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. n original approach is reported, based on random generalized densities (distributions) ل la Dirac-Schwartz, and corresponding mean generalized densities. ove 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. titative 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. s 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).