Prediction of Drug Solubility in Mixed Solvent Systems Using the

Apr 22, 2010 - A simple method is proposed to improve the accuracy in prediction of drug solubility using the COSMO-. SAC model. In this method, the d...
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Ind. Eng. Chem. Res. 2011, 50, 142–147

Prediction of Drug Solubility in Mixed Solvent Systems Using the COSMO-SAC Activity Coefficient Model Chun-Chieh Shu and Shiang-Tai Lin* Department of Chemical Engineering, National Taiwan UniVersity Taipei, Taiwan 10617, Taiwan

A simple method is proposed to improve the accuracy in prediction of drug solubility using the COSMOSAC model. In this method, the data of solubility of drug in a pure solvent is used to determine any error in the COSMO-SAC model for the drug-solvent interactions. These values are then used to correct for the solubility in the mixture of solvents. We have examined this method for the solubility of 33 drug compounds in 127 different solvent mixtures (235 systems, 1955 data points) over a temperature range of 0-50 °C and a wide solubility range of 10-1-10-6 (mole fraction scale). The error in the predicted solubility reduces significantly from 442% (original COSMO-SAC) to only 88% (COSMO-SAC with corrections), making it a practical approach for drug development. The accuracy of this method found to be comparable to another predictive method, NRTL-SAC. 1. Introduction The use of solvents constitutes 80%-90% mass utilization in a typical process for drug manufacturing.1 The chemical synthesis of a drug molecule often involves 3-10 synthetic steps in series, each having some specific drug solubility requirement. For example, high solubility is desired during chemical reactions while low solubility is needed for crystallization. In particular, the role of the solvent in the purification process could very well determine the success or failure of this operation, and therefore, the selection of the appropriate solvent is an important issue.2 The selection of solvent, solvent mixtures, operating temperature, etc. can be a daunting task in a complex synthesis.3 Therefore, having a practical method for estimating solubility of drugs in various combinations of mixed solvent compositions can greatly reduce the cost and increase the success in new drug development. There have been many methods developed for the prediction of solubility,4 most of which are based on the extension of the general solubility equation,5,6 or the Hansen solubility parameter.7 Recently, Mullins et al.8 showed that the solubility of drugs can be predicted within 0.74 log10 units (450% error) using the predictive COSMO-SAC activity coefficient model. While this approach does not require any experimental data (other than the heat of fusion and melting temperature of the drug), a higher level of accuracy is expected for practical applications. Chen and Song developed a nonrandom two-liquid segment activity coefficient (NRTL-SAC)9,10 model for solubility prediction. The species-dependent parameters in the model can be determined using limited amount of experimental data. This method provides a better accuracy and has been shown to be a practical tool for drug process design.2,11,12 In this work, we propose to use COSMO-SAC with the combination of experimental solubility data in pure solvents. We show that the accuracy can be significantly improved even if only some of the needed data are used. Furthermore, any error caused by the uncertainty of the drug conformation is minimized in this approach. As for the original COSMO-SAC model, there is no issue of missing parameters in this newly proposed method. We also show the σ-profile used in the COSMO-SAC calcula* To whom correspondence should be addressed: E-mail [email protected].

tions can provide useful insights to understand the variation of solubility on the solvent compositions. 2. Theory 2.1. Thermodynamics on the Solubility Calculation. The solubility of a solid drug in a liquid solution is its composition under the solid-liquid equilibrium condition. The phase equilibrium criteria require that the fugacity of the drug in the solid phase and that in the liquid phase be the same under constant temperature T and pressure P, i.e., fdsolid(T, P) ) jf liquid (T, P, _x) d

(1)

The solid phase is nearly pure drug, and the liquid phase is a mixture. The fugacity of the drug in the liquid can be expressed in terms of the activity coefficient γd of the drug, jf dliquid(T, P, _x) ) xdγd(T, P, _x)fliquid (T, P) d

