Ultrasonics. Principal component analysis of shear strain effects. Hao Chen, Tomy Varghese * abstract. Contents lists available at ScienceDirect

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1 Ultrasonics 49 (2009) Contents lists available at ScienceDirect Ultrasonics journal homepage: Principal component analysis of shear strain effects Hao Chen, Tomy Varghese * Department of Medical Physics, The University of Wisconsin-Madison, 1111 Highland Avenue, Madison, WI-53706, USA Department of Electrical and Computer Engineering, The University of Wisconsin-Madison, Madison, WI-53706, USA article info abstract Article history: Received 16 September 2008 Received in revised form 18 December 2008 Accepted 18 December 2008 Available online 1 January 2009 Keywords: Displacement Elastography Elastogram Elasticity Elasticity imaging Strain Shear strain Principal component analysis Ultrasound Shear stresses are always present during quasi-static strain imaging, since tissue slippage occurs along the lateral and elevational directions during an axial deformation. Shear stress components along the axial deformation axes add to the axial deformation while perpendicular components introduce both lateral and elevational rigid motion and deformation artifacts into the estimated axial and lateral strain tensor images. A clear understanding of these artifacts introduced into the normal and shear strain tensor images with shear deformations is essential. In addition, signal processing techniques for improved depiction of the strain distribution is required. In this paper, we evaluate the impact of artifacts introduced due to lateral shear deformations on the normal strain tensors estimated by varying the lateral shear angle during an axial deformation. Shear strains are quantified using the lateral shear angle during the applied deformation. Simulation and experimental validation using uniformly elastic and single inclusion phantoms were performed. Variations in the elastographic signal-to-noise and contrast-tonoise ratios for axial deformations ranging from 0% to 5%, and for lateral deformations ranging from 0 to 5 were evaluated. Our results demonstrate that the first and second principal component strain images provide higher signal-to-noise ratios of 20 db with simulations and 10 db under experimental conditions and contrast-to-noise ratio levels that are at least 20 db higher when compared to the axial and lateral strain tensor images, when only lateral shear deformations are applied. For small axial deformations, the lateral shear deformations significantly reduces strain image quality, however the first principal component provides about a 1 2 db improvement over the axial strain tensor image. Lateral shear deformations also significantly increase the noise level in the axial and lateral strain tensor images with larger axial deformations. Improved elastographic signal and contrast-to-noise ratios in the first principal component strain image are always obtained for both simulation and experimental data when compared to the corresponding axial strain tensor images in the presence of both axial and lateral shear deformations. Ó 2008 Elsevier B.V. All rights reserved. 1. Introduction Techniques that image tissue elasticity or local stiffness properties of tissue are relatively newer methods for the non-invasive investigation of tissue mechanical properties [1 7]. Local tissue displacements along the ultrasound beam propagation direction are estimated using classical time delay estimation techniques [1]. Signal processing techniques for estimating local tissue displacements have progressed from the earlier one dimensional (1D) cross-correlation techniques, to two-dimensional (2D) crosscorrelation based tracking and processing [8] and to 2D algorithms with three-dimensional (3D) tracking [9,10]. Significant improvements in the spatial resolution of the axial strain estimates have also been reported, with axial window dimensions on the order * Corresponding author. Address: Department of Medical Physics, The University of Wisconsin-Madison, 1111 Highland Avenue, Madison, WI-53706, USA. Tel.: ; fax: address: tvarghese@wisc.edu (T. Varghese). of 1 2 wavelengths [11], 2D kernels with axial dimensions of less than a wavelength along with several A-lines in the lateral direction [7]. However, most of the strain tensor estimation methods discussed in the literature estimate primarily the axial component of the strain, while the lateral (perpendicular to insonification direction and within the scan plane) and elevational (perpendicular to the insonification direction and scan plane) displacements and strain are generally not estimated. Since, the quasi-static tissue deformation introduces motion and displacements in threedimensions, all strain tensor and displacement vector components are required to completely characterize the applied deformation [6,12]. Several methods have been proposed for the estimation of the displacement vector and the normal and shear strain tensor components [13 16]. Estimation of all the displacement vector and strain tensor components provide a complete depiction of tissue deformation in most situations, however, in certain cases such as for cardiac motion, [17,18] tissue deformations encountered are complex and other approaches for quantifying the displacement X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi: /j.ultras

