  Motivated by models of cancer formation in which cells need to acquire $k$
mutations to become cancerous, we consider a spatial population model in which
the population is represented by the $d$-dimensional torus of side length $L$.
Initially, no sites have mutations, but sites with $i-1$ mutations acquire an
$i$th mutation at rate $\mu_i$ per unit area. Mutations spread to neighboring
sites at rate $\alpha$, so that $t$ time units after a mutation, the region of
individuals that have acquired the mutation will be a ball of radius $\alpha
t$. We calculate, for some ranges of the parameter values, the asymptotic
distribution of the time required for some individual to acquire $k$ mutations.
Our results, which build on previous work of Durrett, Foo, and Leder, are
essentially complete when $k = 2$ and when $\mu_i = \mu$ for all $i$.
