SHEAR ELASTICITY AND SHEAR VISCOSITY IMAGING IN SOFT TISSUE. Yiqun Yang

Transcription

1 SHEAR ELASTICITY AND SHEAR VISCOSITY IMAGING IN SOFT TISSUE By Yiqun Yang A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Electrical Engineering Doctor of Philosophy 2018

2 ABSTRACT SHEAR ELASTICITY AND SHEAR VISCOSITY IMAGING IN SOFT TISSUE By Yiqun Yang In this thesis, a new approach is introduced that provides estimates of shear elasticity and shear viscosity using time-domain measurements of shear waves in viscoelastic media. Simulations of shear wave particle displacements induced by an acoustic radiation force are accelerated significantly by a GPU. The acoustic radiation force is first calculated using the fast near field method (FNM) and the angular spectrum approach (ASA). The shear waves induced by the acoustic radiation force are then simulated in elastic and viscoelastic media using Green s functions. A parallel algorithm is developed to perform these calculations on a GPU, where the shear wave particle displacements at different observation points are calculated in parallel. The resulting speed increase enables rapid evaluation of shear waves at discrete points, in 2D planes, and for push beams with different spatial samplings and for different values of the f-number (f/#). The results of these simulations show that push beams with smaller f/# require a higher spatial sampling rate. The significant amount of acceleration achieved by this approach suggests that shear wave simulations with the Green s function approach are ideally suited for high-performance GPUs. Shear wave elasticity imaging determines the mechanical parameters of soft tissue by analyzing measured shear waves induced by an acoustic radiation force. To estimate the shear elasticity value, the widely used time-of-flight (TOF) method calculates the correlation between shear wave particle velocities at adjacent lateral observation points. Although this method provides accurate estimates of the shear elasticity in purely elastic media, our experience suggests that the TOF method consistently overestimates the shear elasticity values in viscoelastic media because the combined effects of diffraction, attenuation, and dispersion are not considered. To address this problem, we have developed an approach

3 that directly accounts for all of these effects when estimating the shear elasticity. This new approach simulates shear wave particle velocities using a Green s function-based approach for the Voigt model, where the shear elasticity and viscosity values are estimated using an optimization-based approach that compares measured shear wave particle velocities with simulated shear wave particle velocities in the time-domain. The results are evaluated on a point-by-point basis to generate images. There is good agreement between the simulated and measured shear wave particle velocities, where the new approach yields much better images of the shear elasticity and shear viscosity than the TOF method. The new estimation approach is accelerated with an approximate viscoelastic Green s function model that is evaluated with shear wave data obtained from in vivo human livers. Instead of calculating shear waves with combinations of different shear elasticities and shear viscosities, shear waves are calculated with different shear elasticities on the GPU and then convolved with a viscous loss model, which accelerates the calculation dramatically. The shear elasticity and shear viscosity values are then estimated using an optimization-based approach by minimizing the difference between measured and simulated shear wave particle velocities. Shear elasticity and shear viscosity images are generated at every spatial point in a two-dimensional (2D) field-of-view (FOV). The new approach is applied to measured shear wave data obtained from in vivo human livers, and the results show that this new approach successfully generates shear elasticity and shear viscosity images from this data. The results also indicate that the shear elasticity values estimated with this approach are significantly smaller than the values estimated with the conventional TOF method and that the new approach demonstrates more consistent values for these estimates compared with the TOF method. This experience suggests that the new method is an effective approach for estimating the shear elasticity and the shear viscosity in liver and in other soft tissue.

4 Copyright by YIQUN YANG 2018

5 Dedicated to Qianwei Jiang v

6 ACKNOWLEDGEMENTS This thesis is a summary of my Ph.D. work, but more importantly, it is also a milestone for my five years journey at Michigan State University. During the past five years, I ve received a considerable amount of help from several people. Here, I would like to express my sincere thanks to them. First and foremost, I would like to thank my Ph.D. advisor, Prof. Robert McGough. I would like to thank him for providing me this opportunity to work in the Biomedical Ultrasonics and Electromagnetics Lab, and more specifically, on this very interesting project on ultrasound elastography. With his guidance, mentoring, and knowledge, I was able to gradually build up my expertise in medical ultrasound. I ve also learned a lot from his diligence, earnestness, and enthusiasm, which taught me what a researcher should be. I would also like to thank my Ph.D. committee members: Dr. Seungik Baek, Dr. Brian Feeny, and Dr. Lalita Udpa. I would also like to thank them for useful discussions and suggestions on my research. I would like to thank my collaborators at Mayo Clinic. I especially thank Dr. Matthew Urban for sharing knowledge, explaining methods, sending data, and providing insightful discussions for my ultrasound elastography project. I also want to thank Bo Qiang for the great collaborations on the finite element and finite difference simulations. Next, I want to thank all the present and past members in the Biomedical Ultrasonics and Electromagnetics Lab. I thank Xiaofeng Zhao for his help on finite different simulations, and for all of his help since the first day I joined the lab. I especially want to thank him for being the photographer for my wedding, where he took several gorgeous photos of Qianwei and myself. I thank Pedro Nariyoshi for his help on B-mode ultrasound simulations and for answering a lot of my coding questions. I especially want to thank him for his humorous and positive attitude, which makes the research life easier. I would also like to thank Yi Zhu, Peter Beard, Drew Murray, Jim Kelly, and Kenneth Stewart. I thank all of them for the help vi

7 on my projects and the interesting discussions on different aspects of medical ultrasound. Also, I want to thank my graduate student colleagues in the ECE department. I thank Jie Li for tolerating me for a year as a roommate, teaching me what is a Green s function, and providing interesting discussions on computational electromagnetics, which is not quite as interesting as acoustics. I thank Fan Bin for interesting discussions and for his introduction to MEMS, and I would like to thank him for his help inside and outside of graduate school. I especially want to thank the cool guys lunch club (Xiaofeng Zhao, Pedro Nariyoshi, and Fan Bin) for sharing lunch time. I hope that we will have the chance to have lunch together in the future. I thank Xiaorui Wang, Pedro Nariyoshi, Sylmarie Davila, Patrick O Hara, and Yousef Gtat for working as a team in teaching the ECE 331 (microcontroller) lab. I also want to thank Tianlong Song, Runruo Chen, Jiankun Liu, Mingwu Gao, Yu Cheng, Yan Shi, Zhe Wang, Xiaorui Wang, and Biyi Fang for sharing knowledge in their fields that broadened my understanding of electrical engineering. I would like to thank Siemens Healthineers in Issaquah, WA for providing me a valuable internship opportunity during my Ph.D. study so that I could gain some practical experience of medical ultrasound in industry. I thank Liexiang Fan for bringing me into the advanced development team and for sharing his experiences not only in medical ultrasound but also in career building, I thank Yassin Labyed for his great guidance and being the best industry mentor as you can imagine, and I thank Xiaozheng Zeng for recommending this opportunity to me and for providing great help both in work and life. I would like to thank all of the alumni of the Nanjing University Acoustics Alumni Association in America (NJU-AAAA), who shared useful career suggestions with me. I would like to offer special thanks to my undergraduate roommate Tao Sun, who is now at Brigham and Women s Hospital, Harvard Medical School, and Tufts University (yes, that many affiliations), for his discussions on microbubbles and therapeutic ultrasound, and for sharing the latest academic gossip. I also want to thank my two cats, Ramen and Rancho, for their companionship during vii

8 the hardest time of my Ph.D. study. I want to thank my family back in China for their support and care during my Ph.D. study, especially my parents Zhimin Peng and Xiaoning Chen and my grandmom Guozhen Li for their love and encouragement. Lastly and most importantly, I want to thank my wife, Qianwei Jiang, who provided unconditional love and companionship throughout my Ph.D. study. This thesis could not be finished without your love, support, and encouragement. I m deeply thankful that I meet you on orientation day for graduate school (08/16/2012) in the Breslin Center, which then made the days of the following years the best days in my life. Yiqun Yang January, 2018 viii

9 TABLE OF CONTENTS LIST OF TABLES xii LIST OF FIGURES xiii CHAPTER 1 INTRODUCTION Ultrasound elastography Ultrasound elastography techniques Static excitation Natural physiological excitation Transient excitation Acoustic Radiation Force Impulse (ARFI) Imaging Shear Wave Elasticity Imaging (SWEI) Supersonic Shear Imaging (SSI) Spatially Modulated Ultrasound Radiation Force (SMURF) Comb-Push Ultrasound Shear Elastography (CUSE) Harmonic excitation Sonoelasticity imaging Vibro-Acoustography (VA) Harmonic Motion Imaging (HMI) Shear wave Dispersion Ultrasound Vibrometry (SDUV) Review of numerical simulation methods for shear waves induced by an acoustic radiation force Acoustic radiation force simulations Shear wave simulations Motivation and thesis structure CHAPTER 2 BACKGROUND Acoustic radiation force Equation of motion in elastic and viscoelastic soft media Green s function solutions to the equation of motions Green s function for Navier s equation in elastic media Green s function to the Navier s equation in viscoelastic media Shear wave measurements Methods for estimating the mechanical parameters from measured shear waves The time-of-flight (TOF) method The k-space method CHAPTER 3 GPU-BASED GREEN S FUNCTION SIMULATIONS OF SHEAR WAVES GENERATED BY AN APPLIED ACOUSTIC RADIA- TION FORCE IN ELASTIC AND VISCOELASTIC MEDIA Introduction ix

10 3.2 Theory Navier s Equation Green s functions calculations Simulating shear waves with Green s functions Numerical Calculations Acoustic radiation force Parallel implementation Results Simulated shear wave particle displacements at discrete observation points Computation times for serial and parallel calculations Convergence of Green s function calculations Comparisons between simulated and measured shear waves Discussion Diffraction, attenuation, and dispersion Serial vs parallel implementations Convergence Comparisons between measured and simulated shear waves Intended applications Conclusion CHAPTER 4 TIME-DOMAIN ESTIMATION OF SHEAR ELASTICITY AND SHEAR VISCOSITY FOR SHEAR WAVE ELASTICITY IMAG- ING (SWEI) IN VISCOELASTIC SHEAR WAVE PHANTOMS Introduction Theory Acoustic radiation force Shear waves in viscoelastic media Materials and Methods Shear wave measurements Shear wave simulations Phantom experiments Time-of-flight method for shear elasticity estimation Model-based approach for shear elasticity and shear viscosity estimation Results Time of flight parameter estimates evaluated with simulated shear wave data New model-based shear wave parameter estimation approach applied to simulated shear wave data TOF and model-based estimates of shear wave parameters for viscoelastic shear wave phantoms Shear elasticity and shear viscosity imaging in viscoelastic shear wave phantoms Shear elasticity and shear viscosity imaging with different push beams Discussion x

11 4.5.1 Comparisons between the TOF method and the new model-based approach: simulated 3D shear wave data Comparisons between the TOF method and the new model-based approach: viscoelastic shear wave phantoms Shear wave parameter estimation in different phantoms Shear wave parameter estimation with different push beams Conclusion CHAPTER 5 SHEAR ELASTICITY AND SHEAR VISCOSITY ESTIMATION IN LIVER Introduction Theory Acoustic radiation force Navier s equation Green s functions for Navier s equation Methods Acoustic radiation force simulations Calculating shear wave particle velocities induced by an acoustic radiation force using Green s functions Approximate model for shear wave calculation in viscoelastic media Validation of the approximate viscoelastic model Shear wave simulations with the approximate model using GPU Tissue mechanical parameter estimation using a genetic algorithm Validation with In Vivo human data Result Shear wave particle velocities measured in liver Shear elasticity and shear viscosity estimation with the new modelbased approach at a single point in space Shear elasticity and shear viscosity imaging in human liver Shear wave parameter estimates in liver Discussion Shear waves measured with different push beams Shear wave parameter estimates in different locations Shear wave parameter estimates for in vivo liver Conclusion CHAPTER 6 CONCLUSION AND FUTURE WORK Summary of the thesis Future work BIBLIOGRAPHY xi

12 LIST OF TABLES Table 3.1: The FWHM values of the measured and simulated shear wave particle velocities in the x direction evaluated at the peak at 3 ms, 6 ms, and 9 ms for phantom E and E Table 3.2: The FWHM values of the measured and simulated shear wave particle velocities in the x direction evaluated at the peak at 3 ms, 6 ms, and 9 ms for phantom E Table 4.1: Parameters for the three push beams Table 5.1: The average and the standard deviation of the estimated shear elasticity values provided by the new model-based approach for 20 different measurements. The units are kpa for each entry in the table Table 5.2: The average and the standard deviations of the shear elasticity values estimated with the TOF method for 20 different measurements. The units are kpa for each entry in the table Table 5.3: The average and the standard deviations of the shear viscosity values estimated with the new model-based approach for 20 different measurements. The units are Pa s for each entry in the table xii

13 LIST OF FIGURES Figure 3.1: The locations of the observation points relative to a push beam focused at 25 mm in the x-z plane with y = Figure 3.2: Simulated shear wave particle displacements computed with elastic and viscoelastic models for an applied 3D acoustic radiation force push evaluated at the 8 observation points indicated in Figure 3.1. The solid lines describe the normalized shear wave particle displacements calculated for the elastic model with µ = 2.25 kpa and η s = 0 Pa s, and the dashed lines describe the normalized shear wave particle displacements calculated for the viscoelastic model with µ = 2.25 kpa and η s = 1 Pa s. 33 Figure 3.3: Computation times for serial and parallel implementations of time domain shear wave calculations with Green s functions. The solid line with circle markers indicates the computation times for parallel simulations of shear waves with the elastic model, the dashed-dotted line with triangle markers indicates the computation times for parallel simulations of shear waves with the viscoelastic model, the dashed line with diamond markers indicates the computation times for serial simulations of shear waves with the elastic model, and the dotted line with square markers indicates the computation times for serial simulations of shear waves with the viscoelastic model Figure 3.4: 2D cross-sections of 3D ultrasound beams simulated in FOCUS. Figures (3.4a), (3.4b), (3.4c), and (3.4d) show the push beams with f/1.5, f/2, f/3, and f/, respectively. In (3.4a)-(3.4d), each beam is normalized by the maximum intensity value Figure 3.5: Percent differences calculated for shear wave particle displacements simulated with the elastic model. The shear elasticity and shear viscosity values in these simulations are µ = 2.25 kpa and η s = 0 Pa s, where the push beams are calculated with different spatial samplings. The solid line with circle markers indicates the percent difference for the shear wave particle displacements induced by an unfocused push beam with f/, the dotted line with square markers indicates the percent difference for the shear wave particle displacements induced by a focused push beam with f/3, the dash-dotted line with triangle markers indicates the percent difference for the shear wave particle displacements induced by a focused push beam with f/2, and the dashed line with diamond markers indicates the percent difference for the shear wave particle displacements induced by a focused push beam with f/ xiii

14 Figure 3.6: Percent differences calculated for shear wave particle displacements simulated with the elastic model. The shear elasticity and shear viscosity values in these simulations are µ = 2.25 kpa and η s = 1 Pa s, where the push beams are calculated with different spatial samplings. The descriptions of the markers and line types are the same as those given in the caption for Figure Figure 3.7: Measured shear wave particle velocities in phantom E and the corresponding simulated shear wave particle velocities that minimize Eq at three different times. Figs. 3.7(a), 3.7(b), and 3.7(c) show the shear wave particle velocities measured at 3 ms, 6 ms, and 9 ms, respectively. Figs. 3.7(d), 3.7(e), and 3.7(f) show the corresponding simulated shear wave particle velocities computed with µ = 0.3 kpa and η s = 0.1 Pa s at 3 ms, 6 ms, and 9 ms, respectively Figure 3.8: Measured shear wave particle velocities in phantom E and the corresponding simulated shear wave particle velocities that minimize Eq at three different times. Figs. 3.8(a), 3.8(b), and 3.8(c) show the shear wave particle velocities measured at 3 ms, 6 ms, and 9 ms, respectively. Figs. 3.8(d), 3.8(e), and 3.8(f) show the corresponding simulated shear wave particle velocities with µ = 2.3 kpa and η s = 0.1 Pa s at 3 ms, 6 ms, and 9 ms, respectively Figure 3.9: Measured shear wave particle velocities in phantom E and the corresponding simulated shear wave particle velocities that minimize Eq evaluated at three different times. Figs. 3.9(a), 3.9(b), and 3.9(c) show the shear wave particle velocities measured at 2 ms, 4 ms, and 6 ms, respectively. Figs. 3.9(d), 3.9(e), and 3.9(f) show the corresponding simulated shear wave particle velocities computed with µ = 4.9 kpa and η s = 1.5 Pa s at 2 ms, and computed with µ = 4.5 kpa and η s = 1.5 Pa s at 3 ms and 4 ms, respectively Figure 4.1: 2D cross-sections of the three push beams simulated in FOCUS. Figures (4.1a), (4.1b), and (4.1c) show push beams with focal depths of 33 mm, 45 mm, and 70 mm, respectively Figure 4.2: The measured shear wave particle velocity induced by the push beam with a focal depth of 33 mm in phantom E Figures (4.2a), (4.2b), and (4.2c) show the measured shear wave particle velocity at 2 ms, 6 ms, and 10 ms, respectively xiv

15 Figure 4.3: Shear elasticity images generated by the TOF method using noisy 3D shear wave particle velocity data obtained from elastic and viscoelastic simulation models. Fig. 4.3a) shows the estimated shear elasticity for the homogeneous elastic simulation model with µ = 4 kpa and η s = 0 Pa s, and Fig. 4.3b) shows the estimated shear elasticity for the homogeneous viscoelastic simulation model with µ = 4 kpa and η s = 2 Pa s Figure 4.4: Objective functions for the shear wave particle velocities calculated at (x, z) = (10 mm, 30 mm) with the elastic and viscoelastic simulation models. Fig. 4.4a shows the objective function evaluated for the elastic simulation model, and Fig. 4.4b shows the objective function evaluated for the viscoelastic simulation model. The o markers indicate the estimates obtained with the new model-based approach, where µ = 4.00 kpa and η s = 0 in Fig. 4.4a and µ = 4.00 kpa and η s = 1.90 Pa s in Fig. 4.4b. The x markers indicate the estimated values obtained with the TOF method, where µ = 4.11 kpa and η s = 0 in Fig. 4.4a and µ = 5.69 kpa and η s = 0 in Fig. 4.4b Figure 4.5: Shear elasticity and shear viscosity images generated with parameters obtained from the new model-based approach applied to simulated noisy shear wave particle velocities calculated with the elastic and viscoelastic simulation models. Figs. 4.5(a) and 4.5(b) show the shear elasticity and shear viscosity images, respectively, for the elastic simulation model with µ = 4 kpa and η s = 0 Pa s, respectively. Figs. 4.5(c) and 4.5(d) show the shear elasticity and shear viscosity images, respectively, for a viscoelastic simulation model with µ = 4 kpa and η s = 2 Pa s Figure 4.6: The objective function in Eq. 4.9 evaluated for different values of the shear elasticity and shear viscosity for shear wave particle velocities measured at (x, z) = (10 mm, 30 mm) in phantom E The o marker indicates the value obtained with the new model-based estimation approach (µ = 3.60 kpa and η s = 0.80 Pa s), and the x marker indicates the value produced by the TOF method (µ = 6.70 kpa and η s = 0) xv

16 Figure 4.7: Measured and simulated shear wave particle velocities using shear wave parameter values estimated with the time-of-flight (TOF) method and the new model-based estimation approach. In Figs. 4.7a and 4.7b, the blue dashed line indicates the measured shear wave particle velocity induced by the push beam with a focal peak at 33 mm in phantom E2348-1, the black dotted line in Fig. 4.7(a) is the the simulated shear wave particle velocity with the shear wave parameters estimated with the TOF method (µ = 6.70 kpa and η s = 0), and the red solid line in Fig. 4.7(b) is the the simulated shear wave particle velocity with the shear wave parameters (µ = 3.60 kpa and η s = 0.80 Pa s) obtained from the new model-based estimation approach Figure 4.8: Shear wave parameter images generated by the TOF method and the new model-based approach for phantoms E2348-1, E2348-2, and E where the shear wave is induced by a push beam focused at 45 mm. Figs. 4.8(a)-(c) show the shear elasticity image provided by the TOF method, the shear elasticity image provided by the new model-based approach, and the shear viscosity image provided by the new model-based approach evaluated for phantom E2348-1, respectively. Figs. 4.8(d)-(f) show the shear elasticity image provided by the TOF method, the shear elasticity image provided by the new model-based approach, and the shear viscosity image provided by the new model-based approach evaluated for phantom E2348-2, respectively. Figs. 4.8(g)-(i) show the shear elasticity image provided by the TOF method, the shear elasticity image provided by the new model-based approach, and the shear viscosity image provided by the new model-based approach evaluated for phantom E2348-3, respectively Figure 4.9: Shear wave parameter images generated by the TOF method and the new model-based approach for phantom E Figs. 4.9(a), 4.9(e), and 4.9(i) show the shear elasticity images generated by the TOF method with push beams focused at 33 mm, 45 mm, and 70 mm, respectively. Figs. 4.9(b), 4.9(f), and 4.9(j) show the shear elasticity images generated by the new model-based approach with push beams focused at 33 mm, 45 mm, and 70 mm, respectively. Figs. 4.9(c), 4.9(g), and 4.9(k) show the shear viscosity images generated by the new model-based approach with push beams focused at 33 mm, 45 mm, and 70 mm, respectively. Figs. 4.9(d), 4.9(h), and 4.9(l) show the images of the difference between the measured shear wave particle velocity and the simulated shear wave particle velocity for push beams focused at 33 mm, 45 mm, and 70 mm, respectively xvi

17 Figure 4.10: Comparison of the shear elasticity values estimated in three CIRS viscoelastic shear wave phantoms with the new model-based approach and the time-of-flight method. The three groups of bars represent the results for three different phantoms. For each group of bars, the first three bars indicate the result obtained with the new model-based approach, and the last three bars represent the result given by the time-of-flight method. The error bars represent the standard deviation of the estimated elasticity calculated from ten measurements Figure 4.11: Estimated viscosity values obtained with the model-based approach in three CIRS viscoelastic shear wave phantoms. Three different groups of bars describe the results for three different phantoms. The three bars in each group indicate the results generated by pushes at different depths. The error bars indicate the standard deviation of the estimated elasticity from ten measurements Figure 5.1: The results of the individual calculation steps that produce an approximate shear wave particle displacement. Fig. 5.1a shows the push beam, and the red dot indicates the observation point, which is located at (x, z) = (10 mm, 22.8 mm). Fig. 5.1b shows the shear wave particle displacement simulated with the elastic model Eq Fig. 5.1c shows the attenuation term calculated with Eq Fig. 5.1d shows the approximate shear wave particle displacement calculated by convolving the results shown in Fig. 5.1b and Fig. 5.1c Figure 5.2: Comparison between the shear wave particle displacements simulated with the 3D viscoelastic model and approximate model. The blue lines indicate the shear wave particle displacements calculated with the viscoelastic model, and the red dashed lines represent shear wave particle displacements computed with the approximate model Figure 5.3: A 2D image showing the percent difference between shear wave particle displacements calculated with the approximate model in Eq and the viscoelastic model in Eqs The black contour represents a value of 5% for the percent difference Figure 5.4: Two push beams that induce shear waves in human liver. Fig. 5.4a shows the push beam focused at 35 mm, and Fig. 5.4b shows the push beam focused at 50 mm Figure 5.5: The shear wave particle velocities measured in human liver. Figs. 5.5(a)- 5.5(c) show the shear wave particle velocities induced by the push beam focused at 35 mm at 2 ms, 4 ms, and 6 ms, respectively, and Figs. 5.5(d)- 5.5(f) show the shear wave particle velocities induced by the push beam focused at 50 mm at 2 ms, 4 ms, and 6 ms, respectively xvii

