Independent scattering model and velocity dispersion in trabecular bone: comparison with a multiple scattering model

Abstract

Speed of sound measurements are used clinically to assess bone strength. Trabecular bone is an attenuating composite material in which negative values of velocity dispersion have been measured; this behavior remaining poorly explained physically. The aim of this work is to describe the ultrasonic propagation in trabecular bone modeled by infinite cylinders immersed in a saturating matrix and to derive the physical determinants of velocity dispersion. An original homogenization model accounting for the coupling of independent scattering and absorption phenomena allows the computation of phase velocity and of dispersion while varying bone properties. The first step of the model consists in the computation of the attenuation coefficient at all frequencies. The second step of the model corresponds to the application of the general Kramers-Krönig relationship to derive the frequency dependence of phase velocity. The model predicts negative values of velocity dispersion in agreement with experimental results obtained in phantoms mimicking trabecular bone. In trabecular bone, only negative values of velocity dispersion are predicted by the model, which span within the range of values measured experimentally. However, the comparison of the present results with results obtained in Haiat et al. (J Acoust Soc Am 124:4047-4058, 2008) assuming multiple scattering indicates that accounting for multiple scattering phenomena leads to a better prediction of velocity dispersion in trabecular bone.

Determining attenuation properties of interfering fast and slow ultrasonic waves in cancellous bone

“…35,36 Several models have been proposed that predict negative dispersion including multilayer models, 31,37 multiple scattering models, 38 and independent scattering models. 39 The objective of this study was to investigate the potential value of separating the fast waves and slow waves for determining the attenuation properties of cancellous bone and to compare the results of time-domain and frequency-domain methods for the analysis of attenuation.…”

Numerical analysis of ultrasound backscattered waves in cancellous bone using a finite-difference time-domain method: isolation of the backscattered waves from various ranges of bone depths

“…5,8,and 9). Moreover, in several studies [38], [39], it was suggested that multiple scattering could affect the ultrasound behaviors in the bone. The isolation method used for the backscattered waves in this study enables the isolation of every wave reflected from a limited range of depths in the bone, which can include the multiscattered waves [as described above, the third wave observed in the case of bone depth of 2.5 to 3.5 mm in Figs.…”

Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness

“…Moreover, bone properties are known to vary during osseointegration (Mathieu et al, 2011b;Vayron et al, 2012;Vayron et al, 2014b), which was not taken into account. Fourth, bone tissue was modeled as an elastic, homogeneous and isotropic material, similarly to what was done in some previous studies Mathieu et al, 2011a;Vayron et al, 2015Vayron et al, , 2016, whereas real bone tissue is known to be a strongly dispersive medium (Ha€ ıat et al, 2008;Haiat and Naili, 2011). Moreover, although mature bone tissue is known to be anisotropic Sansalone et al, 2012), the anisotropic behavior of newly formed bone tissue remains unknown (Mathieu et al, 2011b;Mathieu et al, 2012).…”