Solubility of palbociclib in supercritical carbon dioxide from experimental measurement and Peng–Robinson equation of state

Abstract

Palbociclib is a poorly water-soluble medicine which acts against metastatic breast cancer cells. Among various techniques to improve the solubility of this medicine, applying supercritical technologies to produce micro- and nano-sized particles is a possible option. For this purpose, extraction of solubility data is required. In this research, the solubility of palbociclib in supercritical carbon dioxide (ScCO₂) at different equilibrium conditions was measured at temperatures between 308 and 338 K and pressures within 12–27 MPa, for the first time. The minimum and maximum solubility data were found to be 8.1 × 10^–7 (at 338 K and 12 MPa) and 2.03 × 10^–5 (at 338 K and 27 MPa), respectively. Thereafter, two sets of models, including ten semi-empirical equations and three Peng–Robinson (PR) based integrated models were used to correlate the experimental solubility data. Bian’s model and PR equation of state using van der Waals mixing rules (PR + vdW) showed better accuracy among the examined semi-empirical and integrated models, respectively. Furthermore, the self-consistency of the obtained data was confirmed using two distinct semi-empirical models. At last, the total and vaporization enthalpies of palbociclib solubility in ScCO₂ were calculated from correlation results of semi-empirical equations and estimated to be 40.41 and 52.67 kJ/mol, respectively.

Introduction

A review of the cancer statistics shows that breast cancer is the most commonly diagnosed among women worldwide, with an estimated 2.3 million cases and more than a quota of 11% of all cancer cases, in 2020¹. Considering this fact, improved access to treatment is one of the main solutions against rising incidence of cancer related deaths. Among various types of cancer treatment, targeted drug therapy is one of the most common biological techniques, i.e., the small-molecule drugs such as cancer growth inhibitor can help to find and target specific parts of cancer cells. Because of low molecular weights, small molecules can penetrate the cell membrane easily to affect protein molecules, such as cyclin-dependent kinase 4 and 6 (or CDK 4 and CDK 6, respectively). Although these proteins play the primary roles in cell cycle progression, their increased activity causes loss of cell cycle control and fast cellular dividing. By blocking these proteins and their major enzymatic actions, the above mentioned drugs aim to put a brake on the spreading of cancer cells². In this regard, palbociclib is a highly selective inhibitor against the proliferation of breast cancer cells, capable of targeting the proteins that stimulate cellular growth and multiplying.

Based on the biopharmaceutics classification system (BCS), the solubility and permeability of drugs are the major factors required for the description of oral drug absorption³. With respect to this fact, improving drug solubility in water, especially for lipophilic drugs, is one of the critical challenges in the pharmaceutical industry. In this regard, particle size manipulation can be helpful in meeting the challenge. For this purpose, the generation of new solubility data on palbociclib, especially in supercritical carbon dioxide (ScCO₂), can clear the way for micro and nanoparticle production via relevant processes such as rapid expansion of supercritical solution (RESS)⁴ and supercritical anti-solvent precipitation (SAS)⁵. The latter is based on fast injection of liquid solution of organic/inorganic solvent and solid solute into the supercritical fluid (SCF), while the former is carried out by depressurization the mixture of supercritical fluid and solid solute through a heated nozzle at high speed. Moreover, the knowledge of solid solubility values as well as its behavior versus temperature and pressure is of high value for optimizing not only particle size but also yield and morphology of the product⁶.

When talking about CO₂, topics such as greenhouse gases, global warming and the relevant environmental considerations maybe occur to someone. However, the magic effects of its supercritical state on the development of numerous fields are undeniable, including extraction of organic compounds^7–11, drug solubility measurement^12–17, particle production^18–20, power generation²¹, polymer synthesis and processing^22,23, impregnation process²⁴, tissue engineering²⁵ and food industry²⁶. Moreover, low toxicity, safety, low environmental impacts and accessible critical conditions make CO₂ a special and the most frequently used supercritical solvent, both on the laboratory scale and in the industrial level processes.

