Analysis of the dynamics of tumour cells along a stemness axis under different oxygen conditions

The concept of cancer stem cells (CSCs) was first introduced to explain intra-tumour heterogeneity. According to the so called ‘CSC’ hypothesis, tumours are organised according to a rigid hierarchical structure where CSCs have the capacity to self-renew. Through asymmetric cell division, CSCs can initiate and maintain tumours that also contain differentiated cells with limited clonogenic potential. Recent studies have challenged this framework and led to the development of the so-called ‘CSC plasticity’ hypothesis. Here, stemness is viewed as a continuous rather than a discrete trait, and it may change in response to micro-environmental signals. In line with this conceptual model, we develop a mathematical model to describe the dynamics of a population of tumour cells structured by their stemness. Cells continuously transition between cancer stem cells (CSC) and terminally differentiated cancer cells. Evolution along the stemness axis is driven by extrinsic (micro-environment) and intrinsic (random epimutation) ``forces'', which are represented by advective and diffusion fluxes respectively. We account for natural selection and competition by introducing a fitness landscape, i.e. phenotypic dependent net growth of the cells. We consider a well-mixed environment in which cells are exposed to a prescribed oxygen environment, and their time-evolution determined by a non-local reaction-advection-diffusion equation, where the non-locality rises from the competition between different phenotypes. We analyse two scenarios, normoxia and hypoxia, in order to capture the different niches present in vivo. In our model, oxygen levels affect not only cell fitness but also act as an extrinsic force, favouring cell maturation (under normoxia) or de-differentiation into CSC (under hypoxia). We show how the qualitative behaviour of the system dynamics and its equilibrium distribution changes as model parameters vary, with tumour extinction predicted for certain regimes. The numerical results are validated by using spectral theory which allow us to characterise the stability property of the trivial steady-state, i.e. extinction. In addition to reproducing a variety of tumour cell distributions characterised by different mean clonogenic capacity, proportion of CSCs and population size, our analysis also gives insight into the role that extrinsic and intrinsic forces play in shaping the organisation of cells in phenotypic space. Finally, we discuss how the model can be extended to incorporate treatment, specifically radiotherapy, accounting for stem-ness dependent radio-resistance.