IEEE TransacTIons on UlTrasonIcs, FErroElEcTrIcs, and FrEqUEncy conTrol, vol. 59, no. 4, aprIl 2012 703

0885–3010/$25.00 © 2012 IEEE

Application of 1-D Transient Elastography for the Shear Modulus Assessment

of Thin-Layered Soft Tissue: Comparison With Supersonic Shear Imaging Technique

Javier Brum, Jean-luc Gennisson, Thu-Mai nguyen, nicolas Benech, Mathias Fink, Mickael Tanter, and carlos negreira

Abstract—Elasticity estimation of thin-layered soft tissues has gained increasing interest propelled by medical applica-tions like skin, corneal, or arterial wall shear modulus assess-ment. In this work, the authors propose one-dimensional tran-sient elastography (1DTE) for the shear modulus assessment of thin-layered soft tissue. Experiments on three phantoms with different elasticities and plate thicknesses were performed. First, using 1DTE, the shear wave speed dispersion curve in-side the plate was obtained and validated with finite difference simulation. No dispersive effects were observed and the shear wave speed was directly retrieved from time-of-flight measure-ments. Second, the supersonic shear imaging (SSI) technique (considered to be a gold standard) was performed. For the SSI technique, the propagating wave inside the plate is guid-ed as a Lamb wave. Experimental SSI dispersion curves were compared with finite difference simulation and fitted using a generalized Lamb model to retrieve the plate bulk shear wave speed. Although they are based on totally different mechanical sources and induce completely different diffraction patterns for the shear wave propagation, the 1DTE and SSI techniques resulted in similar shear wave speed estimations. The main ad-vantage of the 1DTE technique is that bulk shear wave speed can be directly retrieved without requiring a dispersion model.

I. Introduction

Elastography is the common name for many tech-niques developed within the past two decades for non-

invasive assessment of the mechanical properties of bio-logical soft tissues with application to medical diagnosis. These techniques can mainly be divided into two groups: static and dynamic elastography. In static elastography [1], [2] a compression is applied by pressing an ultrasonic probe on the tissue. a strain map is obtained by compar-ing the displacements before and after each compression. However, because of the lack of information on the stress to which the tissue is subjected, this method does not pro-vide a quantitative elasticity estimation. To overcome this limitation, a set of techniques based on shear wave propa-

gation inside the tissue has been developed during the past decade. These techniques can be categorized under the name dynamic elastography. They consist basically of three steps: first, the tissue is mechanically stressed, resulting in shear wave generation; second, the induced displacements are imaged; and, finally, the tissue’s elastic properties are deduced from the measured displacement field. Because shear waves are used, and their speed is related to the tissue’s shear modulus, the dynamic ap-proach provides a quantitative estimation of the tissue’s elasticity.

There are several ways to image the displacement field (e.g., ultrasound [3], [4] or MrI [5]) and to generate the shear waves (e.g., mechanical vibrator [6]–[8] or ultra-sound radiation force [9]–[11]). one-dimensional transient elastography (1dTE) [6] uses a low-frequency vibrator as an external shear wave source, whereas the supersonic shear imaging (ssI) [11] technique consists of generating broadband shear waves inside the sample by using the radiation force created by a focused ultrasonic beam. For both techniques, the shear wave propagation is tracked using an ultrafast ultrasound scanner. The shear wave speed (cT) is retrieved by applying a time of flight algo-rithm to the acquired displacement field. The shear modu-lus (µ) of the medium is then retrieved through the well-known relationship: µ = ρcT

2, where ρ is the medium’s density. Both techniques have been successfully applied to noninvasively determine the mechanical parameters of liv-ing tissues such as breast [12], liver [13], [14], or muscle [15], [16].

The applicability of the ssI technique to viscoelastic assessment of thin soft tissues has recently been demon-strated in the cornea [17] and the arterial wall [18]. In these specific cases, the wavelength of the propagating wave is of the order of the cornea/arterial wall thickness, leading to a propagation which is related to the leaky lamb wave theory of guided waves. The bulk shear wave speed, and thus the shear modulus, are retrieved from the lamb wave dispersion curve using a specific model [18], [19]. The applicability of 1dTE for arterial shear modulus assessment was tested on one arterial phantom at a fixed excitation frequency of 150 Hz [20].

In the present work, the authors propose and validate the use of 1dTE for the quantitative assessment of the

Manuscript received november 4, 2011; accepted december 27, 2011. J. Brum, n. Benech, and c. negreira are with the laboratorio de

acústica Ultrasonora, Instituto de Física, Facultad de ciencias, Monte-video, Uruguay (e-mail: jbrum@fisica.edu.uy).

J.-l. Gennisson, T.-M. nguyen, M. Fink, and M. Tanter are with the Institut langevin ondes et Images, Ecole supérieure de physique et de chemie Industrielles (EspcI, parisTech), centre national de la recher-che scientifique (cnrs) UMr 7587, Inserm Erl U979, paris, France.

doI: http://dx.doi.org/10.1109/TUFFc.2012.2248

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shear modulus of thin-layered tissues. First, 1dTE experi-ments were simulated by finite difference simulation in plates with exactly known thickness and elasticity for a wide range of excitation frequencies (100 to 400 Hz). sec-ond, 1dTE experiments on three different phantoms with different elasticities and layer thicknesses were performed, also for a wide range of excitation frequencies. The experi-mental dispersion curves were compared with the ones ob-tained by 1dTE finite difference simulation. Moreover, to complete the validation process, the ssI technique (con-sidered to be a gold standard) was applied on each sample. For this case, the propagating wave is guided as lamb wave. In previous works [18], [19] the lamb wave model used to retrieve the shear modulus of thin tissue layers as-sumed that the tissue layer was surrounded by an inviscid fluid. In this work, a more general lamb model is pro-posed because the plates under study were surrounded by an elastic medium. Experimental dispersion curves were compared with the ones obtained by finite difference simu-lation and fitted by numerically solving the generalized lamb model. This allows the retrieval of the bulk shear wave speed for each layer. Finally, the results obtained by 1dTE and ssI are discussed, compared, and shown to be in good agreement.

