1

RADIATION FORCE, SHEAR WAVES, AND MEDICAL ULTRASOUND

L. A. Ostrovsky Zel Technologies, Boulder, Colorado, USA, and

Institute of Applied Physics, Nizhny Novgorod, Russia

FNP, July 2007

2

Radiation force

Lord Rayleigh, 1902 Leon Brillouin, 1925 Paul Langevin, 1920s

Robert WoodVilhelm Bjerknes1906

Alfred Lee Loomis

1926-27

3

RADIATION FORCE (RF), RADIATION STRESS, RADIATION PRESSURE- All are average forces generated by sound (ultrasound), acting on a body,boundary, or distributed in space.

Momentum flux in a plane wave:

20' upTxx

2

1

21

A

B

- Nonlinearity parameter In the absence of average mass flux:

Rayleigh radiation pressure:

RR Pc

PP

2

2

In an acoustic beam where Langevin radiation pressure:

0'p

2

2

c

PPL

L

R

P

P

4

Non-dissipative, bulk radiation force Elastic nonlinearity leads to demodulation/rectification effect in modulated ultrasound that can be described in terms of nonlinear, non-dissipative radiation force.

Nonlinear acoustic wave equation first derived by Westervelt (JASA, 1963) for parametric arrays [also suggested by Zverev and Kalachev in Russia in 1959] depends on the Rayleigh force and takes into account physical nonlinearity in the equation of state and “geometrical” nonlinearity:

5

- = -

For a harmonic wave, the forcing in (25) is constant in time:

In non-viscous case,

2

2'

c

P

x

PFS a

= FS. For a damping beam:

6

Shear Wave Elasticity Imaging (SWEI)Shear Wave Elasticity Imaging (SWEI)(Sarvazyan et al, 1998)(Sarvazyan et al, 1998)

Low frequency detector

Pumpin

g and

prob

ing

trans

duce

rs

7

Shear Displacements(Sarvazyan, Rudenko et al, 1998)

Simulated

(dissip. force)MRI

Ultrasound Measured (left) and calculated (right) space-time distribution of shear wave remotely induced in tissue by an ultrasonic pulse

8

Ultrasound-induced displacements in tissue samples (Sutin, Sarvazyan)

Doppler measurement data

Time reversal (TRA)Blue –radiated signalRed –recorded TRA focused signal

9

ikl

l

l

l

i

kik

l

m

m

l

i

l

l

k

l

l

k

iik

l

l

k

l

l

i

l

k

l

i

k

l

i

lik

l

l

i

k

k

iik

x

uC

x

u

x

u

x

u

x

uB

x

u

x

uA

x

u

x

u

x

uB

x

u

x

u

x

u

x

u

x

u

x

uA

x

u

x

u

x

u

22

224

22

4

THEORY: Inhomogeneous MediumELASTIC MEDIA: GENERAL NONLINEAR STRESS(Ostrovsky, Il’inskii, Rudenko, Sarvazyan, Sutin, 2007)

ikN

ikL

ik

Here ui is the displacement vector and σik is the stress tensor.

Then

Linear part:

i

k

k

iikikikllik

L

x

u

x

uuuu

2

1,2

10

G1 − G2 = G3 = + 3μ + A + 2B = Q In fluids and waterlike media,

z

v

x

vG

BAGy

v

x

vG

CBGx

vG

CBAGx

vG

xxzxxz

xxyxxy

xzzyy

xxx

3

33

2

2

2

1

2

1

,23,

,2

,

,332

3,

AVERAGE STRESS COMPONENTS:

zyx vvvzy

kx

,;,

Narrow-angle ultrasonic beam:

ELASTIC MEDIA: GENERAL NONLINEAR STRESS (Ostrovsky et al, JASA, 2007)

Q c l2

11

k

ikik x

F

Shear force component:

Narrow-angle beam: KZK equation:(in terms of Mach number)

Radiation Force:

From here (similar to the known expression but with nonlinear Ma):22

33/2,/

1),,,(

cgcxt

u

czyXM x

a

12

In a smoothly inhomogeneous medium:Nonlinear wave equation for the displacement vector, u

where /)2( lc /tc

are the velocities of linear longitudinal and transverse waves, respectively,and the linear term S is related to spatial parameter variations:

)(2

1udiv

xxx

u

x

uS

iki

k

k

ii

ΦSuu

k

ikN

ltt xdivccuc

])([ 222

WAVES

,k

iki x

u

Medium parameters may slowly depend on coordinate x that is directed along the primary beam axis. Here, S is of the 2nd order and further neglected.

13

ΦSUUU

])([ 222 divccc ltt

21 ΦΦΦ

./1112

21

2

ΦSUU

lct

./2/222

22

2

FΦSUU

tct

AVERAGING

k

Nik

xRF

Φ

Let us represent u as a sum of two vectors, potential, U1 so that x U1 = 0, and solenoidal, U2, for which ( · U2 ) = 0.

As a result,

Potential:

Solenoidal:

ii uU

14

;)()(2

13

2

1

x

u

x

G

z

u

x

u

zxG

x

u

xxG

zxxxxx

Nxz

Nxx

x

z

u

x

u

x

G

x

u

zxG

z

u

x

u

xxG

zxxxxxx

Nzz

Nzx

z13

2

23 )()(

,2)(

)(

1233

2

2

221

2

2

2

2

3

22

21

z

u

x

u

x

G

z

u

x

u

xx

G

x

u

zx

GG

z

u

x

u

xzG

x

u

zxGG

xxxxx

xxx

Hence, for

FOR THE NARROW ACOUSTIC BEAM:

:

xz

rotF zxy

,22

z

F

x

u

zx

Q

z

u

x

u

zx

u

xzQ xxxxx

or

15

Q = - ( + 3μ + A + 2B)

For tissues, Q - c2

21

)2exp(222 x

Q

QxM

c

QU

c

Ua

tzz

t

tt

Non-dissipative radiation force

Dissipative radiation force

= f /17.3 1/cm (f in MHz)

x

QM

zM

zkx

MQUcU aa

azzttt

11 2

0

22

2

AS A RESULT, in a harmonic beam:

ADDING LINEAR LOSSES

16

.1)(1 2

22

RF

Mx

c

r

Ur

rrtcU attt

.1

2

2

2

2

02

T

t

D

rExpM

x

QRF a

CYLINDRICAL (PARAXIAL) BEAM

Beam radius at a half-intensity level near focus: R = 0.3 cm Acoustic pressure in the focus: 2 MPa Length of medium acoustic parameter variation: 0.5 cmShear wave velocity 3 m/s

= 15 Pas, so that = 0.015 m2/s

17

EXAMPLES

= 0.015 m2/s

40 ms20 ms = 0.0015 m2/s

18

-0.020

0.020.04

0.06t,s

1

2

3

4

5

r,cm

0

0.5

1UUmax

-0.020

0.020.04

0.06t,s

3-D PLOT

19

2aM

x

QRF

Spatial distribution of force

20Inhomogeneous/Non-dissipative

LONGITUDINAL DISTRIBUTION

F = 10 cm

D0 =

3 c

m

25

30

35x,cm

-2

-1

0

1

2

r,cm

0

0.25

0.5

0.75

1

UUmax25

30

35x,cm

25

27.5

30

32.5

35

x,cm

-2

-1

0

1

2

r,cm

-1

-0.5

0

0.5

1

UUmax25

27.5

30

32.5

35

x,cm

Homogeneous/Dissipative

21

Tissue Displacement (0.9 mm away from the focal place)

0 0.005 0.01 0.015-2

0

2

4

6

8

10

12

time(s)

tissue d

ispla

cem

ent

(um

)

Before lesion was formed

b 170v for 2.5 msb 280v for 2.5 msb 450v for 2.5 msb 280v for 1.25msb 450v for 1.25 msb 0vfor 0ms

0 0.005 0.01 0.015-2

0

2

4

6

8

10

12

time(s)

tissue d

ispla

cem

ent

(um

)

After lesion was formed

a 170v for 2.5 msa 280v for 2.5 msa 450v for 2.5 msa 280v for 1.25msa 450v for 1.25 msa 0vfor 0ms

Effect increase in lesion can be explained by non-dissipative radiation force.

