1 Introduction

Acetaminophen (ACP, also known as paracetamol is identified with CAS register number: 103-90-2 and PubChem CID: 1983, its condensed molecular formula is: C8H9NO2, its molar mass is: 151.16 g·mol–1, and its molecular structure is shown in Fig. 1), as a p-aminophenol derivative compound, exhibits good analgesic and antipyretic properties but a weak anti-inflammatory activity [1, 2]. This drug is specially indicated in the treatment of several minor diseases presented by pediatric patients [3,4,5]. ACP is normally administered orally or as a rectal suppository for treating mild to moderate pain and fever. It may also be administered by intravenous infusion for the short-term treatment of moderate pain, particularly after surgery, and fever. ACP is often the analgesic or antipyretic of choice, especially in the elderly and also in patients in whom salicylates or other NSAIDs are contra-indicated. Such patients include asthmatics, those with a history of peptic ulcer, and some children.

Fig. 1
figure 1

ACP molecular structure

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The usual oral dose of ACP is from 0.5 to 1.0 g every 4 to 6 h avoiding quantities higher that 4.0 g per day. ACP can also be administered rectally as suppositories in doses varying from 0.5 to 1.0 g every 4 to 6 h up to 4 times per day. In addition, ACP could also administered by intravenous infusion over 15 min; dosage may be calculated by weight as follows: (i) for patients weighing 50 kg or more, single doses of 1.0 g every 4 or more hours, to reach a maximum of 4.0 g per day; (ii) for patients from 33 to 50 kg, single doses of 15 mg/kg every 4 or more hours, to reach a maximum of 60 mg/kg or 3 g per day [3]. It is noteworthy that in the recent COVID-19 pandemic time ACP was commonly used for pain relief in several infected patients [6,7,8].

Regarding injectable products of ACP, it is internationally available under several trademarks for parenteral administration as an aqueous solution in concentration of 10 mg/mL and presented as 1 g/100 mL vials intended for intravenous infusion [9].

In Colombia, this parenteral dosage form is commercially available under the trademark Traucet®, which is produced by “Corpaul Farmacéutica” [10]. In Argentina, some plastic bags of 50 and 100 mL with the same ACP concentration are available [11]. In Spain, ACP injectable is available under the trademark Perfalgan® 10 mg/mL solution intended for perfusion [12]. Main indication of these pharmaceutical dosage forms is the management of acute post chirurgical pain [13,14,15,16]. Nevertheless, up to the best of our knowledge, commercial low-volume injectable forms of ACP presented in ampoules for intra-muscular administration are not yet available. It is important to keep in mind that injectable formulations could supply high doses of drug in small volumes, which could facilitate the transportation and manipulation of medicines [17,18,19]. On the other hand, it is important to remember that owing to the high drug concentrations of these products, they are usually hypertonic in nature. Therefore, their administration commonly imply some local pain after intra-muscular administration, although they are presented as low volume products. This is more complicated if alcohol is present in high concentration as vehicle. However, this problem is partially solved if they are administered slowly by using the intravenous route [9, 15].

For all these reasons, the complete knowledge of several physicochemical properties including the solubility of every solid component and the volumes occupied by the drugs and excipients in the global dissolution are very important because they facilitate the design process of different homogeneous liquid pharmaceutical dosage forms. Moreover, the solubility and dissolution behaviors of drugs in aqueous and non-aqueous cosolvent mixtures is very important because several cosolvent blending is commonly used in optimizing purification procedures, medicines preformulation studies, as well as in the design of low volume parenteral dosage forms, among other chemical applications [17,18,19].

Cosolvents are defined in pharmaceutical sciences as water-miscible organic solvents that are used in liquid medicines to increase the equilibrium solubility of poorly aqueous-soluble substances or to enhance the chemical stability of active pharmaceutical ingredients. Cosolvency refers to the technique of using cosolvents for these purposes and is also referred as solvent blending. Cosolvency has been used successfully for designing liquid drug preparations throughout the history of medicines formulation [20, 21].

