Evidence for Super-Critical Tumour GrowthAuthor(s): Trevor WilliamsSource: Advances in Applied Probability, Vol. 6, No. 2 (Jun., 1974), pp. 237-238Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1426275 .

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Sheffield, 13-17 August 1973 237

Random systems of locally interacting cells

P. TAUTU, German Cancer Research Centre, Heidelberg, and Centre of Mathematical Statistics, Bucharest

This paper deals with a special class of Markov processes studied by Spitzer (1970) and Harris (1972), which may provide stochastic structural models des-

cribing the appearance of malignant cell clones. Let •" be an r-dimensional lattice with sites x, y, .... Each site can be occupied

by a living cell which can assume one of s possible states (the cell cycle). These cells may be differently "coloured" and the transition from one colour to another characterizes the chain of transformations from a normal to a cancer cell.

The corresponding stochastic process can be called a lattice process (see Bartlett (1973)).

Two models are introduced. Model I. The number of cells is constant and the set of occupied sites is equal

to a fixed subset Z C 6', 0 <I Z < oo, which does not depend on time. This characterizes the random evolution of systems of coloured cells (or configurations) on a finite lattice. The number m(x) of cell's x e Z neighbours is always a positive number.

It is supposed that after each division one of the two new cells replaces a neigh- bouring non-dividing cell, as a result of the interaction.

Model II. The number of cells is variable with values increasing in time, and therefore the set of occupied sites is "blowing up". In this model the new cell replaces a non-dividing neighbour or is thrust into an unoccupied neigh- bouring site in g'.

References

BARTLETT, M. S. (1973) The statistical analysis of spatial pattern. This Conference, 188-190. HARRIS, T. E. (1972) Nearest-neighbour Markov interaction processes on multidimensional

lattices. Adv. Math. 9, 66-89. SPITZER, F. (1970) Interaction of Markov processes. Adv. Math. 5, 246-290, 188-190.

Evidence for super-critical tumour growth

TREVOR WILLIAMS, Bristol University

It has been noted (Druckrey (1967)) that if a carcinogen is administered at a

daily dose rate, d, and t is the median induction time of the tumour, then dt" is approximately constant, n being a characteristic of the agent; this is invariably

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238 3RD CONFERENCE ON STOCHASTIC PROCESSES AND APPLICATIONS

greater than 1. The latter property seems to distinguish carcinogenesis from all other actions known.

Suppose that normal cells turn cancerous at a constant rate 6 proportional to d, that cancer cells obey a linear homogeneous birth-death process, with parameters A and p, and that the induction period T is the first-passage time to a cell population of size N, there being n cancer cells initially. Then

#n(O) the

moment generator of T, given n cancer cells initially, satisfies the backward equations,

np/,.+,(0) + {O - [6 + n(A + p)]}A'n(0)

+ (6 + nA)~,+1,(0) = 0,

whose solution is

A'(0) = (6 + nA) ... [6 + (N - 1)A]A(0)/AN(), where the polynomials An(O) have generating function

A(0)O = (1 - Az)-6i/-Ao/I-A)(1 - pz)o/i(-A)

The details here follow Williams (1965) closely. One readily evaluates A' (0) from the last equation, and so obtains

in the case where there are 0 cancer cells to start with. Replacing the fixed N here by a Poisson variate of mean N, we obtain

N [e - 1]-1 d

where

ac = 6/), and y a N((p/A) - 1).

Thus, log T v.

log a is a straight line of slope -1 when y = 0, the birth-death process then being critical; the curves slope down more steeply when y > 0 (the sub-critical case), and less steeply when y < 0 (super-critical). Thus, it is only the latter which accords with Druckrey's observations.

In the critical case, the moment generator is, apart from a numerical factor, the reciprocal of a Bessel function, and may be inverted only in the two cases a = ? and a = ?, to yield the probability density function as a Jacobi theta function.

References

DRUCKREY, H. (1967) Quantitative aspects in chemical carcinogenesis. UICC Monograph Series, 7: Potential Carcinogenic Hazards from Drugs. Springer-Verlag, Berlin.

WILLIAMS, T. (1965) The distribution of response times in a birth-death process. Biometrika 52, 581-585.

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