  Glioblastoma multiforme (GBM) is a fast-growing and highly invasive brain
tumour, it tends to occur in adults between the ages of 45 and 70 and it
accounts for 52 percent of all primary brain tumours. Usually, GBMs are
detected by magnetic resonance images (MRI). Among MRI, Fluid-attenuated
inversion recovery (FLAIR) sequence produces high quality digital tumour
representation. Fast detection and segmentation techniques are needed for
overcoming subjective medical doctors (MDs) judgment. In the present
investigation, we intend to demonstrate by means of numerical experiments that
topological features combined with textural features can be enrolled for GBM
analysis and morphological characterization on FLAIR. To this extent, we have
performed three numerical experiments. In the first experiment, Topological
Data Analysis (TDA) of a simplified 2D tumour growth mathematical model had
allowed to understand the bio-chemical conditions that facilitate tumour
growth: the higher the concentration of chemical nutrients the more virulent
the process. In the second experiment topological data analysis was used for
evaluating GBM temporal progression on FLAIR recorded within 90 days following
treatment (e.g., chemo-radiation therapy - CRT) completion and at progression.
The experiment had confirmed that persistent entropy is a viable statistics for
monitoring GBM evolution during the follow-up period. In the third experiment
we had developed a novel methodology based on topological and textural features
and automatic interpretable machine learning for automatic GBM classification
on FLAIR. The algorithm reached a classification accuracy up to the 97%.
