Experimental and modeling of solubility of sitagliptin phosphate, in supercritical carbon dioxide: proposing a new association model

Abstract

The solubility of an anti-hyperglycemic agent drug, (R)-4-oxo-4-[3-(trifluoromethyl)-5,6-dihydro [1,2,4] triazolo[4,3-a] pyrazin-7(8H)-yl]-1-(2,4,5-trifluorophenyl) butan-2-amine (also known as Sitagliptin phosphate) in supercritical carbon dioxide (scCO₂) was determined by ananalytical and dynamic technique at different temperatures (308, 318, 328 and 338 K) and pressure (12–30 MPa) values. The measured solubilities were in the range of 3.02 × 10^–5 to 5.17 × 10^–5, 2.71 × 10^–5 to 5.83 × 10^–5, 2.39 × 10^–5 to 6.51 × 10^–5 and 2.07 × 10^–5 to 6.98 × 10⁻⁵ in mole fraction at (308, 318, 328 and 338) K, respectively. The solubility data were correlated with existing density models and with a new association model.

Introduction

Diabetes mellitus is a common metabolic disorder in which blood glucose levels are too high during a long period of time, which is increasing rapidly over the world and being considered one of the main threats to public health in the twenty-first century. It is predicted that by 2030, 366 million people worldwide will be affected by diabetes, of which 90% will be type IIof this disease¹. The conventional diabetes treatmentis associated with some side effects such as weight gain, hypoglycemia, digestive problems, and gastric intolerance. For these this reasons, extensive researchhas been conducted to find novel drug delivery systems for this disease.

Sitagliptin phosphate, a dipeptidyl peptidase-4 (DPP-4) inhibitor, is one of the most effective anti-hyperglycemic agents which has been included in the list of diabetes drugs since 2006 with the FDA approval².The use of sitagliptin phosphate effectively reduces the fasting glucose and glycosylated hemoglobin A1C (HbA1c) levels in type II diabetic patients³. However, the biological half life of this drug is short (about 3.6 h in rats), and it is eliminated quickly, implying the use of a high daily dose (prescribed as two doses of 50 mg day⁻¹), which is not favorable for patients. This problem can be solved using an efficient drug delivery system with aprolonged and controlled release rate and increased adsorption efficiency of the drug^{2, 4, 5}.

It has been reported that nano-sized drug delivery systems can deliver the required concentration of the drug to the target site of the body in a reasonable time and, as a result, increase the drug bioavailability and decrease its dosage and side effects⁶. Nanoparticles and the various types of polymericnano formulations are the most commondrug delivery systems which have shown satisfactory results in this regard. However, the method of producing these systems significantly influences their characteristics and, thus,their therapeutic efficacy. The conventional methods of producing drug delivery systems have some disadvantages, including non-uniformity of the size and morphology of the obtained particles, the consumption of a lot of organic solvent and subsequently the need forseveral purification steps to remove residual solvents and to reachpharmaceutical standards, involving the damage of the pharmaceutical compound due to severe operational conditions⁷. Therefore, the design and development of novel techniques to produce efficient drug delivery systems is one of the most attractive research areas.

It has been shown that the scCO₂-based methods can satisfactorily replace the conventional methods used to produce various pharmaceutical formulations. The FDA’s approval for the use of CO₂ as a permitted solvent in the pharmaceutical industry, as well as its unique characteristics, such as abundance, low cost, environmentally friendly nature, and recyclability, are the most important positive features of CO₂⁷, which have its use in the pharmaceutical industries. Different techniques based on supercritical have been developed to produce nanosized drugs, like supercritical antisolvent (SAS)⁸, supercritical solvent impregnation (SSI)⁹, rapid expansion of a supercritical solution (RESS)¹⁰, and some others^{11,12,13,14,15}.

The solubility of the desired medicine in scCO₂ is an important factor that should be known to select the appropriate supercritical fluid-basedmethod for drug designing. For this reason, determining the solubility of different drugs in scCO₂ has become a relevant research topic in recent years. Moreover, the theoretical determination of the solubility of drugs in scCO₂ has attracted much attention due to the complexity and high cost of the experimental process. The empirical models, thermodynamic models based on various equations of state, intelligence models (e.g., artificial neural networks), molecular modeling and machine learning models are the common models used for this purpose. The empirical models and the equations of state-based models are the most widely used ones. The empirical models, also known as density-based models, are the simplest theoretical models that have been used to correlate the experimental solubility data of solutes in scCO₂since 1978¹⁶. These models allow to calculate with acceptable accuracy the solubility of a solute based on the density of the supercritical solvent (scCO₂) at the desired temperature and pressure, without the need of the thermodynamic and chemical properties of the solute. The of these models to predict the supercritical solubility of various drugs has been confirmed by many researchers^{17,18,19,20,21}.

In this study, the solubility of sitagliptin phosphate in scCO₂ was measured at temperatures of 308, 318, 328, and 338 K, and at pressures of 12–30 MPa. In addition, the solubility of the drug was theoretically determined using some well-known empirical models and a new association model.

