IMAGING THE MECHANICAL PROPERTIES OF TISSUE WITH ULTRASOUND: AN INVESTIGATION OF THE RESPONSE OF SOFT TISSUE TO ACOUSTIC RADIATION FORCE

Transcription

1 IMAGING THE MECHANICAL PROPERTIES OF TISSUE WITH ULTRASOUND: AN INVESTIGATION OF THE RESPONSE OF SOFT TISSUE TO ACOUSTIC RADIATION FORCE by Mark L. Palmeri Department of Biomedical Engineering Duke University Date: Approved: Kathryn R. Nightingale, Ph.D., Supervisor Gregg E. Trahey, Ph.D. Tod A. Laursen, Ph.D. Lori A. Setton, Ph.D. Rex C. Bentley, M.D. Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Biomedical Engineering in the Graduate School of Duke University 25

2 abstract (Biomedical Engineering) IMAGING THE MECHANICAL PROPERTIES OF TISSUE WITH ULTRASOUND: AN INVESTIGATION OF THE RESPONSE OF SOFT TISSUE TO ACOUSTIC RADIATION FORCE by Mark L. Palmeri Department of Biomedical Engineering Duke University Date: Approved: Kathryn R. Nightingale, Ph.D., Supervisor Gregg E. Trahey, Ph.D. Tod A. Laursen, Ph.D. Lori A. Setton, Ph.D. Rex C. Bentley, M.D. An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Biomedical Engineering in the Graduate School of Duke University 25

3 Abstract The mechanical characterization of tissues and lesions within tissues has been used by clinicians to determine states of disease. Several pathological states can lead to a change in the mechanical properties of diseased tissues versus healthy tissues: fibrous tissue deposition in breast lesions may allow for masses to be found by manual palpation, replacement of healthy hepatic tissue by fibrosis can lead to stiffening in a cirrhotic liver, and the purposeful destruction of tissue using radiofrequency (RF) ablation leads to the denaturing of proteins and an inflammatory response that stiffens the induced lesion. Additionally, cancers and desmoplastic masses in the colon, breast, and other soft tissues can potentially be distinguished from healthy tissue based on their mechanical properties. The work presented in this dissertation investigates the thermal and mechanical response of soft tissue to a new ultrasonic method, called Acoustic Radiation Force Impulse (ARFI) imaging. ARFI imaging uses impulsive, high-intensity ultrasound pulses to generate acoustic radiation force and material-dependent displacement fields. Displacement magnitudes and dynamics are dependent on tissue stiffness and structure, which may differentiate healthy from diseased tissues. In order to insure patient safety in clinical implementations of ARFI imaging, the thermal response of soft tissue to acoustic radiation force is investigated using Finite Element Method (FEM) models of tissue heating and thermocouple measurements in porcine muscle. The mechanical response of soft tissue to impulsive acoustic radiation force excitation is investigated using an FEM model of elastic materials iii

4 with varying stiffnesses, densities, and acoustic attenuations with spherical inclusions of varying sizes and material properties. These mechanical responses are validated in gelatin and graphite-based tissue-mimicking phantoms, along with a commercial phantom. Finally, the impact of ultrasonically tracking the dynamic displacement fields is evaluated using the FEM models in conjunction with a linear acoustic field simulation package. Thermal FEM models and thermocouple measurements indicate that ARFI imaging is safe to implement clinically, but the implementation of new probes and beam sequences requires additional evaluation. Mechanical studies have revealed that tissue dynamics in response to ARFI excitations are shear wave dominated, and the size and stiffness contrast of spherical inclusions dictates the dynamic response of imaging lesions. Using ultrasound to track the displacement fields introduces jitter and bias due to scatterer shearing immediately after excitation that can alter the interpretation of the dynamic displacement data. These studies indicate that ARFI imaging is safe to perform in vivo and can be useful in characterizing material properties to differentiate healthy from diseased tissues. iv

5 Acknowledgments I would like to thank Dr. Kathy Nightingale for being a superb advisor and a wonderful teacher. I would also like to thank Dr. Gregg Trahey for always supplying a wealth of new ideas and paths to explore. I thank the other members of my dissertation committee for their thoughtful insights and guidance throughout my graduate studies. My thesis work was greatly enhanced by the insights and efforts of my fellow students, especially Kristin Frinkley for her help in making thermocouple measurements. The models developed throughout this thesis presented considerable computational challenges that would not have been overcome without the technical assistance of Joshua Baker-LePain. I thank Dr. Sal Pizzo for allowing me to study in the company of great scientists through the Medical Scientist Training Program. I also thank Dr. Roger Nightingale for his patience in introducing me to the world of finite element analysis and Dr. Stephen McAleavey for providing the perfect combination of intellect and humor each day. Most importantly, none of this would have been possible without the support, encouragement, and guidance of my family and Dee Dee Downie. v

6 Contents Abstract iii Acknowledgments v List of Tables xii List of Figures xiii Introduction 2 Thermal Effects of Radiation Force Imaging Soft Tissue 4 2. Introduction Background Acoustic Radiation Force Impulse (ARFI) Imaging Bio-Heat Transfer Equation Analytic Solutions to the Bioheat Transfer Equation Thermal Expansion Temperature-Dependent Changes in Sound Speed Methods Approach to Thermal Simulation Simulating Intensity Fields and Thermal Model Generation Single Location ARFI Imaging Two-dimensional ARFI Imaging vi

7 2.3.5 ARFI Shear Wave and M-mode Imaging Thermal Expansion Temperature-Dependent Changes in Sound Speed Results Single Location ARFI Imaging Two-dimensional ARFI Imaging ARFI Shear Wave and M-mode Imaging Thermal Expansion Temperature-Dependent Changes in Sound Speed Discussion Thermal Characterization of a Single ARFI Interrogation Thermal Characterization of Two-dimensional ARFI Imaging ARFI Shear Wave and M-mode Imaging Thermal Effects on Displacement Tracking Limitations of the Thermal Model Other Radiation Force-Based Imaging Modalities Conclusions Acknowledgments Experimental Measurement of Radiation Force Imaging Heating Introduction Background Acoustic Radiation Force Impulse (ARFI) Imaging Overview of Thermal Modeling Methods vii

8 3.3. Measurement of Tissue Heating ARFI Imaging Configuration Transducer Face Heating Data Processing FEM Models Results Focal Point Cooling Spatial Maps of Heating Transducer Face Heating Discussion Transient Heating Behavior Elevation and Lateral Heating Axial Heating and Thermocouple Limitations Transducer Face Heating Evaluation of Thermal Safety Conclusions Acknowledgments An FEM Model of Tissue Response to Impulsive Radiation Force Introduction Background Viscoelastic Dynamics Linearly Viscous Fluids and Acoustic Radiation Force Acoustic Radiation Force Impulse (ARFI) Imaging Finite Element Methods viii

9 4.3 Methods FEM Pre-Processing FEM Implementation and Post-Processing Experimental Validation Results Effects of Poisson s Ratio ARFI Imaging Displacement Fields in Homogeneous Media Impact of Poisson s Ratio, Density, and Attenuation Displacement Fields in Stiff Spherical Inclusions Discussion Choice of Modeled Poisson s Ratio & Young s Modulus Dynamic Behavior in the FEM Model and Experiments Limitations of the FEM Model Future Applications of the FEM Model Conclusions Acknowledgments Ultrasonic Tracking of Radiation Force-Induced Displacements Introduction Background Methods Finite Element Method Simulation of Tissue Displacement Simulation of Ultrasonic Displacement Tracking Processing of Tracked Data Results ix

10 5.5 Discussion Conclusions Acknowledgments Mechanical Response of Tissue to Impulsive Radiation Force 6. Introduction Background Methods Results ARFI-Induced Dynamics in Homogeneous Media ARFI-Induced Dynamics in Media with Spherical Inclusions Discussion Conclusions Acknowledgments Conclusions and Future Work Conclusions Future Work Nonlinear Wave Propagation Parallel Tracking Additional Modeling Parameters A Abbreviations 43 B Thermal Notation and Physical Constants 44 C Mechanics Notation 45 x

11 Bibliography 46 Biography 55 xi

12 List of Tables 3. Thermal material properties of biological tissue Transducer face heating Young s moduli of gelatin and CIRS phantom materials Reconstructed shear wave speeds Comparison of FEM and experimental displacement magnitudes Reconstructed Young s moduli xii

13 List of Figures 2. Heat source functions associated with ARFI imaging Temperature increases in the axial-lateral plane for different tissue absorptions after a single ARFI interrogation Heating and cooling at the focus after an ARFI interrogation Cooling as a function of depth in materials with different absorptions Temperature increases along the axial-lateral plane for different absorptions Temperature increases after a single frame of two-dimensional ARFI imaging using an F/.3 focal configuration Temperature increases after a single frame of two-dimensional ARFI imaging using an unfocused, narrow transmit aperture configuration Real-time, two-dimensional ARFI imaging, using an F/.3 focal configuration Real-time two-dimensional ARFI imaging, using an unfocused, narrow transmit aperture configuration Maximum temperature increases generated by real-time ARFI imaging for an absorption of.5 db/cm/mhz using an F/.3 focal configuration Heating associated with ARFI M-mode imaging Impact of thermal expansion and temperature-dependent sound speed changes on ARFI imaging Experimental setup for thermocouple measurements Comparison of thermocouple and FEM data of the cooling behavior at the focal point xiii

14 3.3 Comparison of thermocouple and FEM data of heating in the lateral dimension Comparison of thermocouple and FEM data of heating in the elevation dimension Comparison of thermocouple and FEM data of heating in the axial dimension Impact of Poisson s ratio on axial displacement at the focal point FEM and experimental displacement fields in the axial-lateral imaging plane Comparison of the FEM displacement data in the axial and lateral dimensions with experimental data Impact of Poisson s ratio on dynamic displacement data Impact of density on dynamic displacement data Impact of ultrasonic attenuation on dynamic displacement data Comparison of FEM and experimental displacement data in the axiallateral plane for a stiff spherical inclusion Comparison of FEM and experimental displacement data in the axial and lateral dimensions for a stiff spherical inclusion Speckle SNR as a function of scatterer density Comparison of FEM, tracked FEM, and experimental displacement data in the axial-lateral plane Comparison of FEM and tracked FEM data in the axial dimension Tracking performance as a function of material stiffness Focal point tracking performance as a function of material stiffness Tracking performance as a function of scatterer density Tracking performance as a function of tracking frequency xiv

15 5.8 Transmit/receive lateral beamwidth as a function of transducer bandwidth Tracking performance as a function of displacement magnitude Tracking performance as a function of focal configuration on tracking transmit and receive Displacement profiles in the lateral dimension relative to excitation beamwidth Tracking performance as a function of excitation focal configuration Tracking performance as a function of track transmit focal depth Tracking performance as a function of correlation kernel length Parametric images from a tissue-mimicking phantom Normalized displacement fields through time in the axial-lateral plane for a homogeneous material Focal point displacement as a function of stiffness in a homogeneous material Time-to-peak displacement at the focal point as a function of stiffness and excitation focal configuration Impact of density on focal point dynamics and time-to-peak displacement Impact of acoustic attenuation on tissue dynamics as a function of depth Impact of excitation duration on focal point dynamics and shear wavelengths Impact of stiffness on focal point displacement and recovery time Excitation locations for analysis of shear wave interactions in spherical inclusions Dynamic displacement at the focal depth as a function of excitation location xv

16 6. Impact of ultrasonic displacement estimation on imaging spherical inclusions Shear wave interactions in spherical inclusions Displacement contrast of spherical inclusions as a function of stiffness and inclusion size ( - 3 mm diameters) Displacement contrast of spherical inclusions as a function of stiffness and inclusion size (4-6 mm diameters) Spatial distribution of acoustic radiation force as a function of acoustic attenuation Simulated two-dimensional ARFI image of a spherical inclusion... 4 xvi

17 Chapter Introduction The mechanical characterization of lesions and tissues allows clinicians to determine the presence and progression of disease. There are many pathological states in a variety of organ systems that possess different mechanical properties than healthy tissues. For example, in the breast, malignant masses are associated with fibrous tissue deposition that may allow them to be detected by manual palpation [4,27,89,99]. Cirrhotic livers tend to be stiffer than healthy livers as hepatic tissue is replaced by fibrosis [2]. Changes in the mechanical properties of tissue can also be associated with therapeutic procedures, such as the purposeful destruction of tissue using radio-frequency (RF) ablation that leads to the denaturing of proteins and an inflammatory response that stiffens the induced lesion [93]. Elastography techniques have been developed to characterize the mechanical properties of tissue by externally compressing tissue and tracking the resulting tissue response using ultrasonic methods [9, 39, 4, 53, 67, 68, 76, 82, 84, 86, 9, 93]. External compression of tissues, however, becomes challenging for in vivo applications where it is difficult to control organ location and boundary conditions. Acoustic radiation force imaging modalities remove the need for external tissue compression, and instead rely on the excitation of tissue by high-intensity acoustic pulses that transfer their momentum to the tissue and generate localized displacement fields. Several techniques involve monitoring the steady-state or resonant response of tissue to

18 acoustic radiation force excitations, as is done by Walker et al. [98] in the Kinetic Acoustic Vitroretinal Examination (KAVE) method and Fatemi and Greenleaf [33] in Vibro-acoustography. Another application of acoustic radiation force imaging involves monitoring the dynamic response of tissue to impulsive excitation. The impulsive excitation of tissue leads to the generation of shear waves, and the propagation speed of these waves is dictated by the viscoelastic properties of the tissue outside the region of excitation (ROE), as is done in Shear Wave Elasticity Imaging (SWEI) and supersonic imaging [6, 7, 78]. These methods become challenging in vivo because shear wave attenuation limits the propagation distance in tissue that can be ultrasonically tracked. Acoustic Radiation Force Impulse (ARFI) imaging, the focus of this thesis, allows for the dynamic response of tissue following impulsive excitation with acoustic radiation force to be characterized in the ROE. The clinical goals of ARFI imaging include: Differentiating benign from malignant lesions in the breast, Characterizing the stiffening of arteries and the morphology of arterial plaques, Providing more accurate tumor and lymph node staging in colorectal applications, Characterizing the size and stiffness of thermal ablation lesions relative to adjacent healthy tissue. The purpose of this thesis is to demonstrate that the dynamic response of tissue to ARFI excitation can be used to characterize the mechanical properties of materials with spherical inclusions, as would be the case in the clinical scenarios listed above. 2

19 In order to perform ARFI imaging in clinical settings, studies regarding the thermal safety of ARFI imaging were conducted to insure that this new imaging modality would not expose patients to excessive tissue heating. Chapter 2 covers the use of Finite Element Method (FEM) models to simulate the tissue heating that would be attributed to in vivo ARFI imaging, while Chapter 3 covers validation of these thermal FEM models using thermocouple measurements in porcine muscle. The mechanical response of soft tissue and lesions during ARFI imaging was studied using a linear acoustic field simulation package and dynamic structural FEM models. Chapter 4 reviews the mechanics of homogeneous elastic media and the equations dictating the behavior of acoustic radiation force in biological tissues, in addition to outlining the structural FEM model that was developed and the validation of that model using gelatin and graphite-based tissue-mimicking phantoms. The impact of ultrasonically tracking the displacement fields generated by ARFI imaging are studied using FEM models and a linear acoustic field simulation package in Chapter 5. Chapter 6 uses the structural and tracking models to study the impact of material properties and the presence of spherical inclusions of varying sizes and stiffnesses on the dynamic displacement fields generated during ARFI imaging. Chapter 7 discusses how the results of these studies can guide the implementation of ARFI imaging to characterize the mechanical properties of tissues and lesions, along with outlining future work that has been motivated by these studies. 3

20 Chapter 2 On the Thermal Effects Associated with Radiation Force Imaging of Soft Tissue The work presented in this chapter was published in IEEE Transaction on Ultrasonics, Ferroelectronics, and Frequency Control, 5(5): , 24 [7]. 2. Introduction The use of acoustic radiation force to interrogate the mechanical properties of soft tissues is becoming a widely investigated research area. In general, acoustic radiation force is used to excite tissue, and the tissue response is monitored using either ultrasonic or magnetic resonance methods. For an analytic description of the mechanical response of soft tissue to focused acoustic radiation force, the reader is referred to Sarvazyan et al. [78]. There are many methods using acoustic radiation force currently under investigation. Vibroacoustography uses frequency-shifted, confocal beams to generate an oscillating radiation force within tissues, and the tissue response is monitored with either a hydrophone [2, 33], or ultrasonic methods [48]. The Kinetic Acoustic Vitroretinal Examination (KAVE) method uses radiation force to generate a steady- 4

21 state stress within soft gels and the vitreous of the eye, and ultrasonic displacements are monitored to characterize the steady-state response of these materials [94]. Supersonic imaging [5], and Shear Wave Elasticity Imaging (SWEI) [3,78], monitor the shear waves generated from short duration acoustic radiation force to characterize the shear modulus of the medium. Acoustic Radiation Force Impulse (ARFI) imaging, which is the focus of this thesis, uses a commercial diagnostic scanner to generate multiple, short duration radiation forces to interrogate a two-dimensional region of interest, and monitors the tissue response using ultrasonic, correlation-based methods [3, 56, 58, 63]. Similar work is also under investigation using a High Intensity Focused Ultrasound (HIFU) transducer to generate radiation force in order to monitor treatment [5]. In addition to soft tissue imaging applications, radiation force is being used to manipulate ultrasonic contrast agents in vitro and in vivo [26, 43]. The purpose of the work presented in this chapter is to determine the increase in tissue temperature that is associated with the pulse sequences used in single-location, two-dimensional, shear wave and ARFI M-mode imaging in soft tissue. Additionally, the impact of the heating on ultrasonic displacement tracking is evaluated with respect to thermal expansion and sound speed changes. 2.2 Background A list of the notation and physical constants used throughout this chapter can be found in Appendix B Acoustic Radiation Force Impulse (ARFI) Imaging ARFI imaging is a radiation force-based imaging method that studies the local mechanical properties of tissue [6]. ARFI imaging uses short duration (< ms), 5

22 high-intensity acoustic pulses to generate localized displacements in tissue, and the tissue recovery response is monitored using ultrasonic correlation-based methods [59]. Images of two-dimensional regions of interest (ROIs) are generated by sequentially interrogating multiple lateral locations, as is done in color Doppler imaging. During the application of high-intensity pulses, energy is absorbed by the tissue that results in the generation of acoustic radiation force, tissue displacement, and thermal heating. Herman et al. [44] recently published a study investigating ultrasonic imaging modalities that use short duration, high-intensity pulses, such as streaming detection [57] and shear wave ARFI imaging [58]. This study demonstrates, for moderately absorbing soft tissues ( db/cm/mhz), that repeated interrogations in a given location result in peak heating at the focus versus the tissue surface. Short duration, in situ intensities higher than the currently accepted steady-state limit of.72 W/cm 2 are shown to be reasonable in soft tissue, but the presence of bone interfaces increases the potential for thermal tissue damage Bio-Heat Transfer Equation As an acoustic wave propagates through a dissipative medium, an energy gradient is established in the medium, arising either from absorption or reflection of the wave. This energy gradient applies a force in the direction of wave propagation, and the absorption of energy results in the generation of heat in the tissue. The temperature rise in tissue can be estimated with the linear bio-heat transfer equation [54, 65, 74]: T = κ 2 T T τ + q v c v, (2.) 6

23 where T [ C] is the temperature, T [ C/s] is the time rate change in temperature, κ [.43 cm 2 /s] is the thermal diffusivity for soft tissue, τ [s] is the perfusion time constant, q v [J/cm 3 ] is the rate of heat production per unit volume, and c v [42 mw s/cm 3 / C] is the heat capacity per unit volume for soft tissue. Equation 2. can be solved analytically [65] and/or numerically [44, 54, 87,, ], and it has been validated experimentally by several groups [54,,]. A summary of much of the work done on acoustic heating is provided in NCRP Report No. 3 [54]. For ARFI imaging, the duration of heat application is very short (< ms), thus the effects of perfusion can be neglected [54, 65], simplifying Equation 2. to: T = κ 2 T + q v c v. (2.2) The heat source function for a continuous wave field of ultrasound can be expressed as [54, 65, 87]: q v = αp2 o ρc, (2.3) where α [Np/cm] is the absorption coefficient for soft tissue, ρ [. g/cm] is the density of soft tissue, c [54 m/s] is the speed of sound in soft tissue, and p o [Pa] is the acoustic pressure. For a linearly traveling plane wave, Equation 2.3 reduces to [54, 65]: q v = 2αI, (2.4) where I [W/cm 2 ] is the acoustic beam intensity. For a focused beam, the distribution of the applied heat source (q v ) is dependent upon the focal configuration of the transducer, which is often characterized by the transducer f-number (F/#): F/# = z D, (2.5) 7

24 where z [cm] is the acoustic focal length and D [cm] is the aperture width. This notation will be used throughout this thesis Analytic Solutions to the Bioheat Transfer Equation Analytic steady-state solutions to Equation 2. exist for simple geometries. For example, the steady-state temperature rise for a heated sphere of radius R in an infinite medium is given by the expression [65]: T = αir2 K, (2.6) where K [6. mw/cm/ C] is the thermal conductivity of soft tissue. Equation 2.6 is used to validate our simulation approach. In addition, the transient behavior of our model is validated using a simplified version of the bio-heat transfer equation (Equation 2.), in which heat loss due to perfusion and thermal conduction is neglected. Equation 2. becomes: T = q v c v. (2.7) The solution to Equation 2.7 provides a linear relationship between the increase in temperature, the application time, and the in situ intensity of the acoustic beam, as well as the absorption coefficient of the tissue: T = q v c v t = 2αI c v t, (2.8) where T [ C] is the increase in temperature, and t [s] is the heat application time. Validation of the transient behavior of our thermal model is achieved by comparison with Equation

25 2.2.4 Thermal Expansion Ultrasonic correlation-based methods are used to track radiation force-induced tissue displacements. Tissue displacement arising from thermal expansion can lead to speckle decorrelation and false estimates of displacement that could impact imaging performance. Linear thermal expansion in tissue can be characterized using the expression [8]: d(t ) = d o ( + β(t T o )), (2.9) where T o [ C] is the baseline temperature, d o [cm] is the baseline length of the region of interest, and β [ 3 C ] is the thermal expansion coefficient for soft tissue [54]. Equation 2.9 is used to evaluate the effects of thermal expansion during ARFI imaging Temperature-Dependent Changes in Sound Speed The sound speed of biological tissues is dependent on the temperature of the tissue [79,8]. For temperature changes of less than 6 C, the change in sound speed with respect to temperature ( c c ) is assumed to be linear, with typical values for T T being m/s/ C for breast fat and. m/s/ C for liver [79,8]. False estimates of the sound speed due to heating can lead to over or underestimation of displacements in correlation-based methods [79, 8]. The FEM model developed herein is used to evaluate the impact of sound speed changes on ARFI imaging. 9

26 2.3 Methods 2.3. Approach to Thermal Simulation The FEM model of heating associated with ARFI imaging presented herein estimates increases in temperature resulting from acoustic energy absorption by tissue. Model implementation is performed using a two step approach: first, the spatially distributed acoustic intensity field corresponding to the experimental transducer is simulated and the associated heat source function is computed using Equation 2.4; second, finite element methods are used to determine the thermal response of the tissue Simulating Intensity Fields and Thermal Model Generation FIELD II [46], a linear acoustic field simulation software program, is used to determine the acoustic intensity distribution associated with the high intensity pulses currently used for ARFI imaging (Siemens 75L4 linear transducer: one row of elements, excitation center frequency of 7.2 MHz, aperture width dependent on the number of active elements; Siemens Medical Systems, Ultrasound Group, Issaquah, WA). The transducer is simulated as F/3.8 in elevation and F/.3 laterally, with a focal depth of 2. cm. This corresponds to the experimental setup currently utilized to perform ARFI imaging [62] and is the default transducer focal configuration unless otherwise specified. An unfocused beam (i.e., element transmit aperture fired simultaneously) is also simulated to compare and contrast the tissue heating associated with different focal configurations.

