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**|**PLoS Comput Biol**|**v.13(2); 2017 February**|**PMC5347390

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PLoS Comput Biol. 2017 February; 13(2): e1005391.

Published online 2017 February 14. doi: 10.1371/journal.pcbi.1005391

PMCID: PMC5347390

Natalia L. Komarova, Editor^{}

University of California Irvine, UNITED
STATES

The authors have declared that no competing interests exist.

**Conceptualization:**MPL JHH.**Data curation:**MPL.**Formal analysis:**MPL.**Funding acquisition:**MPL.**Investigation:**MPL.**Methodology:**MPL.**Project administration:**MPL.**Resources:**MPL JHH.**Software:**MPL.**Supervision:**MPL.**Validation:**MPL.**Visualization:**MPL.**Writing – original draft:**MPL JHH.**Writing – review & editing:**MPL JHH.

* E-mail: vog.hin@elttil.kram

Received 2016 August 26; Accepted 2017 January 30.

This is an open access article, free of all copyright, and may be
freely reproduced, distributed, transmitted, modified, built upon, or
otherwise used by anyone for any lawful purpose. The work is made available
under the Creative Commons CC0 public domain
dedication.

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 (*p*<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
(*p*>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
(*p* = 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, *k*, is 3 or 4 and not for smaller
values (1 or 2), but does not strongly depend on the assumed mutagen-associated
mutation rate.

Cancer is thought to arise from a series of mutations in cells. Because mutations are expressed following cell division, more-rapidly renewing tissues could be at higher risk because of the larger number of divisions. Cairns suggested that tissues may reduce cancer risk by partitioning the dividing cell populations into lineages of infrequently-dividing stem cells and frequently-dividing daughter transit cells. We have developed generalizations of three recent cancer models that account for the joint maintenance and renewal of stem and transit cells, with particular focus on assessing the chance of cancer associated with radiation or smoking. All three models demonstrate substantial variation in cancer risks, spanning over twenty orders of magnitude. However, we show that if cancer occurs the chance that it is caused by a dominant mutagenic exposure is less variable, within an order of magnitude. Few cancers arise from mutations occurring solely in stem cells rather than in a combination of stem and transit cells. However, for cancers arising from 2–3 mutations, many have at least one mutation derived from a mutated stem cell. We confirm reports that the probability of a cancer being mutagen-induced is associated with the cumulative number of stem cell divisions in the relevant tissue.

As outlined by Harris [1] (see
also ref. [2]), there are
compelling biological data to suggest that cancer arises from a series of mutations
which affect cell renewal and differentiation processes, and that cancer is largely
unicellular in origin. Because much mutagenic damage is expressed following cell
division, renewing tissues (e.g., the colon) may be at particular risk because of
the large number of cell replications over a lifetime. Cairns [3] suggested that renewing tissues may have a
lower risk of cancer than that based on total cell divisions of all cell types, by
the separation of potential target-cell populations into long-lived stem cells and
short-lived daughter transit cells. Cairns also suggested that there may be a
segregation of old and new DNA strands, such that the old template strands were
retained by the stem cells and the new strands with more errors were passed to the
transit cells [3]. Stem cells
are those cells in each tissue that are controlled by molecular signals and are
responsible for maintaining tissue homeostasis. There is increasing attention
devoted to stem cells and some of their differentiating daughter transit-cells,
regarding their potential role as target cells in the carcinogenic process [4]. Mathematical models of stem
and transit cell development have been constructed that are primarily based on the
ideas of Cairns [3, 5], notably a model developed
by Frank *et al*. [6].

A recent paper of Tomasetti and Vogelstein [7] aroused considerable interest, and suggested
that “the lifetime risk of cancers of many different types [was] strongly correlated
… with the total number of divisions of the normal self-renewing [stem] cells”.
However, this interpretation has been challenged, in particular for the somewhat
heterogeneous biological data used by Tomasetti and Vogelstein, being of rather
variable quality [8]. Little
*et al*. [9] subjected the data used by Tomasetti and Vogelstein [7] to detailed re-analysis, and
concluded that they were “in conflict with predictions of a multistage model of
carcinogenesis, under the assumption of homogeneity of numbers of driver mutations
across most cancer sites”. Little *et al*. found no evidence for
correlations between the extra-risk score developed by Tomasetti and Vogelstein and
radiation- or smoking-related cancer risk [9]. Another reanalysis by Noble *et
al*. of the data used by Tomasetti and Vogelstein [7], suggested that “cancer risk depends not
only on the number of stem cell divisions but varies enormously (approx. 10 000
times) depending on anatomical site” [10]. A recent paper by Wu *et
al*. [11] also
re-analyzed the data of Tomasetti and Vogelstein [7]. Their analysis, combined with insights
gained from a mathematical cancer model that they developed, suggested that
“intrinsic [non-division-related] factors contribute only modestly (less than
~10–30% of lifetime risk) to cancer development”, a strikingly different assessment
from that made by Tomasetti and Vogelstein [7]. The model of Wu *et al*.
[11], which is a type of
Galton-Watson branching process model [12], postulated a number of symmetric stem cell
divisions until a given target stem cell number was attained, after which the stem
cells divided asymmetrically, producing both further stem cells and non-stem transit
cells. Cancer was assumed to arise if a given number of critical mutations arose,
but cancer resulted only if these occurred within the stem cell
population. This model [11] and the previous model of Frank *et
al*. [6], yield
mechanisms for stem and transit cell divisions that protect the critical stem cell
population from excessive numbers of mutations, and yet still generate the necessary
population of normal mature functional cells in each tissue.

In this paper we consider generalizations of the model of Frank *et
al*. [6], and of
Wu *et al*. [11], and a special case of a model developed by Little *et
al*. [13]. We are
particularly concerned with separately assessing the probabilities of mutation and
development of cancer associated with “spontaneous” processes and with those linked
to a specific dominant mutagen, namely ionizing radiation and smoking respectively,
acting in addition to miscellaneous other endogenous and exogenous mutagenic
processes.

The model that we outline is somewhat similar to the model of Frank *et
al*. [6],
although more general. Frank *et al*. [6] did not derive the exact solution of
their model, instead preferring various approximate solutions, which they
compared with Monte Carlo simulations. Likewise we place most emphasis on a
Monte Carlo implementation of a generalization of this model, in which cancer
arises in a specific tissue if *k* critical “driver” mutations in
particular genes are induced in a target cell. Such cells are assumed to arise
from a stem cell that divides asymmetrically *n*_{1}
times. At each division, the cell produces one daughter stem cell in which each
critical gene is subject to mutation of the “spontaneous” (“intrinsic”) sort,
with probability
*u*_{S,s}, or
resulting from some specific mutagen, the “mutagen-induced” (“extrinsic”) sort,
with probability
*u*_{M,s}. The
division also produces one daughter transit cell in which each critical gene is
subject to mutation from some specific dominant mutagen, the “mutagen-induced”
(“extrinsic”) sort, with probability
*u*_{M,t}, or
resulting from the “spontaneous” (“intrinsic”) sort associated with
miscellaneous other endogenous and exogenous mutagenic processes, with
probability *u*_{S,t}.
Each transit cell then undergoes *n*_{2} symmetric
divisions. During each such division each critical gene in each daughter transit
cell is subject to mutation of the “spontaneous” (“intrinsic”) sort, with
probability *u*_{S,t},
or resulting from some specific mutagen, the “mutagen-induced” (“extrinsic”)
sort, with probability
*u*_{M,t}. Each
of the mutational events taking place during stem cell division is statistically
independent of the others, and assumed to be irreversible.

