Mathematical modelling of how the mechanical properties of a host tissue affect tumours kinetics and dynamics.

A common feature of previous mathematical models is that they neglect the mechanic effects produced by the surrounding on tumour growth kinetics and dynamics. The stress field imposed on a tumour by the neighboring tissue (in vivo) or an external matrix (in vitro), as well as that generated internally by cells binds during expansion, is believed to play a pivotal role in tumour regulation. Several models exploited the concepts of oxygen diffusion, growth’s and death’s rate dependence on nutrient levels, porous media theory to predict migration of cells. The occurrence of necrosis is then made dependent on the attainment of a critical oxygen concentration threshold. The assumption that there is a strong correlation between necrosis and nutrient’s lack has been put into question by experimental data. The experimental observations did show several behaviours in different cell types grown in spheroids: some exhibit necrotic regions closely linked to oxygen concentrations, but must do not display this attitude. Cell types, such as EMT6, may not have necrotic regions even in low levels of oxygen concentrations. For this reason, alternative biological mechanisms should be taken into account to investigate the formation and the dynamics of necrosis. A possible approach, inspired by Helmlinger experiments, is to consider the pressure generated by the motion of the extracellular fluid and the intercellular pressures created by cellular migration. The existence of the necrotic region is subjected by the requirement that the nature of forces between cells must be of the compressive type, make them weakly able to give origin to tensile strengths. When the intercellular force drops to zero and tries to become a tensile force, then such a cell die contemporary releasing fluid and filling its previous space. In this way, the remaining live cells will only float in the surrounding dead cells material, no longer able to exert any strength. Hence, the tumour is then comprised of two main regions: an outermost in which cells are at their maximum packing density and reproduce, grow and die; an innermost at a lower packing density where cells float in the extracellular fluid. A mathematical model is then adopted to adequately describe the growth of an avascular tumour spheroid in a deformable gel, with the aim of reproducing Helmlinger experiments.