Abstract
Current diagnosis of bone loss and osteoporosis is based on the measurement of the bone mineral density (BMD) or the apparent mass density. Unfortunately, in most clinical ultrasound densitometers: 1) measurements are often performed in a single anatomical direction, 2) only the first wave arriving to the ultrasound probe is characterized, and 3) the analysis of bone status is based on empirical relationships between measurable quantities such as speed of sound (SOS) and broadband ultrasound attenuation (BUA) and the density of the porous medium. However, the existence of a second wave in cancellous bone has been reported, which is an unequivocal signature of poroelastic media, as predicted by Biot’s poroelastic wave propagation theory. In this paper, the governing equations for wave motion in the linear theory of anisotropic poroelastic materials are developed and extended to include the dependence of the constitutive relations upon fabric—a quantitative stereological measure of the degree of structural anisotropy in the pore architecture of a porous medium. This fabric-dependent anisotropic poroelastic approach is a theoretical framework to describe the microarchitectural-dependent relationship between measurable wave properties and the elastic constants of trabecular bone, and thus represents an alternative for bone quality assessment beyond BMD alone.
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References
Ashman RB, Rho JY (1988) Elastic modulus of trabecular bone material. J Biomech 21: 77–181
Auld B (1973) Acoustic fields and waves in solids, vol. 1. Wiley, New York
Baroud G, Falk R, Crookshank M, Sponagel S, Steffen T (2004) Experimental and theoretical investigation of directional permeability of human vertebral cancellous bone for cement infiltration. J Biomech 37: 189–196
Basillais A, Bensamoun S, Chappard Ch, Brunet-Imbault B, Lemineur G, Ilharreborde B, Ho Ba Tho MC, Benhamou CL (2007) Three-dimensional characterization of cortical bone microstructure by microcomputed tomography: validation with ultrasonic and microscopic measurements. J Orthop Sci 12(2): 141–148
Bear Jacob (1988) Dynamics of fluids in porous media. Dover Publications Inc., Mineola, p 134
Beaudoin AJ, Mihalko WM, Krause WR (1991) Finite element modelling of polymethylmethacrylate flow through cancellous bone. J Biomech 24(2): 127–136
Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12: 155–164
Biot MA (1955) Theory of elasticity and consolidation for a porous anisotropic solid. J Appl Phys 26: 182–185
Biot MA (1956a) Theory of propagation of elastic waves in a fluid saturated porous solid I low frequency range. J Acoust Soc Am 28: 168–178
Biot MA (1956b) Theory of propagation of elastic waves in a fluid saturated porous solid II higher frequency range. J Acoust Soc Am 28: 179–191
Biot MA (1962a) Mechanics of deformation and acoustic propagation in porous media. J Appl Phys 33: 1482–1498
Biot MA (1962b) Generalized theory of acoustic propagation in porous dissipative media. J Acoust Soc Am 28: 1254–1264
Bolotin HH (2007) DXA in vivo BMD methodology: an erroneous and misleading research and clinical gauge of bone mineral status, bone fragility, and bone remodeling. Bone 41(1): 138–154
Bone HG, Santora AC, Chattopadhyay A, Liberman U (2005) Are we treating women with postmenopausal osteoporosis for their low BMD or high fracture risk? J Bone Miner Res 20: 2064–2065
Cardoso L, Meunier A, Oddou C (2008) In vitro acoustic wave propagation in human and bovine cancellous bone as predicted by the Biot’s theory. J Mech Med Biol 8(2): 1–19
Cardoso L, Teboul F, Meunier A, Oddou C (2001) Ultrasound characterization of cancellous bone: theoretical and experimental analysis. IEEE Trans ultrason Symp 2: 1213–1216
Cardoso L, Teboul F, Sedel L, Meunier A, Oddou C (2003) In vitro acoustic waves propagation in human and bovine cancellous bone. J Bone Mineral Res 18(10): 1803–1812
Cowin SC, Mehrabadi MM (2007) Compressible and incompressible constituents in anisotropic poroelasticity: the problem of unconfined compression of a disk. J Mech Phys Solids 55: 161–193
Cowin SC (2004) Anisotropic poroelasticity: fabric tensor formulation. Mech Mater 36: 665–677
