A continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery

Angela Madeo; Tomasz Lekszycki; Francesco dellʼIsola

doi:10.1016/j.crme.2011.07.004

A continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery

Angela Madeo ^1,² ; Tomasz Lekszycki ^3,^2,⁴; Francesco dellʼIsola ^5,^6,²

¹ Laboratoire de génie civil et ingénierie environnementale, université de Lyon–INSA, bâtiment Coulomb, 69621 Villeurbanne cedex, France
² International Center M& MOCS, University of LʼAquila, Palazzo Caetani, Cisterna di Latina, Italy
³ Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawinskiego 5B, 02-106 Warsaw, Poland
⁴ Warsaw Medical University, Lindleya 4, 02-005 Warszawa, Poland
⁵ Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma La Sapienza, Via Eudossiana 18, 00184 Roma, Italy
⁶ Laboratorio Strutture e Materiali Intelligenti, Fondazione Tullio Levi-Civita, Via S. Pasquale snc, Cisterna di Latina, Italy

Comptes Rendus. Mécanique, Volume 339 (2011) no. 10, pp. 625-640.

Abstract

We introduce a two-constituent porous continuum as a model describing the long-term growth/resorption phenomena in bone tissues grafted with bio-resorbable materials as driven by mechanical loads. The proposed model is able to account for the interplay between mechanical and biological phenomena which are known to be important for the bone tissue synthesis and the resorption of both bone tissue and bio-material. In particular, in the presented model the Lagrangian apparent mass densities of the natural bone and of the artificial material evolve in time according to precise ordinary differential equations. These latter are obtained by postulating a growth/resorption law and suitable constitutive equations conceived to account for the influence on bone resorption and synthesis of the action of different applied external loads as mediated by biological stimulus. The considered constitutive equations are chosen on the basis of the known biological phenomena occurring in bone resorption and synthesis. We present some numerical simulations for rod-bones subjected to axial external load. These numerical simulations allow for the description of the most desirable situation in which a gradual resorption of the artificial material takes place together with the contemporary formation of new bone, finally giving rise to an almost complete replacement of the artificial material with natural living tissue.

Received: 2010-10-14
Accepted: 2011-07-19
Published online: 2011-08-18

DOI: 10.1016/j.crme.2011.07.004

Keywords: Biomechanics, ODEs governing growth/resorption, Coupling between mechanical and biological stimuli, Artificial bio-resorbable material, Continuum solid-mixture model, Load-induced replacement of artificial material with natural bone tissue, Numerical simulations

Author's affiliations:

Angela Madeo ^{1, 2}; Tomasz Lekszycki ^{3, 2, 4}; Francesco dellʼIsola ^{5, 6, 2}

¹ Laboratoire de génie civil et ingénierie environnementale, université de Lyon–INSA, bâtiment Coulomb, 69621 Villeurbanne cedex, France
² International Center M& MOCS, University of LʼAquila, Palazzo Caetani, Cisterna di Latina, Italy
³ Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawinskiego 5B, 02-106 Warsaw, Poland
⁴ Warsaw Medical University, Lindleya 4, 02-005 Warszawa, Poland
⁵ Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma La Sapienza, Via Eudossiana 18, 00184 Roma, Italy
⁶ Laboratorio Strutture e Materiali Intelligenti, Fondazione Tullio Levi-Civita, Via S. Pasquale snc, Cisterna di Latina, Italy

@article{CRMECA_2011__339_10_625_0,
     author = {Angela Madeo and Tomasz Lekszycki and Francesco dell'Isola},
     title = {A continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {625--640},
     publisher = {Elsevier},
     volume = {339},
     number = {10},
     year = {2011},
     doi = {10.1016/j.crme.2011.07.004},
     language = {en},
}

