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The impact of radiation on the development of lung cancer
Accepted Manuscript
The Impact of Radiation on the Development of Lung Cancer Lingling Li, Tianhai Tian, Xinan Zhang PII: DOI: Reference:
S0022-5193(17)30290-4 10.1016/j.jtbi.2017.06.020 YJTBI 9116
To appear in:
Journal of Theoretical Biology
Received date: Revised date: Accepted date:
9 December 2016 16 June 2017 19 June 2017
Please cite this article as: Lingling Li, Tianhai Tian, Xinan Zhang, The Impact of Radiation on the Development of Lung Cancer, Journal of Theoretical Biology (2017), doi: 10.1016/j.jtbi.2017.06.020
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Highlights • The probability data which is transformed from the lung cancer incidence rates in the Osaka Cancer Registry (OCR) and Life Span Study (LSS)
mutations.
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cohort of atomic bomb survivors are analyzed by the model with three
• The Chi-square test is utilized to study the mechanism that radiation induces lung cancer development.
• Radiation has more significant impact on the mutations of cells than the
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clonal expansion of cells in the development of lung cancer.
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• There is a gender difference in the progression of lung cancer.
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The Impact of Radiation on the Development of Lung Cancer
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Lingling Lia , Tianhai Tianb , Xinan Zhanga,∗ a School
of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, P.R.China b School of Mathematical Science, Monash University, Melbourne Vic 3800, Australia
Abstract
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Environment factors such as radiation play an important role in the incidence of lung cancer. In spite of substantial efforts in experimental study and mathematical modeling, it is still a significant challenge to estimate lung cancer risk
from radiation. To address this issue, we propose a stochastic model to investigate the impact of radiation on the development of lung cancer. The proposed
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three-stage model with clonal expansion is used to match the data of the male and female patients in the Osaka Cancer Registry (OCR) and Life Span Study (LSS) cohort of atomic bomb survivors in Hiroshima and Nagasaki. Our results
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indicate that the major effect of radiation on the development of lung cancer is to induce gene mutations for both male and female patients. In particular, for male patients, radiation affects the mutation in normal cells and the
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transformation from premalignant cells to malignant ones. However, radiation for female patients increases the mutation rates of the first two mutations in
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the stochastic model. The established relationship between parameters and radiation will provide insightful prediction for the lung cancer incidence in the radiation exposure.
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Keywords: Lung cancer, Gene mutation, Clonal expansion, Three-stage model, Chi-square test
∗ Corresponding author at: School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, P.R.China Email address: [email protected] (Xinan Zhang)
Preprint submitted to Journal of LATEX Templates
June 20, 2017
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1. Introduction The Life Span Study (LSS) cohort of the atomic bomb survivors in Hiroshima and Nagasaki has served as a major source of data used for evaluating the health
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risks from radiation carcinogenesis in human (Kodama et al., 1996; Preston
et al., 2007; Richardson & Hamra, 2010; Ozasa, 2016; Grant et al., 2017). It includes almost 93,000 atomic bomb survivors and 27,000 people who were not
in Hiroshima or Nagasaki at the time of the bombings, but lived in Hiroshima
or Nagasaki in October 1950. Among the atomic bomb survivors, the incidence
of all solid cancers is higher with increasing radiation dose from the bombings. Thus, the LSS cohort of atomic bomb survivors is widely used as a test system
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to explore the impact of radiation on the development of cancer. Lung cancer is the most common cancer world-wide (Parkin et al., 2005) and the second most common cancer in the LSS cohort of the atomic bomb survivors in Hiroshima and Nagasaki. Evidence suggests that environmental exposures such as cigarette smoking and radiation have increased the risks of lung cancer
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(Dela Cruz et al., 2011). Statistical analysis suggests that more than 90% of lung cancer is caused by these extrinsic factors (Song et al., 2015). In addi-
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tion, lung cancer incidence among atomic bomb survivors is strongly associated with radiation, with an estimated excess relative risk per Gy of 0.81 and excess absolute risk per Gy of 7.5 per 10,000 person-year (Preston et al., 2007). There-
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fore, approaches are imperatively needed to explore how radiation affects the development of lung cancer. A pilot study suggests that mutation frequencies
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of certain genes (e.g., the TP53 tumor suppressor gene) and methylation levels (e.g., the retrotransposon LINE1) may be associated with radiation exposure. It is widely accepted that the formation of nearly all sorts of tumors is largely
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owing to the dynamic changes in the genome. There are three types of genes that are responsible for tumorigenesis, which are oncogenes, tumor-suppressor genes and stability genes (Vogelstein & Kinzler, 2004). In the early 1950’s, a multistage model was introduced as an essential tool to understand tumorigenesis
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(Armitage & Doll, 1954). This model describes the tumorigenesis as a process of
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finite number of mutations turning a normal cell into a malignant one. It shows that the logarithm of incidence was a linear function of the logarithm of age. With the advances of molecular biology, clonal expansion was recognized as an
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essential stage in carcinogenesis. In response to these discoveries, the two-stage model with clonal expansion was proposed (Armitage & Doll, 1957; Moolgavkar
& Venzon, 1979). In this model, the first step involves the mutation from a normal cell to an intermediate cell that confers clonal expansion, and the further
mutation of the intermediate cells into a malignant cell is the second step. Since
then the two-stage model with clonal expansion has been widely used to analyze cancer incidence or mortality. For lung cancer, researchers dominantly used this
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model to study the effects of extrinsic factors on this disease (Little et al., 2002; Hazelton et al., 2006; Fakir et al., 2010; Moolgavkar et al., 2012; Hazelton et al., 2014; Moolgavkar et al., 2015). With the development of DNA sequencing technology, however, more and more gene mutations have been identified in tumors. 45
The two-stage model can be readily generalized to the models that incorporate
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more biological and medical discoveries, such as multiple genetic pathways and more number of stages (Little et al., 2008). Recently, multi-stage models have
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been used to fit lung cancer data in the Mayak workers (Z¨ ollner et al., 2015), which indicated that the three-stage model with clonal expansion fitted the 50
data better than the two-stage model and the model with genetic instability.
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Besides, only three driver gene mutations are required for the development of lung cancer (Tomasetti et al., 2015), and the three driver mutations can give a
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more reasonable explanation for cancer development (Song et al., 2015). Thus it is important to study the impact of radiation on lung cancer development by using the model with three mutations.
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In this article, we use the stochastic three-stage model with clonal expan-
sion to analyze the impact of radiation on the development of lung cancer. The model is applied to lung cancer data which are the probability of lung cancer among male and female patients aged 40–79 years from the Osaka Cancer Reg-
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istry (OCR) during 1974-1977 and the LSS cohort of atomic bomb survivors in Hiroshima and Nagasaki during 1958–1987. We first infer the model parameters 4
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by using the data from the OCR. Then the data from the LSS cohort are analyzed to study the impact of radiation on the development of lung cancer. We are particularly interested in the changes of model parameters due to the exposure to radiation. Thus hypotheses are proposed to identify the mechanism of
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radiation action, as well as to lay out which parameters are affected significantly by radiation. These hypotheses are then tested by the Chi-square test.
2. Materials and Methods
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2.1. Data on Lung Cancer Incidence 2.1.1. The OCR data
The OCR has been operated since 1962. It covers nearly 9 000 000 residents in Osaka Prefecture, Japan.
