Determining the mechanical properties of soft biological tissues can be of great importance. For example, the microstructures of many soft tissues, such as those of the human Achilles tendon, have been identified as typical anisotropic materials. This paper proposes an inverse approach that uses guided wave elastography to determine the anisotropic elastic and hyperelastic parameters of thin-walled transversely isotropic biological soft tissues. This approach was developed from the theoretical solutions for the dispersion relations of guided waves, which were derived based on a constitutive model suitable for describing the deformation behavior of such tissues. The properties of these solutions were investigated; in particular, sensitivity to data errors was addressed by introducing the concept of the condition number. To further validate the proposed inverse approach, the guided wave elastography of thin-walled transversely isotropic soft tissues was investigated using numerical experiments. The results indicated that the four constitutive parameters (other than the tensile modulus along the direction of the fibers, EL) could be determined with a good level of accuracy using this method.

1.
Barr
,
R. G.
,
Nakashima
,
K.
,
Amy
,
D.
,
Cosgrove
,
D.
,
Farrokh
,
A.
,
Schafer
,
F.
,
Bamber
,
J. C.
,
Castera
,
L.
,
Choi
,
B. I.
,
Chou
,
Y.-H.
,
Dietrich
,
C. F.
,
Ding
,
H.
,
Ferraioli
,
G.
,
Filice
,
C.
,
Friedrich-Rust
,
M.
,
Hall
,
T. J.
,
Nightingale
,
K. R.
,
Palmeri
,
M. L.
,
Shiina
,
T.
,
Suzuki
,
S.
,
Sporea
,
I.
,
Wilson
,
S.
, and
Kudo
,
M.
(
2015
). “
WFUMB guidelines and recommendations for clinical use of ultrasound elastography: Part 2: Breast
,”
Ultrasound Med. Biol.
41
,
1148
1160
.
2.
Bercoff
,
J.
,
Tanter
,
M.
, and
Fink
,
M.
(
2004
). “
Supersonic shear imaging: A new technique for soft tissue elasticity mapping
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
51
,
396
409
.
3.
Berg
,
W. A.
,
Cosgrove
,
D. O.
,
Doré
,
C. J.
,
Schäfer
,
F. K. W.
,
Svensson
,
W. E.
,
Hooley
,
R. J.
,
Ohlinger
,
R.
,
Mendelson
,
E. B.
,
Balu-Maestro
,
C.
,
Locatelli
,
M.
,
Tourasse
,
C.
,
Cavanaugh
,
B. C.
,
Juhan
,
V.
,
Stavros
,
A. T.
,
Tardivon
,
A.
,
Gay
,
J.
,
Henry
,
J.-P.
,
Cohen-Bacrie
,
C.
, and
Investigators
,
F. T. B.
(
2012
). “
Shear-wave elastography improves the specificity of breast us: The BE1 multinational study of 939 masses
,”
Radiology
262
,
435
449
.
4.
Bernal
,
M.
,
Nenadic
,
I.
,
Urban
,
M. W.
, and
Greenleaf
,
J. F.
(
2011
). “
Material property estimation for tubes and arteries using ultrasound radiation force and analysis of propagating modes
,”
J. Acoust. Soc. Am.
129
,
1344
1354
.
5.
Brum
,
J.
,
Bernal
,
M.
,
Gennisson
,
J. L.
, and
Tanter
,
M.
(
2014
). “
In vivo evaluation of the elastic anisotropy of the human Achilles tendon using shear wave dispersion analysis
,”
Phys. Med. Biol.
59
,
505
523
.
6.
Chadwick
,
P.
, and
Whitworth
,
A. M.
(
1986
). “
Exceptional waves in a constrained elastic body
,”
Q. J. Mech. Appl. Math.
39
,
309
325
.
7.
Chen
,
S.
,
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
.
8.
Couade
,
M.
,
Pernot
,
M.
,
Prada
,
C.
,
Messas
,
E.
,
Emmerich
,
J.
,
Bruneval
,
P.
,
Criton
,
A.
,
Fink
,
M.
, and
Tanter
,
M.
(
2010
). “
Quantitative assessment of arterial wall biomechanical properties using shear wave imaging
,”
Ultrasound Med. Biol.
36
,
1662
1676
.
9.
Dayal
,
V.
, and
Kinra
,
V. K.
(
1989
). “
Leaky Lamb waves in an anisotropic plate. I: An exact solution and experiments
,”
J. Acoust. Soc. Am.
85
,
2268
2276
.
10.
Destrade
,
M.
(
2015
). “
Incremental equations for soft fibrous materials
,” in
Nonlinear Mechanics of Soft Fibrous Materials
, edited by
L.
Dorfmann
and
R. W.
Ogden
(
Springer Vienna
,
Vienna
), pp.
233
267
.
11.
Eby
,
S. F.
,
Song
,
P.
,
Chen
,
S.
,
Chen
,
Q.
,
Greenleaf
,
J. F.
, and
An
,
K.-N.
(
2013
). “
Validation of shear wave elastography in skeletal muscle
,”
J. Biomech.
46
,
2381
2387
.