II. Materials and Methods

A. Phantom Preparation

Experiments on three types of phantoms were per-formed. phantom dimensions were 10 × 15 × 15 cm. phantom a consists of an elastic plate of 1.1 ± 0.1 mm thickness surrounded by polyvinyl alcohol cryogel (pVa-c) [21]. The elastic plate, also made of pVa-c, underwent 4 freeze-thaw cycles, whereas the surrounding gel only un-derwent one cycle. phantom B consists of a three-cycle pVa-c layer of 2.2 ± 0.2 mm thickness surrounded by a one-cycle pVa-c gel. The number of freeze-thaw cycles is directly related to the final phantom’s shear wave speed: the more cycles the sample underwent, the higher is the sample’s shear wave speed [21]. phantom c consists of an agar layer (2% agar diluted in water) of 2.2 ± 0.2 mm thickness surrounded by gelatin-based gel (4% gelatine, 1% agar as ultrasound scatterers diluted in water). layer thicknesses were measured from the B-mode image. For each phantom, a homogenous bulk gel sample of dimen-sions 4 × 4 × 10 cm was made from the same melt follow-ing a procedure identical to that of the corresponding cen-tral layer to measure the bulk shear wave dispersion curve of the material; e.g., bulk sample a is a bulk sample made of the same material and following the same procedure as the central layer of phantom a. no bulk samples were made for the outer layers of the different phantoms. For each phantom, both 1dTE and ssI were applied succes-sively without moving the probe. The experimental setup is presented in Fig. 1.

B. 1-D Transient Elastography

For the 1dTE experiment, a 5-mm-diameter piston at-tached to a mechanical vibrator (mini-shaker, type 4810, Bruel & Kjaer, nærum, denmark) working from 100 to 400 Hz acts on the phantom’s surface, generating shear waves inside the sample [Fig. 1(b)]. For each frequency, the vibrator was excited with a one-cycle sinusoid at the corresponding central frequency. as the piston taps on the phantom’s surface, two types of waves are generated: a longitudinal wave and a shear wave. Because of the cylin-drical symmetry of the problem, both wave propagation directions are parallel to the piston axis. although shear waves are purely transverse in the far field (where they can be considered as plane waves), they have a longitudi-nal component in the near field of the piston because of diffraction effects [6], [7]. This fact, represented in Fig. 2, can be explained using Huygen’s principle: the summed contributions of transversely polarized shear waves com-ing from secondary sources give rise to a globally longi-tudinally polarized shear wave on the piston axis. This longitudinally polarized shear wave propagating along the piston axis is imaged using an ultrafast ultrasound scan-ner at 3000 frames/s (aixplorer, supersonic Imagine, aix en provence, France) and an ultrasonic array (256 ele-ments, 8-MHz central frequency, pitch 0.2 mm; sl 15–4, supersonic Imagine, aix en provence, France) through an rF data cross-correlation algorithm [4], [7].

Fig. 1. (a) Experimental setup. (b) 1dTE: a low-frequency elastic field was created using a mechanical vibrator. The black arrows indicate the propagation direction. (c) ssI: an elastic field was created using ultra-sonic radiation force. In both cases, the displacement field was captured using an ultrafast ultrasound scanner. CT is the shear wave speed in the plate, C T

g is the shear wave speed in the surrounding gel.

brum et al.: 1-dTE for shear modulus assessment of thin-layered soft tissue 705

C. Supersonic Shear Imaging

For the ssI experiment, the ultrasonic radiation force is used as a shear wave source [Fig. 1(c)]. once again, an ultrafast ultrasound scanner (aixplorer) is used to drive a conventional ultrasonic probe by per-channel program-ming in transmit and reception mode. The ssI technique allows ultrasound to be focused on several points (pushing sequence) along the ultrasonic axis to generate a quasi-plane shear wave [11]. In the first step, a pushing sequence consisting of three pushing points (100-μs duration each) at three different depths, spaced 5 mm apart and centered on the central layer position, is used to induce a transient axial displacement. as the sample relaxes, a shear wave propagating perpendicularly to the ultrasonic beam axis is generated. second, the system is switched into an ultra-fast imaging mode to image the shear wave propagation. For large homogeneous media, the elasticity is directly deduced from the shear wave speed. In the case of thin-layered soft tissues, where the wave propagation is guided along the plate because of the successive reflections on the plate boundaries, the relationship between shear wave speed and elasticity is more complex [19] as will be de-scribed in what follows.

D. Generalized Lamb Wave Model

as mentioned in section I, in previous works, a lamb wave model was used to retrieve the shear modulus of ar-teries [18] and thin tissue layers [19] submerged in water. The leaky lamb wave model used in those works assumes that the artery/thin tissue layer is surrounded by an invis-cid fluid. The influence of a non-fluid surrounding medium was qualitatively studied in [18] using numerical simula-tion, but it was not included in the model. In this work, the thin plate is embedded in an elastic medium, so a gen-eralized lamb model is needed which takes this effect into account. Each phantom was considered as a three-layered

structure. Multilayered structures have been widely stud-ied in nondestructive evaluation [22] and seismology [23]. The generalized lamb wave model is achieved by adapting the global matrix technique for modeling elastic waves in multilayered media developed by lowe [22], [24] for the simple case of an isotropic elastic plate embedded in an infinite elastic medium.

