Metastasis is a leading cause of cancer-related deaths. Carcinoma generally initiates at a specific organ as a primary tumor, but eventually metastasizes and forms tumor sites in other organs. In this report, we developed a mathematical model of cancer progression with alterations in metastasis-related genes. In cases in which tumor cells acquire metastatic ability through two steps of genetic alterations, we derive formulas for the probability, the expected number, and the distribution of the number of metastases. Moreover, we investigate practical pancreatic cancer disease progression in cases in which both one and two steps of genetic alterations are responsible for metastatic formation. Importantly, we derive a mathematical formula for the survival outcome validated using clinical data as well as direct simulations. Our model provides theoretical insights into how invisible metastases distribute upon diagnosis with respect to growth rates, (epi)genetic alteration rates, metastatic rate, and detection size. Prediction of survival outcome using the formula is of clinical importance in terms of determining therapeutic strategies.
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