TY  - JOUR
AU  - Angela Madeo
AU  - Tomasz Lekszycki
AU  - Francesco dellʼIsola
TI  - A continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery
JO  - Comptes Rendus. Mécanique
PY  - 2011
SP  - 625
EP  - 640
VL  - 339
IS  - 10
PB  - Elsevier
DO  - 10.1016/j.crme.2011.07.004
LA  - en
ID  - CRMECA_2011__339_10_625_0
ER  -

%0 Journal Article
%A Angela Madeo
%A Tomasz Lekszycki
%A Francesco dellʼIsola
%T A continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery
%J Comptes Rendus. Mécanique
%D 2011
%P 625-640
%V 339
%N 10
%I Elsevier
%R 10.1016/j.crme.2011.07.004
%G en
%F CRMECA_2011__339_10_625_0

Angela Madeo; Tomasz Lekszycki; Francesco dellʼIsola. A continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery. Comptes Rendus. Mécanique, Volume 339 (2011) no. 10, pp. 625-640. doi : 10.1016/j.crme.2011.07.004. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2011.07.004/

References
Cited by

[1] A.S. Greenwald, S.D. Boden, V.M. Goldberg, Y.K. Cato, T. Laurencin, R.N. Rosier, Bone graft substitutes: facts, fictions and applications, in: 68th Annual Meeting American Academy of Orthopaedic Surgeons, San Francisco, California, February 28–March 4, 2001.

[2] R. Kraus; J.-P. Stahl; R. Schnettler Treatment strategies in thoracolumbar vertebral fractures: are there indications for biomaterials?, European Journal of Trauma, Volume 32 (2007), pp. 253-257

[3] J.F. Mano; R.A. Sousa; L.F. Boesel; N.M. Neves Rui; L. Reis Bioinert, biodegradable and injectable polymeric matrix composites for hard tissue replacement: state of the art and recent developments, Composites Science and Technology, Volume 64 (2004), pp. 789-817

[4] M. Jarcho Calcium phosphate ceramics as hard tissue prosthetics, Clinical Orthopaedics and Related Research, Volume 157 (1981), pp. 259-278

[5] J.C. Park; D. Wook; H. Suh A bone replaceable artificial bone substitute: morphological and physiochemical characterizations, Yonsei Medical Journal, Volume 41 (2000) no. 4, pp. 468-476

[6] K.A. Hing Bioceramic bone graft substitutes: influence of porosity and chemistry, International Journal of Applied Ceramics of Technology, Volume 2 (2005) no. 3, pp. 184-199

[7] A.F. Schilling; W. Linhart; S. Filke; M. Gebauer; T. Schinke; J.M. Rueger; M. Amling Resorbability of bone substitute biomaterials by human osteoclasts, Biomaterials, Volume 25 (2004), pp. 3963-3972

[8] H. Burchardt The biology of bone graft repair, Clinical Orthopaedics and Related Research, Volume 174 (1983), pp. 28-42

[9] H.-M. Kim Ceramic bioactivity and related biomimetic strategy, Current Opinion in Solid State and Materials Science, Volume 7 (2003), pp. 289-299

[10] T. Goto; T. Kojima; T. Iijima; S. Yokokura; H. Kawano; A. Yamamoto; K. Matsuda Resorption of synthetic porous hydroxyapatite and replacement newly formed bone, Journal of Orthopaedic Science, Volume 6 (2001), pp. 444-447

[11] Stephen C. Cowin Bone Mechanics Handbook, CRC Press, Boca Raton, FL, 2001

[12] V.I. Sikavitsas; J.S. Temenoff; A.G. Mokos Review: biomaterials and bone mechanotransduction, Biomaterials, Volume 22 (2001), pp. 2581-2593

[13] S. Nomuraa; T. Takano-Yamamoto Molecular events caused by mechanical stress in bone, Matrix Biology, Volume 19 (2000), pp. 91-96

[14] R. Ruimerman; P. Hilbers; B. van Rietbergen; R. Huiskes A theoretical framework for strain-related trabecular bone maintenance and adaptation, Journal of Biomechanics, Volume 38 (2005), pp. 931-941

