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7 Ultrasonic Measurement and Imaging with Lateral Modulation – Echo, Tissue Motion and Elasticity Chikayoshi Sumi Sophia University Japan 1. Introduction It is remarkable that the pathological state of human soft tissues highly correlates with static and low-frequency mechanical properties, particularly shear elasticity (e.g., Sumi, 2005d). Accordingly, we have been developing ultrasonic (US)-strain-measurement-based onedimensional (1D) (Sumi et al., 1993, 1995a, 2010f; Sumi, 1999b, 2005d, 2008a, 2010d; Sumi & Matsuzawa, 2007b), 2D (Sumi et al., 1993, 1995a; Sumi, 1999c, 2005d, 2006b, 2007g, 2008a, 2008c, 2010d) and 3D (Sumi, 1999c, 2005d, 2006b, 2007g, 2010d) shear or Young modulus reconstruction/imaging techniques as differential diagnostic tools for deseases of various in vivo tissues, such as the breast (Sumi, 1999a, 2005b, 2005d; Sumi & Matsuzawa 2007b; Sumi et al., 1995b(strain), 1996, 1997, 1999b, 2000b) and liver (Sumi et al., 2001a, 2001b; Sumi, 2005d), i.e., cancerous deseases etc. Other soft tissues such as heart or blood vessel are also our targets, i.e., myocardinal infraction, atherosclerosis etc. After the first report of the differential type inverse problem of shear modulus by Sumi (1993, 1995a), immediately the results obtained on agar phantoms [e.g., Sumi et al., 1994a(strain & shear modulus), 1994b, 1995d], in vivo breasts (e.g., Sumi et al., 1995b, 1996, 1997; Sumi, 1999a, 1999b) and in vivo liver (e.g., Sumi et al., 2000a, 2001a, 2001b; Sumi 2005d) were reported. For such in vivo tissues, a suitable combination of simple, minimally invasive therapy techniques such as chemotherapy, cryotherapy, and thermal therapy (e.g., Sumi, 2005d; Sumi et al., 2001a) etc with our reconstruction techniques would lead to an innovative, new clinical strategy that would enable differential diagnosis followed by immediate treatment so that overall medical expenses could be substantially reduced (Sumi, 2005d). This is because our developed techniques allow non-invasive confirming of a treatment effectiveness in realtime, i.e., a degeneration. Our early reports on the interstitial rf/micro wave thermal coagulation thrapy are Sumi et al., 2000a, 2001b; Sumi, 2005d, etc. In the respective 1D, 2D and 3D techniques, a 1D (axial) displacement field, and 2D and 3D displacement vector fields generated by compression, vibration, heart motion, radiation force etc are measured to obtain 1D (axial) strain, and 2D and 3D strain tensor fields by partial differentiation. Many other researchers are also developing shear modulus reconstruction methods (e.g., Kallel & Bertrand, 1996; Plewes et al., 2000; Doyley et al., 2005) based on various displacement/strain measurement methods, e.g., conventional 1D Doppler method (Wilson & Robinson, 1982) and 1D autocorrelation method (1D AM) (Kasai et al., www.intechopen.com 114 Ultrasound Imaging 1985; Loupas et al., 1995) for blood flow measurement, and 1D (Ophir et al., 1991) or multidimensional (Yagi & Nakayama, 1988; Bohs & Trahey, 1991) crosscorrelation method (CCM). Now, various tissue motion and elasticity measurement/imaging have been performed over the world (see also references in Sumi et al., 2008i). In our case, other low frequency mechanical properties or quantities can also be reconstrcuted or measured, e.g., Poisson’s ratio or Bulk modulus (Sumi, 2006b), density or inertia (Sumi, 2006b,2010d), viscoelasticity (e.g., Sumi, 2005d; Sumi et al., 2005a) and mechanical source (e.g., Sumi & Suekane, 2009e). Fluid such as blood is also our target (Sumi & Suekane, 2009e). These methods will also be used for a non-destructive evaluation, e.g., food engineering etc. Previosuly, we reported a multidimensional phase matching method (Sumi et al., 1995d; Sumi, 1999a, 2008b) together with three novel methods of measuring a multidimensional displacement vector using a US signal phase, i.e., the multidimensional cross-spectrum phase gradient method (MCSPGM) (Sumi et al., 1995d; Sumi, 1999a, 2008b), multidimensional autocorrelation method (MAM) (Sumi, 2002c, 2005c, 2008b) and multidimensional Doppler method (MDM) (Sumi, 2002c, 2005c, 2008b). These methods can be applied to the measurement of the tissue strain tensor for above-mentioned shear modulus reconstruction (breast, liver, heart etc.), blood flow vector, sonar data, and other target motions. That is, the multidimensional phase matching allows coping with the decorrelation generated by out-ofmotion from a beam or a 2D frame. Although the CCM requires the numerical interpolation of the crosscorrelation function using cosine, parabolic functions, etc to yield analogue displacement vector data, our developed multidimensional methods do