Briefly, the global matrix technique consists of the fol-lowing steps: first, the wave propagation model is con-structed by determining the nature of the bulk waves which can exist in each elastic isotropic layer. once this nature is set, the stresses and displacements within a layer can be expressed in terms of the amplitudes of all of the bulk waves that can exist in that layer. The stresses and displacements at the boundaries of each layer are then combined with the boundary conditions to describe the entire system in one large global matrix that relates the bulk wave amplitudes to the physical constraints. Fig. 3 presents the case for a single layer.

at each interface, it is sufficient to assume six par-tial plane waves (Fig. 3). Inside the plate, there are four waves: two shear waves (s±) and two longitudinal waves (l±), propagating with positive (s+, l+) and negative (s−, l−) x1 components. These four waves create two kinds of waves (one shear and one longitudinal wave) in the surrounding medium. Each partial wave can be de-scribed by

A i k x k x ti i± ⋅ ± + ⋅ −exp( [ ]),1 3 ω (1)

where ki2 = ω 2 2/ci − k2. The subscript i = l, T stands for the longitudinal or shear partial wave, c denotes the phase velocity of each type of wave, and the + and − signs cor-respond to a wave moving with positive (downward) or negative (upward) x1 component. There is no dependence on the x2 direction because of mirror symmetry along the plate plane. By applying the superposition principle and the continuity at each of the interfaces for the two dis-placement components u1 and u3, the normal stress σ11

Fig. 2. schematic representation of diffraction effects present in 1dTE experiment. The summed contributions of transversely polarized shear waves coming from secondary sources give rise to a globally longitudi-nally polarized shear wave on the piston axis.

Fig. 3. sample geometry used for the generalized lamb wave model. In each layer, the partial waves (l±, s±) that combined to generate the guided wave are shown. The longitudinal wave speed, shear wave speed, and medium density are, respectively, Cl, CT, ρ for the plate, and C CL

gTg g, ,ρ for the surrounding medium.

IEEE TransacTIons on UlTrasonIcs, FErroElEcTrIcs, and FrEqUEncy conTrol, vol. 59, no. 4, aprIl 2012706

and the shear stress σ13 of each partial wave amplitude Ai± can be determined by

G A[ ]{ } = 0, (2)

where [G] is the global matrix and {a} is a vector of par-tial wave amplitudes. Further details on the derivation of (2) can be found in [22], [25]. The global matrix represents the complete system. For the particular case treated in this work, it consists of eight equations: four for the cen-tral layer and four for the semi-infinite half-spaces which are treated as layers. Based on [24], the form of the global matrix is (3), see above, where Fl,T = e i k h⋅ ⋅L,T /( ),2 α = ρ (ω2 − 2 ∙ C T

2 ∙ k2), and β = 2ρ ∙ k ∙ C T2, h is the plate thickness,

ρ is the medium’s density, and ω = 2π ∙ f, where f is the wave frequency. The equations for FL,T

g , αg, and βg are similar, but the longitudinal/shear wave speeds and den-sity are replaced by those of the surrounding medium. In (3), each column stands for one partial wave amplitude in each layer, including the two half-spaces. Each row corre-sponds to the continuity of the displacements and stresses applied in each layer interface.

To obtain nontrivial solutions for the vector of partial wave amplitudes A, the determinant of the global matrix must be zero. This constraint will allow only particular lamb wave numbers k for a given frequency, i.e., the dis-persion curves for each mode. By combining columns and rows, the determinant of the global matrix can be calcu-

lated as in (4), see above. as described in (4), the determi-nant of the global matrix is calculated as a product of two determinants. The determinant on the left corresponds to the antisymmetric modes and the one on the right corre-sponds to the symmetric modes. In the framework of ssI, the radiation force mainly induces displacements perpen-dicular to the plate, generating mainly an antisymmetric mode; therefore the attention will be focused on solving only one determinant of (4) using a nelder-Mead minimi-zation method [19], [25], [26].

E. Lamb Wave Empirical Formula

For some particular medical applications (shear wave imaging in media such as the cornea, skin, or arterial wall), it is much simpler to use an empirical formula based on the lamb theory [see (5)] to derive the plate shear wave speed rather than solving (4). In (5), V is the phase velocity of the guided lamb wave:

Vh c

=⋅ ⋅ω T

2 3. (5)

This empirical formula was first proposed by couade et al. [18] for the shear modulus assessment of the arterial wall. The validity of this empirical formula was then discussed in the work of nguyen et al. [19] in the case of thin elastic plates submerged in water. In [19], (5) was tested for plate