To obtain the solubility of solid substances in ScCO₂ within a wide range of temperature and pressure, substantial numbers of correlative and predictive tools have been proposed during recent decades²⁷, which are primarily classified into three categories including equations of state^28–30, empirical and semi-empirical models^12,31 and artificial intelligence models^32–34. This study measured palbociclib solubility in ScCO₂ and correlated the obtained data set by ten semi-empirical models and three integrated Peng–Robinson (PR) equation of state (EoS) based models along with evaluating the accuracy of each approach.

Experiment

Materials

Palbociclib in the form of powder with a purity exceeding 99.0% was purchased from Parsian Pharmaceutical Company (Tehran, Iran). Further information about the molecular structure along with thermodynamic properties of palbociclib are presented in Table 1. CO₂ with a purity of 99% was supplied by Fadak Company (Kashan, Iran). Dimethyl sulfoxide (DMSO) produced by Merck (Darmstadt, Germany) with a minimum purity of 99.9% was used as trapping solvent after the solubility process.

Table 1.

Thermodynamic properties and molecular structure of palbociclib.

	Palbociclib
Molecular Structure
Formula	C24H29N7O2
CAS number	571190-30-2
λmax (nm)	265
Tm (K)68	545.19
Δ $\overline{H}$m (kJ∙mol−1)68	23.7
Tc (K)a	1254.31
Pc (MPa)a	1.6069
$\omega$ (–)a	0.4660
${\overline{V}}^S$(cm3 mol−1)b	303.85

^aThe values of T_c and P_c were estimated by PR + COSMOSAC EoS^70,71 and then rescaled by a factor of 0.88. The rescale factor is determined from regression of experimental solubility data together with k_ij^78,79. The Lee-Kesler equation⁷² was used to estimate the acentric factor.

^bThe molecular cavity volume in the COSMO solvation calculation was used as the solid molar volume.

Experimental apparatus and procedure

The apparatus utilized in this work is depicted in Fig. 1 in which CO₂ storage cylinder, filter (Hylok, 6000 psi), refrigerator unit, air compressor (Ronix, RC 5010), high pressure pump (Haskel pump, MSHP-110), oven (Froilabo AE-60, France) and solubility column composed the major parts of the supercritical unit. 500 mg powder of the sample along with 2 mm-diameter glass beads were loaded to the solubility column with a capacity of 70 ml. Then the column was placed in the oven. By starting the process, CO₂ flowed through a molecular sieve filter with pore sizes of 1 μm from the storage cylinder into the refrigerator, where CO₂ was liquefied by a reduction of temperature to 258 K. Then, liquid CO₂ was entered into the reciprocating pump to reach the desired pressure of the supercritical fluid state. Downstream of the pump, a pressure gauge (WIKA, Germany) with an accuracy of ± 1 bar was installed for measuring and visual observing the supercritical CO₂ pressure. Thereafter, by opening the needle valve of column input, ScCO₂ was passed and brought into direct contact with the powder sample. To keep and control the desired temperature of the column, the oven was equipped with a platinum resistance thermometer (Pt 100 temperature sensor) along with a temperature monitoring system with an accuracy of 0.1 K. In this study, the time required for attaining equilibrium was set to 60 min. After equilibrium, 600 μl of saturated ScCO₂ sample was discharged via a two-state four-port valve into a vial preloaded with a given volume of DMSO.

Figure 1.

Experimental setup for supercritical solubility measurement.

To determine the concentration of palbociclib in each solution sample corresponding to a particular pressure and temperature, a UV–Vis spectrophotometer (Cintra 101, GBC Scientific Equipment Ltd.) was employed along with a set of standard solutions. An analytical balance (LS-220 A SCS, Precisa, Swiss) with an accuracy of 0.0001 g was applied for mass measurement and standard solution preparation. The calibration curve (with a regression coefficient of about 99%) was established by analyzing concentrations of standard solutions with the mentioned UV–Vis spectrophotometer at the wavelength of maximum absorbance (λmax) of 265 nm. Subsequently, the equilibrium solubility (S) in terms of grams per liter and the equilibrium mole fraction (y) were obtained by the following relationships:

$$S_2=\frac{C_2\times V_s}{V_l}$$ 1

$$y_2=\frac{n_2}{n_1+n_2}$$ 2

$$n_1=\frac{{\rho }_1\times V_l}{M_1}$$ 3

$$n_2=\frac{C_2\times V_s}{M_2}$$ 4

where n and M represent the moles number and molecular mass of components in the sampling loop, respectively. In this binary system, subscripts 1 and 2 stand for ScCO₂ and palbociclib as solvent and solute, respectively. C₂ as the concentration of palbociclib (in terms of grams per liter) solved in each vial was obtained from the calibration curve. $V_s$ and $V_l$ refer to the volume, in terms of liter, of the solution and sampling loop, respectively. ${\rho }_1$ is ScCO₂ density obtained from the NIST database at the corresponding conditions of temperature and pressure.

Reliability of the experimental apparatus and procedure was previously examined by measuring the solubility of two well-known substances, namely Alpha-tocopherol and naphthalene at different operational conditions and comparing the obtained results with the data reported in the literature³⁵. In recent years, Sodeifian and coworkers utilized the apparatus frequently for various research purposes related to supercritical fluids processes, so more detailed experimental procedures are available in their published papers^23,36–39.

Results and discussion

Experimental palbociclib solubility data

According to what was mentioned above, solubility values of palbociclib in ScCO₂ were obtained at planned pressures and temperatures, containing a set of 72 experimental data (24 data as primaries and 48 data as replications). These data, including equilibrium solubility (S) and equilibrium mole fraction (y), are presented in Table 2. It should be noted that each reported solubility datum is the average of three distinct experimental runs. As can be found in Table 2 and the corresponding data points presented in different shapes and colors on both Figs. 2a and 4, solubility values increase by increasing pressure along each isotherm. The reason is often attributed to the increment of CO₂ solvating power due to increasing density under higher pressures. Consequently, more intensive molecular interaction between ScCO₂ and solute particles is followed by more solubility of palbociclib. However, the influence of temperature is dualistic. Although the solubility values are adversely affected by temperature elevation at and under the pressure of 18 MPa, a positive relationship between solubility and temperature is observed at and above the pressure of 21 MPa. Therefore, it is concluded that there is a crossover point between the range of 18 and 21 MPa for palbociclib solubility in ScCO₂ in which the isotherm curves intersect each other. As demonstrated by Figs. 2a and 4, higher compactness of experimental values at the right endpoint of this range leads to conclusion that the pressure crossover point must be in the vicinity of 21 MPa. Due to playing two simultaneous different roles at pressures under and above the crossover range, the effects of temperature rising on the solubility values are not as simple as that of pressure to justify. In this regard, temperature elevation makes ScCO₂ density less, resulting in decreasing the solvating power. On the other side, solute’s vapor pressure followed by solubility would be raised, affected by temperature elevation. Depending upon which factor prevails, the type of relationship between solubility and temperature is predictable. In this study, the dominant factor under and above the crossover range is the ScCO₂ density and palbociclib’s vapor pressure, respectively. These results are similar to those reported in other investigations on measuring the solubility of lansoprazole⁴⁰, salsalate⁴¹, chloroquine⁴², glibenclamide⁴³ and haloperidol⁴⁴.

Table 2.

Experimental data of palbociclib solubility in ScCO₂.

