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An optimal control approach to cancer treatment under immunological activity

100%
Ledzewicz U., Naghnaeian M., Schättler H.
Applicationes Mathematicae
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2011
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tom 38
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nr 1
17-31
EN
Mathematical models for cancer treatment that include immunological activity are considered as an optimal control problem with an objective that is motivated by a separatrix of the uncontrolled system. For various growth models on the cancer cells the existence and optimality of singular controls is investigated. For a Gompertzian growth function a synthesis of controls that move the state into the region of attraction of a benign equilibrium point is developed.
2
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Optimal solutions for a model of tumor anti-angiogenesis with a penalty on the cost of treatment

100%
Ledzewicz U., Oussa V., Schättler H.
Applicationes Mathematicae
|
2009
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tom 36
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nr 3
295-312
EN
The scheduling of angiogenic inhibitors to control a vascularized tumor is analyzed as an optimal control problem for a mathematical model that was developed and biologically validated by Hahnfeldt et al. [Cancer Res. 59 (1999)]. Two formulations of the problem are considered. In the first one the primary tumor volume is minimized for a given amount of angiogenic inhibitors to be administered, while a balance between tumor reduction and the total amount of angiogenic inhibitors given is minimized in the second formulation. The optimal solutions to both problems are presented and compared.
3
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Optimal control for a class of compartmental models in cancer chemotherapy

100%
Świerniak A., Ledzewicz U., Schättler H.
International Journal of Applied Mathematics and Computer Science
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2003
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tom 13
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nr 3
357-368
EN
We consider a general class of mathematical models P for cancer chemotherapy described as optimal control problems over a fixed horizon with dynamics given by a bilinear system and an objective which is linear in the control. Several two- and three-compartment models considered earlier fall into this class. While a killing agent which is active during cell division constitutes the only control considered in the two-compartment model, Model A, also two three-compartment models, Models B and C, are analyzed, which consider a blocking agent and a recruiting agent, respectively. In Model B a blocking agent which slows down cell growth during the synthesis allowing in consequence the synchronization of the neoplastic population is added. In Model C the recruitment of dormant cells from the quiescent phase to enable their efficient treatment by a cytotoxic drug is included. In all models the cumulative effect of the killing agent is used to model the negative effect of the treatment on healthy cells. For each model it is shown that singular controls are not optimal. Then sharp necessary and sufficient optimality conditions for bang-bang controls are given for the general class of models P and illustrated with numerical examples.
4
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Singular controls and chattering arcs in optimal control problems arising in biomedicine

87%
Ledzewicz U., Schättler H.
Control and Cybernetics
|
2009
|
tom 38
|
nr 4B
1501-1523
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