Among the data from the OCR, cate-
gories were created for both incident cases of cancer and populations by gender (male and female), calendar year (1974–1997) and 5-year age groups (ages 0–85+). The detailed data can be obtained from the OCR website (http://
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www.iph.pref.osaka.jp/omc/ocr/). Our analysis uses the age-specific lung can-
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cer incidence rates in male and female during the period of 1974–1977, which is reported in the paper (Yoshimi et al., 2003). Rates are expressed as cases per 100, 000 people.
2.1.2. The LSS data
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The LSS cohort comprises 120,321 individuals. Among them, 8666 first
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primary solid cancer patients were diagnosed between 1958 and 1987. Lung cancer accounts for nearly 10% of the cancers in the LSS cohort. The LSS data are available at the Radiation Effects Research Foundation (RERF) website (http://www.rerf.jp/). The data are grouped by city (Hiroshima and Nagasaki),
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gender (male and female), calendar time (5–year intervals for calendar years 1958–1987), attained age (5–year categories from age 15 to 85 and categories for ages less than 15 or 85+), age at exposure (5–year categories from age 0 to 60 and a category for ages 60+), colon dose (10 categories with dose cutoff points
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at 0.01, 0.1, 0.2, 0.5, 1, 1.5, 2, 3, 4 and 4+ Gy ), and other factors. A more detailed description of this dataset can be found in the reference (Thompson et al., 1994). Previous analyses have shown no significant difference between
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patients in Hiroshima and Nagasaki (Thompson et al., 1994; Grant et al., 2017). Therefore, our analyses are based on the incidence rates computed from person95
years and lung cancer cases stratified by attained age and gender during the year 1958–1987 in Hiroshima and Nagasaki. In this study, we omit the lung
cancer incidences for persons aged 0 to 39 years because they are close to zero.
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2.2. Mathematical model
The three-stage model with clonal expansion is designed to radically reduce 100
the complexity of tumorigenesis into three separate mutational steps that are genetic transitions from a pool of normal cells via premalignant to malignant. The premalignant cells are characterized by the symmetric division and differentiation or death processes. Mathematically, we assume that there are N stem
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cells in the lung tissue, and any cell is subject to mutation to a type of cells carrying an irreversible mutation at a rate of µ0 . Thereafter any cell in compart-
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ment Ii divides into two daughter cells at a rate αi and die or differentiate at a rate βi . Each cell may also divide into an equivalent daughter cell and another cell with an additional irreversible mutation at a rate µi . The unit of all rates
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is per year. In addition, the time from a malignant cell to tumor, known as the incubation period, is Ttag . This model is illustrated schematically in Fig. 1. We are interested in the probability, p(t), that at least one malignant cell
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has appeared by time t, which is easier to calculate than the hazard function
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(Moolgavkar et al., 1988). The probability function, p(t), is given by h Z t i p(t) = 1 − exp − h(s)ds ,
(1)
0
where h(s) corresponds to the cancer incidence at age s. This formula is used to
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transform the age-specific incidence of lung cancer into the probability of lung cancer at each age.
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D
D
β2
N
µ0
µ1
I1
µ2
I2
M
Ttag
α2
α1
Tumor
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β1
Fig. 1. The schematic representation of stochastic three-stage model with cell birth,
cell death, and gene mutation. N denotes the normal cell; Ii (i = 1, 2) the com-
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partment of intermediate cell; D the dead or differentiated cell; and M the malignant cell.
For the three-stage model, let N , Yi (t) (i = 1, 2), and Y3 (t) represent, respectively, the numbers of stem, premalignant Ii , and fully malignant cells at time t. For t ≥ τ , we define the following probability generating functions X
n p Y1 (t) = i1 , Y2 (t) = i2 , Y3 (t) = i3 |Y1 (τ ) = 0,
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Ψ(y1 , y2 , y3 ; τ, t) =
i1 ,i2 ,i3
(2)
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o Y2 (τ ) = 0, Y3 (τ ) = 0 y1i1 y2i2 y3i3 ,
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Φ1 (y1 , y2 , y3 ; τ, t) =
X
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and
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i1 ,i2 ,i3
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Φ2 (y2 , y3 ; τ, t) =
n p Y1 (t) = i1 , Y2 (t) = i2 , Y3 (t) = i3 |Y1 (τ ) = 1,
o Y2 (τ ) = 0, Y3 (τ ) = 0 y1i1 y2i2 y3i3 ,
o X n p Y2 (t) = i2 , Y3 (t) = i3 |Y2 (τ ) = 1, Y3 (τ ) = 0
(3)
i2 ,i3
y2i2 y3i3 .
(4)
Functions Ψ, Φ1 and Φ2 satisfy the following Kolmogorov backward equations (Harris, 1963; Christopher et al., 1996) dΨ (y1 , y2 , y3 ; τ, t) = −µ0 N Ψ(y1 , y2 , y3 ; τ, t)(Φ1 (y1 , y2 , y3 ; τ, t) − 1), dτ 7
(5)
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dΦ1 (y1 , y2 , y3 ; τ, t) = (α1 + β1 + µ1 )Φ1 (y1 , y2 , y3 ; τ, t) − α1 Φ21 (y1 , y2 , y3 ; τ, t) dτ −µ1 Φ1 (y1 , y2 , y3 ; τ, t)Φ2 (y2 , y3 ; τ, t) − β1 , (6) and
dΦ2 (y2 , y3 ; τ, t) = (α2 + β2 + µ2 )Φ2 (y2 , y3 ; τ, t) − α2 Φ22 (y2 , y3 ; τ, t) dτ −µ2 y3 Φ2 (y2 , y3 ; τ, t) − β2 .
(7)
Then, by Eqs. (5)–(7), we can get the following equation system at (y1 , y2 , y3 ) =
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(1, 1, 0) dΨ dτ (τ, t) = −µ0 N Ψ(τ, t)(Φ1 (τ, t) − 1) dΦ1 (τ, t) = (α + β + µ )Φ (τ, t) − α Φ2 (τ, t) − µ Φ (τ, t)Φ (τ, t) 1 1 1 1 1 1 1 1 2 dτ , (8) −β1 dΦ2 (τ, t) = (α + β + µ )Φ (τ, t) − α Φ2 (τ, t) − β 2 2 2 2 2 2 2 dτ
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with the boundary conditions
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Ψ(t, t) = 1 Φ1 (t, t) = 1 , Φ (t, t) = 1
(9)
2
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where the dependence of Ψ(τ, t), Φ1 (τ, t) and Φ2 (τ, t) on (1, 1, 0) have been suppressed for convenience in writing. By the definition of the probability generating function Ψ, Ψ(0, t) denotes
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the probability that the tissue is still tumor free (has not yet developed a malignancy) by the time t. Thus, the probability, p(t), of at least one malignant
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cell by the time t, starting with only normal cells at time 0, can be written as
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p(t) = 1 − Ψ(0, t).
(10)
The solution of Ψ(0, t) is required for solving the probability function p(t). However, Ψ(0, t) has not a closed-form solution by system (8). To solve system (8) numerically, we fix t and change variables to s = t − τ . Let A(s, t) = Ψ(τ, t), 8
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B1 (s, t) = Φ1 (τ, t) and B2 (s, t) = Φ2 (τ, t). System (8) can be converted into the
A(0, t) = 1 B1 (0, t) = 1 . B (0, t) = 1
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with initial conditions
(11)
(12)
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following equation system dA ds (s, t) = µ0 N A(s, t)(B1 (s, t) − 1) dB1 (s, t) = −(α + β + µ )B (s, t) + α B 2 (s, t) 1 1 1 1 1 1 ds , +µ1 B1 (s, t)B2 (s, t) + β1 dB2 (s, t) = −(α + β + µ )B (s, t) + α B 2 (s, t) + β 2 2 2 2 2 2 2 ds
2
Then p(t) = 1 − A(t, t).
