Research Papers

Implementing Boundary Conditions in Simulations of Arterial Flows

[+] Author and Article Information
P. Flaud

Laboratoire Matière et Systèmes Complexes,
Université Paris Diderot,
Paris 75251, France

A. Bensalah

Rabat 10102, Morocco

J.-M. Fullana

e-mail: jose.fullana@upmc.fr

M. Rossi

Institut Jean Le Rond D'Alembert,
CNRS UMR 7190,
Paris 75005, France

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received February 11, 2013; final manuscript received June 4, 2013; accepted manuscript posted July 29, 2013; published online September 24, 2013. Assoc. Editor: Ender A. Finol.

J Biomech Eng 135(11), 111004 (Sep 24, 2013) (9 pages) Paper No: BIO-13-1070; doi: 10.1115/1.4025111 History: Received February 11, 2013; Revised June 04, 2013; Accepted July 29, 2013

Computational hemodynamic models of the cardiovascular system are often limited to finite segments of the system and therefore need well-controlled inlet and outlet boundary conditions. Classical boundary conditions are measured total pressure or flow rate imposed at the inlet and impedances of RLR, RLC, or LR filters at the outlet. We present a new approach based on an unidirectional propagative approach (UPA) to model the inlet/outlet boundary conditions on the axisymmetric Navier–Stokes equations. This condition is equivalent to a nonreflecting boundary condition in a fluid–structure interaction model of an axisymmetric artery. First we compare the UPA to the best impedance filter (RLC). Second, we apply this approach to a physiological situation, i.e., the presence of a stented segment into a coronary artery. In that case a reflection index is defined which quantifies the amount of pressure waves reflected upon the singularity.

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Fig. 1

Representation of the mesh. In the fluid, the distribution in size is chosen so as to vary exponentially in the radial direction, with a mesh size near the wall equal to 0.03 times the first mesh size (near the axis); the mesh size is constant in the longitudinal direction and equal to 1.3R. The wall is represented by two cells (upper side).

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Fig. 2

Nondimensional pressure (P/E) as a function of time. Each curve corresponds to different positions z (0, 0.4, 0.8, 1.2, and 1.6 m). Numerical simulations (dashed-dotted line) are compared to analytical solutions (dashed line). The Reynolds number evaluated with the maximum velocity at the entrance, is equal to 23.

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Fig. 3

(a) Pulse wave, propagating towards negative z: first curve from the left contour is the radial displacement. The three figures correspond to three different times. (b) Pressure as a function of time at positions z = 0.4 m and z = 0.1 m. The shear rate of Eq. (23) is evaluated at z = 1 cm, i.e., approximately 7R. (c) Idem than (b) but the shear rate is now evaluated at the tube entrance.

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Fig. 4

SRLC versus SINF: adimensional pressure (P/E) as a function of time at several axial positions (z = 0 (entry point); 0.1; 0.3; 0.5 m (exit point)). Dashed-dotted lines correspond to SRLC, dashed lines to SINF. (a) ΔP/E of 5 × 10−4 and (b) ΔP/E of 5 × 10−4.

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Fig. 5

SUPA versus SINF: adimensional pressure (P/E) as function of the time at several axial positions (z = 0 (entry point); 0.1; 0.3; 0.5 m (exit point)). Dashed-dotted lines correspond to SUPA with UPA outlet conditions (Eq. (17)), dashed lines to SINF (a) SUPA and SINF for ΔP/E = 5 × 10−4 and (b) SUPA and SINF for ΔP/E = 5 × 10−3.

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Fig. 6

Pressure pulse using UPA for a high amplitude (103 Pa) in a 50 cm length tube with Young's modulus E = 106 Pa and viscosity in the physiological range, here μ = 7 mPa s. Wall displacements at successive times (0.05, 0.1, 0.15, 0.2, 0.25, and 0.3 s) are shown: no apparent reflexion occurs. The wave moves from the bottom to the top. The color bar is scaled in 10−2 mm, and the bottom axis in mm.

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Fig. 7

Pressure pulse using UPA for a high amplitude (104 Pa): at each spatial location the maximum of pressure during the simulation is computed. This quantity is represented as function of espace. Diamonds (SINF) and squares (SUPA).

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Fig. 8

Radial displacement waves measured every 10−3 s using UPA: (a) When the tube presents a singularity situated between z = 0.31 m and z = 0.33 m which is a hundred times stiffer than the tube wall and (b) without singularity. Upper continuous line is superimposed, joining the maximum of the displacement wave at each time.

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Fig. 9

Reflection index as a function of the relative Young's modulus of the stent (Er = E/E0), where E0 is the Young's modulus of a normal artery. The results using our numerical separation method are given by upper squares. The results of a linear theory of reflection are represented by lower squares.




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