A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodelling

Comptes Rendus - Mecanique

Volumn 340, Issue 8, 2012, Pages 575-589

  Madeo, Angela ^a,e   George, D ^c   Lekszycki, T ^b,d,e   Nierenberger, Mathieu ^c   Rémond, Yves ^c,e  

^a UNIVERSITÉ DE LYON   (France)

^b WARSAW UNIVERSITY OF TECHNOLOGY   (Poland)

^c UNIVERSITÉ DE STRASBOURG   (France)

^d MEDICAL UNIVERSITY OF WARSAW   (Poland)

^e UNIVERSITY OF L'AQUILA   (Italy)

Author keywords

Bio resorbable material; Biomechanics; Bone resorption and synthesis; Hierarchically heterogeneous materials; Load induced replacement of artificial material with natural bone tissue; Micro structure related size effects; Natural bone tissue; Numerical simulations; Second gradient continuum mixture model

Indexed keywords

BIOMECHANICS; BONE; CONTINUUM MECHANICS; HIERARCHICAL SYSTEM; MICROSTRUCTURE; NUMERICAL MODEL; ONE-DIMENSIONAL MODELING; RECONSTRUCTION; RESORPTION; SIZE EFFECT;

EID: 84864318358     PISSN: 16310721     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.crme.2012.05.003     Document Type: Article

References (81)