On the other hand, although several methods useful for calculating the solubility of a variety drugs are available in the literature, in general these methods are not successful for explaining fully the mechanism of cosolvent action in mixed solvents. Moreover, a significant number of correlation/calculation methods do not take into account the effect of temperature on this fundamental physicochemical property [22, 23].

For all these reasons it is crucial to determine the equilibrium solubility of active ingredients and excipients to expand the required information about physicochemical data for pharmaceutical systems. All this data facilitates widely the duties of research and development of new liquid products at industrial level [24]. This is crucial in the case of drugs exhibiting low aqueous solubilities, in particular those identified as classes II and IV of the Biopharmaceutics Classification System, where the respective doses are not dissolved in 250 mL of simulated gastric fluid owing low solubility. Although this classification system was initially proposed for solid dosage forms considerations, it is useful as general pharmaceutical conceptual framework.

Because the mass/volume percentage (%m/v) solubility of ACP at several temperatures and mixtures compositions, has only been reported in binary cosolvent systems [25], the main goal of this research is to evaluate the effect of the cosolvent composition and temperature on the %m/v solubility of ACP in ternary mixtures of ethanol (EtOH) (component 1), propylene glycol (PG) (component 2) and water (component 3)} as a first step for optimizing the design of concentrated injectable solutions of this drug.

EtOH and PG are the most widely used cosolvents for designing injectable liquid medications presented in low volume [20, 21, 26]. Although all the possible binary mixtures prepared exclusively by using these two cosolvents are not fully considered as suitable vehicles for direct parenteral administration, because they could be significantly painful, they are very good candidates to prepare dissolving ternary mixtures when water is also included [17,18,19]. Thus, this research was performed because the literature ACP solubility studies in binary mixtures have shown the following facts: To administer the required doses of ACP by injectable solution in neat water a huge volume would be required in any dosage form because 1.0 g of ACP requires 70 mL of water for complete drug dissolution [1]. The nearly saturated ACP dissolution in neat EtOH could supply the respective doses but it has been demonstrated that it could cause severe tissue damages even though the ACP solubility allows to dissolve it in small volumes of this cosolvent. Even more, in {EtOH (1) + water (2)}, {PG (1) + water (2)} or {EtOH (1) + PG (2)} binary mixtures, the cosolvent proportions useful for supplying the therapeutical doses of ACP dissolved in low volumes are relatively high.