(2)

where xd is the mole fraction of drug in the liquid phase (i.e., solubility), fdliquid(T,P) is the fugacity of a drug in a hypothetical liquid state under the same T and P. Combining eqs 1 and 2, the solubility is obtained as xd )

solid 1 fd γd f liquid

(3)

d

The ratio of the drug fugacity in the solid phase to that in the (hypothetical) liquid phase can be evaluated from the melting _ df of the drug as13 temperature Tdm, the heat of fusion ∆H ln

fdsolid fdliquid

)

∆Hfd(Tm d) RTm d

(

1-

Tm d T

)

(4)

Note that in eq 4, we have neglected the small corrections from the difference in heat capacity of the drug in the solid and the liquid phases. Substituting eq 4 into eq 3, we have the (natural logarithm of the) solubility as ln xd ) -ln γd(T, _x) +

10.1021/ie100409y  2011 American Chemical Society Published on Web 04/22/2010

∆Hfd(Tm d) RTm d

(

1-

Tm d T

)

(5)

Ind. Eng. Chem. Res., Vol. 50, No. 1, 2011

Tmd

∆H _ fd

In this work, the and of the drug are taken from literature (experiment or estimation). For a solvent (of known composition) at a given temperature, the drug solubility xd is the only unknown in eq 5 and can be obtained provided that the activity coefficient of the drug in the liquid phase is available. 2.2. Activity Coefficient from the COSMO-SAC Model. The conductor-like screening model segment activity coefficient (COSMO-SAC) model development by Lin and Sandler14 is a refinement of the COSMO-RS model of Klamt.15 In this model, the nonideality of a fluid is determined from the interactions between molecules through surface contacts. The strength of surface interactions is measured by the screening charge density (σ) between the two contacting molecular surface segments. The apparent surface screening charges can be determined from quantum mechanical solvation calculations for a molecule dissolved in perfect conductor (COSMO).16,17 The probability distribution of the surface charge density, called the σ profile, pi(σ), represents the electronic nature of the molecule of interest pi(σ) ) Ai(σ)/Ai

(6)

where Ai(σ) is the surface area of species i with a screening charge density of σ, and Ai ) ∑σAi(σ) is the total surface area of species i (note that the summation is over al possible values of σ). For a mixture S, the σ-profile is determined as the mole fraction weighted average of the pure component contributions C

∑ x A (σ)

pS(σ) )

i)1 C

)

C

i i

i)1

where xi is the mole fraction of species i in the mixture of C different species. In the COSMO-SAC model, a solution is considered as a mixture of interacting surface segments possessing certain screening charge density. The segment activity coefficient Γ can be determined from the σ-profile as14,18

σn

[

-∆W(σn, σm) pk(σn)Γk(σn)exp kT

]}

(8)

where the subscript k denotes the solution of interest (k ) i for pure liquid i, k ) S for mixture S) and ∆W measures the energy of interaction between two surface segments of the same area aeff and charge density σm and σn, respectively, ∆W(σm, σn) )

0.096aeff3/2 (σm + σn)2 + chb max[0,σacc ε0 (9) σhb] min[0,σdon + σhb]

where ε0 is the permittivity of free space; chb is a constant for the hydrogen-bonding interaction; σhb is a cutoff value for hydrogen bonding interactions; σacc and σdon are the larger and smaller values of σm and σn; max[...] and min[...] indicate that the larger and smaller values of their arguments are used, respectively. The activity coefficient of species i is determined from the sum of segment contributions as14 ln γCOSMOSAC ) i

Ai aeff

∑ p (σ )[ln Γ (σ ) - ln Γ (σ )] + i

m

S

m

i

m

σm

ln γSG i/S

(10)

j j

(11)

j

(12)

The Margules-type of activity coefficient equation is used here

(7)

