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Contents lists available at ScienceDirect
Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs 4 5
Mathematical modeling the pathway of human breast cancer
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Q1
Xinan Zhang a,⇑, Yingdong Zhao b, Weiming Zheng c a
7 8 9
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, PR China Biometric Research Branch, National Cancer Institute, National Institutes of Health, Bethesda, MD, USA c Department of Computer Science and Technology, Tsinghua University, Beijing 100084, PR China b
10 11
a r t i c l e
1 2 3 4 14 15 16 17 18
i n f o
Article history: Received 19 July 2012 Received in revised form 12 March 2014 Accepted 16 March 2014 Available online xxxx
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Keywords: Breast cancer Tumorigenesis Mathematical model
a b s t r a c t In order to understand the mechanism of human breast cancer we use the growth rates of clonal expansion of intermediate cells and mutation rates as parameters and build two-six stage models to fit the agespecific incidence of breast cancers in the surveillance, epidemiology, and end results (SEER) registry. We propose four types of different mechanisms for the human breast cancer and test those mechanisms by Chi-square test. Our results suggest that loss of functions of instability genes is an early event in the tumorigenesis, which is useful for early diagnosis of breast cancer. The clonal expansion of intermediate cells must depend on the hormone expression level of females, which implies that it may be effective for females to receive hormone blocking therapy for breast cancer before their menopause. Ó 2014 Published by Elsevier Inc.
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1. Introduction Cancer is a genetic disease [1,2], which involves dynamic changes in the genome [3]. Although environmental and other non-genetic factors have roles in many stages of tumorigenesis, it is widely accepted that cancer arises because of mutations in cancer susceptibility genes. Cancer of epithelial tissues is generally thought to develop slowly over many years [4]. Hanahan and Weinberg [5] suggested that the vast catalog of cancer cell genotypes is a manifestation of six essential alterations in cell physiology that collectively dictate malignant growth: self-sufficiency in growth signals, insensitivity to growth-inhibitory signals, evasion of programmed cell death (apoptosis), limitless replicative potential, sustained angiogenesis, and tissue invasion and metastasis. The analysis of age-incidence curves for human cancers using quantitative models has also resulted in claims that 4–8 independent rate-limiting hits are needed [6]. In order to show that many mutations are observed in malignant tumor cells, Loeb et al. [4,7–10] proposed that there exists a mutator phenotype that acts as a mechanism in tumor processes. However, Tomlinson and Bodmer [11–13] have challenged the mutator phenotype hypothesis and claimed that selection for clonal expansion of intermediate cells is sufficient in tumor processes. This is the general debate for the relationship between mutation and selection. A quantitative understanding of cancer biology requires mathematical frameworks to describe the fundamental principles
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Q2
⇑ Corresponding author. Tel.: +86 13016412548; fax: +86 27 67867452. E-mail address: [email protected] (X. Zhang).
of population genetics and evolutionary mechanisms that govern tumor initiation and progression [14,15]. Factors including mutation, selection, and tissue organization determine the dynamics of tumorigenesis [16–20]. The original two-stage model of Armitage and Doll [21] is built to explain the age-dependent incidence curves of human cancer and some multi-stage models claim that 4–8 independent rate-limiting hits are needed [6]. However, those models did not consider the clonal expansion of intermediate cells. Our goal in this paper is to obtain insight into the mechanisms of breast cancer by analyzing the age-specific incidence rate of breast cancer in the surveillance, epidemiology, and end result (SEER) database. This paper follows up to our recent work [22] regarding breast cancer. We use mutation rates per cell per year and clonal expansion rates per cell per year as parameters to deal with biological hypotheses in our mathematical models. We also consider the expression levels of hormones of females as factors to influence the clonal expansion of intermediate cells. It is the first paper to reject some biological assumptions by using the hypothesis and test of statistical inference even if those biological assumptions fit the surveillance, epidemiology, and end result (SEER) database very well. All former mathematical modeling work just considered how to fit the surveillance, epidemiology, and end result (SEER) database and did not test the mathematical results at the 5% significant level.
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2. Materials and methods
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2.1. The SEER data
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Incidence data for breast cancer were obtained from the surveillance, epidemiology, and end result (SEER) registry for the year
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http://dx.doi.org/10.1016/j.mbs.2014.03.011 0025-5564/Ó 2014 Published by Elsevier Inc.
Please cite this article in press as: X. Zhang et al., Mathematical modeling the pathway of human breast cancer, Math. Biosci. (2014), http://dx.doi.org/ 10.1016/j.mbs.2014.03.011
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1973–1999 [23]. For our analyses, we use the reported incidence of breast cancer by gender, race, age, and calendar year in the nine SEER geographic areas, which together represent an estimated 10% of the U.S. population. The population bases were from SEER population files (based on data from U.S. Census Bureau) by sex and race and were cross-tabulated by calendar year (1973–1999) and 5-year age groups (ages 0–85+). Our analyses addressed combined all races for breast cancer in females during the period 1990– Q3 1999. Rates are expressed as cases per 100,000 females.
