Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T09:00:45.565Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical analysis of a piezoelectric bone remodelling problem

Published online by Cambridge University Press:  25 May 2012

J. R. FERNÁNDEZ ,
J. M. GARCÍA-AZNAR  and
R. MARTÍNEZ
J. R. FERNÁNDEZ
Affiliation:
Departamento de Matemática Aplicada I, Universidade de Vigo, ETSI Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain email: [email protected]
J. M. GARCÍA-AZNAR
Affiliation:
Aragon Institute of Engineering Research (I3A), Universidad de Zaragoza, María de Luna, 3, E-50018, Zaragoza, Spain email: [email protected]
R. MARTÍNEZ
Affiliation:
Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, Facultade de Matemáticas, Campus Sur s/n, 15782 Santiago de Compostela, Spain email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Although in recent years bone piezoelectricity has been normally neglected, lately a new interest has appeared to show the importance of bone piezoelectricity in wet bone's complex response to loading. Here we numerically study a problem, including a strain-adaptive bone remodelling and the piezoelectricity. Its variational formulation leads to a coupled system composed of two linear variational equations for displacements and electric potential, and a parabolic variational inequality for the apparent density. Fully discrete approximations are now introduced by using the finite element method to approximate spatial variable and the explicit Euler scheme to discretise time derivatives. Some a priori error estimates are proved and the linear convergence of the algorithm is deduced under additional regularity conditions. Finally, some one- and two-dimensional numerical simulations are described to show the accuracy of the proposed algorithm and the behaviour of the solution.

Keywords

65M1565M6074B2074F99
Type
Papers
Information
European Journal of Applied Mathematics , Volume 23 , Issue 5 , October 2012 , pp. 635 - 657
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ahn, A. C. & Grodzinsky, A. J. (2009) Relevance of collagen piezoelectricity to “Wolff's Law”: A critical review. Med. Eng. Phys. 31 (7), 733–741.CrossRefGoogle ScholarPubMed
[2]Anderson, J. C. & Eriksson, C. (1968) Electrical properties of wet collagen. Nature 218, 167–169.CrossRefGoogle ScholarPubMed
[3]Anderson, J. C. & Eriksson, C. (1970) Piezoelectric properties of dry and wet bone. Nature 227, 491–492.CrossRefGoogle ScholarPubMed
[4]Barboteu, M., Fernández, J. R. & Hoarau-Mantel, T. V. (2005) A class of evolutionary variational inequalities with applications in viscoelasticity. Math. Models Methods Appl. Sci. 15 (10), 1595–1617.CrossRefGoogle Scholar
[5]Batra, R. C. & Yang, J. S. (1995) Saint-Venant's principle in linear piezoelectricity. J. Elast. 38, 209–218.CrossRefGoogle Scholar
[6]Beaupré, G. S., Orr, T. E. & Carter, D. R. (1990) An approach for time-dependent bone modeling and remodeling-theoretical development. J. Orthop. Res. 8, 651–661.CrossRefGoogle ScholarPubMed
[7]Becker, R. O. & Marino, A. A. (1982) Electromagnetism and Life, State University of New York Press, Albany, NY.Google Scholar
[8]Ciarlet, P. G. (1991) The finite element method for elliptic problems. In: Ciarlet, P. G. and Lions, J. L. (editors), Handbook of Numerical Analysis, Vol. 2, North Holland, Amsterdam, Netherlands, pp. 17–352.Google Scholar
