High-fidelity machine learning models for predicting antibacterial effects of cerium oxide nanoparticles across bacterial strains

Abstract

The primary objective of this study is to develop and validate robust data-driven models for accurately predicting bacterial growth inhibition induced by cerium oxide nanoparticles across different bacterial strains and experimental conditions. This study aims to develop and validate data-driven predictive models to quantify bacterial growth inhibition induced by cerium oxide nanoparticles under diverse experimental conditions, with the goal of supporting antibacterial nanotechnology research. To this end, sophisticated AI methods, including Convolutional Neural Networks (CNN), Multi-layer Perceptron Artificial Neural Networks (MLP-ANN), Random Forest (RF), Adaptive Boosting (AdaBoost), and Ensemble Learning (EL), were employed to model bacterial cell concentration (OD600) with high precision. Model hyperparameters were optimized using the Coupled Simulated Annealing (CSA) technique to enhance predictive performance. A comprehensive dataset comprising 484 experimental observations was compiled, with 387 samples allocated for training and 97 for validation. The study considers two bacterial strains, Escherichia coli and Bacillus subtilis, cultivated in media containing cerium oxide nanoparticles with nominal sizes of 6 ± 3.5 nm, 15 ± 4.3 nm, 22 ± 5.7 nm, and 40 ± 10 nm (Samples A–D). Input features included bacterial type, nanoparticle size (medium type), nanoparticle concentration, and exposure time. Monte Carlo sensitivity analysis revealed that exposure time is the dominant factor governing bacterial cell concentration, followed by nanoparticle concentration, nanoparticle size, and bacterial strain. Among the evaluated models, MLP-ANN exhibited the highest predictive accuracy, achieving the greatest R² values and the lowest RMSE and AARE%. Beyond predictive performance, the results provide insight into key drivers of nanoparticle-induced antibacterial activity and demonstrate how data-driven modeling can guide experimental prioritization. Overall, the proposed framework serves as a complementary tool to laboratory experiments, supporting more efficient investigation of antibacterial effects while preserving the necessity of experimental validation. These results demonstrate that AI-based models, particularly MLP-ANN, serve as a powerful complementary tool to laboratory experiments by enabling accurate prediction, guiding experimental prioritization, and reducing experimental burden while maintaining the necessity of experimental validation.

Introduction

Nanoparticles often exhibit chemical and physical characteristics distinct from those of bulk materials. Their exceptionally high surface area-to-volume ratio and their capacity to selectively facilitate chemical processes render them highly effective for diverse functions. These include acting as heterogeneous catalysts in fuel conversion, serving as advanced tools in diagnostic imaging and biosensing, and delivering therapeutic agents. Nanostructures linked with polymers, metals, metal oxides, liposomes, micelles, dendrimers, or metal sulfides are being investigated for therapeutic applications, such as targeting cancer [1, 2] or combating microbial infections [3–6]. Outside of biomedical uses, nanoparticles are already being applied across a range of industrial sectors and consumer products. One prominent example is cerium oxide (CeO₂) nanoparticles, a type of metal oxide nanomaterial used in multiple domains. CeO₂ is employed in semiconductor manufacturing as a polishing compound, in catalytic converters to treat exhaust emissions, in fuel additives to improve combustion, as a UV shield, and in fuel cells as an electrolyte [7, 8]. More recently, CeO₂ nanoparticles have demonstrated antioxidant properties under physiological pH, highlighting their potential for protecting biological tissues from oxidative damage, radiation, and inflammation [9, 10].

However, the same features that contribute to the utility of nanoparticles may also pose environmental and health hazards. The potential for toxicity associated with nanomaterials has been increasingly recognized [11–16], and several reviews and analyses have addressed this concern [14, 17–20]. Gaining a deeper understanding of specific nanomaterial risks is essential to reducing possible harm to ecosystems and human health [21, 22]. Despite limited data on how CeO₂ nanoparticles behave in natural environments, including their movement, persistence, and accumulation, their large-scale use especially as diesel fuel additives (commonly around 5 mg/liter) and polishing agents raises concerns over environmental exposure [22, 23]. The expanding applications and increasing industrial output of CeO₂ nanoparticles have driven more research into their environmental behavior and biological impacts. Notably, contrasting results have been reported. For example, Park et al. found that CeO₂ nanoparticles induced oxidative stress in human lung cells [24], whereas Schubert et al. observed protective, antioxidant effects (43). Other studies suggest that pH levels and environmental conditions may determine whether CeO₂ acts beneficially or detrimentally to cells (2).

Another layer of complexity in toxicity assessments stems from differences in nanoparticle synthesis. Manufacturing techniques can introduce impurities such as residual solvents, detergents, or stabilizers that may not be entirely removed. A case in point is fullerene (C₆₀), initially identified as toxic, but later studies linked the observed toxicity to tetrahydrofuran (THF), a solvent used during its synthesis [25]. As such, the apparent biological effects of nanomaterials may partly depend on associated chemical residues. Moreover, commercial nanoparticle suppliers often omit details about the synthesis process or the presence of capping agents, complicating the interpretation of toxicological findings derived from nanoparticle suspensions.

Most current toxicological investigations into metal oxide nanoparticles have concentrated on mammalian cells, with comparatively limited attention given to bacterial models, especially concerning CeO₂ and related materials [13]. Thill et al. demonstrated that commercial CeO₂ nanoparticles exerted toxic effects on Escherichia coli in KNO₃ solution by inducing oxidative reactions following surface attachment [26]. Conversely, other studies reported negligible or no toxicity to bacteria from CeO₂ exposure [27, 28]. These divergent outcomes are likely influenced by numerous factors, including differences in raw materials, surface stabilizers, nanoparticle size and composition, and variability in experimental toxicity protocols. Such inconsistencies make it challenging to draw definitive conclusions across studies.

Recent thermodynamic and quantum chemical studies have demonstrated how adsorption-driven interactions and molecular electronic properties govern inhibitory behavior in chemical–biological systems, highlighting the value of combining experimental observations with theoretical calculations to interpret surface-mediated inhibition phenomena [29]. Theoretical thermodynamic analyses based on density functional theory (DFT) have further shown that molecular stability, interaction energy, and cavity–target compatibility play a crucial role in capturing aromatic compounds and biological entities, underscoring the relevance of energetics-driven modeling in understanding molecule–bacteria interactions [30]. Complementary experimental and quantum chemical investigations have also linked antimicrobial and antioxidant activities to molecular electronic structure and thermodynamic descriptors, illustrating how physicochemical properties can be used to rationalize bacterial growth inhibition mechanisms [31].

Recent studies have highlighted the versatile physicochemical and biological functionalities of cerium-based nanomaterials across various biomedical applications. For instance, Chandra et al. demonstrated the rapid synthesis of cerium oxide microtubes with high surface area and confirmed their cytocompatibility along with significant reactive oxygen species (ROS) scavenging capability, emphasizing their potential in biological environments [32]. In parallel, multiscale computational investigations revealed that cerium-based metal–organic frameworks (MOFs) exhibit strong and stable adsorption behavior on functionalized carbon nanotubes, governed by π–π interactions and electrostatic forces [33]. Extending these findings, integrated in silico and in vitro studies further showed that Ce-based nanocarriers can enable efficient pH-responsive drug delivery, highlighting the role of nanostructure design in controlling biological performance [34]. Moreover, Ce-based MOF–CNT hybrid systems have been successfully developed for controlled release of therapeutic agents, demonstrating enhanced antifungal activity driven by redox cycling and ROS-mediated mechanisms [35]. Collectively, these studies underscore the significant potential of cerium-based nanomaterials in biomedical and antimicrobial applications and provide a strong foundation for data-driven modeling approaches aimed at predicting their biological effects.

