Theoretical Biology and Medical
Modelling
0
BioMed Central
Research
A statistical model for the identification of genes governing the
incidence of cancer with age
Kiranmoy Das' and Rongling Wu*1,2,3
Address: 'Department of Statistics, University of Florida, Gainesville, FL 32611, USA, 2UF Genetics Institute, University of Florida, Gainesville, FL
32611, USA and 3Department of Operation Research and Financial Engineering, Princeton University, Princeton, NJ 08544, USA
Email: Kiranmoy Das Kiranmoy@stat.ufl.edu; Rongling Wu* rwu@stat.ufl.edu
* Corresponding author
Published: 16 April 2008 Received: 15 September 2007
Theoretical Biology and Medical Modelling 2008, 5:7 doi: 10.1 186/1742468257 Accepted: 16 April 2008
This article is available from: http://www.tbiomed.com/content/5/1/7
2008 Das and Wu; licensee BioMed Central Ltd.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The cancer incidence increases with age. This epidemiological pattern of cancer incidence can be
attributed to molecular and cellular processes of individual subjects. Also, the incidence of cancer
with ages can be controlled by genes. Here we present a dynamic statistical model for explaining
the epidemiological pattern of cancer incidence based on individual genes that regulate cancer
formation and progression. We incorporate the mathematical equations of agespecific cancer
incidence into a framework for functional mapping aimed at identifying quantitative trait loci (QTLs)
for dynamic changes of a complex trait. The mathematical parameters that specify differences in
the curve of cancer incidence among QTL genotypes are estimated within the context of maximum
likelihood. The model provides testable quantitative hypotheses about the initiation and duration
of genetic expression for QTLs involved in cancer progression. Computer simulation was used to
examine the statistical behavior of the model. The model can be used as a tool for explaining the
epidemiological pattern of cancer incidence.
Background
Age is thought to be the largest single risk factor for devel
oping cancer [1,2]. A considerable body of data suggests
that the incidence of cancer increases exponentially with
age [37], although death from cancer may decline at very
old age. This agedependent rise in cancer incidence is
characteristic of multicellular organisms that contain a
large proportion of mitotic cells. For those organisms
composed primarily of postmitotic cells, such as Dro
sophila melanogaster (flies) and Caenorhabditis elegans
(worms), no cancer will develop. Elucidation of the
causes of increasing cancer incidence with age in multicel
lular organisms can help to design a strategy for primary
cancer prevention. The association between cancer and
age can be explained by one or two of the physiological
causes [8], i.e., a more prolonged exposure to carcinogens
in older individuals [9] and an increasingly favorable
environment for the induction of neoplasms in senescent
cells [10]. These two possible causes lead older humans to
accumulate effects of mutational load, increased epige
netic gene silencing, telomere dysfunction, and altered
stromal milieu [2].
As a complex biological phenomena, susceptibility to can
cer and its agedependent increase is thought to include
mixed genetic and environmental components [1113].
The use of candidate gene approaches or association stud
ies has led to the identification of specific genetic variants
for cancer risk and their interactions with other genes and
with environment, such as lifestyle. A more powerful
method for cancer gene identification is to scan the com
plete genome for polymorphisms that confer increased
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Theoretical Biology and Medical Modelling 2008, 5:7
risk [11 ]. Genomewide identification of cancer genes has
been conducted in laboratory mice by mapping individ
ual quantitative trait loci (QTLs) for tumor susceptibility
or resistance [1214]. As a model system for studying
human cancer, mice have been useful for elucidating the
genetic architecture of cancer through the control of envi
ronmental exposure leading to tumorigenesis, which can
not be done with human populations [11]. A recent
success in constructing a haplotype map of the human
genome with single nucleotide polymorphisms (SNPs)
[15] will make it possible to conduct a similar genome
wide search at the DNA sequence level in humans, as long
as a statistical method that can detect the association
between cancer and genes is available.
Unlike a static trait, agerelated progressive changes in
cancer incidence are a dynamic process. For this reason,
traditional methods for QTL mapping of static traits will
not be feasible, at least not be efficient, because the tem
poral pattern of cancer incidence is not considered.
