Original Contribution

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1 PII S (98) Ultrasound in Med. & Biol., Vol. 24, No. 8, pp , 1998 Copyright 1998 World Federation for Ultrasound in Medicine & Biology Printed in the USA. All rights reserved /98 $ Original Contribution A NEW ELASTOGRAPHIC METHOD FOR ESTIMATION AND IMAGING OF LATERAL DISPLACEMENTS, LATERAL STRAINS, CORRECTED AXIAL STRAINS AND POISSON S RATIOS IN TISSUES ELISA KONOFAGOU* and JONATHAN OPHIR* *Ultrasonics Laboratory, Department of Radiology, The University of Texas Medical School, Houston, TX USA; and Department of Electrical Engineering and Program in Biomedical Engineering, University of Houston, Houston, TX USA (Received 31 December 1997; in final form 15 June 1998) Abstract A major disadvantage of the current practice of elastography is that only the axial component of the strain is estimated. The lateral and elevational components are basically disregarded, yet they corrupt the axial strain estimation by inducing decorrelation noise. In this paper, we describe a new weighted interpolation method operating between neighboring RF A-lines for high precision tracking of the lateral displacement. Due to this high lateral-tracking precision, quality lateral elastograms are generated that display the lateral component of the strain tensor. These precision lateral-displacement estimates allow a fine correction for the lateral decorrelation that corrupts the axial estimation. Finally, by dividing the lateral elastogram by the axial elastogram, we are able to produce a new image that displays the distribution of Poisson s ratios in the tissue. Results are presented from finite-element simulations and phantoms as well as in vitro and in vivo experiments World Federation for Ultrasound in Medicine & Biology. Key Words: Correction, Displacement, Elasticity, Elastic modulus, Elastogram, Elastography, Imaging, Interpolation, Lateral, Poisson s ratio, Shear, Strain, Tracking, Ultrasound. INTRODUCTION In elastography, the axial component of the strain tensor is estimated by taking the gradient of the axial (along the beam propagation axis) displacement occurring after a quasistatic tissue compression (Ophir et al. 1991, 1996, 1997). The estimation of the axial displacement is achieved by using time-delay estimation techniques applied to pre- and postcompression radiofrequency (RF) A-lines (Ophir et al. 1991). In general, however, the tissue motion that occurs during compression is threedimensional (3-D). Because the lateral (perpendicular to the beam propagation axis and in the scan plane) and elevational (perpendicular to the beam propagation axis and to the scan plane) motions are not measured, two major drawbacks are encountered. First, the axial elastogram takes into account only a small part of the mechanical tissue motion information. Second, undesirable lateral and elevational motions are the primary causes of signal decorrelation (Kallel and Ophir 1997a; Kallel et Address correspondence to: E. Konofagou, The University of Texas Medical School, Ultrasonics Laboratory, Department of Radiology, Suite 2.100, 6431 Fannin Street, Houston, TX USA.- al. 1997). The lateral or elevational decorrelation can be prevented by appropriate confinement of the tissue under study (Kallel and Ophir 1997a). Confinement may not, however, always be practical in clinical applications, especially when the tissues under study are not easily accessible. Many papers in the literature have dealt with the problem of motion estimation in two dimensions (Trahey et al. 1987; Newhouse et al. 1987; Mailloux et al. 1989; Bonnefous 1988; Bohs and Trahey 1991; Ramamurthy and Trahey 1991; Dotti et al. 1992; Ryan and Foster 1997; Yeung et al. 1998) and in 3-D (Bonnefous 1988; Hein 1993). Currently, most motion estimators are used to estimate the projection of the motion vector onto the ultrasound beam direction, or to measure only the axial component of the total motion (Jensen 1996; page 277). Many researchers have tried to perform 2-D and 3-D time-domain techniques. Initially, multidimensional motion estimation started in the field of blood velocity estimation, to estimate velocity components other than the axial one. Later, in the field of elasticity imaging, some of the methods developed in the flow measurement field were borrowed while others developed new tech- 1183

2 1184 Ultrasound in Medicine and Biology Volume 24, Number 8, 1998 niques, better suitable for elasticity measurements. As detailed below, these fields are different in many ways, and motion estimation algorithms should be shared with caution. Blood-flow velocity estimation Most of the problems in the blood-flow velocity field are due to the complicated motion of the scatterers (mainly, the blood cells) that are fast-moving, with constantly changing local density and orientation (Hein and O Brien 1993). Therefore, the pulse-repetition frequency and the computation speed of the algorithm used for the velocity estimation are extremely crucial. For example, a high acquisition and/or computation time can significantly limit the maximum velocity that can be measured (Hein and O Brien 1993; Trahey et al. 1987). Also, given that the blood scatterers have a low echogenicity, the reflected echoes suffer from very low sonographic signal-to noise ratio (SNR s ). There have been various techniques proposed for 2-D velocity estimation and a few representative ones are mentioned below. Multiple beam approach. Bonnefous (1988) developed a 3-D theoretical blood-flow model that included various correlation algorithms and the use of linear interpolation between adjacent signals to detect all three motion components. Laterally adjacent beams from different pulse excitations are cross-correlated to find the lateral displacement of the scatterers between emissions. The model was tested using only simulated one-bit data. Dotti et al. (1992) used a single-element transducer and extended the 1-D correlation method into 2-D using a two-element transducer. Again, by correlating signals between adjacent elements, they were able to estimate the lateral flow velocity component. However, the range of lateral velocities that could be accurately measured was limited by having only two lateral measurement positions. Hein (1993) developed a system with a triplebeam ultrasound lens forming three parallel beams, that was able to measure high velocities at large depths. The lateral component was calculated when the blood scatterers would move from beam to beam. Speckle tracking method. An alternative to using multiple beams is the speckle tracking (or 2-D kernel) method. Trahey et al. (1987) have suggested using the normalized cross-covariance between small rectangles (or kernels) of two successive B-mode frames. The location of the peak in the covariance estimate indicates the amount of displacement of the speckle pattern between frames, and is converted to velocity from the knowledge of the time lapse between the two frames. A major disadvantage of this technique is the large amount of calculation required, making this an impractical algorithm for