Mathematical models of malignant tumour

I. V. Zhukova ; E. P. Kolpak

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 3 (2014), pp. 5-18

Résumé

The mathematical model of a malignant tumor, representing an initial-boundary problem for the system of three differential equations in partial derivatives, has been developed. In models there are three types of cells that communicate among themselves, dividing, normal and dead. The inhibiting effect of cells on each other, that the growth of continuously dividing cells is accompanied by destruction of normal cells and by formation the dead cells, is taken into account. Kinetic functions are constructed on the principle of pair interactions. In the developed model the growth of tumor cells is accompanied by displacement of normal by dead cells. In the model of immune response the estimation of the critical masses, in excess of which the decision of problem can become an increasing unrestrictedly, has been given. Mathematical analysis of stability of stationary solutions is based on the first method of Lyapunov. In the linear approximation, the estimation of the growth rate of the tumor, growing in the form of thread, has been given. It is proved that the solution of a nonlinear boundary problem can be represented as a spreading wave, the minimum velocity of wave propagation was found. The algorithm of solving a nonlinear boundary value problem for a system of differential equations based on the finite-difference approximation of differential operators has been developed. The numerical experiments were put and a comparison of their results with the results of analytical research was done. Bibliogr. 33. Il. 1.

Détail

Keywords: mathematical modeling, differential equations, boundary value problems, numerical methods, tumor.