Asymptotic behavior of a nonlinear necrotic tumor model with a periodic external nutrient supply

Abstract

In this paper we study a nonlinear free boundary problem for the growth of radially symmetric tumor with a necrotic core. The proliferation of tumor cells depends on the concentration of nutrient which satisfies a diffusion equation within tumor and is periodically supplied by external tissues. The tumor outer surface and the inner interface of the necrotic core are both free boundaries. We give a sufficient and necessary condition for the existence and uniqueness of positive periodic solution, and show it is globally asymptotically stable under radial perturbations. Our analysis implies that tumor growth may finally synchronize the periodic external nutrient supply.

Analysis of a nonlinear necrotic tumor model with angiogenesis and a periodic supply of external nutrients

“…We give a complete classification of asymptotic behavior of solutions to the model (1.1)-(1.5) in Theorem 1.1. It is easy to check that our results are consistent with those in [15,22]. In the model (1.1)-(1.5), besides involving the general nonlinear nutrient consumption rate and the tumor-cell proliferation rate functions, the Robin boundary condition and a periodic supply function of external nutrients, there may exist a necrotic core, and so the tumor may have two different types of free boundaries.…”

“…Precisely speaking, they proved that zero steady state is still globally stable when 1 ω ω 0 φ(t)dt − σ = 0, and a unique positive periodic solution exists if and only if 1 ω ω 0 φ(t)dt − σ > 0, which is also globally stable under radial perturbations. Very recently, Wu-Xu [22] analyzed the model (1.1)-(1.5) with (1.3) replaced by the boundary condition (1.10), and complete existence, uniqueness and stability results were given. Finally, for other related study, we refer the reader to [2,3,6,7,10,14,16,19,24] and the references cited therein.…”

“…All these make the study much more difficult and challenging. The idea of our proof is inspired by that of [15,22,25], which relies essentially on the analysis made in our previous work [20] on the particular case φ(t) ≡ σ.…”

Linear stability for a free boundary tumor model with a periodic supply of external nutrients

“…where I n (r) is the modified Bessel function of order n and P 0 (r) is defined by (2.1). As was analyzed in [34], for 1 T T 0 φ(t)dt > σ, there exists a unique T -periodic positive solution R * (t) for the equation (1.8). Substituting the unique solution R * (t) into the expressions (1.6)-(1.7), one finds that the radially symmetric T -periodic positive solution of the problem (1.1)-(1.5) is uniquely determined.…”

“…where I n (r) is the modified Bessel function of order n and P 0 (r) is defined by (2.1). As was analyzed in [34], for 1 Let us introduce the concept of linear stability/instability and asymptotic stability of the periodic solution (σ * (r, t), p * (r, t), R * (t)). By linear stability/instability, we mean: Linearize the system (1.1)-(1.5) at (σ * (r, t), p * (r, t), R * (t)) by writing…”