On the Williams-Bjerknes tumour growth model: II
On the Williams-Bjerknes tumour growth model: II
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关于 Williams-Bjerknes 肿瘤生长模型：II
DOI：
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发表时间：
1980
影响因子：
0.8
通讯作者：
D. Griffeath
中科院分区：
文献类型：
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作者：
M. Bramson;D. Griffeath
摘要 Williams 和 Bjerknes 于 1972 年引入了癌细胞扩散的随机模型。正常和异常（癌）细胞都位于平面晶格上。每次细胞分裂时，一个子细胞保持原状，而另一个子细胞则篡夺邻居的位置；异常细胞的繁殖速度比正常细胞快。我们对该系统的长期行为感兴趣。我们在“关于 Williams-Bjerknes 肿瘤生长模型 I”(1) 中表明，如果肿瘤永远存活，它最终将包含一个半径线性扩展的球。这里表明，扩张率实际上是线性的，并且感染区域具有由某种（未知）范数给出的渐近形状。为了证明肿瘤包含一个半径线性扩展的球，我们应用了（1）相互作用粒子系统领域常见的某些技术；特别是，我们利用了某些嵌入在我们模型的双过程中的辅助马尔可夫链。这里应用了不同的技术。我们表明，不同位点的首次感染时间几乎是次加性的，并且我们表现出感染时间的各种规律性特性，例如它们的矩的界限。 Williams-Bjerknes 过程的这些属性允许人们遵循理查森在（7）中规定的轮廓，他在其中表明这样的过程确实表现出线性增长，其形状由范数规定。
Abstract Williams and Bjerknes introduced in 1972 a stochastic model for the spread of cancer cells. Cells, normal and abnormal (cancerous), are situated on a planar lattice. With each cellular division, one daughter cell stays put, while the other usurps the position of a neighbour; abnormal cells reproduce at a faster rate than normal cells. We are interested in the long-term behaviour of this system. We showed in ‘On the Williams-Bjerknes Tumour Growth Model I’ (1) that, provided it lives forever, the tumour will eventually contain a ball of linearly expanding radius. Here it is shown that the rate of expansion is actually linear, and that the region of infection has an asymptotic shape which is given by some (unknown) norm. To demonstrate that the tumour contains a ball of linearly expanding radius, we applied in (1) certain techniques common to the field of interacting particle systems; in particular we made use of certain auxiliary Markov chains which are imbedded in the dual processes of our model. Here different techniques are applied. We show that the first infection times of different sites are almost subadditive, and we exhibit various regularity properties of the infection times such as bounds on their moments. These properties of the Williams-Bjerknes process allow one to follow the outline prescribed by Richardson in (7), where he showed that such a process does indeed exhibit a linear growth whose shape is prescribed by a norm.