Multi-scale modelling of elastic moduli of trabecular bone

Journal of the Royal Society Interface

Volumn 9, Issue 72, 2012, Pages 1654-1673

  Hamed, Elham ^a   Jasiuk, Iwona ^a   Yoo, Andrew ^b   Lee, YikHan ^a   Liszka, Tadeusz ^c  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^b ETHICON INC   (United States)

^c ALTAIR ENGINEERING   (United States)

Author keywords

Elastic moduli; Hierarchical structure; MicroCT based finite element modelling; Multi scale modelling; Nanocomposite; Trabecular bone

Indexed keywords

BONE; COMPOSITE MICROMECHANICS; COMPUTERIZED TOMOGRAPHY; ELASTIC MODULI; LAMINATED COMPOSITES; NANOCOMPOSITES;

CONTINUUM MICROMECHANICS; ELASTIC PROPERTIES; FINITE ELEMENT MODELLING; HIERARCHICAL STRUCTURES; LAMINATE THEORY; MESOSCALE; MICRO-SCALES; MINERALIZED COLLAGEN; MULTI-SCALE APPROACHES; MULTI-SCALE MODELLING; NANO SCALE; SINGLE LAMELLA; TRABECULAR BONES;

FINITE ELEMENT METHOD;

COLLAGEN FIBRIL; HYDROXYAPATITE; NANOCOMPOSITE;

ELASTICITY; HUMAN; HUMAN TISSUE; MICRO-COMPUTED TOMOGRAPHY; NANOANALYSIS; REVIEW; TRABECULAR BONE; YOUNG MODULUS;

TRABECULA;

EID: 84863596030     PISSN: 17425689     EISSN: 17425662     Source Type: Journal    
DOI: 10.1098/rsif.2011.0814     Document Type: Review

References (115)

