A new model of time scheme for progression of colorectal cancer
- Shuhao Sun^{1},
- Fima Klebaner^{1} and
- Tianhai Tian^{1}Email author
https://doi.org/10.1186/1752-0509-8-S3-S2
© Sun et al.; licensee BioMed Central Ltd. 2014
Published: 22 October 2014
Abstract
Background
tumourigenesis can be regarded as an evolutionary process, in which the transformation of a normal cell into a tumour cell involves a number of limiting genetic and epigenetic events. To study the progression process, time schemes have been proposed for studying the process of colorectal cancer based on extensive clinical investigations. Moreover, a number of mathematical models have been designed to describe this evolutionary process. These models assumed that the mutation rate of genes is constant during different stages. However, it has been pointed that the subsequent driver mutations appear faster than the previous ones and the cumulative time to have more driver mutations grows with the growing number of gene mutations. Thus it is still a challenge to calculate the time when the first mutation occurs and to determine the influence of tumour size on the mutation rate.
Results
In this work we present a general framework to remedy the shortcoming of existing models. Rather than considering the information of gene mutations based on a population of patients, we for the first time determine the values of the selective advantage of cancer cells and initial mutation rate for individual patients. The averaged values of doubling time and selective advantage coefficient determined by our model are consistent with the predictions made by the published models. Our calculation showed that the values of biological parameters, such as the selective advantage coefficient, initial mutation rate and cell doubling time diversely depend on individuals. Our model has successfully predicted the values of several important parameters in cancer progression, such as the selective advantage coefficient, initial mutation rate and cell doubling time. In addition, experimental data validated our predicted initial mutation rate and cell doubling time.
Conclusions
The introduced new parameter makes our proposed model more flexible to fix various types of information based on different patients in cancer progression.
Background
Carcinogenesis is the transformation of normal cells into cancer cells. This process has been shown to be of a multistage nature. It involves somatic mutations in any cellular genome, either induced by external source or spontaneously occurring during the mitotic replication, and thus it may comprise different types of DNA alterations. Moreover, the cell genome may acquire entire sequences from exogenous sources. The epigenetic changes, which alter chromatin structure, can also be subject to the same selection forces as genetic events [1–4].
Colorectal tumourigenesis proceeds through a number of well defined clinical stages. The process is initiated when a single colorectal epithelial cell acquires a mutation in a gene that inactivates the APC/β-catenin pathway. Mutations that constitutively activate the K-Ras/B-Raf pathway are associated with the growth of a small adenoma to a clinically significant size (namely > 1 cm in diameter).
Subsequent waves of clonal expansion driven by mutations in genes controlling the TGF-β [5], PIK3CA [6], TP53 [7], and other pathways are responsible for the transition from a benign tumour to a malignant tumour. Some tumours eventually acquire the ability to migrate and seed other organs [1]. In general, it is still quite difficult to give a precise definition of the steps in the evolutionary process.
The mathematical investigation of cancer started in early 1950s [8, 9], aiming at deriving the basic laws regarding the tumour dynamics and elaborating a comprehensive framework for testing hypothesis [10–14]. Later on, other types of experimental evidence, based on epidemiological data of cancer mortality, allowed the development of the multistage theory (MST) of cancer development [15–18]. Among them, population genetics models are used extensively to describe tumourigenesis [16, 19–21], since cancer progression is an evolutionary process. Various deterministic and stochastic models have been proposed. Some models addressed specific questions, such as the dynamics of tumour suppressor genes [22–25], genetic instability [26, 27], or tissue architecture [20]. computer Computer simulations [28] and theoretical analysis [29, 30] have been employed to investigate the properties of these models In particular, the stochastic multistage cancer model is a well-known model of cancer development [3, 18, 31, 32]. Models of tumourigenesis have been proposed early on to explain cancer incidence data [33–36]. In a series of studies, Berenblum and Shubik proposed the "two-stage" model of carcinogenesis [37]. These models assumed that cancer is a stochastic multistep process with small transition rates and they have been further developed into the multistage theory of cancer [17, 38–40]. Multistage models provide a natural framework to evaluate the potential benefits of prevention and intervention strategies designed to reduce cancer risk. A comprehensive review on this topic can be found in [41].
where T is the average time of the cell division.
