# American Institute of Mathematical Sciences

May  2019, 24(5): 2039-2052. doi: 10.3934/dcdsb.2019083

## Singularity of controls in a simple model of acquired chemotherapy resistance

 1 Inter-Faculty Individual Doctoral Studies in Natural Sciences and Mathematics, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland 2 Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, Sloneczna 54, 10-710 Olsztyn, Poland 3 Faculty of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

*Corresponding author

Received  December 2017 Revised  January 2019 Published  March 2019

Fund Project: Piotr Bajger and Mariusz Bodzioch were supported by Polish National Science Centre grant 2016/23/N/ST1/01178. Urszula Foryś was supported by Polish National Science Centre grant 2015/17/N/ST1/02564

This study investigates how optimal control theory may be used to delay the onset of chemotherapy resistance in tumours. An optimal control problem with simple tumour dynamics and an objective functional explicitly penalising drug resistant tumour phenotype is formulated. It is shown that for biologically relevant parameters the system has a single globally attracting positive steady state. The existence of singular arc is then investigated analytically under a very general form of the resistance penalty in the objective functional. It is shown that the singular controls are of order one and that they satisfy Legendre-Clebsch condition in a subset of the domain. A gradient method for solving the proposed optimal control problem is then used to find the control minimising the objective. The optimal control is found to consist of three intervals: full dose, singular and full dose. The singular part of the control is essential in delaying the onset of drug resistance.

Citation: Piotr Bajger, Mariusz Bodzioch, Urszula Foryś. Singularity of controls in a simple model of acquired chemotherapy resistance. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2039-2052. doi: 10.3934/dcdsb.2019083
##### References:

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##### References:
Phase portraits for System (1), when (A) the positive steady state exists and is stable (the biologically realistic case), and (B) the zero steady state is stable
Typical choice for a resistance penalty: $G(z) = \tfrac{1}{2}(1 + \tanh(z))$
Singular arcs for different values c of the constant Hamiltonian: (A) c = 3, (B) c = 4, (C) c = 5 and (D) = 10
Optimal solution (A), together with the corresponding control (B), trajectory (C) and the switching function (D)
Nominal parameter values. All the parameters are non-dimensional
 Name Value Role $\gamma_1$ 0.192 Proliferation rate of sensitive cells. $\gamma_2$ 0.096 Proliferation rate of resistant cells. $\tau_1$ 0.002 Mutation rate towards the resistant phenotype. $\tau_2$ 0.001 Mutation rate towards the sensitive phenotype. $T$ 13.5 Therapy duration. $\omega_1$ 60 Weight for sensitive cell volume at the terminal point. $\omega_2$ 120 Weight for the resistant cell volume at the terminal point. $\eta_1$ 3 Weight in the overall tumour burden penalty for sensitive cells. $\eta_2$ 6 Weight in the overall tumour burden penalty for resistant cells. $\xi$ 1 Weight for the resistant phenotype penalty. $\epsilon$ 0.1 Scaling factor in the resistant phenotype penalty function $G$. $\Delta$ $10^{-6}$ Step used in finite differences gradient calculations.
 Name Value Role $\gamma_1$ 0.192 Proliferation rate of sensitive cells. $\gamma_2$ 0.096 Proliferation rate of resistant cells. $\tau_1$ 0.002 Mutation rate towards the resistant phenotype. $\tau_2$ 0.001 Mutation rate towards the sensitive phenotype. $T$ 13.5 Therapy duration. $\omega_1$ 60 Weight for sensitive cell volume at the terminal point. $\omega_2$ 120 Weight for the resistant cell volume at the terminal point. $\eta_1$ 3 Weight in the overall tumour burden penalty for sensitive cells. $\eta_2$ 6 Weight in the overall tumour burden penalty for resistant cells. $\xi$ 1 Weight for the resistant phenotype penalty. $\epsilon$ 0.1 Scaling factor in the resistant phenotype penalty function $G$. $\Delta$ $10^{-6}$ Step used in finite differences gradient calculations.
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