1.1. Related work

Harris Hawks Optimization (HHO) has garnered significant research interest as a nature-inspired metaheuristic owing to its effective balance between exploration and exploitation, guided by an energy-based hunting strategy. In our early efforts, we concentrated on the core structure of the HHO algorithm to improve convergence stability and global search capability. Zhang and colleagues [13] introduce a modified HHO framework to improve convergence behavior in high-dimensional optimization problems.

Qu et al. [14] incorporated information-exchange mechanisms among hawks to preserve population diversity and ameliorate premature convergence. Sihwail et al [15] further increased HHO by: integration of feature selection strategies. Fan et al. [16] proposed a quasi-reflected HHO variant to strengthen exploration efficiency and accelerate convergence. Based on these algorithmic enhancements, HHO has been widely adopted in healthcare applications, particularly for disease diagnosis and risk-prediction. Al-Safi (et al. [17]) used HHO together with artificial neural networks (ANNs) for heart disease diagnosis, reporting improved classification accuracy. Alsafi et al. [18] extended the idea to the diagnosis of general disease with HHO-based ANN frameworks. Balamurugan and colleagues [19] proposed an adaptive HHO integrated

The enhanced genetic algorithm for heart disease classification achieved notable performance improvements. Kumar and Rekha [20] introduced an improved HHO-based learning framework for cardiovascular disease prediction and demonstrated enhanced diagnostic reliability. Prasanna and Challa [21] combined binary HHO with a deep Bi-LSTM architecture for heart risk prediction using HHO (Another Application of HHO in Medical Time Series Data). Maleki et al. [22] Optimized support vector machines based on HHO for coronary artery disease prediction, while Vijayaraj and Pasupathi [23] proposed a context-aware hybrid HHO framework for personalized CAD risk assessment. Beyond medical diagnosis, HHO has been successfully applied to complex engineering optimization tasks. A hybrid differential evolution–Harris hawks optimization (DE–HHO) algorithm was proposed by Fu et al. [24] for optimizing kernel extreme learning machines, achieving superior accuracy and computational efficiency in wind speed forecasting. Moayedi et al. [25] integrated HHO with k-fold cross-validation for slope stability prediction, demonstrating reliable modeling of nonlinear relationships in real-world datasets. Despite these advancements, a common limitation of most existing HHO-based and hybrid approaches is their reliance on static manually tuned control parameters. Fixed exploration coefficients, escape energy settings, and learning-related parameters restrict the ability of the optimizer to adapt to dynamically changing the search landscape. This limitation becomes particularly pronounced when (HHO) is applied to high-dimensional and complex problems (e.g., * -) in deep learning–based coronary artery disease classification, where inappropriate parameter settings can lead to premature convergence, inefficient exploration, or unstable training behavior.

Motivated by these gaps, the proposed Dynamic Optuna Hybrid Harris Hawks Optimization (Dynamic Optuna H-HHO) framework directly extends prior HHO-based research by introducing adaptive and automated parameter tuning into the optimization process. By integrating Optuna’s dynamic hyperparameter sampling strategy with HHO’s biologically inspired hunting mechanism, the proposed method enables the real-time adjustment of critical parameters, such as escape energy, exploration–exploitation coefficients, and learning rates. Unlike existing orthologs and hybrids of HHO [13–25], which use fixed or heuristically correct defined configurations, the Dynamic Optuna H-HHO framework continuously adapts to the evolving search landscape. This design choice provides improved convergence stability, enhanced global search efficiency, and superior generalization performance, making it particularly suitable for deep-learning-based CAD diagnosis in real-world clinical settings.

These studies highlight the versatility of HHO; however, a key limitation remains in that most existing implementations employ static configurations, lacking dynamic adaptation to the evolving search landscape or learning objectives.

