Dynamic Simulation of Viscoelastic Soft Tissue in Acoustic Radiation Force Creep Imaging Xiaodong Zhao Department of Mechanical and Aerospace Engineering, Rutgers, the State University of New Jersey, 98 Brett Road, Piscataway, NJ 08854-8058 e-mail: [email protected]
Assimina A. Pelegri1 Fellow ASME Department of Mechanical and Aerospace Engineering, Rutgers, the State University of New Jersey, 98 Brett Road, Piscataway, NJ 08854-8058 e-mail: [email protected]
Acoustic radiation force (ARF) creep imaging applies step ARF excitation to induce creep displacement o f soft tissue, and the cor responding time-dependent responses are used to estimate soft tis sue viscoelasticity or its contrast. Single degree o f freedom (SDF) and homogeneous analytical models have been used to character ize soft tissue viscoelasticity in ARF creep imaging. The purpose of this study is to investigate the fundamental limitations of the commonly used SDF and homogeneous assumptions in ARF creep imaging. In this paper, finite element (FE) models are developed to simulate the dynamic behavior o f viscoelastic soft tissue sub jected to step ARF. Both homogeneous and heterogeneous models are studied with different soft tissue viscoelasticity and ARF con figurations. The results indicate that the SDF model can provide good estimations for homogeneous soft tissue with high viscosity, but exhibits poor peiformance for low viscosity soft tissue. In addition, a smaller focal region o f the ARF is desirable to reduce the estimation error with the SDF models. For heterogeneous media, the responses of the focal region are highly affected by the local heterogeneity, which results in deterioration of the effective ness o f the SDF and homogeneous simplifications. [DOI: 10.1115/1.4027934]
Introduction Biomechanical imaging techniques with ARF have been devel oped for soft tissue characterization and detection of tumors. These methods use focused ultrasound transducers to generate a highly localized ARF to excite the region of interest noninvasively [1], Based on ARF excitation, shear wave imaging methods have enjoyed great popularity for quantitative imaging of soft tis sue elasticity or viscoelasticity [2-5], A potential drawback to the shear wave imaging methods is that they measure the attenuation of tissue responses outside the region of excitation where the dis placements of soft tissue are much smaller and the corresponding estimation of viscoelasticity is sensitive to jitter due to the low signal-to-noise ratio (SNR) of the measurements [6]. Alterna tively, the ARF induced creep imaging methods use the responses in the region of excitation where the measured displacements have a high SNR. In these methods, SDF models that represent point-by-point characteristics have been used to model the relation
'Corresponding author. Manuscript received October 28, 2013; final manuscript received June 20, 2014; accepted manuscript posted July 1, 2014; published online July 15, 2014. Assoc. Editor: Jeffrey Ruberti.
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between the creep displacement and the step ARF [7-11], Conse quently, the soft tissue viscoelasticity can be estimated by fitting these simplified models with the measured time-dependent creep displacement. The physical basis for the SDF assumption is that the ARF induced motion is highly localized, and the response of the tissue is assumed to be only related to the local distribution of the mechanical properties. It should be noted that the SDF assumption also implies a homogeneous assumption in the analy sis of each response, since the interactions between tissue ele ments are not considered in the SDF models. However, these approximations become challenging when considering the three dimensional (3D) nature of the original dynamic problem and in the presence of heterogeneity [12,13], The purpose of this study is to investigate the fundamental limi tations of the commonly used SDF and homogeneous assumptions in ARF creep imaging. This is the first time, to the authors’ knowledge, that the above assumptions pertinent to the highly localized ARF excitation are evaluated. Numerical models have been commonly used to study the performance of biomechanical reconstruction methods, because it is easy to investigate the recov ery algorithm thoroughly by changing the model configurations [14]. In this study, numerical experiments are performed with a FE method that is capable of modeling complicated geometries, material heterogeneities, ARF configurations, and boundary con ditions of the dynamic system for soft tissue imaging [12,13], For the homogeneous model, the SDF assumption is evaluated quanti tatively by analyzing the inverse problem with SDF models. For the heterogeneous model, a qualitative analysis is performed by comparing the dynamic responses of the homogeneous and the heterogeneous models, which demonstrates the limitations of the SDF and homogeneous assumptions in the presence of model heterogeneity.
