Anal. Chem. 2004, 76, 4856-4865
Validation of the Accuracy of the Perturbation Peak Method for Determination of Single and Binary Adsorption Isotherm Parameters in LC Johan Lindholm,† Patrik Forsse´n,‡ and Torgny Fornstedt*,†
Department of Surface Biotechnology, Uppsala University, BMC Box 577, SE-751 23, Uppsala, Sweden, and Department of IT/Scientific Computing, Uppsala University, Box 337, S-751 05, Uppsala, Sweden
An analytical validation of the precision and accuracy of the perturbation peak (PP) method for determination of single and competitive thermodynamic isotherm parameters was performed using frontal analysis as a reference. The isotherm parameters of 11r-hydroxyprogesterone were determined in an achiral system and the isotherm parameters of (+)-methyl L-mandelate and (-)-methyl D-mandelate were determined in a chiral system, both for the single components and for the competitive binary mixture. The experimental errors in the PP method using different injection techniques were investigated, and we devised a new injection technique for the determination of competitive isotherm parameters that considerably reduced the experimental errors and also made both perturbation peaks detectable. We showed that the PP method with the new injection technique can be used to determine isotherm parameters directly from a racemic mixture. These parameters agreed well with those determined using several enantiomer ratios. Elution-band profiles simulated using the isotherm parameters showed excellent agreement with experimental profiles. In all chromatographic processes of practical importance, we are dealing with multicomponent cases rather than a pure component. In preparative chromatography, mixtures are separated to prepare individual components of the required purity. However, preparative separations are difficult to optimize without computer simulations.1 Essential input data for these computer simulations are the isotherm parameters of the competitive isotherm functions, which account for the band interactions.1-3 In the case of chiral drugs, it is often necessary to determine competitive isotherm parameters from racemic mixtures due to the lack of pure enantiomers, but competitive isotherm parameters are difficult to determine. Frontal analysis (FA) is the traditional method to acquire single-component adsorption data. However, this approach requires large amounts of pure chemicals.1-3 FA can be extended * To whom correspondence should be addressed. E-mail: torgny.fornstedt@ ytbioteknik.uu.se. † Department of Surface Biotechnology. ‡ Department of IT/Scientific Computing. (1) Fornstedt, T.; Guiochon, G. Anal. Chem. 2001, 73, 608A-617A. (2) Poppe, H. J. Chromatogr., A 1993, 656, 19-36. (3) Guiochon, G.; Golshan-Shirazi, S.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography, 1st ed.; Academic Press: Boston, MA, 1994.
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to acquire adsorption data for competitive binary systems, i.e., competitive FA, but this is extremely tedious and time-consuming because it requires continuous fractioning for proper quantification of intermediate plateaus.4,5 In addition, it is necessary to introduce binary plateaus of several different ratios and this demands pure chemicals.5 In some cases, pure chemicals are not available, which makes it impossible to perform the experiments to acquire competitive adsorption data. An alternative to competitive FA is the so-called perturbation peak (PP) method. Here, the chromatographic column is equilibrated with a constant stream of molecules in the mobile phase and a concentration plateau is established. A perturbation is then done by injecting a sample containing an excess or a deficiency of the molecules as compared to the concentration plateau.6-9 The response at the column outlet will be small peaks, known as perturbation peaks, and their retention times are used to determine the isotherm parameters. The most practical and economical approach is to perform the PP method using only a mixture, but most often the PP method has been performed using several mixtures of various relative compositions to obtain high accuracy,6,8 the drawback being that this approach requires pure chemicals. No validation has been made of parameters determined by the PP method using only racemic mixtures. A complication with the PP method, when using the traditional blank perturbation injections, is that the second perturbation peak on a binary plateau will vanish during nonlinear conditions.10 However, a different injection technique, which is firmly supported by theory and has been confirmed experimentally, will give clearly detectable peaks.10 In nonlinear systems, such as the systems in this article, it is necessary to use this injection technique in order to detect the perturbation peaks over the broad concentration range necessary for proper determination of the isotherm param(4) Charton, F.; Nicoud, R. M. J. Chromatogr., A 1995, 702, 97-112. (5) Jacobson, J. M.; Frenz, J. H.; Horva´th, C. Ind. Eng. Chem. Res. 1987, 26, 43-50. (6) Blu ¨ mel, C.; Hugo, P.; Seidel-Morgenstern, A. J. Chromatogr., A 1999, 865, 51-71. (7) Cavazzini, A.; Felinger, A.; Guiochon, G. J. Chromatogr., A 2003, 1012, 139-149. (8) Piatkowski, W.; Antos, D.; Gritti, F.; Guiochon, G. J. Chromatogr., A 2003, 1003, 73-89. (9) Heuer, C.; Ku ¨ sters, E.; Plattner, T.; Seidel-Morgenstern, A. J. Chromatogr., A 1998, 827, 175-191. (10) Forsse´n, P.; Lindholm, J.; Fornstedt, T. J. Chromatogr., A 2003, 991, 3145. 10.1021/ac0497407 CCC: $27.50