Thus, the present research expands those reported in the literature about the equilibrium solubility of ACP in different single and mixed solvent systems reported by Paruta and Irani [27] in dioxane-water mixtures at 25.0 °C, Sheth et al. [28] in sucrose aqueous solutions at 25.0 °C, Grant et al. [29] in neat water from 0.0 to 25.0 °C, Prakongpan and Nagai [30] in aqueous cosolvent mixtures of EtOH, polyethylene glycol (PEG) 400 or PEG 4000 at 30.0 °C, Hamza and Paruta [31] in some aqueous solutions of different non-ionic surfactants at 25.0 °C, Etman and Naggar [32] in aqueous solutions of sorbitol, glucose or sucrose at 20.0 and 37.0 °C, Jiménez et al. [33] in aqueous mixtures of EtOH, PG or glycerol at 20.0 °C, Bustamante et al. [34] in ethyl acetate-EtOH and EtOH-water mixtures from 20.0 to 40.0 °C, Romero et al. [35] in ethyl acetate-EtOH, EtOH-water and dioxane-water mixtures at 25.0 °C, Bustamante et al. [36] in dioxane-water mixtures from 20.0 to 40.0 °C, Pérez et al. [37] in EtOH-water, PG-water and EtOH-PG mixtures at 25.0 °C, Baena et al. [38] in water, 1-octanol, isopropyl myristate, chloroform and cyclohexane, as well as the mutually saturated organic solvents with water from 25.0 to 40.0 °C, Jouyban et al. [39] in EtOH-water, PG and EtOH-PG mixtures at 25.0 °C, Yurquina et al. [40] in PEG 400-water mixtures at 25.0 °C, Jiménez and Martínez [17] in EtOH-water mixtures from 20.0 to 40.0 °C, Jiménez and Martínez [18] in EtOH-PG mixtures from 20.0 to 40.0 °C, Jiménez and Martínez [19] in PG-water mixtures from 20.0 to 40.0 °C, Jouyban et al. [41] in water-glycerol mixtures at 25.0 and 30.0 °C, Ahumada et al. [42, 43] in PEG 400-water mixtures from 20.0 to 40.0 °C, Mehrdad and Miri [44] in aqueous mixtures of the ionic liquid 1-hexyl-3-methyl imidazolium bromide from 20.0 to 40.0 °C, Muñoz et al. [45] in methanol-water mixtures from 20.0 to 40.0 °C, Jesus et al. [46] in some aqueous mixtures of N-acetyl amino acid N-alkyl cholinium-based ionic liquids at 25.0 °C, Pourkarim et al. [47] in 1-propanol-water mixtures from 20.0 to 40.0 °C, Rahimpour et al. [48] in acetonitrile-water mixtures from 20.0 to 40.0 °C, Jesus et al. [49] in aqueous solutions of some N-alkyl cholinium-based ionic liquids at 37.0 °C, and Barzegar-Jalali et al. [50] in aqueous mixtures of a biodegradable betaine/ethylene glycol deep eutectic solvent from 20.0 to 40.0 °C. Some investigations performed about the solubility of ACP in ternary mixtures are those reported by Coronado et al. [51] in alcohol-propylene glycol-water and ethanol-PEG 200-water mixtures at 20.0 °C, Bolaños et al. [52] in alcohol-syrup NF-water and ethanol-sorbitol USP-water mixtures at 20.0 °C, Jouyban et al. [39] in some EtOH-PG-water mixtures at 25.0 °C, Jouyban et al. [53] in some ethanol-propylene glycol-water mixtures at 25.0 and 30.0 °C, Jouyban et al. [54] in ethanol-PEG 200 (or 400)-water mixtures at 25.0 °C and Soltanpour et al. [55] in ethanol-PEG 600-water mixtures at 25.0 °C. More recently, our research group reported the mole fraction and molar solubility of ACP in {EtOH (1) + PG (2) + water (3)} mixtures, including its volumetric behavior and dissolution thermodynamic description [56].

As well-known the equilibrium solubility of ACP is a fundamental parameter to establish better and more effective liquid pharmaceutical dosage forms. Then, such study will lead us to have a good idea about the solubilizing behavior of ternary solvent mixtures and its possible advantage to get higher ACP concentrations. These high-concentration solutions could be administered in low volumes of injectable presented in ampoules with the lower proportions of cosolvents as possible. As described above, in the Colombian, Argentinian and Spanish markets, only vials of 50 or 100 mL of 1.0%m/v ACP, to be administered by infusion, are available as injectable dosage forms. In this way, it is important to investigate the possibility of formulating a cosolvent vehicle providing 500 mg of ACP in 5 mL ampoules in any {EtOH (1) + PG (2) + water (3)} mixture. It is noteworthy that this ternary cosolvent system has been studied earlier for physical stability [57], volumetric properties [58, 59], and some other thermophysical properties of pharmaceutical importance at 20.0 °C [60].

2 Materials and Methods

2.1 Materials and Reagents

ACP (the component 4) was obtained from Fagron Ibérica (Spain; with purity at least 0.990 in mass fraction), EtOH (the solvent component 1, CAS register number: 64-17-5) was obtained from Merck A.R. (Germany; with purity at least 0.998 in mass fraction), PG (the solvent component 2, CAS register number: 57-55-6) was obtained from Dow Chemical Co. (USA; with purity at least 0.995 in mass fraction), and purified water (the solvent component 3, CAS register number: 7732-18-5) was obtained by distillation with electrical conductivity lower than 2.0 µS·cm–1. ACP, EtOH and PG were used as received without further purification, except about keeping EtOH and PG in molecular sieving from Merck (Germany) for removing water.

