Abstract:
An ultrasound machine processes ultrasonic data according to acoustoelastic properties of the materials to obtain strain information without specific assumptions with respect to the material properties of the measured material or a variety of different material properties normally not obtained by ultrasound machines.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a divisional of U.S. application Ser. No. 11/549,865 U.S. Pat. No. 7,744,535 filed Oct. 16, 2006 which is a continuation-in-part of U.S. application Ser. No. 11/192,230 U.S. Pat. No. 7,736,315, filed Jul. 29, 2005, which claims the benefit of U.S. Provisional Application 60/592,746 filed Jul. 30, 2004 hereby incorporated by reference. 
    
    
     STATEMENT REGARDING FEDERALLY-SPONSORED RESEARCH OR DEVELOPMENT 
     This invention was made with United States government support awarded by the following agency: NIH AR049266. The United States has certain rights in this invention. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention relates to ultrasonic imaging and quantitative measurements and, in particular, to an improved apparatus and method for making ultrasonic measurements of material strain and stiffness. 
     Conventional ultrasonic imaging provides a mapping of ultrasonic echo signals onto an image plane where the intensity of the echo, caused principally by relatively small differences in material properties between adjacent material types, is mapped to brightness of pixels on the image plane. While such images serve to distinguish rough structure within the body, they provide limited insight into the physical properties of the imaged materials. 
     Ultrasonic elastography is a new ultrasonic modality that may produce data and images revealing stiffness properties of the material, for example, strain under an externally applied stress, Poisson&#39;s ratio, Young&#39;s modulus, and other common strain and strain-related measurements. 
     In one type of elastography, termed “quasi-static” elastography, two images of a material in two different states of compression, for example, no compression and a given positive compression, may be obtained by the ultrasound device. The material may be compressed by a probe (including the transducer itself) or, for biological materials, by muscular action or movement of adjacent organs. Strain may be deduced from these two images by computing gradients of the relative shift of the material in the two images along the compression axis. Quasi-static elastography is analogous to a physician&#39;s palpation of tissue in which the physician determines stiffness by pressing the material and detecting the amount of material yield (strain) under this pressure. 
     The process of deducing the shift in material under compression may start by computing local correlations between the images, and then evaluating differences in echo arrival time for correlated structures before and after compression. Differences in echo arrival time are converted to material displacement (or strain, which is displacement normalized by length) at different points within the material by multiplying the difference in arrival times by the speed of sound through the material. 
     The amount of material strain indirectly provides an approximate measure of stiffness. Material that exhibits less strain under compression may be assumed to be stiffer, while material that exhibits more strain under compression is assumed to be less stiff. 
     The parent application to the present application provided a new paradigm of strain measurement which, rather than deducing strain by measuring the motion of the material, deduced strain directly from the modification of the ultrasonic signal caused by changes in the acoustic properties of the material under deformation (acoustoelasticity). A similar technique could be used, if the strain is known, to derive the material properties. 
     BRIEF SUMMARY OF THE INVENTION 
     The present inventors have now realized that acoustoelastic analysis can provide both strain and material properties from a set of ultrasonic signals without the need to know one to derive the other. A similar technique can be used to provide a variety of new material property measurements including stiffness gradient (the change in stiffness with strain), tangential modulus, Poisson&#39;s ratio, and wave signal attenuation (all as a function of strain) as well as tissue density that in turn can be used to provide images and novel data describing materials. These measurements, in the medical field, may help differentiate tissue types that otherwise would appear similar in standard or elastographic ultrasound images. 
     Specifically then, one embodiment of the present invention provides an ultrasound system having an ultrasound transducer assembly for transmitting an ultrasound signal to collect a first and second echo signals when the material is at a first and second tension applied across the axis of the ultrasound. A processor receives the first and second echo signals from the ultrasound transducer and processes the signals according to a stored program to deduce both strain of the material and its stiffness. 
     Thus, it is one feature of at least one embodiment of the invention to provide for measurements of strain in materials in situations where the functional loading cannot be easily characterized. It is another feature of at least one embodiment of the invention to allow simultaneous measurement of strain and material properties (stiffness, density, Poisson&#39;s ratio, tangential modulus and wave attenuation) without error-prone precharacterization of the material. 
     The processor may determine strain and material properties (stiffness, Poisson&#39;s ratio, tangential modulus and wave attenuation) from a time of flight of ultrasound between the first and second interfaces in the target material and the reflection coefficients indicating reflected ultrasonic energy at the first and second interfaces. 
     Thus, it is another feature of at least one embodiment of the invention to extract quantitative information from the strength of reflections that allows improved analysis of ultrasonic signals. 
     In another embodiment, the present invention provides an ultrasonic acoustoelastography system that evaluates the first and second echo signals at multiple states of deformation to provide a measure of change in stiffness as a function of deformation and uses this change in material properties (stiffness, density, Poisson&#39;s ratio, tangential modulus and wave attenuation) as a function of deformation to characterize the material. 
     Thus, it is one feature of at least one embodiment of the invention to use a measure of the deviation of stiffness from constant, assumed in other elastographic techniques, in fact, to characterize tissue. 
     The echo signals may be from boundaries of the material in another material, or the reverse, or of the material in a uniform transmission material such as water. 
     Thus, it is one feature of at least one embodiment of the invention to provide a method of obtaining the necessary signals in a variety of situations. 
     The change of stiffness or other material properties (density, Poisson&#39;s ratio, tangential modulus and wave attenuation) may be output as an image. 
     Thus, it is one feature of at least one embodiment of the invention to provide a new imaging mode. 
     In a preferred embodiment, the invention provides a system for measuring material properties with ultrasound, which measures the strength of reflected waves from multiple levels of deformation of the material at a first and second boundary of the material and measures the time of travel of the waves between the first and second boundary. These two measurements are combined to deduce material strain, and then other material properties (stiffness, tangential modulus, Poisson&#39;s ratio, wave attenuation, and density) can be calculated using the deduced strain and measured wave signal sets from multiple states of material deformation. 
     Thus, it is one feature of at least one embodiment of the invention to provide a wealth of new measurable qualities of materials using ultrasound. 
     These particular objects and advantages may apply to only some embodiments falling within the claims, and thus do not define the scope of the invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a simplified block diagram of an ultrasound scanner suitable for use with the present invention having a transducer system providing compression of a measured material along the beam of ultrasound; 
         FIG. 2  is a fragmentary view of an alternative transducer system used with tension on the material applied across the beam of ultrasound; 
         FIG. 3  is an elevational cross-section of material being measured using the devices of  FIG. 1  or  2 , showing the incident and reflected signal and the analysis of the reflected signal to deduce time of flight and reflection coefficients; 
         FIG. 4  is a block diagram of the calculations of the present invention using the system of  FIG. 2  to simultaneously obtain stiffness and strain; 
         FIG. 5  is a detailed view of the calculation block of  FIG. 4  showing the extraction of reflection coefficients and time of flight, per  FIG. 3 , in this process. 
         FIG. 6  is a block diagram of the calculations of the present invention using the system of  FIG. 1  to simultaneously obtain different material properties and tissue strain; 
         FIG. 7  is a block diagram of a mapping system used to provide images and quantitative information from the systems of  FIGS. 4 and 6 ; 
         FIG. 8  is a pair of graphs showing stiffness gradient determined using the present invention and per ex vitro measurement showing the performance of the present invention and the ability of stiffness gradient to distinguish tissue types; and 
         FIG. 9  is a diagram showing ultrasound waves and their magnitudes as they pass through the materials of  FIGS. 2 and 3 . 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     Referring now to  FIG. 1 , an acoustoelastographic ultrasound system  10  suitable for use with the present invention may employ an ultrasonic imaging machine  11  alone or in combination with an external computer  30 . Generally, the ultrasonic imaging machine  11  provides the necessary hardware and/or software to collect and process ultrasonic echo signals by processor  33  held within the ultrasonic imaging machine  11  or in the external computer  30 . 
     An ultrasonic transducer  12  associated with the ultrasonic imaging machine  11  may transmit an ultrasound beam  14  along an axis  15  toward a region of interest  18  within a patient  16  to produce echo signals returning generally along axis  15 . The echo signals may be received by the ultrasonic transducer  12  and converted to an electrical echo signal. For the construction of an image, multiple rays within ultrasound beam  14  and corresponding echo signals will be acquired within a region of interest surrounding for example a tumor  18 . In one embodiment, the transducer  12  may include a force measuring transducer to quantify force applied to the patient from the transducer  12  when the transducer  12  is being used to apply compression to the patient  16 . 
     Multiple acquisitions of echo signals may be obtained with the tissue of the patient  16  in different states of compression, the compression being performed most easily by pressing the ultrasonic transducer  12  into the tissue of the patient  16  along axis  15 . Other compression techniques, including those using independent compressor paddles or muscular action of tissue, may also be used. 
     The electrical echo signals communicated along lead  22  may be received by interface circuitry  24  of the ultrasonic imaging machine  11 . The interface circuitry  24  provides amplification, digitization, and other signal processing of the electrical signal as is understood in the art of ultrasonic imaging. The digitized echo signals are then transmitted to a memory  34  for storage and subsequent processing by a processor  33 , as will be described below. 
     After processing, the echo signals may be used to construct an image displayed on graphical display  36  or may be displayed quantitatively on the graphical display  36 . Input commands affecting the display of the echo signals and their processing may be received via a keyboard  38  or cursor control device  40 , such as a mouse, attached to the processor  33  via interface  24 , as is well understood in the art. 
     Referring now to  FIG. 2 , in an alternative embodiment, the ultrasonic transducer  12  may be placed, for example in a container  42  holding a coupling medium  44  with known material properties, such as water. A portion of the patient&#39;s body, in this case the patient&#39;s heel including a tendon  45 , may be immersed in the coupling medium  44  so that the ultrasonic transducer  12  directs an ultrasound beam  14  generally along a horizontal axis  15  through the tendon  45  which extends vertically across the axis  15 . In this case, the tendon  45  is essentially adjacent to the ultrasonic transducer  12  separated only by a small amount of skin and tissue and a thin layer of the coupling medium  44 . 
     Echo signal, as described above, may be obtained with the tendon in different states of tension, for example by instructing the patient to press down on the ball of the foot down to lift against the patient&#39;s weight during one acquisition set. The tension is applied vertically along axis  46  generally perpendicular and crossing axis  15 . As before the echo signals are provided over lead  22  to the interface circuitry  24 . 
     Referring now to  FIG. 3 , in both the applications of  FIGS. 1 and 2 , incident ultrasound rays  20  of the ultrasound beam  14  are directed toward an internal body structure  48  to cross a front interface  50  and rear interface  52  of the body structure. The front interface  50  and rear interface  52  may, for example, be interfaces between different tissues or materials within a patient  16  or interfaces between a coupling material and the body structure  48 . 
     The incident ultrasound ray  20  will typically be a pulse signal  54  having a first frequency spectrum  56 . The passage of the incident ultrasound ray  20  through the front interface  50  will cause a first reflected ultrasound ray  60  to be returned to the transducer  12  from the front interface  50 . Similarly, the passage of the ray  20  through the rear interface  52  will cause a second reflected ultrasound ray  62  to the transducer  12  from the rear interface  52 . These two reflected ultrasound rays combine with other rays to produce a return signal  64  having two dominant pulses associated with the reflected ultrasound rays  60  and  62 , each having a corresponding frequency spectrum: spectrum  66  for the ray  60 , and spectrum  68  for the ray  62 . 
     These pulses may be readily identified by amplitude peak detection techniques or adaptive filtering, implemented by the processor  33  executing a stored program, so that a travel time t may be determined by measuring the time between these pulses  60  and  62 , such as indicates the travel time of the incident ultrasound ray  20  between the front interface  50  and the rear interface  52 . 
     The processor  33  may further analyze the frequency spectra  56 ,  66 , and  68  to determine reflection coefficients at each of the interfaces  50  and  52  indicating generally how much of the incident ultrasound ray  20  was reflected at each interface  50  and  52 . The reflection coefficients, may, for example, be determined by comparing the peak amplitude of spectra  56  against the peak amplitudes of spectra  66  and  68 , respectively, as will described further below. 
     Acoustoelastic Strain Gauge (ASG) 
     Referring now to  FIG. 4 , the echo data  70  comprising signals  54  and  64  for two states of tension, per  FIG. 1 , may be collected and provided to a processing program  72  contained in the memory  34  and executed by processor  33 . From these data, acceptably without additional measurement or input with respect to the material being measured, a stiffness value or material property  74  and a strain measurement  78  may both be determined, for example, for the tendon  45  of  FIG. 2 . This contrasts to the method of the parent of the present application in which one of the material properties  74  or strain  78  is needed to be known to produce the other. 
     Referring to  FIG. 5 , the program  72 , as described above, generally provides for a determination of the reflection coefficient  75  and the travel time  76 , as described below, for these two states of tension of the tissue, and makes certain general assumptions  79  about the measured materials which are processed as follows. 
     Frequency Domain Analysis 
     First, the transverse stiffness (e.g., along the ultrasound axis  15 ) is assumed to be some function of the applied strain e, for example, a general second order function such as:
 
 {tilde over (C)}   33 ( e )≈ C   2   ·e   2   +C   1   ·e+C   33 .  (1)
 
     As will be discussed later, the order or the form of this function is not critical and is chosen so that it can be fit to the data. 
     By introducing the assumption of near incompressibility (i.e. the material density does not depend on the amount of strain or deformation) it follows that:
 
ρ S ( e )≈ρ S ( e= 0)=ρ 0   S ,  (2)
 
     Under these assumptions, the wave velocity through thickness (in transverse direction) is given by: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       ⁡ 
                       
                         ( 
                         
                           e 
                           i 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         
                           
                             
                               C 
                               ~ 
                             
                             33 
                           
                           ⁡ 
                           
                             ( 
                             e 
                             ) 
                           
                         
                         
                           
                             ρ 
                             S 
                           
                           ⁡ 
                           
                             ( 
                             e 
                             ) 
                           
                         
                       
                     
                   
                   ≈ 
                   
                     
                       
                         
                           
