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Predicting the Solubility of Sulfamethoxypyridazine in Individual Solvents I: Calculating Partial Solubility Parameters
Predicting the Solubility of Sulfamethoxypyridazine in Individual Solvents I: Calculating Partial Solubility Parameters P. BUSTAMANTE*, 8. ESCA?ERA*, A. MARTIN*", AND E. SELLcS* Received May 3, 1988, from the De artamento de Farmacia Galenica, Facultad de Farmacia, Universidad de Alcala de Henares, Madrid, Spain, Accepted for publication December 29, and the *Drug Dynamics Institute, &Nege of Pharmacy, The University of Texas, Austin, TX 78712. 1988. .-
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-
....
Abstract Sulfamethoxypyridazine, a representativemodel of a drug molecule, is used to test the extended Hansen method for estimating partial solubility parameters of solid compounds. Solubilities are determined in polar and nonpolar solvents. The method provides reasonable partial parameters for the sulfonnmide, and it may be useful in obtaining partial parameters for other drug molecules. A four-parameter extended Hansen approach involving proton donor and acceptor parameters is used in fitting the data to a theoretical model. A term, W,,is introduced as an empirical measure of soluie-solvent interactionsdue to hydrogen bonding. The use of the ernpirical term W, allows the researcher to fit experimental solubilities and thus design regression models and equations which provide a reasonable prediction of solubilitiesof a polar drug in a number of very different solvents. A Flow-Huggins size correction term improves the prediction of sulfamethoxypyridazine solubilities in these irregular solutions. -.
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..
.
-.
-
Theoretical Section Martin et a1.10 introduced an adhesive energy density (W) as a measure of solute-solvent interaction. The term W replaces S,S, in the Hildebrand equation" and may be calculated from the following expression:
In (X2/X2) = In a,
U(8:
+ 8;
-
2W)
(2)
where:
and:
-
U Solubility parameters of drug molecules have received considerable interest by the pharmaceutical scientist and have been used in pharmacokinetic and biological studies, as well as in basic pharmaceutics.14 The solubility parameter concept has been suggested as a possible substitute for the partition coefficient in a study of the passage of a drug across living membranes.3 Hansel15 defined three partial solubility parameters (a,, a,, h) to account for dispersion and polar effects, as well as for hydrogen bonding and other Lewis acidbase interactions:
where S,is the total solubility parameter, and is quite similar for most compounds to the original solubility parameter, 6, as defined by Scatchard and Hildebrand.6 Partial solubility parameters have not been inore widely employed in pharmacokinetics and structure-activity studies, perhaps because there are few methods for determining partial parameters which can be applied to pharmaceutical compounds.6 The extended Hansen method:' 9 has been proposed to obtain partial solubility parameters of crystalline compounds. This method may be usefully applied to drug molecules; many of them are solids and their partial parameters cannot be obtained with the methods ordinarily used for solvents. The purpose of this investigation is to test the applicability of the extended Hansen method for a sulfonamide, namely sulfamethoxypyridazine,in order to determine its partial solubility parameters. The method is also used in predicting the solubility of the drug in individual solventsranging from nonpolar to highly polar and hydrogen bonding Another objective of this work is to try a new model for predict mg solubilities based on regression analysis.An empirical parameter W,is introducedand tested for its acceptability in modeling; solute-solvent interactions due to hydrogen bonding. 0022-3549/89/0700-0567$01 .OO/O Q 1989, American Pharmaceutical Association
=
=
V2&RT
(4)
where 2, is the ideal mole fraction solubility of sulfamethoxypyridazine, X , is the observed solubility, In a, is the logarithm of the activity coefficient of the drug, 8, and 8, are the solubility parameters of the solvent and solute, respectively, V , and V, are the molar volumes of the solvent and solute, respectively, 4l is the volume fraction of the solvent W , ( 1 - X,)/[V,(l - X , ) t V&J, R is the gas constant, and T is the absolute temperature. Throughout the paper, subscript 1 refers to solvent and subscript 2 to solute (drug). The parameter K is a solute-solvent interaction parameter12 which relates the geometric mean S,S, to W. This method, called the extended Hildebrand solubility approach,1().12provides good results in solvent mixtures. To account for the deviation of a drug solution from regular solution behavior because of specific solute-solvent interactions and size differences between solute and solvent, eq 2 may be written:
The second right-hand term of eq 5, involving the molar volumes of solute and solvent, is the Flory-Huggins term.6 Rearranging eq 5 and designating the left-hand terms with the letter B, yields the following expression:
B
= 8:
+ 6:
-
2W
(6)
where:
The extended Hansen method7 9 uses the following regression models for calculating the partial solubility parameters of a drug molecule: a three-term model, Journal of Pharmaceutical Sciences I 567 Vol. 78, No. 7, July 1989
and a four-term mode1,Y
where the D and D' coefficients are obtained from multiple regression analysis, and 6, and S, express the acid or proton donor character and the basic or proton acceptor character of the solvent or solute.13 Equations 8 and 9 are used here to calculate the partial solubility parameters of sulfamethoxypyridazine. In order to introduce the Flory-Huggins contribution, B (eq 7) is used as the dependent variable in eqs 8 and 9 instead of (In a,)/U. Equation 8 is a linear combination of dispersion, and polar and hydrogen bonding interactions written as perfect squares in which geometric means are assumed for the three kinds of interactions. However, as Keller et al. suggested,13the original scheme of Hansen can be improved by dividing the polar term 6, into orientation (6,) and induction (S,,) contributions. The authors13 used this correction in their studies on chromatographic retention. These parameters, So and S,,, may also be significant in representing physical-chemical properties of a drug molecule in solution, and consequently may improve the results. According to Keller et al.,13 the interaction of species 1and 2 due to orientation forces is (SIC,- S2J2, and for the induction forces is 2(SId - S2d)(Sl,n - &,n). Applying these ideas to solubility, the Hansen polar term (alp - 2 )S, may be substituted by:
The term involving S, in the Hansen model assumes a geometric mean of 6, and 62,; that is, &,&& = (6~h&)"'. In order to correct this assumption, the hydrogen bonding interaction term is written in this work as (a;, t S&, - 2wh); W, is introduced as an empirical adhesive energy density parameter for the interaction in solution of hydrogen bonded species 1 and 2, and substitutes for the original Hansen term, 81h82h. Applying these arguments, a new model may be written:
regression analyses. The term C , includes the partial parameters of the solute (which are constants and therefore do not appear in the regression) and their associated regression coefficients:
Equation 13 is a regression model used in this work to calculate the mole fraction solubility of sulfamethoxypyridazine; X, is included in the term B. The calculated solubilities using eq 13 may then be compared with those obtained from the extended Hansen method.
Experimental Section The solubilities of sulfamethoxypyridazine14 (Interchimia, Hamburg, Germany, lot no. 4740) were determined in individual solvents14 (analytical or spectrophotometric grade, Panreac, Monplet & Esteban, Barcelona) in a constant temperature bath held at 25 f 0.05 "C. After equilibrium was reached, the solutions were filtered using filters of pore size <1 km (Durapore or Fluoropore), and samples were assayed using a double-beam spectrophotometer (Bausch and Lomb 2000). The densities of the saturated solutions were determined in 10-mLpycnometers. The molar volume of the solute was estimated from apparent densities, according to Gucker et a1.15 to be 174.6 mWmol, which agrees with the molar volume used in this work as calculated by the Fedors method16 (172.5 mLimol). The heat of fusion and temperature of fusion of the solute, determined by differential scanning calorimetry (Mettler TA 3000), are 8110.5 cal/ mol and 180.4 "C,respectively. The results are the average of at least three measurements. The ideal solubility of sulfamethoxypyridazine is calculated from the heat of fusion and temperature of fusion:
1nX;
Now, eq 12 can be written as a regression model, where B is the dependent variable and the partial solubility parameters of the solvents are the independent variables:
-&iRT(T,
- TyT,
(15)
where AHf= 8110.5calimol, T,,, = 453.6K,and T = 298.2 K. The ideal solubility at 25 "C is X i = 9.1484x The solvents used, solubility parameters, and observed mole fractions are listed in Tables I and 11. The activity coefficients were calculated from the ideal mole fraction solubility and the observed mole fraction solubilities. The partial solubility parameters and molar volumes of the solvents are found in the literat~re.6~9
Results and Discussion When the extended Hansen method (eq 8) is applied to sulfamethoxypyridazine, the following regression equation using the three-solubility parameter approach8.9 is obtained:
-4.89416fd + 74.850361d+ 0.28746fp6.450861, + 0.10766fh 0.76088,h - 261.682
(In aJU where the left-hand term, B, includes the activity coefficient of the drug and the Flory-Huggins size correction as defined in eq 7. It will be noted that eq 11is not a regression equation; W, in cq 11is calculated knowing B and the partial solubility parameters of the solute and the solvents. Expanding and rearranging eq 11, one obtains:
=
=
(16)
wheren = 30,s = 4.69,r2 = 0.835,F = 19.46,andF(6,23,0.01) = 3.71. The partial parameters of the solute, calculated from eq 16 according to the extended Hansen method,s are:&, = 7.65, S,, = 11.22, and 62, = 3.58, and the total solubility parameter is S,, = 14.04.Equation 16 is written, according to the model represented by eq 8, as:
(In az)/U = -4.8941(13,~- 7.65)' + 0.2874(6,, 11.22)' 0.1076(6,h - 3.58)' - 12.826 (17)
+
To test the influence of the Flory-Huggins term, B is used instead of (In %)/U as a dependent variable in the threeparameter model (B is defined in eq 7):
B = -2.61736fd + 36.17836,d + 0.34886;, 6.558661, + 0.18926fh - 1.837161, - 95.0541(18) 568 I Journal of Pharmaceutical Sciences Vol. 78, No. 7, July 1989
Table I-Partial and Total Solubility Parameters and Molar Volumes of the Solvents
-
No.
