concentration of the standard is done so the transmutation will convert the response units into concentration units. This is valid when the response is directly proportional to the concentration of the standard.

EXPERIMENTAL AND MODEL DATA The chromatograms of five pure standard solutions (see Table 1) and four mixtures (see Table 2) were recorded with a PerkinElmer LC-235 diode array detector system. Each chromatogram was obtained under identical experimental conditions. Spectra were recorded at 2.05-s intervals from 210.6 to 358.2 s, the region of interest, giving a total of 73 time interval points. Absorbances were recorded at 5-nm intervals from 250 to 385 nm, giving a total of 28 wavelengths. Each chromatogram was corrected for baseline

C(ideal) ) coI

4364

Analytical Chemistry, Vol. 72, No. 18, September 15, 2000

Table 1. Concentrations of Standard Solutions and Data Gathered from Chromatograms (Recorded at 295 nm) Used To Determine the Concentrations of the Components in Mixture 4 by Transmutation Based on the Chromatogram of m-Xylenea concn (mg/mL) c(found) c(expected)

solute

co (mg/mL)

tR (s)

tn (no.)

A(max) (au)

a(max) (au mg-1 mL-1)

f

c(ideal,mix)

toluene naphthalene ethylbenzene m-xylene biphenyl

1.9516 0.1864 1.8824 2.0040 0.0384

245 262 264 274 286

18 26 27 32 38

0.0400 0.4546 0.0183 0.1674 0.0554

0.0205 2.4388 0.0097 0.0835 1.4427

4.0732 0.0342 8.6082 1.0000 0.0579

0.219 2.628

0.890 0.090

0.925 0.569

0.925 0.033

0.958 0.093 0.000 1.002 0.038

a c is the concentration of the solute in the standard solution. t is the retention time of the solute. t is the retention time number. A(max) is o R n the absorbance of the solute at the peak maximum recorded at 300 nm. a(max) is the absorptivity obtained by dividing A(max) by co. f is the relative response function obtained by dividing each absorptivity by the absorptivity of m-xylene. c(ideal, mix) is the relative concentration obtained from Figure 5. c(found) is the concentration of the component in the mixture determined by multiplying c(ideal,mix) by f. c(expected) is the true concentration of the component in the mixture.

Table 2. Concentrations (mg/mL) of Components in Four Mixtures Determined by Transmutation, WFA, RAFA, and MATRA) component mixture 1 toluene ethylbenzene m-xylene mixture 2 toluene naphthalene m-xylene mixture 3 toluene naphthalene ethylbenzene m-xylene mixture 4 toluene naphthalene m-xylene biphenyl

expecteda

transmutation

WFAa

RAFAa

MATRAa

0.976 0.941 1.002

0.954 1.049 1.091

1.073 0.996 0.912

1.025 1.030 1.078

0.917 1.062 1.042

0.958 0.093 1.002

0.688 0.072 0.739

0.755 0.088 0.966

0.804 0.086 0.964

0.763 0.077 1.028

0.958 0.093 0.941 1.002

0.955 0.097b,c 3.107b,d 0.487e

0.931 0.070 0.640 1.066

1.056 0.081 1.617 1.393

0.948 0.098 1.003 1.116

0.958 0.093 1.002 0.038

0.869 0.091 0.940 0.034

0.945 0.073 1.128 0.042

0.991 0.108 1.178 0.043

0.941 0.094 1.359 0.039

a Taken from ref 5. b Insufficient chromatographic separation between naphthalene and ethylbenzene. c Because naphthalene is a strong UV absorber, its signal dominates the signal of ethylbenzene. d The weak signal of ethylbenzene is buried under the strong signal of naphthalene. e The transmuted peak maximum does not coincide with the retention time of the standard.

and for solvent absorption. Details of the experimental procedures are described in a previous paper.5 Model data were constructed from the experimental chromatograms of the pure components described above. These models emulated mixtures of the components. They were constructed for the purpose of studying the effects of sampling rate, band broadening, noise, and limits of detection. PRELIMINARY STUDIES This section describes the results of our investigations involving chromatographic transmutations into Dirac δ functions as well as triangular functions. The studies involved the use of model data, constructed from the chromatograms of the pure components. The models allowed us to investigate errors on the time axis, detector noise, band broadening, and concentration limits. 1. Transmutation into Dirac δ Functions. Our initial studies focused on transmutation into Dirac δ functions. The following is a summary of the results of our findings.

