Advertisement

The Fast and Slow Wave Propagation in Cancellous Bone: Experiments and Simulations

  • Atsushi HosokawaEmail author
  • Yoshiki Nagatani
  • Mami Matsukawa
Chapter

Abstract

Cancellous bone consists of a complex solid trabecular network structure filled with soft bone marrow. The use of a short and broadband ultrasound incident pulse enables the experimental observation of a two longitudinal wave phenomenon, consistently with Biot’s prediction for porous media. This chapter is a review of the experimental studies and discusses theoretical interpretations, including the Biot’s theory and modified Biot’s models. The inhomogeneous nature of cancellous bone often results in some discrepancies between theory and experimental results. However, the two-wave phenomenon may provide detailed information on the structure and characteristics of cancellous bone, beyond conventional quantitative ultrasound (QUS) parameters. In order to understand this complex wave propagation in cancellous bone, numerical simulations offer an interesting and powerful alternative to intractable analytical approaches. Recent progress in computer performances enables the visualization of wave propagation using for example finite difference numerical methods, combined with three-dimensional numerical models of actual cancellous bone structures. In addition, the numerical investigation using virtual trabecular structures brings insightful view into the two-wave phenomenon, which cannot be obtained using the experiments. Finally, this chapter also refers to a new in vivo technique based on the two-wave phenomenon.

Keywords

Angle-dependent Biot’s model (stratified Biot’s model) Anisotropy Artificial model of trabecular structure Bayesian probability theory Biot’s theory Bone mineral density (BMD) Bone volume fraction (BV/TV) Cancellous bone Cancellous bone phantom Clinical application Degree of anisotropy (DA) Erosion/dilation procedure Fast and slow waves Finite element method (FEM) Finite-difference time-domain (FDTD) method Focused (concave) transmitter Frequency-dependent ultrasound attenuation (FDUA) Image processing technique Inhomogeneous In-silico approach LD-100 Mode conversion Modified Biot-Attenborough (MBA) model Numerical simulation Overlapping fast and slow waves Poly(vinylidene fluoride) (PVDF) transducer Scalogram Scattering Short-time Fourier transform SimSonic Spectrograms Stratified model (Schoenberg’s model) Synchrotron radiation microcomputed tomography (SR-μCT) Trabecular length Trabecular microstructure Trabecular orientation Trabecular thickness Two-wave phenomenon Virtual specimen Viscous friction Wave separation technique Wavelet transform X-ray μ CT 

