Abstract:
An ultrasound system visualizes shear wave propagation in real time by slowing down the propagation of the shear wave as seen by the ultrasound probe. The shear wave source propagates shear waves into the medium at a frequency ω. The ultrasound probe is vibrated by a vibrator at the frequency ω−Δω, where Δω is much smaller than ω. The wave propagation as seen by the ultrasound probe is slowed down by a factor Δω/ω. An appropriate value of Δω allows real-time visualization of the wave propagation. Variations include electronically producing a virtual vibration and the use of multiple shear wave sources.

Description:
REFERENCE TO RELATED APPLICATION 
   The present application claims the benefit of U.S. Provisional Patent Application No. 60/653,987, filed Feb. 18, 2005, the disclosure of which is hereby incorporated by reference in its entirety into the present disclosure. 

   STATEMENT OF GOVERNMENT INTEREST 
   The work leading to the present invention was supported by the National Institutes of Health under Grant No. 2R01 AG16317-01A1. The government has certain rights in the invention. 

   FIELD OF THE INVENTION 
   The present invention is directed to a technique for sonoelastography and more particularly to such a technique in which shear waves are caused to interfere to produce a stationary or slowly moving wave from which the shear wave velocity can be determined. 
   DESCRIPTION OF RELATED ART 
   It is widely accepted that the changes in elasticity of tissues are possible markers of certain diseases. In modern medicine, digital palpation is a routine screening method in physical examinations. One approach to examine the tissues&#39; local elasticity is to visualize the local behavior of shear waves excited by internal or external sources. 
   A common approach to doing so is through an ultrasonic imaging modality called sonoelastography. Sonoelastography estimates the peak displacements of particle motion under audio frequency excitations by analyzing the power spectrum variance of the ultrasound echoes, which is proportional to the local vibration amplitude. Vibration fields are then mapped to a commercial ultrasound scanner&#39;s screen. 
   However, since the shear wave speed in soft tissues is on the order of a few meters per second, few imaging modalities are able to follow the propagation on a scale of a centimeter or less. The existing techniques to follow the shear wave propagation, either have a very high temporal sampling rate (high frame rate) or carefully subsample the propagation progressively (also known as the strobe effect). Such available systems either require off-line computing or prolonged data acquisition. Therefore, those techniques are not real-time. 
   SUMMARY OF THE INVENTION 
   It is therefore an object of the invention to develop such a technique which works in real time. 
   It is another object of the invention to develop such a technique which is non-invasive. 
   It is still another object of the invention to develop such a technique which does not require a high frame rate or “strobing.” 
   It is yet another object of the invention to develop such a technique in which the ultrasound scanner can run asynchronously from the vibration, as a normal Doppler imaging sequence. 
   It is yet another object of the invention to develop such a technique which can be implemented with a single shear wave source. 
   It is yet another object of the invention to develop such a technique which does not assume any particular type of propagation of the shear wave in the medium. 
   To achieve the above and other objects, the present invention is directed to a technique for visualizing the shear waves in soft tissues in real time in which wave propagations are virtually “slowed down” for observations. The spatial distribution of the shear wave speed can be reconstructed from the wave propagation video and therefore indicates the size and location of the stiff region in the tissue. The present invention provides many of the advantages of magnetic resonate elastography (MRE), but with the additional advantages of real time visualization of the vibration fields. 
   One shear wave source propagates shear waves into the medium at the frequency ω. An ultrasound (US) probe is held above the surface of the medium. The ultrasound probe is vibrated by a vibrator at the frequency ω−Δω, where Δω is much smaller than ω. 
   The total field estimated by the ultrasound scanner is the shear wave propagation relative to the probe vibration, which is called the modulated field. The modulated field is a representation of the propagating shear wave field, only at a different frequency and thus different velocity. By carefully selecting ω and Δω, the shear wave propagation can be virtually slowed down so that it can be visualized by sonoelastography. 
