Dynamic Mechanical Response of Elastic Spherical Inclusions to ...

NIH Public Access Author Manuscript IEEE Trans Ultrason Ferroelectr Freq Control. Author manuscript; available in PMC 2007 March 7.

NIH-PA Author Manuscript

Published in final edited form as: IEEE Trans Ultrason Ferroelectr Freq Control. 2006 November ; 53(11): 2065–2079.

Dynamic Mechanical Response of Elastic Spherical Inclusions to Impulsive Acoustic Radiation Force Excitation Mark L. Palmeri, Stephen A. McAleavey, Kelly L. Fong, Gregg E. Trahey, and Kathryn R. Nightingale

Abstract


Acoustic Radiation Force Impulse (ARFI) imaging has been used clinically to study the dynamic response of lesions relative to their background material to focused, impulsive acoustic radiation force excitations through the generation of dynamic displacement field images. Dynamic displacement data are typically displayed as a set of parametric images, including displacement immediately after excitation, maximum displacement, time to peak displacement, and recovery time from peak displacement. To date, however, no definitive trends have been established between these parametric images and the tissue's mechanical properties. This work demonstrates that displacement magnitude, time to peak displacement, and recovery time are all inversely related to the Young's modulus of a homogeneous elastic media. Experimentally, pulse repetition frequency during displacement tracking limits stiffness resolution using the time to peak displacement parameter. The excitation pulse duration also impacts the time to peak parameter, with longer pulses reducing the inertial effects present during impulsive excitations. Material density affects tissue dynamics, but is not expected to play a significant role in biological tissues. The presence of an elastic spherical inclusion in the imaged medium significantly alters the tissue dynamics in response to impulsive, focused acoustic radiation force excitations. Times to peak displacement for excitations within and outside an elastic inclusion are still indicative of local material stiffness; however, recovery times are altered due to the reflection and transmission of shear waves at the inclusion boundaries. These shear wave interactions cause stiffer inclusions to appear to be displaced longer than the more compliant background material. The magnitude of shear waves reflected at elastic lesion boundaries is dependent on the stiffness contrast between the inclusion and the background material, and the stiffness and size of the inclusion dictate when shear wave reflections within the lesion will interfere with one another. Jitter and bias associated with the ultrasonic displacement tracking also impact the estimation of a tissue's dynamic response to acoustic radiation force excitation.


I. Introduction Mechanical characterization of tissues and lesions is often used by clinicians to determine states of disease [10]. 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 characterized by manual palpation [7], [2], [32], [36], replacement of healthy hepatic tissue by fibrosis can lead to stiffening of a cirrhotic liver [37], and the purposeful destruction of tissue using radio-frequency (RF) ablation leads to denaturing of proteins and an inflammatory response that stiffens the induced lesion [34]. 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 reconstructing material properties from the resulting strain fields [21], [30], [5], [11], [34], [26], [31], [9], [33], [20], [29], [17]. External

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tissue compression, however, becomes challenging for in vivo applications where organ location and boundary condition control can be difficult. Acoustic radiation force imaging modalities remove the need for external tissue compression and instead rely on the excitation of tissue using high-intensity acoustic pulses that transfer their momentum to the tissue and cause localized displacement fields [19], [27], [3], [24], [8]. Several techniques involve monitoring the steady-state or resonant response of tissue to acoustic radiation force excitations, as is done by Walker et al. [35] in the Kinetic Acoustic Vitroretinal Examination (KAVE) method, Fatemi and Greenleaf [8] in Vibroacoustography, and Konofagou and Hynynen in Harmonic Motion Imaging (HMI) [15]. Another application of acoustic radiation force imaging involves monitoring the dynamic response of tissues 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 [27], [3], [4] measure shear wave speeds to reconstruct shear moduli. These methods become challenging in vivo because viscosity attenuates the shear waves and limits the propagation distance over which these waves can be ultrasonically tracked in tissue and coupled into adjacent structures. Acoustic Radiation Force Impulse (ARFI) imaging, the focus of this manuscript, characterizes the dynamic response of tissue to impulsive excitation within and closely adjacent to the ROE.


In this manuscript, Finite Element Method (FEM) models are used to demonstrate how this dynamic displacement data in ARFI imaging can be used to characterize the local mechanical properties of elastic tissues and lesions (spherical inclusions). The dynamic displacement data are typically displayed as parametric images showing the maximum displacement, time to peak (TTP) displacement, and the time it takes to recover (RT) from the peak displacement. A sample set of parametric images from a tissue-mimicking phantom are shown in Figure 1 [12]. Section II reviews the mechanical response of elastic materials to impulsive acoustic radiation force excitation and the reflection and transmission of shear waves at discrete, elastic material interfaces. Section III outlines the implementation of the FEM to simulate the dynamic mechanical response of elastic materials to ARFI imaging excitations with a linear array, and the simulation of the ultrasonic tracking of these displacements. Section IV shows a parametric analysis of the impact of material properties (stiffness and density) and excitation parameters on the dynamic displacement fields that are generated and tracked during ARFI imaging in elastic media, while Section V correlates these simulated dynamic elastic material responses with elastic inclusion size, stiffness and location relative to the ROE.

II. Background NIH-PA Author Manuscript

The impulsive excitation of tissue with a localized distribution of acoustic radiation force, as is done in ARFI imaging, leads to the generation of shear (transverse) waves that propagate away from the ROE [27], [3], [18]. For details about the generation of acoustic radiation force in ARFI imaging, the reader is referred to Palmeri et al. [24]. In a linear, isotropic, elastic solid, the speed at which these shear waves propagate (cT ) away from the ROE is directly proportional to the shear modulus (μ) or Young's modulus (E) and is indirectly proportional to the density (ρ) and Poisson's ratio (ν) [1], [24]: cT =

μ = ρ

E

2(1 + ν )ρ

.

