# Dose response assessment

## Dose Response

In the QMRA framework, the dose response assessment phase is the quantitative yardstick for the risk estimate, as this phase estimates a risk of response (infection, illness or death) with respect to a known dose of a pathogen. The basis of the dose response phase is the dose response models, which are mathematical functions derived to describe the dose response relationship for specific pathogens. Therefore, for a particular endpoint (response), a specific pathogen and exposure route there is a unique dose response relationship and consequently a dose response model. Dose response models are necessary as it is not possible to perform a direct study (even with animals) to assess dose corresponding to an acceptably low risk.

### Dose Response Models

To be plausible a model should consider the discrete (particulate) nature of organisms, which has a high variability at low dose. It should also be based on the concept of infection from one or more “survivors” of initial dose. Therefore dose response models for QMRA need to be physiologically plausible and be derived from what is known of the general infection process. There are two models which are derived based on these needs for the QMRA dose response relationship, the exponential and beta Poisson models.

#### Types of Models

Exponential Dose Response Model

Assumptions:

• Poisson distribution of organisms among replicated doses (mean number in dose=d).
• One organism is capable of producing an infection if it arrives at an appropriate site.
• Organisms have independent and identical probability (k) of surviving to reach and infect at an appropriate site. Some sources use the letter 'r' instead of 'k' (equation 1). Here we define r=1/k, so the alternative form for equation 1 can be given as P(response) = 1- exp(-dose/r)

Beta-Poisson Model

Assumptions same as the exponential model except:

• Nonconstant survival and infection probabilities
• Survival probabilities (k) are given by the beta distribution

The slope of the beta-Poisson dose response curve is more shallow than the exponential. The exponential model is the same as the beta-Poisson model when alpha approaches infinity. The parameters are alpha and N50. N50 is the dose at which 50% of the population is expected to be affected. The beta-Poisson model is sometimes expressed with a beta parameter instead of an N50 parameter; N50=beta*[2^(1/alpha)-1]. Both the alpha and the beta parameters derive from the use of the beta distribution to model nonconstant pathogen survival probabilities.

The exact form of the beta-Poisson model uses the confluent hypergeometric function, which can be difficult to optimize. However since both the exact and approximate form of the beta Poisson dose response models demonstrate linearity in the low dose range, and there is not a substantial difference between the forms in fitting dose response data, there is not reason to not use the more intuitive form of the beta Poisson. Equation 2 shows the approximate form of the beta Poisson using the N50 parameter, which can be directly optimized using dose response data or estimated using the conversion in equation 4.

## Utility of Using Dose Response Models

An optimized dose response model allows for greater flexibility and a wider range of understanding in the estimated risk. Rather than having a median infectious or lethal dose for a pathogen a model that can describe the full range of probability of response beyond just the median and one that is still accurate at low doses.

## Available Dose Response Models

You can access completed dose response analysis for each microbe by clicking the name of the pathogen in the following table.

