Quantifying metastatic inefficiency: rare genotypes versus rare dynamics

We introduce and solve a 'null model' of stochastic metastatic colonization. The model is described by a single parameter θ: the ratio of the rate of cell division to the rate of cell death for a disseminated tumour cell in a given secondary tissue environment. We are primarily interested in the case in which colonizing cells are poorly adapted for proliferation in the local tissue environment, so that cell death is more likely than cell division, i.e. θ < 1. We quantify the rare event statistics for the successful establishment of a metastatic colony of size N. For N >> 1, we find that the probability of establishment is exponentially rare, as expected, and yet the mean time for such rare events is of the form ~log (N)/(1 - θ) while the standard deviation of colonization times is ~1/(1 - θ). Thus, counter to naive expectation, for θ < 1, the average time for establishment of successful metastatic colonies decreases with decreasing cell fitness, and colonies seeded from lower fitness cells show less stochastic variation in their growth. These results indicate that metastatic growth from poorly adapted cells is rare, exponentially explosive and essentially deterministic. These statements are brought into sharper focus by the finding that the temporal statistics of the early stages of metastatic colonization from low-fitness cells (θ < 1) are statistically indistinguishable from those initiated from high-fitness cells (θ > 1), i.e. the statistics show a duality mapping (1 - θ) --> (θ - 1). We conclude our analysis with a study of heterogeneity in the fitness of colonising cells, and describe a phase diagram delineating parameter regions in which metastatic colonization is dominated either by low or high fitness cells, showing that both are plausible given our current knowledge of physiological conditions in human cancer.