Cancer Turnover at Old Age

Cancer Turnover at Old Age
A thesis presented
by
Francesco Pompei
to
The Division of Engineering and Applied Sciences
In partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
in the subject of
Engineering Sciences
Harvard University
Cambridge, Massachusetts
May, 2002
2002 by Francesco Pompei
All rights reserved
ii
Thesis advisor
Richard Wilson
Author
Francesco Pompei
Cancer Turnover at Old Age
Abstract
It has been commonly assumed for a half century that if a person lives long
enough, he or she will eventually develop cancer. Cancer age-specific incidence data
does not support that assumption, showing instead that incidence flattens at about age 80
and declines thereafter, approaching zero where there is data at about age 100. Previous
cancer models of long standing have been unable to explain the turnover in incidence
beyond age 80, and investigators have tended to ascribe under-reporting of cancers, or
that the oldest persons are somehow not susceptible, as explanations for the data. This
work presents the first model of cancer incidence incorporating the biological process of
cellular replicative senescence. The model provides good fits to human cancer age
distribution data for 40 organ sites from databases from the U.S., Holland, and Hong
Kong. Newly developed mice data from a 24,000 mice study has also been found to be
well fit by the model, confirming that cancers peak at about 80% of lifespan and actually
reach zero for the oldest mice. The model, mathematically a form of Beta function, is
derived by adding to standard multi-stage or clonal expansion models the observation
that in vitro data show aging cells lose their proliferative ability with age. Since
senescent cells cannot produce cancer, the pool of cells available to produce cancer
declines, thus lowering the incidence, reaching zero when all cells are senescent.
Further tests of the model performed against data on interventions that might alter
iii
senescence shows agreement in cancer rate and longevity changes, and also suggests that
longevity might be increased when cancers can be treated. The results also suggest that
studies of cancer associations with various dietary or environmental factors should
include the effect on longevity, since both results depend on senescence.
The most desirable intervention both reduces cancer by reducing cellular damage
causes of carcinogenesis, and reduces cellular damage causes of senescence, thus
achieving both cancer reduction and longer life. This combination is thus far known
only for dietary restriction, but others might be discovered from further research.
iv
Acknowledgements
This thesis, along with the Ph.D. program I was fortunate to be able to enter and
complete, created for me a higher sense of achievement and personal satisfaction than I
could possibly have anticipated. Coming so many years after being away from
academics since earning B.S. and M.S. degrees at MIT, made it all the more satisfying
both intellectually and emotionally. This was especially so, since I had an opportunity
to learn a new field and make an original contribution to the important study of cancer.
First and foremost I would like to thank my thesis advisor Professor Richard
Wilson for first introducing me to this field through his course, then accepting me as his
student and patiently providing guidance and continuous challenge to do my best work,
and finally to introduce me to the academic community to which this work would
contribute. The many hours over the past 7 years we have spent together on the research
leading to this thesis have been among most interesting and stimulating, as well as
instructive, that I have experienced.
I would like to thank the other members of my thesis committee, Professors Peter
Rogers, Richard Kronauer, and Dr. Lorenz Rhomberg: Professor Rogers for his steady
guidance and encouragement on my whole Ph.D. program, and the collegial friendship
we have had since our first meeting 10 years ago; Professor Kronauer for his valuable
guidance in applying the tools of engineering to the medical sciences; and Dr. Rhomberg
for his enthusiasm and guidance, particularly in the biological aspects of the research.
v
I would like to thank Professor Frederick Abernathy for first suggesting the
possibility of earning a Ph.D. in the DEAS, for his encouragement and support over the
program, and for his collegial friendship I have been fortunate to enjoy since we first
met 20 years ago.
In memoriam I would like to thank Professor Thomas McMahon, who through
his friendship and courses reintroduced me to academics after many years away from it,
and resulted in many valuable ideas that I use to this day in my engineering and
scientific work.
As befits the multidisciplinary nature of my thesis, I was fortunate to be able to
learn from many other Harvard faculty who provided guidance and encouragement as
my work progressed. They include Dr. George Gray of the Harvard Center of Risk
Analysis, Professor Howard Stone and Dr. Irvin Schick of the DEAS, Professors Marvin
Zelen and David Harrington of the Biostatistics Dept. of the Harvard School of Public
Health, and Professors James DeCaprio, Myles Brown, and Rakesh Jain of the Harvard
Medical School. I am indebted to all for their interest and encouragement, as well as the
exceptionally useful knowledge I gained from their teaching.
Finally, I would like to thank my wife Marybeth, who was the first to encourage
me to pursue this goal of earning the Ph.D., and provided me with the constant loving
support that made it possible.
vi
Table of Contents
Abstract
iii
Acknowledgements
v
Table of Contents
vii
Citations to Previously Published Work
ix
List of Figures
x
List of Tables
xiii
Chapter 1. Introduction: Thesis Background and Organization
1
Chapter 2. Age Distribution of Cancer: The Incidence Turnover at Old Age
10
2.1.
2.2.
Introduction
Methods
2.2.1. SEER Data
2.2.2. Comparisons to Other Data Sets
2.2.3. Beta Function Selection for Fits
2.2.4. Goodness of Fit
2.3. Results
2.3.1. Fits of the Beta Function to SEER Data
2.3.2. Comparison to Other Data Sets
2.3.3. Comparisons of All Cancer Sites and All Populations
2.4. Discussion
2.4.1. Curve Shape: Comparison to Other Models
2.4.2. Age at Peak Incidence: Comparison to Other Models
2.4.3. Extrapolation of the Beta Distribution Fit
2.4.4. Cumulative Cancer Probability
2.4.5. Modeling Susceptibility and Sensitivity
2.4.6. Biological Hypothesis
2.4.7. Data Reliability
Chapter 3. Age distribution of Cancer in Mice: The Incidence Turnover at Old
Age
3.1. Introduction
3.2. Methods
3.2.1. Data Sources
3.2.2. Analytical Methods
3.2.3. Beta Model
3.3. Results
3.4. Discussion
11
13
13
15
16
18
19
28
33
36
37
39
45
46
46
51
53
56
57
59
60
62
63
71
vii
Chapter 4. Beta-Senescence Model for Cancer Turnover and Longevity:
Interventions by p53, Melatonin, and Dietary Restriction
4.1.
4.2.
4.3.
4.4.
4.5.
75
Introduction
Cellular Senescence
Methods
Results
Discussion
76
79
86
89
95
Chapter 5. Conclusions and Future Work
101
5.1.
5.2.
5.3.
Cancer incidence turnover at old age is likely caused by cellular
senescence reducing the pool of cells available to become
cancerous.
Reducing senescence might appear to be an attractive intervention
to prolong life
Interventions which both decrease cancer and increase longevity are
possible
101
102
102
Appendices
A.
B.
C.
Beta Model Derivation
Commentary: Outliving the Risk for Cancer: Novel Hypothesis or
Wishful Thinking?
Rebuttal to "Outliving the Risk for Cancer: Novel Hypothesis or
Wishful Thinking?"
104
108
116
Acknowledgements in Published Papers
123
References
124
viii
Citations to Previously Published Work
Chapter 2 and Appendix A have been published in their entirety as:
Pompei F, Wilson R. Age distribution of cancer: the incidence turnover at old
age. Human and Ecological Risk Assessment, 7:6, pp 1619-1650 (2001).
Appendix C has been published in its entirety as:
Pompei F, Wilson R. Response to "Outliving the risk for cancer: novel
hypothesis or wishful thinking?". Human and Ecological Risk Assessment, 7:6,
pp 1659-1662 (2001).
Included for clarity is the Commentary to which the Response is directed: HerzPicciotto I, and Sonnenfeld N. Commentary: outliving the risk for cancer: novel
hypothesis or wishful thinking? Human and Ecological Risk Assessment, 7:6, pp
1651-1657 (2001).
Chapter 3 has been published in its entirety as:
Pompei F, Polkanov M, Wilson R. Age distribution of cancer in mice: the
incidence turnover at old age. Toxicology and Industrial Health, 2001; 17:1, pp
7-16. Actual publication date was February 2002.
An abstract of this thesis has been published as:
Pompei F, Wilson R. From mice to men, cancers are not certain at old age. The
Toxicologist, March 2002, Abstract no.756.
ix
List of Figures
Figure
1-1.
2-1.
2-2.
2-3.
2-4.
2-5.
2-6.
2-7.
Title
Page
Result of Pc = (Pb + aDtk)(1-bDt). At low dose (low D and constant t)
cancer is lower than background (top). At high age (constant D and
high t), cancer flattens, turns over, and eventually reduces to zero
(bottom)
2
Age specific incidence vs. age curve shapes for the two major historical
model types, compared to the Beta model and SEER data for combined
male and female cancers.
13
Age specific incidence (per 100,000) vs. age for males and females.
Beta distribution fits of SEER (Reis et al 2000) data for non-genderspecific sites. Parameter values are listed for the Beta function form:
I(t) = (αt)k-1(1-βt)*100,000. The fit values are calculated as the fraction
of the variance of the observed data which are accounted for by the Beta
model with the listed parameter values.
21-23
Age specific incidence (per 100,000) vs. age. Beta distribution fits of
SEER (Reis et al 2000) data for gender-specific sites. Parameter values
are listed for the Beta function form: I(t) = (αt)k-1(1-βt)*100,000, where
t= age-15. The fit values are calculated as the fraction of the variance
of the observed data which are accounted for by the Beta model with the
listed parameter values.
24
Age specific incidence (per 100,000) vs. age data for Holland 19891995 (de Rijke 2000) compared to the SEER data fits with the Beta
function for major cancer sites. Error bars indicate ± 2 SEM.
30
Age specific incidence (per 100,000) vs. age data for Hong Kong 19881992 (Parkin et al 1997) compared to the SEER data fits with the Beta
function for major cancer sites.
32
Age specific incidence (per 100,000) vs. age data for California 19881993 (Saltzstein et al 1998) compared to the SEER data fits with the
Beta function for major cancer sites.
34
Cancer incidence vs. age for all SEER male sites except for childhood
cancers (Hodgkins, thyroid, testes). Each incidence is normalized to the
peak value for that specific cancer. Included for comparison are the data
for Dutch, Hong Kong, and California male sites, and a Beta fit of the
SEER data.
35
x
3-1.
3-2.
3-3.
4-1.
4-2.
4-3.
Liver tumor rates for all TDMS ad libitum controls for mice removed for
natural death or morbidity (solid symbols), and dietary restricted mice
tumor rates of the TDMS scopolamine study controls (open symbols).
A least-squares polynomial curve fit (a0+a1t+a2t2+a3t3) of the data
points is fitted to each data set, for comparison purposes.
64
Age-specific mortality (including morbidity) caused by the three most
common causes of death by neoplasm for ED01 undosed control
animals and data fit by the Beta model. Tests of significant changes
show in all cases that the oldest age group (800-1001 days) has
significantly lower age-specific mortality than the 600-800 days group,
which in turn has significantly higher age-specific mortality than both
the 400-600 and the 200-400 days groups. Calculated age-specific
incidence for the same tumor sites from data by Sheldon et al (1980) are
shown for comparison
66
ED01 age-specific mortality for causes of death (left) and death and
morbidity (right) by all neoplasms vs. dose of 2-AAF. For comparison,
the Beta model fit for the dose=0 data is shown in all curves. Tests of
significant changes show at all doses up to 60 ppm, the oldest age group
(800-1001 days) has significantly lower age-specific mortality than the
600-800 days group, which in turn has significantly higher age-specific
mortality than both the 400-600 and the 200-400 days groups. For the
dose=75, 100 and 150 ppm groups, age-specific mortality continues to
increase beyond the age of turnover observed for the low dose groups.
68-70
Age-specific cancer incidence as modeled by two historically important
models: Armitage-Doll power law model and Moolgavkar-VinsonKnudson clonal expansion model, compared to SEER data and the Beta
model.
78
Cellular senescence evidence in vitro. Increase in number of population
doublings decreases the number of cells which retain replicative
capacity at an approximately linear rate. Lines indicate best linear fit for
each data set
80
Cellular senescence evidence with increase in age of the donor. Increase
in donor age decreases the number of cells which retain replicative
capacity at an approximately linear rate. Lines indicate best linear fir
for each data set.
81
xi
4-4.
4-5.
4-6.
4-7.
4-8.
4-9.
Influence of senescence rate on age-specific cancer incidence in mice.
Beta model fit to ED01 undosed controls is I(t) = (αt)k-1(1-βt), where α
= 0.00115, k-1 = 5, β =0.00108 (Pompei et al 2001). Equivalent MVK-s
model fits shown. Senescence rate is the value of parameter β.
Senescence rate increase by 21% is calculated from Tyner et al (2002)
results of 21% reduction in median lifespan for p53+/m mice compared to
normal p53+/+ mice. Senescence rate of 50% is an assumption for p53+/mice of Tyner et al
90
Probability of tumors in Tyner et al (2002) compared to Beta and MVKs models predictions. Modeled lifetime probability of cancer is
calculated as Prob = 1-exp[−∫ M(t) dt], where M(t) is age specific
mortality. Tyner et al results for p53+/+, p53+/m, and p53+/- are
interpreted as normal senescence, 21% enhanced senescence, and 50%
reduced senescence respectively. Arrow indicate Tyner data reported as
>80% tumor rate
91
Age-specific cancer mortality for female CBA mice dosed with
melatonin vs. controls. Data from Anisimov et al 2001
92
Influence of senescence rate on cancer mortality and lifetime: data from
Tyner et al (2002) for mice with p53+/+, p53+/m, and p53+/- ; compared to
Beta model predictions. Beta model predictions for cancer mortality are
Prob = 1-exp[−∫ M(t) dt]. Beta model predictions for lifetime are
calculated as the lesser of: age at which senescence reaches 100% (t =
1/β), or age at which age-specific cancer mortality reaches 100% [M(t)
= 1]. Human cancer mortality computed from SEER data
93
Liver tumor incidence vs. weight for two studies of control female
B6C3F1 mice. Seilkop data based on body weight measured at 12
months, Haseman data based on maximum weekly average weight.
The Beta-senescence model fit was developed by varying t in inverse
proportion to weight.
94
Results of five rodent studies of the effect of DR on mean lifespan. The
Beta-senescence model comparison line is computed by holding all
variables constant while varying t in inverse proportion to caloric intake.
95
xii
List of Tables
Table
2-1.
2-2.
2-3.
3-1.
Title
Page
Beta Fits to SEER Data for Males: Parameter Values and Their
Implications Compared to SEER Data
25
Beta Fits to SEER Data for Females: Parameter Values and Their
Implications Compared to SEER Data
26
Age-Specific Cancer Incidence for Major Cancers in Other
Countries Compared to Beta Fits of SEER Data: Holland and Hong
Kong
31
Lifetime Cumulative Probability of Mortality from Cancer
71
xiii
Chapter 1
Introduction:
Thesis Background and Organization
As often happens in research, an initial investigation leads to unanticipated
results, as one follows the data to see where they might lead. This thesis was greatly
influenced by such unanticipated directions, starting at the very beginning.
The initial problem investigation was an attempt, by modeling, to understand why
low exposure levels of certain carcinogens appeared to reduce cancer below background
in a number of animal and human studies. Such a possible anti-carcinogenic effect is of
obvious interest in cancer risk assessment, if it is real and testable. Dioxin, radon,
arsenic, and radiation all have shown some evidence of possible low dose anticarcinogenicity, and if the effect were general, it seemed important to understand its
properties.
The preliminary investigation began by exploring a simple mathematical
expression for the probability of cancer that might fit the low dose data:
Pc = (Pb + aDtk)(1-bDt)
(1-1)
where Pb is background cancer probability unrelated to the dose, D is dose, t is time, and
a, b, k are constants. The idea is that some negative slope linear-with-dose biological
mechanism, represented by (1-bDt), reduces both the cancers caused by the dose (aDtk),
and background cancers (Pb), resulting in a "J-shaped" dose-response curve as shown in
Figure 1-1(a).
1
Anti-Carcinogenicity at Low Dose
a
Prob of Cancer: Pc
P c = (P b + aDt k )(1-bDt)
Anti-Carcinogenic Effect
Pb
Dose: D
Cancer Turnover at Old Age
b
Prob of Cancer: Pc
Pc = (P b + aDt k )(1-bDt)
Age: t
4000
c
A-D power law
MVK clonal expansion
SEER (all sites M, F)
5
Age-Specific Incidence (per 10 )
Cancer Incidence in Humans
3000
2000
1000
0
0
20
40
60
80
100
Age (years)
Figure 1-1(a-c). Result of Pc = (Pb + aDtk)(1-bDt). At low dose (low D and constant t)
cancer is lower than background (a). At high age (constant D and high t), cancer flattens,
turns over, and eventually reduces to zero (b). Human cancer data clearly shows turnover
at high age, which previous cancer models (A-D power law and MVK clonal expansion)
failed to fit (c).
2
The curve was then extended to large values of Dt to explore its properties,
producing the Figure 1-1(b) result predicting a marked turnover and negative slope in
cancer probability at old age. This predicted turnover was such a large effect, and
counter to the 50-year paradigm "if you live long enough you will eventually get cancer,"
that it was worth some time to find which way the actual data leads.
The direction was clear, the data shows turnover. Age-specific cancer incidence
data from the Surveillance, Epidemiology, and End Results (SEER) Program at the
National Cancer Institute (Ries et al 2000) showed convincing evidence of flattening and
turnover of cancer incidence, as indicated in Figure 1-1(c). It wasn't until later in the
thesis work that data were found for older age groups than SEER reported, that confirmed
the incidence approaches zero as suggested by the simple function. The two most wellaccepted cancer models, Armitage-Doll multistage (1954) and MVK clonal expansion
(Moolgavkar et al 1981) predict an ever rising incidence with age until cancer becomes
certain. It appeared that the paradigm of the inevitability of cancer, held for a halfcentury, might in fact be wrong. The initial interesting but small anti-carcinogenic effect
at low dose was relegated to future work, and the much larger effect at old age was fully
engaged for this work.
Chapter 2 is the result of the investigation into human data on age distribution of
cancer. Data with flattening of cancer incidence at age above about 75 had long been
observed, but often dismissed as due to less thorough diagnosis conducted on older
people, a suggested most clearly articulated by Sir Richard Doll (Armitage and Doll
1954, Doll 2001). From all available sources, the modern data appear reliable, and data
for 40 organ sites tabulated by SEER appeared to suggest turnover for all cancers. Others
3
had attempted to explain turnover with large variation in individual susceptibility
arguments, which suggest that those who live to the oldest ages are somehow less
susceptible to the cancers (Cook et al 1969, Finkel 1995, Herrero-Jimenez et al 2000).
The susceptibility argument did not appear reasonable, since all human cancer types
tabulated by SEER also tended to turn over at about the same age, despite a factor of 100
range in incidence. This result seemed too improbable to be due to similarity in
subpopulation susceptibility for so many cancers. Further, the evidence of the animal
studies to be discussed in Chapter 3 show that inbred mice raised in identical
environments still produce turnover, thus showing that individual susceptibility is
unlikely to be the cause.
As the human data fit the simple eq. 1-1 function very well (see Ch. 2 for fit test),
a more rigorous derivation from first principles of probability theory was developed,
which combines a model of linear probability of cancer prevention with a model of
probability of cancer creation. The resulting final expression is a form of Beta function
Pc = (α t k-1)(1-β t).
(1-2)
which has well established properties in probability theory. The full derivation is given
in an Appendix A.
The Armitage-Doll power law model of cancer creation, which is an
approximation of the exact solution to the multistage model of cancer, can be shown to
flatten somewhat at old age when made exact (Moolgavkar 1978, Moolgavkar et al
1999), but cannot be made to turn over. Since this approximation is also the first term of
the Beta function eq. 1-2, an exact formulation might result in a somewhat different
function that might be explored.
4
A Commentary (Appendix B) published in the same journal issue with the paper
represented by Chapter 2, challenged some aspects of the interpretation of the data, where
the commenters defended the idea of susceptibility as the cause of the turnover. This
interpretation is a variation on the conventional paradigm, suggesting that if you are
susceptible, then you will get cancer if you live long enough, while if you are immune
you will never get cancer. The Rebuttal to the Commentary (Appendix C) provided an
opportunity to clarify the evidence against susceptibility variations and present new
information that was developed since the paper was first submitted. In particular, new
mice data from genetically inbred mice in identical environments showed turnover, and at
about the same relative age as humans. Also, the candidate biological cause of
senescence was suggested.
Sir Richard Doll questioned the interpretation of turnover, sharing unpublished
data from his long-running physicians cohort study (Doll 2001). The data indicated that
mortality from cancers strongly related to smoking, principally lung cancer, showed
evidence of turnover at old age, while mortality from other cancers showed no turnover.
A preliminary explanation was suggested to Doll, after finding that SEER data also
showed the same pattern observed by Doll, and contained further clues. Lung cancer is a
disease that has little effective treatment, and thus victims usually die shortly after the
cancer is diagnosed. SEER data shows age-specific mortality rates very nearly equal to
age-specific incidence rates, thus "coupling" mortality to incidence (which shows
turnover). Other cancers, however, tend to have more effective medical interventions
that prolong life, thus "decoupling" incidence from mortality. SEER data confirms this,
since mortality rates for non-lung cancers are less than half of incidence rates. Further,
5
there is evidence in the literature that medical intervention is less comprehensive in older
cancer patients than in younger, and that the older patients have more complex diseases
and less ability to fight the cancers, tending to increase age-specific mortality (Yancik et
al 2001). Sir Richard plans to forward data for further ongoing analysis in a collaborative
effort to explain these observations more conclusively.
The excellent fit by the Beta function to human data in Chapter 2 appeared to
suggest an unexplained biological cause of cancer extinction which might cause the
turnover, ought also to produce cancer incidence turnover in animals, which is the subject
of Chapter 3. Only mice studies were found to have sufficient numbers of subjects
living long enough. These studies are rare due to the fact that standard long-term
bioassays are ended at two years, not at the end of the animals' natural lifetime. Some
mice data suggestive of turnover were found in the National Toxicology Program
database, but they were not conclusive due to their small numbers and removal protocol.
