# The epigenetic theory of carcinogenesis: investigation of the model of age-specific incidence

^{*}*Patrick A Riley*

Oncology, Totteridge Institute For Advanced Studies The Grange, Grange Avenue, United Kingdom

^{*}Corresponding Author:

Patrick A Riley

Oncology, Totteridge Institute For Advanced Studies The Grange, Grange Avenue

United Kingdom

** **
**Published on:** 2017-10-03

#### Abstract

To test the applicability of the epigenetic theory of carcinogenesis a comparison has been made between the predictions of a simple two-stage model and a limited set of age-dependent cancer incidence data. The results indicate substantial differences in the relevant variables for different tissue

#### Keywords

Cancer, Epigenetic carcinogenesis, Age-dependent incidence, Mathematical model

#### Mathematical Model

Using the previous approach **[3]** we can state that the ‘instantaneous’ transfer probability (*p1*) from normal stem cells (*S*) **[2]** to pre-malignant cells (*P*) is given by: * *

*P1= [1–exp(μt)]g (i)*

which, for small values of the mutation rate *(μ)*, is equivalent to where* g* = number of relevant genes.

Therefore, at time the size of the pre-malignant compartment is represented by:

*P(t)=S ∫_0^t(μt)^(g ) .dt *

Taking the transfer probability of pre-malignant cells (*P*) to a state of malignancy (*M*) as occurring in a small proportion (*k*) of the stem cell proliferation rate (*R*), we obtain the equivalent of the time-dependent probability density function of malignancy as:

dM/dt =1/(g+1) SkRμ^g t^(g+1)

For comparison with the age-dependent annual incidence data this analysis assumes for the sake of simplicity that the rate of transfer to the malignant compartment given by this function is equivalent to the onset of malignancy as recorded by the clinical statistics.

From the rate of transfer to the malignant compartment the cumulative incidence of malignancy can be obtained by integration:

M(t)= g!/(g+2)! SkRμ^g t^(g+2) (ii)

The computed value for M(t) enables a comparison with the lifetime risk data of **[4]**.

#### Comparison of incidence data

The comparison of data was based on the multistage approach adopted by [5,6] by calculating the best fit power law for the relevant age-dependent cancer incidence data. The relevant clinical data were analysed to obtain values for a matching power law: I = A. t x, where I is the incidence rate, A is a constant, and the best fit exponent of age = x.

In essence, equation (i) gives the probability density function corresponding to the age-specific incidence rate which, in abbreviated form, can be written as:

H = B.t y, where H is the incidence rate, B a constant, the age (in years) and the exponent y = (g+1).

From this analysis, the model parameter (g+1) was matched to an integer value close to the exponent x and other variables in the model adjusted to obtain a similar distribution curve.

#### Comparison with clinical incidence data

In order to investigate the applicability of the model, comparisons were made with the SEER data [7] in selected cases of cancer for which estimates are available for the total stem cell numbers (S) and their annual proliferation rate (R) [4], supplementary material Table S1). The values employed in the analysis are tabulated in Table (1). An annual mutation rate per gene of 10-6 was assumed in all cases. The proportion of division products expressing malignant characteristics (k) was adjusted over a wide range. The number of genes involved in rendering the stem cells susceptible to malignant transformation (g) was based on the exponent (x) of the power analysis of the SEER data (see Table 2).

The power equations calculated for the SEER data (SEER 9, age-adjusted incidence rate for all races and gender, year of diagnosis 2014) are shown as Incidence = A.t x and the corresponding model data as Incidence = B.t y, where y = g+1. The linear correlation coefficients (r2) for the comparison of the age-specific incidence curves are shown in column 6. The notes indicated in column 7 are as follows:

a. The incidence data for age 75 and 85 have been excluded. To improve the fit the calculated constant (B) includes a high value of 8 x 10-2 for k.

b. Age 85 incidence data was excluded. The estimated values for S and R were revised (3 x 1012 and 1 x 102 respectively) together with a k value of 1 x 10-2 to improve the fit.

c. Age 85 incidence data excluded.

d. Age 85 incidence data excluded.

e. Age 85 incidence data excluded. The S, R, and k values were revised as 5 x 1012, 3.5 x 101, and 1 x 10-2.

f. All the available incidence data were included.

