Propagation of Shear Waves Generated by a

Finite-Amplitude Ultrasound Radiation Force

in a Viscoelastic Medium

by

Alexia Giannoula

A thesis submitted in conformity with the requirements for the Degree of Doctor of Philosophy

The Edward S. Rogers Sr. Department of Electrical and Computer Engineering

Collaborative Program with the Institute of Biomaterials and Biomedical Engineering

University of Toronto

© Copyright by Alexia Giannoula (2008)

ii

ABSTRACT Propagation of Shear Waves Generated by a Finite-Amplitude Ultrasound

Radiation Force in a Viscoelastic Medium Alexia Giannoula

Doctor of Philosophy, 2008 The Edward S. Rogers Department of Electrical and Computer Engineering

Collaborative program with the Institute of Biomaterials and Biomedical Engineering University of Toronto

A primary purpose of elasticity imaging, commonly known as elastography, is to extract the

viscoelastic properties of a medium (including soft tissue) from the displacement caused by a stress

field. Dynamic elastography methods that use the acoustic radiation force of ultrasound have several

advantages, such as, non-invasiveness, low cost, and ability to produce a highly localized force field.

A method for remotely generating localized low-frequency shear waves in soft tissue is

investigated, by using the modulated radiation force resulting from two intersecting quasi-CW

confocal ultrasound beams of slightly different frequencies. In contrast to most radiation force-

based methods previously presented, such shear waves are narrowband rather than broadband. As

they propagate within a viscoelastic medium, different frequency-dependent effects will not

significantly affect their spectrum, thereby providing a means for measuring the shear attenuation

and speed as a function of frequency. Furthermore, to improve the detection signal-to-noise-ratio

(SNR), increased acoustic pressure conditions may be needed, causing higher harmonics to be

generated due to nonlinear propagation effects. Shear-wave propagation at harmonic modulation

frequencies does not appear to have been previously discussed in the elastography literature.

The properties of the narrowband shear wave propagation in soft tissue are studied by using

the Voigt viscoelastic model and Green’s functions. In particular, the manner in which the

characteristics of the viscoelastic medium affect their evolution under both low-amplitude (linear)

and high-amplitude (nonlinear) source excitation and conditions that conform to human safety

standards. It is shown that an exact solution of the viscoelastic Green’s function is needed to

properly represent the propagation in higher-viscosity media, such as soft tissue, at frequencies

much beyond a few hundred hertz. Methods for estimating the shear modulus and viscosity in

viscoelastic media are developed based on both the fundamental and harmonic shear components.

iii

ACKNOWLEDGEMENTS

This thesis would never have come to fruition without the help and support of several people.

My heartful thanks go first and foremost to my supervisor, Professor Richard Cobbold, for his

support and guidance throughout the past few years. His willingness to become my official

supervisor in the middle of the academic year (and my Ph.D. program) and his help, advice and

encouragement, kept me motivated to continue my research work at a very difficult moment of my

graduate studies. Words cannot describe my gratitude for this. His enthusiasm to always learn and

explore different scientific areas, his patience and persistence to truth in research, have been an

excellent example to me, both in research and life.

It has been the greatest fortune to me, during the course of my studies at the University of

Toronto, to join the Institute of Biomaterials and Biomedical Engineering and become a member of

the Ultrasound group. I am very grateful to my group mates – Muris Mujagic, Derek Wright, Renee

Warriner, Roozbeh Arshadi, Alfred Yu, Samsher Sidhu, and Al Aly – for supporting a team spirit

and a pleasant working environment.

I will be forever grateful to my parents and my twin sister, Georgia, for their endless love

and support throughout my graduate studies in Toronto. Their frequent phone calls from Greece to

share their everyday news, their encouragement to all my decisions and their help have been

invaluable to me.

Finally, I would like to thank Gerasimos Konstantatos for his everyday encouragement and

understanding. Through his personal example of excellence and devotion to research (in the field of

nanotechnology), he has kept me motivated and has taught me to appreciate not only the educational

but also the everyday benefits of being a graduate student in Toronto. I could not have completed

my Ph.D. thesis without him.

iv

TABLE OF CONTENTS

ABSTRACT.................................................................................................................................. ii

ACKNOWLEDGEMENTS ....................................................................................................... iii

TABLE OF CONTENTS ........................................................................................................... iv

List of Tables ............................................................................................................................... vi

List of Figures............................................................................................................................. vii

List of Symbols and Units in MKS ........................................................................................... xv

CHAPTER 1 INTRODUCTION................................................................................................ 1 1.1 Elastography: Overview and Historical Background ................................................................ 1

1.1.1 Relating Tissue Pathology with Changes in Tissue Stiffness............................................. 2 1.1.2 The Bulk and Shear Elastic Moduli .................................................................................... 4 1.1.3 Static vs. Dynamic Elastography ........................................................................................ 5

1.2 Dynamic Elastography Methods................................................................................................ 6 1.2.1 Stress Generation Using External Mechanical Vibration ................................................... 7 1.2.2 Stress Generation Using Acoustic Radiation Force............................................................ 9

1.3 Principles of the Acoustic Radiation Force ............................................................................. 11 1.4 Narrowband vs. Broadband Shear Wave Generation .............................................................. 13 1.5 Research Objectives................................................................................................................. 14 1.6 Organizational Outline............................................................................................................. 15

CHAPTER 2 Shear-Wave Propagation in Viscoelastic Media ............................................. 16

2.1 Introduction to Viscoelasticity................................................................................................. 16 2.1.1 Viscoelastic Models for Biological Tissue ....................................................................... 17 2.1.2 The Helmholtz Equation for the Voigt and Maxwell Model............................................ 20

2.2 Principles of the Shear Wave Propagation............................................................................... 23 2.3 Green’s Function in an Infinite Viscoelastic Medium............................................................. 24

2.3.1 Green’s function for an elastic medium............................................................................ 24 2.3.2 Green’s function for a viscoelastic medium based on the Voigt model ........................... 25

2.4 Chapter Summary .................................................................................................................... 28

CHAPTER 3 Modulated Finite-Amplitude Acoustic Radiation Force ................................ 30

3.1 Description of the Dual-Beam System Model ......................................................................... 30 3.2 Dual-beam System Model........................................................................................................ 33 3.3 Heating Effects......................................................................................................................... 40 3.4 Effects of the Excitation Conditions ........................................................................................ 41

3.4.1 The Parameters of Nonlinearity, Focusing Gain and Absorption..................................... 42 3.4.2 Effects of the Nonlinearity Parameters on the Harmonic Radiation Force Components . 43 3.4.3 Center Frequency and Source Pressure: Effects on the Harmonic Force Components.... 46

3.5 Chapter Summary .................................................................................................................... 48

CHAPTER 4 Narrowband Shear-Wave Propagation: the Fundamental Component ....... 50 4.1 Generation of Low-Frequency Shear Waves........................................................................... 50 4.2 Analysis of the Shear Displacement Field ............................................................................... 53

4.2.1 Effects of the Emission Duration...................................................................................... 54 4.2.2 Effects of the Coupling Wave........................................................................................... 56

v

4.2.3 Effects of the Shear Viscosity........................................................................................... 57 4.2.4 Effects of the Shear Speed ................................................................................................ 60 4.2.5 Effects of the Spatial Distribution of the Force ................................................................ 62

4.3 Chapter Summary .................................................................................................................... 64

CHAPTER 5 Narrowband Shear-Wave Propagation: the Harmonic Components ........... 66

5.1 Time and Frequency-Domain Analysis ................................................................................... 66 5.1.1 Exact vs. Approximate Viscoelastic Green’s Function .................................................... 66 5.1.2 Effects of the Frequency-Dependent Speed and Attenuation ........................................... 71

5.2 Time-Frequency Analysis........................................................................................................ 74 5.3 Chapter Summary .................................................................................................................... 77

CHAPTER 6 Estimating the Properties of a Viscoelastic Medium ...................................... 79 6.1 Challenges in Assessing the Viscoelastic Properties of Tissue ............................................... 79 6.2 Extracting the Voigt-Model Parameters: based on the Fundamental Component................... 80

6.2.1 Calculation of the Frequency-Dependent Shear Speed and Attenuation.......................... 81 6.2.2 Estimating the Shear Viscosity from the Peak Time Differences..................................... 83

6.3 Extracting the Voigt-Model Parameters: based on the Harmonic Components ...................... 88 6.4 The Inverse-Problem Approach............................................................................................... 96 6.5 Chapter Summary .................................................................................................................. 101

CHAPTER 7 Summary and Conclusions .............................................................................. 102 7.1 Summary ................................................................................................................................ 102 7.2 Thesis contributions ............................................................................................................... 105 7.3 Suggestions for Further Work................................................................................................ 105

REFERENCES......................................................................................................................... 108

APPENDIX A Derivation of the Viscoelastic Green’s Function in k-Space...................... 116

vi

List of Tables

Table 3.1: The maximum amplitudes of the CW fundamental and second-harmonic radiation force at focus for six combinations of the parameters N, A and G for a pulse duration of 5.0 ms. The temperature increase at focus arising from the first two harmonics is also indicated..................................................................................................................................... 44

Table 6.1: Parabolic approximation Y = P0+P1ηs+P2ηs2 of the normalized peak time difference

[ ]),mod( TDrtY Δ= with shear viscosity ηs, at two different radial distances. ...................... 85

Table 6.2: Maximum number of cycles for a temperature increase ≤1°C. ........................................ 90

Table 6.3: Estimated shear modulus and viscosity by applying nonlinear LS-fitting (‘trust-region’ algorithm [90]) to the shear speed calculations of Figure 6.8, for the case of moderate viscosity (1.0 Pa⋅s). .................................................................................................... 92

Table 6.4: Estimated shear modulus and viscosity by applying nonlinear LS-fitting (‘trust-region’ algorithm [90]) to the shear speed calculations of Figure 6.10 for the case of a higher viscosity (5.0 Pa⋅s) and a shear modulus of 2.5 kPa....................................................... 94

Table 6.5: Estimated shear modulus and viscosity by applying nonlinear LS-fitting (‘trust-region’ algorithm [90]) to the shear attenuation calculations of Figure 6.11(a)-(c) for a moderate shear viscosity (1.0 Pa⋅s) and a shear speed of 1.54 m/s (μl ≈ 2.5 kPa). .................... 95

vii

List of Figures

Figure 1.1: From left to right: longitudinal sonogram, elastogram and gross pathology photograph from an ovine kidney in vitro. The elastogram demonstrates structures that are consistent with a stiff (black) renal cortex and medullary pyramids (of which at least seven are seen), softer (white) columns of Bertin and very soft fatty areas at the base of the columns in the renal sinus. Note that the renal sinus is hypoechoic and is not well visualized on the sonogram. Reprinted from J. Med. Ultrasound, 29, Ophir et al. [7], pp. 155-171, ©2002, with kind permission of Springer Science and Business Media...................... 2

Figure 1.2: (a) Sonogram and (b) elastogram of an in-vivo benign breast tumour (fibroadenoma) and (c) sonogram and (d) elastogram of an in-vivo malignant breast tumour (invasive ductal carcinoma). Note that black indicates stiff and white indicates soft tissue. Adapted version, reprinted from Ultrasonics, 38, Konofagou et al. [11], pp. 400-404, ©2000, with permission from Elsevier. ............................................................................................................ 3

Figure 1.3: Summary and comparison of the shear )(μ and bulk moduli (K) of hard and soft tissue. The lower portion shows the shear modulus and the upper, the bulk modulus (N/m2 ≡ Pa). Adapted version, reprinted from Ultrasound Med. Biol., 24, Sarvazyan et al. [13], pp. 1419-1435, ©1998, with permission from Elsevier. ..................................................... 4

Figure 1.4: (a) Sonogram and (b) grayscale elastogram of an in-vivo breast sample. The elastogram clearly indicates the presence of a tumour (invasive ductal carcinoma) as a low-strain (hard) region. (c) Color-coded elastogram superimposed on the B-mode image. Adapted version, reprinted from Proc. IEEE Ultrason. Symp., Nitta et al. [22], pp. 1885-1889, ©2002, with permission from IEEE. ................................................................................. 6

Figure 1.5: Illustrating the principles of the elasticity imaging system proposed by Yamakoshi et al. [30], which was able to measure both the amplitude and phase of the shear waves generated by an external mechanical vibrator. An ultrasonic transducer is simultaneously used as a probing source, transmitting and receiving ultrasound waves, which are then used to extract the amplitude/phase of the vibration based on the Doppler frequency effect...................................................................................................................................................... 8

Figure 1.6: Illustrating the principles of shear wave elasticity imaging (SWEI), proposed by Sarvazyan et al. [13]. A focused transducer generates remotely acoustic radiation force within tissue, which in turn, gives rise to low-frequency shear-wave propagation. The shear waves can be detected using the same or a second transducer that operates in imaging mode (can be placed at several positions), or using a surface acoustic detector and also, MR, or optical methods. ............................................................................................. 10

viii

Figure 2.1: The (a) Voigt and (b) Maxwell viscoelastic models consisting of a spring (described by the elastic modulus E) and a dashpot (described by the coefficient of viscosity η). The corresponding strain-time behaviour under constant stress σ0 at t = 0 are shown in (c) and (d). .................................................................................................................. 18

Figure 2.2: Frequency-dependent (a) shear speed and (b) shear attenuation for frequencies between 10-500 Hz. Both experimental measurements are shown (circles) and also theoretical curves as predicted by the Voigt (indicated by “V”) and Maxwell (indicated by “M”) models, based on (13) and (14). Reprinted with permission from J. Acoust. Soc. Am., 116, Catheline et al. [36], pp. 3734-3741, ©2004, American Institute of Physics.................... 22

Figure 2.3: Normalized total Green’s function cszz

szz

czzzz GGGG ++= at a radial distance r = 3.0

mm from the source (origin O). The medium parameters were assumed cs = 3 m/s, c = 60 m/s, ηs = 0.1 Pa⋅s, ηc = 0 and ρ = 1050 kg/m3. Note that the compressional speed is not representative of that for soft tissues (c ≈ 1500 m/s), but has been selected small for visualization purposes. The times of arrival for the pure compressional and pure shear waves are also demonstrated (r/c = 50 μs and r/cs = 1 ms). ...................................................... 26

Figure 2.4: (a) Speed and (b) attenuation of the shear waves as functions of frequency, based on the Voigt model (solid lines). The shear modulus was assumed μ l = 6.0 kPa and two different viscosities were considered, i.e., 0.5 and 3.0 Pa⋅s. The approximations adopted by Bercoff et al. [69] are also shown (dotted lines)................................................................... 27

Figure 3.1: Illustrating the principles of a simple confocal radiation force imaging system used to generate low-frequency shear waves. .................................................................................... 31

Figure 3.2: Showing (a) a cross-section on the focal plane and (b) an axial view of the source system consisting of two coaxial confocal ultrasound transducers A and B (see Figure 3.1). Transducer A (inner) is a confocal circular disk of aperture radius a1 and transducer B (outer) is a confocal annular disk with inner and outer radii a21 and a22, respectively. The dynamic radiation force FΔ produced by the interference of the two beams is also shown at the common focal point. ........................................................................................................ 32

Figure 3.3: Axial pressure amplitude of the n’th harmonic component (n = 1...4) for (a) beam A (confocal circular disk) and (b) beam B (confocal annular ring). The corresponding slow phases are shown in (c) and (d). Note that close to the source, i.e. z < 3 cm, the nonlinear computational method causes the results to be unreliable. ........................................................ 34

Figure 3.4: (a) Temporal profile of the finite-amplitude dynamic radiation force at the geometric focus (z = 7.0 cm). (b) Normalized frequency spectrum of the total radiation force, where the individual harmonic force components can be clearly observed at 200, 400, 600 and 800 Hz (n = 1…4). (c) Normalized spatiotemporal patterns of the total finite-amplitude radiation force (five harmonics retained) and (d) the fundamental component........ 36

Figure 3.5: Normalized radiation force per unit volume on the geometric focal plane (7.0 cm) for: (a) the fundamental (200 Hz), (b) the second-harmonic (400 Hz), (c) the third-harmonic (600 Hz) and (d) the forth-harmonic (600 Hz) radiation force.................................. 37

Figure 3.6: Normalized radial profiles of the n’th-harmonic dynamic radiation forces ΔnF ,

n = 1…4, at the geometric focus (z = 7.0 cm). The n’th-harmonic force beamwidth was found approximately n-1 times that of the fundamental force beamwidth at -6 dB. The ‘finger’ effect in the pattern of the harmonic sidelobes can be also observed in the figure inset. ........................................................................................................................................... 38

ix

Figure 3.7: Normalized radial profiles of the n’th-harmonic pressure (n = 1...4) for (a) beam A and (b) beam B, at the geometric focal plane (z = 7.0 cm). The extra sidelobes, known as ‘fingers’ can be seen at the higher harmonics. Note that the results for n = 4 in (b) are of limited value due to approximations used in the numerical computations................................ 39

Figure 3.8: Axial pressure amplitudes of the n’th harmonic components (n = 1…3) of beam A (solid, dotted and thick-dotted lines) and beam B (stars, crosses and dots) for various values of the dimensionless parameters of nonlinearity (N), absorption (A) and total focusing gain (G = Ga+Gb). In (a) and (d), the third-harmonic components were very small and are not shown. ..................................................................................................................... 42

Figure 3.9: Normalized radial patterns of the fundamental and second-harmonic radiation force components on the focal plane for various nonlinearity (N), absorption (A) and total focusing gain (G = Ga+Gb) parameters. ..................................................................................... 45

Figure 3.10: (a) Maximum amplitude of the n’th harmonic CW modulated radiation force (n = 1..3) with the center frequency. (b) Total temperature increase at focus (for n = 1..3) with the center frequency for an emission duration of 10 ms. The maximum emission duration, based on (37), for which the temperature increase at focus is 1.0°C, is shown with f0 in (b) inset. The source pressure was assumed to be 450 kPa and the attenuation coefficient was 0.35 dB/(cm⋅MHz 1.1). ....................................................................................... 46

Figure 3.11: Radial beamwidth at -6 dB with the center frequency of the n’th-harmonic modulated force (n = 1…3), on the geometric focal plane at t = 0, 5.0 ms, etc. The source pressure was assumed 450 kPa and the attenuation coefficient 0.35 dB/(cm⋅MHz1.1). ............ 47

Figure 3.12: (a) Maximum amplitude of the n’th harmonic CW modulated radiation force (n = 1..3) with the source pressure. (b) Total temperature increase at focus (for n = 1..3) with the source pressure for an emission duration of 10 ms. The maximum emission duration, based on (37), for which the temperature increase at focus is 1.0°C, is shown with P0 in (b) inset. The center frequency was assumed 2.0 MHz and the attenuation coefficient 0.35 dB/(cm⋅MHz 1.1). .............................................................................................. 48

Figure 4.1: Normalized total displacement csz

sz

czz uuuu ++= at a radial distance r = 3.0 mm

from the source (origin O) for a spatiotemporal impulse (dash-dotted line) and a modulated force at Δf = 500 Hz (solid line) of duration D = 300 μs. The medium parameters assumed are given in the caption of Figure 2.3. The spreading effects of the time convolution are evident. In the purely elastic case, the coupling wave is active for r/cs-r/c+D ≈ 1.45 ms, where r/c and r/cs are the times of arrival for the pure compressional and pure shear waves. ................................................................................................................ 51

Figure 4.2: Showing the two components of the displacement vector produced by the coupling term Greens function on the (r, z) plane. The contributions at three different angles of (a) 0°, (b) 45° and (c) 90° at a distance of 3.0 mm from the geometric focus are shown assuming cs = 3 m/s and c = 1550 m/s. The medium bulk and shear viscosities were taken to be ηc = 0 and ηs = 0.15 Pa⋅s, respectively. Also assumed, was an emission duration of D = 0.5 ms corresponding to 1/4 cycle of the modulated wave (Δf = 500 Hz). (d) Schematic describing the three different locations on the (r, z) plane for (a), (b) and (c)......... 52

Figure 4.3: Normalized polar plots showing the contributions to the z-component shear wave displacement from the shear term and the sum of the shear and coupling terms at two locations on the (r, z) plane from a CW point source at the geometric focal point (0,0). (a)

x

R = 1.1 mm and (b) at R = 8.0 mm. The parameter values assumed are the same as those given in the caption of Figure 4.2, except that D = 20 ms. ........................................................ 53

Figure 4.4: Normalized shear displacement for five emission durations at two locations on the focal plane. (a) The nearfield point A(1.1mm, 0). (b) The near/farfield point B(8.0mm, 0). The displacements, which were normalized with respect to the maximum displacement amplitude at point A, do not include the effects of the coupling term. The parameter values assumed are the same as those given in the caption of Figure 4.2. ................................ 54

Figure 4.5: The effect of the emission duration D on the time difference between the positive and negative peaks, versus the radial distance. The parameter values assumed are the same as those given in the caption of Figure 4.2. ............................................................................... 55

Figure 4.6: Influence of the coupling term on the total displacement at (a) point A and (b) point B. The force duration was taken 1.2 ms. The displacements have been normalized according to the maximum amplitude of the pure shear term. .................................................. 56

Figure 4.7: Normalized shear plus coupling displacement at point B(8.0mm, 0) for three different values of shear viscosity ηs and an emission duration of D = 1.2 ms. The displacement amplitudes have been normalized by the maximum amplitude at point A(1.1mm, 0) for ηs = 0.001 Pa·s (see inset). .............................................................................. 57

Figure 4.8: For an emission duration of D = 1.2 ms and a shear viscosity ηs = 0.6 Pa⋅s showing: (a) The normalized positive peak displacement with the radial distance, for the pure shear component (dotted line) and the shear-plus-coupling component (solid line). For comparison, the 1/r decay is indicated by “x”. The equivalent curves for very low viscosity (0.001 Pa⋅s) are shown in (a) inset. Similar trends were obtained for the negative peak displacement. (b) Demonstrating the behaviour of the peak times with the radial distance. Both τmax, τmin (shear plus coupling) and s

maxτ , sminτ (pure shear) have been

calculated. Also the peak times corresponding to a point-source are shown: pvmaxτ (viscoelastic media based on (42)) and pe

maxτ = r/cs (elastic media). ................................... 59

Figure 4.9: Normalized displacements at point B(8.0 mm, 0) for three different values of the shear speed assuming an emission duration of 1.2 ms and a shear viscosity of 0.15 Pa⋅s. The corresponding displacements induced by a temporal impulse-like force are shown in the inset together with the half-width of each curve. The effects of the coupling term have been accounted for. .................................................................................................................... 60

Figure 4.10: Normalized positive peak displacements versus the shear speed for three different values of shear viscosity at point B(8.0 mm, 0), for (a) a temporal impulse and (b) a force of duration D = 1.2 ms. Both the pure shear components (dashed curves indicated by “S”) and the shear-plus-coupling components (solid curves) are shown. The curves have been normalized according to the maximum of the latter for ηs = 0.001 Pa⋅s. ................................... 61

Figure 4.11: Normalized positive peak displacement at point B(8.0 mm, 0) for (a) several shear speeds and shear viscosities and (b) four different values of the shear speed with viscosity. The force duration was taken to be 1.2 ms and the effect of the coupling term has been included....................................................................................................................... 62

Figure 4.12: Comparison of the normalized displacement (shear plus coupling) waveforms produced at point B(8.0 mm, 0), when account is taken of the 3-D force distribution, to that produced by a point force at the focus. The graphs assume an emission duration 2.0 ms, a shear viscosity 1.0 Pa·s and a shear speed 3.0 m/s. .......................................................... 63

xi

Figure 5.1: Case of a lower shear viscosity with a modulation frequency of 100 Hz. Normalized shear displacement at (a) point A(1.1 mm, 0) and (b) point B(8.0 mm, 0), based on the exact viscoelastic Green’s function (solid lines) of (26) and the approximate Green’s function (dotted lines) [69], for a shear viscosity of 0.5 Pa⋅s and a shear modulus of 2.5 kPa. The source pressure, center frequency, attenuation coefficient and emission duration were taken to be 450 kPa, 2.5 MHz, 0.35 dB/(cm⋅MHz 1.1 ) and 20.0 ms, respectively. The corresponding normalized spectra are shown in (c) and (d), where the harmonic shear content can be observed. .................................................................................. 67

Figure 5.2: Case of a higher shear viscosity for a modulation frequency of 100 Hz. Normalized shear displacement at (a) point A(1.1 mm, 0) and (b) point B(8.0 mm, 0), based on the exact viscoelastic Green’s function (solid lines) of (26) and the approximate Green’s function (dotted lines) [69], for a shear viscosity of 5.0 Pa⋅s. The corresponding normalized spectra are shown in (c) and (d), where the harmonic shear content can be observed. The other assumed parameter values are given in the caption of Figure 5.1. ........... 68

Figure 5.3: Case of a lower shear viscosity with a modulation frequency of 350 Hz. Normalized shear displacement at (a) point A(1.1 mm, 0) and (b) point B(8.0 mm, 0), based on the exact viscoelastic Green’s function (solid lines) of (26) and the approximate Green’s function (dotted lines) [69], for a shear viscosity of 0.5 Pa⋅s. The corresponding normalized spectra are shown in (c) and (d), where the harmonic shear content can be observed. The rest of the parameter values assumed are the same as those given in the caption of Figure 5.1, except that D = 5.71 ms (corresponding to two cycles at Δf = 350 Hz)................................................................................................................................................... 69

Figure 5.4: Case of a higher shear viscosity with a modulation frequency of 350 Hz. Normalized shear displacement at (a) point A(1.1 mm, 0) and (b) point B(8.0 mm, 0), based on the exact viscoelastic Green’s function (solid lines) of (26) and the approximate Green’s function (dotted lines) [69], for a shear viscosity of 5.0 Pa⋅s. The corresponding normalized spectra are shown in (c) and (d), where the harmonic shear content can be observed. The rest of the parameter values assumed are the same as those given in the caption of Figure 5.1, except that D = 5.71 ms (corresponding to two cycles at Δf = 350 Hz)................................................................................................................................................... 70

Figure 5.5: Normalized amplitudes of the first three (n = 1..3) harmonic spectral components of the shear displacement as functions of the source pressure, at point B(8.0 mm, 0). The modulation frequency was assumed 350 Hz and the emission duration 5.71 ms (two cycles). The shear viscosity was taken 0.5 Pa⋅s, but was found not to affect significantly the spectral amplitudes with P0. The rest of the parameter values assumed are given in the caption of Figure 5.1. ................................................................................................................. 71

Figure 5.6: Normalized shear displacements with the coupling term (top row) together with the spectra (bottom row) at point B(8.0 mm, 0). The modulation frequency was taken 350 Hz and the emission duration equal to (from left to right) two, six, twelve and twenty five cycles. The shear modulus and viscosity were assumed 2.5 kPa and 2.0 Pa⋅s, respectively..... 72

Figure 5.7: Normalized shear displacements with the coupling term (top row) together with the spectra (bottom row), for a modulation frequency of 350 Hz and emission duration of 30 cycles. The parameter values were taken μl = 2.5 kPa and (left column) ηs = 2.0 Pa⋅s at point B, (middle column) ηs = 0.001 Pa⋅s at point B and (right column) ηs = 2.0 Pa⋅s at point A(1.1 mm, 0). ................................................................................................................... 73

xii

Figure 5.8: Normalized shear displacement (including the coupling term) together with the Fourier spectrum at point B(8.0 mm, 0). The modulation frequency was taken 100 Hz and the emission duration 30 cycles. The shear modulus and viscosity were assumed 2.5 kPa and 2.0 Pa⋅s, respectively........................................................................................................... 74

Figure 5.9: SPWV distributions of the (a) shear and (b) shear plus coupling displacement at the nearfield point A(1.1 mm, 0). A modulation frequency of 100 Hz and emission duration 20.0 ms were assumed. The shear viscosity was taken 0.001 Pa⋅s and the rest of the parameter values assumed are given in the caption of Figure 5.1. ............................................ 75

Figure 5.10: SPWV distributions of the shear displacement (including the coupling term) at point B(8.0 mm, 0) for shear viscosity of (a) 0.001 Pa⋅s and (b) 2.0 Pa⋅s. A modulation frequency of 100 Hz and emission duration 20.0 ms (2 cycles) were assumed. The rest of the parameter values assumed are given in the caption of Figure 5.1. ...................................... 76

Figure 5.11: (a) Shear displacement (including the coupling term) together with the Fourier spectrum and (b) the corresponding SPWV distribution at point B(8.0 mm, 0) for shear viscosity of 2.0 Pa⋅s. The modulation frequency was assumed 350 Hz and the emission duration 28.6 ms (ten cycles). The rest of the parameter values assumed are given in the caption of Figure 5.1. ................................................................................................................. 77

Figure 6.1: Estimated shear speed (dispersion) for several modulation frequencies. Estimates based on the Voigt model are also shown with the dashed-dotted lines. The coupling term was included and the emission duration was taken D = T/2 = 1/(2Δf), i.e. half cycle for each frequency. The shear viscosity was assumed (a) 0.2 Pa·s and (b) 1.0 Pa·s. Two shear moduli of 2.5 and 7.5 kPa were assumed in each case (the corresponding shear speeds can be obtained from 2

sl cρ≈μ ). ...................................................................................................... 82

Figure 6.2: Estimated shear attenuation for several modulation frequencies. Estimates based on the Voigt model are also shown with the dashed-dotted lines. The coupling term was included and the emission duration was taken D = T/2 = 1/(2Δf), i.e. half cycle for each frequency. A shear viscosity of 1.0 Pa·s was assumed Two shear moduli of 2.5 (top) and 7.5 kPa (bottom) were also assumed (the corresponding shear speeds can be obtained from 2

sl cρ≈μ )........................................................................................................................... 83

