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Solubility Prediction in Supercritical CO2 Using Minimum Number of Experiments
Solubility Prediction in Supercritical CO2 Using Minimum Number of Experiments ABOLGHASEM JOUYBAN,1 MAHBOOB REHMAN,2 BORIS Y. SHEKUNOV,2 HAK-KIM CHAN,3 BRIAN J. CLARK,2 PETER YORK4 1
School of Pharmacy, Tabriz University of Medical Sciences, Tabriz 51664, Iran
2
Bradford School of Pharmacy, University of Bradford, Bradford BD7 1DP, United Kingdom
3
Faculty of Pharmacy, The University of Sydney, Sydney NSW 2006, Australia
4
Bradford Particle Design, 69 Listerhills Science Park, Campus Road, Bradford BD7 1HR, United Kingdom
Received 3 August 2001; revised 12 December 2001; accepted 14 December 2001
ABSTRACT: The correlation ability and solubility prediction in supercritical carbon dioxide of a proposed equation were studied. The work involved the solubilities of nicotinic acid and p-acetoxyacetanilide in supercritical carbon dioxide using a dynamic flow solubility system at 35–758C and 100–200 bar. The generated experimental solubility data together with 21 data sets collected from the literature were used to evaluate the correlation ability of available empirical equations. The average absolute relative deviations (AARD) for the empirical equations are 12.6–24.8%. The prediction capability of the modified empirical relationship was studied with six experimental data points as a training set. Then, solubility at other temperatures and pressures were predicted. The AARD between predicted solubilities and observed values is 17%. ß 2002 Wiley-Liss, Inc. and the American Pharmaceutical Association J Pharm Sci 91:1287–1295, 2002
Keywords:
solubility; prediction; supercritical carbon dioxide; empirical equation
INTRODUCTION Supercritical fluid technologies have received wide-ranging attention in the pharmaceutical industry where they have been applied in separation science and solute extraction. The main areas of pharmaceutical application include production of particulate drugs, extraction and separation of active ingredients, and preparation of microemulsions and sustained drug delivery systems.1 Precise experimental solubility data and accurate solubility calculations are essential in process design and analysis, which can be achieved using computer-aided systems. Modeling solubiCorrespondence to: Abolghasem Jouyban (Telephone: 0098 411 3341315; Fax: 0098 411 3344798; E-mail: [email protected]) Journal of Pharmaceutical Sciences, Vol. 91, 1287–1295 (2002) ß 2002 Wiley-Liss, Inc. and the American Pharmaceutical Association
lity data in supercritical fluids (SCF) is the subject of many published papers during last decade.2–8 Kumar and Johnston developed a thermodynamic formalism to describe the dependence of the solubility of a nonvolatile solute in SCF on the density of the fluid phase and also provided an explanation for linear dependence of the mole fraction solubility of a solute in SCF on density/logarithm of density of SCF.2 Simple empirical correlation of the logarithm of enhancement factor versus density has been recommended when the goal of a study is to correlate solubility isotherms in terms of an adjustable parameters. The hard sphere equation of state (HSE) has been employed to predict solubilities of solids in supercritical carbon dioxide (SC CO2), and an overall prediction error of 18.33% was found; the corresponding error for the Peng–Robinson equation of state (PR) is 24.69% for the same data sets. The
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advantage of the HSE model over PR is that there is no need to know the critical properties of the solids.4 Solubilities of xanthines in SC CO2 have been determined by a static method. Solubilities of caffeine in SC CO2 are about one order of magnitude higher than those of theophylline and about two orders of magnitude higher than those of theobromine.5 The solubilities of phenylacetic acid and vanillan in SC CO2 have been measured and then used to evaluate various models. The results showed that traditional cubic equations of state models, such as PR, have serious limitations when applied to these systems, and density-based equations perform much better.6 Verete presented a predictive method for solubility of solids in SC CO2 using the molecular weight of the solute and its melting temperature as input parameters and reported prediction errors of 6–117% for polar compounds.7 Ashour and co-workers studied three cubic equations of state combined with different mixing rules for prediction of solid–SCF equilibria and found that the Estevez model is superior in comparison with the other models. However, this model would get worse at higher pressures. The authors concluded that no single cubic equation of state currently exists that is equally suitable for the quantitative prediction of all SCF mixtures and also recommended quantitative molecular analysis.8 For collection of experimental solubility data in SCF, researchers are faced with several difficulties including (1) complex and expensive experimental apparatus, (2) time-consuming studies for determination of equilibrium solubility, and (3) scarce and often unreliable literature data. In addition, there is a lack of accurate models to simulate the SCF process design, which presents a major challenge in scale-up utilization of SCF technologies. There are > 40 different forms of equations of state and 15 various mixing rules.9 When dealing with the equations of state to calculate solute solubility in SC CO2, it is necessary to select the correct equation and choose the mixing rule to be applied. Usually, a researcher uses an equation of state and a mixing rule based on his/ her experience and, as already indicated, a single model cannot treat the solubility profile of all compounds of interest successfully. In addition, these models require a degree of knowledge of the physicochemical properties of the solute that are often not readily available. In some cases the estimated physicochemical properties using a group contribution and/or other computational methods are used, but these may cause another error source. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 91, NO. 5, MAY 2002
Hutchenson and Foster10 reported deviations between experimental and calculated solubilities using the PR equation of state11 of > 50% for naphthalene solubility data in SC CO2. Trablesi and co-workers12 employed the PR and Bublik– Mansoori equations of state13,14 to correlate a- and b-naphthol solubility data in SC CO2 and showed that both equations produced an average absolute relative deviation (AARD) of 21%. The authors also assessed the correlation ability of the Bublik– Mansoori equation of state using 13 solubility data sets of different compounds in SC CO2, which resulted in an AARD of 22.6%. To provide a predictive equation, an average value of solute–solvent interaction parameter (a12) was used for nine compounds, which possess a solubility of < 103 mole fraction and high AARD of 40%.12 The a12 parameter is calculated using all experimental points in each data set by an iteration method. The next models to be used to compute the solute solubility in SCF were empirical relationships between the solubility and the independent variables (such as pressure, temperature, and density). These relationships have the advantage that the corresponding calculations are simple and easily available in commercial software. A disadvantage is that they possess a number of curve-fitting parameters that should be calculated using experimental data points. As an alternative solution to predict solubility data, one can employ correlative empirical equations, which can be applied after the equation is trained. To train an empirical equation, a minimum number of experimental data points should be fitted to the model to compute the model constants. From these constants it is suggested that the trained model is able to predict the solute solubility in the SCF at other temperatures and pressures of interest by using an interpolation technique. This method has previously been evaluated to predict solubility in aqueous binary solvents15 and also in electrophoretic mobility predictions in capillary electrophoresis.16 In this work, the solubilities of nicotinic acid (NA) and p-acetoxyacetanilide (PAA) were measured in SC CO2 at temperatures of 35, 55, and 758C and pressures of 100–200 bars. Then a minimum number of experimental solubility points of NA and PAA in SC CO2 were used to compute the proposed model constants. An attempt was then made to predict the solubility at other pressures and temperatures using the trained model. PAA is a synthetic precursor that is structurally similar to paracetamol (acetaminophen) and also a sug-
SOLUBILITY PREDICTION IN SUPERCRITICAL CO2
gested prodrug.17 The COCH3 group chemically discriminates PAA from acetaminophen and is responsible for higher (by a factor of 2–10 depending on pressure and temperature) solubility of this compound relative to the acetaminophen solubility and thus provides a good model substance to test the applicability of the solubility models in a wide dynamic range of mole fraction. NA was used as a model compound to study particle formation process with SC CO2 as an antisolvent.18 It belongs to a group of hydrosoluble B vitamins used for the treatment of hyperlipidemias (reduction of elevated blood lipid levels), and was selected on the basis of its relatively low solubility in pure CO2 (on the order of 107 mole fraction).
