An axisymmetric model of the dynamics of cell tissue density distribution, taking into account the effects of chemotaxis and extracellular matrix deformation

In this paper, an axisymmetric mathematical model of cellular tissue dynamics is considered. The first part describes the initial model, the main provisions and the formulation of the problem in cylindrical coordinates. In the second part, the resulting system is reduced to a dimensionless form, investigated numerically, and the simulation results are presented. It is postulated that cellular tissue is a three-phase medium that consists of a solid skeleton (which is an extracellular matrix), cells, and extracellular fluid. Additionally, the presence of nutrients in the tissue is taken into account. The model is based on the equations of conservation of mass, taking into account mass exchange, the equations of conservation of momentum for each phase, as well as the diffusion equation for nutrients. The equation describing the cellular phase also takes into account the term describing the chemical effect on the tissue, which is called chemotaxis – the movement of cells caused by the concentration gradient of chemicals. The initial system of equations reduces to a system of three equations for finding porosity, cell saturation, and nutrient concentration. These equations are complemented by initial and boundary conditions. In the field of modeling, the presence of a vessel of a certain radius is taken into account, which is modeled as a source of nutrients, as well as the flow of cell concentration at the vessel boundary is zero. At the initial moment of time, two types of conditions are considered: the first is the condition of constant concentration of the cellular phase, the second is the presence of focal areas of increased concentration of the cellular phase. In both cases, the conditions for the matrix and extracellular fluid are the same. As a result of modeling, it was revealed that chemotaxis has a significant effect on tissue growth. In the absence of chemotaxis, the compaction zone extends to the entire modeling area, but with an increase in the effect of chemotaxis on the tissue, a degradation area is formed in which the cell concentration becomes lower than the initial one.