SIAM Journal on Applied Mathematics, Volume 81, Issue 3, Page 1052-1076, January 2021.
At the continuous level, we consider two types of tumor growth models: the cell density model, based on the fluid mechanical construction, is more favorable for scientific interpretation and numerical simulations, and the free boundary model, as the incompressible limit of the former, is more tractable when investigating the boundary propagation. In this work, we aim to investigate the boundary propagation speeds in those models based on asymptotic analysis of the free boundary model and efficient numerical simulations of the cell density model. We derive, for the first time, some analytical solutions for the free boundary model with pressure jumps across the tumor boundary in multidimensions with finite tumor sizes. We further show that in the large radius limit, the analytical solutions to the free boundary model in one and multiple spatial dimensions converge to traveling wave solutions. The convergence rate in the propagation speeds are algebraic in multidimensions as opposed to the exponential convergence in one dimension. We also propose an accurate front capturing numerical scheme for the cell density model, and extensive numerical tests are provided to illustrate the analytical findings.

中文翻译：

---

了解肿瘤生长模型中的边界传播速度

SIAM 应用数学杂志，第 81 卷，第 3 期，第 1052-1076 页，2021 年 1 月。
在连续层面，我们考虑两种类型的肿瘤生长模型：基于流体力学结构的细胞密度模型更利于科学解释和数值模拟，自由边界模型作为前者的不可压缩极限，在研究边界传播时更容易处理。在这项工作中，我们旨在基于自由边界模型的渐近分析和细胞密度模型的有效数值模拟来研究这些模型中的边界传播速度。我们首次推导出了自由边界模型的一些解析解，其中压力跳跃在具有有限肿瘤大小的多维中跨越肿瘤边界。我们进一步表明，在大半径极限下，一维和多维空间中自由边界模型的解析解收敛到行波解。与一维指数收敛相反，传播速度的收敛速度在多维中是代数的。我们还为细胞密度模型提出了一个准确的前端捕获数值方案，并提供了广泛的数值测试来说明分析结果。