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INSTITUTE OF PHYSICS PUBLISHING
PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 52 (2007) 13–28
doi:10.1088/0031-9155/52/1/002
Signal-to-noise ratio, contrast-to-noise ratio and their trade-offs with resolution in axial-shear strain elastography Arun Thitaikumar1,2, Thomas A Krouskop1,3 and Jonathan Ophir1,2 1
The University of Texas Medical School, Department of Diagnostic and Interventional Imaging, Ultrasonics Laboratory, Houston, TX, USA 2 University of Houston, Electrical and Computer Engineering Department, Houston, TX, USA 3 Baylor College of Medicine, Department of Physical Medicine and Rehabilitation, Houston, TX, USA E-mail: [email protected]
Received 10 August 2006, in final form 25 October 2006 Published 4 December 2006 Online at stacks.iop.org/PMB/52/13 Abstract In axial-shear strain elastography, the local axial-shear strain resulting from the application of quasi-static axial compression to an inhomogeneous material is imaged. In this paper, we investigated the image quality of the axial-shear strain estimates in terms of the signal-to-noise ratio (SNRasse) and contrastto-noise ratio (CNRasse) using simulations and experiments. Specifically, we investigated the influence of the system parameters (beamwidth, transducer element pitch and bandwidth), signal processing parameters (correlation window length and axial window shift) and mechanical parameters (Young’s modulus contrast, applied axial strain) on the SNRasse and CNRasse. The results of the study show that the CNRasse (SNRasse) is maximum for axialshear strain values in the range of 0.005–0.03. For the inclusion/background modulus contrast range considered in this study ( 0.01), but the error bar reduces as the window length increases resulting in SNRasse improvements. The axial window shift is not expected to have much influence on the CNRasse (SNRasse) of the axial-shear strain estimates. This is because the axial-shear strain estimator (equation (1)) does not involve the axial shift. The plot shown in figure 9(b) shows such behaviour of the peak axial-shear strain estimates. The plots in figure 9(a) show the dependence of the CNRasse (SNRasse) on axial window shift. It can be observed that the effect of axial shift is weak, although smaller axial shift values yield slightly higher CNRasse values. Trade-offs between CNRasse (SNRasse) and resolution. Figures 10(a)–(d) show the effect of beamwidth, pitch, window length and axial shift on the resolution of axial-shear strain
SNR, CNR and trade-offs in axial-shear strain elastography
(a)
23
(b)
Figure 8. Plots of the (a) SNRasse, CNRasse and (b) estimated peak axial-shear strain as a function of window length. Note that the SNRasse improves with increase in window length while CNRasse does not change.
(a)
(b)
Figure 9. Plots of the (a) SNRasse, CNRasse and (b) estimated peak axial-shear strain as a function of window shift. Observe that as the shift increases the estimated peak axial-shear strain value remains stable. Since the strain estimator is not dependent on shift, the SNRasse or the CNRasse are not expected to be influenced by the shift. However, observe that the values of CNRasse (SNRasse) are slightly higher at smaller shifts.
elastography, respectively, as reported in previous work by Thitaikumar et al (2006). It is seen from these plots that the resolution deteriorates with increase in beamwidth, pitch and window length, while axial shift has only a weak influence. However, figure 7 shows that the CNRasse (SNRasse) improves with increase in pitch. Clearly, this demonstrates the existence of the trade-off. Smaller pitch improves resolution but compromises CNRasse (SNRasse) and vice versa, while no such trade-off is observed with window length or axial shift (see figures 8–10(c) and (d)). Experiments. Figure 11 shows a plot of CNRasse (SNRasse) as a function of applied axial strain. The CNRasse (SNRasse) curves behave similarly to what was shown in simulations. For
24
A Thitaikumar et al
(a)
(c)
(b)
(d)
Figure 10. Plots of resolution as a function of ultrasound system and signal processing parameters: (a) beamwidth at two different window lengths (W), (b) pitch, (c) window length and (d) axial shift. The values were obtained from simulations. The inclusion/background modulus contrast was 3 and the applied axial strain was 2%. The plots are reproduced from Thitaikumar et al (2006).
