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Dunster et al., (2014). The resolution of inflammation: a mathematical model of neutrophil and macrophage interactions.

May 2017, model of the month by Corina Dueñas Roca
Original model: BIOMD0000000616


Introduction

Neutrophils and macrophages are essential players in the immune response. Therefore, there is a special focus to see how they participate and interact in the inflammatory process. The mathematical model presented here consists of a minimal model which is expanded into three phases. The first phase describe interactions between macrophages, active and apoptotic neutrophils after injury under sterile conditions. The second phase is the extension of this model by taking into account the interaction of neutrophils in the presence of pathogens, where anti-inflammatory mediators (as TGF-β) are noted as important players in the resolution of inflammation. Finally in a third stage the model is further extended to include the effect of macrophages and anti-inflammatory mediators. The final extended model is the most complex model, and shows the role of macrophages in the resolution of inflammation towards the healthy steady-state.


The Model

The spatially averaged model of inflammation described by Dunster et al. 2014 [1, BIOMD0000000616] comprises a base model of neutrophils and macrophages interaction after injury in sterile conditions, and two extended models that incorporate the participation of pathogens and anti-inflammatory elements released by macrophages.

The basic model includes a positive feedback loop (active neutrophils arriving in response to pro-inflammatory mediators and later becoming apoptotic) and a negative feedback loop (from active neutrophils that are responsible for the removal of pathogens-not described here) Figure 1. The extension of this model includes the participation of pathogens and the reaction of neutrophils. Therefore, an additional positive feedback is incorporated that links the activity of active and apoptotic neutrophils to the pro-inflammatory mediator. The last stage of this model supports the role of anti-inflammatory mediators released mainly by macrophages. This has been incorporated by an additional negative feedback loop. The additional feedback loop does not substantially alter the systems behavior but supports the original negative feedback loop (which describes the macrophages ability to remove apoptotic neutrophils).


Figure 2

Figure 2 Numerical simulations of basic model (or Model 1). Top row shows the effect of one cycle of damage over a function dependent on time. The damage is removed within a range of time that is considered safe enough for the system to return to the healthy state. Simulation shows the decrease of cells and mediators which promote inflammation, after the damage is removed. Bottom row shows the effect of more cycles of damage (1 to four), where the damage is removed outside the previous mentioned range of time. At this point the effect of pro-inflammatory cells or mediators is too strong, causing an unhealthy response to injury. Figure taken from [1].

Figure 1

Figure 1 Network diagram of the basic model (or Model 1). Feedback loops described in the basic mathematical model are represented here. The positive feedback describes the ability of apoptotic neutrophils to cause tissue damage and the neutrophils mobilization to the damaged site by means of pro-inflammatory mediators. The negative feedback is a consequence of macrophages and their removal of apoptotic neutrophils. Figure taken from [1].

Results

These models highlight two immune cell populations that play an important role during inflammation: neutrophils, promoting inflammation; and macrophages that resolves it.

In the basic model it is noted that the number of cycles of damage is influencing the outcome of inflammation. When the number of damage cycles are increased from one to four, the system reaches a second stable steady-state Figure 2. The levels of all cells and mediators are significantly greater when there are several cycles of damage when compared to the results caused by single damage cycle. In addition, although macrophage levels increase they are unable to remove all apoptotic neutrophils, that is possibly due to several cycles of injury. This second steady-state can be associated to chronic inflammation, where successful healing is not achieved since the inflammatory response does not decrease to a normal values (healthy steady state resolution).

The second positive feedback (from neutrophils to the pro-inflammatory mediator) shows an effect which agrees with other reported results. Moreover, the model describes the influence of the rate between macrophage phagocytosis and levels of neutrophil apoptosis, which enable a healthy or unhealthy condition. Anti-inflammatory mediators are under investigation for being used as therapeutic targets [2].

Furthermore targeting and perturbing these therapeutic targets, and increasing the rate of macrophage phagocytosis and a sufficient rate of neutrophil apoptosis are important for the resolution of the inflammation. In addition, combining the rate of macrophage phagocytosis with an increase in concentration of anti-inflammatory mediators help to achieve health conditions. Therefore treatment protocols applied after the damage can shift the model from an unhealthy to a healthy outcome. This model shows that treatment protocols should target multiple pathways in inflammation, ideally the rate of macrophage phagocytosis alongside that of neutrophil apoptosis.


Conclusion

An increasingly detailed model about inflammation and its resolution is presented here. This model helps to understand the role of the rates of neutrophils and macrophages to maintain healthy condition. In its simple form, inflammation occurs in response to tissue damage in a sterile environment, and the authors remark this model is specialized for such an environment, i.e. there is no presence of pathogens. Therefore a neutrophils' potential to damage healthy tissue is more evident. On the other hand, there are two extensions made to this model that takes into account, the pathogens and anti-inflammatory mediators (mainly by macrophages).

The dynamics of this model can be classified into four regimes: monostability, bistability, excitability and oscillation. Perturbation of some parameters in the model identifies key parameters which govern the behaviour of the system. The authors also emphasize the importance of those parameters as therapeutic targets to control the inflammation resolution. In conclusion, this model strongly suggests that an effective treatment protocols would take an approach targeting multiple pathways in inflammation, i.e., with special focus on controlling the rates of macrophage phagocytosis and neutrophil apoptosis.

References

  1. Dunster. et al. The resolution of inflammation: a mathematical model of neutrophil and macrophage interactions.. Bull Math Biol. 2014 Aug;76(8):1953-80.
  2. Henson, PM. et al. Dampening inflammation. Nat Immunol. 2005 Dec;6(12):1179-81.
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