Elastography: ultrasonic estimation and imaging of the

203

Elastography: ultrasonic estimation and imaging of theelastic properties of tissues

J Ophir1,2*, S K Alam1, B Garra3, F Kallel1, E Konofagou1,2, T Krouskop4 and T Varghese11Department of Radiology, The University of Texas Medical School, Houston, Texas, USA2Bioengineering Program, University of Houston, Texas, USA3Department of Radiology, Georgetown University Medical Center, Washington, DC, USA4Baylor College of Medicine, Houston, Texas, USA

Abstract: The basic principles of using sonographic techniques for imaging the elastic properties oftissues are described, with particular emphasis on elastography. After some preliminaries that describesome basic tissue stiVness measurements and some contrast transfer limitations of strain images arepresented, four types of elastograms are described, which include axial strain, lateral strain, modulusand Poissons ratio elastograms. The strain filter formalism and its utility in understanding the noiseperformance of the elastographic process is then given, as well as its use for various image improve-ments. After discussing some main classes of elastographic artefacts, the paper concludes with recentresults of tissue elastography in vitro and in vivo.

Keywords: elastography, elastogram, elasticity, Youngs modulus, stress, strain, ultrasound, imaging,elastic modulus, Poissons ratio, breast, prostate, kidney

1 INTRODUCTION described in closed-form mathematical expressions,because they are time and moisture dependent. Whenliving, they are metabolically active and exhibit certainPredicting and understanding the behaviour of materialsmechanical properties, which change soon after death.when they are subject to mechanical forces is the basisMoreover, these mechanical properties may be age,for many aspects of modern engineering practice, fromstrain rate and strain range dependent [36 ]. To simplifythe design of structures to the design of solid propellantthe characterization of a tissue when the loading is ofrockets [1]. In the early development of the field of mech-short enough duration that the viscous nature of theanics, studying the behaviour of materials often involvedmaterial can be ignored, the tissue can be assumed toapplying loads to the system until the system failed, andbehave elastically. This means that the state of the tissuethen the failure mechanism was studied and used to inferonly depends on the current loading; there is no eVectthe behaviour of the material prior to failure. Then,from previous loading. By idealizing the tissue as anas technology advanced, it became possible to studyelastic material, the task of describing its behaviour isthe behaviour of complex material systems using non-reduced to a matrix of 81 stiVness constants that mustdestructive testing procedures, e.g. X-ray analysis,be specified [1]. Since obtaining these constants is a veryacoustic behaviour and photoelastic behaviour. Thechallenging process, additional assumptions are oftenresults of the experimental studies have contributed tomade to reduce the complexity of describing the tissuethe development of understanding how materials behavebehaviour [7].and to the development of mathematical models that

It is important to recognize the assumptions that arehelp predict the behaviour of more complex materialoften employed to create a simple mathematical modelsystems that are fashioned in very complex patterns [2].of a tissue system. Tissue has a hierarchical structureIn contrast to engineering materials, biological tissuesand by choosing the scale or size of the tissue samplesare not very well behaved, in the sense of being easilythat are being studied, it is often possible to assume thatthe tissue is orthotropic, so the number of constants is

The MS was received on 27 February 1998 and was accepted afterreduced to 27. Then, by further restricting the scale ofrevision for publication on 27 January 1999.

* Corresponding author: Department of Radiology, Ultrasonics the sample so that the small structures in the system areLaboratory, The University of Texas Medical School, 6431 Fannin Suite randomly and uniformly distributed in the sample, the2100, Houston, TX77030, USA.

assumption of homogeneity is usefully employed so thatPresent address: Riverside Research Institute, New York, USA.Present address: University of Vermont, Burlington, Vermont, USA. only 12 constants are needed to describe the tissue.

H01398 IMechE 1999 Proc Instn Mech Engrs Vol 213 Part H

204 J OPHIR, S K ALAM, B GARRA, F KALLEL, E KONOFAGOU, T KROUSKOP AND T VARGHESE

Again, if the scale of the tissue is picked with care, it from experiments using animal tissues and all of theinformation relates to uniaxial tensile tests of the tissue.may be appropriate to approximate the tissue as an iso-

tropic material so that only two constants are needed to The stiVness parameter cannot be measured directly.A mechanical stimulus of some kind must be propagateddescribe the tissues response to mechanical loads. These

two constants are the Lame constants, or their technical into the tissue, and precision means for detecting theresulting internal tissue motions must be provided. Suchderivatives, Youngs modulus and Poissons ratio.

The elastic properties of soft tissues depend on their means may include ultrasound, magnetic resonanceimaging (MRI) or other diagnostic imaging modalities.molecular building blocks, and on the microscopic and

macroscopic structural organization of these blocks [8]. In the last 15 years, interest has been mounting in theultrasonic imaging of tissue elasticity parameters. AThe standard medical practice of the soft tissue palpation

is based on qualitative assessment of the low-frequency comprehensive literature review of this field can be foundin Ophir et al. [25] and in Gao et al. [17 ], and will notstiVness of tissue. Pathological changes are generally

known to be correlated with changes in tissue stiVness be repeated here. Tissue elasticity imaging methodsbased on ultrasonics fall into two main groups:as well. Many cancers, such as scirrhous carcinoma of

the breast, appear as extremely hard nodules [9]. In (a) methods where a quasi-static compression is appliedto the tissue and the resulting components of the strainmany cases, despite the diVerence in stiVness, the small

size of a pathological lesion and/or its location deep in tensor are estimated [24, 28, 29 ] and (b) methods wherea low-frequency vibration is applied to the tissue andthe body preclude its detection and evaluation by palp-

ation. In general, the lesion may or may not possess the resulting tissue behaviour is inspected [23, 3033].Among the techniques based on the quasi-static esti-echogenic properties which would make it ultrasonically

detectable. For example, tumours of the prostate or the mation of tissue strain, elastography [24] is based onestimating the tissue strain using a correlation algorithm,breast could be invisible or barely visible in standard

ultrasound examinations, yet be much harder than the whereas another elasticity imaging technique [28 ] isbased on estimating such strain using signal phase infor-embedding tissue [10]. DiVuse diseases such as cirrhosis

of the liver are known to significantly increase the stiV- mation. Most importantly, however, in both methods,local tissue displacements are estimated from the timeness of the liver tissue as a whole [9], yet they may

appear normal in conventional ultrasound examination. delays between gated pre- and post-compression echosignals, whose axial gradient is then computed to esti-Since the echogenicity and the stiVness of tissue are gen-

erally uncorrelated, it is expected that imaging tissue mate and display the local strain. Recently, intravascularapplication of elastographic techniques have also beenstiVness or strain will provide new information that is

related to pathological tissue structure. This expectation reported [3436 ]. In this paper, emphasis is placed ondescribing the recent progress in elastography, which ishas now been confirmed [10 ]. In addition to pathology,

there is additional evidence that various normal tissue being developed in the authors laboratory. Among thesecond group of techniques, in sonoelasticity imagingcomponents possess consistent diVerences in their stiV-

ness parameters as well. For example, in the ovine [30, 31 ], the vibration amplitude pattern of the shearwaves in the tissue under investigation is detected usingkidney, the stiVness contrast between the cortex and the

medullary pyramids has recently been measured to be Doppler methods, and a corresponding colour image(similar to a colour Doppler display) is superimposed ononly about 6 dB, and strain images showing easily dis-

cernible contrast have been made [11]. This observation the conventional grey-scale image. An absence ofvibration may signify the presence of tumours. A theoryprovides the basis for imaging the normal anatomy as

well. of sonoelasticity imaging was developed [37] and anin vitro study on excised human prostate showed betterOver the past 20 years there have been numerous

investigations conducted to characterize the mechanical sensitivity and specificity than conventional transrectalultrasound [38 ]. Yamakoshi et al. [32 ] developed aproperties of biological tissue systems [36, 1221 ]

which have been idealized often as homogeneous, iso- method to map both the amplitude and the phase oflow-frequency wave propagation in the tissue. These cantropic elastic materials. Much of the work has focused

on bone, dental materials and vascular tissue. There are be used to derive the velocity and dispersion propertiesof the wave propagation. Krouskop et al. [23 ] used aarticles that discuss methods used to characterize these

tissues and there is a large volume of experimental data single-element pulsed Doppler instrument to measure thetissue flow at points of interest under external vibration.about the mechanical response of these tissues to various

types of loadings [17, 2227]; however, there is a void Somewhat later, a parallel development has occurredin MRI. Plewes et al. [39 ] have developed a method thatin the retrievable literature regarding the mechanical

properties of tissue systems tested in vivo. In fact, there is based on the compression of the tissue and estimationof the resultant strains using MRI. Fowlkes et al. [40]is very limited information about the mechanical proper-

ties of most of the soft tissue systems that make up the have discussed an MRI-based method to measure tissuedisplacements. They have also shown mathematicalbodys organs. Yamadas book [21] presents a relatively

broad range of data, but much of the data are derived reconstruction of the distribution of the moduli for select