where ai, bi are proliferation probability and death (or differentiated) probability of the cells in compartment Ii. The incidence function (or hazard function) can be written in term of W and W0 as
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0
W ð1; . . . ; 1; 0; tÞ hðtÞ ¼ Wð1; . . . ; 1; 0; tÞ
ð2:2Þ
From (2.1), we have
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@W W ð1; . . . ; 1; 0; tÞ ¼ lk ð1; . . . ; 1; 0; tÞ @yk 0
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2.2. Mathematical model We propose a mathematical framework of quantitative models of two-six stages for a sequence of genetic events that define the main pathway of breast cancer according to the findings of Hanahan and Weinberg [5]. The two-stage model with clonal expansion of intermediate cells was set up by Knudson and Moolgavkar et al. [14,15,24]. Luebeck and Moogavkar [25] in 2002 considered models of two-five stages but only with clonal expansion of one compartment of intermediate cells for multi-hit models. Our recent models [22] of two-six stages considered the clonal expansion of each compartment of intermediate cells but did not test how many stages of mutations are needed for breast cancer. Fig. 2.1 shows an example of five-stage of model with clonal expansion in each compartment of intermediate cells. Moolgavkar et al. [14,15,24] and our recent work [22] assumed that the tumor progenitor cells are breast epithelium stem cells and the number of such stem cells increases according to logistic growth curve from an initial value of 10 cells at birth to a maximum of 107 cells by age 20 while the number of progenitor cells decreases after age 45 at a rate of 0.0667 per cell per year because of the expression level of hormone and the menopause, which is 106 cells by age 80. We assume that the latent time from a malignant cell formed in breast tissue to clinical detection is 5 years. For a k-hit (k = 2, . . . , 6) model, we use Y1(t) to denote the number of normal progenitor cells per breast at age t, Yi(t) (i = 2, . . . , k) to denote the number of intermediate cells in compartment Ii per breast at age t, Yk+1(t) to denote the number of fully malignant cells per breast at age t. Let W(y1, y2, . . ., yk+1; t) be the probability generating function at time t starting with a single normal stem cells at time 0 [25]:
Wðy1 ; y2 ; . . . ; ykþ1 ; tÞ ¼
X
Pðy1 ðtÞ ¼ i1 ; . . . ; ykþ1 ðtÞ ¼ ikþ1 jy1 ð0Þ
i1 ;...;ikþ1
¼ 1;
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Then, the Kolmogorov forward differential equation for W is
134 0
W ðy1 ; y2 ; . . . ; ykþ1 ; tÞ ¼ @ W=@t ¼ fl1 y1 y2 þ a1 y21 þ b1 ða1 þ b1 þ l1 Þy1 g@ W=@y1 þ þ flk yk ykþ1 þ ak y2k þ bk ðak þ bk 137
þ lk Þy1 g@ W=@yk
N
γ1
@W @yk
ð1; . . . ; 1; 0; tÞ is
@W @yk
148
evaluate at y1 = 1, . . . , yk = 1, yk+1 = 0.
ð2:1Þ
μ1
I2
γ2
μ2
I3
γ3
μ3
151
ð1; . . . ; 1; 0; tÞ
Wð1; . . . ; 1; 0; tÞ
¼ E yk ðtÞjykþ1 ðtÞ ¼ 0
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where E[Yk(t)|Yk+1(t) = 0] denotes the conditional expectation. We have
hðtÞ ¼ lk E½yk ðtÞjykþ1 ðtÞ ¼ 0
ð2:3Þ
Since cancer is a rare disease, the probability P(t) of there being a malignant cell at time t is close to zero. Thus,
E½Y k ðtÞjY kþ1 ðtÞ ¼ 0 E½Y k ðtÞ:
ð2:4Þ
Then we have the approximation for one breast tissue:
hðtÞ lk E½Y k ðtÞ
dY 1 ðtÞ Y 1 ðtÞ þ Y 2 ðtÞ þ Y 3 ðtÞ þ Y 4 ðtÞ þ Y 5 ðtÞ ¼ c1 Y 1 ðtÞ 1 l1 Y 1 ðtÞ dt K dY 2 ðtÞ ¼ l1 Y 1 ðtÞ þ c2 Y 2 ðtÞ l2 Y 2 ðtÞ dt dY 3 ðtÞ ¼ l2 Y 2 ðtÞ þ c3 Y 3 ðtÞ l3 Y 3 ðtÞ dt dY 4 ðtÞ ¼ l3 Y 3 ðtÞ þ c4 Y 4 ðtÞ l4 Y 4 ðtÞ dt dY 5 ðtÞ ¼ l4 Y 5 ðtÞ þ c5 Y 5 ðtÞ l5 Y 5 ðtÞ dt
γ4
155
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ð2:6Þ
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dY 1 ðtÞ ¼ c01 Y 1 ðtÞ l1 Y 1 ðtÞ dt dY 2 ðtÞ ¼ l1 Y 1 ðtÞ þ c02 Y 2 ðtÞ l2 Y 2 ðtÞ dt dY 3 ðtÞ ¼ l2 Y 2 ðtÞ þ c03 Y 3 ðtÞ l3 Y 3 ðtÞ dt dY 4 ðtÞ ¼ l3 Y 3 ðtÞ þ c04 Y 4 ðtÞ l4 Y 4 ðtÞ dt dY 5 ðtÞ ¼ l4 Y 5 ðtÞ þ c05 Y 5 ðtÞ l5 Y 5 ðtÞ dt μ4
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ð2:5Þ
The incidence rate (or hazard function) is determined by the product between the mutation rate lk and the number of intermediate cells in compartment Ik. In order to estimate the number in each compartment as shown in Fig. 2.1, we set up the model of ordinary differential equations of 5 hit model. To estimate the number of Y5(t) in compartment I5 before and after age 45 for five stage model, Yi(t) (i = 1, . . . , 5) should satisfy the following ordinary differential equations for t 6 45
I4
149 150
and t > 45
i
kþ1 y2 ð0Þ ¼ 0; . . . ; ykþ1 ð0Þ ¼ 0Þyi11 . . . ykþ1
135
where Further
@W @yk
I5
μ5
ð2:7Þ
M
γ5
Fig. 2.1. A five stage model for carcinogenesis. I2, I3, I4, I5 denote the compartments of intermediate cells, l1, l2, l3, l4, l5 are the mutation rates per cell per year of normal progenitor cells, I2 compartment, I3 compartment, I4 compartment and I5 compartment. c1, c2, c3, c4, c5 are the net growth rates per cell per year.
