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Math Med Biol. 2019 Jun 13;36(2):241-267. doi: 10.1093/imammb/dqy010.

Modeling dynamics for oncogenesis encompassing mutations and genetic instability.

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Instituto de Matemática e Computação, UNIFEI, Itajubá, Minas Gerais, Brazil.
Instituto de Matemática, Estatística e Computação Científica, UNICAMP, Campinas, São Paulo, Brazil.


Tumorigenesis has been described as a multistep process, where each step is associated with a genetic alteration, in the direction to progressively transform a normal cell and its descendants into a malignant tumour. Into this work, we propose a mathematical model for cancer onset and development, considering three populations: normal, premalignant and cancer cells. The model takes into account three hallmarks of cancer: self-sufficiency on growth signals, insensibility to anti-growth signals and evading apoptosis. By using a nonlinear expression to describe the mutation from premalignant to cancer cells, the model includes genetic instability as an enabling characteristic of tumour progression. Mathematical analysis was performed in detail. Results indicate that apoptosis and tissue repair system are the first barriers against tumour progression. One of these mechanisms must be corrupted for cancer to develop from a single mutant cell. The results also show that the presence of aggressive cancer cells opens way to survival of less adapted premalignant cells. Numerical simulations were performed with parameter values based on experimental data of breast cancer, and the necessary time taken for cancer to reach a detectable size from a single mutant cell was estimated with respect to some parameters. We find that the rates of apoptosis and mutations have a large influence on the pace of tumour progression and on the time it takes to become clinically detectable.


avascular tumour growth; bifurcations; multi-step tumorigenesis; stability


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