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## Química Nova

##
*Print version* ISSN 0100-4042

### Quím. Nova vol.34 no.9 São Paulo Sept. 2011

#### http://dx.doi.org/10.1590/S0100-40422011000900016

**ARTIGO**

**Indomethacin solubility estimation in 1,4-dioxane + water mixtures by the extended hildebrand solubility approach**

**Miller A. Ruidiaz; Daniel R. Delgado; Fleming Martínez ^{*}**

Departamento de Farmacia, Facultad de Ciencias, Universidad Nacional de Colombia, A.A. 14490, Bogotá D.C., Colombia

**ABSTRACT**

Extended Hildebrand Solubility Approach (EHSA) was successfully applied to evaluate the solubility of Indomethacin in 1,4-dioxane + water mixtures at 298.15 K. An acceptable correlation-performance of EHSA was found by using a regular polynomial model in order four of the *W* interaction parameter vs. solubility parameter of the mixtures (overall deviation was 8.9%). Although the mean deviation obtained was similar to that obtained directly by means of an empiric regression of the experimental solubility vs. mixtures solubility parameters, the advantages of EHSA are evident because it requires physicochemical properties easily available for drugs.

**Keywords:** Extended Hildebrand Solubility Approach; indomethacin; solubility parameter.

**INTRODUCTION**

Indomethacin (IMC, Figure 1) is a non-steroidal anti-inflammatory drug used as analgesic and antipyretic, among other indications.^{1} Although IMC is used in therapeutics, the physicochemical information about its solubility is scarce. It is well known that several physicochemical properties such as the solubility of active ingredients and their respective occupied volumes in useful solutions are very important for pharmaceutical scientists. This kind of information facilitates the drug design processes and the development of new products.^{2}

This work presents a physicochemical study about the solubility prediction of IMC in binary mixtures of 1,4-dioxane and water. The study was done based on the Extended Hildebrand Solubility Approach (EHSA), developed by Martin *et al.* to use it in pharmaceutical systems.^{3} As has been already described, the solubility behavior of drugs in cosolvent mixtures is very important because cosolvent blends are frequently used in purification methods, preformulation studies, and pharmaceutical dosage forms design, among other applications.^{4}

It is known that 1,2-propanediol and ethanol are the cosolvents most widely used in drug formulation design, especially those intended for peroral and parenteral administration and several examples of pharmaceutical formulations using these cosolvents have been presented by Rubino.^{4} 1,2-propanediol and ethanol can act both as hydrogen-donor and hydrogen-acceptor solvents and have relatively large dielectric constants (24 and 32 at 293.15 K, respectively).^{5} Therefore, the behavior of solutes in cosolvent mixtures with low polarities could not be studied by using mixtures of these two solvents with water.

On the other hand, 1,4-dioxane is miscible with water in all possible compositions, although it has a low dielectric constant (2.2 at 293.15 K).^{5} Mixtures of 1,4-dioxane + water can be varied from non-polar to polar since dielectric constant vary from 2 to 80. While 1,4-dioxane acts only as a Lewis base in aqueous solution, 1,2-propanediol and ethanol can act either as Lewis acid or Lewis base. Although 1,4-dioxane is a toxic solvent, it has been widely used as a model cosolvent for solubility studies of drugs by several authors.^{3,6}

This report expands the information presented about the solubility prediction of other analgesic drugs by means of EHSA method,^{7} including the ones developed recently for this drug in ethyl acetate + ethanol and ethanol + water mixtures.^{8} It is remarkable that the solution thermodynamics of IMC in 1,4-dioxane + water mixtures has also been reported earlier.^{9} Accordingly that work, the driving mechanism for IMC solubility in water-rich mixtures is the entropy, probably due to water-structure loss around the drug non-polar moieties by 1,4-dioxane, whereas, above 0.60 mass fraction of 1,4-dioxane the driving mechanism is the enthalpy, probably due to IMC solvation increase by the cosolvent molecules.^{9}

**THEORETICAL**

The ideal solubility (*X*_{2}* ^{id}*) of a solid solute in a liquid solution is calculated adequately by means of the expression,

where, Δ*H _{fus}* is the fusion enthalpy of the solute,

*R*is the universal gas constant (8.314 J mol

^{-1}K

^{-1}),

*T*is the melting point of the solute, and

_{fus}*T*is the absolute temperature of the solution.