not require such interpolation. That is, these methods require only sampled echo data and then do not suffer any artifact errors due to such interpolation. Specifically, in MCSPGM (Sumi et al., 1995d; Sumi, 1999a, 2008b), a local displacement vector is estimated using the local echo phase characteristics, i.e., from the gradient of the phase of the local cross-spectrum evaluated from the local region echo data. In contrast, the other two methods use an instantaneous US phase (Sumi, 2002c, 2005c, 2008b). By performing the multidimensional phase matching using a coarsely measured displacement data by a multidimensional cross-correlation method (MCCM) (Sumi et al., 1995d; Sumi, 1999a, 2008b) or MCSPGM using sampled echo data spatially thinned out (Sumi, 2005d, 2008b), all the methods enable simultaneous axial, lateral and elevational displacement measurements. The multidimensional phase matching method can cope with the decorrelation of local echo data and aliasing that occurs due to a large displacement, i.e., by searching for corresponding local echo data. Significantly, this phase matching method improves the measurement accuracies of multidimensional methods. As shown by simulations (Sumi, 2008b), the accuracies of the multidimensional displacement vector measurement methods are comparable; however, MAM and MDM require less computational time (particularly, MDM) than MCSPGM. Generally, when using such displacement vector measurement methods, the measurement accuracy of lateral displacement was lower than that of axial displacement (Sumi et al, 1995d; Sumi, 1999a, 1999c, 2005c, 2008b; Sumi & Sato, 2007c; Sumi & Ebisawa, 2009a). Even if the target dominantly moves or becomes deformed in the lateral direction, our simultaneous measurements using the multidimensional phase matching result in the accurate measurement of axial displacement (Sumi, 2007f; Sumi et al., 1995b, 1995c). The multidimensional phase matching method also enabled us the high accuracy manual axial strain measurement (e.g., for breast, Sumi, 2005b, 2005d; Sumi & Matsuzawa, 2007b; Sumi et al., 1995b, 1996, 1997, 1999b, 2000b; liver, Sumi et al., 2001a, 2005d; others). By Sumi (1995b), the manual strain measurement was made possible by using multidimensional rf-echo www.intechopen.com Ultrasonic Measurement and Imaging with Lateral Modulation – Echo, Tissue Motion and Elasticity 115 phase matching (cf. the first reports of shear modulus reconstrcution on in vivo beast, Sumi et al., 1996 and 1997; liver, Sumi et al., 2001b). Over the world, such an axial strain measurement/imaging by a manual axial compression using a US transducer has been clinically used. The modality is called as Elastography as named by Ophir (Ophir et al., 1991; Cespedes et al., 1993; Garra et al., 1997). Although the mesurement accuracy is significantly lower, some convetional 1D displacement measurement methods are also used instead of the multidimensional methods (e.g., Sumi 1999c; Sumi & Ebisawa, 2009a), i.e., ones originally used only for an axial displacement measurement along the axial direction. (e.g., Loupas at al., 1995; AM by Yamakawa & Shiina, 2001). In our case, the measuement of axial strains generated by the axial compression or an arbitrary mechanical source are used for a multidimensional imaging of 1D reconstrcution (Sumi et al., 1993, 1995a, 2010f; Sumi, 1999c, 2005d, 2008a, 2010d; Sumi & Matsuzawa, 2007b). On the basis of the calculation of an axial strain ratio, several 1D reconstrcution methods were developed by Sumi. When the measurement accuracy of the axial strain is low, e.g., during thermal treatment (Sumi, 2005d; Sumi et al., 2001a), being dependent of the accuracy at each position, our developed spatially-variant regularization is performed for the strain measurement (Sumi & Sato, 2008c) or shear modulus reconstrcution (Sumi, 2008e; Sumi & Itoh, 2010e), i.e., an application of our developd implicit-integration (Sumi, 1998). That is, the measurement and reconstruction are stabilized to cope with the echo noise and strain measurement noise, respectively. For a focal lesion, by properly setting a reference region of shear modulus for the 1D reconsrcution, the 1D reconstruction allows yielding a higher contrast-to-noise ratio (CNR) than the axial strain (Sumi, 2005d; Sumi & Matsuzawa, 2007b; Sumi et al., 2010f). That is, the reference region should be set in the stressconcentrated or stress weak region in front of or behind the target stiff or soft lesion such that the reference region extends in the direction orthogonal to that of the dominant tissue deformation (Sumi, 2005b; Sumi & Matsuzawa, 2007b; Sumi et al., 1995d, 2010f). In addition, a mechanical source should be realized such that the target tissue