Mathematical modelling of epithelial tissue dynamics

The rapid development of computer technologies and high-performance computing systems has led to the possibility to simulate directly the biomechanical properties of cell tissue. This simulation reproduces both the averaged dynamics of the tissue considered as a continuous medium and the behavior of each cell. In this paper, we propose a mathematical model of epithelial tissue, which is considered to be two-dimensional. In this case, the shape of the basement membrane, on which the epithelium lies, generally, may have a topology that is more complex than the plane. The model is discrete since the tissue consists of cells, each of which evolves according to its scenario. Each cell is a polygon, the number of vertices and shape of which can change during evolution. The model includes two important processes that mimic the properties of real cells. The first one is mitotic cell division, the algorithm of which is written in such a way that the new cell inherits all the properties of the mother cell. Another important process is cell intercalation, which makes the epithelium a mobile elastic medium adapting under the influence of internal and external influences. For each vertex of the cell, we write an equation of motion based on the elastic potential energy. Since the cell resists deviation from the average volume and excessive deformation of the shape, the epithelium as a whole tends to find a state corresponding to a minimum of potential energy. Even though the cellular tissue allows internal movement of elements, it is, generally, a highly dissipative medium. Thus, tissue biomechanics should obey rather Aristotelian dynamics. The model allows a simple generalization to the case of feedback between the biomechanical and chemical properties of the medium (for example, the processes of gene regulation in cells leading to chemoelasticity), the introduction of several competing species of cells (for example, the occurrence of a cancerous tumor), three-dimensional cell tissue, and so on. Specific examples of modelling the dynamic behavior of epithelial tissue are given.