© 2004 American Chemical Society Published on Web 07/03/2004
eters. Another complication is that if the perturbation made on the system is too large, the difference between the initial equilibrium concentration and the concentration at the column outlet is also too large, which introduces an error in the measured retention times of the perturbation peaks. It has been demonstrated that large deficiency perturbations result in increased retention times and that large excess perturbations do the opposite for a Langmuir isotherm.11,12 Interestingly, it was recently demonstrated, for a system described by the liquid-solid multiplayer BET isotherm, that a finite positive perturbation peak has a larger retention than the corresponding negative peak.13 All the reports mentioned above were for the single-component case. According to other reports, the size and direction of the perturbation peaks do not affect the measured retention times.6,7 Obviously, the issue needs to be investigated systematically. This is especially important for the binary experiment, considering that the peak-vanishing problem might lead chromatographers to make too large perturbations. The FA method is usually considered to be the most accurate method to acquire adsorption data.1 Recently, it was stated that the PP method is more reliable,14 but that statement was not based on repetitive comparisons, and in addition, identical systems were not used. The intraday precision (relative standard deviation) of the FA and PP methods has recently been investigated by three repeated measurements,7 but precision should be measured using a minimum of five or six determinations according to the recommendations for validation of analytical procedures, given by the International Conference on Harmonization (ICH), and the bioanalytical methods validation guidelines, given by the U.S. Food and Drug Administration (FDA).15-17 The analytical validation of the PP method in this article will be a first step toward a future comparison with the mass-sensitive SPR technology for determination of protein-drug interactions at the drug discovery stage.18 The aim of this study was threefold: (1) to investigate the experimental errors introduced in the PP method by using both the traditional blank injection technique and the new injection technique, i.e., the technique that makes both perturbation peaks of a binary mixture detectable over the whole concentration range; (2) to compare the precision and accuracy of the PP and the FA methods using five repetitive experiments on an identical singlecomponent system; (3) to compare and validate the accuracy of the isotherm parameters of two enantiomers determined using four different experimental approaches, (I) single FA, (II) single PP, (III) PP with mixtures of various relative compositions, and (IV) PP with racemic mixture. (11) Fornstedt, T.; Guiochon, G. Anal. Chem. 1994, 66, 2116-2128. (12) Sajonz, P.; Yun, T.; Zhong, G.; Fornstedt, T.; Guiochon, G. J. Chromatogr., A 1996, 734, 75-82. (13) Gritti, F.; Piatkowski, W.; Guiochon, G. J. Chromatogr., A 2003, 983, 5171. (14) Jandera, P.; Buncekova´, S.; Mihlbachler, K.; Guiochon, G.; Backovska´, V.; Planeta, J. J. Chromatogr., A 2001, 925, 19-29. (15) Shah, V. P. Guidance for Industry: Bioanalytical Methods Validation. May 2001. http://www.fda.gov/cder/guidance/index.htmL, June 2001. (16) Shah, V. P.; Midha, K. K.; Dighe, S. V.; McGilveray, I. J.; Skelly, J. P.; et al. Conference report. Pharm. Res. 1992, 9. (17) ICH-Topic Q2B: Validation of Analytical Procedures: Methodology, International Conference on Harmonization of Technical Requirements for Registration of Pharmaceuticals for Human Use, Geneva, 1997. http:// www.ich.org/pdfICH/Q2B.pdf, 30 August 2002. (18) Rich, R. L.; Day, Y. S.; Morton, T. A.; Myszka, D. G. Anal. Biochem. 2001, 296, 197-207.
THEORY Assume that the chromatographic column is equilibrated with a constant stream of molecules in the mobile phase such that a concentration plateau is established. A perturbation is then done by injecting a small sample containing an excess or a deficiency of the molecules as compared to the concentration plateau. The response at the column outlet will be small peaks known as perturbation peaks. (1) Single-Component Case. It is possible to show that the mean retention time tR of the perturbation peak will be,
tR ) t0(1 + F(dq/dc)|c)c0)
(1)
where t0 is the column hold-up time, F is the phase ratio, and dq/dc is the derivative of the isotherm function that is evaluated at the plateau concentration c0. One commonly used singlecomponent isotherm function is the Langmuir isotherm,
q ) ac/(1 + bc)
(2)
where q is the solid-phase concentration, c is the mobile-phase concentration, and a and b are isotherm parameters. (2) Binary Case. It is possible to show that the retention times of the two perturbation peaks will be tR,1 ) t0λ1 and tR,2 ) t0λ2, where λ1, λ2 (without loss of generality we assume that λ1 e λ2) are the eigenvalues of the matrix.
