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

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 The traditional view that resolution of inflammation is a passive process is now being superceded by an alternative hypothesis whereby its resolution is an active, anti-inflammatory process that can be manipulated therapeutically.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.

Parameter Values

In this appendix, we discuss the values of the dimensional parameters and nondimensional parameter groups. Obtaining precise values for parameters for such biological processes is difficult for several reasons. Some biological mechanisms in inflammation remain unclear (Haslett 1999); measurements of acute inflammatory markers in vivo are difficult to obtain because acute inflammation is typically short-lived and patients often do not present until the condition has progressed beyond the acute stage. Further the parameter values can vary between tissue types; for example tendon, muscles and lung have different structures and patterns of vascularisation. Where possible, estimates are taken from the cited literature, and extra weight attached to data from humans or human cells and soft-tissue-specific data relative to other sources. If no data are available then order of magnitude estimates that give biologically realistic results are employed.

Decay and death rates are readily available in literature. Extracellular mediator decay rates are reported to lie in the range 0.7–20 per day (Smith et al. 2011; Waugh and Sherratt 2007; Su et al. 2009). Accordingly we fix $\gamma _c=3$ day$^{-1}$ for pro-inflammatory mediator decay. We expect our two generic mediators to have a similar half-life ($\gamma _c\sim \gamma _g$) and neutrophils are reported to undergo secondary necrosis ($\tilde{\gamma _a}$) within a few hours of their death by apoptosis (Haslett 1999). Accordingly we set

$$\begin{aligned} \tilde{\gamma }_a=\frac{\gamma _a}{\gamma _c}=1.0 \text {, and }\quad \tilde{\gamma }_g=\frac{\gamma _g}{\gamma _c}=1.0\text {.} \end{aligned}$$

Under normal conditions, neutrophils are known to have a short life dying within days (Akgul et al. 2001) while macrophages are resistant to apoptotic stimuli therefore living longer (Parihar et al. 2010), often for several weeks, before leaving the inflammatory site via the lymphatics (Serhan and Savill 2005). Accordingly we set

$$\begin{aligned} \tilde{\nu }=\frac{\nu }{\gamma _c}=0.1\text {, and }\quad \tilde{\gamma }_m=\frac{\gamma _m}{\gamma _c}=0.01 \text {.} \end{aligned}$$

We can estimate from the literature the rate at which macrophages engulf cells (Mare et al. 2005; Wigginton and Kirschner 2001) (allowing us to set $\phi =1\times 10^{-3}$ mm$^3$ cell$^{-1}$ day$^{-1}$). We have estimates of the rate of production of TGF$_\beta$ by macrophages of 0.07 pg cell$^{-1}$ day$^{-1}$ (Waugh and Sherratt 2007). The production of a generic anti-inflammatory mediator (such as TGF$_\beta$) is captured in our model by the product of parameters $k_g$ (the rate that macrophages produce anti-inflammatory mediator) and $\phi$ (the rate at which macrophages engulf cells) allowing us to fix $k_g$ so that $k_g\,\phi =0.07$ pg cell$^{-1}$ day$^{-1}$. In the context of this model there is no available data for the rates of influx of neutrophils and macrophages (though we know that neutrophils arrive much quicker than macrophages (Butterfield et al. 2006)) or for the production of generic pro-inflammatory mediators ($k_a$, $k_n$) or saturation constants ($\beta _c$, $\beta _{gc}$, $\beta _g$). The remaining nondimensional parameters are based on groupings of known parameters and those for which no data are available (for example $\tilde{\phi }=\phi \chi _m k_a / \gamma _c^2$). Since such groupings are difficult to estimate we use values based on biologically realistic results so that

$$\begin{aligned} \tilde{\phi }&= \frac{\phi \chi _m k_a}{\gamma _c^2}=0.001\text {,}\quad \tilde{k}_{n}=\frac{k_n k_a}{\gamma _c}=0.01\text {,}\quad \tilde{k}_{g}=\frac{k_g \chi _n k_a}{\beta _{gc} \gamma _c}=0.1\text {,}\quad \tilde{\alpha }=\frac{\alpha }{\gamma _c k_a}=0.05\text {,}\\ \tilde{\beta _a}&= \frac{\beta _a \gamma _c}{\chi _n k_a}=0.1\text {,}\quad \tilde{\beta _n}=\frac{\beta _n\gamma _c}{\chi _n k_a}= 0.1\text {,}\quad \tilde{\beta }_g=\frac{\beta _g}{\beta _{gc}}=0.01 \text {,}\quad \tilde{\beta }_c=\frac{\beta _c}{k_a}= 0.12\text {.}\quad \end{aligned}$$