Recent Progress in the pKa Estimation of Druglike Molecules by the Nonlinear Regression of Multiwavelength Spectrophotometric pH-Titration Data

1. Introduction

Protonation constants, or acid dissociation constants, are very important both in the analysis of drugs and in the interpretation of their mechanisms of action. Spectrophotometry is a convenient method for determination in very dilute aqueous solutions (about 10^-5 to 10^-6 M), provided that the compound possesses pH-dependent light absorption due to the presence of a chromophore in proximity to the ionization centre. Much more information can be extracted if multivariate spectrophotometric data are analyzed by means of an appropriate multivariate data analysis method cf. [1–20]. In previous work [21–33], the authors have shown that the spectrophotometric method can be used in combination with suitable chemometric tools for the determination of protonation constants or acid dissociation constants , even for sparingly soluble drugs.

The Biopharmaceutics Classification system BCS [34, 35] classifies every pharmaceutical active ingredient into one of 4 groups based on two basic characteristics: solubility and permeability. We decided to complete such information and to study the protonation equilibria of a pharmaceutical active ingredient of natural origin, silybin, which possesses low water solubility. While special attention was paid to the methodology, we also studied the protonation equilibria of silybin, in order to verify the validity of the results. Silybin is a flavanolignan belonging to a group of biologically active substances found in the milk (or Saint-Mary) thistle, Silybum marianum (L.) Gaertner. These substances are almost certainly produced in the plant by a radical coupling of flavonoid and coniferyl alcohol [36]. Although the plant has been used since ancient times for the treatment of liver and stomach diseases, the first representative of the group, silybin, was described only in the late 1960s. The group of flavanolignans extracted from the seeds of Saint-Mary thistle was originally called by one common name, silymarin, used interchangeably with the name of its main component, silybin. In 1974 Wagner et al. [37] proposed giving the name silymarin to the whole group of active flavanolignan-like substances and the names silybin, silydianin, and silychristin to its main constituents. The major component of silymarin is silybin, which constitutes 60–70% of the drug [38]. The protonation constants of silybinin at various ionic strengths have been studied at various temperatures [24]. However, in only a few cases has the dependence of the protonation constants on ionic strength been systematically investigated, and the thermodynamic dissociation constant estimated. The reliability of dissociation constants obtained by the regression analysis of potentiometric or pH-spectrophotometric data is dependent upon (i) the calibration of the glass electrode cell, (ii) the algorithm used, (iii) the instrumental method used and the parameters selected for refinement, and (iv) a strategy of efficient experimentation. Much work has been put into developing methods for the resolution of multicomponent spectra but less work has been carried out to reveal the limitations of these methods and in the estimation of the minor components of the resolved spectra. Approaches to determining the rank of the absorbance matrix A are based on pure principal component analysis (PCA). Generally, PCA will extract some of the noise, that is, the experimental and/or random error which will usually be represented by the principal components with smallest size or variance. When no noise is present in the spectra, the number of eigenvalues of the covariance matrix larger than zero is equivalent to the number of components , assuming that the spectra of components in the mixture are linearly independent. As all real data always contain experimental noise, the number of eigenvalues different from zero is usually larger than the number of components, . Experimental and/or random error can mask the identification of the true dimensionality of a given data set. In any study of this type, the level of “experimental noise’’ will be a critical factor. It is therefore necessary to have a consistent definition of the signal-to-noise ratio (SNR), so that the impact of this parameter can be critically assessed. Traditional approaches to SNR are typically based on the ratio of the maximum signal to the maximum noise value. As an alternative, the concept of instrumental error has again been employed, and the signal-to-error ratio (SER) is defined where, as an error level, the instrumental standard deviation of absorbance, is used. Attention should be paid to the methods’ ability to detect a minor component in the presence of major ones. The detection limit or the smallest relative concentration of the minor component present depends on several factors, such as (i) the spectral similarity of the minor component to the others, (ii) instrumental resolution, (iii) noise level and noise type, and (iv) the signal-to-noise ratio SNR with respect to the minor component.