18 Figure 5.6: Comparisons between measured shear wave particle velocities and shear wave particle velocities simulated with the µ and η s values estimated using the new model-based approach at different locations in space. The blue solid lines indicate the measured shear wave particle velocities, and the red dashed lines represent the simulated shear wave particle velocities. Fig. 5.6(a) shows the measured and simulated shear wave particle velocities at (x, z) = (5 mm, 30 mm), Fig. 5.6(b) shows the measured and simulated shear wave particle velocities at (x, z) = (6 mm, 35 mm), Fig. 5.6(c) shows the measured and simulated shear wave particle velocities at (x, z) = (3 mm, 30 mm), and Fig. 5.6(d) shows the measured and simulated shear wave particle velocities at (x, z) = (15 mm, 30 mm) Figure 5.7: Shear wave parameter images for human liver produced by a push beam focused at 35 mm. Fig. 5.7(a) shows the shear elasticity image generated by the TOF method. Fig. 5.7(b) shows the shear elasticity image generated by the new model-based approach. Fig. 5.7(c) shows the shear viscosity image generated by the new model-based approach. Fig. 5.7(d) shows the percent difference between the measured shear waves and the simulated shear waves with the µ and η s values estimated with the new model-based approach Figure 5.8: Shear wave parameter images for human liver produced by a push beam focused at 50 mm. Fig. 5.8(a) shows the shear elasticity image generated by the TOF method. Fig. 5.8(b) shows the shear elasticity image generated by the new model-based approach. Fig. 5.8(c) shows the shear viscosity image generated by the new model-based approach. Fig. 5.8(d) shows the percent difference between the measured shear waves and the simulated shear waves with the µ and η s values estimated with the new model-based approach xviii

19 CHAPTER 1 INTRODUCTION 1.1 Ultrasound elastography For many years, physicians have used the stiffness of biological tissues to identify the pathological conditions of organs to diagnose diseases. As early as 400 B.C., Hippocrates discovered that the normal abdominals are soft, free of pain, and yield when pressed with the finger, are more chronic, while abdominal swellings are painful, hard, and large, indicate danger of speedy death [1]. Clinically, palpation is used to qualitatively detect the stiffness of the tissue for early cancer screening in breast [2] or liver [3]. However, although palpation is a convenient and straightforward diagnostic approach, it has several significant limitations. First, it is only useful in examining shallower tissue, and the effectiveness of this technique drops dramatically when the tissue is deep inside the body. Also, manual palpation is a qualitative technique, so no quantitative information for tissue can be obtained. Thus, the diagnostic accuracy relies on the experience of the physician. All of the above limitations suggest that better diagnostic techniques are needed to assess the stiffness of the tissue. Ultrasound elastography, which has been actively developed for the past two decades, is an enhanced palpation technique that overcomes these limitations. Ultrasound elastography performs remote palpation, which noninvasively detects the mechanical properties of the tissue deep inside the body using ultrasound. The procedure for ultrasound elastography is generally described as [4 10]: 1. Induce a tissue deformation with an external force. The external force can be a static compression force, a natural physiological force such as a pulsating blood vessel, or an acoustic radiation force (ARF) that is generated by a high-intensity ultrasound beam. 2. Measure the tissue deformation using ultrasound. Typically, the measurement is performed by comparing the ultrasound pulse-echo signal before and after the application 1

20 of the external force. Correlation-typed algorithms are applied to detect the movement of local ultrasound speckle [11] and then measure the local movement of tissue. 3. Estimate the tissue mechanical properties using the models that relate the measured tissue deformation to the mechanical properties. In this introductory chapter, multiple ultrasound elastography techniques are reviewed based on different types of excitations. In addition, methods that numerically simulate ultrasound elastography are reviewed. Finally, the motivation for this work is described and the outline of this thesis is given. 1.2 Ultrasound elastography techniques Ultrasound elastography techniques can be classified based on the type of the external force that induces the tissue deformation. In general, four types of excitation sources are used to induce the deformation in the tissue: static excitation, natural physiological excitation, transient excitation, and harmonic excitation. In this section, multiple ultrasound elastography techniques based on these four types of excitation schemes are reviewed Static excitation With static excitation, tissue is deformed by a static external force, which is usually a manual compression applied by the sonographer. A representative example is elastography, which was introduced by Ophir et al. in 1991 [6,12]. In this technique, ultrasound measurement of tissue is first performed at the equilibrium state with no external force applied. Then, the tissue is compressed manually, and ultrasound measurements of the tissue are taken when the tissue reaches a static compressed status. Comparisons are made between the ultrasound pulse-echo signals measured before and after the tissue is compressed, and the images of the local tissue strain, which are also called elastograms, are generated. Typically, lower strains correspond to harder tissue, and higher strains correspond to softer tissue. Clinically, this technique is used for assessing the hardness of muscle [13 16], detecting breast 2

21 tumors [13,17 20], detecting thyroid tumors [21 23], detecting prostate tumors [24 26], characterizing thermal ablation lesions after high-intensity focused ultrasound (HIFU) or radio frequency (RF) treatments [27 29], and detecting vascular plaques with a static excitation deployed by an intravascular balloon [30,31]. The advantage of this technique is that the excitation is relatively easy to apply, but the main disadvantage is that only qualitative images that differentiate between tissues with different stiffnesses are generated but no quantitative information describing the tissue stiffness is provided Natural physiological excitation Natural physiological motion, such as a pulsating blood vessel, heartbeat, or muscle tension can also induce tissue deformation, and ultrasound elastography techniques that exploit this behavior have been developed to assess tissue stiffness. In these techniques, the method for measuring tissue deformation is similar to ultrasound elastography with a static excitation. Ultrasound measurements are performed at different times, while the physiological excitation occurs intermittently, such as the periodic pulsing of a blood vessel. Comparisons are performed between the ultrasound pulse-echo signals measured before and after tissue deformation, and elastograms are generated based on the amplitude of the strain. Clinically, these methods assess the stiffness of vessels and detect vascular plaques [32 35], characterize the stiffness of muscle [36, 37], and detect the stiffness of myocardial tissue [38 40]. The advantage of this technique is that no manual or external excitation is needed to induce tissue deformation, but the obvious drawback is that it is not possible to control the excitation. In addition, this technique cannot provide quantitative information about tissue stiffness Transient excitation In transient ultrasound elastography, tissue deformation is induced using an acoustic radiation force (ARF) for a short period of time (typically several µs to several hundreds of µs) [41, 42]. The ARF is generated by a high-intensity ultrasound beam that is often 3

22 called the push beam. Compared to ultrasound elastography techniques based on static compression or natural physiological motion, ARF-based ultrasound elastography techniques have several advantages including a controllable excitation, so different tissue deformation responses can be induced by changing the type of ARF excitation. Furthermore, ARF can be applied to tissue deep inside the body, which expands its applications beyond superficial targets. Moreover, shear waves can be induced by an ARF push beam in tissue, and quantitative mechanical properties of tissue can be obtained by measuring the shear wave propagation speed. A limitation of this technique is that useful information is only obtained around the focus of the push beam, but this limitation is overcome by applying multiple push beams focused at different depths. Multiple variations of transient ultrasound elastography are reviewed in sections Acoustic Radiation Force Impulse (ARFI) Imaging Acoustic Radiation Force Impulse (ARFI) Imaging technique was introduced by Nightingale et al. [41] in In ARFI, an ultrasound transducer array generates a focused ARF push beam, and the push beam produces tissue deformation. Before and after the push beam is applied to the tissue, multiple frames of radio-frequency (RF) or in-phase/quadrature (IQ) data at the location of the push beam are obtained with pulse-echo ultrasound sequences. Then, the particle displacement is calculated from the IQ data using one-dimensional (1D) autocorrelations between the data measured before and after the push is applied [11, 43]. Then, the push beams and the corresponding ultrasound measurements are scanned through the whole two-dimensional (2D) field-of-view (FOV), and qualitative images for particle displacement are generated. Typically, a large displacement suggests that the tissue is soft, while a small displacement suggests that the tissue is hard. In clinical practice, ARFI imaging has been used to examine tissue stiffness and to detect tumors in liver [44,45], thyroid [46,47], breast [48,49], and prostate [50,51]. ARFI has also been used to assess elasticity properties in vascular tissue [52 54], detecting plaque in vascular tissue [55, 56], and characterize stiffness 4

23 in cardiac tissues [57, 58]. Moreover, ARFI has been used to guide thermal ablation therapy by monitoring the change in stiffness of the tissue during treatments [59]. ARFI imaging only provides qualitative information that describes the relative stiffness, but no quantitative information describing the tissue elasticity is given Shear Wave Elasticity Imaging (SWEI) The shear wave elasticity imaging (SWEI) technique was introduced by Sarvazyan et al. in 1998 [42]. In SWEI, one or more focused or unfocused ultrasound beams are applied for a short period of time to generate tissue deformation. In turn, this deformation produces shear waves. To measure the shear wave particle displacements, multiple frames of RF or IQ data within a 2D FOV are obtained with pulse-echo ultrasound sequences. Then, shear wave particle displacements are calculated from the RF or IQ data using 1D autocorrelations between consecutive frames [11, 42, 60]. Next, the shear wave speed is estimated from the measured particle displacements, and quantitative shear elasticity parameters are then obtained from the estimated shear wave speed. In clinical practice, SWEI has been used to quantitatively measure the stiffness of liver tissue and determine the stage of liver fibrosis [44,61 66]. Other clinical applications for SWEI include quantitative assessment of tissue mechanical properties in thyroid [67, 68], breast [69, 70], kidney [71], and prostate [72, 73] Supersonic Shear Imaging (SSI) The supersonic shear imaging (SSI) technique was introduced by Bercoff et al. in 2004 [74]. SSI is similar to SWEI, but the shear wave generation and detection approaches are different. In SSI, several push beams focused at different depths are sequentially applied to the tissue. The focus of the push beams moves downward faster than the propagation speed of the shear waves, hence the term supersonic. After the sequence of push beams is applied, shear waves are then generated that propagate away from the push location with a coneshaped wavefront. After the shear waves are generated, multiple frames of RF or IQ data 5

24 within a 2D FOV are measured using an ultrafast compound imaging technique [75, 76], which measures the ultrasound signals at a very high frame rate (up to 6000 frames per second). Then, shear wave particle displacements are calculated using 1D autocorrelations between consecutive frames [11]. In clinical practice, SSI has been used to quantitatively assess the shear elasticity of liver [77 80], breast [81, 82], cornea [78], muscle [83 85], and to guide thermal ablation [74] Spatially Modulated Ultrasound Radiation Force (SMURF) The spatially modulated ultrasound radiation force (SMURF) technique was developed by McAleavey et al. in 2007 [86]. As apposed to SWEI and SSI, which excite the tissue in a fixed location and then measure the signal at different lateral positions, the SMURF technique induces shear waves at different lateral positions by spatially moderating the acoustic radiation force transversely. SMURF then measures the induced shear waves at the same position. By measuring the arrival time difference of the shear waves generated by these different pushes located at known positions, the shear wave speed is estimated and the elasticity parameters are obtained. The SMURF technique suffers from fewer problems with correlated noise compared with other transient ultrasound elastography techniques because shear waves are measured at the same location, so the noise and jitter are reduced [87] Comb-Push Ultrasound Shear Elastography (CUSE) The comb-push ultrasound shear elastography (CUSE) technique was developed by Song et al. in 2012 [88]. In CUSE, the push beam is a comb-shaped acoustic radiation force consisting of several unfocused [89] or focused [90] ultrasound beams distributed at equal intervals within the tissue. Each push beam is generated by an individual sub-aperture of a transducer array. The array that generates the push beam is also used as the imaging transducer that measures the shear waves along A-lines outside of the push beam. CUSE provides an estimate of the shear wave speed for the entire field-of-view (FOV) simultaneously 6

25 within a short period of time (less than 35 ms). Thus, this approach provides images of tissue mechanical properties at a relatively high frame rate (up to approximately 30 fps) [90]. In clinical practice, CUSE has been applied to quantitative measurements of elasticity and to detect tumors in breast [91] and liver [92] Harmonic excitation In ultrasound elastography with harmonic excitations, a continuous wave (CW) pressure field with frequency between 10 Hz to 1 khz is applied to the tissue [10, 93]. The CW excitation can be applied either with a mechanical vibrator [93] or with a special distribution of ARF pushes [94, 95]. Compared with some other ultrasound elastography techniques, the advantage of harmonic ultrasound elastography is that quantitative information about tissue mechanical properties in the frequency domain can be obtained. However, a disadvantage is that the acquisition time to obtain the mechanical properties for the entire 2D FOV is much longer than for other ultrasound elastography techniques, so generating a full 2D image using this technique can be challenging. Multiple variations of harmonic ultrasound elastography are reviewed in sections Sonoelasticity imaging Sonoelasticity imaging was developed by Lerner et al. in 1990 [93, 96]. In sonoelasticity imaging, a mechanical vibrator applies an external vibration to generate oscillations at low frequencies (10 Hz - 1 khz) in tissue. The vibratory tissue motion is detected using Doppler ultrasound, and the tissue mechanical properties are obtained using model-based approaches by analyzing tissue motion at different frequencies. The drawback of this technique is that a vibrating source is required to induce tissue deformation, so applications are depth-limited. In clinical practice, sonoelasticity imaging assesses tissue mechanical properties in muscle [97] and detects tumors in the liver [98, 99], kidney [98, 100], breast [98, 101], and prostate [102, 103]. 7

26 Vibro-Acoustography (VA) Vibro-acoustography (VA) was developed by Fatemi et al. in 1998 [94, 104]. In VA, two ARF push beams are simultaneously focused at the same spot with slightly different driving frequencies. The frequency difference, or beat frequency, is on the order of 1 khz. For an ARF push applied in this manner, a sinusoidal vibration at the beat frequency is induced at the focal location, and an acoustic emission is generated by the vibration. The acoustic emission is measured using a hydrophone, and the tissue mechanical properties are obtained by analyzing the amplitude and phase of the measured acoustic emission. The drawbacks of this technique are that VA requires more than one transducer to transmit the excitation and that an additional hydrophone is required to measure the response. However, the former problem can be mitigated by using a single amplitude-modulated (AM) ARF push beam that produces a similar vibration [95]. In addition, the long acquisition time also limits potential in clinical applications that generate a full 2D image. In clinical practice, VA has been used for tissue mechanical imaging in the liver [105], breast [106,107], thyroid [108,109], prostate [110, 111], and in arteries [112] Harmonic Motion Imaging (HMI) Harmonic motion imaging (HMI) was developed by Konofagou and Hynynen in 2003 [113]. HMI uses the same technique as VA to generate tissue vibration, which applies two ARF push beams focused at the same spot with slightly different driving frequencies or a single amplitude-modulated (AM) ARF push beam. However, the technique for tissue vibration measurement in HMI is different from that of VA. In HMI, another imaging ultrasound transducer is used to measure the vibration instead of the hydrophone that is used in VA. The tissue vibration is measured using pulse-echo ultrasound combined with a cross-correlation speckle tracking algorithm [11], which is similar to the approach used in most transient ultrasound elastography techniques. Tissue mechanical properties are obtained by analyzing the measured tissue vibration patterns. Although HMI does not require a hydrophone, the 8

27 long acquisition time associated with HMI is still a disadvantage. In clinical practice, HMI has been used to monitor thermal ablation with high intensity focused ultrasound, where the combined approach for imaging and therapy is called harmonic motion imaging for focused ultrasound (HMIFU) [114] Shear wave Dispersion Ultrasound Vibrometry (SDUV) Shear wave dispersion ultrasound vibrometry (SDUV) was developed by Chen et al. in 2004 [115, 116]. SDUV uses the same excitation method as VA to generate harmonic tissue motion at a specific frequency. However, rather than measuring the acoustic emission from the tissue vibration using a hydrophone, the shear wave induced by the harmonic tissue motion is measured instead. The measurement procedure is similar to that of SWEI, where the shear wave is measured using pulse-echo ultrasound combined with a cross-correlation speckle tracking algorithm [11]. The shear wave propagation speed is measured by tracking the phase shift of the time-harmonic shear wave at different locations. With different driving frequencies, the shear wave propagation speed at multiple frequencies is obtained, and model-based [116] or model-free [117] approaches are used to estimate the tissue mechanical properties. This technique, which is performed with a single commercial ultrasound transducer array, provides quantative measurements of shear wave dispersion for soft tissue. In clinical applications, SDUV has been used to estimate the elasticity parameters in muscle [118], liver [116,119], kidney [120,121], prostate [122], heart [123], and blood vessels [124]. 1.3 Review of numerical simulation methods for shear waves induced by an acoustic radiation force Numerical simulations provide information that is valuable for validating and improving ultrasound elastography techniques. In this section, a review of numerical simulation methods for transient ultrasound elastography is provided. These simulations consist of two main parts. The first part calculates the acoustic radiation force, and the second part computes 9

28 the shear waves induced by the acoustic radiation force Acoustic radiation force simulations Several ultrasound simulation toolboxes such as FOCUS, the Fast Object-oriented C++ Ultrasound Simulator ( fultras-web/) [125], Field II ( [126], Ultrasim ( ultrasim/) [127], and DREAM ( [128] are able to simulate a steady-state focused ultrasound beam. From the output of these, the approximate acoustic radiation force is calculated. Among the above simulation toolboxes, Field II and FOCUS are most widely used by the medical ultrasound research community. In Field II, the ultrasound pressure field is calculated by evaluating the impulse response of the transducer [129]. Because Field II uses a far-field approximation, the calculation for a large transducer element is performed by subdividing the transducer element and superposing the contribution from each small sub-element. In FOCUS, the fast nearfield method (FNM) computes an input pressure plane (usually λ/4 from the front face of the transducer, where λ is the wavelength of the compressional wave) [130, 131], and then the angular spectrum approach (ASA) determines the pressure field throughout the remaining 3D volume [132, 133]. Because the FNM does not utilize the far-field approximation, no subdividing of elements is required. The FNM achieves better accuracy in much less time than the impulse response method [ ]. Furthermore, the calculation time in FOCUS for calculating ultrasound pressure fields in three-dimensions (3D) is significantly smaller than that for impulse response methods [135], since the combination of the FNM and ASA approaches are especially suitable for volumetric calculations. In this thesis, all of the acoustic radiation force push beams are calculated using the FNM combined with the ASA in FOCUS. 10

29 1.3.2 Shear wave simulations At present, most of the numerical simulations of shear waves generated by an applied acoustic radiation force employ the finite element (FEM) method. In 2000, Nightingale et al. developed a finite element model which simulates the tissue response to an acoustic radiation force in breast tumors [137]. In their model, the ARF push beam is calculated using Field II, and the shear wave induced by the push beam is calculated using LS-DYNA3D (Livermore Software Technology Corporation, Livermore, CA). The FEM models are also used to simulate the tissue response in ARFI imaging [138], quantifying the hepatic shear modulus [66], demonstrating the feasibility of ARFI imaging in abdomen examinations [139], modeling the shear wave propagation induced by an acoustic radiation force in a viscoelastic medium [140], and other applications [139, 141, 142]. These FEM models successfully simulate shear waves induced by an acoustic radiation force, which are useful for developing various ultrasound elastography techniques. The main drawbacks of FEM simulations are that FEM models are usually complicated and the computation times are typically very long. Other methods have been developed to simulate shear waves induced by acoustic radiation force push beams. The finite difference (FD) method, which is often easier to implement than the FEM method, is also useful for simulating shear waves [ ]. The finite difference method is sometimes faster than the FEM method, but the ability of finite difference calculations to manage complicated boundaries and peculiar shapes is usually not as good as FEM. The pseudospectral method is also used to simulate shear waves induced by an acoustic radiation force [146]. In the pseudospectral method, much coarser spatial samplings are possible for homogeneous problems relative to FD and FEM methods, which can reduce computation time and memory usage. The Green s function approach is another method for simulating shear waves [147, 148]. The Green s function approach, which is applicable to shear wave modeling in elastic [147] and viscoelastic media [148], is appropriate when calculations at only a few observation points in space are needed, as opposed to other numerical methods (FEM, FD, and pseu- 11

30 dospectral) that require the calculation of an entire 3D field. Thus, shear wave calculations with Green s functions can be much faster than shear wave calculations with other numerical methods in certain circumstances. Furthermore, Green s function calculations are very easily parallelized, since calculations for different observation points are independent of one another. The main disadvantage is that the Green s function approach can only simulate shear wave propagation in homogeneous media. 1.4 Motivation and thesis structure Although significant progress has been made in ultrasound elastography over the past two decades, a substantial amount of work still needed to improve the image quality for ultrasound elastography. Currently, one important difficulty in ultrasound elastography is caused by the viscoelasticity of soft tissue. Most models that are currently used in ultrasound elastography assume that tissue is lossless, even though multiple studies indicate that tissue is always lossy (viscoelastic) [79, 84, 116, 121, ]. Thus, estimation bias is introduced because of this approximation [79, 116]. Quantitative ultrasound methods such as VA [94, ] or SDUV [ ] are able to characterize the viscoelasticity of tissue. However, these methods are limited by long acquisition times and specific hardware requirements. A robust technique that provides accurate quantitative images of viscoelastic tissue properties with a conventional ultrasound system is a critical unmet need in medical ultrasound. To address this problem, this thesis introduces a new approach that generates quantitative images of viscoelastic soft tissue. Rather than analyzing the shear wave in the frequencydomain, as in time-harmonic ultrasound elastography, this new approach estimates tissue mechanical parameters using shear waves that are measured in the time-domain using exactly the same configuration that is employed in traditional transient ultrasound elastography (i.e. SWEI). Thus, this new approach utilizes the same framework that is employed in current commercial ultrasound elastography systems. In this new approach, shear waves are first simulated using a Green s function-based method that evaluates the Voigt model on a GPU, 12

31 and then the shear elasticity and shear viscosity values are estimated using an optimizationbased approach by comparing measured shear wave particle velocities with simulated shear wave particle velocities in the time-domain. This comparison is then performed on a pointby-point basis to generate images. The new approach is evaluated with shear wave data obtained from numerical simulations, viscoelastic shear wave phantoms, and in vivo human livers, and the results are compared with the results of other methods for estimating shear wave parameters. The structure of this thesis is described as follows. In Chapter 2, background information is introduced on acoustic radiation force, elastodynamics, shear wave measurement techniques, and shear wave elasticity estimation methods. In Chapter 3, a GPU-based numerical simulation approach for the calculation of shear waves induced by an acoustic radiation force in elastic and viscoelastic soft tissue models is described. Examples of shear waves simulated using this approach are shown, and the numerical simulation methods are validated by comparing the simulated shear wave particle velocities with shear wave particle velocities measured in several different viscoelastic shear wave phantoms. In Chapter 4, an model-based approach that estimates shear elasticity and shear viscosity values by minimizing the difference between measured and simulated shear wave particle velocities is introduced. The model-based estimation approach is validated with results obtained from numerical simulations, and the model-based estimation approach is also used to estimate shear wave parameters from measured shear wave data collected in viscoelastic shear wave phantoms. Results suggest that the new model-based estimation approach yields more accurate parameter values than the conventional TOF method. In Chapter 5, the new approach is accelerated with an approximation, and then the new approach generates estimates of shear wave parameters with measured data obtained from in vivo human livers. In Chapter 6, the summary and potential future directions are presented. 13

32 CHAPTER 2 BACKGROUND 2.1 Acoustic radiation force In clinical ultrasound elastography, shear waves in soft tissue are induced by an applied acoustic radiation force, which is generated from a focused ultrasound beam. The acoustic radiation force is modeled as [41] f(r, t) = 2αI(r, t) c p, (2.1) where f(r, t) is the body force, r denotes a spatial point in Cartesian coordinates, t denotes the time, α is the attenuation coefficient of the compressional wave, c p is the compressional wave speed, and I(r, t) is the acoustic intensity. The intensity of the ultrasound beam is proportional to the square of the acoustic pressure p, which is defined as I = p 2 /(ρc p ). In transient elastography, the acoustic radiation force is applied to the tissue for a short period of time to produce a shear wave. To account for the temporal component of the acoustic radiation force, f(r, t) is written as f(r, t) = f(r)w(t), where f(r) is the spatial distribution of the acoustic radiation force, and w(t) is a window function that describes the time dependence of the acoustic radiation force. The window function is defined here as w(t) = rect[(t T/2)/T ], where rect( ) is the rectangle function, and T is the duration of the push. 2.2 Equation of motion in elastic and viscoelastic soft media The elastodynamic response induced by an applied acoustic radiation force in elastic and viscoelastic media is described by the equation of motion, which can be found in any classical mechanics textbook such as [152, 153]. The derivation of the equation of motion is based on three fundamental relations: the conservation of momentum, the strain-displacement relations, and the stress-strain relationship. These three relationships are described below. 14