2.2 Cosolvent Mixtures Preparation

All the {EtOH (1) + PG (2) + water (3)} solvent mixtures under research were prepared gravimetrically in quantities of 50.00 g by using an Ohaus Pioneer TM PA214 (USA) analytical balance with sensitivity of ± 0.1 mg. To cover all the mixtures compositions range, the mass fractions of EtOH (w1), PG (w2) and water (w3) varied by 0.10 obtaining 36 ternary cosolvent systems.

2.3 Solubility Determinations

Flask shake method [61, 62] and UV–vis spectrophotometry were used for ACP solubility determinations at different temperatures, as follows: An excess amount of ACP was added to 50.00 g of each ternary solvent system, in well-stoppered amber glass flasks of 60 mL. Then, all the flasks were placed on a Neslab RTE 10 Digital One Thermo Electron Company (USA) thermostatic bath maintained at 40.0 °C and manually stirred as possible for at least three days to reach the ACP saturation equilibrium. This drug saturation time was demonstrated earlier by considering the require time for obtaining constant ACP concentration in neat water [17, 18]. Later, the supernatant solutions were filtered at isothermal conditions using Millipore Corp. Swinnex®-13 (USA) filtration units to ensure the absence of un-dissolved particulate matter before sampling for analysis. ACP concentrations in the saturated solutions were determined after diluting them gravimetrically with neat water (firstly, aliquots of almost 100 mg of saturated solutions were weighted in an Ohaus Pioneer TM PA214 analytical balance and diluted gravimetrically 50 or 100 times with water; secondly, aliquots of almost 100 mg of the first dilution were in turn diluted gravimetrically 300 times with water) by measuring the UV absorbance at the maximum drug absorbance wavelength, namely λmax = 242 nm, using an UV/Vis BioMate 3 Thermo Electron Company (USA) spectrophotometer, followed by the respective interpolation from the respective UV spectrophotometric gravimetric aqueous calibration curve. The UV analytical method validation for ACP quantification involved the following typical validation characteristics: accuracy, precision, specificity, detection and quantitation limits, linearity and range [63]. Thus, the linear equation of the used calibration curve in pure water was: Absorbance = 0.06447 (± 0.00018)·C + 0.0021(± 0.0015), with C (namely, ACP concentration) expressed as µg·g–1. Later, the thermostat temperature was adjusted at 35.0 °C for two days in order to allowing the complete ACP excess precipitation. The ACP composition analysis at this new saturation condition was performed as indicated above. After these stages, the thermostat temperature was sequentially diminished by varying in 5.0 °C steps following the procedures described above to finally reach 20.0 °C. Experiments were performed at least three times in every case and results averaged. It is important to indicate that direct chemical stability studies of ACP in these solvent mixtures were not performed in this research; however, some bottom solid phases in equilibrium with the saturated solutions were analyzed by DSC and XRDA and no changes were observed in comparison with the behavior exhibited by the original untreated ACP sample, as reported in the literature [56]. Moreover, in has also been reported that ACP is stable in aqueous cosolvent mixtures if they are contained in dark recipients [64]. Otherwise, the experimental solubility data of ACP in the respective binary solvent mixtures, namely {EtOH + PG}, {EtOH + water} and {PG + water}, were taken from our previous research work [25].

For transforming the ACP gravimetric concentration obtained from the analytical procedure into a volumetric ACP concentration like the mass/volume percentage is, the density of all the saturated solutions was determined by using a DMA 45 Anton Paar (Austria) digital density meter of four decimal places connected to the same Neslab RTE 10 Digital One Thermo Electron Company (USA) re-circulating thermostatic bath at every temperature under consideration.

