  Tumor recurrence, driven by the evolution of drug resistance is a major
barrier to therapeutic success in cancer. Resistance is often caused by genetic
alterations such as point mutation, which refers to the modification of a
single genomic base pair, or gene amplification, which refers to the
duplication of a region of DNA that contains a gene. Here we investigate the
dependence of tumor recurrence dynamics on these mechanisms of resistance,
using stochastic multi-type branching process models. We derive tumor
extinction probabilities and deterministic estimates for the tumor recurrence
time, defined as the time when an initially drug sensitive tumor surpasses its
original size after developing resistance. For models of amplification-driven
and mutation-driven resistance, we prove law of large numbers results regarding
the convergence of the stochastic recurrence times to their mean. Additionally,
we prove sufficient and necessary conditions for a tumor to escape extinction
under the gene amplification model, discuss behavior under biologically
relevant parameters, and compare the recurrence time and tumor composition in
the mutation and amplification models both analytically and using simulations.
In comparing these mechanisms, we find that the ratio between recurrence times
driven by amplification vs. mutation depends linearly on the number of
amplification events required to acquire the same degree of resistance as a
mutation event, and we find that the relative frequency of amplification and
mutation events plays a key role in determining the mechanism under which
recurrence is more rapid. In the amplification-driven resistance model, we also
observe that increasing drug concentration leads to a stronger initial
reduction in tumor burden, but that the eventual recurrent tumor population is
less heterogeneous, more aggressive, and harbors higher levels of
drug-resistance.
