Plan
Comptes Rendus
no. 5

Novel density-based model for the correlation of solid drugs solubility in supercritical carbon dioxide
Cherif Si-Moussa1  ; Aicha Belghait1  ; Latifa Khaouane1  ; Salah Hanini1  ; Asmaa Halilali1
1 Laboratory of Biomaterials and Transport Phenomena (LBMPT), Faculty of Science and Technology, University Yahia Fares of Medea, Algeria
Comptes Rendus. Chimie, Volume 20 (2017) no. 5, pp. 559-572.

Résumé

A novel density-based model derived by a simple modification of the Jouyban et al. model has been proposed to correlate the solubility of solid drugs in supercritical carbon dioxide. The six-parameter model expresses the solubility only as a function of the solvent density and the equilibrium temperature. This model is in contrast to the Jouyban et al. (J. Superiority. Fluids 24 (2002) 19) model, which gives the solubility as a function of the solvent density and the equilibrium temperature and pressure. The performance of the model has been tested on a database of 100 drugs that account for 2891 experimental data points collected from the literature. The comparison in terms of the mean absolute relative deviation for each solid drug and for the entire database between the proposed model and models that have been suggested to be mostly more accurate demonstrates that the proposed model has the best global correlation performance, exhibiting an overall average absolute relative deviation of 8.13%.

Supplementary Materials:
Supplementary material for this article is supplied as a separate file:

Métadonnées
Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crci.2016.09.009
Mots clés : Solubility, Modeling, Density-based models, Solid drugs, Supercritical carbon dioxide
Cherif Si-Moussa 1 ; Aicha Belghait 1 ; Latifa Khaouane 1 ; Salah Hanini 1 ; Asmaa Halilali 1

1 Laboratory of Biomaterials and Transport Phenomena (LBMPT), Faculty of Science and Technology, University Yahia Fares of Medea, Algeria 
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     author = {Cherif Si-Moussa and Aicha Belghait and Latifa Khaouane and Salah Hanini and Asmaa Halilali},
     title = {Novel density-based model for the correlation of solid drugs solubility in supercritical carbon dioxide},
     journal = {Comptes Rendus. Chimie},
     pages = {559--572},
     publisher = {Elsevier},
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%A Aicha Belghait
%A Latifa Khaouane
%A Salah Hanini
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Cherif Si-Moussa; Aicha Belghait; Latifa Khaouane; Salah Hanini; Asmaa Halilali. Novel density-based model for the correlation of solid drugs solubility in supercritical carbon dioxide. Comptes Rendus. Chimie, Volume 20 (2017) no. 5, pp. 559-572. doi : 10.1016/j.crci.2016.09.009. https://comptes-rendus.academie-sciences.fr/chimie/articles/10.1016/j.crci.2016.09.009/

Version originale du texte intégral

1 Introduction

The level of interest in the technology of supercritical fluids (SCFs) has increased in recent decades because of its economic and environmental impact. Currently, SCF technology has widespread utility ranging from food processing to pharmaceutical applications [1]. Commonly, conventional production processes of many pharmaceuticals comprise a series of separation and purification operations involving a series of organic solvent extraction and precipitation steps and end up very often with the recovery of a large amount of organic solvents. Aside from of being environmentally benign, less energy intensive, and more effective in controlling product specification, SCF processing offers many advantages over conventional methods of extraction, drug particle formation, and drug delivery system design. Reduction of potentially harmful organic solvents and steps required for product fabrication are among the many advantages associated with SCF technology. Being inexpensive, nontoxic, abundant, and environmentally safe, supercritical carbon dioxide (scCO2) can replace toxic and undesirable organic solvents that either require extensive solvent recovery units or remain in very small but still dangerous proportions [1–4].

The importance given to SCF technology is well illustrated by the growing published experimental data on the solubility of solid solutes (e.g., polymers, foods, drugs, nutraceuticals, pesticides, dyes, and metal complexes) in scCO2 with and without cosolvents. Solubility is typically defined as mole fraction (y2) or weight fraction (c2) of solute in the SCF, which is in equilibrium with the bulk solute. Various methods to measure solubility in SCFs have been reviewed [4–9]. These methods can be divided into two major categories: static and dynamic. A compilation of most of the published experimental data before 2004 is provided by Gupta and Shim [4]. A series of review articles [5–9] on high-pressure phase equilibria also present a compilation of experimental investigations on the solubility of solid solutes in SCF. This huge enthusiasm for these experimental measurements is because the most important and decisive factor affecting the efficacy and the correspondent technical and economic success of most SCF processes is the accurate knowledge of the equilibrium solubility of the materials to be processed in the selected SCF solvent and/or the solubility of that SCF in those materials.

Attempts at modeling the solubility of solid solutes in the SCF phase for the purpose of correlation and/or prediction have followed suit experimental investigations. However, these modeling efforts have not yet been satisfactory to the desired level of accuracy, which is one reason why it is still a research subject of interest and more experimental data are continuously being published, and new compounds are being investigated. The solubility of a solid solute in an SCF is a problem of phase equilibrium, and as such, the straightforward and obvious modeling method is the use of an equation of state (EoS) or an activity coefficient method. Simple cubic EoS with various mixing rules has been extensively used for calculation of solid solute solubility in scCO2 [10–14]. The use of cubic EoS requires critical properties, the acentric factor, sublimation pressure, and the molar volume of the solute. In the absence of experimental values of these parameters, which is more often the case for complex pharmaceutical compounds, they are estimated using group contribution (GC) methods. Sublimation pressure may also be considered as an adjusted parameter to be fitted to solubility data. The accuracy of the correlation or prediction using the cubic EoS depends considerably on the method used to estimate these solid solute properties [10,11].

More complex and theoretical-sound state of the art EoS, such as nonrandom hydrogen-bonding (NRHB) theory [15] and the statistical associating fluid theory (SAFT) [16–18], have also been used to predict or to correlate the solubility of solids in SCFs. NRHB theory was applied for the modeling of the solubility of six drugs in scCO2 by Tsivintzelis et al. [15]. The solubility of the six drugs in scCO2 was correlated by this EoS with the one temperature-independent binary parameter fitted to the experimental data. For the application of the NRHB model, physical and hydrogen-bonding fluid-specific parameters are optimized using pure fluid properties. For common fluids, such as alkanols, amines, and others, vapor pressures and liquid densities are often used. Furthermore, the hydrogen-bonding parameters can be determined using calorimetric or spectroscopic data. However, for most pharmaceuticals, extended experimental data on physical properties do not exist [15].

Most SAFT-type models require five parameters for each pure associating compound (three for nonassociation ones). The three parameters for nonassociating fluids are the segment number, the interaction energy, and the hard-core segment diameter, which are either calculated using GC methods, such as proposed by Tihic et al. [19] and used by Abdallah El Hadj et al. [18] to correlate the solubility solid drugs in scCO2, or estimated from vapor pressure and liquid-density data over extended temperature ranges. In the absence of such data for specific compounds, such as polymers and pharmaceuticals, GC methods, such as GC-SAFT [20], GC-SAFT variable range (SAFT-VR) [21–23], and GC-SAFT-γ [24], are used for the calculation of required properties.

