Molecular diffusion of drugs is of major importance for development and understanding drug delivery systems. Indeed, the main phenomenon which is controlling the rate of release is molecular diffusion which can be controlled via different phenomena such as interactions with the drug carrier and solution. In this work, we developed a novel hybrid model based on mass transfer and machine learning for predicting drug diffusion in a 3D space. The mass transfer equation including diffusion is solved in the domain and then the data is extracted for building machine learning models. The present study presents the findings of an analysis conducted with the objective of constructing precise regression models for the prediction of chemical species concentration (C) for a drug diffusion through a three-dimensional space, utilizing coordinates (x, y, z). The dataset comprises over 22,000 data points, with each point containing the coordinates (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_3803_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\(\:x,\:y,\:z\)</EquationSource> </InlineEquation>) and the corresponding concentration (C) in mol/m³. We employ three tree-based ensemble models: Kernel Ridge Regression (KRR), <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_3803_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\:{\upnu\:}\)</EquationSource> </InlineEquation>-Support Vector Regression (<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_3803_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\:{\upnu\:}\)</EquationSource> </InlineEquation>-SVR), and Multi Linear Regression (MLR) for modeling the relationship between spatial coordinates and the concentration. Hyperparameter optimization is performed using the Bacterial Foraging Optimization Algorithm (BFO) to fine-tune the models. The results reveal that <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_3803_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\:\nu\:\)</EquationSource> </InlineEquation>-SVR has the highest performance with a score of 0.99777 in terms of R2, followed by KRR with an R2 score of 0.94296, and MLR with an R2 value of 0.71692. Additionally, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_3803_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\:\nu\:\)</EquationSource> </InlineEquation>-SVR exhibits the lowest RMSE and MAE, showing excellent predictive accuracy compared to KRR and MLR. Overall, our analysis demonstrates the effectiveness of employing tree-based ensemble models coupled with BFO for accurately predicting chemical concentrations in three-dimensional space, with <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_3803_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\:\nu\:\)</EquationSource> </InlineEquation>-SVR emerging as the most promising model for this task. These findings have implications for various applications such as environmental monitoring, pollutant dispersion modeling, and chemical process optimization.

Computational hybrid analysis of drug diffusion in three-dimensional domain with the aid of mass transfer and machine learning techniques

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2025年5月26日

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Mohammed Alqarni,
Ali Alqarni

摘要

Molecular diffusion of drugs is of major importance for development and understanding drug delivery systems. Indeed, the main phenomenon which is controlling the rate of release is molecular diffusion which can be controlled via different phenomena such as interactions with the drug carrier and solution. In this work, we developed a novel hybrid model based on mass transfer and machine learning for predicting drug diffusion in a 3D space. The mass transfer equation including diffusion is solved in the domain and then the data is extracted for building machine learning models. The present study presents the findings of an analysis conducted with the objective of constructing precise regression models for the prediction of chemical species concentration (C) for a drug diffusion through a three-dimensional space, utilizing coordinates (x, y, z). The dataset comprises over 22,000 data points, with each point containing the coordinates ( \(\:x,\:y,\:z\) ) and the corresponding concentration (C) in mol/m³. We employ three tree-based ensemble models: Kernel Ridge Regression (KRR), \(\:{\upnu\:}\) -Support Vector Regression ( \(\:{\upnu\:}\) -SVR), and Multi Linear Regression (MLR) for modeling the relationship between spatial coordinates and the concentration. Hyperparameter optimization is performed using the Bacterial Foraging Optimization Algorithm (BFO) to fine-tune the models. The results reveal that \(\:\nu\:\) -SVR has the highest performance with a score of 0.99777 in terms of R², followed by KRR with an R² score of 0.94296, and MLR with an R² value of 0.71692. Additionally, \(\:\nu\:\) -SVR exhibits the lowest RMSE and MAE, showing excellent predictive accuracy compared to KRR and MLR. Overall, our analysis demonstrates the effectiveness of employing tree-based ensemble models coupled with BFO for accurately predicting chemical concentrations in three-dimensional space, with \(\:\nu\:\) -SVR emerging as the most promising model for this task. These findings have implications for various applications such as environmental monitoring, pollutant dispersion modeling, and chemical process optimization.