Application of machine learning approach to estimate the solubility of some solid drugs in supercritical CO₂

Abstract

Accurate estimation of the solubility of solid drugs (SDs) in the supercritical carbon dioxide (SC-CO₂) plays an essential role in the related technologies. In this study, artificial intelligence models (AIMs) by gene expression programming (GEP) and adaptive neuro-fuzzy inference system (ANFIS) methods were applied to estimate the solubility of SDs in SC-CO₂. Hence, a comprehensive database (1816 datasets) comprising operational conditions (T, P) in the wide ranges (308–348.2 K and 80–400 bar), SD’s molecular weight (MW_SDs), and melting point (MP_SDs) were gathered. Investigation analysis of the models’ strength showed that the model developed by ANFIS exhibited a more satisfactory approximation than the GEP model. According to the optimized ANFIS model, statistical parameters of R², RMSE, MAE, and AARD% were obtained, equivalent to 0.991, 0.260, 0.167, and 13.890% for training and 0.990, 0.256, 0.157, and 15.273% for validation, in that order. Sensitivity analysis showed that the highest effect of independent variables on calculating SDs solubility in SC-CO₂ belong to MW_SDs, P, MP_SDs, and T, respectively. Therefore, MW_SDs is a key factor for modeling the solubility of various SDs in SC-CO₂. Comparing the estimated results obtained from the optimized AIM with previous semi-empirical models showed that the AIMs could be more accurate in modeling the solubility of SDs in SC-CO₂.

Introduction

In recent years, supercritical fluid (SCF) technology has garnered significant attention as an environmentally friendly option due to its unique characteristics such as low viscosity, high diffusion capabilities, strong solvation capacity, and excellent selectivity. Regarding polar compounds, while SC-CO₂ has moderate or even minimal solubility, it remains the most prevalent SCF due to numerous unique traits like low critical temperature and pressure, non-flammability, non-explosiveness, non-toxicity, economic efficiency, and particularly its ease of separation and recycling¹. SC-CO₂ has demonstrated significant capabilities in diverse domains, energy², comprising nanoparticle formation³, food⁴, extraction of essential oil⁵, seed oil⁶, solubility⁷, anti-cancer drugs^8,9, polymer synthesis⁵, impregnation¹⁰, optimization and mathematical modeling¹¹, etc. Several studies showed that the micronization of a drug leads to improving the performance and absorption in the body and reducing side effects initiated from using the drug¹². For this purpose, various methods such as grinding, recrystallization, spray-drying, freeze-drying, and crystallization have been utilized up to now. These prevalent micronization techniques are associated with issues such as significant thermal stress, thermal degradation, broad particle size distribution, and elevated residual solvent in the end product. Recently, new SCF-based technologies have arisen to achieve a high-quality product with a consistent particle size distribution¹³.

The key and fundamental effective factor in SCF extraction processes is the precise understanding of the equilibrium solubility of substances. Despite various methods to experimentally determine drug solubility, a specific set of conditions and tools is necessary due to their intricate and sensitive designs¹⁴. The difficulty and expense of obtaining equilibrium data at high pressures and temperatures increase the need to estimate drug solubility in SC-CO₂¹⁵. The solubility of SDs in SCF can be estimated through thermodynamic equations of state (EoSs), fluid density-based experimental and semi-experimental models, solution activity coefficient-based models, and artificial intelligence techniques¹⁶.

Numerous experimental investigations were conducted in the field of the solubility measurement of SDs in SC-CO₂, which are given in Table 1. Some of these researchers presented experimental and semi-experimental models as follows. Sodeifian et al. measured the solubility of imatinib mesylate in SC-CO₂ and analyzed the data using two sets of procedures involving three EoSs and six semi-experimental models. Their findings highlighted the effectiveness of the Haghtalab et al. EoS with the VdW2 mixing rule over other models. They subsequently introduced a new model, named Sodeifian et al., which incorporates six adjustable parameters and accurately estimates the experimental solubility of imatinib mesylate¹⁷. Housaindokht and Bozorgmehr estimate the solubility of three SDs in SC-CO₂. The estimation was performed by three EoSs (Peng-Robinson, modified Peng-Robinson and Pauzuki) as well as three mixing rules (VdW1, VdW2 and Adachi-Sugi). The optimal outcome was obtained from the Pauzuki equation with the VdW2 mixing rule¹⁸. Huang et al. measured the solubility of aspirin in SC-CO₂ with and without two solvents of ethanol and methanol. Then they estimate experimental data with the Peng-Robinson EoS and VdW mixing rule. Peng-Robinson EoS with an average absolute relative deviation (AARD) of (14–23%) showed good accordance between both experimental and estimated data¹⁹. Ardestani et al. measured the solubility of capecitabine using seven experimental models and three EoSs combined with the mixing rule. The EoS of PR-VdW2 (AARD = 20.32%) could estimate the solubility better⁹.

