Mathematical modelling of the development of a cylindrical tumour

Abstract

The article deals with interaction of tumour cells and leucocytes in the cylindrical cavities. This type of interaction is typical in the cases of development of a tumour in the intestine, blood vessel or in a bone cavity. Two cases are separated: the case of soft and hard tumour. In the case of a solid tumour, leucocytes can interact only with the surface cells of the tumour. This type of interaction is described by the system of two nonlinear first degree differential equations. The expressions of stationary points are obtained and analysis of their stability is performed. In the case of a soft tumour the system of two partial differential equations with first order derivatives and initial and boundary conditions is proposed. An algorithm for computing the numeric solution of the mathematical model is applied. In this case the diffusion of leucocytes and their ability to reach the tumour cells in the whole volume of the tumour is included. The algorithm is constructed and the system is solved numerically. Bifurcation curve is obtained. It separates two qualitatively different areas on the two parameter plane. Under the same initial parameters in the first area development of the tumour cells cannot be stopped, whereas in the second area leukocytes defeat the tumour cells.DOI: http://dx.doi.org/10.15181/csat.v2i1.183