Nonlinear two-dimensional dynamics of interacting cancer cells under extracellular field conditions

Purpose of this work is to theoretically investigate the effect of extracellular field on antiparallel and parallel transfer of interacting cancer cells depending on external parameters of extracellular matrix and parameters of metabolic potential profile in the dynamics of cancer tumor formation and development risks. Methods. In this paper, we use the quasi-classical instanton approximation method (or method of the rarefied «gas» of the instanton - antiinstanton pairs) to consider the mobility of interacting cancer cells in the 2D oscillator potential model under conditions of an external extracellular field. Results. Within framework of the proposed analytical 2D-model, it has been shown that microinteraction in the extracellular matrix of emerging cancer cells through the dynamic metabolic profile significantly affects the dynamics of the formation risks and development of a cancer tumor. It is shown that, depending on the structure of the 2D metabolic profile, a number of characteristic nonlinear features arise, such as 2D bifurcations, beats, randomization superimposed on integral dynamic curves resembling in appearance of the Gompertz function that describe probable risks of the formation and development of a cancerous tumor. Conclusion. Effect of an unstable break in the temperature dependence of the antiparallel dynamics of two cancer cells in a 2D metabolic potential has been theoretically investigated. It is shown that effect of the symmetry breaking is stable for parallel and unstable for antiparallel transfer. A complex fine structure was revealed in the bifurcation region due to fluctuations (beats) for parallel two-dimensional transfer of cancer cells. For the case of antiparallel transfer in the bifurcation region, the contribution of 4, 6, 12, etc. pairs of trajectories becomes significant, which resembles one of the possible regimes of transition to chaos. Additionally, the interaction of a pair of cells with the vibrational modes of the extracellular matrix has been investigated. Such interaction modifies significantly and differently the processes of antiparallel and parallel two-dimensional transfers of cancer cells.