However, HHO faces inherent limitations,

• The algorithm spends significant time in the exploration phase in early iterations, which slows convergence, and the exploitation phase lacks a mechanism to refine the solution, which can lead to suboptimal convergence and stagnation of the algorithm.
• The static nature of the algorithm renders it less suitable for real-time and adaptive scenarios.
• The escaping Energy(E) and movement strategies were static and predefined. This creates optimization problems, and the algorithm is less flexible.
• Poor initialization can lead to slower convergence or an optimal solution.

To address this limitation, the proposed work will explore several improvement strategies, which are briefly described below.

1. We proposed a novel Dynamic Optuna Hybrid Harris Hawks Optimization (Dynamic Optuna H-HHO) algorithm that introduced dynamic adjustment of step size, velocity, and escaping energy, thereby addressing the stagnation and inflexibility of traditional HHO and other metaheuristics.
2. We integrated the proposed optimization framework with four state-of-the-art deep learning classifiers such as ResNet50, VGG16, InceptionV3, and MobileNet. We demonstrated its effectiveness in enhancing the classification performance on a real-world coronary stenosis image dataset.
3. We empirically demonstrated that Optuna H-HHO significantly outperformed both conventional HHO and standard classifiers, achieving up to 97.9% accuracy and showing improved F1 scores and AUC across all the tested models.
4. We compared our proposed approach with traditional optimizers such as Genetic Algorithm, PSO, GWO and demonstrated how the proposed method overcame key limitations, such as static parameters and premature convergence, through dynamic tuning and adaptive control.
5. We highlighted the clinical utility and deployability of the framework, emphasizing its ability to support real-time CAD diagnosis in clinical decision support systems through high accuracy, lightweight model compatibility, and future integration with interpretability tools, such as Grad-CAM, LIME, and SHAP.

Hence, this study proposed a novel hybrid algorithm called Dynamic Optuna H-HHO, which merged Optuna with H-HHO to overcome the limitations of H-HHO. This amalgamated approach integrated three unique strategies to augment its overall effectiveness. The main contributions of this study are as follows.

• Primarily, dynamically adjusting parameters, such as the step size (α) or energy (E), enables the algorithm to adjust progressively to the optimization landscape. Early iterations focused on exploration (broad search), whereas later iterations prioritized exploitation (fine–tuning).
• Secondly, the best solution found so far (Xrab) should be retained and not replaced by inferior solutions, as this helps preserve high-quality results and supports stable convergence.
• Finally, adjusting the exploration and exploitation phases based on fitness improves efficiency by balancing global search to avoid local optima and local refinement for faster and more accurate optimization..

The primary novelty of this work lies in the development of a dynamic hybrid hyperparameter optimization framework that integrates Optuna with Harris Hawks Optimization. Unlike conventional tuning approaches that rely on either probabilistic search or population-based optimization independently, the proposed Dynamic Optuna–H-HHO framework combines adaptive trial-based parameter sampling with the exploration–exploitation capability of HHO to improve optimization efficiency in high-dimensional hyperparameter spaces. The proposed strategy dynamically regulates the search process to enhance convergence stability and reduce the likelihood of premature convergence. Furthermore, the framework is applied to optimize deep learning models for coronary artery disease classification using multiple CNN architectures, including ResNet50, VGG16, InceptionV3, and MobileNet. The effectiveness of the proposed framework is further validated through rigorous statistical analysis using the Friedman Test and Wilcoxon Signed-Rank Test, demonstrating its superiority over the baseline optimization approach.

We conducted extensive experiments to assess the efficacy and robustness of the Dynamic Optuna H-HHO algorithm, employing various classifiers such as ResNet50, VGG16, InceptionV3, and MobileNet. Initially, experimental testing was performed using the stenosis dataset. We employed Dynamic Optuna H-HHO to optimize the classifier parameters, leading to notable enhancements in the overall performance and effectiveness of the model. Our research focuses on using Dynamic Optuna H-HHO to enhance CAD prediction by optimizing classifier parameters and presenting a novel hybrid model that merges features from both Optuna and H-HHO. Fig 1 illustrates the architecture of our proposed method.

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Fig 1. Architecture diagram of the Proposed Dynamic Optuna H-HHO.

