Biomechanics and Modeling in Mechanobiology

, Volume 14, Issue 2, pp 297–313 | Cite as

A combined numerical and experimental framework for determining permeability properties of the arterial media

  • A. Comerford
  • K. Y. Chooi
  • M. Nowak
  • P. D. Weinberg
  • S. J. Sherwin
Original Paper


The medial layer of the arterial wall may play an important role in the regulation of water and solute transport across the wall. In particular, a high medial resistance to transport could cause accumulation of lipid-carrying molecules in the inner wall. In this study, the water transport properties of medial tissue were characterised in a numerical model, utilising experimentally obtained data for the medial microstructure and the relative permeability of different constituents. For the model, a new solver for flow in porous materials, based on a high-order splitting scheme, was implemented in the spectral/hp element library nektar++ and validated. The data were obtained by immersing excised aortic bifurcations in a solution of fluorescent protein tracer and subsequently imaging them with a confocal microscope. Cuboidal regions of interest were selected in which the microstructure and relative permeability of different structures were transformed to a computational mesh. Impermeable objects were treated fictitiously in the numerical scheme. On this cube, a pressure drop was applied in the three coordinate directions and the principal components of the permeability tensor were determined. The reconstructed images demonstrated the arrangement of elastic lamellae and interspersed smooth muscle cells in rat aortic media; the distribution and alignment of the smooth muscle cells varied spatially within the extracellular matrix. The numerical simulations highlighted that the heterogeneity of the medial structure is important in determining local water transport properties of the tissue, resulting in regional and directional variation of the permeability tensor. A major factor in this variation is the alignment and density of smooth muscle cells in the media, particularly adjacent to the adventitial layer.


Permeability Porous media Medial layer Atherosclerosis Spectral/hp element method 



This study was funded by the European Commission Marie Curie Integration Fellowship, the British Heart Foundation and the BHF Centre of Research Excellence.


