Efficient delivery of therapeutic agents to the desired site has been a subject of research owing to the importance of this method in cancer treatment. Drugs might reach other tissues and damage them, while low dosage of drug could reach the cancer cells for treatment. Therefore, the design of targeted drug delivery systems would be of fundamental importance for cancer effective treatment (Kandula et al., 2023; Chen et al., 2024; Lu et al., 2024; Sameer Khan et al., 2024). Drug can be loaded into various carriers such as polymeric nanoparticles and reach the target cells, while its release can be triggered by various means such as pH or temperature change (Ali et al., 2023).

Modeling and computation of drug release from carriers can be utilized for design and optimization of drug delivery systems based on polymeric carriers. Some mathematical models have been developed to simulate mass transfer in polymeric-based drug release (González-Garcinuño et al., 2023; Kubinski et al., 2023; Carr and Pontrelli, 2024). Usually, molecular diffusion is the main mechanism that happens in polymeric-based drug delivery systems where the drug molecules diffuse through the porous structure of polymeric carrier. Some parameters such as pore structure of carrier, molecular interaction, temperature, and pH can affect the release rate of drug molecules. On the other hand, machine learning models can be used for simulation of drug release from polymeric carriers. The method is based upon collection of datasets and building models via appropriate algorithms. This method is indeed fast and possesses higher performance in terms of fitting accuracy.

Machine learning (ML) techniques have shown great potential in the field of drug development by enabling accurate forecasting of drug solubility and density (Abdelbasset et al., 2022; Almehizia et al., 2023). These techniques have the capability to evaluate large amounts of data and extract meaningful patterns and relationships that can be utilized for predictions (Jovel and Greiner, 2021). This paper provides a thorough analysis of three distinct regression models, namely, Decision Tree Regression (DTR), Passive Aggressive Regression (PAR), and Quadratic Polynomial Regression (QPR). These models were carefully evaluated using a comprehensive dataset in the field of drug release from a porous polymeric carrier. The hyper-parameter optimization process is executed using Sequential Model-Based Optimization (SMBO).

Decision Tree Regression is a versatile algorithm that can be utilized in a wide range of regression tasks. Careful tuning of hyperparameters is essential to prevent overfitting and ensure optimal model performance (Talekar and Agrawal, 2020). Passive Aggressive Regression offers a flexible and adaptive approach to regression tasks, particularly in situations where data arrives sequentially or in a streaming fashion (Crammer et al., 2006). Also, Quadratic Polynomial Regression is a valuable tool for capturing quadratic relationships in the data. Careful consideration of model complexity and potential overfitting is crucial for obtaining reliable and meaningful results (Yao and Müller, 2010).

By systematically evaluating Decision Tree Regression (DTR), Passive Aggressive Regression (PAR), and Quadratic Polynomial Regression (QPR) models on a dataset comprising over 15,000 data samples, the study provides valuable insights into the strengths and limitations of each model. The incorporation of Sequential Model-Based Optimization (SMBO) for hyper-parameter tuning enhances the robustness of the models, highlighting the significance of thoughtful parameter optimization.

This research dataset consists of more than 15,000 data points, incorporating three key variables: r measured in meters, z also in meters, and chemical concentration C expressed in mol/m³. The data have been collected from a CFD (Computational Fluid Dynamics) simulation of drug-loaded polymeric matrix. The CFD was utilized to numerically solve time-dependent mass balance per species (COMSOL, 2008) and the generated data was used for building the machine learning models. The correlation heatmap between variables is shown in Figure 1. This step was done as the preliminary data visualization to see how data vary in the domain of drug delivery system.

The z-score, or standard score, is a statistical measure widely employed for outlier detection in various studies, including the present research. When conducting outlier analysis, the z-score is a useful metric that provides a standardized representation of the deviation of a data point from the mean of the dataset. It measures the distance from a data point to the mean in standard deviation units.

The expression for determining the z-score of a data point X within a dataset having a mean of μ and a standard deviation of σ is articulated as follows (Anusha et al., 2019): Z = X − μ σ In this context, Z signifies the z-score of the data point, X represents the individual data value, μ is indicative of the mean within the dataset, and σ denotes the standard deviation.

A high absolute z-score indicates that the data point is far from the mean and is considered a potential outlier. The threshold for identifying outliers using z-scores is often set empirically; commonly, a z-score beyond 2 or 3 standard deviations is considered indicative of an outlier.

