Key Issues in Dose Response: Leadership Forum-Editorial Dose-Response: An International Journal Cancer Risk Assessment and the Biostatistical October-December 2018:1-8 ª The Author(s) 2018 Article reuse guidelines: Revolution of the 1970s—A Reflection sagepub.com/journals-permissions DOI: 10.1177/1559325818806402 journals.sagepub.com/home/dos Kenny Crump1 Abstract Before around 1960, assessment of risk from exposure to toxic substances, including risk of cancer, was generally implemented using the NOAEL-safety factor approach that essentially assumed that an exposure threshold existed and exposures below the threshold carried no risk. In the 1970s there came a realization that cancer could develop from a mutation in a single cell and consequently it was unlikely that a threshold existed for substances that could cause such mutations, and that risk could increase linearly with exposure. During this time the Environmental Protection Agency (EPA) was formed and charged with protecting the public from a perceived high risk of environmental cancer. Faced with this difficult task, EPA decided to assess cancer risk by fitting a statistical model to dose-response cancer data and extrapolating to low dose using the fitted model. After some early experimentation EPA selected the Linearized Multistage Model for this fitting, which predicted risk increased linearly with exposure at low exposures. This approach led to an increased emphasis on statistical issues in risk assessment. Today, cancer risk assessment guidelines allow for different approaches depending upon the understanding of a substance’s mode of action. However, a review of EPA’s experience with current guidelines indicates that most cancer risk assessments still follow proce- dures similar to those initiated more than 40 years ago. Keywords dose–response, LNT, risk assessment, single-hit response, threshold Introduction When based on animal data, as they mostly were, a no- observed-adverse-effect-level (NOAEL) was determined and I was invited to write this reflection because of my involvement in divided by factors intended to account of the differences the development of cancer risk assessment methodology begin- between animals and humans (the “NOAEL-safety factor” ning in the 1970s. Thus, it is a highly personal account. It focusses approach). The NOAEL was typically set at the highest experi- on the Environmental Protection Agency’s (EPA) risk assessment mental dose for which the toxic response was not statistically practices as this agency was most prominent in developing and significantly greater than the response in control animals and applying new approaches for cancer risk assessment during this was effectively assumed to be a dose that was risk-free in the time. This account also emphasizes the low-dose extrapolation animal species. issue since it is probably the most important and controversial There are obvious shortcomings to this approach. It tacitly one, and is the issue I was most involved in. assumes that every toxic response has a threshold dose—a I graduated with my PhD in mathematics in 1968, and for a dose below which exposure has no effect on risk, even though few years continued my research involving the theoretical math- demonstrating the existence of a threshold (which would ematical model that had been the subject of my dissertation. But entail proving a negative) is beyond the ability of science. I began to realize that there were perhaps only 20 people in the world who were interested in this topic, and I resolved to move toward a topic that had promise of making a greater societal 1 Ruston, LA, USA impact. Cancer risk assessment fulfilled this need very nicely. Received 03 July 2018; received revised 10 September 2018; accepted 18 September 2018 The Early Days Corresponding Author: Prior to around 1960, risk assessment for all adverse health Kenny Crump, Ruston, LA 71270, USA. effects, including cancer, were carried out in a similar manner. Email: kennycrump@email.com Creative Commons Non Commercial CC BY-NC: This article is distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 License (http://www.creativecommons.org/licenses/by-nc/4.0/) which permits non-commercial use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 2 Dose-Response: An International Journal In fact, by the 1960s scientists were beginning to doubt that such thresholds existed for certain effects, particularly can- A 0.9 cers that could result from a mutation in a single cell. Also, a 0.8 typical animal carcinogenicity experiment involved exposing no more than around 50 animals at each experimental dose. 0.7 Mantel-Bryan Fracon with Tumor 99% Upper Bound Such small numbers would not provide sufficient power to 0.6 Mantel-Bryan confidently detect a small increase in risk that would be of MLE Fit 0.5 concern in a human population. For example, in an