Cardiovascular disease prediction and mortality risk assessment using a QSAO‑optimized BP neural network

Abstract

Background

Cardiovascular disease remains a leading cause of mortality worldwide, imposing a substantial burden on global healthcare systems. Early detection and accurate mortality risk assessment are critical for timely intervention and improved patient outcomes. However, existing predictive models often suffer from limited clinical applicability due to their restriction to single diagnostic tasks, insufficient prediction accuracy, and a tendency toward overfitting.

Methods

This study proposes a novel hybrid intelligent model, termed QSAO-BP, which integrates a quartile based Snow Ablation Optimizer (QSAO) with a Back Propagation (BP) neural network for simultaneous cardiovascular disease (CVD) diagnosis and mortality risk assessment. The QSAO enhances the SAO by incorporating Gaussian and Lévy random walks through a quartile based strategy, effectively mitigating premature convergence and local optima entrapment of SAO. The QSAO optimizes the hyperparameters of the BP neural network to improve its predictive accuracy and generalization capability. A dual validation framework is established using the Cleveland Heart Disease Dataset and the Heart Failure Clinical Records Dataset. Extensive experiments compare QSAO-BP against nine optimizer based BP variants and four classical machine learning algorithms across ten evaluation metrics.

Results

QSAO-BP achieves superior performance on both datasets, attaining an accuracy of 0.873 on Dataset 1 and 0.854 on Dataset 2, representing improvements of 8.178% and 9.347% over SAO-BP, respectively. The model also achieves the highest accuracy, precision, recall, F1-score, and MCC while maintaining competitive specificity and NPV. SHAP analysis reveals that the key predictors identified by QSAO-BP are consistent with established clinical knowledge.

Conclusions

QSAO-BP delivers robust and accurate predictions across two distinct clinical tasks with satisfactory interpretability, positioning it as a promising tool to support clinical decision making in CVD.

Introduction

According to statistics from the World Heart Federation, approximately 18 million people die from CVD annually, accounting for 31% of total global deaths [1–3]. The majority of CVD-related deaths are caused by stroke and heart disease [4]. These alarming figures and their analysis underscore the necessity for in-depth research into the complex influencing factors of CVD adverse outcomes including disease onset and patient mortality [5]. The complexity of CVD is closely associated with numerous risk factors, such as hypertension, abnormal insulin levels, smoking, work stress, and lifestyle habits, early screening of CVD lesions and accurate judgment of patients’ mortality risk are important clinical demands [6]. Given the complexity and diversity of risk factors that contribute to CVD, understanding these factors is crucial for developing targeted CVD treatment plans and mitigating the global impact of CVD [7]. Research indicates that over 90% of CVD cases can be prevented with early detection [8]. Therefore, both early diagnosis of CVD and precise assessment of CVD patients’ mortality risk are critical for timely clinical intervention [9]. However, the construction of cardiovascular prediction models faces multiple difficulties: CVD clinical datasets contain mixed categorical and continuous heterogeneous features with complex distribution, and there are problems such as redundant indicators and abnormal noisy samples, which greatly increase the difficulty of accurate identification and prognostic judgment [10, 11]. Furthermore, CVD-related datasets often contain irrelevant or redundant data, which can introduce noise into the algorithm’s classification performance and affect classification accuracy [12].

Traditional CVD diagnostic methods primarily rely on clinical signs, symptoms, and subjective judgments, which increases the difficulty of CVD diagnosis and mortality risk assessment [13]. In the medical field, artificial intelligence technologies, particularly machine learning (ML) methods, have been widely applied in the analysis of major diseases such as postoperative delirium [14] and CVD [15]. Researchers have employed various ML classification methods for the analysis of cardiovascular-related events [16]. These methods enable real-time monitoring, classification, and prediction of patients’ health status, thereby issuing risk warnings to patients and physicians when conditions fluctuate [17]. Compared to well-known invasive diagnostic methods for heart disease, AI-based diagnostic methods and non-invasive techniques have wider screening applicability and better safety for both CVD diagnosis and survival risk assessment [18, 19]. Developments in the use of ML in medicine have opened up exciting new avenues for the CVD early detection and mortality risk assessment.

Although ML methods have some advantages over traditional CVD prediction methods, they still suffer from issues such as insufficient accuracy and a tendency to overfit [20, 21]. On the other hand, there are still many challenges in accurately predicting CVD and assessing mortality risk comprehensively [22]. The BP neural network is a multi-layer feedforward neural network whose core is backpropagation and is widely used to solve clinical prediction problems [23]. However, when BP is applied to cardiovascular data prediction, random initialization of hyperparameters will lead to low accuracy and serious overfitting phenomenon [24]. However, most existing studies have two prominent research gaps that motivate this work, which are summarized as follows:

1. Most existing CVD prediction models are primarily designed for single clinical tasks, such as CVD diagnosis. There are relatively few studies that can be used for both CVD diagnosis and mortality risk assessment.

2. When handling medical prediction tasks, BP neural networks face issues such as low prediction accuracy and susceptibility to overfitting due to random hyperparameter settings. Traditional meta-heuristic optimization algorithms tend to prematurely converge and get stuck in local optima when optimizing BP neural network hyperparameters, failing to fully exploit the feature information of CVD datasets, which ultimately reduces the predictive performance of BP neural networks.