Table 1.

Experimental data of SDs solubility in the SC-CO₂ at various operating conditions.

No.	Compound	Formula	MWSDs (g/mol)	Temperature (K)	Pressure (bar)	Density (kg/m3)	MPSDs (K)	Solubility × 102 (g/l)	N*	Refs.
1	Capecitabine		359.35	308-348.15	100–354	234–955	383.15-394.15	0.588–77.6	75	9,32
2	Busulfan		246.304	308–338	120–400	388-973.35	387.15-390.15	0.67-39	32	24
3	Letrozole		285.31	308-348.2	120–360	318.9–955	454.15-456.15	0.3–43.6	65	33,34
4	Sorafenib Tosylate		637.03	308–338	120–270	388–914	513.15-516.15	0.76–14.2	24	35
5	Azathioprine		277.263	308–338	120–270	388–914	516.15-517.15	0.7-9	24	36
6	Sunitinib Malate		532.56	308–338	120–270	388–914	462.15-464.15	5–81	24	37
7	Paclitaxel		853.9	311.1-328.9	141.2-346.2	620.44-943.06	486.15	1.65–12.6	12	38
8	Flutamide		276.1	308–348	122–355	327–955	385.15	0.9–254	45	39
9	Megestrol Acetate		384.5	308–338	122–355	396–955	487.15	1-46.6	36	40
10	Tamoxifen		371.51	308–338	120–400	388-973.35	370.15-371.15	0.5–67.8	32	41
11	Aprepitant		534.4	308.15-338.15	120–330	384–944	517.15-519.15	2.77–77.4	32	42
12	5-Fluorouracil		130.078	313.15-328.15	80–200	203.64-839.81	555.15-559.15	0.095–1.17	12	43
13	Imatinib Mesylate		589.71	308.2-338.2	120–270	388–914	487.15-497.15	0.05–4.6	24	17
14	Docetaxel		807.87	308–348	152–354	477–955	459.15-465.15	54-1046	38	32
15	Anastrazole		293.37	308–348	122–355	327–955	354.15-355.15	0.9–206	45	34
16	Tamsulosin		408.05	308–338	120–270	384–914	499.15-501.15	0.6–73.6	24	44
17	Dutasteride		528.5	308–348	122–355	327–955	515.15-523.15	0.2–165	45	39
18	Finasteride		372.5	308–348	122–355	327–955	526.15	14–288	45	39
19	Decitabine		228.21	308–338	120–400	401.1-962.94	473.15	5.9–477	32	45
20	Temozolomide		194.154	308–338	120–400	401.1-962.94	485.15	76-1976	32	46
21	Exemestane		296.40	308–348	122–355	327–955	465.15-475.15	6.7-1026.7	14	34
22	Silymarin		482.44	308–338	80–220	181.84-881.15	431.15	0.538–63.5	32	21
23	Pazopanib Hydrochloride		474	308–338	120–270	384–914	573.15-577.15	1.5–12	24	47
24	Gefitinib hydrochloride		483.40	308–338	120–270	388–914	392.15-393.15	0. 4-2.9	24	48
25	Triamcinolone acetonide		434.5	308–338	120–270	388–914	566.15	5.2–165	24	49
26	Rivaroxaban		435.90	308–338	120–270	388–914	503.15	0. 81-32.42	24	50
27	Dapagliflozin propanediol		502.98	308–338	120–270	388–914	475.803 + 7	1.7–71	24	51
28	Galantamine		287.359	308–338	120–270	388–914	400.15	0. 2-13.6	24	52
29	Esomeprazole		345.417	308.2-338.2	120–270	382.44-912.88	428.15	3-559	24	53
30	Ceftriaxone sodium		598.5	308–338	120–270	388–914	509.15	0. 5-8.6	24	54
31	Riluzole		234.2	308.2-338.2	120–270	386.82-914.07	392.15	10–62	24	55
32	Metoclopramide hydrochloride		336.26	308–338	120–270	388–914	455.30	4.4-33.29	24	56
33	Teriflunomide		270.2	308–338	120–270	388–914	503	20.98-261.06	24	57
34	Repaglinide		452.595	308–338	120–270	384–914	407.05	1.1–76.7	24	58
35	Oxcarbazepine		252.273	308–338	120–270	384–914	489	0. 02-16.39	24	59
36	Sodium Valproate		166.19	308.15-338.15	120–270	388–914	573.15	0. 1-10.9	24	60
37	Dasatinib monohydrate		506.02	308–338	120–270	384–914	559	0. 4-8.18	24	61
38	Sulfabenzamide		276.31	308–338	120–270	388–914	453.15-457.15	0. 74-10.99	24	62
39	Loratadine		382.88	308.15-338.15	120–270	388–914	407	1.51-888.45	24	63
40	Pantoprazole sodium		432.4	308–338	120–270	388–914	412	1.01–31.63	24	64
41	Palbociclib		447.53	308–338	120–270	388–914	545.19	0. 31-74.3	24	65
42	Codeine phosphate		397.4	308–338	120–270	388–914	428.15	8-45.9	24	66
43	Sertraline hydrochloride		342.69	308–338	120–300	347.04-909.94	521–524	1.06–58.65	28	67
44	2,4,7-Triamino-6-phenylpteridine		253.263	308–338	120–270	384–914	589.15	0. 2–13	24	68
45	Ibrutinib		440.5	308.2-338.2	120–270	388–914	431.15	0. 41-100.2	24	69
46	Amiodarone hydrochloride		681.77	313.2-343.2	120–300	347.04-909.94	427–431	12.7-1169.6	28	70
47	Lansoprazole		369.363	308.2-338.2	120–270	382.44-912.88	439	4-484	24	71
48	Clemastine fumarate		459.96	308–338	120–270	388–914	451.15	1.3–7.71	24	72
49	Empagliflozin		450.9	308–338	120–270	388–914	426.1	2.04–20.79	24	73
50	Hydroxychloroquine sulfate		434	308–338	120–270	388–914	513.15	2.31–42.62	24	74
51	Morphine sulfate		758.83	308–338	120–270	388–914	510.05	5.3–79.7	24	75
52	Oxycodone hydrochloride		351.8	308–338	120–270	388–914	475.803	0. 3-4.3	24	76
53	Fexofenadine hydrochloride		538.10	308.2-338.2	120–270	388–914	472.46	3.4–81.8	24	77
54	Pholcodine		398.5	308–338	120–270	388–914	364	72.1-421.2	24	78
55	Tramadol hydrochloride		299.83	308–338	120–270	388–914	451.15-454.15	6.4–73.7	24	79
56	Buprenorphine hydrochloride		504.10	308.1-338.1	120–270	388–914	533.15-535.15	10.7-748.3	24	80
57	Regorafenib monohydrate		500.83	308–338	120–270	388–914	448.15-451.15	0. 13- 5.53	24	81
58	Minoxidil		209.25	308–338	120–270	384.17-913.69	498.15	0. 046-1.239	24	82
59	Losartan potassium		461	308–338	120–270	388–914	538.51	0. 75-14.12	24	83
60	Amlodipine Besylate		567.05	308–338	120–270	388–914	473.15	4.108–23.205	24	84
61	Quetiapine hemifumarate		441.54	308–338	120–270	388–914	446.62	0. 23-14.2	24	85
62	Ketotifen fumarate		425.5	308.2-338.2	120–300	384.17-929.68	465.15	7.8- 844.2	28	86
63	Ketoconazole		531	308–338	120–300	384.17-929.68	423	0. 1-78.4	28	87
64	Crizotinib		450.3	308–338	120–270	388–914	465.15	0. 6-9.8	24	88
65	Nilotinib hydrochloride		584	308–338	120–270	388–914	480.15	0. 2-9.4	24	30
66	Prazosin hydrochroride		419.9	308–338	120–270	384–914	550.15-553.15	11.7–53.8	24	89