Bacteria
Agent Best fit model* Optimized parameter(s) LD50/ID50 Host type Agent strain Route # of doses Dose units Response Reference
Bacillus anthracis: Dose Response Models exponential k = 1.65E-05 4.2E+04 guinea pig Vollum inhalation 4 spores death Druett 1953
Burkholderia pseudomallei: Dose Response Models beta-Poisson α = 3.28E-01 , N50 = 5.43E+03 5.43E+03 C57BL/6 mice and diabetic rat KHW,316c intranasal,intraperitoneal 10 CFU death Liu, Koo et al. 2002 and Brett and Woods 1996
Campylobacter jejuni and Campylobacter coli: Dose Response Models beta-Poisson α= 1.44E-01 , N50 = 8.9E+02 8.9E+02 human strain A3249 oral (in milk) 6 CFU infection Black et al 1988
Coxiella burnetii: Dose Response Models beta-Poisson α= 3.57E-01 , N50 = 4.93E+08 4.93E+08 C57BL/1OScN mice phase I Ohio intraperitoneal 10 PFU death Williams et al, 1982
Escherichia coli enterohemorrhagic (EHEC): Dose Response Models exponential k=2.18E-04 3.18E+03 pig EHEC O157:H7, strain 86-24 oral (in food) 3 CFU shedding in feces Cornick & Helgerson (2004)
Escherichia coli: Dose Response Models beta-Poisson α = 1.55E-01 , N50 = 2.11E+06 2.11E+06 human EIEC 1624 oral (in milk) 3 CFU positive stool isolation DuPont et al. (1971)
Francisella tularensis: Dose Response Models exponential k = 4.73E-02 1.46E+01 monkeys SCHU S-4 inhalation 4 CFU death Day and Berendt, 1972
Legionella pneumophila: Dose Response Models exponential k = 5.99E-02 1.16E+01 guinea pig Philadelphia 1 inhalation 4 CFU infection Muller et al. (1983)
Listeria monocytogenes (Death as response): Dose Response Models exponential k = 1.15E-05 6.05E+04 C57B1/6J mice F5817 oral 6 CFU death Golnazarian, Donnelly et al. 1989
Listeria monocytogenes (Infection): Dose Response Models beta-Poisson α = 2.53E-01 , N50 = 2.77E+02 2.77E+02 C57Bl/6J mice F5817 oral 10 CFU infection Golnazarian
Listeria monocytogenes (Stillbirths): Dose Response Models beta-Poisson α = 4.22E-02 , N50 = 1.78E+09 1.78E+09 pooled oral 13 CFU stillbirths Smith, Williams2007
Mycobacterium avium: Dose Response Models exponential k = 6.93E-04 1E+03 deer sub sp. Paratuberculosis Bovine oral 3 CFU infection O'Brien et al(1976)
Pseudomonas aeruginosa (Contact lens): Dose Response Models beta-Poisson α = 1.9E-01 , N50 = 1.85E+04 1.85E+04 white rabbit contact lens 10 CFU corneal ulceration Lawin-Brussel et al. (1993)
Pseudomonas aeruginosa (bacterimia): Dose Response Models exponential k = 1.05E-04 6.61E+03 Swiss webster mice (5day old) ATCC 19660 injected in eyelids 12 CFU death Hazlett, Rosen et al. 1978
Rickettsia rickettsi: Dose Response Models beta-Poisson α= 7.77E-01 , N50 = 2.13E+01 2.13E+01 Pooled data R1 and Sheila Smith NA 27 CFU morbidity Saslaw and Carlisle 1966 and Dupont, Hornick et al. 1973
Salmonella Typhi: Dose Response Models beta-Poisson α = 1.75E-01 , N50 = 1.11E+06 1.11E+06 human Quailes oral, in milk 8 CFU disease Hornick et al. (1966),Hornick et al. (1970)
Salmonella anatum: Dose Response Models beta-Poisson α= 3.18E-01 , N50 = 3.71E+04 3.71E+04 human strain I oral, with eggnog 16 CFU positive stool culture McCullough and Elsele,1951
Salmonella meleagridis: Dose Response Models beta-Poisson α= 3.89E-01 , N50 = 1.68E+04 1.68E+04 human strain I oral, with eggnog 11 CFU infection McCullough and Eisele 1951,2
Salmonella nontyphoid: Dose Response Models beta-Poisson α= 2.1E-01 , N50 = 4.98E+01 4.98E+01 mice strain 216 and 219 intraperitoneal 10 CFU death Meynell and Meynell,1958
Salmonella serotype newport: Dose Response Models exponential k = 3.97E-06 1.74E+05 human Salmonella newport oral 3 CFU infection McCullough and Elsele,1951
Shigella: Dose Response Models beta-Poisson α= 2.65E-01 , N50 = 1.48E+03 1.48E+03 human 2a (strain 2457T) oral (in milk) 4 CFU positive stool isolation DuPont et al. (1972b)
Staphylococcus aureus: Dose Response Models exponential k = 7.64E-08 9.08E+06 human subcutaneous 6 CFU/cm2 infection Rose and Haas 1999
TestPage exponential k = 1.65E-05 4.2E+04 guinea pig Vollum inhalation 4 spores death Druett 1953
Vibrio cholerae: Dose Response Models beta-Poisson α= 2.50E-01 , N50 = 2.43E+02 2.43E+02 human Inaba 569B oral (with NaHCO3) 6 CFU infection Hornick et al., (1971)
Yersinia pestis: Dose Response Models exponential k = 1.63E-03 4.26E+02 mice CO92 intranasal 4 CFU death Lathem et al. 2005

*These models are preferred in most circumstances. However, consider all available models to decide which one is most appropriate for your analysis.

Prion
Agent Best fit model* Optimized parameter(s) LD50/ID50 Host type Agent strain Route # of doses Dose units Response Reference
PrP prions: Dose Response Models beta-Poisson α = 1.76E+00 , N50 = 1.04E+05 1.04E+05 hamsters scrapie strain 263k oral 5 LD50 i.c. death Diringer et al. 1998

*These models are preferred in most circumstances. However, consider all available models to decide which one is most appropriate for your analysis.

Protozoa
Agent Best fit model* Optimized parameter(s) LD50/ID50 Host type Agent strain Route # of doses Dose units Response Reference
Cryptosporidium parvum and Cryptosporidium hominis: Dose Response Models exponential k = 5.72E-02 1.21E+01 human TAMU isolate oral 4 oocysts infection Messner et al. 2001
Endamoeba coli: Dose Response Models beta-Poisson α = 1.01E-01 , N50 = 3.41E+02 3.41E+02 human From an infected human oral 5 Cysts infection Rendtorff 1954
Giardia duodenalis: Dose Response Models exponential k = 1.99E-02 3.48E+01 human From an infected human oral 8 Cysts infection Rendtorff 1954
Naegleria fowleri: Dose Response Models exponential k = 3.42E-07 2.03E+06 mice LEE strain intravenous 7 no of trophozoites death Adams et al. 1976 & Haggerty and John 1978

*These models are preferred in most circumstances. However, consider all available models to decide which one is most appropriate for your analysis.