H01398 IMechE 1999Proc Instn Mech Engrs Vol 213 Part H

205ELASTOGRAPHY: ULTRASONIC ESTIMATION AND IMAGING

examples. Muthupillai et al. [41] developed an MRI of elastography, recent results that demonstrate thatmethod similar to the ultrasonic Doppler method quality elastograms may be produced and interpreteddescribed by Yamakoshi et al. [32]. under low and high elastic contrast conditions are pro-

First some basic tissue stiVness results are presented duced for both in vitro and in vivo studies.which demonstrate the existence of stiVness contrastamong normal tissues and between normal and patho-logical tissues in the breast and prostate. These dataprovide the continued motivation for further develop-

2 BASIC DATA ON TISSUE STIFFNESSment of this field. Then the elastographic process isdescribed, starting from the tissue elastic modulus distri-

Very little basic data are available in the literature onbution, progressing through various algorithms for pre-the stiVness of soft tissues. Perhaps this is due to the factcision time delay estimation of echoes from strainedthat, until recently, such data were of no practical conse-tissues and culminating in the production of the elasto-quence. However, the paucity of data does not changegram. Several kinds of elastograms are then described,the well-known fact that tissue elasticity is intimatelyeach displaying a diVerent quantity, which is related torelated to the higher levels of tissue organization. In viewthe elastic properties of the tissue, and each having beenof the present ability to visualize the elastic attributes ofderived using diVerent methodologies. These includetissues, it is inevitable that more data will become avail-axial strain, lateral strain, modulus and Poissons ratioable in the ensuing years. Some data have been collectedelastograms [42]. The use of ultrasound to acquire tissueby Sarvazyan et al. [47 ], Parker et al. [48 ] and Walzmotion information results in certain basic limitationset al. [49 ]. Some of the authors recent results on breaston the attainable elastographic image parameters, whichand prostate tissues in vitro are given in Table 1 [50 ].may be described by the theoretical framework knownThese results indicate that in the normal breast fibrousas the strain filter [43]. The strain filter may be used totissues are stiVer than glandular tissues, which are inpredict and design important improvements to variousturn stiVer than adipose tissues. The two kinds ofelastographic image attributes, such as dynamic rangetumours studied show diVerent behaviours, with theexpansion and improvement in the signal-to-noise ratioinfiltrating ductal carcinomas being significantly stiVerin elastography (SNRe) through multiresolution pro-than the ductal tumours. Some of the tissues exhibitcessing [44]. In combination with certain contrast trans-marked non-linear changes in their stiVness behaviourfer eYciencies (CTEs) inherent in the conversion of thewith applied precompressive strain, while others remainmodulus to strain contrast [45, 46 ], the strain filter for-unchanged. There appear to be opportunities in diVer-malism may be used to predict the upper bound as wellentiating breast tissues based on their stiVness values asas the practically attainable performance of the contrast-well as their non-linear stiVness behaviour. Significantto-noise ratio in elastography (CNRe). After a briefdiVerences are also evident among normal, BPH anddescription of the mechanical, acoustic and signal pro-

cessing artefacts that may be encountered in the practice cancerous tissues of the prostate (Table 2).

Table 1 Results of actual stiVness measurements (in kPa) of normal and abnormal breast tissues in vitro

5% pre-compression 20% pre-compression

Strain rate 1%/s 10%/s 40%/s 2%/s 20%/s 80%/s

Normal fat (n=40) 187 197 229 208 206 235Normal glandular (n=31) 2814 3311 3514 4815 5719 6617Fibrous (n=21) 9733 10732 11883 22088 23359 24583Ductal CA (n=23) 228 254 265 29167 30158 30778Infiltrating ductal CA (n=32) 10632 9333 11243 558180 490112 460178

Table 2 Results of actual stiVness measurements (in kPa) of normal and abnormal prostate tissues in vitro (BPH=benign prostatichypertrophy, CA=prostatic carcinoma)

2% pre-compression 4% pre-compression

Strain rate 0.4%/s 4%/s 16%/s 0.8%/s 8%/s 32%/s

Normal anterior (n=32) 5514 6217 5919 6015 6318 6316Normal posterior (n=32) 6219 6917 6518 6814 7014 7111BPH (n=21) 388 369 388 4012 3611 4113CA (n=28) 9619 10020 9918 23034 22132 24128



3 THE ELASTOGRAPHIC PROCESS by a constant uniaxial stress, all points in the mediumexperience a resulting level of longitudinal strain whoseprincipal components are along the axis of compression.Several years ago the authors introduced a new methodIf one or more of the tissue elements has a diVerenttermed elastography for direct imaging of the strain andstiVness parameter than the others, the level of strain inYoungs modulus of tissues [24, 51, 52 ]. Elastographythat element will generally be higher or lower; a harderdiVers from some of the vibrational methods that weretissue element will generally experience less strain thandescribed above in several important aspects:a softer one. The longitudinal axial strain is estimated

1. The stress applied to the tissue is not vibratory, but in one dimension from the analysis of ultrasonic signalsrather quasi-static. This tends to avoid problems due obtained from standard medical ultrasound diagnosticto reflections, standing waves and mode patterns equipment. This is accomplished by acquiring a set ofwhich may be set up in the tissue and which may digitized radio-frequency (RF) echo lines from the tissueinterfere with quality image formation. region of interest by compressing the tissue with the

2. The applied quasi-static uniaxial stress reduces the ultrasonic transducer (or with a transducer/compressorcomplexity of the generalized one-dimensional dis- combination) along the ultrasonic radiation axis by acrete viscoelastic equation of forced motion of the small amount (about 1 per cent or less of the tissueform depth) and by acquiring a second, post-compression set

of echo lines from the same region of interest. CongruentM

d2xdt2

+Rdxdt

+Kx=F0 ejvt echo lines are then subdivided into small temporal win-dows which are compared pairwise by using cross-corre-

which contains inertial (M), viscous (R) and stiVness lation techniques [54 ], from which the change in arrival(K ) controlled terms, and where x is the displacement, time of the echoes before and after compression can beF0 is the force amplitude and v is the angular estimated. Due to the small magnitude of the appliedvibrational frequency. The reduced form is the much compression, there are only small distortions of the echosimpler Hookean equation Kx=F0 ; since v=0 and lines, and the changes in arrival times are also small.x is a constant, the velocity and acceleration terms The local longitudinal strain is estimated as [24 ]vanish. In principle, this allows the isolation anddirect extraction of the local tissue stiVness parameter

e11,local=(t1bt1a)(t2bt2a)

t1bt1a(K ) from measurement of the diVerential appliedforce (or stress) F0 and the resulting local changes indisplacement x. For the continuous case, the equival- where t1a is the arrival time of the pre-compression echoent equation becomes eE=s, where e is the elastic from the promixal window, t1b is the arrival time of themodulus, E is the strain and s is the applied stress. pre-compression echo from the distal window, t2a is the3. The average levels of strain produced in the tissue are arrival time of the post-compression echo from theusually very small (on the order of 0.01). These strain proximal window and t2b is the arrival time of the post-levels are considered small enough to keep the compression echo from the distal window. The windowsHookean equation well within the linear range, based are usually translated in small overlapping steps alongon the linear stressstrain relationships for gels and the temporal axis of the echo line, and the calculation ishuman muscle in vivo reported by Mridha and repeated for all depths. The fundamental assumptionOdmann [53 ] which were valid up to a strain level of made is that speckle motion adequately represents the0.25. They are kept small also in order to keep the underlying tissue motion for small uniaxial compressionsdistortion in the time-shifted echo signals (before cor- [55 ]. Recently, solutions to the inverse problem haverections) to a minimum, hence maintaining a low level been described in the literature [5658 ] that are capableof decorrelation noise in the elastogram. If proper of converting estimated tissue displacements and/orcorrections can be made, however, it is possible to strains to modulus elastograms, with the assumedincrease the applied strain and gain in image contrast, knowledge of the boundary conditions and under severalto within the limits of the correction methods, and other assumptions. Under the assumption of plane strainperhaps most importantly. state [57 ] and plane stress state [56 ] conditions the

4. Elastography is capable of producing high-resolution reconstruction algorithm proposed in references [56 ] andimages (elastograms) [24, 27 ]. The term elastogram [57 ] requires the knowledge of the lateral tissue strains.is used in this article as a generic descriptor of all the In reference [57 ] the lateral strain information wasdiVerent kinds of images that display some param- simply derived from the estimated axial tissue strainseters that are related to the elastic behaviour or nature based on the assumption of tissue incompressibility. Inof the tissue. In this sense, elastograms will be reference [56 ] only theoretical data derived from the ana-described that display axial or lateral strains, the elas- lytical solution of the elasticity equation of an infinitetic moduli or Poissons ratio distributions in tissues. medium embedding a circular inclusion were used to test

the reconstruction algorithm. In reference [58] only theWhen an elastic medium, such as tissue, is compressed



estimated axial tissue displacements are used for the dynamic range and respective elastographic signal-to-noise ratio (SNRe) at a given resolution in the elasto-reconstruction. It has also become possible to generate

lateral strain elastograms based on incompressibility gram, limited by noise and/or decorrelation. The contri-butions of the signal processing and ultrasound systemprocessing [59] or from direct lateral strain estima-

tions [42 ]. Poissons ratio elastograms have also been parameters and other algorithms are indicated as inputsinto the strain filter. The optimized strain elastogram isreported [42].