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This paper is organized as follows. Section 2 presents a detailed description of the proposed model and its optimization strategy. Section 3 reports the numerical results and compares the proposed approach with existing methods. Finally, Section 4 summarizes the main findings and outlines directions for future research.

2.1. Data acquisition

This study used a publicly available angiographic dataset comprising coronary angiography image series from 100 patients with clinically confirmed one-vessel coronary artery disease The dataset included X-ray angiographic images from 100 individuals who underwent coronary angiography. The dataset consisted of 8,325 grayscale images with dimensions ranging from 512 × 512–1000 × 1000 pixels. These images were captured using two systems: Siemens Coroscop and GE Healthcare Innova. The dataset is publicly available at Mendeley Data (https://doi.org/10.17632/ydrm75xywg.1). This dataset was previously employed in the research titled “Real-time Coronary Artery Stenosis Detection Based on Modern Neural Networks” and now underpins ongoing studies on advanced image analysis methods for detecting coronary artery disease. The dataset was allocated with 70% for 15% for training, 15% for validation and the remaining 15% for testing; all experiments were conducted on a system with an NVIDIA

GPU (16 GB VRAM), Intel i7 CPU, and 32 GB RAM implementation used Python 3.9, TensorFlow/Keras, Optuna, NumPy, OpenCV, and Scikit-learn. The average training time ranged between 2 and 4 h per model, depending on the architecture and optimization iterations.

2.2. Data preprocessing

Preprocessing involved three essential operations: color space conversion, resizing, and normalization. The preprocessing began with converting all grayscale images to RGB format to achieve uniform input channels, which matched the requirements of the CNNs. All images were resized to a uniform size of 224 × 224 pixels, maintaining the same spatial size for all images in the dataset. The values of the pixel intensities were normalized from the original scale [0, 255] to [0, 1]. Normalization facilitates the stability and efficiency of training by keeping the input values within a more standardized range, which supports better optimization during both training and inference of the model. Explicit vessel segmentation and aggressive data augmentation were intentionally not applied to preserve anatomical integrity and to align with established angiography-based CAD

2.3. Architecture of baseline classifiers

In this study, we employed four foundational classifiers to categorize coronary artery disease: ResNet-50, VGG-16, Inception-V3, and Mobile-Net. The primary objective of this study is to evaluate the effectiveness and generalizability of the proposed Dynamic Optuna H-HHO optimization framework across diverse deep learning models. Therefore, four well-established convolutional neural network architectures—ResNet-50, VGG-16, Inception-V3, and MobileNet—were selected as baseline classifiers.

2.4. HHO(Harris Hawks Optimization) algorithm

The Harris Hawks’ collaborative hunting strategies form the basis of this meta-heuristic algorithm, which aims to optimize solutions through a balanced approach. Through exploration and exploitation, the method aims to achieve optimal results. Such a two-phase construction makes the algorithm capable of going through all those search stages and does not focus only on promising domains.

2.4.1. Exploration phase.

In this phase, hunting behavior takes place. The HHO algorithm emulates exploratory action when navigating the search space by Eqn 1

(1)

where

– Hawk’s current position

Hawks randomly chose the present position.

The interval [0,1] was utilized for generating stochastic values.

The computation of the Average position for existing Hawks is facilitated by Eqn 2

(2)

where

– spatial coordinates of hawks at each iteration ‘t’

N- total quantity of Hawks

2.4.2. Progression between two phases Of H-HHO.

It transitions from a broad exploratory to a focused exploitation phase and is driven by an “Escape Energy,” represented by E.

