J Solution Chem (2011) 40:2032–2045DOI 10.1007/s10953-011-9767-2

Solubility of Acetaminophen and Ibuprofenin Polyethylene Glycol 600, N-Methyl Pyrrolidoneand Water Mixtures

Shahla Soltanpour · Abolghasem Jouyban

Received: 24 July 2010 / Accepted: 10 April 2011 / Published online: 17 November 2011© Springer Science+Business Media, LLC 2011

Abstract The solubility of acetaminophen and ibuprofen in binary and ternary mixturesof N-methyl pyrrolidone, polyethylene glycol 600 and water at 25 °C were determined andthe solubilities are mathematically represented by the Jouyban–Acree model. The densityof the solute-free solvent mixtures was measured and employed to train the Jouyban–Acreemodel and then the densities of the saturated solutions were predicted. The overall meanrelative deviations (OMRDs) for fitting the solubility data of acetaminophen and ibuprofenin binary mixtures are 3.2% and 6.0%, respectively. The OMRDs for fitting the solubilitiesin ternary solvent mixtures for acetaminophen and ibuprofen are 15.0% and 28.6%, respec-tively, and the OMRD values for predicting all solubilities of acetaminophen and ibuprofenby a trained version of the Jouyban–Acree model are 9.4% and 17.8%, respectively. Theprediction OMRD for the density of saturated solutions is 1.9%.

Keywords Jouyban–Acree model · Acetaminophen · Ibuprofen · Solubility · N-methylpyrrolidone

1 Introduction

Solutions, especially concentrated solutions, of pharmaceutical compounds have many ad-vantages compared to solid forms. Solutions are easy to use and make excellent vehiclesfor carrying the uniform pharmaceutical compounds. They provide rapid pharmacological

S. SoltanpourLiver and Gastrointestinal Diseases Research Center, Tabriz University of Medical Sciences, Tabriz51664, Iran

Present address:S. SoltanpourFaculty of Pharmacy, Zanjan University of Medical Sciences, Zanjan 45139, Iran

A. Jouyban (�)Drug Applied Research Center and Faculty of Pharmacy, Tabriz University of Medical Sciences, Tabriz51664, Irane-mail: ajouyban@hotmail.com

J Solution Chem (2011) 40:2032–2045 2033

action, because it is not necessary to disintegrate and dissolve the compounds in the gastroin-testinal fluids. Despite these advantages, some pharmaceutical compounds are not marketedin solution form due to their low solubility and/or chemical instability. Many pharmaceuti-cally active compounds have low solubility and require relatively high volumes of the sol-vent for dissolution. Because of safety, compatibility, stability and economic considerations,the number of solvents to be used for making the liquid solutions is limited. Furthermore,using high volumes of solvents for solubilizing pharmaceuticals is not advised because thereare some restrictions, e.g. for injections. One solution for overcoming this solubility problemis the addition of water-miscible co-solvents or surfactants to the formulation [1].

N-methyl pyrrolidone (NMP) is a chemically stable and powerful polar solvent. Theseproperties are very useful in some chemical reactions in which an inert medium is needed.Despite the stability of NMP, it can play an active role in certain reactions such as hydrol-ysis, oxidation, condensation, etc. NMP is a very strong solubilizing agent and is currentlyused in some commercially available pharmaceutical products [2]. NMP has various applica-tions in pharmaceutical and medicinal fields such as a permeation enhancer in transdermalformulations [3, 4], improving the transdermal flux of both hydrophilic and hydrophobicdrugs [5], is a solubilizing agent for poorly soluble drugs [6], provides entrapment of poorlywater-soluble drugs in hybrid nanoparticles [7], enhances aqueous phase transdermal trans-port [5], increases the skin permeation of drugs [8] and is a co-surfactant in microemulsionsystems [9]. Furthermore, various medical devices like self-hardening bone graft substi-tutes [10], dental barrier membranes [11] and subcutaneous drug delivery systems [12, 13]contain small amounts of NMP. The pharmaceutical applications of NMP have been re-viewed in a recent work [14].

Polyethylene glycols (PEGs), called macrogols in the European pharmaceutical industry,are produced by polymerization of ethylene oxide [15, 16]. PEGs with a mean molecularweight up to 400 are non-volatile liquids at room temperature. PEG 600 shows a meltingpoint of 17 to 22 °C, so it may be a liquid at room temperature but a paste at lower temper-atures. Solubility of PEGs in water is an important property, which makes them suitable indifferent applications. Liquid PEGs up to 600 are freely miscible with water [17]. The liquidPEGs have a slightly bitter taste, but it can be adjusted easily by suitable additives (sweet-eners) and solid PEG grades have no taste. Solid PEGs are not soluble in liquid PEGs, butblending them with liquid PEGs leads to a white, pasty ointment with good solubility inwater, good dissolving properties and suitable vehicles for many substances and ointmentbases [18–20]. Solid PEGs are preferred bases for suppositories [21]. Producing tablets re-quires variable excipients with different functions, several of them covered by PEGs. Theymay act as carriers, solubilizers, absorption improvers for active substances, lubricants andbinders [22]. The relatively low melting point favors a sintering or compression technique.At the same time the PEG has a plasticizing effect, which facilitates the shaping of the tabletmass in the compression process and may counteract capping. The flexibility of sugar-coatedtablets is increased by PEGs and, since PEG acts as an anticaking agent, the cores are pre-vented from sticking together. With film formers in sugar-free coating processes, PEGs actas a softener. PEGs 300 and 400 are listed as the active ingredients in ophthalmic demul-cents [23]. With the two OH groups at the end of the PEG molecules, all typical reactions ofalcohols, such as ester, carbonate and carbamate formation are possible. Methylether-cappedPEGs, are available to avoid chain-building reactions, since these PEGs are only able to re-act at one end of the molecule. The PEG conjugation of drugs, e.g. anticancer drugs, makesthem safer therapeutic agents which target malignant tissues with more selectivity [24, 25].

For achieving an optimized solvent mixture for dissolving a certain amount of a drug in agiven volume of the solvent, the trial-and-error approach is usually employed, which is time-consuming and expensive; therefore, employing cosolvency models could be an appropriate

2034 J Solution Chem (2011) 40:2032–2045

solution. Among the developed cosolvency models, the Jouyban–Acree model is one ofthe most versatile. It provides accurate mathematical descriptions and shows how the solutesolubility varies with both temperature and solvent composition. The model for representingthe solubility of a solute in binary solvent mixtures at various temperatures is:

log10 Cm,T = w1 log10 C1,T + w2 log10 C2,T +[

w1w2

T

2∑i=0

Ji(w1 − w2)i

](1)

where Cm,T is the solute’s molar solubility in the binary solvent mixtures at temperature T ,w1 and w2 are the mass fractions of the solvents 1 and 2 in the absence of the solute,and C1,T and C2,T denote the molar solubility of the solute in the neat solvents 1 and 2,respectively. The Ji terms are constants of the model and are computed by regressing

(log10 Cm,T − w1 log10 C1,T − w2 log10 C2,T ) against w1w2T

, w1w2(w1−w2)

T, and w1w2(w1−w2)2

T.

This model was used to calculate multiple solubility maxima [26] and also to correlate otherphysico-chemical properties in solvent mixtures [27–32] and promises accurate mathemati-cal representations.

Equation 1 can be used to model the solubility data of a solute in a binary solvent at var-ious temperatures. It requires C1,T and C2,T data and also a minimum number of solubilitydata in mixed solvents for the training process, which restricts its applications for predictionpurposes. A number of attempts have been made to overcome this limitation. These includesthe presentation of trained versions of the Jouyban–Acree model for specific cosolvents. Thetrained version of the model can be employed for predicting the solutes solubility in aque-ous and non-aqueous solvent mixtures at various temperatures by using only C1,T and C2,T

data. For PEG 400 {1} (to avoid reader confusion the solvent numbers used in the equationsare reported in {})–water {2} mixtures, Eq. 1 was trained using experimental solubilities ofdrugs and the obtained model is [33]:

log10 Cm,T = w1 log10 C1,T + w2 log10 C2,T

+ w1w2

T

[394.82 − 355.28(w1 − w2) + 388.89(w1 − w2)

2], (2)

for propylene glycol {1}–water {2} mixtures is [34]:

log10 Cm,T = w1 log10 C1,T + w2 log10 C2,T + w1w2

T

[37.030 + 319.490(w1 − w2)

], (3)

for ethanol {1}–water {2} mixtures is [35]:

log10 Cm,T = w1 log10 C1,T + w2 log10 C2,T

+ w1w2

T

[724.21 + 485.17(w1 − w2) + 194.41(w1 − w2)

2], (4)

and its updated version [36]:

log10 Cm,T = w1 log10 C1,T + w2 log10 C2,T

+ w1w2

T

[724.21 + 485.17(w1 − w2) + 194.41(w1 − w2)

2]

