Mathematical Modeling and Analysis of Inflammation and Tissue Repair: Lung Inflammation and Wound Healing in Corals Under Stress

A variety of insults, including tissue injury and/or exposure to pathogen, elicit an immune response in many organisms. An improperly regulated immune response can result in deleterious effects to the organism. Here we present models for lung injury in young and old mice and models for wound healing in coral reefs. It is well known that the immune response becomes less effective in older individuals. This is of particular interest in pulmonary insults such as ventilator induced lung injury (VILI) or lung infection. We extended a mathematical model for the inflammatory response to VILI and used experimental data to select parameters and perform model analysis. We then modified this model to include a bacterial insult and specific cytokine populations, performing a similar process for parameter selection and model analysis. In both cases, parameters involved in macrophage activity primarily drove observed differences associated with the young or old data. Coral reefs regularly experience wounding events some corals have exhibited an immune response similar to that of humans. The effects of climate change stresses the reef which may affect wound healing processes. To address this, we formulated a mathematical model for wound healing in corals in normal conditions and heat stressed conditions. We further paired this model with a dynamic energy budget model, to show how corals must balance energy reserves between growth and tissue repair. The models presented here serve as a first step to modeling the immune response to tissue damage in corals in various environments

M2 macrophage equations.

A variety of pulmonary insults can prompt the need for life-saving mechanical ventilation; however, misuse, prolonged use, or an excessive inflammatory response, can result in ventilator-induced lung injury. Past research has observed an increased instance of respiratory distress in older patients and differences in the inflammatory response. To address this, we performed high pressure ventilation on young (2-3 months) and old (20-25 months) mice for 2 hours and collected data for macrophage phenotypes and lung tissue integrity. Large differences in macrophage activation at baseline and airspace enlargement after ventilation were observed in the old mice. The experimental data was used to determine plausible trajectories for a mathematical model of the inflammatory response to lung injury which includes variables for the innate inflammatory cells and mediators, epithelial cells in varying states, and repair mediators.Classification methods were used to identify influential parameters separating the parameter sets associated with the young or old data and separating the response to ventilation, which was measured by changes in the epithelial state variables. Classification methods ranked parameters involved in repair and damage to the epithelial cells and those associated with classically activated macrophages to be influential.Sensitivity results were used to determine candidate in-silico interventions and these interventions were most impact for transients associated with the old data, specifically those with poorer lung health prior to ventilation. Model results identified dynamics involved in M1 macrophages as a focus for further research, potentially driving the age-dependent differences in all macrophage phenotypes. The model also supported the pro-inflammatory response as a potential indicator of age-dependent differences in response to ventilation. This mathematical model can serve as a baseline model for incorporating other pulmonary injuries.</div

Full table modulating parameters prior to ventilation.

Minimum, mean, and maximum change in the variables Eh and Ee from a 10% decrease (indicated by “-”) or a 10% increase (indicated by “+”) in the listed parameters. Values are shaded on a sliding scale where darker colors represent numbers with a larger magnitude and lighter colors represent numbers with a smaller magnitude. For the minimum in each group, induced decreases in the value of Eh and Ee are orange and induced increases in the value of Eh and Ee are blue. (TIF)</p

Model schematic.

The model has two compartments: lung tissue and blood. The various circles and boxes represent the different inflammatory cells, mediators, and epithelial cell states. Black arrows represent upregulation or transition and black lines with bars represent inhibition or down-regulation. The blue arrows represent movement between the two compartments, either diffusion based or at a constant rate. The red arrows represent movement from the blood into the lung compartment as a result of epithelial barrier degradation.</p

Full table modulating parameters after 1 hour of ventilation.

Minimum, mean, and maximum change in the variables Eh and Ee from a 10% decrease (indicated by “-”) or a 10% increase (indicated by “+”) in the listed parameters after 1 hour of ventilation. Values are shaded on a sliding scale where darker colors represent numbers with a larger magnitude and lighter colors represent numbers with a smaller magnitude. For the minimum in each group, induced decreases in the value of Eh and Ee are orange and induced increases in the value of Eh and Ee are blue. (TIF)</p

Neutrophil equations.

A variety of pulmonary insults can prompt the need for life-saving mechanical ventilation; however, misuse, prolonged use, or an excessive inflammatory response, can result in ventilator-induced lung injury. Past research has observed an increased instance of respiratory distress in older patients and differences in the inflammatory response. To address this, we performed high pressure ventilation on young (2-3 months) and old (20-25 months) mice for 2 hours and collected data for macrophage phenotypes and lung tissue integrity. Large differences in macrophage activation at baseline and airspace enlargement after ventilation were observed in the old mice. The experimental data was used to determine plausible trajectories for a mathematical model of the inflammatory response to lung injury which includes variables for the innate inflammatory cells and mediators, epithelial cells in varying states, and repair mediators.Classification methods were used to identify influential parameters separating the parameter sets associated with the young or old data and separating the response to ventilation, which was measured by changes in the epithelial state variables. Classification methods ranked parameters involved in repair and damage to the epithelial cells and those associated with classically activated macrophages to be influential.Sensitivity results were used to determine candidate in-silico interventions and these interventions were most impact for transients associated with the old data, specifically those with poorer lung health prior to ventilation. Model results identified dynamics involved in M1 macrophages as a focus for further research, potentially driving the age-dependent differences in all macrophage phenotypes. The model also supported the pro-inflammatory response as a potential indicator of age-dependent differences in response to ventilation. This mathematical model can serve as a baseline model for incorporating other pulmonary injuries.</div