T (K)	P (MPa)	${\rho }_{{\mathrm{CO}}_2}$a (kg∙m−3)	y × 105 (mole fraction)	Std (y) × 105 (mole fraction)b	U($\overline{y}$) × 105 (mole fraction)c	Solubility (g∙L−1)
308	12.0	769	0.391	0.012	0.027	0.0305
	15.0	817	0.501	0.024	0.053	0.0415
	18.0	849	0.552	0.021	0.047	0.0475
	21.0	875	0.627	0.011	0.034	0.0557
	24.0	896	0.697	0.012	0.036	0.0635
	27.0	914	0.914	0.021	0.056	0.0849
318	12.0	661	0.221	0.010	0.022	0.148
	15.0	744	0.349	0.015	0.033	0.264
	18.0	791	0.419	0.012	0.030	0.337
	21.0	824	0.738	0.011	0.038	0.618
	24.0	851	0.860	0.013	0.043	0.743
	27.0	872	1.242	0.041	0.096	0.1101
328	12.0	509	0.110	0.004	0.009	0.0057
	15.0	656	0.210	0.004	0.012	0.0140
	18.0	725	0.281	0.013	0.030	0.0206
	21.0	769	0.840	0.019	0.053	0.0656
	24.0	802	0.972	0.021	0.059	0.0792
	27.0	829	1.468	0.032	0.091	0.1235
338	12.0	388	0.081	0.003	0.007	0.0031
	15.0	557	0.137	0.002	0.007	0.0078
	18.0	652	0.211	0.003	0.011	0.0139
	21.0	710	0.981	0.041	0.093	0.0708
	24.0	751	1.197	0.011	0.057	0.0914
	27.0	783	2.027	0.091	0.202	0.1613

*Standard uncertainty values (u) for temperature and pressure are u(T) = 0.1 K and u(P) = 0.1 MPa.

^aCO₂ density was obtained from national institute of science and technology (NIST) chemistry webbook (http://webbook.nist.gov/chemistry/).

^bExperimental standard deviation was obtained by $Std\left(y_i\right)=\sqrt{{\sum }_j(y_j-\overline{y})/n-1}$.

^cExpanded uncertainty was determined from $U(\overline{y})=k·u_{\mathrm{comb}}$ with the coverage factor k = 2 (the level of confidence of approximately 95%) and the relative combined standard uncertainty $u_{\mathit{comb}}/\overline{y}=\sqrt{\sum _i{\left(P_i·u_i\left(x_i\right)/x_i\right)}^2}$.

Figure 2.

Palbociclib solubility in ScCO₂ versus (a) pressure and (b) density from experimentation and three semi-empirical models.

Figure 4.

Palbociclib solubility in ScCO₂ from experimentation and PR EoS based thermodynamic models.

Solubility correlation with semi-empirical models

The experimental palbociclib solubility in ScCO₂ was correlated by ten semi-empirical models. Table 3 summarizes the equations and abbreviations of these ten semi-empirical models. The objective function of the average absolute relative deviation in percentage (AARD%) was used for N palbociclib solubility data from experiments (y^exp) and semi-empirical models (y^cal) to obtain adjustable parameter values of these semi-empirical models:

$$\mathrm{AARD}\%=\frac{1}{N}\sum _{i=1}^N\frac{\left|y_i^{\exp },-,y_i^{\mathrm{cal}}\right|}{y_i^{\exp }}\times 100\%$$ 5

Table 3.

Semi-empirical models.

Model	Equations
Chrastil48	$\ln S_i=a_0\ln {\rho }_{{\mathrm{CO}}_2}+\frac{a_1}{T}+a_2$
MST47	$T\ln \left(y_i,\mathrm{P}\right)=a_0+a_1{\rho }_{{\mathrm{CO}}_2}+a_{2}T$
K-J73	$\ln y_i=a_0+a_1{\rho }_{{\mathrm{CO}}_2}+\frac{a_2}{T}$
Bartle49	$\ln \left(y_i,P,/,P_{\mathrm{ref}}\right)=a_0+a_1({\rho }_{{\mathrm{CO}}_2}-{\rho }_{\mathrm{ref}})+\frac{a_2}{T}$
Bian74	$\ln y_i=a_0+\frac{a_1}{T}+\frac{a_2\rho }{T}+(a_3+a_4\rho )\ln \rho$
Garlapati75	$\ln y_i=a_0+(a_1+a_2\rho )\ln \rho +\frac{a_3}{T}+a_4\ln (\rho T)$
Keshmiri76	$\ln y_i=a_0+\frac{a_1}{T}+a_2{P}^2+(a_3+\frac{a_4}{T})\ln \rho$
Khansary77	$\ln y_i=\frac{a_0}{T}+a_{1}P+\frac{a_2{P}^2}{T}+\left(a_3,+,a_4,P\right)\ln \rho$
Sodeifian12	$\ln y_i=a_0+a_1\frac{P^2}{T}+a_2\ln \left(\rho ,T\right)+a_3\left(\rho ,\ln ,\rho \right)+a_{4}P\ln T+a_5\frac{\ln \rho }{T}$
Belghait31	$\ln y_i=a_0+a_1\rho +a_2{\rho }^2+a_3\rho T+a_{4}T+a_5{T}^2+a_6\ln \rho +\frac{a_7}{T}$