[15] C.H. Turner; S.J. Warden; I. Jasiuk; T. Bellido; L.I. Plotkin; N. Kumar; J. Danzig; A.G. Robling Mechanobiology of the skeleton, Science Signaling, Volume 2 (2009) no. 68 (pt3)

[16] A. Yoo; I. Jasiuk Couple-stress moduli of a trabecular bone idealized as a 3D periodic cellular network, Journal of Biomechanics, Volume 39 (2006), pp. 2241-2252

[17] M. Rubin; I. Jasiuk The TEM characterization of the lamellar structure of osteoporotic human trabecular bone, Micron, Volume 36 (2005), pp. 653-664

[18] I. Jasiuk; M. Ostoja-Starzewski Modeling of bone at a single lamella level, Biomechanics and Modeling in Mechanobiology, Volume 3 (2004), pp. 67-74

[19] M. Ostoja-Starzewski; S.D. Boccara; I. Jasiuk Couple-stress moduli and characteristic length of a two-phase composite, Mechanics Research Communications, Volume 26 (1999) no. 4, pp. 387-396

[20] F. Bouyge; I. Jasiuk; S. Boccara; M. Ostoja-Starzewski A micromechanically based couple-stress model of an elastic orthotropic two-phase composite, European Journal of Mechanics A: Solids, Volume 21 (2002) no. 3, pp. 465-481

[21] M. Pawlikowski; M. Klasztorny; K. Skalski Studies on constitutive equation that models bone tissue, Acta of Bioengineering and Biomechanics, Volume 10 (2008) no. 4, pp. 39-47

[22] E. Hamed; Y. Lee; I. Jasiuk Multiscale modeling of elastic properties of cortical bone, New Orleans, LA, June, 2008 (Acta Mechanica), Volume 213 (2010) no. 1–2, pp. 131-154

[23] D. Inglis; S. Pietruszczak Characterization of anisotropy in porous media by means of linear intercept measurements, International Journal of Solids and Structures, Volume 40 (2003) no. 5, pp. 1243-1264

[24] A. Yoo, I. Jasiuk, Modeling of trabecular bone as a couple stress continuum, in: Advances in Bioengineering, ASME International Mechanical Engineering Congress, Washington, DC, November 15–21, 2003, pp. 41–42.

[25] S. Pietruszczak; D. Inglis; G.N. Pande A fabric-dependent fracture criterion for bone, Journal of Biomechanics, Volume 32 (1999) no. 10, pp. 1071-1079

[26] C.Y. Wang; L. Feng; I. Jasiuk Scale and boundary conditions effects on the apparent elastic moduli of trabecular bone modeled as a periodic cellular solid, Journal of Biomechanical Engineering – Transactions of the ASME, Volume 12 (2009), p. 131

[27] S. Pietruszczak; D. Inglis; G.N. Pande Modelling of bone–implant interaction, Computer Methods in Biomechanics & Biomedical Engineering, Volume 2 (1998), pp. 289-298

[28] J. Skoworodko; K. Skalski; W. Cejmer; K. Kwiatkowski Preoperative planning and post-operative estimation of vertebroplasty using CT/CAD/CAE systems, Acta of Bioengineering and Biomechanics, Volume 10 (2008) no. 2, pp. 15-22

[29] M. Dietrich; K. Kedzior; K. Skalski; T. Zagrejek; G. Krzesinski; J. Skoworodko; P. Borkowski; P. Wymyslowski On concurrent engineering and design of an intervertebral disc of lumbar spine (J.A. McGeough, ed.), Computer-Aided Production Engineering, John Wiley & Sons Ltd, Australia, 2003, pp. 209-219

