regional strain and strain rate measurements by cardiac ultrasound: principles, implementation and...

Eur J Echocardiography (2000) 1, 154–170doi:10.1053/euje.2000.0031, available online at http://www.idealibrary.com on

REVIEW ARTICLE

Regional Strain and Strain Rate Measurements byCardiac Ultrasound: Principles, Implementation and

Limitations

J. D’hooge1*, A. Heimdal2, F. Jamal3, T. Kukulski3, B. Bijnens3, F. Rademakers3,L. Hatle3, P. Suetens1 and G. R. Sutherland3

1Medical Image Computing, Department of Electrical Engineering, Katholieke Universiteit Leuven, Leuven,Belgium, 2GE Vingmed Ultrasound, Horten, Norway and 3Department of Cardiology, Katholieke Universiteit,

Leuven, Leuven, Belgium

The non-invasive quantification of regional myocardialfunction is an important goal in clinical cardiology.Myocardial thickening/thinning indices is one method ofattempting to define regional myocardial function. A newultrasonic method of quantifying regional deformation hasbeen introduced based on the principles of ‘strain’ and‘strain rate’ imaging. These new imaging modes introduceconcepts derived from mechanical engineering which mostechocardiographers are not familiar with. In order tomaximally exploit these new techniques, an understandingof what they measure is indispensable. This paper willdefine each of these modalities in terms of physical

principles and will give an introduction to the principles ofdata acquisition and processing required to implementultrasonic strain and strain rate imaging. In addition, thecurrent status of development of the technique and itslimitations will be discussed, together with examples ofpotential clinical applications.(Eur J Echocardiography 2000; 1: 154–170)� 2000 The European Society of Cardiology

Key Words: Strain; strain rate; ultrasound; myocardium;review.

Address for correspondence: Jan D’hooge, University HospitalGasthuisberg, Department of Cardiology, Herestraat 49, B-3000Leuven, Belgium.

Introduction

That regional endocardial motion can be used toattempt to define regional myocardial function is along-standing concept. This tenent has formed the basisof many angiographic, nuclear and two-dimensional(2D) echocardiographic studies[1–4]. A better descriptionof regional myocardial function seems to be given bydetermining local wall thickening and thinning charac-teristics, which are not necessarily directly related toendocardial motion. The optimal data set with which toattempt to describe regional myocardial function woulddefine wall deformation characteristics in three dimen-

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sions and in real time. Currently, three-dimensional (3D)local wall deformation can be acquired by magneticresonance imaging[5,6], while computed tomography andgated single-photon emission computed tomography/positron emission tomography (SPECT/PET) imagingenable 3D acquisition of wall thickening and thin-ning[7,8]. However, none of these are real-time tech-niques and the current temporal resolution used inclinical practice does not resolve all myocardial mechan-ical events. In contrast, real-time acquisition of resolvedlocal thickening/thinning parameters can be obtainedfrom ultrasound by grey-scale M-mode recordings, butonly in one dimension and for a limited number ofregions of the myocardium[9].

Colour Doppler myocardial imaging is a new cardiacultrasound technique which, in its current high framerate format (>120 frames . s�1) can resolve all mean

� 2000 The European Society of Cardiology

Regional Strain and Strain Rate Measurements by Cardiac Ultrasound 155

myocardial velocities along its image lines[10,11]. Initialclinical studies have examined the potential diag-nostic role of this technique in determining regionalmyocardial function from velocity data sets for anumber of disease entities[12–17]. Although encouragingdata were obtained, it was clear that the interrogation ofregional myocardial velocities alone has two majordrawbacks. Firstly, since the amplitude of the estimatedvelocity is dependent on the angle at which the region isimaged, accurate quantification of peak velocities can bedifficult. Secondly, overall heart motion, rotation andcontraction of adjacent myocardial segments will influ-ence regional velocity estimates[18]. In order to overcomesome of these problems, ultrasonic strain rate imaging,or in other words, rate of deformation imaging, has beendeveloped by estimating spatial gradients in myocardialvelocities. From strain rate curves, local strain (i.e.regional deformation curves) can be extracted, resultingin the concept of regional strain imaging.

These new techniques of regional strain rate andstrain imaging introduce concepts derived from mechan-ical engineering which most echocardiographers are notfamiliar with. The purpose of this paper is to give anintroduction to the physical concepts underlying strain(rate) imaging.

Definitions

Figure 1. (a) Deformation (strain) of a one-dimensional object is limited to lengthening or shortening. Strain isthe deformation of an object relative to its original shape. For a 2D object, deformation, i.e. strain, can bedescribed by four strain components: two normal strains [(b1) and (b2)] and two shear strain components [(b3)and (b4)]. The shear strains are also completely characterized by the angles �x and �y.