33 Newton s second law guarantees the conservation of momentum, which relates the stress and the external force to the acceleration. The conservation of momentum equation is written as [152] σ ij x j + f i = ρ 2 u i t 2, (2.2) where σ ij is the stress tensor, f i is the external body force, u i is the displacement, i, j = 1, 2, 3 are the indices for all three dimensions, t is the time, and ρ is the density of the material. Assuming that the strain and displacement are both small so that the behavior is linear, the strain-displacement relationship is written as [152] ε ij = 1 2 ( ui + u ) j, (2.3) x j x i where ε ij is the strain tensor. In elastic soft media, Hooke s law is satisfied, which provides the strain-stress relationship as [152] σ ij = C ijkl ε kl, (2.4) where C ijkl is the compliance matrix which describes the mechanical properties of soft tissue. The engineering notation for Eq. 2.4 is [152] σ x C 11 C 12 C 13 C 14 C 15 C 16 ε x σ y C 12 C 22 C 23 C 24 C 25 C 26 ε y σ z C = 13 C 23 C 33 C 34 C 35 C 36 ε z, (2.5) σ yz C 14 C 24 C 34 C 44 C 45 C 46 2ε yz σ zx C 15 C 25 C 35 C 45 C 55 C 56 2ε zx σ xy C 16 C 26 C 36 C 46 C 56 C 66 2ε xy where the compliance matrix is a 6 6 matrix. For a fully anisotropic medium, the compliance matrix has 21 degrees of freedom. Alternatively, if the medium is isotropic, the compliance matrix C ijkl contains only 2 degrees of freedom, and the compliance matrix is 15

34 written as [152] σ x λ + 2µ λ λ ε x σ y λ λ + 2µ λ ε y σ z λ λ λ + 2µ ε = z, (2.6) σ yz µ 0 0 2ε yz σ xz µ 0 2ε xz σ xy µ 2ε xy where λ and µ are the Lamè constants, also known as the bulk modulus and the shear modulus, respectively. The tensor notation for Eq. 2.6 is σ ij = λε kk δ ij + 2µε ij, (2.7) where δ ij is the Krönecker symbol, which is written as δ ij = 1 for i = j and δ ij = 0 for i j. Combining Eq. 2.2, 2.3, and 2.7, yields the equation of motion in elastic media [152], (λ + 2µ) ( u) µ ( u) + f = ρ u, (2.8) t where u is the particle displacement, f is the induced body force, and t is the time. Eq. 2.8 is also known as Navier s equation. Eq. 2.8 describes elastodynamic waves in a medium without any loss, but the behavior of soft tissue is not accurately characterized by this model, as tissue is always lossy (viscoelastic) [79, 116]. To better describe the mechanical response for an acoustic radiation force in human tissue, a proper loss model is needed. When the Kevin-Voigt model for viscous loss is included in Eq. 2.8, the Lamè constants λ and µ in Eq. 2.7 are replaced by λ + η p ( / t) and µ + η s ( / t), respectively, where η p is the compressional viscosity, and η s is the shear viscosity [148, 154]. The stress-strain relationship then becomes σ ij = ( ) ( ) λ + η p ε t kk δ ij + 2 µ + η s ε t ij. (2.9) 16

35 After Eq. 2.2, 2.3, and 2.9 are combined, the equation of motion in viscoelastic media is given by [153] ( λ + 2µ + (η p + 2η s ) ) ( ) ( u) + µ + η s ( u) + f = ρ u. (2.10) t t t 2.3 Green s function solutions to the equation of motions To solve Eqs. 2.8 and 2.10 with Green s functions that describe the mechanical response to an applied body force in elastic and viscoelastic media, the displacement u and external body force f are separated into two parts through Helmholtz decomposition. One part contains the contributions from scalar potentials that describe the compressional component of the elastodynamic wave, and the other part contains the contributions from vector potentials that describe the shear component of the elastodynamic wave [155]. The Helmholtz decomposition is written as u = φ u + ψ u, (2.11) f = φ f + ψ f, (2.12) where φ u and φ f represent the scalar potentials for the displacement field and the body force, respectively, and ψ u and ψ f represent the vector potentials for the displacement field and the body force, respectively Green s function for Navier s equation in elastic media For an elastic medium, Navier s equation in Eq. 2.8 can be separated into two wave equations that model the scalar and vector components of the wave separately by applying Eqs and 2.12 to Eq. 2.8, which yields 2 φ u t 2 2 ψ u t 2 = φ f ρ + λ + 2µ 2 φ u, (2.13) ρ = ψ f ρ + µ ρ 2 ψ u. (2.14) 17

36 Eqs and 2.14 describe two distinct wave equations with wave propagation speeds of c p = (λ + 2µ)/ρ and c s = µ/ρ, respectively. By applying a point source to Eq. 2.8, the Green s function is obtained. A derivation of this solution is provided in [152]. The Green s function for Eq. 2.8 is given by g ij (r, t) = gij P (r, t) + gs ij (r, t) + gc ij (r, t), (2.15) ) (t rcp, (2.16) gij P 1 1 (r, t) = 4πρc 2 γ i γ j p r δ 0 gij S 1 (r, t) = 4πρc 2 (δ ij γ i γ j ) 1 s r δ 0 gij C 1 (r, t) = 4πρ (3γ iγ j δ ij ) 1 r 3 (t rcs ), (2.17) r/c s τδ 0 (t τ) dτ, (2.18) r/cp where g ij (r, t) represents the Green s function for Navier s equation in elastic media, gij P (r, t), gij S (r, t), and gc ij (r, t) represent the compressional, shear, and the coupling components of g ij (r, t), respectively, r denotes a spatial point in Cartesian coordinates, i = 1, 2, or 3, which represents the x, y, and z directions for the displacement, respectively, and j = 1, 2, or 3, which represents the x, y, and z directions for the external body force, respectively. In the above expressions, r = r is the distance between the source point and the observation point, γ i = x i /r, γ j = x j /r, δ( ) denotes the Dirac delta function, and H( ) denotes the Heaviside (unit step) function. In ultrasound elastography since the ARF push beam is applied along the axial direction, so the body force is applied in the z direction. The displacement of the propagating shear wave is also in the z direction. Thus, for the sake of convenience, Eqs are reduced to the i = 3 and j = 3 case, which represents the shear wave particle displacement in the z direction induced by the external force applied in the z direction. The simplified Green s 18

37 function is then given as g z (r, t) = gz P (r, t) + gz S (r, t) + gz C (r, t), where (2.19) gz P (r, t) = 1 z 2 ) (t 4πρc 2 p r 3 δ rcp, (2.20) gz S (r, t) = 1 ) ) (1 4πρc 2 z2 1 (t s r 2 r δ rcs, (2.21) gz C (r, t) = 1 ( 3z 2 ) [ ) )] 1 4πρ r 2 1 r 3 t H (t rcp H (t rcs. (2.22) In Eqs , g P z (r, t), g S z (r, t), and g C z (r, t) represent the compressional, shear, and the coupling components, respectively, of the Green s function for Navier s equation evaluated in an elastic medium, where z is the distance between the source point and the observation point in the z-direction Green s function to the Navier s equation in viscoelastic media For a viscoelastic medium, similar operations can be applied to obtain the Green s function. Navier s equation for viscoelastic media in Eq can also be separated into two wave equations that model the scalar and vector components separately by applying Eqs and 2.12 to Eq. 2.10, which yields 2 φ u t 2 2 ψ u t 2 = φ ( f ρ + c 2 p + v p t = ψ f ρ + where v p = (η p + 2η s )/ρ and v s = η s /ρ. ( c 2 s + v s t ) 2 φ u, (2.23) ) 2 ψ u, (2.24) By applying a point source to Eq. 2.10, an approximate Green s function is obtained. 19

38 The approximate Green s function for Eq is written as g ij (r, t) = gij P (r, t) + gs ij (r, t) + gc ij (r, t), (2.25) (t r/cp) 2 c 2 gij P p 1 1 (r, t) = 4πρc 2 p 2πvp t γ 1 iγ j r e 2vpt, (2.26) gij S 1 (r, t) = 4πρc 2 s 1 2πvs t (δ ij γ i γ j ) 1 r e (t r/c s) 2 c 2 s 2vst, (2.27) gij C 1 (r, t) = 4πρ (3γ iγ j δ ij ) 1 [ Ip r 3 (r, t) + I s (r, t) ], (2.28) vp t I p (r, t) = e tc2 2vp p e (t r/c p) 2 c 2 [ ( ) ( )] p 2vpt + t c p t c p (t r/c p ) erf erf, 2πcp 2 2vp t 2vp t I s (r, t) = vs t e tc2 s 2vs e (t r/c s) 2 c 2 s 2vst + t [ ( ) cs t erf erf 2πcs 2 2vs t (2.29) ( )] cs (t r/c s ), 2vs t (2.30) where erf( ) is the error function. Eqs are also reduced for the i = 3 and j = 3 case, which represents the particle displacement along the z direction induced by the external 20

39 force applied along the z-direction. The simplified Green s function for Eq is given by g z (r, t) = g P z (r, t) + g S z (r, t) + g C z (r, t), where (2.31) gz P (r, t) = 1 1 z 2 (t r/cp) 2 c 2 p 4πρc p 2πvp t r 3 e 2vpt, (2.32) gz S (r, t) = 1 ) 1 (1 z2 1 (t r/cs) 2 c 2 s 4πρc s 2πvs t r 2 r e 2vst, (2.33) gz C (r, t) = 1 ( 3z 2 ) 1 4πρ r 2 1 [ Ip r 3 (r, t) + I s (r, t) ], (2.34) tc νp t 2 ( p I p (r, t) = 2πcp e 2νp e t cp r ) 2c 2 p ( ) ( ) 2νpt + t c p t erf erf c p t r cp, 2 2νp t 2νp t I s (r, t) = νs t 2πcs tc2 s e 2νs e ( t cs r ) 2c 2 s 2νst + t 2 (2.35) ( ) ( ) cs t erf erf c s t r cs, 2νs t 2νs t (2.36) where gz P (r, t), gz S (r, t), and gz C (r, t) represent the compressional, shear, and the coupling components, respectively, of the Green s function for Navier s equation evaluated in a viscoelastic medium, where z is the distance between the source point and the observation point in the z-direction. 2.4 Shear wave measurements In transient ultrasound elastography, shear waves are measured by processing the pulseecho ultrasound signal with a speckle tracking algorithm. In order to measure the local shear wave particle displacement, multiple frames of RF or IQ data within a 2D FOV are obtained using pulse-echo ultrasound sequences. For each point in the FOV, the IQ data within a small spatial window is compared between consecutive frames. The shear wave particle displacement or velocity is calculated from the RF or IQ data using 1D crosscorrelations [11, 156]. 21

40 2.5 Methods for estimating the mechanical parameters from measured shear waves The time-of-flight (TOF) method In shear wave elasticity imaging (SWEI), an estimate of the shear elasticity is often obtained from the time-of-flight (TOF) of propagating shear waves. In TOF measurements for SWEI, the arrival times of the shear waves are measured at several successive spatial points located at the same depth but in different lateral positions. One approach obtains the arrival times by measuring the time to peak (TTP) [61, 157], which tracks the arrival times of the peaks for each measured shear wave. Another approach obtains the arrival time delays with crosscorrelations [81, 158, 159]. Of these, the cross-correlation approach is preferred because this approach is more robust and much less sensitive to noise. The shear wave propagation speed c s is then calculated from the slope of the linear regression between the lateral locations of the spatial points and the shear wave arrival times. Another approach that improves the robustness of the linear regression uses random sample consensus (RANSAC), which updates the linear fit by randomly generating trial solutions and iteratively updating the model to eliminate the outliers of the arrival time delays [160, 161]. The shear elasticity µ is then obtained from µ = ρc 2 s, where ρ is the density. Although the TOF method often yields accurate results for SWEI in elastic (lossless) shear wave phantoms [61, ], this method is based upon some simplifications and approximations that are responsible for significant bias when applied to shear wave parameter estimates in soft tissue, which is always lossy (viscoelastic) [79,116]. In viscoelastic media, the shear wave speed changes during propagation due to shear wave attenuation and dispersion, so the relationship µ = ρc 2 s no longer holds [115, 148]. Thus, the shear elasticity estimates provided by the TOF method are biased by the viscosity. Another problem is that the TOF method implicitly assumes that the shear wave is a plane wave. However, complicated diffraction effects are observed in all shear waves induced by a three dimensional (3D) acoustic 22

41 radiation force [165]. Also, the TOF method yields different values for the estimated shear elasticity with different transducers, focal depths, and lateral ranges applied to the same viscoelastic shear wave phantom [166], which suggests that the diffraction of the push beam and the diffraction of the shear wave also contribute to the bias observed in shear wave parameter estimates obtained from the TOF method The k-space method Other methods have also been developed for shear wave parameter estimation. One of these is attenuation measuring ultrasound shearwave elastography (AMUSE) [167]. In the AMUSE method, a k-space representation of the shear wave is calculated at several successive spatial points located at the same depth but in different lateral positions using the twodimensional fast Fourier transform (2D-FFT). The k-space representation of the shear wave data is plotted as function of the angular frequency (ω) and the wavenumber in the x- direction (k x ). Then, the shear wave propagation speed c s is obtained at each frequency by determining the location of the peak at each frequency in k-space and dividing the value of ω by the value of k x. The attenuation α s is obtained by measuring the full-width-at-halfmaximum (FWHM) in the k x direction for each angular frequency ω. The AMUSE method provides a model-free estimate of shear wave speed and attenuation. However, this method is only effective in the low-frequency range, and diffraction effects are not considered. 23

42 CHAPTER 3 GPU-BASED GREEN S FUNCTION SIMULATIONS OF SHEAR WAVES GENERATED BY AN APPLIED ACOUSTIC RADIATION FORCE IN ELASTIC AND VISCOELASTIC MEDIA 3.1 Introduction Shear wave elasticity imaging (SWEI) [42] is an emerging modality for noninvasively imaging the mechanical properties of soft tissue including breast [81], liver [61], kidney [121], thyroid [67], heart [168], and blood vessels [150]. In SWEI, an acoustic radiation force push beam is applied to soft tissue, and the transient shear wave particle displacement is then measured using high frame-rate ultrasound [42]. Other methods for ultrasound elastography include acoustic radiation force imaging (ARFI) [41], supersonic shear imaging (SSI) [74], spatially modulated ultrasound radiation force (SMURF) [86], crawling wave spectroscopy (CWS) [169, 170], and comb-push ultrasound shear elastography (CUSE) [88]. Numerical simulation approaches are useful for validating and improving shear wave imaging techniques. At present, most of the numerical simulations of shear waves generated by an applied acoustic radiation force employ the finite element (FEM) method. FEM models have been developed for emulating ARFI imaging [138], simulating the shear wave response to acoustic radiation force in breast [137] and liver [61], modeling the shear wave propagation induced by an acoustic radiation force in a viscoelastic medium [140], studying the thermal effects induced by an acoustic radiation force in soft tissue [171], and other applications [139, 141, 142]. Other simulation options include the finite difference method [143, 144], the pseudospectral method [146], and the Green s function approach [147, 148]. The Green s function approach, which is applicable to shear wave modeling in elastic [147] and viscoelastic media [148], is appropriate when calculations at only a few observation points in space are needed, as opposed to other numerical methods that calculate an entire transient response. 24

43 Despite the advantages for a small number of points, the Green s function approach is very time-consuming for shear wave simulations on a desktop computer. To accelerate these calculations, a more efficient calculation approach is needed. Since Green s functions are readily evaluated in parallel, these calculations are ideal for graphics processing units (GPUs). The GPU has previously been established as an effective approach for accelerating medical ultrasound calculations using the angular spectrum method [172], the k-space simulation method [173], and other calculations [ ]. Similar performance enhancements are anticipated for shear wave calculations on the GPU. After the theory of the Green s function method for the simulation of shear waves generated by an applied acoustic radiation force is introduced, a GPU-based parallel algorithm that calculates the shear wave motion induced by an acoustic radiation force is described for simulations in elastic and viscoelastic media. Shear displacements are calculated at several discrete observation points and also in 2D planes to reveal the effects of shear wave diffraction, attenuation, and dispersion. The computation times of serial and parallel Green s function calculations are also evaluated and compared on a desktop computer and on a high-performance GPU. The convergence of the Green s function calculations is evaluated for push beams with different spatial samplings. In addition, simulated shear waves are compared with shear waves measured in different elastic and viscoelastic shear wave phantoms. The results suggest that the Green s function approach is ideal for shear wave simulations with high-performance GPUs. 3.2 Theory Navier s Equation Shear and compressional wave particle displacements induced by an external body force in an elastic, isotropic, homogeneous, and nearly incompressible model for soft tissue are described 25

44 by Navier s equation [152] (λ + 2µ) ( u) µ ( u) + f = ρ u, (3.1) t where u is the local particle displacement, f is the induced body force, and t is the time. In Eq. 3.1, the bulk modulus and shear modulus are indicated by λ and µ, respectively, and ρ is the density. When the Kevin-Voigt model for viscous loss is included, Eq. 3.1 becomes [148, 154] ( λ + 2µ + (η p + 2η s ) ) ( ) ( u) + µ + η s ( u) + f = ρ u, (3.2) t t t where the parameters η p and η s denote the bulk viscosity and shear viscosity, respectively Green s functions calculations The Green s function for Navier s equation evaluated in an elastic medium calculates the displacement induced by a spatially and temporally impulsive body force. When the acoustic radiation force push is applied in the z-direction, the z-component of the Green s function that describes the resulting displacement in an elastic medium is represented by [152] g z (r, t) = gz P (r, t) + gz S (r, t) + gz C (r, t), where (3.3) gz P (r, t) = 1 z 2 ) (t 4πρc 2 p r 3 δ rcp, (3.4) gz S (r, t) = 1 ) ) (1 4πρc 2 z2 1 (t s r 2 r δ rcs, (3.5) gz C (r, t) = 1 ( 3z 2 ) [ ) )] 1 4πρ r 2 1 r 3 t H (t rcp H (t rcs. (3.6) In Eqs , g P z (r, t), g S z (r, t), and g C z (r, t) represent the compressional, shear, and the coupling components of the Green s function for Navier s equation evaluated in an elastic medium, respectively, where r denotes a spatial point in Cartesian coordinates, the compressional wave speed is c p = (λ + 2µ)/ρ, and the shear wave speed is c s = µ/ρ. In the above expressions, δ( ) denotes the Dirac delta function, H( ) denotes the Heaviside (unit 26

45 step) function, r = r is the distance between the source point and the observation point, and z is the distance between the source point and the observation point in the z-direction. An approximate Green s function for Navier s equation evaluated in a viscoelastic medium is [148] g z (r, t) = g P z (r, t) + g S z (r, t) + g C z (r, t), where (3.7) gz P (r, t) = 1 1 z 2 (t r/cp) 2 c 2 p 4πρc p 2πvp t r 3 e 2vpt, (3.8) gz S (r, t) = 1 ) 1 (1 z2 1 (t r/cs) 2 c 2 s 4πρc s 2πvs t r 2 r e 2vst, (3.9) gz C (r, t) = 1 ( 3z 2 ) 1 4πρ r 2 1 [ Ip r 3 (r, t) + I s (r, t) ], (3.10) tc νp t 2 ( p I p (r, t) = 2πcp e 2νp e t cp r ) 2c 2 p ( ) ( ) 2νpt + t c p t erf erf c p t r cp, 2 2νp t 2νp t I s (r, t) = νs t 2πcs tc2 s e 2νs e ( t cs r ) 2c 2 s 2νst + t 2 (3.11) ( ) ( ) cs t erf erf c s t r cs, 2νs t 2νs t (3.12) where ν p = (η p + 2η s )/ρ, ν s = η s /ρ, and erf( ) denotes the error function Simulating shear waves with Green s functions To simulate shear waves generated by an acoustic radiation force, the Green s function and the acoustic radiation force are convolved in time and space by numerically evaluating the superposition integral u z (r, t) = g z (r r, t τ)f(r, τ)dτdv, (3.13) V τ where u z (r, t) is the displacement at r, g z (r, t) is the Green s function for either the elastic or viscoelastic model, and f(r, t) is the acoustic radiation force. For these calculations, the 27

46 acoustic radiation force is represented by f(r, t) = f(r )w(t), where f(r ) is the steadystate acoustic radiation force obtained from the intensity calculated in FOCUS, and w(t) describes the time dependence of the acoustic radiation force. In this model, w(t) is the same at every point in space, and w(t) is given by w(t) = rect[(t T/2)/T ]. The expression rect[(t T/2)/T ] indicates that the push beam is applied starting at time t = 0 and ending at time t = T, where the variable T describes the duration of the push, and equal weighting is applied at each point in time. For shear wave simulations with the elastic model, the convolution in time between the Green s function g z (r r, t) in Eqs and w(t) is calculated analytically to eliminate the delta functions from these calculations. The results are then given by h z,t (r, t) = g z (r, t τ)w(τ)dτ = h P z,t (r, t) + hs z,t (r, t) + hc z,t (r, t), where (3.14) τ h P 1 z 2 ( t r/cp z,t (r, t) = 4πρc 2 p r 3 rect T/2 T ) ( h S 1 z,t (r, t) = (1 4πρc 2 z2 1 t r/cs s r 2 r rect T/2 T ( h C 1 3z 2 ) 1 z,t (r, t) = 4πρ r 2 1 r [ ( ) 3 ( ) 1 t 2 r2 t r/cp T/2 2 c 2 rect + 1 p T ( r 2 ) ( ) 2 c 2 (t T ) 2 t r/cs T/2 rect 1 s T 2 ), (3.15) ), (3.16) ( 2tT T 2) ( H t r ) T c p ( 2tT T 2) )] H (t rcs, (3.17) where h z,t (r, t) represents the transient displacement of a point body force in the z- direction for the window function w(t) = rect[(t T/2)/T ] in an elastic medium, and h P z,t (r, t), hs z,t (r, t), and hc z,t (r, t) are the compressional, shear, and coupling components of the resulting transient displacement, respectively. The remaining 3D integral h z,t (r r, t)f(r )dv then describes the spatial convolution between h z,t (r, t) and V the steady-state acoustic radiation force f(r ), which is calculated numerically by evaluating u E z (r, t) = Nx Ny m=1 n=1 k=1 Nz [ hz,t (r r, t) ] f(r ) x y z. (3.18) 28

47 In this triple sum, N x, N y, and N z describe the number of spatial points in each direction for the simulated steady-state 3D acoustic radiation force, and x, y, and z are the spatial sampling intervals in the x, y, and z directions for the simulated acoustic radiation force. The superscript in u E z (r, t) indicates that this quantity is the shear wave particle displacement calculated for an elastic medium. For shear wave simulations with the viscoelastic model, no delta functions appear in the Green s function given in Eqs , so no analytical simplifications are needed. For shear wave calculations with Eqs and Eq. 3.13, the convolution in space is evaluated first, then the convolution in time is computed. The discretized representation of Eq for numerical calculations in viscoelastic media is N t u V Nx Ny Nz [ ( z (r, t) = gz r r, t τ )] f(r ) x y z w(τ) τ, (3.19) i=1 m=1 n=1 k=1 where N t is the number of time samples in w(t), and τ is the temporal sampling interval. The superscript in u V z (r, t) indicates that this quantity is the shear wave particle displacement calculated for a viscoelastic medium. 3.3 Numerical Calculations Acoustic radiation force In shear wave elasticity imaging (SWEI), shear waves are generated by focused or unfocused ultrasound beams. For three-dimensional (3D) shear wave calculations in an isotropic, homogeneous simulation model, the pressure field is calculated first, and then the radiation force is obtained from the computed pressure field, where the radiation force then provides the input to Eq. 3.1 or Eq. 3.2 for calculations of the shear wave particle displacements. In each of the following simulations, the pressure, radiation force, and shear wave particle displacements are all evaluated in 3D. The 3D pressure fields are calculated in FOCUS, the Fast Object-oriented C++ Ultrasound Simulator ( fultras-web). In these simulations, the fast 29

