A mathematical model for tumor cords incorporating the flow of interstitial fluid

This work extends a previous model that described the evolution of a tumor cord (a cylindrical arrangement of tumor cells, generally surrounded by necrosis, growing around a blood vessel of the tumor) under the activity of cell killing agents. In the present model we include the most relevant aspects of the dynamics of extracellular fluid, by computing the longitudinal average of the radial fluid velocity and of the pressure field. We still assume that the volume fraction occupied by the cells always keeps the same constant value everywhere in the cord. The necrotic region is treated as a "fluid reservoir". To improve the modelling of therapeutic treatment, we have subdivided the viable cell population into a proliferating and a quiescent subpopulation. The transitions between the two states are both permitted, and are regulated by rates depending on the local oxygen concentration. For simplicity, the rates of death induced by treatment are assumed to be known functions of the radial distance and time. Existence and uniqueness of the stationary state in the absence of treatment has been shown, as well as the existence and uniqueness of the evolution that arises following a cell killing treatment.