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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

CNRSTBP 8027,
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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References

Nerem, R., and Cornhill, J., 1980, “The Role of Fluid Mechanics in Atherogenesis,” ASME J. Biomech. Eng., 102(3), pp. 181–189. [CrossRef]
Zarins, C., Giddens, D., Bharadvaj, B., Sottiurai, V., Mabon, R., and Glagov, S., 1983, “Carotid Bifurcation Atherosclerosis. Quantitative Correlation of Plaque Localization With Flow Velocity Profiles and Wall Shear Stress,” Circ. Res., 53, pp. 502–514. [CrossRef] [PubMed]
Ku, D., Giddens, D., Zarins, C., and Glagov, S., 1985, “Pulsatile Flow and Atherosclerosis in the Human Carotid Bifurcation. Positive Correlation Between Plaque Location and Low Oscillating Shear Stress. Arteriosclerosis,” Thromb. Vasc. Biol., 5, pp. 293–302. [CrossRef]
Giddens, D., Zarins, C., and Glagov, S., 1993, “The Role of Fluid Mechanics in the Localization and Detection of Atherosclerosis,” ASME J. Biomech. Eng., 115(4B), pp. 588–594. [CrossRef]
Tropea, B., Schwarzacher, S., Chang, A., Asvar, C., Huie, P., Sibley, R., and Zarins, C., 2000, “Reduction of Aortic Wall Motion Inhibits Hypertension-Mediated Experimental Atherosclerosis,” Arterioscler. Thromb. Vasc. Biol., 20, pp. 2127–2133. [CrossRef] [PubMed]
Meyerson, S., Skelly, C., Curi, M., Shakur, U., Vosicky, J., Glagov, S., and Schwartz, L., 2001, “The Effects of Extremely Low Shear Stress on Cellular Proliferation and Neointimal Thickening in the Failing Bypass Graft,” J. Vasc. Surg., 34, pp. 90–97. [CrossRef] [PubMed]
Cheng, C., Tempel, D., van Hageren, R., van der Baan, A., Grosveld, F., Daemen, M., Krams, R., and de Crom, R., 2006, “Atherosclerotic Lesion Size and Vulnerability are Determined by Patterns of Fluid Shear Stress,” Circulation, 113, pp. 2744–2753. [CrossRef] [PubMed]
Coskun, A., Chen, C., Stone, P., and Feldman, C., 2006, “Computational Fluid Dynamics Tools Can be Used to Predict the Progression of Coronary Artery Disease,” Physica A, 362, pp. 182–190. [CrossRef]
London, G., Blacher, J., Pannier, B., Guérin, A., Marchais, S., and Safar, M., 2001, “Arterial Wave Reflections and Survival in End-Stage Renal Failure,” Hypertension, 38, pp. 434–438. [CrossRef] [PubMed]
Mitchell, G., Parise, H., Benjamin, E., Larson, M., Keyes, M., Vita, J., Vasan, R., and Levy, D., 2004, “Changes in Arterial Stiffness and Wave Reflection With Advancing Age in Healthy Men and Women the Framingham Heart Study,” Hypertension, 43, pp. 1239–1245. [CrossRef] [PubMed]
Weber, T., Auer, J., O'Rourke, M., Kvas, E., Lassnig, E., Lamm, G., Stark, N., Rammer, M., and Eber, B., 2005, “Increased Arterial Wave Reflections Predict Severe Cardiovascular Events in Patients Undergoing Percutaneous Coronary Interventions,” Eur. Heart J., 26, pp. 2657–2663. [CrossRef] [PubMed]
Botnar, R., Rappitsch, G., Scheidegger, M., Liepsch, D., Perktold, K., and Boesiger, P., 2000, “Hemodynamics in the Carotid Artery Bifurcation: A Comparison Between Numerical Simulations and in Vitro MRI Measurements,” J. Biomech., 33, pp. 137–144. [CrossRef] [PubMed]
Tang, D., Yang, C., Walker, H., Kobayashi, S., and Ku, D., 2002, “Simulating Cyclic Artery Compression Using a 3D Unsteady Model With Fluid-Structure Interactions,” Comput. Struct., 80, pp. 1651–1665. [CrossRef]
Cebral, J., Castro, M., Appanaboyina, S., Putman, C., Milan, D., and Frangi, A., 2005, “Efficient Pipeline for Image-Based Patients Specific Analysis of Cerebral Aneurysm Hemodynamics: Technique and Sensitivity,” IEEE Trans. Med. Imaging, 24, pp. 457–467. [CrossRef] [PubMed]
Li, M., Beech-Brandt, J., John, L., Hoskins, P., and Easson, W., 2007, “Numerical Analysis of Pulsatile Blood Flow and Vessel Wall Mechanics in Different Degrees of Stenoses,” J. Biomech., 40, pp. 3715–3724. [CrossRef] [PubMed]
