Thermodynamic Modeling for Efficient Cocrystal Formation - Crystal

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Crystal Growth & Design

Thermodynamic modeling for efficient cocrystal formation Linda Lange, Gabriele Sadowski* Department of Chemical and Biochemical Engineering, Laboratory of Thermodynamics, TU Dortmund University, Emil-Figge-Str. 70, D-44227 Dortmund, Germany ABSTRACT: The purpose of this work is to increase the efficiency of the cocrystal formation process by thermodynamic modeling using PC-SAFT. By accounting for the thermodynamic non-ideality of the components in the cocrystal system, PC-SAFT is able to model and predict the solubility behavior of pharmaceutical cocrystals based solely on the knowledge of a single cocrystal solubility point in any solvent and at any temperature. Furthermore, the cocrystal solubility in other solvents and for other temperatures can be predicted without the need for additional measurements. The (+)-mandelic acid/(-)-mandelic acid (1:1), caffeine/glutaric acid (1:1) and carbamazepine/nicotinamide (1:1) cocrystal systems were modeled, and the results were in excellent agreement with the experimental data.

1. INTRODUCTION Pharmaceutical cocrystals (CCs) represent an emerging class of solid drugs because they can potentially improve the solubility and dissolution behavior compared to the pure active pharmaceutical ingredients (APIs)1-6. These CCs are crystalline solids that consist of the API and at least one coformer (CF) in a defined stoichiometry7. Due to the weak interactions between the API and CF in the CC, it decomposes into its components upon dissolution. Different methods for CC formation have been reported in the literature. In general, CCs are prepared by grinding8-15, crystallization from melt16, 17 or crystallization from solution18-20. At the industrial scale, CC formation is preferably performed by crystallization from solution19, 21, 22. Specifically for the latter, effective CC formation requires knowledge of the thermodynamic phase diagram19, 23. For a given system of API, CF, and solvent, this diagram provides the concentration range in which stable CCs can form. However, the experimental determination of these diagrams is time consuming and expensive23-27. It is typically based on solubility measurements in a given system of API, CF, and solvent27-29, followed by analysis of the liquid phase, primarily using high-performance liquid chromatography (HPLC)5, 27, 30. Another method uses a calorimetric technique25, 31, 32. In both cases, the solid phase also needs to be characterized, which is primarily performed using powder X-ray diffraction5, 24, 25, 27, 29, 30, 33, 34. ter Horst et al. developed a shortcut to identify the concentration range in an API/CF/solvent system in which stable CCs can be found35. They proposed performing the CC screening in an API/CF/solvent concentration range that spans the solubilities of the API and CF in a common solution that is assumed to be the same as in the single-solute solutions. Thus, this approach is limited to almost ideal solutions in which the solubilities of the API and CF are unaffected by the presence of the other respective components in solution. Other studies have enhanced the efficiency of CC formation by explicitly considering the solubility of the CC itself 3, 20, 25-27, 35-39 . In these studies, CC formation is described as a chemical reaction of the liquid API and the liquid CF, resulting in the solid CC. The equilibrium of this reaction, which also describes the CC solubility, is modeled by the CC solubility product , . However3, 20, 27, 35, 37-41, the CC solubility product , is often calculated neglecting the API and CF activity coefficients,

resulting in , as described by Eq. (1)3, 20, 27, 35, 37-41, or by assuming that the activity coefficients are constant over the entire concentration range25, 26. xi and γi in Eq. (1) are the mole fractions and activity coefficients of the API and CF, respectively, and ai is the activity of these components. The exponents νAPI and νCF correspond to the API and CF stoichiometry, respectively, in the CC. The activity of the solid CC is equal to 1. ∏ ν ,

, = = = ν = ν ∙ ν (1) ∏ ν ∏ ν

Wang et al.42 and Ainouz et al.25 used Eq. (1) to predict CC solubility in various solvents. Wang’s investigation was limited to almost ideal solutions in which activity coefficients could be neglected42. He stated that the ratio of enantiomers at the eutectic points of racemic compound-forming systems is independent of the solvent for highly diluted solutions42. Ainouz et al.25 assumed constant activity coefficients, , and , , (irrespective of other solutes and of concentration) to calculate the apparent CC solubility product Kapp: =

, ν

(2) ν , ∙ , Because the activity coefficients depend on the API and CF concentrations in addition to the solvent, the apparent CC solubility product is only valid for one particular CC solubility point. Ainouz et al., however, used the same to

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calculate the cocrystal solubility points at any API/CF concentration in one solvent. Furthermore, differs from solvent to solvent, whereas Ainouz et al. equated the ratio of values in two different solvents to that obtained from the solubilities of pure API and pure CF in these solvents, again assuming that the activity coefficients did not depend on the presence of other solutes (CF or API) or on concentration. Good and Rodríguez-Hornedo predicted CC solubility at different temperatures based on the melting enthalpy and melting temperature of the CC neglecting the activity coefficients3. This approach considers the CC solubility only at one particular API and CF concentration and not over a concentration range. Consequently, they found significant discrepancies between their model and experiments. Jayansankar et al.40 developed a model that describes CC solubility for different CC stoichiometries. This model again neglected the activity coefficients of the compounds forming the CC. However, the solubility of one component in the presence or absence of other solutes can only be correctly described by accounting for the (concentration-dependent) component activity coefficients and real interactions between the solutes and solvents in solution43. In contrast to the above-mentioned approaches, this work accounts for the thermodynamic non-ideality of components in the CC solution: , = ν = ν ∙ ν

(3)