Methods In order to present a complete picture of the topic, we first dis cuss two SDF models for ARF creep imaging developed by other groups. We follow with a brief description of the ARF model. Finally, the FE models and evaluation methods undertaken in this study are demonstrated in detail. SDF Models in Acoustic Radiation Force Creep Imaging. The soft tissue in this study is assumed to be an isotropic, linear viscoelastic and near-incompressible solid (Poisson’s ratio v = 0.499 [12,13]). The linear viscoelasticity of tissue follows the Voigt model, as it is simple and effective in modeling the visco elastic behavior of soft tissue and tissue mimicking phantoms [5,7,15], In ARF creep imaging, the creep response of viscoelastic soft tissue has been studied with the relation [7,9-11], u( t ) =A{ l - e ~ ‘/T),
x=p
(1)
where u{t) is the creep displacement of the focal point, A is the steady state displacement decided by the configuration of the ARF and shear modulus (fi), tj is the shear viscosity, and t is the time constant that describes the ratio of shear viscosity to shear modu lus of the material. Based on Eq. (1), t can be estimated by fitting to the time-dependent creep displacement, which in practice is measured by the diagnostic ultrasound transducer. Equation (1) is actually an approximation stemming from the high localization of the ARF with the consequent assumption that the inertial effect of the system can be neglected. Here, we will evaluate the errors associated with this approximation. In order to consider the inertial effect in this dynamic problem, Viola and Walker [8] modified the model by adding an inertial component in series with the Voigt model, and solved for the time-dependent displacement induced by a step force. However, as we will illustrate henceforth, this modified model does not improve the estimation accuracy in the inverse problem due to the
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fundamental limitations of the SDF assumption. Our analysis can be extended to other SDF models developed with different repre sentations (or rheological models) of soft tissue viscoelasticity, because the fundamental limitations investigated here are not associated with the constitutive models but with the SDF and ho mogeneous assumptions, which neglect the 3D dynamic system effects and interactions between tissue elements. Modeling Acoustic Radiation Force. ARF is a spatial-varying body force. The focal zone is a Gaussian-shaped beam, and sim plified models, such as a 3D Gaussian function or an ellipsoid of rotation, have been used to model the ARF distribution [16,17]. These models are simplified patterns of the actual ARF distribu tion, but they can adequately capture the main properties of the spatial-varying body force [16]. In our analysis, the step ARF is described by a 3D Gaussian function similar to [17] to mimic ARF generated by a 3.25 MHz focused transducer with focal length 50 mm, diameter of the aperture 60 mm (F-number is 0.83), and inner diameter 22.6 mm (Sonic Concepts, Bothell, WA). Note that F-number indicates the ratio of the transducer’s focal length over the diameter of the aperture. The SDF model is studied for different ARF configurations in the simulation. First, ARFs with the same relative distribution but different magnitude, 0.5/o,/ o, and 2/c (f0 is the spatial peak density of the body force), are assigned as the body force magnitude. Then, for ARFs with the same magnitude, the relative distribution is changed by modeling transducers with different F-numbers, namely, 1.5, 0.83, and 0.5. The first one has a larger focal zone and the third one has a smaller focal zone compared to the one discussed above (Fnutnber = 0.83). The analytical solution for the ARF generated displacement field, uz(z), in an infinite isotropic homogeneous medium can be obtained by convolving the solution for a point load in an infinite isotropic homogeneous solid with the spatial-varying body force field [18], which gives 1
u2{z)
f”
f2n [°° (
16^(1 - V ) J - J o
3— .
3 - 4v
(z - z , f
Jo V ( > / ( z _ ^ 2 + ^ ) 3
\ \f(r,z,)rdrd0dzt
(2)
\j(z —z* )2 + r2) J where z and z* are the axial distances with respect to the origin in the z direction, r is the radial distance from z-axis, f(r,z) is the body force density, and v is the Poisson’s ratio.
Finite Element Models. Axisymmetric FE models are devel oped to simulate the viscoelastic soft tissue using A baqus 6.11 FE software [19], as illustrated in the diagram of Fig. 1(a). The bot tom boundary of the FE model is fixed in all directions; the remaining surfaces of the model are not constrained. For each case studied in this paper, the ARF is always applied along the axial direction (z direction). The origin of the coordinate system of this model is located at the center of the symmetry axis. The axisymmetric model consists of 5366 nodes and 5275 four-node bilinear axisymmetric quadrilateral elements with hybrid formula tion. Convergence of the model is checked by mesh refinement to ensure sufficiency of the mesh density. The FE simulations are implemented with A baqus implicit dynamic analysis. In order to evaluate the effectiveness of the SDF and homogeneous assump tions in describing soft tissue dynamic behavior, as well as to explore their limitations, we simulate a homogenous and a hetero geneous case. Case 1. Homogeneous Model: Different mechanical properties and ARF configurations are investigated. Four shear moduli Gu = 1 kPa, 3 kPa, 9 kPa, and 27 kPa) and three time constants (t = 0.0003 s, 0.0009 s, and 0.0027 s) are first studied with the FE model. The chosen shear moduli are within the range of reported mechanical properties of the normal or tumorous human soft tis sues [20,21]. The time constants used are on the same order as reported values for soft tissues or phantoms [5,7,10], The SDF assumption is evaluated quantitatively by analyzing the inverse problem with the SDF model. FE simulations generate the data of the numerical experiments. Fitting the SDF models to these data formulates an optimization problem in which the mean squared error between displacement computed from FE simulations and the displacement computed from the SDF models fitted to that data should be minimized. Since the purpose of this study is to investigate the fundamental limitations of the SDF and homogeneous assumptions in ARF creep imaging, these data do not contain noise as in practical measurements in order to control variables in the analysis. The results should be considered as the best theoretically achievable ones. Case 2. Heterogeneous Model: In order to demonstrate the limi tations of the SDF and homogeneous assumptions in the presence of tissue heterogeneity, a qualitative analysis is performed by comparing the temporal and steady state responses of the homoge neous model to that of the heterogeneous model. In this study, spherical inclusions are located in the center of the model, as illus trated in Fig. 1(a). There are two cases considered: spheres with diameters of 3 mm and 6 mm. Both the background and inclusion are assigned with the same time constant. Three background and
Fig. 1 Model diagram and validation: (a) Diagram of the axisym m etric model with a spherical inclusion in the center of the model and (b) axial displacem ent induced by ARF for the hom oge neous case with // = 3 kPa. The horizontal axis in (b) is the axial distance from the focal center. 0 9 4 5 0 2 -2
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inclusion shear modulus combinations, (fj.B, /.if), are studied, namely, (3 kPa, 3 kPa), (0.3 kPa, 3 kPa), and (30 kPa, 3 kPa). Under the homogeneous assumption, the steady state displace ment, A, is inversely proportional to the underlying tissue elastic ity (or shear modulus). In the heterogeneous media, the observed elasticity contrast, can be described by
* - '0 —
^background ~~A ^inclusion
where CQ is the observed elasticity contrast of the inclusion to background, A background is the steady state displacement of the background, and Ainclusion is the steady state displacement inside the inclusion [7,22]. Here, /lincluSK)n is measured at the center of the inclusion, where the response is least affected by the heteroge neity. For regions near the boundary of the inclusion and back ground, the response can be more severely affected by heterogeneity [13], which results in a further reduced elasticity contrast. The ideal performance of a biomechanical imaging method is that the observed elasticity contrast equals the true elas ticity contrast.