deforms dominantly in a direction that extends in the direction of much shear-modulus varying (e.g. Sumi et al., 2010f). For the 1D strain measurement/imaging and 1D reconstruction, strain in the dominat deformation direction generated should be measured (e.g. Sumi & Ebisawa, 2009a). Moreover, for the practical imaging of 1D reconstruction, although human perception with respect to gray (negative or positive) scales and color scales must also be considered together with actual tissue shear modulus distributions, optimal displaying could be achieved by determining if the relative shear modulus or the inverse of the relative shear modulus should be imaged on the basis of their CNRs calculated using a stationary statistics of measured strains in the focal lesion and the surrounding region (Sumi et al., 2010f). Although the techniques for shear modulus reconstrcution methods including strain tensor calculations, multidimensional shear modulus reconstructions and the regularizations mentioned (Sumi, 1998, 2005d, 2006b, 2007g, 2008a, 2008e; Sumi & Sato, 2008c; Sumi & Itoh, 2010e) cannot be reviewed in detail in this chapter due to the limitation of the space, the important multidimensional phase matching is reviewed later (section 2.1). However, if the lateral and elevational displacements can be measured with the same degree of an accuracy as that of the axial displacement, manual strain measurement and shear modulus reconstruction can be performed without considering the direction of the beam and target motion or mechanical source with the position (Sumi, 1999c, 2002a, 2002b, 2008a; Sumi et al., 2007e, 2008f, 2008i). That is, for an arbitrary mechanical source, 3D or 2D measurement/reconstruction with only attachment of the US transducer enables such www.intechopen.com 116 Ultrasound Imaging measurement and reconstruction. Clinically, such a measurement will enable the evaluation of the elasticity of more various tissues, e.g., under normal motion such as the heart, arm and leg muscles (during exercise) and even for the deep ROIs such as liver tissues, which are inaccessible from the body surface and normally deformed by heart motion or pulsation. Various possible configurations will increase the applications of the tissue motion measurement and mechanical property reconstrcutions. For the blood flow vector measurement, a high accuracy displacement vector measurement had been performed (Jensen, 1998, 2001; Anderson, 1998, 2000) using a lateral oscillating method obtained by using Fraunhofer approximation (Steinberg, 1976; Goodman, 1996) together with some conventional 1D displacement measurement methods in respective axial and lateral directions, i.e., ones are originally used only for an axial displacement measurement along the axial direction. The measurement of blood flow in vessels running parallel to the surface of the body had been achieved. The method enabled the measurement of the lateral displacement/velocity that was more accurate than the use of the change in bandwidth (Newhouse, 1987). The method falls in a category of the lateral modulation (LM) approach Sumi called (Sumi, 2002c, 2005c, 2005d). The LM was resolved by Sumi as the more simple beamforming that uses a coherent superimposition of steered, crossed beams (Sumi, 2002a, 2008a, 2008b; Sumi et al., 2008f, 2008i). In the field of strain tensor measurement, the LM approach was applied first by Sumi (Sumi, 2004). For our tissue shear modulus reconstruction, to realize comparable high measurement accuracies of axial, lateral and elevational displacements, lateral and elevational modulation frequencies had to be significantly increased (Sumi, 2004, 2005c) compared with that observed in the reported application to the blood flow vector (Jensen, 1998, 2001; Anderson, 1998, 2000) and other tissue strain tensor (Liebgott et al., 2005) measurements (modulation frequency, 2.5 vs 1 mm-1). This is because the strain tensor is obtained by differentiating the measured displacement vector components using a differential filter (i.e., a kind of high pass filter), the displacement vector must be measured with a considerable high accuracy. Deeply situated tissues must also be considered (e.g., liver). By Sumi (2005c, 2008b), a spherical focusing was obtained as a suitable focusing. Moreover, to increase the measurement accuracy, only a digital processing was used for obtaining plural multidimensional analytic signals (Sumi, 2002c, 2005c, 2008b). Moreover, it was confirmed that our developed LM methods are useful for imaging of the spatial difstribution of US reflectivity, i.e., echo imaging (Sumi, 2008a; Sumi et al., 2008f, 2008i). That is, a high resolution can be achieved in lateral and elevational directions as