27 The three-dimensional intensity fields are computed, normalized, and a threshold of 5% of the maximum intensity is imposed to reduce computational overhead. This intensity field is converted to a heat source field using Equation 2.4, and is imported into a finite element mesh generation program (HyperMesh, Altair Computing Inc., Troy, MI) and superimposed onto a rectangular solid mesh with a spatial extent of 7.5 mm (elevation) by 25 mm (lateral) by 37.5 mm (axial). These dimensions are chosen to be much larger than the heat source in order to allow for thermal diffusion without boundary artifacts. Plane-symmetry in elevation is assumed in the model. Figure 2. illustrates the heat source function for ARFI imaging in tissue media with absorptions of.5 and. db/cm/mhz for an F/.3 focal configuration. The transducer is centered on the top, forward surface of the model and spans a total of.52 cm as delimited by the contours of the heat source function on the top surface of the right-hand figure. Since plane-symmetry is assumed, only half of the transducer is modeled in elevation. The front of the figure corresponds to an axial-lateral plane centered in elevation on the transducer face. This plane is the default plane on which temperature profiles are displayed herein, unless otherwise specified. Studies by Duck et al. and Herman et al. have shown that for moderately attenuating tissues (.5 db/cm/mhz), repeated insonification at a given location results in peak heating at the focus, as opposed to the tissue surface [28, 44]. Heat generated at the face of the transducer, along with heat generated during the tracking of tissue displacement, is not included in these simulations. The tissue is modeled as a thermally homogeneous, isotropic solid and is meshed with trilinear dynamic thermal elements. Thermal material properties, such as heat capacity [42 mw s/cm 3 / C] and thermal diffusivity [.43 cm 2 /s], are assumed to be independent of temperature, and therefore can be linearly scaled [65]. There are 2233 nodes and 2776 elements in the mesh, creating a mesh with cubes of

28 α =.5 db/cm/mhz α =. db/cm/mhz Figure 2.: Heat source function associated with ARFI imaging for tissue absorptions of.5 and. db/cm/mhz for an F/.3 focal configuration. The heat source is shown relative to the spatial boundaries of the finite element mesh (7.5 mm by 25 mm by 37.5 mm). The transducer is located on the top, forward surface of the model, as delimited by the contours on the top surface of the figure on the right, for a total lateral dimension of.52 cm. Elevation symmetry is assumed. The front plane represents the axial-lateral plane of the transducer centered in elevation. equal volume and a uniform node spacing of.352 mm. All nodal temperatures in the model are initialized by default to C (the environmental temperature) as relative temperature changes are simulated with respect to baseline temperature. Normalized heat source values applied to elements from the simulated intensity outputs are scaled to a peak in situ intensity value of I = W/cm 2, corresponding to a peak heat source of cal/s/cm 3, when 69 elements are fired in an F/.3 configuration. This value is consistent with in situ values indirectly measured during ARFI imaging [6] using a linear extrapolation of small-signal derated fields, which will overestimate the true focal fields [55]. A peak in situ intensity value of W/cm 2 is also implemented for the unfocused transducer configuration to generate displacements of at least µm that are desirable for ARFI imaging [56]. The top model surface that represents contact with the environment/transducer is held at the environmental 2

29 temperature, while the other boundaries of the model have insulating boundary conditions, simulating a continuum of tissue [87]. LS-DYNA3D (Livermore Software Technology Corporation, Livermore, CA) is used to solve for the dynamic temperature fields and the thermal expansion displacements using a time-domain, explicit iterative solver. Validation of the model is accomplished using the analytic steady-state and transient solutions provided in Equations 2.6 and 2.8. Additional validation of the transient solution of the explicit finite element code is achieved by comparison with implicit three-dimensional dynamic thermal finite element code that was written and implemented in the Finite Element Analysis Program (FEAP, Dr. Robert L. Taylor, Department of Civil and Environmental Engineering, University of California at Berkeley). Run times for the explicit solver are on the order of several hours using.67 GHz dual Athalon processing nodes, with RAM requirements of approximately 2. GB Single Location ARFI Imaging Tissue heating and cooling profiles are characterized for an ARFI interrogation at a single spatial location. A typical ARFI pulse sequence consists of six 28 µs high-intensity pushing pulses with 5 µs between each pulse, for an effective duty cycle of 5.7% and a total insonification time of.92 ms (see bottom of Figure 2.3). This represents the timing of the high-intensity pushing pulses, unless otherwise specified. The models are presented for materials possessing different ultrasonic absorption coefficients ranging from db/cm/mhz. 3

30 2.3.4 Two-dimensional ARFI Imaging Tissue heating and cooling profiles are also characterized for ARFI interrogation of a two-dimensional ROI. As with color Doppler imaging, two-dimensional ARFI images are created by sequentially interrogating lateral locations. A typical ARFI ROI is comprised of up to 5 adjacent beam lines that are separated laterally by a minimum of.35 mm spatially spanning a total of.75 cm. Temporally, each interrogation is separated by 5 ms (i.e., time between initiation of the first pushing pulse in each lateral location) for a total ROI acquisition time of.25 seconds when 5 lines are interrogated. Simulation of two-dimensional ARFI heating is accomplished using convolution [22, 65]. The tissue heating associated with a single ARFI interrogation is characterized for 6 seconds with a sampling rate of 2 Hz. The tissue heating for two-dimensional ARFI imaging of the entire ROI is determined by spatially convolving the individual thermal results for each time step. Results are compared and contrasted for different beam spacings and two focal configurations (F/.3 and an unfocused, narrow transmit aperture). Heating trends associated with real-time ARFI imaging are also simulated by repeatedly interrogating lateral locations at varying frame rates ARFI Shear Wave and M-mode Imaging Radiation force imaging is also used for generating shear waves and tracking their propagation and interaction with structures in tissue [5, 58, 62, 78]. ARFI shear wave sequences involve repeated applications of high-intensity pulses in a given spatial location, while translating the tracking beam index across the desired field of view. Typically, the number of high-intensity interrogations that are applied at a given location is 2, each of which is separated by 5 ms during which displacement 4

31 tracking occurs [58, 62]. The heating associated with ARFI shear wave sequences is characterized as a function of the total number of high-intensity interrogations at a given location. Heating associated with ARFI M-mode imaging can be viewed as an extension of shear wave imaging, where ARFI interrogations also occur at a single location over time; however, the temporal spacing between successive interrogations is much greater (e.g., arterial wall imaging throughout the cardiac cycle) Thermal Expansion Coupled thermal-mechanical simulations are performed to predict the amount of tissue expansion that would occur due to tissue heating during ARFI imaging. The thermal expansion model is validated using the analytic expression in Equation 2.9 to approximate the thermal expansion associated with the heating of a sphere Temperature-Dependent Changes in Sound Speed Changes in sound speed due to tissue heating are simulated for breast and liver using c T values of and. m/s/ C respectively [79, 8]. Errors in displacement due to these changes in sound speed are calculated by determining the mean sound speed change from the transducer face to the focal depth along the center line (laterally) and solving for the error in estimated round-trip pulse-echo time. The error in round-trip time is then converted to a displacement error using the assumed constant sound speed of 54 m/s. 5

32 2.4 Results 2.4. Thermal Characterization of Single Location ARFI Imaging Figure 2.2 shows the temperature increases in the axial-lateral plane (centered in elevation) for different tissue absorptions after a single ARFI interrogation. Note that the temperature scale, in degrees Celsius, is different for each image. The focus is at 2. cm, with an F/.3 focal configuration. The thermal response of tissue is assumed to be linear with intensity; therefore, the thermal profiles shown in Figure 2.2 scale linearly as a function of pulse intensity. Figure 2.3(a) shows the increase in temperature at the focal depth of 2. cm during an ARFI pushing sequence for different absorptions. The bottom figure represents the corresponding timing diagram for the pushing pulses. The maximum temperature value that each simulation reaches at the focus is also the maximum temperature throughout the three-dimensional volume of tissue for that given absorption (Figure 2.2). Figure 2.3(b) illustrates the rate of cooling that occurs due to thermal conductivity at the focal point for absorptions ranging from db/cm/mhz. The cooling rate is the same for all tissue absorptions. The temperature at the focus has cooled to almost 4% of its peak value within.5 s, independent of the near field temperature distribution. Figure 2.4 shows cooling profiles at different depths as a function of different absorptions. Note that the temperature scales are different for each depth. These figures illustrate how cooling times are a function of both position relative to the focus and spatial heat distribution due to tissue absorption. Figure 2.5 shows the temperature increases along the axial-lateral plane (centered 6

33 Axial Position (cm) Axial Position (cm) Lateral Position (cm) (a) α =.3 db/cm/mhz Lateral Position (cm) (b) α =.5 db/cm/mhz Axial Position (cm) Axial Position (cm) Lateral Position (cm) (c) α =. db/cm/mhz Lateral Position (cm) (d) α =.5 db/cm/mhz Figure 2.2: Temperature increases in the axial-lateral plane (centered in elevation) for different tissue absorptions after a single ARFI interrogation. The transducer is centered along the top of the image, spanning a total of.52 cm laterally. The focus is at 2. cm, with an F/.3 focal configuration. Each figure uses a different temperature scale in degrees Celsius. in elevation) for an unfocused, narrow transmit aperture configuration using transmit elements for absorptions of.5 and. db/cm/mhz. These temperature profiles are shown after a single ARFI interrogation (.92 ms, 28 µs pushing pulses, duty cycle of 5.7%). These images can be contrasted with those in Figure 2.2 (F/.3 focal configuration). 7

34 Temperature (deg C) α =.3 db/cm/mhz α =.5 db/cm/mhz α =. db/cm/mhz α =.5 db/cm/mhz Normalized Temperature α =.3 db/cm/mhz α =.5 db/cm/mhz α =. db/cm/mhz α =.5 db/cm/mhz I/I max Time (ms) (a) Heating at Focus Time (s) (b) Cooling at Focus Figure 2.3: Figure 2.3(a) shows heating that occurs at the focus during a single six-cycle ARFI interrogation with an F/.3 focal configuration as a function of tissue absorption. The maximum temperature value that each simulation reaches at the focus is also the maximum temperature throughout the three-dimensional volume of tissue for that given absorption. The bottom of Figure 2.3(a) represents the corresponding timing diagram for the high-intensity pushing pulses. Figure 2.3(b) shows the normalized cooling that occurs at the focus for different absorptions immediately after an ARFI interrogation is complete. The cooling behavior at the focus is identical for all four absorptions. The temperature has cooled to almost 4% of its peak value at the focus within.5 s. Temperature (deg C).3.2. α =.3 db/cm/mhz α =.5 db/cm/mhz α =. db/cm/mhz α =.5 db/cm/mhz Temperature (deg C) α =.3 db/cm/mhz α =.5 db/cm/mhz α =. db/cm/mhz α =.5 db/cm/mhz Time (s) Axial Depth:.5 cm Time (s) Axial Depth:. cm Figure 2.4: Cooling over several seconds that occurs at different axial depths for different absorptions for an F/.3 focal configuration. Near-field locations are laterally centered with respect to the transducer. Note the differences in temperature scales for each plot Thermal Characterization of Two-dimensional ARFI Imaging Figures 2.6(a) and 2.6(b) show the temperature increases after interrogation of a.75 cm ROI with an F/.3 transducer configuration for tissue absorptions of 8

35 Axial Position (cm) Axial Position (cm) Lateral Position (cm) α =.5 db/cm/mhz Lateral Position (cm) α =. db/cm/mhz Figure 2.5: Temperature increases along the axial-lateral plane (centered in elevation) for different absorptions. The transducer is centered along the top of the images, spanning a total of.2 cm. This is an unfocused, narrow transmit aperture configuration using elements..5 and. db/cm/mhz, respectively. Fifty sequential ARFI interrogations are separated by.35 mm spatially and 5 ms temporally from right to left. Again, the temperature scale, in degrees Celsius, is different for these two cases. Note that the maximum temperature increase has shifted from the focal depth of 2. cm (the location of maximum temperature increase for a single ARFI interrogation, shown in Figure 2.2), to.45 cm for an absorption of.5 db/cm/mhz and to.2 cm for an absorption of. db/cm/mhz. Both of these depths correspond to the location where the maximum spatial and thermal overlap occurs between sequential ARFI interrogation locations. Figure 2.6(c) shows the maximum temperature generated in tissue as a function of the number of lateral positions separated by.35 mm spatially and 5 ms temporally. Note that as the ROI increases in size, the maximum temperature increase plateaus. The heating associated with two-dimensional ARFI imaging is also modeled for an unfocused, narrow transmit aperture transducer configuration using elements. The top row in Figure 2.7 shows the temperature profile associated with interrogating a.75 cm ROI with this transducer configuration, where each adjacent interrogation 9

36 Axial Position (cm) Lateral Position (cm) Axial Position (cm) Lateral Position (cm) Maximum Temperature (deg C) α =.5 db/cm/mhz α =. db/cm/mhz Number of Lateral Positions in ROI (a) α =.5 db/cm/mhz (b) α =. db/cm/mhz (c) Maximum Temperature Figure 2.6: Temperature increases after a single frame of two-dimensional ARFI imaging using an F/.3 focal configuration, as shown in Figure 2.2. In the simulation, 5 interrogated locations are laterally spaced.35 mm apart with a time delay of 5 ms between successive locations temporally. The figures on the left and in the middle demonstrate the thermal profiles for absorptions of.5 and. db/cm/mhz, respectively. The temperature scales, in degrees Celsius, are different for each figure. Note that the maximum temperature increase has shifted from the focal depth of 2. cm (the location of maximum temperature increase for a single ARFI interrogation), to.45 cm for an absorption of.5 db/cm/mhz and to.2 cm for an absorption of. db/cm/mhz. The figure on the right shows how the maximum temperature varies as a function of the number of lateral positions in the ROI for this focal configuration, using the same spatial and temporal spacing. line is separated by.35 mm spatially and 5 ms temporally. The bottom row of Figure 2.7 shows a line spacing of.75 mm spatially, effectively eliminating spatial overlap between adjacent interrogations. Note the difference in the temperature scales between the top row and bottom row of figures. For real-time ARFI imaging, the two-dimensional interrogations (shown in Figures 2.6 and 2.7) will be repeated in time. Figure 2.8 shows the temperature distribution for real-time ARFI imaging using an F/.3 focal configuration for absorptions of.5 and. db/cm/mhz. The.75 cm ROI has been imaged with a lateral spacing of.35 mm and a frame rate of 3 frames per second for a total of 3 seconds (i.e., the two-dimensional ROI is interrogated 8 times). Figure 2.9 shows the corresponding real-time implementation for the unfocused, narrow transmit aperture configuration with a line spacing of.7 mm in the top row, and.75 mm in the bottom row. 2

37 Axial Position (cm) Axial Position (cm) Lateral Position (cm) α =.5 db/cm/mhz Lateral Position (cm) α =. db/cm/mhz Axial Position (cm) Axial Position (cm) Lateral Position (cm) α =.5 db/cm/mhz Lateral Position (cm) α =. db/cm/mhz Figure 2.7: Temperature increases after a single frame of two-dimensional ARFI imaging using an unfocused, narrow transmit aperture configuration, as shown in Figure 2.5. In the top row, 5 lines are laterally spaced.35 mm apart with a 5 ms time delay between successive locations temporally, while in the bottom row, lines are laterally spaced.75 mm apart with a 5 ms time delay between successive lateral locations. The figures on the left demonstrate the thermal profile for an absorption of.5 db/cm/mhz, while the figures on the right demonstrate the thermal profile for an absorption of. db/cm/mhz. The temperature scales, in degrees Celsius, are different for each figure. The maximum temperature over the three-dimensional volume of tissue associated with real-time ARFI imaging using an F/.3 focal configuration for an absorption of.5 db/cm/mhz is characterized as a function of the number of lines per frame and the frame rate. Figure 2. shows the maximum temperature (that occurs at a depth of.45 cm), for - 5 lines, spaced.35 mm apart, for frame rates of frame per second and 3 frames per second. For higher numbers of lines, steady-state 2

38 Axial Position (cm) Axial Position (cm) Lateral Position (cm) α =.5 db/cm/mhz Lateral Position (cm) α =. db/cm/mhz Figure 2.8: Real-time, two-dimensional ARFI imaging, using an F/.3 focal configuration, performed at a frame rate of 3 frames per second for a total duration of 3 seconds for absorptions of.5 and. db/cm/mhz. The line spacing is.35 mm with a 5 ms temporal delay between successive locations. Note that the temperature scale, in degrees Celsius, is different for each figure. For an absorption of.5 db/cm/mhz, this maximum temperature occurs at a depth of.45 cm, while for an absorption of. db/cm/mhz, it occurs at a depth of.2 cm. temperature increases are not achieved due to an artifact of the insulating boundary conditions that do not allow for energy to conduct beyond the finite volume of tissue being modeled Thermal Characterization of ARFI Shear Wave and M-mode Imaging Acoustic radiation force is being explored in the generation and imaging of shear waves in tissue [5, 58, 78], which in our implementation involves repeating interrogations at the same spatial location, while sequentially tracking shear wave propagation at different lateral locations [58, 78]. Figure 2.(a) illustrates a linear increase in the maximum temperature, which occurs at the focus, as a function of the number of ARFI interrogations that are applied at a given spatial location over time during ARFI shear wave imaging. Figure 2.(b) shows the maximum temperature increase, which occurs at the focus, in ARFI M-mode imaging for each absorption 22