Solvent
Vl mUmola I
-
132.60 106.90 98.50 89 40 80.80 74.00 85.70 158.40 108.60 80.90 106.00 91.50 92.40 93.00 92.00 76.90 103.90 75.10 77.40 58.70 71.30 89.909 40.70 59.10 95.30g 73.70 55.90 88.60 73.20 18.10
Butyl acetate
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Toluene Ethyl acetate Benzene Chlorofolrm Acetone Dioxane 1-0ctanol 1-Pentanol Pyridine Cyclohsxanol Aniline fsobutyl alcohol N,KDiniethylacetamide Butyl alcohol Isopropyl alcohol Benzyl ,alcohol Propyl alcohol N,NDimethylformeimidc! Ethyl alcohol
I
Dimethylsulfoxide 1,3-Butanediol
Methyl alcohol NMethylformamide Diethylone glycol 1,2-PropanedioI
Ethylene glycol 1,4-Butanediol Glycerol
Water
61T,
6103
( c a ~ / c m ~ ) ” ~ ~ (cal/cm3) 8.49 8.88 9.04 9.07 9.26 9.76 10.01 10.25 10.58 10.68 10.95 11.02 11.11
2.30 0.80 3.66 0.77 2.26 4.58 2.55 3.57 4.83 3.66 4.59 3.75 5.92 5.22 5.79 6.32 4.86 6.76 6.25 8.16 6.85 8.15 10.44 9.06 8.60 9.05 10.67 9.96 11.31 20.26
11.12
11.31 11.50 11.64 12.00 12.14 12.96 13.04 14.149 14.48 14.48 14.64g 14.78 16.03 16.35 17.65 23.39
4 in, (cal/cm3)”‘d 0.49 0.05 0.92 0.04 0.29 1.09 0.39 1.41 1.78 0.76 1.56 0.90 2.27 1.77 2.16 2.15 1.72 2.40 2.12 2.74 2.34 4.18 3.10 3.40 4.93 4.22 4.46 6.15 6.55 5.20
w 7 t 3
whcalc,
~ a l i c r n ~ ~ cal!cm3‘ 24.55 16.30 29.07 16.66 21 .oo 25.36 26.80 30.17 37.30 23.00 35.47 33.31 44.75 36.16 44.18 46.55 40.36 51.72 38.82 64.30 44.31 75.36 88.54 49.29 75.19 89.28 110.99 94.67 135.89 329.60
22.05 18.19 26.06 18.42 20.59 24.20 31.89 34.07 41.89 24.92 38.47 36.57 50.12 32.78 48.18 50.45 40.27 54.93 34.97 65.85 35.35 -h 83.02 37.18
-h
87.90 107.47 94.10 138.98 329.59
a From ref 9. Total solubility parameter calculated from the Hansen partial parameters, (eq 1 and ref 9). From eq 22, using SIT and V , from Table I, columns 3 and 4,and S,, frlom ref 9. (seeAppendix). From eq 23, &,and V, from Table I (seeAppendix). Calculated from eq 11 (seeAppendix). Calculated from eq 11 (see Appendix). Estimated values from eq 26. The molar volumes, V,, of solvents nos. 22 and 25, as well as their partial solubility paramelers for calculating S,, (eq 1) are found in ref 6, p 158. Dashes indicate that W,,,,, cannot be calculated from eq 26 because a,, and S,, for solvents nos. 22 and 25 are not presently available.
‘
wheren = 30, s = 4.26,1! = 01.888, F = 30.47,andF(6,23,0.01) = 3.71. From eq 18,the partial parameters are obtained as S,, = 6.91, , S = ‘9.40, and S,, = 4.85, and the total solubility parameter is 12.63. Equation 18 is written in the form of eq 17 as follows:
B
=
The four-parameter model involving proton donor and acceptor parameters to replace S, (eq 9) is applied here using B as the dependent variable instead of (In cu,)/U.
B
-2.6173(6,d - 6.91)2 + O.34988(61, - 9.4012 + 0.1892(S,h - 4.85)2 - 5.3534 (19)
Partial solubility parameters for sulfamethoxypyridazine calculated using a method of group contributions (Table 111), are S , = 8.84, 8, = 5.95, & , = 5.94, and S,, = 12.2; the S,, value calculated from the Fedors method16 is 12.9. The total solubility parameter, hqP, and the partial parameters, 82, and &&, as calculated are not too different from S,,, S2,, and 8, as estimated by eqs 17 and 19. However, S,, estimated from both equations and 8,, (calculated using the group contribution method (Table 111) are quite different. As observed in eqs 17 and 19, the sign of the term involving the dispersion parameter is negative and its regression coefficient is quite different from unity. The extended Hansen method provides a n 1.2 of 0.835, and this fit is not very satisfactory, probably because the partial paramchter S, does not reflect well the proton-donor electron-accepitor characteristics of complex organic drug mailecules.9 Equation 18, which includes the Flory-Huggins term, improves ? by 5% and provides a 8,,,which agrees better than ,S of eq 16 with the hydrogen bonding parameter calculated from the method of group contributions (Table 111). This could mean that the Flory-Huggins term must be considered to predict reasonable solul~ilitiiesof sulfamethoxypyridazine.
0.71836& - 17.938761, + 0.41716~,,6.942981, - 1.994161, - 2.913861, + 0.66476,,6,b + 127.6507 (20)
=
where n = 28, r2 = 0.929, s = 3.60, F = 37.48, and F(7, 20, 0.01) = 3.71. The partial parameters obtained from an analysis of eq 20 are 8,, = 12.48, ,a, = 8.32, S,, = 4.38, &, = 3.00; and, eq 20 becomes:
B
=
0.7183(8,d - 12.4812 + 0.4171 (SIP - 8.32)2 + O.66479(61, - 4.38)(61b - 3.00) - 30.7147 (21)
The total solubility parameter of sulfamethoxypyridazine obtained by this method is S,, = 15.8. The hydrogen bonding parameter can be calculated from the acidic and basic parameters9 8gh = 26,,82b; therefore, 82, = 5.13. This value agrees with 62, from eq 18 and 82, calculated from group contributions. The use of a system of four parameters together with the Flory-Huggins term has improved the value of 1.2. The regression coefficient of the term involving 8, in eq 21 is nearer to unity than in eqs 17 and 19 and its sign is positive, as it should be for the model of three or four partial parameters.8 Equation 20 improves eq 16 because the term S, is separated into acidic and basic contributions. It also takes into account, by use of the Flory-Huggins term, the size differences between solute and solvent. Journal of Pharmaceutical Sciences I 569 Vol. 78, No. 7, July 7989
Table Il-Observed
Solvent No.