experi- ment involving 50 animals per dose group and 10 animals 0.4 were found with a particular type of cancer in both the control 0.3 group and a dosed group, it can only be concluded with 95% 0.2 probability that the risk in the dosed group was no more than 0.08 greater than the risk in the control group (ie, a 95% upper 0.1 confidence bound for the additional risk in the dosed group 0 compared to the risk in the control group is 0.08.), and this 0 20 40 60 80 100 120 large an increase in risk would be of serious concern in a Dose human population. B 0.06 The Mantel-Bryan (1961) Procedure 0.058 Fracon with Tumor 1,2 As a way around this conundrum, Mantel and coauthors proposed a method for calculating “safe doses” for carcino- 0.056 gens using a high- to low-dose extrapolation procedure that did not involve the assumption of a threshold. Although the 0.054 Mantel-Bryan method apparently was never used by a regu- latory agency to assess risk, it did break new ground by 0.052 assuming a nonzero increase in risk at every dose, no matter how small the dose. There were several conservative assumptions built into the 0.05 0 0.1 0.2 0.3 0.4 0.5 Mantel-Bryan procedure. It employed a log-probit dose– Dose response with no threshold. The probit slope was not estimated but was “conservatively set” at 1 normal deviate per 10-fold Figure 1. A, Illustration of the Mantel-Bryan (1961) dose–response dose increase. A safe dose was defined as one corresponding to extrapolation model fit to data shown (dose, #tumors/#animals): (0, 5/ the tiny increased risk of 1/100 million. And a 99% statistical 50), (25, 4/50), (50, 7/50), (100, 35/50), with 95% confidence intervals lower bound on the safe dose was recommended. However, shown for the response at each dose. Solid curve is the Mantel-Bryan despite these conservative assumptions the Mantel-Bryan maximum likelihood curve. Dotted curve is the Mantel-Bryan upper method cannot be considered conservative.3 It is instructive confidence limit curve. B, Same as A except showing only the upper to examine the method to see why that is the case. confidence limit curve at very low doses. Figure 1A shows a graph of the Mantel-Bryan log-probit dose–response applied to the following data (dose, #tumors/ #animals): (0, 5/50), (25, 4/50), (50, 7/50), (100, 35/50). The Fracon with tumor Mantel-Bryan curve does appear to be conservative in this graph, as its predictions lie above the cancer responses at the lowest 2 doses. It also appears to be curving downward every- where. However, if we use a magnifying glass to examine the Mantel-Bryan curve in the neighborhood of zero dose (Figure 2B), we see that the curve curves upward in the low- dose region (actually for doses, d, such that the additional risk (P(d)  P(0)) is less than 0.01),4 the opposite of what appears to be the case in the unmagnified graph. The first derivative is dose zero at zero dose, so the model is, by definition, not low-dose linear (note 1), which also is not apparent in the unmagnified Figure 2. Illustration of Carcinogen Assessment Group’s (CAG) orig- graph. Not only is the first derivative zero at zero dose, but inal linear extrapolation approach using same data as in Figure 1. A derivatives of all orders are zero at zero dose (note 2). Thus, it straight line was drawn from the response at zero dose to the could be called “super flat” at low dose. The principle problem response at the lowest dose that was significantly greater than the with using the Mantel-Bryan curve to estimate low-dose risk is response at zero dose. Crump 3 that there is no known biological mechanism that would pro- in special cases for the error in approximating the dose– duce a dose–response with this property. response by a completely linear model. Perhaps more importantly, this article introduced the “additive to background” rationale for linearity. This rationale The 1960s and 1970s noted the importance of background carcinogenesis to the There was a lot of environmental activity in the 60s and 70s. shape of the dose–response curve at low doses, and showed Rachel Carson’s book, Silent Spring, which was published in that, “if carcinogenesis by an external agent acts additively 1962, was widely disseminated, and alerted the public to the pos- with any already ongoing process, then under almost any model sible detrimental effects of the indiscriminate use of pesticides. the response will be linear at low dose.” It is important to note Also, during this time, it was estimated that as much as 90% of that this rationale applies, not just to cancer, but to any health human cancers were caused by environmental agents.5 This heigh- effect for which the stated conditions are satisfied. tened public and scientific concern presaged the formation of the At my very last day at NIEHS, upon the completion of my National Institute of Environmental