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Doppler based ultrasound imaging methods for

noninvasive assessment of tissue viability

Andreas Heimdal

A dissertation submitted to

the Norwegian University of Science and Technology

in partial ful�llment of the requirements

for the degree of

Doktor ingeni�r

September� ��

iii

Abstract

Tissue viability can be de�ned as the combination of perfusion� metabolismand function� Perfusion means that blood is supplied to the tissue cells�Metabolism means that the cells are alive and able to utilize the nutrientsand oxygen supplied by the blood to perform a function� The function canbe di�erent depending on the cell type� In this work� the focus will mainlybe on muscle cells� One function of a muscle cell is to contract and relax�

The object of this thesis is to describe non�invasive Doppler methods todetect� measure and visualize two of these viability factors� Blood perfusion�and tissue deformation� First� other non�ultrasonic methods to assess theviability parameters are brie�y described� Next� a review of ultrasoundmethods to assess the parameters is compiled� The review covers the topicsblood �ow detection and velocity estimation� tissue Doppler imaging� andstrain and strain rate imaging�

Blood perfusion can to some extent be assessed through the Doppler mea�surement and detection of blood velocity� As a part of the processing ofthe received Doppler signal is the clutter �lter that is needed to removethe echo from stationary or slowly moving tissue� Unfortunately this �lteralso removes the echo from slowly moving blood and thus prevents mea�surements of blood �ow in the smallest vessels� Signal models for pulsedwave Doppler and color �ow imaging are presented� as well as methods toestimate the lowest detectable velocity� A measured clutter signal from amuscle with involuntary vibrations is used to illustrate the method� Forthis clutter signal and a ultrasound frequency of � MHz� the results show alow velocity limit of �� mms� indicating that capillary blood �ow is notdetectable� regardless of observation time� In color �ow imaging� where theobservation time is limited� the detection limit is even higher� A likelihoodfunction for blood detection and expressions for the probability of detectionare presented�

Tissue deformation can also be assessed by the ultrasound Doppler tech�nique� The strain rate� i�e�� the rate of deformation� of a tissue segmentcan be estimated from the tissue Doppler data by calculating the spatialvelocity gradient� The optimal strain rate estimator and expressions forthe lower bound variance are derived� A simpli�ed estimator suitable forreal�time performance in an ultrasound scanner is investigated in in vitro

and in vivo experiments� A method to estimate the accumulated strainfrom the instantaneous strain rates is also presented�

The in vitro experiments involve cyclic compression of a gel block� Theresults show a bias smaller than �� s−1 and a standard deviation lessthan �� s−1 in the strain rate estimate for strain rates in the range �� s−1�

The strain rate technique is next tested in vivo in a pilot study in � pa�

iv

tients with myocardial infarction and � normal subjects� Apical cardiacimaging was performed on all subjects� and the results show that reducedstrain rates are found in the infarcted muscle segments compared to thenormally functioning segments� The method is also tested in imaging ofthe peristaltic contraction in the stomach muscle� and in imaging of thestrain rate in tumors during external compression�

Several methods to improve the real�time strain rate estimate are presented�These include second harmonic imaging and simple clutter �ltering� It isshown that both stationary reverberations and clutter �ltering introduce abias in the strain rate estimate� and that a clutter �lter therefore only shouldbe used when the reverberation level is high� A method to increase theframe rate by using a sliding window processing technique is also presented�

The angle dependencies of both the strain rate and the strain estimates aredescribed� using a model for the tissue deformation� Furthermore� a methodto estimate the strain rate in other directions than along the ultrasoundbeam is presented� A preliminary test in cardiac short axis imaging indi�cated that the method could measure simultaneously the transmural andthe circumferential strain rates in all parts of the ventricle except wherethese directions were perpendicular to the beam�

Preface

This dissertation is submitted to the Norwegian University of Science and Technology�NTNU� in partial ful�llment of the requirements for the degree of Doktor ingeni�r�The work has been performed at the Department of Physiology and Biomedical

Engineering at the Medical Faculty� NTNU in the period � � � � � My supervi�sor has been professor Hans Torp at the Department of Physiology and BiomedicalEngineering� while professor Jens Hovem at the Department of Telecommunications�NTNU� has been the administrative supervisor� In the period � �� the work wassupported by the Norwegian Research Council� in � � in collaboration between theNorwegian Research Council and GE Vingmed Ultrasound AS and a short period in� by the Medical Faculty� Since early � I have been employed by GE VingmedUltrasound AS�

Acknowledgements

I wish to especially thank my supervisor Hans Torp� He introduced me to the excitingsubject of Doppler ultrasound imaging� and has been of invaluable help during thepreparation of this thesis�When working on the strain rate imaging technique I had very fruitful collaboration

with Asbj�rn St�ylen at the Department of Cardiology at the University Hospital ofTrondheim� His interest for this topic has been a great motivation for the work� andhis clinical examinations using the method have been very valuable�Also many thanks to Steinar Bj�rum� Stein Inge Rabben� Marek Belohlavek� Kai

Thomenius� Bj�rn Angelsen� Vidar S�rhus� Torgrim Lie� �ge Gr�nnings�ter� Stig Sl�r�dahl and Bj�rn Olav Haugen for helpful discussions and for review of my work� and toBj�rn Olstad� Jan D�hooge� Tormod Bakke� Sevald Berg� Johan Kirkhorn� Tor Urdalen�Annette Vanvik Lund and Odd Helge Gilja for indispensable help with computer imple�mentations� practical experiments and clinical examinations� All my colleagues at theDepartment of Physiology and Biomedical Engineering and GE Vingmed Ultrasoundduring the last years are also thanked for a nice and interesting work environment�The �nancial support from the Norwegian Research Council� GE Vingmed Ultra�

sound AS and the Medical Faculty is greatly appreciated�Finally� many thanks to Per and Magnar Heimdal for their support and under�

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standing during the time I worked with the thesis� In fond memory of my motherMarit Agnete Heimdal who died from breast cancer in � �� I miss you very much�

Contents

Nomenclature xi

I �

� Introduction �

�� Tissue viability � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Viability of cardiac muscle � � � � � � � � � � � � � � � � � � � � � � �� Viability in respect to tumors � � � � � � � � � � � � � � � � � � � � �

�� Assessment of tissue viability � � � � � � � � � � � � � � � � � � � � � � � � �� Assessing cardiac muscle viability � � � � � � � � � � � � � � � � � � �� Assessing the nature of tumors and other lesions � � � � � � � � � �

�� Aims of study � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Publications � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Structure of the thesis � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� Summary of papers �

�� Summary of Paper � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Summary of Paper � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Summary of Paper � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� Previous work �

�� Blood �ow detection and velocity estimation � � � � � � � � � � � � � � � �� Physical basis and data acquisition � � � � � � � � � � � � � � � � � �� Signal processing � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Specialized methods � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Tissue Doppler imaging � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Strain rate imaging � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Myocardial velocity gradient � � � � � � � � � � � � � � � � � � � � �� Tracking�based strain rate imaging � � � � � � � � � � � � � � � � � � �� Real�time Doppler strain rate imaging � � � � � � � � � � � � � � � �

�� Strain imaging � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Vibration amplitude sonoelastography � � � � � � � � � � � � � � � �

vii

viii

�� Strain imaging by speckle tracking � � � � � � � � � � � � � � � � � �� Compression RF strain imaging � � � � � � � � � � � � � � � � � � � �� Compression IQ strain imaging � � � � � � � � � � � � � � � � � � � �� Spectral strain imaging � � � � � � � � � � � � � � � � � � � � � � � �� Decorrelation � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Dynamic range � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� One�dimensional strain rate estimation ��

�� Optimal strain rate estimator � � � � � � � � � � � � � � � � � � � � � � � � �� Three component Gaussian signal model � � � � � � � � � � � � � � �� Logarithmic likelihood function for N = 2 � � � � � � � � � � � � � �� Maximum likelihood estimate � � � � � � � � � � � � � � � � � � � � �� Cram�r�Rao bound � � � � � � � � � � � � � � � � � � � � � � � � � � �� Comparing to previous work � � � � � � � � � � � � � � � � � � � � �� Comparing to linear regression � � � � � � � � � � � � � � � � � � � �� A closed form approximation � � � � � � � � � � � � � � � � � � � � �� Conclusions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Real�time strain rate estimator � � � � � � � � � � � � � � � � � � � � � � � �� Basic implementation � � � � � � � � � � � � � � � � � � � � � � � � �� High frame rate � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Second harmonic imaging � � � � � � � � � � � � � � � � � � � � � � ��

�� Estimating strain from strain rate � � � � � � � � � � � � � � � � � � � � � ��

� Angle dependence of strain rate ��

�� Angle dependence of the strain rate and strain estimates � � � � � � � � � �� Coordinate de�nitions � � � � � � � � � � � � � � � � � � � � � � � � �� Assumed relation between strain and strain rate � � � � � � � � � �� Strain rate angle dependence � � � � � � � � � � � � � � � � � � � � �� Assuming incompressible material � � � � � � � � � � � � � � � � � �� Relative errors caused by angle mismatch � � � � � � � � � � � � � � �� Discussion and conclusions � � � � � � � � � � � � � � � � � � � � � ��

�� Estimation of more components of the strain rate tensor � � � � � � � � � �� Discussion of other angle dependencies � � � � � � � � � � � � � � � � � � � ��

� In vitro experiments ��

�� Validation tests for the strain rate estimator � � � � � � � � � � � � � � � � �� Zero strain rate test � � � � � � � � � � � � � � � � � � � � � � � � � �� Gel block compression test � � � � � � � � � � � � � � � � � � � � � �

�� E�ects of stationary reverberations and clutter �ltering � � � � � � � � � �� Theory � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Methods � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Results � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Discussion and conclusions � � � � � � � � � � � � � � � � � � � � � ��

Contents ix

� In vivo examples of strain rate imaging �

�� Cardiac muscle function � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Normal �ndings � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Infarction examples � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Stomach muscle function � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Tumors and cysts � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Breast tumor � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Liver cyst and tumor � � � � � � � � � � � � � � � � � � � � � � � � �

Conclusions and future directions ��

�� Concluding remarks � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Suggestions for future work � � � � � � � � � � � � � � � � � � � � � � � � � �

A The strain strain rate and rate�of�deformation tensors ��

A�� Elementary de�nitions � � � � � � � � � � � � � � � � � � � � � � � � � � � � �A�� Strain tensors � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �A�� Strain rate and rate�of�deformation tensors � � � � � � � � � � � � � � � �

B Fisher information matrix and Cram�r�Rao bound ��

II Papers ��

�� A� Heimdal and H� Torp� �Ultrasound Doppler Measurements of Low VelocityBlood Flow� Limitations Due to Clutter Signal from Vibrating Muscles�� IEEETrans on Ultrason� Ferroelect and Freq Cont� vol� �� pp� ��

�� A� Heimdal and H� Torp� �Detecting Small Blood Vessels in Color Flow Ultra�sound Imaging� A Statistical Approach�� IEEE Trans on Ultrason� Ferroelectand Freq Cont� �Submitted August � ��

�� A� Heimdal� A� St�ylen� H� Torp� and T� Skj�rpe� �Real�time Strain Rate Imagingof the Left Ventricle by Ultrasound�� J Am Soc Echocardiogr� vol� �� pp� ��

x Contents

Nomenclature

Symbols

α Angle between the ultrasound beam and the u� or v�axis in thecardiac muscle

β Magnitude of the complex autocorrelation coe�cient∆φ The phase shift of the echo signal from pulse to pulse∆r Radial distance between two segments or samples∆t Time shift of the RF signal from pulse to pulse

ε Finite strainε Small strainε Small strain rate �velocity gradient�λ Wavelength

ω0 Angular center frequency of the transducerρ Autocorrelation coe�cientσ Standard deviation �square root of the variance�

σ2 VarianceB Bandwidthc Speed of sound in human tissue �approx� �� ms�

C(τ) Cross�correlation functionC Covariance matrix

d(m) Displacement of segment number mf Frequency coordinate

f0 Center frequency of the transducerGR(ω) Fourier transform of the autocorrelation estimate R

IN Identity matrix of size N by Nl Frame number

L Number of framesm Radial sample or segment number in a beamM Number of radial signal samples or segments in a beamµ Mean valuen Pulse number� i� e�� sample number within a packetN Number of samples in a packet� i�e�� number of pulses for each

sample volume

xi

xii Contents

px(x) Probability density function for the complex vector xP PowerQ Perfusion or �ow per unit volumer Spatial coordinate along the ultrasound beam

rs Radial sampling distanceR(∆n) Autocorrelation function

R Autocorrelation estimateS(f) Doppler spectrumS(k) Fourier transform of s(r)

t Temporal coordinate within a pulse echo signalT Time between pulses� equals 1/PRFu Circumferential coordinate� clockwise seen from the apex of the

heartv Meridional coordinate� from apex to base in the heartv Velocityw Transmural coordinate� from endo� to epicard

s(t) RF signal� i�e�� echo from one pulsex(n) Complex IQ signal from one depth and N pulses� n = 1, . . . , N

x Packet� i�e�� complex IQ signal vectorx(m, n) Complex IQ signal from depth m

Abbreviations

DCM Dilated cardiomyopathy �Abnormally large and thin�walled ventri�cle�

FIR Finite impulse responseHCM Hypertrophic cardiomyopathy �Abnormally thick ventricle walls�IIR In�nite impulse responseIQ In�phase and quadrature �real and imaginary parts of the quadra�

ture demodulated signal�LPF Low pass �lter

MSDR Mean to standard deviation ratioMVG Myocardial velocity gradientPRF Pulse repetition frequencyRF Radio frequency �received signal before quadrature demodulation�SNR Signal�to�noise ratioSNRC SNR of the calculated strain imageSRI Strain rate imagingTDI Tissue Doppler imaging

De�nitions

arg Function that returns the argument or phase of a complex variable

Contents xiii

argmax Operation that returns the index that maximizes the submittedfunction

cov Covariance functionE Expected value operator

Im Function that returns the imaginary value of a complex variablepacket Name used for the vector built up of samples from the same depth

in consecutive pulses in the same directionRe Function that returns the real value of a complex variableT Transpose operationvar Variance function

xiv Contents

Part I

�

Chapter �

Introduction

This chapter is meant as a short introduction for engineers without background inphysiology� and does not go into detail about the clinical issues�

�� Tissue viability

Tissue viability is the ability of the tissue to live and grow� In this work� tissue viabilityis de�ned by the following three factors�

• Blood perfusion� The metabolism in the tissue cells can only take place whenenough oxygen and nutrients are available� These substances are supplied by thecapillary blood �ow�

• Metabolism� The tissue cells must be able to absorb and utilize the nutrientsand oxygen delivered by the blood� In this factor cell membrane integrity is alsoincluded�

• Function� The tissue cells must be able to perform some sort of function� Amuscle cell must for instance be able to perform contraction and relaxation�

If one or more of these factors are missing� the tissue cannot be fully viable� Theclinical interest in tissue viability includes detection of growth in tumors� and cardiacmuscle viability�

�� Viability of cardiac muscle

Left ventricular dysfunction occurs when all or some of the myocardial muscle doesnot perform normally� This is usually caused by reduced blood perfusion to all orparts of the cardiac muscle� If the muscle is still viable� normal muscle performancecan be regained by restoring the blood perfusion to normal levels� The state of theviable muscle with reduced contractile function due to reduced blood perfusion hasbeen termed hibernation�

�

� Introduction

Several reperfusion procedures exist� but these are normally only justi�ed if viabilitycan be demonstrated� If no viability can be detected� conservative medical therapy orheart transplantation might be the best choice�

�� Viability in respect to tumors

Viability is not commonly a term used in the description of tumors� but it will be usedbrie�y in this work to describe whether the tumor is growing or not�Tumors can be divided into benign and malignant types �� Benign tumors rep�

resent localized growth that usually remains circumscribed� while malignant tumors�cancer� usually invade surrounding tissue� It is of great clinical interest to distinguishthese two types of tumors� since malignant tumors need di�erent treatment proceduresthan benign tumors� Since some of the treatment procedures can be painful and in�volve some risk� it is important to reduce the number of tumors falsely classi�ed asmalignant�One subtype of benign tumors can be distinguished from malignant tumors by the

fact that they are not viable� i�e�� that they have stopped growing� An indication ofthis is that there is no blood �ow in the tumor� Without the nutrients supplied by theblood� the tumor cells cannot grow� and will shortly die� There are� on the other hand�other types of tumors that do have blood �ow in the tumor� but still are benign�It is known from breast cancer that most large tumors are palpable masses� This

means that the sti�ness of a mass can be used as a marker for the disease� Any sti�masses that are not documented in prior examinations are assumed to be malignanttumors until proven otherwise ��

�� Assessment of tissue viability

There are several known techniques to assess viability in tissue� In this work the focuswill be on noninvasive ultrasound imaging methods that involve the Doppler technique�Noninvasive means that the techniques do not require entering the body or puncturingthe skin� For comparison� some other techniques are presented brie�y in this section�

�� Assessing cardiac muscle viability

Techniques to assess the viability in the cardiac muscle include �� positron emis�sion tomography �PET�� single photon emission computed tomography �SPECT� andechocardiography during stepwise infusion of dobutamine �Stress echo�� Newer tech�niques are magnetic resonance imaging �MRI� and contrast echocardiography� Ta�ble �� shows which factors of the de�nition of viability these techniques can detect ormeasure in the cardiac muscle�With PET� the most common method is to use two di�erent tracers� usually

�uorine�� labeled with deoxyglucose �FDG� and nitrogen�� labeled with ammonia�N��ammonia�� FDG activity indicates metabolism� while N��ammonia activity in�dicates perfusion�

� � Assessment of tissue viability �

Technique Perfusion Metabolism FunctionPET X X �SPECT X X �Stress echo � � XContrast echo X � �

MRI X X X

Table �� The ability of di�erent techniques to detect viability in cardiacmuscle tissue� The term function here means muscle thickening ordeformation�

With SPECT there are several di�erent tracers and protocols commonly used�Thallium�� imaging re�ects cell membrane integrity� Imaging with technetium� m labeled compounds re�ects perfusion and the uptake is also dependent on cellmembrane integrity and mitrochondrial function� Radioiodinated fatty acids measuremetabolism� but this method has still not been validated� Using �� keV collimators�FDG imaging of metabolism has recently also been possible with SPECT�Stress echo is usually based on the use of a low�dose dobutamine infusion that

enhances the systolic contraction in the regions with reduced function� The contractioncan also be enhanced by physically stressing the patient using for instance a treadmill or a stationary exercise bicycle� The increase in contractility is measured byechocardiography as increased wall thickening�With the MRI technique� methods to detect �ow� perfusion� wall motion and car�

diac metabolism have been presented� The same dobutamine procedure as with stressecho has been studied� but is only rarely used� Recent developments include myocar�dial tagging to better quantify the changes in wall motion� and magnetic resonancespectroscopy to assess metabolism�Contrast echocardiography involves venous or arterial administration of gaseous

microbubbles� When insoni�ed by ultrasound pulses these microbubbles produce ahigh intensity scattering� simplifying the detection of blood� The scattering is alsohighly nonlinear� allowing detection by harmonic imaging� i�e�� receiving the echo atthe second or higher harmonic of the transmitted ultrasound frequency� Detection ofperfusion is possible using this method� but quanti�cation is yet to be shown�

�� Assessing the nature of tumors and other lesions

The most common cancer in women is breast cancer� Young female patients �under�� with breast cancer are usually investigated by grayscale ultrasound� and older bymammography �� In young women� the breast is more dense� so the mammographicsensitivity is lower� Tumors are also rare in this age group� Ultrasound is usefulto di�erentiate cystic lesions and solid masses� Cystic lesions are usually consideredbenign� while solid masses are more uncertain� Mammography is X�ray imaging of thebreast� The margins� density and location of a mass is considered in di�erentiatingbenign and malignant tumors�

� Introduction

�� Aims of study

The technical basis for the work presented in this thesis is the extreme sensitivity tomotion of the ultrasound pulsed wave Doppler technique� Miniscule displacements�down to a few microns� in blood and tissue within the human body can be measured�Utilizing this ability of the Doppler technique� the aim of this work is the evaluationand development of Doppler ultrasound methods to detect and quantify two of thetissue viability properties� blood perfusion and tissue deformation�For blood perfusion� the focus will be mainly on the detection problem� Small

blood vessels are often hard to detect� and it is interesting to �nd the lower limit onthe blood velocity that allows detection and measurement� This needs to be evaluatedboth for pulsed Doppler and for color �ow imaging� Also� optimal imaging parametersare important to �nd�The Doppler technique can also be used to measure tissue velocities� The deforma�

tion �strain and strain rate� of the tissue can be estimated from these velocities� butthe variance of such a measurement might be large� Methods to measure the strainand strain rate directly from the Doppler signal is important� To make the methodmore clinically useful� the estimation should be performed in real�time�

�� Publications

References to the papers� posters and abstracts that have been published prior to andduring the preparation of this thesis have been included in the References ��

�� Structure of the thesis

The thesis is divided in two parts� The �rst part consists of unpublished material�except for two conference papers and one abstract that have been rewritten into thetext �� The second part consists of three papers� Summaries of these papersare given in Chapter ��Chapter � reviews several Doppler�based ultrasound imaging methods to detect and

measure blood perfusion and tissue deformation� Thematically next in order� Papers �and � then describe the ability to detect and measure blood �ow using pulsed Dopplerand color �ow imaging�The strain rate imaging technique to measure tissue deformation is introduced with

a technical description in Chapter � and a more clinical description in Paper �� Theangle dependency of the method is discussed in Chapter �� Chapter � describes twoin vitro experiments performed to validate the strain rate estimates� and Chapter �shows several preliminary in vivo results using the method�Finally� the conclusions and suggestions for future work are presented in Chapter ��

It should be noted that a list of symbols� abbreviations and de�nitions is included inthe beginning of the document for reference purposes�

Chapter �

Summary of papers

This chapter gives summaries of the three papers that are submitted as part of thisthesis�

�� Summary of Paper �

The paper titled �Ultrasound Doppler measurements of low velocity blood �ow� Lim�itations due to clutter signals from vibrating muscles� points at the involuntary vi�brations in skeletal muscle cells as an origin for clutter signals of non�zero bandwidth�In the paper� a model of the pulsed wave Doppler signal from vibrating muscles wasdeveloped� An experiment with in vivo data was used to estimate the parameters ofthe model� By comparing this model with previously developed models for the Dopplersignal from blood� a theoretical minimum for detectable blood velocity was found� Thelimit showed a nonlinear relation to the ultrasound frequency� At � MHz� and for themeasured clutter signal� the limit was �� mms�Using the signal model� it was also possible to estimate the radial component of

the muscle vibrations from the phase of the Doppler signal�


The paper titled �Detecting small blood vessels in color �ow ultrasound imaging� A sta�tistical approach� describes a statistical model for the complex demodulated Dopplersignal and a likelihood test for blood detection� In the model� the ultrasound signalwas described as a sum of three independent components� tissue clutter� blood andwhite noise� An approximation of the likelihood test was in the paper shown to involvea clutter �lter and a blood enhancement �lter�An experimental in vivo data set of a human arm muscle was used to illustrate

how the the signal model could be used� The performance of the test was assessedusing receiver operating characteristics for di�erent sets of parameters� including the

�

Summary of papers

blood velocity and the PRF� It was also illustrated how the minimum detectable bloodvelocity could be found�


The paper titled �Real�time strain rate imaging of the left ventricle by ultrasound�describes the �rst clinical pilot study using the strain rate imaging technique� Thistechnique allows real�time imaging of the deformation rate in the cardiac muscle� Thestrain rate is shown to be equal to the spatial velocity gradient� which is measured usingthe tissue Doppler technique� The calculation of the gradient along the ultrasoundbeam is performed in real�time and was presented as color�coded cine�loops and colorM�modes�The method was tested in � healthy subjects and � patients with recent myocardial

infarctions� All the infarcted regions showed up with reduced strain rate� demonstratingthe technique to be useful for imaging regional dysfunction�In this paper� A� Heimdal de�ned the strain rate concept� the description of the

strain rate imaging technique and the discussion on the technical limitations� He alsomade Figures � and � and the general revision of the whole paper� A� St�ylen performedthe patient study and wrote the discussion on the clinical application of the technique�H� Torp and Terje Skjerpe reviewed the technical and clinical parts respectively of thepaper� T� Bakke and T� Urdalen implemented the real�time algorithm� A� V� Lund�B� Olstad and S� Berg implemented the post�processing software and M� Belohlavekhelped with the review�

Chapter �

Previous work

In this chapter� Doppler based ultrasound imaging methods for noninvasive assessmentof tissue viability and elasticity will be reviewed� The focus will be put on methods todetect perfusion and function�Perfusion can be detected� and to some extent also measured� with Doppler ultra�

sound methods� There is� however� a low velocity limit� If the blood �ow has a verylow velocity� the clutter signal from stationary or slowly moving tissue will overshadowthe signal from the blood� This can be overcome by using ultrasound contrast agentsthat enhance the echo from blood� The contrast agents are usually small gas bubbles�some types also with a shell� that are injected into the blood path� Techniques fordetection of contrast agents include the use of non�linear scattering properties or pro�cedures involving destruction of the bubbles using high energy ultrasound� This is alarge �eld of study that is beyond the scope of this thesis and will therefore not bediscussed in more detail�The function of a cell depends on what type of cell it is� The function of a muscle

cell is to perform contractions and relaxations� With Doppler�based methods it ispossible to measure both the velocity of the muscle motion and the deformation of themuscle�The deformation or strain measurement can also be used to study the elasticity of

tumors� This is done by compressing the tissue around the tumor and investigatinghow much this compresses the tumor� Hard tumors will not be deformed as much assoft tumors� The sti�ness of a tumor is to some extent correlated with its malignancy�Most malignant tumors are described as hard in the literature� while the benign tumorsvary in sti�ness ��

�� Blood �ow detection and velocity estimation

The two most widely used Doppler imaging techniques to detect and measure blood�ow are spectral or pulsed Doppler and color �ow imaging� Pulsed Doppler can give lessvariance in the velocity estimate but only gives information for one sample volume at

�� Previous work

a time� Color �ow imaging reduces the examination time by allowing a simultaneousvisualization of a whole two�dimensional region� A pulsed Doppler instrument wasdeveloped in � � by Baker �� In � �� Anglesen and Brubakk �� used the pulsedDoppler technique to measure blood �ow in the aorta� The same year Holen et al�� combined the measured blood velocities with Bernoulli�s equation to measure thepressure gradient in mitral stenosis� Color �ow imaging was introduced in the clinicalsetting in � �� by Omoto �� Hatle and Anglesen published a widely used book onthe Doppler techniques in � �� and � �� Ferrara and DeAngelis published areview of the color �ow imaging techniques in � � �� Color �ow imaging involves both detection of blood �ow and estimation of the

blood velocity� The Doppler signal is used for both these purposes� and since the twooperations are closely related� both will be reviewed�Sections �� and �� give summaries of two papers by the author that are presented

as part of this thesis and that discuss the detection of low velocity blood �ow withpulsed Doppler and color �ow imaging� respectively�

�� Physical basis and data acquisition

The detection and quanti�cation of blood �ow velocity with ultrasound is based onthe scattered echo from the red blood cells� Detection of blood is based on removingthe tissue component of the echo signal by clutter �ltering and estimating the powerin the remaining signal� The velocity of the detected blood can be found from eitherthe time delay or the phase shift between the echo signals from two or more pulses� Itis not possible to detect the Doppler shift using the echo from only one pulse� sincethe frequency resolution of any spectrum estimator is limited by the inverse of theobservation time� To get a reasonable spatial resolution the observation time for eachsample volume is usually smaller than a few µs� This gives a frequency resolutionabove several � kHz� and thus makes it impossible to detect the comparably smallDoppler shift of blood �usually a few kHz��Using the time delay method� the velocity is found as

v =c∆t

2T, ��

where ∆t is the time delay� c is the velocity of sound� and T is the time between thepulses� Using the phase shift method� the velocity is found as

v =c∆φ

2f0T, ��

where ∆φ is the normalized phase shift �|∆φ| < 1/2� and f0 is the center frequency ofthe received signal� Since T is known and f0 and c do not vary much� the velocity canbe estimated from the time delay or the phase shift�The data acquisition is performed by transmitting and receiving typically between

� and �� pulses in each beam direction� The pulse repetition frequency �PRF� mustbe adjusted according to the velocity of the blood� Using too low PRF makes the time

� � Blood �ow detection and velocity estimation ��

Figure �� Block diagram of a typical color �ow imaging system�

delay or the phase shift too large to be estimated without aliasing� and using too highPRF makes it di�cult to estimate low velocities since the observation time becomesvery small�

�� Signal processing

Figure �� shows a block diagram of a typical color �ow imaging system� Electricalpulses are generated in the transmitter and switched to the beamformer� This generatesultrasound pulses that are �red from the transducer array� Echoes received by the sametransducer elements are combined in the beam former to an RF signal that is switchedto the receiver� The receiver usually consists of complex demodulation� illustrated withthe cosine and sine multiplications and the low pass �lters �LPF�� that produces thein�phase and quadrature �IQ� signals� When RF processing is used� as described onpage �� this stage is not included� Finally� the processing for the gray scale imageand the color �ow map is performed� This processing can in principle be performedin parallel� but usually di�erent pulses are used for the gray scale image and the �owmap�Analog to digital conversion of the received signal can be performed in di�erent

stages of the system� If a digital beam former is used� the signals from each elementin the transducer are digitized before they are transferred to the beam former� Othersystems use analog delay lines in the beam former� and digitize the signal later in theprocess�The velocity estimation is performed as shown in Figure �� by �rst storing the

complex IQ data acquired for each pulse in memory� �I and Q are treated as the realand imaginary part of a complex signal�� The data vector from a single depth� in this


Figure �� Block diagram of the �ow estimation�

thesis termed a packet� is then read from memory and processed to extract a powerestimate and a velocity estimate for this position� The processing consists of two steps�First� the strong signal from slowly moving tissue is removed using a clutter �lter� thenthe velocity is estimated� If the time delay method is used� as explained on page �� theclutter �ltered RF signal from several depths is combined for each velocity estimate�while for the phase shift method only the IQ signal from one depth is needed� Thevelocity is found using �� or �� respectively for the two methods� The power ofthe signal after clutter �ltering is also calculated and is compared to a threshold todetect the presence of blood� The power signal can also be used by itself to generateso�called Power Doppler images ��

Clutter �lter

The clutter �lter is a high pass �lter to remove the signal from stationary or slowlymoving tissue� Since only a few packet samples are available for each depth� the designof these �lters is not straightforward� In � �� Jensen �� described a clutter �lter foruse in the time delay method� In � �� Torp �� presented a theoretical descriptionof a general class of linear rejection �lters for use in the phase shift method� Theseincluded �nite impulse response �FIR� �lters� in�nite impulse response �IIR� �lters andregression �lters�The output of a FIR �lter is a weighted sum of the previous inputs x(n) to the

�lter�

y(n) =I∑

i=0

bix(n − i), ��

where n = 1, . . . , N � and N is the packet size� and I is the number of zeros in the �lter�The output of an IIR �lter is on the other hand a weighted combination of both the


previous inputs x(n) and outputs y(n) of the �lter�

y(n) =I∑

i=0

bix(n − i) +J∑

j=1

ajy(n − j), ��

where N and I are as in �� and J is the number of poles ��Regression �lters operate on the assumption that the clutter component of the

signal can be approximated with a polynomial of a given order� This polynomial isfound by a least�squares regression analysis� and is subtracted from the packet�

y(n) = x(n) −D∑

d=0

adnd, ��

where ad are the polynomial coe�cients and D is the order of the polynomial�FIR and IIR �lters have a transient that must be removed before further processing�

This reduces the number of samples available for velocity estimation and will thus limitthe quality of the velocity estimate� FIR �lters have in addition been shown by Hoekset al� �� to produce a signi�cant bias in the estimates of low velocities� IIR �lterscan be initialized by providing values to the unavailable inputs and outputs for n ≤ 0to reduce the e�ect of the transient� An evaluation of step�initialized IIR �lters havebeen performed by Kadi and Loupas �� using simulated signals� They also comparedthe initialized IIR �lter and the regression �lter and found that the regression �lterhad slightly better performance in estimating the correct velocity and power when theclutter component was moving slowly�

Velocity estimation

The time delay method was investigated by Bonnefous and Pesque in � �� Thetime delay ∆t in �� can be found as

∆t = argmaxτ

(C(t, τ)), ��

where argmax indicates an operation that returns the time shift τ that maximizes thecross�correlation function

C(t, τ) =∑n1

∑n2

∫ t+W/2

t−W/2

wn1(t′)wn2 (t

′ + τ)sn1(t′)sn2(t

′ + τ)dt′, ��

where n1 and n2 are pulse indexes� sn(t) is the clutter �ltered RF signal from pulsenumber n� and wn(t) is a window of length W that is applied to signal number n�In � �� Kasai et al� �� presented the use of the autocorrelation technique to

estimate blood velocities� This technique had previously been used in radar velocityestimation� and is the one that has been implemented in most current color �ow imaging


systems� In this method� the phase shift between the pulses ∆φ in �� is found asthe phase of the autocorrelation function of the packet with a lag of one sample�

∆φ = arg(R(1)) = tan−1 RIm(1)RRe(1)

, ��

where RRe(∆n) and RIm(∆n) are the real and imaginary values respectively of theautocorrelation function

R(∆n) =N∑

n=1

x(n + ∆n)x∗(n). ��

This technique corresponds to a �rst order autoregressive �AR� estimate of the meanfrequency� as shown for instance in ��In � �� Torp et al� �� used a parametric model for the autocorrelation function

and found the variance of the phase shift estimate to be

σ2∆φ ≈ 1

2

(1

|ρ(∆m,∆n)| − 1)CS, |∆n| > 0, ��

where ρ(∆m, ∆n) is the autocorrelation coe�cient with radial lag ∆m along the beamand temporal lag ∆n

ρ(∆m, ∆n) =∑

m

∑n x(m + ∆m, n + ∆n)x∗(m, n)∑

m

∑n |x(m, n)|2 , ��

and CS is the fractional variance of the signal power estimate� This showed thatminimum variance was obtained when the correlation amplitude was maximized� Thismaximum could occur for ∆m �= 0� which suggested that using a packet with samplesfrom just one depth could be suboptimal�

Power estimation thresholding and Power Doppler

The power estimation is performed by calculating the remaining power in the signalafter the clutter �lter� A commonly used estimator for the power is

P = R(0) =1N

N∑n=1

|x(n)|2, ��

where R is the autocorrelation function in �� and x(n) here is the IQ signal afterclutter �ltering�In color �ow imaging the thresholding� i�e�� the detection of blood� is performed by

comparing the estimated power with a user controlled threshold� If the power is lessthan this threshold� it means that all the power in the Doppler signal was removed bythe clutter �lter� This is the case when there is only stationary or slowly moving tissuein the sample volume� As described earlier� only a few samples are available at each


depth� and the slope of the clutter �lter will depend on this� By setting the thresholdtoo high� slowly moving blood is not detected� and by setting the threshold too low�fast moving tissue might be detected as blood� To avoid the latter� the intensity of thecorresponding sample in the grayscale image is commonly used to separate blood andtissue� Since blood scatters ultrasound much more weakly than tissue� high intensitiesin the grayscale image indicate tissue� and not blood�In Power Doppler� the estimated power in �� is commonly averaged over several

frames� This reduces the variance of the power estimate and thus increases the sensi�tivity to low velocity blood �ow� but reduces the temporal resolution� Since no meanfrequency is calculated� Power Doppler is less sensitive to changes in beam angle thancolor �ow imaging�

�� Specialized methods

Alternative velocity estimation techniques

As mentioned earlier� the autocorrelation method corresponds to a �rst order autore�gressive estimation� In � � Loupas and McDicken �� used second order autoregres�sive estimation by modeling the blood signal with the two poles� In � �� Ahn andPark �� used one pole to model the clutter signal and the other for the blood and couldthus avoid the clutter �lter� Both methods have signi�cantly reduced performance inthe presence of white noise�As an alternative to the cross�correlation technique presented in section �� Bohs

et al� in � � �� described a sum�absolute�di�erence method� This method performs atwo�dimensional search instead of only a search along the beam� This allows estimationof velocities in other directions than along the ultrasound beam�In � �� Ferrara and Algazi �� presented a wideband maximum likelihood estima�

tor �WMLE� that searched the received echoes for the most likely trajectory causedby moving blood� In � �� Zagar et al� � �� used the WMLE estimator in combinationwith a signal alignment procedure� as an alternative to a clutter �lter� to measureblood with low �ow velocity�

Perfusion quanti�cation

In � �� Eriksson et al� �� and Dymling et al� �� described a method to quantifythe blood perfusion� rather than the blood velocity� Blood perfusion Q was de�ned asthe net in�ow of blood volume to a unit volume of tissue� and was expressed as

Q ∼ N0E{|v|}, ��

where N0 is the number of red blood cells in a unit tissue volume� and E{|v|} is themean velocity magnitude of these blood cells� The perfusion was calculated from theestimated one�sided Doppler spectrum S(f) as

Q ∼∫ ∞

0

fS(f)df. ��


A vascular network where all �ow directions were equally probable was assumed� Theresults of in vivo and in vitro experiments indicated a relation between the estimatedperfusion value and the true perfusion� but that large �uctuations in the estimatedvalue occurred by changes in geometry and ultrasound frequency�In � �� Adler et al� �� found that the speci�c �ow Q/V was given by the temporal

decorrelation of the Doppler power signal p(l)�

Rp(m) = Rp(0)e−QV m, ��

where RP is the frame�to�frame autocorrelation of p(l)� and

p(l) = |x(l)|2 − 1L

L∑l′=1

|x(l′)|2. ��

The method was tested in a �ow phantom and the velocity found from the speci�c�ow through a sample volume correlated well with the result using a standard Dopplershift technique�

Harmonic Doppler

In � �� Burns et al� �� described the principles and preliminary results of harmonicDoppler� In this technique� a band pass �lter on the RF signal is used to accessthe second harmonic component of the signal� This second harmonic signal is thenprocessed as before� taking into account the doubled ultrasound frequency� to detectblood� This technique is especially useful combined with the use of contrast agents�since they give a high amount of nonlinear scattering�

�� Tissue Doppler imaging

The concept of tissue Doppler imaging �TDI� was introduced by McDicken et al� in� � �� and was further developed by Sutherland et al� in � � �� TDI color mapsthe tissue velocities rather than the blood velocities� and is mainly used in the cardiacmuscle� The velocities in the cardiac tissue is considerably lower �� cms� than inthe blood� but the amplitude of the echo signal is approximately � dB larger thanthat of blood �ow ��The signal processing is much like for color �ow imaging in Figure �� only the

clutter �lter is bypassed or greatly weakened so the signal component from the rel�atively slowly moving tissue is not removed� Also the velocity map is usually notthresholded� but just superimposed on the grayscale image� The imaging parametersare also optimized di�erently� the PRF can� for instance� be lower in TDI than in color�ow imaging due to the lower velocities in the tissue compared to the blood�

� � Strain rate imaging ��

�� Strain rate imaging

The term �strain rate� is in this thesis used for the rate of deformation� The rateof deformation is a tensor that describes the rate of change of a length in an object�For small displacements and in�nitesimal strains� as the ones occurring from frame toframe in ultrasound imaging� the rate of deformation tensor is approximately equalto the time derivative of the in�nitesimal strain tensor� thus explaining the choice ofthe term �strain rate�� The strain and rate of deformation tensors are described indetail in Appendix A� As shown in Appendix A� the rate of deformation is in theone�dimensional case given as the spatial velocity gradient�Several methods to measure this velocity gradient have been proposed� and these are

reviewed in the following sections� The myocardial velocity gradient �MVG� method isa post processing method that uses the TDI velocities found by converting the colorsin the digitized TDI image to velocities using the information in the color bar in thesame image� The tracking�based strain rate imaging uses a modi�ed cross�correlationtechnique to track points in the myocardium� while simultaneously measuring theirvelocity� The real�time strain rate imaging �SRI� method calculates the strain ratedirectly from the received IQ signals�

�� Myocardial velocity gradient

In � �� Fleming et al� �� suggested using the gradient of the TDI velocity as ameasure for the relative change in heart wall thickness�

−dW

dt/W =

dv

dr. ��

Here� W is the instantaneous wall thickness and dv/dr is the velocity gradient acrossthe wall� This gradient was found as the slope of the linear regression of the TDIvelocity estimates� Only velocities measured with the ultrasound beam perpendicularto the heart wall were used�In � �� Uematsu et al� �� expanded the method to be used also when the ultra�

sound beam crossed the heart wall at an angle less than degrees� Short axis viewswere used� and the motion was assumed to be towards the center of the ventricle� Theangle θ between the ultrasound beam and the assumed direction of the motion wasthen used to angle correct the velocity data�

vcorrected =vmeasured

cos θ. ��

The velocity pro�le between the endocardium and the epicardium was then obtainedfor each radius from the center of the ventricle� The velocity gradient was �nally foundas the rate of inclination of each velocity pro�le by using least squares linear regression�Using this method on normal volunteers and patients with myocardial infarctions anddilated cardiomyopathy� the peak systolic values in Table �� were found� Signi�cantdi�erences in velocity gradients were found between all the patient groups and thenormal volunteers in the a�ected walls�

� Previous work

Normal Ant MI Post MI DCMAnteroseptal wall 1.69 ± 0.53 0.58 ± 0.41 1.48 ± 0.25 0.72 ± 0.59Posterior free wall 3.28 ± 0.67 2.84 ± 0.37 0.17 ± 0.27 0.93 ± 0.67

Table �� Peak systolic transmural MVG �mean ± standard deviation in s−1�in �� normal volunteers� � patients with old anteroseptal myicar�dial infarction �Ant MI�� patients with old posterior myocardialinfarction �Post MI� and patients with dilated cardiomyopathy�DCM� measured by Uematsu et al� ��

Systole Early diastole Late diastoleYoung normal subjects 4.4 ± 0.8 −9.2 ± 2.0 −1.0 ± 0.9Older normal subjects 4.8 ± 0.8 −3.6 ± 1.5 −3.8 ± 0.9

Athletes 4.6 ± 1.1 −9.9 ± 1.9 −0.9 ± 0.9Patients with hypertension 4.2 ± 1.8 −3.3 ± 1.3 −4.3 ± 1.7Young patients with HCM 3.2 ± 1.1 −3.7 ± 1.5 −1.3 ± 0.8Older patients with HCM 2.9 ± 1.2 −2.6 ± 0.9 −1.4 ± 0.8

Table �� Peak transmural MVG �mean ± standard deviation in s−1� in atotal of � young �30 ± 7 years� and older �58 ± 8 years� nor�mal volunteers� athletes� patients with hypertension� and patientswith hypertrophic cardiomyopathy �HCM� measured by Palka etal� �� The values were determined for early ventricular ejection�systole�� during rapid ventricular �lling �early diastole� and duringatrial contraction �late diastole�� The signs of the values have beenchanged to conform with the de�nition of strain rate used in thisthesis�

Using the same method� Palka et al� �� in � � measured MVG values throughoutthe cardiac cycle in normals� athletes and patients with hypertension and with hyper�trophic cardiomyopathy �HCM�� Their results are summarized in Table �� Signi�cantdi�erences in MVG were found between the HCM patients and all the other groupsin almost all phases of the cardiac cycle� The exceptions were in early diastole withhypertension and in late diastole with athletes and young normal subjects� The au�thors concluded that MVG in early diastole was an accurate variable to discriminatebetween HCM and hypertrophy in athletes�In � �� Tsutsui et al� �� investigated the use of the same MVG method to detect

ischemic myocardium during a submaximal two�step dobutamine challenge �� and �µg/kg body weight per min� in �� patients with and � patients without con�rmedsingle�vessel coronary artery disease� Short axis gray scale and tissue Doppler imageswere obtained� and a visual interpretation of the regional wall motion was done usinga four�point scale� � normal� � hypokinesia� � akinesia and � dyskinesia� Themeasured MVG values are compared to the mean point scores in Table �� In bothanteroseptal and posterior segments the MVG values were signi�cantly increasing withthe dobutamine in the nonischemic segments� In the ischemic segments there were no

� � Strain rate imaging ��

Anteroseptal segments Posterior segmentsDobutamine Dobutamine

Baseline � � Baseline � �Point scoreNonischemic ��

Ischemic �� MVGNonischemic 2.6 ± 0.8 4.6 ± 1.2 6.0 ± 1.0 3.9 ± 0.7 6.1 ± 1.5 7.6 ± 1.8

Ischemic 2.5 ± 0.8 3.1 ± 0.7 2.7 ± 0.7 3.4 ± 1.0 3.5 ± 1.0 4.1 ± 0.9

Table �� Mean point scores and peak systolic MVG �mean ± standard devi�ation in s−1� in nonischemic and ischemic segments during a Dobu�tamine �µg/kg/min� challenge as measured by Tsutsui et al� ��

signi�cant changes in MVG during the dobutamine challenge� The point scores showednormal values throughout the test in all but one patient� which went from normal to hy�pokinesia during the dobutamine challenge� The endocardial tissue Doppler velocitieswere also compared� and showed signi�cant changes during the docutamine challengeonly for the nonischemic posterior segments at high dose� The authors concluded thatMVG may be more sensitive in detecting subtle wall motion changes than a point scoremethod and TDI�

�� Tracking�based strain rate imaging

In � �� Kanai et al� �� presented an o!ine strain rate imaging technique based onIQ cross�correlation� Initially� several points along an ultrasound beam perpendicularto the cardiac wall were manually chosen� The velocity vi(l) of point number i in framenumber l was calculated using an IQ cross�correlation technique where the phase ofthe correlation function at peak magnitude was used to estimate the velocity�

vi(l) =c

2ω0Targ(Ci(δrmax)). ��

Here� δrmax is the radial lag that maximizes |Ci(δr)|� The IQ cross�correlation functionCi(δr) is de�ned as

Ci(δr) =∫ ri+W/2

ri−W/2

x∗(r′, 1)x(r′ + δr, 2)dr′, ��

where x(r, 1) and x(r, 2) were the radial IQ signals derived from pulse numbers 1 and2� and W corresponded to the length of the pulse� The position ri of point number iin the next frame was then found as

ri(l + 1) = ri(l) + T · vi(l), ��


where T is the pulse repetition time� The local strain rate in each frame was �nallycalculated as

εi(l) =vi+1(l) − vi(l)|ri+1(l) − ri(l)| . ��

The strain rate was color coded and superimposed on a grayscale M�mode image ofthe same position� The method was limited to areas where the ultrasound beam couldbe placed perpendicular to the heart wall� and the temporal resolution in the actualsystem was according to the author not so high since temporal averaging over about� received echoes was performed�

�� Real�time Doppler strain rate imaging

The strain rate imaging �SRI� method calculates the strain rate in real�time directlyfrom the complex echo signal� This is in contrast to the MVG method described insection �� where the tissue velocities �rst are calculated using the tissue Dopplertechnique� and the velocity gradient then is calculated o!ine�Section �� gives a summary of a paper by the author that is presented as part

of this thesis and that gives an clinical introduction to the SRI technique �� Thetechnical details and more clinical examples of the method are further described in thefollowing chapters later in this thesis�

�� Strain imaging

Strain imaging by ultrasound is one of several imaging techniques in the �eld of sono�elastography� i�e�� the use of ultrasound to image tissue elastic parameters� Althoughthese are not all Doppler based methods� some are based on similar physical concepts�and will therefore be reviewed� A review of all the sonoelastography techniques wascompiled by Gao et al� in � � �� The methods can be classi�ed in two main groups�The �rst consists of methods involving a low frequency vibration that is externallyapplied� The e�ect on the imaged tissue is then used to estimate the elasticity in thetissue� The second main group consists of methods involving a static compression ofthe tissue� By comparing the ultrasound images from before and after the compression�the strain images are generated� and the strain values can be further analyzed to getother tissue elastic parameters� Strain introduced by the tissue itself� like myocardialthickening and cyclic changes in vessel wall thickness� can in principle also be measuredusing the same techniques�

�� Vibration amplitude sonoelastography

In � �� Lerner et al� �� presented a method where a low�frequency vibration �� Hz� was externally applied to the tissue� Changes in the sti�ness within thetissue caused disturbances in the vibration patterns� which was imaged using Doppler

� � Strain imaging ��

techniques� In � � Lerner et al� �� presented real�time vibration amplitude imagesusing a modi�ed color Doppler system where vibrations over a threshold were colored�A mathematical model for the vibration amplitude sonoelastography was presented

by Gao et al� in � � �� The wave �eld within the tissue was expressed as�

Φtotal = Φi + Φs, ��

where Φi is the incident �eld and Φs the �eld scattered by the inhomogeneities in thetissue� These �elds satis�ed the wave equations

(∇2 + k)Φi = 0 ��

(∇2 + k)Φs = α(r) ��

where α(r) were the inhomogeneity properties� Using a sonoelastic Born approxima�tion� these wave equations could be solved� and the vibration �eld for a inhomogeneitycould be calculated�

�� Strain imaging by speckle tracking

In � � � Meunier et al� �� described a method to compute local myocardial defor�mation from frame�to�frame changes in speckle patterns� In this method� a speckletracking algorithm �� was used to compute the frame�to�frame �D velocity �eld(U(x, y), V (x, y)) for a small region of interest� This velocity �eld was then decomposedinto a translation �T�� a rotation �R� and a deformation �D� �eld�[

UV

]= T + (R + D)

[xy

]. ��

The deformation �eld is in fact the strain �eld� which in this method was furtherdecomposed into the two main directions� perpendicular and parallel to the imagedmuscle segment�In � �� Meunier �� expanded the method to three spatial dimensions� He de�

scribed the deformation through the transformation� x′

y′

z′

= M−1

x

yz

− T

, ��

where T is the translation �eld and M is the combined rotation and deformation �eld�and then calculated the theoretical limitations of the method� To do this� �D grayscale images were simulated as the magnitude of the result of convolving an RF pointspread function with a tissue object function� Then the cross�correlation coe�cientbetween small volumes of interest in a simulated �D image transformed using ��and the corresponding �D image simulated after performing the same transformationon the tissue object function was calculated� The theoretical results showed that lower


ultrasound frequencies and smaller point spread functions were more desirable for aspeckle tracking methodology� i�e�� higher degrees of rotation and deformation couldbe allowed without decorrelation� However� these two aims are in con�ict� When theultrasound frequency is reduced� the point spread function becomes larger and viceversa�In � � Maurice and Bertrand �� studied the speckle�motion artifacts �SMA�

arising from tissue shearing� Rotation had previously been shown to result in lateralspeckle motion �� and Maurice and Bertrand showed that shear deformationsalso produced such artifacts�

�� Compression RF strain imaging

In � �� Ophir et al� �� published a method they called elastography� In this method�the local strain introduced by a small external compression �usually less than � "� wasfound by comparing the pre� and post�compression RF echo signals� The authors alsoestimated the stress �eld in the tissue from stress measurements close to the transducersurface� and could then calculate the elastic modulus pro�le in the tissue�The strain was estimated in the following manner� The RF echo signals were broken

into K small� and possibly overlapping� segments of length T � Segments number mfrom the pre� and postcompression RF signals� s1(t) and s2(t) respectively� were thencross�correlated� The temporal location of the peak of the cross�correlation functionwas then used to estimate the time shift� or corresponding displacement d(m)� of thesegment�

d(m) =c

2argmax

∆t(C(∆t, m)), ��

where argmax indicates an operation that returns the time shift ∆t that maximizesthe cross�correlation function

C(∆t, m) =1T

∫ (m+1)T

mT

s1(τ)s2(∆t + τ)dτ. ��

In the actual implementation� the integral in �� was replaced by a discrete sum�and the peak in the correlation function was found using an interpolation algorithm�The resulting displacements d(m)� m = 0 . . .M − 1� were then used to estimate thestrain between two spatially neighboring segments�

ε(m) =d(m) − d(m − 1)

rs, ��

where rs is the spatial distance between two neighboring segments�The Cram�r�Rao lower bound variance for the strain estimator was in � � ex�

pressed by C�spedes et al� �� as

σ2CRε =

σ2CRd(m) + σ2

CRd(m−1) − 2cov(d(m), d(m − 1))

r2s

, ��


where σ2CRd(m) is the lower bound variance of the displacement estimate� and cov is

the covariance function� By assuming stationarity of the echo signal� the variance ofthe displacement estimates was constant� i�e�� σ2

CRd(m−1) = σ2CRd(m) = σ2

CRd� Thisvariance is found from the lower bound variance of the time delay estimate� which hasbeen presented by several authors� One commonly used expression is ��

σ2CRd =

c2

4σ2

CRτ ≈ c2

41

4π2f20 BTwSNR

, ��

where f0 is the center frequency of the ultraound pulse� B is the signal bandwidth�Tw is the temporal window size used in the estimation of the time delay� and SNR isthe signal�to�noise ratio� It was assumed a wide�band band�pass signal with rectan�gular spectrum� high signal�to�noise ratio �SNR� 1� and uncorrelated additive whiteGaussian noise�Another expression was presented in � � by Walker and Trahey � ��

σ2CRd =

c2

4σ2

CRτ ≈ c2

43

2π2Tw(B3 + 12f20B)

(1ρ2

(1 +

1SNR

)2

− 1

). ��

Here� the e�ect of partly correlated signal segments was taken into account throughthe correlation coe�cient ρ�C�spedes et al� introduced the following limit for the covariance�

cov(d(m), d(m − 1)) ≤ 4c2

σ2CRd

(1 − 2rs

cTw

)for rs ≤ cTw

2

cov(d(m), d(m − 1)) = 0 for rs >cTw

2��

The corresponding Cram�r�Rao lower bound variances for the strain estimator thenbecome�

σ2CRε ≥ c

4π2f20 BT 2

wrsSNRfor rs ≤ cTw

2

σ2CRε ≥ c2

8π2f20 BTwr2

sSNRfor rs >

cTw

2��

using �� and

σ2CRε ≥ 3c

2π2T 2wrs(B3 + 12f2

0B)

(1ρ2

(1 +

1SNR

)2

− 1

)for rs ≤ cTw

2

σ2CRε ≥ 3c2

4π2Twr2s(B3 + 12f2

0B)

(1ρ2

(1 +

1SNR

)2

− 1

)for rs >

cTw

2

��

using ��


In � � Varghese and Ophir �� presented a modi�ed lower bound variance thatwas divided into three regions� This combined bound was referred to as the Ziv�Zakailower bound or the so�called Strain Filter� The three regions were de�ned by the levelof signal�to�noise ratio of the computed strain image �SNRC�� This ratio depends bothon the SNR in the echo signals and the degree of correlation between the pre� and post�compression signal segments� Since the SNRC decreases when the strain increases� theregions correspond to regions in strain value also� For low strains� the variance wasgiven by the Cram�r�Rao bound in �� At strains so large that there are ambiguitiesin the signal phase� but where the strain still can be estimated by correlating only thesignal envelope� the variance was given by the so�called Barankin bound�

σ2BBε = 12

(f0

B

)2

σ2CRε. ��

Finally� at high strains the variance was expressed as a value independent of the SNRC �

σ2CV ε =

s2Tc

12rs. ��

Combining these three bounds� the so�called strain �lter or Ziv�Zakai lower boundbecame�

σ2ZZε ≥

σ2CRε for high SNRC or low strain

σ2BBε for medium SNRC or medium strain

σ2CV ε for low SNRC or high strain

. ��

�� Compression IQ strain imaging

Another strain imaging technique based on the phase information of the signal and amultiple step compression was presented by O�Donnell et al� �� in � �� Here� a step�wise compression was performed on the imaged object� and for each step the recordedRF signals were quadrature demodulated� resulting in the complex signal segmentsx(t, n) for the precompression and x(t, n + 1) for the postcompression� Incrementaldisplacements between neighboring segments were estimated as�

∆dn(m) =c [arg(Cn(0, m)) − arg(Cn(0, m − 1))]

2ω0, ��

where c is the speed of sound� ω0 is the angular center frequency of the ultrasoundpulse and Cn(0, m) is the base�band cross�correlation function