[9]Cowin, S. C. (2006) The exact stimulus of the strain adaptation of bone tissue is unknown. J. Biomech. Sci. Eng. 1 (1), 16–28.CrossRefGoogle Scholar
[10]Cowin, S. C. & Hegedus, D. H. (1976) Bone remodeling I: Theory of adaptive elasticity. J. Elast. 6 (3), 313–326.CrossRefGoogle Scholar
[11]Fernández, J. R., García-Aznar, J. M., Martínez, R. & Viaño, J. M. (2010) Numerical analysis of a strain-adaptive bone remodelling problem. Comput. Methods Appl. Mech. Eng. 199, 1549–1557.CrossRefGoogle Scholar
[12]Fernández, J. R. & Kuttler, K. L. (2009) An existence and uniqueness result for an elasto-piezoelectric problem with damage. Math. Models Methods Appl. Sci. 19 (1), 31–50.CrossRefGoogle Scholar
[13]Fotiadis, D. I., Foutsitzi, G. & Massalas, C. V. (2000) Wave propagation in human long bones of arbitrary cross-section. Int. J. Eng. Sci. 38, 1553–1591.CrossRefGoogle Scholar
[14]Frost, H. M. (1987) Vital biomechanics. Proposed general concepts for skeletal adaptations to mechanical usage. Calcif. Tissue Int. 42, 145–156.CrossRefGoogle Scholar
[15]Fukada, E. & Yasuda, I. (1957) On the piezoelectric effect of bone. J. Phys. Soc. Japan 12, 1158–1162.CrossRefGoogle Scholar
[16]Fukada, E. & Yasuda, I. (1964) Piezoelectric effects in collagen. Japan J. Appl. Phys. 3, 117–121.CrossRefGoogle Scholar
[17]Gjelsvik, A. (1973) Bone remodelling and piezoelectricity II. J. Biomech. 6, 187–193.CrossRefGoogle ScholarPubMed
[18]Glowinski, R. (1984) Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York.CrossRefGoogle Scholar
[19]Guzelsu, N. (1978) A piezoelectric model for dry bone tissue. J. Biomech. 11, 257–267.CrossRefGoogle ScholarPubMed
[20]Han, W. & Sofonea, M. (2002) Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, American Mathematical Society International Press, Berkeley, CA.CrossRefGoogle Scholar
[21]Hegedus, D. H. & Cowin, S. C. (1976) Bone remodeling II: Small strain adaptive elasticity. J. Elast. 6 (4), 337–352.CrossRefGoogle Scholar
[22]Huiskes, R., Weinans, H. & Dalstra, M. (1989) Adaptive bone remodeling and biomechanical design considerations for noncemented total hip arthroplasty. Orthopedics 12, 1255–1267.CrossRefGoogle ScholarPubMed
[23]Huiskes, R., Weinans, H., Grootenboer, H. J., Dalstra, M., Fudala, B. & Sloof, T. J. (1987) Adaptive bone-remodeling theory applied to prosthetic-design analysis. J. Biomech. 20, 1135–1150.CrossRefGoogle ScholarPubMed
[24]Huiskes, R., Weinans, H., Summer, D. R., Fudala, B., Turner, T. M., Grootenboer, H. J. & Galante, J. (1989) Stress-shielding, stress-bypassing and bone resorption around “press-fit” and bone ingrowth THA. Trans. 35th Ann. Meeting Orthop. Res. Soc. 14, 529.Google Scholar
[25]Ideka, T. (1990) Fundamentals of Piezoelectricity, Oxford University Press, Oxford, UK.Google Scholar
[26]Isaacson, B. M. & Bloebaum, R. D. (2010) Bone bioelectricity: What have we learned in the past 160 years? J. Biomed. Mater. Res. A 95 (4), 1270–1279.CrossRefGoogle ScholarPubMed
[27]Kirchner, H. O. K. & Lazar, M. (2008) The thermodynamic driving force for bone growth and remodelling: A hypothesis. J. R. Soc. Interface 5, 183–193.CrossRefGoogle ScholarPubMed
[28]Lakes, R. S. & Katz, J. L. (1977) Dielectric relaxation in cortical bone. J. Appl. Phys. 48, 808–811.CrossRefGoogle Scholar
[29]Martin, R. B., Burr, D. B. & Sharkey, N. A. (1998) Skeletal Tissue Mechanics, Springer, New York.CrossRefGoogle Scholar
