Tumor boundary instability induced by nutrient consumption and supply.

We investigate the tumor boundary instability induced by nutrient consumption and supply based on a Hele-Shaw model. The model describes the geometric evolution of the tumor region and is derived from taking the incompressible limit of a cell density model, which leads to the pressure vanishing on the free boundary. We investigate the boundary behaviors under two different nutrient supply regimes, in vitro and in vivo, where in the former case the supply is adequate and in the latter case the supply is deficient. Our main conclusion is that by investigating typical solutions of the tumor-nutrient model with the asymptotic analysis with respect to the domain perturbation, the tumor boundary in the in vitro regime is shown to be stable regardless of the nutrient consumption rate. However, boundary instability occurs when the nutrient consumption rate exceeds a certain threshold in the in vivo regime, and the bifurcation threshold has a monotonic dependence on the frequency of the domain perturbation. [ABSTRACT FROM AUTHOR]