The theoretical foundations of the modern MMC originate from early observations by Fisher and Hollomon (Fisher and Hollomon, 1951) and Nordling (Nordling, 1953) that the age distribution of cancer death rates follows the sixth power of age. Nordling (Nordling, 1953) then formulated the theory and postulated that cancer develops as a sequence of mutations that transform normal cells into malignant cells. Later, Armitage and Doll (Armitage and Doll, 1954) supported this idea and proposed a mathematical model to substantiate it. As nothing was known about stem cells and the organization of tissue renewal at the time, Armitage and Doll’s model explained the probability of sequential mutation accumulation from a single cell perspective, being unaware of the effects of clonal selection on these probabilities. Armitage and Doll’s model, as they acknowledged, also required that the rate of mutation accumulation is the same throughout lifespan, which was a reasonable assumption at the time. Multiple studies have later challenged this assumption, showing that ~50% of mutations accumulate before maturity (Dollé et al., 2000; Finette et al., 1994; Giese et al., 2002; Horvath, 2013), although this early life pattern of mutation accumulation is not universally observed (Welch et al., 2012; Blokzijl et al., 2016; Osorio et al., 2018). The subsequent deceleration of mutation accumulation is now explained by the considerable slowdown of stem cell division rate upon maturation (Bowie et al., 2006; DeGregori, 2013; Rozhok and DeGregori, 2015; Sidorov et al., 2009). This departure from Armitage and Doll’s original assumptions has been proposed to interfere with the MMC performance (DeGregori, 2013; Rozhok and DeGregori, 2015), but this has never been tested and is not considered by many cancer researchers.

None of the above-mentioned early founders of the theory had an understanding of the diversity of cancer driver mutations at the time. Armitage and Doll therefore assumed a process driven by approximately six sequentially acquired mutations, based on cancer incidence increasing as the sixth power of age, and mentioned that many cancers behave analogously, with a supposedly similar underlying process of carcinogenesis. Further research, however, demonstrated that cancers are driven by very different mutations and require very different numbers of drivers, and yet these cancers still show very similar temporal distribution with age. Later, the theory was solidified by Peter Nowell who developed the concept of clonal selection during carcinogenesis (Nowell, 1976) and put the model within its current general framework. Some fundamental problems, however, were already noticed by early theorists, when Fisher and Hollomon mentioned that childhood cancers drastically deviate from the observed age-dependent pattern and are difficult to explain (Fisher and Hollomon, 1951). An additional complication that early theorists were unaware of is the fact that different cancers originate from very different stem cell systems, with different lifetime stem cell numbers and cell division profiles. More recent models of age-dependent cancer incidence do operate with the stem/progenitor cell paradigm of cancer origins (Calabrese and Shibata, 2010; Gerstung and Beerenwinkel, 2010; Michor et al., 2004; Beerenwinkel et al., 2007; McFarland et al., 2014). However, the analytical models used so far have not been tested for their ability to explain age-dependent incidence for cancers driven by different numbers of mutations and originating in stem cell pools of varying sizes. Some models are limited to the single-cell perspective (Calabrese and Shibata, 2010) and do not account for the effect of clonal expansions in the successive accumulation of mutations. Other models are limited to modeling a typical (generic) cancer process or to a single cancer. Models comparing and unifying the principles of cancer progression of multiple cancer types and across multiple species are lacking. Critically, the current MMC lacks consideration of fundamental evolutionary processes shaping the evolution of animal life history traits (such as longevity, body size, and reproductive strategies) and, as such and despite multiple claims otherwise, has so far not been placed within the framework of evolutionary theory.