Please cite this article in press as: X. Zhang et al., Mathematical modeling the pathway of human breast cancer, Math. Biosci. (2014), http://dx.doi.org/ 10.1016/j.mbs.2014.03.011
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Where K = 107 normal stem cells per breast, li (i = 1, . . . , 5) denote the mutation rates per cell per year, c1 = 1.0074, c01 = 0.0067, ci and c0i (i = 2, . . . , 5) denote the clonal expansion rates per cell per year in compartments Ii (i = 2, . . . , 5) of intermediate cells before and after age 45. Therefore, the incidence function (or hazard function) of one female to a k-stage model should be
hðtÞ 2lk E½Y k ðtÞ
ð2:8Þ
We determine the set of parameters that provide the minimum value of the Chi-square test to the SEER age-specific cancer incidence data using the numerical optimization routine fminsearch in MATLAB. The minimum value of the Chi-square is determined by 16 X 2 ðdatai estimationi Þ =estimationi
ð2:9Þ
i¼1
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In order to test some biological mechanisms such as the effects of mutation, clonal expansion and the hormone level, we have considered the following four cases: Case one: There is a mutator phenotype in the tumorigenesis of breast cancer for multi-hit models, that is the mutation rates per cell per year in k-hit (k = 2, . . . , 6) model should satisfy l1 6 l2 6 6 lk. Case two: There is no mutator phenotype but only selection for clonal expansion in the tumorigenesis of breast cancer for multihit models, that is the mutation rates per cell per year in k-hit (k = 2, . . . , 6) model should satisfy l1 = l2 = = lk, c2 6 c3 6 ck, c20 6 c30 6 6 ck0 (k = 2, . . . , 6). Case three: The clonal expansions in models of two-six stages do not depend on the expression level of hormone in females, that is there are no difference for the clonal expansion in each compartment of intermediate cells before and after menopause or it is the same net growth rate per cell per year for ci (i = 2, . . . , k) in each compartment of intermediate cells before and after age 45 in k-hit (k = 2, . . . , 6) model. It should be c2 = c20 , c3 = c30 , . . . , ck = ck0 . We also assume that the mutation rate in each compartment satisfy l1 6 l2 6 6 lk, (k = 2, . . . , 6). Case four: The clonal expansions of intermediate cells in models of two-six stages depend only on the expression level of hormone in females and there is no difference for the clonal expansion in each compartment before and after age 45, that is there is the same net growth rate per cell per year in each compartment of intermediate cells (c2 = c3 = = ck) before age 45 and (c0 2 = c0 3 = = c0 k) after age 45 in k-hit (k = 2, . . . , 6) model. We also assume that the mutation rate in each compartment satisfy l1 6 l2 6 6 lk , (k = 2, . . . , 6).