12.
Evans
,
A.
,
Whelehan
,
P.
,
Thomson
,
K.
,
McLean
,
D.
,
Brauer
,
K.
,
Purdie
,
C.
,
Jordan
,
L.
,
Baker
,
L.
, and
Thompson
,
A.
(
2010
). “
Quantitative shear wave ultrasound elastography: Initial experience in solid breast masses
,”
Breast Cancer Res.
12
,
R104
.
13.
Ferraioli
,
G.
,
Filice
,
C.
,
Castera
,
L.
,
Choi
,
B. I.
,
Sporea
,
I.
,
Wilson
,
S. R.
,
Cosgrove
,
D.
,
Dietrich
,
C. F.
,
Amy
,
D.
,
Bamber
,
J. C.
,
Barr
,
R.
,
Chou
,
Y.-H.
,
Ding
,
H.
,
Farrokh
,
A.
,
Friedrich-Rust
,
M.
,
Hall
,
T. J.
,
Nakashima
,
K.
,
Nightingale
,
K. R.
,
Palmeri
,
M. L.
,
Schafer
,
F.
,
Shiina
,
T.
,
Suzuki
,
S.
, and
Kudo
,
M.
(
2015
). “
WFUMB guidelines and recommendations for clinical use of ultrasound elastography: Part 3: Liver
,”
Ultrasound Med. Biol.
41
,
1161
1179
.
14.
Fukashiro
,
S.
,
Komi
,
P. V.
,
Järvinen
,
M.
, and
Miyashita
,
M.
(
1995
). “
In vivo achilles tendon loading' during jumping in humans
,”
Eur. J. Appl. Physiol. Occup. Physiol.
71
,
453
458
.
15.
Gasser
,
T. C.
,
Ogden
,
R. W.
, and
Holzapfel
,
G. A.
(
2006
). “
Hyperelastic modelling of arterial layers with distributed collagen fibre orientations
,”
J. R. Soc. Interf.
3
,
15
35
.
16.
Gennisson
,
J.-L.
,
Deffieux
,
T.
,
Macé
,
E.
,
Montaldo
,
G.
,
Fink
,
M.
, and
Tanter
,
M.
(
2010
). “
Viscoelastic and anisotropic mechanical properties of in vivo muscle tissue assessed by supersonic shear imaging
,”
Ultrasound Med. Biol.
36
,
789
801
.
17.
Guo
,
J.
,
Hirsch
,
S.
,
Scheel
,
M.
,
Braun
,
J.
, and
Sack
,
I.
(
2016
). “
Three-parameter shear wave inversion in MR elastography of incompressible transverse isotropic media: Application to in vivo lower leg muscles
,”
Magn. Reson. Med.
75
,
1537
1545
.
18.
Jiang
,
Y.
,
Li
,
G. Y.
,
Qian
,
L. X.
,
Hu
,
X. D.
,
Liu
,
D.
,
Liang
,
S.
, and
Cao
,
Y.
(
2015b
). “
Characterization of the nonlinear elastic properties of soft tissues using the supersonic shear imaging (SSI) technique: Inverse method, ex vivo and in vivo experiments
,”
Med. Image Anal.
20
,
97
111
.
19.
Jiang
,
Y.
,
Li
,
G.
,
Qian
,
L.-X.
,
Liang
,
S.
,
Destrade
,
M.
, and
Cao
,
Y.
(
2015a
). “
Measuring the linear and nonlinear elastic properties of brain tissue with shear waves and inverse analysis
,”
Biomech. Model. Mechanobiol.
14
,
1119
1128
.
20.
Kanai
,
H.
(
2005
). “
Propagation of spontaneously actuated pulsive vibration in human heart wall and in vivo viscoelasticity estimation
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
52
,
1931
1942
.
21.
Kvist
,
M.
(
1994
). “
Achilles tendon injuries in athletes
,”
Sports Med.
18
,
173
201
.
22.
Li
,
G.-Y.
, and
Cao
,
Y.
(
2017
). “
Mechanics of ultrasound elastography
,”
Proc. R. Soc. A: Math Phys. Eng. Sci.
473
,
20160841
.
23.
Li
,
G.-Y.
,
He
,
Q.
,
Mangan
,
R.
,
Xu
,
G.
,
Mo
,
C.
,
Luo
,
J.
,
Destrade
,
M.
, and
Cao
,
Y.
(
2017a
). “
Guided waves in pre-stressed hyperelastic plates and tubes: Application to the ultrasound elastography of thin-walled soft materials
,”
J. Mech. Phys. Solids
102
,
67
79
.
24.
Li
,
G.-Y.
,
He
,
Q.
,
Qian
,
L.-X.
,
Geng
,
H.
,
Liu
,
Y.
,
Yang
,
X.-Y.
,
Luo
,
J.
, and
Cao
,
Y.
(
2016a
). “
Elastic Cherenkov effects in transversely isotropic soft materials—II: Ex vivo and in vivo experiments
,”
J. Mech. Phys. Solids
94
,
181
190
.
