Intelligence analysis of membrane distillation via machine learning models for pharmaceutical separation | Scientific Reports

Scientific Reports volume 14, Article number: 22876 (2024) Cite this article

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This study investigates simulation of pharmaceutical separation via membrane distillation process by computational simulation and machine learning modeling strategy. The efficacy of three regression models, i.e., Multi-layer Perceptron (MLP), Gamma Regression, and Support Vector Regression (SVR) in predicting the solute concentration, C(mol/m³), was evaluated. The hyper-parameters were optimized by fine-tuning the models using the Red Deer Algorithm (RDA). Computational analyses were carried out for removal of pharmaceuticals from solution by membrane distillation in continuous mode. Mass transfer and machine learning models were implemented focusing on concentration of solute in the feed section of membrane. Results indicate that the Multi-layer Perceptron model achieved great accuracy with an R2 of 0.9955, an MAE of 0.0084, and an RMSE of 0.0148, effectively capturing complex nonlinear relationships in the data. Gamma Regression also performed acceptably, with fitting R2 of 0.9214, showing its suitability for positively skewed data. The Support Vector Regression model, while capturing the general trend, showed the lowest performance with an R2 of 0.8710. These findings suggest that the Multi-layer Perceptron is the most accurate model for this dataset, followed by Gamma Regression and Support Vector Regression. This underscores the importance of careful model selection and optimization in regression analysis in combination with computational simulation of membrane processes.

Membrane separation can be applied for pharmaceutical processing in separation and purification of compounds in liquid phase. In solution, both porous and non-porous membranes can be applied to separate a desired compound. The main mechanism for non-porous membranes is solution-diffusion, while a wide range of mechanisms are applied for separation through porous membranes1,2. For pharmaceutical separation, porous membranes can be used as a contactor for application in membrane distillation which is a hybrid unit combining membrane and distillation. Membrane distillation (MD) has been under focus recently due to its inherent attractive properties in separation and purification of organic compounds such as pharmaceutical compounds. The fundamental of MD is similar to the conventional distillation process, but the contact between two phases is carried out via a membrane contactor.

Porous membrane contactors such as hollow-fiber membranes have the privilege of providing high surface area for separation and can enhance the separation efficiency drastically3,4,5. For instance, in direct-contact membrane distillation both phases are brough into contact in the membrane module to exchange the heat between two phases. In this process, understanding the heat and mass transfer would help optimize the processing conditions. The major transport phenomena in membrane distillation can be analyzed via computational fluid dynamics (CFD) which solves the convection-diffusion equations for mass transfer and conduction-convective equations for heat transfer6. The equations can be linked to momentum transfer equations to simulate process7,8,9,10. So, CFD is mainly used for simulation and understanding MD process, however this method is computationally expensive and challenging for implementation at industrial scale.

To further improve the modeling via CFD which is challenging, one can integrate CFD to machine learning (ML) models to obtain hybrid models for simulation of MD process. This strategy makes the modeling easier to implement due to the computational challenges associated with CFD method. The methods of computing based on machine learning regression models are widely used for predicting continuous values and establishing relationships between variables in various fields11. So, these models are applicable to CFD data computed for separation processes such as membrane distillation. Obaidullah and Almehizia6 conducted MD simulation via CFD and regression models, and validated the models via comparison with CFD dataset for concentration distribution. The hybrid methodology was proved to be reliable in simulating membrane distillation.

