Revisiting the Total Hildebrand and Partial Hansen Solubility Parameters of Analgesic Drug Meloxicam

Abstract

The reported total Hildebrand solubility parameter (δ₂) value of meloxicam, as calculated based on the group contribution method proposed by Fedors, was compared with those estimated based on the maximum solubility peaks observed in different aqueous cosolvent systems at T = 298.15 K. Thus, the observed δ₂ values varied from (19.8 to 29.1) MPa^1/2, respectively. Moreover, the Hansen solubility parameters (HSPs) and the total Hildebrand solubility parameter were also determined by using the Bustamante regression method with the reported experimental solubility values of meloxicam in 31 neat solvents (30 organic solvents and water), obtaining the values: δ_d = 19.9 MPa^1/2, δ_p = 16.9 MPa^1/2, δ_h = 5.7 MPa^1/2, and δ_T = 26.7 MPa^1/2. Furthermore, the HSPs of meloxicam were also estimated based on the Hoftyzer–van Krevelen group contribution method, obtaining the values: δ_d = 17.9 MPa^1/2, δ_p = 20.3 MPa^1/2, and δ_h = 9.2 MPa^1/2, and the total solubility parameter as: δ_T = 28.6 MPa^1/2. In addition, the Kamlet–Abboud–Taft linear solvation energy relationship (KAT-LSER) model was also employed to evaluate the role of different intermolecular interactions on the dissolution of meloxicam in different solvents that varied in terms of polarity and hydrogen bonding capability.

1. Introduction

Meloxicam (the molecular structure is shown in Figure 1; IUPAC name: 4-hydroxy-2-methyl-N-(5-methyl-2-thiazolyl)-2H-1,2-benzothiazine-3-carboxamide-1,1-dioxide; molar mass 351.40 g·mol⁻¹; CAS number: 71125-38-7; PubChem CID: 54677470) is a non-steroidal anti-inflammatory drug commonly used for pain and inflammatory treatments, particularly in rheumatic diseases and osteoarthritis. Regarding its action mechanism as an anti-inflammatory drug, it blocks cyclooxygenase-2 more than cyclooxygenase-1, which makes it more specific than other drugs [1,2,3,4,5].

From a thermophysical point of view, meloxicam melts at 254 °C and exhibits very low aqueous solubility [6], which negatively influences its in vivo dissolution rates from solid dosage forms; in turn, this affects its biological performance [7,8,9]. In addition, because of the very low aqueous solubility of meloxicam, all the investigative duties involved in the research and development of homogeneous liquid dosage forms based on this drug, like peroral or injectable medicinal products, are very long and difficult at an industrial pharmaceutical level [10,11,12].

In order to overcome the drawbacks mentioned above, the intention of several investigations has been to increase the aqueous equilibrium solubility of meloxicam by adding some common water-miscible pharmaceutical cosolvents [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. On the other hand, meloxicam solubility in several pure water-immiscible organic solvents, as well as in some non-aqueous mixtures, has also been studied because this information could be useful for improving crystallization procedures during drug purification processes [30,31,32].

As is well known, polarity is one of the most important properties of solutes and solvents involved in the equilibrium solubility magnitudes. Thus, the Hildebrand solubility parameter (δ_T), which corresponds to the square root of cohesive energy density (ΔU°/V°, where ΔU° is the internal energy and V° is the molar volume of the compound), is the most important polarity descriptor for describing the solubility of non-electrolyte or non-ionized weak electrolyte compounds. Moreover, regarding the intermolecular interactions associated with solute and solvent polarities, the calculation of the Hansen solubility parameters (HSPs) was also proposed in order to discriminate between the different intermolecular interactions, which is necessary to researchers involved in the design of new drugs and dosage forms [33,34,35]. Thus, Hansen divided the cohesive energy density into those contributions by non-polar interactions (namely the van der Waals dispersion forces), those by dipole interactions, and those by hydrogen bonding (Equation (1)). It is remarkable that HSP values can be used for polar and hydrogen bonding systems to evaluate the compatibility of different substances. Moreover, HSP values may be useful for preparing drugs and for estimating their miscibility [36,37].

$${\delta }_T=\sqrt{{\delta }_d^2+{\delta }_p^2+{\delta }_h^2}$$

(1)

where the terms δ_d, δ_p, and δ_h denote the partial parameters corresponding to the dispersion, polar, and hydrogen bonding components of the total solubility parameter, δ_T. The value of δ_d of a given solvent was assumed here to be equal to that of a non-polar substance (e.g., hydrocarbon) exhibiting almost the same molecular structure [36,37]. Thus, one of the objectives of this research was to re-evaluate the HSPs of meloxicam based on the reported equilibrium solubility values, as well as by means of the group contribution methods.

