1. Introduction
In a pharmaceutical co-crystal (CC), the active pharmaceutical ingredient (API) and the coformer (CF) are arranged in a common crystal lattice [
1]. The CC might have significantly improved physical properties compared to the pure API, such as increased aqueous solubility or stability [
2,
3]. Due to the change in the regulatory framework of authorities such as the Food and Drug Administration (FDA) [
4] regarding the intellectual property and impact on the product-lifecycle-management, CCs have gained increasing interest in recent years [
5]. An important factor for the approval of CCs is their stability against environmental changes or stress conditions [
6,
7,
8].
CCs have been reported to transform to API or CF solid phases, hydrates, and polymorphs upon exposure to different levels of relative humidity (RH) [
8,
9,
10]. The acetylsalicylic acid/acetamide CC was shown to undergo conversions at RH conditions as low as 75% RH already after four days [
11]. Nevertheless, other CCs, e.g., the theophylline/oxalic acid CC are stable even at 98% RH for 49 days, although the pure theophylline would transform to its hydrate within one day at that RH [
9]. Thakuria et al. [
6] found that the transformation of the caffeine/glutaric acid CC to the caffeine hydrate at 98% RH is a stepwise process. First, the CC deliquesces (liquid phase evolves with the dissolved CC components) and afterwards, the caffeine hydrate crystallizes from the liquid phase. Thus, a critical quality attribute with respect to stability of the CC is the deliquescence RH (DRH).
To prevent time-consuming and expensive stability measurements, Eddleston et al. [
12] developed an experimental evaluation of the CC stability at high RH conditions based on the addition of liquid water to a stoichiometric mixture of the CC components. With this method they investigated, if the phase behavior of the CC in water is congruent or incongruent [
12]. The mechanism by which moisture sorption leads to CC formation was determined by Jayasankar et al. [
13]. It was shown that CCs of carbamazepine/nicotinamide, carbamazepine/saccharin, and caffeine or theophylline with dicarboxilyc acid coformers were generated when solid mixtures of CC components deliquesce. Dissolution of CC components in the deliquesced solution led to supersaturation with respect to CCs and precipitation, even when the phase behavior of these CCs in water was incongruent. CC formation was enhanced by deliquescent formulation additives, such as sugars (fructose and sucrose), and by moisture sorption by polymers (PVP), Good et al. [
14]. The role of critical water activity (equivalent to the critical RH) and CF activity on CC stability has been reported for the case of CFs and additives that modulate water activity, Jayasankar et al. [
15]. CC phase purity was found to be important not only due to its influence on supersaturation, but trace levels of water-soluble CFs can significantly change the critical water activity of the sample and therewith the CC stability [
15].
The prediction of critical water activities such as DRH was already investigated for single crystals and crystal mixtures in a previous work [
16]. This work proposes a thermodynamic in-silico approach to predict the stability of CCs against conversions as a function of moisture uptake. This allows determining the critical RH up to which the CC is stable and predicting the influence of trace level impurities of API, CF, or additional excipients on this critical RH.
2. Theory
The thermodynamic equilibria between the solid phase (CC), the vapor phase (RH), and the liquid phase (present upon deliquescence and CC instability and transformation) must be considered to understand the RH influence on CCs. The solid–liquid equilibrium must be calculated for each present crystalline component (e.g., CC and CF) simultaneously. The vapor–liquid equilibrium must be solved for water being the only component present in the vapor phase.
2.1. Solid–Liquid Equilibrium
The equilibrium between a solid crystal (API or CF) and a liquid is fulfilled when the chemical potential of the components is equal in the solid and in the liquid phase. This is expressed by the following equation:
The resulting mole-fraction solubility of the component i () is calculated from the melting properties (melting temperature , melting enthalpy , and heat capacity difference between solid and liquid component i ) and from the activity coefficient . is the ideal gas constant and is the system temperature. explicitly considers interactions such as van der Waals forces and hydrogen-bond formation between all components present in the liquid phase. At solid–liquid equilibrium conditions, the thermodynamic activity of a given component i only depends on .
