2.1 Metaheuristic optimization in leukemia diagnosis

The diagnostic accuracy improves when natural and biological process-based optimization algorithms fine-tune feature subsets and classifier parameters. Researchers successfully applied WOA and SVM for detecting ALL through a method that resulted in 98.42% accuracy as reported in [15]. The evolutionary algorithm demonstrates how it efficiently searches hyperparameter spaces to enhance conventional classifier outcomes in medical datasets with complex characteristics.

Likewise, the k-means clustering for classifying all subtypes has been applied in conjunction with the Enhanced Grey Wolf Optimizer (EGWO) to achieve an accuracy of 99.22% [16]. These approaches demonstrate the success of nature-inspired optimization in addressing problems related to model architecture selection, reducing overfitting, and selecting salient features, particularly in high-dimensional medical imaging data.

2.2 Deep learning and hybrid architectures

However, Convolutional Neural Networks (CNNs) and other deep learning architectures have become the de facto choice for extracting hierarchical representations from the raw image inputs because they are automatic systems that can learn well. As demonstrated in work by [17], where the classifier and PSO are used, the CSO augmented model is built using a ResNet50 classifier, achieving a high accuracy of 99.84%. It is thus confirmed that the deep architectures complemented by metaheuristic tuning can potentially learn morphology patterns indicative of leukemia.

Another alternative study, combining GoogleNet with an 88-layer deep CNN by [18], utilized binary whale optimization and achieved 99.15% accuracy in all subtype classification. These results suggest that deep networks can learn detailed visual features and adjust their parameters to align with disease-specific discriminative cues.

This further supports the argument because the model proposed by [19] used a Deep Convolutional Neural Network (DCNN) with the addition of a Fuzzy Fusion Hunger Games Optimization (FFHHO) mechanism. The leukemia detection was achieved with a TPR of 91.7% and a TNR of 92.25%, indicating balanced sensitivity and specificity, two key metrics in the medical viability itself.

In addition, feature selection across several CNN architectures was optimized by a firefly algorithm, as shown in [20]. Results of the entropy-controlled firefly mechanism yielded impressive classification accuracies of more than 99%, demonstrating the effectiveness of the hybrid evolutionary method in creating a CNN-based diagnostic system.

2.3 Expanded classification strategies and model integration

In addition to leukemia, several other hematological malignancies have been classified with AI-based classification frameworks beyond leukemia. In [21], a hybrid CNN-ANN model was used to determine a 100% accuracy in identifying Multiple Myeloma, a cancer of plasma cells. This demonstrates the use of such layered neural network architectures to detect subtle variations in the related pathologies.

In this paper, we have applied a dense neural network (DNN) combined with DenseNet201 features extracted via a DenseNet201-based feature extractor and optimized using HHO to reach 99.33% classification accuracy for the lymphoma subtypes. The success of dense architectures and the effective use of features in deciphering the complex yet rich histological signature of cancer cells illustrates this.

Presents another novel approach [22] through a Hybrid CNN model coupled with an Interactive Autodidactic School (HCNN-IAS) learning strategy, for which they achieve a classification of 99%, validating the effectiveness of integrating automated self-learning schemes in the regular deep learning pipeline.

Election-Based Chameleon Swarm Algorithm (E-CSA) is introduced by [23] for the early 183 stage leukemia detection using the MTResUnet3+ segmentation model. The dual layering of this framework combines semantic segmentation and optimization to make better precision in region localization and classification. Like CoTCoNet, proposed in [24], a model that combines convolutionality and attentionality in feature extraction, the CoTCoNet model achieved 98.94% accuracy across different leukemia datasets with attention-based learning in the medical space.

In [25], segmentation and classification were performed with an African Buffalo and African Vulture Optimization (AB-AVO) based on RNNs. Therefore, hybrid evolutionary algorithms and temporal learning structures yielded nearly perfect diagnostic outcomes, achieving 100% accuracy in classifying all IDB datasets.

The last is a comprehensive review from [26] of many different ML strategies used to classify leukemia, establishing a foundation for diverse approaches’ evolution and performance comparison. Finally, the study presents what is becoming a trend of increasing automation, algorithmic interpretability, and multi-stage optimization in medical diagnostics.

2.4 Summary and research gap

Table 1 illustrates that current studies have achieved significant results in leukemia classification using AI-driven techniques, including CNN and various metaheuristic algorithms. Many of these efforts have yielded high diagnostic accuracies exceeding 99%. However, there are several significant constraints that need to be overcome before the further development of more unified and scalable diagnostic frameworks can proceed.

Download:

• PNG
  larger image
• TIFF
  original image

Table 1. Summary of literature review.

https://doi.org/10.1371/journal.pone.0330228.t001

First, most prior work has regarded feature selection and hyperparameter optimization as individual and sequential processes employing different algorithms for each task. Over time, real phenomena create fragmentation, resulting in inconsistent optimization outcomes, increased complexity, and decreased efficiency. Furthermore, although each binary and continuous metaheuristic has demonstrated strong results, few studies have created a single algorithm that handles both discrete and continuous search spaces within a unified structure.

Secondly, a coherent pipeline doesn’t exist that features selection, hyperparameter tuning, and training in any way. However, many reported models optimize individual components in isolation, severely restricting the synergy obtained by jointly optimizing.

The present study introduces the Ocotillo Optimization Algorithm (OcOA) and its binary variant (bOcOA) as a unified solution of feature selection and hyperparameter tuning for a generic standard form of the original HOSSE data. In contrast to independent algorithms as in existing works, the approach uses a consistent metaheuristic framework to navigate both domains. It enables a streamlined optimization process, simplifies model generalization, and reduces the complexity of developing the proposed strategy.

