## Αριθμητικές και αναλυτικές λύσεις των εξισώσεων της ροής του αίματος χρησιμοποιώντας απλουστευμένα και ανώτερης τάξης μαθηματικά μοντέλα (Master thesis)

#### Μαλάτος, Σταύρος

From Womersley to present day, mathematical modeling of blood flow, and particularly the mathematical description of pulse wave propagation in the arterial system, is an interesting topic for the applied mathematician and the biomechanical engineer. Mathematical modeling of blood flow in the arterial system has particular importance for the medical community, since pressure and flow waveforms have important diagnostic significance. A central problem when modeling blood flow, and pressure in the larger systemic arteries, is to determine physiologically based boundary conditions such that the arterial tree be truncated in a few generations e.g. the aorta, iliac, and femoral arteries. In this study, we develope a one-dimensional fluid-structure interaction (FSI) dynamic model, based on non-linear partial differential equations (Navier-Stokes equations), for a Newtonian fluid interacting with the elastic arterial wall. The FSI model predicts flow and pressure in the systemic arteries. The outflow boundary conditions, representing the smaller arteries, are modeled by calculating the impedance and elasticity at the root of the structured tree, which is attached to each terminal branch of the larger arteries and is also described by fluid dynamic principles. Consequently, a mathematic model for the blood flow in the large arteries (aorta and peripheral vessels), in combination with a three-element Windkessel simplified fluid dynamics model representing the arterial bed (smaller arteries, arterioles and capillaries) is proposed and studied. By incorporating, through a peripheral resistance and elasticity, a large number of microvessels, we predict the amount of blood that can be maintained in this part of the blood circulation. This provides a dynamical boundary condition that also accommodates wave propagation effects for all systemic arteries. Additionally, we use the information derived from the aforementioned FSI model to describe the detailed hemodynamic condition of three-dimensional physiological and pathological arterial structures. Thus, from the developed FSI model, we obtain the appropriate dynamic boundary conditions at any part along the arterial tree. We further perform three-dimensional simulations to characterize the basic hemodynamic parameters within the lumen of a threedimensional structure, a stent graft after endovascular repair. Such parameters include wall shear stress (WSS), time averaged wall shear stress (TAWSS) and oscillatory shear index (OSI). Therefore, this study provides important information to the biomechanical and medical community, allowing extraction of blood flow characteristics in the large arteries. The approach is easily adaptable in both physiological and pathological conditions. In conclusion, the main result of this dissertation is the development of a holistic mathematical model which is physiologically adequate and computationally feasible. Thus, we can accurately evaluate flow and pressure at specific parts of the arterial system for additional localized three-dimensional advanced flow simulations, leading to the development of a spatial multi-scale model.
 Institution and School/Department of submitter: Πανεπιστήμιο Ιωαννίνων. Σχολή Θετικών Επιστημών. Τμήμα Μαθηματικών Subject classification: Μαθηματική μοντελοποίηση Keywords: Μαθηματική μοντελοποίηση,Αρτηριακό σύστημα,Ροή αίματος στο καρδιαγγειακό σύστημα URI: http://olympias.lib.uoi.gr/jspui/handle/123456789/29329 Appears in Collections: Διατριβές Μεταπτυχιακής Έρευνας (Masters)

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`http://olympias.lib.uoi.gr/jspui/handle/123456789/29329`