Results and Discussion In this study, numerical experiments are performed by FE simu lations. In order to validate the FE model, the FE simulated steady state responses are first compared to the analytical solution (Eq. (2)) for the homogeneous case with p = 3 kPa. The ARF in the FE model is applied at the origin of the model. The computed axial displacements induced by the ARF are shown in Fig. 1(b). The analytical solution is derived for an infinite medium, while the FE model has boundary to mimic a practical configuration. The results are in good agreement between the FE simulation and analytical solution with slight difference at the two ends of the curves, indicating that the global boundary condition has little effect on the response of inside tissue in the localized ARF imag ing methods [13]. For the homogeneous model with different mechanical proper ties, the creep displacement for each case is shown in
(d)
Figs. 2(a)-2(c). Even though the time constant is different, the near steady state displacement is almost identical for the same elasticity and is inversely proportional to the elasticity of the tis sue, which confirms that the steady state displacement is a good metric to image the elasticity relation between homogeneous models if the same ARF is applied in the focal zone [7-9], In order to show the temporal behavior of the response, the dis placements are normalized by the corresponding steady state dis placement in Figs. 2(d)-2(f). For the same time constant, based on the SDF model of Eq. (1), the temporal behavior is expected to be identical for different shear moduli. However, Eq. (1) is derived from a SDF model without inertial component, in which the stress is proportional to the applied force and the creep strain is propor tional to the creep displacement. This is not the case for a 3D model with the inertial effects considered. It is noticed that the response curve deviates from the response of the SDF model with out inertia as the shear modulus of the model decreases. In addi tion, as the time constant increases, all the response curves converge to the response of the SDF model without inertia. Since the SDF model cannot ideally capture the dynamic behavior of a 3D system with mass, its relative estimation error (REE) is plotted in Fig. 3(a). In this figure, the horizontal axis is in logarithmic scale. The REE can be quite large for models with low time constant and elasticity, e.g., 142% for fi = 1 kPa and t = 0.0003 s. It should be noted that even the REE decreases as the elasticity increases for the same time constant, the REE increases as the elasticity increases for the same viscosity, as shown by the markers in Fig. 3(a). In a Voigt model, the viscosity is proportional to the elasticity for the same time constant. This means that for the same time constant, the REE decrease is due to the increased viscosity, not elasticity. These results imply that the SDF model tends to have better approximation for tissue with high viscosity and low elasticity, in which cases the SDF model may still be preferred in qualitative imaging for its simplicity. Fig ure 3(b) illustrates the REE with the SDF model that included the inertial component. The results demonstrate that adding the iner tial component into the SDF model does not significantly improve the estimation accuracy, because for a 3D dynamic system, not only the focal region but also the surrounding tissue is induced to
(e)
(f)
Fig. 2 Creep displacem ent responses to step ARF for soft tissue with different viscoelasticity. Three tim e constants are studied: (a) r = 0.0003 s, (b) z = 0.0009 s, and (c) z = 0 .0 0 2 7 s. The corre sponding norm alized creep displacem ent responses are shown in (d), (e), and (/).
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(a)
(b)
Fig. 3 REE of t by fitting the SDF models with the FE sim ulated creep displacem ent responses, (a) SDF model w ithout considering the inertial effect and (b) SDF model with the inertial com po nent included. Each m arker represents cases with the same shear viscosity.
motion by the ARF, and the complex system inertia cannot be accurately accounted for by a SDF dynamic system, which is the fundamental limitation of the SDF approach. The effects of the ARF configurations on the dynamic response of soft tissue are illustrated in Fig. 4. The steady state response is
(a)
affected by both the magnitude and distribution of the body force. However, the temporal response of soft tissue is independent of the body force magnitude, as shown in Fig. 4(c). The results con firm that the time constant is a “force-free” parameter in the anal ysis of the SDF model, as indicated in [7,8,10]. In addition, it
(b)
Fig. 4 Creep displacem ent responses to different ARF configurations: (a) ARFs with different m agnitude, but the same distribution, i.e., F nUm ber = 0.83; and ( b) ARFs with different distribu tion, but the sam e m agnitude, i.e., f0. T he corresponding norm alized creep displacem ent responses are shown in (c) and (d). The model is hom ogeneous with n = 3 kPa, and t = 0.0009 s.