COMPUTATIONAL METHODS Chrastil19 correlated the solubility of a solute (S, g L1) in SC CO2 to the density (r, g mL1) and temperature (T, K). The model is: ln S ¼ A0 þ
A1 þ A2 ln r T
ð1Þ
where A0 –A2 are the model constants that can be estimated from experimental solubility data in SC CO2. del Valle and Aguilera20 then added one more term (1/T2) to eq. 1. The resulting equation is: ln S ¼ B0 þ
B1 B3 þ B2 ln r þ 2 T T
ð2Þ
where B0 –B3 are the model constants. For eqs. 1 and 2, it is possible to use mole fraction solubility instead of g L1 concentration, which provides a more accurate correlation.21 It is also possible to correlate the mole fraction solubility of a solute ( y2) with pressure (P, bar) and temperature.22 The model is: 2
y2 ¼ C0 þ C1 P þ C2 P þ C3 PTð1 y2 Þ þ C4 T þ C5 T2
ð3Þ
where C0 –C5 are the curve-fitting parameters and (1 y2) represents the mole fraction of the solvent (SC CO2). Gordillo and co-workers23 proposed another empirical model. The model is: ln y2 ¼ D0 þ D1 P þ D2 P2 þ D3 PT þ D4 T þ D5 T 2
ð4Þ
where D0 –D5 are the model constants. The superiority of eq. 4 over eqs. 1–3 has been shown
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using penicillin G solubility data in SC CO2, where reported mean percentage deviations for eqs. 1–4 were 32.4, 32.4, 22.9, and 14.4%, respectively.23 The authors also compared mean percentage deviation values with those of two equations of state, namely, Redlich–Kwong and Soave–Redlich–Kwong, and found 23 and 21%, respectively. As seen from mean percentage deviation values for penicillin G solubility data, the accuracy order for the equations are eq. 4 > Soave– Redlich–Kwong > eq. 3 > Redlich–Kwong > eqs. 2 and 1. Further comparisons are reported between different empirical relationships throughout this work. An empirical equation based on response surface methodology has also been employed to correlate solubility data in SCF.24 The model is: y2 ¼ E0 þ E1 P þ E2 T þ E3 PT
ð5Þ
where E0 –E3 are the curve-fitting parameters. In contrast, in our earlier work,21 an empirical equation was presented that provides better correlation than those just presented. The model is: ln y2 ¼ M0 þ M1 P þ M2 P2 þ M3 PT þ
M4 T P
þ M5 ln r
ð6Þ
where M0 –M5 are the model constants and r is the density of pure SC CO2 at different pressures and temperatures. In further studies, by replacing P with r, a more accurate model is produced, and this version has been used in this work. The resultant equation is: ln y2 ¼ K0 þ K1 r þ K2 P2 þ K3 PT þ
K4 T P
þ K5 ln r
ð7Þ
where K0 –K5 are the model constants. In these equations, the numerical values of r are calculated using the following equation:21 pffiffiffiffi 3966:170 ln r ¼ 27:091 þ 0:609 T þ T pffiffiffiffi 3:445P þ 0:401 P T
ð8Þ
To test the correlation ability of the models, whole experimental data points in each set have been fitted to the equations and their accuracies have been compared using back-calculated solubilities. A possible application for this type of calculation is the screening of experimental solubility data for possible outliers where re-determination JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 91, NO. 5, MAY 2002
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is required. To evaluate the prediction capability of the proposed model, three solubility data points at the lowest temperature and the lowest, middle, and the highest pressures, with three corresponding data points at the highest temperature have been used to compute the model constants. From these results, the solubility at other temperatures and pressures has been predicted using the trained model. Overall, it is suggested that these computations may provide a useful technique to speed up the pressure and temperature optimization procedure in SCF studies in industry. It can be indicated that in operating the models, as a general rule, the more curve-fitting parameters there are, the more accurate the correlations. To provide a reliable accuracy criterion to compare the models possessing different numbers of curve-fitting parameters, the AARD values were used, which is calculated by: yObserved 100 X ycalculated 2 2 AARD ¼ ð9Þ Observed ðN ZÞ y2 where N is the number of data points in each set and Z is the number of curve-fitting parameters for each model. In addition, the individual absolute relative deviations (IARD) of predicted solubilities from observed values were calculated to evaluate the prediction capability of the proposed model. The IARD was calculated by: calculated ! y yObserved 2 2 IARD ¼ 100 ð10Þ Observed y2
EXPERIMENTAL METHODS Dynamic Flow-Through Solubility System A schematic diagram of the dynamic flow-through solubility system is shown in Figure 1. Initially, the lowest pressure for the solubility study was established using pure SC CO2 pumped through the system at defined flow rates with high-performance liquid chromatography (HPLC) pumps (Jasco PU980/6, Japan). An HPLC ultraviolet (UV) detector (Jasco UV-1575, Japan), equipped with a high-pressure UV flow cell (Jasco UV-975, Japan), was placed between the oven and back pressure regulator (Jasco 880-881, Japan). Both 6-way Rheodyne valves were switched to blank mode to direct the SC CO2 flow along a bypass line and through the high-pressure flow cell. UV data were captured and processed by a computer with JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 91, NO. 5, MAY 2002
Figure 1. A schematic diagram of set up for dynamic solubility measurement.
Borwin version 1.21 software (JMBS developments, France). In the blank mode, the detector response was set to zero to establish a stable base line and then the valves were switched to sample mode, diverting the SC CO2 through the extraction vessel. As NA or PAA extracted into the SC CO2, a characteristic UV response plateau was observed. The pressure was then raised to the next selected level at the back pressure regulator. This procedure was repeated at a range of pressures and temperatures to establish a comprehensive data set at defined conditions. A separate on-line UV detector calibration curve was also prepared using a series of dilutions of NA and PAA solutions (100 mg/100mL, 0.1% w/v) in methanol. For this curve, a wide range of calibration standards were examined at 263 and 254 nm to adequately bracket solubility values in pure SC CO2. From the results, the sample response plateau data were interpolated from the calibration curves to determine the concentration of NA and PAA present in the SC CO2. Sample Preparation Acid washed glass beads (212–300 mm diameter; Sigma Chemicals, Poole, UK) were mixed with NA in a 4:1 ratio by weight of glass beads/NA using a Turbula mixer (Glen Creston, T2C, No. 860961, Stanmore, England) for 30 min to ensure that a homogeneous mixture was achieved. The presence of the glass beads enhances surface area for solute/solvent interaction, prevents caking of the
SOLUBILITY PREDICTION IN SUPERCRITICAL CO2
drug, and eliminates channeling of the SC CO2 through the bed. A 10-mL stainless steel pressure vessel was carefully packed with the glass beads/ NA mixture and placed into a fan-assisted oven (Applied Separation, Allentown, PA) to maintain isothermal conditions. The glass beads/NA mixture was sandwiched between circular paper filters (Schleicher & Schuell, Dassel, Germany) to minimize movement of the packing material and avoid any entrainment of material. All samples were investigated in the pressure range 100–200 bar and at the temperatures of 35, 55, and 758C. For PAA, the same sample preparation was used except that a 3:1 ratio by weight of glass beads/PAA was used. NA (> 99% purity) was purchased from Sigma Chemicals (Poole, Dorset, UK), and solvents were of HPLC grade from Fisher Scientific (Loughborough, Leicestershire, UK). Liquid CO2 (99.99% purity) was supplied by Boc Limited (Guildford, Surrey, UK). PAA was synthesized in house as reported by Bauguess et al.25 at the research laboratories of the University of Bradford, and identification was confirmed by a wide range of analytical techniques.