blood flow measurements in real-time. Therefore, Bohs and Trahey (1991) suggested a faster algorithm that calculates the absolute sum of the pixel differences, or the sum of absolute differences (SAD) algorithm. A comparison of the accuracy of tracking between correlation and SAD showed that both techniques are equally able to track scatterers with comparable accuracy, for both lateral and axial displacements. Also, as expected, it has been shown that RF processing performs more precise tracking than envelope-detected processing (Ramamurthy and Trahey 1991; Bohs and Trahey 1991). More recently, Geiman et al. (1996) have developed a method of ensemble tracking that involves parallel receive processing, 2-D pattern matching, and interpolation of the resulting tracking grid to estimate subpixel speckle translations between successive ultrasonic acquisitions. All the aforementioned 2-D techniques were used to measure displacement and velocity. However, they can also be applied to determine a more complex motion, such as a deformation (Hein and O Brien 1993). Elasticity estimation Tissue elasticity estimation methods based on ultrasonics fall into two main groups: 1) Methods where a quasistatic compression is applied to the tissue and the resulting components of the strain tensor are estimated (Ophir et al. 1991; O Donnell et al. 1994); and 2) methods where a low-frequency vibration is applied to the tissue and the resulting tissue behavior is inspected (Krouskop et al. 1987; Lerner and Parker 1987; Lerner et al. 1990; Yamakoshi et al. 1990, Yeung et al. 1998). Among the techniques based on the quasistatic estimation of tissue strain, elastography (Ophir et al. 1991) is based on estimating the tissue strain using a correlation algorithm to estimate the strain induced in the ultrasonically scanned tissue. The limitations of motion estimation in this field are quite different from those in the blood flow estimation. Fast acquisition and/or computation is not an important factor, because the scatterer motion can be controlled by the applied compression/vibration. Furthermore, the sonographic signal-to noise ratio is higher in tissue than in blood by at least one order of magnitude. Both these factors allow us to aim for high precision of motion estimation and, thus, high elastographic signal-to noise ratio. Although high precision tracking of axial motion has been reported (Ophir et al. 1991; Skovoroda et al. 1994; O Donnell et al. 1994), precision lateral tracking has not been reported. The speckle-tracking method developed by Bohs and Trahey (1991) was used to study the mechanical vibration propagating through tissue (Walker et al. 1992). A similar method, called 2-D companding (Chaturvedi et al. 1998), was recently developed for

3 New elastographic method E. KONOFAGOU and J. OPHIR 1185 elastography to reduce decorrelation noise in elastograms using applied axial strains of up to 5% and correcting for the lateral decorrelation, when out-of plane displacement was considered negligible. This method uses speckle block-matching techniques (Bohs and Trahey 1991) to track the 2-D motion of the scatterers after static compression. It has recently been extended to 3-D companding (Chaturvedi et al. 1997; Insana et al. 1997), in order to compensate for the elevational displacement, as well. The main limitation of this method is that it cannot estimate relatively small lateral displacements with high precision because it is limited by the coarse lateral spacing between adjacent beams 1 to estimate the lateral displacement. Chaturvedi et al. (1998) attempted to perform interpolation using the grid slopes technique described by Geiman et al. (1996), but the gradient calculation involved in the slope estimation introduced large errors. Lubinski et al. (1996) used the speckle-tracking technique, and showed that lateral displacement images obtained by traditional 2-D correlation tracking are noisy. They also showed theoretically that, when envelope data are used to track lateral motion, the variance of the lateral displacement estimation is larger than the variance of the axial displacement estimate by orders of magnitude. Based on these results, they proposed the estimation of lateral displacement by making the assumption of isotropic incompressibility in the tissue and, thus, making use of the precision of the axial measurement. However, not only has compressibility been reported, for example, in the case of biphasic materials, such as cartilage (Jurvelin et al. 1997) and lung tissue (Fung 1989), but also the literature demonstrating total incompressibility in soft tissues, such as breast or prostate, is scarce or nonexistent. Also, the assumption of isotropic behavior of the Poisson s ratio may not hold in some tissues. The limits of the aforementioned methods used for lateral displacement estimation are due to simplifying assumptions, such as tissue incompressibility, and the use of an ensemble (kernel) of A-lines (Bohs and Trahey 1991), which limits the precision of lateral tracking. In fact, the speckle-tracking technique might be the only solution for flow velocity measurements due to the limitations mentioned earlier. However, in a case such as elasticity imaging, where time and Sonographic signalto-noise ratio (SNR s ) are not as important as in bloodflow estimation, cross-correlation of single RF A-lines in the lateral direction and added interpolation techniques 1 Throughout this whole paper, we assume that, for an unsteered linear array, the spacing between the centers of adjacent beams is equal to the spacing between adjacent array elements, defined as pitch. to measure subpitch motion are possible (Konofagou and Ophir 1998). Therefore, in this paper, we show how tracking the phase of single A-lines in the lateral direction through the use of a weighted interpolation method can be used to estimate the lateral component of the strain tensor. The reader should note that this is a method fundamentally different from the speckle-tracking methods (Trahey et al. 1987; Ramamurthy and Trahey 1991; Bohs and Trahey 1991; Chaturvedi et al. 1998) and not a special case of a 2-D kernel with one A-line size. The 2-D kernel methods employ kernels to have a sufficient statistical representation of the signal to be tracked. Therefore, the shorter and wider the kernel, the higher the accuracy of the SAD estimator in the lateral direction (Bohs et al. 1995; Ramamurthy and Trahey 1991). In addition, Bohs and Trahey (1991) reported that, when using RF data, the discrepancy between axial and lateral velocity resolutions yields results with high nonuniformities in the two directions of estimation. This is mainly because there is a lack of RF phase information in the lateral direction (Bohs and Trahey 1991). They finally conclude that, despite the resulting coarser resolution, the use of B-scan data might be more advantageous because the images produced were more uniform than those obtained with RF data. Unlike the 2-D kernel techniques, our goal was to obtain displacement estimates in the lateral direction with the highest tracking resolution