In the Monte Carlo implementation the model starts with a single stem cell. At
the first cycle the stem cell divides into another stem cell and a single
transit cell. At the second cycle the existing transit cell divides and the
daughter stem cell divides into another stem cell and a daughter transit cell
(resulting in a total of 3 transit cells). At the third cycle the 3 transit
cells divide and the stem cell divides into another stem cell and a transit cell
(resulting in a total of 7 transit cells). This carries on, until after
*n*_{1}th cycle all the stem cell divisions have
taken place, and carries on for another *n*_{2} cycles
until all transit cell divisions have also occurred. At the end of the tissue
proliferation process, and at any intermediate stage, there is only a single
stem cell. Implicit in this model is the idea that stem cell and transit cells
have similar cycle times (but see Discussion). Potten and colleagues [14, 15] adduced evidence that the average cycle
time of cycling cells (which would be mostly transit cells) is ~34 hours, and
≥36 hours for stem cells, in human colonic crypts.

The first cell in this division process that carries the *k*
cancer mutations then is deemed to have caused cancer. The mutations can occur
in any of the stem or transit lineages, and each transit cell derives its
mutational burden initially from the particular generation of stem cell it came
from. So if *k* = 3 cancer mutations in total are required, one
could have a single mutation in a stem cell, and then two further transit cell
mutations in the lineage derived from that stem cell (possibly via further stem
cell divisions), or two mutations in the stem cell and a single mutation in a
derived transit cell, or all three in a stem cell, or all three in a transit
cell. The model can be easily generalized to the case in which the numbers of
mutations required by stem and transit cells are different, as for example might
be the case in the colon, as discussed by Frank *et al*. [6]. Such extensions of this
model are not considered here, although we shall assess implications of a
multi-stage cancer model that allows for this possibility (Table A6 in S1
Appendix).

This model is illustrated schematically in Fig 1. Therefore after
*n*_{1} stem cell divisions and
*n*_{2} transit cell divisions there is a single stem
cell and *n*_{1}2^{n2} transit cells, so *n*_{1}2^{n2} + 1 of these cell types in total. We assume
that these cell mutation rates can vary with numbers of cumulative cell
divisions. We accumulate the numbers of each type of mutation in both stem and
transit lineages. The first cell, whether a stem or a transit cell, that
accumulates the necessary *k* critical cancer mutations is used
to label the ensuing cancer that develops. We estimate the total probabilities
of cancer, *C*_{tot}, the probability of
cancer that is due to at least one mutagen-associated mutation,
*C*_{mut}, the probability of cancer
that arises from all mutations in the stem cell lineage,
*C*_{stem–tot},
and the probability of cancer that arises from at least one mutation in the stem
cell lineage,
*C*_{stem–part}.
These probabilities are then used to determine the conditional probability,
given that cancer develops, that it is due to at least one mutation produced by
the specified mutagen:

$${\mathrm{Pr}}_{M}=\frac{{C}_{mut}}{{C}_{tot}}$$

(1)

It may also be of interest to calculate the conditional probability, given that cancer develops, that it is due to the critical mutations developing entirely in the stem-cell lineage, given by:

$${\mathrm{Pr}}_{stem-tot}=\frac{{C}_{stem-tot}}{{C}_{tot}}$$

(2)

or that it is due to at least one of the critical mutations developing in the stem-cell lineage, given by:

$${\mathrm{Pr}}_{stem-part}=\frac{{C}_{stem-part}}{{C}_{tot}}$$

(3)

We also estimate the relative risk (RR), which is the ratio of
the given total cancer probability,
*C*_{tot}, to the total cancer
probability with the mutagen-associated mutation rates set to 0,
*C*_{tot,0}:

$$R{R}_{M}=\frac{{C}_{tot}}{{C}_{tot,0}}$$

(4)

We illustrate with calculations given in Table 1, using a spontaneous stem-cell
mutation rate
(*u*_{S,s}) =
10^{−8}, 10^{−7}, 10^{−6}, 10^{−5} or
10^{−4} per cell division, a spontaneous transition-cell mutation
rate (*u*_{S,t}) =
10^{−6}, 10^{−5} or 10^{−4} per cell division, and
*k* = 1 to 3 critical cancer genes. The mutagen-associated
mutation rates for stem and transit cells are in the range of 0–100% of the
spontaneous rates, and are assumed to apply over the last two thirds of cell
division cycles (i.e., the last two thirds of the *n*_{1}
+ *n*_{2} cycles), the rates before that being 0. The
scenario of spontaneous mutations increasing over the last two-thirds of all
cell cycles corresponds roughly to a person being “unexposed” (to some specific
mutagen) in early life, then “exposed” later in the process of development of
some tissue. The “third” here is somewhat arbitrary, but would be consistent
with occupational exposure to some mutagen, e.g., ionizing radiation, or
exposure in adulthood for some other mutagen, e.g., cigarette smoke. The total
(spontaneous + mutagen-associated) stem-cell and transition-cell mutation rates
are therefore *u*_{s} =
*u*_{S,s} +
*u*_{M,s} =
10^{−8} to 2 x 10^{−4} per cell division and
*u*_{t} =
*u*_{S,t} +
*u*_{M,t} =
10^{−6} to 2 x 10^{−4} per cell division, similar to those
assumed by Frank *et al*. [6], which spanned the range
*u*_{s} = 10^{−10} to
10^{−5} and *u*_{t} =
10^{−6} to 10^{−3} per cell division. So, for example,
taking the third row in Table
1, the stem cell mutation rate is 1 x 10^{−6} and the transit
cell mutation rate is 1 x 10^{−4} in the first third of the
*n*_{1} + *n*_{2} cell cycles,
then for the remaining two thirds of the *n*_{1} +
*n*_{2} cycles the stem cell mutation rate increases
to 1 x 10^{−6} + 5 x 10^{−7} = 1.5 x 10^{−6} and the
transit cell mutation rate increases to 1 x 10^{−4} + 5 x
10^{−5} = 1.5 x 10^{−4}. Frank *et al*.
[6] assumed
*k* = 2 critical cancer genes. We assume a number of
asymmetric stem cell divisions *n*_{1} = 1024 and a
number of symmetric transit cell divisions *n*_{2} = 10,
corresponding to 2^{20} total stem and transit cells (*N*
= 20). *n*_{2} = 10 was used as a generic order of
magnitude, based on steady-state estimates of 5–9 transit cell divisions for
colon, about 3 such divisions in epidermis, and 5–8 transit divisions in the
various lineages in hematopoiesis [4].

