ElastographyGeneral Principles and Clinical Applications

Marvin M. Doyley, PhD*, Kevin J. Parker, PhD

KEYWORDS

� Elastography � Ultrasonic imaging � Ultrasonic elastography � MRI

KEY POINTS

� Like conventional medical imaging modalities, forward and the inverse problems are encounteredin elastography.

� Quasistatic elastography visualizes the strain induced within tissue using either an external or inter-nal source.

� Direct and iterative inversion schemes have been developed to make quasistatic elastograms morequantitative.

� Soft tissues display several biomechanical properties, including viscosity and nonlinearity, whichmay improve the diagnostic value of elastography when visualized alone or in combination withshear modulus. Elastography can characterize the nonlinear behavior of soft tissues and may beused to differentiate between benign and malignant tumors.

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INTRODUCTION

Elastography visualizes differences in the biome-chanical properties of normal and diseased tis-sues.1–4 Elastography was developed in the late1980s to early 1990s to improve ultrasonic imag-ing,5–7 but the success of ultrasonic elastographyhas inspired investigators to develop analogsbased on MRI8–11 and optical coherence tomogra-phy.12–14 This article focuses on ultrasonic tech-niques with a brief reference to approachesbased on MRI.

The general principles of elastography can besummarized as follows: (1) perturb the tissue usinga quasistatic, harmonic, or transient mechanicalsource; (2) measure the resulting mechanicalresponse (displacement, strain or amplitude, andphase of vibration); and (3) infer the biomechanicalproperties of the underlying tissue by applyingeither a simplified or continuum mechanical modelto the measured mechanical response.2,15–18 Thisarticle describes (1) the general principles of qua-sistatic, harmonic, and transient elastography

Department of Electrical and Computer Engineering, Un343, Box 270126, Rochester, NY 14627, USA* Corresponding author.E-mail address: m.doyley@rochester.edu

Ultrasound Clin 9 (2014) 1–11http://dx.doi.org/10.1016/j.cult.2013.09.0061556-858X/14/$ – see front matter Published by Elsevier I

(Fig. 1)—the most popular approaches to elastog-raphy—and (2) the physics of elastography—theunderlying equations of motion that govern themotion in each approach. Examples of clinical ap-plications of each approach are provided.

THE PHYSICS OF ELASTOGRAPHY

Like conventional medical imaging modalities, for-ward and the inverse problems are encountered inelastography. The former problems are concernedwith predicting the mechanical response of amaterial with known biomechanical propertiesand external boundary conditions. Understandingthese problems and devising accurate theoreticalmodels to solve them have been an effective strat-egy in developing and optimizing the performanceof ultrasound displacement estimation methods.The latter problems are concerned with estimatingbiomechanical properties noninvasively using theforward model and knowledge of the mechanicalresponse and external boundary conditions. Acomprehensive review of methods developed to

iversity of Rochester, Hopeman Engineering Building

nc. ultrasound.th

eclinics.

Fig. 1. Schematic representation of current approaches to elastographic imaging: quasistatic elastography (left),harmonic elastography (middle), and transient elastography (right).

Doyley & Parker2

solve inverse problems is given in the article byDoyley19; therefore, this section focuses only onthe forward problem.The forward elastography problem can be des-

cribed by a system of partial differential equations(PDEs) given in compact form20,21:

V� �sij

�5bi (1)

where sij is the 3-D stress tensor (ie, a vector ofvectors), bi is the deforming force, and V is thedel operator. Using the assumption that soft tis-sues exhibit linear elastic behavior, then the straintensor ( 3) maybe related to the stress tensor (s) asfollows22:

sij 5Cijkl 3kl (2)

where the tensor (C) is a rank-four tensor consist-ing of 21 independent elastic constants.16,20,23

Under the assumption that soft tissues exhibitisotropic mechanical behavior, however, thenonly 2 independent constants, l and m (lambdaand shear modulus), are required. The relationshipbetween stress and strain for linear isotropicelastic materials is given by:

sij 5 2m 3ij1ldijQ (3)

whereQ5 V , u5 3111 3221 333 is the compress-ibility relation, d is the Kronecker delta, and thecomponents of the strain tensor are defined as:

3ij 51

2

�vui

vj1vuj

vxi

�(4)

Lame constants (ie, l and m) are related toYoung modulus (E) and Poisson ratio (v), asfollows20,21:

m5E

2ð11vÞ ; l5vE

ð11vÞð1� 2vÞ (5)

