bmi i fs05 – class 8 “ultrasound imaging” slide 1 biomedical imaging i class 8 – ultrasound...

BMI I FS05 – Class 8 “Ultrasound Imaging” Slide 1

Biomedical Imaging IBiomedical Imaging I

Class 8 – Ultrasound Imaging

Physics of Matter-Energy Interactions

11/02/05


Different Forms of EnergyDifferent Forms of Energy

Electromagnetic

Acoustic



ElectromagneticPhotons (quantum description), electromagnetic waves (classical description)

• Connection: classical EM fields and waves are constituted of large numbers of photons

Does not require a material medium through which to propagate• Mechanisms of propagation through material media are

different from that of propagation through free space• In some materials: [see W. C. Chew, Waves and Fields in

Inhomogeneous Media (IEEE Press, Piscataway, NJ, 1995), §1.1]

– electric and magnetic field vectors rotate– Phase shifts develop between electric and magnetic field

vectors

Acoustic



Electromagnetic

AcousticRequires a material medium through which to propagate

Consists of oscillatory motions of the atoms/molecules of which a material is constituted.

Oscillating particles have kinetic energy square of amplitudes of their motions

Through action of intermolecular forces, particles transfer their energy to adjacent particles energy wave traveling through material.


Transfer/Transformation of EnergyTransfer/Transformation of Energy

Light becomes sound — photoacoustic phenomena

Sound becomes light — sonoluminescence

Absorbed electromagnetic (EM) and acoustic energy both become heat

Nevertheless, EM and acoustic energy are FUNDAMENTALLY DISTINCT PHENOMENA!


Acoustic Wave Energy RangesAcoustic Wave Energy Ranges

Just as there are infrared, visible, and ultraviolet ranges in the EM spectrum, 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 is 103, or almost 10 octaves. On the other hand, the ratio of the highest to the lowest frequencies of visible light is a bit less than 2 (i.e., less than one octave).

AudibleInfrasound

Ultrasound

20 Hz

20 kHz


“Mass–and–Spring” Representation of Matter“Mass–and–Spring” Representation of MatterU – particle displacement (10-

10 m (H2O))

ux = dx/dt – particle velocity (10-2 m-s-1 (H2O))

U’ – distance energy propagates during one particle oscillation cycle (~10-5 m (H2O))

c – phase velocity, energy propagation velocity (1.5×103

m-s-1 (H2O))

2Umax

Two

time


Longitudinal Waves of Molecular MotionLongitudinal Waves of Molecular Motion

molecule oscillation time ~ (2×10-10 m)/(10-2 m-s-1) = 2×10-8 s Tdistance wave propagates ~ (2×10 -8 s)·(1.5× 10-3 m-s-1) = 3×10-5 m

Now T/2 ago T ago 3T/2 ago 2T ago …

every Nth particle, N = U’/U, shown greatly enlarged


Definition of StressDefinition of Stressts1s2 force per unit area, acting in direction s1, applied to face that is the s2 axis

s1 = s2: compressional stress acting in s1 direction; by definition, is positive if directed outward from the volume element.

s1 s2: shear stress in s1 direction; positive if || the positive s1 direction.

pressure on any face = -1 × corresponding compressional stress

force = stress × surface area; e.g., fzz = tzzΔxΔy

i.e., tzz(z0) i.e., tzz(z0+Δz)


Definition of StrainDefinition of Strain

W(z0 + Δz)

=

W(z 0

)

U(z0 + Δz)

x

z

yU

V

W

strain: material’s reaction to applied stress(es); how much deformation itundergoes; simply displacing the

cube some distance from its initial location is not strain!

εzz = compressional strain in z direction:

0

0

0 0zz

z

z

W z z W z

zW

zz W

z z

¶¶ ¶

¶

εxz = shear strain in x direction:

0

xzz

Uz

¶¶

x0x0 + Δx


Moduli of ElasticityModuli of Elasticity

Young’s modulus (E) tzz/εzz

Bulk modulus (B) (compressibility)-1 = [-(1/σ)(σ/p)]-1 = -σ/(σ/p)

volume pressure

Lamé constants: μ, ν

— Related to classical moduli via:

3 2 2

,3

E B


Moduli of ElasticityModuli of Elasticity


— Related to classical moduli via:

— Inverting preceding formulae gives:

3 2 2

,3

E B

3 33, .

