Doppler Ultrasound

Pulse-echo ultrasound, as described in the last lesson, tells you where a reflecting boundary is. Doppler ultrasound goes one step further: it measures how fast that boundary is moving towards or away from the transducer. By applying the technique to the blood cells flowing in an artery, clinicians can measure blood flow velocity non-invasively and in real time — enabling diagnosis of narrowed arteries, faulty heart valves, foetal heart activity, and many other conditions.

This lesson, a shorter addition to the ultrasound module, covers Doppler ultrasound as specified in section 6.5.3 of the OCR A-Level Physics A specification (H556).

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The Doppler Effect

You already met the Doppler effect earlier in the course, in the context of waves and sound: when a source or observer moves relative to a wave, the observed frequency differs from the source frequency. In ultrasound, the Doppler shift arises when sound waves reflect off a moving boundary (for example, red blood cells in a blood vessel). The reflected wave reaches the transducer with a different frequency than the one originally emitted, and the shift is proportional to the velocity of the reflector.

For a stationary transducer emitting ultrasound at frequency f_0 and a reflector moving at velocity v along the beam direction, the reflected frequency received back by the transducer is:

Δf/f₀ ≈ (2v cos θ)/c

where c is the speed of sound in the medium (~1540 m s⁻¹ in soft tissue), v is the speed of the reflector, and \theta is the angle between the ultrasound beam and the direction of motion of the reflector. The factor of 2 arises because the wave experiences a Doppler shift twice: once going to the reflector and once coming back.

Rearranging for the velocity:

v = (c Δf)/(2 f₀ cos θ)

This is the central equation of Doppler ultrasound. Measuring \Delta f (the difference between emitted and received frequencies) and knowing f_0, c and the angle \theta, the scanner computes the velocity v.

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Geometry Matters

The factor of \cos\theta is important. If the beam is aimed along the direction of blood flow (\theta = 0, \cos\theta = 1), the measured shift is maximal. If the beam is aimed perpendicular to the flow (\theta = 90°, \cos\theta = 0), there is no Doppler shift at all — the reflector's velocity has no component along the beam. In practice, clinicians aim the beam at an angle of about 45°–60° to the vessel axis, which gives a substantial Doppler shift while still permitting good pulse-echo imaging of the vessel walls.

Because v\cos\theta is what is measured, errors in estimating the angle \theta translate directly into errors in the reported velocity. Ultrasound scanners display a line on the B-scan image that the operator aligns with the vessel axis, allowing the software to compute \theta and hence v.

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Worked Example 1: Blood flow velocity from frequency shift

An ultrasound beam of frequency 5.0 MHz is aimed at a blood vessel at an angle of 60° to the direction of flow. The observed Doppler shift is 300 Hz. Calculate the blood flow velocity. Take the speed of sound in blood as 1560 m s⁻¹.

Solution.

v = (c Δf) / (2 f₀ cos θ)
  = (1560 × 300) / (2 × 5.0 × 10⁶ × cos 60°)
  = (1560 × 300) / (2 × 5.0 × 10⁶ × 0.5)
  = 4.68 × 10⁵ / 5.0 × 10⁶
  ≈ 0.094 m s⁻¹

About 9.4 cm s⁻¹, which is within the normal range for venous blood flow.

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Worked Example 2: Frequency shift from a known velocity

The blood in a patient's carotid artery is flowing at 0.60 m s⁻¹. Ultrasound at 4.0 MHz is incident at 45° to the direction of flow. What Doppler shift is observed?

Solution.