The CC formation is considered similarly to a reaction and the solid–liquid equilibrium is described by the solubility product, which is derived considering the dissolution process of the CC in a solvent [
17,
18]:
The mole fractions of the API and the CF indicating the CC solubility are calculated using the activity coefficients of each component and the stoichiometry of the CC (, ). The CC-specific and solvent-independent CC solubility product () was determined in this work via fitting to the solubility of the CC in one solvent. Afterwards, it was used to predict the solubility in water.
The solid–liquid equilibrium for a hydrate (of the API or CF) is again calculated using the solubility product, which is derived from the dissolution process in water [
16,
19]:
The resulting solubility of the hydrate
depends on the water mole fraction
, the activity coefficients of water and of the hydrate-forming component
and the stoichiometry of the hydrate (
,
). The solubility product of the hydrate
is determined by the following equation:
The solubility product of the hydrate depends on the melting properties of the component i (API or CF) and the hydrate/anhydrate transition properties ( is the temperature, above which the hydrate is not thermodynamically stable anymore, is the enthalpy upon dehydration, and is the transition water activity at which hydrate and anhydrate are in equilibrium).
The solid–liquid equilibrium of a CC hydrate is calculated using a solubility product representing the dissolution process in water:
The solubility product was determined in this work from the solubility of the CC hydrate in water.
2.2. Vapor-Liquid Equilibrium
For the calculation of the vapor–liquid equilibrium it is assumed that the vapor phase behaves like an ideal gas:
Therefore, the water activity in the liquid phase equals RH. The RH is defined as the partial pressure of water divided by the saturation vapor pressure of water . The equilibrium mole fraction of water in the liquid phase is determined by its activity coefficient which depends on all components present in the liquid as well as on their concentrations.
2.3. Deliquescence
DRH is used to determine the stability of CCs and CC hydrates. It is assumed that the CC formation or transformation into its solid components occurs via a dissolution mediated process in the deliquesced liquid phase [
13,
14,
15,
20]. The DRH is calculated by solving the solid–liquid–vapor equilibrium of the present components. The solid–liquid equilibrium must be solved simultaneously for each crystalline component present (Equations (1), (2), (3), or (5)), the vapor–liquid equilibrium (Equation (6)) must be fulfilled at the composition of the saturated liquid phase regarding all crystalline components.
2.4. PC-SAFT
All interactions between the components in the liquid phase are considered by the activity coefficients. The activity coefficients were calculated in this work from the residual Helmholtz energy
obtained from the Perturbed-chain statistical associating fluid theory (PC-SAFT) [
21,
22]. In this work,
is summed up from three contributions:
considers the hard-chain repulsions,
considers the attractions like van der Waals forces, and
considers the hydrogen bond formation:
Every molecule is described as a chain of
segments with the segment diameter
, the dispersion-energy parameter
, the association-energy parameter
, and the association volume
. Berthelot–Lorentz [
23,
24] mixing rules were used to obtain the segment diameter and dispersion energy in mixtures of components
i and
j:
The dispersion energy may be corrected by a binary interaction parameter
to better describe experimental data of the binary mixture. The
may be temperature dependent with a slope of
and a reference
at 0 K.
