**Increasing the Equilibrium Solubility of Meloxicam in Aqueous Media by Using Dimethyl Sulfoxide as a Cosolvent: Correlation, Dissolution Thermodynamics and Preferential Solvation**

**Darío A. Tinjacá 1, Fleming Martínez 2,\*, Ovidio A. Almanza 3, M. Ángeles Peña 4, Abolghasem Jouyban 5,6 and William E. Acree, Jr. 7**

	- 3 Grupo de Física Aplicada, Departamento de Física, Facultad de Ciencias, Universidad Nacional de Colombia, Sede Bogotá, Cra. 30 No. 45-03, Bogotá 111321, Colombia
	- 4 Departamento de Ciencias Biomédicas, Facultad de Farmacia, Universidad de Alcalá, Alcalá de Henares, 28801 Madrid, Spain
	- 5 Pharmaceutical Analysis Research Center, Faculty of Pharmacy, Tabriz University of Medical Sciences, Tabriz<PHONE_NUMBER>, Iran
	- 6 Faculty of Pharmacy, Near East University, 99138 Nicosia, Turkey
	- 7 Department of Chemistry, University of North Texas, Denton, TX 76203, USA
	- **\*** Correspondence<EMAIL_ADDRESS>Tel.: +571-3165000 (ext. 14608)

**Abstract:** Meloxicam is widely prescribed as an analgesic and anti-inflammatory drug in human therapeutics. Owing the very low aqueous solubility of meloxicam, this property has been studied in dimethyl sulfoxide (DMSO)-aqueous solvent systems at several temperatures from 273.15 to 313.15 K to expand the solubility database about analgesic drugs in mixed solvents. The flask shake method followed by ultraviolet-visible (UV-vis) spectrophotometry analysis were used for meloxicam solubility determinations. A number of cosolvency models, including the Jouyban–Acree model, were challenged for solubility correlation/prediction of this drug in these mixtures. The van't Hoff and Gibbs equations were employed to calculate the apparent standard thermodynamic quantities relative to dissolution and mixing processes. The inverse Kirkwood–Buff integral method was employed for calculating the preferential solvation parameters of meloxicam by DMSO in the mixtures. Meloxicam solubility increases with increasing temperature and maximum solubilities are observed in neat DMSO at all temperatures studied. Dissolution processes were endothermic in all cases and entropydriven in the composition interval of 0.40 ≤ *x*1 ≤ 1.00. A nonlinear enthalpy–entropy relationship was observed in the plot of enthalpy vs. Gibbs energy for drug transfer processes. Meloxicam is preferentially solvated by water in water-rich mixtures but preferentially solvated by DMSO in the composition interval of 0.21 < *x*1 < 1.00.

**Keywords:** meloxicam; {DMSO (1) + water (2)} mixtures; cosolvency; Jouyban–Acree model; dissolution thermodynamics; preferential solvation; IKBI

#### **1. Introduction**

Meloxicam (molecular structure shown in Figure 1, IUPAC name: 4-hydroxy-2-methyl-*N*-(5-methyl-2-thiazolyl)-2*H*-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 used commonly for pain and inflammatory treatments [1–5]. From a physicochemical point of view, meloxicam exhibits very low aqueous solubility, which influences negatively in vivo dissolution rates, affecting its biological performance. Otherwise, because of the very low aqueous solubility of this drug, all the duties relative

**Citation:** Tinjacá, D.A.; Martínez, F.; Almanza, O.A.; Peña, M.Á.; Jouyban, A.; Acree, W.E., Jr. Increasing the Equilibrium Solubility of Meloxicam in Aqueous Media by Using Dimethyl Sulfoxide as a Cosolvent: Correlation, Dissolution Thermodynamics and Preferential Solvation. *Liquids* **2022**, *2*, 161–182. https://doi.org/10.3390/ liquids2030011

Academic Editor: Urban Bren

Received: 14 July 2022 Accepted: 9 August 2022 Published: 12 August 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

to research and development of homogeneous liquid dosage forms, such as peroral or injectable products, based on this drug, are very long and hard at an industrial pharmaceutical level. In order to overcome the drawbacks mentioned above, some investigations have been intended to increase the aqueous equilibrium solubility of meloxicam. These investigations were mainly based in the evaluation of the solubilizing effect of some common pharmaceutical cosolvents, as has recently been summarized in a previous communication of our research group [6]. More recently, some other aqueous mixtures involving different cosolvents, including choline-based deep eutectic solvents, have also been studied and reported [7–11]. It is noteworthy that very good meloxicam solubility-increasing has been reported, reaching more than 1000-fold in some cases, as follows: 1144 with Carbitol® [9], 1399 with *N*-methylformamide [6], and 11,233 with *N*,*N*-dimethylformamide [6].

**Figure 1.** Molecular structure of meloxicam.

Dimethyl sulfoxide (molar mass: 78.13 g·mol−1, CAS number: 67-68-5, PubChem CID 679) is a polar, aprotic solvent, miscible with water and with a wide range of organic solvents in all possible compositions, which makes it adequate for dissolving substances of both polar and nonpolar nature [1,12]. It exhibits less toxicity than *N*-methylformamide and *N*-methyl-2-pyrrolidone and has been reported about its good power of penetrating the skin and other membranes without damaging them. This is the reason why it has a high potential to increase the penetration of less soluble active ingredients. Otherwise, DMSO exhibits by itself some analgesic and anti-inflammatory properties [13]. Indeed, there are some references in the literature that suggest the potential use of DMSO as a possible pharmacotherapeutic agent in the management of pain and other conditions [14]. Moreover, owing its high solubilizing power for different active pharmaceutical ingredients, it has been used as an excipient in topical and parenteral medications for human and veterinary use [15]. On the other hand, DMSO has also been considered as solvent model in medicinal chemistry for studying intermolecular effects of different drugs [16]. In this way, aqueous mixtures of DMSO have been studied for solubilizing several drugs, drug-alike compounds, and other organic chemicals, including bergenin [17], *N*-guanylurea dinitramide [18], naringin [19], sinapic acid [20], p-nitrobenzamide [21], d-histidine [22], micoflavin [23], phenformin [24], baricitinib [25], a pyridazinone derivative [26], and nicotinamide [27], among others.

As it is well-described in the specialized chemical and pharmaceutical literature, all the physicochemical data about the equilibrium solubility of drugs or drug-alike compounds in aqueous cosolvent mixtures, as well, as the deep understanding of the respective dissolution mechanisms, are very important for both theoretical and practical points of view in pharmaceutical and chemical sciences. This is because the measured, reported, and analyzed solubility values expand the respective solubility databases, which is very useful for theoretical and practical purposes in both the pharmaceutical and chemical industries, as indicated above [28–31].

Therefore, the main aims of this research were as follows: (i) to determine and analyze the effects of both mixtures' composition and temperature on the solubility of meloxicam in {DMSO (1) + water (2)} mixtures; (ii) to correlate equilibrium solubility data with several well-known thermodynamic models; (iii) to calculate the apparent standard dissolution and mixing thermodynamic parameters; and (iv) to study the preferential solvation parameters of meloxicam in binary mixtures conformed by DMSO and water. Therefore, this research is a continuation of some other similar ones reported earlier in the literature [6–9,11,32] about the meloxicam equilibrium solubility in other aqueous cosolvent systems of pharmaceutical interest for design of dosage forms and its quality control analysis, but using another commonly used cosolvent because of its high solubilizing power.

#### **2. Materials and Methods**

#### *2.1. Materials and Reagents*

Meloxicam (Technodrugs & Intermediates PVT LTD, component 3, purity > 0.995 in mass fraction), DMSO (Panreac, component 1, purity > 0.995 in mass fraction), and distilled water with conductivity < 2 μS·cm−1 (component 2), were used. Purities of meloxicam and DMSO were reported by the suppliers, as determined by high-performance liquid chromatography and gas chromatography, respectively.

#### *2.2. Preparation of Solvent Mixtures*

All the {DMSO (1) + water (2)} binary solvent mixtures were prepared gravimetrically by using an Ohaus Pioneer TM PA214 analytical balance (USA, sensitivity ± 0.1 mg), in quantities of 50.00 g. The mole fractions of DMSO of the nine mixtures prepared, varied by 0.10 in steps from *x*1 = 0.10 to *x*1 = 0.90.