*S_i: solubility in kg∙m⁻³, y_i: solubility in mole fraction, T: temperature in K, ρ: ScCO₂ density in kg∙m⁻³, P: pressure in bar, a₀ ~ a₇: adjustable model parameters, P_ref: reference pressure (= 1 bar), ρ_ref: reference density (= 700 kg∙m⁻³).

Table 4 lists the optimal adjustable parameter values of each semi-empirical model along with the AARD% from these models using the optimized parameters. Bian’s and Belghait’s model had the highest accuracy with the lowest AARD% of 15.06% and 16.21%, respectively, while the other models gave similar AARD% ranging from 23.10 to 25.94%. Figure 2 compares the correlation results from Bian’s, Belghait’s and MST models and experimental solubility. As can be seen, the correlation results are generally consistent with the experimental data. Both Bian’s and Belghait’s models can provide lower overall deviations and showed a wide pressure range (16 ~ 27 MPa) for the crossover region. Although the MST model gave a slightly higher deviation, its results showed a narrow pressure range (around 21 MPa) for the crossover region, which was consistent with the experimental observation. Such a difference on describing the crossover region may be attributed to the function form and considered independent variables of the semi-emiprical model. A wide range of the crossover region makes Bian’s and Belghait’s models better for describing palbociclib solubility behavior at lower pressures and giving lower overall deviations.

Table 4.

Parameter values in semi-empirical models.

Model	Parameters								AARD%
	$a_0$	$a_1$	$a_2$	$a_3$	$a_4$	$a_5$	$a_6$	$a_7$
Chrastil	7.5314	−4860.461	−38.0736	–	–	–	–	–	24.51%
MST	−11,783.097	4.4410	19.0791	–	–	–	–	–	24.20%
K–J	0.4908	0.0108	−6767.2317	–	–	–	–	–	23.73%
Bartle	11.9754	0.0117	−6335.4073	–	–	–	–	–	24.20%
Bian	−20.3979	1.42 $\times$ 104	−24.0799	−6.0178	0.01206	–	–	–	15.60%
Garlapati	−242.7863	−41.7235	0.0017	7231.7370	38.3561	–	–	–	23.10%
Keshmiri	9.8274	1.79 $\times$ 105	2.5895	−1.9618	2256.6849	–	–	–	25.94%
Khansary	−7756.8036	−0.3088	0.0039	2.0840	0.0445	–	–	–	24.07%
Sodeifian	−37.9028	0.0078	1.9762	8.53 $\times$ 10–5	−3.21 $\times$ 10–4	1.9738	–	–	23.71%
Belghait	24.5312	−0.0835	1.61 $\times$ 10–5	$2.145\times$ 10–4	−0.0847	−4.90 $\times$ 10–5	−0.4875	137.3061	16.21%

Due to several reasons such as difficulties related to the measurement of solubility values at small quantities, the reliability of the reported data to be used for process design, simulation and analysis should be confirmed dependently. For this purpose, thermodynamic consistency tests are of common methods to evaluate inherent inaccuracy of experimental data^45,46. In this study, The simple and standard consistency tests using two semi-empirical models, i.e., Chrastil’s and MST models⁴⁷ were conducted to confirm that the measured palbociclib solubility data were reliable. Figure 3 illustrates the consistency test results from the above mentioned models. As shown, the data points under various temperature conditions are approximately collapsed into a straight line, indicating that these data are self-consistent.

Figure 3.

Self-consistency test for experimental data of palbociclib solubility in ScCO₂ using (a) Chrastil and (b) MST models.

