Frequency-dependent wave vectors are derived from multiscale balance, conduction, and bone poroelasticity laws.
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Mega-Hertzian fast and slow wave velocities, and generally also attenuations, are frequency-independent.
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Corresponding wavelengths decrease with increasing frequency, and with decreasing permeability.
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Velocities of both fast and slow waves, and attenuation lengths of slow waves, increase with increasing permeability.
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Fast wave attenuation exhibits a permeability-specific maximum.
Abstract
Ultrasonics is an important diagnostic tool for bone diseases, as it allows for non-invasive assessment of bone tissue quality through mass density–elasticity relationships. The latter are, however, quite complex for fluid-filled porous media, which motivates us to develop a rigorous multiscale poromicrodynamics approach valid across the great variety of different bone tissues. Multiscale momentum and mass balance, as well as kinematics of a hierarchical double porous medium, together with Darcy’s law for fluid flow and micro–poro-elasticity for the solid phase of bone, give access to the so-called dispersion relation, linking the complex wave numbers to corresponding wave frequencies. Experimentally validated results show that 2.25 MHz acoustical signals transmit healthy cortical bone (exhibiting a low vascular porosity) only in the form of fast waves, agreeing very well with experimental data, while both fast and slow waves transmit highly osteoporotic as well as trabecular bone (exhibiting a large vascular porosity). While velocities and wavelengths of both fast and slow waves, as well as attenuation lengths of slow waves, are always monotonously increasing with the permeability of the bone sample, the attenuation length of fast waves shows a minimum when considered as function of the permeability.
