A Model for the Description, Simulation, and ... - ACS Publications

Anal. Chem. 1997, 69, 3822-3831

A Model for the Description, Simulation, and Deconvolution of Skewed Chromatographic Peaks J. R. Torres-Lapasio´, J. J. Baeza-Baeza, and M. C. Garcıá-Alvarez-Coque*

Departamento de Quı´mica Analı´tica, Facultad de Quı´mica, Universitat de Valencia, 46100 Burjassot, Valencia, Spain

A family of models is proposed for the description of skewed chromatographic peaks, based on the modification of the standard deviation of a pure Gaussian peak, by the use of a polynomial function, h(t) ) He-(1/2)([t-tR]/[s0+s1(t-tR)+s2(t-tR)2+...])2, where H and tR are the height and time at the peak maximum, respectively. The model has demonstrated a high flexibility with peaks of a wide range of asymmetry and can be used to accurately predict the profile of asymmetrical peaks, using the values of efficiency and asymmetry factor measured on experimental chromatograms. This possibility permits the simulation of chromatograms and the optimization of the separation of mixtures of compounds producing skewed peaks, where both the position and peak shape are considered. A first-degree polynomial was adequate for peaks of moderate asymmetry, but higher degree polynomials were preferable for peaks showing a high asymmetry, including those with negative skewness. The proposed models can be employed in the resolution of overlapped peaks in binary and ternary mixtures of compounds, or to improve the accuracy in the evaluation of peak shape parameters. The results obtained with the proposed models were comparable or even superior to those achieved with the exponentially modified Gaussian model. The fitting and resolution of peaks is of great importance in the field of analytical chemistry. In the literature, there are a number of reports where numerical methods are used to describe individual peaks and to achieve the deconvolution of overlapped peaks in a chromatogram.1-5 The most simple chromatographic models predict Gaussian elution profiles. However, in practice, skewed peaks with low efficiencies may be obtained, and, in such cases, the assumption of a Gaussian model yields large errors.6,7 The reasons for the deviation from the ideal behavior are diverse, but the main explanation is the slow mass transfer of the solutes between stationary and mobile phases and, to a lesser extent, extracolumn effects. The complexity of the chromatographic process does not facilitate the proposal of a simple function to accurately describe (1) Foley, J. P. J. Chromatogr. 1987, 384, 301. (2) Davis, J. M. Anal. Chem. 1993, 65, 2014. (3) Felinger, A.; Pap, T.; Incze´dy, J. Talanta 1994, 41, 1119. (4) Felinger, A. Anal. Chem. 1994, 66, 3066. (5) Massart, D. L.; Vandeginste, B. G. M.; Deming, S. N.; Michotte, Y.; Kaufman, L. Chemometrics. A Textbook; Elsevier: Amsterdam, 1988. (6) Pauls, R. E.; Rogers, L. B. Sep. Sci. 1977, 12, 395. (7) Kirkland, J. J.; Yau, W. W.; Stoklosa, H. J.; Dilks, H. J. J. Chromatogr. Sci. 1977, 15, 303.

3822 Analytical Chemistry, Vol. 69, No. 18, September 15, 1997

the peak profile. Many of the proposed models lack a physical meaning, such as the bi-Gaussian approach, which describes the peaks using two Gaussian equations with different standard deviation for the leading and tailing halves, the GaussianLorentzian model, which substitutes a Lorentzian function by the Gaussian equation describing the tailing half of the peak,8 and a model that combines the bi-Gaussian approach with an exponential decay.9 Other attemps to explain the shape of chromatographic profiles have employed frequency distributions based on hypothetical equilibrium steps. These models have been applied to slightly distorted peaks, which can approximately be described by binomial and Poisson distributions.10 One of the most popular models used in the literature for chromatographic peaks is the exponentially modified Gaussian model (EMG),11-15 which has a physical justification. The model arises from the fact that the ideal Gaussian peak is distorted by first-order decays, caused by several intra- and extracolumn factors. The equation describing a pure Gaussian peak is

h(t) ) H e-(1/2)([t-tR]/σG)

2

(1)

where t is the time, H and tR are the height and time at the peak maximum, respectively, and σG is the standard deviation. The EMG model considers that, in an asymmetrical peak, the Gaussian model is modified by an exponential function:

f(t) )