Generally, the dissolution of a solute molecule in ScCO₂ is considered as a two-step process: vaporizing solute molecules from the solid state and dissolving solute molecules into the ScCO₂ phase. The total and vaporization enthalpies of this two-step dissolution process can be estimated from the product of the semi-empirical model parameters and ideal gas constant R as follows^48,49:

$${\mathrm{\Delta }H}_{\mathrm{total}}=-a_1^{\mathrm{Chrastil}}·R$$ 6

$${\mathrm{\Delta }H}_{\mathrm{vap}}=-a_2^{\mathrm{Bartle}}·R$$ 7

where $a_1^{\mathrm{Chrastil}}$ and $a_2^{\mathrm{Bartle}}$ are the regressed temperature coefficients of Chrastil’s and Bartle’s models, respectively. Once the total and vaporization enthalpies were obtained, the solvation enthalpy was calculated from

$${{\mathrm{\Delta }H}_{\mathrm{sol}}=\mathrm{\Delta }H}_{\mathrm{total}}-{\mathrm{\Delta }H}_{\mathrm{vap}}$$ 8

The total, vaporization and solvation enthalpies of palbociclib were determined to be 40.41 kJ/mole, 52.67 kJ/mole and −12.26 kJ/mole, respectively. These values can be meaningful in understanding the type and amount of energy needed for solid particles to dissolve in ScCO₂. In this regard, a large value of total enthalpy may indicate that solute particles require more energy to dissolve in ScCO₂ and release more heat when recrystallizing from ScCO₂. Similarly, high value of vaporization enthalpy would show stronger intermolecular forces of solute against ScCO₂ particles. However, these values are not measured experimentally but are determined based on assumptions made for developing the utilized semi-empirical models.

Solubility correlation and prediction using cubic equation of state

Despite their simplicity, Peng–Robinson (PR)⁵⁰ and Soave–Redlich–Kwong (SRK)⁵¹ equations of state are still of interest and have been shown to well correlate phase equilibrium data under high-pressure conditions for CO₂ containing systems such as solid solute solubility in ScCO₂^52,53. These equations of state can apply to predict the phase equilibrium and solubility by combining them with a predictive liquid model^29,54–56, such as UNIQUAC Functional-group Activity Coefficients (UNIFAC)⁵⁷ or Conductor-like Screening Model-Segment Activity Coefficient (COSMOSAC)^58,59. In this study, two correlative approaches based on PR EoS were used to model the palbociclib solubility data:

1. PR EoS using van der Waals mixing rules (PR + vdW),

2. PR EoS integrated with Wong-Sandler mixing rule^60,61 and Wilson model⁶² (PR + WS + Wilson).

Besides, another predictive model based on PR EoS, Huron-Vidal mixing rule⁶³ and COSMOSAC model (PR + HV + COSMOSAC), is applied to estimate the solubility of palbociclib in ScCO₂. The molecular surface screening charges of palbociclib, key information for COSMOSAC, was obtained from the quantum mechanical and COSMO solvation calculations with the “b3lyp/6-31G(d.p)-cosmo” method⁶⁴. Table 5 lists the equations of the above mentioned PR EoS based approaches.

Table 5.

Investigated PR EoS based thermodynamic models.

PR EoS:	$P=\frac{\mathit{RT}}{\overline{V}-b(x)}-\frac{a(T,x)}{\overline{V}\left[\overline{V},+,b,(x)\right]+b\left[\overline{V},-,b,(x)\right]}$
PR + vdWa	$a={\sum }_i{\sum }_j{x_{i}x}_j\sqrt{a_i{a}_j}(1-k_{\mathit{ij}})$ $b={\sum }_i{x}_i{b}_i$
PR + WS + Wilsonb	$\frac{a}{b}={\sum }_i{x}_i\left(\frac{a_i}{b_i}\right)+\frac{{\overline{G}}^E}{C_{\mathrm{WS}}}$ $b=\frac{{\sum }_i{\sum }_j{x_{i}x}_j\left(\frac{b_i+b_j}{2},-,\frac{\sqrt{a_i{a}_j}}{\mathit{RT}}\right)}{1-{\sum }_i{x}_i\left(\frac{a_i}{b_{i}RT}\right)-\frac{{\overline{G}}^E}{C_{\mathrm{WS}}RT}}$
PR + HV + COSMOSACc	$\frac{a}{b}={\sum }_i{x}_i\left(\frac{a_i}{b_i}\right)+\frac{{\overline{G}}^E}{C_{\mathrm{HV}}}$ $b={\sum }_i{x}_i{b}_i$

*a_i and b_i are determined from critical temperature, critical pressure, and acentric factor of component i as described in the literature⁵⁰.

