Cumulative Integral Extrapolation

Taylor expansion of (6.89) is not based on a physical model, and more-complicated nonlinear forms could be used. However, nonlinear extrapolation methods seem to offer little improvement over their linear counterparts [58]. Larger improvements appear to be achieved using the cumulative integral extrapolation method discussed in Sect6.7.3. 

Since the linear and related expansion formulas depend on fits to regions of the curve that are statistically less and less reliable, it makes sense to find a measure for extrapolation that depends on the relative accuracy of the relative free energy estimate for all points along the curve. The cumulative integral extrapolation method is one approach to this idea. 

The general idea of this approach is to consider the relative free energy to be a smooth function of AAn from the inverse x = 1 /nT running from 0 to 1. At the zero limit this reflects infinite sampling and is thus the direction of extrapolation. Note that the value of r that optimizes this transformation must be determined as 

In more detail, the most precise AAn values are obtained for smaller n, or larger X = 1 /nT 1. The linear extrapolation scheme relies exclusively on small x values. More precise free energy estimate can be obtained by including all values of x during the extrapolation [58]. 

consider treating AAn as a smooth function AAn(x), from x = 0 (n = oo) to 1 (n = 1). The area under this function is given by the integral 

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The data passes the baseline with the required cumulative integral in the required time. If the time limit is exceeded, an extrapolated value from the peak difference on the curve is used to find a shorter base line crossing and cumulative integral. 

In estimating the cumulative risk of a chemical in LCA, dose-response extrapolations can be based on toxicological benchmarks. Such a benchmark approach is considered more appropriate for use in comparative assessment contexts, such as in an LCA study. Benchmarks are an exposure measure associated with a consistent change in response, such as the 10% or even the 50% effect level. Regulatory-based measures do not necessarily provide a consistent risk basis for comparison, as they were often never developed for use in such a comparative context or to facilitate low dose-response extrapolation. Other data differences include the use of median, rather than extreme, data in the fate and exposure modeling, as well as the consideration of safety factors only as part of the uncertainty assessment and not as an integral part of the toxicological effects data. 

In Table 4.5.b. the values of the constants obtained from the experiments are given (K in im h ") by extrapolating these data to an integrated exposure period of 30 years, the cumulative corrosion removal from the materials shown in Table 4.5.C. results. 

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