and Calculated Solubilities of SulfamethoxypyridazineIn Individual Solvents at 25 OC
Equation 18 XZB
Xzca,c
1 2 3 4 5 6 7
0.00088 0.00004 0.00205 0.00006 0.00129 0.00918 0.02390 0.00031 0.00057 0.02329 o.oooai 0.06301 0.00043 0.09389 0.00073 0.00054 0.01347
a
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
o.oooa3
0.08807 0.00126 0.20980 0.00301 0.00294 0.08721 0.02891 0.00531 0.00496 0.00422 0.00128 0.00004
0.00016 0.00013 0.00046 0.00017 0.00101 0.00843 0.00200 0.00037 0.00033 0.08042 0.00096 0.04290 0.00038 0.03043 0.00060 0.00063 0.01695 o.oooa7 0.06793 0.00152 0.08842 0.00282 0.00218 0.07096 0.01267 0.00140 0.00372 0.01040 0.00062 4.6 X
Equation 20 Error, %b 81.2 -225.0 77.1 -183.3 21.2 8.1 91.6 - 19.3 42.1 -245.3 - 18.5 31.9 11.6 67.6 17.8 - 16.7 -25.8 -4.8 22.9 -20.6 57.8 6.3 25.8 18.6 56.2 73.6 24.9 - 146.5 51.5 99.9
X2I,C
0.0001a 0.00005 o.ooo5a 0.00005 0.00064 0.00877 0.00845 0.00059 0.00047 0.06134 0.00162 0.05281 0.00033 0.05088 0.00077 0.00070 0.01282 0.00092 0.07907 0.00101 0.10095 -C 0.00353 0.05600
-
C
0.001 14 0.00369 0.01 794
o.oooia
0.000045
Equation 28
Error, %b 79.3 --25.2 71.4 16.6 50.7 4.4 64.7 -90.4 17.5 - 163.4 - 100.0 16.2 23.2 45.8 -5.5 -29.6 4.8 - 10.8 10.2 19.8 51.9
-
C
-20.2 35.8 -C 78.5 25.6 -325.2 85.9 -1.2
Xzca,c
0.00056 0.00003 0.00148 0.00004 0.00127 0.01547 0.01787 0.00033 0.00056 0.041 15 0.00090 0.05004
o.ooo3a
0.03783 0.00062 0.00065 0.01672 o.ooo6a 0.06849 0.00065 0.1068
-
C
0.00109 0.04431
-C
0.00174 0.00418 0.00297 0.00115 0.00005
Error, %b 36.5 25.0 27.3 33.3 1.9 -68.6 25.0 -6.4 1.7 -76.7 -1 1.1 20.6 11.6 59.7 15.0 -20.3 -24.1 18.1 22.2 48.4 49.0
-
C
62.7 49.2
-
C
67.2 15.7 29.6 10.1 -25.0
percent. Dashes indicate that solvents nos. 22 and 25 are not used in eqs 20 and 28 for a Experimental solubilities. Error = (X, - X2ca,.J/Xz calculating X,,, because Sla and S,, are not presently available (see also footnote h on Table I). polarity of the solvent could result from an interaction between the drug and the solvent, a “chameleonic effect” as described by Hoy.18 The solvents used in this work differ in nature, and sulfamethoxypyridazine may self-associate through polar interactions in solvents showing low polarity H 2 N e S 0 2 N H - ( O > OCH, (e.g., benzene, toluene), whereas in the strongly polar and N-N hydrogen bonding solvents (e.g., dimethylsulfoxide, N f l Atom or Number of dimethylformamide), the drug may be solvated by the polar Fdb F,b uIlb Group Groups groups of these solvents. The term w, is used in eq 11 instead of &h&h as an 8400 280 610‘ empirical measure of unsymmetrical solutcsolvent interac160 210 3100 5000 40 1600 tions due to hydrogen bonding and other acid-base interac420 0 0 tions. The terms So and Sin are used instead of Sp to correct the 0 0 1200 geometricmean assumed by the term involving Spin eqs 8 and 280 0 0 9. The term W, may be calculated from eq 11 by assigning 9000 300 1200 invariant partial parameters for sulfamethoxypyridazine cal440 culated from a group contribution method (Table 111). An ZFd = 3120 (ZFZ)”‘ = 2101.5 XUh = 25500 example of such a calculation is shown in the Appendix, and a fi -. ZF,IV = 31201172.5 = 18.09 MPaltZd= 8.84 cal”%n 312; 8, the results are listed in Table I, column 7. The partial = (XFg)i’2/V = 21 01.51172.5 = 12.18 MPa1I2= 5.95 ~ a l ” ~ c m -S,,~ = ’ ~ ; parameters So and Sinfor solvents and solute are calculated (SU,,/V)”* = (25500/172.5)i’z= 12.16 MPal” = 5.94~al”‘cm-~’‘; S, = according to Beerbower et al.9 as follows: (6; + # + %)l1‘ = 12.20 ca11’2cm 3’2. bVan Krevelen and Hoftyzer, from ref6, pp 8586. Hansen and Beerbower, from ref 6, p 85. dThe SI unit MPa”, is related to the cgs units ~al~’‘cm-~’~ for solubility parameters by the equation MPa’” = 2.0455 caI1’‘cm 3’2. Sin = 0.0007SfV (23) The values of the partial parameters obtained can vary The molar volume, V, and the solubility parameters, S, and depending on whether one uses a model of three or four S,, for the solvents were obtained from the literat~re.6,~ parameters. The values of the partial parameters estimated Calculated values of a, and $in for the solvents are found in by regression equations may also change with the nature and Table I; 6,, and S,, for the solute are obtained from S,, = number of solvents used. Martin et al.17 studied the variation 12.20, S,, = 8.84 (Table HI), and V, = 172.5 mL/mol. Sample of the solubility parameter of drug molecules in mixed polar calculations for S,, and Sin are found in the Appendix. As with systems. They suggested that the change in 8, with variable
Table IlI-Group Contribution Method to Calculate Partial Parameters of Sulfamethoxypyridazinea
~~
570 I Journal of Pharmaceutical Sciences Vol. 78,No. 7,July 1989
the W term of the extended Hildebrand approach,lO W, values (Table I, column 7) are not related to S,, by a linear expression, bu.t rather by a polynomial relationship (Figure 1):
Wh
14.869 + 2.73961h 0.0036:,, + 0.0296& 1
-
(24)
where n = 30, .r2 = 0.994, s = 4.91, F = 1440, and F(3,26,0.01) = 4.64. Other empirical relationships from which W, may be obtained involve the acidic and basic parameters: wh =
4.477(81, t 9.48061,
-
17.139
-1.03781, + 0.0716:, + 1.36161b + 0.096:b + 0.8966,,6,b + o.(301(61,61b)2 + 17.696 (26)
where n = 28, r2 = 0.995, s == 4.82, F = 713, andF(6,21,0.01) = 3.81. The Wh(calc)values (i.e., calculated values from eqs 24, 25, and 26) were tested in the regression model suggested by eq 13, and the best fit (highest 3)was provided using Wh(calc) from eq 26 (Table I, column 8). The regression equation is:
B = -1.92086:d + 381.95796,d + 0.34736:, 7.938961, - 4.?1270161d61in+ 35.419061in + 0.69956fh - 0.8929Wh(,,1,, - 92.3671 (27)
350
s f
300
250
i-
C
0 .c 0 200
2 a,
c
S + S
-a>0,
f
150
t
where n = 28, 1-2 = 0.960, s = 2.84, F = 48.54, and F(9, 18, 0.01) = 3.60. The use of UI improves r2 of eq 27 by 3%. Figures 2 and 3 show selected residual plots for the models, eqs 18 and 28; the points are somewhat more scattered in Figure 2. Equation 28 improves the r2 over eq 18 by 7%, and it should be expected that the calculated solubilities from eq 28 would agree better with the observed solubilities. The mole fraction solubility (Xacalc),calculated from the several regression models used (eqs 18, 20, 28) following an iteration procedure,lg as well as the observed mole fractions for sulfamethoxypyridazine, are listed in Table 11. If a 30% error (calculated value within 30% of observed solubility) is taken as a criterion for acceptability,g eq 28 provides the highest number of acceptable calculated solubilities (64% of the cases). The variation of observed solubilities of sulfamethoxypyridazine in the solvents used is very wide (mole fraction values between 0.00003 and 0.2098), and it is difficult to fit such experimental data using a single equation to provide solubilities within 30% error. In addition, the crystalline form of the drug may be altered in highly polar solvents,20721and this will require a determination of the heat of fusion for the crystalline drug in the hydrate form in each solvent. To better compare observed and calculated solubilities, Table IV shows the correlation matrix of experimental and calculated solubilities. It provides a clear and brief summary of how well the experimental and calculated mole fraction solubilities correlate. The highest correlation coefficient corresponds to eq 28. The results show that, in addition to the Flory-Huggins term to correct size differences between solute and solvent, terms to correct the geometric mean assumption to account for specific Lewis acid-base interactions of solute and solvent are desirable in fitting the experimental solubility data of a drug such as sulfamethoxypyridazine in polar and nonpolar solvents using a single equation.