(1)

Compressional waves are not included in this analysis because they are not tracked on the time scales that are feasible with ARFI imaging (0.1 – 5 ms).

IEEE Trans Ultrason Ferroelectr Freq Control. Author manuscript; available in PMC 2007 March 7.

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When elastic spherical inclusions of varying stiffness are present in an otherwise homogeneous medium, shear waves are subject to propagation through materials with different shear wave speeds (Equation 1) and therefore different mechanical impedances (Z), as defined in Equation 2 [6]: Z = ρμ = ρcT .

(2)

Acoustic impedance mismatches at material interfaces lead to the reflection and transmission of shear waves that are incident upon these boundaries. If we consider two elastic, semi-infinite media (A and B) bound together, with a prescribed incident wave from medium A, we know that the displacements and normal stresses at the boundary between A and B must be equal for all times (t). Imposing these boundary conditions and relating stress to displacement through Hooke's law allows the following relationships to be defined for the reflected (R) and transmitted (T) waves in terms of the incident (I) wave [6]:

( ) ( )

Rt+


T t−

1−

ZB

( ) ( )

ZA x x I t+ = , A Z cT B cTA 1+ ZA x 2 x I t− = . B Z cT B cTB 1+ ZA

(3)

(4)

Equations 3 and 4 allow for the characterization of simple wave interactions orthogonal to the interface of semi-infinite media. For ARFI imaging of spherical inclusions, however, such simplifications are not possible because the ARFI excitation is a spatially-varying, threedimensional body force, and the spherical inclusions provide nonorthogonal boundaries relative to the direction of shear wave propagation. Additionally, semi-infinite media cannot be assumed relative to the lesion boundaries since the shear wave lengths are the same order of magnitude as the inclusion diameters. Therefore, the use of finite elements to simulate the mechanical response of tissue to the impulsive application of acoustic radiation force in three dimensions is necessary to study the tissue dynamics associated with ARFI imaging of spherical inclusions.

III. Methods NIH-PA Author Manuscript

FEM modeling has been previously implemented to simulate the tissue dynamics associated with ARFI imaging, as detailed by Palmeri et al. [24]. These models have been used herein to study the impact of elastic material properties (stiffness and density) on ARFI parametric images. A mesh consisting of 0.2 mm, tri-linear, cubic elements was used to simulate an elastic material extending 25 mm axially from the transducer surface, ± 17.5 mm laterally, and 7.5 mm in the elevation dimension. The mesh was generating using LS-PREPOST2 (Livermore Software Technology Corporation, Livermore, CA). Symmetry was assumed in the axiallateral plane, centered in elevation. The nodes in the lateral-elevation planes at depths of 0 (transducer position) and 25 mm were fully constrained [24]. The material was modeled as a linear, isotropic, elastic solid with a Poisson's ratio of 0.499. As described by Palmeri et al., this choice of Poisson's ratio was made because of simulation runtime concerns and because an explicit, time-domain finite element algorithm, as was used for these studies, cannot tolerate nearly incompressible materials [24]. While a Poisson's ratio of 0.499 will not support a compressional wave velocity of 1540 m/s (the approximate sound speed in tissue), the dynamics presented throughout this manuscript are shear-dominated and are minimally


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affected by this slight underestimation in Poisson's ratio, as indicated by Equation 1 [24], [22]. Custom written Perl scripts were used to define the stiffness of inclusions within the mesh based on the location of the center of the inclusion and the inclusion's diameter. No slip was allowed at the interface between inclusions and their background material. Field II1 [14], a linear acoustic field simulation package, was used to simulate the acoustic pressure fields that are generated during ARFI imaging with a linear array (fc = 7.2 MHz, F/ 1.3 lateral focal configuration, fixed elevation focus near 20 mm, variable focal depth) in materials with varying acoustic attenuations. The f-number (F/#) of an imaging array is defined as the ratio of the focal depth (z) to the active aperture width (d): F/ # =

z . d

(5)

Pressure fields simulated using Field II were then converted to intensities that were scaled up to empirically-determined intensity values using linear extrapolation of small-signal hydrophone measurements [24]. Nodal point loads were generated by multiplying the element volumes by the following expression for acoustic radiation force, expressed as a body force [19], [24]: F=

2αI


c

,

(6)

where α represents the acoustic attenuation of the material, c represents the sound speed (1540 m/s), and I represents the time-average intensity at each nodal location. LS-DYNA3D (Livermore Software Technology Corporation, Livermore, CA), an explicit, time-domain finite element analysis package was used to solve for the three-dimensional dynamic displacement fields. This modeling method has been previously validated in calibrated gelatinbased tissue-mimicking phantoms, as detailed by Palmeri et al. [24], and provides realistic three-dimensional data in response to impulsive acoustic radiation force excitation in linear, elastic media.


Ultrasonic estimation of displacements generated by impulsive acoustic radiation force excitations are susceptible to jitter and bias due to scatterer shearing within the point spread function (PSF) of the tracking beams [16], [23]. To evaluate the effects of these tracking artifacts on the observed displacement data, certain simulations incorporated Field II to simulate the ultrasonic tracking of the three-dimensional displacement fields simulated by the FEM. The development of this ultrasonic tracking model and the excitation and tracking parameters that affect jitter and bias in displacement estimates are detailed by Palmeri et al. [23]. Again, a linear array with a center frequency of 7.2 MHz and a fractional bandwidth of 53% was modeled for the tracking simulations. A focal configuration of F/1.0 was simulated on both transmit and receive, along with dynamic receive focusing and receive aperture growth. The transmit tracking focal depth was the same as the excitation focal depth. Field II was used to simulate synthetic RF data for 100 independent, fully-developed speckle realizations with the FEM-simulated displacements used to translate the scatterers in three-dimensions at each time step assuming a 10 kHz pulse repetition frequency (PRF). Normalized cross-correlation, as implemented experimentally (2.5 cycle kernel length, symmetric search, 99% overlap) [25], was applied to the RF data to estimate displacements. The tracking simulation run times are great; therefore the tracking simulations were limited in scope throughout the manuscript. Figure 2 shows a typical set of normalized simulated displacement fields through time in the imaging plane of a 4 kPa elastic material with an attenuation of 0.7 dB/cm/MHz after a 45 μs 1http://www.es.oersted.dtu.dk/staff/jaj/field/ IEEE Trans Ultrason Ferroelectr Freq Control. Author manuscript; available in PMC 2007 March 7.

