Mathematical and Data-Driven Analysis of Atypical Cancer Antigen 15-3 Patterns in Male Metastatic Breast Cancer

Cancer Antigen 15-3 (CA 15-3) is a glycoprotein used to monitor breast cancer progression and treatment response. This study analyzes five years of CA 15-3 measurements from a male patient with metastatic breast cancer using an integrated mathematical and data-driven approach. A phenomenological model combining an oscillatory component, capturing sinusoidal fluctuations, with an exponential term representing clearance, treatment response, or tumor growth, was fitted to the longitudinal data. Key parameters-including baseline level, rates of increase and decay, oscillation frequency, and amplitude-were estimated, providing a structured characterization of the biomarker's temporal dynamics. In parallel, data-driven analysis using Variational Mode Decomposition (VMD) was employed to isolate intrinsic frequency components and detect latent transitions represented in the Hilbert Spectrum (HS) that were not captured by the parametric model. Results revealed a stable sinusoidal pattern with a 170-day period persisting for three cycles-an atypical finding, as CA 15-3 variations typically reflect disease progression or treatment effects rather than a predictable oscillatory rhythm. Oscillations emerged with the start of metastatic-phase therapy, with the sinusoidal pattern linked to Palbociclib, though the underlying biological mechanism remains unclear and warrants further investigation. The model accurately reproduced these periodic fluctuations and overall trends, showing strong agreement with measured values across five years. Complementary VMD/HS analysis identified instantaneous frequency components linked to metastatic onset and therapy intervals, enhancing the temporal resolution of structural changes. Together, the combined modeling approach improves the detection and interpretation of dynamic biomarker behavior and supports more personalized strategies for cancer monitoring.