It is important to emphasize that elastography is a used as an input to the (optional ) inverse problem solu-tion block, where the contrast transfer eYciency (CTE)method that ultimately can generate several new kinds

of images. As such, all the properties of elastograms are is improved, with subsequent reduction of artefacts inthe modulus elastogram. All these blocks are describeddiVerent from the familiar properties of sonograms.

While sonograms convey information related to the local later in this article. A complete description may be foundin Ophir et al. [26 ].acoustic backscatter energy from tissue components,

elastograms relate to its local strains, Youngs moduli orPoissons ratio. In general, these elasticity parametersare not directly correlated with sonographic parameters,

4 CONTRAST-TRANSFER EFFICIENCYi.e. elastography conveys new information about internaltissue structure and behaviour under load that is nototherwise obtainable. Using ultrasonic techniques, it is only possible to meas-

ure some of the components of the local tissue strainThe general process of creating strain and moduluselastograms, beginning with the modulus distribution in tensor. The local components of the stress tensor remain

unknown. If the optional inverse problem solution is nottissue and ending with a corresponding modulus elasto-gram, is shown in Fig. 1, showing a block diagram of used (e.g. because of incomplete knowledge of the

boundary conditions, complexity of the assumptions orthe process. The input to the process is the intrinsic tissuemodulus distribution. The outputs can be either the computational load), the strain elastogram is all that is

available to represent the distribution of tissue elasticstrain (axial and/or lateral ) elastograms, the Poissonelastogram or the modulus elastogram. The tissue strain moduli. This representation may result in mechanical

artefacts and in a limitation of the contrast-transferobtained by a quasi-static tissue compression, restrictedby the mechanical boundary conditions, is measured eYciency (CTE) [45, 46 ] under certain conditions. The

CTE was defined by Ponnekanti et al. [45] as the ratiousing the ultrasound system. The block describing thestrain filter [43 ] embodies the selective filtering of the of the observed (axial ) strain contrast measured from

the strain elastogram and the underlying true modulustissue strains by the ultrasound system and signal pro-cessing parameters. The strain filter predicts a finite contrast, using a plane strain state model. Expressed in

Fig. 1 Block diagram showing the process of generating strain and modulus elastograms from tissues. Notethat the production of the modulus elastogram via the inverse problem solution is an optimal enhance-ment to the process



decibels, it is given by gradient of tissue displacements. The local tissue dis-placements are estimated from the time delays of gatedCTE(dB)=|C0(dB) ||Ct(dB) | pre- and post-compression echo signals [24]. Time delays

where the magnitude is used in order to have the CTE are generally estimated from the location of the peak ofnormalized to the zero dB level; i.e. the maximum the cross-correlation function between the gated pre- andeYciency is reached at 0 dB for both hard and soft post-compression echo signals.inclusions. This property of elastography represents a The quality of elastograms is highly dependent on thefundamental limitation that has been verified by finite quality of the TDE. TDE in elastography is mainly cor-element (FE) simulations [45] and was also corroborated rupted by two factors: firstly, the random noise (elec-theoretically by Kallel et al. [46 ]. Figure 8 in a later sec- tronic and quantization) introduces errors in the TDEtion shows the behaviour of the CTE parameter over an and, secondly, tissue needs to be compressed to produce80 dB dynamic range of true modulus contrast as meas- elastograms. The very same compression of the tissueured from simulated data and as predicted using an ana- distorts the post-compression signal such that it is notlytical model. It is clear from the figure that for low an exact delayed version of the pre-compression signal.modulus contrast levels (a high level of target modulus This decorrelation increases with increasing strain andhomogeneity), the elastographic strain contrast is nearly is independent of the signal-to-noise ratio in the echoequal to the modulus contrast (CTE#1 or 0 dB). This is signals (SNRs). Any phenomenon (such as lateral anda very important observation, since it suggests that for elevational motions) that degrades the precision oflow modulus contrast tissues, the simply computed axial the time delay estimates will also degrade the strainstrain elastogram itself is nearly proportional to the true estimates, thus introducing additional noise into themodulus elastogram. This expected result has been veri-

elastogram.fied experimentally using ex vivo ovine kidneys [11]. StiVinclusions have a relatively high level of contrast-transfereYciency. However, soft inclusions that are completely

5.2 Decorrelation and stretchingsurrounded by harder background material have a lowcontrast-transfer eYciency, and thus may not be well vis-

Echo signal decorrelation is one of the major limitingualized by elastography. The reason for this limitationfactors in strain estimation and imaging. Alam andlies in the fact that due to the incompressible natureOphir [62 ] have demonstrated that for small strains, tem-of many soft tissues (Poissons ratio ~0.5), the softporal stretching [52, 63, 64 ] of the post-compressioninclusion will be so constrained that it will not be able tosignal by the appropriate factor can almost entirely com-deform under pressure as it might otherwise do withoutpensate for the axial decorrelation. When the post-constraints, thus assuming instead elastic properties thatcompression echo signal is stretched, it in eVect realignsare closer to those of the surrounding stiVer material.all the scatterers within the correlation window. In otherLater it will be shown that after considering elastographywords, appropriate temporal stretching removes thewithin the framework of the inverse problem solution, themean intra-window strain. Global stretching (of all win-CTE may be significantly improved.dows equally) was found to significantly improve theSNRe and expand the strain dynamic range in elasto-

5 AXIAL STRAIN ELASTOGRAPHY grams. Moreover, this step is computationally not veryintensive. Thus, a uniform global stretching of the post-

Elastography is inherently a three-dimensional problem. compression A-line prior to the displacement estimationHowever, the early elastograms estimated and displayed is highly advisable. In low-contrast targets and/or lowstrains in the axial direction only [24]. Until recently, all strains, this is a very eYcient displacement estimator. Inthe tissue motion in the lateral and elevational directions these situations, it produces quality elastograms with-were ignored. However, recent work by Lubinski et al. out significantly adding to the computational load.[59], Chaturvedi et al. [60, 61] and Konofagou and However, if the applied strain is large in high-contrastOphir [42 ] have taken into account the non-axial tissue targets, there will be significant overstretching in themovement, and even used the lateral displacement to areas of low strains, which by itself can significantlycompute lateral strain elastograms [42]. In this section, degrade elastograms in these areas. However, stretchinghowever, it will be assumed that there is no non-axial is mandatory in the presence of large strains; otherwisetissue movement, thus concentrating only on the axial the elastograms are so noisy that they are practicallydisplacement and strain estimation. useless. It also enhances the dynamic range of elasto-

graphy. It must be remembered that axial stretching canonly recover most of the decorrelation suVered due to5.1 Time delay estimation (TDE) of signals fromscatterer motion in the axial direction; decorrelation duestrained tissuesto lateral and elevational motions, as well as othersources of decorrelation, cannot be compensated for inTime delay is a very important parameter in elasto-

graphy. The tissue strain is typically estimated from the this way, and require other methods. Alam et al. [65]



have shown that a deconvolution filtering approach may produced with uniform stretching and adaptive stretch-ing respectively. Both elastograms have low noise, butbe useful in reducing the remaining decorrelation that

results due to stretching of the point spread function the elastogram produced using adaptive stretchingappears less noisy. Figure 3 shows the elastograms pro-(PSF) when the post-compression echo is stretched.duced on an in vitro ovine kidney that has a thermallesion in the upper pole. The applied strain was 0.5 percent. Figure 3a was produced with no stretching and5.3 Other estimatorsFig. 3b was produced using adaptive stretching. This last

5.3.1 Adaptive stretching image shows better details and has less visible noise.

Temporal stretching significantly improves TDE in5.3.2 Correlation coeYcient

elastography. However, the proper temporal stretchingfactor is dependent on the local strain, an unknown It has been discussed how the correlation between theparameter one is trying to estimate. In an elastically pre- and post-compression echoes can decrease withinhomogeneous tissue, the strains will vary and thus, applied strain. However, decorrelation itself has beenideally, the stretching factor will have to be varied at used to estimate delay and/or strain. Various researchersdiVerent window locations. Since temporal stretching by used the correlation coeYcient to estimate tissue motionthe factor that compensates for the strain maximizes thecorrelation, an iterative algorithm is indicated. In thisalgorithm, the local temporal stretching factor is adapt-ively varied until a maximum in the correlation isreached. The local strain is then computed directly fromthis temporal stretching factor. Since the axial corre-lation is maximized at each data window between thepre- and post-compression A-lines, this estimator is anoptimal (one-dimensional ) estimator of strain. It is alsowell known that the gradient operation amplifies noisein the displacement estimates. Since adaptive stretchinginvolves only intra-window operations and no inter-window operation, it does not suVer from this type ofdegradation. Overall, adaptive stretching may help inimproving the elastographic performance by a largefactor. Like the uniform global stretching, it alsoincreases the dynamic range of elastography [66 ].