StrainStrain is defined as the deformation of an object,normalized to its original shape. In a one-dimensional

(1D) object (i.e. an infinitesimally thin bar), the onlypossible deformation of the object is lengthening orshortening. This is illustrated in Figure 1(a). The rela-tive amount of deformation is defined as strain. Strain,for which the symbol � is used, can thus be writtenas:

with L the length of the object after deformation and L0

its original length. Since it is the change in length relativeto its initial length, it is a dimensionless quantity (oftenexpressed in percent). By convention, equation (1) isdefined in such a way that lengthening is representedas a positive value for strain, while shortening isrepresented by a negative value.

When the length of the object is not only knownbefore and after deformation but also during thedeformation process, the instantaneous strain can bedefined:

with L(t) the length of the object at time instance tand L(t0)�L0 its initial length. In other words, theinstantaneous deformation is expressed relative to theinitial length; this is the Lagrangian strain. However,the deformation can also be expressed relative to thelength at a previous time instance:

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156 J. D’hooge et al.

with dt an infinitesimally small time interval and d�N(t)the infinitesimally amount of deformation occurringduring this time interval. The total amount of strain canthen be obtained by adding all these infinitesimal straincontributions together:

In other words, in this definition of instantaneous strain,the reference value is not constant over time but changesduring the deformation process; it is called the naturalstrain. It can be shown that the Lagrangian strain � andthe natural strain �N have a fixed (non-linear) relation-ship which is given by[19]:

where ‘ln’ is the natural logarithm. Note that thisrelationship only holds when the rate of deformation isconstant as a function of time. If the strains are small (ofthe order 5–10%) Lagrangian and natural strain valuesare approximately equal. However, for the large defor-mations which can occur during cardiac ejection andrapid filling, the difference becomes significant. For thisreason it is important to define which type of strain ismeasured when studying the heart. For cardiac appli-cations, it has been suggested that it may be moreappropriate to measure the natural strain, since in thatcase the measured values are less dependent on thedefinition of the initial length L0

[19,20].Until now, the assumption was made that the deform-

ing object was one-dimensional. However, for a 2Dobject, the deformation is not limited to lengthening orshortening in one direction: it can lengthen or shortenalong the x- or y-axes [compare Fig. 1(b1) and (b2)] andcan distort by the relative displacement of the upper tothe lower border [compare Fig. 1(b3)] or the right borderto the left border [compare Fig. 1(b4)]. The first type ofdeformation [Fig. 1(b1) and (b2)] is referred to as normalstrain (since its associated motion is normal to the borderof the object), while the latter deformation [Fig. 1(b3)and (b4)] is called shear strain (since its associatedmotion is parallel to the border of the object). In order todescribe the deformation of a 2D object completely, allfour strain components have to be known. Mathemati-cally, these components are recognized by their index:

It is mathematically convenient to write all straincomponents in a single matrix as:


This matrix is refered to as the strain tensor*. The shearstrain components can also uniquely be determined bythe angles �x and �y since their tangent is equal to �y/xand �x/y respectively. Thus:

As in the 1D situation, the instantaneous strain compo-nents can be defined if the deformation process is knownas a function of time. Again, both Lagrangian andnatural strain components can be calculated.

The most general situation is that of a 3D objectwhich deforms (as in a myocardial segment). In thiscase, there are three normal strains (along the x, y andz-axes) and six shear strains (xy, xz, yz, yx, zx and zy).Some of these strain components are illustrated inFigure 2. Defining these nine strain components definesthe deformation of the 3D object completely.†

*Note that in the mechanics literature the shear strain is oftendefined as: �xy= 1

2 (�x/y+�y/x). This implies that both shear straincomponents are identical, i.e. that �xy=�yx. This reduces thenumber of independent strain components to three, which simpli-fies stress-strain analysis. However, it is only appropriate forhomogeneous, isotropic and incompressible media.

†As in the 2D case, defining the shear strain components as infootnote* reduces the number of strain components to six. Again,this is only appropriate for homogeneous, isotropic and incom-pressible media.

‡Note that some authors have used the symbol �� to representstrain rate.

Strain Rate

Strain rate is the speed at which deformation (i.e. strain)occurs. It is represented by the symbol �̇‡ and has theunit s�1. Although s�1 is in fact the same as Hz, it ispreferable to use s�1 since Hz is normally used toexpress a periodic change. Clearly, in the most generalsituation, strain (and thus strain rate) does not necessar-ily have a cyclic nature, as one can only deform an objectonce.

For example, if we assume 20% total (Lagrangian)strain of a 1D object (e.g. the object lengthens from itsoriginal length of 2 cm to 2.4 cm) and if the totaldeformation takes 2 s, the average strain rate �̇ is equalto 0·20/2 s, which is 0·1 s�1: on average, the objectlengthens by 10% every second. When the same amountof deformation occurs in only half this time period (inthis example: 1 s), the average strain rate doubles to0·20/1 s = 0·2 s�1: on average, the object lengthens by20% every second. From this example, it is clear thatstrain rate and strain relate as velocity and distance. Forexample, when a car drives a particular distance (strain)


Figure 2. Deformation of a 3D object is described by three normal and six shear strain components. One normalcomponent (a) and three shear components [�yx (b), �zx (c) and �xy (d)] are illustrated here.

in a certain amount of time, the average velocity(strain rate) of the car can be calculated. Note thatthis does not imply that the instantaneous velocity ofthe car is known over the whole journey. Similarly, thisdoes not imply that the instantaneous strain rate isknown.

Since a 3D object has nine strain components alloccurring at a specific rate, nine strain rates can bedefined. Since both natural and Lagrangian strain canbe defined, both natural and Lagrangian strain rates canbe calculated. For the 2-second deformation in theprevious example, the natural strain value at the end ofdeformation is 18·2% [using equation (5)]. This impliesan average natural strain rate of 0·0912 s�1. However,since the reference value to calculate the natural strain is

not constant over time, the natural strain rate (in thisexample) is a function of time as well. The instantaneousnatural strain rate can be calculated as:

with L�(t) the rate of deformation (e.g. 2·4 cm�2 cm)/2 s = 0·2 cm . s�1) and L(t) the instantaneous length ofthe object. This means that in this example the instan-taneous natural strain rate drops from 0·1 s�1 at theonset of deformation to 0·0833 s�1 at the end of thedeformation. Note the difference between deformationrate (expressed in m . s�1) and strain rate (expressedin s�1).



The Relationship Between Strain and WallThickening

For the heart, wall thickening (WT) is defined as:

with TES and TED end-systolic and end-diastolic wallthickness, respectively. This expression is clearly identi-cal to equation (1). In other words, wall thickening isnothing but myocardial strain measured in one dimen-sion. Since it compares the end-systolic wall thicknessrelative to the (fixed) end-diastolic wall thickness, theclassical wall thickening parameter in fact measuresLagrangian strain. Clearly, the rate at which the myo-cardial wall thickens and thins is thus nothing but the1D strain rate.

Coordinate Systems

In order to uniquely define different positions in space, a3D coordinate system must be constructed. This isdefined to be a set of three different, mutually non-coplanar unit vectors having the same origin. The pos-ition of any point within the concomitant 3D space canbe expressed relative to these unit vectors. In this way,every single spatial point within the coordinate systemcan be ascribed a unique coordinate. However, thecoordinates of each point will depend on the coordinatesystem chosen.


Similarly, the strain component measured will dependon the coordinate system used. Using the most appro-priate coordinate system is important, as this will facili-tate interpretation of the measurements and will reducethe mathematical complexity required to describe thedeformation. The possible coordinate systems which canbe used in ultrasonic cardiac strain imaging are asfollows.

Figure 3. Either a global Cartesian coordinate system (a) or a local heart coordinate system (b) can be defined.However, the latter coordinate system facilitates the physical interpretation of strain measurements of the heart.

The Heart Coordinate System

Rather than defining a global Cartesian coordinatesystem in which the whole heart or the left ventricle isdescribed [Fig. 3(a)], a local heart coordinate system canbe defined. For each point to be interrogated in anymyocardial wall, three mutually perpendicular axes canbe defined:

v The radial (R) axis: perpendicular to the epicardium,pointing outwards, that is, away from the cavity.

v The longitudinal (Lo) axis: perpendicular to the radialaxis, that is, tangent to the epicardium, and pointingtowards the base of the ventricle, away from theapex.

v The circumferential (C) axis: perpendicular to both theradial and longitudinal axis, defined in such a way thatthe R–Lo–C coordinate system is right-handed (if onelooks along the C-axis, the rotation from the R to theLo-axis is clockwise). Thus, the circumferential axis isdirected anticlockwise around the classical echo shortaxis image.

This local coordinate system is illustrated in Fig. 3(b).Note that the spatial orientation of the three local axes


relative to the global coordinate system is different foreach point within the ventricle. Moreover, note that thiscoordinate system cannot be defined for the most distalpoint of the apex of the ventricle, as here neither acircumferential direction nor a longitudinal orientationcan be defined.

The Ultrasound Coordinate System

Within an ultrasound image, an ultrasound coordinatesystem can be defined, with its centre at the position ofthe transducer, by the following three axes:

v The axial (A) axis: along the direction of propagationof the ultrasonic beam, pointing away from thetransducer.

v The lateral (La) axis: within the image plane, perpen-dicular to the axial direction and pointing towards theleft of the image.

v The elevation (E) axis: perpendicular to both the axialand the lateral directions, defined in such a way that

the A–La–E coordinate system is right-handed. Thus,the elevation axis is defined perpendicular to the imageplane.

This coordinate system is illustrated in Fig. 4.

Figure 4. The ultrasound coordinate system has axial,lateral and elevation axes which are, respectively, alongthe image line, perpendicular to image line and withinthe image plane and perpendicular to the image plane.

Figure 5. When the acquisition of the ultrasound data is done in such a way that animage line is parallel (a) or perpendicular (b) to the myocardial wall, coordinatetransformation from the ultrasound coordinate system to the heart coordinate systembecomes straightforward.

Coordinate Transformations

When different coordinate systems exist, it is importantto be able to transform measurements within one systemto measurements made using another system. As will befurther discussed in the next section, at present only theaxial component of myocardial strain can be measured,that is, one component of strain as expressed in theultrasound coordinate system (thus, the componentalong the scan line). However, in order to relate localstrain to regional myocardial function, it is necessaryto transform this component to the heart coordinatesystem.

In general, it is impossible to write closed-formmathematical expressions to describe this transfor-mation since they depend on the echocardiographic viewused and on the myocardial wall studied. However,when the ultrasonic beam is either parallel or perpen-dicular to the myocardial wall under investigation, thistransformation becomes straightforward, since one com-ponent in one coordinate system corresponds directly toone component in the other system. For example, usingthe apical four-chamber view the ultrasound beam (andthus the image line) is normally parallel to the inter-ventricular septum [Fig. 5(a)]. Thus, the axial straincomponent (in the ultrasound coordinate system) corre-sponds to the longitudinal component (in the heartcoordinate system). Similarly, when the ultrasoundbeam is perpendicular to the posterior wall in a para-sternal long-axis view, the axial component in onecoordinate system corresponds to the radial componentin the other [Fig. 5(b)].



However, when the ultrasound beam impinges on themyocardial wall in an oblique way, the axial strain com-ponent corresponds to a combination of the radial,longitudinal and circumferential strains. This implies thatin order to study regional myocardial deformation withcurrent ultrasound techniques (only measuring the axialstrain component), care has to be taken to acquire theappropriate data sets. For example, in a parasternallong-axis view, data sets must be acquired with the ultra-sound beam perpendicular to the myocardium. Withinthe image plane, this can easily be verified visually. How-ever, in the direction perpendicular to the image, theelevation direction, this is not always straightforward. Inother words, when attempting to measure local strain, nooblique cross-sections of the ventricle should be acquired.Due to the motion of the heart during the cardiac cycle,this is often not possible. In such cases it should be keptin mind during any attempted analysis of the data thatother strain components could have contributed to themeasured axial strain.

Measurement of Cardiac Strain

Since the beginning of the 1980s, methods havebeen proposed to estimate strain by means ofultrasound[21–24]. Originally, most methods were devel-oped for static organs such as the liver, breast, kidney orprostate. Part of these techniques deform these organsby means of an external force[25]. In these methods,stiffer tissues will deform less than more elastic tissuesunder the influence of an identical force. Thus, in thissituation strain estimation can be directly related to theelastic properties of the tissue. For this reason, thetechnique has been termed ‘elastography’. Note, how-ever, that this passive compression need not be createdby an external force but can be created by a forcegenerated by the body itself: the intraluminal pulsingbehaviour of the blood under arterial pressure can beused to deform artery walls which can then be imaged byintra-vascular ultrasound[26].

Recently, a series of ultrasound techniques have beendeveloped to measure myocardial strain in vivo. Severaldifferent strain estimation techniques have been pro-posed based on the principles of elastography[27,28].However, only one of them has successfully been appliedfor measuring myocardial strain[29]. The main difficultyin elastographic strain estimation of the heart is the factthat most myocardial segments exhibit a combination ofrelatively large motion and deformation. Additionally,this technique is highly computionally intensive, whichmakes its current application to 2D images in real-time impractical. Therefore, another approach based onDoppler myocardial imaging principles has been at-tempted[30]. Both techniques are briefly discussed in thesubsequent paragraphs.


Figure 6. Schematic illustration of the fundamentalconcept of the cross-correlation method: a radio fre-quent signal line is acquired before and after defor-mation of the object; a local radio frequent signalpattern (resulting from a specific distribution of scatter-ers) is tracked between two subsequent acquisitions bymeans of the cross-correlation function, which is ameasure for the similarity of two signals as a function oftheir relative phase.