∑xA

∑xl

ln γd ) ln γCOSMOSAC + ln γcorr d d

ln γcorr ) d

i)1

i)1

{

φi θi φi + 5qi ln + li xi φi xi

with θi ) xiqi/(∑jxjqj), φi ) xiri/(∑jxjrj), li ) 5(ri - qi) - (ri 1), where xi is the mole fraction of component i; ri and qi are the normalized volume and surface area parameters for i. Equation 10 is the COSMO-SAC activity coefficient model. All the species dependent quantities (pi(σ), Ai, ri, qi, etc.) are obtained from first-principle COSMO calculations. The values of a few universal parameters (aeff, chb, and σhb) have been determined based on experimental vapor-liquid equilibrium (VLE) data and do not need to be changed. 2.3. Improvement of Solubility Predictions in Mixed Solvent Systems via the Use of Solubility Data in Pure Solvents. The COSMO-SAC model provides semiquantitatvie accuracy for the activity coefficient of drug in solution. To improve the accuracy, we propose to include an empirical correction term to the result from the COSMO-SAC model

i i i

∑xA

∑

ln γSG i/S ) ln

∑ x A p (σ)

i i

ln Γk(σm) ) -ln

where ln γi/S is Staverman-Guggenheim combinatorial model which considers the nonideality as a result from the size and shape differences between the species in the mixture19

C

i i

143

SG

1 2RT

C

C

∑ ∑ (B

id

+ Bjd - Bij)xixj

(13)

i)1 j)1

where Bij is the binary interaction parameter between species i and j (note that Bij ) Bij, and Bii ) 0). The drug-solvent interaction parameters (Bid) can be obtained directly using the experimental solubility data xd of the drug in pure solvent i from eqs 5, 12, and 13 as

( ( ) ∆Hfd

Bid RTm d ) RT

Tm d 1- ln xdγCOSMOSAC d T (1 - xd)2

)

(14)

The solvent-solvent interaction parameters Bij are determined via a mixing rule. Two possible choices are examined Bij ) 0

(15a)

Bij ) Bid + Bjd

(15b)

and

It will be shown that setting Bij to zero (no correction to solvent-solvent interactions) usually leads to better accuracy in the solubility predictions. 3. Computational Details The solubility (xd) of a drug in a solution at a given temperature is determined based on eq 5. Experimental values for the heat of fusion (∆H _ df) and temperature of melting (Tdm) of the drug are used. When experimental data are not available (e.g., p-tolylacetic acid), the group contribution method of Chickos20,21 is used to estimate these values. An initial guess of xd is obtained by assuming γd ) 1 in eq 5. Using the guessed solubility, the activity coefficient γd of the drug in solution is then determined from eq 12. The updated value of γd is then

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used in eq 5 to obtain a new value of xd. The calculation is repeated until a self-consistent value of the activity coefficient is obtained. The σ-profiles needed in the COSMO-SAC calculations are taken from the VT-200522 (for solvent molecules) and VT-20068 (for drug molecules) Sigma Profile Database. The universal parameters in COSMO-SAC are taken from the original work of Lin and Sandler (aeff ) 7.5 Å2, chb ) 85580 kcal/(mol Å4 e2), and σhb ) 0.0084 e/Å2). The detailed procedure of can be found elsewhere.14 calculation of γCOSMOSAC i For purely COSMO-SAC predictions, all the binary interaction parameters Bij in eq 13 are set to zero. When experimental solubility data in pure solvent are available, the predictions of solubility in mixture solvent can be improved with the inclusion of γdcorr (eq 13). In such a case, the necessary drug-solvent parameters (Bid) are determined from eq 14 and the solvent-solvent parameters (Bij) are determined from eqs 15.

Table 1. Comparison of RMSE in Solubility Prediction from Different Methods methodb

a

cases

systems/ data points

A 157/1325 B 78/230 overall 235/1955

COSMO-SAC + COSMO-SAC + correction correction COSMO-SAC (eq 15a) (eq 15b) 1.78 1.51 1.69

0.62 0.67 0.63

0.78 0.76 0.77

a

Case A includes systems where solubility data in all needed pure solvents are available, and case B includes systems where at least one of the pure solvent solubility data is not available. b COSMO-SAC represents the results from the original COSMO-SAC model (all Bijs in eq 14 are set to zero). COSMO-SAC + correction utitlizes experimental solubility of a drug in pure solvent to determine Bids.