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shows that the time constant depends on the body force distribu tion. Figure 4 ( d ) exhibits that the induced responses by a smaller focal zone are closer to the output of the SDF model without iner tia. The REEs of t for ARF corresponding to F „ u m b e r = 1.5, 0.83, and 0.5 are 68%, 27%, and 12%, respectively. The results suggest that decreasing the focal zone can reduce the inertial effects on the dynamic responses of soft tissue, which means that when using the SDF model in ARF creep imaging, a smaller focal zone is desired to reduce the REE. This is similar to other imaging methods, e.g., ARFI and HMI, in which smaller focal zones result in better image contrast [22,23], In order to qualitatively illustrate the effect of heterogeneity and the fundamental limitation of the homogeneous assumption in modeling the dynamic behavior of soft tissue to step ARF, the creep displacement responses at the origin of the heterogeneous models (center of the spherical inclusion) are shown in Fig. 5. The ARF is considered as a highly localized force in most ARF based imaging methods and the dynamic response is considered to be only related to the underlying local mechanical properties, which is equivalent to a homogeneous assumption of the imaged tissue. However, the results indicate that the local heterogeneity greatly affects the dynamic response of the focal zone. For inclusion with the same shear modulus, (/q = 3 kPa), the responses vary as the background’s moduli change. In addition, the steady state dis placements can no longer reflect the true elasticity contrast of the
Time (ms)
(a)
Time (ms)
(c)
T able 1 m o d e ls
O b s e rv e d
e la s tic ity
c o n tra s t
fo r
h e te ro g e n e o u s
Observed elasticity contrast Mi (kPa) 3 3 3
(kPa)
True elasticity contrast
3-mm-diameter inclusion
6-mm-diameter inclusion
0.3 3 30
10 1.0 0.10
2.9 1.0 0.18
4.2 1.0 0.13
Mb
inclusion to the background as shown in Table 1. As the dimen sion of the inclusion increases from 3 mm to 6 mm in diameter the observed elasticity contrast gets closer to the true elasticity contrast. The ARF induced displacement is the integration of the normal strain in the axial direction. Figure 6 explains the reduced contrast for heterogeneous media by demonstrating the effects of heteroge neity on the axial normal strain field with the case of the 3-mmdiameter spherical inclusion. The dashed line shows the profile of the inclusion. Even though the inclusions have the same shear modulus, it is shown that the strain fields are quite different for the three cases due to the differences in heterogeneity. Figure 6(a)
Time (ms)
(b)
Time (ms)
(d)
Fig. 5 C re e p d is p la c e m e n t re s p o n s e s a t th e o rig in o f th e h e te ro g e n e o u s m o d e ls w ith d iffe re n t in c lu s io n s ize s : (a) S p h e re w ith d ia m e te r 3 m m a n d (b) S p h e re w ith d ia m e te r 6 m m . T h e c o rre s p o n d in g n o rm a liz e d c re e p d is p la c e m e n t re s p o n s e s a re sh o w n in (c) an d (d ). T h e s o lid b la c k lin e d e n o te s th e h o m o g e n e o u s c a se, z = 0 .0 0 0 3 s is fo r b o th b a c k g ro u n d a n d in c lu s io n .
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+2.9e-03 +2.4e-03 +2.0e-03 +1.5e-03 +1.0e-03 +5.2e-04 +4.1e-05 -4.4e-04 -9.2e-04 -1.4e-03 -1.9e-03 -2.4e-03 -2.8e-03
+3.5e-03 +2.9e-03 +2.4e-03 +1.8e-03 +1.2e-03 +6.5e-04 +7.9e-05 -4.9e-04 -l.le-03 -1.6e-03 -2.2e-03 -2.8e-03 -3.3e-03
(b) Fig. 6 Axial normal strain field near the region of the 3-m m -diam eter spherical inclusion after a 10 ms step ARF excitation. The dim ension of the region is h = 6 mm (axial length) and r = 3 mm (radial length). The tim e constant is 0.0009s and the shear moduli are: (a) /js = 3kP a, and Hi = 3 kPa; (b) h b = 0-3 kPa, and hi = 3 kPa; and (c) hb = 30 kPa, and hi = 3 kPa.
is the case for a homogeneous medium. For Fig. 6(b), when the inclusion is in a soft background, the resulting displacement of the focal zone relates primarily to the strain of the background, and it reflects not only the elasticity of the inclusion but also of the back ground, or the heterogeneous structure of the local area. For an inclusion in a hard background, as shown in Fig. 6(c), the strain field is constrained by the hard background, which results in a reduced displacement compared to the homogeneous case. The temporal responses of soft tissue to step ARF are even more complex in the presence of heterogeneity. For each figure of the normalized displacements in Fig. 5, the background and inclu sion are assumed to have the same time constant, and ideally the responses are expected to be identical according to the SDF assumption. However, for hard inclusions in soft background, the system tends to have larger time constants, i.e., the response needs more time to reach the steady state. At the beginning, before the shear wave reaches the background, the creep displacement results from the strain inside the inclusion. When the shear wave reaches the soft background, the creep will continue in the back ground and the induced strain is still relatively large because the background is much softer than the inclusion, similar to the case in Fig. 6(b). On the contrary, for a soft inclusion in hard back ground the motion inside the inclusion is constrained and the induced strain in the background is relatively small, thus the sys tem tends to have a smaller time constant, i.e., the displacement rises faster to its maximum value than in the homogeneous case. In that case, the shear wave reflection in the boundary leads to an overshoot in some of the responses as shown in Fig. 5. Even though the SDF model has its fundamental limitations, it’s still preferable in many cases involving high viscous soft tis sues because it is simple and minimizes the number of parameters in the optimization, which leads to suboptimal, but stable and ro bust estimation. In addition, this estimation can be improved by reducing the focal zone size. In the cases of soft tissues with low viscosity, performance of the SDF models is quite poor, and inverse procedure based on FE methods may be introduced. Even though lots of the prior information for the FE modeling (local heterogeneity profile, ARF distribution, etc.) cannot be accurately obtained, proper integration of this prior information and its uncertainty could help to improve estimation accuracy, or to enhance the observed contrast, comparing to the SDF models that do not utilize any of that prior information. This topic is currently under investigation by our group. Considering heterogeneity, if the inclusion region is large, the responses in the early stage are not affected by the local heterogeneity as shown in Fig. 5(b), in which case the early responses may still be used under the homo geneous assumption. 0 9 4 5 0 2 -6
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Conclusions Finite element models are developed to study the dynamic behavior of viscoelastic soft tissue subjected to step ARF. Based on the FE simulations, the fundamental limitations of the com monly used SDF and homogeneous assumptions are studied for ARF creep imaging. The results suggest that the SDF model tends to have good approximation for tissue with high viscosity and low elasticity. Especially, for the low viscosity case, the estimation results based on the SDF model can be quite poor. In addition, accounting for the inertial component into the SDF model cannot effectively improve the estimation accuracy, but reducing the size of the focal zone can result in better estimation. For heterogeneous models, both the temporal and steady state responses of the focal region are highly affected by the local heterogeneity, rendering the SDF and homogeneous simplification ineffective. In order to obtain better estimation or enhanced contrast, FE analysis procedure considering the 3D configuration of the dynamic sys tem and model heterogeneity may be necessary for the inverse characterization.