* Number of Data.

Recently, machine learning technology has been widely used as estimation tools across various scientific fields, such as estimating the solubility of solid materials in SC-CO₂. This approach will reduce massive numerical calculations and expenses if a suitable thermodynamic model is provided²⁰. One of the most famous of them is the artificial neural networks (ANNs) operate as a powerful methodology for the comparison and modeling of complex nonlinear problems, facilitating the estimation of solutions for such challenges. Sodeifian et al. studied the solubility of six pharmaceutical compounds in SC-CO₂ using four different models: cubic EoS, empirical and semi-empirical models, regular solution with the Flory-Huggins equation, and ANN. comparing results indicated that the ANN-based model provided the best results in terms of correlating the experimental solubility of the pharmaceutical compounds in SC-CO₂²⁰. Yang et al. measured the solubility of silymarin in pure SC-CO₂ and with a cosolvent. They used three semi-experimental models and one backpropagation artificial neural network (BPANN) model to estimate solubility and compared the results to experimental data. The BPANN model demonstrated higher accuracy, with an AARD of 1.14–2.15%²¹. Mehdizadeh and Movagharnejad estimated the solubility of thirty different compounds in SC-CO₂ using seven semi-experimental models and compared the results to those obtained from an ANN. The ANN method showed greater precision, with an average relative deviation (ARD) of 5.3%, compared to 15.96% for the best semi-experimental equation. Additionally, the ARD of the semi-experimental equations varied significantly across different compounds, whereas the ANN method exhibited more consistent performance regardless of the material type²². Dadkhah et al. estimated the solubility of solid substances in SC-CO₂ by employing the adaptive neuro-fuzzy inference system (ANFIS) model. The model’s input parameters include several thermodynamic properties of the compounds, such as critical temperature, critical pressure, acentric factor, critical molar volume, as well as operational temperature and pressure. The correlation coefficient and The Root Mean Square Error (RMSE) values for all the data were found to be 0.983 and 0.156, respectively. The results of the model comparison showed that the ANFIS model performs superiorly to the FFNN model in estimating the target data²³. Pishnamazi et al. measured the solubility of busulfan, which is an SD to treat blood cancer, in SC-CO₂ using five semi-experimental models, the KJ model (AARD = 7.57%) presented a more suitable estimation of solubility²⁴. Zhu et al. estimated the solubility of busulfan in SC-CO₂ using ANFIS. The results showed that the ANFIS model can be utilized to precisely estimate the solubility of drugs in supercritical solvents⁸.