48 nearfield method computes the input pressure plane [130, 131], and then the angular spectrum approach determines the pressure field throughout the remaining 3D volume [132,133]. For each of the shear wave simulations evaluated here, a Philips L7-4 linear ultrasound transducer array (Philips Healthcare, Andover, MA) is modeled, where the array consists of 128 curved transducer elements with an elevation focus of 25 mm. The width of each ultrasound transducer element is mm, the height is 7 mm, and the kerf between adjacent elements is mm. The transducer array face is centered at z = 0 in the x-y plane, the center of each element is located on the y-axis, and the normal at the center of the array is coincident with the z-axis. After the 3D pressure is calculated, the 3D intensity is computed with I = p 2 /(2ρc p ), and the acoustic radiation force f(r, t) is determined from [41] f(r, t) = 2αI(r, t) c p, (3.20) where α is the ultrasound attenuation coefficient, and I(r, t) is the acoustic intensity Parallel implementation The convolution integral in Eq is very time-consuming when implemented as a serial calculation. For N x N y N z observation points and for N t time samples, Eq is evaluated at N x N y N z N t points in space and time. In practice, this is a problem for calculations on desktop or laptop computers for large values of N x N y N z N t. However, Green s function calculations are independent at each point in time and at each point in space, which suggests that these Green s function calculations are appropriate for parallel computations. Thus, a graphics processing unit (GPU) is ideal for this problem. On the GPU, which contains a large number of independent processors, the jobs are broken into several pieces that are distributed to different processors for parallel computations. When these calculations are implemented properly, GPUs achieve a considerable speedup compared to a single serial CPU. First, the acoustic radiation force, which is pre-calculated 30

49 and stored in a data file, is loaded into the CPU (host) memory. Then, the computed acoustic radiation force is copied into the GPU (device) global memory. The kernel function, which describes the superposition algorithm, is then called and the job is separated into multiple threads that are evaluated on different processors in parallel on the GPU. For each thread, the shear wave particle displacement is calculated at a single observation point and a single moment in time. Each thread obtains the acoustic radiation force from global memory, whereas the calculated shear wave particle displacement result is first stored temporarily in local memory for each thread and then copied into global memory after the calculation is completed. After all of the threads are finished, the results are copied and returned to the host memory. The CPU performs any post-processing that is needed and then stores the result in an output data file. These calculations are implemented using the Compute Unified Device Architecture (CUDA) programming language on an nvidia (Santa Clara, CA) Kepler K20 GPU. 3.4 Results Numerical simulations are performed serially on an Intel(R) Core(TM) i GHz CPU and in parallel on an nvidia Tesla K20 GPU. In these simulations, 3D pressure fields are calculated using the fast nearfield method combined with the angular spectrum approach, and shear waves are then computed for the 3D acoustic radiation force that is obtained from the 3D pressure distribution. In each calculation of the acoustic radiation force, the compressional wave speed c p is 1500 m/s, and the attenuation coefficient α p is 0.5 db/cm/mhz for the compressional wave component. The excitation frequency for the transducer array is f 0 = 4.09 MHz. For the shear wave simulations, the shear elasticity µ s is 2.25 kpa, while the viscosity η s is equal to 0 Pa s for simulations with the elastic model and 1 Pa s for simulations with the viscoelastic model. The duration of the acoustic radiation force push in each simulation is 200 µs. 31

50 0 10 z (mm) x (mm) Figure 3.1: The locations of the observation points relative to a push beam focused at 25 mm in the x-z plane with y = Simulated shear wave particle displacements at discrete observation points To demonstrate the similarities and differences between shear waves simulated with the elastic and viscoelastic models, shear wave particle displacements are calculated at 8 different observation points located at the same depth. The depth selected for each observation point is the same as the depth of the focal peak for the push beam. The observation points are equally spaced in the lateral direction at this depth, where the distance from the peak of the push beam to each observation point in the x-direction ranges from 5 to 40 mm with a 5 mm increment in the y = 0 plane. These simulations are evaluated for an f/2 push beam with 40 transducer elements that is focused on-axis at a depth of 25 mm. The focal peak for this push beam is located at 22.9 mm. The push beam and the locations of the observation points are shown in Figure 3.1. At each observation point, the shear wave particle displacements are calculated for 40 ms with a temporal sampling frequency of 10 khz. Figure 3.2 plots the simulated shear wave particle displacements for the push beam shown in Figure 3.1, where the simulated shear wave particle displacements for the elastic 32

51 1.2 1 x=5mm Elastic Medium 3D Push Result Viscoelastic Medium 3D Push Result Normalized Displacement x=10mm x=15mm x=20mm x=25mm x=30mm x=35mm x=40mm t(ms) Figure 3.2: Simulated shear wave particle displacements computed with elastic and viscoelastic models for an applied 3D acoustic radiation force push evaluated at the 8 observation points indicated in Figure 3.1. The solid lines describe the normalized shear wave particle displacements calculated for the elastic model with µ = 2.25 kpa and η s = 0 Pa s, and the dashed lines describe the normalized shear wave particle displacements calculated for the viscoelastic model with µ = 2.25 kpa and η s = 1 Pa s. and viscoelastic models with µ = 2.25 kpa and η s = 0 or 1 Pa s are evaluated at the observation points shown in Figure 3.1. In Figure 3.2, the peaks for different observation points appear at different times. Figure 3.2 also shows that the peak times for the normalized shear wave particle displacements calculated with the elastic model are different from the peak times for the viscoelastic model. Furthermore, at the same observation point, the peaks calculated for the viscoelastic model arrive later than the peaks for the elastic model. In 33

52 addition, the differences in the peak times between successive observation points are not the same for simulations with the elastic model and the viscoelastic model. For shear wave simulations with the elastic model, the differences in the peak times for adjacent observation points are 3.34 ms, 3.37 ms, 3.36 ms, 3.35 ms, 3.37 ms, 3.35 ms, and 3.38 ms for the observation points with x coordinates equal to (5mm and 10mm), (10mm and 15mm), (15mm and 20mm), (20mm and 25mm), (25mm and 30mm), (30mm and 35mm), and (35mm and 40mm), respectively. For simulations of shear waves with the viscoelastic model, the peak time differences for the same pairs of observation points are 3.69 ms, 3.61 ms, 3.54 ms, 3.42 ms, 3.33 ms, 3.30 ms, and 3.29 ms, respectively. Thus, the peak time differences are nearly constant for simulations of shear waves with the elastic model, whereas the peak time differences for the simulated shear waves with the viscoelastic model decrease monotonically as the pairs of observation points move further away from the push beam. The amplitudes of the peak displacements decrease with distance for the elastic model strictly due to the effect of diffraction, whereas the peak amplitudes for the viscoelastic model decrease due to the combined effects of diffraction and viscous loss. Also, the peak amplitudes decrease much more rapidly with the viscoelastic model than with the elastic model Computation times for serial and parallel calculations To evaluate the computation times of the serial and parallel implementations, the simulations are programmed and evaluated in C and CUDA, respectively. Computation times are determined for shear wave simulations with the elastic and viscoelastic models using the push beam shown in Figure 3.1. In these simulations, the number of observation points ranges from 20 to 200 with an increment of 20. The observation points are sampled uniformly between 5 mm and 40 mm in the x direction, the y-coordinate is equal to zero, and the z coordinate is equal to 22.9 mm. Thus, the end points are the same as in Figure 3.1, but the region in between is more densely sampled. For each observation point, the shear wave particle displacements are calculated at 400 temporal samples for 40 ms, which yields 34

53 a temporal sampling rate of 10 khz. Eq is evaluated once at each temporal sample and at each observation point, so the total number of spatial and temporal points at which the Green s functions are superposed ranges from 8,000 (20 spatial points 400 temporal points) to 80,000 (200 spatial points 400 temporal points). Furthermore, the discretized 3D acoustic radiation force contains = 1,306,910 source points, so the total number of Green s function evaluations with Eqs or Eqs ranges from to in these simulations. The serial and parallel codes are evaluated and the computation times are measured and stored. The serial C code is implemented on a desktop computer with an Intel(R) Core(TM) i GHz CPU, and the parallel CUDA code is implemented on an nvidia Tesla K20 GPU at Michigan State University s high performance computing center ( Figure 3.3 describes the computation time versus the number of observation points for elastic and viscoelastic shear wave simulation models. The results in Figure 3.3 show that the parallel implementation is much faster than the serial implementation. For the shear wave simulations with the elastic model, the parallel implementation is about 45 times faster than the serial implementation, while for shear wave simulations with the viscoelastic model, the parallel implementation is about 700 times faster than the serial implementation. Moreover, the simulations of shear waves with the elastic model are much faster than the simulations of shear waves with the viscoelastic model for both the serial and parallel implementations. In the serial implementation, the simulation of shear waves with the elastic model is about 50 times faster than the simulation of shear waves with the viscoelastic model. In the parallel implementation, the simulation of shear waves with the elastic model is around 2 times faster than the simulation of shear waves with the viscoelastic model Convergence of Green s function calculations To assess the numerical convergence of these Green s function calculations, several simulations of shear waves induced by push beams with different spatial sampling rates and 35

54 Computation time (s) CUDA code - elastic model CUDA code - viscoelastic model C code - elastic model C code - viscoelastic model Number of observation points Figure 3.3: Computation times for serial and parallel implementations of time domain shear wave calculations with Green s functions. The solid line with circle markers indicates the computation times for parallel simulations of shear waves with the elastic model, the dasheddotted line with triangle markers indicates the computation times for parallel simulations of shear waves with the viscoelastic model, the dashed line with diamond markers indicates the computation times for serial simulations of shear waves with the elastic model, and the dotted line with square markers indicates the computation times for serial simulations of shear waves with the viscoelastic model. different f-numbers (f/#) are evaluated. Different ultrasound beams are simulated by selectively activating contiguous groups of transducer elements. For unfocused beam calculations, 12 elements are activated with equal phases and amplitudes. For focused beam calculations, the number of active array elements varies with the f/#. Specifically, 54, 40, and 27 elements are excited and focused on-axis at a depth of 25 mm for values of the f/# equal to 1.5, 2, and 3, respectively. Figure 3.4 depicts four different simulated ultrasound beams in the y=0 plane. Figure 3.4a shows the focused ultrasound beam with f/1.5, Figure 3.4b shows the focused ultrasound beam with f/2, Figure 3.4c shows the focused ultrasound beam with f/3, and Figure 3.4d shows the unfocused beam (f/ ). The f/2 push beam shown in Figure 3.4b describes the same configuration depicted in Figure

55 z (mm) (a) x (mm) z (mm) (b) x (mm) z (mm) (c) x (mm) z (mm) (d) x (mm) Figure 3.4: 2D cross-sections of 3D ultrasound beams simulated in FOCUS. Figures (3.4a), (3.4b), (3.4c), and (3.4d) show the push beams with f/1.5, f/2, f/3, and f/, respectively. In (3.4a)-(3.4d), each beam is normalized by the maximum intensity value. For each push beam, the acoustic radiation force is calculated with spatial samplings of 4λ, 2λ, λ, λ/2, λ/4, λ/8, λ/16, and λ/18, where λ is the wavelength of the compressional wave (λ = c p /f 0 = mm). The push beams with spatial samplings less than or equal to λ/2 are calculated with FOCUS as described in Section 3.3, and the push beams that are more coarsely sampled are obtained by down-sampling the push beam that is computed with λ/2 sampling. The push beam sampled at 4λ, which is the most coarsely sampled push beam, consists of N s x N s y N s z = = 2,457 source points, while the most finely sampled push beam sampled at λ/18 consists of N s x N s y N s z = = 945,520,173 source points. The simulated shear wave particle displacement induced by the λ/18 sampled push beam is used as the reference, and the simulated shear wave particle displacements induced by all of the other push beams are compared with the reference shear wave particle displacement to evaluate the convergence of these Green s function calculations. The percent difference between the two shear wave particle displacement waveforms is defined by % difference = u(r, t) u ref (r, t), (3.21) u ref (r, t) where u(r, t) is the simulated shear wave particle displacement induced by a push beam at a single observation point, u ref (r, t) is the reference shear wave particle displacement induced 37

56 Unfocused push beam Focused push beam with f/# = 3 Focused push beam with f/# = 2 Focused push beam with f/# = 1.5 Percent difference /2 /4 /8 /16 Spatial sampling of the push beam Figure 3.5: Percent differences calculated for shear wave particle displacements simulated with the elastic model. The shear elasticity and shear viscosity values in these simulations are µ = 2.25 kpa and η s = 0 Pa s, where the push beams are calculated with different spatial samplings. The solid line with circle markers indicates the percent difference for the shear wave particle displacements induced by an unfocused push beam with f/, the dotted line with square markers indicates the percent difference for the shear wave particle displacements induced by a focused push beam with f/3, the dash-dotted line with triangle markers indicates the percent difference for the shear wave particle displacements induced by a focused push beam with f/2, and the dashed line with diamond markers indicates the percent difference for the shear wave particle displacements induced by a focused push beam with f/1.5. by the push beam sampled at λ/18 at the same observation point, and denotes the L 2 -norm of the time-domain waveform. For the shear wave particle displacement simulated by each push beam, the percent difference is evaluated at 8 observation points. For each push beam, the observation points 38

57 are all located at the same depth, and the depth selected for each observation point is the same as the depth of the focal peak. For the push beams with f/1.5, f/2, f/3, and f/, these depths are equal to 24.5 mm, 22.9 mm, 20.0 mm, and 16.6 mm, respectively. The observation points are equally spaced in the lateral direction, where the distance from the peak of the push beam to each observation point in the x-direction ranges from 5 to 40 mm with a 5 mm increment in the y = 0 plane. The largest percent difference is retained to represent the worst case value across these 8 locations. The percent difference is evaluated for simulated shear waves with the elastic model using µ s = 2.25 kpa and η s = 0 Pa s and for simulated shear waves with the viscoelastic model using µ s = 2.25 kpa and η s = 1 Pa s for the four push beams with f/1.5, f/2, f/3, and f/ described in section The results are illustrated in Figure 3.5, which shows the percent difference in the shear wave particle displacements calculated using the elastic model for the four push beams evaluated with different spatial samplings. In each of these simulations, the percent difference decreases when the spatial sampling rate increases. For the f/1.5 push beams sampled at λ, the percent difference is 16.4%, the percent difference is 7.04% when the spatial sampling is λ/2, and the percent difference is 2.50% when the spatial sampling of the push beam is λ/16. The percent differences for the f/2 push beams sampled at λ, λ/2, and λ/16 are 13.5%, 4.53%, and 0.98%, respectively. The percent differences for the f/3 push beams sampled at λ, λ/2, and λ/16 are 9.96%, 3.63%, and 0.23%, respectively. The percent differences for the unfocused push beams sampled at 4λ, λ/2, and λ/16 are 6.00%, 2.70%, and 0.29%, respectively. The percent differences in the shear wave particle displacements simulated for the viscoelastic model with µ = 2.25 kpa and η s = 1 Pa s are displayed in Figure 3.6 for the same four push beams. Similar to the Green s function simulations of shear waves with the elastic model, the percent difference decreases when the spatial sampling increases. For the f/1.5 push beam sampled at λ, the percent difference is 6.15%, the percent difference is 4.74% when the spatial sampling is λ/2, and the percent difference is 1.00% when the spatial sampling of the push beam is λ/16. The percent differences for the f/2 push beam sampled at 39

58 10 8 Unfocused push beam Focused push beam with f/# = 3 Focused push beam with f/# = 2 Focused push beam with f/# = 1.5 Percent difference /2 /4 /8 /16 Spatial sampling of the push beam Figure 3.6: Percent differences calculated for shear wave particle displacements simulated with the elastic model. The shear elasticity and shear viscosity values in these simulations are µ = 2.25 kpa and η s = 1 Pa s, where the push beams are calculated with different spatial samplings. The descriptions of the markers and line types are the same as those given in the caption for Figure 3.5. λ, λ/2, and λ/16 are 2.56%, 1.26%, and 0.40%, respectively. The percent differences for the f/3 push beam sampled at λ, λ/2, and λ/16 are 1.15%, 0.92%, and 0.19%, respectively. The percent differences for the unfocused push beam sampled at λ, λ/2, and λ/16 are 1.35%, 0.83%, and 0.34%, respectively. 40

59 3.4.4 Comparisons between simulated and measured shear waves To validate the viscoelastic model, shear waves are measured in viscoelastic shear wave phantoms and compared with the simulated shear waves. Shear waves were measured with a Verasonics ultrasound system (Verasonics Inc., Redmond, WA) and an L7-4 linear ultrasound transducer array, which is the same transducer array that is described and modeled in section To induce shear waves, an f/2 push beam focused at 25 mm with a center frequency of 4.09 MHz is generated when 40 contiguous elements of the transducer array are activated with the same f/2 focused beam shown in Figure 3.1. The push beam is applied for 200 µs, and then the Verasonics system is switched to plane wave imaging mode for shear wave measurements. In-phase/quadrature (IQ) data is measured for 10 ms with a frame rate of khz in a two-dimensional (2D) region ranging from mm to mm in the lateral direction and from 0 to mm in the axial direction with a spatial sampling of mm in each direction. Then, shear wave particle velocities are obtained from the IQ data by calculating one-dimensional (1D) autocorrelations between consecutive frames [11]. A 2D median filter with a 3 by 3 kernel is then applied to the measured shear wave particle velocities for each frame. Three different viscoelastic shear wave phantoms from Computerized Imaging Reference Systems (CIRS), Inc (Norfolk, Virginia, US) are used for these measurements, namely shear wave phantoms E3962-1, E3962-2, and E After the shear waves in the viscoelastic shear wave phantoms are measured, the results are compared to simulated shear waves evaluated within the same 2D region at three different points in time. For phantoms E and E3962-2, shear wave particle displacements are displayed at 3 ms, 6 ms, and 9 ms, while for phantom E3962-3, shear wave particle displacements are displayed at 2 ms, 3 ms, and 4 ms. Different time points are required for phantom E because the shear elasticity for this phantom is much larger, so the shear waves propagate out of the FOV much more quickly. The shear waves are simulated for several different combinations of the shear elasticity and shear viscosity. For phantom E3962-1, the shear elasticity and the shear viscosity are relatively small, so the shear elasticity in the 41

60 simulations ranges from 0 to 0.5 kpa with an increment of 0.1 kpa, and the shear viscosity rages from 0 to 0.5 Pa s with an increment of 0.1 Pa s. For shear wave phantoms E and E3962-3, the shear elasticity in the simulations ranges from 2 to 5 kpa with an increment of 0.1 kpa, and the shear viscosity rages from 0 to 2.5 Pa s with an increment of 0.1 Pa s. For each combination of parameters, the simulation of the shear wave particle velocity in a 2D plane at one point in time takes 25 s on an nvidia Tesla K20 GPU. After the shear wave particle velocities are simulated on the GPU, the peak values of the measured and simulated shear wave particle velocities are both normalized to 1, and the percent difference between the simulated and measured shear wave particle velocities is evaluated with % difference = u sim(r, t) u meas (r, t), (3.22) u meas (r, t) where u sim (r, t) represents the simulated shear wave particle velocity in a 2D plane at time t, u meas (r, t) represents the measured shear wave particle velocity in the same 2D plane at time t, and denotes the L 2 -norm. The percent difference is calculated for each combination of the shear elasticity and shear viscosity parameters, and the combination of shear wave parameters that provides the minimum percent difference is determined from the results. Figure 3.7 shows the measured shear wave particle velocities in phantom E and the corresponding simulated shear wave particle velocities that minimize the percent difference in Eq For all three points in time, the material parameters that provide the minimum percent difference are µ = 0.3 kpa and η s = 0.1 Pa s. In Figure 3.7(a), the location of the peak shear wave particle velocity in the right half of the image is (x, z) = (2.15 mm, mm), while in Figure 3.7(d), the location of the peak shear wave particle velocity in the right half of the image is (x, z) = (2.00 mm, mm). Also, in Figure 3.7(a), the full width at half maximum (FWHM) of the shear wave in the x direction evaluated at the peak on the right is 1.08 mm, and in Figure 3.7(d), the FWHM of the shear wave in the x direction evaluated at the peak on the right is 1.14 mm. In Figure 3.7(b), the location of the peak shear wave particle velocity in the right half of the image is (x, z) = (3.69 mm, mm), while in Figure 3.7(e), the maximum location of the peak shear wave particle velocity in the 42

61 5 time = 3 ms 5 time = 6 ms 5 time = 9 ms z(mm) z(mm) z(mm) (a) x(mm) (b) x(mm) (c) x(mm) 5 10 z(mm) z(mm) z(mm) (d) x(mm) (e) x(mm) (f) x(mm) Figure 3.7: Measured shear wave particle velocities in phantom E and the corresponding simulated shear wave particle velocities that minimize Eq at three different times. Figs. 3.7(a), 3.7(b), and 3.7(c) show the shear wave particle velocities measured at 3 ms, 6 ms, and 9 ms, respectively. Figs. 3.7(d), 3.7(e), and 3.7(f) show the corresponding simulated shear wave particle velocities computed with µ = 0.3 kpa and η s = 0.1 Pa s at 3 ms, 6 ms, and 9 ms, respectively. right half of the image is (x, z) = (3.85 mm, mm). Also, in Figure 3.7(b), the FWHM of the shear wave in the x direction evaluated at the peak on the right is 1.55 mm, and in Figure 3.7(e), the FWHM of the shear wave in the x direction evaluated at the peak on the right is 1.57 mm. In Figure 3.7(c), the location of the peak shear wave particle velocity in the right half of the image is (x, z) = (5.24 mm, mm), while in Figure 3.7(f), the location of the peak shear wave particle velocity in the right half of the image is (x, z) = (5.54 mm, mm). Also, in Figure 3.7(c), the FWHM of the shear wave in the x direction evaluated at the peak on the right is 1.98 mm, and in Figure 3.7(f), the FWHM of the shear wave in the x direction evaluated at the peak on the right is 1.91 mm. In each pair of images, 43

62 5 time = 3 ms 5 time = 6 ms 5 time = 9 ms z(mm) z(mm) z(mm) (a) x(mm) (b) x(mm) (c) x(mm) 5 10 z(mm) z(mm) z(mm) (d) x(mm) (e) x(mm) (f) x(mm) Figure 3.8: Measured shear wave particle velocities in phantom E and the corresponding simulated shear wave particle velocities that minimize Eq at three different times. Figs. 3.8(a), 3.8(b), and 3.8(c) show the shear wave particle velocities measured at 3 ms, 6 ms, and 9 ms, respectively. Figs. 3.8(d), 3.8(e), and 3.8(f) show the corresponding simulated shear wave particle velocities with µ = 2.3 kpa and η s = 0.1 Pa s at 3 ms, 6 ms, and 9 ms, respectively. the peak locations and the FWHM values at these peak locations are similar. The images showing the measured (Figure 3.7a - 3.7c) and simulated (Figure 3.7d - 3.7f) shear wave particle velocities are also visually similar for phantom E Figure 3.8 shows the measured shear wave particle velocities in phantom E and the corresponding simulated shear wave particle velocities that minimize the percent difference in Eq For all three points in time, the material parameters that provide the minimum percent difference are µ = 2.3 kpa and η s = 0.1 Pa s. In Figure 3.8(a), the location of the peak shear wave particle velocity in the right half of the image is (x, z) = (4.93 mm, mm), while in Figure 3.8(d), the location of the peak shear wave particle velocity in the right 44

