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.

Introduction

Supercritical carbon dioxide (ScCO2) 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]. ScCO2 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, ScCO2 is the most desirable solvent[8,13-17]. 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 ScCO2 is readily available[18-30], however, the solubility of empagliflozin has not been reported, therefore in this work for the first time, its solubility in ScCO2 has been measured. This data may be used in the particle micronization process using ScCO2. The molecular formula of empagliflozin is C23H27 ClO7 and its molecular weight is 450.91. The chemical structure is shown in Fig. 1.

Figure 1

Empagliflozin chemical structure.

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 ScCO2. 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[33-35]. 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]. In fact, most of the literature models correlate the solubility of the solid solutes in ScCO2 quite well. A solid–gas equilibrium models need the critical properties and vapour pressure of the solute, while these properties are rarely available in literature, due to this, the group contribution methods are commonly used[38]. On the other hand, the solid–liquid equilibrium (SLE) criterion requires an appropriate model for activity coefficient calculation. A recent study reveals that SLE model in combination with Van Laar activity coefficient model can be a simple approach in model development, but this method resulted in an implicit expression in terms of mole fraction[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 ScCO2 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 CO2 (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.

Table 1

Some physicochemical properties of the used materials.

Compound	Formula	MW (g/mol)	Tm (K)	λmax (nm)	CAS number	Minimum purity by supplier
Empagliflozin	C23H27ClO7	450.9	426.1	276	864070-44-0	99%
Carbon dioxide	CO2	44.01			124-38-9	99.99%
DMSO	C2H6OS	78.13			67-68-5	99%

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 ScCO2 sample was collected via 2-status 6-way port valve in a DMSO preloaded vial. After discharging 600 µL saturated ScCO2, 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 is the moles number of the drug, and is the moles number of CO2 in the sampling loop.

Figure 2

Experimental setup for solubility measurement, E1—CO2 cylinder; E-2—Filter; E-3—Refrigerator unit; E-4—Air compressor; E-5—High pressure pump; E-6—Equilibrium cell; E-7—Magnetic stirrer; E-8—Needle valve; E-9—Back-pressure valve; E-10—Six-port, two position valve; E-11—Oven; E-12—Syringe; E13—Collection vial; E-14—Control panel.

Further, the above quantities are given as:where is the drug concentration in saturated sample vial in g/L. The volume of the sampling loop and vial collection are V1(L) = 600 10–6 m3 and Vs(L) = 5 10–3 m3, respectively. The and are the molecular weight of drug and CO2, respectively. Solubility is also described as

The relation between S and is

A UV–Visible spectrophotometer (Model UNICO-4802) and DMSO solvent were used for the measurement of empagliflozin solubility. The samples were analyzed at 276 nm.

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.

Existing models

Alwi–Garlapati model (three parameters model)[43]

It is one of the latest models for the solubility correlation. It is mathematically explained aswhere are model constants.

Bartle et al., model (three parameters model)[44]

It is an empirical model and mathematically stated as:where are model constants. From parameter , one can estimate sublimation enthalpy using the relation, , in which R is universal gas constant.

Bian et al., model (five parameters model)[45]

It is an empirical model and mathematically formulated as:where are model constants.

Chrastil model (three parameters model)[46]

It is a semi-empirical model and mathematically stated as:where are model constants.

In terms of mole fraction, it is written as[47]:

Garlapati–Madras model (five parameters model)[48]

It is a mathematical model and mathematically formulated aswhere are model constants.

Kumar–Johnstone model (three parameters model)[49]

It is a semi empirical model and mathematically described as:where are model constants.

Mahesh–Garlapati model (three parameters model)[39]

It is one of the latest models. It is based on degree of freedom and mathematically stated as:where are model constants.

Mendez–Teja model (three parameters model)[50]

It is a semi-empirical model and mathematically explained as:where are model constants.

Sodefian et al., model (six parameters model)[40]

It is a mathematical model and stated as:where are model constants.

Reformulated Chrastil model (three parameters model)[47,51]

It is a semi-empirical model and mathematically explained as:where are model constants.

Tippana–Garlapati model (six parameters model)[52]

It is a degree of freedom model and mathematically stated as:where are model constants.

