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In cancer, treatment failure and disease recurrence have been associated withsmall subpopulations of cancer cells with a stem-like phenotype. In this paper,we develop and investigate a phenotype-structured model of solid tumour growthin which cells are structured by a stemness level, which varies continuouslybetween stem-like and terminally differentiated behaviours. Cell evolution isdriven by proliferation and apoptosis, as well as advection and diffusion withrespect to the stemness structure variable. We use the model to investigate howthe environment, in particular oxygen levels, affects the tumour's populationdynamics and composition, and its response to radiotherapy. We use acombination of numerical and analytical techniques to quantify how underphysiological oxygen levels the cells evolve to a differentiated phenotype andunder low oxygen level (i.e., hypoxia) they de-differentiate. Under normoxia,the proportion of cancer stem cells is typically negligible and the tumour mayultimately become extinct whereas under hypoxia cancer stem cells comprise adominant proportion of the tumour volume, enhancing radio-resistance andfavouring the tumour's long-term survival. We then investigate how suchphenotypic heterogeneity impacts the tumour's response to treatment withradiotherapy under normoxia and hypoxia. Of particular interest is establishinghow the presence of radio-resistant cancer stem cells can facilitate a tumour'sregrowth following radiotherapy. We also use the model to show howradiation-induced changes in tumour oxygen levels can give rise to complexre-growth dynamics. For example, transient periods of hypoxia induced by damageto tumour blood vessels may rescue the cancer cell population from extinctionand drive secondary regrowth. Further model extensions to account for spatialvariation are also discussed briefly.

Phenotypic variation modulates the growth dynamics and response to radiotherapy of solid tumours under normoxia and hypoxia