Identifying and characterising the impact of excitability in a mathematical model of tumour-immune interactions

TL;DR

This study develops a new reduced mathematical model of tumour-immune interactions that reveals the excitable nature of the system, providing insights into complex dynamics and potential immunotherapy strategies.

Contribution

A novel two-dimensional approximation of a tumour-immune model that uncovers excitability and its role in tumour-immune dynamics, validated against the full model.

Findings

Model exhibits excitable dynamics not previously associated with tumour-immune systems.

Excitability leads to complex bifurcations and clinically relevant phenomena.

Parameter sensitivity analysis suggests personalized immunotherapy targets.

Abstract

We study a five-compartment mathematical model originally proposed by Kuznetsov et al. (1994) to investigate the effect of nonlinear interactions between tumour and immune cells in the tumour microenvironment, whereby immune cells may induce tumour cell death, and tumour cells may inactivate immune cells. Exploiting a separation of timescales in the model, we use the method of matched asymptotics to derive a new two-dimensional, long-timescale, approximation of the full model, which differs from the quasi-steady-state approximation introduced by Kuznetsov et al. (1994), but is validated against numerical solutions of the full model. Through a phase-plane analysis, we show that our reduced model is excitable, a feature not traditionally associated with tumour-immune dynamics. Through a systematic parameter sensitivity analysis, we demonstrate that excitability generates complex bifurcating dynamics in the model. These are consistent with a variety of clinically observed phenomena, and suggest that excitability may underpin tumour-immune interactions. The model exhibits the three stages of immunoediting - elimination, equilibrium, and escape, via stable steady states with different tumour cell concentrations. Such heterogeneity in tumour cell numbers can stem from variability in initial conditions and/or model parameters that control the properties of the immune system and its response to the tumour. We identify different biophysical parameter targets that could be manipulated with immunotherapy in order to control tumour size, and we find that preferred strategies may differ between patients depending on the strength of their immune systems, as determined by patient-specific values of associated model parameters.