This monograph contains an in-depth and coherent treatment of dimension-reduced modeling of blood flows on the level of large vessels (macrocirculation). The authors reduce the complexity by combining a one-dimensional Navier-Stokes equation and a simplified FSI-concept. The influence of omitted vessels, which are subsequent to the outlets of larger vessels, is accounted for by systems of ordinary differential equations (0D models). The target audience primarily comprises research experts in the field of biomedical engineering, but the book may also be beneficial for graduate students alike.
Author(s): Tobias Köppl, Rainer Helmig
Series: Mathematical Engineering
Publisher: Springer
Year: 2023
Language: English
Pages: 243
City: Cham
Preface
Contents
Acronyms
Symbols
1 Properties of the Cardiovascular System
1.1 The Cardiovascular System
1.1.1 Structure and Function of the Heart
1.1.2 Structure of the Vascular System
1.1.3 Properties of Blood
1.2 Cardiovascular Diseases
1.2.1 Arteriosclerosis
1.2.2 Aneurysms in Arterial Walls
1.2.3 Arteriovenous Malformation
1.2.4 Therapies for Cardiovascular Diseases
1.2.5 Natural Compensation Mechanisms
1.3 Further Literature
References
2 Modeling Approaches for the Macrocirculation
References
3 Dimension Reduced Models for the Macrocirculation
3.1 Blood Flow Within a Single Vessel
3.2 Derivation of the 1D Equations
3.3 Classification of the Model Equations
3.3.1 The Blood Flow Model is a System of Hyperbolic PDEs
3.3.2 The Transport Model is a Hyperbolic PDE
3.4 Existence of Classical Solutions for Flat Velocity Profiles
3.5 Simplification of the Non-Linear 1D Models
3.6 Boundary Conditions for a Complete Vascular Network
3.6.1 Modeling of Inflow Boundaries (IB)
3.6.2 Modeling of Outflow Boundaries (OB)
3.6.3 Modeling of Bifurcations (BB)
3.6.4 Modeling of a Stenosis (SB)
3.7 Validation of the 1D Model
References
4 Numerical Solution Methods
4.1 Construction of the Numerical Methods
4.2 Implementation of Boundary Conditions
4.2.1 Numerical Treatment of Inflow Boundaries
4.2.2 Numerical Treatment of Outflow Boundaries
4.2.3 Numerical Treatment of Bifurcations
4.2.4 Numerical Simulation of a Stenosis
4.3 Stability and Convergence Analysis
4.3.1 Important Mathematical Concepts
4.3.2 Analysis of the Lax-Wendroff Method
4.3.3 Analysis of the Implicit Finite Volume Method
4.4 Numerical Experiments
4.4.1 Choice of the Parameters
4.4.2 Experiment 1: Formation of Shocks
4.4.3 Experiment 2: Stability and Convergence Tests
4.4.4 Experiment 3: Simulation of Realistic Flow and Transport Processes in a Network
References
5 Compensation of Unilateral Carotid Stenoses
5.1 Mathematical Model and Choice of Parameters
5.2 Discussion of the Simulation Results
References
6 Concluding Remarks
References
7 Data and Mathematical Tools
7.1 Leibniz Rule
7.2 Simplification of the Navier-Stokes Equations
7.3 Lemma 1
7.4 Fundamental Lemma of Calculus of Variations
7.5 Lemma 2
7.6 Derivation of Reynold's Transport Theorem
7.7 Fourier Series
7.8 Weak Solutions of a Hyperbolic PDE
7.9 Gaussian Theorem
7.10 Partial Integration in mathbbRn
7.11 Taylor Expansion of a Vector Valued Function
7.12 Discrete Orthogonality
7.13 Network Containing the Circle of Willis
7.14 Data Set for Network 43
References
Appendix Glossary
