A simple but general solution of Navier's equation for axisymmetric shear wave propagation in a homogeneous isotropic viscoelastic medium is presented. It is well-suited for use as a forward model for some acoustic radiation force impulse based shear wave elastography applications because it does not require precise knowledge of the strength of the source, nor its spatial or temporal distribution. Instead, it depends on two assumptions: (1) the source distribution is axisymmetric and confined to a small region near the axis of symmetry, and (2) the propagation medium is isotropic and homogeneous. The model accounts for the vector polarization of shear waves and exactly represents geometric spreading of the shear wavefield, whether spherical, cylindrical, or neither. It makes no assumption about the frequency dependence of material parameters, i.e., it is material-model independent. Validation using measured shear wavefields excited by acoustic radiation force in a homogeneous gelatin sample show that the model accounts for well over 90% of the measured wavefield “energy.” An optimal fit of the model to simulated shear wavefields with noise in a homogeneous viscoelastic medium enables estimation of both the shear storage modulus and shear wave attenuation to within 1%.

1.
S.
Catheline
,
J. L.
Gennisson
,
G.
Delon
,
M.
Fink
,
R.
Sinkus
,
S.
Abouelkaram
, and
J.
Culioli
, “
Measuring of viscoelastic properties of homogeneous soft solid using transient elastography: An inverse problem approach
,”
J. Acoust. Soc. Am.
116
(
6
),
3734
3741
(
2004
).
2.
T.
Deffieux
,
G.
Montaldo
,
M.
Tanter
, and
M.
Fink
, “
Shear wave spectroscopy for in vivo quantification of human soft tissues visco-elasticity
,”
IEEE Trans. Med. Imag.
28
,
313
322
(
2009
).
3.
I. Z.
Nenadic
,
B.
Qiang
,
M. W.
Urban
,
H.
Zhao
,
W.
Sanchez
,
J. F.
Greenleaf
, and
S.
Chen
, “
Attenuation measuring ultrasound shearwave elastography and in vivo application in post-transplant liver patients
,”
Phys. Med. Biol.
62
(
2
),
484
500
(
2017
).
4.
N. C.
Rouze
,
M. L.
Palmeri
, and
K. R.
Nightingale
, “
An analytic, Fourier domain description of shear wave propagation in a viscoelastic medium using asymmetric Gaussian sources
,”
J. Acoust. Soc. Am.
138
(
2
),
1012
1022
(
2015
).
5.
G.
Ferraioli
,
P.
Parekh
,
A. B.
Levitov
, and
C.
Filice
, “
Shear wave elastography for evaluation of liver fibrosis
,”
J. Ultrasound Med.
33
(
2
),
197
203
(
2014
).
6.
J.
Oudry
,
T.
Lynch
,
J.
Vappou
,
L.
Sandrin
, and
V.
Miette
, “
Comparison of four different techniques to evaluate the elastic properties of phantom in elastography: Is there a gold standard?
,”
Phys. Med. Biol.
59
(
19
),
5775
5793
(
2014
).
7.
S.
Franchi-Abella
,
C.
Elie
, and
J.-M.
Correas
, “
Performances and limitations of several ultrasound-based elastography techniques: A phantom study
,”
Ultrasound Med. Biol.
43
(
10
),
2402
2415
(
2017
).
8.
K. J.
Parker
,
M. M.
Doyley
, and
D. J.
Rubens
, “
Imaging the elastic properties of tissue: The 20 year perspective
,”
Phys. Med. Biol.
56
(
1
),
R1
R29
(
2011
).
9.
S.
Chen
,
M. W.
Urban
,
C.
Pislaru
,
R.
Kinnick
,
Y.
Zheng
,
A.
Yao
, and
J. F.
Greenleaf
, “
Shearwave dispersion ultrasound vibrometry (SDUV) for measuring tissue elasticity and viscosity
,”
IEEE Trans. Ultrason., Ferroelectr., Freq. Control
56
,
55
62
(
2009
).
10.
K. R.
Nightingale
,
N. C.
Rouze
,
S. J.
Rosenzweig
,
M. H.
Wang
,
M. F.
Abdelmalek
,
C. D.
Guy
, and
M. L.
Palmeri
, “
Derivation and analysis of viscoelastic properties in human liver: Impact of frequency on fibrosis and steatosis staging
,”
IEEE Trans. Ultrason., Ferroelectr., Freq. Control
62
,
165
175
(
2015
).