^ak_ij is an adjustable binary interaction parameter.

^b $C_{\mathrm{WS}}=\ln \left(\sqrt{2},-,1\right)/\sqrt{2}$^60,61. Excess Gibbs energy is determined from the Wilson model⁶²: ${\overline{G}}^E=-RT{\sum }_i{x}_i\ln \left({\sum }_j,x_j,{\mathrm{\Lambda }}_{\mathit{ij}}\right)$ with ${\mathrm{\Lambda }}_{\mathit{ij}}=\exp (-u_{\mathit{ij}}/RT)$ with $u_{\mathit{ij}}$ being adjustable binary interaction parameters.

^c $C_{\mathrm{HV}}=\ln \left(2\right)$⁶³. Excess Gibbs energy ${\overline{G}}^E$ is calculated from the COSMOSAC model^58,59 without any system-specific adjustable parameter.

The solid solute solubility in ScCO₂ was determined by solving the equifugacity of palbociclib in supercritical fluid (SC) and its pure solid state (S)^65–67:

$$f_{\mathrm{pal}}^{\mathrm{SC}}\left(T,,,P,,,y\right)=f_{\mathrm{pal}}^{\mathrm{S}}\left(T,,,P\right)$$ 9

$$\ln \frac{f_{\mathrm{pal}}^{\mathrm{S}}\left(T,,,P\right)}{f_{\mathrm{pal}}^{\mathrm{L}}\left(T,,,P\right)}=\frac{{\mathrm{\Delta }\overline{H}}_{m,\mathrm{pal}}}{R{T}_{m,\mathrm{pal}}}\left(1,-,\frac{T_{m,\mathrm{pal}}}{T}\right)+\frac{\left({\overline{V}}_{\mathrm{pal}}^{\mathrm{S}},-,{\overline{V}}_{\mathrm{pal}}^{\mathrm{L}}\right)\left(P,-,P_{\mathrm{atm}}\right)}{\mathit{RT}}$$ 10

where ${\mathrm{\Delta }\overline{H}}_{m,\mathrm{pal}}$, $T_{m,\mathrm{pal}}$, ${\overline{V}}_{\mathrm{pal}}^{\mathrm{S}}$, ${\overline{V}}_{\mathrm{pal}}^{\mathrm{L}}$ and $f_{\mathrm{pal}}^{\mathrm{L}}$ are molar enthalpy of fusion, melting temperature, solid molar volume, liquid molar volume and the liquid phase fugacity of palbociclib, respectively. $P_{\mathit{atm}}$ as the atmospheric pressure was set to 101,325 Pa. The values of $T_{m,\mathrm{pal}}$ and ${\mathrm{\Delta }\overline{H}}_{m,\mathrm{pal}}$ was taken from the literature⁶⁸. The required values for ${\overline{V}}_{\mathrm{pal}}^{\mathrm{L}}$, $f_{\mathrm{pal}}^{\mathrm{L}}$ and ${\overline{f}}_{\mathrm{pal}}^{\mathrm{SC}}$ were determined from the studied EoS. In addition, the molecular cavity volume in the COSMO solvation calculation was used as the solid molar volume (${\overline{V}}_{\mathrm{pal}}^{\mathrm{S}}$).