Recent studies have demonstrated the growing importance of nanomaterials and bioactive compounds in antimicrobial applications. Ahmed et al. developed green-synthesized selenium nanoparticles exhibiting significant antifungal activity and gene-level regulation in multidrug-resistant microorganisms [36]. Similarly, Hasoon et al. reported biogenic silver nanoparticles with enhanced antibacterial, antioxidant, and cytotoxic properties driven by reactive oxygen species generation [37]. Hasan et al. investigated plant-derived extracts and confirmed their notable antibacterial and antioxidant activities against pathogenic strains [38]. In addition, Ahmed et al. explored synergistic nanocomposite systems combining chitosan and iron oxide nanoparticles, demonstrating improved antibacterial performance via gene inhibition mechanisms [39]. Furthermore, Ibraheem et al. designed hydrazone-based compounds with strong antimicrobial and antioxidant activities supported by molecular modeling approaches [40]. Collectively, these studies highlight the critical role of nanomaterials, green synthesis, and computational analysis in advancing antimicrobial research, supporting the relevance of data-driven approaches such as those employed in the present study.

Recent advances in deep learning have strengthened their role in translational bioinformatics, particularly for modeling complex biological responses such as antimicrobial activity and resistance. Comprehensive works on deep learning in bioinformatics highlight the use of ensemble and predictive models to support biological interpretation and experimental design, which is consistent with the data-driven modeling approach adopted in the present study to analyze nanoparticle-induced antibacterial effects [41].

Although previous research has explored the antibacterial effects of various nanomaterials, comprehensive datasets covering diverse bacterial strains and nanoparticle characteristics are still scarce. Moreover, a critical gap remains in systematically modeling the influence of nanoparticle physicochemical parameters (e.g., size distribution, concentration, and exposure dynamics) across multiple bacterial species using advanced, hyperparameter-optimized AI architectures. Specifically, prior ML-based efforts often focus on: A single nanoparticle type, Limited bacterial strains or static exposure conditions, and Conventional algorithms (e.g., basic regression or shallow trees) without rigorous hyperparameter optimization. As a result, there is an urgent need to systematically collect extensive data on bacterial growth inhibition across different bacterial types, nanoparticle sizes, concentrations, and exposure times. This study advances the field by: (1) Introducing cerium oxide nanoparticles (CeO₂·NPs), a redox-active, less-studied nanomaterial with dual antioxidant/pro-oxidant behavior, into AI-driven antibacterial modeling, which has not been comprehensively addressed in existing ML literature. (2) Systematically incorporating four distinct nanoparticle sizes (6 ± 3.5–40 ± 10 nm) as categorical input features, enabling the model to learn size-dependent biological responses, that is a dimension often overlooked in prior studies. (3) Employing a diverse ensemble of state-of-the-art AI models (MLP-ANN, CNN, RF, AdaBoost, and Ensemble Learning), all rigorously optimized via Coupled Simulated Annealing (CSA), which is a global optimization technique rarely applied in nanotoxicology ML to maximize predictive fidelity. (4) Quantifying feature importance through Monte Carlo sensitivity analysis, revealing that exposure time dominates over nanoparticle concentration and bacterial type in modulating growth inhibition. Modeling the antimicrobial impacts of cerium oxide nanoparticles with precision represents a formidable obstacle, particularly given the essential function of bacterial cell density in microbiology and nanotechnology fields, stemming from the intricate interactions among variables like bacterial species, medium specimen category, nanoparticle dosage, and duration. Bacterial cell concentration (OD600) is a critical parameter for optimizing antimicrobial strategies, developing nanobiotechnological applications, and designing effective bioprocesses, underscoring the importance of precise predictive models. Traditional laboratory techniques for evaluating microbial suppression, although dependable, usually require extensive manual work and prolonged periods, thereby requiring cutting-edge modeling tools to furnish quick and precise estimations. Development and verification of the models drew from a substantial archive of 484 laboratory-sourced observations, allocating 387 entries to the training phase and 97 to the testing phase. Stringent checks on the data collection verified its viability for forecasting applications, complemented by a Monte Carlo-driven sensitivity examination that gauged the impact of individual variables on microbial proliferation results. Comprehensive reviews of model capabilities employed quantitative indicators and graphical tools, revealing the MLP-ANN variant's exceptional proficiency in forecasting bacterial cell densities. Figure 1 provides a visual summary of the entire investigative framework.

Fig. 1.

Method for identifying the best-performing data-driven model

Previous studies have provided valuable insights into the antibacterial and toxicological effects of metal oxide nanoparticles, including cerium oxide, on microbial systems. However, most available investigations remain limited to isolated experimental conditions, single nanoparticle sizes, or qualitative biological interpretations, with comparatively few efforts devoted to systematic, data-driven modeling across multiple bacterial strains and exposure regimes. In addition, existing machine learning–based studies in nanotoxicology have often relied on conventional algorithms with limited hyperparameter optimization and have rarely incorporated rigorous sensitivity analysis or structured validation strategies.

To address these gaps, the present study introduces a comprehensive AI-driven framework for predicting bacterial growth inhibition induced by cerium oxide nanoparticles. The original contributions of this work can be summarized as follows:

• (i)Development of high-fidelity predictive models for bacterial cell concentration (OD600) by integrating experimental data across two bacterial strains (Escherichia coli and Bacillus subtilis), four distinct cerium oxide nanoparticle size ranges, a wide concentration window (0–150 mg/L), and varying exposure times;
• (ii)Systematic comparison of multiple advanced artificial intelligence techniques (MLP-ANN, CNN, Random Forest, AdaBoost, and Ensemble Learning), all optimized using the Coupled Simulated Annealing (CSA) algorithm, which is rarely applied in nanotoxicology modeling;
• (iii)Quantitative evaluation of input feature importance using Monte Carlo sensitivity analysis, enabling clear identification of the dominant biological and experimental factors governing bacterial growth inhibition; and
• (iv)Robust validation of model generalization through structured cross-validation strategies, demonstrating reliable predictive performance beyond simple interpolation within the training data.

By explicitly positioning the proposed framework within the context of prior experimental and computational studies, this work advances current understanding of nanoparticle–bacteria interactions and provides a practical, efficient alternative to labor-intensive laboratory assays for assessing antibacterial effects of cerium oxide nanoparticles.

Summary of machine learning approaches

In nanobiotechnology studies, predictive analytics powered by artificial intelligence techniques play a vital role, especially in unraveling intricate processes like microbial suppression. This investigation applies five advanced AI frameworks including CNN, MLP-ANN, RF, AdaBoost, and EL to simulate fluctuations in bacterial density (OD600) under the influence of cerium oxide nanoparticles. These strategies prove exceptionally adept at detecting nonlinear, multifaceted patterns in laboratory data, delivering a dependable, efficient, and adaptable replacement for standard testing protocols through their rapid computation, adaptability, and elevated forecasting reliability.

The selection of the models, MLP-ANN, Random Forest, AdaBoost, CNN, and Ensemble Learning, was guided by the need to evaluate diverse machine learning paradigms on a real-world nanotoxicology dataset. MLP-ANN was prioritized for its capacity to capture complex nonlinear dose–response relationships, while tree-based methods (RF, AdaBoost) offered interpretability and robustness to feature scaling. CNNs were included to assess the potential of deep learning architectures on structured tabular data, and ensemble learning was used to combine model strengths. Hyperparameters were optimized using Coupled Simulated Annealing (CSA), a global optimization technique proven effective in avoiding local minima and improving convergence in moderate-sized datasets. This methodology balances predictive accuracy, computational feasibility, and practical utility for experimental scientists [42, 43].

To improve clarity and conciseness of the main manuscript, detailed theoretical descriptions of the employed machine learning algorithms (CNN, MLP-ANN, RF, AdaBoost, and EL) and CSA Optimization Algorithm have been moved to the Supplementary Information, while the main text focuses on model selection rationale, implementation, and comparative performance.

