Estimation of parameters quantifying porosity in random porous structures using ultrasonic attenuation: Solving the inverse problem

The goal of this study is to estimate the porosity (pore size and density) of numerically simulated random porous three-dimensional structures mimicking simplified geometries of cortical bone using attenuation of ultrasonic waves in the MHz range. To do so, we use a physics-based model derived from the Independent Scattering Approximation (ISA) to mathematically model the attenuation of elastic waves in porous structures as a function of the parameters that we wish to identify: pore size and density. Frequency-dependent attenuation data are generated using three-dimensional finite-difference time-domain simulations in the range of 1–8 MHz in mono-disperse structures with the pore diameter and density ranging from 100 to 200 µm and 20–50 pore/mm3, respectively. We then solve an ordinary least squares (OLS) inverse problem to recover the pore size and density by minimizing the sum of squared errors between the simulated data and the model prediction. In doing so, we demonstrate that we can estimate with confidence the parameters quantifying porosity using the ISA model given three-dimensional numerically simulated attenuation data.