The stress tensor is eliminated from the equilib-rium equations (ie, Equation 2) using Equation 3.The strain components are then expressed interms of displacements using Equation 4. The re-sulting equations (ie, the Navier-Stokes equations)are given by:

V,mVu1Vðl1mÞV,u5rv2u

vt2(6)

where r the is density of the material, u is thedisplacement vector, and t is time. For quasistaticdeformations, Equation 6 reduces to:

V,mVu1Vðl1mÞV,u50 (7)

For harmonic deformations, the time-independent (steady-state) equations in the fre-quency domain give10,24:

V,mVu1Vðl1mÞV,u5ru2u (8)

where u is the angular frequency of the sinusoidalexcitation. For transient deformations, the waveequation is derived by differentiating Equation 6with respect to x, y, and z, which gives21:

V2D51

c21

v2D

vt2(9)

where V , u 5 D, and the velocity of the propa-gating compressional wave, c1, is given by:

c1 5

ffiffiffiffiffiffiffiffiffiffiffiffil12m

r

s(10)

The wave equation for the propagating shearwave is given by:

V2z51

c22

v2z

vt2(11)

Elastography 3

where z 5 V , u/2 is the rotational vector, and theshear wave velocity, c2, is given by:

c2 5

ffiffiffim

r

r(12)

Analytical methods have been used to solve thegoverning equations for quasistatic, harmonic,and transient elastographic imaging methods25–28

for simple geometries and boundary conditions.Numeric methods—namely, the finite-elementmethod—are used, however, to solve the govern-ing equations for all 3 approaches to elastographyon irregular domains and for heterogeneous elas-ticity distributions.24,29–36

APPROACHES TO ELASTOGRAPHYQuasistatic Elastography

Quasistatic elastography visualizes the straininduced within tissue using either an external or in-ternal source. A small motion is induced within thetissue (typically approximately 2% of the axialdimension) with a quasistatic mechanical source.The axial component of the internal tissuedisplacement is measured by performing cross-correlation analysis on pre- and postdeformed ra-diofrequency (RF) echo frames6,7,37 and strain isestimated by spatially differentiating the axial dis-placements. In quasistatic elastography, soft tis-sues are typically viewed as a series of 1-Dsprings that are arranged in a simple fashion. Forthis simple mechanical model, the measuredstrain ( 3) is related to the internal stress (s) byHooke’s law:

Fig. 2. Sonogram (A) and strain (B) elastograms obtainedinclusion whose modulus contrast was approximately 6.03

s5 k 3 (13)

where k is the Young modulus (or stiffness) of thetissue. No method can measure the internal stressdistribution in vivo; consequently, the internalstress distribution is assumed to be constant (ie,s z 1); an approximate estimate of Youngmodulus is computed from the reciprocal of themeasured strain. The disadvantage of computingmodulus elastograms in this manner is that itdoes not account for stress decay or stress con-centration; consequently, quasistatic elastogramstypically contain target-hardening artifacts,31,35 asillustrated in Fig. 2.

Despite this limitation, several groups have ob-tained good elastograms in applications where ac-curate quantification of Young modulus is notessential. For example, Fig. 3 shows the resultsof a case study, where quasistatic elastographywas performed on a 73-year-old woman with aphyllodes tumor in the upper outer quadrant ofher left breast. Phyllodes tumors are rare variantsof fibroadenoma, with a rich stromal componentand more cellularity. They grow quickly, devel-oping macroscopically lobulated internal struc-tures and may reach a large size, visibly alteringthe breast profile. Sonography generally shows asolid, moderately hypoechoic nodule with smoothborders and good sound transmission.38 Inho-mogenous structures may be present because ofsmall internal fluid areas. These appearances arenonspecific, and sonography is not currently ableto distinguish between benign and malignantcases, nor can it make a differential diagnosis be-tween fibroadenoma and phyllodes tumors.

from a phantom containing a single 10-mm diameterdB.

Fig. 3. Sonographic (A) and elastographic (B) images of phyllodes breast tumor. (Courtesy of Dr Jeff Bamber, Insti-tute of Cancer Research in London, London, England.)

