Notes on Attenuation, Reflection, Refraction (Lec 3, Ch 6)

Attenuation Overview

• Attenuation is the decrease in intensity, amplitude, and power of a sound wave as it travels through a medium.
• Units: decibels, dB.
• Determined by and directly related to:
  • Distance travelled by the sound → greater distance = greater attenuation
  • Frequency of the sound → higher frequency = greater attenuation
• Attenuation during insonation involves three components:
  • Absorption
  • Scattering
  • Reflection
• All three processes lead to a decrease in amplitude as sound propagates through soft tissue.

Absorption

• Ultrasonic energy is converted into another form of energy (often heat).
• About 80% of attenuation in soft tissue is caused by absorption.

Reflection, Attenuation & Boundaries

• Attenuation includes energy extraction from a wave by reflection, scattering, and absorption.
• Reflection occurs when sound energy returns toward the transducer after striking a boundary/interface.
• Two forms of reflection:
  • Specular reflections
  • Diffuse reflections
• Backscatter refers to echoes that bounce back toward the transducer (e.g., from liver).

Specular Reflection

• Reflections from a large, smooth reflector that “bounce” in one direction.
• The boundary/interface must be longer than the wavelength (λ) of the ultrasound.
• Examples of specular reflectors:
  • Pericardium: fibrous, dense, smooth, strong reflector
  • Diaphragm: smooth muscle/fibrous tissue, strong reflector
  • Glisson’s capsule of the liver: specular reflection
• Conceptual analogy: mirrors—what you see depends on the angle of incidence.
• Longer-than-wavelength specular reflectors produce a single, well-defined reflection.

Diffuse Reflection

• Diffuse reflections come from rough reflectors and scatter echoes in many directions (disorganized, random).
• The wavelength is about the same size as boundary irregularities.
• Echoes are called BACKSCATTER when returning toward the transducer.
• Higher frequencies produce more scatter.
• Rayleigh scattering is a type of diffuse scattering:
  • Occurs when the reflector is smaller than the wavelength (e.g., red blood cells).
  • Higher frequencies result in more Rayleigh scattering.
  • Relationship: Rayleigh scattering ∝ frequency^4
• If frequency doubles, Rayleigh scattering increases by a factor of 16 (2^4).
• Often summarized: Rayleigh scattering = ∝ f^4; higher frequency → more Rayleigh scattering.

Rayleigh Scattering (in detail)

• If reflector is smaller than the wavelength, reflections scatter uniformly in all directions (360°).
• Example: Red blood cells are Rayleigh scatterers.
• Frequency dependence:
  • Amount of scatter ∝ f^4
  • Thus, doubling frequency → 16× more Rayleigh scattering.
• In soft tissues and similar media, scatter increases with frequency^4.

Attenuation Coefficient (AC) & Total Attenuation

• Attenuation Coefficient (AC): amount of attenuation that occurs in 1 cm of travel; units are dB/cm.
• AC is directly related to frequency: higher frequencies yield higher AC.
• In soft tissue:
  • AC_ST ≈ f/2 dB/cm, where f is frequency in MHz.
  • Thus, AC_{ST} = rac{f}{2} ext{ dB/cm}
• Total Attenuation along a beam’s path must include distance:
  • $ext{Total Attenuation (dB)} = AC imes ext{Distance (cm)}$
• Half-value layer thickness (HVLT): depth at which 3 dB of attenuation occurs.
  • Also called penetration depth.
  • In soft tissue: HVLT ≈ 2.0 mm – 10 mm.
  • HVLT increases as frequency decreases.

Attenuation Coefficient in Practice

• AC is the rate of attenuation per cm; it increases with frequency.
• For a beam traveling through soft tissue, AC ≈ f/2 dB/cm (f in MHz).
• Example: At 2 MHz over 5 cm, Total Attenuation = (2/2) dB/cm × 5 cm = 1 × 5 = 5 dB.

Decibels – Notation and Rules

• Decibels express changes in intensity on a logarithmic scale.
• Relationship: $dB = 10 \, \log<em>{10}\left(\frac{I</em>f}{I_i}\right)$
• Positive decibels (+dB) indicate increasing intensity; negative decibels (-dB) indicate decreasing intensity.
• Positive decibels examples:
  • If final intensity is doubled: +3 dB
  • If intensity increases ten-fold: +10 dB
• Negative decibels examples:
  • If intensity is reduced to half: -3 dB
  • If reduced to one tenth: -10 dB

The 3 dB Rule

• A 3 dB increase ≈ final intensity ×2
• A 3 dB decrease ≈ final intensity ×1/2
• Examples (starting from 100):
  • +3 dB: 100 × 2 = 200
  • +6 dB: 100 × 4 = 400
  • +9 dB: 100 × 8 = 800
  • Multiples of 3 dB add up (each 3 dB doubles, cumulatively).
• The 10 dB Rule:
  • -10 dB means final intensity is 1/10 of the original.
  • +10 dB means final intensity is 10× the original, etc.

Attenuation Coefficient (AC) Examples & Calculations

• Example problem reflections in media:
  • If frequency is 2 MHz and distance is 5 cm: Total Attenuation = 5 dB (as above).
  • If frequency is 4 MHz and distance is 5 cm: AC = 4/2 = 2 dB/cm; Total Attenuation = 2 × 5 = 10 dB.
• Quick practice: 2 MHz over 10 cm → 5 dB × 2 = 10 dB; 4 MHz over 5 cm → 10 dB.

