Medical diagnostic systems (Orvosbiológiai képalkotó rendszerek) Fundamental concepts in acoustics (Alapfogalmak az akusztikában) Miklós Gyöngy • Consider 3 methods of acoustic localisation • Through these, learn about concepts in acoustics -propagation of sound -diffraction -reflection -scattering -attenuation • Link back to diagnostic ultrasound throughout Methods of sound localisation 1. Lightning localisation passive (light as reference) 2. Binaural hearing passive (difference in arrival times) 3. Echolocation active (pulse-echo) • Passive method (with light as reference) • Time of arrival (ToA), speed of sound (SoS) › localisation • Analogy with diagnostic ultrasound? • Where does speed of sound come from? • What about propagation in tissue? Analogy with diagnostic ultrasound: localising “flashes of lightning” – photoacoustics • Transmit laser pulse at known time • Optically “dark” tissue absorbs laser preferentially • Localised heating due to laser pulse creates shock wave • Time of arrival depends on location of emission site http://www.ucl.ac.uk/cabi/Photoacustics/Photoacustics.html 5 Propagation of sound • Mechanical vibrations cause travelling waves • Wave can be sustained by normal stress, shear stress, and volumetric compressions • E›. or .›0: c›. (block moves as one) elastic modulus propagation speed C c = .0 undisturbed density 6 Types of elastic moduli • Young’s (E): axial propagation along laterally unconstrained rod • P-wave (M): longitudal propagation, no lateral motion • shear (G): motion transverse to direction of propagation • bulk (K): volumetric propagation (pressure waves) • K=M-4G/3: without shear, equivalent to P-wave • K = -V .p/.V : inverse of compressibility . 7 C axial propagation c = . • C = E (Young’s Modulus) • E = ./. (stress/strain) • cst.steel = .(216×109/7800) . 5300 m/s • E›. or .›0: c›. (block moves as one) 8 C c = . • .= .= 0 xx xx • no diffraction (high frequency) • C = M (P-wave modulus) • M= .zz/.zz (stress/strain) • 5980 m/s cst.steel = 9 C transverse propagation c = . • C = G (shear modulus) • G = ./. (shear stress/shear strain) • cst.steel = .(84×109/7800) . 3300 m/s 10 C c = . volumetric propagation • C = K=1/. (bulk modulus=1/compressibility) • K = -V .p/.V • K = M-4G/3 • : 2.05 GPa at 1 atm • 3.88 GPa at 300 atm Kwater • cwater =.(K/.) . .(2e6) . 1400 m/s 11 Propagation of pressure waves Assuming small pressure and density fluctuations P = p0+ p where p<

4000 Hz: high ., low diffraction: interaural level difference • Passive method without temporal reference • Time differences of arrival (TDoA) in diagnostic ultrasound? • What are diffraction and attenuation? • Role of attenuation and diffraction in diagnostic ultrasond? TDoA in medical ultrasound: Tracking popping bubbles – passive cavitation mapping • Cavitation (bubble activity) often involved in ultrasound therapies • Cavitation may occur at any time (no temporal reference) • Time differences of arrival of shockwaves allows localisation of bubbles f<2000 Hz f>4000 Hz • Point source spreads spherically • Set of point sources interfere with each other • Continuous source region diffracts -analogous to interference of infinite point sources • Level of diffraction decreases with frequency aperture D longitudal axis Diffraction • Consider single frequency f • Pressure field p(r) expressed as complex scalar field of phasors • Small distances |r| (near-field): p(r) = complex interference pattern • Large distances |r|(far-field): |p| = H(.)