possible. Therefore, we tracked the RF pattern of a single A-line segment in the lateral direction. This is made possible through the use of interpolation techniques. The resulting resolution of displacement estimate allows us to obtain a strain image with high signal-to noise ratio as well. The lateral direction is defined perpendicular to the transducer/compressor axis. This is also different from the method used by Chaturvedi et al. (1998) that generates a transverse strain image by computing the strain perpendicular to the compressor axis, but parallel to the transducer axis. Interpolation techniques can help reconstruct an RF signal in the lateral direction provided that lateral sampling is sufficient. This means that, if there is sufficient overlap between the beams in the lateral direction, interpolation methods may be used to overcome the poor ability to track lateral motion that is encountered when using the envelopes of speckle signals (Lubinski et al. 1996; Kallel 1995). Finally, this method does not make any assumptions about the elastic properties of the tissue under study, like the incompressibility method (Lubinski et al. 1996). In fact, the lack of such assumptions allows us to estimate and image an important mechanical parameter of the tissue, the Poisson s ratio. There are three important advantages in the use of high-precision lateral tracking. First, it significantly re-

4 1186 Ultrasound in Medicine and Biology Volume 24, Number 8, 1998 duces the variance of the lateral strain estimates, thereby allowing us to obtain lateral elastograms depicting the lateral strain distribution with good precision. Second, it allows for fine correction of the axial elastogram for high or low strains. This step produces an axial elastogram with significantly reduced overall decorrelation noise. Third, the lateral elastogram may be used together with the axial elastogram to generate an image of the local distribution of an independent elasticity parameter, the Poisson s ratio ( ), which is defined as: l a (1) for a plane-strain state, where l and a are the lateral and axial strains, respectively (Lai et al 1978). The Poisson s ratio describes the coupling between the lateral and axial strains; in plane-strain problems, a value of 0.5 means a maximum coupling, and a value of zero means no coupling. Therefore, the Poisson s ratio varies only within a limited dynamic range. Finally, the shear strain can be derived from the knowledge of axial and lateral displacements (Konofagou and Ophir 1999) but this issue is not discussed in this paper. The Young s modulus (E) and the Poisson s ratio are the two technical parameters that uniquely describe the elastic properties of isotropic materials. They are derived from the Lamé constants and given by (Saada 1989). E E and (2) Displaying the Poisson s ratio of the various tissue elements may allow differentiation among dissimilar normal tissues, as well as between normal and abnormal tissues. Rice and Cleary (1976) defined two types of Poisson s ratios in poroelastic materials: these are the drained and undrained Poisson s ratios, depending on the duration of the compression. A poroelastic material has certain elastic solid characteristics, but is also porous and contains a substance (fluid or gas) in its pores. If the measurement is done immediately after loading, the measured Poisson s ratio is high (close to 0.5). However, if the measurement is made after a relatively short time, allowing some unbound fluid to leave the material, the measured measured Poisson s ratio is reduced according to the amount of of the fluid leaving the tissue after compression (Rice and Cleary 1976; Mak et al. 1987). In the biphasic model (Mak et al. 1987; Jurvelin et al. 1997; Fung 1989), the measured Poisson s ratio corresponds to the Poisson s ratio of the hydrated tissue matrix. If we treat this as a stress-relaxation problem and sustain the compression over a certain period of time, t, ast 3, most of the fluid will leave the matrix and the biphasic material will behave like a compressible elastic solid (Mak et al. 1987). Therefore, the Poisson s ratio of the material will be asymptotically equal to the Poisson s ratio of the matrix. Thus, for example, the ability to image the distribution of Poisson s ratios in tissues over time may allow the visualization of the water-holding capacity of unbound water in various healthy and edematous tissues. Moreover, the changes in the local Poisson s ratios may indicate the quantitative changes in the local fluid volume in the tissue elements. Because the Poisson s ratio is estimated simply from the definition i.e., as the ratio of lateral to axial strain in any given pixel (for a plane strain state), it is unaffected by boundary conditions, inasmuch as both strain components are equally affected. Most papers on the biomechanical properties of tissues (for examaple, Sarvazyan 1975; Skovoroda et al. 1994; Kallel et al. 1996) made the assumption that all tissues are incompressible, with a constant Poisson s ratio of 0.5. Some pathological tissues, such as edematous muscle tissues (Mridha and Ödman 1986), lung tissue (v 0.3; Fung 1989) and cartilage (v 0.18; Jurvelin et al. 1997), however, have been shown to have lower Poisson s ratios and, thus, the simple a priori assumption of incompressibility may result in erroneous imaging of elastic modulus (Skovoroda et al. 1994; Sumi et al. 1995; Kallel and Bertrand 1996). Due to the relatively small dynamic range of Poisson s ratios, the imaging contrast may be limited. However, in elastography, it has been shown that low mechanical contrast is not an important limitation (Kallel and Ophir 1997b). Moreover, the use of the lateral motion correction has been shown by Chaturvedi et al. (1998), and is shown in this paper as well, to improve the contrast-to noise performance of axial elastograms, thereby improving the ability to visualize low-contrast targets. The overall iterative method described in this paper comprises the following main three steps. First, the postcompression RF A-lines are globally stretched axially to compensate for the applied strain (Céspedes and Ophir 1993; Varghese and Ophir 1997a), and an axial elastogram is generated. More A-lines are then created by a process of weighted interpolation and tissue motion is tracked laterally to generate a lateral elastogram (Fig.1, Step A). Second, the tracked segments are shifted to their assumed precompressed position by the estimated amount of lateral displacement (Fig.1, Step B) and new corrected postcompression RF A-lines are constructed. This allows for the axial elastogram to be recomputed. This procedure may be iterated until the correlation co-