                             
                               C 
                               ~ 
                             
                             33 
                           
                           ⁡ 
                           
                             ( 
                             e 
                             ) 
                           
                         
                         
                           ρ 
                           0 
                           S 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     The reflection coefficient at the interface of surrounding medium (e.g., water) and target medium is defined by: 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       ⁡ 
                       
                         ( 
                         e 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             ρ 
                             S 
                           
                           ⁢ 
                           
                             V 
                             ⁡ 
                             
                               ( 
                               e 
                               ) 
                             
                           
                         
                         - 
                         
                           
                             ρ 
                             W 
                           
                           ⁢ 
                           
                             V 
                             W 
                           
                         
                       
                       
                         
                           
                             ρ 
                             S 
                           
                           ⁢ 
                           
                             V 
                             ⁡ 
                             
                               ( 
                               e 
                               ) 
                             
                           
                         
                         + 
                         
                           
                             ρ 
                             W 
                           
                           ⁢ 
                           
                             V 
                             W 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       or 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         re-organized 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       form 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ρ 
                           S 
                         
                         · 
                         
                           V 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               i 
                             
                             ) 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           1 
                           + 
                           
                             R 
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                         
                         
                           1 
                           - 
                           
                             R 
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           ρ 
                           W 
                         
                         · 
                         
                           
                             V 
                             W 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     In these relations, superscripts S and W respectively represent the target tissue (solid) and the surrounding medium (e.g., water). 
     Combining eq. (1)˜(4), a relation between material properties (C 1 , C 2 , C 33 , ρ 0 ) and impedance square (right hand side of equation) for each stretched state is derived as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ρ 
                             0 
                             S 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   C 
                                   2 
                                 
                                 · 
                                 
                                   e 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   C 
                                   1 
                                 
                                 · 
                                 e 
                               
                               + 
                               
                                 C 
                                 33 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             [ 
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       R 
                                       ⁡ 
                                       
                                         ( 
                                         e 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       R 
                                       ⁡ 
                                       
                                         ( 
                                         e 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ρ 
                                 W 
                               
                               ⁢ 
                               
                                 V 
                                 W 
                               
                             
                             ] 
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           IPS 
                           ⁡ 
                           
                             ( 
                             e 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     When the material properties of the surrounding medium (e.g., water) are known, the reflection coefficient can be retrieved from measured echo signals, and the impedance square parameter on the right hand side is known. As will be noted below, however, knowledge of the material properties of the surrounding material is not required. 
     Similarly, the relation between the material property and impedance square at non-stretched state can be derived as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ρ 
                             0 
                             S 
                           
                           ⁢ 
                           
                             C 
                             33 
                           
                         
                         = 
                         
                           
                             [ 
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       R 
                                       0 
                                     
                                   
                                   ) 
                                 
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       R 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ρ 
                                 W 
                               
                               ⁢ 
                               
                                 V 
                                 W 
                               
                             
                             ] 
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             IPS 
                             0 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     By the taking the ratio of reflection information eqs. (5) and (6), the unknown target tissue density cancels and is eliminated as a parameter required to determine the other material properties. This is important because in-vivo tissue density is very difficult to measure. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               
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                                 0 
                                 S 
                               
                               ⁡ 
                               
                                 ( 
                                 e 
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                                   ~ 
                                 
                                 33 
                               
                               ⁡ 
                               
                                 ( 
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                               0 
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                             ⁢ 
                             
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                               33 
                             
                           
                         
                         ≈ 
                           
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                                   C 
                                   2 
                                 
                                 · 
                                 
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                                 · 
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                               33 
                             
                           
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                         = 
                           
                         ⁢ 
                         
                           
                             
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                                     ( 
                                     
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                                         ⁡ 
                                         
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                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
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                                         ( 
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                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
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                               ( 
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                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Here, the left hand side term represents the unknown normalized material property. Again, the right hand side is measured or known information. Since the material properties of the surrounding medium (ρ W V W ) are cancelled out in the same process, the material properties of the surrounding medium that were originally assumed to be known are also not essential to the analysis. 
     Time Domain Analysis: 
     The travel time at e=0 (non-stretched state) can be related to tissue thickness D and wave velocity V 0  by: 
     
       
         
           
             
               
                 
                   
                     T 
                     0 
                   
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         D 
                       
                       
                         V 
                         0 
                       
                     
                     = 
                     
                       2 
                       ⁢ 
                       D 
                       ⁢ 
                       
                         
                           
                             
                               ρ 
                               0 
                               s 
                             
                             
                               C 
                               33 
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     Similarly, the travel time at e≠0 (stretched state) is given by: 
     
       
         
           
             
               
                 
                   
                     T 
                     ⁡ 
                     
                       ( 
                       
                         e 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           d 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               i 
                             
                             ) 
                           
                         
                       
                       
                         V 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             i 
                           
                           ) 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     Here, the wave velocity V and tissue thickness d at a stretched state are given by: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       ≈ 
                       
                         
                           
                             
                               
                                 C 
                                 ~ 
                               
                               33 
                             
                             ⁡ 
                             
                               ( 
                               e 
                               ) 
                             
                           
                           
                             ρ 
                             0 
                             s 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           
                             
                               C 
                               2 
                             
                             ⁢ 
                             
                               e 
                               2 
                             
                           
                           + 
                           
                             
                               C 
                               1 
                             
                             ⁢ 
                             e 
                           
                           + 
                           
                             C 
                             33 
                           
                         
                         
                           ρ 
                           0 
                           s 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
             
               
                 
                   
                     d 
                     ⁡ 
                     
                       ( 
                       
                         e 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       D 
                       
                         
                           1 
                           + 
                           
                             e 
                             i 
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     By taking the ratio of the wave travel time eq. (10) and (11): 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 T 
                                 0 
                               
                               
                                 T 
                                 ⁡ 
                                 
                                   ( 
                                   e 
                                   ) 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         = 
                         
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   D 
                                   / 
                                   
                                     V 
                                     0 
                                   
                                 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                   d 
                                   / 
                                   V 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       
                                         C 
                                         ~ 
                                       
                                       33 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       e 
                                       ) 
                                     
                                   
                                   
                                     C 
                                     33 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   1 
                                   + 
                                   e 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               IPS 
                               ⁡ 
                               
                                 ( 
                                 e 
                                 ) 
                               
                             
                             
                               IPS 
                               0 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               1 
                               + 
                               e 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     the strain e applied to the tissue can be evaluated as: 
     
       
         
           
             
               
                 
                   e 
                   = 
                   
                     
                       
                         
                           IPS 
                           0 
                         
                         
                           IPS 
                           ⁡ 
                           
                             ( 
                             e 
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               T 
                               0 
                             
                             
                               T 
                               ⁡ 
                               
                                 ( 
                                 e 
                                 ) 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                     - 
                     1 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     Since the right hand side of this relation contains parameters that are measured directly from wave signals, the applied strain can be evaluated directly without any supplemental information. Once the applied strains are evaluated, they are used in equation (7) to compute the normalized stiffness. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   C 
                                   2 
                                 
                                 · 
                                 
                                   e 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   C 
                                   1 
                                 
                                 · 
                                 e 
                               
                             
                             
                               C 
                               33 
                             
                           
                           + 
                           1 
                         
                         = 
                         
                           
                             
                               [ 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       R 
                                       ⁡ 
                                       
                                         ( 
                                         e 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       R 
                                       ⁡ 
                                       
                                         ( 
                                         e 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                             2 
                           
                           
                             
                               [ 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       R 
                                       0 
                                     
                                   
                                   ) 
                                 
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       R 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             IPS 
                             ⁡ 
                             
                               ( 
                               e 
                               ) 
                             
                           
                           
                             IPS 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     If the density of the target tissue ρ 0   S  is known prior to the testing, the coefficients in the transverse stiffness function can be directly evaluated by substituting the evaluated strain into relation (5) 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               C 
                               2 
                             
                             · 
                             
                               e 
                               2 
                             
                           
                           + 
                           
                             
                               C 
                               1 
                             
                             · 
                             e 
                           
                           + 
                           
                             C 
                             33 
                           
                         
                         = 
                         
                           
                             
                               1 
                               
                                 ρ 
                                 0 
                                 S 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         R 
                                         ⁡ 
                                         
                                           ( 
                                           e 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       
                                         R 
                                         ⁡ 
                                         
                                           ( 
                                           e 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   ρ 
                                   W 
                                 
                                 ⁢ 
                                 
                                   V 
                                   W 
                                 
                               
                               ] 
                             
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             1 
                             
                               ρ 
                               0 
                               S 
                             
                           
                           ⁢ 
                           
                             IPS 
                             ⁡ 
                             
                               ( 
                               e 
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     5 
                     &#39; 
                   
                   ) 
                 
               
             
           
         
       
     
     Once the transverse stiffness is given as a function of applied strain e, the tissue thickness at any strain level can be calculated from relation (9) 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           d 
                           ⁡ 
                           
                             ( 
                             e 
                             ) 
                           
                         
                         = 
                         
                           
                             
                               T 
                               ⁡ 
                               
                                 ( 
                                 e 
                                 ) 
                               
                             
                             2 
                           
                           ⁢ 
                           
                             V 
                             ⁡ 
                             
                               ( 
                               e 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               T 
                               ⁡ 
                               
                                 ( 
                                 e 
                                 ) 
                               
                             
                             2 
                           
                           ⁢ 
                           
                             
                               
                                 
                                   
                                     C 
                                     2 
                                   
                                   · 
                                   
                                     e 
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     C 
                                     1 
                                   
                                   · 
                                   e 
                                 
                                 + 
                                 
                                   C 
                                   33 
                                 
                               
                               
                                 ρ 
                                 0 
                                 S 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     Implementation 
     Step 1. Measure the reflected echo signal and compute impedance square IPS 0  and wave travel time T 0  through thickness in the non-stretched (unloaded) state (e=0). 
     If the density of target tissue is known or can be assumed with reasonable accuracy, the stiffness at the non-deformed state can be given by (4). 
     
       
         
           
             
               
                 
                   
                     C 
                     33 
                   
                   = 
                   
                     
                       
                         1 
                         
                           ρ 
                           0 
                           S 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   R 
                                   ⁡ 
                                   
                                     ( 
                                     e 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   R 
                                   ⁡ 
                                   
                                     ( 
                                     e 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             ρ 
                             W 
                           
                           ⁢ 
                           
                             V 
                             W 
                           
                         
                         ] 
                       
                     
                     2 
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       1 
                       
                         ρ 
                         0 
                         S 
                       
                     
                     ⁢ 
                     
                       IPS 
                       0 
                     
                   
                 
               
             
           
         
       
     
     Step 2a. Stretch the tissue and measure the reflected echo signal and compute IPS(e 1 ) and wave travel time T(e 1 ). From relation (13), evaluate the applied strain e 1 . 
     Step 2b. Repeat step 2a for the second stretched state to evaluate IPS(e 2 ) and wave travel time T(e 2 ) through thickness. From relation (13), evaluates the applied strain e 2 . 
     Step 3. Set up the following two independent equations, by substituting evaluated strains e 1  and e 2  into relation (7). 
     If density is not known: 
     
       
         
           
             
               
                 
                   For 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   strain 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     e 
                     1 
                   
                   ⁢ 
                   
                     : 
                   
                 
               
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                           · 
                           
                             e 
                             1 
                             2 
                           
                         
                         + 
                         
                           
                             C 
                             1 
                           
                           · 
                           
                             e 
                             1 
                           
                         
                       
                       
                         C 
                         33 
                       
                     
                     + 
                     1 
                   
                   = 
                   
                     
                       IPS 
                       ⁡ 
                       
                         ( 
                         
                           e 
                           1 
                         
                         ) 
                       
                     
                     
                       IPS 
                       0 
                     
                   
                 
               
             
             
               
                 
                   For 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   strain 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     e 
                     2 
                   
                   ⁢ 
                   
                     : 
                   
                 
               
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                           · 
                           
                             e 
                             2 
                             2 
                           
                         
                         + 
                         
                           
                             C 
                             1 
                           
                           · 
                           
                             e 
                             2 
                           
                         
                       
                       
                         C 
                         33 
                       
                     
                     + 
                     1 
                   
                   = 
                   
                     
                       IPS 
                       ⁡ 
                       
                         ( 
                         
                           e 
                           2 
                         
                         ) 
                       
                     
                     
                       IPS 
                       0 
                     
                   
                 
               
             
           
         
       
     
     If the density is known, substitute strains e 1 , e 2  and stiffness C 33  evaluated from previous relation (5) 
     
       
         
           
             
               
                 
                   For 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   strain 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     e 
                     1 
                   
                   ⁢ 
                   
                     : 
                   
                 
               
               
                 
                   
                     
                       
                         C 
                         2 
                       
                       · 
                       
                         e 
                         1 
                         2 
                       
                     
                     + 
                     
                       
                         C 
                         1 
                       
                       · 
                       
                         e 
                         1 
                       
                     
                     + 
                     
                       C 
                       33 
                     
                   
                   = 
                   
                     
                       1 
                       
                         ρ 
                         0 
                         S 
                       
                     
                     ⁢ 
                     
                       IPS 
                       ⁡ 
                       
                         ( 
                         
                           e 
                           1 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   For 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   strain 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     e 
                     2 
                   
                   ⁢ 
                   
                     : 
                   
                 
               
               
                 
                   
                     