Cn(t, m) =1T

∫ (m+1)T

mT

x(τ, n)x∗(t + τ, n + 1)dτ, ��

evaluated at lag zero� Displacements greater than λ/4 will give aliasing of the argumentfunction� but by implementing the phase di�erence in �� as shown in Figure ��the aliasing will only occur if the di�erential displacement is greater than λ/4�


x � LPF �y

yny∗n−1

� LPF �C Cn(0, m)·C∗

n(0, m − 1)� LPF† � �

� iX

c2ω0

��∆dn(m)

Figure �� Computation of the incremental displacement� as described byO�Donnell et al� � �� LPF means low pass �ltering� n is the tem�poral �frame�to�frame� coordinate� and m is the spatial coordinatealong the beam� †The last low pass �lter was only used in the dif�ferential procedure�

Direct procedure

From this incremental displacement� the strain was calculated in two di�erent ways�In the �rst� so�called direct procedure� the displacement at depth m for compressionstep number n was found by the summation

dn(m) =m∑

i=m0

∆dn(i), ��

where i = m0 is a stationary point� for instance at the transducer surface� The totaldisplacement after N compression steps was then found by the sum

d(m) =N∑

n=0

dn(m(n)), ��

where m(n) indicates that the position had to be adjusted from step to step in thecompression� Finally� the strain was estimated as

ε(m) =d(m) − d(m − 1)

rs, ��

where rs is the spatial distance between two neighboring segments� Assuming smallstrains in each compression step� the variance of this estimator is�

σ2ε =

[c2M

2ω20r

2sSNR

], ��

where SNR is the average signal�to�noise ratio for the chosen location over the Ncompression steps�

Di�erential procedure

In the second� so�called di�erential procedure� the di�erential strain for each step wascalculated as�

∆εn(m) =∆dn(m)

rs, ��


where ∆dn(m) is de�ned in �� and rs is the spatial distance between two neigh�boring segments� The total strain for depth m was then calculated as�

εm =N∑

n=0

∆εn(m(n)), ��

where m(n) indicates that the position had to be adjusted from step to step in thecompression� Assuming small strains in each compression step� the variance of thisestimator is the same as in ��

�� Spectral strain imaging

In � � Talhami et al� �� presented a spectral strain imaging technique� They modeledthe RF signal as a convolution of a random valued object function g(r) and a pulsefunction p(r)�

s0(r) = g(r) ⊗ p(r). ��

The corresponding Fourier transform then became�

S0(k) = G(k)P (k), ��

where k is the spatial frequency� By compressing the imaged object with a strainε� the object function became g(εr)� Using the Fourier scaling property� the Fouriertransform of the RF signal could be written�

S(k) =1εG(

k

ε)P (k). ��

As seen� the object function is frequency scaled with 1/ε� Now� since the multiplicationwith P (k) could be considered as a band pass �lter� the strain could be estimated bycomparing the spectrum S(k) with the one obtained before the compression� S0(k)�The strain in an M�mode RF signal s(r, n)� where r is the spatial coordinate and n

is the temporal pulse to pulse coordinate� was estimated by Talhami et al� as�

ε =σk

µk

, ��

where k(n) is a peak spatial frequency estimator over a de�ned bandwidth K�

k(n) = argmaxk∈K

(|S(k)|), ��

and µk and σk its mean and standard deviation respectively over a short time windown = 1, . . . , N �The authors proposed the following understanding of this estimator� If the object

function was assumed to be a random but equi�distant train of delta pulses� simulating


Strain �"� �� Strain error �"�� T = 4µs �� Strain error �"�� T = 2µs ��

Table �� Standard deviation of the estimated strain �strain error� as a func�tion of the strain� when using RF correlation with segment lengths4 and 2µs on simulated RF data � ��

the echo from an object with scatterers regularly spaced with a distanceD� the resultingspectrum |S(k)| would have peaks spaced at ∆k = 1/D� The peak frequency estimatorin �� where K was centered on the �rst peak� would then measure the dominantscatterer spacing� The mean µk would thus estimate the average scatterer spacing overtime and the standard deviation σk would estimate the average change in scattererspacing� Dividing these then gave the strain estimate in ��

�� Decorrelation

An important quality limiting factor in strain imaging techniques is the decorrelationcaused by the compression of the imaged object� Because of the compression� thepost�compression signal segments will also be compressed� and thus will not matchperfectly with the pre�compression signal segments� Lateral motion will also reducethe correlation� Using simulated RF data� O�Donnell et al� �� calculated the standarddeviation of the strain estimate in �� through �� Their results are given intable �� As seen� the strain error at large strains is reduced when a shorter windowis used� At low strains� the decorrelation is less of a problem� and a longer windowyields a better result�Several techniques to reduce the decorrelation have been proposed� including am�

plitude compression� multicompression� signal stretching and lateral correction� andthese will be reviewed in the following sections�

Amplitude compression

In � �� C�spedes and Ophir �� proposed an amplitude compression technique toreduce the decorrelation noise� The idea of this technique is that the values of theRF cross�correlation function will depend on the signal amplitude� Since the pre� andpostcompression signal segments will never match perfectly� a stochastic variations inthe signal amplitude might cause a shift of the peak in the cross�correlation function�In e�ect� this might modulate the strain estimates by the random signal amplitudevariations� and cause artifacts in the strain images�To reduce this artifact� C�spedes and Ophir suggested performing a logarithmic

compression of the RF signals prior to calculating the cross�correlation� An experi�ment was performed using simulated data� The e�ect on the strain image quality wasmeasured by the mean�to�standard deviation ratio �MSDR��

MSDR = µS/σS , ��

� Previous work

No compression Log compression Temporal stretchingMSDR ��

Table �� Average mean�to�standard deviation ratio �MSDR� values of sim�ulated strain images using logarithmic compression or temporalstretching ��

where µS and σS are respectively the mean and standard deviation of the strain in aregion of uniform elasticity� The results are shown in table �� As seen the MSDRis almost doubled when logarithmic compression is used� indicating that the standarddeviation is almost halved�No references to this method have been found in any of the later publications on

elastography�

Multicompression

In � �� Varghese � � �� used a multiple step compression technique to reduce thedecorrelation� The multicompression technique was combined with the signal stretchingtechnique described in the following section� The technique was similar to O�Donnellstechnique described in Section �� only used for RF signals� The idea of the tech�nique was that the decorrelation would be reduced� since the applied strain εs for eachcompression step was small� The �nal strain estimate εf was in this work found byaveraging the strain estimates εn for all the compression steps�

εf =1N

N∑n=1

εn. ��

The total strain εt could also be found by accumulating all the strain estimates�

εt =N∑

n=1

εn. ��

Assuming that the strain estimates for each step were uncorrelated� the standard de�viation of the �nal or total strain estimate could be reduced by

√N � where N is the

number of compression steps�

σ2εf

= σ2εt

=1N

σ2εn

. ��

Signal stretching

In � �� C�spedes and Ophir �� also proposed a signal stretching method to reducethe decorrelation noise� When the imaged object is compressed� the scatterers movecloser to each other� For small strains this was modeled as a temporal compression ofthe RF signal� By stretching the postcompression signal by the same factor� the RFcross�correlation will be improved�


C�spedes and Ophir used a linear interpolation algorithm to stretch the post�compression signal� The stretch factor was chosen equal to the known compressionstrain� The e�ect on the strain image quality was measured by the MSDR as de�nedin �� and the result is shown in the last column in table �� As seen� the MSDRis almost doubled when temporal stretching is used� indicating that the standard de�viation is almost halved�In � �� Varghese and Ophir �� presented a theoretical description of this method�

The RF signal was described as a convolution of the object function g(r) and the pointspread function p(x)� in addition to uncorrelated random noise n1(r)�

s1(r) = g(r) ⊗ p(r) + n1(r). ��

If a strain ε was imposed on the object� the measured RF signal would be�

s2(r) = g(r

1 + ε− r0) ⊗ p(r) + n2(r). ��

The cross�correlation coe�cient between these two signals was termed ρ12� The post�compression RF signal was then stretched by the inverse factor (1 + ε)�

s3(r) = s2((1 + ε)r) = g(r − r0) ⊗ p((1 + ε)r) + n3(r). ��

The cross�correlation function between this signal and the pre�compression signal wastermed ρ13� By modeling the point spread function as a Gaussian modulated cosinepulse� the following simple relation between the two cross�correlation coe�cients wasshown�

ρ13 =ρ12

1 + ε. ��

Since ε ≤ 0 in any compression �in elastography values in the range −0.01 < ε ≤ 0 areused �� ρ13 will always be larger than ρ12� meaning that the stretching operationgenerally improves the correlation�Also in � �� Alam and Ophir �� showed that the e�ect of the stretching tech�

nique could be described as a low�pass �ltering of the pre�compression autocorrelationfunction R11(r) � The expected value of the cross�correlation estimate was written�

E{C13(δr)} = R11(δr) ⊗ h(δr) ⊗ δ(δr − r0), ��

where h(r) is the impulse response of the band pass �lter and r0 is the requiredshift to get peak cross�correlation� The bandwidth of the low�pass �lter was givenby the applied strain� the ultrasound frequency and the pulse length� For small strainsthe �lter approached a Dirac delta pulse indicating that the stretching technique ap�proached complete restoration of the correlation� Still the pre�compression signal andthe stretched signal were not equal due to the di�erent impulse responses p(r) andp((1 + ε)r) embedded in them� as seen in �� and �� The authors proposedusing inverse �ltering on the stretched signal to get a further increase in correlation�In the Fourier domain this was described as�

R3(k)H(k) =1

1 + εR1(k)e−j4πkr0/c + N3(k)H(k), ��


where the �lter H(k) is given by�

H(k) =

{P (k)

P (k/(1+ε)) , G3(k) ≥ N3(k) and P (k/(1 + ε)) �= 00 otherwise.

��

Here R(k)� N(k)� P (k) and G(k) were the spatial Fourier transforms of R(r)� n(r)�p(r) and g(r) respectively�In � �� Alam et al� �� described an adaptive strain estimator where the temporal

stretching of the post�compression signal was adapted using a binary search method�This adaptiveness was needed since the proper stretching is given by the local strain� anunknown and both spatially and temporally varying parameter� Using a global uniformstretching as in �� would therefore be suboptimal� The correct local stretching factorwill maximize the correlation� so this factor could be found by performing a searchover several stretching factors until the peak correlation was found�

ε = argmaxe

ρ13(e), ��

where argmax is a binary search that returns the lag e that maximizes the cross�correlation coe�cient

ρ13(e) =

∫∞−∞ [g(r) ⊗ p(r)]

[g(1+ε

1+er) ⊗ p(r)]dr√∫∞

−∞ [g(r) ⊗ p(r)]2 dr∫∞−∞

[g(1+ε

1+er) ⊗ p(r)]2

dr

. ��

Here g(r) and p(r) are de�ned in �� The stretching of the point spread functionwas ignored� i�e�� p((1 + ε)r) ≈ p(r) was assumed�

Lateral correction

In � �� Konofagou and Ophir �� presented a method to correct the radial strainmeasurements for the lateral displacements caused by the compression� These dis�placements cause a reduction in the correlation between the pre�compression and post�compression signals�The lateral displacement was found by �rst laterally interpolating the stretched

post�compression signals and then searching among the interpolated signals to �ndthe best match� Using this match as the post�compression signal� an increase in thecorrelation was observed�Furthermore� knowing the lateral displacements� the authors were able to calculate

the lateral strain� and could by dividing the lateral strain by the radial strain map thelocal Poisson�s ratio of the tissue�

ν = − εl

εr. ��


�� Dynamic range

In � �� Konofagou et al� �� used variable applied strain to expand the dynamic rangeof the strain measurements� For a certain degree of compression only a limited rangeof strain values can be found using RF elastography� This range is given by the �band�in the strain �lter described in section ��A multiple step compression was performed� but for each step the strain was calcu�

lated using the baseline RF data as reference rather than the data from the previousstep� Thus� elastograms produced by di�erent amounts of strain were gathered� Theseelastograms were �rst normalized by scaling them with the applied strain� then in eachelastogram the regions with strains within the strain �lter band were chosen� while therest were disregarded as noise� Finally a composite elastogram was built up by thechosen regions in each step�elastogram�

Chapter �

One�dimensional strain rate

estimation

This chapter describes techniques to estimate the strain rate in one dimension� that is�along the ultrasound beam� In section �� a maximum likelihood strain rate estimatoris developed� while section �� describes a real�time implementation of a strain rateestimator�Section �� was written in collaboration with my supervisor� professor Hans Torp�

�� Optimal strain rate estimator

In order to �nd the optimal strain rate estimator� a statistical model for the IQ�signalis developed in this section� The signal model is general for any packet size N � but onlythe setting with N = 2 is considered in the rest of the section� A likelihood function isfound from the statistical model� and a maximum likelihood estimator is derived� Thelower bound variance is found and compared to the variance presented in the strainimaging �eld �see Section �� The maximum likelihood estimator is also compared tothe linear regression estimator used in the Myocardial Velocity Gradient method �seeSection �� Finally a closed form expression for the estimator is presented�

�� Three component Gaussian signal model

Consider the complex demodulated IQ signal x(m, n), m = 1 . . . M, n = 1 . . . N �consisting of a total of MN samples from M radial positions along each of N con�secutive beams in the same direction� This signal is reorganized in a complex signalvector

x = [x(1, 1), . . . , x(1, N), x(2, 1), . . . , x(2, N), . . . , x(M, 1), . . . , x(M, N)]T . ��

��

�� One�dimensional strain rate estimation

where T is the transpose operation� This signal vector x is modeled as a zero meancomplex Gaussian process� with a probability density function described by the covari�ance matrix C�

px(x) =1

πMN |C|e−x∗TC−1x ��

where C = E{xx∗T } can be written out as�

C = {cij}i=1...MN,j=1...MN

cij = E{x(m(i), n(i))x∗(m(j), n(j))}m(i) =

⌈i

N

⌉n(i) = [(i − 1)mod N ] + 1

and |C| indicates the determinant of C� ∗ is complex conjugation� E{·} is the expectedvalue operator� �·� is the upwards rounding to the nearest integer� and mod is themodulus operator� The covariance matrix will have a block structure� with M2 blocksof square N by N matrices�To get a simpler expression for the probability density function� only the situation

where there is no correlation between the radial samples is considered� This correspondsto the situation where the radial sampling distance is longer than the radial length ofthe point spread function� which is approximately equal to the pulse length� Thevector x(m) = [x(m, 1), . . . , x(m, N)]T � i�e�� the samples from one range gate� can bede�ned� By assuming that the signal vectors x(m)� m = 1 . . .M � are uncorrelated�the probability density function �pdf� for the long vector x = [x(1), . . . ,x(M)]T canbe found as the product of the pdf for each x(m) �given by setting M = 1 in ��and the logarithmic pdf becomes�

ln px(x) = −MN ln π −M∑

m=1

ln |C(m)| −M∑

m=1

x(m)∗T C(m)−1x(m) ��

where C(m) = E{x(m)x(m)∗T } and ln is the natural logarithm�A three component model for the Doppler signal is used here� a tissue signal

xt(m, n) with varying radial velocity v(m)� a stationary clutter signal xc(m, n)� andwhite noise xn(m, n)� This model is similar to the one used in Paper �� only with di�er�ent components� The three components are modeled as independent complex Gaussianstationary processes� The covariance matrices from each of the signal components areadded to form the total covariance matrix C(m)�

C(m) = σ2t Ct(m) + σ2

cCc(m) + σ2nIN ��

� � Optimal strain rate estimator ��

where

Ct = {ρt(j − i)}i,j =

1 ρt(1)∗ . . . ρt(N − 1)∗

ρt(1) 1 ρt(1)∗

ρt(1) 1��

��

ρt(N − 1) . . . 1

Cc = {ρc(j − i)}i,j

and the correlation coe�cients ρt(l) and ρc(l) are de�ned as

ρt(l) =E{xt(m(i), n(i))x∗

t (m(i − l), n(i − l))}E{|xt(m(i), n(i))|2} ��

and

ρc(l) =E{xc(m(i), n(i))x∗

c(m(i − l), n(i − l))}E{|xc(m(i), n(i))|2} ��

The reverberation signal is often a low�pass process� and one can assume that it istemporally constant� resulting in ρc(l) = 1 for all l� Reverberations from moving tissuewill on the other hand give a reverberation signal with an apparent velocity twice thevelocity of the tissue�

�� Logarithmic likelihood function for N = 2

In the following� the packet size will be assumed to be N = 2� In this situation it isimpossible to reject the reverberation signal� so it will also be assumed that σc = 0�The expected signal amplitude σt is in general a function of m� whereas the white noiseamplitude is constant� and is set to �� The covariance matrix will then take on theform�

C(m) = σ2(m)[

1 ρ(m)∗

ρ(m) 1

]��

where

σ2(m) = σ2t (m) + 1

and the correlation coe�cient is modeled as

ρ(m) =σ2

t (m)β(m)σ2

t (m) + 1ei

4πf0T

c v(m) ��

The real and positive factor β(m) = |ρt(m)| accounts for the decorrelation in thetissue signal� which is caused by transit time e�ect or velocity gradients� and v(m) is


the radial velocity component in radial position m along the beam� In the blood cavity�the tissue signal amplitude is zero� and there is only white noise�The logarithmic pdf for the signal� given the unknown parameters σt(m)� β(m) and

v(m) has the general form�

ln px([x(1), . . . ,x(M)]|σt, β, v) = −2M ln π −M∑

m=1

ln(σ4(m)

(1 − |ρ(m)|2))

−M∑

m=1

2σ2(m) (1 − |ρ(m)|2)

(P (m) − Re

{ρ(m)∗R(m)

})��

where

P (m) =12(|x(m, 1)|2 + |x(m, 2)|2) ��

and

R(m) = x(m, 1)∗x(m, 2) ��

�� Maximum likelihood estimate

If no restrictions are made for the unknown parameters� maximum likelihood �ML�estimates for β(m)� σt(m)� and v(m) can be found independently for each radial pointm� as the maximum point of �� The tissue signal correlation magnitude β(m) mayin general depend on the velocity� but this dependence is in most cases slow comparedto the phase shift of ρ(m)� Then� as seen by inspecting �� maximum likelihood isattained when �ρ = �R� so the maximum likelihood estimator for the velocity v(m)can be found independent of the unknown parameters β(m) and σt(m) as�

vML(m) = a�R(m) ��

where

a =c

4πf0T��

This estimator is identical to the well known autocorrelation method �� Note thatthis ML estimator also applies in a situation with several ultrasonic beams� If thebeams have no overlap� and there is no variation in the signal parameters transversalto the beam direction� the ML estimate of v(m) will have the same form as ��where the estimate of R is averaged over the di�erent beams�When the number of radial points M = 2� the ML estimate for {v(1), v(2)} will

also give a ML velocity gradient estimator

ˆεML =vML(1) − vML(2)

∆r. ��


where ∆r is the spatial distance between two samples in the radial direction� Thevelocity is here de�ned positive towards the transducer� while the index m increasesaway from the transducer� If M > 2� a linear model for the velocity v(r) might beinserted in �� to give a likelihood function for the strain rate� To do this� the tissuesignal amplitude� σt and correlation magnitude |ρ| are assumed to be constant for allthe M radial ranges are considered� The logarithmic pdf can then be written�

ln px(x |v(m), σ, |ρ| ) = −M ln(π2σ4

(1 − |ρ|2))

