Flow spectra from spectral power density calculations for pulsed Doppler

Range gated pulsed Doppler can be used to make localized velocity measurements within a blood vessel. Both the transducer and the sample volume are of finite size, and this prohibits the measurement of velocity at a point. A spectral flow profile can be created by stepping a sufficiently small sample volume across the lumen of a vessel. However no such set of spectra will correspond directly to the true velocity profile. In this study we developed a systematic theoretical treatment which allows Doppler spectral power density (SPD) functions to be calculated under a very wide range of conditions. Simulated flow spectra were created from sets of these spectra. The model is based on the beam intensity weighted volume method and incorporates, through the idea of a spread function, Guidi's individual flow line spectrum. Our method can be applied for different spread functions; with beam profiles which are uniform, Gaussian or arbitrarily narrow (needle beam); with range gated sample volumes which can be maximal (CW-type) or minimal (PW-type); and for beams which intersect the flow tube axis, or are off centre. Under all conditions we find the spread function parameter k, equal to the ratio of the central Doppler shift to half the bandwidth, plays a key role. After formulating the model analytically, we sought simplifications to allow results to be obtained from simple, practical formulae. Spread and unspread SPD functions are in most cases given as single integrals which contain measurable physical parameters and can be easily evaluated numerically. Model results are presented for flow spectra of parabolic flow, illustrating the interplay between different factors in determining the appearance of spectral flow profiles.