Correlation and Prediction of Drug Molecule Solubility with the NRTL-SAC Model Chau-Chyun Chena and Peter A. Craftsb aAspen Technology, Inc., Ten Canal Park, Cambridge, Massachusetts 02141, U.S.A. bAstraZeneca Pharmaceuticals Ltd., Process R&D, Macclesfield, Cheshire, SK10 2NA,U.K.

Abstract The recently proposed Nonrandom Two-Liquid Segment Activity Coefficient model (NRTL-SAC) of Chen and Song (2004) provides a simple and thermodynamically consistent framework to correlate and predict drug solubility in pure solvents and mixed solvents, based on a small initial set of measured solubility data. Used within a process simulator, or through an Excel spreadsheet, the model forms the scientific foundation of an effective solubility modeling tool in support of early stage crystallization process development. The methodology is also applicable to other unit operations where phase equilibrium calculations factor prominently in process design and development. Keywords: Solubility, crystallization, activity coefficient, Nonrandom Two-Liquid theory.

1. Introduction Crystallization is the preferred method of purification in the pharmaceutical industry for both the final drug substance and the isolated intermediates in the synthesis. During the early stages of process development, the quantity of raw material available in support of the laboratory design effort is usually limited, due to the demands of clinical trials, formulation development and the significant cost of manufacture. This problem is compounded by the high rate of drug attrition and the large number of new drug candidates that are concurrently in development and competing for resource. It is common for these factors to constrain the experimental program of solvent selection, which may lead to a sub-optimal design in respect of yield, productivity and manufacturability. Whilst high throughput solubility measurement techniques are improving, they are still time and labor intensive. Where crystallization requires a mixed solvent system it is practically impossible to cover the full range of solvent combinations with sufficient detail to find the optimal solution, even with high throughput techniques. To overcome these obstacles it is highly desirable to have a thermodynamically consistent model to correlate and predict drug solubility in pure and mixed solvent systems based on a small initial data set of measured solubilities. The number of literature references in this area is currently small, particularly with respect to mixed solvent systems where the solubility behavior can be highly non-ideal. Recently Chen and Song (2004) reviewed prior solubility modeling works and proposed the semi-empirical NRTL-SAC model as a thermodynamic framework for the correlation and prediction of drug molecule solubility in pure solvent and mixed solvent systems. They presented satisfactory results with NRTL-SAC in correlating drug solubilities in a few representative pure solvents and in subsequent prediction of drug solubilities in other

and 9th International Symposium on Process Systems EngineeringW. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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pure solvents. Chen and Crafts (2006) further demonstrated robust predictions from NRTL-SAC for drug solubilities in mixed solvent systems. Based on up to four molecule-specific parameters, the model captures the qualitative trends of drug solubilities in common solvents and solvent mixtures. In this short communication we present the semi-empirical NRTL-SAC model and its various industrial applications including support of early stage crystallization process development.

2. Thermodynamic Framework The solubility of a solid organic nonelectrolyte can be described by the expressions:

TB

AT

T

R

SxK mfusSAT

ISATIsp +=⎟

⎠⎞

⎜⎝⎛ −

Δ== 1lnln γ (1)

m

fusfus

RT

H

R

SA

Δ=

Δ= (2)

R

H

R

STB fusfusm Δ

−=Δ

−= (3)

Where spK is the solubility product constant, SATIx is the mole fraction of the solute I

dissolved in the solvent liquid at saturation, SATIγ is the activity coefficient of the solute

I in the solution at saturation, SfusΔ is the entropy of fusion of the solute, mT is the

melting point of the solute, R is the gas constant, T is the temperature, and HfusΔ is

the enthalpy of fusion of the solute. Given a polymorph, HfusΔ and mT are fixed. At a fixed temperature, the solubility is only a function of the activity coefficient of the solute in solution. Clearly, the activity coefficient of the solute in solution plays the key role in determining the solute solubility as the solvent composition changes.

