Deterministic Models of Inhalational Anthrax in New Zealand White Rabbits

Rabbits and B. anthracis

In conducting this research, investigators adhered to the Guide for the Care and Use of Laboratory Animals (NIH publication 86-23). Male certified Pasteurella-free NZW rabbits weighed 2-4 kg. Ames strain bacteria were grown in TSB supplemented with manganese sulfate for approximately 96 hours. Thereafter, spores were washed by centrifugation, resuspended in sterile phosphate-buffered saline (PBS), and stored at 4-6°C.

Anthrax Vaccine Adsorbed (AVA) Vaccination

All rabbits were vaccinated intramuscular (IM) with the human anthrax vaccine or were sham-vaccinated. The vaccination protocol used provided 100% protection against a lethal dose of Ames spores.^61,62 Sham-vaccinated rabbits received vehicle injections (IM) and were considered naïve. As discussed in our previous publications, AVA was 100% protective in the current work.^20,42

Bioaerosol Challenge

Spores were aerosolized in PBS plus 0.01% Triton-X using a 3-jet collison nebulizer. The spore mean mass aerodynamic diameter (MMAD) was estimated at 1.0±0.3 μm. Dose was measured using all glass impingers and a plethysmography system.

Data Collection Postexposure

All rabbits received a lethal dose of spores. Thereafter, rabbits were sacrificed at predetermined time points, 0-36 hours. At 36 hours, 2 of 10 naïve rabbits had succumbed. Without delay following sacrifice, bacteria were quantified in the airways, lung tissue, TBLN, and blood. All quantification used dilution plate analysis from nonheated samples. All samples were kept on ice during processing. To maximize the data collection and minimize animal numbers, the right lung was processed differently from the left lung. The right 3 lobes were clamped and the left 3 lobes were washed (lavaged). Lavage fluid was collected, sonicated, and quantified. Thereafter, the right (nonwashed) and left (washed) middle lung lobe were homogenized, plated, and quantified based on tissue weight. The intent was to collect data to estimate the rate of bacterial uptake into the lung tissue where left lung data represented bacteria in lung tissue only and right lung data represented bacteria in the lung tissue plus bacteria in the lumen (airways). TBLNs were collected, homogenized, and plated. Aliquots of whole blood were plated and the bacterial numbers were adjusted to account for the entire blood volume (≈56 ml kg⁻¹).

Unless otherwise noted, all models, data, and plots were developed, analyzed, and presented using Mathematica, Version 8 (Wolfram Research, Champaign, IL).

Estimation of the Initial Number of Bacteria, B₀

To our knowledge, there are no published data available for spore deposition in the rabbit lung. For this reason, deposition was estimated using a combination of particle deposition data and deposition software. B. anthracis spores have an MMAD of approximately 1 μm. ²⁹ And since deposition is driven in large part by the physical characteristic of the particle, deposition for spore particles can be estimated using data from any rabbit particle deposition study or by using particle deposition software. Given this, spore deposition was estimated using information from a publication ²⁸ and predictions made using Regional Deposited Dose Ratio (RDDR) software.

The objective of Raabe et al ²⁸ was to determine the deposition in several animals, including rabbits. Their work used particles of known sizes (0.18-8.65 μm), but only particles approximating the size of a single spore were used herein (0.97-1.09 μm, which had an average size of 1.03 μm). Their deposition data are shown in Table 1. Pulmonary deposition ranged from 4.0-6.6% (which averaged to 5.13%), with bronchial and tracheal regions estimated at 3.31% and 0.25%, respectively. When these averages were summed, the data suggested 8.7% of the inhaled particle dose was deposited in the rabbit trachea, bronchia, and alveoli.

A second estimate of deposition was obtained using RDDR (Version 2.3) software that was developed by the Environmental Protection Agency using the Raabe et al ²⁸ data. The software is designed to predict regional deposition in several species, including rabbits. Here, the following parameter values were used as model input: 1.0 μm MMAD; 1.30 μm GSD (geometric standard deviation, the default value for monodisperse particles); 3 kg rabbit weight, with all other parameters set to default values (see Table 2). RDDR predicted 4.4% and 5.3% (9.7% total) of the inhaled dose would be deposited in the tracheobronchial (TB) and pulmonary (PUL) regions, respectively (Table 2).

