Anal. Chem. l900, 60,2745-2751
2745
Reliability of Iterative Target Transformation Factor Analysis When Using Multiwavelength Detection for Peak Tracking in Liquid Chromatographic Separations Joost K. Strasters,* Hugo A. H. Billiet, and Leo de Galan
Department of Analytical Chemistry, Delft University of Technology, De Vries van Heystplantsoen 2, 2628 RZ Delft, T h e Netherlands Bernard G. M. Vandeginste and Gerrit Kateman
Department of Analytical Chemistry, Catholic University of Nijmegen, Toernooiveld, 6525 ED, Nijmegen, T h e Netherlands
Rellable UV spectra are needed to asslst In peak tracking In succesdve chromatograms, e.g. for opilmizatlon purposes. If lndlvldual UV spectra of Overlapping components are derived by means of Heratlve target transformath factor anatysls, the results depend on the chromatographk resdutlon, the spectral simllar’ty, and the relative concentratlons of the components. A quantHatlve model Is developed to judge the rellablllty of the derived UV spectra from the observed resolutlon, the observed spectral slmllarfty, and the observed concentration.
When interpretive strategies for mobile phase optimization are applied in high-performance liquid chromatography (HPLC) (1,2), the optimal mobile phase composition is determined on the basis of the retention behavior of the individual components. In order to derive this behavior from a limited number of chromatograms of an unknown sample, corresponding solutes must be recognized in successive chromatograms (peak tracking). Peak recognition can be achieved by a comparison of peak areas (3),peak ratios (4), or complete UV spectra resulting from multiwavelength detection (5). One of the problems encountered is the occurrence of severely overlapping peaks,which hinders a direct determination of the individual component characteristics. However, when multiwavelength detection is applied more conclusions can be drawn by a number of mathematical techniques, which are based on a multivariate approach of the data. The number of components can be determined by an examination of the amount of variation in the data, for instance by means of a principal component analysis (6). Techniques such as multicomponent analysis (7) and target fador analysis (6,8)derive the individual elution profiles by using a set of reference spectra. Self-modeliigcurve resolution and related techniques (9-121, such as iterative target transformation factor analysis (I’M’-FA) ( I I ) , derive both the individual elution profiles and UV spectra of the components by imposing a number of boundary conditions such as nonnegative concentrations in the calculated profiles and nonnegative absorptivities in the calculated spectra. These methods differ mainly in the imposed boundary conditions and in the way in which the spectra and elution profiles that conform to these demands are selected. In a previous publication (13)we described advantages and disadvantages of a number of multivariate techniques used for peak recognition. The multicomponent analysis requires no chromatographic resolution but accurate knowledge of the individual UV spectra, the target factor analysis requires some resolution and approximate knowledge of the spectra, and the 0003-2700/88/0360-2745$0 1.50/0
iterative target transformation demands more resolution but no knowledge of the individual spectra. The complementary characteristics of these techniques suggest a combined approach by way of an expert system. In order to develop such a system, a number of rules must be derived, indicating which technique should be applied in a given instance. These rules must be based on quantitative knowledge regarding the performance of the proposed method(s) under the given circumstances. The results of the iterative target transformation are influenced by the chromatographic resolution, the number of components, the level of noise, the spectral similarity, and the relative amounts of the components (14-16). This paper is concerned with the practical limitations imposed by three of these factors, i.e. the resolution, the spectral similarity, and the relative concentration. A quantification of the quality of the results is used to derive a rule that decides whether or not the calculated UV spectra are accurate enough to be used in the peak-tracking procedure.