(3)

where

– initial energy cover from −1–1

t- current iteration

T – max iteration

2.4.3. Exploitation phase.

In this phase, the chase is accurately emulated using four distinct strategies. One is a random number from 0 to 1, and Escaping energy E

Soft beige: When prey does not have the chance to escape, it makes some effort to escape from the place when |E| ≥ 0.5 and r ≥ 0.5. It is represented by Eqn 4

(4)(5)

where

– Difference between the current entity and the optimal entity of prey

J = 2 × (1 − r₅) – Stochastic leap intensity of rabbit.

r₅ – It ranges from 0 to 1

Hard Beige: When the prey is unable to escape due to insufficient escaping energy when |E| ≤ 0.5 and r ≥ 0.5

It is represented by Eq. 6

(6)

The fitness function of Soft beige and Hard beige is represented as

(7)

where

F- fitness value

(8)(9)(10)

Here, u and V are random numbers ranging from 0 to 1, and constant β is fixed at 1.5

2.5. Proposed dynamic optuna H-HHO algorithm

In our proposed system, the fundamental structure of a conventional HHO algorithm is incorporated with three key strategies: dynamic adjustments, preservation of the best solution, and an adaptive strategy.

2.5.1. Exploration phase.

In the conventional HHO algorithm, this phase calculates the position using randomness in the global search using Eqs. (1) and (2), respectively. However, there was insufficient variability. Larger values enable faster movement, whereas smaller values refine precision. During immediate prey detection, randomness in the step length aids quick adjustments, whereas prolonged searches require a gradually decreasing step length for focused refinement. Early iterations emphasize broad exploration, whereas later iterations shift to detailed local search, balancing exploration and exploitation for improved efficiency and convergence. The modified exploration factor is modeled as follows in eqn 11

(11)

Here, b and rand are dynamically tuned by optuna after adjustment in the exploration phase is given by the eqn 12

(12)

Velocity was introduced into the HHO algorithm, which dynamically guides population exploration, enhancing the capacity of the algorithm to effectively traverse the outcome domain. This modification addresses the challenge of the HHO becoming trapped in local optima by improving global exploration and balancing exploration and exploitation during the exploitation phase. It should be emphasized that this velocity term is not associated with any personal-best or global-best learning mechanisms. Instead, it serves as an auxiliary motion-control variable that regulates the step size and directional continuity of the original HHO energy-based update process. Equations 13 and 14 determine the velocity and position of an object in a soft beige.

(13)(14)

Eqn 15 and 16 determine the velocity and position of the object in hard beige

(15)(16)

where

w, c₁, c₂, r₁, r_{2 –} these factors are dynamically tuned via optuna

Optuna dynamically adjusts the parameters to enhance the adaptability.

V – Velocity of each hawk

P_best is the best position of each hawk.

2.5.2. Escaping Energy Parameter (E).

The shift from exploratory to exploitative behavior is based on the energy required to escape, |E|. When |E| ≥ 1 the algorithm explores the new region(exploration phase) and when |E| < 1 it refines solution near

current best position (exploitation phase). In the conventional HHO, the exploration phase dominates, leaving fewer iterations for exploitation. It focuses on refining the existing solutions. In this study, we propose a novel escape energy parameter (E). It is illustrated in eqn 15

(17)

where

Sinusoidal modulation introduces controlled oscillations in energy decay, enabling periodic reinforcement of exploration in early iterations while progressively favoring exploitation as the search advances. Simultaneously, the monotonically decreasing α(t) acts as a step-size regulator such that the search radius is slowly drained and prevents abrupt convergence.

From a theoretical perspective, this design provides a well-defined and smooth transition from global exploration to local exploitation, which is consistent with annealing-based optimization principles. By dynamically coupling energy decay with adaptive scaling, the proposed mechanism mitigates premature convergence and enhances the ability of the optimizer to escape local optima while maintaining stability during later refinement stages.

2.5.4. Lemma (Exploration – Exploitation Transition).

Let the escaping energy E(t) be defined as

Where

With 0 <α_min <α_max, E_max>E_min ≥ 0 and t ∈ [0, T_max]

Then, the proposed Dynamic Optuna H-HHO algorithm guarantees a finite and deterministic transition from exploration to exploitation during the optimization process.

Proof

Exploration Condition

In Harris Hawks Optimization, the exploration phase is active when

∣E(t)∣ ≥ 1

and exploitation begins when

∣E(t)∣ < 1.