Averaging the Raabe et al ²⁸ data (8.7%) and the RDDR prediction (9.7%), it was estimated that 9.2% of the inhaled B. anthracis spore dose would be deposited in the rabbit lung: B₀=0.092*inhaled dose.

Estimation of Mucociliated Clearance, k_mc

Physical clearance, k_mc, was estimated by quantifying the decrease in bacteria over time in the airways (ie, lavage samples) of nonvaccinated rabbits using a least-squares fit of the natural logarithms of the fraction of bacteria remaining.^42,52 The data are summarized in Table 3 (where exposure 2 had 2 separate 24-hour groups and 2 separate 36-hour groups). k_mc was estimated at 0.0628 hr⁻¹ using the entire pooled data set (Figure 1). To provide an estimate of variability, the clearance rate was 0.0694 hr⁻¹ or 0.0550 hr⁻¹ using only exposure 1 or only exposure 2 data, respectively.

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Figure 1.

Mucociliated (physical) clearance of B. anthracis bacteria from the airways. Data points are the natural log of the fraction of bacteria remaining in the airways over time (hours) for an individual rabbit, and the line is the least squares best fit estimate (slope=−0.0628 hr⁻¹).

Estimation of Immune Killing, k_k

Immune-mediated killing was estimated using data collected from vaccinated rabbits. Vaccination is 100% protective in rabbits, and bacteria do not grow in vaccinated rabbits (see also Tables 4 and 5).^20,62 It is assumed that k_g=0 in these animals. As a result, Eq. 1 only contains k_k and k_mc, and the challenge experimentally was to collect data that allow differentiation between these 2 clearance mechanisms. The most straightforward way was to collect data from vaccinated rabbits under experimental conditions where k_mc was, in effect, also equal to zero. This was accomplished by collecting lung data from rabbits whose airways had been lavaged. Under these conditions, any bacteria remaining in the lung post-lavage would most likely reside in the lung tissue (and not the lung lumen), and as a result these bacteria could not be cleared by the mucociliated mechanism (k_mc is in effect equal to 0) and therefore must be cleared by the immune system. It is noted, as discussed in previous publications, that lavage probably failed to remove 100% of the luminal spores.^20,42 However, the analysis here is most likely not affected by this inefficiency because these time course data are used to estimate a slope (as shown in Figure 2). As an example, if lavage only removed half the luminal spores, then the true number of spores used in Figure 2 would need to be doubled. But multiplying all the data by 2 (a transformation) has no effect on the slope. As a result, Eq. 1 reduces to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}B_t = B_0e^{ - k_kt} \tag{2}\end{align*}\end{document}

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Figure 2.

Biological (immune-mediated) clearance of B. anthracis bacteria. Data points are the natural log of the fraction of bacteria remaining in the animal body over time (hours) for an individual rabbit, and the line is the least squares best fit estimate (slope=−0.0431 hr⁻¹).

Bacterial numbers were quantified in vaccinated rabbit lavaged lung tissue, TBLNs, and blood. The data were summed for each individual rabbit, and the immune-killing rate was estimated using the same regression method described above for mucociliary clearance. The data are shown in Table 4 and the best fit plot in Figure 2. As shown, all data were from exposure 2, which had 2 separate 24-hour and 2 separate 36-hour groups of rabbits (note: TBLN data were not collected during exposure 1, so these data were not available for use in this part of the analysis). The rate of k_k was estimated at 0.0431 hr⁻¹ when all data were used. To help estimate variability, the analysis was run using the 0-, 6-, 12-, 24- (set 1), and 36-hour (set 1) data where k_k was 0.0291 hr⁻¹. When using the 0-, 6-, 12-, 24- (set 2), and 36- (set 2) hour data, k_k was 0.0590 hr⁻¹.

