Incorporating Lesion-Level Heterogeneity in a Branching-Process Model of Ovarian Cancer Metastasis

Source abstract

Abstract Danesh et al. ( J. Theor. Biol. , 2012) estimated the ovarian cancer screening window at roughly 2.9 years, using a multi-type branching process and the hitting time of an aggregate mass threshold. We extend the model of Danesh et al. to incorporate the lesion-level heterogeneity characteristic of intraperitoneal metastasis, replacing the homogeneous secondary compartment by a vector of distinct lesions whose formation follows a Chinese Restaurant Process with concentration parameter $$\alpha$$ α . We prove three results. Because established metastases divide faster than the peritoneal source compartment, the rate of de novo lesion formation is almost surely integrable, so only finitely many lesions ever form and no Yule–Simon power law emerges. Mapping the system to a time-inhomogeneous Hoppe–Polya urn yields a martingale whose almost-sure convergence fixes relative lesion masses at values set by early stochastic events. Kurtz fluid limits give a closed-form trajectory for the largest lesion, whose hitting time of the radiological resolution threshold replaces aggregate mass as the clinical endpoint. A parameter-free criterion follows: the disease is radiologically detectable before it becomes lethal precisely when the largest lesion’s share of the total metastatic mass exceeds the ratio of the imaging resolution limit to the lethal burden. Simulation validates each result and shows the phenotype is governed jointly by $$\alpha$$ α and the peritoneal migration intensity. Calibrating against published surgical series, a fit to the reported prevalence of miliary disease reproduces the observed implant-size dichotomy out of sample and predicts a monotone decline of the screening window with lesion count.