Calculation of Doppler spectral power density functions

A volume integral method for the calculation of Doppler ultrasound spectral power density (SPD) functions is described. Axisymmetric flow in a circular tube with a power law velocity profile is assumed. The SPD function is regarded as a probability density function for scatterer velocity, and the assumptions under which this is justified are considered. It is shown that the SPD function is independent of Doppler angle, except in the presence of wall reflection effects. A coordinate system centered on the beam is used enabling irregularly shaped and nonuniform beams to be treated. For the common flow and beam patterns, which exhibit symmetry, the volume integrals can often be reduced to a single integral and evaluated directly. Results are obtained for uniform rectangular and circular insonating beams, and for nonuniform beams with Gaussian, jinc, and sinc profiles. The effects of narrow beams and wall reflection are shown. The method may be readily applied to more complicated beam and flow patterns.