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On the role of pharmacometrics in mathematical models for cancer treatments

  • * Corresponding author: Urszula Ledzewicz

    * Corresponding author: Urszula Ledzewicz 
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  • We review and discuss various aspects that the modeling of pharmacometric properties has on the structure of optimal solutions in mathematical models for cancer treatment. These include (i) the changes in the interpretation of the solutions as pharmacokinetic (PK) models are added, respectively deleted from the modeling and (ii) qualitative changes in the structures of optimal controls that occur as pharmacodynamic (PD) models are varied. The results will be illustrated with a sample of models for cancer treatment.

    Mathematics Subject Classification: primary: 92C50, 49K15, secondary: 93C15.

    Citation:

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  • Figure 1.  A schematic representation of the interactions described by pharmacometric models

    Figure 2.  $ E_{max} $ model for the PD of a drug

    Figure 3.  Sigmoidal model for the PD of a drug

    Figure 4.  Formulation [CC0]: Example of an optimal control of the type $ \mathbf{u}_{\max}\mathbf{s0} $ as function of time (left) and corresponding controlled trajectory (right) shown with the tumor volume $ p $ on the vertical axis and the vasculature $ q $ on the horizontal axis for initial data $ (p_{0}, q_{0}, A) = (12000, 15000, 300) $. The full dose segment for the control and its corresponding trajectory are shown in red, the singular pieces in blue and the no dose segments in green. While the control is singular, the system follows the singular arc $ S $. The dynamics is very fast horizontally along the full and no-dose segments which generates long segments of the trajectory for small time intervals, but it is much slower vertically along the singular control. This segment is the dominant piece in time and the only segment where a significant reduction of the vasculature is achieved

    Figure 5.  Comparison of the solution to the restricted optimization problem where controls are kept constant for 3 months for year 3 (top row; controls shown on the left and the corresponding states are given on the right) with the solutions to the optimal control problem for year 3 (shown in the bottom row)

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