The regression methods include traditional least-squares curve fitting approaches, based on the previous postulation of a chemical model, that is, the postulation of a set of species defined by their stoichiometric coefficients and formation constants, which are then refined by least-squares minimization. These mathematical procedures require the fulfillment of the mass-balance equations and the mass-action law. The most relevant algorithms are SQUAD(84) [6–11] and SPECFIT32 [14–17, 39].

This paper describes the current status of computational diagnostic tools in the estimation of a drug carried out by the nonlinear regression of the multiwavelength spectrophotometric pH-titration data. The dissociation constants of the drug, silybin, at an ionic strength and at 25°C, are estimated to prove the reliability of the whole regression procedure.

2. Theoretical

2.1. Procedure for the Determination of the Protonation Constants

The protonation equilibria between the anion (the charges are omitted for the sake of simplicity) of a drug and the proton H are considered to form a set of the variously protonated species and so forth, which have the general formula in a particular chemical model and which are represented by the number of species, where index labels their particular stoichiometry; the overall protonation (stability) constant of the protonated species, β _qr, may then be expressed as

(1) where the free concentration and . As each aqueous species is characterized by its own spectrum, for UV/VIS experiments and the th solution measured at the th wavelength, the Lambert-Beer law yields the absorbance, , being defined as

(2) where is the molar absorptivity of the species with the stoichiometric coefficients , measured at the th wavelength for solutions with known total concentrations of basic components, and , at wavelengths. The multicomponent spectra analysing program SQUAD(84) [8] may adjust and for a given absorption spectra set by minimising the residual-square sum function, ,

(3) where represents the element of the experimental absorbance response-surface of size (Figures 1(a), 1(b)) and the independent variables are the total concentrations of the basic components and being adjusted in solutions. This means that the predicted absorbance-response surface is fitted to given spectral data, with one dimension representing the dependent variable (absorbance), and the other two dimensions representing the independent variables, namely. the total component concentrations (or pH) of solutions, at wavelengths. Another popular program is SPECFIT/32 [39], based on singular value decomposition and nonlinear regression modelling using the Levenberg-Marquardt method for the determination of stability constants from spectrophotometric titration data. The experimental and computational schemes for the determination of the protonation constants of the multicomponent system are taken from Meloun et al. [2] and the details for the computer data treatment are collected in the Supporting Information.

Figure 1: The 3D-absorbance-response-surface representing (a) the measured multiwavelength absorption spectral dependence on pH at C and for silybin, and (b) the 3D-residuals map after nonlinear regression performed by SQUAD (and by SPECFIT in brackets) for silybin, which exhibits 780 residuals of the residual bias (-) being close to zero and the residual standard deviation 1 mAU (0.88 mAU) being close to the instrumental standard deviation  mAU (0.30 mAU); the residual skewness () is close to zero and indicates a symmetric distribution of residuals; the residual kurtosis (3.21) is close to 3 and indicates a Gaussian distribution of residuals. The accuracy test of the bias proves that the bias is not significantly different from zero, (S-Plus).

All spectra evaluation may be performed with the factor analysis INDICES algorithm [30] in the -Plus programming environment [40]. Most index methods are functions of the number of principal components ’s, with the use of which the spectral data are usually plotted against an integer index . When the reaches, the value of the instrumental error of the spectrophotometer used, , the corresponding index represents the number of light-absorbing components in a mixture, . In a scree plot the value of PC(k) decreases steeply with increasing PCs as long as the PCs are significant. When is exhausted, the indices fall off, some even displaying a minimum. At this point, for all indices. The index values at this point can be predicted from the properties of the noise, which may be used as a criterion to determine , [30].

2.2. Computational Schema for Protonation Model Building with SPECFIT/32

An experimental and computational scheme for the protonation model building of a multi-component and multiwavelength system was proposed by Meloun et al., compare [2, page 226] or [8, 31], and is here revised with regard to SPECFIT/32 and INDICES application.