The Cross-Correlation Method

The properties of the received backscattered radio fre-quent signal from a distribution of scatterers (the local

inhomogeneities in density and/or compressibility) aredetermined by the precise position of these scatterers:each scatterer will reflect the transmitted pulse as it isreceived; the received signal is the result of the inter-ference between all these reflected wavelets. This impliesthat two different distributions of scatterers, DA and DB,will give rise to a different radio frequent signal patternRFA and RFB, respectively.

Assume these distributions are located at differentdepths x1 and y1 along the same line of an ultrasonicimage as illustrated in Figure 6. If the object containingthe scatterer distributions DA and DB deforms in theaxial direction, the two scattering regions will moverelative to each other and as a consequence the corre-sponding radio frequent signal patterns RFA and RFB

will move in the same manner. This is the fundamentalconcept behind the implementation of cross-correlationmethodology: in this technique a radio frequent signalline is acquired before and after deformation of anobject. A local radio frequent signal pattern (resultingfrom a specific distribution of scatterers) is then trackedbetween the two subsequent acquisitions by means ofthe cross-correlation function.

The cross-correlation function gives a measure of thesimilarity between two radio frequent signal patterns,such as RFA and the second radio frequent signal, as afunction of a relative time shift between the two. Thetime shift for which the cross-correlation function ismaximal corresponds to the time shift for which bothsignals resemble most: that is, to the delay introduced


due to the motion of the radio frequent signal pattern.The change in distance between two specific radio fre-quent signal patterns RFA and RFB, which are L1 and L2

in Figure 6. respectively, can directly be used in equation(1) to estimate the local strain: it is the spatial gradient inthe motion of the radio frequent signal patterns. Notethat this technique can only determine the axial motionof the radio frequent signal patterns. As a result, onlythe axial strain component can be measured.

This method implies an assumption that the radiofrequent signal pattern itself does not change betweenthe two acquisitions. This assumption is only reasonablewhen the deformation of the object is relatively small(when the deformation of the object between the acqui-sition of the two radio frequent signals is small) and ifthere is not too much motion in the lateral and elev-ation directions. Thus, in order for this technique towork, acquisition of data at high temporal resolution isnecessary.

For this reason, the cross-correlation method wasinitially implemented only for M-mode acquisition[29]. Inthis application the initial positions of the endo- andepicardium are manually indicated on the M-modeimage by the user at a specific time point, end-diastole.The wall is then divided into a fixed number of layers. Atall boundary layers the underlying radio frequent signalpattern is tracked as a function of time. Boundariesmoving apart correspond to a positive strain value(compare to lengthening of a 1D bar). The result ofradio frequent signal tracking is an estimation of thedistance between two boundaries as a function of time:L(t). This means that both Lagrangian and naturalstrain can be calculated.

The minimal strain that can be measured with thistechnique is (theoretically) the one corresponding to themotion of a fraction of the wavelength. In other words,a (theoretical) strain resolution in the order of micro-metres can be achieved. Moreover, since the method isbased on radio frequent signal tracking, no aliasing willoccur since a specific radio frequent signal pattern can(theoretically) be tracked over any distance. Finally,since several layers are tracked, a transmural straingradient can be measured.

The major disadvantage of this approach is that it isvery computationally intensive. Although it has recentlybeen implemented to measure strain in real-time[31], it isstill limited to M-mode acquisition. As a consequence,the strain in only one region of myocardium can beinvestigated at a time. Such comparison of strainpatterns in different myocardial regions (for the quanti-fication of regional myocardial deformation) is a tediousand time-consuming task. This method has not yet beenimplemented in commercially available equipment.

The Velocity Gradient Method

Using the definitions in Figure 6, strain can be writtenas:

where x1, y1 and x2, y2 are the positions of a specificradio frequent signal pattern before and after defor-mation, respectively. Dividing both sides of thisequation by �t, the time interval between the twoacquisitions, gives:

In other words, strain rate can be expressed as thedifference in velocities at both ends of an object of initiallength L1: it is the spatial gradient in velocities. Sincelocal instantaneous myocardial velocities can bemeasured by colour Doppler, this means that strain rateinformation can be extracted from real-time, digitallystored myocardial data sets by post-processing. Thisprinciple was recognized from the beginning of Dopplermyocardial imaging and resulted in a series of investi-gations into the radial transmural velocity gradient as anindex of local contraction and relaxation[18,30,32–35]. Allthese studies made use of M-mode acquisition ofvelocity data sets and calculated the velocity gradientoff-line by post-processing. Note, however, that Dopplermyocardial imaging-techniques only estimate the axialcomponent of the velocity vector, that is the componentalong the image line. As a consequence, only the axialstrain rate can be calculated. It can be shown that theresulting strain rate curve is the natural strain rate(Appendix A). Thus, time-integrating this curve resultsin the natural axial strain. Since in this method strainrate is calculated based on the derived velocity datasets, aliasing can induce artifacts in the strain (rate) data(see later section for an example). Recently, real-timeB-mode axial strain rate imaging has been implementedin a prototype clinical scanner[36].

