# 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
##### References:

show all references

##### 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]
 [1] M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014 [2] Walter Alt, Robert Baier, Matthias Gerdts, Frank Lempio. Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 547-570. doi: 10.3934/naco.2012.2.547 [3] Helmut Maurer, Tanya Tarnopolskaya, Neale Fulton. Computation of bang-bang and singular controls in collision avoidance. Journal of Industrial and Management Optimization, 2014, 10 (2) : 443-460. doi: 10.3934/jimo.2014.10.443 [4] Karl Kunisch, Lijuan Wang. The bang-bang property of time optimal controls for the Burgers equation. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3611-3637. doi: 10.3934/dcds.2014.34.3611 [5] Karl Kunisch, Lijuan Wang. Bang-bang property of time optimal controls of semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 279-302. doi: 10.3934/dcds.2016.36.279 [6] Gengsheng Wang, Yubiao Zhang. Decompositions and bang-bang properties. Mathematical Control and Related Fields, 2017, 7 (1) : 73-170. doi: 10.3934/mcrf.2017005 [7] Urszula Ledzewicz, Heinz Schättler. Drug resistance in cancer chemotherapy as an optimal control problem. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 129-150. doi: 10.3934/dcdsb.2006.6.129 [8] Craig Collins, K. Renee Fister, Bethany Key, Mary Williams. Blasting neuroblastoma using optimal control of chemotherapy. Mathematical Biosciences & Engineering, 2009, 6 (3) : 451-467. doi: 10.3934/mbe.2009.6.451 [9] Urszula Ledzewicz, Behrooz Amini, Heinz Schättler. Dynamics and control of a mathematical model for metronomic chemotherapy. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1257-1275. doi: 10.3934/mbe.2015.12.1257 [10] Clara Rojas, Juan Belmonte-Beitia, Víctor M. Pérez-García, Helmut Maurer. Dynamics and optimal control of chemotherapy for low grade gliomas: Insights from a mathematical model. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1895-1915. doi: 10.3934/dcdsb.2016028 [11] M. Soledad Aronna, J. Frédéric Bonnans, Andrei V. Dmitruk, Pablo A. Lotito. Quadratic order conditions for bang-singular extremals. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 511-546. doi: 10.3934/naco.2012.2.511 [12] Urszula Ledzewicz, Heinz Schättler, Mostafa Reisi Gahrooi, Siamak Mahmoudian Dehkordi. On the MTD paradigm and optimal control for multi-drug cancer chemotherapy. Mathematical Biosciences & Engineering, 2013, 10 (3) : 803-819. doi: 10.3934/mbe.2013.10.803 [13] Joseph Malinzi, Rachid Ouifki, Amina Eladdadi, Delfim F. M. Torres, K. A. Jane White. Enhancement of chemotherapy using oncolytic virotherapy: Mathematical and optimal control analysis. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1435-1463. doi: 10.3934/mbe.2018066 [14] Shuo Wang, Heinz Schättler. Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1223-1240. doi: 10.3934/mbe.2016040 [15] Urszula Ledzewicz, Shuo Wang, Heinz Schättler, Nicolas André, Marie Amélie Heng, Eddy Pasquier. On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach. Mathematical Biosciences & Engineering, 2017, 14 (1) : 217-235. doi: 10.3934/mbe.2017014 [16] M. Delgado-Téllez, Alberto Ibort. On the geometry and topology of singular optimal control problems and their solutions. Conference Publications, 2003, 2003 (Special) : 223-233. doi: 10.3934/proc.2003.2003.223 [17] Nacira Agram, Astrid Hilbert, Bernt Øksendal. Singular control of SPDEs with space-mean dynamics. Mathematical Control and Related Fields, 2020, 10 (2) : 425-441. doi: 10.3934/mcrf.2020004 [18] Majid Jaberi-Douraki, Seyed M. Moghadas. Optimal control of vaccination dynamics during an influenza epidemic. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1045-1063. doi: 10.3934/mbe.2014.11.1045 [19] Shuo Wang, Heinz Schättler. Optimal control for cancer chemotherapy under tumor heterogeneity with Michealis-Menten pharmacodynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2383-2405. doi: 10.3934/dcdsb.2019100 [20] Luis A. Fernández, Cecilia Pola. Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2577-2612. doi: 10.3934/dcdsb.2018266

2021 Impact Factor: 1.497