2 H. Chen, T. Varghese / Ultrasonics 49 (2009) and strain, such as the use of principal component analysis (PCA) may be necessary [19]. Both normal and shear strains have been estimated in the presence of only axial deformations [20,21] and with lateral shear deformations [22]. Typically shear strains are observed at the boundaries between a lesion or inhomogeneity and background tissue. The magnitude of the shear strains generated depends on various factors that include the modulus contrast, lesion mobility (i.e. attachment of the lesion to the background) and the applied axial and/or lateral shear deformations. In general the axial deformations applied for strain imaging do not produce large values of the shear strains, and lateral shear deformations have been utilized to enhance shear strain effects to fully characterize lesion mobility assessments for diagnosis [22]. As previously described, shear stresses are always present either due to non-uniform deformation or tissue slippage during the axial deformation of tissue, generally utilized as the mechanical stimulus for quasi-static elastography. Components of the shear stress, both along and perpendicular to the beam direction in the ultrasound image plane may also be present. These shear stresses introduce relative shifts in tissue without introducing changes in the volume. The gradient of these relative shifts are equal to zero in theory due to the minimal volume change. However, strain noise artifacts are introduced in the normal strain tensor (axial and lateral) images with these relative shifts, affecting the quality of the strain tensor images obtained. Principal component analysis (PCA) [23,24] is another method for characterizing the strain distribution in the presence of shear strain artifacts and for geometries where the primary strain tensor components may not lie along the ultrasound insonification direction (i.e. strain distributions in the myocardium). Principal strains are defined as the normal strain components along the deformation axes where the shear strains are also included in principal strains. PCA has been used previously for characterizing myocardial strain distributions [19]. In this paper, we evaluate the impact of shear strains on the normal strain tensor (axial and lateral) and principal component strain images utilizing both simulation and experimental data obtained using tissue-mimicking (TM) phantoms. Shear deformations are quantified using the lateral shear angle for deformed tissue enabling quantification of shear strain effects. For small axial deformations, lateral shear strain artifacts significantly reduce axial and lateral strain tensor image quality. Lateral shear strain artifacts also increase the noise level for axial and lateral strain tensor image with larger axial deformations. In this paper, we demonstrate improved strain estimation performance obtained utilizing PCA. A quantitative comparison of PCA results to that of axial and lateral strain tensor images using elastographic signal-to-noise (SNR e ) [25,26] and contrast-to-noise (CNR e ) [27] ratio parameters is presented. Principal component strain images provide improved SNR e and CNR e estimates even in the presence of large lateral shear deformations. The next section provides theoretical expressions for the normal strain tensor components and the corresponding first and second principal strain components following an axial deformation. Section 3 describes the simulation procedure and the experimental apparatus utilized to apply both the axial and lateral deformations to the TM phantoms. Simulation and experimental results obtained are discussed in the following section. Finally, we discuss the contributions of this paper and summarize our results in Section 5. direction (ultrasound beam direction). The lateral direction or x- axis is defined as the direction orthogonal to the axial direction and located within the imaging plane, while the y-axes represent the elevational direction. Consider the tracking of a displacement vector d at point O in space observed using a 1D linear array ultrasound transducer shown in Fig. 1. The displacement vector d contains orthogonal axial and lateral displacement components d z and d x. Axial and lateral strain estimates obtained from the gradient of the respective displacements, can be written as strain z ¼ e z ; strain x ¼ e x : Shear stresses in the medium introduced by non-uniform deformations introduce sliding forces within the medium. Shear stresses parallel to the insonification direction, are shown in Fig. 1. This component of the introduced shear stress can be quantified by the lateral shear angle. Tissue scatterers located at the same axial depth move in the direction of the applied shear stress. In theory, the relative shift due to the applied shear stress will not change the tissue volume therefore the gradient of the displacement will be equal to zero. However, the lateral shift introduced in the postdeformation data with the applied shear stress introduces noise artifacts in the local displacement estimated from the RF echo signals Principal component analysis Principal component analysis [23,24,28] is a technique utilized to reduce the dimensionality of a data set by transforming them into a new set of uncorrelated variables, namely the principal components of the data set. In many applications, two principal components are sufficient to characterize an entire data set. PCA has also been referred to as the discrete Karhunen Loève transform or the Hotelling transform [24]. The earliest description of PCA was provided by Pearson [23] and Hotelling [24]. The advantage of PCA comes with the increased computational demand for higher dimensionality datasets. PCA does not have a fixed set of basis vectors, with the basis vectors calculated from the dataset. Axial Compression O d Lateral Shear Angle Original Shape ð1þ ð2þ Deformed Shape x 2. Theory z 2.1. Shear strain effects The compressional force or mechanical stimulus for elastography is typically applied along the z-axis also referred to as the axial Fig. 1. Schematic diagram illustrating axial deformation and lateral shear deformation. Tracking of the actual displacement (vector d) at a point O in the displacement field is also shown.