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3. Results
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We use the numerical optimization routine fminsearch in MATLAB to estimate the optimal parameters li (i = 1, . . . , 6), cj, c0 j (j = 2, . . . , 6) by the minimum value of Chi-square between the SEER data and the hazard function that depend on the models satisfied the Hypothesis one-four. Table 1 gives the minimum values of Chi-square test of the goodness of fit and values at the 5% significant level. For Case one, we can only accept the two-four stage models for the null hypothesis of mutator phenotype at the 5% significant level and reject the five and six stage models even if five and six stage models fit the SEER data better than two stage model (see Fig. 3.1). Our recent paper [22] used both two-six stage models with clonal expansion of each compartment of intermediate cells fit the age specific incidence data of breast cancer. We estimated the mutation rates of two-six stage models (see Table 1 in paper
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Table 1 P 2 of Case The minimum Chi-Square values ¼ 16 i¼1 ðdatai estimationi Þ =estimationi one to Case four. Chi2 denotes the value of Chi-Square; df (=16–1-number of parameters) denotes the degrees of freedom; S. 5% denotes the value at 5% significant level. The minimum Chi-Square values were calculated by using the numerical optimization routine fminsearch in MATLAB. 2-hit
3-hit
4-hit
5-hit
6-hit
Hypothesis Chi2 df S. 5%
1 10.4937 11 19.675
6.3151 8 15.507
6.6851 5 11.071
10.6492 2 5.991
10.383
Hypothesis Chi2 df S. 5%
2 10.494 12 21.026
6.2186 10 18.307
19.5339 8 15.507
147.707 6 12.592
561.576 4 9.488
Hypothesis Chi2 df S. 5%
3 92.9915 12 21.026
42.4444 10 18.307
19.3112 8 15.507
20.4038 6 12.592
24.4929 4 9.488
Hypothesis Chi2 df S. 5%
4 10.4937 11 19.675
6.9300 10 18.307
8.5019 9 16.919
11.5148 8 15.507
11.8720 7 14.067
[22]). However, we can only accept those mutation rates of twofour stage models at the 5% significant level, and should reject those mutation rates of five and six stage models at the 5% significant level. For Case two, we can only accept the two and three stage models for the null hypothesis of constant mutation rate per cell per year and clonal expansion of each compartment of intermediate cells at the 5% significant level and reject the four-six stage models. For Case three, we should reject two-six stage models for the null hypothesis of the clonal expansion of each compartment of intermediate cells having the same net growth rate per cell per year before age 45 and after age 45 at the 5% significant level, that is the clonal expansion rate of each compartment of intermediate cells must depend on the expression level of hormone. For Case four, we should accept the two-six stages models at the 5% significant level for the null hypothesis of all compartment of intermediate cells having the same net growth rate before age 45 and after age 45, respectively. That is c2 = c3 = = ck before age 45 and c0 2 = c0 3 = = c0 k after age 45 in k-hit (k = 2, . . . , 6) model. Therefore, the Hypothesis four is the best hypothesis of the four hypotheses because only Hypothesis four is available to two-six stage models, that is the clonal expansion in each compartment of intermediate cells must depend on the expression level of hormone.
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4. Discussion
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Our mathematical models also give the genetic pathway of human breast cancer if a malignant cell is caused by more than three stages of mutations in the genome. The first two or three hits must be occurred on genes that keep the genomic stability, which are called instability genes composed by DNA repair genes or spindle-checking genes. Those genes lost their function can cause genomic instability and then lead to high mutation rate and breast cancer. This is our explanation of mutator phenotype. Breast cancer can be caused by two-six hits in the genome of normal breast progenitor cells. The two-hit and three-hit models should be applicable to those cases for patients who have family history of breast cancer such as BRCA1 and BRCA2 mutation carriers [26]. Four-six hit models should be applicable to those sporadic cases of breast cancer. If breast cancer is caused by the mutations in the genome with more than three hits, the first three hits must be lead to geno-
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Fig. 3.1. The age-specific incidence data of all races per 100,000 females for breast cancer from SEER registry for the year 1973–1999 (black) and rates predicted by two-six stage models.
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mic instability before the cell becomes a malignant cell. Our mathematical results have shown the postulation in reference [27] that genetic instability is an early event for human cancer. Our theoretic result is consistent with experimental evidence regarding the role of BRCA1 for breast cancer [28]. Thus, loss of function of genes that keep the genomic stability can be used as the early diagnosis of breast cancer. Earlier diagnosis of cancer can save millions of lives for all type of cancer cases. The genomic instability means the high mutation rate in the genome among the subtle sequence changes, alterations in chromosome number, chromosome translocations, gene amplifications and loss of heterozygosity. Therefore, the subtle sequence changes are caused by the nucleotide-repairing genes lost their functions such as P53 gene. Alterations in chromosome number, chromosome translocations, gene amplifications and loss of heterozygosity are caused by the spindle-checking genes lost their function. Loss of heterozygosisty is the major phenotype of breast cancer. That is loss of function for spindle-checking genes is the main reason for most of breast cancer cases. Some genes should have the functions both for the nucleotide-repairing and spindle-checking functions such as BRCA1 gene. The most interesting result in this paper is that the clonal expansion of each compartment of intermediate cells must depend on expression level of hormone for breast cancer. Thus, hormone therapy such as tamoxifen should be effective for patients of breast cancer before their menopause of females. Breast cancer may be triggered by more than six hits in the genome of breast stem cells because many mutations have been found by recent large scale genomic analyses of breast cancer genome. Our methods can also be applied to cases with more than six stage models. However, the first two or three hits must be occurred on genes that keep the genomic stability. Therefore, genomic instability is the main cause of many mutations in breast cancer genome. This paper has not discussed the chromosomal instability. Chromosomal instability is also an early event of tumoregeneis [29].
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Acknowledgements
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Q4 This work is supported by the National Natural Science FoundaQ5 tion of China (Nos. 10771081, 11071275, 11228104), and by the
Special Fund for Basic Scientific Research of Central Colleges (CCNU10B01005). XZ thanks Dr. Richard Simon for his guidance when XZ was a postdoctoral fellow in Biometric Research Branch of National Cancer Institute, NIH, USA. We thank the reviewers very much for their very good suggestions on the manuscript.