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excitation focused at 20 mm with an F/1.3 focal configuration. Throughout this manuscript, plots at the focal point show the displacement at the intersection of the two dashed lines in Figure 2 through time, while displacement profiles shown at depths other than the focal depth are plotted along the vertical dashed line in Figure 2.

IV. Results A. ARFI-Induced Dynamics in Homogeneous Media The dynamic displacement fields that are generated in a homogeneous elastic medium after an impulsive excitation with focused acoustic radiation force are dependent on material parameters that affect the speed at which shear waves propagate, namely the material's shear modulus and density (Equation 1), and excitation parameters, including the lateral F/# and the excitation duration. Figures 3 – 7 explore the dependence of the homogeneous elastic material dynamics, including displacement magnitude, TTP displacement, and RT from peak displacement, on these parameters.


A comparison of the simulated dynamic displacement behavior at the focal point with the simulated ultrasonically-tracked displacement is shown in Figure 3. For this simulation, the material was modeled as a homogeneous elastic medium with an attenuation of 0.7 dB/cm/ MHz and a stiffness of 4 kPa. The displacements determined by the FEM models are shown in the solid line, while the simulated tracked displacements are shown in the dashed line, both of which are sampled at an experimentally feasible rate of 10 kHz. The excitation focal configuration was F/1.3 with a focal depth of 20 mm. The displacement dynamics at the focal point are dependent on the stiffness of the material and the lateral F/# of the excitation beam, with greater F/#'s representing larger lateral excitation beamwidths. Figure 4(a) shows the displacement profiles at the focal point in media of varying stiffness (1 kPa and 58 kPa) for two excitation focal configurations (F/1.0 and F/2.0), where the higher F/# represents a broader excitation in the lateral dimension. Figure 4(b) shows how TTP displacement varies as a function of stiffness for the two excitation focal configurations. A material's density will impact the shear wave speed (Equation 1), which impacts the displacement dynamics at the focal point. Figure 5(a) shows the impact of density on the displacement profiles at the focal point, while Figure 5(b) shows the impact of density in TTP displacement in a 4 kPa medium with an attenuation of 0.7 dB/cm/MHz, excited with a focal configuration of F/1.3 focused at 20 mm. Notice how the less dense materials have greater peak displacement magnitudes and peak sooner in time than the more dense materials.


The inertial effects demonstrated by modulating the density of the material, as shown in Figure 5, are also dependent on the duration of the excitation pulse. Figure 6(a) shows the impact of excitation pulse duration on the focal point displacement profiles in an 8.5 kPa medium with an attenuation of 0.7 dB/cm/MHz and an F/1.3 excitation focused at 20 mm. Figures 6(b,c) show the impact of pulse duration on TTP displacement from the initiation and cessation of the excitation pulse, respectively. Figure 6(d) shows how the maximum displacement magnitude at the focal point increases with increasing pulse duration, while Figure 6(e) shows the normalized displacement profiles in the lateral dimension at the focal depth for pulse durations of 22.5 μs (solid line) and 720 μs (dashed line), 1.5 ms after the excitation. In addition to affecting the time it takes to reach peak displacement at the focal point, the stiffness of an elastic material will also affect the time it takes to recover from that peak displacement. Figure 7(a) shows the focal point displacement through time in homogeneous media of varying stiffnesses ranging from 4 – 48 kPa, while Figure 7(b) shows the normalized recovery behavior for each simulated material as a function of time after each material's


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respective peak displacement. The horizontal dotted line represents the recovery level (37%) at which recovery time parametric images are typically generated in ARFI imaging.


B. ARFI-Induced Dynamics in Media with Spherical Inclusions While the dynamic displacement fields generated in homogeneous elastic materials in response to impulsive excitations display fairly intuitive trends with modulation of material and excitation parameters, the presence of an elastic spherical inclusion within or adjacent to the ROE complicates the displacement dynamics. Characterizing all of the effects that an elastic spherical inclusion has on the 3D displacement data is beyond the scope of a single manuscript; therefore, to demonstrate the impact of such inclusions, a subset of case studies are presented. The dynamic displacement data at the focal depth are presented as a function of relative and absolute inclusion stiffnesses for excitations both within and outside an inclusion with a 3 mm diameter. The positions of simulated ARFI excitations relative to the inclusion are shown in Figure 8, with the FEM displacement data in Figures 9 – 11. Finally, since the dynamic behavior demonstrated in these figures is dependent on the size of the inclusion, Figure 12 demonstrates the impact that lesion size, as a function of stiffness contrast, has on the resulting displacement contrast.


Both the stiffness contrast between a spherical inclusion and its background material, along with their absolute stiffnesses, dictate the displacement dynamics. Figure 9 demonstrates that the dynamic behavior of spherical inclusions is a function of the stiffness of the inclusion and background materials. These models simulated a 3 mm diameter inclusion, centered at the focal depth (11.5 mm), excited with an F/1.3 focal configuration. Each plot contains the dynamic displacement profiles at the focal point for five different configurations: (1) homogeneous background material, (2) homogeneous inclusion material, (3) excitation centered in the spherical inclusion, (4) excitation 1 mm off-center, 0.5 mm inside the inclusion, and (5) excitation 2 mm off-center, 0.5 mm outside the inclusion (Figure 8). The inclusions in Figure 9(a,b) have stiffnesses of 12 and 24 kPa, respectively, and a background stiffness of 4 kPa, while the inclusions in Figure 9(c,d) have stiffnesses of 24 and 48 kPa, respectively, and a background stiffness of 8 kPa. The displacement profiles simulated by the FEM models are not directly representative of what would be measured experimentally when ultrasound is used to detect these displacement because ultrasonic displacement estimation is subject to jitter and bias [16], [23]. The FEM displacement profiles shown in Figure 9(b) are normalized in Figure 10(a), while Figure 10 (b–c) show mean and mean ± one standard deviation for the corresponding tracked displacement estimates over 100 speckle realizations.