3 474 H. Chen, T. Varghese / Ultrasonics 49 (2009) PCA technique applied for strain imaging Both the axial and lateral normal strain tensors can be obtained from 2D ultrasound data acquired using a clinical ultrasound scanner and a linear array transducer [13,15,21,29]. Lateral strain estimates are obtained using 2D cross-correlation techniques [8,30,31]. PCA can be utilized to project these axial and lateral normal strain tensors into the first and second principal component strain variables, respectively. The axial and lateral normal strain tensors obtained are dependent on the position of the ultrasound transducer, while the principal strains are defined as the normal strains along the axes of deformation where the shear strains are fully expressed. To apply PCA for strain imaging, the shear strains have to be calculated. Shear strains are defined using l zx ¼ 1 : ð3þ This definition discriminates against rotation artifacts. Similar to cross-correlation matrix of a statistical dataset, the cross-correlation matrix for the strain tensors can be written as C zx ¼ e z l zx : ð4þ l zx e x In general, this is full rank matrix even though the axial and lateral strains are not completely independent. Since the ultrasound data and applied deformation or deformations are in two-dimensions, only two principal strain components are obtained. The cross-correlation matrix can therefore be solved algebraically to obtain the two corresponding eigenvalues or principal strains and corresponding eigenvectors or direction of the principal deformation axes. Axial and lateral strains can therefore be projected into the new deformation coordinate axes, whose principal components are given by e p1 ¼ e x þ e z þ 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e x e 2 z þ l 2 zx; ð5þ 2 e p2 ¼ e rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x þ e z e x e 2 z þ l 2 zx: ð6þ 2 2 Since the applied deformation is primarily along the axial direction of the ultrasound transducer with shear strain effects, the deformation axes or the corresponding eigenvectors would generally be the same as the axial and lateral strain tensors for small shear strains. 3. Materials and methods 3.1. Simulation procedure A numerical TM phantom was constructed using the commercial finite element analysis (FEA) package ANSYS (ANSYS Inc., Pittsburgh, PA). Mechanical stimuli (i.e. the axial and lateral deformation) can be applied to numerical phantom under ANSYS to obtain a deformed phantom and the corresponding axial and lateral displacement information. Uniformly elastic and single inclusion phantoms with dimensions mm were constructed to quantify the SNR e [25,26] and CNR e [27] parameters, respectively. The Poisson ratio for the simulated phantom was set to 0.495, to model tissue incompressibility. The embedded cylindrical inclusion has a diameter of 10 mm and was three times stiffer than the background. Uniform external uniaxial deformations ranging from 0.25 to 3% of the phantom height and lateral shear angles ranging from 0.1 to 5, respectively were applied under ANSYS. Local displacements along the axial and lateral direction obtained under ANSYS are then utilized to displace scatterer positions within the numerical phantom to obtain the deformed state. Ultrasound echo signals are then computed using an ultrasound simulation program using the numerical phantom that includes the randomly distributed scatterers in the pre- and post-deformation state [32]. Pre- and post-deformation ultrasound RF echo signals were obtained using a frequency-domain ultrasound simulation program developed by Li and Zagzebski [32]. A linear array transducer was modeled, which consisted of mm rectangular elements with a 0.1 mm center-to-center separation. Each acoustic beam was formed using 64 identical elements. Incident pulses were modeled to be Gaussian shaped with a 7.5 MHz center frequency and an 80% bandwidth (full-width at half-maximum). The speed of sound in the simulated medium was set to 1540 m/ s, and attenuation coefficient was set to 0.5 db/cm/mhz for both the inclusion and background. Tissue scatterers were modeled using 100 lm radius polystyrene beads, which were randomly distributed in the numerical phantom at a number density of 10 scatterers per cubic millimeter to ensure Rayleigh scattering statistics. The simulated pre- and post-deformation echo signals obtained were then utilized to evaluate the performance of strain tensor estimation and the PCA technique. Ten independent experimental datasets were obtained using the ultrasound simulation program by randomly varying the scatterer locations in the pre-deformed numerical TM phantoms Experimental procedure An uniformly elastic TM phantom with dimensions of mm and Young s Modulus of 30 kpa was used to quantify the strain image quality in terms of the SNR e [33]. A single inclusion TM