^{9}On the other hand, the real solubility (

*X*

_{2}) is calculated by adding the non-ideality term, (log γ

_{2}), to Equation 1 in order to obtain the following expression,

^{10}

The γ_{2} term is the activity coefficient of the solute and it must be determined experimentally for real solutions. Nevertheless, several techniques have been developed in order to obtain reasonable estimates of this term. One of these methods is the referent to regular solutions, in which, opposite to ideal solutions, a little positive enthalpic change is allowed.^{10} The solubility in regular solutions is obtained from,

where, *V*_{2} is the partial molar volume of the solute (cm^{3} mol^{-1}), Φ_{1} is the volume fraction of the solvent in the saturated solution, and δ_{1} and δ_{2} are the solubility parameters of solvent and solute, respectively. The solubility parameter, δ, is calculated as (Δ*H _{v} - RT*)/

*V*)

_{l}^{1/2}, where, Δ

*H*is the vaporization enthalpy and

_{v}*V*is the molar volume of the liquid.

_{l}The vast majority of pharmaceutical dissolutions deviate notoriously of that predicted by the regular solution-theory because strong interactions are present, such as hydrogen bonding, in addition to the differences in molar volumes among solutes and solvents. For these reasons, at the beginning of the 80s of the past century, Martin *et al*. developed the EHSA method, which has been useful to estimate the solubility of several drugs in binary and ternary cosolvent systems.^{3} Accordingly, if the *A* term (defined as *V*_{2}Φ_{1}^{2}/(2.303*RT*)) is introduced in the Equation 3, the real solubility of drugs and other compounds in any solvent can be calculated from the expression,

where *W* is equal to 2*K*δ_{1}δ_{2} and *K* is the Walker parameter.^{11} The *W* factor compensates the deviations observed with respect to the behavior of regular solutions, and it can be calculated from experimental data by means of the following expression,

where, γ_{2} is the activity coefficient of the solute in the saturated solution, and is calculated as *X*_{2}* ^{id}*/

*X*

_{2}.

The experimental values obtained for the *W* parameter can be correlated by means of regression analysis by using regular polynomials in superior order as a function of the solubility parameter of the solvent mixtures, as follows,

These empiric models can be used to estimate the drug solubility by means of back-calculation to resolve this property from the specific *W* value obtained in the respective polynomial regression.^{3,5,7}

**EXPERIMENTAL**

All reagents, materials, and procedures employed in this work have been reported earlier and were as follows.^{9}

**Reagents and materials**

In this investigation the following reagents and materials were used: indomethacin (CAS: [53-86-1], 1-(4-Chlorobenzoyl)-5-methoxy-2-methyl-1*H*-indole-3-acetic acid, purity at least 0.998 in mass fraction)^{1} accomplishing the British Pharmacopoeia quality requirements,^{12} 1,4-dioxane A.R. Scharlau, distilled water with conductivity < 2 µS cm^{-1}, molecular sieve Merck (numbers 3 and 4, pore size 0.3 and 0.4 nm, respectively), and Durapore^{®} 0.45 µm filters from Millipore Corp.

**Solvent mixtures preparation**

All 1,4-dioxane + water solvent mixtures were prepared by mass, using an Ohaus Pioneer TM PA214 analytical balance with sensitivity ± 0.1 mg, in quantities of 50 g. The mass fractions of 1,4-dioxane of the twelve binary mixtures prepared varied by 0.10 from 0.10 to 0.70 and by 0.05 from 0.75 to 0.95.