The inflections present in the displacement profiles at the focal depth shown in Figures 9 and 10 are dependent on the location of the excitation relative to the inclusion boundary (Figure 8), and the absolute and relative stiffnesses of the inclusion and background materials. Figure 11(a–c) shows the FEM displacement profiles in the lateral dimension at the focal depth (11.5 mm) for homogeneous media of 4, 12, and 24 kPa, respectively, taken from the same data set used to generate the corresponding focal point displacement profiles shown in Figure 7. Figure 11(d–f) shows the FEM displacement profiles in the lateral dimensions for a 4 kPa background material with a 3 mm diameter, 12 kPa inclusion centered at a depth of 11.5 mm for excitations that are laterally centered, 1.0 mm off-center, and 2.0 mm off-center, as was shown in Figure 9(a), while Figure 11(g–i) shows the corresponding displacement profiles for a 24 kPa inclusion, as was shown in Figure 9(b). An F/1.3 focal configuration focused at 11.5 mm was used for all of these simulations. All displacement profiles were normalized by their peak displacement magnitudes 0.1 ms after excitation. The inclusion boundaries are indicated by the vertical dashed lines.


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The results presented to this point have been for an inclusion with a 3 mm diameter. However, since the dynamics observed at the excitation focal depths are dependent on the interference of transmitted and reflected shear waves, these behaviors are dependent on the shear wave speed and the size of the inclusion. To demonstrate the effects that lesion size and stiffness have on displacement profiles at different times after radiation force excitation, Figure 12 shows plots of the ratio of mean displacement inside varying diameter inclusions centered at a depth of 11.5 mm compared with the displacement in the otherwise homogeneous background material, 0.2, 0.5, and 0.9 ms after excitation (top, middle, and bottom rows, respectively). Each plot also contains an “ideal” curve representing the expected ratio of displacements in homogeneous media of the inclusion and background stiffnesses for a steady–state excitation (inclusion:background). This curve provides a reference for relative lesion behavior in all of the plots in Figure 12 that is independent of (1) the size of the inclusion and (2) the dynamic nature of this analysis. All ratios are normalized by the displacement at the focal point in the homogeneous medium at the specified time to show displacement trends relative to the absence of an inclusion. Background stiffness of 4 kPa, 8 kPa, and 12 kPa are shown on the plots to demonstrate the dependence of these dynamic displacement trends on background stiffness.

V. Discussion NIH-PA Author Manuscript

The displacement magnitude and the TTP displacement are dependent on an elastic material's stiffness when all other excitation (F/# and focal depth) and material parameters (density and attenuation) are held constant, as shown in the simulated dynamic focal point displacements in Figures 4(a) and 7(a). The tracked displacement profile, shown in the dashed line in Figure 3, demonstrates that considerable jitter and underestimation of magnitude exist in the ultrasonically-estimated displacements [23], reducing the utility of displacement magnitude alone as a sole indicator of material stiffness. As predicted by Sarvazyan et al., TTP displacement appears to be a promising measure of stiffness that would be independent of force magnitude [27]; however, its potential can be limited experimentally by PRF, as shown in Figure 4(b).


As shown in Figure 4, the trends in TTP displacement as a function of stiffness in the focal zone are independent of excitation focal configuration and displacement magnitude. In general, the F/2.0 excitation leads to greater TTP displacements as compared to F/1.0 due to a broader cross-sectional area of the excitation beam. The TTP displacement parameter thus can be expected to be independent of acoustic attenuation within the focal zone, as the lateral and elevation displacement profiles are not attenuation dependent [22]. For a fixed excitation pulse duration, TTP displacement represents a dynamic parameter that reflects material stiffness (Figure 4(b)). In heterogeneous elastic media, TTP displacement, if able to be experimentally realized, would be a better indicator of material stiffness than recovery time since nearby material boundaries cause shear wave reflections within the ROE that influence the measurement of recovery time at the center of the excitation, as shown in Figure 10. While the simulation results shown in Figure 4 demonstrate the benefits of using TTP displacement measurements to estimate material stiffness, experimental realization of this measurement is made challenging by several factors. Jitter, motion artifacts, limited PRFs, and noise in clinical measurements may present significant challenges in making accurate TTP displacement estimates to resolve stiffness with any precision. Displacement underestimation due to scatterer shearing under the track PSF also tends to bias TTP displacement estimates to be later in time [23]. While the times to peak displacement range from 0.1 – 0.5 ms in Figure 4(b), the excitation was only applied for 45 μs in all of these simulations. The difference between TTP displacement and the excitation duration is due to inertial (mass) effects, where the tissue acceleration


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generated by the radiation force is opposed by the elastic restorative forces of the tissue. In stiffer media, these restorative forces are greater, and therefore recovery occurs faster than in more compliant media (Figure 7). Figure 5 demonstrates the impact of mass on the time to peak displacement, where less dense tissues peak sooner, and thus recover sooner, than more dense tissues due to smaller inertial contributions. Figure 5 also demonstrates that variations in density can also create significant errors when correlating TTP displacements with material stiffness if assuming constant density. For the range of densities simulated (0.5 – 1.5 g/cm3) in a 4 kPa background, the TTP displacement varied by 0.12 ms (Figure 5(b)). This could lead to an error of several kilopascals in the stiffness estimate based on this metric. In stiffer media, the error in TTP displacement for this range of densities will decrease, but the times themselves also decrease, leading to larger stiffness estimate errors. Again, in the stiffer media, experimental realization of TTP displacements below 0.1 ms will be challenging, independent of density errors, due to PRF limitations.