(
∂q1 ∂c1
∂q1 ∂c2
1+F
F
∂q2 F ∂c1
∂q2 1+F ∂c2
)|
(3)
c1)c0,1, c2)c0,2
Here ∂q1/∂c1, ∂q1/∂c2, ∂q2/∂c1, and ∂q2/∂c2 are the partial derivatives of the isotherm functions q1, q2 evaluated at the plateau concentrations c0,1, c0,2. Closed expressions for the retention times can be found in ref 10. One commonly used binary isotherm function for enantiomers is the bi-Langmuir isotherm,
qi )
aI,ici aII,ici + , 1 + bI,1c1 + bI,2c2 1 + bII,1c1 + bII,2c2
(4)
where qi is the solid-phase concentration, ci is the mobile-phase concentration, and aI,i, aII,i, bI,i, and bII,i are isotherm parameters. The bi-Langmuir model consists of two Langmuir terms, the first term representing a large number of nonenantioselective interactions, the so-called type-I sites, and the second term representing a smaller number of enantioselective interactions, the so-called type-II sites. Here it should be noted that although we have two perturbation peaks, the same as the number of adsorbed components in the concentration plateau, it is not possible to attribute a peak to an individual component because a peak represents the response to a perturbation of the concentrations of both components. Traditionally the perturbation injection made is a blank sample, but a severe complication is that the second perturbation will vanish during already moderately nonlinear conditions.10 However, assuming equal response factors for the two components and Analytical Chemistry, Vol. 76, No. 16, August 15, 2004
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doing a perturbation injection,
cs,1 ) c0,1 - µ,
cs,2 ) c0,2 + µ
(5)
where cs,1 and cs,2 are the sample concentrations and µ are some number (positive or negative) chosen such that cs,1 g 0 and cs,2 g 0, will always yield two perturbation peaks with equal area but opposite signs, i.e., one positive and one negative.10 (3) Determination of Isotherm Parameters. As opposed to the FA method, the standard PP method do not supply adsorption data directly. In the FA method, the measured adsorption data can subsequently be modeled by an isotherm function, but the standard PP method is used to determine parameters in a preselected isotherm function. The perturbation peak retention times are governed by the derivatives of the isotherm functions at the concentration plateau, and this relation is utilized in the PP method. Usually, isotherm functions are selected and the isotherm parameters are determined from the retention times using a fitting approach. It is often assumed that the PP method, in the binary case, can be used only to determine isotherm parameters in some fixed isotherm functions, and this is indeed true if only the retention times are measured. But, it has been demonstrated that if one also measures the relative areas of the perturbation peaks it is possible to determine adsorption data directly.19,20 Here we will use the fitting approach. In the single-component case, we begin with an isotherm function q(d;c), where the vector d contains the unknown isotherm parameters, e.g., d ) (a, b) if we choose a two parameter model as the Langmuir isotherm; see eq 2. To determine the isotherm parameters, the following leastsquares problem is solved,
minJ1(d) ) min d
d
∑
m i)1
2 |tR(d;c0,i) - texp R (c0,i)|
(6)
where m is the number of experiments at different concentration plateaus c0,i, tR(d;c0,i) are the retention times calculated from eq 1, and texp R (c0,i) are the measured retention times. In the binary case, we begin with two isotherm functions q1(d1;c) and q2(d2;c) and solve the following least-squares problem,
minJ2(d1,d2) ) min d1,d2
d1,d2
∑
m i)1
2 |tR,1(d1,d2;c0,i) - texp R,1 (c0,i)| + 2 |tR,2(d1,d2;c0,i) - texp R,2 (c0,i)| (7)
where c0,i ) (c0,1,c0,2)i are the plateau concentrations, tR,* (d1,d2;c0,i) are the retention times calculated using eq 3, and texp R,* (c0,i) are the measured retention times. Without loss of generality, we require that the measurements are ordered so that texp R,1 (c0,i) e texp R,2 (c0,i). Because it is impossible to relate a specific perturbation peak to a specific component, the solution to the least-squares problem eq 7 is, in general, not unique. If q1(d;c) ) q2(d;c), i.e., the same parametric form of both isotherm functions, then for a solution d/1 , d/2 to eq 7 it holds that J2(d/1,d/2) ) J2(d/2,d/1). As a consequence, the determined isotherm parameters cannot be (19) Tondeur, D.; Kabir, H.; Luo, L. A.; Granger. J. Chem. Eng. Sci. 1996, 51, 3781-3799. (20) Kabir, H.; Grevillot G.; Tondeur, D. Chem. Eng. Sci. 1998, 53, 1639-1654.
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assigned to specific components. To do so we need additional information, e.g., comparing computer simulations with an experimental chromatogram where the peaks can be assigned. Notice that the least-squares problems in eqs 6 and 7 are independent of the sample concentration(s) and that no fractioning of the eluent is required. In a practical situation, there are often a number of isotherm models to choose among.3 To find the one that best suits the experimental data, one should try them all and choose the best one based on some statistical criterion. EXPERIMENTAL SECTION System I. Apparatus. The chromatographic system consisted of a Hewlett-Packard HP 1100 chemstation equipped with an autoinjector, a valve-switching unit with totally 10-port valves, a built-in diode-array UV detector (DAD), two pumps, and a workstation PC (Agilent Technologies, Palo Alto, CA). The mixer, Agilent 1100 (4.6 × 60 mm), was bypassed by a short 0.17-mm PEEK capillary. Thus, the outlets of the two pumps were connected directly to the column by a low dead volume Tconnector. All connections from the system were short sections of 0.17-mm PEEK capillaries. Chemicals. 