Tumour chemotherapy strategy based on impulse control theory

Hai-Peng Ren, Yan Yang, Murilo S. Baptista, Celso Grebogi

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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.
Original languageEnglish
Article number20160221
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences
Issue number2088
Early online date23 Jan 2017
Publication statusPublished - 6 Mar 2017

Bibliographical note

Funding. H.-P.R. was supported in part by NSFC (60804040), Fok Ying Tong Education Foundation Young Teacher Foundation (111065), Innovation Research Team of Shaanxi Province (2013KCT-04) and the Key programme of Natural Science Foundation of Shaanxi Province (2016ZDJC-01). M.S.B. was supported in part
by the EPSRC (EP/I032606/1).


  • chemotherapy
  • impulse control system
  • stability
  • permanence
  • boundedness


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