Modeling Frequency Dependent Ultrasound Attenuation in Cortical Bone: Solving Direct and Inverse Problems

The goal of this work is to quantify the porosity of cortical bone using non-invasive ultrasound attenuation. We do so by formulating a polydisperse version of the Independent Scattering Approximation (ISA) to mathematically model ultrasonic attenuation in cortical bone that is dependent upon the distributed parameter, pore radius. We use analytical Beta distributions to represent two cases of underlying microstructure: one with a relatively low expected value on pore radius, the other with a relatively high value. We simulate data for these cases by inputting the corresponding Beta density functions into a higher-order attenuation model and adding noise. With the polydisperse model and noisy attenuation data, we formulate and solve inverse problems using the Prohorov Metric Framework to reconstruct the input Beta density functions using piecewise linear splines. Furthermore, we use a regularization term to stabilize the inverse problem. We establish that the polydisperse ISA model and inverse problem formulation allow us to reconstruct the true probability density functions on pore radius for cases where the underlying microstructure varies.