Metrics for data collection and assessment

Description of data collection

This investigation's artificial intelligence frameworks were constructed using a compilation of data from laboratory trials focused on examining how cerium oxide nanoparticles suppress microbial expansion across diverse environments. Comprising 484 test-derived records, the set integrates critical predictors including bacterial strain, growth medium category, nanoparticle dosage levels, and duration of exposure (in hours), while using optical density at 600 nm (OD600) to measure cellular density as the target outcome. All experimental data were derived from [44]. The bacterial strains investigated were Escherichia coli and Bacillus subtilis, cultured in media containing cerium oxide nanoparticles with nominal sizes of 6 ± 3.5 nm, 15 ± 4.3 nm, 22 ± 5.7 nm, and 40 ± 10 nm (designated as Samples A, B, C, and D). This robust dataset forms a solid basis for constructing and validating predictive models to simulate bacterial growth inhibition under diverse microbial and nanomaterial conditions [44]. The original dataset from Pelletier et al. (2010) was complete, with no missing entries across the 484 observations (as verified by cross-referencing their supplementary tables and published figures). Consequently, no imputation or deletion strategies were required. All 484 data points were retained for modeling. The CeO₂ nanoparticles used in this study were synthesized via a hydrothermal, surfactant-free method to produce four distinct size ranges (nominally 6 nm, 15 nm, 22 nm, and 40 nm) without organic stabilizers that could confound toxicity assessments [44]. Particle size was validated using TEM, AFM, and DLS, with TEM confirming primary crystallite dimensions and DLS revealing significant aggregation in suspension, especially near the point of zero charge (PZC ≈ 8.0 in water). The PZC shifted to ~ pH 6 in ionic media due to anion adsorption. The absence of surface stabilizers ensured that observed biological effects were attributable to the nanoparticles themselves rather than residual surfactants. Size-dependent antibacterial effects were noted, with aggregation behavior in biological media highlighting the importance of characterizing nanoparticles under relevant experimental conditions.

Table 1 delivers an in-depth overview of the used variables ranges, highlighting their primary attributes along with the testing variables applied in simulating microbial density (OD600). The dataset, comprising 484 observations, includes nanoparticle concentrations ranging from 0 to 150 mg/L, time intervals from 0 to 8 h, and bacterial cell concentrations (OD600) spanning 0 to 2 across different media and bacterial strains. The broad compilation of data, spanning multiple bacterial strains, nanoparticle dimensions, and testing setups, lays a robust groundwork for constructing advanced AI frameworks that reliably estimate the curtailment of microbial expansion in diverse biotechnology and nanoscience contexts.

Table 1.

Overview of experimental findings for E. coli and B. subtilis across different media

Bacteria type	Medium (sample type)	Points number	Concentration range (mg/L)	Range of time (h)	OD600 range
E. coli	CeO2 nanoparticles with nominal sizes of 6 ± 3.5 nm	56	0–150	0.5–7	0–1
E. coli	CeO2 nanoparticles with nominal sizes of 15 ± 4.3 nm	60	0–150	0.5–8	0–1
E. coli	CeO2 nanoparticles with nominal sizes of 22 ± 5.7 nm	56	0–150	0.5–7	0–1
E. coli	CeO2 nanoparticles with nominal sizes of 40 ± 10 nm	56	0–150	0.5–7	0–1
B. subtilis	CeO2 nanoparticles with nominal sizes of 6 ± 3.5 nm	64	0–150	0–7.5	0–2
B. subtilis	CeO2 nanoparticles with nominal sizes of 15 ± 4.3 nm	64	0–150	0–7.5	0–2
B. subtilis	CeO2 nanoparticles with nominal sizes of 22 ± 5.7 nm	64	0–150	0–7.5	0–2
B. subtilis	CeO2 nanoparticles with nominal sizes of 40 ± 10 nm	64	0–150	0–7.5	0–2

Experimental data and materials

All experimental data used in this study were obtained from the comprehensive experimental work reported by Pelletier et al. (2010), in which the antibacterial effects of cerium oxide (CeO₂) nanoparticles were systematically investigated. In that study, CeO₂ nanoparticles with nominal sizes of 6 ± 3.5 nm, 15 ± 4.3 nm, 22 ± 5.7 nm, and 40 ± 10 nm were synthesized using a surfactant-free hydrothermal method to avoid interference from organic stabilizers. Particle size and morphology were characterized using transmission electron microscopy (TEM), atomic force microscopy (AFM), and dynamic light scattering (DLS), while aggregation behavior and surface charge properties were evaluated under relevant aqueous and ionic conditions.

Antibacterial experiments were conducted using two bacterial strains, Escherichia coli and Bacillus subtilis, cultivated in media containing CeO₂ nanoparticles at concentrations ranging from 0 to 150 mg/L. Bacterial growth was monitored over exposure times up to 8 h, and cell concentration was quantified using optical density measurements at 600 nm (OD600). Appropriate nanoparticle-containing blanks were employed to correct for background scattering effects. The resulting dataset comprises 484 experimentally measured observations and serves as the foundation for all predictive modeling and analyses presented in this study.

Model evaluation indices

For assessing and contrasting the forecasting accuracy of the constructed models, key evaluation indicators were calculated for every modeling method employed. This research examines models that utilize sophisticated artificial intelligence methodologies, namely Convolutional Neural Networks (CNN), Multi-layer Perceptron Artificial Neural Networks (MLP-ANN), Random Forest (RF), Adaptive Boosting (AdaBoost), and Ensemble Learning (EL). The algorithms in question were applied for forecasting the optical density at 600 nm (OD600), indicative of bacterial density, within growth media incorporating cerium oxide nanoparticles. The predictive inputs encompassed variables like bacterial strain, category of medium specimen, nanoparticle dosage, and duration. In order to assess the precision and stability of individual models across training and validation stages, essential quantitative measures such as the R² value, RMSE, and AARE% were determined [45–47]:

1

2

3

4

The notations ‘pred’ and ‘exp’ correspond to the model's forecasted and the laboratory-obtained values for bacterial density (measured as OD600), in that order. N symbolizes the full dataset size within the research, comprising 484 laboratory-derived observations on the suppressive impact of cerium oxide nanoparticles on bacterial proliferation [48, 49].

The predictive models incorporated features like bacterial strain (Escherichia coli and Bacillus subtilis), growth medium category (featuring cerium oxide nanoparticles with average particle diameters of 6 ± 3.5 nm, 15 ± 4.3 nm, 22 ± 5.7 nm, and 40 ± 10 nm), nanoparticle dosage, and elapsed duration, aiming to forecast bacterial density via optical density at 600 nm (OD600) as the target outcome. For effective development of the models and credible evaluation of their efficacy, the full dataset underwent random partitioning: 80% (387 entries) designated for the training process, while 20% (97 entries) was set aside for validation. Hyperparameter optimization was conducted exclusively on the training subset; the test set was never involved in any stage of the modeling pipeline including data scaling, hyperparameter tuning, or model training. Consequently, not all data were used for hyperparameter optimization.

Before building the models, an adapted version of the min–max normalization process was utilized to handle the broad variations in scale among the input and output parameters, adjusting every attribute to the consistent interval of [−1, 1]. By diminishing the influence of variables with greater magnitudes, this data preparation phase bolsters the steadiness and training efficiency of the models, ultimately boosting the forecasting dependability of the AI frameworks. The employed normalization procedure is outlined below:

5

In the formula, x_N stands for the scaled value, x refers to raw, unnormalized observation, while X_max and X_min signify the highest and lowest figures in the entire collection of data. This scaling process, which adjusts all attributes to a balanced interval of [−1, 1], promotes uniformity across the dataset and, as a result, enhances the precision and dependability of the AI-based predictive systems created for this investigation. In addition, the MLP-ANN model employs hyperbolic tangent (tanh) activation functions in hidden layers, which operate optimally with inputs centered near zero and bounded within [−1, 1]. This alignment accelerates convergence and improves gradient stability during training. Unlike [0, 1] scaling, [−1, 1] preserves the relative directionality of features (e.g., low vs. high concentration), which can enhance model interpretability and sensitivity, particularly important in physics-informed modeling [50, 51].

Results and analysis

Statistical analysis was conducted to rigorously evaluate the predictive performance and robustness of the developed models. The dataset was randomly divided into training (80%, 387 samples) and testing (20%, 97 samples) subsets to ensure unbiased performance assessment. Model accuracy was quantified using multiple complementary statistical metrics, including the coefficient of determination (R²), root mean square error (RMSE), and average absolute relative error (AARE%). These metrics collectively assess goodness of fit, prediction deviation magnitude, and relative error behavior, respectively.