Excellent agreement with expectations was achieved when each of the component bands was proportionately congruent (i.e., the exact same chromatographic profile but different in intensity) and each peak maximum corresponded exactly to a digitized time point. Quantitative results were obtained even when the overall intensities of the bands were varied, individually. However, shifting any one of the overlapping bands a small fraction along the time axis, so that its peak maximum fell between digitized time points, produced oscillations in the transmuted profiles. Similar oscillations resulted when any one of the profiles was broadened. When the above procedures were applied directly to the HPLC/DAD data reported earlier,5 the results were unsatisfactory due to the large oscillations produced by experimental error along the time axis as well as bandwidth differences. These studies clearly showed that transmutation into a Dirac δ function requires (1) bandwidths that do not vary with retention time and (2) ultraprecision and reproducibility along the time axis. These conditions were not achievable with the instrumentation involved in this study. The Dirac δ function, although most ideal, is not a suitable function for transmutation because it has no bandwidth. Bandwidth is intimately related to the standard deviation of the underlying process. Transmutation into a shape function that has a finite bandwidth is much more realistic because it intrinsically permits and accommodates a degree of uncertainty along the time axis. 2. Transmutation into Triangular Functions. Replacing the Dirac δ function (... 0 0 1 0 0 ...) by a triangular function (... 0 0.5 1 0.5 0 ...) allows a degree of play along the time axis, greatly improving the transmuted profiles. For this reason eq 1 was replaced by eq 5. Here, S is the triangular shape matrix obtained

C(ideal) ) coS

(5)

by adding 0.5 to each of the zero elements adjacent to the diagonal elements of I, the identity matrix. Model studies, similar to those used in the Dirac δ function studies, showed that the triangular function can tolerate errors along the time axis as well as band broadening, yielding transmuted profiles that are relatively free from disturbing oscillations. This is demonstrated in the following section. Analytical Chemistry, Vol. 72, No. 18, September 15, 2000

4365

Figure 3. Chromatogram of mixture 4, containing toluene, naphthalene, m-xylene, and biphenyl, recorded at 295 nm.

Figure 4. Concentration profiles obtained by transmutation of the chromatogram of mixture 4, based on the Dirac δ function and m-xylene standard solution. (The experimental region lies between the dotted lines.)

RESULTS 1. Single-Detector Systems. The quantification of m-xylene in mixture 4 will serve to illustrate the transmutation of an unresolved chromatogram obtained from a single-detector instrument. Figure 3 is the chromatogram of mixture 4 recorded at 295 nm, a single wavelength. The graph shows severe overlap of the absorption signals of the four components: toluene, naphthalene, m-xylene, and biphenyl. The peak maximum of Figure 1, the pure standard solution of m-xylene, appears at time interval 34, corresponding to a retention time of 274 s. Figure 1 was used to generate a transmutation function as described in the Theory section. For pedagogical purposes the mixed chromatogram in Figure 3 was transmuted using the Dirac δ function, eq 1, as well as the three-point triangular transmutation function (3ptTTF), eq 5. When the expanded chromatogram of mixture 4 (expanded by adding 73 zeros to the front and 73 zeros to the end of the chromatographic vector) was postmultiplied by the augmented transmutation matrix, based on the Dirac δ function, the graph shown in Figure 4 was obtained. To return to the experimental time window, the first 73 time points and the last 73 time points of the expanded window must be deleted as shown by the dotted 4366 Analytical Chemistry, Vol. 72, No. 18, September 15, 2000

Figure 5. Concentration profiles obtained by transmutation of the chromatogram of mixture 4, based on the triangular function and m-xylene standard solution. (The experimental region lies between the dotted lines.)