1. Carter D.R., Van der Meulen M.C.H., Beaupré G.S. Mechanical factors in bone growth and development. Bone 1996, 18:S5-S10.
2. Casanova R., Moukoko D., Pithioux M., Pailler-Mattéi C., Zahouani H., Chabrand P. Temporal evolution of skeletal regenerated tissue: What can mechanical investigation add to biological?. Med. Biol. Eng. Comput. 2010, 48:811-819.
3. Huiskes R., Ruimerman R., van Lenthe G.H., Janssen J.D. Effects of mechanical forces on maintenance and adaptation of form in trabecular bone. Nature 2000, 405:704-706.
4. Madeo A., Lekszycki T., dell'Isola F. A continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery. C. R., Méc. 2011, 339:625-640.
5. Lekszycki T. Optimality conditions in modeling of bone adaptation phenomenon. J. Theoret. Appl. Mech. 1999, 37(3):607-624.
6. Lekszycki T. Modeling of bone adaptation based on an optimal response hypothesis. Meccanica 2002, 37:343-354.
7. Lekszycki T. Functional adaptation of bone as an optimal control problem. J. Theoret. Appl. Mech. 2005, 43(3):120-140.
8. Wolff J. Das Gesetz der Transformation der Knochen 1892, Hirschwald Verlag, Berlin.
9. Wolff J. The Law of Bone Remodelling 1986, Springer-Verlag, Berlin.
10. Pawlikowski M., Klasztorny M., Skalski K. Studies on constitutive equation that models bone tissue. Acta Bioeng. Biomech. 2008, 10(4):39-47.
11. Ruimerman R., Hilbers P., van Rietbergen B., Huiskes R. A theoretical framework for strain-related trabecular bone maintenance and adaptation. J. Biomech. 2005, 38:931-941.
12. Hegedus D.H., Cowin S.C. Bone remodeling II: Small strain adaptive elasticity. J. Elast. 1976, 6:337-352.
13. Andreaus U., Colloca M., Iacoviello D., Pignataro M. Optimal-tuning PID control of adaptive materials for structural efficiency. Struct. Multidiscipl. Optim. 2011, 43:43-59.
14. Andreaus U., Colloca M. Prediction of micromotion initiation of an implanted femur under physiological loads and constraints using the finite element method. Proc. Inst. Mech. Eng., H J. Eng. Med. 2009, 223:589-605.
15. Prendergast P.J., Taylor D. Prediction of bone adaptation using damage accumulation. J. Biomech. 1994, 27:1067-1076.
16. Doblaré M., García J.M. Anisotropic bone remodelling model based on a continuum damage-repair theory. J. Biomech. 2002, 35:1-17.
17. Goto T., Kojima T., Iijima T., Yokokura S., Kawano H., Yamamoto A., Matsuda K. Resorption of synthetic porous hydroxyapatite and replacement newly formed bone. J. Orthop. Sci. 2001, 6:444-447.
18. Mano J.F., Sousa R.A., Boesel L.F., Neves Rui N.M., Reis L. Bioinert, biodegradable and injectable polymeric matrix composites for hard tissue replacement: State of the art and recent developments. Compos. Sci. Technol. 2004, 64:789-817.
19. Schilling A.F., Linhart W., Filke S., Gebauer M., Schinke T., Rueger J.M., Amling M. Resorbability of bone substitute biomaterials by human osteoclasts. Biomaterials 2004, 25:3963-3972.
20. Sikavitsas V.I., Temenoff J.S., Mokos A.G. Review: Biomaterials and bone mechanotransduction. Biomaterials 2001, 22:2581-2593.
21. A.S. Greenwald, S.D. Boden, V.M. Goldberg, Y.K. Cato, T. Laurencin, R.N. Rosier, Bone graft substitutes: Facts, fictions & applications, in: 68th Annual Meeting American Academy of Orthopaedic Surgeons, San Francisco, California, February 28-March 4, 2001.
22. Hing K.A. Bioceramic bone graft substitutes: Influence of porosity and chemistry. Int. J. Appl. Ceram. Technol. 2005, 2(3):184-199.
23. Kim H.-M. Ceramic bioactivity and related biomimetic strategy. Curr. Opin. Solid State Mater. Sci. 2003, 7:289-299.
24. Kraus R., Stahl J.-P., Schnettler R. Treatment strategies in thoracolumbar vertebral fractures: Are there indications for biomaterials?. European J. Trauma 2007, 253-257.
25. Park J.C., Wook D., Suh H. A bone replaceable artificial bone substitute: Morphological and physiochemical characterizations. Yonsei Med. J. 2000, 41(4):468-476.
26. Pawlikowski M., Skalski K., Haraburda M. Process of hip joint prosthesis design including bone remodeling phenomenon. Comput. Struct. 2003, 81(8-11):887-893.
27. Pietruszczak S., Inglis D., Pande G.N. Modelling of bone-implant interaction. Comput. Methods Biomech. Biomed. Eng. 1998, 2:289-298.