Figure 2 shows elastograms for various methodsFig. 3 Elastograms of an ovine kidney in vitro with an

discussed in this section for an inhomogeneous tissue- approximately 3 mm diameter induced thermal lesionmimicking phantom. The applied strain was 2 per cent. (black circle) in the upper pole. The applied strain isFigure 2a shows the elastogram produced with no 0.5 per cent. Window size=2 mm, window overlap=stretching. At this strain, the gradient method without 50 per cent. Elastograms: (a) conventional gradientstretching fails to produce a useful elastogram due to method, (b) adaptive stretching. The actual lateral

width of each elastogram is 40 mmsevere decorrelation. Figures 2b and c show elastograms

Fig. 2 Inhomogeneous phantom experiment with 2 per cent applied strain. Window size=3 mm, windowoverlap=50 per cent, no median filtering. Elastograms: (a) gradient, (b) gradient with stretching and(c) adaptive stretching



[6771]. Bamber and Bush [72 ] proposed using the estimate tissue displacement and tissue strain using asubsequent gradient operator. While coherent estimationdecorrelation coeYcient (1 minus the correlation

coeYcient) for the envelope signals for freehand elas- methods [24, 28, 29, 63 ] generally have the advantage ofbeing highly accurate and precise, even relatively smallticity imaging. Varghese and Ophir [73 ] have demon-

strated that the decorrelation coeYcient has poor undesired motions are likely to cause signal decorre-lation, and thus significant degradation of the elasto-precision as a strain estimator. Alam and Ophir [74 ]

have demonstrated using simulated data that changes in gram. For elastography to become more universallypractical in applications such as intravascular andthe centre frequency and SNRs (both the PSF and the

SNR vary due to frequency-dependent attenuation, abdominal imaging, limitations associated with coherentstrain estimation methods that require tissue and systemchanges in the beam, etc., as the ultrasonic pulse propa-

gates through the tissue) introduce unknown variable stability must be overcome. On the other hand, incoher-ent estimators [7983 ] are moderately less precise butbias. Because of its simplicity, this estimator may be a

valuable tool for freehand elasticity imaging. However, far more robust.The principal idea behind the spectral approach isthe disadvantages need to be recognized and care should

be taken when using this estimator. based on the Fourier scaling property. The previouslyused spectral method proposed by Talhami et al. [79]

5.3.3 Phase-based methods relates the relative change in the mean coherent scattererspacing [79, 8490] to the strain, and uses it for intravas-

It is also possible to use phase to measure small tissuecular applications. This method assumes the presence of

displacements [55], and commercial ultrasound scannerscoherent tissue scatterers as well as underlying tissue

use phase change to estimate motion for Doppler pro-periodicities in the tissue structure, which may not hold

cessing. At least one group working on elasticity imagingin most cases. In contrast, the spectral methods pre-

uses a phase-based displacement estimator [28 ]. Sincesented in this section [82, 83] make no assumptions

the phase is only well defined for narrowband systems,regarding the composition or distribution of the tissue

some bandpass filtering is done prior to the computationscatterers.

of the phase, which may introduce a loss in spatialAn important advantage associated with the spectral

resolution.estimators [82, 83 ], as with adaptive stretching [66 ], isthat they may be used to estimate strain directly (i.e.5.3.4 Least-squares strain estimatorwithout using noise-amplifying gradient operators)within a single estimation window. Tissue strain usingA least-squares strain estimator (LSQSE) for elasto-

graphy has been proposed [75]. It was shown that with the spectral shift can be measured via the spectral cen-troid shift [82] or spectral cross-correlation [83 ]. Thesuch an estimator the signal-to-noise ratio in an elasto-

gram (SNRe) was significantly improved due to the spectral centroid [9194] has been widely used in esti-mating the Doppler shift, attenuation [91 ] and back-reduction of the displacement noise amplification due to

the gradient operation. The LSQSE results in an increase scattering [94 ]. While the downshift in the spectrum withfrequency-dependent attenuation complicates the esti-of the elastographic sensitivity (the smallest strain that

could be detected), thereby increasing the strain dynamic mation process for the Doppler shift, in elastographythis problem is not encountered since diVerentialrange that is depicted on the elastogram. Using simu-

lated data, it was shown that a trade-oV exists between estimates are used, which suVer common frequency-dependent attenuation eVects. The upshift in the centrethe improvement in SNRe and the reduction of strain

contrast and spatial resolution. frequency estimate with strain is illustrated with asimulation in Fig. 4.

5.3.5 Butterfly search For the spectral strain estimator using the centroidshift [82 ], it is found that if the bandwidth of the scat-

Alam and Parker [7678] developed the butterflytering noise process is much larger than the bandwidth

search technique for complex envelope signals fromof the system PSF, then

a deterministic analysis, derived using Schwartzsinequality. Since this method can simultaneously analyse fc2 fc1

fc1$Asmore than two successive A-lines, it is a natural candi-

date in multicompression elastography. Preliminarywhere the strain estimate is equal to the fractional shift

results have shown that it may improve the SNR andin the centre frequency of the power spectra, fc1 and fc2 ,the dynamic range in elastography.of the pre- and post-compression signals respectivelywithin a scaling constant A. In the case of spectral cross-5.3.6 Direct, incoherent, spectral strain estimatorscorrelation [83], the ratio of the spectral shift, which was

Elastography has been shown to be capable of producing measured using cross-correlation of the spectra, to thequality strain images in vitro and in vivo. Standard elasto- centre frequency of the PSF is proportional to the tissue

strain. Interestingly, this equation is of the same formgraphy uses a coherent cross-correlation technique to



Fig. 4 The frequency centroid shift due to strain: example of1 per cent compressive tissue strain. The peak of thepower spectrum (solid line), which in this case is veryclose to the actual centroid, is upshifted (broken line)

Fig. 5 Elastograms obtained using coherent and incoherentdue to the applied compressive strainstrain estimation algorithms, for a phantom with aninclusion that is three times stiVer than the back-as the defining equation for the strain itself, which isground. The RF signals were obtained using a 7.5 MHzgiven aslinear array with a 50 per cent bandwidth. The elasto-grams were obtained using (i) 0.5 per cent and (ii) 3L1L0

L0=s per cent applied compression. (a and d) Elastogram

obtained using the coherent cross-section method. (bwhere L0 is the original length of the segment and L1 and e) Elastogram obtained using the centroid method.the new length after compression. Comparison of the (c and f ) Elastogram obtained using the spectral cross-estimation performance using coherent cross-correlation correlation methodand the direct spectral strain estimators are illustratedqualitatively in Fig. 5 using elastograms obtained from

elastography there exists an optimal incrementalan inhomogeneous experimental phantom containing anapplied strain that maximizes the achievable SNRe.inclusion three times stiVer than the background at both

2. The tissue can be mechanically confined to reducelow (0.5 per cent) and high (3 per cent) applied strainslateral motion. This increases the elevational motion,[82, 83]. Note that coherent strain estimation providesbut since the beam is broader in that direction, itsthe least noisy elastogram for the low compression ofeVect can be neglected for the most part [95 ].0.5 per cent [Fig. 5(i)], when compared to the spectral

3. Use incompressibility processing, originally proposedmethods. However, for the large applied compression offor modulus reconstruction purposes [59 ], where the3 per cent [Fig. 5(ii)], the coherent strain estimator failslateral strains are computed from the axial strainswhen compared to the spectral methods that produceunder certain assumptions. However, this methodreasonable elastograms. No stretching techniques wereassumes Poissons ratio to be 0.5, which may notused. Finally, the method of spectral cross-correlationalways be true [53], and the contour of the zero lateralprovides the least noisy elastogram in this case [83 ].displacement must be accurately estimated.

4. Use a method to correct for non-axial motion.Chaturvedi et al. [60, 61] have proposed a compand-5.4 Options in the treatment of non-axial motionsing method that reconstructs the pre-compression

When tissue is compressed, in addition to the axial sonogram by compensating for lateral motion.motions it also undergoes non-axial motions that intro- Konofagou and Ophir [42] have proposed a methodduce additional decorrelations [95]. There are various that can estimate the lateral displacement with highways to handle this problem: precision and produce excellent corrected axial and

lateral strain elastograms. This approach will be1. Use small compression steps (typically


tensor is used to produce the elastogram, while the lat- show consistent improvement following the corrections.The part of the axial elastogram in Fig. 7c most cor-eral and elevational components are basically disre-

garded. However, all three components are needed to rupted by correlation noise coincides with the part ofthe maximum lateral displacement in Fig. 5d. This, alongfully characterize the motion of a three-dimensional

target [98]. Furthermore, the lateral and elevational with the corrected axial elastogram of Fig. 7f, verifiesthe assumption that the decorrelation noise unaccountedcomponents can severely corrupt the axial strain esti-

mation by inducing decorrelation noise [98, 99 ]. A new for in Fig. 7c is due to lateral motion. From Figs 7eand f it may be concluded that axial and lateral elasto-method has therefore been developed that produces high

precision lateral displacements [42]. In reference [42] it grams are diVerent due to existing compressibility and/oranisotropy in the prostate tissue. In a later sectionis shown that the higher the interpolation scheme, the

higher the number of independent displacement esti- (Section 8) a method is proposed to measure compress-ibility through calculation of the local Poisson ratio. Themates and thereby the higher the precision of the esti-

mation. Due to this high precision lateral tracking, iterative method has also been proven to be useful inbreast data in vivo [42].quality lateral elastograms can be generated that display

the lateral component of the strain tensor. The mainrequirements on the transducer specifications are: (a) aminimum of 50 per cent overlap between adjacent beams