**Solubility determination**

An excess of IMC was added to approximately 10 g of each solvent mixture or neat solvent, in stoppered dark glass flasks. Solid-liquid mixtures were placed in re-circulating thermostatic baths (Neslab RTE 10 Digital One Thermo Electron Company) kept at 298.15 (± 0.05) K for at least 7 days to reach the equilibrium. In the case of neat water or water-rich mixtures the equilibration time was 14 days. These equilibrium times were established by measuring the drug concentrations till they became constant. After this time the supernatant solutions were filtered (at isothermal conditions) to ensure that they were free of particulate matter before sampling. Drug concentrations were determined after appropriate dilution by measuring the light absorbance and interpolation from a previously constructed UV spectrophotometry calibration curve (UV/VIS Bio-Mate 3 Thermo Electron Company spectrophotometer). In order to make the equivalence between volumetric and gravimetric concentration scales, the density of the saturated solutions was determined with a digital density meter (DMA 45 Anton Paar) connected to the same recirculating thermostatic bath. All the solubility experiments were run in triplicate at least.

**Estimation of the volumetric contributions **

Because the Equations 3 to 5 require the volume contributions of each component to the saturated solution, in this investigation the IMC apparent specific volume (*Φ*_{V}^{spc}) was used to calculate these contributions. The *Φ*_{V}^{spc} values were calculated according to Equation 7,^{13}

where, *m*_{2} and *m*_{1} are the masses of solute and solvent in the saturated solution, respectively, *VE*_{1} is the specific volume of the solvent, and ρ_{soln} is the solution density. The IMC apparent molar volume is calculated by multiplying the *Φ*_{V}^{spc} value and the molar mass of the solute (357.8 g mol^{-1}).^{1} Otherwise, the calculated molar volume value obtained by means of the Fedors method was used in all calculations and it was taken from the literature (230.0 cm^{3} mol^{-1}).^{8}

**RESULTS AND DISCUSSION**

The information about polarity and volumetric behavior of 1,4-dioxane + water mixtures as a function of the composition is shown in Table 1.^{14} On the other hand, the calorimetric values reported in the literature for IMC were as follows, *T _{fus}* = 432.6 K and Δ

*H*= 39.46 kJ mol

_{fus}^{-1}.

^{15}From these values the calculated ideal solubility for this drug was 7.123 × 10

^{-3}in mole fraction at 298.15 K.

^{8}This value was calculated according to Equation 1.

Table 2 summarizes the experimental IMC solubility expressed in molarity and mole fraction, the density of the saturated mixtures, the apparent molar volume of IMC, and the solvent volume fraction in the saturated solutions, at 298.15 K. The uncertainty in experimental solubility was lower than 1.2% from the mean. Figure 2 shows the experimental solubility and the calculated solubility by using the regular solution model (Equation 3) as a function of the solubility parameter of solvent mixtures. It is clear that experimental solubility is greater than calculated solubility probably due to strong hydrogen-bonding interactions between drug and solvents. This behavior is more evident in mixtures with δ_{1} lower than 32 MPa^{1/2}.

The IMC solubility is greater in the mixture of 0.95 in mass fraction of 1,4-dioxane (*X*_{2} = 3.99 x 10^{-2} with δ_{1} = 21.9 MPa^{1/}2), but this value is slightly greater than the maximum obtained in the mixture of 0.70 in mass fraction of ethyl acetate in mixtures conformed by ethyl acetate + ethanol (*X*_{2} = 3.00 x 10^{-2} with δ_{1} = 20.86 MPa^{1/2}).^{8} It is important to note that the maximum solubility is obtained at similar but slightly different mixtures-solubility parameters in both binary solvent systems. Otherwise, similar but slightly different solubility values were also found in both systems. This result could be interpreted in terms of different Lewis acid or base interactions. Besides, the main reason to obtain the maximum solubility of this drug in the mixture of 0.95 in mass fraction of 1,4-dioxane is referent to similar polarities between drug and solvent mixture in agreement with the expression "like dissolves like".^{5}

From density values of cosolvent mixtures and saturated solutions (Tables 1 and 2), in addition to IMC solubility (Table 2), the solvent volume fraction (Φ_{1}) and apparent molar volume of the solute (*Φ*_{V}^{mol}) in the saturated mixtures, were calculated. These values are also presented in Table 2.

