## 3 Results and Discussion
Thus, with the numerical implementation of the model using the Runge-Kutta, Adams-Bashforth, and backward Euler methods, we observed a high degree of consistency between the approximate and exact solutions. This agreement confirmed both the mathematical validity and computational stability of the proposed differential equation system. Building upon this foundation of numerical reliability, the second phase of the study focused on enhancing predictive capabilities through machine learning. While the initial  
Figure 8: Numerical interpretation of the change in cardiovascular aging by the fourth-order Runge-Kutta method. stage of the study was focused on numerical modeling of cardiovascular risk factors, we further conducted an external validation using a Random Forest classifier to test the generalizability of the model. The model, trained on the 65-74 cohort, was applied to independent subgroups (75-89 and 90+ years) using the same features. Results showed that the model maintained high predictive accuracy without recalibration, achieving AUC = 0.989 in the 75-89 group and 1.000 in the 90+ group, indicating strong extrapolation across age categories. The validation results demonstrated high predictive performance across key evaluation metrics (Table 3). The model achieved an overall accuracy of 98.8% (see Table 4), indicating a high proportion of correct predictions across both classes. The F1-scores of 0.98 for class 0 (absence of condition) and 0.99 for class 1 (presence of condition) reflected strong precision-recall balance and robustness across class distributions. Furthermore, the area under the ROC curve (AUC) was 0.989, demonstrating excellent discrimination between patients with and without the target condition (e.g., ischemic heart disease). These results confirmed that the model retained its predictive capability when applied to an independent population of older individuals. These results support the conclusion that the developed ODE-based mathematical model, when solved using standard numerical methods and trained on representative clinical data, can accurately predict cardiovascular aging trajectories across elderly subpopulations. The consistency of performance across age groups reinforces the model's potential for application in real-world preventive cardiology and geriatric risk stratification. Having validated the model using numerical methods and supervised machine learning approaches, we then proceeded to explore the potential of unsupervised learning for pattern discovery in the data. To complement the supervised analysis, a clustering model based on the k-means algorithm (k-means) was developed and tested for automatic segmentation of clinical data from patients with suspected cardiovascular disease (CVD). The main objective was to test the hypothesis that without predefined labels, the model based on unsupervised learning could categorize patients into groups with different cardiovascular status--with and without CVD. Despite the presence of classes in the original medical records, they were not included in the model, as the task was specifically to independently identify the structure of the data using the k-means algorithm. To ensure robustness of the results and clarity of visualization, the data were first normalized using standardization and then transformed into a two-dimensional space using the principal component analysis (PCA) method. After applying the principal component method (PCA), two-dimensional projections of the training data were visualized on a plane, where color differentiation marked the membership of objects to the corresponding clusters identified by the k-means algorithm. The coordinates of the centroids of the clusters were also marked on the plot, which \begin{table}
\begin{tabular}{l l} \hline \hline
**Metric** & **Value** \\ \hline Accuracy & 98.8\% \\ \hline F1-score (class 0) & 0.98 \\ \hline F1-score (class 1) & 0.99 \\ \hline ROC AUC & 0.989 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: External validation results.