Paradox resolved: The allometric scaling of cancer risk across species

Cancer is not confined to humans, but occurs throughout the animal kingdom, and perhaps beyond [1]. This opens possibilities for deeper understanding of cancer through comparative studies and comparative oncology[2, 3]. The evolutionary thinking driven by species comparisons has been employed previously in understanding tumor growth and metastasis, and is also informative regarding cancer risk. The comparative approach thus has much to contribute to understanding 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 that the 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 some risk of each cell becoming cancerous. Larger body size is correlated with longer lifespan across mammals, and both seemingly should predispose to greater cancer incidence. The apparent lack of this pattern is known as “Peto’s paradox”. The two issues of body mass and lifespan could in principle be addressed separately, but in practice have often been discussed together. In the literature on “Peto’s paradox”, some authors have focused on body mass, (e.g., [6]), while others have 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 fewer cells) would require either a two- to three-fold decrease in the stem cell division rate, or two tumor-suppressor gene mutations. That paper compared mammalian genomes for differences in number of tumor-suppressor gene mutations. Although it reported evidence for genomic amplification of tumor-suppressor genes in several species, ranging in size from microbats to elephants, it did not find a positive correlation of tumor-suppressor genes with increasing body mass.

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 divisions have higher cancer risk [6, 9]. This result further supports the suggestion that larger animals with a greater number of stem cell divisions would be expected to have increased cancer risk, and further demands some explanation for why this does not seem to be true. Absent systematic genetic 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 by considering the scaling consequences of body mass and metabolic rate. Here we pursue this line of thinking, by including the related suggestion that cell-specific cancer risk is a function of cellular access to the resources allowing cell proliferation [11]. We therefore focus on scaling patterns for the cell-specific rate of vascular resource delivery.

Allometric scaling of the cell-specific rate of energy use by somatic tissues has been explained by the observation that, “generic properties of the vascular network constrain resource supply to cells.” [12]. In support of that explanation, empirical evidence indicates that somatic cells intrinsically tend to use energy substrates as rapidly as availability permits. Whereas rates of energy use by somatic cells in vivo are systematically lower in larger mammals, rates for cells in culture converge to a single, much higher, value for cells from all mammals regardless of body mass [13]. This has been interpreted as evidence that somatic-cell energy use and cell proliferation is extrinsically limited by the multicellular organism through control of the vascular delivery of proliferation-limiting energy substrates [11]. These authors suggested that limited vascular resource 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 of resource 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 of proliferation-limiting resources, based on reanalysis of unpublished quantitative simulation results 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 modeling results with consideration of how vascular delivery rates scale with body size, based on the network theory and ontogenetic growth framework of West, Brown et al. [14].

For our analysis we focus on cancers of adulthood, even though rapid cell proliferation during development may explain much of childhood cancer and also contribute to cancers that appear in adults. We focus on adulthood both because adult cancers comprise most cases, and because they have been the focus of the literature on how cancer risk scales with body size. Our analysis considers cancer risk in adulthood in terms of lifetime totals and averages, and thus integrates certain carryover risks from childhood while still focusing on the cancers that appear in adulthood.

The theoretical work suggesting that vascular delivery is rate-limiting for somatic cell proliferation, and thus for oncogenesis, is consistent with substantial empirical evidence. Several different substrates can be rate-limiting for somatic cell proliferation. These include energetic substrates (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 neoplastic cells, making them scarce in neoplastic microenvironments. Each of these scarce and proliferation-limiting substrates is replenished only by vascular delivery. Therefore, vascular delivery is likely to 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-limiting for cell proliferation, because all proliferation-limiting resources arrive only through vascular delivery. These considerations make rates of cell-specific vascular delivery central to questions about cancer risk.

When we consider allometric scaling relationships, which were not part of the original framing of Peto’s paradox, Peto’s observations become less paradoxical. The apparent paradox arose from the assumption that large animals are like small animals in all respects, aside from having more cells and longer life spans, and that all physiological consequences including the risk of cancer naively increase accordingly. Available empirical data do not tell us quantitatively how the cell-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 this dependence does exist, and has a strong effect [11]. Thus, Peto’s apparent paradox may be simply a 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 cell in 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 lifespans which match the waiting times for cancer. Our results indicate that the number of oncogenesis events 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 allometric relationships have only been observed, and explained, between species. Therefore, we do not share the view of Nunney [19] that changes in metabolic rates (or vascular delivery rates) should be discarded as explanations for Peto’s paradox because this explanation is inconsistent with observed intraspecific effects of body size on cancer risk.

The usual functional interpretation of metabolic scaling is that restricted vascular delivery to body tissues of larger animals benefits the organism by reducing the energy costs of vascular delivery [20]. However, given the hypothesis of Wu, Aktipis et al. (2019) regarding the role of resource-dependent evolution of somatic cells, we suggest the observed allometric scaling of vascular delivery provides a second plausible benefit in the form of suppressed per-cell cancer risk in larger mammals. When life span exceeds waiting time to cancer, cancer incidence will be substantial, and its impacts on organismal fitness may drive selection for reduced vascular delivery per cell.