− 0.314w1w2 log10 P (5)

in which log10 P is the logarithm of the octanol–water partition coefficient of the drug.The model for dioxane {1}–water {2} mixtures is [37]:

log10 Cm,T = w1 log10 C1,T + w2 log10 C2,T

+ w1w2

T

[958.44 + 509.45(w1 − w2) + 867.44(w1 − w2)

2]

(6)

J Solution Chem (2011) 40:2032–2045 2035

for glycerol {1}–water {2} mixtures is [38]:

log10 Cm,T = w1 log10 C1,T + w2 log10 C2,T + 1048.364

(w1w2

T

)− 1.232w1w2 log10 P, (7)

and for ethyl acetate {1}–ethanol {2} mixtures the trained model is [39]:

log10 Cm,T = w1 log10 C1,T + w2 log10 C2,T

+ w1w2

T

[382.99 + 125.66(w1 − w2) + 214.58(w1 − w2)

2]. (8)

It should be noted that the solvent composition of the mixtures and also the solute concen-tration can be expressed using different units. Our results (not shown here) revealed thatthese different expressions have no effect on the prediction capability of the model.

The accuracies of Eqs. 2–8 were evaluated using various numbers of data sets (NDS) bycomputing the MRDs. The OMRDs (± SD) for the equations and their NDSs are 39.8%(SD = 46.7, NDS = 80), 24.1% (SD = 15.1, NDS = 27), 49.4% (SD = 88.3, NDS = 26),35.5% (SD = 22.5, NDS = 32), 27.2% (SD = 14.3, NDS = 36), 40.7% (SD = 35.8,NDS = 5) and 13.1% (SD = 8.1, NDS = 26), respectively. As a general point, greaterNDS and less diversity of the investigated solutes resulted in greater accuracy for the model.As it is shown in Eq. 5, addition of a term representing the solute’s structure improves theprediction capability of the model.

The Jouyban–Acree model has theoretical justifications [40], among other cosolvencymodels it shows the most accurate results [41], and by employing the solubility data inmono-solvents, the solubility data in solvent mixtures at various temperatures can be eas-ily predicted [33–39]. In addition to the binary solvent mixtures, the extended models forpredicting the solubility data of drugs in ternary solvent mixtures are:

log10 Cm,T = w1 log10 C1,T + w2 log10 C2,T + w3 log10 C3,T +[

w1w2

T

2∑i=0

Ji(w1 − w2)i

]

+[

w1w3

T

2∑i=0

J ′i (w1 − w3)

i

]+

[w2w3

T

2∑i=0

J ′′i (w2 − w3)

i

](9)

log10 Cm,T = w1 log10 C1,T + w2 log10 C2,T + w3 log10 C3,T

+[

w1w2

T

2∑i=0

Ji(w1 − w2)i

]+

[w1w3

T

2∑i=0

J ′i (w1 − w3)

i

]

+[

w2w3

T

2∑i=0

J ′′i (w2 − w3)

i

]+

[w1w2w3

T

2∑i=0

J ′′′i (w1 − w2 − w3)

i

](10)

where C3,T is the solute molar solubility in the solvent 3 at temperature T , and w3 is themass fraction of the solvent 3 in the absence of the solute. The J ′

i and J ′′i terms are computed

using the same procedure as for the Ji terms. The J ′′′i terms are the ternary solvent interaction

terms and are computed by regressing:⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

log10 CSatm,T − w1 log10 CSat

1,T − w2 log10 CSat2,T − w3 log10 CSat

3,T −[

w1w2

T

2∑i=0

Ji(w1 − w2)i

]

−[

w1w3

T

2∑i=0

J ′i (w1 − w3)

i

]−

[w2w3

T

2∑i=0

J ′′i (w2 − w3)

i

]⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

2036 J Solution Chem (2011) 40:2032–2045

against w1w2w3T

, w1w2w3(w1−w2−w3)

T, and w1w2w3(w1−w2−w3)2

T. The existence of these model con-

stants, which require a number of solubility data in solvent mixtures for the training process,is a limitation for the model when the solubility predictions are the goal of the computationsin early drug discovery studies.

Experimental solubilities of acetaminophen and ibuprofen in PEG 600 {1}–water {3}mixtures were reported in a previous work [42]. In this work, the experimental solubil-ity of these drugs in NMP {2}–water {3}, PEG 600 {1}–NMP {2} and PEG 600 {1}–NMP {2}–water {3} mixtures at 25 °C are reported and the applicability of the Jouyban–Acree model to predict the measured solubility data is shown. In addition, the applicabilityof the Jouyban–Acree model for predicting the density of the saturated solutions by employ-ing density of solute-free solutions of mixed solvents is shown.