Activity coefficient models, such as UNIversal Functional Activity Coefficient (UNIFAC), the regular solution theory based models, and more sophisticated models, such as conductor-like screening model (COSMO), cannot be used for high-pressure phase equilibrium calculations without being coupled with another model, namely, an EoS [15]. Activity coefficient models based on the COSMO method [25] such as COSMO–real solvents [26] and COSMO–surface activity coefficient (COSMO-SAC) [27] have been used to predict the thermodynamics in mixtures without data fitting. Shimoyama and Iwai [28] developed a COSMO-vacancy model to extend the range of applicability of predictive COSMO-SAC liquid model to supercritical conditions. The model was successful in predicting the solubility of 16 solid drugs in scCO2, where the melting points and enthalpies of fusion were the only solute properties used, and the molar volume of scCO2 was calculated using an EoS. Recently, Wang and Lin [29] coupled the well-known Peng-Robinson EoS with the COSMO-SAC model (PR + COSMO-SAC) to predict the solubility of 46 drugs in subcritical and supercritical CO2. In the EoS, the interaction parameter, a, and molecule covolume, b, are determined from the result of first principle solvation calculation instead of using the critical properties and acentric factor of pure substances. Hence, the only experimental data required for drugs are the melting temperature and the melting enthalpy. More recently Wang et al. [30] analyzed the vapor pressure data of 1125 compounds and proposed a modification of the PR + COSMO-SAC model [29] to improve the prediction of saturated pressure that allows the application of the method for the prediction of phase equilibrium containing solid phase, such as solid drugs in supercritical solvents. From this review of the EoS approach in modeling the solubility of solid solutes in supercritical solvents, it is clear that EoS whether predictive or correlative, with the exception of the COSMO model, still rely on experimental data that are required for fitting at least one binary interaction parameter used by mixing rules or the fitting of other required compound properties or for the validation of models. This finding is particularly true for complex multifunctional chemicals, such as pharmaceuticals.

A simpler alternative to EoS models, even if it is also limited to the range of experimental data from which the models are derived, is the use of empirical models based on the density of the SCF, pressure, and/or temperature. The use of empirical and semiempirical models has been extensively reported [31–58]. These models are based on simple error minimization using the least square method to determine the adjustable parameters of the model, and for most of them there is no need to use physical properties of the solute. Since the work of Stahl et al. [31], who proposed a direct linear relationship between the logarithm of the solubility of the solute in a compressed gas and the logarithm of the density of the gas, several models have been proposed by different authors with the aim of improving the quality of correlation of new experimental data for which previous models may present poor correlation. The main differences between the proposed models may be summarized as follows:

  • 1. The number of adjusted parameters of the models, which vary from 3 to 10.
  • 2. Although most proposed models include a linear relationship between the logarithm of the solubility of the solid solute and the logarithm of the SCF density, we may find, in some models, more complex nonlinear relationships between the logarithm of the solubility of the solid solutes and the density of the SCF and equilibrium temperature and pressure.
  • 3. If one analyzes correlation results from previous works [42,44,47,50,53–56], particularly the work of Tabernero et al. [54], who compared the correlation performance of nine semiempirical equations applied on a solubility data set of 27 pharmaceutical compounds in scCO2-containing 774 experimental data points, and the work of Bian et al. [56], who compared the correlation performance of 15 semiempirical equations applied on a solubility data set of 45 compounds in scCO2-containing 1130 experimental data points, one may clearly note that there is no model that presents the best correlation for all the experimental data of all solutes. However, models with five or six adjustable parameters seem to correlate with experimental data more accurately than models with three adjustable parameters.

The aim of this work is to contribute to the improvement of the quality of density-based correlation of solid drug solubility in scCO2. This contribution will be based on a simple modification of a Jouyban et al. [41] model. The proposed model will then be evaluated on a large experimental data set collected from the literature, by comparison with the literature models that have been suggested in several works [42,44,47,50,54–56], to assess the best correlation performance.

2 Modeling solid-SCF phase equilibrium

The solubility of a nonvolatile pure solid (2) in an SCF (1), y2, is determined from standard thermodynamic relationships by equating fugacities in the solid phase and in the supercritical phase for each component. This isofugacity criterion for this binary system is reduced to one equation, because the solvent is assumed insoluble in the solid phase:

f2S=f2SCF(1)

The fugacity of the solid phase is given by

f2S=P2subϕ2satexp[V2SRT(P−P2sub)](2)

HereP2sub, ϕ2sat, and V2S are the sublimation pressure, molar volume, and the fugacity coefficient of the saturated solid.

The fugacity of solid solute in the supercritical phase is given by

f2SCF=y2ϕ2SCFP(3)

Here ϕ2SCF is the fugacity coefficient of the solid solute in the supercritical phase.

Combining Eqs. (1)–(3) gives the expression for the mole fraction solubility:

y2=P2subP·ϕ2satϕ2SCFexp[V2sRT(P−P2sub)](4)

Eq. (4) shows that the solubility is the product of the ideal solubility (P2sub/P) and a correction term known as the enhancement factor that reflects the nonideality of the supercritical phase through the ratio (ϕ2sat/ϕ2SCF) and the effect of pressure on the fugacity of the solid, as it is known for liquids, through the exponential term known as the Poynting factor. In addition, for solids of very low volatility as is the case with many solid drugs, sublimation pressure is very low and values of the saturated fugacity coefficient of the solid, ϕ2sat, are very close to one. The expression for the solid solubility of the solute is then reduced to

y2=P2subP·1ϕ2SCFexp[V2sRT(P−P2sub)](5)

In fact, this equation gives the solubility in an implicit form because ϕ2SCF is a function of y2, T, P, and ZSCF (the compressibility coefficient of the supercritical phase), itself a function of y2, T, and P.

ϕ2SCF=ϕ(T,P,ZSCF,y2)(6)
ZSCF=Z(T,P,y2)(7)

Eqs. (5)–(7) are the basis for calculating the solubility of a solid solute in an SCF by an EoS. The quality of predictions or correlations depends on the EoS, and even for the same EoS, it depends also on the mixing rules. Moreover, this method requires the solute properties, and the calculations are very sensitive to the accuracy of the solute properties [10,11,18].

If one takes the logarithm of Eq. (5),

lny2=lnP2sub−lnP−lnϕ2SCF+V2sR·PT−V2sP2subR·1T(8)

For given T and P the volumetric properties of the solvent are governed by Eq. (9):

P=ZRTρ1(9)

Combining Eqs. (8) and (9), one can relate the solubility of the solute to the density of the solvent and the coefficient of compressibility of the pure solvent at the given temperature and pressure:

lny2=lnP2sub−ln(ZRTρ1)−lnϕ2SCF+V2sZρ1−V2sP2subR·1T(10)

Although Eq. (10) gives the solubility in a complex implicit form as mentioned previously, it shows how the solubility of the solute could be related to the density of the supercritical solvent. A density-based model can be considered a simple empirical approximation to the right-hand side of Eq. (10) that does not require any solute parameter in contrast to EoS models. The next section presents a brief review of density-based modeling.

