A mathematical simulation of the inflammatory response to anthrax infection

Abstract

Bacillus anthracis (anthrax) can trigger an acute inflammatory response that results in multisystem organ failure and death. Previously, we developed a mathematical model of acute inflammation after gram-negative infection that had been matched qualitatively to literature data. We modified the properties of the invading bacteria in that model to those specific to B. anthracis and simulated the host response to anthrax infection. We simulated treatment strategies against anthrax in a genetically diverse population including the following: (1) antibiotic treatment initiated at various time points, (2) antiprotective antigen vaccine, and (3) a combination of antibiotics and vaccine. In agreement with studies in mice, our simulations showed that antibiotics only improve survival if administered early in the course of anthrax infection. Vaccination that leads to the formation of antibodies to protective antigen is anti-inflammatory and beneficial in averting shock and improving survival. However, antibodies to protective antigen alone are predicted not to be universally protective against anthrax infection. Rather, our simulations suggest that an optimal strategy would require both vaccination and antibiotic administration.

Fig. 1. Schematic of the course of anthrax infection

Infection is initiated by inhalation or ingestion of anthrax spores, which are internalized by monocytes/macrophages/dendritic cells. The spores incubate, and therefore, monocytes and immature dendritic cells spread latent anthrax infection throughout the body. Given appropriate environmental conditions, the spores germinate.

Fig. 2. Simulation of a healthy subject in response to gram-negative bacterial infection

A virtual patient is infected by a low dose of gram-negative bacteria (A), which leads to activation of macrophages (B). The macrophages elaborate cytokines (C), including rapidly produced proinflammatory cytokines (Cp; e.g., IL-1 and TNF), more slowly produced proinflammatory cytokines (CpL; e.g., IL-6), and anti-inflammatory cytokines (Ca; e.g., TGF-β1). Tissue damage/dysfunction (D) rises because of the actions of proinflammatory cytokines and effectors but then decreases because of the actions of anti-inflammatory cytokines. Thus, homeostasis is restored after pathogens are cleared.

Fig. 3. Simulation of persistent gram-negative sepsis

A virtual patient is infected by a high dose of gram-negative bacteria (A), which leads to activation of macrophages (B). The macrophages elaborate cytokines (C), including rapidly produced (early) proinflammatory cytokines (Cp; e.g., IL-1 and TNF), more slowly (late) produced proinflammatory cytokines (CpL; e.g., IL-6), and anti-inflammatory cytokines (Ca; e.g., TGF-β1). Tissue damage/dysfunction (D) rises because of the actions of proinflammatory cytokines and effectors and, unlike the case in Figure 1, is not reduced. Thus, the inflammatory response is sustained, with ensuing adverse outcome.

Fig. 4. Modification of the mathematical model of sepsis to account for anthrax-specific effects

Comparison of the predicted effect on immune mediators and tissue damage/dysfunction in anthrax infection. Despite an exuberant immune response, the bacterial load and tissue damage/dysfunction remain high (the bacterial growth parameters are the same as that of the gram-negative infection in Fig. 1).

Fig. 5. Vaccination against protective antigen leads to a decrease in the stimulation of the immune response

Vaccination leads to depletion of protective antigen. The primary stimulants are now elements of the bacterial coat, and the response is insufficient to clear bacteria. Therefore, although a downregulated immune response leads to lower damage, it also results in unbounded growth of bacteria.

Fig. 6. Simulation with a set of parameters where vaccination alone is sufficient in curing anthrax

Varying the parameters in the model could resemble the inherent variations in a population. In the above simulation, pathogen and vaccine properties were kept the same, but immune response parameters were changed such that neutrophil activation after exposure to pathogen was 8 times the base rate, and NO production by neutrophils was 6 times the base rate for comparable stimuli. Tissue damage/dysfunction, in the presence of antiprotective antigen vaccine, in the case of the base parameter values is compared with the modified set.

Fig. 7. Simulation of the efficacy of antibiotics and vaccination in inhalational anthrax

Simulation of tissue damage/dysfunction under the various circumstances. Case 1, anthrax infection, no antibiotics; case 2, anthrax infection, antibiotic administration at t = 0; case 3, anthrax infection, antibiotic administration at t = 48; case 4, anthrax infection in the setting of vaccination leading to pre-existing circulating antibodies to protective antigen; case 5, anthrax infection in the presence of vaccination and antibiotic administration at t = 48 h.

Fig. 8. Predictors of response to combination therapy

A logistic regression model was constructed to determine if variation in host and pathogen factors would impact the response to combination therapy (fully effective vaccine with rescue antibiotics 72 h after the start of clinical infection, compared with vaccine alone; overall rate of response of 60.3%, reducing mortality from 87.1% to 26.8%). Predictors were ranked by quartiles of increasing value (I–IV), with the highest quartile reflecting the highest inoculum size, pathogen growth rate and propensity to mount a proinflammatory or anti-inflammatory response or produce NO. For each predictor, odds ratios associated with quartiles I to III, using quartile IV as reference (odds ratio, 1), are depicted. Note the nonmonotonic relationship between propensity to mount a short proinflammatory response (Cp) and the response to combination therapy. A similar less marked “dose-response” relationship is seen for anti-inflammatory production. Propensity to NO production did not significantly impact response to combination therapy.