A sphere moving back and forth in tissue generates the kinds of complex displacement fields that are used in elastography. The analytical solution of Hans Oestreicher for this phenomenon [(1951). J. Acoust. Soc. Am. 23, 704–714] gives an understanding of the transverse and longitudinal, fast and slow waves that are generated. The results suggest several ways to determine the absorption coefficients of tissues, which together with phase velocity permit the computation of both the real shear modulus and the shear viscosity as functions of frequency.

1.
Carstensen
,
E. L.
, and
Parker
,
K. J.
(
2014
). “
Physical models of tissue in shear fields
,”
Ultrasound Med. Biol.
40
,
655
674
.
2.
Carstensen
,
E. L.
,
Parker
,
K. J.
,
Dalecki
,
D.
, and
Hocking
,
D. C.
(
2015
). “
Biological effects of low frequency shear strain. Part 1. Physical descriptors
,”
Ultrasound Med. Biol.
(in press).
3.
Catheline
,
S.
, and
Benech
,
N.
(
2015
). “
Longitudinal shear wave and transverse dilatational wave in solids
,”
J. Acoust. Soc. Am.
137
,
EL200
EL205
.
4.
Catheline
,
S.
,
Gennisson
,
J. L.
,
Delon
,
G.
,
Fink
,
M.
,
Sinkus
,
R.
,
Abouelkaram
,
S.
, and
Culioli
,
J.
(
2004
). “
Measurement of viscoelastic properties of homogeneous soft solid using transient elastography: An inverse problem approach
,”
J. Acoust. Soc. Am.
116
,
3734
3741
.
5.
Catheline
,
S.
,
Gennisson
,
J. L.
, and
Fink
,
M.
(
2003
). “
Measurement of elastic nonlinearity of soft solid with transient elastography
,”
J. Acoust. Soc. Am.
114
,
3087
3091
.
6.
Catheline
,
S.
,
Thomas
,
J. L.
,
Wu
,
F.
, and
Fink
,
M. A.
(
1999a
). “
Diffraction field of a low frequency vibrator in soft tissues using transient elastography
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
46
,
1013
1019
.
7.
Catheline
,
S.
,
Wu
,
F.
, and
Fink
,
M.
(
1999b
). “
A solution to diffraction biases in sonoelasticity: The acoustic impulse technique
,”
J. Acoust. Soc. Am.
105
,
2941
2950
.
8.
Chanaud
,
R. C.
(
2010
).
Tools for Analyzing Sound Sources
(
CCR Associates, LLC
,
Essex, CT
), p.
4-2
.
9.
Chen
,
S.
,
Fatemi
,
M.
, and
Greenleaf
,
J. F.
(
2002
). “
Remote measurement of material properties from radiation force induced vibration of an embedded sphere
,”
J. Acoust. Soc. Am.
112
,
884
889
.
10.
Chen
,
S.
,
Fatemi
,
M.
, and
Greenleaf
,
J. F.
(
2004
). “
Quantifying elasticity and viscosity from measurement of shear wave speed dispersion
,”
J. Acoust. Soc. Am.
115
,
2781
2785
.
12.
Chen
,
S.
,
Urban
,
M. W.
,
Pislaru
,
C.
,
Kinnick
,
R.
,
Zheng
,
Y.
,
Yao
,
A.
, and
Greenleaf
,
J. F.
(
2009
). “
Shearwave dispersion ultrasound vibrometry (SDUV) for measuring tissue elasticity and viscosity
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
56
,
55
62
.
13.
Chen
,
S. G.
,
Sanchez
,
W.
,
Callstrom
,
M. R.
,
Gorman
,
B.
,
Lewis
,
J. T.
,
Sanderson
,
S. O.
,
Greenleaf
,
J. F.
,
Xie
,
H.
,
Shi
,
Y.
,
Pashley
,
M.
,
Shamdasani
,
V.
,
Lachman
,
M.
, and
Metz
,
S.
(
2013
). “
Assessment of liver viscoelasticity by using shear waves induced by ultrasound radiation force
,”
Radiology
266
,
964
970
.
