Ultrasound physics: frequency, attenuation, impedance

Clinical relevance (why these matter in ultrasound-guided anaesthesia)

• Choosing probe frequency is a trade-off between penetration and resolution.
  • High frequency (e.g. 10–15 MHz): superficial nerves/lines; better detail; poorer depth.
  • Low frequency (e.g. 2–5 MHz): deeper structures (e.g. neuraxial, deep blocks); less detail; better penetration.
• Attenuation limits depth and image quality; it increases with frequency and distance.
  • Compensated by time gain compensation (TGC) and by selecting lower frequency for deeper targets.
• Acoustic impedance mismatch determines reflection strength at tissue boundaries and explains coupling gel use.
  • Air has very low impedance vs tissue → near-total reflection → must eliminate air gap with gel.

Core definitions and relationships

• Ultrasound = mechanical longitudinal wave; requires a medium; typical diagnostic frequencies 2–15 MHz.
• Wave speed in a medium: c (m·s⁻¹). In soft tissue, assume c ≈ 1540 m·s⁻¹.
• Frequency (f, Hz) = cycles per second; Period (T, s) = 1/f.
• Wavelength (λ, m): λ = c/f. Higher f → shorter λ.
  • Example: in soft tissue, f = 5 MHz → λ ≈ 1540/5e6 ≈ 0.308 mm.
  • Example: f = 10 MHz → λ ≈ 0.154 mm.
• Intensity (I): power per unit area (W·m⁻²). Often expressed in dB relative changes.

Frequency: effects on resolution and penetration

• Axial resolution (along beam) depends on spatial pulse length (SPL): smaller SPL gives better axial resolution.
  • SPL = number of cycles per pulse × λ. Higher f → smaller λ → smaller SPL → better axial resolution (if pulse cycles constant).
  • Approx axial resolution ≈ SPL/2.
• Lateral resolution (perpendicular to beam) depends on beam width; improved by focusing and higher frequency (narrower beam for a given aperture).
• Penetration decreases with increasing frequency due to greater attenuation.
• Trade-off summary: high f = high resolution/low penetration; low f = low resolution/high penetration.

Attenuation

• Attenuation = reduction in ultrasound amplitude/intensity with depth due to absorption, scattering and reflection.
• In soft tissue, attenuation coefficient α ≈ 0.5 dB·cm⁻¹·MHz⁻¹ (rule of thumb).
  • Attenuation (one-way, dB) ≈ α × f(MHz) × depth(cm).
  • Pulse-echo systems have two-way travel: total attenuation ≈ 2 × α × f × depth.
• Absorption: conversion of sound energy to heat; increases with frequency (dominant in soft tissue).
• Scattering: redirection from small/rough structures; increases with frequency; contributes to speckle and loss of coherent beam energy.
• Reflection: energy returned at boundaries; depends on impedance mismatch and angle of incidence; contributes to attenuation of transmitted beam.
• Compensation: TGC amplifies deeper echoes to counter depth-dependent attenuation; overall gain amplifies all echoes.
  • TGC corrects for depth; it does not restore lost resolution from high attenuation.
• Worked example (typical FRCA calculation): 5 MHz probe, target at 6 cm. Two-way attenuation ≈ 2 × 0.5 × 5 × 6 = 30 dB.
  • 30 dB intensity loss = 10^(30/10) = 1000-fold reduction in intensity (approx).

Acoustic impedance (Z) and reflection/transmission

• Acoustic impedance Z = ρc (Rayl = kg·m⁻²·s⁻¹), where ρ is density and c is speed of sound in the medium.
• Impedance mismatch determines reflected fraction at normal incidence.
  • Intensity reflection coefficient: R = ((Z2 − Z1)/(Z2 + Z1))^2.
  • Intensity transmission coefficient (ignoring absorption): T = 1 − R.
• Large mismatch (e.g. tissue–air) → R approaches 1 → almost all energy reflected → poor imaging without coupling gel.
• Smaller mismatch (e.g. soft tissue interfaces) → partial reflection → useful echoes for imaging.
• Angle of incidence matters: specular reflection from smooth interfaces is maximal when beam is perpendicular; oblique incidence reflects away from probe → drop-out.

Key numbers and quick recall

• Speed of sound: soft tissue ≈ 1540 m·s⁻¹; air ≈ 330 m·s⁻¹; bone higher (variable, ~3000–4000 m·s⁻¹).
• Attenuation coefficient in soft tissue: ~0.5 dB·cm⁻¹·MHz⁻¹.
• Typical probe frequency selection: linear 6–15 MHz (superficial); curvilinear 2–5 MHz (deep).