Modeling the host response to inhalation anthrax

Quantifying in vitro B. anthracis growth and PA production and decay: a mathematical modelling approach

Protective antigen (PA) is a protein produced by Bacillus anthracis. It forms part of the anthrax toxin and is a key immunogen in US and UK anthrax vaccines. In this study, we have conducted experiments to quantify PA in the supernatants of cultures of B. anthracis Sterne strain, which is the strain used in the manufacture of the UK anthrax vaccine. Then, for the first time, we quantify PA production and degradation via mathematical modelling and Bayesian statistical techniques, making use of this new experimental data as well as two other independent published data sets. We propose a single mathematical model, in terms of delay differential equations (DDEs), which can explain the in vitro dynamics of all three data sets. Since we did not heat activate the B. anthracis spores prior to inoculation, germination occurred much slower in our experiments, allowing us to calibrate two additional parameters with respect to the other data sets. Our model is able to distinguish between natural PA decay and that triggered by bacteria via proteases. There is promising consistency between the different independent data sets for most of the parameter estimates. The quantitative characterisation of B. anthracis PA production and degradation obtained here will contribute towards the ambition to include a realistic description of toxin dynamics, the host immune response, and anti-toxin treatments in future mechanistic models of anthrax infection.

A Stochastic Intracellular Model of Anthrax Infection With Spore Germination Heterogeneity

We present a stochastic mathematical model of the intracellular infection dynamics of Bacillus anthracis in macrophages. Following inhalation of B. anthracis spores, these are ingested by alveolar phagocytes. Ingested spores then begin to germinate and divide intracellularly. This can lead to the eventual death of the host cell and the extracellular release of bacterial progeny. Some macrophages successfully eliminate the intracellular bacteria and will recover. Here, a stochastic birth-and-death process with catastrophe is proposed, which includes the mechanism of spore germination and maturation of B. anthracis. The resulting model is used to explore the potential for heterogeneity in the spore germination rate, with the consideration of two extreme cases for the rate distribution: continuous Gaussian and discrete Bernoulli. We make use of approximate Bayesian computation to calibrate our model using experimental measurements from in vitro infection of murine peritoneal macrophages with spores of the Sterne 34F2 strain of B. anthracis. The calibrated stochastic model allows us to compute the probability of rupture, mean time to rupture, and rupture size distribution, of a macrophage that has been infected with one spore. We also obtain the mean spore and bacterial loads over time for a population of cells, each assumed to be initially infected with a single spore. Our results support the existence of significant heterogeneity in the germination rate, with a subset of spores expected to germinate much later than the majority. Furthermore, in agreement with experimental evidence, our results suggest that most of the spores taken up by macrophages are likely to be eliminated by the host cell, but a few germinated spores may survive phagocytosis and lead to the death of the infected cell. Finally, we discuss how this stochastic modelling approach, together with dose-response data, allows us to quantify and predict individual infection risk following exposure.

Dangerous Pathogens as a Potential Problem for Public Health

Pathogens are various organisms, such as viruses, bacteria, fungi, and protozoa, which can cause severe illnesses to their hosts. Throughout history, pathogens have accompanied human populations and caused various epidemics. One of the most significant outbreaks was the Black Death, which occurred in the 14th century and caused the death of one-third of Europe's population. Pathogens have also been studied for their use as biological warfare agents by the former Soviet Union, Japan, and the USA. Among bacteria and viruses, there are high priority agents that have a significant impact on public health. Bacillus anthracis, Francisella tularensis, Yersinia pestis, Variola virus, Filoviruses (Ebola, Marburg), Arenoviruses (Lassa), and influenza viruses are included in this group of agents. Outbreaks and infections caused by them might result in social disruption and panic, which is why special operations are needed for public health preparedness. Antibiotic-resistant bacteria that significantly impede treatment and recovery of patients are also valid threats. Furthermore, recent events related to the massive spread of Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) are an example of how virus-induced diseases cannot be ignored. The impact of outbreaks, such as SARS-CoV-2, have had far-reaching consequences beyond public health. The economic losses due to lockdowns are difficult to estimate, but it would take years to restore countries to pre-outbreak status. For countries affected by the 2019 coronavirus disease (COVID-19), their health systems have been overwhelmed, resulting in an increase in the mortality rate caused by diseases or injuries. Furthermore, outbreaks, such as SARS-CoV-2, will induce serious, wide-ranging (and possibly long-lasting) psychological problems among, not only health workers, but ordinary citizens (this is due to isolation, quarantine, etc.). The aim of this paper is to present the most dangerous pathogens, as well as general characterizations, mechanisms of action, and treatments.