small applied axial strain, the CNRasse (SNRasse) is low and as the axial strain is increased, the CNRasse (SNRasse) increases but then falls off at even higher strain values. Finally, the effect of signal processing parameters of window length and axial shift on CNRasse (SNRasse) is shown in figures 12(a) and (b), respectively. As with the simulation study, we do not observe any influence of axial shift or window length. Discussion In this paper, we have reported a study on the CNRasse (SNRasse) of axial-shear strain estimates using simulations and experiments. Trade-offs between CNRasse (SNRasse) and the resolution were also investigated using simulations. The results showed that the range of axial compressive strain that results in higher CNRasse (SNRasse) in axial-shear strain elastography is ∼0.5%–3%. This is also the range typically used to acquire RF data for obtaining axial strain elastograms (Srinivasan et al 2003). It must be noted that it is not axial-shear strain, but axial compressive strain that is applied. Therefore, decorrelation is not only due to axial-shear strain but also due to co-existing axial strain, lateral strain and lateral-shear strains. It is
SNR, CNR and trade-offs in axial-shear strain elastography
25
Figure 11. Plots of the SNR and CNR as a function of applied axial strain obtained from experiment. A window length of 2 mm with 80% overlap was used to obtain the axial-shear strain elastograms. Observe the band-pass filter-like curve for CNR in the strain domain, corroborating the simulation results.
(a)
(b)
Figure 12. Plots of the SNR, CNR as a function of (a) window length and (b) window shift. An axial compression strain of 2% was applied. Observe that window shift has a weak influence on the CNR and SNR. However, the SNR showed a significant improvement with window length as observed in simulation.
intuitive to expect that decorrelation noise increases with additional deformations. However, the range of axial strain values which is optimal for the axial-shear strain estimates in terms of CNRasse (SNRasse) in the presence of additional deformations such as lateral-shear strain is not known. In this context, the findings in this paper are important because they suggest that it may be possible to acquire data at compression levels that may be optimal for both kinds of elastograms. The contrast mechanism in axial-shear strain elastography is different from that of axial strain elastography. In axial elastography the fundamental contrast is limited by the inherent inclusion/background modulus contrast, if the effect due to boundary conditions can be neglected. The contrast in axial-shear strain elastography is influenced not only by the
26
A Thitaikumar et al
(a)
(b)
Figure 13. Axial-shear strain elastograms from the gelatin-phantom experiment using (a) a differential strain estimator and (b) a staggered strain estimator. An axial compression strain of 2% was applied. The elastograms shown are an average over 15 realizations. Note that the image in (b) is visibly less noisy compared to the image in (a). The dimension of the above field of view is 40 mm × 40 mm with a 20 mm diameter inclusion.
modulus contrast but also by the applied axial strain. This can be observed from figures 4(b) and 5(a). As the peak axial-shear strain value increases, the contrast also increases because it is contrasted with zero shear strain inside the inclusion. However, the CNRasse drops beyond a certain maximum contrast. We can expect to increase the CNRasse (SNRasse) values above what is obtained in this study by reducing the noise in the estimation processes of at least two estimates, the axial displacement and the axial-shear strain. First, the noise in the axial displacement estimates can be reduced by correcting for the non-normal motion of the scatterer due to shear strain (both axial-shear and lateral-shear). This idea is in principle similar to the idea of correcting for lateral motion of the scatterers to reduce noise in the axial displacement estimates (Konofagou and Ophir 1998). Second, the noise in the axial-shear strain estimation can be reduced by using an estimator that is different from the differential estimator. One alternative is to extend the staggered strain estimation proposed by Srinivasan et al (2002b). This method has been shown to improve CNRe (SNRe) in axial strain estimates without compromising the resolution. Even though performance comparison of different estimators is not the subject of this manuscript, as an example, experimental axial-shear strain elastograms are shown in figure 13 to illustrate the improvement in the image quality, in terms of the CNRasse, by using staggered axial-shear strain estimation over a differential estimator. In this example the CNRasse values were 5.7 and 36.3 for the differential and the staggered estimators, respectively. However, the compromise in the resolution with improvement in CNRasse (SNRasse), if any, needs to be investigated. Among the ultrasound system parameters, results showed that the CNRasse (SNRasse) does not improve with larger beamwidth even though the noise in the axial-shear strain estimates reduces inversely with beamwidth (equation (3)). This is because the peak axial-shear strain value also decreases with increase in beamwidth due to resolution effects of the beamwidth. Note that the resolution decreases with increase in beamwidth. However, the pitch does not influence the estimation of the peak value, as can be observed from figure 7(b). Therefore, CNRasse (SNRasse) improves with increase in pitch. In the signal processing parameters group, the window shift showed a weak influence on the CNRasse (SNRasse). This is reasonable because window shift does not enter the axial-shear strain estimator computation. However, window length does influence the noise in the displacement estimation process and hence the improvement of SNRasse with window length (see figure 8(b)).
