Skip to main content
SpringerLink
Log in

Effect of magnetic field on poroelastic bone model for internal remodeling

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

This paper studies the effects of the magnetic field and the porosity on a poroelastic bone model for internal remodeling. The solution of the internal bone remodeling process induced by a magnetic field is presented. The bone is treated as a poroelastic material by Biot’s formulation. Based on the theory of small strain adaptive elasticity, a theoretical approach for the internal remodeling is proposed. The components of the stresses, the displacements, and the rate of internal remodeling are obtained in analytical forms, and the numerical results are represented graphically. The results indicate that the effects of the magnetic field and the porosity on the rate of internal remodeling in bone are very pronounced.

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

Access this article

Log in via an institution

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. Ahmed, S. M. and Abd-Alla, A. M. Electromechanical wave propagation in a cylindrical poroelastic bone with cavity. Applied Mathematics and Computation, 133, 257–286 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. El-Naggar, A. M., Abd-Alla, A. M., and Mahmoud, S. R. Analytical solution of electro-mechanical wave propagation in long bones. Applied Mathematics and Computation, 119, 77–98 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abd-Alla, A. M., Abo-Dahab, S. M., and Mahmoud, S. R. Wave propagation modeling in cylindrical human long wet bones with cavity. Meccanica, 46, 1413–1428 (2011)

    Article  MathSciNet  Google Scholar 

  4. Hart, R. T. A theoretical study of the influence of bone maturation rate on surface remodeling predictions: idealized models. Journal of Biomechanics, 23, 241–257 (1990)

    Article  Google Scholar 

  5. Qin, Q. H., Qu, C., and Ye, J. Thermoelectroelastic solutions for surface bone remodeling under axial and transverse loads. Biomaterials, 26, 6798–6810 (2005)

    Article  Google Scholar 

  6. Martínez, G. J., Aznar, M. G., Doblaré, M., and Cerrolaza, M. External bone remodeling through boundary elements and damage mechanics. Mathematics and Computers in Simulation, 73, 183–199 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jang, I. G. and Kim, I. Y. Computational simulation of simultaneous cortical and trabecular bone change in human proximal femur during bone remodeling. Journal of Biomechanics, 43, 294–301 (2010)

    Article  Google Scholar 

  8. Tsili, M. C. Theoretical solutions for internal bone remodeling of diaphyseal shafts using adaptive elasticity theory. Journal of Biomechanics, 33, 235–239 (2000)

    Article  Google Scholar 

  9. Cowin, S. C. and Firoozbakhsh, K. Bone remodeling of diaphysial surfaces under constant load: theoretical predictions. Journal of Biomechanics, 14, 471–484 (1981)

    Article  Google Scholar 

  10. Ganghoffer, J. F. A contribution to the mechanics and thermodynamics of surface growth: application to bone external remodeling. International Journal of Engineering Science, 50, 166–191 (2012)

    Article  MathSciNet  Google Scholar 

  11. Ganghoffer, J. F. Extremum principles for biological continuous bodies undergoing volumetric and surface growth. Bulletin of the Polish Academy of Sciences: Technical Sciences, 60, 559–563 (2012)

    Google Scholar 

  12. Zumsande, M., Stiefs, D., Siegmund, S., and Gross, T. General analysis of mathematical models for bone remodeling. Bone, 48, 910–917 (2011)

    Article  Google Scholar 

  13. Malachanne, E., Dureisseix, D., and Jourdan, F. Numerical model of bone remodeling sensitive to loading frequency through a poroelastic behavior and internal fluid movements. Journal of the Mechanical Behavior of Biomedical Materials, 4, 849–857 (2011)

    Article  Google Scholar 

  14. Vahdati, A. and Rouhi, G. A model for mechanical adaptation of trabecular bone incorporating cellular accommodation and effects of microdamage and disuse. Mechanics Research Communications, 36, 284–293 (2009)

    Article  MATH  Google Scholar 

  15. Hazelwood, S. J., Martin, R. B., Rashid, M. M., and Rodrigo, J. J. A mechanistic model for internal bone remodeling exhibits different dynamic responses in disuse and overload. Journal of Biomechanics, 34, 299–308 (2001)

