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.

中文翻译：

---

肿瘤发生动力学建模，包括突变和遗传不稳定性。

肿瘤发生已被描述为一个多步骤过程，其中每个步骤都与遗传改变有关，其方向是逐步将正常细胞及其后代转化为恶性肿瘤。在这项工作中，我们提出了一种癌症发病和发展的数学模型，其中考虑了三种人​​群：正常，恶变前和癌细胞。该模型考虑了癌症的三个特征：生长信号的自给自足，对抗生长信号的敏感性和逃避凋亡。通过使用非线性表达来描述从癌变前到癌细胞的突变，该模型将遗传不稳定性作为肿瘤进展的有利特征。详细进行了数学分析。结果表明凋亡和组织修复系统是阻止肿瘤进展的首要障碍。这些机制之一必须被破坏才能使癌症从单个突变细胞发展而来。结果还表明，侵袭性癌细胞的存在为适应性较差的癌前细胞打开了生存之路。使用基于乳腺癌实验数据的参数值进行数值模拟，并就某些参数估算出癌症从单个突变细胞达到可检测大小所需的时间。我们发现凋亡率和突变率对肿瘤进展的速度和临床检测所需的时间有很大的影响。使用基于乳腺癌实验数据的参数值进行数值模拟，并就某些参数估算出癌症从单个突变细胞达到可检测大小所需的时间。我们发现凋亡率和突变率对肿瘤进展的速度和临床检测所需的时间有很大的影响。使用基于乳腺癌实验数据的参数值进行数值模拟，并就某些参数估算出癌症从单个突变细胞达到可检测大小所需的时间。我们发现凋亡率和突变率对肿瘤进展的速度和临床检测所需的时间有很大的影响。