At the same time, when feature selection and parameter tuning are integrated into a coordinated pipeline, the results show that they can improve diagnostic accuracy and computational efficiency, thereby enhancing the performance of multiple deep learning models, such as CNN, ViT, GAN, SimCLR, and MoCo. Overall, it bridges the gap between the optimization stage and classification methodological divide with a replicable and adaptable framework for high-precision, image-based leukemia diagnosis.

3.1 Dataset

3.1.1 Dataset description.

This study employed a part of the Bone Marrow Cell Classification dataset [28]. This large, carefully crafted biomedical image corpus can simulate the complexities and heterogeneities inherent to hematological diagnostics. A total of 171,375 high-resolution, de-identified images of cellular images derived from bone marrow smears from 945 individual patients form the content of this dataset. Numerous hematopathologists and cytology experts from the Munich Leukemia Laboratory (MLL), an internationally renowned diagnostic and research institute specializing in leukemic and marrow-related disorders, expertly annotate each image.

Given that acquisition protocols conformed to stringently applied standards of morphological uniformity and diagnostic importance, the addition of FIJ samples is not expected to significantly influence overall reliability or impact any results. All smears were stained using a well-established differential staining method, the May-Grünwald-Giemsa/Pappenheim protocol. The dual stain provides comparable contrast for different cellular structures, enabling the clear separation of precursor cells, blasts, erythroid lineage cells, granulocytes, lymphocytes, and other cell types, thus facilitating functional differential diagnosis.

The brightfield microscope was operated at a magnification factor of 40x and oil immersion optics to retain high-resolution morphological detail for image acquisition. There are also reasons this setup was chosen: it can offer a consistent depth of field and color fidelity, which are two key factors in achieving well-defined results within the limits of cytological imaging. A further set of image scanning and digitization procedures was completed using proprietary image scanning equipment designed and developed at Fraunhofer IIS, along with associated post-processing pipelines tailored for Helmholtz Munich to ensure annotation integrity and consistent resolution, lighting, and focus.

The final dataset encompasses a broad diagnostic spectrum, including acute myeloid leukemia (AML), acute lymphoblastic leukemia (ALL), chronic myeloproliferative neoplasms (MPN), myelodysplastic syndromes (MDS), plasma cell disorders, as well as other marrow abnormalities. This diversity makes the dataset a realistic surrogate of the diagnostic workload of large-scale hematology laboratories.

The dataset contains structured metadata for each image (class label to type of bone marrow cell), and all images are also provided in their respective classes. Finally, these are the target variables for the supervised learning tasks. In a multi-class classification framework, these class labels were used for this study, wherein each instance belongs to one and only one of the predefined cellular categories.

To preserve the statistical validity and reproducibility of the experiments, the dataset was stratified and sampled into training, validation, and testing splits. With this strategy, the class distribution is further preserved on all three subsets proportionally, rather than through class imbalance mitigation, so that the entire data distribution is still represented by training and evaluating the models. The split ratio was set empirically to 70% for training, 15% for validation, and 15% for testing, respectively. Then, the images were randomly shuffled, ensuring that all images for partitioning were free from any ordering bias, which helps ensure that the model evaluation results are robust.

To better understand the data being used, Fig 2 shows a sample image from the dataset. This visualization illustrates the format and structure of the data before further analysis or processing.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 2. A sample image from the dataset used in this study.

https://doi.org/10.1371/journal.pone.0330228.g002

3.2 Machine learning models

Utilize diverse deep-learning architectures to address the bone marrow cell classification task. To capture a wide range of modern visual representation learning paradigms, we chose these models to correspond to a broad spectrum of both supervised and self-supervised learning strategies. In particular, we implemented Convolutional Neural Networks (CNNs), Vision Transformers (ViTs), Generative Adversarial Networks (GANs), and two types of contrastive learning frameworks: Simple Contrastive Learning of Representations (SimCLR) and Momentum Contrast (MoCo). Due to their distinctive inductive biases and distinct training methods, each architecture provides a valuable basis for comparative evaluation and ensemble integration in hematological image analysis.

In this research, the Convolutional Neural Network (CNN) [33] is employed, a foundational model used as a baseline to assess other architectures. CNNs have been proven to be highly validated in medical image classification tasks, mainly due to their intrinsic capabilities for local feature extraction, hierarchical representation learning, and translational invariance. The network architecture is based on a standard deep convolutional lineup of consecutive convolutional layers with ReLU activations, max-pooling layers for spatial downsampling, and fully connected output layers, resulting in a softmax output. This structure is very suitable for capturing the morphological characteristics of bone marrow cells, including nuclear size, cytoplasmic granularity, and contour characteristics, which are necessary for differential diagnosis in hematopathology.

The Vision Transformer (ViT) is a model that helps with image understanding tasks by departing from convolutional priors and reinforcing the transformer architecture (designed for natural language processing tasks) for image understanding tasks [34]. While CNNs use overlapping image patches that are flattened and embedded linearly into the CNN layers, and then processed with multiple layers of multi-head self-attention feed-forward networks, ViTs operate on non-overlapping patches, processed with multi-head self-attention and feed-forward networks. In the context of histopathological analysis of cell images, diagonal global contextual relationships are meaningfully distributed over various spatial regions of the cell image, a pattern that the ViT mainly handles well. Moreover, the model’s reliance on attention mechanisms facilitates interpretability, allowing for insight into the decision-making process through attention maps.

A Generative Adversarial Network (GAN) was introduced to incorporate architectural diversity into the modeling pipeline. GANs contain two neural networks, the generator and the discriminator, which are trained together in an adversarial manner. The discriminator is tasked with distinguishing real from synthetic images, whereas the generator aims to produce images that cannot be distinguished from the training data [35]. In this case, while GANs are usually employed for image synthesis, a classification-augmented GAN architecture was utilized, where the discriminator not only classifies the images as fake or real but also provides class probabilities. Providing dual adaptation signals for adversarial training and cell classification enables a discriminator to obtain training benefits while learning discriminative features for cell classification. GANs are especially favorable when synthetic data generation can aid in balancing classes and improving generalization in datasets.