Different clinical applications of ARFI imaging require excitations of variable duration to achieve tissue displacements of at least several microns to be tracked with adequate SNR [23]. Varying the pulse duration, however, also impacts the dynamic parameters than can be extracted from ARFI images, including TTP displacement. As Figure 6(a) shows, longer excitations generate a more steady-state tissue response than more impulsive excitations. As shown in Figure 6(b–c), using excitation pulse durations that are too long (> 100 μs) negates the utility of the TTP displacement parameter as the tissue approaches a steady-state response. The longer pulse durations will also reach a plateau in the maximum displacement achieved at the focal point, as shown in Figure 6(d). Finally, as expected, the longer excitation pulse durations also generate shear waves with longer wavelengths, as shown in Figure 6(e).


As demonstrated in Figures 4 and 7, both TTP displacement and RT are dependent on the stiffness of a homogeneous medium and are dictated by shear wave speed, thus they could be used to quantify material stiffness. These trends are also independent of displacement magnitude, which is beneficial in a clinical arena, where the unknown attenuation of heterogeneous tissue prevents accurate estimation of the acoustic radiation force magnitude used in the excitation. In addition, ultrasonic displacement magnitude estimates can considerably underestimate the true displacement (Figure 3) [23], which reduces the clinical utility of using displacement magnitude alone to quantify tissue stiffness. While ARFIgenerated displacement images of relative differences in estimated displacement magnitude between tissue types should accurately reflect stiffness differences, they are not quantitative. The mechanical information that can be inferred from TTP displacement and RT would allow for quantification of stiffness in homogeneous regions of tissues, but interpreting these parameters becomes more complicated when a spherical inclusion with varying material properties is present in the medium. Figure 9 demonstrates how the recovery of the material at the focal depth is influenced by the stiffness of the medium, the presence of a stiffer spherical inclusion, and the location of an inclusion boundary relative to the excitation. The displacement profiles in the homogeneous media represent the typical response to impulsive excitation (also seen in Figure 3), where the peak displacement magnitude, TTP displacement, and RT are inversely related to the material stiffness [24]. However, when an inclusion is present, the dynamic displacement profile in and near the inclusion is related to the stiffnesses of the inclusion and background materials. For excitations inside the inclusion, the initial tissue response is dictated by the inclusion stiffness, while later in time the tissue response reflects the stiffness of the background material and is impacted by shear wave reflections at the inclusion boundaries. For excitations that occur just outside a stiffer inclusion, the displacement response is dictated by the background stiffness along with inverted shear waves generated by reflections at the inclusion boundaries, resulting


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in a faster recovery than the surrounding material. Figure 9 demonstrates that these trends hold for different ratios of background to inclusion stiffness and absolute background stiffnesses.

NIH-PA Author Manuscript NIH-PA Author Manuscript

Figure 10(a) shows the normalized displacement plots from Figure 9(b), with the horizontal dotted line indicating where recovery time (37%) is typically defined in ARFI imaging. For the excitations inside the inclusion, distinct shear wave reflections off the inclusion boundaries (1.5 mm to each side) can be seen at the center location 1.4 ms after the excitation. The timing of these reflections varies with inclusion stiffness, diameter, and proximity of the excitation to material boundaries (Figure 11). Figure 10(b), which shows the mean tracked displacement behavior at these locations, demonstrates how ultrasonic displacement tracking significantly impacts the estimation of TTP displacement and RT. At the center location, the constructive interference of reflected shear waves at 1.4 ms would be measured as the peak displacement (instead of the actual peak at 0.2 ms in (a)), and the recovery time parameter would not be measured since the material does not recover to 37% of the measured peak displacement during the tracked times. This behavior has been observed in experimental phantom data, as shown in Figure 1, where the stiffer inclusion appears to peak later and recover slower than the more compliant background material, which is opposite the dynamic behavior we would expect in homogeneous media. Such trends have also been observed in some clinical images of breast masses [28]. Also notice that the error bar magnitudes in Figure 10(c) demonstrate that displacement estimates can vary up to 40% between speckle realizations, in the absence of simulated noise, due to jitter and shearing within the track PSF [23]. These variations would also cause variations in the estimates of TTP displacement and RT. Figure 11 shows the corresponding displacement profiles in the lateral dimension at the focal depth to those shown in Figure 9(a,b) and Figure 10. The homogeneous media shown in Figure 11(a–c) (top row) demonstrate how shear wave speed, RT at the center of the ROE, and TTP displacement at the focal point are inversely related to stiffness. Figure 11(d–i) (bottom two rows) shows the distortion of shear waves caused by the presence of a stiffer spherical inclusion. For the excitation centered in the inclusion (g), the displacement at the center grows later in time (1.4 ms) as the shear waves reflecting off of the inclusion boundaries constructively interfere, agreeing with the results shown in Figure 10. For the more compliant 12 kPa inclusion (d), this constructive interference occurs later in time and with decreased magnitude due to the smaller impedance mismatch. This behavior is not observed for excitations closer to the inner boundary of the inclusion (h), while for excitations just outside of the inclusion (f & i), a significant displacement null is created later in time. The displacement gradients in the lateral dimension are less severe fore the more compliant 12 kPa inclusion (e–f) than for the 24 kPa inclusion (h–i).


