COMPARATIVE EVALUATION OF STRAIN-BASED AND MODEL ...
Ultrasound in Med. & Biol., Vol. 31, No. 6, pp. 787– 802, 2005 Copyright © 2005 World Federation for Ultrasound in Medicine & Biology Printed in the USA. All rights reserved 0301-5629/05/$–see front matter
doi:10.1016/j.ultrasmedbio.2005.02.005
● Original Contribution COMPARATIVE EVALUATION OF STRAIN-BASED AND MODEL-BASED MODULUS ELASTOGRAPHY MARVIN M. DOYLEY,*‡ SESHADRI SRINIVASAN,† SARAH A. PENDERGRASS,‡ ZIJI WU,‡ † AND JONATHAN OPHIR *Department of Radiology, Dartmouth-Hitchcock Medical Center, Lebanon, NH, USA; †Ultrasonics Laboratory, University of Texas Medical School, Houston, TX, USA; and ‡Thayer School of Engineering, Dartmouth College, Hanover, NH, USA (Received 30 June 2004, revised 29 January 2005, in final form 3 February 2005)
Abstract—Elastography based on strain imaging currently endures mechanical artefacts and limited contrast transfer efficiency. Solving the inverse elasticity problem (IEP) should obviate these difficulties; however, this approach to elastography is often fraught with problems because of the ill-posed nature of the IEP. The aim of the present study was to determine how the quality of modulus elastograms computed by solving the IEP compared with those produced using standard strain imaging methodology. Strain-based modulus elastograms (i.e., modulus elastograms computed by simply inverting strain elastograms based on the assumption of stress uniformity) and model-based modulus elastograms (i.e., modulus elastograms computed by solving the IEP) were computed from a common cohort of simulated and gelatin-based phantoms that contained inclusions of varying size and modulus contrast. The ensuing elastograms were evaluated by employing the contrast-to-noise ratio (CNRe) and the contrast transfer efficiency (CTEe) performance metrics. The results demonstrated that, at a fixed spatial resolution, the CNRe of strain-based modulus elastograms was statistically equivalent to those computed by solving the IEP. At low modulus contrast, the CTEe of both elastographic imaging approaches was comparable; however, at high modulus, the CTEe of model-based modulus elastograms was superior. (E-mail: [email protected]) © 2005 World Federation for Ultrasound in Medicine & Biology. Key Words: Elastography, Breast imaging, Inverse problems, Model-based inversion, Tissue characterization.
tissue; and, fourth, the spatial variation of the induced internal tissue strains is estimated by performing crosscorrelation analysis on the pre- and postdeformed RF echo frames. The measured strain values can be directly converted to Young’s modulus by using the constitutive elasticity equations (i.e., Hook’s law); however, this requires accurate quantification of all components of the 3-D stress vector. This represents a major limitation because, at present, the internal stress distribution cannot be measured in vivo. However, if the internal stress distribution is constant, then strain elastograms (i.e., images of the axial component of the induced internal tissue strain) can be interpreted as modulus elastograms (relative Young’s or shear modulus images). In practice, the internal stress distribution is not uniform. Accordingly, mechanical artefacts and an incomplete contrast-transfer efficiency are incurred when strain elastograms are interpreted as modulus elastograms based on the assumption of stress uniformity (Konofagou et al. 1996; Ophir et al. 1991; Ponnekanti et al. 1996).
INTRODUCTION Elastography (Ophir et al. 1991) is an emerging imaging modality that can be applied to a broad range of clinical applications. These include assessing plaque vulnerability (Brusseau et al. 2001; de Korte et al. 2000, 2002; Doyley et al. 2001b), guiding minimally invasive therapeutic techniques (Kallel et al. 1999; Righetti et al. 1999; Varghese et al. 2003) and improving the differential diagnosis of breast and prostate cancer (Céspedes 1993; Garra et al. 1997; Hiltawsky et al. 2001; Souchon et al. 2003). The elastographic imaging formation process can be viewed as a four-step process. First, a radio-frequency (RF) echo frame is acquired from the tissue or phantom; second, a small motion is induced within the tissue by using either an external or internal quasistatic mechanical source; third, a second RF frame is acquired from the
Address correspondence to: Marvin Doyley, Ph.D., Thayer School of Engineering, Dartmouth College, 8000 Cummings Hall, Hanover, NH 03755 USA. E-mail: [email protected] 787
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Fig. 2. Schematic diagram of strain and nanoindentation modulus imaging.
Fig. 1. (a) Example of strain elastograms which was obtained from a gelatin-based phantom containing a 10-mm diameter cylindrical inclusions. (a) The original strain elastograms, from which (b) the low-resolution strain elastogram was produced by applying the spatial filter recursively (i.e. five times) to the original strain elastogram (a).
The long-term objective of this work is to enhance the clinical utility of elastography by developing methods to 1. reduce mechanical artifacts and 2. improve the elastographic contrast-transfer efficiency. Considering
elastography within the established framework of solving inverse problems could potentially obviate these limitations, because this approach to elastography does not require accurate quantification of the internal stress distribution to produce modulus elastograms. However, model-based elastography (i.e., solving the inverse problem) is fraught with many problems. Factors such as 1. model-data discrepancy (i.e., the magnitude of error incurred when modeling the forward elasticity problem) and 2. the measurement noise that will undoubtedly compromise the quality and accuracy of the ensuing elastograms. In addition, there is no guarantee of producing unique modulus elastograms with these methods, because of the ill-posed nature of the inverse elasticity problem (IEP) (Barbone and Bamber 2002). Given these problems, there are concerns that model-based elastography may not offer any significant advantage over strain-based elastography. With this view in mind, the aim of the current study was to determine how the quality of model-based modulus elastograms currently compares with that of strain elastograms. In this article, we report the results of computer simulations and experimental studies that were performed on elastically inhomogeneous phantoms to objectively assess the quality of modulus elastograms computed by solving the IEP (model-based modulus elastograms) relative to modulus elastograms computed by simply inverting strain elastograms, based on the assumption of stress uniformity (strain-based modulus elastograms). The ability to discriminate between the physical properties of different tissue types is an important attribute of any diagnostic imaging modality (Hill et al. 1990). Consequently, the contrast discriminations of strain-based and model-based modulus elastograms were assessed by employing the contrast-to-noise ratio (CNRe) performance metric, which is computed in elastography from the means and variances of mechanical properties (i.e., strain or modulus) within an inclusion and surrounding tissue, respectively. Absolute or a good relative estimate of modulus should prove fruitful in several tissue characterization applications. Potential clinical examples include differentiating between different tumor types and monitoring the response of tumors to
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Fig. 3. (a) Modulus elastograms computed by directly inverting strain elastograms and (b) solving the inverse elasticity problem. Elastograms obtained from phantoms containing a single 10-mm diameter inclusion with modulus contrast progressively increased from 0.8 dB to 20 dB (top to bottom). Elastograms in each column are displayed with the same dynamic range. Dashed circle ⫽ circular ROI from which first order statistics (mean and variance) of the inclusion were derived.
therapy. The accuracy (i.e., ability to recover modulus contrast) of modulus elastograms was assessed by using the contrast-transfer efficiency (CTEe) performance metric. Note that this performance metric deals only with the mechanical issues of modulus contrast recovery and does not take estimating of measurement noise into account. Modulus elastograms were also compared with modulus images obtained independently by using the direct me-
chanical indentation measurement technique described in Srinivasan (2003) and Srinivasan et al. (2004). Theory Model-based modulus elastograms can be computed by inverting either the measured displacements (Doyley et al. 2000; Kallel and Bertrand 1996; Oberai et al. 2004) or the measured strains (Raghavan and Yagle
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1994; Skovoroda and Aglyamov 1995; Sumi et al. 1995a; Skovoroda et al. 1995; Sumi et al. 1995b), based on a forward continuum model. In this section, we describe the theory that underpins the iterative model-based inversion technique that we have implemented to solve the IEP. This approach to modulus estimation has previously been described (Doyley et al. 2000; Kallel and Bertrand 1996); therefore, only a brief description of the technique is provided in this section. The IEP can be formulated as a parameter optimization problem with an objective to minimize the difference between the measured axial displacement, um, and that computed by a model description in which tissue mechanical property is parameterized as a set of unknown coefficients representing the observable mechanical behavior. The objective function to be minimized has the following form: ⌽共兲 ⫽ 㛳um ⫺ u兵E其㛳2
(1)
where u{E} represents the axial displacements computed from the Young’s modulus distribution {E} by employing a finite element representation of the forward elastography problem, as described in Reddy (1993). Minimization of eqn (1) with respect to Young’s modulus variations is a nonlinear process that can be realized through an iterative solution for {E}, based upon an initial estimate of the Young’s modulus distribution. The resulting matrix solution at the (k ⫹ 1) iteration has the general form:
兵E其k⫹1 ⫽ 兵⌬E其k ⫹ 关J共Ek兲TJ共Ek兲 ⫹ I兴⫺1 · J共Ek兲T共um ⫺ u兵Ek其兲
(2)
where {⌬E} is a vector of Young’s modulus updates at all coordinates in the reconstruction field and J(Ek} is the Jacobian or sensitivity matrix. This problem usually involves the inversion of a poorly conditioned matrix, [J(Ek)T J(Ek)], which can be regularized by adding an identity matrix, I, multiplied by an arbitrary scalar (i.e., the so-called regularization parameter). Spatial filtering is usually performed as described in Doyley et al. (2000) to suppress local fluctuations in the Young’s modulus estimates and to constrain the reconstruction procedure to a smooth solution. This is realized by computing the weighted average of Young’s modulus at each coordinate location with its surrounding neighbors, according to: E共i兲new ⫽ 共1 ⫺ 兲E共i兲old ⫹
N
N
兺 E共i兲
old
(3)
i⫽1
where the superscripts “old” and “new” represent the unfiltered and filtered Young’s modulus values, respec-
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tively. The filtering weight is chosen so that 0 ⱕ ⱕ 1 and N is the number of neighboring positions that are averaged (i.e., the spatial range of the filter). MATERIALS AND METHODS Simulation study The aim of this investigation was to assess the effect of focal lesions of varying size and modulus contrast on the performance of model-based and strain-based modulus elastograms. Performance was evaluated quantitatively by assessing the contrast discrimination and accuracy of the modulus elastograms. Contrast discrimination was assessed by employing the CNRe performance metric, which is defined in elastography (Bilgen 1997) as: 2共␥L ⫺ ␥B兲2 CNRe ⫽ L2 ⫹ B2
(4)
where ␥L and ␥B represent the mean mechanical properties in the inclusion and background tissues, respectively, and L2 and 2B represent the variance in the respective mechanical properties. The first order statistics (means and variances) were derived from regions-of-interest (ROIs) that were manually selected in the strain-based and model-based modulus elastograms. The ROI in the inclusion was defined by fitting a circle to a contour enclosing the inclusion, as illustrated in Fig. 3. The noise incurred in model-based and strain-based elastography is spatially-dependent. In the case of the model-based technique, this spatial-dependence occurs because of the nonlinear nature of the inverse reconstruction procedure; whereas, in strain-based elastography, this occurs as result of several factors, such as structural decorrelation, the intrinsic noise property of the ultrasonic imaging system and the particulars of the strain-estimation technique. Consequently, ␥B and 2B were computed over the entire background in both cases. The accuracy (i.e., ability to recover modulus contrast completely) was assessed by employing the contrast-transfer efficiency (CTEe) performance metric, which is defined in elastography on a logarithmic scale, as follows (Ponnekanti et al. 1996):
ⱍ
ⱍ ⱍ
共dB兲 ⫽ Co共dB兲 ⫺ Ct共dB兲
ⱍ
(5)
where Co and Ct represent the observed and actual modulus contrast, respectively. Here, ⫽ 0 indicates that modulus contrast is completely recovered (i.e., 100% elasticity contrast recovery) and ⬍ 0 represents an incomplete contrast recovery. The observed modulus contrast was estimated from the mean relative modulus within the ROIs and was manually selected in the inclusion and surrounding background tissue.