7 MODULUS ELASTOGRAPHYand (b) a well-focused beam (on the order of 1 mm).Preliminary studies [42] have shown that the precisionincreases with a beamwidth decrease (possibly due to Elastography based on quantitative strain imaging

suVers from mechanical artefacts (shadowing and targetthe subsequent improvement in the ultrasonic lateralresolution) and this constitutes a topic of current investi- hardening artefact [24 ]) and from limitations to the

contrast-transfer eYciency (CTE). To go beyond suchgations. Briefly, the new method works as follows.Interpolated post-compression A-lines are generated via presumed limitations, a few groups independently con-

sidered elastography as a new, challenging inverse prob-a method of weighted interpolation [42] between neigh-bouring A-lines. Then, pre-compression segments are lem [5658, 101]. The inverse problem (IP) approach is

used extensively in electromagnetics, optics and geophys-cross-correlated with original and interpolated post-compression segments [42] and the location of maximum ics research [102]. In the biomedical field it has been

extensively studied in bioelectricity to determine the dis-correlation indicates the amount of lateral displacementthat resulted from the compression. A least-squares tribution of potentials on the surface of the heart or the

brain from a limited number of peripheral potentialalgorithm [75 ] is used to derive the lateral strain fromthe lateral displacement information. measurements. However, it is relatively new in the field

of continuum mechanics [103105] and until recently itA correction for lateral motion has been described byChaturvedi et al. [61] using a block-matching technique was not applied in the field of biomechanics, to which

elastography belongs.described by Bohs and Trahey [100]. The correctionresults in a significant increase of both the signal-to-noise For many reasons (non-linearity, noise in the data,

lack of complete data set, etc.), most of the IPs are ill-ratio and dynamic range of the axial elastogram at largeapplied strains. This is similar in principle to the one- posed in the sense of Hadamard [106 ]. Therefore, find-

ing a unique and stable solution of an IP is very diYcult.dimensional axial motion compensation and stretchingdescribed by Cespedes and Ophir [63 ] as a means for To this end, during the second half of this century, many

techniques have been proposed [107]. Since these tech-reducing the noise in elastograms. Taking into accountthe coupling between axial and lateral motion, an iterat- niques usually require several big matrix inversions,

they remain complex, consuming computer time andive method has been developed of successive correctionsfor and estimations of axial as well as lateral motion memory. In elastography it is fortunate that the solution

of the IP is an option rather than a necessity (Fig. 1)[42]. This led to a consistent increase of the signal-to-noise ratio of both axial and lateral elastograms. Results unlike in many other fields. Conceivably, the exercise of

this option will be restricted to cases where the simpleof the iterative method are shown in Figs 6 and 7 fromfinite element simulations and in vitro prostate data strain images alone are not suYcient to adequately com-

plete a given task (detection, characterization, etc.).respectively [42]. Figure 6 shows that the technique isable to remove large (curtain noise near the edges of Below is a short summary of the diVerent techniques

proposed to solve the IP in elastography, followed bythe image) as well as smaller (close to the centre of theimage) decorrelation eVects due to lateral motion, which typical results obtained using the technique of Kallel and

Bertrand [101]. The simulation results of the elasticare responsible for poor contrast-to-noise ratios for thehard lesion. This kind of improvement was predicted by modulus reconstruction were obtained after a total of

10 iterations of the linear perturbation technique pro-Kallel et al. [99 ] via the calculation of the correspondingtheoretical strain filters. In a similar fashion, the axial posed in reference [101].

The approach proposed by Skovoroda et al. [57 ] andand lateral elastograms of the prostate in vitro (Fig. 7)



Fig. 6 Finite element plane strain state simulation of an inclusion twice harder than the embedding back-ground (modulus equal to 21 kPa) at 3 per cent axial compression: (a) first axial elastogram (no axialor lateral correction), (b) first lateral elastogram (no axial correction), (c) second axial elastogram(with axial correction, no lateral correction, note the existence of curtain noise resulting fromunaccounted-for high lateral displacement towards the lateral sides of the target), (d) lateral displace-ment image (with axial correction), (e) second lateral elastogram (with axial correction), (f ) thirdaxial elastogram (with axial and lateral corrections) and (g) true axial strain image obtained fromfinite element simulation

Sumi et al. [56 ] to solve this problem consisted of reconstruction results obtained under ideal conditions,where the boundary conditions are perfectly known andrearranging the equations of the forward problem so

that the tissue elastic modulus distribution is the noiseless displacement data are used for the reconstruc-tion. Nevertheless, the displacement data are corruptedunknown, while the strain and displacement fields are

known. With this method, all the components of the by random noise resulting from numerical errors inducedwhen the forward problem is solved using the FEstrain/displacement tensors must be known. The tech-

nique proposed by Kallel and Bertrand [101] is based technique. Since in practice the data are noisier (due todecorrelation noise) and the boundary conditions areon the use of a linear perturbation method. It consists

essentially of minimizing the least-squares error between generally not completely known, all modulus reconstruc-tion techniques will also have their own limitations andthe observed and predicted displacement fields. The pre-

dicted displacement field is obtained using a theoretical artefacts [108]. To date, no systematic comparison ofmodulus reconstruction to simple strain imaging hasmodel of the elasticity equations (constitutive equa-

tions). In this technique both the force distribution under been conducted. It is expected from CTE analysis,however, that the importance of the reconstructionthe compressor and the measured axial displacement

field are used for the reconstruction of the modulus dis- algorithms may be greater in tissues with high stiVnesscontrast and lesser in situations where low contrast istribution. When the force distribution is unknown, a

penalty technique may be used [108]. Using computer encountered, such as in normal tissues.Figure 9a shows an example of a relatively complexsimulations, Kallel [58] has shown that solving the

inverse problem in elastography reduces the inherent simulated modulus distribution. It consists of a homo-geneous background embedding three hard lesions: twostrain artefacts, and results in a significant improvement

of the CTE. This is illustrated in Fig. 8, which shows of them are 3.5 times harder than the background and



Fig. 7 In vitro anteriorposterior view of a canine prostate at 2 per cent axial compression (elevationallyconfined): (a) first axial elastogram (no axial or lateral corrections), (b) first lateral elastogram (noaxial correction), (c) second axial elastogram (with axial correction, no lateral correction), (d) lateraldisplacement image (with axial correction), (e) second lateral elastogram (with axial correction),(f ) third axial elastogram (with axial and lateral corrections) and (g) sonogram (i.e. envelope of theRF sonographic data)

the third one is 4 times harder than the background. block. This artefact is due to the use of the penalty tech-nique, which assumes a pressure that is uniformly dis-Figure 9b shows the axial strain distribution resulting

from the application of a uniform axial pressure on the tributed at the upper surface of the tissue block [58, 95].As shown by the example of Fig. 9, in the case of atop of the tissue block while assuming perfect non-slip

boundary conditions at the upper and lower surfaces of relatively simple arrangement of elastic inhomogeneities,the axial strain image alone is suYcient for detection pur-the tissue. Figure 9c shows the same axial strain distri-

bution but obtained using a higher density finite element poses. However, in more complex situations it is expectedthat the strain image alone may not be suYcient even formesh (3500 elements instead of 1400 elements). Observe

that the increase in the number of elements significantly the simple task of detection. To illustrate this, the recon-struction of the simulated modulus distribution shown inreduces the numerical errors. This is achieved with a

considerable increase in computer time and memory. For Fig. 10a is considered. This modulus distribution isobtained by randomly changing the value of Youngsthe reconstruction, the coarse finite element mesh is usu-

ally used since the present algorithm involves matrix modulus of 25 per cent of the total number of the elementsof the FE mesh (each element was set to be 3 times harderinversion. Both strain images clearly depict the three

lesions, even in the presence of the bright artefacts than the background). As shown in Fig. 10b, the strainimage does not depict the individual elastic inhomogen-resulting from the inherent stress concentrations. As

shown in Figs 9d and e, the artefacts of the strain image eities without ambiguity. However, many of these aredepicted by the reconstructed modulus distributionare completely removed when the uniform pressure is

used as a boundary condition. However, when the uni- (Fig. 10c). Notice that in this complex situation the recon-struction is not perfect even when the boundary con-form displacement is used as a boundary condition, the

shadowing artefact of the strain image is replaced by ditions are completely known. This is a limitation of thealgorithm, and in principle a perfect reconstruction mayanother artefact consisting of the projection of the

lesions towards the upper and lower surfaces of the tissue be possible using an improved algorithm.