Experimental

Materials used

The physicochemical properties of sitagliptin phosphate and the other chemicals that were utilized in the study can be found in Table 1. This includes the chemical's structureas well as its molar mass (M_w), formula, purity, and CAS number. These substances did not require any further processing of any kind before theiruse.

Table 1 The molecular structure and physicochemical properties of the materials examined in this study.

Experimental section

The Fig. 1 shows the apparatususedfor themeasurement of the Sitagliptin solubility in scCO₂. More details about this system can be found in our previous studies^22,23,24. This laboratory setup includes a CO₂ tank, a cooling unit, a high-pressure pump, an equilibrium cell and a magnetic stirrer, which are clearly marked in the Fig. 1.

Figure 1

Schematic diagram of solubility device.

In this solubility measurement method, initially, CO₂ from the tank at 60 bar, after passing through the molecular sieve filter (1 µm pores) and removing pollution, entered the refrigerator unit forits liquefaction by decreasingtemperature from ambient to about − 20 °C. Then liquid CO₂ was pressurized using a high-pressure pump until reached the appropriate pressure. The pressure values were controlled and recorded on both the pressure gauge (WIKA, Germany) and pressure transmitter with an accuracy of u(P) = 0.1 MPa. After adjusting the pressure, the liquefied CO₂ entered to the equilibrium cell whose volume was 70 mL. In the cell, liquefied CO₂ was contacted with the drug (Sitagliptin phosphate) that was already loaded in the cell.

The equilibrium cell was placed in an oven with temperature control with an accuracy of u(T) = ± 1 K. Also, a magnet stirrer was used to achieve a complete saturation of the drug inscCO₂. The time required for the process was 120 min. After equilibrium, using the opening a 2-status 3-way port valve and reducing the pressure,saturated scCO₂ (600 μL)was delivered into the injection loop. Finally, by opening the micrometer valve, saturated scCO₂was collectedinto a vial which was already loaded with 4 mL of methanol. Further, the loopwas washed with 1 mL of methanol through an external line. During the experimentspressurewas controlled with a back pressure valve. Sitagliptin absorbance in methanol was measured with a spectrophotometer- Perkin-Elmer UV–Vis at 268 nm (λ_max) with the calibration curve (with a regression coefficient of 0.998). The experimental runs were performed three times to determine averages. The relationships used to calculate the solubility of Sitagliptin in scCO₂ at different temperatures and pressures, in terms of mole fraction (y) and equilibrium solubility (S (g L⁻¹)), are reported in our previous work²⁵.

Results and discussion

Experimental solubility

The reliability of the solubility device was tested by determining the solubility of naphthalene at a temperature of 308 K anddifferent pressure values was measured by the used setup used in the work and the obtained data was compared with reported data by Iwai et al.²⁶, Yamini et al.²⁷ and Sodeifian et al.²⁸. These data are listed in Table 2.

Table 2 Experimental solubility data of naphthalene in sc-CO₂ at 308 K and comparison with the literature data.

The solubility data for (R)-4-oxo-4-[3-(trifluoromethyl)-5,6-dihydro [1,2,4] triazolo[4,3-a] pyrazin-7(8H)-yl]-1-(2,4,5-trifluorophenyl) butan-2-amine (also known as sitagliptin phosphate) in scCO₂ at different temperature (308, 318, 328 and 338 K) and pressure (12 and 30 MPa) values are shown in Table 3. Crossover points in Fig. 2 are observed for different isotherms between 15 and 16.5 MPa. Below the crossover point’s solubility increase is influenced due to increase in the density of scCO₂, on the other hand above the crossover points the increase in solubility influenced by increase in sublimation pressure of the solute.

Table 3 The experimental data of sitagliptin phosphate solubility in SC-CO2 based on distinct conditions (temperatures (T) and pressures (P) for binary system).

Figure 2

Sitagliptin phosphate solubility vs. pressure.

From Fig. 2, for an isotherm higher solubility is observed at higher pressures and it is due to enhancement of scCO₂ density at higher pressures. The effect of temperature on solubility is typical in nature and crossover points are observed. Below the crossover point’s density of scCO₂ influence the solubility, when density of scCO₂ is higher correspondingly solubility is high even though temperature is lower. On the other hand above the crossover point’s solubility increases with temperature and it is due to increase in sublimation pressure of the solute. Thus, temperature has duel effect on solubility via solvent density and solute sublimation pressure.

Modeling

Solubility of drugs in supercritical fluids was modeled using different approaches that can beclassified indensity, equations of state, solid–liquid equilibrium andintelligence-based models. However, each approach has its own advantages and drawbacks. The density-based models are simple and easy to apply for data correlation due to doesn’t require solute information such as critical properties, acentric factor and sublimation pressure. Sitagliptin phosphate is a typical compound, and its acentric factor, critical properties (Tc and Pc) molar volumes and sublimation pressures can’t be predicted using regular group contribution methods^29,30,31,32, due to the presence of phosphate in its chemical structure. Therefore, equation of sate (EOS) and solid–liquid equilibrium methods cannot be applied to correlate the solubility data of sitagliptin phosphate. Thus, only semi-empirical models (i.e., density-based model) are useful in data correlation. In this work for data correlation Josef Chrastil model³³, Reformulated Chrastil model³⁴,Méndez-Santiago and Teja (MST) model³⁵, Bartle et al. model³⁶ and Kumar and Johnston (KJ) model³⁷ were used. For a better data correlation, a new association model requiring only density, pressure, and temperature of scCO₂ was proposed. Following subsections discuss about the models considered in detail.

