肿瘤是一种复杂且具有侵袭性的疾病，给健康带来重大挑战。了解其进展背后的细胞机制对于开发有效的治疗方法至关重要。在这项研究中，我们开发了一个新颖的数学框架来研究细胞可塑性和异质性在肿瘤进展中的作用。通过利用时间单细胞数据，我们提出了一种反应-对流-扩散模型，可以有效捕获肿瘤微环境中肿瘤细胞和巨噬细胞的时空动态。通过理论分析，我们获得了脉搏波速度的估计并分析了齐次稳态解的稳定性。值得注意的是，我们使用 AddModuleScore 函数来量化细胞可塑性。我们方法的亮点之一是引入脉冲波速度作为定量测量来精确测量细胞表型转变的速率，以及高可塑性细胞状态/低可塑性细胞状态比率的新颖实现作为肿瘤恶性程度的指标。此外，分叉分析揭示了肿瘤细胞群的复杂动态。我们的广泛分析表明，表型转变率的增加与恶性程度的升高相关，这归因于肿瘤探索更广泛表型空间的能力。该研究还探讨了肿瘤细胞的增殖率和死亡率、表型对流速度和表型转变阶段的中点如何影响肿瘤细胞表型转变的速度和进展为腺癌。这些见解和定量测量可以帮助指导制定靶向治疗策略，以有效调节细胞可塑性并控制肿瘤进展。© 2024。作者获得 Springer-Verlag GmbH 德国（Springer Nature 旗下公司）的独家许可。

Tumor is a complex and aggressive type of disease that poses significant health challenges. Understanding the cellular mechanisms underlying its progression is crucial for developing effective treatments. In this study, we develop a novel mathematical framework to investigate the role of cellular plasticity and heterogeneity in tumor progression. By leveraging temporal single-cell data, we propose a reaction-convection-diffusion model that effectively captures the spatiotemporal dynamics of tumor cells and macrophages within the tumor microenvironment. Through theoretical analysis, we obtain the estimate of the pulse wave speed and analyze the stability of the homogeneous steady state solutions. Notably, we employe the AddModuleScore function to quantify cellular plasticity. One of the highlights of our approach is the introduction of pulse wave speed as a quantitative measure to precisely gauge the rate of cell phenotype transitions, as well as the novel implementation of the high-plasticity cell state/low-plasticity cell state ratio as an indicator of tumor malignancy. Furthermore, the bifurcation analysis reveals the complex dynamics of tumor cell populations. Our extensive analysis demonstrates that an increased rate of phenotype transition is associated with heightened malignancy, attributable to the tumor's ability to explore a wider phenotypic space. The study also investigates how the proliferation rate and the death rate of tumor cells, phenotypic convection velocity, and the midpoint of the phenotype transition stage affect the speed of tumor cell phenotype transitions and the progression to adenocarcinoma. These insights and quantitative measures can help guide the development of targeted therapeutic strategies to regulate cellular plasticity and control tumor progression effectively.© 2024. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.