windowed results are shown with dash lines. 111 ...... (2.11). The only parameter that can change the phase of the velocity in the force term is the r...

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A THESIS SUBMITTED TO THE FACULTY OF THE MAYO CLINIC COLLEGE OF MEDICINE MAYO GRADUATE SCHOOL

BY

MATTHEW W. URBAN

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN BIOMEDICAL SCIENCESBIOMEDICAL ENGINEERING

Acknowledgments

There are many people that I would like to thank for their contributions to this work and making my time at Mayo fruitful. I would like to extend my gratitude to my advisor, Dr. James F. Greenleaf for his guidance throughout this research project. I would also like to thank my committee members for their assistance with this thesis work: Dr. Mostafa Fatemi, Dr. Armando Manduca, Dr. Stephen J. Riederer, Dr. Kai-Nan An, and Dr. Hal H. Ottesen. The Ultrasound Research Laboratory has been instrumental in the successful completion of this work. I am indebted to Randy Kinnick for the substantial contributions that he made in helping me with experiments and Tom Kinter for writing software to perform my experiments. I would like to thank Elaine Quarve and Jennifer Milliken for their secretarial assistance. I also would like to acknowledge Dr. Shigao Chen, Dr. Glauber Silva, Dr. Farid Mitri, Dr. Xiaoming Zhang, Dr. Azra Alizad, Dr. Yi Zheng, Dr. Thomas Huber, and Miguel Bernal for their scientific input to my research. I would like to thank my fellow students Heather Argadine, Mambi Madzivire, and Carrie Hruska for making this time at Mayo an enjoyable experience. Lastly, but surely not least, I would like to extend my deepest gratitude to my family for their love and support throughout my education. My mother and father have been instrumental in providing loving encouragement throughout my schooling. I want to thank my wife Joleen for all love and support I could ask for and for the sacrifices that she made to allow me to pursue my education. I dedicate this work to my wife Joleen and my daughter Lily as they have been the source of inspiration to complete this work.

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Contents List of Figures................................................................................................................... vi List of Tables ................................................................................................................ xviii Chapter 1—Introduction.................................................................................................. 1 1.1 Background ............................................................................................................ 1 1.2 Elasticity.................................................................................................................. 2 1.3 Viscoelasticity ......................................................................................................... 5 1.4 Elasticity Imaging .................................................................................................. 6 1.5 Force Excitation ..................................................................................................... 7 1.5.1 External Excitation Sources........................................................................... 7 1.5.2 Internal Excitation Sources............................................................................ 8 1.5.3 Endogenous Excitation Sources................................................................... 10 1.6 Measurement of Deformation............................................................................. 10 1.6.1 Magnetic Resonance Imaging ...................................................................... 10 1.6.2 Ultrasound Imaging ...................................................................................... 11 1.6.3 Optical Signals............................................................................................... 13 1.6.4 Acoustic Signals............................................................................................. 13 1.7 Vibro-acoustography ........................................................................................... 13 1.8 Thesis Description................................................................................................ 17 Chapter 2—Measuring the Phase of Vibration of Spheres in a Viscoelastic Medium as an Image Contrast Modality ..................................................................................... 20 2.1 Introduction........................................................................................................... 20 2.2 Methods.................................................................................................................. 23 2.2.1 Dynamic Motion of a Sphere ........................................................................ 23 2.2.2 Phase of Vibration......................................................................................... 24 2.3 Experiment ............................................................................................................ 30 2.3.1 Experimental Setup ....................................................................................... 30 2.3.2 Experimental Results..................................................................................... 32 2.4 Discussion.............................................................................................................. 36 2.5 Conclusion ............................................................................................................ 39 Chapter 3—Multifrequency Vibro-acoustography and Vibrometry......................... 40 3.1 Introduction.......................................................................................................... 40 3.2 Multifrequency Radiation Stress........................................................................ 43 3.3 Multifrequency Implementation......................................................................... 45 3.4 Point-spread Function Calculations................................................................... 49 3.5 Image Formation.................................................................................................. 50 3.6 Signal-to-Noise Ratio Analysis............................................................................ 51 3.6.1 Dynamic Radiation Force and its Effects ................................................... 52 3.6.2 Multifrequency Vibro-acoustography ........................................................ 53 3.6.3 Signal-to-Noise (SNR) Ratio Analysis ......................................................... 54 3.6.4 Chirp Multifrequency Implementation ...................................................... 55 3.7 Multifrequency Experiments .............................................................................. 56 iii

3.7.1 Point-spread Function Simulation............................................................... 56 3.7.2 Point-spread Function Experimental Validation....................................... 56 3.7.3 Vibrometry Experiment ............................................................................... 58 3.7.4 Vibro-acoustography Experiment............................................................... 58 3.7.5 In Vivo Breast Imaging................................................................................. 59 3.8 Multifrequency Experimental Results ............................................................... 60 3.9 Discussion.............................................................................................................. 69 3.10 Conclusion .......................................................................................................... 76 Chapter 4—Harmonic Motion Detection of a Vibrating Reflective Target.............. 77 4.1 Introduction.......................................................................................................... 77 4.2 Theory ................................................................................................................... 79 4.3 Parameterized Model........................................................................................... 82 4.4 Parameterized Model Results ............................................................................. 86 4.5 Discussion.............................................................................................................. 92 4.6 Conclusion ............................................................................................................ 95 Chapter 5—Harmonic Pulsed Excitation and Experimental Motion Detection of a Vibrating Reflective Target ........................................................................................... 96 5.1 Introduction.......................................................................................................... 96 5.2 Background .......................................................................................................... 96 5.3 Harmonic Pulsed Excitation ............................................................................... 98 5.3.1 Harmonic Pulsed Excitation Experiment ................................................. 101 5.3.2 Harmonic Pulsed Excitation Experimental Results................................. 103 5.4 Motion Detection Experiment........................................................................... 104 5.4.1 Parameter Analysis..................................................................................... 111 5.5 Motion Detection Experimental Results .......................................................... 112 5.6 Discussion............................................................................................................ 122 5.7 Conclusion .......................................................................................................... 125 Chapter 6—Harmonic Motion Detection in a Vibrating Scattering Medium ........ 126 6.1 Introduction........................................................................................................ 126 6.2 Background ........................................................................................................ 126 6.3 Model of Vibrating Scattering Medium........................................................... 128 6.4 Parameterized Model Analysis ......................................................................... 130 6.5 Parameterized Model Results ........................................................................... 131 6.6 Parameterized Model Discussion...................................................................... 139 6.7 Conclusion .......................................................................................................... 141 Chapter 7—Experimental Harmonic Motion Detection in a Vibrating Scattering Medium .......................................................................................................................... 142 7.1 Introduction........................................................................................................ 142 7.2 Background ........................................................................................................ 142 7.3 Motion Detection Experimental Setup............................................................. 143 7.4 Parameter Analysis in Scattering Gelatin Phantom....................................... 146 7.5 Scattering Gelatin Phantom Results ................................................................ 147 iv

7.6 Scattering Gelatin Phantom Discussion........................................................... 154 7.7 Bovine Muscle Section Results.......................................................................... 155 7.8 Bovine Muscle Section Discussion .................................................................... 162 7.9 Imaging of Bovine Muscle Section.................................................................... 163 7.10 Bovine Muscle Section Imaging Discussion................................................... 168 7.11 Conclusion ........................................................................................................ 169 Chapter 8—Discussion and Summary of Thesis........................................................ 170 8.1 Introduction........................................................................................................ 170 8.2 Discussion............................................................................................................ 170 8.3 Multifrequency Radiation Force Excitation.................................................... 171 8.4 Harmonic Motion Detection.............................................................................. 174 8.4 Modeling ............................................................................................................. 177 8.5 Future Directions ............................................................................................... 177 8.6 Significant Academic Achievements ................................................................ 179 8.6.1 Peer-reviewed Papers ................................................................................. 179 8.6.2 Conference Proceedings ............................................................................. 179 8.6.3 Conference Abstracts.................................................................................. 180 8.6.4 Invention Disclosure ................................................................................... 181 8.6.5 Academic Awards ....................................................................................... 181 8.7 Summary............................................................................................................. 181 Appendix 1—Multifrequency Radiation Stress Field Derivation ............................ 183 Appendix 2—Harmonic Pulsed Excitation Radiation Force Function Derivation 187 Appendix 3—List of Acronyms ................................................................................... 190 Appendix 4—List of Symbols ...................................................................................... 192 Bibliography .................................................................................................................. 207

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List of Figures Figure 1.1—Depictions of normal and shear stress and strain, (a) Normal stress is applied to the top of the solid block and a deformation occurs as depicted by the dashed block which changes the height in the vertical dimension, (b) Shear stress is applied parallel to the top surface of the block and a shear strain results.

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Figure 1.2—Ultrasound pressure and radiation force, (a) Ultrasound pressure signal at f0 = 1.00 MHz, (b) Ultrasound pressure at f0 + Δf = 1.10 MHz, (c) Summed ultrasound pressure for pressure signals at f0 and f0 + Δf, (d) Radiation force produced by ultrasound waves at Δf = 100 kHz.

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Figure 1.3—Co-focused beams producing ultrasound radiation force. Ultrasound beams at frequencies f0 and f0 + Δf are co-focused and the radiation force occurs in the interaction zone where the beams overlap. This configuration provides a localized dynamic radiation force to vibrate objects or tissue.

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Figure 2.1—Plot of θZ versus density of sphere for a sphere of radius a = 0.79 mm at Δf = 1.0 kHz.

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Figure 2.2—Plot of change in θZ versus Δf for a sphere of radius a = 0.79 mm and varying the density from 1000-8000 kg/m3.

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Figure 2.3—Plot of θd for spheres of radius a = 0.79 mm made of different materials.

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Figure 2.4—Velocity of spheres for different materials. (a) Normalized magnitude and (b) phase of velocity for five spheres of different materials with radius a = 0.79 mm.

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Figure 2.5—Feature space for simulation results in Figure 2.4. The points within those regions depict simulation results at individual frequencies.

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Figure 2.6—Block diagram for experimental setup.

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Figure 2.7—Measurement results for large and small spheres. (a) Normalized magnitude of the velocity for the large spheres, (b) Phase of velocity for the large spheres, (c) Normalized magnitude of the velocity for the small spheres, (d) Phase of velocity for the small spheres. (◊ - Acrylic, Δ - Soda lime glass,

- Silicon nitride, ο - Stainless steel, * - Brass)

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Figure 2.8—Images of large spheres. (a) Normalized magnitude image of large spheres, (b) Phase image of large spheres. From top to bottom the vi

spheres are made of acrylic, soda lime glass, silicon nitride, stainless steel and brass. Brass is used as a reference.

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Figure 2.9—Images of small spheres. (a) Normalized magnitude image of small spheres, (b) Phase image of small spheres. From top to bottom the spheres are made of acrylic, soda lime glass, silicon nitride, stainless steel and brass. Brass is used as a reference.

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Figure 2.10—Feature spaces for large and small spheres. (a) Feature space for measurements on large spheres for Δf = 400-1250 Hz, (b) Feature space for small spheres for Δf = 700-1250 Hz. (Acrylic – Yellow, Soda lime glass – Red, Silicon nitride – Green, Stainless steel and brass – Blue)

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Figure 2.11—Classification of images using feature spaces for large and small spheres. (a) Classified image for large spheres, (b) Classified image for small spheres.

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Figure 3.1—Plots of maximum number of components created with the MHME implementation of the multifrequency stress field. (a) Maximum number of components, NMM, created using the number of sinusoids, NS, and the number of elements, NE. (b) Maximum number of components, NMM,MH, created using the MH mechanism with NS and NE. (b) Maximum number of components, NMM,ME, created using the ME mechanism with NS and NE. ( -NE = 1, Δ-NE = 2, ◊-NE = 3, {-NE = 4)

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Figure 3.2—Experimental setup for PSF measurements and vibrometry experiment. The arrows labeled x and y depict the scanning directions.

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Figure 3.3—Experimental setup for vibro-acoustography imaging. The arrows labeled x and y depict the scanning directions.

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Figure 3.4—Normalized amplitude profiles for multifrequency PSF components. The simulation results are solid and the experimental measurements are dashed. (a) Δf = 5 kHz, (b) Δf = 10 kHz, (c) Δf = 15 kHz, (d) Δf = 20 kHz, (e) Δf = 25 kHz, (f) Δf = 30 kHz.

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Figure 3.5—Combinations of experimentally measured PSF components. (a) Combination of 5 kHz and 10 kHz components, (b) combination of 10 kHz and 15 kHz components, (c) combination of 5 kHz, 10 kHz, and 15 kHz components. The images are 8 mm by 8 mm with a 50 dB dynamic range.

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Figure 3.6—Normalized magnitude of sphere vibration velocity measured by varying the difference frequency of a MH signal with NS = 2 applied to both elements of the confocal transducer. The velocity measured with the laser vibrometer was processed using a lock-in amplifier.

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Figure 3.7—Multifrequency velocity images of sphere near resonance. (a) Δf = 400 Hz, (b) Δf = 800 Hz, (c) Δf = 1200 Hz, (d) Δf = 1600 Hz, (e) Δf = 2000 Hz, (f) Δf = 2400 Hz.

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Figure 3.8—Normalized magnitude of velocity from images in Figure 3.7 near center of sphere. Squares indicate mean values and error bars indicate ±1 standard deviation.

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Figure 3.9—Curve fitting of resonance curve from multifrequency images with simulation of sphere velocity in viscoelastic medium. The gelatin properties were found to be μ1 = 6750 Pa, and μ2 = 3.0 Pa·s. The dotted curve is the simulated response. The dashed curve is the response shown in Figure 3.5. The solid curve is the mean curve (Figure 3.8) obtained from the multifrequency images shown in Figure 3.7.

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Figure 3.10—Photograph of the breast phantom and field of view used for vibro-acoustic imaging is shown by the dotted line. The field of view has size 60 mm x 80 mm.

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Figure 3.11—Multifrequency vibro-acoustography images of breast phantom. (a) Δf = 10 kHz, (b) Δf = 20 kHz, (c) Δf = 30 kHz, (d) Δf = 40 kHz, (e) Δf = 50 kHz, (f) Δf = 60 kHz. All images are normalized independently. A small bias has been added to images (c)-(f) for visualization purposes.

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Figure 3.12—Combination of vibro-acoustic images, (a) Combination of 30, 40, 50, and 60 kHz components, (b) Combination of 30, 50, and 60 kHz components. Both images are normalized independently, and a small bias has been added for visualization purposes.

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Figure 3.13—In vivo breast vibro-acoustography images for elderly woman with calcified fibroadenoma, (a) X-ray obtained from stereotactic X-ray system, (b) Δf = 20 kHz created by MH mechanism, (c) Δf = 40 kHz created by ME mechanism, (d) Δf = 50 kHz created by ME mechanism, (e) Δf = 60 kHz created by ME mechanism, (f) Δf = 70 kHz created by ME mechanism.

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Figure 4.1—Sample simulated echo data for vibration of a scatterer with D0 = 5000 nm, φs = 0°, Nc = 5, Np = 20, and SNR = 20 dB.

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Figure 4.2—Displacement signal for data in Figure 4.1. The red curve with the data points marked by the open circles is the displacement signal estimated from the data and the blue dashed curve is the true displacement signal. The estimated vibration amplitude and phase are D0 = 4990.2 nm and φs = 0.054° while the true values are D0 = 5000 nm and φs = 0°.

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Figure 4.3—Displacement amplitude results for default parameters in Table 4.1. Each data point represents the mean of the 1000 iterations and the error viii

bars represent one standard deviation. The dashed line represents the target value of D0 = 1000 nm.

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Figure 4.4—Displacement phase results for default parameters in Table 4.1. Each data point represents the mean of the 1000 iterations and the error bars represent one standard deviation. The dashed line represents the target value of φs = 0°.

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Figure 4.5—Displacement amplitude and phase bias and jitter for variation of D0 = 100 ({), 500 (), 1000 (Δ), 5000 (◊), and 10000 (∇) nm. (a) Amplitude bias, (b) Amplitude jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 4.6—Displacement amplitude and phase bias and jitter for variation of Nc = 3 ({), 5 (), 10 (Δ), 15 (◊), and 20 (∇).(a) Amplitude bias, (b) Amplitude jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 4.7—Displacement amplitude and phase bias and jitter for variation of Np = 5 ({), 10 (), 15 (Δ), 20 (◊), 25 (∇), and 30 (*).(a) Amplitude bias, (b) Amplitude jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 4.8—Displacement amplitude and phase bias and jitter for variation of gate length, lg = 0.5 ({), 1.0 () mm. (a) Amplitude bias, (b) Amplitude jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 4.9—Displacement amplitude and phase bias and jitter for variation of sampling frequency of the ultrasound echoes, Fs = 50 ({), 100 () MHz. (a) Amplitude bias, (b) Amplitude jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 4.10—Displacement amplitude and phase bias and jitter for variation of transducer bandwidth, BW = 6.5 ({) and 20.0 () %. (a) Amplitude bias, (b) Amplitude jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 5.1—Timing diagram for harmonic pulsed excitation and pulses used for motion tracking. (a) Ultrasound tonebursts with length Tb and repetition period of Tr, (b) Radiation force produced by ultrasound tonebursts, (c) Transmission gate for ultrasound tracking pulses with an onset delay of td and repetition period of Tprf, (d) Reception gate for echoes of transmitted tracking pulses.

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Figure 5.2—Harmonic pulsed excitation transmission with fr = 200 Hz and Tb = 200 μs, (a) Ultrasound pressure amplitude, (b) Radiation force function.

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Figure 5.3—Magnitude spectrum for HPE transmission with fr = 200 Hz and Tb = 200 μs.

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Figure 5.4—Measured pressure and radiation force for HPE with fr = 500 Hz and Tb = 200 μs, (a) Pressure amplitude, (b) Radiation force function.

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Figure 5.5—Measured and calculated normalized magnitude spectrum of radiation force function for HPE with fr = 500 Hz μs, (a) Tb = 200 μs, normalized to maximum value, (b) Tb = 100 μs, normalized to maximum value measurements for Tb = 200 μs.

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Figure 5.6—Experimental setup for sphere frequency response measurements and motion detection with Doppler laser vibrometer and ultrasound method, (a) Block diagram for measurement of sphere frequency response. Two signal generators produce CW ultrasound signals at f0 and f0 + Δf which are summed together, amplified, and used to drive the transducer. The CW signals are used as inputs to a mixer, and the resulting signal is lowpass filtered and the Δf signal is used as a reference for the lock-in amplifier to compare against the signal from the Doppler laser vibrometer, (b) Experimental setup for excitation and measurement of motion of a stainless steel sphere embedded in a gelatin phantom. The 3.0 MHz transducer creates the radiation force and in a later experiment will also be used to track the motion ultrasonically. The Doppler laser vibrometer provides a calibrated measurement of the sphere’s motion.

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Figure 5.7—Experimental setup for harmonic pulsed excitation and motion tracking. For the excitation, a pulse train with frequency fr = 100 Hz is initiated, and each positive pulse triggers a toneburst of ultrasound at f0 = 3.0 MHz. For the tracking a pulse train at fprf is initiated with specified time delay, td, and each positive pulse triggers a three cycle pulse at ff = 9.0 MHz to be transmitted. The excitation and tracking signals are summed together and amplified before being sent to the 3.0 MHz transducer. For tracking, the gated echoes are filtered with a notch filter centered at 3.0 MHz and a bandpass filter centered at 9.0 MHz before passing through a transmit/receive (T/R) switch. The signal is amplified and filtered again with a bandpass filter centered at 9.0 MHz before being sent to the digitizer (A/D).

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Figure 5.8—Frequency response of 3.0 MHz transducer excited by broadband pulse.

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Figure 5.9—Representative received echo data from the stainless steel sphere for HPE with fr = 100 Hz and Tb = 100 μs.

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Figure 5.10—Velocity of sphere measured by Doppler laser vibrometer and ultrasound based motion detection, (a) Velocity measured by laser vibrometer, (b) Velocity measured with ultrasound based motion detection.

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Figure 5.11—Influence of windowing of displacement signal and the starting sample of the window. The results for the laser are shown in red squares and the ultrasound based results are shown with blue circles. The black dashed line indicates the measurement made by the laser with Fs = 100 kHz. The rectangular windowed results are shown by solid lines, and the Hann windowed results are shown with dash lines.

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Figure 5.12—Displacement frequency response of 1.59 mm diameter stainless steel sphere in gelatin phantom.

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Figure 5.13—Comparison of displacement amplitude measured by laser vibrometer and ultrasound based detection for fv = 200 Hz. Each data point represents the average of five measurements. (a) Comparison for Tb = 50 μs. The blue regression line has equation y = 0.9808x – 5.3618 with R2 = 0.9998, (b) Comparison for Tb = 100 μs. The blue regression line has equation y = 1.0021x – 12.7432, R2 = 0.9989.

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Figure 5.14—Sample data measured using ultrasound method. Displacement and phase data are plotted versus normalized force, (a) Displacement, Tb = 50 μs, (b) Phase, Tb = 50 μs, (c) Displacement, Tb = 100 μs, (d) Phase, Tb = 100 μs.

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Figure 5.15—Displacement and phase bias and jitter for Tb = 50 μs, F0 = 1, and fv = 100 ({), 200 (), 300 (Δ), and 400 (∇) Hz, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 5.16—Displacement and phase bias and jitter for Tb = 100 μs, F0 = 1, and fv = 100 ({), 200 (), 300 (Δ), and 400 (∇) Hz (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 5.17—Displacement and phase bias and jitter for Tb = 50 μs, fv = 200 Hz, and F = F0 ({), F0/2 (), F0/4 (Δ), and F0/8 (∇), (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 5.18—Displacement and phase bias and jitter for Tb = 100 μs, fv = 100 Hz, and F = F0 ({), F0/2 (), F0/4 (Δ), and F0/8 (∇), (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 5.19—Displacement and phase bias and jitter for Tb = 50 μs, F0 = 1, fv = 200 Hz, and fprf = 2.0 ({), 3.0 (), 4.0 (Δ), 5.0 (∇), and 6.0 (◊) kHz, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 5.20—Displacement and phase bias and jitter for Tb = 100 μs, F0 = 1, fv = 200 Hz, and fprf = 2.0 ({), 2.5 (), 3.0 (Δ), 4.0 (∇) kHz, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 5.21—Displacement and phase bias and jitter for Tb = 50 μs, F0 = 1, fv = 200 Hz, and fprf = 2.0 ({), 3.0 (), 4.0 (Δ), 5.0 (∇), and 6.0 (◊) kHz, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 5.22—Displacement and phase bias and jitter for Tb = 100 μs, F0 = 1, fv = 200 Hz, and fprf = 2.0 ({), 2.5 (), 3.0 (Δ), 4.0 (∇) kHz, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 5.23—Displacement and phase bias and jitter for Tb = 50 μs, F0 = 1, fv = 200 Hz, and Ts = 30 ({), 50 (), 100 (Δ), and 150 (∇) ms, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 5.24—Displacement and phase bias and jitter for Tb = 100 μs, F0 = 1, fv = 200 Hz, and Ts = 30 ({), 50 (), 100 (Δ), and 150 (∇) ms, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 5.25—Displacement and phase bias and jitter for Tb = 50 μs, F0 = 1, fv = 200 Hz, and Ts = 30 ({), 50 (), 100 (Δ), and 150 (∇) ms, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 5.26—Displacement and phase bias and jitter for Tb = 100 μs, F0 = 1, fv = 200 Hz, and Ts = 30 ({), 50 (), 100 (Δ), and 150 (∇) ms, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 6.1—Displacement results for default conditions in Table 6.1. Each data point represents the mean and the error bars represent one standard deviation. The black dashed line is the target value of D0 = 1000 nm.

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Figure 6.2—Phase results for default conditions in Table 6.1. Each data point represents the mean and the error bars represent one standard deviation. The black dashed line is the target value of φs = 0°.

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Figure 6.3— Displacement and phase bias and jitter for D0 = 100 ({), 500 (), 1000 (Δ), 5000 (◊), and 10000 (∇) nm, (a) Displacement bias, (b) xii

Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 6.4— Displacement and phase bias and jitter for Nc = 3 ({), 5 (), 10 (Δ), 15 (◊), and 20 (∇), (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 6.5—Displacement and phase bias and jitter for Np = 5 ({), 10 (), 15 (Δ), 20 (◊), 25 (∇), and 30 (*), (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 6.6—Displacement and phase bias and jitter for Ns = 11 ({), 16 (), 21 (Δ), 27 (◊), and 33 (∇), (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 6.7—Displacement and phase bias and jitter for lg = 0.5 ({) and 1.0 () mm, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 6.8—Displacement and phase bias and jitter for Fs = 50 ({) and 100 () MHz, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 6.9—Displacement and phase bias and jitter for BW = 6.5 ({), 20.0 (), and 40.0 (Δ) %, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 6.10—Normalized displacement results for variations of Wx, Wy, and Wz. (a) Wz = 0.1, (b) Wz = 0.5, (c) Wz = 1.0, (d) Wz = 2.18.

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Figure 6.11—Displacement jitter results for variations of Wx, Wy, and Wz. (a) Wz = 0.1, (b) Wz = 0.5, (c) Wz = 1.0, (d) Wz = 2.18.

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Figure 6.12—Phase bias results for variations of Wx, Wy, and Wz. (a) Wz = 0.1, (b) Wz = 0.5, (c) Wz = 1.0, (d) Wz = 2.18.

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Figure 6.13—Phase jitter results for variations of Wx, Wy, and Wz. (a) Wz = 0.1, (b) Wz = 0.5, (c) Wz = 1.0, (d) Wz = 2.18.

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Figure 7.1—Experimental setup for harmonic pulsed excitation and motion tracking. For the excitation, a pulse train with frequency fr = 100 Hz is initiated, and each positive pulse triggers a toneburst of ultrasound at f0 = 3.0 MHz. For the tracking a pulse train at fprf is initiated with specified time delay, td, and each positive pulse triggers a three cycle pulse at ff = 9.0 MHz to be transmitted. The excitation and tracking signals are summed together and amplified before being sent to the 3.0 MHz transducer. For tracking, the gated xiii

echoes are filtered with a notch filter centered at 3.0 MHz before passing through a transmit/receive (T/R) switch and then filtered with a bandpass filter centered at 9.0 MHz. The signal is logarithmically amplified and filtered again with a bandpass filter centered at 9.0 MHz before being sent to the digitizer (A/D).

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Figure 7.2—Experimental setup and phantoms for scattering gelatin and beef experiments, (a) Experimental setup with 3.0 MHz transducer and gelatin phantom containing suspended graphite particles, (b) Photograph of bovine muscle embedded in agar block, (c) Photograph of 3.0 MHz transducer held in water tank above agar block with bovine muscle.

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Figure 7.3—Regression of measured ultrasound displacement versus normalized force. The regression line was computed for the eight highest samples and the resulting equation is y = 3492.3x – 1219.6 with R2 = 0.9778.

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Figure 7.4—Displacement and phase results for default conditions of analysis. Data points represent the mean of five measurements and the error bars represent one standard deviation of those measurements. (a) Displacement results, (b) Phase results.

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Figure 7.5—Mean and standard deviations of displacement and phase measurements for values of fv = 100 ({), 200 (), 300 (Δ), and 400 (∇) Hz, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ). The legend in panel (d) applies to all panels.

150

Figure 7.6—Mean and standard deviations of displacement and phase measurements for values of F = F0 ({), 0.81F0 (), 0.64F0 (Δ), and 0.49F0 (∇), (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ).The legend in panel (d) applies to all panels.

150

Figure 7.7—Mean and standard deviations of displacement and phase measurements for values of fprf = 2.0 ({), 3.0 (), 4.0 (Δ), and 5.0 (∇) kHz, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ).The legend in panel (d) applies to all panels.

151

Figure 7.8—Mean and standard deviations of displacement and phase measurements for values of fprf = 2.0 ({), 3.0 (), 4.0 (Δ), and 5.0 (∇) kHz, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ).The legend in panel (d) applies to all panels.

151

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Figure 7.9—Mean and standard deviations of displacement and phase measurements for values of Ts = 50 ({), 100 (), 150 (Δ), and 200 (∇) ms, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ).The legend in panel (d) applies to all panels.

152

Figure 7.10—Mean and standard deviations of displacement and phase measurements for values of Ts = 50 ({), 100 (), 150 (Δ), and 200 (∇) ms, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ).The legend in panel (d) applies to all panels.

152

Figure 7.11—Mean and standard deviations of displacement and phase measurements for values of lg = 0.5 ({) and 1.0 () mm, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ).The legend in panel (d) applies to all panels.

153

Figure 7.12—Mean and standard deviations of displacement and phase measurements for values of lg = 0.5 ({) and 1.0 () mm, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ).The legend in panel (d) applies to all panels.

153

Figure 7.13—Regression of measured ultrasound displacement versus normalized force for Tb = 50 μs. The regression line was computed for the nine highest samples and the resulting equation is y = 409.0482x – 75.0306 with R2 = 0.9417.

157

Figure 7.14—Regression of measured ultrasound displacement versus normalized force for Tb = 100 μs. The regression line was computed for the eleven highest samples and the resulting equation is y = 811.5938x – 147.3434 with R2 = 0.9761.

158

Figure 7.15—Regression of measured ultrasound displacement versus normalized force for Tb = 200 μs. The regression line was computed for the eleven highest samples and the resulting equation is y = 1653.9x – 222.1 with R2 = 0.9721.

158

Figure 7.16—Displacement and phase results for default conditions of analysis for Tb = 100 μs. Data points represent the mean of five measurements and the error bars represent one standard deviation of those measurements, (a) Displacement results, (b) Phase results.

159

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Figure 7.17—Mean and standard deviations of displacement and phase measurements for values of fv = 100 ({), 200 (), 300 (Δ), and 400 (∇) Hz, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ). Legend in panel (d) applies to all panels.

159

Figure 7.18—Mean and standard deviations of displacement and phase measurements for values of F = F0 ({), 0.81F0 (), 0.64F0 (Δ), and 0.49F0 (∇), (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ). Legend in panel (d) applies to all panels.

160

Figure 7.19—Mean and standard deviations of displacement and phase measurements for values of fprf = 2.0 ({), 3.0 (), 4.0 (Δ), and 5.0 (∇) kHz, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ). Legend in panel (d) applies to all panels.

160

Figure 7.20—Mean and standard deviations of displacement and phase measurements for values of fprf = 2.0 ({), 3.0 (), 4.0 (Δ), and 5.0 (∇) kHz, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ). Legend in panel (d) applies to all panels.

161

Figure 7.21—Mean and standard deviations of displacement and phase measurements for values of Ts = 50 ({), 100 (), 150 (Δ), and 200 (∇) ms, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ). Legend in panel (d) applies to all panels.

161

Figure 7.22—Mean and standard deviations of displacement and phase measurements for values of Ts = 50 ({), 100 (), 150 (Δ), and 200 (∇) ms, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ). Legend in panel (d) applies to all panels.

162

Figure 7.23—Photographs of bovine muscle section. (a) Full phantom with bovine muscle encased in agar, (b) Top surface of muscle section. The muscle was cut in half to show the portion that was scanned. The dashed box depicts the scan area used for imaging, (c) Bottom surface of muscle section shown in (b). The slice has been flipped horizontally compared the image in (b). The dashed box depicts the scan area used for imaging.

164

Figure 7.24—Images of bovine muscle section. (a) Ultrasonic C-scan performed at 9.0 MHz, (b) Vibro-acoustic image acquired with Δf = 50 kHz, xvi

(c) Displacement amplitude for fv = 100 Hz. The units of the colorbar are nanometers, (d) Displacement phase for fv = 100 Hz. The units of the colorbar are degrees.

