Three-dimensional (3D) in vitro tumour spheroid experiments are an important tool for studying cancer progression and potential cancer drug therapies. Standard experiments involve growing and imaging spheroids to explore how different conditions lead to different rates of spheroid growth. These kinds of experiments, however, do not reveal any information about the spatial distribution of the cell cycle within the expanding spheroid. Since 2008, a new experimental technology called fluorescent ubiquitination-based cell cycle indicator (FUCCI) has enabled real-time in situ visualisation of the cell cycle progression. Observations of 3D tumour spheroids with FUCCI labelling reveal significant intratumoural structure, as the cell cycle status can vary with location. Although many mathematical models of tumour spheroid growth have been developed, none of the existing mathematical models are designed to interpret experimental observations with FUCCI labelling. In this work, we adapt the mathematical framework originally proposed by Ward and King (Math Med Biol 14:39–69, 1997. https://doi.org/10.1093/imammb/14.1.39) to produce a new mathematical model of FUCCI-labelled tumour spheroid growth. The mathematical model treats the spheroid as being composed of three subpopulations: (i) living cells in G1 phase that fluoresce red; (ii) living cells in S/G2/M phase that fluoresce green; and (iii) dead cells that are not fluorescent. We assume that the rates at which cells pass through different phases of the cell cycle, and the rate of cell death, depend upon the local oxygen concentration. Parameterising the new mathematical model using experimental measurements of cell cycle transition times, we show that the model can qualitatively capture important experimental observations that cannot be addressed using previous mathematical models. Further, we show that the mathematical model can be used to qualitatively mimic the action of anti-mitotic drugs applied to the spheroid. All software programs required to solve the nonlinear moving boundary problem associated with the new mathematical model are available on GitHub. at https://github.com/wang-jin-mathbio/Jin2021

三维 (3D) 体外肿瘤球体实验是研究癌症进展和潜在癌症药物治疗的重要工具。标准实验涉及生长和成像球体，以探索不同条件如何导致不同的球体生长速率。然而，这些类型的实验并没有揭示关于膨胀球体内细胞周期空间分布的任何信息。自 2008 年以来，一种称为基于荧光泛素化的细胞周期指示器 (FUCCI) 的新实验技术实现了细胞周期进程的实时原位可视化。对带有 FUCCI 标记的 3D 肿瘤球体的观察揭示了显着的瘤内结构，因为细胞周期状态可能随位置而变化。尽管已经开发了许多肿瘤球体生长的数学模型，没有一个现有的数学模型旨在用 FUCCI 标记来解释实验观察。在这项工作中，我们采用了 Ward 和 King 最初提出的数学框架（Math Med Biol 14:39–69, 1997. https://doi.org/10.1093/imammb/14.1.39）来产生一个新的数学模型FUCCI 标记的肿瘤球体生长。数学模型将球体视为由三个亚群组成：(i) G1 期的活细胞发出红色荧光；(ii) S/G2/M 期的活细胞发出绿色荧光；(iii) 没有荧光的死细胞。我们假设细胞通过细胞周期不同阶段的速率以及细胞死亡的速率取决于局部氧浓度。使用细胞周期转换时间的实验测量参数化新的数学模型，我们表明该模型可以定性地捕获使用以前的数学模型无法解决的重要实验观察结果。此外，我们表明该数学模型可用于定性模拟应用于球体的抗有丝分裂药物的作用。GitHub 上提供了解决与新数学模型相关的非线性移动边界问题所需的所有软件程序。在 https://github.com/wang-jin-mathbio/Jin2021 GitHub 上提供了解决与新数学模型相关的非线性移动边界问题所需的所有软件程序。在 https://github.com/wang-jin-mathbio/Jin2021 GitHub 上提供了解决与新数学模型相关的非线性移动边界问题所需的所有软件程序。在 https://github.com/wang-jin-mathbio/Jin2021