The effect of pore size and density on ultrasonic attenuation in porous structures with mono-disperse random pore distribution: A two-dimensional in-silico study

Abstract

This work proposes a power law model to describe the attenuation of ultrasonic waves in non-absorbing heterogeneous media with randomly distributed scatterers, mimicking a simplified structure of cortical bone. This paper models the propagation in heterogeneous structures with controlled porosity using a two-dimensional finite-difference time domain numerical simulation in order to measure the frequency dependent attenuation. The paper then fits a phenomenological model to the simulated frequency dependent attenuation by optimizing parameters under an ordinary least squares framework. Local sensitivity analysis is then performed on the resulting parameter estimates in order to determine to which estimates the model is most sensitive. This paper finds that the sensitivity of the model to various parameter estimates depends on the micro-architectural parameters, pore diameter () and pore density (). In order to get a sense for how confidently model parameters are able to be estimated, 95% confidence intervals for these estimates are calculated. In doing so, the ability to estimate model-sensitive parameters with a high degree of confidence is established. In the future, being able to accurately estimate model parameters from which micro-architectural ones could be inferred will allow pore density and diameter to be estimated via an inverse problem given real or simulated ultrasonic data to be determined.

Quantitative Ultrasound Assessment of Cortical Bone Properties Beyond Bone Mineral Density

“…One route to infer porosity is to use empirical relationships, or material models, relating quantities measured with QUS and porosity. Other routes are currently being explored such as imaging blood perfusion using ultrasound contrast agent [98] and measuring ultrasonic attenuation assuming it has a strong relationship with pore properties [99].…”

Inferring Porosity from Frequency Dependent Attenuation in Cortical Bone Mimicking Porous Media

“…A finite-difference, time domain (FDTD) research freeware, SimSonic (www.simsonic.fr) [18] is used to simulate elastic waves propagating in porous media resembling cortical bone [23][24][25]. The media contains a distribution of fluid-filled pores (with material properties of water) within solid slabs (with material properties of pure bone) [26]. The ability of defining independently tuneable material properties at all points in space enables an understanding of the exclusive effect of the different parameters of the porosity, including pore size and density (number of pores per unit area).…”

“…The attenuation value for the propagated signal within the porous structures is calculated through Time-Distance Matrix Approach (TDMA) described in [26].…”

“…We use a phenomenological model in the form of a power law, developed in [26], to describe frequency dependent attenuation in porous, heterogeneous material mimicking cortical bone. Here model parameters, a and b, are presumed to be functions of the micro-architectural parameters, pore diameter (φ) and pore density (ρ).…”

“…Note here, only a and b are considered to be functionally dependent on diameter and density as the model is most sensitive to these estimates [26].…”