Background
The thermodynamic stability of a solid amorphous dispersion can only be ensured by molecularly dispersing (dissolving) the drug in the polymer below its saturation solubility. Therefore, the prediction of drug-polymer solubility at room temperature is of great academic and industrial interest. By assuming that a drug behaves as a solvent for an amorphous polymer it is possible to describe the solubility of a drug in a polymer using the Flory-Huggins model:
\begin{equation}\frac{\Delta H_m}{R}\left(\frac{1}{T_m}-\frac{1}{T}\right)=\ln(\nu_{\text{drug}})+\left(1-\frac{1}{\lambda}\right)\cdot(1-\nu_{\text{drug}})+\chi(1-\nu_{\text{drug}})^2 \label{eq:flory}\end{equation}
where \(\Delta H_m\) and \(T_m\) are the enthalpy of fusion and melting temperature for the pure drug respectively, \(R\) is the gas constant, \(\lambda\) is the molar volume ratio of the polymer to the drug, \(\chi\) is the Flory-Huggins interaction parameter. \(T\) is the temperature at which the measurement is made and \(\nu_{\text{drug}}\) is the volume fraction of the drug in the polymer.
However, as the majority of pharmaceutically relevant drugs and polymers are solid, measuring the equilibrium solubility at room temperature is unfeasible. Therefore, several differential scanning calorimetry (DSC) protocols have been proposed based on determination of equilibrium thermodynamics at elevated temperature [1]. In a recently proposed protocol, referred to as the recrystallization method, a supersaturated amorphous solid dispersion is annealed at different temperatures above its glass transition temperature (\(T_g\)) to recrystallize excess drug and reach equilibrium solubility. The solubility after annealing is then derived from the \(T_g\) of the annealed material using the Gordon-Taylor equation and extrapolated using the Flory-Huggins model to predict the solubility at room temperature [2]. Other noteworthy approaches include the melting point depression method [3] and an estimation based on the solubility in a liquid low molecular weight analogue of the polymer [4].
Introducing a confidence assessment
In order to assess the confidence of a prediction it is important to realize which variable is subject to experimental noise. Using the recrystallization method, the temperature (\(T\)) is the variable under control and can be regarded as free of error whereas the volume fraction of the drug (\(\nu_{\text{drug}}\)) is subject to error. The optimal estimate for the Flory-Huggins interaction parameter (\(\chi\)) is thus found by minimizing the sum of squares of the residuals between the observed and predicted \(\nu_{\text{drug}}\). Consequently, the confidence of the prediction is expressed as a 95% prediction interval that is dependent on both the inter-replicate variance (reproducibility) and intra-replicate variance (fit to the Flory-Huggins model) [1].
Refining the experimental protocol
The introduction of a formal statistical analysis method enables a comparison of different solubility predictions. In the original method, a milling procedure was used to prepare the supersaturated amorphous solid dispersion [2]. However, as the physical properties and recrystallization behavior of an amorphous material have been reported to be affected by the preparation technique [5], the influence of different preparation techniques (ball milling, spray drying and film casting) on the solubility prediction was investigated [6].
As can be seen in Figure 1, the predicted solubility from the ball milling method is not consistent with those predicted from spray drying and film casting methods, indicating fundamental differences between the three preparation techniques. The most narrow prediction interval was found for spray drying, indicating a combination of a good fit to the Flory-Huggins model and reproducibility of the measurements. The prediction interval for ball milling was wider than that for spray drying, but still relatively narrow. However, as ball milling provided the best reproducibility of the three techniques, the broader prediction interval was a result of a poor fit to the model. In contrast, the broad prediction interval for film casting was a consequence of a poorer reproducibility than for the other two techniques. As previous studies suggest that amorphous mixtures produced by ball milling may still be heterogeneous at the molecular level, the process involved in reaching the equilibrium solubility is most likely driven by dissolution rather than the intentional recrystallization. Therefore, it is recommended that techniques such as spray drying or potentially film casting should be used to prepare the supersaturated amorphous solid dispersions when using this method [6].
Influence of polymer molecular weight on the prediction
One of the most commonly used carriers for amorphous solid dispersions is polyvinylpyrrolidone (PVP). However, despite the widespread use of PVP, the influence of polymer molecular weight (chain length) on drug-polymer solubility has not been sufficiently elucidated. Only a few studies have addressed this issue and none have supported the theoretical considerations and predictions with relevant experimental data [4]. Therefore, the influence of polymer molecular weight on the predicted solubility using spray drying as preparation technique was investigated [7].
As can be seen from Figure 2, it was found that the predicted solubility was independent of the molecular weight of the polymer. This indicates that the solubility of a given drug-polymer system is determined by the strength of the drug-polymer interactions rather than the molecular weight of the polymer. Therefore, during the initial screenings for drug solubility, only one representative molecular weight per polymer is needed. However, it is important to emphasize that this does not mean that the influence of polymer molecular weight on other important factors such as dissolution rate, physical stability and crystallization inhibition should not be considered in the polymer selection process [7].
Conclusions
Knowledge about drug-polymer solubility is an important factor in the development of amorphous solid dispersions as it can ensure the thermodynamic stability of the formulation. Previously, the prediction of drug-polymer solubility was based on a central estimate without an assessment of the confidence of the prediction. By introducing a new dimension to the field in form of formal statistical analysis, we enabled the possibility of comparing different solubility predictions. This can be used advantageously to screen for polymers suitable for amorphous solid dispersions.
Author affiliations
Matthias Manne Knopp and Rene Holm are affiliated with the Department of Pharmaceutical Science and CMC Biologics at H. Lundbeck A/S and Thomas Rades is a Professor in the Department of Pharmacy at the University of Copenhagen. All work has been conducted at H. Lundbeck A/S.
References
PSSRC Facilities
The research group of Prof. Thomas Rades in Copenhagen focusses on the development of amorphous solid dosage forms including amorphous solid dispersions and co-amorphous formulations. Several methods can be applied to obtain the amorphous form and therefore, the group has extensive knowledge and experience in techniques such as ball milling, spray drying and melt-quenching. In order to assess the stability and physicochemical properties of the formulations, the facilities include XRPD, a range of spectroscopic and microscopic techniques, and state of the art thermal analysis equipment including the DSC, TGA, DMA, and IMC.