While different cancers vary significantly in lifetime risk and total incidence, most cancers have very similar temporal incidence dynamics, demonstrating approximately the same fractional increases in incidence with age. Given the vastly different etiologies and tissues of origin, this temporal incidence similarity poses a great problem to the current multi-stage model. Cancers driven by one mutation, such as chronic myeloid leukemia (CML) in chronic phase, exhibit age-dependent kinetics of incidence that are quite similar to cancers that require many more mutations, including leukemias like acute myeloid leukemia (AML) and chronic lymphoblastic leukemia (CLL) thought to also originate in hematopoietic stem cell (HSC) pools and colon cancers which originate in highly fragmented stem cell pools (Figure 1A–C). Moreover, hematopoietic clones driven by a single driver mutation, also thought to originate in HSC pools, have also been shown to increase with age, following a very similar incidence increase as mutationally more complex cancers (Figure 1D). Solutions have been proposed to explain the exponential age-dependent increase of single-mutation cancers (Michor et al., 2006), however the tuning of the incidence of multiple cancers with various numbers of driver mutations and originating from vastly different stem cell pools has not yet been explained. This lack of explanation is further aggravated when another dimension is added – the incidence of different cancers across different species. As in humans, cancer incidence closely follows the physiological aging curve in other animals (Albuquerque et al., 2018). Incidence thus scales to lifespan, but not to body size, a problem demonstrating significant deviations of cancer incidence from cell numbers and known as Peto’s paradox (Peto et al., 1975).

Figure 1

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We have previously proposed that explaining the above mentioned problems of the MMC requires incorporation of evolutionary theory (DeGregori, 2013; Rozhok and DeGregori, 2015; DeGregori, 2011; Rozhok and DeGregori, 2016). Somatic selection in tissues, just like selection in populations, depends on tissue microenvironment. The long co-evolution of stem cells and tissue regulatory processes should have adapted stem cells to the well-regulated microenvironments of a fit body, which should make the normal cells highly fit and should promote stabilizing selection and purifying selection against most phenotype-altering mutations. Aging disrupts tissue ecosystems, and like environmental change for organismal populations, should promote positive selection for random mutants, including oncogenic mutations, that are adaptive to the new microenvironments.

In the present study, we employ the power of the Monte Carlo method to investigate the effect of various factors, such as mutation rate, cell division profiles, the impact of somatic mutations on somatic cell fitness, and the effect of aging on somatic selection and multi-stage carcinogenesis in order to test the modern MMC model and determine if age-dependent alterations in selection are necessary to explain cancer incidence patterns. We assume in this study that the character of age-dependent clonal evolution and clonal expansions is tightly linked with age-dependent cancer incidence, which has been shown for the hematopoietic system (McKerrell et al., 2015; Genovese et al., 2014; Jaiswal et al., 2014; Xie et al., 2014). Our results demonstrate that the MMC model holds generally and is robust against departures from mutation accumulation linearity required by the original Armitage-Doll model. The MMC thus reproduces the general exponential shape of cancer incidence and clonal dynamics regardless of whether stem cell divisions remain stable or change dramatically during body maturation. However, differences in the required number of driver mutations, mutation rate and driver strength cause the MMC to fail to reproduce the expected temporal correlation in the incidence of different cancers. The current MMC is also incapable of explaining the incidence of childhood leukemia. We demonstrate that differential aging-dependent somatic selection improves the performance of the MMC and partially resolves these problems. We propose additionally a theoretical model for how reproductive success and the evolution of life history traits shape the evolution of species-specific and cancer etiology-specific tumor suppressor mechanisms. We argue that the latter mechanism should complement aging-dependent alterations in somatic selection and resolve the observed problems in MMC performance. We generalize the MMC, aging-dependent general shifts in tissue-level tumor suppressive processes, and the evolution of group-specific cellular mechanisms into a theoretical framework that we propose to better explain cancer incidence both across multiple cancer types and in multiple species.

