The importance of chaotic attractors in modelling tumour growth

We examine the importance of chaotic attractors when modelling non-metastatic tumour growth using a model in which cells come in three types: host, immune, and tumour. The relationships between these cell populations are derived from the law of mass action, assuming that a conjugate is formed in the interaction between immune and tumour cells. A nonlinearity in the production of immune cells, based on previous analyses, is introduced and explained. Using previously chosen model parameters, the maximal Lyapunov exponent is calculated numerically as 0.0218, demonstrating the existence of chaotic behaviour. Under the variation of one particular nonlinear rate constant, four distinct types of attractor are observed. Of biological importance, chaotic behaviour is shown to lead to a significantly higher maximum tumour size when compared to non-chaotic behaviour. Counterintuitively, increasing the parameter associated with the killing of tumour cells by immune cells is demonstrated to increase the maximum tumour size when this parameter is below the threshold at which the equilibrium is zero tumour cells.