Predictive analysis of solubility data with pressure and temperature in assessing nanomedicine preparation via supercritical carbon dioxide

Abstract

This work presents a comprehensive study on the prediction of phenytoin solubility at supercritical state using advanced techniques including machine learning analysis. The solubility of small-molecule pharmaceutical was analyzed and calculated to enhance its solubility and bioavailability as well. The models were employed to approximate the solubility at various pressures and temperatures. The dataset comprises temperature (T), pressure (P), and solubility (y) values, along with the corresponding solvent density measurements that were used in the models. Three models, namely Automatic Relevance Determination Regression (ARD), Gaussian process regression (GPR), and Linear Regression (LR) were designed and tuned to build predictive models. The ADABOOST ensemble technique was applied to strengthen the predictive capabilities of the models, while hyperparameter tuning was conducted using the Jellyfish Optimization (JO) algorithm. For phenytoin solubility prediction, the ADA-GPR model demonstrated outstanding accuracy, obtaining an R² of 0.99644. The ADA-LR model also produced competitive results, attaining an R² value of 0.93381, whereas the ADA-ARD model showed robust performance, yielding an R² of 0.95249. In terms of solvent density prediction, the ADA-GPR model once again outperformed the others, with an R² value of 0.9933.

Introduction

Estimating drug solubility and dissolution at different temperatures and times is essential to control the bioavailability of medicines, as the low drug and poor solubility is a major issue in pharmaceutical industry. It has been recognized that some chemical methods such as amorphous solid dispersion (ASD) and co-crystallization would be useful for understanding the interactions between solvent and solute to improve the dissolution as well as solubility of drugs in aqueous solutions^1–3. Thus, the solubility plays an important role for development of novel drug formulations.

Some numerical optimization methods such as machine learning can be integrated to molecular-level models to boost the prediction capability of molecular modeling techniques in design of drugs with enhanced solubility. Machine learning techniques have transformed scientific research by enabling powerful data-driven modeling and predictive capabilities across diverse disciplines. Regression models play a crucial role in this domain which can be also used for prediction of drugs solubility in different solvents^4–6. Compared to the thermodynamic approach for simulation of pharmaceutical solubility, regressive models such as machine learning can be applied for a large number of entities with great accuracy and the estimation of intermolecular interaction parameters are not required for regressive models⁷.

Here, we used several regressive models for simulation and correlation of pharmaceutical solubility in supercritical CO₂. The applied regressive models are based on machine learning (ML) and include Automatic Relevance Determination Regression (ARD), Gaussian process regression (GPR), and Linear Regression (LR) to make models on solubility of small-molecule phenytoin drug and solvent density related to it. The solvent is supercritical carbon dioxide which is green and has great advantages for green processing of nanomedicine^8–10. AdaBoost is employed to enhance the model’s precision, and the Jellyfish Optimizer (JO) is employed for optimizing hyperparameters of the models.

The Adaboost Regression technique is an effective approach to ensemble learning, whereby a robust predictor is constructed by amalgamating several weak regression models. Adaboost Regression operates by sequentially fitting a series of weak regressors to the training data, while dynamically updating the weights assigned to individual samples based on the prediction errors in each iteration. Poorly predicted samples get higher weights, directing later models to focus more on these tough cases. This iterative process gradually improves performance.

Jellyfish Optimizer (JO) is inspired by the movement patterns of jellyfish, offering efficient solutions for regression problems. Automatic Relevance Determination Regression (ARD) determines the relevance of input features in regression models, aiding in feature selection and regularization¹¹. Linear Regression (LR) models the relationship between input variables and a target variable by fitting a linear equation to observed data, assuming the target can be expressed as a weighted combination of inputs plus noise¹². Gaussian process regression (GPR) utilizes Gaussian processes to model continuous outputs, providing a flexible and probabilistic approach to regression tasks^13,14.

Description of solubility

We considered modeling of small molecule in supercritical solvent at various pressures and temperatures to optimize its condition and find out the determining parameters which can impact the change of solubility. The dataset provided contains information about the solubility of phenytoin (y) and the density of carbon dioxide at multiple levels of temperatures (T) and pressures (P). The data is taken from¹⁵ and has been used by Ghazwani et al.^7,9 for machine learning correlation of data, and their method is used in this work. The data is organized into four columns^7,9:

• T (K): Temperature in the ranges: [313–345 K],

• P (bar): Pressure in the range from 95 to 250 bar,

• y: Solubility of phenytoin as the model of drug. It represents the fraction of phenytoin dissolved in the solution,

• Solvent density: Density of carbon dioxide. The density values range from 222.2849 to 858.2483.

