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# Local controllability and optimal control for\newline a model of combined anticancer therapy with control delays

The research presented here was partially supported by the National Science Centre (NCN) in Poland grant DEC-2014/13/B/ST7/00755 (JK, AS). Some preliminary results were presented at the Conference: Micro and Macro Systems in Life Sciences, Bedlewo, 2015. HM is grateful to Urszula Ledzewicz who provided a grant for him to attend the Conference in Bedlewo.
• We study some control properties of a class of two-compartmental models of response to anticancer treatment which combines anti-angiogenic and cytotoxic drugs and take into account multiple control delays. We formulate sufficient local controllability conditions for semilinear systems resulting from these models. The control delays are related to PK/PD effects and some clinical recommendations, e.g., normalization of the vascular network. The optimized protocols of the combined therapy for the model, considered as solutions to an optimal control problem with delays in control, are found using necessary conditions of optimality and numerical computations. Our numerical approach uses dicretization and nonlinear programming methods as well as the direct optimization of switching times. The structural sensitivity of the considered control properties and optimal solutions is also discussed.

Mathematics Subject Classification: Primary: 49J15; Secondary: 93B05, 37N25, 92C50, 93C15, 65K10.

 Citation: • • Figure 5.  Optimal solution for the Hahnfeldt model with logistic growth function and delays $h_1= 10.6$ in $u$ and $h=1.84$ in $v$, objective $J(u) = p(T) + 0.5\, q(T)$ and fixed final time $T=16$. Initial conditions $p(0)=12000, \, q(0) = 15000$ and control bounds $u_{\rm max} = 40, w_{\rm max} = 320, v_{\rm max} = 2, \, z_{\rm max} = 10$. (a) control $u$, (b) control $v$, (c) tumor volume $p$ and vasculature $q$

Figure 1.  Optimal solution for the Hahnfeldt model with Gompertz-type growth function $f(p, q) = - \xi\, p \, \ln (p/q)$, objective $J(u) = p(T)$ and free terminal time $T$. Initial conditions $p(0)=12000, \, q(0) = 15000$ and control bounds $u_{\rm max} = 75, w_{\rm max} = 300, v_{\rm max} = 2, z_{\rm max} = 10$. (a) control $u$, (b) control $v$, (c) tumor volume $p$ and vasculature $q$

Figure 2.  Optimal solution for the Hahnfeldt model with logistic-type growth function $f(p, q)= \xi p (1-p/q)$, objective $J(u) = p(T) + 0.2\, q(T)$ and fixed final time $T=16$. Initial conditions $p(0)=12000, \, q(0) = 15000$ and control bounds $u_{\rm max} = 40, w_{\rm max} = 320, v_{\rm max} = 2, \, z_{\rm max} = 10$. (a) control $u$, (b) control $v$, (c) tumor volume $p$ and vasculature $q$

Figure 3.  Optimal solution for the Hahnfeldt model with logistic growth function$f(p, q) = \xi p (1-p/q)$, objective $J(u) = p(T)$ and free terminal time $T$. Initial conditions $p(0)=12000, \, q(0) = 15000$ and control bounds $u_{\rm max} = 75, w_{\rm max} = 300, v_{\rm max} = 2, z_{\rm max} = 10$. Optimal control $v(t) \equiv 2$ and $T=5$. (a) control $u$ and switching function $\phi_u$ satisfying (41), (b) tumor volume $p$ and vasculature $q$

Figure 4.  Optimal solution for the Hahnfeldt model with logistic growth function $f(p, q) = \xi p (1-p/q)$, objective $J(u) = p(T) + 0.5 q(T)$ and free terminal time $T$. Initial conditions $p(0)=12000, \, q(0) = 15000$ and control bounds $u_{\rm max} = 75, w_{\rm max} = 300, v_{\rm max} = 2, z_{\rm max} = 10$. Optimal control $v(t) \equiv 2$ and $T=5$. (a) control $u$, (b) tumor volume $p$ and vasculature $q$

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