References

  1. 1.
    Z. E. A. Fellah, N. Sebaa, M. Fellah, F. G. Mitri, E. Ogam, W. Lauriks, and C. Depollier, “Application of the Biot model to ultrasound in bone: Direct problem,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 55, 1508–1515 (2008).CrossRefPubMedGoogle Scholar
  2. 2.
    M. A. Biot, “Theory of propagation of elastic waves in fluid-saturated porous solid. I. Low-frequency range,” The Journal of the Acoustical Society of America 28, 168–178 (1956).Google Scholar
  3. 3.
    M. A. Biot, “Theory of propagation of elastic waves in fluid-saturated porous solid. II. Higher frequency range,” The Journal of the Acoustical Society of America 28, 179–191 (1956).Google Scholar
  4. 4.
    M. L. McKelvie and S. B. Palmer, “The interaction of ultrasound with cancellous bone,” Physics in Medicine and Biology 36, 1331–1340 (1991).CrossRefPubMedGoogle Scholar
  5. 5.
    J. J. Kaufman, G. Luo, and R. S. Siffert, “Ultrasound simulation in bone,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 55, 1205–1218 (2008).CrossRefPubMedGoogle Scholar
  6. 6.
    K. Mizuno, M. Matsukawa, T. Otani, M. Takada, I. Mano, and T. Tshujimoto, “Effects of structural anisotropy of cancellous bone on speed of ultrasonic fast waves in the bovine femur,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 55, 1480–1487 (2008).CrossRefPubMedGoogle Scholar
  7. 7.
    E. Bossy, F. Padilla, F. Peyrin, and P. Laugier, “Three-dimensional simulation of ultrasound propagation through trabecular bone structures measured by synchrotron microtomography,” Physics in Medicine and Biology 50, 5545–5556 (2005).CrossRefPubMedGoogle Scholar
  8. 8.
    A. Hosokawa and T. Otani, “Ultrasonic wave propagation in bovine cancellous bone,” The Journal of the Acoustical Society of America 101, 558–562 (1997).CrossRefPubMedGoogle Scholar
  9. 9.
    M. Kaczmarek, J. Kubik, and M. Pakula, “Short ultrasonic waves in cancellous bone,” Ultrasonics 40, 95–100 (2002).CrossRefPubMedGoogle Scholar
  10. 10.
    L. Cardoso, F. Teboul, L. Sedel, C. Oddou, and A. Meunier, “In vitro acoustic waves propagation in human and bovine cancellous bone,” Journal of Bone and Mineral Research 18, 1803–1812 (2003).CrossRefPubMedGoogle Scholar
  11. 11.
    Y. Nakamura and T. Otani, “Frequency response of a piezoelectric polymer film hydrophone and an elastic wave induced on the backing surface,” Japanese Journal of Applied Physics 31 Supplement 31-1, 266–268 (1991).Google Scholar
  12. 12.
    Y. Nakamura, T. Kawabata, and T. Otani, “Anomalous directivity of a piezoelectric polymer film hydrophone,” Japanese Journal of Applied Physics 32, 2288–2290 (1993).CrossRefGoogle Scholar
  13. 13.
    Y. Nakamura and T. Otani, “Study of surface elastic wave induced on backing material and diffracted field of a piezoelectric polymer film hydrophone,” The Journal of the Acoustical Society of America 94, 1191–1199 (1993).CrossRefGoogle Scholar
  14. 14.
    P. H. F. Nicholson, R. Müller, G. Lowet, X. G. Cheng, T. Hildebrand, P. Rüegsegger, G. van der Perre, J. Dequeker, and S. Boonen, “Do quantitative ultrasound measurements reflect structure independently of density in human vertebral cancellous bone?,” Bone 23, 425–431 (1998).CrossRefPubMedGoogle Scholar
  15. 15.
    A. Hosokawa and T. Otani, “Acoustic anisotropy in bovine cancellous bone,” The Journal of the Acoustical Society of America 103, 2718–2722 (1998).CrossRefPubMedGoogle Scholar
  16. 16.
    A. Hosokawa, T. Otani, T. Suzaki, Y. Kubo, and S. Takai, “Influence of trabecular structure on ultrasonic wave propagation in bovine cancellous bone,” Japanese Journal of Applied Physics 36, 3233–3237 (1997).CrossRefGoogle Scholar
  17. 17.
    E. R. Hughes, T. G. Leighton, G. W. Petley, and P. R. White, “Ultrasonic propagation in cancellous bone: A new stratified model,” Ultrasound in Medicine and Biology 25, 811–821 (1999).CrossRefPubMedGoogle Scholar
  18. 18.
    B. K. Hoffmeister, S. A. Whitten, and Y. Rho, “Low-megahertz ultrasonic properties of bovine cancellous bone,” Bone 26, 635–642 (2000).CrossRefPubMedGoogle Scholar
  19. 19.