                       
                         C 
                         2 
                       
                       · 
                       
                         e 
                         2 
                         2 
                       
                     
                     + 
                     
                       
                         C 
                         1 
                       
                       · 
                       
                         e 
                         2 
                       
                     
                     + 
                     
                       C 
                       33 
                     
                   
                   = 
                   
                     
                       1 
                       
                         ρ 
                         0 
                         S 
                       
                     
                     ⁢ 
                     
                       IPS 
                       ⁡ 
                       
                         ( 
                         
                           e 
                           2 
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     Step 4. Evaluate two normalized unknown parameters C 1 /C 33  and C 2 /C 33  for the case of unknown tissue density or stiffness parameters C 1  and C 2  for the case of known tissue density by solving above system of equations simultaneously. 
     With wave signals acquired at two different unknown stretched states (e 1  and e 2 ) and with wave signals from the non-stretched state (e=0), we can evaluate applied strain and two normalized stiffness coefficients (C 1 /C 33  and C 2 /C 33 ) without any prior information or we can evaluate C 1  and C 2  when tissue density is known. 
     In the above example, stiffness was assumed to be a second order function of applied strain. It is possible, however, to use the same technique to evaluate strain and stiffness coefficients even if stiffness is a higher order (or a different type) of function of strain. Unless, it is not possible to acquire wave signals at multiple strain states, a higher order assumption of stiffness does not cause any difficulty for this technique. 
     For example, if stiffness turns out to be a third order function of strain ({tilde over (C)} 33 (e)≈C 3 ·e 3 +C 2 ·e 2 +C 1 ·e+C 33 ), it requires only one extra wave signal measurement to set up the required extra equations. If N+1 sets of signals (N wave signals measured at different stretch levels and one wave signal from non-stretched state) can be acquired, N unknown normalized material coefficients can be evaluated. Therefore, the Nth order normalized transverse stiffness {tilde over (C)} 33 (e)/C 33 ≈(C N /C 33 ·e N + . . . C 1 /C 33 ·e+1) can be found. 
     It is worth noting the differences between the method of “elastography” and the ASG technique presented herein. Since “elastography” uses only the wave equation (and does not use the acoustoelastic equation done in this analysis), changes in tissue acoustic characteristics as a function of strain do not show up in elastographic analysis. Hence the stiffness is always treated as fixed number i.e. {tilde over (C)} 33 (e)=C 33 . As a result, the frequency domain relation (7) becomes, 
     
       
         
           
             
               
                 
                   
                     
                       ρ 
                       0 
                       S 
                     
                     ⁡ 
                     
                       ( 
                       e 
                       ) 
                     
                   
                   ⁢ 
                   
                     
                       
                         C 
                         ~ 
                       
                       33 
                     
                     ⁡ 
                     
                       ( 
                       e 
                       ) 
                     
                   
                 
                 
                   
                     ρ 
                     0 
                     S 
                   
                   ⁢ 
                   
                     C 
                     33 
                   
                 
               
               ≈ 
               
                 
                   
                     ρ 
                     0 
                     S 
                   
                   ⁢ 
                   
                     C 
                     33 
                   
                 
                 
                   
                     ρ 
                     0 
                     S 
                   
                   ⁢ 
                   
                     C 
                     33 
                   
                 
               
             
             = 
             
               
                 1 
                 ≠ 
                 
                   
                     
                       [ 
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 e 
                                 ) 
                               
                             
                           
                           ) 
                         
                         
                           ( 
                           
                             1 
                             - 
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 e 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                       ] 
                     
                     2 
                   
                   
                     
                       [ 
                       
                         
                           1 
                           + 
                           
                             R 
                             0 
                           
                         
                         
                           1 
                           - 
                           
                             R 
                             0 
                           
                         
                       
                       ] 
                     
                     2 
                   
                 
               
               = 
               
                 
                   
                     IPS 
                     ⁡ 
                     
                       ( 
                       e 
                       ) 
                     
                   
                   
                     IPS 
                     0 
                   
                 
                 . 
               
             
           
         
       
     
     Therefore the relationship between reflection coefficients cannot be utilized in “elastography”. In addition, the time domain relation (13) suffers from the same fixed stiffness {tilde over (C)} 33 (e)=C 33  as: 
     
       
         
           
             
               e 
               ^ 
             
             = 
             
               
                 
                   ( 
                   
                     
                       T 
                       0 
                     
                     
                       T 
                       ⁡ 
                       
                         ( 
                         e 
                         ) 
                       
                     
                   
                   ) 
                 
                 2 
               
               - 
               1. 
             
           
         
       
     
     This oversimplified relation in time domain indicates that the applied strain can never correctly account for strain dependent signals within the framework of “elastography”. The differences between the two methods are tabulated in Table 1. 
     
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 Frequency domain 
                 Time domain relation 
                   
               
               
                   
                 Relation (7) 
                 (13) 
                 Variables that can be evaluated 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                 ASG 
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   C 
                                   2 
                                 
                                 ⁢ 
                                 
                                   □ 
                                   
                                     e 
                                     2 
                                   
                                 
                               
                               + 
                               
                                 
                                   C 
                                   1 
                                 
                                 ⁢ 
                                 
                                   □ 
                                   e 
                                 
                               
                             
                             
                               C 
                               33 
                             
                           
                           + 
                           1 
                         
                         = 
                         
                           
                             IPS 
                             ⁡ 
                             
                               ( 
                               e 
                               ) 
                             
                           
                           
                             IPS 
                             0 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         e 
                         = 
                         
                           
                             
                               
                                 IPS 
                                 0 
                               
                               
                                 IPS 
                                 ⁡ 
                                 
                                   ( 
                                   e 
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     T 
                                     0 
                                   
                                   
                                     T 
                                     ⁡ 
                                     
                                       ( 
                                       e 
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           - 
                           1 
                         
                       
                     
                   
                 
                 Normalized coefficients C 1 /C 33 , . . . , C N /C 33  for unknown density or coefficients C 1 , . . . , C N  for known tissue density. &amp; applied strain e 
               
               
                   
               
               
                 Elastography 
                   
                 
                   
                     
                       
                         
                           e 
                           ^ 
                         
                         = 
                         
                           
                             
                               ( 
                               
                                 
                                   T 
                                   0 
                                 
                                 
                                   T 
                                   ⁡ 
                                   
                                     ( 
                                     e 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           1 
                         
                       
                     
                   
                 
                 Approximate strain ê  
               
               
                   
               
             
          
         
       
     
     Stiffness Gradient Identification (SGI) 
     Referring now to  FIG. 6 , the echo data  70  comprising signals  54  and  64  from multiple states of compression per,  FIG. 1 , may be collected and provided to a processing program  72 ′ contained in the memory  34  and executed by processor  33 . The echo data  70  may be processed by a program  72 ′, similar to program  72  described with respect to  FIG. 4 , to produce a variety of different measurements, including stiffness gradient  82 , tangential modulus  91 , initial stiffness  84 , density  86 , Poisson&#39;s ratio  93 , target thickness  95 , and wave attenuation  88 . 
     Referring to  FIG. 5 , the program  72 ′, as described above, generally provides for a determination of the reflection coefficient  75  and the travel time  76 , as described below, for these two states of compression of the tissue, and makes certain general assumptions  79  about the measured materials which are processed as follows. 
     The following analysis will consider a one-dimensional three-layered model (surrounding medium  1 /target medium  2 /surrounding medium  1 ). The material properties in the surrounding medium are assumed to be known. If not, an additional medium (coupling material) with known properties can be added artificially. The method can then analyze the surrounding medium properties first and then use those properties for analysis of the target tissue. Hence this technique can be applied to a multi-layered medium. If the size of the ultrasound wave width is relatively small compared to the size of the target tissue, this assumption can be applied to two dimensions as well. 
     If only the normalized tissue stiffness is required, following relations and steps may be followed. 
     Frequency Domain Analysis: 
     First, based on the theory of acoustoelasticity, the wave velocities in the compressed medium  1  (surrounding tissue) and medium  2  (target tissue) are given by: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         i 
                       
                       ) 
                     
                   
                   ≈ 
                   
                     
                       
                         
                           
                             
                               C 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               t 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                         
                         
                           ρ 
                           1 
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     and 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         V 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           i 
                         
                         ) 
                       
                     
                   
                   ≈ 
                   
                     
                       
                         
                           
                             
                               C 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               t 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                         
                         
                           ρ 
                           2 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   101 
                   ) 
                 
               
             
           
         
       
     
     Here, C(e) and t(e) represent strain (e) dependent stiffness and stress. Subscripts represent the medium numbers. 
     By substituting above relations into the well known definition of a reflection coefficient: 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       ⁡ 
                       
                         ( 
                         
                           e 
                           i 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       ρ 
                                       2 
                                     
                                     · 
                                     
                                       
                                         V 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           e 
                                           i 
                                         
                                         ) 
                                       
                                     
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   
                                     ρ 
                                     1 
                                   
                                   · 
                                   
                                     
                                       V 
                                       1 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         e 
                                         i 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       ρ 
                                       2 
                                     
                                     · 
                                     
                                       
                                         V 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           e 
                                           i 
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     ρ 
                                     1 
                                   
                                   · 
                                   
                                     
                                       V 
                                       1 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         e 
                                         i 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         or 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ρ 
                             2 
                           
                           · 
                           
                             
                               V 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                         
                       
                       = 
                       
                         
                           
                             1 
                             + 
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                           
                             1 
                             - 
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                         
                         ⁢ 
                         
                           
                             ρ 
                             1 
                           
                           · 
                           
                             
                               V 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   102 
                   ) 
                 
               
             
           
         
       
     
     The unknown parameters in target tissues in the i th  compressed state will be given by: 
     
       
         
           
             
               
                 
                   
                     
                       ρ 
                       2 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             C 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               i 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             t 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               i 
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             1 
                             + 
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                           
                             1 
                             - 
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       2 
                     
                     ⁢ 
                     
                       
                         
                           ρ 
                           1 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 C 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 t 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   103 
                   ) 
                 
               
             
           
         
       
     
     With consideration of stress continuity (t 1 (e i )=t 2 (e i )) that is guaranteed in a one-dimensional model, relation (103) can be further simplified to: 
     
       
         
           
             
               
                 
                   
                     
                       ρ 
                       2 
                     
                     ⁢ 
                     
                       
                         C 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           i 
                         
                         ) 
                       
                     
                   
                   ≈ 
                   
                     
                       
                         [ 
                         
                           
                             1 
                             + 
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                           
                             1 
                             - 
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       2 
                     
                     ⁢ 
                     
                       ρ 
                       1 
                     
                     ⁢ 
                     
                       
                         
                           C 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             i 
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   104 
                   ) 
                 
               
             
           
         
       
     
     Here, the material density of mediums  1  and  2  are assumed to be very close ρ 1 ≈ρ 2 . 
     The unknown parameters in target tissues in a non-compressed state are derived to be: 
     
       
         
           
             
               
                 
                   
                     
                       ρ 
                       2 
                     
                     ⁢ 
                     
                       
                         C 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             e 
                             i 
                           
                           = 
                           0 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             1 
                             + 
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     e 
                                     i 
                                   
                                   = 
                                   0 
                                 
                                 ) 
                               
                             
                           
                           
                             1 
                             - 
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     e 
                                     i 
                                   
                                   = 
                                   0 
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       2 
                     
                     ⁢ 
                     
                       ρ 
                       1 
                     
                     ⁢ 
                     
                       
                         
                           C 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               e 
                               i 
                             
                             = 
                             0 
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   105 
                   ) 
                 
               
             
           
         
       
     
     Now assume stiffness in target tissue (medium  2 ) is a second order function of the applied strain e as:
 
 C   2 ( e )≈ a   2   ·e   2   +b   2   ·e+c   2 .  (106)
 
     By taking the ratio of eq. (104) and (105) and with substitution of relation (106), relation between material property in medium  2  and measured reflection coefficient are given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           a 
                           2 
                         
                         ⁢ 
                         
                           e 
                           i 
                           2 
                         
                       
                       + 
                       
                         
                           b 
                           2 
                         
                         ⁢ 
                         
                           e 
                           i 
                         
                       
                       + 
                       
                         c 
                         2 
                       
                     
                     
                       c 
                       2 
                     
                   
                   ≈ 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 1 
                                 + 
                                 
                                   R 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 1 
                                 - 
                                 
                                   R 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                           2 
                         
                         ⁢ 
                         
                           
                             C 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               i 
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             [ 
                             
                               
                                 1 
                                 + 
                                 
                                   R 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         e 
                                         i 
                                       
                                       = 
                                       0 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 1 
                                 - 
                                 
                                   R 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         e 
                                         i 
                                       
                                       = 
                                       0 
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                           2 
                         
                         ⁢ 
                         
                           
                             C 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 e 
                                 i 
                               
                               = 
                               0 
                             
                             ) 
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   107 
                   ) 
                 
               
             
           
         
       
     
     Time Domain Analysis 
     The travel time at e=0 (non-stretched state) is related to tissue thickness D and wave velocity V 0  by: 
     
       
         
           
             
               
                 
                   
                     T 
                     0 
                   
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         D 
                       
                       
                         V 
                         0 
                       
                     
                     = 
                     
                       2 
                       ⁢ 
                       D 
                       ⁢ 
                       
                         
                           
                             
                               ρ 
                               2 
                             
                             
                               C 
                               2 
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   108 
                   ) 
                 
               
             
           
         
       
     
     Similarly, the travel time at e≠0 (stretched state) is given by: 
     
       
         
           
             
               
                 
                   
                     T 
                     ⁡ 
                     
                       ( 
                       
                         e 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           d 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               i 
                             
                             ) 
                           
                         
                       
                       
                         V 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             i 
                           
                           ) 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   109 
                   ) 
                 
               
             
           
         
       
     
     Here, the wave velocity V and tissue thickness d at stretched state is given by: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       ≈ 
                       
                         
                           
                             
                               C 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               e 
                               ) 
                             
                           
                           
                             ρ 
                             2 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           
                             
                               a 
                               2 
                             
                             ⁢ 
                             
                               e 
                               2 
                             
                           
                           + 
                           
                             
                               b 
                               2 
                             
                             ⁢ 
                             e 
                           
                           + 
                           
                             c 
                             2 
                           
                         
                         
                           ρ 
                           2 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   110 
                   ) 
                 
               
             
             
               
                 
                   
                     d 
                     ⁡ 
                     
                       ( 
                       
                         e 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     D 
                     · 
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             e 
                             i 
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   111 
                   ) 
                 
               
             
           
         
       
     
     The ratio of the wave travel time eq. (108) and (109) is given by: 
     
       
         
           
             
               
                 
                   
                     
                       T 
                       0 
                     
                     T 
                   
                   = 
                   
                     
                       
                         
                           D 
                           
                             V 
                             0 
                           
                         
                         
                           d 
                           V 
                         
                       
                       ≈ 
                       
                         
                           D 
                           
                             
                               
