Advertisement

Annals of Biomedical Engineering

, Volume 34, Issue 7, pp 1119–1128 | Cite as

Fluid-Wall Modelling of Mass Transfer in an Axisymmetric Stenosis: Effects of Shear-Dependent Transport Properties

  • Nanfeng Sun
  • Nigel B. Wood
  • Alun D. Hughes
  • Simon A. M. Thom
  • X. Yun Xu
Article

Abstract

Mechanical forces, such as low wall shear stress (WSS), are implicated in endothelial dysfunction and atherogenesis. The accumulation of low density lipoprotein (LDL) and hypoxia are also considered as main contributing factors in the development of atherosclerosis. The objective of this study was to investigate the influences of WSS on arterial mass transport by modelling the flow of blood and solute transport in the lumen and arterial wall. The Navier-Stokes equations and Darcy’s Law were used to describe the fluid dynamics of the blood in the lumen and wall respectively. Convection-diffusion-reaction equations were used to model LDL and oxygen transport. The coupling of fluid dynamics and solute dynamics at the endothelium was achieved by the Kedem-Katchalsky equations. A shear-dependent hydraulic conductivity relation extracted from experimental data in the literature was employed for the transport of LDL and a shear-dependent permeability was used for oxygen. The integrated fluid-wall model was implemented in Comsol Multiphysics 3.2 and applied to an axisymmetric stenosis. The results showed elevated LDL concentration and reduced oxygen concentration in the subendothelial layer of the arterial wall in areas where WSS is low, suggesting that low WSS might be responsible for lipid accumulation and hypoxia in the arterial wall.

Keywords

LDL transport Oxygen transport Lipid accumulation Hypoxia Atherosclerosis Wall stress stress 

GLOSSARY OF TERMS

Principal symbols

c

concentration, mol m−3

D

diffiusivity, m2 s−1

Js

solute flux across the endothelium, mol s−1 m−2

Jv

transmural velocity across the endothelium, m s−1

K

solute lag coefficient

Lp

hydraulic conductivity of the endothelium, m s−1 Pa−1

p

pressure, Pa

P

permeability, m s−1

u

velocity of blood flow, m s−1

κ

Dacian permeability, m2

μ

Pa s

ρ

density, kg m−3

σd

osmotic reflection cofficient

σf

solvent reflection coefficient

τw

wall shear stress, Pa

Subscripts

l

blood lumen

w

arterial wall

LDL

low density lipoprotein

oxy

oxygen

Notes

ACKNOWLEDGMENTS

This work was supported by the Leverhulme Trust (F07 058/AA).