phantom with dimensions of mm, that contains a 10 mm diameter cylindrical inclusion, three times stiffer than the background [33] was used to quantify CNR e variations. The Young modulus of the background material was 40 kpa while that of the inclusion was 120 kpa. Young s modulus measurements were performed using an ELF 3200 mechanical testing system (EnduraTEC, Minnetonka, MN). The lateral shear deformation was introduced experimentally, by placing the TM phantom on a plate with a machined coarse surface, attached to a stepper motor that introduces a lateral translation of the phantom, while keeping the top surface of the phantom fixed. The top surface of the phantom was fixed using a compression plate with a rectangular slot for the transducer and mounted to a linear stage driven by a second stepper motor along the axial direction. The surface of the top and bottom plates was machined to be coarse to prevent slippage between the plate and the phantom. Both compression plates were larger than the phantom surface to provide uniform axial and lateral shear deformations to the phantom. Axial deformations ranging from 0.5 to 5% were applied to the phantom, in conjunction with lateral shear angle deformations ranging from 0 to 9.5, depending on the experimental procedure. Each experiment was repeated 10 times with data acquisition performed at different locations in the phantom to obtain statistically independent pre- and post-deformation data sets. The TM phantoms were scanned using an Aloka SSD 2000 realtime clinical ultrasound scanner (Aloka Inc., Tokyo, Japan), using a 7.5 MHz linear array transducer with an approximate 60% bandwidth. The linear array transducer provided 360 A-lines in a single RF data frame. A single transmit focus was set at a depth of 50 mm in the phantom with dynamic focusing on receive. Echo signals were acquired originating from the top of the phantom to a depth of 6 cm. An initial pre-deformation of 3% was applied to ensure proper contact between the compression plate and the phantom.

4 H. Chen, T. Varghese / Ultrasonics 49 (2009) Ultrasound RF signals were digitized using a 12-bit data acquisition board (Gage Applied Technologies Inc., Quebec, Canada) at a sampling rate of 100 MHz. The RF data was stored in a personal computer for off-line analysis. The pre- and post-deformation RF echo signals were analyzed using a multi-step 2D cross-correlation method that was previously described in the literature [11]. The first correlation step uses 50% overlapped gated kernels that consist of 18 wavelengths along the axial direction and nine beam lines along the lateral direction. The second step uses kernels of 12 wavelengths along the axial dimension and five beam lines with a 50% overlap. Finally, the third step uses kernels of three wavelengths along the axial dimension and five beam lines with a 50% overlap of the kernels to estimate local displacements. The axial and lateral displacement data obtained was then filtered using a 7 7 2D median filter before computing the respective strain images to remove any large displacement noise outliers. Axial, and lateral strains are estimated using a least-squares strain estimator from their respective displacement maps [12], using Eqs. (1) and (2). The first and second principal strain components were obtained using Eqs. (5) and (6), respectively. 4. Simulation and experimental results 4.1. Simulation results Quantitative estimates of the mean and standard deviation of the SNR e estimates obtained using the simulated uniformly elastic phantom for the axial and lateral strain tensor components and the corresponding principal strain components are shown in Fig. 2. SNR e variations for different values of the axial deformation are plotted versus the lateral shear angle in Fig. 2. For the case with no axial deformation or 0% applied axial deformation (Fig. 2a), the SNR e levels of the axial and lateral strain tensor image was less than 0 db. The strain estimates in the axial and lateral strain tensor images are quite small since no axial deformation is applied. At the same time, the noise level in the axial and lateral strain tensor image increases due to the noise artifacts introduced by the lateral shear deformation. The first principal component of the strain tensor images, on the other hand provides significantly improved SNR e levels over the entire range of lateral shear deformations from 0 to 5. The improvement obtained with Fig. 2. Plots of the elastographic SNR e for a simulated uniformly elastic phantom versus the applied lateral shear angle (0 5 ) for the normal (axial and lateral) strain tensors and the first and second principal component strain images. The SNR e results are plotted for (a) 0%, (b) 1%, (c) 3%, and (d) 5% axial deformation. The mean and standard deviation of the SNR e were obtained from 10 independent datasets.

