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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
Article

Abstract

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.

Keywords

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

Notes

Acknowledgements

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.

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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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