In this respect, two self-supervised learning (SSL) models (SimCLR and MoCo) were used to learn feature representation using unlabeled data without direct supervision [36]. The SimCLR is based on the concept of contrastive learning, where the model must distinguish between augmented views of the same image within the same latent space, bringing those views closer together, and opinions of different images further apart. Following a ResNet-based encoder, the architecture follows with a projection head that projects the representations to a contrastive loss space. It can learn the invariant features from the augmented pairs and then fine-tune them on the downstream classification task using a small, labeled subset of the dataset.

Momentum Contrast (MoCo) employs a dynamic dictionary mechanism for negative sampling, utilizing a momentum-updated encoder in another contrastive learning paradigm [37]. In contrast, MoCo approaches the same task using much smaller computational resources than SimCLR by exploiting a queue of encoded features. This approach facilitates the performance of a contrastive learning process in memory, which is particularly valuable for high-resolution medical image datasets. With the flexibility of MoCo in learning discriminative features from unlabeled data, it is an ideal component of the modeling strategy.

Overall, these five models (CNN, ViT, GAN, SimCLR, and MoCo) provide a rich landscape for comparing the two categories of bone marrow cell classification approaches: supervised and self-supervised. Due to their unique architectural designs and learning paradigms, they open the door for a detailed study of the role of different feature extraction and representation learning schemes on classification performance in complex biomedical imaging problems. Both methodological rigor and clinical relevance are ensured through the use of these models, given their state-of-the-art status in the computer vision literature and proven utility in similar medical imaging problems.

3.3 Metaheuristic algorithms

Metaheuristic algorithms have been successfully integrated into machine learning workflows in high-dimensional input spaces with non-convex optimization landscapes. Researchers have increasingly begun to utilize these algorithms across various scientific domains. Metaheuristics offer an alternative to existing feature selection and hyperparameter tuning strategies in the context of medical image analysis, particularly in the field of bone marrow cytology. The rigidity of a heuristic typically restricts such methods, often leading to the exhaustion of an exhaustive search or the use of domain-specific thresholds. Metaheuristic algorithms mimic natural phenomena, such as swarming behavior, evolutionary competition, or gravitational attraction, to enable an efficient search through a vast search space. This study systematically applies both the binary and continuous versions of metaheuristics to address two critical issues in deep learning-based classification: feature selection that is insensitive to dimensionality via dimensionality reduction and performance maximization that is insensitive to hyperparameters within deep learning.

3.3.3 Proposed metaheuristic optimization and feature selection algorithm.

This study proposes to utilize one of the recently developed metaheuristic algorithms inspired by nature, called the Ocotillo Optimization Algorithm (OcOA), as the primary optimization engine for feature selection and hyperparameter tuning in the classification of bone marrow cells. OcOA is emulated from the adaptive behaviors of desert flora, specifically the ocotillo plant (Fouquieria splendens), widely known for its growth and blooming patterns in arid and shifting conditions. The algorithm was first presented to address essential issues in the theory of optimization, such as premature convergence, insufficient exploration, and poor adaptability in high-dimensional search spaces.

OcOA is an exploration and exploitation algorithm that leverages a dual-phase architecture, where stochastic variables and search behaviors control both the exploration phase and exploitation phase are strategically altered between each other (dynamically) in the real world [38]. During the exploration phase, the search space is filled with a wide distribution of candidate solutions using sinusoidal and cosine perturbations. The behavior of these perturbations is shaped by an amplitude factor (A), a distance function (D), and a feedback-informed learning rate (L), which are modulated over time to adapt the sense of convergence state of the algorithm.

As soon as promising regions of the search space can be identified, the OcOA enters the exploitation phase, where local refinements occur using Gaussian-based mutations, convergence factors, and non-linear scaling of previous search attempts. The adaptive feedback mechanism further regulates the convergence trajectory to balance the search between its global and local optima. Most importantly, the algorithm incorporates a convergence-aware control function (K) and a dynamic update rule for the learning factor (L). This allows the exploitation strategy to remain flexible and dynamic, responding to changes in fitness.

Here, we clarify key parameters used throughout the algorithm:

• r₁, r₂, r₃ are uniformly distributed random numbers in [0,1] that introduce stochasticity into the update rules, helping to maintain diversity in the population and prevent premature convergence.
• refers to the current iteration number or a normalization scalar based on the training size, used to scale certain dynamic behaviors as the algorithm progresses.
• K is a convergence control factor computed adaptively to guide exploitation. It adjusts mutation strength based on convergence rate, helping OcOA shift from exploration to exploitation.

In the context of the feature selection task, the binary variant of the OcOA, namely bOcOA, is employed. This is the binarized version of OcOA, where candidate solutions are represented by binary strings, with each bit of a string encoding the presence or absence of a specific feature. OcOA can thus perform the combinatorial space search of feature subsets under the constraint that the resulting subsets are as compact as possible while maintaining high predictive accuracy. bOcOA maintains the core behavioral dynamics of the continuous bOcOA by projecting updates to bOcOA through a probabilistic sigmoid transformation that enforces binary constraints.

For hyperparameter optimization, OcOA was taken in the continuous form. The hyperparameters—learning rate, dropout ratio, number of hidden layers, and batch size—were represented in a constant vector form and optimized based on classification accuracy against a validation set. It progressively explored the hyperparameter space without resorting to local optima. It identified configurations that best aided ROCO in converging quickly and generalizing well across all the deep learning models used.

Finally, as presented in Algorithm 1, outline the complete procedural steps of the continuous OcOA algorithm, namely, the computation of sinusoidal amplitude adjustments, feedback-driven learning factors, and multi-stage Gaussian mutations. Together, these operators improve the search’s flexibility and robustness in nonconvex optimization landscapes with moderate complexity.