63 half of the image is (x, z) = (5.08 mm, mm). Also, in Figure 3.8(a), the full width at half maximum (FWHM) of the shear wave in the x direction evaluated at the peak on the right is 1.23 mm, and in Figure 3.8(d), the FWHM of the shear wave in the x direction evaluated at the peak on the right is 1.17 mm. In Figure 3.8(b), the location of the peak shear wave particle velocity in the right half of the image is (x, z) = (9.70 mm, mm), while in Figure 3.8(e), the location of the peak shear wave particle velocity in the right half of the image is (x, z) = (9.55 mm, mm). Also, in Figure 3.8(b), the FWHM of the shear wave in the x direction evaluated at the peak on the right is 1.62 mm, and in Figure 3.8(e), the FWHM of the shear wave in the x direction evaluated at the peak on the right is 1.59 mm. In Figure 3.8(c), the location of the peak shear wave particle velocity in the right half of the image is (x, z) = (14.17 mm, mm), while in Figure 3.8(f), the location of the peak shear wave particle velocity in the right half of the image is (x, z) = (14.32 mm, mm). Also, in Figure 3.8(c), the FWHM of the shear wave in the x direction evaluated at the peak on the right is 1.80 mm, and in Figure 3.8(f), the FWHM of the shear wave in the x direction evaluated at the peak on the right is 1.94 mm. In each pair of images, the peak locations and the FWHM values at these peak locations are similar. The images showing the measured (Figure 3.8a - 3.8c) and simulated (Figure 3.8d - 3.8f) shear wave particle velocities are also visually similar for phantom E Figure 3.9 shows the measured shear wave particle velocities in phantom E and the corresponding simulated shear wave particle velocities that minimize the percent difference in Eq For the shear waves that are measured and simulated at 2 ms, the material parameters that provide the minimum percent difference are µ = 4.9 kpa and η s = 1.5 Pa s. For the shear waves that are measured and simulated at 3 ms and 4 ms, the material parameters that provide the minimum percent difference are µ = 4.5 kpa and η s = 1.5 Pa s. In Figure 3.9(a), the location of the peak shear wave particle velocity in the right half of the image is (x, z) = (6.00 mm, mm), while in Figure 3.9(d), the location of the peak shear wave particle velocity in the right half of the image is (x, z) = (5.85 mm,

64 5 time = 2 ms 5 time = 3 ms 5 time = 4 ms z(mm) z(mm) z(mm) (a) x(mm) (b) x(mm) (c) x(mm) 5 10 z(mm) z(mm) z(mm) (d) x(mm) (e) x(mm) (f) x(mm) Figure 3.9: Measured shear wave particle velocities in phantom E and the corresponding simulated shear wave particle velocities that minimize Eq evaluated at three different times. Figs. 3.9(a), 3.9(b), and 3.9(c) show the shear wave particle velocities measured at 2 ms, 4 ms, and 6 ms, respectively. Figs. 3.9(d), 3.9(e), and 3.9(f) show the corresponding simulated shear wave particle velocities computed with µ = 4.9 kpa and η s = 1.5 Pa s at 2 ms, and computed with µ = 4.5 kpa and η s = 1.5 Pa s at 3 ms and 4 ms, respectively. mm). Also, in Figure 3.9(a), the full width at half maximum (FWHM) of the shear wave in the x direction evaluated at the peak on the right is 3.04 mm, and in Figure 3.9(d), the FWHM of the shear wave in the x direction evaluated at the peak on the right is 3.37 mm. In Figure 3.9(b), the location of the peak shear wave particle velocity in the right half of the image is (x, z) = (7.39 mm, mm), while in Figure 3.9(e), the location of the peak shear wave particle velocity in the right half of the image is (x, z) = (8.01 mm, mm). Also, in Figure 3.9(b), the FWHM of the shear wave in the x direction evaluated at the peak on the right is 3.55 mm, and in Figure 3.9(e), the FWHM of the shear wave in the x direction evaluated at the peak on the right is 4.24 mm. In Figure 3.9(c), the location 46

65 of the peak shear wave particle velocity in the right half of the image is (x, z) = (10.93 mm, mm), while in Figure 3.9(f), the location of the peak shear wave particle velocity in the right half of the image is (x, z) = (10.47 mm, mm). Also, in Figure 3.9(c), the FWHM of the shear wave in the x direction evaluated at the peak on the right is 4.62 mm, and in Figure 3.9(f), the FWHM of the shear wave in the x direction evaluated at the peak on the right is 4.81 mm. In each pair of images, the peak locations are quite similar, but the FWHM values at these peak locations differ somewhat, since the FWHM for the measured shear wave is consistently smaller than the FWHM for the simulated shear wave. The above results indicate that, although some differences are observed in the FWHM values for the shear wave particle velocities, the peak locations are similar, and these differences and similarities are also observed in visual comparisons between Figures 3.9(a-c) and 3.9(d-f). 3.5 Discussion Diffraction, attenuation, and dispersion The results shown in Figure 3.2 reveal the complicated interaction between diffraction, attenuation, and dispersion in the shear wave particle displacements induced by a 3D acoustic radiation force. Figure 3.2 shows that the peak amplitude of the shear wave particle displacement decreases as the observation point moves away from the push beam. For simulations of shear waves with the elastic model, the amplitude of the shear wave particle displacement decreases solely due to diffraction. The peak time differences between successive observation points are approximately the same, which indicates that the propagation speed of the shear wave remains constant along the line of observation points shown in Figure 3.1. Moreover, the values of the full width at half maximum (FWHM) for the shear wave particle displacement calculated at the observation points shown in Figure 3.1 are 1.06 ms, 1.02 ms, 0.99 ms, 0.97 ms, 1.00 ms, 1.02 ms, 1.06 ms, and 1.08 ms for the pairs of observation points with x coordinates equal to 5 mm, 10 mm, 15 mm, 20 mm, 25 mm, 30 mm, 35 mm, and 40 mm, respectively. Thus, the values of the FWHM are similar at each of these observa- 47

66 tion points, which suggests that there is no significant waveform spreading due to dispersion along this line of points for simulated shear wave particle displacements evaluated with the elastic model. For simulations of shear waves with the viscoelastic model, the amplitude of the shear wave particle displacement decreases much faster than with the elastic model, as the amplitude of the shear wave particle displacement decreases due to the combined effects of diffraction and attenuation. Furthermore, the differences between the times at which the peaks occur for successive observation points are monotonically decreasing as the observation points move away from the push beam. This suggests that the propagation speed of the shear wave particle displacement decreases as the shear wave propagates due to the frequency dependence of the phase velocity. Moreover, the values of the full width at half maximum (FWHM) for the shear wave particle displacements calculated at the observation points shown in Figure 3.1 are 5.50 ms, 6.70 ms, 7.33 ms, 7.68 ms, 7.94 ms, 8.22 ms, 8.54 ms, and 8.89 ms for the observation points with x coordinates equal to 5 mm, 10 mm, 15 mm, 20 mm, 25 mm, 30 mm, 35 mm, and 40 mm, respectively. These values for the FWHM, which increase monotonically, also demonstrate that the shear wave particle velocity spreads out in time as the shear wave propagates, thereby providing additional evidence of shear wave dispersion in the viscoelastic model Serial vs parallel implementations The computation times for the serial and parallel calculations shown in Figure 3.3 indicate that the GPU accelerates these Green s function calculations by factors of 45 and 700 for simulations of shear waves with the elastic and viscoelastic models, respectively. Thus, the GPU achieves a significant reduction in the computation time. For instance, the computation time for the shear waves in a 252 by 257 plane shown in Figs is around 26.3 seconds on an nvidia Tesla K20 GPU, whereas the serial implementation of the same calculation takes 5.27 hours on a desktop computer with an Intel(R) Core(TM) i GHz CPU. This suggests that these Green s function calculations are preferably performed on a high- 48

67 performance GPU. The results shown in Figure 3.3 also reveal that both serial and parallel calculations are faster for simulations of shear waves with the elastic model because the calculation of the error function in Eqs is much more time consuming than the calculation of the Heaviside function in Eqs Convergence Figures 3.5 and 3.6 show that the percent difference is larger for shear wave particle displacements calculated using push beams with lower f/#. Smaller percent differences are achieved in simulations of arrays with larger f/# because for push beams with lower f/#, the energy of the acoustic radiation force is more concentrated in the focal region, so finer spatial samplings are required to capture the high frequency components in the spatial domain. Figures 3.5 and 3.6 also indicate that when the spatial sampling of the push beam is greater than or equal to λ/2, the percent difference for the simulated shear wave particle displacements with the elastic model is usually somewhat higher than for simulations of shear waves with the viscoelastic model. However, for push beams sampled more finely than λ/2, the percent differences are similar for shear wave simulations evaluated with the elastic and viscoelastic models. For most of the results shown in Figures 3.5 and 3.6, the percent difference decreases monotonically as the spatial sampling decreases. Two exceptions to this trend are observed in Figure 3.6, where the percent difference for push beams sampled at λ/2 is smaller than the percent difference for push beams sampled at λ/4 and λ/8. Similar behavior is also observed in the convergence of the fast nearfield method for rectangular and circular pistons [130,131], where the overall trend is that the difference relative to the references is decreasing, but local minima are also observed in several locations. To determine whether the local minima in Figure 3.6 are unique to the particular parameter combinations evaluated here, additional simulations of shear waves in viscoelastic models were performed using f/1.5 push beams evaluated with different values of the shear elasticity and shear viscosity. Four different 49

68 parameter combinations were evaluated: µ s = 2.25 kpa and η s = 2 Pa s, µ s = 2.25 kpa and η s = 0.1 Pa s, µ s = 6 kpa and η s = 1 Pa s, and µ s = 1 kpa and η s = 1 Pa s. For each parameter combination, the percent difference for push beams with spatial samplings of mλ with m = 1 to 4 with an increment of 1 and λ/n with n = 2 to 18 with an increment of 1 were evaluated. In all of these simulations, a local minimum for the percent difference was also observed for the push beam sampled at λ/2. In particular, the percent difference for the shear wave particle displacement simulated using the push beam sampled at λ/2 was consistently smaller than the push beam sampled from 4λ to λ and from λ/3 to λ/8 in each of these results. This suggests that λ/2 is a reasonable default value for the spatial sampling, especially for push beams with smaller f/# evaluated with the viscoelastic model. For shear wave simulations with the elastic model, the percent difference is below 5% for the f/2, f/3, and f/ push beams when the spatial sampling is λ/2, and the percent difference is below 5% for the f/1.5 push beam when the spatial sampling is λ/4. For shear wave simulations evaluated with the viscoelastic model, the percent difference is below 5% for the f/2, f/3, and f/ push beams when the spatial sampling is 2λ, and the percent difference is below 5% for the push beam with f/1.5 when the spatial sampling is λ/2. This information is important for simulations of shear waves because the accuracy of the shear wave particle displacement depends on the sampling of the push beam. If a smaller percent difference is desired, e.g., 1%, then a much finer spatial sampling is required. For shear wave simulations evaluated with the elastic model, the percent difference is greater than 1% for the f/1.5 push beam when the spatial sampling is greater than or equal to λ/16, the percent difference is below 1% for the f/2 push beam when the spatial sampling is λ/16, the percent difference is below 1% for the f/3 push beam when the spatial sampling is λ/8, and the percent difference is below 1% for the f/ push beam when the spatial sampling is equal to λ/4. For shear wave simulations evaluated with the viscoelastic model, the percent difference is 1% for the f/1.5 push beam when the spatial sampling is λ/16, the percent difference is below 1% for the f/2 push beam when the spatial sampling is λ/16, 50

69 Phantom E Phantom E Measured Simulated Measured Simulated FWHM (mm) at 3ms FWHM (mm) at 6ms FWHM (mm) at 9ms Table 3.1: The FWHM values of the measured and simulated shear wave particle velocities in the x direction evaluated at the peak at 3 ms, 6 ms, and 9 ms for phantom E and E and the percent difference is below 1% for the f/3 and f/ push beams when the spatial sampling is λ/ Comparisons between measured and simulated shear waves Fig. 3.7 shows the measured and simulated shear waves for phantom E The lateral positions (x) of the peak shear wave particle velocities at 3 ms, 6 ms, and 9 ms are 2.15 mm, 3.85 mm, and 5.24 mm, respectively, for the measured shear waves, and are 2.00 mm, 3.85 mm, and 5.54 mm, respectively, for the simulated shear waves. The depths (z) of the peak shear wave particle velocities at 3 ms, 6 ms, and 9 ms are mm, mm, and mm, respectively, for the measured shear waves, and are mm, mm, and mm, respectively, for the simulated shear waves. These values indicate that the peak shear wave particle velocity remains at about the same depth during propagation. In addition, as shown in Table 3.1, The FWHM values of the shear wave in the x direction evaluated at the peak at 3 ms, 6 ms, and 9 ms are 1.08 mm, 1.55 mm, and 1.98 mm, respectively, for the measured shear waves, and are 1.14 mm, 1.57 mm, and 1.91 mm, respectively, for the simulated shear waves. These results demonstrate that the FWHM values increase by a significant amount as the shear wave propagates due to shear wave diffraction and waveform spreading despite the relatively small value for the shear viscosity. Fig. 3.8 shows the measured and simulated shear waves for phantom E The lateral positions (x) of the peak shear wave particle velocities at 3 ms, 6 ms, and 9 ms are 4.93 mm, 9.70 mm, and mm, respectively, for the measured shear waves, and are 5.08 mm, 51

70 Phantom E Measured Simulated FWHM (mm) at 2ms FWHM (mm) at 3ms FWHM (mm) at 4ms Table 3.2: The FWHM values of the measured and simulated shear wave particle velocities in the x direction evaluated at the peak at 3 ms, 6 ms, and 9 ms for phantom E mm, and mm, respectively, for the simulated shear waves. The depths (z) of the peak shear wave particle velocities at 3 ms, 6 ms, and 9 ms are mm, mm, and mm, respectively, for the measured shear waves, and are mm, mm, and mm, respectively, for the simulated shear waves. These results indicate that the peak shear wave particle velocity remains at about the same depth during propagation. The above results also indicate that the positions of the peak shear wave particle velocity for the measured and simulated shear waves are similar. In addition, as shown in Table 3.1, the FWHM values for the shear wave particle velocities in the x direction for the peak on the right evaluated at 3 ms, 6 ms, and 9 ms are 1.23 mm, 1.62 mm, and 1.80 mm, respectively, for the measured shear waves, and are 1.17 mm, 1.59 mm, and 1.94 mm, respectively, for the simulated shear waves. These results demonstrate that the FWHM values increase by a significant amount as the shear wave propagates due to shear wave diffraction and waveform spreading despite the relatively small value for the shear viscosity. Fig. 3.9 shows the measured and simulated shear waves for phantom E The lateral positions (x) of the peak shear wave particle velocities at 2 ms, 3 ms, and 4 ms are 6.00 mm, 7.39 mm, and mm, respectively, for the measured shear waves, and are 5.85 mm, 8.01 mm, and mm, respectively, for the simulated shear waves. The depths (z) of the peak shear wave particle velocities at 2 ms, 3 ms, and 4 ms are mm, mm, and mm, respectively, for the measured shear waves, and are mm, mm, and mm, respectively, for the simulated shear waves. These results indicate that the peak shear wave particle velocity is moving upward in Fig. 3.9 during propagation. The above results also indicate that the locations for the peak shear wave particle velocities of measured and 52

71 simulated shear waves have similar positions. In addition, as shown in Table 3.2, the FWHM of the shear wave in the x direction evaluated for the peak value at 2 ms, 3 ms, and 4 ms are 3.04 mm, 3.55 mm, and 4.62 mm, respectively, for the measured shear waves, and are 3.37 mm, 4.24 mm, and 4.81 mm, respectively, for the simulated shear waves. These results show that, for larger values of the shear viscosity, there is a significant difference in the FWHM values. Possible explanations for this are that a different model is required (e.g. a fractional Voigt model) for larger shear viscosity values and/or that a different objective function that emphasizes agreement in the FWHM at the expense of diminished agreement at the peak location is needed. The resolution of these and other related issues is a topic of ongoing research Intended applications These GPU-accelerated shear wave simulations are intended for several different types of calculations. For example, these shear wave simulations are useful for evaluating different push beams and different material parameter combinations. These simulations are also applicable to parametric characterizations of various methods for estimating shear wave parameters, including the time-of-flight approach [61,81, ] and the k-space approach [167]. In addition, these routines can be included in new model-based approachs to minimize the difference between measured and simulated shear waves to obtain more accurate estimates of shear wave parameters. Hardware acceleration is essential for the optimization and for repeated shear wave calculations, because these calculations are otherwise very time-consuming. These calculations could potentially be incorporated into FOCUS, and we also anticipate that other applications will be developed once these routines are made available. 3.6 Conclusion Shear waves induced by an acoustic radiation force are evaluated using Green s functions for simulations with elastic and viscoelastic models. The 3D acoustic radiation force is first 53

72 simulated using the fast nearfield method and the angular spectrum approach in FOCUS, and then shear waves are calculated by superposing the contributions from all of the source points in the 3D push beam using Green s functions. Next, these simulations of shear waves in elastic and viscoelastic models are accelerated using a high-performance GPU. Diffraction effects are observed in the simulated shear wave particle displacements in both elastic and viscoelastic models, while additional dispersion and attenuation effects are produced by the shear viscosity. Comparisons between the computation times for serial and parallel implementations show that the nvidia Tesla K20 GPU accelerates these Green s function calculations by factors of 45 and 700 for shear wave simulations in elastic and viscoelastic models, respectively, relative to serial calculations on a desktop computer with an Intel(R) Core(TM) i GHz CPU. When these Green s functions are evaluated for push beams with different spatial samplings, the results show that the percent difference is larger for shear wave particle displacements calculated by push beams with lower f/#. The results indicate that the percent difference usually decreases monotonically as the spatial sampling decreases, with the exception of the simulated shear waves in a viscoelastic medium using f/1.5 and f/2 push beams, where the percent difference for both of these reach a local minimum when the spatial sampling is λ/2. The results also indicate that effective results are obtained using push beams sampled at λ/2, which yield percent differences less than 5%. Measured and simulated shear waves in elastic and viscoelastic shear wave phantoms are compared, and the results suggest that the Green s function simulation approach provides an effective simulation model for shear waves in elastic and viscoelastic shear wave phantoms. 54

73 CHAPTER 4 TIME-DOMAIN ESTIMATION OF SHEAR ELASTICITY AND SHEAR VISCOSITY FOR SHEAR WAVE ELASTICITY IMAGING (SWEI) IN VISCOELASTIC SHEAR WAVE PHANTOMS 4.1 Introduction Shear wave elasticity imaging (SWEI) detects the mechanical properties of soft tissue by analyzing the shear wave induced by an acoustic radiation force [42]. In SWEI, one or more focused or unfocused ultrasound beams are applied to the soft tissue for a short period of time to generate shear waves. The shear waves are measured by comparing consecutive in-phase/quadrature (IQ) data obtained with a speckle tracking algorithm [11]. Mechanical properties are then obtained by analyzing the measured shear waves. SWEI has significant diagnostic potential to noninvasively characterize tissue mechanical properties, which often correlate with tissue physiology [42, 61]. For example, studies have shown that there is a positive correlation between the stiffness of the liver tissue and the stage of the liver fibrosis [44, 61 65]. Other clinical applications for SWEI include quantitative assessment of mechanical properties in thyroid [68], kidney [71], and prostate [72]. In SWEI, the shear elasticity is often obtained using the time-of-flight (TOF) of propagating shear waves. In TOF measurements for SWEI, the arrival times of the shear waves are measured at several successive spatial points located at the same depth but in different lateral positions. One approach obtains the arrival times by measuring the time to peak (TTP) [61, 157], which tracks the arrival times of the peaks for each measured shear wave. Another approach obtains the arrival time delays with cross-correlations [81, 158, 159]. Of these, the cross-correlation approach is preferred because this approach is more robust and much less sensitive to noise. The shear wave propagation speed c s is then calculated from the slope of the linear regression between the lateral locations of the spatial points and the shear wave arrival times. Another approach that improves the robustness of the linear regression 55

74 uses random sample consensus (RANSAC), which updates the linear fit by randomly generating trial solutions and iteratively updating the model by eliminating the outliers of the arrival time delays [160,161]. The shear elasticity µ is then obtained from µ = ρc 2 s, where ρ is the density. Although the TOF method often yields effective values in elastic (lossless) shear wave phantoms [61, ], this method is based upon some simplifications and approximations that cause significant bias when applied to mechanical parameter estimates in soft tissue, which is always lossy (viscoelastic) [79, 116]. In viscoelastic media, the shear wave speed changes during propagation due to shear wave attenuation and dispersion, so the relationship µ = ρc 2 s no longer holds [115, 148]. Thus, the shear elasticity estimates provided by the TOF method are biased by the shear viscosity. Another problem is that the TOF method implicitly assumes that the shear wave is a plane wave. However, complicated diffraction effects are observed in all shear waves induced by a three dimensional (3D) acoustic radiation force [165]. Also, the TOF method yields different values of the estimated shear elasticity for different transducers, focal depths, and lateral ranges applied to the same viscoelastic shear wave phantom [166], which suggests that the diffraction of the push beam and the diffraction of the shear wave also contribute to the bias observed in shear wave parameter estimates obtained from the TOF method. To address the above problems with the TOF method, we have developed a new approach to estimate the shear elasticity and shear viscosity parameters that considers the effects of the viscoelasticity of the medium, the diffraction of the acoustic radiation force push, and the diffraction of the shear wave. After the mathematical theory for shear wave propagation in viscoelastic media is described, a forward model is provided for the simulation of shear waves induced by an acoustic radiation force push. This model calculates the time-domain Green s function for shear waves propagating in a viscoelastic medium [148] with a GPU-based parallel algorithm [165]. Then, a new approach for estimating the shear elasticity and shear viscosity values of the medium is described, where the new approach minimizes the difference 56

75 between simulated and measured shear wave particle velocities. From measured shear waves in viscoelastic shear wave phantoms, parameters of the shear elasticity and viscosity are estimated and quantitative images are generated. Shear elasticity values estimated with the conventional time-of-flight method are then compared with the values estimated with the proposed model-based estimation approach. In addition, the results provided by the modelbased estimation approach with different push beams in viscoelastic shear wave phantoms are shown and compared. The results indicate that this new model-based approach is an effective method for extracting the values for the shear elasticity and viscosity from viscoelastic shear wave phantoms. 4.2 Theory Acoustic radiation force In shear wave elasticity imaging (SWEI), shear waves in soft tissue are induced by an applied acoustic radiation force, which is produced by a focused ultrasound beam. The acoustic radiation force is modeled as [41] f(r, t) = 2αI(r, t) c p, (4.1) where f(r, t) is the body force, r denotes a spatial point in Cartesian coordinates, t denotes time, α is the attenuation coefficient of the compressional wave, c p is the compressional wave speed, and I(r, t) is the acoustic intensity. The intensity of the ultrasound beam is proportional to the square of the acoustic pressure, which is defined as I = p 2 /ρc p. In SWEI, the acoustic radiation force is applied to the tissue for a short period of time to produce the shear wave. To account for the temporal component of the acoustic radiation force, f(r, t) is written as f(r, t) = f(r)w(t), where f(r) is the spatial distribution of the acoustic radiation force, and w(t) is a window function that describes the time dependence of the acoustic radiation force. The window function is defined here as w(t) = rect[(t T/2)/T ], where rect( ) is the rectangle function, and T is the duration of the push. 57

