Weibull models

A high shape faetor in the 2-parameter model suggests less strength variability. The Weibull model ean also be used to model duetile materials at low temperatures whieh exhibit brittle failure (Faires, 1965). (See Waterman and Ashby (1991) for a detailed diseussion on modelling brittle material strength.)... 

The analysis of the frequeney data is shown in Table 4.12. Note the use of the Median Rank equation, eommonly used for both Weibull distributions. Linear reetifieation equations provided in Appendix X for the 2-parameter Weibull model are used to... 

Whereas in the second approach of the size effects it is also assumed that fracture is controlled by defects, the strength is now considered a statistically distributed parameter rather than a physical property characterised by a single value. The statistical distribution of fibre strength is usually described by the Weibull model [22,23]. In this weakest-link model the strength distribution of a series arrangement of units of length L0 is given by... 

In the work described earlier, the applicability of the Weibull model was further tested by assessing the precision of estimation [expressed by the CV defined as the standard error of estimates divided by the estimated value] and the relative accuracy of estimation of the model parameters (based on the difference of the estimates from the actual value, divided by the actual value). Regarding the precision of estimates, for data with SD = 2 the maximum CV value for Wo, b, and c was 13%, 52%, and 16%, respectively, whereas the corresponding numbers for data with SD = 4 were 33%, 151%, and 34%, respectively. As expected, the precision of the estimates decreases as the SD of the data increases, with the poorest precision for the b estimates and the best for the Wo estimates. Additionally, the maximum CV values were associated with low c values (c = 0.5). 

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).

A comparison of the simulation results and fittings with the Weibull and the power-law model is presented in Figure 4.7. Obviously, the Weibull model describes quite well all release data, while the power law diverges after some time. Of course both models can describe equally well experimental data for the first 60% of the release curve. 

Figure 4.7 Number of particles inside a cylinder as a function of time with initial number of drug molecules n0 = 2657. Simulation for cylinder with height 21 sites and diameter 21 sites (dotted line). Plot of curve n (t) = 2657exp (—0.04910 72), Weibull model fitting (solid line). Plot of curve n(t) = 2657 (l — 0.0941° 45), power-law fitting (dashed line).

Tertiary failure processes akin to this have been modeled by Phoenix and co-workers13-15 in the context of epoxy matrix composites. Indeed, they show that such tertiary failure can occur even when the fiber strength is statistical in nature. This mechanism will not be pursued further in this chapter but some other basic results considered on the assumption are that when there is a sufficient spread in fiber strengths, such tertiary failures can be postponed well beyond the occurrence of first fiber failure or indeed eliminated completely. Thus, attention will be focused on fibers which obey the classical Weibull model that the probability of survival of a fiber of length L stressed to a level oy is given by... 

Considering all the above data, the U.S. EPA (1991) selected the unit risk of 8.5 x 10 per pg/m, derived from the Weibull time-to-tumor model, as the recommended upper bound estimate of the carcinogenic potency of sulfur mustard for a lifetime exposure to HD vapors. However, U.S. EPA (1991) stated that "depending on the unknown true shape of the dose-response curve at low doses, actual risks may be anywhere from this upper bound down to zero". The Weibull model was considered to be the most suitable because the exposures used were long-term, the effect of killing the test animals before a full lifetime was adjusted for, and the sample size was the largest obtainable from the McNamara et al. (1975) data. 

All these quantities are readily determined. However, the determination of lc and Of (/c) necessitates a statistical analysis using the Weibull model. Gf (/c) cannot be measured directly, since /c is usually less than 0.5 mm. Therefore, it is determined from the tensile strength Gf (/) at higher gauge lengths using Equation 20... 

If the hazard is constant, that is, X(t) = X for all t> 0, then S(t) = This distribution is the exponential distribution with hazard equal to X. The Weibull model... 

The available evidence indicates that Druckrey s equation [Eq. (3.10)] with n > 1 can serve as a regulatory basis for risk assessment of genotoxic and nongenotoxic carcinogens. Carlborg (1981) pointed out that this equation is implied by a Weibull model for dose-response functions in carcinogenesis, as follows. The simple form of the Weibull model is... 

Carlborg, F. W. (1981). Dose-response functions in carcinogenesis and the Weibull model. Food Cosmet Toxicol 19, 255-263. 