lines in Figure 4. The expansion and contraction procedure is needed to remove aberrations caused by clipping the chromatograms at the extremes of the expanded time axis. The region between the dotted lines in Figure 4 represents the original experimental time scale. As shown in Figure 4, three of the four components are clearly resolved, but the fourth component, toluene, with a peak maximum located at time interval number 91 (18 + 73) on the expanded axis, is lost in the oscillations. The concentration of m-xylene, determined from the peak maximum at time interval 105 (32 + 73) on the expanded axis, is 0.755 mg/ mL, whereas the true value is 1.0020 mg/mL. Because of the relatively large magnitude of the oscillations these results are not quantitatively reliable. When Figure 3 was transmuted using the triangular function, the results shown in Figure 5 were obtained. This figure shows a marked improvement in comparison to Figure 4. The oscillations produced by band broadening as well as errors along the time axis are greatly reduced. The severe aberration that appears at zero time, caused by truncation of the profiles during construction of C(real) and C(ideal), is discarded by deleting 73 augmented time points from the beginning and the end of Figure 5, as indicated by the vertical dotted lines. As shown in Figure 5, all four components are clearly resolved. The concentration of m-xylene, determined from the peak maximum, is 0.925 mg/mL. This value compares favorably to 1.0020 mg/mL, the true value. In Figure 5 the peak maximums of the other three components do not represent the concentrations of the respective components. They represent the concentrations relative to the response function of m-xylene. To convert these concentrations into true concentrations, the relative response function, f, of each component must be determined. These functions can be obtained from the chromatograms of the standard solutions. Details of the computations involved are found in Table 1. The fifth column of Table 1 lists the measured absorbances, A(max), of each solute at its peak maximum. Dividing these absorbances by their respective concentrations gives a(max), the absorptivities of the respective solutes shown in the sixth column of Table 1. The response function, relative to

m-xylene, is obtained by dividing each absorptivity by the absorptivity of m-xylene (see column seven, Table 1).

f ) a(max,solute)/a(max,m-xylene)

(6)

The concentration, c(found), of any given component is obtained by multiplying the relative concentration of the component, c(ideal,mix) by its relative response function.

c(found) ) f c(ideal,mix)

(7)

The values shown in the ninth column of Table 1 were obtained in this way. They can be compared to the expected concentrations, shown in the tenth column. Alternatively, to determine the concentrations of each component, appropriate transmutation matrices can be generated from the chromatograms of each pure standard solution. From a theoretical viewpoint, the alternative method is preferred because, for each component, a new transmutation matrix is constructed from the chromatogram of the pure component of interest. This procedure provides maximum congruency between the chromatographic profile of the standard and its profile in the mixture, while minimizing the differences in the band shapes of the signals closest to the component of interest. 2. Mutiple-Detector Systems. To compare the quantitative results obtained by the transmutation method to several other popular chemometric methods, the following study was carried out. Although the transmutation method is designed primarily for single-detector systems, data gleaned from instruments equipped with multiple detectors, such as an HPLC/DAD, can be processed by the transmutation method. This can be accomplished in several different ways. One method relies upon judicious choice of a wavelength where the component of interest has maximum sensitivity. This procedure was used in section 1. Another way is to create a composite chromatographic vector by simple elementby-element addition of all of the chromatographic vectors in a given array. For our study involving 28 wavelengths and 73 time points, this addition would compile a 28 × 73 array into a 1 × 73 vector. When mixture 4 was subjected to this procedure, the composite chromatogram shown in Figure 6 was obtained. The same composite procedure was then applied to the digitized chromatograms, the standard which, in this example, is m-xylene. Using a transmutation matrix based on the m-xylene composite chromatogram and an ideal three-point triangular function, Figure 6 was transmuted into Figure 7. At time interval 32, corresponding to 274 s, the retention time of m-xylene, the concentration of m-xylene was determined to be 0.940 mg/mL. Table 2 gives the results of applying this procedure to the four mixtures. Each component in each mixture was quantified by a transmutation matrix constructed from the chromatogram of the pure component standard solution. The results obtained by chromatographic transmutation for mixtures 1, 2, and 4 are in close agreement with the results obtained by WFA, RAFA, and MATRA, three methods used in the earlier study. For mixture 3, the quantification of the components does not agree with expectations. The reason for this disparity is discussed in the following section.

Figure 6. Composite chromatogram of mixture 4, obtained by adding 28 diode array responses at each time interval.

Figure 7. Relative concentrations of mixture 4 obtained by threepoint triangular transmutation of Figure 6 based on the composite chromatogram of m-xylene.

It is important to point out that transmutation requires only a single-detector instrument, whereas WFA, RAFA, and MATRA require two-dimensional data arrays. Although MATRA is the most accurate method, it also requires knowledge of all of the overlapping components and their associated data arrays. The transmutation method is especially useful for quantifying a particular component in a mixture when the other components are anonymous. Furthermore, the quantification can be done with a relatively inexpensive detector system. 3. Noise and Limit of Detection. The effect of noise was investigated by adding normally distributed random error to the absorbance measurements of the pure standards as well as the mixtures. Typical results are displayed in Table 3, which concerns mixture 4. The first row lists the expectation values based on mixture preparation. The second row lists the concentrations determined by three-point triangular transmutation of the unadulterated data; the root-mean-square (rms) error in this case was determined by principal factor analysis.16 The next four rows show the results obtained upon the systematic addition of normally distributed error. The quantifications of both naphthalene and (16) Malinowski, E. R. Factor Analysis in Chemistry, 2nd ed.; Wiley-Interscience: New York, 1991.

Analytical Chemistry, Vol. 72, No. 18, September 15, 2000

4367

Table 3. Example of the Model Studies of Noise: Concentrations (mg/mL) of Components in Mixture 4 Determined by 3-Point Triangular Transmutation after the Addition of Normally Distributed Absorbance Noise to the Chromatograms of the Mixture and the Standards rms noise

toluene

naphthalene

m-xylene

biphenyl

0.0002 0.0010 0.0020 0.0030

0.958a 0.869 1.034b 0.651 0.694

0.093a 0.091 0.094 0.091 0.093b

1.002a 0.940 0.924b 0.491 0.512

0.038a 0.034 0.033 0.030 0.038b

a(max)c

0.0205

2.4388

0.0835

1.4427

a Concentration based on preparation of the mixture. b Concentration corresponding to maximum acceptable rms noise. c Absorptivity (au/ mg/mL) of the solute at 295 nm.

Figure 8. Three-point triangular transmutations with retention times (i, j, k) two time intervals apart. Table 4. Example of the Model Studies of S/N Ratios: Concentrations (mg/mL) of Biphenyl and S/N Ratios Determined by 3-Point Triangular Transmutation of Chromatograms Generated by the Addition of the Chromatogram of Biphenyl to the Chromatogram of Mixture 1 concentrations actual

determined

% difference

S/N

0.00038 0.00077 0.00115 0.00154 0.00192 0.00384 0.00768

0.00023 0.00062 0.00100 0.00139 0.00177 0.00369 0.00753

60 19 13 10 8 4 2

0.5 1.2 2.0 2.8 3.5 7.3 15.0

biphenyl, two strongest UV absorbers, were relatively unaffected by the addition of noise up to 15 times the experimental noise. However, the quantification of toluene and m-xylene, the two much weaker UV absorbers, could only tolerate a 5-fold increase in noise. It should be noted that as noise was added to the chromatograms the spurious oscillations in the baseline became much more pronounced. Similar results were obtained with the other data sets, but the results were inconsistent. For example, the quantification of toluene in mixture 1 was acceptable at noise levels 20 times the experimental noise. The quantification depends not only upon the concentrations of the constituents but also upon their relative absorptivities. Most importantly, noise tolerability depends on the disparity in the relative bandwidths of the constituents. Increasing bandwidth differential gives rise to progressively increasing baseline noise oscillations. Each chemical system as well as each chromatographic instrument has its own unique peculiarities, requiring its own individual noise assessment. The limit of detection was investigated by measuring the signalto-noise (S/N) ratio under controlled conditions. This was accomplished by a series of model studies involving the addition of a pure chromatogram to a mixture chromatogram. A typical example is presented in Table 4. The first column in Table 4 lists the amount of biphenyl added to mixture 1 which originally contained no biphenyl. The concentrations determined by 3ptTTF are shown in the second column. The third column gives the 4368

