Ultrasound

Physics in Medical Imaging

Created by Matt McCormick / G+ / Kitware

Basic Ultrasound Physics

Ultra-sound

  1. ultra - "on the farside of, beyond"
  2. sound - "vibrations transmitted through an elastic solid, liquid, or gas"

Ultrasound range diagram

Main Physical Phenonmena

  1. Speed of sound
  2. Scatter
  3. Attenuation

Speed of sound

$$c = f \lambda$$

  • $c$ - speed of sound
  • $f$ - frequency
  • $\lambda$ - wavelength
wavelength

Speed of sound

$$c = \sqrt{\frac{K}{\rho}}$$

  • $c$ - speed of sound
  • $K$ - bulk modulus (pressure required to compress)
  • $\rho$ - density

Acoustic impedance

$$Z = \frac{p}{u} = \rho c$$

  • $Z$ - acoustic impedance
  • $p$ - acoustic pressure
  • $u$ - wave particle velocity

impedance table

Changes in wave propagation are caused by acoustic impedance gradients.

The behavior of the change depends on the size of the acoustic impedance variation relative to the incident wavelength.

When the structure is much larger than the wavelength, a specular reflection occurs. This happens organ boundaries.

When the structure is much smaller the wavelength, scattering occurs. This occurs in parenchyma where cells are relatively uniform. It manifests as speckle in ultrasound images.

Much content in an ultrasound image are from structures with a size somewhere in between.

Specular reflection

For a plane wave,

$$\frac{P_{-1}}{P_{+1}} = \left( \frac{Z_2 - Z_1}{Z_2 + Z_1} \right)$$

  • $P_{-1}$ - reflected wave amplitude
  • $P_{+1}$ - incident wave amplitude
  • $Z_1$ - first material acoustic impedance
  • $Z_2$ - second material acoustic impedance

$$\frac{P_{+2}}{P_{+1}} = \left( \frac{2 Z_2}{Z_2 + Z_1} \right)$$

  • $P_{+2}$ - transmitted wave amplitude

Refraction

Refraction also occurs for large structures according to Snell's Law, $$\frac{\sin \theta_i}{\sin \theta_t} = \frac{\lambda_1}{\lambda_2} = \frac{c_1}{c_2}$$

Scatter

Scattering is angle dependent.

And frequency dependent.

Attenuation

Attenuation is the fractional loss in plane wave intensity per unit distance travelled.

$$P(x + \Delta x) = P(x) e^{- \alpha \Delta x}$$
  • $P(x + \Delta x)$ - pressure after propagating $\Delta x$ distance
  • $\alpha$ - attenuation coefficient

Attenuation

Two causes of attenuation

  1. scattering - some of the wave is scattered as it propagates. ~10% of the attenuation coefficient.
  2. absorption - the wave energy is converted to thermal energy. ~90% of the attenuation coefficient.

Attenuation

The attenuation coefficient is of expressed in units of Nepers per centimeter per Megahertz (Np / cm / MHz) or Decibels per centimeter per Megahertz (dB / cm / MHz) because an approximate linear dependency with frequency is observed for soft tissues.

Medical Image Formation

Ultrasound Image

Transducer

The transducer is usually a piezoelectric crystal that both transmits and receives the pressure waves.

Transducer

Excitation of the crystal with electricity causes it to vibrate. When pressure wave return to the crystal, its movement creates the electrical signal that is recorded.

Transducer

The transducer has a natural center frequency and bandwidth that result from geometry and mechanical material properties.

Pulse-echo

The transducer is excited with a short-time pulse.

After transmitting, the transducer is immediately put into receive mode to listen for backscattered waves as the pulse propagates.

This is called pulse-echo imaging.

Pulse-echo

The depth of the received echo signal is determined with the range equation $$d = \frac{c t}{2}$$

  • $d$ - depth
  • $t$ - pulse-echo transmit time
  • $c$ - speed of sound, assumed constant 1540 m/s

Beam

The insonified region in roughly confined to a column or beam.

Beam

Directions are defined relative to pulse propagation.

  1. axial - axis of the beam
  2. lateral - orthogonal to the axial direction, in the imaging plane
  3. elevational - orthogonal to the axial direction, out of the imaging plane

Beam

Highly anisotropic resolution.

axial >> lateral > elevational

Time Gain Compensation (TGC)

Beam intensity variations and attenuation cause average signal strength to vary with depth.

A variable gain is applied with depth - time gain compensation (TGC).

Transducer Types

Single element

Array

Single element

Pulse-echo lines are collected as it is mechanically rotated to form an image.

Array

Multiple elements are arranged in a line. By varying the intensity and timing of the element excitation, the beam location and intensity is controlled.

Array

The element yellow dot position is the equivalent effect of pulse time delays.

RF Data

The signal from a single element transducer or the summed signal from all the elements of an array transducer is called the raw RF (radio-frequency) data.

B-mode

The envelope of the RF data forms the B-mode (brightness-mode) intensity image.

Ultrasound Tissue Characterization

B-mode Content

The B-mode intensity image is a mix of
  • Transducer insonification characteristics
  • Tissue acoustic characteristics
  • Plentiful artifacts

Ultrasound Tissue Characterization

Ultrasound tissue characterization techniques attempt to reduce insonification characteristics and present tissue properties in the image.

Parametric Images

Ultrasound tissue characterization images are parametric images derived from acoustic properties, such as
  • Speed of sound
  • Attenuation coefficient
  • Parameters related to the frequency-dependent backscatter coefficient

Backscatter coefficient estimation

Normalize the local speckle spectrum by the RF data from a perfect planar reflector or uniform tissue-mimicking phantom to remove transducer properties.

Spectral characteristics

Fix parametric models of scatterer size, e.g., or measure slope, intercept, integral to normalized spectrum.