Theory of ultrasound Doppler-spectra velocimetry for arbitrary beam and flow configurations

Abstract

Abstract-In conventional ultrasound Doppler systems, the velocity component along the beam axis is derived from the observed frequency shift. Recently, it was verified that by using a pulsed-Doppler system with the beam transversely oriented with respect to the flow, the velocity component transverse to the beam can be derived from the edges of the spectrum. Presently, these results are generalized to take into account arbitrary angles of incidence, effects of velocity gradients, arbitrary apertures, and arbitrary source pulses. For uniform apertures and transverse flow, it has been previously shown that the Doppler output spectrum is symmetrical about zero frequency, with its width depending on the Doppler effect due to the transverse velocity and the geometry of the problem. For a beam direction oblique to the velocity, it is shown that the spectrum is now shifted, and is centered about the classical Doppler frequency. It is shown here that for velocity gradients and transverse flows the spectrum remains symmetrical, with the edges corresponding to the maximal velocity; however, the profile becomes peaked at the center. For oblique incidence, an asymmetrical spectrum is obtained and its edges are related to the maximal and minimal velocities within the sampling volume.For the simple case of a long strip transducer discussed previously, it was shown that the Doppler system output spectrum is essentially obtained by convolving the transmitter and receiver aperture functions. The general discussion given here, even for the single particle, is more complicated. Nevertheless, using the reciprocity theorem it is shown that the output spectrum is obtained by convolving the particle excitation spectrum due to monochromatic excitation, with the receiver input spectrum due to a moving monochromatic source, all this shifted to the classical Doppler frequency mentioned above. It is shown that when the excitation possesses an arbitrary (narrow band) spectrum, this spectrum, replicated in a prescribed manner, has to be convolved with the spectra derived above. Following the single scatterer analysis, the effects produced by an ensemble of particles are considered. Numerical computations of theoretical results are given, supporting the semi-quantitative graphical interpretations for various configurations. To further substantiate the results analytical treatments of simplified models are given.The results contribute to our understanding of Doppler spectra and their use, mainly in medical ultrasound diagnostics.

On the use of Doppler bandwidth for more accurate blood flow estimation

“…(2) can be shown to be given by [2] Bd= 2 v --sin8 w A F where W and F are respectively the effective diameter and focal length of the transducer. the Doppler spectrum due to a flow line of velocity v as Substituting eq.…”

Ultrasonic vector velocity measurement by projection Computed Velocimetry

“…Therefore, these methods will not resolve multiple velocity components of the targets within a sample volume, although these are readily assessed by the conventional pulse Doppler method with spectrum analysis. In the second category LP, a configuration using the width of the Doppler signal spectrum is described [10]. This method will not detect the polarity of the lateral velocity component.…”

Spectrum of Doppler ultrasound signals from nonstationary blood flow

“…Much progress has been made in identifying and quantifying spectral broadening resulting from modulation of the Doppler signal from each scatterer by the beam [14]- [16], deviations from planewave conditions [17], and blood acceleration [18]- [20]. Partial correction for intrinsic broadening when estimating maximum velocity [21], [22] and for the effect of acceleration on spectral width [3], [23], [24] have been proposed.…”