Experimental solubility and thermodynamic modeling of empagliflozin in supercritical carbon dioxide

The solubility of empagliflozin in supercritical carbon dioxide was measured at temperatures (308 to 338 K) and pressures (12 to 27 MPa), for the first time. The measured solubility in terms of mole faction ranged from 5.14 × 10–6 to 25.9 × 10–6. The cross over region was observed at 16.5 MPa. A new solubility model was derived to correlate the solubility data using solid–liquid equilibrium criteria combined with Wilson activity coefficient model at infinite dilution for the activity coefficient. The proposed model correlated the data with average absolute relative deviation (AARD) and Akaike’s information criterion (AICc), 7.22% and − 637.24, respectively. Further, the measured data was also correlated with 11 existing (three, five and six parameters empirical and semi-empirical) models and also with Redlich-Kwong equation of state (RKEoS) along with Kwak-Mansoori mixing rules (KMmr) model. Among density-based models, Bian et al., model was the best and corresponding AARD% was calculated 5.1. The RKEoS + KMmr was observed to correlate the data with 8.07% (correspond AICc is − 635.79). Finally, total, sublimation and solvation enthalpies of empagliflozin were calculated.

Supercritical carbon dioxide (ScCO 2 ) is a fluid above its critical point. It has physical properties (density, diffusivity, viscosity and surface tension) intermediate to that of gas and liquid 1,2 . ScCO 2 has been used as a solvent in various process applications, because it has gas like diffusivity and liquid like density with low viscosity and surface tension 1,3-5 . The major applications are in drug particle micronization, food processing, textile dyeing, ceramic coating, extraction and many more 4,6-12 . Although, several supercritical fluids are utilized as solvent in process industry, ScCO 2 is the most desirable solvent 8,[13][14][15][16][17] . In general, phase equilibrium information is necessary to implement supercritical fluid technology (SFT) 6,7,9 . The solubility is the basic information for the design and development of SFT. In literature, solubility of many drug solids in ScCO 2 is readily available [18][19][20][21][22][23][24][25][26][27][28][29][30] , however, the solubility of empagliflozin has not been reported, therefore in this work for the first time, its solubility in ScCO 2 has been measured. This data may be used in the particle micronization process using ScCO 2 . The molecular formula of empagliflozin is C 23 H 27 ClO 7 and its molecular weight is 450.91. The chemical structure is shown in Fig. 1.
Empagliflozin is an inhibitor of sodium-glucose co-transporter-2 (SGLT2), the transporters primarily responsible for the re-absorption of glucose in the kidney. Further, it is useful in reducing the risk of cardiovascular death in adults with type 2 diabetes mellitus and cardiovascular disease 31 . Sufficient drug dosage is very essential for those treatments and this is achieved through a proper particle size. Therefore, the present study is quite useful in particle micronization using ScCO 2 . Solubility measurement at each desired condition is very cumbersome and hence, there is a great need to develop a model that correlates/predicts the solubility 32 . Recent developments such as machine learning methods may be considered with the improvement of artificial intelligence prediction methods for the data correlation [33][34][35] . However, in general, the solubility models are classified into five types; however, only three are user friendly, and they are equation of state, density-based and mathematical models 36 . Directly or indirectly all of them are derived based on thermodynamic frame work. The derived models make use of the basic concepts related to phase equilibrium criteria (solid-gas or solid-liquid), solvent-solute association theory, dilute solution theory, solution theory and Wilson model or any other model 37 38,39 . Therefore, there is a need to develop an explicit solubility model and hence, this task is taken up in this work.
The main motives of this study were in two levels. In the first level, empagliflozin solubility in ScCO 2 was determined and in the second level, a new explicit solubility model was developed based on solid-liquid equilibrium criterion in combination with Wilson activity coefficient model for the activity coefficient calculation.