39 Axial Position (cm) Axial Position (cm) Lateral Position (cm) α =.5 db/cm/mhz Lateral Position (cm) α =. db/cm/mhz Axial Position (cm) Axial Position (cm) Lateral Position (cm) α =.5 db/cm/mhz Lateral Position (cm) α =. db/cm/mhz Figure 2.9: Real-time two-dimensional ARFI imaging, using an unfocused, narrow transmit aperture configuration, performed at a frame rate of 3 frames per second for a total duration of 3 seconds for absorptions of.5 and. db/cm/mhz. In the top row, the line spacing is.7 mm spatially and 5 ms temporally between successive locations. In the bottom row, the line spacing is.75 mm spatially and 5 ms temporally between successive locations. Note that the temperature scale, in degrees Celsius, is different for each figure. through time. ARFI M-mode imaging is analogous to shear wave imaging, except that the time delay between successive interrogations is extended to 5 ms, instead of 5 ms for shear wave imaging. Figures 2.(c) and 2.(d) illustrate the temperature increases associated with ARFI M-mode imaging for an F/.3 focal configuration for absorptions of.5 and. db/cm/mhz, respectively. These temperature increases occur over 7 seconds while firing an ARFI interrogation every 5 ms (for a total of 4 ARFI interrogations). Note that the temperature scales, in degrees Celsius, are different for the two figures. 23

40 Maximum Temperature (deg C) line 5 lines lines 5 lines Time (sec) Maximum Temperature (deg C) line 5 lines lines 5 lines Time (sec) frame per second 3 frames per second Figure 2.: Maximum temperature increases generated by real-time ARFI imaging for an absorption of.5 db/cm/mhz using an F/.3 focal configuration for two frame rates and a variable number of lines. The line spacing is.35 mm between adjacent ARFI interrogations. There is 5 ms between sequential ARFI interrogations, with the time between frames being a function of the specified frame rate. These maximum temperatures occur at a depth of.45 cm. The 5 line simulation does not reach a steady-state temperature increase due to an artifact of the insulating boundary conditions that do not allow for energy to conduct beyond the finite volume of tissue being modeled Thermal Expansion A coupled thermal-structural simulation for an absorption of.5 db/cm/mhz is implemented to determine how much thermal expansion occurs during ARFI imaging. The coupled simulation is implemented for the same thermal configuration illustrated in Figure 2.2(b). Figure 2.2 shows the magnitude of the maximum displacement (in microns) from thermal expansion, which occurs immediately after completion of the ARFI interrogation. The peak magnitude of displacement is.55 µm, but the greatest axial component of displacement is less than.2 µm Temperature-Dependent Changes in Sound Speed Figures 2.2(b) and 2.2(c) show the change in sound speed and associated displacement errors in breast fat and liver as a function of axial position along the center line of the transducer (where maximum heating occurs) for a single ARFI in- 24

41 Maximum Temperature (deg C) α =.5 db/cm/mhz α =. db/cm/mhz Temperature (deg C) α =.5 db/cm/mhz α =. db/cm/mhz Axial Position (cm) Number of Shear Wave Sequences (a) Shear Wave Imaging Lateral Position (cm) (c) α =.5 db/cm/mhz Axial Position (cm) Time (sec) (b) M-mode Imaging Lateral Position (cm) (d) α =. db/cm/mhz Figure 2.: Figure 2.(a) shows a linear increase in the maximum temperature as a function of the number of ARFI interrogations during ARFI shear wave imaging when interrogations are fired every 5 ms (covering a span of 75 ms). Figure 2.(b) shows the maximum temperature increase, which occurs at the focus, during ARFI M-mode imaging for absorptions of.5 and. db/cm/mhz, with 5 ms between ARFI interrogations. Figures 2.(c) and (d) illustrate the temperature increases associated with ARFI M-mode imaging for 7 seconds while interrogating every 5 ms (for a total of 4 interrogations) for absorptions of.5 and. db/cm/mhz, respectively. All of these models are for an F/.3 focal configuration. terrogation for an absorption of.5 db/cm/mhz. Note that the displacement errors accumulate with axial depth away from the transducer. 25

42 Axial Position (cm) Lateral Position (cm) Sound Speed (m/s) Liver Breast Axial Position (cm) Displacement (µm) Liver Breast Axial Position (cm) (a) (b) (c) Figure 2.2: Figure 2.2(a) shows the magnitude of the displacement due to thermal expansion immediately after an ARFI interrogation for an absorption of.5 db/cm/mhz. Displacements are in microns. The peak magnitude of displacement is.55 µm, but the greatest axial component of displacement is less than.2 µm. Figure 2.2(b) illustrates the change in sound speed in breast fat and liver as a function of axial position along the center line of the transducer (where maximum heating occurs) for an absorption of.5 db/cm/mhz after a single ARFI interrogation. Changes in sound speed are in meters per second. Figure 2.2(c) shows how these changes in sound speed would be reflected as displacement errors. Notice that these errors accumulate with axial distance from the transducer. Negative changes in displacement would result in underestimation of radiation force-induced displacements. 2.5 Discussion 2.5. Thermal Characterization of a Single ARFI Interrogation Temperature distributions arising from ultrasonic energy absorption depend on tissue properties. The absence of blood perfusion effects in these models creates maximum heating conditions and worst-case estimates of tissue temperature increases. The absorption of soft tissue varies significantly. Since absorption is a frequencydependent parameter, modeling a range of absorptions also provides insight into performing ARFI imaging through a range of frequencies in soft tissue. For low-to-moderate tissue absorptions ( db/cm/mhz), there are significant focal gains in intensity, and the majority of energy absorption and heating occurs near the focus (Figure 2.2). More absorbing tissues (. -.5 db/cm/mhz), 26

43 such as the breast [24], absorb energy over a much larger volume in the near field and generate smaller temperature increases. This is demonstrated by the order of magnitude difference in the maximum temperature increase for tissue absorptions that have been modeled ( db/cm/mhz, shown in Figure 2.2). For all tissue absorptions modeled with an F/.3 configuration, however, the maximum temperature for a single ARFI interrogation still occurs at the focus. This is consistent with the findings of Herman and Harris [44]. For a single ARFI interrogation, transducer heating is assumed to be negligible because the total duration of insonification is less than ms. As Figure 2.3 illustrates, the time between the six high-intensity pushing pulses in a single ARFI interrogation is too short for appreciable cooling to occur in the tissue. Therefore, the heating during a single ARFI interrogation is directly related to the total number of highintensity cycles within a few milliseconds. For example, a % duty cycle 68 µs pulse would result in the same temperature increase as six 28 µs pulses transmitted at a 5.7% duty cycle in.92 ms. For ARFI imaging, this finding is important because tissue demonstrates larger maximum displacements for a % duty cycle as compared to a 5.7% duty cycle, and there is no heating penalty for using a longer, single pulse. Cooling effects become appreciable on the order of tenths of a second at the focus, with the maximum focal temperature decaying to 5% of its peak value in less than.5 seconds (Figure 2.3(b)). Figure 2.3(b) also demonstrates that focal point cooling times are independent of tissue absorption for an F/.3 focal configuration. This is due to the similarity in relative heat distribution that occurs at the focus for all tissue absorptions with this focal configuration (Figure 2.2), though maximum temperatures vary due to the total amount of energy delivered to the focal region. The cooling behavior of tissue is more complicated at locations other than the focal 27

44 point since the distribution of thermal energy is a function of tissue absorption. As demonstrated in Figure 2.4, there is a complex relationship between the magnitude and spatial distribution of heat in the near field. At a depth of cm above the focus, temperatures will increase for approximately second as heat from the focal region dissipates through the surrounding tissue. While the magnitude of these temperatures is significantly lower than those experienced at the focus, this heat will accumulate when two-dimensional and real-time ARFI imaging is performed, due to the overlapping thermal profiles in the near field between successive interrogations. This explains why the peak temperature for two-dimensional ARFI occurs at a different depth than for a single interrogation. Figure 2.5 illustrates the temperature increases from an unfocused, narrow transmit aperture configuration ( elements) to compare with the F/.3 temperature distributions shown in Figure 2.2. As is expected, a more uniform distribution of heat deposition is present as compared with the F/.3 focal configuration. Again, for higher tissue absorptions, more heat is deposited in the near-field. While the maximum temperatures are similar to those for the F/.3 focal configuration (with a peak in situ intensity value of I = W/cm 2 used for both configurations), the lateral extent of the heating field is reduced due to the smaller size of the active transmit aperture Thermal Characterization of Two-dimensional ARFI Imaging To achieve two-dimensional ARFI imaging, multiple interrogations are fired in close temporal (5 ms) and spatial ( mm) proximity to one another. As Figure 2.6 illustrates, the thermal effects of interrogating multiple spatial locations 28

45 are not directly cumulative. For higher tissue absorptions, the maximum temperature increase that occurs is located in the near field (e.g., at a depth of.2 cm from the tissue surface for α =. db/cm/mhz). The near-field location of this peak heating is a function of the lateral distribution of heat in an interrogation and the spacing of adjacent interrogations. Therefore, it is important to monitor heating at locations other than the focal depth for two-dimensional ARFI imaging, and these models can be used to determine the locations of maximum heating through time for different imaging implementations. By using a smaller aperture (Figure 2.5) in more absorbing tissues, there is a reduction in the lateral spatial overlap of thermal profiles, and the resulting cumulative heating is reduced (Figure 2.7). For lower tissue absorptions, the temperature effects remain confined to the focal region since there is less near-field heating and less overlap of adjacent heating profiles. Of course, use of a lower ultrasound frequency in the more absorbing tissues would also reduce near-field heating. As Figure 2.6(c) illustrates, for large enough ROIs, a plateau in maximum temperature is reached as the cooling effects of early interrogations balance the heating effects of later interrogations. For higher tissue absorptions, a larger ROI is necessary for this plateau to be reached since the lateral extent of the near-field heating is greater than with lower tissue absorptions. These plateaus in heating also represent when the location of the maximum heating has reached its steady-state location in the near-field. Reduction of the temperature increase during two-dimensional ARFI imaging can be achieved by increasing the spacing between adjacent interrogation locations. This is most clearly demonstrated with the unfocused transducer configuration in which the lateral extent of heating at a single location is uniform through all tissue depths (Figure 2.5). Figure 2.7 illustrates how an increase in the spacing can mini- 29

46 mize the heating between adjacent interrogation locations and reduce the maximum temperature increase associated with two-dimensional ARFI imaging. For real-time implementation of two-dimensional ARFI imaging, a trade-off exists between frame rate (Figure 2.), spacing of adjacent lines (Figure 2.9), and the extent of the ROI (Figure 2.6). The general trends shown in Figures 2.9 and 2. demonstrate that decreasing the frame rate and increasing the spacing between adjacent lines lowers the steady-state temperature that would be achieved in real-time ARFI imaging. Maintaining temperature increases of less than 6 C is possible, however the frame rate, lateral resolution, and duration over which real-time ARFI imaging is performed must best selected accordingly ARFI Shear Wave and M-mode Imaging For ARFI shear wave imaging, the benefit of spatial offset for cooling between successive ARFI interrogations is lost since multiple shear waves must be generated at a single spatial location (in the absence of parallel processing for tracking). Cooling, therefore, only occurs during the temporal delay between successive ARFI interrogations. Tissues experience a greater thermal burden during ARFI shear wave imaging, especially for lower tissue absorptions where thermal energy is concentrated in smaller volumes of tissue. As would be expected, the distribution of heat for shear wave imaging is similar to that for single ARFI interrogations for a given tissue absorption (Figure 2.2). The thermal burden that the tissue experiences in shear wave imaging would be greatly reduced if an ARFI imaging scanner were capable of parallel receive processing, where shear waves generated from a single ARFI interrogation could be tracked across an entire field of view [5]. Multiple instances of shear wave generation at the same spatial location would not be necessary, and temperature increases would be equal to those for a single ARFI interrogation (Figure 2.2). 3

47 ARFI M-mode imaging is used in gated interrogation of blood vessels throughout the cardiac cycle. ARFI M-mode imaging is analogous to shear wave imaging, however, much longer delays occur between interrogations. For ARFI M-mode imaging, the temperature increase is related to the amount of time between successive ARFI interrogations and the amount of heat lost to blood perfusion (which is not included in these models), especially when imaging blood vessels Thermal Effects on Displacement Tracking ARFI imaging uses correlation-based methods to track tissue displacements as small as.2 µm over.3 mm search windows with approximate volumes of.72 mm 3 (.3 mm x.8 mm x.3 mm). The displacements due to thermal expansion are on the order of hundredths of a micron for a single ARFI interrogation (Figure 2.2(a)). The majority of this displacement occurs in the lateral direction, and will not impact ARFI displacement measurements because a new reference line is transmitted every 5 ms for all modes of ARFI imaging. Therefore, cumulative displacements due to thermal expansion over long periods of time are not a concern. Over and underestimation of displacement by a few microns can occur due to changes in sound speed from ARFI-generated tissue heating (Figure 2.2(c)). However, experimental data to date does not appear to demonstrate appreciable heating artifacts. This could be related to an overestimation of c T or, more likely, an overestimation of ARFI in situ intensities determined using linear extrapolation of small-signal derated fields [55]. Changes in sound speed in the lens will also affect displacement estimates. These artifacts would appear as constant displacement offsets at all depths as opposed to the depth dependent accumulation of displacement shown in Figure 2.2(c). Heating artifacts can be corrected using filters in ARFI image processing algorithms. 3

48 2.5.5 Limitations of the Thermal Model Mechanical Waves The models that have been presented have assumed that all energy that is absorbed at a given location will lead to tissue heating at that location. However, with the transfer of momentum associated with radiation force, strain energy is stored in the radiation-force induced tissue displacement fields. After the high-intensity acoustic pulses are applied, the tissue relaxes, and shear and dilatational displacement waves propagate away from the focal region. As these waves disperse, they are attenuated by thermo-viscous mechanisms. Therefore, this mechanical energy is converted to thermal energy, but for materials with low thermo-viscous attenuation, this is not in the focal region. Therefore, the thermal profiles generated by ARFI imaging should be considered functions of tissue stiffness and viscosity, properties that affect mechanical wave velocity and absorption, but these parameters have not been incorporated into these models. These effects will be addressed in future work. Nonlinear Propagation Nonlinear propagation, and its effects on acoustic radiation force, has been extensively treated in the literature [55, 77, 78, 83, 88]. For the same temporal average intensity, a wave with higher pressure amplitude and shorter pulse duration generates a larger radiation force than does a lower amplitude, longer duration acoustic wave. This larger radiation force is due to the higher-order harmonics generated by nonlinear propagation, which result in an increase in energy absorption related to a finite amplitude absorption parameter [23, 55, 77, 78, 83]. In addition to increasing the force magnitude, nonlinear propagation is generally associated with a translation of the location of the peak in the force field and, therefore, the heating closer to 32

49 the transducer. However, this effect decreases with increasing center frequency and finite amplitude absorption that exists in soft tissue as compared to aqueous solutions [55]. ARFI imaging at a center frequency of 7.2 MHz in soft tissues provides an effectively high tissue absorption, which decreases the described effects of nonlinear propagation and, instead, results in a reduction of the linearly estimated focal intensities [55]. Therefore, linear models of tissue intensity fields provide leading order estimates of ARFI imaging tissue heating, which can be refined using nonlinear propagation corrections [55]. These corrections will be more significant for applications of ARFI imaging at greater tissue depths with lower ultrasonic frequencies. For these implementations, nonlinear simulation packages with finite amplitude absorption parameters will need to be utilized to model the tissue intensity fields. Transducer Heating In addition to the temperature increases created by absorption of the ultrasound beam in radiation force imaging, transducer heating may become a concern, but has not been included in these models. While the transducer heating associated with a single ARFI interrogation is assumed to be negligible due to the short insonification time, the cumulative insonification time for two-dimensional real-time and shear wave ARFI imaging warrants further study. Transducer heat can appreciably conduct to two centimeters below the tissue surface during continuous imaging [87] and will have a cumulative effect when coupled with heating due to tissue absorption. Finite Tissue Volume These models simulate a finite volume of tissue with insulating boundaries that do not allow for thermal energy to escape into the environment or surrounding structures later in time. These boundary conditions generate artifacts in the long- 33

50 duration, real-time simulations where heat has time to appreciably conduct to the model boundaries. The models also do not incorporate a radiation boundary at the tissue surface, providing only a conductive mechanism for heat transfer in the lateral and elevation dimensions in the near field. This boundary condition, again, may artificially increase the temperature in the near-field, especially in the realtime models for high tissue absorptions where the maximum heating occurs within a centimeter of the surface. All of the implemented boundary conditions simulate worst-case heating that could occur during ARFI imaging Applications to Other Radiation Force-Based Imaging Modalities The models and results that have been presented in this chapter can be easily extended to the heating associated with other radiation force-based imaging modalities due to energy absorption by soft tissue. Modalities that use lower center frequencies will need to take into account the effects of nonlinear propagation, while modalities that involve contrast agents or imaging of tissue-bone interfaces will need to included heating effects associated with reflective sources. 2.6 Conclusions Acoustic radiation force-based imaging methods appear to have clinical potential for characterizing the local mechanical properties of tissue. The models presented in this chapter characterize the heating that is associated with ARFI imaging as currently implemented as a function of different soft tissue absorptions and focal configurations. The temperature rise at the focus is greater for less absorbing tissues, while more absorbing tissues distribute the thermal energy over greater volumes in 34

51 the near field. By adjusting focal configurations and/or frequency, near-field heating and overlap of adjacent beams can be controlled. Creation of two-dimensional ARFI images is not limited by thermal effects because sufficient cooling occurs spatially and temporally between adjacent ARFI interrogations. Frame rate, lateral resolution, size of the region of interest, and duration of real-time, two-dimensional ARFI imaging are parameters that can be optimized in future development of ARFI beam sequences to control tissue heating. The thermal burden associated with ARFI shear wave imaging can be significantly reduced with parallel receive processing. The effects of thermal expansion are negligible in ARFI image reconstruction, but ARFI images can suffer from artifacts due to sound speed changes in the transducer and the tissue. 2.7 Acknowledgments This work was supported by NIH grant 8 R EB232, the Whitaker Foundation, and the Medical Scientist Training Program grant T 32 GM-77. The authors would also like to thank Siemens Medical Solutions USA, Inc. Ultrasound Division for their technical assistance and Altair Computing, Inc. for providing mesh generation software. 35

52 Chapter 3 Experimental Studies of the Thermal Effects Associated with Radiation Force Imaging of Soft Tissue The work presented in this chapter was published in Ultrasonic Imaging, 2: - 4, 24 [69]. 3. Introduction The use of acoustic radiation force to interrogate the mechanical properties of soft tissues is becoming a widely investigated research area. In general, acoustic radiation force is used to excite tissue, and the tissue response is monitored using either ultrasonic or magnetic resonance methods. Vibro-acoustography uses frequency-shifted, confocal acoustic beams to generate oscillatory radiation force within tissues, and the tissue response is monitored with either a hydrophone [2,33] or ultrasonic methods [48]. The Kinetic Acoustic Vitreoretinal Examination (KAVE) method uses radiation force to generate steady-state stresses within soft gels and the vitreous of the eye and ultrasonic displacements are monitored to characterize the steady-state response of these materials [94]. Supersonic imaging [5] and Shear Wave Elastic- 36

53 ity Imaging (SWEI) [3, 78] monitor the shear waves generated from short duration, acoustic radiation force to characterize the shear modulus of the medium. Acoustic Radiation Force Impulse (ARFI) imaging, which is the focus of this chapter, uses a commercial diagnostic scanner to generate multiple, short duration radiation forces to interrogate a two-dimensional region of interest (ROI), and monitors the tissue response using ultrasonic correlation-based methods [3, 56, 58, 63]. Similar work is also under investigation using a High Intensity Focused Ultrasound (HIFU) transducer to generate radiation force in conjunction with a piston to track displacement in order to monitor ablation treatment [5]. In addition to soft tissue imaging, radiation force is being used to manipulate ultrasonic contrast agents in vitro and in vivo [26, 43]. The generation of micron-scale displacements using acoustic radiation force in tissue requires the use of high-intensity acoustic beams, and the associated heating must be evaluated to ensure safety in performing ARFI imaging in vivo. Finite Element Method (FEM) models of heating due to ultrasonic energy absorption by soft tissue under a variety of ARFI imaging configurations have been presented in Chapter 2 [7]. The purpose of this work is to experimentally characterize the temporal and spatial heating associated with ARFI imaging in porcine muscle to validate these FEM models. Since thermocouple measurements are difficult to perform, these models can then be used to estimate the tissue heating during ARFI imaging using different transducers, beam spacing and focal configurations in tissues with varying material properties. Thermocouples are also used to measure transducer face heating, which has not yet been included in the FEM models. 37