                                 c 
                                 2 
                               
                               
                                 ρ 
                                 2 
                               
                             
                           
                         
                         
                           
                             D 
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   e 
                                   i 
                                 
                               
                               ) 
                             
                           
                           
                             
                               
                                 
                                   
                                     C 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     t 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 ρ 
                                 2 
                               
                             
                           
                         
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               
                                 
                                   C 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   t 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                             
                               c 
                               2 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           1 
                           
                             1 
                             - 
                             e 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   112 
                   ) 
                 
               
             
           
         
       
     
     Since 
                   C   2     ⁡     (     e   i     )       +       t   2     ⁡     (     e   i     )           c   2           
is related to reflection coefficients and parameters in medium  1  by:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           C 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             i 
                           
                           ) 
                         
                       
                       + 
                       
                         
                           t 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             i 
                           
                           ) 
                         
                       
                     
                     
                       c 
                       2 
                     
                   
                   = 
                   
                     
                       
                         ρ 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               C 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               t 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     
                       
                         ρ 
                         2 
                       
                       ⁢ 
                       
                         
                           C 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               e 
                               i 
                             
                             = 
                             0 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         
                           [ 
                           
                             
                               1 
                               + 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                             
                               1 
                               - 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         2 
                       
                       ⁢ 
                       
                         
                           ρ 
                           1 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 C 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 t 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                     
                       
                         
                           [ 
                           
                             
                               1 
                               + 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       i 
                                     
                                     = 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                             
                               1 
                               - 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       i 
                                     
                                     = 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         2 
                       
                       ⁢ 
                       
                         ρ 
                         1 
                       
                       ⁢ 
                       
                         
                           C 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               e 
                               i 
                             
                             = 
                             0 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         
                           [ 
                           
                             
                               1 
                               + 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                             
                               1 
                               - 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               C 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               t 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     
                       
                         
                           [ 
                           
                             
                               1 
                               + 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       i 
                                     
                                     = 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                             
                               1 
                               - 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       i 
                                     
                                     = 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         2 
                       
                       ⁢ 
                       
                         
                           C 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               e 
                               i 
                             
                             = 
                             0 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     the strain e applied to the tissue will evaluated as: 
     
       
         
           
             
               
                 
                   
                     e 
                     i 
                   
                   = 
                   
                     1 
                     - 
                     
                       
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     1 
                                     + 
                                     
                                       R 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           e 
                                           i 
                                         
                                         ) 
                                       
                                     
                                   
                                   
                                     1 
                                     - 
                                     
                                       R 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           e 
                                           i 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               2 
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     C 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     t 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                           
                             
                               
                                 [ 
                                 
                                   
                                     1 
                                     + 
                                     
                                       R 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             e 
                                             i 
                                           
                                           = 
                                           0 
                                         
                                         ) 
                                       
                                     
                                   
                                   
                                     1 
                                     - 
                                     
                                       R 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             e 
                                             i 
                                           
                                           = 
                                           0 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               2 
                             
                             ⁢ 
                             
                               
                                 C 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     e 
                                     i 
                                   
                                   = 
                                   0 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           T 
                           
                             T 
                             0 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   113 
                   ) 
                 
               
             
           
         
       
     
     Since the right hand side of this relation contains parameters that are either known or measured directly from wave signals, the applied strain can be evaluated directly without any supplemental information. Once the applied strains are evaluated, they are used in equation (107) to obtain normalized stiffness properties. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           a 
                           2 
                         
                         ⁢ 
                         
                           e 
                           i 
                           2 
                         
                       
                       + 
                       
                         
                           b 
                           2 
                         
                         ⁢ 
                         
                           e 
                           i 
                         
                       
                       + 
                       
                         c 
                         2 
                       
                     
                     
                       c 
                       2 
                     
                   
                   ≈ 
                   
                     
                       
                         
                           [ 
                           
                             
                               1 
                               + 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                             
                               1 
                               - 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         2 
                       
                       ⁢ 
                       
                         
                           C 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             i 
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           [ 
                           
                             
                               1 
                               + 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       i 
                                     
                                     = 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                             
                               1 
                               - 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       i 
                                     
                                     = 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         2 
                       
                       ⁢ 
                       
                         
                           C 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               e 
                               i 
                             
                             = 
                             0 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   107 
                   ) 
                 
               
             
           
         
       
     
     Implementation 
     Step 1. Compress media with a higher level of known stress for example using a force transducer in the ultrasonic probe.
 
 t   2 ( e   A )= t   1 ( e   A )
 
     This higher stress enhances the contrast between the target medium and the surrounding medium. Since the acoustic contrast is now enhanced, the location and shape of the target tissue can be identified with greater precision. Now, the reflection coefficient for this compressed tissue can be used in the following relation: 
     
       
         
           
             
               
                 ρ 
                 2 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       C 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         A 
                       
                       ) 
                     
                   
                   + 
                   
                     
                       t 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         A 
                       
                       ) 
                     
                   
                 
                 ] 
               
             
             = 
             
               
                 
                   [ 
                   
                     
                       1 
                       + 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             A 
                           
                           ) 
                         
                       
                     
                     
                       1 
                       - 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             A 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
                 2 
               
               ⁢ 
               
                 
                   ρ 
                   1 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         C 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           A 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         t 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           A 
                         
                         ) 
                       
                     
                   
                   ] 
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               or 
             
           
         
       
       
         
           
             
               
                 ρ 
                 2 
               
               ⁢ 
               
                 
                   C 
                   2 
                 
                 ⁡ 
                 
                   ( 
                   
                     e 
                     A 
                   
                   ) 
                 
               
             
             ≈ 
             
               
                 
                   [ 
                   
                     
                       1 
                       + 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             A 
                           
                           ) 
                         
                       
                     
                     
                       1 
                       - 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             A 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
                 2 
               
               ⁢ 
               
                 ρ 
                 1 
               
               ⁢ 
               
                 
                   C 
                   1 
                 
                 ⁡ 
                 
                   ( 
                   
                     e 
                     A 
                   
                   ) 
                 
               
             
           
         
       
     
     Step 2. Repeat the same measurements for different, but compressive stresses that are smaller than the stress used in the previous step. 
     
       
         
           
             
               
                 t 
                 2 
               
               ⁡ 
               
                 ( 
                 
                   e 
                   B 
                 
                 ) 
               
             
             &lt; 
             
               
                 t 
                 2 
               
               ⁡ 
               
                 ( 
                 
                   e 
                   A 
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 ρ 
                 2 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       C 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         B 
                       
                       ) 
                     
                   
                   + 
                   
                     
                       t 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         B 
                       
                       ) 
                     
                   
                 
                 ] 
               
             
             = 
             
               
                 
                   [ 
                   
                     
                       1 
                       + 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             B 
                           
                           ) 
                         
                       
                     
                     
                       1 
                       - 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             B 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
                 2 
               
               ⁢ 
               
                 
                   ρ 
                   1 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         C 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           B 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         t 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           B 
                         
                         ) 
                       
                     
                   
                   ] 
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               or 
             
           
         
       
       
         
           
             
               
                 ρ 
                 2 
               
               ⁢ 
               
                 
                   C 
                   2 
                 
                 ⁡ 
                 
                   ( 
                   
                     e 
                     B 
                   
                   ) 
                 
               
             
             ≈ 
             
               
                 
                   [ 
                   
                     
                       1 
                       + 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             B 
                           
                           ) 
                         
                       
                     
                     
                       1 
                       - 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             B 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
                 2 
               
               ⁢ 
               
                 ρ 
                 1 
               
               ⁢ 
               
                 
                   C 
                   1 
                 
                 ⁡ 
                 
                   ( 
                   
                     e 
                     B 
                   
                   ) 
                 
               
             
           
         
       
     
     Step 3. Measure the reflection information for the non-compressed state. In this state, the acoustic contrast between target medium and surrounding medium is typically lowest. Hence the boundary between the two media is less clear. However, by following the compression-enhanced boundary in the previous two steps, the boundary between the two media can be estimated and the reflection coefficient can be computed. 
     
       
         
           
             
               
                 ρ 
                 2 
               
               ⁢ 
               
                 
                   C 
                   2 
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       e 
                       i 
                     
                     = 
                     0 
                   
                   ) 
                 
               
             
             = 
             
               
                 
                   [ 
                   
                     
                       1 
                       + 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             
                               e 
                               i 
                             
                             = 
                             0 
                           
                           ) 
                         
                       
                     
                     
                       1 
                       - 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             
                               e 
                               i 
                             
                             = 
                             0 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
                 2 
               
               ⁢ 
               
                 ρ 
                 1 
               
               ⁢ 
               
                 
                   C 
                   1 
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       e 
                       i 
                     
                     = 
                     0 
                   
                   ) 
                 
               
             
           
         
       
     
     Step 4. Evaluate applied strain e A  and e B  using the relation (113). 
     Step 5. With evaluated strain e A  and e B , unknown normalized material properties (α=a 2 /c 2  and β=b 2 /c 2 ) can evaluated by solving the following two relations simultaneously. Again, the parameters on the right hand side of the relation are known or measured. 
     
       
         
           
             
               
                 
                   
                     a 
                     2 
                   
                   ⁢ 
                   
                     e 
                     A 
                     2 
                   
                 
                 + 
                 
                   
                     b 
                     2 
                   
                   ⁢ 
                   
                     e 
                     A 
                   
                 
                 + 
                 
                   c 
                   2 
                 
               
               
                 c 
                 2 
               
             
             ≈ 
             
               
                 
                   
                     [ 
                     
                       
                         1 
                         + 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               A 
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               A 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   2 
                 
                 ⁢ 
                 
                   
                     C 
                     1 
                   
                   ⁡ 
                   
                     ( 
                     
                       e 
                       A 
                     
                     ) 
                   
                 
               
               
                 
                   
                     [ 
                     
                       
                         1 
                         + 
                         
                           R 
                           ⁢ 
                           
                             ( 
                             
                               
                                 e 
                                 i 
                               
                               = 
                               0 
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               
                                 e 
                                 i 
                               
                               = 
                               0 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   2 
                 
                 ⁢ 
                 
                   
                     C 
                     1 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         e 
                         i 
                       
                       = 
                       0 
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     a 
                     2 
                   
                   ⁢ 
                   
                     e 
                     B 
                     2 
                   
                 
                 + 
                 
                   
                     b 
                     2 
                   
                   ⁢ 
                   
                     e 
                     B 
                   
                 
                 + 
                 
                   c 
                   2 
                 
               
               
                 c 
                 2 
               
             
             ≈ 
             
               
                 
                   
                     [ 
                     
                       
                         1 
                         + 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               B 
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               B 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   2 
                 
                 ⁢ 
                 
                   
                     C 
                     1 
                   
                   ⁡ 
                   
                     ( 
                     
                       e 
                       B 
                     
                     ) 
                   
                 
               
               
                 
                   
                     [ 
                     
                       
                         1 
                         + 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               
                                 e 
                                 i 
                               
                               = 
                               0 
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               
                                 e 
                                 i 
                               
                               = 
                               0 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   2 
                 
                 ⁢ 
                 
                   
                     C 
                     1 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         e 
                         i 
                       
                       = 
                       0 
                     
                     ) 
                   
                 
               
             
           
         
       
     
     Tissue stiffness often is reasonably linear over a functional range of strain and the slope of the stiffness function depends on the tissue type. For such tissues, parameter β will be more dominant and important than α. Once the slope β is evaluated, tissue identification can be performed by checking a table of stiffness slopes for various tissues. 
     If a full set of material properties (stiffness, stiffness gradient, Poisson&#39;s ratio tangential modulus, and density) are to be evaluated, the following additional relations are further required. 
     In a non-deformed isotropic homogenous material, the Young&#39;s modulus E is known to related the stiffness C via material constants termed Poisson&#39;s ratio (ν) and is given by 
     
       
         
           
             
               
                 
                   E 
                   = 
                   
                     
                       
                         
                           ( 
                           
                             1 
                             + 
                             v 
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             - 
                             
                               2 
                               ⁢ 
                               v 
                             
                           
                           ) 
                         
                       
                       
                         ( 
                         
                           1 
                           - 
                           v 
                         
                         ) 
                       
                     
                     ⁢ 
                     C 
                   
                 
               
               
                 
                   ( 
                   201 
                   ) 
                 
               
             
           
         
       
     
     Now assume this relation also holds in a deformed state if the deformation is one dimensional and relation between tangential modulus and stiffness can be given by: 
     
       
         
           
             
               
                 
                   
                     E 
                     ⁡ 
                     
                       ( 
                       e 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               v 
                               ⁡ 
                               
                                 ( 
                                 e 
                                 ) 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             - 
                             
                               2 
                               ⁢ 
                               
                                 v 
                                 ⁡ 
                                 
                                   ( 
                                   e 
                                   ) 
                                 
                               
                             
                           
                           ) 
                         
                       
                       
                         ( 
                         
                           1 
                           - 
                           
                             v 
                             ⁡ 
                             
                               ( 
                               e 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       C 
                       ⁡ 
                       
                         ( 
                         e 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   202 
                   ) 
                 
               
             
           
         
       
     
     E(e) is termed as tangential modulus instead of Young&#39;s modulus for general finite deformation. Since nearly incompressible materials like biological tissues are considered, the Poisson&#39;s ratio at deformed state can be assumed to be: 
     
       
         
           
             
               
                 
                   
                     v 
                     ⁡ 
                     
                       ( 
                       e 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     - 
                     
                       φ 
                       ⁡ 
                       
                         ( 
                         e 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   203 
                   ) 
                 
               
             
           
         
       
     
     φ(e) is a small variable that governs small volume changes caused by the compression strain e. After substituting relation (203) into (202), expand (202) as function of φ(e) and ignoring the higher order term, then relation (202) simplifies to:
 
 E ( e )≈6φ( e )· C ( e )  (204)
 
     If the stiffness in the target tissue (e.g. tendon  45  in  FIG. 2  or tumor  18  in  FIG. 1 ) C 2  (e) and φ(e) can be assumed to be
 
