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General overview

Bacillus anthracis is a rod shaped gram negative bacterium which is the causative agent of the disease anthrax. Anthrax can be subdivided into three main categories generically named for the route of exposure or affected terminal organs, and are cutaneous, inhalational and gastrointestinal anthrax [1]. Cutaneous anthrax is the most common form of naturally occurring anthrax. Although this type of the disease may be debilitating or scarring to the host, it is not typically fatal especially where modern healthcare is available. Inhalational anthrax is the most lethal form of the disease and the preferred form of anthrax in bioterror scenarios. Inhalation of the spores introduces them to the lower regions of the lungs (alveoli, or air sacs) where oxygen and carbon dioxide are exchanged with the blood. At this location the pathogenesis of B. anthracis occurs where eventually after intracellular transport can result in septicemia and death. Gastrointestinal anthrax is a very rare form of the disease. This form of anthrax is typically observed with immune-compromised individuals or with those exposed to an overwhelming load of pathogens. This form of the disease is potentially lethal due to the development of a systemic infection from the internal infection.

There are three main strains of B. anthracis. The Ames strain garnered wide public attention during the 2001 anthrax postal attacks. The Vollum strain is a weaponized form that is comparatively more infectious than the Ames strain [3]. There is also an attenuated vaccine strain, known as the Sterne strain. The main datasets gathered for B. anthracis are for the Vollum strain.

http://www.bt.cdc.gov/agent/anthrax/ 
In Druett et al (1953)[1] both rhesus macaques and guinea pigs were exposed to an aerosol of spores, while conscious. The macaques and guinea pigs were monitored for mortality, thus allowing for a dose response model estimating the probability of death from aerosol exposure to spores.

In Altboum et al (2002) [5] female Hartley guinea pigs were first rendered unconscious then dosed intranasally with B. anthracis spores. Mortality was monitored for and the lethality rate was recorded, it is this mortality rate that is being modeled with respect to dose. Therefore this dose response model is for a lethal response to inhaled spores. The isolate preparation was not described thoroughly, however the overall dose was estimated in the paper in units of spores inhaled.

Recommended Model

It is recommended that experiment 87 be used as the best dose response model for use. Although guinea pigs are not as representative of humans as the rhesus macaques in 86, the optimized model had the best fit based on the minimized deviance and likelihood ratio test performed. Additionally, the use of the Vollum strain is better able to simulate an attack scenario than the attenuated strain used in experiment 85.

Exponential and betapoisson model.jpg

Advanced Dose Response Model

An advanced dose response model was developed for Bacillus anthracis [6], in which the physiology of the host as well as the pathogenesis of inhaled spores, was modeled and integrated into the does response models. First the fate and transport of inhaled spores was modeled in a stochastic system (Markov chain), and then coupled with a deterministic model of the pathogenesis of inhaled spores.

This first (stochastic) model used independently allows for the estimation of a correction factor for correcting exposed dose to delivered dose [7]. The left figure shows the linear correction factor for human respiratory systems, which remains linear when the same model is adapted to guinea pig or rhesus macaque respiratory systems. The pathogenesis model simulates the survival, germination, and eventual growth of B. anthracis rods. Therefore the coupled models allow for an estimation of the pathogen burden, since the coupled model accounts for fate and transport as well as the pathogenesis. Again as can be seen in the right figure, this is a linear correction as well.

These correction factors can be incorporated into the dose response models by multiplying the dose by the chosen correction factor; ηdd for delivered dose and ηpb for pathogen burden (equations 1 and 2 for the exponential model and 3 and 4 for the beta Poisson). The ηdd allows for a drop in the number of spores, since there are some being entrapped along the path through the respiratory system, with a value for ηdd of 0.000834 for humans [7]. Conversely the pathogen burden accounts for; fate and transport survival and growth of the bacilli in the body therefore, this correction factor allows for an increase between exposed dose and pathogen burden, thus ηpb is 1.528 for humans.

 

Linear correction factor to estimate delivered dose from exposed dose                                 Linear correction factor to estimate pathogen burden from exposed dose

Advanced model Anthrax equations.png

 


References

ID # of Doses Agent Strain Dose Units Host type Μodel Optimized parameters Response type Reference
84 6 Vollum spores guinea pig beta-Poisson a = 5.49E-01

LD50/ID50 = 2.85E+04
N50 = 2.85E+04
death Postexposure Prophylaxis against Anthrax: Evaluation of Various Treatment Regimens in Intranasally Infected Guinea Pigs
85 6 ATCC 6605 spores guinea pig exponential
k = 7.11E-06
LD50/ID50 = 9.75E+04

death Studies on respiratory infection
86 9 Vollum spores monkey exponential
k = 7.16E-06
LD50/ID50 = 9.69E+04

death Studies on respiratory infection
87 4 Vollum spores guinea pig exponential
k = 1.65E-05
LD50/ID50 = 4.2E+04

death Experiments in feeding adult volunteers with Escherichia coli 55, B5, a coliform organism associated with infant diarrhea
Experiment ID:
84
# of Doses:
6
Agent Strain:
Vollum
Dose Units:
spores
Host type:
guinea pig
Μodel:
beta-Poisson
Optimized parameters: a = 5.49E-01

LD50/ID50 = 2.85E+04
N50 = 2.85E+04
Reference:
Guinea pig/Vollum Strain model data [5]
Dose Dead Survived Total
200 0 4 4
2000 0 8 8
2E+04 6 6 12
2E+05 10 2 12
2E+06 7 1 8
2E+07 7 0 7

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom
χ20.95,1 
p-value
χ20.95,m-k 
p-value
Exponential 29.2 26.6 5 3.84 
2.46e-07
11.1 
2.11e-05
Beta Poisson 2.58 4 9.49 
0.631
Beta-Poisson fits better than exponential; cannot reject good fit for beta-Poisson.