Josef Chrastil model

According to this model, the solubility of solutes in SCF is expressed with the following relation:

$${c \mathord{\left/ {\vphantom {c {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }} = \left( {{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}} \right)^{\kappa } \exp \left( {A_{1} + {\raise0.7ex\hbox{${B_{1} }$} \!\mathord{\left/ {\vphantom {{B_{1} } {{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}$}}} \right)$$

(1)

where $\kappa$, $A_{1}$ and $B_{1}$ are model constants.

Equation (1) can be rearranged to mole fraction as follows³⁸

$$\frac{{{c \mathord{\left/ {\vphantom {c {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}}}{{{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}}}\frac{{M_{ScF} }}{{M{}_{Solute}}} = \frac{{M_{ScF} }}{{M{}_{Solute}}}\left( {{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}} \right)^{\kappa - 1} \exp \left( {A_{1} + {\raise0.7ex\hbox{${B_{1} }$} \!\mathord{\left/ {\vphantom {{B_{1} } {{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}$}}} \right)$$

(2)

$$mole\;ratio = \frac{{{c \mathord{\left/ {\vphantom {c {kgmol \cdot m^{ - 3} }}} \right. \kern-0pt} {kgmol \cdot m^{ - 3} }}}}{{{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kgmol \cdot m^{ - 3} }}} \right. \kern-0pt} {kgmol \cdot m^{ - 3} }}}} = \frac{{M_{ScF} }}{{M{}_{Solute}}}\left( {{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}} \right)^{\kappa - 1} \exp \left( {A_{1} + {\raise0.7ex\hbox{${B_{1} }$} \!\mathord{\left/ {\vphantom {{B_{1} } {{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}$}}} \right)$$

(3)

Mole fraction ($y_{2}$) and mole ratio are related as follows

$$\frac{{{c \mathord{\left/ {\vphantom {c {kgmol \cdot m^{ - 3} }}} \right. \kern-0pt} {kgmol \cdot m^{ - 3} }}}}{{{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kgmol \cdot m^{ - 3} }}} \right. \kern-0pt} {kgmol \cdot m^{ - 3} }}}} = \frac{{{{y_{2} } \mathord{\left/ {\vphantom {{y_{2} } {mole\;fraction}}} \right. \kern-0pt} {mole\;fraction}}}}{{1 - {{y_{2} } \mathord{\left/ {\vphantom {{y_{2} } {mole\;fraction}}} \right. \kern-0pt} {mole\;fraction}}}}$$

(4)

$${{y_{2} } \mathord{\left/ {\vphantom {{y_{2} } {mole\;fraction}}} \right. \kern-0pt} {mole\;fraction}} = {{\frac{{{c \mathord{\left/ {\vphantom {c {kgmol \cdot m^{ - 3} }}} \right. \kern-0pt} {kgmol \cdot m^{ - 3} }}}}{{{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kgmol \cdot m^{ - 3} }}} \right. \kern-0pt} {kgmol \cdot m^{ - 3} }}}}} \mathord{\left/ {\vphantom {{\frac{{{c \mathord{\left/ {\vphantom {c {kgmol \cdot m^{ - 3} }}} \right. \kern-0pt} {kgmol \cdot m^{ - 3} }}}}{{{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kgmol \cdot m^{ - 3} }}} \right. \kern-0pt} {kgmol \cdot m^{ - 3} }}}}} {\left[ {1 + \frac{{{c \mathord{\left/ {\vphantom {c {kgmol \cdot m^{ - 3} }}} \right. \kern-0pt} {kgmol \cdot m^{ - 3} }}}}{{{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kgmol \cdot m^{ - 3} }}} \right. \kern-0pt} {kgmol \cdot m^{ - 3} }}}}} \right]}}} \right. \kern-0pt} {\left[ {1 + \frac{{{c \mathord{\left/ {\vphantom {c {kgmol \cdot m^{ - 3} }}} \right. \kern-0pt} {kgmol \cdot m^{ - 3} }}}}{{{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kgmol \cdot m^{ - 3} }}} \right. \kern-0pt} {kgmol \cdot m^{ - 3} }}}}} \right]}}$$