54 3.2 Background 3.2. Acoustic Radiation Force Impulse (ARFI) Imaging Acoustic Radiation Force Impulse (ARFI) imaging is a radiation force-based imaging method that provides information about the local mechanical properties of tissue [6]. ARFI imaging uses short duration (< ms), high-intensity acoustic pulses to generate localized displacements in tissue and the dynamic tissue response is monitored using ultrasonic correlation-based methods [59]. Images of twodimensional regions of interest (ROI) are generated by sequentially interrogating multiple lateral locations, as is done in color Doppler imaging. As the high-intensity pulses propagate through tissue, acoustic energy is absorbed or scattered by tissue in the region of excitation (ROE) that results in the generation of acoustic radiation force, tissue displacement and tissue heating. For more details about the implementation and potential clinical applications of ARFI imaging, the reader is referred to Nightingale et al. [56] Overview of Thermal Modeling A thorough presentation and discussion of the thermal FEM model results can be found in Chapter 2. In general, modeling was performed using a two-step approach: () the spatially distributed acoustic intensity field corresponding to the experimental transducer was simulated using Field II [46], and the associated heat source functions were computed (Equation 2.4); (2) finite element methods were used to determine the tissue thermal response to these heat source functions. The temperature increases generated by the absorption of ultrasonic energy in soft tissue were solved for using the linear bio-heat transfer equation (Equa- 38

55 tion 2.) [54, 65, 74]. Blood perfusion effects were neglected in the models to solve for worst-case heating conditions. As presented in Chapter 2, the models demonstrate that the heating associated with ARFI imaging at the focus is greater in less absorbing materials, while more absorbing materials distribute the thermal energy over a greater volume of tissue in the near field. The focal configuration of a transducer can be described by the F/# (Equation 2.5). Near-field heating and overlap of adjacent beams in two-dimensional ARFI imaging can be controlled by adjusting focal configurations and/or frequency. Creation of two-dimensional ARFI images is not limited by heating because sufficient cooling occurs spatially and temporally between adjacent ARFI interrogations. Frame rate, lateral resolution, size of ROE and duration of imaging are parameters that can be optimized in the development of ARFI beam sequences to limit tissue heating. An assumption of linear acoustic wave propagation was made in the FEM models. Nonlinear propagation, and its effects on acoustic radiation force, have been extensively treated in the literature [55, 77, 78, 83, 88]. Nonlinear propagation is generally associated with a translation of the peak heating location closer to the transducer; however, this effect decreases with increasing center frequency and finite amplitude absorption that exists in soft tissues as compared with aqueous solutions [55]. ARFI imaging at a center frequency of 7.2 MHz in soft tissue provides a relatively high tissue absorption that decreases the impact of nonlinear propagation and results in a reduction of the linearly estimated focal intensities [55]. Linear models of intensity fields in tissue provide leading-order estimates of tissue heating with ARFI imaging, which may be improved using nonlinear propagation corrections [55]. 39

56 3.3 Methods 3.3. Measurement of Tissue Heating Many groups have used thermocouples to measure the heating associated with the absorption of ultrasonic energy [2, 8, 92]. Review of the literature suggests the optimal method for using thermocouples is to place them perpendicular to the beam axis and then adjust them to the position of peak temperature increase [29]. These groups determined the baseline temperature for the first few seconds before firing the ultrasonic sequence of interest [36, 37]. Type-T (Cu/Cu-Ni), 36-gauge thermocouples (Omega Engineering, Stamford, CT) were used to measure tissue temperatures during ARFI imaging of porcine muscle. These thermocouples have a response time on the order of. s, with a noise floor of.4 C and an average signal-to-noise ratio (SNR) of 5.5 db. Porcine muscle was chosen because it has well-characterized thermal properties similar to those reported for clinically relevant human soft tissues [7,5]. The thermocouple data was recorded using a 6-bit data acquisition system with a sampling frequency of 5 khz (SuperLogics, Waltham, MA) that was pre-calibrated with cold junction compensation (CJC) and shielded from the environment with an aluminum cover. To map the heating associated with ARFI imaging in three spatial dimensions, a single thermocouple was placed between two pieces of porcine muscle (Figure 3.), each with a thickness of approximately 2 mm, without any ultrasonic gel in order to prevent shifting of the thermocouple while mechanically translating the transducer. The transducer was acoustically coupled to the porcine muscle using ultrasonic imaging gel. The thermocouple junction was coated with three layers of lacquer to prevent shorting of the junction in the porcine muscle; more layers were not applied to minimize adverse effects on the thermal accuracy and response time of the thermocouple. 4

57 To ensure that the thermocouple was located in the focal zone of the acoustic beam, thermal peaking was performed in all three spatial directions, under the assumption that the peak heating for ARFI imaging occurs in the focal zone. Trying to achieve this placement while relying exclusively on B-mode images was difficult due to lack of tracking in the elevation plane and the inability to distinguish the thermocouple from the porcine muscle speckle patterns and intensities, leading to poor thermocouple boundary definition. The experiments characterizing the heating profiles in each of the three dimensions were performed separately. Thus, the axial depth of the peaked thermocouple varied between the experiments. Therefore, the thermocouple was thermally peaked by measuring temperature increases during ARFI imaging sequences, while stepping the transducer in. mm increments in elevation, laterally, and axially using a servo-controlled translation stage. Once thermally peaked, the thermocouple s axial position was approximated using both B-mode and ARFI imaging displacement data with an accuracy of ± mm. Therefore, a range of axial positions were modeled and matched with experimental data. Transducer Elevation Lateral Axial Focal Depth Region of Excitation (.56 mm) Porcine Muscle Thermocouple (.4 mm) Figure 3.: Schematic diagram illustrating the experimental setup that was used during the thermocouple measurements. The coordinate axes for the three spatial dimensions are shown relative to the focal point, with the elevation plane being into and out of the page. The thermocouple diameter was approximately.4 mm, while the width of the ROE was approximately.56 mm. In this setup, the active aperture was either electronically or mechanically translated relative to the fixed thermocouple position. 4

58 Once the thermocouple was thermally peaked in all three spatial dimensions, it was held in a fixed position while the active aperture was either electronically or mechanically translated relative to the thermocouple position. Spatial maps of heating were then reconstructed from these individual measurements. This experimental setup was chosen () because trying to place multiple thermocouples in porcine muscle, with known relative spacing between them in all spatial dimensions, would be excessively challenging, and (2) thermal artifacts introduced by the presence of thermocouples in the propagation path of the ultrasonic waves are minimized by having only a single thermocouple. A more thorough discussion of these artifacts is provided in the Section of this chapter. Elevation heating was characterized by mechanically translating the transducer from.4 mm out of plane, over the thermocouple tip at the focal point, and then.5 mm past the focal point in. mm increments. The thermocouple was then reset between the two pieces of porcine muscle, thermally re-peaked at the focal point and lateral heating was characterized by electronically walking the aperture across the transducer in.94 mm increments. Temperature measurements were repeated six times at each location without changing the experimental setup between trials. The temperature in the ROE was allowed to reach a steady-state value before acquiring the baseline temperature for the next trial. Axial heating was characterized by changing the acoustic focal depth in.5 mm increments, ranging from - 5 mm, while changing the number of active elements in the aperture to maintain a constant F/.3 focal configuration. Again, temperature measurements were repeated six times without changing the experimental setup for each focal depth and the ROE was allowed to reach a steady-state value before acquiring the baseline temperature for the next trial. 42

59 3.3.2 ARFI Imaging Configuration A 75L4 linear array, with a center frequency of 7.2 MHz, on a Siemens SONO- LINE Elegra TM Scanner (Siemens Medical Solutions USA, Inc., Ultrasound Division, Issaquah, WA), was used for ARFI imaging in porcine muscle, using a constant F/.3 focal configuration. In order to completely encompass the thermocouple (.4 mm diameter) within the ROE while maintaining adequate lateral heating resolution, seven lateral locations, spaced.47 mm apart, were interrogated to achieve an ROE with a total lateral extent of.56 mm at the focal depth (assuming a lateral ultrasound beamwidth of.278 mm). This experimental setup is illustrated in Figure 3.. Each lateral location was interrogated with six 28 µs high-intensity pushing pulses, with 5 µs between each pushing pulse, for an effective duty cycle of 5.7% and a total insonification time of.92 ms in each lateral location. The approximate peak in situ intensity for this configuration was, W/cm 2 [7]. This ultrasound beam configuration represents a typical ARFI imaging configuration, except the lateral locations are spaced four times closer to produce temperature increases greater than the thermocouple noise floor (.4 C) Transducer Face Heating Heating of the transducer face was characterized for a VF-5 linear array, with a center frequency of 6.67 MHz, connected to a Siemens SONOLINE Antares TM Scanner. A thermocouple was centered on the transducer face in a thin gel path between the transducer and a tissue-mimicking phantom consisting of gelatin, graphite, and water [42]. No active cooling mechanisms were in place during ARFI imaging. Temperature increases were measured during acquisition of two-dimensional ARFI data sets consisting of 5 ARFI interrogations per frame, spanning 2 mm laterally, for 43

60 varying transmit powers and pulse durations per interrogation. Pulse durations are much longer than those used for diagnostic Doppler imaging (3 µs vs. 2 µs), and transmit powers are represented as percentages of the maximum transmit power the scanner can produce and are comparable to transmit powers used during Doppler imaging. These parameters yield comparable phantom responses to ARFI excitation to those implemented with the 75L4 transducer on the Siemens Elegra TM scanner Data Processing A 2 ms ( sample) running average was applied to all raw thermocouple data. The baseline temperature was defined for each measurement as the mean temperature over at least a one second window before ARFI excitation of the porcine muscle [36, 37]. The maximum temperature increase was determined once the baseline temperatures were removed from each dataset FEM Models A thorough description of the FEM models developed for modeling ARFI imaging heating is given in Chapter 2. Comparison between the FEM model and thermocouple data was performed over a range of parameters that could vary during the experiments, including thermal diffusivity and the axial depth of the thermocouple during data acquisition. The absorption of the porcine muscle was approximated to be.7 db/cm/mhz [6], which is within the range reported in the literature [5]. The thermal material properties used in the matched simulations are shown in Table

61 Table 3.: Thermal material properties of biological tissue [7, 5]. Thermal Material Property Value K mw/cm/ C c v 2.76 J/g/ C ρ. g/cm Results 3.4. Focal Point Cooling Figure 3.2 shows a normalized experimental cooling curve at the focal point, at a depth of 2 mm, as compared with the FEM data for three different thermal conductivities (solid, dashed and dotted lines). Note that the most conductive material modeled matches the experimental data the best, although it is greater than the range of thermal conductivities quoted in the literature. All three thermal conductivities cool to less than 2% of the maximum temperature increase within one second. As demonstrated in Figure 2.3, the cooling behavior at the focal point is independent of the soft tissue absorption Spatial Maps of Heating Figure 3.3(a) shows the mean, ± one standard deviation, over six trials of the maximum measured temperature increases at lateral positions relative to the focus. The vertical dashed lines in the figure indicate the lateral extent of the ARFI ROE (.56 mm). Note that the maximum temperature increases have decayed to approximately 2% of the peak temperature increase in the porcine muscle within.25 mm of the ROE (dashed lines) and within.4 mm of the focus. Also note that the maximum temperature increase that occurs in the porcine muscle at the focal point is approximately.8 C, while the maximum temperature in the FEM models 45

62 Normalized Temperature Rise T/C Data FEM Data (K = 4.64 mw/cm/ C) FEM Data (K = 6.5 mw/cm/ C) FEM Data (K = 9.28 mw/cm/ C) Time (s) Figure 3.2: Thermocouple data (asterisks) of the cooling behavior at the focal point (depth of 2 mm) as compared with the FEM data (solid, dashed and dotted lines) for three different thermal conductivities. Note that the temperature has decayed to less than 2% of the maximum value within s and that the most conductive material (solid line) matches the thermocouple data the best. was.4 C. Figure 3.3(b) shows the normalized experimental temperature increases (asterisks) at lateral positions relative to the focus (i.e., the center of the ROE) as compared to the corresponding FEM data at a depth of 2. mm (solid line)..8 Temperature Rise (deg C) Normalized Temperature Rise T/C Data FEM Data (2 mm) Lateral Position from Focus (mm) (a) Lateral Thermocouple Data.5.5 Lateral Position from Focus (mm) (b) Comparison with FEM Data Figure 3.3: Figure 3.3(a) shows the experimentally-measured heating profile (mean ± one standard deviation of six trials) in the lateral dimension relative to the focal point. Note that the lateral width of the ARFI ROE is.56 mm, as delimited by the vertical dashed lines, centered about the focus. Figure 3.3(b) compares the mean thermocouple data (asterisks) in the lateral dimension with the FEM data at a depth of 2 mm (solid line) for K = 4.6 mw/cm/ C and α =.7 db/cm/mhz. 46

63 Figure 3.4(a) shows the mean, ± one standard deviation, over six trials of the maximum measured temperature increases at elevation positions relative to the focus, with the elevation extent of the ARFI ROE (.8 mm) indicated by the vertical dashed lines. Note that the temperature increases have decreased to less than 2% of the peak temperature increase within.5 mm of the focus in elevation. Figure 3.4(b) shows the normalized thermocouple data (asterisks) at elevation positions relative to the focus as com pared to the corresponding FEM data at a depth of. mm (solid line). Temperature Rise (deg C) Normalized Temperature Rise T/C Data FEM Data ( mm) Elevation Position from Focus (mm) (a) Elevation Thermocouple Data.5.5 Elevation Position From Focus (mm) (b) Comparison with FEM Data Figure 3.4: Figure 3.4(a) shows the measured heating profile (mean ± one standard deviation of six trials) for the elevation dimension. The elevation ROE (.8 mm) is delimited by the vertical dashed lines. Figure 3.4(b) compares the mean thermocouple data (asterisks) in the elevation dimension with the FEM data at a depth of mm (solid line) for K = 4.6 mw/cm/mhz and α =.7 db/cm/mhz. Figure 3.5(a) shows the mean, ± one standard deviation, over six trials of the maximum measured temperature increases at an axial depth of 2.3 mm for varying focal depths with a constant F/.3 focal configuration. The maximum measured temperature increase was 2. C when the transducer was focused at the thermocouple depth of 2.3 mm. The fact that this maximum temperature increase differs from the lateral and elevation measurements is not surprising since the axial mea- 47

64 surements were made in a different region of the porcine muscle. Figure 3.5(b) shows the comparison between the mean thermocouple data over six trials (asterisks) and the corresponding FEM data (solid line) for the axial heating experiments. 2.2 Temperature Rise (deg C) Focal Depth (mm) (a) Axial Thermocouple Data Normalized Temperature Rise T/C Data FEM Data (2.3 mm) Focal Depth (mm) (b) Comparison with FEM Data Figure 3.5: Figure 3.5(a) shows the maximum temperature increases (mean ± one standard deviation over six trials) that occur at an axial depth of 2.3 mm for different focal depths for a constant F/.3 focal configuration. Figure 3.5(b) compares the normalized mean axial temperature measurements over six trials (asterisks) with the FEM data (solid line). Note the poor agreement between the FEM model and the experimental data when the focal point is deeper than the thermocouple depth Transducer Face Heating Table 3.2 shows the temperature increases measured at the center of the transducer face as a function of the transmit power and pulse duration for each ARFI interrogation. These configurations match those used for ARFI imaging clinical trials in the breast. Note that the temperature increase does not exceed C for any of these configurations. 48

65 Table 3.2: Transducer face heating associated with 2-D ARFI data (5 lateral locations, F/.3) acquired on the Siemens Antares TM Scanner, using a single excitation pulse of varying magnitude and duration. Transmit Power (%) Pulse Length (ms) Temperature Rise ( C) Discussion 3.5. Transient Heating Behavior Figure 3.2 illustrates agreement in the cooling behavior between the FEM model and the thermocouple data at the focal point. Both the experimental and FEM data cool to less than 2% of the maximum temperature increase within one second. The best match between the model and the experimental data occurs for a thermal conductivity greater than those quoted in the literature for porcine muscle. One source of error present in this measurement is that the thermocouple itself has greater thermal conductivity than the porcine muscle, which will create an artificially high overall thermal conductivity near the thermocouple junction. Overall, there is still good agreement between the model and experimental data Elevation and Lateral Heating Figures 3.3 and 3.4 show that the maximum temperature increase generated in the porcine muscle by a 7 line, F/.3 focal configuration ROE with approximate dimensions.56.8 mm in the lateral and elevation dimensions. The models underestimate the heating magnitude by 2-3%. This may be attributed to artifacts associated with the presence of the thermocouple in the propagation path of the acoustic waves, as discussed in Section of this chapter. This may also be due 49

66 to: () an underestimation of the in situ intensities generated during ARFI imaging; (2) the effects of nonlinear propagation, which would cause an increase in the amount of near-field energy absorbed by the tissue; and/or (3) an inaccurate estimation of the thermal diffusivity for porcine muscle. The peak temperature increases that were measured at the focal point in the porcine muscle (between.8 and 2. C) are higher than those that would be achieved in typical ARFI imaging due to the closer lateral spacing of ARFI interrogations that was purposely implemented to exceed the noise floor of the thermocouples. Both the predicted and measured temperature increases in the porcine muscle are below the 6 C temperature increase limit that is considered safe for clinical imaging [44]. The greatest uncertainty in matching the FEM model data with the thermocouple data was the axial position of the thermocouple during the experiments. Since all three spatial experiments were performed after resetting the thermocouple between the pieces of porcine muscle, the thermocouple position varied and could not be considered constant between experiments. The best estimates of the thermocouple position were made using B-mode and ARFI displacement data after the thermocouple was thermally peaked Axial Heating and Thermocouple Limitations The method for characterizing the heating in the axial dimension was chosen after iterating on several alternative methods. Translation of the transducer relative to the thermocouple, as was done in the elevation measurements, led to compression of the thermocouple junction leads and inaccurate temperature measurement since the thermocouple junction gap would change. Mechanically translating the transducer axially also caused the thermocouple to slip out of plane in the elevation and lateral dimensions, causing inaccurate representation of the axial heating profiles. For these 5

67 reasons, it was advantageous to keep the thermocouple at a fixed depth and instead characterize the heating at a fixed depth as the acoustic focal depth was changed. As shown in Figure 3.5, there is good agreement between the normalized thermocouple data and the FEM data for focal depths shallower than the thermocouple depth of 2.3 mm. However, when the focal depth is deeper than the thermocouple (> 2.3 mm), the measured temperatures are greater than the predicted FEM values. This is likely because the presence of the thermocouple in the tissue impacts the heating profiles [2]. Two interactions between ultrasonic waves and the thermocouple are possible: viscous heating and reflection. Viscous heating arises from the passage of the acoustic wave around the thermocouple, while reflection occurs off of the front surface of the thermocouple junction, doubling the heat generated near the junction [2]. Van Baren et al. [92] stopped the ultrasonic pulses during thermocouple measurements to avoid the artifacts created by these interactions; however, for the small temperature rises and quick decay times associated with ARFI imaging, this technique was not feasible. The presence of the thermocouple in the propagation path of the ultrasound waves, therefore, may lead to artificially high temperature measurements Transducer Face Heating Transducer face heating, as shown in Table 3.2, is proportional to the duration and transmit power of the excitation during ARFI imaging. The temperature increases on the transducer face, which are less than. C for all configurations presented, are safe for clinical application and will not damage the transducer. However, in order to perform ARFI imaging over greater time durations or to create greater displacements in stiffer or more attenuating materials, active cooling mechanisms could be introduced to reduce the thermal burden on the transducer. 5

68 3.5.5 Evaluation of Thermal Safety Evaluating the thermal safety of new ARFI beam sequences requires a combination of modeling and experimental efforts. First, intensity measurements must be made using a hydrophone to characterize the I spta for a given transducer and transmit power/duration. Next, thermocouple measurements should be made to measure the heating at the focal point for a typical focal configuration. Finally, these results are then imported and matched with the FEM models, which are then used to evaluate the heating associated with different beam spacings, focal configurations and material properties (e.g., thermal diffusivity, κ, and ultrasonic absorption, α) for the range of values appropriate for the specific clinical application. Thermocouples should also be used to characterize the heating on the transducer face for these beam sequences. 3.6 Conclusions The thermocouple experiments presented in this chapter have validated the transient thermal behavior and the spatial conduction of heat in the FEM models of soft tissue heating associated with ARFI imaging that were presented in Chapter 2. Performing experimental measurements of the spatial heating in soft tissue during ARFI imaging for many different transducers, beam spacing and focal configurations would be prohibitive in terms of time and effort. Therefore, having this validated thermal model will allow for more efficient and accurate estimates of in vivo heating during ARFI imaging. In order for these models to be accurate, however, precise estimates of the peak in situ acoustic intensities associated with a given transducer configuration must be made. Transducer heating has been demonstrated to be below C for current clinical applications. Overall, the FEM models and the thermocou- 52

69 ple data demonstrate that ARFI imaging of soft tissue as currently implemented is safe, although thermal responses must be monitored when developing new acoustic radiation force sequences for different clinical applications. 3.7 Acknowledgments This work was supported by the NIH grant 8 R EB232, the Whitaker Foundation, and the Medical Scientist Training Program grant T 32 GM-77. The authors would also like to thank Amy Congdon for her assistance in developing ARFI imaging beam sequences and Siemens Medical Solutions USA, Inc. Ultrasound Division for their technical assistance. 53