Figure 6.3: Parabolic approximation [ ]),mod( TDrtY Δ= versus the shear viscosity (solid lines), for five different values of the emission duration D at radial distances of (a) 6.0 mm and (b) 10.0 mm from the source. Numerical values are shown with symbols. Note that for D = 2.0, 4.0 & 6.0 ms, Y is not defined since mod(D, T) = 0, and thus, a normalization with T has been instead performed. The coupling term has been accounted for and the shear speed has been taken 3.0 m/s. ........................................................................ 84

Figure 6.4: The parabolic coefficients P0, P1 and P2 versus the emission duration, at radial distances of 6.0 and 10.0 mm (see Table 6.1). It can be observed that P1 and P2 are very close to one another for all emission durations (ranging between 1-3 cycles). Furthermore, the ratio of P0 for the two locations is almost constant for all values of D and approximately equal to the inverse distances (i.e. ≈ 0.6, as shown with ‘square’ symbols). .... 85

Figure 6.5: Calculated (symbols) and linearly approximated (dashed lines) values of the inverse parabolic coefficients 1/P1 and 1/P2 versus the emission duration over the time intervals of (a) 4.2-6.0 ms (i.e. 2-3 cycles) and (b) 2.2-4.0 ms (i.e. 1-2 cycles). P1 and P2 correspond to the parabolic approximation at a radial distance of 6.0 mm (see Table 6.1). ............................ 86

xiii

Figure 6.6: Block diagram describing the proposed method for estimating the shear viscosity by measuring the normalized time differences Y1 and Y2 at two locations r1 and r2, for several emission durations D. .................................................................................................... 88

Figure 6.7: Case of a lower viscosity (0.2 Pa⋅s). Estimated shear speed (dispersion) for several modulation frequencies based on the (a)-(b) fundamental, (c)-(d) second-harmonic and (e)-(f) third-harmonic component. Estimates based on the Voigt model are also shown with the dash-dotted lines. The coupling term was included and two shear speeds of (a), (c), (e) 1.54 m/s (μl ≈ 2.5 kPa) and (b), (d), (f) 2.67 m/s (μl ≈ 7.5 kPa) were assumed. The emission duration was two cycles for each frequency............................................................... 89

Figure 6.8: Case of a moderate viscosity (1.0 Pa⋅s). Estimated shear speed (dispersion) for several modulation frequencies based on the (a)-(b) fundamental and the (c)-(d) second-harmonic component. Estimates based on the Voigt model are also shown with the dash-dotted lines. The coupling term was included and two shear speeds of (a), (c) 1.54 m/s (μl ≈ 2.5 kPa) and (b), (d) 2.67 m/s (μl ≈ 7.5 kPa) were assumed. The emission duration was two cycles for each frequency. ........................................................................................... 91

Figure 6.9: Nonlinear LS-fitting based on the ‘trust-region’ algorithm [90], applied to the shear speed calculations of Figure 6.8 (see also Table 6.3), for both the fundamental and second-harmonic component and two shear speeds of (a) 1.54 m/s (μl ≈ 2.5 kPa) and (b) 2.67 m/s (μl ≈ 7.5 kPa). Note that in (a), the last two values of the shear speed for n = 2 were excluded from the fitting algorithm. ................................................................................. 92

Figure 6.10: Case of a higher viscosity (5.0 Pa⋅s) and modulus of 2.5 kPa. Estimated shear speed (dispersion) for several modulation frequencies based on the (a) fundamental, (b) second-harmonic and (c) LF component. Both the exact (‘star’ symbols) and approximate (‘x’ symbols) solutions of the shear Green’s function were used. Estimates based on the Voigt model are also shown with the ‘circle’ symbols. The emission duration was taken D = 2T, i.e. two cycles for each frequency................................................................................. 93

Figure 6.11: Case of a moderate viscosity (1.0 Pa⋅s) and a shear speed of 1.54 m/s (μl ≈ 2.5 kPa). Estimated shear attenuation for several modulation frequencies based on the (a) fundamental, (b) second-harmonic and (c) third-harmonic component. Estimates based on the Voigt model are also shown with the dash-dotted lines. For n = 2, the exponential fitting was performed over the range d (as for n = 1) and d2 ≈ d-λs (closer to the source). For n = 3, the fitting was performed over d2 and d3 ≈ d-1.2λs. (d) Nonlinear LS-fitting (‘trust-region’ algorithm [90]) applied to the shear speed calculations of (a) and (b) (see also Table 6.5), for the fundamental and second-harmonic component. ................................... 95

Figure 6.12: Case of a higher viscosity (5.0 Pa⋅s) and shear modulus of 2.5 kPa. Estimated shear attenuation for several modulation frequencies based on the fundamental (n = 1) shear component. Both the exact (‘star’ symbols) and approximate (‘x’ symbols) solutions of the shear Green’s function were used. Estimates based on the Voigt model are also shown with the ‘circle’ symbols. The extracted Voigt-model parameters using LS-fitting are also provided. ....................................................................................................................... 96

Figure 6.13: Maps of the estimated local (a), (c) shear modulus and (b), (d) shear viscosity from (a)-(b) the fundamental and (c)-(d) the second-harmonic shear displacement on the (r,z) plane. The modulation frequency was assumed 100 Hz, the emission duration 20.0 ms and the rest of the parameter values are given in the caption of Figure 5.1......................... 98

xiv

Figure 6.14: Maps of the estimated local (a) shear modulus and (b) shear viscosity from the low-frequency (LF) shear component on the (r,z) plane. The modulation frequency was assumed 500 Hz, the emission duration 4.0 ms (2 cycles) and the rest of the parameter values are given in the caption of Figure 5.1. ............................................................................ 99

Figure 6.15: Maps of the estimated local (left column) shear modulus and (right column) shear viscosity from (a)-(b) the fundamental, (c)-(d) the second-harmonic, (e)-(f) the third-harmonic, (g)-(h) the forth-harmonic and (i)-(j) the mean average of the first three harmonic components on the (r,z) plane. The modulation frequency was assumed 100 Hz and the emission duration 20.0 ms........................................................................................... 100

xv

List of Symbols and Units in MKS

αn(f) = α0(nf)γ Attenuation of the medium [Np/m]

α0 Attenuation coefficient [dB/(cm⋅MHzγ) = 11.5×10-6γ Np/(m⋅Hzγ)]

αs Shear attenuation [1/m]

A Absorption parameter [dimensionless]

a1 Outer aperture radius (beam A) [m]

a21, a22 Inner and outer aperture radii (beam B) [m]

β Coefficient of nonlinearity [dimensionless]

γ Power law [dimensionless]

c Speed of sound [m/s]

cs Shear speed [m/s]

cv Heat capacity per unit volume [Watts·s/(m3·°C)]

δij Kronecher delta-function (δij = 1 if i = j and 0 otherwise) [dimensionless]

δ(x) Dirac delta function [1/(units of x)]

Δφn(r) Phase difference of the n’th harmonic radiation force at location r [rads]

Δf Modulation frequency [Hz]

ΔΤn Temperature increase due to the n’th harmonic radiation force [°C]

D Emission duration [s]

E Young’s (elastic) modulus [Pa]

F Geometric focal depth [m]

fa, fb Center frequencies of the beams A and B [Hz]

fj(r,t) Body force acting in the j-direction (j = x,y,z) at location r [Newtons/m3] Δ

nF n’th-harmonic dynamic acoustic radiation force per unit volume [Newtons/m3]

f0 Center frequency [Hz]

xvi

G Focusing gain parameter [dimensionless]

Gij i-component of the Green’s function due to a force acting in the j-direction

H(.) Heaviside function (unit step function)

ηs Shear viscosity (absolute) [Pa⋅s]

ηc Bulk viscosity [Pa⋅s] ΔnI n’th-harmonic dynamic acoustic intensity [Watt/m2]

k =ω/cs(ω)-jαs(ω) Complex Propagation Constant [1/m]

k =ω/cs Shear wave number [1/m]

λl Bulk modulus (first Lame constant) [Pa]

λs Shear wavelength [m]

lp Plane-wave shock formation distance

μl Shear modulus (second Lame constant) [Pa]

MI Mechanical Index [dimensionless]

n Harmonic number

N Nonlinearity parameter [dimensionless]

Nh Number of harmonics

pn,a, pn,b n’th harmonic acoustic pressure (beams A and B) [Pa]

P0 Source pressure amplitude [Pa]

ρ Medium density [kg/m3]

r Radial distance [m]

r0 Rayleigh distance [m]

TI Thermal Index [dimensionless]

Tn Period of the n’th-harmonic radiation force [s]

ui = ui(r,t) i’th (i = x,y,z) component of the particle displacement at location r [m]

uc(r,t) Compressional component of the displacement [m]

ucs(r,t) Coupling component of the displacement [m]

us(r,t) Shear component of the displacement [m]

vn,a, vn,b n’th harmonic particle velocity (beams A and B) [m/s]

vs = ηs/ρ Kinematic shear viscosity [stokes, 1 stoke = 10-4 m2/s]

ω0 = 2πf0 Center angular frequency [rads/s]

z Axial distance [m]

1

CHAPTER 1 INTRODUCTION

Elastography can be defined as the science and methodology of estimating the mechanical

properties of a medium (including soft tissue). Elastography methods generally use an external

source of force to produce a static or dynamic stress distribution on the probed medium. The applied

stress causes a displacement distribution within the medium, which can be measured or imaged by

ultrasound, magnetic resonance (MR), or optical methods. In this chapter, an overview of

elastography and its relation to tissue pathology will be presented. Next, the basic principles of the

static and dynamic methods will be described, and emphasis will be given on the dynamic methods

that rely on the acoustic radiation force of ultrasound. Of interest are the low-frequency narrowband

shear waves that can be generated by a modulated radiation force produced by the interference of

two continuous-wave (CW) ultrasound beams of slightly different frequencies. The advantages of

using narrowband shear waves to estimate the viscoelastic properties of tissue will be discussed.

Finally, the objectives of this thesis will be stated.

1.1 Elastography: Overview and Historical Background

The elasticity of soft tissues depends, to a large extent, on their molecular building blocks

(fat, collagen, etc.), and on the microscopic and macroscopic structural organization of these blocks

[1]. The standard medical practice of soft tissue palpation is based on qualitative assessment of the

low-frequency stiffness of tissue and has been used for centuries by physicians to distinguish

between normal and diseased tissues. Palpation is sometimes used to assess organs such as the liver,

and it is not uncommon for surgeons at the time of laparotomy to palpate tumors that were not

2

detected preoperatively using conventional imaging methods, such as Ultrasound, CT or MRI, since

none of these modalities currently provides the type of information elicited by palpation. Based on

the simple concept of palpation, tissue elastography or elasticity imaging [2] seeks to provide non-

invasive quantitative systems that can measure or image the local mechanical properties of tissue.

Such systems could significantly enhance the accuracy of diagnosis performed by physicians, even

at early stages of disease. An overview of elastography methods can be found in [3]-[7].

Figure 1.1: From left to right: longitudinal sonogram, elastogram and gross pathology photograph from an ovine kidney in vitro. The elastogram demonstrates structures that are consistent with a stiff (black) renal cortex and medullary pyramids (of which at least seven are seen), softer (white) columns of Bertin and very soft fatty areas at the base of the columns in the renal sinus. Note that the renal sinus is hypoechoic and is not well visualized on the sonogram. Reprinted from J. Med. Ultrasound, 29, Ophir et al. [7], pp. 155-171, ©2002, with kind permission of Springer Science and Business Media.

1.1.1 Relating Tissue Pathology with Changes in Tissue Stiffness

Pathological changes are generally correlated with local changes in tissue stiffness (see

Figure 1.1 and Figure 1.2). Many cancers, such as scirrhous carcinoma of the breast, liver

metastases, prostatic carcinoma, and thyroid cancer, appear as extremely hard nodules [8]. Other

types of breast cancers (e.g. intraductal and papillary carcinoma) are soft [9]. Benign fibrocystic

disease has been known to be extremely hard on very rare occasions and the glandular tissues of the

breast are firm in relation to the soft fatty areas. Furthermore, edematous skin tissue incurs an

3

increase in stiffness, which varies with the degree of edema [10]. Other diseases involve fatty and/or

collagenous deposits, which increase or decrease tissue elasticity.

Figure 1.2: (a) Sonogram and (b) elastogram of an in-vivo benign breast tumour (fibroadenoma) and (c) sonogram and (d) elastogram of an in-vivo malignant breast tumour (invasive ductal carcinoma). Note that black indicates stiff and white indicates soft tissue. Adapted version, reprinted from Ultrasonics, 38, Konofagou et al. [11], pp. 400-404, ©2000, with permission from Elsevier.

In many cases, despite the difference in stiffness, the small size of a pathological lesion

and/or its location deep in the body, preclude its detection and evaluation by palpation. In general,

the lesion may or may not possess acoustic backscatter properties, which would make it detectable

using ultrasound. For example, tumors of the prostate or the breast could be invisible or barely

visible in standard ultrasound examinations, yet be much harder than the embedding tissue.

Furthermore, diffuse diseases (e.g. cirrhosis of the liver) are known to significantly increase the

stiffness of the liver tissue as a whole [8]. However, they may appear normal in conventional B-

mode ultrasound examination. Complicated fluid-filled cysts could be invisible in standard

ultrasound, yet be quite softer than the embedding tissue. Since echogenicity and the stiffness of

tissue are generally uncorrelated, it is expected that measuring or imaging tissue stiffness will

4

provide new information that is related to tissue structure and/or pathology and will enhance the

process of diagnosis (see Figure 1.2).

1.1.2 The Bulk and Shear Elastic Moduli

The diagnostic value of a property of tissue depends upon the range of variation of that

property as a function of the state of tissue. In B-mode ultrasound imaging, the ability to

differentiate between various tissues within the body depends on changes in the acoustic properties,

which in turn, depend primarily on the bulk (elastic) modulus K, defined by [12]:

llK μ+λ=κ

=321 (1)

where κ denotes the adiabatic compressibility of the medium and λl, μl are called the first and

second Lamé (elastic) coefficients, which have been traditionally used in the analysis of wave

propagation in solids (their MKS units are Pascals). The second Lamé constant μl is the coefficient

(or modulus) of rigidity, often called the shear modulus and is defined as the ratio of shear stress to

shear strain (angular deformation) that is involved in the passage of a transverse wave. Note that for

a liquid μl →0, while for a perfectly rigid solid μl →∞. An alternative pair of elastic constants is

Young’s modulus and Poisson’s ratio, respectively.

Figure 1.3: Summary and comparison of the shear )(μ and bulk moduli (K) of hard and soft tissue. The lower portion shows the shear modulus and the upper, the bulk modulus (N/m2 ≡ Pa). Adapted version, reprinted from Ultrasound Med. Biol., 24, Sarvazyan et al. [13], pp. 1419-1435, ©1998, with permission from Elsevier.

5

The bulk properties of tissue are determined primarily by the molecular structure, while the

shear properties are determined by the higher level of tissue organization [7]. The range of variation

of the bulk modulus is very small, i.e., significantly less than an order of magnitude. On the other

hand, the shear modulus may change by many orders of magnitude, depending on the tissue [12]. A

comparison of the bulk and shear moduli properties, as summarized from the literature, is shown in

Figure 1.3. This suggests that by imaging one or more characteristics of shear wave propagation,

improved sensitivity to localized changes in elastic properties could be achieved, thereby, providing

useful diagnostic information [13].

1.1.3 Static vs. Dynamic Elastography

Elastography was introduced as an imaging modality capable of producing images of

internal strain, or images of the shear (or Young’s) modulus estimates [2]. During the past two

decades, various methods have been proposed for measuring or estimating the tissue elasticity.

Elastography methods generally use a source of mechanical motion to produce a stress-field

distribution on the probed tissue (tissue excitation). The applied stress then, causes minute

displacements within the tissue, which can be measured using magnetic resonance (MR) [13], [14],

[15], [16], ultrasound [2], [7], [11] or optical techniques [13], [17].

Magnetic Resonance Elastography (MRE) was initially introduced in 1995 for estimating

the tissue elasticity using magnetic resonance. Such methods [14]-[16] have the potential of

obtaining both two-dimensional (2-D) and three-dimentional (3-D) displacement measurements (as

opposed to ultrasound-based methods which generally restrict imaging to 2-D scan planes) virtually

anywhere in the body and in any orientation. However, MRE has proven to be a very time

consuming and costly technique. Optical methods have also received limited interest due to their

requirement to have a “clear” medium in order to accurately detect the induced displacement, a fact

that prohibits its use in real biological media. On the other hand, the use of ultrasound has several

significant advantages, including real-time imaging capabilities, very high resolution in motion

estimation (~1 μm), simplicity, non-invasiveness, and relatively low cost [5], [18].

Elastography methods fall into two general categories, according to the temporal

characteristics of the applied excitation [5]: static (or quasi-static) and dynamic methods. In static

elastography [2], [11], [19]-[22], tissue is compressed slowly and the distribution of its displacement

is measured using one of the aforementioned medical imaging modalities. The measured

distribution of strain is related to the predicted distribution of stress and the resulting parameters of

6

moduli are deduced through elasticity equations. In 1991, Ophir et al. [2] used an external

compressor to form strain images.

The strain image is generally formed by applying one or two-dimensional cross-correlation

techniques to pairs of RF echo fields (e.g. A-lines) acquired before and after the tissue deformation

(compression). Often in these static methods, the strain alone is used as a surrogate for stiffness; that

is, low strain means high stiffness and high strain results from softer, or low-stiffness regions (see

Figure 1.4). The difficulty with the static methods is that they require knowledge of boundary

conditions outside of the region under investigation.

(a)

(b)

(c)

Figure 1.4: (a) Sonogram and (b) grayscale elastogram of an in-vivo breast sample. The elastogram clearly indicates the presence of a tumour (invasive ductal carcinoma) as a low-strain (hard) region. (c) Color-coded elastogram superimposed on the B-mode image. Adapted version, reprinted from Proc. IEEE Ultrason. Symp., Nitta et al. [22], pp. 1885-1889, ©2002, with permission from IEEE.

An alternative means to investigate tissue elastic properties is to generate transient acoustic

waves within the body and measure the associated transient motion in the spatiotemporal domain.

These are known as dynamic elastography methods, an overview of which will be presented in the

following section.

1.2 Dynamic Elastography Methods

In dynamic elastography methods [13], [23]-[33], a stress is created that has a transient

character. Such methods have the advantage of potentially revealing the dynamic properties of the

interrogated medium (such as viscosity) and at the same time, they overcome boundary problems

7

linked to the static methods.

Low-frequency shear waves are known to travel within tissue with a propagation speed that

is several orders of magnitude less than that of compressional waves. In fact, the wavelength of a

compressional wave may be several hundred times the dimensions of the tissue, whereas that for a

low-frequency transverse wave can be less than the dimensions of an organ. With increasing

frequency, the shear wave suffers increasing attenuation and eventually the energy it contributes

becomes small in comparison to the compressional wave. Because of the rapidly increasing

attenuation with frequency, it is not possible to use high-frequency shear waves, which suggests that

the spatial resolution might be rather limited. However, the presence of even a relatively small non-

uniformity in elasticity could create an appreciably larger disturbance to the shear wave propagation:

- much like dropping a small pebble into a pond and observing the resulting surface wave

disturbance [12].

It is possible to create stress in a dynamic method by using external mechanical vibration

[27]-[33] or by using an acoustic radiation force generated by a focused ultrasound beam [13], [24]-

[26], [34]-[38]. It should be also noted, that several attempts have been performed to measure tissue

motion induced by naturally applied stress (e.g. from breathing or cardiac motion), aiming at

correlating the motion patterns with pathological conditions of tissue [39]-[41]. However, the

potential of such methods is limited by the fact that the stress distribution is unknown. An overview

of the first two most significant groups of dynamic methods is provided below.

1.2.1 Stress Generation Using External Mechanical Vibration

In 1987, Krouskop et al. [42] proposed a method to measure non-invasively the elastic

modulus in soft tissue in vivo, by exciting a specific tissue region with an external vibrator

(operating at approximately 10 Hz) and measuring the resulting tissue motion using pulsed Doppler

ultrasound.

In 1988, Lerner et al. [27] proposed an ultrasonic imaging modality called sonoelasticity

imaging, which measured and imaged tissue displacement in response to externally applied

mechanical vibration using color Doppler ultrasound. The vibration fields, produced under low-

frequency monochromatic excitation (10-100 Hz), were mapped to a commercial ultrasound scanner.

Huang et al. [28] established the mathematical relationship between the particle vibration amplitude

and the received Doppler spectrum variance. An advantage of sonoelasticity imaging was its ease of

implementation on modern ultrasound scanners, its low computation requirements, and its ability for

8

real-time implementation. A theoretical analysis together with experimental verification (using liver

samples) of sonoelasticity imaging was performed by Parker’s group [43].

Yamakoshi et al. [30] in 1990, applied low-frequency (less than 200 Hz) sinusoidal vibration

at the surface of the interrogated medium and the resulting motion (both amplitude and phase) was

measured from the Doppler frequency shift of simultaneously transmitted/reflected probing

ultrasound waves (see Figure 1.5). The vibration phase image was used to determine the shear wave

propagation velocity.

Figure 1.5: Illustrating the principles of the elasticity imaging system proposed by Yamakoshi et al. [30], which was able to measure both the amplitude and phase of the shear waves generated by an external mechanical vibrator. An ultrasonic transducer is simultaneously used as a probing source, transmitting and receiving ultrasound waves, which are then used to extract the amplitude/phase of the vibration based on the Doppler frequency effect.

In 1999, Catheline et al. [31] proposed an ultrasound-based technique, which they called

transient elastography, to deal with artifacts induced by diffraction in sonoelastography (e.g. wave

reflection or standing waves). This method used a low-frequency (40–250 Hz) pulsed (transient)

excitation to create displacements in tissue, which were then detected using pulse-echo ultrasound.

The numerical values of elasticity and viscosity were deduced from the wave propagation (using

cross-correlation techniques), which was observed before the echoes from the tissue boundaries

contaminated the data, thereby avoiding problems with standing waves. This technique was

effective, but required frame rates exceeding those available in ultrasound scanners at that time.

9

The work of Catheline et al. [31] was extended by Sandrin et al. [32], [33] with the study of

2-D shear wave propagation. Specifically, with a method entitled time-resolved 2-D pulsed

elastography [32], a low-frequency (50-200 Hz) pulsed shear wave was generated using an external

vibrating device, while an ultra-fast ultrasonic imaging system, specifically designed for this

application, acquired 2-D frames at a very high frame rate (up to 10,000 frames/s). The tissue

displacement that was induced by the slowly propagating shear wave was measured using standard

cross-correlation techniques. A single low-frequency pulsed excitation was necessary to acquire the

full data set. Thus, acquisition times were considerably reduced compared to MRI or Doppler

detection methods. Furthermore, the proposed technique could be applied in the presence of tissue

movement.

1.2.2 Stress Generation Using Acoustic Radiation Force

An alternative solution to external mechanical vibrations is to use the acoustic radiation

force generated by an ultrasound focused beam. The elasticity imaging methods described

previously are characterized by the application of stress throughout the entire interrogated object.

However, it is possible to create a localized stress field within tissue with the acoustic radiation

force of ultrasound. The concept of assessing the mechanical properties of tissue by monitoring its

response to the acoustic radiation force, was first proposed by Sugimoto et al. [24] in 1990, who

attempted to measure tissue hardness by applying a minute deformation in the tissue with the

radiation force of a focused ultrasound beam and measured the induced deformation with a pulse-

echo method as a function of time.

A technique known as shear wave elasticity imaging (SWEI) was introduced by Survazyan

et al. [13] in 1998. In this method, which is illustrated in Figure 1.6, localized shear waves (1 kHz)

were remotely generated in tissue by the radiation force of a focused ultrasound beam using a short

modulating pulse. The propagation of the induced shear waves was detected using a laser-based

optical system and a magnetic resonance imaging (MRI) system. By analyzing the spatiotemporal

characteristics of the shear wave propagation using time-of-flight measurements, they were able to

estimate the shear-wave speed and thereby, the local shear modulus parameter.

Nightingale et al. [26] proposed the acoustic radiation force impulse (ARFI) method. It uses

a short-duration (< 1 ms) acoustic radiation force to generate localized displacements in a 2-D

region-of-interest (ROI) in tissue and monitors the tissue response using ultrasound correlation-

based techniques. The real-time ARFI system is capable of measuring displacements down to the

10

limits projected by the Cramer-Rao lower bound (about 0.2 μm). Furthermore, images of the time

that the tissue needs to reach its maximum displacement and images of the tissue recovery can be

also formed. Nightingale et al. [34] applied also the above process to visualize the propagation of

the induced transient shear waves and used direct inversion methods to estimate the shear elastic

modulus parameter.

Figure 1.6: Illustrating the principles of shear wave elasticity imaging (SWEI), proposed by Sarvazyan et al. [13]. A focused transducer generates remotely acoustic radiation force within tissue, which in turn, gives rise to low-frequency shear-wave propagation. The shear waves can be detected using the same or a second transducer that operates in imaging mode (can be placed at several positions), or using a surface acoustic detector and also, MR, or optical methods.

The supersonic shear imaging (SSI) technique, described in 2004 by Bercoff et al. [35],

provided a new ultrasound method for real-time (less than 30 ms) quantitative mapping of the

viscoelastic properties of soft tissue. It relied on the acoustic radiation force to remotely generate

low-frequency shear waves. A typical ultrasound pulse (~ 100 μs) was used to focus (“push”) the

beam and create a radiation force. By successively focusing the “pushing” beam at different depths

at a supersonic speed, two quasi-plane shear waves of stronger amplitude can be created that

propagate in opposite directions, by the constructive interference of all the resulting shear waves

from each “push”. An ultrafast ultrasound scanner (6000 frames/s), especially developed for this

purpose, was able to generate the supersonic moving shear source and to image the propagation of

11

the resulting transient plane shear waves. Using inversion algorithms, the shear elasticity and

viscosity of the medium was mapped quantitatively from the recorded propagation movie.

In 1998, Fatemi and Greenleaf [25] proposed a technique known as ultrasound-stimulated-

vibro-acoustography (USVA), in which two quasi-CW ultrasound beams of slightly different

frequencies are used to remotely generate a localized dynamic (oscillatory) radiation force at the

difference frequency, typically in the low kHz range. The modulated field that is created in the

intersection zone causes tissue in the focal region to vibrate at the beat frequency. In response, the

region emits low-frequency longitudinal waves (known as acoustic emission), which could be

detected externally by a hydrophone and which depend on the radiation force and the elastic

properties of the medium. In this way, high-resolution qualitative mapping of the local mechanical

properties of tissue can be achieved [25], [44].

By using a similar dual-beam setup, Konofagou and Hynynen [45] proposed a technique

named harmonic motion imaging (HMI), for estimating the local Young’s modulus from the

oscillatory (harmonic) tissue motion induced by a dynamic (harmonically-varying) radiation force.

Similarly to USVA, this method estimates tissue motion during and not after the application of the

force (the latter applies to most shear-based methods, such as these presented above). This could

potentially provide a better estimate of the mechanical properties of the excited lesion or tumour,

unaffected (or less affected) by the periolesional tissue. The technique was successfully tested on

both gel phantoms and samples of porcine muscle tissue [46].

1.3 Principles of the Acoustic Radiation Force

Understanding the acoustic radiation force dates back to 1902, when Rayleigh [47] described

a theory of the acoustic radiation pressure as an acoustic counterpart of that induced by

electromagnetic waves. Since then, several theories have been proposed in order to further explain

the underlying physics; notably the work of Langevin, as reported by Biquard [48], and the

experimental measurements reported by Altberg [49] in 1903. Understanding the physics and

obtaining equations that correctly describe the acoustic radiation pressure, has been the subject of

numerous studies over many years. Controversy has arisen from improperly posed problems,

confusion over definitions and the difficulties associated with nonlinear phenomena [50]-[52].

When ultrasound is incident on an obstacle whose properties differ from that of the

propagation medium, a force will be exerted and this consists of two components: the first is an

oscillatory component with a time-average of zero, arising from the time-varying acoustic pressure

12

acting on the body. The second is a steady component that is known as the radiation pressure. Its

presence is an inherent property of the nonlinear relation between pressure and density in the

propagation media [12], [53]. Thus, in a fluid it seems reasonable to express the total radiation

pressure by:

Radiation Pressure = pressure due to nonlinearity + pressure due to attenuation

which asserts that provided attenuation is present, a radiation pressure will exist even when the

propagation medium is perfectly linear.

In the absence of any obstacle, a finite-amplitude acoustic wave propagating in a lossless

medium, whose density is nonlinearly related to the pressure, will result in a transfer of momentum

from the wave to the medium. If the medium is also viscous, additional momentum will be

transferred and this may cause a bulk steady motion of the medium that is generally known as

acoustic streaming [53], [54]. Acoustic streaming is, therefore, a direct result of the radiation

pressure and it was Westervelt [54] the first to note that it should cause the radiation force to differ

from the total radiation force exerted on an obstacle. Both forms of momentum transfer, i.e., those

resulting from nonlinearity and absorption, contribute to the radiation pressure acting in the

propagation direction. In comparison to the effects of tissue nonlinearity, the effects of attenuation

in soft tissue, at frequencies in the MHz frequency range, can be expected to be dominant and this

will be assumed in this thesis.

Two definitions of the radiation pressure have been introduced [52]: the Rayleigh and the

Langevin radiation pressures, which are measured under two different physical situations. The

former definition considers the force exerted on the wall of a closed fluid-filled cylinder, where the

sound waves are reflected back and forth inside the cylinder. On the other hand, the Langevin

radiation pressure defines the force acting on a wall in a fluid that is not closed but is in contact with

the ambient unperturbed fluid (mass of fluid is conserved). In this case, the hydrostatic pressure acts

equally on both sides of the target and therefore, does not contribute to the radiation force result.

The following simplified differentiation between the two definitions can be made [51]:

The acoustic radiation pressure is denoted as Langevin pressure or force, if it depends only on the

incident acoustic waves.

The acoustic radiation pressure is denoted as Rayleigh pressure or force, if it depends on the

acoustic waves plus a constraint.

