Characterizing non-constant relative potency

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Abstract

Relative potency plays an important role in toxicology. Estimates of relative potency are used to rank chemicals by their effects, to calculate equivalent doses of test chemicals compared to a standard, and to weight contributions of constituent chemicals when evaluating mixtures. Typically relative potency is characterized by a constant dilution factor, even when non-similar dose–response curves indicate that constancy is inappropriate. Improperly regarding relative potency as constant may distort conclusions and potentially mislead investigators or policymakers. We consider a more general approach that allows relative potency to vary as a function of dose, response, or response quantile. Distinct functions can be defined, each generalizing different but equivalent descriptions of constant relative potency. When two chemicals have identical response limits, these functions all carry fundamentally equivalent information; otherwise, relative potency as a function of response quantile is distinct and embodies a modified definition of relative potency. Which definition is preferable depends on whether one views any differences in response limits as intrinsic to the chemicals or as extrinsic, arising from idiosyncrasies of data sources. We illustrate these ideas with constructed examples and real data. Relative potency functions offer a unified and principled description of relative potency for non-similar dose–response curves.

Highlights

► We express relative potency as a function of dose, response, or response quantile. ► These relative potency functions are constructed from dose–response curves for test and reference chemicals. ► All relative potency functions provide the same information if the chemicals have equal response limits. ► The function of response quantile provides distinct information if response limits differ. ► Choice among relative potency functions depends on the application and how differences in response limits arise.

Introduction

Toxicologists use estimates of relative potency for ranking chemicals (e.g., Glass et al., 1991) and for dose–conversion analyses that calculate the equivalent dose of one chemical that produces the same response as a specific dose of another (e.g., Putzrath, 1997). They also combine separate estimates of relative potency, from different studies and endpoints, to determine a single toxic equivalency factor (TEF) (e.g., Van den Berg et al., 2006). The National Toxicology Program may soon use relative potency estimates in prioritizing chemicals via high throughput screening.

Relative potency of a test compound compared to a reference compound is typically thought of as a ratio of doses (reference divided by test) that produce the same mean response for a given endpoint. The notion that relative potency should be constant has roots in analytical dilution assays where all preparations are regarded as dilutions of a standard preparation with an inert diluent (Finney, 1965). In that context, the ratio of doses producing the same mean response should be identical at every response level. Consequently, when dose–response curves for such preparations are plotted on a log–dose axis, the curves should be identical up to a horizontal shift (Fig. 1). Even when preparations are not simple dilutions of a single active compound, there may be biological reasons (e.g., a common mechanism of toxicity) to expect them to behave as though they were and, thus, to have constant relative potencies. Chemicals with constant relative potency are said to have ‘similar’ dose–response curves (i.e., equal up to a horizontal shift on the log–dose axis). Their relative potency is characterized by a single dilution (or concentration) factor, a feature that simplifies ranking chemicals and calculating equivalent doses.

Researchers have long recognized that constant relative potency is not always reasonable because comparative assays often yield dose–response curves that differ in ways other than a horizontal shift (Cornfield, 1964, Cox and Leaverton, 1966, Rodbard, 1974, De Lean et al., 1978, Guardabasso et al., 1987, Guardabasso et al., 1988). When dose–response curves are not similar, regarding them as similar and estimating relative potency accordingly may distort conclusions.

For example, consider the activity of a liver enzyme in response to TCDD, PCB126, PeCDF and their TEF-based mixture (NTP, 2006a, NTP, 2006b, NTP, 2006c, NTP, 2006d). For illustration purposes, we regard the mixture as a fourth chemical. When a separate dose–response model is fit to each chemical and the ratio of median effective doses (ED50s) is used to assess relative potency despite non-similarity (Fig. 2a), the estimated potencies relative to TCDD suggest ranking the chemicals (from most to least potent) as: TCDD > mixture > PCB126 > PeCDF. When the dose–response models are constrained to enforce similarity (Fig. 2b), the constant relative potencies, determined as ratios of revised ED50s, suggest the ranking: mixture > TCDD > PeCDF > PCB126. These rankings differ, in part, because the chemicals vary substantially in their apparent upper limits of response. Both rankings indicate that TCDD and the mixture are more potent than PeCDF and PCB126, but they give opposite results for TCDD versus the mixture and for PeCDF versus PCB126. In both the constrained and unconstrained analyses, the mixture’s fitted curve lies above TCDD’s at all doses (Fig. 2), indicating that the mixture is the more potent and, in turn, demonstrating that the use of ED50s for estimating relative potency is problematic when dose–response curves are not similar. The unconstrained dose–response curves for TCDD and PeCDF cross twice, suggesting that their relative potency changes across response levels and doses; consequently, using any single value of relative potency for ranking or dose conversion would be an oversimplification.