Table 6 summarizes the optimal values of binary interaction parameters in PR + vdW and PR + WS + Wilson, which were obtained from regressing experimental palbociclib solubility, and the AARD% using these optimized parameters, which are 31.1% and 33.3% from PR + vdW and PR + WS + Wilson, respectively. It should be noted that the objective function for optimizing binary interaction parameters in PR + vdW and PR + WS + Wilson is ALD-y ($=\frac{1}{N}{\sum }_{i=1}^N\left|{{\log }_{10}y}_i^{\exp },-,{{\log }_{10}y}_i^{\mathrm{cal}}\right|$, where ALD stands for average absolute logarithmic deviation). Such deviations are about twice those from Bian’s and Belghait’s models and slightly larger than those from the other eight semi-empirical models. Although PR EoS based approaches give slightly larger overall deviations, the obtained binary interaction parameter values could be used for phase behavior prediction of multicomponent systems containing palbociclib and CO₂⁶⁹. The AARD% from PR + HV + COSMOSAC is 79.6% (Table 6) and twice larger than the above two correlation approaches. However, it should be noted that this approach is the only approach without system-specific adjustable binary interaction parameters. Such an AARD% corresponds to 0.781 in terms of ALD-y. Such a deviation is similar to the overall ALD-y of using PSRK EoS and PR + COSMOSAC EoS in predicting solubility of 57 drug-like solid solutes^28,29. This result demonstrates that combining PR EoS and COSMOSAC could at least provide a rough solubility estimation with sufficient data of the studied solid solute.

Table 6.

Parameter values in PR EoS based thermodynamic models.

Model	k12 (-)	u12 (J/mol)	u21 (J/mol)	AARD%	ALD-y* (-)
PR + vdW	−3.783 $\times$ 10–2	–	–	31.1%	0.229
PR + WS + Wilson	–	1.660 $\times$ 103	4.257 $\times$ 104	33.3%	0.242
PR + HV + COSMOSAC	–	–	–	79.6%	0.781

* ALD-y $=\frac{1}{N}{\sum }_{i=1}^N\left|{{\log }_{10}y}_i^{\exp },-,{{\log }_{10}y}_i^{\mathrm{cal}}\right|$ with N being the number of experimental data.

Figure 4 compares the solubility of palbociclib in ScCO₂ from the experiment and the PR EoS based thermodynamic models. In general, the solubility calculation results from the two correlative approaches are similar and agree with the experimental data. For the results from PR + HV + COSMOSAC, the solubility increasing trend due to increasing temperature or pressure is found to be similar to the above two correlative approaches, including the crossover region. However, it underestimates the solubility by nearly an order of magnitude. Since approaches based on combining PR EoS with the COSMOSAC model through a ${\overline{G}}^E$-based mixing rule have been demonstrated to reasonably predict binary vapor–liquid equilibria under high-pressure conditions for systems containing CO₂, the studied predictive approach may be able to roughly estimate the palbociclib solubility in mixtures of ScCO₂ with organic co-solvents.

Conclusion

Reliable measurements of drug solubility in ScCO₂, the most regarded solvent, provide the required information to design and develop the relevant processes in the pharmaceutical industry. In this work, experimental solubility of palbociclib, a cancer growth blocker, in ScCO₂ was measured at temperatures between 308 and 338 K and pressures within 12–27 MPa. Afterwards, these experimental data were correlated by ten semi-empirical equations. The obtained results revealed that the Bian’s model had a closer fitting to the experimental data with the lowest AARD% compared to the other semi-empirical equations. Furthermore, the self-consistency of the palbociclib solubility data was proved by the studied semi-empirical equations. Besides, the capability of three integrated models based on Peng–Robinson EoS in describing the palbociclib solubility data was studied, among which PR + vdW showed better accuracy than the others. Finally, the total and vaporization enthalpies of palbociclib were found to be 40.41 and 52.67 kJ/mole, respectively.

Author contributions

G.S. Conceptualization, Methodology, Validation, Investigation, Supervision, Project administration, Writing-review & editing; C.-M.H. Methodology, Investigation, Software, Writing-original draft; A.T. Writing-original draft; H.-C.W.: Writing-original draft; M.A.N. Measurement, Validation, Resources.

Data availability

The datasets generated and/or analyzed during the current study are not publicly available due to confidential cases and are available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.