Annals of Biomedical Engineering

, Volume 45, Issue 6, pp 1543–1554 | Cite as

Anisotropic Permeability of Trabecular Bone and its Relationship to Fabric and Architecture: A Computational Study

Article
  • 261 Downloads

Abstract

Trabecular bone is a porous, mineralized tissue found in vertebral bodies, the metaphyses and epiphyses of long bones, and in the irregular and flat shaped bones. The pore space is filled with bone marrow, a highly cellular fluid. Together, the bone and marrow behave as a poroelastic solid. In poroelasticity theory, the permeability is the primary material property that governs the momentum transfer between the solid and fluid constituents. In the linearized theory, the permeability of a material depends on the shape and connectivity of the pores. Developing a model of the relationship between trabecular microarchitecture and permeability could lead to improved simulations of trabecular bone mechanical response, which can be used to investigate bone adaptation, mechanobiological signaling, and progression of diseases such as osteoporosis. This study used finite element models of the trabecular pore space to calculate the complete anisotropic permeability tensor of 12 human and 18 porcine femoral trabecular bone samples. The sensitivity of the simulations to model assumptions and post-processing was analyzed to improve confidence in the result. The orthotropic permeability tensor depended on the fabric tensor, trabecular spacing, and structure model index through a power law relationship. Porosity and fabric alone also provided a reasonable prediction, which may be useful in cases where the image resolution is insufficient to obtain detailed measures of architecture.

Keywords

Permeability Poroelasticity Trabecular bone Trabecular architecture Computational modeling Bone marrow Size effects Convergence 

Notes

Acknowledgments

This research was supported by the U.S. National Science Foundation Award CMMI 1435467. We wish to thank an anonymous reviewer for encouraging us to explore the Kozeny–Carman relationship further.

Conflicts of Interest

The authors have no conflicts of interest with the material presented in this paper.