In the near future, multidisciplinary medical research teams comprising medical experts, artificial intelligence specialists, and statisticians will propel advancements in the field of medical diagnosis [25]. By integrating diverse data sources, including electronic health records, imaging data, questionnaire responses, and genetic information, and by extracting and selecting significant features from these datasets, it may be possible to uncover novel cardiovascular risk factors and develop more accurate risk prediction models [26]. Integrating these algorithms into wearable technology, smartphone health apps, and remote monitoring systems facilitates real-time risk evaluation and tailored treatment plans, tailored to individual lifestyles and physiological metrics [27, 28]. In order to predict CVD and assess mortality risk, this research proposes a novel intelligent prediction model that is outfitted with a high accuracy. This work presents three distinct scientific novelties: First, a QSAO is proposed by integrating Gaussian and Lévy random walks, which addresses the premature convergence of the original SAO in high-dimensional CVD data optimization. Second, QSAO is innovatively applied to optimize the hyperparameters of the BP neural network. Third, a dual-validation framework combining CVD prediction and mortality risk assessment is established, making up for the lack of clinical applicability of existing models that only use single dataset. The following is a summary of this work’s main contributions:

1. This study proposes a QSAO to address premature convergence and susceptibility to local optima in the CVD diagnosis and mortality risk assessment of the SAO. The use of QSAO to train hyperparameters in BP neural networks has improved the accuracy and overfitting of CVD diagnosis and mortality risk assessment in BP neural networks, effectively enhancing their predictive ability.

2. The use of QSAO for both the CVD diagnosis and mortality risk assessment in the progressive tasks of CVD has validated the CVD cross scene generalization ability of the BP neural network optimized by QSAO.

3. Compare QSAO-BP with 9 optimizer-tuned BP networks and 4 mainstream ML algorithms. The results indicate that in the two progressive scenarios of CVD, QSAO-BP outperforms the comparative algorithms in all evaluation indicators.

To outline the study’s structure: Section  “Literature review” reviews existing CVD prediction methods and their limitations. Section  “Methodology” details the data preprocessing, the QSAO, BP neural network, and the QSAO-BP workflow. Section  “Experimental results and analysis” presents experimental settings, analyzes the prediction performance of each model on CVD diagnosis and mortality risk datasets respectively. Section  “Conclusion and future works” concludes key findings and outlines future research.

Literature review

This section undertakes a critical analysis of approaches to CVD prediction, emphasizing significant research endeavors and their contributions to the discipline.

Al-Alshaikh et al. [29] employed an undersampling clustering method to address data imbalance, thereby enhancing model predictive capability. They used an adaptive elephant herd optimization approach to train a multi-layered deep convolutional neural network for their classification challenge. The introduced method for predicting CVD demonstrated exceptional performance across various key evaluation metrics. However, the study only targets a single clinical task of CVD diagnosis, and does not expand to patient mortality risk prediction.

Al-Jamimi [30] used the Bayesian classifier to tune the hyperparameters of XGBoost, and improved the accuracy of heart disease predictions. Due to data limitations and the instability of medical datasets, some discrepancies may exist between the study’s prediction results and real-world clinical applications. However, the study only realizes single-task CVD diagnosis and cannot support simultaneous CVD diagnosis and long-term mortality risk assessment.

Liao et al. [31] evaluated two types of heart diseases using a multi-layer perceptron artificial neural network combined with a genetic algorithm and backpropagation mechanism. Compared to other intelligent heart disease prediction methods, the proposed method performed better on a number of evaluation metrics. However, this model only achieves single CVD classification without mortality risk assessment. The model is tailored for IoT sequential physiological signals rather than static tabular electronic medical records.

Geetha and Akila [32] used the grey wolf optimizer (GWO) to tune the parameters of a stacked ensemble algorithm, thereby improving its accuracy in predicting heart disease. However, this study only performs CVD diagnosis without a unified dual-task framework for simultaneous CVD diagnosis and CVD mortality risk assessment. It adopts stacked ensembles technique optimized by GWO without horizontal comparisons with other metaheuristic algorithms including WOA and SAO.

In order to identify and categorize CVD from ECG signals, Rao and Kakollu [33] used a hybrid deep Maxout network model with convolutional block attention. The study preprocessed the MIT-BIH arrhythmia database using an improved finite impulse response filter model, which preserved specific features of the dataset. The proposed model achieved an accuracy rate of 98.81%, precision of 96.18%, recall of 96.87%, and F1 score of 94.39% on the MIT-BIH arrhythmia database. However, this work only performs CVD classification on ECG time-series data without a dual-task framework for CVD diagnosis and mortality risk assessment. This study only applied ECG data, leading to limited generalization.

Dhanushkodi et al. [34] introduced a novel three-kernel support vector Levy Gazelle algorithm aimed at enhancing the accuracy of disease predictions. This proposed algorithm was implemented for forecasting chronic kidney disease and hemodialysis datasets. The predictive outcomes demonstrated the algorithm’s superior performance across both datasets. However, this study only realizes single CVD classification, lacking a dual-task framework for simultaneous CVD diagnosis and CVD mortality risk assessment.

Mansoor et al. [35] introduced an ensemble voting classifier that integrates multiple ML classifiers with an ensemble voting mechanism. Their research employed SMOTE to mitigate dataset imbalance and utilized the chi-square test for selecting features. When compared to other traditional classifiers, the proposed classifier demonstrated superiority across diverse evaluation metrics. However, this study only realizes single CVD classification, lacking a dual-task framework for simultaneous CVD diagnosis and CVD mortality risk assessment.

Ullah et al. [36] extracted features from ECG signals and then used particle swarm optimization to select optimal features. The Extra Tree and Random Forest (RF) classifiers trained on the selected features achieved a remarkable 100% accuracy rate. However, this study only realizes single CVD classification, lacking a dual-task framework for simultaneous CVD diagnosis and CVD mortality risk assessment. This study only employs the PSO for feature selection without any innovation on the optimization mechanism, making the algorithm prone to trapping in local optima.