3 Results and Discussion

3.1 Equilibrium Solubility of ACP in Mixed Solvents

Table 1 summarizes the experimental equilibrium solubility of ACP obtained in the 36 ternary mixtures as expressed in mass/volume percentage at five temperatures from 20.0 to 40.0 °C. Mean relative uncertainty in solubility values was 2.1%. It is worth nothing that Table 1 also summarizes the reported %m/v solubilities of ACP in the respective {EtOH + water}, {PG + water)} and {EtOH + PG} binary mixtures, which were taken from Jiménez et al. [25]. Thus, 66 solvent systems were considered involving 3 neat solvents, 27 binaries, and 36 ternaries. As observed in Table 1 the ACP solubility increases with the temperature arising as well as with the EtOH and/or PG proportion-increasing in the mixtures. ACP solubility reach its maximum values in the mixture w1 + w2 + w3 of 0.80-0.00-0.20 (for instance, 19.60%m/v at 20.0 °C).

Table 1 Experimental equilibrium solubilities of ACP (4) expressed in mass/volume percentage in {EtOH (1) + PG (2) + water (3)} mixtures at several temperatures
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Figures 2 and 3 allow analyzing the effect of the component proportions of the solvent mixtures on the ACP %m/v solubility at 25.0 °C. Figure 2 depicts the ACP solubility dependence on the EtOH mass fraction (w1) for all the fixed mass fractions of PG (w2). Further, Fig. 3 depicts the ACP solubility dependence on the PG mass fraction (w2) for all the fixed mass fractions of EtOH (w1). Moreover, Fig. 4 depicts the Gibbs-Roozeboom ternary graph showing the respective ACP iso-solubility lines at 25.0 °C [65].

Fig. 2
figure 2

Experimental solubility of ACP (4) expressed in mass/volume percentage as function of the EtOH mass fraction (w1) in {EtOH (1) + PG (2) + water (3)} mixtures for different mass fractions of PG (w2) at 25.0 °C. Red circles: w2 = 0.00; red squares: w2 = 0.10; red triangles: w2 = 0.20; blue circles: w2 = 0.30; blue squares: w2 = 0.40; blue triangles: w2 = 0.50; green circles: w2 = 0.60; green squares: w2 = 0.70; green triangles: w2 = 0.80 (Color figure online)

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Fig. 3
figure 3

Experimental solubility of ACP (4) expressed in mass/volume percentage as function of the PG mass fraction (w2) in {EtOH (1) + PG (2) + water (3)} mixtures for different mass fractions of EtOH (w1) at 25.0 °C. Red circles: w1 = 0.00; red squares: w1 = 0.10; red triangles: w1 = 0.20; blue circles: w1 = 0.30; blue squares: w1 = 0.40; blue triangles: w1 = 0.50; green circles: w1 = 0.60; green squares: w1 = 0.70; green triangles: w1 = 0.80 (Color figure online)

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Fig. 4
figure 4

Experimental solubility of ACP (4) expressed in mass/volume percentage in {EtOH (1) + PG (2) + water (3)} mixtures at 25.0 °C. Isometric lines indicate the same mass/volume percentage solubility values as shown numerically in the triangle sides

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As it can be seen in Figs. 2 and 3 several cosolvent mixtures allow obtaining ACP concentrations higher than 10.0%m/v at 25.0 °C. It is worthy to note that ACP solubility profiles could be useful as a basic tool for defining solvent mixtures compositions when designing liquid medicines of this analgesic drug intended for injectable dosage. This is because high ACP concentrations are required for supplying high doses in small volumes of product. Moreover, Gibbs-Roozeboom ternary graph of ACP iso-solubility lines at 25.0 °C shown in Fig. 4 facilitates the visualization of composition mixtures that allow to accomplish the solubility expected in agreement with the products concentration requirements. In the case of a possible injectable solution of 10.0%m/v all the mixtures compositions below the iso-solubility line corresponding to 10.0%m/v could be useful for obtaining this ACP concentration at 25.0 °C. However, owing the decreasing effect of lower temperatures on ACP equilibrium solubility, it is required to choose some mixtures exhibiting higher dissolving capacity, like those allowing 13.0 or 14.0%m/v to avoid drug precipitation in cold room conditions. Nevertheless, other variables like tissue-compatibility and reagents price must be considered in choosing of solvent mixture for product design. Otherwise, similar triangular figures have been reported earlier in the literature for phenobarbital equilibrium solubility in alcohol-glycerin-water [66] and propylene glycol-alcohol-water [67] ternary mixtures at 25.0 °C, as well as for ACP in alcohol-polyol-water [51] and alcohol-sugar-water [52] mixtures at 20.0 °C. On the other hand, equilateral triangle graphs have also been used for locating mixtures composition zones accomplishing some defined physicochemical requirements [68], as well as analytical tools for determining composition of some ternary mixtures based on densities and refractive indices of the mixtures at specified temperature [69].