[ ]G

k F k F k F k F k F k F

k F k F k F

=

− ⋅ − ⋅ − ⋅ ⋅ ⋅ ⋅

⋅ − ⋅ − ⋅

∗ ∗Lg

Lg

Tg

L L L L T T

Lg

Tg

Tg

0 0

LL L T Tg

Lg g

Tg

Tg

L L T

∗ ∗

∗

− ⋅ − ⋅ ⋅

⋅ ⋅ ⋅ − ⋅ − ⋅ ⋅

k F k F k F

i F i k F i F i F i kT T 0 0

α β α α β ⋅⋅ − ⋅ ⋅

− ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅ − ⋅

∗

∗

F i k F

i k F i F i k F i k F i FT T T

gLg

Lg g

Tg

L L L L

β

β α β β α

0 0

TT T

L L L L T T Lg

Lg

Tg

L

∗

∗ ∗

− ⋅

⋅ − ⋅ − ⋅ − ⋅ − ⋅ ⋅

⋅ ⋅

i F

k F k F k F k F k F k F

k F k F

α 0 0

0 0

0 0 LL T T T Lg

Tg

Tg

L L T T

∗ ∗

∗

⋅ − ⋅ − ⋅ − ⋅

⋅ ⋅ − ⋅ ⋅ ⋅ ⋅

k F k F k F k F

i F i F i k F i k FT

T0 0 α α β β TTg

Lg g

Tg

Tg

L L L L T Tg

∗

∗ ∗

− ⋅ ⋅ ⋅

⋅ ⋅ − ⋅ ⋅ ⋅ ⋅ − ⋅

i F i k F

i k F i k F i F i F i

α β

β β α α β0 0 kk F i FLg

Lg g

Tg⋅ − ⋅

α

→ −

→ −

→ −

→ −

→

u h

u h

h

h

u h

1

3

11

13

1

2

2

2

2

2

( )

( )

( )

( )

( )

/

/

/

/

/

σ

σ

→→

→

→

− − + −

u h

h

h

L S L L

3

11

13

2

2

2

( )

( )

( )

/

/

/

ext ext

σ

σ

ext extS S L S+ − + +

(3)

det( )

cos cos

sin

G

k k kk h

kk h

k k i kk h

i k

=

− − ⋅ ⋅

− − ⋅ ⋅ ⋅

( ) ( )( )

Lg

LL T

Tg L

2 2

2 TTT

g gTg L

TT

gLg g

⋅

⋅ ⋅ ⋅ ⋅

− ⋅

( )( ) ( )

sin

sin sin

k h

i i kk h

kk h

i k i

2

2 2α β α β

β α ii kk h

ik h

i kk h

i kk h

⋅ ⋅ ⋅ − ⋅

⋅

⋅ ⋅ ⋅ ⋅

( ) ( )

( ) ( )

β αLL T

LL T

cos cos

sin sin

2 2

2 2−−

⋅ − ⋅ − −

⋅ ⋅ ⋅ ⋅ ⋅

( ) ( )( )

k k

kk h

kk h

k k

ik h

i k

Lg

LT

TTg

LT

cos cos

cos co

2 2

2α β ss

sin sin

k hi i k

kk h k h

i k i

T g gTg

LL T g

Lg g

2

2 2

( )( ) ( )

− ⋅ ⋅

− ⋅ ⋅ ⋅ − ⋅ −

α β

β α β α

== 0

(4)

brum et al.: 1-dTE for shear modulus assessment of thin-layered soft tissue 707

thicknesses and shear wave speeds ranging between 0.5 and 1.5 mm and 5 and 10 m/s, respectively, showing that the relative deviation of the empirical formula from simu-lated dispersion curves is as much as 15% for the thickest and softest plates. However its deviation was less than 5% for plates of 1 mm thickness whose shear wave speeds were greater than 10 m/s. Because of these limitations of the empirical formula, (5) will be used to retrieve the plate shear wave speed only for phantom a. In the case of phantoms B and c, the generalized lamb model proposed in (4) is needed.

F. Finite Differences Simulation

To verify the experimental dispersion curves and the theoretical model used, the ssI and the 1dTE experi-ments were simulated using an in-house finite-differences simulation code (acEl) developed at the langevin Insti-tute [27]. The code is based on the discretization of the equation of motion in an elastic medium by a Virieux scheme.

To model 1dTE experiments, a one-cycle sinusoid ex-citation propagating through the surrounding gel and ar-riving at the thin plate was simulated for different excita-tion frequencies (from 100 to 400 Hz with a 50-Hz step). The transient shear wave propagation inside the plate was isolated. To minimize boundary reflections, a directional filter was applied by performing a two-dimensional Fourier transform and eliminating the components for which the product ω ∙ k < 0 [28]. Then, for each excitation frequency, the phase velocity was calculated, leading to the shear wave dispersion curve. plates of 2.5- and 1-mm thickness with shear wave speeds of 10 and 5 m/s were simulated. For all 1dTE simulations, the shear wave speed of the sur-rounding medium was set to 3 m/s; the density and the longitudinal wave speed were set to be constant through the sample with values 1000 kg/m3 and 1500 m/s, respec-tively. The displacement field was sampled at a rate of 10 kHz.

For the ssI simulations, plates with thicknesses rang-ing from 0.9 to 2.5 mm and transverse speeds from 5 to 12 m/s were investigated. The shear source was defined as a broadband signal (350 to 1400 Hz), in agreement with the experimental conditions. By applying a two-dimen-sional Fourier transform to the simulated spatial-temporal displacement field at a fixed depth inside the layer, the dispersion curve is obtained [29].

III. results

A. Phantom Characterization

The ssI technique was used to obtain the dispersion curves for each bulk sample. The results for each type of bulk sample are presented in Fig. 4. By averaging the shear wave speed over the frequency range (from 500 to 1300 Hz), mean shear wave speeds of 12.1, 10.2, and

5.6 m/s were found for bulk samples a, B, and c, respec-tively, with corresponding standard deviations of 0.2, 0.2, and 0.4 m/s. These results are summarized in Table I; σ is the standard deviation and ΔcT is the mean error of the shear wave speed estimation at a given frequency.

as explained in section II, by using the ssI technique, a shear wave is created using the radiation force of ultra-sound. By focusing the ultrasonic beam on each phan-tom’s surrounding medium, a shear wave speed estimation of the material is obtained. shear wave speed values of 3.8 ± 0.2, 4.2 ± 0.2, and 1.2 ± 0.1 m/s were estimated for the surrounding materials in phantoms a, B, and c, respectively. These values will be used later to calculate/simulate the leaky lamb dispersion curves for the plate.