almost the same as that in the axial direction. Thus, it is expected that LM will lead to a next-generation US diagnosis equipped with various new modes such as displacement/velocity vector, strain tensor measrements and thier applications. Although the LM methods developed by other groups (Jensen, 1998, 2001; Anderson, 1998, 2000; Liebgott, 2005) yield band-unlimited, modulated spectra by using infinite-length apodization functions (e.g., ringing-expressed by sinc functions), our developed lateral Gaussian envelope cosine modulation (LGECM) method realizes band-limited, modulated spectra, i.e., by using a finite length (not ringing) apodization function (Sumi, 2005c, 2008b). This does not cause aliasing. Moreover, for the blood flow vector measurement (Jensen, 1998, 2001; Anderson, 1998, 2000) and other strain tensor measuement (Liebgott et al., 2005), the respective measurements of axial and lateral displacements are performed using a conventional 1D displacement measurement method by realizing point spread functions www.intechopen.com Ultrasonic Measurement and Imaging with Lateral Modulation – Echo, Tissue Motion and Elasticity 117 (PSF’s) oscillating only in the lateral direction and only in the axial direction through a demodulation. Although we also developed a new accurate demodulation method only using digital signal processing (Sumi, 2010g), all the measurements suffer from the decorrelation of echo signals due to displacement orthogonal to the oscillation direction (Sumi, 2008b; Sumi & Shimizu, 2011). However, the highest accuracy in measuring target motions can be achieved by combined use (Sumi, 2002c, 2005c, 2008a, 2008b; Sumi et al., 2008f, 2008i) of the LM approach and our developed displacement vector measurement methods that enable simultaneous axial and lateral displacement measurements (Sumi et al., 1995d, 2002c; Sumi, 1999a, 2008b, 2005c). Not only our developed measurement increases the measurement accuracy of lateral displacements but also that of axial displacement (Sumi, 2005c, 2008b). Our developed LM can be performed by superimposition of the simultaneously or successively transmitted/received, plural steered beams or frames with different steering angles obtained with the multiple transmission method (MTM, Sumi, 2002a, 2005d; Fox, 1978; Techavipoo, 2004). Alternatively, LM can also be realized from a set of received echo data (Sumi, 2008b; Sumi et al, 2008f, 2008i) using not a classical synthetic aperture but our previously developed multidirectional synthetic aperture method (MDSAM, e.g., Sumi, 2002a, 2005d; Tanter, 2002). That is, the aperture is synthesized in multidirections after receiving US signals. Because MDSAM requires less data acquisition time than MTM using successive US beam transmissions, if the transmitted US energies are sufficient, the beamforming suffers the less tissue motion artifact. To obtain high intensity transmitted US signals, a virtual source can be used (Sumi et al., 2010h). However, if tissue motion artifacts do not occur, MTM yields more accurate measurements. With this type of beamforming, multiple transducers can also be used (Sumi, 2008b; Sumi et al, 2008f, 2008i), e.g., when dealing with heart motion due to the existence of the obstacles such as bones. On the evaluations of statitics of measured strain tensor components and reconstructed relative shear modulus in a stiff inclusion of an agar phantom, accurate measurements and reconstructions were obtained (e.g., see Table I in Sumi, 2008f; Table VIII in Sumi, 2008i). Alternatively, with MTM (Fox, 1978; Sumi, 2002a, 2005d; Techavipoo, 2004) and MDSAM (Sumi, 2002a, 2008b; Tanter, 2002), only the most accurately measured axial displacements from the respective beams obtained was used to obtain a displacement vector (i.e., there is no superimposition of beams). Although 1D measurement methods can also be used (Fox, 1978; Sumi, 2002a, 2005d; Tanter, 2002; Techavipoo, 2004) in place of the multidimensional measurement methods (Sumi, 2002a, 2008b), the same decorrelation of local echo signals (mentioned above) occurs due to target displacement in a direction orthogonal to the beams (Sumi et al., 1995d; Sumi, 2008b). Thus, the 1D measurement methods will result in a lower measurement accuracy than the corresponding multidimensional measurement methods, i.e., the 1D cross-spectrum phase gradient method (1D CSPGM) (Sumi et al., 1995d; Sumi, 1999a), conventional 1D AM (Kasai et al., 1985; Loupas et al., 1995), 1D DM (Sumi, 2008b), and 1D CCM (Ophir et al., 1991). Thus, not conventional 1D axial displacement measurement methods (e.g., 1D AM) but multidimensional displacement vector measurement methods should be used. Also for these beamformings, if necessary, separate