   The present invention is validated in both a homogeneous phantom experiment and an inhomogeneous phantom experiment. The present invention provides many of the advantages of magnetic resonate elastography (MRE), but with the additional advantages of real time visualization of the vibration fields. 
   Since the present invention can use the existing Doppler hardware on most modern US scanners, the frame rate of sonoelastography is as high as other Doppler modalities. In one example, regions where the vibration amplitude is low are displayed as dark green, while regions with high vibration are displayed as bright green. 
   The present invention offers a further advantage in that it does not assume any particular type of wave propagation; for example, it does not assume a plane wave. The vibration of the probe is completely temporally dependent and does not interfere with the spatial variation of the propagation of the shear wave. The present invention offers yet another advantage in that only one shear wave source is needed. 
   Variations include electronically producing a virtual vibration of the probe and the use of multiple shear wave sources. 
   The following disclosures disclose aspects of the invention and are hereby incorporated by reference in their entireties into the present disclosure: 
   Zhe Wu et al, “Sonoelastographic imaging of interference patterns for estimation of the shear velocity of homogeneous biomaterials,”  Phys. Med. Biol.  49 (2004) 911-922; and 
   Zhe Wu, “Shear Wave Interferometry and Holography, an Application of Sonoelastography,” submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, Department of Electrical and Computer Engineering, University of Rochester, Rochester, N.Y., 2005. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     A preferred embodiment of the present invention will be set forth in detail with reference to the drawings, in which: 
       FIG. 1  is a schematic diagram of a system according to the preferred embodiment; 
       FIG. 2  is a frame of modulation wave propagation as imaged by the system of  FIG. 1 ; 
       FIG. 3  is an image of a wrapped phase estimation derived in the system of  FIG. 1 ; 
       FIG. 4  is an image of an unwrapped phase estimation derived from the wrapped phase estimation of  FIG. 3 ; 
       FIG. 5  is an image of a reconstructed wave pattern derived from the unwrapped phase estimation of  FIG. 4 ; 
       FIG. 6  is an image of a wave speed distribution derived from the reconstructed wave pattern of  FIG. 5 ; 
       FIG. 7  is a graph showing the time trace of the brightness of a pixel; 
       FIG. 8  is a graph of an ultrasound scanner&#39;s estimated point spread function; and 
       FIG. 9  is a graph of the tradeoff between image resolution and relative error. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
   A preferred embodiment of the invention will be set forth in detail with reference to the drawings, in which like reference numerals refer to like elements or steps throughout. 
     FIG. 1  shows a system  100  according to the preferred embodiment. As shown, the shear wave source  102  vibrates at the frequency ω. Induced by the shear wave source, each particle in the medium  104  oscillates at the same frequency with a spatially dependent phase, the derivative of which is the phase velocity of the shear wave. Assuming that the shear wave source is stationary, the vibration amplitude of each particle in the medium can be written as ƒ(x), where ƒ(x) is a function of x, and x represents the spatial coordinates in two dimensions. In addition, the spatial shape of the shear waves may take any arbitrary form. In other words, the vibration field can be written as:
   U ( x,t )=ƒ( x )*exp[ i (ω t−s ( x ))]  (1) 
   where s(x) is an arbitrary function subject to the wave equation and boundary conditions. 
   A wavefront is determined by assigning (ωt−s(x)) to a constant value:
 
ω t−s ( x )=φ
 
   where φ is a constant phase. 
   Taking derivatives on both sides:
 
ω dt=s ′( x )* dx 
 
   
     
       
         
           
             
               
                 
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   where v s  is the shear wave phase velocity and s′(x) is the first derivative of s(x) with respect to x. s′(x) is the local wave number or the local spatial frequency. Thus, if one can visualize s(x) and its movement over time, then one can determine v s , which is directly linked to the stiffness of tissue: 
   
     
       
         
           
             
               