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 ScCO2. At equilibrium, the solubility may be expressed as[53-57]where is drug activity coefficient at infinitesimal dilution in ScCO2 and and are fugacities of drug in the solid and ScCO2 phases, respectively. The ratio may be expressed as follows:where, is heat capacity difference of the drug in solid phase and that of SCCO2 phase. The terms that include △Cp is much smaller than the term that has [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 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).where and , and are molar volumes of solvent and solute, respectively.

When , the final expression for the infinite dilution activity coefficient is obtained as:

The quantities and 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, and are assumed to have the following form:

Combining Eqs. (22), (23) and (24) with Eq. (20), give the following new explicit solubility model:

Equation (25) has four temperature independent adjustable variables namely ,, and .

Equation of state (EoS) model

The solubility of drug i (solute) in ScCO2 (solvent) is expressed as[59-61]:where is the sublimation pressure of the pure solid at system temperature T, P is the system pressure, is the molar volume of the pure solid, R is the universal gas constant. The fugacity coefficient of the pure solute at saturation () is usually taken to be unity. In this work, the fugacity coefficient of the solute in the supercritical phase is calculated using EoS along with KMmr[57]. The expression used for calculation of is obtained from the following basic thermodynamic relation[60]:

The expression for is

and the associated mixing rules are:

The main reason for considering RKEoS is that it has only two adjustable constants and .

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.

Results and discussion

Table 1 shows some physicochemical properties of the used materials. Empagliflozin solubility in ScCO2 is reported at various temperatures (T = 308 to 338 K) and pressures (P = 12 to 27 MPa). Table 2 indicates the solubility data and ScCO2 density. The reported ScCO2 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[62-65]. 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.

Table 2

Solubility of crystalline empagliflozin in ScCO2 at various temperatures and pressures.

Temperature (K)a	Pressure (MPa)a	Density of ScCO2 (kg/m3)[71]	y2 × 104 (mole fraction)	Experimental standard deviation, S(ȳ) × (104)	S (equilibrium solubility) (g/L)	Expanded uncertainty of mole fraction (104U)
308	12	769	0.0814	0.0021	0.0643	0.0055
	15	817	0.1266	0.0042	0.1060	0.0098
	18	849	0.1327	0.0010	0.1156	0.0062
	21	875	0.1411	0.0051	0.1265	0.0118
	24	896	0.1501	0.0063	0.1378	0.0137
	27	914	0.1806	0.0071	0.1692	0.0161
318	12	661	0.0706	0.0023	0.0479	0.0052
	15	744	0.1182	0.0031	0.0901	0.0081
	18	791	0.1515	0.0032	0.1228	0.0091
	21	824	0.1601	0.0041	0.1353	0.0107
	24	851	0.2040	0.0064	0.1812	0.0151
	27	872	0.2079	0.0093	0.1858	0.0202
328	12	509	0.0611	0.0031	0.0319	0.0066
	15	656	0.1044	0.0023	0.0702	0.0062
	18	725	0.1620	0.0032	0.1203	0.0094
	21	769	0.1860	0.0042	0.1467	0.0115
	24	802	0.2248	0.0091	0.1849	0.0206
	27	829	0.2260	0.0021	0.1920	0.0107
338	12	388	0.0514	0.0023	0.0204	0.0047
	15	557	0.0928	0.0011	0.0530	0.0047
	18	652	0.2002	0.0101	0.1338	0.0219
	21	710	0.2266	0.0112	0.1650	0.0242
	24	751	0.2637	0.0103	0.2030	0.0231
	27	783	0.2590	0.0091	0.2079	0.0213

The experimental standard deviation was obtained by . Expanded uncertainty (U) = k*u and the relative combined standard uncertainty

aStandard uncertainty u are u(T) =  ± 0.1 K; u(p) =  ± 0.1 MPa. The value of the coverage factor k = 2 was chosen on the basis of the level of confidence of approximately 95 percent.

Figure 3

Empagliflozin solubility in ScCO2 vs. pressure.

Table 3

Critical and physical properties of empagliflozin and CO2.

