Microporodynamics of Bones: Prediction of the “Frenkel–Biot” Slow Compressional Wave
Publication: Journal of Engineering Mechanics
Volume 131, Issue 9
Abstract
Understanding of ultrasonic wave propagation in bones is essential for further development of related techniques in clinical practice. As any other saturated porous medium, bone is characterized by different forms of longitudinal wave propagation, either undrained waves or fast and (Frenkel–Biot) slow compressional waves. We here study the wave propagation in the framework of poromicromechanics. A continuum micromechanics model allows for the prediction of the anisotropic poroelastic properties, Biot’s coefficients, and moduli, from tissue-specific composition data, on the basis of tissue-independent (“universal”) elastic properties of the elementary components of all bones. These poroelastic properties enter the governing equations for wave propagation in anisotropic porous media. They allow for the prediction of undrained, fast and slow waves, as is verified by comparison of model results with experimental findings.
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Acknowledgments
The writers are grateful to Luc Dormieux and Jean-Francois Barthélémy, Ecole Nationale des Ponts et Chaussées, Marne-la-Vallée, France, as well as to Eric Lemarchand, Lille University of Science and Technology, France, for various discussions and continuous cooperation in the field of continuum micromechanics.
References
Ashman, R., and Rho, J. (1988). “Elastic modulus of trabecular bone material.” J. Biomech., 21(3), 177–181.
Auriault, J.-L., and Sanchez-Palencia, E. (1977). “Etude du comportement macroscopique d’un milieu poreux saturé déformable [Study of macroscopic behavior of a saturated deformable medium].” J. Mécanique, 16(4), 575–603 (in French).
Bilaniuk, N., and Wong, G. (1993). “Speed of sound in pure water as a function of temperature.” J. Acoust. Soc. Am., 93(3), 1609–1612.
Biot, M. (1956a). “Theory of propagation of elastic waves in a fluid-saturated porous solid, Part I: Low frequency range.” J. Acoust. Soc. Am., 28, 168–178.
Biot, M. (1956b). “Theory of propagation of elastic waves in a fluid-saturated porous solid, Part II: High frequency range.” J. Acoust. Soc. Am., 28, 179–191.
Bonar, L., Lees, S., and Mook, H. (1985). “Neutron diffraction studies of collagen in fully mineralized bone.” J. Mol. Biol., 181, 265–270.
Bryant, J. (1983). “The effect of impact on the marrow pressure of long bones in vitro.” J. Biomech., 16(8), 659–665.
Bryant, J. (1988). “On the mechanical function of marrow in long bones.” Eng. Med., 17(2), 55–58.
Carcione, J. (2000). “Energy balance and fundamental relations in dynamic anisotropic poro-viscoelasticity.” Proc. R. Soc. London, Ser. A, 457, 331–348.
Chateau, X., and Dormieux, L. (1998). “Homogenization of a non-saturated porous medium: Hill’s lemma and applications.” C. R. Acad. Sci., Ser. IIb: Mec., Phys., Chim., Astron., 320, 627–634.
Chateau, X., and Dormieux, L. (2002). “Micromechanics of saturated and unsaturated porous media.” Int. J. Numer. Analyt. Meth. Geomech., 26, 831–844.
Coussy, O. (1995). Mechanics of porous continua, Wiley, Chichester, U.K.
Coussy, O. (2004). Poromechanics, Wiley, Chichester, U.K.
Coussy, O., and Fleureau, J.-M., eds. (2002). Mécanique des sols non saturés [Mechanics of unsaturated solids], Hermes Science, Paris.
Cusack, S., and Miller, A. (1979). “Determination of the elastic constants of collagen by Brillouin light scattering.” J. Mol. Biol., 135, 39–51.
Dormieux, L., and Bourgeois, E. (2003). Introduction à la micromécanique des milieux poreux [Introduction to micromechanics of porous media], Presse de l’École Nationale des Ponts et Chaussées, Paris (in French).
Eshelby, J. (1957). “The determination of the elastic field of an ellipsoidal inclusion, and related problems.” Proc. R. Soc. London, Ser. A, 241, 376–396.
Frenkel, Y. (1944). “On the theory of seismic and seismoelectric phenomena in moist soil.” J. Phys., 8(4), 230–241.
Gong, J., Arnold, J., and Cohn, S. H. (1964). “Composition of trabecular and cortical bone.” Anat. Rec., 149, 325–332.
Hellmich, C., Barthélémy, J.-F., and Dormieux, L. (2004a). “Mineral-collagen interactions in elasticity of bone ultrastructure—A continuum micromechanics approach.” Eur. J. Mech. A/Solids, 23, 783–810.
Hellmich, C., and Ulm, F.-J. (2001). “Hydroxyapatite is uniformly concentrated in the extracollagenous ultrastructure of mineralized tissue.” Proc., 5th International Symp. on Computer Methods in Biomechanics and Biomedical Engineering, J. Middleton, N. Shrive, and M. Jones, eds., Rome.
Hellmich, C., and Ulm, F.-J. (2002a). “Are mineralized tissues open crystal foams reinforced by crosslinked collagen?—some energy arguments.” J. Biomech., 35, 1199–1212.