    M. Pakula and J. Kubik, “Propagation of ultrasonic waves in cancellous bone. Micro and macrocontinual approach,” Poromechanics II, edited by J.-L. Auriault et al. (Swets & Zeitlinger, Lisse, 2002), pp. 65–70.Google Scholar
  20. 20.
    Z. E. A. Fellah, J. Y. Chapelon, S. Berger, W Lauriks, and C. Depollier, “Ultrasonic wave propagation in human cancellous bone: Application of Biot theory,” The Journal of the Acoustical Society of America 116, 61–73 (2004).Google Scholar
  21. 21.
    N. Sebaa, Z. E. A. Fellah, M. Fellah, E. Ogam, A. Wirgin, F. G. Mitri, C. Depollier, and W. Lauriks, “Ultrasonic characterization of human cancellous bone using the Biot theory: Inverse problem,” The Journal of the Acoustical Society of America 120, 1816–1824 (2006).CrossRefPubMedGoogle Scholar
  22. 22.
    N. Sebaa, Z. E. A. Fellah, W. Lauriks, and C. Depollier, “Application of fractional calculus to ultrasonic wave propagation in human cancellous bone,” Signal Processing 86, 2668–2677 (2006).CrossRefGoogle Scholar
  23. 23.
    N. Sebaa, Z. E. A. Fellah, M. Fellah, E. Ogam, F.G. Mitri, C. Depollier, and W. Lauriks “Application of the Biot model to ultrasound in bone: Direct problem,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 55, 1516–1523 (2008).Google Scholar
  24. 24.
    K. Mizuno, M. Matsukawa, T. Otani, P. Laugier, and F. Padilla, “Propagation of two longitudinal waves in human cancellous bone: An in vitro study,” The Journal of the Acoustical Society of America 125, 3460–3466 (2009).CrossRefPubMedGoogle Scholar
  25. 25.
    J. L. Williams, “Ultrasonic wave propagation in cancellous and cortical bone: Prediction of some experimental results by Biot’s theory,” The Journal of the Acoustical Society of America 91, 1106–1112 (1992).CrossRefPubMedGoogle Scholar
  26. 26.
    L. J. Gibson, “The mechanical behaviour of cancellous bone,” Journal of Biomechanics 18, 317–328 (1985).CrossRefPubMedGoogle Scholar
  27. 27.
    D. Ulrich, B. van Rietbergen, A. Laib, and P. Ruegsegger, “The ability of three-dimensional structural indices to reflect mechanical aspects of trabecular bone,” Bone 25, 55–60 (1999).CrossRefPubMedGoogle Scholar
  28. 28.
    T. J. Haire and C. M. Langton, “Biot theory: A review of its application to ultrasound propagation through cancellous bone,” Bone 24, 291–295 (1999).CrossRefPubMedGoogle Scholar
  29. 29.
    K. I. Lee, H. –S. Roh, and S. W. Yoon, “Acoustic wave propagation in bovine cancellous bone: Application of the modified Biot-Attenborough model,” The Journal of the Acoustical Society of America 114, 2284–2293 (2003).Google Scholar
  30. 30.
    K. I. Lee and S. W. Yoon, “Comparison of acoustic characteristics predicted by Biot’s theory and the modified Biot-Attenborough model in cancellous bone,” Journal of Biomechanics 39, 364–368 (2006).CrossRefPubMedGoogle Scholar
  31. 31.
    K. I. Lee, E. R. Hughes, V. F. Humphrey, T. G. Leighton, and M. J. Choi, “Empirical angle-dependent Biot and MBA models for acoustic anisotropy in cancellous bone,” Physics in Medicine and Biology 52, 59–73 (2007).CrossRefPubMedGoogle Scholar
  32. 32.
    H. S. Roh, K. I. Lee, and S. W. Yoon, “Acoustic characteristics of a non-rigid porous medium with circular cylindrical pores,” Journal of the Korean Physical Society 43, 55–65 (2003).Google Scholar
  33. 33.
    D. L. Johnson, J. Koplik, and R. Dashen, “Theory of dynamic permeability and tortuosity in fluid-saturated porous media,” The Journal of Fluid Mechanics 176, 379–402 (1987).CrossRefGoogle Scholar
  34. 34.
    E. R. Hughes, T. G. Leighton, P. R. White, and G. W. Petley, “Investigation of an anisotropic tortuosity in a Biot model of ultrasonic propagation in cancellous bone,” The Journal of the Acoustical Society of America 121, 568–574 (2007).CrossRefPubMedGoogle Scholar
  35. 35.
    M. Schoenberg, “Wave propagation in alternating solid and fluid layers,” Wave Motion 6, 303–320 (1984).CrossRefGoogle Scholar
  36. 36.
    F. Padilla and P. Laugier, “Phase and group velocities of fast and slow compressional waves in trabecular bone,” The Journal of the Acoustical Society of America 108, 1949–1952 (2000).CrossRefPubMedGoogle Scholar
  37. 37.