Health Sciences (NIEHS) in academic-year-long appointment, I had the very good fortune 1969 and the EPA in 1970. President Nixon’s War on Cancer, of meeting my replacement, Harry Guess. That day Harry which was initiated in 1971, followed closely. grilled me about interesting projects I had found at NIEHS that And then in 1977, the Safe Drinking Water Committee of the he might work on. Among those we discussed were statistical National Research Council6 published a highly influential report, problems related to assessment of low-dose risk from carcino- entitled Drinking Water and Cancer, which reflected an emer- gens. This meeting was the beginning of a collaboration ging understanding of how many environmental cancers were between Harry and myself, with Harry usually taking the lead, initiated. This report concluded that for carcinogens “There is no that resulted in several publications on this topic,13-17 and pro- scientific basis for . . . time-honored practice of classical toxicol- duced what became known as the “Linearized Multistage Mod- ogy is the establishment of maximal tolerated (no-effect) doses el” (LMS) for cancer risk assessment. in humans based on finding a no-observed-adverse-effect dose in chronic experiments in animals, and to divide this dose by a ‘safety factor’ of, say, 100, to designate a ‘safe’ dose in humans.” Environmental Protection Agency’s Early Thus, the then-current practice of classical toxicology was Approaches to Quantitative Risk Assessment deemed to be inappropriate for carcinogenic risk. The report also proposed a specific alternative for assessing carcinogenic risk: for Carcinogens For “genetically self-propagating effects, for example, somatic And about this same time the relatively new agency, EPA, was or germ-cell mutation that culminates in a malignant neoplasm grappling with the problem of how to protect the public from a or is transmitted to later generations: Assume no threshold, perceived high risk of environmental cancer. The Agency had assume a linear dose–response at low doses, and estimate risk.” to determine which chemicals posed a cancer risk and how The NRC Committee illustrated this approach to estimating low- stringently to regulate them. This called for so some way to dose risk by applying the multistage model of cancer to animal estimate the cancer risk from specific exposures. This was a data on several chemicals.6,7 Regarding thresholds, the NAS very daunting job, and it still is. It was also a very controversial report concluded that “Methods do not now exist to establish a job, as an aroused agricultural–chemical industry saw the pos- threshold for long-term effects of toxic agents.” Note that this sibility that a zero cancer-risk policy could put them out of conclusion applied to all long-term toxic effects, not just cancer. business.18 The Carcinogen Assessment Group (CAG) was During 1974 to 1975, I spent an academic year as a Visiting formed within EPA and given primary responsibility for deal- Scientist in the Biometry Branch of NIEHS at the invitation of ing with this issue. Roy Albert was the first chairman of CAG. David Hoel, the branch chief. Soon after arriving there, I dis- The CAG decided early on that they could not depend wholly covered that interest was growing in developing new methods on human data to quantify cancer risk. To do so would be equiv- for conducting cancer risk assessments that took account of the alent to using humans as test animals. The only feasible way CAG emerging ideas about the mutation origin of many chemical- could envision to quantify low-dose cancer risk was to fit a math- induced cancers. I concluded that this was an important prob- ematical model to (usually animal) cancer data and extrapolate to lem, and one to which I could profitably devote my attention. low dose using the fitted model. But which model should be used? During that year, David Hoel, Chuck Langley, and myself at The Atomic Energy Commission had previously estimated the NIEHS and with the participation of Richard Peto in England, cancer risk from exposure to Strontium 90 and Iodine 139 using a wrote a paper that examined several mathematical models of linear no-threshold model. Albert argued that EPA would simply carcinogenesis with emphasis on their low-dose risk implica- be following the precedent set by another government agency in tions.8 These models generalized the Armitage and Doll9-10 and selecting a low-dose linear approach.18 Undoubtedly, the emer- Nordling11 cancer models by introducing the effect of dose ging scientific consensus reflected in the soon-to-be-released6 rate into the models following Neyman and Scott.12 It was water report also played a role. So, linear it was! shown that the dose–response for these models would be linear But which linear model should be used? According to the in dose for small dose rates, except