RESULTS AND DISCUSSION The logarithm of mole fraction solubility data for NA and PAA in pure SC CO2 are shown in Table 1.
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The solubility of both compounds increases with an increase in pressure at each temperature used because there is an increase in density and solvent power of pure SC CO2 at each pressure increase. As a result, with increased density, the intermolecular distance is decreased; therefore, the solute–solvent interactions are increased. As a result of these effects, it can be concluded that complex behavior for solubility at different temperature is observed, where the temperature affects the solute vapor pressure, the solvent density, and the intermolecular interactions in the fluid phase. By taking the logarithm of the experimental solubility data, a good linear correlation with respect to the logarithm of the density of pure SC CO2 at each temperature is achieved; this provides a check on the internal consistency of the data.19,26 As already explained, to evaluate the correlation ability of the proposed model, the generated experimental solubility data have been fitted to eq. 7, and the obtained models are shown in eqs. 11 and 12 for NA and PAA, respectively: ln y2 ¼ 20:005 3:472r 0:000151P2 1:656T þ 3:704 ln r þ 0:000225PT þ P ð11Þ
Table 1. Logarithm of Experimental Mole Fraction Solubility of Nicotinic Acid and p-Acetoxyacetanilide, and Density of Pure SC CO2 at Different Temperatures and Pressures T (8K)
P (bar)
Density (g/cm3)a
Nicotinic Acid
p-Acetoxyacetanilide
308.15
100 120 140 160 180 200 100 120 140 160 180 200 100 120 140 160 180 200
0.5286 0.6198 0.7047 0.7818 0.8505 0.9098 0.3633 0.4319 0.4978 0.5599 0.6174 0.6695 0.2684 0.3229 0.3767 0.4289 0.4786 0.5253
13.62 13.51 13.33 13.26 13.15 13.12 13.89 13.21 13.27 12.66 12.46 12.32 13.62 13.33 — 12.35 11.90 11.79
11.75 — 10.77 10.63 10.54 10.47 12.53 11.43 10.28 9.94 9.75 9.65 — 11.77 11.01 9.91 9.55 9.40
328.15
348.15
a
The density of pure SC CO2 was calculated with eq. 8.21 JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 91, NO. 5, MAY 2002
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where R ¼ 0.987, F ¼ 84, AARD ¼ 11.6%, N ¼ 17, and p < 0.0005. ln y2 ¼ 0:219 11:373r 0:000071P2 1:230T þ 6:302 ln r þ 0:000079PT P ð12Þ where R ¼ 0.991, F ¼ 109, AARD ¼ 16.9%, N ¼ 16, and p < 0.0005. The results show that eq. 7 reproduces the best estimate of experimental solubility, where the expected percent deviation has a mean of < 14% which is considered to be an acceptable value when screening for the best conditions. From eqs. 11 and 12 it is possible to calculate solute solubility at temperatures between 35 and 758C and pressures between 100 and 200 bar using an interpolation technique. It can be added that as a general rule, the more training data points included in the training process of the models, the more accurate the solubility predictions. However, collecting a large number of experimental solubility data points in an SCF experiment is not appropriate because it is costly and time consuming. Therefore, as a computational solution to this problem, a minimum number of experimental data points (equal to the number of curve-fitting parameters; that is, six data points) have been employed to train the proposed model. Then, the solubility at other T and P values has been predicted using the trained model. By using only six data points, the produced AARD values for NA and PAA are 17.0 and 15.7%, respectively. These prediction errors are still considered to be quite reasonable, particularly because the reported experimental solubility data at the same conditions are so different. As an example of this variability, the reported solubility data of octacosane in SC CO2 at 358C from four different research groups shows the differences of more than a factor of 10.27 From our experiments and by using an online solubility system, the relative standard deviations for repeated points lie within 15%, indicating that the experimental technique is robust.18 To further investigate the correlation capability of the empirical equations, more sets of solubility data in SC CO2 were collected from the literature. Their detail, references, and the produced AARDs are shown in Table 2. For the data sets for ketoprofen, salicylic acid, sulfadimethoxine, and sulfamerazine, the models produced large AARDs. However, for caffeine, theobromine, theophylline, and vitamin D2, small AARDs were JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 91, NO. 5, MAY 2002
given. It was therefore obvious that a single model could not produce accurate results in all cases. It should also be noted that a possible outlier could affect the regression model and result in a relatively high AARD. Nevertheless, the global (mean) AARDs are < 20% for most models studied, and this was considered to be an acceptable error for empirical equations because there is no need to employ a complicated calculation process or include any physicochemical properties of the solute. But one of the criterion set before starting the work was to obtain the lowest achievable error and thus to provide the most accurate solubility predictions. Equation 7 produced the most accurate solubilities, with global AARD equal to 12.6%, and eq. 3 produced the least accurate results, with 24.8%. It is suggested that percentage errors < 20% are acceptable because the produced correlation error for equations of state (which are used in chemical engineering calculations) is 20%.12 The global AARD for eq. 7 is significantly less than those of eqs. 1–6 (paired t test). By calculating the AARD and its global value, the overall error is presented. In contrast, the IARD shows the error level at each temperature and pressure and is able to detect the possible outliers in each of the data sets. The distribution of the IARD for eqs. 1–7 at four subgroups (I, < 15% error; II, 15–30% error; III, 30–45% error; and IV, > 45% error) is shown in Figure 2. Equations 1 and 2 produced an IARD of subgroup I in 60% of all the cases, whereas 7% of the produced IARD values lie in subgroup IV. The frequency of the cases in subgroups II and III are 25 and 10%, respectively. It is obvious that with an accurate model, the lower frequencies are expected for subgroups II, III, and IV. The corresponding relative frequencies for eq. 7 are 81, 14, 3, and 2%. The IARD distribution for eq. 6 is very close to that of eq. 7. However, for eqs. 3–5, a different pattern was observed. As an overall result, the proposed equation provides acceptable IARD distribution when compared with eqs. 1–5. A minimum number of six experimental data points from the collected data sets were used to train the proposed model. Once trained, the solubility at other temperatures and pressures were predicted by an interpolation technique. Here, the IARD values represent the real percentage differences between predicted (not correlated) values and experimental solubilities using the proposed model (eq. 7), which is shown in Figure 3. The mean IARD for 401 solubility data points (23 solubility data sets mentioned in
