Skip to main content
SpringerLink
Log in

Numerical Methods for Ultrasonic Bone Characterization

  • Chapter
  • First Online:
Book cover Bone Quantitative Ultrasound

Abstract

During the last decade the possibility of investigating the details of wave phenomena with numerical simulation has caused an evolution of the research methodology in ultrasonic bone characterization. Use of numerical simulation as a surrogate of an in vitro or in vivo experiment has been validated in some cases in which the major propagation characteristics observed experimentally could be accurately simulated in trabecular bone as well as in cortical bone. This chapter can be thought of as a guide to numerical modeling for ultrasonic bone characterization, from the definition of the model configuration (geometry, material properties, etc.) to the computation of the solutions with popular finite difference or finite element algorithms. A comprehensive review of the published works in which numerical simulation served to investigate wave phenomena in bone and surrounding structures is provided.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    http://www.simsonic.fr

  2. 2.

    http://www.cyberlogic.org/software.html

  3. 3.

    http://www.simsonic.fr

  4. 4.

    http://www.simsonic.fr

  5. 5.

    http://www.simsonic.fr

References

  1. 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).

    Article  PubMed  Google Scholar 

  2. 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 Part 1-Regular Papers Brief Communications and Review Papers 45, 7186–7190 (2006).

    Google Scholar 

  3. P. Nicholson, P. Moilanen, T. Kärkkäinen, J. Timonen, and S. Cheng, Guided ultrasonic waves in long bones: modelling, experiment and application, Physiological Measurements 23, 755–768 (2002).

    Article  Google Scholar 

  4. E. Bossy, M. Talmant, and P. Laugier, Three-dimensional simulations of ultrasonic axial transmission velocity measurement on cortical bone models, Journal of the Acoustical Society of America 115, 2314–2324 (2004).

    Article  PubMed  Google Scholar 

  5. P. Moilanen, M. Talmant, V. Bousson, P. H. F. Nicholson, S. Cheng, J. Timonen, and P. Laugier, Ultrasonically determined thickness of long cortical bones: two-dimensional simulations of in vitro experiments, Journal of the Acoustical Society of America 122, 1818–1826 (2007).

    Article  PubMed  Google Scholar 

  6. K. Yee, Numerical solutions of initial boundary value problems involving maxwell’s equations in isotropic media, IEEE Transactions on Antennas and Propagation AP-14(3), 302–307 (1966).

    Google Scholar 

  7. A. Hrenikoff, Solution of problems of elasticity by the framework method, Journal of Applied Mechanics A8, 169–175 (1941).

    Google Scholar 

  8. R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bulletin of the American Mathematical Society 49, 1–61 (1943).

    Article  Google Scholar 

  9. B. A. Auld, Acoustic Fields and waves in solids, 2nd edition (Krieger Publishing Company, Malabar) (1990).

    Google Scholar 

  10. D. Royer and E. Dieulesaint, Elastic waves in solids I, 2nd edition (Springer-Verlag, Berlin) (1999).

    Google Scholar 

  11. M. G. Vavva, V. C. Protopappas, L. N. Gergidis, A. Charalambopoulos, D. I. Fotiadis, and D. Polyzos, Velocity dispersion of guided waves propagating in a free gradient elastic plate: application to cortical bone, Journal of the Acoustical Society of America 125 (2009).

    Google Scholar 

  12. B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Mathematics of Computation 31, 629–651 (1977).

    Article  Google Scholar 

  13. G. C. Cohen, Higher-order numerical methods (Springer, Berlin) (2002).

    Google Scholar 

  14. E. Bossy, M. Talmant, and P. Laugier, Effect of bone cortical thickness on velocity measurements using ultrasonic axial transmission: a 2d simulation study, Journal of the Acoustical Society of America 112, 297–307 (2002).

    Article  PubMed  Google Scholar 

  15. G. Haiat, S. Naili, Q. Grimal, M. Talmant, C. Desceliers, and C. Soize, Influence of a gradient of material properties on ultrasonic wave propagation in cortical bone: application to axial transmission, Journal of the Acoustical Society of America 125, 4043–4052 (2009).

    Article  PubMed  Google Scholar 

  16. V. C. Protopappas, D. I. Fotiadis, and K. N. Malizos, Guided ultrasound wave propagation in intact and healing long bones, Ultrasound in Medicine and Biology 32, 693–708 (2006).

