On the mean geometric densities of random closed sets, and their estimation : Application to the estimation of the mean density of inhomogeneous fibre processes

Many biological structures like vessels, membrains, etc., at a suitable scale can be modelled as random sets having Hausdorff dimension lower than the relevant space in which they are observed. In particular, angiogenesis (i.e. the formation of blood vessels from an existing one) is a process under intensive study nowadays in Biomedicine, because of its relation with tumor growth. Generalized densities a'' la Dirac-Schwartz have been shown to provide a useful approach for the mathematical description of such processes. Here an introduction to this mathematical approach is provided, together with related statistical techniques, which may provide a quantitative description of the mean geometric characteristics of the studied phenomenon.