70 Chapter 4 A Finite Element Method Model of Soft Tissue Response to Impulsive Acoustic Radiation Force The work presented in this chapter has been accepted for publication in IEEE Transactions on Ultrasonics, Ferroelectronics, and Frequency Control [72]. 4. Introduction Acoustic radiation force-based imaging modalities are currently being studied by many groups including: vibro-acoustography, Kinetic Acoustic Vitroretinal Examination (KAVE), supersonic imaging, Shear Wave Elasticity Imaging (SWEI), and High Intensity Focused Ultrasound (HIFU) ablation monitoring. Vibro-acoustography uses frequency-shifted, confocal acoustic beams to generate an oscillating radiation force within tissue, and the tissue response is monitored with either a hydrophone [2, 2, 33] or ultrasonic methods [48]. The KAVE method uses radiation force to generate steady-state stresses within soft gels and the vitreous of the eye, and ultrasonic displacements are monitored to characterize the response of these materials [94]. Supersonic imaging [6] and Shear Wave Elasticity Imaging (SWEI) [3,78] 54

71 monitor the shear waves generated by short duration acoustic radiation force to characterize the shear modulus of the medium. HIFU transducers are also being studied to generate radiation force in conjunction with a piston to track displacements in order to monitor ablation treatment [5]. In addition to soft tissue imaging, radiation force is being used to manipulate ultrasonic contrast agents in vitro and in vivo [26, 43, 8]. Acoustic Radiation Force Impulse (ARFI) imaging, which is the focus of this chapter, uses standard, diagnostic ultrasound scanners to generate localized impulsive acoustic radiation forces in tissue. The localized, micron-scale tissue displacements generated by radiation force are tracked through time using ultrasonic, correlation-based methods. The dynamic response of tissue to this impulsive radiation force reflects its mechanical properties, as will be demonstrated in Chapter 6. ARFI imaging is unique from other radiation force modalities in that a spatiallylocalized distribution of radiation force is applied for a short duration (< µs), and the dynamic response of tissue is monitored in the region of excitation (ROE), representing the mechanical impulse response of tissue in that region [56, 58, 63]. Spherical lesions that are stiffer than their surroundings exhibit characteristic dynamic responses to impulsive acoustic radiation force excitation that distinguish them from the surrounding tissue. This chapter presents a Finite Element Method (FEM) model that accurately simulates the dynamic response of homogeneous tissues of varying stiffnesses, with and without stiffer, spherical inclusions, to focused applications of impulsive, acoustic radiation force. The dynamic response of this model has been validated with experimental ARFI data from calibrated tissue-mimicking phantoms. Section 4.2 presents a derivation of the governing elastodynamic equations, along with the derivation of acoustic radiation force in viscoelastic tissues, which are solved us- 55

72 ing the FEM. Section 4.3 provides an overview of the implementation of the FEM to solve the elastodynamic equations, while Section 4.4 presents a comparison between the dynamic FEM results and experimental data from calibrated phantoms, and demonstrates the dependence on the dynamic behavior of tissue to ARFI excitations on ultrasonic attenuation, density, Poisson s ratio, and elasticity. 4.2 Background A list of the notation used throughout this chapter can be found in Appendix C. The Einstein convention for tensor subindices will be used throughout this chapter (i.e., repeated subindices imply summation over that index) Viscoelastic Dynamics To a first-order approximation, at low excitation frequencies (< khz), soft tissues can be described as linear, elastic, isotropic solids. The balance of linear momentum for such a material can be expressed as: σ ij,j + ρb i = ρf i, (4.) where σ ij,j [N/m 3 ] represents the divergence of the stress tensor, ρ [kg/m 3 ] is the material density, b i [m/s 2 ] represents an external steady-state acceleration (e.g., gravity), and f i [m/s 2 ] represents an externally applied acceleration (e.g., radiation force). The constitutive equation that relates stress to strain in a linear, elastic solid is given by: σ ij = λ δ ij + 2µε ij, (4.2) 56

73 where λ and µ represent the Lame constants for the material [49]. The strain (ε ij ) is defined as the symmetric displacement gradient using the following relationship: ε ij = 2 (u i,j + u j,i ), (4.3) where u i represents particle displacement. Note that both σ ij and ε ij are symmetric tensors. represents the material dilatation given by: = u i,i = ε ii. (4.4) Time-independent body forces, such as gravity, can be neglected in Equation 4. because this linear equation can be decomposed into elastostatic and elastodynamic components that are separable [49]. To derive an expression for displacement in the absence of external body forces (b i ), the constitutive relationship (Equation 4.2) is substituted into the balance of linear momentum expression (Equation 4.), with b i = and f i = ü i, to yield: σ ij,j = λ,i + µ (u i,jj + u j,ij ), (4.5) ρü i = (λ + µ) u j,ji + µu i,jj. (4.6) The displacement equation (4.6) can be written in vector form as []: (λ + µ) ( u) + µ 2 u = ρ u. (4.7) The displacement field in a solid can be decomposed such that []: u = ψ + W, (4.8) 57

74 where ψ represents the dilatational (scalar, time-dependent, volume change) components of displacement that occur in the direction of wave propagation (longitudinal), and W represents the equivoluminal components of displacement that occur transversely to the direction of wave propagation. These scalar and vector displacement fields can be determined by using a Helmholtz decomposition []. Substituting Equation 4.8 into Equation 4.6 yields: [ ] [ ] (λ + 2µ) 2 ψ ρ 2 ψ + µ 2 2 W W ρ =. (4.9) t 2 t 2 The displacement components shown in Equation 4.9 are separable and each of them takes the form of a wave equation []. For example, in one direction (ˆx), these equations can be expressed as: = 2 ψ x 2 ψ 2 c 2 L dt, (4.) 2 = 2 W x 2 W 2 c 2 T dt, (4.) 2 (λ + 2µ) c L =, (4.2) ρ µ c T = ρ. (4.3) The constants c L and c T represent the longitudinal and transverse (shear) wave speeds respectively. Assuming constant tissue density, these equations reveal c L to be proportional to the material uniaxial modulus given by λ + 2µ, which in fluids is commonly approximated by the bulk modulus of the medium, and c T to be proportional to µ, where µ is the material shear modulus. The elastic wave speeds, c L and c T, are also commonly expressed in terms of the Young s modulus (E) and 58

75 the Poisson s ratio (ν) as follows []: c L = c T = ( ν)e ( + ν)( 2ν)ρ, (4.4) E 2( + ν)ρ. (4.5) In solids, wave attenuation occurs due to the viscous properties of materials. Discretely, viscoelasticity can be represented by a combination of Maxwell and Voigt lumped-parameter systems. In continua, the time-dependent behavior of the material can be represented by complex Lame constants, λ and µ. These frequencydependent material parameters tend to be empirically determined for soft tissue. For longitudinal waves established in a viscoelastic solid during ultrasonic wave propagation, Equation 4.2 becomes: c λ L = + 2µ. (4.6) ρ A solution to Equation 4. using the expression for the longitudinal wave velocity in Equation 4.6 has the form: ψ = Ae αx+j(ωt kx)ˆx, (4.7) where A is the original wave amplitude, and k represents the wave number (ω/c L ). The attenuation coefficient for the material, when excited at ultrasonic frequencies, is represented by α, and can be represented as a function of λ and µ, as determined empirically by the frequency-dependence of the material. Given these expressions, the particle velocities ( v = jωψ) can be represented by a sinusoid that is phase 59

76 shifted by 9 with respect to the displacement function: v = jωae αx+j(ωt kx)ˆx. (4.8) Linearly Viscous Fluids and Acoustic Radiation Force The generation of acoustic radiation force occurs through a transfer of linear momentum from an acoustic wave to the propagation medium. This process has been thoroughly described in the literature for both gases and fluids in which energy loss mechanisms are present, either through the absorption or reflection of acoustic energy [64]. The derivation of acoustic radiation force cannot be directly obtained from the stress-strain relationships for a purely elastic material since no energy loss mechanisms are described that would allow for a change in wave momentum. Also, at ultrasonic frequencies, soft tissues do not support shear stresses; thus, they are often described as viscous fluids. Therefore, expressions that describe radiation force in linearly viscous fluids are extended to viscoelastic solids herein. Biological tissues have viscoelastic properties that are strongly frequency-dependent: at low excitation frequencies (< khz), the materials can be regarded as purely elastic; and at ultrasonic excitation frequencies, acoustic wave attenuation is appreciable and leads to heating and generation of acoustic radiation force, so the tissues can be regarded as viscous fluids [47]. Linearly Viscous Fluids In a linearly viscous (Newtonian) fluid, the constitutive relationship can be expressed as: σ ij = pδ ij + λ f δ ij + 2µ f D ij, (4.9) 6

77 where D ij represents a rate of deformation tensor defined as: D ij = 2 (v i,j + v j,i ), (4.2) v represents the particle velocity, = D ii, λ f is a fluid Lame constant, and µ f represents the viscosity of the fluid. The latter two terms on the right-hand side of Equation 4.9 represent the viscous stress tensor and are dependent on the rate of deformation (D ij ); p is the scalar pressure that does not depend on the rate of deformation. To arrive at the Navier-Stokes equations of motion for an incompressible Newtonian fluid, the total particle acceleration is defined as [35]: a i = v i + v j v i,j, (4.2) where the first term represents the local particle acceleration, and the second term represents the convective acceleration of the particle. Substituting Equations 4.9 and 4.2 into the balance of linear momentum equation (Equation 4.) yields the Navier-Stokes equation [35]: p,i + µ f v i,jj + ρb i = ρ ( v i + v j v i,j ). (4.22) As was shown in the equations of dynamics for a viscoelastic solid, the solution for the velocity field created by acoustic wave propagation in a linearly viscous fluid (Equation 4.22) can also be represented by Equation 4.8. The dynamics of a viscoelastic solid can, therefore, be viewed as a combination of the properties of an elastic solid and a linearly viscous fluid [25]. 6

78 Acoustic Radiation Force The derivation of radiation force comes from the fact that the viscous (µ f ) terms in Equation 4.22, along with an additional bulk viscosity term (µ f ), represent energy loss described by a frequency-dependent attenuation coefficient (α) [47]. In a linearly viscous fluid, the radiation force generated by a propagating plane wave is proportional to the time-average change of particle velocity in the direction of wave propagation [64]. Using a perturbative expansion of Equation 4. with respect to density (ρ), pressure (p), and particle velocity ( v), radiation force can be related to the change in momentum from the second-order terms in the expansion such that the second-order balance of linear momentum can be expressed as [64]: F = p 2 µ f 2 v 2, (4.23) F = ρ v v + v v, (4.24) where represents the time average quantity, v and v 2 are the first and second-order terms in the expansion of particle velocity, p 2 is the second-order pressure term, and the divergence of particle velocity is assumed to be negligible. F in Equation 4.24 represents a time average transfer of momentum from the wave to the material. For a plane wave, the expression for radiation force in Equation 4.24 can be reduced to: F = 2ρ v v,x. (4.25) The direction of this radiation force occurs in the direction of wave propagation (i.e., the Poynting vector for the acoustic wave) [64]. Substituting Equation 4.8 into Equation 4.25, the magnitude of this acoustic radiation force can be expressed 62

79 as [64]: F = A 2 e 2αx ρα. (4.26) Therefore, at any given spatial location, the magnitude of acoustic radiation force, F [kg/(s 2 cm 2 )], is: F = 2αI c, (4.27) where F is in the form of a body force, c [m/s] is the sound speed, α [Np/m] is the absorption coefficient of the tissue, and I [W/cm 2 ] is the temporal average intensity at that spatial location Acoustic Radiation Force Impulse (ARFI) Imaging Acoustic Radiation Force Impulse (ARFI) imaging is a radiation force-based imaging method that studies the local mechanical properties of tissue. ARFI imaging uses short duration (< µs), high-intensity, acoustic pulses to generate localized displacements in tissue, and the tissue recovery response is monitored using ultrasonic correlation-based methods. Images of two-dimensional regions of interest (ROIs) are generated by sequentially interrogating multiple spatial locations, as is done in color Doppler imaging. During the application of high-intensity pulses, energy is absorbed by the tissue that results in the generation of acoustic radiation force and tissue displacement. The spatial distribution of the radiation force field is determined by the acoustic parameters of the transmitter along with the tissue attenuation and sound speed. In soft tissues, the majority of ultrasonic attenuation occurs through absorptive mechanisms [47]. Equation 4.27 can be used to model radiation force fields associated with complex intensity field geometries by computing the temporal average intensity at points in a three-dimensional ROI. The shape of this intensity field is dependent on 63

80 the transducer focal configuration, which can be characterized by the dimensionless f-number (F/#) of the system (Equation 2.5). The result is a continuous, variable magnitude body force applied throughout the tissue within the geometric shadow of the transducer Finite Element Methods Analytically solving for the dynamic behavior of tissue in response to geometrically complicated distributions of impulsive body forces is non-trivial. Mathematical models using the Khokhlov-Zabolotskaja-Kuznetsov (KZK) equation of the mechanical response of tissue to acoustic radiation force in dissipative media have been developed, but they cannot easily model a variety of transducer configurations and acoustic material properties [2,77,78]. The analysis of stress and displacement fields in continua lends itself to numerical methods. Lumped-parameter systems containing mass, elastic, and viscous components are very useful for modeling the dynamic behavior of discrete masses, but become more cumbersome when used to model deformations in three dimensions. Walker et al. [96] and Bercoff et al. [6] have used simulated acoustic beam intensity patterns and Greens function analysis to simulate the displacement fields that result from acoustic radiation force excitation of solids. In this chapter, the dynamic displacement fields associated with ARFI imaging are modeled using finite element methods (FEM) to solve the three-dimensional weak-form equations of elastodynamics [45]. 64

81 4.3 Methods 4.3. FEM Pre-Processing Mesh Generation A three-dimensional, rectangular, solid mesh was assembled using 8-noded, linear, elastic, brick elements (HyperMesh, Altair Computing Inc., Troy, MI). The mesh extended 7.5 mm in elevation, 35 mm laterally, and 25 mm axially. The elements within the geometric shadow of the transducer (7 mm laterally by 2.7 mm in elevation, for all depths) were uniform cubes with a node spacing of.2 mm, while the elements were biased to be larger farther from the transducer s geometric shadow in the lateral-elevation plane. There were a total of 283,22 nodes and 265, elements in the mesh. The spatial extent of the model allowed displacement and stress fields to decay to negligible levels (<.% of their maximum values in the model) before reaching the mesh boundaries for the duration of the simulation. This insured that waves generated during impulsive tissue excitation were not reflected back into the ROI. The finer mesh density within the ROE allowed accurate simulation of the spatial distribution and the subsequent tissue response to acoustic radiation force excitation, while also allowing for more accurate simulation of ARFI imaging using multiple lateral interrogations. Larger element sizes away from the transducer s geometric shadow allowed for accurate modeling of a tissue continuum, including its inertial contributions, while reducing the model s computational overhead. The model employed elevational plane-symmetry. The bottom surface of the model, opposing the transducer, was fully constrained; the top (transducer) surface of the model was constrained to motion in the lateral-elevation plane. All other 65

82 faces in the model had full degrees of freedom. Tissue and spherical inclusions were modeled with varying Young s moduli, a Poisson s ratio of.499 that approaches incompressibility conditions, and an attenuation of.7 db/cm/mhz to match the calibrated phantoms (Section 4.3.3). A discussion of this choice of Poisson s ratio is provided in Section 4.5. of this chapter. 8-noded brick elements were chosen over 4-noded tetrahedral elements, at the expense of more computational overhead, to reduce artifacts associated with element locking associated with modeling nearly incompressible materials. Intensity Measurement Where possible (e.g., for the tracking beams), intensity measurements were made in accordance with FDA guidelines [4]. However, exact quantification of the pushing beam intensity fields was not possible due to saturation effects in water at these relatively high frequencies (6-7 MHz) and transmit pressures [55]. A comparison method was implemented, using linearly-characterized, low-amplitude pulses quantified with a calibrated hydrophone. ARFI imaging was performed in a compliant phantom that allowed for a displacement of.43 µm when using the low-voltage intensity configuration. Finally, the transmit voltage was maximized and ARFI images were acquired in a typical, high-intensity mode, generating displacements of 48 µm in the same compliant phantom. Assuming a linear relationship between intensity and phantom displacement, the ratio of the displacements reflected the ratio of the intensities. These empirically-determined, linearly-extrapolated, small-signal derated fields have been shown to overestimate the true focal intensity fields [55]. For more discussion about the measurement of these intensity fields, along with the simulation of the three-dimensional distribution of the acoustic energy used in ARFI excitations, the reader is referred to Section

83 Simulating Intensity Fields FIELD II [46] was used to simulate the acoustic intensity fields associated with the high-intensity pulses used for ARFI imaging (Siemens VF-5 linear transducer: one row of acoustic elements, excitation center frequency of 6.67 MHz, aperture width dependent on the number of active elements; Siemens Medical Solutions USA, Inc., Ultrasound Division, Issaquah, WA). The transducer was simulated as F/3.8 in elevation and F/.3 laterally, with a variable lateral focal depth and a fixed elevation focal depth near 2 mm, corresponding to the experimental ARFI imaging setup used in all models, unless otherwise specified. This same method has been successfully implemented to model the heating associated with ARFI imaging and other acoustic radiation force-based imaging modalities in Chapters 2 and 3. Three-dimensional intensity fields were computed, normalized, and thresholded at % of the maximum, computed intensity to reduce computational overhead of the model. These normalized intensities were scaled to a peak in situ, pulse-average intensity value of W/cm 2, when 69 elements were fired in an F/.3 focal configuration for 45 µs (3 cycles at 6.67 MHz) at a focal depth of 2 mm. This in situ intensity is an empirically determined value during ARFI imaging using a linear extrapolation of small-signal, derated fields, as outlined in Section 4.3. [6, 7]. The corresponding radiation body force values, as determined by Equation 4.27, were then converted to nodal point loads by concentrating the body force contributions over an element volume. For locations within ± % of the focal depth, point loads were directed purely in the axial direction. For shallower locations closer to the transducer, point loads were directed toward the focal point, while for deeper locations, point loads were directed away from the focal point, consistent with the wave-propagation Poynting vector. 67

84 4.3.2 FEM Implementation and Post-Processing The balance of linear momentum (Equation 4.) was solved numerically with the finite element analysis package, LS-DYNA3D (Livermore Software Technology Corporation, Livermore, CA), using an explicit, time-domain integration method. Single-point quadrature was performed with hourglassing control to avoid elementlocking and to reduce numerical artifacts. For more detail regarding the specifics of FEM quadrature rules to solve the equations of motion and hourglass numerical damping methods, the reader is referred to the text by Hughes [45]. Post-processing of dynamic displacement and stress fields was performed using LS-PREPOST (Livermore Software Technology Corporation, Livermore, CA) and custom-written MAT- LAB code Experimental Validation Ultrasound Scanner Configuration Siemens SONOLINE Antares TM and Siemens SONOLINE Elegra TM ultrasound scanners (Siemens Medical Solutions USA, Inc., Ultrasound Division, Issaquah, WA), using VF-5 (center frequency of 6.67 MHz) and 75L4 (center frequency of 7.2 MHz) linear arrays were used for all experimental ARFI imaging presented in this chapter. Unless otherwise specified, the transducers were laterally focused at 5 mm with an F/.3 focal configuration, and focused near 2 mm in elevation with an F/3.8 focal configuration. ARFI imaging uses two types of beams to generate and track displacement in tissue: high-intensity pushing beams and conventional B-mode tracking beams. A reference tracking beam is fired, followed by a pushing beam. The pushing beams were typically generated using the maximum system transmit voltage, with pulses 68

85 similar to those used for Color Doppler; however, apertures were unapodized with longer pulse lengths (e.g., 2 cycles at 6.67 MHz). As mentioned in Section 4.3., in situ measurements of I spta values were made using small-signal linear-extrapolation techniques; the spatial distribution of these intensity fields has been explored in more detail in the thermal analysis of ARFI excitation by Palmeri and Nightingale [7]. The pushing beam was followed by a series of tracking beams fired at a pulse repetition frequency (PRF) up to 2.5 khz for up to 5 ms after the ARFI excitation. The pushing and tracking beams were co-located, and adjacent interrogations laterally were separated by.2 mm. Summed radio-frequency (RF) echo data were stored for off-line processing, where -D normalized cross correlation was used between reference and tracking lines with a.3 mm kernel and 99% overlap [6, 9]. Tissue-Mimicking Phantoms A Computerized Imaging Reference Systems (CIRS) tissue-mimicking phantom (CIRS, Norfolk, VA) and gelatin-based tissue-mimicking phantoms [42] were used for experimental validation of the models. The CIRS phantom was designed to have an ultrasonic attenuation of.7 db/cm/mhz and a sound speed of 54 m/s. Additionally, gelatin-based, tissue-mimicking phantoms were fabricated using a combination of -Propanol and water to adjust sound speed to near 54 m/s (the sound speed of biological tissues [54]), and 4239 vein graphite powder (< 2 µm particle diameter, Superior Graphite Co., Chicago, IL) to adjust the attenuation of the phantoms to be approximately.7 db/cm/mhz [42]. Varying gelatin bloom number (Vyse Gelatin Co., Schiller Park, IL) was used to control phantom stiffness, with higher bloom numbers yielding stiffer phantoms [42]. Bloom is defined as the weight (g) required to push a piston of standard shape 4 mm into a gelatin gel that has set for 8 hours at C. 69