Algorithm 1. Ocotillo Optimization Algorithm (OcOA) for Hyperparameter Optimization.

1: Initialize population size N, maximum iterations Maxiter

2: Initialize random positions for

3: Define constants:

4: Set initial best solution:

5: for t = 1 to Maxiter do

6:   for each solution do

7:     Compute distance factor:

8:     Compute amplitude:

9:     Compute learning factor:

10:     Update position (Exploration):

11:   end for

12:   Evaluate fitness of all

13:   Update if better solution found

14:   for each solution do

15:     Compute Gaussian mutation:

16:     Compute:

17:     Update position (Alternative Mutation):

18:   end for

19:   Update population with new solutions

20:   Update if necessary

21:   if stopping criteria met then

22:     Break

23:   end if

24: end for

25: Return best solution

At the same time, bOcOA is used to solve the discrete optimization problem of determining the optimal feature subsets. The binary representation of each candidate solution consists of bits of length d; a bit value of 1 indicates the inclusion (1) of that corresponding feature, and a bit value of 0 means nothing was included or excluded (0). The algorithm adapts the exact overarching two-phase search overall algorithm by adjusting the update rules to suit binary encoding.

The algorithm first uses a sigmoid transfer function, followed by stochastic binarization to transition from real-valued exploratory to binary feature selection. Using this method, real-valued outputs are converted into binary states at probabilistic thresholds, allowing for the effective traversal of a discrete space of possible feature combinations.

The full binary OcOA procedure for feature selection is described in Algorithm 2.

Algorithm 2. Binary Ocotillo Optimization Algorithm (bOcOA) for Feature Selection.

1: Input: Feature dataset X, labels Y, population size N, max iterations Maxiter

2: Output: Best binary feature subset

3: Initialize binary population for

4: Define constants and parameters:

5: Evaluate fitness for each solution using a classifier (e.g., KNN)

6: Set best solution

7: for t = 1 to Maxiter do

8:   for each solution do

9:     Compute:

10:     Compute position update (exploration phase):

11:     Apply Sigmoid transfer function:

12:     Binarize:

13:   end for

14:   Evaluate fitness of all

15:   Update if better solution found

16:   for each solution do

17:     Exploitation phase:

18:     Apply sigmoid and binarization as above to

19:   end for

20:   Replace old solutions with new binary solutions

21:   Update if necessary

22:   if stopping criteria met then

23:     Break

24:   end if

25: end for

26: return

Chained together, these dual modes provide OcOA with a binary encoding for feature subset selection and a continuous encoding for hyperparameter tuning, thereby placing OcOA as a unified, adaptive, and high-performance optimizer. This integrated formulation, with clearly defined stochastic and convergence factors, ensures both global exploration and local exploitation are robustly executed across varying search domains. This represents a significant methodological advancement introduced into this study, enabling efficient operation in both discrete and continuous search domains through a single, biologically inspired framework.

3.4 Evaluation metrics

Several evaluation metrics were employed to rigorously assess the performance of the classification models and the effectiveness of the metaheuristic feature selection strategies used in this study. The selection of these metrics, based on aspects of classification efficacy and optimization robustness, includes predictive accuracy, error reduction, model generalization, and subset stability. The following subsections delineate the metrics used in the evaluation process of the deep learning models and the feature selection algorithms.

3.4.1 Machine learning prediction metrics.

As shown in Table 2, we adopted six standard performance indicators for classifying tasks, including accuracy, sensitivity (true positive rate), specificity (true negative rate), positive predictive value (PPV), negative predictive value (NPV), and F1 score. The confusion matrix is the derivation from which these metrics are obtained. The predictions are based on false positives (FP), false negatives (FN), true positives (TP), and true negatives (TN). Combined, they provide a holistic view of how the classifier performs in terms of correctness, class discrimination, and its balance of precision and recall.

Download:

• PNG
  larger image
• TIFF
  original image

Table 2. Classification evaluation metrics and their mathematical definitions.

https://doi.org/10.1371/journal.pone.0330228.t002

These metrics have different evaluative purposes. Accuracy measures a correct general sense of what is good, but can be deceiving in the presence of class imbalance. The sensitivity indicates how accurately the model can identify positive cases, i.e., cases that the model classifies as negative (false negatives), which would constitute a significant issue in medical diagnostics, as missing disease cases can have dire consequences. Specificity complements sensitivity in measuring the ability to correctly reject negative instances. PPV and NPV are methods for evaluating the reliability of positive and negative predictions. They are used to assess the confidence in model outputs in a clinical decision support setting. The F1 score is the last metric that balances precision and recall, returning a single scalar value for performance that considers both false positives and negatives.

3.4.2 Feature selection metrics.

As shown in Table 3, six key metrics were developed for evaluating feature selection performance, and the evaluation is provided on six metrics to quantify the quality, stability, and relevance of the selected feature subsets. They include the average classification error, the average subset size employed by the optimizer, the average fitness score (as it is an optimization criterion), the best and worst fitnesses seen across iterations, and the standard deviation of fitness values (measuring the degree of stability of the optimizer).

Download:

• PNG
  larger image
• TIFF
  original image

Table 3. Feature selection performance metrics and their mathematical formulations.

https://doi.org/10.1371/journal.pone.0330228.t003

In particular, at iteration, i, let stand for the feature subset, denote as the number of features selected in subset , and write for the fitness score (which usually is a function of classification accuracy and feature subset compactness). The average error metric measures classification performance over the average subdimensions of multiple runs, while the average selected size gives dimensionality reduction. The fitness-related metrics indicate how well the algorithm exploits the search space, and the standard deviation shows the algorithm’s robustness and stability among epochs.

Together, these metrics enable a nuanced and statistically grounded comparison of different feature selection and optimization strategies, supporting reproducibility and facilitating an objective assessment of model and algorithmic performance.

