Characterizing porosity in bone-like porous media from frequency dependent attenuation

Osteoporosis affects both pore size and density in cortical bone. Quantifying levels of osteoporosis by inferring these micro-architectural properties from ultrasonic wave attenuation in cortical bone has yet to be done. Here, we propose a phenomenological power law model that captures the dynamics of frequency dependent attenuation in non-absorbing porous media mimicking a simplified cortical bone structure. We first demonstrate that, although it is frequency dependent, apparent absorption does not depend upon pore density and pore concentration, justifying the use of non-absorbing media for the simulations. We generate scattering attenuation data for various combinations of pore diameters (ranging from 20 to 100 μm) and pore densities (ranging from 3 to 10 pore/[mm]²) using a 2D FDTD package (Simsonic), which simulates the propagation of elastic waves at frequencies of 1–8 MHz. The model is then optimized to fit these datasets by solving an inverse problem under an ordinary least squares (OLS) framework. With this we establish linear, functional relationships between the optimized power law model parameters and the micro-architectural parameters. These relationships showed that ranges of porosity could be inferred from attenuation data. Applying these techniques to attenuation data from cortical bone samples could allow one to characterize the micro-architectural properties of bone porosity.