Skip to main content
SpringerLink
Log in

The influence of different loads on the remodeling process of a bone and bioresorbable material mixture with voids

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

A model of a mixture of bone tissue and bioresorbable material with voids was used to numerically analyze the physiological balance between the processes of bone growth and resorption and artificial material resorption in a plate-like sample. The adopted model was derived from a theory for the behavior of porous solids in which the matrix material is linearly elastic and the interstices are void of material. The specimen—constituted by a region of bone living tissue and one of bioresorbable material—was acted by different in-plane loading conditions, namely pure bending and shear. Ranges of load magnitudes were identified within which physiological states become possible. Furthermore, the consequences of applying different loading conditions are examined at the end of the remodeling process. In particular, maximum value of bone and material mass densities, and extensions of the zones where bone is reconstructed were identified and compared in the two different load conditions. From the practical view point, during surgery planning and later rehabilitation, some choice of the following parameters is given: porosity of the graft, material characteristics of the graft, and adjustment of initial mixture tissue/bioresorbable material and later, during healing and remodeling, optimal loading conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. 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 223(5), 589–605 (2009)

    Article  Google Scholar 

  2. Andreaus U., Colloca M., Iacoviello D.: An optimal control procedure for bone adaptation under mechanical stimulus. Control Eng. Pract. 20(6), 575–583 (2012). doi:10.1016/j.conengprac.2012.02.002

    Article  Google Scholar 

  3. Andreaus, U., Colloca, M., Iacoviello, D.: Modeling of trabecular architecture as result of an optimal control procedure. In: Lecture Notes in Computational Vision and Biomechanics, vol. 4. Springer Netherlands (2013)

  4. Andreaus U., Colloca M., Iacoviello D.: Optimal bone density distributions: numerical analysis of the osteocyte spatial influence in bone remodeling. Comput. Methods Programs Biomed. 113(1), 80–91 (2014). doi:10.1016/j.cmpb.2013.09.00

    Article  Google Scholar 

  5. Andreaus U., Colloca M., Iacoviello D., Pignataro M.: Optimal-tuning pid control of adaptive materials for structural efficiency. Struct. Multidiscip. Optim. 43(1), 43–59 (2011)

    Article  Google Scholar 

  6. Andreaus U., Colloca M., Toscano A.: Mechanical behaviour of a prosthesized human femur: a comparative analysis between walking and stair climbing by using the finite element method. Biophys. Bioeng. Lett. 1(3), 1–15 (2008)

    Google Scholar 

  7. Andreaus, U., Giorgio, I., Lekszycki, T.: A 2-D continuum model of a mixture of bone tissue and bio-resorbable material for simulating mass density redistribution under load slowly variable in time. ZAMM—J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik (2013). doi:10.1002/zamm.201200182

  8. Andreaus, U., Giorgio, I., Madeo, A.: Modeling of the interaction between bone tissue and resorbable biomaterial as linear elastic materials with voids. ZAMP - Zeitschrift für angewandte Mathematik und Physik, 1–29 (2014). doi:10.1007/s00033-014-0403-z

  9. Batra G.: On Hamilton’s principle for thermo-elastic fluids and solids, and internal constraints in thermo-elasticity. Arch. Ration. Mech. Anal. 99(1), 37–59 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bevill G., Eswaran S.K., Farahmand F., Keaveny T.M.: The influence of boundary conditions and loading mode on high-resolution finite element-computed trabecular tissue properties. Bone 44(4), 573-8 (2009). doi:10.1016/j.bone.2008.11.015

    Google Scholar 

  11. Burger E.H., Klein-Nulend J.: Mechanotransduction in bone—role of the lacuno-canalicular network. FASEB J. 13(9001), S101–S112 (1999)

    Google Scholar 

  12. Burger E.H., Klein-Nulend J.: Responses of bone cells to biomechanical forces in vitro. Adv. Dental Res. 13(1), 93–98 (1999)

