Dose response assessment
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
- 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)
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.
Criteria for choosing dose response models
We prefer dose response models with the following criteria, in rough order of importance:
- Statistically acceptable fit (fail to reject goodness of fit, p > 0.05)
- Human subjects, or animal models that mimic human pathophysiology well
- Infection as the response, rather than disease, symptoms, or death
- Exposure route similar/identical to the exposure route of natural infection
- Pathogen strain is similar to strains causing natural infection
- 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)
- 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.
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:
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):
If these approaches are not sufficient to describe the model fit, more complex approaches must be applied.
Bacteria: Bacillus anthracis, Berkholderia mallei and pseudomallei, Campylobacter jejuni, Coxiella burnetii, Escherichia coli, enterohemorrhagic Escherichia coli (EHEC), Francisella tularensis, Legionella pneumophila, Rickettsia rickettsi, Salmonella nontyphoid, Salmonella typhoid, Salmonella anatum, Salmonella newport, Salmonella meleagridis, Shigella sp., Vibrio cholera, Yersinia pestis
Protozoa: Cryptosporidium parvum and Cryptosporidium hominis, Endamoeba coli, Giardia duodenalis, Naegleria fowleri
Virus: Adenovirus-4, Echovirus-12, Enterovirus, Influenza, Lassa virus, Rhinovirus, Rotavirus, SARS