A Markovian Configuration Model for Carcinogenesis

Differential equations for moments of an interacting particle process on a lattice

“…The process is a K-type interacting Markovian branching particle system whose domain S is the lattice Z d, and it is governed by K (positive) rate The model given in [ 11] is flexible enough to include phases of the (classical) cell cycle for cells of each type, but for simplicity, we consider only the special case of a single phase.…”

“…Because of the importance of the role of the parameters in what follows, P ~ _ • ,P will be used to denote the probability measure on W s corresponding to the process with rate vector ~[21 ""2K] and transition matrix p, given that the process begins in state t/c W s. The formal construction is given in [11], but the following is an informal description of the process. Given a configuration q E W s at time t = 0, a realization path t/t ~ W s, t >~ 0 of the process is generated by these rules.…”

“…The models which do take the spatial features of cell growth into account, at least in a limited sense, are the interacting particle systems which originated in the context of theoretical physics. This paper is concerned with certain mathematical aspects of a stochastic interacting particle system which has been proposed for tumor growth [11,9]. These processes are primarily of interest in 2-dimensions (see, for example, [4] in which the radial growth phase of primary melanoma is mainly 2-dimensional) and 3-dimensions. There are several advantages to interacting particle systems as models for carcinogenesis beyond the inclusion of a spatial component.…”

Tumor Growth Rate Approximation-Assisted Estimation

“…A spatial analysis of the tumor cell growth exhibits a differential rate of growth and may be important in accessing the oncogenic status of the tumor as well as its potential to become malignant. Braun and Kulperger (1993) and Braun and Kulperger (1995) have introduced an estimator to estimate the growth parameter of an interacting particle system which is discussed in detail by Schürger and Tautu (1976). The interacting particle system the ory is also dealt with comprehensively by Liggett (1985) for modeling the proliferation of cells in cancer tumors.…”

A class of branching processes on a lattice with interactions

“…We should like to point out that, in the limiting case K = 00 (corresponding to the special case of a Williams-Bjerknes model where recovery of cancerous cells is not possible), only black cells can divide, which, after a change of time scale, is just a contact interaction process such that d" = 0 and c, = i, 1~i~2d. The latter process is contained in a class of cell models (involving types and internal states of cells) which were considered in [31]. The class of branching processes with interactions on Zd we should like to study in this paper, contains the class of all those nearest-neighbour interactions which have the property that once a site is occupied, it remains so forever.…”