Furthermore, although some thermodynamic studies of dissolution in aqueous binary solvent systems have been performed in order to analyze the solute–solvent structural effects on meloxicam solubility, including enthalpy–entropy compensation and preferential solvation effects [17,18,19,20,21,22,23,24,25,26,27,28,29], it is remarkable that no specific attempts to discriminate between the different intermolecular interactions have been reported in the literature. Therefore, the Kamlet–Abboud–Taft linear solvation energy relationship (KAT-LSER) model was applied in this research to explore the solvent effects on the equilibrium solubility of meloxicam in 28 pure solvents, as has recently been reported in the literature for several other drug compounds in pure and mixed solvents, based on the linear solvation energy relationship concept [38,39,40,41,42,43].

To summarize, in this research the HSP values of meloxicam were determined again by considering the former and recently reported solubility values at T = 298.15 K and also by using another method based on the group contributions to every partial parameter; the aim is to amend the evaluation of the Lewis acid–base, polarizability, and volumetric effects on the equilibrium solubility of this drug at 298.15 K, as a complement to that previously reported in the literature [30].

2. Methodology

2.1. Meloxicam Solubility Values

The logarithmic mole fraction solubility values of meloxicam at 298.15 K in 30 organic solvents, as well as in pure water, were taken from the literature [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. These values are summarized in Table 1.

2.2. Physicochemical Properties of Solvents

The total Hildebrand and the partial Hansen solubility parameters, as well as the molar volumes and solvatochromic parameters of all the considered solvents, were taken from the literature, when available [44,45,46,47,48]. These parameters are summarized in Table 1.

Table 1. Logarithmic mole fraction solubility of meloxicam and solvatochromic and Hansen solubility parameters of pure solvents at T = 298.15 K.

Solvent a	ln x2	Solvatochromic Parameters b			V° c (cm3·mol−1)	Hansen Solubility Parameters c			δT (MPa1/2) c
		α	β	π		δd (MPa1/2)	δp (MPa1/2)	δh (MPa1/2)
Hexane	−11.774	0.00	0.00	−0.11	131.6	14.9	0.0	0.0	14.9
Cyclohexane	−12.317	0.00	0.00	0.00	108.7	16.8	0.0	0.0	16.8
Butyl acetate	−7.834	0.00	0.45	0.46	132.5	15.8	3.7	6.3	17.4
CCl4	−9.694	0.00	0.10	0.21	97.1	17.8	0.0	0.6	17.8
Ethyl acetate	−7.701	0.00	0.45	0.45	98.5	15.8	5.3	7.2	18.1
Toluene	−8.681	0.00	0.11	0.49	106.8	18.0	1.4	2.0	18.2
Benzene	−8.733	0.00	0.10	0.55	89.4	18.4	0.0	2.0	18.6
Chloroform	−6.312	0.20	0.10	0.58	80.7	17.8	3.1	5.7	19.0
Acetone	−7.409	0.08	0.48	0.62	74.0	15.5	10.4	7.0	20.0
1,4-Dioxane	−6.576	0.00	0.37	0.49	85.7	19.0	1.8	7.4	20.5
1-Octanol	−8.082	0.77	0.81	0.40	157.7	17.0	3.3	11.9	20.9
1-Heptanol	−8.395				141.8	16.0	5.3	11.7	21.0
1-Hexanol	−8.888	0.80	0.84	0.40	135.8	15.9	5.8	12.5	21.3
PEG400	−4.676				354.5 d	16.6 d	3.7 d	13.3 d	21.6 d
1-Pentanol	−9.124	0.84	0.86	0.40	109.0	16.0	4.5	13.9	21.7
Acetophenone	−6.077	0.04	0.49	0.81	117.4	19.6	8.6	3.7	21.8
Carbitol	−6.645				130.9	16.2	9.2	12.3	22.3
NMP	−3.313	0.00	0.77	0.92	96.5	18.0	12.3	7.2	22.9
1-Butanol	−9.338	0.84	0.84	0.47	91.5	16.0	5.7	15.8	23.1
Benzyl alcohol	−6.299	0.60	0.52	0.98	103.6	18.4	6.3	13.7	23.8
ACN	−9.010	0.19	0.40	0.66	52.6	15.3	18.0	6.1	24.4
1-Propanol	−9.797	0.84	0.90	0.52	75.2	16.0	6.8	17.4	24.5
DMF	−4.361	0.00	0.69	0.88	77.0	17.4	13.7	11.3	24.8
Ethanol	−9.778	0.86	0.75	0.54	58.5	15.8	8.8	19.4	26.5
DMSO	−4.840	0.00	0.76	1.00	71.3	18.4	16.4	10.2	26.7
Methanol	−9.930	0.98	0.66	0.60	40.7	15.1	12.3	22.3	29.6
NMF	−6.444	0.62	0.80	0.90	59.1	17.4	18.8	15.9	30.1
PG	−10.078	0.83	0.78	0.76	73.6	16.8	9.4	23.3	30.2
Glycerol	−12.610	1.21	0.51	0.62	73.3	17.4	12.1	29.3	36.1
Formamide	−7.196	0.71	0.48	0.97	39.8	17.2	26.2	19.0	36.6
Water	−13.687	1.17	0.47	1.09	18.0	15.6	16.0	42.3	47.8

