  The mathematical modeling of tumor growth has a long history, and has been
mathematically formulated in several different ways. Here we tackle the problem
in the case of a continuous distribution using mathematical tools from
statistical physics. To this extent, we introduce a novel kinetic model of
growth which highlights the role of microscopic transitions in determining a
variety of equilibrium distributions. At variance with other approaches, the
mesoscopic description in terms of elementary interactions allows to design
precise microscopic feedback control therapies, able to influence the natural
tumor growth and to mitigate the risk factors involved in big sized tumors. We
further show that under a suitable scaling both the free and controlled growth
models correspond to Fokker--Planck type equations for the growth distribution
with variable coefficients of diffusion and drift, whose steady solutions in
the free case are given by a class of generalized Gamma densities which can be
characterized by fat tails. In this scaling the feedback control produces an
explicit modification of the drift operator, which is shown to strongly modify
the emerging distribution for the tumor size. In particular, the size
distributions in presence of therapies manifest slim tails in all growth
models, which corresponds to a marked mitigation of the risk factors. Numerical
results confirming the theoretical analysis are also presented.
