Transcript Slide 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
1
Radiation force
Lord Rayleigh,
1902
Vilhelm Bjerknes
1906
Leon Brillouin, 1925
Robert Wood
Paul Langevin, 1920s
Alfred Lee Loomis
1926-27
2
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:
B  1
  1

2A
2
Txx  p '  0u 2
In the absence of average mass flux:
Rayleigh radiation pressure:
P2
PR   2 PR
c
In an acoustic beam where
Langevin radiation pressure:
2
P
PL  2
c
- Nonlinearity parameter
p ' 0
PR

PL
3
Non-dissipative, bulk radiation force
Elastic nonlinearity leads to demodulation/rectification effect in
modulated ultrasound that can be described in terms of nonlinear, nondissipative 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:
1 2 p
 2 p2
1  2 PR
p  2 2  
 2
,
4
2
2
c0 t
0c0 t
c0 t
4
In non-viscous case,
= -
-
For a harmonic wave, the forcing in (25) is constant in time:
2

P

P
'
a
= FS. For a damping beam: FS  

x c 2
5
Shear Wave Elasticity Imaging (SWEI)
(Sarvazyan et al, 1998)
6
Shear Displacements
(Sarvazyan, Rudenko et al, 1998)
Simulated
MRI
Ultrasound
(dissip. force)
Measured (left) and calculated (right) space-time
distribution of shear wave remotely induced in tissue by
an ultrasonic pulse
7
Ultrasound-induced displacements in
tissue samples (Sutin, Sarvazyan)
Doppler measurement data
Time reversal
(TRA)
Blue –radiated signal
Red –recorded TRA
focused signal
8
THEORY: Inhomogeneous Medium
ELASTIC MEDIA: GENERAL NONLINEAR STRESS
(Ostrovsky, Il’inskii, Rudenko, Sarvazyan, Sutin, 2007)
 u
u 
 u 

A  u u
u u
u u 
 ik    i  k     l  ik      l l  i k  i l  
4  xi xk xl xl xl xk 

 xk xi 
 xl 
   B   ul

 
 2   xl
2

 ul
ui ul  A uk ul B  ul um
uk ul 
  ik  2
  C 

 
 ik  2
xk xl  4 xl xi 2  xm xl
xi xl 

 xl

Here ui is the displacement vector and
Then
σik is the stress tensor.
 ik   L ik   N ik
Linear part:

L
ik
1  ui uk 

 ull ik  2uik , uik  

2  xk xi 
9
2

  ik

ELASTIC MEDIA: GENERAL NONLINEAR STRESS
(Ostrovsky et al, JASA, 2007)
AVERAGE STRESS COMPONENTS:
Narrow-angle ultrasonic beam:

 
 k  , ; vx  v y , vz
x
y z
  xx
3
 v 
 G1  x  , G1    3  A  3B  C ,
2
 x 
  yy

 v 
  zz  G2  x  , G2   B  C ,
2
 x 
2
2
 v v 
  xy   yx  G3  x x  , G3    3  A  2 B,
 x y 
 v v 
  xz   zx  G3  x x 
 x z 
G1 − G2 = G3 =  + 3μ + A + 2B = Q
In fluids and waterlike media, Q c 2l
10
Radiation Force:
Fik 
  ik
xk
Shear force component:
1 u x
c 
  t  x / c, g  2 / 3c 3
M a ( , X , y, z ) 
Narrow-angle beam: KZK equation:
(in terms of Mach number)
2
2
From here (similar to the known expression but with nonlinear Ma):
11
WAVES
 ik
ui 
,
xk
In a smoothly inhomogeneous medium:
Nonlinear wave equation for the displacement vector, u
N


ik
2
2
2


[u  ct u  (ct  cl )divu]  S 
Φ
xk
where
cl  (  2) / 
ct   / 
are the velocities of linear longitudinal and transverse waves, respectively,
and the linear term S is related to spatial parameter variations:
 u u   1 
Si   i  k 

(  divu)
 xk xi  xk 2 xi
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.
12
U i  ui
AVERAGING

[U ct2 U  (ct2  cl2 )divU]  S  Φ
RF  Φ 
Let us represent u as a sum of two vectors, potential, U1 so that
and solenoidal, U2, for which (  · U2 ) = 0.
As a result,
  ikN
x k
 x U1 = 0,
 Φ  Φ 1  Φ 2
Potential:
 2 U1
2

c
l U1  S1  Φ 1 /  .
2
t
2

U2
Solenoidal:
2

c
t U 2  S 2 /   Φ 2 /   F.
2
t
13
FOR THE NARROW ACOUSTIC BEAM:
 xxN  xzN
  ux 
  ux ux  G1  ux 
x 

 G1 ( x) 
  G3 ( x) 


;
x
z
x  x 
z  x z  x  x 
2
2
 zxN  zzN
  u u 
  u 
G  u u 
z 

 G3 ( x)  x x   G2 ( x)  x    3  x x1 
x
z
x  x z 
z  x 
x  x z 
2
  x  z 
  rotF y  

:
x 
 z
2
 2
 2  u x 
 2  u x u x 
  (G1  G2 )