Conclusions
100
c
3 0
cn
50
0
B = O.22106:, - 6.045261, + 0.38706~, 8.031661, -1.697061dSlin + 14.368761in + 0.60016fh 0.7829Wh,,,,,. + 76.1449 - 5.8100Y (28)
(25)
where n = 27, r2 = 0.962, s = 12.39, F = 309.77, and F(2,24, 0.01) = 5.61. Equation 25 is improved when W, is regressed in a power series on Slrr, Sl,,,and SlaSlb: wh =
where n = 28, r2 = 0.933, s = 3.58, F = 33.28, and F(8, 19, 0.01) = 3.63. The solute-solvent interaction parameter K (ey 3) may be used as a criterion to choose a n indicator variable, W,taking Y = 1whenK > 1and = 0 when K 5 1, in order to improve the correlation coefficient of eq 27. The term K is calculated for each solvent from eq 3. An example of such a calculation is shown in the Appendix. The regression equation involving the indicator variable Y' is:
I'
1
0
5
-
10
15
20
Hydrogen Bonding Parameter, 6, h Figure 1-Relationship of the s.olute-solvent interaction term W, and the solvent partial hydrogen bonding parameter of 6 , , for sulfamethoxypyridazine in 30 solvents. The line was fitted to the experimental points using eq 24. The iiumbers associated with each point correspond to the solvents in Table 1.
The extended Hansen method729 is found useful to obtain partial parameters for a solid drug, particularly when the Flory-Huggins term is introduced into the model. It is also of value to test the results obtained using group contribution methods. A group contribution method6 can be employed to calculate empirical terms such as W, which can then be used to account for unsymmetrical interactions that become important in irregular solutions. Parameterization as demonstrated in this report allows the researcher t o use the fit of experimental solubilities to design regression models and equations which allow a reasonable prediction of solubilities of polar drugs in a range of very different solvents.
Journal of Pharmaceutical Sciences I 571 Vol. 78, No. 7, July 1989
-
Table IV-Correlation Matrix of Observed (X,) and Predicted (X,,) Mole Fraction Solubilities for Sulfarnethoxypyridazinee
Q .
0
Variable 6 -
0
2 3 -
(eq 18) (eq 20) Xzcat (eq 28) %cat
0
0
h
&cat
a
0
-6 -9
I L
-10
=
. 0
L
L
I
I
10
20
30
40
Ql
-
1.000 0.9741
1.000
- &d)
-
0.001,
-
82,)'
+ (810 -
(81in - 82in) + 8fh
820)'
-I-
a&,
+ -
BY2
(All
where & = 8.84, F& = 5.94 (Table 1111, &, = 4.75, hin= 2.72 (see below), a,, = 7.7, a,, = 3.12 a,, = 2.3, hi,,= 0.49 (Table I), and B = 8.1769 (B is defined in eq 7). Therefore, W, = (1.2996 + 6.0025 + 9.61 + 35.2836 - 8.1769)/2 = 24.551 (Table I, column 7). Sample Calculation of, ,a and a,, (eqs 22 and 23)-For sulfamethoxypyridazine, the following equation is used: =
(8& - 8&)/(1+ 0.001482,VcJ
(A2)
where a,, = 12.20, hd= 8.84 (Table III), and V, = 172.5 from the Fedorsmethod.'6Therefore, go= (12.202- 8.84,)/3.13486 = 22.5511, a,, = 4.75, and Szin = O.O0076~,V, = 2.72. Sample Calculation of W and K-From eq 6:
W
=
(Sf,
+ 8&
- B)/2
(A3)
Therefore, for sulfamethoxypyridazine in butyl acetate, a,, = 8.49 (Table I, column 41,,,a = 12.20 (Table III), B = 8.1749, and W = (8.49, + 12.202 - 8.1769)/2 = 106.372. From eq 3, the following can be calculated:
6
h
3
m
(A4)
References and Notes
a S
0
0
$
-
Appendix
8&,
Figure 2-A plot of the residuals versus the predicted B value for sulfamethoxypyridazine in 30 solvents using eq 18.
-
-
1 .oooo 0.9655 0.9228
Sample Calculation of W,-For sulfamethoxypyridazine in butyl acetate, the following equation is used (from eq 11):
2(&
Predicted B Value (BWlc)
i
Significance level, a
wh = [(&d
0
-
1 .ooo 0.8252 0.9091 0.9505
xz
0
0
-3 0
U
-6
-9
B -10
0
I
1
I
10
20
30
40
Predicted B Value (Bcalc) Figure 3-A pbt of the residual versus the predicted B value for sulfarnethoxypyridazinein 28 solvents using eq 28.
Although it would be desired that the calculated solubilities of this study showed closer correspondence with the experimental results, it is gratifying to observe how well the procedures, involving partial cohesive parameters coupled with regression analysis, serve to reproduce the experimentally determined solubilities of sulfamethoxypyridazine in solvents ranging from apolar (benzene and toluene), through amphiprotic (alcohols, glycols, and water), to dipolar aprotic (N-methylformamide and dimethylsulfoxide). 572 I Journal of Pharmaceutical Sciences Vol. 78, No. 7,July 1989
1. Cammarata, A,; Yau, S. J.; Rogers, K. S. J . Med. Chem. 1971,14, 1211. 2. Khalil, S. A,; Ossama, Y.; Moustafa, M. A. Can. J . Pharm. Sci. 1976, 11, 121. 3 . Adjei, A.; Newburger, J.; Stavchansky, S.; Martin, A. J . Pharm. Sci. 1984, 73, 742. 4. Bustamante, P.; Selles, E. J . Pharm. Sci. 1986, 75, 639. 5. Hansen, C. M. J . Paint Tech. 1967,39, 104. 6. Barton, A. F. Handbook of Solubility Parameters and Other Cohesion Parameters; CRC: Boca Raton, FL, 1983. 7. Martin, A.; Wu, P. L.; Adjei, A.; Beerbower, A.; F'rausnitz, J. M. J . Pharm. Sci. 1981, 70,1260. 8. Wu, P. L.; Beerbower, A.; Martin, A. J. Pharm. Sci. 1982, 71, 1285. 9. Beerbower,A.; Wu, P. L.; Martin, A. J . Pharm. Sci. 1984,73,179. 10. Martin, A.; Newburger, J.; Adjei, A. J . Pharm. Sci.1980,69,487. 11. Hildebrand, J.H.; Prausnitz, J.M.; Scott, R.L. Re ular and Related Solutions; Van Nostrand Reinhold: New Yorf, 1970. 12. Martin, A.; Wu, P. L.; Adjei, A,; Mehdizadeh, M.; James, K. C.; Metzler, C. J . Pharm. Sci. 1982,39, 104. 13. Keller, R. A,; Karger, B. L.; Snyder, L. R. Gas Chromatography; R. Stack, Ed.; Institute of Petroleum: London, 1971, pp 125-140. 14. The sulfonamide, sulfamethoxypyridazine, has been withdrawn from the market as a n antibacterial agent in the United States. It is used in this study simply as a model, the solubility characteristics of which are representative of a class of sulfonamides. The solvents emplo ed in this investigation are not suggested for use in clinicaypractice but rather as compounds having a wide range of solubilityfarameters. A number of these solvents are toxic and are not use either internally or externally in humans or animals.
15. Gucker, F. T.;Ford, W I.; Moser, C. E. J . Phys. Chem. 1939,43, 153. 16. Fedors, R. F. Polym. Eng. Sci. 1974, 14, 147. 17. Martin, A.; Wu, P. L.; Liron.,Z.; Cohen, S. J . Pharm. Sci. 1985,74,
Kruif, C. G.; Wilting, J. Int. J . PAarm. 1983, 14, 79.
Acknowledgments -
B3R.
18. H o i , K.J . Paint Technol. 1970,42, 76. 19. Martin, A.; Swarbrick, J.; Cammarata, A. Physical Pharmacy; 3rd Ed.; Lea & Febiger: Philadelphia, PA, 1983; p 285. 20. Bogardus, J. J . Pharm. Sci. 1983, 72,837. 21. Fokkens, J. G.; van Ameliafoort, J. G. M.; de Blaey, C. J.; de
Supported by Comision Interministerial de Ciencia y Tecnologia (CICYT), Spain (project no. PB86-0063) and by a grant provided to P. Bustamante by Instituto Nacional de la Salud (Fis Grant, 87118891. Sup orted in part by an endowed professorship provided to A. Martin by oulter R. Sublett.
