Analysis of a free boundary problem for vascularized tumor growth with a time delay in the process of tumor regulating apoptosis

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.