X. Descovich, G. Pontrelli, S. Melchionna, S. Succi, S. Wassertheurer:
"Modeling Fluid Flows in distensible tubes for Applications in Hemodynamics";
International Journal of Modern Physics C, 24 (2013), 5; S. #.

Kurzfassung englisch:
We present a lattice Boltzmann (LB) model for the simulation of hemodynamic °ows in the presence of compliant walls. The new scheme is based on the use of a continuous bounce-back boundary condition, as combined with a dynamic constitutive relation between the °ow pressure at the wall and the resulting wall deformation. The method is demonstrated for the case of two-dimensional (axisymmetric) pulsatile °ows, showing clear evidence of elastic wave propagation
of the wall perturbation in response to the °uid pressure. The extension of the present two-dimensional axisymmetric formulation to more general three-dimensional geometries is currently under investigation.
1. Introduction
Cardiovascular diseases are the most common cause of death in the European Union1 and in the industrialized world in general. There is considerable evidence that the development of such diseases is, to a great extent, linked to the characteristics of the blood °ow and, in particular, to the arterial wall sti®ening.2 Since, experimental methods in the cardiovascular system are di±cult and invasive,
mathematical models and numerical methods to simulate the hemodynamic processes have gained a growing importance over the years.3 6
Research in the ¯eld includes studies incorporating the whole arterial tree7 as well as studies of only parts of it, e.g. a segment of an artery.8 This paper is aimed at developing a simple method for the simulation of blood °ow in a vessel segment and hence focuses on the investigation of local °ow behavior.
Many applications of °uid dynamics deal with °ows con¯ned by rigid boundaries,such as °ows in ducts and pipes. In some other applications, for instance hemodynamics, it is important to include the e®ects of the wall compliance. Since arteries are
elastic and change in diameter depending on the pulsatile blood pressure inside, it is crucial to include elasticity e®ects in models of physiological °ows in blood vessels.
This means that appropriate models describing the dynamic uid-structure interaction have to be developed. This is one of the major computational challenges in biomechanics. Many papers in the literature deal with °uid-structure interaction problems.9 11 They di®er by the method by which the set of equations for the °uid
component (e.g. Navier Stokes equations) and those for the solid one (e.g. wall
motion equations) are solved. A variety of coupling algorithms have been developed, such as the coupled and the weakly coupled approach, or the uncoupled method where the equations for the °uid and for the structure are computed separately.
A short description of these algorithms can be found in Refs. 12 14.
Some of the above works dealing with °uid-structure interaction problems use independent grids for the °uid part and the solid part. This requires appropriate mappings between the variables describing the two components. A commonly used approach is the Arbitrary Lagrangian Eulerian (ALE) method, which provides a mapping between the Lagrangian system (in which the structure is commonly described)
and the Eulerian system (in which the °uid is generally described).3 In the ALE approach, the physical boundary is moving, while the ¯ctitious boundary (e.g. inlet/outlet section of a blood vessel) is kept ¯xed and undeformed. In such a technique, the grid is moving, which typical calls for re-meshing procedures in order to avoid extreme deformations which would endanger the stability of the simulation.
An alternative method based on a nonmoving grid is the Cartesian cut cells approach. 13 In it, no remeshing is needed, and the wall enclosing the °uid domain moves over a ¯xed grid.
