Abstract We study a system of particles in a two-dimensional geometry that move according to a reinforced random walk with transition probabilities dependent on the solutions of reaction-diffusion equations (RDEs) for the underlying fields. A birth process and a history-dependent killing process are also considered. This system models tumor-induced angiogenesis, the process of formation of blood vessels induced by a growth factor (GF) released by a tumor. Particles represent vessel tip cells, whose trajectories constitute the growing vessel network. New vessels appear and may fuse with existing ones during their evolution. Thus, the system is described by tracking the density of active tips, calculated as an ensemble average over many realizations of the stochastic process. Such density satisfies a novel discrete master equation with source and sink terms. The sink term is proportional to a space-dependent and suitably fitted killing coefficient. Results are illustrated studying two influential angiogenesis models.

中文翻译：

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描述血管生成中活跃生长的血管的积分差异主方程

摘要 我们研究了二维几何中的粒子系统，这些粒子系统根据增强的随机游走移动，其转移概率取决于基础场的反应扩散方程 (RDE) 的解。还考虑了出生过程和历史相关的杀戮过程。该系统模拟肿瘤诱导的血管生成，即由肿瘤释放的生长因子 (GF) 诱导的血管形成过程。粒子代表血管尖端细胞，其轨迹构成了不断增长的血管网络。新船只出现并可能在进化过程中与现有船只融合。因此，系统通过跟踪活动提示的密度来描述，计算为随机过程的许多实现的整体平均值。这种密度满足具有源和汇项的新型离散主方程。汇项与空间相关且适当拟合的杀伤系数成正比。说明了研究两种有影响的血管生成模型的结果。