For most practical situations, the Langevin radiation pressure is appropriate and thus, in the

remaining portions of this thesis, the term acoustic radiation force will be taken to be well

13

approximated by the time-averaged Langevin radiation pressure due to the medium attenuation (see

Chapter 3).

As discussed in the previous section, the acoustic radiation force can be considered as the

basis of remote excitation of tissue and generation of shear waves, whose propagation

characteristics can provide significant information about the viscoelastic properties of the excited

region of tissue. Furthermore, as an ultrasound wave propagates along the radiation path, the

acoustic pressure in the focal zone may be large, thereby, leading to the generation of higher

harmonics due to nonlinear effects. These higher-frequency harmonics are more readily absorbed by

the medium, leading to an enhancement of the generated acoustic radiation force. The transfer of

energy to the nonlinearly generated harmonic components is more pronounced in the prefocal

region and therefore, the radiation force will be increased within the focal zone region. Rudenko et

al. [55] investigated the steady component of the radiation force produced by a focused nonlinear

Gaussian beam in a dissipative medium and showed that the medium nonlinearities can result in a

spatially sharper and increased (by 25%) radiation force in the focal region.

1.4 Narrowband vs. Broadband Shear Wave Generation

An effective way in which localized low-frequency shear waves can be remotely generated

within tissue, is by the dynamic radiation force resulting from the interference of two confocal

quasi-CW ultrasound beams of slightly different frequencies, similar to the concept described by

Fatemi and Greenleaf [25], [44]. In contrast to most radiation force-based methods presented

previously, two interfering ultrasound beams with slightly different frequencies can generate

narrowband low-frequency shear waves. Such waves suffer less from the effects of dispersion,

enabling the frequency-dependent shear speed and attenuation to be estimated at a specific

frequency. This can be achieved by tracking the shear-wave phase delay and change in amplitude

over a specific distance. Measurements at different frequencies can then be fitted to a viscoelastic

model (e.g. the Voigt model, see next chapter), enabling the tissue elasticity and viscosity to be

extracted.

On the other hand, a broadband ultrasound technique relies on the concept of exciting the

entire object (e.g. organ) by a swept frequency signal (short pulse) and measuring the tissue

response of the entire inspected part within a broad frequency band. The result from the inspection

is often a very complicated spectrum containing a considerable amount of information about the

investigated part of tissue. Furthermore, broadband techniques cannot accurately predict the shear

14

speed and attenuation as functions of frequency, since the received spectrum is significantly

distorted and the accuracy of the frequency spectrum may not be sufficient (poor spectral Signal-to-

Noise-Ratio) to resolve small differences in the tissue viscoelastic properties at different frequencies

[56].

If a long narrowband waveform is, instead, used:

The received waveform will be less affected by different kinds of frequency-dependent distortion.

The spectral power will be concentrated in these parts of the signal spectrum, where both the

amplitude and the phase information are not distorted by any frequency-dependent effects,

thereby providing more reliable information about the shear attenuation and dispersion at various

frequencies [57]. Because the shear wave energy is concentrated in a narrow bandwidth, there

should be a significant improvement in the SNR as compared to the broadband ultrasound.

1.5 Research Objectives

Investigating the propagation of narrowband shear waves in viscoelastic media generated by

a modulated finite-amplitude radiation force is important, because they have the potential to provide

estimates of the shear speed and attenuation at a specific frequency, without suffering the effects of

frequency-dependent attenuation and dispersion. To improve the detection signal-to-noise-ratio

(SNR) increased acoustic pressure conditions may be needed, causing higher harmonics to be

generated due to nonlinear propagation effects. Investigating the harmonic shear-field properties and

their dependence on the excitation conditions can give some insight on more realistic scenarios of

tissue excitation for elastography applications and may lead to improved estimates of the

viscoelastic properties of tissue.

The primary objectives of the thesis are to develop a method for the remote generation of

narrowband shear waves by using ultrasound, to investigate their characteristics, and to develop

methods for estimating the viscoelastic properties of tissue. Secondary objectives that follow from

this are:

To model the modulated radiation-force field that can be created in the focal zone by the

interference of the two finite-amplitude quasi-CW ultrasound beams and study its properties for

conditions that conform to safety standards.

To study the properties of the generated narrowband shear waves and specifically, the manner in

which the characteristics of the viscoelastic propagating medium affect their evolution, under

both low-amplitude (linear) and high-amplitude (nonlinear) source excitation.

15

To seek exact solutions of the viscoelastic Green’s function for conditions of increased viscosity

(which are believed to be characteristic of soft tissue) and to explain some of the associated

phenomena due to frequency-dependent effects.

1.6 Organizational Outline

This thesis is organized as follows: Chapter 2 presents a description of the linear theory of

viscoelasticity and the fundamentals of shear wave propagation both in elastic and viscoelastic

media. In Chapter 3, the modulated radiation force produced by interfering two confocal CW

beams at slightly different frequencies is modeled under nonlinear ultrasound propagation. The

dependence of the harmonic force components on the excitation conditions is investigated. Chapter

4 performs a spatiotemporal analysis of the fundamental component of the generated narrowband

shear waves. A spatiotemporal, frequency and time-frequency analysis of the generated harmonic

shear fields is performed in Chapter 5. Estimations of the viscoelastic properties of tissue based on

spatiotemporal and frequency characteristics of the harmonic shear fields are presented in Chapter 6.

The inverse problem approach is also discussed. Finally, Chapter 7 presents a summary and

conclusions of this thesis and recommendations for further work.

16

CHAPTER 2 Shear-Wave Propagation in Viscoelastic Media

Most studies in elastography consider a purely elastic medium to describe the behaviour of

tissue. However, it is well-established that soft biological tissues exhibit a combination of elastic

and viscous behaviour. The fundamentals of the linear theory of viscoelasticity and the two most

commonly used rheological models will be presented, namely, the Voigt and the Maxwell model.

The Voigt model has been shown to be the most appropriate for describing the viscoelastic

properties of tissue in the low kHz range (50-500 Hz). Furthermore, the basic principles of the

shear-wave generation and propagation in a homogeneous, unbounded and isotropic medium due to

a unidirectional point body force acting at a fixed point will be presented. The analytical expressions

of the Green’s functions will be provided both for purely elastic media and for viscoelastic media

based on the Voigt model under weak-viscosity conditions. The limitations of the weak-viscosity

approximation will be demonstrated for higher frequencies in the above range under the higher-

viscosity conditions that are believed to be characteristic of real tissue. A more exact solution of the

shear Green’s function is derived for these conditions.

2.1 Introduction to Viscoelasticity

In the majority of studies in elastography, viscous losses have been ignored and biological

tissue has been considered to be a purely elastic medium [23], [32], [33], [37], [58]. A linear elastic

solid is a solid that obeys Hooke’s law, which states that the stress σ (force per unit area) is linearly

proportional to the strain ε (deformation per unit length), i.e. [59], [60]:

17

ε=σ E , (2)

where E describes the elastic (or Young’s [12]) modulus. In contrast to elastic materials, a perfectly

viscous fluid obeys the Newton’s law, namely, the applied stress is always proportional to the rate

of change in the resulting strain, but is independent of the strain itself, i.e. [61]:

σ = η(dε/dt), (3)

where η denotes the coefficient of viscosity (or dynamic viscosity).

All real materials, however, exhibit some combination of the above properties, i.e., the stress

depends on the strain and the rate of strain (and higher time derivatives of strain) together and as a

result, they are considered to exhibit viscoelastic properties. Biological soft tissues exhibit

viscoelastic behaviour of various degrees. Moreover, soft tissue can be deformed to relatively large

strains, indicating a nonlinear response to the applied stress [61], [62]. It should be noted though,

that in our analysis, the deformation induced by the acoustic radiation force is very small (typically

a few μm [13], [26]) and the linear theory of viscoelasticity can be applied.

From a physical viewpoint, the shear viscosity arises from velocity differences between

adjacent fluid layers. The presence of velocity gradients in the fluid means that adjacent layers move

at differing speeds and as a result, there is a frictional drag force that causes energy to be dissipated.

An additional bulk viscous term (known as bulk viscosity) is often introduced, which accounts for

the effects of energy loss during the passage of a compressional wave. However, for an

incompressible fluid, only the shear viscosity is present [12].

2.1.1 Viscoelastic Models for Biological Tissue

The theory of viscoelasticity is essentially phenomenological and seeks to describe the

mechanical behaviour of all macroscopically homogeneous solid and liquid media. The mainstream

medical and biological ultrasonic literature makes little mention of viscoelasticity. However, since

the earliest measurements of ultrasonic absorption by biological tissues, it has been thought that

similar theoretical descriptions might also be applied to tissue [63].

18

Figure 2.1: The (a) Voigt and (b) Maxwell viscoelastic models consisting of a spring (described by the elastic modulus E) and a dashpot (described by the coefficient of viscosity η). The corresponding strain-time behaviour under constant stress σ0 at t = 0 are shown in (c) and (d).

The Basic Elements: Spring and Dashpot

All linear viscoelastic models are composed of linear springs, which are perfectly elastic and

linear dashpots, which are perfectly viscous. Inertia effects are neglected in such models. A linear

spring element can be described by (2) and exhibits instantaneous elasticity and instantaneous

recovery. On the other hand, a linear viscous dashpot element can be described by (3), which states

that the dashpot will be deformed continuously at a constant rate when it is subjected to a step of

constant stress. Furthermore, when a step of constant strain is imposed on the dashpot, the stress

will have an infinite value at the instant at which the constant strain is imposed and the stress will

then rapidly diminish with time to zero.

In modeling the macroscopic aspects of viscoelasticity, there exist many possible ways of

combining the springs and dashpots. Two simple viscoelastic models that are commonly used in

biological tissue are the Maxwell and the Voigt models shown in Fig. 2.1 and which are described

below.

19

The Voigt and Maxwell Models

The Voigt model, also known as the Kelvin model, consists of a spring and dashpot in

parallel, as shown in Figure 2.1(a), so that they both experience the same deformation or strain and

the total stress is the sum of the stresses in each element. If the spring constant is described by the

elastic modulus E and the dashpot constant by the coefficient of viscosity η, then the stress-strain

relationship for the Voigt model can be written as [59], [60]:

ε⎟⎠⎞

⎜⎝⎛

∂∂

η+=σt

E . (4)

The following stress-time relation, shown in Figure 2.1(c), can be obtained by applying integration

to the above equation together with the initial condition of a constant stress σ = σ0 at time t = 0

( )RteE

t τ−−σ

=ε 1)( 0 , (5)

where τR = η/E is called the relaxation time.

On the other hand, the Maxwell model consists of a spring and a dashpot in series, as shown

in Figure 2.1(b). Therefore, the force or stress is the same in both elements and the total deformation

or strain is the sum of the strains:

t

Et

E∂ε∂

η=σ⎟⎠⎞

⎜⎝⎛

∂∂

η+ . (6)

The stress-time relation under a constant stress σ0 applied at t = 0 can be written as (see

Figure 2.1(d))

⎟⎟⎠

⎞⎜⎜⎝

⎛τ

+σ

=εR

tE

t 1)( 0 . (7)

The Complex Modulus

When a sinusoidally varying stress is applied, the strain is found to vary sinusoidally with

time at the same frequency as that of the stress but generally not in phase with it. Thus, if the stress

is defined as tje ωσ=σ 0 , then the strain can be described by )(0

ϕΔ−ωε=ε tje , where Δφ is the phase

difference. Therefore, the complex modulus can be written as

)()()( 210

0 ω+ω=εσ

=εσ

=ω ϕΔ jMMeM j , (8)

where the real part M1(ω) is called the elastic (or storage) modulus, which is in-phase with a

sinusoidally-varying strain, and the imaginary part M2(ω) is the loss or viscous modulus, which is

20

90° out-of-phase with the strain. Both M1(ω) and M2(ω) depend on the frequency and are connected

by the Kramers-Kronig relationships [12], [64], as will be discussed in the next sub-section.

Based on (4) and (6), the complex moduli for the Voigt and Maxwell models under

oscillating stress are given by [65]

222

2

2222

22

1

21

)(,)(

)(,)(

ηω+ωη

=ωηω+

ηω=ω

ωη=ω=ω

EEM

EEM

MEMMM

VV

. (9)

It should be noted, that M(ω) can describe either the bulk or shear complex modulus, however, the

shear complex modulus is of particular interest in relation to shear wave propagation.

2.1.2 The Helmholtz Equation for the Voigt and Maxwell Model

Frequency-Dependent Speed and Attenuation

The change in the phase velocity of a propagated wave with frequency is known as

dispersion. In an unbounded medium, dispersion is caused by absorption. In fact, absorption is a

necessary and sufficient condition for dispersion to exist and as a result, there exists a direct relation

between the two quantities, described by the Kramers-Kronig (K-K) relationships [12]. For acoustic

waves, the K-K equations can be expressed in terms of the components of a complex propagation

constant, i.e.,

)()(

)()()(0

ωα−ω

ω=ωα−ωβ=ω j

cjk , (10)

in which c0(ω) denotes the phase speed and α(ω) is the attenuation coefficient. The pair of equations

relating these two quantities can be found in [12] and specifically for the Voigt model in [64]. As

discussed below, the viscoelastic properties of soft tissue can be determined by measuring the phase

speed and attenuation using low-frequency CW excitation.

The Shear Dispersion and Attenuation for the Voigt and Maxwell models

One-dimensional shear wave propagation into soft tissue can be modeled by inserting the

Voigt and Maxwell relations described by (4) and (6) into the Helmholtz equation. For a plane wave

propagating in the x-direction, the latter is given by

0),(),( 22

2=+

∂∂ txuk

xtxu

zz , (11)

21

where uz(x,t) denotes the z-component of the displacement vector, sjkk α−= is the complex wave

number (also described by (10) for all acoustic waves) in which k = ω/cs is the shear wave number

and αs is the shear attenuation. For plane monochromatic excitation, the 1-D Helmholtz equation for

the Voigt and Maxwell model, respectively, can be obtained in the frequency domain, as [36]:

[ ]

[ ] 0),()()],([

0),()],([

2

2

2

2

2

2

=ℑηωμ

ωη+μρω+

∂ℑ∂

=ℑωη+μ

ρω+

∂ℑ∂

txuj

jx

txu

txujx

txu

ztsl

slzt

ztsl

zt

(12)

where tℑ [.] denotes the Fourier transform of the displacement with respect to time, μl is the shear

modulus and ηs the shear viscosity. Then, from the complex wave number, by solving

]Re[)(

kcs

ω=ω and αs(ω) = - ]Im[k , the shear speed and attenuation can be obtained for the Voigt

and Maxwell models as [36]:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

ηωμ

++ρ

μω

ηω+μ+μρ

ηω+μω

22

2

2222

222

11

2=)(

)(

)2(=)(

s

l

lMs

sll

slVs

c

c

, (13)

( )

l

s

l

Ms

sl

lslVs

μ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

ηωμ

+ρω

ωα

ηω+μ

μ−ηω+μρωωα

2

11

=)(

)(2=)(

22

22

222

2222

. (14)

The essential differences between the Voigt and Maxwell models are that greater dispersion

is often predicted by the Maxwell model, which also predicts that the attenuation coefficient should

plateau at a maximum high frequency value similar to the curve shown in Figure 2.2(b). Attenuation

according to the Voigt model, however, continues to increase with frequency. While the Maxwell

theory has been found adequate for describing sound propagation in liquids, the additional existence

of a static shear modulus in the Voigt model seems to provide a better description for tissues [61].

Indeed, Ahuja [66] in 1979, using published data on ultrasonic attenuation in tissues, had

successfully modeled tissue as a Voigt body in the diagnostic frequency range of 0.8-10 MHz.

22

(a)

(b)

Figure 2.2: Frequency-dependent (a) shear speed and (b) shear attenuation for frequencies between 10-500 Hz. Both experimental measurements are shown (circles) and also theoretical curves as predicted by the Voigt (indicated by “V”) and Maxwell (indicated by “M”) models, based on (13) and (14). Reprinted with permission from J. Acoust. Soc. Am., 116, Catheline et al. [36], pp. 3734-3741, ©2004, American Institute of Physics.

Catheline et al. [36] investigated the shear viscoelastic behaviour of both models in an agar-

gelatin phantom and bovine muscle in the low-kHz range (50-500 Hz). They found that both models

gave good and similar results for dispersion, but the Voigt model was able to predict the frequency-

dependent shear attenuation much more accurately. For frequencies greater than 100 Hz, the

dashpot becomes ‘harder’ than the spring, therefore, when these two elements are placed in series,

as in the Maxwell model (see Figure 2.1), the viscoelastic properties are entirely driven by the

spring. Thus the speed and attenuation no longer depend on frequency and exhibit a plateau.

However, the experimental measurements gave a continuous increase of the attenuation with

frequency, which is also predicted by the Voigt model, therefore, it was recommended for

describing the viscoelastic properties of tissue in the low-kHz range (see Figure 2.2). It should be

also noted, that in a previous work [31], three sources of error were indicated in the measurements

of shear velocity and attenuation: (a) the finiteness of the medium, causing waves to be reflected at

the boundaries and interfere, (b) the diffraction effects due to the non-negligible size of the vibrating

source compared to the wavelength and (c) the interference between the axial components of shear

and compressional waves.

23

2.2 Principles of the Shear Wave Propagation

To predict the manner in which shear waves are generated and propagate due to a

unidirectional point body force acting at a fixed point in a homogeneous, unbounded, isotropic and

elastic medium, it is necessary to use the Navier-Stokes equation [12]:

fuuu+×∇×∇μ−⋅∇∇μ+λ

∂∂

ρ )()()(=2

2

t (15)

where ∇ denotes the gradient (nabla) operator, ),( trff = is the body force, which in our case, will

be the time-varying acoustic radiation force and ),( truu = is the particle displacement that is

regarded as a function of space and time. The displacement u denotes the vector distance of a

particle at a time t from the position r that it occupies at some reference time t0, often taken as t = 0.

The particle velocity and acceleration are ∂u/∂t and ∂2u/∂t2, respectively. Furthermore, λ and μ

relate the elastic and viscous moduli for the compressional (bulk) and shear wave, respectively. If

the Voigt model described in Section 2.1 is adopted for characterizing the viscoelastic properties of

tissue, the above moduli can be written as follows:

)/(= tcl ∂∂η+λλ and )/(= tsl ∂∂η+μμ , (16)

where λl, μl denote the bulk and shear elastic moduli, respectively, and ηc, ηs, denote the bulk and

shear viscous moduli (or simply bulk/shear viscosities). The initial conditions for (15) are u(r,t) = 0

and ∂u(r,t)/∂t = 0 for r ≠ 0.

The total displacement u(r,t) can be written as a spatiotemporal convolution between the

total Green’s function G(r,t) and the body force (acoustic radiation force) f(r,t) [67]:

∫ ∫∫∫τ

ττ−−τ==k

kkrGkfrGrfru ddtttt ),(),(),(**),(),( (17)

where ∗∗ denotes a 4-D spatiotemporal convolution. It is interesting to note that in 1849, Stokes

[68] published a general solution [see his eqn. (36)] for the lossless displacement field generated by

an arbitrary forcing function (in essence a Green’s function solution). The solution contained three

terms. Two of them describe longitudinal and shear wave components, whose amplitudes vary

inversely with distance from the source. The final term is generally referred to as the coupling term

and it accounts for additional longitudinal and shear wave components that are generated in the

near-field zone.

24

More recently, Aki and Richards [67] rederived analytical expressions of the displacement

field for purely elastic media, known as the Green’s functions ),( trG . Bercoff et al. [69] extended

this model to account for viscous effects and obtained approximate expressions of the Green's

functions. As originally shown by Stokes for an inviscid medium, they also found that the

displacement field could be written as the sum of three types of waves:

1. A pure compressional wave ),( tc ru propagating at a speed c,

2. a pure shear wave ),( ts ru that propagates at a much smaller speed cs and

3. a coupling term ),( tcs ru , that contains both types of waves.

As noted by Sandrin et al. [23], for soft tissue-like media, the bulk modulus is much greater than

the shear modulus (λl >> μl), so that the effects of the compressional wave are negligible compared

to the shear and coupling waves and consequently, will be ignored in the following chapters.

2.3 Green’s Function in an Infinite Viscoelastic Medium

In the following sub-sections, a homogeneous, isotropic, and unbounded medium is

considered and a solution is sought for the displacement u(r,t) that satisfies the Navier-Stokes

equation with a body force ),( trf acting in the j-direction (j = x, y, z) at a fixed point. This can be

achieved by deriving the Green’s function, which corresponds to the solution of (15) for a source

term that behaves as a delta function (impulse) in space and time.

2.3.1 Green’s function for an elastic medium

A perfectly elastic medium is initially considered and the elastodynamic Green’s function

G(r,t) is sought as a function of time. By ignoring viscous losses in (15) and making use of the

relation ∇×(∇×u) = ∇(∇·u)-∇2u, the Navier-Stokes equation becomes

fuuu+∇μ+⋅∇∇μ+λ

∂∂

ρ 22

2

)()(= lllt. (18)

Based on the above elastic-wave equation and the derivations given in [67] and [58], it can be

shown that the ij-component of the pure compressional, shear and coupling terms of the Green’s

function vector can be written, correspondingly, as:

25

⎟⎠⎞

⎜⎝⎛ −δ

πρ

γγ

crt

rctG jic

ij1

4=),( 2r , (19)

⎟⎟⎠

⎞⎜⎜⎝

⎛−δ

πρ

γγ−δ

ss

jiijsij c

rtrc

tG 14

)(=),( 2r , (20)

( )∫ ττ−τδπρ

δ−γγ scr

cr

ijjicsij dt

rtG

/

/3

14

)(3=),(r . (21)

where |=| rr , iri ∂∂γ /= , ρ is the density of the medium, δ(t) is the Dirac delta function and δij is

the Krönecker δ-function with δij= 1 if i = j and 0 otherwise. Note that in the above expressions, a

tensor notation has been used, where i = x, y, z, denotes the direction of the displacement in response

to the force acting in the j-direction.

The coupling term as expressed by (21), contains both longitudinal and shear wave

components. In the direction of the force, the shear contribution is zero and the longitudinal

component is a maximum, while at right angles to the force the opposite is true [23], [58]. Moreover,

it should be noted that the r-3 dependence causes both components to rapidly reduce with distance

(near-field term), as compared to the pure compressional and pure shear components expressed by

(19), (20), which vary as r-1 (far-field terms). The contributions of the far-field terms are Dirac

functions occurring at times r/c and r/cs, respectively. The near-field term is a temporal ramp from

time r/c up to r/cs.

2.3.2 Green’s function for a viscoelastic medium based on the Voigt model

Approximate Green’s Functions for Viscoelastic Media

Bercoff et al. [69] adopted the Voigt model and obtained approximate viscoelastic

expressions of the Green's functions, as mentioned earlier. Inserting the Voigt-model relations

described by (16) to (15), yields:

( ) fuuu+∇⎟

⎠⎞

⎜⎝⎛

∂∂

η+μ+⋅∇∇⎟⎠⎞

⎜⎝⎛

∂∂

η+η+μ+λ∂∂

ρ 22

2

)(=ttt slscll , (22)

By following a similar derivation as in [67], they derived the following approximate viscoelastic

Green’s expressions for the pure compressional, pure shear and coupling waves, respectively:

26

Figure 2.3: Normalized total Green’s function cszz

szz

czzzz GGGG ++= at a radial distance

r = 3.0 mm from the source (origin O). The medium parameters were assumed cs = 3 m/s, c = 60 m/s, ηs = 0.1 Pa⋅s, ηc = 0 and ρ = 1050 kg/m3. Note that the compressional speed is not representative of that for soft tissues (c ≈ 1500 m/s), but has been selected small for visualization purposes. The times of arrival for the pure compressional and pure shear waves are also demonstrated (r/c = 50 μs and r/cs = 1 ms).

tvc

ertvc

tG c

crt

ji

c

cij

22

14

1=),(

22)/( −−

γγ

ππρr , (23)

tvc

ertvc

tG s

sscrt

jiij

ss

sij

2)(

21

41=),(

22)/( −−

γγ−δ

ππρr , (24)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

ττπ

−ττπ

δ−γγ

πρ

τ−−

τ−−

∫∫ dtvc

etv

cdtv

c

etv

cr

tG s

st

scr

s

sc

t

cr

c

ijjicsij

22

22

)(34

1=),(

22)(

/

0

22)(

/

03r , (25)

where ( ) ρη+η sccv 2= and ρηssv = denote the kinematic bulk and shear viscosities,

respectively. An example of the medium total response (z-component) at a radial distance r = 3.0

27

mm to a spatiotemporal impulse acting at the origin r = 0 in the z-direction is demonstrated in

Figure 2.3, where all the above three Green’s function components can be observed.

In deriving the above viscoelastic equations, Bercoff et al. assumed that the attenuation

length is much larger than the wavelength and that dispersion is negligible. Specifically for the shear

component, they assumed that (i) cs → ρμ /l (implying that the shear modulus dominates over the

viscous losses, i.e., μl >> ηsω) and (ii) αs ≈ 3

2

2 s

s

cρηω

(resulting from cs>>kvs ⇔ ssa

λ>>1 , where αs

and λs denote the shear attenuation and wavelength, correspondingly). The limitations of these

assumptions will be investigated in the following sub-section.

Figure 2.4: (a) Speed and (b) attenuation of the shear waves as functions of frequency, based on the Voigt model (solid lines). The shear modulus was assumed μ l = 6.0 kPa and two different viscosities were considered, i.e., 0.5 and 3.0 Pa⋅s. The approximations adopted by Bercoff et al. [69] are also shown (dotted lines).

Exact Green’s Solution for Viscoelastic Media in the k-Space

A great deal of ambiguity exists in the literature as to the range of the shear viscosity values

in soft tissue in the low-kHz range. Furthermore, viscosity values have been reported that are much

larger that those utilized in gel tissue-mimicking phantoms [69], i.e. even up to 10-15 Pa⋅s [36],

[16]-[72]. For such viscosities, the approximations of the previous sub-section may not be accurate

and exact solutions of the viscoelastic Green’s functions are sought. In Figure 2.4, the shear wave

speed and attenuation are shown as functions of the frequency based on (13) and (14), for a shear

modulus of 6.0 kPa and two different viscosities, i.e., 0.5 and 3.0 Pa⋅s. The assumptions adopted in

[69] are also shown for comparison. It can be observed, that for higher viscosities, the shear speed is

28

strongly influenced by the frequency and dispersion becomes important. Furthermore, as viscosity

increases, the approximate expression for the shear attenuation [69] diverges significantly from the

Voigt model, especially for the higher frequencies.

One way to take into account high-viscosity conditions is to solve the viscoelastic Green’s

function in the k-space and then calculate the spatial response numerically via the inverse spatial

Fourier transform. By following a similar derivation as in [67] and [69], the shear term of the

Green’s function in the (k,t) domain can be shown to be given by the following complex function

⎪⎪

⎩

⎪⎪

⎨

⎧

ηρμ

≥⎟⎟⎠

⎞⎜⎜⎝

⎛ρμ−η

ρ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ρη

−

ηρμ

<⎟⎟⎠

⎞⎜⎜⎝

⎛η−ρμ

ρ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ρη

−

=

s

lls

s

s

lsl

s

s

kkkttk

kkktjtk

tkK2

if,42

exp2

exp

2if,4

2exp

2exp

),(22

2

222

, (26)

where [ ]),(),( trrGtkK srs ℑ= denotes a spatial Fourier transform. Details of the above derivation

can be found in APPENDIX A. It should be noted, that the assumptions made in [69] (i.e. cs>>kvs),

enabled the above equation to be approximated by

( )tjkctk

tkK ss

s −⎟⎟⎠

⎞⎜⎜⎝

⎛ρη

−≈ exp2

exp),(2

. (27)

The second term of the above equation is the classical solution of the wave equation in the purely

elastic case (ηs→0). The exact solution Gs(r,t) in the real space can be found by applying the inverse

spatial Fourier transform to (26) or (27). Although an analytical form of the inverse Fourier

transform of (27) can be readily be derived analytically (see (24) and the derivations by Bercoff et

al. [69]), the same cannot be easily done for the exact solution described by (26). Therefore, it will

be numerically evaluated when needed.

The differences between the analytical approximate and numerically calculated exact

Green’s solution will be demonstrated in Chapters 5 and 6. It will become evident that for higher

frequencies and higher-viscosity conditions, it is necessary to use the exact solution, since in such

cases, the shear speed is strongly influenced by the frequency and the viscous losses cannot any

longer be considered weak (see also Figure 2.4).

2.4 Chapter Summary

Most studies in elastography consider a purely elastic medium to describe the behaviour of

tissue. However, it is well-established that soft biological tissues (and all real media) exhibit a

29

combination of elastic and viscous behaviour. In the first part of this chapter, the fundamentals of

the linear theory of viscoelasticity were discussed. Furthermore, the two most commonly used

rheological models for describing the viscoelastic behariour of tissue were presented, namely, the

Voigt and the Maxwell model, with the Voigt model being the most appropriate for low-frequency

(50-500 Hz) shear-wave propagation. The frequency-dependent shear speed and shear attenuation

were provided for both models based on the 1-D lossy Helmholtz equation of motion.

In the second part of this chapter, the basic principles of the shear-wave generation and

propagation in a homogeneous, unbounded, and isotropic medium due to a unidirectional point body

force acting at a fixed point were presented. The total particle displacement, which is a function of

both space and time and can be described by a spatiotemporal convolution between the body force

and the Green’s function, is composed of three components: a pure shear wave (far-field), a pure

compressional wave (far-field) and a coupling wave (near-field) that consists of both types of waves.