11R-Hydroxyprogesterone (11R-OH-PS) of minimum 98% purity was obtained from Sigma (Stockholm, Sweden). The methanol used was of HPLC Gradient Grade from J. T. Baker (Denventer, Holland). The water was obtained from the Milli-Q Academic Gradient A10 Biocel Synthesis system (Millipore AB, Sundbyberg, Sweden). Column and Solutions. The achiral high-capacity column Kromasil KR100-3.5C18 (150 × 4.6 mm; 3.5 µm) was obtained from Eka Chemicals (Bohus, Sweden). The column was placed in a water jacket, and its temperature was kept constant using a MN6 Lauda circulating water bath (Lauda, Ko¨ningshofen, Germany). The mobile phase used was methanol/water (70:30), and the solute was 11R-OH-PS. The stock solutions were filtered through 0.45-µm filters (Kebo, Spånga, Sweden). Procedures. Adsorption data and parameters of the steroid 11ROH-PS were determined using the FA method in the staircase mode. One pump (A) supplied pure mobile phase lacking the component, and the other pump (B) supplied mobile phase containing the component. The concentration of the B solution was increased stepwise (10 steps) in the high-pressure gradient mode, resulting in a staircase chromatogram. The procedure was repeated for the bulk concentrations 0.175, 3.5, and 70 mM, which guaranteed collection of adsorption data over a broad concentration range. Isotherm parameters were also determined using the PP method: the established equilibrium was perturbed at each concentration plateau by injecting a 5.0-µL sample with a small deficiency of the steroid. The resulting perturbation peaks were recorded in the same concentration range as used for the FA method above. The DAD made it possible to use several wavelengths for UV detection; the UV signal was recorded at 245 and 320 nm. The absorbance data from the detector were transformed into concentrations units using a calibration curve derived from absorbance measurements made on the concentration plateaus of the frontal analysis staircase. System Properties. Two different dead volumes must be determined using the FA method in the staircase mode: the column hold-up volume V0 and the total dead volume for the
Table 1. Mean Langmuir Isotherm Parameters and Comparison of Intraday Precision between the FA and PP Methods for System I solute
a
b (mM-1)
qs ) a/b (mM)
Single-Component Frontal Analysis with VT ) 1.950 mL mean 11R-OH-PS 2.539 0.00539 471 precision (CV %) 0.053 0.033 0.28 Single-Component Frontal Analysis with VT ) 1.853 mean 11R-OH-PS 2.563 0.00510 515 precision (CV %) 0.055 0.31 0.26 Single-Component Perturbation Peaks mean 11R-OH-PS 2.530 0.00534 precision (CV %) 0.076 0.34
474 0.37
staircase experiment VT. The latter volume is the entire volume after the T-connector (including V0) and until the column exit. The PP method only requires the column hold-up volume V0. The FA raw data were corrected for VT and the PP raw data were corrected for V0. The V0 value was determined to 1.424 mL by injecting the same sample five times and measuring the first-eluted front disturbance. The VT value was determined to 1.950 mL in one type of experiment and 1.853 mL in another. In the first experiment, one pump was equilibrated with methanol/water 70/ 30 and at time t a steep gradient of a mobile phase with a slightly different methanol concentration was introduced by a second pump. In the second experiment, the second pump was used to give a short (1.2 s) pulse. The two different VT values resulted in two sets of experimental FA adsorption data and isotherm parameters. The volume of stationary phase, VS, was determined by subtracting the column hold-up volume from the geometric volume, VG, of the column; i.e., VS ) VG - V0 ) 2.493-1.424 mL ) 1.069 mL. The column temperature was kept constant at 25.0 °C, the flow rate was 0.60 mL/min, and the phase ratio was 0.753. The Langmuir isotherm parameters are presented in Table 1. The number of plates used in the simulation was 2000 determined by fitting simulated elution profiles to experimental profiles. System II. Apparatus. The chromatographic system consisted of two LC-10AD pumps (Shimadzu, Kyoto, Japan), one of them mastering the other in the gradient programming mode, and the outlets of these two pumps were connected directly with a lowdead-volume PEEK T-connector. All connections between the T-connector and the flow cell were made using 0.17-mm PEEK capillaries. An automatic injector MIDAS (Spark Holland, AJ Emmen, The Netherlands) and a UV detector Lambda 1010 (Bischoff, Leonberg, Germany) were used. The HPLC system was complemented with a fraction collector from Advantec (Advantec Toyo Roshi International, Dublin, CA), and a computer data acquisition system with the software CSW 1.7 (DataApex Ltd, Praha, Czech Republic) was used to record the chromatograms. Chemicals. (+)-Methyl L-mandelate (LM) and (-)-methyl D-mandelate (DM) (purity >99%) were obtained from Fluka (Fluka Chemikal, Buchs, Switzerland). Phosphoric acid 99% crystalline p.a. quality and Titrisol 1 M NaOH were from Merck Sharp and Dohme (Haarlem, Netherlands). 2-Propanol LiChrosolv was from Merck (Darmstadt, Germany). The water was obtained as above for system I. Column and Solutions. The Chiral-AGP column (100 × 4.0 mm, 5 µm), consisted of R1-acid glycoprotein immobilized on silica and
Table 2. Bi-Langmuir Isotherm Parameters for System II site
isomer
I II II
L,D
I II II
a
b (mM-1)
qs ) a/b (mM)
Single-Component Frontal Analysis 1.805 0.1124 L 11.64 4.315 D 16.37 4.923
16.06 2.70 3.32
Single-Component Perturbation Peaks L,D 1.867 0.1166 L 11.46 3.858 D 15.86 4.984
16.02 2.97 3.18
Competitive Perturbation Peaks from All Plateau Ratios: 1/0, 0/1, 1/1, 3/1, 1/3 I L,D 1.83 0.133 13.76 II L 11.5 3.940 2.92 II D 15.9 5.060 3.14
I II II
Competitive Perturbation Peaks from the Racemic Plateau (1/1) L,D 2.08 0.179 L 11.0 3.360 D 15.5 6.030
11.62 3.27 2.57