In the specific context of this study, the z-score method has been employed for outlier detection. By calculating z-scores for the relevant variables or features, the study aims to identify data points that exhibit significant deviations from the norm, facilitating a robust analysis of the dataset and ensuring the reliability of the research findings. The result of z-score analyses is shown in Figure 2.

The evaluation of the Decision Tree Regression (DTR), Passive Aggressive Regression (PAR), and Quadratic Polynomial Regression (QPR) models was conducted on a dataset comprising more than 15,000 data points, with input parameters represented by r(m) and z(m) coordinates, and the output parameter denoted by concentration (C) in mol/m³. The models underwent hyper-parameter optimization using Sequential Model-Based Optimization (SMBO). Table 1 presents a summary of the numeric results obtained from the assessment conducted. This table provides a concise overview of the key metrics and performance measures obtained from the evaluation of the regression models.

The DTR model demonstrates outstanding predictive accuracy, reflected in an impressive R² score of 0.99887, underscoring its capability to discern intricate patterns within the dataset. The negligible RMSE, MAE, and Max Error values further emphasize the precision and reliability of the model in predicting chemical concentrations. Figure 4 showcases a visual comparison between the model predicted values and the true values using the Decision Tree Regression (DTR) model.

While the PAR model demonstrates a slightly lower R² score of 0.94652, its performance remains commendable, with competitive RMSE, MAE, and Max Error values. This indicates the model’s effectiveness in capturing relationships within the dataset, albeit with a nuanced trade-off between accuracy and complexity. In Figure 5, a visual comparison is presented, illustrating the disparities between the values predicted by the PAR model and the actual values.

The QPR model, designed to capture non-linear relationships, achieves a noteworthy R² score of 0.95382. The model’s competitive RMSE, MAE, and Max Error values underscore its proficiency in accommodating the inherent complexities of the chemical engineering dataset. Figure 6 provides a visual representation, demonstrating the distinctions between the values forecasted by the QPR model and the factual values.

Figures 7–9 present three-dimensional representations of concentration in relation to the variables r(m) and z(m), utilizing the three regression models. These visualizations offer a comprehensive view of how concentration varies across different values of r(m) and z(m) for each model. The change in drug concentration which has been obtained by the model could be attributed to the molecular diffusion occurring indie the polymeric matrix. Although both convective and diffusional mass transfer have been considered in the mass transfer model, the contribution of diffusion is significant and controls the release of drug from the carrier.

Leveraging the DTR model, acknowledged as the top-performing model in this investigation, Figures 10, 11 depict the partial dependency of concentration on the variables r(m) and z(m), respectively. These visualizations provide insights into how changes in r(m) and z(m) influence the drug concentration, while keeping the other variable constant at multiple levels. This visualization provides a comprehensive representation of how the concentration varies across different combinations of the input variables, r(m) and z(m). The center of geometry is the drug where its concentration is the highest, while concentration declines beyond the center due to the diffusion as well as chemical reactions.

In conclusion, this paper has presented a rigorous evaluation of three distinct regression models, namely, Decision Tree Regression (DTR), Passive Aggressive Regression (PAR), and Quadratic Polynomial Regression (QPR), within the context of a dataset containing more than 15,000 data points. The dataset has been obtained from mass transfer simulation of drug release from a porous polymeric carrier. The input parameters, r(m) and z(m), were utilized to predict the output concentration (C) in mol/m³. The models underwent hyper-parameter optimization through Sequential Model-Based Optimization (SMBO), ensuring a meticulous exploration of the parameter space.

The results showcase the exceptional predictive capabilities of the Decision Tree Regression model, evidenced by a significant R² score of 0.99887, a negligible RMSE of 9.0092E-06, a minute MAE of 3.51486E-06, and a maximum error of 6.87000E-05. Despite a slightly lower R2 score, the Passive Aggressive Regression model demonstrated commendable performance, while the Quadratic Polynomial Regression model showcased proficiency in capturing non-linear relationships within the dataset.

This comparative analysis not only provides valuable insights into the specific strengths and limitations of each regression model but also serves as a guide for practitioners in selecting an appropriate model tailored to the complexities of chemical engineering datasets. The incorporation of SMBO contributes to the robustness of the models, highlighting the significance of thoughtful hyper-parameter tuning in enhancing predictive accuracy. Overall, this research contributes to the ongoing discourse on regression model selection and optimization techniques in the domain of drug delivery, offering a foundation for further exploration and refinement in predictive modeling methodologies for design of advanced drug delivery systems.