Fluctuaciones aleatorias en la acción quimioterapéutica sobre tumores cancerosos

Abstract

A small patch of a tumor is considered, so that lymphocytes and cancerigenic cell densities are independent of the spatial position. So, a non spatial model with random transition birth and death rates are treated through one-step processes and such that chemotherapy on the physical system is included. Van Kampen expansion is used to separate a macroscopic and a microscopic part. The first one is described through a set of two nonlinear and coupled ordinary differential equations, and the microscopic part is described by using a Fokker-Planck equation. Evolution on time of the mean fluctuations and autocorrelation functions of noise are linear non autonomous systems. The macroscopic part is studied numerically, so that two basins are found, one of fatal results and other of healthy patient. Jacobian matrix has negative eigenvalues, so that there are stable attractor points inside each basin. When a chemotherapy parameter is increased, the final macroscopic state is moved from fatal to healthy basin. While macroscopic stability is found, microscopic results are very different and this is seen by studying the asymptotic behavior of the random fluctuations. This is done by evaluating the eigenvalues of the involved matrices and it is found that random fluctuations has unbounded standard deviations, suggesting that disease could appear again.