The literature includes only a few general semi-empirical models for estimating the solubility of solids in SC-CO₂^25–27. The primary drawback of these models is that the parameters of the model for a particular solid must be determined solely by utilizing its own experimental solubility datasets. This makes it difficult to use easily and restricts the model’s performance to apply for other SDs.

According to the literature, there is no rigorous model to estimate the solubility of a specific SD in SC-CO₂ without needing its sufficient experimental datasets used to fit the coefficients of semi-empirical models. On the other hand, the empirical determination of the solubility of solids under broad conditions is not only often troubled but also requires a significant amount of cost to carry out^28–30. To address this challenge, the present study aimed to estimate the solubility of SDs in SC-CO₂ using AIMs. The selected models include GEP and ANFIS. These models offer several advantages, including reduced time required and the ability to model complex nonlinear processes due to having a number of controllable parameters, which enhances the algorithm compared to other methods³¹. In order to derive the rigorous AIMs, a comprehensive data collection including some SDs properties additional to operating conditions were gathered. Using SDs properties as the input variables enables the model to estimate the solubility of an SD in SC-CO₂ without applying its particular experimental datasets. The input variables of the AIMs were operational temperature and pressure (T, P), SDs’ molecular weight (MW_SDs), and melting point (MP_SDs). Coefficient of determination (R²), root mean square error (RMSE), mean absolute error (MAE), and the percent of Average absolute relative deviation (AARD%), as statistical parameters, were employed to evaluate and validate the results and to select the best model.

Artificial intelligence modeling

Data gathering

The data required for the AIMs was collected by gathering a comprehensive data collection (1816 dataset) from previous studies, as shown in Table 1. Consistent with Fig. 1, the experimental data of T, P, MW_SDs, and MP_SDs variables were selected as the input variables to estimate the solubility of SDs in SC-CO₂. In the current AIMs, a wide range of T, P, MW_SDs, and MP_SDs were investigated, as presented in Table 2.

Fig. 1.

Schematic diagram of the AIMs.

Table 2.

Ranges of input variables of the AIMs.

Variable	T(K)	P(bar)	MWSDs (g/mol)	MPSDs (K)
Range of data	308–348.2	80–400	130.08–853.906	354.65–589.15

Gene expression programming (GEP)

GEP is a combination of genetic algorithms and genetic programming that is based on Darwin’s theory and was developed by Ferreira in 1999⁹⁰. In this method, at the first stage, the initial population of chromosomes are generated randomly or based on input information. The chromosomes are then evaluated as expression trees. If an optimal solution is found, or the number of generations reaches a specified limit, evolution stops, and the best solution is presented. Otherwise, selection is performed, and the process is repeated for several generations to improve the quality of the population. Figure 2 presents the flowchart of GEP⁹⁰. Figure 2 presents the flowchart of GEP.

Fig. 2.

The GEP flowchart⁹⁰.

Development GEP model

The procedure GEP utilizes to present a nonlinear function is mentioned below:

• Entering data: 80% of datasets were selected for training, and 20% of them were selected for validation. The data was randomly selected and introduced to the software. Afterward, the estimator variables and response were identified.

• The preparation of information: Given that the input variables had various ranges and entering raw-form data led to reducing the rate and accuracy of the model, a normalization process was employed to equalize the range of the data, as well as to accomplish better training. The Min-max normalization was conducted by Eq. (1):

1

Where is the normalized data, and are the maximum and minimum values of the data and and are the range in which the datas were normalized, respectively⁹¹.