Furthermore, for soft tissue-like media, the effects of the compressional wave are negligible and can

be ignored. The analytical expressions of the Green’s functions were provided both for purely

elastic media [67] and for viscoelastic media based on the Voigt model under weak-viscosity

conditions [69]. The limitations of the approximations performed by Bercoff et al. [69] are

demonstrated for higher viscosities, such as those reported occasionally in the literature for

biological tissue. Therefore, the exact solution of the shear Green’s function is derived in the k-

space and is proposed for describing the shear-wave propagation under high-viscosity conditions,

where the shear speed is strongly influenced by the frequency and the viscous losses are not weak.

In such cases, the Green’s solution in the real (space and time) domain can be calculated

numerically through the inverse spatial Fourier transform.

30

CHAPTER 3 Modulated Finite-Amplitude Acoustic Radiation Force

A localized modulated radiation force field can be created by using two intersecting CW

ultrasound beams of slightly different frequencies. Because the pressure in the focal zone region

may be large, it is necessary to account for the presence of harmonics generated by nonlinear effects.

In this chapter, the generation of dynamic radiation force components at harmonic modulation

frequencies will be modeled and their properties will be examined for conditions that limit the

associated heating effects. Their dependence on the excitation conditions, described by the

parameters of nonlinearity, focusing gain and absorption, will be also investigated.

3.1 Description of the Dual-Beam System Model

The radiation force exerted in an infinite isotropic, homogeneous, and attenuating medium is

considered here. In addition, nonlinear ultrasound propagation will be assumed, so that a sufficiently

high-intensity force field will be created in the focal zone, leading in turn, to the generation of high-

energy shear waves that can propagate and be detected several wavelengths away from the source.

For a finite-amplitude ultrasound beam propagating in an attenuating medium, it can be shown that

the force per unit volume generated at the location r is given by [73]:

31

Figure 3.1: Illustrating the principles of a simple confocal radiation force imaging system used to generate low-frequency shear waves.

…1,2,=,),(2

=),( nc

tt nn

nrI

rFα

, (28)

where c is the propagation speed, nf0 denotes the n’th harmonic of the fundamental frequency f0 and

In(r,t) is the acoustic intensity vector. Moreover, if the attenuation coefficient is assumed to have a

power-law frequency dependence, then the attenuation can be written as γαα ffn 0=)( , where γ is

typically close to unity in tissue. For a focused acoustic beam, the force is applied throughout the

focal region of the beam and for absorbing media (such as tissue) it is assumed to be in the direction

of the wave propagation, i.e. the z-axis.

To achieve a localized dynamic (oscillatory) radiation stress field, two intersecting CW

ultrasound beams of slightly different frequencies can be used, such that in the intersection zone, a

modulated ultrasound field is produced. Because of this, only the time-varying (dynamic)

component of the incident intensity needs to be considered.

32

Figure 3.2: Showing (a) a cross-section on the focal plane and (b) an axial view of the source system consisting of two coaxial confocal ultrasound transducers A and B (see Figure 3.1). Transducer A (inner) is a confocal circular disk of aperture radius a1 and transducer B (outer) is a confocal annular disk with inner and outer radii a21 and a22, respectively. The dynamic radiation force FΔ produced by the interference of the two beams is also shown at the common focal point.

With reference to Figure 3.1, we consider two coaxial confocal CW ultrasound beams A and

B excited at the angular frequencies /2= 0 ωΔ−ωωa and /2= 0 ωΔ+ωωb , where ω0 and Δω are

the center and difference frequencies, respectively, such that 0 ω<<ωΔ . If the beams propagate in

the z-direction with a common focal point at )0,(0, 0z then, in the region of intersection, the

resulting pressure can be written as:

tepteptp bjnbn

ajnann

ωω + )()(=),( ,, rrr , n = 1,2,… , (29)

where )(, ranp and )(, rbnp are the complex amplitude functions of the ultrasonic beams at the n’th

harmonic. The resultant intensity field at the region of interference will contain terms at not only the

frequencies ωa and ωb, but also at the combination frequencies of n1ωa ± n2ωb, where n1 and n2 are

positive integers [73]. All high-frequency terms can be neglected, since they do not fall within the

low acoustic frequency range. If we restrict our attention to the case of 1n = 2n = n , the intensity

vector and consequently, the radiation force vector at the zone of intersection, will contain low-

33

frequency (dynamic) components at )(= abnn ω−ωωΔ . Denoting the particle velocities by )(, rv an

and )(, rv bn , the dynamic intensity vector can be expressed as

…1,2,=},)]()()()({[21=),( ))((

,,,, neppRet ntnjbnananbnn

rrvrrvrrI ϕΔ+ωΔ∗∗Δ + (30)

where ∗ denotes the complex conjugate, Δφn(r) is the phase difference at which the corresponding

n’th harmonics of the two beams arrive at point r and for convenience we have chosen the time t = 0

to correspond to the time at which the ‘tone bursts’ from the two sources reach the geometric focus.

Based on (28), the dynamic radiation force vector corresponding to the n’th harmonic, is

therefore given by

…1,2,=,),()(2

=),( 0 nc

tffnt nba

nrI

rFΔγγγ

Δ +α (31)

where ),( tn rI Δ is given by (30). The attenuation factor is shown to be affected by the sum of the

incident harmonic frequencies nfa and nfb and for sufficiently focused sources in absorbing media

the direction of the force is primarily in the z-axis.

3.2 Dual-beam System Model

Ultrasound Beam Propagation

For our simulations, a confocal source consisting of two simple coaxial concave radiation

sources with a common geometric focal depth of 7.0 cm was assumed. The outer aperture radius

was 1.3 cm for beam-A, while for beam-B the inner and outer aperture radii were 1.4 cm and 2.1 cm,

respectively. The geometry of the proposed source system is shown in Figure 3.2. The source

pressure amplitude was taken to be 372 kPa: this has been shown to be sufficient for nonlinear

ultrasound propagation [74], [75]. The two transducers were excitated at frequencies of 2.0 MHz ±

100 Hz (i.e., f0 = 2 MHz and Δf = 200 Hz). The medium was assumed to have a propagation speed

of c = 1550 m/s, a density of 1050 kg/m3, a nonlinearity coefficient of 5.0, an attenuation

characterized by γ= 1.1 and α0 = 0.3 dB/(cm MHz1.1).

To calculate the fields generated by the transducers for n = 1,2,…, we made use of a

modified version of the 2nd-order operator splitting nonlinear model of Zemp et al. [74], that

enabled the harmonic field distribution to be calculated up to n = 4 with reasonable accuracy. The

axial pressure amplitudes of the first four harmonics for beam-A and beam-B are shown in

Figure 3.3. Fifty propagation planes were used to capture the axial variations of the harmonics.

34

Close to the transducer, the approximations used in the nonlinear propagation algorithm caused

errors in the calculated pressure profiles, but over the region of the last maxima and beyond, the

accuracy is quite sufficient. It will be noted that for both beams, all the maxima occur somewhat

prior to the geometric focus: a feature that is well established for nonlinear propagation from plane

and focused sources (see Ch.4 in [12]).

Modulated Acoustic Radiation Force

In the region of the geometric focus, where the pressure from the two beams reaches a

maximum, a dynamic component of the radiation force will be present at the fundamental

(a) (b)

(c) (d) Figure 3.3: Axial pressure amplitude of the n’th harmonic component (n = 1...4) for (a) beam A (confocal circular disk) and (b) beam B (confocal annular ring). The corresponding slow phases are shown in (c) and (d). Note that close to the source, i.e. z < 3 cm, the nonlinear computational method causes the results to be unreliable.

35

modulation frequency of 200 Hz, as well as at the harmonic frequencies of nΔf (i.e., at 400 Hz, 600

Hz, etc, for n = 2,3, …). The modulation frequency was selected small enough (i.e. 200 Hz), so that

shear waves of at least up to the second (400 Hz) or third (600 Hz) harmonic could propagate. It

should be noted though, that the modulation frequency could be selected even smaller (e.g., ≤ 100

Hz), in order to take advantage of as much higher-harmonic information as possible. However, it

will be shown in the subsequent sections that the attenuation of the propagating medium (which is

frequency-dependent) can cause an increase of the radiation force (see Section 3.4.2). Therefore, in

order to examine and potentially take advantage of the pronounced effects of the absorption of the

medium, a slightly higher modulation frequency (200 Hz) was finally adopted.

First, we consider the modulated n’th-harmonic radiation force to be a spatial impulse at the

origin (point force) that varies sinusoidally in time at the beat angular frequency nΔω, such that each

harmonic modulation waveform consists of a short cosine wave of duration D starting at t = 0. It

should be noted that as the duration is increased, the bandwidth of the shear waves will decrease,

making it possible to determine the wave propagation characteristics at a specific frequency.

However, heating effects may arise with increasing duration and therefore, an appropriate tradeoff

for the selection of D needs to be made. A more detailed discussion of the associated temperature

increase will be given in Section 3.3. Finally, the origin of our coordinate system will be taken at the

geometric focus (z = 7.0 cm).

Noting that the plane wave assumption enables the particle velocity to be written in terms of

the pressure, (30) and (31) enable the n’th-harmonic component of the radiation force to be written

as

)]0(cos[)()((0)(0))(2

)(=),(0 2,,0

nbnanba

n tntDHtHc

ppffntF ϕΔ+ωΔ−

ρ

+αδ

γγγΔ r , (32)

where H(.) denotes a Heaviside step function and Δφn(0) ≠ 0 is the phase difference at which the

corresponding n’th harmonic pressure components from the two beams intersect at the geometric

focus. The phase term Δφ1(r) for the fundamental component of the dynamic force (see (30), (31)),

can be calculated from the phase difference at which the corresponding fundamental pressure

components from the two beams intersect at point r. However, for tone-burst excitation we can

assume that the fundamental wave components from the confocal source arrive in-phase at the

common focal point. Therefore, without any loss of generality, the phase difference for the

fundamental Δφ1(0) can be set to zero. For the higher harmonics, the phase difference can be

calculated relative to that for the fundamental at focus. In the immediate region surrounding the

36

geometric focus, the primary component of the force is along the z-axis and in the following

analysis, this is assumed to be the only component of significance, i.e., i = z and, for notational

convenience, the subscript z has been dropped from ΔznF , .

The CW waveform of the total finite-amplitude dynamic radiation force at focus is shown in

Figure 3.4(a) and (b) together with the spectrum. The spatiotemporal patterns of the total finite-

amplitude and the fundamental only modulated radiation force are shown in Figure 3.4(c) and (d),

(a) (b)

(c) (d) Figure 3.4: (a) Temporal profile of the finite-amplitude dynamic radiation force at the geometric focus (z = 7.0 cm). (b) Normalized frequency spectrum of the total radiation force, where the individual harmonic force components can be clearly observed at 200, 400, 600 and 800 Hz (n = 1…4). (c) Normalized spatiotemporal patterns of the total finite-amplitude radiation force (five harmonics retained) and (d) the fundamental component.

37

correspondingly. The distortion due to nonlinear effects is evident in the temporal shape of the

generated force field (see Figure 3.4(a) and (c)).

(a) (b)

(c) (d) Figure 3.5: Normalized radiation force per unit volume on the geometric focal plane (7.0 cm) for: (a) the fundamental (200 Hz), (b) the second-harmonic (400 Hz), (c) the third-harmonic (600 Hz) and (d) the forth-harmonic (600 Hz) radiation force.

Phase Difference of the Harmonic Components

The relative phase terms Δφn(0) at the geometric focus (z = 7.0 cm) were found to be

approximately -2π, -6π and -10π, for n = 2, 3 and 4, respectively. The slow phases of the pressure

for the first five harmonics of beams A and beam B are shown in Figure 3.3(c) and (d). The slow

phase of a spectral component is defined as the phase of that component relative to a plane wave of

the same frequency which is in phase on the source and propagates along the acoustic axis [12], [76].

Observation of Figure 3.3(c) reveals a phase shift of roughly π radians for the fundamental pressure

38

component (n = 1) of beam A as it propagates through the focal region. The on-axis slow phases of

the second through forth harmonic components exhibit phase shifts that are at least π/2 greater than

that of the previous harmonics, as pointed out in [76]. The on-axis slow phases of the spectral

components for beam B shown in Figure 3.3(d), exhibit larger phase shifts and a different pattern

than that predicted by the theory of finite-amplitude focused sources and this can be attributed to the

annulus concave geometry of the corresponding transducer (B).

Radial Beamwidths of the Harmonic Force Components

The corresponding spatial force-field patterns on the geometric focal plane at multiples of

the fundamental force period T1 = 1/Δf, i.e., at t = 0, 5.0 ms, etc, are shown in Figure 3.5 for

n = 1…4. For visualization purposes, the forces per unit volume have been normalized with respect

to the maximum fundamental force per unit volume at focus, i.e., 5.7×104 N/m3 (this value has been

obtained from direct substitution in (32)). It is evident that the higher-harmonic force components

exhibit better localization around the geometric focus (narrower mainlobe), lower sidelobes, and

reduced amplitude due to the increased attenuation at the harmonic frequencies. The maximum

amplitude of the second through to the forth harmonic components were found to be approximately

30%, 12% and 5% that of the fundamental. Specifically, for the radial beamwidths, as predicted

Figure 3.6: Normalized radial profiles of the n’th-harmonic dynamic radiation forces

ΔnF , n = 1…4, at the geometric focus (z = 7.0 cm). The n’th-harmonic force beamwidth

was found approximately n-1 times that of the fundamental force beamwidth at -6 dB. The ‘finger’ effect in the pattern of the harmonic sidelobes can be also observed in the figure inset.

39

theoretically by Du and Breazeale [77] for Gaussian beams and verified experimentally for non-

Gaussian sources [78], the harmonic beamwidths (at -6 dB) of the acoustic pressure are simply n-1/2

times that of the fundamental beamwidth. In our case, the relationship between the harmonic and

fundamental beamwidths was found to be approximately n-2/3 for beam A and n-1 for beam B (see

Figure 3.6). Furthermore, the harmonic beamwidth of the generated dynamic radiation force

resulting from the interference of the two beams, was found to behave in a similar manner as beam

B, i.e. n-1 times the fundamental force beamwidth.

Sidelobes of the Harmonic Force Components

As seen in the inset for Figure 3.6, additional sidelobes can be seen in the radial pattern of

the harmonic force components. These extra sidelobes are known as ‘fingers’ in the literature and

are qualitatively similar to those observed in the farfield of unfocused sources [76], [78]. It has been

shown both theoretically and experimentally, that the second-harmonic source pressure pattern has

twice as many sidelobes as the fundamental component, while the third harmonic has three times as

many and so on. This can be also verified in Figure 3.7 for the harmonics of both the circular disk

(beam A) and annular-ring (beam B) transducers of our model. The interference now of the two

beams in the focal zone, results in a more complicated sidelobe pattern for the harmonic radiation-

force components, where some of the predicted ‘fingers’ have been suppressed (see Figure 3.6).

(a) (b) Figure 3.7: Normalized radial profiles of the n’th-harmonic pressure (n = 1...4) for (a) beam A and (b) beam B, at the geometric focal plane (z = 7.0 cm). The extra sidelobes, known as ‘fingers’ can be seen at the higher harmonics. Note that the results for n = 4 in (b) are of limited value due to approximations used in the numerical computations.

40

3.3 Heating Effects

Of considerable importance in the in-vivo application of elastography techniques that

employ high-intensity radiation excitation methods is the associated temperature rise. The passage

of an ultrasound wave through an absorptive medium causes some of the incident energy to be

converted into scattered radiation, some to be converted into shear and longitudinal energy, and

some to be converted into heat, which causes an increase in temperature. Associated with a long

duration modulated tone burst, will be a greater temperature increase and a narrower bandwidth. On

the other hand, a short duration pulse will have a wider bandwidth and a smaller rise in temperature.

In order to determine the speed and attenuation characteristics as a function of frequency of shear

wave propagation in tissue, it would be helpful to generate narrowband shear waves, but since this

implies a longer duration, it is necessary to determine the temperature rise that it can cause.

For in vivo use, the maximum temperature change is limited by regulatory specifications of

the Thermal Index (TI), which provides a measure of the potential for tissue damage by heating [12],

[79], [80]:

refT

TTIΔΔ

= lim , (33)

where ΔTlim is an estimated upper limit to the temperature rise (for a given value of the acoustic

power) in a specified application and ΔTref is a reference value of the temperature rise chosen for its

biological significance. If ΔTref is equated to 1.0 °C, then the thermal index is numerically equal to

ΔTlim [80].

In addition to the thermal heating effect, the potential for tissue cavitation by the negative

peak pressure and frequency must be considered. The Mechanical Index (MI) is a quantity related to

the potential for damage based on mechanical effects during a diagnostic ultrasound examination

and is defined by [12]

(MHz)

(MPa)fC

pMI

MI

−= , (34)

where CMI is 1.0 MPa/MHz1/2 (needed to make MI dimensionless) and p- is the peak value of the

attenuated rarefactional pressure. According to the Food and Drug Administration regulations [81],

the MI should be kept below 1.9, so that negligible risk is imposed to patients. In the subsequent

analysis, emphasis is given on the harmonic radiation force components and therefore, the MI may

41

not be always strictly limited below 1.9. However, in practice, both the MI and TI should be

carefully accounted for.

Under most conditions, a TI value of 1.0 is regarded as posing negligible risk to the patient,

i.e., TI ≤ 1.0 ⇒ ΔTlim ≤ 1.0 °C. If we ignore the effects of heat conduction to surrounding regions

through diffusion and transport by blood circulation, we can arrive at an upper limit to the

temperature rise. It can be shown to be given by [80], [82]

Dc

ITv

α=Δ

2lim , (35)

where cv denotes the heat capacity per unit volume, I is the local time-average acoustic intensity

and D is the heat application time. For the dual-beam system model, the upper limit for the

temperature increase at focus arising from the n’th harmonic, can be written, based on (30)-(35), as

cc

ppfnDT

v

bnann ρ

+α≈Δ

γγ ](0)(0)[(0)

2,

2,00 . (36)

Thus, the temperature rise depends on the sum of the squared acoustic pressures at focus and has a

linear dependence on the duration of the modulation waveform. Therefore, it is necessary that the

duration D in our system must be carefully selected by achieving an appropriate tradeoff, so that the

narrowband nature of the generated shear waves is preserved, while the heating effects do not

exceed the safety limit of 1.0 °C. Based on (36), the maximum emission duration so that the upper

limit of the temperature increase at focus does not exceed 1.0 °C, can be described by

∑∑

+α

ρ≈⇒≤Δ

γγ

nbnan

v

nn ppnf

ccDT

](0)(0)[C1)0(

2,

2,00

max , (37)

where the sum of the heating effects, arising from all generated harmonics for the two beams, has

been considered.

3.4 Effects of the Excitation Conditions

To describe the acoustic characteristics of the concave source, three dimensionless

parameters that are sometimes used [55], [76] will be employed to characterize the excitation

conditions. They all depend on the focal length F and are given by

42

Figure 3.8: Axial pressure amplitudes of the n’th harmonic components (n = 1…3) of beam A (solid, dotted and thick-dotted lines) and beam B (stars, crosses and dots) for various values of the dimensionless parameters of nonlinearity (N), absorption (A) and total focusing gain (G = Ga+Gb). In (a) and (d), the third-harmonic components were very small and are not shown.

plFN /= (Nonlinearity),

FrG /0= (Focusing gain), (38)

FA α= (Absorption),

where lp is the plane-wave shock formation distance and r0 is the Rayleigh distance [12].

3.4.1 The Parameters of Nonlinearity, Focusing Gain and Absorption

The nonlinearity parameter N (not to be confused with the medium parameter of nonlinearity)

is defined as the ratio of the transducer geometric focal length to the plane-wave shock formation

distance lp = ρc3/(βω0P0), where ω0 is the center angular frequency. For the assumed source model

of coaxial concave transducers, the nonlinearity parameters corresponding to beams A and B are

approximately equal (since ωa ≈ ωb ≈ ω0) and can be described by 300 cFPN ρβω≈ . Clearly, N is

proportional to the source pressure, the center frequency, and the coefficient of nonlinearity of the

propagation medium: higher values of N lead to increased nonlinear effects, i.e. increased transfer of

energy to the higher harmonic components. The dimensionless parameter G is defined as the linear

43

focusing gain and can be described by FcaGa 2210ω≈ (beam A, see Figure 3.2) and

FcaaGb 2)( 221

2220 −ω≈ (beam B), where the Rayleigh distance r0 = ω0

2ia /(2c) of an unfocused

source of aperture radius ai [12] has been substituted in (38). Therefore, the total gain G can be

written as:

)(2

221

222

21

0ba aaa

FcGGG −+

ω≈+= (39)

If now the gap between the two transducers is sufficiently small, i.e. 211 aa ≈ , then the above total

focusing gain can be simplified to FcaG 22220ω≈ . Finally, the absorption parameter A depends on

both the center frequency and the attenuation coefficient and can be approximated by FA γωα≈ 00

for both beams. For the simulation conditions performed in Section 3.2, the three dimensionless

parameters are: N = 0.42, Ga = 9.8, Gb = 14.2 (i.e. a total gain of 24.0) and A = 0.52.

3.4.2 Effects of the Nonlinearity Parameters on the Harmonic Radiation Force

Components

Six different combinations of the nonlinearity parameters N, G and A are considered, where

three pairs with the two parameter values are formed and for each pair, the third parameter takes two

values. The axial pressure amplitudes of the first three harmonics of beams A and B are shown in

Figure 3.8 for the six combination cases. In Figure 3.8(a) and (b), the center frequency and

attenuation coefficient were fixed to f0 = 2.0 MHz and α0 = 0.41 dB/(cm⋅MHz1.1) and two values of

the source pressure were considered, i.e., P0 = 178 kPa (N = 0.2) and P0 = 533 kPa (N = 0.6).

Subsequently, in Figure 3.8(c) and (d), the center frequency and source pressure were fixed to f0

= 2.0 MHz and P0 = 267 kPa and the attenuation coefficient was taken 0.06 and 0.58

dB/(cm⋅MHz1.1). Finally, in order to examine the effects of the focusing gain, two different center

frequencies were considered in Figure 3.8(e) and (f), i.e., 1.0 and 3.0 MHz. Furthermore, in order to

keep the nonlinearity and absorption parameters constant for both frequencies, a source pressure of

511 kPa and an attenuation coefficient of 0.62 dB/(cm⋅MHz 1.1) were assumed for the lower center

frequency (1.0 MHz), while for the higher frequency (3.0 MHz), the source pressure and attenuation

coefficient were taken 237 kPa and 0.19 dB/(cm⋅MHz 1.1), respectively.

The maximum amplitudes of the CW fundamental and second-harmonic force components

at the geometric focus are listed in Table 3.1 for the above six combinations. The percent ratio of the

44

second-harmonic amplitude to that of the fundamental is also indicated. It can be observed that

when the focusing gain and absorption are fixed, the three times increase of the nonlinearity

parameter (which is both frequency and pressure-dependent) causes the fundamental force

amplitude to increase by almost a factor of eight. Furthermore, the amplitude of the second-

harmonic force becomes approximately 38.4% of that of the fundamental, indicating the increased

transfer of energy to the higher harmonics due to the pronounced nonlinear absorption in the focal

region, as shown in Figure 3.8(b).

Table 3.1: The maximum amplitudes of the CW fundamental and second-harmonic radiation force at focus for six combinations of the parameters N, A and G for a pulse duration of 5.0 ms. The temperature increase at focus arising from the first two harmonics is also indicated.

When the focusing gain and nonlinearity are kept constant and the absorption parameter

increases ten times, the fundamental force amplitude, surprisingly, increases by approximately 40%

and the second-harmonic force amplitude decreases significantly (five times) and becomes equal to

5.7% of the fundamental. The associated increase of the fundamental force amplitude, despite the

decrease of the fundamental source waves of both beams when A increases, as shown in

Figure 3.8(c) and (d), is attributed to the increased attenuation αn(f)=α0fγ that is involved in the

derivation of the radiation force, as seen in (32). From a physical viewpoint, for a higher medium

attenuation α0, there is more transfer of momentum from the ultrasound wave to the medium,

thereby leading to enhancement of the generated acoustic radiation force [52]. Finally, the increase

by a factor of three of the focusing gain when the absorption and nonlinearity parameters are fixed,

does not significantly alter the fundamental force amplitude at focus, but almost doubles the second-

harmonic force amplitude.

Based on (36), the temperature increase at the focus due to the first two harmonics was

calculated and the results are also given in Table 3.1 for a heat capacity of cv = 4.2×106

Watts⋅s/(m3⋅°C), corresponding to values reported for soft tissues [80]. It can be observed, that the

G = 24, A = 0.7 G = 24, N = 0.3 A = 0.5, N = 0.4 n = 1 1.32 1.49 5.77 Maximum Force

(104 N/m3) n = 2 0.07 (5.3%) 0.60 (40.3%) 1.02 (17.7%)ΔT (°C) 0.01

N = 0.20.02

A = 0.1 0.07

G = 12

n = 1 10.28 2.12 5.32 Maximum Force (104 N/m3) n = 2 3.95 (38.4%) 0.12 (5.7%) 1.98 (37.2%)

ΔT (°C) 0.14 N = 0.6

0.02 A = 1.0

0.07 G = 36

45

maximum temperature increase occurs when the nonlinearity parameter triples, while it remains the

same when the absorption coefficient or the focusing gain change.

The MI value has been calculated in some cases to exceed the safety limit of 1.9 that has

been defined by the FDA [81], e.g., in Figure 3.8(b), (c) and (e). However, the objective of this

section is to demonstrate how the nonlinearity parameters affect the harmonic force components. In

the in-vivo application of elastography, both the TI and MI limits should be considered for negligible

risk to patients. Furthermore, as observed in the above figures, it is the fundamental component that

is produced in the focal zone from the propagation of beam B (see Figure 3.2) that causes the

increase of the MI. A solution to this is to use a different smaller source pressure for beam B or

reduce the effective area of the corresponding annulus transducer (B). In the latter case for example,

if the effective radius (a22-a21) of beam B is reduced by 3 mm, the MI can be decreased from around

2.2 (i.e. slightly beyond the risk region, see Figure 3.8(b)) down to approximately 1.5.

Figure 3.9: Normalized radial patterns of the fundamental and second-harmonic radiation force components on the focal plane for various nonlinearity (N), absorption (A) and total focusing gain (G = Ga+Gb) parameters.

The effects of the six variation cases on the radial profiles of the fundamental and second-

harmonic force components are shown in Figure 3.9. For each case considered, the parameter

variations have little influence on the radial distribution of energy in the focal region. Comparing

Figure 3.9(a) and (b), apart from slightly better definition of the sidelobe structure in the second-

46

harmonic force component, the beam patterns are relatively unaffected by the increase in the source

amplitude. Furthermore, as already discussed in Section 3.2, the extra sidelobes known as ‘fingers’,

can be seen typically at the same amplitude level as the ordinary sidelobes. The sharpest focusing of

both the fundamental and the second-harmonic force component occurs in Figure 3.9(f), i.e., when

the focusing gain (corresponding to a higher source frequency) becomes large. Due to the sharper

focusing, more sidelobe peaks can be seen within the same physical distance (0-4.5 mm). In this

case, it can be more clearly observed that the frequency-dependent attenuation affects the fingers

more than the sidelobes [83]. However, an increase in absorption leaves the fingers in the focal

region relatively unaffected (compare Figure 3.9(c) and (d) where A increases by a factor of ten), as

opposed to the farfield of unfocused systems [76], [83].

3.4.3 Center Frequency and Source Pressure: Effects on the Harmonic Force

Components

In this section, the attenuation coefficient is fixed to 0.35 dB/(cm⋅MHz1.1) and the influence

of the center frequency and source pressure are investigated on both the harmonic modulated force

components (Δf = 200 Hz) and the associated temperature increase at the geometric focus. First, the

(a) (b) Figure 3.10: (a) Maximum amplitude of the n’th harmonic CW modulated radiation force (n = 1..3) with the center frequency. (b) Total temperature increase at focus (for n = 1..3) with the center frequency for an emission duration of 10 ms. The maximum emission duration, based on (37), for which the temperature increase at focus is 1.0°C, is shown with f0 in (b) inset. The source pressure was assumed to be 450 kPa and the attenuation coefficient was 0.35 dB/(cm⋅MHz1.1).

47

center frequency was assumed to vary between 1 and 3.5 MHz (fixed source pressure P0 = 450 kPa)

and the maximum amplitude of the n’th-harmonic (n = 1...3) CW force is shown in Figure 3.10(a). It

can be observed, that as the center frequency increases, the harmonic force amplitudes at focus

increase and in fact, the relative amplitudes behave as n-k, where k is a monotonically decreasing

function of frequency. For example, for f0 = 1.0 MHz, the amplitude of the n’th harmonic force

varies as n-3, while for f0 = 2.0 MHz, the amplitude variation is n-3/2, implying a higher harmonic

content due to the increased nonlinear absorption.

The temperature increase at focus arising from the first three harmonics was calculated

according to (36) and is shown in Figure 3.10(b) with the center frequency for emission duration of

10.0 ms (i.e. two cycles). Furthermore, the maximum emission duration so that the associated

temperature increase is below 1°C is shown in Figure 3.10(b) inset. It can be observed that for

higher source frequencies, the maximum allowed emission duration Dmax is significantly reduced

(approximately exponentially). It should be also noted, that as the source frequency is increased, the

MI is decreased for the same negative peak pressure according to (34).

Figure 3.11: Radial beamwidth at -6 dB with the center frequency of the n’th-harmonic modulated force (n = 1…3), on the geometric focal plane at t = 0, 5.0 ms, etc. The source pressure was assumed 450 kPa and the attenuation coefficient 0.35 dB/(cm⋅MHz1.1).

The effects of the center frequency in the harmonic radial beamwidths (i.e. the width of the

mainlobe at -6 dB) are shown in Figure 3.11 for the first three harmonics on the geometric focal

48

plane. The radial beamwidth was found to vary as 1/f0 for all three harmonic force components.

Furthermore, the n’th-harmonic force beamwidth was found to vary approximately as n-1 for each

center frequency (see Figure 3.6).