The Relation Between Radial StrainEstimation and Grey-scale M-mode Data

The acquisition of grey-scale M-mode images at a hightemporal and spatial resolution allows the definition oflocal myocardial thickening/thinning parameters[9]. Inprinciple, the new ultrasound techniques describedabove can make the same measurements. The questionthen arises as to which extra benefits local strain imagingoffers over conventional M-mode analysis. In summary,strain (rate) imaging has the following advantages overconventional M-mode data:

1. It allows the study of both regional radial thickeningand thinning, and regional longitudinal and circum-ferential shortening and lengthening, and thus canmeasure three aspects of local function as opposed toonly the radial parameters measured by traditionalgrey-scale M-mode.



2. It exploits the phase shift of the radio frequent signalsrather than their amplitude, resulting in a bettersignal-to-clutter ratio (compare with Doppler tech-niques)[37].

3. Its calculation is based on the radio frequentsignal rather than on its envelope. As a result, itsspatial resolution is intrinsically better, as smalldisplacements are more easily detected[38].

4. It does not require image segmentation to define theendo- and epicardial boundaries, and is thus lessuser-dependent and less time-consuming.

Note, however, that M-mode wall thickening/thinningmeasurements cannot directly be compared to strainmeasurements. Firstly, strain imaging measures thestrain occurring within the sample volume (the localstrain) while M-mode analysis determines the overallwall thickening/thinning (the global strain). The localand global strain values are not necessarily the same.Secondly, as explained above, strain imaging based onthe velocity gradient methodology measures the naturalstrain while conventional wall thickening/thinning indi-ces extracted from M-mode analysis correspond to theLagrangian strain. Thus, in order to compare bothmeasurements, equation (5) has to be used to convertone type of strain to the other one§.

The Current Status and Limitations ofthe Techniques

§In fact, this conversion formula only holds when the deformationrate is constant as a function of time. Since this assumption is notvalid for the heart, this equation can only be used as a firstapproximation.


Figure 7. Schematic overview of the operations requiredfor strain and strain rate estimation in the cross-correlation and the velocity gradient methodologies.

Practical Difficulties

As discussed above, two methods have been presented tocalculate myocardial strain and strain rate data:

1. The cross-correlation method initially estimatesregional motion from the radio frequent signaldata. Strain is derived as a spatial gradient in motion.The strain rate can be obtained by calculating thetemporal derivative of the strain curve.

2. The velocity gradient method calculates the strainrate curve as the spatial gradient in velocities. Fromthis strain rate curve, the strain curve can be obtainedby integration over time.

This is summarized in Figure 7. In other words, in orderto obtain information on both the strain and the strainrate, both methods require the calculation of a (tem-poral or spatial) gradient and/or an integral. This causestwo problems:

1. Calculating gradients numerically (i.e. with a digitalcomputer) is very sensitive to noise, as illustratedin Figure 8, where a signal without and with noise

[Fig. 8(a) and (c)] is shown together with its numeri-cally calculated derivative [Fig. 8(b) and (d)]. Notethat the noise level in Fig. 8(c) was very small. As aconsequence, the motion or velocity data sets(depending on which method is used) have to besmoothed in order to obtain strain or strain ratecurves, respectively, which are not too noisy. This canbe done by measuring the average motion or velocityvalue for a region around each pixel within the image.Additionally, smoothing of the extracted strain (rate)data can be performed to improve the resulting strain(rate) image. As a result, the spatial resolution ofthese techniques becomes relatively poor. Typicalspatial resolution values for the cross-correlation andthe velocity gradient method are 1–3 mm and2–5 mm, respectively. This means that no two inde-pendent measurements can be made accurately forpoints closer than these values apart. As a conse-quence, studying local intramural gradients in strainrate or strain might not be possible with the velocitygradient methodology.

