Advancing amorphous solid dispersions through empirical and hybrid modeling of drug–polymer solubility and miscibility: A case study using Ibuprofen

Abstract

This study investigates the solubility and miscibility of ibuprofen (IBU) with four pharmaceutical polymers, KOLVA64®, KOL17PF®, HPMCAS, and Eudragit® EPO, using a combination of empirical and hybrid modeling approaches, supported by differential scanning calorimetry (DSC) experiments. Traditional group contribution methods based on Hildebrand and Hansen solubility parameters (Fedors, Hoftyzer–van Krevelen, and Just–Breitkreutz) showed variability in solubility predictions but consistently classified all polymer–API blends as miscible (Δδ < 7 MPa^½). Bagley plots reinforced these findings, although borderline miscibility was indicated for HPMCAS and EPO depending on the method used. A novel attempt to derive the Flory–Huggins (FH) interaction parameter (χ) from solubility parameters at near-melting temperatures showed poor agreement with experimental data, underscoring the limitations of such extrapolations and the semi-empirical nature of the FH model.

Phase diagrams were constructed from DSC-based melting point depression data using three modeling strategies: FH theory, the empirical approach by Kyeremateng (with two fitting methods), and the perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state, both in pure predictions and with fitted binary interaction parameters (k_ij). The glass transition temperature (T_g) of the mixtures was modeled using the Gordon–Taylor and Kwei equations. All models provided a consistent polymer ranking based on their solubilizing capacity, with KOL17PF as the most compatible and HPMCAS as the least. Demixing zones (liquid-liquid equilibrium – LLE) predicted by FH and PC-SAFT models suggest that for HPMCAS-based ASDs only very low drug loadings (< 5 % w/w) could potentially be stable at room temperature. In contrast, higher drug loadings (> 10 % w/w) fall under a meta-stable zone with the other polymers, making them better candidates for IBU formulation. HPMCAS also exhibited consistently prediction errors across all T_g models, (AARD ∼4.5 %), indicating poorer agreement with experimental data. By integrating empirical and hybrid modeling approaches, this study highlights the strengths and limitations of commonly used solubility prediction methods and advocates for a shift toward a harmonized framework.

Introduction

The current pipeline for API development is dominated by poorly soluble compounds, creating a demand for formulation strategies that improve dissolution and bioavailability in oral dosage forms. An effective approach to improve bioavailability is via amorphization, which enhances the active pharmaceutical ingredient’s (API) apparent solubility converting it into a higher-energy amorphous state (Li et al., 2019; Mantas et al., 2019). However, despite its advantages in performance, the amorphous form tends to be thermodynamically unstable, naturally reverting to the lower-energy crystalline state (Yu, 2001).

In this context, using polymeric matrices to form amorphous solid dispersions (ASDs) is an effective strategy to enhance both stability and supersaturation behavior during dissolution (Vasconcelos et al., 2016). To ensure long-term stability, it is crucial to understand the miscibility between the active pharmaceutical ingredient (API) and polymer, which is primarily driven by non-covalent interactions, such as van der Waals forces, polar interactions, and hydrogen bonding.

At this point, it is important to differentiate between solubility and miscibility. Solubility refers to the maximum concentration of a solute (e.g., API) that can dissolve in a solvent (typically a polymer in this context) under specific conditions, such as temperature and pressure. In a soluble system, equilibrium is achieved when the solid phase of the API and its dissolved form in the solvent reach a balance (solid-liquid equilibrium, SLE), resulting in a solution where the API is uniformly distributed within the solid polymer matrix.

Miscibility, on the other hand, is the ability of two substances to mix and form a single homogeneous phase, especially in the case of liquids or amorphous materials (liquid-liquid equilibrium, LLE). For an API and a polymer, miscibility refers to the extent to which they can integrate at the molecular level to create a homogeneous amorphous blend. This can be assessed using techniques like differential scanning calorimetry (DSC), where a single glass transition temperature (Tg) is observed.

Developing ASD-based formulations is a complex process that requires a comprehensive consideration of both formulation and process variables to ensure the production of safe and effective medicines. Quality by Design (QbD), an approach endorsed by the International Council for Harmonization (ICH) and the Food and Drug Administration (FDA), emphasizes a thorough product understanding. This is achieved by defining a Quality Target Product Profile (QTPP) and linking it to Critical Quality Attributes (CQAs), Critical Process Parameters (CPPs), and Critical Material Attributes (CMAs) (ICH Q8(R2), 2009). The Design of Experiments (DoE) methodology is used to conduct optimized experimental studies, allowing for the evaluation of the main effects and interactions among variables, and ultimately building empirical, data-driven models (Butreddy et al., 2021; Triboandas et al., 2024; Bezerra et al., 2025). Despite the explicit benefits, product development within the QbD framework is experimentally driven, thus multiple empirical data must be generated a priori to model building, often constrained by resources (e.g. expensive materials and time) in the industrial setting.