1. Olszta, M. J., Cheng, X. G., Jee, S. S., Kumar, R., Kim, Y. Y., Kaufman, M. J., Douglas, E. P. & Gower, L. B. 2007 Bone structure and formation: a new perspective. Mater. Sci. Eng. R 58, 77-116. (doi:10.1016/j.mser.2007.05.001)
2. Buehler, M. J. 2008 Nanomechanics of collagen fibrils under varying cross-link densities: atomistic and continuum studies. J. Mech. Behav. Biomed. Mater. 1, 59-67. (doi:10.1016/j.jmbbm.2007.04.001)
3. Rho, J. Y., Kuhn-Spearing, L. & Zioupos, P. 1998 Mechanical properties and the hierarchical structure of bone. Med. Eng. Phys. 20, 92-102. (doi:10.1016/S1350-4533(98)00007-1)
4. Robinson, R. A. 1952 An electron-microscopic study of the crystalline inorganic component of bone and its relationship to the organic matrix. J. Bone Joint Surg. Am. 34-A, 389-435.
5. Jee, W. S. S. 2001 Bone mechanics handbook (ed. S. C. Cowin). Boca Raton, FL: CRC Press LLC.
6. Currey, J. D. 1969 Relationship between stiffness and mineral content of bone. J. Biomech. 2, 477-480. (doi:10.1016/0021-9290(69)90023-2)
7. Katz, J. L. 1971 Hard tissue as a composite material. I. Bounds on elastic behavior. J. Biomech. 4, 455-473. (doi:10.1016/0021-9290(71)90064-9)
8. Akkus, O. 2005 Elastic deformation of mineralized collagen fibrils: an equivalent inclusion based composite model. J. Biomech. Eng. 127, 383-390. (doi:10.1115/1.1894204)
9. Fritsch, A. & Hellmich, C. 2007 'Universal' microstructural patterns in cortical and trabecular, extracellular and extravascular bone materials: micromechanics-based prediction of anisotropic elasticity. J. Theor. Biol. 244, 597-620. (doi:10.1016/j.jtbi.2006.09.013)
10. Hamed, E., Lee, Y. & Jasiuk, I. 2010 Multiscale modeling of elastic properties of cortical bone. Acta Mech. 213, 131-154. (doi:10.1007/s00707-010- 0326-5)
11. Nikolov, S. & Raabe, D. 2008 Hierarchical modeling of the elastic properties of bone at submicron scales: the role of extrafibrillar mineralization. Biophys. J. 94, 4220-4232. (doi:10.1529/biophysj.107.125567)
12. Yoon, Y. J. & Cowin, S. C. 2008 The estimated elastic constants for a single bone osteonal lamella. Biomech. Model. Mechanobiol. 7, 1-11. (doi:10.1007/s10237-006-0072-8)
13. Ji, B. H. & Gao, H. J. 2006 Elastic properties of nanocomposite structure of bone. Compos. Sci. Technol. 66, 1212-1218. (doi:10.1016/j. compscitech.2005.10.017)
14. Siegmund, T., Allen, M. R. & Burr, D. B. 2008 Failure of mineralized collagen fibrils: modeling the role of collagen cross-linking. J. Biomech. 41, 1427-1435. (doi:10.1016/j.jbiomech.2008.02.017)
15. Yuan, F., Stock, S. R., Haeffner, D. R., Almer, J. D., Dunand, D. C. & Brinson, L. C. 2011 A new model to simulate the elastic properties of mineralized collagen fibril. Biomech. Model. Mechanobiol. 10, 147-160. (doi:10.1007/s10237-010-0223-9)
16. Buehler, M. J. 2007 Molecular architecture of collagen fibrils: a critical length scale for tough fibrils. Curr. Appl. Phys. 8, 440-442. (doi:10.1016/j.cap.2007.10.058)
17. Bhowmik, R., Katti, K. S. & Katti, D. R. 2007 Mechanics of molecular collagen is influenced by hydroxyapatite in natural bone. J. Mater. Sci. 42, 8795-8803. (doi:10.1007/s10853-007-1914-1)
18. Bhowmik, R., Katti, K. S. & Katti, D. R. 2009 Mechanisms of load-deformation behavior of molecular collagen in hydroxyapatite-tropocollagen molecular system: steered molecular dynamics study. J. Eng. Mech. 135, 413-421. (doi:10.1061/(ASCE)0733-9399(2009)135:5(413))
