On the williams-bjerknes tumour growth model: Ii

Williams and Bjerknes introduced in 1972 a stochastic model for the spread of cancer cells. Cells, normal and abnormal (cancerous), are situated on a planar lattice. With each cellular division, one daughter cell stays put, while the other usurps the position of a neighbour; abnormal cells reproduce at a faster rate than normal cells. We are interested in the long-term behaviour of this system. We showed in ‘On the Williams-Bjerknes Tumour Growth Model I'(1) that, provided it lives forever, the tumour will eventually contain a ball of linearly expanding radius. Here it is shown that the rate of expansion is actually linear, and that the region of infection has an asymptotic shape which is given by some (unknown) norm. To demonstrate that the tumour contains a ball of linearly expanding radius, we applied in (1) certain techniques common to the field of interacting particle systems; in particular we made use of certain auxiliary Markov chains which are imbedded in the dual processes of our model. Here different techniques are applied. We show that the first infection times of different sites are almost subadditive, and we exhibit various regularity properties of the infection times such as bounds on their moments. These properties of the Williams-Bjerknes process allow one to follow the outline prescribed by Richardson in (7), where he showed that such a process does indeed exhibit a linear growth whose shape is prescribed by a norm.