Multi-layer Perceptron (MLP), Gamma Regression, and Support Vector Regression (SVR) are among the useful types of regression models used in this study for predicting C(mol/m³) using the variables r and z which are the simulation results obtained from CFD simulation of MD process. By fine-tuning the models with the Red Deer Algorithm (RDA), the hyper-parameters of the regression models were tuned to obtain the maximized performance for each. A Multilayer Perceptron (MLP) is a type of neural network architecture that includes a single input layer, hidden layers (one or multiple layers), and one output layer, all made up of neurons. MLP is able to capture complex relationships and dependencies in the data, which makes it well-suited for a range of regression and classification tasks12. SVR is a special case or Support Vector Machines for regression tasks that employs a kernel function to transform input data into a higher-dimensional space. The optimization problem is resolved by SVR, which minimizes a C-insensitive loss function and regularization term. This method allows for a margin of error ε and slack variables to accommodate deviations from the hyperplane13. Gamma Regression represents a particular form of generalized linear model (GLM) used to analyze positive continuous data that follows a gamma distribution. Considering the shape and magnitude of the gamma distribution, the model maximizes the likelihood of the observed data in calculating the parameters. This method empowers Generalized Regression to adeptly handle skewed data, particularly in scenarios where the variance of the dependent variable expands with its mean value. The optimization procedure involves iteratively adjusting the model parameters to reduce deviance, which measures the difference between the observed and estimated values14. So, this study investigates the correlation of CFD data using several machine learning models optimized by Red Deer Algorithm (RDA). Models are applied to vacuum membrane distillation (VMD) process as a case study to separate pharmaceutical molecules from a solution.

The dataset contains 3401 data points with three variables: r, z, and C. The points are the concentration of pharmaceutical compound in each mesh which is obtained by CFD simulation of process. Indeed, CFD was initially executed for the process to generate concentration distribution in the feed channel and membrane section of MD6. The procedure reported by6was applied for data generation and modeling. The main mass transfer equation which is solved in the membrane process is expressed as15:

where C stands for the solute concentration and is determined as a function of membrane coordinates, i.e., r and z. D denotes the diffusivity of the solute. Vz is the axial velocity which is calculated from momentum transfer equation:

Data of concentration distribution were obtained and extracted for use in machine learning correlations. The statistical summary of the dataset is provided in Table 1 and the boxplots of the variables are shown in Fig. 1. This study has developed models with two inputs and a single output.

Boxplots of variables obtained in MD process.

In this study, Cook’s distance was employed to detect and eliminate outliers, ensuring the robustness and reliability of the regression models. Cook’s distance is a measure utilized to identify influential data points that disproportionately affect the parameter estimates of a regression model. By calculating the change in the regression coefficients when a particular data point is excluded, Cook’s distance helps in pinpointing observations that have a significant impact on the model’s predictions. The dataset, comprising 3401 data points with variables r(m), z(m), and C(mol/m3), split into training and testing sets with a ratio of 85–15%. This stratified split allowed for comprehensive model training while reserving a significant portion of data for unbiased evaluation. By identifying and addressing outliers through Cook’s distance, and by ensuring an appropriate split of the dataset, the study optimized model performance and ensured the validity of the predictive outcomes. These methods were integral in enhancing the accuracy and reliability of the regression models used in this research.

RDA is a population-based meta-heuristic optimization algorithm that combines the principles of evolutionary algorithms with heuristic search techniques16. The RDA models the breeding behavior of red deer, where each deer represents a potential solution. The population is split into males and hinds, with males representing superior solutions17.

Workflow of RDA Algorithm.

The algorithm consists of seven phases as shown in Fig. 2: roaring, selecting commanders, fighting between commanders and stags, forming harems, mating commanders, mating stags, and selecting the next generation18. The following items are short description on these phases.

Roaring Phase: males update their positions randomly

Commander Selection Phase: A percentage, \(\:\gamma\:\%\), of males are designated as commanders, while the rest are stags.

Fight phase: commanders and stags update their positions using the following equations17

Here, \(\:{\text{new}}_{1}\) and \(\:{\text{new}}_{2}\) are new solutions, C is the commander’s solution, S represents the stags, and \(\:{b}_{1}\) and \(\:{b}_{2}\) are random numbers uniformly distributed in the range [0, 1].