In order to detect different cardiac problems automatically and early, Vinay [37] used supervised and unsupervised recurrent neural networks based on bidirectional long short-term memory ML methods. However, the evaluation indicators of the proposed model are not comprehensive and systematic enough. This work only realizes single classification of heart sound types without a dual-task framework integrating CVD diagnosis and mortality risk assessment. Besides, all experiments lack comprehensive comparisons among swarm intelligence algorithms, optimized BP models and ensemble tree baselines.

Lenin and Venkatasalam [38] adopted a convolutional neural network combined with principal component analysis, restricted Boltzmann machines, and deep convolutional generative adversarial networks for CVD classification. The experimental outcomes revealed that the proposed method enhanced the predictive accuracy of traditional convolutional neural networks for CVD prediction. Nevertheless, the study was limited in that it did not utilize a broader range of evaluation metrics to fully test the efficacy of the proposed algorithm and only completes single arrhythmia identification without a dual-task framework for simultaneous CVD diagnosis and mortality risk assessment.

Ebraheem et al. [39] used the artificial gorilla troops optimization algorithm and a multilinear regression classification model for CVD classification. The proposed model achieved an accuracy rate of up to 99% across all datasets. However, this study only completes binary classification of CVD without a unified dual-task framework for CVD diagnosis and mortality risk prediction.

Methodology

Data preprocessing

The data preprocessing methods of the CVD-related datasets used in this paper include handling missing values, handling outliers, and data normalization.

Missing values handling

For cases with empty feature values in the dataset, the deletion method [40] is adopted, that is, cases with empty feature values are deleted.

Outliers handling

Outliers refer to feature values that deviate from the overall distribution pattern of the corresponding feature in the dataset [41]. Identify potential outliers using the interquartile range (IQR) method. Review the potential outliers marked by IQR method based on clinical expertise: those that exceed the reasonable physiological range are identified as input errors and removed; Those recognized as true extreme values within the reasonable range of physiology shall be retained. Based on the above strategy identification, no outliers were found in this dataset, and all samples were retained for subsequent modeling.

Data normalization

The purpose of data normalization is to uniformly scale features with different units and different ranges to the same numerical interval, making the model training more stable and accurate. The normalization method used in this paper is as shown in Eq. 1.

1

where, and represent the minimum and maximum values of the feature, respectively.  x is the original feature value, and is the feature value after normalization.

Dataset division

Use missing values handling, outliers handling, and data normalization for the datasets used in this paper, and divide the processed dataset into training set and testing set in an 8:2 ratio. The training set is used to train the BP neural network with 5-fold cross validation, and the testing set is used to test the trained BP neural network.

Quartiles-based snow ablation optimizer

The selection of the QSAO as the core optimization technique is driven by both theoretical rationality and practical adaptability to CVD research scenarios. Theoretically, classical optimizers adopt fixed search strategies, which are prone to local optima in high-dimensional medical data parameter tuning, while QSAO’s quartiles-based strategy dynamically balances global exploration and local exploitation to make up for this inherent defect. Practically, QSAO is customized for CVD and clinical postoperative data characteristics, which can optimize the hyperparameters of BP neural network more accurately and stably than traditional algorithms, and is more in line with the precision requirements of clinical CVD-related prediction.

Then, the original SAO is briefly introduced as follows.

Snow ablation optimizer

A meta-heuristic approach for numerical optimization that mimics the melting of snow is called Snow Ablation Optimization (SAO) [42].

Initialization phase

The SAO begins with randomly generating a population matrix , as shown in Eq. 2.

2

where  L,  Uand  D are the solution space’s lower bound, upper bound and dimension, respectively.  Nis the population size, and is an random number between is the  ith individual in the population as shown in Eq. 3.

3

Dual-population mechanism

This dual-population mechanism effectively simulates the two key factors of mutation and inheritance in natural selection, as shown in Table 1.

Table 1.

Pseudocode for the dual-population mechanism

Pseudo-code of the Dual-population mechanism
1: Initialization:  t=0,.,, where N denotes the population size;
2: while () do
3:      If then
4:          , ;
5:      end if
6:       t=t+1;
7: end while

Based on the dual-population mechanism, the complete position update for SAO is given by Eq. 4.

4

where, is a vector, where each component represents a random number conforming to Brownian motion. The symbol denotes element-wise multiplication. is a random number between . is the current optimal position. is a randomly selected individual from the elite population. is a random number between and  M is the snowmelt rate. Equation 5 illustrates how the snowmelt process is reflected using the degree-day method.

5

where,  T represents the daily mean temperature. is the base temperature, which is usually set to 0.  DDF is as Eq. 6.

6

is the centroid position of the entire population, as shown in Eq. 7.

7

where, represents the number of leaders.

Quartiles-based strategy

In this paper, an improved strategy based on quartiles is used to divide the SAO population into several groups, with different position update strategies applied to each group, as shown in Fig. 1.

Fig. 1.

Improved strategy based on quartiles

In Fig. 1, the first quartile (Q1), second quartile (Q2), and third quartile (Q3) of the SAO population are determined by calculating the fitness values of the individuals in the SAO population. Following that, the SAO population is split into four categories. The first group consists of individuals with fitness values higher than Q1. Individuals in the second group have fitness values in the range of Q1 and Q2. Individuals in the third category had fitness values in the range of Q2 to Q3. Individuals in the fourth category had fitness values lower than Q3. Individuals in the first group are the worst in the population. Individuals in the fourth group have the best fitness values. The performance of individuals in the second and third groups is intermediate between the first and fourth groups. The individuals in the fourth group are closest to the global optimal solution, so a Gaussian random walk [43] is applied to them to enhance the local search capability of SAO. The individuals in the first group have the worst performance, so a Lévy random walk [34] is applied to them to enhance the global search capability. The performance of individuals in the second and third groups is intermediate, so no random walk strategy is applied to them. The improved strategy based on quartiles is given by Eq. 8.