REFERENCES

  1. 1.
    Brooks, A. N., and T. J. Hughes. Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier-stokes euqations. Comput. Methods Appl. Mech. Engrg. 32:199–259, 1982.Google Scholar
  2. 2.
    Caro, C. G., J. M. Fitz-Gerald, and R. C. Schroter. Atheroma and arterial wall shear observation, correlation and proposal of a shear dependent mass transfer mechanism for atherogenesis. Proc. R. Soc. London, Ser. B Biol. Sci. 177:109–159, 1971.Google Scholar
  3. 3.
    Chang, Y. S., A. Yaccino, S. Lakshminarayanan, J. A. Frangos, and J. M. Tarbell. Shear-induced increase in hydraulic conductivity in endothelial cells is mediated by a nitric oxide-dependent mechanism. Arterioscler. Thromb. Vasc. Biol. 20:35–42, 2000.Google Scholar
  4. 4.
    Crawford, D. W., and D. H. Blankenhorn. Arterial wall oxygenation, oxyradicals, and atherosclerosis. Atherosclerosis 9:97–108, 1991.Google Scholar
  5. 5.
    Deng, X., M. King, and R. Guidoin. Localization of atherosclerosis in arterial junctions. modeling the release rate of low density lipoprotein and its breakdown products accumulated in blood vessel walls. ASAIO Journal 39:M489–M495, 1993.Google Scholar
  6. 6.
    Deng, X., M. King, and R. Guidoin. Localization of atherosclerosis in arterial junctions. concentration distribution of low density lipoproteins at the luminal surface in regions of disturbed flow. ASAIO Journal 41:58–67, 1995.Google Scholar
  7. 7.
    Deng, X., Y. Marois, M. King, and R. Guidoin. Uptake of 3H-7-cholesterol along the arterial wall at an area of stenosis. ASAIO Journal 40:186–191, 1994.Google Scholar
  8. 8.
    Ethier, C. R. Computational modeling of mass transfer and links to atherosclerosis. Ann. Biomed. Eng. 30:461–471, 2002.Google Scholar
  9. 9.
    Hillsley, M. V., and J. M. Tarbell. Oscillatory shear alters endothelial hydraulic conductivity and nitric oxide levels. Biochem. Biophys. Res. Commun. 293:1466–1471, 2002.Google Scholar
  10. 10.
    Hoff, H. F., C. L. Heideman, R. L. Jackson, R. J. Bayardo, H. S. Kim, and A. M. J. Gotto. Localization patterns of plasma apolipoproteins in human atherosclerotic lesions. Circ. Res. 37:72–79, 1975.Google Scholar
  11. 11.
    Jo, H., R. O. Dull, T. M. Hollis, and J. M. Tarbell. Endothelial albumin permeability is shear dependent, time dependent, and reversible. Am. J. Physiol. Heart Circ. Physiol. 260:H1992–H1996, 1991.Google Scholar
  12. 12.
    Kaazempur-Mofrad, M. R. and C. R. Ethier. Mass transport in an anatomically realisitic human right coronary artery. Ann. Biomed. Eng. 29:121–127, 2001.Google Scholar
  13. 13.
    Kaazempur-Mofrad, M. R., S. Wada, J. G. Myers, and C. R. Ethier. Mass transport and fluid flow in stenotic arteries: Axisymmetric and asymmetric models. Int. J. Heat. Mass. Tran. 48:4510–4517, 2005.Google Scholar
  14. 14.
    Karner, G., K. Perktold, and H. P. Zehentner. Computational modeling of macromolecule transport in the arterial wall. Comput. Methods. Biomech. Biomed. Engin. 4:491–504, 2001.Google Scholar
  15. 15.
    Karner, G. and K. Perktold. Effect of endothelial injury and increased blood pressure on albumin accumulation in the arterial wall: A numerical study. J. Biomech. 33:709–715, 2000.Google Scholar
  16. 16.
    Kedem, O., and A. Katchalsky. Thermodynamic analysis of the permeaility of biological membranes to non-electrolytes. Biochem. Biophys. Acta. 27:229–246, 1958.Google Scholar
  17. 17.
    Long, Q., X. Y. Xu, K. V. Ramnarine, and P. Hoskins. Numerical investigation of physiologically realistic pulsatile flow through arterial stenosis. J. Biomech. 34:1229–1242, 2001.Google Scholar
  18. 18.
    Ma, P., X. Li, and D. N. Ku. Convective mass transfer at the carotid bifurcation. J. Biomech. 30:565–571, 1997.Google Scholar
  19. 19.
    Meyer, G., R. Merval, and A. Tedgui. Effects of pressure-induced stretch and convection on low-density lipoprotein and albumin uptake in the rabbit aortic wall. Circ. Res. 79:532–540, 1996.Google Scholar
  20. 20.
    Michel, C. C., and F. E. Curry. Microvascular permeability. Physiol. Rev. 79:703–761, 1999.Google Scholar
  21. 21.
