Linear, no-threshold models

Linear, no-threshold model (extrapolated upper bound on low-dose risk)... 

Figure 8.1 Dose-response curves for carcinogens and illustration of low-dose extrapolation using linear, no-threshold model. Benchmark dose (BMD) is also illustrated.

A simple example might make this clearer. Suppose it were known that a 100 mg dose of chemical Z produced an extra 10% incidence of liver tumors in rats. Suppose further that we studied the pharmacokinetics of compound Z and discovered that, at the same 100 mg dose, 10 mg of the carcinogenic metabolite of Z was present in the liver. The usual regulatory default would instruct us to select the 100 mg dose as the point-of-departure for low dose extrapolation, and to draw a straight line to the origin, as in Figure 8.1. We are then further instructed to estimate the upper bound on risk at whatever dose humans are exposed to - let us say 1 mg. If the extra risk is 10% at 100 mg, then under the simple linear no-threshold model the extra risk at 1 mg should be 10% 100 = 0.1% (an extra risk of... 

Linear, No-Threshold Model. This simplest model is based on the assumption that risk is directly proportional to the dose P(d) = ad. When it is assumed that the true dose-response curve is convex, linear extrapolation in the low-dose region may overestimate the true risk. However, it is not known if the experimental dose is in the convex region of the curve. 

A few court decisions, however, have been more skeptical of the linear model. Eor example, the U.S. EPA s use of the linear, no-threshold model in its risk assessment for drinking water chlorinated byproducts was rejected by the court because it was contrary to evidence suggesting a nonlinear model that had been accepted by both the U.S. EPA and its Science Advisory Board (CCC 2000). On the other hand, the U.S. OSHA s departure from the linear, no-threshold model in its formaldehyde risk assessment was likewise rejected by the court (lU 1989). The court held that the U.S. OSHAhad improperly used the maximum likelihood estimate (MLE) rather than the upper confidence limit (UCL) to calculate risk, and the UCL but not the MLE model was consistent with a linear dose-response assumption. The court held that the U.S. OSHA had failed to justify its departure from its traditional linear, no-threshold dose-response assumption. 

Judicial decisions in nonregulatory contexts such as toxic tort and product liability suits are likewise inconsistent in their consideration of the linear, no threshold model. As in the regulatory context, most cases find no problem with an expert s reliance on a risk assessment using the linear model. In a handful of cases, however, the court rejects reliance on a linear dose-response assumption. Eor example, one court in addressing the cancer risks from a low concentration of benzene in Perrier held that there is no scientific evidence that the linear no-safe threshold analysis is an acceptable scientific technique used by experts in determining causation in an individual instance (Sutera 1997). Another court decision concluded that [t]he linear non-threshold model cannot be falsified, nor can it be validated. To the extent that it has been subjected to peer review and publication, it has been rejected by the overwhelming majority of the scientific community. It has no known or potential rate of error. It is merely an hypothesis (Whiting 1995). The inconsistency and unpredictability of judicial review of risk assessments adds an additional element of uncertainty into the risk assessment process. 

From the response point of view, it may be difficult or impossible to detect with certainty a threshold of damage. It is to be expected that where relativefy simple pollutant substances contain essential elements, a threshold model is, a priori, the most Ukety relationship. However, uncertainty boundaries in the dose-response relationship might equally well accommodate both a threshold or a linear no threshold model response. 

A mutated cell may reproduce and begin the formation of a carcinogenic mass (tumor), and mutations may occur after acute or chronic exposure. The specific relationship between acute or chronic exposure rate and cancer risk is hotly debated, although current U.S. regulations conservatively adopted the linear no threshold (LNT) model. This model states that risk is linearly proportional to the total dose even at the smallest possible dose levels (risk is associated with all levels of dose no matter how small). An alternate model theorizes that no measurable adverse health effects appear below doses of about 10 to 25 rem (0.1 to 0.25 Sv). Data supporting both models are limited and, to be conservative, levels of exposure should be kept as low as reasonably achievable (ALARA). Victim and emergency responder doses and dose rate may not be easily controlled in the event of a terrorist attack. However, methods to achieve ALARA exposures are described in Chapters 4 and 5. 

For carcinogens a linear, no-threshold dose-response model (Figure 8.1) is assumed to apply at low dose, unless data are available in specific cases to demonstrate that such a model is inappropriate. 

The linear, no-threshold, dose-response model is accurate for very low exposures. 

Dose B is greater than dose A. This means that under the sublinear, no-threshold model, individuals could tolerate a dose larger than they could tolerate if the linear model, predicting dose A as the maximum allowable, were correct. To put this the other way round, individuals exposed to dose B would experience an extra lifetime risk of 1/100 000 under the sublinear model, but would experience a higher risk (1/10000 orO.OOOl in the example of Figure 5) if the linear model were correct. The linear model thus predicts higher risks at a given dose than does the sublinear model, or, conversely, the linear model requires... 

This chapter deals with correcting the use of conjectures as defaults in regulatory policy, in the context of experimental evidence of hormesis and causation and alternative probabilistic cancer models. Specifically, we summarize how the combination of mode-of-action and weight-of-evidence supports both J-shaped and U-shaped, rather than the linear, no-threshold (LNT) models. The EPA uses the terms nonlinear for the threshold model and low-dose-linear for the LNT models (meaning that the slope is greater than zero at zero dose), which is well-approximated by a straight line, at very low doses and beginning from zero dose (EPA 2005). 

The first conclusion is that the factual and theoretical evidence points to replacing the classical causal regulatory defaults used to deal with low dose-response, the linear no-threshold, and the linear at low-dose-response models, or monotonic functions, with the J- and inverse J-shaped models—or relations. These models have been demonstrated to apply to toxicological and cancer outcomes for a very wide range of substances and diseases. The classical defaults may stiU be applicable on a case-by-case basis. The reasons for changing the defaults include the fact that the J-shaped class of models quantities a wide set of health benefits that are completely excluded from estimations that use monotonic models. We conclude that replacing both a conjecture and an arbitrary model with two theoretically and empirically sound ones leads to rational decision and does not exclude actually demonstrable benefits. Overall, the sum is positive for society. 

Breckow J (2006) Linear-no-threshold is a radiation-protection standard rather than a mechanistic effect model. Ra-diat Environ Biophys 44 257-260 Brenner DJ, Sachs RK (2006) Estimating radiation-induced cancer risks at very low doses rationale for using a linear no-threshold approach. Radiat Environ Biophys 44 253-256... 

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