Paradox resolved: The allometric scaling of cancer risk across species
SSubject Areas:
Theoretical Biology, Physiology,Biophysics
Keywords:
Peto’s Paradox, Biological Scaling,Metabolic Theory, Cancer
Author for correspondence:
Christopher P. Kempese-mail: [email protected]
Paradox resolved: Theallometric scaling of cancerrisk across species
Christopher P. Kempes , Geoffrey B. West and John W. Pepper The Santa Fe Institute, Santa Fe, NM 87501, USA National Cancer Institute, Division of CancerPrevention, Bethesda, MD 20814, USA
Understanding the cross-species behavior of cancer isimportant for uncovering fundamental mechanismsof carcinogenesis, and for translating results ofmodel systems between species. One of the mostfamous interspecific considerations of cancer isPeto’s paradox, which asserts that organisms withvastly different body mass are expected to havea vastly different degree of cancer risk, a patternthat is not observed empirically. Here we showthat this observation is not a paradox at all butfollows naturally from the interspecific scaling ofmetabolic rates across mammalian body size. Weconnect metabolic allometry to evolutionary modelsof cancer development in tissues to show thatwaiting time to cancer scales with body mass ina similar way as normal organism lifespan does.Thus, the expectation across mammals is that lifetimecancer risk is invariant with body mass. Peto’sobservation is therefore not a paradox, but thenatural expectation from interspecific scaling ofmetabolism and physiology. These allometric patternshave theoretical implications for understanding lifespan evolution, and practical implications for usingsmaller animals as model systems for human cancer.1 a r X i v : . [ q - b i o . O T ] N ov
1. Introduction
Cancer is not confined to humans, but occurs throughout the animal kingdom, and perhapsbeyond [1]. This opens possibilities for deeper understanding of cancer through comparativestudies and comparative oncology [2,3]. The evolutionary thinking driven by species comparisonshas been employed previously in understanding tumor growth and metastasis, and is alsoinformative regarding cancer risk. The comparative approach thus has much to contribute tounderstanding cancer epidemiology and to advancing cancer prevention [4].One of the most-discussed open problems in comparative cancer biology is due to Peto [5]who famously remarked that, “no plausible explanation has yet been offered for the fact thatthe risk of cancer in old age is not vastly different in species with very different life-spans”.He also noted here that humans have perhaps 1000 times as many cells as mice, with somerisk of each cell becoming cancerous. Larger body size is correlated with longer lifespan acrossmammals, and both seemingly should predispose to greater cancer incidence. The apparent lackof this pattern is known as “Peto’s paradox”. The two issues of body mass and lifespan couldin principle be addressed separately, but in practice have often been discussed together. In theliterature on “Peto’s paradox”, some authors have focused on body mass, (e.g., [6]), while othershave explicitly considered both body mass and lifespan, (e.g., [7]).Several possible resolutions of Peto’s paradox have been investigated. Caulin, Graham et al. [8]calculated that to reduce a whale’s expected cancer risk to that of a human (with 1000-fold fewercells) would require either a two- to three-fold decrease in the stem cell division rate, or twotumor-suppressor gene mutations. That paper compared mammalian genomes for differencesin number of tumor-suppressor gene mutations. Although it reported evidence for genomicamplification of tumor-suppressor genes in several species, ranging in size from microbats toelephants, it did not find a positive correlation of tumor-suppressor genes with increasing bodymass.In the absence of genomic explanations, physiology has received substantial attention.Comparison of human tissue types has shown that tissues with more tissue stem cell divisionshave higher cancer risk [6,9]. This result further supports the suggestion that larger animals witha greater number of stem cell divisions would be expected to have increased cancer risk, andfurther demands some explanation for why this does not seem to be true. Absent systematicgenetic differences, why is cancer risk not vastly higher for larger animals?Herman, Savage et al. [10] suggested that resolving Peto’s paradox, may be possible byconsidering the scaling consequences of body mass and metabolic rate. Here we pursue this line ofthinking, by including the related suggestion that cell-specific cancer risk is a function of cellularaccess to the resources allowing cell proliferation [11]. We therefore focus on scaling patterns forthe cell-specific rate of vascular resource delivery.Allometric scaling of the cell-specific rate of energy use by somatic tissues has been explainedby the observation that, “generic properties of the vascular network constrain resource supplyto cells.” [12]. In support of that explanation, empirical evidence indicates that somatic cellsintrinsically tend to use energy substrates as rapidly as availability permits. Whereas rates ofenergy use by somatic cells in vivo are systematically lower in larger mammals, rates for cells inculture converge to a single, much higher, value for cells from all mammals regardless of bodymass [13]. This has been interpreted as evidence that somatic-cell energy use and cell proliferationis extrinsically limited by the multicellular organism through control of the vascular