2.1 Review of density-based modeling

Density-based models for the correlation of the solubility of a solid solute in a dense phase have evolved considerably since the work of Stahl et al. [31]. Table 1 summarizes the density-based models that have been proposed since this work. In this table, y2 is the solubility of the solid solute in mole fraction, ρ1 is the density of the SCF (in kg/m3), T is the temperature (in K), P is the pressure (in bar), ai represents the characteristic parameters determined by fitting experimental data of the solute/solvent considered. For the Sparks et al. model [42], the solubility is expressed in a dimensionless form, whereas in the Bain et al. model [47] the solute solubility, c2, is in kilograms per cubic meter. In both models c2 is related to y2 by Eq. (27a), in which Mw1 and Mw2 are the molecular weights of the solvent and the solute, respectively. It is worth mentioning that some of the models do not relate the solubility explicitly to the density of the SCF solvent. The very few models that require solute parameters whose numerical values are not available for compounds of interest, such as solid drugs, were not included in this work. Literature reviews describing the basis and assumptions upon which these models have been proposed, as well as comparisons of the performance of the correlation of the solubility of a variety of solid solutes between the various models, can be found in Refs. [42,44,47,50,53–56]. Therefore, a brief review is given here only for models that have been suggested to have the best correlating performance.

Table 1

Density-based models for the correlation of solid solutes in supercritical fluids.

Table 1
ModelEquationReference
Stahl et al.
lny2=a0+a1lnρ1(11)
[31]
Chrastil
lny2=a0+a1lnρ1+a2T(12)
[32]
Adachi and Lu
lny2=a0+(a1+a2ρ1+a3ρ12)lnρ1+a4T(13)
[33]
del Valle and Aguilira
lny2=a0+a1lnρ1+a2T+a3T2(14)
[34]
Kumar and Johnston
lny2=a0+a1ρ1+a2T(15)
[35]
Bartle et al.
lny2PPref=a0+a1(ρ1−ρ1ref)+a2T(16)
[36]
Yu et al.
y2=a0+a1P+a2P2+a3PT(1−y2)+a4T+a5T2(17)
[37]
Gordillo et al.
lny2=a0+a1P+a2P2+a3PT+a4T+a5T2(18)
[38]
Méndez-Santiago and Teja
Tlny2P=a0+a1ρ1+a2T(19)
[39]
Sung and Shim
lny2=(a0+a1T)lnρ1+a2T+a3(20)
[40]
Jouyban et al.
lny2=a0+a1P+a2P2+a3PT+a4TP+a5lnρ1(21)
[41]
Sparks et al.
c2∗=ρr,1(a0+a1ρr,1+a2ρr,12)exp(a3+a4Tr+a5Tr2)c2∗=c2ρc,1;ρr,1=ρ1ρc,1;Tr=TTc,1(22)
[42]
Garlapati and Madras (2009)
lny2=a0ln(ρ1T)+a1T+a2(23)
[43]
Garlapati and Madras (2010)
lny2=a0+(a1+a2ρ1)lnρ1+a3T+a4ln(ρ1T)(24)
[44]
Jafari Nedjad et al.
lny2=a0+a1P2+a2T2+a3lnρ1(25)
[45]
Ch and Madras
y2=(PPref)(a0−1)exp(a1T+a2ρ1+a3)(26)
[46]
Bian et al. (2011)
c2=ρ1(a0+a1ρ1+a2/lnT)exp(a3+a4ρ1T+a5)(27)
c2=ρ1Mw2y2Mw1(1−y2)(27a)
[47]
Haghbakhsh et al.
y2=10−5(a0+a1P+a2ρ+a3P2+a4ρ2+a5Pρ+a6P3+a7ρ3+a8Pρ2+a9P2ρ)(28)
[48]
Hezave and Lashkarbolooki
ln(P(y2−a3×T)Pref)=a0+a1T+a2(ρ−ρref)(29)
[49]
Keshmiri et al.
lny2=a0+a1T+a2P2+(a3+a4T)lnρ1(30)
[50]
Amooey
lny2=(a0+a1/ρ+a2/ρ2+a3lnT+a4(lnT)21+a5/ρ+a6lnT+a7(lnT)2+a8(lnT))(31)
[51]
Hozhabr et al.
lny2=a0+a1T+a2ρT−a3lnP(32)
[52]
Khansary et al.
lny2=a0T+a1P+a2P2T+(a3+a4P)lnρ1(33)
[53]
Bian et al. (2016)
lny2=a0+a1T+a2ρ1T+(a3+a4ρ1)lnρ1(34)
[56]

The first model that established a linear relationship under isothermal conditions between the logarithm of solubility lny2 of a solute in an SCF and the logarithm of fluid density lnρ1 was proposed by Stahl et al. [31]. On the basis of the theory that at equilibrium a solvato complex is formed between associating solute and solvent molecules, Chrastil [32] added a new term that takes account of the effect of temperature. This led to the model given by Eq. (12) in which a1 is the average association number, a2 is a function of the enthalpy of solvation and enthalpy of vaporization, a0 is a function of the average association number and molecular weight of the solute and solvent.

With the availability of more and newer experimental data, subsequent studies have demonstrated two major drawbacks of Chrastil's equation: it is valid for solubilities less than 100–200 kg/m3 and for a restricted temperature range [33,34]. To account for the effect of solvent density on solubility, Adachi and Lu [33] correlated the average association number to a second-order polynomial of the solvent density, which led to the model given by Eq. (13), achieving a significant decrease in the error between experimental and calculated solubility data for some systems. The effect of temperature on solubility was taken into consideration in the del Valle and Aguilera [34] modification (Eq. (14)) by adding a reciprocal of squared temperature term to Chrastil's equation.

On the basis of a review of published experimental solubility data in scCO2 and previously presented models, Jouyban et al. [41] suggested the existence of

  • (1) a nonlinear relationship between lny2 and pressure in isothermal conditions,
  • (2) a nonlinear relationship between lny2 and temperature in isobaric conditions, and
  • (3) a linear relationship between lny2 and lnρ1 in a certain range of pressure and temperature.

They proposed Eq. (21) as a modification of Chrastil's model in which they introduced the effect of pressure on solubility as a second-order polynomial and a nonlinear relationship that takes account of the combined effect of temperature and pressure. The scarce use of the Jouyban et al. model is the reason why Tabernero et al. [54] excluded it from their comparative study, but Bian et al. [47] recognized the good correlation performance of this model.

Another modification of Chrastil's model was proposed by Sparks et al. [42] (Eq. (22)), in which both the effect of the density in the average association number and the change in the enthalpy of vaporization with the temperature have been considered. In other words, they combined the Adachi and Lu [33] and the del Valle and Aguilera [34] modifications in one model, which improved significantly the results.

On the basis of a review of published experimental solubility data in scCO2 and previously presented models, Garlapati and Madras [44] noted the existence of

  • (1) a nonlinear relationship between ln(y2) and density (ρ1) in isothermal conditions,
  • (2) a nonlinear relationship between ln(y2) and temperature in isopycnic conditions, and
  • (3) a linear relationship between ln(y2) and ln(ρ1T) in a certain range of density and temperature.

They proposed Eq. (24) in which the average association number is approximated by a linear relation of the solvent density, and they added a term that takes account of remark (3).