• Adjusting parameters: The setting of GEP includes some steps as below:

1. The determination of the chromosome’s structure (number of chromosomes; head size; number of genes).

2. The determination of the linking function which determines the relationship between sub-ET (expression tree).

3. The determination of fitness function.

4. The selection of genetic operators, including mutation, IS transposition, RIS transposition, etc., and the determination of their rate.

5. The use of random numerical constants.

The optimal parameters concerning the GEP model settings are given in Table 3. The defaulted values of the software were used for other parameters⁹².

Table 3.

Optimum parameter settings for the GEP model.

General Settings
Sampling
Training Records	Random
Validation Records	Random
Chromosomes	30
Genes	3
Head Size	8
Linking Function	Multiplication
Fitness Function
Function	Enhanced MSE
Genetic Operators
Strategy	Custom
Mutation	0.05
IS Transposition	0.1
RIS Transposition	0.1
One-Point Recombination	0.2
Two-Point Recombination	0.2
Gene Recombination	0.3
Gene Transposition	0.2
Numerical Constants
Constants per Gene	10
Data Type	Floating-Point
Lower Bound	-10
Upper Bound	10

• Selection of functions: as shown Table 4, a collection of functions was selected for modeling. Moreover, the stop condition was MAX fitness. The output of the model is shown in Fig. 3 as tree expression.

Table 4.

Mathematical functions used for the GEP model.

Function	Symbol	Weight	Arity
Addition	+	4	2
Subtraction	-	4	2
Multiplication	*	8	2
Division	/	1	2
Minimum of 2 inputs	Min 2	1	2
Maximum of 2 inputs	Max 2	1	2
x to the power of 3	X3	3	1

Fig. 3.

Expression tree structure for the best GEP model.

The adaptive neuro-fuzzy inference system (ANFIS)

ANFIS is a powerful technique in intelligent computational modeling. This method combines the learning capabilities of ANNs and the logical reasoning of Fuzzy Inference Systems (FIS) to overcome the disadvantages of each. FIS is designed based on “if-then” rules and appropriate membership functions to create an input-output relationship. ANFIS consists of six main sections: data preparation, ANFIS creation, variable selection, training, validation, and output generation, and includes five layers as shown in Fig. 4: Fuzzification Layer, Rule Layer, Normalization Layer, Defuzzification Layer, and Summary Layer. This structure allows ANFIS to effectively process inputs and generate accurate output^93,94. The flowchart of ANFIS is presented in Fig. 5.

Fig. 4.

ANFIS structure.

Fig. 5.

The ANFIS flowchart.

Development ANFIS model

In order to develop AIM based on ANFIS the follow steps were done:

• Entering data: Similar to the GEP model, 80% of datasets were selected for training, and 20% of them were selected for test and check.

• The preparation of information: For the purpose of increasing accuracy and speed of AIM based on ANFIS, all datasets comprising the inputs and the targets were normalized between [− 1, 1] according to Eq. (1).

• The preparation of information: For the purpose of increasing accuracy and speed of AIM based on ANFIS, all datasets comprising the inputs and the targets were normalized between [− 1, 1] according to Eq. (1).

Adjusting parameters:

In order to develop the satisfying AIM based on ANFIS, a Takagi–Sugeno fuzzy inference system was applied by using the fuzzy c-mean (FCM) clustering algorithm. The Gaussian membership function was selected for all independent variables comprising T, P and MW_SDs. The output membership function was chosen linear. Additionally, the specifications of the developed ANFIS model are presented in Table 5.

Table 5.

Specifications of the developed ANFIS model.

Parameter	Descriptions/value
Fuzzy structure	Sugno-type
Initial FIS for training	genfis 3
Number of Clusters	4
Partition Matrix Exponent	2
Number of inputs	4
Number of outputs	1
And method	prod
Or method	probor
Defuzzification method	wtaver
Improvement Method	prod
Number of fuzzy rules	15
Maximum number of iteration	1000
Training maximum epoch number	1000
Minimum Improvement	1e-5
Initial step size	0.04
Step size decrease rate	0.99
Step size increase rate	1.05
Number of nodes	46
Number of linear parameters	20
Number of nonlinear parameters	30
Total number of parameters	50

Model selection

The AIMs were implemented many times to find the optimal setting of the parameters. The best model was selected by a multi-objective strategy as follows:

1. The selection of the simplest model though this parameter is not predominant

2. The training data have the highest value of fitness

3. The validation data have the highest value of fitness

The first objective should be controlled by the user. About the other objectives, the objective function (OBJ) by Eq. (2) was defined as a criterion for the accordance of the output estimated by the model with the output experimentally measured. The selection of the best model was conducted based on the minimum OBJ value that resulted in lower AARD%⁹⁵.

2

3

4

5

6

where No_training and No_validation denote the number of training and validation data, respectively, and P_ij is the value estimated by the individual model i for record j (out of n records). T_j is the target value for record j⁹⁶.