Based on this formula, it was calculated that the actual selective advantage provided by typical somatic mutations in human tumours in situ was 0.004 ± 0.0004. However, formula (2) does not involve the tumour size and the waiting time was independent of the population size. Thus it is not appropriate to use it to calculate the time required for the first mutation appearance.
where F_{ Lad,Aca } is the fraction of mutations in the advanced carcinoma that were not found in the large adenoma and T_{ ACa } is the birthdate of the founder cell of advanced carcinoma.
where T_{ get } = 2.3 days and r = 0.016 per generation, they calculated the elapsed time between the different stages of the tumourigenic process. The results showed an averaged time of 11.7 years from the initiation of tumourigenesis to the birth of the cell giving rise to the parental clone, an averaged time of 6.8 years from then to the birth of the cell giving rise to the index lesion, and an averaged time of 2.7 years from then until the patients' death. This result is consistent with that reported in [1]. Taking these correlations together, the dynamics of the tumour progression offers an opportunity to interfere in the tumour evolution and develop a more customized treatment.
However, the information about the initial mutation rate and selective advantage of tumour cell for each individual patient have not been revealed yet in all studies reviewed above. This information is very important for cancer treatment. Although we conducted studies to describe cancer progression for a number of individuals [44], the results were not satisfactory. Thus in this work we will develop a more general framework that not only maintains the advantage of existing mathematical models but also is able to calculate individual patient's initial mutation rate and selective advantage of tumour cell which has potential applications for medical treatment.
Results and discussion
A general modelling framework
where o(Δt)/Δt → 0 as Δt → 0. In addition, the probability of transformation from state i to state i + j (with j > 1) in time Δt is assumed to be o(Δt). This implies that 1/µ_{ i }+1 is the averaged time required for a cell to go from state i to state i + 1.
The initial condition p_{ i }(0) = 0 (i = 1, 2, · · · ) means that all the cells are normal at time t = 0.
In this section, we first present an explicit formula that includes the two formulae discussed in the previous section as examples. Following the notations introduced in the previous section, we also assume that the mutation rate µ_{ i } = µ(t_{ i }) is an increasing function of time t, where t_{ i } is the time when the i^{ th }-event occurs for i = 1, 2, · · ·, N.
where LambertW is the principal branch of the Lambert W function.
The case of Beerenwinkel's formula
Hence this equation becomes a special case of our equation (5) with a = 1 and µ_{0} = ud.
Note that the innovation od this work is the introduction of parameter a, which is important to make our model to be consistent with clinical data. If restricting a = 1, the predicted result was not supported by simulation [15]. In addition, Beerenwinkel's model assumed that each subsequent mutation has the same incremental effect on the fitness of the cell. It has been widely accepted that the impact of a specific mutation on phenotype will depend on the genetic background. For example, it was pointed out that the subsequent driver mutations appear faster and faster and the cumulative time to have k driver mutations grows with the logarithm of k [42]. Thus our equation depends on four parameters, namely the initial mutation rate µ_{0}, selective advantage coefficient s, transforming factor a, and the number of driver genes. It is still in debate that which gene can be clarified as cancer candidate gene (CAN-gene). Sjö blom et al defined 69 Can-genes [4], while Wood et al defined 142 CAN-genes [48]. If we look at 179854 colorectal cancer mutations in Cosmic v.64 statistics in [49], there were 340 patients who have more than 20 mutations. If we consider the top 100 highest mutation frequency genes as CAN-genes, the average number of the mutated CAN-gene is 29.23 per tumour. Since the average passenger mutation number in this study is around 200 per tumour, we choose the number of CAN-genes as N = 30 which covers the cases of more than 95% patients.