3. NRTL Segment Activity Coefficient Model The semi-empirical NRTL-SAC model computes the activity coefficient for component I from the combinatorial term C

Iγ and the residual term RIγ :

RI

CII γγγ lnlnln += (4)

Here CIγ is calculated with the Flory-Huggins equation for the combinatorial entropy of

mixing and RIγ is calculated with the local composition (lc) interaction contribution

lcIγ of the polymer NRTL model (Chen, 1993). The polymer NRTL equation

incorporates the segment interaction concept and it computes the activity coefficient for component I in solution by summing up contributions to the activity coefficient from all segments that make up component I. The equation is given as follows:

[ ]Ilcm

lcm

mIm

lcI

RI r ,

, lnlnlnln Γ−Γ== ∑γγ (5)

with

C.-C. Chen and P.A. Crafts860

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−+=Γ∑∑

∑∑∑∑

kkmk

jjmjmj

mmm

kkmk

mmm

kkmk

jjmjmj

lcm Gx

Gx

Gx

Gx

Gx

Gx

'

''

'' '

''lnτ

ττ

(6)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−+=Γ∑∑

∑∑∑∑

kkmIk

jjmjmIj

mmm

kkmIk

mmIm

kkmIk

jjmjmIj

Ilcm Gx

Gx

Gx

Gx

Gx

Gx

',

'',

'' ',

','

,

,,ln

ττ

τ (7)

∑∑∑

=

I iIiI

IIjI

j rx

rxx

,

,

(8)

∑=

iIi

IjIj r

rx

,

,, (9)

where I is the component index, ',,,, mmkji are the segment species index, Ix is the

mole fraction of component I, jx is the segment-based mole fraction of segment

species j, Imr , is the number of segment species m contained only in component I, lcmΓ

is the activity coefficient of segment species m, and Ilcm

,Γ is the activity coefficient of

segment species m contained only in component I. G and τ in Eqs. 6 and 7 are local binary quantities related to each other by the NRTL non-random factor parameterα :

)exp( ατ−=G (10) Four pre-defined conceptual segments were proposed by Chen and Song (2004) to account for interacting molecular surface of all types: one hydrophobic (x), one polar attractive (y-), one polar repulsive (y+), and one hydrophilic (z). Chen and Song further suggested values for the various binary segment-segment interaction parameters, i.e., τ andα in Eq. 10. The molecular-specific model parameters for all interacting solvents and solutes, i.e., hydrophobicity X, polarity types Y- and Y+, and hydrophilicity Z, correspond to Imr , (m=x, y-, y+, z) in Eq. 5. The NRTL-SAC model has been further extended for the computation of ionic activity coefficients and solubilities of organic salts in common pure solvents and solvent mixtures (Chen and Song, 2005). In addition to the four molecular parameters, an electrolyte parameter is introduced to characterize both local and long-range ion-ion and ion-molecule interactions attributable to the ionized segments of organic electrolytes. In practice, the NRTL-SAC molecular parameters are first identified for common pure solvents from available experimental binary vapor-liquid or liquid-liquid phase equilibrium data, at room temperature or near room temperature. In this process, hexane (with x =1) and water (with z =1) are treated as the reference hydrophobic solvent and hydrophilic solvent, respectively. A databank of molecular segment parameters for the common pure solvents is thus established.

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Experimental solubility data for the drug solute is measured in four to eight solvents with distinctive surface interaction characteristics, at or near room temperature. These solvents should include hydrophobic solvents such as hexane or heptane, hydrophilic solvents such as water and methanol, and polar solvents such as acetone, acetonitrile, DMSO, DMF, etc. From these drug solubility data, we identify drug molecular parameters including the solubility product constant spK . If drug solubility data are available at multiple temperatures, then the temperature dependency of the solubility product constant spK can be determined, together with the corresponding values for