19. Buehler, M. J. 2007 Molecular nanomechanics of nascent bone: fibrillar toughening by mineralization. Nanotechnology 18, 295102. (doi:10.1088/0957-4484/ 18/29/295102)
20. Dubey, D. K. & Tomar, V. 2008 Microstructure dependent dynamic fracture analyses of trabecular bone based on nascent bone atomistic simulations. Mech. Res. Commun. 35, 24-31. (doi:10.1016/j.mechrescom.2007.10. 011)
21. Jasiuk, I. & Ostoja-Starzewski, M. 2004 Modeling of bone at a single lamella level. Biomech. Model. Mechanobiol. 3, 67-74. (doi:10.1007/s10237-004- 0048-5)
22. Bini, F., Marinozzi, A., Marinozzi, F. & Patane, F. 2002 Microtensile measurements of single trabeculae stiffness in human femur. J. Biomech. 35, 1515-1519. (doi:10.1016/S0021-9290(02)00182-3)
23. Rho, J. Y., Ashman, R. B. & Turner, C. H. 1993 Young's modulus of trabecular and cortical bone material: ultrasonic and microtensile measurements. J. Biomech. 26, 111-119. (doi:10.1016/0021-9290(93)90042-D)
24. Ryan, S. D. & Williams, J. L. 1989 Tensile testing of rodlike trabeculae excised from bovine femoral bone. J. Biomech. 22, 351-355. (doi:10.1016/0021-9290(89)90049-3)
25. Choi, K., Kuhn, J. L., Ciarelli, M. J. & Goldstein, S. A. 1990 The elastic moduli of human subchondral, trabecular, and cortical bone tissue and the size-dependency of cortical bone modulus. J. Biomech. 23, 1103-1113. (doi:10.1016/0021-9290(90)90003-L)
26. Tanaka, K., Kita, Y., Katayama, T. & Matsukawa, M. 2010 Mechanical properties of a single trabecula in bovine femur by the three point bending test. In IFMBE Proc., BME 2010, 11-14 January, Ho Chi Minh City, Vietnam, vol. 27, pp. 235-238. (doi:10.1007/978-3-642-12020-6-59)
27. Turner, C. H., Rho, J. Y., Takano, Y., Tsui, T. Y. & Pharr, G. M. 1999 The elastic properties of trabecular and cortical bone tissues are similar: results from two microscopic measurement techniques. J. Biomech. 32, 437-441. (doi:10.1016/S0021-9290(98)00177-8)
28. Jorgensen, C. S. & Kundu, T. 2002 Measurement ofmaterial elastic constants of trabecular bone: a micromechanical study using a 1GHz acoustic microscope. J. Orthop. Res. 20, 151-158. (doi:10.1016/S0736-0266(01)00061-4)
29. Brennan, O., Kennedy, O. D., Lee, T. C., Rackard, S. M. & O'Brien, F. J. 2009 Biomechanical properties across trabeculae from the proximal femur of normal and ovariectomised sheep. J. Biomech. 42, 498-503. (doi:10.1016/j. jbiomech.2008.11.032)
30. Chevalier, Y., Pahr, D. H., Allmer, H., Charlebois, M. & Zysset, P. K. 2007 Validation of a voxel-based FE method for prediction of the uniaxial apparent modulus of human trabecular bone using macroscopic mechanical tests and nanoindentation. J. Biomech. 40, 3333-3340. (doi:10.1016/j.jbiomech.2007.05. 004)
31. Harrison, N. M., McDonnell, P. F., O'Mahoney, D. C., Kennedy, O. D., O'Brien, F. J. & McHugh, P. E. 2008 Heterogeneous linear elastic trabecular bone modelling using micro-CT attenuation data and experimentally measured heterogeneous tissue properties. J. Biomech. 41, 2589-2596. (doi:10.1016/j. jbiomech.2008.05.014)
32. Rho, J. Y., Tsui, T. Y. & Pharr, G. M. 1997 Elastic properties of human cortical and trabecular lamellar bone measured by nanoindentation. Biomaterials 18, 1325-1330. (doi:10.1016/S0142-9612(97)00073-2)