In the literature the solute molar volume in the saturated solution has been considered as a constant value when EHSA method is used.^{5,11} On this way, for solid compounds this property is generally calculated by means of groups' contribution methods such as the one developed by Fedors.^{16} Nevertheless, this property is not independent on the solvent composition as can be see in Table 2 for apparent molar volume of IMC. This fact would be due to the different intermolecular interactions, depending on the respective solvent proportions. Nevertheless, great variability is found in the experimental *Φ*_{V}^{mol} values without a rational order. In particular, the great difference obtained in water-rich mixtures (greater than 315 cm^{3} mol^{-1}) and those obtained in 1,4-dioxane-rich mixtures (lower than 280 cm^{3} mol^{-1}) is anomalous. For these reasons, in this investigation the calculated molar volume of IMC (230.0 cm^{3} mol^{-1}) was employed in the following calculations as was made with this drug in ethanol + water mixtures.^{8} Otherwise, no significant differences in the predictive character of EHSA were obtained in other investigations when experimental or calculated values of drug molar volumes were interchanged.^{7}

On the other hand, according to the literature the volume fraction of the solvent mixture in the saturated solution has been calculated by means of the expression,^{5,11}

where, *V*_{1} is the molar volume of the solvent which is calculated for solvent mixtures assuming additive volumes as,^{17}

Nevertheless, it is well known that the mixing volumes are not additives in those mixtures with strong presence of hydrogen bonding and great differences in molar volumes. For this reason, the experimental solvent volume fractions were used in this investigation for all the calculations (Table 2). Solvent volume fractions were calculated by subtracting the respective drug volume contributions. The last values were calculated from molar volumes and concentrations of the drug at saturation for each cosolvent mixture.

Ultimately, the activity coefficients of IMC as decimal logarithms are also presented in Table 2. These values were calculated from experimental solubility values (Table 2) and ideal solubility at 298.15 K (*X*_{2} = 7.123 × 10^{-3}).^{8} γ_{2} values in water-rich mixtures were greater than unit because the experimental solubilities are lower than the ideal one.

On the other hand, the parameters *A, K*, and *W* are presented in Table 3. In order to calculate the *W* parameter the experimental solubility parameter of IMC obtained in the mixture of 0.95 in mass fraction of 1,4-dioxane (with δ_{1} = 21.9 MPa^{1/2}) was used. According with the literature, this δ_{2} value is the same of the solvent mixture where the greatest drug solubility was found.^{5,11}

As has been already indicated, the *W* parameter accounts for the deviations presented by real solutions with respect to regular solutions. These deviations are mainly due to specific interactions such as hydrogen bonding. IMC (Figure 1) and both solvents studied can establish these interactions, as hydrogen donors or acceptors because of their polar moieties, in particular due to -OH groups. 1,4-dioxane interacts just as hydrogen acceptor.

Figure 3 shows that the variation of the *W* parameter with respect to the solubility parameter of solvent mixtures, presents deviation from linear behavior. *W* values were adjusted to regular polynomials in orders from 1 to 5 (Equation 6) and their coefficients and statistical parameters are presented in the Table 4 (the empirical regressions were obtained by using MS Excel^{®} and TableCurve 2D v5.01). The obtained *W* values by using the respective regular polynomials are presented in Table 5. It is clear that these values depend on the model used in the *W* back-calculation. Similar behaviors have been reported in the literature for several other compounds.^{3,7,8}

Table 6 summarizes all the calculated drug solubility values. These solubilities were calculated employing the *W* values which were obtained by back-calculation from the polynomial models presented in Tables 4 and 5. Because we are searching the best adjust, the first criterion used to define the polynomial order of *W* as function of δ_{1} was the fitting standard uncertainties obtained, whose values were as follows, 32.6, 1.61, 0.644, 0.403, and 0.427 (Table 4), for orders 1 to 5, respectively. As another comparison criterion, Table 6 also summarizes the percentages of difference between IMC experimental solubility and those calculated by using EHSA.