Subsequently, by fixing the source frequency to 2.0 MHz, the effects of the source pressure

can be seen in Figure 3.12. For the investigated pressure range, the nonlinearity parameter N can be

calculated to vary from 0.23 to 0.68. All harmonics increase with the source pressure; however, the

fundamental force component follows an approximately linear increase while the higher harmonics

exhibit a parabolic behaviour. The relative harmonic force amplitudes at focus increase with the

source pressure, i.e., they behave as n-5/2 for P0 = 300 kPa and n-5/4 for P0 = 500 kPa. The associated

temperature increase for D = 10.0 ms and the maximum permitted emission duration based on (37),

are also shown in Figure 3.12 (b).

3.5 Chapter Summary

The generation of dynamic radiation force components at harmonic modulation frequencies

created by the interference of two finite-amplitude confocal coaxial quasi-CW ultrasound beams of

slightly different frequencies was investigated. The radiation force generates low-frequency

(a) (b) Figure 3.12: (a) Maximum amplitude of the n’th harmonic CW modulated radiation force (n = 1..3) with the source pressure. (b) Total temperature increase at focus (for n = 1..3) with the source pressure for an emission duration of 10 ms. The maximum emission duration, based on (37), for which the temperature increase at focus is 1.0°C, is shown with P0 in (b) inset. The center frequency was assumed 2.0 MHz and the attenuation coefficient 0.35 dB/(cm⋅MHz1.1).

49

narrowband shear waves at the harmonic modulation frequencies and these radiate outwards from

the focal zone. The narrowband characteristics of these waves can be enhanced with increasing

application time of the force. However, an increase of the force application time is also associated

with a temperature rise, which is limited by regulatory specifications of the Thermal Index (TI).

Specifically, a maximum temperature increase of 1.0 °C can be considered not to pose any risk to

patients.

The limitation according to which the Mechanical Index (MI) must be less than 1.9 has not

been strictly followed in this chapter and there were cases where the MI fell into the risk region.

However, the degree of nonlinearity is proportional to the MI [84] and the emphasis of this chapter

was on the harmonic force components and the effects of the nonlinearity parameters. It should be

noted though, that in the in-vivo application of elastography, both the TI and MI should be carefully

accounted for, in order not to pose any risk to patients.

Three dimensionless parameters, that are commonly used to characterize focused sources

were adapted and used to characterize the properties of our finite-amplitude confocal source. They

enabled the effects of nonlinearity, changes in absorption and different focusing gains to be

discussed in a straightforward manner.

50

CHAPTER 4 Narrowband Shear-Wave Propagation: the Fundamental Component

The modulated radiation force described in Chapter 3 generates narrowband shear waves at

harmonic modulation frequencies that can propagate away from the beam axis. In this chapter, the

generation and propagation of short-duration shear waves at the fundamental modulation frequency

are investigated based on the approximate Green's functions for viscoelastic media as derived by

Bercoff et al. [69]. The evolution of these waves is studied in the spatiotemporal domain from a

theoretical perspective. The manner in which the characteristics of the viscoelastic propagation

medium, i.e., the shear viscosity and speed, affect the evolution of the fundamental shear field is

described. The effects of other underlying phenomena, such as the coupling wave, the spatial

distribution of the force and the duration of the modulated wave, are explored.

4.1 Generation of Low-Frequency Shear Waves

As described in Chapter 3, the modulated radiation force, generates shear waves in the focal

zone at the harmonic difference frequencies of nΔf, n = 1,2, etc., and these propagate away from the

beam axis (see Figure 3.1). In this chapter, the effects of just the fundamental component of the

shear displacement field are studied. The harmonic components of the displacement field will be

treated in the following chapter. Parts of the subsequent analysis are presented in [85].

51

Figure 4.1: Normalized total displacement csz

sz

czz uuuu ++= at a radial

distance r = 3.0 mm from the source (origin O) for a spatiotemporal impulse (dash-dotted line) and a modulated force at Δf = 500 Hz (solid line) of duration D = 300 μs. The medium parameters assumed are given in the caption of Figure 2.3. The spreading effects of the time convolution are evident. In the purely elastic case, the coupling wave is active for r/cs-r/c+D ≈ 1.45 ms, where r/c and r/cs are the times of arrival for the pure compressional and pure shear waves.

Based on (17), the i’th-component of the total displacement field due to the radiation (point)

force acting in the j-direction can be expressed as:

[ ]),(),(*)(0,),(),(),( tGtGtFtututu csij

sijj

csi

sii rrrrr +=+≈ Δ , (40)

where the approximation applies to soft tissue media [23] and ∗ denotes a 1-D time convolution. As

already noted in the previous chapter, for sufficiently focused sources, the primary component of the

force is along the z-axis and therefore, only the z-component of the generated shear field will be of

interest (i.e. i = j = z). It should be also noted, that the approximate viscoelastic Green’s functions [69]

described by (24) and (25), will be used in evaluating the above expression.

We shall study the effects of the fundamental component of the modulation force (Δf = 500

Hz) produced by the confocal source, such that the modulation waveform consists of a short cosine

wave of duration D starting at t = 0 as described in (32) for n = 1. From (24) and (40), the shear term

of the displacement field in the z-direction, can be written as

52

∫

∫

ττ−ωΔτππρ

+α=

τττ−=

τ

−τ−

−γγ

Δ

Dsv

sscr

ss

baba

szzz

Dsz

dtc

evrcc

ppff

dGtFtu

0

2

22)/(

2/122

0

0

)]([cos21

4(0)(0))(2

),()(0,),( rr

,(41)

where the subscript 1 was omitted for notational simplicity. In a similar manner, the contribution of

the coupling term to the shear displacement field can also be written down, enabling the effects of

both contributions to the total shear displacement field to be evaluated. Examples of the total

displacement due to a point-force (temporal impulse) and a force of finite duration are given in

Figure 4.1.

(a) (b)

(c)

(d)

Figure 4.2: Showing the two components of the displacement vector produced by the coupling term Greens function on the (r, z) plane. The contributions at three different angles of (a) 0°, (b) 45° and (c) 90° at a distance of 3.0 mm from the geometric focus are shown assuming cs = 3 m/s and c = 1550 m/s. The medium bulk and shear viscosities were taken to be ηc = 0 and ηs = 0.15 Pa⋅s, respectively. Also assumed, was an emission duration of D = 0.5 ms corresponding to 1/4 cycle of the modulated wave (Δf = 500 Hz). (d) Schematic describing the three different locations on the (r, z) plane for (a), (b) and (c).

53

Figure 4.3: Normalized polar plots showing the contributions to the z-component shear wave displacement from the shear term and the sum of the shear and coupling terms at two locations on the (r, z) plane from a CW point source at the geometric focal point (0,0). (a) R = 1.1 mm and (b) at R = 8.0 mm. The parameter values assumed are the same as those given in the caption of Figure 4.2, except that D = 20 ms.

4.2 Analysis of the Shear Displacement Field

A modulation frequency Δf = 500 Hz will be assumed for the simulations of this chapter. The

rest of the parameter values are given in the first paragraph of Section 3.2. As discussed in the

previous chapter, in the subsequent analysis of the fundamental shear field, we shall limit our

discussion to emission durations of a few cycles, thereby limiting the associated peak temperature

rise, while still maintaining a reasonably narrow bandwidth. To investigate the properties of the

generated fundamental shear displacement field, we need to specify particular points. For this

reason, we make use of symmetrical cylindrical coordinates (r, z) and denote the distance of an

observation point from the origin by R (note that on the focal plane R= r). First, we shall examine

the contributions of the longitudinal and shear components of the near-field displacement vector

(coupling wave). These are illustrated in Figure 4.2 for a point on the (r, z) plane, fairly close to the

origin (R = 3 mm) at three different angles, using the parameter values given in the caption.

Subsequent simulation results also use these values. It can be observed, that the longitudinal

component of the coupling wave is a maximum and the shear component zero on the beam axis,

54

while the opposite is true on the radial axis, as already mentioned. At an angle of 45o, both

components contribute equally to the near-field displacement.

To examine displacement contributions of the coupling term and the pure shear term to the

total displacement, two points that lie on the (r, z) plane centered at the geometric focus will be

considered. If the first point is located at a distance R = 1.1 mm and the second point at R = 8.0 mm,

then for a shear wavelength of about 6 mm (corresponding to a frequency of 500 Hz and a

propagation speed of 3 m/s), the first point can be considered to be in the near field and the latter to

be an intermediate point between the near and far field. The polar plots of Figure 4.3 illustrate the

contribution of the coupling term to the total shear displacement at the two points on the (r, z) plane

for a sinusoidal modulation frequency of 500 Hz. It should be noted that for R = 8.0 mm, the effect

of the coupling term on the total shear wave displacement in the z-direction is small and that the

shear displacement in the z-direction is a maximum on the geometric focal plane z = 0. For this

reason, the analysis that follows considers the shear displacement on the focal plane rather than on

the (r, z) plane.

Figure 4.4: Normalized shear displacement for five emission durations at two locations on the focal plane. (a) The nearfield point A(1.1mm, 0). (b) The near/farfield point B(8.0mm, 0). The displacements, which were normalized with respect to the maximum displacement amplitude at point A, do not include the effects of the coupling term. The parameter values assumed are the same as those given in the caption of Figure 4.2.

4.2.1 Effects of the Emission Duration

Normalized temporal variations of the shear displacement induced by the fundamental

dynamic force )(0, tFzΔ at 500 Hz, are shown in Figure 4.4 for the two points, A(1.1 mm, 0) and

55

B(8.0 mm, 0), that lie on the geometric focal plane. Five emission durations D are considered,

ranging from 0.2 to 2.6 ms, corresponding to 1/10th of a cycle up to 1.3 cycles (T = 1/Δf = 2.0 ms).

The spectrum for D = 1.2 ms is also shown. It can be seen that the temporal convolution between the

Green's function and a finite bi-directional sinusoidal radiation force along the z-axis, leads to a

gated shear response whose harmonic content becomes narrower as the emission duration increases.

In the absence of any viscous effects, the relative amplitudes of the waveforms at the two radial

locations could be expected to vary approximately as 1/r. However, the presence of viscous loss

together with the 1/r effect should cause a greater reduction in the displacement amplitude at point

B compared to A, as can be demonstrated from the results shown in Figure 4.4. The effects of

viscous losses together with a proposed measurement method are examined in more detail in the

following sections.

Figure 4.5: The effect of the emission duration D on the time difference between the positive and negative peaks, versus the radial distance. The parameter values assumed are the same as those given in the caption of Figure 4.2.

The total duration of the shear displacement at a specific location can be roughly

approximated by the sum of the duration of the Green’s function at this point and the duration of the

applied force. Therefore, one of the effects of the temporal convolution is to increase the overall

duration of the tissue displacement. Furthermore, as seen in Figure 4.4, an increase of the duration

of the force affects the time difference between two successive peaks. When D = 0.8 ms = 0.4T, the

time difference between the positive and negative peak is significantly smaller (0.55-0.8ms) than the

56

theoretically calculated time of T/2 = 1 ms (according to the cosine-type force). However, when the

force is applied for a 60%-130% of its cycle, the time difference is closer (0.8-0.94ms) to its

theoretical value T/2, especially at larger distances from the source. As shown in Figure 4.5, the

peak time difference as a function of the radial distance initially decreases, reaches a minimum and

then increases monotonically. This behaviour depends on the combined effects of the temporal

convolution and shear viscosity, as will be discussed in the next section.

(a) (b)

Figure 4.6: Influence of the coupling term on the total displacement at (a) point A and (b) point B. The force duration was taken 1.2 ms. The displacements have been normalized according to the maximum amplitude of the pure shear term.

4.2.2 Effects of the Coupling Wave

As noted by Calle et al. [58], the Greens function characterizing the propagation of the

coupling wave consists of a ramp, whose arrival time at a given location corresponds to the

longitudinal propagation speed and whose finish time corresponds to the shear wave speed. The

effects of the coupling wave ),( tu csz r on the temporal shape of the shear wave can be observed in

Figure 4.6 for the fundamental shear displacement. Normalized temporal variations of the

displacement at the referenced radial locations A and B are shown with (solid line) and without

(dash-dotted line) the coupling term. The duration of the force was taken equal to 1.2 ms. In the

near-field (point A), the coupling term has a negative contribution to the displacement of tissue,

which is evident by the decrease of the wave magnitude and the addition of a low-amplitude

negative peak in the beginning of the displacement profile. The effects of the coupling wave are less

evident as the wave propagates away from the source (point B) and become negligible further away.

57

4.2.3 Effects of the Shear Viscosity

Temporal variations of the total peak displacement (shear-plus-coupling) at the near and

near/far-field locations are shown in Figure 4.7 for three different values of shear viscosities and an

emission duration of D = 1.2 ms. A good deal of uncertainty seems to exist at the present time as to

the range of shear viscosity values appropriate for soft tissue. Consequently, we have chosen to

explore a wide range of values, varying from a very low viscosity (0.001 Pa.s) medium, to a

medium value reported by Bercoff et al. [69] for a gel tissue-mimicking phantom (0.3 Pa.s), to a

high value approaching those measured on extracted animal organs using MRI [16] and the much

earlier results reported by von Gierke et al. [70] on the thigh and upper arm of subjects. The latter

results were obtained by observation of surface waves propagating from a vibrating piston. High

values of the shear viscosity, even up to 15 Pa⋅s and higher, have been also reported in [36], [71],

[72].

Figure 4.7: Normalized shear plus coupling displacement at point B(8.0mm, 0) for three different values of shear viscosity ηs and an emission duration of D = 1.2 ms. The displacement amplitudes have been normalized by the maximum amplitude at point A(1.1mm, 0) for ηs = 0.001 Pa·s (see inset).

Peak Displacement and Peak Time: Spatiotemporal Impulse-like Force

As noted by Bercoff et al. [69], viscosity acts like a lowpass filter by attenuating the wave

amplitude, smoothing out the peaks induced by the oscillatory radiation force and broadening the

58

shape of the shear wave, all of which are clearly evident in Figure 4.7. We will first consider the

simple case of a spatiotemporal impulse-like radiation force, i.e., )()(=)(0, 0 tFtFz δδΔ r where F0

(force×time) denotes the force amplitude at the geometric focus and the MKS units for the radiation

force are Newton/m3. For this case, the time at which the pure shear displacement reaches a

maximum at a fixed point on the radial axis, can be obtained from (40) and (23) as

2

222

2

4=

s

ssspvmax

c

vrcv −+τ (42)

and the corresponding maximum displacement is given by

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞⎜

⎝⎛ −+

⎟⎠⎞⎜

⎝⎛ ++−

−×

⎟⎠⎞⎜

⎝⎛ −+ρπ

=ssss

ssss

ssss

pv

vrcvv

rcrcvv

vrcvvr

FU

222

2222

2/12222/3

0max

44

24exp

44

. (43)

It should be noted, that both pvmaxτ and pvUmax are functions of the shear speed and shear viscosity and

consequently, they may have potential for experimental studies of the variations of the local tissue

elasticity and viscosity. However, since narrowband shear waves are desired (for accurate

estimations of the shear speed and attenuation as functions of frequency) and, at the same time, the

heating effects must be limited, then a convolution of the viscoelastic Green’s function with a

dynamic (oscillatory) force of finite-duration must be applied. Due to this tradeoff (i.e. guaranteeing

narrowband characteristics and limiting the associated temperature rise), the complexity of the

resulting equations inevitably increases and becomes very difficult to extract either the tissue

elasticity or viscosity from the peak amplitude and peak time at a given radial location. Alternative

approaches will be considered in Chapter 6.

Peak Displacement and Peak Times: Point Force of Finite Duration

The normalized amplitude of the positive peak displacement is shown as a function of the

radial distance in Figure 4.8(a), for a shear viscosity of 0.6 Pa⋅s. Both the pure shear and the total

(shear plus coupling) components are shown. As the wave propagates away from the point source

the magnitude decreases due to both the inverse distance and viscous effects. For low shear

viscosities, the pure shear-term peak displacement varies as 1/r (Figure 4.8(a) inset). As expected,

the effects of the coupling wave become negligible in the far-field region, moreover, for higher

viscosities, the peak amplitude decreases more rapidly with increasing radial distance. For other

emission durations, a similar behaviour was observed.

59

(a) (b) Figure 4.8: For an emission duration of D = 1.2 ms and a shear viscosity ηs = 0.6 Pa⋅s showing: (a) The normalized positive peak displacement with the radial distance, for the pure shear component (dotted line) and the shear-plus-coupling component (solid line). For comparison, the 1/r decay is indicated by “x”. The equivalent curves for very low viscosity (0.001 Pa⋅s) are shown in (a) inset. Similar trends were obtained for the negative peak displacement. (b) Demonstrating the behaviour of the peak times with the radial distance. Both τmax, τmin (shear plus coupling) and s

maxτ , sminτ (pure shear) have been calculated. Also the peak times

corresponding to a point-source are shown: pvmaxτ (viscoelastic media based on (42)) and

pemaxτ = r/cs (elastic media).

By differentiating (41), the times at which the peaks of the modulation signal for the pure

shear displacement term arrive at the location r can be expressed as

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

ωΔ

ωΔ

ωΔτ

∫∫

+−−

+−−

dsses

dsses

Dsvcsr

Dsvcsr

si

ss

ss

0

2)(2/1

0

2)(

)cos(

)sin(arctan1=

222

2222/1

(44)

where i denotes a minimum or a maximum. The effects of the convolution operation make it

difficult to extract estimates of the shear elasticity and shear viscosity from measurements of the

peak times. For D = 1.2ms, the arrival times are shown as a function of the radial distance in

Figure 4.8(b). For viscosities of 0.6 and 0.001 Pa⋅s, the arrival times for the peaks of the shear-plus-

coupling terms were numerically calculated and can be seen to be very close to those based on the

pure shear term, as given by (44). Furthermore, under low-viscosity conditions (see inset), the

positive peak time (τmax) increases almost linearly with the distance and can be approximated by the

60

equation for a temporal impulse in a purely elastic medium, i.e., spe cr /max =τ , as given by (42) for

vs→0.

4.2.4 Effects of the Shear Speed

Normalized displacements for three different values of the shear speed are shown in

Figure 4.9 at point B(8.0 mm, 0), for a shear viscosity of 0.15 Pa⋅s and an emission duration of 1.2

ms. As intuitively expected, for higher shear speeds (corresponding to ‘stiffer’ media), the peak

times and the duration of the displacements decrease. The latter remark can be explained by the

presence of the shear speed in the exponential term of (23), which causes a compression of the

shape of the shear displacement. This effect is more evident in the absence of the temporal

convolution, as illustrated in Figure 4.9 inset.

Figure 4.9: Normalized displacements at point B(8.0 mm, 0) for three different values of the shear speed assuming an emission duration of 1.2 ms and a shear viscosity of 0.15 Pa⋅s. The corresponding displacements induced by a temporal impulse-like force are shown in the inset together with the half-width of each curve. The effects of the coupling term have been accounted for.

61

In the case of an elastic medium, it can be again intuitively understood (and mathematically

verified [58]) that the peak displacement also decreases monotonically with increasing shear speed.

Based on this observation, Calle et al. [58] suggested estimating the shear modulus of the medium

from the peak displacement. For a viscous medium however, such a monotonic behaviour occurs

only in the case of a temporal impulse-like force, as will be shown below.

The waveform changes seen in Figure 4.9 are examined in more detail in Figure 4.10. The

peak displacement (including the coupling term) was calculated for a temporal impulse for various

shear speeds and the results are shown in Figure 4.10(a). It can be seen that the peak displacement

decreases monotonically with the shear speed (note that the behaviour of the peak displacement with

the shear speed due to the pure shear component is in agreement with (43)). However, as observed

in Figure 4.10(b) for a radiation force with duration 1.2 ms, for higher shear viscosities, the

temporal convolution leads to a non-monotonic behaviour of the peak displacement as a function of

the shear speed. For very low viscosities, the peak displacement varies monotonically, but the

decrease is more rapid than that for the temporal impulse.

Figure 4.11(a) shows the normalized pattern of the positive peak displacement with both the

shear speed and shear viscosity when the effects of the coupling term are included. In addition,

Figure 4.11(b) shows the normalized peak displacement versus shear viscosity for four values of the

(a) (b) Figure 4.10: Normalized positive peak displacements versus the shear speed for three different values of shear viscosity at point B(8.0 mm, 0), for (a) a temporal impulse and (b) a force of duration D = 1.2 ms. Both the pure shear components (dashed curves indicated by “S”) and the shear-plus-coupling components (solid curves) are shown. The curves have been normalized according to the maximum of the latter for ηs = 0.001 Pa⋅s.

62

shear speed (3, 6, 9 and 12 m/s), a finite duration force (1.2 ms), and including the coupling term. It

can be observed, that although the normalized peak displacement decreases with increasing shear

viscosity, the decrease is not the same for each value of the shear speed (in fact, it appears that for

higher shear speeds the peak displacement is less rapidly affected by viscosity). On the other hand,

it can be easily shown that for a temporal impulse-like force, the decrease of the peak displacement

with the shear viscosity does not depend on the value of the speed. Specifically, the decrease due to

the pure shear wave can be predicted by (43).

The above observations, demonstrate once again the increased complexity arising from the

time convolution, which makes it very difficult to extract any viscoelastic information about the

propagation medium from the peak displacements (or peak times discussed in the previous section).

4.2.5 Effects of the Spatial Distribution of the Force

An important advantage of the confocal beam geometry is its high spatial resolution,

enabling the dynamic stress field to be confined within a small spatial region [25], [44], thereby

enabling to improve image resolution. So far, displacement calculations have been performed for

the oscillatory radiation force exerted on the geometric focal point. In practice, however, additional

points of the modulated intensity field in the 3-D neighborhood surrounding this spot, affect the

spatiotemporal pattern of the generated shear waves.

(a) (b) Figure 4.11: Normalized positive peak displacement at point B(8.0 mm, 0) for (a) several shear speeds and shear viscosities and (b) four different values of the shear speed with viscosity. The force duration was taken to be 1.2 ms and the effect of the coupling term has been included.

63

Based on (40) and by using the superposition principle, the z-component of the shear

displacement field produced by a well-defined volumetric source, can be written as a spatiotemporal

convolution given by

kkrkrr

ddtGFtu szzz

Dsz ττ−−τΔ

τ ∫∫ ),(),(=),(0=

. (45)

Because of the source symmetry, we choose a cylindrical volume distribution of 1.0 × 21 mm2

centered at (0, -0.6 mm), i.e., just before the geometric focus (see Figure 3.3). The boundaries

corresponded to about -12 dB in the radial and -6.5 dB in the z-direction. It should be noted, that

while the radial profile of the force is sufficiently localized around the focal point, along the z-axis

the force was found to spread throughout a larger region and has two peaks of similar amplitudes.

The latter remark can be explained by the interference of the fundamental components of the two

source beams whose actual focusing occurred at slightly different locations (approximately 5 mm

apart, see Figure 3.3). The dimensions of the above cylindrical volume distribution were selected as

a tradeoff between accuracy and computational complexity. It is expected that the shear response

will be affected by the volume within which the radiation force contains most of its energy.

Figure 4.12: Comparison of the normalized displacement (shear plus coupling) waveforms produced at point B(8.0 mm, 0), when account is taken of the 3-D force distribution, to that produced by a point force at the focus. The graphs assume an emission duration 2.0 ms, a shear viscosity 1.0 Pa·s and a shear speed 3.0 m/s.

64

The normalized displacement (shear plus coupling) induced by the above volumetric force at

a radial distance of 8.0 mm (point B) is shown in Figure 4.12. The duration of the force was taken

2.0 ms, the shear viscosity 1.0 Pa⋅s and the shear speed 3.0 m/s. For comparison, the corresponding

displacement induced by a point force of the same duration is also shown. It can be observed, that

the 3-D spatial convolution caused a spreading of the response. Nonetheless, the times at which the

waveform peaks occur, remain almost the same, so that the assumption of a point source located at

the geometric focal point (as used in the prior sections) remains valid. Furthermore, as noted earlier,

the assumed 3-D force distribution of dimensions 1.0 × 21 mm2 was selected as the limit up to

which the spatial convolution affects the shear displacement and beyond which, any further increase

in the dimensions of the cylindrical force distribution will not cause any significant changes. A

second larger distribution of dimensions 1.4 × 26 mm2 was also assumed and the corresponding

shear displacement is demonstrated in Figure 4.12. As observed, the increase of the cylindrical

volume dimensions did not change significantly the calculated shear response. Any further increase

from that point led to negligible deviations.

4.3 Chapter Summary

The generation of narrowband shear waves by the dynamic radiation force of two

intersecting confocal quasi-CW ultrasound beams was investigated in this chapter. Based on

approximate Green's functions for viscoelastic media the shear wave generated at the fundamental

modulation frequency was studied in the time and spatial domains. It was pointed out that an

important advantage of generating narrowband shear waves is that their spectrum is not significantly

distorted as they propagate through tissue by different kinds of frequency-dependent distortion.

Furthermore, their spectral power is concentrated within a narrow frequency band around the

modulation frequency, thereby arguing for a more accurate estimation of the shear attenuation and

dispersion at various frequencies in the low-kHz range.

The manner in which the characteristics of the viscoelastic propagation medium, i.e., the

shear viscosity and speed, affect the evolution of the fundamental shear-wave component was

analyzed. It was shown, that a combination of underlying phenomena should be carefully accounted

for, such as the coupling wave and the duration of the modulated wave. It was also noted that the

emission duration should be limited to a few cycles of the modulation frequency, so that heating

effects are minimized, while the transmitted signal still maintains a reasonably narrow bandwidth.

65

It was shown that the above tradeoff (i.e. decreasing the bandwidth of the generated shear

waves and limiting the heating effects) inevitably increases the complexity of the associated

equations, by requiring the application of time convolution of the Green’s function with a

modulated (oscillatory) radiation force of finite duration. This restricts the potential to estimate

shear elasticity or viscosity from measurements of the peak amplitudes or peak times.

66

CHAPTER 5 Narrowband Shear-Wave Propagation: the Harmonic Components

In this chapter, the presence of higher harmonics in the spectrum of the shear field caused by

a finite-amplitude source excitation is investigated. These higher frequencies along with the higher

viscosities believed to be characteristic of biological tissue make it necessary to use the exact

solution of the shear Green’s function described in Section 2.3.2. The properties of the harmonic

shear field are explored both in the time and frequency domains. Furthermore, a time-frequency

analysis will be provided, which is more suitable for transient non-stationary time signals, such as

the investigated shear waves.

5.1 Time and Frequency-Domain Analysis

5.1.1 Exact vs. Approximate Viscoelastic Green’s Function

As pointed out at the end of Chapter 2, the presence of a high shear viscosity together with a

high frequency does not enable an approximate expression of the Green’s function to be used. The

higher frequencies associated with the harmonic components, in combination with the higher shear

viscosity associated with tissue, requires that this matter be examined in more detail.

For simplicity, a point force FΔ(0,t) acting in the z-direction at the geometric focus is

assumed. Furthermore, the coupling term will be ignored since this will enable us to compare the

effects of the exact versus the approximate viscoelastic Green’s function in a straightforward

67

manner. It should be noted though, that the coupling term is expected to cause a decrease in the

displacement amplitude at locations close to the source, as discussed in Section 4.2.2. However, the

qualitative comparison between the exact and approximate Green’s function is expected to be

similar by including the coupling term.

Figure 5.1: Case of a lower shear viscosity with a modulation frequency of 100 Hz. Normalized shear displacement at (a) point A(1.1 mm, 0) and (b) point B(8.0 mm, 0), based on the exact viscoelastic Green’s function (solid lines) of (26) and the approximate Green’s function (dotted lines) [69], for a shear viscosity of 0.5 Pa⋅s and a shear modulus of 2.5 kPa. The source pressure, center frequency, attenuation coefficient and emission duration were taken to be 450 kPa, 2.5 MHz, 0.35 dB/(cm⋅MHz 1.1) and 20.0 ms, respectively. The corresponding normalized spectra are shown in (c) and (d), where the harmonic shear content can be observed.

If Nh harmonics are considered for each beam, then the modulated finite-amplitude radiation

force generated by the interference of these beams (see Figure 3.4), can be written

as ∑=

ΔΔ =hN

nn tFtF

1),0(),0( where ),0( tFn

Δ is given by (32). Similarly to the fundamental displacement

component described by (41), the finite-amplitude shear component can be written as a time

convolution between the total radiation force and the shear Green’s function, i.e.

68

Figure 5.2: Case of a higher shear viscosity for a modulation frequency of 100 Hz. Normalized shear displacement at (a) point A(1.1 mm, 0) and (b) point B(8.0 mm, 0), based on the exact viscoelastic Green’s function (solid lines) of (26) and the approximate Green’s function (dotted lines) [69], for a shear viscosity of 5.0 Pa⋅s. The corresponding normalized spectra are shown in (c) and (d), where the harmonic shear content can be observed. The other assumed parameter values are given in the caption of Figure 5.1.

∫ ∑ τ⎥⎦

⎤⎢⎣

⎡ϕΔ+τ−ωΔ+τ

ρα

==

γγγD N

nnbnanba

szz

s dtnppffnGc

tuh

0 1,,2

0 )]0()(cos[(0)(0))(),(2

),( rr , (46)

where the shear waveform has been taken to consist of a short modulated wave of duration D

starting at t = 0 and szzG is the z-component of the viscoelastic Green’s function.