 C   2 ( e )= a   2   ·e   2   +b   2   ·e+c   2   (106)
 
and
 
φ( e )= s   1   +s   2   ·e,   (205)
 
     the tangential modulus in target tissue (e.g. tendon  45  in  FIG. 1  or tumor  18  in  FIG. 1 ) can be given as the function of strain as
 
 E   2 ( e )≈6φ( e )· C   2 ( e )=6 s   1   c   2 +(6 s   2   c   2 +6 s   1   b   2 ) e +(6 s   2   b   2 +6 s   1   a   2 ) e   2 +6 s   2   a   2   e   3 .  (206)
 
     Stress under these assumptions (one dimensional deformation) is given by integrating the tangential modulus, 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             t 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             e 
                             ) 
                           
                         
                         = 
                         
                           ∫ 
                           
                             
                               E 
                               ⁡ 
                               
                                 ( 
                                 e 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ⅆ 
                               e 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             6 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     s 
                                     1 
                                   
                                   ⁢ 
                                   
                                     c 
                                     2 
                                   
                                   ⁢ 
                                   e 
                                 
                                 + 
                                 
                                   
                                     1 
                                     2 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           s 
                                           2 
                                         
                                         ⁢ 
                                         
                                           c 
                                           2 
                                         
                                       
                                       + 
                                       
                                         
                                           s 
                                           1 
                                         
                                         ⁢ 
                                         
                                           b 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     e 
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     1 
                                     3 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           s 
                                           2 
                                         
                                         ⁢ 
                                         
                                           b 
                                           2 
                                         
                                       
                                       + 
                                       
                                         
                                           s 
                                           1 
                                         
                                         ⁢ 
                                         
                                           a 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     e 
                                     3 
                                   
                                 
                                 + 
                                 
                                   
                                     1 
                                     4 
                                   
                                   ⁢ 
                                   
                                     s 
                                     2 
                                   
                                   ⁢ 
                                   
                                     a 
                                     2 
                                   
                                   ⁢ 
                                   
                                     e 
                                     4 
                                   
                                 
                               
                               ] 
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   207 
                   ) 
                 
               
             
           
         
       
     
     With these relations, the key SGI mathematical relations for evaluating full sets of material properties can be derived in following manner. 
     First, take the ratio of eq. (103) and eq. (105) from previous section as 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             C 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               i 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             t 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               i 
                             
                             ) 
                           
                         
                       
                       ] 
                     
                     
                       
                         C 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             e 
                             i 
                           
                           = 
                           0 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           [ 
                           
                             
                               1 
                               + 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                             
                               1 
                               - 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               C 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               t 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     
                       
                         
                           [ 
                           
                             
                               1 
                               + 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       i 
                                     
                                     = 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                             
                               1 
                               - 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       i 
                                     
                                     = 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         2 
                       
                       ⁢ 
                       
                         
                           C 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               e 
                               i 
                             
                             = 
                             0 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   208 
                   ) 
                 
               
             
           
         
       
     
     By substituting (106) and (207) into this relation, (208) is now changed into 
     
       
         
           
             
               
                 
                   
                     1 
                     + 
                     
                       
                         ( 
                         
                           
                             6 
                             ⁢ 
                             
                               s 
                               1 
                             
                           
                           + 
                           
                             
                               b 
                               2 
                             
                             
                               c 
                               2 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       e 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             3 
                             ⁢ 
                             
                               s 
                               2 
                             
                           
                           + 
                           
                             3 
                             ⁢ 
                             
                               s 
                               1 
                             
                             ⁢ 
                             
                               
                                 b 
                                 2 
                               
                               
                                 c 
                                 2 
                               
                             
                           
                           + 
                           
                             
                               a 
                               2 
                             
                             
                               c 
                               2 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         e 
                         2 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                               s 
                               2 
                             
                             ⁢ 
                             
                               
                                 b 
                                 2 
                               
                               
                                 c 
                                 2 
                               
                             
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               s 
                               1 
                             
                             ⁢ 
                             
                               
                                 a 
                                 2 
                               
                               
                                 c 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         e 
                         3 
                       
                     
                     + 
                     
                       
                         3 
                         2 
                       
                       ⁢ 
                       
                         s 
                         2 
                       
                       ⁢ 
                       
                         
                           a 
                           2 
                         
                         
                           c 
                           2 
                         
                       
                       ⁢ 
                       
                         e 
                         4 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           [ 
                           
                             
                               1 
                               + 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                             
                               1 
                               - 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               C 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               t 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     
                       
                         
                           [ 
                           
                             
                               1 
                               + 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       i 
                                     
                                     = 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                             
                               1 
                               - 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       i 
                                     
                                     = 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         2 
                       
                       ⁢ 
                       
                         
                           C 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               e 
                               i 
                             
                             = 
                             0 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   209 
                   ) 
                 
               
             
           
         
       
     
     Now, this relation (209) can be utilized as frequency domain relation to take the place of relation (107) in previous section. 
     The implementation of this relation is exactly same as the implementation steps of (107). First, evaluate applied strains e i  from relation (113) at multiple deformed configurations (more than 4 sets of data, each at a different strain). Second, feed back the evaluated strain into relation (209) to evaluate four unknown material constants 
               s   1     ,       s   2     ⁢       b   2       c   2       ⁢           ⁢   and   ⁢           ⁢         a   2       c   2       .             
With these evaluated constants, both deformation dependent Poisson&#39;s ratio v (e) and normalized stiffness NC 2 (e) are given by
 
     
       
         
           
             
               v 
               ⁡ 
               
                 ( 
                 e 
                 ) 
               
             
             = 
             
               
                 
                   1 
                   2 
                 
                 - 
                 
                   
                     ( 
                     
                       
                         s 
                         1 
                       
                       + 
                       
                         
                           s 
                           2 
                         
                         · 
                         e 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       NC 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       e 
                       ) 
                     
                   
                 
               
               = 
               
                 
                   
                     
                       a 
                       2 
                     
                     
                       c 
                       2 
                     
                   
                   · 
                   
                     e 
                     2 
                   
                 
                 + 
                 
                   
                     
                       b 
                       2 
                     
                     
                       c 
                       2 
                     
                   
                   · 
                   e 
                 
                 + 
                 1. 
               
             
           
         
       
     
     For simplicity, only a one dimensional layered model (surrounding medium-target medium-surrounding medium) is considered in the current description. Here, the condition of stress continuity t 2 (e)=t 1 (e) is guaranteed at the interface of two different media and t 1 (e) is known from measuring the pressure of the ultrasound transducer associated with each deformation. Using this assumption with equation 207 and the strain evaluated for each loading, a set of equations are generated from the relationship below. From these, the two coefficients describing Poisson&#39;s ratio, and all three material constants (a 2 , b 2  and c 2 ) required to define target medium stiffness C 2  (e) can be determined. 
     
       
         
           
             
               
                 
                   
                     
                       t 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       e 
                       ) 
                     
                   
                   = 
                   
                     
                       t 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       e 
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     6 
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             s 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                           ⁢ 
                           e 
                         
                         + 
                         
                           
                             1 
                             2 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   s 
                                   2 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   s 
                                   1 
                                 
                                 ⁢ 
                                 
                                   b 
                                   2 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             e 
                             2 
                           
                         
                         + 
                         
                           
                             1 
                             3 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   s 
                                   2 
                                 
                                 ⁢ 
                                 
                                   b 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   s 
                                   1 
                                 
                                 ⁢ 
                                 
                                   a 
                                   2 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             e 
                             3 
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             s 
                             2 
                           
                           ⁢ 
                           
                             a 
                             2 
                           
                           ⁢ 
                           
                             e 
                             4 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       6 
                       ⁢ 
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                       ⁢ 
                       e 
                     
                     + 
                     
                       3 
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               s 
                               2 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           + 
                           
                             
                               s 
                               1 
                             
                             ⁢ 
                             
                               b 
                               2 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         e 
                         2 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               s 
                               2 
                             
                             ⁢ 
                             
                               b 
                               2 
                             
                           
                           + 
                           
                             
                               s 
                               1 
                             
                             ⁢ 
                             
                               a 
                               2 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         e 
                         3 
                       
                     
                     + 
                     
                       
                         3 
                         2 
                       
                       ⁢ 
                       
                         s 
                         2 
                       
                       ⁢ 
                       
                         a 
                         2 
                       
                       ⁢ 
                       
                         e 
                         4 
                       
                     
                   
                 
               
             
           
         
       
     
     The unknown target tissue density can be also evaluated by utilizing same relation. 
     First re-organize relation (103) as 
     
       
         
           
             
               
                 
                   
                     
                       ρ 
                       2 
                     
                     ⁢ 
                     
                       
                         C 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           i 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           [ 
                           
                             
                               1 
                               + 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                             
                               1 
                               - 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         2 
                       
                       ⁢ 
                       
                         
                           ρ 
                           1 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 C 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 t 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                     - 
                     
                       
                         ρ 
                         2 
                       
                       ⁢ 
                       
                         
                           t 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             i 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   210 
                   ) 
                 
               
             
           
         
       
     
     By taking the ratio of (210) and (105), the normalized stiffness of target tissue will be given as 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             NC 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             e 
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               
                                 ρ 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   C 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 ρ 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   C 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       i 
                                     
                                     = 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                           
                           = 
                             
                           ⁢ 
                           
                             
                               1 
                               + 
                               
                                 
                                   
                                     b 
                                     2 
                                   
                                   
                                     c 
                                     2 
                                   
                                 
                                 ⁢ 
                                 e 
                               
                               + 
                               
                                 
                                   
                                     a 
                                     2 
                                   
                                   
                                     c 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   2 
                                 
                               
                             
                             ≈ 
                               
                             ⁢ 
                             
                               
                                 
                                   
                                     
                                       [ 
                                       
                                         
                                           1 
                                           + 
                                           
                                             R 
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 e 
                                                 i 
                                               
                                               ) 
                                             
                                           
                                         
                                         
                                           1 
                                           - 
                                           
                                             R 
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 e 
                                                 i 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                     2 
                                   
                                   ⁢ 
                                   
                                     
                                       ρ 
                                       1 
                                     
                                     ⁡ 
                                     
                                       [ 
                                       
                                         
                                           
                                             C 
                                             1 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               e 
                                               i 
                                             
                                             ) 
                                           
                                         
                                         + 
                                         
                                           
                                             t 
                                             1 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               e 
                                               i 
                                             
                                             ) 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     ρ 
                                     2 
                                   
                                   ⁢ 
                                   
                                     
                                       t 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         e 
                                         i 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     [ 
                                     
                                       
                                         1 
                                         + 
                                         
                                           R 
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 e 
                                                 i 
                                               
                                               = 
                                               0 
                                             
                                             ) 
                                           
                                         
                                       
                                       
                                         1 
                                         - 
                                         
                                           R 
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 e 
                                                 i 
                                               
                                               = 
                                               0 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   ρ 
                                   1 
                                 
                                 ⁢ 
                                 
                                   
                                     C 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         e 
                                         i 
                                       
                                       = 
                                       0 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               
                                 
                                   [ 
                                   
                                     
                                       1 
                                       + 
                                       
                                         R 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             e 
                                             i 
                                           
                                           ) 
                                         
                                       
                                     
                                     
                                       1 
                                       - 
                                       
                                         R 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             e 
                                             i 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 2 
                               
                               
                                 
                                   [ 
                                   
                                     
                                       1 
                                       + 
                                       
                                         R 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               e 
                                               i 
                                             
                                             = 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                     
                                       1 
                                       - 
                                       
                                         R 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               e 
                                               i 
                                             
                                             = 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 2 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     C 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     t 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 
                                   C 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       i 
                                     
                                     = 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                           
                           - 
                             
                           ⁢ 
                           
                             
                               
                                 ρ 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   t 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   [ 
                                   
                                     
                                       1 
                                       + 
                                       
                                         R 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               e 
                                               i 
                                             
                                             = 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                     
                                       1 
                                       - 
                                       
                                         R 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               e 
                                               i 
                                             
                                             = 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 2 
                               
                               ⁢ 
                               
                                 ρ 
                                 1 
                               
                               ⁢ 
                               
                                 
                                   C 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       i 
                                     
                                     = 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   211 
                   ) 
                 
               
             
           
         
       
     
     By substituting (209) into (211), the density ratio between unknown target tissue (medium  2 ) and unknown surrounding tissue (medium  1 ) is given as: 
     
       
         
           
             
               
                 
                   
                     ρ 
                     RATIO 
                   
                   = 
                   
                     
                       
                         ρ 
                         2 
                       
                       
                         ρ 
                         1 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               [ 
                               
                                 
                                   1 
                                   + 
                                   
                                     R 
                                     0 
                                   
                                 
                                 
                                   1 
                                   - 
                                   
                                     R 
                                     0 
                                   
                                 
                               
                               ] 
                             
                             2 
                           
                           ⁢ 
                           
                             
                               C 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   e 
                                   i 
                                 
                                 = 
                                 0 
                               
                               ) 
                             
                           
                         
                         
                           
                             t 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             e 
                             ) 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             6 
                             ⁢ 
                             
                               s 
                               1 
                             
                             ⁢ 
                             e 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   3 
                                   ⁢ 
                                   
                                     s 
                                     2 
                                   
                                 
                                 + 
                                 
                                   3 
                                   ⁢ 
                                   
                                     s 
                                     1 
                                   
                                   ⁢ 
                                   
                                     
                                       b 
                                       2 
                                     
                                     
                                       c 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               e 
                               2 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     s 
                                     2 
                                   
                                   ⁢ 
                                   
                                     
                                       b 
                                       2 
                                     
                                     
                                       c 
                                       2 
                                     
                                   
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                     s 
                                     1 
                                   
                                   ⁢ 
                                   
                                     
                                       a 
                                       2 
                                     
                                     
                                       c 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               e 
                               3 
                             
                           
                           + 
                           
                             
                               3 
                               2 
                             
                             ⁢ 
                             
                               s 
                               2 
                             
                             ⁢ 
                             
                               
                                 a 
                                 2 
                               
                               
                                 c 
                                 2 
                               
                             
                             ⁢ 
                             
                               e 
                               4 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   212 
                   ) 
                 
               
             
           
         
       
     
     Note four constants 
             (       s   1     ,       s   2     ⁢       b   2       c   2       ⁢           ⁢   and   ⁢           ⁢       a   2       c   2           )         
on the right hand side have been evaluated in the previous step, hence they are known properties at this stage.
 