28. Piszczatowski S., Skalski K., Swieszkowski W. Load transfer between elastic hip implant and viscoelastic bone. Comput. Methods Biomech. Biomed. Eng. 1998, 2:123-130.
29. Ramaswamy Y., Haynes D.R., Berger G., Gildenhaar R., Lucas H. Bioceramics composition modulate resorption of human osteoclasts. J. Mater. Sci., Mater. Med. 2005, 16:1199-1205.
30. Sanz-Herrera J.A., Boccaccini A.R. Modelling bioactivity and degradation of bioactive glass based tissue engineering scaffolds. Int. J. Solids Struct. 2010, 48:257-268.
31. Sanz-Herrera J.A., García-Aznar J.M., Doblaré M. Micro-macro numerical modelling of bone regeneration in tissue engineering. Comput. Methods Appl. Mech. Eng. 2008, 197:3092-3107.
32. Kuhl E., Holzapfel G.A. A continuum model for remodeling in living structures. J. Mater. Sci. 2007, 42(21):8811-8823.
33. Cowin S.C. Bone poroelasticity. J. Biomech. 1999, 32:217-238.
34. Cowin S.C., Hegedus D.H. Bone remodeling I: Theory of adaptive elasticity. J. Elast. 1976, 6:313-326.
35. Garikipati K., Arruda E.M., Grosh K., Narayanan H., Calve S. A continuum treatment of growth in biological tissue: The coupling of mass transport and mechanics. J. Mech. Phys. Solids 2004, 52:1595-1625.
36. Adachi T., Osako Y., Tanaka M., Hojo M., Hollister S.J. Framework for optimal design of porous scaffold microstructure by computational simulation of bone regeneration. Biomaterials 2006, 27(21):3964-3972.
37. Chen Y., Zhou S., Li Q. Microstructure design of biodegradable scaffold and its effect on tissue regeneration. Biomaterials 2011, 32(22):5003-5014.
38. Buechner P.M., Lakes R.S. Size effects in the elasticity and viscoelasticity of bone. Biomech. Model. Mechanobiol. 2003, 1:295-301.
39. Harrigan T.P., Jasty M., Mann R.W., Harris W.H. Limitations of the continuum assumption in cancellous bone. J. Biomech. 1988, 21:269-275.
40. Lakes R.S. Dynamical study of couple stress effects in human compact bone. J. Biomech. Eng. 1981, 104:6-11.
41. Park H.C., Lakes R.S. Cosserat micromechanics of human bone: Strain redistribution by a hydration sensitive constituent. J. Biomech. 1986, 19(53):85-97.
42. Yang J.F.C., Lakes R.S. Experimental study of micropolar and couple stress elasticity in compact bone in bending. J. Biomech. 1982, 15(2):91-98.
43. Weiner S., Wagner H.D. The material bone: Structure-mechanical function relations. Annu. Rev. Mater. Sci. 1998, 28:271-298.
44. Jasiuk I. Modeling of trabecular bone as a hierarchical material. Comput. Fluid Solid Mech. 2003, 1(2):1727-1728.
45. Rubin M., Jasiuk I. The TEM characterization of the lamellar structure of osteoporotic human trabecular bone. Micron 2005, 36:653-664.
46. Yang J.F.C., Lakes R.S. Transient study of couple stress in compact bone torsion. J. Biomech. Eng. 1981, 103:275-279.
47. Yoo A., Jasiuk I. Modeling of trabecular bone as a couple stress continuum. Adv. Bioeng. ASME 2003, 41-42.
48. Yoo A., Jasiuk I. Couple-stress moduli of a trabecular bone idealized as a 3D periodic cellular network. J. Biomech. 2006, 39:2241-2252.
49. Fatemi J., Van Keulen F., Onck P.R. Generalized continuum theories: Application to stress analysis in bone. Meccanica 2002, 37(4-5):385-396.
50. Papanicolopulos A., Zervos A. Continua with microstructure: Second gradient theory. Theory, examples and computational issues. Eur. J. Environ. Civ. Eng. 2010, 14(8-9):1031-1050.
51. Aifantis E.C. Strain gradient interpretation of size effects. Int. J. Fract. Eur. J. Mech. A, Solids 2002, 21(3):465-481.
52. Exadaktylos G.E., Vardoulakis I. Microstructure in linear elasticity and scale effects: A reconsideration of basic rock mechanics and rock fracture mechanics. Tectonophysics 2001, 335:81-109.
53. Germain P. La méthode des puissances virtuelles en mécanique des milieux continus, premiere partie, théorie du second gradient. J. Méc. 1973, 12(2):234-274.
54. dell'Isola F., Seppecher P. The relationship between edge contact forces, double force and interstitial working allowed by the principle of virtual power. C. R. Acad. Sci. Paris, Ser. IIb 1995, 321:303-308.