4.1 Baseline machine learning performance (before feature selection)

To establish a baseline for the impact of optimization strategies, first ran five selected deep learning models in their default configurations and without feature selection. First, these models (CNN, ViT, GAN, SimCLR, MoCo) were trained using feature sets extracted from bone marrow images obtained from a complete training set, with model hyperparameters fixed at either default values or state-of-the-art values from previous literature.

The performance of each model, in terms of the key evaluation metrics such as accuracy, sensitivity (true positive rate), specificity (true negative rate), precision (PPV), negative predictive value (NPV), and F1 score, is summarized in Table 5. The model’s highest overall classification performance was based on a CNN, with an accuracy of 0.8629, followed by ViT (0.8381) and GAN (0.8205). Raw accuracy was not high for them, but they also served as baseline comparisons for later gains in raw accuracy. The resulting feature and parameter optimization derived from these initial results was then tested on a new dataset as a critical benchmark to evaluate successive improvements in feature and parameter selection.

Download:

• PNG
  larger image
• TIFF
  original image

Table 5. Classification baseline performance.

https://doi.org/10.1371/journal.pone.0330228.t005

To understand how the various evaluation metrics are distributed and compared across models, the Cumulative Distribution Function (CDF) plots of them are used. By taking these plots, the visual understanding of metrics such as Accuracy, Sensitivity (True Positive Rate), Specificity (True Negative Rate), Positive Predictive Value (PPV), Negative Predictive Value (NPV), and F-Score on the tested models is possible. The CDF plots in Fig 6 display the proportion of observations below a certain threshold, providing a comprehensive view of model performance and consistency by illustrating the selected evaluation metrics.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 6. CDF plots representing the distribution of key evaluation metrics across models: Accuracy, Sensitivity (TPR), Specificity (TNR), PPV, NPV, and F-Score.

https://doi.org/10.1371/journal.pone.0330228.g006

A radar chart gives a global performance comparison of multiple evaluation metrics. The Convolutional Neural Network (CNN) model’s strengths and weaknesses are effectively illustrated in a single visualization, which displays Accuracy, Sensitivity (TPR), Specificity (TNR), PPV, NPV, and F-Score in one figure. To validate its performance in bone marrow cell classification tasks, the CNN model exhibits balanced and strong results on all metrics, as shown in Fig 7.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 7. Radar chart depicting the performance of the CNN model across six key evaluation metrics.

https://doi.org/10.1371/journal.pone.0330228.g007

The performance of the model in terms of precision and recall, as well as Sensitivity (True Positive Rate) and F-score, is compared by plotting the model on dual axes. The deep learning models, along with their respective sensitivity values, are represented by the overlaid line graph and the bar plot, which measure the F-score for each model. Running these two models in conjunction allows for assessing each model without considering precision or recall. The results in Fig 8 demonstrate that, metric-wise, the CNN model achieves the closest performance and exhibits a gradual decrease in both metrics compared to the ViT, GAN, SimCLR, and MoCo models.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 8. Dual-axis chart showing F-Score (bars) and Sensitivity (TPR) (line) across various models: CNN, ViT, GAN, SimCLR, and MoCo.

https://doi.org/10.1371/journal.pone.0330228.g008

A stacked bar chart is used to provide a consolidated view of model performance across multiple evaluation criteria. Each bar aggregates the six key metrics—Accuracy, Sensitivity (TPR), Specificity (TNR), positive predictive value (PPV), negative predictive value (NPV), and F-score—for each deep learning model. It’s a format that helps compare total performance across larger datasets and how the metrics that comprise the overall score contribute to it. Comparing Fig 9, can see that the cumulative metric values are the highest among the CNN model, followed by ViT and GAN, while SimCLR and MoCo achieve slightly lower overall values.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 9. Stacked bar chart comparing aggregated performance metrics across CNN, ViT, GAN, SimCLR, and MoCo models.

https://doi.org/10.1371/journal.pone.0330228.g009

These baseline results show that CNN performs more consistently across key metrics such as sensitivity (86.02%) and specificity (86.53%), which suggests its superior capacity to generalize important morphological patterns in bone marrow images. ViT and GAN, although slightly lower in overall metrics, still maintain acceptable classification performance. Self-supervised models, such as SimCLR and MoCo, underperform comparatively, likely due to their heavier reliance on pretext training and less direct supervision, which may not capture domain-specific hematological features as effectively without fine-tuning. These findings justify CNN as the strongest candidate for further optimization and form the foundation for evaluating the impact of feature selection and hyperparameter tuning.

4.2 Feature selection results

Binary metaheuristic algorithms reduced the high-dimensional input space in the second evaluation phase to identify optimal subsets of inputs. The performance of each algorithm was determined by running it multiple times over fitness functions and a constant validation split. It aimed to compare each algorithm’s ability to reduce the dimensionality of the input data without degrading or enhancing the discriminatory power of the resulting feature set.

To ensure statistical robustness and mitigate the effects of randomness inherent in metaheuristic processes, each algorithm was executed 30 times independently. This number of repetitions is a widely accepted standard in optimization literature and allows for reliable estimation of average performance, variance, and stability. Reported metrics, such as average error and fitness, represent the mean results across these 30 independent trials.

To evaluate the quality of the selected feature subsets, K-Nearest Neighbors (KNN) was used as the embedded fitness evaluator during the feature selection phase. KNN was deliberately chosen over CNN at this stage due to its non-parametric and architecture-agnostic nature, which allows for quick and consistent evaluations of feature utility without introducing the confounding effects of CNN training dynamics, such as weight initialization or learning rate scheduling. KNN also avoids excessive computational load during iterative selection, enabling a more tractable evaluation process across multiple metaheuristic runs. The final selected features are later used to retrain the CNN, ensuring consistency in downstream performance evaluation.