B. 1-D Transient Elastography

Because the goal of 1dTE is to obtain the displacement field along the piston axis, an average over the piston surface was made, transforming the two-dimensional dis-placement field into a one-dimensional seismogram. Fig. 5 presents the corresponding displacement fields obtained for 1dTE experiments and the finite-difference simulation for an excitation with a central frequency of 300 Hz. The longitudinal (l) and shear waves (s) created by the piston are clearly visible. note the similarity between the mea-sured and simulated displacement fields. The plate posi-tion is marked in each figure with dotted lines. a trans-mitted and a reflected wave are clearly visible at the plate interface. To minimize boundary reflections, a directional filter was applied by performing a two-dimensional Fourier

TaBlE I. Bulk sample characterization.

phantomcT

(m/s)σ

(m/s)Δ cT (m/s)

a 12.06 0.16 0.05B 10.22 0.19 0.06c 5.64 0.39 0.05

Fig. 4. shear wave speed dispersion curves in bulk samples of the same material as the elastic plates for phantom a (dots), phantom B (squares), and phantom c (triangles).

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transform to the measured/simulated displacement field and eliminating the components for which the product ω ∙ k < 0 [28]. This procedure filters out the waves propagat-ing along the piston axis which are traveling toward the piston.

The transient transmitted part of the shear wave was then isolated within the plate. The phase velocity was calculated by applying a linear regression on the time of flight (ToF) as a function of depth (Fig. 6). The ToF was estimated from the phase estimate using a Fourier transform of the transient wave divided by the angular frequency. Finally, the dispersion curves for the different phantoms were obtained by repeating this procedure for each excitation frequency (Fig. 7). The error provided for each individual shear wave speed estimate is the error on the linear fit.

simulated dispersion curves are presented in Fig. 7(a). By averaging the shear wave speed over the frequency range (from 100 to 400 Hz), mean shear wave speeds of 10.01, 9.97, and 4.79 m/s were found for the 10-m/s 1-mm-thick, 10-m/s 2.5-mm-thick, and 5-m/s 2.5-mm-thick plates, respectively, with corresponding standard deviations of 0.50, 0.14, and 0.13 m/s. The mean error (ΔcT) for each measurement was also calculated for the three cases, yielding 0.58, 0.20, and 0.05 m/s, respectively. This error is due to numerical noise of the simulation and the computation procedure. These results are summarized in Table II.

performing the same calculations for the 1dTE ex-perimental dispersion curves presented in Fig. 7(b), shear wave speeds of 11.44 ± 0.46, 10.37 ± 0.55, and 5.58 ± 0.22 m/s are found for phantoms a, B, and c, respec-

tively, with mean errors of 0.66, 0.39, and 0.49 m/s. The results are summarized in Table III.

C. Supersonic Shear Imaging

The spatial-temporal displacement field obtained for a fixed depth within the layer and its corresponding 2-d Fourier transform are presented in Fig. 8. From the 2-d Fourier transform [Fig. 8(b)], the presence of a highly en-ergetic mode is observed. This mode corresponds to the zero-order antisymmetric lamb mode described by (4). dispersion curves can be obtained from single acquisitions by using the concept of shear wave spectroscopy (sWs) proposed by deffieux et al. [30]. The dispersion curve is obtained by finding, for each frequency, the phase velocity at which the Fourier transform amplitude is maximal. For the final dispersion curve, an average over the plate thick-ness is performed. The results are presented in Fig. 9. The error for each individual measurement is provided by the standard deviation.

The theoretical and simulated dispersion curves are presented in Figs. 9(a) and 9(b), respectively. The theo-retical model was calculated by performing a least-mean-square fit on the theoretical dispersion curves obtained using (4) to the experimental data [Fig. 9(a)]. The model proposed by (4) has seven independent parameters: the density, the shear/longitudinal wave speed for each medi-um (plate and surrounding gel), and the plate thickness. In this work, both densities and longitudinal wave speeds were assumed to be constant throughout the sample (1000 kg/m3 and 1500 m/s, respectively). With this as-sumption (valid for biological tissues), only three param-

Fig. 5. (a) normalized displacement field for a 300-Hz excitation obtained using finite difference simulation (acEl). The plate shear wave speed and thickness were set to 10 m/s and 2.5 mm, respectively. (b) Experimental normalized displacement field for a 300-Hz excitation obtained by 1dTE for phantom B. In both figures, the dotted line indicates the plate position and the piston position corresponds to 0 mm on the depth axis. The longitudinal (l) and shear (s) waves created by the piston are clearly visible.

TaBlE II. summary of 1dTE simulations.

simulation parameters 1dTE simulation results

cT (m/s) h (mm) cT (m/s) σ (m/s) ΔcT (m/s)

10 1 10.01 0.50 0.5810 2.5 9.97 0.14 0.205 2.5 4.79 0.13 0.05

brum et al.: 1-dTE for shear modulus assessment of thin-layered soft tissue 709

eters can be adjusted to minimize the sum of squares: the shear wave speeds (C T

g and CT) and the plate thickness (h). While the parameter CT was allowed to vary freely, C T

g and h were allowed to vary between the upper and lower limit of each measured parameter by taking into ac-count the error provided for each measurement, e.g., for phantom a, C T

g was allowed to vary between 3.6 and 4.0 m/s and h was allowed to vary between 0.9 and 1.3 mm. The resulting plate shear wave speeds are 12.23 ± 0.27, 10.96 ± 0.28, and 5.74 ± 0.07 m/s for phantoms a, B, and c, respectively. The shear wave speed error was calculated by taking into account the error on the experi-mental data. The results are summarized in Table III.