Doyley & Parker4

In the sonogram shown in Fig. 3, the tumorcovers most of the field of view, with the capsuleof the anterior margin visible close to the top ofthe image and the posterior margin visible at thebottom left. Within the tumor, the appearance isheterogeneous on a large scale, with macroscopiclobules separated by echogenic boundaries thatare probably fibrous in nature. The strain elasto-gram (see Fig. 3B) confirms this appearance but

Fig. 4. Modulus elastograms computed from ideal axial anestimates that were corrupted with 4% additive white noisa gaussian modulus distribution that had a peak contrastDepartment of Mechanical and Aeronautic Engineering, B

shows it much more clearly with greater contrastthan the sonogram (see Fig. 3A). The macro-scopic lobules within the tumor are clearly definedas soft regions separated by stiff septa, whichis consistent with the septa being of a fibrousnature.Direct and iterative inversion schemes have

been developed to make quasistatic elastogramsmore quantitative. These techniques compute

d lateral strain estimates (A, left) and (B, right) straine. The simulated phantom contained an inclusion withof 4:1. (Courtesy of Dr P. Barbone, Boston Universityoston, MA.)

Elastography 5

the Young’s or shear modulus from the measureddisplacement or strain using the forward elasticitymodel described in Equation 7. Direct inversionschemes use a linear system of equations derivedby rearranging the PDEs that describe the forwardelastography problem.8,28,39�

vyy � vxx��

3xym�1vxy

�3xym

�50 (14)

Equation 14 contains high-order derivatives thatamplify measurement noise, which compromisesthe quality of ensuing modulus elastograms, asdemonstrated in Fig. 4.

Iterative inversion techniques40,41 overcome thisissue by considering the inverse problem as aparameter-optimization task, where the goal is tofind the Young’s modulus that minimizes the errorbetween measured displacement or strain fieldsand those computed by solving the forward elas-tography problem. The matrix solution at the(k 1 1) iteration that has the general form:

mk115Dmk1hJ�mk�TJ�mk�1rI

i�1

J�mk�T�

um�umk�

(15)

where Dmk is a vector of shear modulus updatesat all coordinates in the reconstruction field andJ is the Jacobian, or sensitivity, matrix. The Hes-sian matrix, ½JðmkÞTJðmkÞ�, is ill conditioned. There-fore, to stabilize performance in the presence ofmeasurement noise, the matrix is regularized us-ing 1 of 3 variational methods: the Tikhonov,41

the Marquardt,42 or the total variationalmethod.43,44 Fig. 5 shows an example of moduluselastograms computed with the iterative inversionapproach.

The contrast-to-noise ratio of the modulus elas-togram is better than that of the strain elastogram,which improved the detection of the boundary be-tween the ablated region and normal tissue to

Fig. 5. Sonogram with arrows indicating ablated tissue (A)RF ex vivo ablated bovine liver. (Courtesy of Drs T.J. HaMadison, Madison, WI.)

enable accurate determination of the size of thethermal zone.

Harmonic Elastography Based on LocalFrequency Estimation

In harmonic elastography,5,5–9,34,45 low-frequencyacoustic waves (typically <1 kHz) are transmittedwithin the tissue using a sinusoidal mechanicalsource.Thephaseandamplitudeof thepropagatingwaves are visualized using either color Doppler im-aging34,45,46 (Fig. 6) or phase-contrast MRI.9–11

Assuming that shear waves propagate withplane wave fronts, then an approximate estimateof the local shear modulus (m) may be computedfrom local estimates of the wavelength:

vshear 5

ffiffiffim

r

r(16)

where vshear is the velocity of the shear wave, and ris the density of the tissue. In a homogeneoustissue, shear modulus can be estimated fromlocal estimates of instantaneous frequency.47,48

Although shear modulus estimated using thisapproach is insensitive to measurement noise,the spatial resolution of the ensuing modulus elas-tograms is limited. A further weakness of theapproach is that the plane wave approximationbreaks down in complex organs, such as thebreast and brain, when waves reflected from inter-nal tissue boundaries interfere constructively anddestructively.

Like quasistatic elastography, solving the in-verse elastography problem improves the perfor-mance of harmonic elastography. Fig. 7 shows arepresentative example of an elastogram obtainedfrom a healthy volunteer by solving the inverse har-monic elastography problem. The resolution of theelastograms was sufficiently high to visualize fibro-glandular tissue from the adipose tissue.49,50

, strain elastogram (B), and modulus elastogram (C) ofll, T. Varghese, and J. Jiang, University of Wisconsin-

Fig. 6. In vivo porcine liver with a thermal lesion. The sonogram (A) shows a lesion with indistinct boundaries. Thesonoelastogram (B) demonstrates a vibration deficit indicating a hard lesion. Yellow arrows point to the lesion.Red arrows point the boundary of the liver.

