Measurement of elastic nonlinearity of soft solid with

Measurement of elastic nonlinearity of soft solid with transient elastography S. Catheline,a) J.-L. Gennisson, and M. Fink Laboratoire Ondes et Acoustique, ESPCI, Universite´ Paris VII, U.M.R. C.N.R.S. 7587, 10 rue Vauquelin, 75231 Paris cedex 05, France

共Received 23 September 2002; revised 11 July 2003; accepted 21 July 2003兲 Transient elastography is a powerful tool to measure the speed of low-frequency shear waves in soft tissues and thus to determine the second-order elastic modulus ␮ 共or the Young’s modulus E兲. In this paper, it is shown how transient elastography can also achieve the measurement of the nonlinear third-order elastic moduli of an Agar-gelatin-based phantom. This method requires speed measurements of polarized elastic waves measured in a statically stressed isotropic medium. A static uniaxial stress induces a hexagonal anisotropy 共transverse isotropy兲 in solids. In the special case of uniaxially stressed isotropic media, the anisotropy is not caused by linear elastic coefficients but by the third-order nonlinear elastic constants, and the medium recovers its isotropic properties as soon as the uniaxial stress disappears. It has already been shown how transient elastography can measure the elastic 共second-order兲 moduli in a media with transverse isotropy such as muscles. Consequently this method, based on the measurement of the speed variations of a low-frequency 共50-Hz兲 polarized shear strain waves as a function of the applied stress, allows one to measure the Landau moduli A, B, C that completely describe the third-order nonlinearity. The several orders of magnitude found among these three constants can be justified from the theoretical expression of the internal energy. © 2003 Acoustical Society of America. 关DOI: 10.1121/1.1610457兴 PACS numbers: 43.25.Dc, 43.25.Ed 关MFH兴

I. INTRODUCTION

Acoustoelasticity is a well-established technique1 to experimentally measure nonlinear third-order elastic constants in solids such as metals,2 crystals,3 or rocks.4 It consists of measuring the speed of ultrasonic waves in stressed solids. More precisely, the third-order moduli are deduced from the slope of the speed as a function of the uniaxial stress. So far no such measurements have been made in soft tissue since it has long been considered a liquid-like medium from an ultrasonic point of view. However, like in all solids, shear waves do propagate in soft tissues at low frequency 共50 Hz typically兲.5,6 As shown in this paper, the speed of these shear waves is modified if the medium is submitted to a uniaxial stress 共which is the evidence of a deviation from the Hooke’s law兲. In such a medium, the uniaxial stress induces anisotropy, the transverse isotropy,7 which remains as long as the uniaxial stress is applied. Thus, a quantitative evaluation of this temporary transverse isotropy with the technique of transient elastography leads to the measurement of the thirdorder constants. In the literature, the quantitative measurement of soft solids nonlinearity lies on the measurement of ultrasound speed8 which cannot lead to the measurement of the Landau moduli A, B, C 共as shown in Sec. II兲 without the complementary measurement of shear wave speeds shown in the present paper. From a more qualitative point of view, it is shown that nonlinear effects can alter the contrast in the elastography images of breast tissues9 and are of fundamental importance in the touch–contact interaction with soft tissues.10 a兲

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J. Acoust. Soc. Am. 114 (6), Pt. 1, Dec. 2003

Pages: 3087–3091

In Sec. II, a brief theoretical recall introduces the set of three equations that gives the speed of the two polarized shear waves 共with a polarization parallel or perpendicular to the stress axis兲 and of the compressional wave 共ultrasonic wave兲 as a function of the applied stress. Section III describes the experiment and explains how polarized shear strain waves can be generated by a rod source. Then, with the help of theoretical Green’s functions, a simulation shows the ability of the transient elastography technique to measure the small speed variations of polarized shear waves induced by nonlinear phenomena. Experimental values of the thirdorder Landau moduli A, B, C are discussed in Sec. III. II. THEORY

Hugues and Kelly2 have established expressions of the speed of elastic waves in a uniaxially stressed solid as function of the second-order Lame´ coefficients 共␭, ␮兲 and the third-order Landau coefficients 共A, B, C兲. The detailed development of the elastic nonlinear theory can be found in numerous textbooks.11,12 Here we summarize the general equations. The equation of motion is