THEORY The iterative target transformation (11)utilizes the results of a principal component analysis (6) performed on the mixture spectra observed during the elution of a chromatographic peak cluster. The original data set is decomposed in two matrices, each consisting of orthogonal vectors in order of diminishing importance with respect to the variance in the data. One of these matrices is related to the spectral characteristics of the components,the other to the elution profiles, thus forming a bilinear system in accordance with Beer’s law. Like most forms of factor analysis, the final goal is the derivation of the transformation matrix [TI, which transforms the abstract matrices into physically meaningful elements [D]= [R][C] = [R’][C’] = [R’][T]-’[T][C’] = [S][E]
(1) [D] is the data matrix of a peak cluster of n components, with nt spectra as columns. Each spectrum consists of nw absorption values, hence the dimensions are (nw X nt). [R] and [C] are the abstract matrices resulting from the principal component analysis (dimensions respectively (nu x nt) and (nt X nt),assuming nw is larger than nt). The rows of matrix [C] are the eigenvectorsof the variance-covariance matrix of [D]. [R’] and [C’] are the reduced matrices, taking only the n significant factors of [R] and [C] into consideration (dimensions respectively (nw x n) and ( n X nt)). The transformation of these matrices results in the physically meaningful matrix [SI,containing the pure component spectra as columns, and the matrix [E] containing the individual elution profiles as rows. The normalization factors that are necessary to present both spectra and elution profiles in a normalized 0 1988 American Chemical Society
2746
ANALYTICAL CHEMISTRY, VOL. 60, NO. 24, DECEMBER 15, 1988
form (vector length of 1) can be used as a measure of concentration [SI[El= [S*l[Wl[E*I (2) [S*]and [E*] contain the normalized spectra and profiles. [W] is a diagonal matrix, dimensions (n x n),containing the normalization factors. The iterative target transformation derives the transformation matrix [TI by using matrix [C’]. First the peak locations of the components are determined by application of a varimax rotation. The first estimate of each elution profile consists of a single delta function at one peak location and is projected onto the hyperspace defined by the rows of [C’]
t = ([C’][C’]T)-l-[C’].el el’ =
~
name
1 2
benzaldehyde acetophenone acetanilide methyl salicylate nitronaphthalene diethyl phthalate benzophenone anisole
7 8
k
[C’IT,t
EXPERIMENTAL SECTION Instrumentation. The UV spectra used in the simulations were registered after elution of the components from a Nova-Pak C18 column (15 cm X 3.9 mm i.d., 5 Fm particles) by means of the HP-1040A fast-scanning LPDA detector (Hewlett-Packard, Waldbronn, F.R.G.). The detector was connected to the HP-1090 liquid chromatograph and an HP-85 desk-top computer, equipped with inputjoutput, plotterlprinter, mass storage and advance programming ROMs, 16-kbyte additional memory, HP-IB IEEE-488interface, and RS-232C serial interface. The data were temporarily stored on 51/4-in.flexible disks using a HP82910M disk drive. The spectra were transferred from the HP-85 to a PDP11jO3 system (Datacare, Zejst, The Netherlands) by means of the serial interfaces on both computers. The multivariate analysis was performed on the PDP11/03, which was equipped with two 8-in. disk drives, a RD51 hard disk unit, and 4006-1 computer display terminal (Tektronix, Beaverton, OR). Further analysis of the results was done on an IBM-PC (International Business Machines Corp., U.S.A.) equipped with two 51j4-in.disk drives, an expansion unit with two 10-Mbyte hard
no.
3 4 5 6
(3)
(4) el is the first estimated target and el’its projection. This projection is adjusted for negative concentrations and secondary maxima, and again subjected to a projection. The iterative procedure continues until one of three conditions is encountered: (a) The projection resembles the last estimate to within a certain degree. The resemblance is expressed as the correlation coefficient between the estimate and its projection and should exceed a critical value, e.g. 0.9999. The occurrence of this condition indicates the number of corrections to the projection has been minimal, hence a realistic chromatographic profile has been derived for this component. (b) The improvement between subsequent estimates is too low. If this condition is not taken into account, the procedure goes on indefinitely when the correlation required in condition (a) cannot be reached. Even though this means there still is a significant deviation in the estimated profile from a true chromatographic profile, the results with respect to the derived spectral and chromatographic characteristics can still be reliable enough to be used for our purposes. (c) The next estimate needs a larger number of corrections than the current one. This means that the procedure is actually “walking away” from an acceptable solution and no reliable profile can be determined at this peak location. This condition is, among other things, encountered when a drifting base line is erroneously considered as an additional component and is treated accordingly. This procedure is repeated for every component (peak location) in the cluster. Since the inverse transformation in eq 1can only be derived after all profiles have been determined, the occurrence of condition (c) at one of the peak locations means that no reliable UV spectra can be calculated for this cluster, given the applied number of components.
