Non-linear dose response

Fig. 3. The way in which an envelope of additivity may be calculated for two pairs of cytotoxic agents (A + B and C + D) where one of each pair gives a non-linear dose-response curve (isobologram analysis from Peckham and Steele ref. 27).

It is assumed that for health effects other than cancer, a threshold or non-linear dose-response relationship exists. This is based on known compensatory and adaptive mechanisms that protect against the toxic effects of childhood exposures, as well as on repair mechanisms at the molecular, cellular, and tissue... 

Figure 2 Linear and Non-Linear Dose-Response Curves...

Figure 11.11 Non-linear dose-responses arising when the assay is carried out brfore all the tubes have reached the arithmetic growth stage...

The similarities between ILs and many cationic surfactants, together with the effect of increasing alkyl chain on the cation, seem to point to the fact that the toxic effect is exerted through a disruption of the phospholipid bilayer. Exposure of Physa snail to sub-lethal doses of ILs causes a reduction in movements as well as grazing rates of the organisms, with a typical non-linear dose-response relationship. The attempt of the snail to escape the toxicant, searching for clean water, may well account for the U-shaped relationships found by the authors [21]. 

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. 

H) side chains are replaced by the pseudoisosteric Cys" —Cys bridging side chain (-CH2-S-S-CH2-) group. [Cys", Cys ]-a-MSH was found to have superpotency in the frog skin assay system, being about 10-100 times more potent than the native hormone in this system over the linear portion of the dose-response curve. In addition, the peptide som etimes exhibits activity at concentrations as low as lO" " M (about 1000 times the minimum effective dose potency of a-MSH), but linear dose-response curves cannot be obtained. The reasons for this are unclear, but may reflect variability with season and from animal to animal, and perhaps variability in compartmentalization, receptor sensitivity, and/or non-specific adsorption properties of the cyclic peptides. 

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... 

The toxic effects model uses concentration-time profiles from the respiratory and skin protection models as input to estimate casualty probabilities. Two approaches are available a simple linear dose-effect model as incorporated in RAP and a more elaborate non-linear response model, based on the Toxic Load approach. The latter provides a better description of toxic effects for agents that show significant deviations of simple Haber s law behaviour (i.e. toxic responses only depend on the concentration-time product and not on each quantity separately). 

Lipophilicity in particular, as reflected in partition coefficients between aqueous and non-aqueous media most commonly water (or aqueous buffer) and Z-octanol,has received much attention [105,141,152,153,176,199,232,233]. Logic )W for the octanol-water system has been shown to be approximately additive and constitutive, and hence, schemes for its a priori calculation from molecular structure have been devised using either substituent tt values or substructural fragment constants [289, 299]. The approximate nature of any partition coefficient has been frequently emphasized and, indeed, some of the structural features that cause unreliability have been identified and accommodated. Other complications such as steric effects, conformational effects, and substitution at the active positions of hetero-aromatic rings have been observed but cannot as yet be accounted for completely and systematically. Theoretical statistical and topological methods to approach some of these problems have been reported [116-119,175,289,300]. The observations of linear relationships among partition coefficients between water and various organic solvents have been extended and qualified to include other dose-response relationships [120-122,160,161,299-302]. 

For non-threshold effects (e.g., sensitization, mutagenicity, genotoxicity, genotoxic carcinogenicity), the dose-response curve, in the absence of mechanistic evidence to the contrary, is assumed to be linear. 

For non-threshold mechanisms of genotoxic carcinogenicity, the dose-response relationship is considered to be linear. The observed dose-response curve in some cases represents a single ratedetermining step however, in many cases it may be more complex and represent a superposition of a number of dose-response curves for the various steps involved in the tumor formation (EC 2003). Because of the small number of doses tested experimentally, i.e., usually only two or three, almost all data sets fit equally well various mathematical functions, and it is generally not possible to determine valid dose-response curves on the basis of mathematical modeling. This issue is addressed in further detail in Chapter 6. 

Another approach is to develop a global model that contains plausible models as special cases, defined by alternative values of particular parameters. This converts model uncertainty into uncertainty about the model parameters. Again this can be done using either Bayesian or non-Bayesian approaches. This approach is favored by Morgan and Henrion (1990), who describe how it can be applied to uncertainty about dose-response functions (threshold versus nonthreshold, linear versus exponential). 

At a fixed dose-rate (45 5 pphm), the time of exposure was varied from zero to 90 min, and treated plants incubated a further 24 hr before chlorophyll assay. This test also revealed non-linear responses (Table I) for each organ on the bases of chlorophyll and fresh weight contents using seedlings of sensitive ages. 

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