In addition to remarks (1) and (2) made by Garlapati and Madras [44], Bian et al. [47] added the following remark.

When the system temperature increases under isobaric conditions, the average association number will certainly decrease because of the increase in the thermal motion of its molecules; on the other hand, when the system pressure increases under isothermal conditions, it will increase as a result of shortening distances and increasing collisions between molecules. Once the average association number changes, the enthalpy of solvation and enthalpy of vaporization will change. Taking account of the latter remark, they proposed Eq. (26) where the average association number is inversely proportional to the logarithm of temperature and linearly proportional to the density of the solvent. They carried a comparison of the correlation performance of their proposed model with 11 previous models, 10 of which correspond to Eqs. (12)–(15),(18),(19),(21),(22),(24),(26), the 11th was a model proposed by Sparks et al. [42] in which the average association number was approximated by a linear relation to solvent density instead of a second-order polynomial as in Eq. (22). The models of Bian et al. [47], Sparks et al. [42], Jouyban et al. [41], and Adachi and Lu [33] were the best at correlating the experimental data of 54 different solid solutes.

On the basis of the same arguments raised by Jouyban et al., Keshmiri et al. [50] proposed, however, a five-parameter model (Eq. (30)), in which the effects of pressure and temperature are assessed differently. Khansary et al. [53] considered that the different empirical linear and nonlinear relationships put forward in previous works could be regarded as rules of thumb for new model developments. They proposed a five adjustable parameters model (Eq. (33)), in which the effects of pressure, temperature, and solvent density on solubility are explicitly expressed.

Recently, Bian et al. [55] proposed a new approach, based on combinations of individual forecasting models in economics, where the proposed model can be viewed as a weighted contribution of models. The models they have selected for combination are those judged from previous works to be the mostly more accurate empirical models (Ch and Madras [46], Garlapati and Madras [44], Adachi and Lu [33], Jouyban et al. [41], Sparks et al. [42], and Bian et al. [47]). They obtained the weights by using the solution of a linear mixed nonnegative programming problem. Bian et al. [55] tested different combinations of two, three, and four models. They found that the three-model combination (Sparks et al. [42], Jouyban et al. [41], and Bian et al. [47]) was the most accurate (average absolute relative deviation [AARD] = 4.37%). However, the difference in the capacity of correlating the 1116 experimental data points between the eight combinations of models tested in terms of AARD was not very significant (4.37% ≤ AARD ≤ 4.77%).

More recently, Bian et al. [56] proposed the five-parameter model (Eq. (34)) shown in Table 1, based on the following empirical observations depending on the system considered:

  • (1) a linear relationship between ln(y2) and lnρ1 in isothermal conditions,
  • (2) a linear relationship between ln(y2) and ρ1/T in isothermal conditions, and
  • (3) a linear relationship between ln(y2) and ρ1lnρ1 in isothermal conditions.

Coelho et al. [58] presented a comparison between seven density-based models (four three-parameter models, one four-parameter model, and two five-parameter models) and the Soave-Kwong-Redlich (SRK) cubic EoS. The comparison concerned a very limited data set of four compounds of interest to the food industry. Although general conclusion cannot be drawn because of the limited data set, the results of Coelho et al. [58] show that the five-parameter models of Garlapati and Madras [44] and Khansary et al. [53] performed better than the other models considered. This study confirmed also the results of previous investigations [10–13,18] that the solubility correlation using the EoS approach is very sensitive to the accuracy of solute properties (critical properties and sublimation pressure) and leads to significantly higher deviations than the density-based models.

2.2 Improved density-based model

Reviewing the previously presented empirical models and the comparative studies [41,47,51,53–56] indicates the following:

  • 1. In most studies, the AARD is used as a criterion of comparison of model performance to correlate the experimental data.
  • 2. The three-parameter models such as [32,35,36,39,43] are the least accurate in correlating experimental data.
  • 3. The models in which the solubility of the solid solute is expressed as a function of temperature, pressure, and the density of the SCF do not take into account the phase rule, because for an SCF only two of these three variables need to be fixed; thus, there is a sort of redundancy in these models.

Considering this, and on the basis of the findings of Jouyban et al. [41], a modification that simplifies their model is proposed based on the following:

  • 1. Previous empirical findings upon which density-based modeling is based.
  • 2. There is a linear relationship between the pressure and density at fixed temperature and a certain range of pressure.
  • 3. Because the volumetric properties (P, T, ρ1) of scCO2 are governed by the phase rule, only two variables need to be fixed.

Just as the density expansion of the virial EoS is more accurate than the pressure expansion at increased pressures [59], replacing the pressure by density in the Jouyban et al. [41] model would lead, a priori, to better correlation of the solubility.

The proposed model is therefore

lny2=a0+a1ρ1+a2ρ12+a3ρ1T+a4Tρ1+a5lnρ1(35)

The parameters a0–a5 are obtained from regression of the experimental data.

To compare the correlative performance of the proposed model with other models, a set of eight models that have been suggested recently [54–56] to have the best correlation performance, was selected. The solubility data of 100 drugs in scCO2 were collected from the literature and were correlated with the eight models. Table 2 displays 100 binary solute scCO2 systems as well as the experimental temperature and pressure ranges and data sources. It should be noted that all the collected data were assumed correct. The AARD was determined to provide a reliable accuracy criterion to compare the accuracy of the models, possessing different numbers of curve-fitting parameters. The AARD can be written as

AARD(%)=100N∑i=1N|y2exp−y2caly2exp|(36)

Table 2

Sources and ranges of solubility data of solid drugs in supercritical carbon dioxide.