␳ u¨ i ⫽

⳵␴ i j , ⳵x j

共1兲

where ␳, ␴, and u¨ designate the density, the stress tensor, and the particle acceleration, respectively. The stress tensor is given by

␴i j⫽

⳵e , ⳵ui ⳵ ⳵x j

冉 冊

0001-4966/2003/114(6)/3087/5/$19.00

共2兲

© 2003 Acoustical Society of America

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where e designates the elastic strain energy which, developed to the third order in strain ␧ in a general elastic medium, is e⫽ 12c i jkl ␧ i j ␧ kl ⫹ 16c i jklmn ␧ i j ␧ kl ␧ mn ,

共3兲

where c i jkl and c i jklmn designate the component of the second-order elastic tensor and the third-order elastic tensor, respectively. In Eq. 共3兲, the components of the Lagrangian strain tensor ␧ are given by 2␧ i j ⫽

⳵ui ⳵u j ⳵uk ⳵uk ⫹ ⫹ . ⳵x j ⳵xi ⳵xi ⳵x j

共4兲

Thus, the general nonlinear elastodynamic equation is

␳ u¨ i ⫽

冋

⳵2 c ⫹ 共 c i jklmn ⫹c i jln ␦ km ⫹c jlmn ␦ ik ⳵ x j ⳵ x l i jkl ⫹c jnkl ␦ im 兲

册

⳵um u . ⳵xn k

共5兲

The speeds V are given by the solutions of

冏兺

Det

jl

冏

Bi jkl N j N l ⫺V 2 ␦ jk ⫽0,

共6兲

where ␦ is the Kronecker symbol, and N j , N l are the components of a unit vector that represents a plane-wave direction of propagation. It should be noted that Bijkl designates the elastic tensor as defined by Hugues and Kelly 关deduced from Eq. 共5兲兴 in isotropic solid and thus contains both second- and third-order elastic constants and the stress along one axis ␴. The stress ␴ comes when replacing the strain in brackets in 共5兲 by means of the formulas of the linear theory 共Hooke’s law兲. Now, in order to use the Landau coefficients, we must recall how they were introduced. A third-order development of the elastic internal energy in an isotropic solid, Eq. 共3兲, is also expressed as e⫽ ␮ ␧ 2ik ⫹

␭ 2 A C ␧ ll ⫹ ␧ ik ␧ il ␧ kl ⫹B␧ 2ik ␧ ll ⫹ ␧ 3ll . 2 3 3

共7兲

The Lame´ and the Landau coefficients as a function of the elastic moduli in Voigt’s notation are, respectively, ␭⫽c 12 , ␮ ⫽c 66 , A⫽4c 456 , B⫽c 144 , C⫽c 123/2. Thus, the three plane-wave solutions that propagate perpendicularly to the static uniaxial stress are associated with the three following eigenvalues:

␳ 共 V p 兲 2 ⫽␭⫹2 ␮ ⫺ ⫺4␭⫺2

冉 冊

冋

␴ ␭ ␭ ⫺ A⫹2B 1⫺ ⫹2C 3␭⫹2 ␮ ␮ ␮

册

␭2 , ␮

冋冉 冉

共8兲

冊

册

␳共 V s 兲2⫽ ␮ ⫺

␴ A ␭ 1⫹ ⫹B⫹␭⫹2 ␮ , 共9兲 3␭⫹2 ␮ 2 2␮

␳ 共 V⬜s 兲 2 ⫽ ␮ ⫺

␴ A␭ ⫺ ⫹B⫺2␭ . 3␭⫹2 ␮ 2␮

储

冊

共10兲

In Eqs. 共8兲, 共9兲, and 共10兲, V P stands for the speed of the 储 compressional wave, V s for the speed of the shear wave with a polarization parallel to the stress axis, and V⬜s for the speed 3088

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of the shear wave with a polarization perpendicular to the stress axis. Let’s compare Eqs. 共8兲, 共9兲, and 共10兲 to the set of equations obtained from a transverse isotropic medium for a propagation direction perpendicular to the plan of isotropy

␳ 共 V p 兲 2 ⫽c 11 , 储

共11兲

␳ 共 V s 兲 2 ⫽c 44 ,

共12兲

␳ 共 V⬜s 兲 2 ⫽c 66 .