2. In order to calculate regional strain in the velocitygradient methodology, the strain rate curves aretime-integrated. However, time integrating the strainrate curve can result in a drifting of the strain curve[as illustrated in Fig. 9(a)]. The main factors whichcan induce this drifting are: a poor frame rate (suchas peak values missing), a bias in the strain rateestimate (for example, the strain rate is systematicallyestimated too high), an inhomogeneous strain distri-bution within the sample volume or angle changesduring the cardiac cycle. However, we know that atthe end of the cardiac cycle strain should be zero, asthe heart muscle returns to its initial shape after acomplete heart cycle: there cannot be a net strain.Thus, using this boundary condition and assumingthat the error introduced is not dependent on the


amplitude of the strain rate nor on the positionwithin the heart cycle, this drifting can automaticallybe compensated for [Fig. 9(b)].

Angle Dependency of Strain Measurements

One of the major criticisms of velocity, strain rate andstrain imaging has been that these techniques areangle-dependent, since only the axial component of thetrue 3D velocity vector or deformation is measured. Asalready discussed above, this problem can be partiallyavoided by making sure that insonation is either per-pendicular or parallel to the myocardial wall duringacquisition of the data sets. Moreover, if some simpleassumptions are made about the way the heart de-forms, the strain rate error induced due to imaging theheart wall under an angle can be estimated[39]. Thismeans that the angle dependency of the strain (rate)technique can be compensate for. Note however that,

although the amplitude of the velocity, strain rate orstrain curves can be influenced by the insonation angle,in clinical practice this angle dependency does prob-ably not have an important influence on the timing ofspecific events nor on the profiles of the curves. How-ever, the importance of the influence of the angledependency needs to be investigated further.

Figure 8. Illustration of the sensitivity of the numerical derivation to noise: a (perfect) straight line (a) results ina precise definition of the derivative (b). However, adding a small amount of (measurement) noise to this line (c)results in a very noisy numerical derivative (d).

Only a Limited Number of StrainComponents can be Measured

Since only the axial strain component can be measuredwith the current cardiac ultrasound strain estimationtechniques, and since the heart can only be visualizedin a limited number of views, not all strain com-ponents (radial, longitudinal and circumferential) canbe measured for all myocardial segments. In general,longitudinal strain can be measured in all left ventricularsegments using the apical two-, three- or four-chamber



views. However, radial and circumferential strain canonly be assessed in some segments, for example,radial strain in the posterior wall or circumferential strainin the lateral wall using a parasternal short-axis view.

Figure 9. (a) Time-integrating the strain rate curve can result in a drifting of the strain curve. However, using thephysical boundary condition that the strain value at the end of the cardiac cycle should be zero (b), this effect canautomatically be compensated for.

Figure 10 (opposite). Colour Doppler B-mode acquisition of a normal volunteer (a) together with its derived strainrate image (b). At specific points within the interventricular septum time profiles of the velocity (a) and strain rate(b) were extracted over one cardiac cycle (middle). Time integrating these curves results in the displacement andstrain curves, respectively (right). The vertical dashed lines indicate the position of aortic valve closure (AVC) andmitral valve opening (MVO).

Clinical Examples

The clinical examples shown in this section to illustratethe potential role of these new modalities are all basedon the velocity gradient method. Figure 10 (a) showsone frame of the original colour Doppler B-mode acqui-sition in an apical four-chamber view from a normalvolunteer. From the digitally stored data sets, velocityprofiles over one cardiac cycle, that is one R–R interval,are extracted for an apical, mid and basal segment of theintra-ventricular septum. The timing of aortic valveclosure and mitral valve opening is indicated by avertical dashed line. The motion during systole, early-and atrial filling (the S, E and A wave) can clearly beidentified. Moreover, a clear base to apex gradient invelocities can be recognized with velocities being higherin the basal segments of the myocardium. Time-integrating the velocity curve results in the displacementcurve shown in the right of Figure 10. As expected in theapical view, there is motion towards the transducerduring systole and away from the transducer duringdiastole. Moreover, the maximal displacement of theapical segments is smaller than that of the basal ones.


Regionally calculating the local spatial gradient inmyocardial velocities results in an estimate of the localstrain rate as explained above. The strain rate imageextracted from the same velocity colour Doppler dataset is shown in Fig. 10(b). Again, the strain rate curvesare extracted from the apical, mid and basal segments ofthe septum and the timing of aortic valve closure andmitral valve opening is indicated. During systole anegative strain rate value is measured, indicatingregional longitudinal shortening of the ventricle, whilethe biphasic relaxation of the wall (rapid filling andfilling with atrial contraction) can also be identifiedbased on the two lengthening phases — positive strainrate phases. Time integration of the strain rate curvesresults in the regional strain curves: it can be seen thatthe wall shortens during systole and lengthens duringdiastole.

