Modeling Elastic Vessels with the LBGK Method in Three Dimensions
The Lattice Bhatnagar Gross and Krook (LBGK) method is widely used to solve fluid mechanical problems in engineering applications. In this work a brief introduction of the LBGK method is given and a new boundary condition is proposed for the cardiovascula
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Austrian Research Centers GmbH - ARC, Donau-City-Straße 1, A-1220 Vienna, Austria 2 Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria 3 Medical University of Graz, Clinical Department for vascular surgery, Auenbruggerplatz 29, 8036 Graz 4 Medical University of Graz, Institute for Medical Informatics, Statistics and Documentation, Auenbruggerplatz 2/V, A-8036 Graz, Austria [email protected]
Abstract. The Lattice Bhatnagar Gross and Krook (LBGK) method is widely used to solve fluid mechanical problems in engineering applications. In this work a brief introduction of the LBGK method is given and a new boundary condition is proposed for the cardiovascular domain. This enables the method to support elastic walls in two and three spatial dimensions for simulating blood flow in elastic vessels. The method is designed to be used on geometric data obtained from magnetic resonance angiography without the need of generating parameterized surfaces. The flow field is calculated in an arbitrary geometry revealing characteristic flow patterns and geometrical changes of the arterial walls for different time dependent input contours of pressure and flow. For steady flow the results are compared to the predictions of the model proposed by Y. C. Fung which is an extension of Poiseuille's theory. The results are very promising for relevant Reynolds and Womersley numbers, consequently very useful in medical simulation applications. Keywords: Simulation, Lattice Boltzmann Model, Haemodynamics, Elasticity, Computer Fluid Dynamics.
1 Introduction In the western industrial countries cardiovascular diseases are the most frequent cause of death. Therefore a lot of research is done to get a better understanding of the cardiovascular system. Of special interest is the simulation of blood flow in three spatial dimensions using the vessel geometry that is obtained from magnetic resonance angiography. This enables an investigation of pressure and flow profiles and shear stress at the vessel wall. The appearing shear stress is important for the risk estimation of arteriosclerosis [1]. A. Holzinger (Ed.): USAB 2007, LNCS 4799, pp. 213–226, 2007. © Springer-Verlag Berlin Heidelberg 2007
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Fig. 1. From a one dimensional model boundary conditions for the more detailed three dimensional model are obtained
In this work a LBGK is used to simulate the blood flow in three spatial dimensions and to solve the incompressible Navier-Stokes equations with the LBGK method [2]. The LBGK method working as a hemodynamical solver on tomographic data has been presented in [3]. For the treatment of elasticity of the vessel walls boundary conditions where proposed by [4] where the vessel wall is represented as a surface. When the vessel walls are represented as voxels, a simpler approach has been proposed in [5], which does not need a parameterized representation of the vessel wall. In this work this approach will be extended to three spatial dimension
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