(1) Instrumental error of absorbance measurements, : the INDICES algorithm, compare. [30], should be used to evaluate . The Cattel’s scree plot of of the Wernimont-Kankare procedure consists of two straight lines intersecting at where is the matrix rank for the system and the instrumental error of the spectrophotometer used, reaching a value of 0.25 mAU in range of 225–360 nm for the Cintra 40 (GBC, Australia) spectrophotometer employed. This value can be used for prediction of the signal-to-error ratio, (SER), for experimental data. It was proven that the indices are able to accurately predict the correct number of components that contribute to a set of absorption spectra for data sets with SER of equal to or higher than 10.

(2) Experimental design: Simultaneous monitoring of absorbance and pH during titrations is used in a titration, when the total concentration of one of the components changes incrementally over a relatively wide range, but the total concentrations of the other components change only by dilution.

(3) Number of light-absorbing species: a qualitative interpretation of the spectra aims to evaluate the quality of the dataset and remove spurious data, and to estimate the minimum number of factors, that is, contributing aqueous species, which are necessary to describe the experimental data. The INDICES [30] determine the number of dominant species present in the equilibrium mixture. The method can detect minor components and predicts the correct number of components for data sets with the signal-to-error ratio SER of at least equal to or higher than 10. For the signal value S in a numerator of the ratio S/E, the absorbance difference for the th-wavelength at the th-spectrum can be used, where is the limiting spectrum of acid form of drug measured. This absorbance change is then divided with the instrumental standard deviation and resulting ratio represents here the signal-to-error ratio SER of the spectra studied. This SER ratio is examined for all absorbance matrix elements in the whole range of wavelength λ and is compared with the limiting SER value. It was proven that when the ratio is equal to or higher than 10, the factor analysis is able to predict the correct number of components in equilibrium mixture.

(4) Choice of computational strategy of regression process: the input data should specify whether or log values are to be refined with an application of two procedures of nonlinear regression.

(5) The initial estimates of predicted parameter from molecular structure: it is wise before starting a regression to analyze actual experimental data, to search for scientific library sources to obtain a good default for the number of ionizing groups, and numerical values for the initial guess as to relevant protonation constants and the probable spectral traces of all the expected components.

(6) Diagnostics indicating a chemical model: when the minimization process of a regression spectra analysis terminates, some efficient diagnostic criteria are examined to determine whether the results should be accepted [32, 33].

1st diagnostic—the physical meaning of the parametric estimates: the physical meaning of the protonation constants, associated molar absorptivities, and stoichiometric indices is examined: and should be neither too high nor too low, and should not be negative. The empirical rule that is often used is that a parameter is considered to be significant when the relation is met and where is equal to 3 at a 99.9% statistical probability level.

2nd diagnostic—the physical meaning of the species concentrations: there are some physical constraints which are generally applied to concentrations of species and their molar absorptivities: concentrations and molar absorptivities must be positive numbers. Moreover, the calculated distribution of the free concentration of the basic components and the variously protonated species of the chemical model should show realistic molarities, that is, down to about 10^-8 M.

3rd diagnostic—parametric correlation coefficients: partial correlation coefficients, , indicate the interdependence of two parameters, that is, the stability constants and , when others are fixed in value.

4th diagnostic—goodness-of-fit test: to identify the “best” or true chemical model when several are possible or proposed, and to establish whether the chemical model represents the data adequately, the residuals e should be carefully analyzed. The goodness-of-fit achieved is easily seen by examination of the differences between the experimental and calculated values of absorbance, . One of the most important statistics calculated is the standard deviation of the absorbance, , calculated at the termination of the minimization process as , where stands for the residuals-square-sum function in minimum and is the degree of freedom. This is usually compared with the standard deviation of absorbance calculated by the INDICES program [30] and the instrumental error of the spectrophotometer used and if it is valid that , or , then the fit is considered to be statistically acceptable. Some realistic empirical limits are employed: for example, when , the goodness-of-fit is still taken as acceptable, while indicates that a good fit has not been obtained. Alternatively, the statistical measures of residuals e can be calculated to examine the following criteria: the residual bias should be a value close to zero; the residual standard deviation being equal to the absorbance standard deviation should be close to the instrumental standard deviation ; the residual skewness should be close to zero for a symmetric distribution of residuals; the residual kurtosis should be close to 3 for a Gaussian distribution of residuals.

