Solving the Fisher Equation to Capture Tumour Behaviour for Patients with Low Grade Glioma
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Typ
Examensarbete för masterexamen
Master Thesis
Master Thesis
Program
Engineering mathematics and computational science (MPENM), MSc
Publicerad
2019
Författare
Larsson, Julia
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
Low Grade Glioma is a slow growing brain tumour, whose size is estimated using Magnetic Resonance Imaging and is treated with a combination of surgery, radiation and chemotherapy. However, cancer cells often remain after surgery leading to recurrence of the tumour and eventually death. To address this problem the Fisher equation has been considered, a partial differential equation describing how the cancer cell density changes over space and time. The Fisher equation can thus be fitted to patient data by comparing the tumour growth rate and the slope of the tumour interface, properties that is hypothesized to affect the survival time of the patients. The aim of this project is to simulate tumour growth to give additional informationaboutLowGradeGliomathatcannotbeestimateddirectlyfromMagnetic Resonance images, but requires mathematical modelling. To do this, tumour propertieshavebeenestimatedfromdataandstatisticallytestedtoinvestigatetheir potential effect on survival. Also, we have compared different methods of solving the Fisher equation and fitted the equation to data through parameter estimation techniques. The results show that two out of three investigated tumour properties have a significant effect on survival. The parameter estimation was successful and the different numerical methods for solving the Fisher equation yielded similar results for most cases. Additional information about the tumour was estimated from the Fisher equation, but the reliability of these results could be questioned. The main caveat is the simplicity of the Fisher equation and the small size of the patient data set. One solution could be to include the effect of surrounding tissue in the Fisher equation, but this requires accurate data containing many measurements for all patients. A second approach could therefore be to create a model using Non Linear Mixed Effect modelling, with the Fisher equation as the framework, in order to make the model more accurately capture the variation among patients.
Beskrivning
Ämne/nyckelord
Matematik , Mathematics