Stochastic geometry and related statistical problems in biomedicine

Many processes of biomedical interest may be modelled as birth-and-growth processes (germ-grain models), which are composed of two processes, birth (nucleation, branching, etc.) and subsequent growth of spatial structures (cells, vessel networks, etc), which, in general, are both stochastic in time and space. These structures induce a random division of the relevant spatial region, known as a random tessellation. A quantitative description of the spatial structure of the tessellation can be given, in terms of the mean densities of interfaces ($n$-facets). In applications to medicine, it is very important for either prevention, or clinical treatment, to describe the relevant phenomenon in terms of a biomathematical model including all significant features (parameters ,variables) of the system. One very significant example in this respect is the mathematical modelling of tumor growth and of tumor-induced angiogenesis. The understanding of the principles and the dominant mechanisms underlying tumor growth is an essential prerequisite for identifying optimal control strategies, in terms of prevention and treatment. Predictive mathematical models which are capable of producing quantitative morphological features of developing tumor and blood vessels can contribute to this. The difficulty arises from the strong coupling of the kinetic parameters of the relevant birth-and-growth (or branching-and-growth) process with interacting underlying fields, and the geometric spatial densities of the existing tumor, or capillary network itself. All these aspects induce stochastic time and space heterogenities, thus motivating a more general analysis of the stochastic geometry of the process. The formulation of an exhaustive evolution model which relates all the relevant features of a real phenomenon dealing with different scales, and a stochastic domain decomposition at different Hausdorff dimensions, is a problem of high complexity, both analytical and computational. Methods for reducing complexity include homogenization at larger scales, thus leading to hybrid models (deterministic at the larger scale, and stochastic at lower scales). As a matter of example we present a couple of simplified stochastic geometric models, for which we discuss how to relate the geometric probability distribution to the kinetic parameters of birth and growth. On the dual side, in order to compare numerical simulations of proposed mathematical models with existing data, we provide methods of statistical analysis for the estimation of geometric densities that characterize the morphology of a real system.