On the other hand, to estimate the approximate dielectric requirement (ADR) of ACP based on the highest drug %m/v solubility obtained at 25.0 °C in these mixtures the following additive expression was used [70, 71]:

$${\varepsilon _{1+2+3}}=\sum\limits_{{i=1}}^{3} {{f_i}{\varepsilon _i}}$$
(1)

Equation 1 requires the dielectric constant of every neat solvent at 25.0 °C, namely ε1 = 24.55 for EtOH (component 1), ε2 = 32.00 for PG (component 2), and ε3 = 78.36 for water (component 3) [72], and the volume fraction (f) of each component in the mixture. Commonly, fi is calculated by assuming additive volumes [70]. Maximum %m/v solubility at 25.0 °C (21.18%m/v, Table 1) is observed in the binary mixture of 0.80-0.00-0.20 in mass fraction (namely, 0.8355-0.0000-0.1645 in volume fraction) that had a ε1+2+3 value of 33.40, which allows proposing an ADR value of 33.4 for ACP.

3.2 ACP Solubility Correlation in Ternary Mixtures

When the solubility of drugs like ACP is experimentally determined as a function of both, the mixtures composition and temperature, the derivation of the respective correlations is very important for the dosage forms design duties since derived models allow the estimation of solubility values at other temperatures and/or other mixtures composition of interest. This is very important in our case because ACP solubility values at temperatures below 20.0 °C, for instance at refrigeration conditions, were not determined and thus this physicochemical property could be estimated. Therefore, a number of mathematical models to estimate the solubility of ACP in solvent mixtures have been reported in the literature [23]. Some of these models have been challenged in the correlation of the equilibrium solubility of several drugs since some years ago [73].

In particular, the Jouyban-Acree model could describe the solubility of a drug (compound 4) in ternary cosolvent mixtures (components 1 + 2 + 3) at various mixtures compositions and temperatures as described by the following [74]:

$$\begin{aligned} \ln x_{{4(1 + 2 + 3),\mathrm T}} = & w_{1} \ln x_{{4(1),\mathrm T}} + w_{2} \ln x_{{4(2),\mathrm T}} + w_{3} \ln x_{{4(3),\mathrm T}} \\ + \left( {\frac{{w_{1} w_{2} }}{T}} \right)\sum\limits_{{i = 0}}^{2} {J_{i} \left( {w_{1} - w_{2} } \right)^{i} } + \left( {\frac{{w_{2} w_{3} }}{T}} \right)\sum\limits_{{i = 0}}^{2} {J_{i}^{{''}} \left( {w_{2} - w_{3} } \right)^{i} } \\ + \left( {\frac{{w_{1} w_{3} }}{T}} \right)\sum\limits_{{i = 0}}^{2} {J_{i}^{'} \left( {w_{1} - w_{3} } \right)^{i} } + \left( {\frac{{w_{1} w_{2} w_{3} }}{T}} \right)\sum\limits_{{i = 0}}^{3} {J_{i}^{{''}} \left( {w_{1} - w_{2} - w_{3} } \right)^{i} } \\ \end{aligned}$$
(2)