Doyley & Parker6

Transient Elastography Based on Arrival TimeEstimation

A major limitation of harmonic elastography is thatshear waves attenuate rapidly as they propagatewithin soft tissues, which limits the depth of pene-tration. The transient approach to elastographyovercomes this limitation by using the acousticradiation force of an ultrasound transducer to per-turb tissue locally.51–53 This elastographic imaging

Fig. 7. Montage of magnitude MRIs (A) and shear moduluusing the subzone inversion scheme. (Courtesy of Drs J.B.School of Engineering, Dartmouth, NH.)

method uses an ultrasound scanner with an ultra-high frame rate (ie, 10,000 frames per second) totrack the propagation of shear waves. As in har-monic elastography, local estimates of shearmodulus are estimated from local estimates ofwavelength. The reflections of shear waves at in-ternal tissue boundaries make it difficult, however,to measure shear wave velocity—this limitationcan be overcome by computing wave speedsdirectly from the arrival times, as discussed by Ji

s elastograms (B) recovered from a healthy volunteerWeaver and K.D. Paulsen, Dartmouth College, Thayer

Fig. 8. Comparison transient shear wave (upper) andB-scan (lower) images of a breast with pathologyconfirmed IDC. The maximum diameter on the longi-tudinal axis on B-mode was 17 mm, whereas bothelastographic techniques indicated a larger footprintof the cancer. (Courtesy of Dr W. Svensson, ImperialCollege, London).

Elastography 7

and colleagues.54 Fig. 8 shows an example ofshear wave elastograms obtained from a breastcancer patient using a commercially availabletransient elastography system.

THE FUTURE OF ELASTOGRAPHY

Soft tissues display several biomechanical proper-ties, including viscosity and nonlinearity, whichmay improve the diagnostic value of elastographywhen visualized alone or in combination with shearmodulus. For example, clinicians could use me-chanical nonlinearity to differentiate betweenbenign and malignant breast tumors.1 Further-more, there is mounting evidence that other me-chanical parameters, namely viscosity55,56 andanisotropy,57 could also differentiate betweenbenign and malignant tissues—similar claimshave also been made for shear modulus.57 Notonly can these mechanical parameters discrimi-nate between different tissue types but also theymay provide value in other clinical areas, includingbrain imaging,58,59 distinguishing the mechanicalproperties of active and passive musclegroups,60–62 characterizing blood clots,63 andgnosing edema.64 Several investigators areactively developing techniques to visualize dif-ferent mechanical properties using quasistatic,harmonic, and transient elastographic imagingapproaches.

Viscoelasticity

In most approaches to elastography, the mechan-ical behavior of soft tissues is modeled using the

theory of linear elasticity (Hooke’s law), which isan appropriate model for linear elastic materials(ie, Hookean materials). It is well known, however,that most materials, including soft tissues, deviatefrom Hooke’s law in various ways. Materials thatexhibit both fluid-like and elastic (ie, viscoelastic)mechanical behavior deviate from Hooke’s law.20

For viscoelastic materials, the relationship be-tween stress and strain is dependent on time.Viscoelastic materials display 3 uniquemechanicalbehaviors: (1) strain increases with time whenstress (externally applied load) is sustained over aperiod of time, a phenomenon known as visco-elastic creep; (2) stress decreases with time whenstrain is held constant, a phenomenon known asviscoelastic relaxation; and (3) during cyclicloading, mechanical energy is dissipated in theform of heat, a phenomenon known as hysteresis.

Several investigators are actively developingelastographic imaging methods to visualize themechanical parameters that characterize linearviscoelastic materials (ie, viscosity, shear mod-ulus, and Poisson ratio). For example, Asbachand colleagues60 developed a multifrequencymethod to measure the viscoelastic properties ofnormal liver tissue versus diseased liver tissuetaken from patients with grades 3 and 4 liverfibrosis. They computed the shear modulus andviscosity variations within the tissue by fitting aMaxwell rheological model to the measured dataand solving the linear viscoelastic wave equationin the frequency domain. They observed thatfibrotic liver tissue had a higher viscosity (14.4 �6.6 Pa s) and elastic modulus (m1 5 2.91 �0.84 kPa and m2 5 4.83 � 1.77 kPa) than normalliver tissue. Their results revealed that althoughliver tissue is dispersive, it appeared as nondisper-sive between the frequency range of 25 Hz to50 Hz. Catheline and colleagues65 computed theshear modulus (m) and viscosity (h) by fitting themeasured speed of sound and attenuation equa-tion to Voigt and Maxwell rheological models.They observed that the recovered shear modulusvalues were independent of the rheological modelused, but viscosity values were highly dependenton the models used.