Attenuation in the Body – Mediums

• Different media have different attenuation and reflection characteristics:
  • Water / Biologic fluids: ~1% reflection, 99% transmission.
  • Very soft tissue (liver, kidney): Intermediate attenuation; greater than water but less than bone.
  • Muscle: Intermediate attenuation; higher than fluid but less than bone.
  • Bone & lungs: Higher attenuation; lungs involve Rayleigh scattering in air-containing regions.
• The amount of reflection depends on impedance mismatch between media.
• In many tissues, 50% absorbed vs 50% reflected vs 0% transmitted at very high impedance differences (e.g., bone vs air) – note: actual values depend on Z1 and Z2.

Boundary Interactions – IRC, ITC, and Impedance

• At interfaces, some energy is reflected and some transmitted.
• Definitions:
  • Intensity Reflection Coefficient (IRC): percentage of intensity that bounces back at the boundary.
  • Intensity Transmission Coefficient (ITC): percentage of intensity that passes forward into the second medium.
  • IRC + ITC = 100%
• Impedance (Z): resistance to sound travel; unit is Rayl.
• Impedance for a medium: $Z = \rho c$ where ρ is density and c is propagation speed.
• At soft tissue interfaces, typical impedances range roughly from $1.25\ \text{Mrayls} \le Z \le 1.75\ \text{Mrayls}$.
• Reflection depends on impedance mismatch between media:
  • Formula for normal incidence (intensity):
  • $IRC = \left|\frac{Z<em>2 - Z</em>1}{Z<em>2 + Z</em>1}\right|^2$
  • $ITC = 1 - IRC$
• Example calculation:
  • If $Z<em>1 = 2\ \text{rayls}$ and $Z</em>2 = 6\ \text{rayls}$, then
  • $IRC = \left|\frac{6 - 2}{6 + 2}\right|^2 = \left(\frac{4}{8}\right)^2 = \left(\tfrac{1}{2}\right)^2 = \tfrac{1}{4} = 25\%.$
• At an interface where IRC = 4%, ITC = 96% (example in notes).

Interfaces at Normal Incidence

• Normal incidence occurs when the incident beam strikes the boundary at 90°.
• At normal incidence, incident, reflected, and transmitted beams are collinear.
• Terminology:
  • Incident beam (I)
  • Reflected beam (R)
  • Transmitted beam (T)
• For normal incidence, c1 is the speed in medium 1 (incident medium) and c2 in medium 2 (transmission medium).
• Reflection occurs in the same medium as the incident beam; transmitted beam travels into medium 2.

Angle Types – Oblique Incidence and Refraction

• Normal incidence: 90° incidence.
• Oblique incidence: incidence at any angle other than 90°.
• Acute incidence: incident angle < 90° relative to the normal.
• Obtuse incidence: incident angle > 90° relative to the normal.
• When a beam strikes at oblique incidence and media have different propagation speeds, refraction occurs.
• Refraction direction is governed by changes in propagation speed between media.

Refraction – Snell’s Law and Critical Angle

• Refraction can only occur if two conditions exist:
  • Oblique incidence (not normal)
  • Media with different propagation speeds (c1 ≠ c2)
• Snell’s Law (describes refraction):
  • $\frac{\sin \theta<em>t}{\sin \theta</em>i} = \frac{c<em>2}{c</em>1}$
  • equivalently, $\sin \theta<em>t = \frac{c</em>2}{c<em>1} \sin \theta</em>i$
• If the transmitted angle becomes tangent to the boundary (i.e., the critical angle is reached), total internal reflection occurs and transmission ceases.
• Critical angle concept illustrated by a rock skipping on a pond; maximum angle before transmission ceases.
• If Medium 2 has a slower propagation speed, transmitted beam refracts toward the normal (θt < θi).
• If Medium 2 has a faster propagation speed, transmitted beam refracts away from the normal (θt > θi).

Practical Notes on Refraction

• Medium 1 (c1) is the incident medium; Medium 2 (c2) is the transmission medium.
• Reflections always occur in the same medium as the incident beam (Medium 1).
• The angle of refraction depends on the relative speeds (c2/c1).

Summary of Key Relationships and Concepts

• Attenuation: $\text{Total Attenuation} = AC \times \text{Distance}$ with $AC = \frac{f}{2} \text{ dB/cm}$ for soft tissue (f in MHz).
• Absorption typically accounts for ~80% of tissue attenuation.
• Reflection at interfaces is influenced by impedance mismatch:
  • Medium 1 impedance: $Z<em>1 = \rho</em>1 c_1$
  • Medium 2 impedance: $Z<em>2 = \rho</em>2 c_2$
  • IRC = \left|\dfrac{Z2 - Z1}{Z2 + Z1}\right|^2; ITC = 1 - IRC.
• Specular reflections require smooth, large reflectors; diffuse/backscatter from rough interfaces or small scatterers (e.g., RBCs).
• Rayleigh scattering increases with frequency as f^4; higher frequencies yield more scatter.
• Impedance is a fundamental factor in reflection; equal impedances yield zero reflection at normal incidence.
• Normal incidence: I, R, T are aligned; angle incidence introduces refraction per Snell’s Law.
• HVLT is the depth where 3 dB attenuation occurs; HVLT decreases with increasing frequency.
• Practical imaging considerations: attenuation reduces beam energy with depth; gain/TGCs compensate for energy loss in imaging.