165

Figure 7.25—Displacement amplitude images of bovine muscle section at different vibration frequencies. The units of the colorbars are nanometers. (a) fv = 100 Hz, (b) fv = 200 Hz, (c) fv = 300 Hz, (d) fv = 400 Hz.

166

Figure 7.26—Displacement phase images of bovine muscle section at different vibration frequencies. The units of the colorbars are degrees. (a) fv = 100 Hz, (b) fv = 200 Hz, (c) fv = 300 Hz, (d) fv = 400 Hz.

166

Figure 7.27—Displacement amplitude images of bovine muscle section at different depths. The units of the colorbars are nanometers. (a) z = 0 mm, (b) z = 4 mm, (c) z = 8 mm, (d) z = 12 mm.

167

Figure 7.28—Displacement phase images of bovine muscle section at different depths. The units of the colorbars are degrees. (a) z = 0 mm, (b) z = 4 mm, (c) z = 8 mm, (d) z = 12 mm.

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List of Tables Table 3.1—Summary of Multifrequency implementations

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Table 3.2—Comparison of PSF Simulation and Experimental Results

60

Table 4.1 – Parameter Study Default Parameters

84

Table 6.1 – Parameter Study Default Parameters

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Chapter 1 Introduction

Therefore, everyone who hears these words of mine and acts on them, may be compared to a wise man who built his house on the rock. And the rain fell, and the floods came, and the winds blew and slammed against that house; and yet it did not fall, for it had been founded on the rock. Matthew 7:24-25

1.1 Background Medical imaging technology takes advantage of different physical properties to form images that have contrast related to the structure and function of many different tissues. The exploitation of new physical parameters has given rise to different imaging modalities to explore the human body. Elasticity imaging is an emerging medical imaging modality which uses tissue elasticity or stiffness as its contrast mechanism. This modality incorporates techniques from existing medical imaging techniques to obtain information about the material characteristics of internal tissues and pathological conditions. For hundreds of years physicians have relied on the art of palpation to identify diseased tissue as in the case of performing a physical examination of a woman’s breast to check for large masses. The premise of using palpation to identify pathology is that diseased tissue feels stiffer than normal tissue. Currently, physicians perform clinical breast examinations [1] and digital rectal examinations [2] to check for breast and prostate cancer because diseased tissue is associated with higher stiffness. Many studies

1

have shown a positive correlation with stiffness and disease state in breast, liver, prostate, and arterial tissues [3-9]. Some of the problems with clinical palpation are that it is subjective, dependent on the proficiency of the clinician, and may be insensitive to small or deep lesions [1012]. The goal of any elasticity imaging modality is to noninvasively acquire high resolution images of the spatial mapping of tissue elasticity or stiffness. The advantages of such an elasticity image should be reproducibility and a quantitative, objective representation of what palpation would observe.

1.2 Elasticity To produce quantitative images, a theoretical basis of elasticity must be understood. In material testing elasticity of a material is typically described by an elastic modulus. To evaluate the elasticity of a material, a stress must be applied, where stress,

σ, is defined as

σ=

F , A

(1.1)

where F is the force applied over an area A. The stress can be applied in a direction normal or tangent to the surface and these types of stress are called normal and shear stresses, respectively. These types of stress are depicted in Figure 1.1.

2

σ τ

L0

L

α

(a)

(b)

Figure 1.1—Depictions of normal and shear stress and strain, (a) Normal stress is applied to the top of the solid block and a deformation occurs as depicted by the dashed block which changes the height in the vertical dimension, (b) Shear stress is applied parallel to the top surface of the block and a shear strain results.

The application of a stress will cause a deformation in the object. The resulting strain to a normal stress is defined as

ε=

L − L0 , L0

(1.2)

where L is the new length of the object and L0 is the original length before deformation. If a shear stress is applied, a shear strain results that is defined as an angle α. The property that relates the stress and the strain is the modulus. This modulus is the desired value for display in elasticity imaging as it is the parameter that relates the applied force and resulting deformation of the material. The Young’s and shear moduli for a Hookean material are defined as [13]

E=

G=

σ ε τ

tan (α )

(1.3)

(1.4)

3

In the Hookean elastic solid, the Young’s and shear modulus are related by the following relationship E = 2G (1 + v ) ,

(1.5)

where v is the Poisson’s ratio. If a stress is applied as in Figure 1(a), it will compress the object in the axial direction and the object will expand in the transverse direction. Poisson’s ratio is the ratio of the axial strain to the transverse strain in this experiment. Soft tissue is often assumed to be nearly incompressible which yields a Poisson’s ratio of nearly 0.5, making E ≈ 3G [13]. In tissue two types of moduli can be identified based on the type of waves that travel through the medium. Compressional wave speed, cl, is governed by the bulk modulus, λ, and shear wave speed, cs, is related to the shear modulus, G, also written as

μ. cl =

E (1 − v ) λ + 2μ = ρ (1 + v )(1 − 2v ) ρ

(1.6)

E (1 + v )(1 − 2v )

(1.7)

λ=

cs =

E = 2 ρ (1 + v )

μ ρ

(1.8)

where ρ is the mass density of the medium [14]. Sarvazyan, et al. showed that for different tissue types, the bulk modulus does not vary significantly, whereas the shear modulus varies over seven orders of magnitude [15, 16].

4

1.3 Viscoelasticity Human tissue is not perfectly elastic, but has a viscous component. Therefore, tissue has been modeled as a viscoelastic material to account for the deviations from elastic behavior. In a viscoelastic material, the stress and strain change on a timedependent basis. Characteristics of a viscoelastic material are relaxation, creep and hysteresis [13]. Relaxation occurs when strain is maintained constant and the corresponding stresses decrease. Creep happens when stress is maintained constant and the strain continues to change. Hysteresis is the phenomenon in which the stress-strain relationship is different during loading and unloading. Different mechanical models have been proposed and adopted for the analysis of different tissues such as the Maxwell, Voigt, and Kelvin. The Voigt model has been the most used for analysis of tissue. This model contains a spring and a dashpot in parallel, where the spring represents an elastic component and the dashpot represents a viscous component [13]. In this model, the bulk and shear moduli are complex to account for losses caused by the viscous components such that λ = λ1 + iωλ2 and μ = μ1 + iωμ2 where λ1 and λ2 are the bulk elasticity and viscosity and μ1 and μ2 are the shear elasticity and viscosity, i is the imaginary number i = −1 and ω is the angular frequency [17]. In a viscoelastic medium, the equation for the compressional sound speed does not change because in tissue λ2 ≈ 0 and λ >> μ, but the shear wave speed changes to [18]

cs =

(

2 ( μ12 + ωμ22 )

ρ μ1 + μ12 + ωμ22

)

.

(1.9)

5

1.4 Elasticity Imaging Over the last two decades, there has been significant development of different methods to perform elasticity imaging. However, every elasticity imaging method involves two common elements, the application of a force or stress and the measurement of a mechanical response. The force or stress source can be generated in three different ways. An external vibrating actuator can be placed on the skin and used to generate a force into superficial and deep tissues. An internal vibration source can be created using an external or internal acoustic or optical excitation method. The body inherently also produces endogenous force such as from the beating heart or pulsating vessels, that can be transmitted into surrounding tissues. The measurement method can be performed using differing physical probes including magnetism, acoustics, or optics. Measurements can be made with magnetic resonance imaging (MRI), ultrasound imaging, optical and acoustic signals. The elasticity of a material relates the stress and the resulting strain. Therefore, different models, whether elastic or viscoelastic, must be applied to extract the elasticity parameters. However, for some methods the stress may be unknown so the strain or some other parameter is used as a surrogate for the elasticity because the strain is generally inversely proportional to the elasticity given the same stress. Each elasticity imaging method can be characterized by the methods used for force excitation and measurement of the tissue response. The state-of-the-art in elasticity imaging will be detailed in the following pages and can also be found in the following review articles [19-21].

6

1.5 Force Excitation To measure the modulus of a material a force or stress must be applied to the material and the response is measured. As detailed above, the force can originate from three sources, external, internal or endogenous.

1.5.1 External Excitation Sources Some of the first studies utilized external actuators in contact with the skin to induce motion into the tissue [18, 22-24]. These actuators were driven with a harmonic signal to induce shear waves into tissue so that shear wave speed could be measured and used to obtain estimates for the shear moduli in (1.9). In the study described by Krouskop, et al., a motorized actuator was placed on the medial side of the thigh to induce shear waves into the muscle tissue and an ultrasound transducer was coupled to the lateral side of the thigh to measure the induced motion using Doppler techniques [22]. Using this system, the elastic modulus of the muscle was measured in different contraction states. Lerner, et al. used an acoustic horn to induce motion into phantoms and excised tissue and used a color Doppler system to measure the resulting motion. In the method proposed by Yamakoshi, et al., a mechanical actuator was coupled to the surface of the phantom or subject’s skin to induce vibration. An ultrasound transducer was placed close to the actuator to insonify the vibrating region and measure the motion. In magnetic resonance elastography, proposed by Muthupillai, et al. a MRI compatible actuator is placed on the phantom surface or the skin and moves in a direction parallel to the surface of the object or subject to induce shear waves.

7

Later, a static compression using either the face of an ultrasound transducer or a plate attached to the transducer was used to induce a deformation that was fairly uniform across the measurement aperture [25-28]. The transducer could be pressed harder or softer to vary the amount of strain induced in the tissue and this compression can be carried out by freehand pressure by the sonographer or by a motorized method [25, 28]. For this method, axial strain is the primary measured quantity, but lateral strain can also be measured [27]. Another method called transient elastography used a vibration actuator to induce a transient force into the tissue [29-31]. This method employs low-frequency vibration induced by a vibrator attached to a piston or vibrating rods attached to an array transducer to create shear waves in the tissue. The various external excitation sources require specialized hardware to cause vibration or static displacement of the tissue. Only elastography, in which a static compression can be performed with the face of the transducer, requires no specialized equipment. In a clinical setting, the use of this external hardware may affect repeatability due to coupling to the subject and variability of compression.

1.5.2 Internal Excitation Sources There are two primary methods used to induce a force from an external source into tissue without an external actuator system. One method to induce a force into tissue is using acoustic radiation force produced by ultrasonic waves [15, 32-40]. The radiation force can be produced by pulsed or continuous wave (CW) ultrasound. The radiation force can be induced as a static, transient force or a sustained harmonic force.

8

Short tonebursts with durations typically less than 1 ms can be used push the tissue [15, 32, 35, 36]. These tonebursts can be used to create a transient force, which is assumed to approximate an impulse, at the focus and the motion at the focus can then be monitored. Also, transient forces induce shear waves that move perpendicular to the radiation force direction. The shear wave propagation can be monitored and the shear wave speed can be estimated to solve for material properties. One method uses tonebursts of ultrasound and moves the focal region at a speed faster than the medium shear wave speed [40]. This supersonic shear imaging (SSI) method creates shear waves that propagate perpendicular to the direction of the radiation force motion and these shear waves are then measured for evaluation of elasticity parameters. If CW ultrasound is used, the pushing force is constant and the resulting displacement increases with time of insonification. However, if the CW ultrasound is modulated, then a local, harmonic vibration can be created at the focal region of the transducer at the modulation frequency [33, 34, 38]. Another method called photoacoustics is used to remotely excite tissue with a laser source which illuminates a region of tissue. The light is absorbed by the tissue and the tissue undergoes thermoelastic expansion and contraction. This expansion and contraction of tissue creates a sound field and the resulting sound is measured by a passive ultrasound transducer [41-44]. One of the advantages of using radiation force is that it can be created with the same transducer used for motion detection; therefore, no special equipment is necessary. For the SSI method, an ultrafast detection system that can image at a rate of 5000 frames/second is used and is not widely available because of this speed requirement. The

9

photoacoustic methods require an external laser and distance to which the light can penetrate into the body limits its applications to superficial tissues. Penetration into internal tissues can be performed with invasive methods using catheter based systems [45].

1.5.3 Endogenous Excitation Sources For most elasticity imaging methods, endogenous motion of tissue is treated as a noise source because the measurements rely on the fact that motion is small, on the order of microns. Other researchers have used the endogenous motion of the beating heart to provide the deformation necessary for analysis of the elasticity measurements [46]. Kanai used measurements of the heart wall motion and a viscoelastic model to measure material properties in the beating heart muscle.

1.6 Measurement of Deformation With the many different methods of inducing deformation into the tissue, there are equally as many ways and algorithms to detect the resulting motion. Different physical probes are used including MRI, ultrasound imaging, and the measurement of optical and acoustic signals.

1.6.1 Magnetic Resonance Imaging Magnetic resonance imaging has been used to measure externally induced shear waves using motion-sensitized gradients in magnetic resonance elastography (MRE). The induced motion can be tracked for shear waves that have amplitudes down to 100 nm

10

[24]. MRE can provide quantitative material property measurements in three-dimensions. Usually, low frequency shear waves are used but specialized equipment can provide measurements at frequencies 1-10 kHz [47]. However, acquisition of the motion data can be slow, and the technique may not be widely available.

1.6.2 Ultrasound Imaging Differences in the acoustic impedance, defined as Z = ρcl, provides the image contrast which is primarily based on changes in the bulk modulus as ρ is relatively constant. These differences in acoustic impedance cause reflections of high frequency ultrasound waves that are transmitted by an ultrasound transducer. The basis of ultrasound imaging is the echo-ranging principle. The echo-ranging principle provides information about the received reflected waves, or echoes, and their depth in the tissue. Assuming an average sound speed in soft tissue as cl = 1540 m/s, the depth of the tissue interface giving rise to the echo can be described as

z=

cl t , 2

(1.10)

where z is the depth, t is the time of flight from transmission to reception and the quantity

clt is divided by two to account for travel to and from the interface [48]. The echo-ranging principle works well for tissue that is stationary, but moving tissue requires the use of Doppler principles to describe the motion. Moving ultrasound scatterers will cause a Doppler frequency shift, fd, in the ultrasound waves described by

fd =

2 f 0 v0 cos (θ ) , cl

(1.11)

11

where f0 is the frequency of the transmitted ultrasound wave, v0 is the velocity of the scatterers, and θ is the angle between the ultrasound beam and the direction of motion of the scatterers. Significant development has been done in the motion detection of small displacements using pulse-echo ultrasound and sophisticated signal processing algorithms. For the methods proposed by Krouskop et al., Lerner et al., and Yamakoshi,

et al., pulsed Doppler ultrasound techniques were used to track the motion that was induced using an external actuator driven in a harmonic fashion [18, 22, 23]. This general type of method has come to be known as sonoelasticity as Lerner and his colleagues named it. Ophir, et al. pioneered the method now known as elastography that uses static compression at the surface of the phantom or subject and cross-correlation between precompression and post-compression echo signals [25]. The location of maximum of the cross-correlation function of two echo signals is used to find the time shift between the two signals and using the longitudinal speed of sound of the medium determine how much motion has occurred. This method of cross-correlation was studied and has been optimized by subsequent studies [26, 49]. One-dimensional cross-correlation has long been used to track motion. However, with the advent of radiation force based excitation, the deformations induced are very small, even sub-micron displacements. More sophisticated signal processing techniques have been introduced to detect the small motion [50-55].

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1.6.3 Optical Signals

Optical signals produced by a laser source have been used to track the sinusoidal motion induced by amplitude modulated ultrasound [56]. However, the inability of light to penetrate limits this method to only a few applications.

1.6.4 Acoustic Signals

The photoacoustic method works by heating tissue locally and the expansion and compression creates acoustic signals that can be detected by ultrasound transducers. The absorption of the tissue varies the amplitude of the acoustic signals [44]. This variable absorption can potentially be used to differentiate different types of tissue. A method called vibro-acoustography uses ultrasound radiation force to induce harmonic motion in tissue and the acoustic emission produced by that vibration is measured by a nearby hydrophone [33, 34].

1.7 Vibro-acoustography

Vibro-acoustography is based on the use of dynamic ultrasound radiation force. This dynamic radiation force is to be differentiated from the static ultrasound radiation force induced by using short tonebursts of ultrasound. The underlying principle of creation of dynamic radiation force is that when ultrasound waves at two frequencies f0 and f0 + Δf interfere with each other, a harmonic radiation force at Δf is produced [33, 34, 57]. This radiation force can be produced in two different ways. The two ultrasound beams can be summed to create a double sideband suppressed carrier amplitude

13

modulated (AM) wave where two ultrasound signals, f0 and f0 + Δf, are added together or an ultrasound signal, f0, is modulated by a signal with frequency Δf as shown in Figure 1.2. The other way to create the dynamic radiation force is to generate two co-focused ultrasound beams. Where the beams interfere with each other, the harmonic radiation force will occur as depicted in Figure 1.3.

Figure 1.2—Ultrasound pressure and radiation force, (a) Ultrasound pressure signal at f0 = 1.00 MHz, (b) Ultrasound pressure at f0 + Δf = 1.10 MHz, (c) Summed ultrasound pressure for pressure signals at f0 and f0 + Δf, (d) Radiation force produced by ultrasound waves at Δf = 100 kHz.

14

f0 + Δf

f0

f0 + Δf

Interaction Zone

Figure 1.3—Co-focused beams producing ultrasound radiation force. Ultrasound beams at frequencies f0 and f0 + Δf are co-focused and the radiation force occurs in the interaction zone where the beams overlap. This configuration provides a localized dynamic radiation force to vibrate objects or tissue.

Vibro-acoustography uses the radiation force of ultrasound to excite a small volume of tissue causing it to vibrate at Δf. This vibration creates a sound field that can be measured by a nearby hydrophone or microphone and that signal can be processed to provide some quantitative measure of the acoustic emission such as its amplitude or phase. If the excitation point is raster scanned over a volume, an image of the acoustic emission can be formed. This harmonically induced motion can also be tracked with ultrasonic or optical methods [58]. This practice of inducing and then measuring motion of the vibrating object will be referred to as vibrometry. Vibro-acoustography is maturing as an imaging modality and has been used for different applications. There has been a significant amount of study in beamforming for vibro-acoustography, specifically concerning the radiation stress produced by different transducer configurations. Many different transducer configurations including AM,

15

confocal, X-focal, sector and linear arrays have been studied extensively investigating the point-spread function (PSF) and the optimization of this imaging modality [59-63]. Vibro-acoustography has been applied in many different types of tissue such as breast [64, 65], calcified arteries in breast tissue [66], calcified deposits on heart valves [67], liver [68], brachytherapy seeds [69], and bone [70]. Vibro-acoustography has also been used for noncontact measurement of elastic parameters of metal rods [71], and monitoring changes in materials and tissue with temperature [72, 73]. Vibrometry has been used in the generation and measurement of propagating waves in arteries. Ultrasound radiation force is used to excite the artery wall and propagating waves are measured at a different points along the artery to monitor pulse wave velocity or some other parameter [58, 74, 75]. Current state-of-the-art practice of vibro-acoustography and vibrometry uses dynamic radiation force to inspect tissues. It has been shown theoretically and experimentally that varying the difference frequency used provides images with different contrast [34, 68, 69]. At different values of the difference frequency, objects in the images can be visualized in unique ways. The frequency response of the object or tissue under investigation is usually not known a priori. Therefore, multiple frequencies distributed over a certain bandwidth may provide images with unique contrast. However, to acquire images at multiple difference frequencies requires scanning the same object multiple times. This can be very tedious and is not desirable for maximizing the information gained during a clinical scan. Acoustic emission images can at times be difficult to interpret because the acquired signal is a function of the ultrasonic, elastic, and acoustic parameters of the

16

tissue [34]. We know that the acoustic emission is proportional to the induced motion due to the dynamic radiation force. To understand the acoustic emission and obtain more quantitative information about the object under inspection, motion detection of the excited region would provide beneficial data. This thesis will describe methods to address these two limitations of state-of-theart practice. Multifrequency radiation force excitation will provide a method to simultaneously cause local vibration of an object at multiple frequencies. The motion detection method proposed by Zheng, et al. [54] will be modified and optimized to perform motion detection of the excited region of the object or tissue. Motion detection during multifrequency excitation will also be performed to obtain quantitative motion data at multiple frequencies.

1.8 Thesis Description

In this dissertation, the focus will be centered in two areas: development of methods to utilize multiple ultrasound frequencies simultaneously to produce multifrequency harmonic radiation force and development of motion detection techniques adapted for vibrometry with simultaneous radiation force excitation. Out of these areas of research, two hypotheses arise: 1) Using an array transducer and multiharmonic ultrasound sources with different frequencies, a stress field with multiple low-frequency components can be created.

17

2) Simultaneous dynamic ultrasound radiation force and motion detection can be implemented with a single transducer to harmonically vibrate tissue and measure the resulting motion. Chapter 2 will discuss a model of the vibration of spheres in a viscoelastic medium and the use of excitation at multiple frequencies to extract information about the material properties of the sphere and the material in which the sphere is embedded. This work provides a model for calcifications in breast tissue. Chapter 3 introduces and develops the concept of multifrequency radiation force excitation. Beamforming for multifrequency vibro-acoustography and vibrometry will be analyzed analytically, numerically, and experimentally. Sample applications in the fields of vibrometry and breast imaging are shown. Also, analysis of the signal-to-noise ratio for this method will be shown. Chapter 4 provides a theoretical basis and a model for harmonic motion detection of a vibrating reflective target. A parametric analysis with this model will be performed to extensively analyze the error in terms of vibration amplitude and phase. This analysis will provide a basis for experimental implementation and analysis. Chapter 5 will introduce another method for multifrequency excitation called harmonic pulsed excitation. A method using harmonic pulsed excitation and harmonic motion detection using a single transducer will be described and experimental results on a reflective target will be shown. Chapter 6 will extend the model in Chapter 4 to motion detection in a vibrating scattering medium. An analysis using this model will be performed involving parameters specific to the experimental system that will be used.

18

Chapter 7 will present experimental results from measurements of motion in a scattering gelatin and a section of ex vivo bovine muscle. A parameterized analysis will be performed to assess how different parameters affect the measurements. This measurement technique will also be extended for image formation of motion amplitude and phase. Chapter 8 will detail the contributions of the work within this dissertation and its impact on elasticity imaging. The chapter will summarize the multifrequency excitation methods described in this work. A study of the motion detection of a vibrating reflective target or a scattering medium will be discussed. Directions for future work will also be explored. Finally, academic accomplishments of this thesis work will be listed.

19

Chapter 2

Measuring the Phase of Vibration of Spheres in a Viscoelastic Medium as an Image Contrast Modality Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution. Albert Einstein

Content taken from [76]: Matthew W. Urban, Randall R. Kinnick, James F. Greenleaf, “Measuring the phase of vibration of spheres in a viscoelastic medium as an image contrast modality,” Journal of the Acoustical Society of America, 118 (6), 3465-3472. 2005.

2.1 Introduction

X-ray mammography is the standard clinical screening modality for breast cancer because of its good sensitivity, high specificity, and relatively low cost. Screening has been shown to reduce breast cancer mortality by 20-35% in women aged 50 to 69 and slightly less in women of ages 40 to 49 [77]. The American Cancer Society currently recommends that women start receiving yearly screening mammograms starting at age 40 [78]. It has been found that breast x-ray density and age are factors that decrease the sensitivity and specificity of the mammography screening process [79]. Increased breast density reduces contrast in mammograms making calcifications and tumors harder to find. Women aged 40 to 49 typically have denser breast tissue than older women, making this group susceptible to mitigated benefits from the screening process.

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The roles of medical ultrasound and magnetic resonance imaging (MRI) techniques are actively being explored to address some of the current limitations of x-ray mammography. Medical ultrasound is often used as an adjunct for women who have suspicious lesions identified on the mammogram and for women with dense breast tissue. However, medical ultrasound is not always able to detect microcalcifications due to a number of factors including the spatial resolution of the imaging system, speckle noise, and phase aberration [80]. MRI has shown increased sensitivity in detection of lesions in studies of women with increased risk of breast cancer but its specificity is lower than xray mammography [77]. Also, the cost of MRI scanning and the need for injection of a contrast agent are other reasons it has not supplanted x-ray mammography as a widespread clinical screening modality. A recently developed ultrasound method called vibro-acoustography is being explored as a new modality for medical imaging with applicability to breast imaging [33, 34, 64]. Vibro-acoustography typically uses two ultrasound beams at slightly different frequencies, f0 and f0 + Δf, where Δf is typically in the kilohertz range. When the two sound beams interfere at the intersecting foci of the two beams, the resulting acoustic radiation force causes the object within the intersection or focal region to vibrate at Δf. When the object vibrates, it creates a sound field which is called acoustic emission. The acoustic emission is measured with a low-frequency hydrophone or microphone. An image of the acoustic emission can be formed by scanning the object in a raster pattern with a transducer. Vibro-acoustography has been used for various medical applications such as imaging of mass lesions in liver tissue and calcified arteries [66, 68]. The potential use of

21

vibro-acoustography in breast imaging has been studied and there have been good results including detection of mass lesions present in the mammogram as well as microcalcifications as small as 100 μm in diameter [65]. Vibro-acoustography overcomes the obstacle of speckle noise that plagues conventional medical ultrasound. Speckle noise arises from ultrasound interactions with scatterers that are roughly the same size as the wavelength of the ultrasound. Since the acoustic emission is at very low frequencies, the wavelength is very large compared to the ultrasound wavelength, thereby avoiding the speckle noise from scattering. The spatial resolution obtained using vibro-acoustography is good. Using a confocal two element 3.0 MHz transducer with an F number of 1.55, the point-spread function in the focal plane is about 0.7 mm wide giving an excitation that is very localized [59]. Breast microcalcifications can be associated with the presence of diseased tissue [81, 82]. The material properties of microcalcifications are very different from those of the surrounding soft tissue. Microcalcifications are composed of apatite, calcite, calcium oxalate, and other materials [83]. These materials have densities ranging from 2200 to 3350 kg/m3 [84], while soft tissue has a density closer to that of water which is 1000 kg/m3. Because these materials are denser and harder than the soft tissue, an imaging modality that is sensitive to density could be used to better detect microcalcifications. There are two reasons to model microcalcifications as small spheres. First, using a sphere simplifies the shape of the calcifications and the model can be extended easily for any sized calcification. Secondly, extensive theoretical development on the motion of spheres in a viscoelastic medium excited by acoustic radiation force has already been

22

carried out [85]. We chose to model the spheres in a viscoelastic medium because soft tissue is inherently viscoelastic. In the vibro-acoustography studies using breast tissue, the difference frequency ranged from 5 to 30 kHz and the magnitude of the acoustic emission signal was used to create the images [64, 65]. As a new contrast mechanism we will explore using the phase of the vibration using very low difference frequencies of 100-3000 Hz.

2.2 Methods 2.2.1 Dynamic Motion of a Sphere

The radiation force resulting from a plane wave incident on a sphere immersed in a fluid is given by F = π a 2Y E where πa2 is the projected area of the sphere, Y is the radiation force function and 〈E〉 is short-term time average of the energy density of the ultrasound [85]. The dynamic radiation force from two interfering plane waves is given by Fd = π a 2Y E0 cos ( (ω1 − ω2 ) t − θ d ) where E0 = 2 P02 ρ c 2 and θd is the phase shift of the radiation force compared to the incident field. The difference frequency is defined as Δω = ω1 - ω2. The velocity of the sphere can be calculated as

V=

Fd . Zr + Zm

(2.1)

The terms Zr and Zm are the radiation and mechanical impedances, respectively given in (2.2) and (2.4), where k = ρΔω 2 ( 2μ + λ ) , h = ρΔω 2 μ , μ = μ1 + iΔωμ2 , and λ = λ1 + iΔωλ2 , a is the radius of the sphere, ρ is the density of the medium, c is the speed of sound of the medium, μ1 and μ2 are the shear elasticity and viscosity of the

23

medium, respectively, and λ1 and λ2 are the bulk elasticity and viscosity of the medium, respectively [17]. Typical values for soft tissues are λ1 ~ 109 Pa, λ2 ~ 0 Pa⋅s, μ1 ~ 104 Pa and μ2 ~ 10-2 Pa⋅s [17, 86]. 3i 3 ⎞ ⎛ i 1 ⎞⎛ a2k 2 ⎞ ⎛ − − − + − 1 2 3 ⎜ ⎜ ⎟ 2 2 ⎟ 2 2 ⎟⎜ aki + 1 ⎠ 4π a 3 ⎝ ah a h ⎠ ⎝ ah a h ⎠ ⎝ ρΔω Z r = −i 3 a2k 2 ⎞ 1 ⎞ a2k 2 ⎛ ⎛ i +⎜2− ⎜ + 2 2⎟ ⎟ aki + 1 ⎠ ⎝ ah a h ⎠ aki + 1 ⎝

(2.2)

For a sphere with mass m vibrating with harmonic velocity VeiΔωt the force required to overcome the inertia of the sphere is given as

F =m

dVeiΔωt = imΔωVeiΔωt . dt

(2.3)

The mechanical impedance is defined as the ratio of the applied force to the resulting velocity given in the following equation, where ρs is the density of the sphere material.

Zm =

−F 4π a 3 ω ρ = − im Δ = − i Δω s 3 VeiΔωt

(2.4)

Using these relationships, we can calculate the response of spheres of varying density over a range of frequencies.

2.2.2 Phase of Vibration

Let us denote Z as the sum of Zr and Zm. Also, we introduce phasor notation for

V, Fd, Zr, Zm, V = V ∠θV ,

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Fd = Fd ∠θ d , Z r = Z r ∠θ r ,

(2.5)

Z m = Z m ∠θ m , Z = Z ∠θ Z .

Equation (2.1) can be rewritten using the phasor notation as

V=

Fd ∠θ d F ∠θ d = d . Z r ∠θ r + Z m ∠θ m Z ∠θ Z

(2.6)

We would like to explore how the density of the sphere changes the phase of the velocity for the sphere. We expand the impedance in the denominator of (2.6). Z = Z r cos (θ r ) + i Z r sin (θ r ) + Z m cos (θ m ) + i Z m sin (θ m )

(2.7)

From (2.4), we know that θm = -90° so we substitute this value in (2.7) and find the magnitude and phase of Z. Z = Z r cos (θ r ) + i Z r sin (θ r ) − i Z m

Z=

Z r − 2 Z r Z m sin (θ r ) + Z m 2

⎛ Z r sin (θ r ) − Z m Z r cos (θ r ) ⎝

θ Z = tan −1 ⎜⎜

(2.8) 2

⎞ ⎟⎟ ⎠

(2.9)

(2.10)

We can calculate the phase for a sphere of radius a = 0.79 mm, in a viscoelastic medium modeling tissue with parameters μ1 = 6.7 × 103 Pa, μ2 = 0.5 Pa⋅s, λ1 = 109 Pa, λ2 = 0 Pa⋅s, c = 1500 m/s, ρ = 1000 kg/m3. We will vary the density of the sphere, ρs, from 1000-8000 kg/m3 and set Δf = 1.0 kHz. The resulting curve is shown in Figure 2.1 and shows a phase change of 54.75° over the range of density given.

25

To find a value for Δf in the range 0-3000 Hz that maximizes the phase change, the phase is calculated over the given density range for different values of Δf. The optimal point for the curve shown in Figure 2.2 is a phase change of 64.5° given at Δf = 600 Hz.

Figure 2.1—Plot of θZ versus density of sphere for a sphere of radius a = 0.79 mm at Δf = 1.0 kHz.

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Figure 2.2—Plot of change in θZ versus Δf for a sphere of radius a = 0.79 mm and varying the density from 1000-8000 kg/m3.