1 -t/τ e τ

(2)

where τ is a constant that quantifies the decay time of the system. In principle, several decay processes can exist in a real chromatographic system, which implies several time constants and a more complex decay function. However, one of the processes is usually dominant, and only one constant can be used to characterize the whole behavior. If t is negative, f(t) is nullified to avoid an infinite increase in the exponential contribution. When τ ) 0, the exponential function is also null, and the EMG model describes a symmetrical peak. Convolution of both functions, the pure (8) Le-Vent, S. Anal. Chim. Acta 1995, 312, 263. (9) Torres Lapasio´, J. R.; Villanueva Caman ãs, R. M.; Sanchis Mallols, J. M.; Medina Hernańdez, M. J.; Garcıá Alvarez-Coque, M. C. J. Chromatogr. 1994, 677, 239. (10) Grushka, E.; Myers, N. M.; Schettler, P. D.; Giddings, J. C. Anal. Chem. 1969, 41, 889. (11) Foley, J. P.; Dorsey, J. G. Anal. Chem. 1983, 55, 730. (12) Foley, J. P.; Dorsey, J. G. J. Chromatogr. Sci. 1984, 22, 40. (13) Hanggi, D.; Carr, P. W. Anal. Chem. 1985, 57, 2394. (14) Berthod, A. Anal. Chem. 1991, 63, 1879. (15) Jeansonne, M. S.; Foley, J. P. J. Chromatogr. Sci. 1991, 29, 258. S0003-2700(97)00223-0 CCC: $14.00

© 1997 American Chemical Society

Gaussian and the exponential one, gives the following:4

y(t) )

∫ h(t′)f(t′ - t) dt′

instead of taking values at 30% or 50% of peak height, since the increased precision of the measurement gives a better definition of the efficiency:

∞

0

[ (

)]

t - tG σG 2 2 A ) e(σG /2τ -[t-tG]/τ) 1 - erf 2τ x2τ x2σG

(3)

where t′ is a dummy variable of integration, A is the area under the peak, σG is the standard deviation of the precursor Gaussian peak, and erf represents the error function, an integral function available in many libraries of programs. The variable tG represents the time of the maximum of the precursor Gaussian peak, which is shorter than the time of the maximum, tR, in peaks with positive skeweness. Several polynomial expansions may substitute the error function to permit its accurate calculation.16 Some simplified models and graphical methods, developed from the EMG model, have also been described. These models utilize easily measurable parameters that facilitate its practical use.11 Fitting of the EMG function may also be performed by changing the signal to the frequency domain and by estimating the parameters through the use of a two-dimensional extended Kalman filter.3 However, the most usual procedures for the determination of the main peak parameters are based on the estimation of statistical moments, which can easily be calculated with a personal computer, although their uncertainty increases with the degree, due to the susceptibility to the noise of the baseline. Upon subtraction of the baseline, the three statistical moments of lower degree are translated to known peak properties: area, centroid, and variance. From these statistical moments, some other important properties can be derived, such as the efficiency of asymmetrical peaks:

N)

tR2 σG2 + τ2

(4)

As indicated above, τ is a time constant responsible for the distortion of the chromatographic peak, which is associated with the exponential function, whereas σG is defined by the h(t) function in eq 1. If all the sources of asymmetry are eliminated, the maximum efficiency can be calculated from eq 4, making τ ) 0:

Nmax )

() tR σG

2

(5)

The relation between tR and tG is given by the following semiempirical equation:12

tR ) tG + σGFB/A ) tG + σG[-0.19(B/A)2 + 1.16B/A - 0.55]

(6)

where FB/A is a function depending on the asymmetry factor measured at 10% of peak height, B/A (B and A are the time distance between the maximum and the tailing and leading edge of the chromatographic peak, respectively), which relates the EMG model with the pure Gaussian model. Foley and Dorsey11 recommended the measurement of B/A at 10% of peak height, (16) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in C; Cambridge University Press: Cambridge, UK, 1992.

N=

41.7[tR2/(A + B)2] 1.25 + B/A

(7)

Equation 6 may be used for the modeling of peaks,17 but it is only an approximation and frequently lacks precision. The description of the EMG model by means of statistical moments is adequate if the peaks are moderately asymmetrical (B/A < 2) and free of noise.11 For larger asymmetry factors, the graphical method of Barber and Carr18 and the method of Anderson and Walters19 offer better results. The models based on Gaussian distributions have the advantage of using very intuitive parameters, related to properties which can be directly measured on the chromatograms (position and height of the maxima, and width of the peaks). A possible modification of the Gaussian model can be the substitution of a mathematical function by one of the constants involved in eq 1. The new function could be modified to fit the Gaussian function to the real skewed peaks, as much as possible. In this work, it is shown how the standard deviation may easily be varied using a polynomial function, to model skewed peaks including those showing a large asymmetry, with a deformation either to the right or to the left. The proposed algorithms were developed to be employed in the simulation and prediction of chromatographic peak profiles and in the resolution of mixtures of compounds showing overlapped peaks. The prediction of the shape of skewed peaks is essential in the optimization studies performed to select the best mobile phase in a given separation. The performance of the proposed polynomial-modified Gaussian model (PMG) is compared with the EMG approach. Most of the examples shown in this paper correspond to the elution of compounds using micellar mobile phases. With these eluents, asymmetrical peaks with low efficiencies are easily achieved. For this work, several examples of extremely distorted peaks have been intentionally selected. EXPERIMENTAL SECTION Apparatus. A Hewlett-Packard HP 1050 (Palo Alto, CA) liquid chromatograph with an isocratic pump and automatic injector, a UV-visible detector, and an HP 3396A integrator was used. The injection volume was 20 µL, and the flow rate was 1 mL/min. The dead volume was determined by the first deviation from the baseline after injection of the solute solutions. Spherisorb ODS-2 columns (5 µm particle size, 125 mm × 4.6 mm i.d.) and precolumns (35 mm × 4.6 mm i.d.) from Scharlau (Barcelona, Spain) were employed. The mobile phase and the solutions to be injected were vacuum filtered through 0.45 and 0.22 µm Nylon membranes, respectively (Micron Separations, Westboro, MA). Data acquisition was made through the PEAK-96 software from Hewlett-Packard (Avondale, PA), and the data treatment was done with MICHROM, a package of programs developed in our laboratory.20 MICHROM takes part in all the stages of the analytical process. It allows the determination of the dead time, (17) Naish, P. J.; Hartwell, S. Chromatographia 1988, 26, 285. (18) Barber, W. E.; Carr, P. W. Anal. Chem. 1981, 53, 1939. (19) Anderson, D. J.; Walters, R. R. J. Chromatogr. Sci. 1984, 22, 353. (20) Torres Lapasio´, J. R.; Garcıá Alvarez-Coque, M. C.; Baeza Baeza, J. J. Anal. Chim. Acta, in press.