For the simulated dispersion curves, a different ap-proach was used: for each phantom, two dispersion curves were simulated corresponding to the upper and lower limit of the shear wave speed estimation from the bulk sample and the plate thickness by taking into account the error provided for these measurements [Fig. 9(b)]. The values of shear wave speed and thickness for phantom a are cT = 12.33 m/s, h = 1.3 mm for the upper limit and cT = 11.90 m/s, h = 0.9 mm for the lower limit. For phantom

B, the values are cT = 10.41 m/s, h = 2.4 mm for the up-per limit and cT = 10.03 m/s, h = 2.0 mm for the lower limit. For phantom c, the values are cT = 6.03 m/s, h = 2.4 mm for the upper limit and cT = 5.25 m/s, h = 2.0 mm for the lower limit. These parameters are sum-marized in Table IV.

In the previous section, the validity of an empirical formula for recovering the bulk shear wave speed was pre-sented (5). a bulk shear wave speed value of 12.86 ± 0.66 m/s was retrieved by fitting this formula in a least square sense for phantom a. For phantoms B and c, the empirical formula could not be applied to retrieve the plate shear wave speed, because for both phantoms the plate thickness is 2.5 mm and the shear wave speed is less than approximately 10 m/s.

IV. discussion

This manuscript presents and compares two different transient elastography techniques for the assessment of thin elastic layers. The main difference between 1dTE

Fig. 6. Time of flight estimation inside the layer for a 300-Hz excitation: (a) finite difference simulation and (b) experimental results.

Fig. 7. dispersion curves obtained by applying 1-d TE inside the plate: (a) finite differences simulation and (b) experimental results.

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and ssI is the way in which the shear wave is generated. Whereas the vibrator used in 1dTE polarizes the shear wave in such a way that it passes through the plate with-out being guided, the acoustic radiation force used in the ssI technique generates a lamb wave strongly guided in-side the plate. These differences in the propagation mode strongly differentiate the inverse problem used in the tech-niques.

By carefully comparing Fig. 7 and Fig. 9, it can be not-ed that no guidance of the shear wave inside the plate was observed for the 1dTE, unlike the lamb wave used in the ssI technique. a good qualitative agreement is found be-tween the simulated and experimental dispersion curves.

For the 1dTE simulations, a general agreement is found between the input and the estimated shear wave speed value within an average relative deviation of 1.5% (Table II). a mean relative error of 3% is found for each indi-vidual set of simulation parameters. It can be observed that the absolute error in each measurement increases as the plate thickness decreases and the shear wave speed increases. There are two main reasons for this. First, the faster the wave propagates, the more difficult it is to iso-late the transmitted transient shear wave. In this case, the shear wave speed estimation is biased by reflected waves that could not be eliminated by the directional filtering along the depth axis [6]. second, the thinner the plate is, the fewer points there are inside the plate for the estima-tion of the shear wave speed from the ToF estimation as a function of depth (Fig. 6).

For the 1dTE measurements (Table III), an agreement within an average relative deviation of 3% is found be-tween the bulk sample and the 1dTE experimental shear

wave speed estimation. The same behavior of the abso-lute error relative to the shear wave speed estimate was observed for the 1dTE experiments and the simulation results. This could represent a limitation of 1dTE, mak-ing it difficult to apply in very thin and hard plates. In the case of 1dTE, the shear wave estimation depends on the ToF measurement through several experimental param-eters: ultrasonic frequency, temporal sampling rate, piston excitation frequency, plate thickness, and elasticity. The higher the ultrasonic frequency and sampling rate are, the higher is the spatial/temporal resolution of the experi-mental displacement field. a higher spatial resolution will result in an increase in the number of points inside the plate for phase estimation. an increase in the sampling rate will help to better isolate the transient shear wave transmitted into the plate, although an increase of the sampling rate alone is usually not enough to isolate the transmitted shear wave because it also depends on the shear wave frequency and wavelength or, equivalently, on the plate elasticity and shear piston excitation frequency. The softer the plate and the higher the excitation fre-quency are, the smaller is the shear wavelength, and the easier it is to isolate the transient part. This has its dis-advantages, because attenuation increases with frequency: for very high frequencies, only very shallow plates could be characterized. For the configuration used in the present work, and based on the experience, the authors suggest that the minimum thickness required to have a reliable shear wave speed estimate is 1 mm for a 10-m/s propagat-ing shear wave, although further studies must be carried out on this point for different choices of the experimental parameters.

TaBlE III. summary of Experimental results.

phantom

Bulk sample ssI on phantom 1dTE on phantom

cT (m/s) σ (m/s) cT (m/s) ΔcT (m/s) cT (m/s) σ (m/s) ΔcT (m/s)

a 12.06 0.16 12.23 0.27 11.45 0.46 0.66B 10.22 0.19 10.96 0.28 10.36 0.55 0.39c 5.64 0.39 5.74 0.07 5.58 0.22 0.48

Fig. 8. (a) normalized displacement field inside the plate for phantom c as function of time (in milliseconds) and distance (in millimeters) for the ssI experience. (b) 2-d Fourier transform (energy) of the displacement field in (a). The energy is normalized to 1 for each frequency. The extracted dispersion curve is also represented in black dots.

brum et al.: 1-dTE for shear modulus assessment of thin-layered soft tissue 711

For the ssI experiment, good agreement is found be-tween the experimental measurements and the theoreti-cal/simulated dispersion curves (Fig. 9). In the case of the theoretical dispersion curves, this is supported by a mean normalized correlation coefficient of 0.97 ± 0.03 and a mean variance of 0.02 m/s between the model and the experimental data. For the simulated dispersion curves, a mean normalized correlation coefficient of 0.97 ± 0.03 and a mean standard deviation of 0.26 m/s are found. For all cases, the experimental dispersion curves lie between the error margins of the simulated curves. This confirms the good agreement between simulation and experiment.