5 476 H. Chen, T. Varghese / Ultrasonics 49 (2009) Fig. 3. Comparison of the axial and lateral strain tensor images of a simulated single inclusion elastic phantom obtained using the multi-step 2D cross-correlation method. Units of the x and y-axes are in mm. The first and second principal component strain images obtained from the normal and shear strain tensors are also shown for different values of the axial deformation, namely (a) 0% and (b) 2% axial deformations. The lateral shear angle used was 4.0.

6 H. Chen, T. Varghese / Ultrasonics 49 (2009) PCA is larger than 30 db when compared to the original axial or lateral strain tensor image for the same lateral shear angle. This result is expected since axial and lateral strain tensor images would be unable to effectively depict a strain distribution in the presence of only lateral shear deformations. In the presence of an axial deformation, the axial strain tensor images provide an effective and accurate depiction of the strain distribution as shown in Figs. 2b d. The SNR e value obtained in the axial strain tensor image is around 30 db for a zero lateral shear angle and shows a decreasing trend in the presence of both an axial deformation and an increase in the lateral shear angle as shown in Fig. 2b for a 1%, (c) for a 3%, and (d) for a 5% axial deformation. Increasing the lateral shear angle introduces additional noise artifacts into the axial and lateral strain tensor images. The first principal component strain image, however, always provides a higher SNR e when compared to the axial strain tensor image to which it corresponds. Increasing the lateral shear angle increases the shear strain contributions to both the normal strain tensors and principal strains. Note that the errorbars between the first principal component strain and axial strain tensor estimates do not overlap for the 0%, 1%, and 3% axial deformation cases. However, with larger applied axial deformations the axial strain estimates and the first principal component strain estimates provide similar results. Fig. 3 presents both the normal strain tensor images (i.e. the axial and lateral strain distribution) along with the first and second principal component strain images for the single inclusion phantom. Observe from Fig. 3a, that the inclusion is not clearly visualized in the normal axial strain tensor image in the presence of only the lateral shear deformation. However, the inclusion can be seen in both the first and second principal strain component images. Note that in the presence of an axial deformation in Fig. 3b, the inclusion is seen in the axial normal strain tensor image. Observe that the inclusion is not seen in the lateral strain tensor image which is corrupted by significant noise artifacts. Observe also that the stiffer inclusion appears softer on the axial strain tensor and principal component images in the presence of only a lateral shear deformation. In the presence of an axial deformation, the inclusion appears stiffer on the axial strain tensor and principal component images, however some reduction in the contrast is apparent when compared to the case with no lateral deformation applied. Possible causes for the differences in the strain contrast are discussed later in the paper. Fig. 4. Plots of the elastographic CNR e for a simulated uniformly elastic phantom versus the applied lateral shear angle (0 5 ) for the normal (axial and lateral) strain tensors and the first and second principal component strain images. The CNR e results are plotted for the cases with (a) 0%, (b) 1%, (c) 3%, and (d) 5% axial deformation. The mean and standard deviation of the CNR e were obtained from 10 datasets.

