Homogenized strain gradient remodeling model for trabecular bone microstructures

  • Zineeddine Louna
  • Ibrahim Goda
  • Jean-François GanghofferEmail author
Original Article


Constitutive models for bone remodeling are established from micromechanical analyses at the scale of individual trabeculae defining the representative unit cell (RUC), accounting for both first- and second-order deformation gradients. On the microscale, trabeculae undergo apposition of new bone modeled by a surface growth velocity field driven by a mechanical stimulus identified to the surface divergence of an Eshelby-like tensor. The static and evolutive first and second gradient effective properties of a periodic network of bone trabeculae are evaluated by numerical simulations with controlled imposed first and second displacement gradient rates over the RUC. The formulated effective growth constitutive law at the scale of the homogenized set of trabeculae relates the (average) first and second growth strain rates to the homogenized stress and hyperstress tensors. The constitutive model is identified relying on the framework of TIP (thermodynamics of irreversible processes), adopting a split of the kinematic and static tensors into their deviator and hydrostatic contributions. The obtained results quantify the strength and importance of strain gradient effects on the overall remodeling process.


External and internal remodeling Trabecular bone Surface growth Homogenized strain gradient growth model Micromechanics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    Alibert, J.-J., Della Corte, A.: Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof. Z. Angew. Math. Phys. 66(5), 2855–2870 (2015)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Altenbach, H., Eremeyev, V.A.: On the linear theory of micropolar plates. ZAMM Z. Angew. Math. Mech. 89(4), 242–256 (2009)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Berkache, K., Deogekar, S., Goda, I., Picu, R.C., Ganghoffer, J.-F.: Construction of second gradient continuum models for random fibrous networks and analysis of size effects. Compos. Struct. 181, 347–357 (2017)Google Scholar
  4. 4.
    Bowman, S.M., et al.: Creep contributes to the fatigue behavior of bovine trabecular bone. J. Biomech. Eng. 120, 647–654 (1998)Google Scholar
  5. 5.
    Buechner, P.M., Lakes, R.S.: Size effects in the elasticity and viscoelasticity of bone. Biomech. Model. Mechanobiol. 1(4), 295–301 (2003)Google Scholar
  6. 6.
    Ciarletta, P., Preziosi, L., Maugin, G.A.: Mechanobiology of interfacial growth. J. Mech. Phys. Solids 61, 852–872 (2013)ADSMathSciNetzbMATHGoogle Scholar
  7. 7.
    Cosserat, E., Cosserat, F.: Théorie des Corps Déformables. Librairie Scientifique A. Hermann et Fils, Paris (1909)zbMATHGoogle Scholar
  8. 8.
    Cowin, S.C., Hegedus, D.H.: Bone remodeling I: theory of adaptive elasticity. J. Elast. 6, 313–325 (1976)MathSciNetzbMATHGoogle Scholar
  9. 9.
    dell’Isola, F., Seppecher, P., Della Corte, A.: The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 471, 2183 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Epstein, M.: Kinetics of boundary growth. Mech. Res. Commun. 37(5), 453–457 (2010)zbMATHGoogle Scholar
  11. 11.
    Epstein, M., Maugin, G.A.: Thermomechanics of volumetric growth in uniform bodies. Int. J. Plast. 16, 951–978 (2000)zbMATHGoogle Scholar
  12. 12.
    Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10(3), 233–248 (1972)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Fernandes, P.R., Folgadoa, J., Jacobs, C., Pellegrini, V.: A contact model with ingrowth control for bone remodelling around cementless stems. J. Biomech. 35, 167–176 (2002)Google Scholar
  14. 14.
    Field, C., Li, Q., Li, W., Thompson, M., Swain, M.: A comparative mechanical and bone remodelling study of all-ceramic posterior inlay and onlay fixed partial dentures. J. Dent. 40(1), 48–56 (2012). Google Scholar
  15. 15.
    Frasca, P., Harper, R., Katz, J.L.: Strain and frequency dependence of shear storage modulus for human single osteons and cortical bone microsamples-size and hydration effects. J. Biomech. 14(10), 679–690 (1981)Google Scholar
  16. 16.