    Article  Google Scholar 

  13. Carcaterra, A., Akay, A.: Dissipation in a finite-size bath. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 84(1), 011121 (2011). doi:10.1103/PhysRevE.84.011121

  14. Carcaterra, A., Roveri, N., Pepe, G.: Fractional dissipation generated by hidden wave-fields. Math. Mech. Solids (2014). doi:10.1177/1081286513518941

  15. Cazzani, A., Malagù, M., Turco, E.: Isogeometric analysis of plane-curved beams. Math. Mech. Solids (2014). doi:10.1177/1081286514531265

  16. Cazzani A., Rovati M.: Sensitivity analysis and optimum design of elastic-plastic structural systems. Meccanica 26(2–3), 173–178 (1991)

    Article  MATH  Google Scholar 

  17. Chun H., Shin H., Han C., Lee S.: Influence of implant abutment type on stress distribution in bone under various loading conditions using finite element analysis. Int. J. Oral Maxillofac. Implants 21(2), 195–202 (2006)

    Google Scholar 

  18. Cowin S.C., Mehrabadi M.M.: Anisotropic symmetries of linear elasticity. Appl. Mech. Rev. 48(5), 247–285 (1995)

    Article  ADS  MATH  Google Scholar 

  19. Cowin S.C., Nunziato J.W.: Linear elastic materials with voids. J. Elast. 13(2), 125–147 (1983)

    Article  MATH  Google Scholar 

  20. dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola. Math. Mech. Solids (2014). doi:10.1177/1081286513509811

  21. dell’Isola F., Kosinski W.: Deduction of thermodynamic balance laws for bidimensional nonmaterial directed continua modelling interphase layers. Arch. Mech. 45, 333–359 (1993)

    MathSciNet  MATH  Google Scholar 

  22. dell’Isola F., Romano, A.: On the derivation of thermomechanical balance equations for continuous systems with a nonmaterial interface. Int. J. Eng. Sci. 25, 1459–1468 (1987)

  23. dell’Isola F., Vidoli S.: Continuum modelling of piezoelectromechanical truss beams: an application to vibration damping. Arch. Appl. Mech. 68(1), 1–19 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. dell’Isola F., Woźniak C.: On continuum modelling the interphase layers in certain two-phase elastic solids. ZAMM—J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 77(7), 519–526 (1997)

    ADS  MathSciNet  MATH  Google Scholar 

  25. dell’Isola F., Woźniak C.: On phase transition layers in certain micro-damaged two-phase solids. Int. J. Fract. 83(2), 175–189 (1997)

    Article  Google Scholar 

  26. Descamps B.: Computational Design of Lightweight Structures: Form Finding and Optimization. Wiley, New York (2014)

    Book  Google Scholar 

  27. Eremeyev V.A., Pietraszkiewicz W.: Phase transitions in thermoelastic and thermoviscoelastic shells. Arch. Mech. 61(1), 41–67 (2009)

    MathSciNet  MATH  Google Scholar 

  28. Eremeyev V.A., Pietraszkiewicz W.: Thermomechanics of shells undergoing phase transition. J. Mech. Phys. Solids 59(7), 1395–1412 (2011). doi:10.1016/j.jmps.2011.04.005

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. Eringen A.C.: Theory of thermo-microstretch elastic solids. Int. J. Eng. Sci. 28(12), 1291–1301 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  30. Farrar, D.: Bioresorbable polymers in orthopaedics. Medical Device Manufacturing & Technology, pp. 36–38 (2005)

  31. Fatihhi S.J., Ardiyansyah S., Harun M.N., Rabiatul A.A.R., Jaafar A., Jaafar A.: Finite element simulation: the effects of loading modes at different anatomical sites of trabecular bone on morphological indices. Adv. Mater. Res. 845, 266–270 (2013)

    Article  Google Scholar 

  32. Federico S.: Volumetric-distortional decomposition of deformation and elasticity tensor. Math. Mech. Solids 15(6), 672–690 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Federico S.: On the linear elasticity of porous materials. Int. J. Mech. Sci. 52(2), 175–182 (2010). doi:10.1016/j.ijmecsci.2009.09.006

