Linearized Multistage model

The most widely used of the many mathematical models proposed for extrapolation of carcinogenicity data from animal studies to low-dose human exposures (i.e., low-dose extrapolation) is the LMS model. This has, in effect, become the default approach for quantitative risk assessment and has been used by, e.g., the US-EPA for many years as well as by the WHO in relation to derivation of drinking-water guideline values for potential carcinogens (WHO 1996) (see Section 9.2.1.2 for drinking-water guideline values). 

The following overview of the LMS model is based primarily on the paper by Lovell and Thomas (1996). 

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 

This model assumes that any dosage effect has the same mechanism as that which causes the background incidence. Low-dose linearity follows directly from this additive assumption, provided that any fraction of the background effect is additive no matter how small. A best fit curve is fitted to the data obtained from a long-term rodent cancer bioassay using computer programs. The estimates of the parameters in the polynomial are called Maximum Likelihood Estimates (MLE), based upon the statistical procedure used for fitting the curve, and can be considered as best fit estimates. Provided the fit of the model is satisfactory, the estimates of these parameters are used to extrapolate to low-dose exposures. 

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. 

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S. Crump, K. S. (1996). The linearized multistage model and the future of quantitative risk assessment. Human Exper. Toxicol, 15, 787-798. 

To.xicity values for carcinogenic effects can be e.xprcsscd in several ways. The slope factor is usually, but not always, the upper 95th percent confidence limit of the slope of the dose-response curve and is e.xprcsscd as (mg/kg-day). If the extrapolation model selected is the linearized multistage model, this value is also known as the ql. That is ... 

This study, like that of Fisher and Allen (1993), incorporated a linear multistage model. However, the mechanism of trichloroethylene carcinogenicity appears to be non-genotoxic, and a non-linear model (as opposed to the linearized multistage model) has been proposed for use along with PBPK modeling for cancer risk assessment. The use of this non-linear model has resulted in a 100-fold increase in the virtually safe lifetime exposure estimates (Clewell et al. 1995). 

Lovell, D.P. and G. Thomas. 1996. Quantitative risk assessment and the hmitations of the linearized multistage model. Hum. Exp. Toxicol. 15 87-104. 

The linearized multistage model (used by the EPA). This determines the cancer slope factor, which can be used to predict cancer risk at a specific dose. It assumes a linear extrapolation to a zero-dose threshold (Fig. 2.10). This factor is an estimate (expressed in mg/kg/day) of the probability that an individual will develop cancer if exposed to the chemical for 70 years. 

The cancer risk values, which these models generate, are of course very different. For example, for the chemical chlordane, the lifetime risk for one cancer death in one million people ranges from exposures of0.03 pg/L of drinking water for the one-hit model, 0.07 pg/L from the linearized multistage model to 50 pg/L for the probit model. 

By using a modified linear multistage model, the NAS (1977) concluded that the nominal lifetime incremental risk of cancer falls between 1.5 and 3 X 10-7 per microgram per liter per day. The range of values obtained is the result of calculations made from several data sets. 

In 1980, the Ambient Water Quality Criteria USEPA Carcinogen Assessment Group (CAG) by using a further modified linear multistage model as well as a one-hit model, arrived at a value of 0.2 /zg/L/day as the upper 95 confidence estimate of the dose in drinking water contributing an excess lifetime risk of 1 in 1 million. 

U.S. regulations for environmental contaminants have generally fallen in the 10-4 to 10-6 lifetime risk range, as calculated from a relatively worst-case linear multistage model. Most of those decisions incorporated consideration of costs and feasibility. 

If RMCLs were to be set at a nonzero level, use of the linearized multistage model would often appear to be more appropriate than others to meet the congressional intent. The conservative nature of the model could actually mean that the real risk of exposure was probably lower (e.g., 10-7 or 10-8) if any risk actually exists (assuming a nonthreshold mechanism was operative) because the model was structured to be conservative and because of the nature of many of the assumptions in the model. 