Table 2
No.Solid drugMw (g/mol)T (K)P (MPa)−log10(y2)NiReference
14-Aminoantipyrine203.2400308.20–328.2010.03–22.043.4353–4.983021[60]
24-Aminosalicylic acid153.1350308.00–328.0011.00–21.004.4449–5.455915[61]
35-Fluorouracil130.0770308.15–328.1512.50–25.004.8356–5.420218[62]
4Anastrozole293.3700308.00–348.0012.20–35.503.4203–5.443745[63]
5Aspirin180.1570308.15–328.1512.00–25.003.4597–4.200724[64]
6Atropine289.3690308.00–348.0012.20–35.502.7773–4.221845[65]
7Atorvastatin558.6400308.00–348.0012.16–35.462.8398–6.000045[66]
8Azadirachtin720.7140308.15–333.1510.00–26.004.8268–5.920854[67]
9Azelaic acid188.2210313.15–333.1510.00–30.004.9948–6.376814[68]
10Benzocaine165.1891308.00–348.0012.20–35.501.9165–3.568640[68]
11Bisacodyl361.4000308.00–348.0012.20–35.503.2343–5.045839[70]
12Budesonide430.5340338.00–358.0021.30–38.504.5361–5.226921[71]
13Caffeine194.1900313.00–353.0019.90–34.902.9469–3.548224[72]
14Capecitabine359.3502308.00–348.0015.20–35.403.7991–5.568640[73]
15Carbamazepine236.26860308.00–348.0012.20–35.504.0269–5.522939[65]
16Cefixime trihydrate507.5000308.00–328.0018.30–33.506.5200–6.795918[74]
17Celecoxib381.3700323.20–343.2015.00–30.003.8182–4.818218[75]
18Cetirizine388.8880308.15–338.1516.00–40.002.3080–4.978828[76]
19Chlormezanone273.7000308.20–328.2011.91–23.993.2636–5.301021[77]
20Chlorpheniramine maleate390.8600308.00–338.0020.00–40.003.3706–4.812524[78]
21Cinnarizine368.5000308.15–328.1513.97–24.003.6882–5.537621[79]
22Climbazole292.7600313.20–333.2010.55–39.892.3116–3.207624[80]
23Clobetasol propionate466.9700308.00–348.0012.20–35.505.4559–7.000044[81]
24Clofenamic acid247.6800313.00–333.0012.00–36.004.4505–5.812524[82]
25Clotrimazole344.8371308.00–348.0012.20–35.503.9722–6.699045[83]
26Clozapine326.8231318.00–348.0012.16–35.464.3778–5.920827[84]
27Codeine299.36420308.00–348.0012.20–35.502.9101–4.397945[65]
28Cortisone acetate402.4800308.15–373.158.24–22.655.3830–6.958630[85]
29Cyproheptadine278.4000308.00–338.0016.00–40.002.5100–4.475028[86]
30Cyproterone acetate416.9380308.00–348.0012.20–35.503.5834–4.886140[87]
31Dexamethasone392.4610308.15–328.1510.00–35.075.5513–5.924530[88,89]
32Diazepam284.7400308.00–348.0012.20–35.502.9547–3.795945[65]
33Diclofenac acid331.3400308.15–338.1512.00–40.002.7033–4.630832[90]
34Diflunisal250.1980308.20–328.209.10–24.605.0930–6.264421[91]
35Docetaxel807.8793308.00–338.0015.20–35.403.2984–4.431832[73]
36Dutasteride528.5000308.00–348.0012.20–35.503.7940–7.000045[92]
37Exemestane296.4000308.00–348.0012.20–35.502.7268–4.899645[63]
38Finasteride372.5000308.00–348.0012.20–35.503.4782–4.473745[92]
39Fluoxetine hydrochloride345.7900308.15–338.1516.00–40.003.0904–4.576828[93]
40Flurbiprofen244.2609303.15–323.158.90–24.503.7059–4.776827[94]
41Flutamide276.1000308.00–348.0012.20–35.503.2930–5.337245[92]
42Fluvastatin411.4700308.00–348.0012.16–35.463.2211–5.301045[66]
43Gabapentin171.2400308.00–338.0016.00–40.002.1331–4.047228[95]
44Gemfibrozil250.3400308.20–328.2010.01–22.022.3778–4.531721[96]
45Ibuprofen206.2808308.15–318.158.00–22.002.1675–4.522929[97]
46Imipramine HCl316.8700313.50–323.5030.00–50.005.0044–5.292410[98]
47Isoniazid137.1393308.00–313.0013.00–18.505.2182–6.275718[99]
48Ketoconazole531.4310308.00–348.0012.20–35.503.7582–6.301045[83]
49Ketoprofen254.2806313.15–328.159.00–25.003.7258–5.481515[100]
50Lamotrigine256.0910318.00–338.0012.16–35.465.3372–6.397927[84]
51Letrozole285.3100308.00–348.0012.20–35.504.0809–6.000045[63]
52Levonorgestrel384.5000308.00–338.0012.20–35.505.5376–6.397936[101]
53Lovastatin404.5500308.00–348.0012.16–35.463.9431–4.958645[66]
54Mandelic acid152.1500308.15–328.1510.10–23.062.5370–4.568621[102]
55Medroxyprogesterone acetate386.2460308.00–348.0012.20–35.502.0458–4.795940[87]
56Mefenamic acid241.2900308.15–338.1516.00–40.002.2262–4.080428[103]
57Megestrol acetate312.5000308.00–338.0012.20–35.504.0600–5.537636[101]
58Meloxicam sodium salt373.3800303.00–323.0014.90–25.504.8941–5.355615[104]
59Metaxalone221.3000308.20–328.2011.95–24.003.8297–6.236621[77]
60Methimazole114.1700308.00–348.0012.20–35.502.7215–4.267640[105]
61Methocarbamol241.2000308.20–328.2011.99–22.053.3125–4.552821[77]
62Methylparaben152.1473308.00–348.0012.20–35.502.9161–3.966640[70]
63Metronidazole benzoate275.7600308.00–348.0012.20–35.502.3420–4.154940[69]
64Nabumetone228.3000308.20–328.2010.00–22.002.5719–4.405621[106]
65Naproxen230.2592313.10–333.108.96–19.314.4976–5.721218[107]
66Nicotinic acid123.1100313.15–373.154.50–30.204.9825–6.508622[108]
67Nifedipine346.3350333.15–373.1512.60–29.604.1495–5.869729[109]
68Niflumic acid282.2200313.20–353.2019.00–31.004.6799–5.148721[75]
69Nimodipine418.4403313.00–333.0010.00–25.004.3737–6.221821[110]
70Nitrendipine360.3600308.00–328.008.00–20.004.1993–6.000015[111]
71Norfloxacin319.3309313.15–323.159.94–30.334.6126–5.913615[112]
72Ofloxacin361.3675318.15–323.1510.11–30.085.8729–6.408910[112]
73O-Nitrobenzoic acid167.1200308.00–328.0010.00–21.004.4484–5.818215[113]
74Oxymetholone332.4770308.00–328.0012.10–30.503.8268–4.795920[74]
75Paclitaxel853.9061308.15–328.1510.00–27.505.2076–5.920821[62]
76Penicillin G334.3901313.15–333.1510.00–35.004.1986–5.376818[44]
77Penicillin V350.3800314.85–334.857.99–28.053.2396–4.263624[114]
78Pentoxifylline278.3000308.15–328.1511.98–24.002.8633–4.522921[79]
79Phenazopyridine213.2385308.00–348.0012.20–35.502.6944–4.356545[106]
80Phenylephrine hydrochloride203.6660308.15–338.1516.00–40.002.5391–3.995728[115]
81Pindolol248.3210298.00–318.008.00–27.503.6498–4.568630[116]
82Piracetam142.2000308.15–328.1512.00–24.004.4283–5.124921[79]
83Piroxicam331.3460308.15–338.1516.00–40.003.2907–4.931828[117]
84Prednisolone360.4400308.15–373.158.24–22.656.1379–9.000028[85]
85Prednisone358.4300308.15–373.158.24–22.656.0491–9.000028[85]
86Propranolol259.3435308.00–348.0012.20–35.501.6205–3.446145[105]
87Pyrocatechol110.1107333.15–348.1510.00–35.002.4149–3.913622[118]
88Rosuvastatin481.5381308.00–348.0012.16–35.463.6126–5.522945[66]
89Salicylamide137.1361308.20–328.2010.00–22.003.6778–5.073121[106]
90Salicylic acid138.1200308.15–318.159.26–15.793.4001–4.013220[119]
91Simvastatin418.5662308.00–348.0012.16–35.463.2716–5.699045[66]
92Spironolactone416.5731308.00–338.0016.00–40.002.3002–4.204128[120]
93Sulindac356.4100308.15–338.1516.00–40.002.0610–4.437728[121]
94Tetramethylpyrazine136.2000318.00–338.0010.00–30.000.8827–2.000015[122]
95Theobromine180.1700313.00–353.0019.30–34.504.3270–5.055523[72]
96Theophylline180.1700313.00–353.0019.90–34.904.4789–4.987224[72]
97Thymidine242.2286308.15–328.1510.00–30.005.0969–5.920825[62]
98Tolfenamic acid461.7100313.00–333.0012.00–36.004.7418–5.793224[82]
99Triclocarban315.5800313.20–333.2010.93–38.963.0605–4.045824[80]
100Zopiclone388.8080313.00–333.0010.00–25.004.6596–5.823921[110]

