Modelling the Immune Response to Cancer: An Individual-Based Approach Accounting for the Difference in Movement Between Inactive and Activated T Cells

Abstract

A growing body of experimental evidence indicates that immune cells move in an unrestricted search pattern if they are in the pre-activated state, whilst they tend to stay within a more restricted area upon activation induced by the presence of tumour antigens. This change in movement is not often considered in the existing mathematical models of the interactions between immune cells and cancer cells. With the aim to fill such a gap in the existing literature, in this work we present a spatially structured individual-based model of tumour–immune competition that takes explicitly into account the difference in movement between inactive and activated immune cells. In our model, a Lévy walk is used to capture the movement of inactive immune cells, whereas Brownian motion is used to describe the movement of antigen-activated immune cells. The effects of activation of immune cells, the proliferation of cancer cells and the immune destruction of cancer cells are also modelled. We illustrate the ability of our model to reproduce qualitatively the spatial trajectories of immune cells observed in experimental data of single-cell tracking. Computational simulations of our model further clarify the conditions for the onset of a successful immune action against cancer cells and may suggest possible targets to improve the efficacy of cancer immunotherapy. Overall, our theoretical work highlights the importance of taking into account spatial interactions when modelling the immune response to cancer cells.

Appendix

See Figs. 9, 10 and Table 2.

Fig. 9

Increasing the number of DCs can lead to longer tumour removal times. Heat maps showing the time evolution of the number of tumour cells for 40 different numbers of immune cells (left panels) and selected samples of the time evolution of the number of tumour cells for 4 different numbers of immune cells (right panels). In all cases under consideration, at the beginning of simulations the tumour contained 1200 cells. Top panels: Increasing the CTL number $N_{C}$, we observe a general decrease in tumour removal time. Middle panels: Increasing the DC number $N_{D}$ above a certain threshold leads to longer tumour removal time. Bottom panels: Increasing both $N_{C}$ and $N_{D}$ causes a decrease in tumour removal time (Color figure online)

Fig. 10

The ratio between the killing rate of tumour cells by CTLs and the tumour cell proliferation rate is a crucial parameter in tumour removal. Heat maps showing the evolution of the total number of tumour cells over time for 40 different values of the rate at which CTLs kill tumour cells, $\mu$, and/or the tumour cell proliferation rate, $\lambda$ (left panels). Sample time evolutions of the tumour cell number for 4 values of $\mu$ and/or $\lambda$ (right panels). In all cases under consideration, at the beginning of simulations the tumour contained 1200 cells. Top panels: Varying $\mu$, we observe a decrease in tumour removal time with increasing values of $\mu$, with little difference between the larger values. Middle panels: Varying $\lambda$ results in an increase in tumour removal time with increasing $\lambda$ and eventually a larger number of tumour cells remaining. Bottom panels: Varying both $\lambda$ and $\mu$ at equal ratios results in a decrease in tumour removal time for increasing values of $\mu$ and $\lambda$, although (in general) is slower than the case where only $\mu$ is altered (Color figure online)

Table 2 Values of parameters used in the model