A Poisson distribution-based general model of cancer rates and a cancer risk-dependent theory of aging

A simple model

In a simple model of exponentially growing cell aggregates, the number of cells doubles with each generation “χ”. In each division, there is a probability “p” that each cell may experience a cancerous mutation. The probability of “m” cells simultaneously undergoing cancerization out of all the cells follows a Poisson Distribution (Figure 1A):

Figure 1. A simple model of cancerization. (A) The model of exponentially expanding cell aggregates; (B) Probability of cancerization (y) vs division times(x).

$$P(m)=\frac{{\lambda }^m}{m!e^{\lambda }};\lambda =\text{n}p;\text{n}=2^x.$$

In the healthy group, m=0. So,

$${\text{P}}_{(\text{health})}=\text{P}(0)=\frac{1}{{\text{e}}^{\lambda }}=\frac{1}{{\text{e}}^{\text{n}p}}=\frac{1}{{\text{e}}^{(2^{x}p)}}.$$ (1)

$${\text{P}}_{(\text{cancer})}=1-\text{P}(0)=1-\frac{1}{{\text{e}}^{(2^{x}p)}}.$$ (2)

The function will produce an S curve for P_(cancer). If we set p=1×10^-15, the curve will jump to 1 around the 50th generation of division (Figure 1B). Under this model, every cellular organism will eventually develop cancer. The likelihood of cancer increases as generations proliferate, with a more rapid increase occurring after a certain age.

An adapted model

Cancer incidence cannot be simply modeled using the formula above because multicellular organisms are not simple cell aggregates that proliferate exponentially without limit, and the “p” value of cancerous mutation is more complex than a constant. In this study we hypothesize that mutations accumulated in proliferating cells are the primary contributors to cancer [21]. Hence, in the updated model, “n” equals the cell turnover number during a certain period, which is not constant but rather a function of age “t”, which is corelated to cell generation. “p” is also a function of “t”. The new formula is now expressed as follows:

$${\text{P}}_{(\text{health})}=\text{P}(0)=\frac{1}{{\text{e}}^{({\text{n}}_{\text{t}}\cdot p_{\text{t}})}}$$ (3)

Based on a recent study, the average daily turnover rate of cells in a standard reference person was 0.33 trillion. Of these cells, 65% were red blood cells that lack a nucleus [10], resulting in a turnover rate of cells with active DNA replication of 0.116 trillion per day. For the purposes of this study, a yearly turnover rate of 42 trillion cells will be used for calculations (Table 1). The parameter “p_t” from formula (3) is further split into two terms: p_constant (p_c) and p_accumulate (p_a). “p_c” represents the background probability of a single cell becoming cancerous with each division, while “p_a” represents the probability of cancerization from a cell that has accumulated mutations over multiple divisions. “p_a” is a function with division generations and is determined based on a raining beads model. In this model, cells are envisioned as infinite bowls into which mutations rain down like beads with each replication. Once the number of mutations exceeds a certain threshold in a bowl, the cell becomes cancerous. The number of mutations in each bowl follows a Poisson distribution (Figure 2). The probability of exceeding the threshold “q” is calculated as the cumulative Poisson distribution in formula (4):

Figure 2. Illustration of modelling P_(accumulate) by cumulative poisson distribution.

$$p_a=1-\text{F}(\text{q})=1-{\sum }_{k=0}^q\frac{{\lambda }^q}{k!{\text{e}}^{\lambda }}$$ (4)

“λ” represents the mean of accumulated mutations per cell. “q” represents the threshold at which a cell becomes cancerous (q > λ). Multiple studies have suggested that somatic mutations increase linearly over the course of an individual’s life [5, 6]. Thus, it is reasonable to assume that with each round of replication, the number of mutations also increases proportionally, resulting in “p_a” increasing as a function of cell division generation or time (Figure 2). From formula (3), a new formula can be derived as follows:

$${\text{P}}_{(\text{health})}=\frac{1}{{\text{e}}^{n_{(\text{t})}\cdot (p_c+p_{\text{a}(\text{t})})}}$$ (5)

Here we set p_a(t) as an internal parameter that does not need to have a specific biological meaning. This internal parameter p_a(t) is used to demonstrate that the overall cancer incidence follows the cumulative Poisson distribution.