5 New elastographic method E. KONOFAGOU and J. OPHIR 1187 p a x Ae x x cos 2 x, (4) and the lateral psf component of an unapodized, boxcar transducer aperture (not considering grating lobes) is: p l y sin Dy x r Dy x r 2 F F sin y y 2, (5) where x is the axial coordinate, x is the pulse length in mm, is the central wavelength of the emitted pulse, D is the azimuthal aperture width, y is the lateral coordinate, x r is the focal length and F is the F-number of the transducer given by: F x r D. (6) Fig. 1. Diagram of the iterative method highlighting the uses of lateral displacement. A generating a lateral elastogram, B correcting for the lateral interpolation in the axial elastogram and C generating a Poisson elastogram. efficient of the axial strain estimation stabilizes at a high value. Finally, we divide the lateral elastogram by the axial elastogram to generate a Poisson s-ratio elastogram (Fig.1, Step C). THEORY Since most clinical ultrasound systems are using lowcost transducers with a small number of channels, the sampling in the lateral direction relative to the amplitude of the lateral displacement is often not sufficiently high to allow precise lateral tracking or, in the case of elastography, estimation of lateral displacements with high precision. However, sufficient beam overlap can allow construction of new interpolated beams between the original beams. In a 2-D scanning model with a 1-D transducer, the point-spread function (psf) may be written as a separable function in the focal area, (Wagner et al. 1983) that is: p x, y p a x p l y, (3) where x and y are the axial and lateral coordinates respectively, and p a (x) and p 1 (y) are the axial and lateral psf components, respectively. If we assume a Gaussian modulated emitted pulse, the axial psf component becomes: Lateral tracking Figure 2a d shows graphically the main steps of the lateral tracking and lateral correction algorithms detailed below. Step 1: Linear interpolation N:1. Figure 2a shows an example of two adjacent A-lines of a precompressed sonogram. The postcompression A-lines (Fig. 2b) are first globally stretched axially by the amount of the axial strain applied (Varghese and Ophir 1997a). Using a scheme of N:1 linear interpolation generates N A-lines (dotted lines in Fig. 2b) between adjacent original postcompression A-lines 1 and 2 (denoted x 1 and x 2, respectively, in Fig. 2) through linear interpolation, described by: z nj a d x 1j 1 a d x 2j, (7) where z nj is the j th sample on the n th A-line generated, x 1j is the j th sample on A-line x 1, x 2j is the j th sample on A-line x 2, a is the distance between x 1j and z nj and d is the pitch (Fig. 2a). The lateral coordinate of the samples generated with interpolation is equal to distance a in eqn (7) and the axial coordinate is the same as the samples of the original A-lines used to perform the interpolation of eqn (7). The interpolation is always performed between two adjacent beams. The pair of beams is chosen according to the magnitude of lateral displacement that was measured. For example, if the lateral displacement that was previously measured is larger than one pitch, the interpolation occurs between the adjacent beams, with

6 1188 Ultrasound in Medicine and Biology Volume 24, Number 8, 1998 Fig. 2. Principle of lateral tracking and correction for lateral decorrelation: (a) Precompression simulated A-lines y 1 and y 2; (b) Interpolation between postcompression simulated A-lines x 1 and x 2 (after axial stretching) and construction of N ( 2) interpolated A-lines z 1 and z N corresponding to a N:1 interpolation scheme, eqn (7); (c) Example of lateral tracking of postcompression A-line x 1, found to have the best match at a distance l; (d) Lateral correction replaces postcompression A-line x, with a new postcompression A-line x 1 that is more correlated with precompression y 1 than x 1. The increased correlation of y 1 with x, compared to y 1 with x 1 ) shows that this leads to a significant increase of the axial correlation coefficient and, subsequently, the signal-to noise ratio of the axial elastogram after correction. the beam closest to the tracked precompressed segment lying one pitch apart from the latter. We note here that other kinds of interpolation (cubic, bandlimited, etc.) are appropriate and have been considered with no noticeable differences in the performance of the algorithm at the absence of sonographic noise. We also note that whether interpolating A-lines and computing the correlation, or interpolating the correlation function itself we should produce the same result. However, in our case, we need the interpolated A-line segments that are used for the lateral correction (detailed in Step 4). Step 2: Computation of cross-correlation coefficient function. The estimated cross-correlation coefficientsquared was computed to be used for weighing the interpolated postcompression A-lines. The cross-correlation coefficient-squared 2 ni of linearly interpolated A- line n with original A-line i is given by (Bendat and Piersol 1986) 2 ni C 2 ni, (8) R nn R ii where C ni is the cross-correlation of the original A-line i with interpolated A-line n and R ii and R nn are the autocorrelation functions of A-lines i and n, respectively. Step 3: Weighted interpolation N:1. The maxima of the correlation coefficients-squared provide the essential information for the characteristics of the linearly interpolated A-lines. The reason for using the correlation coefficient-squared is because it is the proportion of the total variation explained by regression (Wonnacott and Wonnacott 1972, page 342). They are, thus, used as weights for the computation of the final (weight-interpolated) A-lines, so that these have their shape weighted according to their location with respect to the original A-lines, determined by the parameter a in eqn (7). For example, if an A-line z n generated in Step 1 is closer to the original A-line x 1 than A-line x 2, the correlation coefficient of that A-line with x 1 will be greater than x 2 and its shape after weighted interpolation will be more correlated to the shape of x 1 than of x 2. The algorithm performs the weighted interpolation and recomputes A- lines z n with their j th sample given by: z nj x 1j max 2 1n x 2j max 2 2n max max 2 1n, max 2, (9) 2n where max(f(x)) denotes the maximum value of the function f(x). Step 4: Lateral displacement and strain estimation. The purpose of this step is to track the lateral motion of the scatterers after the compression. We thus cross-correlated small segments of each precompression A-line (Fig. 2a) with segments of A-lines on the same lateral level generated via weighted interpolation (Fig. 2c). The lateral displacement is indicated by the location of the maximum of the cross-correlation coefficient function between the precompression A-line and the postcompression interpolated A-lines. The sensitivity (i.e., the smallest lateral displacement that can be measured) is determined by the quantity: pitch N (10) in the case of a N:1 interpolation, providing this method of lateral tracking with high precision as the interpolation scheme increases. This means that this method, unlike other proposed methods is not pitch-limited, but noiselimited. A preliminary simulation study (in Methods) provides more insight into the variance of this estimator, but a more thorough study is the subject of current investigations. The lateral strain is estimated from the lateral displacement using a least-squares algorithm (Kallel and Ophir 1997c).