14.
Dahl
,
J. J.
(
2013
). “
Acoustic radiation force imaging
,” in
Emerging Imaging Technologies in Medicine
, edited by
M. A.
Anastasio
and
R.
La Riviere
(
CRC Press
,
Boca Raton, FL
), Chap. 13, pp.
201
217
.
15.
Deffieux
,
T.
,
Montaldo
,
G.
,
Tanter
,
M.
, and
Fink
,
M.
(
2009
). “
Shear wave spectroscopy for quantification of human soft tissues visco-elasticity
,”
IEEE Trans. Med. Imaging
28
,
313
322
.
16.
Doherty
,
J. R.
,
Trahey
,
G. E.
,
Nightingale
,
K. R.
, and
Palmeri
,
M. L.
(
2013
). “
Acoustic radiation force elasticity imaging in diagnostic ultrasound
,”
IEEE Trans Ultrason Ferroelectr Freq Control
60
,
685
701
.
17.
Domire
,
Z. J.
,
McCullough
,
M. B.
,
Chen
,
Q. S.
, and
An
,
K. N.
Wave attenuation as a measure of muscle quality as measured by magnetic resonance elastography: Initial results
,”
J. Biomech.
42
,
537
540
(
2009
).
18.
Doyley
,
M. M.
(
2012
). “
Model-based elastography: A survey of approaches to the inverse elasticity problem
,”
Phys. Med. Biol.
57
,
R35
R73
.
19.
Gennisson
,
J. L.
, and
Cloutier
,
G.
(
2006
). “
Sol-gel transition in agar-gelatin mixtures studied with transient elastography
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
53
,
716
723
.
20.
Gennisson
,
J. L.
,
Lerouge
,
S.
, and
Cloutier
,
G.
(
2006
). “
Assessment by transient elastography of the viscoelastic properties of blood during clotting
,”
Ultrasound Med. Biol.
32
,
1529
1537
.
21.
Greenleaf
,
J. F.
,
Fatemi
,
M.
, and
Insana
,
M.
(
2003
). “
Selected methods for imaging elastic properties of biological tissues
,”
Ann. Rev. Biomed. Eng.
5
,
57
78
.
22.
Jacob
,
X.
,
Catheline
,
S.
,
Gennisson
,
J. L.
,
Barriere
,
C.
,
Royer
,
D.
, and
Fink
,
M.
(
2007
). “
Nonlinear shear wave interaction in soft solids
,”
J. Acoust. Soc. Am.
122
,
1917
1926
.
23.
Liu
,
Z.
, and
Bilston
,
L.
(
2000
). “
On the viscoelastic character of liver tissue: Experiments and modelling of the linear behaviour
,”
Biorheology
37
,
191
201
.
24.
Mason
,
W. P.
(
1950
). “
Hysteresis losses in solid materials
,” in
Piezoelectric Crystals and their Application in Ultrasonics
(
Van Nostrand
,
New York
), p.
482
.
25.
Morse
,
P. M.
, and
Ingard
,
K. U.
(
1987
).
Theoretical Acoustics
(
Princeton University Press
,
Princeton, NJ
), p.
312
.
26.
Oestreicher
,
H. L.
(
1951
). “
Field and impedance of an oscillating sphere in a viscoelastic medium with an application to biophysics
,”
J. Acoust. Soc. Am.
23
,
707
714
.
27.
Oudry
,
J.
,
Bastard
,
C.
,
Miette
,
V.
,
Willinger
,
R.
, and
Sandrin
,
L.
(
2009
). “
Copolymer-in-oil phantom materials for elastography
,”
Ultrasound Med. Biol.
35
,
1185
1197
.
28.
Palmeri
,
M. L.
,
Wang
,
M. H.
,
Dahl
,
J. J.
,
Frinkley
,
K. D.
, and
Nightingale
,
K. R.
(
2008
). “
Quantifying hepatic shear modulus in vivo using acoustic radiation force
,”
Ultrasound Med. Biol.
34
,
546
558
.
29.
