Speed of sound measurements are widely 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. A homogenization model accounting for the coupling of multiple scattering and absorption phenomena allows the computation of phase velocity and of dispersion while varying bone properties. The present model is adapted from the generalized self-consistent method developed in the work of Yang and Mal [(1994). "Multiple-scattering of elastic waves in a fiber-reinforced composite," J. Mech. Phys. Solids 42, 1945-1968]. It predicts negative values of velocity dispersion, in agreement with experimental results obtained in phantoms mimicking trabecular bone. In trabecular bone, mostly negative and also positive values of velocity dispersion are predicted by the model, which span within the range of values measured experimentally. Scattering effects are responsible for the negative values of dispersion, whereas the frequency dependence of the absorption coefficient in bone marrow and/or in the trabeculae results in an increase in dispersion, which may then become positive.
A theoretical model is proposed for the complete solution of the acoustical problem of the radiation of transient ultrasonic pulses, their scattering by targets of arbitrary shape and acoustic impedance and their reception. The model combines Kirchhoff’s retarded potential formula with the impulse-response approach to the field radiated into fluids by wideband pistonlike sources. Kirchhoff’s formula is made explicit with three geometrical assumptions. Further assumptions and stated approximations allow a generalization of the reciprocity principle from its already-known form for point targets. This leads to a formula for the complete acoustical impulse response which is explicit, very compact and easy to compute, requiring integration only over the insonified area of the target. Throughout, particular attention is given to the hypotheses and approximations made in deriving the model with clear definition of its domain of applicability.
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