An extensive search found a colleague, Dr. Ralph Kodell (2000) with an original data file
from the 24,000 mice "megamouse" ED01 study performed in the 1970's, which had
sufficient numbers for a conclusive determination of turnover. This data showed clearly,
that as in humans, cancers peaked at about 80% of assumed lifespan. Further, for the
oldest mice, cancers reduced to zero as predicted by the Beta model. The conclusiveness
of the mice data greatly increased confidence in the idea that unexplained biology caused
the turnover, rather than cancer reporting bias or susceptibility heterogeneity.
Further attempts to find large cohort animal data living to old age led to Sir
Richard Peto (2001), who led the "two tons of rats" nitrosomine study of 4080 rats (Peto
et al 1991). Since the original data was not recoverable, the published data was examined
6
to find if cancer turnover might be present. Unfortunately, the rats lived only 3 years, an
age sufficiently long for mice, but appears to be too short for rats, which might be
expected to live significantly longer than mice due to their much larger size (10 times by
weight). The data are inconclusive, since within the age range of survival, the cancer
incidence neither turns over, nor does it reach 100%, as would be necessary to disprove
the idea of turnover.
Several ideas on the biological cause of the cancer turnover are discussed during
the work of Chapters 2 and 3, but not until Chapter 4 is a candidate cause clearly
identified along with support data. The evidence points to cellular replicative senescence
as the cause of the turnover. The basic idea is that as we age more of our cells senesce,
and fewer of our cells are left with the ability to divide, although those cells in the
senescent state continue to function normally. These senesced cells cannot produce
cancer. Thus at elevated age there are few cells available to produce cancer, the cancer
incidence turns over and eventually reaches zero when all cells are senescent. Since the
rate of senescence appears from the in vitro cell data to be approximately linear, a good
mathematical fit with the Beta function is found, thus completing the Beta-senescence
model development.
To test the Beta-senescence model more fully, a search for possible interventions
that might alter senescence was conducted. Three candidate interventions were found
and results presented in Chapter 4: genetically modified p53 alleles in mice, which appear
to directly increase or reduce senescence, depending on the alteration; long term
melatonin dose in mice, which might reduce senescence via an antioxidant path; and
dietary restriction, which might appear to reduce both senescence and carcinogenesis by
7
the fundamentally different mechanism of stretching time. The p53 data show that
increased senescence (increased value of β) tends to reduce cancer, as suggested by the
Beta model. The surprising observational result was that longevity also decreased with
increased senescence. This is suggested by the shift in location of the Beta function zero
crossing (see Figure 1-1b). This and other evidence suggests that the age at which there
is zero cancer incidence is not just coincidental with the limit of lifespan, but that both
are caused when all cells reach senescence.
The reverse is also supported by the intervention data: reduced senescence
increases cancer but also increases longevity, to a point. As senescence is lowered
further, cancer mortality then eventually shortens lifespan. An apparent peak value in
lifespan is present, which is estimated to be about 1.3 times normal, at senescence value
of 0.75 of normal.
Chapter 4 explores several of the implications of this very interesting result,
including the possibility of extending life by senescence reduction even at the cost of
increased cancer, since the cancer might be treated. One clear conclusion is that studies
of cancer causation/association with diet or environmental factors, should be re-examined
to determine the effect on longevity, since cancer reduction might have the unintended
consequence of longevity reduction. Interventions that might reduce cancer by effective
reduction of the cellular damage causal to carcinogenesis, might also increase longevity
by reduction of the cellular damage that leads to senescence. The only currently known
reliable method to accomplish this is dietary restriction, although there is some evidence
that selenium may accomplish both. Other such interventions might be found by reexamining the carcinogenicity studies to include longevity.
8
As increased cancer risk seems to be a consequence of aging, so is the reduction
in cancer risk, when age exceeds about 80% of lifespan. Thus, it seems that the old
paradigm of the inevitability of cancer if one lives long enough, ought to be replaced by
the new idea that cancer is a disease primarily of the third quarter of life, and becomes
less of a threat in the final quarter of life.
9
Chapter 2.
Age Distribution of Cancer:
The Incidence Turnover at Old Age
In recent years data on cancer incidence in the USA, the Netherlands and in Hong
Kong indicate a flattening and perhaps a turnover at advanced age, but no model has been
successful in fitting this data and thus providing clues to the underlying biology. In this
work it is assumed these data are reliable and free from bias. A Beta function has been
found to fit SEER age-specific cancer incidence data for all adult cancers extremely well,
and its interpretation as a model leads to the possibility that there is a beneficial cancer
extinction process that becomes important at elevated age. Particularly evident from the
data is the apparent remarkable uniformity of adult cancers peaking in incidence at about
the same age, including cancers in other countries. Possible biological mechanisms
include increasing apoptosis and cell senescence with age. Further, the model suggests
that cancer is not inevitable at advanced age, but reaches a maximum cumulative
probability of affliction with any cancer of about 70% for men and 53% for women in the
US, and much smaller values for individual cancers.
10
2.1 Introduction
It is well known that most cancers arise late in life. There is also substantial
evidence that there is a latency period between the time of the initiation of a cancer to its
observation. An early model was that the cancer cells multiply exponentially, at a slow
but steady rate, and that only when they reach a certain critical number can the cancer be
identified. The latency is then the time for this multiplication to occur. The assumption
that cancers may be initiated throughout life leads naturally to observed age specific
cancer incidence I(t) increasing exponentially with age t as
I(t)=Aebt
(2-1)
The study of the age distribution of cancers began with national mortality data
records of the deaths caused by cancers. It soon became apparent that the cancer death
rate increases less steeply with age than the exponential.
Nordling (1953) and Armitage and Doll (1954), working with national
mortality data in the UK, proposed an alternative multistage theory of cancer to describe
the age distribution data. According to this theory, the cell multiplication is assumed to
be rapid and the time from initiation to cancer observation (the latent period) is assumed
to be the time of passing several discrete stages. This leads to the formula
I(t)=at k-1 or ln I = ln a + (k-1)ln t
(2-2)
where k is the number of stages and a includes various factors representing
environmental exposure, genetic susceptibility, and dietary factors. Armitage and Doll
successfully fitted age specific cancer mortality rates, which they assumed to represent
approximate age specific incidence rates, from several sites and countries and found fits
with values of k between 4 and 8. They omitted death rates above age 75, arguing that at
11
such an advanced age, physicians would tend to assign the nebulous "old age" as the
cause of death rather than make a more careful diagnosis. They therefore ignored the
apparent flattening in age specific mortality rates in the data.
In the years since 1947, cancer registries recording incidence data have much
improved. Also, the increase in survival times and cure rates for many cancers has made
it desirable to examine incidence rates rather than death rates as indicators of biological
mechanisms. Great improvements in the collection of incidence data suggest to us that
the concerns of Armitage and Doll about using data from ages over 75, and using
incidence data at all, may now have been resolved. The reader must be warned that the
conclusions of this paper depend critically on the assumption of validity of the modern
incidence data, which is discussed further later in this paper.
This study follows the same steps as the cancer modelers of the 1950's: 1)
taking the most recent data including the turnover to be considered reliable (SEER data,
Reis et al 2000); 2) attempting to fit them with as simple a model as necessary to obtain a
good fit; 3) comparing curve shapes to other cancers and data from other countries; then
4) discussing possible clues to the underlying biology implied by the model. It is found
that a good fit of all adult cancers can be made with a form of Beta distribution (Olkin et
al 1978) for age-specific incidence:
I(t)=(αt) k-1(1-βt)
(2−3)
which includes 3 arbitrary constants. We extrapolate this distribution to older ages (~100
years) where few data exist and explore the implications of assuming the reliability of the
model implied by the fits. As shown in Figure 2-1, a Beta fit to the SEER data for all
cancers is a very different fit than the curves for the two historically important models,
12
the A-D power law and the MVK clonal expansion models (both to be discussed in more
detail) which have been frequently analyzed for insights to biological causes of cancer.
Since there is a large difference, the Beta fit might be evaluated as a model, and its
Age-Specific Incidence (per 100,000)
biological implications are briefly explored.
5000
A-D power law
4500
MVK clonal expansion
Beta model
4000
SEER (all sites M, F)
3500
3000
2500
2000
1500
1000
500
0
0
20
40
60
80
100
120
Age
Figure 2-1. Age specific incidence vs. age curve shapes for the two major historical
model types, compared to the Beta model and SEER data for combined male and female
cancers.
2.2 Methods
2.2.1 SEER Data
This study takes the most recent age specific incidence data (1993-97) from the
Surveillance, Epidemiology, and End Results (SEER) Program, based within the Cancer
13
Surveillance Research Program at the National Cancer Institute (Ries et al 2000).
Established by the National Cancer Act of 1971, the SEER program routinely collects
cancer incidence and mortality data from designated population based cancer registries in
various areas of the country, representing about 14% of the US population. The cancer
site and histology are coded according to the International Classification of Diseases for
Oncology, second edition (ICD-O-2) (Percy et al 1990).
We emphasize that the reliability of the SEER data is central to this work, and the
conclusions depend on an explicit acceptance of the data as an accurate representation of
the actual incidence in the US population. SEER follows a number of careful procedures
to insure the quality of the data, including "abstracting records for resident cancer
patients seen in every hospital both inside and outside each coverage area; searches of
records of private laboratories, radiotherapy units, nursing homes, and other health
services units that provide diagnostic service to ensure complete ascertainment of cases;
records data on all newly diagnosed cancers, including selected patient demographics,
primary site, morphology, diagnostic confirmation, extent of disease, and first course of
cancer-directed therapy; and conduct periodic quality control studies to correct errors."
Age-specific cancer incidence is defined by SEER as
Cancer incidence= (di/ni)*100,000
(2-4)
where i = the 18 age groups 0-4, 5-9, …, 85+; di = number of new cancers diagnosed in
age group i; ni = person-years in group i. The denominator used by SEER represents the
entire population in the relevant age group, including all who have been diagnosed with
the cancer at an earlier age and have not yet died of that cancer. This point will be
14
important in discussing age-specific incidence data when interpreted as a hazard function
(see Appendix A).
The age-specific incidence data is organized as entries in 5-year age intervals
starting from 0-4 to 80-84, ending with an 85+ category. For all intervals except the last,
the center age was considered as representing that interval. For the 85+ category, a
weighted mean value of 90 was computed from life tables (National Vital Statistics
Report 1999) for persons living beyond 85, and used for the 85+ category. The database
for age-specific cancer incidence includes data for 19 male and 21 female primary cancer
sites in addition to all sites combined. No attempt was made to correct the data for
population cohort effects such as hysterectomy, which would remove people from the
denominator; or smoking status, which would provide variable sensitivities.
2.2.2 Comparisons to Other Datasets
Three other data sets were examined to assess the validity of the model at higher
ages than reported by SEER, and to people from markedly different gene pools, diet, and
environment. A study by de Rijke et al (2000) presenting cancer incidence data to age
95+ for the Dutch population over the period 1989-1995 includes data at higher ages than
SEER and from a different culture and environment. The cancer registries that the
authors rely upon have been confirmed to miss few cases of cancer (96.2% completeness)
even in the highest age groups, and are considered reliable. For each age interval
reported by de Rijke, the weighted mean age for that group was computed and used for
all analyses and figures. Since the absolute numbers of cancer incidences are much
15
smaller than the SEER data, particularly in the higher age groups, error bars representing
±2 SEM are indicated in the figures.
A cancer incidence dataset (Parkin et al 1997) for the Hong Kong population over
the period 1988-1992 provides 35,000 cancer incidences over 50 organ sites. The
population, 98% of whom are Chinese, and 90% from a single nearby province in China,
provides a comparison for a different gene pool, culture, and environment than
Americans or Europeans. Participation in the cancer registry is voluntary, but data
collection processes and checking procedures are believed to be effective in ensuring
reliable data. Parkin et al's analysis indicates that its method of site incidence tabulation
results in incidences within 5% of that employing the SEER method. The data extend
over the same age range as SEER, ending at 85+. Only data for six major sites is
examined.
A study by Saltzstein et al (1998) examined cancer incidence recorded for the 35
million people of California over the period 1988-1993, and reported age-specific cancer
rates in five year age groups from 50-54 to 95-99 and ≥100 years old. This study
included 14,086 cancer patients over age 90, 70.8% of which had histological
confirmation of their cancer diagnoses, compared to 94.5% of those less than 90.
Although cancer cases in California are a major component of the SEER data and thus
are expected to be similar, the investigators specifically designed their study to examine
the incidences for older age groups than SEER report.
2.2.3 Beta Function Selection for Fits
Historically, a good fit to incidence data up to age 60 or 70 is given by
16
I(t)=λt k-1
(2-5)
This suggests a modification of this formula to include a factor to produce the turnover.
One possibility is a form of the Beta probability density function, described in statistics
texts as
f(x)=λx k-1(1-x); 0≤ x≤ 1
(2-6)
We parameterize the Beta function by x =βt, giving
I(t)=(αt) k-1(1-βt)*100,000; 0≤ t≤ β −1
(2−7)
and we find a good fit for the SEER data by adjusting the constants α, k-1, and β to
maximize the fit value of eq. 2-8. Although an additional arbitrary constant always
enables a better fit, we suggest that the additional factor producing the turnover might
represent a cancer extinction process. Unlike the mathematical models (described later)
that have been used in the past to fit the incidence data, the Beta function has value 0 for
t≥β −1, thus suggesting the possibility of a limit to the cumulative probability of cancer
that is less than one. The full derivation is in Appendix A.
The SEER cancer sites are divided into the 17 non-gender-specific sites
(unrelated to reproductive organs), and the 6 gender-specific sites (related to reproductive
organs) to be separately analyzed. The Beta fit to the 17 non-gender sites are performed
with t = age. As first suggested by Armitage and Doll (1954) the sex organs may have
different timing of carcinogenic influences compared to the non-gender-specific sites due
to sexual maturity and activity. The simplest assumption is that the carcinogenic
influences start at sexual maturity, taken as age 15. Thus for the 6 gender-specific sites
the fits are performed with t = (age-15) ≥ 0.
17
2.2.4 Goodness of Fit
We employ a fraction of variance method introduced to cancer modeling by
Cox (1995). The fits were produced by manipulation of the three variables of the Beta
model α, k−1, β to maximize the "Fit" value, which we define as the fraction of the
variance in the observed data points accounted for by the model.
The expression
employed is:
r
Fit = 1 −
E [ (O − M ) ]
2
E [ (O − µ ) ]
2
∑ (oi − mi )2
= 1 − i =1
r
(2-8)
∑ (oi − µ )2
i =1
where O and M are the observations and model results random variables respectively, µ
is the mean of all of the age-group observations for that cancer, oi and mi the observed
and modeled values for each age group for that cancer, and r the number of age group
data points to be fit. As indicated by the equation, a perfect model (Σ(oi-mi)= 0) gives a
Fit value of unity, since the model fully accounts for all variance of the observations from
the mean, and no model at all gives a value of zero, since it accounts for none of the
variance.
No attempt was made to include the effect of the uncertainty in the mean for
each data point value, since the SEER data is a record of about 35 million people (14% of
the U.S. population), about 1% of which is in the smallest (85+) age group. Thus the
maximum sampling error is about 1 per 100,000, which is smaller than the variability of
the data for individual cancers by one to three orders of magnitude. The Beta function
curve fit is extended a few years beyond the end of the SEER data in order to clearly
show the location of the predicted peak in incidence. It should be emphasized that the
18
SEER data by themselves do not show a peak for all sites within the age range reported,
but when data is available to age ~100 (as in the Dutch and California data) a peak occurs
for all organ sites reported.
A possible source of error in modeling the SEER data are birth cohort effects,
in which persons in certain age groups are exposed to a non time-homogeneous cause or
new diagnosis of cancer, such as popularity of smoking, introduction of a new diagnostic
test, or a cataclysmic singularity in exposure to a carcinogen such as Hiroshima and
Nagasaki. We have not attempted to correct any of the data for these effects, and
although they might be important in modeling individual cancers, the main conclusions
of this work are based on all 35 of the adult cancers, which are unlikely to be uniformly
distorted by birth cohort effects.
2.3 Results
2.3.1 Fits of the Beta Function to SEER Data
Figures 2-2(a-q) present the SEER data and Beta fits for each of the 17 nongender-specific cancer sites for both males and females. While 31 of the 34 fits can be
seen to be quite good (Fit values near 1), male and female Hodgkins disease (which
might be interpreted form Fig. 2-2r to be two cancers), and female thyroid cancer appear
to be significantly different cancer types than those which are central to this work. The
Fit values for the 31 cancers range from 0.93 to 1.00 with a mean of 0.97 of the variance
accounted for by the Beta function fit. For comparison, the A-D power law model of
Figure 2-1 produces modeled fraction of variance fit values of 0.99, 0.94, 0.69, and -0.3
for ages 0-74, 0-79, 0-84, and 0-90 respectively for male liver cancer (a 1% cumulative
19
incidence cancer, far enough from unity cumulative incidence such that the A-D
approximation gives good accuracy, as discussed below). The corresponding values for
the Beta fit are 1.00, 1.00, 0.99, and 0.99 for the same cancer over the same age ranges.
Figure 2-2(r) shows the total incidence for all 17 non-gender-specific sites for
males and females separately, created by summing the SEER data and the Beta fits for
each age category (not a true probability but a commonly used approximation). Clearly
the male and female incidence curves have the same shape, both reaching a peak at about
age 80, but a factor of two difference in incidence. Figures 2-3(a-f) present the Beta fits
for the six gender-specific sites, which are all based on t=0 at age 15. All four of the
female site fits are quite good (mean Fit value = 0.97), but the two male sites are not
quite as good (mean Fit value = 0.92). Testicular cancer (Fit = 0.87) clearly is a very
different cancer. Prostate cancer (Fit = 0.96) might be somewhat influenced by the SEER
data itself, which the SEER investigators warn is heavily influenced by the prostate
specific antigen (PSA) test entering into common use over the last few years. The SEER
reported overall age-adjusted incidence rate shows a distinct "bump" in the years 19891996, but the age distribution of this bump is not reported.
20
Lung and bronchus
a
800
Male
α = 0.00755
β = 0.0105
k-1 = 6.6
Fit = 0.99
700
600
500
Female
0.007
0.0108
6.5
0.98
b
Colon rectum
700
Male
α = 0.00732
β = 0.01003
k-1 = 7
Fit = 1.00
600
500
400
Female
0.00717
0.00995
7.3
1.00
400
300
300
200
200
100
100
0
0
0
20
40
60
80
c
Urinary bladder
350
Male
α = 0.00688
β = 0.01007
k-1 = 7.2
Fit = 1.00
300
250
100
Female
0.00525
0.0098
6.7
1.00
0
20
140
Male
a = 0.00509
b = 0.00997
k-1 = 5.7
Fit = 0.99
100
80
150
60
100
40
50
60
80
100
d
Non-Hodgkins lymphoma
120
200
40
Female
0.00481
0.0101
5.7
1.00
20
0
0
20
40
60
80
e
Leukemias
140
Male
α = 0.0048
β = 0.00925
k-1 = 5.9
Fit = 0.99
120
100
100
Female
0.0043
0.009
5.9
0.99
0
0
20
40
60
80
f
Melanomas
120
Male
α = 0.0023
β = 0.0089
k-1 = 3.5
Fit = 1.00
100
80
100
Female
0.00034
0.007
2
0.98
80
60
60
40
40
20
20
0
0
0
20
40
60
80
100
120
0
20
40
60
80
100
Figure 2-2 a-r. Age specific incidence (per 100,000) vs. age for males and females.
Beta distribution fits of SEER (Reis et al 2000) data for non-gender-specific sites.
Parameter values are listed for the Beta function form: I(t) = (αt)k-1(1-βt)*100,000. The
fit values are calculated as the fraction of the variance of the observed data which are
accounted for by the Beta model with the listed parameter values.
21
Stomach
140
g
Male
α = 0.00542
β = 0.00952
k-1 = 6.7
Fit = 1.00
120
100
Female
0.00475
0.00925
6.7
1.00
h
Oral cavity and pharynx
120
Male
α = 0.0038
β = 0.01015
k-1 = 4.6
Fit = 0.99
100
80
Female
0.00305
0.00985
4.6
0.99
80
60
60
40
40
20
20
0
0
0
20
40
60
80
i
Pancreas
120
Male
α = 0.00545
β = 0.00995
k-1 = 6.6
Fit = 1.00
100
80
0
100
40
60
80
Kidney and renal pelvis
100
90
Female
0.00515
0.0095
6.6
1.00
20
Male
α = 0.00435
β = 0.0102
k-1 = 5.2
Fit = 0.99
80
70
60
100
j
Female
0.0038
0.0102
5.2
1.00
50
60
40
40
30
20
20
10
0
0
0
20
40
60
80
k
Multiple myelomas
80
Male
α = 0.00493
β = 0.00998
k-1 = 6.5
Fit = 1.00
70
60
50
100
Female
0.00463
0.01015
6.5
1.00
a
0
20
40
60
80
Esophagus
50
Male
α = 0.00464
β = 0.01035
k-1 = 6
Fit = 0.98
45
40
35
100
l
Female
0.00363
0.0097
6
0.98
30
40
25
30
20
20
15
10
10
5
0
0
20
40
60
80
100
0
0
20
Figure 2-2. (continued)
22
40
60
80
100
60
m
Liver and bile duct
Male
α = 0.00439
β = 0.01025
k-1 = 5.8
Fit = 0.99
50
40
Female
0.00411
0.01
6.3
1.0
50
Larynx
45
Male
α = 0.0047
β = 0.0108
k-1 = 5.9
Fit = 0.96
40
35
30
30
n
Female
0.0031
0.0108
5.4
0.93
25
20
20
15
10
10
5
0
0
0
20
40
60
80
Brain and other nervous
40
Male
α = 0.00295
β = 0.0102
k-1 = 4.5
Fit = 0.94
35
30
25
100
o
Female
0.002655
0.0102
4.5
0.94
0
20
40
60
80
Thyroid
25
Male
α = 0.0002
β = 0.009
k-1 = 2
Fit = 0.96
20
100
p
Female
0.00025
0.0102
1.9
0.71
15
20
10
15
10
5
5
0
0
0
20
40
60
80
Hodgkins disease
Male
Female
0.0000013
α = 0.000008
0.0098
β = 0.0098
k-1 = 1.2
1
Fit = 0.27
0.01
12
10
8
0
100
q
3000
4
1000
2
500
0
0
20
40
60
80
0
100
20
Figure 2-2. (continued)
23
60
80
100
r
Age-specific cancer
incidences for all 17 non-sex
sites summed for each age
interval, for both SEER data
and Beta fits.