7 New elastographic method E. KONOFAGOU and J. OPHIR 1189 Lateral correction This step uses the information of lateral displacement of Step 4 of the lateral tracking method to shift the tracked segments by the amount of displacement estimated. Figure 2d shows an example of lateral correction for this lateral motion. As expected, the performance of the interpolation method depends on the ultrasonic parameters of the system, such as the pitch, the beamwidth and the overlap. This is because all the aforementioned parameters determine the characteristics of the signal produced due to the motion of the scatterers accross the beam. Clearly, the beamwidth should be high enough so that there is sufficient overlap, and low enough so that independent tracking results are allowed and clutter is not significant (see Fig. 6 and Results section). Here we examined the former limitation. The beamwidth (BW), defined here as the distance between the zeros of the main lobe, of the lateral psf, eqn (5), at the focus is equal to (Kino 1987): BW 2F. (11) The percent beam overlap is found by: %overlap 1 pitch BW 100%. (12) Assuming that the lateral psf component is given by eqn (5), then its Fourier transform is a triangular function (Wagner et al. 1988). The maximum spatial frequency of this spectrum is: f 0 1 2F. (13) According to the sampling theorem, the sampling spatial frequency, denoted f s1, should satisfy: f sl 2f 0. (14) Assuming the pitch is the spatial sampling period (or the inverse of the width of the frequency response at zero frequency), and noting from eqns (9) and (12) that the maximum frequency is the inverse of the beamwidth, from eqn (14) we obtain that lateral sampling is sufficient as long as: or, from eqn (12), when: pitch BW 2 (15) %overlap 50%. (16) Fig. 3. Error in lateral displacement estimation (in terms of autocorrelation width, see Fig. 4 for details) vs. sonographic SNR. What allows us to track laterally using interpolation is the fact that the lateral spatial sampling frequency is above the Nyquist frequency as long as the percent beam overlap is over 50%. This result means that there have to be at least two A-lines/beamwidth so that lateral interpolation will be free from artefacts that result from aliasing in the sampling of the echo pattern. The tracking ability of the algorithm may be measured by the 6-dB lateral width of the autocorrelation function of the residuals of the linear regression fit to the estimated lateral displacement line for a simulated homogeneous phantom. In a similar application, Ophir et al. (1989) have shown that the variance ( 2 ) of these residuals is proportional to the square of the autocorrelation width of the residuals (L), or: 2 k l L 2, (17) where k 1 is a constant of proportionality. Thus, the wider the autocorrelation function of the residuals, the poorer the precision of the method in tracking the lateral displacement. METHODS The reader should note that Methods and Results follow the same basic structure in order to facilitate comparison. Lateral elastograms Homogeneous case (finite element stimulation). The lateral tracking algorithm was tested on a finite triangular-element simulated homogeneous phantom, with a Poisson s ratio of and Young s modulus of 21 kpa, generated using the finite element software AL-

8 1190 Ultrasound in Medicine and Biology Volume 24, Number 8, 1998 Fig. 4. (a) True (solid line) and estimated (o) lateral displacement with 4:1 interpolation plotted vs. the lateral distance from the axis of symmetry of the target; (b) autocorrelation of the residual errors (lateral width 2.1 mm). Fig. 5. (a) True (solid line) and estimated (o) lateral displacement with 32:1 interpolation plotted vs. the lateral distance from the axis of symmetry of the target; (b) autocorrelation of the residual errors (lateral width 0.2 mm). GOR. 2 A plane-strain state model was used that prevented any strain in the elevational direction. Furthermore, a strain of 1% in the lateral direction was applied. We defined perfect slip conditions, the target was left to deform freely on the lateral sides and one node in the mesh was fixed so that the entire object did not move sideways. A simulated ultrasound psf was generated from Eqs. (2), (3) and (4) assuming, therefore, that the transducer is in the transmit/receive focus at all depths. This is an approximation of the experimental case where transducers have tracking foci. The simulated sonogram was generated using a convolution model that convolves the psf with a simulated medium containing randomly distributed scatterers. The beamwidth was determined from eqn (11) for a 5-MHz transducer and for a particular F- number. The pitch was kept constant at 0.4 mm (100 A-lines digitized at 48 MHz for scanning a tissue with a lateral dimension of 40 mm) and the beamwidth was equal to 1.5 mm. These parameters were selected to correspond to our experimental Diasonics Spectra ultrasound scanner, operating at 5 MHz. Although no random noise was added to the signals, our studies show that performance of the interpolation method does not degrade significantly, as long as the sonographic SNRs (associated with random noise) is larger than 30 db (Fig. 3), which is the usual case for elastographic experiments. A 64:1 interpolation scheme was used, because no significant improvement was noted with higher interpolation schemes (Fig. 6), possibly due to other noise limitations. In order to generate the 2D lateral displacement and strain images, small A-line segments were tracked using an axial window size of 3 mm and an overlap of 75%. Homogeneous case (experiment). A homogeneous gelatin phantom 3 of size mm that con- 2 ALGOR is a registered trademark of Algor, Inc. 3 The phantom was supplied courtesy of Dr. T. Hall, University of Kansas Medical Center.

9 New elastographic method E. KONOFAGOU and J. OPHIR 1191 Fig. 6. Error in the estimation of lateral displacement for an applied lateral strain of 1% with beam widths and % overlaps: ( ) 1 mm (60%) (ooo) 2 mm (80%), (xxx) 4 mm (90%) and ( ) 8 mm (95%). Note how the error increases with beam width. tains small graphite flakes was used. The phantom was lubricated on all top, bottom and side surfaces with oil to achieve slip boundary conditions. It was also confined in the elevational direction to simulate plane strain state conditions. It was compressed axially by 1% using a circular compressor (90 mm diameter) of a size smaller than the lateral dimension of the phantom, but larger than the lateral width of the transducer (field of view of 40 mm). A 64:1 interpolation scheme was used. Global axial stretching was applied (Varghese and Ophir 1997a) before using the lateral tracking algorithm. The ultrasound system consisted of a Diasonics Spectra II scanner (Diasonics Inc., Santa Clara, CA, USA), used with a 5-MHz linear array and a digitizer (LeCroy Corp., Spring Valley, NY, USA) that digitizes (8 bits) 100 RF A-lines at 48 MHz. The transducer used has a dynamic receive focus. Kallel and Ophir (1997b) measured the beamwidth of this transducer to be on the order of 0.4 mm at the focus, using the method proposed by Wagner et al. (1983). The transducer was operating at 5 MHz, 60% bandwidth, with a maximum SNR s of 30 db. A personal computer controlled the operation of the motion control system to perform the compression and the data acquisition. Iterative method Inhomogeneous case (simulation). The iterative method was applied in the case of an analytic model. The model was derived from the elasticity equation for an infinite medium containing a circular inclusion (Kallel et al. 1996). For the computer simulation, a