Comment/Reply pubs.acs.org/jced
Correct Derivation of Cosolvency Models and Some Comments on “Solubility of Fenofibrate in Different Binary Solvents: Experimental Data and Results of Thermodynamic Modeling” Abolghasem Jouyban,†,‡ Fleming Martinez,§ and William E. Acree, Jr.*,∥ †
Pharmaceutical Analysis Research Center and Faculty of Pharmacy and ‡Kimia Idea Pardaz Azarbayjan (KIPA) Science Based Company, Tabriz University of Medical Sciences, Tabriz 51664, Iran § Grupo de Investigaciones Farmacéutico-Fisicoquímicas, Departamento de Farmacia, Facultad de Ciencias, Universidad Nacional de Colombia − Sede Bogotá, Cra. 30 No. 45-03, Bogotá D.C., Colombia ∥ Department of Chemistry, University of North Texas, Denton, Texas 76203-5070, United States ABSTRACT: The experimental solubility data of fenofibrate in binary aqueous mixtures of ethanol and acetone has been reanalyzed. A correct mathematical derivation is provided for the correlation expression from a combination of the van’t Hoff and Jouyban−Acree models. Sun et al. used the expression in their study. However, it erroneously implied that the expression resulted from a simple transformation of the Jouyban−Acree model.
I
n a recent article appearing in this journal, Sun et al.1 reported the experimental solubility of fenofibrate in aqueous binary solvent mixtures of ethanol and acetone at different temperatures along with some numerical analyses and thermodynamic parameters derived from the generated solubility data. The experimental data were correlated using Apelblat, NRTL, and derived versions of the previously proposed cosolvency models. The accuracy of the computations was evaluated using the root-mean-square deviation (RMSD) and the mean deviation (MD) defined as
reported computations. The aim of this comment is to discuss the derivations of the models and to report the recalculation results. Sun et al.1 derived a previously published general single model4 based on the CNIBS/R-K model and used the derived version to correlate their generated solubility data of fenofibrate in aqueous mixtures of ethanol and acetone at each temperature. A shortcoming in the authors’ published analysis was the use of four curve-fit parameters to describe five experimental values per solution temperature. Not all of the curve-fit parameters can be meaningful in the reported mathematical representations. For example, we have curve-fit the solubility of fenofibrate in aqueous-methanol solvent mixtures at 288.15 K and obtained the following expression
N
RMSD =
MD =
100 N
∑i = 1 (xical − xiexp)2
N
∑ i=1
N
(1)
|x1cal − xiexp| xiexp
(2)
ln xA = − 12.006(0.528) + 9.892(3.447)fb − 0.526(6.362)fb 2 − 2.638(3.495)fb3
xcal i
where N is the number of experimental points, is the calculated solubility, and xexp is the experimental solubility. In i the original article,1 N is defined as 7 for ethanol + water and N = 6 for acetone + water mixtures, which is the case only for solubility computations in a given solvent at various temperatures. In the case of the original version of the CNIBS/R-K model, N = 5, and for the original version of the combined van’t Hoff and Jouyban−Acree model, N = 35 and N = 30 for ethanol + water and acetone + water mixtures, respectively. As noticed above, N is the number of experimental data points employed in the computations. Because both accuracy criteria show parallel variations,2 we prefer to use one of them, and MD is preferred because of its similarity to the relative standard deviations computed for the repeatability of the experimental solubility data. Despite the importance of the reported solubility data of fenofibrate in the pharmaceutical industry and the extension of the available solubility database of pharmaceuticals,3 there are several problems with the model derivation procedure and the © 2017 American Chemical Society
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using the IBM SPSS Statistics version 22 software. In eq 3, xA is the measured mole fraction solubility of fenofibrate, and f b is the mole fraction composition of ethanol in the binary solvent calculated as if the solute were not present. A careful examination of the curve-fit parameters and the standard errors in the respective parameters indicates that the error associated with the f b2 term is 6 times the calculated parameter. The elimination of this term results in a much better mathematical correlation ln xA = −11.965(0.107) + 9.610(0.354)fb − 2.925(0.276)fb3