    Article  MathSciNet  Google Scholar 

  34. Federico S.: Covariant formulation of the tensor algebra of non-linear elasticity. Int. J. Nonlinear Mech. 47(2), 273–284 (2012)

    Article  ADS  Google Scholar 

  35. Federico S., Grillo A., Herzog W.: A transversely isotropic composite with a statistical distribution of spheroidal inclusions: a geometrical approach to overall properties. J. Mech. Phys. Solids 52(10), 2309–2327 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Federico, S., Grillo, A., Imatani, S.: The linear elasticity tensor of incompressible materials. Math. Mech. Solids, p. 1081286514550576 (2014). doi:10.1177/1081286514550576

  37. Federico S., Grillo A., Wittum G.: Considerations on incompressibility in linear elasticity. Il Nuovo cimento della Società à italiana di fisica. C 32(1), 81 (2009)

    ADS  Google Scholar 

  38. Garusi E., Tralli A., Cazzani A.: An unsymmetric stress formulation for reissner-mindlin plates: a simple and locking-free rectangular element. Int. J. Comput. Eng. Sci. 5(3), 589–618 (2004)

    Article  Google Scholar 

  39. Giorgio I., Culla A., Del Vescovo D.: Multimode vibration control using several piezoelectric transducers shunted with a multiterminal network. Arch. Appl. Mech. 79(9), 859–879 (2009)

    Article  ADS  MATH  Google Scholar 

  40. Greco L., Cuomo M.: B-Spline interpolation of Kirchhoff-Love space rods. Comput. Methods Appl. Mech. Eng. 256(0), 251–269 (2013). doi:10.1016/j.cma.2012.11.017

    Article  MathSciNet  ADS  Google Scholar 

  41. Greco L., Cuomo M.: An implicit G1 multi patch B-spline interpolation for Kirchhoff–Love space rod. Comput. Methods Appl. Mech. Eng. 269(0), 173–197 (2014). doi:10.1016/j.cma.2013.09.018

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. Greco L., Impollonia N., Cuomo M.: A procedure for the static analysis of cable structures following elastic catenary theory. Int. J. Solids Struct. 51(7), 1521–1533 (2014)

    Article  Google Scholar 

  43. Grillo A., Federico S., Wittum G.: Growth, mass transfer, and remodeling in fiber-reinforced, multi-constituent materials. Int. J. Nonlinear Mech. 47(2), 388–401 (2012). doi:10.1016/j.ijnonlinmec.2011.09.026

    Article  ADS  Google Scholar 

  44. Grillo, A., Wittum, G.: Growth and mass transfer in multi-constituent biological materials. In: ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010, vol. 1281, pp. 355–358. AIP Publishing (2010)

  45. Grillo A., Wittum G., Federico S., Imatani S., Giaquinta G., Mićunović M.V.: Evolution of a fibre-reinforced growing mixture. Nuovo Cimento C 32C((1), 97–119 (2009). doi:10.1393/ncc/i2009-10356-1

    ADS  Google Scholar 

  46. Grillo, A., Wittum, G., Tomic, A., Federico, S.: Remodelling in statistically oriented fibre-reinforced materials and biological tissues. Math. Mech. Solids (2014). doi:10.1177/1081286513515265

  47. Klein-Nulend J., Bacabac R.G., Mullender M.G.: Mechanobiology of bone tissue. Pathologie biologie 53(10), 576–580 (2005)

    Article  Google Scholar 

  48. Klein-Nulend J., Bakkerl A.D., Bacabac R.G., Vatsa A., Weinbaum S.: Mechanosensation and transduction in osteocytes. Bone 54, 182–190 (2013)

    Article  Google Scholar 

  49. Lekszycki T., dell’Isola F.: A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials. ZAMM—J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 92(6), 426–444 (2012)