     At first glance, in the relation (212), the density ratio may appear to be a function of applied strain e. However, by observing both relation (207) and (212), the density ratio can be shown to a constant as follows. 
     
       
         
           
             
               
                 
                   
                     ρ 
                     RATIO 
                   
                   = 
                   
                     
                       
                         ρ 
                         2 
                       
                       
                         ρ 
                         1 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     1 
                                     + 
                                     
                                       R 
                                       0 
                                     
                                   
                                   
                                     1 
                                     - 
                                     
                                       R 
                                       0 
                                     
                                   
                                 
                                 ] 
                               
                               2 
                             
                             ⁢ 
                             
                               
                                 C 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   = 
                                   0 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               t 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               e 
                               ) 
                             
                           
                         
                         ⁢ 
                         
                           
                             
                               t 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               e 
                               ) 
                             
                           
                           
                             c 
                             1 
                           
                         
                       
                       = 
                       
                         
                           
                             
                               [ 
                               
                                 
                                   1 
                                   + 
                                   
                                     R 
                                     0 
                                   
                                 
                                 
                                   1 
                                   - 
                                   
                                     R 
                                     0 
                                   
                                 
                               
                               ] 
                             
                             2 
                           
                           ⁢ 
                           
                             
                               C 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 = 
                                 0 
                               
                               ) 
                             
                           
                         
                         
                           
                             C 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               = 
                               0 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   213 
                   ) 
                 
               
             
           
         
       
     
     In the process, stress continuity t 2 (e)=t 1 (e) was introduced. 
     This relation can be also derived by re-organizing (105). Hence this indicates that the density ratio given by (213) should result in a constant value regardless of the state of deformation. Once the density ratio is evaluated, the density of target tissue can be simply given by
 
ρ 2 =ρ RATIO □ρ 1   (214)
 
     Once the density of target tissue are given, this can be brought back to equation (102) in SGI to evaluate the wave velocity in target tissue at any deformed state 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         1 
                         + 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               i 
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               i 
                             
                             ) 
                           
                         
                       
                     
                     ⁢ 
                     
                       □ 
                     
                     ⁢ 
                     
                       
                         ρ 
                         1 
                       
                       
                         ρ 
                         2 
                       
                     
                     ⁢ 
                     ⁢ 
                     
                       □ 
                       
                         
                           V 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             i 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   215 
                   ) 
                 
               
             
           
         
       
     
     From the evaluated wave velocity, the size (thickness) of target tissue at any deformed state can be also calculated by following relations.
 
 d ( e   i )= T ( e   i )□ V   2 ( e   i )/2  (216)
 
     In this relation, T (e i ) represents the round trip wave travel time through target medium. 
     This thickness can be used in the evaluation of wave attenuation coefficient. 
     Implementation 
     Step 1. Compress media with a higher level of known stress t 1 (e A ) for example using a force transducer in the ultrasonic probe.
 
 t   2 ( e   A )= t   1 ( e   A )
 
     This higher stress enhances the contrast between the target medium and the surrounding medium. Since the acoustic contrast is now enhanced, the location and shape of the target tissue can be identified with greater precision. Now, the reflection coefficient for this compressed tissue can be used in the following relation: 
     
       
         
           
             
               
                 c 
                 2 
               
               + 
               
                 
                   ( 
                   
                     
                       6 
                       ⁢ 
                       
                         s 
                         1 
                       
                     
                     + 
                     
                       b 
                       2 
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   A 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       3 
                       ⁢ 
                       
                         s 
                         2 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     + 
                     
                       3 
                       ⁢ 
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         b 
                         2 
                       
                     
                     + 
                     
                       a 
                       2 
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   A 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       2 
                       ⁢ 
                       
                         s 
                         2 
                       
                       ⁢ 
                       
                         b 
                         2 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         a 
                         2 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   A 
                   3 
                 
               
               + 
               
                 
                   3 
                   2 
                 
                 ⁢ 
                 
                   s 
                   2 
                 
                 ⁢ 
                 
                   a 
                   2 
                 
                 ⁢ 
                 
                   e 
                   A 
                   4 
                 
               
             
             = 
             
               
                 
                   [ 
                   
                     
                       1 
                       + 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             A 
                           
                           ) 
                         
                       
                     
                     
                       1 
                       - 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             A 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
                 2 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       C 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         A 
                       
                       ) 
                     
                   
                   + 
                   
                     
                       t 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         A 
                       
                       ) 
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Step 2. Repeat the same measurements for different, but compressive stresses (t 1 (e B ), t 1 (e C ) and t 1 (e D )) that are smaller than the stress used in the previous step. 
     
       
         
           
             
                 
             
             ⁢ 
             
               
                 
                   t 
                   1 
                 
                 ⁡ 
                 
                   ( 
                   
                     e 
                     B 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     t 
                     2 
                   
                   ⁡ 
                   
                     ( 
                     
                       e 
                       B 
                     
                     ) 
                   
                 
                 &lt; 
                 
                   
                     t 
                     2 
                   
                   ⁡ 
                   
                     ( 
                     
                       e 
                       A 
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 c 
                 2 
               
               + 
               
                 
                   ( 
                   
                     
                       6 
                       ⁢ 
                       
                         s 
                         1 
                       
                     
                     + 
                     
                       b 
                       2 
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   B 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       3 
                       ⁢ 
                       
                         s 
                         2 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     + 
                     
                       3 
                       ⁢ 
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         b 
                         2 
                       
                     
                     + 
                     
                       a 
                       2 
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   B 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       2 
                       ⁢ 
                       
                         s 
                         2 
                       
                       ⁢ 
                       
                         b 
                         2 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         a 
                         2 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   B 
                   3 
                 
               
               + 
               
                 
                   3 
                   2 
                 
                 ⁢ 
                 
                   s 
                   2 
                 
                 ⁢ 
                 
                   a 
                   2 
                 
                 ⁢ 
                 
                   e 
                   B 
                   4 
                 
               
             
             = 
             
               
                 
                   [ 
                   
                     
                       1 
                       + 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             B 
                           
                           ) 
                         
                       
                     
                     
                       1 
                       - 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             B 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
                 2 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       C 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         B 
                       
                       ) 
                     
                   
                   + 
                   
                     
                       t 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         B 
                       
                       ) 
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               
                 
                   t 
                   1 
                 
                 ⁡ 
                 
                   ( 
                   
                     e 
                     C 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     t 
                     2 
                   
                   ⁡ 
                   
                     ( 
                     
                       e 
                       C 
                     
                     ) 
                   
                 
                 &lt; 
                 
                   
                     t 
                     2 
                   
                   ⁡ 
                   
                     ( 
                     
                       e 
                       A 
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 c 
                 2 
               
               + 
               
                 
                   ( 
                   
                     
                       6 
                       ⁢ 
                       
                         s 
                         1 
                       
                     
                     + 
                     
                       b 
                       2 
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   C 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       3 
                       ⁢ 
                       
                         s 
                         2 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     + 
                     
                       3 
                       ⁢ 
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         b 
                         2 
                       
                     
                     + 
                     
                       a 
                       2 
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   C 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       2 
                       ⁢ 
                       
                         s 
                         2 
                       
                       ⁢ 
                       
                         b 
                         2 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         a 
                         2 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   C 
                   3 
                 
               
               + 
               
                 
                   3 
                   2 
                 
                 ⁢ 
                 
                   s 
                   2 
                 
                 ⁢ 
                 
                   a 
                   2 
                 
                 ⁢ 
                 
                   e 
                   C 
                   4 
                 
               
             
             = 
             
               
                 
                   [ 
                   
                     
                       1 
                       + 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             C 
                           
                           ) 
                         
                       
                     
                     
                       1 
                       - 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             C 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
                 2 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       C 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         C 
                       
                       ) 
                     
                   
                   + 
                   
                     
                       t 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         C 
                       
                       ) 
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               
                 
                   t 
                   1 
                 
                 ⁡ 
                 
                   ( 
                   
                     e 
                     D 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     t 
                     2 
                   
                   ⁡ 
                   
                     ( 
                     
                       e 
                       D 
                     
                     ) 
                   
                 
                 &lt; 
                 
                   
                     t 
                     2 
                   
                   ⁡ 
                   
                     ( 
                     
                       e 
                       A 
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 c 
                 2 
               
               + 
               
                 
                   ( 
                   
                     
                       6 
                       ⁢ 
                       
                         s 
                         1 
                       
                     
                     + 
                     
                       b 
                       2 
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   D 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       3 
                       ⁢ 
                       
                         s 
                         2 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     + 
                     
                       3 
                       ⁢ 
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         b 
                         2 
                       
                     
                     + 
                     
                       a 
                       2 
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   D 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       2 
                       ⁢ 
                       
                         s 
                         2 
                       
                       ⁢ 
                       
                         b 
                         2 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         a 
                         2 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   D 
                   3 
                 
               
               + 
               
                 
                   3 
                   2 
                 
                 ⁢ 
                 
                   s 
                   2 
                 
                 ⁢ 
                 
                   a 
                   2 
                 
                 ⁢ 
                 
                   e 
                   D 
                   4 
                 
               
             
             = 
             
               
                 
                   [ 
                   
                     
                       1 
                       + 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             D 
                           
                           ) 
                         
                       
                     
                     
                       1 
                       - 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             e 
                             D 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
                 2 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       C 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         D 
                       
                       ) 
                     
                   
                   + 
                   
                     
                       t 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         D 
                       
                       ) 
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Step 3. Measure the reflection information for the non-compressed state. In this state, the acoustic contrast between target medium and surrounding medium is typically lowest. Hence the boundary between the two media is less clear. However, by tracking the compression-enhanced boundary in the previous two steps, the boundary between the two media can be accurately estimated and the reflection coefficient can be computed. 
     
       
         
           
             
               
                 ρ 
                 2 
               
               ⁢ 
               
                 
                   C 
                   2 
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       e 
                       i 
                     
                     = 
                     0 
                   
                   ) 
                 
               
             
             = 
             
               
                 
                   [ 
                   
                     
                       1 
                       + 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             
                               e 
                               i 
                             
                             = 
                             0 
                           
                           ) 
                         
                       
                     
                     
                       1 
                       - 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             
                               e 
                               i 
                             
                             = 
                             0 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
                 2 
               
               ⁢ 
               
                 ρ 
                 1 
               
               ⁢ 
               
                 
                   C 
                   1 
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       e 
                       i 
                     
                     = 
                     0 
                   
                   ) 
                 
               
             
           
         
       
     
     Step 4. Evaluate applied strains e A , e B , e C  and e D  using the relation (113). 
     Step 5. With evaluated strains e A , e B , e C  and e D , unknown material properties 
             (       s   1     ,       s   2     ⁢       b   2       c   2       ⁢           ⁢   and   ⁢           ⁢       a   2       c   2           )         
can evaluated by solving the following two relations simultaneously. Again, the parameters on the right hand side of the relation are known or measured.
 
     
       
         
           
             
               1 
               + 
               
                 
                   ( 
                   
                     
                       6 
                       ⁢ 
                       
                         s 
                         1 
                       
                     
                     + 
                     
                       
                         b 
                         2 
                       
                       
                         c 
                         2 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   A 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       3 
                       ⁢ 
                       
                         s 
                         2 
                       
                     
                     + 
                     
                       3 
                       ⁢ 
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         
                           b 
                           2 
                         
                         
                           c 
                           2 
                         
                       
                     
                     + 
                     
                       
                         a 
                         2 
                       
                       
                         c 
                         2 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   A 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       2 
                       ⁢ 
                       
                         s 
                         2 
                       
                       ⁢ 
                       
                         
                           b 
                           2 
                         
                         
                           c 
                           2 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         
                           a 
                           2 
                         
                         
                           c 
                           2 
                         
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   A 
                   3 
                 
               
               + 
               
                 
                   3 
                   2 
                 
                 ⁢ 
                 
                   s 
                   2 
                 
                 ⁢ 
                 
                   
                     a 
                     2 
                   
                   
                     c 
                     2 
                   
                 
                 ⁢ 
                 
                   e 
                   A 
                   4 
                 
               
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         1 
                         + 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               A 
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               A 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   2 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         C 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           A 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         t 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           A 
                         
                         ) 
                       
                     
                   
                   ] 
                 
               
               
                 
                   
                     [ 
                     
                       
                         1 
                         + 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               = 
                               0 
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               = 
                               0 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   2 
                 
                 ⁢ 
                 
                   
                     C 
                     1 
                   
                   ⁡ 
                   
                     ( 
                     
                       e 
                       = 
                       0 
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               1 
               + 
               
                 
                   ( 
                   
                     
                       6 
                       ⁢ 
                       
                         s 
                         1 
                       
                     
                     + 
                     
                       
                         b 
                         2 
                       
                       
                         c 
                         2 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   B 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       3 
                       ⁢ 
                       
                         s 
                         2 
                       
                     
                     + 
                     
                       3 
                       ⁢ 
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         
                           b 
                           2 
                         
                         
                           c 
                           2 
                         
                       
                     
                     + 
                     
                       
                         a 
                         2 
                       
                       
                         c 
                         2 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   B 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       2 
                       ⁢ 
                       
                         s 
                         2 
                       
                       ⁢ 
                       
                         
                           b 
                           2 
                         
                         
                           c 
                           2 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         
                           a 
                           2 
                         
                         
                           c 
                           2 
                         
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   B 
                   3 
                 
               
               + 
               
                 
                   3 
                   2 
                 
                 ⁢ 
                 
                   s 
                   2 
                 
                 ⁢ 
                 
                   
                     a 
                     2 
                   
                   
                     c 
                     2 
                   
                 
                 ⁢ 
                 
                   e 
                   B 
                   4 
                 
               
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         1 
                         + 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               B 
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               B 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   2 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         C 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           B 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         t 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           B 
                         
                         ) 
                       
                     
                   
                   ] 
                 
               
               
                 
                   