A mathematical model shows macrophages delay Staphylococcus aureus replication, but limitations in microbicidal capacity restrict bacterial clearance

S. aureus is a leading cause of bacterial infection. Macrophages, the first line of defence in the human immune response, phagocytose and kill S. aureus but the pathogen can evade these responses. Therefore, the exact role of macrophages is incompletely defined. We develop a mathematical model of macrophage - S. aureus dynamics, built on recent experimental data. We demonstrate that, while macrophages may not clear infection, they significantly delay its growth and potentially buy time for recruitment of further cells. We find that macrophage killing is a major obstacle to controlling infection and ingestion capacity also limits the response. We find bistability such that the infection can be limited at low doses. Our combination of experimental data, mathematical analysis and model fitting provide important insights in to the early stages of S. aureus infections, showing macrophages play an important role limiting bacterial replication but can be overwhelmed with large inocula.

Modeling the macrophage-anthrax spore interaction: Implications for early host-pathogen interactions

Inhalational anthrax, caused by the gram positive bacteria Bacillus anthracis, is a potentially fatal form of anthrax infection. It is initiated after inhaled spores are deposited in the lung, phagocytosed by immune cells, and subsequently transported to nearby lymph nodes. Intracellular spores that successfully germinate and become vegetative bacteria can lyse their host cell and contribute to bacterial outgrowth and toxin production. To better understand the early disease dynamics of the host-pathogen interaction, we develop a mathematical model of ordinary differential Equations and estimate parameters using available data. The model which consists of two subsystems is designed in accordance with an in vitro experimental protocol in which macrophages were challenged with varying doses of spores at spore-to-macrophage ratios of 1:1, 1:2, 1:10, 1:20. Initial modeling results suggested the need to consider two distinct subpopulations of anthrax bacteria: newly germinated bacteria which cannot replicate immediately and fully vegetative bacteria that can. Additional modeling results provide insights into possible reasons why macrophage-induced killing is more effective at the 1:20 ratio.

A Novel Stochastic Multi-Scale Model of Francisella tularensis Infection to Predict Risk of Infection in a Laboratory

We present a multi-scale model of the within-phagocyte, within-host and population-level infection dynamics of Francisella tularensis, which extends the mechanistic one proposed by Wood et al. (2014). Our multi-scale model incorporates key aspects of the interaction between host phagocytes and extracellular bacteria, accounts for inter-phagocyte variability in the number of bacteria released upon phagocyte rupture, and allows one to compute the probability of response, and mean time until response, of an infected individual as a function of the initial infection dose. A Bayesian approach is applied to parameterize both the within-phagocyte and within-host models using infection data. Finally, we show how dose response probabilities at the individual level can be used to estimate the airborne propagation of Francisella tularensis in indoor settings (such as a microbiology laboratory) at the population level, by means of a deterministic zonal ventilation model.

Mathematical Models for Immunology: Current State of the Art and Future Research Directions

The advances in genetics and biochemistry that have taken place over the last 10 years led to significant advances in experimental and clinical immunology. In turn, this has led to the development of new mathematical models to investigate qualitatively and quantitatively various open questions in immunology. In this study we present a review of some research areas in mathematical immunology that evolved over the last 10 years. To this end, we take a step-by-step approach in discussing a range of models derived to study the dynamics of both the innate and immune responses at the molecular, cellular and tissue scales. To emphasise the use of mathematics in modelling in this area, we also review some of the mathematical tools used to investigate these models. Finally, we discuss some future trends in both experimental immunology and mathematical immunology for the upcoming years.