76 4.2.2 Shear waves in viscoelastic media Shear wave propagation in a viscoelastic medium is governed by Navier s equation combined with Voigt s viscosity model [148, 152] ( λ + 2µ + (η p + 2η s ) ) ( ) ( u) + µ + η s ( u) + f = ρ u, (4.2) t t t where u is the shear wave particle displacement, f is the external body force, and t is the time. The bulk modulus and shear modulus are indicated by λ and µ, respectively, the bulk viscosity and shear viscosity are indicated by η p and η s, respectively, and ρ is the density. This equation models the elastodynamic response to an applied body force in a viscoelastic medium. An approximate solution to this equation is given by [148, 165] g z (r, t) = g P z (r, t) + g S z (r, t), (4.3) gz P (r, t) = 1 1 z 2 4πρc p 2πvp t gz S (r, t) = 1 1 4πρc s 2πvs t I p (r, t) = r 3 e (1 z2 r 2 tc νp t 2 p 2πcp e 2νp e (t r/cp) 2 c 2 p 2vpt + 1 4πρ ( 3z 2 ) 1 r 2 1 r 3 I p(r, t), (4.4) ( 3z 2 ) 1 4πρ r 2 1 r 3 I s(r, t), (4.5) ( ) ( ) + t c p t erf erf c p t r cp, 2 2νp t 2νp t ) 1 (t r/cs) 2 c 2 s r e 2vst + 1 ( t r cp ) 2c 2 p 2νpt I s (r, t) = νs t 2πcs tc2 s e 2νs e ( t cs r ) 2c 2 s 2νst + t 2 (4.6) ( ) ( ) cs t erf erf c s t r cs, 2νs t 2νs t (4.7) where gz P (r, t) and gz S (r, t) represent the compressional and shear terms for the Green s function, respectively, and r denotes a point in Cartesian coordinates. The term gz P (r, t) models the propagation of a compressional wave with propagation speed c p = (λ + 2µ)/ρ, and the shear term gz S (r, t) models the propagation of a shear wave with propagation speed 58

77 c s = µ/ρ. In Eqs , ν p = (η p + 2η s )/ρ, ν s = η s /ρ, and erf( ) denotes the error function. 4.3 Materials and Methods Shear wave measurements Shear waves were measured with a Verasonics ultrasound system (Verasonics Inc., Redmond, WA) and a C5-2v curvilinear transducer array. To induce shear waves at different focal depths, three different push beams are generated by the transducer array by changing the number of active elements and the phase of each transducer element. The number of active elements and the focal depth for each push beam are given in Table 4.1. Each push beam is applied for 915 µs with a center frequency of 2 MHz. Active elements Push beam focal depth Push beam mm Push beam mm Push beam mm Table 4.1: Parameters for the three push beams. After the push beam is applied, the Verasonics system is immediately switched to plane wave imaging mode to measure the shear waves induced by the push beam. In plane wave imaging mode, multiple frames of in-phase/quadrature (IQ) data within a two-dimensional (2D) field-of-view (FOV) are obtained for ms with a pulse repetition frequency (PRF) of khz. The 2D FOV ranges from mm to mm in the lateral direction and from 0 to mm in the axial direction with a spatial sampling of mm in each direction. The ultrasound center frequency for plane wave imaging is MHz. The shear wave particle velocities are then calculated from the IQ data using one-dimensional (1D) autocorrelations [11]. 59

78 z (mm) z (mm) z (mm) (a) x (mm) (b) x (mm) (c) x (mm) Figure 4.1: 2D cross-sections of the three push beams simulated in FOCUS. Figures (4.1a), (4.1b), and (4.1c) show push beams with focal depths of 33 mm, 45 mm, and 70 mm, respectively Shear wave simulations To simulate the shear waves induced by the push beams, 3D ultrasound fields are calculated, and then the acoustic radiation force is obtained from Eq The 3D pressure fields are calculated in FOCUS, the Fast Object-Oriented C++ Ultrasound Simulator ( fultras-web/). For these calculations, the fast near field method [130,131] evaluates the pressure in a plane in front of the transducer array, and then the angular spectrum approach [132, 133] computes the 3D pressure field. Fig. 4.1 shows three push beams with push focal depths of 33 mm, 45 mm, and 70 mm (Table 4.1) that are simulated for a C5-2v curvilinear transducer array. In each simulation, the sound speed for the compressional wave is 1540 m/s, and the acoustic attenuation coefficient for the compressional wave is 0.5 db/cm/mhz. The width of each ultrasound transducer element is mm, the element height is 13 mm, the kerf between adjacent elements is mm, and the radius of curvature for each array element in the elevation direction is 60 mm. The radius of curvature of the transducer array is mm. 60

79 The shear wave particle displacements are then calculated by spatially and temporarily convolving the Green s function in Eqs with the acoustic radiation force. This spatiotemporal convolution is given by N t Nx Ny Nz [ ( u z (r, t) = gz r r, t τ )] f(r ) x y z w(τ) τ, (4.8) i=1 m=1 n=1 k=1 where u z (r, t) represents the shear wave particle displacement for the spatial coordinate r at time t. In Eq. 4.8, the triple-summation within the square brackets represents the spatial convolution, where N x, N y, and N z represents the number of spatial points in each direction, and x, y, and z are the spatial increments in each direction. The summation outside of the square brackets represents the temporal convolution, where N t is the number of time samples, and τ is the time sampling interval. After the shear wave particle displacement is computed, the shear wave particle velocity v z (r, t) is obtained by numerically calculating the temporal derivative of the shear wave particle displacement u z (r, t) as v z (r, t) = [u z (r, t + t) u z (r, t)] / t, where t is the time sampling interval. The simulated shear wave particle velocity values are then utilized within the new model-based approach to estimate the shear elasticity and shear viscosity of the viscoelastic shear wave phantoms. To accelerate the simulations that evaluate Eq. 4.8, the calculations are performed on a Graphics Processing Unit (GPU) [165]. For these GPU calculations, the pre-calculated acoustic radiation force, which is stored in a data file, is first loaded into the CPU (host) memory and then copied to the GPU (device) memory. The kernel function is then called to separate the evaluation of the spatial convolution in Eq. 4.8 (the triple-summation within the square brackets) into different threads that are distributed to different processors on the GPU for parallel evaluations. For each thread, the spatial convolution is evaluated at a single point in space and at a single moment in time. Each thread obtains the push beam from the global memory, and the calculated spatial convolution result is also stored in global memory. After all of the threads are finished, the results are copied back into the host memory. The 61

80 temporal convolution in Eq. 4.8 (the summation outside of the square brackets) is then evaluated on the CPU to obtain the shear wave particle displacements, and the shear wave particle velocity is calculated by evaluating the temporal derivative of Eq The simulated shear wave particle velocities are stored in a data file for subsequent evaluations. These calculations are implemented through the Compute Unified Device Architecture (CUDA) programming language on an nvidia (Santa Clara, CA) Kepler K80 GPU Phantom experiments Shear waves are measured in viscoelastic shear wave phantoms provided by Computerized Imaging Reference Systems (CIRS), Inc (Norfolk, Virginia, US) with a Verasonics ultrasound system and a C5-2v transducer. Shear waves are measured in three phantoms (E2348-1, E2348-2, and E2348-3) with different mechanical properties. For each measurement, the shear wave particle velocities are obtained for all of the observation points in a 2D plane. An example of the measured shear waves in phantom E using a push beam with a focal depth of 33 mm is shown in Fig Shear wave measurements are performed ten times on each phantom with all three different push beams. 0 time = 2ms 0 time = 6ms 0 time = 10ms z (mm) z (mm) z (mm) (a) x (mm) (b) x (mm) (c) x (mm) Figure 4.2: The measured shear wave particle velocity induced by the push beam with a focal depth of 33 mm in phantom E Figures (4.2a), (4.2b), and (4.2c) show the measured shear wave particle velocity at 2 ms, 6 ms, and 10 ms, respectively. 62

81 4.3.4 Time-of-flight method for shear elasticity estimation In the TOF method, shear elasticity is estimated using cross-correlations [178]. To estimate the shear elasticity value at point (m, n) where m and n are the indices for the x and y directions, respectively, the cross-correlations of the measured shear wave particle velocities are calculated for multiple pairs of points which are located at either side of point (m, n) at the same depth. Those pairs of points are: {(m 5, n), (m + 1, n)}, {(m 4, n), (m + 2, n)}, {(m 3, n), (m + 3, n)}, {(m 2, n), (m + 4, n)}, and {(m 1, n), (m + 5, n)}. The time delay for the shear wave particle velocities measured at each pair of points are obtained from the cross-correlations, namely t 1, t 2, t 3, t 4, and t 5. The shear wave speed is then calculated for each point pair as V i = 6 x/ t i, where i = 1, 2, 3, 4, 5 and x is the spatial sampling interval in the x direction. The correlation coefficient CC i for each point pair is used as the weight to combine all of the calculated shear wave speeds to obtain the estimated shear wave speed at point (m, n) as V (m, n) = 5 i=1 V i CC i. The shear elasticity µ at point (m, n) is then obtained from µ = ρv 2, where ρ is the density Model-based approach for shear elasticity and shear viscosity estimation Shear elasticity and shear viscosity values are estimated from the measured shear wave particle velocity through the following approach. At each point in space, the reference shear wave particle velocity is represented by v ref (r, t), and the simulated shear wave particle velocity for a specific combination of µ and η s values is represented by v µ,η s (r, t). The estimated shear elasticity and shear viscosity values are obtained by minimizing the difference between the reference shear wave particle velocity and the simulated shear wave particle velocity, arg min µ, ηs v ref (r, t) v µ,η s (r, t), (4.9) v ref (r, t) where. represents the L 2 norm with respect to the time variable. Eq. 4.9 is then minimized with a procedure that determines the values of µ and η s at each point in space. This is 63

82 achieved by first computing the v µ,η s (r, t) waveforms with different shear elasticity µ and shear viscosity η s values calculated on the GPU as described in section The value of the shear elasticity µ ranges from 2 kpa to 12 kpa with an increment of 0.2 kpa, and the value of the shear viscosity η s ranges from 0 to 4 Pa s with an increment of 0.1 Pa s. Eq. 4.9 is then evaluated through an exhaustive search that minimizes the difference between the simulated and the measured shear waves. Finally, this procedure is applied to the timedomain waveforms collected at each point in space, and 2D images of µ and η s are generated with the results. 4.4 Results Time of flight parameter estimates evaluated with simulated shear wave data With the results obtained from 3D shear wave simulations, shear elasticity values are initially estimated using the TOF method. Shear wave particle velocities induced by the push beam shown in Fig. 4.1a), which is focused at a depth of 33 mm, are calculated with shear wave particle velocities obtained from elastic and viscoelastic simulation models. For the elastic simulation model, the shear elasticity is 4 kpa and the shear viscosity is 0 Pa s. For the viscoelastic simulation model, the shear elasticity is 4 kpa and the shear viscosity is 2 Pa s. Both simulation models are homogeneous, linear, and incompressible. The simulated shear wave particle velocities are evaluated in the xz plane with y = 0, which ranges from -20 mm to 20 mm in the x direction and from 0 to 80 mm in the z direction with a spatial increment of 0.2 mm in each direction. After the shear waves are calculated, white noise is added to the shear wave particle velocity at each point in space, where the average noise intensity is 20% of the peak shear wave particle velocity. After the shear waves are calculated, the shear wave elasticity is then estimated using the TOF method described in section Two shear elasticity images obtained with the TOF method are shown in Fig Fig. 4.3a) shows the result obtained from the elastic simulation model with µ = 4 kpa 64

83 Elastic simulation model 0 10kPa Viscoelastic simulation model 0 10kPa 20 8kPa 20 8kPa z (mm) 40 6kPa 4kPa z (mm) 40 6kPa 4kPa 60 2kPa 60 2kPa (a) x (mm) 0kPa (b) x (mm) 0kPa Figure 4.3: Shear elasticity images generated by the TOF method using noisy 3D shear wave particle velocity data obtained from elastic and viscoelastic simulation models. Fig. 4.3a) shows the estimated shear elasticity for the homogeneous elastic simulation model with µ = 4 kpa and η s = 0 Pa s, and Fig. 4.3b) shows the estimated shear elasticity for the homogeneous viscoelastic simulation model with µ = 4 kpa and η s = 2 Pa s. and η s = 0. Near the center of the image, the estimated shear elasticity reaches a maximum value of MPa, and in the region between x = 4 mm and x = 4 mm and between z = 10 mm and z = 65 mm, which is indicated by the red box in Fig. 4.3a), the average elasticity value is kpa. Both of these values differ from the actual value by a significant amount. These results suggest that, in order to obtain realistic values for the shear elasticity µ, the TOF approach should only be applied some distance from the focal peak. Also, within the region indicated by the red box in Fig. 4.3a), the TOF method consistently fails to yield accurate values, so Fig. 4.3a) is thresholded at 10 kpa to emphasize the range of values closer to 4 kpa. The average value of µ evaluated over the entire image is kpa, which is also quite large relative to the actual value for µ. However, on either side of the push beam a short distance from the focal peak, the shear elasticity estimates are more reasonable. In the white boxes in Fig. 4.3 that extend from 6 mm to 18 mm and 6 mm to 18mm in 65

84 the x direction and from 25 mm to 45 mm in the z direction, the average shear elasticity value estimated with the TOF approach is 4.07 kpa, which corresponds to an average error of 1.75% in this region. The standard deviation of the estimated shear elasticity within the white boxes is kpa, so the variation within this region is also relatively small. Fig. 4.3b) shows the shear elasticity estimates obtained from the viscoelastic simulation model using the TOF method, where large errors are observed across the entire image. Near the center of the image, the value of the estimated shear elasticity reaches a maximum value of MPa, and in the region between x = 4 mm and x = 4 mm and between z = 10 mm and z = 60 mm, which is indicated by the red box in Fig. 4.3b), the average elasticity value obtained with the TOF method is MPa. The average value of µ obtained with the TOF method over the entire image is kpa, which differs from the actual value µ = 4 kpa by a significant amount. The values in Fig. 4.3b) are also thresholded at 10 kpa to facilitate comparisons with Fig. 4.3a). In the regions indicated by the white boxes shown in Fig. 4.3b), the estimated shear elasticity is 6.10 kpa, which corresponds to an error of 52.50% in these regions. The standard deviation of the estimated shear elasticity within the white boxes is 0.70 kpa, which is larger than the standard deviation in the same region when the TOF method is applied to the simulation results obtained with the elastic model. These results reveal that, although the TOF method provides accurate shear elasticity estimates in some regions when applied to the elastic simulation model with µ = 4 kpa and η s = 0 Pa s, the TOF method yields significant errors everywhere when applied to the viscoelastic simulation model with µ = 4 kpa and η s = 2 Pa s New model-based shear wave parameter estimation approach applied to simulated shear wave data Shear elasticity and shear viscosity values are also obtained using the new model-based estimation approach applied to the same simulated shear wave particle velocities with white noise added. These are evaluated for the elastic and viscoelastic models described in the 66

85 previous section. To estimate the shear elasticity and shear viscosity values at each point in space, the objective function in Eq. 4.9 is minimized as described in section In calculations with the objective function in Eq. 4.9, v ref (r, t) is the simulated shear wave particle velocity with noise added, and v µ,η s (r, t) is the noise-free shear wave particle velocity calculated on the GPU for a particular combination of µ and η s values. The shear elasticity µ ranges from 2 kpa to 12 kpa with an increment of 0.2 kpa, and the shear viscosity η s ranges from 0 to 4 Pa s with an increment of 0.1 Pa s in the GPU simulations. Fig. 4.4 shows the objective functions for the shear wave particle velocities calculated at (x, z) = (10 mm, 30 mm) with the elastic and viscoelastic simulation models. In Fig. 4.4, the o marker indicates the value obtained when Eq. 4.9 is optimized, and the x marker indicates the estimated value obtained with the TOF method. When applied to the simulated shear wave data produced by the elastic model, the new model-based approach yields a value of µ = 4.00 kpa for the shear elasticity, which is exactly the same as the elasticity in the simulation model, while the estimated shear elasticity provided by the TOF method is 4.11 kpa, which differs slightly from the actual value (by 2.71%). The shear viscosity value estimated with the new model-based approach is 0 Pa s, which is also correct since the simulation model is purely elastic. Fig. 4.4(a) indicates that the objective function for shear waves simulated with an elastic model has a single global minimum and that the new model-based approach successfully finds this global minimum, even in the presence of additive noise. The TOF method yields a value that is close to the global minimum when applied to the simulated shear wave data produced by the elastic model. When applied to the noisy simulated shear wave data produced by the viscoelastic model, the shear elasticity value obtained with the new model-based approach is 4.00 kpa, which is exactly the same as the elasticity from the simulation model, whereas the estimated shear elasticity produced by the TOF method is 5.69 kpa, which differ from the actual value by 42.25%. The shear viscosity value obtained from the noisy simulated shear wave data with the new model-based approach is 1.90 Pa s, which deviates from the actual value by only 67

86 Figure 4.4: Objective functions for the shear wave particle velocities calculated at (x, z) = (10 mm, 30 mm) with the elastic and viscoelastic simulation models. Fig. 4.4a shows the objective function evaluated for the elastic simulation model, and Fig. 4.4b shows the objective function evaluated for the viscoelastic simulation model. The o markers indicate the estimates obtained with the new model-based approach, where µ = 4.00 kpa and η s = 0 in Fig. 4.4a and µ = 4.00 kpa and η s = 1.90 Pa s in Fig. 4.4b. The x markers indicate the estimated values obtained with the TOF method, where µ = 4.11 kpa and η s = 0 in Fig. 4.4a and µ = 5.69 kpa and η s = 0 in Fig. 4.4b. 5%. Figs. 4.4(a) and 4.4(b) indicate that the objective function for the viscoelastic shear wave simulation model also has a single global minimum and that the new model-based approach successfully identifies the location of the global minimum. However, when applied to simulated shear wave data obtained from the viscoelastic model, the parameter estimates produced by the TOF method are located far from the global minimum. Fig. 4.5 shows the shear elasticity and shear viscosity images produced by the new estimation approach applied to simulated noisy shear wave particle velocities calculated with elastic and viscoelastic models. Figs. 4.5(a) and 4.5(b) show the shear elasticity image and shear viscosity image, respectively, for the noisy elastic simulation model with µ = 4 kpa and η s = 0 Pa s. In the center of the image, between x = 4 mm and x = 4 mm and between z = 15 mm and z = 70 mm, the average elasticity value is 3.58 kpa with a standard deviation of 1.16 kpa. The average shear viscosity value in this region is 0.69 Pa s, and the standard deviation is 1.10 Pa s. These results indicate that the new model-based approach 68

87 10 Shear elasticity 8kPa 10 Shear viscosity 4Pa s Elastic simulation model z(mm) Viscoelatic simulation model z(mm) (a) x(mm) (c) x(mm) 6kPa 4kPa 2kPa 0kPa 8kPa 6kPa 4kPa 2kPa 0kPa z(mm) (b) x(mm) z(mm) (d) x(mm) 3Pa s 2Pa s 1Pa s 0Pa s 4Pa s 3Pa s 2Pa s 1Pa s 0Pa s Figure 4.5: Shear elasticity and shear viscosity images generated with parameters obtained from the new model-based approach applied to simulated noisy shear wave particle velocities calculated with the elastic and viscoelastic simulation models. Figs. 4.5(a) and 4.5(b) show the shear elasticity and shear viscosity images, respectively, for the elastic simulation model with µ = 4 kpa and η s = 0 Pa s, respectively. Figs. 4.5(c) and 4.5(d) show the shear elasticity and shear viscosity images, respectively, for a viscoelastic simulation model with µ = 4 kpa and η s = 2 Pa s. also encounters difficulties close to the focal peak. However, on either side of the push beam a short distance from the focal peak, the shear elasticity estimates are much more accurate. In the regions that extend from 6 mm to 18 mm and 6 mm to 18mm in the x direction 69

88 and from 25 mm to 45 mm in the z direction, which are indicated by the white boxes in Figs. 4.5(a) and 4.5(b), the average shear elasticity value is 3.99 kpa, which corresponds to an error of 0.25% in this region. The standard deviation of the estimated shear elasticity within the white boxes is 0.04 kpa, which is significantly smaller than the standard deviation evaluated in the middle of the image. The average shear viscosity value is Pa s with a standard deviation of Pa s, which is very close to the actual shear viscosity value in the simulation model. Figs. 4.5(c) and 4.5(d) show the shear elasticity image and shear viscosity image, respectively, for the noisy viscoelastic simulation model with µ = 4 kpa and η s = 2 Pa s. In the center of the image, between x = 4 mm and x = 4 mm and between z = 15 mm and z = 65 mm, the average shear elasticity value is 3.42 kpa with a standard deviation of 1.39 kpa and the average shear viscosity value is 1.78 kpa with a standard deviation of 1.61 kpa. However, on either side of the push beam a short distance from the focal peak, the shear elasticity estimates with the model-based approach are much more accurate. In the regions that extend from 6 mm to 18 mm and 6 mm to 18mm in the x direction and from 25 mm to 45 mm in the z direction, which are indicated by the white boxes in Figs. 4.5(c) and 4.5(d), the average shear elasticity value is 3.99 kpa, which corresponds to an error of 0.25% in this region. The standard deviation of the estimated shear elasticity within the white boxes is 0.05 kpa, which is significantly smaller than the standard deviation evaluated in the middle of the image. The average shear viscosity value is 1.76 Pa s, which corresponds to an error of 12.2% in this region. The standard deviation of the estimated shear viscosity within the white boxes is 0.14 Pa s, which is also significantly smaller than the standard deviation evaluated in the middle of the image. 70

89 Figure 4.6: The objective function in Eq. 4.9 evaluated for different values of the shear elasticity and shear viscosity for shear wave particle velocities measured at (x, z) = (10 mm, 30 mm) in phantom E The o marker indicates the value obtained with the new model-based estimation approach (µ = 3.60 kpa and η s = 0.80 Pa s), and the x marker indicates the value produced by the TOF method (µ = 6.70 kpa and η s = 0) TOF and model-based estimates of shear wave parameters for viscoelastic shear wave phantoms To compare the new model-based estimation approach with the TOF method using shear wave measured data from viscoelastic shear wave phantoms, the two methods are applied to the shear wave particle velocities measured at one point in space generated in phantom E by a push beam with a focal depth of 33 mm. The TOF method is applied to the measured shear wave particle velocities at (x, z) = (10 mm, 30 mm), and the TOF estimate of the shear elasticity at this location is 6.70 kpa. The model-based estimation approach is applied to the measured shear wave particle velocities in the same location, and the shear elasticity and shear viscosity parameters are estimated by minimizing the difference between the measured shear wave particle velocity and the simulated shear wave particle velocity in Eq The estimated shear elasticity value with the model-based approach is

90 Normalized shaer velocity Measured Simulated with TOF parameters Normalized shaer velocity Measured Simulated with estimated parameters -0.5 (a) time (ms) -0.5 (b) time (ms) Figure 4.7: Measured and simulated shear wave particle velocities using shear wave parameter values estimated with the time-of-flight (TOF) method and the new model-based estimation approach. In Figs. 4.7a and 4.7b, the blue dashed line indicates the measured shear wave particle velocity induced by the push beam with a focal peak at 33 mm in phantom E2348-1, the black dotted line in Fig. 4.7(a) is the the simulated shear wave particle velocity with the shear wave parameters estimated with the TOF method (µ = 6.70 kpa and η s = 0), and the red solid line in Fig. 4.7(b) is the the simulated shear wave particle velocity with the shear wave parameters (µ = 3.60 kpa and η s = 0.80 Pa s) obtained from the new model-based estimation approach. kpa, and the estimated shear viscosity value with the model-based approach is 0.80 Pa s. This result indicates that the shear elasticity value obtained with the TOF method differs by 86.11% from the value estimated with the model-based estimation approach. Fig. 4.6 illustrates the objective function in Eq. 4.9 for different combinations of shear elasticity and shear viscosity values. In Fig. 4.6, the o marker indicates the value obtained with the model-based approach (µ = 3.60 kpa and η s = 0.80 Pa s), and the x marker indicates the value computed with the TOF method (µ = 6.70 kpa and η s = 0). The objective function for the shear wave data measured in phantom E has a single global minimum, and the new model-based approach successfully locates this global minimum. As for the noisy simulated viscoelastic shear wave data, the shear wave parameters estimated with the TOF method are far from the global minimum. Fig. 4.7 demonstrates measured and simulated shear wave particle velocities using the 72