                     [ 
                     
                       
                         1 
                         + 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               = 
                               0 
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               = 
                               0 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   2 
                 
                 ⁢ 
                 
                   
                     C 
                     1 
                   
                   ⁡ 
                   
                     ( 
                     
                       e 
                       = 
                       0 
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               1 
               + 
               
                 
                   ( 
                   
                     
                       6 
                       ⁢ 
                       
                         s 
                         1 
                       
                     
                     + 
                     
                       
                         b 
                         2 
                       
                       
                         c 
                         2 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   C 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       3 
                       ⁢ 
                       
                         s 
                         2 
                       
                     
                     + 
                     
                       3 
                       ⁢ 
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         
                           b 
                           2 
                         
                         
                           c 
                           2 
                         
                       
                     
                     + 
                     
                       
                         a 
                         2 
                       
                       
                         c 
                         2 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   C 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       2 
                       ⁢ 
                       
                         s 
                         2 
                       
                       ⁢ 
                       
                         
                           b 
                           2 
                         
                         
                           c 
                           2 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         
                           a 
                           2 
                         
                         
                           c 
                           2 
                         
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   C 
                   3 
                 
               
               + 
               
                 
                   3 
                   2 
                 
                 ⁢ 
                 
                   s 
                   2 
                 
                 ⁢ 
                 
                   
                     a 
                     2 
                   
                   
                     c 
                     2 
                   
                 
                 ⁢ 
                 
                   e 
                   C 
                   4 
                 
               
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         1 
                         + 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               C 
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               C 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   2 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         C 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           C 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         t 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           C 
                         
                         ) 
                       
                     
                   
                   ] 
                 
               
               
                 
                   
                     [ 
                     
                       
                         1 
                         + 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               = 
                               0 
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               = 
                               0 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   2 
                 
                 ⁢ 
                 
                   
                     C 
                     1 
                   
                   ⁡ 
                   
                     ( 
                     
                       e 
                       = 
                       0 
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               1 
               + 
               
                 
                   ( 
                   
                     
                       6 
                       ⁢ 
                       
                         s 
                         1 
                       
                     
                     + 
                     
                       
                         b 
                         2 
                       
                       
                         c 
                         2 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   D 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       3 
                       ⁢ 
                       
                         s 
                         2 
                       
                     
                     + 
                     
                       3 
                       ⁢ 
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         
                           b 
                           2 
                         
                         
                           c 
                           2 
                         
                       
                     
                     + 
                     
                       
                         a 
                         2 
                       
                       
                         c 
                         2 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   D 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       2 
                       ⁢ 
                       
                         s 
                         2 
                       
                       ⁢ 
                       
                         
                           b 
                           2 
                         
                         
                           c 
                           2 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         
                           a 
                           2 
                         
                         
                           c 
                           2 
                         
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   e 
                   D 
                   3 
                 
               
               + 
               
                 
                   3 
                   2 
                 
                 ⁢ 
                 
                   s 
                   2 
                 
                 ⁢ 
                 
                   
                     a 
                     2 
                   
                   
                     c 
                     2 
                   
                 
                 ⁢ 
                 
                   e 
                   D 
                   4 
                 
               
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         1 
                         + 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               D 
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               D 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   2 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         C 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           D 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         t 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           D 
                         
                         ) 
                       
                     
                   
                   ] 
                 
               
               
                 
                   
                     [ 
                     
                       
                         1 
                         + 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               = 
                               0 
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               = 
                               0 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   2 
                 
                 ⁢ 
                 
                   
                     C 
                     1 
                   
                   ⁡ 
                   
                     ( 
                     
                       e 
                       = 
                       0 
                     
                     ) 
                   
                 
               
             
           
         
       
     
     Step 6. Set up four equations, by substituting evaluated two coefficients (s 1 , s 2 ) for Poisson&#39;s ratio and any given strains say into eq. (207) and solve them simultaneously to retrieve three stiffness constants (a 2 , b 2  and c 2 ): 
     
       
         
           
             
               
                 
                   
                     
                       t 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         A 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       t 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         A 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     6 
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             s 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                           ⁢ 
                           
                             e 
                             A 
                           
                         
                         + 
                         
                           
                             1 
                             2 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   s 
                                   2 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   s 
                                   1 
                                 
                                 ⁢ 
                                 
                                   b 
                                   2 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             e 
                             A 
                           
                         
                         + 
                         
                           
                             1 
                             3 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   s 
                                   2 
                                 
                                 ⁢ 
                                 
                                   b 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   s 
                                   1 
                                 
                                 ⁢ 
                                 
                                   a 
                                   2 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             e 
                             A 
                             3 
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             s 
                             2 
                           
                           ⁢ 
                           
                             a 
                             2 
                           
                           ⁢ 
                           
                             e 
                             A 
                             4 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       t 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         A 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       t 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         A 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     6 
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             s 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                           ⁢ 
                           
                             e 
                             A 
                           
                         
                         + 
                         
                           
                             1 
                             2 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   s 
                                   2 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   s 
                                   1 
                                 
                                 ⁢ 
                                 
                                   b 
                                   2 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             e 
                             A 
                           
                         
                         + 
                         
                           
                             1 
                             3 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   s 
                                   2 
                                 
                                 ⁢ 
                                 
                                   b 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   s 
                                   1 
                                 
                                 ⁢ 
                                 
                                   a 
                                   2 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             e 
                             A 
                             3 
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             s 
                             2 
                           
                           ⁢ 
                           
                             a 
                             2 
                           
                           ⁢ 
                           
                             e 
                             A 
                             4 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       t 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         A 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       t 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         A 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     6 
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             s 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                           ⁢ 
                           
                             e 
                             A 
                           
                         
                         + 
                         
                           
                             1 
                             2 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   s 
                                   2 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   s 
                                   1 
                                 
                                 ⁢ 
                                 
                                   b 
                                   2 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             e 
                             A 
                           
                         
                         + 
                         
                           
                             1 
                             3 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   s 
                                   2 
                                 
                                 ⁢ 
                                 
                                   b 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   s 
                                   1 
                                 
                                 ⁢ 
                                 
                                   a 
                                   2 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             e 
                             A 
                             3 
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             s 
                             2 
                           
                           ⁢ 
                           
                             a 
                             2 
                           
                           ⁢ 
                           
                             e 
                             A 
                             4 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       t 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         A 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       t 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         A 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     6 
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             s 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                           ⁢ 
                           
                             e 
                             A 
                           
                         
                         + 
                         
                           
                             1 
                             2 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   s 
                                   2 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   s 
                                   1 
                                 
                                 ⁢ 
                                 
                                   b 
                                   2 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             e 
                             A 
                           
                         
                         + 
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Step 7 Rebuild stiffness, Poisson&#39;s ratio and tangential modulus by: 
     
       
         
           
             
               
                 
                   C 
                   2 
                 
                 ⁡ 
                 
                   ( 
                   e 
                   ) 
                 
               
               = 
               
                 
                   
                     a 
                     2 
                   
                   · 
                   
                     e 
                     2 
                   
                 
                 + 
                 
                   
                     b 
                     2 
                   
                   · 
                   e 
                 
                 + 
                 
                   c 
                   2 
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 v 
                 ⁡ 
                 
                   ( 
                   e 
                   ) 
                 
               
               = 
               
                 
                   1 
                   2 
                 
                 - 
                 
                   
                     ( 
                     
                       
                         s 
                         1 
                       
                       + 
                       
                         
                           s 
                           2 
                         
                         · 
                         e 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   and 
                 
               
             
           
         
       
       
         
           
             
               
                 E 
                 ⁡ 
                 
                   ( 
                   e 
                   ) 
                 
               
               ≈ 
               
                 6 
                 ⁢ 
                 
                   
                     φ 
                     ⁡ 
                     
                       ( 
                       e 
                       ) 
                     
                   
                   · 
                   
                     C 
                     ⁡ 
                     
                       ( 
                       e 
                       ) 
                     
                   
                 
               
             
             = 
             
               6 
               ⁢ 
               
                 
                   ( 
                   
                     
                       s 
                       1 
                     
                     + 
                     
                       
                         s 
                         2 
                       
                       · 
                       e 
                     
                   
                   ) 
                 
                 · 
                 
                   
                     ( 
                     
                       
                         
                           a 
                           2 
                         
                         · 
                         
                           e 
                           2 
                         
                       
                       + 
                       
                         
                           b 
                           2 
                         
                         · 
                         e 
                       
                       + 
                       
                         c 
                         2 
                       
                     
                     ) 
                   
                   . 
                 
               
             
           
         
       
     
     Step 8. Evaluate material density by: 
     
       
         
           
             
               ρ 
               2 
             
             = 
             
               
                 ρ 
                 1 
               
               ⁢ 
               
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               1 
                               + 
                               
                                 R 
                                 0 
                               
                             
                             
                               1 
                               - 
                               
                                 R 
                                 0 
                               
                             
                           
                           ] 
                         
                         2 
                       
                       ⁢ 
                       
                         
                           C 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               e 
                               i 
                             
                             = 
                             0 
                           
                           ) 
                         
                       
                     
                     
                       
                         t 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         e 
                         ) 
                       
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         6 
                         ⁢ 
                         
                           s 
                           1 
                         
                         ⁢ 
                         e 
                       
                       + 
                       
                         
                           ( 
                           
                             
                               3 
                               ⁢ 
                               
                                 s 
                                 2 
                               
                             
                             + 
                             
                               3 
                               ⁢ 
                               
                                 s 
                                 1 
                               
                               ⁢ 
                               
                                 
                                   b 
                                   2 
                                 
                                 
                                   c 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           e 
                           2 
                         
                       
                       + 
                       
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               
                                 s 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   b 
                                   2 
                                 
                                 
                                   c 
                                   2 
                                 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 s 
                                 1 
                               
                               ⁢ 
                               
                                 
                                   a 
                                   2 
                                 
                                 
                                   c 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           e 
                           3 
                         
                       
                       + 
                       
                         
                           3 
                           2 
                         
                         ⁢ 
                         
                           s 
                           2 
                         
                         ⁢ 
                         
                           
                             a 
                             2 
                           
                           
                             c 
                             2 
                           
                         
                         ⁢ 
                         
                           e 
                           4 
                         
                       
                     
                     ] 
                   
                 
                 . 
               
             
           
         
       
     
     The discussion of ASG can be also applied to this technique. SGI can also use a stiffness function with any order and the form of function will not affect the basic concept of this technique. 
     Evaluation of Wave Attenuation in a Deformed Medium 
     Referring now to  FIG. 9 , considering the passage of an input wave  100  to a target tissue (e.g. tendon  45  in  FIG. 2  or tumor  18  in  FIG. 1 ) and the resulting reflected waves. For simplicity in describing the method, only  FIG. 2  application will be referenced in the remainder of this section with both the tumor and tendon referred to as the target  45  and the patient and coupling medium will be referred to as the medium  44 . The input wave  100  with magnitude M I  is attenuated to be M I e −α     1     d     1    when it arrives at the bottom of surrounding medium  44  (as depicted in  FIG. 9 ) which has known attenuation coefficient α 1 =α 1 (e) and thickness d 1 =d 1 (e). 
     This wave is partially reflected at the near interface between surrounding medium  44  and the target  45  and changes its magnitude to M I e α     1     d     1   R 12  due to strain dependent reflection coefficient R 12 =R 12  (e). The reflected wave travels though the thickness d 1 =d 1 (e) back to the surface. Therefore the wave magnitude is further attenuated to become RW 1 =M I e −2α     1     d     1   R 12  and shows up as the first reflected wave  102  on the recorded data. 
     Now consider the passage of the input wave to generate the second reflected echo  104  in  FIG. 9 . A portion of the wave reaching the bottom of surrounding medium  44  (or 14) is transmitted into target  45 . At this point, the magnitude of the input wave is M I e −α     1     d     1   T 12  at the top of target  45 , where T 12 =T 12  (e) is defined as the transmission coefficient and indicates how much of the ultrasound signal passes from medium  44  into the target  45 . This wave propagates to the bottom medium  44  on the lower side of the target  45  and at this interface reflects back up to the top of target  45 . At the top of target  45 , the wave magnitude is now attenuated to M I e −α     1     d     1   e −2α     2     d     2   T 12 R 21  through the attenuation property of the target  45  and the reflection coefficient at the bottom interface between the target  45  and surrounding medium  44 . Since the wave in now propagating from the target toward the surrounding medium, a strain dependent reflection coefficient R 21 =R 21 (e) is applied. Once again, the wave transmits back into the upper medium  44  and travels back to transducer  12  and shows up as the second reflected echo  104  in the recorded wave signal. Due to the reflection, transmission and attenuation, the second echo  104  has a magnitude of RW 2 =M I e −2α     1     d     1   e −2α     2     d     2   T 12 R 21 T 21 . Again, the wave in now propagating from target toward the surrounding medium a strain dependent transmission coefficient T 21 =T 21 (e) is applied. 
     Since the magnitude of the first reflected echo and second reflected echo are expressed as combination of reflection coefficients and transmission coefficients, the relation between the reflection coefficient and transmission coefficient must be known. The following section defines how these basic relations are formulated in this method of analysis. 
     When a wave is propagating from medium  44  into the target  45 , the reflection coefficient R 12  (e i ) at the interface of two media is given by 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             R 
                             12 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               i 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               
                                 ρ 
                                 2 
                               
                               · 
                               
                                 
                                   V 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 ρ 
                                 1 
                               
                               · 
                               
                                 
                                   V 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 ρ 
                                 2 
                               
                               · 
                               
                                 
                                   V 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 ρ 
                                 1 
                               
                               · 
                               
                                 
                                   V 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             RW 
                             1 
                           
                           
                             IWe 
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               
                                 α 
                                 1 
                               
                               ⁢ 
                               
                                 d 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     300 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
     In this relation, wave propagation velocity in the medium  44  and target  45  are represented by V 1  (e i ) and V 2  (e i ). Material density of both media are indicated by ρ 1  and ρ 2 . 
     IW and RW 1  represent the magnitude of the input wave and the first reflected echo measured from recorded wave signal. Finally, the wave attenuation caused by medium  44  is expressed as e −2α     1     d     1   . If wave attenuation in surrounding medium  44  can be ignored, the reflection coefficient at the near interface simplifies to 
     
       
         
           
             
               
                 R 
                 12 
               
               ⁡ 
               
                 ( 
                 
                   e 
                   i 
                 
                 ) 
               
             
             = 
             
               
                 
                   RW 
                   1 
                 
                 IW 
               
               . 
             