Now we wish to analyze the phase of the velocity, and we calculate the velocity of the sphere using (2.6) as

V=

Fd ∠θ d Z ∠θ Z

=

Fd Z

∠θ d − θ Z .

(2.11)

The only parameter that can change the phase of the velocity in the force term is the radiation force function Y which introduces phase θd. The radiation force function for five different spheres made of acrylic, soda lime glass, silicon nitride, 440-C stainless steel, and brass with densities of 1190, 2468, 3210, 7840, 8467 kg/m3 over a range of Δf = 0-3000 Hz was computed, and the results are shown in Figure 2.3. The phase change of the radiation force function is less than 0.5° for this range of Δf so it can be assumed that any contrast between two different spheres is due to the differences in density of the spheres. The velocity for the five different spheres mentioned above for a frequency

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range of Δf = 0-3000 Hz was calculated. The magnitude and unwrapped phase of the velocity is shown in Figure 2.4, where the magnitude curves are normalized by the maximum value of the brass response.

Figure 2.3—Plot of θd for spheres of radius a = 0.79 mm made of different materials.

Figure 2.4—Velocity of spheres for different materials. (a) Normalized magnitude and (b) phase of velocity for five spheres of different materials with radius a = 0.79 mm.

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Each sphere has a specific resonance frequency at which the magnitude of the velocity is maximal. The resonance frequency is dependent on the values of μ1, μ2, and a. It is important to note that for all values of Δf the phase shift is the least for the sphere of lowest density made of acrylic and the greatest for the most dense sphere made of brass. As density increases, the phase shift increases as shown in Figure 2.1. Therefore, the phase shift of the vibration is proportional to density, providing a predictable density contrast mechanism for evaluating images. One way to view the magnitude and phase data in one plot is to use a feature space. The magnitude is plotted on the x-axis and the phase on the y-axis for the data in Figure 2.4. Before plotting the phase versus the magnitude, the brass response is subtracted at each frequency to use the brass response as a reference. The frequency range is limited from Δf = 750-3000 Hz to avoid the effects of the resonances of the different spheres. The goal of this feature space diagram is to find a pictorial way to separate the contributions from each sphere over a range of frequencies. In the background different grayscales define a discrimination surface to separate the responses from each sphere. The regions in Figure 2.5 depict the discrimination regions for the acrylic, soda lime glass, silicon nitride, and stainless steel in decreasing levels of brightness. The points within those regions depict simulation results at individual frequencies. The discrimination surface adequately separates the responses of the different spheres.

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Figure 2.5—Feature space for simulation results in Figure 2.4. The points within those regions depict simulation results at individual frequencies.

2.3 Experiment 2.3.1 Experimental Setup

Spheres of equal diameters, either 1.5875 mm or 3.175 mm, and different materials including acrylic, soda lime glass, silicon nitride, 440-C stainless steel, and brass were suspended in an optically clear gelatin phantom in a line so that when scanned, they would all be in the same focal plane. The spheres were separated by more than 1 cm to avoid interactions of shear waves in the gelatin and the sidelobes of the point-spread function. The spheres with diameters 1.5875 and 3.175 mm will henceforth be referred to as the small and large spheres, respectively. The spheres were spray painted white to provide a reflective surface for a Doppler laser vibrometer (Polytec, Waldbronn, Germany). The gelatin phantom was made using 300 Bloom gelatin powder (Sigma-

30

Aldrich, St. Louis, MO) with a concentration of 10% by volume. A preservative of potassium sorbate (Sigma-Aldrich, St. Louis, MO) was also added with a concentration of 5%. A confocal, two-element transducer with a nominal frequency of 3.0 MHz and Fnumber of 1.55 was used to excite each sphere. One signal generator (33120A, Agilent, Palo Alto, CA) was set at f0 and the other was set at f0 + Δf, where f0 = 3.0 MHz. A pulse echo technique was used to position the sphere in the focal region of the transducer. A Doppler laser vibrometer measured the velocity of the sphere and this signal was subsequently processed by a lock-in amplifier (Signal Recovery, Oak Ridge, TN) to obtain the measurements of the magnitude and phase of the velocity. The phase was computed using a sinusoidal reference signal at the vibration frequency Δf, created by mixing the two ultrasound drive signals and passing the mixed signal through a low-pass filter. The block diagram of the experimental setup is shown in Figure 2.6.

Figure 2.6—Block diagram for experimental setup.

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The difference frequency was varied from 100-1250 Hz in increments of 50 Hz. The magnitude and phase of vibration at each frequency were measured while the transducer was spatially fixed on each sphere. With the measurement data feature spaces are formed to evaluate the ability to separate the responses of the different spheres. A magnitude and phase image was created for each sphere at Δf = 700 Hz. This difference frequency was chosen because it was higher than the resonance frequencies as to avoid confounding interactions due to resonance and to provide more predictable contrast between the spheres both in the magnitude and phase images. The laser was focused on one sphere while the transducer was raster-scanned across the phantom with at a resolution of 0.2 × 0.2 mm. The same field of view was scanned for each sphere to assure registration all the images.

2.3.2 Experimental Results

The measured magnitude and unwrapped phase for the two phantoms are shown in Figure 2.7.

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Figure 2.7—Measurement results for large and small spheres. (a) Normalized magnitude of the velocity for the large spheres, (b) Phase of velocity for the large spheres, (c) Normalized magnitude of the velocity for the small spheres, (d) Phase of velocity for the small spheres. (◊ - Acrylic, Δ - Soda lime glass, - Silicon nitride, ο - Stainless steel, * - Brass)

The magnitude and phase images from each of the five scans performed at Δf = 700 Hz were combined to create composite images for the phantom. Preprocessing was performed to reduce noise in the background of the images. The magnitude composite images were formed by adding all five magnitude images together. The phase composite images were created using threshold masks made from the magnitude images. The original phase images were multiplied by the mask images and then summed together. As a result of the threshold operation, the phase composite image had a blocky appearance so it was spatially low-pass filtered to make the objects more resemble spheres. This filtering did not affect the image values in terms of comparing the magnitude and phase of the spheres. A bias was added to the background so that the relative phase differences could be better visualized. The composite images are shown in Figures 2.8 and 2.9 for the

33

large and small spheres, respectively. From top to bottom, the spheres are made of acrylic, soda lime glass, silicon nitride, stainless steel, and brass. The brass and stainless spheres have low image intensity values in the phase image because brass was used as the reference for comparison and stainless steel has a phase value very close to the brass.

Figure 2.8—Images of large spheres. (a) Normalized magnitude image of large spheres, (b) Phase image of large spheres. From top to bottom the spheres are made of acrylic, soda lime glass, silicon nitride, stainless steel and brass. Brass is used as a reference.

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Figure 2.9—Images of small spheres. (a) Normalized magnitude image of small spheres, (b) Phase image of small spheres. From top to bottom the spheres are made of acrylic, soda lime glass, silicon nitride, stainless steel and brass. Brass is used as a reference.

For feature space analysis the measurement data were used for frequency ranges of Δf = 400-1250 Hz and 700-1250 Hz for the large and small spheres, respectively. The two different lower cutoff frequencies were chosen to avoid the resonance characteristics of the spheres. The feature spaces for the large and small spheres are shown in Figure 2.10. The feature space results are incorporated back into the images using the discrimination surface created with the feature spaces. In both feature spaces acrylic is identified by yellow, soda lime glass by red, silicon nitride by green, and stainless steel and brass by blue. The raw image data that was not biased was used for classification of the spheres using the feature space from the measurements. The resulting classified images are shown in Figure 2.11.

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Figure 2.10—Feature spaces for large and small spheres. (a) Feature space for measurements on large spheres for Δf = 400-1250 Hz, (b) Feature space for small spheres for Δf = 700-1250 Hz. (Acrylic – Yellow, Soda lime glass – Red, Silicon nitride – Green, Stainless steel and brass – Blue)

Figure 2.11—Classification of images using feature spaces for large and small spheres. (a) Classified image for large spheres, (b) Classified image for small spheres.

2.4 Discussion

Comparing the phase responses in Figures 2.4 and 2.7, we find that the spheres follow the same pattern that was found in the simulations, that is, the phase shift is

36

proportional to the density of the sphere. The magnitude measurements provide no predictable contrast mechanism at a single Δf. The only density contrast evident in the magnitude measurements is the resonance frequency is inversely proportional to the density. If we consider the magnitude image in Figure 2.8(a), we observe that the acrylic, stainless steel, or brass spheres have similar image intensities. Also, the soda lime glass and silicon nitride appear similar. In the phase image, the intensity distinction between the acrylic and stainless steel spheres is very evident. We also note a difference between the soda lime glass and silicon nitride spheres in the phase image. The phase image in Figure 2.8(b) illustrates the same phase shift progression as the measurements, that is, compared to the most dense sphere made of brass, spheres of less dense material have larger phase shifts. For the small spheres, the magnitude image in Figure 2.9(a) would lead us to believe that the soda lime glass, silicon nitride, and stainless steel spheres are similar and that the acrylic and brass spheres may be similar based on image intensity. However, the phase image in Figure 2.9(b) provides evidence of the differences between the five spheres just as the phase image for the large spheres. The feature space analysis in Figure 2.10 provides very good separation of the different spheres for both sizes of spheres used. The discrimination surfaces are slightly different for the different sized spheres due to different relationships between the magnitude and phase for each size. The colors yellow, red, green, and blue represent the divisions for the acrylic, soda lime glass, silicon nitride, and stainless steel and brass spheres. All of the large and small spheres were classified correctly in Figure 2.11. This

37

feature space analysis could be used for in vivo imaging to differentiate between objects of interest in an image. We see potential in this type of analysis being useful in classification of different types of calcifications and disease types but that requires much more additional work. To use this method in vivo a few points must be considered. Vibro-acoustography uses the acoustic emission signal to form images. In this paper, we demonstrated improved contrast from the laser measurement of the velocity of the spheres. To use acoustic emission instead of the velocity we must develop the mathematical relationship relating the velocity to the acoustic emission. As an alternative to measuring the acoustic emission, we could measure the velocity of the sphere using Doppler ultrasound techniques. Motion detection using pulsed Doppler and a Kalman filter designed for detecting the magnitude and phase of motion due to a harmonic source has been developed and has demonstrated that phase and displacements down to 30 nm could be measured reliably [87]. The experiments detailed in this paper were carried out under continuous wave (CW) driving conditions. This type of implementation in vivo could be potentially damaging because of tissue heating effects. The method can be adapted for using tone bursts to prevent undesirable tissue heating. Calcifications in the breast are not necessarily spherical in shape nor are they the same size within a sample. If the shape is close to spherical we could assume an equivalent sphere model where we use some effective radius to analyze objects in an image. To address size variations we can perform simulations using this equivalent sphere concept to better understand the potential contrast in magnitude and phase images.

38

The density of calcifications would probably not exceed 3500 kg/m3 so the contrast in the phase images may only be about 30° but that should be sufficient if the phase of the background is relatively constant. If Δf is kept in the low kilohertz range, phase wrapping in the phase image should not be a significant issue. However, twodimensional phase unwrapping techniques have been developed and could be utilized if the need arose [88].

2.5 Conclusion

The ability to distinguish between different materials in an image is of great import in medical diagnosis. We have shown that we can provide predictable contrast between objects of different density so that when a phase image is presented the density of different objects in the image can be compared. Finding the magnitude and phase responses of small spheres in a viscoelastic medium may give us more insight into how to choose the difference frequency when imaging microcalcifications in the breast. We may be able to provide better contrast in the phase images so that we can identify microcalcifications among other structures in the image, which would aid in diagnosis and treatment.

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Chapter 3 Multifrequency Vibro-acoustography and Vibrometry Fourier’s theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics. Lord Kelvin

Content taken from [89]: Matthew W. Urban, Glauber T. Silva, Mostafa Fatemi, James F. Greenleaf, “Multifrequency vibro-acoustography,” IEEE Transactions on Medical Imaging. vol. 25, no. 10, pp. 1284-1295. 2006.

3.1 Introduction

Elasticity imaging is a burgeoning medical imaging field. Since the beginnings of modern medicine, palpation has been an important method to examine patients. It is known that increased tissue stiffness has been linked to different physiologic states [15]. However, some diseased tissues such as tumors may be too deep in the body for a clinician to assess from hand palpation. Conventional medical imaging modalities often do not detect changes in stiffness. Therefore, conventional imaging modalities have been modified to qualitatively or quantitatively measure changes in tissue stiffness in superficial and deep tissues. Elasticity imaging modalities have two common components, application of a force to the tissue and measurement of the response due to that force. The response of the object may then be used in conjunction with equations relating motion in an elastic or viscoelastic medium to solve for material properties of the tissue under investigation. Modalities can be characterized by their method of applying the force or by the measurement technique utilized [19].

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Static elasticity imaging methods use an external compression of the tissue and then use correlation or speckle tracking of echoes to measure the resulting strain [25, 26]. Some groups have used a dynamic excitation method using an external actuator to induce vibration in the tissue and measure the vibration with Doppler ultrasound [18, 22, 23]. Another dynamic approach to imaging elasticity differences is to measure shear wave propagation in tissue. Magnetic resonance elastography uses an external actuator to induce shear waves in tissue and then measures the propagation using magnetic resonance imaging [24]. A method called transient elastography also uses an external piston to generate transient shear waves and an ultrafast ultrasonic imaging system to measure the transient response of the tissue [90]. The ultrasound community has recently been exploring the use of static acoustic radiation force of ultrasound to move tissue and measure shear wave propagation or tissue motion. Shear wave elasticity imaging uses an amplitude modulated beam of focused ultrasound to locally induce shear waves and another ultrasound transducer to measure the shear wave propagation [15]. Supersonic shear imaging generates a local radiation force in tissue and then moves that stress point at a supersonic speed, compared to the shear wave, to create shear waves [40]. Acoustic radiation force impulse imaging uses radiation force to push tissue and correlation techniques to measure the resulting displacement [36]. Vibro-acoustography is a noninvasive method that uses the dynamic acoustic radiation force to locally vibrate tissue [33, 34]. The radiation force is formed using amplitude modulated ultrasound beams or multiple beams of ultrasound separated by small frequency differences that interfere in a common focal region. The vibrating tissue

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creates a sound field, referred to as “acoustic emission,” which is measured by a nearby hydrophone. We define vibrometry as the use of harmonic radiation force excitation to vibrate tissue or an object accompanied by the measurement of the velocity or displacement of the tissue or object either by a laser vibrometer or Doppler ultrasound. Vibrometry is a complement to vibro-acoustography in which the response of the vibrating region is measured more directly. Vibro-acoustography and vibrometry using radiation force induced vibration has been applied to many areas within the medical field. Vibro-acoustography imaging of calcifications in the breast, arterial wall, and heart valve leaflets, liver lesions, bone, and brachytherapy seeds in the prostate has been accomplished in vitro [20, 65-70, 91, 92]. An example of vibrometry is the use of harmonic radiation force to generate waves in arterial walls, the speed and dispersion of which result in assessment of wall stiffness [58, 74]. Typically, the radiation force is produced using a two-element confocal transducer with ultrasound beams of two different frequencies, f0 and f0 + Δf, where f0 is in the megahertz range and the difference frequency Δf is typically in the kilohertz range. In vibro-acoustography, image information content is intrinsically linked to the value of

Δf. To obtain images with different spectral content, multiple scans of the object are necessary. To more fully understand a tissue or object’s spectral characteristics, it is desirable to obtain images at different values of Δf. In most cases the values of Δf that provide the most useful information are not known a priori. To address this problem we propose a multifrequency vibro-acoustography approach where the radiation stress at

42

several values of Δf are simultaneously generated and used to probe the tissue. In this chapter we present the theory of using multiple ultrasound waves with different frequencies to create a multifrequency radiation stress. Different methods of implementing this multifrequency stress field are analyzed. We evaluate the radiation stress by examining the point-spread function (PSF) and experimentally validate it. We also analyze the signal-to-noise ratio of this multifrequency method. We demonstrate how this method can be used for vibrometry applications using a gelatin phantom with a spherical inclusion and find the material properties of the gelatin. We also show the use of multifrequency vibro-acoustography imaging with results from scanning a breast phantom and from in vivo breast imaging.

3.2 Multifrequency Radiation Stress

We wish to evaluate the spatial distribution of the radiation stress field created by the multifrequency method. It has been previously shown that the amplitude of the PSF for vibro-acoustography is proportional to the radiation stress [34]. In this chapter, we consider multifrequency radiation stress field formation in a lossless fluid with density ρ and sound speed c. We describe the ultrasound waves in terms of the velocity potential,

φˆ ( r, t ) , where r is the position vector, t is time, and the hat denotes a complex variable. The multifrequency dynamic radiation stress is formed using N intersecting ultrasound beams focused at the same spatial location. We assume that in the focal region the incident beams resemble plane waves. Each ultrasound beam has frequency ωi where i =

m, n, where m = 2,…,N, and n = 1, 2,…, N, and we assume that ωm > ωn when m > n. The difference frequency between two frequencies is ωmn = ωm – ωn. Following the theory

43

presented in [57], we can calculate the dynamic radiation force at ωmn on a small sphere of radius a as

f ( r, t ) = π a 2 E0

N

∑

( m ≠ n ) =1

yˆ mnφˆm ( r ) φˆn* ( r ) e jωmnt ,

(3.1)

where E0 = ε 2 ρ c 2 2 is the energy density at the acoustic source, ε is the Mach number,

yˆ mn is the radiation force function for the sphere, and the symbol * denotes the complex conjugate. The force can be thought of as being created by mixing two ultrasound waves with different frequencies, which is mathematically represented as a multiplication of the velocity potential functions. The mixing occurs over the projected surface of the sphere, so to obtain the radiation stress we divide the radiation force by the projected area of the sphere

σ ( r, t ) =

N

∑

( m ≠ n ) =1

σˆ mn ( r ) e jω t , mn

(3.2)

where σˆ mn ( r ) = E0 yˆ mnφˆm ( r ) φˆn* ( r ) . The force on a small sphere is used to introduce the concept of the radiation stress created by the interaction of the acoustic waves. The spatial distribution of the force on a point target, represented here by the sphere, allows us to examine the stress exerted by the ultrasound in terms of the point-spread function. The PSF at each value of ωmn can be computed using the relationship from conventional vibro-acoustography [20, 34] hˆmn = Aφˆm ( r ) φˆn* ( r ) ,

(3.3)

where A is a normalization constant for a → 0, i.e., a point object.

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3.3 Multifrequency Implementation

We can characterize vibro-acoustography excitation implementations in a generalized way. We can characterize vibro-acoustography systems by the number of harmonic ultrasound signals applied to an element or group of elements and the number of elements or groups of elements with different harmonic signals. If we have multiple, i.e., more than two, harmonic ultrasound signals summed together and applied to a single element transducer we would denote this configuration as multiharmonic single element (MHSE) excitation. We can also use an array transducer and apply a different harmonic ultrasound signal to each element or group of elements. We would denote this configuration as single harmonic multielement (SHME), or single harmonic multigroup (SHMG) excitation. We can also apply different multiharmonic signals to different elements or different groups of elements, and these configurations would be denoted as multiharmonic multielement (MHME) or multiharmonic multigroup (MHMG) excitation modes. We can characterize conventional vibro-acoustography methods as special cases of the generalized implementations. Vibro-acoustography with a two-element confocal transducer using one harmonic ultrasound signal per element could be denoted as single harmonic multielement (SHME). We can also use a double sideband suppressed carrier amplitude modulated signal on a single element transducer, and we denote this as multiharmonic single element (MHSE) excitation. For previously described radiation force methods, the application of either multiharmonic or multielement has been limited to two harmonic signals or two different elements or groups of elements. For

45

multifrequency vibro-acoustography we wish to go beyond this paradigm and use more than two harmonic ultrasound signals and/or elements for creating the radiation stress. When using multielement (ME) radiation force, the dynamic radiation force only occurs where the different multiharmonic ultrasound beams intersect at the focal region and interfere with each other. In fact, the information from components created by intersecting beams from different elements brings out local characteristics of the investigated object or region. To evaluate the potential information gain from multifrequency beamforming we must find the number of low-frequency components that could be created using each of the aforementioned methods. A multiharmonic (MH) signal can be decomposed as the NS

sum of multiple sinusoidal terms, x ( t ) = ∑ Gn sin (ωnt + θ n ) , where Gn is the amplitude, n =1

ωn is the angular frequency, θn is the phase for the nth sinusoidal component. We denote the number of sinusoids summed together to create an MH signal as NS. In vibroacoustography, we form a low-frequency radiation stress component by combining any two ultrasound frequencies of different frequencies. To shorten the notation for examining the number of low-frequency components created by this multifrequency method we will use a subscript with two letters, the first denoting a single or multiharmonic signal by S or M, respectively, and the second letter denoting the use of single or multielements or multigroups by S or M, respectively. For multifrequency vibro-acoustography, the maximum number of dynamic components created using the MHSE method, NMS, with NS sinusoids is found by evaluating the number of different combinations of pairs of the NS sources

46

N MS =

NS

Ck =

NS ! , ( N S − k )!k !

(3.4)

where k is the number of ultrasound frequencies used to create one low-frequency component so for all cases k = 2 and NS ≥ 2. For the case k = 2, NMS = NS(NS-1)/2. The SHME case follows the MHSE case closely except that instead of using NS we use the number of ultrasound waves of different frequencies applied to NE elements or NE groups of elements. The maximum number of dynamic components created with the SHME implementation is given by N SM =

N E ( N E − 1) , 2

(3.5)

where NE ≥ 2. In the MHME case, we must distinguish between the maximum number of total components created and the number of components created using the MH or ME mechanisms. The maximum number of components created using MHME, NMM, with different sets of NS sinusoids of different frequency on NE elements or NE groups of elements is given by N MM =

NS NE

Ck =

NS NE ! , ( N S N E − k )!k !

(3.6)

where k = 2 and NE ≥ 1 and NS ≥ 1. For k = 2, the relationship in (3.6) reduces to N MM =

N S N E ( N S N E − 1) . 2

(3.7)

The number of components that are created solely by combination of the MH signals, NMM,MH, for vibro-acoustography are N MM , MH = N E N MS =

N S N E ( N S − 1) . 2

(3.8)

47

The maximum number of components created by the ME mechanism, NMM,ME, is the difference between NMM and NMM,MH N MM , ME

N S2 N E ( N E − 1) = . 2

(3.9)

A summary of the different implementations and the maximum number of components is given in Table 3.1. Figure 3.1 shows plots of the maximum number of components for the MHME multifrequency implementation. Figure 3.1(a)-(c) provide plots of the values of NMM, NMM,MH, and NMM,ME, respectively for NE =1-4 and varying NS from 1-5. Table 3.1—Summary of Multifrequency implementations

Method Multiharmonic Single Element (MHSE) Single Harmonic Multielement (SHME) Multiharmonic Multielement (MHME) MHME, Multiharmonic Mechanism MHME, Multielement Mechanism

Maximum Number of Components (NS ≥ 1, NE ≥ 1)

N S ( N S − 1) 2 N E ( N E − 1) N SM = 2 N N ( N N − 1) N MM = S E S E 2 N N ( N − 1) N MM , MH = S E S 2 2 N N ( N − 1) N MM , ME = S E E 2 N MS =

It should be noted that the maximum number of unique components may not be reached if a difference frequency is repeated by the combination of different ultrasound waves. For example in an MHSE implementation, we could have ultrasound signals with frequencies 2.99, 3.00, and 3.01 MHz added together. This choice creates difference frequencies of 10, 10, and 20 kHz. The repetition of the 10 kHz component means that

48

we have not achieved the maximum number of unique components for this combination of three sinusoids. Therefore, ultrasound frequencies have to be chosen carefully to avoid overlapping of components to achieve the maximum number of independent vibration frequencies.

Figure 3.1—Plots of maximum number of components created with the MHME implementation of the multifrequency stress field. (a) Maximum number of components, NMM, created using the number of sinusoids, NS, and the number of elements, NE. (b) Maximum number of components, NMM,MH, created using the MH mechanism with NS and NE. (b) Maximum number of components, NMM,ME, created using the ME mechanism with NS and NE. ( -NE = 1, Δ-NE = 2, ◊-NE = 3, {-NE = 4)

3.4 Point-spread Function Calculations

For the purposes of this paper we model an MHME implementation with NS = 2 and NE = 2. A two-element confocal transducer with central disc and annular elements is modeled using the notation of [59]. For this case of NS and NE we will have one component created by MH generation from each of the central disc and annular elements. Four other components will be created using ME generation. A derivation of the radiation

49

force for NS = 2 and NE = 2 is provided in Appendix 1. The velocity potential for the central disc and annular elements in the focal plane can be written as [59]

φˆc ( r ) =

⎛ ra ⎞ u0 a12 jinc ⎜ 1 ⎟ , 2R ⎝ λi R ⎠

(3.10)

and

φˆa ( r ) =

⎛ ra22 ⎞ 2 ⎛ ra21 ⎞ ⎤ u0 ⎡ 2 ⎢ a22 jinc ⎜ ⎟ − a21 jinc ⎜ ⎟⎥ , 2R ⎣ ⎝ λi R ⎠ ⎝ λi R ⎠ ⎦

(3.11)

where u0 is the velocity of the transducer element, λi is the wavelength of the ultrasound wave with frequency ωi, r is radial distance to the field point represented on a Cartesian coordinate system as r = x 2 + y 2 where x and y are the azimuthal and elevational coordinates, and R is the focal length of the transducer. The function jinc(x) = J1(2πx)/πx where J1(⋅) is the first order Bessel function of the first kind. The geometric parameters a1, a21, and a22 represent the radius of the central disc element, the inner radius of the

annular element, and the outer radius of the annular element, respectively. We then use appropriate combinations of (3.10) and (3.11) and substitute into (3.3) to find the PSF for the radiation stress for different components.

3.5 Image Formation

In vibrometry, we can make images of the velocity response of an object excited by the multifrequency stress field. Consider an object such as a sphere that is spatially represented by the real function s(r), we can model the magnitude, M, and phase, P, images of the object given the PSF and the frequency response of the object, V(ω), using the following relationships [20]

50

{

}

(3.12)

{

}

(3.13)

M ( r, ωmn ) = Re V (ωmn ) hˆmn ( r ) ⊗ s ( r ) , P ( r, ωmn ) = Re ∠V (ωmn ) hˆmn ( r ) ⊗ s ( r ) ,

where |V(ωmn)| and ∠V(ωmn) are the magnitude and phase of the velocity at ωmn, ⊗ denotes a spatial convolution, and the operator Re{⋅} takes the real part of the argument. For vibro-acoustography, the acoustic emission, Φ(r,ωmn) measured by the hydrophone can be modeled by [34]

{

}

Φ ( r, ωmn ) = Re H mn ( l ) Qmn hˆmn ⊗ s ( r ) ,

(3.14)

where Hmn(l) is the medium transfer function that describes the wave propagation of the acoustic emission from the excitation point to the hydrophone and Qmn is the total acoustic outflow by the object per unit force where acoustic outflow is the volume of the medium in front of the surface of the object that is displaced by the vibrating object [34].

3.6 Signal-to-Noise Ratio Analysis

The implementation of multifrequency vibro-acoustography in biological systems requires a limited root-mean-square (rms) intensity to be used. This limit should not be exceeded in order to prevent excessive heating of tissue. However, since the intensity delivered is limited, the number of multifrequency components generated using multifrequency vibro-acoustography will affect the quality of resulting images. More components will require a fractioning of the limited intensity which will decrease the amount of force produced at any low-frequency and thereby reduce the amount of induced motion and acoustic emission.

51

3.6.1 Dynamic Radiation Force and its Effects

We will analyze the choice of the number of multifrequency components produced and the effects that they have on the SNR in measurements made to construct images. We start with the general definition of the radiation force, F, [34] F = dr S E ,

(3.15)

where dr is the drag coefficient, S is the intercepting surface area of the ultrasound waves, and 〈E〉 is the short-term time average of the energy density. The

radiation

force

of

an

amplitude

modulated

wave

given

by

p ( t ) = PSF ,ω0 cos ( (ω0 + Δω 2 ) t ) + PSF ,ω0 cos ( (ω0 − Δω 2 ) t ) and the resulting radiation

force is FSF ,Δω = PSF2 ,ω0 d r S ρ c 2 ,

(3.16)

where FSF,Δω is the radiation force at the modulation frequency Δω, PSF ,ω0 is the pressure amplitude of the ultrasound waves, ρ is the density of the medium, and c is the sound speed of the medium. To study the effects of this dynamic radiation force we can assume that the force is applied to a solid disk with a mechanical impedance ZΔω and the resulting motion, USF,Δω, is described as

U SF ,Δω = FSF ,Δω Z Δω .

(3.17)

The acoustic emission, PSF,Δω, resulting from this motion can be modeled as PSF ,Δω = ρ c 2 H Δω ( A ) QΔω FSF ,Δω ,

(3.18)

where HΔω is the medium transfer function and QΔω is the total acoustic outflow by the object per unit force.

52

The important points to make in this derivation are that the radiation force induced motion and the acoustic emission amplitude is proportional to the radiation force. Secondly, the radiation force is proportional to the square of the pressure of the ultrasound waves. Lastly, the intensity is proportional to the square of the pressure of the ultrasound waves as well.

3.6.2 Multifrequency Vibro-acoustography

The methods to form the multifrequency radiation force are characterized by the number of different ultrasonic harmonic signals, NS, added together per array element or group of array elements applied to NE elements or groups of elements. The MHME configuration is the most general case and the MHSE case occurs when NE = 1 and the SHME case occurs when NS = 1. With the implementation of multiple frequencies, the total force will be divided by the total number of different ultrasound frequencies used which is calculated as the product NSNE. Therefore, the total force is divided by NSNE. Since the force is proportional to the square of the pressure, the new pressure for the multifrequency implementation is given by FMF ,Δω =

FSF ,Δω NS NE

PMF ,ω0 ∝ FMF ,Δω =

∝

PSF2 ,ω0 NS NE

FSF ,Δω NS NE

∝

,

PSF ,ω0 NS NE

(3.19)

,

(3.20)

where FMF,Δω is the multifrequency radiation force and PMF ,ω0 is the pressure needed to produce FMF,Δω.

53

3.6.3 Signal-to-Noise (SNR) Ratio Analysis

The SNR of a signal is defined as the ratio of the signal power to the noise power and given by [93] SNR =

xrms , nrms

(3.21)

where the root-mean-square (rms) metric is defined by T

yrms =

1 2 y ( t ) dt . T ∫0

(3.22)

We will not be able to calculate the noise directly but we can identify its potential sources. We will assume the presence of white noise associated with thermal fluctuations over the bandwidth Fs/2 where Fs is the sampling frequency used in the experiment. Noise will also be introduced by the digitization process and is dependent on the number of bits used. The signal power for a pure tone is given as A2/2 where A is the amplitude of the signal. The induced velocity or the acoustic emission is proportional to the dynamic radiation force used. The radiation force for multifrequency excitation is reduced by a factor of NSNE. Therefore, the velocity signal resulting from one component of the multifrequency excitation at a specific frequency, Δω is given as U MF ,Δω =

FMF ,Δω Z Δω

=

FSF ,Δω N S N E Z Δω

.

(3.23)

The SNR for the multifrequency and single frequency vibro-acoustography systems can be derived by calculating the signal power for each case. The single frequency case is found be setting NS = 2 and NE = 1 or NS = 1 and NE = 2 such that NSNE = 2.