Estimation of Bacterial Growth Rate, k_g

Because k_g=0 in vaccinated rabbits, Eq. 1 reduced to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}B_{{AVA}_t} = B_{AVA_0}e^{ - ( k_{mc} + k_k ) t} \tag{3}\end{align*}\end{document}

where B_AVA, is the number of bacteria at time t in vaccinated rabbits, and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$B_{AVA_0}$$\end{document} is the initial bacterial number deposited. Based on previous work, ⁶³ bacterial growth over a defined time interval (t₂+t₁=Δt) in nonvaccinated rabbits can be estimated using the following equation, which is obtained after substitution and division of Eq. 1 by Eq. 3: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \frac { B_ { t_2 } } { B_ { AVA_2 } } = \frac { B_ { t_1 } } { B_ { AVA_1 } } e^ { k_g ( \Delta t ) } \tag { 4 } \end{align*}\end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$B_{t_2}$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$B_{t_1}$$\end{document} are bacterial numbers at t₂ and t₁, respectively, in nonvaccinated rabbits. To estimate k_g, bacteria were quantified in the lung (the entire lung, ie, nonwashed lung), TBLNs, and blood in nonvaccinated and vaccinated rabbits. These data were summed for each animal and are presented in Table 5. As shown, there was a clear increase in bacteria in nonvaccinated rabbits at 24 and 36 hours. In addition, bacterial growth was delayed by approximately 12 hours (Table 5). This delay is observed for all types of bacterial infections and for B. anthracis represents the time period required for germination, establishing an infection, and entering a growth phase. For this reason, the time points selected for use in Eq. 4 were 12 and 36 hours, which is the time period that best captured the entire growth period in rabbits. Using these data, k_g was 0.1996 hr⁻¹. To examine variance, using 12-24-hour and 24-36-hour data, k_g was 0.0904 hr⁻¹ and 0.3088 hr⁻¹, respectively.

All parameter values are summarized in Table 6. For the initial model run, the inhaled dose was 6.9×10⁷ spores, which was the average inhaled dose for rabbits sacrificed at 36 hours (see Table 3). Here, Eq. 1 predicted 1.9×10⁸ bacteria in the animal 36 hours postinfection, which is an order of magnitude higher than what was observed (Table 5). However, under these conditions bacterial growth began immediately and did not account for the lag phase. To account for lag, k_g is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}k_k = \begin{cases}0.0000 \ { \rm hr}^{ - 1} , \quad t < { \rm lag \ phase \ time} \\ 0.1996 \ { \rm hr}^{ - 1} , \quad t \geq { \rm lag \ phase \ time}\end{cases} \tag{5}\end{align*}\end{document}

The best estimate of lag time based on data was 12 hours (Table 5), and when this was considered, the model predicted 1.7×10⁷ bacteria at 36 hours, which is in excellent agreement with observations (4.0×10⁷ bacteria at 36 hours, see Table 5). It is noted that when lag phase time is 7.70 hours, Eq. 1 predicts exactly 4.0×10⁷ bacteria at 36 hours postinfection.

The PBBK Model

Building off the results above, a physiological-based biokinetic compartmentalized (PBBK) model was developed that consisted of 2 physiological compartments (Figure 3): the airways (lung lumen) and the body (lung tissue, TBLNs, and blood). As shown, deposited bacteria are removed by mucociliated clearance, k_mc, or transported across the lung into the body, k_tran. It is noted that uptake does not represent a specific mechanism of uptake (ie, Trojan Horse versus jailbreak). Once in the body, bacteria can grow, k_g, or be killed, k_k, by the immune system. Finally, bacteria could be transported back into the airways, k_rev, but this is generally observed only during the late stages of disease (see below). B₀, k_mc, k_k, and k_g have been determined (see above). The remaining 2 parameters were estimated as described below.

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Figure 3.

Schematic of the 2-compartment rabbit PBBK inhalational anthrax model

Estimation of Bacterial Transport Across Lung Epithelium, k_tran

The rate of spore uptake was estimated by quantifying the increase in bacteria in the lung tissue (lavaged lung homogenate) in nonvaccinated rabbits. Here, the 0-, 6-, and 12-hour data, and not the 24- or 36-hour data, were used. The primary reasons for excluding later time points was because clear signs of bacterial growth were observed at 24 and 36 hours (Table 5), and the increase in bacteria at these time points could be from replication in the lung or reseeding the lung from the circulation. The 0-12-hour data (shown in Table 7) were regressed using a linear fit model, and results suggested bacteria were transported across the lung epithelium at 6.7×10³ bacteria hr⁻¹. When divided by the average number of bacteria deposited in the lung for the rabbits used to collect these data (6.2×10⁵ deposited bacteria, 18 rabbits), k_tran was estimated at 0.0107 hr⁻¹.