    A. Hosokawa, “Ultrasonic pulse waves propagating through cancellous bone phantoms with aligned pore spaces,” Japanese Journal of Applied Physics 45, 4697–4699 (2006).CrossRefGoogle Scholar
  38. 38.
    A. Hosokawa, “Influence of minor trabecular elements on fast and slow wave propagations through cancellous bone,” Japanese Journal of Applied Physics 47, 4170–4175 (2008).CrossRefGoogle Scholar
  39. 39.
    A. Hosokawa, “Effect of minor trabecular elements on fast and slow wave propagations through a stratified cancellous bone phantoms at oblique incidence,” Japanese Journal of Applied Physics 48, 07GK07-1-07GK07-7 (2009).Google Scholar
  40. 40.
    A. Hosokawa, “Numerical investigation of ultrasound wave propagation in cancellous bone with oblique trabecular orientation,” Proceedings of 20th International Congress on Acoustics [CD-ROM], 2010, p367 (7 pages).Google Scholar
  41. 41.
    K. R. Marutyan, M. R. Holland, and J. G. Miller, “Anomalous negative dispersion in bone can result from the interference of fast and slow waves,” The Journal of the Acoustical Society of America 120, EL55–EL61 (2006).Google Scholar
  42. 42.
    C. C. Anderson, K. R. Marutyan, M. R. Holland, K. A. Wear, and J. G. Miller, “Interference between wave modes may contribute to the apparent negative dispersion observed in cancellous bone,” The Journal of the Acoustical Society of America 124, 1781–1789 (2008).CrossRefPubMedGoogle Scholar
  43. 43.
    T. Otani, “Quantitative estimation of bone density and bone quality using acoustic parameters of cancellous bone for fast and slow waves,” Japanese Journal of Applied Physics 44, 4578–4582 (2005).CrossRefGoogle Scholar
  44. 44.
    L. Cardoso, A. Meunier, and C. Oddou, “In vitro acoustic wave propagation in human and bovine cancellous bone as predicted by Biot’s theory,” Journal of Mechanics in Medicine and Biology 8, 183–201 (2008).CrossRefGoogle Scholar
  45. 45.
    K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Transactions on Antennas and Propagation 14, 302–307 (1966).CrossRefGoogle Scholar
  46. 46.
    J. Virieux, “P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method,” Geophysics 51, 889–901 (1986).CrossRefGoogle Scholar
  47. 47.
    G. Luo, J. J. Kaufman, A. Chiabrera, B. Blanco, J. H. Kinney, D. Haupt, J. T. Ryaby, and R. S. Siffert, “Computational methods for ultrasonic bone assessment,” Ultrasound in Medicine and Biology 25, 823–830 (1999).CrossRefPubMedGoogle Scholar
  48. 48.
    E. Bossy, M. Talmant, and P. Laugier, “Three dimensional simulations of ultrasonic axial transmission velocity measurement on cortical bone models,” The Journal of the Acoustical Society of America 115, 2314–2324 (2004).CrossRefPubMedGoogle Scholar
  49. 49.
    G. Haïat, F. Padilla, F. Peyrin, and P. Laugier, “Fast wave ultrasonic propagation in trabecular bone: Numerical study of the influence of porosity and structural anisotropy,” The Journal of the Acoustical Society of America 123, 1694–1705 (2008).CrossRefPubMedGoogle Scholar
  50. 50.
    Y. Nagatani, H. Imaizumi, T. Fukuda, M. Matsukawa, Y. Watanabe, and T. Otani, “Applicability of finite-difference time-domain method to simulation of wave propagation in cancellous bone,” Japanese Journal of Applied Physics 45, 7186–7190 (2006).CrossRefGoogle Scholar
  51. 51.
    Y. Nagatani, K. Mizuno, T. Saeki, M. Matsukawa, T. Sakaguchi, and H. Hosoi, “Propagation of fast and slow waves in cancellous bone: Comparative study of simulation and experiment,” Acoustical Science and Technology 30, 257–264 (2009).CrossRefGoogle Scholar
  52. 52.
    L. Goossens, J. Vanderoost, S. Jaecques, S. Boonen, J. D’hooge, W. Lauriks, and G. van der Perre, “The correlation between the SOS in trabecular bone and stiffness and density studied by finite-element analysis,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 55, 1234–1242 (2008).CrossRefPubMedGoogle Scholar
  53. 53.
    G. Haïat, F. Padilla, M. Svrcekova, Y. Chevalier, D. Pahr, F. Peyrin, P. Laugier, and P. Zysset, “Relationship between ultrasonic parameters and apparent trabecular bone elastic modulus: A numerical approach,” Journal of Biomechanics 42, 2033–2039 (2009).CrossRefPubMedGoogle Scholar
