A free boundary problem with unilateral constraints describing the evolution of a tumor cord under the influence of cell killing agents

A system of tumor cords is schematized by an array of identical cords, each one having approximately a rotational symmetry around its central blood vessel. A mathematical model for the evolution of the cord is presented, taking into account the influence of a limiting nutrient on the proliferation and death of the cells, the volume reduction of the necrotic material due to fluid loss from the cord, and the influence of chemotherapy or radiation treatment. Both the steady state and the evolution problem are considered, showing existence and uniqueness of the solution. A peculiar feature of the evolution model is that the boundary conditions for nutrient concentration on the interface between viable cord and the necrotic region may change during the response to treatment, depending on whether or not new cells enter the necrotic region.