33. Zysset, P. K., Guo, X. E., Hoffler, C. E., Moore, K. E. & Goldstein, S. A. 1999 Elastic modulus and hardness of cortical and trabecular bone lamellae measured by nanoindentation in the human femur. J. Biomech. 32, 1005-1012. (doi:10.1016/S0021-9290(99)00111-6)
34. Rice, J. C., Cowin, S. C. & Bowman, J. A. 1988 On the dependence of the elasticity and strength of cancellous bone on apparent density. J. Biomech. 21, 155-168. (doi:10.1016/0021-9290(88)90008-5)
35. Christensen, R. M. 1986 Mechanics of low density materials. J. Mech. Phys. Solids 34, 563-578. (doi:10.1016/0022-5096(86)90037-2)
36. Van Rietbergen, B., Weinans, H., Huiskes, R. & Odgaard, A. 1995 A new method to determine trabecular bone elastic properties and loading using micromechanical finite-element models. J. Biomech. 28, 69-81. (doi:10.1016/0021-9290(95)80008-5)
37. Gibson, L. J. & Ashby, M. F. 1982 The mechanics of 3-dimensional cellular materials. Proc. R. Soc. Lond. A 382, 43-59. (doi:10.1098/rspa.1982. 0088)
38. Gibson, L. J., Ashby, M. F., Schajer, G. S. & Robertson, C. I. 1982 The mechanics of two-dimensional cellular materials. Proc. R. Soc. Lond. A 382, 25-42. (doi:10.1098/rspa.1982.0087)
39. Rajan, K. 1985 Linear elastic properties of trabecular bone - a cellular solid approach. J. Mater. Sci. Lett. 4, 609-611. (doi:10.1007/BF00720046)
40. Gibson, L. J. 1985 The mechanical behavior of cancellous bone. J. Biomech. 18, 317-328. (doi:10.1016/0021-9290(85)90287-8)
41. Gibson, L. J. & Ashby, M. F. 1997 Cellular solids: structure and properties. Cambridge, UK: Cambridge University Press.
42. Gibson, L. J., Ashby, M. F. & Harley, B. A. 2010 Foam-like materials in nature. Cambridge, UK: Cambridge University Press.
43. Kabel, J., Van Rietbergen, B., Odgaard, A. & Huiskes, R. 1999 Constitutive relationships of fabric, density, and elastic properties in cancellous bone architecture. Bone 25, 481-486. (doi:10.1016/S8756-3282(99) 00190-8)
44. Turner, C. H., Cowin, S. C., Rho, J. Y., Ashman, R. B. & Rice, J. C. 1990 The fabric dependence of the orthotropic elastic constants of cancellous bone. J. Biomech. 23, 549-561. (doi:10.1016/0021-9290(90)90048-8)
45. Zysset, P. K. 2003 A review of morphology-elasticity relationships in human trabecular bone: theories and experiments. J. Biomech. 36, 1469-1485. (doi:10.1016/S0021-9290(03)00128-3)
46. Muller, R. & Ruegsegger, P. 1995 Three-dimensional finite element modelling of non-invasively assessed trabecular bone structures. Med. Eng. Phys. 17, 126-133. (doi:10.1016/1350-4533(95)91884-J)
47. Ulrich, D., van Rietbergen, B., Weinans, H. & Ruegsegger, P. 1998 Finite element analysis of trabecular bone structure: a comparison of image-based meshing techniques. J. Biomech. 31, 1187-1192. (doi:10.1016/S0021- 9290(98)00118-3)
48. Bourne, B. C. & van der Meulen, M. C. H. 2004 Finite element models predict cancellous apparent modulus when tissue modulus is scaled from specimen CT-attenuation. J. Biomech. 37, 613-621. (doi:10.1016/j.jbiomech.2003.10.002)
49. Dobson, C. A., Sisias, G., Phillips, R., Fagan, M. J. & Langton, C. M. 2006 Three dimensional stereolithography models of cancellous bone structures from mCT data: testing and validation of finite element results. Proc. Inst. Mech. Eng. H J. Eng. Med. 220, 481-484. (doi:10.1243/09544119H00405)