The performance of metaheuristic optimization algorithms is susceptible to their initial parameter settings. In this study, the configurations for each algorithm were carefully calibrated to ensure robust and consistent convergence behavior across multiple runs. Table 6 summarizes the standard settings, such as population size, number of iterations, number of independent runs, and key algorithm-specific parameters.

Download:

• PNG
  larger image
• TIFF
  original image

Table 6. Initial parameter settings for the optimization algorithms.

https://doi.org/10.1371/journal.pone.0330228.t006

Each optimization algorithm was initialized with a population size of 30 and executed for 500 iterations per run. These parameters were uniformly applied to ensure fairness in comparative evaluation. The use of 30 runs per algorithm enables statistically reliable comparisons while accounting for the stochastic nature of evolutionary processes.

Six binary metaheuristics, including bOcOA, bPSO, bGWO, bFA, bGA, and bWOA, are compared in Table 7 in terms of their comparative performance. Additionally, critical performance statistics (average classification error, average value of the selected feature size as the proportion of total features, average, best, and worst fitness scores) are reported for each row. Notably, the binary Ocotillo Optimization Algorithm (bOcOA) outperformed all other methods in retaining only the most discriminative features, with an average error of 0.4238 and the smallest average subset size of 0.3772. This also showed that runs were more stable (had less variation in their fitness values). bPSO and bFA yielded significantly worse results than bFO, although they selected significantly larger feature subsets and exhibited greater variability.

Download:

• PNG
  larger image
• TIFF
  original image

Table 7. Feature selection results.

https://doi.org/10.1371/journal.pone.0330228.t007

Kernel density estimation (KDE) plots are utilized to evaluate and check the consistency and distribution characteristics of various feature selection algorithms. This visualization provides a roughly smoothed view. In addition to the interquartile range and spread distribution, these visualizations also display scatter. Fig 10 shows the variation in performance density value of all six bio-inspired algorithms bOcOA, bPSO, bGWO, bFA, bGA, and bWAO defined in the metric value. Therefore, it helps determine which algorithm is more stable and performs better under changing conditions.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 10. Density distribution plot of performance metrics for various feature selection algorithms: bOcOA, bPSO, bGWO, bFA, bGA, and bWAO.

https://doi.org/10.1371/journal.pone.0330228.g010

Box plots with swarm overlays are further used to analyze the statistical behavior of different feature selection algorithms more deeply. In addition to the interquartile range and spread distribution, these visualizations also display mean and standard deviation values to provide a more detailed statistical summary. The box plots of six optimization-based algorithms bOcOA, bPSO, bGWO, bFA, bGA, bWAO, and corresponding mean (red dashed line) and standard deviation range (green dotted lines) are shown in Fig 11. This representation helps to understand data variability, skewness, and the presence of outliers, providing deeper insights and aiding in the evaluation of the algorithm’s stability and consistency of performance.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 11. Box plots with swarm overlays for six feature selection algorithms, annotated with mean and standard deviation values.

https://doi.org/10.1371/journal.pone.0330228.g011

A correlation heatmap assesses the relationship and similarity of multiple bio-inspired feature selection algorithms. This visualization condenses all the strengths of the correlations between each algorithm’s performance outcomes. High correlation values indicate that the behavior or performance of the two series is similar, while low correlation values indicate that the results of the two series diverge. Fig 12 shows, for instance, that algorithms like bPSO, bFA, and bGWO have high inter-correlations, and those of bWAO and bOcOA are dissimilar from the others.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 12. Correlation heatmap of performance scores among different feature selection algorithms: bOcOA, bPSO, bGWO, bFA, bGA, and bWAO.

https://doi.org/10.1371/journal.pone.0330228.g012

To fully understand the success of optimization algorithms in feature selection tasks, evaluating their average fitness values is a crucial step in the process. Fitness value denotes the quality of the solutions provided by all the algorithms, and a higher value means better performance while optimizing the selected criteria for the problem. A bar chart showing the average fitness achieved by six bio-inspired algorithms, namely bOcOA, bGWO, bWAO, bGA, bPSO, and bFA, is presented in Fig 13. From the plot, bFA and bPSO are the best in navigating the solution space regarding average fitness, indicating their strong capability of utilizing the solution space.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 13. Bar chart comparing the average fitness values of various feature selection algorithms.

https://doi.org/10.1371/journal.pone.0330228.g013

These results indicate that bOcOA not only achieves better classification performance but also enhances model simplicity by selecting fewer features. Its significantly lower average error compared to bPSO (by 19.24%) and bFA (by 15.44%) suggests superior convergence toward informative feature subsets. Furthermore, the smallest average subset size shows bOcOA’s strong feature compression capability, a desirable property for clinical interpretability and reducing overfitting risks. This performance is likely attributed to bOcOA’s dynamic learning factor and feedback-driven binary updating mechanism, which allow for more adaptive exploration of the feature space. In contrast, algorithms like bPSO and bFA may lack sufficient balancing between exploration and exploitation, resulting in redundant or noisy feature retention. Thus, the obtained outcomes justify the choice of bOcOA as the feature selector for subsequent model refinement.

4.3 Machine learning performance after feature selection

Finally, using the optimized feature subsets of each of the five deep learning models, these models were retrained after applying binary metaheuristic feature selection. This phase aimed to quantify the extent to which the performance improvement is due to dimensionality reduction while assuming no change in hyperparameter configuration.

The post-selection classification Metrics for all models are reported in Table 8. Both statistically and clinically significant performance enhancements are observed in all cases. The CNN increased from 0.8629 to 0.9348 (a gain of more than 7 percentage points) to become the best-performing model, a position it had previously held. The model produced similar improvements for precision, sensitivity, and F1-score, with the support that it is better at determining discriminative and generalization power. The consistency of ViT, GAN, SimCLR, and MoCo in improving all metrics validates the efficacy of feature subset selection guided by metaheuristics in enhancing model learning and reducing overfitting.