#### *2.3. Solubility Determinations*

Equilibrium meloxicam solubilities were determined by using the shake-flask method [33], followed by UV-spectrophotometric analysis, as follows: an excess amount of meloxicam was added to 50.00 g of each binary solvent mixture or neat DMSO in dark glass pharmaceutical flasks. The stoppered flasks were putted in an ultrasonic bath (Elma® E60H Elmasonic, Fremont, CA, USA) during 15 min and were later transferred to thermostatic mechanical shakers (Julabo SW23, Seelbach, Germany) or re-circulating thermostatic baths (Neslab RTE 10 Digital One Thermo Electron Company, Waltham, MA, USA) and kept at 313.15 K for at least four days to ensure that the drug saturation had been achieved. After that, the supernatant solutions were isothermally filtered (Millipore Corp. Swinnex®-13, Burlington, MA, USA) to remove undissolved solid particles before sampling. Meloxicam concentrations were determined after appropriate gravimetric dilution with a 0.10 mol·dm−3 NaOH solution, by measuring the UV light absorbance at the maximum absorbance wavelength, *λ*max = 361 nm (UV/VIS BioMate 3 Thermo Electron Company spectrophotometer, USA), followed by interpolation from a previously validated UV spectrophotometric gravimetric calibration curve prepared in NaOH 0.10 mol·dm−3. The respective equation was: Absorbance = 0.0073 + 52.508·*C*, with *C* expressed as μg·g−1. Later, the thermostatic baths temperature was decreased from 313.15 K to 308.15 K allowing the meloxicam excess precipitation during two days, following with the same procedures mentioned above to determine the new meloxicam concentrations at saturation. All these procedures were performed successively until solid-liquid equilibrium was achieved at 293.15 K. All the solubility experiments were performed at least three times and the respective results were averaged. The density of the saturated solutions was measured by using a digital density meter (DMA 45 Anton Paar, Graz, Austria) connected to a re-circulating thermostatic bath (Neslab RTE 10 Digital One Thermo Electron Company, USA) in order to transform the obtained gravimetric solubility values into volumetric concentration scales. The density meter was calibrated at every temperature by using air and water as standards as indicated in the respective instruction manuals [34].

#### *2.4. Solid Phase Analyses*

#### 2.4.1. X-ray Diffraction (XRD) Analysis

To determine the crystal nature of the solid meloxicam samples both before and after the saturation in neat water, in the mixture of *x*1 = 0.50, and in neat DMSO, the respective X-ray powder diffraction analyses were performed by using a PANalytical Xpert Pro X-ray diffractometer. The equipment is provided with CuKα radiation *λ* = 1.5418 Å. Generator setting: 40 kV and 40 mA and Bragg–Brentano geometry. Data were collected at 2*θ* from 5◦ to 70◦ and angle variation of 0.02◦ with detector data acquisition time of 9.46 min operating under room temperature.

#### 2.4.2. Fourier Transform Infrared (FTIR) Analysis

In additional to XRD analyses, in order to confirm the nature of the solid meloxicam samples, both before and after the saturation in neat water, in the mixture of *x*1 = 0.50, and in neat DMSO, FTIR analyses were also performed. The meloxicam solid samples were ground with quantities from 10 to 100 times its bulk of pure potassium bromide and the resulting mixtures were pressed into discs by using a special mold and a manual hydraulic press (Specac®, Fort Washington, PA, USA). The respective spectra were obtained in an FTIR spectrophotometer (IRAffinity-1, Shimadzu, Kyoto, Japan).

#### **3. Results and Discussion**

#### *3.1. Experimental Mole Fraction and Molarity Solubility*

Tables 1 and 2 summarize the experimental equilibrium solubilities of meloxicam in all {DMSO (1) + water (2)} solvent systems at 293.15 ≤ *T*/K ≤ 313.15, as expressed in mole fraction and molarity (mol·dm−3), respectively. The studied temperature interval includes what is commonly known as "room temperature" for products storage, as well as, the normal human body temperature. It is worth mentioning that all solubility values in neat water were taken from reference [32]. The average relative uncertainty obtained in the reported solubility values was 2.4%, which is in agreement with that commonly observed in this kind of experiments. If the mole fraction scale is considered, at *T* = 298.15 K, Table 1 shows that the meloxicam solubility increased 6956 times from *x*3 = 1.137 × 10−6 in neat water to *x*3 = 7.909 × 10−3 in neat DMSO, where maximum solubility is obtained. A deep comparison about the meloxicam solubility in neat water has been reported and discussed earlier in one of our previous communications [6]. Regarding the meloxicam mole fraction equilibrium solubility in neat DMSO Sathesh-Babu et al. reported a value of *x*3 = 5.496 × 10−3 at *T* = 298.15 K [35], which is in good agreement with the one obtained in this research (i.e., *x*3 = 7.909 × 10−2, Table 1). Moreover, when considering the molarity scale, a value of *C* = 1.516 × 10−2 mol·dm−3 was reported by Castro et al. at *T* = 298.15 K [36], which differs in almost one order of magnitude regarding our value (i.e., *C* = 0.1089 mol·dm−3, Table 2). The observed differences in solubility values could be attributed to several reasons such as different polymorphic states, different saturation times, or different analytical procedures, among others, as described earlier [31]. Up to the best of our knowledge, no solubility values of meloxicam in aqueous mixtures of DMSO have been reported and no more comparisons are possible.

Figure 2 depicts the meloxicam solubility profiles as function of the Hildebrand solubility parameters (*δ*1+2) of {DMSO (1) + water (2)} mixtures at *T* = 298.15 K. As widely described, *δ*1+2 is a very important polarity descriptor of cosolvent mixtures [28–31]. This descriptor was calculated considering the Hildebrand solubility parameter of both pure solvents (*δ*1 = 26.6 MPa1/2 for DMSO and *δ*2 = 47.8 MPa1/2 for water [37,38]), and the volume fraction (*fi*) of each solvent, as [29,39]:

$$
\delta\_{1+2} = \sum\_{i=1}^{2} f\_i \delta\_i \tag{1}
$$

**Table 1.** Experimental mole fraction solubility (*x*3) of meloxicam in {DMSO (1) + water (2)} mixtures at several temperatures and *p* = 96 kPa a,b.

a *p* is the atmospheric pressure in Bogotá, Colombia. *x*1 is the mole fraction of DMSO (1) in the {DMSO (1) + water (2)} mixtures free of meloxicam (3). Mean uncertainty in *x*1, *u*(*x*1) = 0.0005. b Standard uncertainty in *p* is *u*(*p*) = 3.0 kPa. Average relative uncertainty in *x*3, *u*r(*x*3) = 0.024. Standard uncertainty in *T* is *u*(*T*) = 0.10 K. c Data taken from Delgado et al. [32].

**Table 2.** Experimental molar solubility (*C*, mol·dm–3) of meloxicam in {DMSO (1) + water (2)} mixtures at several temperatures and *p* = 96 kPa. a,b

a *p* is the atmospheric pressure in Bogotá, Colombia. *x*1 is the mole fraction of DMSO (1) in the {DMSO (1) + water (2)} mixtures free of meloxicam (3). Mean uncertainty in *x*1, *u*(*x*1) = 0.0005. b Standard uncertainty in *p* is *u*(*p*) = 3.0 kPa. Average relative uncertainty in *C*, *u*r(*C*) = 0.024. Standard uncertainty in *T* is *u*(*T*) = 0.10 K. c Data taken from Delgado et al. [32].

As observed, the solubility curve exhibited maximum in neat DMSO, where *δ*1 is 26.6 MPa1/2. Because solutes normally reach their maximum solubilities in solvent systems exhibiting similar polarity [28,29], it is expected that the meloxicam δ3 value would be lower than 26.6 MPa1/2 at 298.15 K. However, this *δ*3 value is lower compared with the one reported earlier (*δ*3 = 32.1 MPa1/2) [6] calculated by means of the Fedors method [40]. This high discrepancy could be mainly attributed to specific drug solvation processes by DMSO or water, which are not considered in Fedors' calculations, in particular, if considering the structural effects

described for aqueous mixtures of DMSO [41,42]. Otherwise, Figure 2 also compares the logarithmic solubility of meloxicam as function of the Hildebrand solubility parameter in some aqueous-aprotic cosolvent mixtures, namely, {DMSO (1) + water (2)}, {dimethyl formamide (1) + water (2)} [6], and {acetonitrile (1) + water (2)} [7], mixtures at 298.15 K. It is noteworthy that meloxicam solubilities are highest in {dimethyl formamide (1) + water (2)} mixtures followed by {DMSO (1) + water (2)} mixtures of *δ*1+2 < 34.0 MPa1/2 and lowest in {acetonitrile (1) + water (2)} mixtures of *δ*1+2 < 34.0 MPa1/2. Otherwise, in mixtures of *δ*1+2 > 34.0 MPa1/2, the solubilities in aqueous mixtures of DMSO and acetonitrile are similar. This result shows that meloxicam solubility depends not only on the polarity but also on some other physicochemical properties of solute and solvent systems.

**Figure 2.** Logarithmic mole fraction solubility of meloxicam (ln *x*3) as function of the Hildebrand solubility parameter in some {cosolvent (1) + water (2)} mixtures at 298.15 K. •: DMSO (1) + water (2), ♦: Dimethylformamide (1) + water (2) [6], Δ: acetonitrile (1) + water (2) [7].

#### *3.2. Solid Phases' Analyses*

X-ray diffraction (XRD) spectra for meloxicam of the original untreated sample and after its saturation in neat water, neat DMSO, and the aqueous-DMSO mixture of *x*1 = 0.50 are shown in Figure 3. Because of the high similarity among all obtained spectra, it could be concluded that changes of the crystalline form of meloxicam are not observed after its dissolution and saturation in these four solvent systems. Moreover, Table 3 summarizes the position, 2q spacing, peak height (in counts), and the relative intensity of peaks exhibiting values higher than 10% for the original untreated meloxicam sample. These values are in good coincidence with those reported by Wu et al. [43]. Moreover, all the obtained XRD spectra of this research are very similar to that reported earlier for the polymorph I of meloxicam [32,44–47]. Finally, the FTIR spectra of all solid meloxicam samples shown in Figure 4 are also coincident with those reported in the literature, which allows to indicate that the three bottom-solid phases, obtained after drug saturation, have the same nature as the original untreated sample [47–49]. Therefore, meloxicam did not suffer crystal polymorphic transitions or solvates formation after saturation in these dissolution experiments.