Fitting the model to observed cancer incidence

We retrieved the data of the average number of New Cases Per Year and Age-Specific Incidence Rates per 100,000 Population in UK (Cancerstats) [22]. We used these data to fit our proposed model formula (5).

n: Since the turnover number “n” was obtained from the reference Man aged between 20–30 years, we will apply “n” to the group up to age 35 (Table 1). We have no data on cell turnover in children. Since “p” is very low in the early stage of life, the impact of “n” is limited. Furthermore, considering the higher metabolism status and smaller body mass of young children, we will keep “n” the same value before age 35.

p_c: In the early stages of life, “p_a” is insignificant, and we estimated P_(cancer) as 0.05% per year based on Cancerstats data. Based on formula (5) (Supplementary Table 2), “p_c” can be deduced as 2.38E-18.

p_a: As previously discussed, the exact biological meaning of “p_a” cannot be provided at this stage. It is an internal parameter that reflects the increasing probability of cancer incidence, based on the assumption that, on average, each generation of cell division will randomly deposit equal amounts of cancer-related mutations following the Poisson distribution [23, 24]. “λ_Pa” represents the mean number of mutations for each cell, and cells will become cancerous when the number of deposited mutations reaches the threshold “q”. To calculate “p_a” using formula (4), we set “λ” equal to the cells’ generation (Table 1: λ_Pa) and tried different thresholds “q” until the model best matched real cancer rates (with the highest R²). Ultimately, we set “q = 118”, which means that a cell requires 118 mutations to become cancerous, assuming it receives one mutation from each division (Figure 3A). However, we cannot define “q” as the count of physical mutations since it remains an internal parameter. In our current framework, we propose to define “q=118” as effective mutations, which may be related to driver mutations but encompass more than that, although still fewer than the entire spectrum of somatic mutations since many somatic mutations may not be effective for tumorigenesis.

Figure 3. Comparison of model predicted data with real data of cancer incidence vs age. (A) The real data and predicted data were compared in the year group 0-35, considering different values of “q.” (B) The real data and predicted data were compared across all age groups using q=118, with or without considering cell turnover reduction. (C) The predicted data were plotted under log(probability) vs Log(age). Data points from age group 25 (20-25) to group 75 (70-75) (red dots) exhibit a linear trend with a slope of 5.8. (D) The real data and predicted data of cumulative cancer incidence were compared throughout the entire lifespan, with or without considering cell turnover reduction.

Determining the generation of cells at different ages presents a challenge, as cells from various tissues may have different developmental histories. Additionally, differentiated and stem cells may have distinct division cycles. We provided an average estimate of cell generation in different age brackets to assist in building the model and prove that cancer incidence follows our mathematical hypothesis. From the fertilized egg to the newborn infant, cells proliferate exponentially, and the newborn has a total of two trillion cells [25], meaning it has undergone 41 generations of divisions (Supplementary Table 2). For the first five years of life, it requires at least another four generations, and we set λ_Pa = 45 for this age group. For the 5-10 and 10-15 age groups, we set one generation for each stage. Above this age, we set 0.5 generation for each stage until it reached the Hayflick limitation of 55 [26].

By setting n, p_c, p_a, and using formulas (4) and (5), we can model the five-year cancer incidence (Figure 3A and Table 1). We used q=118 for further analysis.

Final adaption of the model to account for reduced cell turnover

The predicted incidence of cancer exceeded the observed data beyond the age of 35 (Figure 3B). This occurred because Formula (5) cannot always use the same “n.” As people age, cell division and turnover rates decrease [5]. As no real data on cell turnover in aging people are available, we determined the turnover decrease rate by assuming the validity of our model. We found a 25% decrease per five years in the 35-75 age group, a 35% decrease per five years in the 75-85 age group, a 40% decrease per five years in the 85-90 age group, and a 50% decrease per five years in the group aged over 90 years (Table 1: n turnover/year). The model accurately fits the observed data since the reduction was reversely deduced (Figure 3B). Therefore, it is feasible to use a general theory-based model to match cancer incidence. This model authentically reflects the decreased cancer incidence in the very aged group [5].