(4)
Received: August 12, 2016 Accepted: January 19, 2017 Published: February 10, 2017 1153
DOI: 10.1021/acs.jced.6b00722 J. Chem. Eng. Data 2017, 62, 1153−1156
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with much smaller standard errors in the calculated parameters. The elimination of f b2 did not adversely affect the mathematical correlation. The standard deviation of the correlation was SD = 0.053 log units for both eqs 3 and 4. We suspect that there may be similar shortcomings in several of the other correlations given in Table 5 of the authors’ published paper. Calculating four adjustable parameters for data sets containing only five experimental values is simply not a good practice. Although both the general single and the CNIBS/R-K models produce the same accuracy for the correlation of the solubility data of a given drug in a certain cosolvent + water mixture, we recommend using the original version of the CNIBS/R-K model in future work. The main reasons for this proposal are (1) the theoretical basis of the CNIBS/R-K model,5 (2) the capability of providing the most accurate correlation/prediction for the solubility of drugs in various cosolvent + water mixtures,6 (3) the possibility of extending the model’s applicability to calculate the solubility in ternary or higher-order solvent mixtures,7 (4) providing generally trained models to predict the solubility of drugs in given cosolvent + water mixtures,8−11 (5) an accurate representation of some commonly observed phenomena in the solutions such as a chameleonic effect12 and the solubility of various polymorphs of a drug in cosolvent + water mixtures,13 and (6) providing globally trained versions of the model using Abraham parameters14 and/or Hansen solubility parameters.15 Regarding the computations in the Jouyban−Acree model,14 Sun and co-workers1 used the model as n
ln x1 = x 2 ln(x1)2 + x3 ln(x1)3 + x 2x3 ∑ i=0
ln x1 = x 2[ln(x1)2 − ln(x1)3 ] + ln(x1)3 +
T ln x1 = x 2T[ln(x1)2 − ln(x1)3 ] + T ln(x1)3 + [J0 x 2 − J0 x 22] + [(J1x 2 − J1x 22)(2x 2 − 1)] + [(J2 x 2 − J2 x 22)(4x 22 − 4x 2 + 1)]
+ [J0 x 2 − J0 x 22] + [2J1x 22 − J1x 2 − 2J1x 23 + J1x 22] +[4J2 x 23 − 4J2 x 22 + J2 x 2 − 4J2 x 24 + 4J2 x 23 − J2 x 22]
(11)
Concerning the constant values of ln(x1)2, ln(x1)3, J0, J1, and J2, further rearrangements yield T ln x1 = M1T + M 2x 2T + M3x 2 + M4x 22 + M5x 23 + M6x 24
(12) 1
A comparison of eq 12 with that derived by Sun et al. reveals that there is no mathematical justification for the A0 term reported by Sun et al. A second major problem with this derivation is that the ln(x1)2 and ln(x1)3 values are not constant values at various temperatures. Rather, ln(x1)2 and ln(x1)3 are temperature-dependent values. As an informational note, eq 12 will reduce to the general single model under isothermal conditions. The correct derivation has been reported in an earlier work17 that is repeated here in a shortened form for the convenience of readers. The original version of the Jouyban−Acree model is
T
in which x1 is the solubility of the solute in the mixed solvent at the temperature of interest, x2 and x3 are the mole fractions of ethanol (or acetone) and water in the absence of the solute, (x1)2 and (x1)2 are the solubilities of the solute in neat ethanol (or acetone) and water at T, and Ji represents the model constants computed using a no intercept least-square analysis.16 Sun et al. defined the mole fraction of ethanol (and acetone) as x2 or f b and x3 or fa. We followed the x2 and x3 definitions in this paper concerning the format of this journal. They replaced x3 with (1 − x2) and derived another equation as
n
ln x1, T = x 2 ln(x1)2, T + x3 ln(x1)3, T + x 2x3 ∑ i=0
Ji (x 2 − x3)i T (13)
in which x1,T is the solubility of the solute in the mixed solvent at the temperature of interest and (x1)2,T and (x1)3,T are the solubilities of the solute in neat ethanol (or acetone) and water at T.16 The derivation procedure includes the replacement of ln(x1)2,T and ln(x1)3,T terms with the van’t Hoff equation in eq 13 with n = 2 as