(5)

$${{y_{2} } \mathord{\left/ {\vphantom {{y_{2} } {mole\;fraction}}} \right. \kern-0pt} {mole\;fraction}} = {{\frac{{M_{ScF} }}{{M{}_{Solute}}}\left( {{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}} \right)^{\kappa - 1} \exp \left( {A_{1} + {\raise0.7ex\hbox{${B_{1} }$} \!\mathord{\left/ {\vphantom {{B_{1} } {{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}$}}} \right)} \mathord{\left/ {\vphantom {{\frac{{M_{ScF} }}{{M{}_{Solute}}}\left( {{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}} \right)^{\kappa - 1} \exp \left( {A_{1} + {\raise0.7ex\hbox{${B_{1} }$} \!\mathord{\left/ {\vphantom {{B_{1} } {{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}$}}} \right)} {\left[ {1 + \frac{{M_{ScF} }}{{M{}_{Solute}}}\left( {{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}} \right)^{\kappa - 1} \exp \left( {A_{1} + {\raise0.7ex\hbox{${B_{1} }$} \!\mathord{\left/ {\vphantom {{B_{1} } {{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}$}}} \right)} \right]}}} \right. \kern-0pt} {\left[ {1 + \frac{{M_{ScF} }}{{M{}_{Solute}}}\left( {{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}} \right)^{\kappa - 1} \exp \left( {A_{1} + {\raise0.7ex\hbox{${B_{1} }$} \!\mathord{\left/ {\vphantom {{B_{1} } {{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}$}}} \right)} \right]}}$$

(6)

In Eq. (6), the model constants are treated as independent of temperature and their values are obtained by regression with experimental data³⁸. The obtained values are reported in Table 4. It is quite evident that a linear plot is observed when the data is depicted as $\ln \left( {{c \mathord{\left/ {\vphantom {c {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}} \right)$ vs. $\ln \left( {{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}} \right)$ (Fig. 3a) and as $\ln \left( {{c \mathord{\left/ {\vphantom {c {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}} \right) - B_{1} /T/K$ vs. $\ln \left( {{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}} \right)$ (Fig. 3b), this confirms the applicability of the Chrastil model to the solubility data³⁹. From the constant $B_{1}$, total heat of reaction is calculated (i.e., $\Delta H_{Total} = B_{1} R$), the obtained values are reported in Table 5

Table 4 The correlation results of the sitagliptin phosphate–CO₂ system provided by semi-empirical models and new association model.

Figure 3

(a) Experimental data of supercritical solubility of sitagliptin phosphate (points) compared to data calculated with the Chrastil model (line). (b) Results of self-consistency analysis for the Chrastil model.

Table 5 Calculated enthalpies of sitagliptin phosphate–CO₂ system provided by semi-empirical models.

Reformulated Chrastil model

According to this model, solubility is a function of $\kappa^{\prime}$(association number), $\rho_{1}$ (solvent density (scCO₂)) and T (temperature). In Eq. (7), it is important to note that depending on the reference fugacity units, R units are selected.

$${{y_{2} } \mathord{\left/ {\vphantom {{y_{2} } {mole\;fraction}}} \right. \kern-0pt} {mole\;fraction}} = \left( {\frac{{{R \mathord{\left/ {\vphantom {R {atm \cdot m^{3} \cdot kgmol^{ - 1} K^{ - 1} }}} \right. \kern-0pt} {atm \cdot m^{3} \cdot kgmol^{ - 1} K^{ - 1} }}{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}}}{{{{M_{ScF} } \mathord{\left/ {\vphantom {{M_{ScF} } {kg \cdot kgmol^{ - 1} }}} \right. \kern-0pt} {kg \cdot kgmol^{ - 1} }} \cdot {{f^{ \cdot } } \mathord{\left/ {\vphantom {{f^{ \cdot } } {1\;atm}}} \right. \kern-0pt} {1\;atm}}}}} \right)^{{\kappa^{\prime} - 1}} \exp \left( {A_{2} + \frac{{B_{2} }}{{{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}}} \right)$$

(7)

where R denotes universal gas constant (0.082057 atm m³ kgmole⁻¹ K⁻¹), $M_{ScF}$ is molecular weight of solvent (For CO₂ 44.01 kg kgmol⁻¹), $f^{ * }$ is reference fugacity (1 atm) and $A_{2} \;$ and $\;B_{2}$ are the reformulated model constants.

In Eq. (7), the model constants were treated as independent of temperature and their values were obtained by regression with experimental data. The obtained values are reported in Table 4 It is quite evident that a linear plot is observed when the data are depicted as $\ln \left( {y_{2} } \right)$ vs. $\ln \left( {{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}} \right)$ (Fig. 4a) and as $\ln \left( {y_{2} } \right) - (\kappa^{\prime} - 1)\ln \left( T \right) - B_{2} /T/K$ vs. ${{\ln \;(\rho_{1} } \mathord{\left/ {\vphantom {{\ln \;(\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }})$ (Fig. 4b), this confirms the applicability of the reformulated Chrastil model to the solubility data. From the constant $B_{2}$, total heat of reaction is calculated (i.e.,$\Delta H_{Total} = B_{2} R$), the obtained values are reported in Table 5.

Figure 4

(a) Experimental data of supercritical solubility of sitagliptin phosphate (points) compared to data calculated with the Reformulated Chrastil Model (line). (b) Results of self-consistency analysis for the Reformulated Chrastil Model.