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Fig. 4. Mean radial profiles computed from modulus elastograms in Fig. 3, showing mean radial profile obtained from (a) 4-mm, (b) 10-mm, and (c) 12-mm diameter inclusion. Error bar represents ⫾ 1 SD computed from 36 independent radial profiles equally distributed in angle between 0° and 360°. Modulus profiles obtained from () strain and (Œ) model-based modulus elastograms. (—-) Mean radial profile obtained from actual modulus distribution used in computer simulations.
Forward displacement computation Two groups of simulated 2-D phantoms (40 mm ⫻ 40 mm) were constructed using a commercial finiteelement package (ALGOR, Inc., Pittsburgh, PA, USA). The first group of phantoms contained a single 10-mm diameter circular inclusion with modulus contrast (Ec) in the range of 0.8 dB to 40 dB, whereas the modulus contrast in the second group of simulation phantoms was held at a constant value of 3.5 dB; the diameter of the inclusion was varied between 1 and 10 mm. Modulus contrast was defined in this study as:
冉冊
EC共dB兲 ⫽ 20 * log
EL EB
(6)
where EL and EB represent the Young’s moduli in the inclusion and background, respectively. A Poisson’s ratio of 0.495 was assigned to the inclusion and surrounding material in both groups of phantoms. A Young’s modulus of 20 kPa was assigned to the surrounding background material of all phantoms; however, the Young’s modulus of the simulated inclusions was varied to give the desired modulus contrast. The Young’s mod-
ulus of the surrounding background material was chosen to be representative of normal fibroglandular breast tissue (Krouskop et al. 1998). The forward elastography problem was solved in all cases by assigning a prescribed displacement of 0.4 mm to all nodes on the upper boundary of the finite element meshes, which corresponded to an applied strain of 1%. The nodes on the lower boundary of the mesh were constrained from moving in the axial direction, but were free to move in the lateral direction to simulate slip boundary conditions. Uniform stress distribution is generally induced within the tissue or phantom under investigation (except at internal tissue boundaries) when elastography is performed with slip boundary conditions (Kallel et al. 1998). Synthesizing ultrasound echo signals The discrete acoustic tissue model described in Kallel and Bertrand (1996) and Ponnekanti et al. (1996) was spatially distorted using the displacements (i.e., axial and lateral) computed by solving the forward elastography problem. Synthetic RF echo frames were generated
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Fig. 5. (a) Mean CNRe and (b) CTE computed from () strain-based and (Œ) model-based modulus elastograms. Error bar represents ⫾ 1 SD computed from 25 independent elastograms at each modulus contrast.
by convolving the pre- and postdeformed acoustic tissue models with the point spread function of a simulated single-element transducer that had a center frequency of 5 MHz and a 60% band width. These parameters were chosen to be representative of the US scanner employed in our experimental and clinical elastographic imaging system. Modulus estimation The noise incurred in ultrasonic elastography is very complex, depending on several variables, such as the ultrasonic imaging parameters (band width, center frequency), frequency dependent attenuation, sonographic SNR, particulars of the strain estimation technique (widow size, overlap ratio) and structural decorrelation noise (Kallel and Bertrand 1996). Strain and displacement images with realistic measurement noise were generated by applying the adaptive-stretching strain-estimation technique described in (Srinivasan et al. 2002b)
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to pre- and postdeformed RF echo frames that were synthesized by using the elastographic image formation model described by Kallel and Bertrand (1996). Adaptive stretching was used in preference to standard crosscorrelation analysis because the quality (peak elastographic SNR, dynamic range, sensitivity, etc.) of the ensuing strain elastograms is superior (Alam et al. 1998). All echo processing was performed using T ⫽ 3 mm (window length) and ⌬T ⫽ 0.6 mm (80% window overlap). Spurious strain estimates (i.e., regions corresponding to an elastically homogeneous tissue where large strain discontinuity was observed) were removed by applying a 2 mm ⫻ 2 mm median filter to the resulting elastograms. Axial displacements were computed by integrating the filtered strain estimates. One of the shortcomings of the iterative inversion technique that we have implemented is that the computational load required to solve the inverse problem increases considerably with the increasing mesh resolution, whereas the error between the computed and measured displacements generally decreases with increasing mesh resolution. Consequently, finite-element representations of the simulated phantoms were constructed by using uniform finite element meshes consisting of 6561 nodes and 12,800 triangular elements to provide a reasonable compromise between computational load and accuracy of the forward computations. The measured strain and displacement images were interpolated onto the coarse meshes by employing a bilinear interpolation function (MATLAB; The MathWorks, Inc. Natick, MA, USA). Modulus elastograms were subsequently computed from the measured strains and displacements using two approaches. In the first approach, modulus elastograms were computed by applying the iterative nodal-based inverse reconstruction technique, previously described, to the measured displacements (Doyley et al. 2000). All reconstructions were performed by assuming that the tissue under investigation had a homogeneous Young’s modulus of 20 kPa. The subsequent computations can be summarized into the following steps: 1. Compute the axial displacement at each coordinate based on the current estimate of Young’s modulus by solving the partial differential equations governing the underlying quasistatic model, given by:
共 ⫹ 兲 ⵜ 共ⵜ · U兲 ⫹ ⵜ2U ⫽ 0
(7)
where U is the computed nodal displacement vector, which contains both the axial (u) and lateral (v) components of displacement (i.e., 兵U其 ⬵ 兵u1v1...uNvN其, where N is the number of nodes in the finite element mesh), and are the Lamé constants and ⵜ; is the Del operator. The
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relationship between Lamé constants and Young’s modulus (E) and Poisson’s ratio () is given by:
⫽
vE E ,⫽ 2共1 ⫹ v兲 共1 ⫹ v兲共1 ⫺ 2v兲
(8)
The measured axial displacements on the upper and lower surface of the finite element meshes were used as known displacement boundary conditions when solving the forward elasticity problem. 2. Compute the residual displacement vector, {⌬; u}, by subtracting the calculated from the measured axial component of displacement at each nodal coordinate. 3. Construct the Jacobian matrix, [J], using the method described in Kallel and Bertrand (1996). 4. Compute an improved estimate of modulus at each coordinate in the uniform finite element mesh by solving eqn (2), as described in Marquardt (1963). 5. Apply the spatial filter to the recovered modulus elastogram, as described in eqn (3). A filtering weight of ⫽ 0.25 was used in all reconstructions (Doyley et al. 2000). 6. Repeat steps 1. to 5. until the inversion technique converges to a stable solution (i.e., 关J共Ek兲TJ共Ek兲 ⫹ I兴⫺1·J共Ek兲T共um ⫺ u兵Ek其兲 ⬵ 0. Note that this inverse technique generally converges to a stable solution within 10 to 15 iterations. In the second approach, relative modulus elastograms were computed by inverting spatially filtered axial strain elastograms (11) based on the simple assumption of stress uniformity (Ophir et al. 1991): E⬵
1 11
(9)
An apparent limitation of this approach is that it is prone to numerical instabilities (i.e., divide by zero error); therefore, only nonzero strain estimates were inverted. It is important to note that spatial filtering is generally not performed in standard strain-based elastographic imaging; however, to facilitate an objective comparison of both elastographic imaging methodologies, the spatial filter (i.e., eqn (3)) was applied recursively to the strain elastograms before inversion so as to 1. suppress local fluctuations in the strain elastograms before inversion and 2. degrade the spatial resolution of the strain elastogram to that of the modulus elastogram. Figure 1 shows examples of strain elastograms obtained from a highcontrast experimental phantom that contained a single 10-mm diameter cylindrical inclusion. The elastogram on the top is the prefiltered strain elastogram, whereas the elastogram on the bottom was created by applying the spatial filter recursively (5 times) to the strain elastogram shown in Fig. 1a to produce approximately the same degree of smoothing as that incurred during the
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model-based inversion process (note that this was deduced empirically). For the purpose of this demonstration, spatial filtering was performed using a filtering weight of ⫽ 0.25, which was chosen to be the same as that used when computing model-based elastograms. The visual appearance of the strain elastogram was significantly improved, which suggest that, in exchange for reduced spatial resolution, there should be a marked improvement in the contrast discrimination of strain elastograms. Experimental study The goal of the experimental investigation was to corroborate the simulation results. More specifically, it was to compare the performance of both elastographic imaging approach under controlled experimental conditions. Phantoms Cubic phantoms (100 mm3) containing cylindrical inclusions with diameters ranging from 2 mm to 25 mm were manufactured in a repeatable and controlled manner from a gelatin-agar suspension. The surrounding background gel of all phantoms was manufactured from gelatin-agar suspensions consisting of 5% by weight porcine skin gelatin (225 bloom, Kind & Knox, Inc., Sioux City, IA, USA), deionized water (18 mol/L⍀), polystyrene granules (⬇ 100 m mean diameter), 2% by weight agar (Sigma Chemical Co., St Louis, MO, USA) and 0.5% by weight ethylenediaminetetraacetic acid (EDTA). The inclusions were manufactured by filling the pores of open-cell sponge cylinders (100 pores/inch2; Anatomic Concepts, Corona, CA, USA) with gelatinagar suspension (5% by weight gelatin, 2% by weight agar, and 0.5% by weight EDTA), as described in Srinivasan et al. (2003). Based on previous independent mechanical testing, it was anticipated that the resulting phantoms would have modulus contrast in the range of 2.3 dB (1.5:1) to 20 dB (10:1). Data acquisition Elastographic images were obtained using the laboratory system described in Srinivasan et al. (2002a), which consists of a modified Philips HDI-1000 commercial US scanner (ATL Philips Inc., Bothell, WA, USA) that was equipped with a 128-element linear-array transducer (5-MHz nominal frequency and 60% fractional band width) and a computer-controlled mechanical deformation system. Elastographic imaging was performed by first placing the phantom between two large compression plates. To ensure proper contact, all phantoms were prestrained by 1% before the upper compression plate was displaced by 1 mm (which corresponds to a total applied strain of
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Fig. 6. (a) Strain-based and (b) model-based modulus elastograms obtained from low-contrast (3.5 dB) elasticity phantoms. The size of the inclusion in each column was increased from 2 mm to 12 mm, top to bottom. Each elastogram represents the average modulus elastogram computed from 25 independent realizations.
