A Modification of the Extended Hildebrand Approach to Predict the Solubility of Structurally Related Drugs in Solvent Mixtures

Abstract— A modification of the extended Hildebrand equation is proposed to estimate the solubility of an organic drug in solvent mixtures. The equation accurately reproduces the solubility of four sulphonamides in dioxane-water mixtures without requiring the heat of fusion of the solute. A single equation is obtained for predicting the solubility of related drugs using the solubilities of the drugs in the pure solvents, dioxane and water, and solute-solvent interaction terms consisting of the solubility parameter, δ₂, of the solute and the solubility parameter, δ₁, and basic partial solubility parameter, δ_1b, of the solvent mixture. By this procedure a single equation was obtained to estimate the solubilities of three xanthines in dioxane-water and another equation to obtain the solubilities of four sulphonamides. The equation obtained for sulphonamides is able to predict the experimental solubilities of two parent compounds, sulphasomidine and sulphathiazole, and the solubilities of a drug of different structure, p-hydroxybenzoic acid. This suggests that the intermolecular solute-solvent interaction of sulphonamides and p-hydroxybenzoic acid are similar. The results indicate that the solubility behaviour of drugs having different structures may be modelled using a common equation provided that they show similar solute-solvent interactions.     

Solubility of drugs in aqueous solutions. Part 4. Drug solubility by the dilute approximation

As in our previous publications in this journal [Int. J. Pharm. 258 (2003a) 193; Int. J. Pharm. 260 (2003b) 283; Int. J. Pharm. 267 (2003c) 121], this paper is concerned with the solubility of poorly soluble drugs in aqueous mixed solvents. In the previous publications, the solubilities of drugs were assumed to be low enough for the so-called infinite dilution approximation to be applicable. In contrast, in the present paper, the solubilities are considered to be finite and the dilute solution approximation is employed. As before, the fluctuation theory of solutions is used to express the derivatives of the activity coefficient of a solute in a ternary solution (dilute solute concentrations in a binary solvent) with respect to the concentrations of the solvent and cosolvent. The expressions obtained are combined with a theoretical equation for the activity coefficient of the solute. As a result, the activity coefficient of the solute was expressed through the activity coefficients of the solute at infinite dilution, solute mole fraction, some properties of the binary solvent (composition, molar volume and activity coefficients of the components) and parameters reflecting the nonidealities of binary species. The expression thus obtained was used to derive an equation for the solubility of poorly soluble drugs in aqueous binary solvents which was applied in two different ways. First, the nonideality parameters were considered as adjustable parameters, determined from experimental solubility data. Second, the obtained equation was used to correct the solubilities of drugs calculated via the infinite dilution approximation. It was shown that both procedures provide accurate correlations for the drug solubility.     

Solubility of drugs in aqueous solutions. Part 2: binary nonideal mixed solvent

As in a previous paper [Int. J. Pharm. 258 (2003) 193–201], the Kirkwood–Buff theory of solutions was employed to calculate the solubility of a solid in mixed solvents. Whereas in the former paper the binary solvent was assumed ideal, in the present one it was considered nonideal. A rigorous expression for the activity coefficient of a solute at infinite dilution in a mixed solvent [Int. J. Pharm. 258 (2003) 193–201] was used to obtain an equation for the solubility of a poorly soluble solid in a nonideal mixed solvent in terms of the solubilities of the solute in the individual solvents, the molar volumes of those solvents, and the activity coefficients of the components of the mixed solvent.

The Flory–Huggins and Wilson equations for the activity coefficients of the components of the mixed solvent were employed to correlate 32 experimental data sets regarding the solubility of drugs in aqueous mixed solvents. The results were compared with the models available in literature. It was found that the suggested equation can be used for an accurate and reliable correlation of the solubilities of drugs in aqueous mixed binary solvents. It provided slightly better results than the best literature models but has also the advantage of a theoretical basis.     

The Estimation of Solubility in Binary Solvents: Application of the Reduced 3-Suffix Solubility Equation to Ethanol–Water Mixtures

The reduced 3-suffix solubility equation (R3SSE) is applied to the characterization of solubility in the ethanol–water system. The data needed are the solubility of the compound in each of the pure solvents and at one ethanol–water composition. This composition has been estimated from solubility data to be 0.56 volume fraction of ethanol. The solubility obtained at this volume fraction is used to estimate the ternary solute–solvent interaction constant, C ₂. The R3SSE, with the C ₂ thus obtained, predicts the mixed solvent solubilities of the compounds tested, as accurately as that obtained from several volume fractions. The superiority of the R3SSE over two related equations—a simple second-degree polynomial equation and a simplified form of the R3SSE which neglects contributions to solubility from the solvent mixture—is also demonstrated for a number of solutes.     