HfusΔ and mT from Eq. 2 and Eq. 3. Alternatively, HfusΔ and mT can be measured experimentally by differential scanning calorimetry. Given the molecular parameters for solvents and solutes, we perform solid-liquid equilibrium calculations at a given temperature and pressure to identify solute solubilities in any mixture of solutes and solvents. The calculations are performed in a process simulator environment that can be linked to an Excel spreadsheet for increased flexibility and ease of use. The simplicity of NRTL-SAC and its ease of use within a process simulator, linked to an Excel spreadsheet, are contributing to the accelerated acceptance of the model in the pharmaceutical industry. Publications on recent industrial applications of the model have emerged. For example, Tung et al. (2005) reported use of NRTL-SAC and COSMO-SAC models to estimate the solubility of two Statin molecules and two Cox molecules. NRTL-SAC demonstrated superior performance to the ab initio COSMO-based model. Crafts (2005) reported the successful implementation and benefits of NRTL-SAC within AstraZeneca.

4. Industrial Applications The use of activity coefficient models in the pharmaceutical industry has been limited to solvent recovery or emission studies due to lack of applicability of existing models to complex pharmaceutical compounds. The UNIFAC model is recognized as perhaps the most successful predictive method available to-date for the chemical industry. However in the pharmaceutical sector UNIFAC suffers from missing functional groups, parameter sets based on data that do not reflect the complexity of pharmaceutical compounds, and the deterioration of the functional group additivity rule for such complex molecules (Gracin et al, 2002). NRTL-SAC offers a promising new activity coefficient model for complex pharmaceutical compounds. By taking solubility measurements in just four to eight representative solvents it is possible to characterize the segment contributions of the drug molecule and then to qualitatively predict the drug solubility in any pre-characterized pure solvent or solvent mixture. This method quickly identifies both good solvent and antisolvent candidates for crystallization process design and provides a first estimate of yield and productivity. The average accuracy of NRTL-SAC has been shown to be ~50% which is considered adequate for the task of solvent selection. Applying NRTL-SAC to the data generated by high throughput solubility measurement techniques can help identify experimental errors, for cases where chemical reaction or decomposition are apparent, or where solid state change, due to structural

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polymorphism, or solvate formation may have occurred. Once identified as outliers these data points can be scrutinized in more detail. NRTL-SAC also brings a new perspective to the design of optimal solvent systems. Instead of finding a solvent molecule or solvent mixture with “optimal” functional groups, the optimization would be performed to find the optimal surface interaction characteristics in terms of conceptual segment makeup. The appearance of a solubility peak in mixed solvent systems is an intriguing and common phenomenon that is of high interest to the industry. The solubility peak represents a minimum in the activity coefficient of the solute. The NRTL-SAC method suggests that the solute activity coefficient would approach that of the ideal solution, i.e., unity, should the conceptual segment makeup of the solvent mixture approach that of the solute. It is also possible that the solute activity coefficient would become less than unity if the solvent system exhibits strong attractive interactions with the solute. Predicting exactly which polymorph of a drug molecule will crystallize from a given solvent system is an unsolved problem that is of great interest to the pharmaceutical industry. Predicting solubility is a bounded problem with respect to polymorphism. The most thermodynamically stable polymorphic form of a solute defines the low solubility limit while the upper solubility limit is defined by the amorphous solid state. NRTL-SAC makes it possible to estimate the amorphous phase solubility by calculating the phase equilibrium between a solute rich pseudo-liquid phase and a solvent rich liquid phase. In pharmaceutical process design the number of available polymorphs and their relative stability is usually determined through milligram scale, high throughput screening experiments, in support of the formulation process design. Such information is available to the manufacturing process design team before they make a final selection of the crystallization solvent. Assuming that the most stable polymorph has been found in the screening experiments then calorimetry measured values of HfusΔ and mT can be used with NRTL-SAC to predict the available solubility window across a range of single and mixed solvent systems. This window is important because it defines the available yield and productivity of the crystallization process. A phase diagram with the solubility of each known polymorph can be generated, by using the respective values of HfusΔ and mT for each of the known polymorphs. This solubility map helps to identify the best crystallization control strategy to obtain a single desired polymorphic form. In addition to solubility modeling, NRTL-SAC can be used with phase equilibrium calculations of vapor-liquid, liquid-liquid, and vapor-liquid-liquid equilibria. In this way NRTL-SAC parameters developed from solubility data for crystallization design can be used in other industrial applications. For example, dissolved solutes can have a significant and often negative effect on the rate of distillation when trying to remove residual water from an organic solvent, such as toluene, prior to a hydrophobic reaction step. A second example is a liquid-liquid extraction step in a batch reactor, in which an aqueous phase is added to an immiscible organic phase to remove reaction by-products such as inorganic salts and unreacted organic coupling reagents Cleaning between manufacturing campaigns in multi-product plant is a time consuming activity, particularly when changing from high potency to high dose products. The