Recent advancements in digitalization, real-time data extraction in pharmaceutical manufacturing, and the transition from batch to continuous processes (Rehrl et al., 2018) have paved the way for Quality by Digital Design (QbDD) as a more efficient approach to product development. By combining physics-based models grounded in thermodynamics, material science, and fluid dynamics with data-driven models, QbDD enables proactive process modeling. This integration reduces the reliance on extensive empirical data, resulting in shorter development timelines and lower costs (Mustoe, 2025). Additionally, QbDD enhances process understanding and expands the flexibility of the design space (DS), leading to more robust and adaptable manufacturing processes. This paper compares different types of models, including empirical and hybrid, to assess the solubility and miscibility of ibuprofen in four polymer matrices. The goal is to shift from purely data-driven empirical models toward more physics-informed, mechanistic, and hybrid modeling approaches.

A range of models has been evaluated to predict the solubility and miscibility of active pharmaceutical ingredients (APIs) within polymer matrices in amorphous solid dispersions (ASDs). These models vary in complexity and theoretical basis, from purely empirical equations derived from experimental data to advanced physics-based models based in molecular theory. Common approaches include the Flory–Huggins theory for polymer–drug miscibility, group contribution methods like Fedors, van-Krevelen, and Just-Breitkreutz methods, and equations of state such as PC-SAFT for phase behavior prediction. Additionally, hybrid models, such as Gordon–Taylor and Kwei equations, are widely used to estimate glass transition temperatures (Tg), which serve as indicators of miscibility and physical stability. The choice of model depends on the available data, the nature of the drug-polymer system, and the desired balance between accuracy and computational efficiency.

The use of Hildebrand solubility parameter is a concept used to estimate the cohesive energy density (CED) of a substance, which is a measure of the strength of intermolecular forces in that substance. However, it does not separate the different types of interactions (polar, hydrogen bonding, etc.), which may limit its ability to model more complex systems. So, the Hansen Solubility Parameter (HSP) is a more detailed and versatile approach than the Hildebrand method, especially in systems where interactions such as polarity and hydrogen bonding are significant.

Based on this principle, group contribution (GC) methods, such as those developed by Fedors (1974) and van Krevelen (1976), calculate solubility parameters by considering the additive contributions of functional groups. The HVK method, also known as the Hoftyzer-van Krevelen method, is a group contribution method used to estimate the Hildebrand solubility parameter (δ), which describes the cohesive energy density (CED) of a substance. The Just-Breitkreutz (JB) method extended the application of the GC methods specifically to the pharmaceutical field, emphasizing solid-state solubility determination rather than liquid-state (Just et al., 2013). This approach is widely used in literature to screen for different drugs and polymers (Kitak et al., 2015; Petříková et al., 2021). For this work, the Fedors, van-Krevelen, and Just-Breitkreutz methods were initially employed to screen a range of polymers with the potential to be a suitable matrix to solubilize ibuprofen. This approach enabled rapid, preliminary screening and served as a hybrid strategy that complemented and informed subsequent verification using thermodynamic properties to describe polymer-solvent phase behavior.

API-polymer solubility can be experimentally determined using differential scanning calorimetry-based methods by examining recrystallization and dissolution behaviors of APIs in polymer matrices (Mahieu et al., 2013; Mathers et al., 2021). How well an active pharmaceutical ingredient dissolves in a polymer is a crucial factor in designing stable amorphous solid drug formulations.

Nishi and Wang (1975) were the first to report melting point depression (MPD) in amorphous-crystalline polymer blends and to link it to miscibility. This phenomenon has since been investigated with poorly soluble APIs and amphiphilic polymers, showing temperature depressions that can vary based on API-polymer interactions, sample preparation, and DSC methodology (Weuts et al., 2004; Jijun et al., 2011; Iemtsev et al., 2020). The end-set evaluation of enthalpic peaks provides the most consistent data for melting analysis, though high polymer content can increase viscosity and thus incorrect MPD values. Based on Flory-Huggins (FH) theory, the obtained near-melting interaction parameter between drug and polymer can be extrapolated to lower temperatures and compositions, enabling the construction of Gibbs free energy of mixing and phase diagrams. While FH theory remains one of the most widely used models in the field of amorphous solid dispersion (ASD) development, it has several limitations. Originally developed to describe the phase behavior of polymers or polymer mixtures in solution, it may not fully capture the complexity of phase behavior in ASDs. Specifically, the simple FH interaction parameter (χ) alone may not accurately represent the interactions within these molecular mixtures, as it does not account for specific molecular forces such as hydrogen bonding, ionic interactions, or van der Waals forces, which play crucial roles in ASDs (Anderson, 2018a, b).