2 Experimental Method

2.1 Materials

Acetaminophen was purchased from Arastoo Pharmaceutical Company (Iran) and ibuprofenwas purchased from Sobhan Pharmaceutical Company (Iran). The purity of the drugs waschecked by determination of their melting points and comparing the measured solubilities inmono-solvents with the corresponding data from the literature [43, 44]. NMP was purchasedfrom Merck (Germany), PEG 600 was a gift from Daana pharmaceutical company (Iran),and double distilled water was used for preparation of the solutions.

2.2 Apparatus and Procedures

The binary solvent mixtures (100.0 g) were prepared by mixing the appropriate weights ofthe solvents with the uncertainty of 0.10 g. The solubilities of acetaminophen and ibuprofenin the solvent mixtures were determined by equilibrating excess amounts of drugs at 25 °Cusing a shaker (Behdad, Tehran, Iran) placed in an incubator equipped with a temperaturecontrolling system maintained constant within ±0.2 °C (Nabziran, Tabriz, Iran). Because ofthe high viscosity of PEG 600, after sufficient time (> 98 h), the saturated solutions of thedrugs were centrifuged at 13,000 rpm for 15 min, diluted with water for acetaminophen andmethanol for ibuprofen, and then assayed at 243 nm and 222 nm, respectively, using a UV–vis spectrophotometer (Beckman DU-650, Fullerton, USA). Concentrations of the dilutedsolutions were determined from the calibration curves. Each experimental data point repre-sents the average of at least three replicate experiments with the measured molar solubilitiesbeing reproducible to within ±3.1%. Densities of the saturated solutions and the solventmixtures in the absence of the solute were measured using a 5 mL pycnometer.

2.3 Computational Methods

The experimental solubility data for each drug, in the binary solvents, were fitted to Eq. 1, themodel constants were computed, and the back-calculated solubilities were used to computethe MRD values. In the next analysis, Eq. 9 was used to calculate the solubility of each drugin the ternary solvents. In order to provide better predictions, the ternary interaction termsof Eq. 10 were calculated using a linear regression analysis.

As discussed above, Eq. 1 should be trained using experimental solubility data. To pro-vide generally trained versions of the model for PEG 600 {1}–water {2} and NMP {1}–water {2} mixtures, the generated data and collected data from the literature [42, 45–48]

J Solution Chem (2011) 40:2032–2045 2037

were used to train the model. In order to show the capability of the Jouyban–Acree modelfor a predicting drug’s solubility in PEG 600 {1}–water {2} and NMP {1}–water {2} mix-tures, one data set was excluded from the training process and its solubility values werepredicted using the trained version of the model. This method is called leave one out cross-validation.

Densities of the saturated solutions are required to convert the molar solubilities intomole fraction solubilities. Any attempt to predict the density of the saturated solutions cansave time and the cost of the experimental efforts. In a previous paper [49], the applica-bility of the Jouyban–Acree model for the prediction of the densities of liquid mixtures atvarious temperatures was shown. The investigated liquid mixtures were solute free, so forshowing the model applicability in predicting the densities of the saturated solutions com-posed of liquid mixtures, the model was fitted to the density of solute-free binary mixturesand the sub-binary constants were calculated for each system. Then, by using these con-stants, the model constants for ternary mixtures were obtained and the trained version ofthe model was used to predict the densities of the saturated solutions and the resulting pre-diction errors were within an acceptable range [42]. Therefore, the same procedure can beused to predict the densities of saturated solutions investigated in this work. The experimen-tal and calculated densities were used to convert the molar solubilities to the mole fractionscale.

The mean relative deviation (MRD) between the calculated and observed (solubil-ity/density) values are used to evaluate the accuracy of the model. The MRD values arecalculated using:

MRD = 100

∑{ |Calculated −Observed|Observed

}N

(11)

where N is the number of data points in each set.

3 Results

Tables 1 and 2 list the experimental solubilities of acetaminophen and ibuprofen in the bi-nary and ternary solvent mixtures along with the measured density of the saturated solutionsat 25 °C, respectively. The densities of the solute-free solvent mixtures are also listed in Ta-ble 1. The minimum solubility of acetaminophen (9.89 × 10−2 mol·L−1) is observed for theaqueous solution and is in good agreement with previous data [50–52]. The maximum solu-bility of acetaminophen (8.60 mol·L−1) in the solvent mixtures is observed for the PEG 600{1}–NMP {2}–water {3} (0.3 + 0.6 + 0.1 mass fractions) mixture. The aqueous solubilityof ibuprofen is 4 × 10−4 mol·L−1 (or 6.72 × 10−6 as mole fraction) and is comparable withthe corresponding values from the literature [53, 54]. The maximum solubility of ibuprofen(8.95 mol·L−1) in the solvent mixtures studied is observed in the PEG 600 {1}–NMP {2}–water {3} (0.1 + 0.6 + 0.3 mass fractions) mixture. There are no published experimentaldata for drugs in the investigated solvent mixtures.