SNR, CNR and trade-offs in axial-shear strain elastography
27
It should be noted that the study reported in this paper modelled the inclusion/background interface as completely bonded. As stated earlier, the magnitude (and thus the contrast) and distribution of the axial-shear strain around the inclusion/background interface depends also on the degree of bonding at the interface. As a result, the effect of bonding may be an important one; however, it is not well understood at this time, and will be a subject of a subsequent publication. Conclusions We have shown that the range of axial compressive strain that is optimal for axial strain elastography, in terms of CNRasse (SNRasse), is also optimal for axial-shear strain elastography. The CNRasse (SNRasse) value was shown to improve with increasing pitch and does not fall with increasing beamwidth. The window length was shown to influence the SNRasse but not the CNRasse. In addition, the axial shift was shown to have a weak influence on the CNRasse (SNRasse). The existence of a trade-off between CNRasse (SNRasse) and the resolution in terms of pitch was also demonstrated. Acknowledgments This work was supported in part by NIH programme project grant P01-EB02105-12 awarded to the University of Texas Medical School at Houston. References Bendat J S and Piersol A G 2000 Random Data: Analysis and Measurement 3rd edn (New York: Wiley) Bilgen M and Insana M F 1997 Error analysis in acoustic elastography: II. Strain estimation and SNR analysis J. Acoust. Soc. Am. 101 1147–54 C´espedes I 1993 Elastography: Imaging of biological tissue elasticity PhD dissertation University of Houston C´espedes I, Insana M and Ophir J 1995 Theoretical bounds on strain estimation in elastography IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42 969–72 Chandrasekhar R, Ophir J, Krouskop T and Ophir K 2006 Elastographic image quality vs tissue motion in vivo Ultrasound Med. Biol. 32 847–55 Chen E J, Adler R S, Carson P L, Jenkins W K and O’Brien W D Jr 1995 Ultrasound tissue displacement imaging with application to breast cancer Ultrasound Med. Biol. 21 1153–62 Kallel F, Varghese T, Ophir J and Bilgen M 1997 The nonstationary strain filter in elastography, part II Lateral and elevational decorrelation Ultrasound Med. Biol. 23 1357–370 Knight M G, de Lacerda L A, Wrobel L C and Henshall J L 2002 Parametric study of the contact stresses around spherical and cylindrical inclusions Comput. Mater. Sci. 25 115–21 Konofagou E E and Ophir J 1998 A new elastographic method for estimation and imaging of lateral displacements, lateral strains, corrected axial strains and Poisson’s ratio in tissues Ultrasound Med. Biol. 24 1183–99 Konofagou EE, Harrigan T and Ophir J 2000 Shear strain estimation and lesion mobility assessment in elastography Ultrasonics 38 400–4 Meunier J and Bertrand M 1995a Ultrasonic texture motion analysis: Theory and simulation IEEE Trans. Ultrason. Ferroelectr. Freq. Control 14 293–300 Meunier J and Bertrand M 1995b Echographic image mean gray level changes with tissue dynamics: a system-based model study IEEE Trans. Biomed. Eng. 42 403–10 Ophir J, C´espedes I, Ponnekanti H, Yazdi Y and Li X 1991 Elastography: a method for imaging the elasticity of biological tissues Ultrason. Imaging 13 111–34 Ophir J, Alam S K, Garra B S, Kallel F, Konofagou E E and Varghese T 1999 Elastography: ultrasonic estimation and imaging of elastic properties of tissues Proc. Inst. Mech. Eng. 213 203–33 Rangayyan R M 2005 Biomedical Image Analysis (Boca Raton, FL: CRC Press) Righetti R 2005 Poroelastography: ultrasonic estimation and imaging of the local time-dependent distribution of the elastic and permeability properties of poroelastic media PhD Dissertation University of Houston
28
A Thitaikumar et al