The details of the computer data treatment are given in the Supporting Information.

3. Experimental

3.1. Chemicals and Solutions

Silybin was generously donated by IVAX-CR, Czech Republic. A silymarin extract of pharmacopoeial quality (DAB IX) was prepared from Silybum marianum, var. Silyb (L.) Gaertn. (Asteraceae). Individual component was isolated and purified by ethylacetate extraction, crystallization, and chromatography. The final purity achieved was Silybin: IVAX-CR company standard AB023, Batch no. 190194, 97.5% (HPLC).

Perchloric acid, 1 M, was prepared from conc. HClO₄ (p.a., Lachema Brno) using redistilled water and standardized against HgO and NaI with a reproducibility of less than 0.20%. Sodium hydroxide, 1 M, was prepared from pellets (p. a., Aldrich Chemical Company) with carbon dioxide-free redistilled water and standardized against a solution of potassium hydrogenphthalate using the Gran Method with a reproducibility of 0.1%. The preparation of other solutions from analytical reagent-grade chemicals has been described previously [31].

3.2. Apparatus and pH-Spectrophotometric Titration Procedure

The apparatus used and the pH-spectrophotometric titration procedure have been described previously [31].

3.3. Software Used

Computations relating to the determination of the dissociation constants were performed by regression analysis of the UV/VIS spectra using the SQUAD(84) [8] and SPECFIT/32 [39] programs. Most of graphs were plotted using ORIGIN 7.5 [41] and S-Plus [40]. Qualitative interpretation of the spectra with the use of the INDICES program [30] aims to evaluate the quality of the dataset and remove spurious data, and to estimate the minimum number of factors, that is, contributing aqueous species, which are necessary to describe the experimental data, and determine the number of dominant species present in the equilibrium mixture.

3.4. Supporting Information Available

Complete experimental and computational procedures, input data specimen, and corresponding output in numerical and graphical forms for the programs INDICES, SQUAD(84), and SPECFIT/32 are available free of charge online at http://meloun.upce.cz/ and in the block DATA.

4. Results and Discussion

Recently, silybin was studied in our laboratory [24] and this drug was therefore taken as an example of a drug acid for the demonstration of the reliability of a protonation model and of protonation constants estimation, because of two problems: the first problem in the evaluation of the protonation equilibria of the drug silybin is the extensively overlapping equilibria, as the difference of two consecutive dissociation constants is less than 3 (here about 1.2). Such close equilibria are always difficult to evaluate and therefore the user should carefully prove the reliability of each dissociation constant estimation. A distribution diagram of the relative concentrations of all of the variously protonated species demonstrates the overlapping protonation equilibria for three close consecutive dissociation constants. The second problem concerns the small differences between the molar absorptivities in the variously protonated species within a spectrum. It may happen that nonlinear regression fails when small differences of absorbance are of the same magnitude as instrumental noise, .

The deprotonated silybin form exhibits several isosbestic points in spectra. Each isosbestic point usually indicates a simple two-state equilibrium. pH-spectrophotometric titration enables absorbance-response data (Figures 1(a) and 2(a)) to be obtained for analysis by nonlinear regression, and the reliability of parameter estimates (pK’s and ’s) can be evaluated on the basis of the goodness-of-fit test of residuals (Figures 1(b) and 2(d)). The A-pH curves at 243, 250, 288, and 327 nm (Figure 2(c)) show that a dissociation constant can be indicated. However, as the changes in spectra are small with deprotonation, the variously protonated species exhibit similar and overlapping absorption bands. The small shift of a band maximum to lower wavelengths in the spectra set is indicated, Figure 2(a). The adjustment of pH value from 6 to 12 causes the absorbance to change, so that the monitoring of the variously protonated components of the protonation equilibrium is rather uncertain. As the changes in spectra are small, very precise measurement of absorbance is required for a reliable estimation of the overlapping deprotonation equilibrium studied.