Quick guide to model

Our model operates with a simulated pool of cells of varying dynamic or stable sizes competing for niche space as one effective population. This design most closely replicates clonal processes in the human HSC system of the bone marrow. Multiple studies demonstrate that HSCs are effectively one population of cells that divide, differentiate and compete for a limited bone marrow niche space (Abkowitz et al., 1996; Abkowitz et al., 2000; Catlin et al., 2011). Their spatial segregation in different bones is compensated by the HSC’s well-established proclivity to migrate (Wright et al., 2001), making their competition to a large extent uniform across the body over time. HSCs represent a good population for modeling somatic evolution, as most leukemias are believed to initiate in the HSC compartment (Fialkow et al., 1967; Jan et al., 2012; Kikushige et al., 2011; Miyamoto et al., 2000; Shlush et al., 2014). The chart of modeled events is shown in Figure 2A and represent a tree of possible scenarios and cell fate decisions during each update of the model. Updates are ‘weekly’ and continue for a total of 100 years of the simulated lifespan. The size of the simulated SC pool increases early in life and reaches its adult size by 18–20 years of age (Figure 2B upper chart). Cell division rates change in a similar age-dependent manner (Figure 2B lower chart) during body growth and maturation (Bowie et al., 2006; Sidorov et al., 2009), or division rates are kept stable (for experimental purposes). While the parameters used were derived from estimates for human HSC, relative changes in pool size and division rates should be similar for other tissues and other animals, given the rapid increase in body size from fetus to adult.

Figure 2 with 1 supplement see all

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We simulated clonal dynamics under two different paradigms: a) driver mutations have a constant age-independent driving potential and always increase cellular somatic fitness, and b) the selective advantage conferred to cells by driver mutations depends on age, being more positive late in life and negative or neutral during the pre- and reproductive periods, as shown in Figure 2C. Each chart in Figure 2C from top to bottom shows three ways of manipulating age-dependent shifting selection: (a) selection curve shift along the fitness effect axis, varying the relative strength of early negative and late positive selection for somatic cell mutants, as this ratio, if the modeled selection shift exists in nature, is unknown; (b) varying the absolute general magnitude of selection, which is unknown as well; (c) age distribution of the selection shift, emulating differences in evolved lifespans.

We tracked the proportions of clones carrying different numbers of oncogenic mutations (1 through 4) under the influence of several primary factors:

(a) cell division speed and profile. HSC and other tissue stem cells are known to become much more quiescent after body maturation (Bowie et al., 2006; Sidorov et al., 2009), resulting in 40–50% of mutations accumulating before maturity (Dollé et al., 2000; Finette et al., 1994; Giese et al., 2002; Horvath, 2013) and thus departing from the assumption of linear mutation accumulation with age made by Armitage and Doll (Armitage and Doll, 1954) (but see also Welch et al., 2012; Osorio et al., 2018; Blokzijl et al., 2016). It has been proposed that such a departure should impact the timing of cancers (DeGregori, 2013; Rozhok and DeGregori, 2015). We simulate both a dynamic cell division profile (resulting in more mutations occurring early in life) and a stable cell division profile (which would lead to more linear mutation accumulation with age).

(b) a range of mutation rates, in order to test the MMC model behavior in tissues with presumably differing mutation loads. Noteworthy, functional mutations in the model incorporate all somatically-heritable phenotypic changes in the cell, not just nucleotide substitutions. Therefore, the simulated mutation rate does not have to match the rates observed for DNA mutations.

(c)the effect of pool size in order to explore the MMC behavior relative to body mass increase (to address Peto’s paradox). Estimates for the pool size for HSC range from about 10,000 to several hundred thousand in humans (Abkowitz et al., 2002; Lee-Six et al., 2018). Notably, we are modeling pool sizes even more generally here, to encompass different animals;

(d) the strength (selective advantage) of driver mutations, which varies for different natural oncogenic mutations. We simulate two different assumptions: (1) fixed fitness effects of mutations as utilized in many modeling studies and by modern MMC theorists (Bozic et al., 2010; Tomasetti et al., 2015; Vogelstein et al., 2013); and (2) age-dependent varying strength of selection acting on oncogenic mutations, which is a theoretical extension of the MMC model with age-dependent Differential Selection, so we designate it as MMC-DS. As we have argued earlier, one of the central paradigms in evolutionary theory posits that selective advantage is not a fixed attribute of genetic change, but is a dynamic property arising at the interface of the resulting phenotype and environment (Rozhok and DeGregori, 2015). Tissue microenvironment, and thus the signaling that controls stem cell fate decisions, change in an aging-dependent manner and thus should impact somatic selection differentially as a function of physiological aging. Additional dynamical patterns of changing fitness effects of mutations are certainly possible (such as reversal of positive selection), but those will not be explored here. Cell intrinsic processes that lead to cell aging should also promote stronger positive selection processes by broadening the distribution of individual cell somatic fitness driven by variations in the accumulation of cellular damage among cells. In combination with microenvironmental changes, these processes should lead to generally increased rates of directional selection in large stem cell pools, such as for hematopoiesis.