Equations 9 and 10 were used to fit the data sets of acetaminophen and ibuprofen; theconstants and MRD values are shown in Table 3. In the binary mixtures of acetaminophenthe lowest MRD value is for PEG 600 {1}–NMP {2} mixtures with 1.0% and the highestMRD value belongs to NMP {2}–water {3} mixtures with 7.4%. For ibuprofen in binarymixtures the lowest MRD is for PEG 600 {1}–NMP {2} mixtures with 0.8% and the highestMRD value belongs to PEG 600 {1}–water {3} mixtures with 9.3%. In the binary mixtures,the overall MRD (OMRD) values are 3.2% and 6.0%, respectively, for acetaminophen andibuprofen. The MRD values for ternary mixtures are 15.0% and 28.6%, respectively, for

2038 J Solution Chem (2011) 40:2032–2045

Table 1 Experimental molar solubilities (Cm,T ) of acetaminophen in PEG 600 {1}–NMP {2}–water {3}mixtures at 25 °C and densities of the saturated and solute-free solutions

w1 w2 w3 Cm,T

(mol·L−1)Density of the saturatedsolutions (g·mL−1)

Density of solute-freesolvent mixtures (g·mL−1)

1.00 0.00 – 1.4531 1.1556 1.1291

0.90 0.10 – 1.5118 1.1536 1.1198

0.80 0.20 – 1.7102 1.1412 1.1119

0.70 0.30 – 2.1035 1.1330 1.1021

0.60 0.40 – 2.4669 1.1248 1.0962

0.50 0.50 – 2.8706 1.1165 1.0863

0.40 0.60 – 3.1281 1.1103 1.0745

0.30 0.70 – 3.4581 1.1021 1.0667

0.20 0.80 – 3.8479 1.0918 1.0509

0.10 0.90 – 4.2377 1.0815 1.0391

0.00 1.00 – 5.0451 1.0703 1.0293

– 0.00 1.00 0.0989 1.0162 0.9837

– 0.10 0.90 0.4595 1.0216 0.9978

– 0.20 0.80 0.7507 1.0288 1.0017

– 0.30 0.70 1.1614 1.0360 1.0056

– 0.40 0.60 1.6455 1.0414 1.0096

– 0.50 0.50 2.1397 1.0486 1.0135

– 0.60 0.40 2.8358 1.0559 1.0155

– 0.70 0.30 3.4999 1.0613 1.0175

– 0.80 0.20 3.7783 1.0649 1.0214

– 0.90 0.10 4.1124 1.0667 1.0253

– 1.00 0.00 5.0451 1.0703 1.0293

0.10 0.10 0.80 0.5670 1.0897 1.0300

0.20 0.10 0.70 0.8370 1.0939 1.0465

0.10 0.20 0.70 0.8788 1.0877 1.0382

0.10 0.30 0.60 1.4698 1.0794 1.0465

0.20 0.20 0.60 1.0592 1.0836 1.0527

0.30 0.10 0.60 1.0209 1.0877 1.0650

0.40 0.10 0.50 1.3097 1.1165 1.0836

0.30 0.20 0.50 1.7168 1.1103 1.0733

0.20 0.30 0.50 1.9882 1.0980 1.0630

0.10 0.40 0.50 2.6145 1.0918 1.0527

0.50 0.10 0.40 1.6271 1.1495 1.0918

0.40 0.20 0.40 1.7983 1.1412 1.0877

0.30 0.30 0.40 2.2297 1.1330 1.0733

0.20 0.40 0.40 2.2437 1.1289 1.0650

0.10 0.50 0.40 2.3550 1.1145 1.0547

0.60 0.10 0.30 2.1810 1.1454 1.1062

0.50 0.20 0.30 2.4107 1.1412 1.1000

0.40 0.30 0.30 2.8282 1.1392 1.0918

0.30 0.40 0.30 3.1205 1.1392 1.0774

J Solution Chem (2011) 40:2032–2045 2039

Table 1 (Continued)

w1 w2 w3 Cm,T

(mol·L−1)Density of the saturatedsolutions (g·mL−1)

Density of solute-freesolvent mixtures (g·mL−1)

