Fedotov Sergei, Iomin Alexander, Ryashko Lev
School of Mathematics, The University of Manchester, Manchester M60 1QD, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Dec;84(6 Pt 1):061131. doi: 10.1103/PhysRevE.84.061131. Epub 2011 Dec 19.
Proliferation and migration dichotomy of the tumor cell invasion is examined within two non-Markovian models. We consider the tumor spheroid, which consists of the tumor core with a high density of cells and the outer invasive zone. We distinguish two different regions of the outer invasive zone and develop models for both zones. In model I we analyze the near-core-outer region, where biased migration away from the tumor spheroid core takes place. We suggest non-Markovian switching between the migrating and proliferating phenotypes of tumor cells. Nonlinear master equations for mean densities of cancer cells of both phenotypes are derived. In anomalous switching case we estimate the average size of the near-core-outer region that corresponds to sublinear growth (r(t)) ~ t(μ) for 0 < μ < 1. In model II we consider the outer zone, where the density of cancer cells is very low. We suggest an integrodifferential equation for the total density of cancer cells. For proliferation rate we use the classical logistic growth, while the migration of cells is subdiffusive. The exact formulas for the overall spreading rate of cancer cells are obtained by a hyperbolic scaling and Hamilton-Jacobi techniques.
在两个非马尔可夫模型中研究了肿瘤细胞侵袭的增殖与迁移二分法。我们考虑肿瘤球体，它由细胞密度高的肿瘤核心和外部侵袭区域组成。我们区分了外部侵袭区域的两个不同区域，并为这两个区域建立了模型。在模型I中，我们分析了近核心外部区域，肿瘤细胞从肿瘤球体核心发生有偏迁移。我们提出肿瘤细胞迁移和增殖表型之间的非马尔可夫切换。推导了两种表型癌细胞平均密度的非线性主方程。在异常切换情况下，我们估计了近核心外部区域的平均大小，对于0 < μ < 1，该区域对应于次线性增长（r(t)）~ t(μ)。在模型II中，我们考虑癌细胞密度非常低的外部区域。我们提出了一个关于癌细胞总密度的积分微分方程。对于增殖率，我们使用经典的逻辑斯蒂增长，而细胞迁移是次扩散的。通过双曲缩放和哈密顿-雅可比技术获得了癌细胞总体扩散率的精确公式。
