# Breast Cancer Survival Analysis Model
## 1 Introduction  
Breast cancer (BC) is one of the most prevalent malignancies in women worldwide. BC affects over two million women each year, with significant mortality rates in both developing and developed nations. BC has surpassed lung cancer as the primary cause of cancer mortality in women (accounting for 15% of all cancer fatalities). The yearly incidence of BC in all countries has been increasing yearly, and what was once a priority in Western countries has become a global issue [1]. Due to numerous factors, such as the consumption of dietary fat, solvent and pesticide exposure have been regarded as potential risk factors for breast cancer. However, the evidence for this is still unconvincing thus far [2]. According to the National Health Agency's cancer registration data, BC was the most common cancer among women in Taiwan in 2017. There were 14,801 patients with BC (per 100,000 population), with a crude incidence rate of 125.64 and an age-standardized incidence rate of 87.54. BC was shown to be more common in females than in males, placing first among the top 10 cancers. Since the patient population for each malignancy differs, age standardization is employed to reduce the disparity in population makeup. As a result, age may be objectively compared on the same basis. Furthermore, an analysis of data from 2010 to 2014 reveals that the overall five-year survival rate of BC in Taiwan is morethan 90%, as is the five-year survival rate of each stage. It has been shown that the higher the stage of BC, the lesser the chance of survival. As a result, if BC is identified early, not only may breast preservation surgery be performed but the five-year survival rate can exceed 90% [3]. However, tumor size, nodal status, and expression of ER, PR, and human epidermal growth factor receptor-2 play a role in the therapy of early-stage BC (HER2) [4].  
To ascertain the survival rate of BC, [PERSON] et al., 2021 [5] used Kaplan-Meier survival analysis to determine the survival time of patients with BC, and they also used [PERSON]'s proportional hazard model and log-rank test in their research. In another advancement, [PERSON] [6] discovered the factors influencing the time to death among patients with BC in his study using the Kaplan-Meier survival curve with the log-rank test to assess survival among predictor variables. The net effect of each independent variable was then calculated using Cox regression. Despite this, significant progress has been achieved in analyzing the survival of patients with BC and BC stages utilizing the aforementioned statistical methods. As a result, early detection of BC demands a precise and trustworthy procedure that allows clinicians to differentiate between benign and malignant tumors [7].  
However, principles and methods related to the evaluation of prognostic factors are not well established due to the fact that assessment of the prognostic factors has been confounded by treatment [8].  
The goal of this study is to establish a collection of BC survival and metastasis analyses. First, the Kaplan-Meier [9], logarithmic test, and Cox proportional hazard models [10] were used to investigate the link between BC stages, different treatment modalities, and the survival rate of patients with BC at various follow-up intervals. Second, we evaluated whether clinical prognostic factors had univariate and multivariate effects on the survival of patients with BC. Finally, we performed a metastasis analysis utilizing the survival transition rate values of BC stages to develop a Markov chain model for BC and a semi-Markov simulation model for BC metastasis analysis, respectively, to establish an objective reference standard for treatment results and provide information to physicians to aid in diagnosis.