Computational fluid dynamics modeling of Bacillus anthracis spore deposition in rabbit and human respiratory airways

Three-dimensional computational fluid dynamics and Lagrangian particle deposition models were developed to compare the deposition of aerosolized Bacillus anthracis spores in the respiratory airways of a human with that of the rabbit, a species commonly used in the study of anthrax disease. The respiratory airway geometries for each species were derived respectively from computed tomography (CT) and μCT images. Both models encompassed airways that extended from the external nose to the lung with a total of 272 outlets in the human model and 2878 outlets in the rabbit model. All simulations of spore deposition were conducted under transient, inhalation-exhalation breathing conditions using average species-specific minute volumes. Two different exposure scenarios were modeled in the rabbit based upon experimental inhalation studies. For comparison, human simulations were conducted at the highest exposure concentration used during the rabbit experimental exposures. Results demonstrated that regional spore deposition patterns were sensitive to airway geometry and ventilation profiles. Due to the complex airway geometries in the rabbit nose, higher spore deposition efficiency was predicted in the nasal sinus compared to the human at the same air concentration of anthrax spores. In contrast, higher spore deposition was predicted in the lower conducting airways of the human compared to the rabbit lung due to differences in airway branching pattern. This information can be used to refine published and ongoing biokinetic models of inhalation anthrax spore exposures, which currently estimate deposited spore concentrations based solely upon exposure concentrations and inhaled doses that do not factor in species-specific anatomy and physiology for deposition.

Animal Models for the Pathogenesis, Treatment, and Prevention of Infection by Bacillus anthracis

This article reviews the characteristics of the major animal models utilized for studies on Bacillus anthracis and highlights their contributions to understanding the pathogenesis and host responses to anthrax and its treatment and prevention. Advantages and drawbacks associated with each model, to include the major models (murine, guinea pig, rabbit, nonhuman primate, and rat), and other less frequently utilized models, are discussed. Although the three principal forms of anthrax are addressed, the main focus of this review is on models for inhalational anthrax. The selection of an animal model for study is often not straightforward and is dependent on the specific aims of the research or test. No single animal species provides complete equivalence to humans; however, each species, when used appropriately, can contribute to a more complete understanding of anthrax and its etiologic agent.

Modeling early events in Francisella tularensis pathogenesis

Computational models can provide valuable insights into the mechanisms of infection and be used as investigative tools to support development of medical treatments. We develop a stochastic, within-host, computational model of the infection process in the BALB/c mouse, following inhalational exposure to Francisella tularensis SCHU S4. The model is mechanistic and governed by a small number of experimentally verifiable parameters. Given an initial dose, the model generates bacterial load profiles corresponding to those produced experimentally, with a doubling time of approximately 5 h during the first 48 h of infection. Analytical approximations for the mean number of bacteria in phagosomes and cytosols for the first 24 h post-infection are derived and used to verify the stochastic model. In our description of the dynamics of macrophage infection, the number of bacteria released per rupturing macrophage is a geometrically-distributed random variable. When combined with doubling time, this provides a distribution for the time taken for infected macrophages to rupture and release their intracellular bacteria. The mean and variance of these distributions are determined by model parameters with a precise biological interpretation, providing new mechanistic insights into the determinants of immune and bacterial kinetics. Insights into the dynamics of macrophage suppression and activation gained by the model can be used to explore the potential benefits of interventions that stimulate macrophage activation.

Deterministic models of inhalational anthrax in New Zealand white rabbits

Computational models describing bacterial kinetics were developed for inhalational anthrax in New Zealand white (NZW) rabbits following inhalation of Ames strain B. anthracis. The data used to parameterize the models included bacterial numbers in the airways, lung tissue, draining lymph nodes, and blood. Initial bacterial numbers were deposited spore dose. The first model was a single exponential ordinary differential equation (ODE) with 3 rate parameters that described mucociliated (physical) clearance, immune clearance (bacterial killing), and bacterial growth. At 36 hours postexposure, the ODE model predicted 1.7×10⁷ bacteria in the rabbit, which agreed well with data from actual experiments (4.0×10⁷ bacteria at 36 hours). Next, building on the single ODE model, a physiological-based biokinetic (PBBK) compartmentalized model was developed in which 1 physiological compartment was the lumen of the airways and the other was the rabbit body (lung tissue, lymph nodes, blood). The 2 compartments were connected with a parameter describing transport of bacteria from the airways into the body. The PBBK model predicted 4.9×10⁷ bacteria in the body at 36 hours, and by 45 hours the model showed all clearance mechanisms were saturated, suggesting the rabbit would quickly succumb to the infection. As with the ODE model, the PBBK model results agreed well with laboratory observations. These data are discussed along with the need for and potential application of the models in risk assessment, drug development, and as a general aid to the experimentalist studying inhalational anthrax.