Cowin, SC, Satake, M (eds) (1978) Continuum mechanical and statistical approaches in the mechanics of granular materials. Gakujutsu Bunken Fukyu-Kai, Tokyo
Cowin SC (1999) Bone poroelasticity. J Biomech 32: 218–238
Cowin SC (1985) The relationship between the elasticity tensor and the fabric tensor. Mech Mater 4: 137–147
Cowin SC (1986) Wolff’s law of trabecular architecture at remodeling equilibrium. J Biomech Eng 108: 83–88
Cowin SC (1997) Remarks on the paper entitled fabric and elastic principal directions of cancellous bone are closely related. J Biomech 30: 1191–1192
Darcy H (1856) Les Fontains Publiques de la Ville de Dijon. Dalmont, Paris
Ericksen JL (1960) Tensor fields. In: Truesdell CA (eds) Encyclopedia of physics. Springer, Berlin, pp 794–858
Formica CA (1998) Standardization of BMD measurements. Osteoporos Int 8: 1–3
Gandolini G, Salvioni PM (2004) Is BMD measurement an adequate surrogate for anti-fracture efficacy? Aging Clin Exp Res 16: 29–32
Glorieux FH, Travers R, Taylor A, Bowen JR, Rauch F, Norman M, Parfitt AM (2000) Normative data for iliac bone histomorphometry in growing children. Bone 26(2): 103–109
Grigorian M, Shepherd JA, Cheng XG, Njeh CF, Toschke JO, Genant HK (2002) Does osteoporosis classification using heel BMD agree across manufacturers? Osteoporos Int 13: 613–617
Grimm MJ, Williams JL (1997a) Assessment of bone quantity and ‘quality’ by ultrasound attenuation and velocity in the heel. Clin Biomech (Bristol, Avon) 12: 281–285
Grimm MJ, Williams JL (1997b) Measurements of permeability in human calcaneal trabecular bone. J Biomech 30: 743–745
Hans D, Fuerst T, Uffmann M (1996) Bone density and quality measurement using ultrasound. Curr Opin Rheumatol 8: 370–375
Harrigan T, Mann RW (1984) Characterization of microstructural anisotropy in orthotropic materials using a second rank tensor. J Mat Sci 19: 761–769
Hengsberger S, Kulik A, Zysset P (2001) A combined atomic force microscopy and nanoindentation technique to investigate the elastic properties of bone structural units. Eur Cell Mater 1: 12–17
Hengsberger S, Kulik A, Zysset P (2002) Nanoindentation discriminates the elastic properties of individual human bone lamellae under dry and physiological conditions. Bone 30: 178–184
Hildebrand T, Laib A, Müller R, Dequeker J, Rüegsegger P (1999) Direct three-dimensional morphometric analysis of human cancellous bone: microstructural data from spine, femur, iliac crest, and calcaneus. J Bone Mineral Res 14: 1167–1174
Hill R (1952) The elastic behaviour of crystalline aggregate. Proc Phys Soc A 65: 349–354
Hilliard JE (1967) Determination of structural anisotropy. In: Stereology—Proceedings of the 2nd International Congress for Stereology, Chicago. Springer, Berlin, p 219
Hoffler CE, Moore KE, Kozloff K, Zysset PK, Goldstein SA (2000) Age, gender, and bone lamellae elastic moduli. J Orthop Res 18: 432–437
Hoffler CE, Moore KE, Kozloff K, Zysset PK, Brown MB, Goldstein SA (2000) Heterogeneity of bone lamellar-level elastic moduli. Bone 26: 603–609
Hosokawa A, Otani T (1997) Ultrasonic wave propagation in bovine cancellous bone. J Acoust Soc Am 101: 558–562
Hosokawa A, Otani T (1998) Acoustic anisotropy in bovine cancellous bone. J Acoust Soc Am 103: 2718–2722
Johnson DL, Koplik J, Dashen R (1987) Theory of dynamic permeability and tortuosity in fluid-saturated porous media. J Fluid Mech 176: 379–402
Jones AC, Sheppard AP, Sok RM, Arns CH, Limaye A, Averdunk H, Brandwood A, Sakellariou A, Senden TJ, Milthorpe BK, Knackstedt MA (2004) Three-dimensional analysis of cortical bone structure using X-ray micro-computed tomography. Physica A: Statistical Mechanics and its Applications 339(1–2):125–130. Proceedings of the International Conference New Materials and Complexity
Jorgensen CS, Kundu T (2002) Measurement of material elastic constants of trabecular bone: a micromechanical analytic study using a 1 GHz acoustic microscope. J Orthop Res 20: 151–158
Kanatani K (1983) Characterization of structural anisotropy by fabric tensors and their statistical test. J Jpn Soil Mech Found Eng 23: 171
Kanatani K (1984a) Distribution of directional data and fabric tensors. Int J Eng Sci 22: 149–164
