Anisotropic density growth of bone - A computational micro-sphere approach

International Journal of Solids and Structures

Volumn 49, Issue 14, 2012, Pages 1928-1946

  Waffenschmidt, Tobias ^a   Menzel, Andreas ^a,b   Kuhl, Ellen ^c,d  

^a TU DORTMUND UNIVERSITY   (Germany)

^b LUND UNIVERSITY   (Sweden)

^c ETH ZURICH   (Switzerland)

^d STANFORD UNIVERSITY   (United States)

Author keywords

Adaptation; Anisotropic growth; Finite element method; Micro sphere; Remodelling

Indexed keywords

ADAPTATION; ANISOTROPIC GROWTH; ANISOTROPIC MECHANICAL PROPERTIES; BONE DENSITY; BONE DENSITY DISTRIBUTION; COMPUTATIONAL MODEL; CONSTITUTIVE RELATIONS; DENSITY DISTRIBUTIONS; DENSITY EVOLUTION; DENSITY PLOTS; ELASTIC PROPERTIES; ENERGY-DRIVEN; EXPERIMENTAL MEASUREMENTS; FINITE ELEMENT ALGORITHMS; FUNCTIONAL ADAPTATION; GROWTH MODELS; GROWTH PROCESS; HETEROGENEOUS DISTRIBUTIONS; HIGH LOADINGS; IN-VIVO; LOCAL DENSITY; LOW LOADING; MECHANICAL LOADING; MICRO-STRUCTURAL; MODELLING AND SIMULATIONS; NUMERICAL STUDIES; OPTIMISATIONS; REMODELLING; SUBJECT-SPECIFIC;

BONE; DENSITY (SPECIFIC GRAVITY); FINITE ELEMENT METHOD; LOADING; SPHERES;

ANISOTROPY;

EID: 84861532381     PISSN: 00207683     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ijsolstr.2012.03.035     Document Type: Article

References (85)