共13兲

c 11 , c 44 , and c 66 stand for the elastic constants of the Christoffel’s tensor using Voigt’s notation. As in the acoustoelasticity experiment, three plane-wave solutions with the same polarization are found. This is the expression of the formal equivalence between transverse isotropic media and stressed media. One difference is that, in the latter configuration, the anisotropy is stress dependent; in an unstressed medium ( ␴ ⫽0), one can easily verify from Eqs. 共8兲, 共9兲, and 共10兲 that the speeds correspond to an isotropic solid. But, at a given stress, one can define a unique set of elastic constants c 11 , c 44 , and c 66 in Eqs. 共11兲, 共12兲, and 共13兲 that perfectly accounts for the temporary nonlinear anisotropy. Moreover, changing the stress in Eqs. 共8兲, 共9兲, and 共10兲 is equivalent to modifying c 11 , c 44 , and c 66 according to a linear law with a slope defined by the second- and third-order elastic constants. In the following part, we are taking advantage of this equivalence to predict the behavior of the transient elastography technique used in an acoustoelasticity experiment. III. EXPERIMENT AND SIMULATION

The experiments presented in the following section are conducted in a model of soft tissues: an Agar-gelatin based phantom 共5% gelatin, 3% Agar兲. Its dimensions are 80⫻80 ⫻80 mm. The mechanical characteristics are highly dependent on temperature and maturation. A 5-MHz transducer is mounted in the middle of a rod 共80 mm long兲 fixed on a vibrator 共Bru¨el & Kjaer, type 4810兲. The whole system 共the shear elasticity probe兲 is applied at the surface of the phantom so that the ultrasonic beam is horizontal 共see Fig. 1兲. A rigid Plexiglas plate is placed on the top, and loads 共a container filled with water兲 can be added to control the uniaxial stress in the sample. The transducer works as a pulse–echo system and backscattered signals sampled at 50 MHz are stored in a 2-MB memory with a repetition frequency of typically 3000 Hz. The low-frequency pulse 共50-Hz central frequency兲 propagates in the medium mainly as a shear wave, and its longitudinal displacements along the ultrasonic beam are measured with a cross-correlation algorithm between successive A lines.13 This technique is known as transient elastography.14 On the seismic-like representation of the displacements 共Fig. 2兲 obtained in an unstressed medium, the maximum amplitude of the 50-Hz pulse is 120 ␮m. A small-amplitude compressional wave 共P兲 first propagates almost instantaneously within the medium 共1500 m s⫺1兲. The shear wave 共S兲 longitudinal component appears at each depth with a phase delay inversely proportional to its speed. Actually, the speed (2.52⫾0.02 m s⫺1 ) is extracted with a simple phase analysis at the central frequency. The standard deviation of the phase Catheline et al.: Nonlinearity using transient elastography

FIG. 1. 共a兲 Experimental setup. A transducer is set in the middle of a rod mounted on a vibrator. A lowfrequency pulse propagates in the medium and the displacements are computed from the A-scan stored in a memory. 共b兲 Picture of the shear elasticity probe. The transducer 共the black disk兲 is in the middle of a Plexiglas rod and the black vibrator is visible in the back ground.

from a linear fit gives the error. We have shown in a recent paper15 that a rod source allows one to generate shear waves with a strain field polarized perpendicularly to the rod in the first few centimeters. Since the speed of shear waves in a uniaxially stressed medium depends on its polarization in regard to the stress, the speed of the shear wave is measured for each amount of stress with the rod in the horizontal and vertical position. Contrary to the theoretical analysis, we are not dealing with plane waves in the experiments, so how close to plane polarized shear waves are the speed measurements obtained with the rod system? Is it accurate enough to detect small variations due to nonlinear effects? The theoretical Green’s functions in transverse isotropic elastic media computed by Vavrycˇuk16 from a higher-order ray theory are used to simulate the displacement field induced by the rod system in an acoustoelasticity experiment. The transverse isotropy is controlled by the following two parameters: the parallel and perpendicular elastic moduli, c 44 and c 66 , respectively 共the compression modulus is kept constant, c 11⫽2.25 GPa). The simulation 共introduced in Ref. 13兲 is run as many time as the number of desired stressed states of the medium. The parameters are adjusted so as to have a comparable value with the experiments 共a typical density for soft tissues is 1000