Sinkus and colleagues56 developed a direct-inversion scheme to visualize the mechanicalproperties of visocelastic materials, in which acurl operation was performed on the time-harmonic displacement field u(x,t) 5 u(x,t)eiut toremove the displacement contribution of thecompressional wave. They derived the governingequation that describes the motion incurred in anisotropic, viscoelastic medium by computing thecurl of the PDEs that describe the motion incurredby both transverse and compressional shear

Doyley & Parker8

waves. The resulting PDEs for transverse wavesare given in compact form by:

rv2t u5 mV2u1hvtV2u (17)

Sinkus and colleagues56 developed a direct-inversion scheme from Equation 17, in which mand h were the unknowns. They evaluated theinversion scheme using (1) computer simulations,(2) phantom studies, and (3) patient studies. Theirsimulation studies revealed that the proposed al-gorithm could accurately recover shear modulusand viscosity from ideal displacement data. Withnoisy displacements, however, a good estimateof shear modulus was obtained only when theshear modulus of the simulated tissue was lessthan 8 kPa. The inversion scheme overestimatedthe shear modulus values when actual stiffnessof the tissue was larger than 8 kPa. A similar effectwas observed when estimating viscosity, albeitmuch earlier (ie, the algorithm provided good esti-mates of viscosity when m <5 kPa). Although theshear modulus affected the bias in the viscositymeasurement, the investigators demonstratedthat the converse did not occur (ie, the viscositydid not affect the bias in shear modulus). Despitethese issues, their phantom studies revealed thatinclusions were discernible in both m and h elasto-grams, and the viscosity values agreed with previ-ously reported values for gelatin (0.21 Pa s). Thepatient studies revealed that the shear modulusvalues of malignant breast tumors were noticeablyhigher than those of benign fibroadenomas, butthere was no significant difference observed inthe viscosity of the tumor types, a result thatseems to contradict results reported by Qiu andcolleagues.55

Nonlinearity

When soft tissues deform by a small amount (aninfinitesimal deformation), their geometry in the un-deformed and deformed states is similar, thus thedeformation is characterized using engineeringstrain. To characterize finite deformation, first areference configuration has to be defined, whichis the geometry of the tissue under investigationin either the deformed or undeformed state. TheGreen-Lagrangian strain is defined as:

3ij 51

2

�vui

vxj1vuj

vxi1vuk

vxi

vuk

vxj

�(18)

The nonlinear term is neglected when themagni-tude of the spatial derivative is small, to producethe linear strain tensor, as defined in Equation 4.The relationship between stress and strain isnonlinear even for a linearly elastic material whenit is undergoing finite deformations. Consequently,

Skovoroda and colleagues66 proposed a direct-inversion scheme to reconstruct the shear mod-ulus distribution of a linear elastic material that isundergoing finite deformation.Some materials exhibit nonlinear material prop-

erties that are typically described using a strain en-ergy density function. Among the strain energyfunctions proposed in the literature, the mostwidely used for modeling tissues are (1) the neo-Hookean hyperelastic model and (2) the neo-Hookean model with an exponential term. Oberaiand colleagues67 used a different model, theVeronda-Westman strain energy density function(W), to describe the finite displacement of a hy-perelastic solid that is undergoing finite deforma-tion, which is defined by:

W5 m0

�egðI1�3Þ � 1

g� I2 � 3

2

�(19)

where the terms I1 and I2 are the first and secondinvariants of the Cauchy-Green strain tensor, m0is the shear modulus, and g denotes the nonline-arity. For the nonlinear case, they proposed aniterative inversion approach to reconstruct anonlinear parameter and the shear modulus atzero strain.Using data obtained from volunteer breast can-

cer patients, one with a benign fibroadenoma tu-mor and another with an invasive ductalcarcinoma (IDC), Oberai and colleagues67

observed that for the fibroadenoma case, the tu-mor was visible in modulus elastograms that hadbeen computed using small strain and largestrain (12%), although the contrast of the elasto-grams computed at large strain (7:1) was lowerthan that computed at smaller strain (10:1). Thefibroadenoma tumor was not visible in nonlinearparameter elastograms. The inclusion in the pa-tient with IDC was discernible in shear moduluselastograms recovered using small and largestrains. The stiffness contrast of the moduluselastograms recovered at both small and largestrains was comparable, however, and the IDCtumor was visible in nonlinear parameter elasto-grams. This result is one of several that havedemonstrated the clinical value of nonlinear elas-tographic imaging. Specifically, elastography cancharacterize the nonlinear behavior of soft tissuesand may be used to differentiate between benignand malignant tumors.