                 
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   where G is the shear modulus of the medium  104  and ρ is its mass density. 
   While transmitting and receiving ultrasound signals, the ultrasound probe  106  is also externally vibrated. The ultrasound probe  106  is carefully positioned above the soft material  104  without touching it. A thick layer of ultrasound gel  108  is applied to couple the acoustic energy into the soft material  104 . The ultrasound probe  106  is carefully positioned so that there is always an ultrasound gel filled gap between the probe and the surface of the medium to make sure the vibration of the US probe does not propagate shear waves into the phantom. Signals received by the US probe  106  are processed in an image processing device  110 , in a manner to be described below, to produce images that are displayed on a video display  112  or output in any other suitable manner. 
   The particle motion relative to the ultrasound probe is estimated by the sonoelastography algorithm. The ultrasound probe is vibrated at a frequency ω−Δω, where Δω&lt;&lt;ω. Therefore, the motion of the ultrasound probe is:
 
 R ( t )= A *exp[ i (ω−Δω) t]   (4)
 
   where A is a constant. 
   Hence the vibration field relative to the ultrasound probe is: 
   
     
       
         
           
             
               
                 
                   
                     
                       
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   where P* is the complex conjugate of P. 
   Since the shear wave is mechanically modulated by the ultrasound probe motion, |P 2 (x,t)| is called the modulation wave. Similarly to equation (2), the phase velocity of the modulation wave is 
   
     
       
         
           
             
               
                 
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                 7 
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   Comparing equation (7) with equation (2) gives: 
   
     
       
         
           
             
               
                 
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                   m 
                 
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                 ( 
                 8 
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   Also notice that the mechanical modulation does not interfere with the spatial component of equation (1), i.e. s(x). Therefore the exact shear wave appearance is preserved. Apart from a DC shift in the amplitude and a change in the velocity, the shear wave propagation is exactly represented by the modulation wave. Since Δω/ω is an arbitrary and controllable factor, the shear wave can thus be slowed by the mechanical modulation to be studied by ordinary ultrasound scanners with the sonoelastography modifications. 
   Experimental validation will now be described. 
   In the validating experiments, a double channel signal generator (Tektronix AFG320) produces two monochrome low frequency signals at slightly different frequencies. One channel of the signals (199.9 Hz) drives a bending piezo elements known as Thunder (Face International Corporation, Norfolk, Va.) which is applied to vibrate the US transducer. The other channel of the signal (200 Hz) drives a shear wave actuator (Piezo System, Massachusetts), which propagates shear waves into a Zerdine (CIRS, Virginia) ultrasound phantom. The phantom has a hard inclusion and is otherwise uniform. A GE Logiq 700, which has been specially modified to implement the sonoelastography functions, is applied to visualize the “modulation wave” propagation. With the realtime visualization, the shear waves are virtually “slowed down” so that the local and subtle behaviors of the waves can be examined closely. The different wave speeds within and outside of the lesion can be perceived by the human eye. One frame of the “modulation wave” propagation is shown in  FIG. 2 . The shear wave wavefronts are visibly distorted by the hard inclusion and thus the size and the location of the lesion is estimated. 
   There are a number of existing techniques from MRE and other methods for calculating v s  from s(x, t). Those can be optionally applied to the present invention as post-processing to create images of local v s . As an example, the local frequency estimator (LFE) filter bank estimates the local spatial frequency which is inversely proportional to the local shear wave speed. Because of the low signal to noise ratio (SNR) nature of ultrasonic image acquisition, a series of procedures are proposed to reduce the noise and to increase the SNR. First of all, the local vibration phase at each pixel is estimated from the wave propagation video: 
   
     
       
         
           
             
               
                 
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                           ⁡ 
                           
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                 ( 
                 9 
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   where              xy  is the Fourier transform of P xy   2 (t):
 
           xy =         ( P   xy   2 ( t ))  (10)

   Because the phase estimation results are always between −π and π, a phase unwrapping procedure is necessary to create a smooth phase surface. A wrapped phase estimation is depicted in  FIG. 3 , and  FIG. 4  depicts an unwrapped phase surface. The phase unwrapping procedure eliminates the sharp transitions in the original phase map to ensure that any subsequent noise reduction procedure does not blur the edges. After noise reduction, the unwrapped phase map is converted to the wave patterns with artificially increased spatial density:
 