    Moore, J. A. and C. R. Ethier. Oxygen mass transfer calculations in large arteries. ASME J. Biomech. Eng. 119:469–475, 1997.Google Scholar
  22. 22.
    Ogunrinade, O., G. T. Kameya, and G. A. Truskey. Effect of fluid stress on the permeability of the arterial endothelium. Ann. Biomed. Eng. 30:430–446, 2002.Google Scholar
  23. 23.
    Ojha, M., R. S. C. Cobbold, K. W. Johnston, and R. L. Hummel. Pulsatile flow through constricted tubes: an experimental investigation using photochromic tracer methods. J. Fluid Mech. 203:173–197, 1989.Google Scholar
  24. 24.
    Prosi, M., P. Zunino, K. Perktold, and A. Quarteroni. Mathematical and numerical models for transfer of low-density lipoproteins through the arterial wall: A new methodology for the model set up with applications to the study of disturbed lumenal flow. J. Biomech. 38:903–917, 2005.Google Scholar
  25. 25.
    Prosi, M. Computer Simulation von Massetransportvorgängen in Arterien. Ph.D. thesis, Technische Universität Graz, 2003.Google Scholar
  26. 26.
    Qiu, Y., and J. M. Tarbell. Numerical simulation of oxygen mass transfer in a compliant curved tube model of a coronary artery. Ann. Biomed. Eng. 28:26–38, 2000.Google Scholar
  27. 27.
    Rappitsch, G., K. Perktold, and E. Pernkopf. Numerical modelling of shear-dependent mass transfer in large arteries. Int. J. Numer. Methods Fluids 25:847–857, 1997.Google Scholar
  28. 28.
    Rappitsch, G., and K. Perktold. Computer simulation of convective diffusion processes in large arteries. J. Biomech. 29:207–215, 1996.Google Scholar
  29. 29.
    Rappitsch, G., and K. Perktold. Pulsatile albumin transport in large arteries: A numerical simulation study. ASME J. Biomech. Eng. 118:511–519, 1996.Google Scholar
  30. 30.
    Ross, R. Atherosclerosis: a denfense mechanism gone awry. Am. J. Pathol. 143:987–1002, 1993.Google Scholar
  31. 31.
    Sill, H. W., Y. S. Chang, J. R. Artman, J. A. Frangos, T. M. Hollis, and J. M. Tarbell. Shear stress increases hydraulic conductivity of cultured endothelial monolayers. Am. J. Physiol. Heart Circ. Physiol. 268:535–543, 1995.Google Scholar
  32. 32.
    Stangeby, D. K. and C. R. Ethier. Computational analysis of coupled blood-wall arterial LDL transport. ASME J. Biomech. Eng. 124:1–8, 2002.Google Scholar
  33. 33.
    Stangeby, D. K. and C. R. Ethier. Coupled computational analysis of arterial LDL transport—effects of hypertension. Comput. Methods. Biomech. Biomed. Engin. 5:233–241, 2002.Google Scholar
  34. 34.
    Tarbell, J. M., M. J. Lever, and C. G. Caro. The effect of varying albumin concentration on the hydraulic conductivity of the rabbit common carotid artery. Microvasc. Res. 35:204–220, 1988.Google Scholar
  35. 35.
    Tarbell, J. M. Mass transport in arteries and the localization of atherosclerosis. Annu. Rev. Biomed. Eng. 5:79–118, 2003.Google Scholar
  36. 36.
    Wada, S. and T. Karino. Theoretical study on flow-dependent concentration polarization of low density lipoproteins at the luminal surface of a straight artery. Biorheology 36:207–223, 1999.Google Scholar
  37. 37.
    Wada, S., and T. Karino. Theoretical prediction of low-density lipoproteins concentration at the luminal surface of an artery with a multiple bend. Ann. Biomed. Eng. 30:778–791, 2002.Google Scholar
  38. 38.
    Wada, S., M. Koujiya, and T. Karino. Theoretical study of the effect of local flow disturbances on the concentration of low-density lipoproteins at the luminal surace of end-to-end anastomosed vessels. Med. Biol. Eng. Comput. 40:576–587, 2002.Google Scholar
  39. 39.
    Zunino, P. Mathematical and Numerical Modeling of Mass Transfer in the Vascular System. Ph.D. thesis, Ecole Polytechnique Federale de Lausanne, 2002.Google Scholar

Copyright information

© Biomedical Engineering Society 2006

Authors and Affiliations

  • Nanfeng Sun
    • 1
  • Nigel B. Wood
    • 1
  • Alun D. Hughes
    • 2
  • Simon A. M. Thom
    • 2
  • X. Yun Xu
    • 1
  1. 1.Department of Chemical EngineeringImperial College LondonLondonUnited Kingdom
  2. 2.National Heart and Lung Institute, International Centre for Circulatory HealthImperial College LondonLondonUnited Kingdom

Personalised recommendations