Modeling tumor growth: a simple individual-based model and its analysis

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.