 I ( x,y )=cos( N *φ( x,y ))  (11)
 
   where N is an arbitrary constant, which is typically chosen from 2 to 20.  FIG. 5  shows a reconstructed wave pattern with N=12. That operation virtually increases the number of wavefronts per unit area. That reconstruction enhances the visibility of the low spatial frequency area which corresponds to the location of the stiff inclusion. Furthermore, the LFE filter bank is applied over the wave patterns to estimate the local spatial frequency. The local wave speed distribution, which is inversely proportional to the local frequency, is then displayed, as shown in  FIG. 6 . 
   The phantom includes an area of high elasticity surrounded by an area of low elasticity. The area of high elasticity is shown in  FIG. 5  as an area of low spatial frequency and in  FIG. 6  as a bright area surrounded by a dark area. In an organ such as a prostate, a lesion is imaged in the same manner, by imaging the change in elasticity between the lesion and surrounding healthy tissues. 
   The estimator accuracy will now be considered. It will be seen from the above that final estimation results rely extensively on the phase estimation of lical vibration. The local vibration phase is estimated by tracking the brightness variation at each pixel, as shown in  FIG. 7 . The lower bound of the wave velocity error will be formulated, and an example with realistic values will be given. 
   According to equation (6), the modulation wave equation is
 
| P ( x,t )| 2 =ƒ( x ) 2   +A   2 −2 A ·ƒ( x )cos(Δω t−s ( x ))  (12)
 
   At a given location x 0 , the pixel value is
 
 B ( t )=ƒ( x   0 ) 2   +A   2 −2 A ·ƒ( x   0 )cos(Δω t−s ( x   0 ))  (13)
 
   Assuming that the signal includes white Gaussian noise, the discrete pixel value over multiple observations (multiple frames of the wave video) can be written as
 
 x[n]=C+D ·cos(Δω n +φ)+ w[n]   (14)
 
   where C=ƒ(x 0 ) 2 +A 2 , D=−2A·ƒ(x), φ=−s(x 0 ) and w[n]=           (0,σ 2 ), a zero mean Gaussian distribution with standard deviation σ. Therefore, the likelihood function is
   
     
       
         
           
             
               
                 
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                 15 
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   Taking the first and second derivatives of the natural logarithm of the likelihood function yields: 
   
     
       
         
           
             
               
                 
                   
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                 ( 
                 16 
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   and 
   
     
       
         
           
             
               
                 
                   
                     
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                       D 
                       
                         σ 
                         2 
                       
                     
                   
                   · 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       [ 
                       
                         
                           
                             ( 
                             
                               
                                 x 
                                 ⁡ 
                                 
                                   [ 
                                   n 
                                   ] 
                                 
                               
                               - 
                               C 
                             
                             ) 
                           
                           · 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   Δω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   n 
                                 
                                 + 
                                 ϕ 
                               
                               ) 
                             
                           
                         
                         - 
                         
                           D 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   Δω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   n 
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   ϕ 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 17 
                 ) 
               
             
           
         
       
     
   
   Taking the negative expected value gives 
   
     
       
         
           
             
               
                 
                   
                     
                       - 
                       
                         E 
                         [ 
                         
                           
                             
                               
                                 ∂ 
                                 2 
                               
                               ⁢ 
                               
                                 p 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     x 
                                     ; 
                                     ϕ 
                                   
                                   ) 
                                 
                               
                             
                             
                               ∂ 
                               
                                 ϕ 
                                 2 
                               
                             
                           
                           = 
                             
                           ⁢ 
                           
                             
                               D 
                               
                                 σ 
                                 2 
                               
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 [ 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           x 
                                           ⁡ 
                                           
                                             [ 
                                             n 
                                             ] 
                                           
                                         
                                         - 
                                         C 
                                       
                                       ) 
                                     
                                     · 
                                     
                                       cos 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             Δω 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             n 
                                           
                                           + 
                                           ϕ 
                                         
                                         ) 
                                       
                                     
                                   
                                   - 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         D 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                 Δω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 n 
                               
                               + 
                               
                                 2 
                                 ⁢ 
                                 ϕ 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           D 
                           