The shear wave speed of the plate for each phantom was retrieved (Table III) by using the generalized lamb wave model. an agreement within 3.5% is found between the bulk sample and the ssI experimental shear wave es-timation. The validity of the lamb model (4) was proven to be valid in plates whose thickness and stiffness ranges correspond to corneal, skin, and vascular applications [19]. The results presented in this work confirm these findings and cross-validate them with a different approach: 1dTE.

For phantom a, the plate shear bulk wave speed was retrieved by fitting the empirical formula (5). The valid-ity of the empirical formula has been demonstrated in a liquid/plate/liquid configuration for very thin plates (≤1.5 mm) [19]. In the case of phantom a, we are dealing with a very thin plate but in a gel/plate/gel configuration. The consequence of the nonliquid surrounding medium is an increased phase velocity at a given frequency [18]. This effect is small when the surrounding medium stiffness

is small compared with plate stiffness [18]. despite these biases, the value obtained agrees, within the margins of error, with the shear wave speed obtained by fitting the lamb model, the bulk sample estimation, and 1dTE.

Finally, both techniques (1dTE and ssI) result in the same bulk shear wave speed estimation (and consequently stiffness) within the margins of error provided in Table III. The absolute and relative error in the bulk shear wave speed estimation is smaller for the ssI technique than for 1dTE. a very good agreement within 5% was found between the 1dTE and the ssI techniques. The main ad-vantage of 1dTE is its simplicity of application, because the true shear wave speed is directly retrieved using a time of flight algorithm. Interestingly, the diffraction pattern of the shear wave in the near field of the generating piston is particularly well suited to such plate stiffness estimation because the wave polarization is parallel to its propaga-tion direction and perpendicular to the plate. This well-adapted geometry optimizes the wave propagation pattern and is not affected by guidance effects, at least along the ultrasonic beam. In addition, the obtained information is only one-dimensional. It is probable that lamb-type modes propagating in a direction parallel to the plate are generated after the shear wave is transmitted into the plate. Further studies involving guided wave propagation inside soft solids must be carried out.

To retrieve 2-d information, the ssI technique uses a completely different propagation mode (lamb wave) for which the inverse problem requires a much more complex acquisition sequence and post-processing analysis: disper-

Fig. 9. dispersion curves obtained by ssI. comparison with (a) theoretical curves obtained by fitting the experimental data to the lamb theory and with (b) curves obtained from finite difference simulation. The shear wave speed values for the calculations were obtained from the bulk sample with its corresponding error. The error in the thickness was also taken into account.

TaBlE IV. summary of supersonic shear imaging simulation parameters.

phantom

Upper limit lower limit

cT (m/s) h (mm) cT (m/s) h (mm)

a 12.22 1.3 11.90 0.9B 10.41 2.4 10.03 2.0c 6.03 2.4 5.25 2.0

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sion effects must be measured using shear wave spectros-copy [30] and used for the recovery of the plate stiffness.

For clinical applications, the ssI technique has already been applied for carotid measurement assessment [18], un-like 1dTE. For arterial stiffness assessment, the several parameters involved in the 1dTE technique (ultrasonic frequency, temporal sampling rate, and piston excitation frequency) should be optimized. The main advantage of 1dTE is its simplicity: only one transducer is used and the shear modulus is directly measured without needing a model. Because the obtained information is local and one-dimensional, the 1dTE technique could be used in preliminary diagnostic studies. The main advantage of the ssI technique is its shear modulus mapping ability (e.g., along the vessel axis), because the obtained information is 2-d. This technique could be useful in the case of more complex diagnostic studies or treatment.

V. conclusion

In this work, the use of 1dTE for the assessment of the elasticity of sub-wavelength layered soft tissues is pro-posed. The good agreement between the 1dTE shear wave speed estimations and the material shear wave speed and the simulations validate the use of this technique. The obtained results were also compared with the results ob-tained by the ssI technique. Both techniques obtained the same shear wave speed estimate, although the physical wave propagation is different. The proposed 1dTE tech-nique has some advantages: it is noninvasive and low cost (only one transducer can be used). Most interestingly, be-cause of optimized diffraction effects in the near field of the vibrating piston, the shear wave speed can be directly retrieved without needing a propagation model.

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Javier Brum was born in Homburg saar, Ger-many, in 1983. He received a degree in physics in 2007 and an M.sc. degree in physics in 2009 from the Universidad de la república, Montevideo, Uruguay. He is currently pursuing his ph.d. de-gree in physics while studying in the laboratorio de acústica Ultrasonora, Facultad de ciencias, Universidad de la república.

currently, he is an assistant professor at the Facultad de ciencias, Universidad de la repúbli-ca. His current research interests include medical

ultrasonic imaging, elasticity imaging of biological tissues, and time re-versal.