8

where  k is a random number between 0 and 1. is a random number with a variance of 1 and a mean of 0 that follows a Gaussian distribution. is a random number with a Lévy distribution, and  l is an adjustment coefficient.

In the improvement strategy of heuristic algorithms, the population is grouped based on the fitness values of individuals in the population, and differentiated update strategies are set for different subpopulations, which can enhance the local search of individuals with better fitness and enhance the global search of individuals with poorer fitness. Thus achieving a good balance between the exploration and exploitation capabilities of the algorithm. This design is particularly important for the progressive tasks of CVD diagnosis and mortality risk assessment carried out in this article, as the clinical data used includes both continuous and categorical data, and the structure and distribution of the dataset are complex. Therefore, the grouping strategy based on fitness values is necessary and reasonable for the proposed algorithm proposed.

The quartiles-based strategy based on fitness values has unique advantages among various grouping strategies, which can divide the population into four equally sized subpopulations based on Q1, Q2, and Q3. Dividing subpopulations in this way can enhance both the local search of a sufficient number of individuals with better fitness and the global search of a sufficient number of individuals with poorer fitness. Using other grouping methods may cause an imbalance in the size of subpopulations. Specifically, the large size of subpopulations with optimal fitness values causes the algorithm to excessively search locally, leading to premature convergence of the algorithm. If the size of the subpopulation with optimal fitness values is too small, it will not be possible to fully train the hyperparameters of the BP neural network that needs to be optimized. Subpopulations with poor fitness can generate redundant global searches and waste computing resources if their size is too large. The small size of subpopulations with poor fitness makes it difficult for the algorithm to escape from local optima. In summary, the imbalance of subpopulation size will reduce the optimization effect of optimization algorithms on BP neural networks, thereby reducing the performance of BP neural networks in CVD diagnosis and mortality risk assessment.

Back propagation neural network

The core principle of the BP neural network is to calculate the output error through forward propagation and then use the chain rule to adjust the weights and biases in the reverse direction to minimize the loss function [44].

The proposed model

This paper employs QSAO to optimize the hyperparameters of BP neural network, which is then utilized for CVD prediction tasks. The proposed model is illustrated in Fig. 2.

Fig. 2.

The flowchart of the proposed model

The QSAO is employed to search for optimal hyperparameters of the BP neural network. The hyperparameter configuration directly determines the training mode and fitting capacity of the BP neural network, and further affects the model’s capability to characterize the relationship between clinical risk factors and prediction outputs including CVD diagnosis and patient mortality risk. Classical optimization algorithms tend to be trapped in local optima during hyperparameter searching, which leads to suboptimal hyperparameter combinations of BP neural network. Poor hyperparameter sets will hinder the BP neural network from accurately reflecting the correlation between clinical risk factors and CVD events, such as CVD diagnosis and mortality risk assessment. In comparison, QSAO maintains a favorable balance between global exploration and local exploitation to find high-quality hyperparameter sets. With well-tuned hyperparameters, the BP neural network can more reliably capture the actual influence of various CVD risk factors, and provide a solid foundation for the following CVD events analysis.

Experimental results and analysis

As elaborated in Section “Methodology”, the proposed QSAO-BP model is specifically designed to address critical limitations of existing CVD prediction and clinical outcome assessment approaches. First, by adopting a quartile-based strategy, QSAO overcomes the premature convergence and local optima issues of traditional optimizers in high-dimensional medical data optimization. Second, the dual-validation framework integrating the heart disease dataset and heart failure clinical records dataset resolves the poor clinical applicability of models relying solely on single dataset. Third, optimizing the hyperparameters of the BP neural network via QSAO effectively improves its prediction accuracy and robustness, addressing the inherent shortcomings of BP in CVD-related tasks.

Simulation details

The experiments in this paper are conducted using MATLAB 2022a. To assess the performance of the proposed model, the datasets are divided into training and testing sets in varying ratios (8:2).

In this study, 5-fold cross-validation is employed for model training and evaluation. This value is selected as it is a standard and widely adopted strategy in medical data analysis and CVD prediction tasks [45], ensuring stable and reliable model performance without excessive computational cost.

Data description

To evaluate cardiovascular disease diagnosis and mortality risk assessment, we selected two cardiovascular datasets with distinct research objectives. The Cleveland Heart Disease Dataset [46] serves as a classic dataset for coronary heart disease screening, which focuses on identifying whether patients suffer from coronary artery disease. The Heart Failure Clinical Records Dataset [47] targets the survival prognosis of patients already diagnosed with heart failure. These two datasets jointly support two core cardiovascular research tasks: cardiovascular disease diagnosis and mortality risk assessment. The detailed descriptions of these datasets are as follows.

• (i)Dataset 1: The Cleveland Heart Disease Dataset contains clinical records of 303 patients and 14 features with suspected coronary heart disease, which includes 138 positive cases with confirmed coronary artery stenosis and 165 negative healthy control cases. It has 6 missing values, and this article uses deletion method to remove these samples with missing values. Each sample is characterized by attributes such as age, chest pain type, resting blood pressure, serum cholesterol, maximum heart rate, ST-segment depression, and exercise-induced angina. Further elaboration on the dataset attributes can be found in Table 2.
• (ii)Dataset 2: The Heart Failure Clinical Records Dataset consists of follow-up data from 299 hospitalized patients fully diagnosed with heart failure, all of whom are positive cardiac patients. Among all samples, 203 patients survived and 96 patients died during follow-up, and there are no healthy control subjects in this dataset. The binary target variable DEATH_EVENT records whether the patient died within the follow-up period, where 0 indicates survival and 1 represents mortality. This dataset records demographic information, comorbidity status, and laboratory test indicators for each patient, including age, anaemia status, diabetes, ejection fraction, serum creatinine, serum sodium, platelet count, hypertension and smoking history. Detailed attribute definitions are provided in Table 3.