Analytical Chemistry, Vol. 69, No. 18, September 15, 1997

3823

Figure 1. (1) Chromatographic peaks of (mobile phase, asymmetry factor, correlation coefficient): (a) chlorthalidone (0.05 M SDS/0.7% pentanol, B/A ) 1.2, r ) 0.999 94), (b) amiloride (0.05 M SDS/3% butanol, B/A ) 3.2, r ) 0.9997), (c) adrenalone (0.09 M SDS/10% propanol, B/A ) 5.1, r ) 0.9998), (d) noradrenaline (0.13 M SDS/9% propanol, B/A ) 8.8, r ) 0.9996). The points are experimental data, and the lines are the fits to eq 10, with a parabolic standard deviation. (2) Standard deviations, measured in seconds, for pure Gaussian peaks passing through each experimental point. The points are the standard deviation of the experimental data, and the lines are the standard deviation of the fitted function.

smoothing of chromatograms, measurement of peak parameters, and fitting of skewed peaks. Tools for the experimental design, optimization of the mobile phase composition to resolve a mixture of analytes, and simulation of chromatograms in several experimental conditions are also implemented. Routines for the graphical representation of chromatograms, resolution surfaces, contour maps, management of data series, optimization, and regression analysis are also included. Reagents. The following compounds were used: the diuretics amiloride (ICI Farma, Madrid, Spain), bendroflumethiazide (Davur, Madrid, Spain), chlorthalidone (Ciba-Geigy, Barcelona, Spain), and triamterene (Sigma, St. Louis, MO); the catecholamines adrenalone hydrochloride and DL-noradrenaline (Fluka, Buchs, Switzerland); the β-blockers atenolol (Zeneca Farma, Madrid, Spain), labetalol chlorhydrate (Glaxo, Madrid, Spain), metoprolol tartrate, oxprenolol chlorhydrate (Ciba-Geigy), nadolol (Squibb, Barcelona, Spain), propranolol chlorhydrate (ICI-Farma), and timolol maleate (Merck, Sharp & Dohme, Madrid, Spain); and the steroids boldenone, methandrostenolone, methyltestosterone (Sigma), dydrogesterone (Kalifarma, Barcelona, Spain), medroxyprogesterone (Frumtos-Zyma, Barcelona, Spain), medroxyprogesterone acetate (Cusı´, Barcelona, Spain), and testosterone (Schering, Madrid, Spain). The compounds, except those from Sigma and Fluka, were kindly donated by the Spanish pharmaceutical industries indicated. 3824


The micellar mobile phases were prepared with sodium dodecyl sulfate (99% purity, Merck, Darmstadt, Germany) and the modifiers 1-propanol (Panreac, Barcelona, Spain), 1-butanol, 1-pentanol, and acetonitrile (Scharlau). A mobile phase of methanol and NaH2PO4 (Scharlau) was also used. RESULTS AND DISCUSSION Fitting of Skewed Peaks. Figure 1 shows four chromatographic peaks with different asymmetries (B/A ) 1.2-8.8). The experimental points of the peaks are represented together with the standard deviation, σt-tR, of pure Gaussian peaks having the same height and position as the skewed peak and going through each experimental point:

σt-tR )

|t - tR|

x-2 ln[h(t)/H]

(8)

As can be seen, the σt-tR vs (t - tR) curves may be described by simple polynomial functions: σt-tR ) s0 + s1(t - tR) + s2(t - tR)2 + ‚‚‚. In the simplest cases, these curves correspond to polynomials of low degree (e.g., straight lines or parabolas). It should be noted that the different regions of a given peak do not have the same significance for the calculation of the σt-tR function. This significance can be calculated through the partial