To ensure reliable comparison among models, all performance metrics were computed consistently for the training set, test set, and the complete dataset. Overfitting was evaluated by comparing metric values between training and testing phases. In addition, cumulative distribution functions (CDFs) of absolute relative error were analyzed to examine the distribution and stability of prediction errors across samples. Scatter plots of predicted versus experimental values were used to visually assess model agreement and systematic bias.

Sensitivity analysis was performed using Monte Carlo simulations to statistically quantify the influence of each input variable on the predicted bacterial cell concentration. Furthermore, outlier detection was carried out using the leverage approach (Williams plot), combining standardized residuals and leverage values to identify influential observations. Together, these statistical analyses provide a comprehensive and quantitative basis for evaluating model performance, reliability, and interpretability.

Outlier detection

The leverage approach is employed to identify outliers by combining the assessment of standardized residuals and leverage values for each data point. The original formula defines the residual Di as detailed below:

6

In this equation, Di denotes the residual for the ith data point, calculated as the difference between the anticipated bacterial cell concentration (X_Pred,i, measured as OD600) and the experimentally observed bacterial cell concentration (X_Exp,i) in media containing cerium oxide nanoparticles. This deviation measures the discrepancy between forecasted and actual values for every entry within the data collection. The formula defining the normalized residual (SDi) is presented as follows:

7

In this formulation, SDi signifies the normalized deviation associated with the ith observation, incorporating Di as the discrepancy derived from the prior computation, N as the aggregate count of observations (484 within the current investigation), and hi as the hat value (leverage metric) tied to that specific entry, extracted via the hat matrix. Figure 2 depicts the outlier identification process for the bacterial density records through the leverage technique, rendered as a Williams graph. The equation's lower component standardizes the deviation via the residual variance, fine-tuned by the leverage h_i, serving as a gauge for each observation's effect on algorithm's calibration. The critical leverage limit, denoted h*, is derived from h* = 3(m + 1)/N, in which m equals the count of predictor elements (four in the present work: bacterial strain, medium category, nanoparticle dosage, and incubation period) and N equals the full data volume. Applying this to the collection yields h* = 3(5)/484 = 0.0310. Entries where hi surpasses h* qualify as influential leverage points, signaling substantial impact potential, whereas deviations with |SDi| beyond standard limits (typically > 3) are designated as anomalies. Overall, this evaluation affirms the data's viability for simulating nanoparticle-induced bacterial suppression and yields essential perspectives on cerium oxide nanoparticle effects within bacterial proliferation patterns across cultivation settings. In Fig. 2, h_i represents the leverage value of the ith observation, which quantifies the influence of each data point on the regression model. Higher h_i values indicate observations with greater potential impact on model fitting and are commonly used in conjunction with standardized residuals to identify influential data points in Williams plot analysis.

Fig. 2.

Detection of anomalies utilizing the Leverage method

Sensitivity analysis

This portion examines the effects of primary input features namely, bacterial strain, category of growth medium sample, nanoparticle dosage, and incubation period on the optical density at 600 nm (OD600) as a measure of bacterial density within cultures incorporating cerium oxide nanoparticles, alongside an evaluation of each element's comparative weight. The role of these predictor variables is measured via correlation metrics, yielding key understandings of their influence on the AI-driven forecasting frameworks established from 484 lab-generated observations.

This research applies the Monte Carlo simulation technique, prized for its straightforwardness and dependability, in order to gauge the comparative effects of predictor factors on the suppression of bacterial proliferation. Through methodical extraction of samples across a wide array of potential input figures, the technique adeptly addresses uncertainties, permitting an unmediated examination of output fluctuations independent of supplemental analytical layers. Under this methodology, the forecasting system integrates several predictor elements, detailed next:

8

To create a thorough array of samples, the span and probabilistic profiles for every predictor element (x) have been defined:

9

In this configuration, k signifies the overall volume of created samples, while n stands for the number of feature inputs. Building the dataset for inputs can involve several extraction methods, such as basic uniform sampling, focused importance-based selection, or Latin hypercube sampling (LHS). Next, the system processes each group of chosen inputs to generate the related output figures:

10

Within this framework, k represents the total count of generated samples, while n denotes the quantity of predictor factors. At this juncture, multiple sampling strategies including even distribution via random picking, weighted importance-driven choice, and LHS can be implemented. Afterward, the forecasting framework is applied to every grouping of chosen inputs, generating the pertinent output results:

11

12

In this formulation, V signifies variance, whereas E stands for the anticipated value. The evaluation of sensitivity relies on the connection between inputs and outputs as outlined in Eq. (13). When it comes to different approaches for graphical representation, producing scatter plots emerges as a particularly reliable and uncomplicated option:

13

Figure 3 delivers an in-depth sensitivity assessment that gauges the comparative effects of factors such as bacterial strain, growth medium category, nanoparticle dosage, and incubation duration on optical density at 600 nm (OD600), a proxy for bacterial density. This evaluation leverages Monte Carlo simulations fused with the AI forecasting systems built within the present investigation. Celebrated for tackling variability, the Monte Carlo technique methodically draws samples from diverse input scenarios to measure their sway over projections of bacterial proliferation. Outputs appear in a correlation matrix or analogous diagrammatic form, illuminating the intensity and polarity of links tying specific input elements to the ultimate bacterial density outcomes.

Fig. 3.

Analysis of the parameters impacting bacterial cell concentration (OD600) in media containing cerium oxide nanoparticles, utilizing advanced predictive models and Monte Carlo simulation

According to the sensitivity evaluation, time emerges as the primary driver affecting bacterial density (measured by OD600), boasting a sensitivity value of 5.3489 that underscores its pivotal function in regulating the suppression of bacterial proliferation when cerium oxide nanoparticles are present. Ranking just behind is nanoparticle dosage, registering a sensitivity of 4.9742, which highlights its notable effect on bacterial behavior patterns. Growth medium category, scoring 4.7521 in sensitivity amount, holds considerable weight as well, implying that the dimensions of nanoparticles serve as a vital element in antimicrobial activity. Lastly, bacterial strain, at a sensitivity of 4.6589, demonstrates a marginally reduced yet noteworthy contribution.

These insights, drawn from 484 experimental data points, highlight the complex interactions among the input parameters, with time and nanoparticle concentration emerging as the primary factors driving bacterial cell concentration in media containing cerium oxide nanoparticles. Incorporating Monte Carlo simulations bolsters the dependability of the current evaluation by tackling variability in predictor elements, laying a solid groundwork for grasping the core processes involved. Researchers can draw on Fig. 3 as a practical guide, gaining practical recommendations to refine antibacterial tactics and nanotechnology-based solutions in fields requiring precise management of microbial expansion.

Model construction

In this section, the process of optimizing different models as well as the performance of various models are fully discussed. All computational modeling was performed using MATLAB R2023a under an institutional academic license. The Statistics and Machine Learning Toolbox, Deep Learning Toolbox, and Optimization Toolbox were employed for model development, training, and hyperparameter tuning. Figure 4 presents the optimization results for the number of base estimators in the Adaptive Boosting (AdaBoost) model and illustrates how model performance varies with ensemble size. As shown, increasing the number of base estimators initially improves predictive accuracy by allowing the ensemble to progressively correct errors made by weak learners. However, beyond a certain point, additional estimators yield diminishing returns and may increase the risk of overfitting. The optimal configuration was achieved with 14 base estimators, representing a balance between learning capacity and generalization. This result indicates that a moderately sized ensemble is sufficient to capture the nonlinear relationships governing bacterial growth inhibition by cerium oxide nanoparticles, without introducing unnecessary model complexity across different bacterial strains and nanoparticle conditions.

Fig. 4.