Srinivasan S, Kallel F, Souchon R and Ophir J 2002a Analysis of an adaptive strain estimation technique in elastography Ultrason. Imaging 24 109–18 Srinivasan S, Ophir J and Alam S K 2002b Elastographic imaging using staggered strain estimates Ultrason. Imaging 25 229–45 Srinivasan S, Righetti R and Ophir J 2003 Trade-offs between the axial resolution and the signal-to-noise ratio in elastography Ultrasound Med. Biol. 29 847–66 Stavros A T, Thickman D and Rapp C L et al 1995 Solid breast nodules: use of sonography to distinguish between benign and malignant lesions Radiology 196 123–34 Thitaikumar A, Ophir J and Krouskop T A 2005 Noise performance and signal-to-noise ratio of shear strain elastograms Ultrason. Imaging 27 145–65 Thitaikumar A, Ophir J, Righetti R and Krouskop T A 2006 Resolution of axial-shear strain elastography Phys.Med. Biol. 51 5245–57 Timoshenko S P and Goodier J N 1970 Theory of Elasticity (New York: McGraw-Hill) pp 8–11 Twiss R J and Moores E M 1992 Structural Geology (New York: Freeman) pp 532 Varghese T and Ophir J 1997a A theoretical framework for performance characterization of elastography: the strain filter IEEE Trans. Ultrason. Ferroelectr. Freq. Control 44 164–72 Varghese T and Ophir J 1997b The nonstationary strain filter in elastography, Part I Frequency dependent attenuation Ultrasound Med. Biol. 23 1343–56 Varghese T and Ophir J 1998 An analysis of elastographic contrast-to-noise ratio Ultrasound Med. Biol. 24 915–24 Wagner R F et al 1983 Statistics of speckle in ultrasound B-scans IEEE Trans. Sonics Ultrason. 30 156–63 Wells P N T 1977 Biomedical Ultrasonics (New York: Academic)
PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 52 (2007) 13–28
doi:10.1088/0031-9155/52/1/002
Signal-to-noise ratio, contrast-to-noise ratio and their trade-offs with resolution in axial-shear strain elastography Arun Thitaikumar1,2, Thomas A Krouskop1,3 and Jonathan Ophir1,2 1
The University of Texas Medical School, Department of Diagnostic and Interventional Imaging, Ultrasonics Laboratory, Houston, TX, USA 2 University of Houston, Electrical and Computer Engineering Department, Houston, TX, USA 3 Baylor College of Medicine, Department of Physical Medicine and Rehabilitation, Houston, TX, USA E-mail: [email protected]
Received 10 August 2006, in final form 25 October 2006 Published 4 December 2006 Online at stacks.iop.org/PMB/52/13 Abstract In axial-shear strain elastography, the local axial-shear strain resulting from the application of quasi-static axial compression to an inhomogeneous material is imaged. In this paper, we investigated the image quality of the axial-shear strain estimates in terms of the signal-to-noise ratio (SNRasse) and contrastto-noise ratio (CNRasse) using simulations and experiments. Specifically, we investigated the influence of the system parameters (beamwidth, transducer element pitch and bandwidth), signal processing parameters (correlation window length and axial window shift) and mechanical parameters (Young’s modulus contrast, applied axial strain) on the SNRasse and CNRasse. The results of the study show that the CNRasse (SNRasse) is maximum for axialshear strain values in the range of 0.005–0.03. For the inclusion/background modulus contrast range considered in this study ( 0.01), but the error bar reduces as the window length increases resulting in SNRasse improvements. The axial window shift is not expected to have much influence on the CNRasse (SNRasse) of the axial-shear strain estimates. This is because the axial-shear strain estimator (equation (1)) does not involve the axial shift. The plot shown in figure 9(b) shows such behaviour of the peak axial-shear strain estimates. The plots in figure 9(a) show the dependence of the CNRasse (SNRasse) on axial window shift. It can be observed that the effect of axial shift is weak, although smaller axial shift values yield slightly higher CNRasse values. Trade-offs between CNRasse (SNRasse) and resolution. Figures 10(a)–(d) show the effect of beamwidth, pitch, window length and axial shift on the resolution of axial-shear strain
SNR, CNR and trade-offs in axial-shear strain elastography
(a)
23
(b)
Figure 8. Plots of the (a) SNRasse, CNRasse and (b) estimated peak axial-shear strain as a function of window length. Note that the SNRasse improves with increase in window length while CNRasse does not change.