Figure 2. Chromatographic signal (points) and significance (lines) for the calculation of σt-tR for the peaks of (mobile phase) (a) amiloride (0.1 M SDS/1.5% butanol) and (b) chlorthalidone (0.05 M SDS/0.7% pentanol).

derivative of eq 1, with respect to the standard deviation:21

Iσ )

(t - tR) ∂ h(t) ) h(t) ∂σ σ3

2

(9)

Figure 2 shows the plot of this function for peaks of amiloride and chlorthalidone. It is obvious that the experimental points having maximal significance for the calculation of the standard deviation are the inflection points of the peaks (maximum of the function), and the less significant regions correspond to the baseline and to the maximum of the peaks (low values of the function), where many different Gaussian functions could be fitted. The simulated peak, represented as a line in Figure 1, was modeled according to 2

2

h(t) ) H e-(1/2)([t-tR]/[s0+s1(t-tR)+s2(t-tR) +‚‚‚])

(10)

with a second-degree polynomial for the σt-tR function. The method of Powell was used to make the nonlinear fitting of the experimental data.22 As observed, real chromatographic signals showing an important asymmetry were reproduced with high accuracy. In spite of the simplicity of the model, excellent correlations were achieved, even for peaks having a large asymmetry. (21) Baeza Baeza, J. J.; Ramis Ramos, G.; Mongay Fernańdez, C. Anal. Chim. Acta 1990, 237, 473. (22) Rao, S. S. Optimization: Theory and Applications; Wiley: New Delhi, India, 1985; pp 274-283.

The coefficient s0 in eq 10 coincides with the standard deviation of a symmetrical Gaussian peak describing the central region, whereas s1 and s2 are coefficients that quantify the skewness of the peak. It was checked that s0 was almost constant, even when new terms were added to the polynomial function. The coefficient s1 depends only on the asymmetry factor, a large value of this coefficient indicating a strong asymmetry and a negative value indicating a left bias. The coefficient s2 and upper terms in higher degree polynomials correct small deviations to better fit the shape of the peaks. However, a model using a polynomial σt-tR function only describes adequately the region that includes the peak being fitted. Outside this region, important deviations can be obtained, since inaccurate estimations of the higher order coefficients produce the abnormal rise of the baseline, far from the peak maxima. This can be detrimental for the simulation of chromatograms of mixtures of compounds. Therefore, with this kind of function, the region of the baseline where the function further increases should be excluded. For this purpose, the minima of the equation at both sides of the peak must be obtained, and the external region must be rejected and substituted by the value of minimum ordinate, or by the ordinate of the baseline. This behavior (the rise of the function outside the fitted region) is observed with any polynomial and may be avoided by using another type of function (e.g., a sigmoidal function); however, this limitation does not affect the practical use of the polynomial functions. Figure 3a,b shows the fitting of the experimental points for the chromatographic peak of atenolol using equations with linear and parabolic standard deviations. As observed, the parabolic function gave a good fit, which could be improved even more with polynomials of higher degree. However, the use of a large number of s coefficients complicates the practical application of the model. Indeed, it has been found that a second- or thirddegree polynomial usually leads to very satisfactory results in most cases. Peaks showing a negative skewness can also be fitted with the proposed model (Figure 4a,b), although achievement of good results requires the use of polynomials of high degree (third degree or higher). The reason for this requirement is the profile of the σt-tR function. For peaks with negative skewness, the σt-tR vs (t - tR) plot is a double wave, which cannot exactly be reproduced with first- or second-degree polynomials. Simulation and Prediction of Peaks in a Chromatogram. A linear standard deviation function in eq 10 satisfactorily approximates the real shape of the peaks in a chromatogram. This model permitted the development of a simple method to simulate chromatograms and to optimize the resolution of mixtures of compounds, using criteria that consider not only the position of the chromatographic peaks but also their shape.9,20 Using a linear standard deviation, the number of chromatographic peak parameters (i.e., position, height, efficiency, and asymmetry factor) coincides with the number of coefficients in the h(t) function (H, tR, s0, and s1). The coefficients s0 and s1 may easily be related to the asymmetry factor and efficiency. The steps to follow in the calculation of these coefficients for the simulation of a peak in a chromatogram are listed below. (i) Prediction of the retention time of the peak in a given mobile phase from equations that describe the retention of solutes. Thus, for example, when micellar mobile phases are used in reversedAnalytical Chemistry, Vol. 69, No. 18, September 15, 1997

3825

Figure 3. Fitting of the peak of atenolol eluted with a mobile phase of methanol/0.05 M phosphate buffer (pH 3) (40:60) (B/A ) 5.2), using the PMG model with (a) linear (r ) 0.9989) and (b) parabolic (r ) 0.999 91) standard deviations and using (c) the EMG model (r ) 0.9988).