deliveryof proliferation-limiting energy substrates [11]. These authors suggested that limited vascularresource delivery is an organismal adaptation that limits somatic-cell proliferation and evolution,and thereby reduces cancer risk. Here we develop that idea by explicitly calculating rates ofresource supply to cells across the full range of mammalian body size.We begin with consideration of how waiting time to cancer scales with the supply ofproliferation-limiting resources, based on reanalysis of unpublished quantitative simulationresults of waiting time to cancer from [11] (Fig. 1). In order to understand cancer risk across diverse organisms and to resolve Peto’s paradox, we combine these computational modelingresults with consideration of how vascular delivery rates scale with body size, based on thenetwork theory and ontogenetic growth framework of West, Brown et al. [14].For our analysis we focus on cancers of adulthood, even though rapid cell proliferationduring development may explain much of childhood cancer and also contribute to cancers thatappear in adults. We focus on adulthood both because adult cancers comprise most cases, andbecause they have been the focus of the literature on how cancer risk scales with body size. Ouranalysis considers cancer risk in adulthood in terms of lifetime totals and averages, and thusintegrates certain carryover risks from childhood while still focusing on the cancers that appearin adulthood.The theoretical work suggesting that vascular delivery is rate-limiting for somatic cellproliferation, and thus for oncogenesis, is consistent with substantial empirical evidence. Severaldifferent substrates can be rate-limiting for somatic cell proliferation. These include energeticsubstrates (e.g., glucose and oxygen), as well as crucial inorganic molecules such as phosphate[15] and iron [16]. Each of these substrates are over-consumed by rapidly dividing neoplasticcells, making them scarce in neoplastic microenvironments. Each of these scarce and proliferation-limiting substrates is replenished only by vascular delivery. Therefore, vascular delivery is likelyto be rate-limiting for cell proliferation, and therefore for oncogenesis and cancer progression[10,11,17]. This is true even though tumors may differ in which specific substrate is rate-limitingfor cell proliferation, because all proliferation-limiting resources arrive only through vasculardelivery. These considerations make rates of cell-specific vascular delivery central to questionsabout cancer risk.
2. Results (a) Effect of resource supply rate on cancer risk
The explicit eco-evolutionary simulations of Wu, Aktipis et al. [11] indicated an inverserelationship between energy supply to tissues and waiting time for cancer, which serves as aproxy for cancer risk within a finite time period. These simulations were built on the assumptionthat in normal tissues without inflammation or wound healing, resting vascular delivery providesenough resources for somatic cells to function and maintain themselves, but not enough toproliferate beyond replacing lost cells. Under this assumption, limited vascular resource supplynormally suppresses somatic cell evolution toward cancer, and any abnormal oversupply ofresources accelerates somatic evolution, thereby increasing cancer risk. By fitting curves to thosequantitative simulation results, we can calculate that the waiting time to cancer ( t c ) in thosesimulation scales with resource delivery rate as: t c ∝ r − . (2.1)where r is the multiple of resource delivery to tissues, and is defined as r = R (cid:48) /R where R and R (cid:48) are the normal and elevated rates of resource delivery respectively. (see Appendices A3-A5, fortechnical details.) This implies that the waiting time for cancer can also be written as t c ∝ R (cid:48)− . . (b) Allometric scaling of cancer risk The results of the preceding section indicate that the waiting time to cancer depends on the cell-specific vascular delivery rate R, which we can compare across organisms of different body size.For mammals of varying body size, we can calculate R as the average adult total cardiovascularoutput, (cid:104) Q (cid:105) , divided by the average adult mass, (cid:104) M (cid:105) , given that cell number N scales linearlywith total body mass, N ∝ M , [12]. It should be noted that the total cardiovascular deliveryrate is proportional to the total metabolic rate, (cid:104) Q (cid:105) ∝ (cid:104) B (cid:105) [14], and we focus our derivationson the metabolic rate because it can be connected with the ontogenetic growth trajectories ofWest, Brown, et al. [14]. In Appendix A2, we use the ontogenetic growth model to calculate the quantities B and M , where we find that (cid:104) B (cid:105) ∝ (cid:104) M (cid:105) / , and thus R = (cid:104) B (cid:105) / (cid:104) M (cid:105) ∝ (cid:104) M (cid:105) − / . (2.2)Thus, the cell-specific vascular delivery rate systematically decreases with increasing body size. Ifwe define as a reference, the mass of the smallest mammal, (cid:104) M (cid:105) , then the reference cell-specificrate of resource delivery, R , for that mass is R = (cid:104) B (cid:105) / (cid:104) M (cid:105) ∝ (cid:104) M (cid:105) − / , (2.3)from which the multiple of the reference cell-specific resource delivery for larger mammals isgiven by r ∝ (cid:104) M (cid:105) − / / (cid:104) M (cid:105) − / . (2.4)Taken together these results combine to predict that the effect on oncogenesis of vascular deliveryrate would influence time to cancer according to t c ∝ (cid:16) (cid:104) M (cid:105) − / (cid:17) − . ∝ (cid:104) M (cid:105) . . (2.5)A similar calculation of the waiting time to cancer (interpreted as the threshold time it takesfor a given amount of energy to be delivered to a cell to stimulate cancerous growth) gives t c ∝(cid:104) M (cid:105) / , which implies that t c ∝ r − , in approximate agreement with the result derived from thesimulation of Wu et al. which gave an exponent of ≈ . .All else being equal, these results show the increase in the waiting time to cancer for tissues oflarger mammals because of slower resource delivery to cells. In isolation, this would suggest thatlarger mammals would have lower lifetime cancer risk owing to lower resource delivery ratesto tissues. However, mammals have radically different lifespans and we need to take this intoconsideration to assess Peto’s paradox.It is known that mammalian lifespans, T , scale as T ∝ (cid:104) M (cid:105) . [18], which is very close to thescaling of time to cancer as t c ∝ M . (see above,also Appendix A4). This implies that total cancerrisk over a lifetime would not vary systematically with body mass. Because waiting times tocancer and to normal end of life vary together with mass, the expectation of cancer arising beforenormal end of life does not change ad a function of mass. Our more detailed derivation in theAppendix also shows that the decreased cell-specific supply rate in larger mammals is counter-balanced by increased lifespans such that cancer risk per cell is expected to be approximatelyinvariant across all mammalian body sizes. Figure 2A shows that at the expected lifespan thetotal energy delivered to a unit of tissue is the same across organisms of different size. Mammalsreach this value at different rates according to lifespan. Another way to present this analysis isto look at the fraction of total energy delivered over a lifespan as a function of the fraction of alifespan, which collapses all the lifespan curves for mammals onto a single universal curve (Figure2B). The scaling of lifespan is the only normalization needed for this collapse.
3. Discussion
When we consider allometric scaling relationships, which were not part of the original framingof Peto’s paradox, Peto’s observations become less paradoxical. The apparent paradox arosefrom the assumption that large animals are like small animals in all respects, aside from havingmore cells and longer life spans, and that all physiological consequences including the risk ofcancer naively increase accordingly. Available empirical data do not tell us quantitatively how thecell-specific risk of malignancy depends on access to proliferation-limiting resources. However,computer simulations of the process of somatic-cell evolution to malignancy have shown that thisdependence does exist, and has a strong effect [11]. Thus, Peto’s apparent paradox may be simplya result of the incorrect assumption of, “all else equal”, with increasing body mass, including per-cell delivery of proliferation-limiting resources. In fact, the lower rate of resource delivery per cellin larger animals reduces cells’ potential for proliferation and clonal evolution, and thus their risk of initiating cancer per unit time. However, this lower risk is compensated for by longer lifespanswhich match the waiting times for cancer. Our results indicate that the number of oncogenesisevents per unit of body mass over a lifetime is expected to be roughly invariant across body mass.Peto (2016) addressed only differences between, and not within, species. Such allometricrelationships have only been observed, and explained, between species. Therefore, we do notshare the view of Nunney [19] that changes in metabolic rates (or vascular delivery rates) shouldbe discarded as explanations for Peto’s paradox because this explanation is inconsistent withobserved intraspecific effects of body size on cancer risk.The usual functional interpretation of metabolic scaling is that restricted vascular delivery tobody tissues of larger animals benefits the organism by reducing the energy costs of vasculardelivery [20]. However, given the hypothesis of Wu, Aktipis et al. (2019) regarding the role ofresource-dependent evolution of somatic cells, we suggest the observed allometric scaling ofvascular delivery provides a second plausible benefit in the form of suppressed per-cell cancerrisk in larger mammals. When life span exceeds waiting time to cancer, cancer incidence willbe substantial, and its impacts on organismal fitness may drive selection for reduced vasculardelivery per cell.How might we explain the striking similarity between the scaling exponents of life span andof cellular waiting time to cancer? If we take as a given the evolved scaling of life span, andtreat the scaling of vascular delivery as an adaptation to accommodate this evolved life span, thissuggests selection to restrict vascular delivery to the point that waiting time to cancer approachesthe typical life span for that body size, rendering cancer unimportant in reducing life span andfitness. Future work should focus on comparing the relative selective effects of hydrodynamicefficiency versus suppression of cancer risk.It should be noted that our work addresses only the increased cancer risk to an individualassociated with maintenance, and excludes the larger number of cell proliferations duringdevelopment of larger animals. Future models could create a more complete picture of cancerrisk by combining this developmental aspect with the risks examined here accompanying cellproliferation for tissue renewal.