SOLUBILITY PREDICTION IN SUPERCRITICAL CO2
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Table 2. Solubility Data in Supercritical Carbon Dioxide, their References, and Average Absolute Relative Deviations (AARD) for Eqs. 1–7 No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Solute Nicotinic acid p-Acetoxyacetanilide Caffeine b-Carotene Ketoprofen Naproxen Nifedipine Nitrendipine Penicillin G Penicillin V Piroxicam Pyrocatechol Resorcinol Retinol Salicylic acid Sulfadimethoxine Sulfamerazine Sulfamethazine Theobromine Theophylline Vitamin D2 Vitamin D3 Vitamin K1
Reference
N
Eq. 1
Eq. 2
Eq. 3
Eq. 4
Eq. 5
Eq. 6
Eq. 7
This work This work 5 28 29 30 31 31 23 32 33 34 34 35 26 36 36 36 5 5 35 35 35 Mean SD
17 16 24 27 15 18 29 42 18 24 9 32 32 20 49 19 18 20 23 24 19 23 24
21.0 35.9 6.3 29.0 54.6 21.5 19.2 20.2 23.4 14.8 23.7 8.8 8.9 9.9 47.0 41.1 37.8 16.6 7.9 7.7 8.6 31.6 12.8 22.1a 13.7
22.5 36.4 5.4 27.6 59.6 23.2 20.4 19.6 24.9 15.5 28.4 9.1 9.3 7.6 48.2 43.9 40.5 17.6 7.6 7.6 8.5 30.2 10.6 22.8a 14.9
16.6 28.9 6.1 108.9 87.2 12.8 39.5 25.0 34.9 8.9 14.6 7.9 5.5 9.2 30.1 17.8 17.7 8.0 4.8 7.6 5.7 67.2 5.1 24.8b 27.6
13.2 20.8 4.5 26.2 39.5 16.0 14.6 19.3 22.9 17.7 22.4 8.6 3.3 6.8 32.2 28.2 36.6 8.7 4.7 6.1 5.7 25.3 9.8 17.1c 10.8
14.8 36.7 5.3 88.0 71.1 11.0 40.0 28.7 29.9 8.0 8.7 8.1 6.5 9.5 28.5 15.4 15.2 7.0 4.7 6.8 8.6 39.2 4.7 21.6d 21.8
14.8 30.3 3.9 23.8 20.6 12.8 13.2 16.0 17.7 6.7 21.6 6.3 2.8 9.6 19.2 28.3 34.0 8.7 4.8 6.2 5.8 26.8 9.9 14.9e 9.2
11.6 16.9 4.3 30.4 11.8 11.2 13.8 18.5 16.2 5.9 15.0 6.0 3.7 9.4 22.0 19.2 18.0 5.7 4.9 5.9 6.2 26.2 7.1 12.6 7.4
a
p < 0.0005; paired t test showed significant difference between global (mean) AARDs with that of eq. 7. p < 0.018. c p < 0.007. d p < 0.021. e p < 0.042. b
Table 2) is 17%. This prediction error could be considered as acceptable because there is huge variations in solubility data reported from different laboratories.27 In addition the reproducibility
of solubility data in SC CO2 from our group is 10% in comparison with the literature values.18 As a general conclusion, the proposed model (eq. 7) produced accurate results and therefore it
Figure 2. Distribution of individual absolute relative deviation (IARD) for eqs. 1–7 in subgroups I–IV.
Figure 3. Distribution of IARD in different subgroups produced by trained proposed model using six experimental data points. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 91, NO. 5, MAY 2002
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is suggested that it could be employed in practice to simulate the solubility data in SC CO2 after conducting a minimum number of experiments where solubility prediction would be helpful to carry out the method development process. An extension of this work has been the use of the model to screen the experimental data points to find any possible outliers. In operating the model in practice. the advantages are its simple calculation procedure and higher accuracy in comparison with other empirical equations and equations of state. However, from this investigation, it is obvious that a single model cannot be appropriate in every case, which could be considered as one possible disadvantage for a model. Of course, this comment also applies to the other models in the literature. Yet, because the proposed model produced the most accurate results among other models, as shown here when using 23 data sets, it should produce accurate solubilities for other data sets and is the best practical model that could be employed in industry.
6.
7.
8.
9.
10.
11.
12.
ACKNOWLEDGMENTS 13.
The authors thank the Australian Department of Education, Training and Youth Affairs, and the University of Sydney for providing the IPRS and IPA scholarships. The authors also thank Dr. Andreas Kordikowski, of Bradford Particle Design Plc, for his help in the synthesis of PAA.
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supercritical carbon dioxide. Fluid Phase Equilibria 95:215–226. Wells PA, Chaplin RP, Foster NR. 1990. Solubility of phenylacetic acid and vanllan in supercritical carbon dioxide. J Supercrit Fluids 3:8–14. Verete A. 1998. A short-cut method to predict the solubilities of solids in supercritical carbon dioxide. Fluid Phase Equilibria 148:83–93. Ashour I, Almehaideb R, Fateen S-E, Aly G. 2000. Representation of solid fluid phase equilibria using cubic equations of state. Fluid Phase Equilibria 167: 41–61. Pfohl O, Petkov S, Brunner G. 2000. Manual for a program to calculate phase equilibria. Downloaded from: www.tu-harburg.de/vt2/pe2000. Hutchenson KW, Foster NR. 1995. Innovations in supercritical fluid science and technology. In: Hutchenson KW, Foster NR, editors. Innovations in supercritical fluids science and technology. Washington, DC: American Chemical Society. pp. 1–31. Smith JM, Van Ness H, Abbott MM. 1996. Introduction to chemical engineering thermodynamics, 5th ed. New York: McGraw-Hill. Trablesi F, Abaroudi K, Recansens F. 1999. Predicting the approximate solubilities of solids in dense carbon dioxide. J Supercrit Fluids 14:151–161. Boublik T. 1970. Hard-sphere equation of state. J Chem Phys 53:471–472. Mansoori GA, Leland TW, Jr. 1972. Statistical thermodynamics of mixtures. New version for the theory of conformal solution. J Chem Soc, Faraday Trans II 68:320–344. Jouyban-Gh A, York P, Hanna M, Clark BJ. 2001. Solubility prediction of salmeterol xinafoate in water-dioxane. Int J Pharm 216:33–41. Jouyban-Gh A, Khaledi MG, Clark BJ. 2000. Calculation of electrophoretic mobilities in water-organic modifier mixtures. J Chromatogr A 868:277:284. Fairbrother JE. 1973. In: Florey K, editor. Analytical profiles of drug substances. New York: Academic Press, vol. 3, p.1. Rehman M, Shekunov B Yu, Colthorpe P, York P. 2001. Solubility and precipitation of nicotinic acid in supercritical carbon dioxide. J Pharm Sci 90: 1570–1582. Chrastil J. 1982. Solubility of solids and liquids in supercritical gases. J Phys Chem 86:3016–3021. del Valle JM, Aguilera JM. 1988. An improved equation for predicting the solubility of vegetable oils in supercritical CO2. Ind Eng Chem Res 27: 1551–1553. Jouyban A, Chan HK, Foster NR. 2002. Mathematical representation of solute solubility in supercritical carbon dioxide using empirical expressions. J Supercrit Fluids, in press. Yu Z, Singh B, Rizvi SSH, Zollewg JA. 1994. Solubilities of fatty acids, fatty acid esters, and fats
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23.