FIG. 2. Typical experimental displacements in an Agar-gelatin-based phantom. The maximum amplitude of the 50-Hz pulse is 120 ␮m. A phase analysis gives the speed of the shear wave: 2.52⫾0.02 m s⫺1 . J. Acoust. Soc. Am., Vol. 114, No. 6, Pt. 1, Dec. 2003

kg m⫺3兲: the parallel and perpendicular elastic moduli range from 6250 to 8000 Pa and from 6250 to 6500 Pa, respectively. The impulse response of the rod is computed from the sum of the Green’s function of point sources uniformly distributed on the contact surface. A 1-mm square grid is sufficient for a 5-cm typical wavelength. This summation is the expression of the Rayleigh–Sommerfeld integral. Finally, a simple convolution of this impulse response with the source excitation 共a one-cycle sinusoid with a 50-Hz central frequency兲 gives the theoretical displacement field sampled at 3000 Hz. As in the experiment, a phase analysis at 50 Hz from the longitudinal displacements in the axis of the rod (3⫻80 mm2 ) is used to compute the speed in Fig. 3. The error bars are deduced from the standard deviation of this phase measurement from a linear fit. It clearly appears that the speeds measured with the two perpendicular positions of the rod relative to the main axis of the transverse isotropy properly reveal the two shear moduli introduced in the simulation. In other words the rod source generates a shear wave which has the speed of a plane shear wave polarized perpendicularly to its bigger dimension. Consequently, any changes on one of the shear moduli modify the speed of either polar-

FIG. 3. In order to simulate an uniaxial stress ranging from 0 to several hundred Pa, we modify the shear elasticities in a transverse isotropic medium. In this model, the increasing stress corresponds to a parallel and perpendicular elastic moduli ranging from 6250 to 8000 Pa and from 6250 to 6500 Pa, respectively. The simulation is run twice for each stress state 共two positions of the rod兲. The rod system shows its ability to recover the right values of elasticity used in the simulation. Catheline et al.: Nonlinearity using transient elastography

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FIG. 4. Experimental shear and compression moduli as function of the uniaxial stress for three Agar-gelatin-based phantom. The nonlinear Landau coefficients A and B are computed from each pair of slopes.

ized strain shear wave. The expected slopes of the moduli 共simulation parameters兲 that characterize their evolution are 8% and 33%. The measured slopes are, respectively, 7% and 20%. The difference is attributed to the fact that the measurements are not performed on purely polarized shear waves and the speed may be influenced by diffraction effects: the phase as a function of depth is not perfectly linear 共error bars兲. However, as a conclusion to this paragraph, one can say that the rod system gives a rather accurate description of the anisotropy of a medium. Thus, it is a reasonable tool to study a nonlinear transverse isotropic medium. IV. RESULTS AND DISCUSSION

Three phantoms are tested. As observed in Fig. 4, all pairs of the experiment, numbered from 1 to 3, show that both elastic moduli 共perpendicular and parallel兲 increase with static axial stress. This is a clear evidence of a nonlinear behavior of the Agar-gelatin-based phantom. Compared to a typical acoustoelasticity experiment in metals where the shear perpendicular modulus decreases, this result constitutes a first particularity of the gelatin-based phantom. Now, from a quantitative point of view, the shear elasticity is extracted from the first point 共null stress兲 of each pair of experiments: ␮ ⫽9.00, 6.35, and 9.67 kPa. The experiment #1 corresponds to our first results. As the experimental setup has then been improved, the stress range increases with experiments #2 and #3. The anisotropy visible on the experimental data #1 at zero applied stress is thought to be a consequence of the phantom’s own weight. It was found experimentally that the