    Article  PubMed  Google Scholar 

  17. S. P. Dodd, J. L. Cunningham, A. W. Miles, S. Gheduzzi, and V. F. Humphrey, Ultrasonic propagation in cortical bone mimics, Physics in Medicine and Biology 51, 4635–4647 (2006).

    Article  CAS  PubMed  Google Scholar 

  18. V. Le Floch, D. J. McMahon, G. Luo, A. Cohen, J. J. Kaufman, E. Shane, and R. S. Siffert, Ultrasound simulation in the distal radius using clinical high-resolution peripheral-ct images, Ultrasound in Medicine and Biology 34, 1317–1326 (2008).

    Article  PubMed  Google Scholar 

  19. J. Grondin, Q. Grimal, K. Engelke, and P. Laugier, Potential of first arriving signal to assess cortical bone geometry at the hip with qus: a model based study, Ultrasound in Medicine and Biology 4, 656–666 (2010).

    Article  Google Scholar 

  20. X. Guo, D. Yang, D. Zhang, W. Li, Y. Qiu, and J. Wu, Quantitative evaluation of fracture healing process of long bones using guided ultrasound waves: a computational feasibility study, Journal of the Acoustical Society of America 125, 2834–2837 (2009).

    Article  PubMed  Google Scholar 

  21. P. Moilanen, M. Talmant, P. H. F. Nicholson, S. Cheng, J. Timonen, and P. Laugier, Ultrasonically determined thickness of long cortical bones: three-dimensional simulations of in vitro experiments, Journal of the Acoustical Society of America 122, 2439–2245 (2007).

    Article  PubMed  Google Scholar 

  22. W. Pistoia, B. van Rietbergen, A. Laib, and P. Rüegsegger, High-resolution three-dimensional-pqct images can be an adequate basis for in-vivo micro-fe analysis of bone, Journal of Biomechanical Engineering 123, 176–183 (2001).

    Article  CAS  PubMed  Google Scholar 

  23. Y. N. Yeni, G. T. Christopherson, X. N. Dong, D. G. Kim, and D. P. Fyhrie, Effect of microcomputed tomography voxel size on the finite element model accuracy for human cancellous bone, Journal of Biomechanical Engineering 127, 18 (2005).

    Article  Google Scholar 

  24. C. S. Rajapakse, J. Magland, X. H. Zhang, X. S. Liu, S. L. Wehrli, X. E. Guo, and F. W. Wehrli, Implications of noise and resolution on mechanical properties of trabecular bone estimated by image-based finite-element analysis, Journal of Orthopaedic Research 27, 1263–1271 (2009).

    Article  PubMed  Google Scholar 

  25. G. M. Luo, J. J. Kaufman, A. Chiabrera, B. Bianco, 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).

    Article  CAS  PubMed  Google Scholar 

  26. A. Hosokawa, Simulation of ultrasound propagation through bovine cancellous bone using elastic and biot’s finite-difference time-domain methods, Journal of the Acoustical Society of America 118, 1782–1789 (2005).

    Article  CAS  PubMed  Google Scholar 

  27. A. Hosokawa, Ultrasonic pulse waves in cancellous bone analyzed by finite-difference time-domain methods, Ultrasonics 44 (Suppl 1), E227–E231 (2006).

    Article  PubMed  Google Scholar 

  28. V. Nguyen, S. Naili, and S. Sansalone, Simulation of ultrasonic wave propagation in anisotropic cancellous bone immersed in fluid, Wave Motion 47, 117–129 (2009).

    Article  Google Scholar 

  29. Q. Grimal, K. Raum, A. Gerisch, and P. Laugier, About the determination of the representative volume element size in compact bone, in Proceedings of the 19 ème Congrès Français de Mécanique (2009), 6 pages.

    Google Scholar 

  30. J. Kabel, B. van Rietbergen, M. Dalstra, A. Odgaard, and R. Huiskes, The role of an effective isotropic tissue modulus in the elastic properties of cancellous bone, Journal of Biomechanics 32, 673–680 (1999).

    Article  CAS  PubMed  Google Scholar 

  31. Q. Grimal, K. Raum, A. Gerisch, and P. Laugier, Derivation of the mesoscopic elasticity tensor of cortical bone from quantitative impedance images at the micron scale, Computer Methods in Biomechanics and Biomedical Engineering 11, 147–157 (2008).