While we consider reducing cell turnover to fit the overall cancer incidence, it is important to acknowledge that different tissues may exhibit varying “np” values due to differences in cell turnover rates or developmental asymmetries in cell lineage trees [27]. Several studies have reported that cancer rates exhibit exponential growth by six powers of “t” [3, 7]. Fisher and Hollomon’s pioneering study of stomach cancer found that ΔLog(p)/ΔLog(age) has a slope of 5.7 between the ages of 20-75 [2]. It is worth noting that the “np” model, without considering cell turnover reduction, also yielded a straight line with a slope of 5.8 from Group 25(20-25) to Group 75 (70-75), which precisely matches Fisher’s case (Figure 3C and Supplementary Table 2). This implies that stomach tissue may not experience an apparent reduction in cell turnover during this age period.

A theory of aging based on the cancer model

If we convert the cancer incidence shown in Figure 3B into cumulative incidence, we get Figure 3D. From this figure, we can see that reduced cell turnover offers advantage in terms of survival. The model indicates that without cell turnover reduction, humans would reach a 50% cancerization rate at age 66, but with cell turnover reduction, the 50% cancerization rate is delayed by two decades to age 87-89 (Figure 3D). This gives us a hint of the ultimate cause of aging, which is based on the unavoidable increase of cancer risk.

Here, we propose an “np” theory of aging. Cells are highly ordered systems, and to maintain cell fitness (youth), the order needs to be maintained, which can be described as an issue of entropy balance [28]. A cell always gains positive entropy, which needs to be reconciled to defy the second law of thermodynamics. Three levels of entropy are postulated here: (1) metabolic entropy; (2) structural entropy; and (3) information entropy. (1) For any living cell, metabolism is the function that maintains energy/matter intake and output. The entropy at this level is balanced biochemically. (2) With time, the microstructure of the cell or cellular organelles experience “wear and tear”. The generation of new cells through division is the final resort to fix this “wear and tear” and reduce structural entropy. (3) However, irreversible random changes accumulated in the genetic material that cannot be fixed will be passed to the progeny cell, leading to an increase in information entropy. The accumulated information entropy will ultimately succumb to the second law of thermodynamics. The increase in information entropy finally destabilizes the regulation of the cell and leads to unregulated proliferation, resulting in cancer [29, 30]. From another perspective, we can categorize cellular information into two arms: pro-proliferation and pro-regulation. Genetic mutations randomly impact either arm, but only the disruption to the pro-regulation arm will be selected for. With the increase of information entropy, the highly regulated eukaryotic cells will return to a more primitive prokaryotic-like status [29]. This theory of information entropy predicts that any multicellular system will eventually develop cancer. As a result, the total number of cells that can be usefully generated from a single zygote is finite. To minimize the risk of cancer, at the later stage of a species’ lifespan, cell turnover is reduced or stopped. The negative entropy introduced into the cells via division cannot balance the positive entropy produced by the system, leading to increased disorder in cellular structure and metabolism. When this happens, the entropy of the whole system increases, the fitness of the organism decreases, and aging occurs.

This theory of aging predicts the ultimate number of cells a given individual can use is “N”. “N” is restricted by “p”. The predetermined number “N” can be plotted as an enclosed area on the “n” and “t” graph (Figure 4A). For the same reason, the quality of reproductive cells is also restricted by the same law [31]. Hence, all species has a limited period of reproduction. Species will develop different ways to use this cell resource strategically, which forms the basis of an organism’s lifespan and aging process. We list three models of survival strategies for species with three typical lifespans.

Figure 4. “np” theory of aging among different species. (A) Theoretical “nt” plot of model I, II, III species; (B) Post reproduction life vs expected life span of 51 mammal species; (C) The percentage of post reproduction life to whole life: human against the other mammals. T test was used to calculate statistical significance.