4. Results and Discussion In this work, we have considered the solubility of 33 drug compounds [with iodine (2 atoms) being the smallest and testosterone (49 atoms) being the largest] in 37 pure solvents and, in particular, their mixtures. We have studied a total of 127 solvent combinations, including 123 binary solvent mixtures (224 systems, each is a specific combination of solvents at some temperature), 3 ternary solvent mixtures (10 systems), and 1 quaternary solvent mixture (1 system). A complete list of all the 235 systems (with temperatures ranging from 273.15 to 323.15 K) studied is provided as the Supporting Information. The accuracy of the predictions is measured by the root-mean square error (RMSE) of ln(xd) defined as follows RMSE )

[∑ 1 n

n

i

]

0.5

cal 2 (ln xexp d - ln xd )

(16)

where superscripts exp and cal denote the experimental or calculated values, and n is the number of data points considered. 4.1. Solubility Prediction from the COSMO-SAC Model. The root-mean-square error (RMSE) in predicted solubility in pure solvents (33 drugs, 37 solvents, 400 data points) is 1.96 (corresponding to 610% in percentage error of solubility). The result is essentially the same as that found in the work of Mullins et al.8 In the case of mixture solvents (127 systems, 1955 data points), the RMSE is found to be 1.69 (442%). This is slightly better compared to the result found by Mullins8 where the RMSE is 2.03 based on 14 solutes in 14 binary solvents (37 solute-solvent systems). We noticed that the molecular geometry of ethanol and butylether in the VT-2005 database were not fully optimized. The improvement observed here is mainly a result of the use of the proper σ-profile for these two solvents (90 systems contain ethanol, 15 systems contain butylether). 4.2. Improved Solubility Predictions in Mixed Solvent Systems via the Use of Data in Pure Solvents. When the solubility of a drug in a pure solvent is available, it can be used to (eq 14) determine the drug-solvent interaction parameter (Bid) in eq 13 and thus improve the predictions in the case of solvent mixtures. Not all the Bid parameters can be determined for the 127 solvent mixtures studied in this work because of the lack of experimental data for some drug-pure solvent pairs. There are 78 (out of the overall 235) systems where at least one of the needed experimental data is missing. In such a case, the missing Bid is set to zero. With the use of experimental solubility data, the predictions in solvent mixtures improves significantly with the RMSE reduced to 0.63 (88%, when Bij

Figure 1. Solubility of benzoic acid in the solvent mixture of cyclohexane and carbon tetrachloride at 25 °C from COSMO-SAC (solid line) and COSMO-SAC with corrections based on solubility of pure solvents (dashed line). Experimental data are shown in open circles.

between solvent species are set to zero (eq 15a)) or 0.77 (116%, when Bij between solvent species are estimated from those between drug and solvent (eq 15b)), compared to that from the original COSMO-SAC, 1.69 (442%). The results are summarized in Table 1. It is interesting to note that the improvement is still significant (1.51 from COSMO-SAC to 0.67 or 0.76 from COSMO-SAC with correction based on pure solvent data) even when part of the experimental data are available. Since setting Bij between solvents to zero (eq 15a) always gives better prediction accuracy, we will focus our future discussions on this approach. Figure 1 illustrates a typical example of the solubility determined from COSMO-SAC with and without corrections based on pure solvent solubility data. It is clear that the original COSMO-SAC model provides good solubility dependency on solvent compositions. Therefore, when the offsets on the two ends (pure solvents) of the figure are removed, very accurate predictions can be achieved for solvent mixtures. This also supports the observation that no additional correction is needed between the solvent molecules (eq 15a) since the COSMO-SAC model has provided the proper description of interactions between solvents. The use of pure solvent data provides the additional benefits of reducing the issue from the uncertainty of the drug conformation. Mullins et al.8 showed that different conformations of the drug used in the COSMO-SAC calculation may lead to significantly different results in the predicted solubility. For example, Figure 2 shows the σ-profiles of paracetamol (aka acetaminophen) in three different conformations (VT-2006 82a, 82b, 82c). We have examined the RMSE of the solubility of paracetamol in 60 systems based on these three conformations. As shown in Table 2, the difference in RMSE as a result of