The solubility product , does not depend on the solvent or concentration; rather, it only varies with temperature. Therefore, it can be calculated based only on one CC solubility data point, regardless of the solvent. The Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT)44 was used to model and predict the activity coefficients of API and CF and the concentration range in which the CC was thermodynamically stable. PC-SAFT has previously been successfully applied to model solubilities in binary and ternary systems44-48, including those with CC formation49, 50. PC-SAFT modeling of CC formation in binary systems of methanol/water (1:1)49, phenol/acetamide (1:1, 2:1)49, and bisphenol A-phenol (1:1)49 and in ternary systems of bisphenol A/phenol (1:1) in water49 and (+)-mandelic acid/(-)mandelic acid (1:1) in water50 showed good agreement with the experimental data. We also describe the CC solubility in API/CF/solvent systems. Ternary phase diagrams were obtained by calculating (1) the solubilities of pure API and pure CF, (2) the solubility of API in the presence of the CF and vice versa, and (3) the CC solubility. Furthermore, we demonstrate how solubility calculations can be used to determine the stoichiometry of an unknown CC and to predict the CC solubilities in different solvents and at different temperatures. The modeling results were validated by comparison with experimental data on the (+)-mandelic acid/(-)-mandelic acid, caffeine/glutaric acid, and carbamazepine/nicotinamide CC systems.

2. TERNARY PHASE DIAGRAMS OF COCRYSTAL SYSTEMS Figures 1a and 1b schematically show ternary phase diagrams for a system consisting of an API, CF and solvent. Figure 1a illustrates a system without CC formation. In the liquid solution (I), the API and CF are completely dissolved in the solvent. Along the solubility lines that separate region I from regions II, III, and IV, the saturated liquid solution is in equilibrium with the respective solid phase, which consists of pure crystalline CF (II), pure crystalline API (III), or a mixture of both (IV). The API/CF ratio in solution determines which of these solid phases is formed. For high API and low CF concentrations (III), pure API crystals are in equilibrium with the liquid solution (I), whereas at high CF and low API concentrations (II), pure CF crystals are formed. The API and CF solubility lines intersect at the eutectic point. Below this point, a heterogeneous mixture of API and CF crystals is formed (IV), which is in equilibrium with the solution at the eutectic point. The dashed line in Fig. 1a illustrates the solubility of the API (or CF) in an ideal solution in which neither the solubility nor the activity coefficient is affected by the addition of CF (or API). In those cases, the solubility of one component is NOT affected by the presence of the other, and the solubility lines of the two components correspond to linear parallels to the two respective triangle axes. In real systems in which the solubility of a component IS affected by the presence of another, the solubilities in the ternary system deviate from the respective pure-component solubilities, leading to the solid lines in Figure 1a. The more CF (or API) added to the solution, the more the ideal and real solubility lines deviate from each other. In the example shown in Figure 1a, the eutectic point in the real solution is much lower than that for the ideal case, thereby also changing the shape of regions II and III (not shown in Figure 1a for clarity). In turn, the concentration ranges in which pure API (or CF) crystallizes differ significantly for the real and the ideal solubility behavior. Figure 1b schematically shows the phase diagram for a system in which API and CF form a 1:1 CC. As in Figure 1a, region I denotes the liquid solution of both the API and CF. Regions II or III again denote the regions where crystals of pure CF (II) or pure API (III) are formed. In addition, region (V) is where pure CCs are formed. The solubility of the CCs can be found along the line separating regions I and V. Because the CC only exists in the solid phase, the liquid solution I contains completely dissolved API and CF molecules in all cases (as in Figure 1a). By analogy to region IV in Figure 1a, there are now two regions in which two solids crystallize: CC and API (VI) and CC and CF (VII). The compositions of the corresponding liquid phases refer to the intersects of the CC solubility line on the one hand and of the API or CF solubility line on the other hand. These two solubility line intersects (CC/API and CC/CF) are again called eutectic points. The dashed lines again illustrate the solubility of API and CF in a hypothetically ideal solution. ter Horst et al.35 suggested using the


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area spanned by these two lines (shaded area in Figure 1b) for CC screening. The case illustrated in Figure 1b would not lead to any CC formation because the CC area V does not intersect with the screening area. Obviously, the size and position of the concentration range in which stable CCs are formed are strongly influenced by thermodynamic non-ideality. Therefore, the solubility behaviors of the API, CF and CC have to be determined by accounting for the real interactions between solutes and solvents in solution. In this study, PC-SAFT modeled and predicted cocrystal solubility considering the activity coefficients of the API and CF.

Figure 1. Schematic ternary phase diagram for an active pharmaceutical ingredient (API)/coformer (CF)/solvent system (a) without cocrystal (CC) formation and (b) with formation of 1:1 CCs. Solid lines are the solubility curves of CF, API, and CC, accounting for thermodynamic non-ideality; dashed lines are the solubility curves neglecting the thermodynamic non-ideality. The phases are denoted as follows: I, liquid mixture consisting of solvent and com-


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pletely dissolved API and CF; II, pure CF crystals in equilibrium with liquid mixture I; III, pure API crystals in equilibrium with liquid mixture I; IV, CF crystals and API crystals in equilibrium with liquid mixture I; V, pure CC in equilibrium with liquid mixture I; VI, CF crystals and CCs in equilibrium with liquid mixture I; and VII, API crystals and CCs in equilibrium with liquid mixture I. Dashed area shows the concentration range for CC screening when neglecting thermodynamic non-idealities.

3. THEORY 3.1. Solubility calculations The solubility of CF or API in the presence of the respective other compound (solubility lines enclosing areas II and III in Figure 1) is calculated considering the thermodynamic equilibrium between a pure solid phase and the liquid solution according to51: ! =

)!

)! 1 ∆h( + ∆.,( +()! +()! #$ %− ,1 − − , − 1 − ln -1 ! *+ * + + +()!