The created objective function considers the variations of R², RMSE, MAE, and AARD%, which are defined in Eqs. (3–6). The higher values of R² and the lower values of RMSE, MAE and AARD% result in reducing the objective function. Consequently, it can present a more precise model. Furthermore, the abovementioned function considers the effects of splitting data sets to the training and validation sets⁹⁵.

Results and discussion

To study the behavior of the AIMs selected based on the optimized parameters, the comparison of output of the models with the training and validation experimental data is shown in Figs. 6 and 7, respectively. The results and the statistical parameters of R², RMSE, MAE, and AARD%, presented in Table 6 indicate that the ANFIS is more satisfactory and reliable in estimating the solubility of SDs in SC-CO₂ than the GEP model. Since the ANFIS model showed the higher R² and lower RMSE, MAE, and AARD% compared to GEP model, it was applied to compare the independent variables importance, to investigate the effect of them on the response, and finally to compare the model with semi-empirical ones.

Fig. 6.

Scattergram of the experimental data versus the estimation by GEP and ANFIS models for training datasets.

Fig. 7.

Scattergram of the experimental data versus the estimation by GEP and ANFIS models for validation datasets.

Table 6.

Performance evaluation of the implemented correlation.

Datasets	AIM	R2	RMSE	MAE	AARD%
	ANFIS	0.991	0.260	0.167	13.890
	GEP	0.900	1.079	0.544	70.091
n	ANFIS	0.990	0.256	0.157	15.273
	GEP	0.690	1.176	0.563	73.092

As before mentioned, to evaluate the model accuracy, 10% of datasets was randomly considered as check data in the ANFIS model. Matching experimental and output values for check data including random SDs at various operating condition is clearly indicated in Fig. 8, The statistical analysis for the accordance investigation showed the R² of 0.991, RMSE of 0.322, MAE of 0.153, and AARD% of 16.820%.

Fig. 8.

Investigation the accordance between experimental and output values for check data.

Variable importance

The sensitivity analysis of independent variables is conducted so as to estimate their relative importance in forecasting the dependent variable, which is significant to design better and optimize the related process⁹⁷. GEP uses a complicated random method to calculate the variable importance of every single input variable. The importance of each independent variable is calculated by randomizing its input values and then by reducing R² between the model output and the target. The results are normalized to nondimensionalize⁹⁶. The relative importance of nondimensionalized numbers related to each independent variable produced in this study is shown in Fig. 9.

Fig. 9.

Variable importance chart for the GEP model.

As can be seen, the independent variables of T, P, MW_SDs, and MP_SDs are denoted by d₀, d₁, d₂, and d₃. According to Fig. 9, MW_SDs, P, MP_SDs, and T have the highest effect in estimating the solubility in SC-CO₂, respectively.

Investigation of the effect of independent variables on the solubility in SC-CO2

Temperature effect

In the equilibrium of a pure solid (component 1) with the SCF, by assuming that SC-CO₂ is not solved in the solid, and the solid molar volume is not as a function of pressure, Eq. (7) can be written as follows:

7

where is the molar fraction of the solvent in the vapor phase, T and P are the operational temperature and pressure, respectively, is the solid fugacity coefficient in the supercritical phase, is the vapor fugacity coefficient originated from the solid, is the solid saturation vapor pressure, is the solid molar volume, P₁ is the solid vapor pressure, and R is the universal gas constant⁹⁸.

Since the temperature enhancement can lead to increasing the solvent vapor pressure, it can also be stated that the solubility of the solid component in SC-CO₂ increases (see Eq. (7)). Despite the fact that increasing temperature will lead to decreasing the density of the solvent, the study of the effect of increasing temperature on the solubility of SDs in SC-CO₂ indicated that the positive effect of vapor pressure of the drug component is dominant. Figure 10a and b indicates the positive effect of temperature on the solubility of SDs in SC-CO₂. The literature research also showed the similar trend in examining the effect of operating temperature on the solubility of solid materials in SC-CO₂ as Ref⁹⁹.

Fig. 10.

Two dimensional plot for Flutamide (a) and three dimensional plot for Galantamine (b) to investigate the effects of operating temperature and pressure on SD solubility in the SC-CO₂.

