Cancer recurrence times and early detection from branching process models
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Cancer is among the leading causes of death worldwide. While primary tumors are often treated effectively, they can spawn secondary cancers called metastases which dramatically decrease chances of survival. In order to develop successful therapies, it is thus crucial to estimate the time until metastases appearance and improve our ability to detect primary tumors before metastases are generated. The estimation of the time to cancer recurrence depends on the dynamics of tumor growth and metastases seeding. For early detection, promising results have recently been obtained with liquid biopsies, id est the analysis of specific biomarker levels in blood samples. This thesis investigates these problems by studying mathematical models of cancer evolution and liquid biopsies based on the theory of branching processes. Firstly, we consider first passage times to a given size in branching birth-death processes. We derive their probability distribution and first moments conditioned on non-extinction, comparing the results obtained for supercritical, critical and subcritical processes. Such results for hitting times are presented both in exact form and in their asymptotic limit for large sizes. In this limit we show that their probability distribution asymptotically converges to extreme value types. Second, we present a semi-stochastic model of cancer recurrence. The primary tumor is described by a deterministically growing population of cells initiating metastases at a rate proportional to its size. Each metastasis is then modelled by a branching birth-death process with the same net growth rate. In this framework we discuss several features of the time to cancer relapse, defined as the first time that any metastasis reaches a given detectable size. We apply this model to different cancer types and compare its predictions with data collected from clinical literature. Third, we present a multi-type branching process model of biomarker shedding. We focus on the case of circulating tumor DNA fragments shed in the bloodstream by both cancerous and healthy cells. We model the population of tumor cells as a supercritical branching birth-death process and take the healthy cells population to be constant in size. As DNA fragments cannot reproduce or divide, their amount is described by a pure death process with immigration. By applying this model, we provide quantitative estimates for the number of circulating tumor DNA fragments detectable in a blood sample, conditioned on the primary tumor size. Comparing our estimates with clinical observations we then discuss the potential of liquid biopsies for early cancer detection.