How Ultrasound Works — Piezoelectricity, B-mode Imaging and Doppler

Ultrasound imagines the inside of the human body using high-frequency sound waves — no ionizing radiation, no magnets, just acoustic pulses sent and received by a piezoelectric crystal. From the timing of echoes it computes distances; from their intensity it distinguishes soft tissue from fluid; from frequency shifts it measures blood velocity. The same physics that guides dolphins and bats now reveals unborn children and coronary artery disease.

1. Piezoelectric Transducers

At the heart of every ultrasound probe is a piezoelectric crystal — typically lead zirconate titanate (PZT) or polyvinylidene fluoride (PVDF). Piezoelectricity describes the conversion between mechanical deformation and electric fields:

Direct piezoelectric effect: mechanical strain → electric charge (receiving mode — detecting returning echoes).
Converse piezoelectric effect: applied voltage → mechanical strain (transmit mode — generating ultrasound pulses).

The crystal resonates at a natural frequency determined by its thickness t and the speed of sound within the crystal material v_crystal:

f_resonance = v_crystal / (2t) PZT: v_crystal \approx 4 000 m/s Medical imaging: f = 2-15 MHz \to t = 0.13-1.0 mm Higher frequency \to better resolution, but more attenuation \to less depth. Frequency tradeoff: 3 MHz for abdominal (15 cm depth); 10-15 MHz for superficial structures (thyroid, tendons, arteries, \leq5 cm).

A brief electrical pulse (~200 ns) excites the crystal, producing a short burst of ultrasound (typically 2–3 wavelengths). A backing layer of tungsten-rubber composite rapidly damps the ringing, keeping pulse length short. A matching layer (~λ/4 thick, impedance √(Z_crystal × Z_tissue)) reduces reflection at the crystal-tissue interface.

2. Sound Propagation in Tissue

Ultrasound propagates as a longitudinal pressure wave. Different soft tissues have slightly different acoustic parameters:

Tissue Speed (m/s) Impedance Z (MRayl) Attenuation (dB/cm/MHz) ────────────────────────────────────────────────────────────────────────── Soft tissue (avg) 1540 1.63 0.5 Blood 1570 1.65 0.2 Fat 1470 1.35 0.6 Bone (cortical) 3500 7.0 10 Air 340 0.0004 high Water 1480 1.48 ~0

Attenuation increases with frequency — at 10 MHz, tissue attenuates about 5 dB/cm, limiting useful depth. The scanner compensates with time-gain compensation (TGC): automatically amplifying signals from deeper structures where attenuation is greater.

Attenuation(dB) \approx α \cdot f \cdot d α \approx 0.5 dB/cm/MHz for typical soft tissue At f = 5 MHz, d = 10 cm: attenuation \approx 25 dB (round trip = 50 dB) TGC adds ~0.5 dB/cm/MHz of gain to compensate.

3. Acoustic Impedance and Reflection

When an ultrasound wave crosses a boundary between two media, the fraction of reflected energy depends on the acoustic impedance mismatch:

Z = ρ \cdot c (acoustic impedance, Rayleigh = Pa\cdots/m) Reflection coefficient (intensity): R = ( (Z₂ - Z₁) / (Z₂ + Z₁) )² Soft tissue \to fat: Z₁=1.63, Z₂=1.35 \to R \approx 0.8% (most light passes through) Soft tissue \to bone: Z₁=1.63, Z₂=7.0 \to R \approx 41% (strong bright echo) Soft tissue \to air: Z₁=1.63, Z₂=0.004 \to R \approx 99.9% (almost total reflection)

The near-total reflection at tissue–air interfaces is why ultrasound gel is essential: it eliminates the air gap between probe and skin. Similarly, gas-filled bowel loops block deeper structures — a full bladder is used as an acoustic window for pelvic scans.

Scattering vs specular reflection: smooth large surfaces (diaphragm, vessel walls) act as specular reflectors; small structures comparable to wavelength (~0.1–0.3 mm at 5 MHz) scatter energy in all directions, producing the characteristic granular speckle texture in ultrasound images.

4. B-mode Image Formation

B-mode (Brightness mode) is the standard 2-D grayscale ultrasound image. Each line in the image corresponds to one pulse-echo measurement along a specific beam direction.