**Figure 3.** X-ray diffraction spectra of meloxicam. From top to bottom: crystallized in water, crystallized in {DMSO (1) + water (2)} (*x*1 = 0.50) mixture, crystallized in DMSO, original sample.

**Table 3.** X-ray diffraction analysis of the original untreated meloxicam sample: position, 2q spacing, peak height (in counts) and relative intensity of peaks with values higher than 10%.

**Figure 4.** FTIR spectra of meloxicam. From top to bottom: original sample, crystallized in DMSO, crystallized in {DMSO (1) + water (2)} (*x*1 = 0.50) mixture, crystallized in water.

#### *3.3. Activity Coefficients in Mixed Solvents*

Table 4 summarizes the asymmetrical activity coefficients (*γ*3) of meloxicam in all the {DMSO (1) + water (2)} solvent systems. These values were calculated as the quotient *x*id 3 /*x*3 from the ideal (*x*id 3 ) and experimental solubilities of meloxicam summarized in Table 1. It is noteworthy that ideal solubilities were taken from the literature [32]. As observed, the *γ*3 values vary from 2708 in neat water (where the lower drug solubilities are observed) to 0.389 in neat DMSO at *T* = 298.15 K, where the maximum drug solubility is observed at this temperature. At all temperatures, meloxicam exhibits *γ*3 values higher than unity in neat water and the mixtures of *x*1 ≤ 0.70, but lower than unity in DMSO-rich mixtures and neat DMSO. On the other hand, in neat water and the mixtures of *x*1 = 0.10 and 0.20, the *γ*3 values increase with an increase in temperature. This implies some distancing from the ideal dissolution behavior with increasing temperature. In mixtures of 0.30 ≤ *x*1 ≤ 0.50, the *γ*3 values are almost independent on temperature. On the contrary, in mixtures of *x*1 ≥ 0.60 and neat DMSO, the *γ*3 values decrease with the increase in temperature.

**Table 4.** Activity coefficients of meloxicam in {DMSO (1) + water (2)} mixtures at several temperatures and *p* = 96 kPa a,b.

a *p* is the atmospheric pressure in Bogotá, Colombia. *x*1 is the mole fraction of DMSO (1) in the {DMSO (1) + water (2)} mixtures free of meloxicam (3). Mean uncertainty in *x*1, *u*(*x*1) = 0.0005. b Standard uncertainty in *p* is *u*(*p*) = 3.0 kPa. Average relative uncertainty in *γ*3 is *u*r(*γ*3) = 0.027. Standard uncertainty in *T* is *u*(*T*) = 0.10 K. c Data taken from Delgado et al. [32].

Moreover, Equation (2) allows a rough estimate of the magnitudes of solute–solvent intermolecular interactions from *γ*3 values [50].

$$
\ln \gamma\_3 = (\varepsilon\_{\rm s5} + \varepsilon\_{\rm 33} - 2\varepsilon\_{\rm s3}) \frac{V\_3 \, q\_s^2}{RT} \tag{2}
$$

where subscript 1 stands for the solvent system that here corresponds to neat solvents or aqueous DMSO binary mixtures, *ess*, *e*33, and *es*3 represent the magnitudes of solvent– solvent, solute–solute, and solvent–solute interaction energies, respectively. However, it is important to keep in mind that in the case of ternary systems, such as DMSO-watermeloxicam studied here, some water–cosolvent interactions are present, which could also play an important role in drug solubilities and dissolution rates. *V*3 denotes the molar volume of the super-cooled liquid meloxicam, whereas, *ϕs* denotes the volume fraction of every solvent system. For low *x*3 solubility values, *V*3*ϕs* 2/*RT* may be considered as constant despite of the solvent system. Hence, *γ*3 values would depend mainly on *ess*, *e*33, and *es*3 [50]. As is well-known, *ess* and *e*33 are unfavorable for drug solubility and dissolution, whereas *es*3 favors the respective drug solubility and dissolution rate increasing. The contribution of *e*33 could be considered as constant in the different solvent systems studied. Thus, from a qualitative viewpoint, based on the energetic quantities described in Equation (2), the following analysis could be established: because *ess* is highest in neat water (*δ*2 = 47.8 MPa1/2) and lowest in neat DMSO (*δ*1 = 26.6 MPa1/2) [37,38], neat water and water-rich mixtures (exhibiting *γ*3 values higher than 2300) would imply high *ess* and low *es*3 values, whereas, in DMSO-rich mixtures (exhibiting *γ*3 values near the unity), the *ess* values are relatively low, and therefore, the *es*3 values would be high. In this way, a higher solvation of meloxicam by DMSO in DMSO-rich mixtures is expected.

#### *3.4. Solubility Modeling*

Among the available cosolvency models presented for calculation of drug solubilities in mixed solvents at isothermal condition or at various temperatures [51,52], the Yalkowsky model is the simplest one [53] and requires only two experimental solubility determinations (in neat solvents) to predict the solubility at other solvent compositions. The Yalkowsky model is commonly represented as:

$$
\ln \mathbf{x}\_{\text{3}-(1+2)} = \mathbf{x}\_1 \ln \mathbf{x}\_{\text{3}(1)} + \mathbf{x}\_2 \ln \mathbf{x}\_{\text{3}(2)} \tag{3}
$$

where *x*3−(1+2) denotes the mole fraction solubility of meloxicam in the aqueous-cosolvent mixtures, *x*3(1) denotes the mole fraction solubility in neat DMSO (component 1), *x*3(2) denotes the mole fraction solubility in neat water (component 2), and *x*1 and *x*2 are the mole fractions of DMSO (1) and water (2) in the cosolvent mixtures in the absence of meloxicam (3). Thus, the obtained mean percentage deviation (*MPD*) values after calculation of the solubility of meloxicam in {DMSO (1) + water (2)} mixtures at *T* = (293.15, 298.15, 303.15, 308.15, and 313.15) K by means of this model were (41.3, 42.9, 44.8, 45.6, and 46.7)%, respectively, with the overall *MPD* of 44.2%. The numerical values of the *MPD* were computed using:

$$MPD = \frac{100}{N} \sum \frac{\left| \mathbf{x}\_3^{calc} - \mathbf{x}\_3 \right|}{\mathbf{x}\_3} \tag{4}$$

where *N* is the number of experimental data points.

As mentioned above, Equation (3) is capable of estimating drug solubility in cosolvent mixtures at individual *T* using only the drug solubility data in the mono-solvents at this *T*. However, it could be extended to obtain:

$$\ln \mathbf{x}\_{3(1+2),T} = \mathbf{x}\_1 \left( A\_1 + \frac{B\_1}{T} \right) + \mathbf{x}\_2 \left( A\_2 + \frac{B\_2}{T} \right) \tag{5}$$

to be applied at various temperatures (*x*3(1+2),*T*) using a single equation. In Equation (5), *A* and *B* terms are the model constants [54]. The trained model for solubility of meloxicam in {DMSO (1) + water (2)} mixtures is:

$$
\ln \mathbf{x}\_{3(1+2),T} = \mathbf{x}\_1 \left( 6.419 - \frac{3357.251}{T} \right) + \mathbf{x}\_2 \left( -10.582 - \frac{924.993}{T} \right) \tag{6}
$$

which resulted in the *MPD* of 44.4%.

The main limitation of the Yalkowsky model is that it does not consider any more interaction term after mixing the solutions and considers the mixing behavior as an ideal one. The Jouyban–Acree model includes as many as required interaction terms (*Ji* terms) to describe the non-ideality of the mixing and produced the most accurate results in correlating drug solubility data in binary solvent mixtures at various temperatures. The model is presented as [51]:

$$
\ln \mathbf{x}\_{3(1+2),T} = \mathbf{x}\_1 \ln \mathbf{x}\_{3(1),T} + \mathbf{x}\_2 \ln \mathbf{x}\_{3(2),T} + \left(\frac{\mathbf{x}\_1 \mathbf{x}\_2}{T}\right) \sum\_{i=0}^{2} f\_i \left(\mathbf{x}\_1 - \mathbf{x}\_2\right)^i \tag{7}
$$

where *Ji* terms are the respective model constants that are computed by using a nonintercept least square analysis [31]. Accordingly, the generated solubility values of meloxicam in {DMSO (1) + water (2)} were fitted to Equation (7) and the obtained trained model was:

$$\ln \mathbf{x}\_{3(1+2),T} = \mathbf{x}\_1 \ln \mathbf{x}\_{3(1),T} + \mathbf{x}\_2 \ln \mathbf{x}\_{3(2),T} + \left(\frac{\mathbf{x}\_1 \mathbf{x}\_2}{T}\right) \left[1707.481 + 1045.950(\mathbf{x}\_1 - \mathbf{x}\_2) - 1264.920(\mathbf{x}\_1 - \mathbf{x}\_2)^2\right] \tag{8}$$

The *F* value of Equation (8) was 791 and the adjusted correlation coefficient (*R*) was 0.979, whereas the correlation and the model constants were significant with *p* < 0.0005. Equation (8) is valid for calculating the solubility of meloxicam in different{DMSO (1) + water (2)} mixtures at various temperatures or mixtures-composition of interest, by employing the solubility data of meloxicam in neat DMSO and neat water at each temperature. The obtained *MPD* for the back-calculated solubility data of meloxicam when using Equation (8) was 9.6%.