in highly specific cases study by Albert,18 CAG’s first risk assessments used a very unlikely to occur in practice. The article also derived bounds simple linear extrapolation approach. They drew a straight line 4 Dose-Response: An International Journal Fracon with tumor Fracon with tumor dose dose Figure 3. Fit of the one-hit model to same data as in Figure 1. Figure 4. Solid curve is the fit of the multistage model to same data as in Figure 1. Dotted curve is the linearized multistage model (LMS) that determines the 95% upper confidence limit on added risk at low dose. from the lowest dose at which the response was significantly greater than that in controls to the response in controls and used this line to estimate low-dose risk (Figure 2). This was certainly 1E+00 a straight-forward method. No computer was needed, only a 1E-01 Mantel-Bryan Upper Bound straight edge and a pencil. But it also had some shortcomings. 1E-02 Linearized Mulstage For one thing, no statistical confidence bounds on risk were 1E-03 Muistage Extra Risk obtained. Also, the method ignored a lot of data. In Figure 2, it 1E-04 gives the impression of overestimating low-dose risk. 1E-05 Carcinogen Assessment Group also tried estimating low- 1E-06 dose risk by fitting the one-hit model to the data. This had the 1E-07 advantage over the strictly linear approach of using all the data, but it often did not fit the data adequately, as illustrated in 1E-08 Figure 3, where it overestimates risk at lower doses. 1E-09 0.0001 0.001 0.01 0.1 1 10 To resolve this problem, CAG organized an informal meet- Dose ing on February 21, 1980 and invited several statisticians, including myself, from around the country, each of whom had a statistical method for low-dose extrapolation. At this meeting Figure 5. Graphs of the multistage, the linearized multistage, and the upper bound Mantel-Bryan model at low dose, all based on data the statisticians were asked to explain their extrapolation shown in Figure 4. method and compare it with those of other participants. As a result of this informal meeting, CAG adopted the Both the multistage model and the Armitage-Doll model have multistage model, which together with the statistical proce- an exponential polynomial form with nonnegative parameters. dure for calculating statistical upper bounds on low-dose risk, However, the multistage model does not contain all the restric- was termed the Linearized Multistage Model, or LMS, as its tions on polynomials that the Armitage-Doll model contains. extrapolation model. The LMS was used by EPA for several The nonnegative restrictions on the parameters make the sta- years to assess risk from chemical carcinogens. tistics associated with the multistage model very unlike standard The multistage dose–response model is defined as regression. For example, it is not necessary to restrict the number 2 k PðdÞ ¼ 1  eðq0 þq1 dþq2 d þþqk d Þ ; of parameters to be less than the number of data points. The statistical theory behind the statistics associated with the LMS where d represents dose of a carcinogen and P(d) the probabil- was worked out in a series of papers in the late 70s.13,14,16 Later it ity of cancer when exposed to dose d, and the q’s are estimated was discovered that confidence intervals derived from the asymp- parameters, all 0. totic w2 distribution of the log-likelihood19,20 had improved prop- The multistage model is a generalization of both the one-hit erties over those based on the asymptotic normal distribution of model, the maximum likelihood estimates that were used in the earlier papers, and these confidence limits were used by EPA.21 PðdÞ ¼ 1  eðq0 þq1 dÞ ; Figure 4 illustrates how the LMS model works. The solid curve is the multistage model, which, unlike the one-hit model and, the9,10 multistage model of cancer, modified to incorpo- (Figure 3), clearly provides a reasonable fit to these data. The rate dose,12 dotted line in Figure 4 is the LMS, which also provides a k PðdÞ ¼ 1  eðpi¼1 ai þbi dÞ ; reasonable fit. The linearized version incorporates a statistical upper upper bound on low-dose risk. It is calculated essentially where the a’s and b’s are all 0. by selecting the largest linear term, q1, consistent with the data, Crump 5 while adjusting the remaining parameters to achieve the best Fracon with tumor fit. Whereas the multistage model itself may not be linear at extrapolaon Apply factors to low dose, the linearized form is always linear. In fact, it con- if the MOA is POD if MOA is tains the largest low-dose slope possible without significantly deemed linear deemed non-linear worsening the fit of the model. Thus, the LMS is not a model per se, but embodies a statistical confidence limit calculated from the multistage model. Figure 5 shows plots of the multistage, LMS, and upper bound Mantel-Bryan models for low doses for which risk is below 