Determination of waiting time λ_{ k } for cancer progression
Values of λ_{ k } for three patients.
k | λ_{ k }1 | λ_{ k }2 | λ_{ k }3 | k | λ_{ k }1 | λ_{ k }2 | λ_{ k }3 |
---|---|---|---|---|---|---|---|
1 | 10^{−10} | 10^{−10} | 10^{−9} | 9 | 0.3211 | 0.3211 | 0.415 |
2 | 1.41 · 10^{−5} | 1.41 · 10^{−5} | 5 · 10^{−5} | 10 | 0.45287 | 0.45287 | 0.57 |
3 | 0.000843 | 0.000843 | 0.00181 | 11 | 0.60523 | 0.60523 | 0.746 |
4 | 0.006999 | 0.006999 | 0.0124 | 12 | 0.776297 | 0.776297 | 0.94 |
5 | 0.0260517 | 0.02605 | 0.04128 | 13 | 0.9642 | 0.9642 | 1.15 |
6 | 0.064499 | 0.0644 | 0.0946 | 14 | 1.16727 | 1.16727 | |
7 | 0.12599 | 0.12599 | 0.175 | 15 | 1.38388 | 1.38388 | |
8 | 0.211684 | 0.2116 | 0.282 | 16 | 1.61264 | 1.61264 |
Tumour stage, waiting time (WT) (years) and number of mutations (NM) for patient Mx34 based on our calculation, and data published in [1].
Selective advantage, initial mutation rate, waiting time for patients Mx34 and Co82.
Patient | s | µ _{0} | a | t _{ k } |
---|---|---|---|---|
Mx34 | 0.01 | 2 ∗ 10^{−5} | 0.316 | 3165 |
Co82 | 0.0075 | 2 ∗ 10^{−7} | 0.309 | 3150 |
which was not revealed in [1]. The determined time required to reach different tumour stages is listed in Table 2 for patient Mx34. For comparison, we also provide the prediction published in [1]. Table 2 suggests that our calculated results are consistent with the published ones.
This proposed time scheme describes how the advanced cancer evolved from the normal cells. However, we note that the first one or two driver mutations may not transform a normal cell into a cancer cell. As Jones et al described in [1], the APC/β-catenin passway mutations may only produce microadenoma. Even when all APC, Coca/Cin, KRAS, BRAF genes were mutated, only a large adenoma was produced. But it was still not yet necessary to be carcinoma.
Tumour stage, time required (years) and number of driver mutations k for patient Co82 from our calculation.
Tumour stages | Time required | k |
---|---|---|
Microadenema | 11 | 3 |
Small-adenoma | 18 | 4 |
Large adenoma | 22 | 5 |
carcinoma | 27-33 | 6-11 |
Estimation of relationship between tumour size and doubling time
The number of driver mutations (k), time required (years) and number of cancer cells (NCCs).
k | time required | NCCs | k | time required | NCCs |
---|---|---|---|---|---|
8 | 13 | 6800 | 2 | 11 | 6.8 |
11 | 16.6 | 2 · 10^{6} | 3 | 18 | 300 |
15 | 19.4 | 10^{8} | 4 | 22 | 10000 |
19 | 21.3 | 10^{9} | 5 | 25 | 10^{5} |
26 | 23.5 | 5.5 · 10^{9} | 7 | 28 | 8 · 10^{5} |
30 | 24.4 | 8 · 10^{9} | 8 | 30 | 10^{7} |
33 | 25 | 9 · 10^{9} | 11 | 36 | 5 · 10^{7} |
Application to pancreas cancer
Estimates of number of mutations (NMs), initial mutation rate (µ_{0}) and selective advantage coefficient s for seven pancreas cancer patients
Patients | NMs | µ _{0} | s |
---|---|---|---|
Pa01c | 49 | 5 · 10^{−5} | 0.007 |
Pa02c | 35 | 9 · 10^{−5} | 0.008 |
Pa03c | 28 | 6 · 10^{−4} | 0.014 |
Pa04c | 34 | 9 · 10^{−5} | 0.008 |
Pa05c | 28 | 4 · 10^{−4} | 0.01 |
Pa07c | 50 | 2.5 · 10^{−4} | 0.008 |
Pa08c | 35 | 5 · 10^{−5} | 0.007 |
Average | 37 | 2.2 · 10^{−4} | 0.0088 |
Conclusions
Cancer progression essentially is a stochastic process. Statistical analysis is an important mathematical tool to analyze the progress of cancer cells based on a large number of cells in a particular position of the human body. In this work we proposed a new approach for analyzing the cancer progression in individuals. The developed model can be used to calculate the waiting time for carcinogenesis. Our model assumes that the expected waiting times depend on the values of three parameters, namely the selective advantage coefficient s, the transform factor a and the initial mutation rate µ_{0}. Comparing with the Novak-Beerenwinkel model [15, 20, 26, 27], we introduced the transform factor as a new parameter which makes our model more flexible. Thus our model is capable of matching with different mutational