In this chapter we review a set of basic mathematical models and tools that have been widely applied to study cancer growth.We start in Section 3.1 with branching processes, simple probabilistic mean-field models for the evolution of a population. Branching processes represent the most widely used approach to model population dynamics of cancer cells. In Section 3.2 we consider probabilistic models for the evolution of mutations in cancer development. These models are sometimes related to branching processes we need to take into account for mutations in expanding clonal populations. In Section 3.3 we discuss models’ gene regulatory networks and signaling networks relevant for cancer, and in particular models for the p53 network. Mean-field models are not adequate to take into account the spatial localization of a tumor or of cancer cell population. To this end, one should introduce individual cell models in which we can follow the dynamics of a set of interacting active cells (Section 3.4). At a more coarse-grained level, the growth dynamics of a tumor can be represented by lattice cellular automata or by continuum phase-field models as discussed in Section 3.5.

Branching Processes

Branching processes (Harris, 1989; Kimmel and Axelrod, 2002) are a class of simple models that have been used extensively to model growth dynamics of stem cells (Vogel et al., 1968; Matioli et al., 1970; Potten and Morris, 1988; Clayton et al., 2007; Antal and Krapivsky, 2010; Itzkovitz et al., 2012) and cancer cells (Kimmel and Axelrod, 1991; Michor et al., 2005; Ashkenazi et al., 2008; Michor, 2008; Tomasetti and Levy, 2010; La Porta et al., 2012). Branching processes can be defined in discrete or continuous time and with evolution rules that may or may not depend on time. A detailed review of the mathematical theory of the branching process is given by Harris (1989) and applications to biology are discussed in Kimmel and Axelrod (2002).