On the Stochastic Geometry of Birth-and-Growth Processes. Application to Material Science, Biology and Medicine

Nucleation and growth processes arise in a variety of natural and technological applications, such as e.g. solidification of materials, semiconductor crystal growth, biomineralization (shell growth), tumor growth, vasculogenesis, DNA replication. All these processes 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 (crystals, vessel networks, etc), which, in general, are both stochastic in time and space. These structures usually induce a random division of the relevant spatial region, known as a random tessellation. A quantitative description of the spatial structure of a random tessellation can be given, in terms of random distributions /à la Schwartz/, and their mean values, known as mean densities of interfaces (n-facets) of the random tessellation, at different Hausdorff dimensions (cells, faces, edges, vertices), with respect to the usual d-dimensional Lebesgue measure. With respect to all fields of applications, predictive mathematical models which are capable of producing quantitative morphological features can contribute to the solution of optimization or optimal control problems. A non trivial difficulty arises from the strong coupling of the kinetic parameters of the relevant birth-and-growth (or branching-and-growth) process with the underlying field, such as temperature, and the geometric spatial densities of the evolving spatial random tessellation itself. Methods for reducing complexity include homogenization at mesoscales, thus leading to hybrid models (deterministic at the larger scale, and stochastic at lower scales); we bridge the two scales by introducing a mesoscale at which we may locally average the microscopic birth-and-growth model in presence of a large number of grains. The proposed approach, also suggests methods of statistical analysis for the estimation of mean geometric densities that characterize the morphology of a real system.