In order to investigate the velocity or strain rateprofiles of several segments within the myocardial wallsimultaneously, a curved M-mode image can be con-structed[40]. Therefore, the velocity or strain rate infor-mation of all points along a manually indicated curvedline within the image is colour-coded and shown as afunction of time. A curved velocity and strain rateM-mode image from a normal volunteer is shown inFigure 11: the vertical axis indicates different points inthe septum while the horizontal axis represents time. Thesynchronous nature of the velocity and strain ratechanges throughout the septal wall can be appreciated.


Figure 11. Example of a curved M-mode velocity (a) and strain rate (b) image ofa normal volunteer: the velocity or strain rate information of all points along amanually indicated curve on the image, is colour coded and shown as a functionof time.

Again, the systolic, early and atrial filling phases can beidentified. Note that the velocity base to apex gradientcan also be seen on the curved M-mode image.

In a patient with severe lateral wall ischaemia the sameimages can be extracted, as shown in Figure 12. Thevelocity image does not accurately characterize the differ-ent contractile state of the lateral wall since global motionand/or rotation of the heart can induce the measuredvelocities. However, the strain rate image clearly demon-strates that the overall rate of deformation (and thus thetotal deformation) of the ischaemic lateral wall is muchlower.


Finally, Figure 13 gives an example of an aliasedstrain rate data set. Aliasing can be identified within astrain rate image by artifacts which appear at theboundaries of the clipped velocity estimates. The effectcan be observed both in the 2D strain rate image andin the individual strain rate curves. It is the resultsof calculating strain rate as the spatial gradient invelocities. Note that aliasing can be compensated for bymeans of, for example, phase unwrapping tech-niques, and thus is not a fundamental problem of themethodology. Moreover, when the cross-correlationmethodology is used aliasing would not occur at all,


since a specific radio frequent signal pattern can(theoretically) be tracked over any distance.

Figure 12. Example of a curved M-mode velocity (a) and strain rate (b) image ofa patient with severe lateral wall ischaemia. The abnormal strain rate profilewithin this wall is obvious. However, the velocity information does not resolvethe abnormal regional function.

ConclusionsStrain rate and strain imaging are potentially importantnew tools for the quantification of regional myocardialfunction. Initial in vitro and in vivo studies have shownpromising results. However, the clinical relevance of thetechnique remains to be proven. The introduction of

these imaging modalities into clinical practice will re-quire the definition of normal segmental values for strainand strain rate (both in the radial, longitudinal andcircumferential direction). At the time of writing thesevalues are not available. As far as pitfalls are concerned,the problem of angle dependency has probably beenoverestimated, since in a clinical situation timing ofregional myocardial events can be represented accu-rately. Moreover, compensation for this angle depen-dency seems possible. A more fundamental problem of



the technique is that currently for a myocardial segmentonly one of the possible nine strain components can bemeasured (unless two different acquisitions are madefrom different echocardiographic views, in which casetwo components can be assessed). It must therefore bedetermined whether such a uni-dimensional data set cansufficiently describe the clinical substrate being investi-gated. Further attempts will have to be made to estimatethe other myocardial strain (rate) components by ultra-sound as well. This again is theoretically feasible byusing information of adjacent scan lines and/or byapplying the principles of conservation of mass[39].Moreover, shear strain estimation has been developedin elastography and should thus, theoretically, also befeasible for cardiac strain[41].

Figure 13. Example of an aliased colour Doppler data set (a) and the resultingstrain rate artifact (b). The strain rate artifact represents itself by the extremely highstrain rate estimates. It can be seen both in the 2D strain rate image (left) and in theextracted strain rate profiles (right).


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Figure 14. Geometry used to derive a relation showingthat the velocity gradient methodology in fact measuresthe natural strain rate.

Appendix A

Take a 1D bar with end-points A and B. Assume that attime instance t these points have an instantaneouscoordinate rA(t) and rB(t), respectively, and that they aremoving with the instantaneous velocities vA(t) and vB(t)(Fig. 14). If we assume linear, uniform strain within thebar, the instantaneous velocity of an arbitrary point


within the bar, having a coordinate r, can be writtenas:

with L(t)�rB(t)�rA(t) the instantaneous length of thebar. Note that the proportionality factor vB(t)�vA(t)/L(t) is in fact the true strain rate of the bar.

Assume that a sample volume with a fixed length LS

and end-points having coordinates r1 and r2, respect-ively, is put inside this (deforming) bar. By using equa-tion (13), this implies that the measured instantaneousstrain rate �̇(t) is given by:

which is the true instantaneous strain rate. Since thedenominator of this expression takes into account theinstantaneous length of the bar, L(t) rather than its initiallength L(t0)�L0, this implies that the natural strain rate ismeasured. Note, however, that this is only true if theassumption of linear, uniform strain within the bar holds.

regional strain and strain rate measurements by cardiac ultrasound: principles, implementation and...

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