[CANCER RESEARCH 38, 1835-1838,
0008-5472/78/0038-OOOOS02.00
June 1978]
Letter to the Editor
Correspondence re: K. S. Crump, D. G. Hoel, C. H. Langley, and R. Peto. Fundamental Carcinogenic Processes and
Their Implications for Low-Dose Risk Assessment. Cancer Res., 36: 2973-2979, 1976, and H. Guess, K. Crump, and
R. Peto. Uncertainty Estimates for Low-Dose-Rate Extrapolations of Animal Carcinogenicity Data. Cancer Res., 37:
3475-3483, 1977.
In recent years 2 papers (4, 7) from the National Institute
of Environmental Health Sciences (including coauthors
from other institutions) have appeared in Cancer Research
on the subject of low-dose risk assessment for carcinogens.
The authors of these papers have made related points in
other publications (3, 5, 6). Related papers with myself as
coauthor (13, 14) have appeared in Cancer Research and
the Journal of the National Cancer Institute.
A main thesis of the first of the papers cited (4) was the
justification of a linear model (at least at very low doses)
relating added cancer risk to dose of carcinogen; cancer
risk above background is proportional to any low level for
the dosage of the agent used. This is shown to apply for
multihit, multistage, and multievent models. For arbitrary
models a linear relationship for small added doses is justi
fied on the basis that the high background rate of cancer in
humans reflects the presence of high levels of carcinogens
that are already in the environment, so that the effect of any
small increment in the dosage should be approximately
proportional to the size of the increment. This argument
would hold on the assumption that the carcinogen being
considered is interchangeable with the carcinogens already
in the environment.
In the more recent article in Cancer Research (7), the
point is pursued still further, and the much lower "safe"
levels given by this approach than those by the M-B1
approach are demonstrated. This actually had already been
anticipated by us (14), and we had indicated that, for the
linear "1-hit" model, the 1/100,000,000 risk level dosage
would be approximately 1/1,000,000 dosage at a risk of 1%.
However, by the unit probit slope M-B model, the corre
sponding dosage fraction would be about 1/2,000 (more
closely, 1/1,830). We suggested that the 1-hit model pro
vided a most conservative rule for extrapolation. [To indi
cate that the M-B model would be conservative for hitness
models higher than 1, we stated that, for a 2-hit or 2-stage
process, the dosage fraction, 1/100,000,000 versus 1%,
would be 1/1,000 (see also Ref. 9)].
While the paper by Guess et al. (7) was under considera
tion for publication, I had some discussions about it with
one of the coauthors, Dr. Richard Peto. Peto indicated that
one of the functions of the paper was to bring out that
under the circumstances the M-B method might fail to be
conservative. Some general impression had arisen that the
M-B method was always conservative and, in Peto's view,
slope under the linear model), it still provided high conserv
atism relative to the threshold model that had earlier pre
dominated. Under a threshold model the zero slope would
apply over a finite dose interval in contrast to the infinitesi
mal interval under the M-B model. Peto did not consider his
work with Guess and Crump (7) a scoring of the M-B
procedure, but only a clarification of one aspect.
However, one aspect of the work of this group leaves me
concerned. They have created the impression that they
have scientifically established the validity of their linear
model, particularly for "direct-acting carcinogens." If hu
mans or mammals were molecules or, possibly, even singlecelled organisms, it could be that they would show a linear
dose response to direct carcinogens. However, humans
and mammals are complex organisms and do not behave
according to molecular theory. It may be that a particular
animal can cope with some moderate amount of an agent,
whether by elimination, repair, or any other process. In a
sense the individual animal has a threshold, but there is no
general threshold since some individual thresholds could
be rather small. It would not be unreasonable for the
response versus, dose slope to be zero in the neighborhood
of zero dose for complex organisms exposed to a direct
carcinogen. Cornfield (1) describes how such response
curves with zero slope at the origin can result even if the
underlying process is a linear one.
If such a scientific basis for a linear model does not apply,
these authors must rely on the justification that the dosage
applied is a small increment to high levels of interchangea
ble carcinogens that are already in the environment. How
ever, this is not a scientifically established fact. Any inves
tigator attempting to establish a threshold for a carcinogen
would be denying this possibility implicitly, for such thresh
olds could not exist for agents that were already at effec
tively high levels (and it was in a milieu of threshold-seeking
attempts that Bryan and I had been working). Indicative of
the noninterchangeability of carcinogens is the fact that
some, like vinyl chloride, induce unique neoplasms that
occur only rarely, if at all, in the general population. It
would seem that in Ref. 7 the authors were only expressing
an opinion and were not establishing a scientific result, and
their demonstration of low safe levels under a linear model
had long before been conceded.