54

SNR MF ,Δω

⎛ FSF ,Δω ⎞ =⎜ ⎟ ⎝ N S N E Z Δω ⎠

SNR SF ,Δω

2

2nrms

⎛ F ⎞ = ⎜ SF ,Δω ⎟ ⎝ ( 2 ) Z Δω ⎠

2 ⎡ 1 ⎛ 1 ⎞2 ⎛ F ⎤ SF , Δω ⎞ =⎢ ⎜ ⎟ ⎜ ⎟ ⎥ nrms ⎢⎣ 2 ⎝ N S N E ⎠ ⎝ Z Δω ⎠ ⎥⎦

2

2nrms

2 ⎡1 ⎛ F ⎤ SF , Δω ⎞ =⎢ ⎜ ⎟ ⎥ nrms ⎢⎣ 8 ⎝ Z Δω ⎠ ⎥⎦

(3.24)

(3.25)

The ratio of SNR for the multifrequency and single frequency implementations is

4 ( N S N E ) . For NS = 2 and NE = 2 the SNR of each multifrequency signal will be ¼ that 2

of a signal acquired with the same ultrasound pressure in single frequency vibroacoustography. This derivation applies similarly to the SNR analysis of acoustic emission. Some of this SNR loss could be regained if the signals from different components could be combined through an additive process. When adding these components together, a weighting must be performed based on the frequency response of the object being inspected and other factors such as parametric amplification of radiation force at higher values of Δω [94].

3.6.4 Chirp Multifrequency Implementation

Multifrequency interrogation could also be implemented by changing Δω linearly with time essentially making the excitation function a linear frequency modulated (FM) or chirp signal. Provided that the bandwidth of the FM signal is small compared to the bandwidth of the transducer, the radiation force will have constant amplitude and the power for the resulting velocity or acoustic emission signals will be A2/2 [93]. The disadvantage of using a chirp is that the energy at any given frequency is very low unless the chirp time is slow or the bandwidth of the chirp is very small. The

55

multifrequency implementation provides energy at distinct frequencies but may not cover the same bandwidth during the course of the excitation.

3.7 Multifrequency Experiments 3.7.1 Point-spread Function Simulation

We wish to validate the theoretical PSF calculations with experimental results. Using (3.3), (3.10), and (3.11), we can compute the PSF components in the focal plane. We will use a different MH signal on the central disc and annular elements. The MH NS

signal applied to the central disc element is xc ( t ) = ∑ Gc ,n sin (ωc ,n t + θ c , n ) and the MH n =1

NS

signal applied to the annular element is xa ( t ) = ∑ Ga ,n sin (ωa ,nt + θ a ,n ) , where the center n =1

element has frequencies fc,1 = 2.9975 MHz and fc,2 = 3.0025 MHz and the annular element has frequencies fa,1 = 3.0175 MHz, and fa,2 = 3.0275 MHz. The MH signals on the central disc and annular elements create components at 5 kHz and 10 kHz, respectively. The ME components are created at frequencies of 15, 20, 25, and 30 kHz. The transducer has center frequency of 3.0 MHz with a focal length of R = 70 mm. The geometric parameters for the transducer are a1 = 14.8 mm, a21 = 16.8 mm, and a22 = 22.5 mm.

3.7.2 Point-spread Function Experimental Validation

A custom-made confocal transducer with the geometric parameters given in the previous section was used for all experiments. Experiments were performed in a large water tank (1.01 m x 0.64 m x 0.37 m) of degassed water. Synchronized waveform

56

generators (Agilent, Palo Alto, CA) were used to create the sinusoidal signals. The MH signals were created by summing the output of two waveform generators using a hybrid junction (M/A-COM, Inc., Lowell, MA). The MH signals were amplified and applied to the appropriate elements of the transducer. The target for the PSF measurements was a 440-C stainless steel sphere with diameter of 0.51 mm. The sphere was embedded in gelatin phantom made from 300 Bloom gelatin powder (Sigma-Aldrich, St. Louis, MO) with a concentration of 10% by volume. A Doppler laser vibrometer [95] (Polytec, Waldbronn, Germany) was used to measure the velocity of the sphere. We measure the velocity because it is proportional to the stress imparted on the sphere. The transducer was scanned across the sphere at an increment of 0.05 mm using a computer controlled motion control system. The signal from the laser vibrometer was digitized (Alazartech, Montreal, QC, Canada) and filtered in custom software in MATLAB (The Mathworks, Inc., Natick, MA). The experimental setup is shown in Figure 3.2. Since the PSF is circularly symmetric for all components, we compared the simulated and measured profiles of the PSF components. The spatial distribution, mainlobe width at the -6 dB level, also known as the full-width half maximum (FWHM), and sidelobe levels were also evaluated.

57

Figure 3.2—Experimental setup for PSF measurements and vibrometry experiment. The arrows labeled x and y depict the scanning directions.

3.7.3 Vibrometry Experiment

First, the frequency response of the sphere embedded in the gelatin was measured. A single continuous wave MH signal was applied to both of the transducer elements and the difference frequency was varied from 100-21,000 Hz. The velocity signal measured by the laser was processed by a lock-in amplifier (Signal Recovery, Oak Ridge, TN). Using the same experimental setup as for the PSF measurements, the values of the ultrasound frequencies were changed to fc,1 = 2.9998 MHz, fc,1 = 3.0002 MHz, fa,1 = 3.0014 MHz, and fa,2 = 3.0022 MHz. This configuration created MH components at 400 and 800 Hz. The ME components had frequencies of 1200, 1600, 2000, and 2400 Hz. A field of view of 10 mm x 10 mm was used, and the scanning increment of the transducer position was 0.1 mm in both dimensions.

3.7.4 Vibro-acoustography Experiment

Using the same transducer, a urethane breast phantom (ATS Laboratories, Bridgeport, CT) was scanned. The breast phantom has dimensions 17 cm x 10 cm x 7 cm. The frequencies used were fc,1 = 2.995 MHz, fc,2 = 3.005 MHz, fa,1 = 3.035 MHz, and fa,2

58

= 3.055 MHz. This configuration created MH components at 10 and 20 kHz. The ME components had frequencies of 30, 40, 50 and 60 kHz. Acoustic emission signals were measured by a nearby hydrophone (International Transducer Corporation, Santa Barbara, CA). The signals were bandpass filtered (Stanford Research Systems, Sunnyvale, CA) with a passband from 5-65 kHz and digitized. The signals were then filtered with zerophase Butterworth filters using MATLAB software. A diagram of the experimental setup is shown in Figure 3.3. A field of view of 60 mm x 80 mm was used, and the scanning increment of the transducer position for the breast phantom images was 0.5 mm in both dimensions.

Figure 3.3—Experimental setup for vibro-acoustography imaging. The arrows labeled x and y depict the scanning directions.

3.7.5 In Vivo Breast Imaging

In vivo breast imaging with multifrequency vibro-acoustography is performed with the system described by Alizad et al., [96]. The system combines a stereotactic mammography system with a vibro-acoustography system. Both X-ray and vibroacoustic images of the same region of breast tissue can be acquired with this system. The multifrequency vibro-acoustography stress field was created using a transducer similar in geometry as used in the vibrometry and vibro-acoustography experiments. 59

The frequencies used were fc,1 = 2.995 MHz, fc,2 = 3.005 MHz, fa,1 = 3.045 MHz, and fa,2 = 3.065 MHz. This configuration created MH components at 10 and 20 kHz. The ME components had frequencies of 40, 50, 60 and 70 kHz. Acoustic emission signals were measured by a hydrophone (International Transducer Corporation, Santa Barbara, CA) coupled to the breast using ultrasound gel. The signals were bandpass filtered with a passband from 10-100 kHz and digitized. The signals were then filtered with zero-phase Butterworth filters using MATLAB software.

3.8 Multifrequency Experimental Results

The simulated and measured amplitude profiles of the PSF components are shown in Figure 3.4. Each of the profiles have been normalized by the value at x = 0. Table 3.2 provides quantitative comparisons of the FWHM resolution and sidelobe levels. The resolution and sidelobe levels for the two MH generated components at 5 and 10 kHz are very different while those created using ME are very similar. The MH components have different resolution and sidelobe levels because the apertures used are different whereas for the ME images, the beamforming is the same. In a few cases, the sidelobe levels in the experimental results are slightly better than the simulation results. Qualitatively and quantitatively, there is good agreement between simulated and measured profiles. Table 3.2—Comparison of PSF Simulation and Experimental Results Frequency (kHz) 5 10 15 20 25 30

FWHM Resolution (mm) Simulation Experiment 1.21 1.18 0.62 0.68 0.77 0.80 0.77 0.80 0.76 0.79 0.77 0.80

Sidelobe levels (dB) Simulation Experiment -35.2 -29.5 -16.7 -16.4 -17.6 -19.6 -17.2 -19.1 -17.1 -18.7 -17.1 -16.7

60

Figure 3.4—Normalized amplitude profiles for multifrequency PSF components. The simulation results are solid and the experimental measurements are dashed. (a) Δf = 5 kHz, (b) Δf = 10 kHz, (c) Δf = 15 kHz, (d) Δf = 20 kHz, (e) Δf = 25 kHz, (f) Δf = 30 kHz.

Note from Figures 3.4(a)-(b) that for the MH components the mainlobe is in phase with the sidelobes and that in Figures 3.4(c)-(f) the ME components have sidelobes that are out of phase with the mainlobe. Because of this phenomenon it is possible to combine different components by addition to improve resolution and sidelobe levels because the positive and negative sidelobes may cancel when added. To combine the different components we must compute the amplitude images that have both magnitude and phase information. It has been shown [20] that for a confocal source (SHME) the amplitude of the normalized PSF can be positive or negative, but its residual phase (in excess of the 180° phase jumps when amplitude sign changes) is approximately zero. Under this condition,

an

amplitude

image

can

be

formed

by

61

{

}

A ( r, ωmn ) = M ( r, ωmn ) sign sin ⎡⎣ P ( r, ωmn ) ⎤⎦ . Before combining, each component was normalized. Figure 3.5 shows three examples of combining different components with experimental profiles and PSF images formed by the combinations. Figure 3.5(a) shows the combination of the two MH components at 5 and 10 kHz which has a FWHM resolution of 0.95 mm and sidelobes at -30.5 dB. Figure 3.5(b) shows the combination of one MH component and one ME component, in this case the 10 and 15 kHz components. The FWHM resolution is 0.77 mm and the sidelobe levels, compared to the ME components, have been reduced to -29.8 dB which are less than the measured sidelobe levels of any of the low-frequency components. In Figure 3.5(c) the 5, 10, and 15 kHz components were combined. This combination made the amplitude of the PSF entirely positive with FWHM resolution of 0.92 mm and the sidelobe levels were measured at -30.5 dB which are lower than any of the measured sidelobe levels of any single component. To examine the properties of a gelatin phantom we employed a vibrometry experiment. Figure 3.6 shows the magnitude of the velocity frequency response of the sphere embedded in the gelatin phantom obtained by varying the vibration frequency and processing the laser signal with the lock-in amplifier. The plot in Figure 3.6 shows a clear resonance present at 1200 Hz. We use this measurement method and curve fitting of the resonance curve using an established model [85] to measure the viscoelastic parameters of a medium. This measurement will be used as our gold standard for the material properties of the gelatin.

62

Figure 3.5—Combinations of experimentally measured PSF components. (a) Combination of 5 kHz and 10 kHz components, (b) combination of 10 kHz and 15 kHz components, (c) combination of 5 kHz, 10 kHz, and 15 kHz components. The images are 8 mm by 8 mm with a 50 dB dynamic range.

Figure 3.6—Normalized magnitude of sphere vibration velocity measured by varying the difference frequency of a MH signal with NS = 2 applied to both elements of the confocal transducer. The velocity measured with the laser vibrometer was processed using a lock-in amplifier.

63

Figure 3.7 shows the images of the stainless steel sphere obtained near its resonance. The resonance is determined by the sphere material properties and the surrounding viscoelastic medium. The brightness of the sphere in the multifrequency images varies depending on the frequency of the component. The sphere is brightest in the 1200 Hz image and decreases in brightness as the frequency increases or decreases. Since these images have information about the resonance characteristic of the sphere, we extracted pixel values from a 5 x 5 window centered on the sphere for all six images and computed the mean and standard deviation of those values. A plot of these mean and standard deviation values is given in Figure 3.8. The error bars show ±1 standard deviation from the mean value.

Figure 3.7—Multifrequency velocity images of sphere near resonance. (a) Δf = 400 Hz, (b) Δf = 800 Hz, (c) Δf = 1200 Hz, (d) Δf = 1600 Hz, (e) Δf = 2000 Hz, (f) Δf = 2400 Hz.

We can estimate the properties of the gelatin using the vibration of the sphere in the following way. Knowing the sphere’s material properties, we can vary parameters

64

related to the gelatin’s material properties to fit the curve in Figure 3.8 [85]. Figure 3.9 shows the results of the curve fitting procedure. The dotted curve is the result from a simulated response. The dashed curve is an overlay of the magnitude response curve in Figure 3.6. The solid curve is the mean curve from Figure 3.8. The theoretical prediction, the measured magnitude response and the response obtained from image analysis all agree very well. The gelatin’s viscoelastic material properties found with the curve fitting were the shear modulus, μ1 = 6750 Pa and the shear viscosity, μ2 = 3.0 Pa·s. The material properties found using this vibrometry method agree well with previously reported results [76], and slight differences in the viscoelastic parameters between the results reported in this paper and those previously published can be attributed to slightly different phantom makeup and age of the phantom at the time of the experiment.

Figure 3.8—Normalized magnitude of velocity from images in Figure 3.7 near center of sphere. Squares indicate mean values and error bars indicate ±1 standard deviation.

65

Figure 3.9—Curve fitting of resonance curve from multifrequency images with simulation of sphere velocity in viscoelastic medium. The gelatin properties were found to be μ1 = 6750 Pa, and μ2 = 3.0 Pa·s. The dotted curve is the simulated response. The dashed curve is the response shown in Figure 3.5. The solid curve is the mean curve (Figure 3.8) obtained from the multifrequency images shown in Figure 3.7.

A photograph of the breast phantom used for vibro-acoustography imaging is shown in Figure 3.10. The multifrequency vibro-acoustic images of the breast phantom are shown in Figure 3.11. The transducer was positioned so that the top two lesions were placed in the focal plane. Figure 3.11(a)-(b) show the images using the two MH components and Figure 3.11(c)-(f) show the images formed using the ME components. All images have been independently normalized. Images in Figure 3.11 (c)-(f) have a small bias added for visualization purposes. The images created using the MH mechanism were observed to have a near uniform gray background. The lesions in the phantom show up but detail is lost. The images created using ME mechanism show different contrast and depict the lesions with different degrees of detail.

66

Figure 3.10—Photograph of the breast phantom and field of view used for vibro-acoustic imaging is shown by the dotted line. The field of view has size 60 mm x 80 mm.

Figure 3.11—Multifrequency vibro-acoustography images of breast phantom. (a) Δf = 10 kHz, (b) Δf = 20 kHz, (c) Δf = 30 kHz, (d) Δf = 40 kHz, (e) Δf = 50 kHz, (f) Δf = 60 kHz. All images are normalized independently. A small bias has been added to images (c)-(f) for visualization purposes.

67

The ME components were combined by incoherent summation of the nonnormalized versions of the images to create a composite images shown in Figure 3.12. The 30, 40, 50, 60 kHz components were combined for Figure 3.12(a), and the 30, 50, 60 kHz components were combined for Figure 3.12(b). These images are independently normalized and have a small bias added for visualization purposes. Some of the images in Figure 3.11 show patterns of dark and bright areas in the background which we believe to be associated with acoustic reverberation in the phantom and the effects of this reverberation have been reduced in the images in Figure 3.12. Both images show an increase in lesion contrast and a suppression of the strong background present in the upper portion of the image. The combination in Figure 3.12(b) seems to provide better contrast due to the exclusion of the 40 kHz component.

Figure 3.12—Combination of vibro-acoustic images, (a) Combination of 30, 40, 50, and 60 kHz components, (b) Combination of 30, 50, and 60 kHz components. Both images are normalized independently, and a small bias has been added for visualization purposes.

68

Figure 3.13 shows the results of in vivo breast vibro-acoustography for an elderly woman with a calcified fibroadenoma in her left breast. The X-ray image and the 20, 40, 50, 60, and 70 kHz components are shown. The 20 kHz component was created by the MH mechanism and the other components were created by the ME mechanism. The images made at different frequencies provide different contrast for the fibroadenoma and the surrounding tissue. Also, tissue structures such as a blood vessel are observed in the images.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3.13—In vivo breast vibro-acoustography images for elderly woman with calcified fibroadenoma, (a) X-ray obtained from stereotactic X-ray system, (b) Δf = 20 kHz created by MH mechanism, (c) Δf = 40 kHz created by ME mechanism, (d) Δf = 50 kHz created by ME mechanism, (e) Δf = 60 kHz created by ME mechanism, (f) Δf = 70 kHz created by ME mechanism.

3.9 Discussion

The PSF is formulated in a lossless fluid, however for radiation force formation requires some loss in the medium or object and in the experimental setting this loss is 69

present in the phantoms used. The presence of absorption serves to decrease the amplitude of the ultrasound waves used, but will not change the shape of the PSF because there is not disparate frequency dependent attenuation of the ultrasound waves because they are separated by small changes in frequency (< 2%). The different methods of implementing the multifrequency stress field afford the user a great deal of flexibility depending on the application. For vibrometry applications, using a MH signal on a single element transducer or using the same MH signal on all the elements of an array transducer will insure that the PSF has the same resolution for all low-frequency components generated. The ME implementation may be optimal for vibroacoustography applications. However, one is limited by the number of elements in the array and all components may have a different PSF shape. For the MHME configuration the maximum number of components increases proportional to square of the product

NSNE as shown in Figure 3.1. The increased number of images to be gained using multifrequency methods can be substantial, a convincing reason to use this method. The information gain for this method depends on the object being investigated. If the object has many features in its frequency response, the information gain from multifrequency inspection could be considerable. A very important advantage is that the information gain comes with no substantial increase in scanning time compared to conventional vibroacoustography with one difference frequency. This advance could decrease patient scan time and increase patient throughput in a clinical setting. However, with an increase in information, the signal-to-noise ratio, SNR, in the images obtained will be less than if only one value of Δf was used for the scan, but that SNR loss may be partially regained by combining images. The increase in number of

70

images must be balanced against maintaining image resolution and contrast.

For

industrial applications, the power may be increased to gain the SNR back while using this multifrequency approach. In medical applications, the power to be deposited is limited, so the number of components gained must be weighed against a loss in SNR. However, by choosing frequencies such that a few difference frequencies are repeated will assist in regaining SNR at the expense of the number of low-frequency components in the multifrequency excitation. Another way to obtain multifrequency information is by encoding a chirp signal into the radiation force. A spectral data set would be obtained but the SNR at any one frequency would be very low. To obtain images with better SNR the response could be integrated over some frequency band and used for image formation. This process serves to reduce the spectral resolution and may be disadvantageous. Also, a short chirp may not be able to excite a high Q object because of the relatively long time it takes to induce resonant vibration, and the information gained from the scan may not be as insightful. It may be advisable to perform a preliminary scan using a chirp excitation and then use the multifrequency method for closer inspection of a certain frequency band that may be of interest. Figure 3.4 shows that qualitatively the simulated and measured PSF components agree very well. Quantitatively, the resolution and sidelobe levels of the PSF components also agree. This result is important because now we can reliably apply this multifrequency approach to any type of array transducer. Stress field calculations can be performed for confocal transducers, X-focal transducer arrangements, linear array transducers, and sector array transducers before implementation to insure optimization of

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the stress field and the transducer parameters using previously published theory [59, 60, 62]. Figure 3.6 shows the magnitude response of the sphere embedded in gelatin and a resonance is observed at 1200 Hz. Figure 3.7 is a manifestation of (3.12) in which the PSF components have been spatially convolved with the sphere and multiplied by discrete points on the magnitude frequency response curve. We showed in Figure 3.8 that this resonance characteristic could be extracted from the brightness of the images and that we could use this response to estimate the material properties of the medium as shown in Figure 3.9. In the data obtained using the lock-in amplifier and the multifrequency images, there is an increase in the velocity magnitude at a frequency lower than the resonance. This trend is not predicted by the theory which assumes an infinite homogeneous medium. The trend found in the experiment may be due to resonance behavior of the gelatin phantom held in a plastic fixture. The curve fitting process operates on the matching of the resonance frequency which depends most on the sphere size and the shear elasticity, μ1, and the width of the response depends on the shear viscosity, μ2. The data point at 400 Hz obtained from the images does not significantly change the curve fitting results if it is included or excluded from the analysis because the curve fitting is less sensitive to viscosity effects at frequencies below resonance as opposed to those frequencies above resonance. A previous study [85] has shown that obtaining this resonance curve and finding parameters for the material properties of the gel is very accurate. We are confident in the result obtained with the images since it matches closely with the curve from Figure 3.6

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and the simulation result. It has also been shown that phase images, modeled by (13), could be used to tell different materials from each other, particularly in imaging of small spheres, which could be used as a model of breast calcifications [76]. The vibrometry example shown in this paper relies on a model of vibration of a sphere in a viscoelastic medium based on a Voigt material. This model requires information about the sphere including density and size and could be extended to breast imaging of calcifications by measurement of the density of a calcification from ultrasound or and X-ray based modality, and the size of the calcification even if not spherical in shape could be used in a model and an equivalent sphere shape could be used to estimate surrounding material properties. This multifrequency approach could be used in other vibrometry applications coupled with modeling for assessment of different organs or systems such as the vasculature. The curve that was extracted from the images shown in Figure 3.8 was crude because only six frequencies were used, but this could be improved by performing subsequent scans by shifting the multifrequency components by a small increment such as 100 Hz. This would provide more points on the resonance curve to make the fit more accurate and precise. In this vibrometry experiment, the velocity of the sphere was measured with a laser vibrometer. Because tissue is opaque except at small depths, another method would have to be used to measure motion such as methods based on the Doppler shift, correlation methods, or phase shift measurements [39, 55, 87]. The vibro-acoustography images in Figure 3.11 show different contrast in the various lesions present in the image. The images at 30, 50, and 60 kHz had the best SNR.

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These three images underscore another subtlety of the multifrequency approach. The frequency response of tissue and other objects may be completely unknown, so imaging at multiple frequencies can provide images with differing SNR and contrast. Acquiring these multiple image datasets allows the user to more efficiently and effectively image and understand the tissue or object under investigation. For inspecting an object without any a priori knowledge of its frequency response, a standard protocol could be introduced that covers different frequencies over a fairly large bandwidth like that detailed in this paper where six frequencies were inspected over a bandwidth of 50 kHz. This will provide for comparison within a given bandwidth and between different patients or samples if the same frequencies are used. This protocol may change based on the application to inspect different frequency bands to obtain the most useful images and information. Multifrequency vibro-acoustography imaging produces images with contrast based on the material properties of the object or tissue and is suitable for medical applications, as most medical imaging methods that are not quantitative. Further research is necessary to solve the inverse problem to quantitatively estimate material properties based on the acoustic emission signal. Figure 3.13 shows results for in vivo breast vibro-acoustography. The calcified fibroadenoma was detected in each of the multifrequency images. The interesting thing is that the contrast varies in the different images. When the MH mechanism is utilized in the 20 kHz image, a shadow image is produced where the surrounding tissue is bright compared to the fibroadenoma. In the other images formed by the ME mechanism, the background tissue is darker than the fibroadenoma except in the 60 kHz image. The

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differences in contrast are not completely understood and could be attributed to reverberation or the frequency response of the tissue. The multifrequency images could also prove useful in artifact reduction in vibroacoustography. Mitri, et al. [97] described a method to remove the ultrasound standing wave artifact caused by continuous wave exposure by adding images that were formed using different ultrasound frequencies while keeping the difference frequency constant. Since multifrequency images are formed with different ultrasound frequencies these images could be used in a similar way by adding them together to reduce this artifact to improve image contrast as well as potentially increasing SNR. Another artifact that is often present is reverberation of acoustic emission inside the object causing patterns of bright and dark regions in the image. The reverberation artifact manifests itself as a low spatial frequency variation in the image. This reverberation pattern is strongly dependent on the temporal frequency of the acoustic emission. Combining images made at different low-frequencies could also reduce this artifact and improve image contrast as shown in Figure 3.12. The contrast was improved in Figure 3.12(b) by the exclusion of the 40 kHz component mostly because the phantom itself seemed to be more reverberant at that frequency and introduced a more uniform background and reduced the contrast of the lesions. Using different combinations of the multifrequency images provides different levels of contrast and image detail. Which images to combine will depend on the object and the low-frequencies used to create the images. Certain images may have better SNR due to being near a resonance of the object and may be better for use in combination with

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other images. Application specific protocols may arise if different combinations are found to improve contrast or detail in images. Using images made at different frequencies or combinations of multifrequency images in a clinical setting may lead the scientist or clinician to visualize and investigate different tissues and objects in a new and insightful manner. If one particular image is interesting or important a certain frequency band may be examined more carefully with subsequent scans with a set of multifrequency images centered around that particular frequency, much like was done in the vibrometry example to examine the resonance. Multispectral image processing techniques such as principal component analysis and classification algorithms could also potentially be used to extract more information from the multifrequency datasets.

3.10 Conclusion

Visualization of objects of different stiffness and tissue composition is vitally important in elasticity imaging. This may allow the clinician to better assess the patient, as well as diagnose and treat that patient. The multifrequency method proposed here can be used for vibro-acoustography or vibrometry imaging applications. The substantial information gain with no increase in scanning time is a compelling advantage of this method over conventional vibro-acoustography. We have shown that using this multifrequency radiation force can provide very rich datasets that can be used to estimate material properties of a tissue-like medium. Also, the use of multiple frequencies provides varying vibro-acoustography image contrast that could be clinically useful for visualization, diagnosis, and treatment planning.

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Chapter 4 Harmonic Motion Detection of a Vibrating Reflective Target Knowledge advances by steps, and not by leaps. Lord Macaulay

4.1 Introduction

The premise of vibro-acoustography is to measure the acoustic emission resulting from radiation force induced vibration. It has been shown that the acoustic emission is proportional to the velocity of the vibrating object [34]. Other ultrasound elasticity imaging methods use the measured displacement after a perturbation to create images of strain, elastic modulus or some other parameter based on a mechanical response [25, 36]. Some applications of vibrometry have used the dynamic ultrasound radiation force excitation such as measurement of propagating waves in tubes and arteries [58, 98], shear wave propagation [99], and monitoring therapeutic ultrasound applications [38, 39]. The measurements reported by Zhang, et al. [58, 98] and Chen, et al. [99] were made using a transducer to produce the dynamic radiation force and the velocity was measured by either a laser vibrometer or a separate transducer operating in a pulse-echo mode to measure the echoes from the moving object. Konofagou, et al. [38, 39] also used two transducers to create the radiation force and track the resulting motion. It would be desirable to measure the motion using a single transducer and with high accuracy and precision. Some methods that use radiation force use a single transducer to excite the tissue and measure the resulting motion including acoustic radiation force impulse (ARFI) imaging [36] and supersonic shear imaging (SSI) [40]. However, neither of these 77

methods creates a harmonic radiation force, so to date harmonic radiation force excitation and motion detection with a single transducer has not been reported. Motion detection would provide complementary information to the acoustic emission information obtained during vibro-acoustic imaging. Information about the velocity or displacement could be used in conjunction with models to extract information about material properties of the object or tissue under inspection. Vibration phase information can also be used for differentiating between different materials as was demonstrated in Chapter 2. Another area in which the vibration phase can be used is in shear wave speed measurement. Chen, et al. [99] demonstrated that shear wave speed, cs, can be measured by measuring the phase difference in the propagating shear wave at two different locations

cs =

ωs Δφ Δr

,

(4.1)

where Δφ = φ(r1) – φ(r2) is the difference in phase at two different locations, r1 and r2 and

Δr = r1 – r2 is the distance between measurement points. The shear wave speed measured at different frequencies can be used to find the shear elasticity and viscosity of the tissue using (1.9). This chapter will present the theory and signal processing methods to perform motion detection of a harmonically vibrating target. A model of a vibrating reflective target will be used to validate theory and optimize the parameters involved in the motion detection scheme. The method is applicable for vibrating reflective targets that may be found in nondestructive testing or use in tracking motion of an artery wall which is often a reflective target compared to surrounding tissue.

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4.2 Theory

Previous studies have examined the Doppler spectrum of backscattered echoes produced by a harmonically vibrating target [100-102]. Huang, et al. describes that if the scatterer is vibrating with a velocity much lower than the wave speed of the backscattered waves and the vibration frequency much lower than the interrogating acoustic waves, then the spectrum of the detected motion can be modeled as a frequency modulated (FM) spectrum [101]. Ultrasonic waves at frequency ωf are used to interrogate scatterers vibrating with frequency ωs, where ωs << ωf. According to the notation introduced by Zheng, et al., the subscript “f” will refer to fast time corresponding to ultrasonic time scale and the subscript “s” will refer to slow time corresponding to the vibration time scale [87]. The displacement and the velocity of the vibrating scatterer is modeled as D ( t ) = D0 sin (ωs t + φs ) ,

(4.2)

v ( t ) = v0 sin (ωs t + φs ) ,

(4.3)

where D0 is the displacement amplitude, φs is the vibration phase, and v0 is the velocity amplitude (v0 = D0ωs). If an ultrasonic pulse at frequency ωf is used, the echo from the vibrating scatterer can be modeled as [87]

r ( t f , ts ) = A ( t f , ts ) cos (ω f t f + φ f + β sin (ωs ts + φs ) ) ,

(4.4)

where tf is fast time, ts is slow time, A is the echo amplitude, φf is the initial phase of the ultrasound signal, and β is defined as

β=

2v0ω f cos (θ )

ωs c

,

(4.5)

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where θ is the Doppler angle, which will always be assumed to be 0°, and c is the longitudinal sound speed of the medium. The parameters that are to be determined by measurements are D0 and φs but both variables are embedded in the phase of the ultrasound echoes. Many groups have approached this problem from the standpoint that a phase shift produces a time shift between consecutive echoes. Therefore, if the time shift can be estimated, the displacement amplitude and vibration phase could also be estimated. Two early methods used a two-dimensional autocorrelation approach [50, 51]. The one striking difference between the algorithm proposed by Loupas, et al. and Kasai,

et al. is that the Loupas method corrects for the mean ultrasound echo center frequency whereas the Kasai method assumes the transmitted ultrasound center frequency. For applications where displacement occurs as a result of impulsive radiation force excitation or static compression, cross-correlation combined with interpolation and spline estimation techniques have been used [26, 49, 52, 53, 103-105]. The interpolation and especially the spline based methods can be computationally expensive. In the method proposed by Zheng, et al. quadrature demodulation is used to obtain the signal y(ts) = β sin(ωsts + φs). However, for the short echo signals the low-pass filtering step in the quadrature demodulation can introduce artifacts because of transient effects of the filters. A different method was used in these studies to obtain the y(ts) signal. Hasegawa and Kanai have proposed a cross-spectrum method that corrects for the center frequency of the ultrasonic echo [55]. If we denote the nth and (n+1)-th received

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echoes as r(n) and r(n+1) and their corresponding frequency spectrums as Rn(f) and

Rn+1(f) then we can calculate the cross-spectrum as Rn* ( f ) Rn +1 ( f ) = Rn ( f ) Rn +1 ( f ) e

j Δθ n ( f )

,

(4.6)

where Δθn(f) is the phase shift between the two echoes and * represents complex conjugation. The motion of the vibrating scatterers can then be extracted by performing this cross-spectral analysis for all echoes after each echo was windowed using a Hann window. The velocity can be estimated by vn =

c ⋅ Δθ n ( f ) 2ω f Tprf

,

(4.7)

f = f0

where the phase shift is evaluated at the center frequency, f0, of the cross-spectrum, and

Tprf is the pulse repetition period of the pulse-echo interrogation. The center frequency of the echo is estimated by finding the frequency at which the maximum occurs in the crossspectrum. The correction for the center frequency is performed because attenuation can downshift the center frequency of the received echo from the transmitted pulse. Hasegawa showed that if no correction is performed for the center frequency the results can be biased [55]. Since the vibration is harmonic at ωs, the displacement signal can be obtained by Dn = vn/ωs. Once Dn has been found, a Kalman filter designed to extract amplitude and phase of the displacement is used [54]. The Kalman filter only requires the vibration frequency,

ωs, as an input. The Kalman filter is a state space based filter that recursively estimates the state variables using a least mean squared error criteria [54]. The Kalman filter is implemented in a digital form and is computationally efficient.