References

  1. 1.
    Abdalrahman, T., S. Scheiner, and C. Hellmich. Is trabecular bone permeability governed by molecular ordering-induced fluid viscosity gain? Arguments from re-evaluation of experimental data in the framework of homogenization theory. J. Theor. Biol. 365:433–444, 2015.CrossRefPubMedGoogle Scholar
  2. 2.
    Arramon, Y. P., and E. A. Nauman. The intrinsic permeability of cancellous bone. In: The Bone Mechanics Handbook, edited by S. C. Cowin. New York: CRC, 2001.Google Scholar
  3. 3.
    Atkin, R. J., and R. E. Craine. Continuum theories of mixtures—basic theory and historical development. Q. J. Mech. Appl. Math. 29:209–244, 1976.CrossRefGoogle Scholar
  4. 4.
    Bathe, K. J., and H. Zhang. A flow-condition-based interpolation finite element procedure for incompressible fluid flows. Comput. Struct. 80:1267–1277, 2002.CrossRefGoogle Scholar
  5. 5.
    Biot, M. A. General theory of three-dimensional consolidation. J. Appl. Phys. 12:155–164, 1941.CrossRefGoogle Scholar
  6. 6.
    Birmingham, E., J. A. Grogan, G. L. Niebur, L. M. McNamara, and P. E. McHugh. Computational modelling of the mechanics of trabecular bone and marrow using fluid structure interaction techniques. Ann. Biomed. Eng. 41:814–826, 2013.CrossRefPubMedGoogle Scholar
  7. 7.
    Birmingham, E., T. C. Kreipke, E. B. Dolan, T. R. Coughlin, P. Owens, L. M. McNamara, G. L. Niebur, and P. E. McHugh. Mechanical stimulation of bone marrow in situ induces bone formation in trabecular explants. Ann. Biomed. Eng. 43:1036–1050, 2014.CrossRefPubMedGoogle Scholar
  8. 8.
    Bryant, J. D., T. David, P. H. Gaskell, S. King, and G. Lond. Rheology of bovine bone marrow. Proc. Inst. Mech. Eng. H 203:71–75, 1989.CrossRefPubMedGoogle Scholar
  9. 9.
    Carballido-Gamio, J., R. Harnish, I. Saeed, T. Streeper, S. Sigurdsson, S. Amin, E. J. Atkinson, T. M. Therneau, K. Siggeirsdottir, X. Cheng, L. J. Melton, 3rd, J. Keyak, V. Gudnason, S. Khosla, T. B. Harris, and T. F. Lang. Proximal femoral density distribution and structure in relation to age and hip fracture risk in women. J. Bone Mineral Res. 28:537–546, 2013.CrossRefGoogle Scholar
  10. 10.
    Carman, P. C. Fluid flow through granular beds. Chem. Eng. Res. Des. 15:415–421, 1937.Google Scholar
  11. 11.
    Chen, X. M., and T. D. Papathanasiou. On the variability of the Kozeny constant for saturated flow across unidirectional disordered fiber arrays. Composites A 37:836–846, 2006.CrossRefGoogle Scholar
  12. 12.
    Coughlin, T. R., and G. L. Niebur. Fluid shear stress in trabecular bone marrow due to low-magnitude high-frequency vibration. J. Biomech. 45:2222–2229, 2012.CrossRefPubMedGoogle Scholar
  13. 13.
    Coughlin, T. R., J. Schiavi, M. Alyssa Varsanik, M. Voisin, E. Birmingham, M. G. Haugh, L. M. McNamara, and G. L. Niebur. Primary cilia expression in bone marrow in response to mechanical stimulation in explant bioreactor culture. Eur. Cell Mater. 32:111–122, 2016.CrossRefPubMedGoogle Scholar
  14. 14.
    Cowin, S. C. The relationship between the elasticity tensor and the fabric tensor. Mech. Mater. 4:137–147, 1985.CrossRefGoogle Scholar
  15. 15.
    Cowin, S. C. Bone poroelasticity. J. Biomech. 32:217–238, 1999.CrossRefPubMedGoogle Scholar
  16. 16.
    Cowin, S. C. A recasting of anisotropic poroelasticity in matrices of tensor components. Transp. Porous Media 50:35–56, 2003.CrossRefGoogle Scholar
  17. 17.