was obtained from ChromTech (Ha¨gersten, Sweden). The column was placed in a laboratory-assembled column jacket, and the temperature was kept constant as above for system I. The mobile phase was phosphate buffer pH 6.15 (ionic strength 0.02 M) with 0.125% 2-propanol. The buffer solutions and stock solutions of enantiomers were filtered through 0.45-µm filters (Kebo, Spånga, Sweden). Procedures. Single adsorption data and parameters of the enantiomers were determined by using the FA method in the staircase mode (see above for system I). Three different bulk concentrations of the respective enantiomers were used successively: 12.5 µM, 0.25 mM, and 5.0 mM, with a total of 30 data points/enantiomer. Both single and competitive isotherm parameters were determined using the PP method (see above for system I) at each concentration plateau by injecting 3.5-µL samples. In the single-component case, we used the same concentration range and number of data points as for the FA method, and the sample had a deficiency of the enantiomer that resulted in a small perturbation peak. In the competitive case, the perturbation peaks were created at three plateau concentration ratios of DM and LM: 1/1, 3/1, and 1/3, the highest ones being 5.0 mM DM/ 5.0 mM LM, 5 mM DM/5/3 mM LM, and 5/3 mM DM/5 mM LM. Totally 90 concentration plateaus were used in the competitive case. The sample composition was designed according to eq 5 and resulted in two small perturbation peaks. All eluent chromatograms and the lowest and middle bulk concentration staircase chromatograms were detected at 225 nm. The highest bulk concentration staircase chromatograms were detected at 250 nm. The absorbance data from the detector at wavelength 225 nm were transformed into concentration units as described above for system I. Fractions from overloaded eluent chromatograms were taken at 6-s intervals, and each fraction was diluted and reinjected. Using the chromatograms from the reinjections, the molar fraction of each enantiomer was calculated by using the peak areas, and the UV trace from the overloaded eluent chromatograms were then used to translate these fractions into enantiomer concentrations. Analytical Chemistry, Vol. 76, No. 16, August 15, 2004
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Figure 1. (a) RRE for the first peak versus the plateau concentration using both the traditional blank injection technique (dotted) and the new injection technique at 50 (solid) and 100% (dashed) disturbance with 50-µL injection volume. (b) The same as (a) but for the second peak. When using the blank injection technique, this peak vanishes at a low plateau concentration. Parameters as in Experimental Section (system II) and in Table 2 (all ratios).
System Properties. All PP data were corrected for V0, and all FA data were corrected for VT. The column hold-up volume, V0, was determined to be 0.928 mL. The total dead volume, VT, was determined to be 0.952 mL by equilibrating the system with mobile phase from one pump and at a time t introducing a steep gradient of a mobile phase with a slightly different 2-propranolol concentration from a second pump. The volume of the stationary phase, VS, was determined to be 0.329 mL. The column temperature was kept constant at 25.0 °C, the flow rate was 0.90 mL/min, and the phase ratio 0.354. The bi-Langmuir parameters are presented in Table 2. The number of plates used in simulations was 3000 and was determined as above for system I. Parameter Determination for Systems I and II. Isotherm parameters were determined by using the PP method as explained in Theory, and by using single-component FA where the measured adsorption data was fitted to an isotherm function. The Langmuir model, eq 2, was used for system I and the bi-Langmuir model, eq 4, was used for system II. We used the Fletcher-Xu hybrid method, as implemented in the TOMLAB Optimization Environment,21 to solve the fitting problems both for single-component FA and for the PP method; see eqs 6 and 7. RESULTS AND DISCUSSION Two model systems of LC, with different capacities, were used. System I was a high-capacity system showing a homogeneous adsorption, and the Langmuir isotherm parameters are presented in Table 1. System II was a low-capacity system showing heterogeneous adsorption, and the bi-Langmuir isotherm parameters are presented in Table 2. (21) TOMLAB Optimization, www.tomlab.biz, 14 January 2004.
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Figure 2. RRE for the new injection technique versus disturbance and injected volume, in (a) for the first peak and in (b) for the second peak. Parameters as in Figure 1; the plateau concentration was 0.0677 mM of both enantiomers.
Effects of Perturbation Size. The importance of the perturbation injection is an issue that has only been sporadically investigated. For example, in ref 6, it is stated that “retention times depend solely on the equilibrium state and not on the type of perturbation”. The authors used only one concentration plateau and in that system it appeared that the size of the perturbation injection did not affect the measured retention time. On the other hand, it has been demonstrated that when the size of the perturbation injection is increased in a single-component system, the response is an increasingly deep negative peak that shows an anti-Langmuir behavior when the isotherm is Langmuirian.11,12 This means that the retention time of the peak minimum increases. The opposite behavior was found for a positive perturbation peak.12 Here it should be noted that eq 1 gives the mean retention time (normalized first-order moment) of the perturbation peak and this is not the same as the retention time of the peak min/max unless the peak is symmetrical. The result of different retention times from a positive or a negative perturbation peaks have also been shown for a BET model by Gritti et al.13 To investigate the importance of the perturbation injection size, we compared the retention times we got from computer simulations with those we got from eq 1. The computer simulations
Figure 3. Experimental adsorption data for 11R-OH-PS determined by the FA method with two different VT and the determined isotherm functions using the FA and PP methods. The symbols are experimental data: FA with VT ) 1.950 mL; (O) and FA with VT ) 1.853 mL (0). The lines are Langmuir isotherm functions calculated using the mean isotherm parameters from five determinations (see Table 1): FA with VT ) 1.950 mL (solid), FA with VT ) 1.853 mL (dashed), and PP (dotted). Notice that the solid line and dotted line coincide perfectly, resulting in only one visible line. The main figure shows the high-concentration range (up to 70 mM), the top left inset shows the low-concentration range (below 0.175 mM), and the bottom right inset shows the medium-concentration range (up to 3.5 mM). Experimental conditions: see Experimental Section for system I.