Figure 2: The nonlinear regression analysis of the protonation equilibria model and factor analysis of silybin: (a) Absorption spectral dependence on pH at C, (b) Cattel’s scree plot of the Wernimont-Kankare procedure for the determination of the number of light-absorbing species in the mixture leads to the actual instrumental error of the spectrophotometer used  mAU (INDICES in S-Plus), (c) The absorbance versus pH curves for 243 nm, 250 nm, 288, and 327 nm in dependence on pH at C, (d) Detecting influential outlying spectra with the use of the goodness-of-fit test and the plot of the residual standard deviation versus pH for 20 spectra in dependence on pH at C and , (e) Pure spectra profiles of molar absorptivities versus wavelengths for variously protonated species , and , (f) Distribution diagram of the relative concentrations of both variously protonated species , and , of silybin as a function of pH at C is calculated from estimates by SQUAD and SPECFIT: pK_a1 = 6.898(0.022) and 6.897(0.002); pK_a2 = 8.666(0.021) and 8.667(0.012); pK_a3 = 9.611(0.010) and 9.611(0.004); pK_a4 = 11.501(0.008) and 11.501(0.007), (SQUAD and SPECFIT, ORIGIN).

In the first step of the regression spectra analysis, the number of light-absorbing species is estimated by the factor analysis INDICES algorithm (Figure 2(b)). The position of the break point on the curve in the factor analysis scree plot is calculated and gives with corresponding coordinate log , that is,  mAU, which also represents the actual instrumental error of the spectrophotometer used. Due to the large variations in the indicator values, these latter are plotted on a logarithmic scale. All other selected methods of the modified factor analysis in the INDICES algorithm estimate the five light-absorbing components and of the protonation equilibrium. The number of light-absorbing species can be predicted from the index function values by finding the point where the slope of the index function changes, or by comparing values to the instrumental error  mAU. This is the common criterion for determining . Very low values of prove that reliable spectrophotometer and experimental techniques were used.

The dissociation constant and two molar absorptivities of silybin calculated for 39 wavelengths of 20 spectra constitute unknown regression parameters which are estimated and refined by SQUAD(84) or SPECFIT32 in the first run. The reliability of the parameter estimates may be tested with the use of following diagnostics.

The first diagnostic value indicates whether all of the parametric estimates and have physical meaning and reach realistic values, for silybin at C and ionic strength with SQUAD (and SPECFIT); that is, pK_a1 = 6.898(0.022) and 6.897(0.002), pK_a2 = 8.666(0.021), and 8.667(0.012); pK_a3 = 9.611(0.010) and 9.611(0.004); pK_a4 = 11.501(0.008) and 11.501(0.007). As the standard deviations of parameters log and of parameters are significantly smaller than their corresponding parameter estimates, all of the variously protonated species are statistically significant at a significance level of . The physical meaning of the dissociation constant, molar absorptivities, and stoichiometric indices is examined. The absolute values of give information about the last -contour of the hyperparaboloid in the neighbourhood of the pit, . For well-conditioned parameters, the last U-contour is a regular ellipsoid, and the standard deviations are reasonably low. High s values are found with ill-conditioned parameters and a “saucer’’-shaped pit. The relation should be met, where is equal to 3 at a 99.9% statistical probability level. The set of standard deviations of for the various wavelengths, , should have a Gaussian distribution; otherwise, erroneous estimates of are obtained. Figure 2(e) shows the estimated molar absorptivities of all of the variously protonated species , of silybin in dependence on wavelength.