~~
~~
Table I. Identity of the Components and the Concentration C Used To Record the Spectral Characteristics Displayed in Figure 1 C, mg/mL
0.17 0.08
0.09 0.19 0.13 0.14 0.20
0.16
1
8
210
nrn
400
210
nm
400
Flgure 1. UV spectra of the components listed in Table I between 210
and
400
nm recorded in 46% acetonitrile.
disks, the Intel 8087 mathematical coprocessor, and an HP-7470A graphics plotter with serial interface. Spectral Data and Simulations. In order to determine the influence of the resolution, spectral similarity, and concentration on the results of the iterative target transformation, a number of simulations was performed. These simulations consisted of the combination of spectral data of two components using Gaussian elution profiles (standard deviation 0.25 min) in an interval of 3 min. Random noise (standard deviation approximately 0.05% of the maximum absorption in a peak) was added. Every cluster was analyzed on the basis of 31 spectra evenly spread over the 3-min interval. The spectral data used for these simulations were recorded in 46% acetonitrile. The identities and concentrationsof the compopentsare listed in Table I; the spectral characteristics are displayed in Figure 1. Before storage in a library, the spectra were normalized in such a way that for each spectrum a summation of the squared absorbances equals one, in order to ensure comparable total UV absorptivity of the components in the simulated clusters. The pairs of components combined in a chromatographic profile and the corresponding spectral similarity expressed by means of a correlation coefficient are listed in Table II. The chromatographicresolution was varied from 0.1 to 0.6 in steps of 0.1 and from 0.6 to 1.2 in steps of 0.2 for equal concentrations of both components. At a number of resolution settings the concentration ratio of the componentswas varied from 0.01 to 99.
ANALYTICAL CHEMISTRY, VOL. 60, NO. 24, DECEMBER 15, 1988
2747
1.2
Table 11. Identity of the Components Combined in the Simulations and Their Spectral Similarity plza
1
component 1
component 2
P12
1 1 4 3 5 1
2 3 5 6 7 8
0.9562 0.9078 0.7379 0.7354 0.4174 0.1260
t
o’8
0.6
R* 0.4
The numbers refer to Table I. 0.2
Software. The software used for the multivariate analysis was written in Fortran IV. The algorithm for the iterative target transformation factor analysis was adapted from the description by Vandeginste et al. (11). Correlation coefficientswere calculated according to Reid and Wong (17). As far as internal memory allowed, the matrix calculations were used from the Scientific Subroutine Package from Digital Equipment Corp. (Marlboro, MA) as was the Varimax subroutine. In all other cases the calculations were performed by simple algorithms. The eigenv&ra and eigenvalues of the covariance matrix of the data matrix were determined by the HQRII algorithm (18). Further analysis of the data on the IBM-PC was performed with the spreadsheet Symphony (Lotus Development Corp.). Nonlinear parameter estimation was performed according to the gradient expansion algorithm described by Bevington (19). The selection algorithm subsequently described was based on the zero-point determination by the Newton-Raphson method. Both algorithms were implemented with the Turbo-Pascal compiler (Borland International).
RESULTS AND DISCUSSION Three factors influencing the results of the iterative target transformation, the resolution, the spectral similarity, and concentration,were varied in a number of simulationsin order to derive a quantitative dependence to be used in a peakrecognition procedure. This investigation was performed in five steps: 1. The Resolution. The first factor investigated was the chromatographic resolution. As described below, the profiles calculated for either of the two components by means of ITT-FA, are asymmetrical and their apparent peak overlap will be lower than the true overlap. In order to define the resolution unambiguously, a bi-Gaussian profile (20)was fitted on each of the two calculated profies. The observed resolution R* was defined by using the retention times and appropriate standard deviations of the fitted profiles (5) where tl and t2are the retention times of respectively the first and second calculated peak profile, IJ1R is the right-hand standard deviation of the first profile, u2L is the left-hand standard deviation of the second profile. In the case of symmetrical Gaussian peaks this equation is equal to the standard defiition, used to defiie the true resolution. The use of fitted bi-Gaussian profiles was preferred to a direct calculation of the resolution by using the calculated concentrations and the statistical moments derived from these concentrations, because of the increased sensitivity of the latter method to the location of the sampled spectra over the profiie. During the simulation the true resolution of different pain of componentswas varied and compared to the resolution observed in the calculated profiles. The relation between the correct resolution R and the observed resolution R* is displayed in Figure 2. As was reported earlier (13),the observed resolution is larger than the true resolution for small values of R. This is caused by a “local optimum” during the iterative approach: when R becomes too small the iteration of the profie reaches a point
0
R-
Figure 2. Observed resolution R versus the true resolution R . The solid line represents eq 6.