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References
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No. of Pages 5, Model 5G
28 March 2014 Mathematical Biosciences xxx (2014) xxx–xxx 1
Contents lists available at ScienceDirect
Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs 4 5
Mathematical modeling the pathway of human breast cancer
3 6
Q1
Xinan Zhang a,⇑, Yingdong Zhao b, Weiming Zheng c a
7 8 9
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, PR China Biometric Research Branch, National Cancer Institute, National Institutes of Health, Bethesda, MD, USA c Department of Computer Science and Technology, Tsinghua University, Beijing 100084, PR China b
10 11
a r t i c l e
1 2 3 4 14 15 16 17 18
i n f o
Article history: Received 19 July 2012 Received in revised form 12 March 2014 Accepted 16 March 2014 Available online xxxx
19 20 21 22 23
Keywords: Breast cancer Tumorigenesis Mathematical model
a b s t r a c t In order to understand the mechanism of human breast cancer we use the growth rates of clonal expansion of intermediate cells and mutation rates as parameters and build two-six stage models to fit the agespecific incidence of breast cancers in the surveillance, epidemiology, and end results (SEER) registry. We propose four types of different mechanisms for the human breast cancer and test those mechanisms by Chi-square test. Our results suggest that loss of functions of instability genes is an early event in the tumorigenesis, which is useful for early diagnosis of breast cancer. The clonal expansion of intermediate cells must depend on the hormone expression level of females, which implies that it may be effective for females to receive hormone blocking therapy for breast cancer before their menopause. Ó 2014 Published by Elsevier Inc.
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1. Introduction Cancer is a genetic disease [1,2], which involves dynamic changes in the genome [3]. Although environmental and other non-genetic factors have roles in many stages of tumorigenesis, it is widely accepted that cancer arises because of mutations in cancer susceptibility genes. Cancer of epithelial tissues is generally thought to develop slowly over many years [4]. Hanahan and Weinberg [5] suggested that the vast catalog of cancer cell genotypes is a manifestation of six essential alterations in cell physiology that collectively dictate malignant growth: self-sufficiency in growth signals, insensitivity to growth-inhibitory signals, evasion of programmed cell death (apoptosis), limitless replicative potential, sustained angiogenesis, and tissue invasion and metastasis. The analysis of age-incidence curves for human cancers using quantitative models has also resulted in claims that 4–8 independent rate-limiting hits are needed [6]. In order to show that many mutations are observed in malignant tumor cells, Loeb et al. [4,7–10] proposed that there exists a mutator phenotype that acts as a mechanism in tumor processes. However, Tomlinson and Bodmer [11–13] have challenged the mutator phenotype hypothesis and claimed that selection for clonal expansion of intermediate cells is sufficient in tumor processes. This is the general debate for the relationship between mutation and selection. A quantitative understanding of cancer biology requires mathematical frameworks to describe the fundamental principles
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Q2
⇑ Corresponding author. Tel.: +86 13016412548; fax: +86 27 67867452. E-mail address: [email protected] (X. Zhang).
of population genetics and evolutionary mechanisms that govern tumor initiation and progression [14,15]. Factors including mutation, selection, and tissue organization determine the dynamics of tumorigenesis [16–20]. The original two-stage model of Armitage and Doll [21] is built to explain the age-dependent incidence curves of human cancer and some multi-stage models claim that 4–8 independent rate-limiting hits are needed [6]. However, those models did not consider the clonal expansion of intermediate cells. Our goal in this paper is to obtain insight into the mechanisms of breast cancer by analyzing the age-specific incidence rate of breast cancer in the surveillance, epidemiology, and end result (SEER) database. This paper follows up to our recent work [22] regarding breast cancer. We use mutation rates per cell per year and clonal expansion rates per cell per year as parameters to deal with biological hypotheses in our mathematical models. We also consider the expression levels of hormones of females as factors to influence the clonal expansion of intermediate cells. It is the first paper to reject some biological assumptions by using the hypothesis and test of statistical inference even if those biological assumptions fit the surveillance, epidemiology, and end result (SEER) database very well. All former mathematical modeling work just considered how to fit the surveillance, epidemiology, and end result (SEER) database and did not test the mathematical results at the 5% significant level.
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2. Materials and methods
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2.1. The SEER data
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Incidence data for breast cancer were obtained from the surveillance, epidemiology, and end result (SEER) registry for the year
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http://dx.doi.org/10.1016/j.mbs.2014.03.011 0025-5564/Ó 2014 Published by Elsevier Inc.