(a)
(b)
Figure 9. Plots of the (a) SNRasse, CNRasse and (b) estimated peak axial-shear strain as a function of window shift. Observe that as the shift increases the estimated peak axial-shear strain value remains stable. Since the strain estimator is not dependent on shift, the SNRasse or the CNRasse are not expected to be influenced by the shift. However, observe that the values of CNRasse (SNRasse) are slightly higher at smaller shifts.
elastography, respectively, as reported in previous work by Thitaikumar et al (2006). It is seen from these plots that the resolution deteriorates with increase in beamwidth, pitch and window length, while axial shift has only a weak influence. However, figure 7 shows that the CNRasse (SNRasse) improves with increase in pitch. Clearly, this demonstrates the existence of the trade-off. Smaller pitch improves resolution but compromises CNRasse (SNRasse) and vice versa, while no such trade-off is observed with window length or axial shift (see figures 8–10(c) and (d)). Experiments. Figure 11 shows a plot of CNRasse (SNRasse) as a function of applied axial strain. The CNRasse (SNRasse) curves behave similarly to what was shown in simulations. For
24
A Thitaikumar et al
(a)
(c)
(b)
(d)
Figure 10. Plots of resolution as a function of ultrasound system and signal processing parameters: (a) beamwidth at two different window lengths (W), (b) pitch, (c) window length and (d) axial shift. The values were obtained from simulations. The inclusion/background modulus contrast was 3 and the applied axial strain was 2%. The plots are reproduced from Thitaikumar et al (2006).
small applied axial strain, the CNRasse (SNRasse) is low and as the axial strain is increased, the CNRasse (SNRasse) increases but then falls off at even higher strain values. Finally, the effect of signal processing parameters of window length and axial shift on CNRasse (SNRasse) is shown in figures 12(a) and (b), respectively. As with the simulation study, we do not observe any influence of axial shift or window length. Discussion In this paper, we have reported a study on the CNRasse (SNRasse) of axial-shear strain estimates using simulations and experiments. Trade-offs between CNRasse (SNRasse) and the resolution were also investigated using simulations. The results showed that the range of axial compressive strain that results in higher CNRasse (SNRasse) in axial-shear strain elastography is ∼0.5%–3%. This is also the range typically used to acquire RF data for obtaining axial strain elastograms (Srinivasan et al 2003). It must be noted that it is not axial-shear strain, but axial compressive strain that is applied. Therefore, decorrelation is not only due to axial-shear strain but also due to co-existing axial strain, lateral strain and lateral-shear strains. It is
SNR, CNR and trade-offs in axial-shear strain elastography
25
Figure 11. Plots of the SNR and CNR as a function of applied axial strain obtained from experiment. A window length of 2 mm with 80% overlap was used to obtain the axial-shear strain elastograms. Observe the band-pass filter-like curve for CNR in the strain domain, corroborating the simulation results.
(a)
(b)
Figure 12. Plots of the SNR, CNR as a function of (a) window length and (b) window shift. An axial compression strain of 2% was applied. Observe that window shift has a weak influence on the CNR and SNR. However, the SNR showed a significant improvement with window length as observed in simulation.
intuitive to expect that decorrelation noise increases with additional deformations. However, the range of axial strain values which is optimal for the axial-shear strain estimates in terms of CNRasse (SNRasse) in the presence of additional deformations such as lateral-shear strain is not known. In this context, the findings in this paper are important because they suggest that it may be possible to acquire data at compression levels that may be optimal for both kinds of elastograms. The contrast mechanism in axial-shear strain elastography is different from that of axial strain elastography. In axial elastography the fundamental contrast is limited by the inherent inclusion/background modulus contrast, if the effect due to boundary conditions can be neglected. The contrast in axial-shear strain elastography is influenced not only by the
26
A Thitaikumar et al
(a)
(b)
Figure 13. Axial-shear strain elastograms from the gelatin-phantom experiment using (a) a differential strain estimator and (b) a staggered strain estimator. An axial compression strain of 2% was applied. The elastograms shown are an average over 15 realizations. Note that the image in (b) is visibly less noisy compared to the image in (a). The dimension of the above field of view is 40 mm × 40 mm with a 20 mm diameter inclusion.