7 478 H. Chen, T. Varghese / Ultrasonics 49 (2009) Variations in the CNR e of strain images depicting the single inclusion phantom are shown in Fig. 4. For the case with no axial deformation or 0% applied deformation along the axial direction (Fig. 4a), the first principal component strain image provides significantly higher CNR e values. The improvement obtained using the PCA technique is larger than 30 db when compared to the original axial or lateral strain image at similar lateral shear angles. This result is expected and similar to the performance visualized using the SNR e parameter in Fig. 2. Note that in the presence of an axial deformation, the axial strain tensor images provide an effective and accurate depiction of the inclusion and the background as quantitated in Fig. 4b d. The CNR e values of the axial strain image vary from 13 to 22 db corresponding to a 1 5% axial deformation for zero lateral shear angle case. In addition, the CNR e shows a decreasing trend with an increase in the lateral shear angle as shown in Fig. 4b for a 1%, (c) for a 3%, and (d) for a 5% axial deformation. Fig. 5 presents plots of the variations in the SNR e and CNR e for a range of applied axial deformations plotted along the x-axes for a lateral shear angle of 1. With a fixed lateral shear angle, the SNR e and CNR e of the axial and lateral strain tensors increase with axial deformation. Note that the SNR e of the first principal strain component is always higher than its corresponding axial strain tensor, with significant improvement obtained for the case with no axial deformation. Similar conclusions are drawn from the variations in the CNR e shown in Fig. 5b. The first principal component strain image provides better image quality in terms of SNR e and CNR e than the original axial and lateral strain tensor images. The image quality of the second principal strain component image is dependent on the axial and lateral deformation applied. Note also the presence of a drop or dip in the SNR e and CNR e values in the curves for the second principal strain component for different combinations of the applied axial and lateral deformation. The exact locations of these minima and their relationships to the applied axial and lateral deformation are currently under investigation Elastograms from experimental data Fig. 6 presents axial and lateral strain tensor images along with the first and second principal component strain images for the single inclusion phantom. The applied axial deformation was 0, and 2% for cases (a), and (b), respectively. The lateral shear angle was 3.18 for all the cases shown in Fig. 6. In the absence of an axial deformation applied to the TM phantom, it is very difficult to differentiate the inclusion from the background in the axial and lateral strain tensor image shown in Fig. 6a. The first and second principal component strain image, on the other hand indicates the presence of an inclusion at the center of the image. The principal component strain image provides better contrast for the stiffer region within the TM phantom. However, the strain contrast of the stiffer inclusion does not correspond to that generally observed on an axial strain tensor image, in that the inclusion appears to be softer i.e. depicts larger strain in the first principal component strain image. This result is primarily caused by the lateral shear deformation applied to the phantom. The shear strain in the phantom depends on the shear modulus and shear stress as shown in the following equation c xy ¼ r xy G xy ; where c xy denotes the shear strain, r xy is the shear stress and G xy represents the shear modulus. The shear stress is introduced by the lateral deformation of the phantom, and the shear modulus is determined by the TM material used to manufacture the phantom. The shear modulus is related to the Young Modulus by ð7þ Fig. 5. Plots of elastographic (a) SNR e and (b) CNR e versus the applied axial deformation for the normal (axial and lateral) strain tensors and the first and second principal component strain images. The lateral shear angle for the simulation was 1. q G xy ¼ 2ð1 þ cþ ; where q represents the Young Modulus of the TM material and c is its Poisson s ratio. Because of the lateral shear forces on the phantom it is tilted, and the center of the phantom incurs increased shear stresses when compared to boundary or outer edges of the phantom. To understand the relative contributions of the lateral shear deformation and axial deformation on the principal component and normal strain tensor images, we utilized ANSYS FEA software to generate the corresponding stress distribution images. The ANSYS software package provides an excellent tool to analyze shear stresses within deformed materials. Both uniformly elastic and single inclusion FEA phantom models previously described were utilized. Ideal images of the axial stress, lateral stress and shear stress for both uniformly elastic and single inclusion phantoms are shown in Fig. 7, for a lateral shear deformation of 4. Observe the brighter region at the center of the shear stress images for the uniform phantom (Fig. 7c (left)), indicating that the center of the uniform phantom would incur increased shear strains when compared to the boundary from Eq. (8), even though the phantom is uniformly elastic. In addition, also note from the shear stress image for the single inclusion phantom (Fig. 7c (right)) that increased ð8þ