  1. Ai L, Vafai K (2006) A coupling model for macromolecule transport in a stenosed arterial wall. Int J Heat Mass Transf 49(9):1568–1591CrossRefzbMATHGoogle Scholar
  2. Tedgui A, Lever MJ (1985) The interaction of convection and diffusion in the transport of 131i-albumin within the media of the rabbit thoracic aorta. Circ Res 57(6):856–863CrossRefGoogle Scholar
  3. Bozsak F, Chomaz J-M, Barakat AI (2014) Modeling the transport of drugs eluted from stents: physical phenomena driving drug distribution in the arterial wall. Biomech Model Mechanobiol 13(2):327–347CrossRefGoogle Scholar
  4. Caro CG, Lever MJ, Lever-Rudich Z, Meyer F, Liron N, Ebel W, Parker KH, Winlove CP (1980) Net albumin transport across the wall of the rabbit common carotid artery perfused in situ. Atherosclerosis 37(4):497–511CrossRefGoogle Scholar
  5. Clark JM, Glagov S (1985) Transmural organization of the arterial media. the lamellar unit revisited. Arterioscler Thromb Vasc Biol 5(1):19–34CrossRefGoogle Scholar
  6. Clarke LA, Mohri Z, Weinberg PD (2012) High throughput en face mapping of arterial permeability using tile scanning confocal microscopy. Atherosclerosis 224(2):417–425CrossRefGoogle Scholar
  7. Cutri E, Zunino P, Morlacchi S, Chiastra C, Migliavacca F (2013) Drug delivery patterns for different stenting techniques in coronary bifurcations: a comparative computational study. Biomech Model Mechanobiol 12(4):657–669CrossRefGoogle Scholar
  8. Dabagh M, Jalali P, Konttinen YT (2009a) The study of wall deformation and flow distribution with transmural pressure by three-dimensional model of thoracic aorta wall. Med Eng Phys 31(7):816–824CrossRefGoogle Scholar
  9. Dabagh M, Jalali P, Tarbell JM (2009b) The transport of ldl across the deformable arterial wall: the effect of endothelial cell turnover and intimal deformation under hypertension. Am J Physiol Heart Circul Physiol 297(3):H983–H996CrossRefGoogle Scholar
  10. Denny WJ, O’Connell BM, Milroy J, Walsh MT (2013) An analysis of three dimensional diffusion in a representative arterial wall mass transport model. Ann Biomed Eng 41(5):1062–1073Google Scholar
  11. Dong S, Liu D, Maxey MR, Karniadakis GE (2004) Spectral distributed lagrange multiplier method: algorithm and benchmark tests. J Comput Phys 195(2):695–717CrossRefzbMATHMathSciNetGoogle Scholar
  12. Durlofsky LJ (2005) Upscaling and gridding of fine scale geological models for flow simulation. 8th International Forum on Reservoir Simulation Iles Borromees. Stresa, Italy, pp 20–24Google Scholar
  13. Düster A, Parvizian J, Yang Z, Rank E (2008) The finite cell method for three-dimensional problems of solid mechanics. Comput Method Appl Mech Eng 197(45):3768–3782CrossRefzbMATHGoogle Scholar
  14. Fothergill JE (1964) Fluorochromes and their conjugation with proteins. In: Nairn RC (ed) Fluorescent protein tracing, 2nd edn. Edinburgh, Livingstone, pp 4–33Google Scholar
  15. Goriely AR, Baldwin AL, Secomb TW (2007) Transient diffusion of albumin in aortic walls: effects of binding to medial elastin layers. Am J Physiol Heart Circul Physiol 292(5):H2195–H2201CrossRefGoogle Scholar
  16. Hansbo A, Hansbo P (2002) An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput Method Appl Mech Eng 191(47):5537–5552CrossRefzbMATHMathSciNetGoogle Scholar
  17. Hou TY, Wu X (1997) A multiscale finite element method for elliptic problems in composite materials and porous media. J Comput Phys 134(1):169–189CrossRefzbMATHMathSciNetGoogle Scholar
  18. Huang ZJ, Tarbell JM (1997) Numerical simulation of mass transfer in porous media of blood vessel walls. American Journal of Physiology-Heart and Circulatory Physiology 273(1):H464–H477Google Scholar
  19. Hwang WR, Advani SG (2010) Numerical simulations of stokes-brinkman equations for permeability prediction of dual scale fibrous porous media. Phys Fluids 22:113101CrossRefGoogle Scholar
  20. Karniadakis GE, Israeli M, Orszag SA (1991) High-order splitting methods for the incompressible Navier-stokes equations. J Comput Phys 97(2):414–443CrossRefzbMATHMathSciNetGoogle Scholar
  21. Karniadakis GE, Sherwin SJ (2005) Spectral/hp element methods for CFD. Oxford University Press, OxfordCrossRefGoogle Scholar
  22. Katora ME, Hollis TM (1975) A simple fluorescent method for quantitative determination of aortic protein uptake. J Appl Phys 39(1):145–149Google Scholar
  23. Khadra K, Angot P, Parneix S, Caltagirone J-P (2000) Fictitious domain approach for numerical modelling of Navier-stokes equations. Int J Numer Methods Fluids 34(8):651–684CrossRefzbMATHGoogle Scholar
  24. Khakpour M, Vafai K (2008) Critical assessment of arterial transport models. Int J Heat Mass Transf 51(3):807–822CrossRefzbMATHGoogle Scholar