Virus
Agent Best fit model* Optimized parameter(s) LD50/ID50 Host type Agent strain Route # of doses Dose units Response Reference
Adenovirus: Dose Response Models exponential k = 6.07E-01 1.14E+00 human type 4 inhalation 4 TCID50 infection Couch, Cate et al. 1966
Echovirus: Dose Response Models beta-Poisson α = 1.06E+00 , N50 = 9.22E+02 9.22E+02 human strain 12 oral 4 PFU infection Schiff et al.,1984
Enteroviruses: Dose Response Models exponential k = 3.74E-03 1.85E+02 pig porcine, PE7-05i oral 3 PFU infection Cliver, 1981
Influenza: Dose Response Models beta-Poisson α = 5.81E-01 , N50 =9.45E+05 9.45E+05 human H1N1,A/California/10/78 attenuated strain,
H3N2,A/Washington/897/80 attenuated strain
intranasal 9 TCID50 infection Murphy et al., 1984 & Murphy et al., 1985
Lassa virus: Dose Response Models exponential k = 2.95E+00 2.35E-01 guinea pig Josiah strain subcutaneous 6 PFU death Jahrling et al., 1982
Poliovirus: Dose Response Models exponential k = 4.91E-01 1.41E+00 human type 1,attenuated oral (capsule) 3 PD50 (mouse paralytic doses) alimentary infection Koprowski
Rhinovirus: Dose Response Models beta-Poisson α = 2.21E-01 , N50 = 1.81E+00 1.81E+00 human type 39 intranasal 6 TCID50 doses infection Hendley et al., 1972
Rotavirus: Dose Response Models beta-Poisson α = 2.53E-01 , N50 = 6.17E+00 6.17E+00 human CJN strain (unpassaged) oral 8 FFU infection Ward et al, 1986
SARS: Dose Response Models exponential k = 2.46E-03 2.82E+02 mice hACE-2 and A/J rSARS-CoV intranasal 8 PFU death DeDiego et al., 2008 & De Albuquerque et al., 2006

*These models are preferred in most circumstances. However, consider all available models to decide which one is most appropriate for your analysis.

## Criteria for choosing dose response models

We prefer dose response models with the following criteria, in rough order of importance:

1. Statistically acceptable fit (fail to reject goodness of fit, p > 0.05)
2. Human subjects, or animal models that mimic human pathophysiology well
3. Infection as the response, rather than disease, symptoms, or death
4. Exposure route similar/identical to the exposure route of natural infection
5. Pathogen strain is similar to strains causing natural infection
6. Pooled model using data from 2 or more experiments, provided the data sets are statistically similar (fail to reject that datasets are from the same distribution, p > 0.05)
7. Low ID50/LD50 (to obtain a conservative risk estimate)

We generally recommend a single dose response model, and we justify the decision in terms of the above criteria. This decision is somewhat subjective, since dose response datasets seldom meet all of these criteria. If all available models are unsatisfactory, we choose a single model to ‘recommend with reservations’. Our recommended model will seldom (if ever) be the best model for all applications. The user should carefully choose the model that is most appropriate for their particular problem.

## Mathematical & Statistical Approaches

Various regression techniques are used to characterize dose-response relationships via a mathematical function. Dose-response relationships are probabilistic and will therefore take a value between 0 and 1. A dose-response analysis begins with a best-fit test of dose-response data. These data are usually provided in the literature as a comparison of the median dose concentration and number of organisms that experienced a given effect (infection, illness, death; known as the endpoint) at that dose. The statistical technique maximum likelihood estimation (MLE) is used to fit the data to theoretical distributions, typically either Beta-Poisson or Exponential due to their biologic plausibility. This process calculates the probability of obtaining the observed data given a theoretical distribution by minimizing deviance (Y) of each of these model fits:

Y=-2(lnM1 - lnM2)

Where L1, L2 are maximized likelihood estimates for the full (L2) and restricted (L1) models. Optimized deviance follows a Χ2 distribution with k – m degrees of freedom, where k is the number of doses and m is the number of dose-response parameters of a given model. This allows the analyst to reject the model if Y > Χ2k-m,α. If both models are significant, the model with the lowest deviance when compared to the full (for example the empirical model with a separate parameter for each dose group) model is chosen. Bootstrapping is performed to characterize the uncertainty of parameter estimates (r, α, N50, etc.) of the distribution, most commonly by generating confidence intervals. The estimates from this approach approximate the uncertainty associated with the “true” distribution by repeatedly sampling the data and re-computing a statistic.

In some cases, it is necessary to pool data from different studies to compare strains or increase confidence in a dose-response model. The ability to pool data is assessed via a hypothesis test (null: no difference in dose-response parameter(s)), where the deviance of the pooled dataset (YT) is added to each individual optimized deviance (Y1, Y2..) and Δ is compared to a Χ2 distribution with df= (number of parameters in each dataset) ‒ (total number of parameters):

Δ=YT - (Y1 + Y2 +...)

If these approaches are not sufficient to describe the model fit, more complex approaches must be applied.

## Dosing Experiments

Besides dose response assessment, the other major components of microbial risk assessment are hazard identification, exposure assessment, and risk characterization.