modulus contrast but also by the applied axial strain. This can be observed from figures 4(b) and 5(a). As the peak axial-shear strain value increases, the contrast also increases because it is contrasted with zero shear strain inside the inclusion. However, the CNRasse drops beyond a certain maximum contrast. We can expect to increase the CNRasse (SNRasse) values above what is obtained in this study by reducing the noise in the estimation processes of at least two estimates, the axial displacement and the axial-shear strain. First, the noise in the axial displacement estimates can be reduced by correcting for the non-normal motion of the scatterer due to shear strain (both axial-shear and lateral-shear). This idea is in principle similar to the idea of correcting for lateral motion of the scatterers to reduce noise in the axial displacement estimates (Konofagou and Ophir 1998). Second, the noise in the axial-shear strain estimation can be reduced by using an estimator that is different from the differential estimator. One alternative is to extend the staggered strain estimation proposed by Srinivasan et al (2002b). This method has been shown to improve CNRe (SNRe) in axial strain estimates without compromising the resolution. Even though performance comparison of different estimators is not the subject of this manuscript, as an example, experimental axial-shear strain elastograms are shown in figure 13 to illustrate the improvement in the image quality, in terms of the CNRasse, by using staggered axial-shear strain estimation over a differential estimator. In this example the CNRasse values were 5.7 and 36.3 for the differential and the staggered estimators, respectively. However, the compromise in the resolution with improvement in CNRasse (SNRasse), if any, needs to be investigated. Among the ultrasound system parameters, results showed that the CNRasse (SNRasse) does not improve with larger beamwidth even though the noise in the axial-shear strain estimates reduces inversely with beamwidth (equation (3)). This is because the peak axial-shear strain value also decreases with increase in beamwidth due to resolution effects of the beamwidth. Note that the resolution decreases with increase in beamwidth. However, the pitch does not influence the estimation of the peak value, as can be observed from figure 7(b). Therefore, CNRasse (SNRasse) improves with increase in pitch. In the signal processing parameters group, the window shift showed a weak influence on the CNRasse (SNRasse). This is reasonable because window shift does not enter the axial-shear strain estimator computation. However, window length does influence the noise in the displacement estimation process and hence the improvement of SNRasse with window length (see figure 8(b)).
SNR, CNR and trade-offs in axial-shear strain elastography
27
It should be noted that the study reported in this paper modelled the inclusion/background interface as completely bonded. As stated earlier, the magnitude (and thus the contrast) and distribution of the axial-shear strain around the inclusion/background interface depends also on the degree of bonding at the interface. As a result, the effect of bonding may be an important one; however, it is not well understood at this time, and will be a subject of a subsequent publication. Conclusions We have shown that the range of axial compressive strain that is optimal for axial strain elastography, in terms of CNRasse (SNRasse), is also optimal for axial-shear strain elastography. The CNRasse (SNRasse) value was shown to improve with increasing pitch and does not fall with increasing beamwidth. The window length was shown to influence the SNRasse but not the CNRasse. In addition, the axial shift was shown to have a weak influence on the CNRasse (SNRasse). The existence of a trade-off between CNRasse (SNRasse) and the resolution in terms of pitch was also demonstrated. Acknowledgments This work was supported in part by NIH programme project grant P01-EB02105-12 awarded to the University of Texas Medical School at Houston. References Bendat J S and