Boundedness of the Escaping Energy

Since the sine function satisfies

The term inside the brackets is bounded as

Because α(t) is strictly decreasing and positive over [0, T_max}, the escaping energy E(t) is bounded and non-negative.

Monotonic Decay of Escaping Energy

The function α(t) decreases linearly with t, while the sinusoidal term is non-increasing in amplitude over the interval [0, T_max]. Consequently,

∣ E (t + 1) ∣ < ∣ E(t) ∣ ∀t ∈ (0, T_max),

which implies that E(t)E(t)E(t) exhibits an overall decreasing trend.

Existence of a Finite Transition Point

At the initial iteration t = 0:

∣E(0)∣ = α_max E_max ≥ 1,

ensuring exploration.

At the final iteration t = T _max:

∣ E (T_max) ∣ = α _min E _min < 1.

By continuity of E(t), there exists a finite iteration t * ∈ (0, T_max) such that

∣ E (t *) ∣ = 1.

Exploitation Phase Dominance

For all t > t*:

∣ E (t) ∣ < 1,

and the algorithm enters the exploitation phase, where soft and hard besiege strategies dominate the search process.

Numerical Example Illustrating Lemma 2

Consider the following parameter settings used in the proposed Dynamic Optuna H-HHO algorithm:

• Maximum iterations:

T_max = 100

• Escaping energy bounds:

E _max = 2, E _min = 0

• Dynamic control parameter:

α_max = 1, α _min = 0

Thus,

α(t) = 1 −

and

Table 1 shows the dynamic behavior of the control parameters between iterations in the proposed optimization strategy: the more iterations, the elasticity of the control parameter 𝛼(𝑡). The linear decrease of (t) from 1. 0–0. 0 suggests that a gradual gradient between global exploration and local exploitation might exist. At the same time the sinusoidal term increases smoothly, modulating the energy parameter E(t) At early iterations (t = 0–20), with higher energy values the search space is explored more thoroughly as iterations progress(𝑡 ≥ 40), because as the energy values move closer to 0, the algorithm finds a more detailed local search around potentially promising solutions. This adaptive transition produces a balancing search–exploration problem which leads to improved convergence performance.

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Table 1. Escaping energy at different iterations.

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Fig 2 illustrates the dynamic escaping energy transition from exploration to exploitation in the proposed DO-HHO framework. The nonlinear decay of the escaping energy E(t) enables effective global exploration in the early iterations and progressive exploitation in the later stages, thereby improving the convergence stability and optimization performance.

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Fig 2. Dynamic Escaping energy transition.

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Although deep learning networks are inherently capable of learning complex patterns and performing classification tasks, their performance is highly sensitive to hyperparameters, such as the learning rate, batch size, dropout rate, optimizer type, and network-specific parameters, such as the number of filters or dense units. These hyperparameters play crucial roles in determining the convergence speed, generalization ability, and final accuracy of the model.

Traditional hyperparameter tuning methods, such as grid search or manual tuning, are often computationally expensive, inflexible, and may not scale well with the increasing complexity of DL models. In response to this challenge, the current study combines Hybrid Harris Hawks Optimization (H-HHO) with Optuna to automatically and effectively explore the hyperparameter space.

2.6. Hyperparameter ranges for proposed method (dynamic optuna H-HHO)

Table 2 explains the hyperparameter-tuned range and the range of configurations in which the proposed method achieved peak performance across the evaluated deep learning models, particularly InceptionV3 and MobileNet.

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Table 2. Optimal hyperparameter ranges for best accuracy using dynamic optuna H-HHO.

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2.7. Performance and evaluation metrics

The performance of the image segmentation and classification system was evaluated. The assessment utilized several metrics, such as accuracy, sensitivity, specificity, and F1 score. Owing to the stochastic nature of deep learning training and metaheuristic optimization, all experiments were repeated five times. To ensure reproducibility, a fixed random seed (seed = 42) was consistently applied across the NumPy, TensorFlow/Keras, and Optuna frameworks. The reported results represent the mean performance across independent runs.