The second diagnostic tests whether all of the calculated free concentrations of the five variously protonated species on the distribution diagram of the relative concentration expressed as a percentage have physical meaning, which proved to be the case (Figure 2(f)). Numerical values of protonation constants and molar absorption coefficients may not seem very interesting, and a graphical presentation is more illustrative; graphs are also efficient diagnostic tools in the search for the most probable chemical model. A distribution diagram makes it easier to judge the contributions of individual species to the total concentration quickly. Since the molar absorptivities will generally be in the range of 10³–10⁵ lmol^-1cm^-1, species present at less than ca. 0.1% relative concentration will affect the absorbance significantly only if their is extremely high. The diagram shows that the protonation equilibria of five species and are strongly overlapping and shows the estimated molar absorptivities of all the variously protonated species , and of silybin in dependence on wavelength. Some spectra overlap considerably here, and such cases may cause some resolution difficulties in a nonlinear regression approach.

The third diagnostic concerning the matrix of correlation coefficients around 0.9 proves that there is an absence of an interdependence of any pair of protonation constants of silybin except for species versus , and versus . The significant correlation of these two pairs may be explained by the too closely overlapping protonation constants, which concern overlapping equilibria.

The fourth diagnostic concerns the goodness-of-fit (Figure 2(d)). The goodness-of-fit achieved is easily seen by examination of the differences between the experimental and calculated values of absorbance, . Examination of the spectra and of the graph of the predicted absorbance response-surface through all the experimental points should reveal whether the results calculated are consistent and whether any gross experimental errors have been made in the measurement of the spectra. One of the most important statistics calculated is the standard deviation of absorbance, , calculated from a set of refined parameters at the termination of the minimization process. It is usually compared with the standard deviation of absorbance calculated by the INDICES program [30], , and if , or , the instrumental error of the spectrophotometer used, the fit is considered to be statistically acceptable. This proves that the value is equal to 0.30 mAU. Numerical values of statistical measures of the residuals now indicate very good fitness, and also prove that the minimum of the elliptic hyperparaboloid U was reached: the residual mean (SPECFIT gives ) proves that there is no bias or systematic error in the spectra fitting. The mean residual = 0.67 (SPECFIT gives 0.68) mAU and the residual standard deviation (SPECFIT gives 0.88) mAU have sufficiently low values. The standard deviation of absorbance after termination of the minimization process is always equal or lower than 1 mAU, and the proposal of a good chemical model and reliable parameter estimates is thus proven. The skewness (SPECFIT gives 0.35) is quite close to zero and proves a symmetric distribution of the residuals set while the kurtosis (SPECFIT gives 3.21) is reasonably close to 3 supporting a Gaussian distribution.

Although this statistical analysis of residuals gives the most rigorous test of the degree-of-fit, realistic empirical limits must be used. The statistical measures of all residuals e prove that the minimum of the elliptic hyperparaboloid is reached: the residual standard deviation always has sufficiently low values of lower than 1 mAU. The criteria of resolution used for the hypotheses were (1) a failure of the minimization process in a divergency or a cyclisation, (2) an examination of the physical meaning of the estimated parameters to see if they were both realistic and positive, and (3) the residuals should be randomly distributed about the predicted regression spectrum, and systematic departures from randomness were taken to indicate that either the chemical model or the parameter estimates were unsatisfactory.

To express changes of absorbance in the spectral set, the absorbance differences for the th wavelength of the th spectrum were calculated so that from the absorbance value of the spectrum measured at the actual pH, the absorbance value of the acidic form was subtracted. The absorbance difference was then divided with an actual instrumental standard deviation of spectrophotometer used, the resulting value representing the signal-to-error value (SER). The left part of Figure 3 brings the graph of the SER in dependence on wavelength in the measured range for silybin. When SER is larger than 10, a factor analysis is able to predict the correct number of light-absorbing components in equilibrium mixture. To prove that nonlinear regression is able to analyze such a data set, the residuals set was compared with the instrumental noise . If the ratio is of similar magnitude, that is, nearly equal to one, it means that sufficient curve fitting by nonlinear regression of spectra set was achieved, and that the minimization process found the minimum of residual-square-sum function . The right part of Figure 3 shows a comparison of the ratio in dependence on wavelength for all three drugs measured. From the figure, it is obvious that most of the residuals are of the same magnitude as the instrumental noise and therefore the regression found reliable estimations of the unknown parameters.