Piersol A G 2000 Random Data: Analysis and Measurement 3rd edn (New York: Wiley) Bilgen M and Insana M F 1997 Error analysis in acoustic elastography: II. Strain estimation and SNR analysis J. Acoust. Soc. Am. 101 1147–54 C´espedes I 1993 Elastography: Imaging of biological tissue elasticity PhD dissertation University of Houston C´espedes I, Insana M and Ophir J 1995 Theoretical bounds on strain estimation in elastography IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42 969–72 Chandrasekhar R, Ophir J, Krouskop T and Ophir K 2006 Elastographic image quality vs tissue motion in vivo Ultrasound Med. Biol. 32 847–55 Chen E J, Adler R S, Carson P L, Jenkins W K and O’Brien W D Jr 1995 Ultrasound tissue displacement imaging with application to breast cancer Ultrasound Med. Biol. 21 1153–62 Kallel F, Varghese T, Ophir J and Bilgen M 1997 The nonstationary strain filter in elastography, part II Lateral and elevational decorrelation Ultrasound Med. Biol. 23 1357–370 Knight M G, de Lacerda L A, Wrobel L C and Henshall J L 2002 Parametric study of the contact stresses around spherical and cylindrical inclusions Comput. Mater. Sci. 25 115–21 Konofagou E E and Ophir J 1998 A new elastographic method for estimation and imaging of lateral displacements, lateral strains, corrected axial strains and Poisson’s ratio in tissues Ultrasound Med. Biol. 24 1183–99 Konofagou EE, Harrigan T and Ophir J 2000 Shear strain estimation and lesion mobility assessment in elastography Ultrasonics 38 400–4 Meunier J and Bertrand M 1995a Ultrasonic texture motion analysis: Theory and simulation IEEE Trans. Ultrason. Ferroelectr. Freq. Control 14 293–300 Meunier J and Bertrand M 1995b Echographic image mean gray level changes with tissue dynamics: a system-based model study IEEE Trans. Biomed. Eng. 42 403–10 Ophir J, C´espedes I, Ponnekanti H, Yazdi Y and Li X 1991 Elastography: a method for imaging the elasticity of biological tissues Ultrason. Imaging 13 111–34 Ophir J, Alam S K, Garra B S, Kallel F, Konofagou E E and Varghese T 1999 Elastography: ultrasonic estimation and imaging of elastic properties of tissues Proc. Inst. Mech. Eng. 213 203–33 Rangayyan R M 2005 Biomedical Image Analysis (Boca Raton, FL: CRC Press) Righetti R 2005 Poroelastography: ultrasonic estimation and imaging of the local time-dependent distribution of the elastic and permeability properties of poroelastic media PhD Dissertation University of Houston
28
A Thitaikumar et al
Srinivasan S, Kallel F, Souchon R and Ophir J 2002a Analysis of an adaptive strain estimation technique in elastography Ultrason. Imaging 24 109–18 Srinivasan S, Ophir J and Alam S K 2002b Elastographic imaging using staggered strain estimates Ultrason. Imaging 25 229–45 Srinivasan S, Righetti R and Ophir J 2003 Trade-offs between the axial resolution and the signal-to-noise ratio in elastography Ultrasound Med. Biol. 29 847–66 Stavros A T, Thickman D and Rapp C L et al 1995 Solid breast nodules: use of sonography to distinguish between benign and malignant lesions Radiology 196 123–34 Thitaikumar A, Ophir J and Krouskop T A 2005 Noise performance and signal-to-noise ratio of shear strain elastograms Ultrason. Imaging 27 145–65 Thitaikumar A, Ophir J, Righetti R and Krouskop T A 2006 Resolution of axial-shear strain elastography Phys.Med. Biol. 51 5245–57 Timoshenko S P and Goodier J N 1970 Theory of Elasticity (New York: McGraw-Hill) pp 8–11 Twiss R J and Moores E M 1992 Structural Geology (New York: Freeman) pp 532 Varghese T and Ophir J 1997a A theoretical framework for performance characterization of elastography: the strain filter IEEE Trans. Ultrason. Ferroelectr. Freq. Control 44 164–72 Varghese T and Ophir J 1997b The nonstationary strain filter in elastography, Part I Frequency dependent attenuation Ultrasound Med. Biol. 23 1343–56 Varghese T and Ophir J 1998 An analysis of elastographic contrast-to-noise ratio Ultrasound Med. Biol. 24 915–24 Wagner R F et al 1983 Statistics of speckle in ultrasound B-scans IEEE Trans. Sonics Ultrason. 30 156–63 Wells P N T 1977 Biomedical Ultrasonics (New York: Academic)
