  Mutation-induced drug resistance in cancer often causes the failure of
therapies and cancer recurrence, despite an initial tumor reduction. The timing
of such cancer recurrence is governed by a balance between several factors such
as initial tumor size, mutation rates and growth kinetics of drug-sensitive and
resistance cells. To study this phenomenon we characterize the dynamics of
escape from extinction of a subcritical branching process, where the
establishment of a clone of escape mutants can lead to total population growth
after the initial decline. We derive uniform in-time approximations for the
paths of the escape process and its components, in the limit as the initial
population size tends to infinity and the mutation rate tends to zero. In
addition, two stochastic times important in cancer recurrence will be
characterized: (i) the time at which the total population size first begins to
rebound (i.e., become supercritical) during treatment, and (ii) the first time
at which the resistant cell population begins to dominate the tumor.
