Annals of Biomedical Engineering

, Volume 47, Issue 11, pp 2178–2187 | Cite as

Nonlinear Inversion of Ultrasonic Dispersion Curves for Cortical Bone Thickness and Elastic Velocities

  • Tho N. H. T. Tran
  • Mauricio D. Sacchi
  • Dean Ta
  • Vu-Hieu Nguyen
  • Edmond Lou
  • Lawrence H. LeEmail author


In this study, a nonlinear grid-search inversion has been developed to estimate the thickness and elastic velocities of long cortical bones, which are important determinants of bone strength, from axially-transmitted ultrasonic data. The inversion scheme is formulated in the dispersive frequency-phase velocity domain to recover bone properties. The method uses ultrasonic guided waves to retrieve overlying soft tissue thickness, cortical thickness, compressional, and shear-wave velocities of the cortex. The inversion strategy requires systematic examination of a large set of trial dispersion-curve solutions within a pre-defined model space to match the data with minimum cost in a least-squares sense. The theoretical dispersion curves required to solve the inverse problem are computed for bilayered bone models using a semi-analytical finite-element method. The feasibility of the proposed approach was demonstrated by the numerically simulated data for a 1 mm soft tissue-5 mm bone bilayer and ex-vivo data from a bovine femur plate with an overlying 2 mm-thick soft-tissue mimic. The bootstrap method was employed to evaluate the inversion uncertainty and stability. Our results have shown that the cortical thickness and wave speeds could be recovered with fair accuracy.


Ultrasonic guided waves Cortical bone Axial transmission technique Nonlinear inversion Grid search 



L. H. Le acknowledges the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant. The work was also supported by the National Natural Science Foundation of China (11525416 and 11827808), the International Scientific and Technological Cooperation Project of Shanghai (17510710700), and Shanghai Municipal Science and Technology Major Project (2017SHZDZX01). Dr. Le is currently a Senior Visiting Scholar at the State Key Laboratory of ASIC and System of Fudan University for the joint work. T.N.H.T. Tran acknowledges Alberta Innovates-Technology Futures (AITF) and the generous supporters of Lois Hole Hospital through Women and Children’s Health Research Institute (WCHRI) for the graduate studentships.

Conflicts of interest

The authors declare no conflict of interest.