In this way, the probability function p(t) is derived by solving numerically the ordinary differential equation system (11, 12) with respect to s (for fixed t)
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2.3. Statistic analysis
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directly.
Before numerical simulation, we firstly transform the incidence rates to the
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probability of lung cancer at each age t by formula (1). Then, the probability values are amplified by using a suitable multiple (i.e. 104 ) to perform the Chi-square test. We regard the cell birth rates, the cell death or differentiation rates, and the cell mutation rates as parameters, as graphically depicted
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in Fig. 1. However, not all of the model parameters can be identifiable from
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epidemiological data. There are two popular approaches to deal with this nonidentifiability problem. The first one is to set some parameters equal to each other and assume a reasonable number of normal cells such as, N = 107 (Z¨ ollner et al., 2015). The other is to use a new set of parameters (Heidenreich et al.,
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1997). Here, we assume that the cell birth rates are equal, namely α1 = α2 ,
since the cancer risk mainly depends on the net proliferation rates (αi − βi ), and the simulated result is insensitive to the cell birth rates. In addition, the incubation period is relatively short, compared to the lifetime of an individual.
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We use the value reported by the Z¨ ollner et al. (Z¨ ollner et al., 2015), about five years. To highlight the influence of radiation exposure on the development of lung
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cancer, we choose the OCR data and LSS data as test systems. They mainly reflect the effect of the exposures to environmental lung cancer risk factors other 165
than radiation exposure. In these systems, the fitting of the data from OCR is
regarded as the control group. To study the impact of radiation exposure on genetic mutation and clonal expansion of cells, the data from LSS cohort are
discussed by modifying values of model parameters in the control group. For the
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fitting of the LSS data, we consider the following two alternative hypotheses:
Hypothesis one: Radiation exposure affects mutation rates of cells in the process of tumorigenesis. Thus, the birth rates and the death or differentiation rates of cells in the model are fixed (determined by the control group) but the mutation rates of cells should be higher than those of the control
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group.
Hypothesis two: Radiation exposure has an effect on the clonal expansion
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of cells in the development of lung cancer. Thus the mutation rates of cells in the model are fixed (determined by the control group), but the net proliferation rates of cells should be higher than those of the control
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group.
The numerical optimization routine fminsearch in MATLAB is used to es-
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timate the optimal parameters by the minimum value of Chi-square statistics between the real data and simulated ones. Goodness of fit is measured by the
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Chi-square test. The value of the Chi-square statistics is determined by Chi2 =
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(datap(i) − estimationp(i) )2 · 104 /estimationp(i) .
(13)
i=40
For the Chi-square test, we set the significance level to be 0.05. If the value of
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the Chi-square statistics is smaller than the value at the 5% significant level, the hypothesis is accepted. Furthermore, a smaller value of the Chi-square statistics represents a better fit. 10
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3. Results The fitting results of the three-stage model with clonal expansion to the data from OCR and LSS cohort are discussed in detail. Furthermore, the changes of parameters due to radiation are described. 3.1. The model fitting
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We firstly fit the data in the control group by the three-stage model with
clonal expansion. Fig. 2 shows the fitting of the age-specific probability of lung 195
cancer in OCR from the three-stage model for male and female patients, respec-
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tively. The fitting of the data to the model are very well for male and female
patients. Table 1 gives the optimal values of net proliferation rates (αi − βi ) and cell mutation rates of the model for both male and female patients. Table 1 suggests that the net proliferation rates of premalignant cells are not very large, 200
and the cells with two mutations have higher net proliferation rates than those
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with one mutation. Furthermore, the net proliferation rates of premalignant cells for female are larger than those for male. However, the mutation rates of cells for male patients are higher than those for female patients. Therefore,
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there is a gender difference in the development of lung cancer. Table 1
Estimated optimal parameters for model (8) using the OCR data for male patients and female patients
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separately.
Parameter
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α1 − β1 [year α2 − β2 [year µ0 [year
−1
]
−1 −1
Definition
Male
Female
]
net proliferation rate in intermediate cells, I1
0.0063
0.0552
]
net proliferation rate in intermediate cells, I2
0.2018
mutation rate in normal cells, N mutation rate in intermediate cells, I1
µ2 [year−1 ]
mutation rate in intermediate cells, I2
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µ1 [year−1 ]
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2.8739 × 10
0.3341 −6
7.3668 × 10−6 7.9592 × 10−6
5.2766 × 10−7
1.7898 × 10−6 2.3973 × 10−6
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a
0.06 Model Data
0.04 0.03 0.02 0.01 0 40
45
50
55
60
65
70
Age (years) 0.016 Model Data
0.014
Probability
0.012 0.01 0.008 0.006 0.004
0 40
45
M
0.002 50
75
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b
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Probability
0.05
55
60
65
70
75
80
Age (years)
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Fig. 2. Cancer probability predicted by the model with three mutations and age-
specific probability of tumor from the OCR in 1974-1977. (a) Prediction for male patients. (b) Prediction for female patients. (Solid-line: prediction of model; star:
3.2. The impact of radiation exposure
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patient data.)
It is well known that radiation exposure can increase the risks of lung cancer.
However, it is not clear how it affects the incidence of lung cancer. To answer this
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question, we consider two alternative hypotheses that are shown in section 2.3 for fitting the data from LSS cohort. Then, these hypotheses are tested by
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the Chi-square test. The Chi-square test result in Table 2 shows that the two hypotheses can be accepted for both male and female patients data at 5% significant level. In addition, the Chi-square statistics values of Hypothesis one
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are smaller than those of Hypothesis two for both male and female patients data. Thus, the model under Hypothesis one describes the data moderately better than the one under Hypothesis two. Table 2
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The Chi-square test results of the two hypotheses for male and female patients. Male
Df
b
c
S.5% a
Hypothesis one
Hypothesis two
Hypothesis one
Hypothesis two
14.997
17.640
4.982
7.347
36
36
36
50.998
50.998
50.998
36
50.998
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Chi2
a
Female
Chi2 is the estimated value of Chi-square (see Eq (13));
b
Df denotes the degree of freedom (= number of data - number of parameters-1);
c
S.5% denotes the value at 5% significant level.
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The optimal values of parameters under Hypothesis one are displayed in Table 3. By comparing the values of parameters in Table 1 and Table 3. we find that the first and third mutation rates are increased while the second mu-
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tation rate remains the same with the control group for male patients. For female patients, the first two mutation rates are increased, but the last mutation rate remains unchanged. These results indicate that for both male and
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female patients, the first mutation rate is increased. Thus, under the condition of radiation exposure, mutation more likely occurs in normal cells, which may
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lead to cancer.
The optimal values of parameters under Hypothesis two are shown in Table 4,
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By observing Table 1 and Table 4, we find that radiation exposure has influence
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on the net proliferation rates of all compartments of intermediate cells in the development of lung cancer. Furthermore, the changes of net proliferation rates of premalignant cells for female patients are larger than that for male patients.
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Therefore, the impact of radiation on clonal expansion of cells for female patients is more significant than that for male patients.
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Table 3
Estimated optimal parameters of model (8) under Hypotheses one by using the LSS cohort for male and female patients, separately. The values of net cell proliferation
Parameters µ0 [year
−1
Male
Female
3.0403 × 10
]
µ1 [year−1 ]
−6
7.3668 × 10−6
µ2 [year−1 ]
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rates are the same as those in Table 1.
9.2831 × 10−6
Table 4
5.4706 × 10−7
2.5239 × 10−6
2.3973 × 10−6
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Estimated optimal parameters of model (8) under Hypotheses two by using the LSS
cohort for male and female patients, separately. The values of cell mutation rates are the same as those in Table 1. Parameters α1 − β1 [year
Male −1
0.0085
α2 − β2 [year−1 ]
0.2073
0.0636 0.3593
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]
Female
To further explore the impact of radiation exposure on the progression of
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lung cancer, we conduct the next test by assuming only one parameter in the model may be affected by radiation. All the other parameters are the same as 235
those determined by the control group. The Chi-square test is implemented to
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evaluate this assumption. For the Chi-square test of the assumption, the degree of freedom is 38 and the value at the 5% significant level is 53.38. Fig. 3 gives the
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estimated minimal values of Chi-square statistics under this assumption. For the case of parameter α2 − β2 , the value of the Chi-square is 56.900 for female
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patients, which is larger than the threshold value at the 5% significant level.
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Therefore, this case should be rejected, and the radiation exposure may not be able to affect the net proliferation rate in intermediate cells, I2 . Compared with
the two cases of net cell proliferation rate (i.e. αi − βi ), the Chi-square statistics values for the cases of three mutation rates (i.e. µ0 , µ1 and µ2 ) are much
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smaller. As a consequence, the major effect of radiation on the progression
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of lung cancer is to induce the mutations of cells for both male and female patients. In particular, the Chi-square statistics value for cell mutation rate µ2 of the male patients is smaller than not only the values for cell mutation rates
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µ0 and µ1 of the male patients, but also the value for cell mutation rate µ2 of the female patents. This observation suggests that, for male patients, the
impact of radiation exposure on the third mutation rate of the model is more significant than that on the first two mutation rates in the development of lung cancer. On the other hand, for female patients, the Chi-square statistics for cell mutation rates µ0 and µ1 have the smallest values in all these tests, which
suggests that the major impact of radiation on the lung cancer development for
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female patients is to change the first two mutation rates of the model. 60
Male Female 50
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Chi2
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30
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10
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0
1
2
3
4
5
test index
Fig. 3. The Chi-square test results under the hypothesis that only one parameter is
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affected by radiation exposure for male and female lung cancer patients in the model.
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(index 1: affected µ0 , 2: µ1 , 3: µ2 , 4: α1 − β1 , 5:α2 − β2 .)
4. Conclusions In this work we have studied the impact of radiation on genetic mutations
and clonal expansion of cells in the development of lung cancer. Instead of using 260
the two-stage stochastic model that has been widely used to study lung cancer, 15
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this work employed the three-stage model with clonal expansion which was a better mathematical model proposed by the recently published research works (Z¨ ollner et al., 2015; Tomasetti et al., 2015; Song et al., 2015). The data from
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the atomic bomb survivors in Hiroshima and Nagasaki are used to infer the key parameters in the model. In particular, the OCR data are used as the control group to infer the parameters of the model for the patients without radiation.
Then the LSS data are used to infer another set of model parameters. The difference between the estimated parameters is used as the evidence to induce
the influence of radiation exposure. Simulations of the stochastic model match the probability of lung cancer from the two datasets very well. Numerical results
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suggest that there is a gender difference in the progression of lung cancer. The inferred key model parameters thus provide insightful predictions regarding the impact of radiation on the development of lung cancer.
To examine the influence of radiation, we considered two alternative hy275
potheses regarding the influence of radiation exposure, namely the influence on
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the mutation rates of cells or on the net proliferation rates of premalignant cells. We used the LSS data to estimate optimal parameters of the three-stage model
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with clonal expansion under different hypotheses. The values of the Chi-square statistics in Table 2 suggest that radiation affects not only the mutation rates of 280
cells but also the net proliferation of cells in lung carcinogenesis. However, the
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radiation exposure has more significant effect on the mutation rates of cells than the net proliferation rates of cells for both female and male patients. Focusing
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on the mutation rates of cells, our results further show that radiation exposure induces the mutation in the normal cells and transformation from premalignant
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cells to malignant ones for male patients. For female patients, however, radia-
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tion exposure increases the rates of the first two mutations in the model. For the clonal expansion of cells, the increasing of net proliferation rates of premalignant cells for female patients is higher than that for male patients.
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We have also examined the effect of radiation exposure on the change of
one single parameter of the model for lung cancer development. Interestly, the major role of radiation exposure on the development of lung cancer is to induce 16
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transformation from premalignant cells to a malignant cell for male patients but to induce the transformation from normal cells to mutated cells with selective advantages for female patients. Thus, radiation exposure can significantly accelerate the mutations that are required to overcome normal homeostatic reg-
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ulation in a tissue, which allows the gradual expansion of premalignant clones for female patients and increases the malignant transformation for male patients.
Although the impact of radiation on the development of lung cancer has
been studied by a number of works (Tokarskaya et al., 1995; Hazelton et al., 300
2006; Egawa et al., 2012; Cahoon et al., 2017), these research works did not
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give the detailed analysis for parameter changes. The innovation of our work is to investigate the effect of radiation exposure on multiple parameters and single parameter by fitting the clinic data as well as to compare the difference between the effect on mutation rates of cells and that on net proliferation rates 305
of cells. The conclusions generated from this work can be used to offer useful guidance for the prediction of lung cancer risks. Although substantial progress
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has been achieved in this work, our results only reflect the difference between the impact of radiation on lung cancer development with radiation exposure and
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that without radiation exposure. It would be important to further consider the impact of radiation intensity on lung cancer, such as the relationship between the amount of radiation dose and parameter values of the model. These interesting
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questions will be the topics of our future research work.
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Acknowledgments
This report makes use of data obtained from the Radiation Effects Research
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Foundation (RERF) in Hiroshima, Japan. RERF is a private foundation funded
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equally by the Japanese Ministry of Health and Welfare and the U.S. Department of Energy through the U.S. National Academy of Sciences. The conclusions in this report are those of the authors and do not necessarily reflect the scientific judgment of RERF or its funding agencies. Our work was partially
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supported by the National Nature Science Foundation of China (No.11071275
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and No.11371161), by self-determined research funds of Central China Normal University from the colleges’basic research and operation of MOE (Grant No.CCNU16JCZX10). We thank the reviewers very much for their very good
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suggestions on the manuscript.
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Legends Fig.1. The schematic representation of stochastic three-stage model with cell growth, cell death, and gene mutation. N denotes the normal cell; Ii (i = 1, 2) the com-
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partment of intermediate cell; D the dead or differentiated cell; and M the malignant cell.
Fig.2. Cancer probability predicted by the model with three mutations and age-
specific probability of tumor from the OCR in 1974-1977. (a) Prediction for male patients. (b) Prediction for female patients. (Solid-line: prediction of model; star:
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patient data.)
Fig.3.The Chi-square test results under the hypothesis that only one parameter is affected by radiation exposure for male and female lung cancer patients in the model.
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(index 1: affected µ0 , 2: µ1 , 3: µ2 , 4: α1 − β1 , 5:α2 − β2 .)