Although Equation (8) provided an accurate correlation for the solubility of meloxicam in these {DMSO (1) + water (2)} mixtures, it requires the experimental solubility data in mono-solvents (i.e., *x*3(1),*T* and *x*3(2),*T*) at any temperature of interest to calculate the solubility of meloxicam in the required binary solvent mixtures. However, one may combine the trained version of Equation (5) with Equation (7) to provide a full predictive model, obtaining the following:

$$\begin{split} \ln \mathbf{x}\_{3(1+2),T} &= \quad \mathbf{x}\_{1} \Big( 6.419 - \frac{3357.251}{T} \Big) + \mathbf{x}\_{2} \Big( -10.582 - \frac{924.993}{T} \Big) \\ &+ \Big( \frac{\mathbf{x}\_{1} \mathbf{x}\_{2}}{T} \Big) \Big[ 1707.332 + 1046.323 \left( \mathbf{x}\_{1} - \mathbf{x}\_{2} \right) - 1265.293 \left( \mathbf{x}\_{1} - \mathbf{x}\_{2} \right)^{2} \Big] \end{split} \tag{9}$$

Equation (9) allows to calculate the solubilities of meloxicam in all these binary mixtures at various temperatures with an *MPD* of 9.9% and it does not require any experimental input data. The *F* and *R* values for Equation (9) were 787 and 0.978, respectively. For practical applications of Equation (9), one may train the model using the minimum number of seven experimental solubility points, and then, predict the rest of required data in any aqueous-DMSO mixture composition and temperature of interest, as has been exemplified in the literature [55]. When the model was trained with the solubility data in DMSO and water at *T* = (293.15 and 313.15) K (i.e., the lowest and highest temperatures) and in *x*1 = 0.3, 0.5 and 0.7 at *T* = 298.15 K (total 7 data points), the rest of the data points were predicted with the MPD of 18.7% (*N* = 48).

In a previous work [56], generally trained version of the Jouyban–Acree–Abraham and Jouyban–Acree–Hansen models were presented for predicting the solubility of meloxicam in various binary solvent mixtures. These models are:

$$\begin{aligned} \ln x\_{m,T} &= \; w\_1 \ln x\_{1,T} + w\_2 \ln x\_{2,T} \\ &+ \left(\frac{w\_1 w\_2}{I}\right) \\ &+ \left(\frac{w\_1 w\_2}{I}\right) \\ &- \left[\frac{-148.762(s\_1 - s\_2)^2 - 230.735(a\_1 - a\_2)^2}{-50.130(b\_1 - b\_2)^2 + 4243.83(v\_1 - v\_2)^2}\right] \\ &+ \left(\frac{-238.740 + 1753.598(c\_1 - c\_2)^2 - 5123.773(c\_1 - c\_2)^2}{I}\right) \\ &+ \left(\frac{w\_1 w\_2 (w\_1 - w\_2)}{I}\right) \\ &+ \left(\frac{w\_1 w\_2 (w\_1 - w\_2)}{I}\right) \\ &+ \left(\frac{w\_1 w\_2 (w\_1 - w\_2)^2}{I}\right) \end{aligned} \tag{10}$$

$$\begin{aligned} \left(\frac{w\_1 w\_2 (w\_1 - w\_2)}{I}\right)^2 \\ &+ \left(\frac{w\_1 w\_2 (w\_1 - w\_2)^2}{I}\right) \begin{bmatrix} 358.925 - 862.281 (c\_1 - c\_2)^2 + 6965.842 (c\_1 - c\_2)^2 \\ &+ 383.849 (s\_1 - s\_2)^2 + 97.860 (a\_1 - a\_2)^2 - 44.224 (b\_1 - b\_2)^2 \\ &+ 162.067 (b\_1 - v\_2)^2 - 13.147 (a\_1 b\_1 - a\_2 b\_2)^2 \end{bmatrix} \end{aligned} \tag{11}$$

and

$$\begin{aligned} \ln x\_{m,T} &= \quad w\_1 \ln x\_{1,T} + w\_2 \ln x\_{2,T} \\ &+ \left(\frac{x\_1 x\_2}{T}\right) \left[\begin{array}{c} 1027.586 - 116.816 \left(\delta\_{d1} - \delta\_{d2}\right)^2 \\ -6.910 \left(\delta\_{p1} - \delta\_{p2}\right)^2 + 1.168 \left(\delta\_{h1} - \delta\_{h2}\right)^2 \end{array} \right] \\ &+ \left(\frac{w\_1 w\_2 \left(w\_1 - w\_2\right)}{T}\right) \left[-717.903 + 11.233 \left(\delta\_{p1} - \delta\_{p2}\right)^2\right] \end{aligned} \tag{11}$$

The *c*, *e*, *s*, *a*, *b*, and *v* are Abraham solvent's coefficients and *δd*1, *δp*1, *δh*1, and *δd*2, *δp*2, and *δh*2 are the Hansen parameters for solvents 1 and 2, respectively [56]. Equations (10) and (11) predicted the solubility of meloxicam in (DMSO + water) mixtures with the MPDs of 74.3 and 35.6%. Although the prediction errors are relatively large, these equations only require the solubility data in the neat solvent.

#### *3.5. Apparent Thermodynamic Functions of Dissolution*

All the apparent standard thermodynamic quantities relative to meloxicam dissolution processes were calculated at the mean harmonic temperature, *T*hm = 303.0 K, which was calculated by using Equation (12) [57].

$$T\_{\rm hm} = \frac{n}{\sum\_{i=1}^{n} \left(\frac{1}{T}\right)}\tag{12}$$

where *n* is the number of temperatures under study (i.e., 5 in this case). Hence, all the apparent standard enthalpy changes relative to dissolution processes (Δsoln*H*◦) were obtained by means of the modified van't Hoff equation, as shown in Equation (13) [58]:

$$\left(\frac{\partial \ln \mathbf{x}\_3}{\partial (1/T - 1/T\_{\rm hm})}\right)\_P = -\frac{\Delta\_{\rm soln} H^\circ}{R} \tag{13}$$

The apparent standard Gibbs energy changes relative to all the meloxicam dissolution processes (Δsoln*G*◦) were calculated by means of [58–60]:

$$
\Delta\_{\text{soln}}G^{\odot} = -RT \cdot \text{intercept} \tag{14}
$$

The intercepts used in Equation (14) were those obtained as the result of the linear regressions of ln *x*3 as function of (1/*T* − 1/*T*hm). Therefore, Figure 5 depicts the meloxicam solubility linear van't Hoff behavior in all the {DMSO (1) + water (2)} mixtures, as well as in both neat solvents. It is noteworthy that linear regressions with *r*2 > 0.993 were observed in all the solvent systems [61–63].

Finally, the apparent standard changes in entropy, for all the studied meloxicam dissolution processes (Δsoln*S*◦) were calculated based on the respective Δsoln*H*◦ and Δsoln*G*◦ values by using [59,60]:

$$
\Delta\_{\text{soln}}S^{\diamond} = \frac{(\Delta\_{\text{soln}}H^{\diamond} - \Delta\_{\text{soln}}G^{\diamond})}{T\_{\text{hm}}} \tag{15}
$$

Table 5 summarizes all the apparent standard thermodynamic quantities for the dissolution processes of meloxicam in all the {DMSO (1) + water (2)} mixtures at *T*hm = 303.0 K, including those corresponding to dissolution processes in neat water and DMSO. It is notable that all apparent standard dissolution thermodynamic quantities for neat water were taken from previous research results [32]. As expected, all the apparent standard Gibbs energies and apparent enthalpies of dissolution of meloxicam are positive in every case in these DMSO-aqueous systems. Otherwise, the apparent standard entropies of dissolution were negative in neat water, as well as in the mixtures of *x*1 ≤ 0.30 but positive from the mixture of *x*1 = 0.40 to neat DMSO. Thus, the global dissolution processes of meloxicam are always endothermic in nature and entropy-driven for those occurring in the composition interval of 0.40 ≤ *x*1 ≤ 1.00; whereas, in neat water and the mixtures of *x*1 ≤ 0.30, neither

entropy or enthalpy-driving are observed because Δsoln*H*◦ > 0 and Δsoln*S*◦ < 0). All Δsoln*G*◦ values decrease continuously from neat water (where highest Δsoln*G*◦ value is obtained) to reach the lowest value in neat DMSO. Otherwise, Δsoln*H*◦ decreases from neat water to reach the lowest value in the mixture of *x*1 = 0.10, and later, it increases with the DMSO proportion to reach a quasi-plateau in the mixtures of *x*1 = 0.30, 0.40, and 0.50 to increase again, reaching the highest value in the mixture of *x*1 = 0.60; after, it decreases to reach a new minimum in neat DMSO. The Δsoln*S*◦ values increase from a lowest negative value in neat water (−87.99 J·mol−1·K−1) to reach the maximum positive value in the mixture of *x*1 = 0.60 (66.49 J·mol−1·K−1), and later, they decrease continuously with the DMSO proportion to reach a lower value in neat DMSO (53.35 J·mol−1·K−1). As observed, the lowest Δsoln*H*◦ and Δsoln*S*◦ values are observed in neat water or in the mixture of *x*1 = 0.00. The negative apparent dissolution entropies observed in neat water and the mixtures of composition *x*1 = 0.10, 0.20, and 0.30 could be a consequence of the possible hydrophobic hydration around the methyl and phenylene groups of meloxicam (Figure 1).