plural transducers are also used simultaneously or successively. However, under conditions in which motion artifacts do not occur, our previous comparison (Sumi, 2008b) of LM (or coherent superimposition) and non-superimposition methods by geometrical evaluations clarifies that the LM has the potential to yield more accurate measurements of axial and lateral displacements with less computational time. www.intechopen.com 118 Ultrasound Imaging However, for practical beamforming applications, the echo SNRs from steered beams must also be considered, i.e., an overly large steered angle makes the echo SNR low (Sumi et al., 2008i). Moreover, LGECM method was improved using parabolic functions or Hanning windows instead of Gaussian functions in the apodization function. Hereafter, the new methods are respectively referred to as the parabolic modulation (PAM) method or Hanning modulation (HAM) method (Sumi, 2008a; Sumi et al., 2008i). Particularly, PAM enables decreases in effective aperture length (i.e., channels) and yield more accurate displacement vector measurements than LGECM. Although the Fourier transform of a parabolic function results in ringing effects, the new modulation yields no ringing effects in the spectra. PAM also yields a high spatial resolution in the reflectivity (echo) imaging with a high echo signal-tonoise ratio (SNR). Thus, we stop using the Fraunhofer approximation for LM. PAM is obtained on the basis of the a priori knowledge of the differences in the focusing scheme and shape between the parabolic function, Hanning window and Gaussian function, and the effects on the decays of the US signals during the propagation. That is, the US energy of the feet is lost during the US propagation and the main lobe contributes to echo signals, the mountains in the apodization functions should have a large full width at half maximum (FWHM) and short feet. Thus, we suceeded in a breakway from the Fraunhofer approximation (Sumi et al., 2006a). Usually, for US imaging, US beam-forming parameters such as frequency, bandwidth, pulse shape, effective aperture size, and apodization function are designed and set appropriately. In addition, US transducer parameters such as the size and materials of the US array element used are also set appropriately. In determining such settings, the US properties of the target are also be considered (e.g., attenuation and scattering). Thus, all parameters are set appropriately for the consideration of a system that involves the US properties of the target. Previously, we proposed to set such parameters in order to realize the required PSF for LM on the basis of optimization theory by using the minimum norm least-squares estimation method (Sumi et al., 2006a; Sumi, 2007d, 2010a). The better envelope shape of the PSF than that of the PA is searched for on the basis of the knowledge of the ideal shape of PSF, i.e., having a large FWHM and short feet (e.g., Sumi et al., 2010c). Nonlinear optimization is also effective to yield such a proper PSF (Sumi et al., 2009c). Although conventional US beam-forming parameters are usually set on the basis of the experience of an engineer, our proposed method realizes the best possible beam-former using optimally determined parameters. Thus, spatial resolution and echo SNR are improved. Although the optimized parameter can also be used, in this chapeter, PAM and LGECM are performed because they can be analytically obtained (Sumi et al., 2008i). As mentioned above, it was confirmed through simulations that when echo SNR is high (SNR ≈ 20 dB), MAM yields a higher accuracy measurement than MDM and vice versa (Sumi, 2008b). Here, the 2D demonstrations are shown on agar phantom that was statically compressed dominantly in a lateral direction (Sumi et al., 2008f), displacement of which cannot be accurately measured by a conventional beamforming. In addition, 2D shear modulus reconstrcutions are also shown together with strain tensor measurements. In this chapeter, after reviewing our developed phase matching, PAM and LGECM, and MAM and MDM (section 2), the images and measurements obtained on an agar phantom are shown (section 3). Comparisons of the spatial resolution of the US images are made and the accuracies of the measured displacement vectors and elasticity (i.e., strain, shear modulus) are determined. Finally, discussions and future problems are provided with conclusions. www.intechopen.com Ultrasonic Measurement and Imaging with Lateral Modulation – Echo, Tissue Motion and Elasticity 119 2. Brief reviews 2.1 Phase matching To cope with the occurrence of decorrelation due to target motion and echo noise, we proposed the iterative phase matching method (Sumi et al., 1995d, 1999a, 2008b), i.e., the iterative search method for corresponding local region echo data. The search can be realized in spatial and spatial frequency domains using the estimate of the displacement vector obtained by a displacement vector measurement method or using a priori data of compression/stretching etc., i.e., by shifting echo data in a spatial domain or by multiplying a complex exponential by echo data in a spatial frequency domain using the estimate. First, this search method was used to increase the measurement accuracy of MCSPGM (e.g., Sumi et al., 1995d; Sumi, 1999a) using the estimate obtained by MCSPGM or MCCM. Here, we note that, strict echo compression/stretching can also be realized in the phase matching by setting a corresponding echo data in the local region using measured displacements and strains (Sumi, 2008b), whereas the effectiveness of the local echo compression/stretching is reported by Srinivasan et al. (2002). In the first phase matching, the estimate obtained at the adjacent point can be used to reduce the number of iterations of phase matching. This phase matching method can also be used for MAM and MDM to increase measurement accuracy (Sumi, 2008b). That is, for MAM, this method enables the increase in the accuracy of instantaneous phase change and instantaneous spatial frequencies by improving the correlation of the local complex correlation function, whereas for MDM, this method enables the increase in their accuracy by increasing that of the temporal derivative in the Taylor expansion of the instantaneous phase of the complex signal. During the iterative phase matching, the moving-average width decreases as the local region size of MCSPGM decreases (Sumi, 1999a). This enables the increase in spatial resolution. Moreover, this increases the accuracy of strain measurement because the estimation accuracies of the instantaneous phase change and instantaneous spatial frequencies are improved due to their being not constant. If there exists no noise in echo data, the moving averaging is not needed and only phase matching should be performed. LM increases the convergence speed of the phase matching (Sumi, 2008b). Thus, various applications of the actual axial strain measurements have been reported using axial (Cespedes et al., 1993; Garra et al., 1997) and multidimensional (Sumi et al., 1995b) displacement measurements, e.g., diagnosis of cancers of human in vivo breasts (Cespedes et al., 1993; Sumi et al., 1995b; Garra et al., 1997) and monitoring various low-invasive treatments such as interstitial rf/micro wave coagulation therapies of an in vivo liver carcinoma (Sumi et al., 2000a, 2001a, 2001b, 2005a, Sumi, 2002b, 2005d, 2007a). By Sumi (1995b), the manual strain measurement was made possible by using multidimensional rf-echo phase matching (Sumi et al., 1995d). These were achieved without any regularizations nor LMs. However, reports of actual shear modulus reconstruction using measured strain tensor distributions are few with the exception of our reports (e.g., Sumi, 2007a, 2008a, 2008e; Sumi & Sato, 2008c; Sumi & Itoh, 2010e using the regularizations for strain measurement or shear modulus reconstrcution; Sumi, 2008a; Sumi et al., 2008f, 2008i using LM) and the reconstruction using a measured axial displacement distribution (e.g., Doyley et al., 2005 using another regularization for Young modulus). Specifically, we reported 2D direct shear modulus reconstruction using regularized strain tensor measurement (Sumi & Sato, 2008c; Sumi & Itoh, 2009b) as well as regularized direct 1D shear modulus reconstruction (Sumi, 2008e; Sumi & Itoh, 2010e; Sumi, 2007a) using raw strain tensor measurement. These www.intechopen.com 120 Ultrasound Imaging reconstructions were stably performed for agar phantoms by using our developed regularization, i.e., spatially variant regularization being dependent of the accuracies at each position of the measured strain tensor components. Because the measurement accuracy depends on the direction of the displacement, according to the accuracies of the respective displacements, they are also properly regularized, i.e., referred to as displacement componentdependent regularization or directional-dependent regularization (Sumi & Sato, 2008c; Sumi & Itoh, 2009b). As briefly reviewed in section 1, various 1D reconstructions using the axial strain were also obtained by Sumi for human in vivo tissues when the targets become deformed in the axial and lateral directions, respectively. By Sumi, in addition to the report (2008e), the regularizations were also performed (2005d; 2007a) for the strain measurement or shear modulus reconstrcution before, during and after the in vivo thermal coagulation treatment. However, simulations revealed that the 1D reconstrcutions such as strain ratio, implicit-integration etc lead to the inaccurate value of reconstrcution and geometrical artifact even if there exists no noise in the axial strain data used (Sumi, 2005d; Sumi & Matsuzawa, 2007b). Moreover, when the target deforms in the lateral direction, the 1D reconstruction further decreases the accuracy in reconstrcution (Sumi, 2007f). Moreover, for the 2D reconstruction, the use of only the regularizations still yields an inaccurate reconstruction (Sumi & Sato, 2008c; Sumi & Itoh, 2009b). Thus, LM was also used later (Sumi, 2008a; Sumi et al., 2008f, 2008i). 2.2 Complex signals with different single-octant and different single-quadrant spectra Both the multidimensional autocorrelation method (MAM) and multidimensional Doppler method (MDM) use the instantaneous US signal phase (Sumi, 2002c, 2005c, 2008b). To measure a three-dimensional (3D) displacement vector (ux,uy,uz), three or four 3D complex signals with different single-octant spectra (Fig. 1a reported by Sumi, 2008b) that extend analytic signals are calculated for respective echo data r1(x,y,z) and r2(x,y,z) obtained before and after a pulse repetition interval Δt , i.e., rc1i(x,y,z) and rc2i(x,y,z) [i = 1,..,3 or 1,..,4]. The multidimensional complex signal having single-orthant spectra was introduced by Hahn (1992) [1D complex signal phase and instantaneous frequency are specifically described in a literature by Bracewell (1986)]. Each 3D complex signal obtained has three instantaneous spatial frequencies, i.e., US frequency fx, lateral frequency fy and elevational frequency fz. Hereafter, (fx,fy,fz) is referred to as a frequency vector. When lateral and elevational modulations are performed, fy and fz are respectively the lateral and elevational modulation frequencies, whereas when lateral and elevational modulations are not performed (i.e., beam-steering is not performed), fy and fz are respectively yielded by synthesizing the lateral (Sumi 2002a, 2002c; Chen et al., 2004) and elevational phases (but, the frequencies are low and then the measurement accuracy is lower than that of LM [Sumi et al., 2010g; Sumi et al., 2010i; 2006c; Sumi & Shimizu, 20011]). Thus, as described next, an equation regarding with the unknown displacement vector (ux,uy,uz) is derived from each pair of complex signals r1ci(x,y,z) and r2ci(x,y,z) having a same frequency vector (fx,fy,fz) [i = 1,..,3 or 1,..,4], and then the displacement vector (ux,uy,uz) can be obtained by simultaneously solving the three or four independent equations having the independent vectors (fx,fy,fz) as the coefficients. The three equations can be arbitrary chosen from the four equations. To mitigate the calculation www.intechopen.com Ultrasonic Measurement and Imaging with Lateral Modulation – Echo, Tissue Motion and Elasticity 121 errors of the instantaneous phases and frequencies, the least-squares method can also be used to solve all the four equations simultaneously. Measuring a 2D displacement vector requires calculating two 2D complex signals with different single-quadrant spectra (Hahn, 1992) (Fig. 1b reported by Sumi, 2008i) and then solving two correspondingly derived simultaneous equations. 2.3 LCMs (Lateral Cosine Modulation Methods) using PAM, HAM and LGECM and optimizations For respective PAM (Sumi, 2007d, 2008a, Sumi et al., 2008f, 2008i) and LGECM (Sumi, 2005c, 2008a, 2008b; Sumi et al., 2008f, 2008i) using a one-dimensional (1D) linear array-type transducer (lateral direction, y), the following apodization functions are used for the transmission or reception of US, i.e., 2λ x 1 {exp[ − (2π )2 ( λx y + f y + f a )2 ( σy a )2 2 ] + exp[ − (2π )2 ( λx y − f y − f a )2 ( σy 2 a )2 ]} (1) and 4 σy 4 σy ⎛ y ⎞ ⎛ y ⎞ {[ − ( )2 π 2 ⎜ + f y + f a ⎟ + 1] + [ − ( )2 π 2 ⎜ − f y − f a ⎟ + 1]}. 2λ x 9 a 9 a ⎝ λx ⎠ ⎝ λx ⎠ 2 1 2 (2) The apodization functions are superimpositions of two Gaussian functions and two parabolic functions. The apodization obtained for 20, 30, 60 and 100 mm depths are shown in Fig. 2 in the report by Sumi (2008b). HAM is also described by Sumi (2008i). These apodization functions are obtained using the Fraunhofer approximation such that the transmitted US energy used are same when realizing the Gaussian-type lateral PSF at a depth x for US with a wavelength λ , i.e., exp( − y2 2σ y2 )cos(2π f y y ). (3) Here, fy is the lateral modulation frequency and σ y corresponds to the lateral beam width for LGECM. fa and a are parameters introduced to regulate lateral modulation frequency and bandwidth, respectively. For comparisons of FWHM and feet length, the Guassian and parabolic functions that have the same area are shown in Fig. 1c in the report by Sumi et al. (2008i). The