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Nonlinear Acoustics for Non-invasive Assessment of Bone Micro-damage

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Bone Quantitative Ultrasound

Abstract

This chapter presents the state of art in the field of nonlinear ultrasound applied to bone micro-damage assessment. An increasing number of groups have been conducting research in the past years on this particular topic, motivated by the particular sensitivity shown by nonlinear ultrasound methods in industrial materials and geomaterials. Some of the results obtained recently on bone damage assessment in vitrousing various nonlinear ultrasound techniques are presented. In particular, results obtained with higher harmonic generation, Dynamic Acousto-Elastic Testing (DAET), Nonlinear Resonant Ultrasound Spectroscopy (NRUS), and Nonlinear Wave Modulation Spectroscopy (NWMS) techniques are detailed. All those results show a very good potential for nonlinear ultrasound techniques for bone damage assessment. They should benefit from a proper quantification of the relationship between micro-damage and nonlinear ultrasound parameters. This could be obtained through a thorough statistical study which remains to be achieved. A practical implementation of an in vivosetup also remains to be conducted.

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Notes

  1. 1.

    M 0is the linear elastic modulus or the second-order (in energy) elastic constant, whereas M 1 M 2are the third-order and fourth-order elastic constants, respectively, which account for nonlinear elasticity. M 1and M 2are negative for most of the materials.

  2. 2.

    Practically, a burst containing at least ten acoustic periods is emitted to facilitate the extraction of the second harmonic amplitude in the frequency domain after the computation of the Fourier transform of the received acoustic signal.

  3. 3.

    In other words, the nonlinearity in the equation of state of the material, relating stress to strain.

  4. 4.

    In an isotropic solid, for a compressional plane wave, \(\beta = -(3/2 + {C}_{111}/(2{C}_{11}))\), where C 111and C 11are elastic constants homogeneous to M 1and M 0, respectively [7]. The value 3 ∕ 2 instead of 1 in the expression of β is related to the difference between Lagrangian and Eulerian descriptions of particle motion. Moreover the negative sign arises from the difference in the definitions of the pressure and the stress. Besides, the reader has to pay attention to the definition of β when comparing values obtained by different studies. Indeed the parameter of quadratic nonlinear elasticity is sometimes defined as \(\beta = -(3 + {C}_{111}/{C}_{11})\), twice the value usually employed in the “fluid” community.

  5. 5.

    The heel bone or calcaneus contains 95% of trabecular surrounded by a thin cortical shell.

  6. 6.

    In the DAET configuration, the convective effect cannot occur because the LF and US beams propagate in perpendicular directions. In this section, we redefine β as \(\beta = B/A\).

  7. 7.

    For most of materials, small relative variations of the density can be neglected compared to small relative variations of the elastic modulus.

  8. 8.

    http://www.biobank.fr/

  9. 9.

    The crack density is expressed in cracks number per square mm of bone tissue.

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Authors and Affiliations

  1. Institut Langevin, Université Denis Diderot, 10 rue Vauquelin, 75005, Paris, France

    Marie Muller

  2. Imagerie et Cerveau, UMR INSERM U930, CNRS ERL3106, Université François Rabelais, 10 boulevard Tonnelle, 37032, Tours, France

    Guillaume Renaud

  3. Laboratoire d’Imagerie Paramétrique UMR CNRS7623, Université Pierre et Marie Curie, 15 rue de L’Ecole de Médecine, 75006, Paris, France

    Guillaume Renaud

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  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

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Muller, M., Renaud, G. (2011). Nonlinear Acoustics for Non-invasive Assessment of Bone Micro-damage. In: Laugier, P., Haïat, G. (eds) Bone Quantitative Ultrasound. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0017-8_15

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