Extrapolation dose-response mathematical models

Dose-Response Extrapolation Models. A dose-response model is simply a hypothetical mathematical relationship between dose-rate and probability of response. For example, the simplest form of such a model asserts that probability of tumor initiation is a linear multiple of dose-rate (provided the dosage is well below the organism s acute effect threshold for the substance in question). In general, we will express dose-response models as follows ... 

Quantitative extrapolation by mathematical modeling of the dose-response curve to estimate the risk at likely human exposures, i.e., low-dose risk extrapolation... 

In the dose-response assessment to determine a dosage that is risk-free for human health, the JFCFA has never used mathematical models to extrapolate risks at low dose and determine a virtually safe dose, on the grounds that the lack of validation would produce very different results. However, the IFCFA could usefully address this matter in its deliberations. When progress in this area permits selection from various validated models, this exercise should no longer be solely associated with risk assessment but will also incorporate an element of risk management. 

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. 

For the most frequently used low-dose models, the multi-stage and one-hit, there is an inherent mathematical uncertainty in the extrapolation from high to low doses that arises from the limited number of data points and the limited number of animals tested at each dose (Crump et al., 1976). The statistical term confidence limits is used to describe the degree of confidence that the estimated response from a particular dose is not likely to differ by more than a specified amount from the response that would be predicted by the model if much more data were available. EPA and other agencies generally use the 95 percent upper confidence limit (UCL) of the dose-response data to estimate stochastic responses at low doses. 

Low-dose extrapolation models are the backbone of dose-response assessments. Because they can play such a dominant role in the regulatory process, it is important to understand some of their characteristics. As shown in Figure 3.10, different extrapolation models usually fit the data in the observable dose region in animal tests about equally well (Krewski et al., 1989), but they often give quite different results in the unobserved low-dose region of interest in assessments of risk to human health. The results obtained by extrapolation of the most commonly used low-dose models usually vary in a predictable manner, because the models use different mathematical equations to describe the chemical s likely behavior in the low-dose region. 

PBPK models are particularly useful for interspecies extrapolations of dose-response data. In using a PBPK model of uptake, distribution, and elimination, an exponential power (e.g., 0.75) of the body weight is used to scale the cardiac output and ventilation rate between the laboratory species (typically rat) and humans. A PBPK model will therefore contain adequate logic to account for routes of administration, storage tissues and residence time therein, elimination rates, and sufficient mathematical detail to mimic the integration of these processes. It is important that the model parameters (e.g., elimination rates) be validated as much as possible by separate kinetic studies in the relevant species. The ultimate test of the model is how the model predictions are for parameters such as blood levels, rate of metabolism, and tissue concentrations relative to real-life animal data for the chemical. 

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. 

Quantitative risk assessment requires extrapolation from results of experimental assays conducted at high dose levels to predicted effects at lower dose levels which correspond to human exposures. The meaning of this high to low dose extrapolation within an animal species will be discussed, along with its inherent limitations. A number of commonly used mathematical models of dose-response necessary for this extrapolation, will be discussed. Other limitations in their ability to provide precise quantitative low dose risk estimates will also be discussed. These include the existence of thresholds incorporation of background, or spontaneous responses modification of the dose-response by pharmacokinetic processes. 

The high to low dose extrapolation problem is conceptually straight-forward. The probability of a toxic response is modeled by a dose-response function P(D) which represents the probability of a toxic response when exposed to D units of the toxic agent. A general mathematical model is chosen to describe this functional relationship, its unknown parameters are estimated from the available data, and this estimated dose-response function P(D) is then used to either (1) estimate the response measure at a particular low dose level of interest or (2) estimate that dose level corresponding to a desired low level of response (this dose estimate is commonly known as the virtually safe dose, VSD). 

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... 

The next question to be addressed was that of the mathematical model to be used for the extrapolation. Most particularly, would one model do for all effects or was more than one required This is obviously particularly a problem with cancer. Various models have been proposed for cancer, but there has been little consideration of the use of dose/response extrapolation for effects other than cancer the safety factor approach is assumed adequate. For reasons given above, the Committee did not agree. 

A number of methods (mathematical models) are available for extrapolating the results at high doses to estimate the frequency of effects at low doses. The data for a given chemical that has been tested on animals are used to fit a mathematical model, using regression techniques. That is, for each dose the total number of animals treated is divided into numbers of animals that showed specific effects (e.g., development of tumor). This fraction or probability of response for each dose comprises the raw data for the model. The results of the available animal studies are collected in various updated databases. 

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