2.7.1. Accuracy.

Accuracy refers to the ability of a classification model or system to correctly identify or dismiss specific situations. This metric measures the ratio of correct outcomes (including true positives and negatives) to all cases assessed, indicating the model’s overall effectiveness in predicting accurate results.

(18)

2.7.2. Sensitivity.

Sensitivity, also known as recall or the likelihood of a positive test result, assesses the effectiveness of a classification method in accurately identifying certain outcomes. This indicates the percentage of actual positive cases that the model correctly detects. Sensitivity is crucial in medical diagnostics and other areas where missing beneficial outcomes can have significant consequences.

(19)

2.7.3. Specificity.

Specificity, often referred to as the likelihood of a negative test result or recall, measures the ability of a classification method to accurately identify positive cases. This reflects the percentage of actual negative cases that the model correctly identifies. Precision is crucial in domains such as medical diagnostics and other tasks where false negatives can be costly.

(20)

2.7.4. F1 Score.

The F1 Score assesses the accuracy of a test by incorporating both precision and recall into a single measure. It employs the harmonic mean to balance these two metrics, particularly when uneven class distributions exist. The F1 Score appears to be advantageous when false positives and negatives must be accounted for simultaneously, offering a single assessment criterion that can effectively analyze classification model performance.

(21)

Precision refers to the ratio of correctly classified positive instances to the total number of instances predicted as positive.

(22)

The ratio of accurately detected positive predictions to the total number of actual positive cases is called recall (or sensitivity).

3.1.1. Accomplishment Of H-HHO.

The study employed multiple DL –based classifiers, including ResNet50, VGG16, InceptionV3, and MobileNet, to perform comparative analysis. The results of employing these classification algorithms are available in Table 3, and the measurements in it include accuracy, F1 score, sensitivity, and specificity. As seen in Table 4, InceptionV3 and MobileNet achieved the highest accuracy in percentage at 97.9%, and ResNet50test and VGG16 at 81.71% and 92.4% respectively. A hybrid Harris hawks optimization (H-HHO) algorithm was used to diagnose the Coronary Artery Disease (CAD), and its prediction using deep learning reached 97.9% accuracy. For greater improvement of these results, an adjusted version named Hybrid HHO (H-HHO) was investigated by integrating Optuna with H-HHO.

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Table 3. Optimization performance of DL models.

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Table 4. Evaluation metrics of H-HHO.

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When employing the Dynamic Optuna H-HHO, InceptionV3 and MobileNet showed the highest accuracy, followed by ResNet50 and VGG16. Table 5 provides a comparative analysis between the proposed Dynamic Optuna H-HHO model and the H-HHO technique using multiple metrics, including accuracy, sensitivity, F-1 score, and specificity. The Dynamic Optuna H-HHO model achieved accuracy rates of 81.73% for ResNet50, 94.49% for VGG16, 97.92% for InceptionV3, and 97.69% for MobileNet classifiers. Compared to the H-HHO model, the Dynamic Optuna H-HHO model demonstrated superior accuracy.

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Table 5. Evaluation metrics of proposed dynamic Optuna H-HHO.

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While the proposed Dynamic Optuna H-HHO framework achieves high classification performance, especially for InceptionV3 and MobileNet, we emphasize that a number of methodological safeguards were taken to ensure that the data were not leakage- and overfitting-resistant. All the data splits were performed at the patient level, i. e., images from the same individual were never found in training, validation, or test images; the test data was not used for optimizing or hyperparameter tuning. As part of the stochastic nature of classification, deep learning and metaheuristic optimization. All experiments were repeated 5 times, and the reported results are the mean results for each run.