Figure 1 displays the frequencies of input and output parameters in histogram plots.

Fig. 1.

Histograms of parameter frequencies for four variables in the dataset.

In the data preprocessing phase and prior to the ML modeling of dataset, normalization was performed using the Min-Max scaling technique to ensure that all input features—namely temperature, pressure, and other continuous variables—were scaled to a uniform range, typically between 0 and 1. This normalization procedure was crucial to avoid bias toward high-magnitude features and improve optimization efficiency. Subsequently, the dataset was randomly divided into training-validation and testing groups using an 80 − 20 split ratio. This random partitioning of dataset ensured that the models were trained on a substantial portion of the data while preserving a hold-out set for unbiased evaluation of model performance.

Methodology

Jellyfish optimizer (JO)

Some numerical optimization methods were employed in this study which are being discussed in the following sections. The JO is a metaheuristic algorithm motivated by the collective behavior of jellyfish. This algorithm is inspired by the natural behavior of jellyfish swarms, particularly their foraging strategies and movement dynamics, to address complex optimization tasks. It is designed to effectively navigate the solution space and converge toward optimal outcomes¹⁶.

The algorithm iteratively improves the solutions through a combination of exploration and exploitation strategies. The positional update for each jellyfish individual is mathematically represented by the following equation¹⁶:

Here, represents the new position of jellyfish in dimension at time , is its current location, and represents the displacement vector¹⁷.

Automatic relevance determination regression (ARD)

ARD Regression is an advanced Bayesian modeling approach that performs simultaneous feature selection and coefficient estimation by inferring the importance of predictors from the data. The model is based on Bayesian principles and provides a principled approach for feature selection and regularization¹⁸.

In ARD Regression, the objective is to predict the target variable based on a given input vector x. The model is written as¹¹:

Inference and learning in ARD are conducted within a Bayesian framework. Variational Bayesian techniques are employed to estimate the posterior distributions of both the model weights and associated hyperparameters¹⁰. These techniques seek to reduce the discrepancy between the exact posterior distribution and its approximate counterpart. The regularization parameters (α) are usually learned by maximizing the model evidence or the posterior probability, utilizing methods like type-II maximum likelihood or empirical Bayes estimation.

ARD regression offers the advantage of automatic feature selection by assigning individual relevance weights to input features, effectively reducing model complexity and improving interpretability. However, it can be computationally intensive for high-dimensional data and may require careful tuning of hyperparameters to avoid overfitting or underfitting.

Linear regression (LR)

LR postulates a linear association between the input feature(s) and the target variable, and aims to find the best-fitting linear equation that describes this relationship^19,20.

Let us consider a dataset including data points, where each sample is denoted by a vector of input variables, and the corresponding target variable is indicated by . The objective of LR is to find a set of coefficients such that the linear equation¹²:

provides the best approximation of the relationship between the input features and the target output. Here, represents the intercept term, (for ) denotes the coefficient associated with the -th input variable, and signifies the error term, which captures the unexplained variability in the target variable²¹.

To estimate the coefficients, the Ordinary Least Squares (OLS) method is frequently utilized in LR. The objective of OLS is to minimize the sum of squared discrepancies between the expected target variable and the model predicted values derived from the linear equation. Mathematically, this can be formulated as¹²:

Once the coefficients are estimated, the Linear Regression model provides a mechanism to the estimation of the target variable for new observations based on their predictor values . The predicted value is calculated as^12,22:

where represents the input values for the new observation.

LR is simple, fast, and easy to interpret, making it a strong baseline for many predictive tasks. Its main limitation is the assumption of linear relationships between variables, which can lead to poor performance on complex, nonlinear data. Additionally, it can be sensitive to multicollinearity and outliers.

Gaussian process regression (GPR)

GP is defined as a group of stochastic variables, where specific variables manifest Gaussian distributions while others do not demonstrate such characteristics^23,24. The employment of mean and covariance functions is a pertinent approach to evaluate the efficacy of a Gaussian Process²⁵. In the course of this procedure, Gaussian distributions (GDs) are subject to expansion. The application of Gaussian process regression models requires a correlation to exist among the past data. When comparing the GP and GD, it can be observed that the former demonstrates a wider spectrum of functionalities¹⁴.