4. Figures
Multiples of Energy Supply Rates W a i t i ngT i m e t o C a n ce r Figure 1.
The scaling of waiting time to cancer with energy supply rates, from computer simulations of cell evolutionwithin tissues. The black line is the best fit to all the data, where the scaling exponent α = − . ± . . Each datapoint represents an independent evolutionary simulation where the x-axis values have an added random graphical factorto show all the data. The data are originally from the study reported by Wu, Aktipis, et al. [11]. A B
Fraction of Lifespan F r ac t i ono f L i f e t i m e E n e r g y D e li ve r e d P e r U n i t M ass ( J / g ) Age ( days ) To t a l E n e r g y D e li ve r e d P e r U n i t M ass ( J / g ) Figure 2. (A) The total energy delivered to a unit of tissue as a function age and body size. The mammalian body sizesrange from g (green curve) to , , g (purple curve). The total energy delivered per unit tissues reaches thesame value at the expected lifespan for each body size. Note that the both axes have logarithmic scales, such that higherparallel lines have higher rates of increase with age, and larger mammals reach the same total energy delivered to tissuesas smaller mammals, but do so at a slower rate. (B) The universal curve for percent of total lifetime energy delivered toeach cell as a function of fraction of lifespan. The dashed line is the expected time to reach maturity. Log (body mass) L og ( d e li v e r y r a t e p e r t i ss u e ) L og ( T i m e ) D e li v e r y R a t e P e r T i ss u e M a m m a li a n li f e s p a n W a i t i n g t i m e f o r c a n c e r Figure 3.
The overall tradeoffs in resource delivery rates, lifespans, and waiting times for cancer as a function ofmammalian body size that together lead to a constant cancer risk across all mammals.
Data Accessibility.
The data used in this paper are a reanalysis of “Wu DJ, Aktipis A, Pepper JW. Energyoversupply to tissues: a single mechanism possibly underlying multiple cancer risk factors. Evolution,Medicine, and Public Health. 2019;2019(1):9-16.”
Authors’ Contributions.
JWP, CPK, and GBW conceived of the study and discussed initial ideas formathematical analysis. CPK and JWP carried out the mathematical analysis, produced figures, and wrote the initial manuscript draft. GBW provided feedback on the analyses contributed to the revision of the finalmanuscript.
Competing Interests.
The authors declare no competing interests.
Funding.
CPK and GBW thank CAF Canada for generously supporting this work. JWP is employed by theNational Cancer Institute.
Acknowledgements.
The authors also thank Van Savage, Alex Herman, and Eric Deeds for insightfulconversations.
Disclaimer.
The opinions expressed by the author are their own and should not be interpreted asrepresenting the official viewpoint of the National Cancer Institute, the National Institutes of Health or theU.S. Department of Health and Human Services.
