In this paper, we study a time-delayed free boundary of tumor growth with Gibbs-Thomson relation in the presence of inhibitors. The model consists of two reaction diffusion equations and an ordinary differential equation. The reaction diffusion equations describe the nutrient and inhibitor diffusion within tumors and take into account the Gibbs-Thomson relation at the outer boundary of the tumor. The tumor radius evolution is described by the ordinary differential equation. It is assumed that the regulatory apoptosis process takes longer than the natural apoptosis and proliferation processes. We first show the existence and uniqueness of the solution to the model. Next, we further demonstrate the existence of the stationary solutions and the asymptotic behavior of the stationary solutions when the blood vessel density is a constant. Finally, we further demonstrate the existence of the stationary solutions and the asymptotic behavior of the stationary solutions when the blood vessel density is bounded. The result implies that, under certain conditions, the tumor will probably become dormant or will finally disappear. The conclusions are illustrated by numerical computations.

In this paper, we study a free boundary problem for vascularized tumor growth with a time delay in the process of tumor regulating apoptosis. The characteristic of this model is that both vascularization and apoptosis regulation is considered. In mathematical form, this model is expressed as a free boundary problem with Robin boundary. We prove the existence and uniqueness of the global solution and their asymptotic behavior. The effects of vascularization parameters and apoptosis regulation parameters on tumor are discussed. Depending on the importance of regulating the apoptosis rate, the tumor will tend to the unique steady state or eventually disappear. For some parameter values, the final results show that the dynamic behavior of the solutions of our model is analogous to the quasi-stationary solutions. Our results are also verified by numerical simulation.