Principles of medical ultrasound

Sound = longitudenal compression/refraction wave that travels in matter. Sound waves can be

• Reflected: at area of acoustic mismatch (boundary between two different speeds of sound). Amplitude of reflected wave \(\uparrow\) with \(\uparrow\) mismatch and \(\uparrow\) angle of incidence (AoI).
• Refracted: at area of acoustic mistmatch when AoI\(\neq\)0
• Scattered: by small acoustic interfaces
• Attenuated: Exponential decease in amplitude by absorption, scattering, and reflection.

Imaging principles

Piezoelectric crystals will \(\Delta \text{shape} \leftrightharpoons \Delta \text{current}\), and therefore applying AC current generates a sound wave, and absorbing sound waves generates AC current \(to\) form ultrasound transducers.

Transducer generates wavepacket \(\to\) listens for reflected wavepacket.

Fourier analysis decomposes reflected packet into contributory waves \(\to\) allows reconstruction of acoustic density in a 1D line.

2D array of crystals \(\to\) 2D image.

Higher wavepacket frequency \(\to\) better axial resolution, worse penetration (more attenuated)

Image depth \(\propto \frac{1}{\text{temporal resolution}}\) because more time taken to acquire each frame

Doppler principles

Reflection off a moving object in same direction as wavepacket changes frequency of wavepacket $$\vec{V} = \frac{c \cdot (f_{\text{reflected}} - f_{\text{transmitted}})}{2 f_{\text{transmitted}} \cos{\theta}}$$ where \(\theta\) is the angle between the object velocity and the wave.

• Lower wavepacket frequencies \(\to\) less aliasing
• Beam parallel with motion \(\to\) accurate \(\vec{v}\) estimate
• Displayed atop 2D image \(\to\) colour doppler
• Continuous-wave doppler - samples higher frequencies, but includes all velocities in the beam direction
• Pulsed-wave doppler - smaller frequency band, selective for only motion in a small gate