Model based extrapolation

When a model-based extrapolation technique is unavailable, it may be possible to work along the axis of Figure 10.2 concerning the choice of data, that is, to choose for the use of physical models. With the term physical model, it is meant that the model itself consists of creating an experimental or observational situation that mimics the situation of concern. This option is often applied in the registration of pesticides, where microcosm, mesocosm, and/or field experiments are used to characterize the impacts of pesticides on nontarget species. When the physical models do not, in some aspects, resemble the situation of concern, 1 or several extrapolation steps may be needed. 

A more general problem is that, at high temperature, a variety of new complexes form that would not be predicted from extrapolation of low temperature data. Sherman et al. (2000) for example, discovered that Sn forms (SnCU) complexes at T > 250°C extrapolation of low temperature data suggest that only SnCb and (SnCls) should be present. Polynuclear complexes appear to be very important near 300°C. The existence of such complexes cannot be easily predicted from a simple Bom-model based extrapolation of stability constants of mononuclear complexes observed at low temperature. 

There are many large molecules whose mteractions we have little hope of detemiining in detail. In these cases we turn to models based on simple mathematical representations of the interaction potential with empirically detemiined parameters. Even for smaller molecules where a detailed interaction potential has been obtained by an ab initio calculation or by a numerical inversion of experimental data, it is usefid to fit the calculated points to a functional fomi which then serves as a computationally inexpensive interpolation and extrapolation tool for use in fiirtlier work such as molecular simulation studies or predictive scattering computations. There are a very large number of such models in use, and only a small sample is considered here. The most frequently used simple spherical models are described in section Al.5.5.1 and some of the more common elaborate models are discussed in section A 1.5.5.2. section Al.5.5.3 and section Al.5.5.4. 

Instead of radical reactions, models based on molecular reactions have been proposed for the cracking of simple alkanes and Hquid feeds like naphtha and gas oil (40—42). However, the vaUdity of these models is limited, and caimot be extrapolated outside the range with confidence. With sophisticated algorithms and high speed computers available, this molecular reaction approach is not recommended. 

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. 

Furthermore, there are two other aspects to the extrapolation problem one structural and one statistical. An illustrative example of these various cases can be found in a dataset of benzamides (S16.1). that one of the present authors (U.N.) published some time ago [44]. If one develops a PLS model based on the same descriptors and the same, experimental design-based, training set (compounds 1-16) augmented by compound 17 (Table 16.8) in order to prove the points raised above [the prediction limit (1.502) set to two times the overall RSD of the model (0.751) which roughly gives 95% confidence interval], one can observe the following with respect to predictions on the remaining test set compounds ... 

Gouy-Chapman, Stern, and triple layer). Methods which have been used for determining thermodynamic constants from experimental data for surface hydrolysis reactions are examined critically. One method of linear extrapolation of the logarithm of the activity quotient to zero surface charge is shown to bias the values which are obtained for the intrinsic acidity constants of the diprotic surface groups. The advantages of a simple model based on monoprotic surface groups and a Stern model of the electric double layer are discussed. The model is physically plausible, and mathematically consistent with adsorption and surface potential data. 

This strategy is absolute in the sense that it employs symmetry properties of perfectly corresponding pairs rather than approximate models based on extrapolation, analogy, homology, or calculations from first principles. No other method is available in chemistry to provide such strong proof of internal consistency. Any measurable... 

Tlie operating ranges of the prediction model based on the calibration data set follow. Predicting future samples from outside this operating i nge is extrapolating and ma produce unreliable results. 

When human data are unavailable or limited, the risk assessment must be based on extrapolation from animal data, short-term tests, and other information. Under these circumstances, the evidence must be weighed with reference to its predictive accuracy, and expert judgment must be exercised in selecting the appropriate model for extrapolation toman. 

Quality by Design The incorporation of functional relationships between inputs, parameters, and product quality (or efficiency), which inherently imply magnitude and directionality, enables the use of a process design space as a tool for multiobjective process optimization. Furthermore, the model-based representation of knowledge is compatible with concepts of risk management, enabling more flexible operation since the risk associated with extrapolation could be predicted. 