0.20 0.50 0.30 3.1692 1.1371 1.0691

0.10 0.60 0.30 3.1831 1.1268 1.0630

0.70 0.10 0.20 2.2048 1.1186 1.1165

0.60 0.20 0.20 2.6780 1.1103 1.1103

0.50 0.30 0.20 3.3043 1.1042 1.1042

0.40 0.40 0.20 3.3600 1.0959 1.0897

0.30 0.50 0.20 3.4852 1.0877 1.0774

0.20 0.60 0.20 4.4595 1.0794 1.0691

0.10 0.70 0.20 4.0656 1.0691 1.0630

0.80 0.10 0.10 2.4833 1.1836 1.1186

0.70 0.20 0.10 2.5263 1.1745 1.1124

0.60 0.30 0.10 3.4901 1.1691 1.1042

0.50 0.40 0.10 3.5548 1.1618 1.0897

0.40 0.50 0.10 7.9486 1.1564 1.0836

0.30 0.60 0.10 8.5972 1.1418 1.0774

0.20 0.70 0.10 6.3619 1.1309 1.0691

0.10 0.80 0.10 6.3202 1.1273 1.0547

Table 2 Experimental molar solubilities (Cm,T ) of ibuprofen in PEG 600 {1}–NMP {2}–water {3} mixturesat 25 °C and density of the saturated solutions

w1 w2 w3 Cm,T (mol·L−1) Density of the saturatedsolutions (g·mL−1)

1.00 0.00 – 1.4425 1.1364

0.90 0.10 – 1.9340 1.1083

0.80 0.20 – 2.3699 1.0980

0.70 0.30 – 2.7695 1.0918

0.60 0.40 – 3.2418 1.0856

0.50 0.50 – 3.5324 1.0753

0.40 0.60 – 3.6501 1.0712

0.30 0.70 – 3.7954 1.0650

0.20 0.80 – 4.0860 1.0609

0.10 0.90 – 4.5582 1.0527

0.00 1.00 – 5.5121 1.0444

– 0.00 1.00 0.0004 0.9873

– 0.10 0.90 0.0201 0.9888

– 0.20 0.80 0.1532 0.9950

– 0.30 0.70 0.3333 1.0032

– 0.40 0.60 0.6057 1.0156

– 0.50 0.50 0.9955 1.0218

– 0.60 0.40 1.4799 1.0279

2040 J Solution Chem (2011) 40:2032–2045

Table 2 (Continued)

w1 w2 w3 Cm,T (mol·L−1) Density of the saturatedsolutions (g·mL−1)

– 0.70 0.30 2.5152 1.0321

– 0.80 0.20 4.1862 1.0362

– 0.90 0.10 5.2942 1.0403

– 1.00 0.00 5.5121 1.0444

0.10 0.10 0.80 0.0479 1.0609

0.20 0.10 0.70 0.0583 1.0671

0.10 0.20 0.70 0.2131 1.0774

0.10 0.30 0.60 0.3148 1.0465

0.20 0.20 0.60 0.2593 1.0506

0.30 0.10 0.60 0.0759 1.0568

0.40 0.10 0.50 0.0941 1.0733

0.30 0.20 0.50 0.2376 1.0568

0.20 0.30 0.50 0.4464 1.0424

0.10 0.40 0.50 0.5118 1.0238

0.50 0.10 0.40 0.2352 1.1042

0.40 0.20 0.40 0.3623 1.0918

0.30 0.30 0.40 0.5939 1.0733

0.20 0.40 0.40 0.7891 1.0568

0.10 0.50 0.40 1.1342 1.0424

0.60 0.10 0.30 1.2976 1.1165

0.50 0.20 0.30 1.4656 1.1062

0.40 0.30 0.30 1.6200 1.0712

0.30 0.40 0.30 1.7925 1.0527

0.20 0.50 0.30 1.9696 1.0341

0.10 0.60 0.30 8.9499 1.6336

0.70 0.10 0.20 1.9396 1.1392

0.60 0.20 0.20 2.5026 1.1227

0.50 0.30 0.20 3.5560 1.1103

0.40 0.40 0.20 3.4915 1.0939

0.30 0.50 0.20 3.0475 1.0815

0.20 0.60 0.20 4.3769 1.0691

0.10 0.70 0.20 7.1919 1.0527

0.80 0.10 0.10 3.2472 1.0873

0.70 0.20 0.10 2.4663 1.0745

0.60 0.30 0.10 2.5953 1.0636

0.50 0.40 0.10 2.9222 1.0509

0.40 0.50 0.10 3.7739 1.0400

0.30 0.60 0.10 3.5923 1.0309

0.20 0.70 0.10 3.0475 1.0218

0.10 0.80 0.10 3.1564 1.0127

J Solution Chem (2011) 40:2032–2045 2041

Table 3 The constants of the Jouyban–Acree model and the mean relative deviations (MRD) of back-calculation for the solubility of acetaminophen and ibuprofen in binary and ternary solvent mixtures

