How does the human brain eliminate waste products? This simple yet crucial question has occupied medical research for decades, in particular with regards Alzheimer's disease, whose onset is closely associated with a failure to remove the cerebral waste product amyloid-β (Aβ). Analytical and numerical modelling of the biomechanical processes underlying waste removal from the brain can play a crucial role in evaluating hypotheses and identifying its driving mechanisms. This talk describes the journey of attempting to resolve some of the questions surrounding the removal of Aβ in healthy individuals, starting from the biomedical evidence, to the analytical and numerical methods to disprove one of the most popular hypotheses and the derivation of new hypotheses. I will argue that in order to fully resolve the onset and development of neurodegenerative diseases we require fully interdisciplinary scientists working side-by-side with experimental researchers and taking into account the multiple scales involved in the cerebral waste transport processes in the human brain.