Recently, Wu and colleagues have developed a series of
statistical models for mapping dynamic traits in which
mathematical functions that specify biological processes
are integrated into a QTL mapping framework (reviewed
in [16]). The basic principle of these models, called func
tional mapping, is to characterize the genetic effects of
QTLs on the formation and process of a biological trait by
estimating and testing genotypespecific mathematical
parameters for dynamic processes. Functional mapping is
now used to map QTLs for growth curves in experimental
crosses through linkage analysis [1720] and for HIV
dynamics and circadian rhythms in natural populations
though linkage disequilibrium analysis [2123].
In this article, we attempt to extend the idea of functional
mapping to detect QTLs that predispose organisms to an
agerelated rise in cancer incidence. Frank [4] proposed a
mathematical model for the agespecific incidence of can
cer based on the molecular processes that lead to uncon
trolled cellular proliferation. This model is defined by two
key parameters, carrying capacity (K) and intrinsic growth
rate (r). Thus, by estimating genotypespecific differences
in these two parameters, the genetic effect of a QTL on age
related increase in cancer incidence can be estimated and
tested. The new model will be designed for mouse sys
tems, in which cancer cells can be counted in lifetime.
Also, by controlling the environment of mouse models,
the new model is able to understand how a QTL interacts
with environmental carcinogens to produce cancer. For
experimental crosses derived from inbred strains of mice,
linkage mapping based on the estimation of the recombi
nation fractions between different loci can serve a
genomewide search for cancer QTLs [24,25]. For outbred
or wild mice that containing multiple genotypes, cancer
QTL identification can be based on linkage disequilib
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rium analysis [26]. The new model for cancer incidence
will be constructed with a random sample drawn from an
experimental or natural population in which genetic
markers are associated with the underlying QTL in terms
of linkage disequilibrium. The new model provides a
number of biologically meaningful hypothesis tests about
the genetic and developmental control mechanisms
underlying cancer risk. Computer simulations were per
formed to investigate the statistical behavior of the new
model and validate its utilization.
Model
Logistic Model
It is well known that the incidence of cancer increases pro
gressively with age [3]. This epidemiological pattern of
cancer incidence is rooted in mutational processes. By
assuming that cancer arises through the sequential accu
mulation of mutations within cell lineages [27], Frank
[4,28] provided a general mathematical (logistic) equa
tion for describing agespecific clonal expansion resulting
from a mutation. Starting with a single cell, the number of
clonal cells due to accumulative mutations after a time
period t is expressed as
rt
y(t) = (1)
K+e t 1
where K is the carrying capacity and r is the intrinsic rate
of increase of the clone. If a QTL affects agedependent
clonal expansion, there will be different carrying capaci
ties and different rates of increase among different QTL
genotypes.
Mapping Population
Suppose there are two groups of mice randomly drawn
from an experimental or natural population at Hardy
Weinberg equilibrium. These two groups are reared in two
different controlled environments, such as case (the mice
exposed to a carcinogen) and control (with no such expo
sure). Let nk be the size of group k (k = 1, 2). For both
groups, molecular markers such as single nucleotide pol
ymorphisms (SNPs) are genotyped throughout the
genome. For each sampled mouse in each group, the
number of cells in the clone due to accumulated muta
tions is counted at a series of equallyspaced ages, (1, 2,...,
T), in lifetime.
Assume that a QTL with alleles A and a affects the clonal
expansion of cells. This QTL is associated with a marker
with alleles M and m. The linkage disequilibrium between
the QTL and marker is denoted as D. Let p, 1 p and q, 1 
q be the frequencies of marker alleles M, m and QTL alleles
A, a, respectively, in the population. The QTL and marker
generate four haplotypes, MA, Ma, mA and ma. The fre
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Theoretical Biology and Medical Modelling 2008, 5:7
quencies of these haplotypes are expressed, respectively,
as
p1 = pq + D,
lo= p(1 ) D,
o = (1 p)q D,
oo= (1 p)( ) + D.