Ind. Eng. Chem. Res., Vol. 50, No. 1, 2011

Figure 2. σ-Profiles for paracetamol in three conformations A (bold solid), B (thin solid), and C (dashed) as listed in Table 2.

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Figure 3. Solubility of benzil in the solvent mixture of cyclohexane and isooctane at 25 °C. The legends are the same as in Figure 1.

Table 2. RMSE in Solubility Prediction of Paracetamol in Different Conformations method

conformationa

systems/ data points

COSMO-SAC

COSMO-SAC + correction (eq 15a)

A B C

60/634 60/634 60/634

1.19 1.15 0.97

0.76 0.78 0.77

a The three conformations of paracetamol correspond to the three σ-profiles in Figure 2.

Table 3. RMSE in Solubility Prediction of 57 Systems from Different Methods method systems/ data points

NRTL-SAC

COSMO-SAC

COSMO-SAC + correction (eq 15a)

57/628

0.58

1.12

0.83

using different conformations for paramcetamol is 0.22 (1.19-0.97) from the original COSMO-SAC model. However, such difference is only 0.02 (0.78-0.76) when the pure solvent solubility is included in the calculation. Therefore, the uncertainty of which drug conformation should be chosen for the calculation becomes a less important issue. We regard this an important benefit of the proposed method for the study of large, flexible drug molecules. 4.3. Comparison with NRTL-SAC. The NRTL-SAC model, recently developed by Chen and Song,9,10 is an attractive approach for the prediction of drug solubility. In this model, a molecule is defined based on four characters: hydrophobicity (x), hydrophilicity (z), polar-attraction (y-), and polar-repulsion (y+). The values of these characters of each molecule are determined from regression to experimental (VLE or solubility) data. Due to the lack of parameters for some species, we are only able to predict solubility of 57 systems (out of the 235 systems considered in this work). The results are summarized in Table 3. While the NRTL-SAC model provides the lowest RMSE, 0.58 or 79%, the inclusion of solubility in pure solvent for COSMO-SAC also gives satisfactory accuracy, 0.83 or 129%. We consider this method a good complement to NRTLSAC when the needed parameters are not immediately available. 4.4. Correlation between σ-Profile and Drug Solubility. From the 235 systems studied in this work, we notice that the variation of drug solubility with solvent composition can be grossly classified in to three types: a linear-shaped solubility

Figure 4. σ-Profiles of benzyl (bold solid), cyclohexane (thin solid), and isooctane (dashed).

Figure 5. Solubility of testosterone in the solvent mixture of chloroform and cyclohexane at 25 °C. The legends are the same as in Figure 1.

curve (Figure 3), significant difference in pure solvent solubility (Figure 5), and the presence of a solubility maximum (Figure 7). The variation of drug solubility with the composition of solution is highly associated with the σ-profile of the drug and solvent species. In Figure 4, we can see that the σ-profiles of cyclohexane and isooctane are very similar, indicating that they have similar solvation capability for the drug, benzil. In such a case, a linear-shaped solubility curve is observed (Figure 3). In Figure 5, we observe that the solubility of testosterone in the mixture of cyclohexane and chloroform can differ, depending on the solvent composition, by nearly 3 orders of magnitude. From Figure 6, we see that the σ-profile of chloroform has a

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Figure 6. σ-Profiles of testosterone (bold solid), cyclohexane (thin solid), and chloroform (dashed).

Figure 8. σ-Profiles of sulfisomidine (bold solid), water (thin solid), and dioxane (dashed).