**Figure 5.** Van't Hoff plot of the solubility of meloxicam (3) in {DMSO (1) + water (2)} solvent systems. -: *x*1 = 0.00 (neat water), Δ: *x*1 = 0.10, : *x*1 = 0.20, ♦: *x*1 = 0.30, ×: *x*1 = 0.40,¼: *x*1 = 0.50, •: *x*1 = 0.60, : *x*1 = 0.70, : *x*1 = 0.80, : *x*1 = 0.90, +: *x*1 = 0.10 (neat DMSO).

**Table 5.** Apparent thermodynamic functions relative to dissolution processes of meloxicam (3) in {DMSO (1) + water (2)} mixtures at *T*hm = 303.0 K and *p* = 96 kPa a,b.

a *p* is the atmospheric pressure in Bogotá, Colombia. *x*1 is the mole fraction of DMSO (1) in the {DMSO (1) + water (2)} mixtures free of meloxicam (3). Mean uncertainty in *x*1, *u*(*x*1) = 0.0005. b Standard uncertainty in *T*hm is *u*(*T*hm) = 0.13 K. Standard uncertainty in *p* is *u*(*p*) = 3.0 kPa. Average relative standard uncertainty in apparent thermodynamic quantities of real dissolution processes are *u*r(Δsoln*G*◦) = 0.027, *u*r(Δsoln*H*◦) = 0.035, *u*r(Δsoln*S*◦) = 0.045, *u*r(*T*Δsoln*S*◦) = 0.045. c *ζH* and *ζTS* are the relative contributions by enthalpy and entropy toward apparent Gibbs energy of dissolution. d Data taken from Delgado et al. [32].

Moreover, to calculate the magnitude contributions by enthalpy (*ζH*) and entropy (*ζTS*) toward the dissolution processes, the following equations were used [64]:

$$\mathcal{Z}\_H = \frac{|\Delta\_{\rm soln} H^\circ|}{|\Delta\_{\rm soln} H^\circ| + |T \Delta\_{\rm soln} S^\circ|} \tag{16}$$

$$\zeta\_{TS} = \frac{|T\Delta\_{\text{soln}}S^{\circ}|}{|\Delta\_{\text{soln}}H^{\circ}| + |T\Delta\_{\text{soln}}S^{\circ}|} \tag{17}$$

As observed in Table 5, the higher contribution to the positive apparent standard molar Gibbs energies of meloxicam dissolution is given by the positive enthalpy. This demonstrates that in almost all the mixtures, the main contributor to this positive standard molar Gibbs energy of solution of meloxicam (reflected in the low meloxicam solubility) is the enthalpy except for neat water and the mixture of *x*1 = 0.10, where *ζH* = 0.224 and 0.177, respectively, and thus, entropy is the dominant function in these two cases.

#### *3.6. Apparent Thermodynamic Quantities of Mixing*

The overall dissolution processes of meloxicam in all {DMSO (1) + water (2)} solvent systems may be represented with the following hypothetical stages:

Solute(Solid state) at *T*hm → Solute(Solid state) at *T*fus → Solute(Liquid state) at *T*fus → Solute(Liquid state) at *T*hm → Solute(Solution state) at *T*hm

> Here, the hypothetical stages are considered as follows: (i) the heating and fusion of meloxicam at *T*fus = 536.7 K, (ii) the cooling of the liquid fused meloxicam to the considered temperature (i.e., *T*hm = 303.0 K), and (iii) the subsequent mixing of both the hypothetical super-cooled liquid meloxicam and the liquid aqueous-DMSO solvent system

at *T*hm = 303.0 K [65]. This treatment allowed us to calculate every individual thermodynamic contribution toward the overall meloxicam dissolution processes by means of:

$$
\Delta\_{\rm soln} H^{\circ} = \Delta\_{\rm fus} H^{T\_{\rm fun}} + \Delta\_{\rm mix} H^{\circ} \tag{18}
$$

$$
\Delta\_{\text{soln}} S^{\circ} = \Delta\_{\text{fus}} S^{T\_{\text{hm}}} + \Delta\_{\text{mix}} S^{\circ} \tag{19}
$$

where Δfus*HT*hm and Δfus*ST*hm represent the thermodynamic quantities relative to meloxicam melting and its cooling at *T*hm = 303.0 K, which, in turn, are calculated by means of [66]:

$$
\Delta\_{\rm fus} H^{\rm fram} = \Delta\_{\rm fus} H^{\rm fras} - \Delta C\_p (T\_{\rm fus} - T\_{\rm lum}) \tag{20}
$$

$$
\Delta\_{\rm fus} S^{T\_{\rm lum}} = \Delta\_{\rm fus} S^{T\_{\rm fus}} - \Delta C\_p \ln \left( \frac{T\_{\rm fus}}{T\_{\rm lum}} \right) \tag{21}
$$

where Δ*Cp* denotes the difference of heat capacities of liquid and solid states at the temperature of melting. Owing the difficulties in Δ*Cp* experimental determinations, the entropy of fusion (Δ*Sf*) is used instead [66]. Table 6 summarizes all the apparent standard thermodynamic quantities of mixing of the hypothetical super-cooled liquid meloxicam with all the studied aqueous-DMSO mixtures and the neat solvents, water, and DMSO, as calculated at *T*hm = 303.0 K. As observed, the Gibbs energies of mixing are positive from neat water to the mixture of *x*1 = 0.70 because the experimental drug solubilities are lower than ideal solubilities; on the contrary, they are negative from the mixture of *x*1 = 0.80 to neat DMSO, because the experimental solubilities are higher than the ideal ones. The contributions by the thermodynamic quantities of mixing subprocesses to the overall dissolution processes of meloxicam are variable and depend on the aqueous-DMSO mixtures' composition. Thus, Δmix*H*◦ values are negative in neat water and the mixtures of *x*1 = 0.10 and 0.20 but positive in the solvent systems in the interval of 0.30 ≤ *x*1 ≤ 1.00. Moreover, Δmix*S*◦ values are negative in the interval 0.00 ≤ *x*1 ≤ 0.50 but positive in the other cases. Thus, the mixing processes in neat water and the mixture of *x*1 = 0.10 and 0.20 are enthalpy-driven because of the exothermic character exhibited. In the mixtures 0.30 ≤ *x*1 ≤ 0.50, neither enthalpy nor entropydriving is observed for mixing. Finally, in the interval of 0.60 ≤ *x*1 ≤ 1.00, entropy-driven is observed. Furthermore, in order to evaluate the relative contributions by enthalpy (*ζH*) and entropy (*ζTS*) to the mixing processes in all these solvent systems, two equations analogous to Equations (16) and (17) were also employed. As observed, in water-rich and DMSO-rich mixtures, the main contributor to Gibbs energies of mixing is the entropy, but in the mixtures' composition interval of 0.30 ≤ *x*1 ≤ 0.70, it is the enthalpy.

As described earlier in the literature, the net variation of Δmix*H*◦ values regarding the aqueous-cosolvent mixtures' composition depends on the contribution of different kinds of intermolecular interactions. Hence, the cavity formation in the solvent system, required for the solute accommodation, is endothermic because some quantity of energy must be supplied against the respective cohesive forces of the solvent. This contribution diminishes the drug solubility as mentioned above. Oppositely, the solvent–solute interactions, resulting mainly from van der Waals and Lewis acid-base interactions, such as hydrogen bonding, are clearly exothermic in nature. This effect increases the drug solubility and dissolution rate as also indicated before. Even more, the structuring of water molecules around the phenylene ring and the methyl group of meloxicam structure (Figure 1) would be contributing to diminish the net Δmix*H*◦ quantity to small or even negative values in water-rich mixtures [67]. This event is clearly observed with meloxicam in aqueous-DMSO mixtures as indicated in Table 5 for solvent systems from neat water to the mixture of *x*1 = 0.50.

**Table 6.** Apparent thermodynamic functions relative to mixing processes of meloxicam (3) in {DMSO (1) + water (2)} mixtures at *T*hm = 303.0 K and *p* = 96 kPa a,b.

a *p* is the atmospheric pressure in Bogotá, Colombia. *x*1 is the mole fraction of DMSO (1) in the {DMSO (1) + water (2)} mixtures free of meloxicam (3). Mean uncertainty in *x*1, *u*(*x*1) = 0.0005. b Standard uncertainty in *T*hm is *u*(*T*hm) = 0.13 K. Standard uncertainty in *p* is *u*(*p*) = 3.0 kPa. Average relative standard uncertainty in apparent thermodynamic quantities of mixing processes are *u*r(Δmix*G*◦) = 0.030, *u*r(Δmix*H*◦) = 0.040, *u*r(Δmix*S*◦) = 0.050, *u*r(*T*Δmix*S*◦) = 0.050. c *ζH* and *ζTS* are the relative contributions by enthalpy and entropy toward apparent Gibbs energy of mixing.