50. Follet, H., Peyrin, F., Vidal-Salle, E., Bonnassie, A., Rumelhart, C. & Meunier, P. J. 2007 Intrinsic mechanical properties of trabecular calcaneus determined by finite-element models using 3D synchrotron microtomography. J. Biomech. 40, 2174-2183. (doi:10.1016/j.jbiomech.2006.10.032)
51. Pahr, D. H. & Zysset, P. K. 2008 Influence of boundary conditions on computed apparent elastic properties of cancellous bone. Biomech. Model. Mechanobiol. 7, 463-476. (doi:10.1007/s10237-007-0109-7)
52. Donaldson, F. E., Pankaj, P., Law, A. H. & Simpson, A. H. 2008 Virtual trabecular bone models and their mechanical response. Proc. Inst. Mech. Eng. H 222, 1185-1195. (doi:10.1243/09544119JEIM408)
53. Ilic, S., Hackl, K. & Gilbert, R. 2010 Application of the multiscale FEM to the modeling of cancellous bone. Biomech. Model. Mechanobiol. 9, 87-102. (doi:10.1007/s10237-009-0161-6)
54. Ruiz, O., Schouwenaars, R., Ramírez, E. I., Jacobo, V. H. & Ortiz, A. 2010 Analysis of the architecture and mechanical properties of cancellous bone using 2D Voronoi cell based models. In Proc. World Congress on Engineering 2010. WCE 2010, 30 June-2 July, London, UK, vol. I, pp. 609-614. Hong Kong: Newswood Limited/International Association of Engineers.
55. van Lenthe, G. H., Stauber, M. & Muller, R. 2006 Specimen-specific beam models for fast and accurate prediction of human trabecular bone mechanical properties. Bone 39, 1182-1189. (doi:10.1016/j.bone.2006.06.033)
56. Wang, C., Feng, L. & Jasiuk, I. 2009 Scale and boundary conditions effects on the apparent elastic moduli of trabecular bone modeled as a periodic cellular solid. J. Biomech. Eng. 131, 121 008-121 018.
57. Yoo, A. & Jasiuk, I. 2006 Couple-stressmoduli of atrabecular bone idealized as a 3Dperiodic cellular network. J. Biomech. 39, 2241-2252. (doi:10.1016/j.jbiomech.2005.07.006)
58. Basillais, A., Bensamoun, S., Chappard, C., Brunet-Imbault, B., Lemineur, G., Ilharreborde, B., Ho Ba Tho, M. C. & Benhamou, C. L. 2007 Three-dimensional characterization of cortical bone microstructure by microcomputed tomography: validation with ultrasonic and microscopic measurements. J. Orthop. Sci. 12, 141-148. (doi:10.1007/s00776-006-1104-z)
59. Stock, S. R. 2009 MicroComputed tomography: methodology and applications. Boca Raton, FL: CRC Press.
60. Huet, C. 1990 Application of variational concepts to size effects in elastic heterogeneous bodies. J. Mech. Phys. Solids 42, 1995-2011.
61. Rubin, C. T. & Lanyon, L. E. 1982 Limb mechanics as a function of speed and gait - a study of functional strains in the radius and tibia of horse and dog. J. Exp. Biol. 101, 187-211.
62. Gupta, H. S., Seto, J., Wagermaier, W., Zaslansky, P., Boesecke, P. & Fratzl, P. 2006 Cooperative deformation of mineral and collagen in bone at the nanoscale. Proc. Natl Acad. Sci. USA 103, 1741-1746. (doi:10.1073/pnas. 0604237103)
63. Cusack, S. & Miller, A. 1979 Determination of the elastic constants of collagen by Brillouin light scattering. J. Mol. Biol. 135, 39-51. (doi:10.1016/0022-2836(79)90339-5)
64. Minary-Jolandan, M. & Yu, M. F. 2009 Nanomechanical heterogeneity in the gap and overlap regions of type I collagen fibrils with implications for bone heterogeneity. Biomacromolecules 10, 2565-2570. (doi:10.1021/bm900519v)
65. Grant, C. A., Brockwell, D. J., Radford, S. E. & Thomson, N. H. 2008 Effects of hydration on the mechanical response of individual collagen fibrils. Appl. Phys. Lett. 92, 1-3.
