Extrapolation using mathematical models

Another major difference between the current approach and the previous approach for evaluating the adverse effects of pesticide exposure involves the way in which the toxicity data are interpreted. In the current system, the carcinogenicity data from the chronic rodent studies are extrapolated using mathematical models which provide a numerical estimate of the upper bound of the cancer risk, and these numbers (Q values) are then used for a variety of regulatory purposes. In essence, this approach substitutes mathematical guidelines for the scientific judgement that was the key element in the earlier approach. 

The primary advantage of extrapolation using mathematical models is that it avoids the necessity of debating a no-effect or threshold level, which cannot be scientifically documented. 

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. 

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. 

In general, short-term tests have limitations. They do not include variables that animal studies do. It becomes difficult to use mathematical models to extrapolate test results to humans. For example, one of the notable short-term tests for carcinogens is the Ames test. The procedure uses Salmonella and the results involve dose, but not time or duration of exposure. Some people criticize the Ames test for a lack of reliability. The Ames test demonstrates high correlations between certain known carcinogens and human experience. For other carcinogens there are low correlations between test results and human experience. 

This requirement also makes good sense. A calibration is nothing more than a mathematical model that relates the behavior of the measureable data to the behavior of that which we wish to predict. We construct a calibration by finding the best representation of the fit between the measured data and the predicted parameters. It is not surprising that the performance of a calibration can deteriorate rapidly if we use the calibration to extrapolate predictions for... 

Mathematical Models. As noted previously, a mathematical model must be fitted to the predicted results shown In each factorial table generated by each scientist. Ideally, each scientist selects and fits an appropriate model based upon theoretical constraints and physical principles. In some cases, however, appropriate models are unknown to the scientists. This Is likely to occur for experiments Involving multifactor, multidisciplinary systems. When this occurs, various standard models have been used to describe the predicted results shown In the factorial tables. For example, for effects associated with lognormal distributions a multiplicative model has been found useful. As a default model, the team statistician can fit a polynomial model using standard least square techniques. Although of limited use for Interpolation or extrapolation, a polynomial model can serve to Identify certain problems Involving the relationships among the factors as Implied by the values shown In the factorial tables. 

Ideally, a mathematical model would link yields and/or product properties with process variables in terms of fundamental process phenomena only. All model parameters would be taken from existing theories and there would be no need for adjusting parameters. Such models would be the most powerful at extrapolating results from small scale to a full process scale. The models with which we deal in practice do never reflect all the microscopic details of all phenomena composing the process. Therefore, experimental correlations for model parameters are used and/or parameters are evaluated by fitting the calculated process performance to that observed. 

The most widely used of the many mathematical models proposed for extrapolation of carcinogenicity data from animal studies to low-dose human exposures (i.e., low-dose extrapolation) is the LMS model. This has, in effect, become the default approach for quantitative risk assessment and has been used by, e.g., the US-EPA for many years as well as by the WHO in relation to derivation of drinking-water guideline values for potential carcinogens (WHO 1996) (see Section 9.2.1.2 for drinking-water guideline values). 

The extrapolation from high to low doses will depend on the type of primary toxic effect. If this is a carcinogenic effect, then a threshold normally cannot be assumed, and a mathematical model is used to estimate the risk at low doses (see above). If the primary toxic effect is noncarcinogenic, then it will normally be assumed that a threshold exists. 

Aging becomes a difficult problem to study in practice, because it proceeds too slow in use conditions (typical lifetime of years). It is then necessary to make accelerated aging tests to build kinetic models that describe the time changes of the material s behaviour, and to use these models to predict the durability from a conventional lifetime criterion. Indeed, the pertinence of the choice of accelerated aging conditions, the mathematical form of kinetic model, and lifetime criterion has to be proved. Empirical models are highly questionable in this domain because they have to be used in extrapolations for which they are not appropriate. 

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. 

What kinds of mathematical models must be used for both the interpolation of data and their extrapolation in terms of time and space with the goals to reduce the frequency and thus the cost of the observations and to increase the reliability of forecasting the environmental behavior of the observed items ... 

Oral bioavailability of aluminum compounds appears to generally parallel water solubility, but current knowledge does not allow a straight extrapolation from solubility in water to bioavailability. Studies of aluminum speciation in the stomach and intestines, including mathematical modeling, would be useful because they could enable such an extrapolation by helping to resolve the critical role of speciation in making aluminum available to uptake mechanisms. 

Because these necessary concepts have not been widely applied in the existing experimental studies, the data that have been collected in the past were reviewed on the basis of existing reviews and the mathematical characteristics of mixture models. From that, it was concluded that the mathematical models that are used in the best-case studies do predict mixture responses relatively well, although the use of some models may not be mechanistically justified, and although the models have peculiar biases that need be taken into account in relation to the objective of the extrapolation. 

Mathematical models have been developed and used to extrapolate toxicity under pulsed exposure conditions (for an overview, see Boxall et al. 2002 Reinert et al. 2002 Ashauer et al. 2006 Jager et al. 2006). Some models consider concentration x time (Meyer et al. 1995) others, uptake and depuration (Mancini 1983) or damage and repair (Breck 1988). Several models are based on the concept of critical body residues, which integrates toxicokinetics and the effects of exposure time on toxicity (McCarty and Mackay 1993 Barron et al. 2002). This approach is promising because several studies showed that toxicity from pulse exposures is largely... 

In order to extrapolate the laboratory results to the field and to make semiquantitative predictions, an in-house computer model was used. Chemical reaction rate constants were derived by matching the data from the Controlled Mixing History Furnace to the model predictions. The devolatilization phase was not modeled since volatile matter release and subsequent combustion occurs very rapidly and would not significantly impact the accuracy of the mathematical model predictions. The "overall" solid conversion efficiency at a given residence time was obtained by adding both the simulated char combustion efficiency and the average pyrolysis efficiency (found in the primary stage of the CMHF). 

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