Application to lesion visualization (E. Ebbini)

22

NONLINEAR PRIMARY BEAM

23

“GEOMETRICAL” STAGE (no diffraction) Implicit form:

Or

cos)(1 rb

1b : shock formation )/1exp( 0* kFMFr

x

24

1

1

1

3

2)(

2

2

0

bbr

FM

crFx

Applicability: until diffraction becomes significant (outside the focal length at the 1st harmonic):

2

4

kRr F

Hence,

2/

)()(

2kFK

FMKRM

Ml

aMlFa

Fx ck2M0

2F2

3r2At small amplitude (b <<1) :

1 2 3 4 510r

4

6

8

10

12

SSrFFx / Fx (r = F) M0 =10-4 , = 15°, F = 10 cm, f = 1 MHz

25

Focal Area (r < RF):Linear, diffracting non-sinusoidal wave (Ostrovsky&Sutin, 1975)

Kirchhoff approximation (from S RF2):

At the focus (r = 0):

M a 12 c

S

1r

t

M St r Fc ds

M F RF 2

2c MS

tt RF F

c

x < 0RF

2

S

2M S

2

2

M 02 F 2 4

2 R F2

sin2 d1 bR F cos 5

M 02 F 2 4

2R F2

4 b2R F 41 b2R F 7/2

.

#

2

)0(

t

MQgrF F

x

26

F x gM 02 4F 2 4

44 b2RF

4 1 b2RF 7/2

0.00002 0.00004 0.00006 0.00008 0.0001M0

100

120

140

160

180

200

220

Fxr0FxrRF

Thus, focal force is

Force growth in the focal area:

(From Sutin, 1978)

Wave profiles:

r = RF

r = 0 (focus)

x = 0, z 0(focal plane)

= 0.7

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RF IN SHOCK WAVESSawtooth stage (b(RF)>1)

)/ln(

,)/ln(

rFkrM

trFkr

tM

s

At the beam axis:

Ostrovsky&Sutin, 1975; Sutin,1978

Shock amplitude:

28

)/(ln66 33

23

rFkr

ccMF s

xs

)4/(ln384)(

233

6

kF

cRF sxs

/1max F

(See also Pishchalnikov et al, 2002)

At geometrical stage

Near (before) the focus:

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NON-DISSIPATIVE, NONLINEAR RF (geometrical stage)

Fx c2kF3M03

r3

2 3b 2 21 b 23/2

b 31 b 23/2

Fx 3 c2 2k2F4M04

4r3lnF/r

At small b:

In spite of a higher power of M0, this force can prevail over the dissipative one

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CONCLUSIONS• Nonlinear distortions in a focused ultrasonic beam can significantly enhance

the resulting shear radiation force

• Diffraction near the focus makes the force even stronger

• The effects are different when the shocks form before the focal area

• Nonlinear distortions can be of importance in biomedical experiments:.

1 2 3 4 5f, MHz

0.5

1

1.5

2

2.5

RF

Mo =10-4 , = 15°, F = 10 cm, f = 1 MHz

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CONCLUSIONS

Acoustic radiation force (RF) is a rather general notion referring to the average action of oscillating acoustic field in the medium.

In water-like media such as biological tissues, shear motions generated by RF are much stronger than potential motions. This effect is used in medical diagnostics.

To generate shear motions, at least one of the following factors must be accounted for: dissipation, inhomogeneity, and nonlinearity of a primary beam. The latter two are the new effects we have considered.