Download:

• PNG
  larger image
• TIFF
  original image

Table 8. Classification performance after feature selection.

https://doi.org/10.1371/journal.pone.0330228.t008

A scatter matrix examines inter-metric relationships and possible correlation of evaluation measures. Heatmap with hierarchical clustering is used to relate evaluation metrics, exploring the relationships between them and identifying any underlying structure in these relationships. Density plots along the diagonal indicate the distribution of each metric. As shown in Fig 14, the visualization enables the identification of groups, outliers, linear relationships, or clustering patterns that may confound the assessment and interpretation of the model.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 14. Pairwise scatter matrix of six evaluation metrics with density plots on the diagonal.

https://doi.org/10.1371/journal.pone.0330228.g014

A heatmap with hierarchical clustering is used to relate evaluation metrics, examining the relationships between them and identifying any underlying structure in these relationships. The combination of correlation analysis and dendrogram-based clustering is used to develop groups of metrics with similar behavior across models. As seen in Fig 15, both the intensity of the relation between correlation values and the hierarchical branches give essential insights into how measurements of Accuracy, F-Score, Sensitivity (TPR), etc., are grouped based on similarity. Clustering is helpful because it reduces metric redundancy and improves model interpretability in evaluation.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 15. Heatmap with hierarchical clustering of model evaluation metrics, showing correlations and metric groupings.

https://doi.org/10.1371/journal.pone.0330228.g015

A dual-axis plot is used to analyze and compare two critical metrics of the performance of Deep Learning models accordingly. This is a plot of the scatter plot for Accuracy and the line plot for Specificity (TNR) for all models combined. The Accuracy and Specificity values for each model can be compared directly with one another on the left and right y-axes, respectively. The Specificity and Accuracy scores of the CNN model, as depicted in Fig 16, are the highest, followed by a gradual decline for ViT, GAN, SimCLR, and MoCo.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 16. Dual-axis plot comparing Accuracy (scatter) and Specificity (TNR) (line) across CNN, ViT, GAN, SimCLR, and MoCo models.

https://doi.org/10.1371/journal.pone.0330228.g016

These results demonstrate the positive effect of binary metaheuristic feature selection. CNN, ViT, and GAN all show an average increase of approximately 7–11 percentage points across accuracy, sensitivity, and specificity. Notably, CNN’s sensitivity improved from 86.02% to 93.31%, and specificity rose from 86.53% to 93.62%, indicating better detection of true positives and true negatives, respectively. These gains can be attributed to the removal of irrelevant or noisy features that previously diluted the model’s ability to learn distinguishing patterns. As a result, models trained on the selected subsets can focus on high-value representations, thereby improving their predictive robustness without requiring hyperparameter retuning. This confirms the metaheuristic-driven selection process as a practical step in streamlining high-dimensional biomedical datasets.

4.4 Optimized model analysis

In addition to the already optimized feature sets, each model was fine-tuned in the final phase of experimentation using continuous metaheuristic optimization to optimize its hyperparameters. Six metaheuristic algorithms were executed over a search across the real-valued parameter space, i.e., OcOA, PSO, GWO, FA, GA, and WOA. All prior phases used the CNN model as the representative architecture and outperformed in all aspects.

The performance of CNN after hyperparameter optimization by each of the six continuous metaheuristics is described in Table 9. Once again, OcOA proved to be the best optimizer, increasing CNN’s accuracy to 0.9824, its sensitivity to 0.9834, and its F1 score to 0.9821, all of which are on par with the current state of the art in image-based cell classification. Also, the PSO and GWO algorithms returned promising results, 0.9638 and 0.9590, respectively. The run performance of FA, GA, and WOA was followed in descending order. All in all, thus in harmony, these findings strongly support the synergistic effect of merging metaheuristic-based feature selection procedures with sophisticated hyperparameter tuning, which is further accentuated by the compactness and the convergence-aware dynamics of the guides to the OcOA framework.

Download:

• PNG
  larger image
• TIFF
  original image

Table 9. Classification performance after hyperparameter optimization.

https://doi.org/10.1371/journal.pone.0330228.t009

To further illustrate the improvements achieved by OcOA, confusion matrices for CNN before and after hyperparameter optimization are presented in Figs 17 and 18. These visualizations demonstrate how the optimized model significantly reduces both false positives and false negatives, thereby enhancing clinical reliability.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 17. Confusion matrix for baseline CNN model before optimization.

Sensitivity and specificity are considerably lower compared to the OcOA-tuned CNN.

https://doi.org/10.1371/journal.pone.0330228.g017

Download:

• PNG
  larger image
• TIFF
  original image

Fig 18. Confusion matrix for CNN after OcOA hyperparameter optimization.

Substantial improvements in classification metrics and class-level balance are evident.

https://doi.org/10.1371/journal.pone.0330228.g018

Key metrics are plotted in line plots, providing an overview of the performance of CNNs where optimization algorithms were combined. The figure presents the trend of specific evaluation metrics, such as Accuracy, Sensitivity (TPR), Specificity (TNR), PPV, NPV, and F-Score, as hybrid models are sampled with each different bio-inspired optimizer combined with a CNN. Additionally, each plot features a mean (red dashed line) and standard deviation bands (green dotted lines) to contextualize the variance in performance of the metric. WOA + CNN performed the worst below all other models in Fig 19, and OcOA+CNN always did better and did so consistently.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 19. Line plots showing performance trends for six evaluation metrics across CNN combined with different bio-inspired optimization algorithms.

