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Velocity dispersion in trabecular bone: Influence of multiple scattering and of absorption
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10.1121/1.3003077
/content/asa/journal/jasa/124/6/10.1121/1.3003077
http://aip.metastore.ingenta.com/content/asa/journal/jasa/124/6/10.1121/1.3003077

Figures

Image of FIG. 1.
FIG. 1.

Description of the three-phase medium model representing the scatterer (corresponding to a trabeculae, radius ), the matrix (marrow or water, radius ), and the effective medium (corresponding to the unknown of the problem). The solid arrows together with two parallel dotted arrows denote a longitudinal wave mode and the solid arrows together with two perpendicular dotted arrows denote a transversal wave mode.

Image of FIG. 2.
FIG. 2.

Bone-mimicking phantom. (Solid line) Variation of the phase velocity as a function of frequency obtained with the model with the parameters corresponding to a bone-mimicking phantom with a scatterer diameter and a scatterer volume fraction . (Dashed line) Linear fit of the variation of the phase velocity vs frequency obtained experimentally with the same phantom (reproduced from Fig. 2 in Wear, 2005 ). The values of the dispersion obtained with the two methods are indicated.

Image of FIG. 3.
FIG. 3.

Variation of the phase velocity at 500 kHz as a function of the properties of the bone-mimicking phantom. The solid and dashed lines, respectively, correspond to the results predicted by the model and measured by Wear (2005) . (a) Phase velocity as a function of trabecular thickness for four phantoms having the same interscatterer spacing equal to , (b) phase velocity as a function of trabecular spacing for four phantoms with the scatterer diameter , and (c) phase velocity as a function of volume fraction for all phantoms. In Fig. 2(a) , the experimental data are given with their error bar; the fitted data are shown in Figs. 2(b) and 2(c) . The experimental results are reproduced from Figs. 3–5 in Wear, 2005 .

Image of FIG. 4.
FIG. 4.

Variation of the velocity dispersion at 500 kHz as a function of the properties of the bone-mimicking phantom. The solid and dashed lines, respectively, correspond to the results predicted by the model and measured by Wear (2005) . (a) Velocity dispersion as a function of trabecular thickness for four phantoms having the same interscatterer spacing equal to , and (b) velocity dispersion as a function of trabecular spacing for four phantoms with a scatterer diameter .

Image of FIG. 5.
FIG. 5.

Variation of the velocity dispersion around 600 kHz as a function of bone volume fraction predicted by the model. The solid and dashed lines, respectively, correspond to results obtained with scatterer diameters and . The black lines correspond to results obtained by neglecting all viscoelastic losses; the dark (light) gray lines correspond to results obtained by considering viscoelastic effects in the trabeculae only (in the trabeculae and in the marrow).

Image of FIG. 6.
FIG. 6.

Variation of the velocity dispersion around 600 kHz as a function of the scatterer diameter predicted by the model for constant scatterer volume fraction value of 5% (dashed lines) and 25% (solid lines). The black lines correspond to results obtained by neglecting all viscoelastic losses; the dark (light) gray lines correspond to results obtained by considering viscoelastic effects in the trabeculae only (in the trabeculae and in the marrow). The black and dark gray dashed lines are almost confounded.

Tables

Generic image for table
TABLE I.

Rheological parameters of the matrix and of the scatterer chosen for trabecular bone and for trabecular bone-mimicking phantoms. The variation of the attenuation coefficient and of phase velocity for the L and SV wave modes as a function of frequency expressed in megahertz are shown.

Generic image for table
TABLE II.

Values of velocity dispersion found by different authors at the calcaneus and with the model developed in the present paper with bone parameters corresponding to the physiological range at the calcaneus.

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/content/asa/journal/jasa/124/6/10.1121/1.3003077
2008-12-01
2014-04-16
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Velocity dispersion in trabecular bone: Influence of multiple scattering and of absorption
http://aip.metastore.ingenta.com/content/asa/journal/jasa/124/6/10.1121/1.3003077
10.1121/1.3003077
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