(4)

In Eq. (4), ! is the mole fraction of component i (API or CF) in the liquid phase (the solubility), T is the temperature of )! the system, and * is the ideal gas constant. +()! and ∆h( are the melting temperature and heat of fusion of component i )! (API or CF), respectively. ∆.,( describes the difference in the solid and liquid heat capacities of component i at its melting point. Eq. (4) is derived from the iso-fugacity condition of a compound in a liquid and a solid phase assuming that )! )! ∆h( and ∆.,( are temperature independent in the considered temperature range51. The activity coefficients ! of the API and CF in the liquid phase depend on the temperature and concentrations of all components (solvent, API and CF) in the mixture. They were calculated in this work by PC-SAFT. CC formation is modeled as a chemical reaction of its components (Eq. (5)), the API and the CF, whereas 3 and 3 correspond to the API and CF stoichiometric coefficients in the CC. 56, (5) 44 789 3 :;< + 3 4>

The equilibrium of this reaction is considered by the CC solubility product , described by Eq. (3). Because the solubility product does not depend on the type or concentration of the solvent, the entire solubility line of a CC in any solvent (and even solvent mixtures) can be modeled based on the knowledge of only a single experimental solubility data point in any solvent, which is used to determine , . The activity coefficients in Eq. (3) were again calculated in this work from PC-SAFT. The CC solubility product , depends on temperature. This temperature dependence can be described based on the Gibbs-Helmholtz equation according to Eq. (6). Thus, , at any temperature can be estimated using a reference CC ?@ ?@ solubility product , at a reference temperature Tref and the reference enthalpy ∆h : ?@

ln , = AB, +

∆CDEF G

H

I

J DEF

I

− K J

(6)

?@

In Eq. (6), the reference enthalpy ∆h corresponds to the enthalpy of fusion of the CC at the reference temperature ?@ Tref , which can also be interpreted as the enthalpy of melting of the CC. However, ∆h is not experimentally available in ?@ most cases. In this work, two different approaches were used to calculate the reference enthalpy ∆h . In the first ap?@ proach, ∆h was estimated from the enthalpy of fusion and the melting temperature of the API and CF weighted by the API and CF stoichiometry in the CC52: )! )! νAPI ∆h νCF ∆h ?@ (7) ∆h = + ?@ , ∙ )! + ∙ )! νAPI + ν + νAPI + ν + Occasionally, however, the melting enthalpies of API and CF are not available. In these cases, a second approach was used, as described by Folas et al.53. As indicated by Eq. (6), the logarithm of the CC solubility product AB, is a linear ref function of the negative inverse temperature −1⁄+. Thus, ∆h can also be derived from the slope and axis intercept of this linear function if at least two CC solubility data points at two different temperatures are known.

3.2. PC-SAFT PC-SAFT is based on perturbation theory using the hard-chain system as reference system. Its derivation has been described in detail in literature44, 45. The residual Helmholtz energy ?M is calculated as the sum of different contributions that account for repulsive forces (hard chain) and attractive forces (dispersion), as well as for hydrogen bonding interactions (association): (8) ?M = C ? NC O + ?PO + PN QPO S In PC-SAFT, each component is characterized by three pure-component parameters: the number of segments R , the segments diameter T and the dispersion-energy parameter U ⁄VW 43. Because almost all components considered in this work are associated, two additional parameters for characterizing the associative interaction were used: the associationenergy parameter X W ⁄VW and the association-volume parameter Y W 44.


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In mixtures of substances Z and [, the segment diameter T\ in Eq. (9) and the dispersion-energy parameter U\ in Eq. (10) are calculated by applying the Berthelot-Lorentz combining rules54. An additional binary interaction parameter V\ is used to adjust the cross-dispersion energy. 1 (9) T\ = T + T\ 2 U\ = 1 − V\ ^U U\ (10) Taking into account the cross-associating interactions between two associating components, the association-energy parameter X W\ ⁄VW in Eq. (11) and the association-volume parameter Y W\ in Eq. (12) are calculated according to the combining rules of Wolbach and Sandler55. 1 (11) X W\ = X W + X \W\ 2 _ ^T T\ (12) Y W\ = ^Y W Y \W\ ,

1⁄2T +T\ For mixtures of a polar but not self-associating component and a self-associating component, induced association was considered as described by Kleiner and Sadowski56. In this approach, the value X W of the not self-associating component is set to zero, whereas the respective association-volume parameter Y W was set to 0.01. The number of associations sites for associating and induced-associating components was defined on the basis of association schemes from Huang and Radosz57 (Tables 1 and 3).

4. RESULTS AND DISCUSSION 4.1. Estimation of PC-SAFT parameters Three different CC systems were considered in this work, namely, (+)-mandelic acid/(-)-mandelic acid (1:1), caffeine/glutaric acid (1:1), and carbamazepine/nicotinamide (1:1). In the first modeling step, the solubilities of the pure API and CF were modeled, using PC-SAFT for the calculation of the respective activity coefficients `Z (Eq. (4)). The application of PC-SAFT requires the pure-component parameters of the APIs, CFs and solvents considered. The pure-component parameters of all solvents and of the API nicotinamide were obtained from the literature and are summarized in Table 1. The pure-component parameters of the other APIs and CFs were fitted to solubility data of the pure APIs or CFs. The applied parameter-fitting procedure has previously been described in detail by Ruether and Sadowski43. Furthermore, binary interaction parameters between the API (or the CF) and the respective solvent must be known for the PC-SAFT modeling. These binary interaction parameters V\ + were assumed to linearly depend on temperature (Eq. (13)). They were fit simultaneously with the pure-component parameters to the solubility data. Fitting of four purecomponent parameters (mi, σi, ui, ε AiBi) plus a temperature-dependent binary interaction parameter (kij,slope, kij,int) requires at least the same number of experimental solubility data points of the pure components (API or CF). V\ + = V\, P + + V\,OQ (13) The pure-component melting properties required for the solubility calculations (Eq. (4)) were taken from the literature. The heat capacities for liquid caffeine and liquid glutaric acid are not available due to decomposition at high temperatures. Thus, these properties were estimated using the group contribution method of Kolska58. Furthermore, the heat capacity of solid glutaric acid was estimated using the group contribution method of Goodman59. All considered purecomponent melting properties are listed in Table 2. The pure-component parameters fitted within this work are summarized in Table 3, and the corresponding binary interaction parameters are given in Table 4. Table 1. Previously published PC-SAFT pure-component parameters for coformers (CFs) and solvents considered within this work component CF nicotinamide(I) solvents water ‡ acetone ‡ acetonitrile ethanol ‡ ethyl acetate methanol 2-propanol