These haplotype frequencies are used to derive the joint
genotype frequencies of the marker and QTL, expressed as
AA Aa aa
MM pj2 2pllP0o P2o
Mm 2p11po0 2p11poo + 2ploPol 2po01oo
mm p21 2PoiPoo PoO
from which we can derive the conditional probabilities of
a QTL genotype, j (j = 0 for aa, 1 for Aa and 2 for AA),
given a marker genotype of subject i, symbolized as owji.
Conditional probability ,j i is a function of = (p, q, D).
Likelihood
For subject i, the number of clonal cells at age t (t = 1, 2,
..., T) under environment k can be expressed in terms of
the underlying QTL as
Y1(t) = iogo(t) + igk(t) + i2g2k(t) + eik(t), (2)
where ij is an indicator variable for a possible QTL geno
type of individual i, defined as 1 if a particular QTL geno
type j is indicated and 0 otherwise;,. (t) is the genotypic
value of QTL genotype j for clonal number at age t, which
can be fit by Frank's [4] logistic model, i.e.,
gj,(t)
KjkeTjkt
Kjk+e'jkt1
specified by a set of parameters
e = {je} = {Kk,rk} jo,k=1 and eik(t) is the resid
ual effect for subject i, distributed as MVN(0, ,i). We
assume that matrix Zi is composed of the two covariance
matrices each under a different environment (k) since cov
ariances between environments are thought not to exist.
The covariance matrix under environment k is fit by a first
order autoregressive (AR(1)) model with variance okf and
correlation p .ar I.l.d in 7= { !}.
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The mixture modelbased likelihood of samples with lon
gitudinal measurements y and marker information M is
formulated as
2 n, [ 2
L(Q, ETfy,M)= 1 J7IXfi(ylk) (3)
h=l i=1 lj=0
where fjk(Yik) is a multivariate normal distribution for the
number of clonal cells with mean vectors specified by Ojk
and covariance matrix specified by the AR(1) model with
4.
Estimation and Algorithm
The likelihood (3) contains three types of parameters (0,
0, 7), which can be estimated by the EM algorithm or
simplex algorithm. Wang and Wu [21] derived a closed
form for the EM algorithm to obtain the maximum likeli
hood estimates (MLEs) of the haplotype frequencies, and
therefore the allele frequencies and linkage disequilib
rium contained in 0. Because agedependent means and
covariances are modeled by nonlinear equations, it is dif
ficult to derive the closed forms for these model parame
ters. Wang and Wu [21] have successfully used the simple
algorithm to obtain the MLEs of parameters contained in
o and q.
Hypothesis Testing
One of the most significant advantages of functional map
ping is that it can ask and address biologically meaningful
questions by formulating a series of statistical hypothesis
tests. Here, we describe the most important hypotheses as
follows:
Existence of a QTL
Testing whether a specific QTL is associated with the logis
tic function of the number of clonal cells is a first step
toward understanding the genetic architecture of clonal
expansion. The genetic control of the entire clonal expan
sion process can be tested by formulating the hypothesis:
H0: D = 0 vs. HI: D 0.
The null hypothesis states that there is no QTL affecting
the clonal expansion of the cells (the reduced model),
whereas the alternative states that such a QTL does exist
(the full model). The statistic for testing this hypothesis is
the loglikelihood ratio (LR) of the reduced to the full
model, i.e.,
LR, = 21[n L(i , ') In L(i,, ,')1,
where the tildes and hats denote the MLEs of the
unknown parameters under the Ho and H1, respectively.
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The LR is asymptotically 2distributed with one degree of
freedom.
A similar test for the existence of a QTL can be performed
on the basis of the hypotheses about genotypicspecific
differences in curve parameters, i.e.,
Ho: Olk (K,r),j = 0,1, 2;k = 2
H1 : At least one of the qualities in Ho does not hold.
(6)
We can compute the LR by calculating the parameter esti
mates under the null and alternative hypotheses above.
However, in this case, it is difficult to determine the distri
bution of the LR because linkage disequilibrium is not
identifiable under the null. An empirical approach to
determine the critical threshold is based on permutation
tests, as suggested by Churchill and Doerge [29].