Figure 4. Fitting of the peaks of bendroflumethiazide eluted with a 0.075 M SDS/0.7% pentanol mobile phase (B/A ) 0.76), using the PMG model with (a) linear (r ) 0.9988) and (b) fifth-degree (r ) 0.999 90) standard deviation and using (c) the EMG model (r ) 0.9985).

phase liquid chromatography, the following equation has proved to give an accurate description of the retention:23

(iii) Calculation of the width of the peak at 10% of peak height:

tR )

(

1 + a4φ

a0 + a1[M] + a2φ + a3[M]φ

)

+ 1 t0

(11)

where [M] is the molar concentration of surfactant forming micelles, φ is the v/v concentration of organic modifier, and t0 is the dead time. (ii) Estimation of N and B/A by interpolation through the fitting of the values of these parameters for experimental mobile phases close to the predicted mobile phase. (23) Garcıá Alvarez-Coque, M. C.; Torres Lapasio´, J. R.; Baeza Baeza, J. J. J. Chromatogr., in press.

3826


W)A+B)

x

41.7(1 + tR)2

N(1.25 + B/A)

(12)

From eq 12, the individual values of A and B from W and B/A are calculated:

W 1 + B/A

(13)

W 1 + 1/(B/A)

(14)

A)

B)

Figure 5. (a) Simulated chromatogram for a mixture of the β-blockers (1) nadolol, (2) timolol, (3) metoprolol, (4) labetalol, (5) oxprenolol, and (6) propranolol, eluted with a mobile phase of methanol/0.05 M phosphate buffer (pH 3) (40:60). A modified Gaussian model with a linear standard deviation was used. The areas of the peaks were not normalized. (b) Experimental chromatogram.

(iv) Evaluation of the height of the normalized peak, which may easily be performed assuming a triangular profile: H ) E/W, E being a normalization constant. If a higher precision is desired, other functions can be used. In the simulation of a real chromatogram, the normalization step can be obviated, and H should be substituted by the real peak height. (v) Finally, the coefficients s0 and s1 of the standard deviation of the skewed peak are calculated for t ) tR - A, and t ) tR + B, using eq 10, where h(t) ) 0.1H. After solving the system of two equations, the following is obtained:

s1 )

s0 )

B/A - 1 B/A - 1 = 0.466 B/A + 1 B/A +1 x2 ln 10 1

B(1 - s1x2 ln 10)

x2 ln 10

= 0.466B(1 - 2.146s1)

(15)

(16)

Figure 5 compares the experimental and simulated chromatograms for a mixture of six β-blockers. As can be observed, the chromatograms are virtually identical, in spite of the simplicity of the linearly modified Gaussian model utilized. Using the above strategy, it is possible to predict the profile of a chromatogram on the basis of a limited number of experiments, even though these experiments were relatively far apart in the variable space. As an example of this prediction capability, two chromatograms are given for the separation of a mixture of two diuretics and five steroids (Figure 6). The position of the maximum of the peaks was calculated using eq 11, and the efficiencies and asymmetry factors were interpolated from the values obtained with the three experimental mobile phases closer to the simulated mobile phase from those available. The experimental mobile phases used for the prediction were 0.075 M SDS, 0.20 M SDS, 0.075 M SDS/10% (v/v) acetonitrile, 0.14 M SDS/ 10% acetonitrile, 0.075 M SDS/20% acetonitrile, and 0.20 M SDS/ 20% acetonitrile. Again, predicted and experimental chromatograms were in good agreement.

Figure 6. (a,c) Predicted and (b,d) experimental chromatograms for the separation of a mixture of diuretics and steroids, eluted with mobile phases of 0.14 M SDS (a,b), and 0.18 M/17% acetonitrile (c,d). The prediction of the position of the peaks was made with eq 11, and the profile of each peak with eqs 12-16 (see text). The area of the peaks were not normalized. Solutes: (1) chlorthalidone, (2) amiloride, (3) boldenone, (4) testosterone, (5) methyltestosterone, (6) medroxyprogesterone acetate, and (7) dydrogesterone.

This prediction capability can be applied to optimize the resolution of complex mixtures. Figure 7 shows the threedimensional plot of resolution for the separation of several steroids and diuretics in the 0.075-0.20 M SDS and 0-20% acetonitrile concentration ranges. The function of resolution used was

Oi

n

r)

∏

∑O /n)

i)1 (

(17) n

i

where Oi is the overlapping extent, given by

Oi ) 1 -

w′i wi

(18)

wi being the total area of each peak and w′i the area of the peak Analytical Chemistry, Vol. 69, No. 18, September 15, 1997

3827

Figure 7. Three-dimensional plot for the resolution of a mixture of two diuretics and five steroids, eluted with mobile phases of SDS and acetonitrile, according to the overlapping extent criterion (eqs 17 and 18).

