title: Tumour chemotherapy strategy based on impulse control theory.
abstract: Chemotherapy is a widely accepted method for tumour treatment. A medical doctor usually treats patients periodically with an amount of drug according to empirical medicine guides. From the point of view of cybernetics, this procedure is an impulse control system, where the amount and frequency of drug used can be determined analytically using the impulse control theory. In this paper, the stability of a chemotherapy treatment of a tumour is analysed applying the impulse control theory. The globally stable condition for prescription of a periodic oscillatory chemotherapeutic agent is derived. The permanence of the solution of the treatment process is verified using the Lyapunov function and the comparison theorem. Finally, we provide the values for the strength and the time interval that the chemotherapeutic agent needs to be applied such that the proposed impulse chemotherapy can eliminate the tumour cells and preserve the immune cells. The results given in the paper provide an analytical formula to guide medical doctors to choose the theoretical minimum amount of drug to treat the cancer and prevent harming the patients because of over-treating.This article is part of the themed issue 'Horizons of cybernetical physics'.
authors.affiliation: Shaanxi Key Laboratory of CSCIIP, Xi'an University of Technology, Xi'an 710048, People's Republic of China renhaipeng@xaut.edu.cn.
authors.forename: Hai-Peng
authors.identifier: http://orcid.org/0000-0003-3834-5103
authors.initials: HP
authors.lastname: Ren
authors.affiliation: Shaanxi Key Laboratory of CSCIIP, Xi'an University of Technology, Xi'an 710048, People's Republic of China.
authors.forename: Yan
authors.initials: Y
authors.lastname: Yang
authors.affiliation: Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, UK.
authors.forename: Murilo S
authors.initials: MS
authors.lastname: Baptista
authors.affiliation: Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, UK.
authors.forename: Celso
authors.initials: C
authors.lastname: Grebogi
mesh_terms: D049109:Cell Proliferation / Q000187:drug effects; D008954:Models, Biological; D009369:Neoplasms / Q000188:drug therapy* / Q000473:pathology
keywords: boundedness; chemotherapy; impulse control system; permanence; stability