66. Hall, R. H. 1951 Variations with pH of the tensile properties of collagen fibres. J. Soc. Leather Trades Chem. 35, 195-210.
67. Yang, L., van der Werf, K. O., Koopman, B., Subramaniam, V., Bennink, M. L., Dijkstra, P. J. & Feijen, J. 2007 Micromechanical bending of single collagen fibrils using atomic force microscopy. J. Biomed. Mater. Res. Part A 82, 160-168. (doi:10.1002/jbm.a.31127)
68. Gilmore, R. S. & Katz, J. L. 1982 Elastic properties of apatites. J. Mat. Sci. 17, 1131-1141. (doi:10.1007/BF00543533)
69. Katz, J. L. & Ukraincik, K. 1971 Anisotropic elastic properties of hydroxyapatite. J. Biomech. 4, 221-227. (doi:10.1016/0021-9290(71)90007-8)
70. Snyders, R., Music, D., Sigumonrong, D., Schelnberger, B., Jensen, J. & Schneider, J. M. 2007 Experimental and ab initio study of the mechanical properties of hydroxyapatite. Appl. Phys. Lett. 90, 1-13.
71. Jager, I. & Fratzl, P. 2000 Mineralized collagen fibrils: a mechanical model with a staggered arrangement of mineral particles. Biophys. J. 79, 1737-1746. (doi:10.1016/S0006-3495(00)76426-5)
72. Pidaparti, R. M. V., Chandran, A., Takano, Y. & Turner, C. H. 1996 Bone mineral lies mainly outside collagen fibrils: predictions of a composite model for osteonal bone. J. Biomech. 29, 909-916. (doi:10.1016/0021-9290(95) 00147-6)
73. Chen, P.-Y., Toroian, D., Price, P. A. & McKittrick, J. 2011 Minerals form a continuum phase in mature cancellous bone. Calcif. Tissue Int. 81, 351-361. (doi:10.1007/s00223-011-9462-8)
74. Hill, R. 1963 Elastic properties of reinforced solidssome theoretical principles. J. Mech. Phys. Solids 11, 357-372. (doi:10.1016/0022-5096(63)90036- X)
75. Budiansky, B. 1965 On elastic moduli of some heterogeneous materials. J. Mech. Phys. Solids 13, 223-227. (doi:10.1016/0022-5096(65)90011-6)
76. Minary-Jolandan, M. & Yu, M. F. 2009 Nanoscale characterization of isolated individual type I collagen fibrils: polarization and piezoelectricity. Nanotechnology 20, 085706. (doi:10.1088/0957-4484/20/8/085706)
77. Hang, F. & Barber, A. H. 2011 Nano-mechanical properties of individual mineralized collagen fibrils from bone tissue. J. R. Soc. Interface 8, 500-505. (doi:10.1098/rsif.2010.0413)
78. Gavazzi, A. C. & Lagoudas, D. C. 1990 On the numerical evaluation of Eshelby's tensor and its application to elastoplastic fibrous composites. Comput. Mech. 7, 13-19. (doi:10.1007/BF00370053)
79. Mori, T. & Tanaka, K. 1973 Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21, 571-574. (doi:10.1016/0001-6160(73)90064-3)
80. Benveniste, Y. 1987 A new approach to the application of Mori-Tanaka theory in composite materials. Mech. Mater. 6, 147-157. (doi:10.1016/0167- 6636(87)90005-6)
81. Van der Linden, J. C., Brikenhager-Frenkel, D. H., Verhaar, J. A. N. & Weinans, H. 2001 Trabecular bone's mechanical properties are affected by its non-uniform mineral distribution. J. Biomech. 34, 1573-1580. (doi:10.1016/S0021-9290(01)00146-4)
82. Katz, E. P. & Li, S. 1973 Structure and function of bone collagen fibrils. J. Mol. Biol. 80, 1-15. (doi:10.1016/0022-2836(73)90230-1)
83. Lees, S., Prostak, K. S., Ingle, V. K. & Kjoller, K. 1994 The loci of mineral in turkey leg tendon as seen by atomic-force microscope and electron microscopy. Calcif. Tissue Int. 55, 180-189. (doi:10.1007/BF00425873)
84. Sasaki, N. & Sudoh, Y. 1997 X-ray pole figure analysis of apatite crystals and collagen molecules in bone. Calcif. Tissue Int. 60, 361-367. (doi:10.1007/s002239900244)