apodization functions for two-dimensional (2D) modulation (i.e., modulations in two directions y and z) using a 2D array-type transducer are also obtained for PAM and LGECM in a similar fashion (Sumi, 2005c; Sumi et al., 2008f, 2008i). According to the type of transducer (e.g., convex), other arbitrary orthogonal coordinates can also be used. When steered beams cannot be transmitted symmetrically in a lateral direction due to the existence of the obstacles such as a bone (for heart, liver etc), the original coordinate can be rotated such that the steered beams become laterally symmetric. However, our developed MCSPGM, MAM and MDM can yield measurement results even if the coordinate is not set in such a way, and the measurement accuracy in a displacement vector will be increased by the fact that the measurement accuracy of a lateral displacement can be significantly www.intechopen.com 122 Ultrasound Imaging improved (Sumi, 2010g; 2010i). Such a rotation also allows the use of our developed demodulation method with one of conventional 1D displacement measurement methods. When carrying out PAM, HAM or LGECM, Methods 1 and 2 developed for transmission/reception focussing are used (Sumi, 2005c, 2007d, 2008a, 2008b; Sumi et al., 2008i, 2008f). The Methods used for LCM are also reviewed in this section. Both Methods 1 and 2 yield twofold lateral modulation frequency (Sumi, 2004, 2008b) compared with a method that performs only a receiving modulation, i.e., the method using a non-steered plane wave for a transmission with a rectangular window, Bingham window, Hanning window or Gaussian function as the apodization function (Jensen et al., 1998; Anderson, 1998. Also both Methods 1 and 2 enable decreases in effective aperture length (i.e., channels). When using LGECM method together with Method 1, the following PSF would be realized, i.e., ⎛ y2 exp ⎜ − ⎜ 2(σ y / 2 )2 ⎝ ( ) ⎞ ⎟ cos ⎡ 2π 2 f y y ⎤ . ⎣ ⎦ ⎟ ⎠ (4) Different from Method 2, Method 1 also enables an increase in lateral bandwidth compared with the method that performs only a receiving modulation. For both methods, a lowfrequency envelope signal must also be removed. By Basarab (2007), a twofold frequency sine modulation (i.e., not LCM) is carried out. However, the modulation is not appropriate for US imaging and measurement of displacement. This can be easily understood by assuming the existence of a point reflection target in the region of interest (ROI). Method 1: (i) When a point of interest is dealt with, twofold frequency modulation can be performed using the same lateral modulation apodization (i.e., PAM) and spherical focusing in transmitting/receiving beam-forming as in conventional beam-forming (Sumi, 2004, 2008b). For 2D displacement vector measurement, two steered beams are used, whereas for 3D displacement vector measurement, three or four steered beams are used. These beams can be simulataneously transmitted. Alternatively, they can be superimposed after transmitting and receiving the respective beams successively. To obtain the steered beams, mechanical scans can also be performed. Plural transducers can also be used. When a finite ROI is dealt with, multiple transmitting modulations may also be useful for so-called multiple transmitting focusing. (ii) When performing a twofold frequency modulation over a finite ROI, the classical synthesis of an aperture (i.e., a monostatic or multistatic synthetic aperture) can also be carried out. However, if the target motion is rapid, a motional artifact may occur due to the low US energy transmitted from an element. In such a case, low-SNR echo modulation may be achieved. To increase the echo SNR, virtual sources can be used (e.g., Sumi and Uga, 2010h). By performing a modulation using (i) or (ii), LGECM theoretically yields 2 times as wide a lateral bandwidth as that yielded by a method that performs only a receiving modulation (theoretically, the beam width becomes σ y/ 2 ). Similarly, when performing other modulations such as PAM and HAM, an increase in lateral bandwidth is also achieved. Thus, using the same effective aperture width (i.e., the same number of channels), Method 1 realizes twofold lateral modulation frequency and a wide lateral bandwidth. The same lateral modulation frequency and lateral bandwidth can also be obtained by using a small effective aperture width (i.e., fewer channels). Method 2: When dealing with a finite ROI, transmissions of steered two laterally wide plane waves that are realized simultaneously or successively with the same steering angles as those used in the receiving modulation can also be performed, of which apodizations are also properly performed. Although Method 2 cannot increase the lateral bandwidth, it is www.intechopen.com