/|r| • Transition on longitudal axis at |r|=D2f/4c [Olympus 2006] • Transition depends on aperture D as well as frequency! Diffraction in diagnostic ultrasound • Typical abdominal 1D array: the L10-5 from Zonare medical systems • Focusing in imaging plane using acoustic lens • z=17.5 mm elevational focus • z=60–100mm: roughly constant, ~10 mm sensitivity in elevational direction • scattering object 5 mm out of imaging plane may be seen! [Gyöngy 2010] 20 Attenuation • Consider a planar wave travelling in the z-direction • Without any attenuation, the wave will maintain its amplitude: p=A g(t-z/c) • In reality, some of wave redirected in other direction (scattering) and some is converted to microscopic random motion – heat (absorption) • If attenuation is uniform over distance: p=A exp(-.z) g(t-z/c) where . is attenuation coefficient in Nepers • What if attenuation is caused by a single object? Attenuation in diagnostic ultrasound • For plane wave travelling in z-direction, attenuation coefficient . describes “weakening” of pressure with distance: p=A exp{j(kz-.t)}exp(-.z) |p|=A exp(-.z) where . is in Nepers (Np for short). • For tissue, .dB . 1 dB/cm/MHz [Brunner 2002] • Therefore, at 6 MHz -pressure amplitude halves for every cm travelled -pressure received from perfectly reflecting target 10 cm deep (consider two-way propagation)? • Exercise: show that 1 dB . 0.115 Np • What is origin of attenuation? 3. Echolocation [Au et al. 2007] • Active method: time of transmission acts as reference • Two-way travel time, speed of sound (SoS) › localization 22 • Analogy with diagnostic ultrasound? • How accurate is the localization? • How do echoes form from the fish (scattering)? Diagnostic echolocation: pulse-echo B-mode imaging • Most widespread form of diagnostic ultrasound imaging • Very simple conceptually: 1. transmit pulse along different lines 2. convert timeline of recorded echoes to distance (d=t/2c) 3. convert amplitude of echoes to brightness on a screen Transmit and Received A-line receive data envelope beamforming amplitude along thin ‘line’ (A-line) Multiple A-line envelopes create B-mode image (here, porcine liver in water imaged using Terason t2000) Medical diagnostic systems – Fundamental concepts in acoustics Localisation accuracy Determined by width of transmit pulse autocorrelation pulse trace frequency spectrum autocorrelation 1 1 1 0.5 0 sinusoid 0 -1 0 -1 0 10 20 30 40 0 0.1 0.2 0.3 0.4 0.5 -20 -10 0 10 20 0 10 20 30 40 0 0.1 0.2 0.3 0.4 0.5 -20 -10 0 10 20 frequency time lag 25 Localisation accuracy .t .f .1; 2.355×0.375=0.883 #oscillations . f0.t . f0/.f = Q (=1/0.375=2.667) Approximation better for Q>>1 (underdamping) 1 1 0.5 0.75 0 0.5 -0.5 0.25 -1 0 -5 0 5 00.5 11.5 2 time frequency 26 Scattering • Caused by inhomogeneities of the medium (variations in compressibility . and density .) • Total pressure field modelled as sum of incident and scattered field: p(r,t) = pi (r,t) + ps (r,t) • Hence, scattering creates Surface wave scattered in bath tub by 27 mm object “virtual sources” Regimes of echo formation (scattering): sub-wavelength scattering resonant scattering reflective scattering “diffusive” “diffractive” “specular” (speculum, mirror) . ka<<1 ka ~ 1 ka>>1 • k =2./.: angular wavenumber • a: characteristic size of scatterer (for sphere, equals radius) • ka: number (dimensionless): characterises scattering behaviour • reflection a limiting case of scattering Sub-wavelength scattering (ka << 1) [Lighthill 2001] • Changes in compressibility . and density . has different effects: -.. causes angle-independent (monopolar) scattering -.. causes dipolar scattering equivalent to two opposing monopoles .-. 3(. -. ) 3 s 0 s 0 ps (r,t).+ cos .= {fixed, incompressible}-1+ cos . . .+ 2. 2 00 s -.: direction relative to direction of propagation monopolar scattering dipolar scattering dipole ~ . volumetric changes . momentum changes 2 anti-phase mpoles • Amplitude of scattered pressure increases with k and a -how to quantify “scattering ability” of object? Resonant scattering (ka ~ 1) [Lighthill 2001] • Incident pressure varies over object • Interference between scattering wavefronts at different locations causes complicated scattered field – backscattered wavefronts from front and back of scatterer in phase ›resonance • Mode conversion at boundary (pressure wave - shear wave) also causes resonance peaks • By definition, in far-field of scatterer, pressure