School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK.
Math Med Biol. 2023 Dec 15;40(4):327-347. doi: 10.1093/imammb/dqad008.
We employ the multiphase, moving boundary model of Byrne et al. (2003, Appl. Math. Lett., 16, 567-573) that describes the evolution of a motile, viscous tumour cell phase and an inviscid extracellular liquid phase. This model comprises two partial differential equations that govern the cell volume fraction and the cell velocity, together with a moving boundary condition for the tumour edge, and here we characterize and analyse its travelling-wave and pattern-forming behaviour. Numerical simulations of the model indicate that patterned solutions can be obtained, which correspond to multiple regions of high cell density separated by regions of low cell density. In other parameter regimes, solutions of the model can develop into a forward- or backward-moving travelling wave, corresponding to tumour growth or extinction, respectively. A travelling-wave analysis allows us to find the corresponding wave speed, as well as criteria for the growth or extinction of the tumour. Furthermore, a stability analysis of these travelling-wave solutions provides us with criteria for the occurrence of patterned solutions. Finally, we discuss how the initial cell distribution, as well as parameters related to cellular motion and cell-liquid drag, control the qualitative features of patterned solutions.
我们采用 Byrne 等人（2003，应用数学快报，16，567-573）的多相、移动边界模型，该模型描述了运动、粘性肿瘤细胞相和无粘性细胞外液相的演变。该模型由两个偏微分方程组成，用于控制细胞体积分数和细胞速度，以及肿瘤边缘的移动边界条件，我们在这里对其进行特征描述和分析，研究其传播波和图案形成行为。该模型的数值模拟表明，可以得到图案化的解，这些解对应于高细胞密度区域和低细胞密度区域之间的多个区域。在其他参数区域中，模型的解可以发展成向前或向后传播的传播波，分别对应于肿瘤的生长或灭绝。传播波分析使我们能够找到相应的波速，以及肿瘤生长或灭绝的准则。此外，对这些传播波解的稳定性分析为图案化解的发生提供了准则。最后，我们讨论了初始细胞分布以及与细胞运动和细胞-液体阻力相关的参数如何控制图案化解的定性特征。
