We consider a model for tumor cells that are growing around a cylindrical blood vessel of diameter db as shown in the figure below. In this figure P and Q respectively denote the proliferating and the quiescent layer. We suppose that oxygen diffuses into the tumor from the surface of the blood vessel in the radial direction. To simplify the problem we also assume that the oxygen concentration quickly reaches equilibrium.

(a) If live cells inside the tumor consume oxygen at a constant rate of M kg/m3/s write down the differential equation that is satisfied by the equilibrium oxygen concentration at the proliferating layer. Justify your answer.

(b) Solve the equation in (a) assuming that oxygen concentration at the blood vessel surface is a fixed value cb.

(c) Write down the differential equation that is satisfied by the equilibrium oxygen concentration at the quiescent layer. Justify your answer.

blood do P Q