The glass transition temperature (Tg) is the temperature at which an amorphous material transitions from a hard, glassy state to a more rubbery, flexible state which can be measure by differential scanning calorimetry (DSC). This transition is sensitive to the molecular interactions in the system. In a miscible system (where the drug and polymer are well-integrated at the molecular level), the Tg will typically shift compared to the individual components. A single Tg observed for ASD suggests that the drug and polymer are miscible and have formed a homogeneous mixture. If the Tg of the ASD is lower or higher than the Tg of the pure components (drug or polymer), that can provide insights into the strength and nature of interactions between the drug and polymer. For example, lower Tg in the ASD compared to the pure polymer may indicate good interaction between the drug and polymer, which helps to reduce rigidity and improve dissolution properties. A higher glass transition temperature (Tg) may indicate that the drug is more tightly “locked” within the polymer matrix, potentially influencing its release profile. The physical stability of an amorphous solid dispersion (ASD) is closely related to its Tg. A higher Tg generally suggests greater stability, as reduced molecular mobility lowers the risk of crystallization. In contrast, a lower Tg may imply increase susceptibility to phase separation or crystallization over time, which can compromise drug performance. A system Tg can be experimentally measured and theoretically calculated using various models and approaches depending on the system under study and the available data. For simpler systems, equations like Gordon-Taylor and Kwei models may suffice, while more complex systems might require simulation-based methods such as molecular dynamics. These methods are particularly useful when experimental data is difficult or impossible to obtain, as they offer predictions based on molecular-level interactions, free volume, and thermodynamic principles.

In recent years, the Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) has gained attention as a more advanced model for API-polymer phase behavior modeling (Prudic et al., 2014; Lehmkemper et al., 2017b; Luebbert et al., 2017). PC-SAFT is a thermodynamic model that calculates Helmholtz energy by summing contributions from molecular repulsion, dispersion and association. Although PC-SAFT requires additional initial effort to define the complex molecular parameters, they more accurately represent chemical interactions and molecular size differences in API-polymer compositions.

Constructing a thermodynamic phase diagram is a valuable approach for analyzing the solubility and miscibility of an API in a polymer. This enables, for instance, the identification of critical formulation parameters for hot melt extrusion (HME) and can be used to predict stability through mathematical modeling. The solubility curve (solid-liquid equilibrium, SLE) represents the API’s equilibrium solubility in the polymer and defines the maximum stable drug loading at a given temperature. Beyond the solubility limit, liquid-liquid phase separation (LLPS), also called amorphous-amorphous phase separation (AAPS), may occur, explained by the binodal and spinodal curves (Qian et al., 2021). The binodal curve delineates the phase boundary between a homogeneous single-phase region and a two-phase region, where one phase is drug-rich and the other is polymer-rich. The area between the binodal and spinodal curves represents the metastable region, where phase separation is thermodynamically favored but kinetically hindered. In this region, the system remains temporarily stable unless triggered by external factors such as temperature and humidity changes (Klueppelberg et al., 2023; Moseson et al., 2024). The spinodal curve defines the boundary beyond which spontaneous phase separation occurs.

In this work, ibuprofen (IBU) was selected as model API, a class II API according to the biopharmaceutical classification system (BCS) due to its low aqueous solubility and high permeability. Because of its inherent low melting temperature (Tm) and glass-transition temperature (Tg) resulting in high mobility and low configurational entropy, IBU is deemed class IV in the recently proposed amorphous classification system (ACS), representing a challenging API for amorphous–based formulations, (Zhou et al., 2019). Binary blends containing Kollidon® VA64, Kollidon® 17PF, HPMCAS, and Eudragit® EPO were systematically investigated to evaluate the solubility and miscibility of these systems using empirical and hybrid models.

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Materials

IBU, KOL VA64 (Mw 65 kDa) and KOL 17PF (Mw 11 kDa) were kindly supplied by BASF (Ludwigshafen, Germany). HPMCAS (AQOAT- AS-LMP) was donated by Harke Chemlink (UK) and EPO (Eudragit EPO®) by Evonik Pharma Polymers (Darmstadt, Germany). API and polymers physicochemical properties are represented on Table 1 and their chemical structure in Fig. 1. The true density (ρ), glass transition temperature (Tg), melting transition temperature (Tm), molar enthalpy of fusion (ΔfusH) and change in heat capacity (ΔCp) were determined experimentally.

Matheus de Castro, Ana Sara Cordeiro, Mingzhong Li, Christian Lübbert, Catherine McColl, Jatin Khurana, Mark Evans, Walkiria S. Schlindwein, Advancing amorphous solid dispersions through empirical and hybrid modeling of drug–polymer solubility and miscibility: A case study using Ibuprofen, International Journal of Pharmaceutics: X, Volume 10, 2025, 100373, ISSN 2590-1567, https://doi.org/10.1016/j.ijpx.2025.100373.

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