Bulletin of Mathematical Biology

, Volume 81, Issue 10, pp 4144–4173 | Cite as

Tumor Clearance Analysis on a Cancer Chemo-Immunotherapy Mathematical Model

  • Paul A. ValleEmail author
  • Luis N. Coria
  • Yolocuauhtli Salazar
Original Article


Mathematical models may allow us to improve our knowledge on tumor evolution and to better comprehend the dynamics between cancer, the immune system and the application of treatments such as chemotherapy and immunotherapy in both short and long term. In this paper, we solve the tumor clearance problem for a six-dimensional mathematical model that describes tumor evolution under immune response and chemo-immunotherapy treatments. First, by means of the localization of compact invariant sets method, we determine lower and upper bounds for all cells populations considered by the model and we use these results to establish sufficient conditions for the existence of a bounded positively invariant domain in the nonnegative orthant by applying LaSalle’s invariance principle. Then, by exploiting a candidate Lyapunov function we determine sufficient conditions on the chemotherapy treatment to ensure tumor clearance. Further, we investigate the local stability of the tumor-free equilibrium point and compute conditions for asymptotic stability and tumor persistence. All conditions are given by inequalities in terms of the system parameters, and we perform numerical simulations with different values on the chemotherapy treatment to illustrate our results. Finally, we discuss the biological implications of our work.


Tumor clearance Tumor persistence Global stability Chemo-immunotherapy 



This work is fulfilled within the TecNM Project Number 6575.18-P “Ingeniería aplicada mediante el modelizado matemático para comprender la evolución del cáncer, la respuesta inmunológica y el efecto de algunos tratamientos,” the TecNM Project Number 6178.17-P “Modelos matemáticos para cáncer, VIH y enfisema,” the TecNM Project “Biomatemáticas: modelizado y análisis para comprender fenómenos del mundo real” and the PRODEP research group ITTIJ-CA-6.


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Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Postgraduate Program in Engineering Sciences, BioMath Research GroupTijuana Institute of TechnologyTijuanaMexico
  2. 2.Department of Electric and Electronic EngineeringDurango Institute of TechnologyDurangoMexico

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