Dose-response relationships, mathematical

For non-threshold mechanisms of genotoxic carcinogenicity, the dose-response relationship is considered to be linear. The observed dose-response curve in some cases represents a single ratedetermining step however, in many cases it may be more complex and represent a superposition of a number of dose-response curves for the various steps involved in the tumor formation (EC 2003). Because of the small number of doses tested experimentally, i.e., usually only two or three, almost all data sets fit equally well various mathematical functions, and it is generally not possible to determine valid dose-response curves on the basis of mathematical modeling. This issue is addressed in further detail in Chapter 6. 

Mathematical modelling of the dose-response relationship is an alternative approach to quantify the estimated response within the experimental range. This approach can be used to determine the BMD or benchmark concentration (BMC) for inhalation exposure, which can be used in place of the LOAEL or NOAEL (Crump, 1984). The BMD (used here for either BMD or BMC) is defined as the lower confidence limit on a dose that produces a particular level of response (e.g., 1%, 5%, 10%) and has several advantages over the LOAEL or NOAEL (Kimmel Gaylor, 1988 Kimmel, 1990 USEPA, 1995 IPCS, 1999). For example, (1) the BMD approach uses all of the data in fitting a model instead of only data indicating the LOAEL or NOAEL (2) by fitting all of the data, the BMD approach takes into account the slope of the dose-response curve (3) the BMD takes into account variability in the data and (4) the BMD is not limited to one experimental dose. Calculation and use of the BMD approach are described in a US EPA... 

For more detail and the mathematical basis and treatment of the relationship between the receptor-ligand interaction and dose-response relationship, the reader is recommended to consult one of the texts indicated at the end of this chapter (5-7). 

There are two possible approaches to estimating the human safe dose for chemicals that cause deterministic effects the use of safety and uncertainty factors and mathematical modeling. The former constitutes the traditional approach to dose-response assessment for chemicals that induce deterministic effects. Biologically-based mathematical modeling approaches that more realistically predict the responses to such chemicals, while newer and not used as widely, hold promise to provide better extrapolations of the dose-response relationship below the lowest dose tested. 

Statistical models. A number of statistical dose-response extrapolation models have been discussed in the literature (Krewski et al., 1989 Moolgavkar et al., 1999). Most of these models are based on the notion that each individual has his or her own tolerance (absorbed dose that produces no response in an individual), while any dose that exceeds the tolerance will result in a positive response. These tolerances are presumed to vary among individuals in the population, and the assumed absence of a threshold in the dose-response relationship is represented by allowing the minimum tolerance to be zero. Specification of a functional form of the distribution of tolerances in a population determines the shape of the dose-response relationship and, thus, defines a particular statistical model. Several mathematical models have been developed to estimate low-dose responses from data observed at high doses (e.g., Weibull, multi-stage, one-hit). The accuracy of the response estimated by extrapolation at the dose of interest is a function of how accurately the mathematical model describes the true, but unmeasurable, relationship between dose and response at low doses. 

Generally, mathematical models are used to extrapolate the data on the exposure- or dose-response relationship derived from carcinogenicity bioassays to estimate the risk at concentrations to which the general population is exposed in the absence of more biologically based kinetic or dynamic models. There are numerous uncertainties in such approaches, which often involve linear extrapolation of results over several orders of magnitude, commonly in the absence of relevant data on mode of action for tumor induction or differences in toxico-kinetics and -dynamics between the relevant experimental animal species and humans. 

Many mathematical models of this dose-response relationship have been proposed for this problem. The following section describes the models currently being used. One of the major difficulties inherent in this high to low dose extrapolation... 

There are basically two types of mathematical models commonly used to represent dose response relationships. One type which is referred to as a dichotomous or yes-no modelis concerned with whether or not a specific response, such as cancer, has occurred with increased incidence in the treated pwpulation. The second type, which is a time-to-response model, attempts to relate dose levels to the time of appearance of the measured effect. 

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