amplitude varies reciprocally with distance for constant angle: H (.) ps (r,t) = | r | Reflective scattering (ka >> 1) [Lighthill 2001] • Scatterer very large: meetings of pressure wave with object boundary independent of each other (no phase information). (In reality, if transmitted pulse is long enough and attenuation does not extinguish a wave before it hits a new boundary, standing waves will be set up) • At each boundary, mismatch in characteristic acoustic impedance (=.c) creates reflection (as well as refraction) • Laws of geometric acoustics used for ray tracing (cf. optics) • Rays describe direction of high-frequency acoustic beams that undergo negligible diffraction or interference reflected ray Fish as (resonant) scatterers .=3 cm a.10 cm ka.20 [Ye and Farmer 1996] Water Swimbladder Fish Mass density (kg m3) 1026 1.24 1560 Bulk modulus (MPa) 2200 0.15 2600 Echolocation of airborne objects • Air-water boundary creates great impedance mismatch • Most sound is reflected from boundary How does a fish school scatter? • Multiple scattering inside fish school: diffusion of sound • School fish as bulk inhomogeneous material: reflection • As fish (parts) made smaller -diffusion (causing attenuation) decreases (eventually) -fish school becomes homogeneous medium Acoustic concepts covered so far... and their relevance to diagnostic ultrasound • propagation of sound: .1540 m/s in soft tissue • diffraction: focussing of mm-thick beams • reflection and refraction: organ boundaries • scattering: cells, collagen, elastin • attenuation: .1 dB/cm/MHz Let us review these concepts again... and provide some additional notes Propagation of pressure waves [Coussios 2005] • Derivation of wave equation from the governing equations of acoustics: Eqn. of state (pressure function of density): P(R) Continuity eqn.: (mass rate of change in dV = flux in/out dV): .R/.t =-.·(Rv) Momentum eqn. (Newton’s second law of motion): -.P = . .v/.t • Assuming small pressure and density fluctuations P = p0+ p where p<1), both effects cause an increase of c with p: 1. Medium non-linearity: medium less dense than expected 2. Convective non-linearity: particle with forward motion carries pressure quicker c = c0+ ßv=c0 + (1+B/2A)v where ß is the coefficient of non-linearity (water:5.0 blood:6.3 liver:7.8 pig fat:11.1*) • Pressure dependent wave speed causes distortion of waveform with distance • As a result, waveform accumulates harmonics as it travels (think of a loudspeaker placed in castor oil... what would happen as you increased the frequency?) original waveform shocked waveform why does this not happen? Non-linear processes in diagnostic ultrasound • Non-linear propagation of ultrasound introduces harmonics into the wave as it propagates towards reflector/scatterer and back towards array, the degree of non­linear propagation being highest at the highest amplitude (focus) • Pulse-echo imaging of such harmonics is called tissue harmonic imaging • Air bubbles are highly non-linear scatterers, scattering sound at harmonics of the incident wave (for high enough amplitudes, they will scatter sound at the subharmonics, ultraharmonics and even in the broadband frequency range [Neppiras 1980]) • By introducing stabilised bubbles (ultrasound contrast agents) into bloodstream, perfusion can be imaged (contrast agent imaging) • Harmonics can be recovered in several ways: • send one pulse and extract harmonic component of echo • send two pulses, one inverse of other, and consider difference between two echoes (pulse inversion) Diffraction • Huygen’s principle: each point of non-zero pressure field (such as wavefront) is itself a superposition of point sources soft baffle direction of transducer propagation • But: consider a single planar source. As it spreads in two directions, the source won’t keep splitting in two! • Modified Huygen’s principle: point sources have directivity given by obliquity factor (maximum at propagation