Peto's acceptance of the contributions by Bryan and
this required correction. Peto recognized that, although the
M-B model called for a response versus dose slope of zero
in the neighborhood of zero dose (in contrast to a positive
myself would seem not to be shared by all his coauthors.
Thus, when Salsburg (16) published an open query about
theoretical problems in the modified M-B procedure,
Crump (2) responded by raising his own questions about
the M-B procedure while answering none of Salsburg's
Supported by USPHS Grant CA-15686 from the National Cancer Institute.
1 The abbreviation used is: M-B. Mantel-Bryan.
questions. (Although Salsburg and Crump may be in con
cert in faulting M-B, they are in serious disagreement with
each other. Salsburg was motivated by a concern that the
Received December
22, 1977; accepted
March 17, 1978.
JUNE 1978
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1835
N. Mantel
M-B procedure is grossly overconservative, whereas Crump
is concerned that it is insufficiently conservative.) Were all
issues raised by Crump resolved, the causes for Salsburg's
concern would still remain, and to these I have provided my
own reply (12).
Crump's questions relate primarily to the poor fit to data
provided by the M-B model and the arbitrariness of the
model. Crump contends that, relative to a model incorpo
rating reasonable biological assumptions, use of the M-B
would be decidedly nonconservative. He further questions
the M-B device of removing the data successively, starting
from the highest dose down and then selecting the highest
resulting safe dose (except in special situations as indicated
in Ref. 10). However, corresponding issues arise with meth
ods that Crump and Guess advocate.
The arbitrariness and poor fit to data for the M-B probit
model are conceded properties of the M-B procedure. It
was never claimed to provide a fit to the data; it was only
claimed to provide an upper limit, under stated assump
tions, to the true dose response at low dosage levels, and
the plausible exception for 1-hit models was indicated.
Crump cannot verify any better fit for the polynomial model
he uses; any goodness of fit that he may observe reflects
only that he has fitted so many parameters as to force the
fit. If it is a model advocated by others, such advocacy has
been based largely on mathematical rather than biological
considerations. One justification of the polynomial model
could even be that, whatever the true response curve might
be, it can be mimicked as closely as desired by a polynomial
of sufficiently high degree; however, use of a high-degree
polynomial on such a basis would not consequently make
valid the safe dose estimates resulting. For that matter, if
the fitted polynomial were one in the square root of dose (a
possibility raised in the following discussion) (6), the result
ing safe levels would make those obtained with the polyno
mial seem nonconservative by comparison. I would not so
term them, however, since some way can be found for
characterizing any safe level except zero as nonconservative.
If the M-B procedure involves successive removal of data
at higher doses, the polynomial approach of Crump and
Guess does the same effectively but subtly. The M-B ap
proach can be applied by substituting the 1-hit model for
the probit model. If we did so apply it and found that, at
higher doses, responses observed rose more rapidly than
called for by the 1-hit model, the procedure of removing
higher dose data would discount such observations. Since
our interest is in what might happen at an extremely lowdose level, sharp increases in what happens on the right
would be irrelevant, and only the lower dose observations
would matter. However, if one specifically used a polyno
mial model, sharp increases on the right would lead to
fitted positive coefficients for the higher powers of dose.
Yet, in going down to extremely low-dose levels, only the
linear term in the polynomial would really matter, and the
higher-degree positive coefficients would play virtually no
role. Much the same safe levels would result if we were to
drop deliberately the high response data at the higher doses
or if we explain such data by positive high-degree coeffi
cients that do not influence the safe dose estimate. (I note
that, when Crump and Guess use polynomial fitting, they
1836
put certain limitations on the fitted coefficients to make for
convexity. Thus, the fitted linear coefficient may not be
negative, although with unrestricted fitting it might be. The
statistical evaluation of such restricted fitting is not com
pletely straightforward.)
Whereas Crump has questioned the conservatism of the
M-B probit model, Jerome Cornfield in private discussions
with me has expressed concern that it is too conservative.
Cornfield and Crump come to diametrically opposite posi
tions, starting from similar multihit models. For a true high
multihit model, the safe dose could be much higher than
called for by the M-B probit model (except at fantastically
low levels of risk). Even for a 2-hit model, the safe dose
would be moderately higher. Cornfield would be willing to
use whatever degree of hitness the data suggest in extrap
olating to safe levels, foregoing any low or linear coeffi
cients unless clearly indicated by the data. However, by
Crump's approach, even if the linear coefficient is not
clearly indicated in the data, that linear coefficient and
other low-order coefficients might still be positive, and lim
its on those coefficients, more precisely on the fitted curve,
must be considered. The setting of limits in a way that
allows for positive linear coefficients makes for the much
lower safe levels by Crump's interpretation than by Corn
field's interpretation of multihit models. (Crump's interpre
tation is also questionable by the concept of interchangeability of carcinogens, to which I alluded previously.)
To understand what is involved in the setting of limits,
consider that we have performed an experiment involving
several dosage levels, in which the response measured has
the same known or unknown variability at all levels. The
variability of the fitted curve and the fitted coefficients
would depend only on the experimental design and could
be relatively high for some designs. If the true linear
coefficient were indeed zero or, if not zero, comparatively
small, any statistical upper limit on that coefficient would
then depend primarily on its statistical variability and thus
primarily on the experimental design. Since, at doses close
to zero, it is virtually only the linear coefficient that matters,
the calculated safe dose would depend on the calculated
upper limit on the linear coefficient and so also would
depend only on the experimental design. It could even be
predictable without experimentation if the response vari
ance is known in advance.
All of this, however, must be altered for the more real
case in which there is a yes/no response to whether cancer
is present or absent in a treated animal. For such binomial
responses, the variability depends on the true response
rate. However, it will be the outcome, among control
animals and among animals receiving the smallest nonzero
dosage, that will effectively determine the limiting value for
the linear coefficient in a polynomial model. Conceptually,
if we had several different response curves that agreed at
the lowest nonzero dose level used and also agreed by
definition at the zero dosage, all of these would tend to
yield the same safe level by Crump's approach. Whether the
fraction of the 1% dosage needed to get a 1/100,000,000
risk should be 1/3.16 for a pure although true 12-hit model,
1/10 for a true 6-hit model, 1/100 for a true 3-hit model, or
1/1,000 for a true 2-hit model, the Crump approach would
yield the fraction 1/1,000,000 corresponding to the 1-hit
CANCER
RESEARCH
VOL. 38
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Letter fo the Editor
model, which is the only one of these models relative to
which the M-B probit model is not conservative. That
Crump's safe levels invariably correspond to the 1-hit model
can be verified in Fig. 2 of his response to Salsburg (2) and
in various figures by Crump and Guess in other publica
tions. Invariably, there is about a 45°line shown for the
log-log relationship between added risk and dose for the
safe dose curve, signifying simple proportionality between
the 2.
In one of his papers with Guess as senior author (6),
Crump takes a position that would suggest incomplete
understanding of the concepts. Since the 1-hit model would
yield impractically low safe levels, as stated by Mantel and
Schneiderman (15), Guess and Crump suggested that their
use of a polynomial form model would be remedial. At very
low doses the increased risk over background would be
dependent essentially on the lowest power of dosage with a
positive coefficient in the polynomial expression and, if that
power is sufficiently large, higher safe levels would result.
Such safe levels, they state, might be as low as the ones for
the 1-hit model or as high as the ones for the probit model.
(Actually, if the power is 2 or greater, the safe levels would
be higher than the ones yielded by the M-B probit model.) It
would seem at first that Guess and Crump are taking a
position similar to that of Cornfield, which was indicated
previously. Instead, in the illustrations given by Guess and
Crump, all of the safe level curves end up of necessity as
curves corresponding to the 1-hit model. Perhaps these
authors did intend to convey that higher than 1-hit safe
levels could result only if the lowest power of dosage with
positive coefficient was stipulated in advance to take some
particular value in excess of unity, but I cannot read this
into their text. They do seem to make a promise of possibly
coming up with such higher safe levels, yet they cannot do
so once they admit that the linear coefficient may be
positive and, if this is denied, they must concede practical
conservatism for the M-B method. Also, if the concept of
interchangeability of environmental carcinogens with the
test carcinogen is held, a concept with which Guess and
Crump, among others, are identified, these authors were
not in a position to promise higher safe levels. Under the
interchangeability concept, any safe level would be virtually
independent of the mathematical model for dose response,
as Richard Peto understands.
A word on the approach of Hartley and Sielken (8) is in
order here. Hartley and Sielken suggest the use of a
polynomial model that is not unlike that of Guess and
Crump, but they use a different conditioning when fitting to
impose convexity. If statistical variation is taken into ac
count, that too should reduce to getting safe levels corre
sponding to the 1-hit model, so that the results should be
essentially the same as those obtained by Guess and Crump
Professor Mantel has raised a number of issues with
regard to the estimation of risks from low doses of carcino
gens. My point of view and that of various coauthors with
regard to some of these issues has been stated elsewhere
(2-7). Nevertheless, there are a few points made here by
Mantel for which I feel further comments could possibly be
useful.
or those that would be obtained if the M-B approach were
coupled with the 1-hit model. If Hartley and Sielken have
derived different safe levels, it is because they have used a
different and, to my mind, questionable way of taking
statistical variation into account (11). As a matter of interest,
Hartley discussed in advance, with myself and Cornfield,
his plans for using a polynomial regression form. We
predicted that the results would be essentially equivalent to
those for a 1-hit model. Crump and Guess have effectively
verified this prediction.
