Aristotelous Andreas C, Durrett Richard
Department of Mathematics, Duke U., Box 90320, Durham, NC 27708-0320.
Exp Math. 2014;23(4):465-474. doi: 10.1080/10586458.2014.947053. Epub 2014 Nov 14.
Motivated by the widespread use of hybrid-discrete cellular automata in modeling cancer, two simple growth models are studied on the two dimensional lattice that incorporate a nutrient, assumed to be oxygen. In the first model the oxygen concentration (, ) is computed based on the geometry of the growing blob, while in the second one (, ) satisfies a reaction-diffusion equation. A threshold value exists such that cells give birth at rate ((, ) - ) and die at rate ( - (, ). In the first model, a phase transition was found between growth as a solid blob and "fingering" at a threshold = 0.5, while in the second case fingering always occurs, i.e., = 0.
受混合离散细胞自动机在癌症建模中广泛应用的启发，我们在二维晶格上研究了两个简单的生长模型，该模型纳入了一种假定为氧气的营养物质。在第一个模型中，氧气浓度（，）是根据生长的细胞团的几何形状计算得出的，而在第二个模型中，（，）满足一个反应扩散方程。存在一个阈值，使得细胞以速率（（，） - ）产生，并以速率（ - （，））死亡。在第一个模型中，发现在阈值 = 0.5 时，生长从作为固体团块转变为“指状生长”，而在第二种情况下，总是会出现指状生长，即 = 0。