Kanatani K (1984b) Stereological determination of structural anisotropy. Int J Eng Sci 22: 531–546
Kanatani K (1985) Procedures for stereological estimation of structural anisotropy. Int J Eng Sci 23: 587–596
Kaptoge S, Benevolenskaya LI, Bhalla AK, Cannata JB, Boonen S, Falch JA, Felsenberg D, Finn JD, Nuti R, Hoszowski K, Lorenc R, Miazgowski T, Jajic I, Lyritis G, Masaryk P, Naves-Diaz M, Poor G, Reid DM, Scheidt-Nave C, Stepan JJ, Todd CJ, Weber K, Woolf AD, Roy DK, Lunt M, Pye SR, O’neill TW, Silman AJ, Reeve J (2005) Low BMD is less predictive than reported falls for future limb fractures in women across Europe: results from the European prospective osteoporosis study. Bone 36: 387–398
Kleerekoper M, Nelson DA (2005) Is BMD testing appropriate for all menopausal women? Int J Fertil Womens Med 50: 61–66
Kohles SS, Roberts JB (2002) Linear poroelastic cancellous bone anisotropy: trabecular solid elastic and fluid transport properties. J Biomech Eng 124: 521–526
Kohles SS, Roberts JB, Upton ML, Wilson CG, Bonassar LJ, Schlichting AL (2001) Direct perfusion measurements of cancellous bone anisotropic permeability. J Biomech 34: 1197–1202
Li GP, Bronk JT, An KN, Kelly PJ (1987) Permeability of cortical bone of canine tibiae. Microvasc Res 34(3): 302–310
Lim TH, Hong JH (2000) Poroelastic properties of bovine vertebral trabecular bone. J Orthop Res 18: 671–677
Link TM, Vieth V, Matheis J, Newitt D, Ying L, Rummeny EJ, Majumdar S (2002) Bone structure of the distal radius and the calcaneus vs BMD of the spine and proximal femur in the prediction of osteoporotic spine fractures. Eur Radiol 12: 401–408
Mason WP (1958) Physical acoustics and the properties of solids. Van Nostrand Reinhold, Princeton
Matsuura M, Eckstein F, Lochmüller E-M, Zysset PK (2008) The role of fabric in the quasi-static compressive mechanical properties of human trabecular bone from various anatomical locations. Biomech Model Mechanobiol 7: 27–42
Morgan EF, Bayraktar HH, Keaveny TM (2003) Trabecular bone modulus-density relationships depend on anatomic site. J Biomech 36: 897–904
Nauman EA, Fong KE, Keaveny TM (1999) Dependence of intertrabecular permeability on flow direction and anatomic site. Ann Biomed Eng 27: 517–524
Nicholson PH, Cheng XG, Lowet G, Boonen S, Davie MW, Dequeker J, Vander Perre G (1997) Structural and material mechanical properties of human vertebral cancellous bone. Med Eng Phys 19: 729–737
Nicholson PHF, Müller R, Lowet G, Cheng XG, Hildebrand T, Rüegsegger P, Vander Perre G, Dequeker J, Boonen S (1998) Do quantitative ultrasound measurements reflect structure independently of density in human vertebral cancellous bone? Bone 23: 425–431
Nielsen SP (2000) The fallacy of BMD: a critical review of the diagnostic use of dual X-ray absorptiometry. Clin Rheumatol 19: 174–183
Njeh CF, Fuerst T, Diessel E, Genant HK (2001) Is quantitative ultrasound dependent on bone structure? A reflection. Osteoporos Int 12: 1–15
Oda M (1976) Fabrics and their effects on the deformation behaviors of sand. Department of Foundation Engineering, Saitama University, Japan
Oda M, Konishi J, Nemat-Nasser S (1980) Some experimentally based fundamental results on the mechanical behavior of granular materials. Geotechnique 30: 479
Oda M, Nemat-Nasser S, Konishi J (1985) Stress induced anisotropy in granular masses. Soils Found 25: 85
Odgaard A (1997a) Three-dimensional methods for quantification of cancellous bone architecture. Bone 20: 315–328
Odgaard A, Kabel J, van Rietbergen B, Dalstra M, Huiskes R (1997b) Fabric and elastic principal directions of cancellous bone are closely related. J Biomech 30: 487–495
Odgaard A (2001) Quantification of cancellous bone architecture. In: Cowin SC (eds) Bone mechanics handbook. CRC Press, Boca Raton
Parfitt AM, Mathews CH, Villanueva AR, Kleerekoper M, Frame B, Rao DS (1983) Relationships between surface, volume, and thickness of iliac trabecular bone in aging and in osteoporosis. Implications for the microanatomic and cellular mechanisms of bone loss. J Clin Invest 72(4): 1396–1409
Perrot C, Chevillotte F, Panneton R, Allard J-F, Lafarge D (2008) On the dynamic viscous permeability tensor symmetry. J Acoust Soc Am Express Lett 124: EL210–EL217