24.
25.
26.
27.
28.
29.
and oils in supercritical carbon dioxide. J Supercrit Fluids 7:51–59. Gordillo MD, Blanco MA, Molero A, Martinez de la Ossa E. 1999. Solubility of the antibiotic penicillin G in supercritical carbon dioxide. J Supercrit Fluids 15:183–190. Garcia J, Rodriguez F, Revenga JA. Modelling solubility of solids in supercritical fluids using response surface methodology. J Chem Technol Biotechnol 75:245–251. Bauguess CT, Sadik F, Fincher JH, Hartman CW. 1975. Hydrolysis of fatty acid esters of acetaminophen in buffered pancreatic lipase system I. J Pharm Sci 64:117–120. Gurdial GS, Foster NR. 1991. Solubility of o-hydroxybenzoic acid in supercritical carbon dioxide. Ind Eng Chem Res 30:575–580. Bush D, Eckert CA. 1997. Estimation of solid solubilities in supercritical carbon dioxide from solute solvatochromic parameters. In: Abraham MA, Sunol AK, editors. Supercritical fluids: Extraction and pollution. Washington, DC: American Chemical Society. pp. 37–50. Subra P, Castellani S, Ksibi H, Garrabos Y. 1997. Contribution to the determination of the solubility of b-carotene in supercritical carbon dioxide and nitrous oxide: Experimental data and modeling. Fluid Phase Equilibria 131:269–286. Stassi A, Bettini R, Gazzaniga A, Giordano F, Schiraldi A. 2000. Assessment of solubility of
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ketoprofen and vanillic acid in supercritical CO2 under dynamic conditions. J Chem Eng Data 45: 161–165. Ting SST, MacNaughton SJ, Tomasko DL, Foster NR. 1993. Solubility of naproxen in supercritical carbon dioxide with and without cosolvents. Ind Eng Chem Res 32:1471–1481. Knez Z, Skerget M, Sencar-Bozic P, Rizner A. Solubility of nifedipine and nitrendipine in supercritical CO2. J Chem Eng Data 40:216–220. Ko M, Shah V, Bienkowski PR, Cochran HD. 1991. Solubility of the antibiotic penicillin V in supercritical CO2. J Supercrit Fluids 4:32–39. MacNaughton SJ, Kikic I, Foster NR, Alessi P, Cortesi A, Colombo I. 1996. Solubility of antiinflammatory drugs in supercritical carbon dioxide. J Chem Eng Data 41:1083–1086. Yamini Y, Fat’hi MR, Alizadeh N, Shamsipur M. 1998. Solubility of dihydroxybenzene in supercritical carbon dioxide. Fluid Phase Equilibria 152: 299–305. Johannsen M, Brunner G. 1997. Solubilities of the fat-soluble vitamins A, D, E and K in supercritical carbon dioxide. J Chem Eng Data 42:106– 111. Hampson JW, Maxwell RJ, Li S, Shadwell RJ. 1999. Solubility of three veterinary sulfonamides in supercritical carbon dioxide by a recirculating equilibrium Method. J Chem Eng Data 44:1222– 1225.
JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 91, NO. 5, MAY 2002
School of Pharmacy, Tabriz University of Medical Sciences, Tabriz 51664, Iran
2
Bradford School of Pharmacy, University of Bradford, Bradford BD7 1DP, United Kingdom
3
Faculty of Pharmacy, The University of Sydney, Sydney NSW 2006, Australia
4
Bradford Particle Design, 69 Listerhills Science Park, Campus Road, Bradford BD7 1HR, United Kingdom
Received 3 August 2001; revised 12 December 2001; accepted 14 December 2001
ABSTRACT: The correlation ability and solubility prediction in supercritical carbon dioxide of a proposed equation were studied. The work involved the solubilities of nicotinic acid and p-acetoxyacetanilide in supercritical carbon dioxide using a dynamic flow solubility system at 35–758C and 100–200 bar. The generated experimental solubility data together with 21 data sets collected from the literature were used to evaluate the correlation ability of available empirical equations. The average absolute relative deviations (AARD) for the empirical equations are 12.6–24.8%. The prediction capability of the modified empirical relationship was studied with six experimental data points as a training set. Then, solubility at other temperatures and pressures were predicted. The AARD between predicted solubilities and observed values is 17%. ß 2002 Wiley-Liss, Inc. and the American Pharmaceutical Association J Pharm Sci 91:1287–1295, 2002
Keywords:
solubility; prediction; supercritical carbon dioxide; empirical equation
INTRODUCTION Supercritical fluid technologies have received wide-ranging attention in the pharmaceutical industry where they have been applied in separation science and solute extraction. The main areas of pharmaceutical application include production of particulate drugs, extraction and separation of active ingredients, and preparation of microemulsions and sustained drug delivery systems.1 Precise experimental solubility data and accurate solubility calculations are essential in process design and analysis, which can be achieved using computer-aided systems. Modeling solubiCorrespondence to: Abolghasem Jouyban (Telephone: 0098 411 3341315; Fax: 0098 411 3344798; E-mail: [email protected]) Journal of Pharmaceutical Sciences, Vol. 91, 1287–1295 (2002) ß 2002 Wiley-Liss, Inc. and the American Pharmaceutical Association
lity data in supercritical fluids (SCF) is the subject of many published papers during last decade.2–8 Kumar and Johnston developed a thermodynamic formalism to describe the dependence of the solubility of a nonvolatile solute in SCF on the density of the fluid phase and also provided an explanation for linear dependence of the mole fraction solubility of a solute in SCF on density/logarithm of density of SCF.2 Simple empirical correlation of the logarithm of enhancement factor versus density has been recommended when the goal of a study is to correlate solubility isotherms in terms of an adjustable parameters. The hard sphere equation of state (HSE) has been employed to predict solubilities of solids in supercritical carbon dioxide (SC CO2), and an overall prediction error of 18.33% was found; the corresponding error for the Peng–Robinson equation of state (PR) is 24.69% for the same data sets. The
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advantage of the HSE model over PR is that there is no need to know the critical properties of the solids.4 Solubilities of xanthines in SC CO2 have been determined by a static method. Solubilities of caffeine in SC CO2 are about one order of magnitude higher than those of theophylline and about two orders of magnitude higher than those of theobromine.5 The solubilities of phenylacetic acid and vanillan in SC CO2 have been measured and then used to evaluate various models. The results showed that traditional cubic equations of state models, such as PR, have serious limitations when applied to these systems, and density-based equations perform much better.6 Verete presented a predictive method for solubility of solids in SC CO2 using the molecular weight of the solute and its melting temperature as input parameters and reported prediction errors of 6–117% for polar compounds.7 Ashour and co-workers studied three cubic equations of state combined with different mixing rules for prediction of solid–SCF equilibria and found that the Estevez model is superior in comparison with the other models. However, this model would get worse at higher pressures. The authors concluded that no single cubic equation of state currently exists that is equally suitable for the quantitative prediction of all SCF mixtures and also recommended quantitative molecular analysis.8 For collection of experimental solubility data in SCF, researchers are faced with several difficulties including (1) complex and expensive experimental apparatus, (2) time-consuming studies for determination of equilibrium solubility, and (3) scarce and often unreliable literature data. In addition, there is a lack of accurate models to simulate the SCF process design, which presents a major challenge in scale-up utilization of SCF technologies. There are > 40 different forms of equations of state and 15 various mixing rules.9 When dealing with the equations of state to calculate solute solubility in SC CO2, it is necessary to select the correct equation and choose the mixing rule to be applied. Usually, a researcher uses an equation of state and a mixing rule based on his/ her experience and, as already indicated, a single model cannot treat the solubility profile of all compounds of interest successfully. In addition, these models require a degree of knowledge of the physicochemical properties of the solute that are often not readily available. In some cases the estimated physicochemical properties using a group contribution and/or other computational methods are used, but these may cause another error source. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 91, NO. 5, MAY 2002
