Moving mesh partial differential equations modelling to describe oxygen induced effects on avascular tumour growth

The urokinase plasminogen activator (uPA) is a proteolytic enzyme, which over expression by cancer cells is recognized as promoting tumour growth and proliferation. In the early stage of formation, cancer cells grow in the avascular phase and oxygen supply come from surrounding healthy tissue. Using parameters estimated from in vivo experiments on human tumours, we simulate the effects of hypoxic conditions on cancer cell interacting with the uPA system, in the avascular phase. The resulting system of six-coupled partial differential equations is solved over one-dimensional domain implementing an adaptive grid technique, using the finite element method. Our results predict that changes of both diffusion and uptake/decay coefficients for oxygen, because of possible microenvironment changes of cancer cells, induce variations of the invasion velocity, with crowding effects during cell growth and proliferation.