^a PEG: polyethylene glycol; NMP: N-methyl-2-pyrrolidone; ACN: acetonitrile; DMF: N,N-dimethylformamide; DMSO: dimethyl sulfoxide; NMF: N-methylformamide; PG: propylene glycol. ^b From Marcus (1998) [45]. ^c From Barton (1991) [46] and Hansen (2007) [47]. ^d From Liu et al. (2000) [48].

Table 1. Logarithmic mole fraction solubility of meloxicam and solvatochromic and Hansen solubility parameters of pure solvents at T = 298.15 K.

Solvent a	ln x2	Solvatochromic Parameters b			V° c (cm3·mol−1)	Hansen Solubility Parameters c			δT (MPa1/2) c
		α	β	π		δd (MPa1/2)	δp (MPa1/2)	δh (MPa1/2)
Hexane	−11.774	0.00	0.00	−0.11	131.6	14.9	0.0	0.0	14.9
Cyclohexane	−12.317	0.00	0.00	0.00	108.7	16.8	0.0	0.0	16.8
Butyl acetate	−7.834	0.00	0.45	0.46	132.5	15.8	3.7	6.3	17.4
CCl4	−9.694	0.00	0.10	0.21	97.1	17.8	0.0	0.6	17.8
Ethyl acetate	−7.701	0.00	0.45	0.45	98.5	15.8	5.3	7.2	18.1
Toluene	−8.681	0.00	0.11	0.49	106.8	18.0	1.4	2.0	18.2
Benzene	−8.733	0.00	0.10	0.55	89.4	18.4	0.0	2.0	18.6
Chloroform	−6.312	0.20	0.10	0.58	80.7	17.8	3.1	5.7	19.0
Acetone	−7.409	0.08	0.48	0.62	74.0	15.5	10.4	7.0	20.0
1,4-Dioxane	−6.576	0.00	0.37	0.49	85.7	19.0	1.8	7.4	20.5
1-Octanol	−8.082	0.77	0.81	0.40	157.7	17.0	3.3	11.9	20.9
1-Heptanol	−8.395				141.8	16.0	5.3	11.7	21.0
1-Hexanol	−8.888	0.80	0.84	0.40	135.8	15.9	5.8	12.5	21.3
PEG400	−4.676				354.5 d	16.6 d	3.7 d	13.3 d	21.6 d
1-Pentanol	−9.124	0.84	0.86	0.40	109.0	16.0	4.5	13.9	21.7
Acetophenone	−6.077	0.04	0.49	0.81	117.4	19.6	8.6	3.7	21.8
Carbitol	−6.645				130.9	16.2	9.2	12.3	22.3
NMP	−3.313	0.00	0.77	0.92	96.5	18.0	12.3	7.2	22.9
1-Butanol	−9.338	0.84	0.84	0.47	91.5	16.0	5.7	15.8	23.1
Benzyl alcohol	−6.299	0.60	0.52	0.98	103.6	18.4	6.3	13.7	23.8
ACN	−9.010	0.19	0.40	0.66	52.6	15.3	18.0	6.1	24.4
1-Propanol	−9.797	0.84	0.90	0.52	75.2	16.0	6.8	17.4	24.5
DMF	−4.361	0.00	0.69	0.88	77.0	17.4	13.7	11.3	24.8
Ethanol	−9.778	0.86	0.75	0.54	58.5	15.8	8.8	19.4	26.5
DMSO	−4.840	0.00	0.76	1.00	71.3	18.4	16.4	10.2	26.7
Methanol	−9.930	0.98	0.66	0.60	40.7	15.1	12.3	22.3	29.6
NMF	−6.444	0.62	0.80	0.90	59.1	17.4	18.8	15.9	30.1
PG	−10.078	0.83	0.78	0.76	73.6	16.8	9.4	23.3	30.2
Glycerol	−12.610	1.21	0.51	0.62	73.3	17.4	12.1	29.3	36.1
Formamide	−7.196	0.71	0.48	0.97	39.8	17.2	26.2	19.0	36.6
Water	−13.687	1.17	0.47	1.09	18.0	15.6	16.0	42.3	47.8

^a PEG: polyethylene glycol; NMP: N-methyl-2-pyrrolidone; ACN: acetonitrile; DMF: N,N-dimethylformamide; DMSO: dimethyl sulfoxide; NMF: N-methylformamide; PG: propylene glycol. ^b From Marcus (1998) [45]. ^c From Barton (1991) [46] and Hansen (2007) [47]. ^d From Liu et al. (2000) [48].

