U-Shaped Dose-Response Curves for Carcinogens

Christopher J. Portier, Ph.D. and Frank Ye, Ph.D.

Laboratory of Computational Biology and Risk Analysis,

National Institute of Environmental Health Sciences,

Research Triangle Park, NC USA 27709

Tel: (919) 541-4999

Fax: (919) 541-1479


Is there theoretical justification that supports the concept that there are some situations in which, for a fixed age and pattern of exposure, increasing doses of a chemical or physical agent will result in decreasing cancer risk at the lowest doses followed by increasing cancer risks at the higher doses (hormesis or U-shaped dose-response)? Yes. There is little doubt that numerous justifications for such patterns could be developed; the three examples developed for this exercise (inducible, but saturable, repair, cytotoxicity and lesion grouping with multiple dose-response patterns) have been discussed previously in the literature. Other examples include competition for receptor by endogenous versus exogenous ligands (Kohn and Portier, 1993) and alteration of differentiation pathways (Luster et al, 1993). Yet the question of existence may not be the most noteworthy from the point of view of risk assessment.

Public health professionals tend to be fairly conservative in their application of data to the assessment of risks from exposure to environmental agents. They tend to err in favor of protecting the public at the possible expense of small additional costs and loss of marketability of a product. It is unlikely that this attitude could be altered by theoretical arguments alone; stronger support would be needed. However, there are at least two very daunting statistical and scientific challenges beyond the development of a plausible mechanism in support of the existence of U-shaped dose-response relationships to overcome in clearly demonstrating a U-shaped dose-response relationship; one must demonstrate that the relationship is supported by data and that this is the predominant relationship for all of the cancer endpoints observed. If either of these two criteria are not shown, it is unlikely the U-shaped response will dominate the assessment. In addition, one must determine if a protective effect for one endpoint is increasing the potential for risk in some other toxic endpoint.


All of the data we observe in the assessment of risks from exposure to chemical agents is subject to statistical variation. A careful application of statistical methods can be used to assess the degree to which the data support the parameter estimates describing a U-shaped dose-response curve (versus, for example, a dose-response curve with non-decreasing response). In the three cases presented in this volume, none of the authors formally described the probability that the data could be explained by a monotonic curve. However, in the paper by Bogen (1998), it is clearly possible to derive such a test from the given estimates. In the paper by Anderson and Conolly (1998), no attempt was made to use formal statistical methods for estimation of the model parameters making it impossible to formulate any statistical tests within the context of the models proposed. This comment is not applicable to the paper by Downs and Frankowski (1998) since their approach was simply to do a sensitivity analysis of the model with regard to U-shaped dose-response behavior avoiding the complexities of applying data to the model.


It has been demonstrated by numerous authors that, given multiple dose-response curves for any one environmental agent, the model(s) yielding the highest risk at low exposures will generally include only the models with the slope of the dose-response curve positive when evaluated at the control dose. In general, models demonstrating U-shaped dose-response would have a negative slope when evaluated at the control dose. These U-shaped dose-response curves will generally yield the lowest estimates of risk for a given environmental agent. Hence, the difficult issue will be when to utilize a U-shaped dose-response curve for risk assessment in light of variability which encompasses more traditional models (such as the linear model) and 1 or 2 dose-response curves for other cancers which do not appear to follow a U-shaped pattern.

The conservative (for public health) approach is generally to choose the cancer response with the highest low-exposure risk as the basis for the risk assessment. However, in light of strong evidence for U-shaped response in most of the other observed cancers, the burden of proof may be on showing that the one or two remaining cancers cannot be fit by a U-shaped curve. In essence, we should be prepared to alter our typical null hypothesis (e.g. linearity in the low-dose range) to a different null hypothesis in light of convincing evidence of the predominance of another pattern.

One must also be careful that the observed U-shaped response for one cancer is not at the expense of some other cancer or some different toxicity endpoint. One of the best designed studies for evaluating multiple endpoints for dose-response relationships is in the study done by Luster et al (1993) on cyclophosphamide. Using a design with eight exposure levels, they measured 9 immunological endpoints (including tumor cell development) in all of their animals. There was a clear U-shaped dose-response curve for survival after exposure to Listeria monocytogenes with maximal survival given a dose of 50 mg/kg cyclophosphamide (validated by statistical analysis). In contrast, resistance to PYB6 tumor cell challenge was significantly reduced at this same dose. This is likely to be due to the greater sensitivity of T-suppresser cells to cyclophosphamide at low doses allowing greater resistance to infection at the expense of resistance to tumor cells.

There is a wide variety of data on cancer and non-cancer endpoints for dioxins. Clearly, as demonstrated by Portier (1998), these data have a variety of dose-response patterns with an average response appearing to be linear. We are not sufficiently familiar with radon to make the same assertion.


Three issues are key in the investigation of any dose-response relationship for cancer; the data, the underlying biological complexity and the mathematics used to study the issue. All three of these papers addressed some component of these three issues; no one paper addressed them all. In all cases, the models are developed under reasonable biology and theoretical mathematical models have been developed to address this biology. Downs and Frankowski did not go beyond this development and no further comment will be given.