Despite recognition of these issues, approaches to analysis that explicitly allow non-constant relative potencies are rare. Cornfield (1964) relaxed the similarity constraint, but he assumed that response was a linear function of log–dose (see also, Cox and Leaverton, 1966); typically linearity is reasonable for only a portion of the dose–response curve. Under a more general dose–response model, DeVito et al. (2000) assumed relative potency was constant for doses below some point and otherwise was linear in the reciprocal of dose. Several researchers suggested accommodating non-similar dose–response curves by simply reporting estimates of relative potency at a few selected doses (Putzrath, 1997, Villeneuve et al., 2000). Recently, Ritz et al. (2006) formulated relative potency as a function of response. These proposals notwithstanding, toxicologists often estimate relative potency by the ratio of ED50s despite non-similarity of the dose–response curves (Rodbard and Frazier, 1975, Villeneuve et al., 2000).

This article builds on previous proposals for quantifying relative potency for non-similar dose–response curves. We define functions that describe relative potency as depending on dose, response, or response quantile. In general, the ratio of doses used to define relative potency depends on where along the dose–response curves the ratio is taken; these functions reflect that dependence. We develop our proposals conceptually and graphically, describing their utility and limitations, and we defer technical details to an appendix. We postpone a full discussion of the statistical analysis of relative potency functions as beyond our present scope. We illustrate our proposals with the NTP enzyme activity data introduced earlier.

Section snippets

Modeling dose–response

A dose–response model expresses the mean (or average) response as a mathematical function of dose. For a given endpoint, we use the notation f(d; θ) to represent the mean response elicited by dose d of the chemical of interest, where f is a monotone function of d that depends on a vector θ of unknown parameters. We consider dose–response curves where the mean response increases from a lower limit, L, at a dose value of zero (d = 0) to an upper limit, U, at an infinite dose (d = ∞). If the opposite

Concepts

When similar dose–response curves are plotted with dose on a logarithmic scale, the magnitude of the logarithm of ρ equals the horizontal distance between the curves at any mean response level μ; and the sign of the logarithm depends on the direction from the test chemical’s curve to the reference chemical’s curve (left being negative and corresponding to relative potencies less than 1) (Fig. 1). (Relative potency is not defined for values of μ below L or above U.) For non-similar dose–response

Application to enzyme activity data

The US National Toxicology Program (NTP) recently evaluated the relative potency of dioxin-like compounds with respect to toxicity and carcinogenicity endpoints (NTP, 2006a, NTP, 2006b, NTP, 2006c, NTP, 2006d). They studied 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD), 3,3′,4,4′,5-pentachlorobiphenyl (PCB126), 2,3,4,7,8-pentachlorodibenzofuran (PeCDF), and a TEF-based mixture of the three. We used a subset of these data (Appendix B) to illustrate several concepts introduced earlier. Specifically,

Discussion

The relative potency of chemicals is a fundamental concern of toxicology. The usual approach is to report a single constant value for relative potency. This approach is sound only when the chemicals have similar dose–response curves. Investigators have, however, long recognized that the ideal of similarity is not always realized in comparative assays. Some analysts simply enforce similarity in fitting curves so that relative potency is constant and estimated as the ED50 ratio for the similar

Funding and conflict of interest statement

This research was supported by the Intramural Research Program of the NIH, National Institute of Environmental Health Sciences (Z01-ES-102685). The NIH had no involvement in the study design; collection, analysis and interpretation of data; writing of the manuscript; or decision to submit the manuscript for publication. The authors declare that there are no conflicts of interest.

Acknowledgements

We are grateful to M. DeVito, J. Haseman, S. Peddada, and N. Walker for their constructive comments and discussions. We thank S. Harris for programming the Hill Viewer application.

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