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Quantitative implications of the
approximate irrelevance of mammalian
body size and lifespan to lifelong
cancer risk
Preface
Richard Peto
Cite this article: Peto R. 2015 Quantitative
implications of the approximate irrelevance of
mammalian body size and lifespan to lifelong
cancer risk. Phil. Trans. R. Soc. B 370:
20150198.
http://dx.doi.org/10.1098/rstb.2015.0198
Nuffield Department of Population Health, Oxford University, Oxford, UK
Accepted: 13 May 2015
One contribution of 18 to a theme issue
‘Cancer across life: Peto’s paradox and the
promise of comparative oncology’.
The probability of cancer developing within the normal lifespan of a species
seems to be of comparable magnitude for small, short-lived mammals such as laboratory mice and for large, long-lived mammals such as humans, or even whales.
From an evolutionary point of view, of course, this observation makes good sense;
evolutionary pressure will tend to ensure that cancer incidence rates are relatively
low up to the reproductive and child-caring years, but not beyond (as avoiding
cancer in old age by excessive checks on cell division at earlier ages could
retard growth, delaying reproduction).
Turning, however, from evolutionary perspectives to biochemical mechanisms, there are two reasons why this superficially unremarkable comparability
of lifelong cancer risks demands some extremely remarkable mechanistic explanation(s) [1,2]. The first reason is obviously the 10-million-fold difference
in body mass between a 20 g mouse and a 200-ton blue whale (so each mousesized piece of blue whale tissue must in some sense be about ten million times
more cancer-proof than a mouse-sized piece of mouse tissue).
The second and somewhat less obvious reason is that the lifespan of a laboratory mouse is only about 2.5 years while that of a human or whale is about
80 years, which represents five doublings of the mouse lifespan. At first sight
this 32-fold difference in lifespan may seem less important than the 3000fold difference in body mass between mouse and human or the 3000-fold
difference in body mass between human and blue whale, but in some ways it
is numerically even more important.
This is because for many types of cancer ( particularly carcinomas of epithelial cells in organs such as the stomach or intestine that are common to
both sexes, and are therefore little affected by age-related changes in sex hormone levels), the risk of developing the disease before a given age increases
steeply with age, and for some common types of cancer it is proportional to
about the sixth power of age. Thus, it is given approximately by the formula
Ct 6, in which a constant of proportionality C that determines susceptibility to
cancer induction is multiplied by the sixth power of the age in years, t.
(N.B. By differentiating this formula, it can be seen that the incidence rate is
approximately proportional to the fifth power of age.) This sixth-power formula
for risk means that doubling the time available for such a cancer to arise produces not just a twofold difference in risk but about six twofold differences
in risk, i.e. about a 64-fold difference in risk. As the 80-year lifespan of longlived animals such as blue whales or humans differs by five doublings from
the 2.5-year lifespan of a laboratory mouse, the constant C relating risk to the
sixth power of time has to be about a billion-fold smaller (5 6 ¼ 30 powers
of 2) in humans than in mice for the two species to have comparable lifelong
risks of developing cancer.
Combining this approximately billion-fold factor reflecting the effects of
differences in lifespan between humans and mice, and the approximately
3000-fold factor reflecting differences in body size, if the risk per gram of
tissue is written as k times t 6 for humans and K times t 6 for mice, then k is
about 3 trillion times smaller than K. These constants of proportionality are
& 2015 The Author(s) Published by the Royal Society. All rights reserved.
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Competing Interests. I declare I have no competing interests.
Funding. I received no funding for this paper.
References
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2.
Peto R. 1976 Epidemiology, multi-stage models and
short-term mutagenicity tests. In Origins of human
cancer (eds HH Hiatt, JD Watson, JA Winsten), pp.
1403 –1428. New York, NY: Cold Spring Harbor
Laboratory.
Peto R, Parish SE, Gray RG. 1986 There is no
such thing as ageing, and cancer is not related to it.
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cancer and a multi-stage theory of carcinogenesis.
Br. J. Cancer 8, 1–12. (doi:10.1038/bjc.1954.1)
Armitage P, Doll R. 1957 A two-stage theory of
carcinogenesis in relation to the age distribution of
human cancer. Br. J. Cancer 11, 161–169. (doi:10.
1038/bjc.1957.22)
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Peto R, Roe FJC, Lee PN, Levy L, Clack J.
1975 Cancer and ageing in mice and men.
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2
Phil. Trans. R. Soc. B 370: 20150198
necessarily imply any effect of ageing itself on the separate
cellular processes leading to cancer. This conclusion has
been confirmed by animal experiments in which carcinogenic
treatments were started at different ages. In some (despite
a power-law relation of risk to the duration of regular treatment), age was of no independent relevance to the production
of cancer [1,2,5], and in others carcinogenic treatments actually
elicited cancer somewhat less rapidly in older than in younger
animals [1,2,6,7].