AiBi

M [g/mol]

mi [-]

σi [Å]

ui [K]

ε

[K]

122.12

4.6485

1.7143

166.25

1056.2

18.015 58.08 41.052 46.069 88.105 32.04 60.096

1.2047 2.8913 2.3290 2.3827 3.5370 1.5255 3.0929

2.7927 3.2279 3.1898 3.1771 3.3080 3.23 3.2085

353.94 247.42 311.31 198.24 230.80 188.90 208.42

2424.67 0 0 2653.4 0 2899.5 2253.9


κ

AiBi

[-]

assoc scheme

ref

0.002

1/1

60

0.045 0.01 0.01 0.03284 0.01 0.035176 0.024675

1/1 1/1 1/1 1/1 1/1 1/1 1/1

61 62 63 45 44 45 45

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‡

Consideration of induced association in mixtures with associating components

Table 2. Melting properties of active pharmaceutical ingredients (APIs) and coformers (CFs) considered in this work SL

component API (+)-/(-)-mandelic acid caffeine carbamazepine(III) CF glutaric acid nicotinamide(I)

SL

SL

T0i [K]

∆h0i [kJ/mol]

∆cp.0i [J/mol]

ref

404.05 509.15 466.35

26.2 21.6 26.2

57.76 101.52 65.37

64

370.95 401.15

20.9 28.0

107.45 78.12

58, 59, 69

58, 65, 66 67, 68

70

Table 3. Pure-component PC-SAFT parameters for active pharmaceutical ingredients (APIs) and coformers (CFs) fitted within this work and the average relative deviations (ARDs) of calculated and experimental solubilities component

API (+)- /(-)-mandelic acid caffeine carbamazepine(III) CF glutaric acid

M [g/mol]

mi [-]

σi [Å]

ui [K]

ε

152.15 194.19 236.27

5.4294 9.5552 9.9778

2.4605 2.9590 2.6583

202.64 428.51 151.55

132.12

4.4371

2.7990

257.67

AiBi

[K]

AiBi

κ [-]

assoc scheme

ref for exp data

ARD [%]

1976.3 827.7 1094.0

0.02 0.02 0.02

2/2 2/2 1/1

71

9.56 2.46 5.07

1762.5

0.02

2/2

28, 73

72 20

4.34

Table 4. PC-SAFT binary interaction parameters for the binary sub-systems of active pharmaceutical ingredients (APIs) (or coformers (CFs)) and solvents considered in this work and the average relative deviations (ARDs) of calculated and experimental solubilities. Binary parameters should be used together with pure-component parameters from Tables 1 and 3 binary parameter

API/solvent (+)-/(-)-mandelic acid/water caffeine/acetone caffeine/acetonitrile caffeine/methanol carbamazepine/2-propanol carbamazepine/ethanol carbamazepine/ethyl acetate carbamazepine/methanol CF/solvent glutaric acid/acetonitrile nicotinamide/methanol nicotinamide/2-propanol nicotinamide/ethanol nicotinamide/ethyl acetate

temperature range of the experimental data [K]

ref for parameters

ref for exp data

ARD [%]

9.56 2.29 0.43 4.65 3.81 5.89 0.002 1.79

kij,slope [-]

kij,int [-]

-4

-0.0397 -0.048 -0.0313 -0.1375 0.1337 0.0796 0.0791 0.0321

273.15 - 372.15 298.00 - 323.00 283.15 - 308.15 298.00 - 323.00 285.10 - 337.54 278.80 - 338.16 298.15 276.8 - 326.8

this work this work this work this work this work this work this work this work

71

0.0185 -0.2212 0.0119 0.0183 -0.1307

283.15 - 308.15 278.15 - 323.15 289.93 - 320.62 289.96 - 318.79 283.05 - 302.45

this work this work

28

70

70

70

70

60

58

1.44 · 10 -4 -1.82 · 10 -5 -2.53 · 10 -4 -1.53 · 10 -4 -2.54 · 10 -4 -1.48 · 10 0 -4 -1.22 · 10 -5

-5.99 · 10 -4 7.68 · 10 -4 4.58 · 10 -4 2.13 · 10 -3 1.45 · 10

72 28 72 74 74 20 74

39

1.39 15.49 -

As an example, Figure 2 compares the modeling results for the solubility of caffeine to experimental literature data72. The pure-component parameters for caffeine and the corresponding binary interaction parameters were adjusted to the solubility data in methanol, acetone, and acetonitrile. This demonstrates that PC-SAFT allows for satisfactory correlation of the solubility data and the pure-component parameters of caffeine (and the other solutes) can be used for near quantitative solubility calculations in a variety of solvents.


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Figure 2. Solubility of caffeine in methanol (diamonds), acetone (triangles) and acetonitrile (circles). Lines corre28, 72 spond to the PC-SAFT correlations; symbols represent the experimental data points .