Please cite this article in press as: X. Zhang et al., Mathematical modeling the pathway of human breast cancer, Math. Biosci. (2014), http://dx.doi.org/ 10.1016/j.mbs.2014.03.011
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1973–1999 [23]. For our analyses, we use the reported incidence of breast cancer by gender, race, age, and calendar year in the nine SEER geographic areas, which together represent an estimated 10% of the U.S. population. The population bases were from SEER population files (based on data from U.S. Census Bureau) by sex and race and were cross-tabulated by calendar year (1973–1999) and 5-year age groups (ages 0–85+). Our analyses addressed combined all races for breast cancer in females during the period 1990– Q3 1999. Rates are expressed as cases per 100,000 females.
where ai, bi are proliferation probability and death (or differentiated) probability of the cells in compartment Ii. The incidence function (or hazard function) can be written in term of W and W0 as
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0
W ð1; . . . ; 1; 0; tÞ hðtÞ ¼ Wð1; . . . ; 1; 0; tÞ
ð2:2Þ
From (2.1), we have
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@W W ð1; . . . ; 1; 0; tÞ ¼ lk ð1; . . . ; 1; 0; tÞ @yk 0
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2.2. Mathematical model We propose a mathematical framework of quantitative models of two-six stages for a sequence of genetic events that define the main pathway of breast cancer according to the findings of Hanahan and Weinberg [5]. The two-stage model with clonal expansion of intermediate cells was set up by Knudson and Moolgavkar et al. [14,15,24]. Luebeck and Moogavkar [25] in 2002 considered models of two-five stages but only with clonal expansion of one compartment of intermediate cells for multi-hit models. Our recent models [22] of two-six stages considered the clonal expansion of each compartment of intermediate cells but did not test how many stages of mutations are needed for breast cancer. Fig. 2.1 shows an example of five-stage of model with clonal expansion in each compartment of intermediate cells. Moolgavkar et al. [14,15,24] and our recent work [22] assumed that the tumor progenitor cells are breast epithelium stem cells and the number of such stem cells increases according to logistic growth curve from an initial value of 10 cells at birth to a maximum of 107 cells by age 20 while the number of progenitor cells decreases after age 45 at a rate of 0.0667 per cell per year because of the expression level of hormone and the menopause, which is 106 cells by age 80. We assume that the latent time from a malignant cell formed in breast tissue to clinical detection is 5 years. For a k-hit (k = 2, . . . , 6) model, we use Y1(t) to denote the number of normal progenitor cells per breast at age t, Yi(t) (i = 2, . . . , k) to denote the number of intermediate cells in compartment Ii per breast at age t, Yk+1(t) to denote the number of fully malignant cells per breast at age t. Let W(y1, y2, . . ., yk+1; t) be the probability generating function at time t starting with a single normal stem cells at time 0 [25]:
Wðy1 ; y2 ; . . . ; ykþ1 ; tÞ ¼
X
Pðy1 ðtÞ ¼ i1 ; . . . ; ykþ1 ðtÞ ¼ ikþ1 jy1 ð0Þ
i1 ;...;ikþ1
¼ 1;
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Then, the Kolmogorov forward differential equation for W is
134 0
W ðy1 ; y2 ; . . . ; ykþ1 ; tÞ ¼ @ W=@t ¼ fl1 y1 y2 þ a1 y21 þ b1 ða1 þ b1 þ l1 Þy1 g@ W=@y1 þ þ flk yk ykþ1 þ ak y2k þ bk ðak þ bk 137
þ lk Þy1 g@ W=@yk
N
γ1
@W @yk
ð1; . . . ; 1; 0; tÞ is
@W @yk
148
evaluate at y1 = 1, . . . , yk = 1, yk+1 = 0.
ð2:1Þ
μ1
I2
γ2
μ2
I3
γ3
μ3
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ð1; . . . ; 1; 0; tÞ
Wð1; . . . ; 1; 0; tÞ
¼ E yk ðtÞjykþ1 ðtÞ ¼ 0
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where E[Yk(t)|Yk+1(t) = 0] denotes the conditional expectation. We have
hðtÞ ¼ lk E½yk ðtÞjykþ1 ðtÞ ¼ 0
ð2:3Þ
Since cancer is a rare disease, the probability P(t) of there being a malignant cell at time t is close to zero. Thus,
E½Y k ðtÞjY kþ1 ðtÞ ¼ 0 E½Y k ðtÞ:
ð2:4Þ
Then we have the approximation for one breast tissue:
hðtÞ lk E½Y k ðtÞ
dY 1 ðtÞ Y 1 ðtÞ þ Y 2 ðtÞ þ Y 3 ðtÞ þ Y 4 ðtÞ þ Y 5 ðtÞ ¼ c1 Y 1 ðtÞ 1 l1 Y 1 ðtÞ dt K dY 2 ðtÞ ¼ l1 Y 1 ðtÞ þ c2 Y 2 ðtÞ l2 Y 2 ðtÞ dt dY 3 ðtÞ ¼ l2 Y 2 ðtÞ þ c3 Y 3 ðtÞ l3 Y 3 ðtÞ dt dY 4 ðtÞ ¼ l3 Y 3 ðtÞ þ c4 Y 4 ðtÞ l4 Y 4 ðtÞ dt dY 5 ðtÞ ¼ l4 Y 5 ðtÞ þ c5 Y 5 ðtÞ l5 Y 5 ðtÞ dt
γ4
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ð2:6Þ
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dY 1 ðtÞ ¼ c01 Y 1 ðtÞ l1 Y 1 ðtÞ dt dY 2 ðtÞ ¼ l1 Y 1 ðtÞ þ c02 Y 2 ðtÞ l2 Y 2 ðtÞ dt dY 3 ðtÞ ¼ l2 Y 2 ðtÞ þ c03 Y 3 ðtÞ l3 Y 3 ðtÞ dt dY 4 ðtÞ ¼ l3 Y 3 ðtÞ þ c04 Y 4 ðtÞ l4 Y 4 ðtÞ dt dY 5 ðtÞ ¼ l4 Y 5 ðtÞ þ c05 Y 5 ðtÞ l5 Y 5 ðtÞ dt μ4
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ð2:5Þ
The incidence rate (or hazard function) is determined by the product between the mutation rate lk and the number of intermediate cells in compartment Ik. In order to estimate the number in each compartment as shown in Fig. 2.1, we set up the model of ordinary differential equations of 5 hit model. To estimate the number of Y5(t) in compartment I5 before and after age 45 for five stage model, Yi(t) (i = 1, . . . , 5) should satisfy the following ordinary differential equations for t 6 45
I4
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and t > 45
i
kþ1 y2 ð0Þ ¼ 0; . . . ; ykþ1 ð0Þ ¼ 0Þyi11 . . . ykþ1
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where Further
@W @yk
I5
μ5
ð2:7Þ
M
γ5
Fig. 2.1. A five stage model for carcinogenesis. I2, I3, I4, I5 denote the compartments of intermediate cells, l1, l2, l3, l4, l5 are the mutation rates per cell per year of normal progenitor cells, I2 compartment, I3 compartment, I4 compartment and I5 compartment. c1, c2, c3, c4, c5 are the net growth rates per cell per year.