T ln x1 = A 0 + A1T + A 2 Tx 2 + A3Tx 2 + A4 x 22 + A5x 23 (6)
⎛A ⎞ ⎛A ⎞ ln x1, T = x 2⎜ 1 + B1⎟ + x3⎜ 2 + B2 ⎟ + ⎝T ⎠ ⎝T ⎠ 1 + {[J0 x 2x3] + [J1x 2x3(x 2 − x3)] T
When the algebraic manipulations are performed in a step-bystep fashion, ln x1 = x 2 ln(x1)2 + (1 − x 2)ln(x1)3
+ [J2 x 2x3(x 2 − x3)2 ]}
1 {[J x 2(1 − x 2)] + [J1x 2(1 − x 2)(x 2 − (1 − x 2))] T 0 + [J2 x 2(1 − x 2)(x 2 − (1 − x 2))2 ]}
(14)
followed by the replacement of x3 with (1 − x2) in eq 14: ⎛A ⎞ ⎛1⎞ ln x1, T = A1x 2⎜ ⎟ + B1x 2 + (1 − x 2)⎜ 2 + B2 ⎟ ⎝T ⎠ ⎝T ⎠
(7)
ln x1 = x 2 ln(x1)2 + ln(x1)3 − x 2 ln(x1)3 1 + {[J0 x 2 − J0 x 22] + [(J1x 2 − J1x 22)(2x 2 − 1)] T + [(J2 x 2 − J2 x 22)(2x 2 − 1)2 ]}
(10)
T ln x1 = x 2T[ln(x1)2 − ln(x1)3 ] + T ln(x1)3
(5)
+
(9)
By multiplying both sides of eq 9 by T, one obtains
Ji (x 2 − x3)i
+ A 6x 24
1 {[J x 2 − J0 x 22] + [(J1x 2 − J1x 22)(2x 2 − 1)] T 0 + [(J2 x 2 − J2 x 22)(4x 22 − 4x 2 + 1)]}
+ (8)
1 {[J x 2(1 − x 2)] + [J1x 2(1 − x 2)(x 2 − (1 − x 2))] T 0 + [J2 x 2(1 − x 2)(x 2 − (1 − x 2))2 ]} (15)
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As an informational note, there is no requirement that the summation in eq 13 must go to n = 2. One can use as few or as many adjustable curve-fit parameters as needed to satisfactorily describe the experimental solubility data. When eq 13 was first proposed, the systems described covered modest ranges in experimental mole fraction (and molar) solubilities; however, in some of the more recent applications, the model has been used to describe systems that cover mole fraction ranges of several thousand-fold. Naturally, more curve-fit parameters may be needed in describing systems covering larger solubility ranges. The terms in eq 15 are then multiplied out and algebraically rearranged to finally give ln x1, T = B2 − B2 x 2 + B1x 2 + +
T ln x1 = −5928.238 + 8.002T + 7.543Tx 2 + 1570.983x 2 − 1639.838x 22
with MDs of 26.6, 8.2, and 8.4%, respectively. The main reason for the poor descriptive ability of eq 19 likely results from the treatment of the mole fraction solubilities in the two neat solvents as temperature-independent constants. The recalculated MD for eq 18 and the model constants reported in Table 6 of Sun et al. are 7.9%. It should be noted that the MD value reported in Table 6 of Sun et al. is 10.15%! The corresponding equations for correlating the solubility data of fenofibrate in water + acetone mixtures are
1 (A1x 2 + A 2 − A 2 x 2) T
T ln x1, T = − 3042.888x 2 − 13.107T + 34.690x 2T − 5069.550x 22 + 1187.436x 23
1 {[J0 x 2 − J0 x 22] + [(J1x 2 − J1x 22)(2x 2 − 1)] T + [(J2 x 2 − J2 x 22)(4x 22 − 4x 2 + 1)]}
Both sides of eq 17 are now multiplied by T and like terms are combined to yield the following expression:
+ 909.873x 2x3
T ln x1, T = A 2 + (A1 − A 2 + J0 − J1 + J2 )x 2 + B2 T + (B1 − B2 )x 2T
Because all of the A, B, and J terms in eq 20 are constant values, one may rewrite the mathematical equation in its simplified version as
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with MDs of 24.1, 10.0, and 10.1%, respectively. The recalculated MD for eq 18 and the model constants reported in Table 6 of Sun et al. are 89.9%, which is reported as 13.45% in Table 6 of Sun et al. In conclusion, discussions regarding the correct derivation of a mathematical representation based on a combination of the van’t Hoff and Jouyban−Acree models were given. The mathematical representation has been used by several research groups17−20 in reporting their measured solubility data; however, the authors incorrectly state that the final equation is a simple transformation of the Jouyban−Acree model. Research groups often fail to mention that one must describe the solubility in the two mixture cosolvents with the van’t Hoff model in order to correctly arrive at eq 6. To prevent such occurrences from happening in the future, referring to the original references is strongly recommended. Nonoriginal references may contain mathematical errors that may go undetected by both manuscript reviewers and journal readers unless one carefully goes through the algebraic manipulations and transformations in going from one equation to another.