0.01, all based on the data shown in Figures 1–4. The dose NOAEL BMDL or POD multistage model approaches zero as dose cubed (*q3  dose3 because q1 and q2 are both estimated as zero with these data). The LMS approaches zero linearly (*q1  dose) because it Figure 6. Illustration of low-dose cancer risk assessment under Envi- ronmental Protection Agency’s (EPA) 2005 guidelines using the same contains the upper bound on q1, which must be positive. data as in Figure 1. Nevertheless, despite the divergence of these models at low dose, both fit the data adequately, as shown in Figure 4. The dose corresponding to a prescribed increase in risk, is not sub- Mantel-Bryan upper bound lies above the LMS for doses cor- ject to this interpretation. responding to risk larger than around 105 but drops off much Physiologically based pharmacokinetic models of the more rapidly than the LMS at lower doses. In fact, at still lower distribution, metabolism, and elimination of toxic chemicals doses the Mantel-Bryan eventually fall below the multistage in animals and humans were recommended to be used in extra- due to its extreme flatness at low doses. polating risks from animals to humans and across exposure To obtain an estimate of low-dose human risk from an ani- patterns.22,23 mal study, it is necessary, in addition to performing low-dose extrapolation, to convert animal doses to equivalent human doses. By equivalent human doses are meant dose measures Environmental Protection Agency’s 2005 that are estimated to produce the same cancer risk in humans as were estimated in the experimental animals. The CAG accom- Cancer Guidelines plished this by assuming that mg/surface area per day, approxi- In 2005 EPA produced cancer guidelines25 that made some mated by mg/[body weight]2/3/day is an equivalent exposure in fundamental changes in how cancer risk assessments are con- animals and humans.21 ducted by the agency. According to these guidelines, the pre- ferred approach for cancer risk assessment is to use a toxicodynamic model, also known as a biologically based Later Modifications dose–response (BBDR) model, of the agent’s mode of action In the 1990s, EPA produced guidelines for risk assessment of (MOA) and use that model for extrapolation to lower doses, if a several health effects. Among these were guidelines for repro- suitable BBDR model is available. Biologically based dose– ductive toxicity22 and neurotoxicity.23 Both of these documents response models provide estimates of the probability of an allowed replacing the NOAEL in the NOAEL-safety factor adverse response, expressed as a function of biological vari- approach with the benchmark lower bound (BMDL).24 The ables involved in the response, such as cell division rates, death benchmark is defined as the dose corresponding to a prescribed rates, and so on, that have physiologic meaning and, at least in increase (eg, 0.1) in the risk of an adverse outcome and is theory, could be measured. computed using a fit of a dose–response model to the dose– Lacking a suitable BBDR model, a statistical dose–response response data. The BMDL has several advantages over the model (such as the LMS) is fit to the data and used to determine NOAEL in safety assessment: Unlike the BMDL, the NOAEL a “point of departure” (POD). The POD dose typically is a depends only on the dose that is the NOAEL and does not BMDL corresponding to a predetermined excess risk, for incorporate information on the slope of the dose–response example, 0.1. Below the POD dose, risk is extrapolated to low curve or the variability in the data. Smaller experiments (those dose using either linear extrapolation or “nonlinear extra- with fewer animals per dose) will tend to give larger NOAELs, polation.” Linear extrapolation involves extrapolating line- while the opposite is true of the BMDL. Thus, the BMDL more arly to low doses less than the POD dose and is used when appropriately accounts for the greater evidence of safety result- there are data to indicate the MOA involves a dose–response ing from more data. The NOAEL is limited to one of the having a linear component below the POD dose or as the experimental doses and consequently the number and spacing default when the MOA is not established. Nonlinear extrapo- of doses can have an untoward effect upon the NOAEL. The lation is used when there is sufficient data to ascertain the NOAEL is often interpreted as a risk-free dose and, as noted MOA and to conclude that the dose–response is not linear at earlier, this interpretation does not properly account for the low doses. Nonlinear extrapolation involves dividing the POD power of the data to detect a response, and therefore is not by safety or adjustment factors to obtain a reference dose defensible. The BMDL, however, being a lower bound on a (RfD) or reference concentration (RfC), and thus does not 6 Dose-Response: An International Journal involve extrapolating a dose–response to low doses and does nonlinear MOA appear to have focused on showing that the not provide an estimate of low-dose risk. Thus, the guidelines chemical is nonmutagenic. However, other MOAs could also mandate a bifurcated approach, in which the extrapolation lead to linearity at low dose. In particular, Crump et al8 pro- approach depends upon whether MOA information indicates posed the “additive to background” rationale for low-dose lin- a linear or nonlinear MOA. earity. If a carcinogen acts by adding to a mechanism that is Figure 6 illustrates the how this bifurcated approach is already producing background cancers, then the response will implemented. First, a BMDL is estimated using a statistical be linear at low dose. This rationale for low-dose linearity dose–response model (often apparently the LMS which was applies, not only to cancer, but also to any toxic effect that is used for this step in Figure 6). This BMDL is a lower bound adding to a mechanism that is causing the effect in background. on the dose corresponding to a prescribed increase in risk (0.1 For example, see Crump27 (Appendix B) for a description of was used in Figure 6). This point (BMDL, 0.1) is the POD. If how this mechanism could produce a linear dose–response in a the MOA is determined to be linear or there is not sufficient population in which individuals have a threshold response information to determine the MOA, low-dose risk is estimated mediated by inactivation of acetylcholinesterase molecules using the straight line shown in the figure. If the MOA is by covalent binding to an organophosphorus pesticide (In the considered nonlinear, low-dose risk is not estimated. Instead last equation in this Appendix, the second P should be P0 .). the POD dose is divided by factors to arrive at an RfD or RfC. Seemingly, very little study has been directed toward this potential mechanism for low-dose linearity. As noted by a Thoughts About this History committee of the National Research Council, “EPA practices do not call for systematic evaluation of endogenous and exo- It is interesting to see how the EPA 2005 cancer guidelines genous exposures or mechanisms that can lead to linearity.28 have been implemented in the 13 years they have been opera- In the 2005, EPA cancer guidelines, MOA information is tional as of this writing. There has been deemed sufficient used only to inform which of 2 risk assessment tracks to take, information to support a nonlinear MOA of action for only 2 linear or nonlinear extrapolation. There are several conceptual chemicals, carbon tetrachloride and chloroform, and for carbon problems with this approach. It is beyond science to determine tetrachloride it was an alternative, and not the preferred conclusively whether a dose–response is low-dose linear or approach (personal communication from EPA staff). Thus, it nonlinear. Mode of action information is not treated quantita- appears that, for the most part, chemical carcinogens continue tively. Other than its use in deciding whether to classify a to be regulated using a linear approach that is very similar to carcinogen as linear or nonlinear, there is no relation between the one used by the EPA CAG in 1980. a MOA and the resulting RfD or RfC. Seemingly, a factor Although the EPA 2005 cancer risk assessment guidelines should be used to reduce the dose from the POD dose, which permit use of a BBDR model, up to now no such model has clearly cannot qualify as a risk-free dose, to a dose that is “safe been deemed reliable for low-dose risk estimation. Biologically based dose–response models replace the uncertainty in the enough” or below a threshold. Rather than treating all carcino- shape of the dose–response for apical cancer at low doses with gens deemed to have a nonlinear MOA the same, MOA infor- the corresponding uncertainty in the dose–responses for one or mation could be used in determining such a factor. more intermediate steps in the cancer process, such as the It is noteworthy that current cancer risk assessment practices division rate of cells. The dose–responses for these latter rates bare close agreement with approaches employed 40 to 60 years typically are no less uncertain than that for apical cancer. The in the past. On the one hand, if the MOA is deemed nonlinear, data needed to develop a BBDR model, if available at all, risk is assessed very much like it was assessed prior to 1970, the generally cannot be obtained from the same group of animals, main difference being that the 2005 guidelines permit replace- which causes problems associated with heterogeneity. The ment of the NOAEL with the BMDL. However, proposals have MOA being assumed will frequently be in question, as will the been made for using probabilistic methods to replace the safety relevance of measurements to this