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Overview of Ultrasound Physics & Interactions

Overview of Ultrasound Physics & Interactions

R. Jason Stafford, Ph.D.Department of Imaging Physics

The University of Texas MD Anderson Cancer Center

R. Jason Stafford, Ph.D.Department of Imaging Physics

The University of Texas MD Anderson Cancer Center

Medical Physics I - Fall 2014Medical Physics I - Fall 2014

Email: jstafford@mdanderson.org

Office: FCT14.6092

Phone: 713-563-8052

Email: jstafford@mdanderson.org

Office: FCT14.6092

Phone: 713-563-8052

Reading assignmentsReading assignments

1. Bushberg JT, et al, The Essential Physics of Medical Imaging: Ch. 14.1, 14.2, & 14.11

2. Harpen MD, Basic nonlinear acoustics: An introduction for radiological physicists, Med. Phys., 33(9):3241–3247, 2006.

• More on relaxation for those interested:– Szabo TL, Diagnostic Ultrasound Imaging: Inside Out

What is sound?What is sound?

• Propagation of mechanical disturbance through a medium

– Longitudinal/shear wave propagates energy, not particles

– elastic properties of medium determines aspects propagation

– tissue is viscoelastic with various non-linear propagation effects

compression

‘rarefaction’

high pressure

low pressure

longitudinal shear pressure pulse

wavelength () = distance between wave crests ~ 1.54mm @ 1MHz in tissuefrequency (f) = oscillations per second ~ 1-15 MHz for medical USvelocity (c=*f) = rate of wave propagation ~ 1540 m/s in soft-tissuedisplacement () = degree of particle motion ~ 5.7 nmpressure (p) = excess pressure created ~ 100 kPaintensity (I) = power density ~ 100-700 mW/cm2

attenuation () = decrease in beam intensity with depth ~ 0.5 dB/cm/MHz

spatial pulse “length” ~ 2

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Mechanical properties: elasticityMechanical properties: elasticity

• Elastic– returns to original size/shape after applied forces/torques

removed w/o dissipation of energy

• Basis for theory of elasticity (Hooke’s Law generalized):Stress = (Strain) X (Modulus)

• Stress = applied force per unit area (pressure)

• Strain = fractional change (x/x0 or V/V0) in conformation

• Modulus = A constant of proportionality (i.e., K)

Aside: stress & strain are tensorsAside: stress & strain are tensors

Stress Tensor:

Strain Tensor:

xx xy xz

ij yx yy yz

zx zy zz

xx xy xz

ij yx yy yz

zx zy zz

Cauchy Stress Tensor

• Constituitive equation for elastic material: ij = sijklkl

• Modulus tensor (s) represents physical properties of the material

1 1

2 2

3 3

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1

1E

Linear Isotropic material => & principle axes coincideLame’ constants: relationship between physical parameters in sE = Young’s modulus; G=Shear modulus; K = Bulk modulus; = Poisson’s ratio = contraction/extension (loosely)~ ½(E/G) – 1 ~ ½ - E/6K

Review+Applications: Hall TJ, RadioGraphics 2003; 23:1657–1671

Keeping it simple:bulk modulusKeeping it simple:bulk modulus

• Volume elasticity – assume linear change in volume for P– strain quantified as V/V0

• Volume or adiabatic bulk modulus (K) relation:P = -K·V/V0

• As P 0:

• Note– K => material more stiff (i.e., metal or bone)– K => material more malleable (i.e., adipose)

0 0

dP dPK V

dV d(continuity or conservation of mass)

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1D linear acoustic wave equation1D linear acoustic wave equation

• Propagation due to local “neighbor-to-neighbor” interactions (transfer of momentum and kinetic energy)

– For a range of clinically relevant frequencies and power densities, can be modeled as if in water

• Assume perfectly elastic medium– i.e., linear, lossless and no dispersion!

