Doppler Ultrasound

Spec mapping: OCR H556 Module 6.5 — Medical Imaging (the Doppler effect applied to reflected ultrasound; the two-way reflection geometry doubles the single-shift formula to $\Delta f / f_0 \approx 2 v \cos\theta / c$ for a reflector moving at velocity $v$ at angle $\theta$ to the beam; measurement of blood-flow velocity in arteries and veins; colour-Doppler imaging; the angle-dependence trade-off in clinical practice). Refer to the official OCR H556 specification document for exact wording.

Pulse-echo ultrasound, as described in the previous lesson, tells you where a reflecting boundary is — it reads off the depth from the round-trip pulse time. 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 red blood cells flowing in an artery or vein, clinicians can measure blood-flow velocity non-invasively and in real time. The diagnostic implications are enormous: narrowed arteries, leaky or stenotic heart valves, foetal heart activity, deep-vein thrombosis and the perfusion of transplanted organs are all routinely assessed by Doppler ultrasound.

This lesson covers the Doppler-ultrasound section of Module 6.5 — Medical Imaging of the OCR A-Level Physics A specification (H556) — the physics of the frequency shift, the central formula $\Delta f / f_0 \approx 2 v \cos\theta / c$ , the geometry, clinical applications and the colour-Doppler imaging mode.

The Doppler Effect for Reflected Waves

You met the Doppler effect earlier in the course, in the context of audible-frequency sound: when a source or observer moves relative to a medium, the observed frequency differs from the source frequency. The same physics applies to ultrasound — and with one important extra twist when the wave is reflected by a moving target rather than directly emitted by a moving source.

Consider a stationary transducer emitting ultrasound at frequency $f_0$ . A reflector (a red blood cell, say) moves at velocity $v$ along the beam direction. There are two Doppler shifts in series:

Going to the reflector. The moving reflector sees the incident wave at a Doppler-shifted frequency $f_0' \approx f_0 (1 + v/c)$ (for $v$ towards the transducer).
Coming back from the reflector. The reflector now acts as a moving source re-emitting at $f_0'$ ; the stationary transducer receives a further Doppler-shifted frequency $f_0'' \approx f_0'(1 + v/c) \approx f_0(1 + 2v/c)$ (to first order in $v/c$ ).

The net shift detected at the transducer is therefore approximately twice the single-stage shift:

$\dfrac{\Delta f}{f_0} \approx \dfrac{2 v \cos\theta}{c}$

where $c$ is the speed of sound in the medium ( $\approx 1540$ m s $^{-1}$ in soft tissue, $\approx 1560$ m s $^{-1}$ in blood) and $\theta$ is the angle between the ultrasound beam and the direction of motion of the reflector. The factor of 2 is the signature of a reflected-wave Doppler measurement — it is not present in a one-way Doppler shift (e.g. the redshift of a distant galaxy).

Rearranging for the velocity:

$v = \dfrac{c\,\Delta f}{2 f_0 \cos\theta}.$

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

Geometry Matters: The Role of cos θ

The factor of $\cos\theta$ is essential. The reflector's velocity $\vec v$ has a component $v\cos\theta$ along the beam direction, and only this component produces a Doppler shift; the perpendicular component does not change the round-trip path length and so does not shift the frequency.

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 — even though the blood is flowing at full speed.
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 (which prefers shallower angles).

Because $v\cos\theta$ is what is actually measured, errors in estimating the angle translate directly into errors in the reported velocity. Ultrasound scanners display a steering-line overlay on the B-scan image that the operator aligns with the vessel axis; the software then knows $\theta$ and computes $v$ from $\Delta f$ .

The diagram shows the central idea: the beam is incident at angle $\theta$ to the flow direction; only the component $v\cos\theta$ along the beam contributes to the Doppler shift; the reflected wave returns to the transducer with a wavelength shorter (and frequency higher) than the emitted wave when the reflector is moving towards the transducer.