86 An MTS hydraulic actuator (Mechanical Testing Systems, Minneapolis, MN) was used to quantify Young s modulus of phantom materials. One circular aluminum plate was attached to a hydraulic actuator, while another plate was attached to a lb Sensotec load cell (Honeywell Sensotec, Columbus, OH). Cylindrical phantom samples, 2.5 cm in height and 5.2 cm in diameter, were placed between the two plates coated with glycerol, with assumed frictionless boundary conditions between the sample and plates. A 5% pre-compression strain was applied to the samples to insure uniform contact between plates and sample. Strains up to 5% were applied to samples in six independent ramp functions, ranging from. - 8 seconds. Linear regressions were performed on the stress-strain data during the ramp, and Young s moduli were determined by the slope of these regressed lines. All phantom samples were measured in a single session to reduce variations in temperature between samples. Table 4. shows the measured Young s moduli of the gelatin-based phantoms and homogeneous CIRS phantom samples. Shear wave speeds can be reconstructed to measure the Young s modulus of the phantoms (Equation 4.5), and the accuracy of that reconstruction can be validated using the FEM models. To reconstruct the shear wave speeds (c T ) from the experimental and ARFI data, axial displacement data is substituted into Equation 4.. This method has been demonstrated by Bercoff et al. [6] and Nightingale et al. [58]. Table 4.: MTS-Measured Young s Moduli of Gelatin and CIRS Phantom Materials Phantom Young s Modulus (kpa) Phantom A (Gelatin) 4. ±. Inclusion A (Gelatin) 5 ±. Bloom Gelatin 8.5 ±.2 5 Bloom Gelatin 8 ±. 2 Bloom Gelatin 23 ±.4 CIRS Phantom 4. ±. 7

87 4.4 Results 4.4. Effects of Poisson s Ratio The effects of varying Poisson s ratio (ν) for the model were evaluated, as illustrated in Figure 4.. As ν increases, the peak displacement decreases. At ultrasonic excitation frequencies (> MHz), the Poisson s ratio must be very close to.5 to support a sound speed of 54 m/s. However, at lower excitation frequencies (< khz), as are used with ARFI imaging, the Poisson s ratio has been measured in muscle and prostate to range from.49 to.499 [73]. ν =.499 ν =.4999 Relative Axial Displacement Time (ms) Figure 4.: Comparison of the simulated axial displacement at the focal point through time in response to a 45 µs ARFI excitation. Notice that there is a difference in the magnitude of the peak displacement, but there are not significant differences in the loading and relaxation times of the tissue ARFI Imaging Displacement Fields in Homogeneous Media Figure 4.2(a) shows the simulated distribution of acoustic radiation force in the axial-lateral plane, centered in elevation, for α =.7 db/cm/mhz and an F/.3 focal configuration focused at 2 mm, where the transducer would be centered on the 7

88 top of the image, and brighter pixels correspond to greater radiation force values. Figure 4.2(b-d) shows normalized displacement profiles in this same plane for an 8.5 kpa material.6,., and 2.2 ms after excitation, with brighter pixels indicating greater displacement away from the transducer. Figure 4.2(e-g) shows the corresponding normalized experimental displacement fields in the bloom phantom for the same time steps. Throughout the rest of this chapter, displacement profiles will be shown as a function of axial position, centered laterally, and as a function of lateral position at the focal depth. ARFI imaging displacement fields were compared between FEM models and experimental data in 4 different homogeneous media: CIRS, bloom, 5 bloom, and 2 bloom phantoms. The MTS-measured Young s moduli of these phantoms are summarized in Table 4.. Figure 4.3, row A shows data from the CIRS phantom (focal depth of 5 mm, F/.3 focal configuration), while rows B-D show data from the bloom, 5 bloom, and 2 bloom gelatin phantoms, respectively (focal depth of 2 mm, F/.3 focal configuration). Figure 4.3i-iii shows normalized axial displacement profiles, laterally centered in the ROE as a function of depth away from the transducer; while Figure 4.3(iv-vi) shows normalized axial displacements at the focal depth of 5 mm, spanning ± 6 mm laterally from the center of the ROE for different specified times after cessation of a 45 µs ARFI excitation pulse. The solid lines represent the FEM data, while the dashed lines represent experimental data (mean of six independent trials) acquired with the VF-5 transducer. The shear wave speeds measured from the FEM and experimental ARFI data [58], along with the reconstructed Young s moduli using these shear wave speeds, are summarized in Table 4.2. Table 4.3 shows a comparison of the maximum displacements for the FEM and experimental data for each of the phantoms, along with the normalization factors that were used to generate the plots in Figure

89 (a) Simulated Acoustic Radiation Force Field 5 Axial Position (mm) Lateral Position (mm) (b) FEM, t =.6 ms (c) FEM, t =. ms (d) FEM, t = 2.2 ms Axial Position (mm) 5 Axial Position (mm) 5 Axial Position (mm) Lateral Position (mm) (e) Experiment, t =.6 ms Lateral Position (mm) (f) Experiment, t =. ms Lateral Position (mm) (g) Experiment, t = 2.2 ms Axial Position (mm) 5 Axial Position (mm) 5 Axial Position (mm) Lateral Position (mm) Lateral Position (mm) Lateral Position (mm) Figure 4.2: Figure 4.2(a) shows the simulated distribution of acoustic radiation force in the axial-lateral plane, centered in elevation, for an F/.3 focal configuration, focused at 2 mm, with α =.7 db/cm/mhz. The transducer would be centered at the top of these images. Brighter pixels indicate greater radiation force magnitude. Figure 4.2(b-d) shows the normalized FEM displacement fields resulting from this acoustic radiation force distribution,.6,. and 2.2 ms after the excitation, respectively, in an 8.5 kpa medium. Figure 4.2(e-g) shows the normalized experimental ARFI displacement data from the bloom gelatin phantom, again, for the same time steps. Brighter pixels in the displacement images represent greater displacement away from the transducer. 73

90 Table 4.2: Shear Wave Speeds Phantom FEM c T (m/s) Experimental c T (m/s) CIRS..2 ±.3 Bloom.7.5 ±.2 5 Bloom ±.2 2 Bloom ±.3 Table 4.3: Comparison of FEM and Experimental Displacement Magnitudes Phantom FEM Max. Disp. (µm) ARFI Max. Disp. (µm) Norm. Factor CIRS ±.3.8 Bloom.3. ± Bloom ± Bloom ± Impact of Poisson s Ratio, Density, and Attenuation In addition to elastic modulus, there are several other parameters in this model that could affect the dynamic behavior of tissue in response to an ARFI excitation, including Poisson s ratio, density and ultrasonic attenuation. Figure 4. shows the effects of Poisson s ratios on displacement behavior at the focal point. Figure 4.4 shows Poisson s ratios ranging from affect the displacement profiles in the axial and lateral dimensions, as were shown in Figure 4.3. An F/.3 focal configuration, focused at 2 mm, was simulated, with E = 8.5 kpa, ρ =. g/cm 3, and α =.7 db/cm/mhz. Figure 4.5 shows the impact of small variations in density from.9 -. g/cm 3 on the dynamic behavior of tissue in response to an ARFI excitation, in the axial and lateral dimensions. An F/.3 focal configuration, focused at 2 mm, was simulated, with E = 8.5 kpa, ν =.499, and α =.7 db/cm/mhz. 74

91 (A.i) t =.8 ms (A.ii) t =.6 ms (A.iii) t = 3.4 ms (A.iv) t =.6 ms (A.v) t =.6 ms (A.vi) t = 3.4 ms Axial Position (mm) Normalized Axial Dispplacement (B.i) t =.6 ms (B.ii) t =. ms (B.iii) t = 2.2 ms (B.iv) t =.6 ms (B.v) t =. ms (B.vi) t = 2.2 ms Axial Position (mm) Normalized Axial Displacement (C.i) t =.6 ms (C.ii) t =. ms (C.iii) t = 2.2 ms (C.iv) t =.6 ms (C.v) t =. ms (C.vi) t = 2.2 ms Axial Position (mm) Normalized Axial Displacement (D.i) t =.6 ms (D.ii) t =. ms (D.iii) t =.5 ms (D.iv) t =.6 ms (D.v) t =. ms (D.vi) t =.5 ms Axial Position (mm) Normalized Axial Displacement Normalized Axial Displacement Lateral Position (mm) Figure 4.3: Comparison of FEM models and experimental normalized displacement profiles in the CIRS and gelatin phantoms. Row A shows the CIRS phantom, with a focal depth of 5 mm and an F/.3 focal configuration, while rows B-D show, 5, and 2 bloom gelatin phantoms, respectively, with a focal depth of 2 mm with an F/.3 focal configuration. The solid lines represent FEM data and the dashed lines represents experimental phantom data (mean of six independent trials). Figure 4.3(i-iii) represents displacement profiles laterally centered in the ROE as a function of axial depth away from the transducer, for the indicated times after cessation of a 45 µs ARFI excitation. Note the different time steps being represented by the figures between rows. Figure 4.3(iv-vi) represents normalized axial displacement profiles at the focal depth (5 mm for row A, 2 mm for rows B-D), spanning ± 6 mm laterally from the center of the ROE. The shear wave speeds and reconstructed Young s moduli from this data are summarized in Table

92 Figure 4.6 shows the impact of ultrasonic attenuations ranging from.5 -. db/cm/mhz on the dynamic behavior of tissue in response to an ARFI excitation, in the axial and lateral dimensions. An F/.3 focal configuration, focused at 2 mm, was simulated, with E = 8.5 kpa, ρ =. g/cm 3, and ν =.499. (a) t =.6 ms (b) t =. ms (c) t = 2.2 ms (d) t =.6 ms (e) t =. ms (f) t = 2.2 ms Axial Position (mm) Normalized Displacement Normalized Displacement Lateral Position (mm) Figure 4.4: Comparison of normalized FEM displacement values laterally-centered in the ROE as a function of axial depth away from the transducer (a-c) and spanning ± 7 mm laterally from the center of the ROE at the focal depth of 2 mm (d-f), for varying material Poisson s ratios (solid line =.499, dashed line =.4999, dotted line =.49, dashed-dotted =.48). The times shown refer to times after the ARFI excitation, and the material was modeled with E = 8.5 kpa, ρ =. g/cm 3 and α =.7 db/cm/mhz. Notice that the displacement profiles for ν =.49 and ν =.48 are almost identical. (a) t =.6 ms (b) t =. ms (c) t = 2.2 ms (d) t =.6 ms (e) t =. ms (f) t = 2.2 ms Axial Position (mm) Normalized Displacement Normalized Displacement Lateral Position (mm) Figure 4.5: Comparison of normalized FEM displacement values laterally-centered in the ROE as a function of axial depth away from the transducer (a-c) and spanning ± 7 mm laterally from the center of the ROE at the focal depth of 2 mm (d-f), for varying material densities (solid line =. g/cm 3, dashed line =.9 g/cm 3, dotted line =. g/cm 3 ). The times shown refer to times after the ARFI excitation, and the material was modeled with E = 8.5 kpa, ν =.499, and α =.7 db/cm/mhz. 76

93 (a) t =.6 ms (b) t =. ms (c) t = 2.2 ms (d) t =.6 ms (e) t =. ms (f) = t = 2.2 ms Axial Position (mm) Normalized Displacement Normalized Displacement Lateral Position (mm) Figure 4.6: Comparison of normalized FEM displacement values laterally-centered in the ROE as a function of axial depth away from the transducer (a-c) and spanning ± 7 mm laterally from the center of the ROE at the focal depth of 2 mm (d-f), for varying material ultrasonic attenuations (solid line =.5 db/cm/mhz, dashed line =.7 db/cm/mhz, dotted line =. db/cm/mhz). The times shown refer to times after the ARFI excitation, and the material was modeled with E = 8.5 kpa, ν =.499, and ρ =. g/cm ARFI Imaging Displacement Fields with a Stiff Spherical Inclusion The dynamic displacement fields generated in response to ARFI excitation in a medium with a stiff, 3 mm diameter, spherical inclusion were both modeled and measured experimentally in gelatin phantom A. Row A of Figure 4.7 shows the B- mode image, the experimental ARFI image, and the FEM axial displacement image (from left to right).4 ms after the cessation of the ARFI excitation with the 75L4 transducer. The displacement images are on the axial-lateral plane, centered in elevation, with the transducer located at the top of the images. Rows B and C show the corresponding images for.9 ms and.8 ms after cessation of the ARFI excitation. Figure 4.8(a-c) shows corresponding normalized axial displacement profiles laterally centered in the ROE as a function of axial depth away from the transducer, while Figure 4.8(d-e) shows normalized axial displacements laterally at a depth of 5.5 mm (center of the spherical inclusion). FEM data is represented by solid lines and experimental data is represented by dashed lines. The dotted lines represent the boundaries of the spherical inclusion. 77

94 (A) t =.4 ms B mode Experiment FEM Axial Position (mm) Lateral Position, (mm) (B) t =.9 ms 2 2 B mode Experiment FEM Axial Position (mm) Lateral Position, (mm) (C) t =.8 ms 2 2 B mode Experiment FEM Axial Position (mm) Lateral Position, (mm) 2 2 Figure 4.7: Matched simulation and experimental displacement profiles in the axiallateral plane, centered in elevation, demonstrating similar temporal behavior for a stiff spherical inclusion in an otherwise homogeneous medium. The images at each time step are normalized by the maximum displacement value for that time step. Each row represents a different time step:.4,.9, and.8 ms for the top, middle and bottom rows, respectively. The Young s moduli for the background and lesion in both simulation and experiment were 4. and 5 kpa, respectively, with a lesion diameter of 3 mm. Initially, the displacements are indicative of differences in stiffness; however, later in time, wave propagation effects are portrayed, and a reversal in contrast and apparent spreading of the lesion is observed in the displacement profiles. Note that the simulation images do not include the effects of speckle tracking the ARFI-induced displacements. 4.5 Discussion 4.5. Choice of Modeled Poisson s Ratio & Young s Modulus Figure 4. shows that there is a 25% increase in the magnitude of the peak displacement at the focus for a.9 increase in Poisson s ratio, but there are not 78

95 (a) t =.4 ms (b) t =.9 ms (c) t =.8 ms. (d) t =.4 ms. (e) t =.9 ms. (f) t =.8 ms Axial Position (mm) Normalized Axial Displacement Normalized Axial Displacement Lateral Position (mm) Figure 4.8: Figure 4.8(a-c) compares FEM model (solid line) and experimental (dashed line) normalized axial displacement profiles in gelatin phantom A with spherical inclusion A. The displacement profiles are laterally centered in the ROE as a function of axial depth away from the transducer, for.4 ms,.9 ms, and.8 ms after the cessation of a 45 µs ARFI excitation. The ARFI excitation was focused at 5 mm with an F/.3 configuration. Figure 4.8(d-f) shows normalized axial displacement profiles at the focal depth of 5.5 mm, spanning ± 2.5 mm laterally from the center of the ROE. The dotted lines represent the boundaries of the spherical inclusion. significant differences in the loading and relaxation times of the tissue. The greater Poisson s ratio, though, demands a much smaller explicit time step, as determined by the increase in longitudinal wave velocity (Equation 4.4). This results in run times that are approximately 3.3 times longer for the ν =.4999 model than for ν =.499. Therefore, a ν =.499 was chosen for all of these models, with the knowledge that there may be some discrepancy in the absolute displacement values due to this choice of Poisson s ratio. Additionally, the relaxation of tissue after ARFI excitation is closely related to shear wave speed. As Equation 4.5 shows, varying ν from only varies the shear wave speed by approximately.3 m/s for materials with Young s moduli ranging from - kpa. There were several variables present in making the MTS measurements of the Young s moduli of the phantom samples shown in Table 4.. Because all measurements were made in the same session, variations in temperature between samples 79

96 were minimized. Variations in sample geometry were minimized by using the same phantom molds for all samples. The errors introduced by assumptions in slip boundary conditions at the plates and incompressibility of the phantom materials were constant between all samples; therefore, these assumptions would not have an impact on the relative stiffnesses measured by the MTS system. ARFI imaging measurements of shear wave speeds were also made within 24 hours of the MTS measurements to reduce the potential for changes in phantom stiffness that can occur with time. The stress-strain relationships determine by the MTS system were made under strain rates of Hz, with standard deviations of <. kpa for all phantom samples, indicating little frequency-dependence in the Young s modulus at these strain rates. While ARFI excitations occur in less than 5 µs, the dynamic behavior of the phantoms after the ARFI excitation appears to be dictated by the shear wave speed of the tissue. The absence of viscous material properties in the model may cause overestimation of the maximum displacement achieved after ARFI excitation in tissue. The presence of viscosity in tissue may also lead to phase dispersion in the propagating shear wave that the models also would not predict Comparison of Dynamic Behavior in FEM Model and Experimental Studies Figure 4.3 demonstrates good temporal agreement between FEM models and experimental ARFI data in homogeneous phantoms of varying stiffness. Notice that immediately after ARFI excitation, displacement is localized around the focal depth and is an accurate representation of the radiation force field, as demonstrated in Figure 4.2. Later in time (Figure 4.3(ii-iii)) greater relative displacements are observed in the near-field as compared with the focal point, again with good agreement 8

97 between models and experiments. This shift of displacement into the near field is dictated by shear wave propagation away from the ROE. As shown in Figure 4.2(b,e), the lateral extent of the displacement field is larger in the near field than at the focus. As shear waves propagate away from the ROE, the tissue recovers quicker at the focus than it does in the near field due to these differences in lateral extent of the initial displacement field. Within.5 ms after ARFI excitation, the axial position of the peak displacement in the FEM models and the experimental data agree to within mm. For the later time steps portrayed in Figure 4.3(A-D.iii), the locations of the peak displacements differ by up to 3 mm, but the overall shape of the displacement curves still agree. Absolute magnitudes of axial displacement in the FEM data (Table 4.3) deviated by a mean of 32.% from the experimental ARFI data for the four phantoms, with the greatest error being.9 µm in the 2 bloom gelatin phantom and the smallest error being.4 µm in the CIRS phantom. Figure 4.3(iv-vi) shows that as tissue relaxes at the focal point, shear waves propagate away from the ROE. The Young s moduli reconstructed from these shear wave speeds using Equation 4.5 are summarized in Table 4.4. Notice that the relative decay in shear wave magnitude as they propagate away from the ROE in the FEM and experimental data sets are similar, suggesting that this decay is due to geometric spreading of the waves, versus viscous effects that are not taken into account in the FEM models. The FEM data sets agree within a mean of -.8 kpa of the specified Young s moduli, with the greatest deviation being 2. kpa in the 5 bloom gelatin phantom. The experimentally reconstructed Young s moduli from the shear wave speeds deviated by a mean value of.7 kpa from the MTS measurements, with the greatest deviation being 3.9 kpa in the 5 bloom gelatin phantom. 8

98 Table 4.4: Differences in Shear Wave-Reconstructed Young s Moduli Phantom MTS E (kpa) FEM E (kpa) ARFI E (kpa) CIRS Bloom Bloom Bloom Figure 4.4 reinforces the trend shown in Figure 4. that Poisson s ratio can have an impact on the peak magnitude of displacement, but overall the dynamics of the displacement field in the axial dimension are not changed (Figure 4.4(a-c)). In the lateral dimension (Figure 4.4(d-f)), again, the displacement magnitudes are impacted by the Poisson s ratio, with less compressible materials displacing less; however, as expected, the speed at which the shear waves propagate away from the ROE is not affected. Figure 4.5 shows that small variations in density do not greatly impact the displacement profiles. As expected, the less dense material is able to displace slightly more than the more dense materials in response to the same excitation, but the overall dynamic response of the material in the axial dimension (a-c) remains the same. As expected by Equation 4.3, shear wave speed is impacted by density; when compared with the shear wave speed for ρ =. g/cm 3. there is an expected decrease in c T of 4.66% for ρ =. g/cm 3, and an increase of 5.4% for ρ =.9 g/cm 3. The computed shear wave speeds changes from Figure 4.5(d-f) are ± 7.7% for ρ =.9 g/cm 3 and. g/cm 3, respectively. Upon further inspection, changes of 7.7% in the shear wave speed in the FEM models correspond to displacement differences in the lateral dimension of.2 mm, which is the node spacing. This represents a limitation in the resolution of tracking the shear wave speeds from the FEM data since the peaks will always occur at nodal locations. Improved accuracy, therefore, could be achieved by refining the mesh to have smaller node spacing. 82