Based on (46), temporal variations of the shear displacement at points A(1.1 mm, 0) and

B(8.0 mm, 0) of the geometric focal plane are shown together with the spectra in Figure 5.1 and

Figure 5.2, for viscosities of 0.5 and 5.0 Pa⋅s, respectively. The latter value is closer to those

measured for biological tissue [16], [70]. The shear modulus was assumed to be 2.5 kPa, which is in

accord with the shear moduli reported in [36], [69]-[70]. A finite-amplitude modulated radiation

force at 100 Hz of two cycle-duration (D = 20.0 ms) was considered and the first five harmonic

components (Nh = 5) were retained. The source pressure, center frequency and attenuation

coefficient were assumed 450 kPa, 2.5 MHz and 0.35 dB/(cm⋅MHz 1.1 ), respectively. The

69

waveforms have been calculated both theoretically, based on the approximate viscoelastic shear

Green’s function described by (24) (where the shear speed was approximated by ρμ≈ lsc ) and

numerically, based on the exact solution described by (26). The corresponding waveforms together

with their frequency spectra for a modulation frequency of 350 Hz are shown in Figure 5.3 and

Figure 5.4.

Figure 5.3: Case of a lower shear viscosity with a modulation frequency of 350 Hz. Normalized shear displacement at (a) point A(1.1 mm, 0) and (b) point B(8.0 mm, 0), based on the exact viscoelastic Green’s function (solid lines) of (26) and the approximate Green’s function (dotted lines) [69], for a shear viscosity of 0.5 Pa⋅s. The corresponding normalized spectra are shown in (c) and (d), where the harmonic shear content can be observed. The rest of the parameter values assumed are the same as those given in the caption of Figure 5.1, except that D = 5.71 ms (corresponding to two cycles at Δf = 350 Hz)

According to (38), the parameters of nonlinearity, focusing gain and absorption were

computed N = 0.63, G = 12.2+17.7= 29.9 and A = 0.77, respectively (independent of the modulation

frequency). Based on (36), the temperature increase, arising from the first five harmonics, was

found approximately 0.7 °C for Δf = 100 Hz and 0.2°C for Δf = 350 Hz. Furthermore, the

Mechanical Index (MI) is approximately 1.5 according to (34) (since the pressure at the focal point

is around 2.9 MPa), a value which is below the FDA [81] limit of 1.9.

70

Figure 5.4: Case of a higher shear viscosity with a modulation frequency of 350 Hz. Normalized shear displacement at (a) point A(1.1 mm, 0) and (b) point B(8.0 mm, 0), based on the exact viscoelastic Green’s function (solid lines) of (26) and the approximate Green’s function (dotted lines) [69], for a shear viscosity of 5.0 Pa⋅s. The corresponding normalized spectra are shown in (c) and (d), where the harmonic shear content can be observed. The rest of the parameter values assumed are the same as those given in the caption of Figure 5.1, except that D = 5.71 ms (corresponding to two cycles at Δf = 350 Hz)

For the lower modulation frequency (100 Hz), the results obtained numerically based on the

exact solution as described by (26), are very close to those obtained from the approximate

viscoelastic solution derived by Bercoff et al. [69], as shown in Figure 5.1 and Figure 5.2. In the

high-viscosity case, there is significant disagreement only at the nearfield point A. At a higher

modulation frequency (350 Hz), the lowpass and dispersive viscous characteristics become

significant and, as shown in Figure 5.4, the exact viscoelastic Green’s function needs to be used.

From the frequency spectra of Figure 5.1, the harmonic shear components can be clearly

observed under low-viscosity conditions (0.5 Pa⋅s), particularly for a point close to the source.

When viscosity increases (see Figure 5.2), the higher-harmonics are decreased but can still be

observed close to the source. Further away (point B), the third-through-fifth harmonic components

have been suppressed, and only the second-harmonic component is clearly visible. On the other

71

hand, when the modulation frequency is increased to 350Hz, the higher-harmonic shear content

suffers greater attenuation even close to the source, as shown in Figure 5.3 and Figure 5.4. As

expected, further away (point B), all harmonics have been almost completely suppressed, even for

the lower viscosity value.

Figure 5.5 shows the behaviour of the fundamental and the first two harmonic spectral

components with the source pressure at point B. A linear increase of all three components can be

observed with the source pressure, where the slope of the lines increases with increasing harmonic

frequency. It should be also noted, that the above behaviour was found not to be affected by the

shear viscosity.

5.1.2 Effects of the Frequency-Dependent Speed and Attenuation

By observing Figure 5.1 through to Figure 5.4, a low-frequency (LF) component can be seen

for conditions of increased viscosity. This LF effect is more pronounced for the higher modulation

frequency (350 Hz) and at larger distances from the source. The likely cause of this LF component

will be investigated in this section.

Figure 5.5: Normalized amplitudes of the first three (n = 1..3) harmonic spectral components of the shear displacement as functions of the source pressure, at point B(8.0 mm, 0). The modulation frequency was assumed 350 Hz and the emission duration 5.71 ms (two cycles). The shear viscosity was taken 0.5 Pa⋅s, but was found not to affect significantly the spectral amplitudes with P0. The rest of the parameter values assumed are given in the caption of Figure 5.1.

72

Figure 5.6: Normalized shear displacements with the coupling term (top row) together with the spectra (bottom row) at point B(8.0 mm, 0). The modulation frequency was taken 350 Hz and the emission duration equal to (from left to right) two, six, twelve and twenty five cycles. The shear modulus and viscosity were assumed 2.5 kPa and 2.0 Pa⋅s, respectively.

In the following simulations, the modulation frequency will be assumed 350 Hz and point

B(8.0 mm, 0) will be considered. Normalized temporal variations of the shear displacement together

with the associated spectra are shown in Figure 5.6 for four different values of the emission duration,

i.e. D equal to two, six, twelve, and twenty five cycles (all heating effects have been ignored). The

shear modulus and viscosity were taken 2.5 kPa and 2.0 Pa⋅s, respectively. The excitation values

assumed are given in the caption of Figure 5.1. It should be also noted, that the coupling term has

been included and the waveforms have been calculated based on the approximate viscoelastic

Green’s functions described by (24), (25).

The appearance of LF spectral components (with Nc-1 peaks for Nc cycles) can be observed

in the above figure, whose relative amplitudes decrease with increasing emission duration.

Furthermore, by looking at the corresponding time signals, a ‘slow’ component appears to arrive at

the end of each waveform. This ‘slow’ component is most likely caused by the effects of dispersion

due to viscosity and the frequency-dependent attenuation. It is well established that in a viscous

medium, the lower frequencies of a propagating shear wave will move much more slowly and will

73

be less attenuated than the higher frequencies, thereby appearing as a ‘slow’ wave at the end of the

total waveform. Therefore, the LF spectral components observed for higher modulation frequencies

under higher-viscosity conditions, is most probably attributed to this ‘slow’ component caused by

the frequency-dependent shear speed and absorption.

Figure 5.7: Normalized shear displacements with the coupling term (top row) together with the spectra (bottom row), for a modulation frequency of 350 Hz and emission duration of 30 cycles. The parameter values were taken μl = 2.5 kPa and (left column) ηs = 2.0 Pa⋅s at point B, (middle column) ηs = 0.001 Pa⋅s at point B and (right column) ηs = 2.0 Pa⋅s at point A(1.1 mm, 0).

The above conclusion is reinforced by the shear displacement and spectrum shown in Figure

5.7 for very low viscosity (i.e. ηs = 0.001 Pa⋅s) at point B, where the ‘slow’ wave cannot be

observed at the end of the shear waveform and thereby, the LF components have been completely

disappeared and only the fundamental and higher harmonics (at 350n Hz) can be seen. Furthermore,

the shear displacement and spectrum for higher viscosity (ηs = 2.0 Pa⋅s) but closer to the source

(point A) are shown. As expected, very close to the source the viscous and dispersion effects have

not had sufficient time to distort the waveform. Finally, as demonstrated in Figure 5.8, if the

modulation frequency is assumed 100 Hz, the above phenomena do not appear especially for greater

74

emission durations. In this case, the effects of frequency-dependent speed and attenuation are seen

to have a smaller influence at lower frequencies.

Figure 5.8: Normalized shear displacement (including the coupling term) together with the Fourier spectrum at point B(8.0 mm, 0). The modulation frequency was taken 100 Hz and the emission duration 30 cycles. The shear modulus and viscosity were assumed 2.5 kPa and 2.0 Pa⋅s, respectively.

5.2 Time-Frequency Analysis

As described previously, because the shear waves have a transient nature, they are non-

stationary [86]. For such signals, the Fourier analysis of the previous section provided a global

energy-frequency distribution. However, if an acceptable estimate of the signal frequency content

with time is needed, then the usual Fourier analysis is insufficient and a time-frequency

representation is more appropriate, such as the Spectrogram, the Wigner-Ville, the Morlet-wavelet

distribution, and the Hilbert spectrum. In studying the propagation of shear waves, such a time-

frequency representation could help to localize a signal in both time and frequency domains and

thus, reveal more detailed information about their dispersive viscous properties, the waveform

distortion and the energy-frequency distribution.

75

There exist several time-frequency representations to characterize non-stationary signals,

however, in this section, we will focus on the Smoothed Pseudo Wigner-Ville (SPWV) distribution

in order to explore the time-frequency properties of the finite-amplitude modulated shear waves.

The Wigner-Ville distribution is a time-frequency representation with optimized resolution in both

domains and particularly suitable for dispersive signals. For a signal x(t), it can be written as [87]:

τ⎟⎠⎞

⎜⎝⎛ τ

+⎟⎠⎞

⎜⎝⎛ τ

−=ω ωτ−+∞

∞−

∗∫ detxtxtV j

22),( (47)

Time and frequency smoothing can be applied in the above representation in order to reduce the

interference from the cross terms (they usually appear as ‘ghost’ patterns in the spectrum), thereby

leading to the Smoothed Pseudo Wigner-Ville (SPWV) distribution. Selection of the window is a

compromise between the joint time-frequency resolution and the level of cross-term interference.

The SPWV can be loosely interpreted as an energy density over the time-frequency plane and

possesses some convenient properties for practical applications, such as energy conservation, time

and frequency shift invariance, preservation of time duration and bandwidth [87].

(a) (b) Figure 5.9: SPWV distributions of the (a) shear and (b) shear plus coupling displacement at the nearfield point A(1.1 mm, 0). A modulation frequency of 100 Hz and emission duration 20.0 ms were assumed. The shear viscosity was taken 0.001 Pa⋅s and the rest of the parameter values assumed are given in the caption of Figure 5.1.

76

Figure 5.9 shows SPWV distributions of the shear displacement with and without the

coupling term at the nearfield point A(1.1 mm, 0), for a modulation frequency of 100 Hz and

emission duration of 20.0 ms. Two Hamming [88] windows, one time-smoothing of 10%-length and

one frequency-smoothing of 25%-length were used. The shear viscosity was taken 0.001 Pa⋅s and

the other assumed parameter values are given in the caption of Figure 5.1. It can be seen that the

inclusion of the coupling term results in the addition of high-frequency content at the beginning

(due to the negative peak, see Section 4.2.2) and end of the waveform.

Smoothed Pseudo Wigner-Ville distributions of the shear displacement (including the

coupling term) at point B(8.0 mm, 0) are shown in Figure 5.10 for viscosities of 0.001 and 2.0 Pa⋅s.

The existence of higher harmonics can be observed in Figure 5.10(a). However, the energy

distribution map appears smoothed and continuous and is spread up to almost the fifth harmonic.

This can be attributed to the smoothing windows used, which suffer from the limitations of a

windowed Fourier analysis (e.g. reduced frequency resolution). The lowpass and dispersive effects

of viscosity, i.e., elimination of the higher harmonics, smoothing and distortion of the energy of the

waveform, are shown in Figure 5.10(b) localized on the time axis. Furthermore, introduction of low-

frequency content due to viscosity can be observed, as discussed in the previous sub-section.

(a) (b) Figure 5.10: SPWV distributions of the shear displacement (including the coupling term) at point B(8.0 mm, 0) for shear viscosity of (a) 0.001 Pa⋅s and (b) 2.0 Pa⋅s. A modulation frequency of 100 Hz and emission duration 20.0 ms (2 cycles) were assumed. The rest of the parameter values assumed are given in the caption of Figure 5.1.

77

The appearance of lower frequencies (localized in time) due to the ‘slow’ component caused

by increased viscosity can be more clearly seen in Figure 5.11 for a modulation frequency of 350 Hz

and emission duration 28.6 ms (ten cycles).

The group delay of the time-frequency distribution can be estimated, for a given frequency,

by the first-order moment of the SPWV along the time axis, as follows:

dttV

dtttV

g

∫

∫∞+

∞−

+∞

∞−

ω

ω=ωτ

),(

),()( (48)

The above equation can be useful, since it enables estimation of the shear speed at various

frequencies from the group delays at two locations.

5.3 Chapter Summary

In this chapter, modulated radiation force components at harmonic modulation frequencies

were considered and the finite-amplitude shear field was investigated both in time and frequency

domain, for excitation conditions that limit the associated temperature increase. The approximate

viscoelastic expressions [69] used in Chapter 4 were shown not to be accurate for higher frequencies

(a) (b) Figure 5.11: (a) Shear displacement (including the coupling term) together with the Fourier spectrum and (b) the corresponding SPWV distribution at point B(8.0 mm, 0) for shear viscosity of 2.0 Pa⋅s. The modulation frequency was assumed 350 Hz and the emission duration 28.6 ms (ten cycles). The rest of the parameter values assumed are given in the caption of Figure 5.1.

78

(in the low-kHz range) under the higher-viscosity conditions that are believed to be characteristic of

real tissue, since for these conditions, dispersion becomes important and can cause significant

changes in the finite-amplitude shear waveform. On the other hand, the exact solution of the shear

Green’s function, obtained numerically from the inverse spatial Fourier transform (see Section

2.3.2), was shown to describe more accurately the dispersive characteristics of viscosity and be

more appropriate under high-viscosity conditions.

In the frequency analysis it was shown that low-frequency (LF) components were present

under higher-viscosity conditions, whose relative amplitudes decreased with increasing emission

duration. These lower frequencies were more pronounced for higher modulation frequencies and

further away from the source. The most likely cause of this LF spectral content was shown to be the

‘slow’ component arriving at the end of the shear waveform, caused by the effects of dispersion due

to viscosity and the frequency-dependent attenuation. The potential of using the LF spectral

information to extract estimates of the shear modulus and viscosity needs will be discussed in the

following chapter.

A time-frequency analysis of the generated shear waves was also performed. The Smoothed

Pseudo Wigner-Ville (SPWV) distribution was applied and the local distribution of energy was

demonstrated in a continuous and smoothed form. The dispersive and lowpass characteristics of

viscosity were demonstrated localized in time and frequency.

79

CHAPTER 6 Estimating the Properties of a Viscoelastic Medium

Estimating or measuring the properties of viscoelastic media, such as soft tissue, has been

the subject of numerous studies in the field of elastography, due to its potential to provide new

information for enhancing the process of diagnosis. However, such assessment appears to be very

challenging in real biological tissue, especially in-vivo. Different approaches have been adopted by

various researchers, among which, the shear-wave methods have proven to be promising. In this

chapter, the calculation of the frequency-dependent shear speed and attenuation will be investigated,

based on both the fundamental and higher-harmonic spectral components. A novel approach will be

also presented for estimating the shear viscosity of a medium, by measuring the difference between

the arrival times of two peaks. An inverse-problem approach will be also described and local spatial

maps of the shear modulus and viscosity will be obtained from the fundamental and higher

harmonics of the shear wave.

6.1 Challenges in Assessing the Viscoelastic Properties of Tissue

As discussed in Chapter 1 (Section 1.1.1), local changes in tissue viscoelasticity are

generally related to tissue pathology and thus, their assessment could provide new information that

will enhance the process of diagnosis. Quantitative estimation or measurement of the properties of

viscoelastic media, such as soft tissue, has been the subject of numerous theoretical and

experimental studies within the general field of elastography during the past two decades. The shear

80

wave methods have shown the potential to provide reliable estimates of tissue viscoelasticity [13],

[33]-[35], [89]. However, measuring the shear modulus or viscosity in real tissue has proven to be a

very challenging task, especially in-vivo. In part, this can be attributed to the complex nature of

biological tissue (e.g. inhomogeneity, nonlinearity, and anisotropy) and to SNR issues associated

with detecting the shear waves. Indicative of the above difficulties, appears to be the ambiguity

existing in the literature as to the range of the shear viscosity values in soft tissue in the low-kHz

range, as discussed in Section 2.3.2.

Calle et al. [58] suggested that in purely elastic medium (ηs → 0), the local shear elasticity

could be determined from the peak amplitude or peak time of the shear waveform. It was shown in

Chapter 4 (Section 4.2.3), that if viscous losses are also accounted for (ηs ≠ 0), the assumption of a

temporal impulse-like force leads to fairly simple equations of the peak amplitude and peak time as

functions of both the shear viscosity and speed (see (42) and (43)). However, in order to achieve

narrowband excitations (without imposing heating effects), it is required to apply a time convolution

between the Green’s function and the excitation waveform. This inevitably complicates the resulting

equations making it difficult to extract reliable estimates of either shear elasticity or viscosity from

measurements of the peak amplitudes and peak times. The above authors realized this difficulty and

suggested that a time deconvolution process could compensate for the deformation effect. However,

such a signal processing operation is not always feasible in real-tissue conditions and has not yet

been experimentally demonstrated.

6.2 Extracting the Voigt-Model Parameters: based on the

Fundamental Component

In this section, it will be shown how the Voigt-model parameters (i.e., the shear modulus and

viscosity) can be extracted from the fundamental shear component in a viscoelastic medium.

Furthermore, a novel approach will be presented for estimating the shear viscosity by measuring the

difference between the arrival times of two peaks. The higher harmonics will not be taken into

account for estimating the tissue viscoelasticity in the following two sub-sections.

For the subsequent simulations, the source pressure, center frequency and attenuation

coefficient will be assumed 372 kPa, 2.0 MHz and 0.3 dB/(cm⋅MHz 1.1), respectively, according to

the values defined in Chapter 3-Section 3.2. The rest of the parameter values assumed can be also

81

found in Section 3.2. Furthermore, weak-viscosity conditions will be assumed, thereby enabling the

approximate viscoelastic Green’s functions [69] to be used.

6.2.1 Calculation of the Frequency-Dependent Shear Speed and Attenuation

As discussed in Chapter 2 (Section 2.1), measurements of the shear speed and attenuation at

different modulation frequencies can be fitted to the Voigt model, which has been shown to be

appropriate for describing the shear viscoelastic response in tissue and tissue-like phantoms over the

range from 50 to 500 Hz [30], [36], [71]. Specifically, by fitting measurements of the frequency-

dependent shear speed and shear attenuation to (13) and (14), estimates of the shear modulus μl and

viscosity ηs can be extracted.

Frequency-Dependent Shear Speed

For a monochromatic shear wave propagating at an angular modulation frequency

Δω= 2πΔf, the shear speed can be derived from the phase delay Δφ between two observation points,

i.e.:

ϕΔΔωΔ

=ωΔrcs )( , (49)

where Δr is the distance between the points (see Section 2.1.2).

Examples of estimating the shear speed cs(Δω) from the phase delay of the generated

narrowband shear waves, are demonstrated in Figure 6.1. The coupling term was accounted for and

the emission duration was taken to vary with the modulation frequency, so that each time it was

equal to half cycle, i.e., D = T/2 = 1/(2Δf). Such emission durations guaranteed heating effects

below the safety limit, as verified from (36).

Two shear moduli of 2.5 and 7.5 kPa were assumed (note that the corresponding shear

speeds used in the approximate Green’s functions (24) and (25) were found 1.54 and 2.67 m/s,

respectively, according to the weak-viscosity approximation ρμ≈ lsc ). The selection of the

radial distances, where the two observation points should be located, has been performed according

to the method suggested by Chen et al. [17]. Specifically, the phase of the propagated shear waves

was measured at seven different radial distances ranging between 3.0 and 9.0 mm from the source

(at 1.0-mm intervals). Subsequently, six different estimations of the shear speed were obtained from

the corresponding phase delays for Δr = 1.0 mm. The averaged shear speed is plotted with the

82

modulation frequency in Figure 6.1, for two different values of the shear viscosity. The

corresponding shear speed based on the Voigt model is also shown.

It can be observed, that for low viscosity (0.2 Pa⋅s), a very good agreement is achieved

between the estimated cs(Δω) and predicted )( ωΔVsc shear dispersion, for both values of the shear

modulus. When viscosity is increased (1.0 Pa⋅s), the estimated dispersion slightly diverges from that

predicted by the Voigt model and this is probably attributed to the weak-viscosity approximation

[69]. Overall, there is still a quite good agreement between cs(Δω) and )( ωΔVsc . It should be finally

noted, that the above positioning method was found not to be affected by the shear speed (or

modulus).

Frequency-Dependent Shear Attenuation

Estimations of the shear attenuation αs(Δω) for several modulation frequencies are shown in

Figure 6.2, for the above two shear moduli (2.5 and 7.5 kPa) and a shear viscosity of 1.0 Pa⋅s. For

these estimations, the plane-wave assumption was adopted, enabling the shear attenuation to be

extracted by fitting the displacement spectral amplitudes (at the modulation frequency Δω) to an

exponential function of the radial distance r (i.e., rse α− ), similarly to the methods presented in [36],

[69]. The range of radial distances over which the fitting can be performed, should be carefully

(a) (b) Figure 6.1: Estimated shear speed (dispersion) for several modulation frequencies. Estimates based on the Voigt model are also shown with the dashed-dotted lines. The coupling term was included and the emission duration was taken D = T/2 = 1/(2Δf), i.e. half cycle for each frequency. The shear viscosity was assumed (a) 0.2 Pa·s and (b) 1.0 Pa·s. Two shear moduli of 2.5 and 7.5 kPa were assumed in each case (the corresponding shear speeds can be obtained from 2

sl cρ≈μ ).

83

selected, such that the plane-wave assumption will be sufficiently accurate (i.e., not too close to the

source) and at the same time, the received waveforms will not be significantly attenuated (i.e., not

too far from the source). Specifically, the fitting was found to be more accurate after one-

wavelength distance and up to about four wavelengths away (where fcss Δ≈λ is the wavelength)

A very good agreement between αs(Δω) and )( ωΔVsa can be observed in Figure 6.2 for both values

of μl (the shear viscosity did not affect the results).

6.2.2 Estimating the Shear Viscosity from the Peak Time Differences

As discussed in the previous chapters, the shear viscosity of a medium causes dispersion to

appear and distort the propagated signal. The extent of spreading of the shear waveform (e.g. based

on the time difference between two peaks) could be used as a criterion for extracting the shear

viscosity. A method for estimating the shear viscosity by measuring the differences in the arrival

times of the peaks at two locations is presented in this sub-section, as described in [85]. In the

following simulations a modulation frequency of 500 Hz will be assumed.

The normalized time difference between the first two positive peaks can be written as

( )TDt ,modΔ , where ( ) ⎥⎦⎥

⎢⎣⎢−=

TDTDTD,mod denotes a modulus operation between the emission

duration and the modulation period and is itself periodic with the same period T = 2.0 ms. If the

normalized time difference is divided by the radial distance r, then the resulting

[ ]),mod( TDrtY Δ= can be fitted to the parabolic curve Y = P0+P1ηs+P2ηs2 (see Figure 6.3). A

Figure 6.2: Estimated shear attenuation for several modulation frequencies. Estimates based on the Voigt model are also shown with the dashed-dotted lines. The coupling term was included and the emission duration was taken D = T/2 = 1/(2Δf), i.e. half cycle for each frequency. A shear viscosity of 1.0 Pa·s was assumed Two shear moduli of 2.5 (top) and 7.5 kPa (bottom) were also assumed (the corresponding shear speeds can be obtained from

2sl cρ≈μ ).

84

possible physical interpretation of the function Y is that it measures the waveform spreading

(distortion) due to the dispersive effects of viscosity. It also incorporates the 1/r effect (see Section

4.2.3) and the periodicity of the modulated radiation force.

Two observation points have been considered at radial distances of 6.0 and 10.0 mm from

the source (Δr = 4 mm) and the corresponding parabolic coefficients are shown in Figure 6.4 for

several emission durations, ranging between 2.0 and 6.0 ms (i.e. 1-3 cycles at 500 Hz). Specific

values of the coefficients for seven emission durations are also shown in Table 6.1. The coupling

term has been accounted for and the shear speed has been taken 3.0 m/s (μl ≈ 9450 Pa). It should be

noted, that the emission duration values used were selected greater than one cycle (D ≥ 1 cycle), so

that the generated shear waveforms consisted of at least three peaks (positive-negative-positive),

thereby enabling measurements between the first two positive peaks to be performed. Such

measurements would be more reliable than measuring the smaller time difference between two

successive (positive-negative) peaks (for e.g. D < 1 cycle).

Furthermore, by observing Figure 6.4, it can be seen that the above parabolic coefficients

exhibit an approximately periodic behaviour with the emission duration (period of 2.0 ms): this

arises from the periodicity of the modulus operation. It should be noted that an advantage of

normalizing with mod(D, T) is that the proposed fitting process can be restricted within easily

controlled time intervals of duration T. Therefore, the range of values of D that can be used in the

(a)

(b)

Figure 6.3: Parabolic approximation [ ]),mod( TDrtY Δ= versus the shear viscosity (solid lines), for five different values of the emission duration D at radial distances of (a) 6.0 mm and (b) 10.0 mm from the source. Numerical values are shown with symbols. Note that for D = 2.0, 4.0 & 6.0 ms, Y is not defined since mod(D, T) = 0, and thus, a normalization with T has been instead performed. The coupling term has been accounted for and the shear speed has been taken 3.0 m/s.

85

fitting algorithm should be selected between either 2.0 and 4.0 ms (i.e. 1-2 cycles) or 4.0 and 6.0 ms

(i.e. 2-3 cycles). Similarly, the fitting could be performed in the ranges of 6.0-8.0 ms, 8.0-10.0 ms,

etc. However, for such high emission durations, heating effects may arise and cause tissue damage

(as discussed in Section 3.3).

Table 6.1: Parabolic approximation Y = P0+P1ηs+P2ηs2 of the normalized peak time difference

[ ]),mod( TDrtY Δ= with shear viscosity ηs, at two different radial distances.

r = 6 mm r = 10 mm Emission Duration,

D P0 P1 P2

Corr.

Coeff. P0 P1 P2

Corr.

Coeff.

2.2 ms 1.44 0.32 -0.032 0.999 0.88 0.32 -0.029 0.999

2.6 ms 0.50 0.10 -0.010 0.999 0.30 0.10 -0.009 0.999

3.0 ms 0.32 0.02 -0.004 0.964 0.19 0.03 -0.004 0.999

3.6 ms 0.20 0.01 -0.002 0.932 0.12 0.01 -0.002 0.995

4.2 ms 1.61 0.08 -0.012 0.924 0.97 0.09 -0.014 0.978

4.8 ms 0.40 0.02 -0.003 0.924 0.24 0.02 -0.004 0.972

5.4 ms 0.23 0.01 -0.002 0.930 0.14 0.01 -0.003 0.946

It can be also observed in Figure 6.4, that the values of P1 and P2 at the two locations are

very close to one another for each value of D (see also Table 6.1) and the ratio of the coefficients P0

for the two locations varies inversely with r, i.e., 106

)6()10(

0

0 ≈mmPmmP . This can be mathematically

Figure 6.4: The parabolic coefficients P0, P1 and P2 versus the emission duration, at radial distances of 6.0 and 10.0 mm (see Table 6.1). It can be observed that P1 and P2 are very close to one another for all emission durations (ranging between 1-3 cycles). Furthermore, the ratio of P0 for the two locations is almost constant for all values of D and approximately equal to the inverse distances (i.e. ≈ 0.6, as shown with ‘square’ symbols).

86

proven in Box 6.1 below. It is also shown in Figure 6.5, that the inverse coefficients 1/P2 and 1/P1

exhibit almost linear characteristics with the emission duration.

To show how the above observations can be used to determine the shear viscosity, we note

that the normalized time differences measured at two radial distances r1 and r2 can be fitted to

(a) (b) Figure 6.5: Calculated (symbols) and linearly approximated (dashed lines) values of the inverse parabolic coefficients 1/P1 and 1/P2 versus the emission duration over the time intervals of (a) 4.2-6.0 ms (i.e. 2-3 cycles) and (b) 2.2-4.0 ms (i.e. 1-2 cycles). P1 and P2 correspond to the parabolic approximation at a radial distance of 6.0 mm (see Table 6.1).

Box 6.1: Proof of P01/P02 ≈ r2/r1.

Let P01, P02 be the constant parabolic coefficients of Υ1 and Y2 at two radial distances r1 and r2. The parabolic approximations for very low viscosity (ηs→0) can be written as

21

12

02

01divide

022

2

011

1

022

21022

22

012

21011

11

),mod(

),mod(

00 ),mod(

00 ),mod(

trtr

PP

PTDr

t

PTDr

t

PPPPTDr

tY

PPPPTDr

tY

ΔΔ

≈⇒

⎪⎪⎭

⎪⎪⎬

⎫

≈Δ

≈Δ

⇒

⎪⎪⎭

⎪⎪⎬

⎫

=×+×+≈Δ

=

=×+×+≈Δ

=

Now, in the absence of viscosity, it is expected that the time difference between two peaks will not be significantly changed at two locations sufficiently close together. Therefore, we

can assume that Δt1 ≈ Δt2, so that the above equation approximates to: 1

2

02

01

rr

PP

≈ .

87

Y 1 = P01+P1ηs+P2ηs2 and Y 2 = P02+P1ηs+P2ηs

2, as shown in Figure 6.3. In addition, since

P01 ≈ P02 r2/r1, the coefficients P01 and P02 can be found from ( )( )210 YYrrP ji −Δ= , where i, j = 1,2

and i ≠ j. Finally, because 1/P1 and 1/P2 vary nearly linearly with D (see Figure 6.5) they can be

written as qpDP +=11 and lkDP +=21 , where p, q, k, and l are linear coefficients, then

1,2for 11 2ss0 =η

++η

++≈ i

lkDqpDPY ii . (50)

The above equation can be further simplified by noting from Figure 6.5, that the ratio of the

absolute values of the coefficients P1 and P2 at each location, can be approximated by a constant Ci

over the investigated range of emission durations, i.e.,

qClpCkPCPCPP iiii −=−≈⇒×−=⇒≈ ,11 1221 .

