Shrestha Pujan, Fan Yijia, George Jason T
bioRxiv. 2025 Aug 21:2025.08.15.670571. doi: 10.1101/2025.08.15.670571.
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.
肿瘤-免疫相互作用是癌症进展和治疗反应的核心，通过免疫介导的杀伤和资源有限的竞争驱动细胞死亡。在疾病早期或有效治疗后，癌症群体通常较小且难以直接观察到。因此，疾病监测依赖于检测生物标志物，如循环肿瘤DNA（ctDNA），作为癌症大小的嘈杂替代指标。然而，当群体接近检测阈值时，现有方法缺乏从这些信号推断肿瘤负荷的稳健框架。为了解决这个问题，我们提出了一个耦合的确定性-随机框架，将肿瘤-免疫动力学与生物标志物释放联系起来。一个双猎物、单捕食者的Lotka-Volterra模型捕捉了在共享资源限制下免疫细胞与竞争肿瘤亚群之间的相互作用。生物标志物的产生使用线性随机微分方程进行建模，纳入了凋亡死亡（免疫介导）和坏死死亡（由于肿瘤内竞争）释放的ctDNA。我们推导了所得生物标志物轨迹的解析解，包括在最小检测阈值下的平均检测时间。这些结果表明，测量的生物标志物信号的波动性、疾病异质性和免疫压力如何共同塑造信号的出现和持续。最后，为了模拟观察者可以获得未来信息的情况——例如回顾性研究中的终端生物标志物信号或采样时间——我们采用了预期理论视角。使用预期随机微积分，我们推导了所得预期随机微分方程的路径解，捕捉了未来观察如何影响推断的生物标志物动态。这种方法将潜在的肿瘤-免疫相互作用的动态与相应的可检测生物标志物水平联系起来，对早期检测、免疫监测和疾病进展的回顾性重建具有重要意义。