The mixing rules of Wolbach and Sandler [
25] were used to determine the association energy and association volume of mixtures:
3. Phase Diagrams
The stability regions of a CC can be determined from phase diagrams. CC phase diagrams can be categorized into either congruently dissolving or incongruently dissolving CCs. Congruently dissolving CCs can establish a thermodynamically stable solution with stoichiometric composition. Contrary incongruently dissolving CCs cannot establish a thermodynamically stable solution with CC stoichiometric composition. The schematic phase diagrams for a congruently dissolving CC are shown in
Figure 1. This system shows an almost symmetrical phase behavior (API solubility similar to CF solubility). The CC stoichiometry line crosses CC solubility line, which is the definition of a congruently dissolving CC. The API solubility line (separating region API and L), CC solubility line (separating region CC and L), and the CF solubility line (separating CF and L) separate the region of an unsaturated liquid phase (L) from the regions where crystals are present (below the solubility lines). Iso-RH lines are shown in the phase diagram to illustrate the influence of RH on the phase behavior. The RH (and therefore the water activity according to Equation (6)) along these lines is constant. DRH
CC is the RH above which a pure CC will start to deliquesce, and this value is determined by the water activity at the CC solubility line at CC stoichiometric composition (solid–liquid–vapor equilibrium). The DRH
CC/API is determined by the water activity at the eutectic point of the CC and the API (intersection of the CC solubility line and the API solubility line).
The DRH
CC/CF is determined by the water activity of the eutectic point of the CC and the CF (intersection of the CC solubility line and the CF solubility line). The iso-RH lines are only experimentally accessible in the liquid region (L) as both, vapor and liquid are required for the water to be in vapor–liquid equilibrium. Nevertheless, the iso-RH lines are also drawn in the metastable regions to improve their visibility. CC storage at DRH
CC will lead to the formation of a liquid phase with the composition on the solubility line (upper circle in
Figure 1). This liquid phase is in equilibrium with the solid CC (lower circle in
Figure 1a) and the vapor phase (indicated by the iso-RH line)
Figure 1b shows the influence of RH on the congruent phase behavior of the water-free system. The deliquescence lines separate the liquid phase (L) from regions, where crystals exist. As the transformation of a CC is assumed to occur through a liquid phase, it’s worth considering all regions with liquid phases (possibly besides crystals) [
15,
20]. Depending on the composition of the mixture, the liquid-phase formation (deliquescence) and therewith also the possible CC transformation occurs at different RH levels. Pure CC (on the CC stoichiometry line) is thermodynamically stable below DRH
CC. A liquid phase will form next to the CC stored at the DRH
CC (compare circles in
Figure 1a). This critical RH decreases when API crystals (even at trace levels) are present next to the CC. The composition moves slightly to the left of the CC stoichiometry and a liquid phase occurs at DRH
CC/API. CF crystals next to the CC decrease this critical RH even further to DRH
CC/CF in this case. For low amounts of API or CF next to the CC, the CC will be present next to the liquid phase upon deliquescence (region CC + L). For higher amounts of API or CF next to the CC, either API or CF will coexist with the liquid phase and CC will completely dissolve in thermodynamic equilibrium (regions API + L or CF + L). If the RH is subsequently lowered below the critical RH values, the initially present crystals will crystallize again.
A schematic ternary phase diagram of an incongruently dissolving CC is shown in
Figure 2a. Compared to congruently dissolving CC in
Figure 1, the API solubility is significantly lower than the CF solubility. This leads to a shift of the CC region to the CF side of the diagram. Incongruently dissolving CCs cannot establish a thermodynamically stable solution with CC stoichiometric composition (CC stoichiometry line only crosses the metastable part of the CC solubility line). Therefore, the metastable CC solubility line is used to determine DRH
CC. Analogously to the congruent phase behavior, DRH
CC is determined by the water activity on the CC solubility line at the stoichiometric composition. If an incongruently dissolving CC is stored at DRH
CC, a liquid phase will evolve. This liquid phase (intersection of DRH
CC iso-RH line with the metastable CC solubility line) is supersaturated regarding the API in this exemplary case. Therefore, the API will crystallize and the liquid phase with the composition indicated as the circle on the API solubility line will evolve. In thermodynamic equilibrium, the CC is not stable and will be fully transformed into the API. Of course, the CC stored above DRH
CC, might also transform to the CF or other polymorphs of the CC, the API, or the CF. These transformations depend on the inclination of the CC region in
Figure 2a and on the kinetics of the crystallization. In this work, only the thermodynamically stable crystal form is evaluated.
