A method for parametric estimation of the number and size distribution of cell clusters from observations in a section plane

The problem of finding the number and size distribution of cell clusters that grow in an organ or tissue from observations of the number and sizes of transections of such cell clusters in a planar section is considered. This problem is closely related to the well-known corpuscle or Wicksell problem in stereology, which deals with transections of spherical objects. However, for most biological applications, it is unrealistic to assume that cell clusters have spherical shapes since they may grow in various ways. We therefore propose a method that allows for more general spatial configurations of the clusters. Under the assumption that a parametric growth model is available for the number and sizes of the cell clusters, expressions are obtained for the probability distributions of the number and sizes of transections of the clusters in a section plane for each point in time. These expressions contain coefficients that are independent of the parametric growth model and time but depend on which model is chosen for the configuration of the cell clusters in space. These results enable us to perform estimation of the parameters of the growth model by maximum likelihood directly on the data instead of having to deal with the inverse problem of estimation of three-dimensional quantities based on two-dimensional data. For realistic choices of the configuration model, it will not be possible to obtain the exact values of the coefficients, but they can easily be approximated by means of computer simulations of the spatial configuration. Monte Carlo simulations were performed to approximate the coefficients for two particular spatial configuration models. For these two configuration models, the proposed method is applied to data on preneoplastic minifoci in rat liver under the assumption of a two-event model of carcinogenesis as the parametric growth model.