2%). The upper and lower surfaces of the phantoms were thoroughly lubricated with corn oil to approximate slip boundary conditions (Kallel et al. 1999). Independent elastographic imaging A thin section (3-mm thick) that coincided with the imaging plane was carefully removed from all
phantoms before mechanical testing was performed, using a nanoindenter™ (Testworks Inc, Nashville, TN, USA) that was equipped with a 2-mm diameter cylindrical punch. The modulus imaging procedure is shown schematically in Fig. 2; however, a more detailed description of this procedure is given in Srinivasan et al. (2004). Nanoindenter modulus images
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Fig. 7. Mean radial profiles computed from the modulus elastograms shown in Fig. 6. Error bar represents ⫾ 1 SD computed from 36 independent radial profiles equally distributed between 0° and 360°. Modulus profiles obtained from () strain- and (Œ) model-based modulus elastograms .
were obtained by performing measurements over a 16 ⫻ 16 rectangular grid. All mechanical measurements were performed using strain rates of 1% per s. Modulus estimation Modulus elastograms were computed by inverting 1. strain elastograms based on the assumption of stress uniformity and 2. displacement images by using the model-based inversion technique, as described in the simulation study. The adaptive stretching strain-estimation technique used in the simulation study was applied to the digitized RF echo frames to generate strain and displacement images that corresponded with the highest achievable image quality (i.e., peak SNR, dynamic range, sensitivity, etc.). All processing parameters were identical to those used in the simulation study (i.e., T ⫽ 3 mm and ⌬T ⫽ 0.6 mm). RESULTS Simulations Effects of modulus contrast. Composite elastograms obtained from phantoms with modulus contrast in the range of 3.5 dB to 15.5 dB are shown in Fig. 3. Each composite elastogram was computed from 25 statistically- independent elastograms. Paired strain-based and model-based modulus elastograms are shown in Figs. 3a
and b, respectively. Modulus elastograms within the same row are displayed on the same color scale, to facilitate visual comparison between elastograms generated by both approaches. The 10-mm diameter inclusion is discernible in both groups of modulus elastograms at the correct location; however, an artefact in the form of an upright cross is visible in strain-based modulus elastograms (Fig. 3a), which was expected because stress is known to accumulate on the periphery of uniform cylindrical inclusions (Kallel et al. 2001). In general, modelbased elastography takes proper account of the internal stress distribution when computing modulus elastograms; however, artefacts would appear to manifest in two forms. First, artefacts are apparent proximal and distal to the inclusion in the model-based modulus elastograms (Fig. 3b), an expected consequence of the nonunique nature of the inverse elasticity problem, particularly when reconstruction is performed using known displacement boundary conditions (Barbone and Bamber 2002). Second, spatially periodic noise is apparent in model-based modulus elastograms. The exact mechanism of this noise is not fully understood. However, modulation artifacts could occur because of the nonunique nature of the inverse problem. Furthermore, it is well known that a critical phase in the development of any diagnostic imaging technique is the need to optimize the system variables to maximize the sensitivity and
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the iterative model-based inversion approach. The large SD on the estimated CNRe was due primarily to large variation in mean modulus in the background tissue from one independent realization to another. However, it is apparent from Fig. 5b that, at high modulus contrast (Ec ⬎ 6 dB), the CTEe of modulus elastograms computed by employing the model-based inversion approach was superior.
Fig. 8. (a) The mean CNRe and (b) CTE computed from () strain and (Œ) modulus elastograms in Fig. 6. Error bar represents ⫾ 1 SD computed from 25 independent elastograms at each lesion size.
accuracy, and minimizing noise and artifacts that compromise image quality. No attempt was made objectively to optimize the reconstruction process; therefore, the observed modulation artifacts could be the consequence of computing modulus elastograms with a suboptimum reconstruction procedure. We plan to conduct a further study to optimize the reconstruction process and better to understand the mechanism that leads to this modulation. Despite the artifacts, it is apparent, from the mean normalized modulus profiles shown in Fig. 4, that there was good agreement between modulus elastograms computed based on the assumption of stress uniformity and those computed by employing the model-based inversion approach. The mean CNRe and CTEe computed from the elastograms shown in Fig. 3 are presented in Fig. 5a and b, respectively. The CNRe of modulus elastograms computed by simply inverting strain elastograms based on the assumption of stress uniformity would appear to be statistically equivalent to those computed by employing
Effects of lesion size. The ability to detect lowcontrast lesion could potentially lead to earlier tumor detection. Figure 6 shows the mean of 25 modulus elastograms obtained from contrast phantoms (i.e., 3.5 dB) containing inclusions with diameters varying from 2 mm to 12 mm. The paired strain-based and modelbased modulus elastograms are shown in Fig. 6a and b, respectively. The elastogram of the 1-mm diameter inclusion is not shown in Fig. 6, because it proved difficult to visualize such small inclusions with either approach. Note that the modulus modulation artefact observed in Fig. 3b is also apparent in the model-based modulus elastograms shown in Fig. 6b. Nonetheless, the tumor-like inclusions were discernible by both modulus-estimation techniques. Noticeable smoothing of the gradient between the tumor and background tissue was apparent in modulus elastograms produced by both modulus estimation techniques (Fig. 7). The mean CNRe and CTEe metrics are plotted as a function of actual inclusion size in Fig. 8a and b, respectively. It is apparent that, at low modulus contrast, there was statistically no significant difference in either the CNRe or the CTEe recovered from modulus elastograms computed by employing the model-based inversion approach and those computed by simply inverting strain elastograms. Despite the apparent diminution in the spatial resolution of the elastograms, both approaches to elastographic imaging could discern inclusions as small as 2 mm in diameter, with good CNRe (Fig. 8a). Experimental Figure 9 shows representative examples of sonograms obtained from gelatin-based phantoms containing localized inclusions of varying sizes. The position and extent of the inclusions can clearly be visualized in the sonograms, due to differences in acoustic contrast between the inclusion and the surrounding background tissues. Figure 10 shows representative examples of strain-based (Fig. 10a) and model-based (Fig. 10b) modulus elastograms and nanoindenter modulus images (Fig. 10c), corresponding to the sonograms shown in Fig. 9. This figure demonstrates several interesting results. First, the inclusions are discernible in all three images, with the exception of small inclusions (i.e., inclusions with diam-
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Fig. 9. Sonograms obtained from gelatin phantoms containing (a) 2 mm, (b) 5 mm, and (c) 10 mm diameter cylindrical inclusions. Rectangles overlayed on the sonograms denote the ROI from which the elastograms in Fig. 10a and b were computed.
eter ⬍ 5 mm) that were poorly visualized in the low spatial resolution (i.e., the spatial resolution of nanoindentor modulus images was low compared with that of the model-based and strain-based modulus elastograms) nanoindentor modulus image. Second, the modulus contrasts recovered from the strain-based modulus elastograms were lower than those observed in the modelbased modulus elastograms and the nanoindentor modulus images; this was anticipated because, at constant stress, the strain contrast of a linear elastic material is nonlinearly related to the actual modulus contrast. Miller and Bamber (2000) demonstrated theoretically that strain contrast increases asymptotically from zero to unity as the magnitude of the actual modulus contrast of a linear elastic material increases linearly from zero to infinity, which represents a fundamental limitation in contrasttransfer efficiency of strain-based elastography, as discussed in Ponnekanti et al. (1996). It has been demonstrated experimentally that strain contrast and modulus contrast are equivalent only in low-contrast phantoms
(i.e., phantoms with a modulus contrast ⱕ 6.02 dB) (Kallel et al. 1998). Third, a stress concentration artifact (in the form of an upright Maltese cross) are clearly visible in the strain-based modulus elastograms (for the two larger lesions); however, this artefact is not discernible in the model-based modulus elastograms. This was expected because reconstructing modulus elastograms within the framework of inverse problem solution generally takes proper account of the internal stress distribution. Fourth, there was generally good correlation between the modulus contrast recovered using the modelbased inversion technique and those measured using the mechanical indentation technique. The mean CNRe and CTE from the elastograms shown in Fig. 10 are plotted as a function of the lesion size in Fig. 11. As in computer simulations, the CNRe of strain-based and model-based elastograms were statistically equivalent. However, the CTEs of the model-based modulus elastograms were, in general, superior to those obtained by simply inverting strain elastograms.