– direct proportion between applied acoustic stress and particle displacement

• Assume no shear component of the modulus– “water like“

• Assume homogeneous medium– scalar modulus

Now, go get out your spherical chickens …

• Consider instantaneous acceleration (a) at point x+x/2:

1D linear acoustic wave equation1D linear acoustic wave equation

x-x x x+x x+2x

(x-x) (x) (x+x) (x+2x)

P1 P3A

m=A0x

(x,t) = instantaneous density 0 = undisturbed equilibrium density of the medium

P(x,t) = instantaneous pressure P0 = undisturbed equilibrium pressure of the medium

p = P-P0 = instantaneous excess pressure (x,t) = displacement of point at x along x axis

2

2 2

x x xa

t

‘particle position’

‘particle displacement’

1D linear acoustic wave equation1D linear acoustic wave equation

• Use Newton’s 2nd to express in terms of applied stress so we can relate to the bulk modulus:

• Note– P1 & P3 : change in excess pressure in volume elements for [x:x-x]

and [x+x:x+2x] (respectively) due to deformation

– K: adiabatic bulk modulus

2

1 32

1 3

2

2

2

2

2

2

x x xF M a m A P P

t

x x x x x x xwith P K and P K

x x

x x x x x x x x x xm A K

t x x

4

1D linear acoustic wave equation1D linear acoustic wave equation

• This can be simplified to:

• Taking the limit of x 0 (implying j 0) gives:

• Solutions are of the form:

• Displacement (), particle instantaneous velocity (u or v), pressure (p) and density () obey same form of wave equation

– Remember Newton’s 2nd (Linear Force Equation):

• Interesting exercise– Derive 1D wave equation using bulk modulus and continuity (cons. mass)

equation:

2

0 1 22

( ) ( )

2 x x x

x x x x xx K

t x x

2 2 22

2 2 20 0 0

x x xK K dPc with c

t x x d

( , ) right leftp x t f kx t f kx t

0 0p u

x t

0u

x t

Acoustic properties of tissueAcoustic properties of tissue

(kg/m3)

K (Pa x 109)

c (m/s)

Fat 928 1.96 1450 Water 1000 2.25 1500 Soft-Tissue 1050 2.49 1540 Tendon 1110 3.40 1750 Bone 1990 20.4 3200

Wavelength @ 1MHz 1540/1e+6 = 1.54 mm

Typical US wavelengths are then 1.54 mm - .154 mm

Wave equation: general solutionsWave equation: general solutions

Table 1.1 – Hill-Bamber-ter Haar – Physical Principles of Ultrasound (2006)

((r,t)/0)

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Quick note on dispersion ...Quick note on dispersion ...

• Dispersionless system: c = /k = constant

• In this case, group velocity = d/dk = c

– note waves in absorptive mediums, nonlinear waves,

non-planar waves in waveguides, etc. will have (k)

• group velocity can differ from the phase velocity

Phase velocity:

Group velocity:

Acoustic energyAcoustic energy

• Kinetic Energy (KE) = 1/2 mv2

• Potential Energy (PE) =

• For our volume element:

21 2

dKE dV u

21 1

2 2

bb b

aa a

F d k d k F

2

2

1 1 1 1

2 2 2 2

d p pdPE p A d p A dx p dV dV

dx K c

(recalling: p = -K·(/x)

Acoustic intensityAcoustic intensity

• Total energy per unit volume (energy density):

• Acoustic intensity (I) = average rate of energy flow through a unit area normal to propagation:

• Plane wave: u = u0*exp[-i(kx-t)] and p = cu:

22

2

1

2 ( )

dE pu

dV c

22

20

1

2 ( )

c pI u dt

c

222 2 0

0 020

1 1

2 ( ) 2 2

pc pI u dt cu

c c 2

0 0

1

2

dEu

dV

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Acoustic intensity (plane waves)Acoustic intensity (plane waves)• Higher pressures (p) give larger displacements

– intensity a measure of energy in beam

– higher wave “intensity” (I) => I ~ p2

– decibel (dB) in measurement of relative intensity

• dB = 10 x log(I/I0) = 20 x log(p/p0)

– Attenuation => -dB

– Amplification => +dB

– “Full width half maximum” (I/I0 = 0.5) ~ -3 dB (i.e., ~ HVL)

– Beam widths often quoted as -3dB or -6dB

Acoustic intensity measurementsAcoustic intensity measurements

(see Bushberg 14.11 for details)

FDA guidance for vendors

ISATA= ISA averaged over the PRP at the transducer face

ISAPA= ISATA/duty cycle= spatially averaged intensity averaged over pulse duration

– duty cycle = PRP fraction transducer transmits over

= pulse duration/PRP

ISATP = temporal peak of spatially averaged intensity

ISPTA = ISATA x ISP/ISA= peak beam intensity (ISP) averaged over PRP– good metric for estimating worst case heating (FDA: derated ISPTA<720 W/cm2)

ISPPA = ISATA x (ISP/ISA)/duty cycle = maximum intensity in the beam (ISP) averaged over the pulse duration

