Mathematical models of tissue stem and transit target cell divisions and the risk of radiation- or smoking-associated cancer

<div><p>There is compelling biological data to suggest that cancer arises from a series of mutations in single target cells, resulting in defects in cell renewal and differentiation processes which lead to malignancy. Because much mutagenic damage is expressed following cell division, more-rapidly renewing tissues could be at higher risk because of the larger number of cell replications. Cairns suggested that renewing tissues may reduce cancer risk by partitioning the dividing cell populations into lineages comprising infrequently-dividing long-lived stem cells and frequently-dividing short-lived daughter transit cells. We develop generalizations of three recent cancer-induction models that account for the joint maintenance and renewal of stem and transit cells, also competing processes of partially transformed cell proliferation and differentiation/apoptosis. We are particularly interested in using these models to separately assess the probabilities of mutation and development of cancer associated with “spontaneous” processes and with those linked to a specific environmental mutagen, specifically ionizing radiation or cigarette smoking. All three models demonstrate substantial variation in cancer risks, by at least 20 orders of magnitude, depending on the assumed number of critical mutations required for cancer, and the stem-cell and transition-cell mutation rates. However, in most cases the conditional probabilities of cancer being mutagen-induced range between 7–96%. The relative risks associated with mutagen exposure compared to background rates are also stable, ranging from 1.0–16.0. Very few cancers, generally <0.5%, arise from mutations occurring solely in stem cells rather than in a combination of stem and transit cells. However, for cancers with 2 or 3 critical mutations, a substantial proportion of cancers, in some cases 100%, have at least one mutation derived from a mutated stem cell. Little difference is made to relative risks if competing processes of proliferation and differentiation in the partially transformed stem and transit cell population are allowed for, nor is any difference made if one assumes that transit cells require an extra mutation to confer malignancy from the number required by stem cells. The probability of a cancer being mutagen-induced correlates across cancer sites with the estimated cumulative number of stem cell divisions in the associated tissue (<i>p</i><0.05), although in some cases there is sensitivity of findings to removal of high-leverage outliers and in some cases only modest variation in probability, but these issues do not affect the validity of the findings. There are no significant correlations (<i>p</i>>0.3) between lifetime cancer-site specific radiation risk and the probability of that cancer being mutagen-induced. These results do not depend on the assumed critical number of mutations leading to cancer, or on the assumed mutagen-associated mutation rate, within the generally-accepted ranges tested. However, there are borderline significant negative correlations (<i>p</i> = 0.08) between the smoking-associated mortality rate difference (current vs former smokers) and the probability of cancer being mutagen-induced. This is only the case where values of the critical number of mutations leading to cancer, <i>k</i>, is 3 or 4 and not for smaller values (1 or 2), but does not strongly depend on the assumed mutagen-associated mutation rate.</p></div

Schematic diagram of generalized cancer model with <i>k</i> mutations, allowing for mutations in stem cell and transit cell compartments.

<p>This is a special case of the fully-stochastic destabilization model developed by Little <i>et al</i>. [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005391#pcbi.1005391.ref013" target="_blank">13</a>].</p

Schematic diagram of stem cell model corresponding to that of Wu <i>et al</i>. [11].

<p>The single initial stem cell divides symmetrically <i>n</i>₁ times to produce stem cells. Each cell then divides asymmetrically <i>n</i>₂ times.</p

Probabilities of cancer for various sites, conditional probability of a cancer being mutagen induced (expression (9)), and the relative risk (expression (10)), assuming a spontaneous mutation rate (<i>u</i>_<i>S</i>) = 10⁻⁸ per cell division, and <i>k</i> = 2 to 4 critical cancer genes, mutagen-associated rates increase from 0 after the first third of stem cell divisions, using a generalization of the model of Wu <i>et al</i>. [11].

<p>Assumptions as to the number of symmetric (<i>n</i>₁) and asymmetric cell divisions (<i>n</i>₂) are as for the paper of Wu <i>et al</i>. [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005391#pcbi.1005391.ref011" target="_blank">11</a>].</p

Probability [at least one mutation is mutagen-induced | cancer occurs] versus smoking-associated cancer risk (using data taken from Doll <i>et al</i>. [18]).

<p>The conditional probability is evaluated (via expression (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005391#pcbi.1005391.e016" target="_blank">9</a>)) using a generalization of the model of Wu <i>et al</i>. [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005391#pcbi.1005391.ref011" target="_blank">11</a>] using <i>k</i> = 2 or <i>k</i> = 4 cancer stages, a spontaneous mutation rate of 10⁻⁸ per cell division, and a mutagen-induced mutation rate of 2 x 10⁻⁹, 5 x 10⁻⁹ or 1 x 10⁻⁸ per cell division, and mutagen-associated rates increase from 0 after the first third of stem cell divisions. The data used are given in Table A1 in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005391#pcbi.1005391.s001" target="_blank">S1 Appendix</a>.</p

Linear regression analysis of probability[at least one mutation is mutagen-induced | cancer occurs] (dependent variable) versus smoking-associated cancer risk (using data taken from Doll <i>et al</i>. [18]) (independent variable).