T ln x1, T = W0 + W1x 2 + W2T + W3x 2T + W4x 22 + W5x 23 (18)
Equations 6 and 18 have identical mathematical forms. As noted above, Sun et al.1 described their measured solubility data using eq 6. The mathematical steps needed to obtain eq 6 (or eq 18), however, are much more involved than what the authors implied in their paper. The statement made by the authors is that when n = 2 and x3 = (1 − x2) the model can be rewritten as eq 6. The mathematical manipulations suggested by the authors lead to eq 12, not to eq 6. It is only by assuming that the van’t Hoff equation for how the solubility in the two neat organic solvents varies with temperature does one get the A0 term. When the solubility data in water + ethanol mixtures at various temperatures were fitted to eqs 12, 13, and 18, the obtained models (after excluding nonsignificant parameters) are
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AUTHOR INFORMATION
Corresponding Author
T ln x1, T = −6614.896x 2 − 12.122T + 35.112x 2T
*E-mail:
[email protected]. Fax: 940-565-4318. ORCID
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Abolghasem Jouyban: 0000-0002-4670-2783 William E. Acree Jr.: 0000-0002-1177-7419
⎛ 5880.593 ⎞⎟ ln x1, T = x 2⎜8.163 − ⎝ ⎠ T ⎛ 5974.001 ⎞⎟ 1 + x3⎜15.453 − + 1417.766x 2x3 ⎝ ⎠ T T (x − x3) − 325.504x 2x3 2 T
(23)
− 6350.361x 22 + 1868.878x 23
(17)
−
(x 2 − x3) T
T ln x1 = −6325.770 + 8.275T + 5.707Tx 2 + 6095.462x 2
+ ( −J0 + 3J1 − 5J2 )x 22 + ( −2J1 + 8J2 )x 23 + ( −4J2 )x 24
1560.698x 22
(22)
⎛ 6311.442 ⎞⎟ ln x1, T = x 2⎜8.277 − ⎝ ⎠ T ⎛ 4700.507 ⎞⎟ 1 + x3⎜13.950 − + 3502.883x 2x3 ⎝ ⎠ T T
(16)
+ W6x 24
(21)
Notes
The authors declare no competing financial interest.