– for pulses of non-constant amplitude (FDA: <190 W/cm2)

– this or mechanical index (MI) more appropriate for mechanical effects

Acoustic impedance (plane waves)Acoustic impedance (plane waves)

• Acoustic analogy with electricity– Voltage => Pressure (force)

– Current => Velocity (flow)

• Specific acoustic impedence (Z) ~ force/flow

• Z = p/u (units = Rayl = Pa*s/m = kg/m2/s)

• For plane wave:

Z = c

0 0Z = c= K

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Summary of U/S tissue interactionsSummary of U/S tissue interactions

• Specular (mirror) reflection– fraction of incident beam intensity directed

back from interface with dimensions >

• Refraction– fraction of incident beam transmitted,

medium change alters propagation angle w.r.t normal in new medium

• Diffuse Scattering– low amplitude diffusive scattering from

particles – can have incoherent & coherent diffuse

scatter– backscattered echoes provide valuable

diagnostic info

• Absorption– deposition of US energy into the

propagation medium that is converted into other forms of energy (e.g., heat)

• Attenuation– quantification of US beam intensity decay

with depth

Specular ReflectionSpecular Reflection

• Specular = “mirror-like” (Latin)– Occurs at interfaces with dimensions greater than the

wavelength of the sound beam

• Can reflect a high percentage of the ultrasound beam back to the transducer – air / tissue => diaphragm / pericardium

– soft-tissue / bone interfaces

Propagation across a boundaryPropagation across a boundary

• In order to understand specular reflection, let’s investigate the propagation of a simple 1-D plane wave across a boundary (Medium 1 => Medium 2)

• Plane wave expressions for the pressure:

exp[ ( )] ( )

exp[ ( )] ( )

exp[ ( )] ( )

i i i

r r r

t t t

p A i t k z incident wave

p A i t k z reflected wave

p A i t k z transmitted wave

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Propagation across a boundaryPropagation across a boundary

• We know ki = kr= k1 because both in Medium 1 (ratio of c1/f1does not change)

• Pressure reflection/transmission coefficients:

• Boundary conditions– (1) pressures equal at boundary:

– (2) velocities normal to interface equal:

0

0

( .)

( .)

r r

i iz

t t

i iz

p Ar pressure reflectioncoeff

p A

p At pressuretransmissioncoeff

p A

( 0) ( 0) ( 0)i r tp z p z p z

( 0) ( 0) ( 0)i r tu z u z u z

Propagation across a boundaryPropagation across a boundary

• Assume plane waves separable (i.e., harmonic)

• Recall linear force relation:

to obtain particle velocity expression

– n=normal component along propagation

1

( , ) exp[ ( )]n nn n

n n n

p Au z t i t k z

i z c

0 0p u

z t

Normal incidenceNormal incidence

• Boundary relations (just like optics)– Pressure conserved across boundary– Normal component of velocity

• Solve to get coefficients:

1 1 1 1 2 2

( )

( )

i r t

ti r

A A A pressure

AA Avelocity

c c c

2 2 1 1 2 1

2 2 1 1 2 1

2 2 2

2 2 1 1 2 1

( )

2 2( )

c c Z Zr reflection

c c Z Z

c Zt transmission

c c Z Z

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Propagation across a boundaryPropagation across a boundary

• NOTE: This results is for amplitudes & is NOT the expression for reflected & transmitted intensity!

• NOTE: 1+ r = t

• To get reflected (R) and transmitted (T) intensities, use definition:

2

2

2 1

2

1(planewave)

2

(reflectioncoeffient)

(transmissioncoeffient)

r

i

t

i

I cu

IR r

I

I ZT t

I Z

Reflection and Transmission Reflection and Transmission

• Reflection/transmission at a normal interface

• Often expressed logarithmically in dB as:

2

2 1

2 1

1 22

2 1

1

(reflectioncoeffient)

4(transmissioncoeffient)

R T

Z ZR

Z Z

Z ZT

Z Z

2

2( ) 10log 10log 20logi

t

i i

t t

pI pR dB

I p p

Incidence at an angle: ReflectionIncidence at an angle: Reflection

Medium 1 (1, c1) Medium 2 (2, c2< c1)

1

i

xii

r

t

1i r

i r

sin( ) = sin( )x

xr

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Incidence at an angle: RefractionIncidence at an angle: Refraction

Medium 1 (1, c1) Medium 2 (2, c2< c1)

1

2

i

x

t

r

t

i

t 2 2

i 1 1

sin( )(Snell'sLaw)

sin( )

c

c

Aside: mode conversionAside: mode conversion

• For non-normal incidence, part of the longitudinal wave undergoes mode conversion at an interface

• Conversion between modes is more prominent for interfaces with large acoustic impedance differences (i.e., tissue bone).