region of interest (ROI) of mm was selected and the inclusion diameter was 10 mm. The inclusion with Poisson s Fig. 7. Simulated homogeneous phantom for a laterally applied strain of 1%. True: (a) Lateral displacement; (b) Lateral strain. Estimated: (c) Lateral displacement (absolute value); (d) Lateral strain for a simulated homogeneous target. Estimated (e) Lateral displacement (absolute value) and experimental homogeneous phantom for an axially applied strain of 1%; (f) Estimated lateral strain. The isodisplacement lines outline various levels of displacement variation. ratios of either 0.3 or was twice as hard as the embedding medium or background (Poisson s ratio of and Young s modulus of 21 kpa). The applied strain was relatively large (3%) so as to clearly demonstrate the importance of the correction. To form the lateral elastogram, an axial window of size equal to 3 mm with 75% (axial) overlap was used in all cases to perform lateral tracking (64:1 interpolation). The goals were to obtain a high-precision lateral elastogram and to correct the axial elastogram for lateral decorrelation. The lateral correction was performed as shown in Fig. 2. The segments tracked laterally were shifted by the measured amount of lateral displacement to synthesize postcompression A-lines as similar to the precompression A-lines as possible (Fig. 2). This was expected to improve the axial correlation coefficient significantly and, therefore, the signal-to noise ratio of the corrected axial elastogram. Inhomogeneous case (experiment). The same ultrasonic, signal processing and mechanical parameters as for the homogeneous case were used to image a phantom 3 containing a circular inclusion of a 5-mm radius

10 1192 Ultrasound in Medicine and Biology Volume 24, Number 8, 1998 Poisson elastograms One of the aims was to investigate the ability of the Poisson elastogram to show new mechanical information that could not be seen on either axial or lateral elastograms. For this purpose, we considered two separate cases of the analytic model of an inclusion embedded in a homogeneous background: these were one without and one with a Poisson s ratio contrast between the inclusion and the background. In the first case, the inclusion and the background have equal Poisson s ratios of In biomechanical terms, both inclusion and the background were considered incompressible. In the second case, the inclusion was no longer incompressible, having a Poisson s ratio of 0.3, but the background remained incompressible. In both cases, the Young s modulus contrast was set to be equal to 2, with the inclusion being twice as hard as the background (the background Young s modulus was 21 kpa). The mechanical, as well as ultrasonic, parameters were identical to the previously described case with, again, an axial applied strain of 3%. The Poisson elastograms were generated by dividing the lateral elastograms by the corrected axial elastograms, as indicated by eqn (1). The axial elastograms used were corrected for lateral decorrelation (as described in the previous section), and generated using the same axial window of 3 mm and with a window overlap of 75%. Least-squares smoothing (Kallel and Ophir 1997c) with a kernel size of 10 points was used to smooth both the axial and lateral elastograms. Because the pitch is equal to 0.4 mm, using a 10-point leastsquares approximation is equivalent to a 4-mm lateral window. This lateral window size is on the same order as the one typically used for the generation of axial elastograms. Fig. 8. Iterative method. Simulated inhomogeneous phantom with one inclusion under 3% applied axial strain. (a) First axial elastogram (no correction); (b) First lateral elastogram (no correction); (c) Second iteration axial elastogram (after global stretching); (d) Lateral displacement image; (e) Second lateral elastogram (after global stretching); (f) Third iteration axial elastogram (after global stretching and lateral correction); and (g) The true axial strain image. (See Fig. 1 for correspondence.) The loop indicates that the same steps can be repeated until no substantial increase in the signal-to noise ratio of the images is noted. twice harder than the background. The pre- and postcompression sonograms were averaged over eight frames (by keeping the same compressor position), in order to reduce the random noise on the sonographic data before using the iterative method. RESULTS Lateral elastograms Figures 4a, b and 5a, b show the relationship between the width of the autocorrelation with the precision of lateral tracking using two examples of 4:1 and 32:1 interpolation, respectively, in the case of a 1-mm beam width and applied lateral strain of 0.5%. The reader should note the substantial increase in the precision of the lateral tracking inherent in this method, due to the refinement of the interpolation scheme. Figure 6 shows the results for each beamwidth case (for 1, 2, 4 and 8 mm) at an applied lateral strain of 1%. From the results shown in Fig. 6, we can draw some preliminary conclusions. First, if the appropriate interpolation scheme is used, the narrower the beam width, the smaller the error in the lateral displacement estimation. Figure 6 also shows that larger beam widths (also incurring larger overlap) increase the variance in the displacement estimation, possibly because there are fewer independent tracking estimates as well as more clutter. However, large overlap assures lateral interpolation without aliasing. Therefore, there is clearly an optimum interpolation scheme and an optimal beam width that need to be investigated. Homogeneous case (simulation). Figure 7a f shows the absolute value of the lateral displacement image (to show the symmetry of the lateral displacement with respect to the axis of the target) and the lateral elastogram, both for the true (Fig. 7a and b) and the estimated cases (Fig. 7c and d). The high correlation between the true and estimated lateral elastograms is evident. Homogeneous case (experiment). Figure 7e and f show the lateral displacement image and the lateral elastogram, in the case of the homogeneous phantom, respectively, after an axial compression of 1% and an axial stretching by the same amount. The lines of constant displacement (or isodisplacement lines) of Fig. 7a, c and e lead to a very important observation. In Fig. 7a is shown the true

11 New elastographic method E. KONOFAGOU and J. OPHIR 1193 Fig. 9. Plots of the elastographic signal-to noise ratios (SNR e ) corresponding to the images of Fig. 8. The iteration steps (a), (c) and (e) are those used to produce the axial elastograms of Fig. 8a, c and e, respectively. The signal-to noise ratio was computed in a pixel image region in the background, assumed homogeneous. The size of the error bars is equal to two standard deviations. displacement image for the case of perfect slip boundary conditions and a compression symmetrically applied around the vertical axis of the tissue. Therefore, the isodisplacement lines are perfectly parallel. Figure 7e indicates the existence of nonperfect slip conditions in the experimental case. First, the compression may not have been uniformly applied. Therefore, zero lateral displacement did not occur on the vertical axis of the tissue. Also, as a result of this nonuniform compression, larger maximum lateral displacement occurred on the left side of the tissue than on the right. Last, the isodisplacement lines are now curved, indicating the existence of nonslip conditions at the top and bottom surfaces of the phantom. Iterative method As mentioned earlier, another use for the lateral strain information