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. However, existing approaches lack robust frameworks to infer tumor burden from these signals when populations approach detection thresholds. To address this, 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 subpopulation under shared resource constraints. Biomarker production is modeled using a linear stochastic differential equation, incorporating ctDNA release from both apoptotic death (immune-mediated) and necrotic death (due to intratumor competition). We derive analytical solutions for the resulting biomarker trajectories, including mean detection time under a minimal detection threshold. These results show how volatility in measured biomarker signal, disease heterogeneity, and immune pressure jointly shape signal emergence and persistence. Finally, to model situations in which the observer has access to future information -- such as the terminal biomarker signal or sampling time in retrospective studies -- we adopt an anticipative theoretic perspective. Using anticipative stochastic calculus, we derive path solutions to the resulting anticipating stochastic differential equation, capturing how future observations influence the inferred biomarker dynamics. This approach links the dynamics of underlying tumor-immune interactions to the corresponding detectable biomarker levels, with implications for early detection, immune monitoring, and retrospective reconstruction of disease progression.