3 Results and discussion

Data from experimental measurements of the solubility of solid solutes in SCFs are reported in the literature, more often by the intensive state variables: temperature, pressure, and solubility. The solubility is expressed either in units of mass of soluble solid solute in a volume of solution in equilibrium with the pure solid solute or in mole fraction of the solute in the supercritical phase. Note that the values of the solubility are very low when expressed in molar fractions. As mentioned previously, the last three decades have been marked by a significant number of studies on the measurement of the solubility of solid solutes in scCO2, including numerous solid drugs.

The details of the experimental data used in this work and their references are listed in Table 2. Considering the experimental data collected, it is worth mentioning that there may be some differences between experimental data for a given solute from different sources. With regard to the density of CO2, we have used the data provided in the experimental solubility data source. Otherwise, the densities of CO2 for different sets of temperature and pressure were calculated using the carbon dioxide property data available through the National Institute of Standards and Technology web database [123].

Adjustments of the parameters of the selected literature models (Eqs. 13,21,22,24,27,30,33,34) together with the proposed model (Eq. 35) were carried out using genetic algorithms (ga MATLAB function) to obtain initial estimates of the parameters. To avoid convergence to local minima, optimization using genetic algorithm is run 20 times. The set of model parameters with best fit is saved and then used as an initial approximation in the least square fitting (lsqcurvefit MATLAB function) with the use of the Levenberg–Marquardt algorithm as a computation option. In both cases, Eq. (36) was used as the objective function to be minimized. The results of parameter estimations for the proposed model are shown in Table 3. Parameters of the other models can be accessed in the supplementary material.

Table 3

Estimated model parameters (ai) of the model proposed in this work (Eq. (35)) for the 100 drugs considered.

Table 3
No.a0a1a2a3a4a5
1−333.91−0.072511.55 × 10−5−6.12 × 10−630.3411154.33453
2−300.287−0.09312.22 × 10−54.77 × 10−526.7801548.91598
3−2856.04−1.274810.00039−1.01 × 10−538.36335538.8038
4−262.11−0.081451.61 × 10−57.07 × 10−523.325341.77072
543.832140.088543−4.05 × 10−54.19 × 10−520.38589−17.3615
6131.4653−0.01148−2.05 × 10−70.000118−15.8007−23.1136
7−275.142−0.083841.60 × 10−58.94 × 10−524.145443.38118
8−222.326−0.090772.42 × 10−55.63 × 10−515.9273336.9064
9305.18770.013857−1.66 × 10−60.00013−37.0519−51.8162
1010.05584−0.019596.60 × 10−77.72 × 10−5−3.80024−2.88614
1177.77971−0.00529−9.19 × 10−79.28 × 10−5−9.39991−15.3031
121128.9140.572145−0.00026.81 × 10−5−8.13235−222.418
13−120.552−0.0067−2.79 × 10−62.14 × 10−518.0204115.99816
1413.756670.071958−3.30 × 10−56.06 × 10−520.72101−12.7081
15135.03070.147712−5.71 × 10−53.51 × 10−526.05973−36.9962
1621.93157−0.018334.88 × 10−64.85 × 10−5−6.07003−5.32296
17−341.078−0.078991.66 × 10−51.99 × 10−531.6307554.73179
18−317.281−0.02262−6.88 × 10−68.14 × 10−549.4708543.37128
19−234.084−0.00654−1.22 × 10−55.57 × 10−535.5572231.4782
20106.8562−0.160265.22 × 10−50.000204−57.2256−7.74003
21−265.5960.217796−9.68 × 10−5−4.10 × 10−6115.333114.7502
22−154.130.029543−1.69 × 10−5−1.48 × 10−529.1485119.06914
23287.37060.068205−2.19 × 10−59.25 × 10−5−23.9234−53.2504
24−293.208−0.062971.13 × 10−52.61 × 10−529.9068745.82619
25−217.757−0.058311.05 × 10−55.93 × 10−518.5042133.48974
26647.73290.110936−3.43 × 10−50.000181−62.8639−111.833
27118.6692−0.013465.16 × 10−69.36 × 10−5−15.2758−20.5678
28−108.467−0.06842.98 × 10−52.76 × 10−53.42969418.37946
29211.87690.106825−3.94 × 10−50.0001087.237829−46.6076
30238.17770.034356−5.19 × 10−67.32 × 10−5−23.4978−42.0553
3182.64781−0.010545.20 × 10−64.65 × 10−5−14.8406−14.5072
3281.453960.002502−1.32 × 10−66.43 × 10−5−7.30592−15.5991
33−118.329−0.067041.33 × 10−50.0001036.79940418.81183
34117.96320.030546−1.65 × 10−58.32 × 10−5−9.32795−24.2007
35281.03990.116924−4.38 × 10−58.95 × 10−5−1.4997−56.5648
36−228.403−0.043224.26 × 10−65.48 × 10−521.9579333.90646
37−192.596−0.087181.74 × 10−50.00011815.7347330.75498
38302.81090.068543−2.52 × 10−50.000106−22.7291−55.2465
39−537.598−0.259837.68 × 10−58.41 × 10−59.81252898.95053
40−205.394−0.047177.95 × 10−65.91 × 10−528.6846930.29182
41186.550.023145−1.09 × 10−50.000102−20.3335−33.7139
425.167317−0.00819−4.40 × 10−69.47 × 10−51.10902−4.5213
43−357.553−0.00124−1.54 × 10−52.92 × 10−558.0573549.37206
44−704.118−0.230747.07 × 10−5−1.99 × 10−547.82098123.2081
45−256.998−0.060974.92 × 10−66.99 × 10−526.2644840.30101
4612,3556.579034−0.00187−0.00068591.9505−2467.39
47−6515.35−1.286130.000329−0.00068744.21331077.031
48−167.186−0.03992.64 × 10−70.00010120.6533822.94356
49141.58790.083332−4.05 × 10−57.95 × 10−5−3.70116−31.4735
50−22.4241−0.0045−3.33 × 10−64.30 × 10−51.1580430.538126
5135.109670.02185−1.96 × 10−50.00012.786269−11.7478
52−230.88−0.065391.85 × 10−52.18 × 10−521.60236.4095
53−102.45−0.021951.07 × 10−62.79 × 10−59.01476214.73499
54−295.747−0.100192.36 × 10−59.01 × 10−528.3798347.8683
55−95.8311−0.036731.30 × 10−53.32 × 10−58.15163114.35616
56−214.513−0.064361.40 × 10−58.02 × 10−520.4806632.96468
57169.579−0.00923.88 × 10−60.000108−24.798−28.871
58−1624.16−0.905270.0002810.000119−44.0891320.7279
59−35.05980.113437−5.70 × 10−58.54 × 10−540.24724−10.0175
60226.59720.017148−4.48 × 10−60.000104−24.9282−39.1657
61−400.299−0.068157.74 × 10−64.59 × 10−551.3368261.25143
62−84.0022−0.03727.92 × 10−65.46 × 10−57.56235112.46796
63−257.96−0.061761.43 × 10−52.36 × 10−525.5546441.10095
64−105.454−0.061541.42 × 10−56.90 × 10−51.36221118.14281
65−240.522−0.072681.91 × 10−53.42 × 10−521.2039938.7047
66−33.1442−0.034611.01 × 10−55.73 × 10−50.6550123.899973
6729.08865−0.025986.47 × 10−68.73 × 10−5−5.54546−6.5935
6813.75492−0.029528.16 × 10−65.63 × 10−5−10.3614−2.57467
69−57.4741−0.055321.21 × 10−59.20 × 10−5−1.500318.993218
70−119.148−0.073762.00 × 10−58.41 × 10−55.66116719.71953
711912.3280.39439−0.000150.000556−183.351−330.995
72−1359.79−0.431590.000132−2.54 × 10−5122.7584233.9027
73−388.333−0.128163.04 × 10−56.89 × 10−534.4447364.19007
7441.8925−0.0033−3.70 × 10−66.54 × 10−5−8.25917−8.88317
75−1006.32−0.119183.93 × 10−5−0.00017168.8394155.5943
76224.67890.025636−1.15 × 10−50.000127−21.0055−40.8712
77−101.416−0.063532.10 × 10−54.66 × 10−54.66716817.49912
78−352.552−0.051793.30 × 10−63.26 × 10−541.9494753.78755
79−33.1213−0.018841.94 × 10−66.68 × 10−53.8476843.057963
80−355.568−0.177414.98 × 10−58.99 × 10−58.97540364.4528
8168.746980.0044562.15 × 10−64.08 × 10−5−7.00159−13.5184
82−343.133−0.0182−4.59 × 10−6−2.39 × 10−550.3995850.2686
83567.25050.09314−2.65 × 10−50.000178−54.843−98.5152
84−193.026−0.120815.31 × 10−54.68 × 10−57.5246133.68521
85−152.842−0.122854.76 × 10−50.0001225.69656625.7895
8650.99776−0.035358.35 × 10−60.000119−7.18326−9.33889
87−91.473−0.020813.87 × 10−61.62 × 10−57.18623913.81712
88−138.121−0.02273−9.61 × 10−74.67 × 10−514.938519.30491
89−146.614−0.066571.62 × 10−55.46 × 10−57.14981324.518
90−276.323−0.051228.56 × 10−61.63 × 10−534.4318942.82894
91−93.9377−0.00999−4.93 × 10−65.92 × 10−511.3347411.3662
92522.9610.077963−3.12 × 10−50.000217−41.6145−91.6123
93−239.605−0.02929−9.98 × 10−60.00013244.7762531.26755
94−266.645−0.125433.07 × 10−59.66 × 10−520.7482846.44233
95−54.2958−0.159515.90 × 10−59.94 × 10−5−27.118517.68296
96−4.17623−0.056831.86 × 10−55.79 × 10−5−13.20332.525788
97−1037.74−0.434520.000143−2.61 × 10−544.22912189.9748
98−99.9916−0.039544.07 × 10−67.86 × 10−59.56486513.88758
99−177.1470.009528−9.73 × 10−62.50 × 10−631.6602423.13632
100−314.429−0.123333.85 × 10−53.21 × 10−521.6337453.90277