How might we explain the striking similarity between the scaling exponents of life span and of cellular waiting time to cancer? If we take as a given the evolved scaling of life span, and treat the scaling of vascular delivery as an adaptation to accommodate this evolved life span, this suggests selection to restrict vascular delivery to the point that waiting time to cancer approaches the typical life span for that body size, rendering cancer unimportant in reducing life span and fitness. Future work should focus on comparing the relative selective effects of hydrodynamic efficiency versus suppression of cancer risk.

It should be noted that our work addresses only the increased cancer risk to an individual associated with maintenance, and excludes the larger number of cell proliferations during development of larger animals. Future models could create a more complete picture of cancer risk by combining this developmental aspect with the risks examined here accompanying cell proliferation for tissue renewal.

The data used in this paper are a reanalysis of “Wu DJ, Aktipis A, Pepper JW. Energy oversupply to tissues: a single mechanism possibly underlying multiple cancer risk factors. Evolution, Medicine, and Public Health. 2019;2019(1):9-16.”

JWP, CPK, and GBW conceived of the study and discussed initial ideas for mathematical 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 final manuscript.

The authors declare no competing interests.

CPK and GBW thank CAF Canada for generously supporting this work. JWP is employed by the National Cancer Institute.

The authors also thank Van Savage, Alex Herman, and Eric Deeds for insightful conversations.

The opinions expressed by the author are their own and should not be interpreted as representing the official viewpoint of the National Cancer Institute, the National Institutes of Health or the U.S. Department of Health and Human Services.

Several previous efforts have derived growth rate from metabolic rate by considering that the total energy budget of an organism is partitioned between growth and maintenance purposes:

where $E_{m}$ is the energy required to synthesize a unit of new biomass, and $B_{m}$ is the metabolic rate required to maintain an existing unit of biomass. This equation can be solved for the growth trajectory of a single organism (e.g. [14, 21, 22]), and it can also be used for finding the growth rate of an organism. If we define $a=B_{0}/E_{m}$ and $b=B_{m}/E_{m}$, then we can rewrite equation 6 as

This equation can be solved [22] to give the size trajectory of an organism

where for mammals the metabolic scaling is given by $\alpha\approx 3/4$. It should be noted that the asymptotic mass corresponds to $dm/dt=0$ in Equation 7, leading to $M^{1-\alpha}=a/b$. This asymptotic mass is achieved only as time goes to infinity. Typically, $a$ is taken to be a constant as $B_{0}$ 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 term shifts across organisms of different adult size, because $b=a/M^{1-\alpha}=a/M^{1/4}$. Taken together this implies that $B_{m}\propto M^{-1/4}$ and that larger organisms spend less metabolic power on maintenance per unit mass during any portion of organism growth. The above metabolic rates and ontogenetic growth curves allow us to calculate all relevant lifetime totals and averages for the argument presented in the main text and detailed below.

Our argument rests on the time to cancer given the resource supply rate, $\left<R\right>$, 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

where $t_{death}$ is the lifespan of an organism. Similarly, the lifetime average mass is given by

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 asymptotic mass in Equation 8 : $\epsilon M=m\left(t_{death}\right)$. For any fixed value of $\epsilon$ this relationship scales like

This derived scaling of lifespan compares well with observations, where empirically it is known that $t_{death}=dM^{\beta}$ with $d=2.34$ (years g^-β) and $\beta=0.21$ [18, 23, 24]. Using the scaling of lifetime it is possible to show that the solutions to Equations 9 and 10 are given by

and

Taken together these equations imply that

which is the main result needed for the calculations performed in the main text to find that $\left<R\right>\propto M^{-1/4}$.

The most accurate assessment of average lifetime energy supply to tissues is given by considering resource supply rates at each point in ontogeny and averaging over these. This is given by

This integral combined with the scaling for $t_{death}$ can be shown to produce

which is the same scaling result found from taking the ratio $\left<m\right>/\left<B\right>$.

We can also turn these arguments around and ask what lifespan would result from all organisms dying 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

where $\alpha=-0.89$ (see an analysis of [11] below), and this allows us to find the appropriate timescale for death as

Solving this equation for $t_{death}$ gives

which again scales similarly to known lifespans across mammals [18, 23, 24]. This result shows that one plausible argument for what sets lifespan scaling is the timescale for cancer formation. In this scenario the energetics associated with different body sizes leads to different times at which cancer forms and eventually kills an organism. It should be noted that this result still follows from the underlying scaling of metabolism with body size and all of the arguments for what sets that scaling (e.g. [20]).

The explicit evolutionary simulations performed in [11] provide the connection between the latency to cancer and the energy supply rate. We reanalyzed the data to find the scaling relationships

The overall fit of this power-law to the data gives $\alpha=-0.89\pm 0.01$. It should be noted that for low 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 $\alpha=-0.72\pm 0.01$. For our analysis we used the value of $\alpha=-0.89\pm 0.01$ found from fitting all of the data.