Experimental
Materials. Gaseous CO 2 (purity > 99.9%) was obtained from Fadak company, Kashan (Iran), empagliflozin (CAS Number: 864070-44-0, purity > 99%) was purchased from Amin Pharma company, and dimethyl sulfoxide (DMSO, CAS No. 67-68-5, purity > 99%) was provided from Sigma Aldrich company. Table 1 indicates all the information about the chemicals utilized in this work. Experiment details. The detailed discussion of the solubility apparatus and equilibrium cell has been presented in our earlier studies ( Fig. 2) 19,25,40,41 . However, a brief description about the apparatus is presented in this section. This method may be classified as an isobaric-isothermal method 42 . Each measurement was carried out with high precision and temperatures and pressures were controlled within ± 0.1 K and ± 0.1 MPa, respectively. For all measurement, 1 g of empagliflozin drug was used. As mentioned in our previous works, the equilibrium was observed within 60 min. After equilibrium, 600 µL saturated ScCO 2 sample was collected via 2-status 6-way port valve in a DMSO preloaded vial. After discharging 600 µL saturated ScCO 2 , the port valve was washed with 1 ml DMSO. Thus, the total saturation solution was 5 ml. Each measurement was repeated thrice and their average values were reported. Mole fraction is obtained as follows: where n solute is the moles number of the drug, and n CO 2 is the moles number of CO 2 in the sampling loop.
Further, the above quantities are given as: where C s is the drug concentration in saturated sample vial in g/L. The volume of the sampling loop and vial collection are V 1 (L) = 600 × 10 -6 m 3 and V s (L) = 5 × 10 -3 m 3 , respectively. The M s and M CO 2 are the molecular weight of drug and CO 2 , respectively. Solubility is also described as The relation between S and y 2 is

Existing and new models and their correlations
In this section, the details of various solubility models are presented along with a new explicit solubility model. 43 . It is one of the latest models for the solubility correlation. It is mathematically explained as where A 1 − C 1 are model constants. 44 . It is an empirical model and mathematically stated as:

Bartle et al., model (three parameters model)
where A 2 − C 2 are model constants. From parameter B 2 , one can estimate sublimation enthalpy using the relation, sub H = −B 2 R , in which R is universal gas constant. 45 . It is an empirical model and mathematically formulated as:

Bian et al., model (five parameters model)
Chrastil model (three parameters model) 46 . It is a semi-empirical model and mathematically stated as: where κ, A 4 and B 4 are model constants.
In terms of mole fraction, it is written as 47 :   48 . It is a mathematical model and mathematically formulated as Kumar-Johnstone model (three parameters model) 49 . It is a semi empirical model and mathematically described as: where A 6 − C 6 are model constants. 39 . It is one of the latest models. It is based on degree of freedom and mathematically stated as:

Mahesh-Garlapati model (three parameters model)
Mendez-Teja model (three parameters model) 50 . It is a semi-empirical model and mathematically explained as: where 40 . It is a mathematical model and stated as:

Sodefian et al., model (six parameters model)
where A 9 − F 9 are model constants. 47,51 . It is a semi-empirical model and mathematically explained as:

Reformulated Chrastil model (three parameters model)
where κ ′ , A 10 and B 10 are model constants. 52 . It is a degree of freedom model and mathematically stated as:
New model. According to solid-liquid phase equilibrium criteria, the fugacity of the solute in the solid phase and liquid phase is equal at equilibrium. The liquid phase is considered as an expanded liquid phase of ScCO 2 . At equilibrium, the solubility may be expressed as [53][54][55][56][57] where γ ∞ 2 is drug activity coefficient at infinitesimal dilution in ScCO 2 and f S 2 and f L 2 are fugacities of drug in the solid and ScCO 2 phases, respectively. The f S 2 /f L 2 ratio may be expressed as follows: www.nature.com/scientificreports/ where, C p is heat capacity difference of the drug in solid phase and that of SCCO 2 phase. The terms that include △Cp is much smaller than the term that has H m 2 58 , thus leaving △Cp term yields a much simpler expression for fugacity ratio as: Combining Eq. (19) with Eq. (17) give the expression for the solubility model (Eq. (20)).
In order to use Eq. (20), the appropriate model for γ ∞ 2 is essential. In this work, the required activity coefficient is obtained from Wilson activity coefficient model 56 at infinite dilution and it is given by the Eq. (21).
and V 2 are molar volumes of solvent and solute, respectively.
When ρ 1 = 1/V 1 , the final expression for the infinite dilution activity coefficient is obtained as: The quantities a 12 and a 21 are assumed to be functions of reduced solvent density 57 , and molar volume of the solute is assumed as a constant value. In this work, a 12 and a 21 are assumed to have the following form: Combining Eqs. (22), (23) and (24)  where P s i is the sublimation pressure of the pure solid at system temperature T, P is the system pressure,V s is the molar volume of the pure solid, R is the universal gas constant. The fugacity coefficient of the pure solute at saturation ( φ S i ) is usually taken to be unity. In this work, the fugacity coefficient of the solute in the supercritical phase φ  www.nature.com/scientificreports/ The main reason for considering RKEoS is that it has only two adjustable constants k ij and l ij . All the models (density-based, new and RKEoS models) are correlated with the following objective function 58 : The regression ability of a model is indicated in terms of an average absolute relative deviation percentage (AARD %).
For the regression, fminsearch (MATLAB 2019a ® ) algorithm has been used. Table 1 shows some physicochemical properties of the used materials. Empagliflozin solubility in ScCO 2 is reported at various temperatures (T = 308 to 338 K) and pressures (P = 12 to 27 MPa). Table 2 indicates the solubility data and ScCO 2 density. The reported ScCO 2 density is obtained from the NIST data base. Figure 3 shows the effect of pressure on various isotherms. The cross over region is observed at 16.5 MPa. From Fig. 3, below the cross over region, solubility decreases with increase in temperature, and on the other hand, above the cross over region, the solubility increases with increase in temperature. The EoS model requires critical properties which are computed with standard group contribution methods based on the chemical structure [62][63][64][65] . The summary of the critical properties computed are shown in Table 3. Figure 4 presents the self-consistency of the measured data with MT model.

Results and discussion
The density-based models considered in this work have different number of adjustable parameters. These parameters range from three to six numbers. The regression results of all the models are indicated in Tables 4  and 5. The correlating ability of the models is depicted in Figs. 5,6,7,8,9,10,11. From the results, it is clear that all the models are able to correlate the data reasonably well and maximum AARD% is observed to be 10.4%. It is believed that, more parameter models are able to correlate the data more accurately. Sodefian et al., model is able to correlate the data with AARD = 5.84% and Akaike's information criterion (AIC = − 637.59) (more relevant information is presented in the following section). Among density models, Bian et al., model (five parameters model) is able to correlate the data well and corresponding AARD% is 5.1%. Interestingly, Chrastil (three parameters model) and Reformulated Chrastil models (three parameters model) are also able to correlate the data quite well. Further, Chrastil and Reformulated Chrastil models are able to provide the total enthalpy. Whereas, Bartle et al., model parameters are able to provide sublimation enthalpy of the empagliflozin drug. From the magnitude difference between the total and sublimation enthalpies, a solvation enthalpy is calculated. These results are reported in Table 6.
A new explicit solubility model based on solid-liquid equilibrium criteria combined with Wilson activity coefficient model corresponding to infinitesimal dilution is derived.  Table 3 (temperature independent correlations). The optimization results of the new solubility and RKEoS models are indicated in Table 5.
In order to examine the ability of models for correlating the solubility data, AIC is applied [67][68][69][70] . When the data number is less than < 40, the corrected AIC (AIC c ) is used.

Conclusion
Solubilities of empagliflozin in ScCO 2 at temperatures (T = 308-338 K) and pressures (P = 12-27 MPa) were reported for the first time. The measured solubility in terms of mole faction ranged from 5.14 × 10 -6 to 25.9 × 10 -6 . The data was successfully correlated with several models, Bian et al., model (AARD = 5.1%) was observed to be the best model in correlating the solubility data. All the models are able correlate the data reasonable. However,

Data availability
The datasets generated and/or analysed during the current study are not publicly available due to confidential cases are available from the corresponding author on reasonable request.  Figure 11. Empagliflozin solubility vs. pressure. Solid lines are calculated solubilities with RKEoS + KM mixing rule.