Nonlinear dose-response relationship

W. K. Lutz, S. Vamvakas, A. Kopp-Schneider, J. Schlatter and H. Stopper, Deviation from additivity in mixture toxicity relevance of nonlinear dose-response relationships and cell line differences in genotoxicity assays with combinations of chemical mutagens and g-radiation. Environmental Health Perspectives Supplements, 2002,110(6), 915-918. 

For experimental studies of mixtmes, consideration is given to the possibility of changes in the physicochemical properties of the test substance during collection, storage, extraction, concentration and delivery. Chemical and toxicological interactions of the components of mixtmes may result in nonlinear dose-response relationships. 

Nonlinear dose-response relationship Any dose-response relationship that is not linear. Eor example, a dose-response relationship in which the probability of a specified response changes either faster (supralinear) or slower (sublinear) than linear as the dose increases (i.e. probability either not increasing, or not decreasing at all, or changing slower or faster than in direct proportion to the increase in dose) (Sielken, Ch. 8). 

If all processes are linear, then the concentration rate of the toxic substance at its site of action ( effective dose ) will be proportional to the external exposure rate ( administered dose ). However, saturation phenomena may produce different results depending upon the processes affected if elimination and/or detoxification pathways are saturable, then the effective dose will increase more rapidly with the administered dose than linear kinetics would suggest if the distribution and/or activation pathways are saturable, then the effective dose will increase less rapidly with the administered dose. These simplified pharmacokinetic models may provide more realistic explanations of observed nonlinear dose-response relationships than other dose-response models currently in use. 

Note that in actuality, dose-response relationships are often not linear and instead we must use either a transform (to linearlize the data) or a nonlinear regression method (Gallant, 1975). 

Dose-response assessment today is generally performed in two steps (1) assessment of observed data to derive a dose descriptor as a point of departure and (2) extrapolation to lower dose levels for the mmor type under consideration. The extrapolation is based on extension of a biologically based model (see Section 6.2.1) if supported by substantial data. Otherwise, default approaches that are consistent with current understanding of mode of action of the agent can be applied, including approaches that assume linearity or nonlinearity of the dose-response relationship, or both. The default approach is to extend a straight line to the human exposure doses. 

The order in which the summations over the responses (r) and substances (i) of concern are executed in the second and third steps above is arbitrary. However, these steps must be executed before the MAX and INTEGER functions are applied to the result. If the risk index for substances causing deterministic responses were based on calculations of health risk per se, rather than dose, the INTEGER function in Equation 6.5 would not be necessary, because the risk would be zero whenever a dose is below the threshold. Again, however, evaluation of the risk index for substances that cause deterministic responses based on dose is recommended when the dose-response relationship is assumed to have a threshold. The use of dose is supported by the observation that the dose-response relationship above the threshold generally is nonlinear. 

The boundaries between different waste classes would be quantified in terms of limits on concentrations of hazardous substances using a quantity called the risk index, which is defined in Equation 6.1. The risk index essentially is the ratio of a calculated risk that arises from waste disposal to an allowable risk (a negligible or acceptable risk) appropriate to the waste class (disposal system) of concern. The risk index is developed taking into account the two types of hazardous substances of concern substances that cause stochastic responses and have a linear, nonthreshold dose-response relationship, and substances that cause deterministic responses and have a threshold dose-response relationship. The risk index for any substance can be expressed directly in terms of risk, but it is more convenient to use dose instead, especially in the case of substances that cause determinstic responses for which risk is a nonlinear function of dose and the risk at any dose below a nominal threshold is presumed to be zero. The risk index for mixtures of substances that cause stochastic or deterministic responses are given in Equations 6.4 and 6.5, respectively, and the simple rule for combining the two to obtain a composite risk index for all hazardous substances in waste is given in Equation 6.6 or 6.7 and illustrated in Equation 6.8. The risk (dose) that arises from waste disposal in the numerator of the risk index is calculated based on assumed scenarios for exposure of hypothetical... 