8
Journal of Pharmaceutical Sciences I 573 Vol. 78, No. 7, July 1989
-
-
-
....
Abstract Sulfamethoxypyridazine, a representativemodel of a drug molecule, is used to test the extended Hansen method for estimating partial solubility parameters of solid compounds. Solubilities are determined in polar and nonpolar solvents. The method provides reasonable partial parameters for the sulfonnmide, and it may be useful in obtaining partial parameters for other drug molecules. A four-parameter extended Hansen approach involving proton donor and acceptor parameters is used in fitting the data to a theoretical model. A term, W,,is introduced as an empirical measure of soluie-solvent interactionsdue to hydrogen bonding. The use of the ernpirical term W, allows the researcher to fit experimental solubilities and thus design regression models and equations which provide a reasonable prediction of solubilitiesof a polar drug in a number of very different solvents. A Flow-Huggins size correction term improves the prediction of sulfamethoxypyridazine solubilities in these irregular solutions. -.
-
..
.
-.
-
Theoretical Section Martin et a1.10 introduced an adhesive energy density (W) as a measure of solute-solvent interaction. The term W replaces S,S, in the Hildebrand equation" and may be calculated from the following expression:
In (X2/X2) = In a,
U(8:
+ 8;
-
2W)
(2)
where:
and:
-
U Solubility parameters of drug molecules have received considerable interest by the pharmaceutical scientist and have been used in pharmacokinetic and biological studies, as well as in basic pharmaceutics.14 The solubility parameter concept has been suggested as a possible substitute for the partition coefficient in a study of the passage of a drug across living membranes.3 Hansel15 defined three partial solubility parameters (a,, a,, h) to account for dispersion and polar effects, as well as for hydrogen bonding and other Lewis acidbase interactions:
where S,is the total solubility parameter, and is quite similar for most compounds to the original solubility parameter, 6, as defined by Scatchard and Hildebrand.6 Partial solubility parameters have not been inore widely employed in pharmacokinetics and structure-activity studies, perhaps because there are few methods for determining partial parameters which can be applied to pharmaceutical compounds.6 The extended Hansen method:' 9 has been proposed to obtain partial solubility parameters of crystalline compounds. This method may be usefully applied to drug molecules; many of them are solids and their partial parameters cannot be obtained with the methods ordinarily used for solvents. The purpose of this investigation is to test the applicability of the extended Hansen method for a sulfonamide, namely sulfamethoxypyridazine,in order to determine its partial solubility parameters. The method is also used in predicting the solubility of the drug in individual solventsranging from nonpolar to highly polar and hydrogen bonding Another objective of this work is to try a new model for predict mg solubilities based on regression analysis.An empirical parameter W,is introducedand tested for its acceptability in modeling; solute-solvent interactions due to hydrogen bonding. 0022-3549/89/0700-0567$01 .OO/O Q 1989, American Pharmaceutical Association
=
=
V2&RT
(4)
where 2, is the ideal mole fraction solubility of sulfamethoxypyridazine, X , is the observed solubility, In a, is the logarithm of the activity coefficient of the drug, 8, and 8, are the solubility parameters of the solvent and solute, respectively, V , and V, are the molar volumes of the solvent and solute, respectively, 4l is the volume fraction of the solvent W , ( 1 - X,)/[V,(l - X , ) t V&J, R is the gas constant, and T is the absolute temperature. Throughout the paper, subscript 1 refers to solvent and subscript 2 to solute (drug). The parameter K is a solute-solvent interaction parameter12 which relates the geometric mean S,S, to W. This method, called the extended Hildebrand solubility approach,1().12provides good results in solvent mixtures. To account for the deviation of a drug solution from regular solution behavior because of specific solute-solvent interactions and size differences between solute and solvent, eq 2 may be written:
The second right-hand term of eq 5, involving the molar volumes of solute and solvent, is the Flory-Huggins term.6 Rearranging eq 5 and designating the left-hand terms with the letter B, yields the following expression:
B
= 8:
+ 6:
-
2W
(6)
where:
The extended Hansen method7 9 uses the following regression models for calculating the partial solubility parameters of a drug molecule: a three-term model, Journal of Pharmaceutical Sciences I 567 Vol. 78, No. 7, July 1989
and a four-term mode1,Y
where the D and D' coefficients are obtained from multiple regression analysis, and 6, and S, express the acid or proton donor character and the basic or proton acceptor character of the solvent or solute.13 Equations 8 and 9 are used here to calculate the partial solubility parameters of sulfamethoxypyridazine. In order to introduce the Flory-Huggins contribution, B (eq 7) is used as the dependent variable in eqs 8 and 9 instead of (In a,)/U. Equation 8 is a linear combination of dispersion, and polar and hydrogen bonding interactions written as perfect squares in which geometric means are assumed for the three kinds of interactions. However, as Keller et al. suggested,13the original scheme of Hansen can be improved by dividing the polar term 6, into orientation (6,) and induction (S,,) contributions. The authors13 used this correction in their studies on chromatographic retention. These parameters, So and S,,, may also be significant in representing physical-chemical properties of a drug molecule in solution, and consequently may improve the results. According to Keller et al.,13 the interaction of species 1and 2 due to orientation forces is (SIC,- S2J2, and for the induction forces is 2(SId - S2d)(Sl,n - &,n). Applying these ideas to solubility, the Hansen polar term (alp - 2 )S, may be substituted by:
The term involving S, in the Hansen model assumes a geometric mean of 6, and 62,; that is, &,&& = (6~h&)"'. In order to correct this assumption, the hydrogen bonding interaction term is written in this work as (a;, t S&, - 2wh); W, is introduced as an empirical adhesive energy density parameter for the interaction in solution of hydrogen bonded species 1 and 2, and substitutes for the original Hansen term, 81h82h. Applying these arguments, a new model may be written:
regression analyses. The term C , includes the partial parameters of the solute (which are constants and therefore do not appear in the regression) and their associated regression coefficients:
Equation 13 is a regression model used in this work to calculate the mole fraction solubility of sulfamethoxypyridazine; X, is included in the term B. The calculated solubilities using eq 13 may then be compared with those obtained from the extended Hansen method.
Experimental Section The solubilities of sulfamethoxypyridazine14 (Interchimia, Hamburg, Germany, lot no. 4740) were determined in individual solvents14 (analytical or spectrophotometric grade, Panreac, Monplet & Esteban, Barcelona) in a constant temperature bath held at 25 f 0.05 "C. After equilibrium was reached, the solutions were filtered using filters of pore size <1 km (Durapore or Fluoropore), and samples were assayed using a double-beam spectrophotometer (Bausch and Lomb 2000). The densities of the saturated solutions were determined in 10-mLpycnometers. The molar volume of the solute was estimated from apparent densities, according to Gucker et a1.15 to be 174.6 mWmol, which agrees with the molar volume used in this work as calculated by the Fedors method16 (172.5 mLimol). The heat of fusion and temperature of fusion of the solute, determined by differential scanning calorimetry (Mettler TA 3000), are 8110.5 cal/ mol and 180.4 "C,respectively. The results are the average of at least three measurements. The ideal solubility of sulfamethoxypyridazine is calculated from the heat of fusion and temperature of fusion:
1nX;
Now, eq 12 can be written as a regression model, where B is the dependent variable and the partial solubility parameters of the solvents are the independent variables:
-&iRT(T,
- TyT,
(15)
where AHf= 8110.5calimol, T,,, = 453.6K,and T = 298.2 K. The ideal solubility at 25 "C is X i = 9.1484x The solvents used, solubility parameters, and observed mole fractions are listed in Tables I and 11. The activity coefficients were calculated from the ideal mole fraction solubility and the observed mole fraction solubilities. The partial solubility parameters and molar volumes of the solvents are found in the literat~re.6~9
Results and Discussion When the extended Hansen method (eq 8) is applied to sulfamethoxypyridazine, the following regression equation using the three-solubility parameter approach8.9 is obtained:
-4.89416fd + 74.850361d+ 0.28746fp6.450861, + 0.10766fh 0.76088,h - 261.682
(In aJU where the left-hand term, B, includes the activity coefficient of the drug and the Flory-Huggins size correction as defined in eq 7. It will be noted that eq 11is not a regression equation; W, in cq 11is calculated knowing B and the partial solubility parameters of the solute and the solvents. Expanding and rearranging eq 11, one obtains:
=
=
(16)
wheren = 30,s = 4.69,r2 = 0.835,F = 19.46,andF(6,23,0.01) = 3.71. The partial parameters of the solute, calculated from eq 16 according to the extended Hansen method,s are:&, = 7.65, S,, = 11.22, and 62, = 3.58, and the total solubility parameter is S,, = 14.04.Equation 16 is written, according to the model represented by eq 8, as:
(In az)/U = -4.8941(13,~- 7.65)' + 0.2874(6,, 11.22)' 0.1076(6,h - 3.58)' - 12.826 (17)
+
To test the influence of the Flory-Huggins term, B is used instead of (In %)/U as a dependent variable in the threeparameter model (B is defined in eq 7):
B = -2.61736fd + 36.17836,d + 0.34886;, 6.558661, + 0.18926fh - 1.837161, - 95.0541(18) 568 I Journal of Pharmaceutical Sciences Vol. 78, No. 7, July 1989
Table I-Partial and Total Solubility Parameters and Molar Volumes of the Solvents
-
No.