is for correcting the axial measurements and thereby improving the signal-to noise ratios in the axial and Poisson elastograms. Thus, the same lateral tracking algorithm went through a second iteration, this time using a set of corrected postcompression RF A-lines (Fig. 2d). Inhomogeneous case (simulation). Figure 8a f shows the results of the iterative method for an applied strain of 3%, in the case of the simulated phantom, where the inclusion is twice harder than the background, with a Poisson s ratio of 0.3. Figure 8a shows how decorrelation noise corrupts the axial elastogram. The highest lateral displacement and, therefore, decorrelation occurs near the lateral sides of the phantom. Figure 8b shows the lateral elastogram where the postcompression sonogram has not been globally stretched to correct for axial decorrelation. The decorrelation noise associated with the axial motion is so significant in this case that the lateral Fig. 10. Figure provided courtesy of Kallel et al. (1997). It shows the strain filter variation at different lateral positions (denoted by L) of the ultrasound beam relative to the axis of symmetry of the target. Note how the highest (i.e., largest SNR e ) and widest (i.e., largest dynamic range) strain filter is obtained for zero lateral displacement (i.e., with the ultrasonic beam around the target s axis of symmetry). The case of zero lateral displacement is an equivalent case of the finer correction for lateral motion, achieved with the lateral correction method, thereby explaining the improvement in Figs. 8 and 9. elastogram is too noisy to provide any kind of useful information. Therefore, axial correction or axial stretching is applied. Figure 8c displays the axial elastogram after axial stretching. Stretching can partly compensate for large axial decorrelations, which correspond to regions of lesser lateral displacement (e.g., around the axis of symmetry of the target). By comparison of Fig. 8a and c, we observe that the contrast between the inclusion and the background is increased due to axial stretching. However, axial stretching is not sufficient to correct for the large lateral displacements toward the lateral sides of the phantom. Figure 8d depicts the lateral elastogram that has been corrected for axial decorrelation noise (i.e., after stretching). The significance of the correction is obvious by comparison of Fig. 8b and d. Figure 8e shows the importance of the high-precision correction of axial measurements; not only are the decorrelation artifacts eliminated, but also the axial elastogram appears to be smoother and closer to the true strain image (Fig. 8f). A simple observation of Fig. 8 shows how the high-precision correction increases both the overall signal-to noise ratio and the dynamic range of the axial elastogram. Figure 9 shows graphically the improvement in the elastographic signal-to noise ratio (SNR e ): SNR e e e, (18)

12 1194 Ultrasound in Medicine and Biology Volume 24, Number 8, 1998 lateral correction. The loop indicates that the same steps can be repeated until no substantial increase in the signal-to noise ratio of the images is noted. We note here that for subsequent iteration steps adaptive stretching (Alam et al. 1998) can be used or the axial strain estimated in the preceding iteration serves as the information used by a local stretching algorithm. This extension of the iterative method is currently being investigated. Fig. 11. Iterative method. Experimental inhomogeneous phantom with one inclusion under 3% applied axial strain. ( a) Axial elastogram (after global stretching); (b) Lateral displacement image (after global stretching); (c) Lateral elastogram (after global stretching); (d) Second axial elastogram (after global stretching and lateral correction); and (e) The corresponding sonogram. where e and e are the estimated axial strain and the axial strain standard deviation, respectively. These results are in total agreement with Kallel et al. s (1997) theoretical predictions using the Strain Filter, which plots the SNR e versus the estimated strain (Fig. 10). The Strain Filter corresponding to no corrections has a lower SNR e and a lesser upper strain range (i.e., smaller dynamic range). The zero lateral displacement case or, equivalently, the fine correction for lateral displacement at all strains, fixes both shortcomings, as is the case of this high-precision lateral correction method. This dramatic improvement is achieved through the reduction of the strain variance due to the increase of the lateral correlation coefficient (i.e., the correlation coefficient between the pre- and postcompressed A-lines used for the estimation of the lateral displacement). In fact, there are two levels of improvement, as shown by comparison of Fig. 8c and e, both due to a two-level increase of the lateral correlation coefficient and, subsequently, the axial signal-to noise ratio (Kallel and Ophir 1997a; Varghese and Ophir 1997b). The high decorrelation noise (highintensity artefacts of Fig. 8c), where large lateral displacements occur (near the edges of the phantom), is eliminated due to lateral correction. The subsequent increase of the axial correlation coefficient helps reduce the strain variance from the Barankin bound to Cramér Rao lower bound level (Kallel et al. 1997). Additionally, the lower decorrelation noise in Fig. 8c is also reduced in Fig. 8e because the lateral correlation coefficient is further improved through this high-precision Inhomogeneous case (experiment). Figure 11a e shows the experimental results for the phantom containing an inclusion twice harder than the background. Only the most important steps of the iterative method are shown. Figure 11e shows the sonogram where the inclusion also has a different echogenicity, as well as stiffness, from that of the background. Figure 11a shows the axial elastogram after stretching; Fig. 11b shows the lateral displacement image (note how the zero lateral displacement zone was shifted toward the left, showing the existing experimental boundary conditions) that was used to estimated the lateral elastogram (Fig. 11c), and to correct the axial elastogram (Fig. 11d). Poisson elastograms Figures 12a f and 13a f show the true (theoretical) and estimated axial, lateral and Poisson s elastograms for the targets without and with Poisson s ratio contrast, respectively. In both cases, to create the Poisson elastograms, the axial and lateral elastograms have been normalized to the same grey-scale and interpolated so that they have the exact same dimensions. The stress concentration artefacts in both images should cancel out during the division of eqn (1). In fact, from Figs. 12e and 13e, it is obvious that the contrast-to noise ratio has improved in Fig. 13e in the case where there was additional Poisson s ratio contrast. By comparing Fig. 12d and e to Fig. 13d and e, respectively, we find that it is impossible to ascertain from the lateral or axial elastograms alone whether the contrast between the inclusion and the background is due to differences in the Young s moduli and/or in the Poisson s ratios between the inclusion and the background. However, when the Poisson elastograms are computed (Figs. 12f and 13f), it is evident that, in the first case, there is no apparent difference in the Poisson s ratio of the inclusion and the background (so that the strain contrast can only be due to Young s modulus contrast) while, in the second case, there is an obvious contrast in the Poisson s ratio between the background and the inclusion. This is also demonstrated in the true Poisson elastograms (Figs. 12c and 13c).