Drug N Solvent system J0 J1 J2 MRD%

Acetaminophen 11 PEG 600 {1}–NMP {2} 25.595 21.946 −190.757 1.0

Acetaminophena 11 PEG 600 {1}–water {3} 525.287 178.733 110.211 1.3

Acetaminophen 11 NMP {2}–water {3} 573.236 −520.733 480.948 7.4

Acetaminophen 36 PEG 600 {1}–NMP {2}–water {3} 1549.527 4400.519 3805.727 15.0

OMRD 6.2

Ibuprofen 11 PEG 600 {1}–NMP {2} 108.590 −180.307 −77.088 0.8

Ibuprofena 11 PEG 600 {1}–water {3} −196.127 693.573 2546.005 9.3

Ibuprofen 11 NMP {2}–water {3} 1549.194 −1575.793 1968.898 8.0

Ibuprofen 36 PEG 600 {1}–NMP {2}–water {3} 2867.939 3246.069 7430.691 28.6

OMRD 11.7

aSolubility data taken from a previous work [42]

acetaminophen and ibuprofen. The MRD values for ternary solvent mixtures are higherthan those of the binary solvent mixtures. After calculating the sub-binary and ternary con-stants for each drug (details are shown in Table 3), the trained versions of the Jouyban–Acree model were used to predict the solubility of acetaminophen and ibuprofen in bothbinary and ternary solvent mixtures. The OMRD values for predicting the acetaminophenand ibuprofen solubility in binary and ternary solvent mixtures are 6.2% and 11.7%, respec-tively.

In addition to those trained versions of the Jouyban–Acree model, two generally trainedmodels for PEG 600 {1}–water {2} and NMP {1}–water {2} mixtures, using data setstaken from previous work [42, 45–48], are presented in this work. The trained version forPEG 600 {1}–water {2} mixtures is:

log10 Cm,T = w1 log10 C1,T + w2 log10 C2,T + 213.21w1w2

T(12)

and the trained version for NMP {1}–water {2} mixtures is:

log10 Cm,T = w1 log10 C1,T + w2 log10 C2,T

+ w1w2

T

[668.67 − 678.59(w1 − w2) + 1220.13(w1 − w2)

2]. (13)

The back-calculated MRDs of Eqs. 12 and 13 and the references of the data sets are shownin Table 4. In PEG 600 {1}–water {2} mixtures the lowest (18.4%) and highest (127.9%)MRDs are observed for lamotrigine and clonazepam, respectively. Also in NMP {1}–water {2} mixtures the lowest and highest MRDs belong to lamotrigine (25.6%) and clon-azepam (149.9%), respectively. The OMRD values for Eqs. 12 and 13 are 56.9% and 58.1%,respectively. Comparing the MRD values of Eqs. 12 and 13 with those of Eq. 1 (listed inTable 3) reveal that the MRD values of Eq. 1 are less than those of Eqs. 12 and 13. But themain advantage of Eqs. 12 and 13 is that they need experimental solubility data of drugsin mono-solvents, but for computing the constants of Eq. 1 at least three solubility data inmixed solvents are required. Therefore, with the advantage and weakness of Eqs. 12 and 13,the MRD values of these equations are acceptable.

In the leave one out cross-validation method, one drug was excluded from the trainingsets and Eq. 1 was trained with the rest of the data sets, and then by using the trained ver-

2042 J Solution Chem (2011) 40:2032–2045

Table 4 The details of data sets and the mean relative deviations (MRD) of the back-calculation and leaveone out methods

Drug N Solvent system Reference MRD%(back-calculation)

MRD%(leave oneout)