2.3. Meloxicam Parameter Calculations

The multivariable correlations of the logarithmic mole fraction solubility of meloxicam at 298.15 K as a function of the partial Hansen solubility parameters or as a function of the solvatochromic parameters of the solvents were performed using the regression application of MS Excel.

3. Results and Discussion

3.1. Meloxicam Mole Fraction Solubility in Pure Solvents at 298.15 K

As mentioned above, Table 1 summarizes the equilibrium solubility of meloxicam at T = 298.15 K, as reported in the literature [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. To the best of our knowledge, in all cases the shake-flask method, followed by UV or gravimetric composition analysis, was performed for the solubility determinations [49]. As reported earlier in the literature, the solubility values obtained by following these experimental methods involve relative uncertainties lower than 7.0% in the majority of cases and differences lower than 12.0% when compared with other reported solubility values obtained under the same conditions [6]. Moreover, as described in all cases, the bottom solid phases in equilibrium with the saturated solutions corresponded to polymorph I of meloxicam, as demonstrated by DSC and XRDA analyses [50,51,52,53,54,55,56]. It is noteworthy that the listed solvents covered a wide range of polarities, as is shown when considering that they vary from pure hexane (δ₁ = 14.9 MPa^1/2) to pure water (δ₁ = 47.8 MPa^1/2).

It is well known that drug solubility depends on the polarity of solvents and that similar solubility values are expected when using solvents of similar polarity [6,10]. However, this is apparently not completely the case with meloxicam in the considered solvents, as can be seen in Table 1, where the solvents are ordered from lower to higher polarity, as defined by their total Hildebrand solubility parameters. Thus, the meloxicam mole fraction solubility of meloxicam at this temperature diminishes in the following order: N-methyl-2-pirrolidone (x₂ = 3.64 × 10⁻²) [29] > N,N-dimethylformamide (x₂ = 1.28 × 10⁻²) [22] > PEG400 (x₂ = 9.32 × 10⁻³) [21] > dimethyl sulfoxide (x₂ = 7.91 × 10⁻³) [28] > acetophenone (x₂ = 2.29 × 10⁻³) [30] > benzyl alcohol (x₂ = 1.84 × 10^–3) [30] > chloroform (x₂ = 1.81 × 10⁻³) [30] > N-methylformamide (x₂ = 1.59 × 10⁻³) [22] > 1,4-dioxane (x₂ = 1.39 × 10⁻³) [20] > Carbitol^® (x₂ = 1.30 × 10⁻³) [25] > formamide (x₂ = 7.49 × 10⁻⁴) [22] > acetone (x₂ = 6.06 × 10⁻⁴) [30] > ethyl acetate (x₂ = 4.53 × 10⁻⁴) [31] > butyl acetate (x₂ = 3.96 × 10⁻⁴) [30] > 1-octanol (x₂ = 3.09 × 10⁻⁴) [30] > 1-heptanol (x₂ = 2.26 × 10⁻⁴) [30] > toluene (x₂ = 1.70 × 10⁻⁴) [30] > benzene (x₂ = 1.61 × 10⁻⁴) [30] > 1-hexanol (x₂ = 1.38 × 10⁻⁴) [30] > acetonitrile (x₂ = 1.22 × 10⁻⁴) [23] > 1-pentanol (x₂ = 1.09 × 10⁻⁴) [30] > 1-butanol (x₂ = 8.80 × 10⁻⁵) [30] > carbon tetrachloride (x₂ = 6.17 × 10⁻⁵) [30] > ethanol (x₂ = 5.67 × 10⁻⁵) [17] > 1-propanol (x₂ = 5.56 × 10⁻⁵) [27] > methanol (x₂ = 4.87 × 10⁻⁵) [24] > propylene glycol (x₂ = 4.20 × 10⁻⁵) [18] > hexane (x₂ = 7.70 × 10⁻⁶) [30] > cyclohexane (x₂ = 4.47 × 10⁻⁶) [30] > glycerol (x₂ = 3.34 × 10⁻⁶) [30] > water (x₂ = 1.14 × 10⁻⁶) [18]. These different trends of solvent polarity and meloxicam solubility demonstrate that drug solubilities depend on solvent properties other than just polarity. It is noteworthy that meloxicam solubility increases almost 32 thousand times when passing from pure water to pure N-methyl-2-pirrolidone, demonstrating the wide dissolving power exhibited by the different pure solvents evaluated.