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Fig. 10. (a) Strain-based, (b) model-based modulus and (c) nanoindenter elastograms corresponding to sonogram in Fig. 9.
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Fig. 10. Continued.
DISCUSSION Elastography based on quantitative strain imaging is well established in the literature; however, it currently suffers from an incomplete CTEe and mechanical artefacts (shadowing and target hardening) (Ophir et al. 1991). Solving the IEP could potentially circumvent these problems; however, this approach to elastography is fraught with many problems, because of the ill-posed nature of the IEP. Consequently, there are concerns that the quality of elastograms computed by solving the IEP is likely to be inferior to those produced using conventional strain imaging methodology. In this paper, we report the first of a series of planned studies that are designed to evaluate the quality of modulus elastograms produced by solving the IEP relative to those produced using conventional elastographic imaging modality in terms of standard measures of imaging quality, such as spatial and contrast resolution, dynamic range, etc. In addition to performing computer simulations, experiments were also conducted on gelatin-based phantoms to facilitate direct comparison between strain-based and modelbased modulus elastograms and modulus images obtained independently by using a direct mechanical indentation technique. To our knowledge, this is the
first reported study where model-based and strainbased modulus elastograms are compared with modulus images obtained by independent mechanical testing. Although the reported results pertain only to quantitative strain imaging as described in Ophir et al. (1991) and the iterative model-based inversion approach described in Doyley et al. (2000), the analysis is applicable to all elastographic imaging techniques. It is important to note that the boundary conditions used in this study represent the best-case scenario for both approaches to elastographic imaging. Therefore, it is reasonable to assume that the results presented in this study are likely to be valid only for slip boundary conditions. We imagine that a different choice in boundary conditions may be conducive for one elastographic imaging approach and not the other. For instance, boundary conditions that produce nonuniform internal stress distribution are likely to favor model-based elastography; similarly, discontinuity in the intrinsic tissue mechanical properties at internal tissue boundaries will undoubtedly produce erroneous results and artefacts in both approaches. However, the ensuing artefacts are likely to be more acute in the model-based inversion approach. Consequently, we propose to conduct further studies to assess the effects
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Fig. 11. (a) The mean CNRe and (b) CTE computed from (●) strain-based and (●) model-based modulus elastograms in Fig. 10. Error bar represents ⫾ 1 SD computed from 8 independent elastograms at each lesion size.
of boundary conditions on the quality of strain-based and model-based modulus elastograms. To summarize our current findings, solving the IEP minimizes stress concentration artefacts incurred when strain elastograms are interpreted as modulus elastograms, based on the assumption of stress uniformity, as demonstrated in Fig. 3; however, artefacts are also incurred in the model-based approach to elastographic imaging. Various investigators have reported similar results by employing iterative and direct model-based inversion techniques. The CNR of both approaches to elastographic imaging increases linearly with increasing modulus, as demonstrated in Fig. 5b. Doyley et al. (2003) observed a similar trend when evaluating the performance of their magnetic resonance imaging-based elastographic imaging system. The main drawback of developing elastography based on model-based modulus imaging is that the spatial resolution of the resulting elastograms is gener-
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ally inferior to that of strain elastograms. This is not surprising because the performance of most inverse reconstruction techniques is stabilized by imposing constraints on the reconstruction procedure, particularly, when reconstruction is performed in the presence of measurement noise. It is apparent, from Figs. 5a, 8a and 11a, that, at a constant spatial resolution, the CNR of strain-based modulus elastograms is statistically equivalent to that computed by using the model-based inversion technique. In general, considerable improvement in the CTEe is achieved when modulus elastograms are computed by using the model-based inversion technique, as demonstrated in Fig. 5b. This observation corroborates the prediction of Ophir and colleagues that solving the inverse problem should improve the CTEe (Kallel et al. 1998; Ponnekanti et al. 1996). Low-contrast lesions as small as 2 mm in diameter are discernible in strain-based and model-based elastograms, with good CNR, as demonstrated in Figs. 6 and 11. The performance metrics increase with increasing lesion diameter in both cases, as demonstrated in Figs. 8 and 11. In addition, it is demonstrated in Fig. 8 that the CTEe of both elastographic imaging approaches are statistically equivalent at low modulus contrast. This result corroborates the experimental results of Kallel et al. (1998), that revealed that strain-based elastography could potentially provide a good relative estimate of modulus in low-contrast medium. Visually, there was good correlation between both groups of modulus elastograms (i.e., model-based and strain-based modulus elastograms) and the nanoindenter modulus images. However, it is important to note that Fig. 11b represents the combined effect of lesion size and modulus contrast on the CTEe, which was due to an oversight in the phantom manufacturing process (i.e., the inclusions were manufactured using the same concentration of gelatin and agar; however, the inclusions were unintentionally manufactured with different sponge material that had different stiffness). Nonetheless, it is apparent that the CTEe of both approaches were comparable for the small lowcontrast inclusions; however, for large high-contrast inclusions, better CTEe was achieved form the modelbased modulus elastograms. A complete comparative evaluation of strain-based and model-based elastographic imaging is clearly beyond the scope of this article and will be the focus of our research for some time. A limitation of the present study is that no attempt was made to compare the performance of both elastographic imaging approaches with multiple heterogeneities and/or complex distributions. In addition, no attempt was made to compare the performance of both approaches within the clinical environment, where there are likely to be complex and continuing varying
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boundary conditions, especially if elastography is performed using the freehand scanning approach described in Doyley et al. (2001a); Hall et al. (2003), and Hiltawsky et al. (2001). These issues will be the focus of future communications.
CONCLUSIONS This paper describes the first reported study to objectively evaluate the performance of model-based elastographic imaging relative to quantitative strain imaging. The results of this preliminary study demonstrate that solving the inverse problem will 1. reduces mechanical artifacts; (however, some artifacts will also be incurred in model-based approaches to elastography, due the illposed nature of the IEP), and 2. substantially improve the CTEe, which would imply that that this approach to elastography should be more suited to tissue characterization applications. In both elastographic imaging approaches, there is a trade-off between spatial and contrast resolution; therefore, contrast resolution can only be compared at a fixed spatial resolution (and vice versa). The results presented in this study demonstrate that, for simple boundary conditions, no advantage will be gained in terms of improving the contrast resolution of elastography by solving the inverse problem. Acknowledgements—The National Cancer Institute (grant P01CA64597) has funded this work.
REFERENCES Alam SK, Ophir J, Konofagou EE. An adaptive strain estimator for elastography. IEEE Trans Ultrason Ferroelec Freq Control 1998;45: 461– 472. Barbone PE, Bamber JC. Quantitative elasticity imaging: What can and cannot be inferred from strain images. Phys Med Biol 2002;47(12): 2147–2164. Bilgen M. Target detectability in acoustic elastography. IEEE Trans Ultrason Ferroelec Freq Control 1997;46(5):1128 –1133. Brusseau E, Fromageau J, Finet G, Delachartre P, Vray D. Axial strain imaging of intravascular data: Results on polyvinyl alcohol cryogel phantoms and carotid artery. Ultrasound Med Biol 2001;27(12): 1631–1642. Céspedes EI. Elastography: Imaging of biological tissue elasticity. Ph.D. dissertation. Houston, Texas: University of Houston, 1993. de Korte CL, Carlier SG, Mastik F, et al. Morphological and mechanical information of coronary arteries obtained with intravascular elastography. Feasibility study in vivo. Eur Heart J 2002;23(5): 405– 413. de Korte CL, Pasterkamp G, van der Steen AF, Woutman HA, Bom N. Characterization of plaque components with intravascular ultrasound elastography in human femoral and coronary arteries in vitro. Circulation 2000;102(6):617– 623. Doyley MM, Bamber JC, Fuechsel F, Bush NL. A freehand elastographic imaging approach for clinical breast imaging: System development and performance evaluation. Ultrasound Med Biol 2001a;27(10):1347–1357.