Although the two hypotheses (4 and 6) can be used to test
the existence of a QTL in association with a genotyped
marker, they have a different focus. The null hypothesis of
(4) proposes that a QTL may exist, but it is not associated
with the marker. The null hypothesis of (6) states that no
significant QTL exists, regardless of its association with the
marker. Because of this difference, the critical value for the
LR calculated under Hypothesis (4) can be determined
from a X2distribution, whereas permutation tests are used
to determine the critical value under Hypothesis (6)
because the LR distribution is unknown.
Pleiotropic Effect of the QTL
If a significant QTL is found to exist, the next test is for a
pleiotropic effect of this QTL on clonal expansion under
two different environments. The effects of this QTL
expressed in environments 1 and 2 are tested by
Ho: Ejl (KI,r),j = 0,1,2
H1 : At least one of the qualities in Ho does not hold,
(7)
and
Ho: Ej2 (K2,2),j= 0,1,2
H1 : At least one of the qualities in Ho does not hold.
(8)
If both the null hypotheses above are rejected, this means
that the detected QTL exerts a pleiotropic effect on clonal
expansion in the two environments considered. The
thresholds for these tests can be determined from permu
tation tests separately for different environments.
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QTL by Environment Interaction
If the QTL shows a significant effect only in one environ
ment, this means that a significant QTL by environment
interaction exists. However, a pleiotropic QTL may also
show significant QTL by environment interactions,
depending on whether there is a difference in agespecific
genetic effects between the two environments. This can be
tested by formulating the following hypotheses:
Ho: 01 + 021 = 02 + 022 and 281 (801 + 21) = 2812 (002 + 22)
H1 : At least one of the qualities in Ho does not hold,
(9)
The critical value for the testing QTL by environment
interactions can be based on simulation studies.
Testing for Individual Parameters
Our hypotheses can also be based on individual parame
ters (K and r) that determine agerelated changes for the
numbers of clonal cells. We can test how a QTL affects
each of these two parameter, and whether there is a signif
icant QTL by environment interaction for each parameter.
The critical values for these tests can be based on simula
tion studies.
Computer Simulations
We perform simulation experiments to examine the statis
tical properties of the model proposed to detect QTLs
responsible for clonal expansion. We assume an experi
mental or natural mouse population that is at Hardy
Weinberg equilibrium. A molecular marker with two alle
les M and m is associated with a QTL with two alleles A
and a that determines the clonal expansion of a cancer
with age. The allele frequencies of marker allele M and
QTL allele Q are assumed to be p = 0.5 and q = 0.6, respec
tively, and there is a positive value of linkage disequilib
rium (D = 0.08) between the marker and the QTL. Using
these allele frequencies and linkage disequilibrium, the
distribution and frequencies of markerQTL genotypes in
the population can be simulated.
In order to study the genetic control of cancer incidence,
we select a panel of mice randomly from the population
and divide them into two groups, each (with nk = 100 or
200 mice) reared under a different environmental condi
tion. This design allows QTL by environment interaction
tests. For each mouse from each study group, the number
of cancer cells is simulated at eight successive ages (T = 8)
by assuming a multivariate normal distribution with envi
ronmentspecific mean vectors specified by the logistic
equation (1) and environmentspecific covariance matri
ces specified by the AR(1) model. The parameters that fit
the logistic equations and AR(1)structured matrices are
given in Table 1. Although the markerQTL genotype fre
quencies are identical for the two groups, the effects of the
QTL may be different because of the impact of environ
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ment on gene expression. Thus, the two groups are
assumed to have different curve parameters for the same
QTL genotype (Table 1). The residual variance is deter
mined on the basis ofheritability. For each group, two lev
els of heritability, 0.1 and 0.4, are assumed for the
number of cancer cells at a middle time point.
