Analysis of necrotic core formation in angiogenic tumor growth. (February 2020) - Xu, Shihe; Su, Dan
Abstract: In this paper we study a free boundary problem for tumor growth with angiogenesis and a necrotic core. The problem has two free boundaries: the outer boundary R ( t ) and the necrotic core boundary ρ ( t ) . In order to receive nutrients σ, the tumor attracts blood vessels at a rate proportional to α, so that the Robin condition ∂ σ ∂ r + α ( σ − σ ̄ ) = 0 holds on the outer boundary, where σ ̄ is the nutrients concentration outside the tumor. The sufficient conditions for the existence, uniqueness and stability of the stationary solution to the model are given. The results show that there exists a positive constant σ 0 such that if the external nutrient supply σ ̄ is greater than or equal to σ 0, there exists a unique stationary solution with a necrotic core and this stationary solution is stable. Meanwhile, we also discuss the asymptotic behavior of the transient solutions when the external nutrient supply is below the threshold value σ 0 . The effects of the external nutrients supply and the connection between the non-necrotic stage and the necrotic stage are also discussed.