The optimal value for the hyperparameter (number of base estimators) in the AdaBoost method, which in this case is 14

Figure 5 summarizes the optimization of the maximum tree depth in the Random Forest model, a hyperparameter that directly controls the expressiveness of individual decision trees. Shallow trees were unable to fully capture the complex, nonlinear effects of nanoparticle concentration, size, and exposure time, leading to underfitting. Conversely, excessively deep trees tended to fit noise within the experimental dataset. An optimal maximum depth of 10 was identified, yielding the highest predictive accuracy while maintaining robust generalization across training and testing subsets. This outcome suggests that bacterial growth inhibition induced by cerium oxide nanoparticles can be effectively described by hierarchical decision rules of moderate complexity, reinforcing the suitability of ensemble tree-based models for nanotoxicological datasets.

Fig. 5.

The optimal value for the hyperparameter (number of estimators) in the random forest method, which in this case is determined to be 10

Figure 6 depicts the optimization of the number of neurons in the second hidden layer of the MLP-ANN model. The network architecture employed in this study consists of an input layer, a first hidden layer with 20 neurons using the tansig activation function, a second hidden layer with a variable number of neurons, and a linear output layer. The results show that model performance improves as the number of neurons increases up to an optimal value of 13, beyond which predictive accuracy begins to plateau or slightly degrade. This behavior reflects the trade-off between representational power and overfitting. The optimized architecture enables the MLP-ANN to effectively model the highly nonlinear dose–time–size interactions underlying bacterial growth inhibition, which explains its superior performance relative to other models examined in this study.

Fig. 6.

The optimal value for the hyperparameter (number of neurons in hidden layer) in the MLP-ANN method, which in this case is determined to be 13

In addition to individual models, an ensemble learning (EL) framework was constructed by combining Support Vector Machine (SVM), Decision Tree (DT), and K-Nearest Neighbors (KNN) algorithms to leverage complementary modeling strengths. The optimized KNN component employed five neighbors using the Euclidean distance metric, while the SVM utilized an RBF kernel with tuned hyperparameters (C = 120, ε = 0.001, γ = 0.02) to balance bias and variance. This hybrid ensemble improved predictive stability by integrating distance-based, margin-based, and rule-based learning paradigms. In parallel, the convolutional neural network (CNN) was configured with three convolutional layers, a pooling layer, and a fully connected network. Although CNNs are traditionally designed for spatial data, the optimized architecture demonstrated the ability to extract higher-level feature representations from structured experimental inputs, capturing complex interactions among bacterial type, nanoparticle size, concentration, and exposure time.

Overall, these optimization results highlight that model performance is strongly influenced by the balance between complexity and generalization. Models with insufficient capacity fail to capture the nonlinear antibacterial effects of cerium oxide nanoparticles, whereas overly complex configurations risk overfitting the experimental dataset. The optimized hyperparameters identified here provide mechanistic insight into the underlying data structure and contribute to the robust predictive performance observed across all models, particularly the MLP-ANN.

Table 2 summarizes the predictive performance of the five artificial intelligence models including Random Forest (RF), Ensemble Learning (EL), Adaptive Boosting (AdaBoost), Multi-Layer Perceptron Artificial Neural Network (MLP-ANN), and Convolutional Neural Network (CNN) using three complementary evaluation metrics (R², RMSE, and AARE%) computed for the training set, testing set, and the full dataset. These results reflect the effectiveness of the CSA-based hyperparameter optimization strategy in achieving high accuracy while limiting overfitting across diverse bacterial strains, nanoparticle sizes, concentrations, and exposure times.

Table 2.

The evaluation index values achieved for all created models regarding

Model	R2			RMSE			AARE%
	Training	Test	Total	Training	Test	Total	Training	Test	Total
EL	0.99	0.98	0.99	0.05	0.08	0.06	5.86	10.12	6.72
Adaboost	0.99	0.99	0.99	0.03	0.06	0.04	5.60	9.20	6.32
MLP-ANN	0.99	0.99	0.99	0.03	0.06	0.04	5.05	8.81	5.80
RF	0.99	0.99	0.99	0.03	0.06	0.04	5.31	8.55	5.96
CNN	0.99	0.99	0.99	0.04	0.07	0.04	5.86	9.99	6.68

A clear performance hierarchy emerges from Table 2, with the MLP-ANN model consistently outperforming all other approaches. Its superior accuracy (R² = 0.9972 for the complete dataset), combined with the lowest RMSE and AARE%, indicates a strong ability to capture the highly nonlinear and coupled relationships between bacterial type, nanoparticle size, concentration, and exposure duration. This finding suggests that the flexible, multilayer structure of MLP-ANN is particularly well suited for modeling complex dose–time–size interactions characteristic of nanoparticle–bacteria systems.

The Random Forest model ranks closely behind MLP-ANN, achieving comparably high R² values and low error metrics. This strong performance highlights the effectiveness of ensemble tree-based methods in handling heterogeneous experimental data and suggests that bacterial growth inhibition can be reasonably approximated through hierarchical decision rules. In contrast, AdaBoost, CNN, and Ensemble Learning models, while still demonstrating robust predictive capability (R² > 0.994 and AARE% < 6.8), exhibit slightly higher error levels. These differences may reflect limitations in capturing subtle nonlinear dependencies or increased sensitivity to noise in the experimental dataset (Fig. 7).

Fig. 7.

MSE, R² and AARE% for every developed model in this document (evaluation stage)

Figure 8 complements the numerical results in Table 2 by providing a visual comparison of model performance on the validation dataset. The graphical representation clearly shows the superior predictive consistency of the MLP-ANN model, evidenced by the tightest agreement between predicted and experimental OD600 values and the lowest dispersion in error metrics. Models such as CNN and EL display marginally broader deviations, indicating reduced precision when extrapolating across the validation samples. Together, Table 2 and Fig. 8 demonstrate that although all CSA-optimized models achieve strong predictive performance, neural network–based approaches particularly MLP-ANN offer a distinct advantage in capturing the complex, nonlinear dynamics of bacterial growth inhibition induced by cerium oxide nanoparticles.

Fig. 8.

Cumulative distribution function (CDF) of absolute relative error (ARE%) for artificial intelligence models predicting bacterial cell concentration (OD600) in media containing cerium oxide nanoparticles

Overall, the integrated analysis of numerical metrics and graphical results confirms that CSA-optimized AI models provide a reliable and efficient alternative to traditional laboratory-based evaluations. Among them, the MLP-ANN framework stands out as the most accurate and robust tool for predicting bacterial cell concentration (OD600), supporting its potential application in nanobiotechnology and antimicrobial design where precise control of microbial proliferation is required.

Figure 8 provides a cumulative distribution function (CDF) analysis of the absolute relative error (ARE%) for all predictive models, offering a probabilistic assessment of their accuracy in estimating bacterial cell concentration (OD600) under cerium oxide nanoparticle exposure. Unlike single-point error metrics, the CDF representation enables evaluation of both the magnitude and distribution of prediction errors across the entire validation dataset.

The MLP-ANN model exhibits the most favorable error distribution, with nearly all predictions falling below an ARE% of approximately 5.8. This steep ascent toward 100% cumulative frequency indicates not only high average accuracy but also strong consistency across diverse experimental conditions, including variations in bacterial strain, nanoparticle size, concentration, and exposure time. Such behavior reflects the MLP-ANN’s ability to robustly capture complex, nonlinear interactions governing bacterial growth inhibition.

The Random Forest and Adaptive Boosting models also demonstrate strong predictive performance, with their CDF curves closely following that of MLP-ANN but displaying slightly broader error distributions. This suggests that while ensemble tree-based methods effectively model general trends in the data, they may be less sensitive to subtle nonlinear dependencies or interaction effects compared to multilayer neural networks.

In contrast, the Convolutional Neural Network (CNN) and Ensemble Learning (EL) models exhibit wider ARE% ranges, indicating reduced precision and greater variability in prediction errors. This outcome may be attributed to architectural mismatches between these models and the structured, tabular nature of the experimental dataset, as well as their comparatively limited ability to generalize across all combinations of nanoparticle size, dose, and exposure duration.

Overall, the CDF-based analysis reinforces the conclusions drawn from numerical performance metrics by demonstrating that the MLP-ANN model not only minimizes average prediction error but also maintains superior reliability across the full spectrum of experimental conditions. This level of predictive stability is particularly critical for nanobiotechnological applications where precise control of bacterial growth dynamics is required, further supporting the suitability of MLP-ANN as a robust surrogate modeling tool for nanoparticle-induced antibacterial effects.