Acetaminophen 11 PEG 600 {1}–water {3} [42] 30.0 34.4

Ibuprofen 11 PEG 600 {1}–water {3} [42] 52.4 53.2

Pioglitazone HCl 12 PEG 600 {1}–water {3} [48] 54.9 59.7

Diazepam 11 PEG 600 {1}–water {3} [46] 57.9 74.3

Lamotrigine 11 PEG 600 {1}–water {3} [46] 18.4 20.4

Clonazepam 11 PEG 600 {1}–water {3} [46] 127.9 172.9

OMRD 56.9 69.2

Acetaminophen 11 NMP {2}–water {3} This work 34.1 40.9

Ibuprofen 11 NMP {2}–water {3} This work 58.6 62.1

Pioglitazone HCl 14 NMP {2}–water {3} [45] 56.9 70.2

Diazepam 11 NMP {2}–water {3} [47] 41.8 49.2

Lamotrigine 11 NMP {2}–water {3} [47] 25.6 30.1

Clonazepam 11 NMP {2}–water {3} [47] 149.9 198.8

Phenobarbital 11 NMP {2}–water {3} [47] 39.7 44.0

OMRD 58.1 70.8

sion, the solubility of the excluded drug was predicted. The details of the MRD values forthe leave one out method are shown in Table 4. In PEG 600 {1}–water {2} mixtures thelowest and highest MRDs belong to lamotrigine (20.4%) and clonazepam (172.9%), respec-tively, and for NMP {1}–water {2} mixtures the lowest and highest MRDs belong again tolamotrigine (30.1%) and clonazepam (198.8%), respectively. The OMRD values for PEG600 {1}–water {2} and NMP {1}–water {2} mixtures are 69.2% and 70.8%, respectively.

For predicting the density of the saturated solutions, the densities of solute free binaryand ternary solvent mixtures were measured. Then, by the density of the binary mixtures,the constants of Eq. 14 were computed:

log10 ρm,T = w1 log10 ρ1,T + w2 log10 ρ2,T + w3 log10 ρ3,T +[

w1w2

T

2∑i=0

Ji(w1 − w2)i

]

+[

w1w3

T

2∑i=0

J ′i (w1 − w3)

i

]+

[w2w3

T

2∑i=0

J ′′i (w2 − w3)

i

]. (14)

Then by using these sub-binary constants, the ternary constants of Eq. 15 were computed:

log10 ρm,T = w1 log10 ρ1,T + w2 log10 ρ2,T + w3 log10 ρ3,T

+[

w1w2

T

2∑i=0

Ji(w1 − w2)i

]+

[w1w3

T

2∑i=0

J ′i (w1 − w3)

i

]

+[

w2w3

T

2∑i=0

J ′′i (w2 − w3)

i

]+

[w1w2w3

T

2∑i=0

J ′′′i (w1 − w2 − w3)

i

]. (15)

J Solution Chem (2011) 40:2032–2045 2043

The final trained version is:

log10 ρm,T = w1 log10 ρ1,T + w2 log10 ρ2,T + w3 log10 ρ3,T + 0.290(w1w2)

T

+ 1.403(w1w3)

T+ w2w3

T

[0.266 − 0.504(w2 − w3) + 0.336(w2 − w3)

2]

+ w1w2w3

T

[3.108 − 7.474(w1 − w2 − w3)

]. (16)

Equation 16 was used to predict the densities of the saturated solutions. The predictionMRDs were 1.4% and 2.4%, respectively, for acetaminophen and ibuprofen and the OMRDwas 1.9%. The experimental and calculated densities were used to convert the molar solubil-ity to the mole fraction solubility, and the OMRD value for the difference of mole fractionsolubilities from experimental and predicted densities was 1.9%.

4 Discussion

Experimental solubilities of acetaminophen and ibuprofen are reported in aqueous and non-aqueous mixtures of PEG 600 and NMP. Aqueous mixtures data could be used in liquiddrug formulation whereas non-aqueous data could be used in other formulations such assoftgels. The Jouyban–Acree model fits well to the experimental solubility data for drugsat all compositions of the solvent mixtures. Also, it fits well to the experimental solubili-ties of drugs in ternary solvents with given fractions of the cosolvents. These findings arealso supported by the small MRD values of the back-calculated and experimental solubil-ity data. Although the MRDs are very low, especially for sub-binary solvents, it should bekept in mind that the constants are computed using the solubility of acetaminophen and/oribuprofen, which need experimental measurements. Generally, the overall MRDs observedin these predictions show that the Jouyban–Acree model provided more accurate predictionsin the presence of one or two cosolvents. According to the density prediction results, it’s notnecessary to measure the density of all saturated solutions, and by measuring the density ofthe solute-free solvent mixtures and with trained version of the Jouyban–Acree model, thedensity of the saturated solutions could be predicted within an acceptable MRD.

Acknowledgements The authors would like to thank the Research Affairs of Tabriz University of MedicalSciences for the partial financial support under grant No. 5/4/5452, and also Daana pharmaceutical companyfor supplying the drug powders and materials used in this work.

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