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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)
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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]
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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.
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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
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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)