curves. In addition, our new model can be used to reveal the values of initial mutation rate, the selective advantage coefficient of tumour cells and the subsequent clonal expansion for individual patients. Our approach showed that, if the averaged mutational rate is known, a number of other biological parameters, such as the initial mutational rate and the mutational function, can be determined by using our proposed model. These parameters are important for constructing appropriate clinical treatment schemes for individual patients. The predicted values of parameters from our proposed model are consistent with the published ones in literature, which partially validated our model. The first example is the mean doubling time. We showed that the mean doubling time is one year for patient Mx34 and 1.5 years for patient Co82, which are consistent with the predictions in [1]. These results suggested that the mean doubling time in metastases, which is generally 2-4 months, should be much shorter than that in adenomas and carcinomas [1]. The second example is the selective advantage coefficient. Our model suggested that the selective advantage coefficient in colon cancer is about 0.01 ~ 0.0075 in Tables 3 and about 0.007 ~ 0.014 for pancreas caner in Table 6 which is close to but a little higher than the value of 0.004 that was estimated in [42].
Finding the values of initial mutation rate and the selective advantage coefficient also have potential applications in other related issues. For example, a mathematical approach has been designed to investigate the targeted cancer therapy recently [37, 47, 51, 52]. The targeted cancer therapies use drugs that interfere with specific molecular structures implicated in tumour development [53]. The majority of the targeted therapies are either small-molecule drugs that act on targets inside the cell (usually protein tyrosine kinases) or monoclonal antibodies directed against tumour-specific proteins on the cell surface [54]. It has been showed that the overall probability P of tumour eradication as P = P_{1}P_{2}P_{3}. Here P_{1}, P_{2} and P_{3} are the probabilities that no resistance mutation leading to treatment failure arises during expansion, during steady state and during treatment, respectively [55].
The key parameters in this probability model are the number of tumour cells at steady state, time that the tumour remains at steady state before treatment, initial rate of cell division, initial death rate of tumour cells, the rate of resistance mutations, and the division and death rates (r′ and d′, respectively) of sensitive cells under treatment, in the absence of density constraints [55]. Note that (d′ − r′) is the selective advantage coefficient. This coefficient and the initial mutation rate can be calculated by using our formula. Hence our proposed model may have potential applications to design the targeted cancer therapy.
Declarations
Acknowledgements
This work is in part supported by grants from the Australian Research Council (ARC) (DP120104460). T.T. is also in receipt of an ARC Future Fellowship (FT100100748). The publication costs for this article were funded by the corresponding author.
Declarations
The publication costs for this article were funded by the corresponding author. This article has been published as part of BMC Systems Biology Volume 8 Supplement XXX, 2014: Selected articles from the IEEE International Conference on Bioinformatics and Biomedicine 2013: Systems Biology. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcsystbiol/supplements/8/SXXX.
This article has been published as part of BMC Systems Biology Volume 8 Supplement 3, 2014: IEEE International Conference on Bioinformatics and Biomedicine (BIBM 2013): Systems Biology Approaches to Biomedicine. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcsystbiol/supplements/8/S3.
Authors’ Affiliations
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