A dynamical model of tumour immunotherapy

Math Biosci. 2014 Jul:253:50-62. doi: 10.1016/j.mbs.2014.04.003. Epub 2014 Apr 20.


A coupled ordinary differential equation model of tumour-immune dynamics is presented and analysed. The model accounts for biological and clinical factors which regulate the interaction rates of cytotoxic T lymphocytes on the surface of the tumour mass. A phase plane analysis demonstrates that competition between tumour cells and lymphocytes can result in tumour eradication, perpetual oscillations, or unbounded solutions. To investigate the dependence of the dynamic behaviour on model parameters, the equations are solved analytically and conditions for unbounded versus bounded solutions are discussed. An analytic characterisation of the basin of attraction for oscillatory orbits is given. It is also shown that the tumour shape, characterised by a surface area to volume scaling factor, influences the size of the basin, with significant consequences for therapy design. The findings reveal that the tumour volume must surpass a threshold size that depends on lymphocyte parameters for the cancer to be completely eliminated. A semi-analytic procedure to calculate oscillation periods and determine their sensitivity to model parameters is also presented. Numerical results show that the period of oscillations exhibits notable nonlinear dependence on biologically relevant conditions.

Keywords: Cancer; Dynamical systems; Finite-time tumour elimination; Immunotherapy; Ordinary differential equations.

Publication types

  • Research Support, Non-U.S. Gov't

MeSH terms

  • Cancer Vaccines / therapeutic use
  • Humans
  • Immunotherapy*
  • Mathematical Concepts
  • Models, Immunological*
  • Neoplasms / immunology*
  • Neoplasms / pathology
  • Neoplasms / therapy*
  • Nonlinear Dynamics
  • T-Lymphocytes, Cytotoxic / immunology


  • Cancer Vaccines