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The simulated data were analyzed by the model, which
was repeated 100 times to estimate the means and sample
errors of the MLEs of parameters. The estimation results
are tabulated in Table 1. It can be seen that the QTL con
trolling agedependent clonal expansion can be detected
using the marker in association with the QTL. As expected,
Table I: Maximum likelihood estimates of the parameters describing the clonal expansion, each corresponding to a QTL, and marker
allele frequency, QTL allele frequency and markerQTL linkage disequilibrium with 8 time points. Numbers in parentheses are the
sampling errors of the estimates
n= 400
Parameters
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True value
H2= 0.1
0.48(0.023)
0.62(0.014)
0.074(0.0067)
102.50(0.452)
0.097(0.00069)
108.56(0.492)
0.147(0.0011)
152.89(0.865)
0.209(0.0015)
163.09(0.106)
0.244(0.0016)
198.26(0.756)
0.247(0.0053)
209.56(0.685)
0.311(0.002)
0.61(0.0078)
0.593(0.0022)
1.322(0.0052)
1.322(0.0072)
n = 200
H2= 0.4
0.49(0.021)
0.61(0.01 1)
0.075(0.003)
102.31(0.449)
0.097(0.00062)
109.20(0.481)
0.147(0.0011)
152.35(0.862)
0.21(0.0014)
162.52(0.105)
0.244(0.0016)
199.63(0.752)
0.248(0.0050)
209.79(0.682)
0.311(0.001)
0.61(0.0076)
0.595(0.0021)
0.539(0.0026)
0.53(0.0018)
100
0.60
0.60
1.3 1
0.538
1.3 1
0.53
H2= 0. I
0.49(0.014)
0.61(0.01 1)
0.079(0.004)
101.65(0.450)
0.098(0.00061)
109.98(0.486)
0.149(0.0010)
151.72(0.862)
0.21(0.0014)
161.08(0.102)
0.248(0.0012)
199.90(0.743)
0.248(0.0051)
209.98(0.681)
0.308(0.001)
0.61(0.0071)
0.595(0.0021)
1.319(0.0045)
1.320(0.0056)
H2= 0.4
0.50(0.012)
0.60(0.010)
0.079(0.004)
100.59(0.441)
0.099(0.00012)
110.25(0.479)
0.150(0.0007)
150.06(0.858)
0.20(0.001 1)
160.32(0.098)
0.250(0.001 1)
200.03(0.735)
0.249(0.0046)
210.22(0.668)
0.302(0.0008)
0.61(0.0070)
0.598(0.0020)
0.538(0.0019)
0.53(0.001 1)
Theoretical Biology and Medical Modelling 2008, 5:7
the frequencies of marker alleles can be estimated more
precisely than those of QTL alleles. The precision of esti
mating QTL allele frequencies and markerQTL linkage
disequilibrium increases with increasing sample size and
increasing heritability (Table 1). The curve parameters
that describe agespecific cancer incidence can be gener
ally well estimated, with increasing precision when sam
ple size and heritability increase. A similar trend was
found for the AR(1) parameters that model the structure
of the covariance matrices.
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Figures 1 and 2 illustrate the shapes of estimated age
dependent cancer incidence curves for each QTL geno
type, comparing with those of given curves. In general, the
estimated curves are consistent with those given curves
even when the heritability (0.1) and sample size (200) are
modest, suggesting that the model can reasonably detect
the genetic control of cancer incidence curves. In practice,
our model can formulate a number of meaningful
hypotheses, e.g., (7)(9). In this study, these hypothesis
tests were not performed because no real data are pres
ently available.
Figure I
Curves for the number of cancer clones changing with age, determined by three different QTL genotypes AA,
Aa, and aa, using given parameter values (solid) and estimated values (broken) with different heritabilities (H2)
for a sample size of n = 200. (A) Group I, H2 = 0.1, (B) Group I, H2 = 0.4, (C) Group 2, H2 = 0.1, and (D) Group 2, H2 =
0.4.
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Theoretical Biology and Medical Modelling 2008, 5:7
 AA actual
 AA fitted
 Aa actual
 Aa fitted
 aa actual
 aa fitted
2 3 4 5 6 7
Time
Time
Figure 2
Curves for the number of cancer clones changing with age, determined by three different QTL genotypes AA,
Aa, and aa, using given parameter values (solid) and estimated values (broken) with different heritabilities (H2)
for a sample size of n = 400. (A) Group I, H2 = 0.1, (B) Group I, H2 = 0.4, (C) Group 2, H2 = 0.1, and (D) Group 2, H2 =
0.4.