MOA. Crump et al26 dis- or adjustment factors in the nonlinear approach (eg, IPCS cussed difficulties with developing such models and concluded 2017).29 Probabilistic methods have also been proposed for that “Difficulties in using BBDR models for [estimating low- addressing human variability (eg, Chiu and Slob 2015).30 dose risk] are conceptually the same as those faced when fitting Alternatively, if a nonlinear MOA cannot be established, empirical models to data on apical responses in intact animals. risk is assessed using a linear model in a manner very similar Moreover, these difficulties are exacerbated by problems inher- to that used by CAG. The current approach calls for using a ent in complex models.” Their overall conclusion was “BBDR dose–response model (apparently often the same LMS model models are unlikely to be fruitful in reducing uncertainty in used by CAG in the 1980s) to estimate a POD and then extra- quantitative estimates of human risk from low-level polating linearly downward from the POD. This should result exposures.” Therefore, it should not be surprising that, to date, in low-dose animal risks very similar to those obtained by CAG no BBDR model has yet been developed that is deemed reliable based on the LMS (although perhaps a bit higher, see Figure 6). for use in setting exposure guidelines (note 3). Subramaniam et al31 applied both approaches to 104 data sets Since cancer via a mutational mechanism has been generally and concluded that the 2 approaches provide estimates that are agreed to likely have a linear dose–response, arguments for a very similar. Crump 7 Despite millions of dollars that have been spent on risk 3. Crump KS. Low-dose extrapolations of animal carcinogenicity research, that effort has failed to resolve the shape of dose– data (Reply to the letter of Nathan Mantel). Cancer Res. 1978; response curves at low doses. After 70 or so years of research 38(6):1837-1838. on radiation and cancer, there is still sharp disagreement on the 4. Crump KS. Open query: theoretical problems in the modified shape of dose–response curves for ionizing radiation (eg, Mantel-Bryan procedure. Biometrics. 1977;33(4):752-755. Crump).27 As noted above, current risk assessment practices 5. Higginson J. Population studies in cancer. Acta Unio Int Contra are very similar to those employed many years ago. I believe Cancrum. Washington, DC: National Academies Press; 1960;16: that this state of affairs reflects a fundamental limitation in the 1667-1670. ability of science to resolve critical questions regarding low- 6. National Research Council. Drinking Water and Health. dose risk. Before meaningful progress can be made in improv- Washington, DC: National Academies Press; 1977;1. ISBN: 0- ing risk assessment procedures, this fundamental limitation 309-55400-4. must be acknowledged and accommodated. 7. National Research Council. Drinking Water and Health. National Academies Press; 1980;3. ISBN: 978-0-309-02932-2. Acknowledgments 8. Crump KS, Hoel DG, Langley CH, Peto R. Fundamental I am very thankful for many people with whom I have had close carcinogenic processes and their implications for low dose risk associations over the years and from whom I received inspiration and assessment. Cancer Res. 1976;36(9):2973-2979. encouragement. I would like to particularly acknowledge the contri- 9. Armitage P, Doll R. The age distribution of cancer and a multi- butions of 4 people: Charles Mode, who gave me guidance and stage theory of carcinogenesis. Br J Cancer. 1954;8(1):1-12. encouragement in my graduate studies; David Hoel, who made it possible for me to become acquainted with the problems of risk assess- 10. Armitage P, Doll R. Stochastic models for carcinogenesis. In: ment; Jerry Stara (deceased), who invited me to numerous free- Proceedings of the Fourth Berkeley Symposium on Mathematical wheeling meetings at his EPA office in Cincinnati, where there were Statistics and Probability, Berkeley and Los Angeles, CA: Uni- invigorating discussions regarding problems in risk assessment, and versity of California Press; 1961;4:19-38. Harry Guess (deceased) whose outstanding technical skills, intense 11. Nordling CO. A new theory on the cancer-inducing mechanism. focus on his work and engaging personality made him the best colla- Br J Cancer. 1953;7(1):68-72. borator anyone could ever wish for. 12. Neyman J, Scott EG. Statistical aspects of the problem of carci- nogenesis. In: Proceedings of the Fifth Berkeley Symposium on Declaration of Conflicting Interests Mathematical Statistics and Probability, Berkeley and Los The author(s) declared no potential conflicts of interest with respect to Angeles, CA: University of California Press; 1965:4:745-776. the research, authorship, and/or publication of this article. 13. Guess H, Crump KS. Low-dose extrapolation of data from animal Funding carcinogenesis experiments—analysis of a new statistical tech- nique. Math Biosci. 