Microbicidal power of alpha radiation in sterilizing germinating Bacillus anthracis spores

Radioimmunotherapy (RIT) takes advantage of the specificity and affinity of the antigen-antibody interaction to deliver microbicidal radioactive nuclides to a site of infection. In this study, we investigated the microbicidal properties of an alpha particle-emitting 213Bi-labeled monoclonal antibody (MAb), EA2-1 (213Bi-EA2-1), that binds to the immunodominant antigen on Bacillus anthracis spores. Our results showed that dormant spores were resistant to 213Bi-EA2-1. Significant spore killing was observed following treatment with EA2-1 labeled with 300 μCi 213Bi; however, this effect was not dependent on the MAb. In contrast, when spores were germinating, 213Bi-EA2-1 mediated MAb-specific killing in a dose-dependent manner. Dormant spores are very resistant to RIT, and RIT should focus on targeting vegetative cells and germinating spores.

Modeling low-dose mortality and disease incubation period of inhalational anthrax in the rabbit

There is a need to advance our ability to conduct credible human risk assessments for inhalational anthrax associated with exposure to a low number of bacteria. Combining animal data with computational models of disease will be central in the low-dose and cross-species extrapolations required in achieving this goal. The objective of the current work was to apply and advance the competing risks (CR) computational model of inhalational anthrax where data was collected from NZW rabbits exposed to aerosols of Ames strain Bacillus anthracis. An initial aim was to parameterize the CR model using high-dose rabbit data and then conduct a low-dose extrapolation. The CR low-dose attack rate was then compared against known low-dose rabbit data as well as the low-dose curve obtained when the entire rabbit dose-response data set was fitted to an exponential dose-response (EDR) model. The CR model predictions demonstrated excellent agreement with actual low-dose rabbit data. We next used a modified CR model (MCR) to examine disease incubation period (the time to reach a fever >40 °C). The MCR model predicted a germination period of 14.5h following exposure to a low spore dose, which was confirmed by monitoring spore germination in the rabbit lung using PCR, and predicted a low-dose disease incubation period in the rabbit between 14.7 and 16.8 days. Overall, the CR and MCR model appeared to describe rabbit inhalational anthrax well. These results are discussed in the context of conducting laboratory studies in other relevant animal models, combining the CR/MCR model with other computation models of inhalational anthrax, and using the resulting information towards extrapolating a low-dose response prediction for man.

Inhalational anthrax (Ames aerosol) in naïve and vaccinated New Zealand rabbits: characterizing the spread of bacteria from lung deposition to bacteremia

There is a need to better understand inhalational anthrax in relevant animal models. This understanding could aid risk assessment, help define therapeutic windows, and provide a better understanding of disease. The aim here was to characterize and quantify bacterial deposition and dissemination in rabbits following exposure to single high aerosol dose (> 100 LD₅₀) of Bacillus anthracis (Ames) spores immediately following exposure through 36 h. The primary goal of collecting the data was to support investigators in developing computational models of inhalational anthrax disease. Rabbits were vaccinated prior to exposure with the human vaccine (Anthrax Vaccine Adsorbed, AVA) or were sham-vaccinated, and were then exposed in pairs (one sham and one AVA) so disease kinetics could be characterized in equally-dosed hosts where one group is fully protected and is able to clear the infection (AVA-vaccinated), while the other is susceptible to disease, in which case the bacteria are able to escape containment and replicate uncontrolled (sham-vaccinated rabbits). Between 4–5% of the presented aerosol dose was retained in the lung of sham- and AVA-vaccinated rabbits as measured by dilution plate analysis of homogenized lung tissue or bronchoalveolar lavage (BAL) fluid. After 6 and 36 h, >80% and >96%, respectively, of the deposited spores were no longer detected in BAL, with no detectable difference between sham- or AVA-vaccinated rabbits. Thereafter, differences between the two groups became noticeable. In sham-vaccinated rabbits the bacteria were detected in the tracheobronchial lymph nodes (TBLN) 12 h post-exposure and in the circulation at 24 h, a time point which was also associated with dramatic increases in vegetative CFU in the lung tissue of some animals. In all sham-vaccinated rabbits, bacteria increased in both TBLN and blood through 36 h at which point in time some rabbits succumbed to disease. In contrast, AVA-vaccinated rabbits showed small numbers of CFU in TBLN between 24 and 36 h post-exposure with small numbers of bacteria in the circulation only at 24 h post-exposure. These results characterize and quantify disease progression in naïve rabbits following aerosol administration of Ames spores which may be useful in a number of different research applications, including developing quantitative models of infection for use in human inhalational anthrax risk assessment.