EPA carried out the analysis using the polynomial model and the Weibull model. The results of the two models were within 3% of each other. EPA based its analysis on the Weibull model due to goodness of fit and history of use. The Benchmark dose is an estimate of an experimental dose associated with a specified low incidence of adverse effects. According to the Integrated Risk Information System (IRIS), the following uncertainty factors were applied 3 for the variability in human population (variability in the half-life of methylmercury and in hair-to-blood ratio) and 3 for the lack of a two-generation reproductive study and data on the effect of exposure duration on sequelae of the developmental neurotoxicity effects and on adult paresthesia. 

Pharmacokinetic models involving nonlinear kinetics of the Michaelis-Menten form have the important extrapolation characteristic of being linear at low dose levels. This low dose linearity contrasts with the low dose nonlinearity of the multihit and Weibull models. Each model, pharmacokinetic, multihit, and Ifeibull, has the desirable ability to describe either convex (upward curvature) or concave (downward curvature) dose-response relationships. Other models, stich as the log normal or multistage, are not consistent with concave relationships. However, the pharmacokinetic model differs from the multihit and Heibull in that it does not assume the nonlinear behavior observed at high dose levels will necessarily correspond to the sane nonlinear behavior at low dose levels. 

Although the Weibull model allows qualitative description of the sensor reliability and thus the comparison Q15 to other variants, more specific experiments were needed to identify and understand the parameters of the failure mechanism itself. Thus, dedicated experiments, accompanied by simulations, were designed. With this setup, it became feasible to test several membranes differing in thickness, layer composition, and length/width ratio in a reasonable amount of time. The effects of particle size, velocity, and intensity Q16 on lifetime were also investigated. 

Still, even with following Wagner s guidelines it may be that many different models fit the data equally well. The situations becomes more complex when variables other than time are included in the model, such as in a population pharmacokinetic analysis. Often then it is of interest to compare a number of different models because the analyst is unclear which model is the more appropriate model when different models fit almost equally to the same data. For example, the Emax model is routinely used to analyze hyperbolic data, but it is not the only model that can be used. A Weibull model can be used with equal success and justification. One method that can be used to discriminate between rival models is to run another experiment, changing the conditions and seeing how the model predictions perform, although this may be unpractical due to time or fiscal constraints. Another alternative is to base model selection on some a priori criterion. 

Buzrul et al. (2008) described high-pressure (400-600 MPa, 22°C) inactivation kinetics of E. colt and L. innocua using the Weibull model. A parameter Zp was defined as an increase in pressure resulfing in 1-log reduction in microbial population. 

The use of Weibull plots for design purposes has to be handled with extreme care. As with all extrapolations, a small uncertainty in the slope can result in large uncertainties in the survival probabilities, and hence to increase the confidence level, the data sample has to be sufficiently large N > 100). Furthermore, in the Weibull model, it is implicitly assumed that the material is homogeneous, with a single flaw population that does not change with time. It further assumes that only one failure mechanism is operative and that the defects are randomly distributed and are small relative to the specimen or component size. Needless to say, whenever any of these assumptions is invalid, Eq. (11.23) has to be modified. For instance, bimodal distributions that lead to strong deviations from a linear Weibull plot are not uncommon. 

All the models are based upon standard statistical distributions. The Multi-Hit model is based upon a Poisson distribution. The Probit model is based upon a normal distribution, and the One Hit, Multistage and Weibull models rely upon linear probabilities. Such distributions have proven applicability in dealing with substantial percentages of the population (up to 1 in 20). However, the models lose precision when they are pushed to extremes such as 1 in 10, such as encountered in risk assessment. Furthermore, homeostatic mechanisms such as DMA repair and immunological survellance may be poorly evaluated in risk assessment. Such high doses are administered to achieve the siaximum tolerated dose, that these mechanisms are surely overwhelmed in the animal studies. Additionally it should be noted that a risk of one in a million does not mean one tumor in the lifetime of a million people, it means that each individual has one chance in a million of developing a tumor in a lifetime. 

Infant-mortality period. T3fpical infant-mortality distributions for state-of-the art semiconductor chips follow a Weibull model with a jS in the range of O.l-O.h." ... 

The Weibull model cannot describe the volume dependence of strength data [22], although aWeibull modulus (m) can be extracted from the statistical distribution of strengths m is in the range 20-29. This value provides an evaluation of the scatter in strength data. 

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