As in the above model, this generalized model also assumes a division into stem
and transit lineages, and postulates a model in which cancer arises in a
specific tissue if there arises a cell with *k* critical driver
mutations in particular genes, but only within the stem cell
population. As with the model of Wu *et al*.
[11], it is assumed
that the stem cell divisions are symmetric, each producing two daughter stem
cells, and also that there are asymmetric divisions, each producing one daughter
stem cell and a daughter non-stem cell; the latter cells are assumed to be
irrelevant to the carcinogenic process because of the premises of a limited
lifespan and no competition with stem cells for residence in the stem-cell
niche. Wu *et al*. [11] assumed that the symmetric stem cell
divisions happen first, in this respect contrasting with other models of
stem-cell carcinogenesis [6]; however, there is nothing in the mathematical development of Wu
*et al*. [11] that makes use of this assumption. We shall not assume that this
is necessarily the case either. There are *n*_{1}
symmetric stem cell divisions and *n*_{2} asymmetric
stem-cell divisions, each stem cell resulting in a total population after the
*N* = *n*_{1} +
*n*_{2} divisions in the tissue of *T*_{s+t} = 2^{n1+n2} cells, containing a subpopulation of
*T*_{s} = 2^{n1} stem cells. In contrast to the model of Wu
*et al*. [11], at each stem cell division *i*, whether
symmetric or asymmetric, the daughter stem cell is subject to mutation in each
critical gene, either from some specific dominant mutagen, the “mutagen-induced”
(“extrinsic”) sort, with probability
*u*_{M,i}, or
resulting from the “spontaneous” (“intrinsic”) sort associated with
miscellaneous other endogenous and exogenous mutagenic processes, with
probability *u*_{S,i}.
This is illustrated schematically in Fig 2. It should be noted that since the
number of stem cell divisions, *n*_{1}, is generally
assumed to be somewhat less than the number of transit cell divisions,
*n*_{2} (Table 2), at least in the scheme outlined by
Wu *et al*. [11] with stem cell divisions occurring first, over much of the life
of the individual the stem cell population is constant.

The model of Wu *et al*. [11] does not distinguish the two types of
mutation, and as is appropriate for a model using only the “intrinsic” mutation
rate, does not allow for variations in mutation rate over time (by cell division
generation or loss/competition of mutated cells). Each of the mutational events
taking place during stem cell division is statistically independent of the
others, conditional on the mutations that have already taken place in a
particular lineage, and assumed to be irreversible. Let
*S*_{g} and
*M*_{g} denote the number of
spontaneous and mutagen-induced mutations that have accumulated in a given
lineage in generation *g*. Then we have the recurrence relation:

$$\begin{array}{l}P\left({S}_{g+1}=s,{M}_{g+1}=m|{\left({u}_{S,i}\right)}_{i=1}^{g+1},{\left({u}_{M,i}\right)}_{i=1}^{g+1}\right)\\ ={\displaystyle \sum _{j=0}^{s}{\displaystyle \sum _{l=0}^{m}\left[\begin{array}{l}\frac{(k-j-l)!}{(s-j)!(m-l)!(k-s-m)!}{u}_{S,g+1}{}^{s-j}{u}_{M,g+1}{}^{m-l}{(1-{u}_{S,g+1}-{u}_{M,g+1})}^{k-s-m}\text{x}\\ P\left({S}_{g}=j,{M}_{g}=l|{\left({u}_{S,i}\right)}_{i=1}^{g},{\left({u}_{M,i}\right)}_{i=1}^{g}\right)\end{array}\right]}}\end{array}$$

(5)

This assumes that at each cell division, two daughter cells are produced, whether
in the stem cell lineage or a combination of stem and transit cell lineages,
with independently produced sets of mutations in the stem cell daughter(s). So
that if the stem cell in generation *g* carries
*s* spontaneous mutations and *m*
mutagen-induced mutations, the numbers of each sort of mutation in the
*k* − *s* − *m* remaining
non-mutated critical genes is multinomially distributed (~
*Multinom*(*k* − *s* −
*m*,*u*_{S,g+1},*u*_{M,g+1},1
– *u*_{S,g+1} −
*u*_{M,g+1})). We
also have that:

$$P\left({S}_{g}=s,{M}_{g}=m|{\left({u}_{S,i}\right)}_{i=1}^{0},{\left({u}_{M,i}\right)}_{i=1}^{0}\right)=\{\begin{array}{c}0\phantom{\rule{0.25em}{0ex}}\text{if}\phantom{\rule{0.25em}{0ex}}s\ge 1\phantom{\rule{0.25em}{0ex}}\text{or}\phantom{\rule{0.25em}{0ex}}m\ge 1\\ 1\phantom{\rule{0.25em}{0ex}}\text{if}\phantom{\rule{0.25em}{0ex}}s=m=0\end{array}$$

(6)

The probability of cancer is then given by:

$$\begin{array}{l}{C}_{k}\left({n}_{1},{n}_{2}|{\left({u}_{S,i}\right)}_{i=1}^{N},{\left({u}_{M,i}\right)}_{i=1}^{N}\right)=1-{\left[1-P\left({S}_{N}+{M}_{N}=k|{\left({u}_{S,i}\right)}_{i=1}^{N},{\left({u}_{M,i}\right)}_{i=1}^{N}\right)\right]}^{{T}_{s}}\\ =1-{\left[1-{\displaystyle \sum _{s=0}^{k}P\left({S}_{N}=s,{M}_{N}=k-s|{\left({u}_{S,i}\right)}_{i=1}^{N},{\left({u}_{M,i}\right)}_{i=1}^{N}\right)}\right]}^{{2}^{{n}_{1}}}\end{array}$$

(7)

Also of interest is the probability that cancer develops in which at least one of the critical mutations is mutagen-induced:

$${C}_{k}{}^{m}\left({n}_{1},{n}_{2}|{\left({u}_{S,i}\right)}_{i=1}^{N},{\left({u}_{M,i}\right)}_{i=1}^{N}\right)=1-{\left[1-{\displaystyle \sum _{s=0}^{k-1}P\left({S}_{N}=s,{M}_{N}=k-s|{\left({u}_{S,i}\right)}_{i=1}^{N},{\left({u}_{M,i}\right)}_{i=1}^{N}\right)}\right]}^{{2}^{{n}_{1}}}$$

(8)

Therefore the conditional probability, given that cancer develops, that it is due to at least one mutation produced by the specified mutagen is:

$${\mathrm{Pr}}_{M}=\frac{{C}_{k}{}^{m}\left({n}_{1},{n}_{2}|{\left({u}_{S,i}\right)}_{i=1}^{N},{\left({u}_{M,i}\right)}_{i=1}^{N}\right)}{{C}_{k}\left({n}_{1},{n}_{2}|{\left({u}_{S,i}\right)}_{i=1}^{N},{\left({u}_{M,i}\right)}_{i=1}^{N}\right)}=\frac{1-{\left[1-{\displaystyle \sum _{s=0}^{k-1}P\left({S}_{N}=s,{M}_{N}=k-s|{\left({u}_{S,i}\right)}_{i=1}^{N},{\left({u}_{M,i}\right)}_{i=1}^{N}\right)}\right]}^{{2}^{{n}_{1}}}}{1-{\left[1-{\displaystyle \sum _{s=0}^{k}P\left({S}_{N}=s,{M}_{N}=k-s|{\left({u}_{S,i}\right)}_{i=1}^{N},{\left({u}_{M,i}\right)}_{i=1}^{N}\right)}\right]}^{{2}^{{n}_{1}}}}$$