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Doyley MM, Mastik F, de Korte CL, et al. Advancing intravascular ultrasonic palpation toward clinical applications. Ultrasound Med Biol 2001b;27(11):1471–1480. Doyley MM, Meaney PM, Bamber JC. Evaluation of an iterative reconstruction method for quantitative elastography. Phys Med Biol 2000;45(6):1521–1540. Doyley MM, Weaver JB, Van Houten EE, Kennedy FE, Paulsen KD. Thresholds for detecting and characterizing focal lesions using steady-state MR elastography. Med Phys 2003;30(4):495–504. Garra BS, Cespedes EI, Ophir J, et al. Elastography of breast lesions: Initial clinical results. Radiology 1997;202(1):79 – 86. Hall TJ, Zhu Y, Spalding CS. In vivo real-time freehand palpation imaging. Ultrasound Med Biol 2003;29(3):427– 435. Hill CR, Bamber JC, Cosgrove DO. Performance criteria for quantitative ultrasonology and image parameterisation. Clin Phys Physiol Meas 1990;11:57–72. Hiltawsky KM, Kruger M, Starke C, et al. Freehand ultrasound elastography of breast lesions: Clinical results. Ultrasound Med Biol 2001;27(11):1461–1469. Kallel F, Bertrand M. Tissue elasticity reconstruction using linear perturbation method. IEEE Trans Med Imaging 1996;15:299 –313. Kallel F, Ophir J, Magee K, Krouskop T. Elastographic imaging of low-contrast elastic modulus distributions in tissue. Ultrasound Med Biol 1998;24(3):409 – 425. Kallel F, Prihoda CD, Ophir J. Contrast-transfer efficiency for continuously varying tissue moduli: Simulation and phantom validation. Ultrasound Med Biol 2001;27(8):1115–1125. Kallel F, Stafford RJ, Price RE, et al. The feasibility of elastographic visualization of HIFU-induced thermal lesions in soft tissues. Image-guided high-intensity focused ultrasound. Ultrasound Med Biol 1999;25(4):641– 647. Konofagou E, Dutta P, Ophir J, Cespedes I. Reduction of stress nonuniformities by apodization of compressor displacement in elastography. Ultrasound Med Biol 1996;22(9):1229 –1236. Krouskop TA, Wheeler TM, Kallel F, Garra BS, Hall T. Elastic moduli of breast and prostate tissues under compression. Ultrason Imaging 1998;20(4):260 –274. Marquardt DW. An algorithm for least-squares estimation of nonlinear parameters. J Soc Indust Appl Math 1963:431– 441. Miller NR, Bamber JC. Thresholds for visual detection of Young’s modulus contrast in simulated ultrasound image movies. Phys Med Biol 2000;45(8):2057–2079. Oberai AA, Gokhale NH, Doyley MM, Bamber JC. Evaluation of the adjoint equation based algorithm for elasticity imaging. Phys Med Biol 2004;49(13):2955–2974. Ophir J, Cespedes I, Ponnekanti H, Yazdi Y, Li X. Elastography: A quantitative method for imaging the elasticity of biological tissues. Ultrason Imaging 1991;13(2):111–134. Ponnekanti H, Ophir J, Huang YJ, Cespedes I. Fundamental mechanical limitations on the visualization of elasticity contrast in elastography. Ultrasound Med Biol 1996;22:173–178. Raghavan KR, Yagle AE. Forward and inverse problems in elasticity imaging of soft-tissues. IEEE Trans Nuclear Sci 1994;41:1639 – 1648. Reddy JN. An introduction to the finite element method. Hightstown, NJ: McGraw-Hill, 1993. Righetti R, Kallel F, Stafford RJ, et al. Elastographic characterization of HIFU-induced lesions in canine livers. Ultrasound Med Biol 1999;25(7):1099 –1113. Skovoroda AR, Aglyamov SR. On reconstruction of elastic properties of soft biological tissues exposed to low-frequencies. Biofizika 1995;40:1329 –1334. Skovoroda AR, Emelianov SY, O’Donnell M. Tissue elasticity reconstruction based on ultrasonic displacement and strain images. IEEE Trans Ultrason Ferroelec Freq Control 1995;42(4):747–765. Souchon R, Rouviere O, Gelet A, et al. Visualisation of HIFU lesions using elastography of the human prostate in vivo: Preliminary results. Ultrasound Med Biol 2003;29(7):1007–1015.
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Srinivasan S. Comparing elastographic strain images with modulus images obtained using a nano-indenter: Applications in characterizing the mechanical properties of tissues. Electrical and computer engineering. Ph.D. dissertation. Houston, University of Texas, 2003. Srinivasan S, Kallel F, Ophir J. The effects of digitization on the elastographic signal-to-noise ratio. Ultrasound Med Biol 2002a; 28(11–12):1521–1534. Srinivasan S, Kallel F, Souchon R, Ophir J. Analysis of an adaptive strain estimation technique in elastography. Ultrason Imaging 2002b;24(2):109 –118. Srinivasan S, Krouskop T, Ophir J. Comparing elastographic strain images with modulus images obtained using nanoindentation: Pre-
Volume 31, Number 6, 2005 liminary results using phantoms and tissue samples. Ultrasound Med Biol 2004;30(3):329 –343. Sumi C, Suzuki A, Nakayama K. Estimation of shear modulus distribution in soft-tissue from strain distribution. IEEE Trans Biomed Eng 1995a;42:193–202. Sumi C, Suzuki A, Nakayama K. Phantom experiment on estimation of shear modulus distribution in soft-tissue from ultrasonic measurement of displacement vector field. Ieice Trans Fundam Electron Communic Comput Sci 1995b;E78A:1655–1664. Varghese T, Techavipoo U, Liu W, et al. Elastographic measurement of the area and volume of thermal lesions resulting from radiofrequency ablation: Pathologic correlation. AJR Am J Roentgenol 2003;181(3):701–707.
doi:10.1016/j.ultrasmedbio.2005.02.005
● Original Contribution COMPARATIVE EVALUATION OF STRAIN-BASED AND MODEL-BASED MODULUS ELASTOGRAPHY MARVIN M. DOYLEY,*‡ SESHADRI SRINIVASAN,† SARAH A. PENDERGRASS,‡ ZIJI WU,‡ † AND JONATHAN OPHIR *Department of Radiology, Dartmouth-Hitchcock Medical Center, Lebanon, NH, USA; †Ultrasonics Laboratory, University of Texas Medical School, Houston, TX, USA; and ‡Thayer School of Engineering, Dartmouth College, Hanover, NH, USA (Received 30 June 2004, revised 29 January 2005, in final form 3 February 2005)
Abstract—Elastography based on strain imaging currently endures mechanical artefacts and limited contrast transfer efficiency. Solving the inverse elasticity problem (IEP) should obviate these difficulties; however, this approach to elastography is often fraught with problems because of the ill-posed nature of the IEP. The aim of the present study was to determine how the quality of modulus elastograms computed by solving the IEP compared with those produced using standard strain imaging methodology. Strain-based modulus elastograms (i.e., modulus elastograms computed by simply inverting strain elastograms based on the assumption of stress uniformity) and model-based modulus elastograms (i.e., modulus elastograms computed by solving the IEP) were computed from a common cohort of simulated and gelatin-based phantoms that contained inclusions of varying size and modulus contrast. The ensuing elastograms were evaluated by employing the contrast-to-noise ratio (CNRe) and the contrast transfer efficiency (CTEe) performance metrics. The results demonstrated that, at a fixed spatial resolution, the CNRe of strain-based modulus elastograms was statistically equivalent to those computed by solving the IEP. At low modulus contrast, the CTEe of both elastographic imaging approaches was comparable; however, at high modulus, the CTEe of model-based modulus elastograms was superior. (E-mail: [email protected]) © 2005 World Federation for Ultrasound in Medicine & Biology. Key Words: Elastography, Breast imaging, Inverse problems, Model-based inversion, Tissue characterization.
tissue; and, fourth, the spatial variation of the induced internal tissue strains is estimated by performing crosscorrelation analysis on the pre- and postdeformed RF echo frames. The measured strain values can be directly converted to Young’s modulus by using the constitutive elasticity equations (i.e., Hook’s law); however, this requires accurate quantification of all components of the 3-D stress vector. This represents a major limitation because, at present, the internal stress distribution cannot be measured in vivo. However, if the internal stress distribution is constant, then strain elastograms (i.e., images of the axial component of the induced internal tissue strain) can be interpreted as modulus elastograms (relative Young’s or shear modulus images). In practice, the internal stress distribution is not uniform. Accordingly, mechanical artefacts and an incomplete contrast-transfer efficiency are incurred when strain elastograms are interpreted as modulus elastograms based on the assumption of stress uniformity (Konofagou et al. 1996; Ophir et al. 1991; Ponnekanti et al. 1996).
INTRODUCTION Elastography (Ophir et al. 1991) is an emerging imaging modality that can be applied to a broad range of clinical applications. These include assessing plaque vulnerability (Brusseau et al. 2001; de Korte et al. 2000, 2002; Doyley et al. 2001b), guiding minimally invasive therapeutic techniques (Kallel et al. 1999; Righetti et al. 1999; Varghese et al. 2003) and improving the differential diagnosis of breast and prostate cancer (Céspedes 1993; Garra et al. 1997; Hiltawsky et al. 2001; Souchon et al. 2003). The elastographic imaging formation process can be viewed as a four-step process. First, a radio-frequency (RF) echo frame is acquired from the tissue or phantom; second, a small motion is induced within the tissue by using either an external or internal quasistatic mechanical source; third, a second RF frame is acquired from the
Address correspondence to: Marvin Doyley, Ph.D., Thayer School of Engineering, Dartmouth College, 8000 Cummings Hall, Hanover, NH 03755 USA. E-mail: [email protected] 787
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Fig. 2. Schematic diagram of strain and nanoindentation modulus imaging.
Fig. 1. (a) Example of strain elastograms which was obtained from a gelatin-based phantom containing a 10-mm diameter cylindrical inclusions. (a) The original strain elastograms, from which (b) the low-resolution strain elastogram was produced by applying the spatial filter recursively (i.e. five times) to the original strain elastogram (a).
The long-term objective of this work is to enhance the clinical utility of elastography by developing methods to 1. reduce mechanical artifacts and 2. improve the elastographic contrast-transfer efficiency. Considering
elastography within the established framework of solving inverse problems could potentially obviate these limitations, because this approach to elastography does not require accurate quantification of the internal stress distribution to produce modulus elastograms. However, model-based elastography (i.e., solving the inverse problem) is fraught with many problems. Factors such as 1. model-data discrepancy (i.e., the magnitude of error incurred when modeling the forward elasticity problem) and 2. the measurement noise that will undoubtedly compromise the quality and accuracy of the ensuing elastograms. In addition, there is no guarantee of producing unique modulus elastograms with these methods, because of the ill-posed nature of the inverse elasticity problem (IEP) (Barbone and Bamber 2002). Given these problems, there are concerns that model-based elastography may not offer any significant advantage over strain-based elastography. With this view in mind, the aim of the current study was to determine how the quality of model-based modulus elastograms currently compares with that of strain elastograms. In this article, we report the results of computer simulations and experimental studies that were performed on elastically inhomogeneous phantoms to objectively assess the quality of modulus elastograms computed by solving the IEP (model-based modulus elastograms) relative to modulus elastograms computed by simply inverting strain elastograms, based on the assumption of stress uniformity (strain-based modulus elastograms). The ability to discriminate between the physical properties of different tissue types is an important attribute of any diagnostic imaging modality (Hill et al. 1990). Consequently, the contrast discriminations of strain-based and model-based modulus elastograms were assessed by employing the contrast-to-noise ratio (CNRe) performance metric, which is computed in elastography from the means and variances of mechanical properties (i.e., strain or modulus) within an inclusion and surrounding tissue, respectively. Absolute or a good relative estimate of modulus should prove fruitful in several tissue characterization applications. Potential clinical examples include differentiating between different tumor types and monitoring the response of tumors to
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Fig. 3. (a) Modulus elastograms computed by directly inverting strain elastograms and (b) solving the inverse elasticity problem. Elastograms obtained from phantoms containing a single 10-mm diameter inclusion with modulus contrast progressively increased from 0.8 dB to 20 dB (top to bottom). Elastograms in each column are displayed with the same dynamic range. Dashed circle ⫽ circular ROI from which first order statistics (mean and variance) of the inclusion were derived.