Table 2.

Description and raw range of attributes used for Dataset 1

Feature	Description	Range
age	Patient age (years)	(29–77)
sex	Gender. 0 = female, 1 = male	(0–1)
cp.	Chest pain type. 1 = typical angina, 2 = atypical angina, 3 = non-anginal pain, 4 = asymptomatic	(1–4)
trestbps	Resting blood pressure (mm Hg)	(94–200)
chol	Serum cholesterol (mg/dL)	(126–564)
fbs	Fasting blood sugar > 120 mg/dL. 0 = false, 1 = true	(0–1)
restecg	Resting electrocardiographic results. 0 = normal, 1 = ST-T wave abnormality, 2 = probable or definite left ventricular hypertrophy by Estes’ criteria	(0–2)
thalach	Maximum heart rate achieved	(71–202)
exang	Exercise induced angina. 0 = no, 1 = yes	(0–1)
oldpeak	ST depression induced by exercise relative to rest	(0–6.2)
slope	Slope of peak exercise ST segment. 1 = upsloping, 2 = flat, 3 = downsloping	(1–3)
ca.	number of major vessels (0–3) colored by flourosopy	(0–3)
thal	3 = normal, 6 = fixed defect, 7 = reversable defect	(3–7)
Class	Target variable. 0 = no coronary heart disease, 1 = confirmed coronary heart disease	(0–1)

Table 3.

Description and raw range of attributes used for Dataset 2

Feature	Description	Range
age	Patient age (years)	(40–95)
anaemia	decrease of red blood cells or hemoglobin. 0 = no, 1 = yes	(0–1)
creatinine_phosphokinase	level of the CPK enzyme in the blood (mcg/L)	(23–7861)
diabetes	If the patient has diabetes. 0 = negative, 1 = positive	(0–1)
ejection_fraction	percentage of blood leaving the heart at each contraction (%)	(14–80)
high_blood_pressure	if the patient has hypertension. 0 = no hypertension, 1 = hypertension	(0–1)
platelets	platelets in the blood (kiloplatelets/mL)	(25.1–850.0)
serum_creatinine	level of serum creatinine in the blood (mg/dL)	(0.5–9.4)
serum_sodium	level of serum sodium in the blood (mEq/L)	(113–148)
sex	Patient gender. 0 = female, 1 = male	(0–1)
smoking	if the patient smokes or not. 0 = no, 1 = yes	(0–1)
time	follow-up period (days)	(4–285)
DEATH_EVENT	if the patient died during the follow-up period. 0 = alive during follow-up, 1 = death during follow-up	(0–1)

Evaluation of the proposed model’s and traditional models’ performance

To evaluate whether missing value processing approaches affect model performance, we carried out ten independent repeated tests under median imputation and listwise deletion respectively. The detailed results are summarized in Table 4.

Table 4.

Sensitivity analysis of different missing value processing strategies

	Accuracy	AUC	Precision	Recall	F1-score	MCC	Specificity	NPV	FPR	FNR
Median imputation	0.858 ± 0.030	0.903 ± 0.039	0.855 ± 0.034	0.891 ± 0.056	0.871 ± 0.030	0.716 ± 0.061	0.819 ± 0.051	0.868 ± 0.056	0.181 ± 0.051	0.109 ± 0.056
Deletion method	0.873 ± 0.035	0.921 ± 0.030	0.862 ± 0.048	0.916 ± 0.059	0.886 ± 0.032	0.748 ± 0.073	0.822 ± 0.069	0.897 ± 0.064	0.178 ± 0.069	0.084 ± 0.059

As shown in Table 4, the predictive metrics of the QSAO-BP model were very close under the two missing value processing methods. Taking accuracy as a representative indicator, the QSAO-BP obtained 0.873 ± 0.035 via deletion method and 0.858 ± 0.030 via median imputation. The fluctuation range of this indicator shows significant overlap in the ten independent runs of QSAO-BP, indicating that the choice of missing value strategy has little impact on model performance. Although the deletion method reduces sample size by excluding incomplete records, it did not result in significant performance loss in our experiment. In order to maintain a consistent experimental setup in all subsequent comparative experiments, the deletion method was used for all experiments in this paper.

To quantitatively evaluate the diagnostic performance of the proposed QSAO optimized BP model on Dataset 1, we compare it against nine state of the art optimizer based BP models: Hiking Optimization Algorithm (HOA) [48], Hippopotamus optimization algorithm (HO) [49], Puma optimizer (PO) [50], Brain Storm Optimization (BSO) [51], Artificial Bee Colony (ABC) [52], Social Spider Optimization (SSO) [53], whale optimization algorithm (WOA) [54], GWO [55], and SAO, each combined with a BP neural network, as well as four classical ML algorithms: Decision Tree (DT), RF, BP, and Extreme Gradient Boosting (XGBoost) [56].

Ten widely adopted metrics are reported, namely accuracy, AUC, precision, recall, F1-score, Matthews correlation coefficient (MCC), specificity, negative predictive value (NPV), false positive rate (FPR), and false negative rate (FNR). Table 5 summarizes the mean and standard deviation (std) of each metric over ten independent runs. To further validate the statistical significance of the observed improvements, we conduct the Wilcoxon signed rank test between QSAO-BP and each competing model based on the ten independent runs. In Tables 5 and 6, the symbol * indicates that the corresponding model achieves a significantly lower value than QSAO-BP for that specific metric at the significance level of p < 0.05.