Hutchenson and Foster10 reported deviations between experimental and calculated solubilities using the PR equation of state11 of > 50% for naphthalene solubility data in SC CO2. Trablesi and co-workers12 employed the PR and Bublik– Mansoori equations of state13,14 to correlate a- and b-naphthol solubility data in SC CO2 and showed that both equations produced an average absolute relative deviation (AARD) of 21%. The authors also assessed the correlation ability of the Bublik– Mansoori equation of state using 13 solubility data sets of different compounds in SC CO2, which resulted in an AARD of 22.6%. To provide a predictive equation, an average value of solute–solvent interaction parameter (a12) was used for nine compounds, which possess a solubility of < 103 mole fraction and high AARD of 40%.12 The a12 parameter is calculated using all experimental points in each data set by an iteration method. The next models to be used to compute the solute solubility in SCF were empirical relationships between the solubility and the independent variables (such as pressure, temperature, and density). These relationships have the advantage that the corresponding calculations are simple and easily available in commercial software. A disadvantage is that they possess a number of curve-fitting parameters that should be calculated using experimental data points. As an alternative solution to predict solubility data, one can employ correlative empirical equations, which can be applied after the equation is trained. To train an empirical equation, a minimum number of experimental data points should be fitted to the model to compute the model constants. From these constants it is suggested that the trained model is able to predict the solute solubility in the SCF at other temperatures and pressures of interest by using an interpolation technique. This method has previously been evaluated to predict solubility in aqueous binary solvents15 and also in electrophoretic mobility predictions in capillary electrophoresis.16 In this work, the solubilities of nicotinic acid (NA) and p-acetoxyacetanilide (PAA) were measured in SC CO2 at temperatures of 35, 55, and 758C and pressures of 100–200 bars. Then a minimum number of experimental solubility points of NA and PAA in SC CO2 were used to compute the proposed model constants. An attempt was then made to predict the solubility at other pressures and temperatures using the trained model. PAA is a synthetic precursor that is structurally similar to paracetamol (acetaminophen) and also a sug-
SOLUBILITY PREDICTION IN SUPERCRITICAL CO2
gested prodrug.17 The COCH3 group chemically discriminates PAA from acetaminophen and is responsible for higher (by a factor of 2–10 depending on pressure and temperature) solubility of this compound relative to the acetaminophen solubility and thus provides a good model substance to test the applicability of the solubility models in a wide dynamic range of mole fraction. NA was used as a model compound to study particle formation process with SC CO2 as an antisolvent.18 It belongs to a group of hydrosoluble B vitamins used for the treatment of hyperlipidemias (reduction of elevated blood lipid levels), and was selected on the basis of its relatively low solubility in pure CO2 (on the order of 107 mole fraction).
COMPUTATIONAL METHODS Chrastil19 correlated the solubility of a solute (S, g L1) in SC CO2 to the density (r, g mL1) and temperature (T, K). The model is: ln S ¼ A0 þ
A1 þ A2 ln r T
ð1Þ
where A0 –A2 are the model constants that can be estimated from experimental solubility data in SC CO2. del Valle and Aguilera20 then added one more term (1/T2) to eq. 1. The resulting equation is: ln S ¼ B0 þ
B1 B3 þ B2 ln r þ 2 T T
ð2Þ
where B0 –B3 are the model constants. For eqs. 1 and 2, it is possible to use mole fraction solubility instead of g L1 concentration, which provides a more accurate correlation.21 It is also possible to correlate the mole fraction solubility of a solute ( y2) with pressure (P, bar) and temperature.22 The model is: 2
y2 ¼ C0 þ C1 P þ C2 P þ C3 PTð1 y2 Þ þ C4 T þ C5 T2
ð3Þ
where C0 –C5 are the curve-fitting parameters and (1 y2) represents the mole fraction of the solvent (SC CO2). Gordillo and co-workers23 proposed another empirical model. The model is: ln y2 ¼ D0 þ D1 P þ D2 P2 þ D3 PT þ D4 T þ D5 T 2
ð4Þ
where D0 –D5 are the model constants. The superiority of eq. 4 over eqs. 1–3 has been shown
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using penicillin G solubility data in SC CO2, where reported mean percentage deviations for eqs. 1–4 were 32.4, 32.4, 22.9, and 14.4%, respectively.23 The authors also compared mean percentage deviation values with those of two equations of state, namely, Redlich–Kwong and Soave–Redlich–Kwong, and found 23 and 21%, respectively. As seen from mean percentage deviation values for penicillin G solubility data, the accuracy order for the equations are eq. 4 > Soave– Redlich–Kwong > eq. 3 > Redlich–Kwong > eqs. 2 and 1. Further comparisons are reported between different empirical relationships throughout this work. An empirical equation based on response surface methodology has also been employed to correlate solubility data in SCF.24 The model is: y2 ¼ E0 þ E1 P þ E2 T þ E3 PT
ð5Þ
where E0 –E3 are the curve-fitting parameters. In contrast, in our earlier work,21 an empirical equation was presented that provides better correlation than those just presented. The model is: ln y2 ¼ M0 þ M1 P þ M2 P2 þ M3 PT þ
M4 T P
þ M5 ln r
ð6Þ
where M0 –M5 are the model constants and r is the density of pure SC CO2 at different pressures and temperatures. In further studies, by replacing P with r, a more accurate model is produced, and this version has been used in this work. The resultant equation is: ln y2 ¼ K0 þ K1 r þ K2 P2 þ K3 PT þ
K4 T P
þ K5 ln r
ð7Þ
where K0 –K5 are the model constants. In these equations, the numerical values of r are calculated using the following equation:21 pffiffiffiffi 3966:170 ln r ¼ 27:091 þ 0:609 T þ T pffiffiffiffi 3:445P þ 0:401 P T
ð8Þ
To test the correlation ability of the models, whole experimental data points in each set have been fitted to the equations and their accuracies have been compared using back-calculated solubilities. A possible application for this type of calculation is the screening of experimental solubility data for possible outliers where re-determination JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 91, NO. 5, MAY 2002
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is required. To evaluate the prediction capability of the proposed model, three solubility data points at the lowest temperature and the lowest, middle, and the highest pressures, with three corresponding data points at the highest temperature have been used to compute the model constants. From these results, the solubility at other temperatures and pressures has been predicted using the trained model. Overall, it is suggested that these computations may provide a useful technique to speed up the pressure and temperature optimization procedure in SCF studies in industry. It can be indicated that in operating the models, as a general rule, the more curve-fitting parameters there are, the more accurate the correlations. To provide a reliable accuracy criterion to compare the models possessing different numbers of curve-fitting parameters, the AARD values were used, which is calculated by: yObserved 100 X ycalculated 2 2 AARD ¼ ð9Þ Observed ðN ZÞ y2 where N is the number of data points in each set and Z is the number of curve-fitting parameters for each model. In addition, the individual absolute relative deviations (IARD) of predicted solubilities from observed values were calculated to evaluate the prediction capability of the proposed model. The IARD was calculated by: calculated ! y yObserved 2 2 IARD ¼ 100 ð10Þ Observed y2
EXPERIMENTAL METHODS Dynamic Flow-Through Solubility System A schematic diagram of the dynamic flow-through solubility system is shown in Figure 1. Initially, the lowest pressure for the solubility study was established using pure SC CO2 pumped through the system at defined flow rates with high-performance liquid chromatography (HPLC) pumps (Jasco PU980/6, Japan). An HPLC ultraviolet (UV) detector (Jasco UV-1575, Japan), equipped with a high-pressure UV flow cell (Jasco UV-975, Japan), was placed between the oven and back pressure regulator (Jasco 880-881, Japan). Both 6-way Rheodyne valves were switched to blank mode to direct the SC CO2 flow along a bypass line and through the high-pressure flow cell. UV data were captured and processed by a computer with JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 91, NO. 5, MAY 2002
Figure 1. A schematic diagram of set up for dynamic solubility measurement.