    Ganghoffer, J.F., Sokolowski, J.: A micromechanical approach to volumetric and surface growth in the framework of shape optimization. Int. J. Eng. Sci. 74, 207–226 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ganghoffer, J.F.: Mechanical modeling of growth considering domain variation-part II: volumetric and surface growth involving Eshelby tensors. J. Mech. Phys. Solids 58(9), 1434–1459 (2010)ADSzbMATHGoogle Scholar
  18. 18.
    Ganghoffer, J.F.: A contribution to the mechanics and thermodynamics of surface growth, application to bone remodeling. Int. J. Eng. Sci. 50(1), 166–191 (2012)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Ganghoffer, J.F., Plotnikov, P.I., Sokolowski, J.: Mathematical modeling of volumetric material growth. Arch. Appl. Mech. 84(9–11), 1357–1371 (2014)ADSzbMATHGoogle Scholar
  20. 20.
    Giorgio, I., Andreaus, U., dell’Isola, I., Lekszycki, T.: Viscous second gradient porous materials for bones reconstructed with bio-resorbable grafts. Extreme Mech. Lett. 13, 141–147 (2017)Google Scholar
  21. 21.
    Goda, I., Assidi, M., Belouettar, S., Ganghoffer, J.-F.: A micropolar anisotropic constitutive model of cancellous bone from discrete homogenization. J. Mech. Behav. Biomed. Mater. 16, 87–108 (2012)Google Scholar
  22. 22.
    Goda, I., Assidi, M., Ganghoffer, J.-F.: A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure. Biomech. Model. Mechanobiol. 13, 53–83 (2014)Google Scholar
  23. 23.
    Goda, I., Ganghoffer, J.-F.: 3D plastic collapse and brittle fracture surface models of trabecular bone from asymptotic homogenization method. Int. J. Eng. Sci. 87(58–82), 2015 (2015b)zbMATHGoogle Scholar
  24. 24.
    Goda, I., Ganghoffer, J.-F.: Construction of first and second order grade anisotropic continuum media for 3D porous and textile composite structures. Compos. Struct. 141, 292–327 (2016)Google Scholar
  25. 25.
    Goda, I., Ganghoffer, J.F., Maurice, G.: Combined bone internal and external remodeling based on Eshelby stress. Int. J. Solids Struct. 94–95, 138–157 (2016a)Google Scholar
  26. 26.
    Goda, I., Ganghoffer, J.-F.: Identification of couple-stress moduli of vertebral trabecular bone based on the 3D internal architectures. J. Mech. Behav. Biomed. Mater. 51, 99–118 (2015a)Google Scholar
  27. 27.
    Goda, I., Rahouadj, R., Ganghoffer, J.-F.: Size dependent static and dynamic behavior of trabecular bone based on micromechanical models of the trabecular. Int. J. Eng. Sci. 72, 53–77 (2013)zbMATHGoogle Scholar
  28. 28.
    Goda, I., Rahouadj, R., Ganghoffer, J.-F., Kerdjoudj, H., Siad, L.: 3D couple-stress moduli of porous polymeric biomaterials using \(\mu \)CT image stack and FE characterization. Int. J. Eng. Sci. 100, 25–44 (2016b)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Harrigan, T.P., Jasty, M.J., Mann, R.W., Harris, W.H.: Limitations of the continuum assumption in cancellous bone. J. Biomech. 21, 269–275 (1988)Google Scholar
  30. 30.
    Kröner, E.: Elasticity theory of materials with long range cohesive forces. Int. J. Solids Struct. 3(5), 731–742 (1976)zbMATHGoogle Scholar
  31. 31.
    Lacroix, D., Prendergast, P.J.: A mechano-regulation model for tissue differentiation during fracture healing: analysis of gap size and loading. J. Biomech. 35, 1163–1171 (2002)Google Scholar
  32. 32.
    Lakes, R.: Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. In: Muhlhaus, H.-B. (ed.) Continuum Models for Materials with Microstructure, pp. 1–22. Wiley, New York (1995)Google Scholar
  33. 33.
    Lemaitre, J., Chaboche, J.L.: Mécanique des matériaux solides. Dunod, Paris (2009)Google Scholar
  34. 34.
    Louna, Z., Goda, I., Ganghoffer, J.F., Benhadid, S.: Formulation of an effective growth response of trabecular bone based on micromechanical analyses at the trabecular level. Arch. Appl. Mech. 87(3), 457–477 (2016)Google Scholar
  35. 35.