therapy. The accuracy (i.e., ability to recover modulus contrast) of modulus elastograms was assessed by using the contrast-transfer efficiency (CTEe) performance metric. Note that this performance metric deals only with the mechanical issues of modulus contrast recovery and does not take estimating of measurement noise into account. Modulus elastograms were also compared with modulus images obtained independently by using the direct me-
chanical indentation measurement technique described in Srinivasan (2003) and Srinivasan et al. (2004). Theory Model-based modulus elastograms can be computed by inverting either the measured displacements (Doyley et al. 2000; Kallel and Bertrand 1996; Oberai et al. 2004) or the measured strains (Raghavan and Yagle
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1994; Skovoroda and Aglyamov 1995; Sumi et al. 1995a; Skovoroda et al. 1995; Sumi et al. 1995b), based on a forward continuum model. In this section, we describe the theory that underpins the iterative model-based inversion technique that we have implemented to solve the IEP. This approach to modulus estimation has previously been described (Doyley et al. 2000; Kallel and Bertrand 1996); therefore, only a brief description of the technique is provided in this section. The IEP can be formulated as a parameter optimization problem with an objective to minimize the difference between the measured axial displacement, um, and that computed by a model description in which tissue mechanical property is parameterized as a set of unknown coefficients representing the observable mechanical behavior. The objective function to be minimized has the following form: ⌽共兲 ⫽ 㛳um ⫺ u兵E其㛳2
(1)
where u{E} represents the axial displacements computed from the Young’s modulus distribution {E} by employing a finite element representation of the forward elastography problem, as described in Reddy (1993). Minimization of eqn (1) with respect to Young’s modulus variations is a nonlinear process that can be realized through an iterative solution for {E}, based upon an initial estimate of the Young’s modulus distribution. The resulting matrix solution at the (k ⫹ 1) iteration has the general form:
兵E其k⫹1 ⫽ 兵⌬E其k ⫹ 关J共Ek兲TJ共Ek兲 ⫹ I兴⫺1 · J共Ek兲T共um ⫺ u兵Ek其兲
(2)
where {⌬E} is a vector of Young’s modulus updates at all coordinates in the reconstruction field and J(Ek} is the Jacobian or sensitivity matrix. This problem usually involves the inversion of a poorly conditioned matrix, [J(Ek)T J(Ek)], which can be regularized by adding an identity matrix, I, multiplied by an arbitrary scalar (i.e., the so-called regularization parameter). Spatial filtering is usually performed as described in Doyley et al. (2000) to suppress local fluctuations in the Young’s modulus estimates and to constrain the reconstruction procedure to a smooth solution. This is realized by computing the weighted average of Young’s modulus at each coordinate location with its surrounding neighbors, according to: E共i兲new ⫽ 共1 ⫺ 兲E共i兲old ⫹
N
N
兺 E共i兲
old
(3)
i⫽1
where the superscripts “old” and “new” represent the unfiltered and filtered Young’s modulus values, respec-
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tively. The filtering weight is chosen so that 0 ⱕ ⱕ 1 and N is the number of neighboring positions that are averaged (i.e., the spatial range of the filter). MATERIALS AND METHODS Simulation study The aim of this investigation was to assess the effect of focal lesions of varying size and modulus contrast on the performance of model-based and strain-based modulus elastograms. Performance was evaluated quantitatively by assessing the contrast discrimination and accuracy of the modulus elastograms. Contrast discrimination was assessed by employing the CNRe performance metric, which is defined in elastography (Bilgen 1997) as: 2共␥L ⫺ ␥B兲2 CNRe ⫽ L2 ⫹ B2
(4)
where ␥L and ␥B represent the mean mechanical properties in the inclusion and background tissues, respectively, and L2 and 2B represent the variance in the respective mechanical properties. The first order statistics (means and variances) were derived from regions-of-interest (ROIs) that were manually selected in the strain-based and model-based modulus elastograms. The ROI in the inclusion was defined by fitting a circle to a contour enclosing the inclusion, as illustrated in Fig. 3. The noise incurred in model-based and strain-based elastography is spatially-dependent. In the case of the model-based technique, this spatial-dependence occurs because of the nonlinear nature of the inverse reconstruction procedure; whereas, in strain-based elastography, this occurs as result of several factors, such as structural decorrelation, the intrinsic noise property of the ultrasonic imaging system and the particulars of the strain-estimation technique. Consequently, ␥B and 2B were computed over the entire background in both cases. The accuracy (i.e., ability to recover modulus contrast completely) was assessed by employing the contrast-transfer efficiency (CTEe) performance metric, which is defined in elastography on a logarithmic scale, as follows (Ponnekanti et al. 1996):
ⱍ
ⱍ ⱍ
共dB兲 ⫽ Co共dB兲 ⫺ Ct共dB兲
ⱍ
(5)
where Co and Ct represent the observed and actual modulus contrast, respectively. Here, ⫽ 0 indicates that modulus contrast is completely recovered (i.e., 100% elasticity contrast recovery) and ⬍ 0 represents an incomplete contrast recovery. The observed modulus contrast was estimated from the mean relative modulus within the ROIs and was manually selected in the inclusion and surrounding background tissue.
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Fig. 4. Mean radial profiles computed from modulus elastograms in Fig. 3, showing mean radial profile obtained from (a) 4-mm, (b) 10-mm, and (c) 12-mm diameter inclusion. Error bar represents ⫾ 1 SD computed from 36 independent radial profiles equally distributed in angle between 0° and 360°. Modulus profiles obtained from () strain and (Œ) model-based modulus elastograms. (—-) Mean radial profile obtained from actual modulus distribution used in computer simulations.
Forward displacement computation Two groups of simulated 2-D phantoms (40 mm ⫻ 40 mm) were constructed using a commercial finiteelement package (ALGOR, Inc., Pittsburgh, PA, USA). The first group of phantoms contained a single 10-mm diameter circular inclusion with modulus contrast (Ec) in the range of 0.8 dB to 40 dB, whereas the modulus contrast in the second group of simulation phantoms was held at a constant value of 3.5 dB; the diameter of the inclusion was varied between 1 and 10 mm. Modulus contrast was defined in this study as:
冉冊
EC共dB兲 ⫽ 20 * log
EL EB
(6)
where EL and EB represent the Young’s moduli in the inclusion and background, respectively. A Poisson’s ratio of 0.495 was assigned to the inclusion and surrounding material in both groups of phantoms. A Young’s modulus of 20 kPa was assigned to the surrounding background material of all phantoms; however, the Young’s modulus of the simulated inclusions was varied to give the desired modulus contrast. The Young’s mod-
ulus of the surrounding background material was chosen to be representative of normal fibroglandular breast tissue (Krouskop et al. 1998). The forward elastography problem was solved in all cases by assigning a prescribed displacement of 0.4 mm to all nodes on the upper boundary of the finite element meshes, which corresponded to an applied strain of 1%. The nodes on the lower boundary of the mesh were constrained from moving in the axial direction, but were free to move in the lateral direction to simulate slip boundary conditions. Uniform stress distribution is generally induced within the tissue or phantom under investigation (except at internal tissue boundaries) when elastography is performed with slip boundary conditions (Kallel et al. 1998). Synthesizing ultrasound echo signals The discrete acoustic tissue model described in Kallel and Bertrand (1996) and Ponnekanti et al. (1996) was spatially distorted using the displacements (i.e., axial and lateral) computed by solving the forward elastography problem. Synthetic RF echo frames were generated
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Fig. 5. (a) Mean CNRe and (b) CTE computed from () strain-based and (Œ) model-based modulus elastograms. Error bar represents ⫾ 1 SD computed from 25 independent elastograms at each modulus contrast.