Pressure effect

Operating pressure typically demonstrates a dual influence in investigating the solid solubility in SC-CO₂. In assessing the impact of pressure, one must consider two significant variables: SC-CO₂ density and solubilized vapor pressure. For this objective, there exists a cross-over region for drug solubility. This phenomenon transpires when the magnitude of the two determinants of SC-CO₂ density and solute vapor pressure are equivalent^73,75. In other word, as pressure rises at constant temperature constant, the density of the solvent rises, whereas the vapor pressure of the solute diminishes. At higher pressures, the extent of this density alteration becomes less pronounced, and the change in solute vapor pressure becomes more significant, effectively outweighing the impact of solvent density changes on the solubility of the solute. Since many drugs were used in this study and not all of them necessarily exhibited cross-over pressure, it is not possible to select a specific pressure or range as the crossover pressure. However, this phenomenon is clearly observable at a pressure of 215 bar and a temperature range of 308–328 K in Fig. 10b drawn for Galantamine. At pressures less than this region, the density effect dominates, and solubility increases with decreasing temperature. In contrast, above this region, the vapor pressure of the solute becomes a more significant factor, and higher solubility is observed at higher temperatures. The dual effect of temperature on the solubility of various drugs in supercritical carbon dioxide has been repeatedly studied in the literature^58,62,79,81. In fact, the increase of pressure leads to reducing the intermolecular distance in SCF and as a result increasing the density, which heightens the solubility. Furthermore, the solvation power of SC-CO₂ increases under the influence of increasing pressure. In the simplest binary solutions of SCF, the solvation phenomenon requires the soluble molecules to be surrounded by solvent because of local clustering of solvent molecules. In a cluster, the highest number of solvent molecules is aggregated around a soluble. The increase of the number of solvent-soluble interactions created by the displacement of solvent molecules towards the potential range of solvent-soluble interaction locally increases the average density of the solvent compared to the bulk, leading to enhancing the solubility¹⁰⁰. Qingzhao Shi et al.¹⁰¹ also reached the similar results with this study during investigating the effect of pressure on the solubility of n-alkanes in SC-CO₂. Figure 10a indicates the positive effect of increasing operating pressure and temperature on the solubility of Flutamide as an SD in SC-CO₂. While the effect of pressure is more significant compared to temperature, which is also reported in Refs^24,44,45,99. The results obtained from the sensitivity analysis of the independent variables of the model in Fig. 9 also confirms this finding. Furthermore, its three-dimensional surface plot was drawn as Fig. 10b to take a broader view of the operating condition effects on the ACDs solubility in the SC-CO₂. As can be seen in Fig. 10, the maximum solubility of Flutamide and Galantamine is obtained in the highest operating conditions. Therefore, it is suggested that these two variables be at the highest possible value. On the other hand, due to the increase in the process cost, the cost of the supercritical process should be considered to determine the optimal operating conditions.

It is noteworthy, Fig. 10b confirms the results reported in the literature like the surface description of the estimated solubility data for salsalate based on ANN modeling reported by Nguyen et al.¹⁰² and for busulfan based on ANFIS modeling reported by Zhu et al.⁸. Since both of these models are provided only for a specific SD, the operating temperature and pressure were only their independent variables. While in the current model, using the physical properties of SDs as input model variables cause to the model applicability for solubility estimate of various SDs in the SC-CO₂.

The effect of physical properties of SDs: MWSD and MPSD

The drugs with low molecular weight usually have higher solubility in SC-CO₂. This is because smaller molecules can easily penetrate and interact with the molecular networks of carbon dioxide. Also, lower molecular weight usually means more surface area in contact with the solvent. Materials with higher molecular weight may have lower solubility in SC-CO₂ due to their larger size and structural complexity^28,103. In addition, the solubility of the substance depends on the type and chemical structure of the substance. SC-CO₂ is a nonpolar solvent. Therefore, nonpolar drugs with high molecular weight are usually better dissolved in SC-CO₂. On the other hand, polar drugs or drugs with polar groups may have lower solubility in SC-CO₂. The type of bonds and molecular structure can also have a great effect on solubility. For example, drugs with strong hydrogen bonds may not dissolve in SC-CO₂⁸⁷. In general, although molecular weight is one of the factors affecting solubility, it should be noted that other factors such as chemical structure, presence of functional groups, pressure and temperature also play an important role in estimating drug solubility in SC-CO₂¹⁰³.

The melting point can indicate the type of intermolecular bonds and the chemical structure of the drugs. Drugs with complex structures and strong bonds usually have higher melting points and may have lower solubility in SC-CO₂^28,104. On the other hand, if the drug has functional groups that can interact with SC-CO₂, it may have more solubility in SC-CO₂, even if its melting point is high. Therefore, the solubility of a drug in SC-CO₂ is strongly influenced by the type of interactions between drug molecules and SC-CO₂¹⁰³. Also, drugs with lower melting points usually have higher solubility in SC-CO₂. This is because, as the temperature decreases, the drugs can easily turn into a gas state and thus dissolve in SC-CO₂⁴⁷.