#### *3.7. Enthalpy–Entropy Compensation Analysis*

Classical extra-thermodynamic studies, in particular those based on the enthalpy– entropy compensation analyses, provide a powerful physicochemical tool to find or identify similar mechanisms responsible for some physical and chemical processes involving organic compounds [68,69]. Some well-known literature reports demonstrated the presence of nonlinear-enthalpy–entropy compensation effects in the dissolution processes of many drugs and drug-alike compounds in different aqueous cosolvent binary systems. These extra-thermodynamic studies have usually been performed by different research groups to identify the main mechanisms involved in the cosolvent action for solubility increasing or decreasing, depending on the mixtures' composition [70–72]. As shown in Figure 6, meloxicam exhibits a nonlinear Δsoln*H*◦ vs. Δsoln*G*◦ trend with negative slopes from neat water to the mixture of *x*1 = 0.10 and from the mixture of *x*1 = 0.60 to neat DMSO, whereas, in the interval of 0.10 ≤ *x*1 ≤ 0.30 and from *x*1 = 0.50 to *x*1 = 0.60, positive slopes are observed; finally, in the interval of 0.30 ≤ *x*1 ≤ 0.50, a plateau is observed. In the first cases, the driving mechanism for transferring meloxicam from the most polar solvent systems to less polar solvent systems is the entropy. For the intervals exhibiting positive slopes, the drug transfer is driven by the enthalpy. Otherwise, in the interval of 0.30 ≤ *x*1 ≤ 0.50, the function driving the transfer is not clear. Nevertheless, it is not easy to identify the molecular effects involved, owing the complexity of aqueous-DMSO mixtures as indicated earlier [41,42].

**Figure 6.** Δsoln*H*◦ vs. Δsoln*G*◦ enthalpy–entropy compensation plot for the solubility of meloxicam (3) in {DMSO (1) + water (2)} mixtures at *T*hm = 303.0 K. The points represent the mole fraction of DMSO (1) in the {DMSO (1) + water (2)} mixtures in the absence of meloxicam (3).

#### *3.8. Preferential Solvation Analysis*

The preferential solvation parameter of meloxicam (component 3) by DMSO (component 1) in the {DMSO (1) + water (2)} mixtures at saturation is defined as:

$$
\delta \mathbf{x}\_{1,3} = \mathbf{x}\_{1,3}^L - \mathbf{x}\_1 = -\delta \mathbf{x}\_{2,3} \tag{22}
$$

where *xL* 1,3 is the local mole fraction of DMSO in the molecular environment around meloxicam and *x*1 is the bulk mole fraction of DMSO in the initial aqueous-DMSO mixture in the absence of meloxicam. If *δx*1,3 values were positive, meloxicam would be preferentially solvated by DMSO, but if they were negative, meloxicam would be preferentially solvated by water. Thus, the respective *δx*1,3 values were obtained by means of the inverse Kirkwood–Buff integrals (IKBI) for the solvent components based on [73–75]:

$$\delta \mathbf{x}\_{1,3} = \frac{\mathbf{x}\_1 \mathbf{x}\_2 (G\_{1,3} - G\_{2,3})}{\mathbf{x}\_1 G\_{1,3} + \mathbf{x}\_2 G\_{2,3} + V\_{\text{Cor}}} \tag{23}$$

with,

$$G\_{1,3} = RT\kappa\_T - \overline{V}\_3 + \mathbf{x}\_2 \overline{V}\_2 \left(\frac{D}{\overline{Q}}\right) \tag{24}$$

$$G\_{2,3} = RT\kappa\_T - \nabla\_3 + \mathbf{x}\_1 \nabla\_1 \left(\frac{D}{Q}\right) \tag{25}$$

$$V\_{\rm cor} = 2522.5 \cdot \left\{ r\_3 + 0.1363 \cdot \left( \mathbf{x}\_{1,3}^L \nabla\_1 + \mathbf{x}\_{2,3}^L \nabla\_2 \right)^{1/3} - 0.085 \right\}^3 \tag{26}$$

Here, *κT* represents the isothermal compressibility of every {DMSO (1) + water (2)} mixture. *V*1 and *V*2 denote the partial molar volumes of DMSO and water in the aqueous-DMSO mixtures. *V*3 denotes the partial molar volume of meloxicam. The function *D* corresponds to the first derivative of the variation of standard molar Gibbs energies of transfer of meloxicam from neat water to {DMSO (1) + water (2)} mixtures regarding the DMSO-proportion in the mixtures free of solute, as shown in Equation (27). The function *Q* involves the second derivative of the variation of excess molar Gibbs energy of mixing of DMSO and water (*GExc* 1+2) regarding the water-proportion in the aqueous-DMSO mixtures, as shown in Equation (28). *V*cor is the correlation volume and *r*3 is the hydrodynamic

molecular radius of meloxicam, which is commonly calculated by means of Equation (29), where *N*Av is the number of Avogadro.

$$D = \left(\frac{\partial \Delta\_{\text{tr}} G^{\text{o}}\_{3,2\to1+2}}{\partial x\_1}\right)\_{T,p} \tag{27}$$

$$Q = RT + x\_1 x\_2 \left(\frac{\partial^2 G\_{1+2}^{Exc}}{\partial x\_2^2}\right)\_{T,p} \tag{28}$$

$$r\_3 = \left(\frac{3 \cdot 10^{21} V\_3}{4 \pi N\_{\text{Av}}}\right)^{1/3} \tag{29}$$

As exposed in the literature, the definitive *V*cor values require iteration because they depend on the local mole fractions of DMSO and water around the meloxicam molecules in the equilibrated solutions. Hence, these iterations are performed by substituting *δx*1,3 and *V*cor values in Equations (22), (23), and (26) in order to recalculate the *xL* 1,3 value until almost invariant values of *V*cor are obtained.

Figure 7 depicts the apparent Gibbs energies of transfer of meloxicam from neat water to {DMSO (1) + water (2)} mixtures (Δtr*G*o 3,2→1+2) at *T* = 298.15 K. These Δtr*G*o 3,2→1+2 values were calculated from the mole fraction solubilities shown Table 1, by using:

$$
\Delta\_{\text{tr}} G^{\text{O}}\_{3,2\to1+2} = RT \ln \left( \frac{\varkappa\_{3,2}}{\varkappa\_{3,1+2}} \right) \tag{30}
$$

Δtr*G*o 3,2→1+2 values were correlated according to the regular fourth degree polynomial presented as Equation (31), with adjusted *r*2 = 0.998, typical error = 0.317, and *F* value = 1484.

$$\begin{aligned} \Delta \mathbf{t} \mathbf{G}\_{3,2\to1+2}^{\circ} &= -0.19(\pm 0.30) - 15.72(\pm 4.68) \mathbf{x}\_1 - 63.14(\pm 20.49) \mathbf{x}\_1^2 \\ &+ 96.50(\pm 31.53) \mathbf{x}\_1^3 - 39.34(\pm 15.64) \mathbf{x}\_1^4 \end{aligned} \tag{31}$$

**Figure 7.** Gibbs energy of transfer of meloxicam (3) from neat water (2) to {DMSO (1) + water (2)} mixtures at *T* = 298.15 K.

In this way, the *D* values shown in Table 7 were calculated from the first derivative of Equation (31) by considering the variation of aqueous-DMSO mixtures composition in incremental *x*1 = 0.05 steps through all the mixtures' composition interval. Otherwise, the required *Q*, *RTκT*, *V*1, and *V*2 values corresponding to {DMSO (1) + water (2)} mixtures were taken from the literature [76].

**Table 7.** Some properties associated to preferential solvation of meloxicam (3) in {DMSO (1) + water (2)} mixtures at *T* = 298.15 K.

a *x*1 is the mole fraction of DMSO (1) in the {DMSO (1) + water (2)} mixtures free of meloxicam (3).

Because *V*3 values are not available for meloxicam in {DMSO (1) + water (2)} mixtures, these values were considered as the one calculated based on Fedors' method, i.e., 183.3 cm3·mol−1 [6]. *G*1,3 and *G*2,3 shown in Table 7 are negative in all cases, indicating the affinity of meloxicam by DMSO and water. The approximated hydrodynamic radius of meloxicam (*r*3) was calculated as 0.417 nm by means of Equation (29). In turn, the preferential solvation parameters of meloxicam by DMSO molecules are also summarized in Table 7. According to Figure 8, initially adding of DMSO to water makes the *δx*1,3 values of meloxicam negative in the interval from neat water to the mixture of *x*1 = 0.21. The maximum negative value of this parameter is obtained in the mixture of *x*1 = 0.10, with *δx*1,3 = −1.36 × 10−2, which is higher than 1.00 × 10−2 if the absolute value is considered; therefore, it could be a consequence of real preferential solvation effects by water on meloxicam, rather than a consequence of the uncertainty propagation in the respective IKBI calculations [77,78]. The cosolvent action of DMSO for increasing the meloxicam solubility in these water-rich mixtures could be associated to the breaking of the ordered structure exhibited by water molecules, such as "icebergs", around the non-polar moieties of meloxicam, which, in turn, would be increasing the meloxicam solubility and solvation.