Table 5.

Comparison of 14 classifiers in 10 evaluation metrics for Dataset 1 diagnosis

	Accuracy	AUC	Precision	Recall	F1-score	MCC	Specificity	NPV	FPR	FNR
HOA-BP	0.807 ± 0.051*	0.876 ± 0.053	0.811 ± 0.068	0.759 ± 0.073	0.782 ± 0.056	0.612 ± 0.103	0.847 ± 0.063	0.808 ± 0.052	0.153 ± 0.063	0.241 ± 0.073
HO-BP	0.814 ± 0.042*	0.866 ± 0.047	0.834 ± 0.065	0.744 ± 0.054	0.785 ± 0.047	0.626 ± 0.086	0.872 ± 0.056	0.802 ± 0.036	0.128 ± 0.056	0.256 ± 0.054
PO-BP	0.800 ± 0.048*	0.869 ± 0.047	0.797 ± 0.054	0.759 ± 0.094	0.775 ± 0.059	0.599 ± 0.098	0.834 ± 0.051	0.808 ± 0.064	0.166 ± 0.051	0.241 ± 0.094
BSO-BP	0.822 ± 0.027*	0.886 ± 0.039	0.841 ± 0.065	0.763 ± 0.063	0.796 ± 0.031	0.645 ± 0.057	0.872 ± 0.062	0.816 ± 0.036	0.128 ± 0.062	0.237 ± 0.063
ABC-BP	0.788 ± 0.054*	0.846 ± 0.044	0.777 ± 0.084	0.767 ± 0.087	0.768 ± 0.059	0.579 ± 0.111	0.806 ± 0.086	0.808 ± 0.062	0.194 ± 0.086	0.233 ± 0.087
SSO-BP	0.820 ± 0.044*	0.882 ± 0.042	0.832 ± 0.066	0.767 ± 0.072	0.796 ± 0.053	0.640 ± 0.089	0.866 ± 0.063	0.817 ± 0.044	0.134 ± 0.063	0.233 ± 0.072
WOA-BP	0.803 ± 0.026*	0.874 ± 0.039	0.792 ± 0.038	0.778 ± 0.065	0.783 ± 0.034	0.606 ± 0.051	0.825 ± 0.045	0.817 ± 0.038	0.175 ± 0.045	0.222 ± 0.065
GWO-BP	0.825 ± 0.057*	0.890 ± 0.055	0.832 ± 0.073	0.781 ± 0.091	0.803 ± 0.063	0.652 ± 0.117	0.863 ± 0.069	0.828 ± 0.070	0.138 ± 0.069	0.219 ± 0.091
SAO-BP	0.807 ± 0.059*	0.880 ± 0.060	0.806 ± 0.086	0.767 ± 0.061	0.785 ± 0.065	0.612 ± 0.122	0.841 ± 0.073	0.810 ± 0.048	0.159 ± 0.073	0.233 ± 0.061
DT	0.776 ± 0.048*	0.812 ± 0.048	0.791 ± 0.050	0.803 ± 0.068	0.795 ± 0.044	0.552 ± 0.096	0.744 ± 0.083	0.765 ± 0.062	0.256 ± 0.083	0.197 ± 0.068
RF	0.827 ± 0.035*	0.920 ± 0.028	0.827 ± 0.048	0.866 ± 0.053	0.844 ± 0.031	0.655 ± 0.071	0.781 ± 0.075	0.835 ± 0.056	0.219 ± 0.075	0.134 ± 0.053
BP	0.807 ± 0.041*	0.876 ± 0.032	0.810 ± 0.061	0.853 ± 0.053	0.828 ± 0.029	0.616 ± 0.075	0.752 ± 0.121	0.818 ± 0.051	0.248 ± 0.121	0.147 ± 0.053
XGBoost	0.836 ± 0.031*	0.917 ± 0.031	0.833 ± 0.041	0.875 ± 0.044	0.852 ± 0.028	0.671 ± 0.062	0.789 ± 0.065	0.845 ± 0.043	0.211 ± 0.065	0.125 ± 0.044
QSAO-BP	0.873 ± 0.035	0.921 ± 0.030	0.862 ± 0.048	0.916 ± 0.059	0.886 ± 0.032	0.748 ± 0.073	0.822 ± 0.069	0.897 ± 0.064	0.178 ± 0.069	0.084 ± 0.059

Table 6.