where C4(1+2+3),T, C4(1),T, C4(2),T and C4(3),T are the %m/v solubilities of ACP in the mixed ternary and mono-solvents 1, 2 and 3 at temperature T expressed in kelvin (K), w1, w2 and w3 denote the mass fractions of EtOH (solvent 1), PG (solvent 2) and water (solvent 3) in the absence of ACP (component 4), and finally, J terms are the model constants that are calculated by using a non-intercept least square analysis [75]. Thus, considering all the solubility data summarized in Table 1 the obtained model is:

$$\begin{gathered} \ln C_{{4(1 + 2 + 3),\mathrm T}} = w_{1} \ln C_{{4(1),\mathrm T}} + w_{2} \ln C_{{4(2),\mathrm T}} + w_{3} \ln C_{{4(3),\mathrm T}} \hfill \\ \quad \quad \quad + \left( {\frac{{w_{1} w_{2} }}{T}} \right)\left( {143.649 + 70.908\left( {w_{1} - w_{2} } \right) + 151.889\left( {w_{1} - w_{2} } \right)^{2} } \right) \hfill \\ \quad \quad \quad + \left( {\frac{{w_{1} w_{3} }}{T}} \right)\left( {581.004 + 537.909\left( {w_{1} - w_{3} } \right)} \right) \hfill \\ \quad \quad \quad + \left( {\frac{{w_{2} w_{3} }}{T}} \right)\left( {1527.121 + 201.217\left( {w_{2} - w_{3} } \right) - 153.913\left( {w_{1} - w_{2} } \right)^{2} } \right) \hfill \\ \quad \quad \quad + \left( {\frac{{w_{1} w_{2} w_{3} }}{T}} \right)\left( {1534.794} \right) \hfill \\ \end{gathered}$$
(3)

This equation back-calculates the %m/v solubility of ACP in {EtOH (1) + PG (2) + water (3)} cosolvent mixtures obtaining a mean percentage deviation (MPD) value of 3.8% (N = 330). As it is evident from Eq. (3), the equilibrium solubilities of ACP in the three mono-solvents at each temperature of interest are required as input values. This fact limits the applicability of the model only to temperatures where the ACP solubilities in mono-solvents is available. Otherwise, the MPD values were computed from the following relation:

$${\text{MPD}}=\frac{{100}}{N}\sum {\frac{{|{\text{Calculated}} - {\text{Observed}}|}}{{{\text{Observed}}}}}$$
(4)

where N is the number of experimental data points. On the other hand, it is noteworthy that ACP has been studied as a model drug for predicting its solubility in some other cosolvent systems [76]. Finally, similar correlations have been reported earlier for the solubility of naproxen [77] and mesalazine [78] in the same ternary cosolvent system. As indicated above, the requirement of the experimental solubility data in the mono-solvents at each temperature of interest, namely C4(1),T, C4(2),T, and C4(3),T is a limitation for Eq. (3). Therefore, one may solve this problem by combining the previous correlation model with the van’t Hoff equation, as follows [79]:

$$\begin{gathered} \ln C_{{4(1 + 2 + 3),\mathrm T}} = w_{1} \left( {\alpha _{1} + \frac{{\beta _{1} }}{T}} \right) + w_{2} \left( {\alpha _{2} + \frac{{\beta _{2} }}{T}} \right) + w_{3} \left( {\alpha _{3} + \frac{{\beta _{3} }}{T}} \right) \hfill \\ \quad \quad \quad + \left( {\frac{{w_{1} w_{2} }}{T}} \right)\left( {J_{0} + J_{1} \left( {w_{1} - w_{2} } \right) + J_{2} \left( {w_{1} - w_{2} } \right)^{2} } \right) \hfill \\ \quad \quad \quad + \left( {\frac{{w_{1} w_{3} }}{T}} \right)\left( {J_{0}^{'} + J_{1}^{'} \left( {w_{1} - w_{3} } \right) + J_{1}^{'} \left( {w_{1} - w_{3} } \right)^{2} } \right) \hfill \\ \quad \quad \quad + \left( {\frac{{w_{2} w_{3} }}{T}} \right)\left( {J_{0}^{{''}} + J_{0}^{{''}} \left( {w_{2} - w_{3} } \right) + J_{0}^{{''}} \left( {w_{1} - w_{2} } \right)^{2} } \right) \hfill \\ \quad \quad \quad + \left( {\frac{{w_{1} w_{2} w_{3} }}{T}} \right)\left( {J_{0}^{{'''}} + J_{1}^{{'''}} \left( {w_{1} - w_{2} - w_{3} } \right) + J_{2}^{{'''}} \left( {w_{1} - w_{2} - w_{3} } \right)^{2} } \right) \hfill \\ \end{gathered}$$
(5)

