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

Arturo Alvarez-Arenas; Konstantin E. Starkov; Gabriel F. Calvo; Juan Belmonte-Beitia

doi:10.3934/dcdsb.2019082

Article Contents

2019, Volume 24, Issue 5: 2017-2038. Doi: 10.3934/dcdsb.2019082

This issue Previous Article Preface Next Article Singularity of controls in a simple model of acquired chemotherapy resistance

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
^* Corresponding author: Juan Belmonte-Beitia

Received: January 2018

Revised: January 2019

Early access: March 2019

Published: March 2019

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Abstract

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.

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Mathematics Subject Classification: Primary: 49K15, 49M37; Secondary: 37C10, 92B05.

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Figure 1. 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

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Figure 2. 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

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Figure 3. 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

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Figure 4. 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

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Figure 5. 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

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Figure 6. 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

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Figure 7. 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

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Figure 8. 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

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Table 1. 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]

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