solved the equilibrium dispersive model of chromatography with a finite difference scheme,3 and we “measured” the retention time at the peak min/max. To quantify the error, we use the relative retention error (RRE) defined to be RRE ) |tR,sim - tR|/tR, where tR,sim is the simulated retention time and tR is the retention time calculated using eq 1 or 3. For system I, the RRE was studied for 50-µL blank injections and the maximal RRE was found to be ∼9.0%. For the single-component case in system II, with 50-µL blank injections the maximal RRE is ∼8.7% for both LM and DM. For the racemic binary case in system II, the RRE was investigated versus the plateau concentrations at 50-µL injections for both the traditional blank injection technique and for the new injection technique at 100% and at 50% disturbance, where the disturbance is defined to be |cs,* - c0,*|/c0,*, see eq 5; i.e., for 50% disturbance on a racemic plateau, the sample concentration are 0.5 and 1.5 times the plateau concentration of the first and second components, respectively. For 100% disturbance, the sample concentrations are 0 of the first component and double the plateau concentration of the second component. For the first peak, the RRE is much smaller with the new injection technique than for the blank one, as illustrated in Figure 1a. For the blank injection technique, the largest RRE is ∼9.0%, and for the new injection technique, at 50% disturbance, the largest RRE is 0.9%. The largest RRE takes place at the plateau concentration with the largest isotherm curvature. The RRE for the second peak is illustrated in Figure 1b. With the blank technique, the peak already vanishes at a quite low plateau concentration where the largest RRE is 1.7%, and for the new injection technique at 50% disturbance, the largest
RRE is slightly lower, 1.6%. This shows that the new injection technique results in a drastic decrease of the RRE for the first peak and that the RRE is about the same for the second peak, which does not vanish at high plateau concentrations so that all retention times can be measured. The dependence of the RRE for the optimal injection technique on the disturbance and the injected volume was studied for several racemic plateau concentrations. A typical plot is shown for the first peak in Figure 2a and for the second peak in Figure 2b at a plateau concentration of 0.0677 mM concentration of each enantiomer. As seen, the RRE depends almost linearly on the disturbance and the injected volume. It is clear that the largest RRE occurred when using large disturbances, large injection volumes, or both. With the new injection technique it is possible not only to detect both perturbation peaks but also to considerably reduce the disturbance and the injection volume and thus to minimize the retention time error. For example, in this study, we used the new injection technique with 3.5-µL injections and 100% disturbances on 1/1 concentration plateaus to determine the isotherm parameters for system II. The largest RRE is only 0.22%, which is very satisfactory. We also performed some experiments for system II using plateau concentrations 0.4 and 5 mM that confirmed our calculations. System I: Validation of Precision and Accuracy. The precision of the parameters for system I determined by the FA and PP methods was validated through five repetitive experiments, as recommended for validation of precision.15,16 The intraday Analytical Chemistry, Vol. 76, No. 16, August 15, 2004
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Figure 4. Comparison between experimental (O) and calculated elution band profiles of 11R-OH-PS with parameters from the FA with VT ) 1.950 mL (solid), FA with VT ) 1.853 mL (dashed), and PP methods (dotted). The elution profiles were simulated using the mean Langmuir isotherm parameters from five determinations (see Table 1) and 2000 theoretical plates. The solid line and dotted line coincide perfectly, resulting in only one visible line. Experimental conditions: see Experimental Section for system I; but the injection was 100 µL of 70 mM 11R-OH-PS.
precision (coefficient of variation, CV %) of the a, b, and qs Langmuir terms are reported in Table 1 and are excellent, with errors smaller than 0.4%. These values are slightly better than those in ref 7 that were determined by three repetitive experiments. Two different total dead volumes, VT, resulted in two sets of FA experimental adsorption data and isotherm parameters. The adsorption data acquired by the FA method and the corresponding isotherm functions are compared with the isotherm function determined by the PP method in Figure 3. The five repetitive experimental symbols for each plateau concentration coincide
almost perfectly, so that the five symbols seem to be only one i.e., the precision is excellent. The Langmuir adsorption isotherms for the PP (dotted line) and FA method with VT ) 1.950 mL (solid line) coincide, whereas the isotherm for FA with VT ) 1.853 mL (dashed line) differs considerably. Therefore, the larger VT value seems to be the more correct. The accuracies of the PP isotherm parameters were first validated by comparing them with the isotherm parameters determined by the FA method, since the latter method is considered to be the most accurate today.1,14 The difference between the mean values of the Langmuir isotherm a term determined by the FA (VT ) 1.950 mL) and PP methods was 0.35%, and the difference between the mean values of the b term was 0.93%, which means great agreement between the parameters. These differences are slightly smaller than the ones obtained in ref 7. A t-test shows that there are significant differences (P ) 0.05) for the a term but not for the b term between the two methods. To further validate the accuracy of the isotherm parameters determined by the PP and FA methods we used them to computer simulate elution profiles and compared those profiles with the experimental ones. To quantify how well the simulation fits the experimental data we defined the overlap to be
∫
∞
0
min [csim(t),cexp(t)] dt
∫
∞
0
(8)
csim(t) dt
where cexp(t) and csim(t) are the experimental and simulated responses at time t. When the experimental and simulated elution
Figure 5. Single-component adsorption data for LM and DM determined by the FA method and the determined isotherm functions. The symbols are experimental data: LM (∆) and DM (0). The lines are bi-Langmuir isotherm functions calculated using isotherm parameters determined by the FA method (see Table 2): LM (solid) and DM (dashed). The main figure shows the medium-concentration range (up to 0.25 mM), the top left inset shows the low-concentration range (below 12.5 µM), and the bottom right inset shows the high-concentration range (up to 5.0 mM). Experimental conditions: see Experimental Section for system II. 4862
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Figure 6. Perturbation peak retention times at different plateau concentrations of LM and DM. The symbols are experimental data: LM (∆) and DM (0). The lines are calculated retention times using the single bi-Langmuir isotherm parameters determined by the PP method (see Table 2): LM (solid) and DM (dashed). The main figure shows the medium concentration range (up to 0.25 mM), the top left inset shows the low-concentration range (below 12.5 µM), and the bottom right inset shows the high-concentration range (up to 5.0 mM). Experimental conditions: see Experimental Section for system II.