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. Take the speed of sound in blood as $c = 1560$ m s $^{-1}$ . Calculate the blood-flow velocity.

Solution.

\begin{aligned} v &= \dfrac{c\,\Delta f}{2 f_0 \cos\theta} \\ &= \dfrac{(1560)(300)}{2 (5.0 \times 10^{6})(\cos 60°)} \\ &= \dfrac{4.68 \times 10^{5}}{2 (5.0 \times 10^{6})(0.5)} \\ &= \dfrac{4.68 \times 10^{5}}{5.0 \times 10^{6}} \\ &\approx 0.094\,\text{m s}^{-1}. \end{aligned}

About $9.4$ cm s $^{-1}$ — well within the normal range for venous blood flow.

Worked Example 2: Frequency shift from a known velocity

The blood in a patient's carotid artery is flowing at $v = 0.60$ m s $^{-1}$ . A 4.0 MHz ultrasound beam is incident at $45°$ to the direction of flow. Take $c = 1560$ m s $^{-1}$ in blood. What Doppler shift is observed?

Solution.

\begin{aligned} \Delta f &= \dfrac{2 f_0 v \cos\theta}{c} \\ &= \dfrac{2 (4.0 \times 10^{6})(0.60)(\cos 45°)}{1560} \\ &= \dfrac{2 (4.0 \times 10^{6})(0.60)(0.707)}{1560} \\ &\approx 2175\,\text{Hz} \approx 2.2\,\text{kHz}. \end{aligned}

A shift of about $2$ kHz. Doppler shifts in clinical ultrasound typically range from $\sim 100$ Hz (slow venous flow) to several kHz (fast arterial flow, especially near a narrowing). The shift falls within the audible range — and most Doppler scanners play it through a loudspeaker so the operator can hear the flow.

Worked Example 3: Sensitivity to angle error

In Worked Example 1, an operator misreports the beam–flow angle as $50°$ when the true angle is $60°$ . What is the percentage error in the reported velocity?

Solution. The reported velocity is $v_\text{reported} = c\,\Delta f / (2 f_0 \cos 50°)$ , while the true velocity is $v_\text{true} = c\,\Delta f / (2 f_0 \cos 60°)$ . The ratio is

$\dfrac{v_\text{reported}}{v_\text{true}} = \dfrac{\cos 60°}{\cos 50°} = \dfrac{0.500}{0.643} \approx 0.778.$

The reported value is about 22% low. Conversely, an over-estimate of $\theta$ produces an over-estimate of $v$ . Doppler-ultrasound velocity measurements are inherently sensitive to angle, especially near $\theta \to 90°$ where $\cos\theta \to 0$ and the formula diverges. This is why scanners refuse to compute $v$ at $\theta > 70°$ and why clinicians keep the beam at $45°$ – $60°$ wherever possible.

Clinical Applications

Doppler ultrasound is routinely used to:

Measure blood-flow velocity in arteries and veins. Narrowed vessels (atherosclerotic stenoses) produce high velocities as the flow is squeezed through the constriction — by mass conservation, the flow speed in a vessel narrowed to half its area must double. A high-velocity Doppler signal at a particular vessel position is a key indicator of stenosis.
Assess heart-valve function (echocardiography). Leaky (regurgitant) valves allow backward flow during the wrong part of the cardiac cycle, producing a reversed Doppler signal; narrowed (stenotic) valves force blood to accelerate as it passes through, producing high-velocity jets.
Monitor foetal heart rate. The pulsing of the foetal heart produces a clear Doppler signal that can be tracked throughout pregnancy, even when the foetus is too small for direct anatomical imaging. The hand-held foetal Doppler used at routine antenatal appointments is the simplest Doppler-ultrasound instrument in clinical use.
Detect deep-vein thrombosis. A clot blocks venous return; the absence of an expected Doppler signal indicates obstruction.
Assess perfusion of transplanted organs. Healthy perfusion gives a normal Doppler signature; obstruction or rejection produces a characteristic abnormal pattern.