2000
1500
40
Total non-sex sites
2500
6
0
20
40
60
80
100
a
Prostate
1400
600
α = 0.00375
β = 0.0115
k-1 = 2.8
Fit = 1.00
500
1200
α = 0.0085
β = 0.0122
k-1 = 4.8
Fit = 0.96
1000
800
b
Breast (F)
400
300
600
200
400
100
200
0
0
0
20
140
Corpus Uteri
120
α = 0.0038
β = 0.0124
k-1 = 3.7
Fit = 0.98
100
80
40
60
80
0
100
c
20
40
60
80
100
d
Ovary
80
70
α = 0.00142
β = 0.0108
k-1 = 2.6
Fit = 1.00
60
50
40
60
30
40
20
20
10
0
0
0
20
40
60
80
e
Cervix Uteri
25
100
α = 0.0000065
β = 0.01
k-1 = 1
Fit = 0.91
20
15
0
20
20
40
60
80
100
f
Testis
18
α = 0.000035
β = 0.029
k-1 = 1.1
Fit = 0.87
16
14
12
10
8
10
6
5
4
2
0
0
20
40
60
80
0
100
0
20
40
60
80
100
Figure 2-3 a-f. Age specific incidence (per 100,000) vs. age. Beta distribution fits of
SEER (Reis et al 2000) data for gender-specific sites. Parameter values are listed for the
Beta function form: I(t) = (αt)k-1(1-βt)*100,000, where t= age-15. The fit values are
calculated as the fraction of the variance of the observed data which are accounted for by
the Beta model with the listed parameter values.
24
Table 2-1. Beta Fits to SEER Data for Males:
Parameter Values and Their Implications Compared to SEER Data
α
Site
k-1
(x102)
Non-gender-specific
sites
Lung and bronchus
Age at
Fit
Peak agezero
(fraction
specific Age at peak
β
incidence
incidence
of
incidence
variance
(per
100k)
(x102) modeled)
I(tp);[SEER] tp; [SEER] β−1 = to
Cumulative
probability*
over lifespan
Pc; [SEER]++
0.755
6.6
1.05
0.986
588; [560]
83; [78]
95
0.165; [0.153]++
Colon and rectum
0.732
7
1.003
1.000
541; [519]+
87; [90]+
100
0.153; [0.132] ++
Urinary bladder
Non-Hodgkins
lymphoma
Leukemias
0.688
7.2
1.007
0.998
308; [296]+
87; [90]+
99
0.085; [0.074] ++
0.509
5.7
0.997
0.985
129; [127]
85; [83]
0.48
5.9
0.925
0.994
120; [117]+
92; [90]+
108
0.041; [0.029] ++
Stomach
0.542
6.7
0.952
0.998
117; [117]+
91; [90]+
105
0.036; [0.026] ++
Pancreas
0.545
6.6
0.995
0.998
98; [94]+
87; [90]+
101
0.029; [0.025] ++
Melanomas of skin
Oral Cavity and
Pharynx
Kidney and renal
pelvis
Multiple myeloma
0.23
3.5
0.89
0.999
81; [80]+
87; [90]+
112
0.040; [0.029] ++
0.38
4.6
1.015
0.986
79; [76]
81; [83]
0.435
5.2
1.02
0.990
77; [71]
82; [78]
0.493
6.5
0.998
0.998
54; [51]
87; [85]
100
0.016; [0.013] ++
Esophagus
0.464
6
1.035
0.981
46; [42]
83; [83]
97
0.014; [0.013] ++
Liver and bile duct
0.439
5.8
1.025
0.991
43; [39]
83; [83]
98
0.013; [0.013] ++
Larynx
Brain and other
nervous
Thyroid
0.47
5.9
1.08
0.960
42; [41]
79; [73]
93
0.013; [0.011] ++
0.295
4.5
1.02
0.942
28; [28]
80; [78]
0.02
2
0.9
0.956
7; [9]
74; [73]
111
0.005; [0.004] ++
0.0008
1.2
0.98
0.267
4; [6]
56; [28]
102
0.003; [0.003] ++
Prostate
0.9500
5.8
1.250
0.957
1227; [1149]
83; [73]
97
0.367; [0.329] ++
Testis
0.0035
1.1
2.900
0.874
14; [14]
33; [33]
49
0.003; [0.004] ++
Hodgkins disease
100
99
98
98
0.042; [0.039] ++
0.029; [0.027] ++
0.026; [0.025] ++
0.010; [0.011] ++
Gender-specific sites
At least one cancer
∑ I (t p ) =
all
3603; [3436]
At least one non-gender-specific cancer
∑ I (t p ) =
2362; [2273]
non − sex
*P
c =
+
1 − ∏ (1 − Pc ) =
0.704; [0.652] ++
∏ (1 − Pc ) =
0.531; [0.480] ++
all
1−
non− sex
β −1
k −1
∫ (αt ) (1 − βt )dt
0
Indicates SEER data that does not record a peak prior to the highest age category, which is mean 90 years.
++
Indicates integration of SEER data up to age 90.
25
Table 2-2. Beta Fits to SEER Data for Females:
Parameter Values and Their Implications Compared to SEER Data
α
Site
k-1
(x102)
Age at
Peak agezero
Goodness specific Age at peak
β
incidence incidence
of
incidence
Fit
(per 100k)
(x102)
I(tp);[SEER] tp; [SEER] β−1 = to
Cumulative
probability*
over lifespan
Pc; [SEER]++
Non-gender-specific sites
Colon and rectum
0.717
7.3
0.995
0.997
432; [423]+
88; [90]+
101
0.119; [0.097] ++
Lung and bronchus
0.7
6.5
1.08
0.978
314; [287]
80; [77.5]
93
0.087; [0.084]++
Pancreas
Non-Hodgkins
lymphoma
Urinary bladder
0.515
6.6
0.95
0.998
91; [91]+
91; [90]+
105
0.028; [0.021] ++
0.481
5.7
1.01
0.996
87; [85]
84; [82.5]
0.525
6.7
0.98
0.995
78; [73]+
89; [90]+
102
0.023; [0.019] ++
Leukemias
0.43
5.9
0.9
0.987
74; [71]+
95; [90]+
111
0.026; [0.017] ++
Stomach
Kidney and renal
pelvis
Melanomas of skin
Oral Cavity and
Pharynx
Multiple myeloma
0.475
6.7
0.925
0.997
59; [57]+
94; [90]+
108
0.019; [0.012] ++
0.38
5.2
1.02
0.995
38; [38]
82; [82.5]
0.034
2
0.7
0.977
35; [36]+
95; [90]+
0.305
4.6
0.985
0.993
33; [33]
83; [82.5]
0.463
6.5
1.015
0.997
32; [31]
85; [82.5]
99
0.009; [0.009] ++
Liver and bile duct
Brain and other
nervous
Esophagus
0.411
6.3
1
0.996
20; [20]
86; [82.5]
100
0.006; [0.005] ++
0.2655 4.5
1.02
0.943
17; [18]
80; [77.5]
0.363
6
0.97
0.985
16; [15]+
88; [90]+
103
0.005; [0.004] ++
Thyroid
0.025
1.9
1.02
0.706
13; [15]
64; [47.5]
98
0.008; [0.008] ++
Larynx
0.31
5.4
1.08
0.928
7; [8]
78; [67.5]
93
0.002; [0.002] ++
0.0001
1
0.98
0.008
3; [5]
51; [22.5]
102
0.002; [0.002] ++
Breast
0.375
2.8
1.15
0.995
486; [482]
79; [77.5]
102
0.207; [0.187] ++
Corpus uteri
0.38
3.7
1.24
0.980
110; [109]
79; [77.5]
96
0.038; [0.033] ++
Ovary
0.142
2.6
1.08
0.996
61; [62]
82; [77.5]
108
0.029; [0.024] ++
Cervix uteri
0.0007
1
1
0.911
16; [17]
65; [47.5]
115
0.011; [0.010] ++
Hodgkins disease
99
98
143
102
98
0.028; [0.026] ++
0.013; [0.012] ++
0.028; [0.016] ++
0.012; [0.011] ++
0.006; [0.007] ++
Gender-specific sites
At least one cancer
∑ I (t p ) =
all
2022; [1976]
At least one non-gender-specific cancer
∑ I (t p ) =
1349; [1306]
non − sex
*P =
c
+
1 − ∏ (1 − Pc ) =
0.526; [0.475] ++
∏ (1 − Pc ) =
0.354; [0.305] ++
all
1−
non− sex
β −1
k −1
∫ (αt ) (1 − βt )dt
0
Indicates SEER data that does not record a peak prior to the highest age category, which is mean 90 years.
++
Indicates integration of SEER data up to age 90.
26
Tables 2-1 and 2-2 present the tabulation of the Beta parameters for the fits for
males and females respectively ranked by peak incidence, and calculated implications
compared to the SEER data. The α parameter varies with the ranking of peak incidence
when (k-1) values are similar, departing somewhat from this ranking when (k-1) is
different. The β parameter is remarkably consistent, varying by only about 20% for the
35 adult cancers, even as the peak incidences vary by a factor of 100. Also, the value of
α is always less than the value of β, suggesting that the probabilities of the (k-1) uniform
random variables representing cancer creation are always less than the probability of the
one random variable representing cancer extinction (see Appendix A).
The Beta calculated peak incidence I(tp) and age at peak incidence tp are
compared to the SEER values. Several of the entries for the SEER data are noted to
indicate that a peak was not recorded for those cancers in the age intervals reported, thus
providing an uncertain SEER value for peak incidence and age at peak. For these cases it
is best to refer to the figures to judge the adequacy of the estimate of SEER peak and age
at peak. Since tp is derived from the Beta function as tp = (k-1)/kβ, there is no dependence
on α and only weak dependence on k. Accordingly, the age at peak incidence, which can
be described as the turnover age, is almost entirely dependent on β, and as shown in the
tables is consistent over all adult cancers and over a factor of 100 in incidence. The age
at zero incidence represents the upper bound of the Beta function, and is equal to
β −1. None of the SEER data extends to high enough age to test this model prediction, but
cancers of the lung, larynx, brain, and corpus uteri show marked downturn of incidence
within the age range reported. The Dutch and California data with older age groups show
data tending to zero incidence at β −1 age.
27
The final column of each of the two tables present calculated cumulative
probability of each cancer, based on the Beta fit, and assuming the individual lives to at
least age β −1. The SEER comparison is the sum of the age specific incidence over all age
groups. Since the SEER data does not extend to zero incidence, the SEER result should
be somewhat lower than the Beta result, particularly if β −1 is higher than 90, which is the
case. Individual cancer site probabilities rank approximately in order of the peak
incidences, indicating that the incidence curve shapes are not too different from cancer to
cancer, which can also be concluded from the constancy of k and β. For males, the
maximum lifetime probability of an individual cancer ranges from 0.3% for Hodgkins
disease to 36.7% for prostate cancer. For females, the range is 0.2% for Hodgkins
disease to 20.7% for breast cancer. The calculated upper limit to the lifespan probability
of any cancer for males at 70% and for females at 53%.
2.3.2 Comparison to Other Datasets
Figures 2-4(a-f) and Table 2-3 presents the de Rijke data for six major cancer
sites, compared to the SEER data curve fits with the Beta function. Colorectal cancer
incidence for Dutch males and females is not very different than the values for
Americans, and closely agrees with the predicted turnover age and shape, even though
the model was fit to SEER data that did not actually reach a peak (Figure 2-2b). Lung
cancer shows different levels of incidence, but the location of the peak and curve shapes
are similar to SEER. Of interest, the oldest male lung cancer group has incidence almost
zero at age 100, which is close to the predicted β −1 value. Prostate cancer also shows
about equal incidence and similar curve shape as the SEER fits, although the reported
28
incidence appears to occur about 10 years later in the Dutch than in the SEER population.
Female breast cancer appears to be a good match to the SEER fit curve shape at a
somewhat lower incidence level, and suggests near-zero incidence at an age not too
different from β −1. Bladder cancer and stomach cancer also show similar curve shapes to
the SEER data fits, with the age at peak incidence correctly predicted. The model is
particularly accurate in predicting stomach cancer peak, since the SEER data does not
show a peak in its age range.
Figures 2-5(a-f) and Table 2-3 present Hong Kong age-specific incidence data
for six major cancer sites, both male and female, with comparisons to the SEER model
fits. Colorectal cancer incidence is about three-fourths of the SEER value, but the shape
is similar, along with the age at peak incidence, to the SEER. Lung cancers are very
close to SEER data in shape and age at peak incidence, with levels about one-third
higher. Stomach cancer incidence is about twice that of the US, but appears to peak at
about the same age. For bladder cancer, the incidence is lower than SEER for men in
Hong Kong, but appears similar in age at peak incidence for both sexes. Conversely,
prostate cancer is only one-sixth the SEER value, but with peak incidence appearing at
about the same age. Breast cancer incidence appears quite different for Hong Kong
women than their US counterparts, for reasons that are unknown.
29
Colorectal
800
a
Male (Rijke 2000)
Male (Model Fit to SEER)
Female (Rijke 2000)
Female (Model Fit to SEER)
700
600
Lung
1000
b
900
800
700
500
600
400
500
300
400
300
200
200
100
100
0
0
0
1400
20
40
60
80
0
100
c
Prostate
20
600
1200
40
60
80
100
d
Breast
500
1000
400
800
300
600
200
400
100
200
0
0
0
400
20
40
60
80
100
e
Bladder
0
20
40
60
80
f
Stomach
250
100
350
200
300
250
150
200
100
150
100
50
50
0
0
0
20
40
60
80
100
0
20
40
60
80
100
Figure 2-4 a-f. Age specific incidence (per 100,000) vs. age data for Holland 1989-1995
(de Rijke 2000) compared to the SEER data fits with the Beta function for major cancer
sites. Error bars indicate ± 2 SEM.
30
Table 2-3. Age-Specific Cancer Incidence for Major Cancers in Other Countries
Compared to Beta Fits of SEER Data: Holland*and Hong Kong+
Site
Peak age-specific
incidence
(per 100k)
Male
Age at peak
incidence
Male
Peak agespecific
incidence
(per 100k)
Female
Data; [Beta]
Data; [Beta]
Data; [SEER]
Data; [Beta]
449; [541]
85-94; [87]
321; [432]
85-94; [88]
75; [314]
65-74; [80]
349; [486]
75-84; [79]
Age at peak
incidence
Female
Holland
Colorectal
Lung
741; [588]
75-84; [83]
Prostate
939; [1227]
85-94; [83]
Breast
Bladder
235; [308]
85-94; [87]
51; [78]
85-94; [89]
Stomach
207; [117]
85-94; [91]
98; [59]
85-94; [94]
Lymphomas
76; [129]
85-94; [85]
58; [87]
85-94; [84]
Bronchus, lung
827; [588]
80-84; [83]
427; [314]
80-84; [80]
Colon and rectum
437; [541]
85+; [87]
285; [432]
80-84; [88]
Bladder
224; [308]
85+; [87]
68; [78]
80-84; [89]
140; [59]
85+; [94]
150; [486]
85+; [79]
Hong Kong
Stomach
221; [117]
85+; [91]
Prostate
219; [1227]
80-84; [83]
Breast
*deRijke et al (2000), + Parkin et al (1997)
31
a
Colon rectum
600
400
b
Bronchus, lung
900
Hong Kong (M)
Model Fit to SEER (M)
Hong Kong (F)
Model Fit to SEER (F)
500
1000
800
700
600
500
300
400
200
300
200
100
100
0
0
0
300
20
40
60
80
c
Stomach
0
100
20
40
60
80
d
Bladder
400
100
350
250
300
200
250
150
200
150
100
100
50
50
0
0
0
20
40
60
80
e
Prostate
1400
0
100
20
40
60
80
f
Breast
600
100
500
1200
1000
400
800
300
600
200
400
100
200
0
0
0
20
40
60
80
100
0
20
40
60
80
100
Figure 2-5 a-f. Age specific incidence (per 100,000) vs. age data for Hong Kong 19881992 (Parkin et al 1997) compared to the SEER data fits with the Beta function for major
cancer sites.
32
Figures 2-6(a-d) present the Saltzstein et al (1998) data compared to the Beta
fit of the SEER data for six cancers. There is the expected good agreement for the age
range up to about 90, which is the range reported by SEER. However, as we observed in
the Dutch data, the turnover in incidence, and the continued decrease in incidence to age
100 predicted by the Beta model, is present. The slight rise in incidence for the oldest
age group is ascribed by the investigators to be due to under-reporting of the population
of the ≥100 population over the relevant time period.
2.3.3 Comparisons of All Cancer Sites and All Populations
By normalizing each cancer vs. age data point to the peak value for that
particular cancer and age group, we can plot the results on a single chart. Figure 2-7
shows all of the SEER incidences for adult male cancers (leaving out Hodgkins disease,
thyroid, testes since these appear at young age and are very different cancers than those
modeled by an A-D power law) plotted together, along with the mean value of all of the
SEER incidences at each age. The Beta model fit to the SEER data is included, extending
to age 101. Also plotted are the Dutch, Hong Kong and California incidence data. The
Dutch and California data are particularly valuable because they extend to age 97 and 102
respectively, where the SEER data ends at 90. By inspection of Figure 2-7, all of the male
adult cancers incidences fall into a well defined band, despite the factor of 100 variation in
peak incidence for the range of cancers considered. The band scatter standard deviation
averages approximately ± 8% of the peak incidence about the mean at each age group of
each cancer, and all but brain cancer and leukemia (which have significant incidence at
young age) are near zero normalized incidence up to about age 30.
33
a
Colorectal
600
California (M)
Model Fit to SEER (M)
California (F)
Model Fit to SEER (F)
500
400
800
b
Bronchus, lung
700
600
500
400
300
300
200
200
100
100
0
0
0
20
40
60
80
c
Prostate
1400
0
100
600
20
40
60
80
100
d
Breast
500
1200
1000
400
800
300
600
200
400
100
200
0
0
0
20
40
60
80
100
0
20
40
60
80
100
Figure 2-6 a-d. Age specific incidence (per 100,000) vs. age data for California 19881993 (Saltzstein et al 1998) compared to the SEER data fits with the Beta function for
major cancer sites.
34
Brain (M)
Colo-rectal (M)
Esophagus (M)
Kidney (M)
Larynx (M)
Βeta parameters
α = 0.01655
k−1 = 5.1
β = 0.0098
1
0.9
Leukemias (M)
Liver (M)
Lung (M)
Melanomas (M)
Age -Specific Cancer Incidence Normalized to Peak
Myelomas (M)
Lymphoma (M)
0.8
Oral (M)
Pancreas (M)
0.7
Stomach (M)
Bladder (M)
Prostate
0.6
Mean (SEER-M)
Beta model of SEER
Colorectal (Dutch)
0.5
Lung (Dutch)
Prostate (Dutch)
Stomach (Dutch)
0.4
Lymphoma (Dutch)
Bladder (Dutch)
0.3
Esophagus (HK)
Stomach (HK)
Colorectal (HK)
0.2
Lung (HK)
Prostate (HK)
Bladder (HK)
0.1
Colorectal (Calif)
Lung (Calif)
0
Prostate (Calif)
0
20
40
Age
60
80
100
Figure 2-7. Cancer incidence vs. age for all SEER male sites except for childhood
cancers (Hodgkins, thyroid, testes). Each incidence is normalized to the peak value for
that specific cancer. Included for comparison are the data for Dutch, Hong Kong, and
California male sites, and a Beta fit of the SEER data.
35
2.4 Discussion
There are five alternative ways of describing the modeling of the cancer incidence
data (all of them assuming the data at elevated age are valid, a point discussed further
later):
1. The simple Beta function fits the age distribution including the turnover at
elevated age well, while previous biologically based models have been unable to
do so, which in turn leads to a search for a biological basis for the Beta model.
2. The curve shape for adult cancers, including the turnover, appears consistent from
male to female, from culture to culture, and even from cancer to cancer, varying
only in level of incidence.
3. There is apparent remarkable uniformity of the age at peak incidence across all
adult cancers, despite a factor of 100 difference in peak incidence in these
cancers.
4. Extrapolation of the Beta function fits beyond the age for which there are data
allows us to calculate the age at which incidence is expected to be zero. Then we
may integrate the incidence to calculate a lifespan cumulative probability for each
cancer, and all cancers combined.
5. The cumulative probability of a person contracting any cancer is less than one,
and of each individual cancer it is much less than one. The conventional wisdom
that everyone will eventually contract any or a specific cancer if he or she does
not die of some other cause, may be incorrect.
36
2.4.1 Curve Shape: Comparison to Other Models
The earlier models directed to explaining age distribution of cancer are of two
general types: multistage, and clonal expansion. The earliest derivations of the
multistage view approximated the model as the product of independent probabilities of
stage transitions, µ1t…µk-1t. Specifying the order of the transitions resulted in the age
distribution of cancer incidence as
I(t)=t k-1(µ1µ2… µk)/(k-1)!
(2-9)
where µi are the transition rates for each stage, a result first proposed by Armitage and
Doll (1954). This form we refer to as the A-D power law model. Although highly
successful in fitting the rising side of cancer incidence data (up to age 74), it is obvious
by inspection that this model cannot fit the turnover, and thus cannot produce the desired
shape. Moreover it became clear that the formula was only an approximation, valid only
for low cancer rates. A mathematically exact form is discussed later.
