Cancer recurrence times and early detection from branching process models
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Date
28/11/2019Author
Avanzini, Stefano
Metadata
Abstract
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.