  Despite initial success, cancer therapies often fail due to the emergence of
drug-resistant cells. In this study, we use a mathematical model to investigate
how cancer evolves over time, specifically focusing on the state of the tumor
when it recurs after treatment. We use a two-type branching process to capture
the dynamics of both drug-sensitive and drug-resistant cells. We analyze the
clonal diversity of drug-resistant cells at the time of cancer recurrence,
which is defined as the first time the population size of drug-resistant cells
exceeds a specified proportion of the initial population size of drug-sensitive
cells. We examine two clonal diversity indices: the number of clones and the
Simpson's Index. We calculate the expected values of these indices and utilize
them to develop statistical methods for estimating model parameters.
Additionally, we examine these two indices conditioned on early recurrence in
the special case of a deterministically decaying sensitive population, with the
aim of addressing the question of whether early recurrence is driven by a
single mutation that generates an unusually large family of drug-resistant
cells (corresponding to a low clonal diversity), or if it is due to the
presence of an unusually large number of mutations causing drug resistance
(corresponding to a high clonal diversity). Our findings, based on both
indices, support the latter possibility. Furthermore, we demonstrate that the
time of cancer recurrence can serve as a valuable indicator of clonal
diversity, offering new insights for the treatment of recurrent cancers.