The clonal expansion model is based on the hypothesis that no more than two
stages were supported by biological evidence, and that a cell need undergo only two
transitions to become malignant, with the first transition conferring a survival advantage
causing exponential growth of the cell by division. The second transition is required in
order to release the cell from control completely and become malignant. Accordingly the
incidence may be approximated as
bt
I(t)=µ1µ2e
(2-10)
where µ1, µ2 are the rates of the two transitions, and b a growth factor, as Armitage and
Doll noted (1957). Although useful for fitting incidence at small values of t, this model,
37
which we can refer to as the simplified A-D clonal expansion form, cannot produce the
age turnover. A more complete derivation results in
I(t) = Nµ1{1-exp[-µ2( eat-1)/b]}
(2-11)
where N is the mean number of cells per person exposed to the first transition. This
incidence function increases monotonically and approaches Nµ1 as t → ∞, thus avoiding
the fate of limitless growth of incidence in the simpler A-D clonal expansion expression,
but it likewise cannot produce a turnover.
By adding additional features to the clonal expansion model: that the number of
cells at risk might be a variable, and that transformed cells have a death rate as well as a
proliferation rate, the incidence may be approximated as
t
I ( t ) ≈ µ1µ 2 ∫ N ( s )e (α 2 − β 2 )(t − s )ds
(2-12)
0
with α2 and β2 growth and death rate of transformed cells respectively, and N(s) is a
variable cell number function. Holding N(s) constant, we integrate to:
I ( t ) ≈ µ1µ 2
(α 2 − β 2 )−1
e
N
(α 2 − β 2 )
(2-13)
which produces a convex monotonically increasing exponential curve if α2 >β2, or a
concave asymptotically limited curve if α2<β2 as t → ∞. This approach, well known as
the simplified MVK model, was developed by Moolgavkar and colleagues (1981) and
has been very successful in modeling many cancers. It is clear the model can produce an
age turnover by applying a suitable function N(s), and specifying that α2<β2 , which
Moolgavkar et al (1981) proposed, but was never applied to model the turnover at old
age. Although quite successful, this simplified form is known to have limitations and
38
therefore the exact form of the two-stage clonal expansion model is currently
recommended, although it requires numerical procedures, with no closed form of solution
readily accessible (Moolgavkar et al 1999).
As shown in Appendix A, the Beta model results from adding a factor to the A-D
power law model representing the probability of a cancer extinction step, which is
modeled as a uniformly distributed random variable over the interval (0, βt). Applying
the same cancer extinction step to the simplified MVK clonal expansion model with
constant parameters might fit the cancer incidence data with turnover as well as the Beta
model, as can be inferred from Figure 2-1. The simplified MVK form includes an
explicit deterministic factor (α2 - β2) modeling the difference between birth and death
rates of initiated cells. Later work on exact models suggest that this deterministic
approach is incorrect since initiated cells appear to have a stochastic character: the
probability of initiated cells being present in the tissue and consequently the probability
of cancer per unit time, is greater than zero even if α2<β2 for long times (Moolgavkar et
al 1999 p.197). This stochasticity assumption is an important part of the exact form of
the MVK model. The Beta model prediction that the probability of cancer per unit time
goes to zero with certainty at t≥β −1 is clearly different.
2.4.2 Age at Peak Incidence: Comparisons to Other Models
The tabulations of age at peak incidence of Tables 2-1, and 2-2, derived from the
Beta fits, also evident in Figure 2-7, are quite uniform for the adult cancers: male 85.0
mean ± 3.7 s.d., and female 84.5 ± 7.1 (the s.d. indicating the standard deviation of the
age at peak incidence over all cancers). The Beta distribution formula for the age at peak
39
incidence, tp=(k-1)/kβ , has no dependence on the cancer creation coefficient α , is only
weakly dependent on the number of stages k, and is almost entirely dependent on the
value of the cancer extinction factor β. Earlier models have not produced this constancy
of tp , and when tested tend to predict a much different result as discussed below.
Armitage and Doll (1954) fitted a power law to age-specific mortality data to age
74, which they assumed was a good representation of age-specific incidence since cancer
victims quickly died, and found I(t) = at k-1 . This fit should be interpreted as a hazard
function,
h( t ) =
f (t )
1 − F (t )
(2-14)
the probability of dying at age t (a probability distribution function, pdf, which is f(t)), is
conditioned on survival to age t (one minus the associated cumulative distribution
function, cdf, which is F(t)); since the victims' death removes them from the denominator
when computing the ratio of cancer deaths to population at risk. We denote age-specific
mortality m(t) to indicate this hazard function. For individual cancer mortality data, the
cumulative probability of death from any specific cancer F(t) by end of normal life is of
order a few percent, and the A-D approximation that the incidence I(t)=m(t)≈ f(t) is quite
good. It is only when considering the turnover in I(t), which must occur to the pdf by the
unitarity criterion (must integrate to one), do we need consider the exact pdf expression
derived from the data fits.
Accordingly we write
I (t ) = m (t ) = at k −1 =
40
f (t )
1 − F (t )
(2-15)
exactly, and note that for small values of cancer cumulative probability F(t), I(t)=at k-1≈
f(t), which is the usual approximation taken, resulting in a pdf that appears to grow
without limit. However to consider age at peak incidence we must consider high values
of F(t), and thus use only the exact hazard function implied by the A-D power law fit:
at k-1= f(t)/[1-F(t)], to derive the exact pdf implied by that power law fit.
Rewriting the hazard function in its differential form and integrating, we obtain
the exact probability of cancer based on the fit as
F (t ) = 1 − e
− ∫ h(t )dt
=1− e
−
at k
k
(2-16)
which clearly gives F(t)= 1 as t→ ∞ , and thus is a cdf (this is so for any h(t)>0).
Differentiating with respect to time, we write the pdf associated with the cdf as:
F (t ) = at k −1e
−
at k
k
(2-17)
Noting that f(0)= 0 and f(∞)= 0 , there is a peak in f(t), which we derive as
1
 k (k − 1)  k
tp = 

 a 
(2-18)
Since the incidence level at any age is determined by a and k, and k ~ 7 for most
cancers, a variation of a factor of 100 in incidence from rare cancers to common cancers
implies a shift in the value in tp of a factor of two, which is a very different result than for
the Beta model and the SEER data.
The exact pdf derived from the A-D power law hazard function fit to age-specific
mortality data, is the unconditional age-specific incidence, which would be measured if
all cancer victims remained in the population. This is based entirely on the assumption
41
that the mortality data is accurately fitted by at k-1. This is clearly not the case above age
75, but its failure to fit at high age does not appear to be due to mathematical
approximations, but due to additional biology not modeled by the power law.
Armitage and Doll (1954) inferred a detailed multistage model for cancer from
their biological interpretation of the power law fit (and other evidence), for which the
power law model is an approximation. Their model can be made mathematically exact
by solving the system of differential equations describing the probability of finding a cell
in each of the stages in its transitions to cancer. Moolgavkar (1978, 1999) found a method
of expressing this exact pdf as a MacLaurin series expansion as
f (t ) =
t k −1µ o µ1 ...µ k −1
1 − µt + f (µ , t )
(k − 1)!
[
]
(2-19)
where µi are the transition rates for each stage, and µ is the mean of the transition rates.
It is important to note that the above expression was derived as a pdf, and the
approximation taken that I(t)≈ f(t), valid for small values of I(t), as was assumed by
Armitage and Doll for the power law model. The reader is carefully alerted by
Moolgavkar to the fact that there is a difference between epidemiological data from
which one infers a hazard function, and theoretical derivations inferring pdf's.
We can immediately note that the first term is the A-D power law as a first order
approximation to the multistage model, or the exact hazard function of the power law fit.
Adding the second term results in an expression that is very similar to the Beta function,
but with constants that are not arbitrary. Since this derivation is based on the exact
multistage pdf, and there is similarity of form to the Beta function, we might investigate
its properties further below.
42
Assuming that the two terms are adequate to test the model for its prediction of
the shift of age at peak incidence with incidence level, we can then derive: tp=(k-1)/(µk).
Since incidence is proportional to the product of the transition rates µi , and if those rates
are all approximately equal such that µ varies with incidence to the k -1 power, and k ~ 7,
it is easy to see that that tp shifts by a factor of two for 100-fold change in peak
incidence. However, this two-term expansion form of the exact multistage model leaves
open the mathematical possibility that large changes in incidence may be produced by
making one or more µi very much smaller than the others. The average of the µi then
becomes constant, while permitting unlimited change in incidence by changing the small
µi , thus making tp constant. Accordingly we cannot rule out this model from also
producing the SEER data fits observed with the Beta function, but observe that all of the
µi for all adult cancer sites (probably in all countries) must conspire to produce exactly
the same average value, within a few percent, while producing peak cancer incidences for
each of those sites that vary over a factor of 100.
The Beta distribution model is, as are the A-D and Moolgavkar models, derived
as a pdf, f(t)=(αt)k-1(1-βt); 0≤ t≤ β -1, with the approximation taken that the pdf is a good
model of the SEER incidence data: I(t)≈ f(t). This is clearly accurate for small values of
cumulative cancer probability F(t), but leads to the question as to whether the SEER data
should be considered a hazard function (all people with that cancer removed from the
denominator) or a pdf (all people with that cancer remain in the denominator) when
modeling high values of F(t) at the turnover age of common cancers (lung, colorectal,
prostate, and breast cancers have cumulative probabilities greater than 10%: see Tables 21 and 2-2). Since the overall mortality rate from cancers in the SEER data is about one43
half of the overall incidence, the SEER data suggests an interpretation about midway
between a hazard function and a pdf, i.e. about halfway between I(t)≈ m(t) and I(t)≈ f(t).
One possibility is to carefully account for the survival fractions for each cancer
for each age group and construct a fit to f(t)=[I(t)][1-M(t)], where M(t) is the cumulative
number of people to die of the cancer. For example, mortality is high for lung cancer,
thus M(t)~F(t), and prostate cancer mortality is low, thus M(t)~0. It should be noted that
for prostate cancer, the high cumulative incidence with low mortality would tend to
increase the fraction with the cancer in the population in the SEER data, thus reducing the
pool without prostate cancer, and causing a turnover in reported incidence as the
cumulative incidence approaches unity. However maximum cumulative incidence is
only 37%, which too far from unity to cause the marked turnover observed, particularly
in the Dutch and California data.
The Beta distribution proves to be very robust when modeling data with uncertain
removal by death, giving the same results in curve shape, quality of curve fit, and
constant tp, with either M(t)=F(t) or M(t)=0 interpretation of the data. This observation
results from writing the exact hazard function bh(t)=b(t)/[1-B(t)], where b(t) is the Beta
distribution, B(t) its integral, and bh(t) the hazard function associated with the Beta
distribution. Then
bh (t ) =
(αt )k −1 (1 − βt ) ;
1 − (at )k (1 − bt )
0 ≤ t ≤ β −1
(2-20)
where a=[α/k1/(k-1)](k-1)/k and b=kβ/(k+1). Since b<β, then bh(t)→0 as b(t)→0, thus
predicting the identical age at zero incidence, which is a critical feature of the model. If
the SEER data is fitted as I(t)= bh(t), then the parameters α, β and (k-1) will change
44
slightly, but produce the same curve shapes with same fit quality, and the same age at
peak incidence. Accordingly, we can conclude that the Beta distribution, b(t) models the
incidence data in a robust way, and is not sensitive to mortality rates for its major
features.
2.4.3 Extrapolation of the Beta Distribution Fit
Since the Beta distribution is a successful fit we venture to extrapolate the
distributions beyond the turn over where data are non-existent for the SEER data or
limited for the Dutch and California data. Figure 2-7 suggests that all adult cancers might
share a uniform characteristic of power law or exponential growth at about the same rate
to about age 70, where the incidences level off and eventually reduce toward zero at age
ca. 100. Accordingly, the Beta fit equation: I(t)=(αt)k-1(1-βt); 0≤ t≤ β-1, with
α=0.01655, (k-1)=5.1, β=0.0098 provides a useful general formula for the age
distribution of any adult cancer as a fraction of its peak value. When considering
absolute values of incidence, the cancer creation coefficient α scales the curve to the
appropriate level.
The extrapolation of the beta distribution yields the interesting parameter which is
the age at predicted zero incidence, which is simply t0 = β −1. This discussion is clearly
more speculative, but if we make the obvious interpretation, after about age 100 cancer
incidence falls to zero. There is general agreement in the literature that cancer is a less
threatening disease for persons living to age near 100 (Smith 1996, Stanta 1997,
Saltzstein 1998), but the Beta prediction that cancer incidence (both the pdf and hazard
function) will fall to zero with probability one is new. These observations suggest a new
45
view that is different from the conventional wisdom, which was largely based on the
historically important models described above: that cancer probability continues to
increase with age until it reaches certainty.
2.2.4 Cumulative Cancer Probability
Tables 2-1 and 2-2 show the cumulative probabilities, calculated from the beta
distribution fit I(t)=b(t), of each cancer and all cancers over a lifespan (defined as
surviving to age>β −1). For males, these range from 0.3% to 31% for individual cancers,
and 70% for at least one cancer of any type. For females the range is 0.2% to 20% for
individual cancers, and 53% for any cancer. The simple and obvious conclusion is
contrary to the common understanding that: "if a person lives long enough he or she will
get cancer," which is a result of the historical success of the simple power law and clonal
expansion models, both of which imply a rising probability that always reaches unity at
large enough values of t.i The data, as interpreted with the Beta model, suggest that "if a
person lives long enough, he or she may avoid cancer entirely," with about a one in three
chance for men and an even chance for women.
2.4.5 Modeling Susceptibility and Sensitivity
Both the multistage and clonal expansion hazard function models, whether
approximations or exact, have the characteristic that the pdf of any cancer integrates to
one over sufficiently long time {∫f(t)dt=F(t)=1-exp[-∫h(t)dt]=1, integration limits
0→∞ ; valid for any positive function h(t)≠ 0 as t→∞}. As the data (including the
i
This applies also to the mathematically more exact versions of the multistage model.
46
extrapolation with the Beta function) indicate, however, the cancer incidence yields
cumulative probabilities much lower than one, and range over a factor of 100, while
maintaining similar curve shape. One simple and obvious assumption is that only a
fraction C of the population is susceptible, which leads to modification of the Beta
function model as
b(t)≈ I(t)=C(γ t)k-1(1-βt)
(2-21)
where
C=∫(αt)k-1(1-βt)dt, and ∫(γ t)k-1(1-βt)dt = 1 ; 0≤ t≤ β -1
Numerically, the susceptibles fraction is identical to the cumulative probability over
lifespan tabulated in Tables 2-1 and 2-2. Inherent in this interpretation is that the fraction
of susceptible people is different from one cancer site to another by about a factor of 100.
Cook et al (1969) added a limited pool of susceptibles expression to the A-D
power law, which produced a turnover, but they found that the location of the age of peak
incidence varied markedly with the incidence level. Since the data did not support
variation in the age at peak incidence, they deemed this hypothesis unsupported.
Herrero-Jimenez et al (1998, 2000) employed a biologically detailed modified clonal
expansion model to examine colon cancer mortality turnover, which might avoid the
Cook problem with their hypothesis of an exposure factor in addition to a susceptibles
factor. This allows the shape to be held constant with one factor, with the level
controlled by the other factor, but relies on the same assumption: turnover occurs because
we "run out of candidates" beyond about age 80.
Finkel (1995) raises the importance of distribution in susceptibility in risk
assessment and formulates an interesting analytical method of modeling susceptibility by
47
combining a lognormal distribution assumption with a modified Armitage-Doll cancer
model. The basic idea is that there is an asymmetric distribution in susceptibility of
individuals, which allows for a long "tail" in the distribution for the least susceptible. He
shows that including this distribution assumption causes the modeled age-specific
incidence to plateau at an elevated age, thus improving the fit to colon cancer mortality
data compared to an unmodified A-D model. The biological basis for a distribution in
susceptibility is certainly plausible, given the heterogeneity in genetic, environmental,
and life style influences on cancers. Finkel clearly supports the idea that the data
indicating flattening at old age is not artifactual, but like the Cook and Herrero-Jimenez
models, the flattening occurs in his model because the susceptibles pool is being
depleted. However, Finkel's model cannot predict an actual decrease in cancer incidence
until the cumulative incidence approaches unity, which no individual cancer approaches.
The weight of the evidence seems to argue against a distribution of susceptibles
view, since the distribution would have to be quite similar for each of the 35 adult
cancers to peak at close to the same age over an incidence range of a factor of 100. This
suggests a biological mechanism which is uniform in its genetic or environmental
influence, opposite to the Finkel view that requires heterogeneity. Additionally, in a
study performed in parallel with this work, Pompei et al (2001) included as Chapter 3,
analyzed animal data for a single species of inbred mice living their lives in a controlled
uniform environment, which show similar curve shapes and turnover in cancer incidence.
Interestingly the turnover for mice also occurred at about 80% of the lifespan. These new
observations clearly tend to weaken the susceptibles view based on heterogeneity, and
strengthen the view that a biological process, yet to be modeled, must be considered.
48
A further consideration for the susceptibles hypothesis is data on persons heavily
exposed occupationally. In one well-documented case, exposure to β-naphthylamine, 15
out of 15 persons exposed developed bladder cancer (Case et al 1966). ) These data
suggest that everyone is susceptible to cancer if the dose is suitably high, although
variations in basic sensitivity by a factor of 3 are possible, as discussed by Finkel. The
Beta model does not preclude rates of cancer approaching unity at high doses, but does
require that they occur at younger ages, where the age-related slowing of the cancer
process suggested by the (1-βt) term is less important.ii The Case et al paper is
instructive on this point, showing convincing data that chemical workers with high
bladder cancer rates contracted the cancer at ages 20 years younger than the general
population contracted the same cancer. This observation suggests that susceptibility and
exposure might be a valid method of modeling the rising side (and perhaps flattening, as
Finkel suggested) of the age distribution. But at age greater than about 80% of lifespan,
the SEER, Dutch, Hong Kong and California data suggest a uniform cancer extinction
biology may dominate.
Decreased stage sensitivity at old age might produce both a flattening of the
cancer incidence and a turnover if sensitivity approaches zero. Consider the A-D
multistage model for a relatively uncommon cancer such as liver cancer (cumulative
lifespan incidence ~ 1%), where the A-D approximation is accurate: I(t) = at k-1. If the
stage probabilities are not equal and constant, the A-D model becomes
I(t)= (p1t)(p2t)... ( pk-1t)pk /(k-1)!
(2-22)
Thus it is not in contradiction to the observation that 100% of the highly exposed βnaphthylamine workers developed cancer.
ii
49
where (pit) are the transition probabilities for each stage. Since this model
includes the requirement that the stages (1,2,…,k) must occur in order, then it is the later
stages that are of interest, since the earlier stages have already occurred if they are going
to, and a change in early stage probability at larger t will not alter the overall probability
(Armitage and Doll 1954). By inspection, it is clear that if pk-1 or pk approaches value 0
at some time t approaching age ~100 years, then I(t)→ 0, and thus produces a turnover.
Thus if a decrease in sensitivity is interpreted as a reduction in probability of a late stage
transition, a flattening will be produced, followed by turnover as the probability of the
late stage approaches zero. A similar argument can be made for the clonal expansion
models. Accordingly, the Beta model cancer extinction factor might be equally
interpreted as a linear sensitivity decrease with age of a late stage: pk = µk(1-ct), with no
loss of generality or goodness of fit.
Other forms for this sensitivity reduction factor, such as e-ct might appear also to
work adequately (since the first two terms of its expansion are also (1-ct)). This produces
a Gamma function form (αt)k-1e-ct when combined with the A-D power law, and avoids
the slight mathematical discomfort of negative incidence when t>β−1 with the Beta form.
However the fit is not nearly as satisfactory as the Beta function form, and the values of
the constants become seemingly unrealistic when as good a fit as possible is forced. For
example, (k-1) is about 5 for the Beta, but is about 15 for the Gamma to fit the SEER
data, suggesting an unrealistically large number of stages. Also the extinction
coefficient, β, has to be much larger in the Gamma, and seriously distorts the fit at low
values of age. We conclude that the Beta form, though having an abrupt limit at t=β−1,
nonetheless is what the data suggests. The exponential form of extinction factor might,
50
however, work well if applied to a different cancer creation model, particularly an exact
form, but this has not yet been explored.
2.4.6 Biological Hypotheses
The fact that the Beta distribution fits the data well, and that the multistage and
clonal expansion models appear not to do so, even when made mathematically exact,
suggests that we call the Beta distribution a "model" and enquire about its possible
biological plausibility. The Beta model suggests a very different cause of the turnover:
the active involvement of a cancer extinction step such that the probability of a cancerous
cell survival (or proliferative ability) approaches zero at ages corresponding to
approximately a human lifespan. The model can be derived from the first order
multistage model (power law fit hazard function) by multiplying by a "cancer extinction
factor" (1-βt ) by which the transformed cells are eventually destroyed or deactivated at a
rate greater than their creation (derived in Appendix A). However, it might be incorrect
to interpret the constant β in the fits as this factor, since an exact multistage or clonal
expansion model with cancer extinction might have a somewhat different formulation.
The Beta model applied to the biology can be viewed as a simple combination of
two factors: (1) cancer creation, which is most simply modeled with a power law
multistage assumption, although it would fit equally well with an exponential clonal
expansion assumption or most any other rapidly increasing function; and (2) cancer
extinction, which is modeled as a cumulative probability that linearly increases to
certainty at age ~100. The first factor may be interpreted in the same way as all of the
relevant historical models: caused by mutations and promotion steps from genetic,
51
environmental, etc. exposures. The extinction factor is new, and its biology must be
carefully considered.