The AARDs (%) of each model equation with each drug compound are shown in Table 4, in which the best AARD for the correlation of the solubility data of each binary system is shown in bold.

Table 4

Average absolute relative deviation (AARD, %) for 100 drugs calculated for each of the 09 studied correlations.

Table 4
No.Equation
(13)(21)(22)(24)(27)(30)(33)(34)(35)
16.337.136.387.447.627.598.847.826.73
22.424.155.271.661.732.263.537.831.92
38.226.7416.956.045.767.698.1715.186.72
47.2210.153.987.387.419.2610.065.516.97
53.005.652.684.714.455.395.443.433.31
617.6619.4521.5416.9815.3816.6816.8020.2216.24
74.0411.9411.175.905.449.0910.6411.163.84
82.675.259.993.363.432.603.7011.023.05
921.5117.695.6821.8619.8216.8115.808.5717.04
1011.998.3312.2211.136.2611.0512.5515.789.67
1110.857.170.7012.246.1012.9615.061.959.16
1210.118.552.7112.0711.868.6910.044.549.83
133.432.8824.022.952.863.205.4218.752.90
145.8711.7112.729.098.5611.0711.4412.436.95
1513.0815.047.4015.8315.4317.6318.024.5811.85
162.333.015.400.700.612.111.814.242.22
172.252.8713.021.801.441.752.9311.201.42
185.368.125.914.884.625.8411.187.634.11
194.124.769.155.144.795.617.3410.454.13
208.8312.855.166.626.086.916.515.407.35
215.354.808.135.895.975.638.356.284.12
224.143.7211.573.833.372.835.5110.862.39
2314.3812.028.5315.829.5914.8016.7111.8610.95
244.996.7612.955.325.184.975.9912.944.80
252.838.636.914.764.587.038.345.533.71
2624.0917.777.6124.8217.1322.7225.8612.6815.02
2712.6710.698.3512.2610.7214.2813.318.3812.68
2814.4623.843.4217.3212.7121.6637.5310.7915.40
297.246.934.028.467.608.909.934.957.10
3011.0410.253.7410.697.6710.329.703.625.55
317.724.394.208.297.506.485.895.787.42
327.475.0014.287.543.427.938.1112.243.53
336.6013.195.105.314.225.815.845.845.01
3412.4710.059.3013.2910.6015.5414.5214.8511.05
356.146.342.948.346.408.339.093.825.83
363.9810.803.817.236.7210.0110.263.745.05
3718.1626.3821.1610.8510.1619.9117.7119.3720.60
3816.4315.287.3818.8512.0119.0021.227.6811.57
399.6713.6329.6610.2410.1310.459.4813.599.82
405.155.858.835.395.366.195.979.005.23
4119.7713.392.8519.6513.1115.9018.843.7715.63
428.286.873.8310.094.4011.4212.193.003.71
4310.3111.7716.6510.8210.4910.1912.5810.859.61
4410.6715.427.4211.7911.1113.9816.487.5511.35
458.4011.3515.8211.9011.6910.639.7313.226.22
4612.957.206.4513.0011.367.377.647.447.67
478.336.513.186.895.215.706.737.005.37
489.3218.402.6911.9111.5318.8914.673.5712.20
498.496.203.6110.848.389.319.863.616.18
502.977.326.3811.686.6024.6023.479.594.31
5114.8618.757.4219.8718.2719.8419.509.2213.96
529.6411.022.309.9910.089.5110.523.379.54
534.087.9013.884.954.824.884.2523.634.29
543.904.495.953.583.367.577.755.483.56
5524.1326.214.7525.6922.4927.7726.216.0423.64
563.615.965.703.823.493.896.986.053.42
5722.3823.883.0520.0420.8019.9918.963.9721.10
584.294.957.735.715.734.895.747.533.20
5922.9611.684.1023.9325.5722.1819.934.9722.89
6014.059.0823.6914.1210.1012.4313.4922.599.46
614.015.964.024.364.124.285.134.213.82
625.855.0210.215.153.727.247.4710.455.21
6315.5514.437.199.969.6214.8019.487.5112.86
642.899.024.173.723.475.595.685.431.75
653.834.941.564.083.673.844.342.293.77
6621.4116.0214.6121.2814.1118.8326.2812.4817.68
6716.1110.0019.7716.3611.1513.1517.1417.5412.16
681.913.214.081.171.040.931.876.940.86
697.388.4516.358.086.486.6510.1413.144.80
707.799.117.679.388.447.559.688.708.25
7129.3213.421.1831.3312.8719.6627.331.0514.93
728.8510.501.809.178.437.808.421.746.64
733.666.0619.623.073.524.105.2321.313.12
743.833.8710.033.742.542.923.799.652.83
7516.8616.776.7116.527.9212.4916.956.548.92
7623.0011.043.4122.6316.4924.0519.194.1115.57
773.765.422.639.079.137.108.304.654.97
785.696.548.395.996.045.756.618.415.61
797.437.2014.308.316.5911.0810.8913.606.67
8010.8113.1422.3011.3511.2011.2511.5716.3210.41
818.6414.0810.758.219.1916.3613.7611.677.67
824.273.372.923.233.243.665.283.242.75
8316.0315.175.3214.0413.6014.4214.405.2613.86
8419.6038.043.7529.7525.2629.6655.774.6119.03
8527.0436.816.2533.8624.7122.9148.317.6521.04
8615.837.4715.2315.7410.7314.0515.1718.3311.29
876.5810.3410.758.347.149.948.5217.056.57
883.717.698.387.206.928.849.838.853.82
892.757.699.803.553.435.014.3112.291.78
903.613.7025.743.643.413.833.8724.973.63
916.579.3810.249.738.9811.2912.958.625.92
9212.0120.0416.369.847.6111.4113.7514.499.69
938.4412.693.698.846.749.4610.979.116.70
942.9010.5417.972.742.832.992.9030.383.63
954.484.4112.144.434.433.833.2821.313.17
965.344.1926.645.334.244.825.0935.294.61
9723.0222.893.3114.5914.5022.4122.113.3722.88
984.537.915.824.713.704.314.815.343.60
995.833.565.015.884.715.176.055.045.15
1005.965.324.486.565.285.457.263.705.77
Minimum AARD1.912.870.700.700.610.931.811.050.86
Maximum AARD29.3238.0429.6633.8625.5729.6655.7735.2923.64
Average9.5710.448.9610.178.4010.4711.949.618.13