profiles coincide perfectly, the overlap is 100% and when they are totally separated the overlap is 0%. Comparison between an overloaded experimental elution profile of 11R-OH-PS and simulated elution profiles with parameters from the FA (with two different VT) and PP methods are shown in Figure 4. The simulated elution profiles from the FA method with VT ) 1.950 mL (solid line) and from the PP method (dotted line) coincide almost perfectly, showing that the two methods gave nearly identical results. Both simulated profiles deviate slightly from the experimental profile (O) probably due to small errors in the determined isotherm parameters. However, the overlaps are quite good, 95.1% for the FA and 94.1% for the PP method. The simulated elution profile for the FA method with VT ) 1.853 mL (dashed line) is considerably different from the experimental profile and the overlap is only 74.1%. These results confirm the importance of measuring the correct total dead volume VT when using the FA method. This can be a difficult task and is a drawback of the FA method compared to the PP method, where VT is not required. System II: Validation of Accuracy. Single-component isotherm parameters were determined using both the FA and PP methods. Figure 5 shows the experimental adsorption data acquired by the FA method for the enantiomers LM (∆) and DM (0). The lines, LM (solid) and DM (dashed), are the isotherm functions calculated using the single-component bi-Langmuir isotherm parameters (see Table 2), and they are in excellent agreement with the experimental adsorption data. Figure 6 shows the experimental perturbation peak retention times acquired by the PP method for the enantiomers LM (∆) and DM (0) at different concentration plateaus. The lines, LM (solid) and DM
(dashed), are calculated retention times using the singlecomponent bi-Langmuir isotherm parameters determined by the PP method (see Table 2) and are in excellent agreement with the experimental retention times. The values of the singlecomponent bi-Langmuir parameters determined by the PP method are close to these determined by the FA method (see Table 2). Binary perturbation peaks were generated using plateau concentrations of the LM and DM enantiomer at three different ratios: 3/1, 1/1, and 1/3. For each LM/DM ratio, the total binary plateau concentration was increased in 30 steps from 0.2 µM to more than 6.0 mM; see Experimental Section for system II. Since 6.0 mM is 20 times higher than 0.3 mM, which was the total binary concentration when the second perturbation peak vanished using the blank injection technique in ref 10, it is clearly necessary to use the new injection technique (see eq 5) to get detectable perturbation peaks over the whole concentration range. In Figure 7, the retention times of the perturbation peaks were plotted versus the plateau concentrations of the respective enantiomer. The symbols are experimental retention times, first peak (∆) and second peak (0), and the surfaces are retention times calculated using the bi-Langmuir isotherm parameters determined by the PP method (all enantiomer ratios used; see Table 2). Note that the two surfaces do not correspond to the two enantiomers but instead represent a fast- and a slow-moving perturbation peak. Furthermore, there is perfect coincidence between the experimental and calculated values over the whole concentration range. The left top inset in Figure 7 shows the low-concentration range. The corresponding single perturbation peak plot was shown in the upper left inset of Figure 6. The main plot in Figure 7 shows the medium-concentration range. Interestingly, the two surfaces Analytical Chemistry, Vol. 76, No. 16, August 15, 2004
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Figure 7. Perturbation peak retention times at different binary plateaus with LM/DM ratios 1/0, 3/1, 1/1, 1/3, and 0/1. The symbols are experimental data: first peak (∆) and second peak (0). The surfaces are calculated retention times using the competitive bi-Langmuir isotherm parameters determined by the PP method (see Table 2, all plateau ratios): first peak (bottom surface) and second peak (top surface). The main figure shows the medium-concentration range (up to 0.25 mM), the upper left inset shows the low-concentration range (up to 0.012 mM), and the upper right inset shows the high-concentration range (up to 5.0 mM). Experimental conditions: see Experimental Section for system II.