The transducer emits a focused pulse in direction θ.
The returning echo is sampled at the ADC — amplitude vs time encodes depth via d = c·t/2 (dividing by 2 because sound travels to the reflector and back).
Signal processing: bandpass filter, envelope detection (Hilbert transform), log compression (to fit 50 dB dynamic range into displayable 8-bit grayscale), TGC.
This A-line is placed as a column in the image at angle θ.
Steps 1–4 repeated for all beam angles, building up the 2-D image (typically 256–512 lines) in ~30 ms for a 30 fps refresh rate.

Depth-time conversion: d = c \cdot t / 2 (c \approx 1540 m/s assumed constant) Maximum imaging depth d_max = c / (2 \cdot PRF) PRF = pulse repetition frequency; for d_max = 20 cm: PRF \leq 1540 / (2 \times 0.2) = 3850 Hz (echo must return before next pulse)

5. Resolution: Axial vs Lateral

Ultrasound resolution has two distinct components:

Axial resolution (along beam): determined by pulse length. Shorter pulse = better axial resolution. Typical: 0.1–0.5 mm at common medical frequencies. Axial resolution ≈ c × pulse duration / 2 ≈ λ (for 2-cycle pulses).
Lateral resolution (perpendicular to beam): determined by the beam width at the focus depth. A focused transducer creates a beam waist at the focal point; lateral resolution = beam width ≈ λ × f/D (where f = focal distance, D = aperture diameter). Typically 1–3 mm — worse than axial resolution.

At f_center = 5 MHz: λ = c/f = 1540 / 5\times10⁶ \approx 0.31 mm Axial resolution \approx 0.3-0.6 mm (2-4 cycles) Lateral resolution at focus \approx 1-2 mm (depends on f-number) At 15 MHz: λ \approx 0.1 mm \to axial res \approx 0.1-0.2 mm (high-freq dermatology probe)

6. Doppler Ultrasound — Blood Flow

Moving blood cells shift the frequency of reflected ultrasound — the Doppler effect. A cell moving toward the transducer at velocity v compresses the wavefronts, increasing the received frequency:

Doppler shift: Δf = 2 \cdot f₀ \cdot v \cdot cos(θ) / c f₀ = transmitted frequency (Hz) v = blood velocity (m/s) θ = angle between beam and flow direction c = speed of sound (~1540 m/s) Example: f₀=5 MHz, v=0.5 m/s (normal carotid artery), θ=60° Δf = 2 \times 5\times10⁶ \times 0.5 \times cos(60°) / 1540 \approx 1623 Hz (audible!) Aliasing in pulsed Doppler: maximum measurable velocity = c/(4\cdotf₀\cdotdepth)

Color Doppler encodes mean flow velocity at each pixel as a color (red = toward probe, blue = away). Power Doppler encodes flow energy regardless of direction — more sensitive for slow flow. Spectral Doppler displays velocity vs time across all red cells in a sample volume — used to characterize cardiac valve stenosis (Bernoulli equation: ΔP = 4v²).

7. Phased Array Beam Steering

Modern probes use arrays of 128–512 individual piezoelectric elements instead of a single crystal. By firing each element with a slight time delay, the wavefronts from adjacent elements constructively interfere in a steerable direction — analogous to a phased array radar antenna.

Steering angle θ from an N-element array with spacing d: Time delay between adjacent elements: Δt = d \cdot sin(θ) / c For d = 0.3 mm (λ/2 at 2.5 MHz), steering to θ = 30°: Δt = 0.3\times10⁻³ \times sin(30°) / 1540 \approx 97 ns All N beams can be formed simultaneously using receive beam-forming (delay-and-sum), enabling real-time 3-D imaging (4-D ultrasound).

The large aperture created by the array allows dynamic focusing at different depths during reception — every depth is received with optimal focus. Transmit focusing is fixed at one depth per pulse; multiple transmit foci multiply frame time.

Harmonic imaging: tissue transmits sound nonlinearly and generates harmonic frequencies (2f₀, 3f₀…) of the transmitted frequency. Receiving only the 2nd harmonic dramatically reduces reverberation artefacts and improves contrast, especially in difficult-to-image patients. Standard in modern scanners.

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