where a slight change in the relative contributions of the eigenvectors causes a negligible change in the corrected projections. Consequently the same projection is repeated and the procedure stops on the basis of condition (b) mentioned in the theoretical section. Geometrically speaking the plane through the projection and its correction is perpendicular to the hyperspace defined by the rows of [C’]. Because the procedure is stopped before the correct profiles are reached the calculated peaks will be narrower than the actual chromatographic peaks. This effect manifests itself more strongly for smaller values of R; hence reasonable profiles with a resolution smaller than 0.25 will never be observed (assuming the presence of two or more components can be detected). It is important to realize that the relation between R and R* presented in Figure 2 is independent of the spectral similarity of the components. This can be explained as follows: The vectors of matrix [C’] form a basis of a hyperspace containing the elution profiles. Consequently these profiles can also be regarded as a basis of this hyperspace. Although the vectors in matrix [C’] are themselves dependent on the UV spectra of the components, the hyperspace that is defined by these vectors is not, under the condition that the profiles of the components do not change. Hence the subsequent projections observed during the iterative procedure will follow the same course and the resulting profiies will be independent of the spectral similarity. Furthermore, a similar argument is valid for profiles of components present in different quantities: the shape of the derived profiles is independent of the relative concentrations of the components. When the spectral similarity or difference in concentration becomes extremely large, the above conclusionsare still valid, but the experimental noise will start to play a role as well. This has not yet been examined. The relation in Figure 2 can be described by R* = R
+ 0.25 exp(-4.2R)
(6)
This equation was chosen because of the asymptotic approach to the line R* = R. The conclusion is that we have to base our evaluation on an observed resolution rather than the true one. 2. Influence of the Resolution on the Observed Concentrations. In order to describe the concentrations of the components in the two-component mixtures, we define the fraction ni wii
ni = Wll
+ w22
(7)
wI1 and w22 are the diagonal elements of the normalization matrix W (eq 2). The observed fraction ni*,derived from the profiles and spectra calculated by means of ITT-FA, is com-
2748
ANALYTICAL CHEMISTRY, VOL. 60, NO. 24, DECEMBER 15, 1988 1
R
A -I
/-
I
p
11
0.94
0.92
0.9 0
0.2
0.4
0.6
0.8
R-
1
n--+ Figure 3. Observed fraction n versus the true fraction n at various
values of the resolution R . The lines represent eq 8.
pared with the true fraction ni and the resulting relation is displayed in Figure 3. Although there is a slight dependence on the spectral similarity of the components, this dependence could be neglected in the remainder of the analysis. As discussed in the previous section, the calculated profiles will be distorted at low resolutions. In fact, part of the profile of one component is described by erroneous contributions of the second component. When both components are present in equal concentrations, these errors will be compensated by a similar error in the second profile; hence the central point n = n* = 0.5 in Figure 3. However, when the concentration of one component is significantly lower, the positive contribution from the second component is no longer compensated by an equal loss of the first one and the observed fraction will be higher than the true fraction (n< 0.5). On the other hand, when n is larger than 0.5, the observed fraction will be lower than the true fraction. This effect will be more noticeable when the resolution is smaller. With the second-ordereffects neglected at very low and high fractions, the following relation could be derived for the observed fraction n*: n* = 0.5 ( n -0.5)(1 - exp(-5.2R)) (8)
+
3. Influence of the Resolution on the Spectral Quality. Since we will be using the spectra derived from the transformation in a peak-recognition procedure, the central point in this discussion is how well the calculated spectra resemble the true spectra of the componentspresent in the peak cluster. Consequently, we have to define the quality of a calculated spectrum. Because we are using a vector description of the spectra, this quality is expressed by means of pll, the correlation coefficient between the true and the calculated spectrum of the same component. This quantity is directly related to the angle between two vectors or the distance between two averaged and normalized vectors (13)and as such describes the effects of the observed distortions in the profiles on the derived spectra. It follows from eq 1 that errors in the estimated profiles will be transformed into errors in the calculated spectra because of matrix [TI.In practice this means that the relative contributions of the columns of [R']to the calculated spectra are in error and will thus result in mixtures of the true spectra of the components. However, this mixing will be more pronounced when the true spectra of the two overlappingsolutes differ more strongly and hence the resulting relation between the resolution and the quality of the spectra will depend on the spectral similarity of the components involved in the peak cluster. This similarity is defined by means of the correlation coefficient p12 between the true spectra of the components. The results of the simulations with respect to components in
Flgure 4. Correlation pI1 between the spectrum calculated by means
of IlT-FA and the true spectrum as a function of the resolution R and the spectral similarity p,* at a constant concentrationratlo of 1 (n = 0.5). Identical plots are observed for the second component @=). The solid curves refer to eq 9. 1
t
0.9
o'890 0.7
0
0.2
0.4
0.6
0.8
nFlgure 5. Correlation p,, between the spectrum calculated by means of I T - F A and the true spectrum as a function of the resolution R and the fraction n at a fixed spectral slmilarlty of p,2 = 0.7379. Identical
plots are observed for the second component @22). The simulations were performed at R = 0.20 (O), 0.35( O ) ,0.50(X ), 0.75 (+), and 1.00 (A). The curves refer to eq 9.