coincide at a point around 0.07 mM DM and 0 mM LM. To explain this, we consider the situation where we have a concentration plateau containing only DM and make a perturbation injection containing both LM and DM. The response to this injection will be two perturbation peaks: at a low DM plateau concentration, the peaks will be separated but the difference in retention time will gradually become smaller as the DM plateau concentration is increased until there is no difference, i.e., the peaks overlap, and this corresponds to the point where the two surfaces coincide in Figure 7. Increasing the DM plateau concentration further from this point will increase the difference in retention time; i.e., the peaks are again separated. It should be noted that this is not a retention shift, because it is impossible to assign the perturbation peaks to specific components. It is also interesting to compare 4864
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this situation with the single-component situation, i.e., making an injection containing only DM on a DM plateau, as is shown in the left face of the main Figure 7 by the experimental symbols (0). The response will then be a single perturbation peak. At plateau concentrations before the point where the two surfaces coincide, the retention time will be equal to that of the slower moving peak in the previous case, but at plateau concentrations after the intersection point, the retention time will be equal to that of the faster moving peak. The right top inset in Figure 7 shows the high-concentration range. The two surfaces are completely horizontal and parallel except for a small region in the lowconcentration region where the surfaces point sharply upward. It is interesting that the two surfaces do not converge to the same limit at increasing plateau concentrations, in contrast to the
DM (dashed line). The agreement between the experimental data and the simulations is excellent in all cases; i.e., the parameters from the PP method are very accurate. Four sets of isotherm parameters were determined using different techniques: (I) single-component FA, (II) single-component PP, (III) competitive PP using all plateau ratios, and (IV) competitive PP using only the racemic plateaus. The accuracy of the isotherm parameters was evaluated by calculating the overlap between the simulated and experimental elution profiles (see eq 8). Although there are differences between the isotherm parameters, the effect on the simulated elution profiles is less pronounced. The mean overlap of the total elution profiles for the three injections was over 90% for all four sets of parameters, the PP parameters from all plateaus ratios gave the largest mean overlap, 92.9%; and the PP parameters using only racemic plateaus are almost as accurate with a mean overlap of 91.1%.
Figure 8. Experimental and simulated elution profiles for different sample compositions. The symbols are experimental data: LM (∆) and DM (0). The lines are simulated elution profiles using the competitive bi-Langmuir isotherm parameters determined by the PP method (see Table 2, all plateau ratios) and 3000 theoretical plates: LM (solid) and DM (dotted). (a) Sample, 20 µL of 5.0 mM LM and 1.67 mM DM, i.e., a 3:1 ratio. (b) Sample, 20 µL of 5.0 mM LM and 5.0 mM DM, i.e., a 1:1 ratio. (c) Sample, 20 µL 1.67 mM LM and 5.00 mM DM, i.e., a 1:3 ratio. Experimental conditions: see Experimental Section for system II.
retention times of the corresponding single-component perturbation peaks, which converge to virtually identical retention times at the plateau concentration 0.25 mM (see Figure 6; main plot and lower right inset). Here it should be noted that the retention data shown in Figure 6 belong to the lower surface in Figure 7 (for DM > 0.07 mM). The retention times of the binary perturbation peaks were used to determine isotherm parameters, and as in the single-component case, the bi-Langmuir model fitted best to the experimental data. First, all perturbation experiments were used, i.e., the LM/DM concentration plateaus with ratios 3/1, 1/1, 1/3, 1/0, and 0/1. The determined parameters are given in Table 2 and are very similar to those determined using the single perturbation experiments, the only significant difference being that the bI term increased from 0.1166 to 0.133 mM-1. The parameters were also determined using only retention times from the binary LM/DM concentration plateau with the ratio 1/1. These parameters can be determined directly without prior purification of the two enantiomers, and the parameters can be seen in Table 2. The best way to judge which parameters are the most accurate is to use them to computer simulate elution profiles and to compare the simulations with the experimental profiles. Figure 8 shows individual overloaded experimental elution profiles of LM (∆) and DM (0) at different sample compositions together with simulated elution profiles using parameters from the PP method (all enantiomer ratios used; see Table 2), for LM (solid line) and
CONCLUSIONS A new injection technique to solve the problem with vanishing perturbation peaks in the PP method was outlined previously10 and was used with success in this study. We showed that considerable experimental errors occurred when using the traditional blank injection technique and that the error was significantly reduced with the new injection technique (a 10-fold reduction). We further show that it is important to use small injection volumes and to use samples that deviate insignificantly in composition from that of the concentration plateau. The precision of the determined single-component data for the steroid 11R-OH-PS was excellent using both the PP and FA methods. The accuracy of the determined isotherm parameters was validated by comparing those determined by the FA and PP methods, and the difference was smaller than 1%. The accuracy was further validated by comparing experimental and simulated profiles using the isotherm parameters determined by the two methods. The accuracy was good in both cases if the correct value of VT for FA was used. However, if the measured VT value used in the FA method differed slightly from the true value, the isotherm function and the simulated elution profiles changed considerably. This is an important drawback of the FA method compared to the PP method where VT is not required for parameter evaluation. In the binary case, the isotherm parameters for the two enantiomers, LM and DM, were determined by the PP and FA methods and used to simulate elution band profiles of mixtures with different ratios of the enantiomers. Excellent agreement between simulated and experimental profiles was observed, which confirms the accuracy of the isotherm parameters determined by the PP method. We further demonstrated how to experimentally obtain the isotherm parameters for the enantiomers directly from a racemic mixture. This is an important result, allowing major savings in time and money when determining competitive isotherm parameters. Specifically, it opens the possibility to use the method in the pharmaceutical industry to quantify proteinenantiomer interactions at the drug discovery stage, e.g., as an alternative to the mass-sensitive SPR technology.18 Received for review February 15, 2004. Accepted May 16, 2004. AC0497407 Analytical Chemistry, Vol. 76, No. 16, August 15, 2004
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