91 shear wave parameter values estimated with the TOF method and the new model-based estimation approach. The shear wave particle velocity is induced by a push beam with a focal peak at 33 mm in phantom E and is then measured at (x, z) = (10 mm, 30 mm), which is the same (x, z) coordinate that is used in Fig Fig. 4.7(a) shows the measured shear wave particle velocity and the simulated shear wave particle velocity using the parameters estimated with the TOF method (µ = 6.70 kpa and η s = 0). This result shows that, although the peaks arrive at approximately the same time, the two waveforms differ by a significant amount. Overall, the percent difference between these two waveforms is 111.5%. Fig. 4.7(b) shows the measured and simulated shear wave particle velocity and the simulated shear wave particle velocity using parameters obtained with the new model-based approach (µ = 3.60 kpa and η s = 0.80Pa s). The two waveforms in Fig. 4.7(b) demonstrate much better alignment, where the percent difference between the two waveforms in Fig. 4.7(b) is 42.1%. This suggests that the new model-based approach provides more effective estimates of the shear elasticity and shear viscosity than the TOF method when evaluated at this location in viscoelastic shear wave phantom E Shear elasticity and shear viscosity imaging in viscoelastic shear wave phantoms Fig. 4.8 shows the shear elasticity images provided by the TOF method and the shear elasticity and shear viscosity images provided by the new model-based approach for shear waves induced by a push beam focused at 45 mm in three different phantoms (E2348-1, E2348-2, and E ). In the regions that extend from 6 mm to 18 mm and 6 mm to 18mm in the x direction and from 40 mm to 55 mm in the z direction, which are indicated by the white boxes in each of the figures, the new model-based estimation approach provides relatively homogeneous values for µ and η s. Figs. 4.8(a)-(c) show the shear wave parameter images obtained with the TOF method and the new model-based approach for phantom E In Fig. 4.8(a), which shows the shear elasticity image generated by the 73

92 Phantom E z(mm) Phantom E z(mm) Phantom E z(mm) TOF method (kpa) x (mm) (a) (d) x (mm) x (mm) (g) Model-based (kpa) z(mm) z(mm) z(mm) (b) x(mm) (e) x(mm) (h) x(mm) Model-based (Pa.s) 10 4 z(mm) z(mm) z(mm) (c) x(mm) (f) x(mm) (i) x(mm) Figure 4.8: Shear wave parameter images generated by the TOF method and the new modelbased approach for phantoms E2348-1, E2348-2, and E where the shear wave is induced by a push beam focused at 45 mm. Figs. 4.8(a)-(c) show the shear elasticity image provided by the TOF method, the shear elasticity image provided by the new model-based approach, and the shear viscosity image provided by the new model-based approach evaluated for phantom E2348-1, respectively. Figs. 4.8(d)-(f) show the shear elasticity image provided by the TOF method, the shear elasticity image provided by the new model-based approach, and the shear viscosity image provided by the new model-based approach evaluated for phantom E2348-2, respectively. Figs. 4.8(g)-(i) show the shear elasticity image provided by the TOF method, the shear elasticity image provided by the new model-based approach, and the shear viscosity image provided by the new model-based approach evaluated for phantom E2348-3, respectively. 74

93 TOF method, the average shear elasticity value within the white boxes is 5.24 kpa with a standard deviation of 1.27 kpa. In Fig. 4.8(b), which shows the shear elasticity image generated by the new model-based approach, the average shear elasticity value within the white boxes is 3.72 kpa with a standard deviation of 0.23 kpa. These results show that the shear elasticity value estimated with the TOF method is significantly larger than the value estimated with the new model-based approach and that the standard deviation is also larger with the TOF method. In Fig. 4.8(c), which shows the shear viscosity image generated by the new model-based approach, the average shear viscosity value within the white boxes is 1.04 Pa s with a standard deviation of 0.13 Pa s. Figs. 4.8(d)-(f) show the shear wave parameter images for phantom E In Fig. 4.8(d), which shows the shear elasticity image generated by the TOF method, the average shear elasticity value within the white boxes is 9.15 kpa with a standard deviation of 3.06 kpa. In Fig. 4.8(e), which shows the shear elasticity image generated by the new model-based approach, the average shear elasticity value within the white boxes is 6.31 kpa with a standard deviation of 0.38 kpa. These results show that the shear elasticity value estimated with the TOF method is significantly larger than the value estimated with the new model-based approach, and the standard deviation is also larger with the TOF method. In Fig. 4.8(f), which shows the shear viscosity image generated by the new model-based approach, the average shear viscosity value within the white boxes is 1.51 Pa s with a standard deviation of 0.27 Pa s. Figs. 4.8(g)-(i) show the shear wave parameter images for phantom E In Fig. 4.8(g), which shows the shear elasticity image generated by the TOF method, the average shear elasticity value within the white boxes is kpa with a standard deviation of 3.30 kpa. In Fig. 4.8(h), which shows the shear elasticity image generated by the new model-based approach, the average shear elasticity value within the white boxes is 8.14 kpa with a standard deviation of 0.47 kpa. These results show that the shear elasticity value estimated with the TOF method is significantly larger than the value estimated with the 75

94 new model-based approach, and the standard deviation is also larger with the TOF method. In Fig. 4.8(i), which shows the shear viscosity image generated by the new model-based approach, the average shear viscosity value within the white boxes is 1.87 Pa s with a standard deviation of 0.35 Pa s Shear elasticity and shear viscosity imaging with different push beams Fig. 4.9 shows the estimated shear wave parameter images generated by the TOF method and the new model-based approach evaluated in phantom E with three different push beams. Figs. 4.9(a)-(d) show the estimated shear wave parameter images for a push beam focused at 33 mm. In the regions that extend from 6 mm to 18 mm and 6 mm to 18mm in the x direction and from 25 mm to 40 mm in the z direction, which are indicated by the white boxes, the model-based estimation approach provides relatively homogeneous values for µ and η s. For the region outside of the white boxes, greater variation is observed in µ and η s. This suggests that the model-based estimation approach should only be applied on either side of the focal peak some distance away from the push beam. In Fig. 4.9(a), which shows the shear elasticity image generated by the TOF method for a push beam focused at z = 33 mm, the average shear elasticity value within the white boxes is 9.41 kpa with a standard deviation of 3.04 kpa. In Fig. 4.9(b), which shows the shear elasticity image generated by the new model-based approach for a push beam focused at z = 33 mm, the average shear elasticity value within the white boxes is 6.14 kpa with a standard deviation of 0.43 kpa. These results show that shear elasticity value estimated with the TOF method is significantly larger than the value estimated with the new model-based approach, and the standard deviation is also larger with the TOF method. In Fig. 4.9(c), which shows the shear viscosity image generated by the new model-based approach for a push beam focused at z = 33 mm, the average shear viscosity value within the white boxes is 1.45 Pa s with a standard deviation of 1.92 Pa s. In Fig. 4.9(d) shows an image that describes the percent difference between the measured shear wave particle velocity and the simulated shear wave 76

95 33 mm push beam z(mm) 45 mm push beam z(mm) 70 mm push beam z(mm) TOF method (kpa) (a) x (mm) (e) x (mm) x (mm) (i) Model-based (kpa) z(mm) z(mm) z(mm) (b) x(mm) (f) x(mm) (j) x(mm) Model-based (Pa.s) 10 4 z(mm) z(mm) z(mm) (c) x(mm) (g) x(mm) (k) x(mm) z (mm) z (mm) z (mm) % difference (d) x (mm) (h) x (mm) (l) x (mm) 200% 150% 100% 50% 0 200% 150% 100% 50% 0 200% 150% 100% 50% 0 Figure 4.9: Shear wave parameter images generated by the TOF method and the new modelbased approach for phantom E Figs. 4.9(a), 4.9(e), and 4.9(i) show the shear elasticity images generated by the TOF method with push beams focused at 33 mm, 45 mm, and 70 mm, respectively. Figs. 4.9(b), 4.9(f), and 4.9(j) show the shear elasticity images generated by the new model-based approach with push beams focused at 33 mm, 45 mm, and 70 mm, respectively. Figs. 4.9(c), 4.9(g), and 4.9(k) show the shear viscosity images generated by the new model-based approach with push beams focused at 33 mm, 45 mm, and 70 mm, respectively. Figs. 4.9(d), 4.9(h), and 4.9(l) show the images of the difference between the measured shear wave particle velocity and the simulated shear wave particle velocity for push beams focused at 33 mm, 45 mm, and 70 mm, respectively. particle velocity with the optimized shear wave parameters obtained from the model-based approach. The average percent difference within the white boxes is 38.96% with a standard deviation of 6.44%. Figs. 4.9(e)-(h) show the shear wave parameter images for a push beam focused at 45 mm. 77

96 In the regions that extend from 6 mm to 18 mm and 6 mm to 18mm in the x direction and from 40 mm to 55 mm in the z direction, which are indicated by the white boxes in all the figures in Fig. 4.9(e)-(h), the model-based approach provides relatively homogeneous values for µ and η s. For the region outside of the white boxes, the variation in µ and η s is much greater. Fig. 4.9(e) shows the shear elasticity images generated by the TOF method for a push beam focused at 45 mm. The average shear elasticity value within the white boxes is 9.15 kpa with a standard deviation of 3.06 kpa. Fig. 4.9(f) shows the shear elasticity image generated by the new model-based approach for a push beam focused at 45 mm. The average shear elasticity value within the white boxes is 6.31 kpa with a standard deviation of 0.38 kpa. These results show that shear elasticity value estimated with the TOF method is significantly larger than the value estimated with the new model-based approach, and the standard deviation is also larger with the TOF method. Fig. 4.9(g) shows the shear viscosity image generated by the new model-based approach for a push beam focused at 45 mm. The average shear viscosity value within the white boxes is 1.51 Pa s with a standard deviation of 0.27 Pa s. Fig. 4.9(h) contains and image that describes the percent difference between the measured shear wave particle velocity and the simulated shear wave particle velocity with shear wave parameters obtained from the model-based approach. The average percent difference within the white boxes is 36.90% with a standard deviation of 6.06%. Figs. 4.9(i)-(l) show the shear wave parameter images for a push beam focused at 70 mm. In the regions that extend from 6 mm to 18 mm and 6 mm to 18mm in the x direction and from 60 mm to 78 mm in the z direction, which are indicated by the white boxes in all the figures in Fig. 4.9(i)-(l), the new model-based estimation approach provides relatively homogeneous values for µ and η s. For the region outside of the white boxes, the variation in µ and η s is much greater. Fig. 4.9(i) shows the shear elasticity images generated by the TOF method for a push beam focused at 70 mm. The average shear elasticity value within the white boxes is 9.50 kpa with a standard deviation of 2.81 kpa. Fig. 4.9(j) shows the shear elasticity image generated by the new model-based approach for a push beam focused at 70 78

97 mm. The average shear elasticity value within the white boxes is 6.30 kpa with a standard deviation of 0.38 kpa. These results show that shear elasticity value estimated with the TOF method is significantly larger than the value estimated with the new model-based approach, and the standard deviation is also larger with the TOF method. Fig. 4.9(k) shows the shear viscosity image generated by the new model-based approach for a push beam focused at 70 mm. The average shear viscosity value within the white boxes is 1.49 Pa s with a standard deviation of 0.25 Pa s. Fig. 4.9(l) shows the image for the percent difference between the measured shear wave particle velocity and the simulated shear wave particle velocity with the shear wave parameters obtained from the model-based approach. The average percent difference within the white boxes is 37.95% with a standard deviation of 6.13%. 4.5 Discussion Comparisons between the TOF method and the new model-based approach: simulated 3D shear wave data Figs. 4.3, 4.4, and 4.5 show the shear wave parameters estimated with the TOF method and the new model-based approach. For all of the estimated shear elasticity and shear viscosity images in Figs. 4.3 and 4.5, the values at the center of the image show significant bias relative to the actual values in the simulation. This is expected in the center of the image, where the shear wave is still very close to the peak of the push beam. However, at some distance from the peak of the push beam ( x from 6mm to 18mm and z from 25 mm to 45 mm), more reasonable estimates are obtained. For shear waves generated by the elastic simulation model, the average shear elasticity value inside of the white boxes provided by the TOF method shown in Fig. 4.3(a) is 4.07 kpa with a standard deviation of kpa. This corresponds to an average error of only 1.75%. The standard deviation is also small. The average shear elasticity value estimated with the new model-based approach in Fig. 4.5(a) is 3.99 kpa with a standard deviation of 0.04 kpa. This corresponds to an average error of only 0.25% with a small standard deviation. Thus, for the shear waves generated by the elastic simulation 79

98 model, both methods provide accurate estimates for the shear elasticity. However, for the shear waves produced by the viscoelastic simulation model, the average shear elasticity value estimated with the TOF method in Fig. 4.3(b) is 6.10 kpa with a standard deviation of 0.70 kpa. This corresponds to an average error of 52.50% with a higher standard deviation. The average shear elasticity value estimated with the new model-based approach in Fig. 4.5(c) is 3.99 kpa with a standard deviation of 0.05 kpa. This corresponds to an average error of only 0.25% with a small standard deviation. Thus, when applied to shear waves generated by the viscoelastic simulation model, the new model-based approach provides an accurate shear elasticity estimate, but the estimated provided by the TOF method contains significant bias. In addition, the average shear viscosity value estimated with the new model-based approach in Fig. 4.5(c) is 1.76 Pa s, which corresponds to an error of 12.22%. Although the shear viscosity estimate provided by the new model-based approach is not as accurate as the shear elasticity estimate, the error and the standard deviation (0.14 Pa s) are still relatively small. Fig. 4.4 contains another example that compares the performance of the TOF method and the model-based approach applied to elastic and viscoelastic simulation results. Fig. 4.4(a) shows that, for shear waves simulated with the elastic model, both the TOF method and the new model-based approach provide accurate shear elasticity estimates that are close to the global minimum of the objective function. However, Fig. 4.4(b) shows that, when applied to shear waves generated by the viscoelastic simulation model, the new model-based estimation approach provides accurate shear elasticity and shear viscosity values, but the shear elasticity estimate provided by the TOF method contains significant bias Comparisons between the TOF method and the new model-based approach: viscoelastic shear wave phantoms Figs. 4.6 and 4.7 contain shear wave parameter estimation results for the TOF method and the new model-based approach applied to a viscoelastic shear wave phantom at one point in space. Fig. 4.6 shows that the new model-based approach yields effective results, but the 80

99 shear elasticity estimate provided by the TOF method contains significant bias. Fig. 4.7 compares the measured shear wave particle velocities to the simulated shear wave particle velocities obtained from the shear elasticity and shear viscosity values estimated with the TOF method and the new model-based approach. Fig. 4.7(a) reveals that the simulated shear wave particle velocity using the µ and η s values estimated with the TOF method demonstrates poor agreement with the measured shear wave particle velocity, as evidenced by the large percent difference between the two waveforms (111.5%). Fig. 4.7(b) shows that the simulated shear wave particle velocity using the µ and η s values estimated with the new model-based approach demonstrates much better alignment with the measured shear wave particle velocity, which is supported by the significantly smaller value of the percent difference between the two waveforms (42.1%). Also, the above results indicate that the new model-based approach consistently outperforms the TOF method, since the µ and η s values estimated with the new model-based approach achieve much better alignment with the measured shear wave particle velocity Shear wave parameter estimation in different phantoms Fig. 4.8 contains images of shear wave parameters generated by the TOF method and the new model-based approach in a viscoelastic shear wave phantom. For phantom E2348-1, the average µ value estimated with the TOF method is 5.24 kpa with a standard deviation of 1.27 kpa, and the average µ value estimated with the new model-based approach is 3.72 kpa with a standard deviation of 0.23 kpa. This indicates that the estimated shear elasticity provided by the TOF method differs by 40.9% relative to the estimated shear elasticity provided by the new model-based approach. For phantom E2348-2, the average µ value estimated with the TOF method is 9.15 kpa with a standard deviation of 3.06 kpa, and the average µ value estimated with the new model-based approach is 6.31 kpa with a standard deviation of 0.38 kpa. This reveals that the estimated shear elasticity provided by the TOF method differs by 45.0% relative to the estimated shear elasticity provided by the new model-based approach. 81

100 Shear Elasticity (kpa) Model-based approach - 33 mm push beam Model-based approach - 45 mm push beam Model-based approach - 70 mm push beam TOF method - 33 mm push beam TOF method - 45 mm push beam TOF method - 70 mm push beam 5 0 E E E Figure 4.10: Comparison of the shear elasticity values estimated in three CIRS viscoelastic shear wave phantoms with the new model-based approach and the time-of-flight method. The three groups of bars represent the results for three different phantoms. For each group of bars, the first three bars indicate the result obtained with the new model-based approach, and the last three bars represent the result given by the time-of-flight method. The error bars represent the standard deviation of the estimated elasticity calculated from ten measurements. For phantom E2348-3, the average µ value estimated with the TOF method is kpa with a standard deviation of 3.30 kpa, and the average µ value estimated with the new model-based approach is 8.14 kpa with a standard deviation of 0.47 kpa. This indicates that the estimated shear elasticity provided by the TOF method differs by 47.7% relative to the estimated shear elasticity provided by the new model-based approach. Fig shows a bar plot that compares the estimated shear elasticity for these three phantoms. Fig indicates that, in all of the phantoms, the shear elasticity estimate provided by the TOF method is significantly higher than the estimate of the shear elasticity provided by the new model-based approach and that the standard deviation is also much higher with the TOF method. Fig. 4.8 shows several images of the shear elasticity and the shear viscosity in three phantoms. For phantom E2348-1, the average η s value estimated with the new model-based 82

101 Shear Viscosity (Pa.s) mm push beam 45 mm push beam 70 mm push beam E E E Figure 4.11: Estimated viscosity values obtained with the model-based approach in three CIRS viscoelastic shear wave phantoms. Three different groups of bars describe the results for three different phantoms. The three bars in each group indicate the results generated by pushes at different depths. The error bars indicate the standard deviation of the estimated elasticity from ten measurements. approach is 1.04 Pa s with a standard deviation of 0.13 Pa s. For phantom E2348-2, the average η s value estimated with the new model-based approach is 1.51 Pa s with a standard deviation of 0.27 Pa s. For phantom E2348-3, the average η s value estimated with the new model-based approach is 1.87 Pa s with a standard deviation of 0.35 Pa s. Fig also contains a bar plot that compares the estimated shear viscosity of the three phantoms Shear wave parameter estimation with different push beams Fig. 4.9 contains images of shear wave parameter estimates with the TOF method and the new model-based approach for three different push beams evaluated in phantom E For the white boxes in Figs. 4.9(a), 4.9(e), and 4.9(i), the average shear elasticity values estimated for the TOF method with push beams focused at 33 mm, 45 mm, and 70 mm are 9.41 kpa, 9.15 kpa, and 9.50 kpa, respectively. For the white boxes in Figs. 4.9(b), 83

102 4.9(f), and 4.9(j), the average shear elasticity values estimated with the new model-based approach for push beams focused at 33 mm, 45 mm, and 70 mm are 6.14 kpa, 6.31 kpa, and 6.30 kpa, respectively. The estimated shear elasticity values for all three phantoms and all three push beams are compared in Fig. 4.10, which shows that for all of the phantoms, the variance of the shear elasticity estimate provided by the new model-based approach is smaller than that of the TOF method for all three push beams. In Figs. 4.9(c), 4.9(g), and 4.9(k), the shear viscosity value estimated in the white boxes with the new model-based approach for push beams focused at 33 mm, 45 mm, and 70 mm are 1.45 Pa s, 1.51 Pa s, and 1.49 Pa s, respectively. The estimated shear viscosity values for all three phantoms and all three push beams are compared in Fig. 4.11, which shows that for different phantoms, the variance of the shear viscosity estimate provided by the new model-based approach is larger than the variance of the shear elasticity estimate provided by the TOF method. In Figs. 4.9(d), 4.9(h), and 4.9(l), the average percent difference between the measured and simulated shear wave particle velocities using the µ and η s values estimated with the modelbased approach for push beams focused at 33 mm, 45 mm, and 70 mm are 38.96%, 36.90%, and 37.95%, respectively. This result indicates that for all three push beams, the new modelbased approach yields effective estimates for the shear elasticity and shear viscosity on either side of the focal peak away from the push beam. 4.6 Conclusion A new model-based approach is introduced that provides estimates for the shear elasticity and shear viscosity in viscoelastic shear wave phantoms. The shear wave particle velocity is first calculated with different combinations of the shear elasticity and shear viscosity on a GPU, then a new model-based approach is employed to determine the parameter combination that yields the smallest percent difference between the measured and simulated shear waves. The shear wave parameter estimates provided by the new model-based approach are compared with the TOF method in simulation models, and the results show that the 84

103 new model-based approach provide more accurate result than the TOF method when applied to simulation data obtained from a 3D viscoelastic shear wave model. Shear wave parameters are then estimated in measured shear wave data collected from three different phantoms for three different push beams. The results indicate that the shear elasticity value estimated with the TOF method is significantly higher than the shear elasticity value estimated with the new model-based approach, and the standard deviation is also higher with the TOF approach. The results also indicate that the shear elasticity values estimated with the new model-based approach consistently demonstrate much smaller standard deviations for different push beams relative to those produced by the TOF method. 85

104 CHAPTER 5 SHEAR ELASTICITY AND SHEAR VISCOSITY ESTIMATION IN LIVER 5.1 Introduction Liver fibrosis and cirrhosis are wound healing responses to chronic liver injury from a variety of causes including viral, autoimmune, drug induced, cholestatic, and metabolic diseases [179]. Liver cirrhosis affects many patients. In the United States, an estimated 900,000 patients are suffering from liver cirrhosis, and approximately 30,000 deaths are caused by liver cirrhosis per year [179]. Quantitative bio-markers are critically needed for more accurate liver fibrosis diagnostics and staging. Recently, transient ultrasound elastography techniques have proven to effectively provide a positive correlation between the stiffness of the liver tissue and the stage of the liver fibrosis [44, 61 65, 77, 79]. Noninvasive ultrasound techniques that determine liver stiffness include shear wave elasticity imaging (SWEI) [44, 61 65] and supersonic shear imaging (SSI) [77, 79]. In SWEI, one or more focused or unfocused ultrasound beams are applied for a short period of time to generate tissue deformation. In turn, this deformation produces shear waves. To measure the shear wave particle displacements, multiple frames of radio-frequency (RF) or in-phase/quadrature (IQ) data within a two-dimensional (2D) field-of-view (FOV) are obtained with pulse-echo ultrasound sequences. Then, shear wave particle displacements are calculated from the RF or IQ data using one-dimensional (1D) autocorrelations between consecutive frames [11, 42, 60]. In SSI, several push beams focused at different depths are sequentially applied to the tissue. The focus of the push beams moves downward faster than the propagation speed of the shear waves, hence these are supersonic relative to the shear wave speed. After the sequence of push beams is applied, shear waves are generated that propagate away from the push location with a cone-shaped wavefront. After the shear waves are generated, multiple frames of RF or IQ data within a 2D FOV are measured using an 86

