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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
27
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:
29
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
30
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
31
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.
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