Harem formation phase: Hinds are grouped based on the strength of their commanders

Commander Mating Phase: Commanders mate with hinds within and outside their group, using the equation18:

Stag mating phase: each stag mates with the nearest Hind

Next Generation Selection Phase: The next generation is formed, enhancing exploration during harem formation and commander mating, while exploitation occurs during the roaring, commander selection, fighting, and stag mating phases. The process ensures escape from local optima by selecting the fittest for the next generation.

Here are the configurations of the RDA which used for hyper-parameter tuning in this study:

Population Size: Number of Red Deers (= 80).

Percentage of Commanders (γ%): Proportion of males as commanders (= 20%).

Fight Phase Parameters (b1, b2): Random values influencing position updates ([0, 1]).

Mating Proportion in Own Harem (α%): Commander’s mating with its own hinds (= 60%).

Mating Proportion in Other Harems (β%): Commander’s mating with hinds from other harems (= 20%).

Max Iterations: Algorithm termination point (= 200).

Exploration vs. Exploitation Ratio: Balance between exploring new solutions and refining known ones (70%-30%).

Fitness Function: Mean RMSE value of the model developed on the candidate solution by 3-fold CV.

The Multi-layer Perceptron (MLP) regression model is a robust and adaptable method that is employed in machine learning for regression applications. It is a member of the artificial neural networks (ANNs) family and is especially adept at capturing intricate connections between input features and output objectives12. The perceptron, a computational representation of a biological neuron, is central to the MLP. In fact, the Multilayer Perceptron (MLP) consists of an array of perceptrons, structured hierarchically, where each layer is linked to the subsequent one. Perceptrons have input-specific activation functions and weights. The Multilayer Perceptron (MLP) computes weights depending on data to optimize prediction accuracy. The MLP architecture has input, output, and hidden layers19,20. In this study, logistic activation functions and hyperbolic tangent (tanh) are used to provide nonlinearity to the MLP model and allow it to capture complex patterns. The function ϕ(x) that represents the hyperbolic tangent activation is defined as21:

The logistic activation function, often represented as σ(x), is defined as22:

These activation functions are crucial for the MLP’s ability to learn and model intricate relationships in the data.

Gamma regression method is usually used to model and analyze data with a response variable that follows the gamma distribution. Gamma regression is commonly used when there is a correlation between the mean of the output variable and a set of input features23. This regression algorithm assumes that the output Yfollows a gamma distribution, which is defined by scale and shape parameters24. The connection between the mean response (\({\upmu}\)) and the inputs (X) is modeled using a log link function. The regression equation is formulated as follows25,26:

In this equation, \(\:{\text{b}}_{0},\:{\text{b}}_{1},\:\dots\:,\:{\text{b}}_{\text{p}}\) are the regression coefficients corresponding to the inputs \(\:{X}_{1},{X}_{2},\dots\:,{X}_{p}\) and \(\:g\left({\upmu\:}\right)\)indicates the log link function. MLE is often utilized to estimate model parameters by maximizing data observation likelihood. Newton-Raphson or Fisher scoring are used for numerical optimization. The interpretation of coefficients in gamma regression is done in a logarithmic manner. The coefficients represent the percentage change in the expected mean output variable when the input variable increases by one unit. The direction of the correlation between the input and the variable of interest is shown by the sign of the coefficient24.

Support Vector Machines (SVMs) are a significant category of machine learning techniques commonly utilized in regression analysis. A regression task is performed using a type of Support Vector Machine known as Support Vector Regression (SVR). SVR seeks to identify a function f(x) that maintains a deviation from the true target values of no more than \(\epsilon\)for each training point, while also striving for a minimal slope27. Given a set of training data \(\:\{\left({x}_{i},{y}_{i}\right){\}}_{i=1}^{n}\) where \(\:{x}_{i}\in\:{R}^{d}\) indicates the input data and \(\:{y}_{i}\in\:R\)represents the target values, the SVR model is formulated as follows28,29:

Objective Function: The goal is to minimize the model complexity and the prediction error outside the \({\epsilon}\)-tube, which is achieved by minimizing the following objective function13:

Where \(\:w\) denotes the weight vector, b refers to the bias term, \(\:{{\upxi\:}}_{i}\)and \(\:{{\upxi\:}}_{i}^{*}\)are slack variables representing the deviations of the predictions outside the \(\epsilon\)-tube, and C stands for a regularization coefficient that governs the balance between the flatness of \(\:f\left(x\right)\) and the amount up to which deviations larger than \(\epsilon\) are tolerated.

Constraints: The constraints ensure that the prediction errors are within the \(\epsilon\)-tube, except for the slack variables \(\:{{\upxi\:}}_{i}\) and \(\:{{\upxi\:}}_{i}^{*}\):

Here, \(\:{\upvarphi\:}\left(x\right)\) is a function that transforms the input space into a feature space with a higher number of dimensions.

The concentration distribution of drug was obtained using CFD and then it was used as reference for machine learning models. The obtained 2D surface distribution of drug concentration in the feed side of membrane distillation is illustrated in Fig. 3. The separation of solute can be observed in the process which is due to diffusional mass transfer of drug towards the membrane and then permeation through the membrane’s pores. It is perceived that high separation efficiency can be achieved in the membrane module which is significant considering the short residence time of feed in the membrane module.

2D surface concentration distribution of drug calculated from CFD. Feed channel distribution is represented.

For final evaluation, the parameters of the three introduced models are optimized using the Red Deer Algorithm (RDA). For SVR, the parameter C = 3.134 regulates the balance between model’s complexity and the extent to which deviations greater than the margin are tolerated. The radial basis function (RBF) kernel is applied for capturing nonlinear relationships, while the tolerance tol = 0.001752 controls the precision of the stopping criterion. For the MLP model, the model has 213 hidden units, employing the ‘relu’ activation function, and is solved using the ‘lbfgs’ optimizer, with a high iteration limit of 20,000 to ensure convergence with a tolerance of 0.0003811. Finally, the Gamma regression model uses an \(\:\alpha\:\) parameter of 0.0315, without fitting an intercept, to control regularization, and a tol = 0.0028533, ensuring convergence after up to 4,368 iterations. These hyperparameters were refined to maximize the model performance in predicting concentration values in membrane distillation.

The findings of ML analysis are succinctly presented in Table 2. Data set divided to train and test subsets in ratio of 80 to 20% before obtaining final results. Out of the three models, the MLP demonstrated superior performance, attaining a R² score of 0.9955, a MAE of 0.0084, and an RMSE of 0.0148. These metrics indicate that the MLP model has an excellent fit to the data, with very low prediction errors.

Gamma Regression also performed well, with an R² score of 0.9214, a MAE of 0.0325, and an RMSE of 0.0550. While not as accurate as the MLP, Gamma Regression still provided a strong predictive capability, making it a viable option for this type of data. On the other hand, SVR demonstrated the lowest performance among the three models, with a R² score of 0.8710, a MAE of 0.0676, and an RMSE of 0.0723. Although SVR was able to capture the general trend of the data, its predictive accuracy was significantly lower compared to the MLP and Gamma Regression models.

Figs 4, 5, 6, and 7 illustrate the results of the analysis using three different regression models. Figure 4 compares the observed and predicted values of drug concentration C(mol/m³) using the MLP, Gamma Regression, and SVR models, showing the MLP providing the closest fit to the actual data. Figure 5 shows a 3D plot of concentration C as a function of r and z using the MLP model, capturing a complex nonlinear relationship. The Gamma Regression model’s efficacy in analyzing positively skewed data is illustrated in Fig. 6. The 3D depiction of the SVR model is illustrated in Fig. 7, which illustrates its weaker accuracy in comparison to the Gamma Regression and MLP models. In a nutshell, MLP demonstrated the maximum level of accuracy and the lowest rates of error in its predictions of the concentration C(mol/m³). Although the SVR model was less accurate, it has the potential to be improved through further optimization, whereas the Gamma Regression model produced relatively accurate predictions (R2 score of 0.9214).