11.
S.
Kazemirad
,
S.
Bernard
,
S.
Hybois
,
A.
Tang
, and
G.
Cloutier
, “
Ultrasound shear wave viscoelastography: Model-independent quantification of the complex shear modulus
,”
IEEE Trans. Ultrason., Ferroelectr., Freq. Control
63
(
9
),
1399
1408
(
2016
).
12.
E.
Budelli
,
J.
Brum
,
M.
Bernal
,
T.
Deffieux
,
M.
Tanter
,
P.
Lema
,
C.
Negreira
, and
J.-L.
Gennisson
, “
A diffraction correction for storage and loss moduli imaging using radiation force based elastography
,”
Phys. Med. Biol.
62
(
1
),
91
106
(
2016
).
13.
N. C.
Rouze
,
C. A.
Trutna
,
Y.
Deng
,
M. L.
Palmeri
, and
K. R.
Nightingale
, “
Comparison of swei methods for measuring the frequency dependent phase velocity and attenuation in viscoelastic materials
,” in
2017 IEEE International Ultrasonics Symposium (IUS)
(
2017
), pp.
1
4
.
14.
H.
Zhao
,
P.
Song
,
M. W.
Urban
,
R. R.
Kinnick
,
M.
Yin
,
J. F.
Greenleaf
, and
S.
Chen
, “
Bias observed in time-of-flight shear wave speed measurements using radiation force of a focused ultrasound beam
,”
Ultrasound Med. Biol.
37
(
11
),
1884
1892
(
2011
).
15.
K. F.
Graff
,
Wave Motion in Elastic Solids
(
Dover
,
New York
,
1991
), pp.
274
275
.
16.
W.
Findley
,
J.
Lai
, and
K.
Onaran
,
Creep and Relaxation of Nonlinear Viscoelastic Materials: with an Introduction to Linear Viscoelasticity
(
Dover
,
New York
,
1989
), Chaps. 5 and 6, pp.
90
91
, 95–96, 108–109.
17.
A. D.
Pierce
,
Acoustics: An Introduction to its Physical Principles and Applications
(
McGraw-Hill
,
New York
,
1981
), pp.
177
178
.
18.
M.
Abramowitz
and
I. A.
Stegun
,
Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables
(
U.S. Government Printing Office
,
Washington, DC
,
1964
), pp.
358
359
.
19.
A. K.
Mal
and
S. J.
Singh
,
Deformation of Elastic Solids
(
Prentice Hall
,
Englewood Cliffs, NJ
,
1991
), pp.
81
85
, 138–142, 312–316.
20.
J.
Domínguez
and
R.
Abascal
, “
On fundamental solutions for the boundary integral equations method in static and dynamic elasticity
,”
Eng. Anal.
1
(
3
),
128
134
(
1984
).
21.
T.
Nordenfur
, “
Comparison of pushing sequences for shear wave elastography
,” Master's thesis, KTH,
School of Technology and Health (STH)
,
2013
.
22.
T. J.
Hall
,
M.
Bilgen
,
M. F.
Insana
, and
T. A.
Krouskop
, “
Phantom materials for elastography
,”
IEEE Trans. Ultrason., Ferroelectr., Freq. Control
44
,
1355
1365
(
1997
).
23.
T.
Loupas
,
J. T.
Powers
, and
R. W.
Gill
, “
An axial velocity estimator for ultrasound blood flow imaging, based on a full evaluation of the Doppler equation by means of a two-dimensional autocorrelation approach
,”
IEEE Trans. Ultrason., Ferroelectr., Freq. Control
42
,
672
688
(
1995
).
24.
R. J.
Dewall
and
T.
Varghese
, “
Improving thermal ablation delineation with electrode vibration elastography using a bidirectional wave propagation assumption
,”
IEEE Trans. Ultrason., Ferroelectr., Freq. Control
59
,
168
173
(
2012
).
25.
K. J.
Parker
,
A.
Partin
, and
D. J.
Rubens
, “
What do we know about shear wave dispersion in normal and steatotic livers?
,”
Ultrasound Med. Biol.
41
(
5
),
1481
1487
(
2015
).
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