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X. Descovich, G. Pontrelli, S. Succi, S. Melchionna, M. Bammer:
"Modeling Elastic Walls in Lattice Boltzmann Simulations of Arterial Blood Flow";
Vortrag: MATHMOD 2012 - 7th Vienna Conference on Mathematical Modelling, Wien; 14.02.2012 - 17.02.2012; in: "Preprints Mathmod 2012 Vienna - Full Paper Volume", F. Breitenecker, I. Troch (Hrg.); Argesim / Asim, 38 (2012), S. 265 - 266.

Introduction. An essential part in the simulation of blood flow in arteries is the incorporation of the arterial
elasticity by modeling the vessel wall and its interaction with the fluid inside the vessel. Common numerical
methods for blood flow simulations with elastic walls are complex. We suggest a simple approach for modeling
elastic walls in lattice Boltzmann (LB) simulations of arterial blood flow. Our model fulfills the essential properties
of an elastic wall and respects the basic conservation laws.
Modeling elastic walls in lattice Boltzmann simulations. In simulations of blood flow, it is important to consider
the compliance of the vessel. Therefore, a model for the vessel wall is needed that describes its spatial
displacement as it interacts with the flow dynamics. Based on [1], we have developed a model which does not need
a parametrization of the wall. The method acts strictly locally, like the LB method, so that the complexity of the
algorithm is low.
The model uses a lattice of nodes that can have two different states: fluid, representing the blood inside the vessel,
and solid, describing the vessel wall. The compliance of the wall is modeled by changing the type of a node - from
solid to fluid in the case of expansion and vice versa in the case of contraction of the vessel. This change of node
type is dependent on the local pressure of the surrounding fluid nodes. Pressure thresholds are assigned to each
node, increasing with the radius of the vessel segment (based on a linear relationship between the pressure and the
radius).
Improved method to model elastic vessel wall. In our model, contrary to [1], the wall of the vessel is not
situated on the solid nodes but is imagined to be located between last fluid and first solid node in a given direction.
All nodes that are not fluid are by default solid. Thus, the problem of rupture of the vessel wall does not occur and
our approach does not require the use of cellular automata. Created fluid nodes need to be initialized, which is done
by averaging the LB populations from the fluid nodes surrounding the new fluid node. Compared to the method of
[1], where new fluid nodes are initialized with an equilibrium distribution function, this approach includes also the
non-equilibrium part of the populations, which is not negligible for nodes in proximity of the wall.
Furthermore, mass is a priori not conserved when the total number of fluid nodes increases or decreases (its
circumvention is not detailed in [1].) In order to ensure mass conservation, we developed two methods that rescale
the LB populations when a node type change occurs. The method of local rescaling takes into account only the
nearest neighbors of the node changing its state. It redistributes mass between nodes that change state and their
neighboring nodes. The method of rescaling `by columns´ takes into account the whole column of nodes - the
vessel can be considered as a sequence of `rings´ adjoint to each other - in which a node type change occurs. When
a node changes its state, mass is redistributed along all nodes in the same column.
Simulation and preliminary results. We implemented a simulation software for the lattice Boltzmann algorithm
in two dimensions combined with our elastic wall model using the programming language C. The program includes
the rescaling methods and the pressure threshold algorithm described in the full paper. Using our software program,
we conducted numerical experiments to show the feasibility of our approach.
First, we compared the computed velocity profile in a straight channel to the analytical solution of a Poiseuille flow
showing that both velocity profiles coincide. Second, we tested our modeling of elasticity using local rescaling,
showing that mass is properly conserved and initial values of the density and the velocity are recovered after
one cycle of expansion and subsequent contraction. Third, we tested the modeling of elasticity using rescaling
`by columns´ which minimizes local perturbations but exhibits the drawback that node type changes at one wall
boundary affect the flow field within the whole channel, which does not correspond to real fluid dynamics.
Outlook. The aim of the future work is the simulation of blood flow in stented arteries. The approach for the
modeling of elastic walls presented in this work has the advantage that it can also be extended to include stents.
This enhanced model will be implemented in our simulation software and further elaboration of our approach will
be reported in a later work.