Figures 9, 10, 11, 12, 13 present crossplots comparing predicted versus experimentally measured bacterial cell concentrations (OD600) for all five artificial intelligence models across the full dataset of 484 observations, including both training (387 samples) and validation (97 samples) subsets. These plots provide an intuitive assessment of model accuracy and bias, where optimal performance is indicated by a tight clustering of data points along the 45° reference line, corresponding to perfect agreement between predictions and experimental values.

Fig. 9.

Crossplots of predicted versus real values for all segments for ensemble learning model

Fig. 10.

Crossplots of estimated versus actual values for all segments for adaptive boosting model

Fig. 11.

Crossplots of estimated versus actual values for all segments for MLP-ANN model

Fig. 12.

Crossplots of estimated versus actual values for all segments for CNN model

Fig. 13.

Crossplots of estimated versus actual values for all segments for random forest model

Across all experimental conditions including two bacterial strains (Escherichia coli and Bacillus subtilis), four cerium oxide nanoparticle size ranges, concentrations spanning 0–150 mg/L, and exposure times up to 8 h the MLP-ANN model exhibits the closest alignment between predicted and measured OD600 values. The minimal scatter around the reference line indicates both high accuracy and low systematic bias, demonstrating the model’s capacity to capture complex, nonlinear dependencies among nanoparticle size, dose, exposure duration, and bacterial type. This graphical behavior is consistent with the superior numerical performance metrics reported earlier and confirms the robustness of the MLP-ANN framework across diverse biological and experimental regimes.

The Random Forest and Adaptive Boosting models also show strong predictive agreement, with most points closely distributed around the 45° line, albeit with slightly increased dispersion compared to MLP-ANN. This suggests that ensemble tree-based methods effectively approximate the dominant trends in bacterial growth inhibition but may be less adept at resolving subtle nonlinear interactions or higher-order feature couplings present in the dataset. In contrast, the Convolutional Neural Network and Ensemble Learning models display visibly wider scatter, particularly at higher OD600 values, indicating reduced precision and greater variability in predictions under certain conditions.

Importantly, the consistency observed across Figs. 9, 10, 11, 12, 13 supports the reliability of the experimental dataset and confirms the effectiveness of the applied normalization strategy in stabilizing model training across variables with differing scales. The integrated interpretation of these crossplots demonstrates that while all evaluated AI models are capable of capturing the general behavior of bacterial growth inhibition induced by cerium oxide nanoparticles, the MLP-ANN approach provides the most accurate and stable representation of the underlying nonlinear biological processes.

Overall, these results reinforce the suitability of MLP-ANN as a high-fidelity surrogate model for predicting bacterial responses to nanomaterial exposure and highlight its potential utility in applications such as antimicrobial strategy optimization, nano-biotechnological design, and process-level control of microbial systems.

Figure 14 illustrates the distribution of relative prediction errors for all five AI-based models across both the training and validation stages, providing insight into model stability and generalization rather than average accuracy alone. Concentration of points near the zero-error line indicates consistent predictive performance across the full dataset of 484 observations, encompassing diverse bacterial strains, nanoparticle sizes, concentrations, and exposure durations.

Fig. 14.

Percentage of relative error during training and testing sections of all algorithmd developed in this research

Among the evaluated models, the MLP-ANN demonstrates the most compact error distribution in both training and validation sets, reflecting its ability to generalize without significant performance degradation. This behavior is consistent with its lowest AARE% and highest R2 values and suggests that the network effectively balances representational capacity and regularization when modeling nonlinear antibacterial effects of cerium oxide nanoparticles. The Random Forest model exhibits similarly stable behavior with slightly broader dispersion, indicating strong robustness but marginally reduced sensitivity to complex feature interactions. Adaptive Boosting follows closely, maintaining acceptable error bounds while showing increased variability in some regions of the dataset. In contrast, the Convolutional Neural Network and Ensemble Learning models display noticeably wider error distributions, particularly in the validation phase, which may indicate greater sensitivity to data heterogeneity and reduced extrapolation capability. This increased dispersion suggests that these models are less effective at consistently capturing the coupled influence of bacterial type, nanoparticle size, dosage, and exposure time across all experimental regimes.

Overall, the relative error analysis reinforces the conclusions drawn from numerical performance metrics and crossplot evaluations by demonstrating that the MLP-ANN model not only achieves superior accuracy but also maintains the highest level of predictive stability across unseen data. Such robustness is essential for practical deployment in nanobiotechnological applications, including antimicrobial strategy optimization, nano-enabled process design, and controlled bioprocess engineering where reliable prediction of bacterial growth dynamics is critical.

Leave-one-out cross validation (LOO-CV)

To ensure the highest reliability in predicting the antibacterial effects of cerium oxide (CeO₂) nanoparticles, we employed Leave-One-Out Cross-Validation (LOO-CV) across the best model, MLP-ANN. Given the finite size of our experimental dataset (n = 484 observations from E. coli and B. subtilis exposed to four CeO₂ nanoparticle sizes at varying concentrations and exposure times), LOO-CV provides a nearly unbiased estimate of model performance while fully utilizing the available data. Instead of randomly omitting individual data points, we deliberately held out entire groups of measurements that correspond to distinct, physically relevant experimental conditions during training. The model was then evaluated solely on these unseen regimes to assess its ability to extrapolate beyond the range of data it was trained on. Three specific validation scenarios were designed for this purpose: (1) Concentration-Exclusion Test: All measurements obtained at the highest nanoparticle concentration (150 mg/L) were entirely excluded from the training set and used exclusively for testing. This evaluates the model’s capacity to generalize to high-dose antibacterial conditions not encountered during training, (2) Nanoparticle Size (Medium) Exclusion: All data corresponding to the largest cerium oxide nanoparticles (40 ± 10 nm) were withheld from training and reserved for validation, assessing the model’s ability to predict responses to unobserved particle size regimes, and (3) Time-Band Exclusion: A contiguous late-exposure interval (6–8 h) was removed from the training data and used solely for evaluation, testing the model’s performance in forecasting bacterial growth dynamics under extended exposure scenarios beyond the trained temporal window.

To uphold strict methodological integrity and eliminate any risk of data leakage, hyperparameter optimization was performed independently within each training fold using the CSA algorithm. This ensured that tuning relied exclusively on information available during training, thereby preserving the validity of performance evaluation on truly unseen conditions. As summarized in Table 3, this rigorous approach yields a modest, but expected decline in predictive metrics compared to the more optimistic random train/test split. This reduction reflects the greater challenge of extrapolating to the excluded physical regimes (e.g., unobserved concentrations, nanoparticle sizes, or time intervals) rather than merely interpolating within the training distribution. Despite this increased difficulty, the MLP-ANN model maintains exceptionally high accuracy and stability across all structured hold-out scenarios. These results strongly indicate that the model does not merely memorize or interpolate locally; rather, it learns robust, generalizable functional relationships that remain reliable under stringent out-of-distribution conditions, closely mirroring the demands of real-world applications in nanomaterial safety and antimicrobial design.

Table 3.

Results of LOO-CV analysis using the proposed MLP-ANN model

Validation strategy	R2 (Test)	RMSE (Test)	AARE % (Test)
Random 80/20 split	0.9933	0.0633	8.8199
Leave-one-concentration-out	0.9814	0.0912	9.4857
Leave-one-nanoparticle size-out	0.9835	0.0836	9.3795
Leave-high-time-band-out	0.9840	0.0748	9.1662

Ahmed et al. demonstrated that synergistic nanocomposites such as chitosan–iron oxide nanoparticles can effectively inhibit multidrug-resistant bacteria through gene-level suppression and in silico interaction analysis, highlighting the therapeutic relevance of nanoparticle–bacteria interactions guided by computational modeling [52]. Alghurabi et al. reported that structurally engineered Ag@TiO₂ core–shell nanoparticles exhibit enhanced antibacterial activity, supported by molecular docking results, emphasizing the importance of nanoparticle composition and structure in improving therapeutic efficacy [53]. Sami et al. showed that laser-synthesized platinum nanoparticles possess strong antibacterial and anti-biofilm activity against oral pathogens, with complementary in silico analysis providing insight into nanoparticle–target interactions relevant to therapeutic applications [54]. Najm et al. demonstrated that combining antibiotics with titanium dioxide nanoparticles significantly enhances antibacterial and anti-biofilm performance, supported by in silico binding energy analysis, illustrating how nanomaterials can potentiate therapeutic outcomes [55].