4.2. API solubility in the presence of CF and vice versa In the second modeling step, the API solubility in the presence of the CF and vice versa was calculated via Eq. (4). Thus, the corresponding binary interaction parameters between the API and the CF had to be determined. These parameters were fitted to solubility data in systems containing the two solutes dissolved in a common solvent. Fitting a temperaturedependent binary parameter required consistently two solubility data points at different temperatures. All binary interaction parameters between the APIs and the CFs are reported in Table 5. Table 5. PC-SAFT binary interaction parameters between the active pharmaceutical ingredients (API) and coformers (CF) fitted within this work and the average relative deviations (ARDs) of calculated and experimental solubilities binary parameter

API/CF (+)-/(-)-mandelic acid caffeine/glutaric acid carbamazepine/nicotinamide

kij,slope [-]

kij,int [-]

-4

0.3 -0.3084 0.2091

9.76 · 10 -4 8 · 10 -3 -1 · 10

temperature range of the experimental data [K]

ref for exp data

ARD [%]

298.15 - 308.15 283.15 - 308.15 278.15 - 323.15

71

3.61 8.18 5.82

28 20

For example, Figure 3 compares the modeling results for the solubilities of caffeine and glutaric acid commonly dissolved in acetonitrile with experimental data28. The solubility of glutaric acid strongly increases by adding small amounts of caffeine, whereas the solubility of caffeine is only slightly affected by adding glutaric acid. Furthermore, the solubility of glutaric acid in acetonitrile is approximately three times higher at 308.15 K compared to 288.15 K, whereas the solubility of caffeine is only marginally influenced by the temperature increase. Because the glutaric acid solubility strongly depends on the presence of caffeine, the experimental data show a larger deviation from the solubility curves in a hypothetically ideal solution. This deviation is larger at 308.15 K than at 288.15 K. In contrast, the calculations using PC-SAFT accounting for thermodynamic non-ideality show a high degree of conformity with the experimental data at both temperatures. This example again confirms the high modeling accuracy of PC-SAFT for correlating the solubility data of real systems.


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Figure 3. Solubility of glutaric acid (solubility line on the left-hand side) and caffeine (solubility line on the righthand side) in the ternary system glutaric acid/caffeine/acetonitrile at 288.15 K (triangles) and 308.15 K (diamonds) with shortened axes in mole fractions. Continuous lines correspond to the PC-SAFT calculations; dashed lines represent the solubility curves neglecting the thermodynamic non-ideality; symbols refer to the experimental data 28 points for glutaric acid (black) and caffeine (white) .

4.3. CC solubility Finally, the CC solubility is calculated using Eq. (3). Figure 4 shows as an example the modeling results for the caffeine/glutaric acid (1:1) system in acetonitrile at 298.15 K. As illustrated in Figure 3, the solubility of glutaric acid strongly increases by adding small amounts of caffeine, whereas the CC solubility strongly decreases by adding caffeine. Thus, the two solubility lines of caffeine and the CC shown in Figure 4 meet at a tapered eutectic point. In contrast, the solubility of caffeine is only slightly affected by adding glutaric acid, and thus, the intersection of the glutaric acid solubility line with the CC solubility line is difficult to see in Figure 4. The CC solubility product , was calculated based on only one CC solubility data point (marked as star in Figure 4). This value and all other solubility products , determined within this work are listed in Table 6. Figure 4 shows the modeling results for the solubility of pure API and CF, the solubility of each of these components in the presence of the others and the CC solubility line. The modeling results are in very good agreement with the experimental data28.


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Figure 4. Ternary phase diagram of glutaric acid and caffeine forming a 1:1 cocrystal (CC) in acetonitrile at 298.15 K with shortened axes in mole fractions. Lines correspond to the PC-SAFT calculations; circles represent the experimental solubility data points for glutaric acid (black), caffeine (white) and the CC (grey); the star is the CC solubility 28 point used to calculate the solubility product , . def

Table 6. Cocrystal (CC) solubility products ab,cc and reference enthalpies ∆h of CCs consisting of an active pharmaceutical ingredient (API) and a coformer (CF) calculated within this work; all reference CC solubility def products ab,cc are marked with (*) CC API/CF

CC stoichiometry API:CF

, [-] -3

(+)-/(-)-mandelic acid

1:1

caffeine/glutaric acid

1:1

caffeine/glutaric acid caffeine/glutaric acid

1:2 2:1

carbamazepine/nicotinamide

1:1

1.99 · 10 -3 ( ) 3.63 · 10 * -3 6.24 · 10 -2 2.12 · 10 -2 ( ) 2.61 · 10 * -2 3.16 · 10 -3 6.29 · 10 -3 2.82 · 10 -4 2.58 · 10 -4 ( ) 7.26 · 10 * -3 2.21 · 10

temperature [K] 288.15 298.15 308.15 288.15 298.15 308.15 298.15 298.15 278.15 298.15 323.15

?@

∆h [kJ/mol]

Source ?@ ∆h

42.11

Eq. (6)

14.72

Eq. (7)

-

-

27.60

3

for

4.4. Estimation of CC stoichiometry from solubility data Using the above-described thermodynamic framework for modeling CC solubility, the stoichiometry of an unknown CC can be determined if two CC solubility data points are known. First, the CC solubility product , is calculated via Eq. (3) assuming different CC stoichiometries, defined by the coefficients νAPI and νCF . The same solubility point was used to calculate the CC solubility products corresponding to the different stoichiometries. Afterwards, the CC solubilities resulting from the different solubility products , were calculated and compared with at least one additional CC solubility data point. Only with the correct CC stoichiometry (and the corresponding correct solubility product) do the two solubility data points lie on the same calculated CC solubility line. Otherwise, they do not.