One distinguishing characteristic of GPR in comparison to other regression models is its lack of necessity for a precise specification of a fitting function. Field observations can be understood as a probabilistic framework similar to a randomly selected portion taken from a multivariate normal distribution^26,27.

The output corresponds to a set of multidimensional pairs , where each target value and the input vector represents the feature data¹⁴.

A GP is defined in terms of a latent function as¹⁴:

Mean operator is indicated by m(x) and kernel covariance is denoted by K, which depends on the input values and kernels²⁸.

GPR provides flexible, nonparametric modeling with built-in uncertainty quantification, making it well-suited for capturing complex, nonlinear relationships. Optimal implementation requires balancing computational demands against model flexibility - the kernel matrix inversion becomes computationally intensive for large data sets, while suboptimal kernel specifications degrade generalization capability.

Modeling results

This section presents the comprehensive modeling results obtained from the application of ADA-ARD, ADA-GPR, and ADA-LR models on the provided dataset.

For the prediction of solubility, the models were boosted using ADABOOST, and hyper-parameter tuning was performed using the JO algorithm. The modeling results for solubility prediction are summarized as follows:

• ADA-GPR: The ADA-GPR model achieved outstanding performance with an impressive R² value of 0.99644. The MSE was calculated as 7.4007E-02, indicating the model’s high accuracy. Furthermore, the MAPE was remarkably low at 4.60427E-02.

• ADA-LR: The ADA-LR method offered great accuracy, achieving an R² of 0.93381. MSE for this model is estimated as 8.8473E-02, indicating reasonable accuracy. The MAPE for ADA-LR was determined as 5.1172E-02.

• ADA-ARD: This method showed favorable performance for data fitting, obtaining an R² of 0.95249. The MSE for ADA-ARD was calculated as 8.1420E-02, indicating good accuracy. The MAPE value for this model was 4.94415E-02.

For solvent density prediction, we employed the same ensemble of models (ADA-GPR, ADA-ARD, and ADA-LR), with comparative results detailed in the following items:

• ADA-GPR: The ADA-GPR model showed excellent precision in determination of density, with score of 0.9933. The MSE was determined as 2.9764E + 02, indicating a low level of error. The MAPE value for this model was 3.09560E-02.

• ADA-ARD: The ADA-ARD model also performed well, obtaining an R² of 0.9493. The MSE for this model was calculated as 1.9115E + 03, indicating reasonable accuracy. The MAPE value for ADA-ARD was determined as 7.5128E-02.

• ADA-LR: The ADA-LR model demonstrated a satisfactory performance in predicting solvent density, returning accuracy with R² = 0.92743. The MAPE value for ADA-LR was determined as 9.36355E-02.

Based on the above information, the ADA-GPR model used for rest of analyses in this study. Additionally, 5-fold cross-validation was performed to evaluate the generalization ability of the models (see Table 1). The ADA-GPR model showed excellent consistency, having an R² of 0.99452 and MAPE of 0.04879 for solubility, and R² of 0.99191 and MAPE of 0.03296 for density prediction, with low standard deviations across folds. These results confirm the model’s robustness and absence of overfitting.

Table 1.

Cross validation values of the models using 5-fold method.

Output	Model	Metric	Mean	Std
Solubility	ADA-GPR	R²	0.99452	0.00183
		MSE	0.08281	0.00712
		MAPE	0.04879	0.00297
	ADA-LR	R²	0.92841	0.01345
		MSE	0.09560	0.01128
		MAPE	0.05398	0.00674
	ADA-ARD	R²	0.94897	0.00936
		MSE	0.08759	0.00859
		MAPE	0.05137	0.00521
Density	ADA-GPR	R²	0.99191	0.00294
		MSE	311.025	63.2145
		MAPE	0.03296	0.00167
	ADA-ARD	R²	0.94587	0.01182
		MSE	2000.23	217.4839
		MAPE	0.07874	0.00635
	ADA-LR	R²	0.92236	0.01769
		MSE	2455.04	284.917
		MAPE	0.09661	0.00892

The performance of the ADA-GPR model for both outputs was visualized using residual plots, as presented in Figs. 2 and 3. These figures show the high performance and closeness of actual and predicted values. The density is decreased with temperature which is due to the expansion of the solvent and can result in less solubility in the solvent⁷.