Fig. 8 Contrast-transfer eYciency curve. The zero dB ordi- Fig. 9 Simulated elastic modulus distribution and corre-nate is the ideal contrast-transfer eYciency. Observe sponding axial strain and reconstructed imagesthe close match between theory and simulation: obtained for both force and displacement boundary(CCC ) predicted observed contrast-transfer eY- conditions. (a) Cross-section from the simulated modu-ciency (from Kallel et al. [46 ]), (( ) contrast- lus distribution. The size of the ROI is 4040 mm2.transfer eYciency measured from simulated The homogeneous medium is embedding three circularelastograms (from Ponnekanti et al. [45]), (&) (5 mm diameter) inclusions; the left two upper andcontrast-transfer eYciency after solving the inverse lower inclusions are 3.5 times harder than the back-problem in elastography as obtained using FE simu- ground and the right one is 4 times harder than thelation (from Kallel [58 ]) background. (b) Axial strain image when a constant

pressure is used as boundary conditions obtained fora coarse FE mesh. (c) Axial strain image obtained fora finer FE mesh. (d) Reconstructed modulus distri-8 POISSONS RATIO ELASTOGRAPHY bution when uniform pressure is used as boundary con-ditions. (e) Reconstructed modulus distribution when

Poissons ratio (n) for a plane strain state under uniaxial uniform surface displacement is used as boundary con-stress conditions is defined as ditions. (f ) Inverse of panel (c). Note the various forms

of aretfacts shown in the images

n=elea

(1)a Poissons ratio contrast, elastic modulus contrast orboth.where el and ea are the lateral and axial strains respect-

ively. Poissons ratio is an important mechanical param- Poisson elastograms may have interesting applicationsin assessing the degree of unbound water content in tis-eter that describes the degree of material compressibility,

or the change in volume following an applied com- sues [110]. The Poisson elastogram or the time sequenceof Poisson elastograms may be used for quantitativepression. Poissons ratio equals 0.5 for totally incom-

pressible materials and 0 for totally compressible ones. assessment and imaging of fluid transport in local regionsof oedema, inflammation or other hydrated poro-By measuring and imaging the distribution of Poissons

ratio in tissues, it may be possible to estimate the amount elastic tissues [111]. Another interesting and potentiallyvery useful property of the Poisson elastogram is that,of relocatable water contained in diVerent tissue

regions [109]. as long as the tissue isotropy assumption holds, mechan-ical stress concentration artefacts due to geometricalFrom the definition of Poissons ratio [equation (1)],

it is possible to produce a new image that displays the boundary conditions should cancel out (Fig. 12c). Thismeans that unlike earlier methods for quantifying tissuespatial distribution of Poissons ratios in the tissue,

namely the Poissons ratio elastogram, or Poisson elasto- fluid transport that were highly dependent on thegeometry [111], it may be possible to produce images ofgram [42 ], by simply dividing the negative of the lateral

elastogram by the axial elastogram. Figures 11 and 12 this basic tissue parameter that are free from geometricalartefacts. Finally, the knowledge of both lateral strainshow simulation results of the axial, lateral and Poisson

elastograms for cases without and with Poissons ratio and Poissons ratio in addition to the axial strainis in general necessary for reconstruction algorithmscontrast respectively. In cases where there is a strain

contrast between an inclusion and the background, the [56, 57, 108]; this implies that the final modulus elasto-gram could become more accurate if the lateral andPoisson elastogram is able to indicate whether that strain

contrast (on the axial and lateral elastograms) is due to Poisson elastograms are computed first.



Fig. 10 Simulated elastic modulus distribution and corresponding axial strain image and reconstructedmodulus image obtained for uniform applied force boundary conditions. (a) Original modulus distri-bution. The brighter areas are 3 times harder than the background. (b) The inverse of the axialstrain image. (c) Reconstructed modulus distribution

Fig. 12 True: (a) axial strain image, (b) negative of lateralFig. 11 True: (a) axial strain image, (b) negative of lateralstrain image, (c) Poissons ratio image. Estimated:strain image, (c) Poissons ratio image. Estimated:(d) axial elastogram, (e) negative of lateral elasto-(d) axial elastogram, (e) negative of lateral elasto-gram, (f ) Poisson elastogram for a Youngs modulusgram, (f ) Poisson elastogram for a Youngs moduluscontrast of two and a Poissons ratio contrast of 1.65.contrast of two and a Poissons ratio contrast of one.In this case, the Poisson elastogram shows that theNote how the uniformity of the Poisson elastogramstrain contrast in the strain elastograms is also duedenotes the lack of Poissons ratio contrast betweento the Poissons ratio contrast, possibly revealingthe inclusion and the backgrounddiVerent poroelastic properties between the inclusionand the background

9 THE STRAIN FILTERrange of strains is due to the limitations of the ultrasoundsystem and of the signal processing parameters. The SFThe elastographic system has been characterized using

both theoretical [43, 44] and experimental [112] is obtainable as the ratio between the mean strain esti-mate and the appropriate lower bound on its standardmethods. Recently, a general theoretical framework

referred to as the strain filter (SF) was proposed by deviation. The SF may be derated due to eVects such astissue attenuation [113] and speckle decorrelation dueVarghese and Ophir [43]. The SF describes the relation-

ship among the resolution, dynamic range (DRe), sensi- to undesired lateral tissue motion [99 ]. The SF is basedon well-known lower bounds on the TDE variance, pre-tivity (Smin) and elastographic SNR (SNRe), and may

be plotted as a graph of the upper bound of the SNRe sented in the literature [114, 115]. The low-strain behav-iour of the SF is determined by the variance as computedversus the strain experienced by the tissue, for a given

elastographic axial resolution as defined by the data from the CramerRao lower bound (CRLB) (modifiedfor partially correlated signals [116 ]); the high-strainwindow length at a fixed overlap. The SF is a statistical

upper bound of the transfer characteristic that describes behaviour of the SF is determined by the rate of decorre-lation of a pair of congruent signals due to tissue distor-the relationship between actual tissue strains and the

corresponding strain estimates depicted on the elasto- tion, as shown in Fig. 13. Figure 14 illustrates the generalappearance of the SF, demonstrating the trade-oVsgram. It describes the filtering process in the strain

domain, which allows quality elastographic depiction of among SNRe and resolution for all strains. An import-ant extension to the SF is its combination with the CTEonly a limited range of strains from tissue. This limited



Fig. 13 The theoretical strain filter (SF) [43] along with the experimental strain response (ESR) [112 ]obtained with () global temporal stretching. The ESR is obtained around a small region(1 cm) around the focus of the transducer from 2 to 3 cm in the elastographic phantom. The ESRand SF are computed using a transducer with a 5 MHz centre frequency and 60 per cent bandwidth,using a window length (Z) of 3 mm with a 50 per cent overlap between data segments. The strainestimates for the ESR is computed from 20 independent experimental realizations of the RF pre-and post-compression signals. DRe is the strain dynamic range measured at the SNRe=4. Smin isthe sensitivity, defined as the lower strain bound of the DRe

Fig. 14 The general three-dimensional appearance of the SF, showing the trade-oVs among strain dynamicrange and sensitivity, elastographic SNRe and resolution (defined here as the window length at afixed overlap). It is generally convenient to plot a cross-section of the three-dimensional SF ata fixed resolution and refer to this cross-section as the SF; the theoretical SF curve of Fig. 13 isan example

formalism to produce elastographic contrast-to-noise demonstrated, using a controlled simulation experiment,that the axial resolution may be expressed as a bilinearratio (CNRe) versus strain curves (Fig. 16) [117]. This

allows the description of the CNRe of simple elastic function of window size and window shift, the latterbeing more important [118].inclusions or layers in terms of both the mechanical

strain contrast limitations in the target and the noise The elastographic system is characterized experimen-tally by estimating the strain response of the system usingproperties of the apparatus. The authors have recently



an homogeneous uniformly elastic phantom [112]. The Here ms denotes the statistical mean strain estimate andss denotes the standard deviation for the strain noiseexperimental strain response (ESR) is evaluated by sub-

jecting the uniformly elastic phantom to a wide range of estimated from the elastogram. When the statisticalmean value of the strain is replaced by the ideal tissueinput strains while computing the accuracy and precision

of the estimated strain. The plot of the SNRe estimate strain, and the minimum standard deviation by thetheoretical lower bound on the standard deviation, anversus the estimate of the strain incurred in the phantom

over the entire range of compressions generates the ESR, upper bound on the performance of the strain estimatoris obtained, referred to as the strain filter [43]:as illustrated in Fig. 13.