25.
Li
,
G.-Y.
,
He
,
Q.
,
Xu
,
G.
,
Jia
,
L.
,
Luo
,
J.
, and
Cao
,
Y.
(
2017b
). “
An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves
,”
J. Biomech.
51
,
97
104
.
26.
Li
,
G.-Y.
,
Zheng
,
Y.
,
Liu
,
Y.
,
Destrade
,
M.
, and
Cao
,
Y.
(
2016b
). “
Elastic Cherenkov effects in transversely isotropic soft materials—I: Theoretical analysis, simulations and inverse method
,”
J. Mech. Phys. Solids
96
,
388
410
.
27.
Makarov
,
S.
,
Chilla
,
E.
, and
Fröhlich
,
H. J.
(
1995
). “
Determination of elastic constants of thin films from phase velocity dispersion of different surface acoustic wave modes
,”
J. Appl. Phys.
78
,
5028
5034
.
28.
Merodio
,
J.
, and
Ogden
,
R. W.
(
2005
). “
Mechanical response of fiber-reinforced incompressible non-linearly elastic solids
,”
Int. J. Nonlinear Mech.
40
,
213
227
.
29.
Murphy
,
J. G.
(
2013
). “
Transversely isotropic biological, soft tissue must be modelled using both anisotropic invariants
,”
Eur. J. Mech. A–Solids
42
,
90
96
.
30.
Nam
,
N. T.
,
Merodio
,
J.
,
Ogden
,
R. W.
, and
Vinh
,
P. C.
(
2016
). “
The effect of initial stress on the propagation of surface waves in a layered half-space
,”
Int. J. Solids Struct.
88–89
,
88
100
.
31.
Nenadic
,
I.
,
Mynderse
,
L.
,
Husmann
,
D.
,
Mehrmohammadi
,
M.
,
Bayat
,
M.
,
Singh
,
A.
,
Denis
,
M.
,
Urban
,
M.
,
Alizad
,
A.
, and
Fatemi
,
M.
(
2016
). “
Noninvasive evaluation of bladder wall mechanical properties as a function of filling volume: Potential application in bladder compliance assessment
,”
PLoS One
11
,
e0157818
.
32.
Nguyen
,
T. M.
,
Couade
,
M.
,
Bercoff
,
J.
, and
Tanter
,
M.
(
2011
). “
Assessment of viscous and elastic properties of sub-wavelength layered soft tissues using shear wave spectroscopy: Theoretical framework and in vitro experimental validation
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
58
,
2305
2315
.
33.
Ogden
,
R. W.
(
2007
). “
Incremental statics and dynamics of pre-stressed elastic materials
,” in
Waves in Nonlinear Pre-Stressed Materials
, edited by
M.
Destrade
and
G.
Saccomandi
(
Springer Vienna
,
Vienna
), pp.
1
26
.
34.
Otténio
,
M.
,
Destrade
,
M.
, and
Ogden
,
R. W.
(
2007
). “
Acoustic waves at the interface of a pre-stressed incompressible elastic solid and a viscous fluid
,”
Int. J. Nonlinear Mech.
42
,
310
320
.
35.
Palmeri
,
M. L.
, and
Nightingale
,
K. R.
(
2011
). “
Acoustic radiation force-based elasticity imaging methods
,”
Interf. Focus
1
,
553
564
.
36.
Prikazchikov
,
D. A.
, and
Rogerson
,
G. A.
(
2003
). “
Some comments on the dynamic properties of anisotropic and strongly anisotropic pre-stressed elastic solids
,”
Int. J. Eng. Sci.
41
,
149
171
.
37.
Rose
,
J. L.
(
2014
).
Ultrasonic Guided Waves in Solid Media
(
Cambridge University Press
,
New York
), pp.
209
244
.
38.
Rouze
,
N. C.
,
Wang
,
M. H.
,
Palmeri
,
M. L.
, and
Nightingale
,
K. R.
(
2013
). “
Finite element modeling of impulsive excitation and shear wave propagation in an incompressible, transversely isotropic medium
,”
J. Biomech.
46
,
2761
2768
.
39.
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
.
40.
Tanter
,
M.
,
Bercoff
,
J.
,
Athanasiou
,
A.
,
Deffieux
,
T.
,
Gennisson
,
J.-L.
,
Montaldo
,
G.
,
Muller
,
M.
,
Tardivon
,
A.
, and
Fink
,
M.
(
2008
). “
Quantitative assessment of breast lesion viscoelasticity: Initial clinical results using supersonic shear imaging
,”
Ultrasound Med. Biol.
34
,
1373
1386
.
41.
Van Beers
,
B. E.
,
Daire
,
J.-L.
, and
Garteiser
,
P.
(
2015
). “
New imaging techniques for liver diseases
,”
J. Hepatol.
62
,
690
700
.
42.
Yeh
,
C.-H.
, and
Yang
,
C.-H.
(
2011
). “
Characterization of mechanical and geometrical properties of a tube with axial and circumferential guided waves
,”
Ultrasonics
51
,
472
479
.
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