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4.3 Parameterized Model

There are many parameters that enter into modeling the motion detection of a vibrating reflective target. First, it is assumed that there will be only one scatterer that acts similar to a point reflector. The parameters that are expected to most affect performance of this method are the displacement amplitude, D0, signal-to-noise ratio, SNR, of the ultrasound echoes, the number of cycles of vibration used, Nc, and the number of points sampled per vibration cycle, Np. Other parameters that may affect performance such as bandwidth, gate length, and sampling frequency will also be explored. Displacement amplitude is determined by the radiation force amplitude. Since radiation force is proportional to the ultrasound intensity, this parameter is limited in practice because of bioeffect concerns. The intensities that lie within the limits of the Food and Drug Administration produce small displacements, <10 μm, in tissue. Therefore, the lower limit of displacement amplitude necessary to obtain good results needs to be determined. The SNR of the ultrasound echoes will primarily be determined by scatterer backscatter strength and the electronic noise introduced in the pulse-echo system. For a reflective target the SNR should be 30 dB or higher. In a scattering medium, it could be lower. The number of vibration cycles sampled over will directly affect acquisition time for a measurement, T = Nc/fv, where T is the acquisition time and fv is the vibration frequency. For static point measurements, this may not have much impact, but if this

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method is employed for imaging, the acquisition time for the image will be governed by

Nc. Also, increasing Nc will increase the processing time for the displacement estimate. The number of points sampled per vibration cycle, Np, will reflect on the pulse repetition frequency, fprf, used in the experiment as Np = fprf/fv. To satisfy the Shannon sampling theorem, Np ≥ 3. The value of fprf is limited by the distance of the vibrating scatterer from the transducer, where fprf,m = c/2z where c is the sound speed of the medium and z is the axial depth of the vibrating scatterer from the transducer. Increasing the value of Np will also increase processing time for the displacement estimate. Since Nc and Np are dimensionless, the results of this model can be extended for any value of fv and fprf within the limits described above. For this parameter study, D0 will vary from 100-10,000 nm, SNR will vary from 0-60 dB, Nc will vary from 3-20, and Np will vary from 5-30. Default parameters for the study are given in Table 4.1. A default value of D0 = 1000 nm was chosen because this is a typical value seen in experimentation. In vibrometry experiments, low frequency vibration on the order of a few hundred Hertz is used so fv = 200 Hz. The values of Nc = 5 and Np = 20 give values of T = 25 ms and fprf = 4.0 kHz. The interrogating pulses have been windowed with a Gaussian window to reflect the bandwidth of the transducer, BW. The gate length, lg, is the spatial extent used for comparison of consecutive echoes and phase shift calculation. The sampling frequency, Fs, is the fast time sampling frequency for the ultrasound echoes.

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Table 4.1 – Parameter Study Default Parameters D0

φs

Nc Np fv ff Fs BW lg c N

Displacement amplitude Vibration phase Cycles of vibration Sampled points per vibration cycle Vibration frequency Ultrasound frequency Sampling frequency Transducer bandwidth Gate length Speed of sound Iterations

1000 nm 0° 5 20 200 Hz 9.0 MHz 100 MHz 6.5 % 1.0 mm 1480 m/s 1000

To evaluate the performance of the method, we will introduce two metrics, bias and jitter. These metrics originate from estimating time delays. The bias, xB, is the mean of the error, and the jitter, σJ, is the standard deviation of the error and they are calculated as

xB =

σJ =

1 N

N

∑( x n =1

n

− xT ) ,

1 N 2 ( xn − xT − x ) , ∑ N − 1 n =1

(4.7)

(4.8)

where N is the number of data samples, xT is the true value, and x is the mean of the data samples [49]. Minimal bias and jitter is desired. The bias reflects on the accuracy of the motion detection, and the jitter reflects on the precision of the motion detection. Bias and jitter measures will be evaluated on both displacement amplitude and phase for 1000 iterations with different initial conditions for the noise added to adjust the SNR. The added noise was normally distributed. Figure 4.1 shows sample data for D0 = 5000 nm, φs = 0°, Nc = 5, Np = 20, SNR = 20 dB, f0 = 9.0 MHz, BW = 6.5 %, lg = 1.0 mm, and Fs = 100 MHz.

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Figure 4.1—Sample simulated echo data for vibration of a scatterer with D0 = 5000 nm, φs = 0°, Nc = 5, Np = 20, and SNR = 20 dB.

The displacement signal after the phase estimation is shown in Figure 4.2. The dashed line represents the true displacement signal. The estimates for the vibration amplitude and phase for this case were D0 = 4990.2 nm and φs = 0.054° which are very close to the true values.

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Figure 4.2—Displacement signal for data in Figure 4.1. The red curve with the data points marked by the open circles is the displacement signal estimated from the data and the blue dashed curve is the true displacement signal. The estimated vibration amplitude and phase are D0 = 4990.2 nm and φs = 0.054° while the true values are D0 = 5000 nm and φs = 0°.

4.4 Parameterized Model Results

Figures 4.3 and 4.4 show the displacement amplitude and phase results, respectively, for the default conditions in Table 4.1. Each data point represents the mean of the 1000 iterations and the error bars represent one standard deviation. The dashed lines in these figures show the target values.

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Figure 4.3—Displacement amplitude results for default parameters in Table 4.1. Each data point represents the mean of the 1000 iterations and the error bars represent one standard deviation. The dashed line represents the target value of D0 = 1000 nm.

Figure 4.4—Displacement phase results for default parameters in Table 4.1. Each data point represents the mean of the 1000 iterations and the error bars represent one standard deviation. The dashed line represents the target value of φs = 0°.

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Figure 4.5 shows the displacement amplitude and phase bias and jitter while varying D0. As SNR of the ultrasonic echoes increases the bias and jitter stabilize to certain value. For the amplitude and phase jitter, increasing SNR yields lower jitter values. The amplitude bias increases as D0 increases, but as a percent of D0 it remains relatively low. The amplitude jitter does not seem to change significantly with increasing values of D0. As D0 increases, the phase bias and jitter decrease. The phase bias is very low for all cases except D0 = 100 nm. The phase jitter requires higher values of D0 to decrease to levels comparable to the bias.

Figure 4.5—Displacement amplitude and phase bias and jitter for variation of D0 = 100 ({), 500 (), 1000 (Δ), 5000 (◊), and 10000 (∇) nm. (a) Amplitude bias, (b) Amplitude jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

Figure 4.6 shows the results of the amplitude and phase bias and jitter for varying Nc. The amplitude bias does improve with increasing Nc but not to a significant degree. Increasing Nc does serve to decrease the amplitude bias at low SNR values but does not

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have a large effect at higher SNR. The phase bias is consistently small for all values of Nc, while the phase jitter does decrease with increasing values of Nc.

Figure 4.6—Displacement amplitude and phase bias and jitter for variation of Nc = 3 ({), 5 (), 10 (Δ), 15 (◊), and 20 (∇).(a) Amplitude bias, (b) Amplitude jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

Figure 4.7 shows the results for amplitude and phase bias and jitter for varying Np. The value of Np has a significant effect on decreasing the amplitude bias, and has a less dramatic effect on decreasing the amplitude jitter. The phase bias is consistently low, but increasing Np does serve to decrease the phase jitter. Figure 4.8 shows the results of varying the gate length on the amplitude and phase bias and jitter. The gate length was set to 0.5 and 1.0 mm. This corresponds to 3λ and 6λ of 9.0 MHz ultrasound waves. Increasing gate length does not drastically affect either the amplitude or phase bias. The amplitude and phase jitter is reduced by using a longer gate length.

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Figure 4.9 provides the results of the amplitude and phase bias and jitter for varying the sampling frequency of the ultrasound echoes, Fs. Increasing Fs had similar effects as using a longer gate length. Amplitude and phase bias was largely unaffected, but amplitude and phase jitter were decreased by an increased value of Fs.

Figure 4.7—Displacement amplitude and phase bias and jitter for variation of Np = 5 ({), 10 (), 15 (Δ), 20 (◊), 25 (∇), and 30 (*).(a) Amplitude bias, (b) Amplitude jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 4.8—Displacement amplitude and phase bias and jitter for variation of gate length, lg = 0.5 ({), 1.0 () mm. (a) Amplitude bias, (b) Amplitude jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

Figure 4.9—Displacement amplitude and phase bias and jitter for variation of sampling frequency of the ultrasound echoes, Fs = 50 ({), 100 () MHz. (a) Amplitude bias, (b) Amplitude jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 4.10 gives the results of varying the transducer bandwidth, BW, on the amplitude and phase bias and jitter. Increasing bandwidth has no effect on the results.

Figure 4.10—Displacement amplitude and phase bias and jitter for variation of transducer bandwidth, BW = 6.5 ({) and 20.0 () %. (a) Amplitude bias, (b) Amplitude jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

4.5 Discussion

When varying the displacement amplitude it was found that increasing amplitude yielded lower jitter values and served to decrease phase bias. The absolute value of the amplitude bias increased as D0 increased, but if the bias was considered as a fraction of D0, then the error can be considered small. The bias and jitter at D0 = 100 nm are large enough to cause considerable error for that measurement to be trusted. The phase bias and jitter are only considerable for SNR < 20 and D0 < 500 nm. Increasing the value of Nc minimizes the amplitude and phase bias and jitter almost universally. This occurs because the Kalman filter acts as an averaging filter and

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as the results show, the extraction of the correct values improves as the filter is given more data to work with because of its recursive nature. By increasing Np, the results show a decrease in errors; however, for a given increase in Np the bias or jitter does not decrease as much as in the case of increasing Nc. By sampling more points per vibration cycle, the harmonic function becomes more sinusoidal in shape and the Kalman filter can operate on the improved data set to extract the amplitude and phase with better accuracy and precision. In almost all results, for a given value of SNR, and varying any parameter besides transducer bandwidth, the amplitude or phase jitter had larger values than the corresponding amplitude or phase bias. The results from this parametric model study are useful for serving to optimize the implementation of this method in an experimental setting. The results show that to decrease displacement bias, D0 should decrease and Nc, Np, SNR, Fs, and lg should increase. To minimize displacement jitter and the phase bias and jitter, all parameters should be increased. However, in an experimental setting, maximizing all of the parameters will have to be weighed against tradeoffs associated with each parameter. To maximize D0, the maximum safe ultrasonic intensity should be used to achieve maximal radiation force. The increase of Nc will need to be weighed against time constraints of the experiment. If point measurements are being performed, then it may worthwhile to use 10-20 cycles of vibration, but in an imaging situation for a biological subject, 5 cycles may be sufficient. Increasing the value of Np will not affect acquisition time but will increase data size and processing time. However, increasing Np above 15 does not change the results

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significantly. Increasing the gate length will require sampling over more of the echoes in fast time and will affect the data size. Increasing sampling frequency will also increase the size of the data taken. For a reflective target, transducer bandwidth had no effect on the results. The dimensionless nature of Nc and Np allow this analysis to be extended to different values of fv. If a value of fv is chosen, based on desired specifications for allowable bias and jitter, proper values of Nc and Np can be chosen. The use of the method proposed by Hasegawa and Kanai for phase shift estimation with cross-spectral analysis is fast and does not suffer from finding false peaks in a cross-correlation function. This method also avoids the errors introduced by low-pass filter transient effects in quadrature demodulation. The method also corrects for the center frequency of the ultrasound echo that may change due to attenuation processes during wave propagation. This correction reduces potential bias errors. The Kalman filter based on harmonic vibration at a known frequency, ωs, is very powerful and provides a robust estimate even in the face of high noise as shown in Figure 4.2. Another advantage of the Kalman filter is that it could be used to process multifrequency vibration data by processing the same data with different values of ωs used as an input. These simulations on a reflective target provide a model for motion detection in applications such detecting wave motion on vessels [58], for phase aberration correction methods based on tissue vibration[106], nondestructive evaluation, and model-based calculations for objects or tissue regions that are ultrasonically reflective.

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This model also provides a method to analyze the vibration phase and the associated error in measurements when the phase measurements are used to estimate shear wave speed as discussed in Chapter 1. In the next chapter, this technique will be extended to experimental practice and its performance will be assessed and compared with the results in this chapter.

4.6 Conclusion

The theory and model for motion detection on a vibrating reflective target were presented. Results from a parametric study show that the method works well and these results can be used for optimization of the method. It was shown that by maximizing displacement amplitude, SNR, Nc, and Np will provide results that have low bias and jitter in estimation of the vibration amplitude and phase.

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Chapter 5 Harmonic Pulsed Excitation and Experimental Motion Detection of a Vibrating Reflective Target When he was six, he believed that the moon overhead followed him. By nine, he deciphered the illusion, trading magic for fact, no trade-backs. So this is what it's like to be an adult? If he only knew now what he knew then. Pearl Jam

5.1 Introduction

In the previous chapter the theory and algorithm for motion detection of a vibrating reflective scatterer was introduced. Also, a parameterized model was used to explore how different parameters affected the performance of this method. In this chapter, a new excitation method that uses repetitive pulses of ultrasound to generate radiation force is presented. Also, experimental results of motion detection on a reflective target will be presented.

5.2 Background

The motion of a harmonically vibrating scatterer causes a Doppler shift in interrogating ultrasonic waves [101]. The Doppler spectrum can be modeled as a puretone frequency modulated process. Many groups have used this model to estimate motion with knowledge about the measured spectrum [100-102, 107, 108] using continuous wave (CW) ultrasound. However, it was noted that nonlinear propagation of CW ultrasound could produce effects that mimic motion in the Doppler spectrum and confuse the results [102, 107, 108]. Because the nonlinear propagation produces effects similar to 96

harmonic motion, the frequency range for which motion detection is possible is limited. This problem can largely be alleviated by using pulse-echo ultrasound and comparison between consecutive echoes to detect the motion of harmonically vibrating scatterers. Ultrasound radiation force can be characterized as either static or dynamic. A static radiation force can be generated with continuous wave ultrasound. In the acoustic radiation force impulse imaging method (ARFI), a quasi-static radiation force can be generated using toneburst of ultrasound [36] where the tissue is pushed in a static manner and then the tissue relaxes. Dynamic radiation force can be produced by using an amplitude modulated ultrasound wave or by the interaction of two ultrasound beams separated by a small frequency difference [34, 57]. Another method to produce dynamic radiation force is to use gated tonebursts of amplitude modulated ultrasound [109]. To perform motion detection in the ARFI method, the radiation force “push” pulse is transmitted and then the scanner switches to B-mode imaging using pulse-echo ultrasound [36]. The raw radiofrequency (RF) data is then used to perform crosscorrelation to estimate the displacement of the tissue. The push pulse is then repeated at other positions and the motion tracking ensues from the corresponding radiation force application. The pushing pulse never occurs while motion tracking is being performed. Localized harmonic motion imaging, proposed by Konofagou, et al., uses either amplitude modulated ultrasound or the interaction of two ultrasound beams to produce dynamic radiation force and a separate transducer confocal with the transducer(s) producing the radiation force is used to perform pulse-echo ultrasound for motion detection [38]. Since motion detection is performed simultaneously with radiation force excitation, the radiation force is produced using ultrasound at 2.27 MHz and the pulse-

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echo motion detection is performed at 1.1 MHz so that signals from the radiation force could be separated from those used for motion tracking. Michishita, et al., proposed using gated tonebursts of amplitude modulated ultrasound to produce dynamic radiation force. In between these excitation tonebursts, pulse-echo ultrasound was performed to detect the induced motion. In their study, separate 5.0 MHz transducers were used for radiation force excitation and motion detection [109].

5.3 Harmonic Pulsed Excitation

Harmonic pulsed excitation (HPE) is a new method that combines attributes of ARFI and the method proposed by Michishita and his colleagues. In this method, gated tonebursts of ultrasound are applied in a repetitive manner to produce a dynamic radiation force. In between the tonebursts, pulse-echo ultrasound is performed to obtain radio-frequency (RF) data to be used for motion detection. This method does not require amplitude modulation of the ultrasound which eliminates the need for a modulating signal. Another, attribute is that radiation force excitation and motion detection can be performed with the same transducer as in the ARFI method. A timing diagram of the method is shown in Figure 5.1. The transmission of the ultrasound tonebursts for radiation force excitation is shown in Figure 5.1(a) and the radiation force waveform is shown in Figure 5.1(b). The gates for the transmitted and received pulses for motion tracking are Figures 5.1(c)-(d). The excitation tonebursts have length Tb and are repeated at a rate of fr, where fr = 1/Tr. The motion detection pulses are

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transmitted with a pulse repetition frequency of fprf where fprf = 1/Tprf. There may also be a delay, td, for the onset of the transmission of the motion detection pulses.

Tr

Tb (a)

(b) td

Tprf (c)

(d)

Figure 5.1—Timing diagram for harmonic pulsed excitation and pulses used for motion tracking. (a) Ultrasound tonebursts with length Tb and repetition period of Tr, (b) Radiation force produced by ultrasound tonebursts, (c) Transmission gate for ultrasound tracking pulses with an onset delay of td and repetition period of Tprf, (d) Reception gate for echoes of transmitted tracking pulses.

The radiation force function for this type of excitation is proportional to a lowpass filtered version of the square of the ultrasound pressure, so the function becomes a rectified rectangular pulse train as shown in Figure 5.2 for fr = 200 Hz and Tb = 200 μs.

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Figure 5.2—Harmonic pulsed excitation transmission with fr = 200 Hz and Tb = 200 μs, (a) Ultrasound pressure amplitude, (b) Radiation force function.

Using Bracewell’s conventions, we can describe this radiation force function, f(t), as the convolution of an impulse train, III(t), with a time-offset rectangle function, II(t), [110] ⎛ t − Tb / 2 ⎞ f ( t ) = f r III ( f r t ) ⊗ a II ⎜ ⎟, ⎝ Tb ⎠

(5.1)

where a is the radiation force amplitude, which for this derivation will be unity and ⊗ indicates convolution. The radiation force function can be simplified as F ( f ) = af rTb

∞

∑e

n =−∞

− iπ Tb nf r

sinc (Tb nf r ) δ ( f − nf r ) ,

(5.2)

and a full derivation of the radiation force function is found in Appendix 2. A few things should be noted about the spectrum described in (5.2). First, there are components at all multiples of fr. Therefore, this method is a multifrequency

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excitation method. The velocity or displacement at vibration frequencies fv = nfr can be analyzed with the method detailed in Chapter 4 by processing the data with the Kalman filter and choosing an appropriate vibration frequency input. Secondly, the magnitude of those components is modulated by a sinc shaped envelope with zeros at frequencies that are multiples of 1/Tb. The magnitude spectrum for the radiation force function in Figure 5.2(b) is shown in Figure 5.3.

Figure 5.3—Magnitude spectrum for HPE transmission with fr = 200 Hz and Tb = 200 μs.

5.3.1 Harmonic Pulsed Excitation Experiment

To verify the shape of the radiation force function and its spectrum, an experiment was performed. The ultrasonic pressure from the transducer was measured from a twoelement confocal 3.0 MHz transducer with outer diameter of 45 mm and a focal length of 70 mm. The two elements were driven with the same signal. The pressure was measured using a needle hydrophone (NTR Systems, Seattle, WA) in a large water tank. The needle hydrophone was placed at the focus of the transducer for these pressure measurements. 101

A HPE sequence was used with fr = 500 Hz and Tb = 200 μs. The radiation force is proportional to the intensity of the ultrasound beam which is proportional to the square of the pressure. To find the low-frequency radiation force, the squared pressure waveform was low-pass filtered and its frequency spectrum was calculated. These waveforms are shown in Figure 5.4. The spectrum of the measured radiation force function is compared with a theoretical calculation using the given parameters, fr and Tb. Figure 5.5 shows the differences in the spectrum of using tonebursts with Tb = 100 μs. The two spectra are normalized with respect to the maximum value in the spectrum for the Tb = 200 μs.

Figure 5.4—Measured pressure and radiation force for HPE with fr = 500 Hz and Tb = 200 μs, (a) Pressure amplitude, (b) Radiation force function.

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Figure 5.5—Measured and calculated normalized magnitude spectrum of radiation force function for HPE with fr = 500 Hz μs, (a) Tb = 200 μs, normalized to maximum value, (b) Tb = 100 μs, normalized to maximum value measurements for Tb = 200 μs.

5.3.2 Harmonic Pulsed Excitation Experimental Results

The results in Figure 5.4 show that there is some reverberation between the transducer and needle hydrophone, extending the toneburst length to be longer than specified. However, the amplitude is reduced during the reverberation. There is the presence of a standing wave that increases the amplitude of the pressure slightly for the second 100 μs of the toneburst. Figures 5.4 and 5.5 show that the measured results match very well with the theoretical calculations of the radiation force function. Figure 5.5 shows that when the toneburst length is decreased by a factor of two, the magnitude of the radiation force function is reduced by a factor of two as (5.4) predicts. Also, the zero positions of the sinc function occur at different frequency positions. For Tb = 200 μs, the

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zeros occur at multiples of 5.0 kHz, and in the case of Tb = 100 μs, the zeros occur at multiples of 10.0 kHz.

5.4 Motion Detection Experiment

To assess the performance of the HPE and motion detection methods, an experiment was performed. The target for this experiment was a 440-C stainless steel sphere of diameter 1.59 mm embedded in a gelatin phantom made using 300 Bloom gelatin powder (Sigma-Aldrich, St. Louis, MO) with a concentration of 10% by volume. A preservative of potassium sorbate (Sigma-Aldrich, St. Louis, MO) was also added with a concentration of 10 g/L. The first part of the experiment included measuring the frequency response of the sphere in the gelatin phantom. To perform this measurement, two signal generators (33120A, Agilent, Palo Alto, CA) were used to create continuous wave signals of frequency f0 and f0 + Δf where f0 = 3.0 MHz and Δf was varied in increments of 50 Hz over the range 50-1200 Hz. These signals were added together and amplified and applied to the transducer described in the HPE experiment. The two ultrasound signals were also mixed, which serves to multiply the signals, and lowpass filtered (fc = 100 kHz) to provide a reference signal of Δf for a lock-in amplifier (Signal Recovery, Oak Ridge, TN). The sphere was placed in the focal region of the transducer. The resulting motion was detected with a Doppler laser vibrometer (Polytec, Waldbronn, Germany) and the lock-in amplifier. A block diagram of this experimental setup is shown in Figure 5.6.

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100 kHz Lowpass Filter

Mixer

f0

~

~

Δf

Lock-in Amplifier

Doppler Laser Vibrometer

f0 + Δf

Σ

3.0 MHz Transducer 40 dB Amplifier (a) Stainless Steel Sphere

3.0 MHz Transducer

Gelatin Phantom

Doppler Laser Vibrometer

(b) Figure 5.6—Experimental setup for sphere frequency response measurements and motion detection with Doppler laser vibrometer and ultrasound method, (a) Block diagram for measurement of sphere frequency response. Two signal generators produce CW ultrasound signals at f0 and f0 + Δf which are summed together, amplified, and used to drive the transducer. The CW signals are used as inputs to a mixer, and the resulting signal is lowpass filtered and the Δf signal is used as a reference for the lock-in amplifier to compare against the signal from the Doppler laser vibrometer, (b) Experimental setup for excitation and measurement of motion of a stainless steel sphere embedded in a gelatin phantom. The 3.0 MHz transducer creates the radiation force and in a later experiment will also be used to track the motion ultrasonically. The Doppler laser vibrometer provides a calibrated measurement of the sphere’s motion.

The HPE method was initiated using a trigger signal from a signal generator producing one cycle of a TTL pulse. This master trigger signal initiated a signal generator to produce a 20 cycle rectangular pulse train with a frequency of fr = 100 Hz. This pulse train triggered a signal generator that produced a toneburst of length Tb = 50 or 100 μs. The voltage amplitude of this toneburst was varied to change the radiation force

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amplitude. The master trigger was also applied to another signal generator (33250A, Agilent, Palo Alto, CA) to produce a rectangular pulse train with a frequency of fprf. The value of the fprf was varied during the experiment to assess its effects on the results. For Tb = 50 μs, fprf = 2.0, 3.0, 4.0, 5.0, and 6.0 kHz, and for Tb = 100 μs, fprf = 2.0, 2.5, 3.0, 4.0, and 5.0 kHz. This pulse train was used as a trigger input to the analog-to-digital converter (ADC) board in a personal computer. The ADC produced a trigger signal for every pulse in the pulse train and this triggered a signal generator to generate a three cycle pulse at 9.0 MHz. The radiation force toneburst and tracking pulse were added together using a hybrid junction (M/A-COM, Inc., Lowell, MA) and this signal was amplified with a 40 dB amplifier. This signal passed through a diode bridge to eliminate low-level noise from the amplifier and through a matching transformer to the transducer. The echoes were received and filtered with a 3.0 MHz notch filter with a 50% bandwidth and a 9.0 MHz bandpass filter with 33% bandwidth. The echoes then passed through a transmit/receive (T/R) switch, amplified by a broadband amplifier and finally filtered by another 9.0 MHz bandpass filter before being digitized at a sampling frequency of 100 MHz. A block diagram of the experimental setup of HPE and motion detection is shown in Figure 5.7.

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9.0 MHz BPF

ff

40 dB Amplifier

fr

fprf

~

~

f0

A/D

3.0 MHz Transducer

Σ

3.0 MHz Notch Filter

T/R Switch 9.0 MHz BPF

40 dB Amplifier Figure 5.7—Experimental setup for harmonic pulsed excitation and motion tracking. For the excitation, a pulse train with frequency fr = 100 Hz is initiated, and each positive pulse triggers a toneburst of ultrasound at f0 = 3.0 MHz. For the tracking a pulse train at fprf is initiated with specified time delay, td, and each positive pulse triggers a three cycle pulse at ff = 9.0 MHz to be transmitted. The excitation and tracking signals are summed together and amplified before being sent to the 3.0 MHz transducer. For tracking, the gated echoes are filtered with a notch filter centered at 3.0 MHz and a bandpass filter centered at 9.0 MHz before passing through a transmit/receive (T/R) switch. The signal is amplified and filtered again with a bandpass filter centered at 9.0 MHz before being sent to the digitizer (A/D).

The pulse-echo motion detection is performed at 9.0 MHz to separate the motion detection signals from the excitation in frequency. This is possible because the transducer is air-backed and therefore responds well at its third harmonic. The frequency response, shown in Figure 5.8, of the transducer was measured using the needle hydrophone after excitation from a broadband pulser (5050PR, Panametrics, Waltham, MA). With reference to the peak at 3.0 MHz, the 9.0 MHz frequency component is only attenuated by 16 dB. This provides enough sensitivity to perform the pulse-echo measurements. Also, it is advantageous to perform the motion detection using a higher ultrasound frequency because small displacements can be detected without interpolation or other computationally expensive signal processing. Another group has reported performing a

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similar technique using a phased array and performed radiation force excitation at 1.1 MHz and tracked the motion near the fifth harmonic (4.86 MHz) [111].

Figure 5.8—Frequency response of 3.0 MHz transducer excited by broadband pulse.

A representative sample of the echo data is shown in Figure 5.9. The vibration signal derived from the data in Figure 5.9 is shown in Figure 5.10. It is observed that the displacement response has a very short rise time, the sphere oscillates and the amplitude decays to rest until another excitation pulse is applied. Also, there is a 180° change between the measurements made by the laser and the ultrasound method because the phase is being measured on opposite sides of the sphere. This relationship will be used to assess phase bias. For each toneburst length and each voltage level for the excitation toneburst, the laser vibrometer signal was digitized at sampling frequency Fs = 100 kHz preceded by an anti-aliasing filter with a cutoff frequency of 20 kHz. This measurement was performed

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because in the experiment the laser vibrometer signal will be sampled at the fprf used for the ultrasound based measurements. At these values of fprf, aliasing may occur in the measurements, so the measurements with a sampling frequency of 100 kHz insures minimal to no aliasing. It was found through experimentation that a bandpass filter implemented before the Kalman filter was beneficial in obtaining better results.

Figure 5.9—Representative received echo data from the stainless steel sphere for HPE with fr = 100 Hz and Tb = 100 μs.

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Figure 5.10—Velocity of sphere measured by Doppler laser vibrometer and ultrasound based motion detection, (a) Velocity measured by laser vibrometer, (b) Velocity measured with ultrasound based motion detection.

To process, the data a windowed version of the displacement signal is used as the input to the Kalman filter. However, when a rectangular window is used, the starting sample of the window can cause large amounts of variation in the answer. Therefore, a Hann window has been employed to window the processing window. The use of the Hann window decreases the strength of the signal by a factor of two as given by the equation for a Hann window, ⎧1 ⎡ ⎛ 2π n ⎞ ⎤ ⎪ ⎢1 + cos ⎜ ⎟⎥ , w [ n] = ⎨ 2 ⎣ ⎝ 2M + 1 ⎠ ⎦ ⎪ 0 , ⎩

−M ≤n≤ M

.

(5.3)

otherwise

As a result the final displacement result acquired after the Kalman filter is multiplied by two. An example of the variation of the results with rectangular and Hann windowing is shown in Figure 5.11 for Tb = 100 μs, fv = 100 Hz, and fprf = 2.0 kHz.

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Figure 5.11—Influence of windowing of displacement signal and the starting sample of the window. The results for the laser are shown in red squares and the ultrasound based results are shown with blue circles. The black dashed line indicates the measurement made by the laser with Fs = 100 kHz. The rectangular windowed results are shown by solid lines, and the Hann windowed results are shown with dash lines.

5.4.1 Parameter Analysis

To assess the effects of different parameters on the results, different variations of experimental parameters were performed. Results for the two different values of Tb will be shown and compared. Since different values of applied voltage for the excitation toneburst were used, the results can be compared versus the radiation force applied. For the purposes of reporting the results, the radiation force will be normalized based on the maximum force produced by the highest voltage setting used and denoted as F0. For each value of Tb, the value of fprf was varied to explore the differences in the results. Also, for each value of Tb, the value of Ts, the length of the temporal window used for analysis in slow time was varied, such that the product NcNp was constant where Nc is the number of cycles analyzed and Np is the number of samples per vibration cycle. For example if Ts =

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50 ms, fv = 100 Hz, fprf = 4.0 kHz, then Nc = 5 and Np = 40 and NcNp = 200. If fv is increased to 200 Hz, and all other values are not changed, Np = 20 but Nc = 10 because twice as many cycles of vibration will occur in the same slow time window and NcNp = 200. With this understanding of the parameter analysis, we can compare the results from the experiment with the simulation results in Chapter 4. For the parameter analysis, the default values for analysis are F0 = 1, fv = 200 Hz, fprf = 4.0 kHz.