equal concentrations are displayed in Figure 4. As was to be expected, the quality of the spectra is close to 1 (a perfect match) when the resolution is reasonably large but starts to deteriorate when the resolution drops below approximately 0.4. The extent of the deterioration depends on the spectral similarity pI2. The influence of the resolution is very prominent for highly dissimilar components, but less noticeable for components with similar spectra (plz close to 1). The extent of the deviation from pn = 1is limited by the correlation between the true spectrum of component 1and the mixture of the two pure-component spectra. This is dictated by the correlation between the two spectra (p12)and by the relative concentrations of the two components. This can be compared to the situation where the resolution is too low to detect two components and the cluster is represented by one hybrid spectrum. The influence of the relative concentration is examined in Figure 5 for one set of spectra with a spectral similarity of 0.7379. When the fraction n is high, good quality spectra are derived, irrespective of the resolution of the components. However, when both the fraction of the component and the resolution are low, strong deviations are observed. In this case the lowest value of the spectral quality will be equal to the spectral similarity of the components: the calculated spectrum will be almost equal to the spectrum of the second component, present in excess, instead of the (correct) spectrum of the
ANALYTICAL CHEMISTRY, VOL. 60, NO. 24, DECEMBER 15, 1988
2749
1
0.9
0.8
0.2
1
0.126
0.2
0
0.6
0.4
0.8
1
0.7
1.2
RFigure 6. Observed correlation p12*between the spectra calculated by means of ITT-FA as a function of the resolution R and the true melation p12between the correct spectra at a constant concentration ratio of 1 (n = 0.5). The solM curves refer to eq 10.
component present in short measure. Hence, the determination of the spectral quality will be equal to the comparison of the spectra of the two components in the mixture. When both effects, resolution and concentration, are taken into account, the following relation for the spectral quality can be derived:
1 - Pll --
-
exp((-18.1R - 3 . 3 ) n ( - ” ~ ~ .~) 1 ~ + ~(9) 1 - Pl2 The important characteristics are the asymptotic approach to 1 for large values of either n or R and the extent of the deviation from pll = 1when both R and n are small, ultimately resulting in the spectral similarity p12 when the fraction n approaches 0. Although we observed a slight dependence of the values pll on the shape of the spectra, this influence was negligible with respect to the overall deviation from 1. 4. Spectral Similarity Between the Components. We just showed that the spectral similarity between the two overlapping solutes is an important factor when considering the quality of the spectra calculated for each one. However in the case of unknown solutes, this similarity, p12, is not exactly known: because of the mixing of pure component spectra in the case of strongly overlapping peaks, the calculated spectra will resemble each other more closely than the true spectra, and the observed spectral similarity will be closer to 1than the real similarity. This is illustrated in Figure 6, displaying the observed spectral similarity, p12*, as a function of the resolution and the correct spectral similarity, p12, for two components present in equal concentrations. As was to be expected, the value of p12* asymptotically approaches the true correlation plz for large values of R but shows a dramatic increase when R drops to lower values. This increase is expressed in the following proposed equation:
1 - P12 --
1 - Pl2*
-1+
0.021
~
~- p12)0.34 ~ (
1
(10)
Again, the dependence on the actual shape of the spectra is low, as is illustrated by the virtually coinciding data corresponding to respectively methyl salicylate/nitronaphthalene (pI2 = 0.7379) and acetanilide/diethyl phthalate (pE = 0.7354). Although the spectral characteristics of both pairs of components are completely different (Figure 1,components 4 and 5 or 3 and 6, respectively), the results on the basis of the spectral correlation exhibits similar behavior because the relative positions of the vectors describing the spectra are comparable. The errors introduced with diminishing resolution are further emphasized when the relative concentration of one
3 0 0.2 0.4 0.6 0.8 1
n Ftgure 7. Observed correlation pI2* between the spectra calculated by means of ITT-FA as a function of the resolution R and the fraction n at a flxed spectral slmllarlty of p12= 0.7379. The curves refer to eq 11.
