A model for hepatocarcinogenesis treating phenotypical changes in focal hepatocellular lesions as epigenetic events
A stochastic carcinogenesis model (called color-shift multistage model) is formulated to incorporate the biological hypothesis that foci of altered cells in an organ are formed randomly in space and time, that these colonies expand and that they irreversibly change their phenotype as an entity rather than by mutation of a single cell in the colony. A Poisson process assumed to be homogeneous in location and to have rate \(\mu(s)\) which may vary with the age \((s)\) generates in a unit cube \([0,1]^3\) the centers of spheres with radius \(r_0\). Every sphere that is generated grows according to an exponential law with rate parameter \(B\). At the time point of its appearance a sphere starts in color 1 and changes its color sequentially after exponential times. Hence every sphere has four attributes: (i) location \(X\in[0,1]^3\), (ii) time point of appearance \(T\), (iii) color \(C\) and (iv) radius \(R\). It is assumed that location, \(X\), is independent of the time point of formation \(T\), the color \(C\) and the radius \(R\) of the focus. Once a focus is generated, its color and size change over time, i.e. \(C(t)\) and \(R(t)\) are both stochastic processes in time. It is assumed that color and radius are conditionally independent given \(T\). The waiting time for each color change is described by the single parameter \(\lambda\) (per time and colony), and a sphere starts in color 1 and sequentially passes through the colors. The number of available colors for the foci is \(m\), where \(m\) may be infinite. The authors achieve to calculate the probability that a colony has color \(j\) and derive the respective distribution of the number of colonies of color \(j\). Assuming independent and replicate observations of the number of colonies, and size and color of each colony at sacrifice times \(t_i\) are available, the likelihood function is constructed and MLE theory is developed. Development of the model for the case in which three-dimensional data are available is given first, followed by the alteration of the mathematics to account for two-dimensional data from tissue slides. This model is applied to data from a study on histopathological lesions in the liver.
scientific article; zbMATH DE number 1350874