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

  • * Corresponding author: Juan Belmonte-Beitia

    * Corresponding author: Juan Belmonte-Beitia
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  • 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.

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

    Citation:

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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

    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

    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

    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

    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

    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

    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

    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

    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]
     | Show Table
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