REFERENCES

1. Krouskop TA, Wheeler TM, Kallel F, et al. Elastic

moduli of breast and prostate tissues under

compression. Ultrason Imaging 1998;20:260–74.

Elastography 9

2. Parker KJ, Doyley MM, Rubens DJ. Imaging the

elastic properties of tissue: the 20 year perspec-

tive. Phys Med Biol 2011;56:R1–29.

3. Samani A, Zubovits J, Plewes D. Elastic moduli of

normal and pathological human breast tissues: an

inversion-technique-based investigation of 169

samples. Phys Med Biol 2007;52:1565–76.

4. Sarvazyan AP, Skovoroda AR, Emelianov SY, et al.

Biophysical bases of elasticity imaging. Acoust Im-

aging 1995;21:223–40.

5. Lerner RM, Parker KJ, Holen J, et al. Sono-elastic-

ity: medical elasticity images derived from ultra-

sound signals in mechanically vibrated targets.

Acoust Imaging 1988;16:317–27.

6. O’Donnell M, Skovoroda AR, Shapo BM, et al. Inter-

nal displacement and strain imaging using ultra-

sonic speckle tracking. IEEE Trans Ultrason

Ferroelectrics Freq Contr 1994;41(3):314–25.

7. Ophir J, Cespedes I, Ponnekanti H, et al. Elastog-

raphy: a quantitative method for imaging the elas-

ticity of biological tissues. Ultrason Imaging 1991;

13:111–34.

8. Bishop J, Samani A, Sciarretta J, et al. Two-dimen-

sional MR elastography with linear inversion recon-

struction: methodology and noise analysis. Phys

Med Biol 2000;45:2081–91.

9. Muthupillai R, Lomas DJ, Rossman PJ, et al. Mag-

netic-resonance elastography by direct visualiza-

tion of propagating acoustic strain waves.

Science 1995;269:1854–7.

10. Sinkus R, Lorenzen J, Schrader D, et al. High-res-

olution tensor MR elastography for breast tumour

detection. Phys Med Biol 2000;45:1649–64.

11. Weaver JB, Van Houten EE, Miga MI, et al. Mag-

netic resonance elastography using 3D gradient

echo measurements of steady-state motion. Med

Phys 2001;28:1620–8.

12. Khalil AS, Chan RC, Chau AH, et al. Tissue elastic-

ity estimation with optical coherence elastography:

toward mechanical characterization of In vivo soft

tissue. Ann Biomed Eng 2005;33:1631–9.

13. Kirkpatrick SJ, Wang RK, Duncan DD. OCT-based

elastography for large and small deformations. Opt

Express 2006;14:11585–97.

14. Ko HJ, Tan W, Stack R, et al. Optical coherence

elastography of engineered and developing tissue.

Tissue Eng 2006;12:63–73.

15. Bamber JC, Barbone PE, Bush NL, et al. Progress

in freehand elastography of the breast. IEICE Trans

Inf Sys 2002;E85d:5–14.

16. Greenleaf JF, Fatemi M, Insana M. Selected

methods for imaging elastic properties of biological

tissues. Annu Rev Biomed Eng 2003;5:57–78.

17. Manduca A, Dutt V, Borup DT, et al. Inverse

approach to the calculation of elasticity maps for

magnetic resonance elastography. SPIE Med Im-

aging Proceedings 1998;3338:426–36.

18. Ophir J, Garra B, Kallel F, et al. Elastographic

imaging. Ultrasound Med Biol 2000;26(Suppl 1):

S23–9.

19. Doyley MM. Model-based elastography: a survey

of approaches to the inverse elasticity problem.

Phys Med Biol 2012;57:R35–73.

20. Fung YC. Biomechanics: mechanical properties of

living tissue. New York: Springer; 1981.

21. Timoshenko SP, Goodier JN. Theory of elasticity.

Singapore: McGraw-Hill; 1970.

22. Landau LD, Lifshitz EM, Kosevich AM, et al. Theory

of elasticity. Oxford (United Kingdom): Elsevier But-

terworth-Heinemann; 1986.

