Population Dose-Response Models

Individual Dose-Response Models

In toxicology, there are two general classes of dose-response models.  Benchmark dose modeling software calls one set “continuous”, while the other set is “quantal”.  This mathematical categorization doesn’t convey the essential difference: While continuous models represent what happens in an individual, a quantal model represents the frequency of occurrence of an effect in a population (WHO, 2014).  Since pharmacology is closed associated with the practice of Medicine, the continuous “target theory” models discussed previously are what pharmacologists are generally interested in; the magnitude of an effect in an individual as it relates to dose.  Toxicologists use the same models for the same purpose.

Frequency of Occurrence Models

While some branches of toxicology are also primarily concerned with individuals (e.g. clinical toxicology), the better part of toxicology is concerned with public health where the “client” is a population rather than an individual.  In addition, the data used to characterize a dose-response relationship often come from a set of observations from a population of either laboratory animals or people, so the most straightforward use of the data is to characterize the dose-response relationship in a population.  The models that are used for this purpose come from two different traditions:

• Cancer-Risk Assessment.  These biophysical models are largely based on radiation theory, where it is theorized that biological variability in the outcome is entirely attributable to quantum variation in the interaction between a genotoxic carcinogen and a DNA target.  Given what is known about the complexity and variability of physiology in general and the etiology of cancer in particular, the theory is quite silly.  The allusion to quantum physics also brought the Copenhagen Interpetation to cancer risk assessment, which is also silly.   But, as there are also other potential justifications for it (Crawford and Wilson, 1996), at the heart of these models is a simple exponential equation that isn’t silly at all.
• Probit Analysis.  The other approach employs the concept of a threshold, at least in one sense of the word.  In probit analysis, it is theorized that for any given subject (animal or human), there is a specific dose at which any given effect occurs, but the threshold dose varies among the members of a population (Finney, 1954).   The most widely used distributions that used for descriptive purposes are the normal and lognormal distributions.  Probit analysis is often used to estimate the dose at which a specific percentage of subjects that will respond.  For example, an LD50 is the dose that is lethal to 50% of the population, while an ED10 (aka a BMD10) is the dose at which 10% of the population will exhibit a particular response.  Since essentially any statistical distribution may be used as a descriptive tool, variations of the concept behind probit analysis are virtually infinite.  For example, if there are large differences in specific subpopulations, a bimodal or multimodal distribution may provide a better prediction.  Using a distribution that is skewed towards the low end, such as a Weibull distribution, will produce a population dose-response curve that is in the same family if models as those used in cancer risk assessment.  

So, in summary, here is a graphical synopsis of a few potential theoretical impacts of a chemical agent on a population:

2D "Hybrid" Models

Population data is also often used to characterize individual dose response relationships, with the more or less inevitable result that the dose-response relationship will characterize what will happen in an “average” person.  Since average people don’t really exist at all, it is a quite reasonable to wonder about what the range of variation in a continuous response will be in a population.  The difficulty with answering that question is intimately tied to the question of causality: Variation in effect of the substance of interest diminishes the ability to distinguish it from other causes.  While this is often even true even in a laboratory animal experiment, it is an even bigger problem when trying to interpret epidemiological data.  However, when there are obvious population differences, they can be accounted for; when the effect gets large relative to other causal influences, characterizing variability becomes possible. 

Nonetheless, hybrid dose-response models can be developed from animal data and sometimes from epidemiological data as well.  The simplest way to do this is to combine a descriptive statistical model with a continuous individual dose-response model, which is the approach generally taken for the deriving benchmark dose software (Crump, 1995).   Using a hybrid model to derive a BMD requires a two-dimensional definition of the benchmark response; one for the continuous effect and another for the population frequency dimension.  There are far more complicated methods as well.  Population pharmacokinetic models have also been developed for some drugs that require keeping track of correlations between different pharmacokinetic parameters. 

Using Population Models to Make Statements About Individuals

When population data is being used to identify a dose that has no measurable effect (i.e. a NOAEL or a BMD) then the distinction between a continuous and quantal endpoint doesn’t matter so much: Either way, you can more-or-less reasonably claim that at much lower levels of exposure nothing much is going to happen.  But if you are using the same analysis to actually predict effects (i.e. a real risk assessment instead of a safety assessment), the distinction is much more important.  A continuous effect model is very easy to translate into a statement about individual effect, with or without some acknowledgement that your mileage may vary, since that is essentially what it is about in the first place. 

But a quantal population model is a whole ‘nother story, and it doesn’t really matter whether it concerns cancer or some other endpoint.  Even if the population model is a stone cold statistical fact, when someone asks “What will happen to me?” something dreadful happens: Population variability suddenly becomes uncertainty, and a person becomes a chance.   I blame Copenhagen for that.

References

Crawford, M. and R. Wilson. (1996). Low dose linearity: The rule or the exception? Human and Ecological Risk Assessment 2: 305-330.

Crump K. (1995).  Calculation of benchmark doses from continuous data. Risk Anal. 15:79–89.
Finney DJ (1954).  Probit Analysis. Cambridge University Press, Cambridge.

World Health Organization (2014). International Programme on Chemical Safety, Harmonization Project Document 11.; Guidance Document on Evaluating And Expressing Uncertainty in Hazard Characterization  See chapter 3 for discussion of continuous vs quantal models.

Official Post Soundtrack

Cure, The (1980).  A Forest.  In: Seventeen Seconds, Track 7

Post Notes

Thesis Post #19.  Goes in dose-response modeling thread; mostly background material for non-toxicologists.