Comparison of 14 classifiers in 10 evaluation metrics for Dataset 2 diagnosis

	Accuracy	AUC	Precision	Recall	F1-score	MCC	Specificity	NPV	FPR	FNR
HOA-BP	0.781 ± 0.056*	0.820 ± 0.039	0.690 ± 0.121	0.611 ± 0.112	0.642 ± 0.089	0.493 ± 0.132	0.863 ± 0.068	0.825 ± 0.045	0.138 ± 0.068	0.389 ± 0.112
HO-BP	0.753 ± 0.030*	0.768 ± 0.059	0.649 ± 0.067	0.537 ± 0.153	0.573 ± 0.092	0.417 ± 0.086	0.855 ± 0.062	0.800 ± 0.047	0.145 ± 0.062	0.463 ± 0.153
PO-BP	0.769 ± 0.057*	0.801 ± 0.075	0.665 ± 0.114	0.637 ± 0.104	0.641 ± 0.068	0.481 ± 0.112	0.833 ± 0.094	0.830 ± 0.040	0.168 ± 0.094	0.363 ± 0.104
BSO-BP	0.780 ± 0.050*	0.821 ± 0.067	0.713 ± 0.126	0.553 ± 0.194	0.601 ± 0.147	0.477 ± 0.145	0.888 ± 0.069	0.813 ± 0.060	0.113 ± 0.069	0.447 ± 0.194
ABC-BP	0.742 ± 0.050*	0.765 ± 0.053	0.631 ± 0.115	0.479 ± 0.141	0.537 ± 0.120	0.377 ± 0.140	0.868 ± 0.053	0.781 ± 0.046	0.133 ± 0.053	0.521 ± 0.141
SSO-BP	0.758 ± 0.062*	0.798 ± 0.074	0.672 ± 0.115	0.521 ± 0.143	0.574 ± 0.124	0.425 ± 0.143	0.870 ± 0.082	0.795 ± 0.046	0.130 ± 0.082	0.479 ± 0.143
WOA-BP	0.756 ± 0.054*	0.784 ± 0.069	0.679 ± 0.117	0.489 ± 0.111	0.560 ± 0.098	0.414 ± 0.126	0.883 ± 0.065	0.785 ± 0.038	0.118 ± 0.065	0.511 ± 0.111
GWO-BP	0.802 ± 0.044*	0.813 ± 0.060	0.771 ± 0.138	0.579 ± 0.089	0.652 ± 0.072	0.535 ± 0.108	0.908 ± 0.067	0.820 ± 0.029	0.093 ± 0.067	0.421 ± 0.089
SAO-BP	0.781 ± 0.035*	0.798 ± 0.049	0.719 ± 0.100	0.553 ± 0.067	0.619 ± 0.050	0.482 ± 0.081	0.890 ± 0.056	0.808 ± 0.022	0.110 ± 0.056	0.447 ± 0.067
DT	0.785 ± 0.045*	0.770 ± 0.053	0.829 ± 0.032	0.863 ± 0.070	0.844 ± 0.037	0.503 ± 0.102	0.621 ± 0.095	0.698 ± 0.112	0.379 ± 0.095	0.138 ± 0.070
RF	0.824 ± 0.030	0.894 ± 0.026	0.853 ± 0.026	0.895 ± 0.044	0.873 ± 0.023	0.590 ± 0.069	0.674 ± 0.069	0.760 ± 0.074	0.326 ± 0.069	0.105 ± 0.044
BP	0.761 ± 0.035*	0.781 ± 0.046	0.810 ± 0.022	0.848 ± 0.057	0.827 ± 0.030	0.444 ± 0.075	0.579 ± 0.070	0.653 ± 0.081	0.421 ± 0.070	0.153 ± 0.057
XGBoost	0.824 ± 0.031	0.897 ± 0.025	0.851 ± 0.023	0.898 ± 0.043	0.873 ± 0.024	0.589 ± 0.070	0.668 ± 0.061	0.763 ± 0.075	0.332 ± 0.061	0.103 ± 0.043
QSAO-BP	0.854 ± 0.035	0.863 ± 0.039	0.875 ± 0.033	0.918 ± 0.046	0.895 ± 0.026	0.663 ± 0.082	0.721 ± 0.082	0.814 ± 0.081	0.279 ± 0.082	0.083 ± 0.046

As shown in Table 6, the QSAO-BP outperforms other comparison methods in most indicators. The QSAO-BP achieved the highest accuracy, AUC, precision, recall, F1-score, MCC, and NPV of 0.873 ± 0.035, 0.921 ± 0.030, 0.862 ± 0.048, 0.916 ± 0.059, 0.886 ± 0.032, 0.748 ± 0.073, and 0.897 ± 0.064, respectively. The QSAO-BP achieved the lowest FNR of 0.084 ± 0.059. Meanwhile, the specificity and FPR are 0.822 ± 0.069 and 0.178 ± 0.069, respectively, indicating good competitiveness. Compared with SAO-BP and BP, the accuracy of QSAO-BP has increased by 8.178%, indicating that QSAO can solve the premature convergence and easy to fall into local optima of SAO in CVD diagnosis, and through QSAO, it can improve the accuracy and overfitting problems of BP neural network in CVD diagnosis, effectively enhancing its predictive ability. The Wilcoxon signed rank test further confirms that the superiority of QSAO-BP over all competing classifiers is statistically significant. In addition, the QSAO-BP has a relatively small std compared to other comparative algorithms in the vast majority of cases, verifying its good stability and robustness.

To provide a more intuitive comparison among the 14 classifiers, Fig. 3 visualises the bar charts of the ten evaluation metrics for all 14 models on Dataset 1. This graphical representation highlights the relative strengths and weaknesses of each model, particularly emphasising the multi-dimensional superiority of QSAO-BP.

Fig. 3.

Comparison of CVD diagnosis by 10 classifiers under Dataset 1

We employ SHapley Additive exPlanations (SHAP), to analyse the contribution of each input feature to the QSAO-BP’s predictions on Dataset 1. Figure 4 presents the feature importance bar plot, Fig. 4a, and the SHAP beeswarm summary plot, Fig. 4b.

Fig. 4.

Feature contribution analysis based on SHAP under Dataset 1

Figure 4 presents SHAP interpretative results for CVD diagnosis. The beeswarm plot reveals clear correlation differences between features and CVD diagnosis. Features including sex, trestbps, chol,, restecg, exang, oldpeak, slope, ca, and thal are positively correlated with CVD diagnosis, while thalach shows negative correlations with CVD diagnosis.The feature patterns captured by QSAO-BP align with the inherent relationships reflected in the clinical dataset, verifying the satisfactory interpretability of the proposed model.