direction) • Application to ultrasound transducers: pressure field result of sum of (directional) point sources across transducer surface Reflection and refraction • Reflection and refraction governed by change in characteristic acoustic impedance Z=.c across boundary. • Ratio of pressure reflected: (Z2-Z1)/(Z1+Z2) • Z has units of Rayls • For planar waves, p/|v|=Z, where v is velocity field [Kaye&Laby] Air Water Blood Bone Z (MRayl) 4e-4 1.5 1.1 3.5–4.6 • Over 99.9% of pressure is reflected at air-water boundary! • Refraction governed by Snell’s law: sin.1/sin.2 = c1/c2 Attenuation in simple conceptual terms • Ordered vibrations of a wave gradually • re-transmitted in other directions (scattering) • turned into unordered, random mechanical (i.e. thermal) fluctuations (absorption) • Simple model of wave propagation: particles held together by springs • Wave propagation due to reaction force of springs and inertia of particles • Scattering caused by variations in particle mass and spring stiffness • Absorption: addition (series or parallel) of dashpots to springs [Gao et al. 1996] scattering by stiffer springs scattering by larger mass Maxwell model Voigt model Attenuation in tissue [Sehgal and Greenleaf 1984] • Scattering from density and compressibility changes (cf. mass-spring model) • Classical thermoviscous model: absorption arises from phase difference between p, . [Lighthill 2001 pp. 78-79] p=c2. + .../.t; leading to ../.t–c-2 ../.t+ .c-2.3./(.z2.t)=0 • Such phase difference may arise from [Cobbold 2007, pp. 84-86] • heat conduction • viscosity • molecular (thermal and structural) relaxation • Scattering: diffuse to diffractive single particules (ka.1) .s ~ f 2-4 predicted • Absorption: thermoviscous model predicts .a ~ f 2 (sim. to Kelvin-Voigt model) In contrast, .s, .a both ~ f 1.1-1.2 in tissue! Modify models: • .s : spatial auto-correlation for ..,.. [Sehgal and Greenleaf 1984] • .a: [Szabo 2004, pp. 77-83]; large mass-spring-dashpot arrangements [Gao et al. 1996] Attenuation by single objects [Cobbold 2007, pp. 270-271] • Consider intensity I plane wave impinging on object with cross-section (c.s.) A • If object removes all incident intensity (“full attenuator”) , Premoved = IA • Object with c.s. A removes e.g. half of I acts like full attenuator of c.s. A/2 • Define acoustic c.s. as equivalent c.s. of full attenuator • Total acoustic c.s. (area) sum of attenuation c.s. and scattering c.s. . = .+ . a s = Premoved/ I • Differential scattering c.s.(area/solid angle) P(.)/I (unlike attenuation, scattering .-dependent) .ds(.) =s • Differential backscattering c.s. (area /solid angle) .dbs = .ds(.=[. 0]) (arises in pulse-echo ultrasonics) • Backscattering coefficient (area/solid angle/volume) [Cobbold 2007, p. 308] .BSC = .ds(.=[. 0])/V (gives “density” of scattering) [Au et al. 2007] Modeling the detection range of fish by echolating bottlenoise dolphins and porpoises [Brunner 2002] Ultrasound system considerations and their impact on front-end components [Cobbold 2007] Foundations of biomedical ultrasound [Coussios 2005] Biomedical ultrasonics lecture notes [Gao et al. 1996] Imaging of the elastic properties of tissue – a review [Gyöngy 2010] Passive cavitation mapping for monitoring ultrasound therapy [Hill et al. 2004] Physical principles of medical ultrasonics [Kaye and Laby] Tables of physical and chemical constants. http://www.kayelaby.npl.co.uk/ [Lighthill 2001] Waves in fluids [McLaughlin and Renzi 2006] Shear wave speed recovery in transient elastography and supersonic imaging using propagating fronts ... ... [Neppiras 1980] Acoustic cavitation [Olympus 2006] Ultrasonic transducers technical notes. http://www.olympus­ ims.com/data/File/panametrics/UT-technotes.en.pdf [Sehgal and Greenleaf] Scattering of ultrasound by tissues [Sekuler and Blake 1994] Észlelés [Ye and Farmer 1996] Acoustic scattering by fish in the forward direction [Wells 1999] Ultrasonic imaging of the human body