The above simplifications enable (50) to be written as

1,2for 11 - 2ss

2s

1s10 =⎟⎟

⎠

⎞⎜⎜⎝

⎛η−η

+=η−η≈ i

CqpDCP

PPYii

ii . (51)

The linear coefficients and the shear viscosity can be determined from measurements of Δt for at

least three different emission durations. If there are M measurements of Υi such that M ≥ 3 and D(m)

denotes the emission duration of the m’th measurement, then from (51) it can be readily shown that

qpDqpD

CqpD

CqpDPYPY

m

m

m

m

mi

mi

mi

mi

++

=⎟⎠⎞

⎜⎝⎛ η−η

+

⎟⎠⎞

⎜⎝⎛ η−η

+=

−

− +

+

++ )(

)1(

2ss)1(

2ss)(

)1(0

)1(

)(0

)(

11

11

, m = 1,2,…,M (52)

The linear coefficients p and q can be estimated by applying a curve fitting algorithm to the

above formula for either of the observation points. The estimated coefficients p, q can be substituted

in (51), thereby leading to an equation with two unknowns, ηs and Ci. Subsequently, by solving (51)

for a specific location and two values of D, the shear viscosity can then be extracted. A block

diagram describing the proposed algorithm for estimating the shear viscosity can be seen in Figure

6.6.

As noted earlier, the fitting process can be performed for values of the emission duration in

the range (vT, (v+1)T], where T is the period of the modulated radiation force and v ≥ 1 denotes an

integer (this must be selected small enough, such that heating effects are minimized). The accuracy

of the proposed method can be improved by increasing the number of measurements Yi (performed

in the above or a smaller range) that will be fed to the curve fitting algorithm described by (52).

Furthermore, the two observations points should be selected sufficiently far apart and at such

88

distances, where viscosity will have time to act by spreading the shape of the shear waveforms and

at the same time, the received waveforms will not be significantly attenuated. Similar results were

obtained for higher shear speeds (i.e. higher shear moduli), except that in such cases, it was

empirically found that the observation points should be selected slightly further away from the

source in order to guarantee sufficient fitting accuracy. Finally, it should be noted that the proposed

algorithm and the fitting process do not depend on the excitation conditions (e.g. the source pressure,

center frequency, and absorption coefficient).

Figure 6.6: Block diagram describing the proposed method for estimating the shear viscosity by measuring the normalized time differences Y1 and Y2 at two locations r1 and r2, for several emission durations D.

6.3 Extracting the Voigt-Model Parameters: based on the Harmonic

Components

This section describes a method whereby the harmonic shear components can be used to

calculate the frequency-dependent shear speed and attenuation. According to Section 5.1, a finite-

amplitude radiation force is considered at modulation frequencies between 100-500 Hz and the first

five harmonic components (Nh = 5) are retained. The source pressure, center frequency and

attenuation coefficient assumed are given in the caption of Figure 5.1. Furthermore, emission

89

durations of exactly two cycles were assumed for each modulation frequency, i.e., D = 2T1=2/Δf. It

should be noted, that to avoid exceeding the human safety limit the emission duration must be

selected so that the temperature rise does not exceed 1°C. Based on (37), Table 6.2 shows the

maximum permitted number of cycles for several modulation frequencies and source pressures. The

maximum number of cycles corresponding to the assumed source pressure (P0 = 450 kPa) is

highlighted.

Figure 6.7: Case of a lower viscosity (0.2 Pa⋅s). Estimated shear speed (dispersion) for several modulation frequencies based on the (a)-(b) fundamental, (c)-(d) second-harmonic and (e)-(f) third-harmonic component. Estimates based on the Voigt model are also shown with the dash-dotted lines. The coupling term was included and two shear speeds of (a), (c), (e) 1.54 m/s (μl ≈ 2.5 kPa) and (b), (d), (f) 2.67 m/s (μl ≈ 7.5 kPa) were assumed. The emission duration was two cycles for each frequency.

90

Table 6.2: Maximum number of cycles for a temperature increase ≤1°C.

Modulation Frequency Δf (Hz) P0 (kPa) 50 100 200 300 400 500 Dmax (ms)

350 3.0 6.1 12.1 18.2 24.2 30.3 60.6 400 2.2 4.5 8.9 13.4 17.9 22.3 44.6 450 1.7 3.4 6.8 10.2 13.6 16.9 33.9 500 1.3 2.6 5.3 7.9 10.5 13.2 26.3 550 1.0 2.1 4.2 6.3 8.4 10.4 20.9

Frequency-Dependent Shear Speed

In the first set of simulations, the approximate viscoelastic Green’s functions [69] will be

used. For a case of low viscosity (0.2 Pa⋅s), the estimated shear speed based on the fundamental,

second-harmonic and third-harmonic shear component is shown in Figure 6.7. The coupling term

was accounted for and two shear speeds of 1.54 and 2.67 m/s were considered (the corresponding

shear moduli were approximately 2.5 and 7.5 kPa, according to 2sl cρ≈μ ). Moreover, since the

effects of frequency-dependent attenuation and dispersion likely cause significant waveform

distortion at higher frequencies (>1200 Hz), data beyond this point was excluded. The positioning

system by Chen et al. [17] was again adopted (see also Section 6.2.1), where the phase delay

between two observation points (according to (49)) was calculated at six different intervals of

Δr = 1.0 mm between 3.0 and 9.0 mm from the source and the averaged shear speed is finally

estimated and plotted versus the modulation frequency. The corresponding shear speed based on the

Voigt model is also shown.

For the low-viscosity case, a very good agreement is achieved between the estimated cs(nΔω)

and predicted )( ωΔncVs shear dispersion, both for the fundamental and the higher harmonics (n = 1,2

& 3), and both values of the shear modulus. However, for the lower modulus (2.5 kPa), the shear

speed based on the third-harmonic component started to deviate at higher frequencies (Δf > 400 Hz

or 3Δf > 1200 Hz, not shown in Figure 6.7(e)). This can be explained by the fact that for such low

moduli and high frequencies, the shear wavelength is decreased ( )( )fnls Δρμ≈λ and thus, the

above radial distances correspond to several wavelengths away from the source, where the third

harmonic is expected to be severely attenuated and the waveform distorted even for this low value

of viscosity (0.2 Pa⋅s). For example, for a modulation frequency of 500 Hz and a shear modulus of

2.5 kPa, the farthest observation point at r = 9.0 mm corresponds to about three wavelengths (λs ≈ 3

91

mm) away from the source. For the higher shear modulus (7.5 kPa), it corresponds to about 1.7

wavelengths (λs ≈ 5.4 mm).

As shown in Figure 6.8, when the viscosity is increased (1.0 Pa⋅s), the estimated dispersion

exhibits small deviations from that predicted by the Voigt model, both for the fundamental and

second-harmonic component, but overall, the agreement is quite reasonable. It should be noted, that

for the above value of viscosity, the calculations were performed at four 1.0mm-intervals between

3.0 and 7.0 mm from the source and the third-harmonic component was completely ignored (it was

severely attenuated especially at higher frequencies). Here again, because the effects of frequency-

dependent attenuation and dispersion probably cause significant waveform distortion at higher

frequencies (>800 Hz), data beyond this point was excluded.

Nonlinear Least-Squares (LS) fitting, using the ‘trust-region’ algorithm [90], was performed

on the fundamental and second harmonic, as shown in Figure 6.9. The extracted values of shear

modulus and viscosity, with 95% confidence bounds, are shown in Table 6.3. It can be observed,

Figure 6.8: Case of a moderate viscosity (1.0 Pa⋅s). Estimated shear speed (dispersion) for several modulation frequencies based on the (a)-(b) fundamental and the (c)-(d) second-harmonic component. Estimates based on the Voigt model are also shown with the dash-dotted lines. The coupling term was included and two shear speeds of (a), (c) 1.54 m/s (μl ≈ 2.5 kPa) and (b), (d) 2.67 m/s (μl ≈ 7.5 kPa) were assumed. The emission duration was two cycles for each frequency.

92

that the shear modulus is more accurately estimated from the fundamental and the shear viscosity

from the second-harmonic component. The second harmonic overestimates the shear modulus,

while the fundamental overestimates the shear viscosity.

Table 6.3: Estimated shear modulus and viscosity by applying nonlinear LS-fitting (‘trust-region’ algorithm [90]) to the shear speed calculations of Figure 6.8, for the case of moderate viscosity (1.0 Pa⋅s).

Assumed Shear Speed and Viscosity (Calculated Shear Modulus) Estimated

Voigt Parameters

1.54 m/s 1.0 Pa⋅s (μl ≈ 2.5 kPa)

2.67 m/s 1.0 Pa⋅s (μl ≈ 7.5 kPa)

n = 1 1.3 ± 0.2 Pa⋅s 1.6 ± 0.7 Pa⋅s Estimated Shear Viscosity n = 2 1.1 ± 0.4 Pa⋅s 1.1 ± 0.3 Pa⋅s

n = 1 2.6 ± 0.8 kPa 7.6 ± 1.2 kPa Estimated Shear Modulus n = 2 3.2 ± 2.3 kPa 8.4 ± 1.2 kPa

Estimations of the shear speed for a case of higher shear viscosity (5.0 Pa⋅s) and modulus of

2.5 kPa are shown in Figure 6.10, based on both the approximate (according to (24)) and the exact

shear-wave Green’s solution obtained numerically from (26). The coupling term was ignored in

(a)

(b) Figure 6.9: Nonlinear LS-fitting based on the ‘trust-region’ algorithm [90], applied to the shear speed calculations of Figure 6.8 (see also Table 6.3), for both the fundamental and second-harmonic component and two shear speeds of (a) 1.54 m/s (μl ≈ 2.5 kPa) and (b) 2.67 m/s (μl ≈ 7.5 kPa). Note that in (a), the last two values of the shear speed for n = 2 were excluded from the fitting algorithm.

93

order to accurately compare the above Green’s functions. The calculations were performed at three

1.0mm-intervals between 5.0 and 8.0 mm. It can be observed, that if the exact numerical solution of

the Green’s function is used, the shear speed calculations with frequency based on the fundamental

component are in very good agreement with the Voigt model, while the approximate viscoelastic

solution deviates quite significantly especially for higher frequencies. The exact solution describes

also much more accurately the dispersion based on the second harmonic (deviations at frequencies

greater than 400 Hz were observed, as expected). Here again, these observations indicate the need to

use the exact viscoelastic Green’s solution for conditions of increased viscosity, coupled with

moderate or high modulation frequencies.

Figure 6.10: Case of a higher viscosity (5.0 Pa⋅s) and modulus of 2.5 kPa. Estimated shear speed (dispersion) for several modulation frequencies based on the (a) fundamental, (b) second-harmonic and (c) LF component. Both the exact (‘star’ symbols) and approximate (‘x’ symbols) solutions of the shear Green’s function were used. Estimates based on the Voigt model are also shown with the ‘circle’ symbols. The emission duration was taken D = 2T, i.e. two cycles for each frequency.

94

Furthermore, as discussed in Section 5.1.2, under higher-viscosity conditions and higher

modulation frequencies, low-frequency (LF) spectral components appear that propagate further

away from the source. Under such conditions and especially at larger distances from the source,

where the higher harmonics attenuate and the fundamental can be of comparable energy to the LF

component (see Figure 5.6), the latter could provide an alternative means of estimating the local

shear modulus and viscosity. Specifically, the propagation characteristics of the ‘slow component’

of the time waveform (the end component of each time waveform shown in the top row of Figure

5.6) can be used. In fact, Figure 6.10(c) shows the dispersion obtained from the LF component,

which is in excellent agreement with the Voigt model if the exact solution is used. This indicates the

potential to use the LF spectral components in cases of increased viscosity for estimating the shear

modulus and viscosity, since these components travel further away from the source and still remain

at detectable levels when viscosity increases (see Figure 5.4), while the higher-harmonic

components rapidly attenuate. Specific estimates of the shear modulus and viscosity using LS-fitting,

with 95% confidence bounds, are shown in Table 6.4, for all three spectral components (n = 1, 2 &

LF). For Figure 6.10(b), the two highest-frequency estimated components were considered as

‘outliers’.

Table 6.4: Estimated shear modulus and viscosity by applying nonlinear LS-fitting (‘trust-region’ algorithm [90]) to the shear speed calculations of Figure 6.10 for the case of a higher viscosity (5.0 Pa⋅s) and a shear modulus of 2.5 kPa.

Estimated Shear Modulus

Estimated Shear Viscosity

n = 1 2.1 ± 0.9 kPa 5.3 ± 0.6 Pa⋅s n = 2 3.5 ± 2.4 kPa 5.9 ± 3.1 Pa⋅s LF 2.5 ± 0.05 kPa 5.0 ± 0.03 Pa⋅s

Frequency-Dependent Shear Attenuation

Estimations of the shear attenuation for several modulation frequencies using exponential

fitting (see Section 6.2.1) are shown in Figure 6.11, based on the first three harmonic components.

The coupling term was included and the emission duration was taken two cycles at each frequency.

A case of moderate viscosity (1.0 Pa⋅s) was assumed and the approximate Green’s functions were

used. The shear speed was taken 1.54 m/s (μl ≈ 2.5 kPa). Very good agreement with the Voigt model

was achieved for the fundamental component, for which, the exponential fitting was performed over

a range d of radial distances between 1.5 and 3 wavelengths from the source. Good agreement with

the Voigt model was also achieved with the second and third harmonics. For the higher harmonics,

95

however, the fitting algorithm should be applied at distances closer to the source in order to avoid

severe attenuation, i.e., over the range d2 ≈ d-λs for the second harmonic and d3 ≈ d-1.2λs for the third

harmonic. Specific values of the extracted shear modulus and viscosity for all three components

(n = 1, 2, & 3) using LS-fitting (see Figure 6.1(d)) are shown in Table 6.5.

Table 6.5: Estimated shear modulus and viscosity by applying nonlinear LS-fitting (‘trust-region’ algorithm [90]) to the shear attenuation calculations of Figure 6.11(a)-(c) for a moderate shear viscosity (1.0 Pa⋅s) and a shear speed of 1.54 m/s (μl ≈ 2.5 kPa).

Shear Modulus Shear Viscosity

n = 1 2.6 ± 0.4 kPa 1.2 ± 0.5 Pa⋅s n = 2 2.2 ± 0.3 kPa 0.9 ± 0.3 Pa⋅s n = 3 2.7 ± 0.4 kPa 1.1 ± 0.2 Pa⋅s

Figure 6.11: Case of a moderate viscosity (1.0 Pa⋅s) and a shear speed of 1.54 m/s (μl ≈ 2.5 kPa). Estimated shear attenuation for several modulation frequencies based on the (a) fundamental, (b) second-harmonic and (c) third-harmonic component. Estimates based on the Voigt model are also shown with the dash-dotted lines. For n = 2, the exponential fitting was performed over the range d (as for n = 1) and d2 ≈ d-λs (closer to the source). For n = 3, the fitting was performed over d2 and d3 ≈ d-1.2λs. (d) Nonlinear LS-fitting (‘trust-region’ algorithm [90]) applied to the shear speed calculations of (a) and (b) (see also Table 6.5), for the fundamental and second-harmonic component.

96

The frequency-dependent attenuation for a case of increased viscosity (5.0 Pa⋅s) and shear

modulus of 2.5 kPa is shown in Figure 6.12, for n = 1. Both the approximate and the exact shear-

wave Green’s solutions were used (the coupling term was ignored). It is evident that a much better

agreement to the Voigt model is obtained with the exact numerical solution, indicating once again

the importance of using the latter solution in cases of increased viscosity. The extracted Voigt-

model parameters using LS-fitting, with 95% confidence bounds, are also provided. The fitting was

performed over radial distances between 1.5 and 3 wavelengths from the source. Higher harmonics

did not provide reliable results due to high attenuation and dispersion.

Figure 6.12: Case of a higher viscosity (5.0 Pa⋅s) and shear modulus of 2.5 kPa. Estimated shear attenuation for several modulation frequencies based on the fundamental (n = 1) shear component. Both the exact (‘star’ symbols) and approximate (‘x’ symbols) solutions of the shear Green’s function were used. Estimates based on the Voigt model are also shown with the ‘circle’ symbols. The extracted Voigt-model parameters using LS-fitting are also provided.

6.4 The Inverse-Problem Approach

From measurements of the shear displacement on an imaging plane as a function of time, it

is possible to derive maps of the local shear modulus and viscosity of the medium, based on inverse

algorithms [33], [35], [36], [89]. The inverse problem approach in elastography is based on the

shear-wave propagation equation. Based on (A-1) of the APPENDIX A, the propagation equation in

the frequency domain for a viscoelastic, isotropic, piece-wise homogeneous solid can be written as

follows:

( ) [ ]),,(1),,(2

2

tzxujt

tzxutslt Δℑωη+μ

ρ−⎥

⎦

⎤⎢⎣

⎡∂

∂ℑ 0= (53)

97

where ),( tu r is the shear displacement and Δ = 2

2

2

2

2

2

zyx ∂∂

+∂∂

+∂∂ denotes the Laplacian operator in

the 3-D space. If the imaging area is specified to be the (x, z) plane, then only two of the above three

second-order spatial derivatives can be experimentally measured. Therefore, a strong assumption,

that the out-of-plane spatial derivative ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

2

2

yu is negligible, needs to be made [33], [36], [91], i.e.,

2

2

2

2

zx ∂∂

+∂∂

≈Δ . Finally, (53) yields the following expressions for the local shear modulus and

viscosity:

[ ][ ]

[ ][ ] ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Δℑ∂∂ℑ

ωρ

=ωη

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Δℑ∂∂ℑ

ρ=ωμ

ω

ω

ω

ω

),,(),,(

Im):,(

),,(),,(

Re):,(

22

22

tzxuttzxu

zx

tzxuttzxu

zx

t

ts

t

tl

(54)

where Re{.} and Im{.} denote the real and imaginary parts.

The above equations can be averaged over a range of frequencies around the excitation

frequency (i.e. around the harmonic modulation frequency nΔf) and have the potential to provide

quantitative estimates of the local shear modulus and shear viscosity based on either the

fundamental or the higher-harmonic shear components. However, they do not provide accurate

results in the region where the radiation force is applied (the inverse algorithm is based on a free-

source wave equation and thus, there exist errors along the beam axis, where the shear term is

extremely weak and the coupling term does not provide sufficient displacement information to

ensure relevant shear-modulus estimation [35], [91]).

In the forward problem, a circular inclusion of negligible thickness with a radius of 6.0 mm,

centered at (15.0 mm, 15.0 mm) was assumed to have a shear modulus and viscosity 6.0 kPa and 5.0

Pa⋅s, respectively. The corresponding shear modulus and viscosity of the background medium were

taken 2.0 kPa and 0.1 Pa⋅s. Furthermore, a finite-amplitude modulated force at 100 Hz was assumed

to have duration 20.0 ms (two cycles). The approximate viscoelastic Green’s functions (including

the coupling term) were used. The displacement field was sampled throughout a 30×30 mm2 field in

order to determine ),,( tzxuΔ and 22 ),,( ttzxu ∂∂ . Examples of the reconstructed maps of the local

shear modulus and viscosity on the (r,z) plane are shown in Figure 6.13, based on the fundamental

and second-harmonic component of the calculated displacement fields. Except within the immediate

98

focal zone, the reconstruction results are a reasonable approximation to the assumed distribution for

both the fundamental and the second harmonic.

As discussed in Section 5.1.2, under higher-viscosity conditions and higher modulation

frequencies, low-frequency (LF) spectral components appear that propagate further away from the

source (‘slow’ components). An example of reconstructing the shear modulus and shear viscosity on

the (r,z) plane from the LF spectral components is shown in Figure 6.14, for a modulation frequency

of 500 Hz, an emission duration of 4.0 ms (two cycles) and the same assumed 2-D circular

distribution described in the last paragraph. Furthermore, the exact solution of the shear-wave

Green’s function obtained by (26) was used (note that the coupling term was approximated by (25)).

A very good estimation has been obtained for both medium parameters, indicating once again the

Figure 6.13: Maps of the estimated local (a), (c) shear modulus and (b), (d) shear viscosity from (a)-(b) the fundamental and (c)-(d) the second-harmonic shear displacement on the (r,z) plane. The modulation frequency was assumed 100 Hz, the emission duration 20.0 ms and the rest of the parameter values are given in the caption of Figure 5.1.

99

potential to use the LF components under conditions of increased viscosity and frequency. It should

be also noted, that the corresponding estimates from the fundamental component were not found to

be very accurate further away from the focal zone and within the circular inclusion due to the high

attenuation and the effects of dispersion. The higher harmonics were severely attenuated at even

smaller distances from the source.

In Figure 6.15, reconstructed maps are shown under noisy conditions, based on the first four

harmonic components and the average of the first three harmonic components, for a modulation

frequency of 100 Hz. Specifically, Gaussian noise of zero mean and (normalized) variance 0.2 was

added to the calculated total shear displacement field and the inverse equations described by (54)

were next applied. The shear viscosity within the circular inclusion region was assumed 2.0 Pa⋅s.

Here again, the approximate viscoelastic Green’s expressions (including the coupling term) were

used.

Good estimation results have been obtained especially from the fundamental and second-

harmonic component. For the higher harmonics (n = 3, 4), noise amplification is observed especially

at larger distances from the source and this is probably attributed to the derivatives involved in the

inverse algorithm, known as highpass operations which tend to amplify noise. However, taking the

average of the first three reconstructed maps gives better reconstruction results by smoothing out

outliers.

(a)

(b)

Figure 6.14: Maps of the estimated local (a) shear modulus and (b) shear viscosity from the low-frequency (LF) shear component on the (r,z) plane. The modulation frequency was assumed 500 Hz, the emission duration 4.0 ms (2 cycles) and the rest of the parameter values are given in the caption of Figure 5.1.

100

Figure 6.15: Maps of the estimated local (left column) shear modulus and (right column) shear viscosity from (a)-(b) the fundamental, (c)-(d) the second-harmonic, (e)-(f) the third-harmonic, (g)-(h) the forth-harmonic and (i)-(j) the mean average of the first three harmonic components on the (r,z) plane. The modulation frequency was assumed 100 Hz and the emission duration 20.0 ms.

101

6.5 Chapter Summary

In this chapter, the shear speed and attenuation were calculated versus the modulation

frequency, using both the fundamental and higher-harmonic shear information. It was shown how

the Voigt-model parameters can be extracted from the above spectral components by applying

nonlinear LS-fitting to the frequency-dependent shear speed and/or attenuation.

Conditions of increased viscosity were also examined and it was shown that the exact

solution of the shear Green’s function (obtained numerically, see Section 2.3.2) provides much

better agreement to the Voigt model than the approximate viscoelastic solution, since it describes

more accurately the dispersive effects of viscosity (see also Section 5.1.1). Furthermore, as

discussed in Section 5.1.2, the low-frequency (LF) component that is present under higher-viscosity

conditions, provides the potential to extract the Voigt-model parameters in a more accurate manner,

since these low components travel further away from the source and are not significantly attenuated

when viscosity increases.

The extracted values from the higher-harmonics or the LF component can be used in

complement or correctively to those obtained from the fundamental component. For example, the

second-harmonic component provided a more accurate estimate of viscosity than the fundamental in

Table 6.3 and the LF component provided more accurate estimates for both the shear modulus and

viscosity in Table 6.4. Furthermore, the estimates obtained in Table 6.5 from the first three

harmonics (n = 1, 2 & 3) can be averaged to achieve a more accurate result.

Based on the fundamental component, a novel approach was also presented for estimating

the shear viscosity by measuring the difference between the arrival times of two peaks. Furthermore,

the inverse-problem approach was described, where a circular inclusion of increased modulus and

viscosity was initially synthesized. It was subsequently shown how local spatial maps of both

parameters can be obtained from several spectral components (i.e., the fundamental, higher-

harmonics, or LF component). Both noise-free and noisy cases were considered.

102

CHAPTER 7 Summary and Conclusions

This thesis has focused on studying the properties of narrowband shear waves propagating in

a viscoelastic medium created by the interference of two finite-amplitude confocal quasi-CW

ultrasound beams. It has emphasized potential methods of estimating the shear modulus and

viscosity of viscoelastic media, such as soft tissue.

7.1 Summary

The observation that local changes in tissue stiffness are generally correlated with tissue

pathology has resulted in several studies within the field of elastography in the past two decades,

aiming at measuring or imaging the elastic properties of tissue in order to enhance the process of

diagnosis. An overview of elastography and its major categories was provided in Chapter 1. The

advantages of using ultrasound for the detection of the induced tissue displacement, as opposed to

magnetic resonance or optical methods, were clearly stated. It was next discussed that dynamic

elastography methods have the potential of revealing the dynamic properties of the interrogated

medium (such as viscosity) and at the same time, they overcome boundary problems linked to the

static methods (see Section 1.2). The principles of the acoustic radiation force were also presented

in the same chapter. The dynamic elastography methods that use the acoustic radiation force of

ultrasound have been recently receiving growing interest, due to their non-invasive character and

their ability to create a highly localized force field.

The concept of the acoustic radiation force for the generation of shear waves was adopted in

this thesis, and in particular, the modulated radiation force that can be created in the focal zone by

103

the interference of two finite-amplitude confocal quasi-CW ultrasound beams of slightly different

frequencies. Nonlinear ultrasound propagation was assumed, so that a high-intensity force field is

created in the focal zone, leading in turn, to the generation of shear waves of sufficient energy that

can propagate and be detected at several wavelengths away from the source. The issue of shear-

wave propagation at harmonic modulation frequencies does not appear to have been previously

discussed in the literature.

The above modulated source can generate low-frequency narrowband shear waves, an

important advantage of which is that their spectrum is not significantly distorted as they propagate

through tissue by different kinds of frequency-dependent distortion, and their spectral power is

concentrated within a narrow frequency band around the modulation frequency. In contrast to most

elastography methods presented so far, which make use of a broadband excitation, the

aforementioned narrowband excitations argue for a more accurate estimation of the shear

attenuation and dispersion at various frequencies in the low-kHz range. This hypothesis was

investigated in this thesis for both the fundamental and the higher-harmonic shear components.

The generation of dynamic radiation force components at harmonic modulation frequencies

was modeled in Chapter 3 and their properties were examined for conditions that conformed to

safety standards. Specifically, it was shown that the force application time is associated with a

temperature rise, which is limited below 1.0 °C by regulatory specifications of the Thermal Index

(TI). The Mechanical Index (MI) limitation (i.e. MI < 1.9 [81]) was not strictly followed in this

analysis and there were cases of increased acoustic pressure in the focal zone leading to values of

the MI beyond the risk region (see Figure 3.8). However, such high pressures permitted us to more

clearly examine harmonic effects. In elastography applications in-vivo, both the TI and MI should be

carefully accounted for, in order not to pose any risk to patients. The dependence of the harmonic

force components on the excitation conditions (i.e. the source pressure, center frequency, and

absorption coefficient) was demonstrated in a straightforward manner, using the dimensionless

parameters of nonlinearity, focusing gain and absorption, which are commonly used to characterize

focused sources.

In Chapter 4, the propagation and evolution of the generated narrowband shear waves at the

fundamental modulation frequency was studied in the spatiotemporal domain, based on approximate

Green's functions for viscoelastic media [69] (assuming weak-viscosity conditions). The coupling

wave was included and the effects of the shear viscosity and speed on the fundamental shear

component were analyzed. For the selection of the emission duration, an appropriate tradeoff must

be performed, so that a sufficiently narrow bandwidth of the transmitted signal is achieved without

104

violating the TI limitation according to (37). However, this required the application of time

convolution of the Green’s function with a modulated (oscillatory) radiation force of finite duration,

which inevitably increased the complexity of the associated equations and restricted the potential to

estimate the viscoelastic properties from the peak amplitudes and/or peak times.

Modulated radiation force components at harmonic modulation frequencies were considered

in Chapter 5 and the induced finite-amplitude shear field was investigated both in time and

frequency domains, for excitation conditions that limited the associated temperature increase. The

approximate viscoelastic expressions used in Chapter 4 were shown not to be accurate for higher

frequencies and increased viscosities. Such high viscosities are believed to be characteristic of real

tissue (values even up to 10-15 Pa⋅s have been reported [36], [70], [71]) and can cause dispersion to

become important and distort the shear waveform. The exact solution of the shear Green’s function,

whose expression in the k-space (see (26)) was derived in Chapter 2 (see also APPENDIX A), was

shown to describe more accurately the dispersive characteristics of viscosity and be more

appropriate under high-viscosity conditions. Another important finding of the provided frequency

analysis is that low-frequency (LF) components appeared with increased viscosity and were

pronounced for higher modulation frequencies and further away from the source. The most likely

cause of this LF spectral content was shown to be the ‘slow’ component arriving at the end of the

shear waveform, caused by the effects of dispersion due to viscosity and the frequency-dependent

attenuation.

In Chapter 6, the shear speed and attenuation were calculated versus the modulation

frequency, based on both the fundamental and higher-harmonic shear components. Examples of

extracting the shear modulus and viscosity of the propagated medium were provided, by fitting the

frequency-dependent shear speed and attenuation to the Voigt model, which was shown to describe

most appropriately the viscoelastic properties of tissue in the low-kHz range (see Chapter 2).

Conditions of increased viscosity were also examined and it was shown that the exact solution of the

shear Green’s function (described by (26)) provided much better agreement to the Voigt model than

the approximate viscoelastic solution.

The low-frequency (LF) component appearing with increased viscosity was shown to have

the potential of estimating the Voigt-model parameters more accurately. The extracted values of the

shear modulus and viscosity from the higher-harmonics or the LF component can be used in

complement or correctively to those obtained from the fundamental component. Furthermore, a

novel approach was presented for estimating the shear viscosity by measuring the difference

between the arrival times of two peaks. Finally, the inverse-problem approach was described and

105

local spatial maps of the shear modulus and viscosity were derived based on the fundamental, the

higher-harmonics and the LF component.