Commonly accepted, but not entirely understood, inexorable changes due to
ageing might lead to clues. As a first possibility, apoptosis is a candidate for the
mechanism of "cancer extinction". Although we added this term by assuming a process
that is uniform with age, an age dependence might also be included. In vitro human cell
studies by Schindowski et al (2000), Lechner et al (1996), Potestio et al (1998) found
that apoptosis increases with age due to reduced defense from oxidative attack. Higami
et al (2000) suggest apoptosis increases in vivo with level of accumulated injury related
to ageing. Ogawa et al (2000) found apoptosis rates low in the young, and increased in
the old from bone marrow samples from newborns to age 100. Lee et al (2000) found
that rat colon epithelial cells were more sensitive to apoptosis stimulation with advancing
age. It is likely, however, that apoptosis is a small effect compared to senescence, as
discussed below and in more detail in Chapter 4.
A second possibility is cell senescence, or loss of proliferative ability, which may
be interpreted as a loss of sensitivity. This point has been discussed by Faragher (1998,
2000) who suggest that cell senescence, like apoptosis, occurs as an anti-cancer
mechanism, and that a large body of evidence suggests cell senescence contributes to a
variety of pathological changes seen in the aged. Hayflick (2000), Jennings et al (2000),
Oloffson et al (1999), and Rubelj et al (1999) all suggest cell senescence or the related
observation of telomere shortening increases with age, and thus may profoundly
influence the cancer process. Rubelj further raises the interesting possibility that
52
telomeres may shorten abruptly by a stochastic process, thus producing senescence in
some cells even at young age.
If the probability of abrupt shortening were uniform with time, this mechanism
could be modeled exactly as causing cell senescence with probability of βt, and thus the
cancer extinction factor becomes the (1- βt) proposed in the Beta model. Such a process
is suggested by observations beginning decades ago showing that replicative ability of
cells markedly decreases as they age, which ultimately defines senescence (Hart et al
1976, 1979). Further, the loss of replicative ability appears to reduce approximately
linearly with age, thus suggesting a factor such as (1- βt). This approach is proposed and
discussed in Pompei et al (2001).
2.4.7 Data Reliability
The conclusions of this work rely critically on the assumption that the modern
cancer registry data is truly representative of cancer incidence, particularly above age 80.
The concern first expressed by Armitage and Doll (1954), that less extensive workups are
performed in diagnosing cancer for older persons than for younger, cannot be completely
dismissed. However, there are accumulating evidence via autopsy of the oldest old that
cancer prevalence indeed reduces with increasing age. The Stanta autopsy study
mentioned earlier, which included 507 people who died between the ages of 75 and 106,
was designed to investigate this very question. The authors state "… our autoptic
population may be considered representative of the general population" and "We
discovered a cancer in 36% of the people between 75 and 90 years of age, but only in
22% of those over 95, and in 16% of the centenarians." The details of the histologic
53
examinations were not reported. However the authors do find substantial and increasing
underreporting of cancer with age when the autopsy results are compared to the original
clinical diagnoses. Imaida et al (1997) studied autopsies of 871 patients aged 48 to 113
at death and also found that prevalence of malignancies reduce at the older ages, but also
found increasing prevalence of latent cancers with age, latent cancers defined as those
visible at autopsy but not diagnosed clinically.
The de Rijke study reported histologically or cytologically confirmed cancer
diagnoses in 98% of males and 97% of females in the 55-64 age group, and 87% and
84% for those in the ≥95 age group, suggesting the possibility of a reduction in
thoroughness for the older ages. Referring to Figure 1, we see that without some
important effect(s) at age > ~75 (these effects might be depletion of susceptibles,
increased apoptosis, increased senescence, slowing of proliferation, and/or
underreporting of cancers), cancer incidence is expected to continue to increase strongly,
by the historical paradigm, until the cumulative incidence reaches unity. At about age
75, 10-15% deficit in cancer diagnosis might be sufficient to account for the deficit
between expected and observed incidence. At age ≥ 95, however, deficits of a factor of
two or more from the aforementioned effects are required to reduce incidence to the
observed values. Referring to figures 4 and 5 for the Dutch and California data for
individual cancers, we observe that the deficit in incidence, compared to even a
conservative straight line trajectory extended from the 60, 70, 80 year-old incidence rates,
is a factor of two or much more by age 95.
SEER have not specifically addressed the issue of the reliability of the cancer
incidence reporting for the oldest age groups, but believe the data is at least as reliable as
54
that reported by other countries (Ries 2001 personal communication). SEER themselves
seem to accept the data are reliable enough to describe a turnover, and further have
observed, "Whatever is occurring, fortunately cancer is not inevitable for all older
persons" (Yancik and Ries 1995). We have also not considered other possible influences
such as altered diet, lifestyle or environment for the oldest, which may tend to reduce
cancers by mechanisms other than age, which suggests further study. Although we are
not yet able to rule out concern about cancer reporting (or lifestyle effects) in the oldest,
the weight of the evidence, including the previously discussed mice data, is tending to
support the validity of the reported data.
55
Chapter 3
Age Distribution of Cancer in Mice:
The Incidence Turnover at Old Age
This chapter studies cancer incidence in mice as a function of age in those cohorts
where the rodents are allowed to live very close to their full natural lifetime. The data
shows that the incidence rises as a function of age, but then flattens and turns over at an
age of about 800 days. This behavior is similar to that which observed in Pompei and
Wilson (2001) and Chapter 2, in the SEER data where the age distribution of human
cancer incidence turns over at about age 80. Although other fits are possible, the 3
parameter Beta function model fits both the mouse data and the human data well. The
Beta model implies, and the data do not deny, the interpretation that cancer is not a
certainty, and mice may also outlive their cancers, although high-dose cohort results
suggest cancer might be certain if dose is sufficiently high. Limited data suggest that the
cancer age distribution, including the turnover, may be time-shifted by dietary restriction.
56
3.1 Introduction
In Chapter 2, data on the incidence of cancer in humans as a function of age is
examined in detail. These are primarily the Surveillance, Epidemiology, and End Results
(SEER) data of the National Cancer Institute (Ries et al. 2000) but also two cohorts from
Hong Kong (Parkin et al 1997) and from the Netherlands (de Rijke et al 2000). The
common feature is that the incidence increases steadily with age up to age 70, appears to
flatten off for all cancers at about age 80, and falls thereafter. The age of maximum
incidence is remarkably consistent for all adult cancers, considering incidences vary over
two orders of magnitude. Two well known theories of cancer, the multistage model
developed by Armitage and Doll (1954) and Armitage (1985) and the 2-stage clonal
expansion model discussed by Armitage and Doll (1957) and in more detail by
Moolgavkar (1978) and Moolgavkar and Knudsen (1981), can easily be modified to
allow a flattening of the age-incidence curve but cannot be easily modified to allow a turn
over at high ages. The data, taken at face value, imply an interpretation different from
the traditional view that “if you don’t die of anything else you will die of cancer” may not
be accurate and might be replaced by “if you live to 90, you will have beaten cancer”.
One explanation of the turnover data observed at old age was suggested by Sir
Richard Doll (2001) who emphasized that at older ages, records of cancer are less
reliable, since attending physicians often used the nebulous cause of death “old age”.
This, indeed was a stated reason that in their seminal work Armitage and Doll (1954)
stopped their analysis at age 74. While fifty years later the collectors of the SEER data
claim that their data are more reliable, other verifications of the turnover seem highly
57
desirable. In support of his suggested explanation, Sir Richard pointed out that all or
most members of several cohorts of persons occupationally exposed to high levels of
certain pollutants died of cancer (e.g. β-naphthylamine, Case et al 1966). Noting that
pathologists who examine the data on cancers in rodents may have biases, they are
unlikely to have Doll’s suggested bias. Therefore a process of examining all data sets of
cancers in animals to see whether an age turnover of incidence is present is appropriate.
In many bioassays, including most of the bioassays of the National Toxicology
Program (NTP), animals are killed before the end of the full normal lifetime in a
“terminal sacrifice” which makes most of these bioassays unsuited for this study. A few,
however, remain. More importantly, Dr. R.L. Kodell of National Center for
Toxicological Research kindly made available the original data of the ED01 study where
24,000 mice were exposed to various amounts of 2-acetylaminoflourene (2-AAF). This
chapter shows the age turnover in the incidence rate of cancers, particularly fatal cancers,
in that bioassay. This turnover tends to substantiate the idea that the age turnover in
people is unlikely to be an artifact as suggested by Doll. As pointed out in Chapter 2 for
human cancer turnover, these conclusions depend critically on the reliability of the data at
elevated age. It should also be noted that neither of these animal studies were designed to
examine the turnover in cancer incidence with age, and thus might be subject to unknown
biases.
58
3.2 Methods
3.2.1 Data Sources
Initial evaluation of cancer incidence turnover in rodents was performed with the
Toxicology Data Management System (TDMS) database, obtained from National
Toxicology Program (NTP) contractor Analytical Sciences, Inc. The files contain data
for mice and rat bioassays published in technical reports TR-341 through TR-491, which
were issued from September 1989 to July 1999. The data was subsequently edited to
remove results from dietary restriction studies, which affect the age distribution of tumors
differently than ad libitum feeding of all other studies (Haseman 2001). The database's
control animals were searched for high prevalence tumor rates for animals classified as
"natural death" or "moribund sacrifice" for each 100-day age interval. The tumor rates
were calculated as the number of animals with a specific tumor divided by the total
animals dying by natural death in each age group. Since nearly all ad libitum studies
were terminated at 2 years (the exception being TR-440: Ozone), only data for <800 days
age was included.
A separate analysis of the dietary restricted data for Scopolamine Hydrobromide
Trihydrate (TR-460, TMDS TR-445 Study No. 0512108) controls was conducted,
following the same method as above. These animals were fed a restricted diet to
maintain 85% of the weight of the ad libitum group.
Since the TMDS database lacked statistical power, particularly for older animals
in any one study for direct age-specific incidence measurement, a dataset for a much
larger single study, the ED01 study, (Cairns 1980) was obtained from FDA National
Center for Toxicology Research (courtesy RL Kodell) and used for the bulk of the data in
59
this work. Designed to detect the effective dose of 2-acetylaminoflourene (2-AAF)
required to produce 1% tumor rate, the original study included 24,192 female BALB/c
mice, and 23,419 were included in the database we obtained.
Importantly, the ED01 data included cause of death from neoplasms by type of
neoplasm, as a pathology entry. This made it possible to produce an objective
examination of age-specific mortality caused by each type of cancer, which was
conventionally calculated as
M(t) = % mortality/100 animal-days
= 100*(No. of animals dying of tumor in the 100 day period)/
(No. of animal-days at risk in the 100 day period).
(3-1)
Additionally, the survival rate for the animals were such that the ED01 study was
extended to 33 months, compared to the more typical 24 months, thus producing
significant data for older animals. The data were searched both for deaths caused by
neoplasms, and for morbidity caused by neoplasms. It was noted that for the groups of
dose = 30, 35, 60 ppm, large numbers of animals were apparently misclassified as dead or
moribund from neoplasms about a month before final termination at day 1001, but ought
to have been classified as terminally sacrificed. These data are assumed to be in error,
and not included in natural death or morbidity from neoplasms. All dosed animals
considered were dosed continuously over their lifetimes.
3.2.2 Analytical Methods
The method of developing age-specific mortality rate data was designed to
emulate as precisely as possible the method of obtaining age-specific mortality in
60
humans: natural deaths were tabulated with cause of death as determined by pathology.
In the ED01 experiments, animal cages were examined twice daily, and those that died
were immediately removed to be autopsied to establish cause of death. Moribund
animals were treated in the same manner as dead animals, and listed as a separate
removal category. Since the animals were obviously not treated for cancer, age-specific
cancer mortality is believed to be a good estimate of age-specific cancer incidence, as it
was in human cancer studies (Armitage and Doll 1954) before development of our
current successful interventions for many human cancers.
Independent estimates of age-specific tumor incidence were developed with
cumulative incidence data for spontaneous neoplasms in untreated groups for the ED01
study published by Sheldon et al (1980). Animals were examined at the scheduled
terminal sacrifice periods, and those dying of natural causes were combined with the
nearest scheduled sacrifice group. The age-specific incidence was calculated as
I(t) = [(Cumulative incidence % for the group sacrificed at age t2) (Cumulative incidence % for the group sacrificed at age t1)] / [t2 - t1]
(3-2)
where t is the average age at death. The cumulative incidence is defined as the
proportion of animals with the tumor in the sacrifice group at age t1, t2, … tn. This
method of calculation assumes that each group of mice is identical to the others, which is
a fundamental result of the randomization of the animals and uniformity of facilities of
the ED01 study design. This implies that cumulative incidence is different between age
groups only due to age and random effects.
Statistical analysis of the turnover for the ED01 data was performed by comparing
the mean of the incidence for a given age group to the mean of the incidence for the
61
immediately younger age group with a standard two-sample z-test. By comparing the
age-specific incidence for a given age group to the age-specific incidence of a similar
group at younger age, the test provides a method of evaluating the key elements of the
age distribution statistically: are there tumor sites for which incidences clearly increase at
middle age, and also clearly decrease at older age at the same sites? The test statistic is:
z = [(Observed age-specific incidence of group in age range 2)
- (Observed age-specific incidence of group in age range 1)]
/√[(St. dev. of group in age range 1)2 + (St. dev. of group in age range 2)2] (3-3)
The null hypothesis is that there is no difference in incidence between the two ages, with
the alternate hypothesis that incidence increases or decreases as age increases, employing
the one-tailed p-value as the test of significance. Significance is accepted at the p<0.05
level.
Dividing the age-specific mortality data into 4 age groups for statistical analysis:
200-400 day, 400-600 day, 600-800 day, and 800-1001 day group, the test produces
results of the form: M(400-600)>M(200-400); p=0.01, denoting that mortality due to
tumors in the older age group exceeds that in the younger group, with probability 0.01
that the result was random. The notation M(t) is used to denote age-specific mortality,
I(t) denotes age-specific incidence, and CI(t) denotes cumulative incidence, throughout.
Error bars indicate ± 1 SEM throughout.
3.2.3 Beta Model
62
The Beta model for fitting the mice data is the same as used for age distribution of
cancer data in humans including the turnover at ages > 80. The derivation is presented in
Appendix A
Since mortality removes the animals with tumors from the population at risk, the
age-specific mortality data is properly interpreted as a hazard function, with the general
form h(t) = f(t)/[1-F(t)], where f(t) is the probability density function (pdf) of the
modeled (assumed) cancer-causing mechanism, and F(t) its time integral. Using the Beta
function as derived in the Appendix, b(t)= (αt)k-1(1-βt) as the pdf, we write the exact
hazard function associated with the Beta pdf as bh(t)=b(t)/[1-B(t)], where B(t) is the
integral of b(t). Then bh(t)= [(αt)k-1(1-βt)]/[1−(at)k(1-bt)] ; 0≤ t≤ β -1 , where
a=[α/k1/(k-1)](k-1)/k and b=kβ/(k+1). Since b<β, then bh(t)→0 as b(t)→0, thus predicting
the identical age at zero incidence for both the hazard function and pdf, which is a critical
feature of the Beta model. Further, it can be shown that the shapes of the exact hazard
function bh(t) and the Beta function pdf b(t) are the same, thus providing robustness when
there is uncertainty to the composition of the animals at risk.
To compute lifetime cumulative probability of death or morbidity by cancer, the
time integral of the implied pdf f(t) from the M(t) hazard function data (not the Beta pdf)
was evaluated as: F(t) = ∫ f(t)dt = 1-exp{-∑M(t)}, where the age-specific mortality are
summed over assumed lifetime of 1001 days.
3.3 Results
Of 2093 tabulated combined male and female B6C3F1 mice TDMS controls
removed for natural death or moribund sacrifice at age <800 days, 621 mice had
63
hepatocellular carcinoma and 432 had hepatocellular adenoma, the most prevalent
neoplasms. The age distribution results are given in Figures 3-1(a,b).
Percent with Tumor
Liver Hepatocellular Carcinoma Rate
in B6C3F1 Mice Controls:
Natural Death or Moribund Sacrifice
100
90
80
70
60
50
40
30
20
10
0
a
All TDMS (Ad Libitum) Controls
Dietary Restricted (Scopolamine study)
3rd order polynomial fit to data points
0
200
400
600
800
1000
Age at Death
Liver Hepatocellular Adenoma Rate
in B6C3F1 Mice Controls:
Natural Death or Moribund Sacrifice
100
90
Percent with Tumor
b
All TDMS (Ad Libitum) Controls
Dietary Restricted (Scopolamine study)
3rd order polynomial fit to data points
80
70
60
50
40
30
20
10
0
0
200
400
600
800
1000
Age at Death
Figure 3-1(a, b). Liver tumor rates for all TDMS ad libitum controls for mice removed
for natural death or morbidity (solid symbols), and dietary restricted mice tumor rates of
the TDMS scopolamine study controls (open symbols). A least-squares polynomial
curve fit (a0+a1t+a2t2+a3t3) of the data points is fitted to each data set, for comparison
purposes. Vertical dashed line indicates terminal sacrifice for ad libitum mice.
It must be noted that we don't know when the cancers actually occurred, only that
they occurred before the time of death. If it is assumed that the neoplasms did not cause
the natural deaths (as specifically stated in TR-421), the natural death data might
64
represent cumulative incidence of the neoplasm. In principle, the time derivative of the
cumulative incidence curve would yield the desired age-specific incidence, which clearly
tends to zero as implied by the flattening observed. However, in other studies the
neoplasms were a direct cause of removal for morbidity (e.g. TR-390), or the effect of
neoplasms on death or morbidity was uncertain (e.g. TR-391), which suggests that the
results of Figure 3-1 are an uncertain mixture of hazard function and cumulative
incidence, thus making interpretation and modeling via the Beta function equivocal. For
purposes of interpretation of the general trends of the data, a least-squares polynomial fit
of the data points is shown.
For the liver neoplasms, it seems clear that tumor rates as defined for autopsied
mice dying from natural causes appear to level off in the range of about 500 to 800 days.
The polynomial fits leave some impression of a turnover, but without data at elevated age
and clearer definition of the death process relation to tumor incidence, it is difficult to
place significant further weight on this evidence.
The dietary restriction data, which included 51 animals, 8 with liver carcinoma
and 8 with liver adenoma, is also plotted in Figures 3-1. The data and curve fit are
suggestive that the main effect of such restriction is to shift (or possibly stretch) the time
scale, without appreciably influencing the peak value of the cancer rates.
Figures 3-2(a-c) and 3-3(a-p) present the age-specific mortality of the ED01 mice
study. In contrast to the TDMS data, natural deaths were known to be caused by the
neoplasms, ranging from 84% to 92% for controls and dosed cohorts (Kodell et al 1980), and
the pathology data identified each animal for which death or morbidity was caused by
neoplasm. The reticulum cell sarcoma, lymphoma, and lung alveoli tumor mortality
65
RCSTY_B: Reticulum Cell Sarcomas
% per 100 animal-days at risk
8
Age-specific mortality M(t)
7
Beta model fit to M(t)
6
Age-specific incidence I(t) (from Sheldon)
5
M(400-600) > M(200-400); p=5E-8
M(600-800) > M(400-600); p<1E-10
M(800-1001) < M(600-800); p<1E-10
4
3
2
1
a
0
0
200
400
600
800
1000
Age (days)
Lymphomas
4.0
Age-specific mortality M(t)
Beta model fit to M(t)
Age-specific incidence I(t) (from Sheldon)
% per 100 animal-days at risk
3.5
3.0
2.5
M(400-600) > M(200-400); p=0.01
M(600-800) > M(400-600); p=0.0001
M(800-1001) < M(600-800); p<1E-10
2.0
1.5
1.0
0.5
b
0.0
0
200
400
600
800
1000
Age (days)
Lung Alveoli Tumors
Age-specific mortality M(t)
Beta model fit to M(t)
Age-specific incidence I(t) (from Sheldon)
% per 100 animal-days at risk
5.0
4.5
4.0
3.5
M(400-600) > M(200-400); p=0.0004
M(600-800) > M(400-600); p=2E-10
M(800-1001) < M(600-800); p=0.0004
3.0
2.5
2.0
1.5
1.0
0.5
c
0.0
0
200
400
600
800
1000
Age (days)
Figures 3-2(a-c). Age-specific mortality (including morbidity) caused by the three most
common causes of death by neoplasm for ED01 undosed control animals and data fit by
the Beta model. Tests of significant changes show in all cases that the oldest age group
(800-1001 days) has significantly lower age-specific mortality than the 600-800 days
group, which in turn has significantly higher age-specific mortality than both the 400-600
and the 200-400 days groups. Calculated age-specific incidence for the same tumor sites
from data by Sheldon et al (1980) are shown for comparison.
66
data of Figure 3-2 show age-specific mortality falling to near zero at age between 900
and 1000 days, as predicted by the Beta model fit shown for reference. In all cases the
turnover is highly significant (p<<0.05). The age-specific incidence data estimated from
Sheldon et al (1980) data show turnover at about the same age as the mortality data,
confirming that incidence as well as mortality tends toward zero. The unusually high
mortality by lymphomas at a young age (Figure 3-2c) was reported by Sheldon, but no
reason was given for this result.
The age-specific mortality for all sites combined for all dose groups of Figures 33(a-p) show good agreement with the general shape of the Beta model shown for
reference, for all doses except 75, 100 and 150 ppm, where dose related effects become
dominant. The Beta curve shown for reference is the fit for the dose=0 cohort, providing
a graphical indication of the effect of dose on the age distribution of cancer mortality.
The data for death by neoplasms, and death or morbidity by neoplasms is shown
separately, confirming that the age distribution of morbidity is very similar in shape to
the mortality. In all cases except the high doses, the turnover is statistically significant
(p<0.05). It is worth noting that though no turnover is evident for the high doses, there
were at least some mice surviving to 1001 days without dying of cancer. Of interest is
the observation that cancer mortality and morbidity are lower for the 30 ppm dose cohort
than for undosed controls at all ages.
Table 3-1 tabulates the cumulative lifetime mortality caused by neoplasms for
each of the ED01 cohorts examined. Except for the high dose groups, age-specific
mortality and morbidity falls well short of certainty, suggesting that incidence also falls
well short of certainty.