speed measurements at 4 cm beneath the top surface of the phantom were not subjected to the nonlinear effect of a stress induced by its weight. Indeed, in experiments #2 and #3, the same speed is found for the two polarized strain shear waves in the unstressed medium. When the uniaxial stress is linearly increased within the phantoms #1, #2, #3, the perpendicular and parallel elastic moduli increase with slopes of 1.27, 0.13, 3.41, and 2.05, 3.12, 4.17, respectively. From each pairs of slopes and using the set of equations 共9兲 and 共10兲, we have found the following values: ⫺64, ⫺101, ⫺68 kPa and ⫺12, ⫺14, ⫺26 GPa for the Landau coefficients A and B, respectively. The huge difference between these third-order moduli is striking since in more conventional media such as metal, rocks, or crystals they are of the same order. Now, the last Landau coefficient can be deduced from the results found in the literature. Indeed, in Ref. 8, with a thermodynamic experimental setup 共the pressure is changed while the temperature is kept constant兲, Everbach measured the nonlinear parameter ␤ ⫽3.64 in a gelatin-based phantom from speed measurements of ultrasound. This value of the nonlinear parameter does not significantly change from one sample to another and is in fact close to the parameter found in the water, ␤ ⫽3.50. Since ␤ is expressed as function of the Landau coefficients11 as ␤ ⫽⫺(3/2)⫺ 关 (A⫹3B⫹C)/ ␳ o c 2l 兴 ⫽3.64, and using the experimental values of A and B, one finally obtains C⫽24, 31, and 67 GPa. The experimental errors in these measurements are difficult to quantify. They may emerge from the diffraction biases cited earlier, although the error bar on each individual speed measurement 共less than 2%兲 is too small to be represented in Fig. 4. An unperfected experimental uniaxial stress may cause wrong estimations of nonlinear moduli. Finally, the linear relationship between elastic modulus and stress may be a hazardous approximation if higher order nonlinear moduli are involved. Thus, one can take as an error estimate the standard deviation of the experimental measurements from a linear fit. The results are summarized in Table I. The gap between the first 共A兲 and the two last Landau coefficient 共B and C兲 is huge for the three samples. This result can be justified by the following considerations: for soft solids, in Eq. 共7兲, the first coefficient ␮ in front of a shear strain is 106 smaller than the coefficient ␭ in front of the compression strain 共one has to keep in mind the order of magnitude of the second-order moduli in the Agar-gelatin-based phantom; typically ␭ is in GPa, whereas ␮ is in kPa兲. Thus, it is not surprising for the third-order coefficient A in front of shear strain terms to present a very small value compared to the coefficients B and C in front of terms that contain compres-

TABLE I. Elastic moduli measured in three Agar-gelatin-based phantoms. Linear second-order elastic moduli 共Lame´ coefficients兲

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Nonlinear third order elastic moduli 共Landau coefficients兲

Phantom #

␭ 共GPa兲

␮ 共kPa兲

A 共kPa兲

B 共GPa兲

C 共GPa兲

1 2 3

2.25 2.25 2.25

9.0⫾0.2 6.35⫾0.04 9.67⫾0.06

⫺64⫾13 ⫺101⫾7 ⫺68⫾3

⫺12⫾3 ⫺14⫾2 ⫺26⫾2

24⫾6 31⫾3 67⫾4

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Catheline et al.: Nonlinearity using transient elastography

sion strain. This intuitive justification does not hold for a theoretical explanation; some efforts still have to be done to completely understand the results of Table I. V. CONCLUSION

In this paper, we have used the transient elastography technique in an acoustoelasticity experiment in order to characterize the nonlinear behavior of a model of soft solid: the gelatin-based phantom. The results clearly show the particularity of soft solids. Indeed, the shear Lame´ coefficient ␮ is roughly 106 times smaller than the compression Lame´ coefficient ␭. The contribution of this paper is to show that the Landau coefficient A is 106 times smaller than the two Landau coefficients B and C. These first quantitative results on nonlinearity of soft solids encourage us to pursue the experiments on in vitro and in vivo biological tissues. Our point of view is that the nonlinear behavior of living tissues may be as important as its linear behavior. 1

F. D. Murnaghan, ‘‘Finite deformation of an elastic solid,’’ Am. J. Math. 49, 235 共1937兲. 2 D. S. Hugues and J. L. Kelly, ‘‘Second-order elastic deformation of solids,’’ Phys. Rev. 92, 1145–1149 共1953兲. 3 T. Bateman, W. P. Mason, and H. J. McSkimin, ‘‘Third-order elastic moduli of germanium,’’ J. Appl. Phys. 32, 928 –936 共1961兲. 4 F. Birch, ‘‘Compressibility; elastic constants,’’ in Handbook of Physical Constants, edited by S. P. Clark, Jr., Mem. Geol. Soc. Am. 97, 97–174 共1966兲.

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