The numerical values of Eq. (5) coefficients could be computed by employing the minimum number of experimental solubility data points [80, 81]. The trained model by using the solubility data of ACP in the mono-solvents at 20.0 and 40.0 °C (293.15 and 313.15 K) (N = 6 points), as well as the ACP solubility in the binary solvent mixtures of w1 or w2 = 0.30, 0.50, and 0.70, and in the ternary solvent mixtures of w1 + w2 + w3 (0.10-0.10-0.80, 0.10-0.80-0.10 and 0.80-0.10-0.10) at 25.0 °C (298.15 K) (N = 12 points) is:

$$\begin{gathered} \ln C_{{4(m),\mathrm{T}}} = w_{1} \left( {8.696 - \frac{{1884.804}}{T}} \right) + w_{2} \left( {7.213 - \frac{{1374.347}}{T}} \right) + w_{3} \left( {11.276 - \frac{{3235.438}}{T}} \right) \hfill \\ \quad \quad \quad + \left( {\frac{{w_{1} w_{2} }}{T}} \right)\left( {190.595 + 25.995\left( {w_{1} - w_{2} } \right) + 471.497\left( {w_{1} - w_{2} } \right)^{2} } \right) \hfill \\ \quad \quad \quad + \left( {\frac{{w_{1} w_{3} }}{T}} \right)\left( {583.530 + 742.038\left( {w_{1} - w_{3} } \right)} \right) \hfill \\ \quad \quad \quad + \left( {\frac{{w_{2} w_{3} }}{T}} \right)\left( {1529.493 + 221.499\left( {w_{2} - w_{3} } \right) - 165.649\left( {w_{1} - w_{2} } \right)^{2} } \right) \hfill \\ \quad \quad \quad + \left( {\frac{{w_{1} w_{2} w_{3} }}{T}} \right)\left( { - 342.206} \right) \hfill \\ \end{gathered}$$
(6)

which predicts the rest of solubility data points in the mono-, binary and ternary solvent systems with a MPD value of 8.5% (N = 312).

4 Conclusions

Mass/volume percentage equilibrium solubilities and correlating models of ACP in some {EtOH (1) + PG (2) + water (3)} solvent systems at temperatures from 20.0 to 40.0 °C were reported. ACP solubility increases when temperature arises, as well, as with the cosolvents-proportion increasing in the solvent mixtures. Maximum ACP %m/v solubility values were in the binary mixture w1 + w2 + w3 of 0.80-0.00-0.20 at all temperatures, namely in the aqueous mixture of 80% w/w of ethanol, which could be very painful after intra-muscular administration. However, several ternary mixtures could be useful for designing an injectable solution of 10.0%m/v of ACP, for instance the one corresponding to w1 + w2 + w3 of 0.20-0.30-0.50, i.e. with 20% w/w of ethanol and 30% w/w of PG, among others, that could be more easily tolerated after injection. Moreover, ACP solubility data were adequately correlated by means of the Jouyban-Acree model. Finally, reported ACP solubilities expand the available solubility database on pharmaceutical compounds in aqueous ternary mixtures of EtOH and PG at several temperatures.