Ever since multi-stage models for the process of changing
a normal cell into the seed of a growing cancer were proposed in the 1950s [3,4,8], this trillion-fold difference, or
something roughly equivalent to it, has puzzled many scientists (R. Doll, personal communication). I came across it in the
late 1960s when Doll, investigating the age distribution of
human cancer, was arguing that age itself did not materially
affect the processes of carcinogenesis [9,10], a null result that
we confirmed for mouse skin carcinogenesis [5] and for
smoking and lung cancer [11,12]. The lack of any direct
effect of ageing, whatever that might mean, on the mechanisms of production of the common carcinomas served,
however, to sharpen discussion of what did cause the trillion-fold species differences in cancer susceptibility. As my
name began with P, the problem of the approximate irrelevance of mammalian body size and lifespan to cancer
risk happened to become known alliteratively as Peto’s paradox, but could more appropriately have become known as
Doll’s dilemma [10] or Cairns’ conundrum [13], as both of
them had emphasized how important it was to understand
this properly.
Although the issue has been discussed for decades, it
could not previously be investigated properly because far
too little was known about normal cell biology, cancer cell
genetics, mammalian stem cell kinetics and the many mechanisms for avoiding accumulation of permanently fixed
mutations in stem cells. Now, however, as the papers in
this theme issue show, the question is starting to be
addressed successfully by observation and experiment,
informed by increasingly detailed species-specific knowledge
of genotypes, DNA repair mechanisms and stem cell biology.
But, although the papers in the present issue go some way
towards explaining the enormous differences between
species in the susceptibility of their tissues to carcinogenesis,
there is still far to go if the numerical effects of both body size
and lifespan on cancer incidence rates are to be fully
accounted for.
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directly proportional to cancer incidence rates, so, gram for
gram, human tissue is in some real sense about 3 trillion
times less susceptible to cancer induction than mouse tissue.
To illustrate that these constants of proportionality do
directly determine real cancer incidence rates, consider a 60 kg
person with risk kt6 per gram of tissue by age t years, and
consider the effects of twofold variation in k. If k ¼ 10217, then
the risk by age 80 would be 1 in 6, if k ¼ 2 10217 then this
risk would be 1 in 3, and if k ¼ 0.5 10217 then this risk
would be 1 in 12. Thus, twofold differences in k really matter;
doubling k doubles the risk of getting cancer, and halving k
halves it. Yet, k for human tissue differs from K for mouse
tissue not by a factor of 2, but by a factor of about a trillion.
By what biological mechanisms does evolution produce trillion-fold differences between species in this constant of
proportionality, which are essential for large body size
and long lifespan to exist? Do those mechanisms operate
within the cell, within the local stem cell architecture or in
the whole animal?
It has long been recognized that a power-law relationship
between risk and age could be produced by a multi-stage
model in which the process of changing a normal epithelial
stem cell into the seed of a growing cancer involves several
consecutive changes in the genetic material of that cell,
with the rate-constants governing progression of a partially
altered cell from one stage to the next being largely unaffected by age [3,4].
If there are six stages that are rate-limiting (i.e. improbable
in the time available), then the risk of developing cancer
before a certain age would be expected to be approximately
proportional to the sixth power of age [1,3], which is consistent with what is observed for some types of cancer. (Note
that if six cellular changes all have to happen to get cancer
by a certain age, then by half that age each of those six
changes would have had only half as long to happen, and
the probability of all of them happening would be halved
six times.) It was soon recognized, however, that roughly
the same sixth-power relation with age would be predicted
by many different hypothetical models [1], some having
fewer stages but some selective advantage of partially altered
cells over their unaltered neighbours [4], and some having
more stages but susceptibility to neoplastic change varying
substantially between individuals [2]. Hence, these early
multi-stage models could not even predict the number of
rate-limiting steps needed to go from a normal to a fully neoplastic cell, let alone what heritable cellular changes those
steps actually involved. They did, however, suffice to show
that the 1000-fold differences in cancer rates between old
and young adults that had already been noted did not
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9.
10. Doll R. 1971 The age distribution of cancer:
implications for models of carcinogenesis.
J. R. Stat. Soc. A 134, 133–166. (doi:10.2307/
2343871)
11. Doll R, Peto R. 1981 The causes of cancer:
quantitative estimates of avoidable risks of cancer in
the United States today. J. Natl Cancer Inst. 66,
1193 –1308.
12. Peto R, Doll R. 1997 There is no such
thing as aging: old age is associated
with disease, but does not cause it (editorial). Br.
Med. J. 315, 1030. (doi:10.1136/bmj.315.7115.
1030)
13. Cairns J. 1975 Mutation selection and the natural
history of cancer. Nature 255, 197–200. (doi:10.
1038/255197a0)
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8.
the age of starting exposure. Cancer Res. 51,
6470 –6491.
Nordling CO. 1953 A new theory on the cancerinducing mechanism. Br. J. Cancer 7, 68 –72.
(doi:10.1038/bjc.1953.8)
Cook PJ, Doll R, Fellingham SA. 1969 A mathematical
model for the age distribution of cancer in man.
Int. J. Cancer 4, 93–112. (doi:10.1002/ijc.2910040113)
Phil. Trans. R. Soc. B 370: 20150198