    Cowin, S. C. Anisotropic poroelasticity: fabric tensor formulation. Mech. Mater. 36:665–677, 2004.CrossRefGoogle Scholar
  18. 18.
    Dickerson, D. A., E. A. Sander, and E. A. Nauman. Modeling the mechanical consequences of vibratory loading in the vertebral body: microscale effects. Biomech. Model. Mechanobiol. 7:191–202, 2008.CrossRefPubMedGoogle Scholar
  19. 19.
    Downey, D. J., P. A. Simkin, and R. Taggart. The effect of compressive loading on intraosseous pressure in the femoral head in vitro. J. Bone Joint Surg. Am. 70:871–877, 1988.CrossRefPubMedGoogle Scholar
  20. 20.
    Ferguson, S. J., K. Ito, and L. P. Nolte. Fluid flow and convective transport of solutes within the intervertebral disc. J. Biomech. 37:213–221, 2004.CrossRefPubMedGoogle Scholar
  21. 21.
    Grimm, M. J., and J. L. Williams. Measurements of permeability in human calcaneal trabecular bone. J. Biomech. 30:743–745, 1997.CrossRefPubMedGoogle Scholar
  22. 22.
    Gurkan, U. A., and O. Akkus. The mechanical environment of bone marrow: a review. Ann. Biomed Eng. 36:1978–1991, 2008.CrossRefPubMedGoogle Scholar
  23. 23.
    Harrigan, T. P., M. Jasty, R. W. Mann, and W. H. Harris. Limitations of the continuum assumption in cancellous bone. J. Biomech. 21:269–275, 1988.CrossRefPubMedGoogle Scholar
  24. 24.
    Hildebrand, T., A. Laib, R. Muller, J. Dequeker, and P. Ruegsegger. Direct three-dimensional morphometric analysis of human cancellous bone: microstructural data from spine, femur, iliac crest, and calcaneus. J. Bone Miner. Res. 14:1167–1174, 1999.CrossRefPubMedGoogle Scholar
  25. 25.
    Keaveny, T. M., E. F. Morgan, G. L. Niebur, and O. C. Yeh. Biomechanics of trabecular bone. Annu. Rev. Biomed. Eng. 3:307–333, 2001.CrossRefPubMedGoogle Scholar
  26. 26.
    Keaveny, T. M., and O. C. Yeh. Architecture and trabecular bone—toward an improved understanding of the biomechanical effects of age, sex and osteoporosis. J. Musculoskelet. Neuronal Interact. 2:205–208, 2002.PubMedGoogle Scholar
  27. 27.
    Kilinc, S., U. A. Gurkan, S. Guven, G. Koyuncu, S. Tan, C. Karaca, O. Ozdogan, M. Dogan, C. Tugmen, E. E. Pala, U. Bayol, M. Baran, Y. Kurtulmus, I. Pirim, E. Kebapci, and U. Demirci. Evaluation of epithelial chimerism after bone marrow mesenchymal stromal cell infusion in intestinal transplant patients. Transplant. Proc. 46:2125–2132, 2014.CrossRefPubMedGoogle Scholar
  28. 28.
    Kim, Y. J., and J. Henkin. Micro-computed tomography assessment of human alveolar bone: bone density and three-dimensional micro-architecture. Clin. Implant Dent. Relat. Res. 17:307–313, 2015.CrossRefPubMedGoogle Scholar
  29. 29.
    Kohles, S. S., and J. B. Roberts. Linear poroelastic cancellous bone anisotropy: trabecular solid elastic and fluid transport properties. J. Biomech. Eng. 124:521–526, 2002.CrossRefPubMedGoogle Scholar
  30. 30.
    Kohles, S. S., J. B. Roberts, M. L. Upton, C. G. Wilson, L. J. Bonassar, and A. L. Schlichting. Direct perfusion measurements of cancellous bone anisotropic permeability. J. Biomech. 34:1197–1202, 2001.CrossRefPubMedGoogle Scholar
  31. 31.
    Krishnamoorthy, D., D. M. Frechette, B. J. Adler, D. E. Green, M. E. Chan, and C. T. Rubin. Marrow adipogenesis and bone loss that parallels estrogen deficiency is slowed by low-intensity mechanical signals. Osteoporos. Int. 27:747–756, 2016.CrossRefPubMedGoogle Scholar
  32. 32.
    Lorentzon, M., and S. R. Cummings. Osteoporosis: the evolution of a diagnosis. J. Intern. Med. 277:650–661, 2015.CrossRefPubMedGoogle Scholar