5.5 Motion Detection Experimental Results

The displacement frequency response of the sphere is shown in Figure 5.12. The response shows a resonance at 300 Hz, and exhibits a bandpass nature as the displacement magnitude decreases with higher frequencies. Figure 5.13 compares the ultrasound based displacement versus the measured laser displacement for Tb = 50, 100 μs. Regression lines were calculated for each case and the slopes were 0.9808 and 1.0021 for Tb = 50, 100 μs, respectively. These slopes indicate good correlation between the values measured with the ultrasound method and the gold standard provided by the laser vibrometer. The regression lines appear to have a knee only because of the logarithmic plotting scales. At around the 100 nm, the data points start to deviate from the regression line. This represents a lower threshold for accurate measurement. Sample data plotted versus normalized force for the default cases of analysis are shown in Figure 5.14, where the data points are the average of five measurements and the error bars represent one standard deviation. The error bars in the phase plots only increase for low values of force indicating a loss in the ability to make repeatable

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measurements. This provides a good metric to assess the lower threshold of accurate detectability. The phase estimates had to be corrected for a frequency dependent phase shift because of the constant time delay associated with wave propagation between the transducer and sphere.

Figure 5.12—Displacement frequency response of 1.59 mm diameter stainless steel sphere in gelatin phantom.

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Figure 5.13—Comparison of displacement amplitude measured by laser vibrometer and ultrasound based detection for fv = 200 Hz. Each data point represents the average of five measurements. (a) Comparison for Tb = 50 μs. The blue regression line has equation y = 0.9808x – 5.3618 with R2 = 0.9998, (b) Comparison for Tb = 100 μs. The blue regression line has equation y = 1.0021x – 12.7432, R2 = 0.9989.

Figure 5.14—Sample data measured using ultrasound method. Displacement and phase data are plotted versus normalized force, (a) Displacement, Tb = 50 μs, (b) Phase, Tb = 50 μs, (c) Displacement, Tb = 100 μs, (d) Phase, Tb = 100 μs.

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Figures 5.15 and 5.16 show the displacement and phase bias and jitter for Tb = 50, 100 μs, respectively. The results are plotted versus measured laser displacement for fv = 100, 200, 300, 400 Hz. The results show that displacement bias decreases in measurements with large displacement amplitudes. The displacement jitter does not change significantly for Tb = 50 μs but decreases as D0 increases for Tb = 100 μs. The displacement jitter is on the same order of the displacement bias across all values of measured displacement amplitude. The phase bias is nearly constant for varying displacement amplitude for a given vibration frequency. The phase jitter decreases to very small values as displacement amplitude increases. Figures 5.17 and 5.18 show the displacement and phase results for Tb = 50, 100 μs, respectively, while varying the normalized radiation force. For Tb = 50 μs, the displacement bias is larger for larger radiation force, probably because the displacement is larger. The displacement jitter, phase bias, and phase jitter do not change significantly for different levels of applied force. The phase bias exhibits a negative linear trend with increasing frequency. This increase may be due to decreased motion amplitude at higher frequencies and a time delay that may not have been accounted for. The phase jitter increases with increasing frequency. When Tb = 50 μs, the displacement bias and jitter decrease with lower values of applied radiation force, but the phase bias and jitter do not change very much. The value of fprf was varied in Figures 5.19-5.22. As a function of displacement amplitude, the value of fprf did not significantly affect the results. As a function of frequency, the use of fprf = 2.0 kHz yielded results with larger displacement and phase bias for both lengths of the excitation toneburst. In the case where Tb = 100 μs, the

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increase in fprf provided graded differences in the displacement and phase bias. However, the use of fprf = 4.0 kHz, gave much higher displacement and phase jitter values. This may be due to interference between the excitation and tracking pulses. Figures 5.23-5.26 show the results of varying the length of the slow time processing window. The length of the window was varied to be Ts = 30, 50, 100, 150 ms. As a function of the displacement amplitude, the displacement and phase bias and jitter do not vary significantly. Only for the case of Ts = 30 ms, the displacement bias and jitter was higher compared to the other three values of Ts. When compared over frequency, Ts = 30 ms yields larger values of displacement bias. There is graded decrease in the displacement and phase jitter as the value of Ts increases. Phase bias seems unaffected by varying this parameter at low frequencies but shows graded differences at higher vibration frequencies.

Figure 5.15—Displacement and phase bias and jitter for Tb = 50 μs, F0 = 1, and fv = 100 ({), 200 (), 300 (Δ), and 400 (∇) Hz, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 5.16—Displacement and phase bias and jitter for Tb = 100 μs, F0 = 1, and fv = 100 ({), 200 (), 300 (Δ), and 400 (∇) Hz (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

Figure 5.17—Displacement and phase bias and jitter for Tb = 50 μs, fv = 200 Hz, and F = F0 ({), F0/2 (), F0/4 (Δ), and F0/8 (∇), (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 5.18—Displacement and phase bias and jitter for Tb = 100 μs, fv = 100 Hz, and F = F0 ({), F0/2 (), F0/4 (Δ), and F0/8 (∇), (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

Figure 5.19—Displacement and phase bias and jitter for Tb = 50 μs, F0 = 1, fv = 200 Hz, and fprf = 2.0 ({), 3.0 (), 4.0 (Δ), 5.0 (∇), and 6.0 (◊) kHz, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 5.20—Displacement and phase bias and jitter for Tb = 100 μs, F0 = 1, fv = 200 Hz, and fprf = 2.0 ({), 2.5 (), 3.0 (Δ), 4.0 (∇) kHz, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

Figure 5.21—Displacement and phase bias and jitter for Tb = 50 μs, F0 = 1, fv = 200 Hz, and fprf = 2.0 ({), 3.0 (), 4.0 (Δ), 5.0 (∇), and 6.0 (◊) kHz, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 5.22—Displacement and phase bias and jitter for Tb = 100 μs, F0 = 1, fv = 200 Hz, and fprf = 2.0 ({), 2.5 (), 3.0 (Δ), 4.0 (∇) kHz, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

Figure 5.23—Displacement and phase bias and jitter for Tb = 50 μs, F0 = 1, fv = 200 Hz, and Ts = 30 ({), 50 (), 100 (Δ), and 150 (∇) ms, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 5.24—Displacement and phase bias and jitter for Tb = 100 μs, F0 = 1, fv = 200 Hz, and Ts = 30 ({), 50 (), 100 (Δ), and 150 (∇) ms, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

Figure 5.25—Displacement and phase bias and jitter for Tb = 50 μs, F0 = 1, fv = 200 Hz, and Ts = 30 ({), 50 (), 100 (Δ), and 150 (∇) ms, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

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Figure 5.26—Displacement and phase bias and jitter for Tb = 100 μs, F0 = 1, fv = 200 Hz, and Ts = 30 ({), 50 (), 100 (Δ), and 150 (∇) ms, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (d) applies to each panel.

5.6 Discussion

Harmonic pulsed excitation is an effective method to provide multifrequency radiation force excitation. With an analytic description, the method can be tailored to experimental situations and the radiation force function can be accurately estimated as shown by the results in Figures 5.4 and 5.5. The HPE method could also be used to produce images acquired using the ARFI method by evaluating the response for the first pulse of the pulse sequence. However, this assumes that fr is low enough to provide relaxation of the object or tissue so that analysis of the relaxation may be performed. Along with the newly characterized excitation method, motion detection has been performed at multiple frequencies with the same data. This provides the prospect for dispersive measurements for applications that require such measurements. Figure 5.13

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shows that the results from the ultrasound based method provide results close to those measured by a laser vibrometer. This comparison provides confidence in the measurements made in the rest of the experiment. A parameterized analysis of the method with experimental data was performed. A few general conclusions can be made from this analysis. The displacement bias and phase jitter decreased with increased value of displacement amplitude. The displacement bias seemed to be sensitive to the amount of radiation force used. An increased value of fprf led to decreases in the displacement and phase bias and jitter. The use of longer slow time processing windows lowered the displacement and phase jitter values in a graded fashion, but overall, the error was not sensitive to this parameter. These conclusions agree with those made in Chapter 4 in that an increase in D0 lowered the phase jitter, an increase in Np decreased the displacement and phase bias and jitter, and an in increase in Nc caused a decreased in the displacement and phase jitter values. There are a few sources of error that may be encountered using HPE and ultrasound based motion detection on a reflective target. First, there is the risk of aliasing. If there is significant motion at high frequencies and the value of fprf is low, then motion information from frequencies above the Nyquist sampling limit, fprf/2, could alias down to corrupt the information at frequencies of interest. This can be prevented by increasing fprf and/or reducing the radiation force used. When varying the value of fprf, it was found that fprf = 2.0 kHz yielded results much different from those with higher values of fprf. Aliasing errors may have been a factor since the Nyquist frequency in this case would only by 1.0 kHz. For most cases in biomedical applications, there is a viscous component to the tissue

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that acts as a low-pass filter, which essentially makes the tissue a physical anti-aliasing filter. McAleavey, et al., have reported that in ARFI imaging measurement bias of the displacement can occur because of beam shapes used for excitation and motion detection [112]. They model the beam shapes with Gaussian functions and describe how only an average displacement estimate will be gained based on how much of the tracking beams intercepts the excitation beam. If the tracking beam is wide with respect to the excitation beam, then more decorrelation or bias will result in the measurements. Ideally, the tracking beam should be much thinner with respect to the excitation beam so that the tracking beam only intercepts the peak displacement of the scatterer(s). In this experiment, the full width at half maximum (FWHM) of the excitation beam is 0.80 mm and the FWHM of the tracking beam is 0.365 mm. These differences in the FWHM are accomplished with the same aperture, but the excitation beam is created using 3.0 MHz ultrasound while the tracking beam uses 9.0 MHz ultrasound, resulting in the decrease in the size of the beam. Using, the results from McAleavey, et al., with a ratio of excitation to tracking width of Wx = Wy = 2.19, we should expect that the best results attainable would be tracking 92% of the peak displacement or would have at the least an 8% error [112]. However, the results presented in this chapter show that the motion detection method can obtain errors less than this theoretical threshold. The reflective spherical target serves to scatter the ultrasound from the tracking pulses directly back to the transducer and effectively decreases the tracking FWHM such that the motion detected using the

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ultrasound method is closer to the peak displacement as measured by the laser vibrometer. One glaring error is the large phase bias that occurs with increasing frequency. The fact that the bias increases with frequency, linearly in most cases, leads us to believe that some systematic error in processing or data acquisition may be present. The laser signal was filtered with a low-pass Bessel filter with a cutoff frequency of 20.0 kHz, which would produces a linear phase shift with a 5.7° bias at 1.0 kHz, [113]. The linear nature of the remaining phase bias may indicate a constant time delay in the processing electronics that could not be identified.

5.7 Conclusion

Harmonic pulsed excitation was introduced as a multifrequency radiation force excitation method that allows motion detection tracking to be implemented easily between excitation tonebursts. Motion detection results show that motion can measured with low displacement and phase bias and jitter. A parameterized analysis provided insight on how to optimize acquisition of the displacement data for minimization of errors.

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Chapter 6 Harmonic Motion Detection in a Vibrating Scattering Medium If I have seen further it is by standing on the shoulders of giants. Isaac Newton

6.1 Introduction

For the development of dynamic radiation force excitation and motion detection for biomedical applications, implementation in a scattering medium must be considered. Chapters 4 and 5 demonstrated a theoretical model, signal processing methods, and the use of harmonic pulsed excitation (HPE) and simultaneous motion detection using pulseecho ultrasound to measure the motion of a vibrating reflective target. While this may be a good model for applications such as vessels and nondestructive evaluation, it is not adequate for analysis of most soft tissues. Therefore, the motion detection model for a single reflective target is extended to accommodate a volume of diffuse scatterers.

6.2 Background

There has been significant study in the ultrasound field in understanding how scatterers in tissue give rise to the signature speckle seen in diagnostic ultrasound images [114, 115]. To understand this process, algorithms have been devised to simulate the speckle that results from insonification of a group of scatterers [115, 116]. For the purposes of the presented model, the scatterers are uniformly distributed within a volume with unity reflectivity. The density of the scatterers is specified as scatterers per

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resolution cell, where the resolution cell of the ultrasound beam is defined as the volume encompassed by the ultrasound beam that has magnitude -6 dB below the peak value. To achieve fully developed speckle, that is, speckle with a signal-to-noise ratio of 1.91, Palmeri, et al., demonstrated that a minimum of 11 scatterers/resolution cell volume was necessary [103]. As was mentioned in Chapter 5, McAleavey, et al., have reported that in acoustic radiation force impulse (ARFI) imaging measurement bias of the displacement can occur because of the beam shapes used for excitation and motion detection [112]. McAleavey and his colleagues modeled the beam shapes as two-dimensional Gaussian functions, and made the argument that the tracking beam samples a finite amount of the motion created by the excitation beam. However, because this distribution is not uniform, some averaging will occur and measured result will be less than the peak displacement. When the tracking beam is wide with respect to the excitation beam, then more decorrelation or negative bias will result in the measurements. Ideally, the tracking beam would be much thinner with respect to the excitation beam so that the tracking beam only intercepts the peak displacement of the scatterer(s). In this model, the beam shape dimensions in the azimuthal and elevation or x- and y-directions are considered. They assume that axially, or in the z-direction, the excitation, and therefore the motion will be constant. This however, will not be the case in practice as the excitation beam’s magnitude will vary with axial distance. The beam shapes can be characterized by the full width at half maximum (FWHM) in each dimension for both the excitation and tracking beams. The FWHM of the excitation beam is scaled to fit a Gaussian model. In the x-, y-, and z-directions, the

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excitation beam parameters will be denoted as Ex, Ey, and Ez. Likewise, tracking beam parameters in the x-, y-, and z-directions will be denoted as Tx, Ty, and Tz. McAleavey, et al., compares the size of the excitation and tracking beams using a ratio where Wi = Ei/Ti, and i = x, y, or z. For the circular, 3.0 MHz transducer used for experiments with 45 mm diameter and focal length of 70 mm, Ex = Ey = 0.17 mm and Ez = 1.80 mm, which correspond to FWHM values of 0.80, 0.80, and 8.48 mm in the x-, y-, and z-directions. Since tracking pulses are transmitted at 9.0 MHz using the same transducer, Tx = Ty = 0.078 mm and Tz = 0.83 mm, which corresponds to FWHM values of 0.365, 0.365, and 3.89 mm in the x-, y-, and z-directions. For these beam shape parameters, Wx = Wy = 2.19 and Wz = 2.18. McAleavey, et al., derived an analytic formula describing the effects of different values of Wx and Wy on the ability to track the peak displacement caused by a radiation force excitation toneburst. With the values for this transducer, the best results attainable would be tracking 92% of the peak displacement or at the least an 8% error [112]. With the addition of the axial beam shape, the error should increase although the error should only be represented as a displacement bias. The inclusion of the beam shape weighting in the z-direction is important to completing the model.

6.3 Model of Vibrating Scattering Medium

We can extend the model presented in Chapter 4 to calculate the motion and resulting echoes obtained from vibrating scatterers where the motion and tracked motion is weighted by the beam shapes of the excitation and tracking ultrasound beams. Keeping with the conventions from Chapter 4 concerning fast time, tf, and slow time, ts, we can

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model the first received echo from a volume with N scatterers and the nth scatterer has position (xn, yn, zn) as r1 ( t f , ts ) = ∑ I ( xn , yn , zn ) cos (ω0t f + φ f ) , N

(6.1)

n =1

where I(xn, yn, zn) is the weighting function for the tracking beam focused at an axial distance of zF defined as I ( xn , yn , zn ) = e

−

xn2 y2 ( z − z ) − n − n 2F 2Tx2 2Ty2 2Tz

2

.

(6.2)

The kth echo in the sequence can be written as

(

)

rk ( t f , ts ) = ∑ I ( xn , yn , zn ) cos ω0 ( t f − tk ,n ) + φ f , N

n =1

(6.3)

where tk , n =

2u z ,k ( xn , yn , zn ) c

u z ,k ( xn , yn , zn ) = A ( kts ) e

−

xn2 2 Ex2

,

−

(6.4) yn2

2 E y2

−

( zn − z F )2 2 Ez2

,

A ( kts ) = D0 sin ( kωs ts + φs ) .

(6.5) (6.6)

Equation (6.4) describes the time delay of the kth echo encountered for the nth scatterer and c is the longitudinal speed of sound in the medium. The function uz,k(xn, yn, zn) is the weighting function for the excitation and A(kts) is the vibration amplitude with displacement amplitude D0, phase φs, at frequency ωs for the kth echo. The cross-spectral analysis proposed by Hasegawa and Kanai [55] and the Kalman filter for harmonic motion described by Zheng, et al. [87] is used for analysis of the echoes and extraction of the displacement amplitude and phase.

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6.4 Parameterized Model Analysis

To evaluate the effects of different parameters on the results of motion detection in a scattering medium, a parameterized analysis was performed. The default parameters for the parameter study are listed in Table 6.1. The displacement amplitude will varied to values of D0 = 100, 500, 1000, 5000, 10000 nm. The number of cycles of vibration used in the analysis will be varied to values of Nc = 3, 5, 10, 15, 20. The number of points sampled per vibration cycle will be varied to values of Np = 5, 10, 15, 20, 25, 30. The number of scatters/resolution cell volume was varied to values of Ns = 11, 16, 21, 27, 33. The gate length was varied to lg = 0.5, 1.0 mm. The sampling frequency of the echoes was varied to Fs = 50, 100 MHz. The bandwidth of the tracking beam was varied to BW = 6.5, 20.0, 40.0%. The beam shape parameters, Wx, Wy, were varied from 0.1-10 and Wz, were varied to Wz = 0.1, 0.5, 1.0, and 2.18. Table 6.1 – Parameter Study Default Parameters D0

φs

Nc Np fv ff Fs BW lg c Ns Ex Ey Ez Tx Ty Tz xv yv zv N

Displacement amplitude Vibration phase Cycles of vibration Sampled points per vibration cycle Vibration frequency Ultrasound frequency Sampling frequency Transducer bandwidth Gate length Speed of sound Scatterers/resolution cell volume Excitation beam width in x direction Excitation beam width in y direction Excitation beam width in z direction Tracking beam width in x direction Tracking beam width in y direction Tracking beam width in z direction x dimension of scattering volume y dimension of scattering volume z dimension of scattering volume Iterations

1000 nm 0° 5 20 200 Hz 9.0 MHz 100 MHz 6.5% 1.0 mm 1540 m/s 11 0.17 mm 0.17 mm 1.80 mm 0.078 mm 0.078 mm 0.83 mm 1.0 mm 1.0 mm 5.0 mm 1000

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6.5 Parameterized Model Results

Figures 6.1 and 6.2 show the displacement and phase results for the default conditions in Table 6.1, respectively. Each data point represents the mean and the error bars represent one standard deviation. The black dashed line is the target value. As SNR increases the displacement estimates level out to a constant value and the error bar height decreases. For the phase, the same occurs, but no bias is evident in the results. Figure 6.3 shows the results of varying the displacement amplitude. The displacement bias and jitter decrease as D0 decreases. The phase bias and jitter decrease as the displacement amplitude increases. Figure 6.4 shows the effects of varying the number of vibration cycles used for the motion detection. Displacement bias and jitter decrease as Nc increases. The phase bias and jitter is not affected significantly by this parameter. In Figure 6.5, the variation of the number of points sampled per vibration cycle shows that an increase in Np decreases the displacement bias but increases the displacement jitter while having minimal effects on changing the phase bias and jitter. As the results in Figure 6.6 show that increasing the number of scatterers contained in a resolution cell volume decreases the error in the displacement and phase estimates. Figure 6.7 shows that lengthening the processing gate improves the displacement and phase results. Increasing the sampling frequency of the echoes also improves the displacement and phase results as shown in Figure 6.8. Figure 6.9 shows that increasing the bandwidth of the tracking beam increases the error significantly for both the displacement and phase. Figures 6.10-6.13 show the normalized displacement, displacement jitter, phase bias, and phase jitter while varying Wx, Wy, and Wz. An 11 x 11 grid of Wx and Wy was

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simulated with 100 iterations for each point. The results were interpolated to a 101 x 101 grid for display purposes. The normalized displacement is referenced to D0 = 1000 nm. As was shown by McAleavey, et al. [112], as Wx and Wy increase the normalized displacement increase and the displacement jitter, phase bias, and phase jitter decrease. It was also shown that as Wz increases, the same trends occur as for increases in Wx and Wy. For the case Wx = Wy = 2.19, which describe the parameters for the experimental implementation, the normalized displacement is approximately 0.73, which is very close to the results found through the other simulations. The phase bias and jitter are less than 5° for values of Wz as low as 0.5.

Figure 6.1—Displacement results for default conditions in Table 6.1. Each data point represents the mean and the error bars represent one standard deviation. The black dashed line is the target value of D0 = 1000 nm.

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Figure 6.2—Phase results for default conditions in Table 6.1. Each data point represents the mean and the error bars represent one standard deviation. The black dashed line is the target value of φs = 0°.

Figure 6.3— Displacement and phase bias and jitter for D0 = 100 ({), 500 (), 1000 (Δ), 5000 (◊), and 10000 (∇) nm, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 6.4— Displacement and phase bias and jitter for Nc = 3 ({), 5 (), 10 (Δ), 15 (◊), and 20 (∇), (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

Figure 6.5—Displacement and phase bias and jitter for Np = 5 ({), 10 (), 15 (Δ), 20 (◊), 25 (∇), and 30 (*), (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 6.6—Displacement and phase bias and jitter for Ns = 11 ({), 16 (), 21 (Δ), 27 (◊), and 33 (∇), (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

Figure 6.7—Displacement and phase bias and jitter for lg = 0.5 ({) and 1.0 () mm, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 6.8—Displacement and phase bias and jitter for Fs = 50 ({) and 100 () MHz, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

Figure 6.9—Displacement and phase bias and jitter for BW = 6.5 ({), 20.0 (), and 40.0 (Δ) %, (a) Displacement bias, (b) Displacement jitter, (c) Phase bias, (d) Phase jitter. The legend in (b) applies to each panel.

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Figure 6.10—Normalized displacement results for variations of Wx, Wy, and Wz. (a) Wz = 0.1, (b) Wz = 0.5, (c) Wz = 1.0, (d) Wz = 2.18.

Figure 6.11—Displacement jitter results for variations of Wx, Wy, and Wz. (a) Wz = 0.1, (b) Wz = 0.5, (c) Wz = 1.0, (d) Wz = 2.18.

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Figure 6.12—Phase bias results for variations of Wx, Wy, and Wz. (a) Wz = 0.1, (b) Wz = 0.5, (c) Wz = 1.0, (d) Wz = 2.18.

Figure 6.13—Phase jitter results for variations of Wx, Wy, and Wz. (a) Wz = 0.1, (b) Wz = 0.5, (c) Wz = 1.0, (d) Wz = 2.18.

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6.6 Parameterized Model Discussion

The variation of displacement amplitude showed that as D0 increases, the displacement bias and jitter increased. This agrees with expected results as the decorrelation due to different beam shapes dictated that the maximum detected amplitude would be less than 92% of the peak value. In these simulations at high SNR, the percent error turned out to be 26.6 ± 13.2 %. If we just take into account the bias, the inclusion of the weighting of the excitation and tracking beam in the z-direction provides amplitude estimates that are 73.4 % of the peak amplitude. Palmeri, et al., reported that in studies involving finite element simulations of displacement induced by radiation force, that ultrasonic tracking of the motion could only achieve displacement estimates that were between 50-75% of the maximum amplitude [103]. The results from this simulation study fall at the higher range of those results which holds promise for this method being implemented in a scattering medium such as tissue. As in the results in Chapter 4 for the reflective target, increasing Nc served to decrease the displacement bias and jitter. However, in the scattering medium increasing Np decreased the displacement bias but increased the displacement jitter whereas in the model for the reflective target, both the bias and jitter decreased. As the density of the scatterers was increased, the error decreased for both the displacement and phase. Experimentally, this means that tissues with increased density of scatterers will provide a better medium for motion tracking. However, this increase in scatterer density may increase the attenuation, leading to lower radiation force for inducing motion.

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Increasing parameters such as the processing gate length and sampling frequency decreased the error in the results. The longer processing gate takes into account the contributions from more scatterers and a longer section of the echoes to perform the cross-spectral analysis. Other studies have shown that lengthening the processing gate for motion detection can improve the displacement bias and jitter [49, 103, 105]. The increased sampling frequency also provides for higher temporal resolution for correlative analysis between consecutive echoes. A study by Pinton and Trahey showed that increasing the sampling rate through interpolation and then using cross-correlation could decrease the displacement bias and jitter [53]. The increase in the transducer bandwidth increased the error present in the results. This may be due to increased resolving of the scatterers, providing echoes that were not suitable for use in tracking motion. Another explanation is that the larger bandwidth allows more noise into the signal and corrupts the phase shift estimation and ultimately the amplitude and phase results. The analysis of variation of the beam shapes can apply to any transducer that produces point-spread functions that can be modeled as three-dimensional Gaussian function. It was found that even as Wz increased, the normalized displacement could only achieve at most 73% of the peak value whereas with McAleavey’s theoretical development that ignored the variation of Wz predicted that 92% of the peak displacement could be obtained with Wx = Wy = 2.19. It was found but not shown that as Wz was increased to 5 and 10, that the results did not change significantly from those with Wz = 2.18. The phase bias and jitter were not as sensitive to Wz and remained relatively constant for Wz greater than or equal to 0.5.

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This simulation model provided insights into implementing motion detection for use in tissue after radiation force excitation. Accurate displacement results will not be possible due to the weighting of the beam shapes. However, displacement could still be used in model-based approaches with this limitation taken into account. In applications involving shear wave speed estimation, this model can also be useful. Shear wave speed of a harmonic shear wave can be calculated after measuring the phase at two different locations. This model provides insights on how to minimize the phase bias and jitter in the estimates. The most important parameter for optimizing phase estimation is to increase the displacement amplitude. The simulation model at present assumes a stationary system. However, in an in vivo setting there will be physiological motion due to the beating heart, pulsatile blood flow in tissue, and breathing. These gross motions are much larger than the small motion that we are trying to detect. Zheng, et al. has proposed modifications to the Kalman filter used to address these concerns and even with large gross motion and acceleration, the small displacement amplitude and the vibration phase was recovered [117].

6.7 Conclusion

The model of a vibrating scattering medium has direct application to modeling motion detection in tissue. A parameterized analysis was performed to assess the effects of different parameters had on the vibration displacement and phase results. The use of excitation and tracking beam shapes introduced a bias in the displacement results. The phase errors were decreased by increasing the displacement amplitude. This model provides a platform for testing different experimental settings with different parameters.

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Chapter 7 Experimental Harmonic Motion Detection in a Vibrating Scattering Medium The true worth of a researcher lies in pursuing what he did not see in his experiment as well as what he sought. Claude Bernard

7.1 Introduction

Chapter 6 provided a model for harmonic motion detection in a scattering medium. Harmonic pulsed excitation will be used to induce motion in a scattering medium and the same transducer will be used to track the motion. Experimental results from implementation of the HPE method in a scattering gelatin and a section of bovine muscle will be shown. A parameter analysis using these experimental results will be performed to assess which parameters of the motion detection method provide optimal results.

7.2 Background

Many research groups that use radiation force or some other excitation method have used different methods for tracking of the motion induced by a deforming force. Phase shift [50, 51, 55] and time delay estimation [26, 49, 52, 53] methods have been utilized in analyzing small amplitude motion in tissue. In this chapter, we will use the motion detection method described in Chapter 4 to perform motion detection in tissue and tissue-like phantoms. Experimental evaluation of

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the results found in Chapter 6 will assess how well the model of the vibrating scattering medium resembles the results obtained in practice.

7.3 Motion Detection Experimental Setup

To assess the performance of the HPE and motion detection methods, two experiments were performed with different media. The media for the first experiment was a scattering gelatin phantom made using 300 Bloom gelatin powder (Sigma-Aldrich, St. Louis, MO) with a concentration of 8% by volume. A preservative of potassium sorbate (Sigma-Aldrich, St. Louis, MO) was also added with a concentration of 10 g/L. Graphite scatterers (Sigma-Aldrich, St. Louis, MO) with size < 45 μm were added with a concentration of 5% by volume. The media for the second experiment was a piece of beef roast that was cut into 6.5 x 7.0 x 2.5 cm section. The beef muscle was embedded in an agar mixture (BactoTM Agar, Becton, Dickinson Company, Sparks, MD) made with a concentration of 3% by volume. Agar was used so that the water temperature in the water tank could be raised to approximately 37°C to mimic body temperature. Gelatin was not used because of its lower boiling point. The HPE method was initiated using a trigger signal from a signal generator producing one cycle of a TTL pulse. This master trigger signal triggered a signal generator to produce a 20 cycle rectangular pulse train with a frequency of fr = 100 Hz. This pulse train triggered a signal generator that would produced a toneburst of length Tb = 50 or 100 μs. The voltage amplitude of this toneburst was varied to change the radiation force amplitude. The master trigger was also applied to another signal generator

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(33250A, Agilent, Palo Alto, CA) that produced a rectangular pulse train with a frequency of fprf. The value of the fprf was varied during the experiment to assess its effects on the results. For Tb = 50 μs, fprf = 2.0, 3.0, 4.0, and 5.0 kHz, and for Tb = 100 μs, fprf = 2.0, 2.5, 3.0, and 4.0 kHz. This pulse train was used as a trigger input to the analogto-digital (A/D) converter board in a personal computer. The A/D converter produced a trigger signal for every pulse in the pulse train and this triggered a signal generator to generate a three cycle pulse at 9.0 MHz. The radiation force toneburst and tracking pulse were added together using a hybrid junction (M/A-COM, Inc., Lowell, MA) and this signal was amplified with a 40 dB amplifier. This signal passed through a diode bridge to eliminate low-level noise from the amplifier and through matching transformer to the transducer. The echoes were received and filtered with a 3.0 MHz notch filter with a 50% bandwidth. The echoes then passed through a transmit/receive (T/R) switch, filtered with a 9.0 MHz bandpass filter with 33% bandwidth, amplified by a logarithmic amplifier and finally filtered by another 9.0 MHz bandpass filter before being digitized at a sampling frequency of 100 MHz. A block diagram of the experimental setup of HPE and motion detection is shown in Figure 7.1. Diagrams and pictures of the phantoms are shown in Figure 7.2. The cross-spectral analysis proposed by Hasegawa and Kanai was applied to the measured echoes [55]. A bandpass filter was used in software to isolate the information at the frequency of interest. A Hann windowed version of the displacement signal is used as the input to the Kalman filter [54]. The use of the Hann window decreases the strength of the signal by a factor of two so the final displacement result acquired after the Kalman filter is multiplied by two as was described in Chapter 5.

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9.0 MHz BPF fprf

ff

Logarithmic Amplifier

fr

~

~

f0

A/D

3.0 MHz Transducer

Σ

3.0 MHz Notch Filter

9.0 MHz BPF T/R Switch

40 dB Amplifier Figure 7.1—Experimental setup for harmonic pulsed excitation and motion tracking. For the excitation, a pulse train with frequency fr = 100 Hz is initiated, and each positive pulse triggers a toneburst of ultrasound at f0 = 3.0 MHz. For the tracking a pulse train at fprf is initiated with specified time delay, td, and each positive pulse triggers a three cycle pulse at ff = 9.0 MHz to be transmitted. The excitation and tracking signals are summed together and amplified before being sent to the 3.0 MHz transducer. For tracking, the gated echoes are filtered with a notch filter centered at 3.0 MHz before passing through a transmit/receive (T/R) switch and then filtered with a bandpass filter centered at 9.0 MHz. The signal is logarithmically amplified and filtered again with a bandpass filter centered at 9.0 MHz before being sent to the digitizer (A/D).