105 ultrafast compound imaging technique [75, 76], which measures ultrasound signals at a very high frame rate (up to 6000 frames per second). Then, shear wave particle displacements are calculated using 1D autocorrelations between consecutive frames [11]. In SWEI and SSI, the shear elasticity value, which represents the stiffness of the tissue, is usually obtained using the time-of-flight (TOF) method. The TOF method calculates the differences in the arrival times of the shear waves at several spatial points located at the same depth but in different lateral positions to determine the shear wave propagation speed. The differences in the arrival times are determined from the time to peak (TTP), which identifies the peak of each measured shear wave [61, 157], or from cross-correlations that calculate the arrival time differences [81, 158, 159]. The shear wave propagation speed c s is then estimated from the slope of the linear fit between the lateral positions of the observation points and the differences in the arrival times of the shear waves measured at these points. The shear elasticity µ is then obtained from µ = ρc 2 s, where ρ is the density. The TOF method provides accurate shear elasticity estimates in purely elastic media [61, ]. However, because the TOF method neglects the viscosity, this method yields significant bias when applied to estimates of µ in soft tissue, which is viscoelastic [79,116]. Furthermore, the TOF method assumes that the shear wave is a plane wave. However, complicated diffraction patterns are observed in shear waves induced by a three-dimensional (3D) acoustic radiation force [165], and the TOF method provides different values of the estimated shear elasticity for different transducers, focal depths, and lateral ranges as a consequence of shear wave propagation [166]. Thus, these effects are also important when estimating the mechanical properties of soft tissue. Other methods have also been developed for shear wave parameter estimation. One of these is attenuation measuring ultrasound shearwave elastography (AMUSE) [167]. With the AMUSE method, a k-space representation of the shear wave is calculated at several successive spatial points located at the same depth but in different lateral positions using the two-dimensional fast Fourier transform (2D-FFT). The k-space representation of the 87

106 shear wave data is plotted as function of the angular frequency (ω) and the wavenumber in the x-direction (k x ). Then, the shear wave propagation speed c s is obtained at each frequency by determining the location of the peak at each frequency in k-space and dividing the value of ω by the value of k x. The attenuation α s is obtained by measuring the full-width-at-halfmaximum (FWHM) in the k x direction for each angular frequency ω. The AMUSE method provides a model-free estimate of shear wave speed and attenuation. However, this method is only effective in the low-frequency range, and diffraction effects are not considered. The above problems are addressed with a new model-based approach that estimates the shear elasticity and shear viscosity of soft tissue. In contract to the frequency-domain analysis with AMUSE, the new approach estimates tissue mechanical parameters in the time-domain. In this new approach, shear wave particle velocities are first simulated using an approximate viscoelastic Green s function model on the GPU [165]. The shear elasticity and shear viscosity values are then estimated using a model-based approach that minimizes the difference between measured and simulated shear wave particle velocities. Shear elasticity and shear viscosity images are generated by evaluating this approach for all of the spatial points in a two-dimensional (2D) field-of-view (FOV). The new approach is applied to measured shear waves data from in vivo human livers, and the estimation results are compared with the TOF method. The results indicate that this new model-based approach successfully generates shear elasticity and shear viscosity images with the measured shear wave data from in vivo human livers. 5.2 Theory Acoustic radiation force In transient ultrasound elastography, shear waves in biological tissue are induced by an applied acoustic radiation force (push beam). The acoustic radiation force is generated by 88

107 a focused ultrasound beam and is modeled as f(r, t) = f(r)w(t) = 2αI(r) c p w(t), (5.1) where α is the acoustic attenuation coefficient for the compressional wave, c p is the sound speed for the compressional wave, r is the spatial coordinate, t is the time, f(r, t) is the spatio-temporal acoustic radiation force at location r and time t, I(r) is the time-average ultrasound intensity at location r, and w(t) is the temporal window that describes the time duration of the applied acoustic radiation force. In clinical ultrasound elastography, the window function is described by w(t) = rect[(t T/2)/T ], where rect( ) is the rectangle function, and T is the push duration Navier s equation Shear wave propagation in elastic, linear, and isotropic biological tissue is modeled by Navier s equation [152] (λ + 2µ) ( u) µ ( u) + f = ρ u, (5.2) t where u represents the particle displacement, f represents the applied acoustic radiation force, t represents the time, ρ represents the density, λ and µ represent the bulk modulus and shear modulus, respectively, and ρ represents the density. Eq. 5.2 models the elastodynamic wave induced by the acoustic radiation force in a purely elastic medium. If the viscoelasticity of the biological tissue is considered, Voigt s model is incorporated into to Navier s equation [148, 154], yielding ( λ + 2µ + (η p + 2η s ) ) ( ) ( u) + µ + η s ( u) + f = ρ u, (5.3) t t t where η p and η s represent the bulk viscosity and shear viscosity, respectively. 89

108 5.2.3 Green s functions for Navier s equation The Green s function for Navier s equation evaluated in an elastic medium calculates the displacement induced by a spatially and temporally impulsive body force. When the acoustic radiation force push is applied in the z-direction, the z-component of the Green s function that describes the resulting displacement in an elastic medium is represented by [152] g z (r, t) = gz P (r, t) + gz S (r, t) + gz C (r, t), where (5.4) gz P (r, t) = 1 z 2 ) (t 4πρc 2 p r 3 δ rcp, (5.5) gz S (r, t) = 1 ) ) (1 4πρc 2 z2 1 (t s r 2 r δ rcs, (5.6) gz C (r, t) = 1 ( 3z 2 ) [ ) )] 1 4πρ r 2 1 r 3 t H (t rcp H (t rcs. (5.7) In Eqs , g P z (r, t), g S z (r, t), and g C z (r, t) represent the compressional, shear, and the coupling components of the Green s function for Navier s equation evaluated in an elastic medium, respectively, where r denotes a spatial point in Cartesian coordinates, the compressional wave speed is c p = (λ + 2µ)/ρ, and the shear wave speed is c s = µ/ρ. In the above expressions, δ( ) denotes the Dirac delta function, H( ) denotes the Heaviside (unit step) function, r = r is the distance between the source point and the observation point, and z is the distance between the source point and the observation point in the z-direction. An approximate Green s function for Navier s equation evaluated in a viscoelastic medium 90

109 is [148] g z (r, t) = g P z (r, t) + g S z (r, t) + g C z (r, t), where (5.8) gz P (r, t) = 1 1 z 2 (t r/cp) 2 c 2 p 4πρc p 2πvp t r 3 e 2vpt, (5.9) gz S (r, t) = 1 ) 1 (1 z2 1 (t r/cs) 2 c 2 s 4πρc s 2πvs t r 2 r e 2vst, (5.10) gz C (r, t) = 1 ( 3z 2 ) 1 4πρ r 2 1 [ Ip r 3 (r, t) + I s (r, t) ], (5.11) tc νp t 2 ( p I p (r, t) = 2πcp e 2νp e t cp r ) 2c 2 p ( ) ( ) 2νpt + t c p t erf erf c p t r cp, 2 2νp t 2νp t I s (r, t) = νs t 2πcs tc2 s e 2νs e ( t cs r ) 2c 2 s 2νst + t 2 (5.12) ( ) ( ) cs t erf erf c s t r cs, 2νs t 2νs t (5.13) where ν p = (η p + 2η s )/ρ, ν s = η s /ρ, and erf( ) denotes the error function. 5.3 Methods Acoustic radiation force simulations In SWEI and SSI, shear waves are generated by focused or unfocused ultrasound beams. The pressure field is calculated first, and then the radiation force is obtained from the computed pressure field, where the radiation force then provides the input to Eq. 5.2 or Eq. 5.3 for calculations of the shear wave particle displacements. The 3D acoustic pressure fields are calculated in FOCUS, the Fast Object-oriented C++ Ultrasound Simulator ( fultras-web). FOCUS calculates the full 3D acoustic pressure field with the fast near field method (FNM) [130, 131] combined with the angular spectrum approach (ASA) [132, 133]. After the 3D acoustic pressure field is computed, the acoustic 91

110 radiation force is calculated from Eq Calculating shear wave particle velocities induced by an acoustic radiation force using Green s functions With the models described in Eqs and Eqs , the shear wave particle displacement induced by an acoustic radiation force is calculated by evaluating the spatio-temporal convolution between the Green s function and the push beam as u z (r, t) = g z (r r, t τ)f(r, τ)dτdv, (5.14) V τ where u z (r, t) is the induced shear wave particle displacement at a spatial point r, g z (r, t) is the Green s function for the elastic or viscoelastic model, and f(r, t) = f(r )w(t) is the acoustic radiation force. This convolution is evaluated numerically by superposing the contributions from all points in the 3D push beam with N t Nx Ny Nz [ ( u z (r, t) = gz r r, t τ )] f(r ) x y z w(τ) τ, (5.15) i=1 m=1 n=1 k=1 where N x, N y, and N z are the number of spatial points in each direction. Although Eq effectively calculates the shear wave particle displacement induced by the push beam in elastic or viscoelastic media, this calculation is very time-consuming when the number of observation points is large, especially when shear waves are evaluated in 3D. In addition, the computation time increases further if multiple combinations of shear elasticity and shear viscosity parameters are evaluated. For instance, when calculating the shear wave particle displacements for N shear elasticity values and M shear viscosity values, Eq is evaluated N M times. In order to reduce the computational effort, an approximate model for shear wave calculation in viscoelastic media is needed Approximate model for shear wave calculation in viscoelastic media In the far field where r x, the Green s function g V z (r, t) for the viscoelastic model (Eqs ) is approximated with a convolution between the Green s function g E z (r, t) 92

111 for the elastic model (Eqs ) and an attenuation term A(r, t) as g V z (r, t) g E z (r, t) t A(r, t), (5.16) where A(r, t) is the approximate causal solution to the 1D Stokes wave equation [180], given by ρc s A(x, t) = H(t) 4 [F (t, x) + F (t, x)], (5.17) 2πγt ( F (x, t) = 1 + x ) exp [ (c st x) 2 ] c s t 2γc 2. (5.18) st In Eq and 5.18, H(t) is the Heaviside function, and γ = η/ρc 2. When calculating the shear wave particle displacement induced by an acoustic radiation force with the approximate viscoelastic model, Eq is represented by u V z (r, t) V τ g E z (r r, t τ)a(r r, t τ)f(r, τ)dτdv. (5.19) In the far-field, A(r r, t τ) is approximated by A(x 0, t τ), so Eq is approximated by u V z (r, t) τ A(x 0, t τ) V g E z (r r, t τ)f(r, τ)dτdv. (5.20) Since the triple integral in Eq is exactly the same as Eq. 5.14, Eq is represented by a temporal convolution between the shear wave particle displacement induced by a push beam with an elastic model and the attenuation term, which is given by u V z (r, t) u E z (r, t) t A(x 0, t). (5.21) The expression for u E z (r, t) described in Eq is calculated with u E z (r, t) = Nx Ny m=1 n=1 k=1 Nz [ hz,t (r r, t) ] f(r ) x y z. (5.22) In this triple sum, N x, N y, and N z describe the number of spatial points in each direction, and x, y, and z are the spatial sampling intervals in the x, y, and z directions for 93

112 the simulated 3D acoustic radiation force. The superscript in u E z (r, t) indicates that this quantity is the shear wave particle displacement calculated with an elastic model. In Eq. 5.22, h z,t (r, t) describes the temporal convolution between the Green s function for the elastic model g E z (r, t) and the window function w(t), which is given by h z,t (r, t) = h P 1 z,t (r, t) = 4πρc 2 p [ ( 1 2 h S 1 z,t (r, t) = [ 1 2 gz E (r, t τ)w(τ)dτ = h P z,t (r, t) + hs z,t (r, t), (5.23) τ z 2 ( ) t r/cp r 3 rect T/2 1 ( 3z 2 ) 1 T 4πρ r 2 1 r 3 ) ( ) t 2 r2 t r/cp T/2 c 2 rect + 1 ( 2tT T 2) H p T 2 ) ( ) (1 4πρc 2 z2 1 t r/cs s r 2 r rect T/2 1 ( 3z 2 ) 1 T 4πρ r 2 1 r 3 ( r 2 ) ( ) c 2 (t T ) 2 t r/cs T/2 rect 1 ( 2tT T 2) H s T 2 ( t r c p T ) ], (5.24) (t rcs )]. (5.25) The shear wave particle displacement that is predicted by this approximate viscoelastic model is calculated with the following steps: 1. Calculate the shear wave particle displacement induced by a push beam with the elastic model using Eq Obtain the attenuation term from Eq Numerically convolve the shear wave particle displacement calculated with the elastic model and the attenuation term, which yields the approximate shear wave particle displacement in a viscoelastic medium. Fig. 5.1 shows an example of a calculation with the approximate viscoelastic model. Fig. 5.1a) shows a 2D cross section of the 3D push beam, which is the same push beam shown in Fig This result is computed in FOCUS for a Philips L7-4 linear ultrasound transducer array focused at 25 mm. The red dot in Fig. 5.1a) indicates the observation point, which is located at x = 10 mm and z = 22.8 mm. In this simulation, the shear elasticity is 2.25 kpa, and the shear viscosity is 1.0 Pa s. Fig. 5.1b) shows the shear wave particle displacement computed with the elastic model in Eq. 5.22, Fig. 5.1c) shows the attenuation 94

113 term calculated with Eq. 5.18, and Fig. 5.1d) shows the approximate shear wave particle displacement that is calculated by convolving the attenuation term with the shear wave particle displacement computed with the elastic model. z (mm) Push Beam (a) x (mm) Normalized Displacement Elastic model (b) t (ms) Normalized Displacement Attenuation term (c) t (ms) Normalized Displacement Approximate model (d) t (ms) Figure 5.1: The results of the individual calculation steps that produce an approximate shear wave particle displacement. Fig. 5.1a shows the push beam, and the red dot indicates the observation point, which is located at (x, z) = (10 mm, 22.8 mm). Fig. 5.1b shows the shear wave particle displacement simulated with the elastic model Eq Fig. 5.1c shows the attenuation term calculated with Eq Fig. 5.1d shows the approximate shear wave particle displacement calculated by convolving the results shown in Fig. 5.1b and Fig. 5.1c. This approximate viscoelastic model significantly reduces the computation time. For example, to calculate the shear wave particle displacements for N shear elasticity values and M shear viscosity values, the spatial convolution in the conventional viscoelastic model in Eq is evaluated N M times. However, with the approximate viscoelastic model, the spatial convolution in Eq is only evaluated N times when calculating the shear wave particle displacements with the elastic model using different µ values. The approximate shear wave particle displacements in viscoelastic media are then obtained from temporal convolutions, which take much less time than spatial convolutions Validation of the approximate viscoelastic model To validate the approximate viscoelastic model, shear wave particle displacements calculated with the viscoelastic model in Eqs and with the approximate model in Eq are compared. Shear wave particle displacements are calculated for the push beam shown in Fig. 5.1(a) at the eight observation points shown in Fig The depth of each observation 95

114 point is the same as the depth of the focal peak for the push beam, which is 22.9 mm. The observation points are equally spaced in the lateral direction at this depth, where the distance from the peak of the push beam to each observation point in the x-direction ranges from 5 to 40 mm with a increment of 5 mm in the y = 0 plane. The shear wave particle displacements calculated with the two models are shown in Fig The percent difference between the shear wave particle displacements calculated with the two models is evaluated with %difference = u approx(r, t) u visco (r, t), (5.26) u approx (r, t) where u approx (r, t) is the shear wave particle displacement calculated with the approximate model in Eq. 5.21, u visco (r, t) is the shear wave particle displacement calculated using the viscoelastic model in Eqs , and. represents the L 2 norm in time. For the eight observation points, the percent differences between the shear wave particle displacements calculated with the two models are 3.07%, 2.97%, 3.10%, 3.28%, 3.14%, 3.35%, 3.40%, and 3.56%. These results indicate that the shear wave particle displacements calculated with the two models are very similar. The percent difference is also evaluated for observation points in a 2D plane ranging from -20 mm to 20 mm in the x-direction and from 0 mm to 40 mm in the z-direction, where the spatial sampling interval in both direction is 1 mm. A 2D image of the percent difference is shown in Fig. 5.3, which also contains a contour that indicates where the percent difference is equal to 5%. For the region close to the push beam ( x < 2 mm), the percent difference is much larger because of the far field approximation that is employed in the model. The percent difference is also much larger for observation points located close to the transducer (z < 10 mm). However, when x > 2 mm, and when the depth of the observation point is similar to the depth of the peak of the push beam, the percent difference is usually less than 5%. This suggests that the approximate model is an effective approximation for the viscoelastic model on either side of the push beam at the depth of the focal peak. 96

115 Normalized Displacement D Viscoelastic Model Approximate Model time (ms) Figure 5.2: Comparison between the shear wave particle displacements simulated with the 3D viscoelastic model and approximate model. The blue lines indicate the shear wave particle displacements calculated with the viscoelastic model, and the red dashed lines represent shear wave particle displacements computed with the approximate model Shear wave simulations with the approximate model using GPU To accelerate shear wave simulations with Eq. 5.21, these calculations are performed on a Graphics Processing Unit (GPU) [165]. For each calculation on the GPU, the pre-calculated acoustic radiation force, which is stored in a data file, is first loaded into the CPU (host) memory and then copied to the GPU (device) memory. The kernel function is then used to separate the evaluation of the spatial convolution in Eq into different threads that are distributed to different processors on the GPU for parallel evaluations. For each thread, the spatial convolution is evaluated at a single point in space and at a single moment in time. Each thread obtains the push beam from the global memory, and the calculated spatial convolution result is also stored in global memory. After all of the threads are finished, the results are copied back into the host memory. The temporal convolution in 97

116 0 10% z (mm) % 6% 4% 2% x (mm) 0% Figure 5.3: A 2D image showing the percent difference between shear wave particle displacements calculated with the approximate model in Eq and the viscoelastic model in Eqs The black contour represents a value of 5% for the percent difference. Eq is then evaluated on the CPU to obtain the shear wave particle displacements, and the shear wave particle velocity is calculated by evaluating the temporal derivative of u V z (r, t) in Eq The simulated shear wave particle velocities are stored in a data file for subsequent evaluations. These calculations are implemented through the Compute Unified Device Architecture (CUDA) programming language on an nvidia (Santa Clara, CA) Kepler K80 GPU Tissue mechanical parameter estimation using a genetic algorithm The above model provides the capability to rapidly simulate shear wave particle velocities induced by an acoustic radiation force, which then enables the estimation of shear wave parameters from the measured shear wave particle velocities. The shear elasticity and shear viscosity values are estimated by minimizing the difference between the simulated and measured shear waves described by arg min µ, ηs v measured (r, t) v µ,η s (r, t), (5.27) v measured (r, t) 98

117 where v measured (r, t) is the measured shear wave particle velocity, and v µ,η s (r, t) is the shear wave particle velocity simulated with the approximate model in Eq In chapter 4, this minimization problem is solved by exhaustively searching the 2- parameter space to obtain the global minimum. In this chapter, the global minimum of the objective function in Eq is determined with a genetic algorithm. This is achieved through the following steps: 1. Calculate the acoustic radiation force as described in section Calculate the shear wave particle velocity induced by the acoustic radiation force with the elastic model using Eqs as described in section Shear wave particle velocities are calculated with shear elasticities ranging from 0.5 kpa to 4 kpa with an increment of 0.1 kpa on GPU and are then stored. 3. Use a genetic algorithm to find the global minimum of the objective function in Eq First, the pre-calculated shear particle velocities simulated with the elastic model in step 2 are loaded into memory. Then, the minimization problem in Eq is solved using a genetic algorithm to find the µ and η s values that minimize the difference between the simulated and the measured shear wave particle velocities. In Eq. 5.27, v µ,η s (r, t) is calculated using the approximate viscoelastic model in Eq. 5.21, which evaluates a temporal convolution between the pre-calculated shear particle velocities simulated with the elastic model and the attenuation term. For the optimization, the shear elasticities range from 0.5 kpa to 4 kpa with an increment of 0.1 kpa, the shear viscosities range continuously from 0.1 Pa s to 4 Pa s, the population size for the genetic algorithm is 30, and the stopping criteria is 0.1%, which means that if the difference between the best µ and η s values for the consecutive generations are less than 0.1%, the optimization will stop. The genetic algorithm is applied implemented through the Global Optimization Toolbox in MATLAB (Mathworks, MA). 99

118 z (mm) 40 z (mm) (a) x (mm) 80 (b) x (mm) Figure 5.4: Two push beams that induce shear waves in human liver. Fig. 5.4a shows the push beam focused at 35 mm, and Fig. 5.4b shows the push beam focused at 50 mm Validation with In Vivo human data Human liver data was collected at Mayo Clinic under the approval of the Institutional Review Board at Mayo Clinic in accordance with the Health Insurance Portability and Accountability Act. Shear wave measurements were performed on healthy subjects with written consent. The Verasonics Vantage ultrasound system was used in these experiments, and a Philips P4-2 linear transducer array was used to generate and measure the shear waves, where the array contains 64 transducer elements with an approximate elevation focus of 70 mm. The height of each transducer element is 13 mm, the width is 315 µm, and the kerf between adjacent transducer elements is 20 µm. During each measurement, the ultrasound probe was inserted into one of the intercostal spaces above the liver, and a homogeneous region of the liver was located with B-mode ultrasound imaging. Then, shear waves were induced and measured within that region. Figure 5.4 shows the two push beams that are used to induce the shear waves. The first push beam in Figure 5.4(a) is electronically focused at 35 mm and 100

119 is excited at 2.8 MHz. The second push beam in Figure 5.4(b) is electronically focused at 50 mm and is excited at 2.7 MHz. During the experiment, both push beams are excited three times consecutively for a duration of 267 µs with an interval of 15 µs. Immediately after the push sequences are completed, the Verasonics ultrasound system is switched to pulse-echo imaging mode, in which focused ultrasound pulses with a center frequency of 2.78 MHz are transmitted and the pulse-echo IQ data for the 2D lateral plane are quickly collected. For the push beam focused at 35 mm, the IQ data is measured for 15.7 ms with a repetition frequency of 4.78 khz. For the push beam focused at 50 mm, the IQ data is measured for 17.6 ms with a repetition frequency of 4.27 khz. Shear wave displacement are then calculated from the IQ data using the 1D autocorrelation method [11]. For both push beams, shear wave particle velocities are measured in two different locations: the 7th intercostal space and the 8th intercostal space. At each location, the measurement is repeated five times. Thus, there are 2 (push beams) 2 (positions) 5 (repetition times) = 20 datasets in total. 5.4 Result Shear wave particle velocities measured in liver Fig. 5.5 shows the shear wave particle velocities measured in human liver. Figs. 5.5(a)-5.5(c) show the shear wave particle velocities induced by the push beam focused at 35 mm at 2 ms, 4 ms, and 6 ms, respectively. All three figures are normalized to the maximum value in Fig. 5.5(a). In Fig. 5.5(a), the maximum value is located at (x, z) = (3.35 mm, mm), and the amplitude is normalized to 1. At x = 3.35 mm, the full width at half maximum (FWHM) along the z-direction is mm. In Fig. 5.5(b), the maximum value is located at (x, z) = (5.72 mm, mm), and the normalized amplitude is At x = 5.72 mm, the FWHM along the z-direction is mm. In Fig. 5.5(c), the maximum value is located at (x, z) = (8.08 mm, mm), and the normalized amplitude is At x = 8.08 mm, the FWHM along the z-direction is mm. Also, the amplitude of the shear waves decreases as the shear wave propagates due to the combined effects of diffraction and attenuation in 101

120 z (mm) time = 2ms z (mm) time = 4ms (a) (b) (c) x (mm) x (mm) x (mm) z (mm) z (mm) z (mm) time = 6ms (d) (e) (f) x (mm) x (mm) x (mm) Figure 5.5: The shear wave particle velocities measured in human liver. Figs. 5.5(a)-5.5(c) show the shear wave particle velocities induced by the push beam focused at 35 mm at 2 ms, 4 ms, and 6 ms, respectively, and Figs. 5.5(d)-5.5(f) show the shear wave particle velocities induced by the push beam focused at 50 mm at 2 ms, 4 ms, and 6 ms, respectively. z (mm) viscoelastic soft tissue. Figs. 5.5(d)-5.5(f) show the shear wave particle velocities induced by the push beam focused at 35 mm at 2 ms, 4 ms, and 6 ms, respectively. All three figures are normalized to the maximum value in Fig. 5.5(d). In Figs. 5.5(d), the maximum value is located at (x, z) = (3.35 mm, mm), and the amplitude is normalized to 1. At x = 3.35 mm, the FWHM along the z-direction is mm. In Fig. 5.5(e), the maximum value is located at (x, z) = (5.32 mm, mm), and the normalized amplitude is At x = 5.32 mm, the FWHM along the z-direction is mm. In Fig. 5.5(f), the maximum value is located at 102