Borwin version 1.21 software (JMBS developments, France). In the blank mode, the detector response was set to zero to establish a stable base line and then the valves were switched to sample mode, diverting the SC CO2 through the extraction vessel. As NA or PAA extracted into the SC CO2, a characteristic UV response plateau was observed. The pressure was then raised to the next selected level at the back pressure regulator. This procedure was repeated at a range of pressures and temperatures to establish a comprehensive data set at defined conditions. A separate on-line UV detector calibration curve was also prepared using a series of dilutions of NA and PAA solutions (100 mg/100mL, 0.1% w/v) in methanol. For this curve, a wide range of calibration standards were examined at 263 and 254 nm to adequately bracket solubility values in pure SC CO2. From the results, the sample response plateau data were interpolated from the calibration curves to determine the concentration of NA and PAA present in the SC CO2. Sample Preparation Acid washed glass beads (212–300 mm diameter; Sigma Chemicals, Poole, UK) were mixed with NA in a 4:1 ratio by weight of glass beads/NA using a Turbula mixer (Glen Creston, T2C, No. 860961, Stanmore, England) for 30 min to ensure that a homogeneous mixture was achieved. The presence of the glass beads enhances surface area for solute/solvent interaction, prevents caking of the
SOLUBILITY PREDICTION IN SUPERCRITICAL CO2
drug, and eliminates channeling of the SC CO2 through the bed. A 10-mL stainless steel pressure vessel was carefully packed with the glass beads/ NA mixture and placed into a fan-assisted oven (Applied Separation, Allentown, PA) to maintain isothermal conditions. The glass beads/NA mixture was sandwiched between circular paper filters (Schleicher & Schuell, Dassel, Germany) to minimize movement of the packing material and avoid any entrainment of material. All samples were investigated in the pressure range 100–200 bar and at the temperatures of 35, 55, and 758C. For PAA, the same sample preparation was used except that a 3:1 ratio by weight of glass beads/PAA was used. NA (> 99% purity) was purchased from Sigma Chemicals (Poole, Dorset, UK), and solvents were of HPLC grade from Fisher Scientific (Loughborough, Leicestershire, UK). Liquid CO2 (99.99% purity) was supplied by Boc Limited (Guildford, Surrey, UK). PAA was synthesized in house as reported by Bauguess et al.25 at the research laboratories of the University of Bradford, and identification was confirmed by a wide range of analytical techniques.
RESULTS AND DISCUSSION The logarithm of mole fraction solubility data for NA and PAA in pure SC CO2 are shown in Table 1.
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The solubility of both compounds increases with an increase in pressure at each temperature used because there is an increase in density and solvent power of pure SC CO2 at each pressure increase. As a result, with increased density, the intermolecular distance is decreased; therefore, the solute–solvent interactions are increased. As a result of these effects, it can be concluded that complex behavior for solubility at different temperature is observed, where the temperature affects the solute vapor pressure, the solvent density, and the intermolecular interactions in the fluid phase. By taking the logarithm of the experimental solubility data, a good linear correlation with respect to the logarithm of the density of pure SC CO2 at each temperature is achieved; this provides a check on the internal consistency of the data.19,26 As already explained, to evaluate the correlation ability of the proposed model, the generated experimental solubility data have been fitted to eq. 7, and the obtained models are shown in eqs. 11 and 12 for NA and PAA, respectively: ln y2 ¼ 20:005 3:472r 0:000151P2 1:656T þ 3:704 ln r þ 0:000225PT þ P ð11Þ
Table 1. Logarithm of Experimental Mole Fraction Solubility of Nicotinic Acid and p-Acetoxyacetanilide, and Density of Pure SC CO2 at Different Temperatures and Pressures T (8K)
P (bar)
Density (g/cm3)a
Nicotinic Acid
p-Acetoxyacetanilide
308.15
100 120 140 160 180 200 100 120 140 160 180 200 100 120 140 160 180 200
0.5286 0.6198 0.7047 0.7818 0.8505 0.9098 0.3633 0.4319 0.4978 0.5599 0.6174 0.6695 0.2684 0.3229 0.3767 0.4289 0.4786 0.5253
13.62 13.51 13.33 13.26 13.15 13.12 13.89 13.21 13.27 12.66 12.46 12.32 13.62 13.33 — 12.35 11.90 11.79
11.75 — 10.77 10.63 10.54 10.47 12.53 11.43 10.28 9.94 9.75 9.65 — 11.77 11.01 9.91 9.55 9.40
328.15
348.15
a
The density of pure SC CO2 was calculated with eq. 8.21 JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 91, NO. 5, MAY 2002
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where R ¼ 0.987, F ¼ 84, AARD ¼ 11.6%, N ¼ 17, and p < 0.0005. ln y2 ¼ 0:219 11:373r 0:000071P2 1:230T þ 6:302 ln r þ 0:000079PT P ð12Þ where R ¼ 0.991, F ¼ 109, AARD ¼ 16.9%, N ¼ 16, and p < 0.0005. The results show that eq. 7 reproduces the best estimate of experimental solubility, where the expected percent deviation has a mean of < 14% which is considered to be an acceptable value when screening for the best conditions. From eqs. 11 and 12 it is possible to calculate solute solubility at temperatures between 35 and 758C and pressures between 100 and 200 bar using an interpolation technique. It can be added that as a general rule, the more training data points included in the training process of the models, the more accurate the solubility predictions. However, collecting a large number of experimental solubility data points in an SCF experiment is not appropriate because it is costly and time consuming. Therefore, as a computational solution to this problem, a minimum number of experimental data points (equal to the number of curve-fitting parameters; that is, six data points) have been employed to train the proposed model. Then, the solubility at other T and P values has been predicted using the trained model. By using only six data points, the produced AARD values for NA and PAA are 17.0 and 15.7%, respectively. These prediction errors are still considered to be quite reasonable, particularly because the reported experimental solubility data at the same conditions are so different. As an example of this variability, the reported solubility data of octacosane in SC CO2 at 358C from four different research groups shows the differences of more than a factor of 10.27 From our experiments and by using an online solubility system, the relative standard deviations for repeated points lie within 15%, indicating that the experimental technique is robust.18 To further investigate the correlation capability of the empirical equations, more sets of solubility data in SC CO2 were collected from the literature. Their detail, references, and the produced AARDs are shown in Table 2. For the data sets for ketoprofen, salicylic acid, sulfadimethoxine, and sulfamerazine, the models produced large AARDs. However, for caffeine, theobromine, theophylline, and vitamin D2, small AARDs were JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 91, NO. 5, MAY 2002
