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Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics

  • * Corresponding author: Urszula Ledzewicz

    * Corresponding author: Urszula Ledzewicz 
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  • We analyze the structure of optimal protocols for a mathematical model of tumor anti-angiogenic treatment. The control represents the concentration of the agent and we consider the problem to administer an a priori given total amount of agents in order to achieve a minimum tumor volume/maximum tumor reduction. In earlier work, this problem was studied with a log-kill type pharmacodynamic model for drug effects which does not account for saturation of the drug concentration. Here we study the effect of incorporating a Michaelis-Menten (MM) or $ E_{\max} $-type pharmacodynamic model, the most commonly used model in the field of pharmacometrics. We compare the formulations of both problems and the resulting solutions. The reformulated problem with $ E_{\max} $ pharmacodynamics is no longer linear in the control. This results in qualitative changes in the structure of optimal controls which, in line with an interpretation as concentrations, now are continuous while discontinuities exist if the log-kill model is used which is more in line with an interpretation of the control as dose rates. In spite of these qualitative differences, similarities in the structures of solutions can be observed. Both aspects are discussed theoretically and illustrated numerically.

    Mathematics Subject Classification: Primary: 92C50, 49K15; Secondary: 93C15.


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  • Figure 1.  $ E_{\max} $ pharmacodynamic model.

    Figure 2.  Geometric shape of the sets $ {\mathcal L}_0 $ and $ {\mathcal L}_m $ and definition of the associated regions $ R_i $ and $ S_i $ for ${i}{\rm{ = 1, 2, 3, 4}}{\rm{.}} $

    Figure 3.  The region $ S_2 $ (inside the loop $ {\mathcal L}_m $ and above the diagonal) is not positively invariant for $ u = u_{\max} $.

    Figure 4.  Graph of the value $ V(\varepsilon) $ for the one-parameter family $ {\mathcal F}_{\varepsilon} $ near its minimum value.

    Figure 5.  Optimal control (left) and corresponding trajectory (right) for the initial condition $ (p_0, q_0) = (4600, 5800) $ and $ A = 12 $. The graphs are shown with $ UC_{50} = 50 $, $ U_{\max} = 1.5UC_{50} = 75 $ and $ A = 12UC_{50} = 600 $. The optimal controlled trajectory crosses the diagonal along a full dose segment and the optimal concatenation sequence is full dose $ \rightarrow $ $ \tilde{u} $ $ \rightarrow $ no dose. The initial full dose segment is shown in black, the intermediate interior control segment in blue and the final no dose segment in black; junctions are marked with an asterisk. The loop $ {\mathcal L}_m $ is shown as the dashed red curve.

    Figure 6.  Optimal control (left) and corresponding trajectory (right) for the initial condition $ (p_0, q_0) = (12000, 15000) $ and $ A = 2 $. The graphs are shown with $ UC_{50} = 50 $, $ U_{\max} = 1.5UC_{50} = 75 $ and $ A = 2UC_{50} = 100 $. The optimal controlled trajectory crosses the diagonal using intermediate doses along the control $ \tilde{u} $ and the optimal concatenation sequence is of the shortened form $ \tilde{u} $ $ \rightarrow $ no dose. The optimal controlled trajectory is shown in red and as a comparison the corresponding full dose trajectory is shown in blue. Here there is a significant difference in the final tumor volumes.

    Table 1.  Parameter values used in numerical computations.

    interpretation numerical value dimension
    $ p $ tumor volume $ mm^{3} $
    $ q $ carrying capacity of the vasculature $ mm^{3} $
    $ \xi $ tumor growth parameter $ 0.192 $ per day
    $ \mu $ natural loss of endothelial support $ 0.02 $ per day
    $ b $ stimulation parameter, 'birth' $ 5.85 $ per day
    $ d $ inhibition parameter, 'death' $ 0.00873 $ $ mm^{-2} $
    per day
    $ G $ anti-angiogenic killing coefficient at limiting concentration $ 10 $ per day
    $ UC_{50} $ concentration with 50% effect $ 1 $
    $ u_{\max} $ normalized maximum dose rate $ 1.5 $
    $ A $ normalized total dose $ 2 $; $ 12 $
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