by convolving the pre- and postdeformed acoustic tissue models with the point spread function of a simulated single-element transducer that had a center frequency of 5 MHz and a 60% band width. These parameters were chosen to be representative of the US scanner employed in our experimental and clinical elastographic imaging system. Modulus estimation The noise incurred in ultrasonic elastography is very complex, depending on several variables, such as the ultrasonic imaging parameters (band width, center frequency), frequency dependent attenuation, sonographic SNR, particulars of the strain estimation technique (widow size, overlap ratio) and structural decorrelation noise (Kallel and Bertrand 1996). Strain and displacement images with realistic measurement noise were generated by applying the adaptive-stretching strain-estimation technique described in (Srinivasan et al. 2002b)
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to pre- and postdeformed RF echo frames that were synthesized by using the elastographic image formation model described by Kallel and Bertrand (1996). Adaptive stretching was used in preference to standard crosscorrelation analysis because the quality (peak elastographic SNR, dynamic range, sensitivity, etc.) of the ensuing strain elastograms is superior (Alam et al. 1998). All echo processing was performed using T ⫽ 3 mm (window length) and ⌬T ⫽ 0.6 mm (80% window overlap). Spurious strain estimates (i.e., regions corresponding to an elastically homogeneous tissue where large strain discontinuity was observed) were removed by applying a 2 mm ⫻ 2 mm median filter to the resulting elastograms. Axial displacements were computed by integrating the filtered strain estimates. One of the shortcomings of the iterative inversion technique that we have implemented is that the computational load required to solve the inverse problem increases considerably with the increasing mesh resolution, whereas the error between the computed and measured displacements generally decreases with increasing mesh resolution. Consequently, finite-element representations of the simulated phantoms were constructed by using uniform finite element meshes consisting of 6561 nodes and 12,800 triangular elements to provide a reasonable compromise between computational load and accuracy of the forward computations. The measured strain and displacement images were interpolated onto the coarse meshes by employing a bilinear interpolation function (MATLAB; The MathWorks, Inc. Natick, MA, USA). Modulus elastograms were subsequently computed from the measured strains and displacements using two approaches. In the first approach, modulus elastograms were computed by applying the iterative nodal-based inverse reconstruction technique, previously described, to the measured displacements (Doyley et al. 2000). All reconstructions were performed by assuming that the tissue under investigation had a homogeneous Young’s modulus of 20 kPa. The subsequent computations can be summarized into the following steps: 1. Compute the axial displacement at each coordinate based on the current estimate of Young’s modulus by solving the partial differential equations governing the underlying quasistatic model, given by:
共 ⫹ 兲 ⵜ 共ⵜ · U兲 ⫹ ⵜ2U ⫽ 0
(7)
where U is the computed nodal displacement vector, which contains both the axial (u) and lateral (v) components of displacement (i.e., 兵U其 ⬵ 兵u1v1...uNvN其, where N is the number of nodes in the finite element mesh), and are the Lamé constants and ⵜ; is the Del operator. The
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relationship between Lamé constants and Young’s modulus (E) and Poisson’s ratio () is given by:
⫽
vE E ,⫽ 2共1 ⫹ v兲 共1 ⫹ v兲共1 ⫺ 2v兲
(8)
The measured axial displacements on the upper and lower surface of the finite element meshes were used as known displacement boundary conditions when solving the forward elasticity problem. 2. Compute the residual displacement vector, {⌬; u}, by subtracting the calculated from the measured axial component of displacement at each nodal coordinate. 3. Construct the Jacobian matrix, [J], using the method described in Kallel and Bertrand (1996). 4. Compute an improved estimate of modulus at each coordinate in the uniform finite element mesh by solving eqn (2), as described in Marquardt (1963). 5. Apply the spatial filter to the recovered modulus elastogram, as described in eqn (3). A filtering weight of ⫽ 0.25 was used in all reconstructions (Doyley et al. 2000). 6. Repeat steps 1. to 5. until the inversion technique converges to a stable solution (i.e., 关J共Ek兲TJ共Ek兲 ⫹ I兴⫺1·J共Ek兲T共um ⫺ u兵Ek其兲 ⬵ 0. Note that this inverse technique generally converges to a stable solution within 10 to 15 iterations. In the second approach, relative modulus elastograms were computed by inverting spatially filtered axial strain elastograms (11) based on the simple assumption of stress uniformity (Ophir et al. 1991): E⬵
1 11
(9)
An apparent limitation of this approach is that it is prone to numerical instabilities (i.e., divide by zero error); therefore, only nonzero strain estimates were inverted. It is important to note that spatial filtering is generally not performed in standard strain-based elastographic imaging; however, to facilitate an objective comparison of both elastographic imaging methodologies, the spatial filter (i.e., eqn (3)) was applied recursively to the strain elastograms before inversion so as to 1. suppress local fluctuations in the strain elastograms before inversion and 2. degrade the spatial resolution of the strain elastogram to that of the modulus elastogram. Figure 1 shows examples of strain elastograms obtained from a highcontrast experimental phantom that contained a single 10-mm diameter cylindrical inclusion. The elastogram on the top is the prefiltered strain elastogram, whereas the elastogram on the bottom was created by applying the spatial filter recursively (5 times) to the strain elastogram shown in Fig. 1a to produce approximately the same degree of smoothing as that incurred during the
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model-based inversion process (note that this was deduced empirically). For the purpose of this demonstration, spatial filtering was performed using a filtering weight of ⫽ 0.25, which was chosen to be the same as that used when computing model-based elastograms. The visual appearance of the strain elastogram was significantly improved, which suggest that, in exchange for reduced spatial resolution, there should be a marked improvement in the contrast discrimination of strain elastograms. Experimental study The goal of the experimental investigation was to corroborate the simulation results. More specifically, it was to compare the performance of both elastographic imaging approach under controlled experimental conditions. Phantoms Cubic phantoms (100 mm3) containing cylindrical inclusions with diameters ranging from 2 mm to 25 mm were manufactured in a repeatable and controlled manner from a gelatin-agar suspension. The surrounding background gel of all phantoms was manufactured from gelatin-agar suspensions consisting of 5% by weight porcine skin gelatin (225 bloom, Kind & Knox, Inc., Sioux City, IA, USA), deionized water (18 mol/L⍀), polystyrene granules (⬇ 100 m mean diameter), 2% by weight agar (Sigma Chemical Co., St Louis, MO, USA) and 0.5% by weight ethylenediaminetetraacetic acid (EDTA). The inclusions were manufactured by filling the pores of open-cell sponge cylinders (100 pores/inch2; Anatomic Concepts, Corona, CA, USA) with gelatinagar suspension (5% by weight gelatin, 2% by weight agar, and 0.5% by weight EDTA), as described in Srinivasan et al. (2003). Based on previous independent mechanical testing, it was anticipated that the resulting phantoms would have modulus contrast in the range of 2.3 dB (1.5:1) to 20 dB (10:1). Data acquisition Elastographic images were obtained using the laboratory system described in Srinivasan et al. (2002a), which consists of a modified Philips HDI-1000 commercial US scanner (ATL Philips Inc., Bothell, WA, USA) that was equipped with a 128-element linear-array transducer (5-MHz nominal frequency and 60% fractional band width) and a computer-controlled mechanical deformation system. Elastographic imaging was performed by first placing the phantom between two large compression plates. To ensure proper contact, all phantoms were prestrained by 1% before the upper compression plate was displaced by 1 mm (which corresponds to a total applied strain of
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Fig. 6. (a) Strain-based and (b) model-based modulus elastograms obtained from low-contrast (3.5 dB) elasticity phantoms. The size of the inclusion in each column was increased from 2 mm to 12 mm, top to bottom. Each elastogram represents the average modulus elastogram computed from 25 independent realizations.
2%). The upper and lower surfaces of the phantoms were thoroughly lubricated with corn oil to approximate slip boundary conditions (Kallel et al. 1999). Independent elastographic imaging A thin section (3-mm thick) that coincided with the imaging plane was carefully removed from all
phantoms before mechanical testing was performed, using a nanoindenter™ (Testworks Inc, Nashville, TN, USA) that was equipped with a 2-mm diameter cylindrical punch. The modulus imaging procedure is shown schematically in Fig. 2; however, a more detailed description of this procedure is given in Srinivasan et al. (2004). Nanoindenter modulus images
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Fig. 7. Mean radial profiles computed from the modulus elastograms shown in Fig. 6. Error bar represents ⫾ 1 SD computed from 36 independent radial profiles equally distributed between 0° and 360°. Modulus profiles obtained from () strain- and (Œ) model-based modulus elastograms .
were obtained by performing measurements over a 16 ⫻ 16 rectangular grid. All mechanical measurements were performed using strain rates of 1% per s. Modulus estimation Modulus elastograms were computed by inverting 1. strain elastograms based on the assumption of stress uniformity and 2. displacement images by using the model-based inversion technique, as described in the simulation study. The adaptive stretching strain-estimation technique used in the simulation study was applied to the digitized RF echo frames to generate strain and displacement images that corresponded with the highest achievable image quality (i.e., peak SNR, dynamic range, sensitivity, etc.). All processing parameters were identical to those used in the simulation study (i.e., T ⫽ 3 mm and ⌬T ⫽ 0.6 mm). RESULTS Simulations Effects of modulus contrast. Composite elastograms obtained from phantoms with modulus contrast in the range of 3.5 dB to 15.5 dB are shown in Fig. 3. Each composite elastogram was computed from 25 statistically- independent elastograms. Paired strain-based and model-based modulus elastograms are shown in Figs. 3a
and b, respectively. Modulus elastograms within the same row are displayed on the same color scale, to facilitate visual comparison between elastograms generated by both approaches. The 10-mm diameter inclusion is discernible in both groups of modulus elastograms at the correct location; however, an artefact in the form of an upright cross is visible in strain-based modulus elastograms (Fig. 3a), which was expected because stress is known to accumulate on the periphery of uniform cylindrical inclusions (Kallel et al. 2001). In general, modelbased elastography takes proper account of the internal stress distribution when computing modulus elastograms; however, artefacts would appear to manifest in two forms. First, artefacts are apparent proximal and distal to the inclusion in the model-based modulus elastograms (Fig. 3b), an expected consequence of the nonunique nature of the inverse elasticity problem, particularly when reconstruction is performed using known displacement boundary conditions (Barbone and Bamber 2002). Second, spatially periodic noise is apparent in model-based modulus elastograms. The exact mechanism of this noise is not fully understood. However, modulation artifacts could occur because of the nonunique nature of the inverse problem. Furthermore, it is well known that a critical phase in the development of any diagnostic imaging technique is the need to optimize the system variables to maximize the sensitivity and
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the iterative model-based inversion approach. The large SD on the estimated CNRe was due primarily to large variation in mean modulus in the background tissue from one independent realization to another. However, it is apparent from Fig. 5b that, at high modulus contrast (Ec ⬎ 6 dB), the CTEe of modulus elastograms computed by employing the model-based inversion approach was superior.
