In the study of cancer evolution and therapeutic strategies, scientific evidence shows that a key dynamics lies in the tumor-environment interaction. In particular, oxygen concentration plays a central role in the determination of the phenotypic heterogeneity of cancer cell populations, whose qualitative and geometric characteristics are predominant factors in the occurrence of relapses and failure of eradication. We propose a mathematical model able to describe the eco-evolutionary spatial dynamics of tumour cells in their adaptation to hypoxic microenvironments. As a main novelty with respect to the existing literature, we combine a phenotypic indicator reflecting the experimentally-observed metabolic trade-off between the hypoxia-resistance ability and the proliferative potential with a 2d geometric domain, without the constraint of radial symmetry. The model is settled in the mathematical framework of phenotype-structured population dynamics and it is formulated in terms of systems of coupled non-linear integro-differential equations. The computational outcomes demonstrate that hypoxia-induced selection results in a geometric characterization of phenotypic-defined tumour niches that impact on tumour aggressiveness and invasive ability. Furthermore, results show how the knowledge of environmental characteristics provides a predictive advantage on tumour mass development in terms of size, shape, and composition.

翻译：在癌症进化与治疗策略研究中，科学证据表明肿瘤与微环境相互作用是关键动力学过程。其中，氧浓度在决定癌细胞表型异质性中起核心作用，而细胞群的定性与几何特征是复发和根治失败的主导因素。我们提出一个数学模型，能够描述肿瘤细胞在适应低氧微环境过程中的生态进化空间动力学。与现有文献相比，主要创新在于：将反映实验观察到的低氧抵抗能力与增殖潜力间代谢权衡的表型指标，与二维几何域相结合，且不受径向对称性约束。该模型基于表型结构群体动力学的数学框架，采用耦合非线性积分微分方程组表述。计算结果表明，低氧诱导的选择导致表型定义的肿瘤微环境产生几何特征，进而影响肿瘤侵袭性与浸润能力。此外，结果展示了环境特征知识如何为预测肿瘤肿块的大小、形态和组成提供优势。