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Physics of UltrasoundImagingYao WangPolytechnic University, Brooklyn, NY 11201Based on J. L. Prince and J. M. Links, Medical Imaging Signals andSystems, and lecture notes by Prince. Figures are from the textbookexcept otherwise noted.

Lecture Outline Ultrasound imaging overview General characterization of acoustic wave Wave equation– 3D– Plane wave– Spherical wave Reflection of waveAbsorption and scattering of waveDoppler effectField pattern of a transducerEL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn2

Ultrasound Imaging Measure the reflectivity of tissue to sound wavesCan also measure velocity of moving objects, e.g. blood flow (Dopplerimaging)No radiation exposure, completely non-invasive and safeFastInexpensiveLow resolutionMedical applications: imaging fetus, heart, and many othersEL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn3

What is Acoustic Wave Pressure waves that propagate through matter viacompression and expansion o the material– Generated by compressing and releasing a small volume oftissue Longitudinal wave– Particles in the medium move back and force in the samedirection that the wave is traveling Shear Wave– Particles move at right angles to the direction of the wave– Not used for medical ultrasound imagingEL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn4

Longitudinal WaveFrom Graber: Lecture note for BMI F05From Prince, Lecture noteEL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn5

EM vs Acoustic Wave Electromagnetic– Self propagating, consisting of electric and magnetic componentsoscillating at right angles to each other, and to the direction ofpropagation– Does not require a material medium through which to propagate– Classification (increasing in frequency, decreasing in wavelength): radio, microwave, infrared, visible light, ultraviolet, x-ray, gamma ray Acoustic– Pressure waves that propagate through matter via compression andexpansion of the material– Requires a material medium through which to propagate– Classification (increasing in frequency): Infra sound, audible sound, ultrasoundEL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn6

Transfer and Transformation of Energy Light becomes sound — photoacoustic phenomena Sound becomes light — sonoluminescence Absorbed electromagnetic (EM) and acoustic energyboth become heat Nevertheless, EM and acoustic energy areFUNDAMENTALLY DISTINCT PHENOMENA!EL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn7

Acoustic Wave Energy RangesInfrasoundAudible20 HzUltrasound20 kHz Just as there are infrared, visible, and ultraviolet ranges in the EMspectrum, so there are infrasound (“infra” “below,” “beneath”),audible (i.e., sound) and ultrasound (“ultra” “beyond,” “above”)ranges of acoustic wave frequencies Note that the ratio of the highest to the lowest audible frequencies is103, or almost 10 octaves. On the other hand, the ratio of thehighest to the lowest frequencies of visible light is a bit less than 2(i.e., less than one octave).EL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn8

Characterization of Acoustic Wave Speed of sound in a medium depends on the medium propertyc 1κρ ; κ :compressibility; ρ : density Air: 330 m/s; water, c 1480; most tissue: 1500 m/s; bone: 4080 m/s Particle displacement velocity (v)– Note that particle speed v is different from sound speed c! Acoustic pressure (p):p Zv– Analogy: p: voltage, v: current, Z: impedance ρCharacteristic impedance of a medium: Z ρc κ– Unit: kg/m 2s or rayls (after Lord Rayleigh)EL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn9

Acoustic energy and intensity Particles in motion have kinetic energy; those poised for motionhas potential energy1 2wk ρv Kinetic energy density:2 Potential energy density: Acoustic energy density:1w p κp 22w wk w p Acoustic intensity (acoustic energy flux):p2I pv Z– Analogy: I (power), p (voltage), v (current)EL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn10

Acoustic Properties of Common MaterialNotice how similarthese values areto each other andto that for water,metalgasacrylicand how differentthey are fromthese.soft tissueshard tissueFrom [Graber: Lecture note for BMI F95]Table 10.1 in [Prince] gives more information, includingdensity and absorption coefficientEL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn11

3D Wave EquationEL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn12

Plane WaveForward traveling waveBackward traveling waveOne of them may be zeroEL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn13

Example An ultrasound transducer ispointing down the z axis.Starting at time t 0, itgenerates an acoustic pulsewith formφ (t ) (1 e t /τ )e t /τ12 Assume the speed of thesound in the tissue is c 1540m/s. What is the forwardtraveling wave down the zaxis? At what time does theleading edge of the impulse hitthe interface 10 cm away fromthe transducer?EL5823 Ultrasound Physicsφ f ( z, t ) φ (t z / c) (1 e (t z / c ) /τ 1 )e ( t z / c /τ 2T 0.1m / 1540 64.9 µsYao Wang, Polytechnic U., Brooklyn14

Harmonic Waves Harmonic plane wave:p( z , t ) cos(k ( z ct )) Viewed at a fixed particle, the pressure changes in time with frequencyft kc/2π (cycles/s)Viewed at a fixed time, the pressure changes in z with frequency fz k/2π– k is called wavenumber Wavelength is the spacing between peak or valleys of the wave at anytime– λ 1/fz 2π/k c/ft (approximately) Harmonic wave are widely used in ultrasound imagingGiven ft, the wavelength depends c, which in turn depends on the tissueproperty!– Wavelength determines the resolution of ultra sound imaging– Ex: ft 3.5 MHz, c 1540m/s (most tissue), λ 0.44mmEL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn15