Whereas there is little doubt that the method of moments, as the procedure is called, is basically sound, it is obvious that for reliable results high-quality experimental data over a broad range of frequencies and temperatures are desirable. As importantly, reliable models of the interaction potential must be known. Since these requirements have rarely been met, ambiguous dipole models have sometimes been reported, especially if for the determination of the spectral moments substantial extrapolations to high or to low frequencies were involved. Furthermore, since for most works of the kind only two moments have been determined, refined dipole models that attempt to combine overlap and dispersion contributions cannot be obtained, because more than two parameters need to be determined in such case. As a consequence, empirical dipole models based on moments do not attempt to specify a dispersion component, or test theoretical values of the dispersion coefficient B(7) (Hunt 1985). 

The PLS model for fed-batch fermentation performs well when the fermentation is operating within conditions that are represented in the 20 training batches used in developing the model. As with many model-based systems, model performance is poor when extrapolating outside of the operating conditions of the training set. 

Significant advances have been made in recent years in the understanding of the mechanisms of liver carcinogenicity of DEHP in rats and mice. However, there is increasing mechanistic evidence suggesting that rats and mice are not an appropriate model for extrapolating to humans (Cattley et al. 1998 Doull et al. 1999 Huber et al. 1996). Therefore, speculation on how to prevent liver cancer in humans based on information in rodents seems without scientific basis. 

When a model (e.g., concentration addition or response addition) is considered appropriate for describing the mixture effects observed in experiments, it can serve predictive purposes (such as formulating a scientific null hypothesis for an experiment), or for practical extrapolation and for risk assessment. There are, however, limitations associated with the concepts and the associated models based on pharmacodynamic reasoning. These limitations were first recognized by Plackett and Hewlett (1952), yet have mainly gone unnoticed by followers of the mechanistic school of mixture toxicity. Three main limitations are identified, and extrapolation solutions are provided. 

A proposal for tiering was suggested, based on an array of simple and more complex modeling approaches, consisting of an uncertainty factor-based approach (Tier-1), point-estimate-based extrapolation (Tier-2), and full-curve-based extrapolation (Tier-3) to curve-based extrapolation with special emphasis on the link between mode of action and ecological receptors (Tier-4, only for assemblages). 

Recently, metapopulation models have been successfully applied to assess the risks of contaminants to aquatic populations. A metapopulation model to extrapolate responses of the aquatic isopod Asellus aquaticus as observed in insecticide-stressed mesocosms to assess its recovery potential in drainage ditches, streams, and ponds is provided by van den Brink et al. (2007). They estimated realistic pyrethroid concentrations in these different types of aquatic ecosystems by means of exposure models used in the European legislation procedure for pesticides. It appeared that the rate of recovery of Asellus in pyrethroid-stressed drainage ditches was faster in the field than in the isolated mesocosms. However, the rate of recovery in drainage ditches was calculated to be lower than that in streams and ponds (van den Brink et al. 2007). In another study, the effects of flounder foraging behavior and habitat preferences on exposure to polychlorinated biphenyls in sediments were assessed by Linkov et al. (2002) using a tractable individual-based metapopulation model. In this study, the use of a spatially and temporally explicit model reduced the estimate of risk by an order of magnitude as compared with a nonspatial model (Linkov et al. 2002). 

Finally, the diversity of extrapolation techniques relates to the diversity of technical solutions that have been defined in the face of the various extrapolation problems. Methods may range from simple to complex, or from empirical-statistical methods that describe sets of observations (but do not aim to explain them) to mechanism-based approaches (in which a hypothesized mechanism was guiding in the derivation of the extrapolation method). In addition, they may range from those routinely accepted in formal risk assessment frameworks to unique problem-specific approaches, and from laboratory-based extrapolations consisting of 1 or various kinds of modeling to physical experiments that are set up to mimic the situation of concern (with the aim to reduce the need for extrapolation modeling). 

When sufficient data are available, use of the benchmark dose (BMD) or benchmark concentration (BMC) approach is preferable to the traditional health-based guidance value approaches (IPCS, 1999a, 2005 USEPA, 2000 Sonich-Mullin et al 2001). The BMDL (or BMCL) is the lower confidence limit on a dose (the BMD) (or concentration, BMC) that produces a particular level of response or change from the control mean (e.g. 10% response rate for quantal responses one standard deviation from the control mean for a continuous response) and can be used in place of the NOAEL. The BMD/BMC approach provides several advantages for dose-response evaluation 1) the model fits all of the available data and takes into account the slope of the dose-response curve 2) it accounts for variability in the data and 3) the BMD/BMC is not limited to one experimental exposure level, and the model can extrapolate outside of the experimental range. 

Figure 12.21 Plot of/, vs. x, showing extrapolation to x, = 1. The straight lines represent ideal-solution models based on Henry s law and the Lewis/Randall rule.

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