9 9

B B EBEB E B E


Shear Modulus vs. State of MatterShear Modulus vs. State of Matter


— μ = shear modulus

— When displacements U, V, and W all are small:

— tzz (ν + 2μ)εzz = (ν + 2μ)(W/z)

— tyz μεyz = μ(V/z)

— txz μεxz = μ(U/z)

— In fluids (liquid or gas), μ 0: E 0, B ν

— i.e., fluids (including most biological soft tissues) can not support shear waves!


Inability of Fluid Media To Support Shear WavesInability of Fluid Media To Support Shear Waves

Direction of wave propagation

Directions of particle displacement

Why do soft tissues behave more like fluids than like solids, with respect to (lack of) shear wave propagation?


Non–Rigorous Derivation of Linearized Equation of MotionNon–Rigorous Derivation of Linearized Equation of Motion

1) Estimate force, in z direction, on each face by multiplying corresponding stress and area: fzz = tzzΔxΔy, fzx = tzxΔyΔz, …

2) Sum differences between forces on each pair of parallel faces (because only non-zero force difference produces deformation)3) Estimate net acceleration, in z direction, on cube as second temporal derivative of displacement W: acc. 2W/t 2

4) Mass of cube is its density, η, times its volume, ΔxΔyΔz

5) Equating result of step 2 with product of cube’s mass (step 4) and acceleration (step 3) gives:

2zyzx zz

tt t Wx y z t

2¶¶ ¶ ¶¶ ¶ ¶ ¶


Compressional and Shear WavesCompressional and Shear Waves

2zyzx zz

tt t Wx y z t

2¶¶ ¶ ¶¶ ¶ ¶ ¶

Special case #1 — tzx = tzy = 0 (i.e., no shear stress, medium is fluid):

2 2

2 22

zz

zzW

tz

t Wz t

W Wz t

2

¶¶

2 2

¶ ¶¶ ¶

¶ ¶¶ ¶ 2

1c

Special case #2 — tzz = tzy = 0 (i.e., shear wave, so medium is a solid):

2 2 2zx

zxW

tx

t W W Wx t x t

2 2 2

¶¶

¶ ¶ ¶ ¶¶ ¶ ¶ ¶


Sinusoidal Solutions to Wave EquationsSinusoidal Solutions to Wave Equations

02 22

2 (angular frequency)

2 (phase velocity)

(wave number)

i t kxW WW W e

z t

f

c

kc

2 2¶ ¶¶ ¶

02 2

2 (angular frequency)

(phase velocity)

(wave number)

ti t k x

t

tt

W WW W e

x t

f

c

kc

2 2¶ ¶¶ ¶

Definition: Acoustic impedance pressure/(particle velocity)

Z± = p±/uz± = ±ηc (uz = W/t = iωW)

[Compare electric circuit analogue, impedance = voltage/current]


Sinusoidal Solutions to Wave EquationsSinusoidal Solutions to Wave Equations

metal gasacrylic

soft tissueshard tissue

Notice how similar these values are to each other and to that for water,and how different they are from these.


Frequency-domain Form of Wave EquationFrequency-domain Form of Wave Equation

One-dimensional:

22 0

Wk W

z

2¶¶

More generally:

2 2 0W k W

1, 2, or 3-dimensional

Laplacian operator

Formally identical to homogeneous potential equation (Zwillinger, p. 418) or homogeneous

Helmholtz equation (Kak & Slaney, p. 210)

For solutions that are not identically zero, addition of a source term is required:

2 204W k W r r

Same dimensionality as 2

Source location


pulse average

temporal average

spatial peak

spatial average

peak

average

Intensity, Radiation ForceIntensity, Radiation Force

IntensityAverage Intensity: I = (½)ηcω2W0

2 = (½) p0u0

• Assumes sinusoidal, plane-wave irradiationTemporal AverageSpatial AverageCombined Spatial and Temporal Averaging

• Spatial Average Temporal Average, I(SATA)• Spatial Peak Temporal Average, I(SPTA)• Spatial Peak Pulse Average, I(SPPA)