Vignon, I., and Taylor, C., 2004, “Outflow Boundary Conditions for One-Dimensional Finite Element Modeling of Blood Flow and Pressure Wave in Arteries,” Wave Motion, 39, pp. 361–374. [CrossRef]
Vignon-Clementel, I., Figueroa, C., Jansen, K., and Taylor, C., 2006, “Outflow Boundary Conditions for Three-Dimensional Finite Element Modeling of Blood Flow and Pressure in Arteries,” Comput. Methods Appl. Mech. Eng., 195, pp. 3776–3796. [CrossRef]
Alastruey, J., Parker, K., Peiro, J., Byrd, S., and Sherwin, S., 2007, “Modelling the Circle of Willis to Assess the Effects of Anatomical Variations and Occlusions on Cerebral Flows,” J. Biomech., 40, pp. 1794–1805. [CrossRef] [PubMed]
Marchandise, E., Willemet, M., and Lacroix, V., 2009, “A Numerical Hemodynamic Tool for Predictive Vascular Surgery,” Med. Eng. Phys., 31, pp. 131–144. [CrossRef] [PubMed]
Kim, H., Vignon-Clementel, I., Coogan, J., Figueroa, C., Jansen, K., and Taylor, C., 2010, “Patient-Specific Modeling of Blood Flow and Pressure in Human Coronary Arteries,” Ann. Biomed. Eng., 10(38), pp. 3195–3209. [CrossRef]
Olufsen, M., 1999, “Structured Tree Outflow Condition for Blood Flow in Larger Systemic Arteries,” Am. J. Physiol. Heart Circ. Physiol., 276, pp. 257–268. Available at: http://ajpheart.physiology.org/content/276/1/H257
Olufsen, M., Peskin, C., Kim, W., Pedersen, E., Nadim, N., and Larsen, J., 2000, “Numerical Simulation and Experimental Validation of Blood Flow in Arteries With Structured-Tree Outflow Conditions,” Ann. Biomed. Eng., 28, pp. 1281–1299. [CrossRef] [PubMed]
Willemet, M., Lacroix, V., and Marchandise, E., 2011, “Inlet Boundary Conditions for Blood Flow Simulations in Truncated Arterial Networks,” J. Biomech., 44, pp. 897–903. [CrossRef] [PubMed]
Bokov, P., 2011, “Description Expérimentale rt Numérique de L'interaction Entre un Stent Biodégradable et la Paroi Artérielle,” Ph.D. Thesis, Université Paris Diderot.
McDonaldA. R., 1960, Blood Flow in Arteries, Williams and Wilkins, Baltimore, MD.
Ozolanta, I., Tetere, G., Purinya, B., and Kasyanov, V., 1998, “Changes in the Mechanical Properties, Biochemical Contents and Wall Structure of the Human Coronary Arteries With Age and Sex,” Med. Eng. Phys., 20, pp. 523–533. [CrossRef] [PubMed]
Seo, T., Schachter, L., and Barakat, A., 2005, “Computational Study of Fluid Mechanical Disturbance Induced by Endovascular Stents,” Ann. Biomed. Eng., 33(4), pp. 444–456. [CrossRef] [PubMed]
LaDisa, J. F. J., Olson, L., Guler, I., Hettrick, D., Kersten, J., Warltier, D., and Pagel, P., 2005, “Circumferential Vascular Deformation After Stent Implantation Alters Wall Shear Stress Evaluated With Time-Dependent 3D Computational Fluid Dynamics Models,” J. Appl. Physiol., 98, pp. 947–957. [CrossRef] [PubMed]
Gijsen, F., Migliavacca, F., Schievano, S., Socci, L., Petrini, L., Thury, A., Wentzel, J., van der Steen, A., Serruys, P., and Dubini, G., 2008, “Simulation of Stent Deployment in a Realistic Human Coronary Artery,” BioMed. Eng. OnLine, 7(23), pp. 1–11. [CrossRef] [PubMed]
Nicoud, F., Vernhet, H., and Dauzat, M., 2005, “A Numerical Assessment of Wall Shear Stress Changes After Endovascular Stenting,” J. Biomech., 38, pp. 2019–2027. [CrossRef] [PubMed]
Stergiopulos, N., Tardy, Y., and Meister, J., 1993, “Nonlinear Separation of Forward and Backward Running Waves in Elastic Conduits,” J. Biomech., 26, pp. 201–209. [CrossRef] [PubMed]
Rogova, I., 1998, “Propagation d'Ondes en Hémodynamique Artérielle: Application a l’Évaluation Indirecte des Parametres Physiopathologiques,” Ph.D. Thesis, Université Paris Diderot, Paris.

Figures

Grahic Jump Location
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).

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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).

Grahic Jump Location
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.

Grahic Jump Location
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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