99 Ultrasonic attenuation of the material did not have an impact on the shear wave speed, as shown in Figure 4.6(d-f), but does show differences in the displacement profiles in the axial dimension (a-c). As Figure 4.6(a-c) shows, higher attenuations lead to greater relative displacements in the near field compared to the displacement at the focal depth, though the absolute magnitude of the displacement field is less. As demonstrated in Figure 4.2, the near-field displacement field has a greater lateral extent than at the focal depth, which causes a greater relative displacement in the near-field at later times for higher attenuations, as demonstrated by Figure 4.6(b-c). Regardless of the relative displacement profiles in the axial dimension, the relaxation rate is determined by the shear wave speed and is independent of ultrasonic attenuation. The temporal behavior of gelatin phantom A with stiff spherical inclusion A is complicated (Figures 4.7 and 4.8). Initially, the displacements are inversely related to the material stiffness (i.e., the stiffer lesion does not move as far as the softer background) and the lesion appears to be the actual diameter (3 mm), as shown in Figures 4.8(a) and 4.7(A). As waves propagate away from the ROE, reflections and interference within the lesion and at its boundaries make the lesion boundaries appear to spread, as shown in Figures 4.7B-C and 4.8(b-c) [58]. These wave interactions are covered in more detail in Chapter 6. The plots of axial displacement laterally through the center of the inclusion (Figure 4.8(d-f)) also qualitatively match the experimental data, including the inversion of the location of the peak displacement from outside of the lesion to within the lesion by.8 ms after ARFI excitation. Qualitatively, the models match the shapes of the experimental displacement fields through time in Figure

100 4.5.3 Limitations of the FEM Model The use of single point quadrature and the addition of hourglassing control, instead of using full 8-point quadrature rules, reduces computation overhead considerably, but does introduce some degree of numerical damping into the model. This numerical damping, though, is believed to have minimal impact on the dynamic response of the model, as suggested by extrapolations performed to study the impact of varying degrees of numerical damping in the model. The magnitudes of peak displacement in these models, in general, over-estimated the experimentally measured displacements. This could be related to the choice of Poisson s ratio, or the peak in situ intensity of W/cm 2 that was chosen to scale the intensity fields. Therefore, while the relative trends presented in this chapter are not affected by inaccurate choice of these values for the models, the absolute values of displacement may be impacted and could be corrected in a future version of the model if more accurate parameters become available. Additionally, the displacements measured by ultrasonic, correlation-based tracking methods have a negative bias in the presence of displacement gradients occurring within the imaging Point Spread Function (PSF). Bias, along with the other effects associated with ultrasonically tracking the displacement fields, are covered in detail in Chapter 5. The limited spatial extent of the mesh restricts the maximum time that can be modeled. Once displacement waves reach the boundary closest to the ROE, they are reflected back into the ROI and corrupt the displacement fields. Therefore, stiffer materials and larger ROEs will suffer from reduced times over which they can be modeled. This can be overcome by increasing the size of the mesh, but with a computational penalty. 84

101 4.5.4 Future Applications of the FEM Model Ultrasonic tracking of the FEM displacement fields will be added to future versions of this model to evaluate the degree of displacement underestimation (bias) that occurs due to lateral speckle shearing under the track (PSF), and to evaluate jitter magnitudes (see Chapter 5). These models assumed that spherical inclusions, in an otherwise homogeneous background medium, share the same attenuation (α) and sound speed (c) as the background medium. This assumption cannot be made for accurate modeling of clinical lesions (e.g., tumors and ablated tissue). Instead, simulation of intensity fields and force distributions using Field II must be modified to accommodate heterogeneous distributions of ultrasonic attenuation and sound speed in the ROE. Viscoelastic material properties consistent with soft tissue also need to be incorporated into the model to evaluate viscosity s impact on tissue dynamics associated with ARFI imaging. These models will also be extended to incorporate biological structures other than spherical inclusions, such as layered organs and blood vessels with fluid flow. The impact of boundary conditions between inclusions and layers will also be studied in future models. 4.6 Conclusions The FEM model presented in this chapter allows for the simulation of tissue s dynamic response to an impulsive excitation with acoustic radiation force, as is done in ARFI imaging. The model is able to take into account acoustic attenuation and tissue material properties, in addition to the transducer parameters, for generating the acoustic radiation force. The dynamic response of this model has been validated using four calibrated, tissue mimicking phantoms, with and without spherical inclusions, providing good agreement in reconstructed shear wave speeds. Shear wave 85

102 speed, and therefore elasticity, appears to be the dominant factor affecting tissue relaxation following ARFI excitation within the ROE, while Poisson s ratio and density do not significantly alter relaxation rates in tissue. Increased acoustic attenuation in tissue increases the relative amount of tissue displacement in the near-field as compared with the focal depth, but relaxation rates are not altered. Applications of these validated ARFI imaging models include improving image quality and distilling material and structural information from tissue s dynamic response to ARFI excitation. These models will be extended to include viscous material properties, variable attenuation and sound speed in inclusions, and ultrasonic displacement tracking. 4.7 Acknowledgments This work was supported by NIH grant 8 R EB232, the Whitaker Foundation, and the Medical Scientist Training Program grant T 32 GM-77. The authors would also like to thank Siemens Medical Solutions USA, Inc. Ultrasound Division for their technical assistance, Altair Computing, Inc., for providing mesh generation software, Paul Toomey for writing post-processing code, and Joshua Baker-LePain for his technical assistance. 86

103 Chapter 5 Ultrasonic Tracking of Acoustic Radiation Force-Induced Displacements in Homogeneous Media The work presented in this chapter has been submitted for publication in IEEE Transactions in Ultrasonics, Ferroelectronics, and Frequency Control [7]. 5. Introduction Ultrasonic methods are used in the field of elastography to track tissue displacements that result from either the external compression/vibration of tissue [9, 39, 4, 53, 67, 68, 76, 82, 84, 86, 9, 93] or the excitation of the tissue using acoustic radiation force [6, 32, 34, 48, 5, 56, 78, 85, 98]. Ultrasonic tracking of speckle patterns has been well established in the literature, specifically for the tracking of blood motion [,,3] and local strain estimation in solids [66,68]. Ultrasonic displacement tracking in solid media, like fluids, can suffer from underestimation of the peak displacement when the scatterer distribution within the point spread function (PSF) of the tracking beam is distorted. This distortion is known as shearing [8,5,3,38]. An analytic study performed by McAleavey et al. [52] demonstrated such an un- 87

104 derestimation when tracking a steady-state Gaussian distribution of displacement broader than the PSF of the tracking beam. Ultrasonic tracking also suffers from jitter (or tracking inaccuracies), the magnitude of which is related to tracking frequency, transducer bandwidth, signal-to-noise ratio (SNR), kernel length, and the correlation coefficient between radio-frequency (RF) lines being tracked [6, 95, 97]. In elastography, the external compression of tissue provides a relatively uniform compression of the scatterer distribution within the PSF of the tracking beams. This compression causes minimal shearing in the lateral and elevation dimensions, but leads to decorrelation of the RF signal in the axial dimension. This problem has been addressed in elastography with companding [8, 9]. In contrast, when focused acoustic radiation force is used to excite tissue, as is done in Acoustic Radiation Force Impulse (ARFI) imaging, the deformation of tissue in the focal zone can introduce significant shearing in the lateral and elevation dimensions that can result in signal decorrelation and introduce jitter in tissue displacement estimates. In contrast with elastography, signal decorrelation in the axial dimension is not significant since acoustic radiation force induces much smaller displacements than those applied in elastography. This chapter presents studies investigating the impact of tracking parameters on the accuracy of ultrasonic tracking of tissue displacements generated by temporally short (45 µs), focused acoustic radiation force, as is used in ARFI imaging, in homogeneous media with varying elastic constants. The conclusions drawn from these studies can be applied to other acoustic radiation force imaging modalities in addition to ultrasonic tracking used in elastography. 88

105 5.2 Background The mechanical response of tissue to acoustic radiation force excitation or external compression/vibration can be used to image tissue s structural and material properties. ARFI imaging is one imaging modality that excites tissue using acoustic radiation force generated by a commercial diagnostic scanner [6]. The radiation force is generated using short-duration (< µs), high-intensity (I spta.5 3 W/cm 2 ) acoustic pulses. A thorough description of the generation of radiation force in ARFI imaging is presented in Section 4.2. To track the displacements generated by ARFI imaging, a reference A-line tracking pulse is initially fired, followed by a high-intensity pushing pulse, and then a series of A-line tracking pulses to monitor the resulting tissue displacement for up to 5 ms after excitation. Normalized cross-correlation of the tracked echoes through time is used to estimate the tissue displacements toward and away from the transducer, as described by Nightingale et al. [6], McAleavey et al. [52] and Pinton et al. [75]. The volume of tissue interrogated by the tracking pulse is not uniformly displaced, but instead contains a range of displacements (i.e., shearing is present) due to several factors. The tissue displacement generated by acoustic radiation force is non-uniform and is strongly dependent on tissue properties and ultrasound beam characteristics. The displacement is a dynamic event, generating shear waves that propagate away from the region of excitation (ROE). The spatial distribution of the pushing and tracking beams are of similar scale because the same transducer generates both beams. In particular, for a D linear array, the fixed elevation focus results in identical elevation beamwidth (neglecting bandwidth effects) for the pushing and tracking beams. These effects make it difficult (impossible in the D array case) to 89

106 confine the tracking beam within a region of uniform displacement. Two practical consequences arise from shearing within the tracked volume: displacement underestimation due to the averaging of displacements within the tracking beam PSF, and reduced echo signal correlation that increases the displacement estimate variance (i.e., jitter) [8, 9, 97]. McAleavey et al. previously reported an expression for estimating the displacement measured using ultrasonic cross-correlation of a random scattering target under a steady-state Gaussian deformation [52]. The results indicate that the estimator, on average, yields the mean beam-weighted displacement, which can be significantly less than the peak displacement. Furthermore, due to the fixed elevation focus of the transducer, arbitrarily fine focusing in the lateral dimension will not yield a measure of the true peak displacement, but only 2 2 of that amount. An in vitro experiment in a tissue-mimicking phantom showed generally good agreement between the observed and theoretically predicted results. An expression was derived for the expected value of the cross-correlation function at the beam-weighted mean displacement, which agrees well with simulations and provides an estimate of the expected jitter, as determined by the Cramer Rao Lower Bound (CRLB) [3, 97]: [ Jitter 3 2fc 3 π 2 T (B 3 + 2B) ρ 2 ( + ) 2 SNR 2 ], (5.) where SNR represents the signal-to-noise ratio, B represents the fractional bandwidth of the transducer, T represents the kernel size, ρ represents the correlation coefficient between the reference and the tracked RF-data, and f c represents the center frequency of the transmitted track beam. The work presented by McAleavey et al. [52] was limited to static Gaussian displacement distributions in the focal zone. The present work investigates the 9

107 dynamic measured displacement and cross-correlation values for clinically realistic displacement fields and tracking beam profiles. Acoustic radiation force excitation volumes were calculated for a linear array used in experimental ARFI implementations (Siemens VF-5, Siemens Medical Solutions USA, Inc., Ultrasound Division, Issaquah, WA). These forcing functions were applied to a Finite Element Method (FEM) model to calculate the resulting displacement of tissue through time [72]. This calculated displacement data was used to translate scatterers in a simulated uniform speckle phantom. The RF-data from this simulation was then tracked to provide estimates of displacement and correlation. 5.3 Methods 5.3. Finite Element Method Simulation of Tissue Displacement The Finite Element Method (FEM) model of tissue displacement in response to acoustic radiation force excitation that was presented in Chapter 4 was used to determine the three-dimensional displacement fields in response to acoustic radiation force excitations corresponding to those used in ARFI imaging [72]. For the analysis presented in this chapter, radiation force excitations were applied for 45 µs. Field II [46] was used to simulate excitation focal configurations with varying focal depths and F/#s (Equation 2.5). The excitation frequency was 6.7 MHz and the elevation focus was fixed at 9 mm. Pressure values determined by Field II were then scaled up to empirically-calibrated intensity values (e.g., I spta.5 = 3 W/cm 2 focused at 2 mm, F/.3) and converted to body force values using Equation The radiation force field was then expressed as point loads, as described in Section 4.3., 9

108 and superimposed onto the model mesh (HyperMesh, Altair Computing Inc., Troy, MI). LS-DYNA3D (Livermore Software Technology Corporation, Livermore, CA) was used to solve the dynamic equations of motion for tissue displacement [72] Simulation of Ultrasonic Displacement Tracking A uniform scattering phantom was represented as randomly positioned point targets of equal echogenicity. These point locations were taken as the initial, predisplacement scatterer positions. A reference RF-data tracking line was generated from these scatterer locations, using Field II to model the appropriate aperture geometry, focus and apodization functions [46]. To simulate the echoes generated from this phantom after the acoustic radiation force impulse was applied, displacement vector field data from the FEM simulations were used to reposition the scatterers. For each scatterer in the phantom, the eight surrounding nodes were determined, and the displacement components of the scatterer were linearly interpolated from the displacement vectors at each node. Although correlation is performed in only the axial dimension, scatterer motion in all three dimensions was simulated to evaluate the impact of scatterer shearing in all dimensions. After repositioning all scatterers, the corresponding RF line was simulated using Field II. The process was repeated for each time step in the simulation at an experimentally-realistic pulse repetition frequency (PRF) of khz. Windowed cross-correlation of the simulated RF signals was used to estimate the displacements and correlation coefficients as in normal ARFI imaging [75]. 92

109 5.3.3 Processing of Tracked Data The simulated RF-data was tracked using the same normalized cross-correlation code used in experimental ARFI data acquisitions. Unless otherwise specified, all simulations used a transducer center frequency of 6.7 MHz with a fractional bandwidth of 53% and a kernel length of 2.5 cycles. To compute displacement underestimation and jitter magnitudes, a running average with a 3.85 mm kernel was performed on the tracked data. Displacement underestimation was measured at the focal depth as the difference between the input FEM displacement value and the running-averaged tracked value at the focal depth, where the greatest underestimation occurs. The running-averaged tracked data was subtracted from the tracked data at each axial position to generate the jitter plots over axial positions centered laterally in the region of excitation. The running window length was chosen as the smallest length to maintain a zero-mean on the jitter as a function of depth. Characteristic images for these representations are shown in Figure 5.3. All plots of displacement underestimation represent the mean and standard deviation of independent random scatterer distribution simulations at the focal depth, while plots of jitter represent the standard deviation of the displacement estimate between the tracked and running-averaged displacement data over all depths - 25 mm from the transducer face. 5.4 Results Before running a parametric analysis on the variables affecting ultrasonic displacement tracking, a constant scatterer density was determined and used for the simulations where the track frequency was held constant. Figure 5. shows the speckle SNR in the focal zone of the tracking beam as a function of number of 93

110 scatterers in the simulated volume of material. As Figure 5. shows, a minimum of scatterers per -6 db resolution cell volume can be used to reach the theoretical speckle SNR ceiling of.9 [4]. This minimum scatterer density was chosen to maintain fully developed speckle while minimizing the computational overhead of our simulations Speckle SNR Number of Scatterers / Resolution Cell Volume Figure 5.: Calculated speckle SNR in the focal zone as a function of the number of scatterers in the -6 db resolution cell volume. The theoretical speckle SNR ceiling of.9 is approached when approximately scatterers exist per resolution cell volume. Figure 5.2 shows characteristic normalized displacement fields in the axial-lateral plane, centered in elevation, for a material with a Young s modulus of 8.5 kpa and an acoustic attenuation (α) of.7 db/cm/mhz. The top row corresponds to.8 ms after initiation of the ARFI excitation (45 µs, F/.3, 2 mm focal depth), while the center and bottom rows correspond to.5 and.82 ms respectively. Brighter pixels indicate greater displacement away from the transducer, which is located at the top of each image. The left column shows the FEM simulated displacement fields, while the center column shows those displacement fields after being ultrasonically-tracked at 7 MHz, F/.5 focal configuration on transmit and dynamic receive and a transmit focal depth of 2 mm. The right column shows the corresponding experimental ARFI displacement data in a gelatin phantom [42] with similar acoustic and me- 94

111 chanical parameters to those simulated. Statistical analysis was performed on the characteristic line plots shown in Figure 5.3, representing displacement as a function of depth, centered laterally. The rows in Figure 5.3 correspond to the same time steps shown in Figure 5.2. The solid lines in left column (a,d,g) represent the FEM displacement data at the center lateral position, while the dashed line represents the tracked displacement data along these center lines. These dashed lines are repeated in the center column plots (b,e,h) (again representing the same tracked data as in (a,d,g)), while the solid lines in the center column now represent the tracked data after a 3.85 mm kernel running average. The plots in the right column (c,f,i) represent the jitter in the tracked displacement data after the running average data was subtracted from the tracked data (solid and dashed lines, respectively, in (b,e,h)). Note that the underestimation of maximum displacement is worse in the early time step (a), and the focal zone jitter is worse immediately after excitation (c) than later in time. Figure 5.4 shows how the maximum displacement estimate in the focal zone (a) and the jitter magnitudes over all depths (b) vary as functions of time after the ARFI excitation for materials with Young s moduli of 8.5, 23, and 58 kpa, using the same tracking configuration as described for Figure 5.2 and 5.3. Figure 5.5 shows the input and tracked displacement through time at the focal point for 4 (a), 8.5 (b), and 23 kpa (c) materials. The solid lines represent the FEM displacement data, while the dashed lines represent the tracked displacement data. Since biological tissues can be underpopulated by scatterers and may not have fully developed speckle, simulations were run to determine the impact of underdeveloped speckle on displacement estimate accuracy. Figure 5.6 shows that underestimation of maximum displacement in the focal zone (a) and jitter magnitude (b) increase as scatterer density decreases. 95

112 (a) FEM (t =.8 ms) (b) Tracked FEM (c) Experiment Depth (mm) 5 Depth (mm) 5 Depth (mm) Lateral Position (mm) Lateral Position (mm) Lateral Position (mm) (d) FEM (t =.5 ms) (e) Tracked FEM (f) Experiment Depth (mm) 5 Depth (mm) 5 Depth (mm) Lateral Position (mm) Lateral Position (mm) Lateral Position (mm) (g) FEM (t =.82 ms) (h) Tracked FEM (i) Experiment Depth (mm) 5 Depth (mm) 5 Depth (mm) Lateral Position (mm) Lateral Position (mm) Lateral Position (mm) Figure 5.2: Normalized displacement fields following ARFI excitation for the FEM data (left column), tracked FEM data (center column), and phantom data (right column) for three different time steps (.8 ms top row,.5 ms center row,.82 ms bottom row). The modeled material and phantom were 8.5 kpa with an attenuation of.7 db/cm/mhz. Ultrasonic tracking was performed at a center frequency of 7 MHz, a transmit focal depth of 2 mm, and an F/.5 focal configuration on transmit and dynamic receive. 96

113 (a) Center Disp. (t =.8 ms) (b) Tracked FEM (c) Jitter Depth (mm) 5 Depth (mm) 5 Depth (mm) Displacement (µm) Displacement (µm) Jitter (µm) (d) Center Disp. (t =.5 ms) (e) Tracked FEM (f) Jitter Depth (mm) 5 Depth (mm) 5 Depth (mm) Displacement (µm) Displacement (µm) Jitter (µm) (g) Center Disp. (t =.82 ms) (h) Tracked FEM (i) Jitter Depth (mm) 5 Depth (mm) 5 Depth (mm) Displacement (µm) Displacement (µm) Jitter (µm) Figure 5.3: The solid lines in (a,d,g) represent the FEM displacement data, centered laterally,.8,.5, and.82 ms after ARFI excitation, respectively, as was shown in Figure 5.2(a,d,g). The dashed line represents the tracked displacement data along these center lines. The dashed line in (b,e,h) represents the same tracked data as in (a,d,g), while the solid line represents the tracked data after a 3.85 mm running average. The plots in (c,f,i) represent the jitter in the tracked displacement data after the solid lines were subtracted from the dashed lines in (b,e,h). Note that the underestimation of maximum displacement is worse in the early time step (a), and the focal zone jitter is worse immediately after excitation (c) than later in time (i). 97

114 Percent Maximum Displacement Tracked Time (ms) (a) RMS Jitter (µm) E = 8.5 kpa E = 23 kpa E = 58 kpa Time (ms) (b) Figure 5.4: Figure 5.4(a) shows the percentage of maximum displacement tracked in the focal zone in materials with Young s moduli of 8.5 (solid), 23 (dashed) and 58 (dotted) kpa at times ranging from ms after radiation force excitation. Note that accuracy of maximum displacement estimation improves more rapidly for the stiffer material (E = 58 kpa) than for the more compliant media. The jitter (b) decreases with time for all media and is greater in the more compliant media due to greater displacement magnitudes and slower shear wave speeds. 35 (a) E = 4 kpa 8 (b) E = 8.5 kpa 2 (c) E = 23 kpa Displacement (µm) Displacement (µm) Displacement (µm) Time (ms) Time (ms) Time (ms) Figure 5.5: Focal point displacement through time for 4. (a), 8.5 (b) and 23 kpa (c) materials. FEM displacement data is represented by the solid line with squares, while the tracked data is represented by the dashed line with circles. Note that the greater displacement underestimation occurs at the time of peak displacement. Also note that the jitter is greatest at the first time step, not the time of peak displacement. As the CRLB (Equation 5.) indicates, increasing the center frequency of the tracking beams should reduce jitter magnitude. Figure 5.7(a-b) shows the accuracy of maximum displacement estimates and jitter magnitude with increasing tracking frequency for a transducer with a fixed center frequency of 7.2 MHz and 53% fractional bandwidth, with a fixed.35 µs tracking kernel length for all frequencies. To 98