Hellmich, C., and Ulm, F.-J. (2002b). “Micromechanical model for the ultrastructural stiffness of mineralized tissues.” J. Eng. Mech., 128(8), 898–908.
Hellmich, C., and Ulm, F.-J. (2003). “Average hydroxyapatite concentration is uniform in extracollageneous ultrastructure of mineralized tissue.” Biomech. Model. Mechanobiol., 2, 21–36.
Hellmich, C., and Ulm, F.-J. (2005). “Drained and undrained poroelastic properties of healthy and pathological bone: A poro-micromechanical investigation.” Transp. Porous Media, 58, 243–268.
Hellmich, C., Ulm, F.-J., and Dormieux, L. (2004b). “Can the diverse elastic properties of trabecular and cortical bone be attributed to only a few tissue-independent phase properties and their interactions?—Arguments from a multiscale approach.” Biomech. Model. Mechanobiol., 2, 219–238.
Hosokawa, A., and Otani, T. (1997). “Ultrasonic wave properties in bovine cancellous bone.” J. Acoust. Soc. Am., 101(1), 558–562.
Hosokawa, A., and Otani, T. (1998). “Acoustic anisotropy in bovine cancellous bone.” J. Acoust. Soc. Am., 103(5), 2718–2722.
Hughes, E., Leighton, T., Petley, G., and White, P. (1999). “Ultrasonic propagation in cancellous bone: a new stratified model.” Ultrasound Med. Biol., 25(5), 811–821.
Kafka, V. (1983). “On hydraulic strengthening of bones.” Biorheology, 20, 789–793.
Kafka, V. (1993). “On hydraulic strengthening of bones.” J. Biomech., 26, 761–762.
Katz, J., and Ukraincik, K. (1971). “On the anisotropic elastic properties of hydroxyapatite.” J. Biomech., 4, 221–227.
Knothe Tate, M. (2003). “Whither flows the fluid in bone?—An osteocyte’s perspective.” J. Biomech., 36, 1409–1424.
Kohles, S., Roberts, J., Upton, M., Wilson, C., Bonassar, L., and Schlichting, A. (2001). “Direct perfusion measurements of cancellous bone anisotropic permeability.” J. Biomech., 34, 1197–1202.
Lakes, R., Yoon, H., and Katz, J. (1983). “Slow compressional wave propagation in wet human and bovine cortical bone.” Science, 220, 513–515.
Laws, N. (1985). “A note on penny-shaped cracks in transversely isotropic materials.” Mech. Mater., 4, 209–212.
Lee, K., Roh, H.-S., and Yoon, S. (2003). “Correlations between acoustic properties and bone density in bovine cancellous bone from .” J. Acoust. Soc. Am., 113(5), 2933–2938.
Lees, S. (1987). “Considerations regarding the structure of the mammalian mineralized osteoid from viewpoint of the generalized packing model.” Connect. Tissue Res., 16, 281–303.
Lees, S., Bonar, L., and Mook, H. (1984a). “A study of dense mineralized tissue by neutron diffraction.” Int. J. Biol. Macromol., 6, 321–326.
Lees, S., Heeley, J., and Cleary, P. (1979). “A study of some properties of a sample of bovine cortical bone using ultrasound.” Calcif. Tissue Int., 29, 107–117.
Lees, S., Pineri, M., and Escoubes, M. (1984b). “A generalized packing model for type I collagen.” Int. J. Biol. Macromol., 6, 133–136.
Lees, S., Prostak, K., Ingle, V., and Kjoller, K. (1994). “The loci of mineral in turkey leg tendon as seen by atomic force microscope and electron microscopy.” Calcif. Tissue Int., 55, 180–189.
Luppé, F., Conoir, J.-M., and Franklin, H. (2002). “Scattering by a fluid cylinder in a porous medium: application to trabecular bone.”J. Acoust. Soc. Am., 111(6), 2573–2582.
Luppé, F., Conoir, J.-M., and Franklin, H. (2003). “Multiple scattering in a trabecular bone: Influence of the marrow viscosity on the effective properties.” J. Acoust. Soc. Am., 113(5), 2889–2892.
Schoenberg, M. (1984). “Wave propagation in alternating solid and fluid layers.” Wave Motion, 6, 303–320.
Suquet, P., ed. (1997). Continuum micromechanics, Springer, New York.
Thompson, M., and Willis, J. (1991). “A reformation of the equations of anisotropic poroelasticity.” J. Appl. Mech., 58, 612–616.
Zaoui, A. (1997). “Structural morphology and constitutive behavior of micro-heterogeneous materials.” Continuum micromechanics, P. Suquet, ed., Springer, New York, 291–347.
Zaoui, A. (2002). “Continuum micromechanics: Survey.” J. Eng. Mech., 128(8), 808–816.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 131 • Issue 9 • September 2005
Pages: 918 - 927
Copyright
© 2005 ASCE.
History
Received: Oct 20, 2003
Accepted: Jun 4, 2004
Published online: Sep 1, 2005
Published in print: Sep 2005
Notes
Note. Associate Editor: Alexander H.-D. Cheng
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