of the components is lowered, as illustrated in Figure 7. At both low and high values of n,the observed similarity plz* will be closer to 1,since the spectrum of the component present in short measure will be heavily contaminated with the spectrum of the other component. The highest value, observed at low resolutions and low and high concentrations, is 1, because both calculated spectra will be equal to the spectrum of the component present in excess. When the fraction n is taken into account, the following equation can be applied to describe the observed deviations:
1 - P12* --
n’
(11) + 0.39) + n’ 1 - p12/ p l i refers to the observed spectral correlation when both components are present in equal concentrations, as expressed by eq 10. The adjusted fraction n’ is defined by exp(-5.6R
n’ = (n/(0.5 - n))-”.40R+1.6 ( n < 0.5) n’ = ((1 - n ) / ( n- 0.5))4.40R+1.6 ( n > 0.5)
(12a) (12b)
Equation 11 will approach 0 if n approaches either 0 or 1, corresponding with p12* = 1 or identical spectra. If n is approximately 0.5, eq 11predicts the value derived from eq 10. 5. Practical Implementation of the Results. Until now, we looked at the data as though the true characteristics (i.e. R, p12, and n) of the system are known. In reality we are confronted with unknown samples, and peak clusters with unknown UV spectra and profiles. This means that from the above variables only three are known, i.e. the observed resolution, R*, the observed correlation between the spectra, p12*, and the observed fractions of the components nl* and n2*. However by combining the results of eq 6,8,9, and 11we can still derive an estimate of the quality of the calculated spectra. Once we have estimated the true resolution R from eq 6, which will be more accurate for larger values of R*, the true fractions can be derived from eq 8. The observed correlation p12* combined with this resolution and these fractions can be used to solve 11 for plz. Inserting the values for the resolution, concentration, and the correct spectral similarity in eq 9 finally results in an approximation of the reliability of the calculated spectra. This is illustrated in Figures 8-10. Figure 8 shows a three-dimensional view of the situation when both components are present in equal concentrations. The quality of the spectra, expressed by pll, is presented as a function of R* and p12*and shows a strong decrease for low resolutions and dissimilar spectra. In other words, the more dissimilar the spectra, the higher the demands with respect to the resolution when we want to use the results for a
2750
ANALYTICAL CHEMISTRY, VOL. 60, NO. 24, DECEMBER 15, 1988
1
00
\
\
0 60
I\
-03.0
0.05
-T,
o.88 0.95
O!
Flgure 8. Three-dimensional picture of the dependence of the spectral quality p i l of the spectra calculated by means of I n - F A on the observed resolutlon R * of the calculated profiles and the observed correlation pi2* between the calculated spectra at equal concentrations of both components.
I
0.40
0.40
0.30
p
0.30
I
0.50
Flgure 9, Isoquality lines of constant p f l in dependence of the observed resolution R' and the observed correlation p12*between the two components In the peak-cluster present in equal concentrations.
peak-recognition procedure. This apparently illogical result follows from the fact that ITT-FA uses the full matrix of mixture spectra and calculates the spectra on the basis of the derived profiles. Obviously if we know the two dissimilar spectra beforehand, we can obtain much better results (asfar as the elution profiles are concerned) by a judicious choice of the wavelengths or techniques which make full use of the available knowledge such as a straightforward target factor analysis. Figure 9 shows a more practial approach of the problem which gives a better idea on the limitations of the method. The quality of the calculated spectra is expressed by means of "isoquality" lines in a plane defined by R* and plz*. The actual values for R* and plz* determine the quality of the calculated spectra and the user can judge whether or not this quality is sufficient to use the spectra in a peak-tracking procedure. Spectra corresponding to points located on the right side of a given line (pCrit)will have a reliability that is larger than the selected value (pll > pcAt). Finally, Figure 10 displays the influence of the actual concentration of a component on the lines in the previous figure. All lines correspond to a desired spectral quality of 0.99. As expected from the previous discussion, the amount of resolution required for components in short measure will be much higher to derive reliable spectra. On the other hand, very little resolution is required for a component in excess,
0.50
R*-Flgure 10. Isoquality lines of pl, = 0.99 at dlfferent fractions n In dependence of the observed resolution R * and the observed correlation p i 2 * .
especially because the observed correlation will be almost equal to 1. The required quality will be dictated by several factors. On the one hand it is not practical to demand a higher reliability than that which is feasible given the experimental reproducibility. In the practice of optimization for reversed-phase HPLC this is especially relevant because different organic modifiers are used that may cause slight shifts in the pure component spectra (13). Thus even for one and the same component the spectrum will vary to some extent. In chromatographic practice it seems unrealistic to demand pcrit > 0.995. On the other hand the spectra need only be accurate enough to discriminate within a given set of pure component spectra. When a sample only contains components with highly dissimilar spectra, the demands on the quality of the calculated spectra can be much lower than for a sample with two or more components with similar spectra.
CONCLUSIONS As indicated before (13-16) the resolution, spectral similarity, and concentration are important factors influencing the results of the iterative target transformation. By investigating a number of relations between these factors and the spectral quality, we were able to derive practical quantitative rules to evaluate the results in the case of truly unknown peak clusters. Especially important is the realization that the derived elution profiles are in principle independent of the spectral similarity of the components, thus simplifying the derived models for the reliability of the calculated spectral results. However, the exact form of these relations is limited to the case of clusters containing two components. Although the general influence of the factors will be the same for more complicated situations, the quantitative conclusionsmust be applied with caution. Currently we are investigating the results for clusters containing more than two components. The above rules have been implemented for the interpretive optimization of an LC separation, which was presented elsewhere (21). LITERATURE CITED (1) Schoenmakers. P. J. J. of Chromatog. Libr. 1086, 35. ( 2 ) Drouen, A. C. J. H.;Billiet, H. A. H.;Schoenmakers, P. J.; De Galan, L. Chromatograph& j@82, 16, 48. (3) Otto,M.; Wegscheider, W.; Lankmayr, E. P. Anal. Chem. 1088, 6 0 , 517. (4) Drouen, A. C. J. H.; Billiet, H. A. H.;De Galan, L. Anal. Chem. 1084, 56, 971. (5) Drouen. A. C. J. H.;Billiet, H. A. H.; De Galan, L. Anal. Chem. 1085. 5 7 , 962. (6) Malinowsky, E. R.; Howery, D. G. Factor Analysis in Chemistry; Wlley: New York, 1980.
275 1
Anal. Chem. 1988, 60, 2751-2755
(7) (8) (9) (10) (11) (12) (13) (14) (15)
Blackburn, J. A. Anal. Chem. 1965, 37, 1000. McCue, M.; Mallnowsky,E. R. Appl. Spechosc. 1983, 37, 463. Lawton, W. H.; Sylvestre, E. A. Technomehics 1971, 13, 617. Vandeginste, 8. 0. M.; Essers, R.; Bosman. T.; Reljnen. J.; Kateman, G. Anal. Chem. 1985, 57, 971. Vandeginste, B. G. M.; Derks, W.; Kateman, G. Anal. Chim. Acta 1985, 173, 253. Maeder, M. Anal. Chem. 1987, 59, 530. Strasters, J. K.; Billlet, H. A. H.; De Galan, L.; Vandeginste. B. G. M.; Kateman, G. J . Chromatogr. 1987, 385, 530. Vandeglnste, 8. G. M.; Leyten, F.; Gerrltsen, M.; Noor, J. W.; Kateman, G.; Frank, J.; J . Chemom. 1987. 1 , 57. Gemperline, P. J. Anal. Chem. 1986, 58, 2656.
(16) (17) (18) (19)
Seaton, G. G. R.; Fell, A. F. Chromatographla 1967, 24, 208. Reid, J. C.; Wong, E. C. Appl. Spechosc. 1966, 20, 220. Beppu, Y.; Nlnomlya, I. Comput. Chem. 1982. 6, 87. Bevington, P. R. Data Reduction and Error Analysis for the Physicel Sciences; McGfaw-Hill: New York, 1969; Chapter 11. (20) Grushka, E.; Myers, M. N.; Giddlngs, J. C. Anal. Chem. 1970, 42, 21. (21) Strasters, J. K.; Billiet, H. A. H.; De Galan, L.; Vandeglnste, B. 0 . M.; Kateman. G. Abstracts 1Ith International Symposium on Column Llquld Chromatography,Amsterdam, 1987; Th-P-23.