3.2. Hildebrand Solubility Parameter of Meloxicam

As is widely known, the maximum drug solubility in blended solvents is commonly observed in the cosolvent mixtures exhibiting almost the same polarity as the drug under consideration [9,44,57]. In turn, the Hildebrand solubility parameter of the mixture exhibiting the maximum peak of the drug solubility is used indirectly for assigning δ values to solid drugs. In this way, several maximum solubility peaks of meloxicam in cosolvent mixtures have been reported in the literature, implying that different solubility parameter values were obtained for this drug; these values are summarized in

Table 2. Hildebrand solubility parameters (δ_T/MPa^1/2) of meloxicam as defined by the maximum solubility peaks (w₁) in different blended solvent systems.

δT (MPa1/2)	Solvent Mixture	Mixture Composition (w1) of Maximum Meloxicam Solubility
29.1 a	Ethanol (1) + water (2)	0.85
21.2 b	1,4-Dioxane (1) + water (2)	0.975
26.0 c	Acetonitrile (1) + water (2)	0.80
25.9 d	2-Propanol (1) + water (2)	0.70
19.8 e	Ethyl acetate (1) + Ethanol	0.70

^a From Delgado et al. (2011) [17]. ^b From Jiménez et al. (2014) [20]. ^c From Tinjacá et al. (2021) [23]. ^d From Tinjacá et al. (2022) [27]. ^e From Cristancho and Martinez (2014) [31].

. Thus, the meloxicam δ_T values vary from (19.8 to 29.1) MPa^1/2, which is clearly a wide drug polarity interval. On the other hand, the meloxicam δ_T value calculated by means of the group contribution method proposed by Fedors is 32.1 MPa^1/2 [22,58]; this value is higher than all those obtained by considering the polarity of the maximum solubility peaks (i.e., from 19.8 MPa^1/2 to 29.1 MPa^1/2) in the cosolvent mixtures. Moreover, it is worth mentioning here that the molar volume of meloxicam used for the δ_T calculation was the same as the one calculated by the Fedors group contribution method (i.e., 183.3 cm³·mol^–1) [22,58]. However, if the molar volume is calculated by considering the molar mass (351.39 g·mol⁻¹ [50]) and the estimated density (1.564 g·cm⁻³ [50]) of the enol form of meloxicam (i.e., 224.7 cm³·mol⁻¹), a Fedors δ_T value of 29.0 MPa^1/2 is thereby obtained, which is closer to the δ_T value observed from the maximum meloxicam solubility peak in the {ethanol (1) + water (2)} mixtures (i.e., 29.1 MPa^1/2, Table 2) [17].

3.3. Hansen Solubility Parameters of Meloxicam

Furthermore, as indicated above, Hansen solubility parameters are commonly used in the study of polar and hydrogen-bonded systems in order to evaluate the compatibility of different substances in aqueous and non-aqueous solutions. In turn, in order to evaluate the different drug–solvent interactions, the difference between the solubility parameters of the solute (compound 2) and solvent (compound 1), namely Δδ, which is calculated by means of Equation (2), was employed here to understand the mixing processes between meloxicam and all the solvents analyzed [31].

$$\Delta \delta =\sqrt{{\left({\delta }_{d2}-{\delta }_{d1}\right)}^2+{\left({\delta }_{p2}-{\delta }_{p1}\right)}^2+{\left({\delta }_{h2}-{\delta }_{h1}\right)}^2}$$

(2)

It has been reported that Δδ values lower than 5.0 MPa^1/2 indicate a high possibility of miscibility between the hypothetically liquid solute and the respective liquid solvent system [59].

Bustamante et al. [60,61] demonstrated the possibility of calculating the partial Hansen solubility parameters of the solute by directly regressing ln x₂ against the corresponding values of the three partial solubility parameters of the considered solvents (δ_d1, δ_p1, and δ_h1), based on improving the significance of every one of the regression coefficients. The modified Bustamante regression model is commonly presented as:

$$\ln x_2=C_0+C_1{\delta }_{d1}^2+C_2{\delta }_{d1}+C_3{\delta }_{p1}^2+C_4{\delta }_{p1}+C_5{\delta }_{h1}^2+C_6{\delta }_{h1}^{}$$

(3)

Thus, Equation (3) was used here to calculate the three partial solubility parameters of meloxicam by using the ratio of the respective coefficients, as indicated in the following equations:

$${\delta }_{d2}=-\frac{C_2}{2C_1}$$

(4)

$${\delta }_{p2}=-\frac{C_4}{2C_3}$$

(5)

$${\delta }_{h2}=-\frac{C_6}{2C_5}$$

(6)

As indicated above, Table 1 summarizes the logarithmic mole fraction solubilities of meloxicam at T = 298.15 K; these solubilities were taken from the literature [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. This table also includes the solvatochromic parameters (α, β, and π) as well as the molar volumes and the Hansen solubility parameters of the different solvents considered in this research; these were also reported in the literature at T = 298.15 K.