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Where K = 107 normal stem cells per breast, li (i = 1, . . . , 5) denote the mutation rates per cell per year, c1 = 1.0074, c01 = 0.0067, ci and c0i (i = 2, . . . , 5) denote the clonal expansion rates per cell per year in compartments Ii (i = 2, . . . , 5) of intermediate cells before and after age 45. Therefore, the incidence function (or hazard function) of one female to a k-stage model should be
hðtÞ 2lk E½Y k ðtÞ
ð2:8Þ
We determine the set of parameters that provide the minimum value of the Chi-square test to the SEER age-specific cancer incidence data using the numerical optimization routine fminsearch in MATLAB. The minimum value of the Chi-square is determined by 16 X 2 ðdatai estimationi Þ =estimationi
ð2:9Þ
i¼1
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In order to test some biological mechanisms such as the effects of mutation, clonal expansion and the hormone level, we have considered the following four cases: Case one: There is a mutator phenotype in the tumorigenesis of breast cancer for multi-hit models, that is the mutation rates per cell per year in k-hit (k = 2, . . . , 6) model should satisfy l1 6 l2 6 6 lk. Case two: There is no mutator phenotype but only selection for clonal expansion in the tumorigenesis of breast cancer for multihit models, that is the mutation rates per cell per year in k-hit (k = 2, . . . , 6) model should satisfy l1 = l2 = = lk, c2 6 c3 6 ck, c20 6 c30 6 6 ck0 (k = 2, . . . , 6). Case three: The clonal expansions in models of two-six stages do not depend on the expression level of hormone in females, that is there are no difference for the clonal expansion in each compartment of intermediate cells before and after menopause or it is the same net growth rate per cell per year for ci (i = 2, . . . , k) in each compartment of intermediate cells before and after age 45 in k-hit (k = 2, . . . , 6) model. It should be c2 = c20 , c3 = c30 , . . . , ck = ck0 . We also assume that the mutation rate in each compartment satisfy l1 6 l2 6 6 lk, (k = 2, . . . , 6). Case four: The clonal expansions of intermediate cells in models of two-six stages depend only on the expression level of hormone in females and there is no difference for the clonal expansion in each compartment before and after age 45, that is there is the same net growth rate per cell per year in each compartment of intermediate cells (c2 = c3 = = ck) before age 45 and (c0 2 = c0 3 = = c0 k) after age 45 in k-hit (k = 2, . . . , 6) model. We also assume that the mutation rate in each compartment satisfy l1 6 l2 6 6 lk , (k = 2, . . . , 6).