Méndez-Santiago and Teja (MT) model

Internal consistency of the measured solubility data was checked with this model. It is stated as Eq. (8) and when $T\ln \left( {y_{2} P} \right) - C_{3} T$ vs. $\rho_{1}$ is established, all data points fall around a single straight line.

$${T \mathord{\left/ {\vphantom {T {K \cdot \ln \left( {y_{2} {P \mathord{\left/ {\vphantom {P {bar}}} \right. \kern-0pt} {bar}}} \right)}}} \right. \kern-0pt} {K \cdot \ln \left( {y_{2} {P \mathord{\left/ {\vphantom {P {bar}}} \right. \kern-0pt} {bar}}} \right)}} = A_{3} + B_{3} \cdot {{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} + C_{3} \cdot {T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}}} \right. \kern-0pt} {kg \cdot m^{ - 3} + C_{3} \cdot {T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}}$$

(8)

where A₃ to C₃ are the model constants.

In Eq. (8), the model constants were treated as independent of temperature and their values were obtained by regression with experimental data. The obtained values were reported in Table 4. The experimental data obtained in this work is checked for consistency with the help of Mendez-Santiago and Teja model (MT model). According to the MT model, the solubility data at different temperatures collapsed into a single line. It is quite evident that linear plots are observed when the data are depicted as $\ln \left( {y_{2} \cdot P/bar} \right)$ vs. ${{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}$ (Fig. 5a) and as ${T \mathord{\left/ {\vphantom {T {K\ln \left( {y_{2} \cdot P/bar} \right)}}} \right. \kern-0pt} {K\ln \left( {y_{2} \cdot P/bar} \right)}} - C_{3} T/K$ vs. ${{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}$(Fig. 5b), this confirms the applicability of the MT model to the solubility data.

Figure 5

(a) Experimental data of supercritical solubility of sitagliptin phosphate (points) compared to data calculated with the MT Model (line). (b) Results of self-consistency analysis for the MT Model.

Bartle et al., model

According to the model the solubility was expressed as Eq. (9)

$$\ln \left( {\frac{{y_{2} \cdot P}}{{P_{ref} }}} \right) = A_{4} + \frac{{B_{4} }}{{{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}} + C_{4} \left( {{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }} - {{\rho_{ref} } \mathord{\left/ {\vphantom {{\rho_{ref} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}} \right)$$

(9)

where reference pressure is 0.1 MPa or 1 bar, reference density is 700 kg m⁻³ and A₄ to C₄ are the model constants. From the constant $B_{4}$, sublimation enthalpy is calculated (i.e.,$\Delta_{sub} H = - B_{4} R$).

In Eq. (9), the model constants were treated as independent of temperature and their values were obtained by regression with experimental data. The obtained values were reported in Table 4 it is quite evident that linear plots are observed when the data are depicted as $\ln \left( {\frac{{y_{2} P}}{{P_{ref} }}} \right)$ vs. ${{\left( {\rho_{1} - \rho_{ref} } \right)} \mathord{\left/ {\vphantom {{\left( {\rho_{1} - \rho_{ref} } \right)} {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}$(Fig. 6a) and as $\ln \left( {\frac{{y_{2} P}}{{P_{ref} }}} \right) - \frac{{B_{4} }}{{{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}}$ vs. ${{\left( {\rho_{1} - \rho_{ref} } \right)} \mathord{\left/ {\vphantom {{\left( {\rho_{1} - \rho_{ref} } \right)} {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}$ (Fig. 6b) this confirms the applicability of the Bartle et al. model to the solubility data.

Figure 6

(a) Experimental data of supercritical solubility of sitagliptin phosphate (points) compared to data calculated with the Bartle et al. Model (line). (b) Results of self-consistency analysis for the Bartle et al. Model.

Kumar and Johnston (KJ) model

According to the model the solubility was expressed as Eq. (10)

$$\ln \left( {y_{2} } \right) = A_{5} + B_{5} \left( {{{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}} \right) + \frac{{C_{5} }}{{{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}}$$

(10)

where A₅ to C₅ are the model constants.

In Eq. (10), the model constants were treated as independent of temperature and their values were obtained by regression with experimental data. The obtained values were reported in Table 3 it is quite evident that a linear plots are observed when the data are depicted as $\ln \left( {y_{2} } \right) - \frac{{C_{5} }}{{{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}}$ vs.${{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}$(Fig. 7a) and as $\ln \left( {y_{2} } \right) - \frac{{C_{5} }}{{{T \mathord{\left/ {\vphantom {T K}} \right. \kern-0pt} K}}}$ vs. ${{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}$(Fig. 7b), this confirms the applicability of the KJ model to the solubility data.

Figure 7

(a) Experimental data of supercritical solubility of sitagliptin phosphate (points) compared to data calculated with the KJ Model (line). (b) Results of self-consistency analysis for the KJ Model.