Classification performance and robustness analysis: The proposed Dynamic Optuna–HHO framework was evaluated using four deep convolutional architectures, including ResNet50, VGG16, InceptionV3, and MobileNet, on the stenosis dataset. To ensure fair evaluation and prevent optimistic bias, all models were trained and tested using a strict data partitioning protocol, where training and testing sets were mutually exclusive. No augmentation artifacts or duplicated samples were shared across splits. Minimal preprocessing (image resizing and normalization only) was applied to avoid artificially enhancing vessel contrast or introducing handcrafted bias.

Among the evaluated architectures, InceptionV3 achieved the highest classification performance with an accuracy of 97.92%, F1-score of 97.95%, sensitivity of 97.92%, and specificity of 98.06%. While such performance levels are comparatively high for medical imaging tasks, several factors support the reliability of these results.

First, model stability was examined through five independent experimental runs with different random initializations. The performance variance across runs remained consistently low, indicating convergence stability rather than performance arising from a favorable initialization or split. This stability analysis (Fig 5) demonstrates that the observed performance is reproducible and not attributable to stochastic variation.

Second, non-parametric statistical testing was conducted to validate that performance improvements were systematic. The Friedman test revealed statistically significant differences between the proposed Dynamic Optuna–HHO and baseline HHO across classifiers (p < 0.05). Subsequent Wilcoxon signed-rank tests confirmed that improvements were significant for each backbone network individually. Because no assumption of normality was imposed, the statistical conclusions remain robust under distribution-free conditions.

Third, both sensitivity and specificity values remained balanced and consistently high, suggesting that performance was not driven by class imbalance or majority-class dominance. The ROC curves further demonstrated smooth and well-separated decision boundaries, with AUC values approaching 0.98, reinforcing discriminative capability across threshold settings.

Collectively, the combination of strict data separation, repeated-run stability analysis, balanced sensitivity–specificity metrics, and non-parametric statistical validation indicates that the reported 97.92% accuracy reflects genuine optimization-driven improvement rather than overfitting or dataset bias. Nevertheless, external multi-center validation will be considered in future work to further confirm generalizability in broader clinical settings.

The strong accuracy, sensitivity, and specificity values are therefore attributed to pretrained deep models, fine-grained hyperparameter optimization, and balanced dataset characteristics, and not variance-driven inflation or memorization.

The convergence behavior of GA, PSO, conventional HHO, and Dynamic Optuna H-HHO is illustrated in Fig 4. GA exhibits slow convergence and stabilizes at a higher fitness value. PSO tends to converge faster initially, but stagnates in later iterations. Conventional HHO improves convergence but plateaus due to static escaping energy. Dynamic Optuna H-HHO converges faster and reaches the lowest fitness value, mentions positive aspects of dynamic parameter adaptation, and Optuna-guided tuning.

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Fig 4. Convergence behavior of optimization algorithms over iterations.

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To assess robustness, stability analysis was conducted across repeated runs are shown in Fig 5. The distribution of accuracy values obtained from five independent experiments for each deep learning model. The narrow interquartile ranges and high median accuracies for InceptionV3 and MobileNet indicate stable and consistent performance under the proposed Dynamic Optuna H-HHO optimization framework.

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Fig 5. Stability analysis of classification accuracy across repeated runs.

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Fig 6 shows that MobileNet achieves high accuracy with significantly lower computational cost, making it suitable for resource-constrained or real-time clinical applications. InceptionV3 attains the highest accuracy at the expense of increased training time.

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Fig 6. Accuracy versus computational cost across deep learning models.

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Fig 7 depicts the receiver operating characteristic (ROC) curves of ResNet50, VGG16, InceptionV3 and MobileNet models optimized using the proposed Dynamic Optuna–Hybrid Harris Hawks Optimization (H-HHO) framework. It can be seen from ROC curves clearly separate from the diagonal reference line, which indicates that effective discriminative ability is achieved across the investigated architectures. Some models (InceptionV3 and MobileNet) show better true positive rate when comparing comparatively low false positive rate, which suggests superior classification behavior, while VGG16 shows competitive performance and ResNet50 is showing lower discrimination rate. The performance trend is similar to the quantitative evaluation metrics reported in Table 5 accuracy, sensitivity, and specificity. In addition to the visual analysis, numerical AUC values are also used as descriptive markers of the performance of the models (higher AUC values being related to higher robustness and better discrimination accuracy). The inclusion of AUC values helps in clarifying and contextualizing the observed performance differences among the models. The stability and coherence of the proposed Dynamic Optuna–H-HHO optimization algorithm is proposed.