Spherical WaveEL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn16

Reflection and Refraction: GeometricCharacteristicsEL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn17

Reflection and Refraction: EnergyCharacteristics Particle motion conservation– Tangential particle motion caused by the incident wave sum of particlemotions of transmitted and reflected wavesvi cos θ i vr cos θ r vt cos θ t Pressure conservationpt pr pi Based on above equations, and the relation p Zv, one can deriverelation of pr and pt with pi, and consequently pressure reflectivityand pressure transmitivity defined in the next slideEL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn18

Reflected and Refracted WaveEL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn19

Example Layered medium, determine total reflected signal frommultiple layersZ1pt3pr1 piZ2pt1Z3pt2EL5823 Ultrasound PhysicsTotal reflected signal pr1 pt3pr2Yao Wang, Polytechnic U., Brooklyn20

Energy Attenuation due to Absorptionand Scattering The pressure of an acoustic wave decreases as thewave propagates due to– Absorption The wave energy is converted to thermal energy Two forms– Classical: due to frictions between particles as the wave propagates– Relaxation: due to particle motion to return (relax) to original positionafter displacement by the wave pressure– Scattering When the sound wave hits an object much larger than itswavelength, reflection occurs. When the object size wavelength, scattering occur (reflection inall directions) Note the difference between reflection and scattering!EL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn21

Overall Attenuation Absorption and scattering together causes the pressure and intensity of asound wave to decrease exponentially in the propagation distance zSuppose p(0, t ) A0 f (t )No attenuation : p ( z , t ) A0 f (t c 1 z )With attenuation : p( z , t ) A0 e µ a z f (t c 1 z )µ a : Amplitude attenuation factor [cm -1 ]Attenuation coefficient in dB :α 20(log10 e )µ a 8.7 µ a [dB/cm] The attenuation coefficient depends on the frequency of the wave,bgenerally α afRough approximation (1 MHz f 10MHz): b 1, α afSee Table 10.2 in textbook for “a” value for biological tissues(much larger for bone and lung)EL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn22

Example Same layered medium as before, but considering theattenuationZ1L2pt3pr1 piZ2,µ2pt1Z3pt2EL5823 Ultrasound PhysicsTotal reflected signal pr1 pt3pr2Yao Wang, Polytechnic U., Brooklyn23

Scattered Signal When the sound wave hits an object much smaller than sound wavelength,scattering occur (reflection in all directions), giving rise to spherical wavesConsider a plane wave traveling in z, hitting a small target at z d Source wavep ( z , t ) A0 f (t c 1 z )EL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn24

Doppler Effect: Moving source Doppler effect: change in frequency of sound due to the relativemotion of the source and receiver Case 1: moving source (scatterer), stationary receiver (transducer)– Source moving away, wavelength longer, lower freq– Source moving closer, wavelength shorter, higher freq.source (freq f o )moving with speed v opposit the wave direction :one period T 1/f ocrest in wave moves a distance of cT c/f o without source motionsource moves a distance of vT v/f oWith source motion, crest moves a distance (equivalenth wavelength λT ) of cT vTEquivalent temporal frequency isfT cλT cfoc vGeneral case :source moving in a direction θ :(angle between source- receiver vector and source motion vector)cfOc v cos θDoppler frequency :fT v cos θv cos θfO fOc v cos θcθ 0 : source moving away from receiver, f D 0f D fT f O θ 0 : source moving towards receiver, f D 0EL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn25

Doppler Effect: Moving Receiver Case 2: stationary source (transducer), moving receiver (target)––––Transducer transmitting a wave at freq fs, wavelength c/fsObject is a moving receiver with speed v, with angle θTarget moving away (θ 0), sound moves slowerTarget moving closer (θ 0), sound moves fasterc v cos θfScDoppler frequency :v cos θf D fO f S fScθ 0 : source moving away from receiver, f D 0fO θ 0 : source moving towards receiver, f D 0EL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn26

Doppler Effect for Transducer Transducer:–––––Transmit wave at freq fs to objectObject moves with velocity v at angle θObject receives a wave with freq fo (c vcosq)/c fsThe object (scatterer) reflects this wave (acting as a moving source)Transducer receives this wave with freqcc v cos θfT fO fSc v cos θc v cos θ– Doppler freq 2v cos θ2v cos θf D fT f S fS fSc v cos θcDopper-shift velocimeter: as long as θ not eq 90 0, can recover objectspeed from doppler freq.Doppler imaging: display fD in space and time–EL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn27

Schematic of an Ultrasound ImagingSystemEL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn28

Functions of transducer Used both as Transmitter And Receiver Transmission mode: converts an oscillating voltage intomechanical vibrations, which causes a series of pressurewaves into the body Receiving mode: converts backscattered pressurewaves into electrical signalsEL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn29

Single Crystal Transducer (Probe)(damping)EL5823 Ultrasound PhysicsYao Wang, Polytechnic U., Brooklyn30