As expected, Figure 12 demonstrates that inclusion size, stiffness contrast between inclusion and background material, and stiffness of the background material dictate the dynamic mechanical response of an inclusion to an ARFI excitation since all of these parameters affect shear wave interference patterns. This is demonstrated as deviations from the “ideal” displacement to stiffness ratios curve that is independent of lesion size and tissue dynamics. Immediately after excitation (Figure 12(a,b)), the initial displacements across the inclusion are largely independent of these variables. Later in time (0.5 ms, Figure 12(c,d)), the larger inclusion (d) nicely matches the ideal curve, but the smaller inclusion (c) never reaches the ideal curve since the closer proximity of the inclusion boundaries to the excitations generates appreciable interference of reflected shear waves soon after the excitation. As the smaller inclusion increases in stiffness (Figure 12(e)), repeated shear wave reflections at the lesion boundaries during the observation time cause the displacement ratio (inside:outside) to increase, causing deviation from the shape of the ideal curve. In contrast, for the larger inclusion (Figure 12(f)), the trends were consistent with the ideal curve.


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A limitation of these models is that they assume a linear elastic material and do not incorporate viscoelastic effects. Future models will be developed to quantify the impact of viscoelasticity on the dynamic responses presented in this manuscript, however some general conclusions can be made. Viscoelasticity of biological tissues is characterized by a stiffness relaxation over a given time domain, using a “short-time” shear modulus (Go), a “long-time” shear modulus (G∞), and a decay constant (β), such that the dynamic shear modulus of the viscoelastic tissue can be represented by G(t) = G∞ + (Go − G∞)e−βt [13]. In general, however, the appropriate viscoelastic parameters for soft tissues are difficult to characterize. Tissue relaxation of at least several kilopascals (Go − G∞ > 1 kPa) within several milliseconds (β > 1 kHz) would cause a dynamic softening of the tissue while displacements were being tracked after a radiation force excitation. Such tissue relaxation would delay the TTP displacement and extend the relaxation times of the tissue. Tissue relaxations in this range would also lead to kinetic and strain energy loss that would reduce displacement magnitudes and reduce the distance over which trackable shear wave displacements would propagate, and potentially reduce the shear wave interference patterns observed within the simulated elastic lesions. The shear waves may also experience phase distortion that would change the shape of the shear waves observed later in time. Tissue relaxation that occurred over larger time domains (β < 1 kHz) would behave like an elastic material with G = Go, while tissue relaxation that occurred much faster (β >> 1 kHz) would respond like an elastic material with G = G∞. Viscoelastic tissues that fell within these ranges would be adequately modeled using the elastic material definitions used in this manuscript.


The shear wave reflections and transmissions demonstrated in these inclusion models were for discrete material boundaries between the elastic background material and the inclusion. In the presence of a stiffness gradient, the impedance mismatch would decrease, and the reflection magnitudes would decrease in favor of shear wave transmission. These models also do not take into account slip boundary conditions that may exist between inclusions and their backgrounds. The presence of a slip condition would not allow the inclusion interface to support a shear stress, and the continuity boundary conditions imposed to derive Equations 3 and 4 would no longer be valid.

VI. Conclusions


In homogeneous elastic media, the dynamic displacement profiles tracked within the ROE are dependent on the underlying material stiffness and density, with time to peak displacement, displacement magnitude, and recovery time all being inversely related to tissue stiffness. Near the focal depth, these parameters are also dependent on excitation focal configuration and duration. Spherical inclusions with different material properties than the background material generate shear wave reflections and transmissions at material interfaces. The magnitude and timing of shear wave interference within an elastic inclusion are related to the absolute and relative stiffnesses of the inclusion and background materials, along with the diameter of the inclusion. These shear wave dynamics, combined with ultrasonic displacement tracking artifact, contribute to the the dynamic displacement behaviors that have been experimentally observed when imaging stiff, spherical inclusions during phantom and clinical studies. Overall, these studies correlate the dynamic displacement behaviors generated in response to impulsive acoustic radiation force with the radiation force excitation parameters and the elastic material properties of structures in the imaged media. Acknowledgements This work was supported by NIH grants R01 EB002132, R01 CA114075, R01 CA114093 and the Coulter Foundation. The authors would also like to thank Siemens Medical Solutions USA, Inc. Ultrasound Division for their technical assistance, and Joshua Baker-LePain for his technical assistance.


