Optimal control for a class of compartmental models in cancer chemotherapy

Andrzej Świerniak; Urszula Ledzewicz; Heinz Schättler

International Journal of Applied Mathematics and Computer Science (2003)

  • Volume: 13, Issue: 3, page 357-368
  • ISSN: 1641-876X

Abstract

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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.

How to cite

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Świerniak, Andrzej, Ledzewicz, Urszula, and Schättler, Heinz. "Optimal control for a class of compartmental models in cancer chemotherapy." International Journal of Applied Mathematics and Computer Science 13.3 (2003): 357-368. <http://eudml.org/doc/207650>.

@article{Świerniak2003,
abstract = {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.},
author = {Świerniak, Andrzej, Ledzewicz, Urszula, Schättler, Heinz},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {necessary and sufficient conditions for optimality; cancer chemotherapy; optimal control; compartmental models; compartment models; bang-bang control; simulations},
language = {eng},
number = {3},
pages = {357-368},
title = {Optimal control for a class of compartmental models in cancer chemotherapy},
url = {http://eudml.org/doc/207650},
volume = {13},
year = {2003},
}

TY - JOUR
AU - Świerniak, Andrzej
AU - Ledzewicz, Urszula
AU - Schättler, Heinz
TI - Optimal control for a class of compartmental models in cancer chemotherapy
JO - International Journal of Applied Mathematics and Computer Science
PY - 2003
VL - 13
IS - 3
SP - 357
EP - 368
AB - 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.
LA - eng
KW - necessary and sufficient conditions for optimality; cancer chemotherapy; optimal control; compartmental models; compartment models; bang-bang control; simulations
UR - http://eudml.org/doc/207650
ER -

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