                             σ 
                             2 
                           
                         
                         · 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             [ 
                             
                               
                                 D 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     cos 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         Δω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         n 
                                       
                                       + 
                                       ϕ 
                                     
                                     ) 
                                   
                                 
                               
                               - 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         D 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ω 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                             + 
                             
                               2 
                               ⁢ 
                               ϕ 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             D 
                             2 
                           
                           
                             σ 
                             2 
                           
                         
                         · 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             [ 
                             
                               
                                 1 
                                 2 
                               
                               + 
                               
                                 
                                   1 
                                   2 
                                 
                                 ⁢ 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         Δω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         n 
                                       
                                       + 
                                       ϕ 
                                     
                                     ) 
                                   
                                 
                               
                               - 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ω 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                             + 
                             
                               2 
                               ⁢ 
                               ϕ 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 ( 
                 18 
                 ) 
               
             
           
         
       
     
   
   If an integer or half integer number of cycles is acquired in experiments by choosing ΔωN=mπ, m being an integer, then the expected value of the cosine term is zero:
 
 E [cos(2Δω n+ 2φ)]=0  (19)
 
   Thus, 
   
     
       
         
           
             
               
                 
                   - 
                   
                     E 
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             ∂ 
                             2 
                           
                           ⁢ 
                           
                             p 
                             ⁡ 
                             
                               ( 
                               
                                 x 
                                 ; 
                                 ϕ 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           
                             ϕ 
                             2 
                           
                         
                       
                       ] 
                     
                   
                 
                 = 
                 
                   
                     ND 
                     2 
                   
                   
                     2 
                     ⁢ 
                     
                       σ 
                       2 
                     
                   
                 
               
             
             
               
                 ( 
                 20 
                 ) 
               
             
           
         
       
     
   
   The inverse of equation (20) gives the Cramer-Rao lower bound of the phase estimation: 
   
     
       
         
           
             
               
                 
                   
                     var 
                     ⁡ 
                     
                       ( 
                       
                         ϕ 
                         ^ 
                       
                       ) 
                     
                   
                   ≥ 
                   
                     1 
                     
                       - 
                       
                         E 
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 ∂ 
                                 2 
                               
                               ⁢ 
                               
                                 p 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     x 
                                     ; 
                                     ϕ 
                                   
                                   ) 
                                 
                               
                             
                             
                               ∂ 
                               
                                 ϕ 
                                 2 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
                 = 
                 
                   
                     2 
                     ⁢ 
                     
                       σ 
                       2 
                     
                   
                   
                     ND 
                     2 
                   
                 
               
             
             
               
                 ( 
                 21 
                 ) 
               
             
           
         
       
     
   
   The local shear wave velocity estimation is equivalent to estimating the local slope of the phase function. At that stage, the tradeoff of image resolution and estimation accuracy has to be considered. If the image resolution is set to be M pixels, the accuracy of the slope estimation of bounded by a function of M. If the problem is modeled as a line fitting problem, and the phase function is assumed to be in the form of:
 
φ[ m]=F+G·m+w[m]   (22)
 
   where w[m] is a zero mean Gaussian distribution with variance determined by equation (21), with independent observations at those M pixels, than the slope estimation G can be obtained with variance 
   
     
       
         
           
             
               
                 
                   var 
                   ⁡ 
                   
                     ( 
                     
                       G 
                       ^ 
                     
                     ) 
                   
                 
                 ≥ 
                 
                   
                     12 
                     · 
                     
                       var 
                       ⁡ 
                       
                         ( 
                         
                           ϕ 
                           ^ 
                         
                         ) 
                       
                     
                   
                   
                     M 
                     ⁡ 
                     
                       ( 
                       
                         
                           M 
                           2 
                         
                         - 
                         1 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 23 
                 ) 
               
             
           
         
       
     
   
