Coupled SDE-ODE Modeling of Tumor-Immune Dynamics to Infer Biomarker Release.

Tumor-immune interactions are central to cancer progression and treatment response, driving cell death through immune-mediated killing and resource-limited competition. In early-stage disease or following effective treatment, cancer populations are often small and difficult to observe directly. Disease monitoring therefore relies on the detection of biomarkers such as circulating tumor DNA (ctDNA) as noisy proxies to cancer size. Existing approaches lack robust frameworks to infer tumor burden from these signals when populations approach detection thresholds. We present a coupled deterministic-stochastic framework that links tumor-immune dynamics to biomarker release. A two prey, one predator Lotka-Volterra model captures interactions between immune cells and competing tumor subpopulations under shared resource constraints. Biomarker production is modeled using stochastic differential equations driven by tumor cell death arising from immune-mediated apoptosis and intratumor competition-induced necrosis. We incorporate both square-root (CIR-type) noise, representing count-limited fluctuations near the detection limit, and multiplicative (geometric-type) noise, representing proportional variability at higher concentrations. Within this framework, we derive analytical expressions for biomarker trajectories and first-passage statistics, including mean detection times under minimal detection thresholds. Our results demonstrate how tumor heterogeneity, immune pressure, and stochastic variance structure jointly shape biomarker emergence, persistence, and detectability. Together, this work provides a mechanistically interpretable and mathematically tractable foundation for inferring tumor burden from noisy biomarker signals across clinically relevant concentration regimes.