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Gerontology , 90–98. AppendicesA1. Lifetime Average Tissue Supply Rates
Several previous efforts have derived growth rate from metabolic rate by considering that thetotal energy budget of an organism is partitioned between growth and maintenance purposes: B m α = E m dmdt + B m m (A1.1)where E m is the energy required to synthesize a unit of new biomass, and B m is the metabolicrate required to maintain an existing unit of biomass. This equation can be solved for the growthtrajectory of a single organism (e.g. [14,21,22]), and it can also be used for finding the growth rateof an organism. If we define a = B /E m and b = B m /E m , then we can rewrite equation A1.1 as dmdt = am α − bm. (A1.2)This equation can be solved [22] to give the size trajectory of an organism m ( t ) = (cid:20) − (cid:18) − ba m − α (cid:19) e − b (1 − α ) t (cid:21) / (1 − α ) (cid:16) ab (cid:17) / (1 − α ) , (A1.3)where for mammals the metabolic scaling is given by α ≈ / . It should be noted that theasymptotic mass corresponds to dm/dt = 0 in Equation A1.2, leading to M − α = a/b . Thisasymptotic mass is achieved only as time goes to infinity. Typically, a is taken to be a constantas B applies to an entire class of organisms conforming to one metabolic scaling relationship,and E m is found to be invariant across a variety of organisms. Thus, the maintenance termshifts across organisms of different adult size, because b = a/M − α = a/M / . Taken together thisimplies that B m ∝ M − / and that larger organisms spend less metabolic power on maintenanceper unit mass during any portion of organism growth. The above metabolic rates and ontogeneticgrowth curves allow us to calculate all relevant lifetime totals and averages for the argumentpresented in the main text and detailed below. A2. Simple Lifetime Totals
Our argument rests on the time to cancer given the resource supply rate, (cid:104) R (cid:105) , to a unit of tissue.The simplest calculation considers the simple ratio of the lifetime average metabolic rate and organism mass. The lifetime average metabolic rate is given by (cid:104) B (cid:105) = 1 t death (cid:90) t death B m ( t ) / dt (A2.1)where t death is the lifespan of an organism. Similarly, the lifetime average mass is given by (cid:104) m (cid:105) = 1 t death (cid:90) t death m ( t ) dt. (A2.2)For these calculations it is essential to understand how the lifespan changes with organism size,which can be found by considering the time that it takes to reach a given fraction of the asymptoticmass in Equation A1.3 : (cid:15)M = m ( t death ) . For any fixed value of (cid:15) this relationship scales like t death ∝ M / . (A2.3)This derived scaling of lifespan compares well with observations, where empirically it is knownthat t death = dM β with d = 2 . (years g − β ) and β = 0 . [18,23,24]. Using the scaling of lifetimeit is possible to show that the solutions to Equations A2.1 and A2.2 are given by (cid:104) B (cid:105) ∝ M / (A2.4)and (cid:104) m (cid:105) ∝ M. (A2.5)Taken together these equations imply that (cid:104) B (cid:105) ∝ (cid:104) m (cid:105) / (A2.6)which is the main result needed for the calculations performed in the main text to find that (cid:104) R (cid:105) ∝ M − / . A3. Detailed Lifetime Averages
The most accurate assessment of average lifetime energy supply to tissues is given by consideringresource supply rates at each point in ontogeny and averaging over these. This is given by (cid:104) R (cid:105) = 1 t death (cid:90) t death B m ( t ) / m ( t ) dt (A3.1) = 1 t death (cid:90) t death B m ( t ) / dt. (A3.2)This integral combined with the scaling for t death can be shown to produce (cid:104) R (cid:105) ∝ M − / (A3.3)which is the same scaling result found from taking the ratio (cid:104) m (cid:105) / (cid:104) B (cid:105) . A4. Timescale Optimization for Constant Cancer Risk
We can also turn these arguments around and ask what lifespan would result from all organismsdying of cancer. If cancer sets the timescale for death then we are interested in finding the point where t death = t c . The waiting time for cancer is given by t c = cR α (A4.1)where α = − . (see an analysis of [11] below), and this allows us to find the appropriatetimescale for death as t death = c (cid:104) R (cid:105) α = c (cid:34) t death (cid:90) t death B m ( t ) / dt (cid:35) α . (A4.2)Solving this equation for t death gives t death ∝ M . (A4.3)which again scales similarly to known lifespans across mammals [18,23,24]. This result showsthat one plausible argument for what sets lifespan scaling is the timescale for cancer formation. Inthis scenario the energetics associated with different body sizes leads to different times at whichcancer forms and eventually kills an organism. It should be noted that this result still follows fromthe underlying scaling of metabolism with body size and all of the arguments for what sets thatscaling (e.g. [20]). A5. Estimating the scaling of cancer waiting times
The explicit evolutionary simulations performed in [11] provide the connection between thelatency to cancer and the energy supply rate. We reanalyzed the data to find the scalingrelationships t c ∝ R α . (A5.1)The overall fit of this power-law to the data gives α = − . ± . . It should be noted that forlow energy supply rates the data are slightly more noisy than for higher energy supply rates,and that the higher energy supply rates converge to a different power law characterized by α = − . ± . . For our analysis we used the value of α = − . ± .01