1. V. Alastrué, M. Martínez, M. Doblaré, and A. Menzel Anisotropic micro-sphere-based finite elasticity applied to blood vessel modelling Journal of the Mechanics and Physics of Solids 57 2009 178 203
2. V. Alastrué, M. Martínez, A. Menzel, and M. Doblaré On the use of non-linear transformations for the evaluation of anisotropic rotationally symmetric directional integrals. Application to the stress analysis in fibred soft tissues International Journal for Numerical Methods in Engineering 79 2009 474 504
3. D. Ambrosi, G. Ateshian, E. Arruda, S. Cowin, J. Dumais, A. Goriely, G. Holzapfel, J. Humphrey, R. Kemkemer, E. Kuhl, J. Olberding, L. Taber, and K. Garikipati Perspectives on biological growth and remodeling Journal of the Mechanics and Physics of Solids 59 2011 863 883
4. R. Ashman, J. Rho, and C. Turner Anatomical variation of orthotropic elastic moduli of the proximal human tibia Journal of Biomechanics 22 1989 895 900 (Pubitemid 20087390)
5. A. Baliunas, D. Hurwitz, A. Ryals, A. Karrar, J. Case, J. Block, and T. Andriacchi Increased knee joint loads during walking are present in subjects with knee osteoarthritis Osteoarthritis and Cartilage 10 2002 573 579 (Pubitemid 34919073)
6. Z. Bažant, and B. Oh Microplane Model for Progressive Fracture of Concrete and Rock Journal of Engineering Mechanics 111 1985 559 582 (Pubitemid 15479255)
7. Z. Bažant, and B. Oh Efficient numerical integration on the surface of a sphere Zeitschrift für angewandte Mathematik und Mechanik 66 1986 37 49 (Pubitemid 16571699)
8. G. Beaupré, T. Orr, and D. Carter An approach for time-dependent bone modelling and remodelling Journal of Orthopeadic Research 8 1990 651 670
9. T. Böhlke, and C. Brüggemann Graphical representation of the generalized Hooke's law Technische Mechanik 21 2001 145 158
10. R.D. Carpenter, and D.R. Carter Computational simulation of spontaneous bone straightening in growing children Biomechanics and Modeling in Mechanobiology 9 2010 317 328
11. D. Carter, and W. Hayes Compressive behavior of bone as a 2-phase porous structure Journal of Bone and Joint Surgery 59 1977 954 962 (Pubitemid 8218257)
12. D. Carter, T. Orr, and D. Fyhrie Relationships between loading history and femoral cancellous bone architecture Journal of Biomechanics 22 1989 231 244 (Pubitemid 19102191)
13. D.R. Carter, and G.S. Beaupré Skeletal Function and Form: Mechanobiology of Skeletal Development, Aging, and Regeneration 2001 Cambridge University Press
14. A. Cazzani, and M. Rovati Extrema of Young's modulus for cubic and transversely isotropic solids International Journal of Solids and Structures 40 2003 1713 1744 (Pubitemid 36357497)
15. P. Coelho, P. Fernandes, H. Rodrigues, J. Cardoso, and J. Guedes Numerical modeling of bone tissue adaptation - a hierarchical approach for bone apparent density and trabecular structure Journal of Biomechanics 42 2009 830 837
16. P.G. Coelho, P.R. Fernandes, J.M. Guedes, and H.C. Rodrigues A hierarchical model for concurrent material and topology optimisation of three-dimensional structures Structural and Multidisciplinary Optimization 35 2008 107 115
17. S. Cowin The relationship between the elasticity tensor and the fabric tensor Mechanics of Materials 4 1985 137 147
18. S. Cowin, and S. Doty Tissue Mechanics 2007 Springer
19. S. Cowin, and D. Hegedus Bone remodelling I: Theory of adaptive elasticity Journal of Elasticity 6 1976 313 326
20. S.C. Cowin Wolff's Law of Trabecular Architecture at Remodeling Equilibrium Journal of Biomechanical Engineering 108 1986 83 88 (Pubitemid 16165152)
21. M. Doblaré, and J. García Anisotropic bone remodelling model based on a continuum damage-repair theory Journal of Biomechanics 35 2002 1 17 (Pubitemid 34017346)
22. A.E. Ehret, M. Itskov, and H. Schmid Numerical integration on the sphere and its effect on the material symmetry of constitutive equations - a comparative study, microplane International Journal for Numerical Methods in Engineering 81 2010 189 206
23. P. Fernandes, H. Rodrigues, and C. Jacobs A model of bone adaptation using a global optimisation criterion based on the trajectorial theory of Wolff Computer Methods in Biomechanics and Biomedical Engineering 2 1999 125 138