<p>The conditional probability is evaluated (via expression (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005391#pcbi.1005391.e016" target="_blank">9</a>)) using the model of Wu <i>et al</i>. [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005391#pcbi.1005391.ref011" target="_blank">11</a>] using <i>k</i> = 1 to 4 critical cancer mutations, a spontaneous mutation rate of <i>u</i>_<i>S</i> = 10⁻⁸ per cell division, and a mutagen-induced mutation rate, <i>u</i>_<i>M</i> = 2 x 10⁻⁹, 5 x 10⁻⁹ or 1 x 10⁻⁸ per cell division, mutagen-associated rates increase from 0 after the first third of stem cell divisions. The data used in the regression are given in Table A1 in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005391#pcbi.1005391.s001" target="_blank">S1 Appendix</a> and in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005391#pcbi.1005391.t002" target="_blank">Table 2</a> of Little <i>et al</i>. [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005391#pcbi.1005391.ref009" target="_blank">9</a>].</p

Dataset for: Correlated Poisson Models for Age-Period-Cohort Analysis

Age-period-cohort (APC) models are widely used to analyze population-level rates, particularly cancer incidence and mortality. These models are used for descriptive epidemiology, comparative risk analysis, and extrapolating future disease burden. Traditional APC models have two major limitations: 1) they lack parsimony because they require estimation of deviations from linear trends for each level of age, period, and cohort; and 2) rates observed at similar ages, periods, and cohorts are treated as independent, ignoring any correlations between them that may lead to biased parameter estimates and inefficient standard errors. We propose a novel approach to estimation of APC models using a spatially-correlated Poisson model that accounts for over-dispersion and correlations in age, period, and cohort, simultaneously. We treat the outcome of interest as event rates occurring over a grid defined by values of age, period, and cohort. Rates defined in this manner lend themselves to well-established approaches from spatial statistics in which correlation among proximate observations may be modeled using a spatial random effect. Through simulations, we show that in the presence of spatial dependence and over-dispersion: 1) the correlated Poisson model attains lower AIC; 2) the traditional APC model produces biased trend parameter estimates; and 3) the correlated Poisson model corrects most of this bias. We illustrate our approach using brain and breast cancer incidence rates from the Surveillance Epidemiology and End Results Program of the United States. Our approach can be easily extended to accommodate comparative risk analyses and interpolation of cells in the Lexis with sparse data

Trends of linear regression model (13) of log₁₀[US lifetime natural cancer incidence risk] vs log₁₀[cumulative stem-cell divisions]among cancer sites with available radiation risk (Table 1) or smoking risk (Table 2) data, and in the dataset of Tomasetti and Vogelstein [2], and in some cases omitting tumors with short latency (leukemia, bone, thyroid).

<p>Trends of linear regression model (13) of log₁₀[US lifetime natural cancer incidence risk] vs log₁₀[cumulative stem-cell divisions]among cancer sites with available radiation risk (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0150335#pone.0150335.t001" target="_blank">Table 1</a>) or smoking risk (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0150335#pone.0150335.t002" target="_blank">Table 2</a>) data, and in the dataset of Tomasetti and Vogelstein [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0150335#pone.0150335.ref002" target="_blank">2</a>], and in some cases omitting tumors with short latency (leukemia, bone, thyroid).</p

Linear regression fits of model (9) to data of Tomasetti and Vogelstein [2]^a, and in some cases omitting tumors with short latency (leukemia, bone, thyroid).

<p>Linear regression fits of model (9) to data of Tomasetti and Vogelstein [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0150335#pone.0150335.ref002" target="_blank">2</a>]<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0150335#t003fn001" target="_blank">^a</a>, and in some cases omitting tumors with short latency (leukemia, bone, thyroid).</p

Lack of Correlation between Stem-Cell Proliferation and Radiation- or Smoking-Associated Cancer Risk

<div><p>Background</p><p>A recent paper by Tomasetti and Vogelstein (<i>Science</i> 2015 <b>347</b> 78–81) suggested that the variation in natural cancer risk was largely explained by the total number of stem-cell divisions, and that most cancers arose by chance. They proposed an extra-risk score as way of distinguishing the effects of the stochastic, replicative component of cancer risk from other causative factors, specifically those due to the external environment and inherited mutations.</p><p>Objectives</p><p>We tested the hypothesis raised by Tomasetti and Vogelstein by assessing the degree of correlation of stem cell divisions and their extra-risk score with radiation- and tobacco-associated cancer risk.</p><p>Methods</p><p>We fitted a variety of linear and log-linear models to data on stem cell divisions per year and cumulative stem cell divisions over lifetime and natural cancer risk, some taken from the paper of Tomasetti and Vogelstein, augmented using current US lifetime cancer risk data, and also radiation- and tobacco-associated cancer risk.</p><p>Results</p><p>The data assembled by Tomasetti and Vogelstein, as augmented here, are inconsistent with the power-of-age relationship commonly observed for cancer incidence and the predictions of a multistage carcinogenesis model, if one makes the strong assumption of homogeneity of numbers of driver mutations across cancer sites. Analysis of the extra-risk score and various other measures (number of stem cell divisions per year, cumulative number of stem cell divisions over life) considered by Tomasetti and Vogelstein suggests that these are poorly predictive of currently available estimates of radiation- or smoking-associated cancer risk–for only one out of 37 measures or logarithmic transformations thereof is there a statistically significant correlation (<i>p</i><0.05) with radiation- or smoking-associated risk.</p><p>Conclusions</p><p>The data used by Tomasetti and Vogelstein are in conflict with predictions of a multistage model of carcinogenesis, under the assumption of homogeneity of numbers of driver mutations across most cancer sites. Their hypothesis that if the extra-risk score for a tissue type is high then one would expect that environmental factors would play a relatively more important role in that cancer’s risk is in conflict with the lack of correlation between the extra-risk score and other stem-cell proliferation indices and radiation- or smoking-related cancer risk.</p></div