Hypoxia and acidity act as environmental stressors promoting selection for cancer cells with a more aggressive phenotype. As a result, a deeper theoretical understanding of the spatio-temporal processes that drive the adaptation of tumour cells to hypoxic and acidic microenvironments may open up new avenues of research in oncology and cancer treatment. We present a mathematical model to study the influence of hypoxia and acidity on the evolutionary dynamics of cancer cells in vascularised tumours. The model is formulated as a system of partial integro-differential equations that describe the phenotypic evolution of cancer cells in response to dynamic variations in the spatial distribution of three abiotic factors that are key players in tumour metabolism: oxygen, glucose and lactate. The results of numerical simulations of a calibrated version of the model based on real data recapitulate the eco-evolutionary spatial dynamics of tumour cells and their adaptation to hypoxic and acidic microenvironments. Moreover, such results demonstrate how nonlinear interactions between tumour cells and abiotic factors can lead to the formation of environmental gradients which select for cells with phenotypic characteristics that vary with distance from intra-tumour blood vessels, thus promoting the emergence of intra-tumour phenotypic heterogeneity. Finally, our theoretical findings reconcile the conclusions of earlier studies by showing that the order in which resistance to hypoxia and resistance to acidity arise in tumours depend on the ways in which oxygen and lactate act as environmental stressors in the evolutionary dynamics of cancer cells.

缺氧和酸度作为环境压力源促进对具有更具攻击性表型的癌细胞的选择。因此，对推动肿瘤细胞适应低氧和酸性微环境的时空过程的更深入的理论理解可能会开辟肿瘤学和癌症治疗研究的新途径。我们提出了一个数学模型来研究缺氧和酸度对血管化肿瘤中癌细胞进化动力学的影响。该模型被表述为一个部分积分微分方程系统，描述了癌细胞的表型进化，以响应作为肿瘤代谢关键参与者的三种非生物因素的空间分布的动态变化：氧气、葡萄糖和乳酸。基于真实数据的模型校准版本的数值模拟结果概括了肿瘤细胞的生态进化空间动力学及其对缺氧和酸性微环境的适应。此外，这些结果表明肿瘤细胞和非生物因素之间的非线性相互作用如何导致环境梯度的形成，环境梯度会选择具有随与肿瘤内血管距离而变化的表型特征的细胞，从而促进肿瘤内表型异质性的出现. 最后，