# The Resolution of Inflammation: A Mathematical Model of Neutrophil and Macrophage Interactions

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## Abstract

There is growing interest in inflammation due to its involvement in many diverse medical conditions, including Alzheimer’s disease, cancer, arthritis and asthma. The traditional view that resolution of inflammation is a passive process is now being superceded by an alternative hypothesis whereby its resolution is an active, anti-inflammatory process that can be manipulated therapeutically. This shift in mindset has stimulated a resurgence of interest in the biological mechanisms by which inflammation resolves. The anti-inflammatory processes central to the resolution of inflammation revolve around macrophages and are closely related to pro-inflammatory processes mediated by neutrophils and their ability to damage healthy tissue. We develop a spatially averaged model of inflammation centring on its resolution, accounting for populations of neutrophils and macrophages and incorporating both pro- and anti-inflammatory processes. Our ordinary differential equation model exhibits two outcomes that we relate to healthy and unhealthy states. We use bifurcation analysis to investigate how variation in the system parameters affects its outcome. We find that therapeutic manipulation of the rate of macrophage phagocytosis can aid in resolving inflammation but success is critically dependent on the rate of neutrophil apoptosis. Indeed our model predicts that an effective treatment protocol would take a dual approach, targeting macrophage phagocytosis alongside neutrophil apoptosis.

## Keywords

Inflammation Resolution Mathematical modelling Bifurcation analysis## Notes

### Acknowledgments

JLD gratefully acknowledges support from the Engineering and Physical Sciences Research Council, the Health and Safety Laboratory and the industrial mathematics KTN for this work in the form of a CASE studentship. JRK acknowledges the funding of the Royal Society and Wolfson Foundation. The work of HMB was supported in part by Award No. KUK-013-04, made by King Abdullah University of Science and Technology (KAUST).

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