The paper by Anderson and Conolly is creative and insightful, proposing a model in which there are two populations of altered hepatic foci which could be precursors to tumor onset. The first focal lesion type (type A) has decreasing net growth rate as the exposure to TCDD increases whereas the second lesion type (type B) has increasing growth rate as a function of exposure to TCDD. The U-shaped dose-response behavior seen with this model results in cases where the loss of type-A cells at low exposures exceeds the increase in growth of type-B cells in the same exposure region. Eventually the type-A cells die out altogether and the type-B cells dominate the dose-response relationship for the focal lesions. The resulting U-shaped response for the premalignant lesions would lead to a U-shaped response for tumors as well if both phenotypes can proceed to tumors. In the original work of Jirtle et al. (1991), it was not clear that the focal lesions whose growth is inhibited by presence of phenobarbital could ever proceed to tumors. If this is not the case for TCDD, the shape of the tumor dose-response relationship would be driven by the shape for only the TCDD sensitive focal lesions, not the combined set. The possibility of a U-shaped dose-response in this case is clear and avenues for exploring the possibility are obvious.

As mentioned earlier, it is impossible to judge the quality of the fits from this model to the available data. The authors discuss comparisons of their model with the competing model of Portier et al (1996); these modeling exercises are not easily compared . The analysis performed by Portier et al used objective statistical methods for parameter estimation and demonstrated that a model without an effect of TCDD on the rate of formation of initiated cells fit the data significantly worse than a model with this effect included. The authors also used the model to formally adjust for the differences in the ages of the control versus the treated animals (2 months) and corrected for stereology in the context of their model in a rigorous fashion previously documented. None of these steps were taken with the model described here making it impossible to answer the question of whether a model with one focal lesion type versus two would fit these data equally well. This is crucial to the argument given the lack of data on individual markers of different lesions.

The model developed by Bogen is more akin to a standard two-stage model but with replenishment of lost cells from an unexposed base population of cells; similar but not identical to multipathway models of carcinogenesis (Sherman and Portier, 1997). Bogen has used rigorous means to estimate the parameters in his model, but did not go on to test the hypothesis that the U-shape is statistically necessary for the explaination of the data. A cursory look at the standard errors presented in the paper would suggest that the data could be adequately fit by a model without the U-shape, but this fit would be slightly poorer than the fit shown by Bogen. In addition, the details given in the Bogen paper were insufficient for these authors to replicate the model making it difficult for us to comment on other issues of concern such as identifiability of parameter estimates common to these types of problems (Sherman and Portier, 1997).

In summary, these three papers have presented excellent theoretical arguments for the existence of U-shaped dose-response curves and limited data/analyses illustrating the need to use U-shaped curves for effects from selected agents. Not addressed is the key issue of when to know that a U-shaped dose-response curve is the dominant, and probably correct, pattern for low-dose behavior.


Andersen, M and Conolly, R. (1998) Mechanistic modeling of rodent liver tumor promotion at low levels of exposure: an example related to dose-response relationships for 2,3,7,8-tetrachlorodibenzo-p-dioxin. (BELLE Newsletter, current issue)

Bogen, K. (1998) Mechanistic model predicts a U-shaped relation of radon exposure to lung cancer risk reflected in combined occupational and U.S. residential data (BELLE Newsletter, current issue)

Downs, T and Frankowski, R. (1998) A cancer risk model with adaptive repair. (BELLE Newsletter, current issue)

Kohn, M. and Portier, C. (1993) Effects of the mechanism of receptor-mediated gene expression on the shape of the dose-response curve. Risk Analysis,13 (5), 565-572.

Luster, M., Portier, C., Pait, G., Rosenthal, G., Germolec, D., Comment, C., Corsini, E., Blaylock, B., Pollock, P., Kouchi, Y., Craig, W., White, K. and Munson, A. (1993) Risk assessment in immunotoxicology. II: Relationships between immune and host resistance tests. Fundamental and Applied Toxicology, 21, 71-82.

Portier, C. (1998) Risk ranges for various endpoints following exposure to 2,3,7,8-TCDD. In WHO-ECEH / IPCS Consultation On Assessment Of The Health Risk Of Dioxins; Re-Evaluation Of The Tolerable Daily Intake, World Health Organization, Geneva, Switzerland

Portier, C.J., Kohn, M.C., Kopp-Schneider, A., Sherman, C.D., Maronpot, R., and Lucier, G.W. (1996) Modeling the number and size of hepatic focal lesions following exposure to 2,3,7,8-TCDD. Toxicology and Applied Pharmacology 138, 20-30.

Sherman, C. and Portier, C. (1997) The two-stage model of carcinogenesis: overcoming the nonidentifiability dilemma. Risk Analysis 17: 367-374.

Sherman, C.D. and Portier, C.J. (1994) The multipath/multistage model of carcinogenesis. Informatik Biometrie und Epidemiologie in Medizin und Biologie 25(4), 250-254.