(9)

It may also be of interest to calculate the RR due to the specified mutagen, which is given by the quantity:

$$R{R}_{M}=\frac{{C}_{k}\left({n}_{1},{n}_{2}|{\left({u}_{S,i}\right)}_{i=1}^{N},{\left({u}_{M,i}\right)}_{i=1}^{N}\right)}{{C}_{k}\left({n}_{1},{n}_{2}|{\left({u}_{S,i}\right)}_{i=1}^{N},0\right)}=\frac{1-{\left[1-{\displaystyle \sum _{s=0}^{k}P\left({S}_{N}=s,{M}_{N}=k-s|{\left({u}_{S,i}\right)}_{i=1}^{N},{\left({u}_{M,i}\right)}_{i=1}^{N}\right)}\right]}^{{2}^{{n}_{1}}}}{1-{\left[1-{\displaystyle \sum _{s=0}^{k}P\left({S}_{N}=s,{M}_{N}=k-s|{\left({u}_{S,i}\right)}_{i=1}^{N},0\right)}\right]}^{{2}^{{n}_{1}}}}$$

(10)

It should be noted that the model does not accommodate re-hits of the driver
mutations that have already occurred, in other words the addition of further
mutations in the same critical genes that have already been mutated (whether
spontaneously or mutagen-induced), resulting from various types of endogenous
and exogenous mutagens. These can happen, but the labelling of the particular
stem cell lineage would not alter, and in particular a cell that already has
*s* spontaneous and *m* mutagen-induced
mutations would be deemed still to have those numbers of mutations if some of
these were re-hit by new mutations, whether spontaneous or mutagen-associated.
Nevertheless, such re-hits would be expected to be rare occurrences, and
arguably of little relevance practically. The model allows for the mutation
rates to vary depending on whether the divisions are symmetric or asymmetric, or
equivalently whether the divisions take place occur before or after the first
*n*_{1} stem cell divisions. However, to the best of
our knowledge there are insufficient data to suggest that the mutations rates
vary depending on whether cell division is symmetric or asymmetric, so we shall
not investigate this possibility further.

We illustrate the effect of assuming a spontaneous mutation rate of
*u*_{S} = 10^{−8} per cell
division, and mutagen associated rates in the range
*u*_{M} = 0–10^{−8} per
cell division in Table 2,
for various specific cancer sites, assuming between *k* = 2 to 4
critical cancer mutations [4]. Table A2 in S1 Appendix presents additional calculations
assuming a somewhat higher spontaneous mutation rate,
*u*_{S} = 10^{−6} per
cell division, and correspondingly higher mutagen-associated rates,
*u*_{M} = 0–10^{−6} per cell
division. [At least four genetic targets, including *Ras* and
*FAP*, have been identified for colon cancer, although not
all of them have mutation or sequence loss that are present in every cancer
[16], so that this
range for *k* appears reasonable to us.] Table A3 in S1 Appendix
shows similar calculations, but with the mutagen-associated mutation rate
increasing from 0 at birth, rather than after the first third of stem-cell
divisions. The mutagen-associated mutation rate is in the range of 0–100% of the
spontaneous mutation rate. This range was chosen to yield the range of relative
risks (of between 1.2 and 16) commonly observed for many carcinogens, in
particular ionising radiation where relative risks tend to be lower [17], or cigarette smoke
where the relative risks [18] approach the upper range of 16 yielded by our modelling
assumptions (Table 2,
Tables A2 and A3 in S1 Appendix). These assumptions also imply
that the total mutation rate *u*_{S} +
*u*_{M} =
10^{−8}–10^{−7} per cell division, which is similar to
values assumed by Wu *et al*. [11], spanning the range
10^{−10}–10^{−6} per cell division. We assume a range of
numbers of critical mutations required for cancer of *k* = 2 to 4
(Table 2, Tables A2
and A3 in S1
Appendix) or *k* = 1 to 4 (Tables (Tables33 and and4,4, and Tables A4, A5 in S1
Appendix), similar to the range assumed by Wu *et al*.
[11]. Larger values
of the critical number of mutations required for cancer (up to
*k* = 7) were also evaluated, but these did not suggest any
markedly different findings, so are not reported further. All other parameters
(*n*_{1},*n*_{2}) are the same
as assumed by Wu *et al*. [11]. For simplicity we show the
calculations for a subset of the more environmentally modifiable cancer types
considered by Wu *et al*. [11], in Table 2 and Tables A2 and A3 in S1
Appendix. In Table A1 in S1 Appendix we estimate risks for all cancer
sites considered by Wu *et al*. [11], using essentially the cancer site data
of Tomasetti and Vogelstein [7].

We also assessed predictions of a variety of multistage carcinogenesis models, in
order to determine the likely effect of intermediate (partially transformed)
cell proliferation and death/differentiation, also variations made by assuming
that excess mutations affect only the stem cell or transition cell compartment.
Stem cells or transit cells can acquire up to *k* successive
mutations, at which point they are assumed to become malignant. The model is
illustrated schematically in Fig
3. Cells at different stages of the process are labelled by
*I*_{(α,β)}, where
the first subscript, *α*, represents the number of cancer
mutations that the cell has accumulated, the second subscript,
*β*, represents whether the cell is a stem cell
(*β* = 0) or a transit cell (*β* = 1). At all
stages stem or transit cells are allowed to divide symmetrically or
differentiate (or undergo apoptosis) at rates
*G*(*α*,*β*) and
*D*(*α*,*β*), respectively.
Each stem or transit cell can asymmetrically divide into an equivalent daughter
cell and another cell with an extra cancer mutation at rate
*M*(*α*,*β*). Likewise, stem
cells can also asymmetrically divide into an equivalent daughter cell and a
transit cell at rate *A*(*α*,0). The model assumes
that at age 0 there is a single stem cell, and no transit cells. This model is a
special case of models developed by Little *et al*. [13] and used to fit to
population retinoblastoma data. This differs from the otherwise very similar
models of Little *et al*. [19] and Little and Wright [20] only in that the
previous models assumed a deterministic (non-stochastic) untransformed stem cell
population. In Fig 3 stem
cells correspond to the upper horizontal axis, whereas transit cells are given
by the lower horizontal axis. The acquisition of carcinogenic mutations amounts
to moving horizontally (left to right) via successive symmetric division
processes both for stem (upper axis) and transit cells (lower axis) in Fig 3, whereas the asymmetric
division that produces for each stem cell a single daughter stem and transit
cell, and which can happen in principle to a stem cell with any number of
accumulated mutations, amounts to moving vertically (top to bottom) in this
figure. Further details on the mathematical assumptions and the numerical
solution of the governing partial differential equations are given in Little
*et al*. [13]. We shall assume that during gestation (assumed to be of length
*L*_{g} = 0.728 years [38 weeks])
the stem cell population divides at a rate:

(11)

per cell per year, with cell differentiation/apoptosis rate
*D*(0,0) = 0, so that at the end of gestation the expected
number of stem cells ≈ 2^{n1}. The lifetime of the individual is assumed
to be of duration *L*_{t} = 80 years. We
assume that the transit cell population has a slightly faster growth rate:

(12)

and again the cell differentiation/apoptosis rate is 0. After
gestation we generally assume that all
*G*(*α*,*β*) and
*D*(*α*,*β*) are 0; in
particular this implies that after birth the stem-cell and transit-cell
populations are approximately constant, although both are random processes, so
that there will be modest fluctuations in the size of each cell population.
However, to allow for the possibility of intermediate (partially transformed)
stem and transit cells being subject to competing processes of proliferation and
differentiation/apoptosis, a process that is incorporated in many recently
developed mathematical cancer models [13, 19–23], we conduct sensitivity analysis in
Table 5 in which we
assume a birth/death process for all intermediate cell compartments, with
*G*(*α*,*β*) = 1.1 /cell / year
and *D*(*α*,*β*) = 0.71 /cell /
year for (*α*,*β*) {(0,0),(0,1)}; these values
are derived from analysis of lung cancer mortality data [24]. Assuming that the transit cell
population at the end of gestation is ≈ *T*_{g} = *p*2^{n1}, implies that the stem cell→transit cell
transition rate must be:

(13)

We assume a ratio of transit:stem cells at the end of gestation of 1:100, i.e.,
*p* = 0.01, throughout. In order to derive the mutation rates
*M*(*α*,*β*), we note that the
probability of a mutation per asymmetric cell division, whether in stem or
transit cells, over the expected duration,
*L*_{t} /
*n*_{2}, between asymmetric cell divisions, is
*u* = 1 −
exp[−*M*(*α*,*β*)*L*_{t}
/ *n*_{2}]. Therefore:

(14)

Frank *et al*. discuss the possibility that
“transit cells [in the colon] may require mutations to avoid sloughing to cause
cancer. For example, an additional mutation that makes a transit cell surface
sticky may prevent it from shedding” [6]; however, Frank *et al*.
[6] did not actually
fit such a model. We conduct additional sensitivity analysis in Table A6 in
S1
Appendix in which we assess the implications of a model in which
there is an additional mutational stage required for transit cells than for stem
cells.

Transmissible genomic instability, for which there is experimental evidence after
radiation exposure [25,
26], implies that
certain sorts of mutation can result in a long-lasting increase in mutation
rate. The role of genomic instability is particularly well established for colon
cancer; chromosomal instability is present in about 85% of non-familial colon
cancers, and microsatellite instability is associated with most of the remaining
carcinomas [27–29]. We therefore assess
the effect of non-homogenous mutation rates on the relative risk of colon
adenocarcinoma, whereby the ratio between successive mutation rates, whether for
the stem cells or transit cells is given by *d* =
*M*(*α* + 1,*β*) /
*M*(*α*,*β*), so that:

(15)

The case in which *d* > 1 implies that with
each acquired mutation the mutation rate (per cell per unit time) increases, so
that after the first mutation the second mutation is acquired somewhat faster
(per cell per unit time), and after the second mutation, the third mutations is
acquired even faster (per cell per unit time), somewhat analogous to
transmissible genomic instability, whereas *d* < 1 implies
that with each acquired mutation the mutation rate (per cell per unit time)
decreases, the opposite of this process. We plot the relative risks implied by
such a process in Fig 4,
varying *d* over the range from 0.5 to 1.5 with values of the
number of critical mutations, *k*, between 2 and 4; a lower
mutation rate, both for stem and transit cells, is used for *k* =
2 than for *k* = 3,4 in order to avoid the probability of cancer
saturating (at 1).

A test can be made of the assumption that Pr_{M}, as
given by expression (1) (for the generalization of the model of Frank *et
al*. [6]) and
by expression (9)
(for the generalization of the model of Wu *et al*. [11]), may be correlated
with the susceptibility of a tissue to environmental mutagenic factors, using
radiation-associated and smoking-associated cancer risk as examples of such
factors, both being mutagens that induce a large number of types of cancer
[17, 18]. We consider
radiation-exposure induced solid cancer incidence risk (REIC) at 1 Gy evaluated
by the United Nations Scientific Committee on the Effects of Atomic Radiation
(UNSCEAR) [Table 70 in Annex A of [17]] for various cancer sites, and as also
quoted by Little *et al*. [Table 1 in [9]]; for leukemia we use radiation-exposure
induced cancer death risk (REID) evaluated by UNSCEAR [Table 65 in Annex A of
[17]]; mortality risk
is used because leukemia incidence was not evaluated in the latest Japanese
atomic bomb survivor Life Span Study (LSS) cancer incidence report [30], a preliminary version
of which formed the basis of the UNSCEAR evaluations [17]. We use both the International
Commission on Radiological Protection (ICRP) recommended [31] cancer-site specific weighting of
excess absolute risk (EAR) *vs* excess relative risk (ERR) models
(Table A4 in S1
Appendix), and the Biological Effects of Ionizing Radiation (BEIR
VII) committee recommended [32] cancer-site specific weighting of EAR vs ERR models are employed
(Table A5 in S1
Appendix). While these risks may be taken as representing those
associated with exposure at high doses and high dose rates, they are likely to
be proportional to risks associated with environmental and occupational levels
of radiation exposure [31, 33, 34]. Proceeding along the
lines of analysis conducted by Little *et al*. [9] we tested whether the
conditional probability, Pr_{M}, of cancer being
mutagen-induced, as given by expression (9), was associated with estimates of the
cumulative number of stem-cell divisions in that tissue. In order to do this we
regress the conditional probability Pr_{M}, in relation
to the log of the cumulative number of cell divisions, ln[*D*],
given by:

Pr_{M} = *α*_{0} + *α*_{1}ln[*D*] + *ε*

(16)

For REIC, we fit a model in which:

(17)

Likewise, we assess the correlations of smoking-associated cancer risk using data
on differences in mortality rates between current and former smokers,
*Sm*_{diff}, in the British doctors’
cohort [18], as also
shown in Table 2 of Little
*et al*. [9], by fitting a model in which:

(18)

Tables Tables3
and3
and and44 and Tables A4 and A5
in S1
Appendix record the results of these regression analyses, based on
the data referred to above and also in Table A1 in S1
Appendix. We are mainly interested in the significance of the regression
coefficient *α*_{1} in expressions (16)-(18). Linear
regressions are performed via ordinary least squares [35], using R [36]. The *p*-values shown in
Tables Tables33 and and44 and Tables A4 and A5 in
S1
Appendix are estimated using an *F*-test [35], and are in relation to
the trend parameter (*α*_{1}). We also estimate Pearson
and Spearman correlation coefficients between Pr_{M}
and these other variables, ln[*D*], *REIC*, and
*Sm*_{diff}. We assess the influence
of high-leverage datapoints by assessing the difference made by removing those
points from each regression with Cook’s distance [37] exceeding 4 / [*n* −
*p* − 1], where *n* = number of relevant
datapoints, *p* = number of fitted parameters, a generally used
threshold [38].

Table 1 demonstrates that if
the generalization of the model of Frank *et al*. is employed, there
is a substantial variation in cancer risks, by at least 4 orders of magnitude,
depending on the assumed number of critical mutations required for cancer,
*k*, and the stem-cell and transition-cell mutation rates. If
only a single cancer mutation, *k* = 1, is assumed the total
probability of cancer is between 88–100%, irrespective of the assumed mutation
rates, but the cancer probability is lower, generally less than 80%, if
*k* = 2 or *k* = 3 cancer mutations. If
*k* = 3 and stem cells have 100-fold lower mutation rates than
transit cells (upper part of Table
1), the cancer probability is less than 0.6%. However, in most cases the
conditional probabilities, Pr_{M}, as given by expression
(1), of cancer
being mutagen-induced, are within a factor of about 10, ranging between 7–81%. The
only situations in which this is not the case are when a single mutation is required
for cancer and the baseline mutation rates are sufficiently high that cancer
develops during the first third of cell divisions, before any external mutagen is
assumed to be present. The relative risks,
*RR*_{M}, as given by expression
(4), are also
moderately stable, spanning the range 1.0–4.5 (Table 1). In general, conditional on cancer
developing, the probability is low (<0.5%) of all the relevant mutations being
derived from stem cell-associated mutations (Table 1). However, particularly 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.

If now the generalization of the model of Wu *et al*. is employed, the
results of Table 2, Tables A2
and A3 in S1
Appendix again demonstrate that cancer risks vary quite substantially, by
about 12 orders of magnitude, depending on the assumed number of critical mutations
required for cancer, *k*, and the spontaneous and mutagen-associated
mutation rates. However, again the conditional probabilities,
Pr_{M}, as given by expression (9), are reasonably
stable, ranging between 22–98% (Table
2, Tables A2 and A3 in S1 Appendix). Relative risks,
*RR*_{M}, as given by expression
(10), are also
fairly consistent, spanning the range 1.2–16.0 (Table 2, Tables A2 and A3 in S1
Appendix).

The analysis of Table 3
demonstrates that in general, using the generalization of the model of Wu *et
al*. [11], the
probability of a cancer being mutagen-induced, Pr_{M},
correlates significantly with the cumulative number of stem cell divisions
(*p*<0.05). This correlation does not depend on the assumed
value of the number of critical cancer mutations, *k*, between 1 and
4, and neither does it depend on the mutagen-assumed mutation rate. However, the
correlation is slightly less significant when *k* = 2 mutations are
assumed (Table 3). The trend
is also shown in six particular (typical) cases in Fig 5. When *k* = 3 or
*k* = 4 increasing the cumulative number of cell divisions leads
to a reduction in the probability of cancer being mutagen induced, but for
*k* = 1 or *k* = 2 increasing the cumulative
number of cell divisions leads to an increase in the probability of cancer being
mutagen induced (Table 3,
Fig 5, Table A1 in S1 Appendix).
As can be seen (Table 3, Fig 5, Table A1 in S1 Appendix)
the correlations, although quite striking, in some cases, for example with
*k* = 3 or *k* = 4, account for only a very modest
variation in probability, by about 2%; the significant findings are also in some
cases sensitive to removal of high-leverage datapoints (with Cook’s distance > 4
/ [*n* − *p* − 1]) (Table 3).

The analyses of Tables A4 and A5 in S1 Appendix demonstrate that there are no
significant correlations (*p*>0.3) between lifetime cancer-site
specific population radiation risk (REIC) and the probability of that cancer being
mutagen-induced, Pr_{M}. This is the case whether the ICRP
or BEIR VII recommended cancer-site specific weighting of EAR *vs*
ERR models are used (Tables A4 and A5 in S1 Appendix respectively). These null results are
also insensitive to the variations tested in the assumed critical number of
mutations leading to cancer, *k*, in the assumed mutagen-associated
mutation rate, or to removal of high-leverage datapoints (with Cook’s distance >
4 / [*n* − *p* − 1]).

The analysis of Table 4
suggests that there are borderline-significant positive correlations
(*p* = 0.08) between smoking-associated mortality rate difference
(current vs former smokers), *Sm*_{diff},
and Pr_{M}. This is only the case when values of the
critical number of mutations leading to cancer, *k*, is 3 or 4 and
not 1 or 2), but does not strongly depend on the assumed mutagen-associated mutation
rate (varying between 20–100% of the underlying rate). The trend is also shown in
six particular (typical) cases in Fig
6. However, the significant findings are in all cases sensitive to
removal of high-leverage points (with Cook’s distance > 4 / [*n* −
*p* − 1]), resulting in loss of significance
(*p*>0.3) (Table
4).

The analysis of Table 5 shows
that very similar relative risks are produced to those of Table 2 if instead of the model of Wu *et
al*. [11] a
multistage cancer model is used. Although cancer risks span at least 23 orders of
magnitude, with few exceptions relative risks generally span the range of 1–10.
Little difference is made if the intermediate cells (whether stem or transit cells)
that have acquired one or more mutations undergo competing processes of
proliferation and differentiation/apoptosis, or if the excess mutational load falls
entirely on either stem cells or transit cells (Table 5). Likewise, little difference is made if
models allowing for an extra mutation for transit cells, whether in the case
*k* = 2 and *k* = 3 for stem cells and transit
cells respectively, or in the case *k* = 3 and *k* = 4
for stem cells and transit cells respectively (Table A6 in S1 Appendix).
Fig 4 demonstrates that the
relative risk tends to decrease when the value of the mutation rate multiplier,
*d*, increases. This decrease is most striking when the critical
number of mutations *k* = 3.

We have developed three separate cancer models, which share certain features, and
yield similar predictions, namely substantial variation in cancer risks, by over 20
orders of magnitude (a range that is arguably highly implausible), depending on the
assumed numbers of various model parameters (numbers of mutations required for
cancer, mutation rates etc). However, in most cases the conditional probabilities of
cancer being induced by some dominant mutagen are similar. The relative risks
associated with mutagen exposure compared to background rates are also fairly
stable. It was calculated that very few cancers, generally <0.5%, would arise
from mutations occurring solely in stem cells rather than in a combination of stem
and transit cells. However, particularly for cancers with 2 or 3 critical mutations,
a substantial proportion of cancers, in some cases 100%, would have at least one
mutation derived from a mutated stem cell. It should be noted that while for many
common epithelial cancers of adulthood 3 or more critical mutations are plausible,
the number of critical mutations may be less than this, 1 or 2, for leukemia and for
childhood cancer [4]. Indeed,
a recent ICRP report suggested that “it is tempting to speculate that childhood
thyroid cancer requires two mutations, and a linear dose response with the short
latency occurred for those carrying the pre-existing *RET/PTC*
rearrangement, with radiation responsible for inducing the second hit necessary for
conversion of the cells to full malignancy” [4].

We have also shown that the probability of a cancer being mutagen-induced correlates
significantly (across cancer sites) with the estimated cumulative number of stem
cell divisions in the associated tissue (*p*<0.05), so that at
least when the number of cancer mutations, *k*, takes values of 3 or
4, values that are generally more likely than values of 1 or 2, increasing the
cumulative number of cell divisions leads to a reduction in the probability of
cancer being mutagen induced (Table
3, Fig 5, Table A1 in
S1
Appendix). This correlation does not depend on the assumed value of the
number of critical cancer mutations, or with the mutagen-associated mutation rate,
in the tested range. Intuitively, one might expect a negative correlation, because
when the number of stem cell divisions is very high, cancer is more likely to occur
in the first third of life, before the mutagen becomes effective. Although there is
indeed a negative correlation, when *k* = 3 or *k* =
4, the effect size shown in Fig
5 and Table A1 in S1 Appendix is small. Increasing the number of
stem cell divisions by a factor of one million decreases the probability of
mutagen-associated cancer by about 2–3%. There are also borderline significant
negative correlations (*p* = 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, *k*, is 3 or 4, and not smaller values,
of 1 or 2, but does not strongly depend on the assumed mutagen-associated mutation
rate. Both of these findings provide a measure of support to the controversial
findings of Tomasetti and Vogelstein [7], which suggested that cancer risk for a
tissue may be associated with the total number of stem cell divisions.

We have considered a modest range in the excess dominant mutagen-associated rate,
doubling the “spontaneous” mutation rate. It is probably the case that if one could
subtract out all mutagens that may cause cancer a wider range of multipliers of
“spontaneous” mutation rate could be considered, but it is difficult to know where
one would draw the line. For example, there are endogenous mutagenic processes such
as the myriad of oxidative processes within a cell that cause single (and some
double) strand break damage; subtracting these off from the baseline “spontaneous”
mutation rate is arguably artificial. Considering a wider range of relative increase
of the “spontaneous” mutation rate would scarcely change the conclusions of the
present calculations. Although in the simulations using the model of Frank
*et al*. [6] (Table 1) we
assume that the relative increases in mutations in the transit and stem cell
populations are the same, sensitivity analysis using a special case of the
multistage carcinogenesis model developed by Little *et al*. [13] in which the additional
mutations were assumed to only apply either to the stem cell population or to the
transit cell population (Table
5) did not suggest markedly different relative risks.

A slight limitation of the model of Frank *et al*. [6] is that stem cells and
transit cells are assumed to have similar cycle times. However, by varying the
mutation rates of stem and transit cells one can largely circumvent this
restriction–the effect of stem cells for example having longer cycle times is
essentially accounted for by the fact that mutation rates (per single division
cycle) are lower for this type of cell. This model could be easily generalized to
the case in which the numbers of mutations required by stem and transit cells are
different, as for example might be the case in the colon if transit cells required
an extra mutation to make them “sticky” to avoid being sloughed off into the lumen,
as discussed by Frank *et al*. [6], although no associated calculations were
carried out by them. The analysis we have conducted (Table A6 in S1 Appendix)
suggests that little difference would be made by assuming that transit cells require
an extra mutation from the total required to confer malignancy on stem cells.

Both the models of Frank *et al*. [6] and of Wu *et al*. [11], generalizations of which
are used here, assume that stem cells never die and never fully differentiate (i.e.,
they never divide symmetrically into two transit cells). However, there is ample
experimental evidence that stem cells can and do produce differentiated daughter
cells, particularly in the intestinal crypt [39, 40], and mathematical modeling indicates that
this has important effects on cancer risk [41, 42]. Many recently developed mathematical
cancer models allow for such birth/death processes in the partially transformed cell
compartment, although they do not generally specifically allow for this in stem and
transit cells [13, 19–23]. However, the analysis we have performed
(Table 5) allowing for
such competing processes of cell proliferation and differentiation/apoptosis in the
partially transformed stem and transit cell populations, using a special case of a
multi-stage cancer model developed elsewhere [13], does not materially alter our
conclusions.

All of the three models employed here assume a stem cell population that is either
constant, in particular comprising a single stem cell, as in the generalization of
the model of Frank *et al*. [6], or eventually constant, as in the
generalization of the model of Wu *et al*. [11], or the special case of the multistage
cancer model of Little *et al*. [13]. However, the precise age at which the stem
cell population becomes constant differs between the three models, and would
generally be largest in certain cases of the generalization of the model of Wu
*et al*. [11] in which the number of stem cell divisions
*n*_{1}, is large in comparison with the number of
transit cell divisions, *n*_{2}, specifically for lung
adenocarcinoma, osteosarcoma, and thyroid carcinoma. Tomasetti and Vogelstein [7] in their analysis of stem
cell divisions and cancer make a similar assumption to Wu *et al*.
[11], namely that after a
period of symmetric stem cell division, there is a period of largely asymmetric
division of stem cells into stem and transit cells; this assumption was made to
derive the total number of stem cell divisions in each organ/tissue. The assumed
absence of symmetric stem cell divisions in the model of Frank *et
al*. [6] is
arguably implausible. The model is mainly concerned with the consequences of
establishment of a tissue niche from a single stem cell. As discussed by Frank
*et al*. [6] it is possible that mutations associated with symmetric stem cell
divisions during tissue growth and development may contribute substantially to
lifetime cancer risk. As such the model of Frank *et al*. [6], although valid for study of
cancers that arise in a renewing tissue such as the colon or skin in adulthood, may
not be a good model of the carcinogenic process over the full lifespan.

As we make clear in the Methods the generalization of the model of Wu *et
al*. [11] that we
have developed allows for the mutation rates to vary depending on whether the
divisions are symmetric or asymmetric. There have been a number of theoretical
investigations that assess patterns of accumulation of mutational or epigenetic
damage associated with asymmetric or symmetric cell divisions over the lifetime of
an individual, although they do not allow for variation of mutation rate by the
symmetry vs asymmetry of the division process [41, 43]. We are not aware of any data that suggest
that the mutation rate per cell division varies depending on whether cell division
is symmetric or asymmetric. Nevertheless this could be investigated further using
this generalized model.