Analytical Chemistry, Vol. 72, No. 18, September 15, 2000

percent difference between the actual and the determined concentrations of biphenyl in each of the simulated mixtures. The fourth column contains the signal-to-noise ratios. These were determined by dividing the peak intensity of the biphenyl signal after 3ptTTF by the rms of the baseline noise outside the triangulated profiles. At concentrations below 0.0016 mg/mL, the percent error in the determination was found to be greater than 10% and the S/N ratio was less than 2.8. At concentrations greater than 0.0016 mg/ mL, the percent error was less than 10% and the S/N ratio was greater than 3. Similar studies indicated that the limits of detection are dependent not only upon the concentration by also upon many variable factors such as the measured absorbance of the component relative to the absorbance signals of the overlapping components, the concentration of the component, the retention times of the overlapping components, the digitization process, etc. These results are unique to this investigation and are not directly transferable to other chemical systems (and instrumentation) because each system (and instrumentation) has its own unique characteristics and sensitivities. DISCUSSION 1. Three-Point Triangular Transmutation Function. In order for the transmutation method to be successful, a degree of chromatographic separation between the components is required. For the three-point triangular transmutation function (... 0 0.5 1 0.5 0 ...) used in this study, the retention times of any two components must be at least two digitized time intervals apart.

tRi + 2∆t e tRj g tRk - 2∆t

(8)

In eq 8, tRj is the retention time of the component of interest and tRi and tRk are the retention times of the overlapping components. Figure 8 depicts the situation where these conditions are just fulfilled. Notice that at the retention time of each component there is no overlap from adjacent bands. The peak maximum is unaffected by the overlapping components. This means that the relative concentrations, corresponding to the respective retention times of the components, are true representations of concentrations.

Figure 9. Three-point triangular transmutations with relative times (j, k) one time interval apart.

However, if the difference in retention time between any two components is only one digitized time interval apart, the triangular bands of the offending components will contribute to the total intensities at each of their respective retention times. This condition is illustrated in Figure 9. In this case, the intensities of the offending components at their respective retention times will be composed of contributions from both components and will not represent the concentrations of the individual components. The transmutation technique cannot be used to quantify components when this situation arises. The retention times of the five components are listed in Table 1. All of the components have retention times that differ by at least two time intervals except naphthalene and ethylbenzene, which differ by only one time interval. Thus, transmutation of the chromatogram of mixture 3 cannot be used to quantify these two components but can be used to quantify the other two components, toluene and m-xylene. Some further comments concerning mixture 3 are warranted. Because naphthalene is a strong UV absorber, its signal dominates the signal of ethylbenzene. For this reason, the determined concentration of naphthalene, 0.097 mg/mL, is close to the expectation value, 0.093 mg/mL (see Table 2). In other words, the ideal concentration profile of ethylbenzene has a negligible contribution at the retention time of naphthalene, but the ideal concentration profile of naphthalene contributes a sizable amount at the retention time of ethylbenzene. The determined value of ethylbenzene, 3.107 mg/mL, illustrates this behavior. Although the retention time of m-xylene is greater than two retention intervals from any of the other three components, its determined concentration, 0.487 mg/mL, does not agree with 1.002 mg/mL, the true value. The fact that the peak maximum of the transmuted profile of m-xylene appeared at time interval 33 instead of 32 indicates that a glitch in time occurred during the recording of mixture 3. It is interesting to note here that the transmuted peak at time interval 33 has an intensity of 1.101 mg/ mL, which compares favorably to the expected value, 1.002 mg/ mL. In conclusion, however, if the peak maximum of the transmuted profile does not correspond to the retention time of the component, the analysis is unreliable and should be discarded. 2. Multiple-Point Triangular Functions and Sampling Frequency. Studies were conducted to explore the use of broader

