Mathematical Analysis of the Dynamic Modeling of Heart Rate

Most existing heart rate monitoring systems rely on fixed threshold values to trigger alerts, resulting in reactive responses that often fail to capture early physiological deterioration. Such approaches neglect the temporal structure of heart rate signals and their dynamic behavior. In this work, we propose a predictive framework for heart rate monitoring based on robust temporal analysis. The heart rate signal is modeled as a timedependent function and analyzed through discrete approximations of its first and second derivatives computed via finite-difference methods. These derivatives quantify the rate of change and acceleration of heart rate dynamics, enabling the identification of emerging risk patterns. Anomalies are detected using Z-scores calibrated to an individualized baseline. To ensure robustness under noisy real-world sensing conditions, the median and the Median Absolute Deviation (MAD) are employed as statistical estimators. The scaling factor 1.4826⋅MAD is used to obtain a dispersion measure consistent with normal variability. The time-to-critical state is estimated by extrapolating the temporal evolution of the Z-score, using linear prediction in steady regimes and a quadratic formulation, solved analytically, when positive acceleration is observed. The proposed framework identifies risk trajectories that remain undetected by static threshold methods, particularly in scenarios involving rapidly accelerating heart rate. The time-to-critical estimation provides a quantitative prediction of imminent risk, enabling anticipatory monitoring rather than delayed alerting. The results demonstrate that integrating calculus-based temporal analysis with robust statistical modeling yields a transparent and mathematically consistent predictive framework for physiological monitoring. The approach also shows strong potential for extension to other biosignals and contributes to applied mathematics and statistical analysis of health-related time series.