  1. 1.
    Anast, G. T., T. Fields, and I. M. Siegel. Ultrasonic technique for the evaluation of bone fractures. Am. J. Phys. Med. 37:157–159, 1958.CrossRefGoogle Scholar
  2. 2.
    Beaty, K. S., D. Schmitt, and M. D. Sacchi. Simulated annealing inversion of multimode Rayleigh wave dispersion curves for geological structure. Geophys. J. Int. 151:622–631, 2002.CrossRefGoogle Scholar
  3. 3.
    Bernard, S., V. Monteiller, D. Komatitsch, and P. Lasaygues. Ultrasonic computed tomography based on full-waveform inversion for bone quantitative imaging. Phys. Med. Biol. 62:7011–7035, 2017.CrossRefGoogle Scholar
  4. 4.
    Bochud, N., Q. Vallet, Y. Bala, H. Follet, J. G. Minonzio, and P. Laugier. Genetic algorithms-based inversion of multimode guided waves for cortical bone characterization. Phys. Med. Biol. 61:6953–6974, 2016.CrossRefGoogle Scholar
  5. 5.
    Bochud, N., Q. Vallet, J. G. Minonzio, and P. Laugier. Predicting bone strength with ultrasonic guided waves. Sci. Rep. 7:43628, 2017.CrossRefGoogle Scholar
  6. 6.
    Chiras, D. D. Human Biology (Chapter 10), Evergreen, CO: Jones & Bartlett Learning, 2013.Google Scholar
  7. 7.
    Efron, B. Bootstrap methods: another look at the Jackknife. Ann. Stat. 7:1–26, 1979.CrossRefGoogle Scholar
  8. 8.
    Foiret, J., J. G. Minonzio, C. Chappard, M. Talmant, and P. Laugier. Combined estimation of thickness and velocities using ultrasound guided waves: a pioneering study on in vitro cortical bone samples. IEEE Trans. Ultrason. Ferroelectr. Freq. Control. 61:1478–1488, 2014.CrossRefGoogle Scholar
  9. 9.
    Hesse, D. and P. Cawley. A single probe spatial averaging technique for guided waves and its application to surface wave rail inspection. IEEE Trans. Ultrason. Ferroelectr. Freq. Control. 54:2344–2356, 2007.CrossRefGoogle Scholar
  10. 10.
    Le, L. H., Y. J. Gu, Y. P. Li, and C. Zhang. Probing long bones with ultrasonic body waves. Appl. Phys. Lett. 96:114102, 2010.CrossRefGoogle Scholar
  11. 11.
    Li, G. Y., Q. He, L. Jia, P. He, J. Luo, and Y. Cao. An inverse method to determine the arterial stiffness with guided axial waves. Ultrasound. Med. Biol. 43:505–516, 2017.CrossRefGoogle Scholar
  12. 12.
    Lowe, M. J., P. Cawley, and A. Galvagni. Monitoring of corrosion in pipelines using guided waves and permanently installed transducers. J. Acoust. Soc. Am. 132:1932, 2012.CrossRefGoogle Scholar
  13. 13.
    Menke, W. Geophysical Data Analysis: Discrete Inverse Theory. Oxford: Academic Press, Elsevier, 2012.Google Scholar
  14. 14.
    Moilanen, P. Ultrasonic guided waves in bone. IEEE Trans. Ultrason. Ferroelectr. Freq. Control. 55:1277–1286, 2008.CrossRefGoogle Scholar
  15. 15.
    Moreau, L., C. Lachaud, R. Thery, M. V. Predoi, D. Marsan, E. Larose, J. Weiss, and M. Montagnat. Monitoring ice thickness and elastic properties from the measurement of leaky guided waves: a laboratory experiment. J. Acoust. Soc. Am. 142:2873–2880, 2017.CrossRefGoogle Scholar
  16. 16.
    Nguyen, V. H. and S. Naili. Simulation of ultrasonic wave propagation in anisotropic poroelastic bone plate using hybrid spectral/finite element method. Int. J. Numer. Method. Biomed. Eng. 28:861–876, 2012.CrossRefGoogle Scholar
  17. 17.
    Nguyen, V. H. and S. Naili. Ultrasonic wave propagation in viscoelastic cortical bone plate coupled with fluids: a spectral finite element study. Comput. Methods. Biomech. Biomed. Eng. 16:963–964, 2013.CrossRefGoogle Scholar
  18. 18.
    Nguyen, T. K. C., L. H. Le, T. N. Tran, M. D. Sacchi, and E. Lou. Excitation of ultrasonic Lamb waves using a phased array system with two array probes: phantom and in-vitro bone studies. Ultrasonics. 54:1178–1185, 2014.CrossRefGoogle Scholar
  19. 19.
    Nguyen, V. H., T. N. Tran, M. D. Sacchi, S. Naili, and L. H. Le. Computing dispersion curves of elastic/viscoelastic transversely-isotropic bone plates coupled with soft tissue and marrow using semi-analytical finite element (SAFE) method. Comput. Biol. Med. 87:371–381, 2017.CrossRefGoogle Scholar
  20. 20.
