Mathematical Analysis of Cancer Recurrence

Abstract

A significant challenge in cancer treatment is that cancer cells can develop resistance to therapy. We consider a situation where an effective therapy leads to a substantial decline in tumor burden in the early stage of the treatment. However, cancer cells develop drug resistance during the treatment through mutation and eventually lead to cancer recurrence. In this dissertation, we present a two-type branching process model of this phenomenon. Based on this model, we can obtain a good approximation to cancer recurrence time. We then analyze the event of early recurrence such that cancer recurs some units of time earlier than the approximated time. We show that early recurrence is a rare event and then obtain the corresponding large deviations rate. Through the expression of the large deviations rate, we can form a deeper understanding of the dynamics of cancer recurrence. An important consideration in the treatment of recurrent cancers is the clonal diversity present, i.e., how many genetically distinct populations are present at the time of recurrence. In our model, each drug-resistant cell mutated from drug-sensitive cells initializes a clone of mutants. We investigate two indices related to clonal diversity: the number of clones and the Simpson’s Index at cancer recurrence. We obtain the expectation of these two indices with and without conditioning on the event of early recurrence. Our results connect clonal diversity with cancer recurrence time and could bring helpful insights into the treatment of recurrent cancers. In this dissertation, we focus on cancer recurrence. However, our results can be applied to various settings such as pest control, the treatment of parasitic infection, and the treatment of many other diseases caused by viruses or bacteria.