115 Percent Maximum Displacement Tracked Speckle SNR (a) RMS Jitter (µm) Speckle SNR (b) Figure 5.6: These plots demonstrate the impact of scatterer density, as represented by speckle SNR, on displacement tracking accuracy. The plot on the left indicates that lower speckle SNR is associated with greater displacement underestimation, while the plot on the right shows that jitter decreases with increasing speckle SNR. Note that fully developed speckle has an SNR of.9. explore the effects of bandwidth on different tracking frequencies, Figure 5.7(c-d) shows the same plots for transducers with 53% fractional bandwidth, but with a transducer center frequency at the specified tracking frequency. Again, the tracking kernel length was held at.35 µs for all frequencies. The impact of varying the center frequency of the transducer on the lateral transmit/receive beamwidth can be seen in Figure 5.8, where (a) shows the beamplots for a fixed 7.2 MHz center frequency, and (b) shows the beamplots for the variable center frequencies. This same transducer configuration is shown in Figure 5.7(e-f), except the kernel length was held at 2.5 cycles for each tracking frequency instead of.35 µs. The CRLB theoretical curves are shown in the dashed lines in Figure 5.7(d,f). Figure 5.9 shows the displacement underestimation at the focal depth (a) and the jitter (b) in an 8.5 kpa material where the force magnitude was scaled to achieve the maximum displacement values (at the focal point) shown on the x-axis. The effects of using different F/#s for the track beams were evaluated for an F/ push configuration. Figure 5. shows the displacement underestimation (a) and 99

116 Percent Maximum Displacement Tracked Percent Maximum Displacement Tracked Percent Maximum Displacement Tracked (a) Fixed Transducer Frequency Excitation Frequency (MHz) (c) Fixed.35 µs Kernel Length Excitation Frequency (MHz) (e) Fixed 2.5 Cycle Kernel Length Excitation Frequency (MHz) RMS Jitter (µm) RMS Jitter (µm) RMS Jitter (µm) (b) Fixed Transducer Frequency Excitation Frequency (MHz) (d) Fixed.35 µs Kernel Length Simulation CRLB Excitation Frequency (MHz) (f) Fixed 2.5 Cycle Kernel Length Simulation CRLB Excitation Frequency (MHz) Figure 5.7: These plots show the impact of transducer center frequency and bandwidth on displacement underestimation (left column) and jitter (right column) for multiple excitation frequencies. The plots in the top row had a fixed transducer center frequency of 7.2 MHz, with a fractional bandwidth of 53% and a constant kernel length of.35 µs. The center row maintains a fractional bandwidth of 53%, but the transducer center frequency was set to the tracking frequency. The bottom row shows the same configuration as the center row, except the kernel length was fixed at 2.5 cycles instead of a constant.35 µs. The CRLB is shown by the dashed line with circles in plots (d) and (f).

117 .8 5 MHz 7 MHz 9 MHz.8 5 MHz 7 MHz 9 MHz Normalized Intensity.6.4 Normalized Intensity Lateral Position (mm) (a) Lateral Beamwidth (Tx/Rx, Narrow BW).5.5 Lateral Position (mm) (b) Lateral Beamwidth (Tx/Rx, Broad BW) Figure 5.8: Plots of the lateral transmit/receive (Tx/Rx) beamwidths at the focal depth for a fixed fractional bandwidth of 53%. Plot Figure 5.8(a) represents a fixed transducer center frequency of 7.2 MHz, while Figure 5.8(b) represents a transducer center frequency that varies with the transmitted frequency. The solid lines represent a transmitted beam at 7 MHz, while the dashed and dotted lines represent frequencies of 5 and 9 MHz respectively. Note that in Figure 5.8(a), the beamwidth are almost identical, highlighting the limitation of the fractional bandwidth, which is overcome in Figure 5.8(b). Percent Maximum Displacement Tracked Maximum Displacement (µm) (a) RMS Jitter (µm) Maximum Displacement (µm) (b) Figure 5.9: Figure 5.9(a) shows that the relative displacement underestimation is not significantly affected by the magnitude of the displacement field, while Figure 5.9(b) shows that for larger displacements, jitter is directly related to displacement magnitude. jitter (b) immediately after excitation for track beams of varying transmit/receive track F/# configurations. Figure 5. shows the displacement profiles in the lateral dimension for materials with Young s moduli of (top row), 8.5 (middle row) and 23 kpa (bottom row) for

118 Percent Maximum Displacement Tracked ,,2 2, 2,2 Track F/#: Tx,Rx (a) RMS Jitter (µm) ,,2 2, 2,2 Track F/#: Tx,Rx (b) Figure 5.: Displacement underestimation (a) and jitter (b) immediately after excitation with an F/ push configuration for varying transmit (Tx) / receive (Rx) F/# combinations used for the tracking beams. F/ and F/2 pushing focal configurations,. ms after excitation. Figure 5.(a) shows the lateral beamwidth of the ARFI push beams (which are not apodized). The left and center columns show comparisons of the push beam lateral beamwidths (dashed lines) relative to the simulated displacement profiles (solid lines) in the lateral dimension for the F/ and F/2 excitations, respectively, by the dashed lines. The right column shows the comparison of the simulated F/ (solid) and F/2 (dashed) displacement profiles. Note that as the media stiffness increases, the differences in the displacement profiles between F/ and F/2 excitations decrease (g,j). Figure 5.2 shows the impact that a broader F/# focal configuration on the excitation has on the displacement underestimation (a) and jitter (b).2 ms after excitation for 4 (solid line), 8.5 (dashed line), and 23 kpa (dotted line) media. The tracking transmit and receive focal configurations were held at F/ for all pushing configurations. In applications of ARFI imaging where multiple focal depths for excitation are employed, it is experimentally advantageous to maintain a constant tracking transmit focal depth. Figure 5.3 shows how displacement underestimation (a) and jitter 2

119 (a) Lateral Beamplots F/ F/2 Normalized Intensity (b) F/, E = kpa.5.5 Lateral Position (mm) (c) F/2, E = kpa (d) Displacement Plots, E = kpa Normalized Displacement/Intensity Displacement Beamplot Normalized Displacement/Intensity Displacement Beamplot Normalized Displacement F/ F/2.5.5 Lateral Position (mm) (e) F/, E = 8.5 kpa.5.5 Lateral Position (mm) (f) F/2, E = 8.5 kpa.5.5 Lateral Position (mm) (g) Displacement Plots, E = 8.5 kpa Normalized Displacement/Intensity Displacement Beamplot Normalized Displacement/Intensity Displacement Beamplot Normalized Displacement F/ F/2.5.5 Lateral Position (mm) (h) F/, E = 23 kpa.5.5 Lateral Position (mm) (i) F/2, E = 23 kpa.5.5 Lateral Position (mm) (j) Displacement Plots, E = 23 kpa Normalized Displacement/Intensity Displacement Beamplot Normalized Displacement/Intensity Displacement Beamplot Normalized Displacement F/ F/2.5.5 Lateral Position (mm).5.5 Lateral Position (mm).5.5 Lateral Position (mm) Figure 5.: Displacement profiles in the lateral dimension for materials with Young s moduli of (top row), 8.5 (middle row) and 23 kpa (bottom row) for F/ and F/2 push focal configurations,. ms after the ARFI excitation. Figure 5.(a) shows the lateral beamwidth of the ARFI excitation beams (which are not apodized). The left and center columns show comparisons of the lateral beamwidths (dashed) relative to the resulting displacement profiles. ms after excitation (solid) in the lateral dimension for the F/ and F/2 excitations respectively, while the right column shows the comparison of the F/ (solid) and F/2 (dashed) displacement profiles. Note that the displacement profiles are more similar between F/ and F/2 excitation configurations in the stiffer media at this time due to the faster shear wave speed (g,j). 3

120 8.8 Percent Maximum Displacement Tracked kpa 8.5 kpa 23 kpa F/ F/.3 F/.5 F/.7 F/2. Push F/# (a) RMS Jitter (µm) F/ F/.3 F/.5 F/.7 F/2. Push F/# (b) Figure 5.2: This figure shows the reduction in displacement underestimation (a) when using higher F/# pushing focal configurations, while not significantly affecting the jitter levels (b). The tracking F/# focal configuration was F/ for both transmit and receive. The material was modeled as 4 (solid line), 8.5 (dashed line) and 23 kpa (dotted line), respectively. (b).2 ms after excitation vary as a function of tracking transmit focal depth for an ARFI excitation focused at mm. Percent Maximum Displacement Tracked Transmit Focal Depth (mm) (a) RMS Jitter (µm) Transmit Focal Depth (mm) (b) Figure 5.3: This figure shows that displacement estimation accuracy (a) and jitter (b) are not affected by tracking deep to the excitation focal depth. In this case, the excitation focal depth was mm, while the transmit focal depth on tracking was varied from - 2 mm, always using dynamic receive. To evaluate the impact of axial shearing within the tracking kernel, the data presented in Figure 5.2 and 5.3 was tracked using kernel lengths of.7 µs (.25 4

121 cycles) and.7 µs (5 cycles), and compared with the standard.35 µs (2.5 cycle) kernel length, as shown in Figure 5.4. Percent Maximum Displacement Tracked Kernel Size (µs) (a) RMS Jitter (µm) Kernel Size (µs) (b) Figure 5.4: This figure shows displacement underestimation (a) and jitter levels (b).2 ms after excitation for varying kernel lengths. Note that the larger kernel size reduces the jitter level, but for the sizes evaluated, the kernel length does not appear to impact accuracy of the displacement estimate. 5.5 Discussion Displacement underestimation and jitter in ultrasonic cross-correlation estimated displacements (Figure 5.3) is significant, where up to a 5% displacement underestimation can occur within the focal zone. As past work by McAleavey et al. [52] and the simulation results presented herein have shown, this is attributed to shearing in the lateral and elevation dimensions. This underestimation is dependent on material stiffnesses because shear wave propagation reduces shearing in the lateral and elevation dimensions relative to the track beam PSF, and this occurs more quickly for higher shear wave speeds (i.e., stiffer media). This result is confirmed in Figure 5.4, where in the stiffer medium (58 kpa) tracking accuracy improves sooner after excitation than in the more compliant media. Figure 5.4 also shows that the more compliant medium (E = 8.5 kpa) suffers from greater displacement underes- 5

122 timation later in time, concurrent with the later peak displacement in this more complaint media, as shown in Figure 5.5. Also notice in Figure 5.4(b) and Figure 5.5 that while the greatest displacement underestimation occurs at the time of peak displacement, the greatest jitter occurs at the first time step after excitation, independent of the material stiffness, and the jitter is greater for the more compliant media. Figure 5.3(a,c) also shows that the greatest displacement underestimation and jitter occurs in the focal zone, where shearing and displacement magnitude are the greatest. This indicates that jitter levels are a function of the displacement magnitude (Figure 5.9(b)) and the amount of shearing under the track PSF as dictated by shear wave propagation away from the ROE. Jitter increases with greater displacement magnitude, consistent with the findings of McAleavey et al. [52], and is due to a lower correlation coefficient during tracking. Displacement magnitude, however, does not appreciably change the relative amount of displacement underestimation that occurs, as shown in Figure 5.9(a), which is again consistent with the findings of McAleavey et al. [52]. Greater signal decorrelation also occurs in the presence of underdeveloped speckle, and as shown in Figure 5.6, results in greater displacement underestimation and jitter. Biological tissues, such as breast tissue, do not always exhibit fully developed speckle [89]. Even within a single field-of-view (FOV), structures of varying scatterer density may be present that could impact the relative accuracy of displacement tracking within those structures (e.g., fat versus muscle versus connective tissue). This would be most evident in the in vivo setting, where tissues such as the breast can be very heterogeneous. Given that shearing impacts the degree of displacement underestimation in the focal zone, it would be expected that using broader push beams relative to the tracking beams would reduce this underestimation, which is shown in Figure 5.2 where 6

123 improved tracking accuracy is achieved by using higher F/# push focal configurations. In contrast to the anticipated improvement in displacement tracking with more tightly focused track beams, Figure 5.(a) shows that using an F/ versus an F/2 tracking focal configuration does not provide a tracking benefit. Further insight is provided by the results in Figure 5.. Figure 5.(a) demonstrates that, as expected, the F/2 excitation beamwidth is broader in the lateral dimension than the F/ excitation beamwidth; however, as the comparison of the resulting displacement profiles in the lateral dimension from excitation using these two focal configurations show (d,g,j,m), the -6 db displacement widths are very similar for the stiffer media (E = 23 kpa). This demonstrates that the lateral displacement profile broadens with time after excitation at a rate that is dictated by the shear wave speed. Therefore, considerable tracking improvement is possible by using broader F/# focal configurations on the push (Figure 5.2). The modeled transducers had a fixed elevation focus, but the use of.5d or 2D arrays to control the elevation beamwidth in a similar manner to the lateral beamwidth would also allow for more accurate displacement tracking by reducing shearing in the elevation dimension. The benefits of increasing the tracking frequency, as predicted by the CRLB (Equation 5.), are only fully evident if the transducer has adequate bandwidth to take advantage of the higher excitation frequency. As shown in Figure 5.8(a), a fractional bandwidth of 53% does not allow the transducer to effectively operate at frequencies other than the center frequency. This was evident in Figure 5.7(a-b), where there was no appreciable difference in tracking accuracy nor jitter levels when using different tracking frequencies. By changing the center frequency to match the tracking frequency, as shown in Figure 5.8(b), the changes in beamwidth that are expected with changing frequency (Lateral Beamwidth λ F/#) are now evident and anticipated improvement in tracking are apparent in Figure 5.7(c-d). 7

124 The fixed kernel length of.35 µs in these plots caused larger variances for the lower frequencies where fewer number of cycles were included in each kernel as compared with the higher track frequencies. By holding a fixed 2.5 cycles per kernel, as is done experimentally, the jitter variances for the lower frequencies improved, as shown in Figure 5.7(f). This occurs because more samples are fed into the cross correlation algorithm when compared with the fixed.35 µs kernel length that includes 3.2 cycles at 9 MHz versus.8 cycles at 5 MHz. The lower mean jitter predicted by the simulations compared with the CRLB (Figure 5.7(d,f)) may be due to the fact that estimator bias leads to displacement underestimation since the tracking window is symmetric in both dimensions of motion, centered around zero displacement [75]. The CRLB does not take into account a biased estimator, which can lead to lower jitter than predicted by CRLB in the case of displacement underestimation [75]. Experimentally, it is advantageous to use a single track focal depth when multiple pushing focal depths are used, as is described by Nightingale et al. [6]. Tracking displacement deeper than the excitation focal depth is also advantageous to observe mechanical wave propagation and displacement fields that are generated deeper than the excitation focal depth. As Figure 5.3 shows, there is no penalty in using a deeper focal depth on track than on push. This is made possible by the use of dynamic receive to modify the receive focal configuration, and therefore the receive beamwidth, remains constant as a function of depth. Figure 5.4(a) demonstrates that the minimal axial shearing that occurs within the kernel lengths used for ARFI displacement tracking does not appreciably impact tracking accuracy. Jitter levels, however, were improved (Figure 5.4(b)) due to improved correlation achieved by using larger data ensembles in the cross correlation algorithms. 8

125 The FEM displacement field shown in Figure 5.2(a) does not exhibit near-field interference patterns that would be expected in the pressure fields generated during the ARFI excitations. The Field II output does reflect these interference patterns, as shown Figure 4.2(a); however, the continuum nature of the tissue and the spatial gradients present due to the excitation focal configurations cause the loss of these interference patterns before the first experimentally tracked time step. Therefore, the displacement variations present in the near-field in the tracked simulation and experimental data should not be mistaken for these interference patterns, but are attributed to the jitter associated with the displacement tracking algorithms. 5.6 Conclusions These simulation studies have demonstrated that lateral and elevation shearing of scatterers within the track PSF causes a significant underestimation in the displacements in response to ARFI excitations, while the minimal axial shearing over the kernel lengths used in ARFI imaging does not significantly impact displacement estimates. Jitter increases with increasing tissue displacement and increasing lateral and elevation shearing due to decreased correlation coefficients. Displacement underestimation and jitter decrease with time after the ARFI excitation as shear waves propagate away from the ROE, reducing the amount of shearing within the track PSF. In stiffer media with higher shear wave speeds, this improvement occurs sooner than in more compliant media. Using higher F/# excitations leads to less lateral shearing within the tracking beams and greater displacement tracking accuracy. Displacement underestimation and jitter are higher in materials with underdeveloped speckle (SNR <.9). With high-bandwidth transducers, displacement tracking with higher frequencies results in decreased jitter and reduced displacement under- 9

126 estimation. Finally, there is no loss in tracking accuracy in using a transmit track focus deeper than the ARFI excitation focus when using dynamic receive focusing. 5.7 Acknowledgments This work was supported by NIH grants R EB232 and R CA475, and the Medical Scientist Training Program grant T 32 GM-77. The authors would like to thank Jeremy Dahl and Gianmarco Pinton for their insights, Siemens Medical Solutions USA, Inc. Ultrasound Division for their technical assistance, and Joshua Baker-LePain for his technical assistance.

127 Chapter 6 Dynamic Mechanical Response of Spherical Inclusions to Impulsive Acoustic Radiation Force Excitation 6. Introduction Mechanical characterization of tissues and lesions within tissues has been used by clinicians to determine states of disease. Several pathological states can lead to a change in the mechanical properties of diseased tissues versus healthy tissues: fibrous tissue deposition in breast lesions may allow for masses to be found by manual palpation [4, 27, 89, 99], replacement of healthy hepatic tissue by fibrosis can lead to stiffening of a cirrhotic liver [2], and the purposeful destruction of tissue using radio-frequency (RF) ablation leads to the denaturing of proteins and an inflammatory response that stiffens the induced lesion [93]. Additionally, cancers and desmoplastic masses in the colon, breast, and other soft tissues can potentially be distinguished from healthy tissue based on their mechanical properties. Elastography techniques have been developed to characterize the mechanical properties of tissue by externally compressing tissue and tracking the resulting tissue response using ultrasonic methods [9, 39, 4, 53, 67, 68, 76, 82, 84, 86, 9, 93]. External

128 compression of tissues, however, becomes challenging for in vivo applications where organ location and control of boundary conditions become difficult. Acoustic radiation force imaging modalities remove the need for external tissue compression, and instead rely on the excitation of tissue by high-intensity acoustic pulses that transfer their momentum to the tissue and cause localized displacement fields. Several techniques involve monitoring the steady-state or resonant response of tissue to acoustic radiation force excitations, as is done by Walker et al. [98] in the Kinetic Acoustic Vitroretinal Examination (KAVE) method and Fatemi and Greenleaf [33] in Vibro-acoustography. Another application of acoustic radiation force imaging involves monitoring the dynamic response of tissue to impulsive excitation. The impulsive excitation of tissue leads to the generation of shear waves, and the propagation speed of these waves is determined by the viscoelastic properties of the tissue outside the region of excitation (ROE). Shear Wave Elasticity Imaging (SWEI) and supersonic imaging [6, 7, 78] monitor the shear wave speed to reconstruct shear modulus. These methods become challenging in vivo because wave attenuation limits the propagation distance that can be ultrasonically tracked in tissue. Acoustic Radiation Force Impulse (ARFI) imaging, the focus of this chapter, allows for the characterization of the dynamic response of tissue to impulsive excitation within the ROE. Finite Element Method (FEM) models are used to demonstrate how this localized dynamic information can be used to determine the local mechanical properties of the tissues and lesions (spherical inclusions) that are interrogated. This dynamic information is typically displayed as a set of parametric images showing the maximum displacement, time-topeak displacement, and the time it takes to recover 63% from the peak displacement. A sample set of parametric images from a tissue-mimicking phantom are shown in Figure 6. [42]. 2

129 2 B mode displ, Displacement t=.7ms, ~2µm Depth depth, (mm) Depth depth, (mm) Lateral Position (mm) time to peak,~.5ms recovery, ~.8ms Time to Peak Recovery Time displacement,µm Lateral lateral Position position, (mm) Figure 6.: Typical ARFI parametric images from a tissue-mimicking phantom. The conventional B-mode image is shown in the top left. The brighter pixels in the maximum displacement image (top right) represent greater displacement away from the transducer (top of the image), and range from - 2 µm. The brighter pixels in the time-to-peak displacement (bottom left) and recovery time (bottom right) images represent greater times from their respective references (initiation of the excitation pulse and time of peak displacement). 3