Solvent
Vl mUmola I
-
132.60 106.90 98.50 89 40 80.80 74.00 85.70 158.40 108.60 80.90 106.00 91.50 92.40 93.00 92.00 76.90 103.90 75.10 77.40 58.70 71.30 89.909 40.70 59.10 95.30g 73.70 55.90 88.60 73.20 18.10
Butyl acetate
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Toluene Ethyl acetate Benzene Chlorofolrm Acetone Dioxane 1-0ctanol 1-Pentanol Pyridine Cyclohsxanol Aniline fsobutyl alcohol N,KDiniethylacetamide Butyl alcohol Isopropyl alcohol Benzyl ,alcohol Propyl alcohol N,NDimethylformeimidc! Ethyl alcohol
I
Dimethylsulfoxide 1,3-Butanediol
Methyl alcohol NMethylformamide Diethylone glycol 1,2-PropanedioI
Ethylene glycol 1,4-Butanediol Glycerol
Water
61T,
6103
( c a ~ / c m ~ ) ” ~ ~ (cal/cm3) 8.49 8.88 9.04 9.07 9.26 9.76 10.01 10.25 10.58 10.68 10.95 11.02 11.11
2.30 0.80 3.66 0.77 2.26 4.58 2.55 3.57 4.83 3.66 4.59 3.75 5.92 5.22 5.79 6.32 4.86 6.76 6.25 8.16 6.85 8.15 10.44 9.06 8.60 9.05 10.67 9.96 11.31 20.26
11.12
11.31 11.50 11.64 12.00 12.14 12.96 13.04 14.149 14.48 14.48 14.64g 14.78 16.03 16.35 17.65 23.39
4 in, (cal/cm3)”‘d 0.49 0.05 0.92 0.04 0.29 1.09 0.39 1.41 1.78 0.76 1.56 0.90 2.27 1.77 2.16 2.15 1.72 2.40 2.12 2.74 2.34 4.18 3.10 3.40 4.93 4.22 4.46 6.15 6.55 5.20
w 7 t 3
whcalc,
~ a l i c r n ~ ~ cal!cm3‘ 24.55 16.30 29.07 16.66 21 .oo 25.36 26.80 30.17 37.30 23.00 35.47 33.31 44.75 36.16 44.18 46.55 40.36 51.72 38.82 64.30 44.31 75.36 88.54 49.29 75.19 89.28 110.99 94.67 135.89 329.60
22.05 18.19 26.06 18.42 20.59 24.20 31.89 34.07 41.89 24.92 38.47 36.57 50.12 32.78 48.18 50.45 40.27 54.93 34.97 65.85 35.35 -h 83.02 37.18
-h
87.90 107.47 94.10 138.98 329.59
a From ref 9. Total solubility parameter calculated from the Hansen partial parameters, (eq 1 and ref 9). From eq 22, using SIT and V , from Table I, columns 3 and 4,and S,, frlom ref 9. (seeAppendix). From eq 23, &,and V, from Table I (seeAppendix). Calculated from eq 11 (seeAppendix). Calculated from eq 11 (see Appendix). Estimated values from eq 26. The molar volumes, V,, of solvents nos. 22 and 25, as well as their partial solubility paramelers for calculating S,, (eq 1) are found in ref 6, p 158. Dashes indicate that W,,,,, cannot be calculated from eq 26 because a,, and S,, for solvents nos. 22 and 25 are not presently available.
‘
wheren = 30, s = 4.26,1! = 01.888, F = 30.47,andF(6,23,0.01) = 3.71. From eq 18,the partial parameters are obtained as S,, = 6.91, , S = ‘9.40, and S,, = 4.85, and the total solubility parameter is 12.63. Equation 18 is written in the form of eq 17 as follows:
B
=
The four-parameter model involving proton donor and acceptor parameters to replace S, (eq 9) is applied here using B as the dependent variable instead of (In cu,)/U.
B
-2.6173(6,d - 6.91)2 + O.34988(61, - 9.4012 + 0.1892(S,h - 4.85)2 - 5.3534 (19)
Partial solubility parameters for sulfamethoxypyridazine calculated using a method of group contributions (Table 111), are S , = 8.84, 8, = 5.95, & , = 5.94, and S,, = 12.2; the S,, value calculated from the Fedors method16 is 12.9. The total solubility parameter, hqP, and the partial parameters, 82, and &&, as calculated are not too different from S,,, S2,, and 8, as estimated by eqs 17 and 19. However, S,, estimated from both equations and 8,, (calculated using the group contribution method (Table 111) are quite different. As observed in eqs 17 and 19, the sign of the term involving the dispersion parameter is negative and its regression coefficient is quite different from unity. The extended Hansen method provides a n 1.2 of 0.835, and this fit is not very satisfactory, probably because the partial paramchter S, does not reflect well the proton-donor electron-accepitor characteristics of complex organic drug mailecules.9 Equation 18, which includes the Flory-Huggins term, improves ? by 5% and provides a 8,,,which agrees better than ,S of eq 16 with the hydrogen bonding parameter calculated from the method of group contributions (Table 111). This could mean that the Flory-Huggins term must be considered to predict reasonable solul~ilitiiesof sulfamethoxypyridazine.
0.71836& - 17.938761, + 0.41716~,,6.942981, - 1.994161, - 2.913861, + 0.66476,,6,b + 127.6507 (20)
=
where n = 28, r2 = 0.929, s = 3.60, F = 37.48, and F(7, 20, 0.01) = 3.71. The partial parameters obtained from an analysis of eq 20 are 8,, = 12.48, ,a, = 8.32, S,, = 4.38, &, = 3.00; and, eq 20 becomes:
B
=
0.7183(8,d - 12.4812 + 0.4171 (SIP - 8.32)2 + O.66479(61, - 4.38)(61b - 3.00) - 30.7147 (21)
The total solubility parameter of sulfamethoxypyridazine obtained by this method is S,, = 15.8. The hydrogen bonding parameter can be calculated from the acidic and basic parameters9 8gh = 26,,82b; therefore, 82, = 5.13. This value agrees with 62, from eq 18 and 82, calculated from group contributions. The use of a system of four parameters together with the Flory-Huggins term has improved the value of 1.2. The regression coefficient of the term involving 8, in eq 21 is nearer to unity than in eqs 17 and 19 and its sign is positive, as it should be for the model of three or four partial parameters.8 Equation 20 improves eq 16 because the term S, is separated into acidic and basic contributions. It also takes into account, by use of the Flory-Huggins term, the size differences between solute and solvent. Journal of Pharmaceutical Sciences I 569 Vol. 78, No. 7, July 7989
Table Il-Observed
Solvent No.
