Motivated by the widespread use of hybrid-discrete cellular automata in modeling cancer, we study two simple growth models on the two-dimensional lattice that incorporate a nutrient, assumed to be oxygen. In the first model, the oxygen concentration u(x, t) is computed based on the geometry of the growing blob, while in the second one, u(x, t) satisfies a reaction–diffusion equation. A threshold θ value exists such that cells give birth at rate β(u(x, t) − θ)+ and die at rate δ(θ − u(x, t)+. In the first model, a phase transition was found between growth as a solid blob and “fingering” at a threshold θc = 0.5, while in the second case, fingering always occurs, i.e., θc = 0.

中文翻译：

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随机增长模型中的指法

受到混合离散元胞自动机在癌症建模中的广泛使用的推动，我们研究了二维晶格上的两个简单生长模型，这些模型包含了一种假定为氧气的养分。在第一个模型中，氧浓度 u(x, t) 是根据生长斑点的几何形状计算的，而在第二个模型中，u(x, t) 满足反应扩散方程。存在一个阈值 θ 值，使得细胞以 β(u(x, t) − θ)+ 的速率出生并以 δ(θ − u(x, t)+ 的速率死亡。在第一个模型中，发现了相变在 θc = 0.5 的阈值下生长为实体斑点和“指法”之间，而在第二种情况下，指法总是发生，即 θc = 0。