24. J.M. García, M. Doblaré, and J. Cegoňino Bone remodelling simulation: a tool for implant design Computational Materials Science 25 2002 100 114 (Pubitemid 35019368)
25. J.M. García, M.A. Martínez, and M. Doblaré An anisotropic internal-external bone adaptation model based on a combination of CAO and continuum damage mechanics technologies Computer Methods in Biomechanics and Biomedical Engineering 4 2001 355 377
26. I. Gitman, H. Askes, E. Kuhl, and E. Aifantis Stress concentrations in fractured compact bone simulated with a special class of anisotropic gradient elasticity International Journal of Solids and Structures 47 2010 1099 1107
27. S. Göktepe, O.J. Abilez, and E. Kuhl A generic approach towards finite growth with examples of athlete's heart, cardiac dilation, and cardiac wall thickening Journal of the Mechanics and Physics of Solids 58 2010 1661 1680
28. S. Göktepe, and C. Miehe A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage Journal of the Mechanics and Physics of Solids 53 2005 2259 2283 (Pubitemid 41165641)
29. S. Goldstein The mechanical properties of trabecular bone: dependence on anatomic location and function Journal of Biomechanics 20 1987 1055 1061 (Pubitemid 18010105)
30. T. Harrigan, and J. Hamilton An analytical and numerical study of the stability of bone remodelling theories: dependence on microstructural stimulus Journal of Biomechanics 25 1992 477 488 Corrigendum 26 (3) (1993) 365-366
31. T. Harrigan, and J. Hamilton Bone strain sensation via transmembrane potential changes in surface osteoblasts: loading rate and microstructural implications Journal of Biomechanics 26 1993 183 200 (Pubitemid 23067447)
32. T. Harrigan, and J. Hamilton Finite element simulation of adaptive bone remodelling: a stability criterion and a time stepping method International Journal for Numerical Methods in Engineering 36 1993 837 854
33. T. Harrigan, and J. Hamilton Necessary and sufficient conditions for global stability and uniqueness in finite element simulations of adaptive bone remodelling International Journal of Solids and Structures 31 1994 97 107
34. M. Harrysson, M. Ristinmaa, M. Wallin, and A. Menzel Framework for deformation induced anisotropy in glassy polymers Acta Mechanica 211 2010 195 213
35. Y.M.A. Hashash, J.I.C. Yao, and D.C. Wotring Glyph and hyperstreamline representation of stress and strain tensors and material constitutive response International Journal for Numerical and Analytical Methods in Geomechanics 27 2003 603 626 (Pubitemid 36803796)
36. S. Heo, and Y. Xu Constructing fully symmetric cubature formulae for the sphere Mathematics of Computation 70 2001 269 279 (Pubitemid 33574345)
37. G. Himpel, E. Kuhl, A. Menzel, and P. Steinmann Computational modelling of isotropic multiplicative growth Computer Modeling in Engineering and Sciences 8 2005 119 134 (Pubitemid 40819843)
38. G. Himpel, A. Menzel, E. Kuhl, and P. Steinmann Time-dependent fibre reorientation of transversely isotropic continua - finite element formulation and consistent linearization International Journal for Numerical Methods in Engineering 73 2008 1413 1433 (Pubitemid 351338054)
39. R. Huiskes, H. Weinans, H. Grootenboer, M. Dalstra, B. Fudala, and T. Slooff Adaptive bone-remodeling theory applied to prosthetic-design analysis Journal of Biomechanics 20 1987 1135 1150 (Pubitemid 18010111)
40. C. Hulet, J. Sabatier, D. Souquet, B. Locker, C. Marcelli, and C. Vielpeau Distribution of bone mineral density at the proximal tibia in knee osteoarthritis Calcified Tissue International 71 2002 315 322 (Pubitemid 36929612)
41. D. Hurwitz, D. Sumner, T. Andriacchi, and D. Sugar Dynamic knee loads during gait predict proximal tibial bone distribution Journal of Biomechanics 31 1998 423 430 (Pubitemid 28408340)
42. C.R. Jacobs, M.E. Levenston, G.S. Beaupré, J.C. Simo, and D.R. Carter Numerical instabilities in bone remodeling simulations: the advantages of a node-based finite element approach Journal of Biomechanics 28 1995 449 459
