Effective particle methods for Fisher–Kolmogorov equations: Theory and applications to brain tumor dynamics

• We find a simple finite-dimensional description of the front dynamics in FK equations.
• The finite-dimensional description allows to compute tumor parameters.
• A discrete map is found describing the dynamics under radiotherapy.
• The methodology provides explicit formulae for metrics of practical relevance.

Extended systems governed by partial differential equations can, under suitable conditions, be approximated by means of sets of ordinary differential equations for global quantities capturing the essential features of the systems dynamics. Here we obtain a small number of effective equations describing the dynamics of single-front and localized solutions of Fisher–Kolmogorov type equations. These solutions are parametrized by means of a minimal set of time-dependent quantities for which ordinary differential equations ruling their dynamics are found. A comparison of the finite dimensional equations and the dynamics of the full partial differential equation is made showing a very good quantitative agreement with the dynamics of the partial differential equation. We also discuss some implications of our findings for the understanding of the growth progression of certain types of primary brain tumors and discuss possible extensions of our results to related equations arising in different modeling scenarios.