8 H. Chen, T. Varghese / Ultrasonics 49 (2009) Fig. 6. Comparison of the axial and lateral strain tensor images of a single inclusion tissue-mimicking phantom obtained using the multi-step 2D cross-correlation method. The unit of the x and y-axes are in mm. The first and second principal component strain images obtained from the normal and shear strain tensors are also shown for different values of the axial deformation, namely (a) 0% and (b) 2% axial deformations. The lateral shear angle used was shear stresses are observed at the center of the phantom. Thus even when the inclusion has a higher Young s modulus, the inclusion will incur high shear strains due to the increased shear stresses at the center of the phantom. Therefore for low values of the axial deformation, shear strain effects dominate the principal component strain images. Therefore the strain contrast of the stiffer inclusion would not correspond to the strain contrast observed based on an axial strain tensor image. The inclusion in these cases appear to be softer i.e. depicts larger strain in the first principal component strain image. However, the strain contrast of the stiffer inclusion becomes closer to that observed in an axial strain image when the axial deformation is more dominant as shown in Fig. 6c. As the axial deformation is increased from 0% to 2%, the quality of the axial and lateral strain tensor images improve as shown in Fig. 6a and b for the same shear angle of However, only a partial visualization of the inclusion is observed in the axial and lateral strain tensor images due to the presence of the lateral shear deformation. On the other hand, the first and second principal component strain images provide improved contrast. Quantitative experimental results that demonstrate the improvement in the SNR e and CNR e obtained using PCA are shown in Figs. 8 and 9. Fig. 8 presents the variation in the SNR e obtained from the axial and lateral strain tensor estimates and the first and second principal component strain estimates obtained using the uniformly elastic TM phantom. The mean and standard deviation of the SNR e estimates are calculated from 10 independent datasets to obtain statistically significant results. Fig. 8, plots the SNR e values for the normal strain tensors and PCA strain estimates for different lateral shear angles in the presence of (a) 0%, (b) 1% and (c) 2% applied axial deformation, respectively. For the case with no axial deformation or 0% applied compression along the axial direction, the first principal component strain image provides a SNR e level that is 11.8 db better than that obtained from the axial normal strain tensor image. However, in the presence of an axial deformation, the strain tensor images provide an effective and accurate depiction of the strain distribution as shown in Fig. 8b and c. The SNR e values estimated, however, show a decreasing trend in the presence of both an axial deformation and an increase in the lateral shear angle as shown in Fig. 8b for a 1%

9 480 H. Chen, T. Varghese / Ultrasonics 49 (2009) Fig. 7. Images of (a) axial stress (b) lateral stress and (c) shear stress for a uniformly elastic TM phantom on the left and the single inclusion phantom in the right column. Note the presence of increased shear stresses at the center of phantom for both the uniformly elastic and single inclusion phantoms constructed using FEA. The lateral shear angle used was 4. and (c) for a 2% axial deformation. The first principal component strain image always provides a higher SNR e value when compared to its corresponding axial strain tensor image. The errorbars between the first principal component strain estimate and axial strain tensor estimates do not overlap for the 0% and 1% axial deformation cases. However, with the increased applied axial deformations the axial strain estimates and the first principal component strain estimates provide similar results. The SNR e of the lateral strain tensor estimates also reduces with increased shear angle as shown in Fig. 8b and c. The second principal component that corresponds to the lateral strain tensor for small deformations also reduces initially with an increase in the shear angle, but gradually levels off and then increases steadily with an increase in the shear angle. After a crossover point that depends on the applied shear deformation, the second principal component consistently provides higher SNR e estimates when compared to the lateral strain tensor component. The SNR e value of the second principal component strain estimates also dips when the second principal component strain value reaches the crossover point as shown in Fig. 4b and c. The principal component strains are the projection of the original axial, lateral and shear strain to the new coordinate system. The SNR e of the first principal component strain is always higher than the normal axial strain tensor image. However, the second principal component strain values do change gradually following the lateral and shear strain. The second principal strain component values vary in Fig. 8a and b as was previously shown in Fig. 6a and b. The SNR e of the second principal strain component strain value also indicate the presence of minima whose position changes with the applied axial compression and lateral shear deformation. In a similar manner, the CNR e results obtained using the single inclusion phantom also present similar trends as that described for the SNR e estimates and is shown in Fig. 9. For the case with no applied axial deformation (0%), both the first and