Numerical solution of an optimal control model of dendritic cell treatment of a growing tumour

Abstract

A new optimal control model of the interactions between a growing tumour and the host immune system, along with an immunotherapy treatment strategy, is presented. The model is based on an ordinary differential equation model of interactions between the growing tumour and the natural killer, cytotoxic T lymphocyte and dendritic cells of the host immune system, extended through the addition of a control function representing the application of a dendritic cell treatment to the system. The numerical solution of this model, obtained from a multi species Runge,Kutta forward-backward sweep scheme, is described. We investigate the effects of varying the maximum allowed amount of dendritic cell vaccine administered to the system and find that control of the tumour cell population is best effected via a high initial vaccine level, followed by reduced treatment and finally cessation of treatment. We also found that increasing the strength of the dendritic cell vaccine causes an increase in the number of natural killer cells and lymphocytes, which in turn reduces the growth of the tumour. © Austral. Mathematical Soc. 2013.