    Rose, J. L. Ultrasonic Guided Waves in Solid Media. New York: Cambridge University Press, 2014.CrossRefGoogle Scholar
  21. 21.
    Sen, M. and P. L. Stoffa. Global Optimization Methods in Geophysical Inversion. Amsterdam: Elsevier Science, 1995.Google Scholar
  22. 22.
    Tasinkevych, Y., J. Podhajecki, K. Falinska, and J. Litniewski. Simultaneous estimation of cortical bone thickness and acoustic wave velocity using a multivariable optimization approach: bone phantom and in-vitro study. Ultrasonics. 65:105–112, 2016.CrossRefGoogle Scholar
  23. 23.
    Tran, T. N., L. Stieglitz, Y. J. Gu, and L. H. Le. Analysis of ultrasonic waves propagating in a bone plate over a water half-space with and without overlying soft tissue. Ultrasound. Med. Biol. 39:2422–2430, 2013.CrossRefGoogle Scholar
  24. 24.
    Tran, T. N., L. H. Le, M. D. Sacchi, V. H. Nguyen and E. Lou. Multichannel filtering and reconstruction of ultrasonic guided wave fields using time intercept-slowness transform. J. Acoust. Soc. Am. 136:248–259, 2014.CrossRefGoogle Scholar
  25. 25.
    Tran, T. N., K. T. Nguyen, M. D. Sacchi, and L. H. Le. Imaging ultrasonic dispersive guided wave energy in long bones using linear Radon transform. Ultrasound. Med. Biol. 40:2715–2727, 2014.CrossRefGoogle Scholar
  26. 26.
    Tran, T. N., L. H. Le, M. D. Sacchi, and V. H. Nguyen. Sensitivity analysis of ultrasonic guided waves propagating in tri-layered bone models: a numerical study. Biomech. Model. Mechanobiol. 17:1269–1279, 2018.CrossRefGoogle Scholar
  27. 27.
    Ulrych, T. J. and M. D. Sacchi. Information-Based Inversion and Processing with Applications. Oxford: Elsevier Science, 2005.Google Scholar
  28. 28.
    Vallet, Q., N. Bochud, C. Chappard, P. Laugier, and J. G. Minonzio. In vivo characterization of cortical bone using guided waves measured by axial transmission. IEEE Trans. Ultrason. Ferroelectr. Freq. Control. 63:1361–1371, 2016.CrossRefGoogle Scholar
  29. 29.
    Werner, P. Knowledge about osteoporosis: assessment, correlates and outcomes. Osteoporosis. Int. 16:115–127, 2005.CrossRefGoogle Scholar
  30. 30.
    Wilcox, P., M. J. Lowe, and P. Cawley. Omnidirectional guided wave inspection of large metallic plate structures using an EMAT array. IEEE Trans. Ultrason. Ferroelectr. Freq. Control. 52:653–665, 2005.CrossRefGoogle Scholar
  31. 31.
    Yeh, C. H. and C. H. Yang. Characterization of mechanical and geometrical properties of a tube with axial and circumferential guided waves. Ultrasonics. 51:472–479, 2011.CrossRefGoogle Scholar
  32. 32.
    Yu, X., M. Ratassepp, and Z. Fan. Damage detection in quasi-isotropic composite bends using ultrasonic feature guided waves. Compos. Sci. Technol. 141:120–129, 2017.CrossRefGoogle Scholar
  33. 33.
    Zhang, Z. D., G. Schuster, Y. Liu, S. M. Hanafy, and J. Li. Wave equation dispersion inversion using a difference approximation to the dispersion-curve misfit gradient. J. Appl. Geophys. 133:9–15, 2016.CrossRefGoogle Scholar
  34. 34.
    Zheng, R., L. H. Le, M. D. Sacchi, and E. Lou. Imaging internal structure of long bones using wave scattering theory. Ultrasound. Med. Biol. 41:2955–2965, 2015.CrossRefGoogle Scholar

Copyright information

© Biomedical Engineering Society 2019

Authors and Affiliations

  • Tho N. H. T. Tran
    • 1
  • Mauricio D. Sacchi
    • 2
  • Dean Ta
    • 1
    • 3
    • 4
  • Vu-Hieu Nguyen
    • 5
  • Edmond Lou
    • 1
    • 6
  • Lawrence H. Le
    • 1
    • 2
    • 3
    Email author
  1. 1.Department of Radiology and Diagnostic ImagingUniversity of AlbertaEdmontonCanada
  2. 2.Department of PhysicsUniversity of AlbertaEdmontonCanada
  3. 3.State Key Laboratory of ASIC and SystemFudan UniversityShanghaiChina
  4. 4.Department of Electronic EngineeringFudan UniversityShanghaiChina
  5. 5.Laboratoire Modélisation et Simulation Multi Echelle UMR 8208 CNRSUniversité Paris-EstCréteilFrance
  6. 6.Department of Electrical and Computer EngineeringUniversity of AlbertaEdmontonCanada

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