Estimation of Bacterial Transport from Lung Tissue to Airways, k_rev

Movement of bacteria from lung tissue back into the conducting airways is not thought to occur except toward the later stages of disease when the lungs are leaking and there is an overwhelming number of bacteria in the blood. The number of bacteria in the blood at the time of death is as high as 10¹² per ml of blood, and there is significant lung hemorrhage in rabbits at the time of death (see Discussion). As a result, k_rev was set at 0.0001 hr⁻¹.

Now that all of the parameters have been estimated (summarized in Table 6), the following system can be analyzed: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}a{ \prime} [ t ] = - k_{tran}a - k_{mc}a + k_{rev}b \tag{5}\end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} b{ \prime} [ t ] = - k_{rev}b + k_{tran}a - k_kb + k_g b \tag{6}\end{align*}\end{document}

where a is the airway compartment and b is the body compartment. In matrix form, this system is governed by: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}A = \left( \begin{matrix} - 0.07345 + 0.00001 \\ + 0.01070 + 0.15952\end{matrix} \right) \tag{7}\end{align*}\end{document}

The eigenvalues for matrix A are {0.1487, −0.0735} with the following eigenvectors: {−0.000045, −1} and {−0.9988, 0.0480}. Trajectories will not oscillate, which is in line with disease kinetics for inhalational anthrax where there is no chronic or recurring disease. However, because 1 eigenvalue is positive and 1 is negative, the trajectory will head toward the origin prior to going to infinity. A biological interpretation of this would be that immediately following exposure the total number of bacteria will initially decrease, presumably because physical clearance begins immediately whereas bacterial growth has a lag period, but eventually bacterial growth will overcome clearance and total bacterial numbers will monotonically increase to infinity (Figure 4), which also makes sense conceptually because the data were collected from rabbits that received a 100% lethal dose. The next step was to compare model predictions and data during the course of disease (Table 8). As shown, the model predicted 6.5×10⁷ bacteria at 36 hours, which agreed well with actual experimental data (4.0×10⁷ bacteria). There was also good agreement with the other time points.

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Figure 4.

Trajectories from the 2D PBBK model. 4A and 4C are trajectories in the airways. 4B is the body compartment trajectory. The black circle is the bacterial starter value (ie, the deposited dose of bacterial spores).

On a final note, the model shows bacteria numbers decreasing in the airways through 36 hours, and then shortly thereafter they increase (see Figure 4C). By setting the first derivative of the solution to a(t) in Eq. 5 equal to 0 and solving, the model suggested bacteria stop decreasing and start increasing in the airways 45 hours following exposure. An interpretation of this model result would be to suggest that at 45 hours all clearance mechanisms become saturated, in which case bacterial accumulation increases sharply at 45 hours. This observation is relevant for inhalational anthrax because a hallmark of the disease in all animal models, and even humans, is an extremely rapid progression from a clinically normal state to death. It can occur within a few hours in all animal models of disease, and the speed of disease progression once a threshold number of bacteria is reached in the body is faster in rabbits than any other animal model. ²³ The model suggests the animal will succumb to the infection very quickly beyond 45 hours, and it is noted that the mean time to death for NZW rabbits exposed to lethal doses of Ames spores is 46.9 hours. ²²

In summary, an exponential ODE model and a 2D PBBK model were developed. Both models agreed well with the laboratory data. The data and model results are summarized in Figure 5. Figure 5A shows the experimental data for nonvaccinated rabbits exposed to lethal doses of spores (black circles) with the exponential ODE model results when a 12-hour lag phase was used (dashed line) along with the 2D PBBK model results (solid line). Figure 5B shows the same PBBK model results overlaid with the exponential ODE model results when the lag phase was set to 7.7 hours.

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Figure 5.

Summary of bacteremia for rabbit inhalational anthrax. 5A shows experimental data (circles), results from the exponential ODE model when lag phase time was set to 12 hours (dashed line), and results from the 2D PBBK model (solid line). 5B shows the same model result from the 2D PBBK model (solid line) and model results when the exponential ODE model was run using a 7.7-hour lag phase time.