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cleaning activity can often take several months to complete. The identification of optimal solvent mixtures for cleaning is another area that NRTL-SAC could support. Mixed solvent reflux is often used to remove material from overhead condensers and their associated pipe-work. NRTL-SAC is well suited for predicting solubility in this operation. Product drying is often the bottleneck step in a production process and is notoriously difficult to scale up. NRTL-SAC allows the effect of solute composition on vapor pressure and drying rate to be predicted. Adsorption and capillary effects are still very important towards the end of drying and will require experimental characterization. Chromatography is frequently used in the early stages of process development to recover drug substance that has been contaminated with an unacceptable level of impurities during scale-up. The dissolved solutes must be kept in solution throughout the chromatography operation to prevent blockages. It is common for binary and ternary solvent mixtures to be used in chromatography and the use of NRTL-SAC to predict solubility behavior should make the solvent selection task much quicker. Further applications of NRTL-SAC can be found in the prediction of octanol-water partition coefficients (Log P) to characterize drug bioavailability, solvent selection for formulations, and in drug discovery and pharmacokinetics, where partitioning between the gut, blood and body tissues is of critical importance in predicting dose levels and pharmocological effects.

5. Conclusions Correlation and prediction of drug molecule solubilities play a critical role in the development of pharmaceutical processes, especially in the area of crystallization. The recently developed NRTL-SAC model offers a practical thermodynamic framework for solubility modeling of complex pharmaceutical molecules. Used with process simulators and Excel spreadsheets, the model finds growing acceptance in the industry as an effective engineering tool for crystallization process development. The model may find applications in many other important pharmaceutical operations.

References 1. C.-C. Chen, Y. Song, 2004, “Solubility Modeling with a Non-Random Two-Liquid

Segment Activity Coefficient Model,” Ind. Eng. Chem. Res., 43, 8354. 2. C.-C. Chen, P.A. Crafts, 2006, “Correlation and Prediction of Drug Molecule Solubility in

Mixed Solvent Systems with the NRTL-SAC Model,” paper submitted for publcation in Ind. Eng. Chem. Res.

3. C.-C. Chen, 1993, “A Segment-Based Local Composition Model for the Gibbs Energy of Polymer Solutions,” Fluid Phase Equilibria, 83, 301.

4. C.-C. Chen, Y. Song, 2005, “Extension of Nonrandom Two-Liquid Segment Activity Coefficient Model for Electrolytes,” Ind. Eng. Chem. Res., 44, 8909.

5. H.-H. Tung, J. Tabora, N. Variankaval, D. Bakken, C.-C. Chen, 2005, “Prediction of Pharmaceuticals Solubility via NRTL-SAC and COSMO,” paper presented at the 16th International Symposium of Industrial Crystallization, Dresden, Germany.

6. P.A. Crafts, 2005, “Solubility Modeling in AstraZeneca with Aspen’s NRTL-SAC Method,” paper presented at Aspen User Group Meeting, Amsterdam, Netherlands.

7. S. Gracin, T. Brinck, A.C. Rasmuson, 2002, “Prediction of Solubility of Solid Organic Compounds in Solvents by UNIFAC,” Ind. Eng. Chem. Res., 41, 5114.

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