The Multistage Model

What is the physical background of the linear dependencies showing up in the T/djT -diagram of Fig. 5.33, i.e., the meaning of the crystallization and the recrystallization lines First of all, the difference between the crystallization and the melting lines demonstrates that different laws control crystallization and melting in bulk polymers. Here, crystallization and melting are not reverse 

The thermodynamic conditions under which such a mesomorphic phase can interfere and affect the crystallization process are described in Fig. 5.40. The schematic plot shows the difference of the chemical potential (per monomer) to that of the melt ( a ) for both the crystalline phase (label c ) and the mesomorphic phase ( m )  

Coming from high temperatures the chemical potential of the crystalhne phase drops below the value of the melt when crossing the equilibrium melting point, now denoted. The mesomorphic phase requires a lower temperature, T, to fall with its chemical potential below that of the melt. The plot also includes a temperature represents the temperature of a virtual transition, 

It is possible to construct a thermodynamic scheme that shows the features of Fig. 5.33, i.e., a crystallization hne, a reerj taUization hne, both being unaffected by co-units and stereo-defects, and a melting line. It deals with four different phases  

The scheme is displayed in Fig. 5.41 and delineates the stability ranges and transition lines for these phases. 

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Crump, K.S., and R.B.Howe. 1984. The multistage model with a time-dependent dose pattern Applications to carcinogenic risk assessment. Risk Anal 4 163-176. 

To adjust for uncertainties in assessing potential cancer risks for short-term exposures under the multistage model, the 24-h exposure is divided by an adjustment factor of 6 (Crump and Howe 1984). 

In its 1986 Guidelines for Carcinogen Risk Assessment, the US-EPA introduced the LMS model into the U.S. regulatory framework. The multistage model was chosen for regulatory... 

The linear component of the LMS model, qi (i.e., one of the parameters of the polynomial), is approximately equivalent to the slope at low doses of the dose-response relationship between the tumor incidence and the dose. This linearity at low dose is a property of the formulation developed for the multistage model and is considered by proponents to be one of its important properties. This linear component of the polynomial, qi, is used to carry out low-dose extrapolation. The linear response at low doses is considered to be conservative with regard to risk, as the dose-response relationship at low doses may well be sublinear. Although supralinearity at low doses cannot be excluded, it is usually considered to be unlikely. 

Because relatively low intakes (compared to those experienced by test animals) are most likely from environmental exposure at Superfund hazardous waste sites, it generally can be assumed that the dose-response relationship will be linear on the low-dose portion of the multistage model dose-response curve. The equation above can apply to these linear low-dose situations. This linear equation is valid only at low risk levels (i.e., below the estimated risk of 0.01). For risk above 0.01 the one-hit equation should be used ... 

For the above reasons, the linearized multistage model is used as a default when estimating risks for short-term exposures from lifetime carcinogenesis bioassays. In all of the above referenced publications on the multistage model, the maximum number of stages modeled was six. 

Conventionally, an extractor is designed to separate a specific compound in the most economical way. However, for multi-product applications the extractor should contain sufficient stages for each component of (future) interest. It was found with the multistage model that if a multi-purpose extractor with 50 stages is built, all systems with oCop >1.5 can be succesfully separated, and with 30 stages, all systems with oCop > 2.0. Any excess stages for a certain component opens possibilities to increase the capacity or reduce costs in that specific separation. The multistage model can be used as a tool for optimization. 

The equation for the multistage model is as follows, estimating the probability of developing cancer from exposures equivalent to a daily dose d ... 

Little, M. P. (1995). Are two mutations sufficient to cause cancer Some generalizations of the two-mutation model of carcinogenesis of Moolgavkar, Venzon, and Knudson, and of the multistage model of Armitage and Doll. Biometrics 51, 1278-1291. 

I also (Crouch 1996) evaluated the data of Gold et al. (see above) while illustrating the use of uncertainty distributions for interspecies extrapolations, but used the multistage model (rather than the one-hit model used to derive TDsqs) and a different adjustment to a standard lifetime. I demonstrated the lognormality of rat-to-mouse, mouse-to-hamster, and hamster-to-rat interspecies ratios of a carcinogenic potency parameter, and I also demonstrated that the available information was inconsistent with allometric scaling between these rodents with any power law. 

In order to avoid the disadvantages, seen or inferred, of the simple addition of q values, various analysts have either calculated or assumed a distribution (for each tumor type) representing the likelihood for the plausible range of estimates of the linear term (q ) of the multistage model (qi), and then they used the Monte Carlo procedure to add the distributions rather than merely adding specific points on the distributions such as the maximum likelihood estimate (MLE) or 95% confidence limit. A combined potency estimate (q for all sites) is then obtained as the 95% confidence limit on the summed distribution. This resembles the approach for multiple carcinogens by Kodell et al. (1995) noted above. 

A partial approach to the problem of adding potency values without imdertak-ing the Monte Carlo addition of the entire distributions of estimates has used the addition of fixed points on the distributions otho- than the 95% confidence limit. Unlike confidence limits, the MLE values of qi can be simply added to generate an MLE for the combined distribution [see US EPA (2002), for example]. However, it is generally recognized that the MLE is an unsatisfactory parameter for describing estimated potency slopes, as will be demonstrated below. This value is unstable for polynomial fits of variable order such as those used in the multistage model. Except... 

It should be noted that both the single-hit model and the multistage model (when Aj >0) become approximately linear at low dose levels. This low dose linearity is an important aspect of "conservative" extrapolation models. Many researchers believe that the true dose-response at low exposure levels is convex, i.e. may have some degree of upward curvature. Therefore, linearity, which represents an "upper bound" to the general class of convex functions, provides conservative extrapolated risk estimates at low doses ("conservative" in the sense of producing higher estimated risks than other convex functions). 

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