Jawad et al. reported that eco-friendly synthesized MgS@CuS nanocomposites exhibit synergistic antibacterial effects against multidrug-resistant bacteria, with computational docking studies elucidating their therapeutic binding mechanisms [56]. Al-Azawi et al. demonstrated that newly synthesized bioactive compounds exhibit strong antibacterial, antioxidant, and anticancer activities, supported by in silico interaction studies, underscoring the growing role of computational analysis in therapeutic agent design [57]. Mohammed et al. showed that biosynthesized graphene oxide nanoparticles possess broad-spectrum antimicrobial, antioxidant, and anticancer activities, with molecular docking analyses reinforcing their potential as multifunctional therapeutic nanomaterials [58].

Temporal examination of bacterial proliferation and pH variations in LB and M63 media via machine learning modeling

A crucial aspect of forecasting systems lies in their proficiency at accurately estimating discrete observations alongside the general pattern of the response metric. Within the present investigation, the leading model MLP-ANN was applied to project the progression of bacterial density (OD600) across various strains in nanoparticle-enriched cultivation environments featuring cerium oxide particles. Results demonstrate that the MLP-ANN effectively replicates this progression, highlighting its prowess in replicating the evolving suppression of bacterial expansion. Moreover, this methodology serves as an emulation of operational processes via cutting-edge cognitive frameworks, supplying a dependable mechanism for deciphering and anticipating microbial reactions to nanoscale materials.

Figure 15 showcases the bacterial growth trends for Escherichia coli and Bacillus subtilis in media containing cerium oxide nanoparticles (6 ± 3.5 nm, labeled as Medium#1), with nanoparticle concentrations of 50 mg/L for E. coli and 100 mg/L for B. subtilis, over a 8-h period. The blue line illustrates the MLP-ANN-predicted bacterial cell concentration (OD600) for E. coli, increasing from 0.12 at 0.5 h to 0.897 at 7 h, while the red dots represent the corresponding experimental OD600 values, ranging from 0.095 to 0.907 for 0.5 to 7 h. Similarly, the green line depicts the predicted OD600 for B. subtilis, rising from 0.09 at 0.5 h to 1.273 at 7.5 h, with the yellow dots showing experimental values ranging from 0.0745 to 1.218. The close alignment between the predicted trends (lines) and experimental data (dots) highlights the MLP-ANN model's accuracy in capturing the growth inhibition dynamics of both bacterial strains, Escherichia coli and Bacillus subtilis, under the influence of cerium oxide nanoparticles.

Fig. 15.

Time-based patterns of bacterial cell concentration (OD600) for Escherichia coli and Bacillus subtilis in medium containing cerium oxide nanoparticles (6 ± 3.5 nm) with concentrations of 50 mg/L for E. coli and 100 mg/L for B. Subtilis

Figure 16 displays the progression of bacterial cell concentration (OD600) for Escherichia coli and Bacillus subtilis in a medium containing cerium oxide nanoparticles (15 ± 4.3 nm, labeled as Medium#2), with nanoparticle concentrations of 150 mg/L for E. coli and 50 mg/L for B. subtilis. The cyan line represents the MLP-ANN-predicted OD600 for E. coli, escalating from 0.0604 at 0.5 h to 0.223 at 8 h, while the red dots indicate the experimental OD600 values, ranging from 0.0643 to 0.217. Likewise, the green line shows the predicted OD600 for B. subtilis, increasing from 0.149 at 0.5 h to 1.788 at 7.5 h, with the yellow dots reflecting experimental values spanning 0.105 to 1.877. The strong correspondence between the predicted curves (lines) and observed data points (dots) underscores the MLP-ANN model's precision in modeling the inhibitory effects on bacterial growth induced by cerium oxide nanoparticles.

Fig. 16.

Time-based patterns of bacterial cell concentration (OD600) for Escherichia coli and Bacillus subtilis in medium containing cerium oxide nanoparticles (15 ± 4.3 nm) with concentrations of 150 mg/L for E. coli and 50 mg/L for B. Subtilis

Figure 17 illustrates the bacterial growth dynamics for Escherichia coli and Bacillus subtilis in a medium containing cerium oxide nanoparticles (22 ± 5.7 nm, labeled as Medium#3), with nanoparticle concentrations of 50 mg/L for E. coli and 150 mg/L for B. subtilis. The blue line depicts the MLP-ANN-predicted bacterial cell concentration (OD600) for E. coli, rising from 0.0803 at 0.5 h to 0.549 at 7 h, while the red dots represent the corresponding experimental OD600 values, ranging from 0.080 to 0.529. Likewise, the green line shows the predicted OD600 for B. subtilis, increasing from 0.0868 at 0.5 h to 0.976 at 7.5 h, with the yellow dots indicating experimental values spanning 0.0833 to 0.894. The tight correspondence between the predicted curves (lines) and observed data points (dots) demonstrates the MLP-ANN model's precision in capturing the inhibitory effects on bacterial growth caused by cerium oxide nanoparticles.

Fig. 17.

Time-based patterns of bacterial cell concentration (OD600) for Escherichia coli and Bacillus subtilis in medium containing cerium oxide nanoparticles (22 ± 5.7 nm) with concentrations of 50 mg/L for E. coli and 150 mg/L for B. Subtilis

Figure 17 illustrates the bacterial growth dynamics for Escherichia coli and Bacillus subtilis in a medium containing cerium oxide nanoparticles (40 ± 10 nm, labeled as Medium#4), with nanoparticle concentrations of 100 mg/L for E. coli and 50 mg/L for B. subtilis. The blue line depicts the MLP-ANN-predicted bacterial cell concentration (OD600) for E. coli, rising from 0.0804 at 0.5 h to 0.839 at 7 h, while the red dots represent the corresponding experimental OD600 values, ranging from 0.080 to 0.789. Likewise, the green line shows the predicted OD600 for B. subtilis, increasing from 0.0868 at 0.5 h to 1.418 at 7.5 h, with the yellow dots indicating experimental values spanning 0.0813 to 1.417. The tight correspondence between the predicted curves (lines) and observed data points (dots) demonstrates the MLP-ANN model's precision in capturing the inhibitory effects on bacterial growth caused by cerium oxide nanoparticles. This graphical representation emphasizes the model’s effectiveness in simulating microbial responses to nanomaterials, providing critical insights for enhancing antimicrobial approaches and supporting progress in nano-biotechnological research (Fig. 18).

Fig. 18.