As is evident from the data in Table (2), it is possible to generate reasonable statistical correspondence between the model and the SEER data as indicated by the values of the linear correlation coefficient in column 6. However, to permit reasonable correlation the values assigned to the various parameters that comprise the constant B differ greatly (Table 1 and notes to Table 2).

Table (3) shows the extent of the correspondence with the lifetime risk assessed [4] employing the parameter values derived from the analysis summarised in Table (2).

#### Discussion

It might have been supposed that if the essential mechanism leading to malignancy followed a standard path, that all the incidence data would be explicable by a single basic model. However, even for the tissues for which presumed stem cell numbers and proliferation rates have been estimated, the data do not readily fit a single version of the proposed model and detailed interpretation of the results summarised in Table (2) turns out to be quite problematic. Moreover, the estimates of life-time risk derived from equation (ii) differ substantially in the case of four of the six tissues examined suggesting that there are substantial variations in the parameter values adopted.

**Stem Cell Population Size (S)**

Some of the difficulty may be ascribable to the applicability of the simple stem cell model employed. It is unlikely that all the available stem cells are predetermined, since in conditions such as wound-healing and grafts it is obvious that new tissue stem cells are made available. There is also the problem of loss of cells that become defective for various reasons (including, as proposed, by elimination due to epigenetic error) so that the dynamics of stem cell turnover and regulation are likely to complicate the issue of total stem cell numbers. Thus, oversimplification of stem cell pattern imposes a serious constraint on the proposed model. Moreover, it is not clear whether clonal expansion of defective stem cells is a matter that needs to be embraced, but it is likely that some modifying factors need to be included in the model to simulate the real situation regarding the total numbers of the stem cell population.

** Stem Cell Proliferation Rate (R) **

Another important criterion is the proliferation rate of stem cells. This is a very difficult area for accurate estimates. Clearly there are many factors that will influence proliferation such as age, hormonal influences, and local tissue disturbances including inflammation, etc., and obviously such effects will influence factors such as the observed prolonged latency of some malignancies. The reduction in stem cell proliferation rate as a function of age has been investigated by [8] who showed that a linear reduction of stem cell proliferation rate with age accounts for the diminished incidence rate of cancers in the upper age groups. In the majority of cases shown in Table 2 the >75 yrs. incidence data have been excluded to take this effect in to account.

**Crucial Genes (g)**

The question of the number of genes necessary to bring about malignant behaviour poses further problems. It is highly probable that there will be a wide spectrum of genes involved in the manifestation of malignant properties across different tissues, although certain core properties may always be involved, such as migratory behaviour and proliferation. However, the present approach postulates that the specific abnormalities occur as the result of a basic fault in epigenetic pattern transmission so that it is likely that there is a limited number of crucial genes involved in all cancers. These genes, which regulate the vertical transmission of epigenetic pattern, could be directly involved in the mechanism for transmission of the DNA methylation pattern [9,10] or perhaps more likely an editing facility that excludes defective copies, such as p53 [2], [11-18], In either case it might reasonably be expected that there are one or two crucial loci involved in mutations that produce pre-malignant stem cell clones. Such a model would predict cancer incidence based on two, or more probably, four mutated genes. Variations of this would depend on pre-existing mutations.

** Mutation rate (μ)**

Another important factor is the mutation rate. It is known that some genes are more prone to mutation and there a potentially many factors influencing the mutation rate, either generally or for specific tissues that are exposed to particular environmental hazards, such as UV irradiation or products of gut organisms or hepatic metabolism.

**Probability of acquisition of malignant phenotype (k)**

Finally, given the existence of pre-malignant cells in which abnormal epigenetic patterns are expressed what is the likelihood of acquisition of malignant characteristics? It might vary according to tissue according to the ease of re-expression of certain genes. For example, melanocytes might be susceptible to re-expression of migratory characteristics, so that the probability of malignant transformation is high in comparison to osteocytes. Hence there may be variations in the anticipated probability of malignant transformation according to tissue type.

In conclusion, despite the ability to demonstrate reasonable concordance with the limited cancer incidence data so far analysed, it is not possible to conclude that the proposed model is fully supported since there are many areas of uncertainty that require clarification and possible modification. Notwithstanding this limitation, the results do not exclude the validity of the general argument advanced by the epigenetic theory of carcinogenesis.

#### Acknowledgements

I am grateful to my colleagues in the Quintox Group and to John Vince and Roger Dean for encouragement, help and advice. I thank Charles Harding for useful discussion and helpful comments concerning the mathematical model.

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