Transport in Porous Media

, Volume 96, Issue 2, pp 255–270 | Cite as

Numerical Approach of the Permeability of a Macroporous Bioceramic with Interconnected Spherical Pores

  • Trong-Khoa Nguyen
  • Olivier CarpentierEmail author
  • Philippe Herin
  • Philippe Hivart


Manufacturing a hybrid bone substitute requires a dynamic culture of the cells preliminarily seeded in a scaffold through a flow of physiological fluid. The velocity, pressure, and the distribution of fluid flow in this kind of macroporous medium are the important keys. Because of the difficulties in determining these parameters by experiment, a numerical approach has been chosen. One of the primary step of this study consists in the determination of permeability K. In this article, two types of structure of macroporous bioceramics are concerned. One is the interconnected pore spheres arranged either simple cubic, body-centered cubic or face-centered cubic systems. The other is the interconnected pore spheres randomly arranged. Based on Darcy’s law, the permeability K was calculated for many cases (type, porosity) by simulating the fluid flow through a small representative volume. These results are compared with some previous models such as Ergun, Carman–Kozeny, Rumpf–Gupte, and Du Plessis. The limits of Darcy’s law and the above-mentioned models have been determined using numerical simulation. The result showed that the porous media with spherical interconnected pores of BCC systems can be used to replace a complex random system in a range of porosity from 0.71 to 0.76 (i.e., porosity of our scaffolds). This assumption is validated for a pressure gradient lower the 1,000 Pa m–1 and a simple polynomial relation linking permeability and porosity (0.71–0.76) has been established.


Bioceramics Macroporosity Laminar flow Permeability Unit cell Numerical simulation 