23. Ophir J, Alam SK, Garra B, et al. Elastography: ul-

trasonic estimation and imaging of the elastic prop-

erties of tissues. Proc Inst Mech Eng H 1999;213:

203–33.

24. Van Houten EE, Miga MI, Weaver JB, et al. Three-

dimensional subzone-based reconstruction algo-

rithm for MR elastography. Magn Reson Med

2001;45:827–37.

25. Bilgen M, Insana M. Elastostatics of a spherical in-

clusion in homogeneous biological media. Phys

Med Biol 1998;43:1–20.

26. Kallel F, Bertrand M, Ophir J. Fundamental limita-

tions on the contrast-transfer efficiency in elastog-

raphy: an analytic study. Ultrasound Med Biol

1996;22:463–70.

27. Love A. The stress produced in a semi-infinite solid

bypressureonpart of theboundary. In: Philosophical

transactions of the Royal Society of London, vol 228.

London: The Royal Society; 1929. p. 377–420.

28. Sumi C, Suzuki A, Nakayama K. Estimation of

shear modulus distribution in soft-tissue from

strain distribution. IEEE Trans Biomed Eng 1995;

42:193–202.

29. Brigham JC, Aquino W, Mitri FG, et al. Inverse esti-

mation of viscoelastic material properties for solids

immersed in fluids using vibroacoustic techniques.

J Appl Phys 2007;101:023509–1–14.

30. Hall TJ, Bilgen M, Insana MF, et al. Phantom mate-

rials for elastography. IEEE Trans Ultrason Ferro-

electrics Freq Contr 1997;44:1355–65.

31. Konofagou E, Dutta P, Ophir J, et al. Reduction of

stress nonuniformities by apodization of com-

pressor displacement in elastography. Ultrasound

Med Biol 1996;22:1229–36.

32. McLaughlin J, Renzi D. Shear wave speed recov-

ery in transient elastography and supersonic imag-

ing using propagating fronts. Inverse Probl 2006;

22:681–706.

33. Miga MI. A new approach to elastography using

mutual information and finite elements. Phys Med

Biol 2003;48:467–80.

34. Parker KJ, Huang SR, Musulin RA, et al. Tissue-

response to mechanical vibrations for sonoelastic-

ity imaging. Ultrasound Med Biol 1990;16:241–6.

Doyley & Parker10

35. Ponnekanti H, Ophir J, Cespedes I. Ultrasonic-im-

aging of the stress-distribution in elastic media

due to an external compressor. Ultrasound Med

Biol 1994;20:27–33.

36. Samani A, Bishop J, Plewes DB. A constrained

modulus reconstruction technique for breast can-

cer assessment. IEEE Trans Med Imaging 2001;

20:877–85.

37. Bamber JC, Bush NL. Freehand elasticity imaging

using speckle decorrelation rate. New York: Plenum

Press; 1995.

38. Rizzatto G, Chersevani R, Solbiati L. High resolu-

tion ultrasound assists in breast diagnosis. Diagn

Imaging Int 1993;9:42–5.

39. Skovoroda AR, Aglyamov SR. On reconstruction of

elastic properties of soft biological tissues exposed

to low-frequencies. Biofizika 1995;40:1329–34.

40. Doyley MM, Bamber JC, Shiina T, et al. Recon-

struction of elasticity modulus distribution from en-

velope detected B-mode data. Proc IEEE Ultrason

Symp 1996;2:1611–4.

41. Kallel F, Bertrand M. Tissue elasticity reconstruc-

tion using linear perturbation method. IEEE Trans

Med Imaging 1996;15:299–313.

42. Doyley MM, Meaney PM, Bamber JC. Evaluation of

an iterative reconstruction method for quantitative

elastography. Phys Med Biol 2000;45:1521–40.

43. Jiang J, Varghese T, Brace CL, et al. Young’s

modulus reconstruction for radio-frequency abla-

tion electrode-induced displacement fields: a

feasibility study. IEEE Trans Med Imaging 2009;

28:1325–34.

44. Richards MS, Barbone PE, Oberai AA. Quantitative

three-dimensional elasticity imaging from quasi-

static deformation: a phantom study. Phys Med

Biol 2009;54:757–79.

45. Yamakoshi Y, Sato J, Sato T. Ultrasonic imaging of

internal vibrtation of soft tissue under forced vibra-

tion. IEEE Trans Ultrason Ferroelectrics Freq Contr

1990;37:45–53.