Parker
,
K. J.
(
2014
). “
A microchannel flow model for soft tissue elasticity
,”
Phys. Med. Biol.
59
,
4443
4457
.
30.
Parker
,
K. J.
,
Doyley
,
M. M.
, and
Rubens
,
D. J.
(
2011
). “
Imaging the elastic properties of tissue: The 20 year perspective
,”
Phys. Med. Biol.
56
,
R1
R29
.
31.
Sandrin
,
L.
,
Catheline
,
S.
,
Tanter
,
M.
,
Hennequin
,
X.
, and
Fink
,
M.
(
1999
). “
Time-resolved pulsed elastography with ultrafast ultrasonic imaging
,”
Ultrason. Imaging
21
,
259
272
.
32.
Sandrin
,
L.
,
Tanter
,
M.
,
Gennisson
,
J. L.
,
Catheline
,
S.
, and
M.
Fink
(
2002
). “
Shear elasticity probe for soft tissues with 1-D transient elastography
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
49
,
436
446
.
33.
Sarvazyan
,
A. P.
,
Rudenko
,
O. V.
,
Swanson
,
S. D.
,
Fowlkes
,
J. B.
, and
Emelianov
,
S. Y.
(
1998
). “
Shear wave elasticity imaging: A new ultrasonic technology of medical diagnostics
,”
Ultrasound Med. Biol.
24
,
1419
1435
.
34.
Sarvazyan
,
A. P.
,
Urban
,
M. W.
, and
Greenleaf
,
J. F.
(
2013
). “
Acoustic waves in medical imaging and diagnostics
,”
Ultrasound Med. Biol.
39
,
1133
1146
.
35.
Schmitt
,
C.
,
Hadj Henni
,
A.
, and
Cloutier
,
G.
(
2011
). “
Characterization of blood clot viscoelasticity by dynamic ultrasound elastography and modeling of the rheological behavior
,”
J. Biomech.
44
,
622
629
.
36.
Sinkus
,
R.
,
Tanter
,
M.
,
Xydeas
,
T.
,
Catheline
,
S.
,
Bercoff
,
J.
, and
Fink
,
M.
(
2005
). “
Viscoelastic shear properties of in vivo breast lesions measured by MR elastography
,”
Magn. Reson. Imaging
23
,
159
165
.
37.
Urban
,
M. W.
,
Chen
,
S. G.
, and
Greenleaf
,
J. F.
(
2009
). “
Error in estimates of tissue material properties from shear wave dispersion ultrasound vibrometry
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
56
,
748
758
.
38.
Urban
,
M. W.
,
Fatemi
,
M.
, and
Greenleaf
,
J. F.
(
2010
). “
Modulation of ultrasound to produce multifrequency radiation force
,”
J. Acoust. Soc. Am.
127
,
1228
1238
.
39.
Urban
,
M. W.
, and
Greenleaf
,
J. F.
(
2009
). “
A Kramers-Kronig-based quality factor for shear wave propagation in soft tissue
,”
Phys. Med. Biol.
54
,
5919
5933
.
40.
Urban
,
M. W.
,
Nenadic
,
I. Z.
,
Mitchell
,
S. A.
,
Chen
,
S.
, and
Greenleaf
,
J. F.
(
2011
). “
Generalized response of a sphere embedded in a viscoelastic medium excited by an ultrasonic radiation force
,”
J. Acoust. Soc. Am.
130
,
1133
1141
.
41.
Wochner
,
M. S.
,
Hamilton
,
M. F.
,
Ilinskii
,
Y. A.
, and
Zabolotskaya
,
E. A.
(
2008
). “
Cubic nonlinearity in shear wave beams with different polarizations
,”
J. Acoust. Soc. Am.
123
,
2488
2495
.
42.
Zabolotskaya
,
E. A.
,
Ilinskii
,
Y. A.
, and
Hamilton
,
M. F.
(
2007
). “
Nonlinear surface waves in soft, weakly compressible elastic media
,”
J. Acoust. Soc. Am.
121
,
1873
1878
.
You do not currently have access to this content.