Age-structured mechanical models for tumor growth

In this paper, we introduce and analyze a mechanical model for tumor growth that takes into account the life cycle of a tumor cell. The underlying process for tumor growth is the same as in classical mechanical models: the spatial expansion of the tumor is driven by the proliferation of the cells (mitosis) which is only limited by the pressure inside the tissue. The natural incompressibility of the cells, which leads to a movement of the cells away from regions of high pressure, is taken into account via a nonlinear Darcy's law. Compared to similar models studied recently, we include an additional variable, which represents the age of the cells. The various phases of the life of a cell (growth, mitosis and death) are then dependent on this age variable. We prove the existence of weak solutions and investigate their behavior numerically, focusing on the age distribution of the cells inside the tumor, the convergence to traveling wave solutions and the existence of a threshold for the death rate for expansion/regression of the tumor.