triangular functions such as five-point (... 0 1/3 2/3 1 2/3 1/3 0 ...), seven-point (... 0 1/4 2/4 3/4 1 3/4 2/4 1/4 0 ...), and ninepoint (... 0 1/5 2/5 3/5 4/5 1 4/5 3/5 2/5 1/5 0 ...) triangular functions. The chromatograms of the four mixtures described in Table 2 were transmuted using the above functions. As the number of points in the triangular function was increased, the baseline oscillations increased and the transmuted profiles became less resolved, making the quantification unreliable. This is to be expected because the transmuted profiles, in these situations, are insufficiently resolved. This can be verified by applying the arguments described in section 1 to the broader triangular functions. To investigate the effect of sampling frequency, defined as the number of absorbance points (or absorbance spectra) recorded per unit time, model chromatograms were constructed with a sampling frequency twice that used to obtain the chromatograms of the mixtures reported in Table 2. The overall chromatograms were identical to the original in all respects except for the doubling of the number of acquisition time points. Large baseline oscillations appeared in the transmuted profiles resulting from a threepoint triangular transmutation function (3ptTTF). The oscillations decreased considerably when a 5ptTTF was employed. The oscillations almost disappeared when a 7ptTTF was used and then progressively increased when 9ptTTF and 11ptTTF were used. As the nptTTF function was increased from 3 to 11, the transmuted profiles became progressively broader, as expected. For the model that emulated mixture 4 in Table 2, the analysis of naphthalene yielded the following quantifications (in mg/mL): 0.2059 (3ptTTF), 0.1936 (5ptTTF), 0.1869 (7ptTTF), 0.1991 (9ptTTF), and 0.1935 (11ptTTF). In this case, the 7ptTTF yields a concentration value closest to the true value, 0.1864, listed in Table 1. The 7ptTTF profile also exhibited minimum baseline oscillations. This limited study indicates that the best triangular transmutation function depends on the number of time points as well as the retention times and the bandwidths. The arguments presented in section 1 are applicable to this situation. For the 7ptTTF, the digitized time interval numbers in Figures 8 and 9 should be doubled. Thus the 7ptTTF is sufficient to resolve naphthalene in mixture 4 if the sampling frequency is doubled. However, when the sampling frequency is increased, the transmutation becomes more sensitive to the inequality of the band shapes of the components, which produce undesirable baseline oscillations. Thus, the analyst must seek a tradeoff between oscillation noise and resolution. The process of choosing an appropriate transmutation function is analogous to the use of apodization functions in Fourier transform spectroscopy. One function produces sharp bands and a wavy baseline. Another function produces a smoother baseline but broader bands. We have also investigated other functions such as Gaussian, Lorentzian, rectangular, etc. Albeit, in our limited studies, we found that the triangular function outperformed all of the alternatives. Using an excessive number of time points has other disadvantages besides increasing the baseline noise. For example, as the number of points is increased, the computational time increases dramatically, primarily due to the inverse calculation. Furthermore, an excessive number of points can exceed the computer’s capability to carry out the inverse computation with Analytical Chemistry, Vol. 72, No. 18, September 15, 2000

4369

sufficient numerical precision. In particular, as the ratio ∆t/∆w, where ∆t is the digitized time interval and ∆w is the peak width, gets smaller, the inverse computation will become numerically unstable. A full theoretical treatment of these aspects is the subject of future studies. SUMMARY This study shows how the three-point triangular transmutation method can be used to separate and quantify the components of mixtures that exhibit overlapping chromatographic bands. The method is applicable to data gleaned from single-detector as well as multiple-detector instruments. The following sequence summarizes the steps involved in the transmutation technique. 1. The method requires recording the chromatogram of a pure standard solution of the component of interest using the same experimental conditions used to record the mixture. 2. For multiple-detector instruments, a single wavelength, judiciously chosen to best represent the component of interest, may be selected for the transmutation analysis. Alternatively, a composite vector may be composed from an element-by-element addition of the multiple chromatograms. 3. A real matrix, C(real), is constructed from the chromatogram of a standard solution of the component of interest, using a 3-fold expansion of the time axis. 4. An ideal matrix, C(ideal), is constructed using a triangular function with a maximum set equal to the concentration of the standard and a time axis identical to that used to generate C(real). 5. The transmutation matrix is constructed by premultiplying C(real) by the inverse of C(ideal). 6. The chromatographic vector of the mixture is augmented by adding an equal number of zero elements to the beginning and to the end of the vector, tripling the number of elements in the vector. 7. A transmuted chromatogram of the mixture is obtained by postmultiplying the augmented chromatogram of the mixture by the transmutation matrix and then deleting the augmented regions of the time axis. 8. If the retention times of the adjacent overlapping components are equal to or greater than two digitized time intervals, the height of the transmuted profile at the retention time of the component will indicate the concentration of the component. (The heights of the other components at their respective retention times will not represent their concentrations but will represent their concentrations relative to their response factors and the response factor of the standard used to generate C(real). Quantification of these components will be less accurate because their band shapes, due to band broadening, will not be exactly the same as the standard used to generate the transmutation matrix.) 9. If the component peak maximum of the transmuted profile does not coincide with the retention time of the component, the analysis should be discarded. A computer program, written in MATLAB,17 illustrating all of the above steps, is presented in the Appendix II. APPENDIX I: DERIVATION OF EQ 4 Consider eq 2 of the Theory section, which, for convenience, can be expressed as (17) MATLAB is a matrix laboratory, copyrighted by The Mathworks, Inc., Cochituate Place, 24 Prime Way, Natick, MA 01760.