New association model

If one molecule of a solute (A) associates with $\kappa^{\prime\prime}$ molecules of solvent (B) to form one molecule of a solvato complex $AB_{\kappa }$ in equilibrium with the gaseous system³³,

$$A + \kappa^{\prime\prime}B \Leftrightarrow AB_{\kappa }$$

(11)

Equation (12) represents the equilibrium constant in terms of the individual component’s fugacity

$$K_{f} = \frac{{\left( {{\raise0.7ex\hbox{${\hat{f}_{{AB_{\kappa } }} }$} \!\mathord{\left/ {\vphantom {{\hat{f}_{{AB_{\kappa } }} } {f_{{AB_{\kappa } }}^{ * } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${f_{{AB_{\kappa } }}^{ * } }$}}} \right)_{ScP} }}{{\left( {\frac{{\hat{f}_{A} }}{{f_{A}^{ * } }}} \right)_{S} \;\left( {\left( {\frac{{\hat{f}_{B} }}{{f_{B}^{ * } }}} \right)^{{\kappa^{\prime\prime}}} } \right)_{ScP} }}$$

(12)

where ScP represents the supercritical phase, S represents the solute phase and $f^{ * }$ is reference fugacity.

The fugacity for each component can be calculated by the following equations^40,41,42.

$$\hat{f}_{A} = y_{A} \hat{\phi }_{A} P$$

(13)

$$\hat{f}_{B} = y_{B} \hat{\phi }_{B} P$$

(14)

$$\hat{f}_{{AB_{\kappa } }} = y_{{AB_{\kappa } }} \hat{\phi }_{{AB_{\kappa } }} P$$

(15)

$$f_{{AB_{\kappa } }}^{ * } = \phi_{{AB_{\kappa } }}^{ * } \;P^{ * }$$

(16)

$$f_{A}^{ * } = \phi_{A}^{ * } \;P^{ * }$$

(17)

$$f_{B}^{ * } = \phi_{B}^{ * } \;P^{ * }$$

(18)

Here the main assumption is fluid-phase component does not dissolve in the solid. i.e., the solid is pure. Solute A exists in an associated state in ScP

$$y_{B} + y_{{AB_{\kappa } }} = 1$$

(19)

where $y_{B} ,y_{{AB_{\kappa } }}$ are mole fraction of solvent and solvato complex respectively.

Since the solute A mainly exists in an associating state, the solubility of solute A in ScP is^43,44,45

$$y_{2} = \frac{{y_{{AB_{\kappa } }} }}{{1 + \kappa^{\prime\prime}\;y_{{AB_{\kappa } }} }}$$

(20)

when standard state of the solute A is treated as pure solute at system pressure (P) and temperature (T), then

$$\hat{f}_{A} = f_{A}$$

(21)

The fugacity of pure solute can be written as

$$f_{A} = P_{A}^{sub} \exp \left( {\frac{{V_{A} (P - P_{A}^{sub} )}}{RT}} \right)$$

(22)

where $P_{A}^{sub}$ is the sublimation pressure of the pure solid, and $V_{A}$ is the molar volume of the pure solid at system temperature(T), and pressure, (P).

Substituting Eqs. (13)–(18) and Eq. (22) in Eq. (12) gives Eq. (23)

$$K_{f} = \frac{{\left( {\frac{{y_{{AB_{\kappa } }} \hat{\phi }_{{AB_{\kappa } }} P}}{{\phi_{{AB_{\kappa } }}^{ * } P^{ * } }}} \right)}}{{\left( {\frac{{P_{A}^{sub} \exp \left( {\frac{{V_{A} \left( {P - P_{A}^{sub} } \right)}}{RT}} \right)}}{{\phi_{A}^{ * } P^{ * } }}} \right)\;\left( {\frac{{y_{B} \hat{\phi }_{B} P}}{{\phi_{B}^{ * } P^{ * } }}} \right)^{{\kappa^{\prime\prime}}} }}$$

(23)

$$\begin{aligned} \ln \left( {K_{f} } \right) & = \ln \left( {y_{{AB_{\kappa } }} } \right) + \ln \left( {{\raise0.7ex\hbox{${\hat{\phi }_{{AB_{\kappa } }} }$} \!\mathord{\left/ {\vphantom {{\hat{\phi }_{{AB_{\kappa } }} } {\phi_{{AB_{\kappa } }}^{ * } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\phi_{{AB_{\kappa } }}^{ * } }$}}} \right) + \ln \left( {{\raise0.7ex\hbox{$P$} \!\mathord{\left/ {\vphantom {P {P^{ * } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${P^{ * } }$}}} \right) + \ln \left( {\phi_{A}^{ * } } \right) - \ln \left( {{\raise0.7ex\hbox{${P_{A}^{sub} }$} \!\mathord{\left/ {\vphantom {{P_{A}^{sub} } {P^{ * } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${P^{ * } }$}}} \right) - \frac{{V_{A} \left( {P - P_{A}^{sub} } \right)}}{RT} \\ & \quad \quad - \kappa^{\prime\prime}\ln \left( {y_{B} } \right) - \kappa^{\prime\prime}\ln \left( {{\raise0.7ex\hbox{${\hat{\phi }_{B} }$} \!\mathord{\left/ {\vphantom {{\hat{\phi }_{B} } {\phi_{B}^{ * } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\phi_{B}^{ * } }$}}} \right) - \kappa^{\prime\prime}\ln \left( {{\raise0.7ex\hbox{$P$} \!\mathord{\left/ {\vphantom {P {P^{ * } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${P^{ * } }$}}} \right) \\ \end{aligned}$$

(24)