Fig. 8. (a) The mean CNRe and (b) CTE computed from () strain and (Œ) modulus elastograms in Fig. 6. Error bar represents ⫾ 1 SD computed from 25 independent elastograms at each lesion size.
accuracy, and minimizing noise and artifacts that compromise image quality. No attempt was made objectively to optimize the reconstruction process; therefore, the observed modulation artifacts could be the consequence of computing modulus elastograms with a suboptimum reconstruction procedure. We plan to conduct a further study to optimize the reconstruction process and better to understand the mechanism that leads to this modulation. Despite the artifacts, it is apparent, from the mean normalized modulus profiles shown in Fig. 4, that there was good agreement between modulus elastograms computed based on the assumption of stress uniformity and those computed by employing the model-based inversion approach. The mean CNRe and CTEe computed from the elastograms shown in Fig. 3 are presented in Fig. 5a and b, respectively. The CNRe of modulus elastograms computed by simply inverting strain elastograms based on the assumption of stress uniformity would appear to be statistically equivalent to those computed by employing
Effects of lesion size. The ability to detect lowcontrast lesion could potentially lead to earlier tumor detection. Figure 6 shows the mean of 25 modulus elastograms obtained from contrast phantoms (i.e., 3.5 dB) containing inclusions with diameters varying from 2 mm to 12 mm. The paired strain-based and modelbased modulus elastograms are shown in Fig. 6a and b, respectively. The elastogram of the 1-mm diameter inclusion is not shown in Fig. 6, because it proved difficult to visualize such small inclusions with either approach. Note that the modulus modulation artefact observed in Fig. 3b is also apparent in the model-based modulus elastograms shown in Fig. 6b. Nonetheless, the tumor-like inclusions were discernible by both modulus-estimation techniques. Noticeable smoothing of the gradient between the tumor and background tissue was apparent in modulus elastograms produced by both modulus estimation techniques (Fig. 7). The mean CNRe and CTEe metrics are plotted as a function of actual inclusion size in Fig. 8a and b, respectively. It is apparent that, at low modulus contrast, there was statistically no significant difference in either the CNRe or the CTEe recovered from modulus elastograms computed by employing the model-based inversion approach and those computed by simply inverting strain elastograms. Despite the apparent diminution in the spatial resolution of the elastograms, both approaches to elastographic imaging could discern inclusions as small as 2 mm in diameter, with good CNRe (Fig. 8a). Experimental Figure 9 shows representative examples of sonograms obtained from gelatin-based phantoms containing localized inclusions of varying sizes. The position and extent of the inclusions can clearly be visualized in the sonograms, due to differences in acoustic contrast between the inclusion and the surrounding background tissues. Figure 10 shows representative examples of strain-based (Fig. 10a) and model-based (Fig. 10b) modulus elastograms and nanoindenter modulus images (Fig. 10c), corresponding to the sonograms shown in Fig. 9. This figure demonstrates several interesting results. First, the inclusions are discernible in all three images, with the exception of small inclusions (i.e., inclusions with diam-
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Fig. 9. Sonograms obtained from gelatin phantoms containing (a) 2 mm, (b) 5 mm, and (c) 10 mm diameter cylindrical inclusions. Rectangles overlayed on the sonograms denote the ROI from which the elastograms in Fig. 10a and b were computed.
eter ⬍ 5 mm) that were poorly visualized in the low spatial resolution (i.e., the spatial resolution of nanoindentor modulus images was low compared with that of the model-based and strain-based modulus elastograms) nanoindentor modulus image. Second, the modulus contrasts recovered from the strain-based modulus elastograms were lower than those observed in the modelbased modulus elastograms and the nanoindentor modulus images; this was anticipated because, at constant stress, the strain contrast of a linear elastic material is nonlinearly related to the actual modulus contrast. Miller and Bamber (2000) demonstrated theoretically that strain contrast increases asymptotically from zero to unity as the magnitude of the actual modulus contrast of a linear elastic material increases linearly from zero to infinity, which represents a fundamental limitation in contrasttransfer efficiency of strain-based elastography, as discussed in Ponnekanti et al. (1996). It has been demonstrated experimentally that strain contrast and modulus contrast are equivalent only in low-contrast phantoms
(i.e., phantoms with a modulus contrast ⱕ 6.02 dB) (Kallel et al. 1998). Third, a stress concentration artifact (in the form of an upright Maltese cross) are clearly visible in the strain-based modulus elastograms (for the two larger lesions); however, this artefact is not discernible in the model-based modulus elastograms. This was expected because reconstructing modulus elastograms within the framework of inverse problem solution generally takes proper account of the internal stress distribution. Fourth, there was generally good correlation between the modulus contrast recovered using the modelbased inversion technique and those measured using the mechanical indentation technique. The mean CNRe and CTE from the elastograms shown in Fig. 10 are plotted as a function of the lesion size in Fig. 11. As in computer simulations, the CNRe of strain-based and model-based elastograms were statistically equivalent. However, the CTEs of the model-based modulus elastograms were, in general, superior to those obtained by simply inverting strain elastograms.
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Fig. 10. (a) Strain-based, (b) model-based modulus and (c) nanoindenter elastograms corresponding to sonogram in Fig. 9.
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Fig. 10. Continued.
DISCUSSION Elastography based on quantitative strain imaging is well established in the literature; however, it currently suffers from an incomplete CTEe and mechanical artefacts (shadowing and target hardening) (Ophir et al. 1991). Solving the IEP could potentially circumvent these problems; however, this approach to elastography is fraught with many problems, because of the ill-posed nature of the IEP. Consequently, there are concerns that the quality of elastograms computed by solving the IEP is likely to be inferior to those produced using conventional strain imaging methodology. In this paper, we report the first of a series of planned studies that are designed to evaluate the quality of modulus elastograms produced by solving the IEP relative to those produced using conventional elastographic imaging modality in terms of standard measures of imaging quality, such as spatial and contrast resolution, dynamic range, etc. In addition to performing computer simulations, experiments were also conducted on gelatin-based phantoms to facilitate direct comparison between strain-based and modelbased modulus elastograms and modulus images obtained independently by using a direct mechanical indentation technique. To our knowledge, this is the
first reported study where model-based and strainbased modulus elastograms are compared with modulus images obtained by independent mechanical testing. Although the reported results pertain only to quantitative strain imaging as described in Ophir et al. (1991) and the iterative model-based inversion approach described in Doyley et al. (2000), the analysis is applicable to all elastographic imaging techniques. It is important to note that the boundary conditions used in this study represent the best-case scenario for both approaches to elastographic imaging. Therefore, it is reasonable to assume that the results presented in this study are likely to be valid only for slip boundary conditions. We imagine that a different choice in boundary conditions may be conducive for one elastographic imaging approach and not the other. For instance, boundary conditions that produce nonuniform internal stress distribution are likely to favor model-based elastography; similarly, discontinuity in the intrinsic tissue mechanical properties at internal tissue boundaries will undoubtedly produce erroneous results and artefacts in both approaches. However, the ensuing artefacts are likely to be more acute in the model-based inversion approach. Consequently, we propose to conduct further studies to assess the effects
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Fig. 11. (a) The mean CNRe and (b) CTE computed from (●) strain-based and (●) model-based modulus elastograms in Fig. 10. Error bar represents ⫾ 1 SD computed from 8 independent elastograms at each lesion size.
of boundary conditions on the quality of strain-based and model-based modulus elastograms. To summarize our current findings, solving the IEP minimizes stress concentration artefacts incurred when strain elastograms are interpreted as modulus elastograms, based on the assumption of stress uniformity, as demonstrated in Fig. 3; however, artefacts are also incurred in the model-based approach to elastographic imaging. Various investigators have reported similar results by employing iterative and direct model-based inversion techniques. The CNR of both approaches to elastographic imaging increases linearly with increasing modulus, as demonstrated in Fig. 5b. Doyley et al. (2003) observed a similar trend when evaluating the performance of their magnetic resonance imaging-based elastographic imaging system. The main drawback of developing elastography based on model-based modulus imaging is that the spatial resolution of the resulting elastograms is gener-
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ally inferior to that of strain elastograms. This is not surprising because the performance of most inverse reconstruction techniques is stabilized by imposing constraints on the reconstruction procedure, particularly, when reconstruction is performed in the presence of measurement noise. It is apparent, from Figs. 5a, 8a and 11a, that, at a constant spatial resolution, the CNR of strain-based modulus elastograms is statistically equivalent to that computed by using the model-based inversion technique. In general, considerable improvement in the CTEe is achieved when modulus elastograms are computed by using the model-based inversion technique, as demonstrated in Fig. 5b. This observation corroborates the prediction of Ophir and colleagues that solving the inverse problem should improve the CTEe (Kallel et al. 1998; Ponnekanti et al. 1996). Low-contrast lesions as small as 2 mm in diameter are discernible in strain-based and model-based elastograms, with good CNR, as demonstrated in Figs. 6 and 11. The performance metrics increase with increasing lesion diameter in both cases, as demonstrated in Figs. 8 and 11. In addition, it is demonstrated in Fig. 8 that the CTEe of both elastographic imaging approaches are statistically equivalent at low modulus contrast. This result corroborates the experimental results of Kallel et al. (1998), that revealed that strain-based elastography could potentially provide a good relative estimate of modulus in low-contrast medium. Visually, there was good correlation between both groups of modulus elastograms (i.e., model-based and strain-based modulus elastograms) and the nanoindenter modulus images. However, it is important to note that Fig. 11b represents the combined effect of lesion size and modulus contrast on the CTEe, which was due to an oversight in the phantom manufacturing process (i.e., the inclusions were manufactured using the same concentration of gelatin and agar; however, the inclusions were unintentionally manufactured with different sponge material that had different stiffness). Nonetheless, it is apparent that the CTEe of both approaches were comparable for the small lowcontrast inclusions; however, for large high-contrast inclusions, better CTEe was achieved form the modelbased modulus elastograms. A complete comparative evaluation of strain-based and model-based elastographic imaging is clearly beyond the scope of this article and will be the focus of our research for some time. A limitation of the present study is that no attempt was made to compare the performance of both elastographic imaging approaches with multiple heterogeneities and/or complex distributions. In addition, no attempt was made to compare the performance of both approaches within the clinical environment, where there are likely to be complex and continuing varying
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boundary conditions, especially if elastography is performed using the freehand scanning approach described in Doyley et al. (2001a); Hall et al. (2003), and Hiltawsky et al. (2001). These issues will be the focus of future communications.
CONCLUSIONS This paper describes the first reported study to objectively evaluate the performance of model-based elastographic imaging relative to quantitative strain imaging. The results of this preliminary study demonstrate that solving the inverse problem will 1. reduces mechanical artifacts; (however, some artifacts will also be incurred in model-based approaches to elastography, due the illposed nature of the IEP), and 2. substantially improve the CTEe, which would imply that that this approach to elastography should be more suited to tissue characterization applications. In both elastographic imaging approaches, there is a trade-off between spatial and contrast resolution; therefore, contrast resolution can only be compared at a fixed spatial resolution (and vice versa). The results presented in this study demonstrate that, for simple boundary conditions, no advantage will be gained in terms of improving the contrast resolution of elastography by solving the inverse problem. Acknowledgements—The National Cancer Institute (grant P01CA64597) has funded this work.
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