  In cancer, treatment failure and disease recurrence have been associated with
small subpopulations of cancer cells with a stem-like phenotype. In this paper,
we develop and investigate a phenotype-structured model of solid tumour growth
in which cells are structured by a stemness level, which varies continuously
between stem-like and terminally differentiated behaviours. Cell evolution is
driven by proliferation and apoptosis, as well as advection and diffusion with
respect to the stemness structure variable. We use the model to investigate how
the environment, in particular oxygen levels, affects the tumour's population
dynamics and composition, and its response to radiotherapy. We use a
combination of numerical and analytical techniques to quantify how under
physiological oxygen levels the cells evolve to a differentiated phenotype and
under low oxygen level (i.e., hypoxia) they de-differentiate. Under normoxia,
the proportion of cancer stem cells is typically negligible and the tumour may
ultimately become extinct whereas under hypoxia cancer stem cells comprise a
dominant proportion of the tumour volume, enhancing radio-resistance and
favouring the tumour's long-term survival. We then investigate how such
phenotypic heterogeneity impacts the tumour's response to treatment with
radiotherapy under normoxia and hypoxia. Of particular interest is establishing
how the presence of radio-resistant cancer stem cells can facilitate a tumour's
regrowth following radiotherapy. We also use the model to show how
radiation-induced changes in tumour oxygen levels can give rise to complex
re-growth dynamics. For example, transient periods of hypoxia induced by damage
to tumour blood vessels may rescue the cancer cell population from extinction
and drive secondary regrowth. Further model extensions to account for spatial
variation are also discussed briefly.