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Gelatin Graphite Phantom

3.0 MHz Transducer

(a)

(b)

(c)

Figure 7.2—Experimental setup and phantoms for scattering gelatin and beef experiments, (a) Experimental setup with 3.0 MHz transducer and gelatin phantom containing suspended graphite particles, (b) Photograph of bovine muscle embedded in agar block, (c) Photograph of 3.0 MHz transducer held in water tank above agar block with bovine muscle.

7.4 Parameter Analysis in Scattering Gelatin Phantom

To assess the effects of different parameters on the results, different variations of experimental parameters were performed. Because different values of applied voltage for the excitation toneburst were used, the results can be compared versus the radiation force applied. For the purposes of reporting the results, the radiation force will be normalized based on the maximum force produced by the highest voltage setting used and denoted as F0. The value of fprf was varied to explore the differences in the results. Also, the value of Ts, the length of the temporal window used for analysis in slow time was varied, such that the product NcNp was constant where Nc is the number of cycles of vibration and Np is the 146

number of samples taken for each vibration cycle. For example if Ts = 100 ms, fv = 100 Hz, fprf = 4.0 kHz, then Nc = 10 and Np = 40 and NcNp = 400. If fv is increased to 200 Hz, and all other values are not changed, Np = 20 but Nc = 20 because twice as many cycles of vibration will occur in the same slow time window and NcNp = 400. For the parameter analysis, the default values for analysis are Tb = 50 μs, F0 = 1, fv = 200 Hz, fprf = 4.0 kHz, and Ts = 100 ms.

7.5 Scattering Gelatin Phantom Results

Figure 7.3 shows the regression of measured displacement versus normalized force. The regression was performed for the eight highest samples and yielded the regression equation, y = 3492.3x – 1219.6 with R2 = 0.9778. The samples below 0.4F0 are results of noise because of low amplitude motion. This provides a lower threshold between 100 and 200 nm for display of future results. Figure 7.4 shows the displacement and phase for the default conditions. The data points represent the mean of five measurements and the error bars represent one standard deviation of those measurements. The phase estimates had to be corrected for a frequency dependent phase shift because of the constant time delay associated with wave propagation between the transducer and focal region of the transducer. Figure 7.5 shows the mean and standard deviation of the displacement and phase measurements for fv = 100, 200, 300, 400 Hz. The mean and standard deviation of the measurements are reported because there is no reference to compare with for estimates of bias and jitter. The mean displacement, μd, varies linearly with the normalized force. The

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standard deviation of the displacement, σd, remains relatively constant versus the normalized force. To evaluate measurement limits, the σd must compared to the value of

μd because when the two metrics are nearly equal, the accuracy of the measurement may be compromised. The mean of phase, μφ, remains constant versus normalized force for different values of fv. The mean value of phase is determined by the frequency response of the gelatin. The standard deviation of phase, σφ, decreases to a constant level as normalized force increases. Figure 7.6 shows the mean and standard deviation of the displacement and phase versus vibration frequency for different values of F = F0, 0.81F0, 0.64F0, 0.49F0. The mean displacement decreases almost monotonically as frequency increases. For each frequency, the mean phase measured for different values of F was very close. There is also a negative trend with frequency for μφ. The σd and σφ remain relatively constant versus frequency; however, σφ decreases as F increases. Figures 7.7 and 7.8 shows results for variation of the fprf with values fprf = 2.0, 3.0, 4.0 and 5.0 kHz. The μd, μφ, and σφ results match well for all values of fprf except fprf = 2.0 kHz. This discrepancy may be due to aliasing effects of higher frequency information being aliased down to lower frequencies. The σd remains fairly constant for all values of fprf.. Figures 7.9 and 7.10 provide results for variation of the slow time processing window, Ts = 50, 100, 150, 200 ms. The results for Ts > 100 ms do not vary significantly indicating that above this threshold, the minimization of error may not be sensitive to this parameter. Though even for Ts = 50 ms, the results are not drastically different from those for Ts > 100 ms, but there is some improvement above this threshold.

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Figures 7.11 and 7.12 show results of using different processing gate lengths. The shorter gate length causes only a slight difference in the results for σd and σφ.

Figure 7.3—Regression of measured ultrasound displacement versus normalized force. The regression line was computed for the eight highest samples and the resulting equation is y = 3492.3x – 1219.6 with R2 = 0.9778.

Figure 7.4—Displacement and phase results for default conditions of analysis. Data points represent the mean of five measurements and the error bars represent one standard deviation of those measurements. (a) Displacement results, (b) Phase results.

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Figure 7.5—Mean and standard deviations of displacement and phase measurements for values of fv = 100 ({), 200 (), 300 (Δ), and 400 (∇) Hz, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ). The legend in panel (d) applies to all panels.

Figure 7.6—Mean and standard deviations of displacement and phase measurements for values of F = F0 ({), 0.81F0 (), 0.64F0 (Δ), and 0.49F0 (∇), (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ).The legend in panel (d) applies to all panels.

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Figure 7.7—Mean and standard deviations of displacement and phase measurements for values of fprf = 2.0 ({), 3.0 (), 4.0 (Δ), and 5.0 (∇) kHz, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ).The legend in panel (d) applies to all panels.

Figure 7.8—Mean and standard deviations of displacement and phase measurements for values of fprf = 2.0 ({), 3.0 (), 4.0 (Δ), and 5.0 (∇) kHz, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ).The legend in panel (d) applies to all panels.

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Figure 7.9—Mean and standard deviations of displacement and phase measurements for values of Ts = 50 ({), 100 (), 150 (Δ), and 200 (∇) ms, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ).The legend in panel (d) applies to all panels.

Figure 7.10—Mean and standard deviations of displacement and phase measurements for values of Ts = 50 ({), 100 (), 150 (Δ), and 200 (∇) ms, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ).The legend in panel (d) applies to all panels.

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Figure 7.11—Mean and standard deviations of displacement and phase measurements for values of lg = 0.5 ({) and 1.0 () mm, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ).The legend in panel (d) applies to all panels.

Figure 7.12—Mean and standard deviations of displacement and phase measurements for values of lg = 0.5 ({) and 1.0 () mm, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ).The legend in panel (d) applies to all panels.

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7.6 Scattering Gelatin Phantom Discussion

Motion detection was performed in a gelatin phantom with graphite scatterers a multiple frequencies. The gelatin had a low-pass characteristic, which served to decrease errors due to aliasing. Only the data obtained at fprf = 2.0 kHz was much different than data acquired at higher values of fprf which indicates that aliasing artifacts may have entered the signals. A level of the normalized force was established for analysis because the displacement values did not change below this threshold. A parameterized analysis was performed with the data to compare with the results in Chapter 6 and optimize experimental implementation of measurements. An increase in normalized force increased the mean displacement and reduced the variations in the phase measurements. As was previously mentioned, the value of fprf did not change the results significantly except in the case of fprf = 2.0 kHz. The slow time processing window length served to reduce the standard deviations in the displacement and phase measurements. Lengthening the processing gate length provided a slight reduction in the displacement and phase standard deviation. The displacement standard deviations were relatively constant at different levels of the normalized force and frequency, and some points were on the same order as the mean displacement values. The phase standard deviations were fairly low for vibration with large amplitudes and processed with high fprf and Ts. For these acquisition parameters, the standard deviation was usually less than 20°. The results for motion detection in the scattering gelatin phantom agree well with the results in Chapter 6. The results in Chapter 6 show that by decreasing fprf and

154

increasing Ts, which correspond to Np and Nc would decrease the variation or jitter in the displacement measurements. However, in the scattering gelatin phantom, the σd remains constant versus normalized force and frequency. In most cases except fprf = 2.0 kHz, the mean phase values agree across parameter variation which indicates that phase bias is relatively low. However, this can not be quantified because no reference measurement is available to compare against as in the measurements with the Doppler laser vibrometer in Chapter 5. The σφ decreases as the normalized force increases and as frequency decreases since a lower vibration frequency correlates with larger displacement amplitudes. The increase of fprf or Ts did not show a consistent trend towards decreasing σφ.

7.7 Bovine Muscle Section Results

Figures 7.13-7.15 shows the regression for Tb = 50, 100, and 200 μs, respectively. For Tb = 50, 100, and 200 μs, fprf = 4.0, 4.0, and 2.5 kHz, respectively, while F = F0, fv = 200 Hz, and Ts = 100 ms. Regressions were performed with the 9, 11, and 13 highest samples for Tb = 50, 100, and 200 μs, respectively. When the toneburst length doubled, the slope of the regression line fitting the data also nearly doubled as expected with slopes of 409, 812, and 1654 for Tb = 50, 100, and 200 μs, respectively. The threshold at which the data points level off was between 100 and 200 nm for each toneburst length. The data points level off because the displacement signal contains too much noise to make a reliable measurement. The normalized force at which this threshold was reached decreased as the toneburst length was increased, but the threshold represents a nearly constant value of force and resulting motion amplitude.

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For a parameterized analysis, the data for Tb = 100 μs was analyzed and parameters such as normalized force, vibration frequency, fprf, and Ts were varied. Figure 7.16 shows the displacement and phase results for the default conditions with each data point representing the mean of five measurements and the error bars representing one standard deviation of the measurements. The phase estimates were corrected for a frequency dependent phase shift because of the constant time delay associated with wave propagation between the transducer and focal region of the transducer. Figure 7.17 shows mean and standard deviations of the displacement and phase measurements for fv = 100, 200, 300, 400 Hz. The mean displacement, μd, varies linearly with the normalized force. The displacement amplitude decreases with increasing frequency, indicating a low-pass filter characteristic. The standard deviation of the displacement, σd, remains relatively constant versus the normalized force and is reduced by almost an order of magnitude compared to the scattering gelatin phantom. That reduction also comes with motion amplitudes that are more than a factor of two less. The mean phase, μφ, remains at a stable value versus normalized force for different values of fv. The standard deviation of phase, σφ, decreases as normalized force increases and never rises above 25°. Figure 7.18 shows the mean and standard deviation of the displacement and phase versus vibration frequency for different values of F = F0, 0.81F0, 0.64F0, 0.49F0. The mean displacement decreases monotonically as frequency increases, indicating that the beef acts as a low-pass filter for motion. For each frequency, the mean phase measured for different values of F was very close, except at fv = 1000 Hz. There is also a negative trend with frequency for μφ. The σd remained relatively constant versus frequency;

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however, σφ increases as frequency increases probably because μd is simultaneously decreasing to the threshold for accurate detection. Figures 7.19 and 7.20 shows results for variation of the fprf with values fprf = 2.0, 2.5, 3.0, and 4.0 kHz. The μd results match well for all values of fprf which indicates no aliasing. The μφ varied with fprf with the results for fprf = 3.0 and 4.0 kHz matching well while fprf = 2.0 and 2.5 kHz either underestimated or overestimated the mean phase values, respectively, assuming that increasing fprf provides better estimates. The σd remains fairly constant for all values of fprf.. The σφ results match fairly well, showing a decreasing trend as F increases and an increasing trend as frequency increases. Figures 7.21 and 7.22 provide results for variation of the slow time processing window, Ts = 50, 100, 150, 200 ms. The μd and μφ do not seem to change whereas a noticeable decreases in the σd and σφ were observed as Ts increased.

Figure 7.13—Regression of measured ultrasound displacement versus normalized force for Tb = 50 μs. The regression line was computed for the nine highest samples and the resulting equation is y = 409.0482x – 75.0306 with R2 = 0.9417.

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Figure 7.14—Regression of measured ultrasound displacement versus normalized force for Tb = 100 μs. The regression line was computed for the eleven highest samples and the resulting equation is y = 811.5938x – 147.3434 with R2 = 0.9761.

Figure 7.15—Regression of measured ultrasound displacement versus normalized force for Tb = 200 μs. The regression line was computed for the eleven highest samples and the resulting equation is y = 1653.9x – 222.1 with R2 = 0.9721.

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Figure 7.16—Displacement and phase results for default conditions of analysis for Tb = 100 μs. Data points represent the mean of five measurements and the error bars represent one standard deviation of those measurements, (a) Displacement results, (b) Phase results.

Figure 7.17—Mean and standard deviations of displacement and phase measurements for values of fv = 100 ({), 200 (), 300 (Δ), and 400 (∇) Hz, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ). Legend in panel (d) applies to all panels.

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Figure 7.18—Mean and standard deviations of displacement and phase measurements for values of F = F0 ({), 0.81F0 (), 0.64F0 (Δ), and 0.49F0 (∇), (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ). Legend in panel (d) applies to all panels.

Figure 7.19—Mean and standard deviations of displacement and phase measurements for values of fprf = 2.0 ({), 3.0 (), 4.0 (Δ), and 5.0 (∇) kHz, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ). Legend in panel (d) applies to all panels.

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Figure 7.20—Mean and standard deviations of displacement and phase measurements for values of fprf = 2.0 ({), 3.0 (), 4.0 (Δ), and 5.0 (∇) kHz, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ). Legend in panel (d) applies to all panels.

Figure 7.21—Mean and standard deviations of displacement and phase measurements for values of Ts = 50 ({), 100 (), 150 (Δ), and 200 (∇) ms, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ). Legend in panel (d) applies to all panels.

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Figure 7.22—Mean and standard deviations of displacement and phase measurements for values of Ts = 50 ({), 100 (), 150 (Δ), and 200 (∇) ms, (a) Mean of displacement (μd), (b) Standard deviation of displacement (σd), (c) Mean of phase (μφ), (d) Standard deviation of phase (σφ). Legend in panel (d) applies to all panels.

7.8 Bovine Muscle Section Discussion

Results for excitation and measurement of motion in a section of bovine muscle were shown. The results indicate that small harmonic motion can be measured with good precision. The standard deviations of the displacement are nearly an order of magnitude less than encountered in the scattering gelatin phantom, and the mean displacements in the beef were less than half in the muscle tissue. This provides confidence in the displacement measurements. The phase measurements also exhibited high precision. As the acquisition parameters were optimized towards large amplitude vibration and high fprf, the σφ could be reduced to less than 5°. When varying fprf, the mean phase values do not agree and indicates that phase bias may be about ±10°.

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The decrease in the variations of the displacement and phase measurements compared to those made in the scattering gelatin phantom could be due to the density of scatterers. As was shown in Chapter 6, as the density of scatterers/resolution cell volume increases, the jitter for the displacement and phase decreased dramatically. The same trend was seen when comparing the results in the gelatin phantom and muscle tissue which indicates that the muscle had more ultrasonic scatterers per resolution cell than the gelatin phantom. The results for motion detection in the bovine muscle also agree well with the results in Chapter 6. In Chapter 6, it was shown that by decreasing fprf and increasing Ts, which correspond to Np and Nc would decrease the variation or jitter in the displacement measurements. As in the scattering gelatin phantom, the results in the muscle show that

σd remains constant versus normalized force and frequency. The σφ decreases as the normalized force increases and as frequency decreases since a lower vibration frequency correlates with larger displacement amplitudes. The increase of fprf did not show a consistent trend towards decreasing σφ, whereas increasing Ts did show a decrease in σφ.

7.9 Imaging of Bovine Muscle Section

The same experimental system described for the point measurements in the scattering gelatin and bovine muscle phantoms was used for imaging another phantom containing bovine muscle. This phantom and dissected muscle are shown in Figure 7.23. A dashed box depicts the region displayed in the images,

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(b)

(a)

(c)

Figure 7.23—Photographs of bovine muscle section. (a) Full phantom with bovine muscle encased in agar, (b) Top surface of muscle section. The muscle was cut in half to show the portion that was scanned. The dashed box depicts the scan area used for imaging, (c) Bottom surface of muscle section shown in (b). The slice has been flipped horizontally compared the image in (b). The dashed box depicts the scan area used for imaging.

The muscle was scanned at a resolution of 0.2 mm x 0.2 mm. Four different constant depth scans (C-scans) were performed with 4 mm between each plane. An ultrasonic C-scan and a vibro-acoustic image were acquired for each imaging plane. The ultrasonic C-scan was acquired at 9.0 MHz and the vibro-acoustic image was formed using a SHME formation with NS = 1 and NE = 2 and Δf = 50 kHz. Figure 7.24 shows the ultrasonic C-scan, the vibro-acoustic image, and images of the displacement amplitude and phase at fv = 100 Hz. The ultrasonic C-scan and vibro-acoustic images were independently normalized and the motion detection images have units of nanometers and degrees for the amplitude and phase. The motion detection images were median filtered with a 3 x 3 window because there was some impulsive noise due to some pixels having noisy estimations of the motion.

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Figure 7.24—Images of bovine muscle section. (a) Ultrasonic C-scan performed at 9.0 MHz, (b) Vibroacoustic image acquired with Δf = 50 kHz, (c) Displacement amplitude for fv = 100 Hz. The units of the colorbar are nanometers, (d) Displacement phase for fv = 100 Hz. The units of the colorbar are degrees.

Figures 7.25 and 7.26 show the displacement amplitude and phase images for acquisition of images of the same plane scanned in Figure 7.23 for of fv = 100, 200, 300, and 400 Hz. Figures 7.27 and 7.28 show the displacement amplitude and phase images for acquisition of images at different depths for fv = 100 Hz and depths z = 0, 4, 8, and 12 mm.

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Figure 7.25—Displacement amplitude images of bovine muscle section at different vibration frequencies. The units of the colorbars are nanometers. (a) fv = 100 Hz, (b) fv = 200 Hz, (c) fv = 300 Hz, (d) fv = 400 Hz.

Figure 7.26—Displacement phase images of bovine muscle section at different vibration frequencies. The units of the colorbars are degrees. (a) fv = 100 Hz, (b) fv = 200 Hz, (c) fv = 300 Hz, (d) fv = 400 Hz.

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Figure 7.27—Displacement amplitude images of bovine muscle section at different depths. The units of the colorbars are nanometers. (a) z = 0 mm, (b) z = 4 mm, (c) z = 8 mm, (d) z = 12 mm.

Figure 7.28—Displacement phase images of bovine muscle section at different depths. The units of the colorbars are degrees. (a) z = 0 mm, (b) z = 4 mm, (c) z = 8 mm, (d) z = 12 mm.

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7.10 Bovine Muscle Section Imaging Discussion

Figure 7.24 shows motion of greater than 5 μm along what appears to be two muscle fibers running vertically. The fibers are partially resolved in the ultrasonic C-scan but not in the vibro-acoustic image. It should be noted that the vibro-acoustic image was acquired using 3.0 MHz ultrasound waves so the resolution is different. The vibration phase is relatively uniform across the whole plane. This is not largely unexpected because the tissue is fairly homogenous except for the anisotropy of the muscle fibers. When the vibration frequency was varied, the displacement amplitude decreased as frequency increased, indicating a low-pass filter characteristic. The displacement phase is relatively constant for a given frequency, but the mean value decreases at higher frequencies. When the depth of focus was moved deeper into the muscle, the displacement amplitude decreased. This reduction of motion can be attributed to attenuation of the ultrasound waves used for excitation. The displacement phase is fairly constant but does have more variation at deeper depths. The images are able to show features that the ultrasonic C-scan and vibro-acoustic images can not. This added motion information may aid in understanding the tissue structure and complement the information contained in the other two images. One limitation to scanning with motion detection is the time that it takes to acquire the data. Each pixel took 50 ms to acquire so for an image of 100 x 100 pixels, the total acquisition time was 500 seconds plus overhead for transferring and storing data and controlling the motors that moved the transducer. This acquisition could be sped up by using a higher value of fr and scanning at coarser resolution so that the motor speed

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could be increased. The motor speed was limited to prevent motion artifacts within a pixel.

7.11 Conclusion

Experimental multifrequency radiation force excitation and motion detection was performed. A parameterized analysis of motion detection in a scattering gelatin phantom and a section of bovine muscle were performed. As in Chapter 6, it was found that having vibration with larger amplitudes reduces the variation in both displacement and phase measurements. The motion detection results in the bovine muscle showed much less variation in the measurements for lower values of displacement amplitude as compared to results in the gelatin phantom. These experimental results provide confidence in measuring displacement and phase with high precision in tissue. Lastly, this measurement technique has been extended to an imaging method for displacement amplitude and phase at multiple frequencies.

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Chapter 8 Discussion and Summary of Thesis

We are drowning in information, while starving for wisdom. The world henceforth will be run by synthesizers, people able to put together the right information at the right time, think critically about it, and make important choices wisely. E.O. Wilson

8.1 Introduction

This chapter will discuss and summarize the work presented in this thesis. The discussion will center on the advances made in this work and what these advances mean for current practice in ultrasound research and elasticity imaging. The discussion will also present future directions for the techniques presented. The summary will address how the advances affect current and future applications of ultrasound in medicine and engineering.

8.2 Discussion

The main advances in this thesis work have been contributions to performing vibro-acoustography and vibrometry in a new, multiplexed manner. This work has produced techniques and tools for excitation and measurement of small amplitude harmonic motion. Also, models for simulating vibration responses have been presented, which serve to further understanding of current and future applications.

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8.3 Multifrequency Radiation Force Excitation

Multifrequency radiation force excitation provides a method to induce motion with multiple frequency components simultaneously. The acoustic emission or motion information can be acquired and filtered properly to obtain the information at each of the frequency components that were excited. This multiplexing of frequency information into one measurement can provide more information about an object that would otherwise require multiple excitations or scans to acquire the same information. Because the motion has very small amplitude, it is reasonable to assume that the force and motion are linear. Because of this linearity, Fourier based methods may be utilized to extract information at different frequencies. Four different unique methods have been presented to perform multifrequency ultrasound radiation force excitation. The first three methods were presented in Chapter 3 and involved the application of sums of harmonic signals and use of different transducer apertures. Multiharmonic single element (MHSE) excitation can be performed by adding ultrasound signals of slightly different frequencies together and applying this signal to a single element transducer. Single harmonic multielement (SHME) excitation is produced by using ultrasound signals of slightly different frequencies on different elements or groups of elements in an array transducer. Multiharmonic multielement (MHME) excitation is performed when different sums of ultrasound signals of slightly different frequencies are applied to different elements or groups of elements of an array transducer. The other method used to produce multifrequency radiation force is the harmonic pulsed excitation (HPE) method in which tonebursts of ultrasound are repeated at frequency fr, which produces radiation force at harmonics of fr. These frequency

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components are weighted by a sinc function that depends on the length of the ultrasound toneburst. There are certain advantages for each of these methods which make them more attractive for different applications. MHSE and HPE create a multifrequency radiation force that has the same point-spread function (PSF) for each low-frequency component. These methods are most useful for measurements and imaging based on the velocity or displacement of the radiation force induced motion because using the whole aperture of a transducer will provide the smallest beam width, the lowest sidelobe levels, and the highest radiation force. The MHSE and HPE methods will produce radiation force in the path from the transducer to the focus of the transducer. This may not be desirable for vibroacoustography because acoustic emission may be created from objects along the path. Also, in the HPE method the ultrasound is turned on and off very quickly at the beginning and end of the toneburst which may create acoustic noise that can corrupt or overwhelm acoustic emission signals. The SHME method provides localized dynamic radiation force where the different elements or regions of the transducer that are being excited with different ultrasound signals are co-focused. This provides the optimal way to perform multifrequency vibro-acoustography because the radiation force is localized and acoustic emission signals should only originate from vibration occurring in that local region. However, this method requires a transducer with multiple elements or enough elements that can be split into regions. The low-frequency components produced will have different PSF’s because different apertures are used to form each component.

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The MHME method is a combination of MHSE and SHME and therefore combines some of the advantages and disadvantages of each method. For the MHSE, SHME, and MHME methods, multiple ultrasound generators are necessary and hardware constraints may prevent applicability in some circumstances. However, these methods allow the most flexibility in create different low-frequency components provided that ultrasound signals are carefully chosen. In most cases in medical imaging, the ultrasonic intensity that can be deposited is limited, so that intensity must be fractionated to produce different low-frequency components. The more low-frequency components produced comes with a loss in signalto-noise ratio (SNR). However, information from different components may be combined to regain some of the SNR loss. The hydrophones used are very sensitive so fractioning can be done without severe degradation of the images [33]. The HPE method has direct potential to move quickly from an experimental method to implementation in a commercial scanner without significant reconfiguration of the scanner. The method alternates between transmitting excitation tonebursts and tracking pulses. However, the vibration frequencies that the method produces are restricted to multiples of fr. As was mentioned previously, HPE is really only suitable for vibrometry and not vibro-acoustography because of the acoustic noise produced by the switching on and off of the tonebursts. It could be argued that multifrequency excitation is performed with acoustic radiation force impulse (ARFI) imaging since the entire frequency response of an object subjected to an impulsive force can be measured. However, the SNR of this frequency response may be very low. The multifrequency methods introduced in this thesis provides

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samples of the frequency response, but each of the components that are excited and measured have very high SNR because the detection methods are very sensitive if the vibration frequency is known. These multifrequency methods have been used in both vibrometry and vibroacoustography and can be used in point measurements and imaging. The increase in information in the same time for a single scan is very attractive for improving visualization. Models based on dispersive quantities would most benefit from these excitation methods as measurements can be made at many different frequencies simultaneously. Analysis methods that involve using this multifrequency data are key to realizing the method’s full potential. However, the analysis may be very application specific and data would have to be obtained to create a method that would extract important information about the object or tissue under investigation.

8.4 Harmonic Motion Detection

Many methods have been proposed for motion detection using ultrasound. The methods proposed by Zheng, et al., [87] and Hasegawa and Kanai [55] were combined to make a hybrid method. This new method was used in models for motion detection of a harmonically vibrating reflective target and a harmonically vibrating scattering medium. These models provide platforms for optimization of the motion detection methods. The models were parameterized to provide a basis for dimensionless analysis so that results could be extended to different values of vibration frequency. Lastly, experiments were performed to evaluate the conclusions of the models and the implementation of the motion detection for real applications.

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The new hybrid method uses the fast phase shift detection method from Hasegawa and Kanai that corrects for the center frequency of the echo. This technique provides an estimate of the motion. The Kalman filter proposed by Zheng, et al. is a very powerful filter that extracts the amplitude and phase information from a harmonic signal. The power of this filter lies in knowing the vibration frequency as it acts as a narrowband phase-locked loop. Because the filter is tuned to the vibration frequency, it is robust and sensitive to the information at the vibration frequency. With a multifrequency radiation force, multifrequency motion is produced assuming a linear system. The same data can be processed with different values of vibration frequency to extract amplitude and phase information at those frequencies. This method was implemented in a model that tracked the motion of a single reflective scatterer and a group of vibrating scatterers. The results from the single reflective target model can be used in applications with very reflective targets such as an artery wall, calcifications, or in nondestructive evaluation with materials that have large acoustic impedance differences from the surrounding medium. The creation and analysis of propagating waves in the artery for evaluation of stiffness is an emerging area of research. Motion detection has been performed using both a laser vibrometer and Doppler ultrasound. The use of the pulse-echo method with the Kalman filter has demonstrated that sensitive measurements can be made [54]. The motion detection model for a reflective target could be useful in optimization of this method. The vibrating scattering medium model provides a platform for optimization of the motion detection in soft tissue. Analysis of the parameters that lead to better results

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was performed. The results corresponded well with the results from the model with a vibrating reflective target. The parameters that were found to have the most effect on minimizing error were increasing the values of the displacement amplitude, the number of vibration cycles sampled, signal-to-noise ratio of the ultrasound echoes, and the number of points sampled per vibration cycle. However, optimizing each of these parameters is limited in practice. The displacement amplitude is limited by the allowable ultrasound intensity to comply with safety regulations. An increase in the number of vibration cycles used will increase acquisition time of the measurements or images. For point measurements, this may not be critical, but for imaging, this may become substantial. The SNR is limited by the echogenecity of the material and the noise of the motion detection system. The number of points sampled per vibration cycle is limited by focal depth and increasing this parameter will increase the data size and processing time. Performance specifications for different applications can be established and parameters to meet these specifications could be found using the results of the simulation studies or future studies. Experimental results for excitation and motion detection of the stainless steel sphere, the scattering gelatin phantom, and the bovine muscle phantom confirm some of the conclusions found in the models. The most consistent findings in the experimental results were that increasing the displacement amplitude and slow time sampling time was found to decrease variation and error in the displacement and phase measurements. These findings were also predicted by the model results. The effects of the pulse repetition frequency for the tracking pulses was more ambiguous and only when this parameter was

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very low were the results significantly different, which may be attributed to aliasing artifacts.

8.4 Modeling

With the new multifrequency radiation force excitation and harmonic motion detection methods, we have the ability to make measurements at multiple frequencies simultaneously. This amplitude and phase data could be coupled with different models of involving stiffness, viscosity, relaxation, and resonance to extract material properties and understand the object or tissue of interest. Vibrometry of targeted calcifications coupled with the model and results from Chapter 2 could be used for local stiffness and viscosity measurements. Acquisition of the data could be performed with multifrequency excitation and the motion could be processed with the method proposed in this thesis.

8.5 Future Directions

The use of multifrequency radiation force and accompanying motion detection could be used in a number of applications. Shear wave speed is dispersive in viscoelastic materials, and it has been demonstrated that phase measurements can be used to measure the local shear wave speed [99]. With the ability to measure phase in tissue with excellent precision as demonstrated by the results in Chapter 7, we could make multifrequency shear wave speed measurements using a multifrequency radiation force and motion detection at

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offset locations to measure shear wave propagation. This would allow quantitative estimation of the local shear elasticity and viscosity of different tissues. Phase aberration of ultrasound beams results in media with inhomogeneities of tissue sound speed. Recently, a method has been proposed to use radiation force and vibrometry optimization to focus an ultrasonic beam [106]. Future development of this method will rely on the use of a model of a vibrating scattering medium presented in Chapter 6 and the motion detection method described in Chapter 4. Recently, there has been a large amount of growth in the prospects for delivering therapeutic doses of ultrasound to ablate cancerous lesions. However, a monitoring strategy has not been well established. Evidence has been shown that as the tissue coagulates it becomes stiffer and as result moves less when probed with radiation force or deformed with some other stress source [118-120]. The multifrequency excitation and motion detection methods could be used to monitor this therapy with appropriate modeling. Transducer modeling could be further utilized to simulate and explore the use of multifrequency excitation fields in emerging applications for vibro-acoustography and vibrometry. Signal processing methods for use in motion detection methods are constantly being introduced and refined. Future study in the implementation and testing of these methods could provide even more sensitive amplitude and phase measurements.

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8.6 Significant Academic Achievements

This thesis work has produced significant academic achievements including three published peer-reviewed papers, seven conference proceedings papers on which I was an author, six conference abstracts on which I was an author, an invention disclosure, and various student awards. These accomplishments are listed below.

8.6.1 Peer-reviewed Papers

The peer reviewed papers encompass the material presented in Chapters 2 and 3 and an application related to this thesis work. Matthew W. Urban, Miguel Bernal, James F. Greenleaf, “Phase aberration correction using ultrasound radiation force and vibrometry optimization,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control. In press. Matthew W. Urban, Glauber T. Silva, Mostafa Fatemi, James F. Greenleaf, Multifrequency vibro-acoustography. IEEE Transactions on Medical Imaging. vol. 25, no. 10, pp. 1284-1295. 2006. Matthew W. Urban, Randall R. Kinnick, James F. Greenleaf, Measuring the phase of vibration of spheres in a viscoelastic medium as an image contrast modality. Journal of the Acoustical Society of America. 118 (6), 3465-3472. 2005.