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Fig 7. ROC Curves of Deep Learning Models with Optimization Technique.

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Overall, the ROC analysis results showed that the proposed optimization framework provides reliable and consistent performance across different deep learning architectures, which contributes to the comparative evaluation of our baseline.

3.1.2. Statistical comparison of model performance.

For each classifier, performance metrics were obtained from five independent experimental runs, yielding five paired observations per method for statistical comparison. These repeated measurements constitute the sample size used in the non-parametric tests. Since the distribution of performance metrics obtained from deep learning models cannot be assumed to follow a normal distribution, no parametric distribution assumptions were imposed. Therefore, non-parametric statistical tests were selected. A Friedman test was used to compare multiple optimization methods across repeated runs, followed by pairwise Wilcoxon signed-rank tests to assess statistical significance between the proposed method and each baseline classifier.

Statistical comparisons were conducted using results obtained from five independent runs for each model. Each run produced one performance value per classifier, resulting in five paired samples per method. As no assumption of normality was made for the performance distributions, non-parametric tests were employed. The Friedman test was applied for overall comparison, followed by Wilcoxon signed-rank tests for pairwise analysis, with a significance level of 0.05.

To examine the influence of the proposed Dynamic Optuna H-HHO on classification accuracy against the traditional H-HHO, we performed a Friedman test using the classification accuracy of four deep learning models, namely ResNet50, VGG16, InceptionV3, and MobileNet. For this task, we conducted a non-parametric analysis and classified the performance of each method according to the average rank of four different models. As shown in Fig 8, the proposed Dynamic Optuna H-HHO demonstrated a decreased average rank of 1. 25, but the comparison model H-HHO reached an overall average rank of 2. 75. The non-parametric Friedman test result shows the statistically significant improvement (p0. 05).

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Fig 8. Average Rank (Accuracy) Comparison between H-HHO and Dynamic Optuna H-HHO.

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To further confirm the statistical reliability of performance improvements, a Wilcoxon signed-rank test was conducted for each classifier. This non-parametric test compares paired accuracy values from H-HHO and Dynamic Optuna H-HHO across the same dataset. The results are shown in Table 6.

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Table 6. Wilcoxon Signed-Rank Test (Accuracy).

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4.1. Ablation study

An ablation study was conducted to evaluate the impact of integrating Optuna with Harris Hawks Optimization (HHO). In the Optuna-only configuration, hyperparameters were selected using Bayesian trial-level sampling, which efficiently explores the search space based on previous trial outcomes. This approach required approximately 38 minutes to complete the hyperparameter optimization process for the selected deep learning architecture.

In contrast, the proposed Optuna–HHO framework incorporates a population-based HHO search within each Optuna trial to further refine candidate solutions. Due to the additional iterative search performed by the HHO population, the optimization time increased slightly to approximately 44 minutes. Despite this modest increase in computational cost, the hybrid approach demonstrated improved optimization robustness and convergence stability across repeated experimental runs.

The reason for this improvement lies in the complementary roles of the two optimization mechanisms. While Optuna performs efficient global sampling of hyperparameter configurations, the HHO algorithm enhances the exploration–exploitation balance through cooperative population search. This allows the optimizer to escape local optima that may occur during purely Bayesian sampling and move toward more promising regions of the search space. Consequently, the proposed Optuna–HHO framework is more capable of identifying better near-global optimal hyperparameter configurations, rather than simply producing different parameter combinations. These findings confirm that the slight increase in optimization time is justified by the improved classification performance and optimization stability achieved by the proposed hybrid strategy.