4370

Analytical Chemistry, Vol. 72, No. 18, September 15, 2000

CrealT ) Cideal

(A-1)

Recall that each row of Creal represents a pure (unmixed) chromatographic vector with a retention time that corresponds to its respective row number. Each vector is identical in shape and intensity but differs with respect to peak position. According to (A-1), the jth chromatographic vector can be expressed as

Creal(j,:) T ) Cideal(j,:)

(A-2)

Multiplying (A-2) by an arbitrary scalar, mj, associated with the jth vector, gives

mjCreal(j,:)T ) mjCideal(j,:)

(A-3)

Null chromatographic vectors, where all of the elements are essentially zero, are those vectors for which mj equals zero. According to (A-2), transmutation of any sum of real chromatographic vectors, each having the same shape but different intensity and different retention time, will yield a vector that is the sum of the individual ideal vectors, as expressed by

∑m C j

real(j,:)T

)

j

∑m C j

ideal(j,:)

(A-4)

j

A chromatogram that exhibits overlapping signals, c(real,mix), is simply the sum of individual chromatograms and null chromatograms, where each such chromatogram is associated with a specific row. In other words,

c(real,mix) )

∑m C j

real(j,:)

(A-5)

j

In this case, mj represents the relative amounts of the components having retention times corresponding to their respective rows. Null vectors are associated with those rows that do not correspond to the retention times of any of the chemical components. Inserting (A-5) into (A-4), and recognizing that

c(ideal,mix) )

∑m C j

ideal(j,:)

(A-6)

j

gives (A-7), which is presented as eq 4 in the Theory section.

c(real,mix) T ) c(ideal,mix)

(A-7)

APPENDIX II: COMPUTER PROGRAM The following is a computer program, written in MATLAB,17 designed to carry out three-point triangular transmutation of a mixed chromatogram recorded with a single or multiple detector instrument. function [cpred, cidealmix] ) transmut(chrmix,chrstd, cstd) % chrmix ) chromatogram with overlapping bands % chrstd ) chromatogram of pure component in pure standard solution % cstd ) concentration of pure component in standard solution

% cpred ) concentration of component in mixture determined by transmutation % cidealmix ) concentration profile obtained by transmutation [r,c] ) size(chrmix); if r > l chrmix ) sum(chrmix); chrstd ) sum(chrstd); end plot(chrmix), shg xlabel(‘Data Column (Time Axis)’) ylabel(‘Intensity’) pause % triangle ) an ideal triangular chromatographic profile triangle ) [0.5 1 0.5]; timepts ) length(chrmix); zt ) zeros(1,timepts); augmix ) [zt chrmix zt]; augstd ) [zt chrstd zt]; [dummy mxastd] ) max(augstd); [dummy mxtri] ) max(triangle); [dummy mxstd] ) max(chrstd); times3 ) timepts*3; z ) zeros(1,times3);

realv(i,:) ) [z augstd z]; idealv(i,:) ) [z triangle z]; for i ) l:times3 realm(i,:) ) realv(1,times3+mxastd+l-i:times3+mxastd+ times3-i); idealm(i,:) ) cstd*idealv(1,times3+mxtri+l-i:times3+mxtri+ times3-i); end T ) pinv(realm)*idealm; cidealmixx ) augmix*T; plot(cidealmixx),shg pause cidealmix ) cidealmixx(1,timepts+l:2*timepts); plot(cidealmix), shg hold plot(mxstd, 0,′+′,mxstd,max(cidealmixx),′+′),shg pause hold off cpred ) cidealmix(1,mxstd)

Received for review April 10, 2000. Accepted July 3, 2000. AC000407M

Analytical Chemistry, Vol. 72, No. 18, September 15, 2000

4371