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Abstract

A trabecular bone consists of trabeculae whose mechanical properties differ significantly from the surrounding marrow, therefore an ultrasonic wave propagating within the bone structure is strongly scattered. The aim of this paper was to evaluate the contribution of the first, second and higher order scattering (multiple scattering) into the total scattering of ultrasound in a trabecular bone. The scattering due to the interconnections between thick trabeculae, usually neglected in trabecular bone models, has been also studied. The basic element in our model of the trabecular bone was an elastic cylinder with a various finite-length and diameter as well as orientation. The applied model was taking into account variation of both, elements size and their spatial configuration. The field scattered on the bone model was evaluated by solving numerically the integral form of the generalized Sturm-Liouville equation describing a scalar wave in inhomogeneous and lossy media. For the scattered fields calculated numerically the effective cross-sections were determined. The influence of absorption on the scattering coefficients was demonstrated. The results allowed to conclude that within the frequency range from 0.5 to 1.5 MHz contribution of the second order scattering to the effective backscattering cross-section is at least 500 times lower than that due to the first order scattering. It was noticed that for a frequency higher than 1.5 MHz fast growth of the backscattering (reflection) coefficients, calculated for the second order scattering, occurs.

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Authors and Affiliations

Andrzej Nowicki
Janusz Wójcik
Jerzy Litniewski
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Abstract

The paper considers the application of statistical properties of backscattered ultrasonic signal for assessment of the trabecular bone status.

Computer simulations were conducted to investigate the properties of the ultrasound pulse-echo signal, as it is received on the transducer surface after scattering in trabecular bone. The micro-architecture of trabecular bone was modeled by a random distribution of long and thin cylindrical scatterers of randomly varying diameters and mechanical properties, oriented perpendicular to the ultrasound beam axis. The received echo signal was calculated as a superposition of echoes from all the scatterers present in the scattering volume.

The simulated signal envelope was used for statistical processing to compute various parameters like the mean amplitude, the amplitude MSR defined as the ratio of the mean to the standard deviation and the amplitude histogram.

Results indicated that while for the well-defined trabeculae properties within the simulated bone structure the signal envelope values are Rayleigh distributed the significant departures from Rayleigh statistics may be expected as the thickness of trabeculae become random. The influence of the variation of mechanical properties of the bone tissue building the trabeculae on the bone backscattered signal parameters was not observed.

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Authors and Affiliations

Jerzy Litniewski

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