Frequency dependence of backscatter from thin, oblique, finite-length cylinders measured with a focused transducer - with applications in cancellous bone

A model is presented for the echo from a thin, oblique, finite-length cylinder. The echo is calculated from the line integral of the transducer directivity pattern along the cylinder axis. The model was validated with broadband pulse-echo measurements from 1) a perpendicular (to the ultrasound beam) nylon wire as a function of lateral displacement from the beam center, 2) a tilted nylon wire as a function of the angle of inclination relative to the ultrasound beam, and 3) a quasi-parallel-nylon-wire phantom, which mimicked the scattering properties of cancellous bone. The transducer directivity pattern (as a function of position and frequency) was measured with a membrane hydrophone. The model predicts an approximately cubic frequency dependence of backscatter coefficient from the phantom, as has been observed experimentally in cancellous bone. The model also predicts the relationship between cylinder length and the exponent of a power law fit to backscatter coefficient vs. frequency, which is 4 for very short (compared to a wavelength) cylinders and asymptotically approaches 3 for very long cylinders.