RECEIVED for review October 20, 1987. Resubmitted August 10, 1988. Accepted September 13, 1988.
Structural Basis for Enantiomeric Resolution of Pseudoephedrine and the Failure To Resolve Ephedrine by Using P-Cyclodextrin Mobile Phases Edward A. Mularz,’ Linda J. Cline-Love, and Matthew Petersheim* Chemistry Department, Seton Hall University, South Orange, New Jersey 07079
Resolution of the enantlomers of pseudoephedrlne was accomplished by the addition of @-cyciodextrinto a llquid chromatographlc mobile phase, but attempts to resolve (+)and (-)-ephedrine falled. ‘H NMR experiments demonstrate that ail four Isomers of ephedrine and pseudoephedrlne include in the cavlty of @-cyciodextrlnwith nearly identical positlons and orlentations of their phenyl rings. These NMR studies also provlde evidence that the drug’s ammonlum and hydroxyl groups hydrogen bond wlth hydroxyls of @-cyciodextrin. The arrangement of these hydrogen bonds appears to be responsible for the observed chromatographic behavior.
INTRODUCTION I t is often possible to radically improve chromatographic performance by introducing a component to the mobile phase that differentially interacts with the analytes, altering the free analyte concentration for partitioning into the stationary phase. These intentional mobile phase interactions are referred to as secondary chemical equilibria, and there are several recent reviews that discuss the various approaches which have been taken (1-3). The work presented here has to do with the use of cyclodextrins in a mobile phase with the intent of chromatographically resolving analyte enantiomers without prior derivatization or the use of specially prepared columns. In these experiments the secondary chemical equilibrium involves the formation of analyte-cyclodextrin inclusion complexes. Cyclodextrins are toroidal-shaped oligomers of a-D-glucosethat are widely noted for the inclusion complexes they form with other molecules. These complexes have proven useful in directing stereospecific chemistry ( 4 , 5 )and stabilization of pharmaceutical and cosmetic preparations, and have been previously employed as chiral resolving agents (6-8). The hexamer, heptamer, and octamer forms are referred to as a-, p-, and y-cyclodextrin, respectively, and all three are commercially available, stable under a wide range of chromatographic conditions, and relatively inexpensive. Present address: Schering Plough Corp., Galloping Hill Rd., Kenilworth, NJ 07033. 0003-2700/88/0360-2751$01.50/0
The primary objective of these studies was to resolve the enantiomers of ephedrine and pseudoephedrine, which happen to be diastereomers: CHI CH3NH.C
I
FH3
*H
H & NI H C H ~
CH3
CH3 H L C dNHCH3
I
C H ~ N H L ~ ~ H HBC40H
I
0000 HOC C 4 H
\
HLClOH
\
(+I-
(-)-
ephedrine
\
HOBCIH
\
(-)-
(+I-
pseudoephedrine
(-)-Ephedrine and (+)-pseudoephedrine are two widely used adrenergics whose pharmacological efficacy depends on enantiomeric purity. Consequently, resolution of the corresponding pairs of enantiomers is important both in commercial production and in quality control. Formerly, this was accomplished by derivitizing the drugs with a chiral agent and separating the new diasteriomeres by conventional chromatographic methods (9, 10). In the chromatographic experiments presented here, &cyclodextrin provided the greatest resolution for the pseudoephedrines but none of the cyclodextrins proved capable of distinguishing the enantiomers of ephedrine. NMR studies of the drug-cyclodextrin complexes were performed in an effort to understand these chromatographic results.
EXPERIMENTAL SECTION Materials. (-)-Ephedrine, (+)-pseudoephedrine, and (-)pseudoephedrine were purchased from Sigma Chemical Co. as the hydrochlorides and (+)-ephedrine hydrochloride was generously donated by Schering-Plough Corp. P-Cyclodextrin and 99.8% deuterium oxide were purchased from Aldrich Chemical Co. and the HPLC mobile phase was prepared from triethanolamine (analytical grade), acetic acid (analytical grade), acetonitrile (HPLC grade), and water (HPLC grade). Chromatographic Studies. The mobile phase was prepared from 96:4 (v/v) water-acetonitrile. To this mixture was added 0.8% (v) triethanolamine and the pH adjusted to 5 with acetic 0 1988 American Chemical Society