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1. Achenbach, J. Wave Propagation in Elastic Solids. Elsevier Science Publishers; The Netherlands: 1999. 2. Bassett, L.; Gold, R.; Kimme-Smith, C. Hand-Held and Automated Breast Ultrasound. Slack, Thorofare; New Jersey: 1986. 3. Bercoff J, Tanter M, Fink M. Supersonic shear imaging: A new technique for soft tissue elasticity mapping. IEEE Trans Ultrason, Ferroelec, Freq Contr 2004;51(4):396–409. 4. Bercoff J, Tanter M, Muller M, Fink M. The role of viscosity in the impulse diffraction field of elastic waves induced by the acoustic radiation force. IEEE Trans Ultrason, Ferroelec, Freq Contr 2004;51 (11):1523–1536. 5. Bilgen M, Insana MF. Elastostatics of a spherical inclusion in homogeneous biological media. Phys Med Biol 1998;43(1):1–20. [PubMed: 9483620] 6. Davis, J. Introduction to Dynamics of Continuous Media. Macmillan Publishing Company; New York: 1987. 7. Donegan, W.; Spratt, J. Cancer of the Breast. Saunders; Philadelphia: 2002. 8. Fatemi M, Greenleaf J. Vibro-acoustography: An imaging modality based on utrasound-stimulated acoustic emission. Proc Natl Acad Sci 1999;96:6603–6608. [PubMed: 10359758] 9. Fu D, Levinson S, Gracewski S, Parker K. Noninvasive quantitative reconstruction of tissue elasticity using an iterative forward approach. Phys Med Biol 2000;45(6):1495–1509. [PubMed: 10870706] 10. Greenleaf JF, Fatemi M, Insana M. Selected methods for imaging elastic properties of biological tissues. Annu Rev Biomed Eng 2003;5:57–78. [PubMed: 12704084] 11. Hall T, Zhu Y, Spalding C. In vivo real-time freehand palpation imaging. Ultrasound Med Biol 2003;29(3):427–435. [PubMed: 12706194] 12. Hall TJ, Bilgen M, Insana MF, Krouskop TA. Phantom materials for elastography. IEEE Trans Ultrason, Ferroelec, Freq Contr 1997;44(6):1355–65. 13. Herrmann, L.; Peterson, F. A numerical procedure for viscoelastic stress analysis; In Proceedings of the Seventh Meeting of ICRPG Mechanical Behavior Working Group; 1968. 14. Jensen J, Svendsen N. Calculation of pressure fields from arbitrarily shaped, apodized, and excited ultrasound transducers. IEEE Trans Ultrason, Ferroelec, Freq Contr 1992;39(2):262–267. 15. KonofagouEHynynenKLocalized harmonic motion imaging: theory, simulations and experimentsUltrasound Med Biol2910140514132003 [PubMed: 14597337] 16. McAleaveySNightingaleKTraheyGEstimates of echo correlation and measurement bias in acoustic radiation force impulse imagingIEEE Trans Ultrason, Ferroelec, Freq Contr5066316412003 17. McKnight A, Kugel J, Rossman P, Manduca A, Hartmann L, Ehman R. MR elastography of breast cancer: Preliminary results. AJR 2002;178(6):1411–1417. [PubMed: 12034608] 18. Nightingale K, McAleavey S, Trahey G. Shear wave generation using acoustic radiation force: in vivo and ex vivo results. Ultrasound Med Biol 2003;29(12):1715–1723. [PubMed: 14698339] 19. Nyborg, W. Acoustic streaming. In: Mason, W., editor. Physical Acoustics. IIB. Academic Press Inc; New York: 1965. p. 265-331.chapter 11 20. Oliphant T, Manduca A, Ehman R, Greenleaf J. Complex-valued stiffness reconstruction fro magnetic resonance elastography by algebraic inversion of the differential equation. Magnetic Resonance in Medicine 2001;45:299–310. [PubMed: 11180438] 21. Ophir J, Alam S, Garra B, Kallel F, Konofagou E, Krouskop T, Varghese T. Elastography: Ultrasonic estimation and imaging of the elastic properties of tissue. Proc Inst Mech Engrs 1999;213:203–233. 22. Palmeri, M. Imaging the mechanical properties of tissue with ultrasound: An investigation of the response of soft tissue to acoustic radiation force. Duke University; 2005. PhD thesis 23. M, Palmeri; S, McAleavey; G, Trahey; K, Nightingale. Ultrasonic tracking of acoustic radiation forceinduced displacements in homogeneous media. IEEE Trans Ultrason, Ferroelec, Freq Contr. in press 24. Palmeri M, Sharma A, Bouchard R, Nightingale R, Nightingale K. A finite-element method model of soft tissue response to impulsive acoustic radiation force. IEEE Trans Ultrason, Ferroelec, Freq Contr 2005;52(10):1699–1712. 25. Pinton, G.; McAleavey, S.; Dahl, J.; Nightingale, K.; Trahey, G. Real-time acoustic radiation force impulse imaging; Proceedings of the 2005 SPIE Conference on Medical Imaging; 2005.


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26. Plewes D, Bishop J, Samani A, Sciarretta J. Visualization and quantification of breast cancer biomechanical properties with magnetic resonance elastography. Phys Med Biol 2000;45(1):1591– 1610. [PubMed: 10870713] 27. Sarvazyan A, Rudenko O, Swanson S, Fowlkes J, Emelianov S. Shear wave elasticity imaging: A new ultrasonic technology of medical diagnostics. Ultrasound Med Biol 1998;24(9):1419–1435. [PubMed: 10385964] 28. Sharma, A.; Soo, M.; Trahey, G.; Nightingale, K. Acoustic radiation force impulse imaging of in vivo breast masses. In Proceedings of the 2004 IEEE Ultrasonics, Ferroelectrics and Frequency Control Joint Symposium; 2004. 29. Sinkus R, Lorenzen J, Schrader D, Lorenzen M, Dargatz M, Holz D. High-resolution tensor MR elastography for breast tumour detection. Phys Med Biol 2000;45(6):1649–1664. [PubMed: 10870716] 30. Steele D, Chenevert T, Skovoroda A, Emelianov S. Three-dimensional static displacement, stimulated echo NMR elasticity imaging. Phys Med Biol 2000;45(1):1633–1648. [PubMed: 10870715] 31. Taylor L, Porter B, Rubens D, Parker K. Three-dimensional sonoelastography: Principles and practices. Phys in Med Biol 2000;45:1477–1494. [PubMed: 10870705] 32. E, Tohno; Cosgrove, D.; Sloane, J. Ultrasound Diagnosis of Breast Diseases. Churchill Livingstone Inc., 650 Avenue of the Americas; New York, NY 10011: 1994. 33. Van Houten E, Weaver J, Miga M, Kennedy F, Paulsen K. Elasticity reconstruction from experimental MR displacement data: initial experience with an overlapping subzone finite element inversion process. Medical Physics 2000;27(1):101–107. [PubMed: 10659743] 34. Varghese T, Zagzebski J, Lee F. Elastographic imaging of thermal lesions in the liver in vivo following radiofrequency ablation: Preliminary results. Ultrasound Med Biol 2002;28:1467–1473. [PubMed: 12498942] 35. Walker W, Fernandez F, Negron L. A method of imaging viscoelastic parameters with acoustic radiation force. Phys Med Bio 2000;45(6):1437–1447. [PubMed: 10870702] 36. Wolfe, J. Xeroradiography of the Breast. Thomas; Springfield, IL: 1972. 37. Yeh W, Li P, Jeng Yea. Elastic modulus measurements of human liver and correlation with pathology. Ultrasound Med Biol 2002;28(4):467–474. [PubMed: 12049960]

NIH-PA Author Manuscript IEEE Trans Ultrason Ferroelectr Freq Control. Author manuscript; available in PMC 2007 March 7.