13 New elastographic method E. KONOFAGOU and J. OPHIR 1195 Fig. 12. Simulated inhomogeneous phantom with one inclusion with a Young s modulus contrast of 2 and a Poisson s ratio contrast of 1 under 3% applied axial strain. True: (a) Axial elastogram; (b) Lateral elastogram; (c) Poisson elastogram, and estimated: (after iterative method correction) (d) Axial elastogram; (e) Lateral elastogram; (f) Poisson elastogram. Iterative method: In vitro and in vivo results Figure 14 shows a typical in vitro result of the iterative method as used in the case of a canine prostate that was embedded in a gel background and scanned with the same ultrasound system as the one used in the case of the experimental phantoms. Before processing, the sonographic data were averaged over six frames each. The applied strain was 2% and the prostate was confined elevationally for best results (since it minimizes elevational motion and doubles the lateral one). The window size and overlaps were 2 mm, and 80%, respectively for Fig. 13. Simulated inhomogeneous phantom with one inclusion with a Young s modulus contrast of 2 and a Poisson s ratio contrast of 1.65 under 3% applied axial strain. True: (a) Axial elastogram; (b) Lateral elastogram; (c) Poisson elastogram, and estimated: (after iterative method correction) (d) Axial elastogram; (e) Lateral elastogram; (f) Poisson elastogram. Fig. 14. In vitro results of a canine prostate at an applied axial strain of 2%: a) First iteration axial elastogram, b) first iteration lateral elastogram (note the axial decorrelation noise that increases with depth), c) second iteration axial elastogram (after stretching), d) second iteration lateral displacement image (note that the high displacement regions also correspond to the regions on the second iteration axial elastogram most affected by noise), e) second iteration lateral elastogram, f) third iteration axial elastogram (after lateral correction, to be compared to c). The rough contour sketch of the prostate shows: 1) the posterior lobe (possibly the prostatic utricle), 2) the lateral lobes, 3) the verumontanum with the urethral stroma underlying it, 4) the isthmus and 5) the prostatic capsule. both lateral and axial elastograms. Figs. 14a and b show how no correction (neither for axial nor for lateral decorrelation) yields noisy axial and lateral images, respectively. However, stretching (Fig. 14c and 14e) and lateral correction (Fig. 14f) result in high quality axial and lateral elastograms. Figure 15 corresponds to the case of a pathologically confirmed breast fibroadenoma in vivo in a 49-year old patient. The applied strain was 0.6%. The breast was not confined. The window size and overlap were 3 mm and 80% for both lateral and axial elastograms. Figs. 15a and b show the first iteration axial and lateral elastograms, respectively and the elastogram of Fig. 15c depicts the second iteration axial elastogram after stretching. Note how stretching is not as significant in this case due to the low axial decorrelation associated with a small applied compression. However, Fig. 15c shows a high strain area on the left side of the lesion causing a poor definition of the lesion. There are many potential factors leading to this high strain area, such as the presence of a soft region creating high strain, lateral or elevational motion of the lesion causing a decorrelation artefact, or a pulsating vessel. However, both the lateral elastogram of Fig. 15b and the lateral displacement image (Fig. 15d) reveal the cause as being a lateral decorrelation artefact due to a large local lateral displacement, also shown on the second lateral elastogram (Fig. 15e). Therefore, lat-

14 1196 Ultrasound in Medicine and Biology Volume 24, Number 8, 1998 Fig. 15. In vivo results of a pathologically confirmed breast fibroadenoma of a 49-year old female at an applied axial strain of 0.6%: a) first iteration axial elastogram, b) first iteration lateral elastogram (showing a high lateral strain area on the left side of the lesion), c) second iteration axial elastogram (containing a lateral decorrelation artefact corresponding to the same high lateral strain region in b) that causes poor definition of the lesion), d) second iteration lateral displacement image (with the highest lateral displacement corresponding to the decorrelation artefact of c), e) second iteration lateral elastogram, f) third iteration axial elastogram (after lateral correction, both decorrelation artifacts on top left of the image and near the lesion were successfully eliminated, thereby significantly improving the visualization of the lesion on the elastogram). eral correction eliminates this artefact as well as other artefacts, such as the one near the top left part of the image, thus significantly improving the visualization of the lesion in the third iteration axial elastogram (Fig. 15f). The latter shows that the region that looked like a soft area in Fig. 15c is actually hard. From these preliminary results, we may conclude that lateral motion estimation not only can provide sufficient information for both the detection and correction of lateral decorrelation, but also offers important insight in the amount of lesional and perilesional tissue mobility. Since the origin of breast cancer diagnosis, the amount of lesion mobility has been an important factor in the classification of breast lesions. For example, Chen (1995) reported that perilesional tissues in the case of benign breast tumors displayed a shear motion opposite to the direction of palpation while in the case of the malignant lesions the perilesional tissue was pulled towards the same direction. This can be explained by the fact that, unlike benign lesions, the malignant tissues are associated with infiltrative growth, otherwise known as desmoplastic reactions (Chen 1995). In the future, we plan to examine this issue by estimating and imaging the shear strain for different types of lesions. DISCUSSION Elastography is a method dealing with 3-D mechanical problems and, therefore, a 3-D treatment is necessary if optimal imaging performance is desired. All methods used previously for lateral tracking are mainly pitch-limited (i.e., the minimum measurable lateral displacement is limited by the pitch of the transducer used). In this paper, we studied the lateral motion and proposed a new method for overcoming these limitations. By performing a weighted interpolation between adjacent RF A-lines, it is possible to perform a high-precision (subpitch) lateral tracking. This high precision is primarily due to the preservation of the axial phase information in the lateral interpolation. The theoretical study of this estimator is currently being developed to assess its potential and limitations. The lateral motion information was used in three distinct ways. First, due to the resulting high precision of its estimation, the lateral displacement was used to generate a new image displaying the lateral strain experienced by the tissue, called lateral elastogram. From the experimental results, it can be concluded that the lateral displacement image provides us with information about the mechanical conditions existing during the experiment, such as slip or nonslip boundary conditions, position of the compressor relative to the axis of symmetry of the target and the amount of uniformity of compression. The boundary conditions are crucial parameters in designing an elastographic experiment and they may be indicated by the lateral (displacement and/or strain) image. In addition, in freehand elastography, the lateral displacement image can provide essential correction factors to the relatively poorly controlled compressor motion. Acquiring a high-precision lateral displacement image may also be important in other modalities as well, such as transverse blood-flow imaging that is not limited by the Doppler angle. Lastly, it can also be used to precisely estimate and image shear strain. Second, the lateral displacement information was used to shift the laterally displaced A-lines and to compute a corrected axial elastogram that suffers from fewer decorrelation errors than an uncorrected one. An important finding of this work was that there are two kinds of decorrelation noise that are corrected using a finer correction method. One advantage is the increase of the signal-to noise ratio for higher strains. The most important advantage, however, is the improvement of the signal-to noise ratio at all strains due to the ability of the algorithm to correct for all lateral motions, whether fine or coarse. This leads to a significant increase in the overall signal-to noise ratio of the corrected axial elastogram, both near the edge of the elastogram and elsewhere. The improvement described here due to the lat-