    Article  PubMed  Google Scholar 

  32. W. R. Taylor, E. Roland, H. Ploeg, D. Hertig, R. Klabunde, M. D. Warner, M. C. Hobatho, L. Rakotomanana, and S. E. Clift, Determination of orthotropic bone elastic constants using fea and modal analysis, Journal of Biomechanics 35, 767–773 (2002).

    Article  CAS  PubMed  Google Scholar 

  33. C. Baron, M. Talmant, and P. Laugier, Effect of porosity on effective diagonal stiffness coefficients (c(ii)) and elastic anisotropy of cortical bone at 1 MHz: a finite-difference time domain study, Journal of the Acoustical Society of America 122, 1810–1817 (2007).

    Article  PubMed  Google Scholar 

  34. H. S. Yoon and J. L. Katz, Ultrasonic wave propagation in human cortical bone–ii. measurements of elastic properties and microhardness, Journal of Biomechanics 9, 459–464 (1976).

    Google Scholar 

  35. H. Yoon and J. Katz, Ultrasonic wave propagation in human cortical bone-i. theoretical considerations for hexagonal symmetry, Journal of Biomechanics 9, 407–412 (1976).

    Google Scholar 

  36. A. A. Espinoza Orias, J. M. Deuerling, M. D. Landrigan, J. E. Renaud, and R. K. Roeder, Anatomic variation in the elastic anisotropy of cortical bone tissue in the human femur, Journal of Biomedical Materials Research 2, 255–263 (2009).

    Google Scholar 

  37. J. M. Crolet, M. Racila, R. Mahraoui, and A. Meunier, A new numerical concept for modeling hydroxyapatite in human cortical bone, Computer Methods in Biomechanics and Biomedical Engineering 8, 139–43 (2005).

    CAS  PubMed  Google Scholar 

  38. C. Baron, Q. Grimal, M. Talmant, and P. Laugier, Investigation of the porous network as a determinant of the overall stiffness of cortical bone: Mori-tanaka model vs. ultrasound propagation, Journal of the Acoustical Society of America 123, 3514 (2008).

    Google Scholar 

  39. W. J. Parnell and Q. Grimal, The influence of mesoscale porosity on cortical bone anisotropy. investigations via asymptotic homogenization, Journal of the Royal Society Interface 6, 97–109 (2009).

    Google Scholar 

  40. K. Kunz, The finite difference time domain method for electromagnetics (CRC Press, Boca Raton) (1993).

    Google Scholar 

  41. K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical methods for physics and engineering, 3 edition (Cambridge University Press, Cambridge, UK) (2006).

    Google Scholar 

  42. R. Madariaga, Dynamics of an expanded circular fault, Bulletin of the Seismological Society of America 66, 639–666 (1976).

    Google Scholar 

  43. J. Virieux, P-sv wave propagation in heterogeneous media: velocity-stress finite-difference method., Geophysics 51, 889–901 (1986).

    Google Scholar 

  44. F. Collino and C. Tsogka, Application of the pml absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media, Geophysics 66, 294–307 (2001).

    Article  Google Scholar 

  45. M. A. Hakulinen, J. S. Day, J. Toyras, H. Weinans, and J. S. Jurvelin, Ultrasonic characterization of human trabecular bone microstructure, Physics in Medicine and Biology 51, 1633–1648 (2006).

    Article  PubMed  Google Scholar 

  46. G. Haiat, 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).

    Article  PubMed  Google Scholar 

  47. O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu, The finite element method: its basis and fundamentals, 6 edition (Elsevier, Burlington) (2005).

    Google Scholar 

  48. C. Desceliers, C. Soize, Q. Grimal, G. Haiat, and S. Naili, A time-domain method to solve transient elastic wave propagation in a multilayer medium with a hybrid spectral-finite element space approximation, Wave Motion 45, 383–399 (2008).

    Article  Google Scholar 

  49. S. Modak and E. Sotelino, The generalized method for structural dynamics applications, Advances in Engineering Software 33, 565–575 (2002).

    Article  Google Scholar 

  50. L. Cagniard, Reflection and Refraction of Progressive Seismic Waves (Gauthier-Villars, Paris) (1962), translation and revision by Flinn, E.A., and Dix, C.H., of Cagniard, L., 1939. Réflexion et Réfraction des Ondes Séismiques Progressives.

    Google Scholar 

  51. A. de Hoop, A modification of cagniard’s method for solving seismic pulse problems, Applied Scientific Research 8, 349–356 (1960).