Jean-Luc Gennisson was born in 1974 in France. He received the d.E.a. degree in electron-ics in 2000 from the University of paris VI. In 2003, he received the ph.d. degree in physics (acoustics) from the University of paris VI for his work on elastography. From 2003 to 2005, he worked at the laboratory of Biorheology and Medical Ultrasound in Montreal, canada, for postdoctoral research with prof. Guy cloutier. In 2005, he became a research scientist of the French national research center (cnrs). He is current-

ly working at the langevin Institute, directed by prof. Mathias Fink. His current research interests include medical ultrasonic imaging, shear wave propagation in soft tissues for cancer detection, elastography, and non-linear shear waves.

Thu-Mai Nguyen was born in May 1986 in Vil-leurbanne, France. she received an engineering degree from Ecole supérieure de physique et chi-mie Industrielles de paris (EspcI parisTech) with a specialization in physics. In 2009, she received an M.s degree in acoustics from paris VII Univer-sity. she is currently working toward a ph.d. de-gree at the Institut langevin in paris. Her current research activities are centered on elasticity imag-ing of the cornea and skin layers.

Nicolás Benech was born in Montevideo, Uru-guay, in February 1976. He received a physics de-gree in 2000 from the science Faculty, Universidad de la república. He received the M.s. and ph.d. degrees in acoustics in 2004 and 2008, respective-ly, from the same university.

currently, he is an assistant professor at the science Faculty, Universidad de la república. His current research interests are in the areas of elas-tography, ultrasonic medical imaging, signal pro-cessing, tissue biomechanics, and physical acous-tics.

Mathias Fink received the M.s. degree in math-ematics from paris University, France, in 1967, and the ph.d. degree in solid state physics in 1970. He then moved to medical imaging and re-ceived the doctorat ès sciences degree in 1978 from paris University. His doctorat ès sciences research was in the area of ultrasonic focusing with transducer arrays for real-time medical imag-ing.

dr. Fink is a professor of physics at the Ecole superieure de physique et de chimie Industri-

elles de la Ville de paris (EspcI parisTech), paris, France. In 1990, he founded the laboratory ondes et acoustique at EspcI that became the langevin Institute in 2009. In 2002, he was elected to the French acad-emy of Engineering, in 2003 to the French academy of science and in 2008 to the college de France on the chair of Technological Innovation.

His area of research is concerned with the propagation of waves in complex media and the development of numerous instruments based on this basic research. His current research interests include time-reversal in physics, super-resolution, metamaterials, medical ultrasonic imaging, ultrasonic therapy, multiwave imaging, acoustic smart objects, acoustic tactile screens, underwater acoustics, geophysics, and telecommunica-tions. He has developed different techniques in medical imaging (ultra-fast ultrasonic imaging, transient elastography, and supersonic shear im-aging), wave control, and focusing in complex media with time-reversal mirrors. He holds more than 55 patents, and he has published more than 350 articles. Four start-up companies have been created from his research (Echosens, sensitive object, supersonic Imagine, and Time re-versal communications).

Mickaël Tanter is a research professor of the French national Institute for Health and Medical research (InsErM). For five years, he has been heading the team Inserm Erl U979, Wave phys-ics for Medicine, at langevin Institute, EspcI parisTech, France. He was awarded the ph.d. de-gree in physics from paris VII University in 1999.

His main activities are centered on the devel-opment of new approaches in wave physics for medical imaging and therapy. His current research interests a wide range of topics: elastography us-

ing shear wave imaging, high-intensity focused ultrasound, ultrasonic imaging using ultrafast ultrasound scanners, adaptive beamforming, and the combination of ultrasound with optics and MrI. In 2009, he received the Frederic lizzi Early career award of the International society of Therapeutic Ultrasound and the Montgolfier prize of the national so-ciety for Industry Valorization (s.E.I.n.) in 2010. Mickael Tanter is the recipient of 17 patents in the field of ultrasound imaging and the author of more than 110 technical peer-reviewed papers and book chapters. He is associate Editor of the IEEE Transactions on Ultrasonics, Ferroelec-trics, and Frequency Control, Technical program committee member of the IEEE International Ultrasonics symposium, and member of the Brain advisory board of the Focused Ultrasound surgery Foundation. In 2005, he, along with M. Fink, J. souquet, c. cohen-Bacrie, and J. Bercoff, founded supersonic Imagine, an innovative French company positioned in the field of medical ultrasound imaging and therapy; in 2009, it launched a new-generation ultrasound imaging platform called aixplorer, with a unique shear wave imaging modality.

Carlos A. Negreira was born in Montevideo, Uruguay, in october 1951. He received the d.E.a. (M.sc. degree) in physics (acoustics) in 1981 from paris VII University, France. In 1984, he received the ph.d. in physics (acoustics option) from the strasbourg I University of sciences, France. From 1986 to 1989, he was assistant professor of phys-ics, and since 1992 he has been professor at Uni-versity of sciences (Universidad de la república, Uruguay). In 1994, he founded the Ultrasound laboratory in the Institute of physics at the same

university, and has since guided many M.a. and ph.d. students in the field of ultrasound.

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His current research interests include biomedical applications, elas-tography, ultrasonic imaging and therapy, nondestructive evaluation in the food industry, piezocomposites transducers, acousto-optics applica-tions, waves in scattering media, and time-reversal physics in cavities.

He is engaged in several scientific cooperation projects with Europe-an groups (for example, loa, EspcI-France; acoustics Institute-csIc, Madrid, spain) and several south american groups (for example, phys-

ics department, University of são carlos, são paulo, Brazil ; cITEI national Institute of Industrial Technology, Buenos aires, argentina; Biomedical Engineering cInVEsTaV, Mexico; and the Institute of Bio-medical Engineering, University of rio de Janeiro, Brazil).

With his team, he has developed several acoustic and ultrasonic tech-niques to characterize materials and processes. He has published more than 55 reviewed articles.