The equilibrium constant,$K_{f} ,$ may be expressed as $\ln \left( {K_{f} } \right) = \frac{{\Delta H_{s} }}{RT} + q_{s}$.

where $\Delta H_{s}$, the heat of solvation and q_sis a constant and ${\raise0.7ex\hbox{${V_{A} P}$} \!\mathord{\left/ {\vphantom {{V_{A} P} {RT}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${RT}$}}$ may be expressed as ${\raise0.7ex\hbox{${ZV_{A} \rho }$} \!\mathord{\left/ {\vphantom {{ZV_{A} \rho } M}}\right.\kern-0pt} \!\lower0.7ex\hbox{$M$}}$ where $\rho$ is the density of the supercritical phase. At the supercritical state, $\rho$ is a function of three variables namely pressure, temperature and composition. Thus, the fugacities in Eq. (24) are a very complex function of pressure, temperature and composition.

Then Eq. (24) may be expressed as

$$\begin{aligned} \ln \left( {y_{{AB_{\kappa } }} } \right) - \kappa^{\prime\prime}\ln \left( {y_{B} } \right) + \left( {1 - \kappa^{\prime\prime}} \right)\ln \left( {{\raise0.7ex\hbox{$P$} \!\mathord{\left/ {\vphantom {P {P^{ * } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${P^{ * } }$}}} \right) & = - \ln \left( {{\raise0.7ex\hbox{${\hat{\phi }_{{AB_{\kappa } }} }$} \!\mathord{\left/ {\vphantom {{\hat{\phi }_{{AB_{\kappa } }} } {\phi_{{AB_{\kappa } }}^{ * } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\phi_{{AB_{\kappa } }}^{ * } }$}}} \right) - \ln \left( {\phi_{A}^{ * } } \right) + \ln \left( {P_{A}^{sub} } \right) - \ln \left( {P^{ * } } \right) \\ & \quad \quad + \frac{{ZV_{A} \rho }}{M} - \frac{{V_{A} P_{A}^{sub} }}{RT} + \kappa^{\prime\prime}\ln \left( {{\raise0.7ex\hbox{${\hat{\phi }_{B} }$} \!\mathord{\left/ {\vphantom {{\hat{\phi }_{B} } {\phi_{B}^{ * } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\phi_{B}^{ * } }$}}} \right) + \frac{{\Delta H_{s} }}{RT} + q_{s} \\ \end{aligned}$$

(25)

The sublimation pressure of the solid solute may be expressed as

$$R\ln \left( {P_{A}^{sub} } \right) = \beta + \frac{\gamma }{T} + \Delta_{sub} \delta \ln \left( {\frac{T}{298.15}} \right)$$

(26)

where $\beta$, $\gamma$ and $\Delta_{sub} \delta$ are temperature independent parameters.

When $\frac{{V_{A} P_{A}^{sub} }}{RT}$(~ 10^–9) term is neglected (since the sublimation pressures (~ 10^–4) and molar volume of solid solutes (~ 10^–4) are very low) and density of solution is treated as approximately as density of supercritical fluid

$$\ln \left( {y_{{AB_{\kappa } }} } \right) - \kappa^{\prime\prime}\ln \left( {y_{B} } \right) + \left( {1 - \kappa^{\prime\prime}} \right)\ln \left( {{\raise0.7ex\hbox{$P$} \!\mathord{\left/ {\vphantom {P {P^{ * } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${P^{ * } }$}}} \right) = \frac{{A_{6} }}{T} + B_{6} \ln \left( T \right) + C_{6} \rho_{1} + D_{6}$$

(27)

where $A_{6} = \frac{{\Delta H_{s} }}{R} + \frac{\gamma }{R}$, $B_{6} = {{\Delta_{sub} \delta } \mathord{\left/ {\vphantom {{\Delta_{sub} \delta } R}} \right. \kern-0pt} R}$, $C_{6} = \frac{{ZV_{A} }}{M}$ and $D_{6} = - \ln \left( {\frac{{\left( {\phi_{A}^{ * } } \right)\left( {{\raise0.7ex\hbox{${\hat{\phi }_{{AB_{\kappa } }} }$} \!\mathord{\left/ {\vphantom {{\hat{\phi }_{{AB_{\kappa } }} } {\phi_{{AB_{\kappa } }}^{ * } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\phi_{{AB_{\kappa } }}^{ * } }$}}} \right)}}{{\left( {{\raise0.7ex\hbox{${\hat{\phi }_{B} }$} \!\mathord{\left/ {\vphantom {{\hat{\phi }_{B} } {\phi_{B}^{ * } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\phi_{B}^{ * } }$}}} \right)^{\kappa } }}} \right) - \ln P^{ * } + q_{s} + \frac{\beta }{R} - \frac{{\Delta_{sub} \delta \ln \left( {298.15} \right)}}{R}$

Equation (27) may be written as Eq. (28)

$$y_{{AB_{\kappa } }} = \left( {y_{B} } \right)^{{\kappa^{\prime\prime}}} \left( {\frac{P}{{P^{ * } }}} \right)^{{\left( {\kappa^{\prime\prime} - 1} \right)}} \exp \left( {\frac{{A_{6} }}{T} + B_{6} \ln (T) + C_{6} \rho_{1} + D_{6} } \right)$$