Substance	Tc (K)	Pc(Pa)	ω	Vs × 10–6 (m3/mol)	T(K)
					Psub (Pa)e
					308	318	328	338
Empagliflozin	870.367a	18.7565b	0.479c	184.397d	0.0034	0.0089	0.022	0.0508
CO2	304.18	73.8	0.225

Tc: critical temperature; Pc: critical pressure; : acentric factor; Vs: solid molar volume; T: temperature.

aEstimated by Fedors method.

bEstimated by the Joback and Reed method.

cEstimated by Lee-Kesler vapour pressure relations (the required normal boiling temperature (at 1.0 atm), Tb is estimated with Klincewicz relation, Tc = 50.2–0.16 M + 1.41 Tb, where M is molecular weight).

dEstimated by Immirzi, A., Perini, B method.

eEstimated by Lee-Kesler vapour method.

Figure 4

Self-consistency plot based on MT model.

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.

Table 4

Correlation constants for the exiting empirical models.

Name of the empirical model	Correlation parameters	AARD%	R2	R2adj
Alwi–Garlapati model	A_1 = − 1.8293; B_1 = − 14.218; C_1 = 2.8519	6.58	0.941	0.932
Bartle et al., model	A_2 = 12.195; B_2 = − 5972.3; C_2 = 7.7336 × 10–3	10.4	0.922	0.910
Bian et al., model	A_3 = − 0.062205; B_3 = − 5.7629 × 10–4; C_3 = − 6230.8; D_3 = 2.9473; E_3 = 4.5582	5.1	0.951	0.938
Chrastil model	κ = 3.9083; A_4 = − 18.97; B_4 = − 3674.3	9.21	0.943	0.934
Garlapati–Madras model	A_5 = − 750.2; B_5 = 852.82; C_5 = 1.0855; D_5 = − 7397.7; E_5 = − 11.163	7.09	0.946	0.930
Kumar–Jonstone model	A_6 = − 14.274; B_6 = − 0.53652; C_6 = 2.1216	27.3	0.902	0.892
Mahesh_Garlapati model	A_7 = − 14.266; B_7 = − 0.52714; C_7 = 2.0972	8.14	0.931	0.921
Mendez–Teja model	A_8 = − 7775.4; B_8 = 2.3557; C_8 = 12.694	9.95	0.924	0.912
Sodefian et al., model	A_9 = − 23.94; B_9 = 1.6043 × 10–3; C_9 = 2.4939; D_9 = 2.6639 × 10–4; E_9 = − 9.5238 × 10–3; F_9 = − 1.037 × 103	5.84	0.956	0.940
Reformulated Chrastil model	κ^′ = 3.8748; A_10 = − 33.58; B_10 = − 2705.8	9.14	0.943	0.935
Tippana–Garlapati model	A_11 = − 6.4027 × 10–5; B_11 = − 1.1813 × 10–5; C_11 = 2.0367 × 10–5; D_11 = 4.4544 × 10–5; E_11 = 3.5989 × 10–5; F_11 = − 2.4670 × 10–5	6.63	0.927	0.924

Table 5

Calculated result for the new model and RKEoS + Kwak-Mansoori mixing rule model.

Model	Correlation parameters	AARD%	R2	R2adj
New model	A = 36,634; B = − 0.096039; C = − 9673.6; D = − 0.16480	7.22	0.949	0.941
RKEoS + Kwak Mansoori mixing rule model	k_ij = 0.32061; l_ij = 0.1949	8.07	0.951	0.946

Figure 5

Empagliflozin solubility vs. ScCO2 density. Solid lines and broken lines are calculated solubilities with Chrastil and Reformulated Chrastil models, respectively.

Figure 6

Empagliflozin solubility vs. ScCO2 density. Solid lines and broken lines are calculated solubilities with KJ and Bartle et al., models, respectively.

Figure 7

Empagliflozin solubility vs. ScCO2 density. Solid lines and broken lines are calculated solubilities with Alwi–Garlapati and Mahesh–Garlapati models, respectively.

Figure 8

Empagliflozin solubility vs. ScCO2 density. Solid lines and broken lines are calculated solubilities with Bian et al., and Garlapati–Madras models, respectively.

Figure 9

Empagliflozin solubility vs. ScCO2 density. Solid lines and broken lines are calculated solubilities with Tippana–Garlapati and Sodeifian et al., models, respectively.

Figure 10

Empagliflozin solubility vs. ScCO2 density. Solid lines are calculated solubilities with new model.