overlapped by other peaks. This optimization criterion considers not only the position but also the shape (efficiency and asymmetry) of all the peaks in the chromatogram.9 As depicted in Figure 7, the peaks of the steroids and diuretics will be well resolved in a wide range of concentrations of SDS and acetonitrile. However, a mobile phase of large eluent strength, such as 0.2 M SDS/20% acetonitrile, is preferable for the analysis of the mixture, because of the faster elution (the retention time of the last-eluted compound, medroxyprogesterone acetate, will be 11.1 min for this eluent). For mobile phases with low acetonitrile contents, the deterioration of the resolution was due to a decrease in the efficiencies of the peaks of the steroids and to a change in the elution order of medroxyprogesterone acetate and dydrogesterone. On the other hand, the deep valley crossing the diagram for concentrations of acetonitrile larger than 10% was produced by a change in the order of elution of the peaks of amiloride and boldenone. Comparison with the EMG Model. A more complete evaluation of the results obtained with the proposed PMG models needs comparison with other approaches commonly used in the experimental practice. Among these alternative models, the EMG model gives the most reliable description of peak profiles and is widely accepted. It is thus obligatory to compare the results achieved with both approaches, the EMG and the PMG models. The EMG model convolutes a Gaussian function with an exponential one; therefore, a hypothetical Gaussian peak, often called the precursor peak, is attenuated, deformed, and shifted, while its area is maintained constant to fit the experimental peak. For tailing peaks, the precursor Gaussian peak appears at a lower retention time and is narrower than the experimental peak. For fronting peaks, the fits are poorer, and the retention time of the maximum of the precursor peak is larger than the time of the experimental peak. The application of the EMG model has the drawback of performing the evaluation of the parameters of the real peak using ideal values from a pure Gaussian peak. In contrast, the parameters of the PMG model are directly related to the properties of the experimental peak, such as its height and retention time. The parameters of the PMG model can approximately be related to the parameters of the EMG model. Thus, the efficiency 3828


and τ can be related to s0, s1, and tR, calculated from eq 10; the asymmetry factor can be evaluated from s1; and σG can be related to s0 and s1, and so on. The constant term in the standard deviation function, s0, corresponds to the minimum standard deviation. In tailing peaks, this value is found close to the initial rise of the peak (see Figure 1), a value somewhat larger than the standard deviation in the EMG model, but close to it. The values of tG (minutes) and σG (seconds), according to the EMG model, for the peaks shown in Figure 1a-d, were (4.53, 5.73), (12.47, 9.71), (10.39, 4.20), and (3.36, 8.35), respectively. The position (minutes) and the value (seconds) of the minimum standard deviation, corresponding to the fitted parabolic PMG functions, were, for the same figures, (4.44, 6.24), (12.08, 11.11), (10.25, 5.00), and (3.10, 11.41), respectively. Another drawback of the EMG model, cited in the literature, is the complexity of its mathematical formulation. A usual strategy for the evaluation of the parameters is the use of properties derived from direct measurement on the chromatographic peak. However, a correct comparison between both models (EMG and PMG) requires the accurate evaluation of the peak parameters, which needs the fitting of the whole set of experimental points of the peak. Several formulations of the EMG model have been reported that can be adapted to a least-squares regression algorithm. In this work, eq 3 was used to describe the chromatographic peaks. For fronting peaks, tG - t was substituted by t - tG. The error function was implemented by means of an approximation based on Chebyshev polynomials, which yields results with a precision better than 1.2 × 10-7.16 Fitting of the experimental points in the chromatograms was done using the iterative method of Powell,22 after subtraction of the baseline and bracketing of the region of the chromatogram including the peak. Using initial values similar to the real parameters, a rapid convergence of the iterative process was achieved. Figures 3 and 4 allow a comparison of the performance of the EMG and PMG models. For the EMG model, an initial pure Gaussian peak having the retention time of the real peak was taken. As observed, the PMG model with a linear standard deviation yielded results similar to those of the EMG model, but the use of a higher degree function improved greatly the peak fitting. Fronting peaks were more problematic (Figure 4), but again the performance of the PMG model with linear standard deviation was similar to the EMG approach. In the example given, a fifth-degree polynomial yielded an accurate fitting. In the iterative process, the EMG model always required initial values closer to the real ones, to assure the convergence, than the PMG models. This is caused by the presence of an exponential term in the EMG model, which can produce numerical errors during the fitting process. Also, the risk of achieving a local minimum is larger. Resolution of Overlapped Peaks in Binary and Ternary Mixtures. The statistical theory for peak overlapping indicates that a disordered chromatogram must be approximately 95% vacant to have 90% probability for a given peak of being isolated.24 The necessity of solving overlapped peaks is, thus, not uncommon. The PMG model can be used for this task. Application of eq 10 to the deconvolution of two or three overlapped peaks is immediate. For this, the overlapped signal is considered as the sum of two or three modified Gaussian functions, and the problem is (24) Davis, J. M.; Giddings, J. C. Anal. Chem. 1983, 55, 418.

