Mitochondrial DNA deletions accumulate over the life course in post-mitotic cells of many species and may contribute to aging. Often a single mutant expands clonally and finally replaces the wild-type population of a whole cell. One proposal to explain the driving force behind this accumulation states that random drift alone, without any selection advantage, is sufficient to explain the clonal accumulation of a single mutant. Existing mathematical models show that such a process might indeed work for humans. However, to be a general explanation for the clonal accumulation of mtDNA mutants, it is important to know whether random drift could also explain the accumulation process in short-lived species like rodents. To clarify this issue, we modelled this process mathematically and performed extensive computer simulations to study how different mutation rates affect accumulation time and the resulting degree of heteroplasmy. We show that random drift works for lifespans of around 100 years, but for short-lived animals, the resulting degree of heteroplasmy is incompatible with experimental observations.