85. Sasaki, N., Tagami, A., Goto, T., Taniguchi, M., Nakata, M. & Hikichi, K. 2002 Atomic force microscopic studies on the structure of bovine femoral cortical bone at the collagen fibril-mineral level. J. Mater. Sci. Mater. Med. 13, 333-337. (doi:10.1023/A:1014079421895)
86. Fritsch, A., Dormieux, L. & Hellmich, C. 2006 Porous polycrystals built up by uniformlyand axisymmetricallyoriented needles: homogenization of elastic properties. C. R. Mec. 334, 151-157. (doi:10.1016/j.crme.2006.01.008)
87. Hellmich, C., Barthelemy, J. F. & Dormieux, L. 2004 Mineral-collagen interactions in elasticity of bone ultrastructure - a continuum micromechanics approach. Eur. J. Mech. A. Solids 23, 783-810. (doi:10.1016/j.euromechsol.2004. 05.004)
88. Hellmich, C. & Ulm, F. 2002 Micromechanical model for ultrastructural stiffness of mineralized tissues. J. Eng. Mech. 128, 898-908. (doi:10.1061/(ASCE)0733-9399(2002)128:8(898))
89. Remaggi, F., Canè, V., Palumbo, C. & Ferretti, M. 1998 Histomorphometric study on the osteocyte lacunocanalicular network in animals of different species. I. Woven-fibered and parallel-fibered bones. Ital. J. Anat. Embryol. 103, 145-155.
90. Sun, C. T. & Li, S. 1988 Three-dimensional effective elastic constants for thick laminates. J. Compos. Mater. 22, 629-639. (doi:10.1177/002199838802200703)
91. Rubin, M. A. 2001 Multiscale characterization of the ultrastructure of trabecular bone in osteoporotic and normal humans and in two inbred strains of mice. M.Sc. Thesis, Georgia Institute of Technology, Atlanta, USA.
92. Rubin, M. A. & Jasiuk, I. 2005 The TEM characterization of the lamellar structure of osteoporotic human trabecular bone. Micron 36, 653-664. (doi:10.1016/j.micron.2005.07.010)
93. Giraud-Guille, M. M. 1988 Twisted plywood architecture of collagen fibrils in human compact bone osteons. Calcif. Tissue Int. 42, 167-180. (doi:10.1007/BF02556330)
94. Daniel, I. M. & Ishai, O. 2006 Engineering mechanics of composite material. New York, NY: Oxford University Press.
95. Guo, X. E. & Kim, C. H. 1999 Effects of age-related bone loss: a 3D microstructural simulation. In Proc. of Bioengineering Conference, ASME BED, 6-20 June, Big Sky, Montana, USA, vol. 42, pp. 327-328.
96. Silva, M. J. & Gibson, L. J. 1997 Modeling the mechanical behavior of vertebral trabecular bone: effects of age-related changes in microstructure. Bone 21, 191-199. (doi:10.1016/S8756-3282(97)00100-2)
97. Vajjhala, S., Kraynik, A. M. & Gibson, L. J. 2000 A cellular solid model for modulus reduction due to resorption of trabeculae in bone. J. Biomech. Eng. 122, 511-515. (doi:10.1115/1.1289996)
98. Guo, X. E. & Kim, C. H. 2002 Mechanical consequences of trabecular bone loss and its treatment: a three dimensional model simulation. Bone 30, 404-411. (doi:10.1016/S8756-3282(01)00673-1)
99. Keaveny, T. M., Morgan, E. F., Niebur, G. L. & Yeh, O. C. 2001 Biomechanics of trabecular bone. Annu. Rev. Biomed. Eng. 3, 307-333. (doi:10.1146/annurev.bioeng.3.1.307)
100. Zysset, P. K., Guo, X. E., Hoffler, C. E., Moore, K. E. & Goldstein, S. A. 1998 Mechanical properties of human trabecular bone lamellae quantified by nanoindentation. Technol. Health Care 6, 429-432. (Pubitemid 29090465)