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Fig. 1.

Typical ARFI parametric images from a tissue-mimicking phantom with a 3 MM diameter spherical inclusion that is four times stiffer than the background material, centered at a depth of 15 MM. The conventional B-mode image is shown in the top left and does not delineate the inclusion. The brighter pixels in the maximum displacement image (top right) represent greater displacement away from the transducer (top of the image), and displacements range from 0 – 12 μM. As expected, the more compliant background material moves farther than the stiffer inclusion material in the displacement image. The brighter pixels in the time to peak (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), providing additional dynamic metrics to delineate the inclusion.


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NIH-PA Author Manuscript Fig. 2.


Normalized displacement fields through time in the imaging plane of a 4 kPa elastic material with an attenuation of 0.7 dB/cm/MHz after a 45 μs excitation focused at 20 MM with an F/ 1.3 focal configuration. The transducer would be located at the top of each plot, and brighter pixels indicate greater displacements away from the transducer. Notice the shear waves that propagate away from the ROE through time. Plots shown throughout this manuscript at the focal point are shown at the intersection of the dashed lines, while displacement profiles at other depths are shown along the vertical dashed line.


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Displacement at the focal point through time in a 4 kPa homogeneous medium with an attenuation of 0.7 dB/cm/MHz. The FEM data is shown in the solid line, while tracked FEM data is shown in the dashed line (mean ± one standard deviation of 100 independent speckle realizations). The data points correspond to an experimentally-feasible PRF of 10 KhZ.


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(A) Normalized, dynamic, focal point displacement for two stiffnesses (1 kPa and 58 kPa, α = 0.7 dB/cm/MHz) and two excitation focal configurations (F/1.0 and F/2.0). (B) Time to peak displacement as a function of stiffness for two excitation focal configurations (F/1.0 and F/ 2.0).


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(A) Displacement through time at the focal point (20 MM) as a function of tissue density in a 4 kPa material with an attenuation of 0.7 dB/cm/MHz. (B) Time to peak displacement at the focal point as a function of tissue density. Note that the less dense materials have greater peak displacement magnitudes and peak earlier in time than the more dense materials.


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NIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author Manuscript

Fig. 6.

The impact of pulse duration on the focal point displacement in an 8.5 kPa medium with an attenuation of 0.7 dB/cm/MHz and an F/1.3 excitation focused at 20 MM is shown in a, while (B) and (C) show the impact of pulse duration on the time to peak parameter relative to the initiation and cessation of the excitation pulse, respectively. The effect of pulse duration on the maximum displacement in the focal zone is shown in (D), while the impact of pulse duration on normalized displacement profiles in the lateral dimension at the focal depth of 20 MM, 1.5 MS after the excitation is shown in (E).


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Fig. 7.

(A) Focal point displacements through time for homogeneous media of 4, 8, 12, 24, and 48 kPa. The attenuation for all simulated materials was 0.7 dB/cm/MHz, and an F/1.3 focal configuration excitation focused at 11.5 MM and applied for 45 μs was used for the simulated ARFI excitation. (B) Normalized recovery behavior for the plots shown in (A), with the time axis referring to time after peak displacement. The horizontal dotted line represents where the recovery level (37%) at which recovery time parametric images are usually generated in ARFI imaging.


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Excitation locations relative to the inclusion position for the simulations shown in Figures 9 and 10.


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NIH-PA Author Manuscript NIH-PA Author Manuscript Fig. 9.


Displacement through time at the focal depth (11.5 MM) for excitations at different lateral positions relative to the inclusion boundaries. The model simulated a 3 MM diameter inclusion centered at a depth of 11.5 MM, excited with an F/1.3 focal configuration. The background material was set at 4 kPa in the top row and 8 kPa in the bottom row, with the inclusions being three times stiffer than the background in the left column, and six times stiffer than the background in the right column.


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Fig. 10.

Normalized FEM displacement profiles from Figure 9(B). The mean ultrasonically-tracked displacement profiles are shown in (B), and the mean tracked displacement profiles ± 1 standard deviation are shown in (C). Notice the change in time to peak and recovery time of the center displacement profile (circles) between the FEM data (A) and the tracked displacement profiles (B & C). The horizontal dotted line represents where the recovery level (37%) at which recovery time parametric images are usually generated in ARFI imaging.


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Fig. 11.

Top Row: Shear waves propagating away from the ROE in homogeneous media of (A) 4, (B) 12, and (C) 24 kPa. Middle Row: Shear waves propagating away from the ROE for excitations that are (D) centered, (E) 0.5 MM from the inner inclusion boundary and (F) 0.5 MM outside the inclusion boundary, in a 3 MM diameter, 12 kPa inclusion, with a background stiffness of 4 kPa. Bottom Row: Shear waves propagating away from the ROE for excitations that are (G) centered, (H) 0.5 MM from the inner inclusion boundary and (I) 0.5 MM outside the inclusion boundary, in a 3 MM diameter, 24 kPa inclusion, with a background stiffness of 4 kPa. All displacement profiles are normalized to their respective displacement profiles at 0.1 MS.


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Fig. 12.

Mean displacement ratio (inside:outside) for a 1 MM diameter inclusion (left column) and a 6 MM diameter inclusion (right column), centered at the focal depth of 11.5 MM, versus the input stiffness ratio (inclusion:background) for background stiffnesses of 4, 8, and 12 kPa. The displacement ratios are normalized to the homogeneous displacements at the focal point for each time step, 0.2, 0.5, and 0.9 MS after excitation (top, middle, and bottom rows, respectively). The “ideal” curves (solid lines) represent the expected displacement ratios in homogeneous media under steady-state excitations (inclusion:background).


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