• Head waves => local heating

• Shear wave c ~ 0.5 clong

• Angles still obey Snell’s law:

Scatter of ultrasound wavesScatter of ultrasound waves

• Molecules and objects with dimensions < – “Rayleigh” type scattering

– diverging spherical wavefront

• Small percentage backscatters to transducer

– but provides majority of diagnostic U/S information

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Rayleigh ScatteringRayleigh Scattering

• Medium with random scatterers• Scatter varies strongly with frequency (~ f4)• Higher frequency => more scatter• Rayleigh scattering cross-section for a wave with

wavenumber (k) off a particle of radius (R):

– fractional difference in compressibility (=K-1) between particle & medium: /

– difference between densities:

• Backscattered intensity (IBS):

2242 2 34 1

9 3 4R kR

264

0 4 2

16 3, 1 cos

9 2BS

RI r I

rr=scatterer-to-observation distance=angle of scatter

ScatterScatter

• Pressure amplitude pattern of monopole radiation from a scatterer is isotropic

• Pattern from a scatterer is highly directional (nearly dipolar)

=> monopole => dipole

(most U/S radiation is mixed mode)

Faran JJ, Journal of the Acoustical Society of America, 23:405-418, 1951.

Multiple Scatterering eventsMultiple Scatterering events• Diffusive scatter

– randomly oriented scatterers• random “phase” at the transducer face from coherent/incoherent echoes (‘speckle’)

– Rayleigh distributed

• Coherent scatter– Quasi-periodic distribution of scatterers

• i.e., irregular tissue interface or collagen layers

– coherent backscatter translates into useful diagnostic information– Rician distributed

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Ultrasound ‘speckle’Ultrasound ‘speckle’

• Speckle is sometimes just considered noise with regard to low contrast detectability of objects

• If motion tracking or diagnosis based on tissue heterogeneity is not the focus, speckle “noise” can be reduced

• Speckle contains a large component of correlated noise and so simple averaging is not effective at removal

– must find means to de-correlate if need to reduce for contrast

– in imaging, temporal, spatial and frequency averaging can all be used to reduce speckle

speckle fluids

(low backscatter, homogenous)

oblique tissue boundaries

Absorption of the ultrasound beamAbsorption of the ultrasound beam

• Fraction of the ultrasound beam energy is imparted to the medium as the beam travels– absorption accounts for majority of beam attenuation

• Absorption can lead to– heating– structural changes– changes in vibrational or rotational energy states– ionic disassociations, etc

• Two physical mechanisms of absorption– viscous forces– relaxation

Absorption - viscous forcesAbsorption - viscous forces

• ‘Classic’ absorption primarily due to viscous shearing (frictional forces)

• Energy spent overcoming tissue viscosity => increase in disordered motion and therefore increased kinetic energy

• 1D linear force equation in a medium:

– pressure (force) => time varying momentum

– time varying momentum => time varying displacement– small displacements still modeled by Hooke’s law in elastic materials

• However, realistically, part of the force goes into overcoming viscous/frictional drag forces in the medium

• First order modeling via a viscosity dependent pressureterm (p) that acts to reduce the net pressure in the wave:

0

p u

z t

0

( )p p u

z t

Harpen MD, Basic nonlinear acoustics: An introduction for radiological physicists, Med. Phys., 33(9):3241–3247, 2006.

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Absorption - viscous forcesAbsorption - viscous forces

• The viscosity related pressure is estimated as:

• Shear viscosity coefficient = s

• Bulk viscosity coefficient = b

• Note term increases with increasing spatial gradients in the displacement

4

3 s b

up

z

Absorption - viscous forcesAbsorption - viscous forces

• 1D wave equation + 1st order viscous effects:

• Navier-Stokes originally developed and presented solutions of the form:

2 3 2

02 2 2

40

3 s b

p p p

z z t t

0

vi t kz zp p e

143

v

s b

2

222 0

1 1v

phase

v v

kck ic

i i

2

2

2 20 2

1

2( )

1 1

vphase

v

c creal k

1/ 22

2

0

1 1

21

vv

v

c

Define a characteristic frequency:

*The equation for a full treatment including diffraction, absorption and nonlinearity is referred to as the KZK (Khokholv-Zabolotskaya-Kuznetsov) equation

Absorption - viscous forcesAbsorption - viscous forces

• Under condition << v , absorption coefficient due to viscous forces is approximated by– implies /k << 1 (not much absorption per wavelength)

0

vi t kz zp p e

2

30

4

3 2v s b c

Effect is frequency dependent!