67
Age-Specific Mortality for Dose = 0:
Death Caused by Neoplasms
M(400-600) > M(200-400); p=2E-5
M(600-800) > M(400-600); p=1E-5
M(800-1001) < M(600-800); p=4E-6
2.5
a
Age-specific mortality %
Beta model fit
2
1.5
1
0.5
0
0
200
400
600
Age (days)
800
Percent of Population at Risk (per
100 days)
Percent of Population at Risk
(per 100 days)
3
Age-Specific Mortality for Dose = 0:
Death or Moribidity Caused by Neoplasms
12
8
Dead or Moribund
Beta model fit
6
4
2
0
0
1000
Animals Dead
Beta model fit for dose=0
1.5
1
0.5
0
0
200
400
600
800
Percent of Population at Risk (pe
100 days)
Percent of Population at Risk (per
100 days)
2
12
10
8
Animals Dead or Moribund
1000
d
Beta model fit for Dose=0
6
4
2
0
1000
0
200
400
600
800
1000
e
Animals Dead
Beta model fit for dose=0
1
Age-Specific Mortality for Dose = 35 ppm:
Death or Moribidity Caused by Neoplasms
Percent of Population at Risk
(per 100 days)
Percent of Population at Risk
(per 100 days)
M(400-600) > M(200-400); p=0.006
M(600-800) > M(400-600); p<1E-10
M(800-900) < M(600-800); p=0.02
2
800
Age (days)
Age-Specific Mortality for Dose = 35 ppm:
Death Caused by Neoplasms
3
600
M(400-600) > M(200-400); p<1E-10
M(600-800) > M(400-600); p<1E-10
M(800-900) < M(600-800); p=0.0001
Age (days)
4
400
Age-Specific Mortality for Dose = 30 ppm:
Death or Moribidity Caused by Neoplasms
c
M(400-600) > M(200-400); p=4E-9
M(600-800) > M(400-600); p<1E-10
M(800-900) < M(600-800); p=0.04
2.5
200
Age (days)
Age-Specific Mortality for Dose = 30 ppm:
Death Caused by Neoplasms
3
b
M(400-600) > M(200-400); p<1E-10
M(600-800) > M(400-600); p<1E-10
M(800-1001) < M(600-800); p<1E-10
10
f
M(400-600) > M(200-400); p<1E-10
M(600-800) > M(400-600); p<1E-10
M(800-900) < M(600-800); p=0.0003
14
12
10
Animals Dead or Moribund
Beta model fit at Dose=0
8
6
4
2
0
0
0
200
400
600
800
0
1000
200
400
600
Age (days)
Age (days)
Figure 3-3. continued
68
800
1000
3.5
M(400-600) > M(200-400); p=0.0001
M(600-800) > M(400-600); p=2E-9
M(800-1001) < M(600-800); p=9E-6
3
Age-Specific Mortality for Dose = 45 ppm:
Death or Moribidity Caused by Neoplasms
g
14
Percent of Population at Risk (per
100 days)
Percent of Population at Risk (per
100 days
Age-Specific Mortality for Dose = 45 ppm:
Death Caused by Neoplasms
2.5
Animals Dead
2
Beta model fit for Dose=0
1.5
1
0.5
0
0
200
400
600
800
10
1000
Animals Dead or Moribund
Beta model fit for dose=0
8
6
4
2
0
0
200
Age (days)
2.5
i
Beta model fit for dose=0
1.5
1
0.5
0
1000
j
M(400-600) > M(200-400); p<1E-10
M(600-800) > M(400-600); p<1E-10
M(800-900) < M(600-800); p=2E-8
12
10
Animals Dead or Moribund
8
Beta model fit for dose=0
6
4
2
0
0
200
400
600
800
1000
0
200
Age (days)
40
30
k
Percent of Population at Risk (per
100 days)
M(400-600) > M(200-400); p=0.0006
M(600-800) > M(400-600); p=2E-7
M(800-1001) > M(600-800); p=0.01
35
Animals Dead
25
Beta model fit for dose=0
20
15
10
5
0
0
200
400
600
400
600
800
1000
Age (days)
Age-Specific Mortality for Dose = 75 ppm:
Death Caused by Neoplasms
Percent of Population at Risk (per
100 days
800
14
Animals Dead
2
600
Age-Specific Mortality for Dose = 60 ppm:
Death or Moribidity Caused by Neoplasms
Percent of Population at Risk (per
100 days)
Percent of Population at Risk (per
100 days
M(400-600) > M(200-400); p=0.0007
M(600-800) > M(400-600); p=1E-6
M(800-1001) < M(600-800); p=0.001
3
400
Age (days)
Age-Specific Mortality for Dose = 60 ppm:
Death Caused by Neoplasms
3.5
h
M(400-600) > M(200-400); p<1E-10
M(600-800) > M(400-600); p<1E-10
M(800-1001) < M(600-800); p<1E-10
12
800
1000
Age-Specific Mortality for Dose = 75 ppm:
Death or Moribidity Caused by Neoplasms
M(400-600) > M(200-400); p<1E-10
90
M(600-800) > M(400-600); p<1E-10
80
M(800-1001) > M(600-800); p<1E-10
70
60
Animals Dead or Moribund
50
Beta model fit for dose=0
40
30
20
10
0
0
200
400
600
Age (days)
Age (days)
Figure 3-3. continued
69
l
800
1000
40
70
M(400-600) > M(200-400); p=0.0002
M(600-800) > M(400-600); p=9E-5
M(800-900) > M(600-800); p=0.02
M(900-1001) < M(800-900); p=0.15
35
30
25
Animals Dead
20
Beta model fit for dose=0
15
10
5
0
0
200
400
600
Age-Specific Mortality for Dose = 100:
Death or Moribidity Caused by Neoplasms
m
Percent of Population at Risk (per
100 days)
Percent of Population at Risk (per
100 days)
Age-Specific Mortality for Dose = 100 ppm:
Death Caused by Neoplasms
800
1000
M(400-600) > M(200-400); p<1E-10
M(600-800) > M(400-600); p<1E-10
M(800-900) > M(600-800); p=0.002
M(900-1001) < M(800-900); p=0.08
60
50
40
Animals Dead or Moribund
Beta model fit at Dose=0
30
20
10
0
0
200
Age (days)
100
Percent of Population at Risk (per
100 days)
Percent of Population at Risk (per
100 days)
40
35
30
Animals Dead
25
Beta model fit for dose=0
600
800
Age-Specific Mortality for Dose = 150:
Death or Moribidity Caused by Neoplasms
o
M(400-600) > M(200-400); p=7E-5
M(600-800) > M(400-600); p=2E-5
M(800-1001) > M(600-800); p=0.17
45
400
1000
Age (days)
Age-Specific Mortality for Dose = 150:
Death Caused by Neoplasms
50
n
20
15
10
5
0
p
M(400-600) > M(200-400); p<1E-10
M(600-800) > M(400-600); p<1E-10
M(800-1001) > M(600-800); p=3E-6
90
80
70
Animals Dead or Moribund
Beta model fit for dose=0
60
50
40
30
20
10
0
0
200
400
600
800
1000
0
Age (days)
200
400
600
800
1000
Age (days)
Figure 3-3 (a-p). ED01 age-specific mortality for causes of death (left) and death and
morbidity (right) by all neoplasms vs. dose of 2-AAF. For comparison, the Beta model fit for
the dose=0 data is shown in all curves. Tests of significant changes show at all doses up to 60
ppm, the oldest age group (800-1001 days) has significantly lower age-specific mortality than
the 600-800 days group, which in turn has significantly higher age-specific mortality than both
the 400-600 and the 200-400 days groups. For the dose=75, 100 and 150 ppm groups, agespecific mortality continues to increase beyond the age of turnover observed for the low dose
groups.
70
Table 3-1. Lifetime1 Cumulative Probability2 of Mortality from Cancer
Dose
Site(s) of Fatal
Tumors
Cumulative
Lifetime Mortality
From Fatal Tumors
(%)
Controls
30 ppm
35 ppm
45 ppm
60 ppm
75 ppm
100 ppm
150 ppm
All
7
All
4
All
8
All
7
All
7
All
43
All
50
All
58
Reticulum Cell
Controls
Sarcomas
Controls
Lymphomas
Lung Alveoli
Controls
Tumors
1
Assuming natural lifetime of 1001 days.
2
Calculated as 1-exp{-∑M(t)}.
Cumulative Lifetime
Mortality or Morbidity
From Fatal Tumor (%)
25
16
29
22
23
87
84
93
15
5
10
3.4 Discussion
The data from the ED01 study show unequivocally that cancer age-specific
incidence in mice, assumed to be well correlated with age-specific cancer mortality and
morbidity, turns over after about 800 days for all tumor sites and doses discussed except
for the highest doses. Because of the reliance on mortality and morbidity data, the data
are limited to those sites where the cancer is fatal and near fatal. The age (about 800
days) of maximum incidence is about 80% of the maximum age. This may be compared
71
to the age at peak incidence of people of about 85 years, which is also about 80% of
maximum lifetime.
The 75, 100 and 150 ppm results of Figure 3-3 and Table 3-1 support the
paradigm (Doll 2001) that sufficiently high doses of carcinogens must produce near
certainty of cancer. However, as importantly, this data also suggests that the medical
pathology bias toward under-reporting cancer as cause of death in the oldest humans
(also Doll 2001) is absent in the ED01 data. For the lower doses, the data support the
similar conclusion drawn from human data, that cancer is not inevitable as the animal
ages.
As shown in Figures 3-2 and 3-3, the Beta function, which provides an accurate
fit to human incidence data (Chapter 2), also well represents the features of the ED01
data: non-linear increase to a peak incidence value during the first 3/4 of a lifetime,
followed by leveling and sharp decrease during the final 1/4 of lifetime. This clearly
suggests a similar biological mechanism in both species.
As mentioned in Appendix A and discussed in Chapter 2, one interpretation of the
Beta function is the probability of achieving k-1 stages in any order for cancer creation,
before achieving the one step that would prevent the precancerous cell from becoming a
malignancy. The cancer creation process may be an exponential as opposed to the power
law expression, but the resultant fit and interpretation are unchanged: the data supports
the existence of an important cancer extinction process which dominates near end of life.
Such biological mechanisms discussed in the human work include increasing apoptosis
with age and increasing cell senescence with age. More recently, a slowing of
72
microscopic tumor angiogenesis with age might be linked to a possible explanation
(Folkman 2001).
A promising biologically and mathematically consistent model of the underlying
cancer extinction process may be proposed by considering the demonstration by Hart and
Setlow (1976) that DNA synthesis in human cells, both scheduled (normal) synthesis and
unscheduled (to repair damage from UV radiation) synthesis, markedly reduce with age.
Young cells were found to undergo normal synthesis with near 100% probability, with
the proportion reducing roughly linearly to about only 10% of the oldest cells able to
synthesize DNA. Since repair synthesis is also reduced with age at about the same rate as
normal synthesis, the authors suggest that lack of DNA repair is not a determining
characteristic with age, but rather that cells lose the capacity for any DNA synthesis as
they age, and thus cannot replicate. As further discussed by Hart et al (1979), lack of
replicative ability defines senescence, which allows the cell to function normally, but
inability to maintain genome integrity eventually leads to its death.
Recently, Rubelj and Vondracek (1999) and Rubelj et al (2000) have proposed
that cell senescence may be produced by a stochastic process which abruptly shortens
DNA telomeres, thus causing immediate (within one cell cycle) loss in replicative ability,
instead of gradual loss of telomere length. This causes cells at any age to suddenly
switch from replicative to senescent, thus arresting DNA synthesis and proliferation by
those cells. When, as proposed, the probability of this sudden senescence is uniformly
distributed, and its cumulative probability approaches certainty of senescence for the
oldest cells, this process may well model the results observed by Hart and Setlow, and
might mathematically be similar to the cancer extinction factor (1-βt) of the Beta model.
73
Accordingly, cell senescence might be a significant causal factor in the incidence
turnover by a process which may approximate the Beta model derivation assumption of
uniformly distributed loss of proliferative ability (see Appendix A), i.e. linearly
increasing probability of loss of proliferative ability with age, reaching certainty at
approximately maximum lifespan. Since we believe that an explanation of the cancer
incidence turnover necessarily involves the inclusion of some biological process or
processes not included in the historically dominant models of cancer induction – the
multistage and clonal expansion models, further work is required to determine if the
candidate process is as easily added as the above implies.
74
Chapter 4
Beta-Senescence Model for Cancer Turnover and Longevity:
Interventions by p53, Melatonin, and Dietary Restriction
The evidence of the previous chapters are consistent with the idea that turnover in
the age distribution of cancer at old age might be caused by cellular replicative
senescence. In vitro studies suggest senescence reduces the number of proliferative cells
to near zero by the end of a lifetime, thus eventually removing all cells from the pool
available to cause cancer. By adding the assumption that a limited pool of cells falls with
time to the first order Armitage-Doll multistage model, a Beta function is derived by that
fits the cancer age distribution data well. Denoted as a Beta-senescence model, the model
is further tested by comparing it to results of interventions that might be altering
senescence in mice: 1) altered p53; 2) long term melatonin dose; and 3) caloric
restriction. Increased senescence by enhanced p53 activity reduces cancer but reduces
longevity by premature aging, while reduced senescence by either reduced p53 activity or
melatonin dose increases longevity but also increases cancer. Reducing senescence
might increase longevity to a peak value about 1.3 times normal (lifetime with normal
senescence) before increased cancer rate causes premature death. Interventions such as
antioxidants, which might reduce senescence, might be an attractive strategy of extending
longevity, at the possible cost of increased cancers. Caloric restriction accomplishes both
cancer postponement and increased longevity, probably by slowing both senescence and
75
carcinogenesis. These tentative results further suggest that in any full exact mathematical
modeling, senescence must be included and may be dominant.
4.1 Introduction
In earlier chapters, human epidemiological data and mice bioassay data both
indicate that cancer incidence rates flatten and reduce markedly if the person or animal
lives sufficiently long (>80 years for humans and >800 days for BALB/c mice).
Although one is not able to rule out under-reporting of cancer incidence at old age for
humans (Doll 2001), the weight of the human data and the corroborating mice data
suggest that the turnover might be at least in part a real biological effect. Further, the
human data suggest that incidence rates for all cancers, over incidences ranging a factor
of 100, peak at approximately the same age (mean 85.0 years ± 3.7 s.d. for males and
84.5 ± 7.1 for females, see Chapter 2), suggesting that the unknown biology is strongly
related to age and applicable to all cancers. Accordingly a modeling investigation was
conducted to learn more about the possible properties of this proposed biological effect.
Beginning with the Armitage-Doll (1954) multistage model for cancer incidence
I(t)=at k-1, derived as a fit to early 1950's cancer mortality data for age range 25 to 74, it is
recognized that this model is only a first order approximation of the exact mathematics
describing the modeled cellular steps to produce cancer, and is valid only for small values
of incidence (Moolgavkar 1978, Moolgavkar et al 1999, Pompei and Wilson 2001a). From
the Armitage-Doll model an expression was derived (Appendix A) which resulted in
adding the factor (1-βt), producing the formula
I(t) = (αt) k-1(1-βt)
76
(4-1)
Recognizing this formula as a Beta function f(x)=λt r-1(1-x) over the interval 0≤ x≤ 1, where
x=βt, it has the mathematical interpretation f(x) is the probability density function for the
(r-1)th largest of r uniform random variables. This can be restated as the probability density
function (pdf) for achieving (r-1) stages (cancer creation) without achieving the rth stage
(cancer prevention).
Whereas the textbook Beta function f(x) is usually assumed to integrate to one as
a proper pdf should, the derived Beta function I(t) does not, and its integral varies over a
range of about 0.002 to 0.526 for human cancers (see Chapter 2, Tables 2-1 and 2-2). One
possible interpretation is that a coefficient C, representing a susceptible sub-population,
might be applied for each cancer, as discussed in Appendix A. However, for the reasons
extensively detailed in Appendix C, in response to a well-presented Commentary in the
published paper (reproduced as Appendix B), the susceptibility idea does not seem correct.
The major evidence against this idea is that susceptibility requires heterogeneity
in the population, certainly reasonable for genetic and exposure differences for humans.
However the 24,000 mice of the ED01 study of Chapter 3 were a single inbred strain,
carefully housed and maintained such that they were as little different from each other as
possible. As asked by the Commenters regarding the ED01 data, (on p. B-6) "Did all
animals in this study develop cancer simultaneously? If not, then even genetically
homogeneous animals provide evidence of heterogeneous susceptibility, which cannot,
from such a study, be distinguished from stochastic variation." The point made in the
Response (Appendix C) is that "This stochasticity assumption is inherent in any ArmitageDoll or Moolgavkar type of causality model." Accordingly, the data seem to support the
long standing idea that for equal genetics and exposure, cancer risk is largely stochastic.
77
Age-Specific Incidence (per 100,000)
Age-Specific Cancer Incidence in Humans
5000
A-D power law
4500
MVK clonal expansion
Beta model
4000
SEER (all sites M, F)
3500
3000
2500
2000
1500
1000
500
0
0
20
40
60
80
100
120
Age
Figure 4-1. Age-specific cancer incidence as modeled by two historically important
models: Armitage-Doll power law model and Moolgavkar-Vinson-Knudson clonal
expansion model, compared to SEER data and the Beta model. The Beta curve fit is I(t)
=100,000 (αt) k-1(1-βt), with α = 0.00833, β =0.01, and k = 6.1. (Reproduced for
convenience from Figure 2-1)
The Beta function was shown to fit the human and mice data well, and thus
might be considered a model. Figure 4-1 compares the fit of the Beta function to
Surveillance, Epidemiology, and End Results (SEER) data (Ries et al 2000), compared to
two historically important cancer models: the Armitage-Doll multistage, and the
Moolgavkar-Vinson-Knudson (MVK) two-stage clonal expansion model (Moolgavkar and
Knudsen 1981). The SEER data beyond age 80, and particularly datasets from Holland (de
Rijke et al 2000) and California (Saltzstein et al 1998) which extend to age 100 for a
number of cancers, exhibit marked turnover in incidence not anticipated by either model
78
(see Chapter 2). This suggests a biological cause or causes not explained by previous
work, even if a full "exact" model were used.
4.2 Cellular Senescence
Several suggestions for this unknown biology were explored briefly in the
earlier chapters, including cellular replicative senescence, which might be interpreted as a
late stage cancer-limiting step, since a senesced cell cannot produce cancer. Senescence
appears to be a good candidate, since it is widely accepted that: 1) cellular replicative
capacity is limited; 2) this limitation has been observed in vitro and in vivo, both animal
and human; 3) it is closely related to the ageing process; 4) it is a dominant phenotype
when fused with immortal tumor-derived cells; 5) it is considered to be an important antitumor mechanism, since a senescent cell cannot produce cancer; 6) cells senesce by
fraction of population, rather than all at the same time; and 7) senescent cells continue to
function normally, but are unable to repair or renew themselves (Wynford-Thomas 1999;
Faragher and Kipling 1998; Campisi 1997, 2001; Reddel 2000).
The discovery of the now well accepted existence of cellular senescence is
usually credited to Hayflick (Hayflick and Moorhead 1961, Hayflick 1965), who found
that cells had finite and predictable number of doublings that can be achieved in vitro.
This limit might be directly related to aging by reduced capacity for repair, and different
gene expression by senescent cells (Kipling 2001). Investigators have found that cells do
not all reach their limit in population doublings simultaneously, but rather the number of
non-replicating cells gradually increases as a fraction of the total cells (Cristafalo and
Sharf 1973, Hart and Setlow 1976, Dimri et al 1995, Campisi et al 1996, Rubelj and
79
Vondracek 1999, Campisi 2000, Faragher 2000, Rubelj et al 2000, Paradis et al 2001).
That there is a relationship between cellular senescence and aging has been firmly
established (Campisi 1997, Jennings et al 2000, Campisi 2000, Leung et al 2001, Paradis
et al 2001, Campisi 2001, Tyner et al 2002). That senescence is an important tumor
suppressing mechanism is also well established (Campisi 2000, Campisi 2001, Campisi
1997, Faragher 2000, Sager 1991).
Normal fibroblasts (Hart et al
1976)
Percent of cells able to proliferate
100
90
UV irradiated fibroblasts (Hart et
al 1976)
80
Normal fibroblasts (WynfordThomas 1999)
70
AGO7086A (Thomas et al 1997)
60
DD1 (Thomas et al 1997)
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
In vitro population doublings
Figure 4-2. Cellular senescence evidence in vitro. Increase in number of population
doublings decreases the number of cells which retain replicative capacity at an
approximately linear rate. Lines indicate best linear fit for each data set.
Experimental evidence for increasing senescence with population doublings is
shown in Figure 4-2, which suggest the fraction of cells senescing with population
doublings is approximately linear (Hart and Setlow 1976, Thomas et al 1997, WynfordThomas 1999). There appears to be no evidence that the rate of senescence is related to
the remaining fraction of proliferating cells, which would produce an exponential decay
in the number of proliferating cells. Rather, there is a finite limit of doublings that any
80
individual cell can achieve, as Hayflick had observed. That population doublings and in
vivo age are linearly related is less easily observed, but the data of Figure 4-3, where in
vitro observations of cells from a range of donor age were conducted (Ruiz-Torres et al
Replicative capacity (normalized to highest value
measured)
1999, Yang et al 2001), suggest an approximately linear relationship.
1.0
Vascular smooth muscle
cells (Ruiz-Torres et al
1999)
0.9
0.8
Adrenocortical cells (Yang
et al 2001)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
20
40
60
80
100
Donor age (years)
Figure 4-3. Cellular senescence evidence with increase in age of the donor. Increase in
donor age decreases the number of cells which retain replicative capacity at an
approximately linear rate. Lines indicate best linear fir for each data set.
