A new approach to modelling the formation of necrotic regions in tumours

We present a mathematical model of the growth of tumours. The cells in the tumour are taken to proliferate and die at rates determined by the concentration of oxygen which diffuses into the tumour across its surface. Tumour cells are assumed to be composed primarily of water while the extracellular water is taken to move through the tumour as a porous media Bow between the cells. Exchange of water between the two is governed by cell proliferation. We model the mass of cells as an inviscid fluid with the pressure in the fluid, keeping the cells loosely packed together. Cells move in response to pressure in both fluids until the extracellular water pressure exceeds the cell pressure, resulting in the rupture of the tumour cells as they are ripped from one another. The resulting model is one of porous media flow with distributed sources and sinks determined by the oxygen concentration. The boundary conditions change type depending on whether the tumour surface is retreating or advancing. Retreating interfaces leave ruptured cells creating necrotic regions. An example of the model behaviour in one dimension is presented.