    Louna, Z., Goda, I., Ganghoffer, J.F.: Identification of a constitutive law for trabecular bone samples under remodeling in the framework of irreversible thermodynamics. Thermodyn. Contin. Mech. (2018).
  36. 36.
    Madeo, A., George, D., Lekszycki, T., Nierenberger, M., Rémond, Y.: A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodeling. C. R. Méc. 340(8), 575–589 (2012)Google Scholar
  37. 37.
    Madeo, A., Lekszycki, T., dell’Isola, F.: Continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery. C. R. Méc. 339(10), 625–682 (2011)Google Scholar
  38. 38.
    Maire, E., Withers, P.J.: Quantitative X-ray tomography. Int. Mater. Rev. 59, 1–43 (2014)Google Scholar
  39. 39.
    McNamara, L.M., Prendergast, P.J.: Bone remodelling algorithms incorporating both strain and microdamage stimuli. J. Biomech. 40, 1381–1391 (2007)Google Scholar
  40. 40.
    Olivares, L., Lacroix, D.: Computational methods in the modeling of scaffolds for tissue engineering. In: Geris, L. (ed.) Computational Modeling in Tissue Engineering, pp. 107–126. Springer, Berlin (2013)Google Scholar
  41. 41.
    Olive, M., Auffray, N.: Isotropic invariants of a completely symmetric third-order tensor. J. Math. Phys. American Institute of Physics (AIP) 55(9), 1.4895466 (2014)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Park, H.C., Lakes, R.S.: Cosserat micromechanics of human bone: strain redistribution by a hydration sensitive constituent. J. Biomech. 19(5), 385–397 (1986)Google Scholar
  43. 43.
    Ramézani, H., El-Hraiech, A., Jeong, J., Benhamou, C.-L.: Size effect method application for modeling of human cancellous bone using geometrically exact Cosserat elasticity. Comput. Methods Appl. Mech. Eng. 237, 227–243 (2012)ADSGoogle Scholar
  44. 44.
    Reda, H., Goda, I., Ganghoffer, J.F., L’Hostis, G., Lakiss, H.: Dynamical analysis of homogenized second gradient anisotropic media for textile composite structures and analysis of size effects. Compos. Struct. 161, 540–551 (2017)Google Scholar
  45. 45.
    Sanz-Herrera, J., Garcia-Aznar, J., Doblaré, M.: On scaffold designing for bone regeneration: a computational multiscale approach. Acta Biomater. 5(1), 219–229 (2009)Google Scholar
  46. 46.
    Skalak, R., Farrow, D.A., Hoger, A.: Kinematics of surface growth. J. Math. Biol. 35, 869–907 (1997)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Taylor, M., Cotton, J., Zioupos, P.: Finite element simulation of the fatigue behaviour of cancellous bone. Meccanica 37, 419–429 (2002)zbMATHGoogle Scholar
  48. 48.
    Wagner, D.W., Lindsey, D.P., Beaupre, G.S.: Deriving tissue density and elastic modulus from microCT bone scans. Bone 49(5), 931–938 (2011)Google Scholar
  49. 49.
    Wang, C., Han, J., Li, Q., Wang, L., Fan, Y.: Simulation of bone remodelling in orthodontic treatment. Comput. Methods Biomech. Biomed. Eng. 17(9), 1042–1050 (2012)Google Scholar
  50. 50.
    Yang, J.F.C., Lakes, R.S.: Transient study of couple stress effects in compact bone: Torsion. J. Biomech. Eng. 103, 275–279 (1981)Google Scholar
  51. 51.
    Yang, J.F.C., Lakes, R.S.: Experimental study of micropolar and couple stress elasticity in compact bone in bending. J. Biomech. 15(2), 91–98 (1982)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Zineeddine Louna
    • 1
  • Ibrahim Goda
    • 2
    • 3
  • Jean-François Ganghoffer
    • 4
    Email author
  1. 1.LMFTA, Faculté de physiqueUSTHBBab Ezzouar AlgiersAlgeria
  2. 2.LPMTUniversité de Haute-AlsaceMulhouse CedexFrance
  3. 3.Department of Industrial Engineering, Faculty of EngineeringFayoum UniversityFayoumEgypt
  4. 4.CNRSLEM3– Université de LorraineMetzFrance

Personalised recommendations