Solubility of drugs in aqueous solutions. Part 3: multicomponent mixed solvent

The results obtained previously by Ruckenstein and Shulgin [Int. J. Pharm. 258 (2003a) 193; Int. J. Pharm. 260 (2003b) 283] via the fluctuation theory of solutions regarding the solubility of drugs in binary aqueous mixed solvents were extended in the present paper to multicomponent aqueous solvents. The multicomponent mixed solvent was considered to behave as an ideal solution and the solubility of the drug was assumed small enough to satisfy the infinite dilution approximation.An expression derived for the activity coefficient of a solid solute in a multicomponent solvent was used to obtain an equation for the solubility of a drug in terms of its solubilities in two subsystems of the multicomponent solvent and their molar volumes. Ultimately the solubility can be expressed in terms of those in binary or even in individual solvents and their molar volumes.The method was applied to the solubility of tioconazole and 19-Nor-1alpha,25-dihydrovitamin D(2) in several ternary and in a quaternary aqueous mixed solvents. The predicted solubilities were compared with experimental data and good agreement was found.     

Solubility properties in polymers and biological media. 2. The correlation and prediction of the solubilities of nonelectrolytes in biological tissues and fluids

Solubilities of a range of nonelectrolyte solutes in biological systems, such as blood, plasma, brain, lung, liver, kidney, muscle tissue, and human fat, are correlated and predicted through an equation that takes the form log Ltissue = c + w log Lwater + o log Loil, where L is the Ostwald solubility coefficient (or gas/liquid partition coefficient). The ratio of the constants o and w gives a measure of the "oiliness" of a given biological tissue or fluid. The strong possibility exists that, for many types of nonelectrolyte solutes, simple measurements of solubilities in water and oil (gas/liquid partition coefficients) will allow accurate predictions of solubilities in the above biological solvents, as well as tissue/blood partition coefficients. The solubility of rare gases and the inorganic gases H2, N2, CO, and O2 may be correlated through the simpler equation log Ltissue = l'RG + d', where l' and d' are constants that characterize the phase, and RG is a known parameter, obtained by normalizing and averaging solubilities over a range of solvent systems, that characterizes the solute. Both of the above equations allow prediction of L in biological solvents to within about 20%, which compares well with the precision of the experimental measurements.     

Solubility of drugs in aqueous solutions. Part 5. Thermodynamic consistency test for the solubility data

This paper is devoted to the verification of the quality of experimental data regarding the solubility of sparingly soluble solids, such as drugs, environmentally important substances, etc. in mixed solvents. A thermodynamic consistency test based on the Gibbs-Duhem equation for ternary mixtures is suggested. This test has the form of an equation, which connects the solubilities of the solid, and the activity coefficients of the constituents of the solute-free mixed solvent in two mixed solvents of close compositions. The experimental data regarding the solubility of sparingly soluble substances can be verified with the suggested test if accurate data for the activity coefficients of the constituents of the solute-free mixed solvent are available. The test was applied to a number of systems representing the solubilities of sparingly soluble substances in mixed solvents. First, the test was scrutinized for four nonaqueous systems for which accurate solubility data were available. Second, the suggested test was applied to a number of systems representing experimental data regarding the solubility of sparingly soluble substances in aqueous mixed solvents.     

Dependence of solute solubility parameters on solvent polarity

In nonpolar solvents a solute may self-associate through polar interactions, exposing its nonpolar surface to a solvent with a low solubility parameter, delta 1. In polar solvents a solute is solvated, presumably, by the polar groups of the solvent. This "chameleonic" effect results in different solubility parameters for a solute, depending on the polarity of the solvent. This report presents data for solute solubility parameters in solvents of variable polarity and gives suggestions for dealing with the chameleonic effect associated with solute-solvent interaction.     

The effect of temperature on the solubilities of testosterone propionate in low polarity solvents

The solubilities of testosterone propionate have been determined between 25 and 100° in 15 solvents and compared with theoretical values obtained from regular solution theory. Entropy considerations show that solvent-solute interactions occur in some solvents, increasing the solubility and resulting in deviations from regular solution behaviour. “Regular solutions” are obtained only in saturated hydrocarbon solvents, and the solubility is more accurately predicted as the temperature approaches the melting point, and as the molar volumes of solvent approach that of the solute.     

