A holistic multi-scale mathematical model of the murine extracellular fluid systems and study of the brain interactive dynamics

Recent advances in medical science regarding the interaction and functional role of fluid compartments in the central nervous system have attracted the attention of many researchers across various disciplines. Neurotoxins are constantly cleared from the brain parenchyma through the intramural periarterial drainage system, glymphatic system and meningeal lymphatic system. Impairment of these systems can potentially contribute to the onset of neurological disorders. The goal of this thesis is to contribute to the understanding of brain fluid dynamics and to the role of vascular pathologies in the context of neurological disorders. To achieve this goal, we designed the first multi-scale, closed-loop mathematical model of the murine fluid system, incorporating: heart dynamics, major arteries and veins, microcirculation, pulmonary circulation, venous valves, cerebrospinal fluid (CSF), brain interstitial fluid (ISF), Starling resistors, Monro-Kellie hypothesis, brain lymphatic drainage and the modern concept of CSF/ISF drainage and absorption based on the {\em Bulat-Klarica-Orešković} hypothesis. The mathematical model relies on one-dimensional Partial Differential Equations (PDEs) for blood vessels and on Ordinary Differential Equations (ODEs) for lumped parameter models. The systems of PDEs and ODEs are solved through a high-order finite volume ADER method and through an implicit Euler method. The computational results are validated against literature values and magnetic resonance flow measurements. Furthermore, the model is validated against {\em in-vivo} intracranial pressure waveforms acquired in healthy mice and in mice with impairment of the intracranial venous outflow. Through a systematic use of our computational model in healthy and pathological cases, we provide a complete and holistic neurovascular view of the main murine fluid dynamics. We propose a hypothesis on the working principles of the glymphatic system, opening a new door towards a comprehensive understanding of the mechanisms which link vascular and neurological disorders. In particular, we show how impairment of the cerebral venous outflow might potentially lead to accumulation of solutes in the parenchyma, by altering CSF and ISF dynamics. This thesis also concerns the development of a high-order ADER-type numerical method for systems of hyperbolic balance laws in networks, based on a new implicit solver for the junction-generalized Riemann problem. The resulting ADER scheme can deal with stiff source terms and can be applied to non-linear systems of hyperbolic balance laws in domains consisting of networks of one-dimensional sub-domains. Also, we design a novel one-dimensional mathematical model for collecting lymphatics coupled with a Electro-Fluid-Mechanical Contraction (EFMC) model for dynamical contractions. The resulting mathematical model gives each lymphangion the autonomous capability to trigger action potentials based on local fluid-dynamical factors.