REFERENCES
1. Cornfield. J. Carcinogenic Risk Assessment. Science, 798. 693-699,
1977.
2. Crump, K. S. Response to Open Query: Theoretical Problems in the
Modified Mantel-Bryan Procedure. Biometrics, 33. 752-755, 1977.
3. Crump, K. S., Guess, H. A., and Deal, K. L. Confidence Intervals and
Tests of Hypotheses concerning Dose Response Relations Inferred from
Animal Carcinogenicity Data. Biometrics, 33. 437-451, 1977.
4. Crump, K. S., Hoel. D. G., Langley, C. H., and Peto, R. Fundamental
Carcinogenic Processes and Their Implications for Low-Dose Risk As
sessment. Cancer Res., 36. 2973-2979. 1976.
5. Guess, H. A., and Crump, K. S. Low Dose-Rate Extrapolation of Data
from Animal Carcinogenicity Experiments —Analysisof a New Statistical
Technique. Math. Biosci., 30: 15-36, 1976.
6. Guess, H. A., and Crump, K. S. Can We Use Animal Data to Estimate
"Safe" Doses for Chemical Carcinogens? In: A. S. Whittemore (ed.),
Environmental Health. Quantitative Methods, pp. 13-30. Philadelphia:
Society for Industrial and Applied Mathematics, 1977.
7. Guess. H., Crump, K.. and Peto, R. Uncertainty Estimates for Low-DoseRate Extrapolations of Animal Carcinogenicity Data. Cancer Res.. 37.
3475-3483, 1977.
8. Hartley. H. O., and Sielken, R. L., Jr. Estimation of "Safe Doses" in
Carcinogenic Experiments. Biometrics. 33. 1-30, 1977.
9. Mantel. N. The Concept of Threshold in Carcinogenesis. Clin. Pharmacol. Therap.,4. 104-109. 1963.
10. Mantel, N. Estimating Limiting Risk Levels from Orally Ingested DDT and
Dieldrin Using an Updated Version of the Mantel-Bryan Procedure.
Report Prepared for the Office of Toxic Substances, Environmental
Protection Agency. April 9, 1974. Supplementary Report, December 27,
1974. Washington, D. C.: Environmental Protection Agency.
11. Mantel, N. Aspects of the Hartley-Sielker Approach for Setting "Safe
Doses" of Carcinogens. In: H. H. Hiatt, J. D. Watson, and J. A. Winsten
12.
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(eds.). Origins of Human Cancer. Book C, Human Risk Assessment, pp
1397-1401. Cold Spring Harbor, N. Y.: Cold Spring Harbor Laboratory.
1977.
Mantel, N. Response to Open Query: Theoretical Problems in the
Modified Mantel-Bryan Procedure. Biometrics, 33: 755-757, 1977.
Mantel, N.. Bohidar, N. R., Brown, C. C., Ciminera, J. L., and Tukey, J.
W. An Improved Mantel-Bryan Procedure for "Safety" Testing of Carcin
ogens. Cancer Res., 35: 865-872, 1975.
Mantel, N., and Bryan, W. R. "Safety" Testing of Carcinogens. J. Nati.
Cancer Inst.. 27: 455-470, 1961.
Mantel, N., and Schneiderman, M. A. Estimating "Safe" Levels, a
Hazardous Undertaking. Cancer Res., 35: 1379-1386, 1975.
Salsburg, D. S. Open Query: Theoretical Problems in the Modified
Mantel-Bryan Procedure. Biometrics, 33: 419-421, 1977.
Nathan Mantel
Biostatistics Center
George Washington University
7979 Old Georgetown Road
Bethesda, Maryland 20014
First, I would like to acknowledge the real pioneering
contribution of Mantel through the introduction of the
Mantel-Bryan procedure (8) in 1961. This procedure, pro
posed in an age when threshold seeking was very much in
vogue, broke new and significant ground by utilization of a
model in which no dose could be considered completely
without risk.
JUNE 1978
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1837
Correspondence re: K. S. Crump, D. G. Hoel, C. H. Langley, and
R. Peto. Fundamental Carcinogenic Processes and Their
Implications for Low-Dose Risk Assessment. Cancer Res., 36:
2973−2979, 1976, and H. Guess, K. Crump, and R. Peto.
Uncertainty Estimates for Low-Dose-Rate Extrapolations of
Animal Carcinogenicity Data. Cancer Res., 37: 3475−3483, 1977.
Nathan Mantel
Cancer Res 1978;38:1835-1837.
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