Search

This paper deals with a mathematical model for tumor growth with angiogenesis and a necrotic core. The problem is a nonlinear problem which has two free boundaries: the external boundary of the tumor and the external boundary of the internal necrotic core. Using the implicit function theorem, the relationship between the two free boundaries is studied. The asymptotic stability of the unique steady-state solution is proved. The effect of nutrient concentration on tumor growth is studied. The relationship between the stage with a necrotic core and the stage without necrotic core is also studied. The results show that whether the steady-state of tumor growth has a necrotic core or not is determined by the concentration of external nutrients. The stage of tumor with necrotic core and the stage without necrotic core could transform into each other by adjusting the concentration of external nutrients. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model for solid avascular tumor growth with a time delay in regulatory apoptosis is studied. In the model, two types of cell apoptoses are considered, one is natural apoptosis and the other is regulatory apoptosis. The process of regulatory apoptosis is delayed compared to the processes of proliferation and natural apoptosis. The existence and uniqueness of a solution to the model are proved. The long-time asymptotic behavior of the solutions is studied. The results show that the dynamical behavior of solutions to this mathematical model is similar to that of the corresponding quasi-stationary problem for some special parameter values. Numerical simulations of some special parameter values are also given to verify our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a time-delayed free boundary problem with time dependent Robin boundary conditions. The special case where n = 3 is a mathematical model for the growth of a solid nonnecrotic tumor with angiogenesis. In the problem, both the angiogenesis and the time delay are taken into consideration. Tumor cell division takes a certain length of time, thus we assume that the proliferation process leg behind as compared to the process of apoptosis. The angiogenesis is reflected as the time dependent Robin boundary condition in the model. Global existence and uniqueness of the nonnegative solution of the problem is proved. When c > 0 is sufficiently small, the stability of the steady state solution is studied, where c is the ratio of the time scale of diffusion to the tumor doubling time scale. Under some conditions, the results show that the magnitude of the delay does not affect the final dynamic behavior of the solutions. An application of our results to a mathematical model for tumor growth of angiogenesis is given and some numerical simulations are also given. [ABSTRACT FROM AUTHOR]

In this paper we study a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays. It is assumed that the process of proliferation is delayed compared with apoptosis. The delay represents the time taken for cells to undergo mitosis. By employing stability theory for functional differential equations, comparison principle and some meticulous mathematical analysis, we mainly study the asymptotic behavior of the solution, and prove that in the case c (the ratio of the diffusion time scale to the tumor doubling time scale) is sufficiently small, the volume of the tumor cannot expand unlimitedly. It will either disappear or evolve to one of two dormant states as t → ∞. The results show that dynamical behavior of solutions of the model are similar to that of solutions for corresponding nonretarded problems under some conditions. [ABSTRACT FROM AUTHOR]

In this paper, a free boundary problem for a solid avascular tumor growth under the action of periodic external inhibitors with time delays in proliferation is studied. Sufficient conditions for the global stability of tumor-free equilibrium are given. Moreover, if external concentration of nutrients is large, we also prove that the tumor will not disappear and determine the conditions under which there exist periodic solutions to the model. The results show that the periodicity of the inhibitor may imply periodicity of the size of the tumor. More precisely, if (the concentration of external nutrients) is greater than , where are two constants; ; is a periodic function which can be interpreted as a treatment and is the period of . Results are illustrated by computer simulations. [ABSTRACT FROM AUTHOR]

A time-delayed mathematical model for tumor growth with the effect of periodic therapy is studied. The establishment of the model is based on the reaction-diffusion dynamics and mass conservation law and is considered with a time delay in cell proliferation process. Sufficient conditions for the global stability of tumor free equilibrium are given. We also prove that if external concentration of nutrients is large the tumor will not disappear and the conditions under which there exist periodic solutions to the model are also determined. Results are illustrated by computer simulations. [ABSTRACT FROM AUTHOR]