肿瘤生长模型：一个简单的基于个体的模型及其分析
Modeling tumor growth: a simple individual-based model and its analysis
摘要：
恶性肿瘤的发生和发展是一个复杂的现象，具有决定其长期行为的关键阶段。这一现象在数学上是通过各种模型来描述的：从简单的启发式模型到采用随机过程的模型。在本章中，我们通过分析一个简单的基于个体的模型来讨论这种建模的一些方面，在该模型中，肿瘤细胞被表示为在$\mathbf{R}_{+}:=[0,+\infty)$中以单位速度向原点漂移的点粒子。在原点，它们中的每一个都分裂成两个新的粒子，它们立即出现在$\mathbf{R}_{+}$中的随机位置。在漂移过程中，粒子在分裂之前会随机死亡。在该模型中，给定细胞的性状$x\in \mathbf{R}_{+}$对应于其分裂的时间，并且死亡由治疗因素引起。在此基础上，我们展示了如何推导出一个条件-涉及治疗相关的死亡率和细胞周期分布参数-在此条件下，肿瘤大小在时间上保持有界，这实际上意味着与疾病作斗争。
英文摘要：
Initiation and development of a malignant tumor is a complex phenomenon that has critical stages determining its long time behavior. This phenomenon is mathematically described by means of various models: from simple heuristic models to those employing stochastic processes. In this chapter, we discuss some aspects of such modeling by analyzing a simple individual-based model, in which tumor cells are presented as point particles drifting in $\mathbf{R}_{+}:=[0,+\infty)$ towards the origin with unit speed. At the origin, each of them splits into two new particles that instantly appear in $\mathbf{R}_{+}$ at random positions. During their drift the particles are subject to a random death before splitting. In this model, trait $x\in \mathbf{R}_{+}$ of a given cell corresponds to time to its division and the death is caused by therapeutic factors. On its base we demonstrate how to derive a condition -- involving the therapy related death rate and cell cycle distribution parameters -- under which the tumor size remains bounded in time, which practically means combating the disease.
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