    Article  Google Scholar 

  52. Q. Grimal and S. Naili, A theoretical analysis in the time-domain of wave reflection on a bone plate, Journal of Sound and Vibration 298, 12–29 (2006).

    Article  Google Scholar 

  53. K. Macocco, Q. Grimal, S. Naili, and C. Soize, Elastoacoustic model with uncertain mechanical properties for ultrasonic wave velocity prediction; application to cortical bone evaluation, Journal of Acoustical Society of America 119, 729–740 (2006).

    Article  Google Scholar 

  54. E. Camus, M. Talmant, G. Berger, and P. Laugier, Analysis of the axial transmission technique for the assessment of skeletal status, Journal of Acoustical Society of America 108, 3058–65 (2000).

    Article  CAS  Google Scholar 

  55. J. Diaz and P. Joly, Robust high order non-conforming finite element formulation for time domain fluid-structure interaction, Journal of Computational Acoustics 13, 403–431 (2005).

    Article  Google Scholar 

  56. R. W. Graves, Simulating seismic wave propagation in 3d elastic media using staggered-grid finite differences, Bulletin of the Seismological Society of America 86, 1091–1106 (1996).

    Google Scholar 

  57. N. A. Kampanis, V. Dougalis, and V. Ekaterinaris, Effective computational methods for wave propagation (Chapman & Hall/CRC, Boca Raton) (2008).

    Google Scholar 

  58. J. F. Aubry, M. Tanter, M. Pernot, J. L. Thomas, and M. Fink, Experimental demonstration of noninvasive transskull adaptive focusing based on prior computed tomography scans, The Journal of the Acoustical Society of America 113, 84–93 (2003).

    Article  CAS  PubMed  Google Scholar 

  59. F. Marquet, M. Pernot, J. F. Aubry, G. Montaldo, L. Marsac, M. Tanter, and M. Fink, Non-invasive transcranial ultrasound therapy based on a 3d CT scan: protocol validation and in vitro results, Physics in Medicine and Biology 54, 2597–2613 (2009).

    Article  CAS  PubMed  Google Scholar 

  60. R. Barkmann, S. Lüsse, B. Stampa, S. Sakata, M. Heller, and C.-C. Glüer, Assessment of the geometry of human finger phalanges using quantitative ultrasound in vivo, Osteoporosis International 11, 745–755 (2000).

    Article  CAS  PubMed  Google Scholar 

  61. S. P. Dodd, J. L. Cunningham, A. W. Miies, S. Gheduzzi, and V. F. Humphrey, An in vitro study of ultrasound signal loss across simple fractures in cortical bone mimics and bovine cortical bone samples, Bone 40, 656–661 (2007).

    Article  CAS  PubMed  Google Scholar 

  62. G. Haiat, F. Padilla, F. Peyrin, and P. Laugier, Fast wave ultrasonic propagation in trabecular bone: numerical study of the influence of porosity and structural anisotropy, Journal of the Acoustical Society of America 123, 1694–1705 (2008).

    Article  CAS  PubMed  Google Scholar 

  63. 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).

    Article  PubMed  Google Scholar 

  64. F. Padilla, E. Bossy, and P. Laugier, Simulation of ultrasound propagation through three-dimensional trabecular bone structures: comparison with experimental data, Japanese Journal of Applied Physics Part 1-Regular Papers Brief Communications and Review Papers 45, 6496–6500 (2006).

    Google Scholar 

  65. E. Bossy, P. Laugier, F. Peyrin, and F. Padilla, Attenuation in trabecular bone: a comparison between numerical simulation and experimental results in human femur, Journal of the Acoustical Society of America 122, 2469–2475 (2007).

    Article  PubMed  Google Scholar 

  66. S. Gheduzzi, S. P. Dodd, A. W. Miles, V. F. Humphrey, and J. L. Cunningham, Numerical and experimental simulation of the effect of long bone fracture healing stages on ultrasound transmission across an idealized fracture, Journal of the Acoustical Society of America 126, 887–894 (2009).

    Article  CAS  PubMed  Google Scholar 

  67. V. C. Protopappas, I. C. Kourtis, L. C. Kourtis, K. N. Malizos, C. V. Massalas, and D. I. Fotiadis, Three-dimensional finite element modeling of guided ultrasound wave propagation in intact and healing long bones, Journal of the Acoustical Society of America 121, 3907–3921 (2007).

