Semigroup Approaches of Cell Proliferation Models

In this chapter, we provide a mathematical framework for the qualitative analysis of the behavior of two types of cell cycle proliferation models. The first type uses an integral equation to model the dynamics of the cell density, and the second type uses partial differential equations for modeling. As examples of the second type, we consider a multi-transition phase cell proliferation model and also a two-compartment cell cycle model with proliferating and quiescent cell population. The framework considers two approaches that both are based on the theory of semigroups of operators. The first approach is based on the use of the theory developed for the class of translation semigroups that are associated with a core operator ϕ and are solutions of equations of the type m(t) = ϕ(m t). The second approach uses tools based on perturbation theory and the duality method of “suns and stars” to transform the models to abstract integral equations. Within this framework, properties for the cell cycle models such as existence, uniqueness, positivity, compactness, and spectral properties of their associated solution semigroups are derived in order to conclude the asynchronous exponential growth property for the models. The framework makes it possible, based on each of the two considered approaches, to conclude such properties by only using assumptions on model parameters defining their associated core operators ϕ.