Intensity, Radiation ForceIntensity, Radiation Force

IntensityAverage Intensity: I = (½)ηcω2W0

2 = (½) p0u0

• Assumes sinusoidal, plane-wave irradiationTemporal AverageSpatial AverageCombined Spatial and Temporal Averaging

• Spatial Average Temporal Average, I(SATA)• Spatial Peak Temporal Average, I(SPTA)• Spatial Peak Pulse Average, I(SPPA)

Radiation Force (fr) per Unit Area

Ultrasound exerts pressure at interfaces within the mediumfr = D(I/c), for plane-wave irradiation

• D = 2 at strongly reflecting interface, D = 1 at strongly absorbing interface


Reflection and RefractionReflection and RefractionBehavior or ultrasound at an interface between materials of different Z is analogous to behavior of light at interface between materials of different refractive index.

Fraction of pressure reflected =

Reflection Coefficient, R; fraction

of pressure transmitted =

Transmission Coefficient, T

Intensity reflection and transmission coefficients are derived from the preceding equations and p = Zu and I = p0

2/(2Z):

Z1, u1

Z2, u2

pr

pt

pi

2 1

2 1

2

2 1

cos cos,

cos cos

2 cos.

cos cos

i tr

i i t

t i

i i t

Z ZpR

p Z Z

p ZT

p Z Z

2 22 1 1 2

22 1 2 1

cos cos 4 cos, .

cos cos cos cosi t tr i

i i t i i t

Z Z II Z Z

I Z Z I Z Z


AttenuationAttenuation

For plane wave propagating through homogeneous medium,

β = pressure attenuation coefficient:

Attenuation is caused by absorption and scattering• Two mechanisms for absorption: viscosity (drag) and relaxation• Scattering is really the same phenomenon as reflection, at

interfaces of volumes whose dimensions are small relative to the ultrasound wavelength

0 zp z p e

01ln

p

z p z

Orders of magnitude larger than the value for water, because of absorption and scattering by other constituents of tissue.


Clinical Potential of Attenuation MeasurementsClinical Potential of Attenuation MeasurementsNote, overall attenuation coefficient β, not only absorption or only (back)scattering

Healthy myocardium

Infarcted myocardium

That is, ultrasound attenuation and backscatter measurements can be used (among many other things) to assess extent of tissue death in myocardial infarction


Absorption MechanismsAbsorption Mechanisms

Viscosity-induced absorptionSome of the energy in the propagatingbeam is transferred, in the transversedirection, to adjacent portions of thematerialViscosity coefficient (ζ):

Attenuation coefficient:Relaxation

Critical parameter is ratio of particle relaxation time to wave period U´/c.

Negligible energy loss occurs if the ratiois either <<1 or >>1; substantial losscan occur when it is ~1.Attenuation coefficient:

xz

x

t

u z¶ ¶

22

3v c

20

21

R

R

B f

ff


Tissue AbsorptionTissue Absorption

Experimentally determined values of β:

Curves do not resemble homogeneous

single-component result from previous

slide; why not?

Tissue is a mixture of many components,

each with its own fR:

Overall net attenuation curve is the sum

of those for all of the components

2 21

i

iiR

BA

f ff


Scattering (Far–Field)Scattering (Far–Field)

iip e k rr

0,ikR

s i

ep f p

Rr o i r

2

22

2

1,

2

,12

ii

ss i

pI

Z

fpI I

Z R

i o

2

2

, , , differential scattering cross section

i.e., power scattered into direction from direction

, per unit solid angle, per unit incident intensity

, , , cross section

d

d

s

f

f backscattering

i o i o

o

i

i i i i

4

, , scattering cross sectiond di o

Note, this is microscopic, or single-particle cross section


Tissue ScatteringTissue ScatteringIf all particle dimensions are << ultrasound wavelength, then σs ω 4 or f 4.

If particle is cylindrical with radius << ultrasound wavelength, then σs ω3 or f 3.

(Healthy Tissue)

(Infarcted Myocardium)

(Healthy Myocardium)

time-integrated measurementsmeasurement gated to heart cycle

bmi i fs05 – class 8 “ultrasound imaging” slide 1 biomedical imaging i class 8 – ultrasound...

Documents