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Again, the example system of caffeine/glutaric acid (1:1) in acetonitrile at 298.15 K was used for this purpose. Aside from the previously modeled solubility line for a 1:1 CC (compare Figure 4), the solubility lines of hypothetical CCs with 1:2 and 2:1 ratios of caffeine and glutaric were calculated based on the same CC solubility point, as shown in Figure 5. The CC solubility products , obtained when assuming the different CC stoichiometries are listed in Table 6. Figure 5 indicates that only the modeling assuming a 1:1 CC allows for the correct representation of the CC solubility data. In contrast, the modeling results that assume a 1:2 or 2:1 ratio significantly differ from the experimental data. Thus, the combination of experiments and thermodynamic modeling can reliably determine the stoichiometry of an unknown CC, based on two CC solubility points only.

Figure 5. Ternary phase diagram of glutaric acid and caffeine with different cocrystal (CC) compositions in acetonitrile at 298.15 K with shortened axes in mole fractions. Lines correspond to the PC-SAFT calculations for 1:1 (solid line), 1:2 (dotted line) and 2:1 (dashed line) CC stoichiometry; circles represent the experimental data points for glutaric acid (black), caffeine (white) and the CC (grey); the star is the CC solubility data point used to calculate the 28 solubility product , .

4.5. Prediction of CC solubilities in different solvents

The CC solubility product , does not depend on the solvent, and it can therefore be used to predict the CC solubilities in different solvents. This means that the solubility product , for a specific CC can be used to predict the CC solubilities in different solvents using Eq. (3). For example, , of the carbamazepine/nicotinamide (1:1) CC system (listed in Table 6) was calculated using one solubility data point in ethanol at 298.15 K (marked in Figure 6). Subsequently, the same CC solubility product , was used to predict the CC solubility in 2-propanol and in ethyl acetate without using any additional information. Figure 6 shows the solubilities of carbamazepine, nicotinamide and the respective 1:1 cocrystal in ethanol, 2-propanol and ethyl acetate20. The solubility of carbamazepine is highest in ethanol, followed by ethyl acetate and finally 2-propanol. In contrast, the solubility of nicotinamide and the respective CC is highest in ethanol, followed by 2-propanol and ethyl acetate. The phase diagrams for ethanol and 2-propanol systems are highly asymmetrical because the solubility of nicotinamide in these solvents is approximately ten and hundred times higher than that of carbamazepine. In contrast, the solubilities of these compounds in ethyl acetate are quite similar, resulting in an almost symmetrical phase diagram. Using the , for the carbamazepine/nicotinamide (1:1) CC system from Table 6 and the PC-SAFT-predicted activity coefficients of carbamazepine and nicotinamide in the different solvent systems, the CC solubilities could be predicted in almost quantitative agreement with the experimental data.


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Figure 6. Ternary phase diagram of carbamazepine and nicotinamide and a 1:1 cocrystal (CC) in ethanol (diamonds), 2-propanol (triangles) and ethyl acetate (circles) at 298.15 K with shortened axes in mole fractions. Lines correspond to the PC-SAFT predictions; symbols represent the experimental data points for carbamazepine (black) 20 and for CC (grey). The star is the CC solubility point used for the calculation of the solubility product , .

4.6. Prediction of CC solubilities at different temperatures In this section, CC solubilities in the caffeine/glutaric acid/acetonitrile system and in the (+)-/(-)-mandelic acid/water system were predicted for 288.15 K and 308.15 K. As indicated by Eq. (6), the logarithm of CC solubility product AB, is a ?@ linear function of the negative inverse temperature −1⁄+, assuming that ∆h is does not depend on temperature. For the caffeine/glutaric acid (1:1) system, the solubility product at 298.15 K (see Table 6) was taken as reference value in ?@ ?@ Eq. (6): Tghi = 298.15 K; , = 2.61 · 10-2 . The reference melting enthalpy ∆h for this CC was calculated applying the first approach using the melting enthalpies of caffeine and glutaric acid from Table 2 in Eq. (7) which resulted in a value of 14.72 kJ mol-1. This calculation did not require additional CC solubility points. ?@ Besides this, ∆h was also calculated using the second approach (compare section 3.1). In contrast to the first approach, this calculation using Eq. (6) requires at least one additional CC solubility point at another temperature. Thus, a ?@ CC solubility point at 293.15 K was used resulting in ∆h = 23.73 kJ mol-1. Afterwards, AB, values were calculated from experimental CC solubility data at 283.15 K, 288.15 K, 293.15 K, 298.15 K, 303.15 K, and 308.15 K taken from literature28 and using PC-SAFT activity coefficients. These calculations were performed ?@ via Eq. (3) and thus without using ∆h . Figure 7 shows the course of these AB, as function of −1⁄+ compared to the ?@ correlations derived from the Gibbs-Helmholtz equation (Eq. 6) together with the above-mentioned ∆h from the first approach on the one hand, and from the second approach on the other hand. As to be seen, the two approaches are in satisfactory agreement with the experimentally-found AB, . The second approach (Eq. (6)) obviously ensures calculat?@ ing a more-appropriate value of ∆h than the first approach (Eq. (7)). This is due to the fact that the second approach is based on two CC solubility points at different temperatures (Eq. (6)). In cases where no additional CC solubility points are available, the first approach leads to slightly worse but still satisfactory results. To demonstrate this, the CC solubilities ?@ were predicted at 298.15 K and 308.15 K, using the AB, and ∆h resulting from the two approaches. The ARD for the ?@ predicted CC solubility using ∆h from the first approach (Eq. (7)) accounts to 14.6 %, whereas the prediction using ?@ ∆h from the second approach (Eq. (6)) deviates by 11.8 %. Thus, Eq. (7) can be used without significant loss of accuracy to calculate the solubility products at 288.15 K and 308.15 K (given in Table 6).