Thus, Equation (3), when applied to meloxicam solubility in 31 solvents of different polarities, leads to Equation (7) (with the following statistical parameters: n = 31, r = 0.826, typical error = 1.550, F = 8.557, and critical F value = 4.96 × 10⁻⁵) [62,63,64].

$$\begin{array}{l}\ln x_2=-65.2(\pm 60.6)-0.144(\pm 0.210){\delta }_{d1}^2+5.729(\pm 7.155){\delta }_{d1}-1.31(\pm 0.65)\cdot {10}^{-2}{\delta }_{p1}^2\\ +0.442(\pm 0.166){\delta }_{p1}-6.46(\pm 2.44)\cdot {10}^{-3}{\delta }_{h1}^2+7.4(\pm 10.8)\cdot {10}^{-2}{\delta }_{h1}^{}\end{array}$$

(7)

From the obtained coefficients of Equation (7), the following partial solubility parameters were calculated for meloxicam: δ_d = 19.9 MPa^1/2, δ_p = 16.9 MPa^1/2, and δ_h = 5.7 MPa^1/2, whereas the total solubility parameter of meloxicam obtained by using Equation (1) is δ_T = 26.7 MPa^1/2. These new values are higher than those reported earlier by Sathesh-Babu et al., namely δ_T = 23.81 MPa^1/2 (δ_d = 19.16 MPa^1/2, δ_p = 13.22 MPa^1/2, δ_h = 5.0 MPa^1/2), which were obtained by considering 26 solvents [30]. The observed differences in the solubility parameter magnitudes thus estimated could be due to the different number of solvents studied in both cases.

As also indicated above, the HSP values can also be estimated by means of different methods that are based on group contribution [65,66]. Thus, the Hoftyzer–van Krevelen method is the most commonly employed in physical pharmacy and pharmaceutics for drugs characterization. Nevertheless, it is important to keep in mind that all the methods based on group contribution are only approximate. However, they are useful in obtaining good estimates of the magnitude of the solubility parameters of organic compounds like meloxicam. The partial solubility parameters are calculated as defined by the following expressions:

$${\delta }_d=\frac{{\displaystyle \sum n{F}_d}}{{\displaystyle \sum nV}}$$

(8)

$${\delta }_p=\frac{\sqrt{{\displaystyle \sum n{F}_p^2}}}{{\displaystyle \sum nV}}$$

(9)

$${\delta }_h=\sqrt{\frac{{\displaystyle \sum n{U}_h}}{{\displaystyle \sum nV}}}$$

(10)

where the F_d, F_p and U_h values for the different groups were determined and reported by Hoftyzer–van Krevelen [14,36,37,38]. Moreover, Just et al. reported F_d, F_p and U_h values for some other functional groups involving sulfur atoms that had not been studied previously [67]. F_d represents the contribution by London dispersion forces, F_p stands for the contribution by polar forces, and U_h stands for the contribution by the hydrogen bonding interaction energy. Furthermore, V is the molar volume of meloxicam that could be estimated from the molar mass and density values of the drug if they were available; this is the case for meloxicam. Sometimes V is calculated by using the Fedors method, which is also based on the group contributions, as indicated above.

Table 3. Hansen solubility parameters [(δ_d, δ_p, and δ_h)/MPa^1/2)] of meloxicam as calculated by the Hoftyzer–van Krevelen group contribution method.

