Ultrasound imaging
Ultrasound is sound above 20 kHz. Medical scans use about 1–15 MHz. Higher frequency gives finer detail (shorter wavelength) but is absorbed faster, so it cannot go as deep.
The piezoelectric transducer
A piezoelectric crystal changes shape when a voltage is applied, and makes a voltage when it is squeezed. A short voltage pulse makes it vibrate and send out an ultrasound pulse. Between pulses the same crystal acts as a receiver and turns the returning echoes into voltages. A backing block damps the crystal so pulses are short.
Acoustic impedance and reflection
Acoustic impedance Z = ρc (density × speed of sound), unit kg m⁻² s⁻¹. When ultrasound meets a boundary, the fraction of intensity reflected is
R = I_r ÷ I₀ = (Z₂ − Z₁)² ÷ (Z₂ + Z₁)²
A big difference in Z means strong reflection. Soft tissue to soft tissue reflects only about 1%, so most sound goes on and shows deeper layers. Air reflects almost 100%, which is why coupling gel (Z close to skin) is spread on the skin. Bone reflects strongly and absorbs, so it is hard to see behind bone.
A-scan and B-scan
- A-scan (amplitude): one beam; echo strength is plotted against time. Depth d = c t ÷ 2. Used to measure distances, such as the length of the eyeball.
- B-scan (brightness): the probe has many crystals or is swept across; each echo becomes a bright dot at the right depth, building a 2D picture, as in pregnancy scans.
Advantages: no ionising radiation, cheap, portable, real-time. Limits: cannot pass through air (lungs, gut gas) or bone well; finer detail is limited by wavelength.
Fibre optics and endoscopy
An optical fibre has a glass core of higher refractive index surrounded by cladding of lower index. Light hitting the core–cladding boundary at an angle bigger than the critical angle c is totally reflected: total internal reflection (TIR). sin c = n₂ ÷ n₁. The cladding protects the core surface and stops light jumping into touching fibres.
An endoscope is a thin flexible tube put into the body. It holds:
- an incoherent bundle: fibres in any order, used just to carry light in to light up the inside.
- a coherent bundle: each fibre keeps the same position at both ends, so the pattern of light, the image, arrives in the right order. Thinner fibres give finer detail.
- an objective lens at the tip, an eyepiece or camera at the top, and often channels for water, air and small tools (keyhole surgery).
The magnetic resonance (MR) scanner
The body is mostly water, full of hydrogen nuclei (protons). Each proton spins and acts like a tiny magnet.
- A superconducting magnet makes a strong, uniform field (about 1.5–3 T). The protons line up with or against it and precess (wobble round the field) at the Larmor frequency, about 42.6 MHz per tesla (64 MHz at 1.5 T).
- A pulse of radio waves at exactly that frequency is sent in. The protons absorb it (resonance) and tip over.
- When the pulse stops, the protons relax back and send out radio waves, which coils detect. Different tissues relax at different rates, so they look different.
- Gradient coils make the field slightly different at each point, so the frequency tells the computer where the signal came from. It builds slices and 3D images.
Advantages: no ionising radiation; superb soft-tissue detail (brain, spinal cord, joints); any slice angle. Limits: expensive, slow, noisy, patient must keep still; not for people with some metal implants or pacemakers.
Try it: an echo you can time
Stand about 50 m from a tall wall and clap. Ask a friend with a stopwatch to time from the clap to the echo. Use d = v t ÷ 2 with v = 340 m s⁻¹ to find the distance, and check it by pacing. That is an A-scan with your hands! For fibres: shine a torch into a clear plastic bottle of water with a hole near the bottom; the light stays inside the falling stream of water by TIR.
Key formulas and definitions
- Depth from echo: d = c t ÷ 2
- Acoustic impedance: Z = ρ c
- Reflected fraction: R = (Z₂ − Z₁)² ÷ (Z₂ + Z₁)²
- Critical angle: sin c = n₂ ÷ n₁ (n₁ > n₂)
- Larmor frequency ≈ 42.6 MHz × B (in tesla) for hydrogen
Worked examples
1. An echo returns 65 µs after the pulse. c = 1540 m s⁻¹. How deep is the boundary?
d = c t ÷ 2 = 1540 × 65 × 10⁻⁶ ÷ 2 = 0.050 m = 5.0 cm.
2. Muscle has ρ = 1075 kg m⁻³ and c = 1590 m s⁻¹. Find Z.
Z = 1075 × 1590 = 1.71 × 10⁶ kg m⁻² s⁻¹.
3. Find the fraction reflected at a fat (Z = 1.38 × 10⁶) to muscle (Z = 1.70 × 10⁶) boundary.
R = (1.70 − 1.38)² ÷ (1.70 + 1.38)² = 0.1024 ÷ 9.49 = 0.011, about 1.1%.
4. Why is gel needed? Air Z = 430 kg m⁻² s⁻¹, skin Z = 1.7 × 10⁶.
R = (1.7 × 10⁶ − 430)² ÷ (1.7 × 10⁶ + 430)² ≈ 0.999. Almost all the sound would bounce off the skin. Gel has Z close to skin, so little reflects.
5. A fibre core has n = 1.52 and cladding n = 1.40. Find the critical angle.
sin c = 1.40 ÷ 1.52 = 0.921, c = 67°.
6. Find the Larmor frequency of protons in a 3.0 T scanner.
f = 42.6 MHz × 3.0 = 128 MHz (radio waves).
Common mistakes
- Forgetting the ÷ 2 in d = c t ÷ 2: the time is for the trip there and back.
- Thinking a bigger Z means more reflection. It is the difference in Z between the two sides that matters.
- Mixing up the bundles: the coherent bundle carries the image, the incoherent one only carries light.
- Saying MRI uses X-rays or is ionising. It uses a magnetic field and radio waves only.