Mean and standard deviation bands provide statistical context.

https://doi.org/10.1371/journal.pone.0330228.g019

Two heatmaps are presented to summarize the performance metrics across models: one displays the mean value for each metric, while the other shows the standard deviation of each metric. Through this dual visualization, it is straightforward to compare which metrics are high-performing and stable across different experimental runs. The NPV metric has the highest mean value (see Fig 20), and its standard deviation is the lowest. Therefore, this metric is both reliable and balanced. In contrast, metrics such as PPV and Sensitivity show slightly higher variation.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 20. Left: Heatmap of mean values for each performance metric.

Right: Heatmap of standard deviations for the same metrics.

https://doi.org/10.1371/journal.pone.0330228.g020

Histograms with overlaid normal distribution curves are used for the distributional behavior of evaluation metrics across models. These plots indicate the shape, spread, and normality of metrics such as Accuracy, Sensitivity (TPR), Specificity (TNR), positive predictive value (PPV), negative predictive value (NPV), and F-score. Through the visual comparison of the actual distribution with the fitted standard curve, it is possible to determine if the metric is sufficiently distributed, ensuring stable and predictable behavior. The modeling approach rests on the assumption that most metrics have symmetric distributions, as shown in Fig 21, where most metrics exhibit reasonable symmetric distributions, indicating the reliability of the modeling approach.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 21. Histograms with overlaid normal distribution curves for six evaluation metrics across models.

https://doi.org/10.1371/journal.pone.0330228.g021

Andrews Curves are used to visualize multivariate performance profiles of multiple hybrid models and compare their multivariate performance profiles. This technique reduces the dimensionality of high-dimensional data into a two-dimensional curve, making model similarity and anomalies more detectable. Each of these has a curve for a model built with a continuously functioning function based on its feature values, in this case, evaluation metrics such as accuracy, sensitivity, specificity, and more. Fig 22 shows how the curves for all hybrid CNN-based models are nearly on top of each other, indicating minimal sensitivity between metrics between optimization strategies and a broad consistency in performance across optimization strategies.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 22. Andrews curves for visual comparison of multivariate metric profiles across CNN-based hybrid models.

https://doi.org/10.1371/journal.pone.0330228.g022

From these results, it is evident that hyperparameter tuning plays a critical role in unlocking the full potential of deep learning models. The OcOA-optimized CNN model achieved a nearly 5% increase in accuracy and an over 5% increase in sensitivity compared to its post-feature-selection performance, rising from 93.48% to 98.24% and from 93.31% to 98.34%, respectively. Such improvements are not only statistically significant but clinically important in minimizing false negatives. The OcOA’s superior performance can be attributed to its dynamic balance between exploration and exploitation, driven by feedback-informed sinusoidal and Gaussian updates. Unlike PSO and GWO, which may converge prematurely or require more iterations for fine-tuning, OcOA consistently converged faster and more precisely, leading to better diagnostic metrics across the board. This supports OcOA’s suitability for tuning non-convex, high-dimensional neural network landscapes in clinical imaging tasks.

The empirical evidence across all phases demonstrates that the presented integrated optimization framework substantially outperforms baseline configurations. Classification accuracies, model generalization, and computational efficiency are improved, consistent, and strongly resistant to algorithmic variations.

4.5 Optimization statistical analysis

To validate the significance of observed performance improvements among optimizers, statistical tests were conducted to assess whether OcOA’s superiority is statistically supported. We first applied one-way ANOVA across six optimization methods using CNN as the base classifier. The ANOVA test (see Table 10) yielded an F-statistic of 158.6 with p < 0.0001, indicating significant differences between optimizer performance distributions.

Download:

• PNG
  larger image
• TIFF
  original image

Table 10. ANOVA test result.

https://doi.org/10.1371/journal.pone.0330228.t010

To confirm the pairwise superiority of OcOA over each baseline optimizer, the Wilcoxon signed-rank test was used as a non-parametric comparison, given the relatively small sample size (n = 10 per group). Table 11 summarizes the test results. Across all comparisons, OcOA achieved a significantly higher actual median accuracy (0.9824) than all competing methods, with a consistent p-value of 0.002 (two-tailed), indicating statistically significant performance gains at .

Download:

• PNG
  larger image
• TIFF
  original image

Table 11. Wilcoxon signed-rank test results comparing OcOA+CNN vs. Other optimizers.

https://doi.org/10.1371/journal.pone.0330228.t011

To visually support these results, Fig 23 presents multiple comparative plots. The top left and top right subplots display the actual metric distributions, while the bottom left subplot shows the fitted regression of measured versus expected performance. The bottom right heatmap highlights the superiority of OcOA across key performance dimensions, showing strong contrast in favor of OcOA’s consistency and accuracy.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 23. Statistical diagnostic plots: (top left, top right) metric spread and clustering; (bottom left) regression fit of expected vs. observed accuracy; (bottom right) performance heatmap highlighting optimizer differences.

https://doi.org/10.1371/journal.pone.0330228.g023

Fig 24 illustrates the histogram of metric distributions across all optimizers, showing how OcOA’s values cluster consistently near the upper boundary, reinforcing its statistical dominance. The rightward shift and narrow spread reflect both its high median and low variance.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 24. Histogram showing distribution of model performance metrics across optimization strategies.

OcOA yields tightly grouped, high-value results.

https://doi.org/10.1371/journal.pone.0330228.g024

Lastly, Fig 25 combines boxplots with violin overlays to capture the dispersion and density of optimizer performance. OcOA exhibits a compact distribution with minimal spread and outliers, while other optimizers show broader or skewed distributions. This further underscores OcOA’s convergence consistency and statistical robustness.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 25. Box-violin plots comparing optimizer performance distributions.

OcOA shows the highest median with the least variance.

https://doi.org/10.1371/journal.pone.0330228.g025

These results collectively confirm that the observed improvements offered by OcOA are not only numerically superior but also statistically significant, reinforcing the robustness of OcOA’s optimization behavior over traditional metaheuristics in tuning CNN classifiers for bone marrow cytology.