Acknowledgment This research was supported by NSF CMMI-0900596. The authors kindly acknowledge the support of NSF CMMI division and the program managers Dr. Demitris Kouris and Dr. Dennis R. Carter.
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Assimina A. Pelegri1 Fellow ASME Department of Mechanical and Aerospace Engineering, Rutgers, the State University of New Jersey, 98 Brett Road, Piscataway, NJ 08854-8058 e-mail: [email protected]
Acoustic radiation force (ARF) creep imaging applies step ARF excitation to induce creep displacement o f soft tissue, and the cor responding time-dependent responses are used to estimate soft tis sue viscoelasticity or its contrast. Single degree o f freedom (SDF) and homogeneous analytical models have been used to character ize soft tissue viscoelasticity in ARF creep imaging. The purpose of this study is to investigate the fundamental limitations of the commonly used SDF and homogeneous assumptions in ARF creep imaging. In this paper, finite element (FE) models are developed to simulate the dynamic behavior o f viscoelastic soft tissue sub jected to step ARF. Both homogeneous and heterogeneous models are studied with different soft tissue viscoelasticity and ARF con figurations. The results indicate that the SDF model can provide good estimations for homogeneous soft tissue with high viscosity, but exhibits poor peiformance for low viscosity soft tissue. In addition, a smaller focal region o f the ARF is desirable to reduce the estimation error with the SDF models. For heterogeneous media, the responses of the focal region are highly affected by the local heterogeneity, which results in deterioration of the effective ness o f the SDF and homogeneous simplifications. [DOI: 10.1115/1.4027934]
Introduction Biomechanical imaging techniques with ARF have been devel oped for soft tissue characterization and detection of tumors. These methods use focused ultrasound transducers to generate a highly localized ARF to excite the region of interest noninvasively [1], Based on ARF excitation, shear wave imaging methods have enjoyed great popularity for quantitative imaging of soft tis sue elasticity or viscoelasticity [2-5], A potential drawback to the shear wave imaging methods is that they measure the attenuation of tissue responses outside the region of excitation where the dis placements of soft tissue are much smaller and the corresponding estimation of viscoelasticity is sensitive to jitter due to the low signal-to-noise ratio (SNR) of the measurements [6]. Alterna tively, the ARF induced creep imaging methods use the responses in the region of excitation where the measured displacements have a high SNR. In these methods, SDF models that represent point-by-point characteristics have been used to model the relation
'Corresponding author. Manuscript received October 28, 2013; final manuscript received June 20, 2014; accepted manuscript posted July 1, 2014; published online July 15, 2014. Assoc. Editor: Jeffrey Ruberti.
Journal of Biomechanical Engineering
between the creep displacement and the step ARF [7-11], Conse quently, the soft tissue viscoelasticity can be estimated by fitting these simplified models with the measured time-dependent creep displacement. The physical basis for the SDF assumption is that the ARF induced motion is highly localized, and the response of the tissue is assumed to be only related to the local distribution of the mechanical properties. It should be noted that the SDF assumption also implies a homogeneous assumption in the analy sis of each response, since the interactions between tissue ele ments are not considered in the SDF models. However, these approximations become challenging when considering the three dimensional (3D) nature of the original dynamic problem and in the presence of heterogeneity [12,13], The purpose of this study is to investigate the fundamental limi tations of the commonly used SDF and homogeneous assumptions in ARF creep imaging. This is the first time, to the authors’ knowledge, that the above assumptions pertinent to the highly localized ARF excitation are evaluated. Numerical models have been commonly used to study the performance of biomechanical reconstruction methods, because it is easy to investigate the recov ery algorithm thoroughly by changing the model configurations [14]. In this study, numerical experiments are performed with a FE method that is capable of modeling complicated geometries, material heterogeneities, ARF configurations, and boundary con ditions of the dynamic system for soft tissue imaging [12,13], For the homogeneous model, the SDF assumption is evaluated quanti tatively by analyzing the inverse problem with SDF models. For the heterogeneous model, a qualitative analysis is performed by comparing the dynamic responses of the homogeneous and the heterogeneous models, which demonstrates the limitations of the SDF and homogeneous assumptions in the presence of model heterogeneity.
Methods In order to present a complete picture of the topic, we first dis cuss two SDF models for ARF creep imaging developed by other groups. We follow with a brief description of the ARF model. Finally, the FE models and evaluation methods undertaken in this study are demonstrated in detail. SDF Models in Acoustic Radiation Force Creep Imaging. The soft tissue in this study is assumed to be an isotropic, linear viscoelastic and near-incompressible solid (Poisson’s ratio v = 0.499 [12,13]). The linear viscoelasticity of tissue follows the Voigt model, as it is simple and effective in modeling the visco elastic behavior of soft tissue and tissue mimicking phantoms [5,7,15], In ARF creep imaging, the creep response of viscoelastic soft tissue has been studied with the relation [7,9-11], u( t ) =A{ l - e ~ ‘/T),
x=p
(1)
where u{t) is the creep displacement of the focal point, A is the steady state displacement decided by the configuration of the ARF and shear modulus (fi), tj is the shear viscosity, and t is the time constant that describes the ratio of shear viscosity to shear modu lus of the material. Based on Eq. (1), t can be estimated by fitting to the time-dependent creep displacement, which in practice is measured by the diagnostic ultrasound transducer. Equation (1) is actually an approximation stemming from the high localization of the ARF with the consequent assumption that the inertial effect of the system can be neglected. Here, we will evaluate the errors associated with this approximation. In order to consider the inertial effect in this dynamic problem, Viola and Walker [8] modified the model by adding an inertial component in series with the Voigt model, and solved for the time-dependent displacement induced by a step force. However, as we will illustrate henceforth, this modified model does not improve the estimation accuracy in the inverse problem due to the
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fundamental limitations of the SDF assumption. Our analysis can be extended to other SDF models developed with different repre sentations (or rheological models) of soft tissue viscoelasticity, because the fundamental limitations investigated here are not associated with the constitutive models but with the SDF and ho mogeneous assumptions, which neglect the 3D dynamic system effects and interactions between tissue elements. Modeling Acoustic Radiation Force. ARF is a spatial-varying body force. The focal zone is a Gaussian-shaped beam, and sim plified models, such as a 3D Gaussian function or an ellipsoid of rotation, have been used to model the ARF distribution [16,17]. These models are simplified patterns of the actual ARF distribu tion, but they can adequately capture the main properties of the spatial-varying body force [16]. In our analysis, the step ARF is described by a 3D Gaussian function similar to [17] to mimic ARF generated by a 3.25 MHz focused transducer with focal length 50 mm, diameter of the aperture 60 mm (F-number is 0.83), and inner diameter 22.6 mm (Sonic Concepts, Bothell, WA). Note that F-number indicates the ratio of the transducer’s focal length over the diameter of the aperture. The SDF model is studied for different ARF configurations in the simulation. First, ARFs with the same relative distribution but different magnitude, 0.5/o,/ o, and 2/c (f0 is the spatial peak density of the body force), are assigned as the body force magnitude. Then, for ARFs with the same magnitude, the relative distribution is changed by modeling transducers with different F-numbers, namely, 1.5, 0.83, and 0.5. The first one has a larger focal zone and the third one has a smaller focal zone compared to the one discussed above (Fnutnber = 0.83). The analytical solution for the ARF generated displacement field, uz(z), in an infinite isotropic homogeneous medium can be obtained by convolving the solution for a point load in an infinite isotropic homogeneous solid with the spatial-varying body force field [18], which gives 1
u2{z)
f”
f2n [°° (
16^(1 - V ) J - J o
3— .