5. Conclusion The combination of COSMO-SAC model and the drug solubility in pure solvents is an effective approach for the prediction of drug solubility in solvent mixtures. The error in the prediction improves from 442% (COSMO-SAC) to 88% (COSMO-SAC with pure solvent data) based on analysis of 33 drugs in 127 mixture solvents (1955 data points). Furthermore, the potential error caused by uncertainty in the drug conformation becomes negligible in this approach. We believe that the level of accuracy would make it a practical tool for drug discovery. Acknowledgment Figure 7. Solubility of sulphisomidine in the mixture solvent of water and dioxane at 25 °C. The legends are the same as in Figure 1.

much greater degree of overlap with testosterone than that of cyclohexane, especially in the region of negatively charged surfaces (almost ideal distributions between -0.015 and -0.005 e/Å2). Furthermore, chloroform has a large amount of positively charged surfaces (e.g., the peak at around 0.004 e/Å2) that could help solvate the polar surfaces (large positive and negative values of σ) of testosterone. In contrast, surface segments of cyclohexane are nearly neutral (centered around 0). The much higher degree of overlap in the σ-profile of testosterone and chloroform implies that a much higher solubility of testosterone in chloroform. Therefore, we observe a strong solubility variation with solvent composition when the σ-profiles of the solvents are very dissimilar. Sulphisomidine shows an interesting solubility maximum in the mixture of water and dioxane (Figure 7). While the σ-profiles of water and dioxiane are quite different, their combination is surprisingly similar to the σ-profile of sulphisomidine. This implies that the mixture of water and dioxiane would have a better solvation capability for sulphisomidine compared to either pure solvent. As a result, we observe a maximum in the solubility curve in Figure 8. Based on these observations, we see that solubility is highly correlated with the similarity between the σ-profile of the drug and that of the solvent (either pure or mixture). High drug solubility can be achieved by mixing different solvents that gives the most similar σ-profile of the drug.

One of the authors (Shiang-Tai Lin) would like to acknowledge Prof. Stan Sandler for his guidance during his graduate studies, during which the original COSMO-SAC model was developed. This work was partially supported by grants from the National Science Council of Taiwan (NSC 97-2221-E-002-085 and NSC 98-2221-E-002-087-MY3). The computation resources from the National Center for HighPerformance Computing of Taiwan and the Computing and Information Networking Center of the National Taiwan University are acknowledged. Supporting Information Available: Solubility systems used in the research. This material is available free of charge via the Internet at http://pubs.acs.org. Literature Cited (1) Constable, D. J. C.; Jimenez-Gonzalez, C.; Henderson, R. K. Perspective on Solvent Use in the Pharmaceutical Industry. Org. Process Res. DeV. 2007, 11, 133–137. (2) Modarresi, H.; Conte, E.; Abildskov, J.; Gani, R.; Crafts, P. ModelBased Calculation of Solid Solubility for Solvent Selection - a Review. Ind. Eng. Chem. Res. 2008, 47, 5234–5242. (3) Kokitkar, P. B.; Plocharczyk, E.; Chen, C. C. Modeling Drug Molecule Solubility to Identify Optimal Solvent Systems for Crystallization. Org. Process Res. DeV. 2008, 12, 249–256. (4) Jouyban, A. Review of the Cosolvency Models for Predicting Solubility of Drugs in Water-Cosolvent Mixtures. J. Pharm. Pharm. Sci. 2008, 11, 32–57. (5) Ran, Y. Q.; Yalkowsky, S. H. Prediction of Drug Solubility by the General Solubility Equation (Gse). J. Chem. Inf. Comput. Sci. 2001, 41, 354–357. (6) Ran, Y. Q.; Jain, N.; Yalkowsky, S. H. Prediction of Aqueous Solubility of Organic Compounds by the General Solubility Equation (Gse). J. Chem. Inf. Comput. Sci. 2001, 41, 1208–1217.

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ReceiVed for reView February 24, 2010 ReVised manuscript receiVed April 4, 2010 Accepted April 9, 2010 IE100409Y