Tumour chemotherapy strategy based on impulse control theory. (English) Zbl 1404.92103
Philos. Trans. R. Soc. Lond., A, Math. Phys. Eng. Sci. 375, No. 2088, Article ID 20160221, 15 p. (2017).
Summary: 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.
MSC:
| 92C50 | Medical applications (general) |
| 93C95 | Application models in control theory |
| 92-08 | Computational methods for problems pertaining to biology |
References:
| [1] | Mamat M, Subiyanto X, Kartono A. (2013) Mathematical model of cancer treatments using immunotherapy, chemotherapy and biochemotherapy. Appl. Math. Sci. 7, 247-261. (doi:10.12988/ams.2013.13023) · doi:10.12988/ams.2013.13023 |
| [2] | Carl Panetta J. (1996) A mathematical model of periodically pulsed chemotherapy: tumour recurrence and metastasis in a competitive environment. Bull. Math. Biol. 58, 425-447. (doi:10.1007/BF02460591) · Zbl 0859.92014 · doi:10.1007/BF02460591 |
| [3] | Ledzewicz U, Naghnaeian M, Schttler H. (2012) Optimal response to chemotherapy for a mathematical model of tumour-immune dynamics. J. Math. Biol. 64, 557-577. (doi:10.1007/s00285-011-0424-6) · Zbl 1284.92042 · doi:10.1007/s00285-011-0424-6 |
| [4] | Lakmeche A, Arino O. (2001) Nonlinear mathematical model of pulsed therapy of heterogeneous tumours. Nonlinear Anal. 2, 455-465. (doi:10.1016/S1468-1218(01)00003-7) · Zbl 0982.92016 · doi:10.1016/S1468-1218(01)00003-7 |
| [5] | Solis FJ, Delgadillo SE. (2013) Discrete modeling of aggressive tumour growth with gradual effect of chemotherapy. Math. Comput. Model. 57, 1919-1926. (doi:10.1016/j.mcm.2011.12.032) · Zbl 1305.92044 · doi:10.1016/j.mcm.2011.12.032 |
| [6] | Lakshmi Kiran K, Lakshminarayanan S. (2013) Optimization of chemotherapy and immunotherapy: in silico analysis using pharmacokinetic-pharmacodynamic and tumour growth models. J. Process Control 23, 396-403. (doi:10.1016/j.jprocont.2012.12.006) · doi:10.1016/j.jprocont.2012.12.006 |
| [7] | Villasana M, Radunskaya A. (2003) A delay differential equation model for tumour growth. J. Math. Biol. 47, 270-294. (doi:10.1007/s00285-003-0211-0) · Zbl 1023.92014 · doi:10.1007/s00285-003-0211-0 |
| [8] | Wang X, Tao Y, Song X. (2011) Global stability of a virus dynamics model with Beddington-DeAngelis incidence rate and CTL immune response. Nonlinear Dyn. 66, 825-830. (doi:10.1007/s11071-011-9954-0) · Zbl 1242.92045 · doi:10.1007/s11071-011-9954-0 |
| [9] | Borges FS, Iarosz KC, Ren HP, Batista AM, Baptista MS, Viana RL, Lopes SR, Grebogi C. (2014) Model for tumour growth with treatment by continuous and pulsed chemotherapy. BioSystems 116, 43-48. (doi:10.1016/j.biosystems.2013.12.001) · doi:10.1016/j.biosystems.2013.12.001 |
| [10] | Song X, Guo H, Shi X(2011) Theory and application of impulsive differential equations. Beijing, China: Science Press. |
| [11] | Pei Y, Zeng G, Chen L. (2008) Species extinction and permanence in a prey-predator model with two-type functional responses and impulsive biological control. Nonlinear Dyn. 51, 71-81. (doi:10.1007/s11071-007-9258-6) · Zbl 1169.92048 · doi:10.1007/s11071-007-9258-6 |
| [12] | Dagbovie AS, Sherratt JA. (2014) Absolute stability and dynamical stabilisation in predator-prey systems. J. Math. Biol. 68, 1403-1421. (doi:10.1007/s00285-013-0672-8) · Zbl 1284.92109 · doi:10.1007/s00285-013-0672-8 |
| [13] | Ma J, Zhang Q, Gao Q. (2012) Stability of a three-species symbiosis model with delays. Nonlinear Dyn. 67, 567-572. (doi:10.1007/s11071-011-0009-3) · Zbl 1242.92059 · doi:10.1007/s11071-011-0009-3 |
| [14] | Huang M, Liu S, Song X, Chen L. (2013) Periodic solutions and homoclinic bifurcation of a predator-prey system with two types of harvesting. Nonlinear Dyn. 73, 815-826. (doi:10.1007/s11071-013-0834-7) · Zbl 1281.92069 · doi:10.1007/s11071-013-0834-7 |
| [15] | Zhang Y, Liu B, Chen L. (2003) Extinction and permanence of a two-prey one-predator system with impulsive effect. IMA J. Math. Med. Biol. 20, 309-325. (doi:10.1093/imammb/20.4.309) · Zbl 1046.92051 · doi:10.1093/imammb/20.4.309 |
| [16] | Xiang Z, Song X. (2006) Extinction and permanence of a two-prey two-predator system with impulsive on the predator. Chaos Solitons Fractals 29, 1121-1136. (doi:10.1016/j.chaos.2005.08.076) · Zbl 1142.34306 · doi:10.1016/j.chaos.2005.08.076 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
