On modeling the immune competition with Darwinian dynamics

Mathematical and computational models are increasingly used in this century to help modeling of living systems. Mathematical modeling presents many methods for studying and analyzing the behavior of biological systems, in particular, cellular systems. As Bellomo (2008), Bellouquid and Delitala (2006), suggest " The modeling of living systems is not an easy task, it requests technically complex mathematical methods to deal with the inner complexity of biological systems which exhibit features and behaviors very different from those of inert matter". The mathematical approach used in this dissertation is based on the Kinetic Theory of Active Particles (KTAP), that has been specifically developed to model a variety of complex systems, Bellomo et al. (2009, 2010); Bellomo and Forni (2006); Brazzoli et al. (2010). The aim of this thesis is the detailed mathematical study of the immune competition with Darwinian dynamics, and its modeling and simulations. In particular, the competition between the immune system and cells carrier of a pathological state. As a theoretical background, we say that the immune system has the ability to activate a defence of immune cells against infectious agents and mutated cells. The derivation of models must deal with the analysis of microscopic interactions, due to the presence of proliferation, mutations and/or destruction of cells. The thesis is made up of four chapters: Chapter 1 is the starting point to enlightening the topic under study. This chapter is focused on a general presentation of the mathematical tools of the kinetic theory of active particles and presents a concise description of the mathematical tools "concepts" and "definitions" used in this thesis. In Chapter 2, we provide a concise introduction to the immune system and cancer cells. This chapter provides a phenomenological description of some aspects of the biology of the system we are dealing with, while the following chapters are based on studying of a model describing the competition between the immune system and cancer cells. Chapter 3, deals with the modeling of interactions between the immune system and cancer cells. The model was proposed in the paper by Bellouquid et al. (2013) Bellouquid et al. (2013), which assumed discrete values of the activity of cancer and immune cells. Further, in this chapter, we have made a number of simulations with the aim to investigate how the state of the various cell populations evolves in time depending on the choice of the free parameters. In addition, we present a proposal for modify the parameter that characterizes the probability density function. In Chapter 4, we present a generalization of the model proposed by Bellouquid et al. (2013) Bellouquid et al. (2013). The model is obtained by replacing the discrete activity values by continuous values, choosing suitable relations to describe the interactions between active particles. This chapter also deals with the derivation from the model at the cellular scale of a model at the macroscopic scale, that considers as variables quantities obtained by local averages of the microscopic state. In the introduction of each chapter, we explain the specific motivations for such analysis, and we discuss the state of the art in research. During each chapter, we make some remarks, that are recalled in the final discussions of this thesis.