    Article  PubMed  Google Scholar 

  68. A. Mahmoud, D. Cortes, A. Abaza, H. Ammar, M. Hazey, P. Ngan, R. Crout, and O. Mukdadi, Noninvasive assessment of human jawbone using ultrasonic guided waves, IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control 55, 1316–1327 (2008).

    Article  Google Scholar 

  69. V. C. Protopappas, M. G. Vavva, D. I. Fotiadis, and K. N. Malizos, Ultrasonic monitoring of bone fracture healing, IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control 55, 1243–1255 (2008).

    Article  Google Scholar 

  70. G. Haiat, F. Padilla, and P. Laugier, Sensitivity of qus parameters to controlled variations of bone strength assessed with a cellular model, IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control 55, 1488–1496 (2008).

    Article  CAS  Google Scholar 

  71. A. S. Aula, J. Toyras, M. A. Hakulinen, and J. S. Jurvelin, Effect of bone marrow on acoustic properties of trabecular bone-3d finite difference modeling study, Ultrasound in Medicine and Biology 35, 308–318 (2009).

    Article  CAS  PubMed  Google Scholar 

  72. M. Pakula, F. Padilla, and P. Laugier, Influence of the filling fluid on frequency-dependent velocity and attenuation in cancellous bones between 0.35 and 2.5 MHz, Journal of the Acoustical Society of America 126, 3301–3310 (2009).

    Google Scholar 

  73. G. Haiat, F. Padilla, R. Barkmann, C. C. Gluer, and P. Laugier, Numerical simulation of the dependence of quantitative ultrasonic parameters on trabecular bone microarchitecture and elastic constants, Ultrasonics 44, E289–E294 (2006).

    Article  PubMed  Google Scholar 

  74. A. Hosokawa, Effect of minor trabecular elements on fast and slow wave propagations through a stratified cancellous bone phantom at oblique incidence, Japanese Journal of Applied Physics 48 (2009).

    Google Scholar 

  75. C. Desceliers, C. Soize, Q. Grimal, M. Talmant, and S. Naili, Determination of the random anisotropic elasticity layer using transient wave propagation in a fluid-solid multilayer: model and experiments, Journal of the Acoustical Society of America 125, 2027–2034 (2009).

    Article  PubMed  Google Scholar 

  76. P. Moilanen, M. Talmant, V. Kilappa, P. Nicholson, S. L. Cheng, J. Timonen, and P. Laugier, Modeling the impact of soft tissue on axial transmission measurements of ultrasonic guided waves in human radius, Journal of the Acoustical Society of America 124, 2364–2373 (2008).

    Article  PubMed  Google Scholar 

  77. M. G. Vavva, V. C. Protopappas, L. N. Gergidis, A. Charalambopoulos, D. I. Fotiadis, and D. Polyzos, The effect of boundary conditions on guided wave propagation in two-dimensional models of healing bone, Ultrasonics 48, 598–606 (2008).

    Article  PubMed  Google Scholar 

  78. 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).

    Article  Google Scholar 

  79. 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).

    Article  CAS  Google Scholar 

  80. 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).

    Article  Google Scholar 

  81. G. Haiat, 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).

    Article  CAS  PubMed  Google Scholar 

Download references

Acknowledgements

The authors are grateful to Lucie Fréret for providing helpful comments and suggestions.

Author information

Authors and Affiliations

  1. Institut Langevin, ESPCI ParisTech, CNRS UMR 7587, INSERM ERL U979, 10 rue Vauquelin, 75231, Paris Cedex 05, France

    Emmanuel Bossy

  2. Laboratoire d’Imagerie Parametrique, Université Pierre et Marie Curie, CNRS, F-75006, Paris, France

    Quentin Grimal

Authors
  1. Emmanuel Bossy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Quentin Grimal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Quentin Grimal .

Editor information

Editors and Affiliations

  1. UMR CNRS, Labo. d'Imagerie Parametrique, Université Paris VI, rue de L'Ecole de Medecine 15, Paris, 75006, France

    Pascal Laugier

  2. , Labortatoire Modélisation Simulation Mul, CNRS, Avenue du Général de Gaulle, Créteil, 94010, France

    Guillaume Haïat

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Netherlands

About this chapter

Cite this chapter

Bossy, E., Grimal, Q. (2011). Numerical Methods for Ultrasonic Bone Characterization. In: Laugier, P., Haïat, G. (eds) Bone Quantitative Ultrasound. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0017-8_8

Download citation

Publish with us

Policies and ethics

Search

Navigation