  Originally developed to elucidate the mechanisms of natural selection in
bacteria, the Luria-Delbr\"uck model assumed that cells are intrinsically
capable of dividing an unlimited number of times. This assumption however, is
not true for human somatic cells which undergo replicative senescence.
Replicative senescence is thought to act as a mechanism to protect against
cancer and the escape from it is a rate-limiting step in cancer progression.
Here we introduce a Luria-Delbr\"uck model that explicitly takes into account
cellular replication limits in the wild type cell population and models the
emergence of mutants that escape replicative senescence. We present results on
the mean, variance, distribution, and asymptotic behavior of the mutant
population in terms of three classical formulations of the problem. More
broadly the paper introduces the concept of incorporating replicative limits as
part of the Luria-Delbr\"uck mutational framework. Guidelines to extend the
theory to include other types of mutations and possible applications to the
modeling of telomere crisis and fluctuation analysis are also discussed.