and Calculated Solubilities of SulfamethoxypyridazineIn Individual Solvents at 25 OC
Equation 18 XZB
Xzca,c
1 2 3 4 5 6 7
0.00088 0.00004 0.00205 0.00006 0.00129 0.00918 0.02390 0.00031 0.00057 0.02329 o.oooai 0.06301 0.00043 0.09389 0.00073 0.00054 0.01347
a
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
o.oooa3
0.08807 0.00126 0.20980 0.00301 0.00294 0.08721 0.02891 0.00531 0.00496 0.00422 0.00128 0.00004
0.00016 0.00013 0.00046 0.00017 0.00101 0.00843 0.00200 0.00037 0.00033 0.08042 0.00096 0.04290 0.00038 0.03043 0.00060 0.00063 0.01695 o.oooa7 0.06793 0.00152 0.08842 0.00282 0.00218 0.07096 0.01267 0.00140 0.00372 0.01040 0.00062 4.6 X
Equation 20 Error, %b 81.2 -225.0 77.1 -183.3 21.2 8.1 91.6 - 19.3 42.1 -245.3 - 18.5 31.9 11.6 67.6 17.8 - 16.7 -25.8 -4.8 22.9 -20.6 57.8 6.3 25.8 18.6 56.2 73.6 24.9 - 146.5 51.5 99.9
X2I,C
0.0001a 0.00005 o.ooo5a 0.00005 0.00064 0.00877 0.00845 0.00059 0.00047 0.06134 0.00162 0.05281 0.00033 0.05088 0.00077 0.00070 0.01282 0.00092 0.07907 0.00101 0.10095 -C 0.00353 0.05600
-
C
0.001 14 0.00369 0.01 794
o.oooia
0.000045
Equation 28
Error, %b 79.3 --25.2 71.4 16.6 50.7 4.4 64.7 -90.4 17.5 - 163.4 - 100.0 16.2 23.2 45.8 -5.5 -29.6 4.8 - 10.8 10.2 19.8 51.9
-
C
-20.2 35.8 -C 78.5 25.6 -325.2 85.9 -1.2
Xzca,c
0.00056 0.00003 0.00148 0.00004 0.00127 0.01547 0.01787 0.00033 0.00056 0.041 15 0.00090 0.05004
o.ooo3a
0.03783 0.00062 0.00065 0.01672 o.ooo6a 0.06849 0.00065 0.1068
-
C
0.00109 0.04431
-C
0.00174 0.00418 0.00297 0.00115 0.00005
Error, %b 36.5 25.0 27.3 33.3 1.9 -68.6 25.0 -6.4 1.7 -76.7 -1 1.1 20.6 11.6 59.7 15.0 -20.3 -24.1 18.1 22.2 48.4 49.0
-
C
62.7 49.2
-
C
67.2 15.7 29.6 10.1 -25.0
percent. Dashes indicate that solvents nos. 22 and 25 are not used in eqs 20 and 28 for a Experimental solubilities. Error = (X, - X2ca,.J/Xz calculating X,,, because Sla and S,, are not presently available (see also footnote h on Table I). polarity of the solvent could result from an interaction between the drug and the solvent, a “chameleonic effect” as described by Hoy.18 The solvents used in this work differ in nature, and sulfamethoxypyridazine may self-associate through polar interactions in solvents showing low polarity H 2 N e S 0 2 N H - ( O > OCH, (e.g., benzene, toluene), whereas in the strongly polar and N-N hydrogen bonding solvents (e.g., dimethylsulfoxide, N f l Atom or Number of dimethylformamide), the drug may be solvated by the polar Fdb F,b uIlb Group Groups groups of these solvents. The term w, is used in eq 11 instead of &h&h as an 8400 280 610‘ empirical measure of unsymmetrical solutcsolvent interac160 210 3100 5000 40 1600 tions due to hydrogen bonding and other acid-base interac420 0 0 tions. The terms So and Sin are used instead of Sp to correct the 0 0 1200 geometricmean assumed by the term involving Spin eqs 8 and 280 0 0 9. The term W, may be calculated from eq 11 by assigning 9000 300 1200 invariant partial parameters for sulfamethoxypyridazine cal440 culated from a group contribution method (Table 111). An ZFd = 3120 (ZFZ)”‘ = 2101.5 XUh = 25500 example of such a calculation is shown in the Appendix, and a fi -. ZF,IV = 31201172.5 = 18.09 MPaltZd= 8.84 cal”%n 312; 8, the results are listed in Table I, column 7. The partial = (XFg)i’2/V = 21 01.51172.5 = 12.18 MPa1I2= 5.95 ~ a l ” ~ c m -S,,~ = ’ ~ ; parameters So and Sinfor solvents and solute are calculated (SU,,/V)”* = (25500/172.5)i’z= 12.16 MPal” = 5.94~al”‘cm-~’‘; S, = according to Beerbower et al.9 as follows: (6; + # + %)l1‘ = 12.20 ca11’2cm 3’2. bVan Krevelen and Hoftyzer, from ref6, pp 8586. Hansen and Beerbower, from ref 6, p 85. dThe SI unit MPa”, is related to the cgs units ~al~’‘cm-~’~ for solubility parameters by the equation MPa’” = 2.0455 caI1’‘cm 3’2. Sin = 0.0007SfV (23) The values of the partial parameters obtained can vary The molar volume, V, and the solubility parameters, S, and depending on whether one uses a model of three or four S,, for the solvents were obtained from the literat~re.6,~ parameters. The values of the partial parameters estimated Calculated values of a, and $in for the solvents are found in by regression equations may also change with the nature and Table I; 6,, and S,, for the solute are obtained from S,, = number of solvents used. Martin et al.17 studied the variation 12.20, S,, = 8.84 (Table HI), and V, = 172.5 mL/mol. Sample of the solubility parameter of drug molecules in mixed polar calculations for S,, and Sin are found in the Appendix. As with systems. They suggested that the change in 8, with variable
Table IlI-Group Contribution Method to Calculate Partial Parameters of Sulfamethoxypyridazinea
~~
570 I Journal of Pharmaceutical Sciences Vol. 78,No. 7,July 1989
the W term of the extended Hildebrand approach,lO W, values (Table I, column 7) are not related to S,, by a linear expression, bu.t rather by a polynomial relationship (Figure 1):
Wh
14.869 + 2.73961h 0.0036:,, + 0.0296& 1
-
(24)
where n = 30, .r2 = 0.994, s = 4.91, F = 1440, and F(3,26,0.01) = 4.64. Other empirical relationships from which W, may be obtained involve the acidic and basic parameters: wh =
4.477(81, t 9.48061,
-
17.139
-1.03781, + 0.0716:, + 1.36161b + 0.096:b + 0.8966,,6,b + o.(301(61,61b)2 + 17.696 (26)
where n = 28, r2 = 0.995, s == 4.82, F = 713, andF(6,21,0.01) = 3.81. The Wh(calc)values (i.e., calculated values from eqs 24, 25, and 26) were tested in the regression model suggested by eq 13, and the best fit (highest 3)was provided using Wh(calc) from eq 26 (Table I, column 8). The regression equation is:
B = -1.92086:d + 381.95796,d + 0.34736:, 7.938961, - 4.?1270161d61in+ 35.419061in + 0.69956fh - 0.8929Wh(,,1,, - 92.3671 (27)
350
s f
300
250
i-
C
0 .c 0 200
2 a,
c
S + S
-a>0,
f
150
t
where n = 28, 1-2 = 0.960, s = 2.84, F = 48.54, and F(9, 18, 0.01) = 3.60. The use of UI improves r2 of eq 27 by 3%. Figures 2 and 3 show selected residual plots for the models, eqs 18 and 28; the points are somewhat more scattered in Figure 2. Equation 28 improves the r2 over eq 18 by 7%, and it should be expected that the calculated solubilities from eq 28 would agree better with the observed solubilities. The mole fraction solubility (Xacalc),calculated from the several regression models used (eqs 18, 20, 28) following an iteration procedure,lg as well as the observed mole fractions for sulfamethoxypyridazine, are listed in Table 11. If a 30% error (calculated value within 30% of observed solubility) is taken as a criterion for acceptability,g eq 28 provides the highest number of acceptable calculated solubilities (64% of the cases). The variation of observed solubilities of sulfamethoxypyridazine in the solvents used is very wide (mole fraction values between 0.00003 and 0.2098), and it is difficult to fit such experimental data using a single equation to provide solubilities within 30% error. In addition, the crystalline form of the drug may be altered in highly polar solvents,20721and this will require a determination of the heat of fusion for the crystalline drug in the hydrate form in each solvent. To better compare observed and calculated solubilities, Table IV shows the correlation matrix of experimental and calculated solubilities. It provides a clear and brief summary of how well the experimental and calculated mole fraction solubilities correlate. The highest correlation coefficient corresponds to eq 28. The results show that, in addition to the Flory-Huggins term to correct size differences between solute and solvent, terms to correct the geometric mean assumption to account for specific Lewis acid-base interactions of solute and solvent are desirable in fitting the experimental solubility data of a drug such as sulfamethoxypyridazine in polar and nonpolar solvents using a single equation.
