Studies on somatic mutations in cancer cells DNA, their roles in tumour growth and progression between successive stages are of high importance for improving understanding of cancer evolution. Important insights into scenarios of cancer growth, roles of somatic mutations and types/strengths of evolutionary forces they introduce are gained by using mathematical and computer modelling. Previous studies developed mathematical models of cancer cell evolution with driver and passenger somatic mutations. Driver mutations were assumed to have a strong advantageous effect on cancer cell population growth, while passenger mutations were considered as fully neutral or mildly deleterious. In this paper, following several experimental results, we develop models of cancer evolution with somatic mutations introducing weakly advantageous force to the evolution of cancer cells. Our models belong to two classes, deterministic and stochastic. Deterministic models are systems of differential master equations of balances of average numbers of cells and mutations in evolution. We, additionally introduce a modification in equations for balances aimed at accounting for effects on the finite size of the cancer cell population. The novelty of our deterministic modelling is deriving parameters of travelling waves of advantageous mutations and quantification of the pattern of the population size growth. A stochastic model based on the Gillespie algorithm is used to verify the results of deterministic modelling. We confront predictions of modelling with some observational data on cancer evolution. We propose the scenario of evolution driven by a large number of weakly advantageous mutations in transition from the early stage to the invasive form of breast cancer. We also fit our models to observed patterns of rates of growth of tumours.

翻译：对癌细胞DNA中体细胞突变及其在肿瘤生长和阶段间演进中作用的研究，对于加深对癌症进化机制的理解至关重要。通过数学建模与计算机模拟，可获得关于癌症生长情景、体细胞突变角色及其引入进化力量类型/强度的重要见解。此前研究建立了含驱动型和乘客型体细胞突变的癌细胞进化数学模型，其中驱动突变被认为对癌细胞群体增长具有强优势效应，而乘客突变则被视为完全中性或轻微有害。本文基于多项实验结果，构建了引入弱优势进化驱动力的体细胞突变癌症进化模型。我们的模型分为确定性模型与随机模型两类：确定性模型采用微分主方程系统来描述进化过程中细胞与突变平均数量的平衡。此外，我们在平衡方程中引入修正项，用以考虑癌细胞群体有限大小效应。该确定性建模的创新之处在于推导了优势突变的行波参数，并量化了群体规模增长模式。基于Gillespie算法的随机模型被用于验证确定性建模结果。我们将模型预测与部分癌症进化观测数据进行了对比，提出从乳腺癌早期阶段向侵袭性转变过程中，大量弱优势突变驱动的进化情景，并将模型拟合至观察到的肿瘤生长速率模式。