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. 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