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REFERENCES
(1) Sun, H.; Liu, B.; Liu, P.; Zhang, J.; Wang, Y. Solubility of Fenofibrate in Different Binary Solvents: Experimental Data and
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Results of Thermodynamic Modeling. J. Chem. Eng. Data 2016, 61, 3177−3183. (2) Jouyban, A. Review of the cosolvency models for predicting solubility of drugs in water-cosolvent mixtures. J. Pharm. Pharm. Sci. 2008, 11, 32−57. (3) Jouyban, A. Handbook of Solubility Data for Pharmaceuticals; CRC Press: Boca Raton, FL, 2010. (4) Barzegar-Jalali, M.; Jouyban-Gharamaleki, A. A general model from theoretical cosolvency models. Int. J. Pharm. 1997, 152, 247− 250. (5) Acree, W. E., Jr. Mathematical representation of thermodynamic properties. Part II. Derivation of the combined nearly ideal binary solvent (NIBS)/Redlich-Kister mathematical representation from a two-body and three-body interactional mixing model. Thermochim. Acta 1992, 198, 71−79. (6) Jouyban-Gharamaleki, A.; Valaee, L.; Barzegar-Jalali, M.; Clark, B. J.; Acree, W. E., Jr. Comparison of various cosolvency models for calculating solute solubility in water-cosolvent mixtures. Int. J. Pharm. 1999, 177, 93−101. (7) Jouyban, A.; Chan, H. K.; Chew, N. Y. K.; Khoubnasabjafari, M.; Acree, W. E., Jr. Solubility prediction of paracetamol in binary and ternary solvent mixtures using Jouyban-Acree model. Chem. Pharm. Bull. 2006, 54, 428−431. (8) Jouyban, A.; Acree, W. E., Jr. In silico prediction of drug solubility in water-ethanol mixtures using Jouyban-Acree model. J. Pharm. Pharmaceut. Sci. 2006, 9, 262−269. (9) Jouyban, A. Solubility prediction of drugs in water-PEG 400 mixtures. Chem. Pharm. Bull. 2006, 54, 1561−1566. (10) Jouyban, A. In silico prediction of drug solubility in waterdioxane mixtures using Jouyban-Acree model. Pharmazie 2007, 62, 46−50. (11) Jouyban, A. Prediction of drug solubility in water-propylene glycol mixtures using Jouyban-Acree model. Pharmazie 2007, 62, 365− 367. (12) Jouyban-Gharamaleki, A.; Acree, W. E., Jr. Comparison of models for describing multiple peaks in solubility profiles. Int. J. Pharm. 1998, 167, 177−182. (13) Jouyban, A.; Clark, B. J. Describing solubility of polymorphs in mixed solvents by CNIBS/R-K equation. Pharmazie 2002, 57, 861− 862. (14) Jouyban, A.; Soltanpour, Sh.; Soltani, S.; Tamizi, E.; Fakhree, M. A. A.; Acree, W. E., Jr. Prediction of drug solubility in mixed solvents using computed Abraham parameters. J. Mol. Liq. 2009, 146, 82−88. (15) Jouyban, A.; Shayanfar, A.; Panahi-Azar, V.; Soleymani, S.; Yousefi, B. H.; Acree, W. E., Jr.; York, P. Solubility prediction of drugs in mixed solvents using partial solubility parameters. J. Pharm. Sci. 2011, 100, 4368−4382. (16) Jouyban-Gharamaleki, A.; Hanaee, J. A novel method for improvement of predictability of the CNIBS/R-K equation. Int. J. Pharm. 1997, 154, 245−247. (17) Jouyban, A.; Martinez, F.; Acree, W. E., Jr. Correct derivation of a combined version of the Jouyban-Acree and van’t Hoff model and some comments on ‘Determination and correlation of the solubility of myricetin in ethanol and water mixtures from 288.15 to 323.15 K’. Phys. Chem. Liq. 2017, 55, 131−140. (18) Fan, S.; Yang, W.; Guo, Q.; Hao, J.; Li, H.; Yang, S.; Hu, Y. Thermodynamic models for determination of the solubility of boc-(r)3-amino-4-(2, 4, 5-trifluorophenyl) butanoic acid in different pure solvents and (tetrahydrofuran + n-butanol) binary mixtures with temperatures from 280.15 to 330.15 K. J. Chem. Eng. Data 2016, 61, 1109−1116. (19) Yang, W.; Fan, S.; Guo, Q.; Hao, J.; Li, H.; Yang, S.; Zhao, W.; Zhang, J.; Hu, Y. Thermodynamic models for determination of the solubility of 4-(4-aminophenyl)-3-morpholinone in different pure solvents and (1,4-dioxane + ethyl acetate) binary mixtures with temperatures from (278.15 to 333.15) K. J. Chem. Thermodyn. 2016, 97, 214−220.
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