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The Impact of Radiation on the Development of Lung Cancer Lingling Li, Tianhai Tian, Xinan Zhang PII: DOI: Reference:
S0022-5193(17)30290-4 10.1016/j.jtbi.2017.06.020 YJTBI 9116
To appear in:
Journal of Theoretical Biology
Received date: Revised date: Accepted date:
9 December 2016 16 June 2017 19 June 2017
Please cite this article as: Lingling Li, Tianhai Tian, Xinan Zhang, The Impact of Radiation on the Development of Lung Cancer, Journal of Theoretical Biology (2017), doi: 10.1016/j.jtbi.2017.06.020
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights • The probability data which is transformed from the lung cancer incidence rates in the Osaka Cancer Registry (OCR) and Life Span Study (LSS)
mutations.
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cohort of atomic bomb survivors are analyzed by the model with three
• The Chi-square test is utilized to study the mechanism that radiation induces lung cancer development.
• Radiation has more significant impact on the mutations of cells than the
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clonal expansion of cells in the development of lung cancer.
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• There is a gender difference in the progression of lung cancer.
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The Impact of Radiation on the Development of Lung Cancer
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Lingling Lia , Tianhai Tianb , Xinan Zhanga,∗ a School
of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, P.R.China b School of Mathematical Science, Monash University, Melbourne Vic 3800, Australia
Abstract
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Environment factors such as radiation play an important role in the incidence of lung cancer. In spite of substantial efforts in experimental study and mathematical modeling, it is still a significant challenge to estimate lung cancer risk
from radiation. To address this issue, we propose a stochastic model to investigate the impact of radiation on the development of lung cancer. The proposed
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three-stage model with clonal expansion is used to match the data of the male and female patients in the Osaka Cancer Registry (OCR) and Life Span Study (LSS) cohort of atomic bomb survivors in Hiroshima and Nagasaki. Our results
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indicate that the major effect of radiation on the development of lung cancer is to induce gene mutations for both male and female patients. In particular, for male patients, radiation affects the mutation in normal cells and the
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transformation from premalignant cells to malignant ones. However, radiation for female patients increases the mutation rates of the first two mutations in
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the stochastic model. The established relationship between parameters and radiation will provide insightful prediction for the lung cancer incidence in the radiation exposure.
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Keywords: Lung cancer, Gene mutation, Clonal expansion, Three-stage model, Chi-square test
∗ Corresponding author at: School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, P.R.China Email address: [email protected] (Xinan Zhang)
Preprint submitted to Journal of LATEX Templates
June 20, 2017
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1. Introduction The Life Span Study (LSS) cohort of the atomic bomb survivors in Hiroshima and Nagasaki has served as a major source of data used for evaluating the health
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risks from radiation carcinogenesis in human (Kodama et al., 1996; Preston
et al., 2007; Richardson & Hamra, 2010; Ozasa, 2016; Grant et al., 2017). It includes almost 93,000 atomic bomb survivors and 27,000 people who were not
in Hiroshima or Nagasaki at the time of the bombings, but lived in Hiroshima
or Nagasaki in October 1950. Among the atomic bomb survivors, the incidence
of all solid cancers is higher with increasing radiation dose from the bombings. Thus, the LSS cohort of atomic bomb survivors is widely used as a test system
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to explore the impact of radiation on the development of cancer. Lung cancer is the most common cancer world-wide (Parkin et al., 2005) and the second most common cancer in the LSS cohort of the atomic bomb survivors in Hiroshima and Nagasaki. Evidence suggests that environmental exposures such as cigarette smoking and radiation have increased the risks of lung cancer
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(Dela Cruz et al., 2011). Statistical analysis suggests that more than 90% of lung cancer is caused by these extrinsic factors (Song et al., 2015). In addi-
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tion, lung cancer incidence among atomic bomb survivors is strongly associated with radiation, with an estimated excess relative risk per Gy of 0.81 and excess absolute risk per Gy of 7.5 per 10,000 person-year (Preston et al., 2007). There-
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fore, approaches are imperatively needed to explore how radiation affects the development of lung cancer. A pilot study suggests that mutation frequencies
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of certain genes (e.g., the TP53 tumor suppressor gene) and methylation levels (e.g., the retrotransposon LINE1) may be associated with radiation exposure. It is widely accepted that the formation of nearly all sorts of tumors is largely
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owing to the dynamic changes in the genome. There are three types of genes that are responsible for tumorigenesis, which are oncogenes, tumor-suppressor genes and stability genes (Vogelstein & Kinzler, 2004). In the early 1950’s, a multistage model was introduced as an essential tool to understand tumorigenesis
30
(Armitage & Doll, 1954). This model describes the tumorigenesis as a process of
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finite number of mutations turning a normal cell into a malignant one. It shows that the logarithm of incidence was a linear function of the logarithm of age. With the advances of molecular biology, clonal expansion was recognized as an
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essential stage in carcinogenesis. In response to these discoveries, the two-stage model with clonal expansion was proposed (Armitage & Doll, 1957; Moolgavkar
& Venzon, 1979). In this model, the first step involves the mutation from a normal cell to an intermediate cell that confers clonal expansion, and the further
mutation of the intermediate cells into a malignant cell is the second step. Since
then the two-stage model with clonal expansion has been widely used to analyze cancer incidence or mortality. For lung cancer, researchers dominantly used this
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model to study the effects of extrinsic factors on this disease (Little et al., 2002; Hazelton et al., 2006; Fakir et al., 2010; Moolgavkar et al., 2012; Hazelton et al., 2014; Moolgavkar et al., 2015). With the development of DNA sequencing technology, however, more and more gene mutations have been identified in tumors. 45
The two-stage model can be readily generalized to the models that incorporate
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more biological and medical discoveries, such as multiple genetic pathways and more number of stages (Little et al., 2008). Recently, multi-stage models have
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been used to fit lung cancer data in the Mayak workers (Z¨ ollner et al., 2015), which indicated that the three-stage model with clonal expansion fitted the 50
data better than the two-stage model and the model with genetic instability.
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Besides, only three driver gene mutations are required for the development of lung cancer (Tomasetti et al., 2015), and the three driver mutations can give a
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more reasonable explanation for cancer development (Song et al., 2015). Thus it is important to study the impact of radiation on lung cancer development by using the model with three mutations.
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In this article, we use the stochastic three-stage model with clonal expan-
sion to analyze the impact of radiation on the development of lung cancer. The model is applied to lung cancer data which are the probability of lung cancer among male and female patients aged 40–79 years from the Osaka Cancer Reg-
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istry (OCR) during 1974-1977 and the LSS cohort of atomic bomb survivors in Hiroshima and Nagasaki during 1958–1987. We first infer the model parameters 4
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by using the data from the OCR. Then the data from the LSS cohort are analyzed to study the impact of radiation on the development of lung cancer. We are particularly interested in the changes of model parameters due to the exposure to radiation. Thus hypotheses are proposed to identify the mechanism of
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radiation action, as well as to lay out which parameters are affected significantly by radiation. These hypotheses are then tested by the Chi-square test.
2. Materials and Methods
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2.1. Data on Lung Cancer Incidence 2.1.1. The OCR data
The OCR has been operated since 1962. It covers nearly 9 000 000 residents in Osaka Prefecture, Japan.