1976;32(1-2):15-36. The author(s) received no financial support for the research, author- ship, and/or publication of this article. 14. Crump KS, Guess H, Deal K. Confidence intervals and tests of hypotheses inferred from animal carcinogenicity data. Biometrics. Notes 1977;33(3):437-451. 15. Guess H, Crump KS, Peto R. Uncertainty estimates for low-dose 1. A low-dose linear dose–response is one for which the first deriva- tive is positive at zero dose, and consequently can be approximated extrapolations of animal carcinogenicity data. Cancer Res. 1977; by a straight line with positive slope at low doses. 37(10):3475-3483. 2. Thus, the Mantel-Bryan curve doesn’t have a Maclaurin series 16. Guess H, Crump KS. Maximum likelihood estimation of dose- expansion and consequently is not analytic. response functions subject to absolutely monotonic constraints. 3. Perhaps the most ambitious attempt to date at BBDR modeling is Ann Stat. 1978;6(1):101-111. the model for cancer risk from formaldehyde exposure.32,33 This 17. Guess H, Crump KS. Best-estimate low-dose extrapolation of model was subjected to a sensitivity analysis that found that the carcinogenicity data. Environ Health Perspect. 1978;22:149-152. formaldehyde model was so exquisitely sensitive to small changes 18. Albert RE. Carcinogen risk assessment in the U.S. environmental in estimated parameters that it would not be useful in quantitative protection agency. Crit Rev Toxicol. 1994;24(1):75-85. risk assessment.34,35 A Committee of the National Research Coun- 19. Crump KS. An improved procedure for low-dose carcinogenic cil (NRC 20) criticized certain aspects of the sensitivity analysis and recommended that EPA use the Conolly et al model to assess risk assessment from animal data. J Environ Pathol Toxicol low-dose risk from formaldehyde. As of this date, EPA has not Oncol. 1980;5(4-5):675-684. acted on this recommendation. 20. Crump KS, Crockett P. Improved confidence limits for low-dose carcinogenic risk assessment from animal data. J Haz Matr. 1985; References 10(2-3):419-431. 1. Mantel N, d’Bryan WR. “Safety” testing of carcinogenic agents. 21. Anderson EL. Quantitative approaches in use to assess cancer J Natl Cancer Inst. 1961;27:455-470. risk. Risk Anal. 1983;3(4):277-295. 2. Mantel N, Bohidar NR, Brown CC, Ciminera JL, Tukey JW. An 22. United States Environmental Protection Agency. Guidelines for improved Mantel-Bryan procedure for “Safety” testing of carci- Reproductive Toxicity Risk Assessment. Washington, DC: U. S. nogens. Cancer Res. 1975;35(4):865-872. Environmental Protection Agency, EPA/630/R-96/009; 1996. 8 Dose-Response: An International Journal 23. United States Environmental Protection Agency. Guidelines for 30. Chiu WA, Slob W. A unified probabilistic framework for dose- Neurotoxicity Risk Assessment. Washington, DC: U. S. Environ- response assessment of human health effects. Environ Health mental Protection Agency, EPA/630/R-95/001F; 1998. Perspect. 2015;123(12):1241-1254. 24. Crump KS. A new method for determining allowable daily 31. Subramaniam RP, White P, Cogliano VJ. Comparison of cancer intakes. Fundam Appl Toxicol. 1984;4(5):854-871. slope factors using different statistical approaches. Risk Anal. 25. United States Environmental Protection Agency. Guidelines for 2006;26(3):825-830. Carcinogen Risk Assessment. Washington, DC: U. S. Environ- 32. Conolly RB, Kimbell JS, Janszen D, et al. Biologically motivated mental Protection Agency, EPA/630/P-03/001B; 2005. computational modeling of formaldehyde carcinogenicity in the 26. Crump KS, Chen C, Chiu W, et al. What role for biologically- 344 rat. Toxicol Sci. 2003;75(2):432-447. based dose-response models in estimating low-dose risk? Environ 33. Conolly RB, Kimbell JS, Janszen D, et al. Human respiratory tract Health Perspect. 2010;118(5):585-588. cancer risks of inhaled formaldehyde: dose-response predictions 27. Crump KS. Use of threshold and mode of action in risk assess- derived from biologically motivated computational modeling of a com- ment. Crit Rev Toxicol. 2011;41(8):637-650. bined rodent and human dataset. Toxicol Sci. 2004;82(1):279-296. 28. National Research Council. Science and Decisions: Advancing Risk 34. Crump KS, Chen C, Fox JF, Van Landingham C, Subramaniam R. Assessment, Washington, DC: National Academies Press; 2009. Sensitivity analysis of biologically motivated model for 29. International Programme on Chemical Safety. Guidance Docu- formaldehyde-induced respiratory cancer in humans. Ann Occup ment on Evaluating and Expressing Uncertainty in Hazard Char- Hyg. 2008;52(6):481-495. acterization (Second Addition). Geneva, Switzerland: World 35. Crump KS, Chen C, Fox JF, Van Landingham C, Subramaniam R. Health Organization; 2017:160. Letter to the editor: Reply. Ann Occup Hyg. 2009;53:184-189.