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Comput Math Methods Med. 2014;2014:982978. doi: 10.1155/2014/982978. Epub 2014 Aug 14.

Delay differential model for tumour-immune response with chemoimmunotherapy and optimal control.

Author information

1
Department of Mathematical Sciences, College of Science, UAE University, P.O. Box 15551, Al-Ain, UAE ; Department of Mathematics, Faculty of Science, Helwan University, Cairo 11795, Egypt.
2
Department of Mathematical Sciences, College of Science, UAE University, P.O. Box 15551, Al-Ain, UAE.
3
Zayed bin Sultan Al Nahyan Center for Health Sciences, College of Medicine and Health Sciences, UAE University, P.O. Box 17666, Al-Ain, UAE.
4
Institut für Angewandte Mathematik, LS III, TU Dortmund, Vogelpothsweg 87, 44227 Dortmund, Germany ; Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt.
5
Department of Mathematics, Faculty of Science, Helwan University, Cairo 11795, Egypt.

Abstract

We present a delay differential model with optimal control that describes the interactions of the tumour cells and immune response cells with external therapy. The intracellular delay is incorporated into the model to justify the time required to stimulate the effector cells. The optimal control variables are incorporated to identify the best treatment strategy with minimum side effects by blocking the production of new tumour cells and keeping the number of normal cells above 75% of its carrying capacity. Existence of the optimal control pair and optimality system are established. Pontryagin's maximum principle is applicable to characterize the optimal controls. The model displays a tumour-free steady state and up to three coexisting steady states. The numerical results show that the optimal treatment strategies reduce the tumour cells load and increase the effector cells after a few days of therapy. The performance of combination therapy protocol of immunochemotherapy is better than the standard protocol of chemotherapy alone.

PMID:
25197319
PMCID:
PMC4147205
DOI:
10.1155/2014/982978
[Indexed for MEDLINE]
Free PMC Article

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