43. C.R. Jacobs, J.C. Simo, G.S. Beaupré, and D.R. Carter Adaptive bone remodeling incorporating simultaneous density and anisotropy considerations Journal of Biomechanics 30 1997 603 613 (Pubitemid 27214192)
44. K.I. Kanatani Distribution of directional data and fabric tensors International Journal of Engineering Science 22 1984 149 164
45. U. Kocks, C. Tomé, and H.R. Wenk Texture and Anisotropy-Preferred Orientations in Polycrystals and their Effect on Material Properties 2000 Cambridge University Press
46. Kriz, R.; Glaessgen, E.; MacRae, J.; 1995. Eigenvalue-eigenvector glyphs: visualizing zeroth, second, fourth and higher order tensors in a continuum. In: Workshop on Modeling the Development of Residual Stresses During Thermalset Composite Curing. University of Illinois, Champaign-Urbana.
47. N. Krstin, U. Nackenhorst, and R. Lammering Zur konstitutiven Beschreibung des anisotropen beanspruchungsadaptiven Knochenumbaus Technische Mechanik 20 2000 31 40
48. E. Kuhl, R. Maas, G. Himpel, and A. Menzel Computational modeling of arterial wall growth Biomechanics and Modeling in Mechanobiology 6 2007 321 331
49. E. Kuhl, A. Menzel, and K. Garikipati On the convexity of transversely isotropic chain network models Philosophical Magazine 86 2006 3241 3258 (Pubitemid 43862406)
50. E. Kuhl, A. Menzel, and P. Steinmann Computational modeling of growth - a critical review, a classification of concepts and two new consistent approaches Computational Mechanics 32 2003 71 88
51. E. Kuhl, E. Ramm, and K. Willam Failure analysis of elasto-plastic material models on different levels of observation International Journal of Solids and Structures 37 2000 7259 7280
52. E. Kuhl, and P. Steinmann Mass- and volume specific views on thermodynamics for open systems Proceedings of the Royal Society A 459 2003 2547 2568 (Pubitemid 38333335)
53. E. Kuhl, and P. Steinmann Theory and numerics of geometrically non-linear open system mechanics International Journal for Numerical Methods in Engineering 58 2003 1593 1615 (Pubitemid 37453907)
54. E. Kuhl, and P. Steinmann Material forces in open system mechanics Computer Methods in Applied Mechanics and Engineering 193 2004 2357 2381 (Pubitemid 38503360)
55. E. Kuhl, P. Steinmann, and I. Carol A thermodynamically consistent approach to microplane theory. Part II. Dissipation and inelastic constitutive modeling International Journal of Solids and Structures 38 2001 2933 2952 (Pubitemid 32257382)
56. V. Lubarda, and A. Hoger On the mechanics of solids with a growing mass International Journal of Solids and Structures 39 2002 4627 4664
57. V. Lubarda, and D. Krajcinovic Damage tensors and the crack density distribution International Journal of Solids and Structures 30 1993 2859 2877
58. D. Ma, M. Friák, J. Neugebauer, D. Raabe, and F. Roters Multiscale simulation of polycrystal mechanics of textured β-Ti alloys using ab initio and crystal-based finite element methods Physica Status Solidi (b) 245 2008 2642 2648
59. A. Menzel Modelling of anisotropic growth in biological tissues - a new approach and computational aspects Biomechanics and Modeling in Mechanobiology 3 2005 147 171 (Pubitemid 40454314)
60. A. Menzel Anisotropic remodelling of biological tissues G. Holzapfel, R. Ogden, Mechanics of Biological Tissue 2006 Springer 91 104
61. A. Menzel A fibre reorientation model for orthotropic multiplicative growth -configurational driving stresses, kinematics-based reorientation, and algorithmic aspects Biomechanics and Modeling in Mechanobiology 6 2007 303 320 (Pubitemid 47325972)
62. A. Menzel, M. Harrysson, and M. Ristinmaa Towards an orientation- distribution-based multi-scale approach for remodelling biological tissues Computer Methods in Biomechanics and Biomedical Engineering 11 2008 505 524
63. A. Menzel, and T. Waffenschmidt A micro-sphere-based remodelling formulation for anisotropic biological tissues Philosophical Transactions of the Royal Society A 367 2009 3499 3523
64. A. Menzel, T. Waffenschmidt, and V. Alastrué An anisotropic micro-sphere approach applied to the modelling of soft biological tissues G. Kreiss, P. Lötstedt, A. Målqvist, M. Neytcheva, Numerical Mathematics and Advanced Applications 2010 Springer 637 644