  G3  2  2 

xz  x 
x  x z 
 z
Hence, for
G3   u x u x   G3  u x u x1 
 (G1  G2 )  2  u x 


2



 2 
,
2
x
z  x 
x x  x z  x  x z 
2
or
2
2
Fx
    u x 
  u x u x  Q   u x 
 Q  



,





z  x  x  z  x z  x z  x 
z
14
AS A RESULT, in a harmonic beam:
2


M
Q
1   2  
2
2 1 Q
a
U tt  ct U zz  

Ma
  M a
  x k z 
z 
 x

0
ADDING LINEAR LOSSES
 1 Q

U tt
Q
2
 U zz  2 M a exp(2x)
 2 
2
ct
ct
 Q x

 = f /17.3 1/cm (f in MHz)
Non-dissipative
radiation force
Dissipative
radiation force
Q = - ( + 3μ + A + 2B)
For tissues, Q - c2
15
CYLINDRICAL (PARAXIAL) BEAM
2

1


U

(

c
) 1 2 RF
 2



U tt   ct  
Ma 
.
r

t  r r  r 
x 


 r2 t2 
Q 1 2
RF 
M a 0 Exp  2  2 .
x 
 D T 
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 cm
Shear wave velocity 3 m/s
 = 15 Pas, so that  = 0.015 m2/s
16
EXAMPLES
 = 0.015 m2/s
 = 0.0015 m2/s
20 ms
40 ms
17
3-D PLOT
1
5
U U max
0.5
4
0
3
-0.02
2
0
0.02
r,cm
1
t,s 0.04
0.06
18
Spatial distribution of force
Q 2
RF 
Ma
x
19
D0 = 3 cm
LONGITUDINAL DISTRIBUTION
F = 10 cm
1
0.5
1
U U max
0.75
U U max
0.5
2
2
0
-0.5
1
-1
0.25
1
0
0
25
-1
30
x,cm
27.5
r,cm
-1
30
x,cm
32.5
35
35
-2
Homogeneous/Dissipative
0
25
-2
20
Inhomogeneous/Non-dissipative
r,cm
Application to lesion visualization (E. Ebbini)
BeforeDisplacement
lesion was formed
After lesion
formed
Tissue
(0.9 mm away
from the
focalwasplace)
12
12
b
b
b
b
b
b
10
170v for 2.5 ms
280v for 2.5 ms
450v for 2.5 ms
280v for 1.25ms
450v for 1.25 ms
0vfor 0ms
10
170v for 2.5 ms
280v for 2.5 ms
450v for 2.5 ms
280v for 1.25ms
450v for 1.25 ms
0vfor 0ms
8
tissue displacement (um)
tissue displacement (um)
8
6
4
6
4
2
2
0
0
-2
a
a
a
a
a
a
0
0.005
0.01
time(s)
0.015
-2
0
0.005
0.01
0.015
time(s)
Effect increase in lesion can be explained by non-dissipative radiation force.
21
NONLINEAR PRIMARY BEAM
22
“GEOMETRICAL” STAGE (no diffraction)
Implicit form:

Or
x

b  1 : shock formation

1  b(r ) cos
r*  F exp(1 / M 0kF )
23
Hence,
2   FM 0 
Fx (r ) 


3c  br 
At small amplitude (b <<1) :
2
 1




1


2
1

b


F x 
c
k 2 M 20 F 2
3r 2
Applicability: until diffraction becomes significant (outside the focal length
at the 1st harmonic):
4
r  RF 
k 2
Fx / Fx (r = F)
M0 =10-4 ,  = 15°, F = 10 cm, f = 1 MHz
M a ( RF )  K Ml M a ( F )
K Ml   2 kF / 2
24
Focal Area (r < RF):
Linear, diffracting non-sinusoidal wave
(Ostrovsky&Sutin, 1975)
x<0
RF
2
S
Kirchhoff approximation (from S  RF2):
Ma 
1
2
c
S
1 
rF
M

t

S
r 
c
t

ds
At the focus (r = 0):
MF 
 M F 
Fx (r  0)  Qg 

 t 
MS
RF 2 
2c

t
t
RF F
 c
2
2M

S
2


2
sin 2 d
 1 bR cos 5 
F
2
4
4 b 2 
RF 
F
2
M0
.
2
7/2
2
2R F 4
1 b 
R F 
M 20
F 2 4
2
R 2F

25
Thus, focal force is
Fx
4 F 2 4
2
 gM
Wave profiles:
0
4
4
b2
RF 
4 1b 2 
RF 
7/2
Force growth in the focal area:
r = RF
x = 0, z  0
(focal plane)
r=0
(focus)
 = 0.7
(From Sutin, 1978)
26
RF IN SHOCK WAVES
Sawtooth stage (b(RF)>1)
M
t
kr ln(F / r )
Shock amplitude:
Ms 
,  t  

kr ln(F / r )
At the beam axis:
Ostrovsky&Sutin, 1975; Sutin,1978
27
At geometrical stage
cM
 c
Fxs 

3
3
6
6kr  ln ( F / r )
3
s
2
(See also Pishchalnikov et al, 2002)
c
Fxs ( Rs ) 
3
3
2
384 ln ( kF / 4 )
6
Near (before) the focus:
Fmax  1 /
28
NON-DISSIPATIVE, NONLINEAR RF
(geometrical stage)
Fx 
c 2 kF 3 
M 30
23b 2 2
1b 2 3/2
r3
b 3
1b 2 3/2
At small b:
2 2 4 4
3
c 2
k F M0
F x 
4r
3
ln
F/r
In spite of a higher power of M0, this
force can prevail over the dissipative one
29
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:.
Mo =10-4 ,  = 15°, F = 10 cm, f = 1 MHz
30
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.
31