Model I: Species with short lifespan and short post-fecundity life. Low fitness is not acceptable for these species. Model I species have a very short half-life of survival in the natural environment, so there is not much evolutionary pressure for longevity. Their natural lifespan is compatible with their survival rate, with “nt” curve has a small area on the plot. The model species are rodents.

Model II: Species with medium to long lifespan and short post-fecundity life. Low fitness is not acceptable. If the species adapt to a strategy where longevity is favored, they are allowed to have more “N”, which enlarges the enclosed area on the “nt” plot (Figure 4A). This process can continue under evolutionary pressure until the advantage of longevity is canceled out by the cancer risk. These species are stronger and have a higher chance of survival for a longer period, so evolution gives them more predetermined cells in their lifespan. However, lifespan is still restricted by the risk of cancer. Eventually, the organism will shut down cell proliferation quickly and no longer compete for survival. The model species are large carnivores.

For Models species I and II, after the reproductive period, the organism undergoes aging, leading to a rapid decline in fitness, which typically results in death in the wild. Their lifespan matches the disposable soma theory [20].

Model III: Species with long lifespan and a long post-fecundity life. Low fitness is acceptable. Few species are extremely favored by longevity, however, a longevity strategy may be evolutionarily favored by the “grandma effect” [32–34], where longevity may provide community benefit. We hypothesize that the “N” reaches an evolutionary limit, but the Model III species develop another strategy for using the available “N” by reducing cell turnover at the cost of lower fitness. This type of species has an elongated senescence period among all species. All of them are social and intelligent species, where survival with low independent fitness is possible in the context of a community. This also offers an explanation for the brain weight theory, which found that lifespan was positively related to species’ brain weight [35].

To support this theory, we re-explored the data from Samuel Ellis and Darren P. Croft about the reproductive lifespan and post-reproduction lifespan of 51 mammals [36]. The post-reproduction lifespan vs. total expected lifespan was plotted (Figure 4B). If we divide the species into three groups based on their expected lifespan on the x-axis and two groups based on post-reproduction life on the y-axis, 49 out of 51 species fall into three groups (Supplementary Table 1). These three groups represent aging strategy models I, II, and III, respectively. We note that humans have the highest post-reproductive lifespan and the highest percentage of post-reproductive time (Figure 4C), suggesting that humans have a unique position in evolution and that longevity is highly favored in this species.

Images and graphs

Figures 1A, 2 were plotted by Biorender. Figure 1B was graphed using the Desmos Graphing Calculator (https://www.desmos.com/calculator).

Calculation

The Keisan online calculator (https://keisan.casio.com/exec/system/1180573179) was used to calculate the cumulative value of the Poisson distribution p_(a) for Table 1 and the Supplementary Table 2. The coefficient of determination was calculated as R² = 1- (RSS/TSS). RSS was the sum of squares of residuals, while TSS was the sum of squares of theoretical incidence. $\text{RSS}={\sum }_i^n{({\text{P}}_{\text{observed}}\text{i}-{\text{P}}_{\text{model}}\text{i})}^2;\text{TSS}={\sum }_i^n{({\text{P}}_{\text{model}}\text{i})}^2$.

The calculation methods for Table 1: In the Poisson distribution calculator, percentile x=118, mean λ=λ_Pa (data from Table 1). p_(a) of specific generation was calculated from the difference of neighboured “upper cumulative Q”. For the “0-5” group, put “percentile x” =118 (q), which is the constant threshold. Generation 45 is the “mean λ”. 1.18E-19 is output as upper cumulative Q₍₄₅₎. For next group (5-10), mean λ of 46 is used to get Q₍₄₆₎ = 5.86E-19. The cancerization probability in each age group is (Q_n+1 - Q_n)/(1 - Q_n). Since Q is very small, (Q_n+1 - Q_n)/(1 - Q_n) ≈ Q_n+1 – Q_n = 5.86E-19 - 1.18E-19 = 4.69E-19, which is the p(a) for “5-10” group, so forth, to calculate p_(a) for every group. Put “p_(a)”, “p_(c)” and “n” into formula (5) to get P(0). P_(cancer) /year= 1-P(0). P_(cancer)/5 years (%) = [1-(1-P_(cancer) /year)⁵]× 100.