Memory effects and of the killing rate on the tumor cells concentration for a one-dimensional cancer model

In this article, the fractional models of cancer tumor are studied using the Laplace transform and numerical inversion. The generalized model with Caputo time-fractional derivative is considered for two different cases of the killing rate of the cancer cells. The models based on the time-fractional derivatives give a better description of the tumor evolution because, in such models, the history of the tumor cells concentration influences the time evolution of the tumor. To solve the initial-boundary value problems, some adequate transforms of variables and functions have been considered, together with the Laplace transform and the Bessel equation. A numerical method, namely the Stehfest's algorithm is used to determine the inverse Laplace transforms. The effect of fractional parameter on the tumor cells concentration has been highlighted by numerical simulations and graphical illustrations. The time variation of tumor cell concentration and its dependence of the memory parameter could provide useful information in choosing an appropriate treatment.