Abstract
As cancer advances, cells often spread from the primary tumor to other parts of the body and form metastases. This is the main cause of cancer related mortality. Here we investigate a conceptually simple model of metastasis formation where metastatic lesions are initiated at a rate which depends on the size of the primary tumor. The evolution of each metastasis is described as an independent branching process. We assume that the primary tumor is resected at a given size and study the earliest time at which any metastasis reaches a minimal detectable size. The parameters of our model are estimated independently for breast, colorectal, headneck, lung and prostate cancers. We use these estimates to compare predictions from our model with values reported in clinical literature. For some cancer types, we find a remarkably wide range of resection sizes such that metastases are very likely to be present, but none of them are detectable. Our model predicts that only very early resections can prevent recurrence, and that small delays in the time of surgery can significantly increase the recurrence probability.
Highlights
As cancer advances, cells often spread from the primary tumor to other parts of the body and form metastases
In this paper we model the size of the primary tumor as a deterministic function, while allow the seeded metastases to evolve stochastically according to branching processes
We introduced a model of metastasis formation where metastases are initiated at a time dependent rate, in the simplest case proportional to the size of a growing primary tumor
Summary
Cells often spread from the primary tumor to other parts of the body and form metastases. Many generalizations of the Luria-Delbruck model [9] have been employed to study specific traits of tumor evolution, such as the development of drug resistance [10, 11, 12, 13], the role of driver mutations [14, 15] and metastasis formation [5, 6, 16, 17] Another line of research focused on temporal features, after the first stochastic model for the time to tumor onset was proposed by Armitage and Doll in their pioneering work on carcinogenesis [18]. A few decades later authors began to investigate stochastic models of tumor latency time These works led to mathematical descriptions of optimal schedules of cancer surveillance [19, 20], cure rates [21] and cancer recurrence [22]. In this paper we model the size of the primary tumor as a deterministic function (focusing on exponential and logistic growth as examples), while allow the seeded metastases to evolve stochastically according to branching processes
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