Among the data from the OCR, cate-
gories were created for both incident cases of cancer and populations by gender (male and female), calendar year (1974–1997) and 5-year age groups (ages 0–85+). The detailed data can be obtained from the OCR website (http://
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www.iph.pref.osaka.jp/omc/ocr/). Our analysis uses the age-specific lung can-
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cer incidence rates in male and female during the period of 1974–1977, which is reported in the paper (Yoshimi et al., 2003). Rates are expressed as cases per 100, 000 people.
2.1.2. The LSS data
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The LSS cohort comprises 120,321 individuals. Among them, 8666 first
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primary solid cancer patients were diagnosed between 1958 and 1987. Lung cancer accounts for nearly 10% of the cancers in the LSS cohort. The LSS data are available at the Radiation Effects Research Foundation (RERF) website (http://www.rerf.jp/). The data are grouped by city (Hiroshima and Nagasaki),
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gender (male and female), calendar time (5–year intervals for calendar years 1958–1987), attained age (5–year categories from age 15 to 85 and categories for ages less than 15 or 85+), age at exposure (5–year categories from age 0 to 60 and a category for ages 60+), colon dose (10 categories with dose cutoff points
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at 0.01, 0.1, 0.2, 0.5, 1, 1.5, 2, 3, 4 and 4+ Gy ), and other factors. A more detailed description of this dataset can be found in the reference (Thompson et al., 1994). Previous analyses have shown no significant difference between
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patients in Hiroshima and Nagasaki (Thompson et al., 1994; Grant et al., 2017). Therefore, our analyses are based on the incidence rates computed from person95
years and lung cancer cases stratified by attained age and gender during the year 1958–1987 in Hiroshima and Nagasaki. In this study, we omit the lung
cancer incidences for persons aged 0 to 39 years because they are close to zero.
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2.2. Mathematical model
The three-stage model with clonal expansion is designed to radically reduce 100
the complexity of tumorigenesis into three separate mutational steps that are genetic transitions from a pool of normal cells via premalignant to malignant. The premalignant cells are characterized by the symmetric division and differentiation or death processes. Mathematically, we assume that there are N stem
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cells in the lung tissue, and any cell is subject to mutation to a type of cells carrying an irreversible mutation at a rate of µ0 . Thereafter any cell in compart-
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ment Ii divides into two daughter cells at a rate αi and die or differentiate at a rate βi . Each cell may also divide into an equivalent daughter cell and another cell with an additional irreversible mutation at a rate µi . The unit of all rates
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is per year. In addition, the time from a malignant cell to tumor, known as the incubation period, is Ttag . This model is illustrated schematically in Fig. 1. We are interested in the probability, p(t), that at least one malignant cell
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has appeared by time t, which is easier to calculate than the hazard function
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(Moolgavkar et al., 1988). The probability function, p(t), is given by h Z t i p(t) = 1 − exp − h(s)ds ,
(1)
0
where h(s) corresponds to the cancer incidence at age s. This formula is used to
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transform the age-specific incidence of lung cancer into the probability of lung cancer at each age.
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D
D
β2
N
µ0
µ1
I1
µ2
I2
M
Ttag
α2
α1
Tumor
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β1
Fig. 1. The schematic representation of stochastic three-stage model with cell birth,
cell death, and gene mutation. N denotes the normal cell; Ii (i = 1, 2) the com-
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partment of intermediate cell; D the dead or differentiated cell; and M the malignant cell.
For the three-stage model, let N , Yi (t) (i = 1, 2), and Y3 (t) represent, respectively, the numbers of stem, premalignant Ii , and fully malignant cells at time t. For t ≥ τ , we define the following probability generating functions X
n p Y1 (t) = i1 , Y2 (t) = i2 , Y3 (t) = i3 |Y1 (τ ) = 0,
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Ψ(y1 , y2 , y3 ; τ, t) =
i1 ,i2 ,i3
(2)
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o Y2 (τ ) = 0, Y3 (τ ) = 0 y1i1 y2i2 y3i3 ,
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Φ1 (y1 , y2 , y3 ; τ, t) =
X
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and
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i1 ,i2 ,i3
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Φ2 (y2 , y3 ; τ, t) =
n p Y1 (t) = i1 , Y2 (t) = i2 , Y3 (t) = i3 |Y1 (τ ) = 1,
o Y2 (τ ) = 0, Y3 (τ ) = 0 y1i1 y2i2 y3i3 ,
o X n p Y2 (t) = i2 , Y3 (t) = i3 |Y2 (τ ) = 1, Y3 (τ ) = 0
(3)
i2 ,i3
y2i2 y3i3 .
(4)
Functions Ψ, Φ1 and Φ2 satisfy the following Kolmogorov backward equations (Harris, 1963; Christopher et al., 1996) dΨ (y1 , y2 , y3 ; τ, t) = −µ0 N Ψ(y1 , y2 , y3 ; τ, t)(Φ1 (y1 , y2 , y3 ; τ, t) − 1), dτ 7
(5)
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dΦ1 (y1 , y2 , y3 ; τ, t) = (α1 + β1 + µ1 )Φ1 (y1 , y2 , y3 ; τ, t) − α1 Φ21 (y1 , y2 , y3 ; τ, t) dτ −µ1 Φ1 (y1 , y2 , y3 ; τ, t)Φ2 (y2 , y3 ; τ, t) − β1 , (6) and
dΦ2 (y2 , y3 ; τ, t) = (α2 + β2 + µ2 )Φ2 (y2 , y3 ; τ, t) − α2 Φ22 (y2 , y3 ; τ, t) dτ −µ2 y3 Φ2 (y2 , y3 ; τ, t) − β2 .
(7)
Then, by Eqs. (5)–(7), we can get the following equation system at (y1 , y2 , y3 ) =
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(1, 1, 0) dΨ dτ (τ, t) = −µ0 N Ψ(τ, t)(Φ1 (τ, t) − 1) dΦ1 (τ, t) = (α + β + µ )Φ (τ, t) − α Φ2 (τ, t) − µ Φ (τ, t)Φ (τ, t) 1 1 1 1 1 1 1 1 2 dτ , (8) −β1 dΦ2 (τ, t) = (α + β + µ )Φ (τ, t) − α Φ2 (τ, t) − β 2 2 2 2 2 2 2 dτ
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with the boundary conditions
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Ψ(t, t) = 1 Φ1 (t, t) = 1 , Φ (t, t) = 1
(9)
2
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where the dependence of Ψ(τ, t), Φ1 (τ, t) and Φ2 (τ, t) on (1, 1, 0) have been suppressed for convenience in writing. By the definition of the probability generating function Ψ, Ψ(0, t) denotes
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the probability that the tissue is still tumor free (has not yet developed a malignancy) by the time t. Thus, the probability, p(t), of at least one malignant
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cell by the time t, starting with only normal cells at time 0, can be written as
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p(t) = 1 − Ψ(0, t).
(10)
The solution of Ψ(0, t) is required for solving the probability function p(t). However, Ψ(0, t) has not a closed-form solution by system (8). To solve system (8) numerically, we fix t and change variables to s = t − τ . Let A(s, t) = Ψ(τ, t), 8
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B1 (s, t) = Φ1 (τ, t) and B2 (s, t) = Φ2 (τ, t). System (8) can be converted into the
A(0, t) = 1 B1 (0, t) = 1 . B (0, t) = 1
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with initial conditions
(11)
(12)
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following equation system dA ds (s, t) = µ0 N A(s, t)(B1 (s, t) − 1) dB1 (s, t) = −(α + β + µ )B (s, t) + α B 2 (s, t) 1 1 1 1 1 1 ds , +µ1 B1 (s, t)B2 (s, t) + β1 dB2 (s, t) = −(α + β + µ )B (s, t) + α B 2 (s, t) + β 2 2 2 2 2 2 2 ds
2
Then p(t) = 1 − A(t, t).
