Modelling cell populations with spatial structure: steady state and treatment-induced evolution of tumour cords

Tumour cells growing around blood vessels form structures called tumour cords. We review some mathematical models that have been proposed to describe the stationary state of the cord and the cord evolution after single-dose cell killing treatment. Whereas the cell population has been represented with age or maturity structure to describe the cord stationary state, for the response to treatment a simpler approach was followed, by representing the cell population by means of the cell volume fractions. In this latter model, where transport of oxygen is included and its concentration is critical for cell viability, some constraints to be imposed on the interface separating the tumour from the necrotic region have a crucial role. An analysis of experimental data from untreated tumour cords, which involves modelling by cell age and by volume fractions, and some results about the cord response to impulsive cell killing, are also presented.