Set-valued analysis of anti-angiogenic therapy and radiotherapy

Abstract

The aim of the paper is to study a cancer model based on anti-angiogenic therapy and radiotherapy. A set-valued analysis is carried out to control the tumor and carrying capacity of the vasculature, so in order to reverse tumor growth and augment tumor repair. The viability technique is used on an augmented model to solve the control problem. Obtained control is a selection of set-valued map of regulation and reduces tumor volume to around zero. A numerical simulation scheme with graphical representations and biological interpretations are given.

Viability control of chemo-immunotherapy and radiotherapy by set-valued analysis

“…1, is reversed to an upper exponential decreasing y(t) ≤ y 0 e −αt (11) with α > α 1 , towards null limit (7), see Fig. 2, by the therapy (v I (t), v M (t), h m (t)) = u(t, x) solution to the minimization problem (20), which is numerically solved in the Section 4 by Uzawa method (26-27) of parameter (λ, σ), to get numerical solution (y, x) of discretized model (25) by Euler method of step h, see Fig. 3.…”

“…Remark 3. The variant of an infeasible interior-point algorithm presented in [48], may be also used to solve the problem of minimization (20).…”

“…[17] Checks the controllability and observability for a mathematical model of immunotherapy and chemotherapy for cancer. [18] Uses set-valued analysis to approach the problem of decreasing and asymptotically stabilizing tumor cells, by anti-angiogenic therapy combined with radiotherapy, as well as in in [19] by tumor-immune with chemotherapy, so as in [20,21] by anti-angiogenic therapy with chemotherapy, while in [22] chemotherapy is used alone. [23] Studies the optimal effects of immuno-chemotherapy in the presence of gene therapy.…”

“…In this paper we propose to extend the set-valued method of papers [18][19][20][21] into non autonomous model of three mixed therapies of immunotherapy, chemotherapy, and radiotherapy. Section 2 presents the model and states the associated control problem in the viability form.…”