In this way, the probability function p(t) is derived by solving numerically the ordinary differential equation system (11, 12) with respect to s (for fixed t)
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2.3. Statistic analysis
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directly.
Before numerical simulation, we firstly transform the incidence rates to the
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probability of lung cancer at each age t by formula (1). Then, the probability values are amplified by using a suitable multiple (i.e. 104 ) to perform the Chi-square test. We regard the cell birth rates, the cell death or differentiation rates, and the cell mutation rates as parameters, as graphically depicted
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150
in Fig. 1. However, not all of the model parameters can be identifiable from
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epidemiological data. There are two popular approaches to deal with this nonidentifiability problem. The first one is to set some parameters equal to each other and assume a reasonable number of normal cells such as, N = 107 (Z¨ ollner et al., 2015). The other is to use a new set of parameters (Heidenreich et al.,
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1997). Here, we assume that the cell birth rates are equal, namely α1 = α2 ,
since the cancer risk mainly depends on the net proliferation rates (αi − βi ), and the simulated result is insensitive to the cell birth rates. In addition, the incubation period is relatively short, compared to the lifetime of an individual.
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We use the value reported by the Z¨ ollner et al. (Z¨ ollner et al., 2015), about five years. To highlight the influence of radiation exposure on the development of lung
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cancer, we choose the OCR data and LSS data as test systems. They mainly reflect the effect of the exposures to environmental lung cancer risk factors other 165
than radiation exposure. In these systems, the fitting of the data from OCR is
regarded as the control group. To study the impact of radiation exposure on genetic mutation and clonal expansion of cells, the data from LSS cohort are
discussed by modifying values of model parameters in the control group. For the
170
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fitting of the LSS data, we consider the following two alternative hypotheses:
Hypothesis one: Radiation exposure affects mutation rates of cells in the process of tumorigenesis. Thus, the birth rates and the death or differentiation rates of cells in the model are fixed (determined by the control group) but the mutation rates of cells should be higher than those of the control
175
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group.
Hypothesis two: Radiation exposure has an effect on the clonal expansion
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of cells in the development of lung cancer. Thus the mutation rates of cells in the model are fixed (determined by the control group), but the net proliferation rates of cells should be higher than those of the control
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group.
The numerical optimization routine fminsearch in MATLAB is used to es-
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timate the optimal parameters by the minimum value of Chi-square statistics between the real data and simulated ones. Goodness of fit is measured by the
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Chi-square test. The value of the Chi-square statistics is determined by Chi2 =
79 X
(datap(i) − estimationp(i) )2 · 104 /estimationp(i) .
(13)
i=40
For the Chi-square test, we set the significance level to be 0.05. If the value of
185
the Chi-square statistics is smaller than the value at the 5% significant level, the hypothesis is accepted. Furthermore, a smaller value of the Chi-square statistics represents a better fit. 10
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3. Results The fitting results of the three-stage model with clonal expansion to the data from OCR and LSS cohort are discussed in detail. Furthermore, the changes of parameters due to radiation are described. 3.1. The model fitting
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190
We firstly fit the data in the control group by the three-stage model with
clonal expansion. Fig. 2 shows the fitting of the age-specific probability of lung 195
cancer in OCR from the three-stage model for male and female patients, respec-
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tively. The fitting of the data to the model are very well for male and female
patients. Table 1 gives the optimal values of net proliferation rates (αi − βi ) and cell mutation rates of the model for both male and female patients. Table 1 suggests that the net proliferation rates of premalignant cells are not very large, 200
and the cells with two mutations have higher net proliferation rates than those
M
with one mutation. Furthermore, the net proliferation rates of premalignant cells for female are larger than those for male. However, the mutation rates of cells for male patients are higher than those for female patients. Therefore,
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there is a gender difference in the development of lung cancer. Table 1
Estimated optimal parameters for model (8) using the OCR data for male patients and female patients
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separately.
Parameter
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α1 − β1 [year α2 − β2 [year µ0 [year
−1
]
−1 −1
Definition
Male
Female
]
net proliferation rate in intermediate cells, I1
0.0063
0.0552
]
net proliferation rate in intermediate cells, I2
0.2018
mutation rate in normal cells, N mutation rate in intermediate cells, I1
µ2 [year−1 ]
mutation rate in intermediate cells, I2
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µ1 [year−1 ]
11
2.8739 × 10
0.3341 −6
7.3668 × 10−6 7.9592 × 10−6
5.2766 × 10−7
1.7898 × 10−6 2.3973 × 10−6
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a
0.06 Model Data
0.04 0.03 0.02 0.01 0 40
45
50
55
60
65
70
Age (years) 0.016 Model Data
0.014
Probability
0.012 0.01 0.008 0.006 0.004
0 40
45
M
0.002 50
75
80
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b
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Probability
0.05
55
60
65
70
75
80
Age (years)
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Fig. 2. Cancer probability predicted by the model with three mutations and age-
specific probability of tumor from the OCR in 1974-1977. (a) Prediction for male patients. (b) Prediction for female patients. (Solid-line: prediction of model; star:
3.2. The impact of radiation exposure
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patient data.)
It is well known that radiation exposure can increase the risks of lung cancer.
However, it is not clear how it affects the incidence of lung cancer. To answer this
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question, we consider two alternative hypotheses that are shown in section 2.3 for fitting the data from LSS cohort. Then, these hypotheses are tested by
210
the Chi-square test. The Chi-square test result in Table 2 shows that the two hypotheses can be accepted for both male and female patients data at 5% significant level. In addition, the Chi-square statistics values of Hypothesis one
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are smaller than those of Hypothesis two for both male and female patients data. Thus, the model under Hypothesis one describes the data moderately better than the one under Hypothesis two. Table 2
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The Chi-square test results of the two hypotheses for male and female patients. Male
Df
b
c
S.5% a
Hypothesis one
Hypothesis two
Hypothesis one
Hypothesis two
14.997
17.640
4.982
7.347
36
36
36
50.998
50.998
50.998
36
50.998
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Chi2
a
Female
Chi2 is the estimated value of Chi-square (see Eq (13));
b
Df denotes the degree of freedom (= number of data - number of parameters-1);
c
S.5% denotes the value at 5% significant level.
M
The optimal values of parameters under Hypothesis one are displayed in Table 3. By comparing the values of parameters in Table 1 and Table 3. we find that the first and third mutation rates are increased while the second mu-
220
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tation rate remains the same with the control group for male patients. For female patients, the first two mutation rates are increased, but the last mutation rate remains unchanged. These results indicate that for both male and
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female patients, the first mutation rate is increased. Thus, under the condition of radiation exposure, mutation more likely occurs in normal cells, which may
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lead to cancer.
The optimal values of parameters under Hypothesis two are shown in Table 4,
225
By observing Table 1 and Table 4, we find that radiation exposure has influence
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on the net proliferation rates of all compartments of intermediate cells in the development of lung cancer. Furthermore, the changes of net proliferation rates of premalignant cells for female patients are larger than that for male patients.
230
Therefore, the impact of radiation on clonal expansion of cells for female patients is more significant than that for male patients.