8.6.2 Conference Proceedings

I was first author on four of the seven conference proceedings of which I was an author. Matthew W. Urban, Miguel Bernal Restrepo, James F. Greenleaf, Phase aberration correction using ultrasound radiation force and vibrometry optimization. 2006 IEEE Ultrasonics Symposium. p. 132-135. 2006. James F. Greenleaf, Mostafa Fatemi, Silva, Glauber T., Matthew W. Urban, Vibroacoustography: the most promising approaches and inferred needs for transducers and arrays. 2006 IEEE Ultrasonics Symposium. p. 2322-2324. 2006.

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Randall R. Kinnick, Matthew W. Urban, James F. Greenleaf, Sum frequency imaging during vibro-acoustography of tissue. 2006 IEEE Ultrasonics Symposium. p. 1-4. 2006. Matthew W. Urban, Glauber T. Silva, Randall R. Kinnick, Mostafa Fatemi, James F. Greenleaf, Stress field formation for multifrequency vibro-acoustography. 2005 IEEE Ultrasonics Symposium. p. 2275-2278. 2005. Matthew W. Urban, James F. Greenleaf, Dynamic signal arrival correction for vibroacoustography image formation. 2005 IEEE Ultrasonics Symposium. p. 26-29. 2005. Glauber T. Silva, Matthew W. Urban, Image formation of multifrequency vibroacoustography: Theory and computational simulations. Brazilian Symposium on Computer Graphics and Image Processing (SIBGRAPI), p. 5-12. 2005. Matthew W. Urban, Randall R. Kinnick, James F. Greenleaf, Measuring the phase of vibration of spheres in a viscoelastic medium using vibrometry. Acoustical Imaging 28. Springer: Dordrecht, The Netherlands. p. 119-126. 2007.

8.6.3 Conference Abstracts

I was first author on five of the six conference abstracts of which I was an author. Matthew W. Urban, James F. Greenleaf, Phase aberration correction for a linear array transducer using ultrasound radiation force and vibrometry optimization: simulation study. Journal of the Acoustical Society of America. 120 (5) Pt. 2, p. 3271. 2006. 152nd Meeting of the Acoustical Society of America, Honolulu, HI. 2006. Azra Alizad, Dana H. Whaley, Matthew W. Urban, Randall R. Kinnick, James F. Greenleaf, Mostafa Fatemi, Spectral characteristics of breast vibro-acoustography images. Journal of the Acoustical Society of America. 120 (5) Pt. 2, p. 3269. 2006. 152nd Meeting of the Acoustical Society of America, Honolulu, HI. 2006. Matthew W. Urban, James F. Greenleaf, Motion detection for vibro-acoustography. Journal of the Acoustical Society of America. 119 (5) Pt. 2, p. 3465. 2006. 151st Meeting of the Acoustical Society of America, Providence RI. 2006. Matthew W. Urban, Mostafa Fatemi, James F. Greenleaf, Multifrequency vibroacoustography: theory and imaging applications. Ultrasonic Imaging. vol. 28, (1), p. 59. Jan. 2006. 31st International Symposium on Ultrasonic Imaging and Tissue Characterization. Arlington, VA. 2006. Matthew W. Urban, James F. Greenleaf, Coded excitation for identification of reverberant environment and vibro-acoustography spectral measurements. 2006 AIUM Annual Convention, Washington, D.C. 2006

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Matthew W. Urban, Mostafa Fatemi, James F. Greenleaf, Stress field formation for multifrequency vibro-acoustography: a simulation study. Journal of the Acoustical Society of America. 118 (3) Pt. 2, p. 2006. 2005. 150th Meeting of the Acoustical Society of America, Minneapolis, MN. 2005.

8.6.4 Invention Disclosure

An invention disclosure was submitted to Mayo Medical Ventures for a method that utilized portions of this thesis work. Phase aberration correction using ultrasound radiation force and vibrometry optimization. Urban, Matthew W. Urban, Miguel Bernal, James F. Greenleaf, Mayo Medical Ventures. Jan. 2006.

8.6.5 Academic Awards

In conjunction with presenting portions of this thesis work, I was awarded a Student Travel Award and was a Student Paper Competition Winner at the 2005 IEEE International Ultrasonics Symposium. The paper was on multifrequency vibroacoustography and is the fourth reference in section 8.6.2. I was also the recipient of a Best Student Paper Award given by the Biomedical Ultrasound/Bioresponse to Vibration Technical Committee at the 151st Meeting of the Acoustical Society of America. The paper was about motion detection and is the third reference in section 8.6.3.

8.7 Summary

The multifrequency radiation force excitation and harmonic motion detection methods presented in this thesis provide tools that can be extended to many different applications for flexible spectroscopic studies of materials and tissues. The multifrequency excitation can save time in acquiring data that would have required multiple measurements or scans. The motion detection method is very sensitive in

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extracting amplitude and phase of small amplitude harmonic motion and could be utilized in a wide range of medical and engineering applications. The methods presented in this thesis offer advances in the area of elasticity imaging with a new way to deform tissue with a multifrequency radiation force and techniques to perform sensitive measurements of the induced motion. The ability to make these measurements coupled with sophisticated models could aid in the diagnosis and understanding of material properties in viscoelastic soft tissues.

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Appendix 1 Multifrequency Radiation Stress Field Derivation

The purpose of this appendix is to derive how the frequency components of multifrequency radiation force are determined. This analysis will take the case of NS = 2 and NE = 2 used in Chapter 3. The results can be extended to other combinations of NS and NE with appropriate considerations. We will consider using four different ultrasound signals with frequencies ω1, ω2,

ω3, and ω4 that have associated starting phases φ1, φ2, φ3, and φ4 such that a = ω1t + φ1, b = ω2t + φ2, c = ω3t + φ3, and d = ω4t + φ4. Using these defined arguments we can define two pressure signals produced by adding two ultrasound signals of different frequencies p1 ( t ) = A cos ( a ) + B cos ( b ) ,

(A1.1)

p2 ( t ) = C cos ( c ) + D cos ( d ) ,

(A1.2)

where the coefficients A, B, C, and D are dependent on the geometry of the transducer. The total pressure experienced at the focus is a sum of p1(t) and p2(t), p ( t ) = p1 ( t ) + p2 ( t ) .

(A1.3)

The dynamic radiation force, F, produced by these waves can be computed using [34] F = dr S E ,

(A1.4)

where dr is the drag coefficient, S is the projected area of the ultrasound waves and 〈E〉 is the short-term time average of the energy density. The energy density is given as

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p2 (t ) E= , ρ c02

(A1.5)

where ρ is the density of the medium and c0 is the sound speed of the medium and the short-term time average can be computed as [34]

ξ (t )

t +T 2 T

=

∫ ξ (τ ) dτ ,

(A1.6)

t −T 2

where T is chosen such that 2π/ω1 << T << 4π/(ω2-ω1). Essentially, the short-term time average acts as a low-pass filter function to extract the low-frequency components from the energy density. So to find the multifrequency radiation force we must find 〈E〉. We start by evaluating p 2 ( t ) = ( p1 ( t ) + p2 ( t ) ) = p12 ( t ) + 2 p1 ( t ) p2 ( t ) + p22 ( t ) ,

(A1.7)

p12 ( t ) = A2 cos 2 ( a ) + 2 AB cos ( a ) cos ( b ) + B 2 cos 2 ( b ) ,

(A1.8)

2

Using the property that cos(a)cos(b) = 1/2[cos(a – b) + cos(a + b)], the first and third terms of (A1.7) are p12 ( t ) = A2 cos 2 ( a ) + AB cos ( a − b ) + AB cos ( a + b ) + B 2 cos 2 ( b ) ,

(A1.9)

p22 ( t ) = C 2 cos 2 ( c ) + CD cos ( c − d ) + CD cos ( c + d ) + D 2 cos 2 ( d ) ,

(A1.10)

The second term of (A1.7) can be written and then expanded as 2 p1 ( t ) p2 ( t ) = 2 ( A cos ( a ) + B cos ( b ) ) ( C cos ( c ) + D cos ( d ) ) , 2 p1 ( t ) p2 ( t ) = 2 ⎡⎣ AC cos ( a ) cos ( c ) + AD cos ( a ) cos ( d ) + BC cos ( b ) cos ( c ) + BD cos ( b ) cos ( d ) ⎤⎦

,

(A1.11)

(A1.12)

184

⎛1⎞ 2 p1 ( t ) p2 ( t ) = 2 ⎜ ⎟ ⎡⎣ AC cos ( a − c ) + AC cos ( a + c ) + AD cos ( a − d ) + ⎝2⎠ , AD cos ( a + d ) + BC cos ( b − c ) + BC cos ( b + c ) +

(A1.13)

BD cos ( b − d ) + BD cos ( b + d ) ⎤⎦

2 p1 ( t ) p2 ( t ) = ⎡⎣ AC cos ( a − c ) + AC cos ( a + c ) + AD cos ( a − d ) + AD cos ( a + d ) + BC cos ( b − c ) + BC cos ( b + c ) +

,

(A1.14)

BD cos ( b − d ) + BD cos ( b + d ) ⎤⎦ The full expansion of (A1.7) is given as p 2 ( t ) = A2 cos 2 ( a ) + AB cos ( a − b ) + AB cos ( a + b ) + B 2 cos 2 ( b ) + C 2 cos 2 ( c ) + CD cos ( c − d ) + CD cos ( c + d ) + D 2 cos 2 ( d ) + ⎡⎣ AC cos ( a − c ) + AC cos ( a + c ) + AD cos ( a − d ) + AD cos ( a + d )

,

(A1.15)

+ BC cos ( b − c ) + BC cos ( b + c ) + BD cos ( b − d ) + BD cos ( b + d ) ⎤⎦ Using the property cos2(a) = (1 + cos(2a))/2, (A1.15) can be further expanded as ⎛ 1 + cos ( 2a ) ⎞ 2 ⎛ 1 + cos ( 2b ) ⎞ p 2 ( t ) = A2 ⎜ ⎟ + AB cos ( a − b ) + AB cos ( a + b ) + B ⎜ ⎟ 2 2 ⎝ ⎠ ⎝ ⎠ ⎛ 1 + cos ( 2c ) ⎞ 2 ⎛ 1 + cos ( 2d ) ⎞ + C2 ⎜ ⎟ + CD cos ( c − d ) + CD cos ( c + d ) + D ⎜ ⎟ , (A1.16) 2 2 ⎝ ⎠ ⎝ ⎠ + ⎡⎣ AC cos ( a − c ) + AC cos ( a + c ) + AD cos ( a − d ) + AD cos ( a + d ) + BC cos ( b − c ) + BC cos ( b + c ) + BD cos ( b − d ) + BD cos ( b + d ) ⎤⎦

Performing the short-term time average eliminates all terms with a sum frequency or a doubling of any particular frequency, reducing them to zero. This elimination occurs because the average of a high frequency signal over the time course of a cycle of one of the difference frequencies, tends towards zero. The short-term time average of the energy density is

185

E =

1 ⎛ A2 + B 2 + C 2 + D 2 ⎞ 1 ⎜ ⎟ + 2 ⎡⎣ AB cos ( a − b ) + CD cos ( c − d ) + 2 ρ c02 ⎝ , ⎠ ρ c0

(A1.17)

AC cos ( a − c ) + AD cos ( a − d ) + BC cos ( b − c ) + BD cos ( b − d ) ⎤⎦ This can be substituted into (A1.4) to find the dynamic radiation force along with the expressions for a, b, c, and d F=

d r S ⎛ A2 + B 2 + C 2 + D 2 ⎞ d r S ⎜ ⎟ + 2 AB cos ⎡⎣(ω1 − ω2 ) t + φ1 − φ2 ⎤⎦ + ρ c02 ⎝ 2 ⎠ ρ c0

{

CD cos ⎡⎣(ω3 − ω4 ) t + φ3 − φ4 ⎤⎦ + AC cos ⎡⎣(ω1 − ω3 ) t + φ1 − φ3 ⎤⎦ +

, (A1.18)

AD cos ⎡⎣(ω1 − ω4 ) t + φ1 − φ4 ⎤⎦ + BC cos ⎡⎣(ω2 − ω3 ) t + φ2 − φ3 ⎤⎦ +

}

BD cos ⎡⎣(ω2 − ω4 ) t + φ2 − φ4 ⎤⎦

This radiation force equation would be the same if NS = 4 and NE = 1 or NS = 1 and NE = 4 and only the coefficients A, B, C, and D would change because of the different configurations used for the transducer. Also, the expression for dynamic radiation force with NS = 1 and NE = 2 could be found by setting C = D = 0. This would reduce (A1.18) to ⎞ d r S ⎛ A2 + B 2 + AB cos ⎡⎣(ω1 − ω2 ) t + φ1 − φ2 ⎤⎦ ⎟ , F= 2⎜ ρ c0 ⎝ 2 ⎠

(A1.19)

186

Appendix 2 Harmonic Pulsed Excitation Radiation Force Function Derivation

Working with Bracewell’s notation and conventions, the radiation force function can be modeled as a time convolution of a impulse train, III(t), with a time-shifted rectangular function, II(t) [110]. The time-shifted rectangular function can be modeled as ⎛ t − Tb 2 ⎞ g ( t ) = a II ⎜ ⎟ , ⎝ Tb ⎠

(A2.1)

where a is the amplitude of the rectangular function, Tb is the width of the rectangular function, and the time shift is half the width of the function. The impulse train, repeated with period Tr, or at a rate fr = 1/Tr, with unit height can be written as h (t ) =

1 ⎛ t ⎞ III ⎜ ⎟ = f r III ( f r t ) , Tr ⎝ Tr ⎠

(A2.2)

The radiation force function is f(t) = g(t)⊗h(t) ⎛ t − Tb 2 ⎞ 1 ⎛ t f ( t ) = a II ⎜ ⎟ ⊗ III ⎜ ⎝ Tb ⎠ Tr ⎝ Tr

⎞ ⎟, ⎠

(A2.3)

The following Fourier properties will be used in solving this problem, where f(x) is the function of x and F(s) is the Fourier transform of f(x) and if g(x) is a function of x then G(s) is the Fourier transform of g(x). Equations (A2.4) and (A2.5) provide Fourier transforms of II(x) and III(x) where ℑ is the symbol over the double arrow is used to

187

denote a Fourier transform. Equations (A2.6)-(A2.8) are known as the similarity, shift, and convolution theorems. ℑ II( x) ←⎯ → sinc ( s ) ,

(A2.4)

ℑ III( x) ←⎯ → III ( s ) ,

(A2.5)

ℑ → f ( ax ) ←⎯

1 ⎛s⎞ F ⎜ ⎟, a ⎝a⎠

(A2.6)

ℑ f ( x − a ) ←⎯ → e − i 2π as F ( s ) ,

(A2.7)

ℑ f ( x ) ⊗ g ( x ) ←⎯ → F (s)G (s) ,

(A2.8)

The function III(x) can also be written as ∞

∑ δ ( x − n) .

III ( x ) =

(A2.9)

n =−∞

If we use the similarity theorem, III ( ax ) =

1 a

∞

⎛

n⎞

∑ δ ⎜⎝ x − a ⎟⎠ .

(A2.10)

n =−∞

To find the Fourier transform of g(t), G(f), we will use the transform of II(x), the similarity theorem, and the shift theorem. g1 ( t ) = a II ( t ) , g 2 ( t ) = g1 ( t − Tb 2 ) , ⎛ t g3 ( t ) = g 2 ⎜ ⎝ Tb

⎞ ⎟, ⎠

G1 ( f ) = a sinc ( f ) ,

G2 ( f ) = e

− i 2π

Tb f 2

G1 ( f ) = e −iπ Tb f G1 ( f ) ,

(A2.11) (A2.12)

(A2.13)

(A2.14)

(A2.15)

188

G3 ( f ) = TbG2 (Tb f ) ,

(A2.16)

G3 ( f ) = TbG2 (Tb f ) = Tb e− iπ Tb f G1 (Tb f )

.

(A2.17)

= aTb e− iπ Tb f sinc (Tb f ) To find the Fourier transform of h(t), H(f), we will use the transform of III(x) and the similarity theorem, 1 ⎛ t ⎞ III ⎜ ⎟ , Tr ⎝ Tr ⎠

(A2.18)

Tr III (Tr f ) = III (Tr f ) , Tr

(A2.19)

h (t ) =

H(f )=

H ( f ) = III (Tr f ) =

1 Tr

∞

⎛

n =−∞

⎝

n⎞ ⎟, r ⎠

∑ δ ⎜ f −T

(A2.20)

To calculate F(f), we use the convolution theorem and multiply the Fourier transforms of g(t) and h(t) F ( f ) = G( f )H ( f ).

(A2.21)

The III(f) function acts as a sampling function. When multiplying any function

Y(f) with III(f) yields a sampling of Y(f). F ( f ) = aTb e − iπ Tb f sinc (Tb f ) III (Tr f ) , F( f )=

aTb Tr

∞

∑e

− iπ Tb n Tr

n =−∞

F ( f ) = af rTb

∞

∑e

n =−∞

− iπ Tb nf r

(A2.22)

⎛ T ⎞ ⎛ n⎞ sinc ⎜ n b ⎟ δ ⎜ f − ⎟ , Tr ⎠ ⎝ Tr ⎠ ⎝

(A2.23)

sinc (Tb nf r ) δ ( f − nf r ) ,

(A2.24)

This function in the frequency domain will be a set of impulses located at integral multiples of fr and modulated by af rTb e −iπ Tb nfr sinc (Tb nf r ) .

189

Appendix 3 List of Acronyms

MRI

Magnetic resonance imaging

CW

Continuous wave

SSI

Supersonic shear imaging

MRE

Magnetic resonance elastography

AM

Amplitude modulated

PSF

Point-spread function

F number

Focal number

MHSE

Multiharmonic single element

SHME

Single harmonic multielement

SHMG

Single harmonic multigroup

MHME

Multiharmonic multielement

MHMG

Multiharmonic multigroup

ME

Multielement

MH

Multiharmonic

S

Single

M

Multi

rms

root-mean-square

SNR

Signal-to-noise ratio

FM

Frequency modulated

190

FWHM

Full-width half maximum

ARFI

Acoustic radiation force impulse

RF

Radiofrequency

HPE

Harmonic pulsed excitation

TTL

Transistor-transistor logic

T/R

Transmit/receive

BPF

Bandpass filter

A/D

Analog-to-Digital

C-scan

Constant depth scan

191

Appendix 4 List of Symbols

Symbols are listed in the order of their appearance

Chapter 1

σ

Normal stress, Pa

F

Force, N

A

Area, m2

ε

Normal strain, %

L0

Original length, m

L

Deformed length, m

τ

Shear stress, Pa

α

Shear strain, rad

E

Elastic or Young’s Modulus, Pa

G

Shear Modulus, Pa

v

Poisson’s ratio

cl

Longitudinal sound speed, m/s

λ

Bulk modulus, Pa

cs

Shear wave speed, m/s

μ

Shear modulus, Pa

ρ

Mass density, kg/m3

λ1

Bulk elasticity, Pa 192

λ2

Bulk viscosity, Pa⋅s

ω

Angular frequency, rad/s

i

Imaginary number,

μ1

Shear elasticity, Pa

μ2

Shear viscosity, Pa⋅s

Z

Acoustic impedance, Rayl

t

Time, s

z

Axial depth, m

fd

Doppler frequency shift, Hz

f0

Center frequency, Hz

v0

Velocity, m/s

θ

Doppler angle, rad

Δf

Difference frequency, Hz

−1

Chapter 2

f0

Center frequency, Hz

Δf

Difference frequency, Hz

F

Force, N

a

Radius of sphere, m

Y

Radiation force function

E

Energy density, J/m3

〈⋅〉

Short-term time average

E0

Energy density of incident pressure field, J/m3

193

P0

Incident pressure, Pa

ρ

Mass density, kg/m3

c

Longitudinal sound speed, m/s

Fd

Dynamic radiation force, N

θd

Phase shift of the radiation force compared to the incident field, rad

Δω

Angular difference frequency, rad/s

ω1

Angular frequency of wave 1, rad/s

ω2

Angular frequency wave 2, rad/s

V

Velocity, m/s

Zr

Radiation impedance, kg/s

Zm

Mass impedance, kg/s

k

ρΔω 2 ( 2μ + λ )

h

ρΔω 2 μ

λ1

Bulk elasticity, Pa

λ2

Bulk viscosity, Pa⋅s

i

Imaginary number,

μ1

Shear elasticity, Pa

μ2

Shear viscosity, Pa⋅s

t

Time, s

m

Mass, kg

ρs

Mass density of the sphere, kg/m3

Z

Total impedance, kg/s

−1

194

|⋅|

Absolute value

∠

Angle operator

θV

Phase of velocity, rad

θr

Phase of radiation impedance, rad

θm

Phase of mass impedance, rad

θZ

Phase of total impedance, rad

Chapter 3

f0

Center frequency, Hz

Δf

Difference frequency, Hz

ρ

Mass density, kg/m3

c

Longitudinal sound speed, m/s

φ

Velocity potential

r

Position vector, m

t

Time, s

^

Complex variable

N

Number of ultrasound beams

ωi

Angular frequency of the ith wave, rad/s

ωmn

Difference frequency between wave m and wave n, rad/s

ωm

Angular frequency of wave m, rad/s

ωn

Angular frequency wave n, rad/s

a

Radius of sphere, m

ymn

Radiation force function for ωmn 195

E0

Energy density of incident field, J/m3

〈⋅〉

Short-term time average

ε

Mach number

*

Complex conjugate

σ

Radiation stress, Pa

σmn

Radiation stress for ωmn, Pa

hmn

Point-spread function for ωmn

A

Normalization constant

Gn

Amplitude of nth sinusoidal component

θn

Phase for the nth sinusoidal component, rad

NS

Number of sinusoidal signals summed together

NMS

Number of low-frequency components created using MHSE

NE

Number of elements in transducer array

NSM

Number of low-frequency components created using SHME

NMM

Number of low-frequency components created using MHME

NMM,MH

Number of low-frequency components created by multiharmonic mechanism using MHME

NMM,ME

Number of low-frequency components created by multielement mechanism using MHME

φc

Velocity potential from center element

φa

Velocity potential from annular element

u0

Velocity of the transducer element, m/s

λi

wavelength of the ultrasound wave with frequency ωi, m

196

r

Radial distance to the field point represented on a Cartesian coordinate system, m

x

Azimuthal coordinate, m

y

Elevational coordinate, m

R

Focal length of the transducer, m

J1(⋅)

First order Bessel function of the first kind

a1

Radius of the central disc element, m

a21

Inner radius of the annular element, m

a22

Outer radius of the annular element, m

s

Spatial function of object

M

Magnitude of point-spread function

P

Phase of point-spread function, rad

V(ω)

Frequency response of object, m/s

⊗

Convolution operator

Re{⋅} Real part of argument |⋅|

Absolute value

∠

Angle operator

Φ

Acoustic emission, Pa

Hmn

Medium transfer function for path length l

l

Path length of acoustic propagation, m

Qmn

Total acoustic outflow by the object per unit force

SNR

Signal-to-noise ratio, dB

F

Radiation force, N

dr

Drag coefficient

197

S

Intercepting surface area of the ultrasound waves, m2

p

Pressure, Pa

PSF ,ω0 Pressure for single frequency vibrometry at ω0, Pa FSF,Δω Radiation force for single frequency vibrometry at Δω, N ZΔω

Radiation impedance at Δω, kg/s

USF,Δω Velocity for single frequency vibrometry at Δω, m/s HΔω

Medium transfer function at Δω

QΔω

Acoustic outflow at Δω

FMF,Δω Multifrequency radiation force at Δω, N PMF ,ω0 Pressure for multifrequency vibrometry at ω0, Pa xrms

rms value for signal x(t)

nrms

rms value for noise signal n(t)

yrms

rms value for signal y(t)

T

Averaging time, s

Fs

Sampling frequency, Hz

A

Amplitude of signal

xc

Multiharmonic signal applied to center element

Gc,n

Amplitude of nth wave applied to center element

ωc,n

Angular frequency of nth wave applied to center element, rad/s

fc,n

Frequency of nth wave applied to center element, Hz

θc,n

Phase of nth wave applied to center element, rad

xa

Multiharmonic signal applied to annular element

Ga,n

Amplitude of nth wave applied to annular element 198

ωa,n

Angular frequency of nth wave applied to annular element, rad/s

fa,n

Frequency of nth wave applied to annular element, Hz

θa,n

Phase of nth wave applied to annular element, rad

A(r,ωmn)

Amplitude image

sign{⋅}

Sign function

μ1

Shear elasticity, Pa

μ2

Shear viscosity, Pa⋅s

Chapter 4

cs

Shear wave speed, m/s

ωs

Shear wave frequency, rad/s

Δφ

Difference in phase at two different locations, rad

Δr

Distance between measurement points r1 and r2, m

r1

Position of first phase measurement, m

r2

Position of second phase measurement, m

ωf

Fast time angular frequency of ultrasound waves, rad/s

ωs

Slow time angular vibration frequency, rad/s

D

Displacement, m

D0

Displacement amplitude, m

tf

Fast time, s

ts

Slow time, s

φs

Slow time vibration phase, rad

v

Velocity, m/s

199

v0

Velocity amplitude, m/s

A

Echo amplitude

r

Echo function

φf

Phase of the fast time ultrasound signal, rad

β

Dimensionless amplitude functional

θ

Doppler angle, rad

c

Longitudinal sound speed, m/s

R

Frequency domain representation of r

Δθn

Phase shift between the two echoes, rad

*

Complex conjugation

f0

Center frequency, Hz

Tprf

Pulse repetition period of the pulse-echo interrogation, s

SNR

Signal-to-noise ratio of the ultrasound echoes, dB

Nc

Number of cycles of vibration

Np

Number of points sampled per vibration cycle

T

Acquisition time, s

fv

Vibration frequency, Hz

fprf

Pulse repetition frequency of the pulse-echo interrogation, Hz

z

Axial depth of the vibrating scatterer from the transducer, m

BW

Bandwidth of the transducer, %

lg

Fast time gate length, m

Fs

Sampling frequency, Hz

ff

Fast time ultrasound frequency, Hz

200

xB

Measurement bias

σJ

Measurement jitter

N

Number of data samples

xT

True measurement value

x

Mean of the data samples

λ

Wavelength of fast time ultrasound wave, m

Chapter 5

Tb

Toneburst length in harmonic pulsed excitation, s

fr

Repetition rate of ultrasound tonebursts in harmonic pulsed excitation, Hz

Tr

Repetition period of ultrasound tonebursts in harmonic pulsed excitation, s

Tprf

Pulse repetition period of the pulse-echo interrogation, s

fprf

Pulse repetition frequency of the pulse-echo interrogation, Hz

td

Time delay for onset of the transmission of the motion detection pulses, s

t

Time, s

f(t)

Radiation force function, N

III(t)

Impulse train

II(t)

Rectangular function

a

Amplitude

⊗

Convolution operator

δ

Impulse function

F(f)

Frequency domain representation of radiation force function, N

f0

Center frequency, Hz

201

Δf

Difference frequency, Hz

Fs

Sampling frequency, Hz

w[n]

Hann window function

F0

Normalized force

R

Regression goodness of fit coefficient

Wx

Beam shape parameter in x-direction

Wy

Beam shape parameter in y-direction

Chapter 6

Ex

Excitation beam width in the x-direction, m

Ey

Excitation beam width in the y-direction, m

Ez

Excitation beam width in the z-direction, m

Tx

Tracking beam width in the x-direction, m

Ty

Tracking beam width in the y-direction, m

Tz

Tracking beam width in the z-direction, m

Wx

Beam shape parameter in x-direction

Wy

Beam shape parameter in y-direction

Wz

Beam shape parameter in z-direction

tf

Fast time, s

ts

Slow time, s

N

Number of scatterers

r1

First measured echo

rk

kth measured echo

202

xn

Position of nth scatterer in x-direction, m

yn

Position of nth scatterer in y-direction, m

zn

Position of nth scatterer in z-direction, m

zF

Axial focal depth of transducer, m

I

Weighting function for tracking beam

k

Number of echoes

ω0

Fast time angular frequency of ultrasound waves, rad/s

φf

Phase of the fast time ultrasound signal, rad

tk,n

Time delay of nth scatterer of the kth echo, s

uz,k

Displacement in z-direction measured at the kth echo, m

A

Displacement amplitude, m

D0

Displacement amplitude, m

φs

Slow time vibration phase, rad

ωs

Slow time angular vibration frequency, rad/s

SNR

Signal-to-noise ratio of the ultrasound echoes, dB

Nc

Number of cycles of vibration

Np

Number of points sampled per vibration cycle

Ns

Number of scatterers/resolution cell

BW

Bandwidth of the transducer, %

lg

Fast time gate length, m

Fs

Sampling frequency, Hz

N

Iterations

c

Longitudinal sound speed, m/s

203

Chapter 7

Tb

Toneburst length in harmonic pulsed excitation, s

fr

Repetition rate of ultrasound tonebursts in harmonic pulsed excitation, Hz

fprf

Pulse repetition frequency of the pulse-echo interrogation, Hz

F0

Normalized force

Ts

Slow time processing window, s

Nc

Number of cycles of vibration

Np

Number of points sampled per vibration cycle

R

Regression goodness of fit coefficient

fv

Vibration frequency, Hz

μd

Mean displacement, m

σd

Standard deviation of displacement, m

μφ

Mean phase estimates, rad

σφ

Standard deviation of phase estimates, rad

F

Radiation force, N

z

Axial depth, m

Chapter 8

fr

Repetition rate of ultrasound tonebursts in harmonic pulsed excitation, Hz

Appendix 1

NS

Number of sinusoidal signals summed together

NE

Number of elements

204

ω1

Angular frequency of ultrasound wave 1, rad/s

ω2

Angular frequency of ultrasound wave 2, rad/s

ω3

Angular frequency of ultrasound wave 3, rad/s

ω4

Angular frequency of ultrasound wave 4, rad/s

φ1

Phase of ultrasound wave 1, rad/s

φ2

Phase of ultrasound wave 2, rad/s

φ3

Phase of ultrasound wave 3, rad/s

φ4

Phase of ultrasound wave 4, rad/s

p1

Pressure signal 1, Pa

p2

Pressure signal 2, Pa

t

Time, s

A

Coefficient for ultrasound wave 1, Pa

B

Coefficient for ultrasound wave 2, Pa

C

Coefficient for ultrasound wave 3, Pa

D

Coefficient for ultrasound wave 4, Pa

a

Argument for ultrasound wave 1, rad

b

Argument for ultrasound wave 2, rad

c

Argument for ultrasound wave 3, rad

d

Argument for ultrasound wave 4, rad

p

Total pressure, Pa

F

Radiation force, N

dr

Drag coefficient

S

Intercepting surface area of the ultrasound waves, m2

205

E

Energy density, J/m3

〈⋅〉

Short-term time average

c0

Longitudinal sound speed, m/s

ρ

Mass density, kg/m3

Appendix 2

Tb

Toneburst length in harmonic pulsed excitation, s

fr

Repetition rate of ultrasound tonebursts in harmonic pulsed excitation, Hz

Tr

Repetition period of ultrasound tonebursts in harmonic pulsed excitation, Hz

t

Time, s

g(t)

Time-shifted rectangular function

h(t)

Scaled impulse train

f(t)

Radiation force function, N

III(t)

Impulse train

II(t)

Rectangular function

a

Amplitude

⊗

Convolution operator

ℑ

Fourier transform

δ

Impulse function

G(f)

Frequency domain representation of time-shifted rectangular function

H(f)

Frequency domain representation of scaled impulse train

F(f)

Frequency domain representation of radiation force function, N

206

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