(28)

Because the solubility of drug in scCO₂ are very dilute, therefore for a binary system we may assume $y_{B}$ is unity. Then Eq. (28) reduced to Eq. (29)

$$y_{{AB_{\kappa } }} = \left( {\frac{P}{{P^{ * } }}} \right)^{{\left( {\kappa^{\prime\prime} - 1} \right)}} \exp \left( {\frac{{A_{6} }}{T} + B_{6} \ln (T) + C_{6} \rho_{1} + D_{6} } \right)$$

(29)

Combining Eq. (20) and Eq. (29) gives expression for solubility Eq. (30)

$$y_{2} = \frac{{\left( {\frac{P}{{P^{ * } }}} \right)^{{\left( {\kappa^{\prime\prime} - 1} \right)}} \exp \left( {\frac{{A_{6} }}{T} + B_{6} \ln (T) + C_{6} \rho_{1} + D_{6} } \right)}}{{1 + \kappa^{\prime\prime}\left( {\frac{P}{{P^{ * } }}} \right)^{{\left( {\kappa^{\prime\prime} - 1} \right)}} \exp \left( {\frac{{A_{6} }}{T} + B_{6} \ln (T) + C_{6} \rho_{1} + D_{6} } \right)}}$$

(30)

From Eq. (30),it is clear that solubility is a function of density, temperature and association number (i.e.,$y_{2} = y_{2} (\rho_{1} ,T,\kappa^{\prime\prime})$) and further all equations are dimensionally consistent. Hereafter, this equation may be called as the new association model. In Eq. (30), the model constants were treated as independent of temperature and their values were obtained by regression with experimental data. The obtained values are reported in Table 4 it is quite evident that a better fit is observed when the data is plotted as Sitagliptin Phosphate solubility, $y_{2}$ vs. $\rho_{1} /kg \cdot m^{ - 3}$(Fig. 8a). It is important to note that Fig. 8a is not linear due to its functionality (i.e., $y_{2} = y_{2} (\rho_{1} ,T,\kappa^{\prime\prime})$). However, a linear plot is observed when the data are depicted as $\ln ({{y_{2} } \mathord{\left/ {\vphantom {{y_{2} } {\left( {1 - \kappa^{\prime\prime}y_{2} } \right)}}} \right. \kern-0pt} {\left( {1 - \kappa^{\prime\prime}y_{2} } \right)}} - (\kappa^{\prime\prime} - 1)\ln \left( {{P \mathord{\left/ {\vphantom {P {P^{ * } }}} \right. \kern-0pt} {P^{ * } }}} \right) - {{A_{6} } \mathord{\left/ {\vphantom {{A_{6} } {T - B_{6} \ln \left( T \right) - D_{6} }}} \right. \kern-0pt} {T - B_{6} \ln \left( T \right) - D_{6} }}$ vs. ${{\rho_{1} } \mathord{\left/ {\vphantom {{\rho_{1} } {kg \cdot m^{ - 3} }}} \right. \kern-0pt} {kg \cdot m^{ - 3} }}$(Fig. 8b),which confirms the applicability of the new association model to the solubility data. Solubility of solids substances in scCO₂ are best understood in terms of solvato-complex formation, thus the interactions between sitagliptin phosphate (solute) and supercritical carbon dioxide (solvent) is visualized as formation of a solvato-complex. The new association model (solvato-complex model) proposed in this study is able to capture solubility phenomena with least AARD% (i.e., 2.53%). Thus, present study confirms solvato-complex theory holds good for this sitagliptin phosphate–scCO₂ system.

Figure 8

(a) Sitagliptin phosphate solubility,$y_{2}$ vs. $\rho_{1} /kg \cdot m^{ - 3}$. (b) Results of self-consistency analysis for the New Association Model.

Conclusion

Thesolubility of (R)-4-oxo-4-[3-(trifluoromethyl)-5,6-dihydro [1,2,4] triazolo[4,3-a] pyrazin-7(8H)-yl]-1-(2,4,5-trifluorophenyl) butan-2-amine (i.e., sitagliptin phosphate) was measured by a static method in the pressure range of 12–30 MPa for different temperature values (30, 318, 328 and 338 K). The measured solubilities range from (0.2074 to 0.698) × 10^–4 mol fraction of (R)-4-oxo-4-[3-(trifluoromethyl)-5,6-dihydro [1,2,4] triazolo[4,3-a] pyrazin-7(8H)-yl]-1-(2,4,5-trifluorophenyl) butan-2-amine. Further, measured solubilities are reasonably correlated with the Chrastil model, the reformulated Chrastil model,the Méndez-Santiago and Teja (MST) model, the Bartle et al., model, the Kumar and Johnston (KJ) model. The newly proposed association model was able to correlate the solubility data with the lowest absolute relative deviation (2.53%). Calculated sublimation and solvation enthalpies of Sitagliptin phosphate in scCO₂ are 40.416 kJ mol⁻¹, − 24.535 kJ mol⁻¹ for Bartle et al., model and Bartle et al., model + Chrastil model combination, respectively.