Conclusions
100
c
3 0
cn
50
0
B = O.22106:, - 6.045261, + 0.38706~, 8.031661, -1.697061dSlin + 14.368761in + 0.60016fh 0.7829Wh,,,,,. + 76.1449 - 5.8100Y (28)
(25)
where n = 27, r2 = 0.962, s = 12.39, F = 309.77, and F(2,24, 0.01) = 5.61. Equation 25 is improved when W, is regressed in a power series on Slrr, Sl,,,and SlaSlb: wh =
where n = 28, r2 = 0.933, s = 3.58, F = 33.28, and F(8, 19, 0.01) = 3.63. The solute-solvent interaction parameter K (ey 3) may be used as a criterion to choose a n indicator variable, W,taking Y = 1whenK > 1and = 0 when K 5 1, in order to improve the correlation coefficient of eq 27. The term K is calculated for each solvent from eq 3. An example of such a calculation is shown in the Appendix. The regression equation involving the indicator variable Y' is:
I'
1
0
5
-
10
15
20
Hydrogen Bonding Parameter, 6, h Figure 1-Relationship of the s.olute-solvent interaction term W, and the solvent partial hydrogen bonding parameter of 6 , , for sulfamethoxypyridazine in 30 solvents. The line was fitted to the experimental points using eq 24. The iiumbers associated with each point correspond to the solvents in Table 1.
The extended Hansen method729 is found useful to obtain partial parameters for a solid drug, particularly when the Flory-Huggins term is introduced into the model. It is also of value to test the results obtained using group contribution methods. A group contribution method6 can be employed to calculate empirical terms such as W, which can then be used to account for unsymmetrical interactions that become important in irregular solutions. Parameterization as demonstrated in this report allows the researcher t o use the fit of experimental solubilities to design regression models and equations which allow a reasonable prediction of solubilities of polar drugs in a range of very different solvents.
Journal of Pharmaceutical Sciences I 571 Vol. 78, No. 7, July 1989
-
Table IV-Correlation Matrix of Observed (X,) and Predicted (X,,) Mole Fraction Solubilities for Sulfarnethoxypyridazinee
Q .
0
Variable 6 -
0
2 3 -
(eq 18) (eq 20) Xzcat (eq 28) %cat
0
0
h
&cat
a
0
-6 -9
I L
-10
=
. 0
L
L
I
I
10
20
30
40
Ql
-
1.000 0.9741
1.000
- &d)
-
0.001,
-
82,)'
+ (810 -
(81in - 82in) + 8fh
820)'
-I-
a&,
+ -
BY2
(All
where & = 8.84, F& = 5.94 (Table 1111, &, = 4.75, hin= 2.72 (see below), a,, = 7.7, a,, = 3.12 a,, = 2.3, hi,,= 0.49 (Table I), and B = 8.1769 (B is defined in eq 7). Therefore, W, = (1.2996 + 6.0025 + 9.61 + 35.2836 - 8.1769)/2 = 24.551 (Table I, column 7). Sample Calculation of, ,a and a,, (eqs 22 and 23)-For sulfamethoxypyridazine, the following equation is used: =
(8& - 8&)/(1+ 0.001482,VcJ
(A2)
where a,, = 12.20, hd= 8.84 (Table III), and V, = 172.5 from the Fedorsmethod.'6Therefore, go= (12.202- 8.84,)/3.13486 = 22.5511, a,, = 4.75, and Szin = O.O0076~,V, = 2.72. Sample Calculation of W and K-From eq 6:
W
=
(Sf,
+ 8&
- B)/2
(A3)
Therefore, for sulfamethoxypyridazine in butyl acetate, a,, = 8.49 (Table I, column 41,,,a = 12.20 (Table III), B = 8.1749, and W = (8.49, + 12.202 - 8.1769)/2 = 106.372. From eq 3, the following can be calculated:
6
h
3
m
(A4)
References and Notes
a S
0
0
$
-
Appendix
8&,
Figure 2-A plot of the residuals versus the predicted B value for sulfamethoxypyridazine in 30 solvents using eq 18.
-
-
1 .oooo 0.9655 0.9228
Sample Calculation of W,-For sulfamethoxypyridazine in butyl acetate, the following equation is used (from eq 11):
2(&
Predicted B Value (BWlc)
i
Significance level, a
wh = [(&d
0
-
1 .ooo 0.8252 0.9091 0.9505
xz
0
0
-3 0
U
-6
-9
B -10
0
I
1
I
10
20
30
40
Predicted B Value (Bcalc) Figure 3-A pbt of the residual versus the predicted B value for sulfarnethoxypyridazinein 28 solvents using eq 28.
Although it would be desired that the calculated solubilities of this study showed closer correspondence with the experimental results, it is gratifying to observe how well the procedures, involving partial cohesive parameters coupled with regression analysis, serve to reproduce the experimentally determined solubilities of sulfamethoxypyridazine in solvents ranging from apolar (benzene and toluene), through amphiprotic (alcohols, glycols, and water), to dipolar aprotic (N-methylformamide and dimethylsulfoxide). 572 I Journal of Pharmaceutical Sciences Vol. 78, No. 7,July 1989
1. Cammarata, A,; Yau, S. J.; Rogers, K. S. J . Med. Chem. 1971,14, 1211. 2. Khalil, S. A,; Ossama, Y.; Moustafa, M. A. Can. J . Pharm. Sci. 1976, 11, 121. 3 . Adjei, A.; Newburger, J.; Stavchansky, S.; Martin, A. J . Pharm. Sci. 1984, 73, 742. 4. Bustamante, P.; Selles, E. J . Pharm. Sci. 1986, 75, 639. 5. Hansen, C. M. J . Paint Tech. 1967,39, 104. 6. Barton, A. F. Handbook of Solubility Parameters and Other Cohesion Parameters; CRC: Boca Raton, FL, 1983. 7. Martin, A.; Wu, P. L.; Adjei, A.; Beerbower, A.; F'rausnitz, J. M. J . Pharm. Sci. 1981, 70,1260. 8. Wu, P. L.; Beerbower, A.; Martin, A. J. Pharm. Sci. 1982, 71, 1285. 9. Beerbower,A.; Wu, P. L.; Martin, A. J . Pharm. Sci. 1984,73,179. 10. Martin, A.; Newburger, J.; Adjei, A. J . Pharm. Sci.1980,69,487. 11. Hildebrand, J.H.; Prausnitz, J.M.; Scott, R.L. Re ular and Related Solutions; Van Nostrand Reinhold: New Yorf, 1970. 12. Martin, A.; Wu, P. L.; Adjei, A,; Mehdizadeh, M.; James, K. C.; Metzler, C. J . Pharm. Sci. 1982,39, 104. 13. Keller, R. A,; Karger, B. L.; Snyder, L. R. Gas Chromatography; R. Stack, Ed.; Institute of Petroleum: London, 1971, pp 125-140. 14. The sulfonamide, sulfamethoxypyridazine, has been withdrawn from the market as a n antibacterial agent in the United States. It is used in this study simply as a model, the solubility characteristics of which are representative of a class of sulfonamides. The solvents emplo ed in this investigation are not suggested for use in clinicaypractice but rather as compounds having a wide range of solubilityfarameters. A number of these solvents are toxic and are not use either internally or externally in humans or animals.
15. Gucker, F. T.;Ford, W I.; Moser, C. E. J . Phys. Chem. 1939,43, 153. 16. Fedors, R. F. Polym. Eng. Sci. 1974, 14, 147. 17. Martin, A.; Wu, P. L.; Liron.,Z.; Cohen, S. J . Pharm. Sci. 1985,74,
Kruif, C. G.; Wilting, J. Int. J . PAarm. 1983, 14, 79.
Acknowledgments -
B3R.
18. H o i , K.J . Paint Technol. 1970,42, 76. 19. Martin, A.; Swarbrick, J.; Cammarata, A. Physical Pharmacy; 3rd Ed.; Lea & Febiger: Philadelphia, PA, 1983; p 285. 20. Bogardus, J. J . Pharm. Sci. 1983, 72,837. 21. Fokkens, J. G.; van Ameliafoort, J. G. M.; de Blaey, C. J.; de
Supported by Comision Interministerial de Ciencia y Tecnologia (CICYT), Spain (project no. PB86-0063) and by a grant provided to P. Bustamante by Instituto Nacional de la Salud (Fis Grant, 87118891. Sup orted in part by an endowed professorship provided to A. Martin by oulter R. Sublett.
8
Journal of Pharmaceutical Sciences I 573 Vol. 78, No. 7, July 1989