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Local controllability and optimal control for\newline a model of combined anticancer therapy with control delays
February  2017, 14(1): 217-235. doi: 10.3934/mbe.2017014

## On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach

 1 Institute of Mathematics, Lodz University of Technology, 90-924 Lodz, Poland 2 Dept. of Mathematics and Statistics, Southern Illinois University, Edwardsville, Il, 62026-1653, USA 3 Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo, 63130, USA 4 Service d'hématologie et Oncologie Pédiatrique, Centre Hospitalo-Universitaire Timone Enfants, AP-HM, Marseille, UMR S_911 CRO2 Aix Marseille Université, Marseille, France 5 Metronomics Global Health Initiative, Marseille, France 6 Childrens Cancer Institute Australia, Lowy Cancer Research Centre, University of New South Wales, Randwick, NSW, Australia

* Corresponding author

Received  December 2015 Accepted  June 30, 2016 Published  October 2016

Effects that tumor heterogeneity and drug resistance have on the structure of chemotherapy protocols are discussed from a mathematical modeling and optimal control point of view. In the case when two compartments consisting of sensitive and resistant cells are considered, optimal protocols consist of full dose chemotherapy as long as the relative proportion of sensitive cells is high. When resistant cells become more dominant, optimal controls switch to lower dose regimens defined by so-called singular controls. The role that singular controls play in the structure of optimal therapy protocols for cell populations with a large number of traits is explored in mathematical models.

Citation: 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
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Schematic illustration of the distribution of traits around dominant steady states
Dose rates (left) for the control $u_{\tau}$ defined by equation (26) for $\tau=1$ (top row), $\tau=1.75$ (middle row) and $\tau =2$ (bottom row), and corresponding time evolutions of the concentrations $c$ (middle) and the states $S$ and $R$ (right)
Evolution of the states S and R if the therapy horizon is extended to 42 days for the control strategies uτ for τ = 1:75 (left) and τ = 2 (right)
An administration protocol which gives full dose until time $\tau$ and then switches to the dose given by the singular control over the interval $[\tau, 10]$ optimized over $\tau$. The weights in the objective are chosen equal using $\bar{\alpha} = \bar{\beta} = \frac{1}{21}$ and $\gamma = 100$. Shown are the control (top, left), corresponding evolution of the total population $T=\sum_{i=1}^{21} N_i(t)$ (top, right), evolution of the traits $N_i(t)$ for $i=1, \ldots, 21$ (middle) and a comparison of the initial density $n(0, x)\equiv \frac{200}{21}$ and the terminal density $n(10, x)$ shown as red curve (bottom)
An administration protocol which gives full dose until time $\tau$ and then switches to the dose given by the singular control over the interval $[\tau, 10]$ optimized over $\tau$. The weights in the objective are chosen equal using $\bar{\alpha} = \frac{1}{21}$, $\bar{\beta} = \frac{8}{21}$ and $\gamma = 100$. Shown are the control (top, left), corresponding evolution of the total population $T=\sum_{i=1}^{21} N_i(t)$ (top, right), evolution of the traits $N_i(t)$ for $i=1, \ldots, 21$ (middle) and a comparison of the initial density $n(0, x)\equiv \frac{200}{21}$ and the terminal density $n(10, x)$ shown as red curve (bottom)
Values for the initial data and parameters used in numerical computations
 parameters interpretation value S0 initial condition of sensitive cells 9:4051 × 109 R0 initial condition of resistant cells 0:5949 × 109 r1 growth rate of sensitive population 3:5 r2 growth rate of resistant population 1 θ1 rate at which sensitive cells become resistant 0:15 θ2 rate at which resistant cells become resensitized 0:02 ψ1 log-kill coefficient for sensitive population 5 ψ2 log-kill coefficient for resistant population 1 k clearance rate of drug [Taxol] 2:9706
 parameters interpretation value S0 initial condition of sensitive cells 9:4051 × 109 R0 initial condition of resistant cells 0:5949 × 109 r1 growth rate of sensitive population 3:5 r2 growth rate of resistant population 1 θ1 rate at which sensitive cells become resistant 0:15 θ2 rate at which resistant cells become resensitized 0:02 ψ1 log-kill coefficient for sensitive population 5 ψ2 log-kill coefficient for resistant population 1 k clearance rate of drug [Taxol] 2:9706
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