Fig 1.
Schema of the spatial domain used in our simulations.
Inset: the Moore neighbourhood of a generic cell, showing the indexing system used to number neighbouring grid spaces when defining and calibrating our model of cell chemotaxis.
Fig 2.
Flow chart illustrating the overarching cell recruitment and update structure of the model.
Here, m and n represent the total number of macrophages and neutrophils in the domain respectively.
Fig 3.
Flow chart illustrating the actions of each agent of the macrophage class.
Here, represents the total concentration of mediator c in the neighbourhood of the agent.
Fig 4.
Flow chart illustrating the actions of each neutrophil agent.
Fig 5.
Flow chart illustrating the actions of each agent of the apoptotic class.
Table 1.
Model parameters, together with baseline values used in the simulations below.
PDE parameters are inferred from existing literature as described in the text. Chemotaxis parameters are computed via comparison with the experimental data of [10]. The remaining parameters are mostly unavailable in existing literature, and are thus estimated based on simulation. The model’s sensitivity to these parameter choices is investigated in more detail in the Results section below.
Fig 6.
Plots of the mean chemotactic index attained for each combination of kgrad and σmem.
In (A), we calibrate our model against the healthy control data of [10] using a mean cell velocity of 3.77 μm/min; the black line represents a mean chemotactic index of 0.39. In (B), we calibrate our model against the COPD-affected data of [10] using a mean cell velocity of 0.09 μm/min; the black line represents a mean chemotactic index of 0.04.
Fig 7.
Comparison of the in vitro cell tracking experiments of [10] (A,C) with simulations of our calibrated chemotaxis model (B,D).
In (A,B), we illustrate the healthy case, with simulations calibrated using a mean cell velocity of 3.77 μm/min and a mean chemotactic index is 0.39, providing kgrad = 80 and σmem = 1. In (C,D), we illustrate the inflamed (COPD) case, with simulations calibrated using a mean cell velocity of 0.09 μm/min and a mean chemotactic index is 0.04, providing kgrad = 8 and σmem = 1.2. (A,C) Reprinted with permission of the American Thoracic Society. Copyright ©2019 American Thoracic Society. Cite: Sapey et al. (2011) ‘Behavioral and structural differences in migrating peripheral neutrophils from patients with chronic obstructive pulmonary disease’, American Journal of Respiratory and Critical Care Medicine 183(9), 1176-1186. The American Journal of Respiratory and Critical Care Medicine is an official journal of the American Thoracic Society.
Fig 8.
A typical simulation of the inflammation model showing a chronic, globally inflamed outcome, for healthy choices of chemotactic parameters (kgrad = 80, σmem = 1) and all other model parameters as given in Table 1.
(A) Successive snapshots of the simulation in which high levels of pro-inflammatory mediators are shown in dark reds in the background; active/apoptotic neutrophils are shown in green/orange respectively and macrophages are shown in blue. (B–D) Global cell counts for active neutrophils, apoptotic neutrophils and macrophages respectively. (E,F) Maximal concentrations of mediators c and g across the domain. Solid lines represent the mean result over 100 simulations of the model; shaded areas represent plus/minus one standard deviation.
Fig 9.
Snapshots of a typical healthy response, for healthy choices of chemotactic parameters (kgrad = 80, σmem = 1), αmr = 0.25 and all other model parameters as given in Table 1.
High levels of pro-inflammatory mediators are shown in dark reds in the background; active/apoptotic neutrophils are shown in green/orange respectively and macrophages are shown in blue. Corresponding time-courses showing global cell counts and mediator concentrations are shown in green in Fig 10.
Fig 10.
Simulations resulting in healthy, resolved outcomes for c0 = 0.5 (blue), r = 5 (red), αmr = 0.25 (green), αncr = 0.1 (magenta) and αngr = 0.0015 (cyan), with healthy choices of chemotaxis parameters (kgrad = 80, σmem = 1) and all other parameters as in Table 1.
The chronic inflammation of Fig 8 is shown in black. (A–C) Global cell counts for active neutrophils, apoptotic neutrophils and macrophages respectively. (D,E) Maximal concentrations of mediators c and g across the domain. Solid lines represent the mean result over 100 simulations of the model; shaded areas represent plus/minus one standard deviation.
Fig 11.
Simulations for αmr = 0.25 and all other parameters as given in Table 1, with healthy neutrophil chemotaxis (kgrad = 80, σmem = 1, shown as dashed blue lines) and impaired neutrophil chemotaxis (kgrad = 8, σmem = 1.2, shown as solid red lines).
(A–C) Global cell counts for active neutrophils, apoptotic neutrophils and macrophages respectively. (D,E) Maximal concentrations of mediators c and g across the domain. Dotted lines demark the thresholds for neutrophil recruitment (c = αncr, g = αngr); the dash-dotted line demarks the threshold for macrophage recruitment (c = αmr). Impaired neutrophil chemotaxis causes a previously healthy outcome to become chronic.
Fig 12.
Percentage change in the maximal level of pro-inflammatory mediator c at t = 5000 on varying each individual parameter by + 50% (green) and −50% (red), from the baseline parameter values given in Table 1, with healthy choices of chemotaxis parameters (kgrad = 80, σmem = 1).
Triangles denote the mean percentage change in the response; error bars denote plus/minus one standard deviation. (For probabilities for which an increase of 50% results in a choice greater than one, the green results correspond to simulations with unit probability.) Note that a change in response of −100% corresponds to a switch from a chronic to a healthy outcome.