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 WilliamsBjerknes Tumour Growth Model I ' d ) 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 (l) 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.
IntroductionIn a 1972 paper, Williams and Bjerknes(9) proposed a simple stochastic model for the spread of cancer cells. Based on biological considerations, they restricted attention to the basal layer of an epithelium, thereby obtaining a two-dimensional setting. The cells, normal and abnormal, are situated on a planar lattice. With each cellular division, one daughter cell stays put while the other usurps the position of a neighbour. Splitting of each normal cell is assumed to occur at rate 1, whereas due to 'carcinogenic advantage', each cancer cell splits at rate K > 1.Assuming the lattice to be Z 2 , these axioms give rise to a simple continuous time Markov chain on the state space iS 0 = {finite subsets of Z 2 }. (Actually, the hexagonal lattice was preferred in (9), though the square and triangular lattice were considered as well. Our results apply to all three lattices; Z 2 is chosen largely for notational convenience.) If we let E,f denote the set of sites occupied by cancer cells at time t, given that the original cancerous population occupies A eS 0 , then the processes (£,i) t > 0 are strong Markov (see, for example, (2)). Their jump rates are given by