given. It was therefore obvious that a single model could not produce accurate results in all cases. It should also be noted that a possible outlier could affect the regression model and result in a relatively high AARD. Nevertheless, the global (mean) AARDs are < 20% for most models studied, and this was considered to be an acceptable error for empirical equations because there is no need to employ a complicated calculation process or include any physicochemical properties of the solute. But one of the criterion set before starting the work was to obtain the lowest achievable error and thus to provide the most accurate solubility predictions. Equation 7 produced the most accurate solubilities, with global AARD equal to 12.6%, and eq. 3 produced the least accurate results, with 24.8%. It is suggested that percentage errors < 20% are acceptable because the produced correlation error for equations of state (which are used in chemical engineering calculations) is 20%.12 The global AARD for eq. 7 is significantly less than those of eqs. 1–6 (paired t test). By calculating the AARD and its global value, the overall error is presented. In contrast, the IARD shows the error level at each temperature and pressure and is able to detect the possible outliers in each of the data sets. The distribution of the IARD for eqs. 1–7 at four subgroups (I, < 15% error; II, 15–30% error; III, 30–45% error; and IV, > 45% error) is shown in Figure 2. Equations 1 and 2 produced an IARD of subgroup I in 60% of all the cases, whereas 7% of the produced IARD values lie in subgroup IV. The frequency of the cases in subgroups II and III are 25 and 10%, respectively. It is obvious that with an accurate model, the lower frequencies are expected for subgroups II, III, and IV. The corresponding relative frequencies for eq. 7 are 81, 14, 3, and 2%. The IARD distribution for eq. 6 is very close to that of eq. 7. However, for eqs. 3–5, a different pattern was observed. As an overall result, the proposed equation provides acceptable IARD distribution when compared with eqs. 1–5. A minimum number of six experimental data points from the collected data sets were used to train the proposed model. Once trained, the solubility at other temperatures and pressures were predicted by an interpolation technique. Here, the IARD values represent the real percentage differences between predicted (not correlated) values and experimental solubilities using the proposed model (eq. 7), which is shown in Figure 3. The mean IARD for 401 solubility data points (23 solubility data sets mentioned in
SOLUBILITY PREDICTION IN SUPERCRITICAL CO2
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Table 2. Solubility Data in Supercritical Carbon Dioxide, their References, and Average Absolute Relative Deviations (AARD) for Eqs. 1–7 No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Solute Nicotinic acid p-Acetoxyacetanilide Caffeine b-Carotene Ketoprofen Naproxen Nifedipine Nitrendipine Penicillin G Penicillin V Piroxicam Pyrocatechol Resorcinol Retinol Salicylic acid Sulfadimethoxine Sulfamerazine Sulfamethazine Theobromine Theophylline Vitamin D2 Vitamin D3 Vitamin K1
Reference
N
Eq. 1
Eq. 2
Eq. 3
Eq. 4
Eq. 5
Eq. 6
Eq. 7
This work This work 5 28 29 30 31 31 23 32 33 34 34 35 26 36 36 36 5 5 35 35 35 Mean SD
17 16 24 27 15 18 29 42 18 24 9 32 32 20 49 19 18 20 23 24 19 23 24
21.0 35.9 6.3 29.0 54.6 21.5 19.2 20.2 23.4 14.8 23.7 8.8 8.9 9.9 47.0 41.1 37.8 16.6 7.9 7.7 8.6 31.6 12.8 22.1a 13.7
22.5 36.4 5.4 27.6 59.6 23.2 20.4 19.6 24.9 15.5 28.4 9.1 9.3 7.6 48.2 43.9 40.5 17.6 7.6 7.6 8.5 30.2 10.6 22.8a 14.9
16.6 28.9 6.1 108.9 87.2 12.8 39.5 25.0 34.9 8.9 14.6 7.9 5.5 9.2 30.1 17.8 17.7 8.0 4.8 7.6 5.7 67.2 5.1 24.8b 27.6
13.2 20.8 4.5 26.2 39.5 16.0 14.6 19.3 22.9 17.7 22.4 8.6 3.3 6.8 32.2 28.2 36.6 8.7 4.7 6.1 5.7 25.3 9.8 17.1c 10.8
14.8 36.7 5.3 88.0 71.1 11.0 40.0 28.7 29.9 8.0 8.7 8.1 6.5 9.5 28.5 15.4 15.2 7.0 4.7 6.8 8.6 39.2 4.7 21.6d 21.8
14.8 30.3 3.9 23.8 20.6 12.8 13.2 16.0 17.7 6.7 21.6 6.3 2.8 9.6 19.2 28.3 34.0 8.7 4.8 6.2 5.8 26.8 9.9 14.9e 9.2
11.6 16.9 4.3 30.4 11.8 11.2 13.8 18.5 16.2 5.9 15.0 6.0 3.7 9.4 22.0 19.2 18.0 5.7 4.9 5.9 6.2 26.2 7.1 12.6 7.4
a
p < 0.0005; paired t test showed significant difference between global (mean) AARDs with that of eq. 7. p < 0.018. c p < 0.007. d p < 0.021. e p < 0.042. b
Table 2) is 17%. This prediction error could be considered as acceptable because there is huge variations in solubility data reported from different laboratories.27 In addition the reproducibility
of solubility data in SC CO2 from our group is 10% in comparison with the literature values.18 As a general conclusion, the proposed model (eq. 7) produced accurate results and therefore it
Figure 2. Distribution of individual absolute relative deviation (IARD) for eqs. 1–7 in subgroups I–IV.
Figure 3. Distribution of IARD in different subgroups produced by trained proposed model using six experimental data points. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 91, NO. 5, MAY 2002
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is suggested that it could be employed in practice to simulate the solubility data in SC CO2 after conducting a minimum number of experiments where solubility prediction would be helpful to carry out the method development process. An extension of this work has been the use of the model to screen the experimental data points to find any possible outliers. In operating the model in practice. the advantages are its simple calculation procedure and higher accuracy in comparison with other empirical equations and equations of state. However, from this investigation, it is obvious that a single model cannot be appropriate in every case, which could be considered as one possible disadvantage for a model. Of course, this comment also applies to the other models in the literature. Yet, because the proposed model produced the most accurate results among other models, as shown here when using 23 data sets, it should produce accurate solubilities for other data sets and is the best practical model that could be employed in industry.
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ACKNOWLEDGMENTS 13.
The authors thank the Australian Department of Education, Training and Youth Affairs, and the University of Sydney for providing the IPRS and IPA scholarships. The authors also thank Dr. Andreas Kordikowski, of Bradford Particle Design Plc, for his help in the synthesis of PAA.
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