The expansion of mitochondrial DNA molecules with deletions has been associated with aging, particularly in skeletal muscle fibers; its mechanism has remained unclear for three decades. Previous accounts have assigned a replicative advantage (RA) to mitochondrial DNA containing deletion mutations, but there is also evidence that cells can selectively remove defective mitochondrial DNA. Here we present a spatial model that, without an RA, but instead through a combination of enhanced density for mutants and noise, produces a wave of expanding mutations with speeds consistent with experimental data. A standard model based on RA yields waves that are too fast. We provide a formula that predicts that wave speed drops with copy number, consonant with experimental data. Crucially, our model yields traveling waves of mutants even if mutants are preferentially eliminated. Additionally, we predict that mutant loads observed in single-cell experiments can be produced by de novo mutation rates that are drastically lower than previously thought for neutral models. Given this exemplar of how spatial structure (multiple linked mtDNA populations), noise, and density affect muscle cell aging, we introduce the mechanism of stochastic survival of the densest (SSD), an alternative to RA, that may underpin other evolutionary phenomena.