Analysis of a free boundary problem modeling spherically symmetric tumor growth with angiogenesis and a periodic supply of nutrients

Abstract

Abstract In this paper, we study a free boundary problem modeling spherically symmetric tumor growth with angiogenesis and a periodic supply of nutrients. The mathematical model is a free boundary problem since the external radius of the tumor denoted by R ( t ) $R(t)$ changes with time. The characteristic of this model is the consideration of both angiogenesis and periodic external nutrient supply. The cells inside the tumor absorb nutrient u ( r , t ) $u(r,t)$ through blood vessels and attracts blood vessels at a rate proportional to α. Thus on the boundary, we have u r ( r , t ) + α ( u ( r , t ) − ψ ( t ) ) = 0 , r = R ( t ) , t > 0 , $$\begin{aligned} u_{r}(r,t)+\alpha \bigl(u(r,t)-\psi (t)\bigr)=0,\quad r=R(t),t>0, \end{aligned}$$ where ψ ( t ) $\psi (t)$ is the nutrient concentration provided externally. Considering that the nutrient provided externally to the tumor are generally provided periodically, in this paper, we assume that ψ ( t ) $\psi (t)$ is a periodic function. Sufficient conditions for a tumor to disappear are given. We investigate the existence, uniqueness, and stability of solutions. The results show that when the nutrient concentration exceeds a certain value and c is sufficiently small, the solutions of the model can be arbitrarily close to the unique periodic function as t → ∞ $t\rightarrow\infty$ .

Keywords

• Tumor growth
• Free boundary problem
• Angiogenesis
• Asymptotic behavior
• Periodic solution