A linear relationship between age and fraction of cells capable of proliferating
implies that each individual cell has an approximately uniform probability density
function (p(t) = constant) of senescing over a specified time interval, which integrates to
one (∫ p(t)dt=1) over the time period. This suggests a stochastic process that might be
related to cellular damage, as suggested by many investigators (Reddel 1998, Rubelj and
Vondracek 1999, Rubelj et al 2000, Duncan et al 2000, Toussaint et al 2000, Suzuki et al
81
2001). As suggested by Figure 4-3, the cumulative probability of senescence for any
individual cell and for all cells, appears to reach one over a lifetime.
The data of Figures 4-2 and 4-3 are not conclusive in supporting a linear
senescence rate, and might fit an exponential decay. This possibility was discussed in
Chapter 2 (p. 2-41) by replacing (1-βt) in the Beta function with e-ct, thus producing the
Gamma function I(t) = (αt)k-1 e-ct. Data fits to the SEER data were attempted with this
Gamma function, but the fits were poor compared to the Beta function. Accordingly, it
appears that a finite limit to every cell's proliferative lifetime is the assumption that best
fits the data.
Another alternative to the linear senescence model is to assume that senescence
is slow at young age, then accelerates as age increases, producing a non-linear downward
curving shape. A method of modeling this assumed property is to replace the factor
(1-βt) with (1-βt)/(1-ct), adding an additional adjustable constant (Kronauer 2002). Tests
of fit on several cancers suggest that the fit can be about as precise as the linear model
with a maximum value of c of approximately 0.007 when β ≈0.01). This results in a
senescence rate of about 0.3% per year at young age gradually increasing to about 3% per
year near the end of lifespan, compared to the linear assumption of a constant 1% per
year.
Biologically, this hypothesis might be attractive, since the process of apoptosis
continually removes cells as they are damaged beyond functioning or repair, which are
then replaced by daughter cells of proliferating normal cells. This results in fewer cells
dividing more times to maintain cell count homeostasis, thus accelerating the rate at
82
which cells reach the Hayflick limit of divisions, and thus accelerating the rate of cells
reaching senescence.
Cellular apoptosis is a related phenomenon to senescence as a cancer control
mechanism by preventing damaged non-reparable DNA from reproducing. The
difference is that apoptosis destroys the damaged cell, which is then replaced by a normal
cell from a proliferating neighbor, which is able to reproduce. Senescence leaves the cell
in a functional state that is not replaced by a proliferating neighbor, but it is unable to
reproduce (Ran et al 2000, Faragher 2000, Campisi et al 2001b). Senescence appears to
be a much more common event than apoptosis, as recent experimental evidence with
arsenic indicates a factor of 100 difference in rates (Liao et al 2001).
In addition to the characteristic peak in incidence occurring at about age 85, the
Beta-senescence model suggests, and both human and mice data appear to support, that
cancer incidence for all organ sites might approach zero at an age approximating a natural
human lifespan: 102.3 ± 8.4 (mean ± 1 s.d.) years for 39 of the 40 SEER human male and
female cancers, leaving out testicular cancer as a childhood cancer (see Chapter 2, Tables
2-1, 2-2). The corresponding values for β are mean 0.01006 ± 9% (s.d). For the BALB/c
mice, lifespan and age of zero cancer incidence is approximately 1000 days (Chapter 3,
Kodell et al 1980). This apparent relationship between zero cancer incidence rate and
end of lifespan is simply accepted at face value, implying that when cellular replicative
capacity reaches zero (100% senescence), death from natural causes is near. This
observation also answers the question as to what happens to I(t) if t>β-1. Since all of the
cells are senesced, I(t) remains at zero for the (assumed short) remaining lifespan.
83
The Beta function derivation is just one possible mathematical interpretation of
the effect of senescence on cancer incidence, a result of adding senescence as a ratelimiting step to the multistage Armitage-Doll power law cancer model. To consider a
second mathematical interpretation based on an entirely different but also highly
successful cancer model, the often-used approximate form of the MVK two-stage clonal
expansion model (Moolgavkar and Knudsen 1981) is modified in this work to include
senescence. The result, discussed below, is very similar to the Beta model result,
suggesting a robustness to the senescence interpretation.
To explore whether senescence might be the hypothesized biological
mechanism causing the turnover in cancer incidence at old age, we consider experiments
which alter senescence in some way. 1) Recent experiments with mice with genetically
altered p53 expression, and 2) mice with long term melatonin dose, suggest altered
senescence and might be studied. In addition, 3) dietary restriction (DR) has long been
known to significantly effect both cancer and longevity, and also might be altering
senescence in some way.
The recent experiment by Tyner and colleagues (2002) with mice genetically
engineered with truncated p53 in one allele produced over-expression of p53 function.
This enhanced p53 function produces increased senescence that is not only associated
with markedly reduced cancer, but also markedly reduced longevity with extensive signs
of premature aging. It is generally accepted that p53 is an important tumor suppressor
gene, since almost half of human cancers examined have mutated p53 (Venkatachalam et
al 1998). Further, a condition with an inherited mutated p53 in one allele (either of the
two genes at the same locus in a specific chromosome), known as the Li-Fraumeni
84
syndrome, is known to cause cancer predisposition. A person with this syndrome will
develop cancer with 50% probability by age 30 (Venkatachalam et al 1998). p53 is also
known to influence senescence (Blagosklonny 2002, Bargonetti et al 2002). The Tyner
experiment stimulated us to examine the role of p53 more closely.
A recent experiment by Anisimov et al (2001) in which mice were dosed with
melatonin for most of their lives showed evidence of reduced senescence: increased
longevity and increased cancer, and delayed signs of aging. Melatonin is a naturally
occurring hormone considered a chronobiotic, due to its strong connection to circadian
periodicity, both as a marker, and as an influence (Armstrong et al 1991). Melatonin has
been found to be protective against cellular oxidative damage (Reiter 1999, Beckman et
al 1998), and influences senescence and aging (Pierpaoli et al 1994, Oxenkrug et al
2001). The Anisimov experiment led us to examine melatonin more closely also.
In contrast, dietary restriction is known to simultaneously significantly reduce
or postpone cancers while extending lifespan, a very different result than p53 or
melatonin intervention. DR has not previously been discussed in the context of
senescence, but there is ample and long standing evidence in the literature that lifespan is
extended with this intervention (Masoro et al 1982, Hart et al 1999, Hansen et al 1999,
Sheldon et al 1995), and thus might have a significant effect on senescence. Further, it
has been shown that cells retain the properties of the DR intervention, even when
removed and cultured in vitro, suggesting a heritable alteration (Hass et al 1993). One
long-held interpretation of the effect of DR is timescale stretching (Masoro 1982,
Greenburg 1999) for both longevity and cancer. This interpretation may be, but has not
yet been applied mathematically, not just in the Beta-senescence model, but to any other
85
cancer model to test against the experimental data. It is clear that DR has a very different
relationship to senescence than other interventions considered.
4.3 Methods
Data sources for modeling and model comparisons for possible variations in
senescence in mice are from published work: 1) age-specific cancer mortality from the
ED01 study of 24,000 female BALB/c mice (Pompei et al 2001); 2) effect of p53
mutation on cancer mortality and longevity on genetically modified mice from Tyner et
al (2002); 3) effect of melatonin on cancer mortality and longevity on female CBA mice
from Anisimov et al (2001); and 4) effect of dietary restriction on longevity and cancer in
seven rodent studies (Fernandes et al 1976, Weindruch et al 1982, Masoro et al 1982,
Weindruch et al 1986, Haseman 1991, Seilkop 1995, Sheldon et al 1995, Pompei et al
2001). Changes in senescence were modeled by changing the value of β in the Beta
function eq. 4-1, i.e. increased senescence forces cells to senesce more quickly, thus
reaching zero proliferating cells at a younger age, and vice versa.
The ED01 study was designed to detect the effective dose of 2acetylaminoflourene (2-AAF) required to produce 1% tumor rate. The study's undosed
controls' cumulative cancer mortality (including morbidity) was about the same as the
Tyner wild type p53+/+ (normal p53 in both alleles) cancer rates. Tyner produced mice
with one p53 allele mutated (p53+/m) which the authors believe enhanced senescence, a
third group with p53 absent from one allele (p53+/-) are believed to reduce senescence,
and a fourth group with p53 absent from both alleles (p53-/-) which are believed to reduce
senescence further. In this modeling work, experimental variation in senescence is
86
computed from maximum longevity for the p53+/+ and p53+/m groups, since lifetime was
limited by causes unrelated to the cancers. For the p53+/- and p53-/- groups, senescence
was arbitrarily taken as 0.5 of normal and 0 respectively. The senescence variations
assumed to be caused by p53 variations in Tyner are applied mathematically to the ED01
results, to compare the model results against the observations of Tyner.
Anisimov et al (2001) dosed female CBA mice with 20 mg/L in drinking water
for 5 consecutive days each month from age 6 months until their natural deaths,
comparing cancer incidence, longevity, and physiological markers to undosed controls.
The increase in longevity and increase in cancer is interpreted in the present work as
caused by a reduction in senescence associated with melatonin. The relative senescence
is computed from the maximum longevity ratio between dosed and control groups. The
results are compared to the model predictions for altered senescence.
The senescence rate is modeled as the value β and has units of t-1, where t is
age. We make the simplest assumption throughout: that no cells are senescent at t=0 and
all cells are senescent at t=β−1. Normal senescence is taken as the value of β necessary
to fit the cancer mortality data for normal mice, and corresponds to the inverse of the age
at which modeled cancer incidence reaches zero. Relative senescence is modeled by the
relative longevity of the mice compared to normal in the three experiments studied, when
the longevity is not limited by cancers. Where cancer data is given as age-specific
mortality, it is defined as animals dying of cancer in the time period, divided by the
animal-days at risk. Since the age-specific mortality M(t) is a hazard function (animals
dying previously are not in the denominator), the cumulative probability of mortality is
computed as Prob = 1-exp[−∫ M(t) dt]. A model of longevity vs. senescence is
87
constructed by assuming death occurs at the age at which senescence reaches 100%, or
the age at which age-specific cancer mortality reaches 80%, whichever occurs first. A
model of probability of cancer mortality vs. senescence is constructed by varying the
value of β in the Beta-senescence model.
The Beta function is derived in Appendix A. The MVK model with
senescence, denoted here as the MVK-s model, is derived from the commonly used
approximate version I(t) ≈ µ1µ2∫ N(s)exp[(α2 - β2 )(t -s)]ds (Moolgavkar and Knudson
1981). The integration is taken from 0 to t, µ1 and µ2 are the rates of the two transitions
(initiation and malignancy), α2 and β2 are growth and death rate of initiated cells
respectively [(α2 - β2 ) assumed positive], and N(s) is a variable normal cell number
function. For the simplest case of constant cell numbers, the integration yields:
I(t) = (Nµ/γ) [eγt - 1]
(4-2)
where γ = (α2 - β2 ), and µ = µ1µ2 and produces the curve indicated in Figure 4-1. The
simplest method of adding senescence is to assume it is a limiting stage with stage
probability (1-βt), yielding the relation
I(t) = (µ/γ) [eγt - 1](1-βt)
(4-3)
for the MVK-s model.
The time stretching effect of DR is modeled by applying the assumption that t in
the Beta function eq. 4-1: I(t) = (αt) k-1(1-βt), changes in proportion to caloric intake, or
in proportion to weight, which is assumed to be a reasonable measure of caloric intake.
To model total probability of tumors, the cumulative distribution function (cdf) is
required, which is the integral of the Beta function eq. 4-1, and is denoted as B(t):
B(t) = (at)k(1-bt)
88
(4-4)
where a = [α/k1/(k-1)](k-1)/k and b = kβ/(k+1); 0≤ t≤β−1. The cdf eq. 4-4 is similar to the
Beta function, but has different constants a and b in place of α and β, and exponent k
instead of k-1. Since the value of t is still limited to β−1, B(t) never reaches a negative
slope, ending at the peak value of probability with zero slope at t=β−1.
In contrast to the p53 and melatonin interventions, in which the modeled
senescence rate is altered by altering β, the DR intervention is modeled by altering t.
Replacing t with ct', the Beta function for DR becomes:
I(t') = (αct') k-1(1-βct')
(4-5)
The values of α, β, and k are determined for the normal case with c=1, then are held
constant as c is varied in proportion to caloric intake. Note that a fourth constant has
been added to fit the data, and that the coefficient c could be applied to α and β
equivalently.
4.4 Results
Figure 4-4 shows the result of varying the value of the senescence parameter β on
age-specific cancer mortality, for both the Beta and MVK-s models. The particular
values of normal, 1.21 times normal, and 0.5 of normal were chosen to correspond to the
senescence values calculated from the Tyner data, presented in further detail below. The
ED01 controls data of Figure 3-3 (Chapter 3) is taken as normal senescence data, to
which both models are fit. As shown, the Beta and MVK-s model curves are only
slightly different in shape, and give essentially the same result with variation in β.
The cumulative probability of cancer resulting from variations in senescence is
presented in Figure 4-5. As indicated, normal senescence is assumed for the p53-normal
89
(p53+/+) mice, 1.21 times normal for the p53-enhanced (p53+/m) mice and 0.5 for the p53deficient (p53+/-) mice. The 1.21 value is calculated as the ratio of the median longevity
of the p53+/+ group compared to the p53+/m group, etc. As shown, the two approximate
models produce cancer rates predictions which are in good agreement with the Tyner
data.
Age-Specific Cancer Mortality: Beta and MVK/s Models of
Senescence Effects
Age specific mortality (percent of population at risk
per 100 days)
25
ED01 mice controls (Pompei et al 2001)
Beta model
20
MVK/s model
Normal senescence x 0.5
15
10
Normal senescence
5
Normal senescence x 1.21
0
0
200
400
600
800
1000
Age (days)
Figure 4-4. Influence of senescence rate on age-specific cancer incidence in mice. Beta
model fit to ED01 undosed controls is I(t) = (αt)k-1(1-βt), where α = 0.00115, k-1 = 5,
β =0.00108 (Pompei et al 2001). Equivalent MVK-s model fits shown. Senescence rate
is the value of parameter β. Senescence rate increase by 21% is calculated from Tyner et
al (2002) results of 21% reduction in median lifespan for p53+/m mice compared to
normal p53+/+ mice. Senescence rate of 50% is an assumption for p53+/- mice of Tyner et
al.
Anisimov et al (2001) data for mice dosed with melatonin are shown in Figure 4-6
as age-specific mortality vs. age. There is a marked difference in the curves between
dosed and controls, with the controls showing turnover, and the dosed with no turnover.
90
However, since there were only 50 mice in each group, three deaths by tumor in the
controls, and 13 deaths by tumor in the dosed group, the error bars are large. Anisimov
report that the difference in cancer mortality between the two groups is statistically
significant (p < .001).
Effect of Senescence on Tumor Probability in Mice
100
Percent of mice with tumors
90
Normal senescence
Enhanced senescence
Reduced senescence
80
70
60
50
40
30
20
10
0
p53+/+
(Tyner
et al
2002)
Beta
MVK-s
p53+/m
(Tyner
et al
2002)
Beta
MVK-s
p53+/(Tyner
et al
2002)
Beta
MVK-s
Figure 4-5. Probability of tumors in Tyner et al (2002) compared to Beta and MVK-s
models predictions. Modeled lifetime probability of cancer is calculated as
Prob = 1-exp[−∫ M(t) dt], where M(t) is age specific mortality. Tyner et al results for
p53+/+, p53+/m, and p53+/- are interpreted as normal senescence, 21% enhanced
senescence, and 50% reduced senescence respectively. Arrow indicate Tyner data
reported as >80% tumor rate.
Cancer mortality and lifetime vs. senescence rate are shown in Figure 4-7,
combining the data from all of the sources and comparing them to Beta-senescence
model predictions. The predictions for cancer are the direct calculation of cumulative
cancer mortality vs. normalized senescence rate. The predictions for lifetime are the
91
lesser of the age at which t = β-1 (age at which cancer incidence drops to zero), or cancer
age-specific mortality reaches >80%, the reported cancer rate by Tyner for p53+/- mice.
Effect of Melatonin Dose on Cancer Mortality
Age-specific mortality (per 90 animaldays at risk)
1.0
0.9
Controls
Melatonin Dosed
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
200
400
600
800
1000
Average age at death (days)
Figure 4-6. Age-specific cancer mortality for female CBA mice dosed with melatonin
vs. controls. Data from Anisimov et al 2001.
The lifetime prediction curve shows a peak value of about 1.3 at a value of 0.75
for normalized β. As shown cancer mortality follows the model prediction's trends, with
cancer rates approaching zero at senescence value >1.2, and approaching certainty at
senescence <0.6. The lifetime data follow the model predictions for senescence >0.8,
since these points were used to "calibrate" the value of senescence (lifetime ends by
reaching 100% senescence). For senescence <0.8, the model departs from the data for
lifetime limited by cancer mortality, and a curve fit is shown for clarity. Human (SEER)
cancer mortality is shown for comparison.
92
Mice Cancer Mortality and Lifetime vs. Senescence
p53+/+ mice cancer mortality
p53+/m mice cancer mortality
1.4
Cancer moratlity or relative lifetime
p53+/- mice cancer mortality
p53-/- mice cancer mortality
1.2
p53+/+ mice lifetime
p53+/m mice lifetime
1
p53+/- mice lifetime
0.8
p53-/- mice lifetime
Melatonin controls cancer mortality
0.6
Melatonin dosed cancer mortality
Melatonin controls lifetime
0.4
Melatonin dosed lifetime
ED01 mice cancer mortality
0.2
Human cancer mortality
Beta model of cancer mortality
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Normalized senescence
1.6
Beta model of lifetime
------- Curve fit for lifetime data
Figure 4-7. Influence of senescence rate on cancer mortality and lifetime: data from
Tyner et al (2002) for mice with p53+/+, p53+/m, and p53+/- ; compared to Beta model
predictions. Beta model predictions for cancer mortality are Prob = 1-exp[−∫ M(t) dt].
Beta model predictions for lifetime are calculated as the lesser of: age at which
senescence reaches 100% (t = 1/β), or age at which age-specific cancer mortality reaches
80% [M(t) = 0.8]. Human cancer mortality computed from SEER data.
Figure 4-8 shows the result of two investigations into the relationship between
weight and mice liver tumors from the National Toxicology Program (NTP) database. A
Beta-senescence model fit is shown for comparison, where the fit is developed by varying
time t in proportion to weight while holding all other variables constant, in accordance
with the interpretation that DR stretches time. The assumption is made that weight is a
reasonable approximation to caloric intake.
93
Liver Tumors vs. Weight for
Female Control B6C3F1 Mice
1
Haseman 1991
Seilkop 1995
Beta-senescence-time model fit
0.9
0.8
Liver tumor rate
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
Weight (g)
50
60
70
Figure 4-8. Liver tumor incidence vs. weight for two studies of control female B6C3F1
mice. Seilkop data based on body weight measured at 12 months, Haseman data based
on maximum weekly average weight. The Beta-senescence model fit was developed by
varying t in proportion to weight.
Figure 4-9 shows the results of five rodent studies of the effect of DR on mean
lifespan. The Beta-senescence model with the modified time variable comparison line is
computed by holding all variables constant while varying t in proportion to caloric intake.
These data suggest that the model can be fit accurately by adding only a coefficient of
about 0.9 to the inverse proportionality, suggesting that about 10% of the causes of death
might be attributable to unrelated mechanisms.
94
Rodent Longevity vs. Deitary Restriction
Weindruch et al 1986
Weindruch et al 1982
Masoro et al 1982
Fernandes et al 1976
Sheldon et al 1995
Ad libitum
Beta-senescence-time model
2
Relative Longevity
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.6
0.8
1
1.2
Caloric intake relative to ad libitum
Figure 4-9. Results of five rodent studies of the effect of DR on mean lifespan. The
Beta-senescence model comparison line is computed by varying t in proportion to caloric
intake.
4.5 Discussion
The central hypothesis of this work is that the turnover observed in agespecific cancer incidence, as illustrated by Figure 4-1, is caused by increasing cellular
replicative senescence: as age increases fewer cells are available to become cancerous
because only non-senescent cells retain proliferative ability. Figures 4-2 and 4-3 show
seven examples of generally accepted in vitro data supporting the reduction in the
number of proliferating cells with age. A linear senescence vs. age assumption leads to
the (1-βt) factor added to the Armitage-Doll multistage power law model, which can be
interpreted as a limiting last stage, becoming the Beta function I(t) = (αt) k-1(1-βt) used
to successfully fit human and mice age distribution cancer data including the turnover at
95
old age. The senescence hypothesis may also be applied to the two stage clonal
expansion MVK model with the same result, as shown in Figure 4-4.
Although the data fits are very good, as extensively documented in Chapters 2
and 3, and the model arguments seem strong, as reported above, we cannot yet rule out
the observation that the reduced incidence might be caused by reduced diagnosis in older
people (Doll 2001). This will have to await further research into the quality of the data at
elevated age.
A further test of the senescence hypothesis is to compare the model predictions
to available data on animals which have been subjected to treatment which may be
altering senescence in some way. As p53 is known to induce senescence, the Tyner et al
(2002) experiment with genetically altered mice showed that increased p53 activity,
which leads to increased senescence, results in shorter lifespan (with extensive symptoms
of premature aging) but decreased cancers. With mice with 50% reduced senescence
(assumed, with one allele missing p53), cancers increased substantially and directly
caused shorter lifespan by cancer mortality. Mice with senescence reduced to zero
(assumed, with both alleles missing p53), lifespan decreased still further due to even
earlier onset of lethal cancers. The comparisons with the Beta and MVK-s model
predictions of Figure 4-5 suggest that the main features of the proposed senescence
hypothesis on cancer are well supported by the p53 data.
Melatonin is not usually considered a modifier of senescence, but in addition to
its chronobiotic properties, is well known as an antioxidant that reduces damage to DNA.