  54. 54.
    A. Hosokawa, “Simulation of ultrasound propagation through bovine cancellous bone using elastic and Biot’s finite-difference time-domain methods,” The Journal of the Acoustical Society of America 118, 1782–1789 (2005).CrossRefPubMedGoogle Scholar
  55. 55.
    A. Hosokawa, “Ultrasonic pulse waves in cancellous bone analyzed by finite-difference time-domain method,” Ultrasonics 44, e227–e231 (2006).CrossRefPubMedGoogle Scholar
  56. 56.
    V. –H. Nguyen, S. Naili, and V. Sansalone, “Simulation of ultrasonic wave propagation in anisotropic cancellous bone immersed in fluid,” Wave Motion 47, 117–129 (2010).Google Scholar
  57. 57.
    E. Bossy, P. Laugier, F. Peyrin, and F. Padilla, “Attenuation in trabecular bone: A comparison between numerical simulation and experimental results in human femur,” The Journal of the Acoustical Society of America 122, 2469–2475 (2007).CrossRefPubMedGoogle Scholar
  58. 58.
    Y. Nagatani, K. Mizuno, T. Saeki, M. Matsukawa, T. Sakaguchi, and H. Hosoi, “Numerical and experimental study on the wave attenuation in bone –FDTD simulation of ultrasound propagation in cancellous bone,” Ultrasonics 48, 607–612 (2008).CrossRefPubMedGoogle Scholar
  59. 59.
    G. Haïat, F. Padilla, F. Peyrin, and P. Laugier, “Variation of ultrasonic parameters with microstructure and material properties of trabecular bone: A 3D model simulation,” Journal of Bone and Mineral Research 22, 665–674 (2007).CrossRefPubMedGoogle Scholar
  60. 60.
    A. Hosokawa, “Numerical analysis of variability in ultrasound propagation properties induced by trabecular microstructure in cancellous bone,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 56, 738–747 (2009).CrossRefPubMedGoogle Scholar
  61. 61.
    A. Hosokawa, “Effect of porosity distribution in a propagation direction on ultrasound waves through cancellous bone,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 57, 1320–1328 (2010).CrossRefPubMedGoogle Scholar
  62. 62.
    A. Hosokawa, “Effect of trabecular irregularity on fast and slow wave propagations through cancellous bone,” Japanese Journal of Applied Physics 46, 4862–4867 (2007).CrossRefGoogle Scholar
  63. 63.
    A. Hosokawa, “Development of a numerical cancellous bone model for finite-difference time-domain simulations of ultrasound propagation,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 55, 1219–1233 (2008).CrossRefPubMedGoogle Scholar
  64. 64.
    F. Padilla, Q. Grimal, and P. Laugier, “Ultrasonic propagation through trabecular bone modeled as a random medium,” Japanese Journal of Applied Physics 47, 4220–4222 (2008).CrossRefGoogle Scholar
  65. 65.
    S. Hasegawa, Y. Nagatani, K. Mizuno, and M. Matsukawa, “Wavelet transform analysis of ultrasonic wave propagation in cancellous bone,” Japanese Journal of Applied Physics 49, 07HF28-1-07HF28-5 (2010).Google Scholar
  66. 66.
    K. R. Marutyan, G. L. Bretthost, and J. G. Miller, “Bayesian estimation of the underlying bone properties from mixed fast and slow mode ultrasonic signals,” The Journal of the Acoustical Society of America 121, EL8–EL15 (2007).Google Scholar
  67. 67.
    E. T. Jaynes, Probability Theory: The Logic of Science, edited by G. L. Bretthorst (Cambridge, University Press, Cambridge, UK, 2003).Google Scholar
  68. 68.
    I. Mano, K. Horii, S. Takai, T. Suzaki, H. Nagaoka, and T. Otani, “Development of novel ultrasonic bone densitometry using acoustic parameters of cancellous bone for fast and slow waves,” Japanese Journal of Applied Physics 45, 4700–4702 (2006).CrossRefGoogle Scholar
  69. 69.
    T. Otani, I. Mano, T. Tsujimoto, T. Yamamoto, R. Teshima, and H. Naka, “Estimation of in vivo cancellous bone elasticity,” Japanese Journal of Applied Physics 48, 07GK05-1-07GK05-5 (2009).Google Scholar

Copyright information

© Springer Netherlands 2011

Authors and Affiliations

  • Atsushi Hosokawa
    • 1
    Email author
  • Yoshiki Nagatani
    • 2
  • Mami Matsukawa
    • 3
  1. 1.Akashi National College of TechnologyAkashiJapan
  2. 2.Kobe City College of TechnologyKobeJapan
  3. 3.Doshisha UniversityKyotanabeJapan

Personalised recommendations