1/ 22

2

0

1 1

21

vv

v

c

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Absorption – relaxation effectsAbsorption – relaxation effects

• Relaxation time (R) of a medium– describes rate molecules/particles return to equilibrium

positions after displacement

– molecules pushed forward during compression phase and fall back during rarefaction phase

– particle motion and wave arrival out of phase => kinetic energy removed from wave to stop and reverse direction of molecule motion

– property of the medium

• Frequency dependence (rate of arrival versus relaxation time)

• Several models exist– Generally results in an absorption form similar to viscous treatment,

but with m<2 for fm

Harpen MD, Basic nonlinear acoustics: An introduction for radiological physicists, Med. Phys., 33(9):3241–3247, 2006.

Absorption - relaxation effectsAbsorption - relaxation effects

• Relaxation processes in tissue can be modeled as a summation of independent events (in simple systems)

• An expression for absorption due to relaxation:

• where R is inversely proportional to the relaxation rate (R)

• Note form is similar to viscous absorption coefficient, but the source of absorption different

22

1i

ir

i

R

A

Net absorption coefficientNet absorption coefficient

• Net absorption coefficient may be expressed as:

• Overall frequency dependence is ~fm (1<m<2)– no relaxation effects (i.e., water or air) => m=2– some relaxation effect => fm, m>1– relaxation dominance => m ~ 1– At high frequencies, relaxation effects are reduced– At low frequencies relaxation processes dominate

22

230

4

3 21

i

iv r s b

i

R

A

c

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Attenuation of the ultrasound beamAttenuation of the ultrasound beam

• Amplitude attenuation cumulative effect of – absorption (dominant) and scatter

– amplitude attenuation: = A + S

• Attenuation coefficients do not include effects from specular reflections or geometric effects

• Generally concerned with intensity (I ~ p2)– intensity attenuation = 2 = A + S

– normally given in Nepers (natural log) per cm (Np/cm) or dB/cm– (dB/cm) =(20/ln(10)) = 8.686 (Np/cm)

z (cm) z (cm)

|p/p

0|

|I/I 0

|

01ln

I

d I

Duck FA, Ultrasound in Medicine

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Non-linear propagationNon-linear propagation• Viscoelastic tissue properties also impact the

wave velocity and waveform via viscous losses

• Assume instead our volume element deforms in x-direction by at x and + at x+x

• Using continuity equation and the equation of motion we can arrive at:

• Pressure wave equation is a little more difficult

2 2 2

22 2

1

c

t x

x

(Non-linear wave equation)

(for displacement ONLY!)

Harpen MD, Basic nonlinear acoustics: An introduction for radiological physicists, Med. Phys., 33(9):3241–3247, 2006.

• Still assume homogeneous medium …

• Power series expansion of pressure as adiabatic process (i.e., P1/P2=(1/2)) with constant entropy (S):

• Ratio of B/A = parameter of nonlinearity– Note A~K in this formulation (adiabatic bulk modulus)

• New “effective” speed of sound:

Non-linear propagationNon-linear propagation

0 0

0

( , ) 1 ( , )2

dP Bc c u x t c u x t

d A

0 0

22

0 2, ,

2

0 0

1...

2

2

S S

p pp p

Bp A

Non-linear propagationNon-linear propagation

• Propagation now depends on local displacement• Large displacements propagate faster than

small displacements, distorting the waveform• As relative amplitude of 0 declines => relative

strength of harmonics increased

Note: Relative amplitude of harmonics increas with depth, but decrease f0.

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Non-linear effectsNon-linear effects

2

0(0.5 * / 1)

cd

B A u

Mean distance until “shock wave”

Duck FA, Nonlinear acoustics in diagnostic ultrasound, Ultrasound in Medicine & Biology, 28(1):1-18, 2002.Harpen MD, Basic nonlinear acoustics: An introduction for radiological physicists, Med. Phys., 33(9):3241–3247, 2006.

Shock wave at focus