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Originally developed to elucidate the mechanisms of natural selection inbacteria, the Luria-Delbr\"uck model assumed that cells are intrinsicallycapable of dividing an unlimited number of times. This assumption however, isnot true for human somatic cells which undergo replicative senescence.Replicative senescence is thought to act as a mechanism to protect againstcancer 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 accountcellular replication limits in the wild type cell population and models theemergence of mutants that escape replicative senescence. We present results onthe mean, variance, distribution, and asymptotic behavior of the mutantpopulation in terms of three classical formulations of the problem. Morebroadly the paper introduces the concept of incorporating replicative limits aspart of the Luria-Delbr\"uck mutational framework. Guidelines to extend thetheory to include other types of mutations and possible applications to themodeling of telomere crisis and fluctuation analysis are also discussed.

Cellular replication limits in the Luria-Delbr\"uck mutation model