3 - 4v
(z - z , f
Jo V ( > / ( z _ ^ 2 + ^ ) 3
\ \f(r,z,)rdrd0dzt
(2)
\j(z —z* )2 + r2) J where z and z* are the axial distances with respect to the origin in the z direction, r is the radial distance from z-axis, f(r,z) is the body force density, and v is the Poisson’s ratio.
Finite Element Models. Axisymmetric FE models are devel oped to simulate the viscoelastic soft tissue using A baqus 6.11 FE software [19], as illustrated in the diagram of Fig. 1(a). The bot tom boundary of the FE model is fixed in all directions; the remaining surfaces of the model are not constrained. For each case studied in this paper, the ARF is always applied along the axial direction (z direction). The origin of the coordinate system of this model is located at the center of the symmetry axis. The axisymmetric model consists of 5366 nodes and 5275 four-node bilinear axisymmetric quadrilateral elements with hybrid formula tion. Convergence of the model is checked by mesh refinement to ensure sufficiency of the mesh density. The FE simulations are implemented with A baqus implicit dynamic analysis. In order to evaluate the effectiveness of the SDF and homogeneous assump tions in describing soft tissue dynamic behavior, as well as to explore their limitations, we simulate a homogenous and a hetero geneous case. Case 1. Homogeneous Model: Different mechanical properties and ARF configurations are investigated. Four shear moduli Gu = 1 kPa, 3 kPa, 9 kPa, and 27 kPa) and three time constants (t = 0.0003 s, 0.0009 s, and 0.0027 s) are first studied with the FE model. The chosen shear moduli are within the range of reported mechanical properties of the normal or tumorous human soft tis sues [20,21]. The time constants used are on the same order as reported values for soft tissues or phantoms [5,7,10], The SDF assumption is evaluated quantitatively by analyzing the inverse problem with the SDF model. FE simulations generate the data of the numerical experiments. Fitting the SDF models to these data formulates an optimization problem in which the mean squared error between displacement computed from FE simulations and the displacement computed from the SDF models fitted to that data should be minimized. Since the purpose of this study is to investigate the fundamental limitations of the SDF and homogeneous assumptions in ARF creep imaging, these data do not contain noise as in practical measurements in order to control variables in the analysis. The results should be considered as the best theoretically achievable ones. Case 2. Heterogeneous Model: In order to demonstrate the limi tations of the SDF and homogeneous assumptions in the presence of tissue heterogeneity, a qualitative analysis is performed by comparing the temporal and steady state responses of the homoge neous model to that of the heterogeneous model. In this study, spherical inclusions are located in the center of the model, as illus trated in Fig. 1(a). There are two cases considered: spheres with diameters of 3 mm and 6 mm. Both the background and inclusion are assigned with the same time constant. Three background and
Fig. 1 Model diagram and validation: (a) Diagram of the axisym m etric model with a spherical inclusion in the center of the model and (b) axial displacem ent induced by ARF for the hom oge neous case with // = 3 kPa. The horizontal axis in (b) is the axial distance from the focal center. 0 9 4 5 0 2 -2
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inclusion shear modulus combinations, (fj.B, /.if), are studied, namely, (3 kPa, 3 kPa), (0.3 kPa, 3 kPa), and (30 kPa, 3 kPa). Under the homogeneous assumption, the steady state displace ment, A, is inversely proportional to the underlying tissue elastic ity (or shear modulus). In the heterogeneous media, the observed elasticity contrast, can be described by
* - '0 —
^background ~~A ^inclusion
where CQ is the observed elasticity contrast of the inclusion to background, A background is the steady state displacement of the background, and Ainclusion is the steady state displacement inside the inclusion [7,22]. Here, /lincluSK)n is measured at the center of the inclusion, where the response is least affected by the heteroge neity. For regions near the boundary of the inclusion and back ground, the response can be more severely affected by heterogeneity [13], which results in a further reduced elasticity contrast. The ideal performance of a biomechanical imaging method is that the observed elasticity contrast equals the true elas ticity contrast.
