Doppler Ultrasound

The Doppler effect in ultrasound imaging exploits frequency shifts of backscattered waves from moving scatterers—primarily red blood cells—to quantify flow velocities. When an ultrasonic wave at frequency (f_0) encounters a scatterer moving at velocity (v), the received frequency (f_r) is shifted by:

$$f_D = f_r - f_0 = \frac{2\,v\,f_0\,\cos\theta}{c},$$

where (c\approx1540,\text{m/s}) is the speed of sound in tissue and (\theta) is the angle between the beam and flow direction. Cerebral blood velocities range from 0.1 to 20 cm/s, yielding Doppler shifts of several hundred Hz to a few kHz at clinical frequencies (1–10 MHz).

Power Doppler imaging enhances sensitivity by integrating the squared amplitude of Doppler‐shifted echoes across multiple pulse emissions:

$$P_{\mathrm{PD}} = \sum_{n=1}^{N} \lvert s_n \rvert^2,$$

with (s_n) the quadrature‐demodulated echo for each ensemble of (N) emissions (typically (N=100)–1000). This approach resolves slow flows (<1 mm/s) and microvascular perfusion, with ensemble frame rates exceeding 500 Hz.