Comparison of reference and estimated drug concentration values using multi-layer perceptron (MLP), gamma regression, and support vector regression (SVR) models.

Multi-layer perceptron: drug concentration as a function of r and z.

Gamma regression: drug concentration as a function of inputs.

Support vector regression: drug concentration versus inputs.

Figure 8 shows the partial dependence of concentration (C) on the variable r. This figure illustrates how changes in r impact the predicted concentration, holding other variables constant. Figure 9 depicts the partial dependence of concentration (C) on the variable z. Similar to Fig. 8, this figure demonstrates how variations in z affect the predicted concentration while keeping other variables constant. The same trend for concentration profile was observed in the previous study6 which reveals the validity of the models in simulation of concentration profile. Finally, contour plot of concentration (C) is displayed in Fig. 10. The results obtained from ML models are in agreement with the CFD results for concentration distribution of drug in the feed side of MD (see Fig. 3). The changes in radial and axial directions can be observed because of diffusion and convection in both directions in the feed and membrane as well. In the CFD model, both directions were considered for mass transfer including diffusion and convection. It should be pointed out that for higher removal efficiency, the diffusion should be enhanced in radial direction which can be implemented by enhancing the driving force, i.e., pressure difference across the membrane, or adjusting other processing parameters such as feed flow rate and temperature. Moreover, the physical properties of membrane such as porosity, tortuosity, material, and length can alter the mass transfer rate of drug which can be considered in the future study by combination of CFD and machine learning models for MD process.

Dependence plot of drug concentration on r.

Dependence plot of drug concentration on z.

Contour plot of drug concentration based on r and z.

A hybrid computational methodology was utilized in this study for analysis of separation process in purification of drugs. Vacuum membrane distillation (VMD) process was evaluated using mass transfer modeling and machine learning. CFD was employed for mass transfer modeling and the drug concentration on each mesh was extracted for machine learning modeling. With a dataset consisting of 3,401 data points and the variables r(m) and z(m), this study evaluated the performance of three regression models: MLP, Gamma Regression, and SVR—in predicting the concentration C(mol/m³). The hyper-parameters were optimized by fine-tuning the models using the Red Deer Algorithm (RDA). The MLP model demonstrated exceptional performance, as evidenced by an R2 score of 0.9955, an MAE of 0.0084, and an RMSE of 0.0148. This model effectively captured the intricate nonlinear relationships present in the data. Gamma Regression also demonstrated its suitability for positively skewed data, as evidenced by an R2 score of 0.9214. The SVR model demonstrated the lowest performance, with an R2 score of 0.8710, despite capturing the general trend. The significance of model selection and optimization in regression analysis is underscored by these findings, which indicate that MLP is the most accurate model for this dataset, followed by Gamma Regression and SVR. Future studies can look into the influence of process parameters and membrane physical properties on the maximization of separation efficiency by enhancing molecular diffusion towards the membrane.

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The author would like to thank the Deanship of Scientific Research at Shaqra University for supporting this work.

Department of Pharmacology, College of Pharmacy, Shaqra University, Shaqra, 11961, Saudi Arabia

Abdullah Alkhammash

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Abdullah Alkhammash: Conceptualization, Formal analysis, Investigation, Validation, Writing - Original Draft, Visualization.

Correspondence to Abdullah Alkhammash.

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Alkhammash, A. Intelligence analysis of membrane distillation via machine learning models for pharmaceutical separation. Sci Rep 14, 22876 (2024). https://doi.org/10.1038/s41598-024-74616-w

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Received: 18 July 2024

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DOI: https://doi.org/10.1038/s41598-024-74616-w

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