Plona TJ, Johnson DL (1983) Acoustic properties of porous systems: I. phenomenological description. In: Johnson DL, Sen PN (eds) Physcis and chemistry of porous media, AIP conference proceedings No. vol. 107, pp 89–104
Rehman MT, Hoyland JA, Denton J, Freemont AJ (1994) Age related histomorphometric changes in bone in normal British men and women. J Clin Pathol 47(6): 529–534
Rho JY, Roy ME 2nd, Tsui TY, Pharr GM (1999) Elastic properties of microstructural components of human bone tissue as measured by nanoindentation. J Biomed Mater Res 45: 48–54
Rho JY, Tsui TY, Pharr GM (1997) Elastic properties of human cortical and trabecular lamellar bone measured by nanoindentation. Biomaterials 18: 1325–1330
Roy ME, Rho JY, Tsui TY, Evans ND, Pharr GM (1999) Mechanical and morphological variation of the human lumbar vertebral cortical and trabecular bone. J Biomed Mater Res 44: 191–197
Sakata S, Barkmann R, Lochmuller EM, Heller M, Gluer CC (2004) Assessing bone status beyond BMD: evaluation of bone geometry and porosity by quantitative ultrasound of human finger phalanges. J Bone Miner Res 19: 924–930
Satake M (1982) Fabric tensor in granular materials. In: Vermeer PA, Lugar HJ (eds) Deformation and failure of granular materials. Balkema, Rotterdam, p 63
Sharma MD (2005) Propagation of inhomogeneous plane waves in dissipative anisotropic poroelastic solids. Geophys J Int 163: 981–990
Sharma MD (2008) Propagation of harmonic plane waves in a general anisotropic porous solid. Geophys J Int 172(3): 982–994
Siffert R, Kaufman J (2006) Ultrasonic bone assessment: the time has come. Bone 40(1): 5
Steiger P (1995a) Standardization of measurements for assessing BMD by DXA. Calcif Tissue Int 57: 469
Steiger P (1995b) Standardization of postero-anterior (PA) spine BMD measurements by DXA. Committee for Standards in DXA. Bone 17: 435
Thompson M, Willis JR (1991) A reformation of the equations of anisotropic poroelasticity. J Appl Mech 58: 612–616
Turner CH, Cowin SC (1987) On the dependence of the elastic constants of an anisotropic porous material upon porosity and fabric. J Mater Sci 22: 3178–3184
Turner CH, Cowin SC, Rho JY, Ashman RB, Rice JC (1990) The fabric dependence of the orthotropic elastic properties of cancellous bone. J Biomech 23: 549–561
Turner CH, Rho J, Takano Y, Tsui TY, Pharr GM (1999) The elastic properties of trabecular and cortical bone tissues are similar: results from two microscopic measurement techniques. J Biomech 32: 437–441
Van Rietbergen B, Odgaard A, Kabel J, Huiskes R (1996) Direct mechanical assessment of elastic symmetries and properties of trabecular bone architecture. J Biomech 29: 1653–1657
Van Rietbergen B, Odgaard A, Kabel J, Huiskes R (1998) Relationships between bone morphology and bone elastic properties can be accurately quantified using high-resolution computer reconstructions. J Orthop Res 16: 23–28
Whitehouse WJ (1974a) The quantitative morphology of anisotropic trabecular bone. J Microsc 101: 153–168
Whitehouse WJ, Dyson ED (1974b) Scanning electron microscope studies of trabecular bone in the proximal end of the human femur. J Anat 118: 417–444
Williams JL (1992) Ultrasonic wave propagation in cancellous and cortical bone: prediction of some experimental results by Biot’s theory. J Acoust Soc Am 91: 1106–1112
Yang G, Kabel J, van Rietbergen B, Odgaard A, Huiskes R, Cowin SC (1999) The anisotropic Hooke’s law for cancellous bone and wood. J Elast 53: 125–146
Zysset PK, Guo XE, Hoffler CE, Moore KE, Goldstein SA (1999) Elastic modulus and hardness of cortical and trabecular bone lamellae measured by nanoindentation in the human femur. J Biomech 32: 1005–1012
Acknowledgments
This work was supported by the National Institutes of Health (AG34198 & HL069537-07 R25 Grant for Minority BME Education), the National Science Foundation (NSF 0723027, PHY-0848491), and the PSC-CUNY Research Award Program of the City University of New York.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Cowin, S.C., Cardoso, L. Fabric dependence of wave propagation in anisotropic porous media. Biomech Model Mechanobiol 10, 39–65 (2011). https://doi.org/10.1007/s10237-010-0217-7
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DOI: https://doi.org/10.1007/s10237-010-0217-7