Group	Number	Fd (J·cm3/2·mol−1) a	${\mathit{F}}_{\mathit{p}}^2$ (J·cm3·mol−2) a	Uh (J·mol−1) a
–CH3	2	673.2	0	0
=CH–	1	255.0	1444	0
=C<	4	−226.8	1600	0
Phenylene	1	1173.0	4057.69	40.4
–OH	1	76.5	1,500,625	6060
–CO-NH–	1	225.0	160,000	11,000
–N<	1	30.0	22,500	750
–N=	1	380.0	10,000	250
–S–	1	815.9	38,416	297.5
–SO2–	1	295.8	19,018,321	200
Ring ≥ 5	2	285.6	0	0
Double bond	3	45.0	613.47	250.5
	Σ	4028.2	20,757,577	18,848.4
			δd	17.9 MPa1/2
			δp	20.3 MPa1/2
			δh	9.2 MPa1/2
			δT	28.6 MPa1/2

^a Energy values taken from Just et al. (2013) [67].

summarizes the partial Hansen solubility parameters of meloxicam as δ_d = 17.9 MPa^1/2, δ_p = 20.3 MPa^1/2, and δ_h = 9.2 MPa^1/2 and the total solubility parameter as δ_T = 28.6 MPa^1/2; the molar volume of meloxicam is again considered to be 224.7 cm³·mol⁻¹. As observed, the total Fedors (i.e., 29.0 MPa^1/2) and the Hoftyzer–van Krevelen (i.e., 28.6 MPa^1/2) δ_T values are similar. However, both of them differ by more than 2.0 MPa^1/2 from the one obtained from the coefficients of Equation (7) (i.e., δ_T = 26.7 MPa^1/2). These small differences are expected, as indicated above, owing to the low number of solvents studied (e.g., 31), as well as the quality of the reported F_d, F_p and U_h energy values for all the groups or fragments; this quality depends on the number of compounds studied in its determinations. Therefore, all these partial solubility parameters could be more reliable as more meloxicam solubility values in other organic solvents are determined, and thus become available in physicochemical databases, and more robust F_d, F_p, and U_h energy values appear in the literature.

On the other hand, the Hansen solubility parameters obtained here for meloxicam could be useful in improving the predictive power of some models based on the Abraham and Hansen solvation parameters that have recently been reported in the literature for estimating the meloxicam solubility in aqueous cosolvent systems at different temperatures [68].

3.4. Solvent Effects: KAT-LSER Model

As indicated above, the KAT-LSER model was also applied here to the experimental meloxicam solubility values in order to evaluate the solute–solvent Lewis acid or base and the polarization effects upon the improvement of this relevant physicochemical property for this drug in solution. The classical KAT-LSER model is presented in the form of Equation (11) [38,39,40,41,42,43].

$$\ln x_2=c_0+c_1\alpha +c_2\beta +c_3\pi +c_4\left(\frac{V_2{\delta }_1^2}{100RT}\right)$$

(11)

where c₁α and c₂β denote the energetic terms associated with the specific solute–solvent Lewis acid and base interactions, respectively; in turn, c₃π represents the energetic term related to non-specific interactions; finally, the last term in Equation (11) denotes the cavity effect, which defines the energetic requirement for solvent–solvent interactions, i.e., the energy required to overcome the cohesive forces and, thus, to separate the solvent molecules in order to create the cavity for the accommodation of the solute molecules. The last term designates the meloxicam accommodation energy, which is defined as the product of the total Hildebrand solubility parameter of the solvent (δ₁) and the molar volume of meloxicam (V₂ = 224.7 cm³·mol⁻¹). The universal gas constant (R = 8.3145 J·mol⁻¹·K^–1) and experimental temperature (T/K = 298.15) are considered here in the quotient denominator to obtain a dimensionless magnitude relative to the cavity term. Furthermore, c₀ represents the meloxicam–meloxicam interactions and measures the intercept when α = β = π = δ₂ = 0. c₁ and c₂ are a measure of the meloxicam susceptibility to drug–solvent interactions based on specific hydrogen bonding, while c₃ and c₄ represent the solute sensitivity to the non-specific electrostatic meloxicam–solvent and solvent–solvent interactions. Thus, the KAT-LSER model obtained here is shown as Equation (12) (with the following statistical parameters: n = 28, r = 0.936, typical error = 0.945, F = 40.38, and critical F value = 4.40 × 10⁻¹⁰).

$$\begin{array}{l}\ln x_2=-10.58(\pm 0.46)-2.15(\pm 0.85)\alpha +2.33(\pm 1.04)\beta \\ +7.22(\pm 1.10)\pi -4.31(\pm 0.99)\left(\frac{V_2{\delta }_1^2}{100RT}\right)\end{array}$$

(12)

The obtained positive values of c₂ (2.33) and c₃ (7.22) demonstrate the favorable contribution of the Lewis base and polarizability effects to the equilibrium meloxicam solubility. On the other hand, the negative values of c₀ (−10.58), c₁ (−2.15), and c₄ (−4.31) demonstrate the unfavorable contribution of the solute–solute interactions, the Lewis acid behavior, and the cavity energy requirements to the solubility of this drug in the considered solvents. The negative contribution of the Lewis acid behavior to the equilibrium meloxicam solubility could be attributed to the unfavorable solute–solute interactions between the hydroxyl group of one meloxicam molecule and the heterocyclic nitrogen atom of another meloxicam molecule in the solution state. Furthermore, if the absolute values of c₁, c₂, c₃, and c₄ are compared together, the following contribution percentages are obtained: 13.44% for α, 14.52% for β, 45.11% for π, and 26.93% for $V_2{\delta }_1^2/100RT$, respectively. Thus, the polarization effects make the higher contribution to the drug dissolution, followed by the solvent volumetric effects, whereas the Lewis acid and base effects contribute in almost in the same proportion to this thermodynamic property of meloxicam (Figure 1).