  33. 33.
    Lynch, M. E., D. Brooks, S. Mohanan, M. J. Lee, P. Polamraju, K. Dent, L. J. Bonassar, M. C. van der Meulen, and C. Fischbach. In vivo tibial compression decreases osteolysis and tumor formation in a human metastatic breast cancer model. J. Bone Miner. Res. 28:2357–2367, 2013.CrossRefPubMedPubMedCentralGoogle Scholar
  34. 34.
    Lynch, M. E., A. Chiou, M. J. Lee, S. C. Marcott, P. V. Polamraju, Y. Lee, and C. Fischbach. 3D mechanical loading modulates the osteogenic response of mesenchymal stem cells to tumor-derived soluble signals. Tissue Eng. A 22:1006–1015, 2016.CrossRefGoogle Scholar
  35. 35.
    Lynch, M. E., and C. Fischbach. Biomechanical forces in the skeleton and their relevance to bone metastasis: Biology and engineering considerations. Adv. Drug Deliv. Rev. 79:119–134, 2014.CrossRefPubMedGoogle Scholar
  36. 36.
    Mantila Roosa, S. M., Y. Liu, and C. H. Turner. Gene expression patterns in bone following mechanical loading. J. Bone Miner. Res. 26:100–112, 2011.CrossRefPubMedGoogle Scholar
  37. 37.
    Mantila Roosa, S. M., C. H. Turner, and Y. Liu. Regulatory mechanisms in bone following mechanical loading. Gene Regul. Syst. Biol. 6:43–53, 2012.CrossRefGoogle Scholar
  38. 38.
    McCloskey, E. V., A. Oden, N. C. Harvey, W. D. Leslie, D. Hans, H. Johansson, R. Barkmann, S. Boutroy, J. Brown, R. Chapurlat, P. J. Elders, Y. Fujita, C. C. Gluer, D. Goltzman, M. Iki, M. Karlsson, A. Kindmark, M. Kotowicz, N. Kurumatani, T. Kwok, O. Lamy, J. Leung, K. Lippuner, O. Ljunggren, M. Lorentzon, D. Mellstrom, T. Merlijn, L. Oei, C. Ohlsson, J. A. Pasco, F. Rivadeneira, B. Rosengren, E. Sornay-Rendu, P. Szulc, J. Tamaki, and J. A. Kanis. A meta-analysis of trabecular bone score in fracture risk prediction and its relationship to FRAX. J. Bone Miner. Res. 31:940–948, 2016.CrossRefPubMedGoogle Scholar
  39. 39.
    Metzger, T. A., T. C. Kreipke, T. J. Vaughan, L. M. McNamara, and G. L. Niebur. The in situ mechanics of trabecular bone marrow: the potential for mechanobiological response. J. Biomech. Eng. 137:011006, 2015.CrossRefGoogle Scholar
  40. 40.
    Metzger, T. A., S. A. Schwaner, A. J. LaNeve, T. C. Kreipke, and G. L. Niebur. Pressure and shear stress in trabecular bone marrow during whole bone loading. J. Biomech. 48:3035–3043, 2015.CrossRefPubMedGoogle Scholar
  41. 41.
    Metzger, T. A., J. M. Shudick, R. Seekell, Y. Zhu, and G. L. Niebur. Rheological behavior of fresh bone marrow and the effects of storage. J. Mech. Behav. Biomed. Mater. 40C:307–313, 2014.CrossRefGoogle Scholar
  42. 42.
    Morgan, E. F., and T. M. Keaveny. Dependence of yield strain of human trabecular bone on anatomic site. J. Biomech. 34:569–577, 2001.CrossRefPubMedGoogle Scholar
  43. 43.
    Nauman, E. A., K. E. Fong, and T. M. Keaveny. Dependence of intertrabecular permeability on flow direction and anatomic site. Ann. Biomed. Eng. 27:517–524, 1999.CrossRefPubMedGoogle Scholar
  44. 44.
    Parfitt, A. M. The cellular basis of bone turnover and bone loss: a rebuttal of the osteocytic resorption–bone flow theory. Clin. Orthop. Relat. Res. 127:236–247, 1977.Google Scholar
  45. 45.
    Parfitt, A. M. Quantum concept of bone remodeling and turnover: implications for the pathogenesis of osteoporosis. Calcif. Tissue Int. 28:1–5, 1979.CrossRefPubMedGoogle Scholar
  46. 46.