Figure 3: The plot of absorbance changes in the spectrum (a) means that the value of the absorbance difference for the th-wavelength of the th-spectrum is divided by the instrumental standard deviation and the resulting ratios are plotted as a function of wavelength λ for all absorbance matrix elements where means the limiting spectrum of the acid form of the drug measured. This ratio is compared to the limiting SER value for silybin to test if the absorbance changes are significantly larger than the instrumental noise . The plot of the ratio , that is, the ratio of residuals divided by the instrumental standard deviation as a function of wavelength λ for all residual matrix elements (b) for silybin tests if the residuals e are of the same magnitude as the instrumental noise .

As additivity of absorbance should be proven, the sum of all the absorbances of the variously protonated species at a given wavelength should be equal to the experimental absorbance. Resolution of each experimental spectrum into the spectra for the individual variously protonated species shows whether the experimental design, that is, the pH range was efficient. If for a particular pH value the spectrum consists of just a single component, further spectra for that pH or similar value would be redundant even though they should improve only the precision. In concentrations or pH ranges where more components contribute significantly to the spectrum, several spectra should be measured. Figure 4 proves that deconvolution of the spectra into absorbance increments for the individual species helps in planning future efficient experimentation, and that the selected pH values have been chosen efficiently. Spectrum deconvolution seems quite a useful tool in the proposal of a strategy for efficient experimentation. Such a spectrum provides sufficient information for a regression analysis which monitors at least two species in equilibrium, where none of them is a minor species. A minor species has a relative concentration in a distribution diagram of less than 5% of the total concentration of the basic component . When, on the other hand, only one species is prevalent in solution, the spectrum yields quite poor information into the regression analysis, and the parameter estimate is rather unsure and definitely not reliable enough.

Figure 4: Deconvolution of the experimental absorption spectrum of silybin for 39 wavelengths into spectra of the individual variously protonated species in solution (above) and the statistical analysis of the residuals (below) of each particular absorption spectrum for a selected value of pH equal to: (a) 12.2, (b) 11.8, (c) 10.2, (d) 9.0, (e) 7.4, and (f) 6.3. The charges of the species are omitted for the sake of simplicity (SQUAD, ORIGIN).

5. Conclusions

When drugs are very poorly soluble, then pH-spectrophotometric titration may be used with the non-linear regression of the absorbance-response-surface data instead of a potentiometric determination of dissociation constants. Regression diagnostics represent procedures for examination of the regression triplet (data, model, method) for the identification of (a) the data quality for a proposed model (b) the model quality for a given set of data, and (c) the fulfillment of all least-squares assumptions. The reliability of the dissociation constants of the drug silybin may be proven with goodness-of-fit tests of the absorption spectra measured at various pH. The dissociation constant pK_a was estimated by nonlinear regression of data at C with SQUAD (and SPECFIT), that is, pK_a1 = 6.898(0.022) and 6.897(0.002); pK_a2 = 8.666(0.021) and 8.667(0.012); pK_a3 = 9.611(0.010) and 9.611(0.004); pK_a4 = 11.501(0.008) and 11.501(0.007) at . Goodness-of-fit tests for various regression diagnostics enabled the reliability of the parameter estimates to be determined. Most indices always predict the correct number of components, and even the presence of a minor one when the signal-to-error ratio (SER) is higher than 10. The Wernimont-Kankare procedure in the factor analysis program INDICES performs reliable determination of the instrumental standard deviation of spectrophotometer used and correctly predicts the number of light-absorbing components present. This procedure also solves ill-defined problems with severe collinearity in the spectra, very small changes in spectra, and with overlapping equilibria.

Acknowledgments

The financial support of the IGA Grant Agency (Grant no. NR9055-4/2006 and NR9831-4/2008) and of the Czech Ministry of Education (Grants no. MSM0021627502 and MSM0021627501) is gratefully acknowledged.