# American Institute of Mathematical Sciences

May  2019, 24(5): 2017-2038. doi: 10.3934/dcdsb.2019082

## Ultimate dynamics and optimal control of a multi-compartment model of tumor resistance to chemotherapy

 1 Department of Mathematics & MȏLAB-Mathematical Oncology Laboratory, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain 2 Instituto Politecnico Nacional-CITEDI, Av. de IPN 1310, Nueva Tijuana, Tijuana 22435, B.C., Mexico

* Corresponding author: Juan Belmonte-Beitia

Received  January 2018 Revised  January 2019 Published  May 2019 Early access  March 2019

In this paper, a non-trivial generalization of a mathematical model put forward in [35] to account for the development of resistance by tumors to chemotherapy is presented. A study of the existence and local stability of the solutions, as well as the ultimate dynamics of the model, is addressed. An analysis of different chemotherapeutical protocols using discretization and optimization methods is carried out. A number of objective functionals are considered and the necessary optimality conditions are provided. Since the control variable appears linearly in the associated problem, optimal controls are concatenations of bang-bang and singular arcs. A formula of the singular control in terms of state and adjoint variables is derived analytically. Bang-bang and singular controls from the numerical simulations are obtained where, in particular, singular controls illustrate the metronomic chemotherapy.

Citation: Arturo Alvarez-Arenas, Konstantin E. Starkov, Gabriel F. Calvo, Juan Belmonte-Beitia. Ultimate dynamics and optimal control of a multi-compartment model of tumor resistance to chemotherapy. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2017-2038. doi: 10.3934/dcdsb.2019082
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##### References:
Phase portrait of the orbits of system (4). Examples of convergent trajectories to $P_2$ for $x_{0}+y_{0}+z_{0}\leq K$. Parameters used to calculate the phase portrait are given in Table 1
Optimal solutions for the objective $J_{0, 1}(u)$ and $M = 1/6$. a) Optimal control $u^*(t)$ (blue curve) and switching function $\phi$ (dashed red line) satisfying the bang-bang control law (12). b) Sensitive cells $x(t)$ c) Damaged cells $y(t)$ d) Resistant cells $z(t)$. The time horizon was fixed to $T = 1$ month
Optimal solutions for the objective $J_{0, 1}(u)$ and $M = 5$. a) Optimal control $u^*(t)$ (blue curve) and switching function $\phi$ (dashed red line) satisfying the bang-singular-bang control law (24). b) Sensitive cells $x(t)$ c) Damaged cells $y(t)$ d) Resistant cells $z(t)$. The time horizon was fixed to $T = 30$ months
Suboptimal protocol for the objective $J_{0, 1}(u)$ and $M = 5$. a) Suboptimal bang-bang control. b) Sensitive cells $x(t)$ c) Damaged cells $y(t)$ d) Resistant cells $z(t)$. The time horizon was fixed to $T = 30$ months
Optimal solutions for the objective $J_{1, 0}(u)$ and $M = 1/6$. a) Optimal control $u^*(t)$ (blue curve) and switching function $\phi$ (dashed red line) satisfying the bang-bang control law (12). b) Sensitive cells $x(t)$ c) Damaged cells $y(t)$ d) Resistant cells $z(t)$. The time horizon was fixed to $T = 1$ month
Optimal solutions for the objective $J_{1, 0}(u)$ and $M = 5$. a) Optimal control $u^*(t)$ (blue curve) and switching function $\phi$ (dashed red line) satisfying the bang-singular-bang control law (26). b) Sensitive cells $x(t)$ c) Damaged cells $y(t)$ d) Resistant cells $z(t)$. The time horizon was fixed to $T = 30$ months
Suboptimal protocol for the objective $J_{1, 0}(u)$ and $M = 5$. a) Suboptimal bang-bang control. b) Sensitive cells $x(t)$ c) Damaged cells $y(t)$ d) Resistant cells $z(t)$. The time horizon was fixed to $T = 30$ months
Suboptimal protocol for the objective $J_{1, 0}(u)$ and $M = 5$. a) Suboptimal bang-bang control. b) Sensitive cells $x(t)$ c) Damaged cells $y(t)$ d) Resistant cells $z(t)$. The time horizon was fixed to $T = 30$ months
Values of the biological parameters for the system (4)
 Variable Value Units Reference $\rho_s$ 0.000385 day$^{-1}$ [35] $\alpha_s$ 0.0382 L day/g [35] $\mu_d$ 0.00219 day$^{-1}$ [35] $\mu_r$ 0.000544 day$^{-1}$ [35] $\rho_r$ 0.000385 day$^{-1}$ [35] $\gamma$ 0.000136 day$^{-1}$ Estimated $k_s$ 0.474 [35] $P_0$ 40 mm [35] $x(0)$ $k_s P_0$ mm [35] $y(0)$ 0 mm [35] $z(0)$ 0 mm [35] $K$ 120 mm [35]
 Variable Value Units Reference $\rho_s$ 0.000385 day$^{-1}$ [35] $\alpha_s$ 0.0382 L day/g [35] $\mu_d$ 0.00219 day$^{-1}$ [35] $\mu_r$ 0.000544 day$^{-1}$ [35] $\rho_r$ 0.000385 day$^{-1}$ [35] $\gamma$ 0.000136 day$^{-1}$ Estimated $k_s$ 0.474 [35] $P_0$ 40 mm [35] $x(0)$ $k_s P_0$ mm [35] $y(0)$ 0 mm [35] $z(0)$ 0 mm [35] $K$ 120 mm [35]
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