    MathSciNet  ADS  MATH  Google Scholar 

  50. Li L.H., Kommareddy K.P., Pilz C., Zhou C.R., Fratzl P., Manjubala I.: In vitro bioactivity of bioresorbable porous polymeric scaffolds incorporating hydroxyapatite microspheres. Acta Biomater. 6(7), 2525-31 (2010). doi:10.1016/j.actbio.2009.03.028

    Article  Google Scholar 

  51. Luongo A.: Perturbation methods for nonlinear autonomous discrete-time dynamical systems. Nonlinear Dyn. 10(4), 317–331 (1996)

    Article  MathSciNet  Google Scholar 

  52. Luongo A., Paolone A.: Perturbation methods for bifurcation analysis from multiple nonresonant complex eigenvalues. Nonlinear Dyn. 14(3), 193–210 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  53. Luongo A., Piccardo G.: A continuous approach to the aeroelastic stability of suspended cables in 1:22 internal resonance. JVC/J. Vib. Control 14(1–2), 135–157 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  54. Luongo A., Zulli D., Piccardo G.: A linear curved-beam model for the analysis of galloping in suspended cables. J. Mech. Mater. Struct. 2(4), 675–694 (2007)

    Article  Google Scholar 

  55. Madeo A., Djeran-Maigre I., Rosi G., Silvani C.: The effect of fluid streams in porous media on acoustic compression wave propagation, transmission, and reflection. Contin. Mech. Thermodyn. 25(2–4), 173–196 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  56. Madeo A., George D., Lekszycki T.: A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodelling. Comptes Rendus Mécanique 340(8), 575–589 (2012)

    Article  ADS  Google Scholar 

  57. 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 339(10), 625–640 (2011)

    Article  Google Scholar 

  58. Madeo A., Placidi L., Rosi G.: Towards the design of metamaterials with enhanced damage sensitivity: second gradient porous materials. Res Nondestruct. Eval. 25(2), 99–124 (2014)

    Article  ADS  Google Scholar 

  59. Maugin G.A.: Material Inhomogeneities in Elasticity, vol. 3. CRC Press, Boca Raton, FL (1993)

    Book  MATH  Google Scholar 

  60. Mindlin R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  61. Misra A., Singh V.: Micromechanical model for viscoelastic-materials undergoing damage. Contin. Mech. Thermodyn. 25, 1–16 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  62. Misra A., Yang Y.: Micromechanical model for cohesive materials based upon pseudo-granular structure. Int. J. Solids Struct. 47, 2970–2981 (2010)

    Article  MATH  Google Scholar 

  63. Olsson T., Klarbring A.: Residual stresses in soft tissue as a consequence of growth and remodeling: application to an arterial geometry. Eur. J. Mech. A/Solids 27(6), 959–974 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  64. Peng L., Bai J., Zeng X., Zhou Y.: Comparison of isotropic and orthotropic material property assignments on femoral finite element models under two loading conditions. Med. Eng. Phys. 28(3), 227–233 (2006)

    Article  Google Scholar 

  65. Pietraszkiewicz W., Eremeyev V., Konopińska V.: Extended non-linear relations of elastic shells undergoing phase transitions. ZAMM—J. Appl. Math. Mech. 87(2), 150–159 (2007). doi:10.1002/zamm.200610309

    Article  MathSciNet  MATH  Google Scholar 

  66. Placidi, L.: A variational approach for a nonlinear 1-dimensional second gradient continuum damage model. Contin. Mech. Thermodyn., 1–16 (2014). doi:10.1007/s00161-014-0338-9

  67. Placidi L., dell’Isola F., Ianiro N., Sciarra G.: Variational formulation of pre-stressed solid–fuid mixture theory, with an application to wave phenomena. Eur. J. Mech. A Solids 27(4), 582–606 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  68. Placidi L., Rosi G., Giorgio I., Madeo A.: Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials. Math. Mech. Solids 19(5), 555–578 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  69. Porfiri M., dell’Isola F., Frattale Mascioli F.M.: Circuit analog of a beam and its application to multimodal vibration damping, using piezoelectric transducers. Int. J. Circuit Theory Appl. 32(4), 167–198 (2004). doi:10.1002/cta.273