Figure 11

Empagliflozin solubility vs. pressure. Solid lines are calculated solubilities with RKEoS + KM mixing rule.

Table 6

Computed thermodynamic properties.

Model	Thermodynamic property
	Total entalpy, ΔHtotal (kJ/mol)	Enthalpy of sublimation, ΔHsub (kJ/mol)	Enthalpy of solvation,ΔH_sol (kJ/mol)
Chrastil model	30.548a		− 19.105d
Reformulated Chrastil model	22.496b		− 27.157e
Bartle et al., model		49.653c (approximate value)

aObtained with Chrastil model

bObtained with Reformulated Chrastil model

cObtained with Bartle et al.

dObtained as a result of difference between the ΔHsubc and ΔHtotala.

eObtained as a result between the ΔHsubc and ΔHtotalb.

A new explicit solubility model based on solid–liquid equilibrium criteria combined with Wilson activity coefficient model corresponding to infinitesimal dilution is derived. The new model has four parameters , , and . While regression, new model parameters are treated as temperature independent and solid molar volume is kept constant. The new model requires melting point, melting enthalpy and molar volume of empagliflozin drug, and these values are obtained from literature and group contribution methods. From literature[31], the melting point of empagliflozin drug (426.1 K), molar volume (3.2699 × 10–4 m3/mol) and melting enthalpy (60.238 kJ/mol) are calculated based on literature, Immirzi and Perini[63] and Jain et al., methods[66], respectively. The new model makes use of objective function given in Eq. (33). Similarly, RKEoS along KMmr correlations are established with the help of critical properties given in 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[67-70]. When the data number is less than < 40, the corrected AIC (AICc) is used.where AIC, N, and SSE are , the number of observations, the number of adjustable parameters of the model and the error sum of squares, respectively. According to AICc criterion, the best model has the least AICc value. Table 7 shows AICc values for various models considered in this study. In terms of AICc, all the models are able to correlate the data closely. However, Reformulated Chrastil model has AICc value (− 637.02), hence it is treated as the best model and at the same time, Tippana–Garlapati model has the highest AICc value (− 621.69), therefore, it is considered as a poorly correlating model. Three parameters models namely Chrastil, Alwi–Garlapati and Mendez–Teja models have AICc values − 636.95, − 635.3 and − 635.4, respectively. The new model which has four parameters, indicating comparable performance with the best model (AICc value of − 637.24).

Table 7

Statistical quantities (SSE, RMSE, AIC and AICc) of various models.

Model	SSE.1011	RMSE.106	N	Q	AIC	AICc
Existing density models
Alwi–Garlapati model	5.673	1.537	3	24	− 636.5	− 635.30
Bartle et al., model	7.391	1.755	3	24	− 630.15	− 628.95
Bian et al., model	4.338	1.345	5	24	− 638.94	− 635.6
Chrastil model	5.297	1.486	3	24	− 638.15	− 636.95
Garlapati–Madras model	4.828	1.418	5	24	− 636.37	− 633.04
Kumar–Jonstone model	6.537	1.650	3	24	− 633.1	− 631.9
Mahesh_Garlapati model	6.337	1.625	3	24	− 633.84	− 632.64
Mendez–Teja model	5.650	7.51	3	24	− 636.60	− 635.4
Sodefian et al., model	4.222	1.326	6	24	− 637.59	− 632.65
Reformulated Chrastil model	5.280	1.483	3	24	− 638.22	− 637.02
Tippana–Garlapati model	6.659	1.666	6	24	− 627.0	− 621.69
New model
New solid–liquid equilibrium model	4.635	1.389	4	24	− 639.35	− 637.24
EoS model
RKEoS model + Kwak-Mansoori mixing rule	6.201	1.607	2	24	− 636.36	− 635.79

Conclusion

Solubilities of empagliflozin in ScCO2 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, the correlating ability in ascending order of various models in terms of the lowest AICc values is as follows: Bian et al., Reformulated Chrastil, Chrastil, new solid–liquid equilibrium, Mendez–Teja, RKEoS + KMmr, Alwi–Garlapati, Sodefian et al., Mahesh–Garlapati, Bartle et al., Tippana–Garlapati models. The new model proposed in this work may be useful for correlating solids solubility in any SCF.