Recent experimental work has revealed that interstitial fluid flow can mobilize two types of tumor cell migration mechanisms. One is a chemotactic-driven mechanism where chemokine (chemical component) bounded to the extracellular matrix (ECM) is released and skewed in the flow direction. This leads to higher chemical concentrations downstream which the tumor cells can sense and migrate toward. The other is a mechanism where the flowing fluid imposes a stress on the tumor cells which triggers them to go in the upstream direction. Researchers have suggested that these two migration modes possibly can play a role in metastatic behavior, i.e., the process where tumor cells are able to break loose from the primary tumor and move to nearby lymphatic vessels. In Waldeland and Evje (J Biomech 81:22–35, 2018), a mathematical cell–fluid model was put forward based on a mixture theory formulation. It was demonstrated that the model was able to capture the main characteristics of the two competing migration mechanisms. The objective of the current work is to seek deeper insight into certain qualitative aspects of these competing mechanisms by means of mathematical methods. For that purpose, we propose a simpler version of the cell–fluid model mentioned above but such that the two competing migration mechanisms are retained. An initial cell distribution in a one-dimensional slab is exposed to a constant fluid flow from one end to the other, consistent with the experimental setup. Then, we explore by means of analytical estimates the long-time behavior of the two competing migration mechanisms for two different scenarios: (i) when the initial cell volume fraction is low and (ii) when the initial cell volume fraction is high. In particular, it is demonstrated in a strict mathematical sense that for a sufficiently low initial cell volume fraction, the downstream migration dominates in the sense that the solution converges to a downstream-dominated steady state as time elapses. On the other hand, with a sufficiently high initial cell volume fraction, the upstream migration mechanism is the stronger in the sense that the solution converges to an upstream-dominated steady state.

中文翻译：

---

间质液流驱动的两种竞争性癌细胞迁移机制的数学分析

最近的实验工作表明，组织液流动可以动员两种类型的肿瘤细胞迁移机制。一种是趋化驱动的机制，其中结合到细胞外基质（ECM）的趋化因子（化学成分）被释放并在流动方向上偏斜。这导致肿瘤细胞可以感知并向其迁移的下游更高的化学浓度。另一个是一种机制，其中流动的流体在肿瘤细胞上施加应力，从而触发它们向上游方向移动。研究人员建议，这两种迁移模式可能在转移行为中起作用，即肿瘤细胞能够从原发肿瘤中脱离出来并转移到附近的淋巴管的过程。在Waldeland和Evje（J Biomech 81：22–35，2018），提出了基于混合理论公式的数学细胞-流体模型。结果表明，该模型能够捕获两种竞争性迁移机制的主要特征。当前工作的目的是通过数学方法寻求对这些竞争机制的某些定性方面的更深入了解。为此，我们提出了上述细胞-流体模型的简化版本，但保留了两个相互竞争的迁移机制。与实验设置一致，一维平板中的初始细胞分布暴露于从一端到另一端的恒定流体流。然后，我们通过分析估计的方式探索两种竞争迁移机制在两种不同情况下的长期行为：（i）当初始细胞体积分数低时，和（ii）当初始细胞体积分数高时。特别地，从严格的数学意义上证明，对于足够低的初始细胞体积分数，下游迁移占主导地位，因为随着时间的流逝，溶液收敛于下游主导的稳态。另一方面，在初始细胞体积分数足够高的情况下，从溶液收敛到上游为主的稳态的意义上讲，上游迁移机制更强。下游迁移占主导地位，因为随着时间的流逝，解决方案收敛到下游主导的稳态。另一方面，在初始细胞体积分数足够高的情况下，从溶液收敛到上游为主的稳态的意义上讲，上游迁移机制更强。下游迁移占主导地位，因为随着时间的流逝，解决方案收敛到下游主导的稳态。另一方面，在初始细胞体积分数足够高的情况下，从溶液收敛到上游为主的稳态的意义上讲，上游迁移机制更强。