65. H. von Meyer Die Architektur der Spongiosa Reichert's und Du Bois Reymoiid's Archiv für Anatomie und Physiologie 34 1867 615 628
66. C. Miehe, and S. Göktepe A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of rubber viscoelasticity Journal of the Mechanics and Physics of Solids 53 2005 2231 2258 (Pubitemid 41165640)
67. C. Miehe, S. Göktepe, and F. Lulei A micro-macro approach to rubber-like materials - Part I: the non-affine micro-sphere model of rubber elasticity Journal of the Mechanics and Physics of Solids 52 2004 2617 2660
68. P. Moreo, M. Pérez, J. García-Aznar, and M. Doblaré Modelling the mechanical behaviour of living bony interfaces Computer Methods in Applied Mechanics and Engineering 196 2007 3300 3314 (Pubitemid 46927844)
69. U. Nackenhorst, N. Krstin, and R. Lammering A constitutive law for anisotropic stress adaptive bone remodeling ZAMM Journal of Applied Mathematics and Mechanics/ Zeitschrift für Angewandte Mathematik und Mechanik 80 2000 399 400
70. S. Nemat-Nasser, and M. Hori Micromechanics: Overall Properties of Heterogeneous Materials Applied Mathematics and Mechanics 1993 Elsevier
71. H. Pang, A. Shiwalkar, C. Madormo, R. Taylor, T. Andriacchi, and E. Kuhl Computational modeling of bone density profiles in response to gait: a subject-specific approach Biomechanics and Modeling in Mechanobiology 11 2012 379 390
72. H. Pettermann, T. Reiter, and F. Rammerstorfer Computational simulation of internal bone remodeling Archives of Computational Methods in Engineering 4 1997 295 323 (Pubitemid 127716110)
73. E. Reina-Romo, M.J. Gómez-Benito, J.M. García-Aznar, J. Domínguez, and M. Doblaré Growth mixture model of distraction osteogenesis: effect of pre-traction stresses Biomechanics and Modeling in Mechanobiology 9 2010 103 115
74. J.Y. Rho, L. Kuhn-Spearing, and P. Zioupos Mechanical properties and the hierarchical structure of bone Medical Engineering & Physics 20 1998 92 102 (Pubitemid 28318322)
75. H. Rodrigues, C. Jacobs, J.M. Guedes, and M.P. Bendso Global and local material optimization models applied to anisotropic bone adaptation P. Pedersen, M.P. Bendso, IUTAM Symposium on Synthesis in Bio Solid Mechanics Solid Mechanics and Its Applications vol. 69 2002 Springer Netherlands 209 220
76. L. Taber Biomechanics of growth, remodelling, and morphogenesis American Society of Mechanical Engineers - Applied Mechanics Reviews 48 1995 487 545
77. R.E. Taylor, C. Zheng, R.P. Jackson, J.C. Doll, J.C. Chen, K.R.S. Holzbaur, T. Besier, and E. Kuhl The phenomenon of twisted growth: humeral torsion in dominant arms of high performance tennis players Computer Methods in Biomechanics and Biomedical Engineering 12 2009 83 93
78. T. Waffenschmidt, and A. Menzel Application of an anisotropic growth and remodelling formulation to computational structural design Mechanics Research Communications 2012 10.1016/j.mechrescom.2011.12.004
79. H. Weinans, R. Huiskes, and H. Grootenboer The behavior of adaptive bone-remodeling simulation models Journal of Biomechanics 25 1992 1425 1441
80. S. Weng Ein anisotropes Knochenumbaumodell und dessen Anwendung Technische Mechanik 18 1998 173 180
81. J.L. Williams, and J.L. Lewis Properties and an anisotropic model of cancellous bone from the proximal tibial epiphysis Journal of Biomechanical Engineering 104 1982 50 56
82. J. Wolff Ueber die innere Architectur der Knochen und ihre Bedeutung für die Frage vom Knochenwachsthum Virchows Archiv für Pathologische Anatomie und Physiologie und für Klinische Medizin 50 1870 389 453
83. J. Wolff Das Gesetz der Transformation der Knochen 1892 Hirschwald Berlin
84. J. Zhang, M. Michalenko, E. Kuhl, and T. Ovaert Characterization of indentation response and stiffness reduction of bone using a continuum damage model Journal of the Mechanical Behavior of Biomedical Materials 3 2010 189 202
85. P. Zysset, and A. Curnier An alternative model for anisotropic elasticity based on fabric tensors Mechanics of Materials 21 1995 243 250 See also Technical Note by Cowin, S.C.; 1998. Journal of Biomechanics 31, 759-762