Results and Discussion In this study, numerical experiments are performed by FE simu lations. In order to validate the FE model, the FE simulated steady state responses are first compared to the analytical solution (Eq. (2)) for the homogeneous case with p = 3 kPa. The ARF in the FE model is applied at the origin of the model. The computed axial displacements induced by the ARF are shown in Fig. 1(b). The analytical solution is derived for an infinite medium, while the FE model has boundary to mimic a practical configuration. The results are in good agreement between the FE simulation and analytical solution with slight difference at the two ends of the curves, indicating that the global boundary condition has little effect on the response of inside tissue in the localized ARF imag ing methods [13]. For the homogeneous model with different mechanical proper ties, the creep displacement for each case is shown in
(d)
Figs. 2(a)-2(c). Even though the time constant is different, the near steady state displacement is almost identical for the same elasticity and is inversely proportional to the elasticity of the tis sue, which confirms that the steady state displacement is a good metric to image the elasticity relation between homogeneous models if the same ARF is applied in the focal zone [7-9], In order to show the temporal behavior of the response, the dis placements are normalized by the corresponding steady state dis placement in Figs. 2(d)-2(f). For the same time constant, based on the SDF model of Eq. (1), the temporal behavior is expected to be identical for different shear moduli. However, Eq. (1) is derived from a SDF model without inertial component, in which the stress is proportional to the applied force and the creep strain is propor tional to the creep displacement. This is not the case for a 3D model with the inertial effects considered. It is noticed that the response curve deviates from the response of the SDF model with out inertia as the shear modulus of the model decreases. In addi tion, as the time constant increases, all the response curves converge to the response of the SDF model without inertia. Since the SDF model cannot ideally capture the dynamic behavior of a 3D system with mass, its relative estimation error (REE) is plotted in Fig. 3(a). In this figure, the horizontal axis is in logarithmic scale. The REE can be quite large for models with low time constant and elasticity, e.g., 142% for fi = 1 kPa and t = 0.0003 s. It should be noted that even the REE decreases as the elasticity increases for the same time constant, the REE increases as the elasticity increases for the same viscosity, as shown by the markers in Fig. 3(a). In a Voigt model, the viscosity is proportional to the elasticity for the same time constant. This means that for the same time constant, the REE decrease is due to the increased viscosity, not elasticity. These results imply that the SDF model tends to have better approximation for tissue with high viscosity and low elasticity, in which cases the SDF model may still be preferred in qualitative imaging for its simplicity. Fig ure 3(b) illustrates the REE with the SDF model that included the inertial component. The results demonstrate that adding the iner tial component into the SDF model does not significantly improve the estimation accuracy, because for a 3D dynamic system, not only the focal region but also the surrounding tissue is induced to
(e)
(f)
Fig. 2 Creep displacem ent responses to step ARF for soft tissue with different viscoelasticity. Three tim e constants are studied: (a) r = 0.0003 s, (b) z = 0.0009 s, and (c) z = 0 .0 0 2 7 s. The corre sponding norm alized creep displacem ent responses are shown in (d), (e), and (/).
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(a)
(b)
Fig. 3 REE of t by fitting the SDF models with the FE sim ulated creep displacem ent responses, (a) SDF model w ithout considering the inertial effect and (b) SDF model with the inertial com po nent included. Each m arker represents cases with the same shear viscosity.
motion by the ARF, and the complex system inertia cannot be accurately accounted for by a SDF dynamic system, which is the fundamental limitation of the SDF approach. The effects of the ARF configurations on the dynamic response of soft tissue are illustrated in Fig. 4. The steady state response is
(a)
affected by both the magnitude and distribution of the body force. However, the temporal response of soft tissue is independent of the body force magnitude, as shown in Fig. 4(c). The results con firm that the time constant is a “force-free” parameter in the anal ysis of the SDF model, as indicated in [7,8,10]. In addition, it
(b)
Fig. 4 Creep displacem ent responses to different ARF configurations: (a) ARFs with different m agnitude, but the same distribution, i.e., F nUm ber = 0.83; and ( b) ARFs with different distribu tion, but the sam e m agnitude, i.e., f0. T he corresponding norm alized creep displacem ent responses are shown in (c) and (d). The model is hom ogeneous with n = 3 kPa, and t = 0.0009 s.
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shows that the time constant depends on the body force distribu tion. Figure 4 ( d ) exhibits that the induced responses by a smaller focal zone are closer to the output of the SDF model without iner tia. The REEs of t for ARF corresponding to F „ u m b e r = 1.5, 0.83, and 0.5 are 68%, 27%, and 12%, respectively. The results suggest that decreasing the focal zone can reduce the inertial effects on the dynamic responses of soft tissue, which means that when using the SDF model in ARF creep imaging, a smaller focal zone is desired to reduce the REE. This is similar to other imaging methods, e.g., ARFI and HMI, in which smaller focal zones result in better image contrast [22,23], In order to qualitatively illustrate the effect of heterogeneity and the fundamental limitation of the homogeneous assumption in modeling the dynamic behavior of soft tissue to step ARF, the creep displacement responses at the origin of the heterogeneous models (center of the spherical inclusion) are shown in Fig. 5. The ARF is considered as a highly localized force in most ARF based imaging methods and the dynamic response is considered to be only related to the underlying local mechanical properties, which is equivalent to a homogeneous assumption of the imaged tissue. However, the results indicate that the local heterogeneity greatly affects the dynamic response of the focal zone. For inclusion with the same shear modulus, (/q = 3 kPa), the responses vary as the background’s moduli change. In addition, the steady state dis placements can no longer reflect the true elasticity contrast of the
Time (ms)
(a)
Time (ms)
(c)
T able 1 m o d e ls
O b s e rv e d
e la s tic ity
c o n tra s t
fo r
h e te ro g e n e o u s
Observed elasticity contrast Mi (kPa) 3 3 3
(kPa)
True elasticity contrast
3-mm-diameter inclusion
6-mm-diameter inclusion
0.3 3 30
10 1.0 0.10
2.9 1.0 0.18
4.2 1.0 0.13
Mb
inclusion to the background as shown in Table 1. As the dimen sion of the inclusion increases from 3 mm to 6 mm in diameter the observed elasticity contrast gets closer to the true elasticity contrast. The ARF induced displacement is the integration of the normal strain in the axial direction. Figure 6 explains the reduced contrast for heterogeneous media by demonstrating the effects of heteroge neity on the axial normal strain field with the case of the 3-mmdiameter spherical inclusion. The dashed line shows the profile of the inclusion. Even though the inclusions have the same shear modulus, it is shown that the strain fields are quite different for the three cases due to the differences in heterogeneity. Figure 6(a)
Time (ms)
(b)
Time (ms)
(d)
Fig. 5 C re e p d is p la c e m e n t re s p o n s e s a t th e o rig in o f th e h e te ro g e n e o u s m o d e ls w ith d iffe re n t in c lu s io n s ize s : (a) S p h e re w ith d ia m e te r 3 m m a n d (b) S p h e re w ith d ia m e te r 6 m m . T h e c o rre s p o n d in g n o rm a liz e d c re e p d is p la c e m e n t re s p o n s e s a re sh o w n in (c) an d (d ). T h e s o lid b la c k lin e d e n o te s th e h o m o g e n e o u s c a se, z = 0 .0 0 0 3 s is fo r b o th b a c k g ro u n d a n d in c lu s io n .