− 2M

σ2 (1 − |ρ|2)

(P − Re

{1M

M∑m=1

ρ(m)∗R(m)

})��

where R(m) is given in �� and

P =1

2M

M∑m=1

(|x(m, 1)|2 + |x(m, 2)|2)

The linear velocity pro�le is given by the start velocity v1� and the strain rate ε� as

v(m) = v1 − (m − 1)εrs, m = 1, . . . , M, ��

where rs is the radial sampling distance� When �� is combined with �� and�� the likelihood function for {v1, ε} takes on the simple form�

ln px(x |v1, ε) = −M ln(π2σ2b

)− 2M

bP +

2b|ρ|Re

{e−i

v1a GR(−rs

aε)}

��

where a is given in �� and

b = σ2(1 − |ρ|2)

GR(ω) =M∑

m=1

R(m)e−i(m−1)ω

Maximum of �� occurs when the real value function in the last term is maximized�If ε is chosen to maximize the magnitude of the Fourier transform GR(ω)� the startvelocity v1 can be chosen so the phase angle of the product in the real value functionbecomes zero� The ML estimate of {v1, ε} is thus�

v1ML = a�GR(ωmax) ��

ˆεML = − a

rsωmax ��

where a is given in �� and

ωmax = argmaxω

|GR(ω)| ��

� One�dimensional strain rate estimation

�� Cramr�Rao bound

The Cram�r�Rao lower variance bounds for the velocity and the strain rate whenN = 2 are now found by inverting the Fisher information matrix �� p�� as shownin Appendix B�

var(v1) ≥ a2(1 − |ρ|2)|ρ|2

2M − 1M(M + 1)

��

and

var(ˆε) ≥ a2

(rs)2(1 − |ρ|2)

|ρ|26

M(M2 − 1)��

In the following� the radial distance between the sampling points rs is chosen to beequal to the pulse length ∆� the smallest value for which there is no correlation betweenadjacent signals� From �� one can see that the correlation magnitude is given bythe signal amplitude and the decorrelation factor β� In � �� Torp �� presented aparametric model for the decorrelation factor that was given by the signal to noiseratio� the pulse length ∆� and the beam opening angle Θ�

β = e−32 ( vrT

∆ )2− 32 ( vtT

Θ )2

��

where vr was the radial velocity component� and vt was the transversal component�which for simplicity will be chosen equal to zero for the rest of this chapter� As seenfrom �� the correlation magnitude can be written

|ρ| =1

1 + σ−2t

β. ��

The Cram�r�Rao bound in �� can be rewritten

var(ˆε) ≥(

12πNbT

)2 (1 − |ρ|2)|ρ|2

6M(M2 − 1)

��

where

Nb =2f0∆

c��

is the number of half periods used in the ultrasound pulse� Figure �� shows a plotof the Cram�r�Rao bound as a function of PRF found using �� and ��The transversal velocity component was set to zero and the radial velocity componentwas chosen to � cms in this example� Since the noise power was set to 1� the signalto noise ratio is SNR = σ2

t �For high SNR �and PRF � vr/∆� the Cram�r�Rao bound approximates to

var(ˆε) ≥( √

3vr

πN2b λ

)26

M(M2 − 1)≈ 9λ

4π2NbL3v2

r ��


0 100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PRF [Hz]

Str

ain

rate

SD

[s−

1 ]

SNR=5dB

SNR=30dB

Figure �� Lower Cram�r�Rao bound on the strain rate standard deviation�SD� as a function of the pulse repetition frequency �PRF�� Thefollowing values were used in this example� c = 1540 m�s� f0 =4 MHz� Nb = 3� vr = 6 cm�s� vt = 0 cm�s� M∆ = 10 mm� Curvesfor SNR � � �� and �� dB are shown�


where the last approximation applies when M2 � 1� The constant L is the spatial sizeof the sample vector�

L =MNbλ

2��

From �� one can see that for high signal to noise ratios the Cram�r�Rao bound isindependent of the PRF �assuming PRF � vr/∆� and inversely proportional to thecenter frequency f0 and the pulse length ∆� as long as the radial resolution �given byNb� is kept constant� The variance decreases with the third power of the total radialsample volume length L�

�� Comparing to previous work

In Section �� the Cram�r�Rao lower bound for estimating the small strain ε using theelastography technique was presented� To compare this with the Cram�r�Rao boundfor strain rate presented above� a relation between the small strain and the strain ratemust be de�ned� Since only two pulses were used in the elastography method� therelation is given as

ε =ε

T, ��

where T is the time between the two pulses� The variance of the strain rate estimatorcan then be related to the variance of the strain estimator as

var(ε) ≈ 1T 2var(ε). ��

The Cram�r�Rao bounds presented for the strain estimates in Section �� can then becompared to the Cram�r�Rao bound for the strain rate estimator� Setting the numberof radial samples to M = 2 �� is rewritten�

var(ε) ≥ c2

16π2f20 T 2r2

s

(|ρ|−2 − 1). ��

Combining with �� and �� and writing σ−2t = σ2

n/σ2t = 1/SNR the following

strain variance is found�

var(ε) ≈ c2

16π2f20 r2

s

(1β2

(1 +

1SNR

)2

− 1

). ��

Comparing this with the strain Cram�r�Rao bound in �� one can see that theyare equal if rs > cTw

2 � i�e�� the samples are separated by at least one pulse length� ifTw = 1/B and if B2 12f2

0 � i�e�� the pulse is longer than 1/√

12 ≈ 0.3 periods� Noticethat the correlation coe�cient magnitude β is written ρ in �� When the samplesare closer than one pulse length� as they can be in the elastography technique� theCram�r�Rao bound in �� becomes smaller than the one in ��


�� Comparing to linear regression

The maximum likelihood �ML� strain rate estimator can be compared to the one ob�tained by linear regression applied to the ML velocity estimates from individual ranges�{vML(m), m = 1, . . . , M}� The ML estimate of v(m) is given by the phase angle ofR(m) as shown in �� in the previous section� This means the autocorrelationestimate can be written

R(m) = |R(m)|ei 1a vML . ��

Using the model in �� for the velocity� the phase angle error of the autocorrelationestimate is

∆θ =vML(m) − v1 + (m − 1)εrs

a. ��

Assuming that the ML estimates of v(m) are close to the linear model� i�e�� small errors|∆θ| 1� the exponential function of the phase angle error can be approximated tothe second order Taylor expansion�

Re{e−i∆θ

} ≈ 1 − 12(∆θ)2. ��

By inserting this in �� one gets the following approximate form of the logarithmiclikelihood function

ln px(x |v1, ε ) = c1 +2b|ρ|Re

{M∑

m=1

|R(m)|e−i∆θ

}

≈ c1 +2b|ρ|

M∑m=1

|R(m)|(

1 − 12(∆θ)2

)

= c2 +|ρ|a2b

M∑m=1

|R(m)| (vML(m) − v1 + (m − 1)εrs)2 ��

where c1 and c2 are constants� The linear mean square error �MSE� functional for vML

has the form

MSE =M∑

m=1

(vML(m) − v1 + (m − 1)εrs)2. ��

By comparing �� and �� one can recognize that the ML estimate of {v1, ε} isa linear weighted mean square �t to the model� where the weights are the magnitudeof the corresponding correlation estimates� By inspecting the joint probability of thephase and magnitude of the autocorrelation estimator �see Figure �� one can observethat the phase angle has decreasing variance for increasing magnitude of R� This meansthat the likelihood function in �� gives increased weight to the velocity estimates


0 1 2 3 4 5

x 104

−3

−2

−1

0

1

2

3

Magnitude of R estimate

Pha

se a

ngle

of R

est

imat

e

Figure �� Scatter plot of the phase angle versus the magnitude of R� Thefollowing values were used in this simulation� c = 1540 m�s� f0 =4 MHz� Nb = 3� vr = 6 cm�s� SNR = 40 dB� PRF = 300 Hz

which have lowest variance� In contrast� the linear regression method in �� givesuniform weight to all the velocity estimates�

Figure �� was generated by simulating the complex data vector in �� for M = 1and N = 2 and calculating the autocorrelation as in �� This was repeated �times to generate the scatter plot�

The two estimation methods have been compared by computer simulations� Signalsincluding noise were generated� with a velocity gradient of 1.0 msm� The velocity ineach depth range was estimated� and the regression line was found by the two methods�A typical outcome is shown in Figure �� Note that the two outlayer points give alarge error in the linear regression line� while the e�ect on the weighted regression lineis much less since the weights associated with these points are lower�

In Figure �� strain rates estimated by the two methods are compared for �independent simulations� showing less variance for the ML method�


0 0.005 0.01 0.0150

0.005

0.01

0.015

0.02

0.025

Depth [m]

Est

imat

ed v

eloc

ity [m

/s]

Figure �� Linear regression �t �dashed line� and weighted linear regression �t�solid line� to simulated velocity estimates �circles�� The followingvalues were used in this simulation� c = 1540 m�s� f0 = 4 MHz�Nb = 3� SNR = 20 dB� PRF = 300 Hz� v1 = 1.0 cm�s� ε =1.0 s−1


0 5 10 15 20 25 30 35 40 45 500.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Simulation number

Est

imat

ed s

trai

n ra

te [s

−1 ]

Figure �� Strain rates found by linear regression �stars� and weighted linearregression �circles� in � independent simulations� The followingvalues were used in the simulations� c = 1540 m�s� f0 = 4 MHz�Nb = 3� SNR = 20 dB� PRF = 300 Hz� v1 = 1.0 cm�s� ε =1.0 s−1


�� A closed form approximation

In Section �� it was shown that the maximum likelihood �ML� estimate of thestrain rate was found by maximizing the magnitude of the Fourier transform of theautocorrelation estimate� An alternative form of the likelihood function will now bedeveloped from the magnitude square of the Fourier transform�

∣∣∣GR(ω)∣∣∣2 = MS(0) + 2MRe

{M−1∑m=1

(1 − m

M)S(m)e−imω

}��

where

S(m) =1

M − m

M−m∑k=1

R(k)∗R(k + m) ��

By doing this� the number of statistics is reduced by one� since S(0) is independentof the unknown strain rate� Note that the expected values of the strain correlation

estimates S(m), m = 1, . . . , M − 1 have the following properties�

E{

S(m)}

=1

M − m

M−m∑k=1

E{R(k)∗R(k + m)

}

=1

M − m

M−m∑k=1

E{R(k)∗

}E{

R(k + m)}

= |R|2eimωs ��

where

ωs ≡ −4πf0Trs

cε. ��

The strain correlation estimates can also be written�

S(m) = |S(m)|eimωs(m) ��

where

ωs(m) ≡ 1m�S(m). ��

As explained in section �� the maximum likelihood estimate for the strain rateis found from the frequency that maximizes the magnitude of GR(ω)� To �nd themaximum� �� is di�erentiated� By combining with �� this can be written�

∂

∂ω

∣∣∣GR(ω)∣∣∣2 = −2M

M−1∑m=1

m(1 − m

M)|S(m)| sin(m(ωs(m) − ω))

≈ −2M

M−1∑m=1

m2(1 − m

M)|S(m)|(ωs(m) − ω) ��


where the last approximation is valid when the estimation error for all the phase angleestimates ωs(m) are small� By setting �� equal to zero and combining with �� one gets the following explicit form of the maximum likelihood estimator for strainrate�

ˆεML = − c

4πf0Trs

∑M−1m=1 am|S(m)|ωs(m)∑M−1

m=1 am|S(m)| ��

where

am ≡ m2(1 − m

M

). ��

Note that the ML estimator in this form is a weighted average of the strain correlationangle estimates ωs(m) with radial lagsm = 1, . . . , M−1� The weights are the productsof the correlation magnitudes and lag�dependent constants am� As seen in �� thestrain correlation magnitudes all have the same expected value� and since they all arecalculated by sums of products from the set {R(1), . . . , R(m)}� they will probably notdi�er much in magnitude� The weighting function am is plotted in Figure �� andshow a peak at m/M = 2/3� which means that this lag has the highest weight in thelikelihood function� In order to save computations in a real�time implementation� itwill probably be a good idea to simplify the ML estimator by just calculating S(m)for one� or maybe a few m�values close to the peak of am�

�� Conclusions

Under certain assumptions� the maximum likelihood estimate for strain rate can befound as the peak of the Fourier transform of a vector of complex valued temporalcorrelation estimates� An analytical expression for the Cram�r�Rao bound was found�Simulation results indicate that the ML estimator is unbiased� but the criterion forestimator e�ciency has not been demonstrated� so the Cram�r�Rao bound is not nec�essarily attained by the ML estimator� Comparison with linear regression methodsshowed that the ML estimator gave higher weights to the velocity estimates with low�est variance� which explained the signi�cant improvement compared to simple linearregression�The closed form approximation to the ML estimator developed in the last section

showed a close relationship to a simpler algorithm which is suitable for real�time im�plementation� The algorithm extracts the phase shift of the autocorrelation estimatesin the radial direction by a radial correlation operation� The closed form ML estimatorexpression indicates an optimum value for the radial lag to be used in the simpli�edestimator� which should equal 2/3 of the total radial sampling volume�

�� Real�time strain rate estimator

The principles of a real�time implementation of the strain rate estimation is describedin this section� This imaging technique is for simplicity termed Strain Rate Imaging�SRI��

� � Real�time strain rate estimator ��

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

m

Wei

ghts

am

Figure �� The weight function used in the maximum likelihood estimatorfor strain rate� M = 30 in this example�

� One�dimensional strain rate estimation

Color Strain rate Apical views PLAXDark blue Large positive Rapid lengthening Rapid thickeningCyan Medium positive Slow lengthening Slow thickeningGreen Near zero No length change No thickness changeYellow Medium negative Slow shortening Slow thinningRed Large negative Rapid shortening Rapid thinning

Table �� Color map used for strain rate imaging� The interpretations whenthe ultrasound beam is along the muscle� as in the apical views� andperpendicular to the muscle� as in the parasternal long axis view�PLAX�� are described� In the parasternal short axis view �SAX�the angle between the beam and the muscle varies� and thus alsothe interpretation�

�� Basic implementation

A simpli�ed implementation of the strain rate estimation in �� involves calculatingS(m) only for the m�value that maximizes a(m) in �� m′ = 2M/3� The simpleestimator then becomes�

ˆεSRI = − c

4πf0Trs

am′ |S(m′)|ωs(m′)am′ |S(m′)|

= − c

4πf0T (m′rs)1

M − m′�(M−m′∑

k=1

R(k)∗R(k + m′))

= − c

4πf0T (2Mrs/3)1

M/3�(

M/3∑k=1

R(k)∗R(k + 2M/3)) ��

A constant o�set of 2M/3 samples �equivalent to a spatial distance of ∆r =2Mrs/3� is used in the multiplication of the correlation functions� The result is aver�aged over the remaining M/3 samples�The visualization can be performed in the same fashion as in tissue Doppler� by

coloring the gray scale image according to the estimated strain rate� In the strain rateimages presented in this thesis� the color map described in Table �� has been used�A di�erence from tissue Doppler is the smooth transition at strain rates near zero asopposed to the sharp transition from negative to positive velocities in tissue Doppler�The reason for this smooth transition is to suppress the apparent noise caused by theestimator variance�

�� High frame rate

To achieve high frame rate in color Doppler applications� two techniques are commonlyused� multi line acquisition �MLA� and interleaving� These two techniques make itpossible to acquire more data than in the basic mode� The time to acquire one frame

� � Real�time strain rate estimator ��

Figure �� Illustration of the pulse order and beam directions in the interleav�ing method for three di�erent interleave group sizes Nint� Numberof beams Nb = 8 and packet size N = 2 in this example�

of Doppler data is in the basic mode

tD0 = NbNT, ��

where Nb is the number of beams in the image� N is the number of pulses in eachdirection and T is the pulse repetition time� The relatively small extra delay relatedto the change in setup of the transmitter and beamformer between pulses is ignoredhere�In the MLA method� a broad beam is transmitted� When receiving the echo�

the signals from all the transducer elements are processed in parallel in two or morebeamformers� Each beamformer time delays the element signals di�erently to generatedi�erent receive beams� This way� two or more beams can be acquired during the timefor one pulse�echo cycle� and the frame rate can be increased correspondingly� UsingMLA� the time to acquire a frame of Doppler data is

tDMLA =Nb

NMLA

NT, ��

where NMLA is the number of beams that are processed in parallel�In the interleaving technique� the waiting time T from one pulse to the next in the

same direction is utilized to send pulses in other directions� as illustrated in Figure ��There is however a minimum waiting time T0 where no other pulses can be �red in anydirection� This is given by the time for the pulse to travel to the maximum depth andback� T0 > 2d/c� The number of directions that pulses are �red during the time T iscalled the interleave group size� Nint� This obviously has to be an integer number� andT = NintT0� Using interleaving� the time to acquire a frame of Doppler data becomes

tDint =Nb

NintNMLA

NT. ��

A typical scanning procedure for a tissue Doppler application is illustrated in Figure�� A tissue frame is �rst captured� using high beam density� The PRF used for tissueDoppler is usually so low that only one interleave group is necessary� So N Doppler


Figure �� Illustration of the scanning procedure for a tissue Doppler ap�plication� The packet size N = 3 and the interleave group sizeNint = Nb in this example� T is the pulse repetition time� TT andTD are the times needed to acquire a tissue frame and a Dopplerframe respectively� and TF is the total acquisition time for onetissue Doppler frame�

Figure �� Illustration of the scanning procedure for a high frame rate SRIapplication� The packet size N = 3 and the interleave group sizeNint = Nb in this example� T is the pulse repetition time� tT andtD are the times needed to acquire a tissue frame and a Dopplerframe respectively� and tF is the total acquisition time for one SRIframe�

subframes are captured separately� usually using fewer beams than in the tissue frame�The velocity is calculated from the N subframes� is color coded and mapped onto thetissue frame� The time to acquire a tissue Doppler frame then becomes

tF = tT +Nb

NMLA

NT, ��

where tT is the time required to acquire the tissue frame�The same scanning procedure can be used for strain rate imaging� but since even

lower PRF is optimal for SRI� as illustrated by Figure �� earlier in this Chapter� adi�erent procedure can be used� Instead of collecting a tissue frame� the number ofbeams in the Doppler subframes can be increased to allow tissue visualization basedon only these frames� As illustrated in Figure �� the Doppler frame is still generatedfrom N subframes� but a sliding window technique can be used� so the time for oneframe will be only

tFSRI = tT , ��

� � Estimating strain from strain rate ��

assuming that the time to acquire one Doppler subframe is equal to the time to acquireone tissue frame� Comparing �� and �� one can see that the time for one frameis greatly reduced and thus allowing a high frame rate�If the same method were to be tried in color �ow imaging or tissue Doppler imaging�

the low PRF would lead to a lot of aliasing and make clutter �ltering impossible� thusmaking the interpretation of the measured velocities very di�cult�

�� Second harmonic imaging

Second harmonic imaging has recently been shown by many authors to produce muchbetter image quality in gray scale images �� The technique involves usinga band�pass �lter for the second harmonic component of the received signal beforefurther processing of the data�This technique can also be used in strain rate imaging� First of all the background

gray scale image can be generated using second harmonic imaging� but also the strainrate can be estimated using the second harmonic part of the signal� When usingsecond harmonic data for Doppler applications the center frequency parameter f0 inthe estimators for velocity or strain rate must be replaced by 2f0� Except from thisthe estimators are as before�The bene�t by doing this is a strain rate estimate with less bias and smaller variance�

since there are less reverberations in the second harmonic signal� The 2f0 factor willalso double the phase shift of each of the correlation function estimates in �� andthus also the phase shift of the product� Since stationary reverberations have a phaseshift of zero� using second harmonic causes the tissue signal to be separated more fromthe stationary clutter�A drawback is that the Nyquist limit for strain rate

εNyq =c

4f0T∆r��

will be halved when 2f0 is used rather than f0� However� when using second harmonicimaging� f0 is usually reduced� So in e�ect� the Nyquist strain rate might not bea�ected very much�

�� Estimating strain from strain rate

As explained in Section �� the estimated strain rate is in fact a velocity gradient�The strain rate is equal to the temporal derivative of the strain only for in�nitesimalstrains� It will be shown that the calculation of a possible large accumulated strainafter a certain time from the strain rate data must take this into account�The relation between the �nite strain and the strain rate can be developed by way

of an example� Consider an in�nitesimal one�dimensional object of length L(t) thatexperiences a strain rate ε(t)�


Figure �� Example of an object that changes length L(t) as a function oftime�

� � Estimating strain from strain rate ��

The change in length over a small time step ∆t can then be estimated as

L(t + ∆t) − L(t) ≈ ∆tε(t)L(t). ��

Letting ∆t → 0 one gets the temporal derivative of the length�

L(t) = lim∆t→0

L(t + ∆t) − L(t)∆t

= ε(t)L(t). ��

The solution to this di�erential equation is

L(t) = L0 exp(∫ t

t0

ε(τ)dτ

)��

where L0 = L(t0)� The accumulated �nite strain is �nally found as

ε(t) =L(t) − L0

L0= exp

(∫ t

t0

ε(τ)dτ

)− 1. ��

Just integrating the strain rate values would in comparison give a strain

ε(t) =∫ t

t0

ε(τ)dτ. ��

This is in fact the �rst order Taylor expansion of the expression in ��

exp(∫ t

t0

ε(τ)dτ

)− 1 =

∞∑n=0

(∫ t

t0ε(τ)dτ

)n

n!− 1

≈1∑

n=0

(∫ t

t0ε(τ)dτ

)n

n!− 1 =

∫ t

t0

ε(τ)dτ. ��

This approximation is only good when∣∣∣∫ t

t0ε(τ)dτ

∣∣∣ 1� which might not always bethe case� Figure �� shows a comparison of the two methods to calculate the strain�As seen� just integrating the strain rates overestimates the strain� The e�ect is clearlyvisible at strains in the range found in the cardiac muscle�


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.18

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

Time [s]

Str

ain

Figure �� The estimated strain of the object in Figure �� The start timewas chosen to t0 = 0� The solid line shows the strain estimatedusing �� while the dashed line shows the approximated strainestimated using ��

Chapter �

Angle dependence of strain rate

The concepts of strain and strain rate in principle relate to deformation of three�dimensional objects� Appendix A gives an introduction to the strain and strain ratetensors suitable for describing such deformations� We will in this chapter not go furtherinto the tensor description� but instead describe the angle dependence of the one�dimensional strain rate and strain estimates and how to calculate the strain rate indirections other than along the ultrasound beam�

�� Angle dependence of the strain rate and strain

estimates

As all other Doppler based imaging methods� strain rate imaging as described in theprevious chapter is also angle dependent� The method measures the strain rate as thegradient of the velocity component along the ultrasound beam� If the desired strainrate is in another direction� the measured strain rate� and the calculated strain� willdepend on the angle between the beam and the direction of the desired strain�In this section we give a theoretical description of the angle dependence of systolic

strain measurements� We base this on a simpli�ed model of how the muscle deforms�

�� Coordinate de�nitions

Locally for each muscle segment� we de�ne the coordinates�

r � along the ultrasound beam� positive away from the transducer

u � circumferential� clockwise seen from the apex

v � meridional �longitudinal�� from apex to base

w � transmural� from endocardium to epicardium

��

�� Angle dependence of strain rate

Figure �� De�nition of the coordinates r� u� v� and w� and the insonationangle α�

where u� v and w will be approximately perpendicular� as shown in Figure �� Theorigin (u, v,w) = (0, 0, 0) does not need to be de�ned in relation to the macroscopicventricle geometry� and can be chosen anywhere in the imaged muscle segment�Furthermore we de�ne α as the angle between the v�axis and the r�axis� so that zero

degrees corresponds to measuring along the muscle in the meridional direction� Weassume that the angle is in the v�w�plane �long axis or apical views�� so the problembecomes two�dimensional� Notice that the angle is negative in Figure ��

�� Assumed relation between strain and strain rate

As shown in section �� the strain ε after a time t can be estimated from the measuredstrain rate as

ε(t0, t) = exp(∫ t

t0

ε(τ)dτ

)− 1. ��

The starting time t0 is here speci�ed separately� The inverse relation becomes

ε(t) =d ln(ε(t0, t) + 1)

dt. ��

� � Angle dependence of the strain rate and strain estimates ��

For small strains |ε(t0, t)| 1 one can use the approximate formulas�

ε(t0, t) =∫ t

t0

ε(τ)dτ ��

and

ε(t) =dε(t0, t)

dt

≈ ε(t0, t + ∆t) − ε(t0, t)∆t

=ε(t, ∆t)

∆t��

where the approximation is valid for ∆t 1�

�� Strain rate angle dependence

Without losing generality� we can assume that the point (v,w) = (0, 0) is not moving�If the strain rate is spatially homogeneous in the muscle segment� the muscle point(v,w) will then move with the velocity components�

vv = vεv ��

and

vw = wεw ��

These velocity components are shown in Figure ��We want to �nd the strain rate along the ultrasound beam

εr =dvrdr

��

Notice that� for simplicity� the velocity vr is de�ned as being positive away from thetransducer� i�e�� in positive r�direction� This is opposite of the usual de�nition for thevelocity sign in Doppler imaging� Since the center of the muscle is not moving� thevelocity derivative can be simpli�ed to

εr =vr∆r

��

where ∆r is the distance over which the strain rate is measured� Since the beam hasthe angle α to the v�axis� the velocity along the ultrasound beam in position (v,w)becomes

vr = vεv cosα + wεw sin α ��

� Angle dependence of strain rate

Figure �� Illustration of the velocity components vv� vw and vr� the distance∆r and the angle α in a small muscle segment� All the parametersare drawn positive� but notice that the angle α is usually negativewhen imaging from the apex� and that vv� and consequently vr�normally are negative during systole� The lateral �beam�to�beam�l�axis is included for reference in Section ��

The coordinates can also be written

v = ∆r cosα ��

and

w = ∆r sin α ��

so the measured strain rate becomes�

εr = εv cos2 α + εw sin2 α ��

Notice that when imaging from the apex� the angle α will be close to zero for most ofthe ventricle�The same formulas will apply if one is imaging in the u�w�plane �short axis view�

by interchanging u and v� and rede�ning α as the angle between the u�axis and ther�axis� In that case α will have values from −π to π�

�� Assuming incompressible material

Since the cardiac muscle tissue can be considered incompressible� the following relationholds if u� v and w are the principal strain direction �see Appendix A��

(εu + 1)(εv + 1)(εw + 1) = 1 ��


Mean SD Method Value derived from Referenceεu �� MR tagging Local Lagrangian strains � ��

�� Cineangiogram Midwall short axis FS ��εv �� MR tagging Local Lagrangian strains � ��

�� D M�mode echo Long axis FS ��εw �� Short axis �D echo Wall thickening ��

Table �� Normal values for systolic strain in the human myocardium� SD�Standard deviation� FS� Fractional shortening�

By using �� we can derive the corresponding relation for strain rate�

εu + εv + εw = 0. ��

This relation is also given in the literature in �� p� �� and �� p� �� There areunlimited solutions that satisfy these equations� but each parameter has only a smallrange that is clinically relevant�Normal systolic strain values can be found in the literature� εu has been measured

directly using MR tagging� but since the circumference of the ventricle has an approx�imately linear relationship to the radius� the systolic midwall fractional shortening ofthe minor axis in the left ventricle can also be used� εv has also been measured directlyusing MR tagging� but the fractional shortening of the major axis can also be usedas an indication� εw is systolic wall thickening� and has been measured in numerousstudies� The values and references are given in Table �� In some publications� thestrain was given as the Greens strain tensor E �see Appendix A�� The strains in Table�� were then converted from the diagonal components of E using�

εx =√

2Exx + 1 − 1 ��

Several studies have showed that there is a di�erence between εw in the di�erent walls�� which can explain the large standard deviation of this value in Table ��These normal values do not exactly satisfy the incompressibility equation� but by

introducing a small o�set in each value the relation holds� One solution is for exampleεu = −0.19� εv = −0.17� εw = 0.48� These are all values at a distance of �� standarddeviations from the normal mean value�Normal values for strain rate have not been found in the literature�

�� Relative errors caused by angle mismatch

One might propose that there is a linear relationship between εu and εv�

εu(t) = kuvεv(t). ��

Using the incompressibility equation �� we then get the relation

εw(t) = −kwvεv(t), ��


−40 −30 −20 −10 0 10 20 30 40−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

kwv

= 1

Angle of insonation [deg]

Rel

ativ

e st

rain

rat

e er

ror

[%]

kwv

= 2

kwv

= 3

Figure �� Relative strain rate error caused by the angle α� The parameterkwv is the linearity constant in the assumed linear relationshipbetween εw and εv�

where kwv = 1 + kuv�The relation between the measured strain rate εr(t) and the desired strain rate εv(t)

in �� can then be rewritten

εr(t) = εv(t)(cos2 α − kwv sin2 α

)= εv(t) (1 + ∆ε(α, kwv)) , ��

where

∆ε(α, kwv) =εr(t) − εv(t)

εv(t)

= cos2 α − kwv sin2 α − 1 ��

is the relative strain rate error caused by the angle α when assuming the linearityconstant kwv� Figure �� shows the relation for kuv � � and �� or correspondinglykwv �� and ��A relative error for accumulated strain measurements can also be derived� Inserting

�� into �� one gets

εr(t) =(eR

t0 εv(τ)dτ

)1+∆ε(α,kwv) − 1

= (εv(t) + 1)1+∆ε(α,kwv) − 1. ��


−40 −30 −20 −10 0 10 20 30 40−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

kwv

= 1

kwv

= 2

kwv

= 3

Angle of insonation [deg]

Rel

ativ

e st

rain

err

or [%

]

εv = −5.0 %

εv = −15.0 %

εv = −25.0 %

Figure �� Relative accumulated strain error caused by the angle α for di�er�ent values for the meridional systolic shortening εv� The parame�ter kwv is the linearity constant in the assumed linear relationshipbetween εw and εv�

By rearranging this equation the relative strain error is found as

∆ε(α, kwv) =εr(t) − εv(t)

εv(t)

=(εv(t) + 1)1+∆ε(α,kwv) − 1

εv(t)− 1. ��

Figure �� shows this relationship for di�erent values of εv for kwv �� and �� Asseen� the shape of ∆ε(α, kwv) is very similar to ∆ε(α, kwv)� and a �rst order Taylorexpansion of �� assuming small εv(t) can be shown to be equal to ∆ε(α, kwv)�

�� Discussion and conclusions

Several assumptions have been made in the calculations� and these have to be takeninto account when discussing the results� A constant linear relationship between twoof the strain rate components has been assumed� Combined with the assumption ofincompressibility and no shear strains� a linear relationship will exist between any twoof the three strain rate components� To be able to get simple expressions we have alsoassumed that the strain rate is spatially homogeneous�


The formula for incompressibility assumes that u� v and w are the principal straindirections� This is not necessarily true� especially since the ventricle is known to haveabout �� degrees rotation from the base to the apex around the major axis duringthe systole� When we insert values from the literature� we see that the formula is notperfectly ful�lled� Therefore the relations between Sw and Sv are probably somewhatdi�erent than what we have used� and thus the curves will be di�erent�These uncertainties make it di�cult to exactly predict the angle dependence� but

still the curves give an impression of the behaviour�

�� Estimation of more components of the strain rate

tensor

Using velocity�information from more than one beam at the time it is possible tocalculate the strain rate in other directions than along the beam� Using the samecoordinate de�nitions as in Section �� but also including the lateral direction lperpendicular to the beams in the image plane� The beams are assumed to be parallelin the region of interest�The vw�axis system is then a rotation of the lr�axis system by an angle of (α−π/2)�

and one can write

v = r cosα + l sin α ��

w = r sin α − l cosα. ��

Substituting this in �� one gets

vr = εv(r cosα + l sin α) cosα + εw(r sin α − l cosα) sin α. ��

Taking the derivatives in the two directions r and l� one gets the two equations

∂vr∂r

= εv cos2 α + εw sin2 α ��

∂vr∂l

= εv sin α cosα − εw sin α cosα. ��

Solving for εv and εw gives

εv =∂vr∂r

+∂vr∂l

tan α ��

εw =∂vr∂r

− ∂vr∂l

cotα. ��

This means the strain rates in the anatomical directions v �meridional� and w �transmu�ral� can be found from the radial and lateral gradients of the measured radial velocity�as long as the angle α is known� The image plane lr must coincide with the vw plane�which is the case for all apical views and the parasternal long axis view �PLAX��

� � Discussion of other angle dependencies ��

The same formulas apply if one substitutes v with u� so the strain rate in the u�direction �circumferential� can also be found� The image plane lr must then coincidewith the uw plane� which is the case for the short axis view �SAX��There will be some angles where the strain rates are unavailable� though� For the

u or the v directions these are the angles where tan α approaches in�nite values� Forthe w�direction these are the angles where cotα approaches in�nite values�In the SAX view and using a sector scan� an approximation of α can automatically

be found if the user de�nes the centre of the ventricle� By assuming that the SAXcross section of the ventricle is circular� α at a particular location is given as

α =3π

2− θb + θc ��

where θb is the angle of the ultrasound beam that intersects the point �θb = 0 isde�ned as the center beam�� and θc is the angle between the center ultrasound beaman imaginary beam from the centre of the ventricle through the point�A preliminary test has been performed using this method� A velocity data set

from a healthy volunteer was obtained using tissue Doppler imaging with high beamdensity� The short axis view was used� Figure �� shows the estimated circumrferentialand transmural strain rate components in three phases of the cardiac cycle� Themyocardium was segmented manually�As expected� the radial strain rate is equal to the transmural strain rate at twelve

and six o�clock in the images� and the circumferential strain rate at two and ten o�clock�Except from where the cotα or tanα approach in�nity� the apparent noise in the imagesdoes not seem to be increased by this procedure�

�� Discussion of other angle dependencies

Both skeletal and cardiac muscles are highly anisotropic due to the ordered arrangementof the muscle �bers� An in vitro study on bovine tendon� which also is anisotropic�by Holland et al� �� has shown that the ultrasonic backscatter intensity is reduced24.6± 1.1 dB when imaged parallel to the tendon compared to imaging perpendicularto it�In the cardiac muscle the �ber orientation angle changes up to �� degrees from

endocardium to epicardium �� so the angle dependency will be more complex� In

vivo short�axis imaging on healthy subjects has been shown by Holland et al� �� togive di�erences in integrated backscatter values in di�erent parts of the muscle� Theirresults showed that the intensity was highest in the anterior septum and was decreasedby 15.9 ± 3.5 dB in the lateral wall� by 17.7 ± 3.5 dB in the inferior septum and by8.1 ± 3.8 dB in the posterior wall�A reduction in the backscattered intensity will increase the variance in the strain

rate estimate� since other signal components� like white noise and reverberation noise�will become more dominant� This must be kept in mind when analyzing the strainrates measured in di�erent parts of the muscle�


εr

εu

εw

�a� �b� �c�

Figure �� Color coded images of the radial strain rate εr �top row�� the cir�cumferential strain rate εu �mid row� and the transmural strainrate εw �bottom row� from �a� mid systole� �b� early diastolicrelaxation and �c� mid diastole� Positive strain rate �thicken�ing�lengthening� is colored blue and negative strain rate �thin�ning�shortening� red� The noisy areas at twelve and six o�clockin the mid row and at two and ten o�clock in the bottom row arecaused by the tan α and the cot α factors respectively�

Chapter �

In vitro experiments

In this chapter� two strain rate validation experiments are described� The collectedIQ�data from one of the experiments was also used to study the e�ect of stationaryreverberations and clutter �ltering on the strain rate imaging�

�� Validation tests for the strain rate estimator

When a new imaging technique is introduced it is always interesting to know if it reallymeasures the correct value� The real�time strain rate imaging technique has thereforebeen tested in two experiments� First� the relatively simple test for zero strain rate wasperformed� This was done in collaboration with A� V� Lund� MS� and was part of herMaster�s thesis �� The second test was for more clinically relevant strain rates up to �� s−1� and was performed by compressing a gel block� This experiment was performedin collaboration with J� D�Hooge� MS� B� Bijnens� PhD� and G� Sutherland� PhD� atthe University Hospital Gasthuisberg in Leuven� Belgium� A similar experiment waslater performed by M� Belohlavek� PhD� at the Mayo Clinic in Rochester� Minnesota�USA� and some of his results are included for comparison�

�� Zero strain rate test

To test what e�ect a pure motion without deformation of an object would give on theestimated strain rate� the following experiment was performed� A �×�×� cm block of�oral foam �Smithers�Oasis Company� Cuyahoga Falls� Ohio� USA� was drenched inwater and boiled under vacuum to remove all air bubbles� The block was fastened toa movable rod as shown in Figure �� and imaged using a �� MHz phased array probeand a specially programmed System Five digital ultrasound scanner �GE VingmedUltrasound AS� Horten� Norway�� Both IQ�data and real�time estimated strain ratedata were gathered� The setup parameters used in the experiment are listed in Table��

��

�� In vitro experiments

Speed of sound c �� msTransmit frequency f0 �� MHzPulse repetition time T �� s

Packet size N �Sample distance ∆r � mmSample distance N∆r � samplesRadial averaging Nr � samplesLateral averaging Nl � beams

Table �� Parameters used in the experiment�

Figure �� Illustration of the experiment setup showing how a foam blockwas moved up and down in a water bath ��

Three di�erent motion patterns were used� Motion towards the probe� motionlaterally in the image and motion perpendicular to the image plane� In one experimenta thin plastic plate was put between the probe and the foam block to create stationaryreverberations inside the block� The speed of the block was measured with pulsedDoppler to be �� mms� This is higher than the normal tissue velocities in the heart�� mms�� Second harmonic strain rate imaging was also performed in some ofthe experiments�Figure �� shows SRI images in the fundamental and second harmonic mode� and

also shows the e�ect of stationary reverberations�The collected strain rate images were processed in Matlab� Here� the mean value

and standard deviation of the measured strain rate values within the foam block werecalculated� Values were taken at distances larger than the size of the point spreadfunction to get statistically independent measures� Since the expected strain rate was

� � Validation tests for the strain rate estimator ��

�a� �b�

�c� �d�

Figure �� Strain rate imaging �SRI� of a block that is moving but not de�forming �� The yellow rectangle indicates the position of theblock in the image� �a� Fundamental SRI� �b� Fundamental SRIwhen reverberations were introduced� �c� Second harmonic SRIwith no reverberations� �d� Color map used for the strain rate val�ues� A low Nyquist strain rate has been used to better visualizethe variance�

� In vitro experiments

Mean bias Standard deviationMotion towards probe

SRI� no reverberations �� s−1 �� s−1

Octave SRI� no reverberations �� s−1 �� s−1

SRI� with reverberations �� s−1 �� s−1

Octave SRI� with reverberations �� s−1 �� s−1

Motion perpendicular through image planeSRI �� s−1 �� s−1

Octave SRI �� s−1 �� s−1

Motion laterally in imageOctave SRI �� s−1 �� s−1

Table �� Mean bias and standard deviation of the measured strain rateswithin a non�deforming foam block moving at a speed of mm�s�Both fundamental strain rate imaging �SRI� and second harmonic�Octave� SRI was used� ∆r was � cm in all cases�

zero� the mean value corresponded to the mean bias of the measurements� The resultsare summarized in Table ��The estimated variances �square of the standard deviations� are not directly com�

parable with the Cram�r�Rao variance lower bound in Section �� since the packetsize was assumed to be N = 2 in the development of the lower bound� while N = 3 inthe experiment� The Cram�r�Rao for N = 2 is found by inserting the parameters inTable �� in �� The correlation coe�cient magnitude |ρ| for the experiment withmotion towards the probe was estimated from the IQ�data using spatial averaging of�� to be � �� and the number of estimation samples was set as M = N∆r + Nr�

σˆε ≥√

c2

16π2f20 T 2r2

s

(|ρ|−2 − 1)6

M(M2 − 1)= 0.112 s−1. ��

The bound in �� was further divided by√

Nl to account for the lateral averaging�assuming non�overlapping beams� To account for the increase in packet size fromN = 2 to N = 3� the bound was further divided by

√2� assuming independent signal

in the additional echo� These two assumptions are not realistic but give a limit for thelowest possible variance� The standard deviation lower bound was this way calculatedto�

σˆε ≥√

c2

32π2f20 T 2r2

sNl(|ρ|−2 − 1)

6M(M2 − 1)

= 0.0459 s−1. ��

The correct lower bound would be between the values in �� and ��As seen in Table �� the standard deviation was increased when a stationary re�

verberation was introduced� but reduced when second harmonic strain rate imagingwas used� There seemed to be little dependence on the direction of the motion� and


Figure �� Experiment setup with a cyclically compressed gel block� �Photoscourtesy of J� D�hooge��

the bias was always less than �� s−1 in magnitude when there were no reverbera�tions� Both the biases and the standard deviations were small compared to the normalpeak magnitude strain rates found in cardiac imaging �up to � s−1 in Table �� inSection �� so even if the standard deviations were almost twice as large as theCram�r�Rao bound� the e�ect of reducing the variance would be small�

�� Gel block compression test

To test whether the strain rate imaging technique measured correct strain rates in therange that is found in the cardiac muscle� the following experiment was performed�A specially programmed System Five digital ultrasound scanner �GE Vingmed

Ultrasound AS� Horten� Norway� was used to collect IQ�data and real�time strain ratedata from a dynamic in vitro model of myocardial tissue� The model consisted of a�×�×� cm homogeneous gel block �� that was exposed to � " cyclic compressionwith a frequency of � Hz� This simulated the systolic phase of the cardiac contraction�The model was imaged in the direction of the compression� as shown in Figure ��Both surfaces were oiled with ultrasound gel to avoid shear forces in the gel block�Because of this and the homogeneity of the phantom� a spatially constant strain ratecould be assumed throughout the phantom� The scanning parameters that were usedare listed in Table ��Strain rate images from peak compression rate and peak relaxation rate are shown

in Figure ��The distance L(t) to the back plate was continuously measured by tracking the

peak intensity from the compressing plate in a reconstructed M�mode from the middle


�a� �b�

Figure �� Strain rate images of the gel being released in �a� and being com�pressed in �b�� The compressing plate and its direction of motionis drawn in for clarity� Notice the spatially uniform strain ratealong the middle beam in the images�

Speed of sound c �� msTransmit frequency f0 �� and �� MHzPulse repetition time T �� s

Packet size N �Sample distance ∆r �� mmSample distance N∆r �� samplesRadial averaging Nr � samplesLateral averaging Nl � beams

Table �� Parameters used in the experiment�


Figure �� Reconstructed M�mode from the middle beam in consecutivegrayscale �D images� In the lower panel the echo from the backplate has been magni�ed and the tracked curve has been plotted�

beam� Figure �� shows this M�mode�The instantaneous strain imposed on the gel was then estimated as

ε(t) =L(t) − L0

L0��

where L0 was the maximum distance from the probe to the back plate� The assumedstrain rate along the middle beam in the gel was then estimated as

ˆεa =dε(t)dt

≈ ε(t + ∆t) − ε(t)∆t

��

where ∆t here was the time from frame to frame� This strain rate was then comparedto the one measured by real time SRI� Figure �� shows a comparison of the assumedstrain rate and the mean strain rate found by SRI� A scatter plot of all the valuesfound by SRI is also included to illustrate the variance�The peak negative assumed strain rate was then compared to the strain rate mea�

sured with SRI at the same time in the compression cycle� The peak negative strain


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

Str

ain

rate

[1/s

]

Figure �� Comparison of the assumed strain rate �dashed line� and the meanstrain rate measured using real time SRI �solid line�� A scatterplot of all the strain rate values along the beam is also includedto give an impression of the variance�

� � E�ects of stationary reverberations and clutter �ltering ��

Ultrasound frequency �� MHz �� MHz �� MHzRelative bias ��" ��" ��"Relative SD ��" ��" ��"

Table �� Bias and standard deviation �SD� of the measured strain rate inpercentage of the assumed strain rate �� s−1��

rate for this experiment was estimated to �� s−1� Table �� shows the bias andstandard deviation of the SRI values in percentage of this strain rate value�As in the previous section� the Cram�r�Rao bound for N = 2 was found by inserting

the parameters in Table �� in �� The correlation coe�cient magnitude |ρ| for the�� MHz experiment was estimated from the IQ�data using spatial averaging of ��In the area of highest motion� i�e�� near the back plate at peak strain rate� |ρ| wasfound to be � �� The number of estimation samples was again set asM = N∆r +Nr�

σˆε ≥√

c2

16π2f20 T 2r2

s

(|ρ|−2 − 1)6

M(M2 − 1)= 0.0501 s−1. ��

The bound in �� was further divided by√

2Nl to account for the lateral averagingand the increase in packet size� The standard deviation lower bound was this waycalculated to�

σˆε ≥√

c2

32π2f20 T 2r2

sNl(|ρ|−2 − 1)

6M(M2 − 1)

= 0.0205 s−1. ��

The correct lower bound would be between the values in �� and �� The ��"relative standard deviation for the �� MHz experiment corresponds to a standarddeviation of �� s−1 for comparison�From the results one can conclude that the implemented strain rate imaging tech�

nique measured the strain rate in real time with a bias less than � percent of thenormal peak negative systolic strain rate value� The standard deviation of the strainrate value was �� percent or less� This is several times higher than the Cram�r�Raobound� indicating that further improvements in the strain estimation are possible�The same experiment was performed for lower strain rates by Marek Belohlavek at

the Mayo Clinic� His results are shown in Figure �� for comparison� Notice that thesign of the strain rate has been reversed in his presentation�

�� Eects of stationary reverberations and clutter

�ltering

Stationary reverberations are often present in clinical ultrasound imaging� In Dopplerimaging stationary reverberations give a bias towards zero and an increased variance in


Figure �� Comparison of measured strain rate and assumed �calculated�strain rate� The correlation �r�� the regression line �y� and andthe t�test p�value are given� This �gure was made by Marek Be�lohlavek�


the velocity magnitude estimates� Since SRI is an extension of the tissue Doppler tech�nique �� it will thereby be a�ected by the stationary reverberations� The stationaryreverberations can be removed using a clutter �lter on the Doppler signal� For lowblood velocities the �lter introduces a bias and an increased variance in the velocityestimate �� In this section� the e�ect of stationary reverberations and polynomialregression type clutter �ltering on SRI is studied in an in vitro experiment� to �nd outwhether the clutter �lter used in Doppler imaging is useful also in SRI�

�� Theory

First� v is de�ned as the velocity component and r as the position along the ultrasoundbeam axis� Both are in this section de�ned positive away from the transducer� In theideal case of a velocity �eld linearly dependent on spatial position the strain rate ε isfound from two point velocities v1 and v2 at a radial distance ∆r from each other as

ε =v1 − v2

∆r��

If there are velocity dependent biases bv(v) for the velocity estimates

v = v + bv(v), ��

the estimated strain rate� ˆε� will be

ˆε =v1 − v2 + bv(v1) − bv(v2)

∆r

≈ ε

(1 +

∂bv(v1)∂v

)��

resulting in a strain rate bias

bε(v) = ε∂bv(v)

∂v. ��

If in an experimental setup� the velocity is linearly dependent on the distance fromthe transducer� r� the strain rate will be

ε(v) =v

r��

which is a spatially constant value� The strain rate bias is then

bε(v) =v

r

∂bv(v)∂v

. ��

�� Methods

The same experiment setup of a cyclically compressed gel block as described in Section�� was used� only IQ�data were gathered rather than real time processed strain rate


data� The IQ�data were collected for � ms and transferred to Matlab for post�processing� The way the experiment was set up� the local strain rate and velocity inthe gel block were as described in ��Stationary reverberations were simulated by adding a white Gaussian noise vector

to each of the beams in the IQ�data before processing� The signal�to�clutter ratio�SCR�� de�ned as the power of the IQ�data divided by the power of the Gaussiannoise� was varied in di�erent simulations� The resulting IQ�data beams were bandpass�ltered to a bandwidth of �� MHz to make them more suitable for the autocorrelationmethod� The velocity in each sample was estimated using the autocorrelation method�� on data from � consecutive pulses� The strain rate was then estimated from thevelocity di�erence over � cm along the ultrasound beam� The estimated strain ratedata were �nally smoothed by averaging over � cm along the ultrasound beam� Thisway Ne �� strain rate samples with and without added reverberation noise weregenerated� The samples were collected in vectors called erev and e respectively� Byplotting the elements in these vectors against each other in a scatter plot� one gets animpression of the variance and bias caused by the reverberations� Increased strain rateestimate variance when introducing stationary reverberations could then be indicatedby a reduced correlation coe�cient �ρe� between erev and e�

ρe =

∑Ne

n1=1 e(n1)erev(n1)√∑Ne

n2=1 e2(n2)∑Ne

n3=1 e2rev(n3)

. ��

The regression curve of erev on e was found as

erev = αe+ β ��

and least squares estimators for the slope α and the o�set β were found as

α =

∑Ne

n1=1 (erev(n1) − erev) (e(n1) − e)∑Ne

n2=1 (e(n2) − e)��

and

β = erev − αe ��

where erev and e are the mean values of erev and e respectively� The regression curvecould give an impression of the bias in the strain rate estimate caused by the rever�berations� If β �= 0 there was a bias independent of the strain rate value� and if α �= 1there was a fractional bias� i�e�� a bias that depends on the strain rate�Later� the same IQ�data were passed through a zero order polynomial regression

clutter �lter before the velocity and strain rate were calculated� This �lter involvedsimply to subtract the mean of the complex signal in each sample� The velocities andstrain rates found from the clutter �ltered data were presented in the same fashionas earlier� Since the clutter �lter removes all the stationary clutter� the result isindependent of the SCR�


SCR ρe α β� dB �� ± �� ± �� ± �� dB � � ± �� ± �� ± �� dB �� ± �� ± � � �� ± ��

Table �� Correlation coe�cient ρe� regression line slope α and o�set β �meanvalue ± standard deviation� for di�erent signal to clutter ratios�SCR� when clutter �ltering was not used�

SCR ρe α β�� dB ��

Table �� Correlation coe�cient ρe� regression line slope α and o�set β �meanvalue�� equal for all signal to clutter ratios �SCR� when clutter�ltering was used�

To reduce the variance in the estimates of ρe� α and β� the same calculations wereperformed � times with di�erent realizations of the white Gaussian noise each time�The presented values for ρe� α and β� are the mean values and standard deviationsfrom these � calculations�

�� Results

Figure �� shows the e�ect of stationary reverberations and a zero order polynomialregression clutter �lter on the measured velocity� Notice that the reverberations causethe velocity magnitude to be under�estimated� while the clutter �ltering causes over�estimation� The theoretical bias according to �� is included in panel �b� for compar�ison�Figure �� shows scatter plots of erev versus e for three di�erent SCRs� when no

clutter �ltering was performed� The correlation coe�cient and the mean fractionalbias are presented in Table �� Notice that the correlation coe�cient is reduced as theSCR is reduced� indicating that the estimate variance is increasing� Also notice thatthe mean fractional bias is increasing as the SCR is reduced�Figure �� shows a scatter plot of erev versus e when the IQ�data was clutter

�ltered before further processing� Notice the large estimation error for strain ratesnear zero� The correlation coe�cient and the mean fractional bias are presented inTable �� The results were independent of the SCR and had zero standard deviation�

Notice that for dB SCR� the correlation coe�cient is larger with clutter �lteringthan without� At � and � dB SCR� though� the correlation coe�cient is largest whenno clutter �lter is used� Similarly� for dB SCR the regression line slope α is closer tounity with clutter �ltering than without� while at � and � dB the slope is closest tounity when clutter �ltering is used� The correlation line o�set β was close to zero inall the settings�


�a�

�b�

Figure �� Scatter plots of the measured velocity after introducing stationaryreverberations at SCR � � dB in �a� and after clutter �ltering in�b� versus the actual velocity� The velocities are measured relativeto the Nyquist limit� The solid line is the theoretical response�


−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5

Actual strain rate, SR

Str

ain

rate

with

rev

erbe

ratio

ns, S

Rre

v

�a�

−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5


Str

ain

rate

with

rev

erbe

ratio

ns, S

Rre

v

�b�

−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5


Str

ain

rate

with

rev

erbe

ratio

ns, S

Rre

v

�c�

Figure �� Scatter plots of erev �SRrev� versus e �SR� for �a� SCR � �� dB��b� SCR � �� dB and �c� SCR � � dB when no clutter �lteringwas performed� The solid line is a linear regression of the data�and the dashed line is a line with unity slope for comparison�


−5 −4 −3 −2 −1 0 1 2 3 4 519

19.5

20

20.5

21

21.5

22

22.5

23

23.5

24

−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5

Str

ain

rate

with

rev

erbe

ratio

ns, S

Rre

v

−5 −4 −3 −2 −1 0 1 2 3 4 5−24

−23

−22

−21

−20

−19

−18

−17

−16

−15

−14


Figure �� Scatter plot of erev �SRrev� versus e �SR� when a zero orderclutter �lter was used� The solid line is a linear regression of thedata� and the dashed line is a line with unity slope for compar�ison� Notice the vertical axis gaps included to show the largevariance around zero strain rate�

� � E�ects of stationary reverberations and clutter �ltering �

�� Discussion and conclusions

Since stationary reverberations introduce an increased variance in the velocity esti�mates� it seems reasonable that the variance of the velocity gradient� i�e�� the strainrate� is also increased� Stationary reverberations also introduce a velocity bias with anegative slope� as seen in Figure �� a�� Equation �� then shows that this resultsin a bias towards zero in the strain rate estimate as seen in Figure �� For velocities that di�er from zero� clutter �ltering reduces the variance of the

velocity estimates caused by stationary reverberations� and thereby also improves thestrain rate estimate as seen by comparing Figure �� c� and Figure �� If theactual velocity is zero� the clutter �lter removes everything except the white noise�The velocity estimate� and thus also the velocity gradient� will then be randomlydistributed� In our experiment zero velocity corresponds to zero strain rate as in�� so this explains the large variance for zero strain rate in Figure ��In itself� however� clutter �ltering gives a velocity bias with a negative slope� es�

pecially for low velocity magnitudes� as seen in Figure �� b�� Thus� from �� onecan see that clutter �ltering results in a bias towards zero in the strain rate estimate�as seen in Figure ��The sign of the velocity bias slope� and thereby the sign of the strain rate bias� will

depend on the type of clutter �lter and the number of pulses� N� In this work only thesetting with a zero order clutter �lter and a signal of N � has been investigated�In conclusion� when there are high clutter levels and velocities di�ering from zero�

a zero order clutter �lter could be used to lower the variance and fractional bias of thestrain rate estimate� while at low clutter levels� this type of clutter �ltering should notbe performed�

Chapter �

In vivo examples of strain rate

imaging

In this chapter some preliminary clinical strain rate images are presented� The tech�nique has been tested on healthy and infarcted hearts� and on breast and liver withtumors and cysts� Asbj�rn St�ylen� M�D�� at the University Hospital of Trondheimand Odd Helge Gilja� M�D�� Ph�D�� at Haukeland University Hospital in Bergen haveaquired most of the data presented in this chapter�

�� Cardiac muscle function

Strain rate imaging from the apical view gives an estimate of the local meridional�longitudinal� strain rate in the muscle� This is because the beam is parallel or closeto parallel to the meridional axis in most of the ventricle� In the apex other angles arepresent� but the basal parts of the apical segments are usually accessible without toolarge angle�The measured strain rate has been found to be most easy to interpret by using

curved M�mode analysis�

�� Normal �ndings

Figure �� shows the result of a curved M�mode along a normal interventricular septumfor one full heart beat at a relaxed heart rate� The �gure also illustrates the normal�ndings in the left ventricle walls�

�� A shortening �yellow� in the systole that propagates from the base to the apex�

�� A quick lengthening �blue� in the early diastole�

�� When the heart rate is low� a period of no length change �green��

��

� In vivo examples of strain rate imaging

�a� �b�

Figure �� a� Real�time strain rate image from mid systole of a normal leftventricle� �b� Curved M�mode through the septum for one heartcycle� The yellow curve in �a� indicates where the curved M�modein �b� is taken� The numbers correspond to the normal �ndingsin section �� The data was aquired by Asbj�rn St�ylen�

�� A lengthening �blue� during the atrial contraction�

�� Some recoiling �yellow� after the quick lengthening �blue� occurrences�

The amount of systolic shortening and early diastolic relaxation strain rates in amuscle segment might give information on the viability of the segment� The lenghteningduring atrial contraction is not expected to give much information on the performanceof the left ventricle� since it is only an e�ect of the atrium pumping blood into the leftventricle cavity�

�� Infarction examples

In recent myocardial infarctions� parts of the left ventricle do not contract normally�This can show up as reduced or delayed systolic strain rate� Examples from recentapical infarction� recent inferior infarction and recent lateral infarction are shown inFigures ��

�� Stomach muscle function

The peristaltic motion of the stomach can be visualized with strain rate imaging�Figure �� shows a �D view with little information� while Figures �� and �� showthe timing and position of the peristalitc motion� Figure �� shows curved anatomicalM�modes along the upper and lower walls in Figure �� The curves were adjusted foreach time step so they always stayed inside the muscle� Figure �� shows four M�modesperpendicular to both walls� The walls have been manually segmented out�

� � Stomach muscle function �

�a� �b�

Figure �� a� A ��chamber view �D frame from mid systole of a patient withapical infarction� Notice only contraction �yellow� in the healthybasal parts of the walls� �b� A curved M�mode through the septalwall with apical infarction� Notice akinesia �green� in the apicalpart of the wall during systole� The data was aquired by Asbj�rnSt�ylen�

�a� �b�

Figure �� a� A ��chamber view �D frame from mid systole of a patient withbasal inferior infarction� Notice only contraction �yellow� in thehealthy apical part of the wall� �b� A curved M�mode through theinferior wall with basal infarction� Notice akinesia �green� in thebasal part of the wall during systole� The data was aquired byAsbj�rn St�ylen�

� In vivo examples of strain rate imaging

�a� �b�

Figure �� a� A ��chamber view �D frame from mid systole of a patient withlateral infarction� Notice only contraction �yellow� in the septumand the healthy apical part of the lateral wall� �b� A curved M�mode through a lateral wall with basal infarction� Notice akinesia�green� or hypokinesia �mottled yellow and green� in the basalpart of the wall during systole� The data was aquired by Asbj�rnSt�ylen�

Figure �� Strain rate image of the stomach at the beginning of a peristalticmotion� The walls are manually schetched� The inward peristalticbulging at position A moves distally down to D in seconds� Thelines A� B� C and D indicate where the corresponding anatomicalM�Modes in Figure �� are taken� The data was aquired by OddHelge Gilja�

� � Tumors and cysts �

Figure �� Curved anatomical M�modes along the upper �upper panel� andlower �bottom panel� stomach walls in Figure �� Notice how thecontraction �blue� starts at position A and propagates to positionD� A wave of relaxation �yellow� follows� The vertical lines areprobably caused by the nearby pulsating aorta� The color gainhas been increased compared to Figure ��

�� Tumors and cysts

Strain rate imaging can perhaps be used in a way similar to elastography� which isdescribed in Section �� The idea is to detect di�erences in sti�ness within the tissueby measuring the strain rate during external compression� This procedure was tested inthree preliminary studies� one involving a breast tumor� one involving a liver metastasisand one involving a liver cyst�

�� Breast tumor

The strain rate imaging technique was tried out on a young healthy volunteer and anold patient that had discovered a lump in the breast� In x�ray mammography� the lumpwas clearly visible� as seen in Figure �� A biopsy was later performed that showedthe lump to be a malignant tumor�Both persons were imaged using the strain rate imaging technique� The probe was

positioned on the skin surface� in the patient directly above the tumor� and pushedinwards to compress the breast� The probe was pushed approximately � mm� and thenthe pressure was released�Strain rate images from mid�compression� and M�modes during compression and

relaxation are shown in Figure �� for the breast with the tumor� and Figure �� forthe healthy breast� Notice that the strain rate is spatially homogeneous in the healthybreast� while in the breast with a tumor the strain rate is lower within the tumor thanin the surroundings� indicating that it resisted the compression�

In vivo examples of strain rate imaging

A

B

C

D

Figure �� Anatomical M�Modes through the stomach� The walls are man�ually segmented out� The color gain has been increased com�pared to Figure �� to see the color in the walls� Notice that thecontraction happens later in the more distal M�modes� and thatthe walls are greenish �not changing� before the contraction� blue�thickening� during the contraction� and yellow �thinning� afterthe contraction�

� � Tumors and cysts �

Figure �� X�ray mammography of a breast with a tumor�

Figure �� Strain rate image during compression �left� and M�mode duringcompression and relaxation �right� of a breast with a tumor� TheM�mode is taken through the center of the tumor� The verticalaxes are scaled in cm� and the color map shows strain rate in s−1�

�� In vivo examples of strain rate imaging

Figure �� Strain rate image during compression �left� and M�mode duringcompression and relaxation �right� of a normal breast� Verticalaxis is scaled in cm� and color map shows strain rate in s−1�

�� Liver cyst and tumor

Since tumors are usually more sti� than the surrounding tissue while cysts usuallyare less sti� than the surrounding tissue� their appearances in compression strain rateimages will be di�erent� During compression� a sti� object might resist the compressionand have lower strain rate than the surroundings� This is to some degree seen in thecompression strain rate image of a liver tumor in Figure ��A soft object might be compressed more than its surroundings during external

compression� This can be seen in the compression strain rate image of a liver cyst inFigure ��

� � Tumors and cysts ��

Figure �� Tissue �left� and strain rate �right� images during compressionof a liver with a tumor� Data was acquired by Odd Helge Gilja�

Figure �� Tissue �left� and strain rate �right� images during compressionof a liver with a cyst� Data was acquired by Odd Helge Gilja�

�� In vivo examples of strain rate imaging

Chapter �

Conclusions and future

directions

�� Concluding remarks

The goal of this thesis was to develop and evaluate Doppler based methods to detectand quantify two of the tissue viability properties�The �rst property was blood perfusion� The detection of blood using pulsed Doppler

and color �ow Doppler was investigated in Papers � and �� In Paper � it was foundthat involuntary skeletal muscle vibrations in the hand of the operator or in the patientitself produce low frequency side bands in the Doppler signal and thus limit the possi�bility of detecting the low velocity blood �ow� An in vivo measurement of a normallyvibrating muscle was performed� and used to model the Doppler clutter signal� Forthis measurement and a model for the Doppler signal from blood� it was found thatcapillary blood �ow was not detectable with any Doppler method� regardless of ob�servation time� In Paper �� the color �ow imaging technique with limited observationtime was considered� The demodulated Doppler signal used in color �ow imaging wasmodeled as a complex Gaussian signal� and a likelihood test for the presence of bloodwas developed� An approximation of the likelihood function was shown to describeclutter �ltering and blood enhancement� Using an in vivo measured clutter signal� itwas illustrated how this model could be used to optimize the detection�The second viability property was the tissue function� This was limited to the

self�induced deformation of muscle tissue and the response in tumors to externallyintroduced deformation� To measure the deformation� a strain rate imaging techniquewas developed�A maximum likelihood estimator for the strain rate was developed for packet size

N = 2� together with an analytical expression for the lower bound variance� Thevariance was shown to depend on both the velocity and the velocity gradient in thetissue� It was shown that the maximum likelihood estimator gave improved strain rate

�

�� Conclusions and future directions

estimates as compared to the linear regression method used in the Myocardial VelocityGradient method� published by other authors� A simpli�ed estimator was implementedfor real�time performance in an ultrasound scanner� and was tested in in vitro and in

vivo experiments�

The in vitro experiments involved cyclic compression of a gel block� and was per�formed to validate the method� Strain rates in the range �� s−1 were tested� Theresults showed a bias smaller than �� s−1 in magnitude and a standard deviationless than �� s−1 for all the experiments� This standard deviation was almost twicethe Cram�r�Rao lower bound for the same settings� but still small compared to thenormal peak strain rates found in a normally contracting cardiac muscle� which hasbeen reported as high as � s−1 in magnitude� depending on the position and direction�

Paper � describes a pilot study� where the strain rate imaging technique was usedin vivo to image � patients with myocardial infarction and � normals� All the a�ectedregions in the patients showed up with reduced strain rate� demonstrating the techniqueto be useful for imaging regional dysfunction�

Other in vivo experiments presented in Section � showed some of the potentials ofthe method� In cardiac imaging� regions of myocardial infarction showed up as delayedor reduced systolic strain rate� In imaging of the stomach muscle� the normal peristalticcontraction was measurable� In tumor imaging� the resistance to external pressure�indicated by reduced strain rate compared to the surroundings� was demonstrated fora breast tumor and a liver tumor� A liver cyst showed increased strain rate comparedto the surrounding tissue�

Methods to increase the frame rate and to improve the strain rate estimator qualitythrough the use of second harmonic imaging� have been presented� The e�ect of sta�tionary reverberations and a simple clutter �lter have been investigated� It was shownthat both the reverberations and the clutter �lter introduced a bias in the strain rateestimate� and that a clutter �lter therefore only should be used when the reverberationlevel is high�

It has been shown that� in one dimension� the accumulated strain can be estimatedfrom the instantaneous strain rate� and the two�dimensional angle dependencies ofboth the strain rate and the strain measurements have been described� Furthermore� amethod to estimate the strain rate in directions in the image not necessarily along theultrasound beam was developed� The method involved the calculation of the velocitygradient both along the ultrasound beam� and from beam to beam laterally in theimage� From these gradients� the strain rate in any direction except perpendicular tothe beam could be found� A preliminary test in cardiac short axis imaging indicatedthat the method could measure simultaneously the transmural and the circumferentialstrain rates in all parts of the ventricle except where these directions were perpendicularto the beam�

� Suggestions for future work ��

�� Suggestions for future work

Detecting and quantifying capillary blood perfusion by ultrasound seems to be a dif��cult task� Still� using ultrasound contrast agents� this might be possible� The echofrom the contrast agent in the blood will then be increased� and might be detectableeven in the presence of clutter signal from sources like vibrating muscles�Paper � also showed that it was possible to measure the muscle vibration pattern�

This opens for the possibility of using adaptive clutter �lters that might be more narrowthan ordinary clutter �lters� Blood of lower velocity might then be detectable�For the strain rate imaging technique� the real�time estimation can be improved

using an estimator closer to the optimal� This might be possible given increased signalprocessing capabilities and data transfer speeds in the ultrasound scanner� Implement�ing the high frame rate and second harmonic methods described in Sections �� and�� will probably also improve the estimator quality and thus the clinical usefulnessof this method�The use of strain rate imaging in the clinical setting needs further studies� Other

suggestions for the strain rate imaging technique are to acquire �D data sets to allowa fast quanti�cation of the strain rate in all parts of the left ventricle� The angledependency and correction in �D would then need to be investigated�A method to detect the third viability property� metabolism� might be to use spe�

cially designed contrast agents which can bind to either viable or non�viable tissue�

�� Conclusions and future directions

Appendix A

The strain� strain rate and

rate�of�deformation tensors

This appendix contains some basic de�nitions from continuum mechanics� based on atextbook by L� E� Malvern ��

A�� Elementary de�nitions

If an object of initial length L0 changes length to L� the conventional strain can bemeasured as

ε =L − L0

L0. �A��

The change in angle� α� between two line segments that originally were perpendiculardescribes the shear strain� A common measure for the shear strain is

γ =12

tan α. �A��

Also� the change in volume can be measured by the volume strain

εV =V − V0

V0, �A��

where V0 is the initial volume and V is the volume after deformation� An incompressiblematerial will thus have a volume strain of zero�

A�� Strain tensors

The strain de�nitions in the previous section are impractical for describing deforma�tions in more than one direction� A tensor formalism is commonly used to describe

�

� The strain strain rate and rate�of�deformation tensors

Figure A�� Displacement u� stretch and rotation of an initial segment dX toa new position dx�

multi�dimensional deformations� This tensor de�nes a quadratic form that describesthe change in quadratic length of a material segment�

|dx|2 − |dX|2 = 2dX ·E · dX, �A��

in Lagrangian formulation and

|dx|2 − |dX|2 = 2dx · e · dx, �A��

in Eulerian formulation� Here dX is an initial material vector that is displaced�stretched and rotated to a new position dx� as shown in Figure A�� and E and eare two de�nitions of the strain tensor� The Green�s strain tensor or Lagrangian strain

tensor Eij can be written

Eij =12

(∂uj

∂Xi+

∂ui

∂Xj+

∂uk

∂Xi

∂uk

∂Xj

)�A��

while the Almansi�s strain tensor or Eulerian strain tensor eij can be written

eij =12

(∂uj

∂xi+

∂ui

∂xj+

∂uk

∂xi

∂uk

∂xj

)�A��

where uk is the displacement� Xk is the original material position� and xk is the materialposition after the deformation� for the spatial directions k = 1, 2, 3� In these equationsand the rest of the section the summation convention which states that in Cartesiancoordinates whenever the same letter subscript occurs twice in a term� that subscriptis to be given all possible values and the results added together� is adopted�The two tensors are symmetric� i�e��

Eij = Eji, eij = eji. �A��

Because of this� there always exist three orthogonal principal strain directions whereonly principal strains and no shear strains are present� This property applies to all thetensors described in this appendix�In one dimension the two strain tensors reduce to the strains

εX = E11 =∂u1

∂X1+

12

(∂u1

∂X1

)2

�A� �

A � Strain rate and rate�of�deformation tensors ��

and

εx = e11 =∂u1

∂x1+

12

(∂u1

∂x1

)2

. �A��

For small strains both tensors in �A�� and �A�� reduce to the Cauchy�s in�nites�imal strain tensor �

εij =12

(∂uj

∂xi+

∂ui

∂xj

). �A��

This can be written out as

εxx = ∂ux

∂x , εxy = 12

(∂ux

∂y + ∂uy

∂x

)= εyx,

εyy = ∂uy

∂y , εxz = 12

(∂ux

∂z + ∂uz

∂x

)= εzx,

εzz = ∂uz

∂z , εyz = 12

(∂uy

∂z + ∂uz

∂y

)= εzy.

�A��

The expressions in the �rst column represent strains and the expressions in the lastcolumn� if doubled� represent shear strains� In one dimension� the small strain tensorreduces to the small strain�

ε = εxx =∂ux

∂x. �A��

A�� Strain rate and rate�of�deformation tensors

The strain rate tensor is the time derivative of the strain tensor

E =dEdt

, e =dedt

. �A��

Notice that the derivative operator is de�ned as

d

dt=(

∂

∂t+ v · grad

), �A��

where v is the velocity �eld� so both the strain and the strain rate tensors depend onthe initial con�guration of the object through the displacement u in �A�� and �A��To describe motion in linear viscosity theory �� the rate�of�deformation tensor is

commonly used

Dij =12

(∂vj

∂xi+

∂vi

∂xj

)�A��

where vk is the velocity and xk is the spatial direction number k = 1, 2, 3� Note thatthis tensor is given by the instantaneous velocity �eld� which can be measured byultrasound Doppler techniques�

�� The strain strain rate and rate�of�deformation tensors

The strain rate tensors can then be written�

E = FT ·D ·F, �A��

and

e = D − (e · L + LT · e), �A��

where F is the deformation gradient tensor given by

Fij =∂xi

∂Xj�A��

and L is the spatial velocity gradient tensor given by

Lij =∂vi

∂xj. �A��

From this one can see that it generally is not possible to derive the strain tensor E ore from the rate�of�deformation D only�As seen by using �A�� the strain rate tensor is approximately equal to the rate�

of�deformation tensor when the displacements and displacement gradients are small�i�e�� for small strains�

Eij ≈ eij ≈ εij =12

(∂uj

∂xi+

∂ui

∂xj

)

=12

(∂vj

∂xi+

∂vi

∂xj

)= Dij �A��

In one dimension the small strain rate or rate�of�deformation becomes the spatial ve�locity gradient�

ε =∂vx

∂x. �A��

The temporal integral of the small strain rate

ε =∫ t

t0

∂vx

∂xdt. �A��

only has a physical signi�cance if the strain rate is small� or if a material integrationis used� i�e�� evaluating the integral by following each point through its motion� Thelatter case is often a formidable problem� but an approximation can be achieved byassuming spatially constant strain rate� The conventional strain ε in �A�� can befound as �� p ��

ε = eε − 1. �A��

Appendix B

Fisher information matrix and

Cram�r�Rao bound

The Cram�r�Rao bound is found as the diagonal elements of the inverted Fisher matrixF �� p� � �

F = −E

{[∂2

∂v21

ln px(x |v1, ε) ∂2

∂v1 ε ln px(x |v1, ε)∂2

∂v1 ε ln px(x |v1, ε) ∂2

∂ε2 ln px(x |v1, ε )

]}. �B��

First the partial derivatives found in F are calculated�

∂

∂v1ln px(x |v1, ε) =

2|ρ|ab

Im

{M∑

m=1

R(m)e−iv1+(m−1)∆rε

a

}�B��

∂

∂εln px(x |v1, ε) =

2|ρ|∆r

abIm

{M∑

m=1

(m − 1)R(m)e−iv1+(m−1)∆rε

a

}�B��

E

{∂2

∂v21

ln px(x |v1, ε )}

= −2|ρ|a2b

Re

{M∑

m=1

E{R(m)

}e−i

v1+(m−1)∆rεa

}

=2|ρ|2

a2(1 − |ρ|2)M �B��

E

{∂2

∂ε2ln px(x |v1, ε )

}= − 2|ρ|2(∆r)2

a2(1 − |ρ|2)M∑

m=1

(m − 1)2

=2|ρ|2(∆r)2

a2(1 − |ρ|2)(M − 1)M(2M − 1)

6�B��

��

�� Fisher information matrix and Cram�r�Rao bound

E

{∂2

∂v1εln px(x |v1, ε)

}= − 2|ρ|2∆r

a2(1 − |ρ|2)M∑

m=1

(m − 1)

=2|ρ|2∆r

a2(1 − |ρ|2)(M − 1)M

2�B��

In these calculations it has been utilized that

E{R(m)

}= σ2ρ(m) = σ2|ρ|ei

v1+(m−1)∆rεa . �B��

Thus the Fisher information matrix is

F =[

F11 F12

F21 F22

]=

2M |ρ|2a2(1 − |ρ|2)

[1 M−1

2 ∆rM−1

2 ∆r (M−1)(2M−1)6 (∆r)2

]. �B��

The inverse of F is found as

F−1 =1|F |

[F22 −F12

−F21 F11

]�B� �

where the determinant |F | is

|F | =(

2M |ρ|2a2(1 − |ρ|2)

)2

(∆r)2M2 − 1

12. �B��

The Cram�r�Rao bounds are then �nally found as

var(v1) ≥ F22

|F | =a2(1 − |ρ|2)

|ρ|22M − 1

M(M + 1)�B��

and

var(ε) ≥ F11

|F | =a2(1 − |ρ|2)(∆r)2|ρ|2

6M(M2 − 1)

�B��

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Part II

Papers

��

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