 

Optimized parameters for the beta-Poisson model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 5.49E-01 2.55E-01 3.03E-01 3.32E-01 5.51E+00 2.45E+03 6.16E+03
N50 2.85E+04 1.00E+04 1.25E+04 1.43E+04 6.57E+04 7.85E+04 1.09E+05

 

Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters.

beta Poisson model plot, with confidence bounds around optimized model

Experiment ID:
85
# of Doses:
6
Agent Strain:
ATCC 6605
Dose Units:
spores
Host type:
guinea pig
Μodel:
exponential
Optimized parameters:
k = 7.11E-06
LD50/ID50 = 9.75E+04

Reference:
Guinea pig/ATCC 6605 Strain model data [5]
Dose Dead Survived Total
30 0 6 6
300 1 5 6
3000 0 10 10
3E+04 3 7 10
3E+05 8 2 10
3E+06 10 0 10

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom
χ20.95,1 
p-value
χ20.95,m-k 
p-value
Exponential 8.56 1.81 5 3.84 
0.178
11.1 
0.128
Beta Poisson 6.75 4 9.49 
0.15
Exponential is preferred to beta-Poisson; cannot reject good fit for exponential.

 

Optimized parameters for the exponential model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
k 7.11E-06 2.60E-06 3.52E-06 3.87E-06 1.61E-05 1.97E-05 2.85E-05
ID50/LD50/ETC* 9.75E+04 2.43E+04 3.52E+04 4.30E+04 1.79E+05 1.97E+05 2.66E+05
*Not a parameter of the exponential model; however, it facilitates comparison with other models.

Parameter histogram for exponential model (uncertainty of the parameter)

 

Exponential model plot, with confidence bounds around optimized model

Experiment ID:
86
# of Doses:
9
Agent Strain:
Vollum
Dose Units:
spores
Host type:
monkey
Μodel:
exponential
Optimized parameters:
k = 7.16E-06
LD50/ID50 = 9.69E+04

Reference:
Rhesus Macaques/Vollum Strain model data [4]
Dose Dead Survived Total
70300 1 7 8
77000 4 4 8
109000 5 3 8
138000 6 2 8
156000 5 3 8
161000 3 5 8
240000 8 0 8
3E+05 7 1 8
398000 8 0 8

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom
χ20.95,1 
p-value
χ20.95,m-k 
p-value
Exponential 11.3 -0.000503 8 3.84 
1
15.5 
0.188
Beta Poisson 11.3 7 14.1 
0.128
Exponential is preferred to beta-Poisson; cannot reject good fit for exponential.

 

Optimized parameters for the exponential model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
k 7.16E-06 4.99E-06 5.47E-06 5.71E-06 9.09E-06 9.51E-06 1.05E-05
ID50/LD50/ETC* 9.69E+04 6.63E+04 7.29E+04 7.63E+04 1.21E+05 1.27E+05 1.39E+05
*Not a parameter of the exponential model; however, it facilitates comparison with other models.

Parameter histogram for exponential model (uncertainty of the parameter)

 

Exponential model plot, with confidence bounds around optimized model

 

Best Fit
Experiment ID:
87
# of Doses:
4
Agent Strain:
Vollum
Dose Units:
spores
Host type:
guinea pig
Μodel:
exponential
Optimized parameters:
k = 1.65E-05
LD50/ID50 = 4.2E+04

Reference:
Guinea pig/Vollum Strain model data [4]
Dose Dead Survived Total
19800 8 24 32
40800 18 14 32
76200 21 11 32
118000 28 4 32

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom
χ20.95,1 
p-value
χ20.95,m-k 
p-value
Exponential 1.43 -0.000184 3 3.84 
1
7.81 
0.699
Beta Poisson 1.43 2 5.99 
0.49
Exponential is preferred to beta-Poisson; cannot reject good fit for exponential.

 

Optimized parameters for the exponential model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
k 1.65E-05 1.20E-05 1.30E-05 1.35E-05 2.03E-05 2.12E-05 2.28E-05
ID50/LD50/ETC* 4.2E+04 3.04E+04 3.27E+04 3.41E+04 5.14E+04 5.35E+04 5.80E+04
*Not a parameter of the exponential model; however, it facilitates comparison with other models.

 

Parameter histogram for exponential model (uncertainty of the parameter)

Exponential model plot, with confidence bounds around optimized model

Bacillus Graph 2.png