A Quantitative Model to Evaluate Solubility Relationship of Polymorphs from Their Thermal Properties

Abstract

The objective of the study is to develop a model to estimate the solubility ratio of two polymorphic forms based on the calculation of the free energy difference of two forms at any temperature. This model can be used for compounds with low solubility (a few mole percent) in which infinite dilution can be approximated. The model is derived using the melting temperature and heat of fusion for apparent monotropic systems, and the solid-solid transition temperature and heat of transition for apparent enantiotropic systems. A rigorous derivation also requires heat capacity (Cp) measurement of liquid and two solid forms. This model is validated by collecting thermal properties of polymorphs for several drugs using conventional or modulated differential scanning calorimetry. From these properties the solubility ratio of two polymorphs is evaluated using the model and compared with the experimental value at different temperatures. The predicted values using the full model agree well with the experimental ones. For the purpose of easy measurement, working equations without Cp terms are also applied. Ignoring Cp may result in an error of 10% or less, suggesting that the working equation is applicable in practice. Additional error may be generated for the apparent enantiotropic systems due to the inconsistency between the observed solid-solid transition temperature and the true thermodynamic transition temperature. This inconsistency allows the predicted solubility ratios (low melt/high melt) to be smaller. Therefore, a correction factor of 1.1 is recommended to reduce the error when the working equation is used to estimate the solubility ratio of an enantiotropic system. The study of the free energy changes of two crystalline forms of a drug allows for the development of a model that successfully predicts the solubility ratio at any temperature from their thermal properties. This model provides a thermodynamic foundation as to how the free energy difference of two polymorphs is reflected by their equilibrium solubilities. It also provides a quick and practical way of evaluating the relative solubility of two polymorphs from single differential scanning calorimetry runs.