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+2.9e-03 +2.4e-03 +2.0e-03 +1.5e-03 +1.0e-03 +5.2e-04 +4.1e-05 -4.4e-04 -9.2e-04 -1.4e-03 -1.9e-03 -2.4e-03 -2.8e-03
+3.5e-03 +2.9e-03 +2.4e-03 +1.8e-03 +1.2e-03 +6.5e-04 +7.9e-05 -4.9e-04 -l.le-03 -1.6e-03 -2.2e-03 -2.8e-03 -3.3e-03
(b) Fig. 6 Axial normal strain field near the region of the 3-m m -diam eter spherical inclusion after a 10 ms step ARF excitation. The dim ension of the region is h = 6 mm (axial length) and r = 3 mm (radial length). The tim e constant is 0.0009s and the shear moduli are: (a) /js = 3kP a, and Hi = 3 kPa; (b) h b = 0-3 kPa, and hi = 3 kPa; and (c) hb = 30 kPa, and hi = 3 kPa.
is the case for a homogeneous medium. For Fig. 6(b), when the inclusion is in a soft background, the resulting displacement of the focal zone relates primarily to the strain of the background, and it reflects not only the elasticity of the inclusion but also of the back ground, or the heterogeneous structure of the local area. For an inclusion in a hard background, as shown in Fig. 6(c), the strain field is constrained by the hard background, which results in a reduced displacement compared to the homogeneous case. The temporal responses of soft tissue to step ARF are even more complex in the presence of heterogeneity. For each figure of the normalized displacements in Fig. 5, the background and inclu sion are assumed to have the same time constant, and ideally the responses are expected to be identical according to the SDF assumption. However, for hard inclusions in soft background, the system tends to have larger time constants, i.e., the response needs more time to reach the steady state. At the beginning, before the shear wave reaches the background, the creep displacement results from the strain inside the inclusion. When the shear wave reaches the soft background, the creep will continue in the back ground and the induced strain is still relatively large because the background is much softer than the inclusion, similar to the case in Fig. 6(b). On the contrary, for a soft inclusion in hard back ground the motion inside the inclusion is constrained and the induced strain in the background is relatively small, thus the sys tem tends to have a smaller time constant, i.e., the displacement rises faster to its maximum value than in the homogeneous case. In that case, the shear wave reflection in the boundary leads to an overshoot in some of the responses as shown in Fig. 5. Even though the SDF model has its fundamental limitations, it’s still preferable in many cases involving high viscous soft tis sues because it is simple and minimizes the number of parameters in the optimization, which leads to suboptimal, but stable and ro bust estimation. In addition, this estimation can be improved by reducing the focal zone size. In the cases of soft tissues with low viscosity, performance of the SDF models is quite poor, and inverse procedure based on FE methods may be introduced. Even though lots of the prior information for the FE modeling (local heterogeneity profile, ARF distribution, etc.) cannot be accurately obtained, proper integration of this prior information and its uncertainty could help to improve estimation accuracy, or to enhance the observed contrast, comparing to the SDF models that do not utilize any of that prior information. This topic is currently under investigation by our group. Considering heterogeneity, if the inclusion region is large, the responses in the early stage are not affected by the local heterogeneity as shown in Fig. 5(b), in which case the early responses may still be used under the homo geneous assumption. 0 9 4 5 0 2 -6
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Conclusions Finite element models are developed to study the dynamic behavior of viscoelastic soft tissue subjected to step ARF. Based on the FE simulations, the fundamental limitations of the com monly used SDF and homogeneous assumptions are studied for ARF creep imaging. The results suggest that the SDF model tends to have good approximation for tissue with high viscosity and low elasticity. Especially, for the low viscosity case, the estimation results based on the SDF model can be quite poor. In addition, accounting for the inertial component into the SDF model cannot effectively improve the estimation accuracy, but reducing the size of the focal zone can result in better estimation. For heterogeneous models, both the temporal and steady state responses of the focal region are highly affected by the local heterogeneity, rendering the SDF and homogeneous simplification ineffective. In order to obtain better estimation or enhanced contrast, FE analysis procedure considering the 3D configuration of the dynamic sys tem and model heterogeneity may be necessary for the inverse characterization.
Acknowledgment This research was supported by NSF CMMI-0900596. The authors kindly acknowledge the support of NSF CMMI division and the program managers Dr. Demitris Kouris and Dr. Dennis R. Carter.
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