Time-based patterns of bacterial cell concentration (OD600) for Escherichia coli and Bacillus subtilis in medium containing cerium oxide nanoparticles (40 ± 10 nm) with concentrations of 100 mg/L for E. coli and 50 mg/L for B. Subtilis

Although this work focuses on AI-driven prediction, the model’s sensitivity to specific input features aligns well with known biological mechanisms of nanoparticle–bacteria interactions: (1) Nanoparticle Size Effects: The observed stronger inhibitory effect of smaller CeO₂ nanoparticles (e.g., Sample A: 6 ± 3.5 nm) compared to larger ones (Sample D: 40 ± 10 nm) is consistent with established nanotoxicological principles. Smaller nanoparticles exhibit: (a) Higher surface-area-to-volume ratios, increasing reactive oxygen species (ROS) generation via Ce³⁺/Ce⁴⁺ redox cycling on the particle surface, (b) Enhanced membrane adhesion and penetration, facilitating physical disruption of cell envelopes, (c) Greater cellular internalization, particularly in Gram-negative bacteria like E. coli. These mechanisms collectively lead to more pronounced growth inhibition (lower OD600), which our model correctly captures as a function of size. (2) Bacterial Strain Differences: The model identified Bacillus subtilis (Gram-positive) as generally less susceptible to CeO₂ NPs as Escherichia coli (Gram-negative), a trend reported in multiple studies. This difference arises from structural variations in cell envelopes: (a) E. coli possesses an outer membrane containing lipopolysaccharides (LPS) that can bind CeO₂ NPs, promoting ROS-induced peroxidation and permeability, (b) B. subtilis has a thick peptidoglycan layer that may act as a partial barrier, reducing nanoparticle internalization and oxidative damage. However, under prolonged exposure or high concentrations, both strains show significant inhibition, reflecting the dose- and time-dependent nature of nanotoxicity. (3) OD600 as a Proxy for Viability: While OD600 primarily measures optical density (cell density/turbidity), it correlates strongly with viable cell counts in controlled batch cultures. In the presence of CeO₂ NPs, reduced OD600 likely reflects: (a) Growth arrest due to oxidative stress damaging DNA, proteins, and lipids, (b) Loss of membrane integrity, leading to cell lysis and reduced scattering, (c) Aggregation of nanoparticles with cells, which may artifactually lower OD, but this was mitigated in the source study by using NP-containing blanks for baseline correction.

Importantly, our Monte Carlo sensitivity analysis ranked exposure time as the dominant factor influencing OD600, followed by nanoparticle concentration, which is consistent with the cumulative nature of oxidative damage. This biological coherence reinforces the reliability of our AI models not just as statistical tools, but as interpretable surrogates for complex nano-bio interactions.

While this study demonstrates the power of AI to predict bacterial responses to CeO₂ nanoparticles using limited experimental inputs, a more comprehensive risk–benefit profile requires integration of mechanistic biology, expanded microbial panels (including multidrug-resistant pathogens), multi-endpoint toxicity assays, and environmental interaction studies. Future work will focus on building hybrid experimental-computational platforms that combine high-throughput screening with interpretable AI to accelerate the safe design of antimicrobial nanomaterials.

While experimental assays remain essential for validating antibacterial mechanisms and ensuring biological relevance, AI-based predictive models should be viewed as complementary tools rather than replacements for laboratory experiments. In this context, machine learning frameworks can guide experimental design by prioritizing promising nanoparticle configurations, reducing the number of required assays, and accelerating data-driven hypothesis generation. Such synergistic integration of computational modeling and experimentation enhances efficiency without compromising scientific rigor.

Conclusions

This investigation successfully formulated and verified a reliable array of forecasting systems leveraging state-of-the-art AI methodologies encompassing MLP-ANN, RF, AdaBoost, CNN, and EL aimed at estimating bacterial density (OD600) in cultivation environments doped with cerium oxide nanoparticles. Drawing from an extensive collection of 484 lab-generated observations, the systems integrated essential predictor elements like microbial species (Escherichia coli and Bacillus subtilis), growth medium variant (nanoparticles sized at 6 ± 3.5 nm, 15 ± 4.3 nm, 22 ± 5.7 nm, and 40 ± 10 nm), nanoparticle levels (0–150 mg/L), and exposure period (0–8 h). The core objective centered on crafting dependable simulation tools for microbial suppression amid diverse nanotechnology scenarios. Predictive efficacy was refined via precise tuning of parameters through (CSA technique, yielding superior forecasting fidelity. Thorough assessments spanning correlation examinations, scatter visualizations, error accumulation profiles, and deviation ratio evaluations singled out MLP-ANN as the leading performer, with metrics of R² at 0.997202025, RMSE at 0.041472425, and AARE% at 5.809784325, affirming its prowess in replicating elaborate microbial expansion patterns. Monte Carlo-based sensitivity probing uncovered time as the paramount influencer (sensitivity value of 5.3489), trailed by nanoparticle dosage (4.9742), medium category (4.7521), and microbial variant (4.6589), furnishing vital perspectives on the core drivers of microbial restraint. Such outcomes spotlight the superior merits of AI-driven strategies relative to conventional lab protocols, delivering a meticulously accurate, streamlined, and economical platform for anticipating biological reactions to nanoscale agents. This framework bears substantial potential for nanotechnology in biotechnology, aiding the refinement of anti-infective protocols and biological engineering advancements. Subsequent studies might amplify predictive sharpness by integrating extra variables, like nanoparticle coatings or ambient factors, to broaden scope over wider microbial-nanomaterial pairings. While this study establishes a robust AI framework for predicting bacterial responses to CeO₂ nanoparticles, future work could integrate multi-scale data, from molecular dynamics simulations of nanoparticle–membrane interactions to X-ray spectroscopic characterization of surface reactivity, to build hybrid physics-AI models. AI-based approaches offer a powerful complementary framework to conventional laboratory methods by enabling rapid prediction, guiding experimental prioritization, and supporting interpretation of complex nano–bio interactions. Future studies could employ molecular dynamics to model CeO₂ nanoparticle interactions with bacterial membranes and molecular docking to identify high-affinity binding sites on essential proteins. Such multi-scale approaches, linking quantum/atomistic phenomena to population-level responses, would bridge the gap between empirical observation and molecular mechanism, ultimately enabling rational design of safer and more effective antimicrobial nanomaterials. Furthermore, extending predictive modeling to non-target organisms (e.g., mammalian cells, soil microbes, or aquatic species) would enable holistic environmental and health impact assessments. In addition to antibacterial effects, cerium oxide nanoparticles have been reported to exhibit a range of other bioactive properties, including antioxidant, anti-inflammatory, and cytoprotective effects under specific physicochemical conditions. While the present study focuses on modeling bacterial growth inhibition, the proposed AI-driven framework could be extended in future work to investigate additional therapeutic effects by integrating relevant biological endpoints and experimental datasets. Such extensions may support the rational design of multifunctional nanomaterials for biomedical and therapeutic applications.

Abbreviations

AI

Artificial intelligence

ANN

Artificial neural network

MLP

Multi-layer perceptron

MLP-ANN

Multi-layer perceptron artificial neural network

CNN

Convolutional neural network

RF

Random forest

AdaBoost

Adaptive boosting

EL

Ensemble learning

CSA

Coupled simulated annealing

SVM

Support vector machine

KNN

K-nearest neighbors

DT

Decision tree

RBF

Radial basis function

OD600

Optical density measured at 600 nm

RMSE

Root mean square error

AARE

Average absolute relative error

R²

Coefficient of determination

CDF

Cumulative distribution function

ARE

Absolute relative error

NP

Nanoparticle

CeO₂

Cerium oxide

TEM

Transmission electron microscopy

AFM

Atomic force microscopy

DLS

Dynamic light scattering

PZC

Point of zero charge

ROS

Reactive oxygen species

LOO-CV

Leave-one-out cross-validation

Author contributions

Omar Almomani contributed to the conceptualization of the study, machine learning methodology design, and analysis of model performance. Magdi E. A. Zaki contributed to the chemical interpretation of nanoparticle–bacteria interactions and supported the discussion of physicochemical and antibacterial mechanisms. Raed Alfilh participated in data preprocessing, implementation of computational models, and validation of results. Gadug Sudhamsu contributed to the development and optimization of artificial intelligence algorithms and assisted in comparative model evaluation. Prabhat Kumar Sahu contributed to data analysis, statistical evaluation of model outputs, and interpretation of predictive accuracy metrics. Murari Devakannan Kamalesh assisted in algorithm implementation, model training, and performance benchmarking across datasets. Sumit Sharma contributed to software implementation, computational experiments, and visualization of results. Sobhi M. Gomha contributed to the chemical and biological interpretation of the results and supported the discussion of antibacterial and potential therapeutic implications. Samim Sherzod led the study design, supervised the research workflow, coordinated contributions among authors, and finalized the manuscript. All authors contributed to writing, reviewing, and editing the manuscript and approved the final version.

Funding

None.

Data availability

Data is available on request from the corresponding author.

Declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Clinical trial number

Not applicable.

Competing interests

The authors declare no competing interests.