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Figure 7. Logarithm of cocrystal (CC) solubility product AB, of the CC caffeine/glutaric acid (1:1) system as function of the negative inverse temperature −1⁄+. Solid and dashed lines correspond to the prediction using the GibbsHelmholtz equation (Eq. 6) and using Eq. (7) and a second CC solubility point at 293.15 K for the calculation of the ?@ reference melting enthalpy ∆h , respectively; symbols represent the AB, values calculated from the experi28 mental solubility data from literature and PC-SAFT activity coefficients at 283.15 K, 288.15 K, 293.15 K, 298.15 K, 303.15 K and 308.15 K.

Figure 8 displays the so-predicted solubility lines compared to experimental data at 288.15 K and 308.15 K28 for the CC system of caffeine/glutaric acid (1:1) in acetonitrile. The solubility lines for caffeine and glutaric acid were adopted from section 4.2.. The CC solubility lines were predicted using the solubility products at 288.15 K and 308.15 K (given in Table 6), which were in excellent agreement with the experimental data. It can be seen that , , increases with increasing temperature and so does the CC solubility.

Figure 8. Ternary phase diagram of glutaric acid and caffeine with a 1:1 cocrystal (CC) in acetonitrile at 288.15 K (triangles) and 308.15 K (diamonds) with shortened axes in mole fractions. Lines correspond to the PC-SAFT calculations using the solubility products from Table 6; symbols represent the experimental data points for glutaric acid


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Crystal Growth & Design 28

(black), caffeine (white) and for CC (grey) .

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As a second example, the solubility of the (+)-/(-)-mandelic acid (1:1) CC in water was predicted as a function of temperature. This CC system is particular because the two enantiomers have the same melting properties64 and solubilities71, which in turn results in the same PC-SAFT parameters listed in Table 3. The pure-component parameters and the binary interaction parameter between (+)- or (-)-mandelic acid and water were fitted to the solubility data71 of (+)-mandelic acid in water. However, (+)- and (-)-mandelic acid affect each other in solution, and thus, a binary interaction parameter between these two enantiomers had to be determined. The binary interaction parameters are summarized in Table 4 and Table 5. To determine the temperature-dependent , , the reference temperature was again set to T ?@ = 298.15 K. The corre?@ sponding reference solubility product is , = 3.63 · 10-3 (Table 6). The reference melting enthalpy however could not be determined using Eq. (7) because Lorenz and Seidel-Morgenstern observed decomposition of mandelic acid during the melting process, which prevented a reliable determination of the melting properties75. Therefore, the second approach (Eq. (6)) was used to determine the reference enthalpy based on second CC solubility point (here: at 303.15 K), resulting in ?@ ∆h = 42.11 kJ mol-1. Figure 9 displays the modeled and experimentally-determined71 solubility data for the CC system (+)-/(-)-mandelic acid (1:1) in water at 288.15 K, 298.15 K, and 308.15 K. The CC solubility increases with increasing temperature. Due to the identical pure-component parameters of (+)- and (-)-mandelic acid, these pure components have the same solubility in water, and the ternary phase diagram is completely symmetrical with respect to the left- and right-hand side. This phenomenon is also called congruent dissolution19. The CC solubility lines at 288.15 K and 308.15 K were predicted using the previously determined , at 298.15 K (see Table 6). All modeled and predicted lines are again in very good agreement with the experimental data.

Figure 9. Ternary phase diagram of (+)- and (-)-mandelic acid with a 1:1 cocrystal (CC) in water at 288.15 K (circles), 298.15 K (triangles) and 308.15 K (diamonds) with shortened axes in mole fractions. Lines correspond to the PCSAFT calculations; symbols represent the experimental data points for (+)-mandelic acid (black), (-)-mandelic acid 71 (white) and CC (grey) . The star is the CC solubility point at 298.15 K used to calculate the solubility product , . The CC lines at 288.15 K and 308.15 K were fully predicted.

4.7. Prediction of CC solubilities in different solvents and at different temperatures Finally, the thermodynamic model was used to predict the CC solubility by varying both the solvent and temperature. Figure 10 shows the predicted CC solubility and experimental data for the carbamazepine/nicotinamide (1:1) cocrystal system in methanol at 278.15 K, 298.15 K and 323.15 K. The experimental solubilities of carbamazepine, nicotinamide, and the CC increase with temperature.


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The predictions were performed using the above-determined CC solubility product at 298.15 K derived from one CC solubility point in ethanol at 298.15 K (see Table 6) without the need of additional measurements. The reference melting ?@ ?@ enthalpy ∆h equals the melting enthalpy of the CC measured by DSC3: ∆h = 27.60 kJ mol-1, assuming it is constant over the observed temperature range of 45 K. As shown in Figure 10, the predictions are in excellent agreement with the experimental data. The highest deviation between the predicted and experimental data was observed at 323.15 K, likely ?@ due to the use of a constant value for ∆h . However, the correlation of Lee and Kim39 also showed a significant deviation from the experimental data based on one solubility point at this temperature. Thus, the deviation might also be caused by experimental uncertainties.

Figure 10. Ternary phase diagram of carbamazepine and nicotinamide and a 1:1 cocrystal (CC) in methanol at 278.15 K (circles), 298.15 K (triangles) and 323.15 K (diamonds) with shortened axes in mole fractions. Lines correspond to the PC-SAFT calculations; symbols represent the experimental data points for carbamazepine (black), nicotinamide 39 acid (white) and CC (grey) . All CC lines were fully predicted.