Our results indicate that the general principle of successive cell transformations proposed by Nordling, Armitage and Doll (Nordling, 1953; Armitage and Doll, 1954) and later developed into the modern Multi-Stage Model of Carcinogenesis (MMC) recapitulates the late-life exponential increases in the rates of somatic evolution regardless of the non-linear pre-maturity shifts in stem cell division rates. The model, however, requires the incorporation of differential aging-dependent somatic selection acting on somatic cell variants, a principle earlier proposed as a key postulate of the Adaptive Oncogenesis model (DeGregori, 2011), in order to universally postpone increased rates of somatic evolution to late ages, independent of the number of driver mutations and the magnitude of the somatic selective value of these mutations. This principle of aging-dependent somatic selection can potentially explain the scaling of increased cancer incidence to the species-specific lifespan, such that most cancers occur after 1.5 years for mice and 50 years for humans.

The current MMC does not explain a very puzzling phenomenon – that most cancers show strikingly similar late life patterns of incidence, despite the fact that these cancers require very different numbers of driver mutations and originate in very different sized/organized stem cell pools (whether for different tissues or in different species). Previous attempts to explain the age-dependence of multi-stage carcinogenesis have mostly used analytical modeling, with fixed effects of mutations on cellular fitness (Calabrese and Shibata, 2010; Gerstung and Beerenwinkel, 2010; Michor et al., 2004; Beerenwinkel et al., 2007; McFarland et al., 2014). However, fitness is a dynamic phenomenon that arises at the interface of phenotype and environment and is determined by the mode of selection acting on a phenotype, the latter being typically defined by environment (Wade and Kalisz, 1990). While with analytical modeling parameters can be derived that fit the late-life curve for a particular cancer, these studies do not explain what evolutionary process/mechanism fine-tuned the occurrence of multiple cancers of vastly different etiology and harboring different numbers and spectra of mutations.

The MMC-DS model also demonstrates early life increases in the rates of somatic evolution, mirroring the risk of early childhood cancers, which does not arise from the standard MMC model. As we have argued before (Rozhok et al., 2016), these early-life processes should be driven by drift due to the small initial pool of stem cells in a fetus and right after birth. Such domination of drift has also been shown experimentally for tissues in which stem cell compartments are fragmented into small local populations (Vermeulen et al., 2013). Later suppression of somatic evolution until older ages can be explained by purifying somatic selection that acts in larger adult stem cell pools during reproductive periods of life-spans. Consistent with this logic, early childhood cancers mostly comprise malignancies originating in large stem cell systems, such as hematopoietic, while childhood carcinomas, which originate in the small fragmented epithelial stem cell compartments where drift dominates, are rare (www.seer.cancer.gov) (Rozhok et al., 2016). Unlike large pool SC systems, the strong presence of drift should limit positive somatic selection in fragmented epithelial SC compartments throughout lifespan. As modeled in the MMC-DS setup, oncogenic mutations that are negatively selected in youth can be positively selected late in life, most likely due to alterations in microenvironments which alter the adaptive value of mutations. Indeed, previous studies have demonstrated how the somatic fitness effects of oncogenic mutations can be dramatically affected by the age of the host (Henry et al., 2010; Henry et al., 2015; Parikh et al., 2018; Vas et al., 2012).