According to Table 6 it follows that, as more complex the polynomial used is, better the agreement found between experimental and calculated solubility is. This fact is confirmed based on the mean deviation percentages (8.9 and 8.8%, for orders 4 and 5, respectively). In similar way to that found in other similar investigations,^{3,7,8} in this case, the most important increment in concordance is obtained passing from order 1 to order 2 (from 1.88 x 10^{7} to 47.8% in mean deviation). It is remarkable that mole fractions greater than unit are found between 0.40 and 0.75 in mass fraction of 1,4-dioxane by using order 1, which of course is not logic. Otherwise, significant increment in concordance is also obtained by passing from order 2 to order 3 (from 47.8 to 15.7%). Therefore, in the following calculations the model with lowest fitting uncertainty was used (order 4, Table 4).

An important consideration about the usefulness of the EHSA method is the one referent to justify the complex calculations involving any other variables of the considered system (Equation 4, Tables 1 to 5), instead of the simple empiric regression of the experimental solubility as a function of the solvent mixtures' solubility parameters (Table 2, Figure 4). For this reason, in the Table 7 the experimental solubilities are confronted to those calculated directly by using a regular polynomial in grade 4 of log *X*_{2} as a function of δ_{1} values (Equation 10, with determination coefficient r^{2} = 0.999 and fitting standard uncertainty = 0.054) and are also confronted to those calculated involving the *W* parameters obtained from Equation 6 adjusted to order 4 (Table 4). The respective difference percentages are also presented in Table 7.

Based on mean deviation percentages presented in Table 7 (8.1 and 8.9% for direct calculation and EHSA method, respectively) it follows that non-significant differences are found between the values obtained by using both methods. In similar way with that found for IMC in ethyl acetate + ethanol and ethanol + water mixtures,^{8} the present results would be showing non-significant usefulness of EHSA method for practical purposes. The last point apparently could be controversial considering that EHSA method implies additional experimentation including density determinations and thermal characterization of the solid-liquid equilibrium for the solid compound. Nevertheless, it is necessary keep in mind that EHSA method considers the drug solubility from a systematic physicochemical point of view. Moreover, it would be just necessary to found an effective method to calculate the Walker *K* parameter in order to calculate the *W* term according to the expression 2*K*δ_{1}δ_{2}, because the δ_{1} and δ_{2} terms would be known, and thus, the drug experimental solubility could be calculated in any mixture in particular. Otherwise, as it was already said EHSA method has been widely employed to estimate drugs solubilities since beginning 80s of the past century and it proved to be a powerful technique in pharmaceutical sciences because it uses physicochemical properties that are easily available for several kinds of drugs.

**CONCLUSION**

In this investigation the EHSA method has been adequately used to study the solubility of IMC in 1,4-dioxane + water mixtures by using calculated values of molar volume and estimated Hildebrand solubility parameter of this analgesic drug. In particular, a good predictive character has been found by using a regular polynomial in order 4 of the interaction parameter *W* vs. Solubility parameter of the solvent mixtures free of solute. Finally, this work expands the number of analgesic drugs successfully evaluated in front to EHSA method.

**ACKNOWLEDGEMENTS**

The authors thank to the DIB the Universidad Nacional de Colombia (UNC) for the financial support and the Department of Pharmacy of UNC for facilitating the equipments and installations used in the experimental solubility determinations.

**REFERENCES**

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Recebido em 19/1/11; aceito em 9/5/11; publicado na web em 29/6/11

* e-mail: fmartinezr@unal.edu.coo