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Table 3
Estimated optimal parameters of model (8) under Hypotheses one by using the LSS cohort for male and female patients, separately. The values of net cell proliferation
Parameters µ0 [year
−1
Male
Female
3.0403 × 10
]
µ1 [year−1 ]
−6
7.3668 × 10−6
µ2 [year−1 ]
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rates are the same as those in Table 1.
9.2831 × 10−6
Table 4
5.4706 × 10−7
2.5239 × 10−6
2.3973 × 10−6
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Estimated optimal parameters of model (8) under Hypotheses two by using the LSS
cohort for male and female patients, separately. The values of cell mutation rates are the same as those in Table 1. Parameters α1 − β1 [year
Male −1
0.0085
α2 − β2 [year−1 ]
0.2073
0.0636 0.3593
M
]
Female
To further explore the impact of radiation exposure on the progression of
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lung cancer, we conduct the next test by assuming only one parameter in the model may be affected by radiation. All the other parameters are the same as 235
those determined by the control group. The Chi-square test is implemented to
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evaluate this assumption. For the Chi-square test of the assumption, the degree of freedom is 38 and the value at the 5% significant level is 53.38. Fig. 3 gives the
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estimated minimal values of Chi-square statistics under this assumption. For the case of parameter α2 − β2 , the value of the Chi-square is 56.900 for female
240
patients, which is larger than the threshold value at the 5% significant level.
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Therefore, this case should be rejected, and the radiation exposure may not be able to affect the net proliferation rate in intermediate cells, I2 . Compared with
the two cases of net cell proliferation rate (i.e. αi − βi ), the Chi-square statistics values for the cases of three mutation rates (i.e. µ0 , µ1 and µ2 ) are much
245
smaller. As a consequence, the major effect of radiation on the progression
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of lung cancer is to induce the mutations of cells for both male and female patients. In particular, the Chi-square statistics value for cell mutation rate µ2 of the male patients is smaller than not only the values for cell mutation rates
250
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µ0 and µ1 of the male patients, but also the value for cell mutation rate µ2 of the female patents. This observation suggests that, for male patients, the
impact of radiation exposure on the third mutation rate of the model is more significant than that on the first two mutation rates in the development of lung cancer. On the other hand, for female patients, the Chi-square statistics for cell mutation rates µ0 and µ1 have the smallest values in all these tests, which
suggests that the major impact of radiation on the lung cancer development for
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255
female patients is to change the first two mutation rates of the model. 60
Male Female 50
M
Chi2
40
30
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20
10
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0
1
2
3
4
5
test index
Fig. 3. The Chi-square test results under the hypothesis that only one parameter is
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affected by radiation exposure for male and female lung cancer patients in the model.
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(index 1: affected µ0 , 2: µ1 , 3: µ2 , 4: α1 − β1 , 5:α2 − β2 .)
4. Conclusions In this work we have studied the impact of radiation on genetic mutations
and clonal expansion of cells in the development of lung cancer. Instead of using 260
the two-stage stochastic model that has been widely used to study lung cancer, 15
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this work employed the three-stage model with clonal expansion which was a better mathematical model proposed by the recently published research works (Z¨ ollner et al., 2015; Tomasetti et al., 2015; Song et al., 2015). The data from
265
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the atomic bomb survivors in Hiroshima and Nagasaki are used to infer the key parameters in the model. In particular, the OCR data are used as the control group to infer the parameters of the model for the patients without radiation.
Then the LSS data are used to infer another set of model parameters. The difference between the estimated parameters is used as the evidence to induce
the influence of radiation exposure. Simulations of the stochastic model match the probability of lung cancer from the two datasets very well. Numerical results
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270
suggest that there is a gender difference in the progression of lung cancer. The inferred key model parameters thus provide insightful predictions regarding the impact of radiation on the development of lung cancer.
To examine the influence of radiation, we considered two alternative hy275
potheses regarding the influence of radiation exposure, namely the influence on
M
the mutation rates of cells or on the net proliferation rates of premalignant cells. We used the LSS data to estimate optimal parameters of the three-stage model
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with clonal expansion under different hypotheses. The values of the Chi-square statistics in Table 2 suggest that radiation affects not only the mutation rates of 280
cells but also the net proliferation of cells in lung carcinogenesis. However, the
PT
radiation exposure has more significant effect on the mutation rates of cells than the net proliferation rates of cells for both female and male patients. Focusing
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on the mutation rates of cells, our results further show that radiation exposure induces the mutation in the normal cells and transformation from premalignant
285
cells to malignant ones for male patients. For female patients, however, radia-
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tion exposure increases the rates of the first two mutations in the model. For the clonal expansion of cells, the increasing of net proliferation rates of premalignant cells for female patients is higher than that for male patients.
290
We have also examined the effect of radiation exposure on the change of
one single parameter of the model for lung cancer development. Interestly, the major role of radiation exposure on the development of lung cancer is to induce 16
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transformation from premalignant cells to a malignant cell for male patients but to induce the transformation from normal cells to mutated cells with selective advantages for female patients. Thus, radiation exposure can significantly accelerate the mutations that are required to overcome normal homeostatic reg-
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295
ulation in a tissue, which allows the gradual expansion of premalignant clones for female patients and increases the malignant transformation for male patients.
Although the impact of radiation on the development of lung cancer has
been studied by a number of works (Tokarskaya et al., 1995; Hazelton et al., 300
2006; Egawa et al., 2012; Cahoon et al., 2017), these research works did not
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give the detailed analysis for parameter changes. The innovation of our work is to investigate the effect of radiation exposure on multiple parameters and single parameter by fitting the clinic data as well as to compare the difference between the effect on mutation rates of cells and that on net proliferation rates 305
of cells. The conclusions generated from this work can be used to offer useful guidance for the prediction of lung cancer risks. Although substantial progress
M
has been achieved in this work, our results only reflect the difference between the impact of radiation on lung cancer development with radiation exposure and
310
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that without radiation exposure. It would be important to further consider the impact of radiation intensity on lung cancer, such as the relationship between the amount of radiation dose and parameter values of the model. These interesting
PT
questions will be the topics of our future research work.
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Acknowledgments
This report makes use of data obtained from the Radiation Effects Research
315
Foundation (RERF) in Hiroshima, Japan. RERF is a private foundation funded
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equally by the Japanese Ministry of Health and Welfare and the U.S. Department of Energy through the U.S. National Academy of Sciences. The conclusions in this report are those of the authors and do not necessarily reflect the scientific judgment of RERF or its funding agencies. Our work was partially
320
supported by the National Nature Science Foundation of China (No.11071275
17
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and No.11371161), by self-determined research funds of Central China Normal University from the colleges’basic research and operation of MOE (Grant No.CCNU16JCZX10). We thank the reviewers very much for their very good
325
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suggestions on the manuscript.
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Legends Fig.1. The schematic representation of stochastic three-stage model with cell growth, cell death, and gene mutation. N denotes the normal cell; Ii (i = 1, 2) the com-
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partment of intermediate cell; D the dead or differentiated cell; and M the malignant cell.
Fig.2. Cancer probability predicted by the model with three mutations and age-
specific probability of tumor from the OCR in 1974-1977. (a) Prediction for male patients. (b) Prediction for female patients. (Solid-line: prediction of model; star:
440
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patient data.)
Fig.3.The Chi-square test results under the hypothesis that only one parameter is affected by radiation exposure for male and female lung cancer patients in the model.
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M
(index 1: affected µ0 , 2: µ1 , 3: µ2 , 4: α1 − β1 , 5:α2 − β2 .)
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