Nevertheless, the key question that arises from our results and within the general MMC model is what mechanism underlies the temporal coincidence of the increased cancer rates among cancers of vastly different etiology driven by different driver mutations and different numbers of driver mutations? The general MMC-DS architecture resolves the problem of early postnatal elevated rates of somatic evolution and the scaling of somatic evolution rates to lifespan, with increased rates of somatic evolution delayed until late in life. However, the general MMC-DS principle is not sufficient to overcome the problem of different numbers of driver mutations for different cancers with similar age-dependent patterns. For example, and as shown in Figure 3A, the incidence curve for a 1-driver cancer is significantly earlier than for a 4-driver cancer, while the same is not observed for actual cancer incidence curves (Figure 1C). While it is difficult to imagine what evolutionary forces could have ‘tuned’ the incidence of vastly discrepant cancers within the standard MMC paradigm, we propose a theoretical generalization to explain this phenomenon, which integrates the current MMC theory, the principle of aging-dependent shifts in somatic selection, and the group-specific evolution of cellular machinery that depends on the evolution of life history traits. This generalized theory can also explain the early childhood cancer incidence peaks and Peto’s paradox. We postulate that the following principles should operate in shaping the general character of somatic evolution in order to resolve the current problems of the MMC:

The principle of a mutual feedback between germline and somatic selection

Based on the above presented results, we propose that somatic selection in the body and germline selection in a population are directly linked with mutual feedback acting on each other through reproductive success as shown in Figure 6. Somatic selection, as the main determinant of somatic evolution in a tissue, impacts age-dependent tissue and individual fitness, and thus provides the fuel for germline selection operating on individual fitness. In this way, the effect of somatic mutations on cell somatic fitness is subject to germline selection based on an individual’s fitness/health risks conferred by particular somatic mutations. Reproductive success provides a quantitative effect whereby the strength and directionality of germline selection acting on particular somatic cell alterations is determined by the overall reduction in reproductive success of an individual. As demonstrated in Figure 6, such cost diminishes with age proportionally to the likelihood that an individual of a particular species will die at a specific age of causes less unrelated to health (such as predation). Therefore, germline selection acts on the effects of somatic mutations on cellular physiology depending on a species-specific general age-dependent mortality profile.

Figure 6

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Accordingly, the somatic fitness effects of different mutations should differ within a species. Principle #1 can help explain the similar age-dependent incidence curves shown in Figure 1, despite the fact that these cancers originate in very different sized stem cell pools and require very different numbers of driver mutations. For example, in the MMC-DS(b) setup in Figure 4A, the expansion of a 1-mutation clone (light blue curve in the leftmost chart) will approximate the expansion of a 4-mutation clone (green curve in the rightmost chart), if germline selection ‘equalizes’ the effect of the single mutation in the first case and the combined effects of the four mutations in the second based on their overall health risk, following the logic presented in Figure 6. Basically, germline selection acting to determine the fitness effects of each somatic mutations functions as the ‘invisible hand’ that pushes cancer risk to older ages where reproductive success is low.

Primary data related to this submission are publicly available at Dryad: https://dx.doi:10.5061/dryad.6ff5k07

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Author details

1. Andrii Rozhok

   Department of Biochemistry and Molecular Genetics, University of Colorado Anschutz Medical Campus, Aurora, United States

   Contribution

   Conceptualization, Data curation, Software, Formal analysis, Investigation, Methodology, Writing—original draft, Writing—review and editing

   For correspondence

   andrii.rozhok@ucdenver.edu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-9475-0472

2. James DeGregori

   1. Department of Biochemistry and Molecular Genetics, University of Colorado Anschutz Medical Campus, Aurora, United States
   2. Department of Immunology and Microbiology, University of Colorado Anschutz Medical Campus, Aurora, United States
   3. Department of Pediatrics, University of Colorado Anschutz Medical Campus, Aurora, United States
   4. Department of Medicine, Section of Hematology, University of Colorado Anschutz Medical Campus, Aurora, United States

   Contribution

   Conceptualization, Data curation, Funding acquisition, Project administration, Writing—review and editing

   For correspondence

   james.degregori@ucdenver.edu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-1287-1976

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