Multi-stage model

The most widely-accepted dose response model at the present time is the multi-stage model, which has great flexibility in curve-fitting, and also has a strong physiological justification. Although it is difficult to implement, there are already computer codes in existence that estimate the model parameters (13). The two most widely-used models, until recently, were the one-hit model and the log-probit model. They are both easy to implement, and represent opposite extremes in terms of shape - the former represents the linear non-threshold assumption, whereas the latter has a steep threshold-like curvature. In numerous applications with different substances it has been found that these three... 

Mathematical optimization models that explicitly consider such a multi-stage structure belong to the class of multi-stage stochastic programs. A deterministic optimization model with uncertain parameters is extended to a multi-stage model by three measures ... 

In this fashion, we extend our deterministic model with a prediction horizon of H = 2 to a multi-stage model. The multi-stage tree of the possible outcomes of the demand within this horizon (starting from period i = 1) with four scenarios is shown in Figure 9.5. Each scenario represents the combination fc out of the set of all combinations of the demand outcomes within the horizon. The production decision x has to be taken under uncertainty in all future demands. The decision xj can react to each of the two outcomes of d i, but has to be taken under uncertainty in the demand di. The corrective decisions are explicitly modeled by replacing xj by two variables 2,1 and 2.2 ... 

However, the description of the tree structure of a multi-stage model leads to complicated constraints. To simplify the original multi-stage model, it is approximated by a model with two stages. It consists of only one sequence of decisions-observation-decisions. The two-stage structure leads to considerably simpler optimization problems. It is also adequate from a practical point of view in the moving horizon scheme, only the first decision x is applied to the plant while all the remaining variables are used to compute the estimated performance only. 

Multi-stage model (with quadratic term) 5 3.5... 

Fig. 8.6 Estimated risk of liver cancer, P(d), in relation to dose of aflatoxin, d, as determined with different dose-incidence models. The models for the different curves. are as follows OH. one-hit model MS, multi-stage model W, Weibull model MH, multihit model MB, Mantel-Bryan (log-probit model) (from Krewski and Van Ryzin, 1981).

For the purpose of classifying waste that contains radionuclides, NCRP reaffirms use of the nominal probability coefficient for fatal cancers (i.e., the probability of a fatal cancer per unit effective dose) of 0.05 Sv 1 normally assumed in radiation protection of the public. For chemical carcinogens, NCRP believes that MLEs of probability coefficients obtained from the linearized, multi-stage model should be used in classifying waste, in order to provide reasonable consistency with the probability coefficient for radionuclides. The use of MLEs for chemical carcinogens usually will result in substantially lower probability coefficients than the use of upper 95 percent confidence limits. 

The one-hit and linearized multi-stage models usually will predict the highest response rates and the probit model the lowest (Paustenbach, 1989a 1989b). 

For chemicals that cause stochastic responses, NCRP believes that a linearized multi-stage model should be used to estimate risks at low doses based on data at high doses in humans or animals. Furthermore, NCRP believes that best estimates (MLEs) of response probabilities obtained from that model should be used for purposes of risk-based waste classification, rather than UCLs that are used nearly... 

An electrokinetic model of breakdown in polymers aims to explain the kind of breakdown process that occurs over a relatively long timescale in high-quality (uncontaminated, defect-free) materials, in contrast to the quick and violent modes of breakdown that are often experienced, when materials are overstressed. Undoubtedly, we require a multi-stage model, which will vary somewhat, depending on the exact composition of the material and its environment. 

Dawson, S. V., and Alexeeff, G. V. (2001). Multi-stage model estimates of lung cancer risk from exposure to diesel exhaust, based on a U.S. railroad worker cohort. Risk Anal 21(1), 1-18. Comment in Risk Anal 21(1), 19-23 Risk Anal 21(2), 213-216. 

For oncogenic materials the NACA committee recommended a calculated risk of one in a million using the Multi-Stage model. The NACA committee further indicated that Health Advisories are not intended to encourage practices which will result in health advisory levels [10]. 

U.S. EPA (Ref. 2 year rat study Kociba a1. (Ref. 30) Non Threshold, Linearized, Multi-stage model (dose/surface area) V.S.D. for upper-limit cancer risk of 10" =0.0064 pg/kg/day... 

For dioxin, there Is evidence that Its mechanism of action Is receptor-mediated ( ) and, to describe this mechanism, a low dose extrapolation model other than the linearized multi-stage Is more appropriate. The linearized multi-stage model forces linearity at low doses - a trait which should not be forced for chemicals which act through a receptor mediated event. Further, the linearized models do not account for the reversible behavior of promoters or the possibility of a dose at which no Increased risk would be expected. 

The decision to use an amortized exposure value or a peak exposure value has a profound Impact on the outcome of the quantitative risk assessment. To Illustrate this point, data from an actual field exposure study were used. The average dally dermal exposure level as measured by the patch technique was used to calculate the amortized exposure level and the peak exposure level (Table VIII). Estimates of risk at low doses were obtained using linear extrapolation from the 11 excess risk point based on a fitted Welbull model (32) and the Armltage-Doll multi-stage model (33). While both models gave similar results, the effect of the exposure estimates had a dramatic effect on the risk estloiates. The amortized exposure estloiates lowered the estloiates of risk substantially. 

Multi-stage Model 1.3x10"8 3.6x10 7 9.5x10- 1.2x10-2... 

Table III. For example, if a worker hae a urine concentration of 100 ppb of chlordimeform, the extra risk of acquiring cancer due to chlordimeform exposure is approximately 1 in 240,000 according to the multi-stage model. For the majority of the monitored workers whose urinary chlordimeform levels were below 50 ppm, their increased risk might be as low as 1 in 500,000.

An additional limitation of the Multi-Stage model, in fact of all the models discussed, is that generally the maximal incidence is assumed to be 100%. This, in practice, may not be possible since increasing administration may result in death rather than tumor formation. When the Multi-Stage model is used to evaluate the dose response for dimethylnitrosarnine, hepatocar-cinogenicity in the rat (16), it shows a better fit than the One-Hit, but the data does not fit the curve particularly well (Figure 3). The less than perfect fit is possibly because the model forces the maximum incidence to be 100% -- a condition not apparent in the data. 

When used with the saccharin data (20), the Multi-Stage model appears to fit the dose response very well (Figure 4). 

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