Matematisk modellering av hjärntumörer Dataanalys och utvidgning av modell av glioblastom

Abstract

Glioblastoma may because of its aggressive behaviour be modelled mathematically. A frequently used model for simulation of cancer growth is based on the Fisher-equation @tu = Dr2u + u(1 u). One such model for specific glioblastoma growth based on phenotypic switching was postulated 2012, and later analysed 2016 by mathematician Philip Gerlee and cancer biologist Sven Nelander. Via the addition of a linear term corresponding to apoptosis of the cancer due to treatment on the proliferating cells, said model is adjusted to produce slightly different behaviour characterised by its underlying parameters. The solution of this new system is analysed numerically and methods to calculate the solution's corresponding propagation speed and slope has been found. Furthermore a perturbation approach is applied to find a closed form analytic approximation of the solution, while phase space analysis allows for the discovery of a limiting wave speed. We bring these results to scrutiny under the lens of a set of real world data and find that an optimal set of parameters provides for a good fit between model and reality. The fit foremost indicates that the parameters for over 60% of the total number of patients decreases or are kept constant over the treatment phases.