The manner in which the proportion of animals developing a response changes as the dose level changes is the dose-response relationship. If the proportion decreases in parallel with decreasing dose (e.g., halving the dose halves the proportion), then the dose-response relationship is linear. However, if the proportion decreases faster than linearly (e.g., halving the dose results in either one-fourth the proportion or no occurrences of the adverse effect), then the dose-response relationship is sublinear (one type of nonlinearity). 

This type of correlative approach is widespread, as only a few marine studies involving inducible defenses (and none with mobile invertebrates) have directly demonstrated that the induction results in a decrease in the susceptibility of the organism to predation.71,72 Statistically significant differences in shell thickness or concentrations of defensive chemicals may or may not meaningfully affect predator preferences in ecologically relevant field situations. For chemical defenses, compound dose-response relationships may be nonlinear, and threshold levels of defense could be sufficient to deter predators so that further induction has little additional benefit. Thus, future studies should focus on directly demonstrating whether an induced response reduces predation on prey organisms. 

Lutz WK. 1998. Dose-response relationships in chemical carcinogenesis superposition of different mechanisms of action, resulting in linear-nonlinear curves, practical thresholds, J-shapes. Mutat. Res. 405 117-24... 

Figure 33-10 Dose-response curves. L/ne A illustrates the linear relationship between serum drug concentration and total daily dose of a drug that displays first-order kinetics typical of most drugs. Line B illustrates the dose-response relationship for a drug that displays capacity-limited kinetics because of a saturable enzyme or transport mechanism in this situation, serum concentration becomes independent of total daily dose, and the relationship of drug concentration to dose becomes nonlinear. (Modified from Pippenger CE. Practical pharmacokinetic appiications. Syvo Monitor, Son Jose Syva Co, January, i 979 1-4.)...

The potential for a nonhnear extrapolation from human data should also be considered. Some risk assessors have argued that benzene-induced leukemia and lymphoma is a threshold phenomenon, based on mechanistic considerations, and that the human data demonstrate such a threshold (Cox and Ricci 1992 Cox 1996 Yokley et al. 2006). However, regulatory risk assessors have not yet accepted these arguments for benzene (Bailer and Hoel 1989 ERA 1998 OEHHA 2001). Considerations include the multiple genotoxic metabolites of benzene, a nonlinear production of protein and DNA adducts in humans (saturated at higher benzene doses), and the difficulty of establishing dose-response relationships at lower doses with the available animal and human data (Bailer and Hoel 1989 Henderson et al. 1992 Turteltaub and Mani 2003 Rappaport et al. 2005 Lin et al. 2007). 

Lutz, W. K., Gaylor, D. W, Conolly, R. B., and Lutz, R. W. (2005). Nonlinearity and thresholds in dose-response relationships for carcinogenicity due to sampling variation, Ic arithmic dose scaling, or small differences in individual susceptibility. Toxicol Appl Pharmacol 207, S565-S569. 

Additionally, more recently, some scientists have also suggested that even when compounds exhibit nonlinearity on the cellular level, they are likely to manifest as a linear dose-response relationship on the population level. This hypothesis, as presented by White et al. (2009), holds that modifying factors (such as nutritional status) and biological variability cause the dose-response relationship to be linear across the population (White et al. 2009). 

Over the past decade there has been a movement to harmonize cancer and noncancer risk assessment (Gaylor 1997 Bogdanffy et al. 2001) based on the premise that cancer and noncancer events share similar pharmacokinetic dependencies and overlapping MOAs and thus have similar dose-response relationships. The benchmark dose approach lends itself to the evaluation of both linear and nonlinear dose-response. In fact, one of the stated purposes of EPA s formalization of the benchmark dose process was to provide a standardized approach to chemical dose-response assessment, regardless of whether the chemical is a carcinogen. 

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. 

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