nonlinearly solved. Five parameters are needed for each component in the mixture when the model with a parabolic standard deviation is used, and six parameters are required with a cubic function. The accuracy of the model was evaluated by using fragments of experimental chromatograms that included the peaks of two diuretics: triamterene eluted with 0.15 M SDS/3% pentanol (tR ) 4.84 min, N ) 800, B/A ) 1.9) and amiloride eluted with 0.10 M SDS/3% pentanol (tR ) 5.01 min, N ) 600, B/A ) 2.3). The baseline of the chromatograms was subtracted, and the true position of the peak of triamterene was maintained, while the peak of amiloride was shifted in diverse extent to modify the overlapping of both peaks. The variation of the position of amiloride by only 10 s through the use of a mobile phase of larger eluent strength will scarcely affect the shape of the chromatographic peak. In Figure 8a, the retention time of amiloride was increased by 10 s to diminish the overlapping between both peaks. Figure 8b shows the peaks at their real retention times (separated by 10 s). Finally, in Figure 8c, the retention time of amiloride was decreased to make it coincident with the retention time of triamterene. This example was selected due to its high complexity, since the shape parameters of the peaks (s coefficients) were very similar. Equation 10, with a cubic standard deviation, was used for the deconvolution of the peaks of triamterene and amiloride. In Figure 8, the sum of the signals of the real peaks was fitted to obtain the equation of the individual peaks in such a way that the calculated sum reproduced the experimental chromatogram of the mixture. For this calculation, an algorithm based on the methods of Powell and Newton-Raphson with Marquardt modification16,22 was programmed. For very close peaks, the determination of the optimum set of parameters describing the shape of the peaks is complex, due to the high number of parameters. In this case, several combinations of peaks can be obtained, whose sum gives a similar chromatogram of the mixture. First, the fittings were performed without imposing any restriction on the values of the parameters (i.e., tR and H). The peaks could thus change in position and shape with total freedom during the iterations, to fit the data. When the overlapping of the peaks was not excessive (Figure 8a), the original signal was perfectly reproduced. On the contrary, when the overlapping was considerable (Figure 8b), the fitted individual peaks could differ significantly from the corresponding experimental peaks. To avoid this indetermination, at least one of the parameters of the fitted equation should be fixed, to restrict the variation range of the solutions. In some examples, where deconvolution techniques are applied, the nature of the compounds in the mixture is known, allowing the quantification of each component. In other cases, one of the components can be identified, and its shape parameters facilitate the solution of the problem. In Figure 8, the retention time of each solute was fixed, and a modified Gaussian function with a cubic standard deviation having 10 parameters, instead of 12, was nonlinearly fitted to resolve the mixture. As observed, the fits were satisfactory. Figure 8c, where the peak of amiloride was shifted to coincide with the peak of triamterene, represents an extreme example. Both peaks are almost identical, only differing in the efficiencies and asymmetry factors. Deconvolution of these peaks requires, thus, knowledge of the retention times or peak shape parameters. Without this information, the fitting will lead to only one peak.

Figure 8. Resolution of an artificial binary mixture of triamterene (peak 1) and amiloride (peak 2), using the PMG model with a cubic standard deviation and fixed retention times. The chromatograms in (a), (b), and (c) correspond to different shifts of the peak of amiloride. The sum of the experimental peaks, 1 + 2, was fitted to obtain the peaks 1′ and 2′, whose sum, 1′ + 2′, should coincide ideally with 1 + 2. The relative errors in the determination of the peak areas for peaks 1 and 2 were (a) -3.2%, +2.0%; (b) +1.21%, -1.07%; (c) -14.4%, +13.6%.

The figure shows that the PMG model found a solution very close to the real peaks when the retention time was fixed. Figure 9 shows the performance of the model to resolve a real mixture of two (medroxyprogesterone and methandienone) and three (boldenone, methandienone, and testosterone) steroids. The binary mixture was resolved using the cubic function, and, for the ternary mixture, three parabolic functions were added. The figure shows how, in both cases, the sum of the resolved peaks coincided satisfactorily with the experimental chromatogram, taking into account the complexity of the problem: clearly distorted peaks were overlapped, and the concentration of the Analytical Chemistry, Vol. 69, No. 18, September 15, 1997

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was left to evolve with total freedom, until no further improvement was achieved in any of the explored patterns. The errors achieved should be partially attributed to the preparation of the standard solutions and to the instrumental noise. The same mixtures were treated with the EMG model. It was observed that the initial values should be even closer to the real solution than in the case of fitting individual peaks, to assure convergence. The errors obtained using the EMG model with no restrictions were, for the binary mixture, 81.8% and 33.5% for medroxyprogesterone and methandienone, respectively, and, for the ternary mixture, 9.2%, 51.3%, and 86.9% for boldenone, methandienone, and testosterone, respectively. When the values of the retention times obtained from the standards were fixed for the EMG model, the errors for the binary mixture were 8.8% and 5.4% for medroxyprogesterone and methandienone, respectively, and, for the ternary mixture, 0.8%, 46.9%, and 50.1% for boldenone, methandienone, and testosterone, respectively.

