Wavelet decomposition of transmitted ultrasound wave through a 1-D muscle-bone system

In the attempt for using ultrasound as a diagnostic device for osteoporosis, several authors have described the result of the in vitro experiment in which ultrasound is passed through a cancellous bone specimen placed in a water tank. However, in the in vivo setting, a patient's cancellous bone is surrounded by cortical and muscle layers. This paper considers in the one-dimensional case (1) what effect the cortical bone segments surrounding the cancellous segment would have on the received signal and (2) what the received signal would be when a source and receiver are placed on opposite sides of a structure consisting of a cancellous segment surrounded by cortical and muscle layers. Mathematically this is accomplished by representing the received signal as a sum of wavelets which go through different reflection-transmission histories at the muscle-cortical bone and cortical-cancellous bone interfaces. The muscle and cortical bone are modeled as elastic materials and the cancellous bone as a poroelastic material described by the Biot-Johnson-Koplik-Dashen model. The approach presented here permits the assessment of which possible paths of transmission and reflection through the cortical-cancellous or muscle-cortical-cancellous complex will result in significant contributions to the received waveform. This piece of information can be useful for solving the inverse problem of non-destructive assessment of material properties of bone. Our methodology can be generalized to three-dimensional parallelly layered structure by first applying Fourier transform in the directions perpendicular to the transverse direction.