   Because stiff regions are generally more important than the normal background, more attention is to be paid to the estimator accuracy in the stiff regions. The vibration amplitude is low in those regions because of the sonoelastography effect; thus, the signal-to-noise ratio (SNR) is also low. An empirical estimate of the SNR in the stiff regions is one. In the experiments, a typical number of frames of the shear wave propagation video is 60. Thus, in equation (21), the variance of the phase estimation is approximately 1/30. 
   M in equation (23) refers to the number of independent measurements. The ultrasound scanner determines that only one independent measurement can be made within the width of the point spread function. A point spread function is simulated with the Field II tool box. The imaging system parameters are selected from a typical experimental setting and are summarized in Table I below: 
   
     
       
             
             
             
           
         
             
                 
               TABLE I 
             
             
                 
                 
             
             
                 
               Parameter 
               Value 
             
             
                 
                 
             
           
           
             
                 
               Sampling frequency 
               150 × 10 6  Hz 
             
             
                 
               Speed of sound 
               1540 m/s 
             
             
                 
               Central frequency 
               7.5 × 10 6  Hz 
             
             
                 
               Relative bandwidth 
               30% 
             
             
                 
               Number of elements 
               128 
             
             
                 
               F number 
               3 
             
             
                 
                 
             
           
        
       
     
   
   The simulation shows that the 6 dB width of the point spread function in the lateral direction is approximately 0.5 mm, as shown in  FIG. 8 . Assuming a realistic shear wave speed of 4 m/s and a driving frequency of 200 Hz, the relation between the elasticity estimation resolution and the estimation relative error will be discussed. 
   If the resolution is chosen to be 2 mm, there are four independent measurements within that length. According to equation (23), 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         var 
                         ⁡ 
                         
                           ( 
                           
                             G 
                             ^ 
                           
                           ) 
                         
                       
                       ≥ 
                         
                       ⁢ 
                       
                         
                           12 
                           · 
                           
                             var 
                             ⁡ 
                             
                               ( 
                               
                                 ϕ 
                                 ^ 
                               
                               ) 
                             
                           
                         
                         
                           M 
                           ⁡ 
                           
                             ( 
                             
                               
                                 M 
                                 2 
                               
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           12 
                           · 
                           
                             1 
                             / 
                             30 
                           
                         
                         
                           4 
                           ⁢ 
                           
                             ( 
                             
                               
                                 4 
                                 2 
                               
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       0.0067 
                     
                   
                 
               
             
             
               
                 ( 
                 24 
                 ) 
               
             
           
         
       
     
   
   Since the phase increase is 2π over one shear wave wavelength, the phase slope may be estimated in terms of the wavelength λ by: 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         slope 
                         ϕ 
                       
                       = 
                         
                       ⁢ 
                       
                         2 
                         ⁢ 
                         
                           π 
                           / 
                           λ 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         2 
                         ⁢ 
                         
                           π 
                           / 
                           
                             ( 
                             
                               20 
                               * 
                               2 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       0.1571 
                     
                   
                 
               
             
             
               
                   
               
             
           
         
       
     
   
   Thus, the relative error is 4%. 
   The tradeoff between the elasticity image resolution and the estimation relative error is plotted in  FIG. 9 . Equation (23) provides a lower bound of the estimation accuracy. In practice, that lower bound may not be achievable, and the error in practice may be higher. 
   Variations of the preferred embodiment are possible. For instance, physical vibration of the ultrasound probe  106  can be replaced with “virtual probe motion,” in which the image processing device  110  electronically processes the ultrasound signals from the ultrasound probe  106  at an adjustable frequency to simulate the vibration of the ultrasound probe  106 . Such processing can include complex rotation or temporal shifting on the ultrasound signals. In another variation, in addition to or instead of the probe motion, whether physical or virtual, multiple shear wave sources  102  are provided to introduce multiple shear waves at the same or different frequencies to create shear wave interference in the tissue. 
   While a preferred embodiment and variations thereof have been set forth in detail, those skilled in the art who have reviewed the present disclosure will readily appreciate that other embodiments can be realized within the scope of the invention. For example, numerical values are illustrative rather than limiting, as are disclosures of specific equipment and technology. Therefore, the present invention should be construed as limited only by the appended claims.