In mixtures of 0.21 < *x*1 < 1.00, the *δx*1,3 values are positive indicating preferential solvation of meloxicam by DMSO. Maximum *δx*1,3 value was obtained in the mixture of *x*1 = 0.45 (*δx*1,3 = 3.34 × 10−2). This maximum positive *δx*1,3 value is also higher than |1.00 × 10−2| being a consequence of real preferential solvation effects by DMSO [77,78]. From a mechanistic viewpoint, in the mixtures' composition region of 0.19 < *x*1 < 1.00, it is adequately conjecturable that meloxicam could be acting as a Lewis acid in front of

the DMSO molecules owing the unshared electrons of the sulfoxide oxygen atom of this cosolvent. Notably, this cosolvent is more basic than water, as remarkable by the magnitude of their Kamlet–Taft hydrogen bond acceptor parameters, namely *β* = 0.76 for DMSO and 0.47 for water [38]. Moreover, Figure 8 allows the comparison of preferential solvation of meloxicam by DMSO and *N*,*N*-dimethylformamide in their respective aqueous mixtures [6]. As observed, the cosolvent regions of preferential solvation are similar, as well as the magnitudes of preferential solvation by both cosolvents. Nevertheless, preferential hydration of meloxicam is higher with *N*,*N*-dimethylformamide, which could be a consequence of its lower polarity (*δ*1 = 24.1 MPa1/2) regarding DMSO (*δ*1 = 26.6 MPa1/2) [37,38]. These interesting behaviors could be a consequence of the high water-association effects around the non-polar groups of this drug, which could be favored by the more hydrophobic moieties present in the cosolvents, as they exhibit less polar nature. In turn, the hydrophobic groups present in the cosolvents could also be acting as water-association promotors depending on their respective molecular sizes [11]. Finally, from all the physicochemical analyses reported, it is noteworthy to mention that this investigation expands the equilibrium solubility database about non-steroidal anti-inflammatory drugs in commonly used aqueous cosolvent mixtures [79].

**Figure 8.** Preferential solvation parameters of meloxicam (3) in some {cosolvent (1) + water (2)} mixtures at *T* = 298.15 K. •: DMSO (1) + water (2), ♦: Dimethylformamide (1) + water (2) [6].

#### **4. Conclusions**

Equilibrium molar and mole fraction solubilities of analgesic drug meloxicam in different {DMSO (1) + water (2)} mixtures at five temperatures from 293.15 to 313.15 K were determined by using the shake flask method followed by UV-vis drug quantification, reported and analyzed. Meloxicam mole fraction solubility in these mixtures was adequately correlated with some well-known correlation models obtaining mean percentage deviations (MPDs) of 9.6 to 9.9%. In addition, a number of predictive models which were already trained using published datasets and by employing the minimum number of measured experimental data from this project were produced with MPDs of 35.6 to 74.3%. Apparent standard thermodynamic quantities of dissolution and mixing processes were calculated observing endothermal dissolution processes in all cases and favored in DMSO-rich mixtures. Nonlinear enthalpy-entropy compensation was observed indicating different mechanisms for the cosolvent action. IKBI treatment demonstrated preferential hydration of meloxicam in water-rich mixtures but preferential solvation by DMSO in mixtures of 0.21 < *x*1 < 1.00.

**Author Contributions:** Conceptualization, D.A.T., F.M., O.A.A., M.Á.P., A.J. and W.E.A.J.; methodology, D.A.T., F.M. and O.A.A.; software, D.A.T., F.M., M.Á.P. and A.J.; validation, D.A.T., F.M., A.J. and W.E.A.J.; formal analysis, D.A.T., F.M., O.A.A., M.Á.P., A.J. and W.E.A.J.; investigation, D.A.T., F.M., O.A.A., M.Á.P., A.J. and W.E.A.J.; resources, D.A.T., F.M. and O.A.A.; data curation, D.A.T. and F.M.; writing—original draft preparation, D.A.T., F.M., M.Á.P. and A.J.; writing—review and editing, D.A.T., F.M., O.A.A., M.Á.P., A.J. and W.E.A.J.; visualization, D.A.T., F.M., O.A.A., M.Á.P., A.J. and W.E.A.J.; supervision, F.M., M.Á.P., A.J. and W.E.A.J.; project administration, D.A.T., F.M., A.J. and W.E.A.J.; funding acquisition, D.A.T., F.M. and O.A.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Minciencias (formerly Colciencias), grant number 785-2019.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** We thank the Departments of Pharmacy and Physics of the Universidad Nacional de Colombia for facilitating the equipment and laboratories used. Financial support of Minciencias (formerly Colciencias) is also highly appreciated.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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*Risk Management and Assessment*


**1. Introduction**

Faults play a major role in lateral sealing of thin reservoirs and accumulation of the remaining oil in conventional and unconventional reservoirs onshore in China [1]. Almost all developed onshore oil and gas fields in China are distributed in rift basins which are rich in oil and gas resources with highly developed and very complex fault systems [2–4]. At present, there are many kinds of fault recognition techniques with different principles, but there are still great difficulties in fine fault imaging. This is because the rift basin experienced a variety of external forces during its growth, and developed a variety of faults, such as normal faults, normal oblique-slip faults, oblique faults, and strike–slip faults, etc. According to their different combinations in plane and section, they also present many forms such as broom shaped, comb shaped, goose row shaped, and parallel interlaced in planes. In rift basins, the filling of sediments, the development and distribution of sedimentary sequences, the formation, distribution and evolution of oil and gas reservoirs (including the formation and effectiveness of traps, and the migration and accumulation of oil and gas) are closely related to the distribution and activity of faults [5]. Therefore, fine detection and characterization of faults in rift basins in China has become a key basic geological problem for oil and gas exploration and development efforts and has become the key topic of basin tectonic research [6].

Continuous and regular event breakpoints constitute faults in seismic imaging data. However, because the accuracy, resolution and signal-to-noise ratio of seismic imaging data cannot reach the theoretical limit and the geological situation is complicated, it is a great challenge for petroleum engineers to describe the spatial distribution of faults from seismic data. In the past, fault characterization has been regarded as an interpretative task, followed by seismic data processing and imaging, because it requires extensive geological

**Citation:** Wu, J.; Shi, Y.; Wang, W. Fault Imaging of Seismic Data Based on a Modified U-Net with Dilated Convolution. *Appl. Sci.* **2022**, *12*, 2451. https://doi.org/10.3390/ app12052451

Academic Editor: Chiara Bedon

Received: 5 January 2022 Accepted: 23 February 2022 Published: 26 February 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

knowledge and experience. In recent years, researchers use convolutional neural network to identify faults, focusing on the construction of network architecture, network parameter debugging and optimization and model training. They are less and less constrained by geological knowledge and personal experience, and the processing and mining of seismic data are becoming more and more important. Therefore, it is very reasonable to attribute fault identification via deep learning to the research field of seismic data processing and imaging, and it is also the development trend in the future. Based on this concept, our research employs seismic imaging data to realize the description of fault characteristics through a new neural network model, that is, to realize fault imaging.

In the past 30 years, with the continuous development of computer hardware and software, fault identification has made great progress in efficiency and accuracy. From the perspective of method evolution, fault interpretation has experienced from the initial manual interpretation to the emergence of various identification methods, such as coherence method, curvature attribute method and ant colony algorithm, which describe faults by calculating transverse discontinuities of seismic data. In the past five years, with the rapid development of artificial intelligence [7–9], various fault identification methods based on deep learning have gained remarkable achievements. In 2014, Zheng et al. [10] used deep learning tools to conduct fault identification tests on prestack synthetic seismic records. Araya-Polo et al. [11] applied machine learning and deep neural network algorithms to automate fault recognition, which greatly improved the efficiency and stability of fault interpretation. Waldeland et al. [12] and Guitton et al. [13] successively used a Convolutional Neural networks (CNN) model to make progress in fault description. Xiong et al. [14] used results of the skeletonized seismic coherent self-correction method as training samples to train a CNN model to identify seismic faults. In 2019, Wu et al. [15] realized the identification of small-scale faults by using U-Net network. Wu et al. [6] used a full-convolutional neural network, FCN, to achieve a better characterization of faults. Among these networks, the U-Net architecture is currently very popular, due to its shortcut operation which concatenates attribute maps from the low-level feature (shallow layer) to maps of high-level feature (deep layer), and it can be seen as a special kind of CNN [16–18]. In addition, the U-Net does not have strict requirements on the size of training sets, and smaller training sets can also provide satisfactory results. However, most networks including the U-Net uniformly process all feature maps of the same layer, resulting in the same receptive field of the network in the same layer, thus obtaining relatively single local information. Moreover, with the continuous down-sampling of the network and the convolution operation with step size, the defect that only a single size information can be obtained at the same layer becomes more and more obvious, resulting in the inaccurate identification of faults by the neural network.