5. APPLICATION OF THERMODYNAMIC MODELING FOR EFFICIENT CC FORMATION Figure 11 displays the modeled and experimental data71 for the CC system of (+)-/(-)-mandelic acid (1:1) in water at 308.15 K. These modeling results are compared to the solubilities calculated for an ideal solution in which the solubility of one component is not affected by the other (activity coefficients being unity). The lines illustrating ideal-solution behavior remarkably deviate from the experimental data points. The CC solubility line cannot be modeled at all because the solubility lines of the single mandelic acid components intersect above the CC solubility points. Based only on the ideal solubilities, one would search for the (+)-/(-)-mandelic acid (1:1) CC in the dashed concentration region of Figure 11. This region does not even intersect the concentration range of the real CC solubility, and thus, a screening procedure neglecting thermodynamic non-idealities would not detect any concentration ranges of successful CC formation. However, accounting for thermodynamic non-ideality using PC-SAFT predicted the CC solubilities as shown in Figure 9. These modeling results agreed well with the experimental data, which enables a reliable determination of the concentration range in which CCs form.


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Figure 11. Ternary phase diagram of (+)- and (-)-mandelic acid with a 1:1 cocrystal (CC) in water at 308.15 K (diamonds) with shortened axes in mole fractions. Solid lines correspond to the PC-SAFT calculations accounting for thermodynamic non-ideality; dashed lines correspond to calculated solubility curves neglecting the thermodynamic non-ideality. Symbols represent the experimental data points for (+)-mandelic acid (black), (-)-mandelic acid 71 (white) and CC (grey) .

6. CONCLUSIONS Knowledge of the API/CF/solvent concentration range in which the CC is thermodynamically stable is the key to efficient CC formation. Conventional methods for CC formation consist of time-consuming and expensive experiments. Existing theoretical approaches are limited to almost ideal solutions in which the solubilities of the API and CF are unaffected by the presence of the other respective components. In contrast, this study presents a thermodynamically correct description of the CC solubility behavior. The CC solubility was modeled by allowing the CC solubility product to account for thermodynamic non-idealities using the activity coefficients of all components. The CC solubility product was determined using only one experimental CC solubility data point in any solvent. With this approach, the API/CF/solvent concentration range in which the CC is thermodynamically stable could be predicted for all investigated systems in very good agreement with the experimental data. Furthermore, using the Gibbs-Helmholtz equation, the model can precisely predict the CC solubility at any temperature based on the knowledge of a reference CC solubility product at a reference temperature and the reference enthalpy. The latter can be derived from the enthalpy of melting of either the CC or the API and CF, depending on which enthalpies are available. If no melting enthalpies are experimentally available, it can be estimated on the basis of at least two CC solubility data points. The model is even able to predict the CC solubility when varying both solvent and temperature. Finally, the model enables the determination of the stoichiometry of an unknown CC based only on the knowledge of at least two CC solubility data points. In summary, the proposed approach reliably determines the concentration range in which a CC can be found based on a minimum of experimental data. In contrast to existing approaches, it is able to more realistically describe the solubility behavior of real CC-forming systems, which in turn provides a reliable basis for more efficient CC formation.

Notation

-1

molar Helmholtz energy [J mol ] activity of component i [-]


15


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)! ∆.,

Page 16 of 19

difference of the heat capacity of the solid and the liquid component i at its −1 −1 melting point [kJ K kg ]

u

-1

Gibbs energy [J mol ] ?@

∆h

-1

reference enthalpy [kJ kg ]

)!

∆h

heat of fusion of component i [kJ kg ]

V\, P

slope of the temperature-dependent binary interaction parameter [K ]

,

cocrystal solubility product [-]

-1

V\ + V\,OQ

,

S

binary interaction parameter [-] -1

intercept of the temperature-dependent binary interaction parameter [K] cocrystal solubility product in an ideal solution [-] apparent cocrystal solubility product [-]

R

number of segments of component i [-]

+

temperature [K]

+ ?@

reference temperature [K]

*

-1

-1

gas constant [J mol K ]

+)!

melting temperature of component i [K] stoichiometric coefficient of component i [-]

νi

mole fraction of component i [-]

Abbreviations API

active pharmaceutical ingredient

CC

Cocrystal

CF

coformer

PC-SAFT

perturbed-chain statistical associating fluid theory

ARD

average relative deviation

Greek symbols

U ⁄VW X

W ⁄

Y

W

activity coefficient of component i [-] VW

σ

dispersion energy parameter [K] association energy parameter [K] association volume parameter [-] segment diameter [Å]

Subscripts Z, [

component i, component j

AUTHOR INFORMATION ACS Paragon Plus Environment

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Corresponding Author *E-mail: [email protected]. Tel: ++49 (0)231 755 2635. Fax: ++49 (0)231 755 2572.

Notes The authors declare no competing financial interest.

ACKNOWLEDGMENT The authors gratefully acknowledge financial support from the CLIB-Graduate Cluster Industrial Biotechnology.

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(45) (46) (47) (48) (49) (50) (51) (52) (53) (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) 2013. (71) (72) (73) (74) (75)

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For Table of Contents Use Only Manuscript title: Thermodynamic modeling for efficient cocrystal formation Author list: Linda Lange, Gabriele Sadowski

Table of Contents artwork Synopsis: The efficiency of cocrystal formation processes can be increased by thermodynamic modeling using PC-SAFT. The solubility behavior of pharmaceutical cocrystals in any solvent and at any temperature can be modeled and predicted based solely on the knowledge of a single cocrystal solubility point. Furthermore, the model enables the determination of the stoichiometry of an unknown cocrystal.


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Thermodynamic Modeling for Efficient Cocrystal Formation - Crystal

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