The Application Of Stochastic Model In Cancer Study

Cancer is still very difficult to conquer problem in the medical field. In order to understand the mechanism of cancer, scholars have made a lot of researches in recent years. We build a mathematical stochastic model to fit the age-specific incidence of breast cancers in the SEER (surveillance, epidemiology, and end results) registry. In this paper, we mainly discuss the deterministic-stochastic model. And the growth rates, differentiation rates of intermediate cells and mutation rates of cells are regarded as parameters in the model. We also improve the approximate solution and propose the exact solution of the deterministic-stochastic model. This method constitutes a system of several coupled ordinary differential equations, which derive from probability generating functions. Then, the probability of at least one malignant cell at any time, and the corresponding hazard function are given by solving the system of several coupled ordinary differential equations. We use the exact solution of the deterministic-stochastic multistage model to fit the SEER data. The results show that the 2-13 stage models are well. Especially, the 3-12 stage models fit the data very well and 13-stage model is not quite as good but still works well. By parameters values of the fitting results, we find that the loss of function of genes that keep the genomic stability is an early event in the tumorigenesis, which is very useful for early diagnosis of breast cancer. In addition, we also compare the simulation results of the exact solution and of the approximate solution. The results reveal that the two fitting results are similar. This indicates that the approximate solution can be used to simulate the SEER data, which is acceptable for breast cancer. Finally, we make use of the chi-square test to test our hypothesis and fitting results. The results demonstrate that the clone expansion of intermediate cells must depend on the hormone expression level of females; 2-13 mutations is acceptable for human breast cancer, which are consistent with the results of biology.