Understanding the complex dynamics of tumor growth to develop more efficient therapeutic strategies is one of the most challenging problems in biomedicine. Three-dimensional (3D) tumor spheroids, reflecting avascular microregions within a tumor, are an advanced in vitro model system to assess the curative effect of combinatorial radio(chemo)therapy. Tumor spheroids exhibit particular crucial pathophysiological characteristics such as a radial oxygen gradient that critically affect the sensitivity of the malignant cell population to treatment. However, spheroid experiments remain laborious, and determining long-term radio(chemo)therapy outcomes is challenging. Mathematical models of spheroid dynamics have the potential to enhance the informative value of experimental data, and can support study design; however, they typically face one of two limitations: while non-spatial models are computationally cheap, they lack the spatial resolution to predict oxygen-dependent radioresponse, whereas models that describe spatial cell dynamics are computationally expensive and often heavily parameterized, impeding the required calibration to experimental data. Here, we present an effectively one-dimensional mathematical model based on the cell dynamics within and across radial spheres which fully incorporates the 3D dynamics of tumor spheroids by exploiting their approximate rotational symmetry. We demonstrate that this radial-shell (RS) model reproduces experimental spheroid growth curves of several cell lines with and without radiotherapy, showing equal or better performance than published models such as 3D agent-based models. Notably, the RS model is sufficiently efficient to enable multi-parametric optimization within previously reported and/or physiologically reasonable ranges based on experimental data. Analysis of the model reveals that the characteristic change of dynamics observed in experiments at small spheroid volume originates from the spatial scale of cell interactions. Based on the calibrated parameters, we predict the spheroid volumes at which this behavior should be observable. Finally, we demonstrate how the generic parameterization of the model allows direct parameter transfer to 3D agent-based models.
理解肿瘤生长的复杂动态以制定更有效的治疗策略，是生物医学领域最具挑战性的问题之一。三维肿瘤球体作为反映肿瘤内无血管微区域的先进体外模型系统，可用于评估联合放（化）疗的疗效。肿瘤球体展现出关键的病理生理学特征，例如径向氧梯度会显著影响恶性细胞群体对治疗的敏感性。然而，球体实验仍较为繁琐，且确定长期放（化）疗效果具有挑战性。球体动态数学模型有望提升实验数据的信息价值，并支持研究设计；但这类模型通常面临两大局限：非空间模型虽计算成本低，却缺乏预测氧依赖性放射反应的空间分辨率；而描述空间细胞动态的模型则计算成本高昂且参数化程度高，难以实现与实验数据所需的校准。本文提出一种基于径向球体内外细胞动态的有效一维数学模型，通过利用肿瘤球体的近似旋转对称性，完整呈现其三维动态特性。研究证明，该径向壳层模型能够复现多种细胞系在有/无放疗条件下的实验性球体生长曲线，其表现与已发表的模型（如三维基于代理的模型）相当或更优。值得注意的是，该模型具有足够高的计算效率，可根据实验数据在先前报道和/或生理合理范围内实现多参数优化。模型分析表明，实验中观察到的球体体积较小时动态特征变化源于细胞相互作用的空间尺度。基于校准参数，我们预测了这种行为可被观测到的球体体积范围。最后，我们展示了该模型的通用参数化如何实现参数向三维基于代理模型的直接转移。
Efficient Radial-Shell Model for 3D Tumor Spheroid Dynamics with Radiotherapy
肿瘤(癌症)患者之家