[30]Migorski, S., Ochal, A. & Sofonea, M. (2009) Weak solvability of a piezoelectric contact problem. Eur. J. Appl. Math. 20, 145–167.CrossRefGoogle Scholar
[31]Migorski, S., Ochal, A. & Sofonea, M. (2010) Variational analysis of fully coupled electro-elastic frictional contact problem. Math. Nachr. 283 (9), 1314–1335.CrossRefGoogle Scholar
[32]Migorski, S., Ochal, A. & Sofonea, M. (2011) Analysis of a quasistatic contact problem for piezoelectric materials. J. Math. Anal. Appl. 382, 701–713.CrossRefGoogle Scholar
[33]Mindlin, R. D. (1968) Polarisation gradient in elastic dielectrics. Int. J. Solids Struct. 4, 637–663.CrossRefGoogle Scholar
[34]Mindlin, R. D. (1969) Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin dielectric films. Int. J. Solids Struct. 4, 1197–1213.CrossRefGoogle Scholar
[35]Mindlin, R. D. (1972) Elasticity, piezoelasticity and crystal lattice dynamics. J. Elast. 4, 217–280.CrossRefGoogle Scholar
[36]Morro, A. & Straughan, B. (1991) A uniqueness theorem in the dynamical theory of piezoelectricity. Math. Methods Appl. Sci. 14 (5), 295–299.CrossRefGoogle Scholar
[37]Pollack, S. R. (2001) Streaming potentials in bone. In: Cowin, S. C. (editors), Bone Mechanics Handbook, Taylor & Francis, Boca Raton, FL, pp. 24.1–24.17.Google Scholar
[38]Qin, Q. H. & Ye, J. Q. (2004) Thermoelectroelastic solutions for internal bone remodeling under axial and transverse loads. Int. J. Solids Struct. 41, 2447–2460.CrossRefGoogle Scholar
[39]Ramtani, S. (2008) Electro-mechanics of bone remodelling. Int. J. Eng. Sci. 46, 1173–1182.CrossRefGoogle Scholar
[40]Saha, S. & Williams, P. A. (1992) Electric and dieletric properties of wet human cortical bone as a function of frequency. IEEE Trans. Biomed. Eng. 39 (12), 1298–1304.CrossRefGoogle Scholar
[41]Singh, S. & Saha, S. (1984) Electrical properties of bone. Clin. Orthop. Relat. Res. 186, 249–271.CrossRefGoogle Scholar
[42]Smit, T. H., Burguer, E. H. & Huyghe, J. M. (2002) A case for strain-induced fluid flow as a regulator of BMU-coupling and osteonal alignment. J. Bone Mineral Res. 17 (11), 2021–2029.CrossRefGoogle ScholarPubMed
[43]Toupin, R. A. (1956) The elastic dielectrics. J. Ration. Mech. Anal. 5, 849–915.Google Scholar
[44]Toupin, R. A. (1960) Stress tensors in elastic dielectrics. Arch. Ration. Mech. Anal. 5, 440–452.CrossRefGoogle Scholar
[45]Toupin, R. A. (1963) A dynamical theory of elastic dielectrics. Int. J. Eng. Sci. 1, 101–126.CrossRefGoogle Scholar
[46]Turbé, N. & Maugin, G. A. (1991) On the linear piezoelectricity of composite materials. Math. Methods Appl. Sci. 14 (6), 403–412.CrossRefGoogle Scholar
[47]Wang, L., Fritton, S. P., Cowin, S. C. & Weinbaum, S. (1999) Fluid pressure relaxation depends upon osteonal microstructure: Modeling an oscillatory bending experiment. J. Biomech. 32, 663–672.CrossRefGoogle ScholarPubMed
[48]Weinans, H., Huiskes, R. & Grootenboer, H. J. (1989) Convergence and uniqueness of adaptive bone remodeling. Trans. 35th Ann. Meeting Orthop. Res. Soc. 14, 310.Google Scholar
[49]Weinans, H., Huiskes, R. & Grootenboer, H. J. (1992) The behavior of adaptive bone-remodeling simulation models. J. Biomech. 25 (12), 1425–1441.CrossRefGoogle ScholarPubMed
[50]Xinhua, Y., Chuanyao, C., Yuantai, H. & Cheng, W. (2005) Combined damage fracture criteria for piezoelectric ceramics. Acta Mech. Sin. 18 (1), 21–27.Google Scholar
[51]Wolff, J. (1986) The Law of Bone Remodelling, Springer, Berlin.CrossRefGoogle Scholar