The generalizations of the models of Frank *et al*. [6] and Wu *et
al*. [11] that we
have developed assume that after a period of latency or lack of exposure, in the
first third of cell divisions, when there is no effect, the mutagen acts equally at
each stage of carcinogenesis. There are known to be variations between the mutagenic
effect of particular agents on different stages of the carcinogenic process, so that
for example ionizing radiation is thought to act at a relatively early stage, and
cigarette smoke at a much later, although not final, stage [44, 45]. More generally, it is possible that
mutation rates associated with endogenous (spontaneous) and exogenous (specific
mutagen-associated) processes could exhibit quite complicated patterns of
heterogeneity, rather than a simple early/late stage dependence, as we now discuss.
There is evidence of genomic destabilization in many types of solid cancer [29], and for colon cancer the
evidence is particularly strong for the involvement of chromosomal and
microsatellite instability in most cancers [27, 28]. There is a large body of experimental data
suggesting that transmissible genomic destabilization is associated with ionizing
radiation exposure, with effects on a number of biological endpoints [25, 26]. Mutagen-associated genomic destabilization
implies that the probability (per cell division) of an unaffected cell acquiring a
mutation may not be the same as the probability (per cell division) of the same cell
acquiring a subsequent mutation from that mutagen. As such radiation-associated
genomic destabilization implies marked non-linearity in dose response, for which
there is no strong evidence in the radioepidemiologic cancer literature [17, 46, 47]. This implies a minimal role for
radiation-associated genomic destabilization associated with cancer induction in
humans. Nevertheless, the evidence we present in Fig 4 is that with increasing levels of genomic
destabilization (corresponding to larger values of the multiplier,
*d*, in particular with *d* > 1) the relative
risk reduces. Genomic destabilization and other similar types of non-linearity of
mutagenic effect could also be modelled slightly differently than here, using
generalizations of the multi-stage cancer model given here, and some others [13, 19, 48]; however these types of model would not
allow for genomic stabilization, corresponding to the values of *d*
< 1 in our model. The modifications required to incorporate genomic
destabilization in the generalizations of the models of Frank *et
al*. [6] and Wu
*et al*. [11] developed here would be non-trivial.

In the colon there is abundant evidence that stem cells divide many times and remain
at the base of the mucosal crypts [14, 49]. Thus,
each stem cell division gives rise to one stem cell that remains at the basal
location, and one transit cell. The transit cell divides a limited number of times
(likely 5–9), producing more differentiating cells that move away from the basal
position, maturing and eventually being terminally differentiated and sloughing off
from the lumenal surface [49]. The idea that transit cells as well as stem cells could be target cells
for colon cancer has been discussed previously, largely on the basis that tumors
arise which are composed predominantly, or exclusively, of mucin-secreting cells,
endocrine cells or even Paneth cells [50, 51]. Recently, this aspect has been emphasised,
with the suggestion that in the colon adenomas likely arise from stem cells and
microadenomas arise from transit amplifying cells [52]. Also, histological sections of small
adenomatous polyps showed some crypts with the bottom half apparently normal and an
abrupt transition to the upper mutated dysplastic half [53]. In addition, recently a small population
of radioresistant Krt19 (intermediate filament keratin-19) labelled stem cells has
been detected in the presumed transit cell zone of mouse colonic crypts, capable of
giving rise to Lgr5^{+} stem cells with the suggestion that the former could
be the target cells for colon cancer [54]. The increasing evidence for plasticity of
stem and other cells in the lineage, potentially may increase the target number,
especially at radiation doses which induce some cytotoxicity which transiently
disturbs the homeostatic locations of particular cell types [55]. In contrast, higher doses causing more
cytotoxicity may reduce the surviving target cell number. These features would add
further complexity to the current models.

There is also evidence in other tissues that transit cells rather than stem cells are
the more likely target. For basal cell carcinoma the normal renewal process is
slower than in the colon, but the process is similar, with basal stem cells that
divide and give rise to more-rapidly dividing transit lineages, each transit lineage
comprising three to five rounds of cell replication [56]. A model for human skin cancer proposed
that stem cells were the likely target cells for basal cell carcinomas, early
progenitor cells for squamous cell cancers, and late progenitor cells for papillomas
[57]. Also, there is
evidence in mice that the initial radiation-induced acute myeloid leukemia (AML)
stem cell may originate not only from irradiated hematopoietic stem cells (HSC), but
also from multipotent and common myeloid progenitor cells [58]. In addition, in a
*Cre*-knock-in mouse cancer model, lung type II cells, partially
differentiated Clara cells of the terminal bronchioles, and bronchioalveolar stem
cells all were identified as the cells of origin for *K-ras*-induced
lung hyperplasia [59].
Interestingly, only type II cells progressed to adenocarcinoma [59].

There is a considerable body of evidence supporting the existence of what have been termed cancer stem cells (CSC), that is to say a subpopulation of cancer cells that have stem-like tumor-initiating properties. The evidence is particularly strong for leukemia [60, 61], and for colon cancer [62–65], but somewhat less for cancers of the breast [66] and brain [67]. This idea is still controversial, and the data in support of its application to certain solid cancer sites somewhat contradictory [68]. Irrespective of that, it is not clear for all cancer sites what may be the origin of the CSC. It is possible that the CSC derives from a mutated stem cell, and evidence for this is strongest for AML, where the associated CSCs have been shown to comprise distinct, hierarchically-arranged classes, similar to those observed with HSC, that dictate distinct fates [61]. It is also plausible in the light of our analysis (Table 1) since, as discussed above, AML is likely to have a small number of critical mutations. However, CSC may also arise from what we term transit cells, that already have undergone one or more stages of differentiation, via some process of de-differentiation and relocation into the stem cell niche. Although it is not known whether this does occur, in most tissues there are very many more differentiated than stem cells, and the number of steps involved in de-differentiating human adult somatic cells into pluripotent human stem cells is modest [69, 70], suggesting that this process may be more likely than the alternative, of mutation of stem cells.

Evidence has been found supporting Cairns’ hypothesis of DNA strand-segregation,
which would reduce the mutation rate in the stem cell population from chronic
exposures, in a number of experimental systems, including small intestinal crypts,
mammary epithelium, some muscle satellite cells and progenitor cells, some central
nervous system cells, although not in haematopoietic stem cells [4]. In particular, Potten
*et al*. [71] using a pulse/chase experiment with tritiated thymidine
(^{3}HTdR), found long-term label-retaining cells in the intestinal
crypts of neonatal mice. Potten *et al*. [71] hypothesized that long-term incorporation
of ^{3}HTdR occurred because neonatal mice have undeveloped small
intestines, and that pulsing ^{3}HTdR soon after the birth of the mice
allowed the “immortal” DNA of adult stem cells to be labeled during their formation.
These long-term (stem) cells were demonstrated to be actively cycling, as
demonstrated by incorporation and release of BrdU [71]. Another mechanism for reducing the
mutation rate from chronic exposures is stem cell competition for residence in the
niche, between advantaged/undamaged stem cells and disadvantaged/damaged stem cells
from radiation [72–74]. Presumably, in this
scenario stem cells would also be advantaged over transit cells, in steady state
conditions.

In summary, the analysis we have presented suggests that the probability of a cancer being mutagen-induced correlates significantly with the cumulative number of stem cell divisions, confirming an earlier report [7]; in some cases the effect is of quite modest size, and in some cases the findings are also sensitive to removal of high-leverage datapoints, but these issues do not affect the validity of the findings. Our analysis also suggests that the relative contribution to total cancer risk from mutated transit cells (as opposed solely to mutated stem cells) is relatively large, so that almost no cancers arise solely from stem-cell mutations. However, particularly for cancers with 2 or 3 critical mutations, such as leukemia, a substantial proportion of cancers, in some cases 100%, will have at least one mutation deriving from a mutated stem cell.

(ZIP)

Click here for additional data file.^{(3.7M, zip)}

This work was supported by the Intramural Research Program of the National Institutes of Health, the National Cancer Institute, Division of Cancer Epidemiology and Genetics. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Data Availability

All relevant data are within the paper and its Supporting Information files.

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