The immersed ¯bers method proposed by Peskin in 1977 (Ref. 15) can be regarded as the ¯rst attempt to study °uid-structure interaction problems in the cardiovascular system. Since then, other methods have been developed. Conventional numerical schemes for °uid-structure interaction problems couple the ¯nite element method for solving the governing equations of the structure, with a ¯nite di®erence,
¯nite element or ¯nite volume method for the °uid description. The solid and °uid
subsystems can be solved simultaneously16 or separately.17
A recent option, which o®ers a competitive alternative to the aforementioned numerical methods, especially on parallel computing architectures, is provided by the
lattice Boltzmann (LB) method.18,19 This method has been applied to a broad variety
of complex °ows,20 including °uid-structure interactions problems.9,10,12,14,21 23
Within the LB framework, the method proposed by Fang et al.21,22 provides conditions for elastic and moving boundaries. In this approach, virtual distribution functions at the boundary are introduced. The velocity at boundary nodes required to compute these virtual distribution functions is obtained by quadratic interpolation
or extrapolation. This method has been successfully applied to LB simulations in two-dimensional elastic tubes by Hoekstra et al.23 It is accurate, but not appropriate for large simulations running in parallel because of the quadratic extrapolations and the fact that nodes change from the solid domain to the °uid domain
and vice versa.10 Another modi¯cation of the LB scheme to model coupled °uidstructure problems has been proposed by Krafczyk et al.14 It is based on the work of Ladd, who presented a general technique for the simulation of °uid-particle interactions.
24 The method is a combination of the LB equation for the °uid domain and Newtonian dynamics of the solid particles. It allows to determine the interaction between °uid and solid boundary by directly using the LB variables. Krafczyk et al.
applied the method to study the °uid-structure system of moving lea°ets of an arti¯cial heart valve driven by physiological blood °ow.14 Other numerical simulations using the method of Ladd have been carried out.25,26 Chopard and Marconi9 modeled the °uid-wall interaction based on local exchange of momentum between solid
and °uid particles. Their method di®ers from that in Ref. 24, in which the solid particles immersed in the °uid are rigid, and allows to model solid, deformable particles suspended in a °uid by using the LB method, for both the °uid and the solid phase. An approach coupling the LB model (for the °uid) with a lattice spring model (for the compliant wall) has been proposed by Buxton et al.12A network of \springs" which are connected to each other describes the interaction.Amajor advantage of a lattice spring
model is its computational e±ciency.12 E±cient computing is also provided by the approach of Doctors et al.10 who performed LB simulations of pulsatile °uid °ow in three-dimensional elastic pipes. Their method is \based on estimating the distances from sites at the edge of the simulation box to the wall along the lattice directions from the displacement of the closest point on the wall and the curvature there, followed by application of a nonequilibrium extrapolation method."10 Quite recently, Leitner8 proposed a simple method for modeling elastic vessel walls in LB simulations of arterial blood °ow. The method is similar to the Cartesian
cut cells approach mentioned above. The wall displacement is based on the local pressure and involves transmural pressure corresponding to the pressure needed to balance the restoring forces from the elastic wall and to maintain this wall in equilibrium.
The method presented by Leitner does not treat a realistic °uid-structure interaction, since it does not include the feedback from the wall to the °uid. Nevertheless, it provides satisfactory results for simulations of blood °ow in artery segments,
by keeping the numerical algorithm simple and robust. Inevitably, the
method presents some drawbacks, for instance the fact that the wall cannot be displaced by less than one single lattice unit of the underlying grid. This stepwise wall displacement is a source of signi¯cant discretization errors, which may compromise
the accuracy of the simulation. This paper presents a simple and accurate method for incompressible °ow through distensible tubes. Most of the material included here has been developed in
an extended form in Ref. 27, and the reader is referred to it for more details. The present work introduces a novel boundary condition allowing a continuous displacement of the wall, as opposed to the aforementioned step-wise motion. The modeling of the elastic wall is kept fully local, hence basically as simple as Leitner's method, thereby preserving the e±ciency of the LB algorithm.
A signi¯cant advantage of the present approach is that it can also be used for simulating the blood °ow through stents, i.e. wire metal meshes inserted into a narrowed artery to prevent its occlusion.28 Due to the geometry and the di®erent sti®ness properties of the stent, separation and local recirculation can occur near the stented region. It is well recognized that such a recirculation may cause a re-narrowing of the artery, the so-called in-stent restenosis, which is a pathological process often occurring after stent implantation.29 E±cient methods to simulate the blood °ow through stents are therefore of particular interest, to gain a deeper understanding
of the physiological features of stented arteries

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