Strategies of applied mathematics towards an immuno-mathematical theory on tumors and immune system interactions.

*(English)*Zbl 0947.92014From the introduction: This paper is divided into eight sections. The first section is the introduction which defines the aims of the paper and its contents. Section 2 analyzes the interplay between immunology and mathematics and defines the role of applied mathematics to support both, the developments of theories and simulations of real systems. Section 3 deals with an introduction to the necessary steps towards a mathematical theory. We start with the definition of the natural scales of the system from the subcellular scale to the macroscopic behavior. Section 4 deals with the selection of the cell populations which are involved in the game of life and death between tumor cells and the immune system. Section 5 deals with the modelling of the cellular interactions. Section 6 proposes a mathematical model for the description of the behavior of the system at the cellular scale. This section also deals with simulation problems. Section 7 analyzes the link between the cellular description and macroscopic observations. This section also provides a discussion of simplified models, which can either be phenomenological competition models resembling population dynamics or discretizations of continuous models. Section 8 deals with the strategy to be developed in order to obtain an immuno-mathematical theory starting from the analysis of models developed at a cellular level. This section is also devoted to the discussion and to the indication of research perspectives.

##### MSC:

92C50 | Medical applications (general) |