To address these issues, this paper introduces a new neural network model, which takes U-Net as the basic network and uses inter-group channel dilated convolution module (GCM) to connect each cross-connection layer between encoding path and decoding path, and uses inter-group space dilated convolution module (GSM) to connect layers after each deconvolution layer in decoding path. Both GCM and GSM use dilated convolution. Dilated Convolution, also known as hole convolution or expanded convolution, is to inject cavities into the standard convolution kernel to increase the reception field of the model. In the CNN structure, convolution and pooling drive most layers, convolution performs feature extraction, while pooling performs feature aggregation. For an image classification task, this network structure has good feature extraction capability, among which the most classical structure is VGGNet (a convolutional neural network was developed by the University of Oxford's Visual Geometry Group and Google DeepMind in 2014). However, this structure has some problems for target detection and image segmentation. For example, the size of receptive field is very important in target detection and image segmentation, and the guarantee of receptive field depends on down-sampling. However, down-sampling will make small targets difficult to detect. If down-sampling is not completed and the number of convolutional layers is only increased, the computation of the network will

increase. In addition, if pooling is not carried out for features, the final feature extraction effect will also be affected, and there will be no change in the receptive field. In order to solve these problems in CNN, this paper introduces dilated convolution to increase the receptive field without sacrificing the size of the feature map. Compared with the conventional convolution operation, dilation rate is added to the dilated convolution, which refers to the number of intervals of points in the convolution kernel [19–21]. When dilatation rate is 1, the dilated convolution will degenerate into conventional convolution. The similarity between dilated convolution and conventional convolution lies in that the size of convolution kernel is the same, that is, the number of parameters of neural network remains unchanged [22,23]. The difference lies in that dilated convolution has a larger receptive field and can preserve the structure of internal data [24,25].

### **2. Illustration of Dilated Convolution**

We use a set of pictures to illustrate the principle of dilated convolution. Figure 1a is the conventional convolution kernel, and the dilated convolution is obtained by adding intervals to this basic convolution kernel. Figure 1b corresponds to the convolution of 3 × 3, with dilation rate of 2 and interval 1, that is, corresponding to 7 × 7 image blocks. It can be understood that the kernel size becomes 7 × 7, but only 9 points have parameters, and the rest have parameters of 0. Convolution calculation was performed for the 9 points in Figure 1b and the corresponding pixels in the feature map, and the other positions were skipped. Figure 1b,c are similar, except that the dilation rate is 4, which is equivalent to a 15 × 15 convolution kernel. As the convolution kernel becomes larger, the receptive field becomes larger naturally.

**Figure 1.** The dilated convolution with dilation rate of 1, 2 and 4, respectively. (**a**) dilation rate = 1; (**b**) dilation rate = 2; (**c**) dilation rate = 4.

In practical application, when the same dilation rate is used for all convolutional layers, a problem called grid effect will appear. Since the convolution calculation points on the feature map are discontinuous, for example, if we repeatedly accumulate 3 × 3 kernel of dilation rate 2, this problem will occur.

The blue square in Figure 2 is the convolution calculation points participating in the calculation, and the depth of the color represents the calculation times. As can be seen, since the dilation rates of the three convolutions are consistent, the calculation points of the convolution will show a grid expansion outward, while some points will not become calculation points. Such kernel discontinuities, that is, not all pixels are used for calculations, will result in the loss of continuity of information, which is very detrimental for tasks such as pixel-level dense prediction. The solution is to discontinuously use dilated convolution with the same dilation rate, but this is not comprehensive enough. If the dilation rate is multiple, such as 2,4,8, then the problem still exists. Therefore, the best way is to set the dilation rate of continuous dilated convolution as "jagged", such as 1,2,3, respectively, so that the distribution of convolution calculation points will become like Figure 3 without discontinuity.

**Figure 2.** The dilated convolution with the same dilation rate of 2, respectively. There are grid effects in all three graphs. (**a**) the first convolution with dilation rate of 2; (**b**) the second convolution with dilation rate of 2; (**c**) the third convolution with dilation rate of 2.

**Figure 3.** The dilated convolution with dilation rate of 1, 2 and 3, respectively. (**a**) dilation rate = 1; (**b**) dilation rate = 2; (**c**) dilation rate = 3.

## **3. The Architecture of the Modified U-Net**

The proposed neural network adopts a U-Net of a 4-layer structure as the basic network. In the coding path, feature maps of each layer are connected to the corresponding decoding layer by GCM, in which each layer adopts two 3 × 3 convolution layers and maximum pooling layer for feature extraction. Then, in the decoding path, feature maps of each layer are connected to the corresponding decoding layer by GSM. Each layer adopts a 3 × 3 convolution layer, up-sampling layer and 1 × 1 convolution layer to restore, and the output layer adopts a 3 × 3 convolution layer and 1 × 1 convolution layer to output. In this modified U-Net, the batch regularization (BN) and modified linear units (ReLU) are added to all convolution layers to correct data distribution, except for the output layer. GCM and GSM modules play a key role in the modified U-Net, and their operation mechanism is similar. GCM module can divide the input feature map into four groups on average, and then carry out the dilated convolution operation with dilation rates of 1, 2, 3 and 5, respectively. In addition, the module extracts and outputs the features of the input feature map through pooling, convolution, batch regularization, activation, softmax and other conventional operations, and finally obtains the channel information of all groups. For the GSM module, it can divide the input feature map into three groups on average, and each group carries out the dilated convolution operation with dilation rates of 1, 2 and 4, respectively. This module carries out feature extraction and output in sequence by down-sampling, convolution, batch regularization, activation, up-sampling, convolution, batch regularization, activation and softmax, and finally obtains the spatial information of all groups.

Figure 4 shows the structure of GCM in the modified U-Net. This module divides the input feature map into four groups evenly, and each group performs dilated convolution operations with dilation rates of 1, 2, 3, and 5. The size of the target area of fault identification determines the value of the dilation rate. After dilated convolution, four groups of feature maps with different scales are obtained. Besides, four groups of channel information are returned by softmax, which were taken as weights and multiplied by four groups of feature maps with different scales obtained via dilated convolutions to acquire new feature maps. The receptive field corresponding to the group with the largest weight contributed the most to the final network prediction. Finally, the four groups of new feature maps are spliced together and then a residual operation is performed with the input feature map to obtain the final prediction result. The GCM module uses the idea of grouping and realizes the automatic selection of inter-group multi-scale information under the guidance of channel information.

**Figure 4.** The architecture of GCM.

Figure 5 shows the structure of GSM, which realizes the selection of multi-scale information between groups in another way and enhances the consistency of receptive field and target region recognition. In this module, the input feature map was divided into three groups, and then three groups of feature maps with different scales are obtained by the dilated convolution with dilation rates of 1, 2 and 4. At the same time, three feature maps with spatial weights are cropped from the input feature map through a series of conventional operations. In these operations, the purpose of down-sampling is to obtain more global information, the purpose of up-sampling is to restore the size of feature maps, and the purpose of softmax is to enable the module to automatically select multi-scale information. Three feature maps with spatial weights are multiplied by three feature maps of different scales obtained by dilated convolution to get three new feature maps. Finally, after splicing the three groups of new feature maps, a residual operation is performed with the input feature map to acquire the final prediction results. In summary, under the guidance of spatial information, the GSM module can select multi-scale information among a group of feature maps.

**Figure 5.** The architecture of GSM.

The proposed neural network is based on the U-Net network and has two functional modules, the GCM and GSM, which can finely describe faults of different scales. Its architecture is shown in Figure 6. Due to the powerful multi-scale information selection ability of GCM and GSM modules, this paper only uses a 4-layer U-Net based on a coding– decoding structure as the basic network. In the coding path, only two 3 × 3 convolution and maximum pooling are used to quickly obtain feature maps with different resolutions. In the decoding path, multiple simple decoding blocks are used to quickly and effectively recover feature maps with high resolution. In this neural network, the data distribution after convolution is corrected by BN and ReLU, and the GCM module is placed at the connection layer of the network to automatically select multi-scale information, which makes up for the lack of transmitting single information to the decoder by the encoder of conventional U-Net. At the same time, the GSM module between groups is placed in the path of decoder to realize the function of multi-scale information selection, which makes up for the disadvantage of losing global information in up-sampling process.

**Figure 6.** The architecture of the modified U-Net.


**3. Materials and methods**

This study is based on the stratigraphic and petrographic analysis of Paleocene sedimentary sequences of the well-exposed section in the Badelyan area NE Iraq (**Figure 2**). A total of 170 m thicknesses were studied and logged, which is located directly beneath the suture of the Zagros thrust belt. The field works cover the measurement of bed thicknesses and the total thickness of section. Moreover, litho and facies types, sedimentary structures, sedimentary sequences, and bed extensions are identified and measured. The field works of the measured section, lithological description, thickness variations, and stratigraphy of the sequences are studied and summarized in **Figure 2**. Descriptions of field features accompanied by the results of microscopic studies have been used for facies analysis and paleoenvironmental construction. For subsequent studies, a total of 50 hand specimens were collected. Sampling intervals were generally between 1 and 3 m. Additional data, including macroscopic rock descriptions and ichonological data, have been studied as well as trace fossils in the field work.

Petrographic facies analyses were carried out in 45 thin sections and reported all mineralogical constituents, matrix, and cement types. Detailed photomicrographs

#### **Figure 2.**

*The stratigraphic section shows different stratigraphic units of the Kolosh formation in Badelyan area, NE Iraq.*

are picked up for all mineralogical details. Abbreviations for names of rock-forming minerals used to point to the types of minerals in the photomicrographs are followed [12]. The sandstone classification of Ref. [13] as well as the provenance and tectonic setting of Ref. [14] cannot be applied because of the high content cf. 25% of olivine and pyroxene grains.