Aqueous solubility of drugs is one of the key factors in developing a new drug and the blending of different solvents is a common method to increase the solubility. Apart from experimental determinations of a solute solubility in water-cosolvent mixtures, many mathematical models have been established to describe solute solubility in mixed solvents. [1][2][3][4][5][6][7][8][9] Some of these models are theoretical, while others are semitheoretical or empirical. While the empirical ones are mainly used to correlate between experimental solubilities and independent variables such as the volume fraction of the cosolvent, the theoretical ones can improve the understanding of solubility behaviour for drugs in mixed solvents.It has been found that the solute solubility in mixed solvents can be mathematically represented by a single equation. There, however, is a number of equations that can be considered which usually produce comparable results. The question is then posed-is there a mathematical difference between these models? To address this point, it has been demonstrated in this work that all the suggested cosolvency models could be made equivalent by using algebraic manipulations. Based on these manipulations, a unified cosolvency model has been proposed in the present study.
Theoretical TreatmentThe log-linear relationship, 2) extended Hildbrand solubility approach, 1) excess free energy equations, 3) the simplest form of the mixture response surface method, 4) and the combined nearly ideal binary solvent/RedlichKister (CNIBS/R-K) model 5) have been converted to a general single model, GSM.8) GSM correlates the logarithm of a solute solubility as a polynomial function of cosolvent's volume fraction as:Where X m is the mole fraction solubility of the solute, f 1 is volume fraction of cosolvent in the absence of the solute and M 0 -M 3 are the model constants. Before the unified cosolvency model derived in this study is discussed, different non-linear mathematical models on solubility was first reviewed. Mixture Response Surface Model Statistically based mixture response surface methods, MRS, 4) have also been proposed for correlative purposes and these models are as follows:in which b 1 -b 3 , b 1 Ј-b 4 Ј and b 1 Љ-b 5 Љ are the model's parameters and f 1 Ј and f 2 Ј, are given by f 1 Јϭ0.96f 1 ϩ0.02 and f 2 Јϭ0.96f 2 ϩ0.02 in which f 2 is volume fraction of water.
4)
Modified Wilson Model The modified Wilson model (MWM), is another possibility which is shown below:where X 1 and X 2 denote the mole fraction solubility in neat cosolvent and water, respectively.5) It was shown that a simplified form of the modified Wilson model, SMW, 9) is able to calculate solute solubility in water-cosolvent mixtures more accurate than MWM, although this simplification is not successful in the case of solubility prediction in non-aqueous binary solvents.5) Thus the SMW is:where L 12 adj , L 21 adj , l 12 adj and l 21 adj are adjustable parameters of the models which can be evaluated via a nonlinear least squares analysis. Organic...
