  In this work we explore the temporal dynamics of spatial heterogeneity during
the process of tumorigenesis from healthy tissue. We utilize a spatial
stochastic process model of mutation accumulation and clonal expansion in a
structured tissue to describe this process. Under a two-step tumorigenesis
model, we first derive estimates of a non-spatial measure of diversity:
Simpson's Index, which is the probability that two individuals sampled at
random from the population are identical, in the premalignant population. We
next analyze two new measures of spatial population heterogeneity. In
particular we study the typical length scale of genetic heterogeneity during
the carcinogenesis process and estimate the extent of a surrounding
premalignant clone given a clinical observation of a premalignant point biopsy.
This evolutionary framework contributes to a growing literature focused on
developing a better understanding of the spatial population dynamics of cancer
initiation and progression. Although initially motivated by understanding
questions in cancer, these results can be applied more generally to help
understand the dynamics of heterogeneity and diversity in a variety of
spatially structured, evolving populations.