It is through its damage protective properties that the action of melatonin might be
interpreted to influence senescence, since oxidation damage is known to be a cause of
96
senescence. The experiment of Anisimov et al (2001) resulted in melatonin-dosed mice
exhibiting maximum longevity 17% longer than controls, but with 5 times the lethal
tumors than controls, although total tumors were about the same (22 vs. 20). The age
distribution of mortality due to cancer plotted in Figure 4-6 shows the same features as
the model predicts in Figure 4-4: normal senescence results in turnover in cancer, while
reduced senescence eliminates the turnover.
Ferbeyre and Lowe (2002) observed that there is a balance between cancer and
aging in their commentary on the Tyner paper, sketching a curve of lifespan vs. p53
activity, wherein the curve shows lifetime peak at normal p53 activity. With the Betasenescence model we quantify such a curve, and compare the model to the data. These
results, shown in Figure 4-7, confirm Ferbeyre's observation that there ought to be a peak,
but also suggest the intriguing possibility that a longevity peak is higher than normal for
lower values of senescence than normal: about 1.3 times normal longevity at 0.75 of
normal senescence. This results from accepting higher levels of cancer as the cost of
longer life, an attractive strategy, since many cancers are successfully treated by modern
medicine.
The left part of the lifetime data of Figure 4-7 drops considerably more rapidly
than predicted by the model, which raises questions about the validity of some of the
assumptions. The low senescence data is entirely based on the assumption that reduced
p53 reduces longevity only by increasing cancer, which in turn occurs only because of
reduced senescence. This is clearly a gross simplification, since p53 is known to be very
important in DNA repair as well as causing apoptosis, both of which affect cancer rate
without necessarily involving senescence. A second gross simplification might be the
97
cancer creation assumptions represented by the Armitage-Doll multistage and MVK
clonal expansion models, since these formulations were based on biological assumptions
that did not include senescence, but data that they were fit to, did. It is instructive to
consider the aforementioned Li-Fraumeni syndrome, which causes cancer with 50%
probability by age 30, a difficult point to reconcile with either model. Even with the
removal of senescence, the cancer incidence rate does not increase appreciably from that
shown in Figure 4-1 at age 30, since the value of βt is still small.
The link between cancer and longevity, which appears to be a cardinal
characteristic of senescence, leads to testable hypotheses. One possibility is the activity
of arsenic, a known human carcinogen at high doses, but recently shown to be a strong
inducer of senescence in vitro (about 100 times the rate of apoptosis induction), which
might be a reason arsenic rarely exhibits carcinogenicity in animal models (Liao et al
2001). Accordingly, an epidemiological study of longevity vs. low levels of arsenic
ingestion might show both longevity reduction and cancer reduction, as predicted by
Figure 4-7. Similarly epidemiological studies on many environmental or diet influences
that might include longevity data with cancer data might be re-examined to find if the
expected correlations are observed.
A confounding effect on cancer rate might be the possible action of
antioxidants to directly reduce cancers by reducing DNA damage (Beckman and Ames
1998). However, studies have shown that this is not a consistent result, and dietary
supplementation may increase cancer (Potter 1997). It is possible that observations of
increased cancer with antioxidant supplementation might be due to the action of the
antioxidant in reducing senescence. The issue might be settled in such studies by
98
longevity data. Of particular interest are agents that might reduce damage to DNA
sufficiently to both increase longevity and reduce cancer, a combination so far observed
most clearly for DR (Hart et al 1999, Roth et al 2001).
That DR intervention may alter senescence, perhaps through time stretching,
has not yet been directly measured by in vitro studies of cells taken from DR donors
compared to ad libitum donors. However the comparisons between the Beta-senescence
model results, and cancer and longevity data of Figures 4-8, 4-9, provide support for this
model interpretation. The data of Figure 3-1 of Chapter 3 is not modeled, but appears to
support the idea that DR might stretch time, as it relates to carcinogenesis.
DR intervention creates very complex biochemical responses, and most but not
all of them are consistent with the time-stretching hypothesis. As noted by Anisimov
(2001): "It was calculated that 80-90% from 300 various parameters studied in rodents
maintained on the calorie restricted diet (including behavioral and learning capacity,
immune response, gene expression, enzyme activity protein synthesis rate, effects of
hormones, glucose tolerance, DNA repair efficacy) revealed features of slow aging."
Accordingly, the alternative Beta-senescence model interpretation that α and β vary in
proportion to caloric intake, may be a more precise interpretation. This suggests that DR
increases longevity by decreasing the rate of senescence β, and simultaneously reduces
cancer by reducing the rate of each stage of carcinogenesis, as represented by the value of
α, by the same proportion. Further evaluation of this alternative to time-stretching will
have to await exact models of carcinogenesis derived with senescence, and more
extensive data to test such models.
99
Dietary restriction is the only consistently effective intervention we know of
that both increases longevity and reduces cancer, but there may be others. For example,
selenium has shown some promise in this regard in certain experiments (Anisimov 2001).
In searches for life-extending interventions, clearly those similar to DR are the most
desirable. The characteristics to be sought are reduction or slowing of damage to DNA
which causes both carcinogenesis and senescence.
100
Chapter 5
Conclusions and Future Work
Three tentative conclusions appear to be appropriate as a result of this work, each
of which suggests future work as follows:
5-1.
Cancer incidence turnover at old age is likely caused by cellular senescence
reducing the pool of cells available to become cancerous. Accordingly if one
lives long enough, cancer will be avoided, although death may come earlier from
other causes. Future work is related to improving the data and modeling to find
if this idea is as general as it appears to be from this work.
a. Improve our confidence in the reliability of the human data above age 80, and
any variation of that reliability with age.
b. Conduct toxicological and carcinogenicity rodent bioassays designed to last
the full natural lifetime instead of the standard two years, in order to build a
data base to study the cancer rate turnover and longevity, features that are
missing from the available large databases.
c. Exact mathematical modeling of cancer mechanisms with senescence. An
exact form of the Armitage-Doll multistage model would take into account the
fact that if the later transition stages proceed quickly, the number of cells
which are available for proceeding to cancer will be affected. This depletion
of cells available to proceed to later stages of cancer is common to several
101
biological effects, including senescence. It is an important next step to
perform precise mathematical modeling (which probably cannot have a closed
algebraic form, and inherently will have to be numerical, as found by
Moolgavkar et al (1999) for the clonal expansion model. This modeling
should include all possibilities of reduction of the pool of cells, including by
senescence; apoptosis; slowing of biological processes at older ages; and
effects on biological processes by dietary restriction.
5-2.
Reducing senescence might appear to be an attractive intervention to
prolong life, even if cancer is increased, since modern medical science has been
successful at treating many cancers. In future work:
a. Studies of the effects of environment or diet on cancer should be re-examined
to include effects on longevity. An apparent reduction in cancer associated
with a particular agent might be accompanied by reduction in longevity,
which might suggest a contraindication rather than a recommendation.
b. New studies should be considered with prospective design to uncover any
relation between the two endpoints.
c. Such studies on rodents should be accompanied by cellular-level bioassays to
test for proliferative ability alterations by the interventions.
5-3.
Interventions which both decrease cancer and increase longevity are
possible, by reducing the cellular damage that causes carcinogenesis and
102
increases senescence. The only presently known intervention which
accomplishes this is dietary restriction.
a. Both long term rodent, and human epidemiological studies of candidate
interventions should be launched to find other possible agents.
b. A direct test of the Beta-senescence hypothesis may be conducted by in vitro
measurement of proliferative ability vs. donor caloric intake.
103
Appendix A
Beta Model Derivation
The selection of the Beta distribution for the data fits arises from the
observation that the power law equation I(t) = at k-1 well fits many cancer site incidence
data up to about age 74 (ignoring childhood cancers) At older ages, the incidence data
markedly flatten, and show reduction at sufficiently elevated age. Accepting the validity
of the power law fits at younger ages (but not necessarily the validity of the power law
model itself), we add the hypothesis that a "cancer extinction" term is influencing the
carcinogenesis process (which is proposed to be cellular senescence), eventually
becoming dominant at sufficiently elevated age.
Adding this cancer extinction term to the power law is accomplished directly
by forming the probability statement
probability of cancer = the probability of reaching k stages
and the cancerous cell does not die (or lose its proliferative ability).
(A-1)
We write this probability and expand as
Pc = P(bt k ∩ not death) = P(bt k|not death)P(not death) = bt k * P(not death).
(A-2)
The simplest assumption for a probability density for a cell losing proliferative ability is a
uniform distribution, leading to P(not death) = 1-∫c dt = 1-ct. Then the total probability
is
Pc=bt k(1-ct)
where c is a constant. Taking the time derivative to convert the probability to a
probability density function for a single cell, then
104
(A-3)
f(t)=αt k-1(1-βt)
(A-4)
where α and β are constants. We immediately recognize the Beta distribution
f(x)=λt r-1(1-x)
(A-5)
over the interval 0≤ x≤ 1, where x=βt. A textbook interpretation of f(x) is the density for
the (r-1)th largest of r uniform (0,1) random variables (Larson 1982), which can be
restated as the probability density function for achieving (r-1) stages (cancer creation)
without achieving the rth stage (cancer extinction).
Expanding from consideration of a single cell to N cells in an organ, and
denoting f(t)=F'(t), the probability of cancer is
G(t)=1-[1-F(t)]N
(A-6)
For large N, this simplifies to
G(t)=1-e-NF(t)
(A-7)
which is accurate to 10-10 for N=108 cells.
As discussed by both Moolgavkar (1978) and Armitage (1985), the agespecific incidence function for the organ tissue is not the density function G'(t) itself, but
the associated hazard function, given by
hc(t)=G'(t)/[1-G(t)]
(A-8)
which represents the incremental risk of cancer per unit time given that the tissue has
been cancer-free to time t. Completing the derivation,
hc(t)=e-NF(t) Nf(t)/ e-NF(t) = Nf(t)
(A-9)
We note that the age-specific cancer incidence for a site tissue is related to the
probability density function for one cell by the constant N, thus leaving the Beta model as
f(t)=αtk-1(1-βt)
105
(A-4)
modified by only by a constant (absorbed into α) to apply to a multicellular organ site.
The final expression chosen immerses the α constant into the k-1 power in order to
preserve the historical view of k-1 stages, each with its own transition rate (assumed to be
equal in this case), thus denoting the final form as
b(t)=(αt)k-1(1-βt)
(A-10)
To apply the Beta model to fit epidemiological age-specific incidence data for
a specific cancer, I(t), we consider whether the data is properly interpreted as a hazard
function or a pdf. Since the hazard function is given by I(t)=fe(t)/[1-Fe(t)], the subscript
e denoting a pdf and cdf derived from epidemiological data, which is the number of new
cancer cases divided by the population at risk; the question reduces to whether the data
set modeled has in the denominator only the population at risk for that cancer or the
entire population for that group. For the SEER data, the denominator includes all
members of an age group still alive at time t, which includes all who have been diagnosed
with a cancer still alive. If the mortality due to cancer were zero, then the usual
approximation I(t)≈ fe(t)=b(t) (valid for small Fe(t)) would be exactly I(t)=fe(t)=b(t).
Since mortality rate is about one-half of incidence rate overall for the SEER data, the
exact statement cannot be made, and the approximation must be taken. It should be noted
that since only one-half of the Fe(t) are removed from the denominator, this
approximation is considerably more accurate than if incidence is inferred by age-specific
mortality in which the approximation is taken that I(t)≈ fe(t). Further discussion of this
point is in the main text.
As employed for the fits, the Beta model does not integrate to 1, as a correct
density function must, but integrates to the cumulative probability for that cancer site,
106
which is always less than 1 from the data. The Beta model may be converted into a
density by writing
b(t)=C(γ t)k-1(1-βt)
(A-11)
where
C=∫(αt)k-1(1-βt)dt, and ∫( γ t)k-1(1-βt)dt = 1 ; 0≤ t≤ β-1
The factor C might be interpreted as a susceptibility factor, suggesting that a fraction C of
the population will contract the site cancer with probability unity if they live long
enough. This is mathematically indistinguishable from an interpretation of C as the
certain result of a stochastic process creating the cancers in a large cohort, and is the
subject of some discussion in Appendices B and C as well as the main text.
107
Appendix B
Commentary: Outliving the Risk for Cancer:
Novel Hypothesis or Wishful Thinking?
Appendix B, a written commentary published in Human and Ecological Risk
Assessment together with the paper represented by Chapter 2, is reproduced here for
clarity, since the commenters raised important interpretation issues for the work.
Appendix C is the response to the commentary, also published in the same issue of
HERA, and follows this reproduction, as it did in the original journal.
108
109
110
111
112
113
114
115
Appendix C
Rebuttal to "Outliving the Risk for Cancer:
Novel Hypothesis or Wishful Thinking?"
We thank Professors Hertz-Picciotto and Sonnenfeld for their carefully
considered comments, and for the opportunity to clarify our results and interpretation. We
follow the same organization of their comments, in our response.
Model Development and Fit
As emphasized by the commenters, the SEER data for bladder, leukemias, colon,
stomach, and pancreas cancers do not show actual incidence turnover for the age range
reported. However, in each of those sites where data from other data sets exist for older
ages (Dutch and California data for bladder, colon, stomach), the incidence turnover is
present, and the peak is very close to the Beta model predicted age. It is worth noting
that for the above 5 cancer sites not achieving turnover in the age range reported by
SEER, the Beta model explains 1.00/1.00, 0.99/0.99, 1.00/1.00, 1.00/1.00, 1.00/1.00 of
the male/female data variance (according to the Cox criterion) for those 5 SEER cancers
respectively. The near perfect model fit values, along with the fact that three of the five
were subsequently proven to show turnover very near the predicted age, gives us some
degree of confidence that the other two will also show a peak near the predicted age when
data become available.
116
As for testicular and Hodgkins disease, we consider those cancers to be outside
the range of our Armitage-Doll cancer creation modeling assumption, since they appear
to be from quite different age-dependent mechanisms than the other 30 or so cancers. We
conducted the model fit for completeness of data reporting. These cancers do, however
share a common characteristic of incidence trending down at the oldest ages reported, and
thus, we believe, are subject to the same biological mechanism (whatever it may be)
causing the turnover for all of the other cancers.
Regarding the fits to the Dutch and California data for ages beyond that reported
by SEER, the Beta model would not be expected to fit as precisely as it does the data for
which the fit was actually derived. We consider the Beta model to be an adequate first
approximation in its predictions at ages > 90, since it is able to correctly predict the
turnover location, and considering the simplicity of the model, it does remarkably well in
predicting the approximate downward slope. Since the SEER data are the most complete
below age 85, we chose to use extrapolations to the best fits to SEER to compare to the
turnover in the other data sets. Our comment regarding the Dutch lung cancer incidence
reaching almost zero at age 100 is in comparison to its peak value at age 80.
Our claim that the age at peak incidence is remarkably uniform between sites
must be considered in relation to alternative models, which all appear to predict the age to
change as the incidence changes. At our reported standard deviation of 3.7 and 7.1 years
for males and females respectively, this is approximately ± 4 to 8% variation of the age at
peak incidence, compared to a factor of two change predicted by previous models, all of
them appearing to be based on one variation or another of the "running out of candidates"
117
approach (mathematically, the pdf must approach zero as the cdf approaches one for
whatever group or subgroup is considered).
Cook et al described their results in terms of curvature as opposed to age at peak
incidence, but it is different words for the same mathematical result. Their model is a
modified Armitage-Doll power law, which they derived as
I(t) = a + k ln(t) - ln[C+(1-C)eF/C
(C-1)
where F = eatk+1/(k+1), to include a susceptibility fraction C. This expression produces
a factor of two variation in age at peak incidence with factor of 100 variation in
susceptibles fraction - a similar result as the unmodified A-D power law. They did not
find the degree of curvature (and hence the location of the peak) to change with incidence
as suggested by their model, and thus Cook et al themselves rejected the susceptibles
hypothesis.
Data Quality
We cannot prove the reliability of the incidence data at the oldest ages. We
simply accept them at face value while alerting the reader to their importance, providing
relevant evidence where we can, and suggesting that science only advances by believing
data and seeing where they might lead. Importantly, we cite the small, but growing body
of direct pathology evidence that cancer prevalence flattens or reduces at the oldest ages,
which appears consistent with incidence reducing at those ages, and not consistent with
incidence increasing at the oldest ages. Although the commenters question the validity of
the turnover data, they also point out that many authors have published models
attempting to explain this turnover with a susceptibles depletion hypothesis. We accept
118
this as support for our position, since those many authors explicitly share our tentative
acceptance of the turnover data.
Biological Plausibility
We concur with the commenters that cancer is a complex group of diseases, but
nonetheless if there are undiscovered biological principles applicable to all of them, it is
worth our effort to attempt to find those principles (and the effort of commenters to
challenge them). Variation in susceptibility is one candidate for the turnover, which the
commenters propose as a perfectly valid hypothesis. Unfortunately, it does not appear
that this hypothesis had settled the matter of the turnover in incidence.
Amongst the reasons we looked beyond the susceptibles hypothesis were opinions
on our results expressed to us by two of the foremost investigators in this field, Sir
Richard Doll and Suresh Moolgavkar. Both were troubled by the notion of groups who
are immune to cancer, noting that cancer probability must approach certainty if: a) the
dose is sufficiently high (Doll 2001); or b) time is sufficiently long (Moolgavkar 2001).
Further, while the various models might all be "fixed" by making the number of nonsusceptibles vary appropriately with site, this appeared to us to be a somewhat arbitrary
"ad hoc" explanation.
The Cook et al study is in many ways similar to ours, in their attempt to identify
general principles for cancer creation to include the oldest ages, by examining the age
distribution of 31 types of cancers in 11 populations and applying a modified version of
the Armitage-Doll model. They conclude: "No evidence was found to suggest that the
shape of the observed relationship could be attributed to attenuation of a limited pool of
119
susceptibles." As noted by the commenters, they proceed by suggesting variation in
carcinogen exposure as a possible cause, but avoid the inclusion of this hypothesis as part
of a "susceptibles" mechanism. Although not stated by them, including exposure in the
susceptibles hypothesis suggests that if a person is not exposed to the carcinogen, they
are immune to the cancer. In the end, Cook et al essentially concede the limitations of
mathematical models to shed light on the underlying biology, a concession we propose
might be premature.
As an example for discussing the commenters' concerns about Finkel's model, if
25% of the population were immune, there would be a turnover when the cumulative
incidence begins to approach 75%, as occurs with the Cook model. Finkel's model does
not appear to have any persons completely non-susceptible, and therefore, one might
expect it to be necessary to approach 100% cumulative incidence before turnover must
necessarily occur. We are not aware of any analyses of the Finkel model that shows
actual downturn in incidence when cumulative incidence is less than one, but we cannot
completely rule it out on purely mathematical grounds.
We consider the mice data to be quite important in weighing the evidence for and
against susceptibility. Quoting the commenters, many other authors have published
models proposing depletion of a pool of susceptibles based on those who have "both
genetic susceptibility and environmental exposure." The mice were bred and housed to
have as little variation as possible in either, yet show statistically significant turnover in
incidence at about 80% of lifespan. Indeed as the commenters suggest, the mice did
develop cancer at different times in their lifespans, which leaves only stochastic variation.
120
This stochasticity assumption is inherent in any Armitage-Doll or Moolgavkar type of
causality model.
In the time since our paper was accepted, we have concentrated on studying the
growing field of cell replicative senescence, a possible mechanism we suggest briefly in
our paper. It appears that this may be the additional biological phenomenon for which we
were searching, since it seems widely accepted that: 1) cellular replicative capacity is
limited, a fact known for 40 years; 2) it has been observed in vitro and in vivo for many
cell types, both animal and human; 3) it is closely related to the ageing process; 4) it is a
dominant phenotype when fused with immortal tumor-derived cells; 5) it is considered to
be an important anti-tumor mechanism, since a senescent cell cannot produce cancer; 6)
cells senesce by fraction of population, rather than all at the same time; and 7) senescent
cells continue to function normally, but are unable to repair or renew themselves.
(Wynford-Thomas 1999; Faragher and Kipling 1998; Campisi 1997, 2001; Reddel 2000)
As outlined in our paper, senescence leads to the Beta function if the cumulative
probability of finding a given cell is not senesced reduces linearly as age increases, to a
value near zero at the oldest ages. This is mathematically the same as finding the
population fraction of non-senesced proliferating cells reduces to near zero at the oldest
ages. The literature suggests just such properties: a) percent of non-senesced cells
decreases linearly with number of cell divisions to near zero (Hart and Setlow 1976,
Thomas et al 1997, Wynford-Thomas 1999); and b) replicative capacity of human cells in
culture reduces approximately linearly as a function of donor age (Yang et al 2001,
Ruiz-Torres 1999).
121
Senescence appears to have all of the necessary properties to fit the data, and to be
a biological endpoint of sufficient certainty and overwhelming effect to stop the
carcinogenesis process. To the best of our knowledge no other modeler has yet included
senescence in an attempt to explain the incidence turnover. Work is proceeding to
explore this hypothesis, which we intend to present in a new paper. Since epidemiologic
data at old ages is inevitably sparse and less reliable than data at younger ages, we
believe that the solution or solutions to understanding of these data will only come from
combining epidemiology with animal and in vitro data.
Again our sincerest appreciation to Professors Hertz-Picciotto and Sonnenfeld for
their thoughtful discussion.
Francesco Pompei and Richard Wilson
Harvard University
Cambridge, MA
122
Acknowledgements in Published Papers
In the development of the work of Chapter 2, the detailed reviews and discussions
with Dr. Suresh Moolgavkar of the Fred Hutchinson Cancer Research Center and Sir
Richard Doll of the University of Oxford were particularly helpful, coming from two of
the foremost authorities in cancer modeling. Dr. James Wilson and Dr. Dan Kammen
both of UC Berkeley, and Dr. Chiu Weihsueh of the US EPA also provided helpful
comments on the work. In the development of the work of Chapter 3, Drs. R.W. Hart of
FDA NCTR and J.K. Haseman of the NIEHS, provided valuable references and
comments that aided the development of the data and search for biological causality.
These acknowledgements were published with the papers represented by these chapters,
and I am grateful to all for their help and guidance.
123
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