7.2 Thesis contributions

The work described in this thesis represents the first detailed study of narrowband shear

wave propagation in a viscoelastic medium created by a modulated finite-amplitude ultrasound

radiation force. Specifically, the contributions can be summarized as follows:

Proposing narrowband shear waves using a modulated finite-amplitude acoustic radiation force

for estimating the viscoelastic properties of tissue.

Modeling the modulated (oscillatory) acoustic radiation force created in the focal zone by the

interference of two finite-amplitude confocal coaxial quasi-CW ultrasound beams of slightly

different frequencies under nonlinear ultrasound propagation.

Studying the properties of the generated low-frequency narrowband shear waves at both the

fundamental and harmonic modulation frequencies, in the time, frequency, and time-frequency

domains, for conditions that conform to safety standards.

Deriving and pointing out the importance of an exact solution of the shear term of the Green’s

function in the k-space. Demonstrating its ability to describe more accurately the dispersive

effects of viscosity under increased-viscosity conditions and higher modulation frequencies.

Revealing the low-frequency (LF) spectral component that appears with increased viscosity and

demonstrating its relation to the ‘slow’ component arriving at the end of the shear waveform,

most likely caused by the effects of dispersion and the frequency-dependent attenuation.

Estimating the local shear modulus and viscosity by fitting the frequency-dependent shear speed

and attenuation to the Voigt model, based on both the fundamental and the higher-harmonic or

LF spectral components. It was demonstrated that the higher-harmonic or LF-based estimates can

be used in complement or correctively to those based on the fundamental.

Proposing a novel algorithm for estimating the shear viscosity of a propagating viscoelastic

medium from measurements of the normalized arrival times between two peaks.

7.3 Suggestions for Further Work

Dynamic elastography, and in particular, the radiation force-based elastography methods

(described in Section 1.2.2) are of growing interest due to their promise of providing quantitative

106

stiffness estimates in a locally excited region of interest within tissue in a non-invasive manner. In

order for this perspective to come to fruition, future work is required.

Experimental validation of the proposed model of narrowband radiation-force excitations is

an important future direction that needs to be addressed. Validation of the generated shear-field

properties at both the fundamental and harmonic modulation frequencies must be performed under

clinically realistic scenarios. For this reason, tissue-mimicking phantoms can be initially used, but

ultimately, the proposed model needs to be tested in real biological soft tissues.

Specifically for agar-gelatin phantoms, the concentration of the gelatin and the viscous

additive (e.g. xanthane gum solution [69]) can be appropriately selected in order to fix the elasticity

(stiffness) and viscosity of the phantom. Two focused transducers operating at MHz frequencies

(±Δf/2, where Δf should be selected a few hundred Hertz) will be needed that can be focused at a

depth of a few centimeters. A moderate source pressure -common for both transducers- can be

initially assumed (e.g. equal to 372 kPa: this has been shown to be sufficient for nonlinear

ultrasound propagation [74], [75]). The propagated shear waves (their phase and amplitude) can be

detected at two specific locations away from the source by using a low-frequency hydrophone,

acoustic surface wave detector [13], or probing ultrasonic pulses [30], [46]. Furthermore, by using

an ultrafast imaging system (e.g. pulse-echo ultrasound with frame rates of up to 5-10 kHz) [26],

[69], frames of the shear-wave propagation can be acquired on a 2-D plane (e.g. x-z) sampled with a

grid of e.g. 1×1 mm2. The system of the two inverse equations at specific frequencies described by

(54) can be subsequently applied on the shear displacement movie and maps of the local shear

modulus and viscosity could be finally obtained.

As discussed in this thesis, nonlinear propagation of the two ultrasound beams must be

assumed, if shear waves of sufficient energy and thus, increased detection sensitivity, are desired.

However, in applications in-vivo, the limit up to which the acoustic pressure in the focal zone can be

increased must be tested, in order to produce detectable shear waves at both the fundamental and

harmonic modulation frequencies without, at the same time, exceeding the TI and MI safety

regulations for negligible risk to patients [81]. Another challenge often encountered in-vivo, is the

physiological motion that many organs (such as the liver [92]) experience due to the respiration

and/or the cardiac cycle. The effects of such motion on the proposed model and possible ways of

compensating for it need also to be examined.

Further analysis of the inverse-problem approach is another direction for future work. The

performance of the generated harmonic maps of the local shear modulus and viscosity can be

107

assessed based on image quality metrics that are commonly used in elastography. Such metrics

involve the observed contrast, the contrast transfer efficiency (CTE), contrast-to-noise-ratio (CNR)

[93], [94] and also, the SNR and resolution. Furthermore, fusion algorithms for combining

efficiently the generated local maps from several spectral components (e.g. from the fundamental

and the higher harmonic or the low-frequency component) must be also sought, in order to generate

a single output elastogram of improved quality (e.g. increased CTE and SNR).

Another area of pursuit is the time-frequency analysis of the generated shear waves. As

shown in Section 5.2, a time-frequency representation can localize such signals (i.e. non-stationary)

in both time and frequency domains, and thus, provide more detailed description of the shear wave

properties and the energy-frequency distribution with time. The potential of extracting the Voigt-

model parameters from the dispersion obtained by the group delays )()( ωτ ig (see (48)) at two

different locations ri (i = 1, 2) needs to be examined, according to ( ))()()( )2()1( ωτ−ωτΔ=ω ggs rc [87].

However, preliminary results have shown that although a better physical interpretation of the

underlying physics can be provided with a time-frequency distribution, the calculated dispersion

using the SPWV distribution does not improve significantly that based on the Fourier phase

spectrum. Alternative time-frequency representations, such as the wavelet-based, should be also

examined.

Another direction for further research is to provide a more detailed analysis of the low-

frequency (LF) component effect that appears under high-viscosity conditions (see Section 5.1.2).

The potential to use the LF component for estimating more accurately the shear modulus and

viscosity under certain conditions (as shown in Table 6.4 and Figure 6.14) needs further

investigation and experimental validation.

Finally, the proposed model should be ultimately expanded and tested under more realistic

tissue conditions, such as, anisotropic [95] and inhomogeneous [96] tissue.

108

REFERENCES

[1] Y. C. Fung, Ch. 4 in Biomechanical properties of living tissues, New York, USA: Springer

Verlag, 1981.

[2] J. Ophir, I. Cespedes, H. Ponnekanti, Y. Yazdi, and X. Li., “Elastography: a quantitative

method for imaging the elasticity of biological tissues”, Ultrasonic Imaging, vol. 13, pp. 111-134,

1991.

[3] J. Ophir, I. Cespedes, B. Garra, H. Ponnekanti, Y. Huang and N. Maklad, “Elastography:

Ultrasonic imaging of tissue strain and elastic modulus in vivo”, European J. Ultrasound, vol. 3, no.

1, pp. 49-70, 1996.

[4] L. S. Wilson, D. E. Robinson, and M. J. Dadd, “Elastography- the movement begins”, Phys.

Med. Biol., vol. 45, pp. 1409-1421, 2000.

[5] J. F. Greenleaf, M. Fatemi, and M. Insan, “Selected methods for imaging elastic properties

of biological tissues”, Ann. Rev. Biomed. Eng., vol. 5, pp. 57-78, 2003.

[6] K. J. Parker, L. S. Taylor, and S. Gracewski, “A unified view of imaging the elastic

properties of tissue”, J. Acoust. Soc. Am., vol. 117, pp. 2705-2712, 2005.

[7] J. Ophir, S. K. Alam, B. S. Garra, F. Kallel, E. Konofagou, T. Krouskop, C. R. B. Merritt, R.

Righetti, R. Souchon, S. Srinivasan, and T. Varghese, “Elastography: imaging the elastic properties

of soft tissues with ultrasound”, J. Med. Ultrasound, vol. 29, pp. 155-171, 2002.

[8] W. A. D. Anderson, Pathology (8’th ed.), Saint Louis, Mo, USA: CW Mosby Co., 1984.

[9] I. M. Ariel and J. B. Cleary, Breast cancer diagnosis and treatment, New York, USA:

McGraw-Hill, 1987.

[10] M. Mridha and S. Ödman, “Characterization of Subcutaneous Edema by Mechanical

Impedance Measurements”, J. Investig. Dermatology, vol. 85, pp. 575-578, 1985.

[11] E. Konofagou, T. Hannigan, and J. Ophir, “Shear strain estimation and lesion mobility

assessment in elastography”, Ultrasonics, vol. 38, pp. 400-404, 2000.

109

[12] R. S. C. Cobbold, Foundations of Biomedical Ultrasound, New York: Oxford Univ. Press,

2007.

[13] A. P. Sarvazyan, O. V. Rudenko, D. Scott, J. Swanson, B. Fowlkes, and S. Y. Emelianov,

“Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics”, Ultrasound

Med. Biol., vol. 24, no. 9, pp. 1419-1435, 1998.

[14] R. Muthupillai, D. J. Lomas, P. J. Rossman, J. F. Greenleaf, A. Manduca, and R. L. Ehman,

“Magnetic resonance elastography by direct visualization of propagating acoustic strain waves”,

Science, vol. 269, pp. 1854-57, 1995.

[15] J. Bishop, G. Poole, M. Leitch, and D. B. Plewes, “Magnetic resonance imaging of shear

wave propagation in excised tissue”, J. Magn. Reson. Imaging, vol. 8, pp. 1257–1265, 1998.

[16] S. A. Kruse, J. A. Smith, A. J. Lawrence, M. A. Dresner, A. Manduca, J. F. Greenleaf, and

R. L. Ehman, “Tissue characterization using magnetic resonance elastography: preliminary results”,

Phys. Med. Biol., vol. 45, 1579-1590, 2000.

[17] S. Chen, M. Fatemi, and J. F. Greenleaf, “Quantifying elasticity and viscosity from

measurements of shear wave speed dispersion”, J. Acoust. Soc. Am., vol. 115, no. 6, pp. 2781-2785,

2004.

[18] J. Ophir, “A review of theoretical and experimental aspects of imaging the elastic attributes

of tissue in-vivo”, Ch. 2, pp. 3-6 in: Research and development in breast ultrasound, Tokyo:

Springer, 2005.

[19] E. Konofagou and J. Ophir, “A new elastographic method for estimation and imaging of

lateral displacements, lateral strains, corrected axial strains and Poisson’s ratios in tissues”,

Ultrasound Med. Biol., vol. 24, no. 8, pp. 1183-1199, 1998.

[20] B. S. Garra, E. I. Cespedes, J. Ophir, S. R. Spratt, R. A. Zuurbier, C. M. Magnant, and M. F.

Pennanen, “Elastography of breast lesions: initial clinical results”, Radiology, vol. 202, no. 1, pp.

79-86, 1997.

[21] C. L. de Korte, A. F. W. van der Steen, E. I. Cespedes, G. Pasterkamp, S. G. Carlier, F.

Mastik, A. H. Schoneveld, P. W. Serruys, and N. Bom, “Characterization of plaque components and

vulnerability with intravascular ultrasound elastography”, Phys. Med. Biol, vol. 45, pp. 1465-1475,

2000. [22] N. Nitta, T. Shiina, and E. Ueno, “Quantitative assessment and imaging of viscoelastic

properties of soft tissue”, Proc. IEEE Ultrason. Symp., pp. 1885-1889, 2002.

[23] L. Sandrin, D. Cassereau, and M. Fink, “The role of the coupling term in transient

elastography”, J. Acoust. Soc. Am., vol. 115, pp. 73-83, 2004.

110

[24] T. Sugimoto, S. Ueha, and K. Itoh, “Tissue hardness measurement using the radiation force

of focused ultrasound”, Proc. IEEE Ultrason. Symp., pp. 1377-1380, 1990.

[25] M. Fatemi and J. Greenleaf, “Ultrasound-stimulated vibro-acoustic spectrography”, Science,

vol. 280, pp. 82-85, 1998.

[26] K. Nightingale, M. S. Soo, R. Nightingale, and G. Trahey, “Acoustic radiation force impulse

imaging (ARFI): In vivo demonstration of clinical feasibility”, Ultrasound Med. Biol., vol. 28, no.

2, pp. 227-235, 2002.

[27] R. M Lerner, K. J. Parker, J. Holen, R. Gramiak, and R. C. Waag, “Sono-elasticity: medical

elasticity images derived from ultrasound signals in mechanically vibrated targets”, Acoustical

Imaging, vol. 16, L.W. Kessler, (Ed.), Plenum Press, New York, pp. 317-327, 1988.

[28] S. R. Huang, R. M. Lerner, and K. J. Parker, “On estimating the amplitude of harmonic

vibration from the doppler spectrum of reflected signals”, J. Acoust. Soc. Am., vol 88, pp. 2702–

2712, 1990.

[29] Z. Wu, K. Hoyt, D. J. Rubens, and K. J. Parker, “Sonoelastographic imaging of interference

patterns for estimation of shear velocity distribution in biomaterials”, J. Acoust. Soc. Am., vol. 120,

no. 1, pp. 535-545, 2006.

[30] Y. Yamakoshi, J. Sato, and T. Sato, “Ultrasonic imaging of internal vibration of soft tissue

under forced vibration”, IEEE Trans. Ultrason., Ferroelect. Freq. Contr., vol. 37, pp. 45-53, 1990.

[31] S. Catheline, F. Wu, and M. Fink, “A solution to diffraction biases in sonoelasticity: The

acoustic impulse technique”, J. Acoust. Soc. Am., vol. 105, pp. 2941–2950, 1999.

[32] L. Sandrin, S. Catheline, X. Hennequin, and M. Fink, “Time-resolved 2D pulsed

elastography”, Ultrasonic Imaging, vol. 21, pp. 259-272, 1999.

[33] L. Sandrin, M. Tanter, S. Catheline, and M. Fink, “Shear modulus imaging with 2-D

transient elastography”, IEEE Trans. Ultrason., Ferroelect. Freq. Contr., vol. 49, pp. 426 – 435,

2002.

[34] K. Nightingale, S. McAleavey, and G. Trahey, “Shear-wave generation using acoustic

radiation force: In vivo and ex vivo results”, Ultrasound Med. Biol., vol. 29, pp. 1715-1723, 2003.

[35] J. Bercoff, M. Tanter, and M. Fink, “Supersonic shear imaging: a new technique for soft

tissue elasticity mapping”, IEEE Trans. Ultrason., Ferroelect. Freq. Contr., vol. 51, pp. 396-409,

2004.

[36] S. Catheline, J. L. Gennisson, G. Delon, M. Fink, R. Sinkus, and S. Abouelkaram,

“Measurement of viscoelastic properties of homogeneous soft solid using transient elastography: An

111

inverse problem approach”, J. Acoust. Soc. Am., vol. 116, pp. 3734-3741, 2004.

[37] J. Bercoff, S. Chaffai, M. Tanter, L. Sandrin, S. Catheline, M. Fink, J. L. Gennisson, and M.

Meunier, “In Vivo breast tumor detection using transient elastography”, Ultrasound Med. Biol., vol.

29, no. 10, pp. 1387-1396, 2003.

[38] M. Fatemi, A. Manduca, and J. F. Greenleaf, “Imaging elastic properties of biological tissues

by low-frequency harmonic vibration”, Proc. of the IEEE, vol. 91. no. 10, pp. 1503-1519, 2003.

[39] R. J. Dickinson and C. R. Hill, “Measurement of soft tissue motion using correlation

between A-scans”, Ultrasound Med. Biol, vol. 8, no. 3, pp. 263-271, 1982.

[40] M. Tristam, D. C. Barbosa, D. O. Cosgrove, D. K. Nassiri, J. C. Bamber, and C. R. Hill,

“Ultrasonic study of in vivo kinetic characteristics of human tissues”, Ultrasound Med. Biol., vol. 12,

no. 12, pp. 927-937, 1986.

[41] K. Sunagawa and H. Kanai, “Measurement of shear wave propagation and investigation of

estimation of shear viscoelasticity for tissue characterization”, J. Med. Ultrasonics, vol. 32, pp. 39-

47, 2005.

[42] T. A. Krouskop, D. R. Dougherty, and F. S. Vinson, “A pulsed Doppler ultrasonic system

for making noninvasive measurements of the mechanical properties of soft tissue”, J. Rehabil. Res.

Dev., vol. 24, no. 2, pp. 1-8, 1987.

[43] L. Gao, K. J. Parker, and S. K. Alam, “Sonoelasticity imaging: theory and experimental

verification”, J. Acoust. Soc. Am., vol. 97, no. 6, pp. 2875-3886, 1995.

[44] M. Fatemi and J. Greenleaf, “Probing the dynamics of tissue at low frequencies with the

radiation force of ultrasound”, Phys. Med. Biol., vol. 45, pp. 1449-1464, 2000.

[45] E. E. Konofagou and K. Hynynen, “Localized Harmonic Motion Imaging: Theory,

Simulations and Experiments”, Ultrasound Med. Biol., vol. 29, pp. 1405-13, 2003.

[46] E. E. Konofagou, M. Ottensmeyer, S. Agabian, S. L. Dawson, and K. Hynynen, “Estimating

localized oscillatory tissue motion for assessment of the underlying mechanical modulus”, Elsevier

Ultrasonics, vol. 42, pp. 951-956, 2004.

[47] L. Rayleigh, “On the pressure of vibrations”, Phil. Mag., vol. 3, pp. 338-346, 1902.

[48] P. Biquard, “Les ondes ultra-sonores”, Rev. d’Acoust., vol. 1, pp. 93-109, 1932.

[49] W. Altberg, “Uber die Druckkrafte der Schallwellen und die absolute Messung der

Schallintensitat” (On the force due to sound waves and the absolute measurement of sound

intensity), Annalen der Physik, vol. 11, pp. 405-420, 1903.

[50] R. T. Beyer, “Radiation pressure – the history of a mislabeled tensor”, J. Acoust. Soc. Am.,

vol. 63, pp. 1025-1030, 1978.

112

[51] C. P. Lee and T. G. Wang, “Acoustic radiation pressure”, J. Acoust. Soc. Am., vol. 94, pp.

1099-1109, 1993.

[52] G. R. Torr, “The acoustic radiation force”, Am. J. Phys., vol. 52, pp. 402-408, 1984.

[53] F. A. Duck, A. C. Baker, and H. C. Starritt (Eds.), “Radiation pressure and acoustic

streaming”, Ch. 3 in: Ultrasound in Medicine, Bristol, UK: Inst. of Physics Pub., 1998.

[54] P. J. Westervelt, “The theory of steady forces caused by sound waves”, J. Acoust. Soc. Am.,

vol. 23, pp. 312-315, 1951.

[55] O. V. Rudenko, A. P. Sarvazyan, and S. Y. Emelianov, “Acoustic radiation force and

streaming induced by focused nonlinear ultrasound in a dissipative medium”, J. Acoust. Soc. Am.,

vol. 99, pp. 2791–2798, 1996.

[56] F. M. J. Linssen, A. P. G. Hoeks, and P. J. Brands, “The use of narrowband ultrasound for

characterization of tissue in a thin layer”, Phys. Med. Biol, vol. 36, no. 10, pp. 1319-1330, 1991.

[57] A. Migliori and J. L. Sarrao, Resonant Ultrasound Spectroscopy, New York, USA: John

Wiley & Sons, Inc, 1997.

[58] S. Calle, J.-P. Remenieras, and O. B. Matar, “Temporal analysis of tissue displacement

induced by a transient ultrasound radiation force”, J. Acoust. Soc. Am., vol. 118, pp. 2829-2840,

2005.

[59] R. S. Lakes, Viscoelastic Solids, Florida, USA: CRC Press LLC, 1999.

[60] W. N. Findley, J. S. Lai, and K. Onaran, Creep and relaxation of nonlinear viscoelastic

materials, New York, USA: North-Holland Publishing Company, 1976.

[61] J. C. Bamber, “Attenuation and Absorption”, Ch. 4, pp.93-156 in: Physical Principles of

Medical Ultrasonics, C. R. Hill, J. C. Bamber, and G. R. ter Haar (Eds.), West Sussex, UK: John

Wiley & Sons Ltd, 2004.

[62] Y. C. Fung, “Stress strain history relations of soft tissues in simple elongation”, in

Biomechanics, its Foundations and Objective, New Jersey, USA: Prentice Hall, Englewood Cliffs,

1970.

[63] T. Hueter, “Visco-elastic losses in tissue in the ultrasonic range”, Tech. Rept. No. 57-706,

ASTIA, Doc. No. AD142171, Wright Air Development Centre, Wright-Patterson AFB, Ohio, 1958.

[64] D. R. Raichel, “Sound propagation in Voigt fluid”, J. Acoust. Soc. Am., vol. 52, no. 1B, pp.

395-398, 1971.

[65] M. Qaisar, “Attenuation properties of viscoelastic material”, in Pure and Applied

Biophysics, vol. 131, no. 4, pp. 703-713, 1989.

113

[66] A. S. Ahuja, “Tissue as a Voigt body for propagation of ultrasound”, Ultrasonic Imaging,

vol. 1, pp. 136-143, 1979.

[67] K. Aki and P. G. Richards, Quantitative Seismology, Sausalito, California: University

Science Books, 2002.

[68] G. G. Stokes, “On the dynamical theory of diffraction”, Camb. Phil. Soc., Trans., vol. 9,

pp. 1-62, 1849.

[69] J. Bercoff, M. Tanter, M. Muller, and M. Fink, “The role of viscosity in the impulse

diffraction field of elastic waves induced by the acoustic radiation force”, IEEE Trans. Ultrason.,

Ferroelect. Freq. Contr., vol. 51, pp. 1523-1536, 2004.

[70] H. E. von Gierke, H. L. Oestreicher, E. K. Frank, H. O. Parrack, and W. W. von Wittern,

“Physics of vibrations in living tissues”, J. Appl. Physiol., vol 4, pp. 886-900, 1952.

[71] H. L. Oestreicher, “Field and impedance of an oscillating sphere in a viscoelastic medium

with an application to biophysics”, J. Acoust. Soc. Am., vol. 23, pp. 707-714, 1951.

[72] R. O. Potts, D. A. Chrisman, and E. M. Buras “The dynamic mechanical properties of

human skin in vivo”, J. Biomechanics, vol. 16, no. 6, pp. 365-372, 1983.

[73] W. L. Nyborg, “Acoustic streaming”, Ch. 7, pp.207-231 in: Nonlinear Acoustics, M.F.

Hamilton and D.T. Blackstock (Eds.), New York: Academic Press, 1998.

[74] R. J. Zemp, J. Tavakkoli, and R. S. C. Cobbold, “Modeling of nonlinear ultrasound

propagation in tissue from array transducers”, J. Acoust. Soc. Am., vol. 113, pp. 139-152, 2003.

[75] M. A. Averkiou, D. N. Roundhill, and J. E. Powers, “A new imaging technique based on the

nonlinear properties of tissues”, Proc. IEEE Ultrason. Symp., pp. 1561-1566, 1997.

[76] T. S. Hart and M. F. Hamilton, “Nonlinear effects in focused sound beams”, J. Acoust. Soc.

Am., vol. 84, no. 4, pp. 1488-1496, 1988.

[77] G. Du and M. A. Breazeale, “Theoretical description of focussed gaussian ultrasonic beam in

a nonlinear medium”, J. Acoust. Soc. Am., vol. 81, pp. 51-57, 1987.

[78] C. R. Reilly and K. J. Parker, “Finite-amplitude effects on ultrasound beam patterns in

attenuating media”, J. Acoust. Soc. Am., vol. 86, pp. 2339-2348, 1989.

[79] J. G. Abbott, “Rationale and derivation of MI and TI – a review”, Ultrasound Med. Biol., vol

25, pp. 431-441, 1999.

[80] NCRP, “Report No. 113: Exposure Criteria for Medical Diagnostic Ultrasound: I. Criteria

Based on Thermal Mechanisms”, National Council on Radiation Protection and Measurements,

NCRP Publications, Bethesda, MD 20814, 1992.

114

[81] Food and Drug Administration, “Information for manufacturers seeking marketing clearance

of diagnostic ultrasound systems and transducer”, U. S. Dept. Health and Human Services, Food

and Drug Administration, Center for Devices and Radiological Health, 1997.

[82] M. L. Palmeri and K. R. Nightingale, “On the thermal effects associated with radiation force

imaging of soft tissue”, IEEE Trans. Ultrason., Ferroelect. Freq. Contr., vol. 51, pp. 551-565,

2004.

[83] M. F. Hamilton, J. N. Tjotta, and S. Tjotta “Nonlinear effects in the farfield of a directive

sound source”, J. Acoust. Soc. Am., vol. 78, no. 1, pp. 202-216, 1985.

[84] M. A. Averkiou, “Tissue Harmonic Imaging”, Proc. IEEE Ultrason. Symp., pp. 1563-1572,

2000.

[85] A. Giannoula and R. S. C. Cobbold, “Narrowband shear wave generation by a finite-

amplitude radiation force: the fundamental component”, IEEE Trans. Ultrason., Ferroelect. Freq.

Contr., in press.

[86] A. Papoulis, Probability, random variables, and stochastic processes, N. Y, USA: McGraw-

Hill, 1991.

[87] W. Mecklenbrauker and F. Hlawatsch (Eds.), The Wigner distribution, The Netherlands:

Academic Press, New Elsevier Science B. V., 1997.

[88] A. V. Oppenheim and R. W. F. Schafer, Digital Signal Processing, Englewood Cliffs, N. J,

USA: Prentice-Hall, 1989.

[89] E. Park and A. Maniatty, “Shear modulus reconstruction in dynamic elastography: time

harmonic case”, Phys. Med. Biol., vol. 51, pp. 3697-3721, 2006.

[90] J. J. More, “Recent developments in algorithms and software for trust region methods”, pp.

258-287 in: Mathematical Programming: The State of the Art, A. Bachem, M. Grotschel, and B.

Korte (Eds.), Berlin, Germany: Springer, 1984.

[91] J. Bercoff, M. Tanter, and M. Fink, “Local inversion of transient shear wave propagation for

elasticity and viscosity mapping in soft tissues: theoretical and experimental analysis”, Proc. IEEE

Ultrason. Symp., pp. 2149-2152, 2004.

[92] B. J. Fahey, S. J. Hsu, P. D. Wolf, R. C. Nelson, and G. E. Trahey, “Liver ablation guidance

with acoustic radiation force impulse imaging: challenges and opportunities”, Physics in Med. Biol.,

vol. 51, pp. 3785-3808, 2006.

[93] H. Ponnekanti, J. Ophir, Y. Huang, and I. Cespedes, “Fundamental mechanical limitations

on the visualization of elasticity contrast in elastography”, Ultrasound Med. Biol., vol. 21, pp. 533-

543, 1995.

115

[94] K. Nightingale, M. Palmeri, and G. Trahey, “Analysis of contrast in images generated with

transient acoustic radiation force”, Ultrasound Med. Biol., vol. 23, no. 1, pp. 61-72, 2006.

[95] J.-L. Gennisson, S. Catheline, S. Chaffai, and M. Fink, “Transient elastography in

anisotropic medium: application to the measurement of slow and fast shear wave speeds in

muscles”, J. Acoust. Soc. Am., vol. 114, no. 1, pp. 536-541, 2003.

[96] P. Bai, M. Wan, S. Wang, D. Zhang, and J. Dang, “A novel method for evaluating tissue

inhomogeneity in elastography”, Progress in Natural Science, vol. 16, no. 5, pp. 474-479, 2006.

116

APPENDIX A Derivation of the Viscoelastic Green’s Function in k-Space

By following a similar derivation as in [67] and [69], the Green’s shear-term solution Gs(r,t)

of the following equation is sought:

)()(),(1),( 22

2

ttGtt

tGssl

s δδ=∇⎟⎠⎞

⎜⎝⎛

∂∂

η+μρ

−∂

∂rr

r (A-1)

By symmetry, the spatial dependence of the solution can be only on the distance |=| rr from the

source, therefore, the functional form of Gs = Gs(r,t) is sought. Expressing ∇2 as a differential

operator in spherical polar coordinates, i.e., 2

22 )(1

rrG

rG s

s ∂∂

=∇ , it follows that for r ≠ 0, the

function Grs = r Gs satisfies the one-dimensional wave equation:

012

2

2

2

=∂

∂⎟⎠⎞

⎜⎝⎛

∂∂

η+μρ

−∂

∂rG

ttG rs

slrs (A-2)

By applying the spatial Fourier transform, the above equation yields in the (k,t) domain:

0222

2

=μ+∂

∂η+

∂∂

ρ rslrs

srs Kk

tK

ktK

(A-3)

where [ ]),(),( trGtkK rsrrs ℑ= is the spatial Fourier transform of Grs(r,t).

The solution of (A-3) can be shown to be described by the following complex function:

117

⎟⎟⎠

⎞⎜⎜⎝

⎛ρμ−η

ρ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ρη

−= lss

rs kkttktkK 4

2exp

2exp),( 22

2

(A-4)

It should be noted, that the assumptions made in [69], enabled the second term of the above

equation to be written as tjkcse− , which is also the classical solution of the wave equation in the

purely elastic case (ηs→0). The above equation can be analyzed as follows:

⎪⎪

⎩

⎪⎪

⎨

⎧

ηρμ

≥⎟⎟⎠

⎞⎜⎜⎝

⎛ρμ−η

ρ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ρη

−

ηρμ

<⎟⎟⎠

⎞⎜⎜⎝

⎛η−ρμ

ρ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ρη

−

=

s

lls

s

s

lsl

s

s

kkkttk

kkjkttk

tkK2

if,42

exp2

exp

2if,4

2exp

2exp

),(22

2

222

(A-5)

The solution Gs(r,t) in the real space can be found by applying the inverse spatial Fourier transform

to (A-5) and normalizing with a factor C0 so that (A-1) is satisfied, i.e.:

[ ]),(),( 10 tkKr

CtrG rsks

−ℑ= (A-6)