ToxSci Advance Access originally published online on October 19, 2005
Toxicological Sciences 2006 90(1):33-38; doi:10.1093/toxsci/kfj026
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Dose-Incidence Relationships Derived from Superposition of Distributions of Individual Susceptibility on Mechanism-Based Dose Responses for Biological Effects
* Department of Toxicology, University of Würzburg, 97078 Würzburg, Germany; Seminar for Statistics, Swiss Federal Institute of Technology, 8092 Zürich, Switzerland; and CIIT Centers for Health Research, Research Triangle Park, North Carolina 27709
Received July 18, 2005; accepted October 10, 2005
Dose-response relationships for incidence are based on quantal response measures. A defined effect is either present or not present in an individual. The dose-incidence curve therefore reflects differences in individual susceptibility (the "tolerance distribution"). At low dose, only the more susceptible individuals manifest the effect, while higher doses are required for more resistant individuals to be recruited into the affected fraction of the group. Here, we analyze how such dose-incidence relationships are related to mechanism-based dose-response relationships for biological effects described on a continuous scale. As an example, we use the quantal effect "cell division" triggered by occupancy of growth factor receptors (R) by a hormone or mitogenic ligand (L). The biologically effective dose (BED) is receptor occupancy (RL). The dose-BED relationship is described by the hyperbolic Michaelis–Menten function, RL/Rtot = L / (L + KD). For the conversion of the dose-BED relationship to a dose–cell division relationship, the dose-BED curve has to be combined with a function that describes the distribution of susceptibilities among the cells to be triggered into mitosis. We assumed a symmetrical sigmoid curve for this function, approximated by a truncated normal distribution. Because of the supralinear dose-BED relationship due to the asymptotic saturation of the Michaelis–Menten function, the composite curve that describes cell division (incidence) as a function of dose becomes skewed to the right. Logarithmic transformation of the dose axis reverses this skewing and provides a nearly perfect fit to a normal distribution in the central 95% incidence range. This observation may explain why dose-incidence relationships can often be described by a cumulative normal curve using the logarithm of the administered dose. The dominant role of the tolerance distribution for dose-incidence relationships is also illustrated with the example of a linear dose-BED relationship, using adducts to protein or DNA as the BED. Superimposed by a sigmoid distribution of individual susceptibilities, a sigmoid dose-incidence curve results. Linearity is no longer observed. We conclude that differences in susceptibility should always be considered for toxicological risk assessment and extrapolation to low dose.
Key Words: dose-response relationship; models; logarithm; tolerance distribution.