Comparison the selected AIM with semi-empirical models

To compare the accuracy and ability of estimating SDs solubility in the SC-CO₂ by the ANFIS model and previously reported models, the Flutamide SD was selected as a case study. As Table 7, the statistical parameters including R², MAE, RMSE and AARD% showing the ANFIS model has more satisfactory results with higher accuracy than the other semi-empirical models.

Table 7.

Comparison the ANFIS model with the previously semi-empirical models.

Model	Statistical parameters
	R-square	MAE	RMSE	AARD%
ANFIS	0.997	0.240	0.187	15.290
Chrastil [26]	0.996	0.08008	0.09248	16.426
Garlapati-Madras [25]	0.996	0.08358	0.09446	18.005
Méndez -Santiago-Teja [27]	0.994	0.11629	0.14381	19.420

Conclusion

The solubility estimation in SCF is an important step in the development of micro- and nanoscale particle production processes, as well as the extraction and purification of drugs. Alternatively, the time-consuming and costly access to solubility equilibrium data and the sensitivity of some drugs to heat, upsurge the necessity to model the solubility of drugs in SC-CO₂. In this research, the solubility of SDs in SC-CO₂ was modeled in a wide range of temperature and pressure (308–348.2 K and 80–400 bar) using AIMs based on GEP and ANFIS. The independent variables were T, P, MW_SDs, and MP_SDs. The collection of data for SDs in different operational conditions contains 1816 datasets, 80% of them were randomly selected for training, and the rest of them were for validation and test. The AIM output was selected through a multi-objective strategy that resulted in lower AARD% based on the simplest model having the highest fitness. The ANFIS model outperformed the GEP model in terms of accurate estimation, with statistical parameters such as R², RMSE, MAE, and AARD% obtained in training and validation data equivalent to 0.991, 0.260, 0.167 and 13.890% and 0.990, 0.256, 0.157 and 15.273%, in that order. The RMSE, MAE and AARD% values not only are low, but they are very close together as much as possible, indicating the selected model has high precision in estimating the solubility. The sensitivity analysis of the optimized GEP model concerning the input variables was conducted. The results revealed that MW_SDs, P, MP_SDs, and T had the most influence in determining the solubility of SDs in SC-CO₂, in that order. Increasing MW_SDs as a consequence of increasing the number of carbon atoms leads to moving SDs towards nonpolarity. Hence, given the nonpolarity nature of SC-CO₂, the increase of MW_SDs enhances SDs solubility. Furthermore, the results showed that the positive effect of increasing temperature is dominant in such a way as to result in increasing the vapor pressure of SDs and consequently increasing the solubility in SC-CO₂. Moreover, operating pressure significantly influences SDs solubility in SC-CO₂, considering SC-CO₂ density and solute vapor pressure. A cross-over region exists for drug solubility when the two determinants are equivalent. Other factors like chemical structure, functional group presence, and MP_SDs also can play a significant role. Lower MP_SDs exhibiting higher solubility due to gas-state dissolution. Eventually, in order to estimate the nonlinear relation between the input and output variables of the current research, the AIMs developed by ANFIS provides satisfactory results. Comparison the performance of the ANFIS model with the other semi-empirical models showed that the current model in addition to providing generalizability for various SDs by using MW_SDs and MP_SDs as the input model variables indicates more satisfactory results with higher accuracy.

Abbreviations

AARD

Average absolute relative deviation

AIMs

Artificial intelligence models

ANFIS

Adaptive neuro-fuzzy inference system

ANN

Artificial neural network

ARD

Average relative deviation

BPANN

Backpropagation artificial neural network

EoSs

Equations of state

ET

Expression tree

Exp

Exponential

FCM

Fuzzy c-mean

FFNN

Feed forward neural network

FIS

Fuzzy inference system

GEP

Gene expression programming

Inv

Inverse

IS transpozition

Insertion sequence transpozition

MAE

Mean absolute error

MFs

Membership functions

MP_SD

Solid Drug Melting Point

MSE

Mean square error

MW_SDs

Solid Drugs Molecular Weight

OBJ

Objective

PR-VdW

Peng-Robinson- van der waals

RIS Transposition

Root insertion sequence transpozition

RMSE

Root mean square error

SC-CO₂

Supercritical carbon dioxide

SCF

Supercritical fluid

SDs

Solid drugs

Tanh

Hyperbolic tangent

VdW

Van der waals

Author contributions

Zahra Bahrami: Data curation management, Developing GEP modeling, Managed the data curation process, Drafted the initial manuscript, Reviewed and edited the manuscriptFatemeh Bashipour: Conceptualization ideas, Developing GEP and ANFIS modeling, Reviewed and edited the manuscript, SupervisionAlireza Baghban: Development methodology, Reviewed and edited the manuscript.

Data availability

The datasets used during the current study available from the corresponding author (F. Bashipour) on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.