second principal component strain images provide a CNR e level that is at least 16.3 db higher than that obtained with the axial and lateral strain tensor images. In the presence of the axial applied deformation, the CNR e estimates obtained with PCA reduce with increasing shear angle, and the CNR e levels of the axial and lateral strain tensor images decrease gradually. However, improvements in the CNR e values with PCA analysis are still observed for both the applied axial deformation cases (1% and 2%). Note that the maximum lateral shear angle utilized for the results presented in Fig. 9 (i.e. 5.5 ), is smaller than the lateral shear angle utilized in Fig. 8 (i.e. 9.5 ). Although, we would have preferred to demonstrate the variations in the CNR e over the larger shear deformation in Fig. 9, we were unable to apply large shear deformations for the single inclusion phantom. Two factors precluded the application of the larger lateral shear deformations to the inclusion phantom, the first being that the dimensions of the

10 H. Chen, T. Varghese / Ultrasonics 49 (2009) Fig. 8. Plots of elastographic SNR e versus the applied lateral shear angle (0 9.5 ) for the normal (axial and lateral) strain tensors and the first and second principal component strain images. Three cases with (a) 0%, (b) 1% and (c) 2% axial deformation are shown. The mean and standard deviation of the SNR e are obtained from 10 independent experimental datasets. Fig. 9. Plots of elastographic CNR e versus the applied lateral shear angle (0 5.1 ) for the normal (axial and lateral) strain tensors and first and second principal component strain images. Three cases with (a) 0%, (b) 1% and (c) 2% axial deformation are shown. The mean and standard deviation of the CNR e are obtained from 10 independent experimental datasets.

11 482 H. Chen, T. Varghese / Ultrasonics 49 (2009) uniformly elastic phantom ( mm) were larger than that of the inclusion phantom ( mm). The uniformly elastic phantom therefore provided a larger surface area to apply the lateral deformation with the compression plate that was machined to be coarse, to provide a better grip on the phantom. Secondly, the Young modulus of the uniformly elastic phantom (30 kpa) was lower than that of the inclusion phantom (40 kpa). A combination of both these factors caused the surface of the inclusion phantom to slip for larger applied lateral shear angles (>5.5 ). 5. Discussion and conclusions In this paper, we demonstrate the impact of lateral shear deformations on both normal strain tensor and principal component strain images utilizing both simulated and experimental data acquired on TM phantoms. Lateral shear deformations introduce strain noise artifacts into the normal axial and lateral strain tensor images. These noise artifacts are reduced in the first and second principal component strain images. We also compare the first principal strain component to the normal axial strain tensor image in the paper. This is due to the fact that currently the normal axial strain tensor is the most commonly imaged strain tensor component in elastography. Note that the first principal component would approach the shear stress with larger lateral deformations, especially when they are not accompanied by large axial deformations. In this paper, PCA has been utilized to project axial and lateral strain tensors to the new principal coordinate system. Simulation and experimental results demonstrate that the first and second principal component strain images improve the strain image quality in terms of the SNR e and CNR e when compared to the axial and lateral strain tensor images. Experimental results show that PCA can provide reliable strain images with small lateral shear angles even under small axial deformations. Principal component strain images provide better contrast between the stiffer inclusion and softer background regions of the TM phantom. However, contrast in the principal component strain images do not correspond to the Young modulus contrast of the TM phantom especially in the absence of axial deformations. Stiffer regions appear with brighter gray scale levels than softer regions in the principal component strain image when an axial deformation is not present. These results were justified using Fig. 6 and 7, where we demonstrate that increased shear stresses are present at the center of the phantom due to the lateral shear deformation. However, in the presence of an axial deformation the stiffer inclusion is clearly demarcated as a stiff region in both the axial and lateral strain tensor images and the principal component strain images. Quantitative results obtained using both simulation and experimental data indicate that the first principal component image always presents with an increased SNR e and CNR e value when compared to its corresponding axial strain tensor image in the presence of both lateral shear and axial deformations. In the presence of only lateral shear deformations, experimental results indicate that the first principal component strain image provides a SNR e level that is 11.8 db higher than that obtained from the axial strain tensor image under experimental conditions. The SNR e value obtained was over 20 db with simulations and 10 db under experimental conditions for both the first and second PCA components over the entire range of shear deformations when compared to the axial and lateral strain tensor images for the 0% axial deformation case. 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