Solubility prediction of salicylic acid in water-ethanol-propylene glycol mixtures using the Jouyban-Acree model

To show the applicability of a solution model, i.e. the Jouyban-Acree model, for predicting the solubility of a solute in ternary solvent systems based on model constants computed using solubility data of the solute in binary solvent systems, the solubility of salicylic acid in water-ethanol, water-propylene glycol, ethanol-propylene glycol mixtures was determined. A minimum number of three data points from each binary system was used to calculate the binary interaction parameters of the model. Then the solubility in other binary solvent compositions and also in a number of ternary solvents was predicted, and the mean percentage deviation (MPD) was calculated as an accuracy criterion. The overall MPD (+/-SD) was 7.3 (+/-7.3)% and those of a similar predictive model was 15.7 (+/-11.5)%. The mean difference between the proposed and a previous model was statistically significant (paired t-test, p < 0.004).     

Hydrophobic and solvation effects on the solubility of hydroxysteroids in various solvents: Quantitative and qualitative assessment by application of the mobile order and disorder theory

Following the preliminary discussion outlining the relative difficulty of the experimental determination of solubilities versus the lack of general applicability and sound theoretical basis of most current predictive approaches for solubility, we look closely at the recently developed pure thermodynamic model for solubility in real solutions: that derived from mobile order and disorder theory. With successful estimates of the solubility of 62hydroxysteroids and related drugs in common organic solvents of differing polarities and H-bonding capacity, the model has proved to be a valuable tool in predicting the solubility of complex solutes such as polyfunctional drugs. Free of any adjusted regression coefficients and based on a limited number of readily available parameters, the proposed model is a time-saving alternative procedure to experimentation. By properly quantifying the enthalpic and entropic contributions involved in the overall solubility process, the model furthermore assesses the factors that determine solubility differences between steroids and solubility changes upon solvent properties. Therefore, the poor solubility of hydroxysteroids in aliphatic hydrocarbons results from the negative effects due to the change in non-specific cohesion forces upon mixing and due to steroid self-association in solution. In water, the low solubilities are mainly due to the large negative value of the hydrophobic effect which cannot be overcome by steroid–water functional group associations, i.e., the solvation effect. The relatively good solubility of hydroxysteroids in polar non-associated solvents (ketones, ethers, esters) and in alcohols is explained by the fact that, in both kinds of solvents, steroid self-association is rather well counterbalanced by the formation of a more or less important number of steroid–solvent interactions without being penalized by a strong negative hydrophobic effect in the case of alcohols. Some practical rules regarding how some parameters like the molar volume or the substitution may affect solubility are finally derived, which might help the pharmaceutical scientist to orient the choice of a solvent for liquid pharmaceutical forms. This revised version was published online in August 2006 with corrections to the Cover Date.     

Measurement and correlation of solute solubility in HFA-134a/ethanol systems

The goals of this study were to determine the solubility values of solid organic solutes in pure HFA-134a and in HFA-134a/ethanol cosolvent systems (0-20%, w/w), and to investigate the relationship between these solubilities and a solute's physico-chemical properties. A direct inject on-line HPLC method was used to determine the solubility of 21 solutes in HFA-134a/ethanol. The samples were allowed to equilibrate for at least 48h. The filtered sample was injected directly on an analytical HPLC column through a manual injector interface, and analyzed at an appropriate solute wavelength via HPLC. The solutes display diverse physico-chemical properties and yielded solubility values that ranged over four orders of magnitude. In general, a linear-linear solubility relationship was observed as the fraction of ethanol increased. The effects on solubilization ranged from 1.3 to 99.4 times when 20% (w/w) ethanol was introduced, relative to pure HFA-134a. A regression equation utilizing a solute's hydrogen bonding potential resulted in a significant correlation to the slope obtained from a linear model for solubility in HFA-134a with 0-20% (w/w) ethanol, and may be useful for pre-formulation studies.     