The ESR will in general vary with the use of diVerentultrasound scanners, boundary conditions associated SNReUB=

sts(s)ZZLB,r

(3)with the experiment and the strain estimation algorithmused. The accuracy and precision of the strain estimation where st is the tissue strain and s(s)ZZLB,r* is the modifiedalgorithm depends on the transducer used to scan the ZivZakai lower bound (ZZLB) [122, 123] on the stan-phantom. In addition, accurate tracking of the scatterer dard deviation of the strain estimator. The ZZLB pro-motion is obtained around the focus of the transducer vides the tightest lower bound for the displacementwhen compared to either the near- or far-field regions estimator. The modified ZZLB expression for the strainof the transducer. For an elastically homogeneous iso- estimation variance [43, 44] is complicated, and is nottropic phantom, when the scan plane is on the axis of reproduced here.symmetry of the phantom in the elevational direction,the motion of the scatterers out of the scan plane isminimized (along the axis of symmetry the scatterers 9.2 Non-stationarity of the strain filtermove only in the axial direction under the conditions

Estimation of tissue strains is inherently a non-stationarydescribed above). In addition, the ultrasound beamprocess, since the pre- and post-compression RF echoalong the elevational direction is broader, allowing scat-signals are jointly non-stationary (due to signal defor-terers to stay longer within the beam during com-mation caused by straining tissue). However, the pre-pression. It is for this reason that the characterizationand post-compression signals can be assumed toof the elastographic system is performed along the axisbe jointly stationary if the tissue strain is estimatedof symmetry of the phantom. The optimal ESR of theusing small windowed data segments in conjunctionelastographic system is therefore obtained at the focuswith temporal stretching of the post-compression sig-of the transducer and along the scan plane containingnal. Frequency-dependent attenuation adds additionalthe axis of symmetry of the elastographic phantom. The(axial ) non-stationarity into the strain estimation pro-ESR at any other position in the phantom is aVectedcess versus depth [113], while lateral and elevationalby boundary conditions, lateral and elevational signalsignal correlation introduce non-stationarities in thedecorrelation and the attenuation properties of thestrain estimation process along the lateral and elev-phantom.ational directions respectively [99 ]. The eVect of theseThe SF allows the design and synthesis of elasto-non-stationarities on the elastogram can be predictedgraphic systems [43, 44 ], while the ESR allows an evalu-by the SF (Fig. 15). For example, the eVect of lateralation of noise performance of any elastographic systemdecorrelation contributes predominantly to the non-[112]. The SFs illustrated in this paper incorporate thestationary variation in the SNRe [95, 99 ]. Both theuse of uniform temporal stretching for strain estimation.SNRe and the dynamic range are reduced with anBilgen and Insana [119121] also derived theoreticalincrease in lateral decorrelation. As long as any station-closed-form expressions for the strain variances using aary or non-stationary noise source can be described, itsTaylors series expansion. Their method uses a noiseeVect may be incorporated into the strain filter formal-figure analysis (ratio of the SNRe to SNRs), and whenism, resulting in a more realistic, derated strain filterplotted as a function of applied strain it is similar to the[99, 113].SF. However, these expressions are valid only in cases

of non-ambiguous detection of the cross-correlationpeak.

9.3 Contrast-to-noise ratio in elastography

The contrast-to-noise ratio in elastography (CNRe) is9.1 Theoretical frameworkan important quantity which is related to the detect-

A criterion that quantitatively measures the accuracy ability of a lesion [117]. The properties of the ultrasoundand precision of the strain estimate is the SNRe [46, 63 ],defined by * The lower bounds on the strain estimation variance are denoted with

an additional subscript r to illustrate that these variances are computedfor partially correlated signals. These lower bounds converge to theSNRe=

msss

(2)classical bounds when r=1.



Fig. 15 (a) The non-stationary evolution of the strain filter with linear frequency-dependent attenuation.The bandwidth remains constant with depth. However, the downshift of the centre frequency andthe reduction in the SNRs with depth causes the derating in the SF. Note the progressive narrowingof the SF (both in width and height) with depth (increasing frequency-dependent attenuation). Thecentre frequency downshift induces the shift in the SF towards higher strains, as observed withincreasing depths, while the reduction in the SNRs contributes to the dramatic reduction in thedynamic range and sensitivity. (b) Non-stationary variation in the SF for diVerent lateral positionsalong the transducer aperture and away from its centre. Lateral decorrelation increases with anincrease in the beam lateral position (increased lateral tissue scatterers motion), reducing the strainestimation performance with lateral position

imaging system and signal processing algorithms modulus contrasts the improvement in the CNRe is duedescribed by the SF can be combined with the elas- to the large diVerence in the mean strains. Note fromtic contrast properties (CTE) of tissues with simple the three-dimensional visualization of the CNRe curvesgeometries, enabling prediction of the elastographic in Fig. 16 that when the diVerences in the mean straincontrast-to-noise ratio (CNRe) parameter. This com- values are small (the region around the middle of thebined theoretical model enables prediction of the CNRe graph at low contrasts), the CNRe value obtained isfor simple geometries such as layered (one-dimensional almost zero. In addition, the regions with large strainsmodel ) or circular lesions (two-dimensional model ) (corresponding to large variances in the strain estimateembedded in a uniformly elastic background. An upper due to signal decorrelation) also contribute to low CNRebound on the CNRe [117] is obtained using values. Knowledge of the theoretical upper bound on

the CNRe in elastography is crucial for determining theability to discriminate between diVerent regions in theCNRe=

2(s1s2)2s2s1+s2s2

(4)elastograms. The CTE for the elasticity models andthe elastographic noise characterized by the SF deter-The CNRe for specific geometries that possess an ana-mine the CNRe in elastography. The three-dimensionallytic or experimental CTE description can be obtainedvisualization of the CNRe curves illustrates the strainby substituting the strains obtained using the elasticitydependence of the CNRe [117]. The three-dimensionalmodel and their respective variances from the SF intoplot provides a means of maximizing the CNRe in theequation (4).elastogram for the given ultrasound system and signalFigure 16 illustrates the general appearance of theprocessing parameters.upper bound on the CNRe, demonstrating the trade-oVs

Figures 17a and b show two-dimensional cross-among CNRe and modulus contrast for all appliedsectional cuts of Fig. 16 as well as simulation andstrains [117]. Note from Fig. 16 and equation (4) thatexperimental measurements at a fixed applied strainthe highest values of the CNRe are obtained where two(1 per cent) and a fixed modulus contrast (10 dB)conditions are satisfied: firstly, the diVerences in meanrespectively. Figure 17a compares the behaviour of thestrain values must be large and, secondly, the sum of theupper bound of the CNRe using both one- and two-variances of the strain estimates should be small. Thedimensional models for the tissue geometries. For the one-improvement of the CNRe at low modulus contrasts is

primarily due to the small strain variances, while at high dimensional model, strain is modelled by considering an



Fig. 16 Three-dimensional plot of the CNRe curves, illustrating the variation in the upper bound of theCNRe with the modulus contrast with varying applied strains for the two-dimensional tissue elasticitymodel. The curves were obtained using a transducer with a 7.5 MHz centre frequency with a 60per cent bandwidth and a window length Z=2 mm with a 50 per cent overlap in the data segments

equivalent one-dimensional spring system [52], while the model (soft lesion within a hard background), the strainswithin the lesion are quite small, even at large contrasts.two-dimensional analytic solution is used for the two-

dimensional model [46 ]. In addition, observe that the In the case of hard lesions, one- and two-dimensionalmodels provide similar results. Using the two-simulation and theoretical results follow a similar trend.

Note that for soft lesions, the increased signal decorre- dimensional model, a diVerence of about 10 dB can beobserved in the CNRe between the soft lesion and thelation that occurs at large contrasts using the one-

dimensional model drastically reduces the CNRe. hard background due to the lower value of the CTE.Note that the suboptimal CTE for soft lesions in theHowever, due to the geometry of the two-dimenional

(a) (b)

Fig. 17 (a) Theoretical and simulation results illustrating the variation in the CNRe with contrast for a 1per cent applied strain for the one-dimensional () and two-dimensional (#) tissue models. Thegraphs with the error bars and symbols denote the simulation results. The same parameters as inFig. 16 were used. (b) Theoretical (solid line) and experimental () results illustrating the vari-ation in the CNRe with applied strain using a phantom with an inclusion that is three times stiVerthan the background. The theoretical results are derated for strain estimation at the edges due tolateral signal decorrelation (20 mm lateral position). The CNRe is computed between a region withinthe inclusion and the background regions (the four corners of the elastogram shown in Fig. 2). Theerror bars represent the CNRe variation about these four estimates. The curves were obtained usinga transducer with a 7.5 MHz centre frequency and a 60 per cent bandwidth, and using a windowlength Z=3 mm with a 50 per cent overlap in the data segments



two-dimensional model is responsible for the improve- significant increase in these two essential image qualityparameters of elastography is evident by comparison ofment of the CNRe when compared to the one-

dimensional model. Figure 17b clearly shows that, as pre- Figs 18 and 19. They show simulated axial strain elasto-grams of a finite element phantom containing three cir-dicted by the theory and shown on experimental data,

elastography can reach the highest levels of CNRe (on cular inclusions that are 10, 20 and 40 dB stiVer thanthe background (whose elastic modulus is a constantthe order of 40 dB) at relatively low strains on the order

of 0.5 and 1 per cent, where also motion resulting from 21 kPa). The single compression elastograms of Fig. 18are not suYcient to display the high strain dynamic rangethe compression is the least complex. The discrepancy

between theory and experiment at higher strains is most of the target. However, the combination of dynamicrange expansion, envelope information and averagingprobably due to the unknown elevational decorrelation

that has not been accounted for in this theoretical result. show a systematic improvement of the image signal-to-noise ratio from Figs 19b to d respectively. The com-posite elastogram of Fig. 19d depicts the high dynamic

9.4 Applications range of the target by displaying the diVerence in strainbetween the inclusions and by showing the high strain