101. Follet, H. et al. 2011 Effects of preexisting microdamage, collagen cross-links, degree of mineralization, age, and architecture on compressive mechanical properties of elderly human vertebral trabecular bone. J. Orthop. Res. 29, 481-488. (doi:10.1002/jor.21275)
102. Keaveny, T. M., Pinilla, T. P., Crawford, R. P., Kopperdahl, D. L. & Lou, A. 1997 Systematic and random errors in compression testing of trabecular bone. J. Orthop. Res. 15, 101-110. (doi:10.1002/jor.1100150115)
103. Glimcher, M. J. 1984 Recent studies of the mineral phase in one and its possible linkage to the organic matrix by protein-bound phosphate bonds. Phil. Trans. R. Soc. Lond. B 304, 479-508. (doi:10.1098/rstb.1984.0041)
104. Lee, D. D. & Glimcher, M. J. 1989 The three-dimensional spatial relationship between the collagen fibrils and the inorganic calcium-phosphate crystals of pickerel and herring fish bone. Connect. Tissue Res. 21, 247-257. (doi:10.3109/03008208909050014)
105. Jaasma, M. J., Bayraktar, H. H., Niebur, G. L. & Keaveny, T. M. 2002 Biomechanical effects of intraspecimen variations in tissue modulus for trabecular bone. J. Biomech. 35, 237-246. (doi:10.1016/S0021-9290(01)00193-2)
106. van der Linden, J. C., Day, J. S., Verhaar, J. A. N. & Weinans, H. 2004 Altered tissue properties induce changes in cancellous bone architecture in aging and diseases. J. Biomech. 37, 367-374. (doi:10.1016/S0021-9290(03)00266- 5)
107. Keaveny, T. M. & Hayes, W. C. 1993 A 20-year perspective on the mechanical properties of trabecular bone. J. Biomech. Eng. 115, 534-542. (doi:10.1115/1.2895536)
108. Roy, M., Rho, J. Y., Tsui, T. Y. & Pharr, G. M. 1996 Variation of Young's modulus and hardness in human lumbar vertebrae measured by nanoindentation. In Proc. Bioengineering Conf., ASME BED, 17-22 November, Atlanta, GA, USA, vol. 33, pp. 385-386.
109. van Eijden, T. M. G. J., van Ruijven, L. J. & Giesen, E. B. W. 2004 Bone tissue stiffness in the mandibular condyle is dependent on the direction and density of the cancellous structure. Calcif. Tissue Int. 75, 502-508. (doi:10.1007/s00223-004-0295-6)
110. Ferreri, S. L., Hu, B. & Qin, Y. X. 2010 Nanoindentation measurements of viscoelastic material properties are sensitive to preparation techniques. In Proc. 2010 IEEE 36th Annual Northeast Bioengineering Conf., NEBEC 2010, 26-28 March, New York, NY, USA, art. no. 5458218.
111. Bayraktar, H. H., Morgan, E. F., Niebur, G. L., Morris, G. E., Wong, E. K. & Keaveny, T. M. 2004 Comparison of the elastic and yield properties of human femoral trabecular and cortical bone tissue. J. Biomech. 37, 27-35. (doi:10.1016/S0021-9290(03)00257-4)
112. Morgan, E. F. & Keaveny, T. M. 2001 Dependence of yield strain of human trabecular bone on anatomic site. J. Biomech. 34, 569-577. (doi:10.1016/S0021-9290(01)00011-2)
113. van Ruijven, L. J., Giesen, E. B. W., Farella, M. &van Eijden, T. M. G. J. 2003 Prediction of mechanical properties of the cancellous bone of the mandibular condyle. J. Dent. Res. 82, 819-823. (doi:10.1177/154405910308201011)
114. Giesen, E. B. W., Ding, M., Dalstra, M. & van Eijden, T. 2001 Mechanical properties of cancellous bone in the human mandibular condyle are anisotropic. J. Biomech. 34, 799-803. (doi:10.1016/S0021-9290(01)00030-6)
115. Ding, M., Dalstra, M., Danielsen, C. C., Kabel, J., Hvid, I. & Linde, F. 1997 Age variations in the properties of human tibial trabecular bone. J. Bone Joint Surg. Br. 79B, 995-1002. (doi:10.1302/0301-620X.79B6.7538)