Mathematical models for the evolution of tumour cords

Tumour cords grow around blood vessels and therefore they are characterized by the fact that oxygen and nutrients are supplied through their inner surface, differently from the so-called spheroidal tumours that have instead an inner necrotic core. Here we consider a fully developed system of cords, schematized as a regular array of identical elements, each one having approximately a rotational symetry around its blood vessel. A mathematical model for the evolution of the cord is presented in which the following facts are taken into account: the influence of a limiting nutrient on the proliferation rate of viable cells, the presence of totally quiescent regions and of a necrotic region, the presence of a death rate possibly induced by chemical or radiative treatments, the volume reduction rate of the necrotic material due to fluid loss from the cord, the role of the interstitial fluid in the overall mass balance. We point out in which conditions it is not necessary to include in the model the study of the flow of interstitial fluids, without neglecting its important effect. Both the steady state and the evolution problem are considered, showing existence and uniqueness of a solution.