To test the performance of the QSAO-BP in assessing the mortality risk from CVD, we evaluate it on the heart failure clinical records dataset, Dataset 2, where the task is to predict in mortality risk among confirmed heart failure patients. Table 6 reports the ten evaluation metrics for all 14 classifiers on this more challenging prognostic task.

As shown in Table 6, the QSAO-BP outperforms other comparison algorithms in most indicators. The QSAO-BP achieved the highest accuracy, precision, recall, F1-score, and MCC of 0.854 ± 0.035, 0.875 ± 0.033, 0.918 ± 0.046, 0.895 ± 0.026, and 0.663 ± 0.082, respectively, while the FNR achieved the lowest of 0.083 ± 0.046. Meanwhile, AUC, specificity, NPV, and FPR are 0.863 ± 0.039, 0.721 ± 0.082, 0.814 ± 0.081, and 0.279 ± 0.082, respectively, indicating good competitiveness. Compared with SAO-BP and BP, the accuracy of QSAO-BP has increased by 9.347% and 12.221% respectively, indicating that QSAO can solve the premature convergence and easily fall into local optima in the mortality risk assessment of SAO in CVD, and through QSAO, it can improve the accuracy and overfitting problems of BP neural network in the mortality risk assessment of CVD, effectively enhancing its predictive ability. Wilcoxon signed-rank tests show QSAO-BP significantly outperforms most competitors on Dataset 2, with only RF and XGBoost showing no statistical difference. The strong performance of ensemble tree models narrows the performance gap, yet QSAO-BP still holds clear overall advantages. In addition, the QSAO-BP algorithm has a relatively small std compared to other comparative algorithms in the vast majority of cases, verifying its good stability and robustness.

The bar chart comparison for Dataset 2 is depicted in Fig. 5, offering a visual summary of the relative performance of all classifiers in the mortality risk assessment task.

Fig. 5.

Comparison of CVD diagnosis by 10 classifiers under Dataset 2

We employ SHAP to analyse the contribution of each input feature to the QSAO-BP’s predictions on Dataset 2. Figure 6 presents the feature importance bar plot, Fig. 6a, and the SHAP beeswarm summary plot, Fig. 6b.

Fig. 6.

Feature contribution analysis based on SHAP under Dataset 2

Figure 6 presents SHAP interpretative results for CVD mortality risk prediction. The beeswarm plot reveals clear correlation differences between features and mortality risk. Features including smoking, serum_creatinine, high_blood_pressure, diabetes, anaemia, and age are positively correlated with CVD mortality, while ejection_fraction and platelets show negative correlations with CVD mortality.The feature patterns captured by QSAO-BP align with the inherent relationships reflected in the clinical dataset, verifying the satisfactory interpretability of the proposed model.

Conclusion and future works

This study proposed QSAO‑BP that integrates a QSAO with a BP neural network, to simultaneously perform CVD diagnosis and mortality risk assessment. The Quartile‑based strategy, which introduces Gaussian and Lévy random walks via population fitness, effectively balances global exploration and local exploitation. By optimizing the hyperparameters of the BP neural network, QSAO‑BP significantly improves prediction accuracy and generalization of BP neural network, alleviating the overfitting problems in medical prediction tasks. A dual‑validation framework was established on the Cleveland Heart Disease Dataset and the Heart Failure Clinical Records Dataset. Extensive experiments demonstrated that QSAO‑BP consistently outperformed nine optimizer‑based BP variants and four classical machine learning algorithms, achieving the highest accuracy, precision, recall, F1‑score, and MCC, along with the lowest FNR, on both tasks. The improvements over SAO‑BP in accuracy reached 8.178% and 9.347%, respectively, and were statistically significant according to the Wilcoxon signed‑rank test. SHAP analysis further verified that the key predictors identified by the model are consistent with established clinical knowledge, endorsing the interpretability and clinical plausibility of QSAO‑BP.

Based on the valuable findings of this study, we have planned the following clear and progressive directions for future research work: (1) QSAO-BP will be compared with more ML baselines and deep learning baselines, such as LightGBM and CatBoost, to further validate its superiority in CVD diagnosis and mortality risk assessment tasks. (2) Collect a larger, independent, multi-center CVD related clinical dataset to further validate the generalization ability and stability of QSAO-BP in CVD diagnosis and mortality risk assessment tasks. (3) Systematic ablation experiments will be carried out to quantify the specific performance gain brought by the quartile mechanism compared with the pure random walk strategy.

Author contributions

C.M.M. and Y.Y.N. conceived the research framework, screened, sorted and organized the public experimental datasets used in this study. C.M.M. and X.Y.G. proposed the improved QSAO optimization algorithm, constructed the prediction model, and completed all simulation experiments and result analysis. W.Y.T. acquired research funding and project support, organized the full-text structure, and refined academic wording. C.M.M. and X.Y.G. jointly drafted the initial manuscript. W.Y.T., X.Y.G. and C.M.M. repeatedly revised and optimized the manuscript throughout the revision process. All authors have read, reviewed, and approved the final revised version for submission.

Funding

The work is partially supported by the Zhejiang Provincial Medical and Health Science and Technology Plan Project (Grant No. 2023KY901) and the project of the Wenzhou Science and Technology Bureau (No.Y20220571).

Data availability

Data availability: at link (1) https://archive.ics.uci.edu/dataset/45/heart+disease. (2) https://archive.ics.uci.edu/dataset/519/heart+failure+clinical+records.

Code availability

Code is available on request.

Declarations

Ethics approval and consent to participate

Not applicable. The data leveraged in the study were obtained from open‑source databases, no additional approval is required.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.