[30] S. Piszczatowski; K. Skalski; W. Swieszkowski Load transfer between elastic hip implant and viscoelastic bone, Barcelona, Spain, May, 1997 (J. Middleton; M.L. Jones; G.N. Pande, eds.) (Computer Methods in Biomechanics & Biomedical Engineering), Volume 2 (1998), pp. 123-130

[31] P. Germain Cours de Mécanique des Milieux Continus, Masson, Paris, 1973

[32] J. Salencon Mécanique des milieux continus, Ellipse, 1988

[33] T. Lekszycki Optimality conditions in modeling of bone adaptation phenomenon, Journal of Theoretical and Applied Mechanics, Volume 37 (1999) no. 3, pp. 607-624

[34] T. Lekszycki Modeling of bone adaptation based on an optimal response hypothesis, Meccanica, Volume 37 (2002), pp. 343-354

[35] T. Lekszycki Functional adaptation of bone as an optimal control problem, Journal of Theoretical and Applied Mechanics, Volume 43 (2005) no. 3, pp. 120-140

[36] J.Y. Rho; R.B. Ashman; C.H. Turner Youngʼs modulus of trabecular and cortical bone material: ultrasonic and microtensile measurements, Journal of Biomechanics, Volume 26 (1993) no. 2, pp. 111-119

[37] J. Engelbrecht; M. Vendelin Microstructure described by hierarchical internal variables, Rendiconti del Seminaro Matematica Univ. Pol. Torino, Volume 58 (2001) no. 1, pp. 83-91

[38] J. Engelbrecht; M. Vendelin; G. Maugin Hierarchical internal variables reflecting microstructural properties: application to cardiac muscle contraction, Journal of Non-Equilibrium Thermodynamics, Volume 25 (2000), pp. 119-130

[39] G.A. Maugin; W. Muschik Thermodynamics with internal variables. 1. General concepts, Journal of Non-Equilibrium Thermodynamics, Volume 19 (1994), pp. 217-249

[40] A. Luongo; A. Paolone Perturbation methods for bifurcation analysis from multiple nonresonant complex eigenvalues, Nonlinear Dynamics, Volume 14 (1997), pp. 193-210

[41] A. Luongo; A. Paolone Multiple scale analysis for divergence-Hopf bifurcation of imperfect symmetric systems, Journal of Sound and Vibration, Volume 218 (1998) no. 3, pp. 527-539

[42] A. Luongo; A. Paolone; A. Di Egidio Multiple timescales analysis for 1:2 and 1:3 resonant Hopf bifurcations, Nonlinear Dynamics, Volume 34 (2003), pp. 269-291

[43] A. Di Egidio; A. Luongo; A. Paolone Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams, International Journal of Non-Linear Mechanics, Volume 42 (2007), pp. 88-98

[44] F. dellʼIsola; P. Seppecher Commentary about the article “Hypertractions and hyperstresses convey the same mechanical information”, Continuum Mechanics and Thermodynamics, Volume 22 (2010), pp. 163-176 (by Prof. Podio Guidugli and Prof. Vianello and some related articles on higher gradient theories, hal-00495800)

[45] F. dellʼIsola; A. Madeo; P. Seppecher Boundary conditions at fluid-permeable interfaces in porous media: a variational approach, International Journal of Solids Structures, Volume 46 (2009), pp. 3150-3164

[46] I. Jasiuk Modeling of trabecular bone as a hierarchical material, MIT Cambridge, MA, June 17–20, 2003 (K.J. Bathe, ed.) (Computational Fluid and Solid Mechanics), Volume 1 (2003) no. 2, pp. 1727-1728

[47] M. Pawlikowski; K. Skalski; M. Haraburda Process of hip joint prosthesis design including bone remodeling phenomenon, Computers & Structures, Volume 81 (2003) no. 8–11, pp. 887-893

[48] J.C.J. Webb; J. Tricker A review of fracture healing, Current Orthopaedics, Volume 14 (2000), pp. 457-546

Cited by Sources:

Comments - Policy