Discussion
Aging is associated with a number of molecular, cellular,
and physiological events that affect carcinogenesis and
subsequent cancer growth [8]. In both humans and labo
ratory animals, the incidence of cancer is observed to
increases with age [1,2,6]. A clear understanding of the
genetic and developmental control of agerelated cancer
incidence is needed to design an optimal drug for cancer
prevention based on an patient's genetic makeup.
Although cellular and molecular explanations for this
phenomenon are available [30,31], knowledge about its
genetic causes is very limited. In this article, we derive a
computational model for mapping quantitative trait loci
(QTLs) that control an agerelated rise in cancer incidence.
The model was founded on the idea of functional map
ping [1621,23,32] by implementing a logistic equation
for the agerelated progression of cancer cells that is
derived from molecular and cellular processes related to
the pathway of cancer formation [4,33].
Our model for QTL mapping was constructed for mouse
models for two reasons. First, it is possible to count cancer
cells of an experimental mouse in lifetime, which is cru
cial for studying the association between cancer and cellu
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Time
Time
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Theoretical Biology and Medical Modelling 2008, 5:7
lar senescence. Second, environmental exposure for the
mouse that leads to tumorigenesis can be controlled so
that the effects of QTL by environment interactions on
cancer incidence can be characterized. The model is built
on the premise of linkage disequilibrium (i.e., nonran
dom association between different loci) that has proven
useful for finescale mapping of QTLs [34]. A recent survey
about linkage disequilibria with a natural population of
mice in Arizona suggests that this population is suitable
for finescale QTL mapping and association studies [26].
In humans, it is not possible to count cancer cells in a per
son's lifetime. However, the idea of our model can be
modified for human cancer studies by sampling people
with different ages ranging from young (e.g., 10 years) to
old (e.g., 75 years). For each subject in such a sampling
design, the number of cells in the clone due to accumu
lated mutations is counted at several subsequent ages (at
least three years). Thus, we will have an incomplete data
set in which cell numbers for all subjects are missing at
some particular ages. Hou et al.'s [35] functional mapping
model, which takes into account unevenly spaced time
intervals and missing data, can be used to manipulate
such an incomplete data set.
We model the effects of environment including those
related to lifestyle exposures on agespecific increases in
cancer incidence. When the sexes are viewed as different
environments, it will be interesting to incorporate sexspe
cific differences in haplotype frequencies, allele frequen
cies and linkage disequilibrium [36]. Also, as a general
framework, we model the association between one
marker and one QTL, which is far from the reality in
which multiple QTLs interact with each other in a compli
cated network to affect a phenotype [24]. However, our
model can be easily extended to consider these possible
genetic interactions and fully characterize the detailed
genetic architecture of cancer incidence. Bayesian
approaches that have been shown to be powerful for solv
ing highdimensional parameter estimation [37] will be
useful for implementing genetic interactions between dif
ferent QTLs into our model for mapping agerelated accel
eration of cancer incidence.
With the availability of highdensity SNPbased maps in
humans and experimental crosses of mice, QTL mapping
has developed to a point at which genetic variants for
complex traits can be specified at the DNA sequence level.
Wu and colleagues developed a handful of computational
models for associating the haplotypes constructed by a
series of SNPs and complex traits [22,3841]. By incorpo
rating these haplotypebased mapping strategies into the
model proposed here, we can characterize specific combi
nations of nucleotides that encode an agerelated increase
in cancer incidence. Although our model has not been
used in a practical project because no real data are availa
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ble for now, specific experimental designs can be
launched to establish and test new hypotheses about can
cer progression. All in all, our model should stimulate
new empirical tests and help to perform cuttingedge stud
ies of carcinogenesis by integrating the epidemiological
pattern of cancer incidence, molecular processes that
derive cancer formation and development, mathematical
modeling of cellular dynamics and statistical analyses of
DNA sequences.
Authors' contributions
KD derived the equation, programmed the algorithm and
performed computer simulations. RW conceived the idea
and wrote the manuscript.
Acknowledgements
The preparation of this manuscript is supported by NSF grant (0540745) to
RW.
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