Figure 9. (a) Resolution of a real binary mixture of (1) medroxyprogesterone (tR ) 14.52 min, B/A ) 1.8) and (2) methandienone (tR ) 14.88 min, B/A ) 2.5). (b) Resolution of a real ternary mixture of (1) boldenone (tR ) 12.21 min, B/A ) 2.8), (2) methandienone, and (3) testosterone (tR ) 16.07 min, B/A ) 2.6). The mobile phase was 0.2 M SDS/3% acetonitrile. The PMG model with parabolic standard deviation was used. All the peak parameters were allowed to evolve during the fittings, without any restriction. Only one of each four experimental points is shown to facilitate the observation of the fitting capability of the model. Peaks 1, 2, and 3 are the peaks resolved.

steroids was very low for detection; therefore, the noise of the signal was considerable and affected the fitting process. The errors obtained in the resolution of the peaks (measured as the coefficient of variation of the areas under the peaks) when no restriction was applied, that is, when all the parameters were supposed to be unknown (including the retention time), were, for the binary mixture, 10.7% and 1.8% for medroxyprogesterone (0.7 µg/mL) and methandienone (1.5 µg/mL), respectively, and, for the ternary mixture, 2.9%, 5.0%, and 17.8% for boldenone (1.0 µg/mL), methandienone (1.5 µg/mL), and testosterone (0.9 µg/ mL), respectively. When the known values of retention times were utilized in the model, the errors decreased for the binary mixture to 8.7% and 4.5% for medroxyprogesterone and methandienone, respectively, and for the ternary mixture to 1.4%, 9.0%, and 5.3% for boldenone, methandienone, and testosterone, respectively. The evaluation of the errors was made through the use of standard solutions of each component having the same concentration as in the mixture. These standards were injected in the chromatograph before and after the mixture solutions. The iterative process 3830 Analytical Chemistry, Vol. 69, No. 18, September 15, 1997

CONCLUSIONS It has been demonstrated that the accurate description of peaks showing a large asymmetry is possible by using a modified Gaussian model, where the standard deviation of a pure Gaussian peak varies with the distance to the maximum of the peak. Several functions were examined for the standard deviation. Real chromatographic signals may be reproduced using a linear function, but the accuracy of the fitting can easily be improved by increasing the degree of the polynomial. The proposed models showed a large flexibility and offered very accurate results in the fitting of real chromatographic peaks, even those showing a large distortion or negative skewness. The linear PMG model may be used, with good results, in the simulation and optimization of chromatograms that include skewed peaks. For this purpose, the values of the efficiency and asymmetry factor of the peaks are required. The speed in the calculations performed to simulate a chromatogram allows, using an appropriate software (such as MICHROM),20 the graphical following of the transformation of the chromatogram shape, as a cursor is moved by the user inside a space representing the mobile phase composition range. The proposed approach has been demonstrated to be equivalent or superior to the exponentially modified Gaussian model. The practical application of the EMG model in the simulation of asymmetrical peaks and optimization of the resolution in a chromatographic separation is somewhat problematic, since the parameters of the model are not directly related to the properties of the real peaks, and their evaluation requires the use of approximations. The PMG model has also shown to be appropriate for the resolution of overlapped peaks. For this application, parabolic or cubic standard deviation functions should be used. However, when the peaks are highly overlapped, the fitting of the experimental peaks can yield diverse sets of parameters, describing very different individual peaks that, by addition, reproduce accurately the experimental signal. In such cases, the retention time of each solute in the mixture should be given, in order to restrict the number of possible solutions. Nevertheless, the results shown for the resolution of binary and ternary mixtures of steroids is an example of the impressive capability of the PMG model in the

resolution of overlapped peaks, even when no standards are available. Fitting of the chromatographic signals to eq 10, using a linear or parabolic standard deviation, is suitable to refine peak parameters, estimated by direct measurement. The retention time and peak height are given directly by eq 10, and the area is obtained through the integration of the function. The area, A, can also be obtained directly if A/[σt-tR(2π)1/2] is substituted by H in eq 10. Finally, the asymmetry factor and width at 10% of peak height can be evaluated from eqs 15 and 16, and the efficiency can be

calculated with eq 7. The accuracy of the values obtained depends on the validity of the description of the peak with the linear PMG model. ACKNOWLEDGMENT This work was supported by the DGICYT, Project PB94/967. Received for review February 27, 1997. Accepted June 3, 1997.X AC970223G X

Abstract published in Advance ACS Abstracts, July 1, 1997.


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