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

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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.

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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.

Author information

Authors and Affiliations

  1. Department of Radiology and Diagnostic Imaging, University of Alberta, Edmonton, AB, T6G 2B7, Canada

    Tho N. H. T. Tran, Dean Ta, Edmond Lou & Lawrence H. Le

  2. Department of Physics, University of Alberta, Edmonton, AB, T6G 2E1, Canada

    Mauricio D. Sacchi & Lawrence H. Le

  3. State Key Laboratory of ASIC and System, Fudan University, Shanghai, China

    Dean Ta & Lawrence H. Le

  4. Department of Electronic Engineering, Fudan University, Shanghai, China

    Dean Ta

  5. Laboratoire Modélisation et Simulation Multi Echelle UMR 8208 CNRS, Université Paris-Est, Créteil, France

    Vu-Hieu Nguyen

  6. Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, T6G 2W3, Canada

    Edmond Lou

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  1. Tho N. H. T. Tran
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  2. Mauricio D. Sacchi
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  6. Lawrence H. Le
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Correspondence to Lawrence H. Le.

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Associate Editor Mona Kamal Marei oversaw the review of this article.

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Tran, T.N.H.T., Sacchi, M.D., Ta, D. et al. Nonlinear Inversion of Ultrasonic Dispersion Curves for Cortical Bone Thickness and Elastic Velocities. Ann Biomed Eng 47, 2178–2187 (2019). https://doi.org/10.1007/s10439-019-02310-4

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  • DOI: https://doi.org/10.1007/s10439-019-02310-4

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