Fig. 2.

Residuals results for solubility.

Fig. 3.

Residuals results for CO₂ density.

Finally, the 2D plots of Figs. 4 and 5 are the individual effects of T on both outputs. These figures show that solubility increases on increase of both inputs. But solvent density increases on increase of Temperatures. It can be understood that the molecular interactions between drug and the solvent is more responsible for the solubility enhancement with increasing T, so that the solubility is increased with enhancing T, despite the reduction in the solvent density²⁹. Moreover, pressure has direct relationship with both solubility and density such that increasing pressure is favorable for supercritical processing due to rising solubility with P⁹. This fact has been analyzed and reported by Ghazwani et al.^7,9 for the solubility of phenytoin with similar results.

Fig. 4.

Temperature effect on solubility on multiple constant pressures.

Fig. 5.

Temperature effect on CO₂ density on multiple constant pressures.

To demonstrate the generality of the best-performing model, ADA-GPR, its predictive capabilities were extended to a diverse group of drugs, as detailed in Table 2 and visualized in Fig. 6. The model was applied to a dataset comprising over 450 data points across multiple pharmaceutical compounds, including Azathioprine, Sunitinib malate, Levonorgestrel, Docetaxel, Anastrozole, Exemestane, Tamsulosin, Nilotinib hydrochloride monohydrate, Crizotinib, Ketoconazole, Ketotifen Fumarate, Quetiapine hemifumarate, Amlodipine Besylate, Losartan Potassium, and Regorafenib monohydrate. The ADA-GPR model achieved high R² scores for these drugs, ranging from 0.94841 for Docetaxel to 0.99469 for Levonorgestrel, illustrating its robust performance across varied chemical structures and conditions in supercritical processing. Figure 6 illustrates the close alignment between actual and predicted solubility values for these compounds, confirming the model’s versatility and effectiveness in predicting solubility for a broad range of pharmaceuticals in supercritical carbon dioxide, thus supporting its potential for widespread application in nanomedicine development.

Table 2.

Generalizability of the best model for a group of different drugs.

Drug	R2 score
Azathioprine	0.99218
Sunitinib malate	0.98192
Levonorgestrel	0.99469
Docetaxel	0.94841
Anastrozole	0.98009
Exemestane	0.99107
Tamsulosin	0.95894
Nilotinib hydrochloride monohydrate	0.98783
Crizotinib	0.95137
Ketoconazole	0.99188
Ketotifen fumarate	0.96311
Quetiapine hemifumarate	0.95329
Amlodipine besylate	0.97587
Losartan potassium	0.96218
Regorafenib monohydrate	0.97420

Fig. 6.

Actual and predicted values comparison on a group of different drugs with more than 450 data points.

Conclusion

In this study, we used ARD, GPR, and LR models to determine phenytoin solubility and solvent density using temperature (T) and pressure (P). The model was a hybrid comprehensive model combining numerical optimizer and molecular-level analysis for understanding the solvent-drug interactions and finding the controlling step in drug dissolution. This would help one determine and control the rate of drug dissolution in aqueous phase and design new drug delivery system with desired solubility and rate of dissolution. We achieved outstanding modelling results by using ADABOOST for model boosting and the JO algorithm for hyper-parameter tuning. Both solubility and solvent density were estimated using the developed hybrid model in this study.

All three models performed well when it came to predicting solubility. ADA-GPR stood out with an R² of 0.99644, indicating an exceptional level of accuracy. ADA-LR and ADA-ARD performed admirably as well, with R² equal to 0.93381 and 0.95249, respectively. These findings demonstrate the models’ ability to accurately estimate solubility, providing valuable insights for drug formulation processes.

In terms of solvent density prediction, ADA-GPR performed admirably, achieving an impressive R² of 0.9933. ADA-ARD also produced positive results, with an R² of 0.9493. Despite having a slightly lower R² of 0.92743, ADA-LR demonstrated reasonable predictive capabilities. These findings show that these models have the potential to accurately estimate solvent density, which can have significant implications in environmental studies.

Author contributions

H.O.A.: Writing, Methodology, Investigation, Software, Supervision, Funding. Y.S.A.: Writing, Conceptualization, Investigation, Software, Validation, Visualization.All authors reviewed the manuscript.

Data availability

The datasets used and analyzed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.