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3. Results
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We use the numerical optimization routine fminsearch in MATLAB to estimate the optimal parameters li (i = 1, . . . , 6), cj, c0 j (j = 2, . . . , 6) by the minimum value of Chi-square between the SEER data and the hazard function that depend on the models satisfied the Hypothesis one-four. Table 1 gives the minimum values of Chi-square test of the goodness of fit and values at the 5% significant level. For Case one, we can only accept the two-four stage models for the null hypothesis of mutator phenotype at the 5% significant level and reject the five and six stage models even if five and six stage models fit the SEER data better than two stage model (see Fig. 3.1). Our recent paper [22] used both two-six stage models with clonal expansion of each compartment of intermediate cells fit the age specific incidence data of breast cancer. We estimated the mutation rates of two-six stage models (see Table 1 in paper
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Table 1 P 2 of Case The minimum Chi-Square values ¼ 16 i¼1 ðdatai estimationi Þ =estimationi one to Case four. Chi2 denotes the value of Chi-Square; df (=16–1-number of parameters) denotes the degrees of freedom; S. 5% denotes the value at 5% significant level. The minimum Chi-Square values were calculated by using the numerical optimization routine fminsearch in MATLAB. 2-hit
3-hit
4-hit
5-hit
6-hit
Hypothesis Chi2 df S. 5%
1 10.4937 11 19.675
6.3151 8 15.507
6.6851 5 11.071
10.6492 2 5.991
10.383
Hypothesis Chi2 df S. 5%
2 10.494 12 21.026
6.2186 10 18.307
19.5339 8 15.507
147.707 6 12.592
561.576 4 9.488
Hypothesis Chi2 df S. 5%
3 92.9915 12 21.026
42.4444 10 18.307
19.3112 8 15.507
20.4038 6 12.592
24.4929 4 9.488
Hypothesis Chi2 df S. 5%
4 10.4937 11 19.675
6.9300 10 18.307
8.5019 9 16.919
11.5148 8 15.507
11.8720 7 14.067
[22]). However, we can only accept those mutation rates of twofour stage models at the 5% significant level, and should reject those mutation rates of five and six stage models at the 5% significant level. For Case two, we can only accept the two and three stage models for the null hypothesis of constant mutation rate per cell per year and clonal expansion of each compartment of intermediate cells at the 5% significant level and reject the four-six stage models. For Case three, we should reject two-six stage models for the null hypothesis of the clonal expansion of each compartment of intermediate cells having the same net growth rate per cell per year before age 45 and after age 45 at the 5% significant level, that is the clonal expansion rate of each compartment of intermediate cells must depend on the expression level of hormone. For Case four, we should accept the two-six stages models at the 5% significant level for the null hypothesis of all compartment of intermediate cells having the same net growth rate before age 45 and after age 45, respectively. That is c2 = c3 = = ck before age 45 and c0 2 = c0 3 = = c0 k after age 45 in k-hit (k = 2, . . . , 6) model. Therefore, the Hypothesis four is the best hypothesis of the four hypotheses because only Hypothesis four is available to two-six stage models, that is the clonal expansion in each compartment of intermediate cells must depend on the expression level of hormone.
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4. Discussion
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Our mathematical models also give the genetic pathway of human breast cancer if a malignant cell is caused by more than three stages of mutations in the genome. The first two or three hits must be occurred on genes that keep the genomic stability, which are called instability genes composed by DNA repair genes or spindle-checking genes. Those genes lost their function can cause genomic instability and then lead to high mutation rate and breast cancer. This is our explanation of mutator phenotype. Breast cancer can be caused by two-six hits in the genome of normal breast progenitor cells. The two-hit and three-hit models should be applicable to those cases for patients who have family history of breast cancer such as BRCA1 and BRCA2 mutation carriers [26]. Four-six hit models should be applicable to those sporadic cases of breast cancer. If breast cancer is caused by the mutations in the genome with more than three hits, the first three hits must be lead to geno-
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Fig. 3.1. The age-specific incidence data of all races per 100,000 females for breast cancer from SEER registry for the year 1973–1999 (black) and rates predicted by two-six stage models.
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mic instability before the cell becomes a malignant cell. Our mathematical results have shown the postulation in reference [27] that genetic instability is an early event for human cancer. Our theoretic result is consistent with experimental evidence regarding the role of BRCA1 for breast cancer [28]. Thus, loss of function of genes that keep the genomic stability can be used as the early diagnosis of breast cancer. Earlier diagnosis of cancer can save millions of lives for all type of cancer cases. The genomic instability means the high mutation rate in the genome among the subtle sequence changes, alterations in chromosome number, chromosome translocations, gene amplifications and loss of heterozygosity. Therefore, the subtle sequence changes are caused by the nucleotide-repairing genes lost their functions such as P53 gene. Alterations in chromosome number, chromosome translocations, gene amplifications and loss of heterozygosity are caused by the spindle-checking genes lost their function. Loss of heterozygosisty is the major phenotype of breast cancer. That is loss of function for spindle-checking genes is the main reason for most of breast cancer cases. Some genes should have the functions both for the nucleotide-repairing and spindle-checking functions such as BRCA1 gene. The most interesting result in this paper is that the clonal expansion of each compartment of intermediate cells must depend on expression level of hormone for breast cancer. Thus, hormone therapy such as tamoxifen should be effective for patients of breast cancer before their menopause of females. Breast cancer may be triggered by more than six hits in the genome of breast stem cells because many mutations have been found by recent large scale genomic analyses of breast cancer genome. Our methods can also be applied to cases with more than six stage models. However, the first two or three hits must be occurred on genes that keep the genomic stability. Therefore, genomic instability is the main cause of many mutations in breast cancer genome. This paper has not discussed the chromosomal instability. Chromosomal instability is also an early event of tumoregeneis [29].
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Acknowledgements
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Q4 This work is supported by the National Natural Science FoundaQ5 tion of China (Nos. 10771081, 11071275, 11228104), and by the
Special Fund for Basic Scientific Research of Central Colleges (CCNU10B01005). XZ thanks Dr. Richard Simon for his guidance when XZ was a postdoctoral fellow in Biometric Research Branch of National Cancer Institute, NIH, USA. We thank the reviewers very much for their very good suggestions on the manuscript.
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References
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