    Article  MATH  Google Scholar 

  70. Porfiri M., dell’Isola F., Santini E.: Modeling and design of passive electric networks interconnecting piezoelectric transducers for distributed vibration control. Int. J. Appl. Electromagn. Mech. 21(2), 69–87 (2005)

    Google Scholar 

  71. Quiligotti S., Maugin G.A., dell’Isola F.: An Eshelbian approach to the nonlinear mechanics of constrained solid-fluid mixtures. Acta Mech. 160(1-2), 45–60 (2003)

    Article  MATH  Google Scholar 

  72. Shi X., Wang X., Niebur G.L.: Effects of loading orientation on the morphology of the predicted yielded regions in trabecular bone. Ann. Biomed. Eng. 37(2), 354-62 (2009). doi:10.1007/s10439-008-9619-4

    Article  Google Scholar 

  73. Soltész U.: The influence of loading conditions on the life-times in fatigue testing of bone cements. J. Mater. Sci. Mater. Med. 5(9–10), 654–656 (1994)

    Article  Google Scholar 

  74. Turco E., Caracciolo P.: Elasto-plastic analysis of kirchhoff plates by high simplicity finite elements. Comput. Methods Appl. Mech. Eng. 190(5), 691–706 (2000)

    Article  ADS  MATH  Google Scholar 

  75. Vidoli S., dell’Isola F.: Vibration control in plates by uniformly distributed PZT actuators interconnected via electric networks. Eur. J. Mech. A/Solids 20(3), 435–456 (2001). doi:10.1016/S0997-7538(01)01144-5

    Article  ADS  MATH  Google Scholar 

  76. Walpole L.J.: Elastic behavior of composite materials: theoretical foundations. Adv. Appl. Mech. 21, 169–242 (1981)

    Article  MATH  Google Scholar 

  77. Walpole L.J.: Fourth-rank tensors of the thirty-two crystal classes: multiplication tables. Proc. R. Soc. Lond. A Math. Phys. Sci. 391(1800), 149–179 (1984)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  78. Yeremeyev V., Freidin A., Sharipova L.: The stability of the equilibrium of two-phase elastic solids. J. Appl. Math. Mech. 71(1), 61–84 (2007). doi:10.1016/j.jappmathmech.2007.03.007

    Article  MathSciNet  Google Scholar 

  79. Zhu, X., Hao, H., Fan, K., Wang, Y., Ou, J.: Debond detection in RC structures using piezoelectric materials. In: Alexander, M.G., Beushausen, H-D., Dehn, F., Moyo, P. (eds.) Rehabilitation and Retrofitting, 2nd International Conference on Concrete Repair, ICCRRR-2, pp. 261–262. CRC Press, Cape Town, South Africa (2008). doi:10.1201/9781439828403.ch93

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria Strutturale e Geotecnica, Università Sapienza di Roma, Via Eudossiana 18, 00184, Rome, Italy

    Ivan Giorgio & Ugo Andreaus

  2. International Research Center for the Mathematics and Mechanics of Complex Systems ‘MeMoCS’, Cisterna di Latina, Italy

    Ivan Giorgio & Angela Madeo

  3. Institut National des Sciences Appliquées (INSA), Université de Lyon, 20, Avenue Albert-Einstein, 69621, Villeurbanne Cedex, France

    Angela Madeo

Authors
  1. Ivan Giorgio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ugo Andreaus
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Angela Madeo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ivan Giorgio.

Additional information

Communicated by Victor Eremeyev, Peter Schiavone and Francesco dell'Isola.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Giorgio, I., Andreaus, U. & Madeo, A. The influence of different loads on the remodeling process of a bone and bioresorbable material mixture with voids. Continuum Mech. Thermodyn. 28, 21–40 (2016). https://doi.org/10.1007/s00161-014-0397-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-014-0397-y

Keywords

Search

Navigation