    Article  Google Scholar 

  16. Qu, C., Qin, Q. H., and Kang, Y. A hypothetical mechanism of bone remodeling and modeling under electromagnetic loads. Biomaterials, 27, 4050–4057 (2006)

    Article  Google Scholar 

  17. Papathanasopoulou, V. A., Fotiadis, D. I., Foutsitzi, G., and Massalas, C. V. A poroelastic bone model for internal remodeling. International Journal of Engineering Science, 40, 511–530 (2002)

    Article  Google Scholar 

  18. Mengoni, M. and Ponthot, J. P. Isotropic continuum damage/repair model for alveolar bone remodeling. Journal of Computational and Applied Mathematics, 234, 2036–2045 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Boyle, C. and Kim, I. Y. Three-dimensional micro-level computational study ofWolff’s law via trabecular bone remodeling in the human proximal femur using design space topology optimization. Journal of Biomechanics, 44, 935–942 (2011)

    Article  Google Scholar 

  20. Wang, X., Erickson, A. M., Allen, M. R., Burr, D. B., Martin, R. B., and Hazelwood, S. J. Theoretical analysis of alendronate and risedronate effects on canine vertebral remodeling and microdamage. Journal of Biomechanics, 42, 938–944 (2009)

    Article  Google Scholar 

  21. Peterson, M. C. and Riggs, M. M. A physiologically based mathematical model of integrated calcium homeostasis and bone remodeling. Bone, 46, 49–63 (2010)

    Article  Google Scholar 

  22. Qin, Q. H. and Ye, J. Q. Thermoelectroelastic solutions for internal bone remodeling under axial and transverse loads. International Journal of Solids and Structures, 41, 2447–2460 (2004)

    Article  MATH  Google Scholar 

  23. Boyle, C. and Kim, I. Y. Comparison of different hip prosthesis shapes considering micro-level bone remodeling and stress-shielding criteria using three-dimensional design space topology optimization. Journal of Biomechanics, 44, 1722–1728 (2011)

    Article  Google Scholar 

  24. Cowin, S. C. and Buskirk, W. C. V. Surface bone remodeling induced by a medullary pin. Journal of Biomechanics, 12, 269–276 (1979)

    Article  Google Scholar 

  25. Biot, M. A. General theory of three-dimensional consolidation. Jouranl of Applied Physics, 12, 155–165 (1941)

    Article  MATH  Google Scholar 

  26. Biot, M. A. Theory of elasticity and consolidation for a porous anisotropic solid. Journal of Applied Physics, 26, 182–185 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  27. Biot, M. A. and Willis, D. G. The elastic coefficients of the theory of consolidation. Journal of Applied Mechanics, 79, 594–601 (1957)

    MathSciNet  Google Scholar 

  28. Biot, M. A. General solutions of the equations of elasticity and consolidation for a porous material. Journal of Applied Mechanics, 78, 91–96 (1956)

    MathSciNet  Google Scholar 

  29. Nowinski, J. L. and Davis, C. F. The flexure and torsion of bones viewed as anisotropic poroelastic bodies. International Journal of Engineering Science, 10, 1063–1079 (1972)

    Article  MATH  Google Scholar 

  30. Hegedus, D. M. and Cowin, S. C. Bone remodeling, II, small-strain adaptive elasticity. Journal of Elasticity, 6, 337–352 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  31. Johnson, M. W., Chakkalakal, D. A., Harper, R. A., Katz, J. L., and Rouhana, S.W. Fluid flow in bone in vitro. Journal of Biomechanics, 15, 881–885 (1982)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, Taif University, Taif, 888, Saudi Arabia

    A. M. Abd-Alla & S. M. Abo-Dahab

  2. Department of Mathematics, Faculty of Science, Sohag University, Nasser, 82524, Sohag, Egypt

    A. M. Abd-Alla

  3. Department of Mathematics, Faculty of Science, South Valley University, Qen, 83523, Egypt

    S. M. Abo-Dahab

Authors
  1. A. M. Abd-Alla
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. M. Abo-Dahab
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. M. Abd-Alla.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abd-Alla, A.M., Abo-Dahab, S.M. Effect of magnetic field on poroelastic bone model for internal remodeling. Appl. Math. Mech.-Engl. Ed. 34, 889–906 (2013). https://doi.org/10.1007/s10483-013-1715-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-013-1715-6

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation