Brain venous haemodynamics, neurological diseases and mathematical modelling. A review

Behind Medicine (M) is Physiology (P), behind Physiology is Physics (P) and behind Physics is always Mathematics (M), for which I expect that the symmetry of the quadruplet MPPM will be compatible with the characteristic bias of hyperbolic partial differential equations, a theme of this paper. I start with a description of several idiopathic brain pathologies that appear to have a strong vascular dimension, of which the most prominent example considered here is multiple sclerosis, the most common neurodegenerative, disabling disease in young adults. Other pathologies surveyed here include retinal abnormalities, Transient Global Amnesia, Transient Monocular Blindness, Ménière's disease and Idiopathic Parkinson's disease. It is the hypothesised vascular aspect of these conditions that links medicine to mathematics, through fluid mechanics in very complex networks of moving boundary blood vessels. The second part of this paper is about mathematical modelling of the human cardiovascular system, with particular reference to the venous system and the brain. A review of a recently proposed multi-scale mathematical model then follows, consisting of a one-dimensional hyperbolic description of blood flow in major arteries and veins, coupled to a lumped parameter description of the remaining main components of the human circulation. Derivation and analysis of the hyperbolic equations is carried out for blood vessels admitting variable material properties and with emphasis on the venous system, a much neglected aspect of cardiovascular mathematics. Veins, unlike their arterial counterparts, are highly deformable, even collapsible under mild physiological conditions. We address mathematical and numerical challenges. Regarding the numerical analysis of the hyperbolic PDEs, we deploy a modern non-linear finite volume method of arbitrarily high order of accuracy in both space and time, the ADER methodology. In vivo validation examples and brain haemodynamics computations are shown. We also point out two, preliminary but important new findings through the use of mathematical models, namely that extracranial venous strictures produce chronic intracranial venous hypertension and that augmented pressure increases the blood vessel wall permeability.