Solubilities of testosterone propionate in non-polar solvents at 100°

THE solubilities of the formate to valerate esters of testosterone in non-polar solvents at 25° were determined by James & Roberts (1968) who also compared them with ideal mole fraction solubilities (X₂), calculated from the equation, (Hildebrand & Scott 1962). ΔH^F is the heat of fusion of the solute and T_M the melting point. Changes in solubility as the homologous series is ascended were predicted by equation (1), but the individual experimental results did not agree with the calculated values. ΔH^F was calculated from the heat of fusion at the melting point, ΔH^F_M, by correcting for the differences in heat capacity of the solid and the supercooled liquid between T_M and T. The correction was estimated with a differential scanning calorimeter by extrapolating the liquid enthalpy line back to 25° and measuring the area between the extrapolation and the enthalpy line of the solid. The method was considered questionable, however, because it assumed that the enthalpy line of the supercooled liquid decreased linearly over the whole range of temperature. This theory is tested below by comparing the measured and calculated solubilities of testosterone propionate at a temperature just below its melting point, where the heat capacity correction is small and ΔH^F_M can be used for ΔH^F. The solvents examined by James & Roberts (1968) had smaller molar volumes than the testosterone esters, and it was suggested that the difference in molar volume between solute and solvent could prevent the random distribution assumed by regular solution theory. Prediction of solubility would thus improve as the molar volume of the solvent approached that of the solute. The test is applied below by determining the solubility of testosterone propionate in a range of solvents.     

Solubility Prediction of Satranidazole in Propylene Glycol-Water Mixtures Using Extended Hildebrand Solubility Approach

Extended Hildebrand solubility approach is used to estimate the solubility of satranidazole in binary solvent systems. The solubility of satranidazole in various propylene glycol-water mixtures was analyzed in terms of solute-solvent interactions using a modified version of Hildebrand-Scatchard treatment for regular solutions. The solubility equation employs term interaction energy (W) to replace the geometric mean (δ₁δ₂), where δ₁ and δ₂ are the cohesive energy densities for the solvent and solute, respectively. The new equation provides an accurate prediction of solubility once the interaction energy, W, is obtained. In this case, the energy term is regressed against a polynomial in δ₁ of the binary mixture. A quartic expression of W in terms of solvent solubility parameter was found for predicting the solubility of satranidazole in propylene glycol-water mixtures. The expression yields an error in mole fraction solubility of ~3.74%, a value approximating that of the experimentally determined solubility. The method has potential usefulness in preformulation and formulation studies during which solubility prediction is important for drug design.     

Prediction of solubility in nonideal multicomponent systems using the UNIFAC group contribution model

There is a need to identify suitable blends of solvents to dissolve drugs. Empirical approaches, such as trial-and-error and response surface, require several solubility measurements. In this study the UNIFAC method was used to predict solubility in highly nonideal multicomponent systems in which only the solute enthalpy of fusion and melting point must by measured. UNIFAC combines a group contribution approach with the UNIQUAC model for activity coefficients. Parameters characterizing interactions among constituent groups of a molecule have been previously determined from binary vapor pressure data. These tabulated group parameters are used to predict activity coefficients for newly synthesized compounds. These coefficients, together with the ideal solubility, permit a prediction of solubility. The solubility of 4-hexylresorcinol in ethyl acetate, ethyl myristate, and hexane mixtures was both measured and calculated using UNIFAC. The predicted solubilities were within 10% of the experimental solubilities for all but 3 of 21 mixtures. Since the method accounted for positive and negative deviations from ideality in a hydrogen-bonding system of molecules having different sizes, it shows great potential for use in pharmacy.     

Solubility of drugs in aqueous solutions. Part 1. Ideal mixed solvent approximation

The present paper deals with the application of the fluctuation theory of solutions to the solubility of poorly soluble drugs in aqueous mixed solvents. The fluctuation theory of ternary solutions is first used to derive an expression for the activity coefficient of a solute at infinite dilution in an ideal mixed solvent and, further, to obtain an equation for the solubility of a poorly soluble solid in an ideal mixed solvent. Finally, this equation is adapted to the solubility of poorly soluble drugs in aqueous mixed solvents by treating the molar volume of the mixed solvent as nonideal and including one adjustable parameter in its expression. The obtained expression was applied to 32 experimental data sets and the results were compared with the three parameter equations available in the literature.     

Solubility in binary solvent systems. IV. Prediction of naphthalene solubilities using the UNIFAC group contribution model

This work extends the UNIFAC group contribution method of solid-liquid equilibria to binary solvent mixtures, and compares its predictions to experimental solubilities for naphthalene in 16 different solvent mixtures. Deviations between experimental and calculated values are of the order of 10–20% for most solvent systems, and are comparable in magnitude to deviations noted in the pure solvents. The ability of the UNIFAC model to provide reasonable estimates of naphthalene solubilities based only on heat of fusion data and group contribution parameters suggests that the model may be useful in the area of drug design.