  25. Khaled ARA, Vafai K (2003) The role of porous media in modeling flow and heat transfer in biological tissues. Int J Heat Mass Transf 46(26):4989–5003CrossRefzbMATHGoogle Scholar
  26. Lever MJ, Jay M (1990) Albumin and cr-edta uptake by systemic arteries, veins, and pulmonary artery of rabbit. Arterioscler Thromb Vasc Biol 10(4):551–558CrossRefGoogle Scholar
  27. Lever MJ, Jay MT, Coleman PJ (1996) Plasma protein entry and retention in the vascular wall: possible factors in atherogenesis. Can J Physiol Pharmacol 74(7):818–823CrossRefGoogle Scholar
  28. Levick JR (1987) Flow through interstitium and other fibrous matrices. Exp Phys 72(4):409–437CrossRefGoogle Scholar
  29. Liu X, Fedkiw RP, Kang M (2000) A boundary condition capturing method for Poisson’s equation on irregular domains. J Comput Phys 160(1):151–178CrossRefzbMATHMathSciNetGoogle Scholar
  30. Masud A (2007) A stabilized mixed finite element method for darcy-stokes flow. Int J Numer Methods Fluids 54(6–8):665–681CrossRefzbMATHMathSciNetGoogle Scholar
  31. Nielsen LB (1996) Transfer of low density lipoprotein into the arterial wall and risk of atherosclerosis. Atherosclerosis 123(1):1–15CrossRefGoogle Scholar
  32. Parker KH, Winlove CP (1988) The macromolecular and ultrastructural basis of the permeability properties of the vascular wall. Eng Med 17(4):175–180CrossRefGoogle Scholar
  33. Prosi M, Zunino P, Perktold K, Quarteroni A (2005) Mathematical and numerical models for transfer of low-density lipoproteins through the arterial walls: a new methodology for the model set up with applications to the study of disturbed lumenal flow. J Biomech 38(4):903–917CrossRefGoogle Scholar
  34. Schillinger D, Düster A, Rank E (2012) The hp-d-adaptive finite cell method for geometrically nonlinear problems of solid mechanics. Int J Numer Methods Eng 89(9):1171–1202CrossRefzbMATHGoogle Scholar
  35. Sherwin SJ, Casarin M (2001) Low-energy basis preconditioning for elliptic substructured solvers based on unstructured spectral/hp element discretization. J Comput Phys 171(1):394–417CrossRefzbMATHGoogle Scholar
  36. Sun N, Torii R, Wood NB, Hughes AD, Thom SA, Xu XY (2009) Computational modeling of ldl and albumin transport in an in vivo ct image-based human right coronary artery. J Biomech Eng 131(2):021003–021003CrossRefGoogle Scholar
  37. Tada S, Tarbell JM (2000) Interstitial flow through the internal elastic lamina affects shear stress on arterial smooth muscle cells. Am J Physiol Heart Circ Physiol 278(5):H1589–H1597Google Scholar
  38. Tada S, Tarbell JM (2002) Flow through internal elastic lamina affects shear stress on smooth muscle cells (3d simulations). Am J Physiol Heart Circ Physiol 282(2):H576–H584CrossRefGoogle Scholar
  39. Tada S, Tarbell JM (2004) Internal elastic lamina affects the distribution of macromolecules in the arterial wall: a computational study. Am J Physiol Heart Circ Physiol 287(2):H905–H913CrossRefGoogle Scholar
  40. Tarbell JM (2003) Mass transport in arteries and the localization of atherosclerosis. Ann Rev Biomed Eng 5(1):79–118 Google Scholar
  41. Tarbell JM (2010) Shear stress and the endothelial transport barrier. Cardiovasc Res 87(2):320–330CrossRefGoogle Scholar
  42. Vairo G, Cioffi M, Cottone R, Dubini G, Migliavacca F (2010) Drug release from coronary eluting stents: a multidomain approach. J Biomech 43(8):1580–1589CrossRefGoogle Scholar
  43. Vos PEJ, van Loon R, Sherwin SJ (2008) A comparison of fictitious domain methods appropriate for spectral/hp element discretisations. Comput Method Appl Mech Eng 197(25):2275–2289CrossRefzbMATHGoogle Scholar
  44. Wang DM, Tarbell JM (1995) Modeling interstitial flow in an artery wall allows estimation of wall shear stress on smooth muscle cells. J Biomech Eng 117:359Google Scholar
  45. Warboys CM, Berson RE, Mann GE, Pearson JD, Weinberg PD (2010) Acute and chronic exposure to shear stress have opposite effects on endothelial permeability to macromolecules. Am J Physiol Heart Circ Physiol 298(6):H1850–H1856CrossRefGoogle Scholar
  46. Weinberg PD (2004) Rate-limiting steps in the development of atherosclerosis: the response-to-influx theory. J Vasc Res 41(1):1–17CrossRefGoogle Scholar
  47. Yang C, Grattoni CA, Muggeridge AH, Zimmerman RW (2002) Flow of water through channels filled with deformable polymer gels. J Colloid Interface Sci 250(2):466–470Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • A. Comerford
    • 1
  • K. Y. Chooi
    • 2
  • M. Nowak
    • 1
  • P. D. Weinberg
    • 2
  • S. J. Sherwin
    • 1
  1. 1.Department of AeronauticsImperial College LondonLondonUK 
  2. 2.Department of BioengineeringImperial College LondonLondonUK

Personalised recommendations