Example of Structure Modeling and Analysis of Ultrasound Scattering for Trabecular Bone

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.

Statistical properties of trabecular bone backscatter: Experimental and simulations results

“…The calculations were performed by solving numerically the integral form of the Sturm-Liouville equation that describes scalar wave in inhomogeneous media [5]. The results, obtained for the frequency range used in the bone scanner measurements and the finite length cylinder with diameters equaled to the averaged diameters of thin trabecula, indicate similarity to the field scattered on the sphere (Fig.1).…”

Statistics of the envelope of ultrasonic backscatter from human trabecular bone

“…The calculations were performed by solving numerically the integral form of the Sturm-Liouville equation that describes scalar wave in inhomogeneous media. 19 In calculations the 3D geometry of cylinder and multiple scattering were taken into account. The results, obtained for the frequency range used in the bone scanner measurements indicated similarity to the field scattered on the sphere (Fig.…”