Stochastic geometric models, and related statistical issues in tumour-induced angiogenesis

In the modelling and statistical analysis of tumor-driven angiogenesis it is of great importance to handle random closed sets of different (though integer) Hausdorff dimensions, usually smaller than the full dimension of the relevant space. Here an original approach is reported, based on random generalized densities (distributions) á la Dirac-Schwartz, and corresponding mean generalized densities. The above approach also suggests methods for the statistical estimation of geometric densities of the stochastic fibre system that characterize the morphology of a real vascular system. A quantitative description of the evolution of tumor-driven angiogenesis requires the mathematical modelling of a strongly coupled system of a stochastic branching-and-growth process of fibres, modelling the network of blood vessels, and a family of underlying fields, modelling biochemical signals. Methods for reducing complexity include homogenization at mesoscales, thus leading to hybrid models (deterministic at the larger scale, and stochastic at lower scales); in tumor-driven angiogenesis the two scales can be bridged by introducing a mesoscale at which one locally averages the microscopic branching-and-growth process, in presence of a sufficiently large number of vessels (fibers).