Table 4 shows that the AARD ranges between 0.61% and 55.77%, but when considering the best AARDs only (bold) they range between 0.61% and 15.38% with an average of 4.83%. The minimum AARD (0.61%) and the maximum AARD (15.38%) are both obtained by the Bian et al. [47] model (Eq. (27)) for the systems 6 (scCO2–atropine) and 16 (scCO2–cefixime trihydrate), respectively. This demonstrates that for the 100 systems considered, empirical modeling as a whole correlates with the experimental data quite satisfactorily. However, the global comparison between the performances of different models indicates that the model proposed in this work gives the best correlating performance, where the AARD ranges between 0.86% and 23.64% with an average of 8.13%. The second best correlating performance is given by the Bian et al. [47] model, where the AARD ranges between 0.61% and 25.57% with an average of 8.40%. The third best correlating performance is given by the Sparks et al. [42] model (Eq. (22)), where the AARD ranges between 0.70% and 29.66% with an average of 8.96%. As shown in Fig. 1, these three models together with the recent model of Bian et al. [56] represent the best correlation of 82% of the 100 systems considered. Beside this, the proposed model for calculating the solubility is simpler as it requires only the temperature and CO2 density. Furthermore, Table 4 confirms the results of previous comparative studies [41,47,51,53–56], insofar as there is not a model that can best correlate all of the systems of the experimental data set.

Fig. 1

The number of systems best correlated by each model equation.

Validation and performance of the models selected in this study are shown as scatter plots for the 2891 experimental points of the database. Fig. 2 shows the scatter plot of the proposed model. These plots are generated using the postreg function of MATLAB, in which the calculated solubility in logarithmic scale is plotted against experimental solubility. The scatter plot shows the dispersion of the cloud of points of the whole data set around the first bisector (log(ycal) = log(yexp)), the degree of dispersion can be evaluated from the equation for the best linear fit (log(ycal) = α. Log(yexp) + β), where α is the slope and β is the intercept, which together with the correlation coefficient R are used as a validation agreement vector. The ideal performance is achieved when [α = 1, β = 0, and R = 1]. The values of α, β, and R for the seven selected models and the proposed model are summarized in Table 5. According to the α value close to 1, the β value close to 0, and R close to 1, clearly the proposed model has the best validation agreement vector and as such the best overall correlation performance.

Fig. 2

Scatter plot of the entire data set of the drug solubility calculated using the model proposed in this work (Eq. (35)) versus experimental solubility.

Table 5

Validation agreement vector (α [slope], β [y intercept], and R [correlation coefficient]) calculated for the nine studied models.

Table 5
Validation agreement vectorEquation
(13)(21)(22)(24)(27)(30)(33)(34)(35)
α0.99490.99470.99530.99460.99600.99470.99360.99510.9960
β−0.0219−0.0226−0.0199−0.0232−0.0172−0.0228−0.0274−0.0209−0.0171
R0.99740.99740.99770.99730.99800.99730.99660.99760.9980

4 Conclusions

The novel density-based model proposed in this work has been derived by simple modification of the six-parameter Jouyban's model to correlate the solubility of solid drugs in scCO2. The modification considers the phase rule insofar as the solubility of the solid solute in the supercritical phase is expressed only as a function of the solvent density and the equilibrium temperature, in contrast to Jouyban's model that gives the solubility as a function of the solvent density and the equilibrium temperature and pressure. The performance of the model has been tested on a database of 100 drugs that account for 2891 experimental data points collected from the literature. The correlative performance of the proposed model has been compared with that of seven previous models that have been selected on the basis of being suggested in previous works to possess the best correlating performance. This study demonstrates that for the data set considered, empirical modeling as a whole correlates with the experimental data quite satisfactorily with an AARD range of 0.61%–15.38% and an overall average of 4.83%. The global comparison between the performances of different models indicates that the model proposed in this work gives the best correlating performance, where the AARD ranges between 0.86% and 23.64% with an average of 8.13%.

Acknowledgments

The authors gratefully acknowledge the Algerian Ministry of Higher Education and Scientific Research (CNEPRU Projects: No. J0102620110007 and No. J0102620140015) and the University Yahia Fares of Medea.

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