Body centered cubic


Face centered cubic


β-Tricalcium phosphate


Representative elementary volume


Simple cubic


Unit cell


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  1. Bachmat Y., Bear J.: Macroscopic modelling of transport phenomena in porous media. 1: the continuum approach. Transp. Porous Media 1, 213–240 (1986)CrossRefGoogle Scholar
  2. Bashoor-Zadeh M., Baroud G., Bohner M.: Geometric analysis of porous bone substitutes using micro- computed tomography and fuzzy distance transform. Acta Biomater. 6(3), 75–867 (2010)CrossRefGoogle Scholar
  3. Belluci D., Cannillo V., Sola A.: A new approach towards high strength bioceramic scaffolds for bone generation. Mater. Lett. 64(2), 203–206 (2010)CrossRefGoogle Scholar
  4. Berthier J.S.P: Microfluidics for Biotechnology. Artech House, Boston (2006)Google Scholar
  5. Bird R., Stewart W., Lightfoot E.: Transport Phenomena. Wiley, New York (2002)CrossRefGoogle Scholar
  6. Bohner M., Lenthe G.V., Grnenfelder S., Hirsiger W., Evison R., Mller R.: Synthesis and characterization of porous beta-tricalcium phosphate blocks. Biomaterials 26(31), 6099 (2005)CrossRefGoogle Scholar
  7. Boomsma K., Poulikakos D., Ventikos Y.: Simulations of flow through open cell metal foams using an idealized periodic cell structure. Int. J. Heat Fluid Flow 24, 825–834 (2003)CrossRefGoogle Scholar
  8. Botchwey E., Pollack S., El-Amin S., Levine E., Tuan R.L. C.T: Human osteoblast-like cells in three-dimensional culture with fluid flow. Biorheology 40, 299–306 (2003)Google Scholar
  9. Carman P.: Fluid flow through granular beds. Chem. Eng. Res. Des. 75, 32–48 (1997)CrossRefGoogle Scholar
  10. Costa U., Andrade Jr, J., Makse H.S.H.E: The role of inertia on fluid flow through disordered porous media. Phys. A Stat. Mech. Appl. 266, 420–424 (1999)CrossRefGoogle Scholar
  11. Damien E., Hing K., Sareed S., Revell P.: A preliminary study on the enhancement of the osteointegration of a novel synthetic hydroxyapatite scaffold in vivo. J. Biomater. Mater. Res. 66(2), 241–246 (2003)CrossRefGoogle Scholar
  12. Darcy H.: Les fontaines publiques de la ville de Dijon. Dalmont, Paris (1856)Google Scholar
  13. Descamps M., Duhoo T., Monchau F., Lu J., Hardouin P., Hornez J., Leriche A.: Manufacture of macroporous-tricalcium phosphate bioceramics. J. Eur. Ceram. Soc. 28, 149–157 (2008a)CrossRefGoogle Scholar
  14. Descamps M., Richart O., Hardouin P., Hornez J., Leriche A.: Synthesis of macroporous [beta]-tricalcium phosphate with controlled porous architectural. Ceram. Int. 34, 1131–1137 (2008b)CrossRefGoogle Scholar
  15. Dias M., Fernandes P., Guedes J., Hollister S.: Permeability analysis fo scaffolds for bone tissue engineering. J. Biomech. 45, 938–944 (2012)CrossRefGoogle Scholar
  16. Dong J., Kojima H., Uemura T., Kikuchi M., Tateishi T., Tanaka J.: In vivo evaluation of a novel porous hydroxyapatite to sustain osteogenesis of transplanted bone narrow-derived osteoblastic cells. J. Biomater. Mater. Res. 57(2), 208–216 (2001)CrossRefGoogle Scholar
  17. Du Plessis J.P.: Analytical quantification of coefficients in the ergun equation for fluid friction in a packed bed. Transp. Porous Media 16, 189–207 (1994)CrossRefGoogle Scholar
  18. Dullien F.: Porous media Fluid Transport and Pore Structure. Academic Press, London (1979)Google Scholar
  19. Ergun S.: Fluid flow through packed columns. Chem. Eng. Process. 48, 89–94 (1952)Google Scholar
  20. Flautre B., Descamps M., Delecourt C., Blary M., Hardouin P.: Porous ha ceramic for bone replacement: role of the pores and interconnections experimental study in the rabbit. J. Mater. Sci. 12, 679–682 (2001)CrossRefGoogle Scholar
  21. Getachew D., Minkowycz W., Poulikakos D.: Macroscopic equations of non-Newtonian fluid flow and heat transfer in a porous matrix. J. Porous Media 1, 273–283 (1998)Google Scholar
  22. Harris M., Doraiswamy A., Narayan R., Patz T., Chrisey D.: Recent progress in cad/cam laser direct-writing of biomaterials. Mater. Sci. Eng. C 28(3), 359–365 (2008)CrossRefGoogle Scholar
  23. Hsu C., Cheng P.: Thermal dispersion in a porous medium. Int. J. Heat Mass Transf. 33, 1587–1597 (1990)CrossRefGoogle Scholar
  24. Karageorgiou V., Kaplan D.: Porosity of 3d biomaterial scaffolds and osteogenesis. Biomaterials 26, 5474–5491 (2005)CrossRefGoogle Scholar
  25. Kasten P., Beyen I., Niemeyer P., Luginbühl R., Bohner M., Richter W.: Porosity and pore size of β-tricalcium phosphate scaffold can influence protein production and osteogenic differentiation of human mesenchymal stem cells: an in vitro and in vivo study. Acta Biomater. 4, 1904–1915 (2008)CrossRefGoogle Scholar
  26. Kececioglu I., Jiang Y.: Flow through porous media of packed spheres saturated with water. J. Fluids Eng. 116, 164–170 (1994)CrossRefGoogle Scholar
  27. Khaled A., Vafai K.: The role of porous media in modeling flow and heat transfer in biological tissues, Int. J. Heat Mass Transf. 46, 4989–5003 (2003)CrossRefGoogle Scholar
  28. Kruyt M., Bruijin J., Wilson C., Oner F., Blitterswijk C.V., Verbout A.: Viable osteogenic cells are obligatory for tissue-engineered ectopic bone formation goats. Tissue Eng. 9(2), 327–336 (2003)CrossRefGoogle Scholar
  29. Lu J., Flautre B., Anselme K., Hardouin P., Gallur A., Descamps M., Thierry B.: Role of interconnections in porous bioceramics on bone recolonization in vitro and in vivo. J. Mater. Sci. 10, 111–120 (1999)CrossRefGoogle Scholar
  30. Olivier V., Faucheux N., Hardouin P.: Biomaterial challenges and approaches to stem cell use in bone reconstructive surgery. Drug Discov. Today 9, 803–811 (2004)CrossRefGoogle Scholar
  31. Robinson D., Friedman S.: Electrical conductivity and dielectric permittivity of sphere packings: Measurements and modelling of cubic lattices, randomly packed monosize spheres and multi-size mixtures. Phys. A Stat. Mech. Appl. 358, 447–465 (2005)CrossRefGoogle Scholar
  32. Rumpf H., Gupte A.: Einflsse der porositt und korngrenverteilung im widerstandsgesetz der porenstrmung. Chem. Ing. Tech. 43, 367–375 (1971)CrossRefGoogle Scholar
  33. Shenoy A.: Non-Newtonian fluid heat transfer in porous media. Adv. Heat Transf. 24, 101–190 (1994)CrossRefGoogle Scholar
  34. Singh R., Lee P., Lindley T.C., Dashwood R., Ferrie E., Imwinkelried T.: Characterization of the structure and permeability of titanium foams for spinal fusion devices. Acta Biomater. 5(1), 477–487 (2009)CrossRefGoogle Scholar
  35. Takagi K., Takahashi T., Kikuchi K., Kawasaki A.: Fabrication of bioceramic scaffolds with ordered pore structure by inverse replication of assembled particles. J. Eur. Ceram. Soc. 30(10), 2049–2055 (2010)CrossRefGoogle Scholar
  36. Tian W., Finehout E.: Microfluidics for Biological Applications. Springer Science + Business Media, New York (2008)Google Scholar
  37. Truscello S., Kerckhofs G., Bael S.V., Pyka G., Schrooten J., Oosterwyck H.V.: Prediction of permeability of regular scaffolds for skeletal tissue engineering: a combined computational and experimental study. Acta Biomater. 8(4), 1648–1658 (2012)CrossRefGoogle Scholar
  38. Valdes-Parada F., Ochoa-Tapia J.A., Alvarez-Ramirez J.: On the effective viscosity for the darcybrinkman equation. Phys. A Stat. Mech. Appl. 385, 69–79 (2007)CrossRefGoogle Scholar
  39. Valdes-Parada F., Ochoa-Tapia J., Alvarez-Ramirez J.: Validity of the permeability Carman–Kozeny equation: a volume averaging approach. Phys. A Stat. Mech. Appl. 388, 789–798 (2009)CrossRefGoogle Scholar
  40. Whitaker S.: The Method of Volume Averaging. Kluwer, Dordrecht (1999)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Trong-Khoa Nguyen
    • 1
  • Olivier Carpentier
    • 1
    Email author
  • Philippe Herin
    • 1
  • Philippe Hivart
    • 1
  1. 1.Laboratoire de Génie-Civil etgéo-Environnement, PRES Lille Nord de France, EquipeBiomatériaux ArtoisUniversité d’ArtoisBéthuneFrance

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