Figure 2b shows a schematic phase diagram indicating the influence of RH on the dry mixture for an incongruently dissolving CC. The pure CC stored below the DRH
CC is thermodynamically stable although the phase diagram shows liquid-phase formation starting from DRH
CC/API. These API and CF deliquescence lines are not valid for the pure CC but only the CC deliquescence line must be considered for the deliquescence. Thus, a liquid phase can only occur at storage above DRH
CC. The liquid phase occurring above DRH
CC is supersaturated with respect to API (evolving liquid phase lies in the API + L region) and API will crystallize in thermodynamic equilibrium. The evolving phases in thermodynamic equilibrium at DRH
CC are indicated as circles in
Figure 2b. When small amounts of API crystals are present next to the CC, a liquid phase can occur at DRH
CC/API. The evolving liquid phase is again supersaturated with respect to API and API will crystallize. Thereby, CC crystals will transform into the API crystals in thermodynamic equilibrium. Small amounts of CF next to the CC will even lower the RH above which a liquid phase can occur to DRH
CC/CF. Above this RH, the liquid phase is in equilibrium with the CC and therefore CF will completely dissolve (for compositions left of the eutectic composition of CC and CF). For compositions right of the eutectic point of CC and CF, CF will be stable above DRH
CC/CF and CC will completely dissolve.
To conclude, CCs with incongruent phase behavior in water can transform to either API or CF above their DRHCC (depending on the location of the CC region). CC to API transformations are likely to occur with pharmaceutical CCs, since the API is the least water-soluble component of the CC. Congruent phase behavior may lead to unwanted CC dissolution above DRHCC but as RH is decreased, CC is likely to crystallize again. This dissolution and subsequent recrystallization might alter the crystal surface area and therefore the performance of the CC. The knowledge of DRHCC, DRHCC/API, and DRHCC/CF allows to determine the stability region for the CC depending on its purity and phase behavior (congruently dissolving or incongruently dissolving).
6. Conclusions
Phase diagrams (crystal solubility, deliquescence relative humidity (DRH)) predicted via PC-SAFT allow to determine the critical storage conditions at which CCs become unstable. The phase diagrams can be predicted only based on simple solubility measurements of the CC in water or any other solvent. The stability of a CC against RH depends on three influencing factors:
First, if the CC might form CC hydrates, the critical CC-hydrate RH needs to be considered. The CC anhydrate is stable below the critical RH of the CC hydrate, whereas the CC hydrate forms above this RH.
The second factor is the DRH of the CC, which significantly depends on the presence of the crystal form (CC anhydrate or CC hydrate) and on the presence of other coexisting crystal forms (active pharmaceutical ingredient (API), coformer (CF), or excipient crystals). The investigations showed that even small excess amounts of API crystals or CF crystals next to the CC significantly lower the DRH and therewith decrease the stability of the CC. Contrary, formation of API hydrates or CF hydrates does not affect the CC stability aside from the DRH decrease. In fact, hydrate formation of either API or CF can be even prevented by formulating the API as a CC. Sugar excipients considered in this work were predicted to significantly decrease the DRH. The DRH of the pure excipient crystal is a suitable indicator for the DRH decrease in the CC/excipient system. A liquid phase forms (deliquescence) for RHs above the DRH of the crystal mixture, which is a prerequisite for the unwanted transformation of the CC into any other stable crystal form. This transformation is only occurring if the CC is incongruently dissolving.
That’s why the third factor is the congruence of the CC/water phase diagram. For congruently dissolving CCs, deliquescence is reversible, and the CC crystallizes again when RH is decreased below DRH. Incongruently dissolving CCs might irreversibly transform to another crystal form upon deliquescence. All predictions were validated by stability measurements of CCs (with excess API/CF or additional excipients). The PC-SAFT predictions were in excellent agreement with the CC stability measurements at humid conditions. Thus, this approach is considered as being able to predict the stability of any CC in combination with any crystalline excipient.