The three sets of risk values listed were calculated by two versions of the linear multistage model generally from the same data, except for vinyl chloride for which CAG 1984 calculation used a different animal... 

ADI-type calculations were performed for noncarcinogens. For potential carcinogenic risk calculations, a linearized multistage model or a one-hit model was employed and water concentrations equivalent to calculated risks of 10"5, 10 6, and 10r were reported. No selection was made as the specific criterion however, 10-5 was suggested as a reasonable value. 

Concentration values in micrograms per liter are provided in the box on page 728 for 40 substances at the calculated 10-5 risk level at the upper bound. Both a linearized multistage model and the one-hit model (in parentheses) were used (45). Many of these values are now being updated. 

In the proposed criteria, the linear model was used to calculate the concentrations associated with incremental lifetime risks of 10-5. However, in response to public comment, the USEPA ultimately decided to adopt the linearized multistage model to make full use of all available data. Comparison of the values reported in the box indicates that, for most cases, the concentrations calculated by either model for a given nominal risk are very close. 

The one-hit or linear multistage models should always be among the models employed because they are usually among the more conservative procedures. Reasonable worst-case assumptions must often be made to err on the side of safety. 

The use of MLEs of probability coefficients for radionuclides but UCLs for chemicals that induce stochastic responses is the most important issue that would need to be resolved to achieve a consistent approach to estimating risks for the purpose of waste classification. For some chemicals, the difference between MLE and UCL can be a factor of 100 or more. The difference between using fatalities or incidence as the measure of response is unlikely to be important. Use of the linearized, multistage model to extrapolate the dose-response relationship for chemicals that induce stochastic effects, as recommended by NCRP, should be reasonably consistent with estimates of the dose-response relationship for radionuclides, and this model has been used widely in estimating probability coefficients in chemical risk assessments. The difference in the number of organs or tissues that are taken into account, although it cannot be reconciled at the present time, should be unimportant. 

The EPA uses the linearized multistage model (LMS)—illustrated in Figure 9.34—to conduct its cancer risk assessments. It yields a cancer slope factor, known as the ql (pronounced Ql-star), which can be used to predict cancer risk at a specific dose. The LMS assumes a linear extrapolation with a zero dose threshold from the upper confidence level of the lowest dose that produced cancer in an animal test or in a human epidemiology study. 

In this example, the distributions of EDioS were derived for atrazine and simazine. The 5th percentiles in these distributions have a straightforward interpretation. However, this would not be the case if lower bounds (LEDiqs) based on the linearized multistage model were used instead of EDioS. In addition, the use of multiple lower bounds introduces an unknown amount of compounded conservatism which is not readily interpretable and is potentially very misleading in a regulatory setting. 

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. 

The consideration of mode of action in carcinogen risk assessment is becoming standard practice. When data are adequate to demonstrate use of the standard default low dose extrapolation models such as the linearized multistage model is not appropriate, alternate approaches, including threshold approaches are now being used. 

The assessment of liver cancer risks associated with human exposure to trichloroethylene (TCE) was initially conducted by Fisher and Allen (1993) using a PBPK-modeling approach. The use of the amount of TCE metabolized per day as dose metric used in the linearized multistage model led to lOppb in air and 7 ag/L in water as acceptable concentrations—that is, environmental levels corresponding to a population cancer risk of 1 in 10 (Fisher and Allen 1993). Corresponding values based on circulating levels of the metabolite, trichloroacetic acid, were 10 times and twice lower than those based on the amount of TCE metabolized per unit time, whereas the acceptable TCE concentration in air as defined by the EPA at the time was 90 times lower. A number of authors subsequently investigated the dose metrics and cancer risks associated with TCE [e.g., Bois (2000), CleweU and Andersen... 

In the final analysis it seems evident that the time-dependent version of the linearized multistage model is the best way of fitting a complex dataset such as that for... 

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