    Rahmoun, J., F. Chaari, E. Markiewicz, and P. Drazetic. Micromechanical modeling of the anisotropy of elastic biological composites. Multiscale Model. Simul. 8:326–336, 2009.CrossRefGoogle Scholar
  47. 47.
    Sander, E. A., and E. A. Nauman. Permeability of musculoskeletal tissues and scaffolding materials: experimental results and theoretical predictions. Crit. Rev. Biomed. Eng. 31:1–26, 2003.CrossRefPubMedGoogle Scholar
  48. 48.
    Sander, E. A., D. A. Shimko, K. C. Dee, and E. A. Nauman. Examination of continuum and micro-structural properties of human vertebral cancellous bone using combined cellular solid models. Biomech. Model. Mechanobiol. 2:97–107, 2003.CrossRefPubMedGoogle Scholar
  49. 49.
    Sandino, C., P. Kroliczek, D. D. McErlain, and S. K. Boyd. Predicting the permeability of trabecular bone by micro-computed tomography and finite element modeling. J. Biomech. 47:3129–3134, 2014.CrossRefPubMedGoogle Scholar
  50. 50.
    Soves, C. P., J. D. Miller, D. L. Begun, R. S. Taichman, K. D. Hankenson, and S. A. Goldstein. Megakaryocytes are mechanically responsive and influence osteoblast proliferation and differentiation. Bone 66C:111–120, 2014.CrossRefGoogle Scholar
  51. 51.
    Syahrom, A., M. R. Abdul Kadir, J. Abdullah, and A. Ochsner. Permeability studies of artificial and natural cancellous bone structures. Med. Eng. Phys. 35:792–799, 2013.CrossRefPubMedGoogle Scholar
  52. 52.
    Vaughan, T. J., M. Voisin, G. L. Niebur, and L. M. McNamara. Multiscale modeling of trabecular bone marrow: understanding the micromechanical environment of mesenchymal stem cells during osteoporosis. J. Biomech. Eng. 137:011003, 2015.CrossRefGoogle Scholar
  53. 53.
    Wang, X., X. Liu, and G. L. Niebur. Preparation of on-axis cylindrical trabecular bone specimens using micro-CT imaging. J. Biomech. Eng. 126:122–125, 2004.CrossRefPubMedGoogle Scholar
  54. 54.
    Whyne, C. M., S. S. Hu, K. L. Workman, and J. C. Lotz. Biphasic material properties of lytic bone metastases. Ann. Biomed. Eng. 28:1154–1158, 2000.CrossRefPubMedGoogle Scholar
  55. 55.
    Widmer, R. P., and S. J. Ferguson. On the interrelationship of permeability and structural parameters of vertebral trabecular bone: a parametric computational study. Comput. Methods Biomech. Biomed. Eng. 16:908–922, 2013.CrossRefGoogle Scholar
  56. 56.
    Wu, Z., A. J. Laneve, and G. L. Niebur. In vivo microdamage is an indicator of susceptibility to initiation and propagation of microdamage in human femoral trabecular bone. Bone 55:208–215, 2013.CrossRefPubMedPubMedCentralGoogle Scholar
  57. 57.
    Zeiser, T., M. Bashoor-Zadeh, A. Darabi, and G. Baroud. Pore-scale analysis of Newtonian flow in the explicit geometry of vertebral trabecular bones using lattice Boltzmann simulation. Proc. Inst. Mech. Eng. H 222:185–194, 2008.CrossRefPubMedGoogle Scholar
  58. 58.
    Zysset, P. K. A review of morphology-elasticity relationships in human trabecular bone: theories and experiments. J. Biomech. 36:1469–1485, 2003.CrossRefPubMedGoogle Scholar
  59. 59.
    Zysset, P. K., and A. Curnier. An Alternative model for anisotropic elasticity based on fabric tensors. Mech. Mater. 21:243–250, 1995.CrossRefGoogle Scholar

Copyright information

© Biomedical Engineering Society 2017

Authors and Affiliations

  1. 1.Tissue Mechanics Laboratory, Bioengineering Graduate Program, and Department of Aerospace and Mechanical EngineeringUniversity of Notre DameNotre DameUSA
  2. 2.147 Multidisciplinary Research BuildingNotre DameUSA

Personalised recommendations