The SF can be used to analyse the trade-oVs in the strain concentration artefact occurring between the top andestimation performance for diVerent ultrasound systemparameters, namely with bandwidth and centre fre-quency [43, 44 ], algorithms like multicompression[96, 97], temporal stretching [96 ], multiresolution [44 ]and extension of the dynamic range in the elastogram[81, 124, 125]. In general, the area under the strain filtercomputed for a given resolution can be thought of as avery general measure of elastographic quality; the widthof the SF describes the quantity of information, whilethe height describes the quality of that information. Thusimprovements in the elastograms may be achieved when-ever the width and/or the height of the SF are increaseddue to any procedure. Two examples of such proceduresare described below. Fig. 18 (a) Ideal axial strain elastogram (from a noiseless

finite element simulation) of a three-lesion simulated9.4.1 Dynamic range expansion phantom with a strain dynamic range DRe=40 dB,

(b) elastogram of a single compression at 0.75The strain filter [43] shows the range of strains that canper cent applied strain and (c) elastogram of a singlebe displayed in the elastogram. For diVerent com-compression at 2 per cent applied strain. Note that

pressions, however, diVerent tissue elements will experi- the full dynamic range of the ideal target cannot beence the strains that fall within the strain filter. The adequately reproduced in either (b) or (c)method of variable applied strains [124, 125] is based onthe idea of appropriately combining data from diVerentcompressions in order to cover the entire strain dynamicrange in the tissue. These data are selected accordingto their value of SNRe. By appropriately scaling thesevalues, depending on the known amounts of appliedstrain they correspond to, a composite elastogram canbe formed containing all the strain values with the high-est obtainable SNRe. This corresponds to a shift of thestrain filter in logarithmic units by an amount deter-mined by the magnitude of the applied strain. Moreover,due to the slower decorrelation inherent in the strainestimation using the envelope, the use of envelope datahelps recover higher strains. By combining these data in

Fig. 19 (a) Ideal strain image for DRe=40 dB and (b) com-the composite elastogram, the strain filter is furtherposite elastogram resulting from the variable applied

extended to the high strain side. Lastly, due to the large strain method which extends the dynamic range.amount of data available, averaging can be performed (c) Case (b) with added averaging and (d) case (c)so as to further increase the SNRe, i.e. the amplitude of with additional envelope processing, further increas-the strain filter. Therefore, a composite strain filter is ing the dynamic range. Note the similarity betweencreated that has both a wide elastographic dynamic (a) and (d) and the vast improvement of all images

as compared to those in Figs 18b and crange (DRe) and a high SNRe (see Fig. 20). This



9.4.2 Multiresolution elastography

Multiresolution elastography is also described using theSF formulation where the strain estimates with the high-est SNRe are obtained by processing the pre- and post-compression waveforms at diVerent window lengths togenerate a composite elastogram [44 ]. A comparisonof the elastograms obtained using multiresolutionprocessing as opposed to the traditional method of usinga fixed window length is illustrated in Fig. 22.Quantitative values of the SNRe (computed in a smalluniform region near the top left corner of the elasto-gram) and the average resolution (Zavg) in the elastogramare also presented. Zavg is obtained by computing aFig. 20 The composite strain filter (bold curve) is constructedweighted sum of the window length times the percentagefrom strain-shifted replicas of the strain filter (brokenof the strain estimates computed using that window.lines) whose SNRe levels are modulated by the aver-Note that although the SNRe values for both theseaging process. The rightmost strain filter is due toelastograms are in the same range, a significant improve-signal envelope processing. The combined dynamicment in Zavg is observed in the multiresolution elasto-range is shown as DReC. Actual implementation of

this strain filter is shown in Fig. 19 gram. For a realistic comparison of the SNReimprovement, the multiresolution elastogram in Fig. 22dshould be compared to the fixed resolution elastogramobtained using Z=1.7 mm (Fig. 22c).

bottom inclusions in the first half of the image. Similar The evolution of the SNRe with Zavg for the two tech-results have been produced experimentally, using a three- niques is illustrated in Fig. 23. Comparing the techniqueslayered phantom of 40 dB total stiVness dynamic range, at the same Zavg (2.5 mm) indicates a significantand are shown in Fig. 21. Finally, a similar method has improvement in the SNRe (denoted as the increase frombeen developed more recently by Lubinski et al. [135]. A to B). In addition, comparing the elastograms with

similar SNRe illustrates the improvement in Zavg(denoted by the improvement from A to C).

The enhancement of the elastographic parameters mayalso be described in terms of the SF formulation, asillustrated in Fig. 24. In addition to the SNRe enhance-ment, the dynamic range and sensitivity also improvedwith multiresolution elastography. Comparing the SFs,it can be seen that DRe=BD is obtained for the elasto-gram using the fixed window when compared to DRe=BE with multiresolution processing [44 ].

The properties of the ultrasound imaging system andsignal processing algorithms described by the SF can becombined with the elastic contrast properties (CTE) oftissues with simple geometries, enabling prediction of theelastographic contrast-to-noise ratio (CNRe) parameter.The CNRe in elastography is an important quantity thatis related to the detectability of a lesion [117].Fig. 21 Experimental results: (a) sonogram of a three-layered

phantom with the middle and bottom layers havingstiVnesses 20 and 40 dB larger that the top layer 10 ELASTOGRAPHIC ARTEFACTSrespectively (thus, having a total stiVness dynamicrange of 40 dB), (b) elastogram obtained with a single

Like any other imaging modality, elastography has vari-compression of 0.5 per cent, (c) elastogram obtainedous artefacts. The elastographic artefacts can be dividedwith a single compression of 2 per cent and (d) com-

posite elastogram obtained with the method of vari- into three primary categories:able applied strains using eight compressions at a step

1. Mechancial artefacts. Strains in the tissue depend notof 0.25 per cent. All images are displayed on the sameonly on the modulus distribution in the tissue but alsoscale as shown. Notice how the composite elastogramon boundary conditions, both internal and external.allows the total stiVness dynamic range to be vis-Unlike other artefacts, mechanical artefacts representualized and the diVerences in stiVness between the

three layers to be appreciated true variations in strain; other artefacts generally



Fig. 22 Sonogram (a) and elastograms (bd), with 1 per cent total compression, of a phantom with aninclusion that is three times stiVer than the background. The sonogram was obtained using a 7.5 MHzlinear array with a 50 per cent bandwidth. The elastograms in (b) and (c) were obtained using single3 mm and 1.7 mm windows respectively, while (d) represents the elastogram obtained using multi-resolution elastography [44], with the largest window length used=3 mm. The image size is4040 mm2

Fig. 23 Comparison of the elastograms obtained using multiresolution () and fixed resolution(###) elastography. The elastographic SNRe and the average resolution are computed for theelastograms in Fig. 22c (fixed resolution) and Fig. 22d (multiresolution)

hinder an accurate depiction of the strain in the tissue. 10.2 Acoustic artefactsThe mechanical artefacts may sometimes be ben-

The ultrasonic scanner used to collect the data has aeficial, and facilitate diagnosis by highlighting thesignificant eVect on the strain estimation and can intro-targets.duce various artefacts. Parameters such as large acoustic2. Acoustic artefacts. Estimation of true strain in thecontrast, changes in the beam, reverberation and phasetissue is also aVected by the ultrasound system.aberrations can all introduce errors in the elastograms.3. Signal processing artefacts. The signal processingHowever, elastography is conventionally a method basedalgorithms can also introduce significant artefactson diVerential measurement, i.e. the pre-compressioninto elastograms.and post-compression signals generally undergo similardegradations, tending to minimize these artefacts.However, these eVects have to date not been thoroughly

10.1 Mechanical artefacts studied. Some specific acoustic artefacts have beendescribed by Ophir et al. [25].Several mechanical artefacts may be present in elasto-

grams [25 ]. These may include stress concentrations anddilutions due to internal and external boundaries, target 10.3 Signal processing artefactshardening artefacts and artefacts due to CTE limitations.In principle, all these could disappear if the perfect Elastography in its present form of development has two

major signal processing artefacts, dubbed zebras andmodulus reconstruction is made.



Fig. 24 The multiresolution SFs predicted theoretically for the elastograms shown in Fig. 22. Note theimprovement in the sensitivity (minimum strain at the DRe level ) with a decrease in elastographicresolution (defined here as the window length)

worms. Elastography is performed on digital signals. elastogram has horizontal bands of black and whitegoing through the image. In the region of larger strains,Time delay is not generally an integral multiple of the

sampling period. Thus, correlation functions can be the bands are more closely packed than in the hardinclusion where the strain is lower.interpolated to improve the precision of the time delay

estimates [126128]. Elastography commonly uses a The other signal processing artefact appears whenlarge signal overlaps are used, which produce correlatedparabolic or cosine fit. It is computationally simple, but

can introduce a cyclic bias error [52 ]. If the displacement noise patterns. This artefact also appears as horizontalstructures like the zebras. However, there are someestimates have a cyclic error, the strain, obtained by

taking a gradient of the displacements, will also contain important diVerences. T

Elastography: ultrasonic estimation and imaging of the

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