Mathematical modelling and simulation of the human circulation with emphasis on the venous system: application to the CCSVI condition

Recent advances in medical science regarding the role of the venous system in the development of neurological conditions has renewed the attention of researchers in this district of the cardiovascular system. The main goal of this thesis is to perform a theoretical study of Chronic CerebroSpinal Venous Insufficiency (CCSVI), a venous pathology that has been associated to Multiple Sclerosis. CCSVI is a condition in which main cerebral venous drainage pathways are obstructed. Its impact in cerebral hemodynamics and its connection to Multiple Sclerosis is subject of current debate in the medical community. In order to perform a credible study of the haemodynamical aspects of CCSVI, a sufficiently accurate mathematical model of the problem under investigation must be used. The venous system has not received the same attention as the arterial counterpart by the medical community. As a consequence, the mathematical modeling and numerical simulation of the venous system lies far behind that of the arterial system. The venous system is a low-pressure system, formed by very thin-walled vessels, if compared to arteries, that are likely to collapse under the action of gravitational or external forces. These properties set special requirements on the mathematical models and numerical schemes to be used. In this thesis we present a closed-loop multi-scale mathematical model of the cardiovascular system, where medium to large arteries and veins are represented as one-dimensional (1D) vessels, whereas the heart, the pulmonary circulation, capillary beds and intracranial pressure are modeled as lumped parameter models. A characteristic feature of our closed-loop model is the detailed description of head and neck veins. Due to the large inter-subject variability of the venous system, we perform a patient-specific characterization of major veins of the head and neck using MRI data collected in collaboration with the Magnetic Resonance Research Facility of the Wayne State University, Detroit (USA). Computational results are carefully validated using published data for the arterial system and most regions of the venous system. For head and neck veins validation is carried out through a detailed comparison of simulation results against patient-specific Phase-Contrast MRI flow quantification data. Regarding the development of novel numerical schemes, we construct high-order accurate, robust and efficient numerical schemes for 1D blood flow in elastic and viscoelastic vessels, as well as a solver for vessel networks. The solver is validated in the context of an in vitro network of vessels for which experimental and numerical results are available. After validation of both, the mathematical model and the numerical methodology, we use our theoretical tool to study the influence of different CCSVI patterns on cerebral hemodynamics. CCSVI patterns are defined by the medical literature as combinations of venous obstructions at different locations. Here we used two strategies. First, we take a venous configuration corresponding to a healthy control and explore the effect of different CCSVI patterns by modifying this network. Then, we characterize our venous network with the geometry of a real CCSVI patient and compare results with the ones obtained for the healthy control. The presented model provides a powerful tool to study still unresolved aspects of cerebral blood flow physiology, as well as several venous pathologies. Furthermore, it constitutes an ideal platform for improving currently used algorithms and for integrating fundamental physiological processes, such as detailed hemodynamics, regulatory mechanisms and transport of substances.