3.5. Comparison between Hansen Solubility Parameters and KAT-LSER Model for Intermolecular Interactions Analyses

Although some conceptual differences between both physicochemical approaches are observed, particularly with regard to the cavity volume effects that are not directly considered in the solubility estimations based on HSPs, a comparison between both approaches could be made, as shown in

Table 4. Contribution percentages by different kind of interactions based on HSPs and KAT-LSER approaches.

Interaction	HSP (Bustamante et al.) a	HSP (Hoftyzer–Van Krevelen) b	KAT-LSER c
Dispersion	46.7%	37.9%	26.9%
Dipolar	39.8%	42.8%	45.1%
Hydrogen bonding	13.5%	19.3%	28.0%

^a Calculated directly from the partial parameters obtained from coefficients of Equation (7) [60,61]. ^b Calculated directly from the partial parameters shown in Table 3 [67]. ^c Calculated by adding c₁ and c₂ to obtain the hydrogen bonding, c₃ for dipolar, and c₄ for dispersion effects, from Equation (12) [38,39,40,41,42,43].

. Here, the contribution percentages by different kind of interactions, namely the London dispersion forces, dipolar interactions, and hydrogen bonding, are summarized.

As observed, in all cases significant differences are observed among the respective contribution percentages. Moreover, some differences between the HSP values estimated by Bustamante et al.’s equation and the Hoftyzer–van Krevelen group contribution method are also observed. Therefore, it is assumed that more refinement would be required in both methods to understand the molecular mechanisms involved in the drug dissolution processes. However, from a qualitative point of view these physicochemical approaches are useful and could be more effective if more solubility values in other solvents were available in the future.

4. Conclusions

The Hildebrand solubility parameter of meloxicam was reanalyzed when it was calculated using the Fedors group contribution method, obtaining the value of 29.0 MPa^1/2. It was also obtained from the polarity of those cosolvent mixtures exhibiting peaks of maximum meloxicam solubility, which varied from (19.8 to 29.1) MPa^1/2. Moreover, the partial Hansen solubility parameters of meloxicam were estimated by following Bustamante et al.’s regression model method with the available experimental solubility values of meloxicam in 31 pure solvents (corresponding to 30 organic solvents and water) at T = 298.15 K. Thus, the following values were obtained: δ_d = 19.9 MPa^1/2, δ_p = 16.9 MPa^1/2, δ_h = 5.7 MPa^1/2, and δ_T = 26.7 MPa^1/2. In addition, the partial Hansen solubility parameters of meloxicam were also calculated based on the respective group contributions, as established in the Hoftyzer–van Krevelen method, obtaining the following values: δ_d = 17.9 MPa^1/2, δ_p = 20.3 MPa^1/2, δ_h = 9.2 MPa^1/2, and δ_T = 28.6 MPa^1/2. Finally, the KAT-LSER model was also employed to evaluate the role of different intermolecular interactions in the dissolution processes of meloxicam in the considered solvents. Thus, the positive effects of Lewis base behavior and polarization on the solubility of this drug in different solvents, which varied widely in terms of polarity and hydrogen bonding capability, were observed.

As previously mentioned, the understanding of the intermolecular interactions involved in drug dissolution processes could be useful for improving purification procedures and the design of liquid medicines and even for understanding the pharmacokinetics and the drug–receptor interactions in the respective biophases just before the beginning of the pharmacodynamic effects as an anti-inflammatory drug. In general terms, it could be stated that this research contributes to the understanding of the different intermolecular interactions present when meloxicam is dissolved in solvents of different kinds with differing polarity and variable hydrogen bonding capabilities. Thus, this new information expands what was previously proposed in the literature about meloxicam behavior based on the apparent thermodynamic quantities of dissolution as a function of the mixture polarity, including enthalpy–entropy compensation and preferential solvation analyses in different aqueous–cosolvent mixtures, in relation to the water–structural effects and Lewis acid or base behaviors for both the drug and the solvents in blended systems [17,18,19,20,21,22,23,24,25,26,27,28,29].