55. dell'Isola F., Gouin H., Seppecher P. Radius and surface tension of microscopic bubbles by second gradient theory. C. R. Acad. Sci. Paris, Ser. IIB 1995, 320:211-216.
56. dell'Isola F., Gouin H., Rotoli G. Nucleation of spherical shell-like interfaces by second gradient theory: Numerical simulations. Eur. J. Mech. B, Fluids 1996, 15(4):545-568.
57. dell'Isola F., Seppecher P. Edge contact forces and quasi-balanced power. Meccanica 1997, 32:33-52.
58. dell'Isola F., Sciarra G., Vidoli S. Generalized Hooke's law for isotropic second gradient materials. Proc. R. Soc. A 2009, 465(2107):2177-2196.
59. Toupin R.A. Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 1962, 11:385-414.
60. Quiligotti S., Maugin G., dell'Isola F. An Eshelbian approach to the nonlinear mechanics of constrained solid-fluid mixtures. Acta Mech. 2003, 160:45-60.
61. Sciarra G., dell'Isola F., Ianiro N., Madeo A. A variational deduction of second gradient poroelasticity I: General theory. J. Mech. Mater. Struct. 2008, 3(3):507-526.
62. Luongo A., Paolone A. Perturbation methods for bifurcation analysis from multiple nonresonant complex eigenvalues. Nonlinear Dyn. 1997, 14:193-210.
63. Luongo A., Paolone A. Multiple scale analysis for divergence-Hopf bifurcation of imperfect symmetric systems. J. Sound Vib. 1998, 218(3):527-539.
64. Luongo A., Paolone A., Di Egidio A. Multiple time scales analysis for 1:2 and 1:3 resonant Hopf bifurcations. Nonlinear Dyn. 2003, 34:269-291.
65. Di Egidio A., Luongo A., Paolone A. Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams. Int. J. Non-Linear Mech. 2007, 42:88-98.
66. Alessandroni A., dell'Isola F., Porfiri M. A revival of electric analogs for vibrating mechanical systems aimed to their efficient control by PZT actuators. Int. J. Solids Struct. 2002, 39:5295-5324.
67. Andreaus U., dell'Isola F., Porfiri M. Piezoelectric passive distributed controllers for beam flexural vibrations. J. Vib. Control 2004, 10:625.
68. dell'Isola F., Vidoli S. Damping of bending waves in truss beams by electrical transmission lines with PZT actuators. Arch. Appl. Mech. 1998, 68:626-636.
69. Hamed E., Lee Y., Jasiuk I. Multiscale modeling of elastic properties of cortical bone. Acta Mech. 2010, 213(1-2):131-154.
70. Inglis D., Pietruszczak S. Characterization of anisotropy in porous media by means of linear intercept measurements. Int. J. Solids Struct. 2003, 40(5):1243-1264.
71. Rho J.Y., Ashman R.B., Turner C.H. Young's modulus of trabecular and cortical bone material: Ultrasonic and microtensile measurements. J. Biomech. 1993, 26(2):111-119.
72. dell'Isola F., Romano A. On the derivation of thermomechanical balance equations for continuous systems with a nonmaterial interface. Int. J. Eng. Sci. 1987, 25:1459-1468.
73. dell'Isola F., Rosa L., Woźniak C. A micro-structured continuum modelling compacting fluid-saturated grounds: The effects of pore-size scale parameter. Acta Mech. 1998, 127:165-182.
74. dell'Isola F., Guarascio M., Hutter K. A variational approach for the deformation of a saturated porous solid. A second-gradient theory extending Terzaghi's effective stress principle. Arch. Appl. Mech. 2000, 70:323-337.
75. dell'Isola F., Rosa L. An extension of Kelvin and Bredt formulas. Math. Mech. Solids 1996, 1:243-250.
76. Forest S., Trinh D.K. Generalized continua and non-homogeneous boundary conditions in homogenization methods. ZAMM 2011, 91:90-109.
77. Forest S., Aifantis E.C. Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua. Int. J. Solids Struct. 2010, 47:3367-3376.
78. Forest S. Milieux continus généralisés et matériaux hétérogènes Avril 2006, Les presses de l'Ecole des Mines, Paris.
79. dell'Isola F., Romano A. A phenomenological approach to phase transition in classical field theory. Int. J. Eng. Sci. 1987, 25:1469-1475.
80. Wang C.Y., Feng L., Jasiuk I. Scale and boundary conditions effects on the apparent elastic moduli of trabecular bone modeled as a periodic cellular solid. J. Biomech. Eng., Trans. ASME 2009, 131(12):121008.
81. Mindlin R.D. Second gradient of strain and surface tension in linear elasticity. Int. J. Solids Struct. 1965, 1:417-438.