46. Lerner RM, Huang SR, Parker KJ. “Sonoelasticity”

images derived from ultrasound signals in mechan-

ically vibrated tissues. Ultrasound Med Biol 1990;

16:231–9.

47. Manduca A, Oliphant TE, Dresner MA, et al. Mag-

netic resonance elastography: non-invasive map-

ping of tissue elasticity. Med Image Anal 2001;5:

237–54.

48. Wu Z, Hoyt K, Rubens DJ, et al. Sonoelastographic

imaging of interference patterns for estimation of

shear velocity distribution in biomaterials. J Acoust

Soc Am 2006;120:535–45.

49. Doyley MM, Srinivasan S, Pendergrass SA, et al.

Compartive evaluation of strain-based and model-

based modulus elastography. Ultrasound Med

Biol 2004;31:787–802.

50. Van Houten EE, Doyley MM, Kennedy FE, et al.

Initial in vivo experience with steady-state sub-

zone-based MR elastography of the human breast.

J Magn Reson Imaging 2003;17:72–85.

51. McAleavey S, Collins E, Kelly J, et al. Validation of

SMURF estimation of shear modulus in hydrogels.

Ultrason Imaging 2009;31:131–50.

52. Nightingale K, McAleavey S, Trahey G. Shear-wave

generation using acoustic radiation force: in vivo

and ex vivo results. Ultrasound Med Biol 2003;29:

1715–23.

53. Sarvazyan AP, Rudenko OV, Swanson SD, et al.

Shear wave elasticity imaging: a new ultrasonic

technology of medical diagnostics. Ultrasound

Med Biol 1998;24:1419–35.

54. Ji L, McLaughlin JR, Renzi D, et al. Interior elasto-

dynamics inverse problems: shear wave speed

reconstruction in transient elastography. Inverse

Probl 2003;19:S1–29.

55. Qiu YP, Sridhar M, Tsou JK, et al. Ultrasonic

viscoelasticity imaging of nonpalpable breast tu-

mors: preliminary results. Acad Radiol 2008;15:

1526–33.

56. Sinkus R, Tanter M, Xydeas T, et al. Viscoelastic

shear properties of in vivo breast lesions measured

by MR elastography. Magn Reson Imaging 2005;

23:159–65.

57. Sinkus R, Tanter M, Catheline S, et al. Imaging

anisotropic and viscous properties of breast tissue

by magnetic resonance-elastography. Magn Reson

Med 2005;53:372–87.

58. Hamhaber U, Klatt D, Papazoglou S, et al. In vivo

magnetic resonance elastography of human brain

at 7 T and 1.5 T. J Magn Reson Imaging 2010;32:

577–83.

59. Sack I, Beierbach B, Wuerfel J, et al. The impact of

aging and gender on brain viscoelasticity. Neuro-

image 2009;46:652–7.

60. Asbach P, Klatt D, Hamhaber U, et al. Assess-

ment of liver viscoelasticity using multifrequency

MR elastography. Magn Reson Med 2008;60:

373–9.

61. Hoyt K, Castaneda B, Parker KJ. Two-dimensional

sonoelastographic shear velocity imaging. Ultra-

sound Med Biol 2008;34:276–88.

62. Perrinez PR, Kennedy FE, Van Houten EE, et al.

Modeling of soft poroelastic tissue in time-

harmonic MR elastography. IEEE Trans Biomed

Eng 2009;56:598–608.

63. Schmitt C, Soulez G, Maurice RL, et al. Noninva-

sive vascular elastography: toward a complemen-

tary characterization tool of atherosclerosis in

carotid arteries. Ultrasound Med Biol 2007;33:

1841–58.

64. Righetti R, Garra BS, Mobbs LM, et al. The feasi-

bility of using poroelastographic techniques for

Elastography 11

distinguishing between normal and lymphedema-

tous tissues in vivo. Phys Med Biol 2007;52:

6525–41.

65. Catheline S, Gennisson J, Delon G, et al. Measure-

ment of viscoelastic properties of homogeneous

soft solid using transient elastography: an inverse

problem approach. J Acoust Soc Am 2004;116:

3734–41.

66. Skovoroda AR, Lubinski MA, Emelianov SY, et al.

Reconstructive elasticity imaging for large defor-

mations. IEEE Trans Ultrason Ferroelectrics Freq

Contr 1999;46:523–35.

67. Oberai AA, Gokhale NH, Goenezen S, et al. Linear

and nonlinear elasticity imaging of soft tissue

in vivo: demonstration of feasibility. Phys Med Biol

2009;54:1191–207.