Modern scanners combine Doppler with B-scan imaging to produce colour-Doppler displays, in which the velocity is shown as a colour overlay on the grey-scale anatomical image: red for flow towards the transducer, blue for flow away. This makes vascular anatomy vivid and intuitive — and shows turbulent or reversed flow as a multi-colour mosaic, instantly highlighting clinically interesting regions.

Where Does Doppler Fit Among the Imaging Modes?

flowchart TB
    Q["Ultrasound question:<br/>position or velocity?"]
    Pos["Position<br/>(structure)"]
    Vel["Velocity<br/>(motion)"]
    PE["Pulse-echo<br/>A-scan, B-scan<br/>d = ct/2"]
    Dop["Doppler<br/>Δf/f₀ = 2v cosθ/c"]
    Col["Colour Doppler<br/>(B-scan + Doppler overlay)"]
    Q --> Pos
    Q --> Vel
    Pos --> PE
    Vel --> Dop
    Dop --> Col

The choice is clean: pulse-echo for static structure, Doppler for moving reflectors, and colour Doppler when you want both at once.

Synoptic Links

Doppler effect for sound and light (Module 4.4 / Module 6.5 astrophysics): The Doppler-ultrasound formula is the same physics that gives redshift and blueshift in the spectra of stars and galaxies — but with two important differences. (1) Ultrasound Doppler uses reflection, giving the factor of 2 from the round-trip; one-way astronomical Doppler does not. (2) Astronomical Doppler is relativistic ( $v$ comparable to $c$ ); ultrasound Doppler is firmly non-relativistic ( $v \sim 1$ m s $^{-1}$ vs $c \sim 1500$ m s $^{-1}$ ).
Continuity equation in fluid flow: Blood flow through a vessel obeys mass conservation, $A_1 v_1 = A_2 v_2$ for an incompressible fluid. A vessel narrowed to half its cross-sectional area must have flow doubled in velocity — directly observable as a higher Doppler shift, and the basis for non-invasive diagnosis of arterial stenoses.
Pulse-echo ultrasound (Lesson 11, this module): Doppler uses exactly the same hardware as pulse-echo B-scan — a piezoelectric transducer emitting pulses and receiving reflections — but extracts the frequency-shift of the reflected pulse rather than just its time-of-arrival. A modern scanner switches between the two modes (or interleaves them) electronically.

Specimen question modelled on the OCR H556 paper format

Question (8 marks): A vascular sonographer uses Doppler ultrasound at $f_0 = 4.0$ MHz to measure the blood-flow velocity in a patient's carotid artery. The beam is incident at an angle of $\theta = 60°$ to the vessel axis. The speed of sound in blood is $c = 1560$ m s $^{-1}$ . The observed Doppler shift is $\Delta f = 1500$ Hz.

(a) Derive (or state, with reason) the Doppler-shift formula $\Delta f / f_0 = 2 v \cos\theta / c$ for ultrasound reflected from a moving target. Explain why the formula contains a factor of 2 that is absent from a one-way Doppler shift. [3]

(b) Use the data to calculate the blood-flow velocity. [2]

(c) The sonographer realises she has measured the angle wrong; the true angle is $50°$ , not $60°$ . Calculate the corrected velocity, and comment on the magnitude of the error. [3]

Doppler Ultrasound

Doppler Ultrasound

The Doppler Effect for Reflected Waves

Geometry Matters: The Role of cos θ

Worked Example 1: Blood-flow velocity from frequency shift

Worked Example 2: Frequency shift from a known velocity

Worked Example 3: Sensitivity to angle error

Clinical Applications

Where Does Doppler Fit Among the Imaging Modes?

Synoptic Links

Specimen question modelled on the OCR H556 paper format

AO breakdown

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