           
         
       
     
     The transmission coefficient T 12  (e i ) at the same interface is given by 
     
       
         
           
             
               
                 
                   
                     
                       T 
                       12 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       2 
                       · 
                       
                         ρ 
                         2 
                       
                       · 
                       
                         
                           V 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             i 
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           ρ 
                           2 
                         
                         · 
                         
                           
                             V 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               i 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           ρ 
                           1 
                         
                         · 
                         
                           
                             V 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               i 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     300 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     In the reverse case (wave propagates from the target  45  into the medium  44 ), the reflection coefficient R 21  (e i ) and transmission coefficient T 21  (e i ) are given by 
     
       
         
           
             
               
                 
                   
                     
                       
                         R 
                         21 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           i 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             ρ 
                             1 
                           
                           · 
                           
                             
                               V 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                         
                         - 
                         
                           
                             ρ 
                             2 
                           
                           · 
                           
                             
                               V 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             ρ 
                             2 
                           
                           · 
                           
                             
                               V 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             ρ 
                             1 
                           
                           · 
                           
                             
                               V 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   
                     300 
                     ⁢ 
                     c 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       T 
                       21 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       2 
                       · 
                       
                         ρ 
                         2 
                       
                       · 
                       
                         
                           V 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             i 
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           ρ 
                           2 
                         
                         · 
                         
                           
                             V 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               i 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           ρ 
                           1 
                         
                         · 
                         
                           
                             V 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               i 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     300 
                     ⁢ 
                     d 
                   
                   ) 
                 
               
             
           
         
       
     
     By manipulating these relations, R 21  (e i ), T 12  (e i ) and T 21  (e i ) can be all expressed as the function of R 12  (e i ) in following manner.
 
 R   21 ( e   i )=− R   12 ( e   i ),  T   12 ( e   i )=1 +R   12 ( e   i ) and  T   21 ( e   i )=1 −R   12 ( e   i )  (300e)
 
     These relations are then utilized as follows. 
     Evaluation of Wave Attenuation by Comparing First and Second Echo 
     The straight forward way of evaluating wave attenuation in the target  45  can be achieved by taking the ratio of the magnitudes of first reflected echo and second reflected by 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               RW 
                               1 
                             
                             
                               RW 
                               2 
                             
                           
                           = 
                           
                             
                               
                                 M 
                                 I 
                               
                               ⁢ 
                               
                                 ⅇ 
                                 
                                   
                                     - 
                                     2 
                                   
                                   ⁢ 
                                   
                                     α 
                                     1 
                                   
                                   ⁢ 
                                   
                                     d 
                                     1 
                                   
                                 
                               
                               ⁢ 
                               
                                 R 
                                 12 
                               
                             
                             
                               
                                 M 
                                 I 
                               
                               ⁢ 
                               
                                 ⅇ 
                                 
                                   
                                     - 
                                     2 
                                   
                                   ⁢ 
                                   
                                     α 
                                     1 
                                   
                                   ⁢ 
                                   
                                     d 
                                     1 
                                   
                                 
                               
                               ⁢ 
                               
                                 ⅇ 
                                 
                                   
                                     - 
                                     2 
                                   
                                   ⁢ 
                                   
                                     α 
                                     2 
                                   
                                   ⁢ 
                                   
                                     d 
                                     2 
                                   
                                 
                               
                               ⁢ 
                               
                                 T 
                                 12 
                               
                               ⁢ 
                               
                                 R 
                                 21 
                               
                               ⁢ 
                               
                                 T 
                                 21 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               R 
                               12 
                             
                             
                               
                                 
                                   ⅇ 
                                   
                                     
                                       - 
                                       2 
                                     
                                     ⁢ 
                                     
                                       α 
                                       2 
                                     
                                     ⁢ 
                                     
                                       d 
                                       2 
                                     
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       R 
                                       12 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   - 
                                   
                                     R 
                                     12 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     R 
                                     12 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               - 
                               1 
                             
                             
                               
                                 ⅇ 
                                 
                                   
                                     - 
                                     2 
                                   
                                   ⁢ 
                                   
                                     α 
                                     2 
                                   
                                   ⁢ 
                                   
                                     d 
                                     2 
                                   
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     R 
                                     12 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   or 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           
                              
                             
                               
                                 RW 
                                 1 
                               
                               
                                 RW 
                                 2 
                               
                             
                              
                           
                           = 
                           
                              
                             
                               
                                 - 
                                 1 
                               
                               
                                 
                                   ⅇ 
                                   
                                     
                                       - 
                                       2 
                                     
                                     ⁢ 
                                     
                                       α 
                                       2 
                                     
                                     ⁢ 
                                     
                                       d 
                                       2 
                                     
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       R 
                                       12 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                              
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             1 
                             
                               
                                 ⅇ 
                                 
                                   
                                     - 
                                     2 
                                   
                                   ⁢ 
                                   
                                     α 
                                     2 
                                   
                                   ⁢ 
                                   
                                     d 
                                     2 
                                   
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     R 
                                     12 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       due 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       to 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             - 
                             2 
                           
                           ⁢ 
                           
                             α 
                             2 
                           
                           ⁢ 
                           
                             d 
                             2 
                           
                         
                       
                     
                     &gt; 
                     
                       0 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             R 
                             12 
                             2 
                           
                         
                         ) 
                       
                     
                     &gt; 
                     0. 
                   
                 
               
               
                 
                   ( 
                   301 
                   ) 
                 
               
             
           
         
       
     
     In this derivation, relations (300e) were used. 
     From this ratio, the wave attenuation in the target  45  at any deformed configuration is 
     
       
         
           
             
               
                 
                   
                     
                       α 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       e 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         
                           - 
                           2 
                         
                         ⁢ 
                         
                           
                             d 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             e 
                             ) 
                           
                         
                       
                     
                     ⁢ 
                     
                       log 
                       ⁡ 
                       
                         [ 
                         
                           
                              
                             
                               
                                 
                                   RW 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   e 
                                   ) 
                                 
                               
                               
                                 
                                   RW 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   e 
                                   ) 
                                 
                               
                             
                              
                           
                           ⁢ 
                           
                             ( 
                             
                               1 
                               - 
                               
                                 
                                   R 
                                   12 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   e 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   302 
                   ) 
                 
               
             
           
         
       
     
     In this relation, if the reflection is evaluated to be very small, this relation may be further simplified to 
     
       
         
           
             
               
                 
                   
                     
                       α 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       e 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         
                           - 
                           2 
                         
                         ⁢ 
                         
                           
                             d 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             e 
                             ) 
                           
                         
                       
                     
                     ⁢ 
                     
                       log 
                       ⁡ 
                       
                         [ 
                         
                            
                           
                             
                               
                                 RW 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 e 
                                 ) 
                               
                             
                             
                               
                                 RW 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 e 
                                 ) 
                               
                             
                           
                            
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   303 
                   ) 
                 
               
             
           
         
       
     
     If attenuation coefficient can be assumed to be a function of applied strain with a form of
 
α 2 ( e )=γ 1   ·e   2 +γ 2   ·e+γ   3   (304)
 
     then eq. (302) becomes: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             α 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             e 
                             ) 
                           
                         
                         = 
                         
                           
                             
                               γ 
                               1 
                             
                             · 
                             
                               e 
                               2 
                             
                           
                           + 
                           
                             
                               γ 
                               2 
                             
                             · 
                             e 
                           
                           + 
                           
                             γ 
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             1 
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   e 
                                   ) 
                                 
                               
                             
                           
                           ⁢ 
                           
                             log 
                             ⁡ 
                             
                               [ 
                               
                                 
                                    
                                   
                                     
                                       
                                         RW 
                                         1 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         e 
                                         ) 
                                       
                                     
                                     
                                       
                                         RW 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         e 
                                         ) 
                                       
                                     
                                   
                                    
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       
                                         R 
                                         12 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         e 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   305 
                   ) 
                 
               
             
           
         
       
     
     In relation (304), γ 1 , γ 2  and γ 3  represents the unknown attenuation constants to be determined from recorded wave signals. 
     Evaluation of Attenuation by Comparing Second Echo Measured at Different Strain e i    
     The attenuation coefficient of target  45  can also be derived by comparing the second reflected echo measured at different strain levels. Say, second reflected echo is measured at strain level e i  and e j .
 
 RW   2 ( e   i )= M   I   e   −2α     1     (e     i     )d     1     (e     i     )   e   −2α     2     (e     i     )d     2     (e     i     ) (− R   12 ( e   i )+ R   12   3 ( e   i ))  (306)
 
 RW   2 ( e   j )= M   I   e   −2α     1     (e     j     )d     1     (e     j     ) e −2α     2     (e     j     )d     2     (e     j     ) (− R   12 ( e   j )+ R   12   3 ( e   j ))  (307)
 
     The ratio of these two echoes are given by, 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           RW 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             i 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ⅇ 
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 α 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   j 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 d 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   j 
                                 
                                 ) 
                               
                             
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               - 
                               
                                 
                                   R 
                                   12 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     j 
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 R 
                                 12 
                                 3 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   j 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           RW 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             j 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ⅇ 
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 α 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 d 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               - 
                               
                                 
                                   R 
                                   12 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 R 
                                 12 
                                 3 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         ⅇ 
                         
                           
                             - 
                             2 
                           
                           ⁢ 
                           
                             
                               α 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               d 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                         
                       
                       
                         ⅇ 
                         
                           
                             - 
                             2 
                           
                           ⁢ 
                           
                             
                               α 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 j 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               d 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 j 
                               
                               ) 
                             
                           
                         
                       
                     
                     = 
                     
                       ⅇ 
                       
                         
                           
                             - 
                             2 
                           
                           ⁢ 
                           
                             
                               α 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               d 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                             
                               α 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 j 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               d 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 j 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   308 
                   ) 
                 
               
             
           
         
       
     
     If the attenuation and the thickness of the target medium can be considered small, then this relation can be further simplified to be 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           RW 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             i 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ⅇ 
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 α 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   j 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 d 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   j 
                                 
                                 ) 
                               
                             
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               - 
                               
                                 
                                   R 
                                   12 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     j 
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 R 
                                 12 
                                 3 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   j 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           RW 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             j 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ⅇ 
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 α 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 d 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               - 
                               
                                 
                                   R 
                                   12 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 R 
                                 12 
                                 3 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   ≈ 
                   
                     1 
                     - 
                     
                       2 
                       ⁢ 
                       
                         
                           α 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             i 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         d 
                         2 
                       
                       ⁢ 
                       
                         ( 
                         
                           e 
                           i 
                         
                         ) 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           α 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             j 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           d 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             j 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   309 
                   ) 
                 
               
             
           
         
       
     
     Applying a logarithmic operation to eq. (308) yields 
     
       
         
           
             
               
                 
                   
                     
                       
                         - 
                         2 
                       
                       ⁢ 
                       
                         
                           α 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             i 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           d 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             i 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           α 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             j 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           d 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             j 
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                       
                   
                   ⁢ 
                   
                     log 
                     [ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           
                             RW 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               i 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             ⅇ 
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   α 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     j 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   d 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     j 
                                   
                                   ) 
                                 
                               
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 - 
                                 
                                   
                                     R 
                                     12 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       j 
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 
                                   R 
                                   12 
                                   3 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     j 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             RW 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               j 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             ⅇ 
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   α 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   d 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 - 
                                 
                                   
                                     R 
                                     12 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 
                                   R 
                                   12 
                                   3 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   310 
                   ) 
                 
               
             
           
         
       
     
     Substitution of eq. (304) into (310) results in the final relation to evaluate wave attenuation for this case, that is 
     
       
         
           
             
               
                 
                   
                     
                       
                         γ 
                         1 
                       
                       · 
                       
                         [ 
                         
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 e 
                                 i 
                                 2 
                               
                             
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     j 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 e 
                                 j 
                                 2 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         γ 
                         2 
                       
                       · 
                       
                         [ 
                         
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 e 
                                 i 
                               
                             
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     j 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 e 
                                 j 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         γ 
                         3 
                       
                       · 
                       
                         [ 
                         
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 d 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               
                                 d 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   j 
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     log 
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               RW 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ⅇ 
                               
                                 
                                   - 
                                   2 
                                 
                                 ⁢ 
                                 
                                   
                                     α 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       j 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     d 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       j 
                                     
                                     ) 
                                   
                                 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   - 
                                   
                                     
                                       R 
                                       12 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         e 
                                         j 
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     R 
                                     12 
                                     3 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       j 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               RW 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 j 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ⅇ 
                               
                                 
                                   - 
                                   2 
                                 
                                 ⁢ 
                                 
                                   
                                     α 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     d 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   - 
                                   
                                     
                                       R 
                                       12 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         e 
                                         i 
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     R 
                                     12 
                                     3 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   309 
                   ) 
                 
               
             
           
         
       
     
     In this relation, the parameters on the right hand side either can be measured or can be evaluated from the steps described in previous sections. 
     If the magnitude of reflection coefficient R 12  (e j ) is very small and the higher term can be ignored, eq. (309) can be further simplified to: 
     
       
         
           
             
               
                 
                   
                     
                       
                         γ 
                         1 
                       
                       · 
                       
                         [ 
                         
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 e 
                                 i 
                                 2 
                               
                             
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     j 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 e 
                                 j 
                                 2 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         γ 
                         2 
                       
                       · 
                       
                         [ 
                         
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 e 
                                 i 
                               
                             
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     j 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 e 
                                 j 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         γ 
                         3 
                       
                       · 
                       
                         [ 
                         
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 d 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               
                                 d 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   j 
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   ≈ 
                   
                     log 
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               RW 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ⅇ 
                               
                                 
                                   - 
                                   2 
                                 
                                 ⁢ 
                                 
                                   
                                     α 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       j 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     d 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       j 
                                     
                                     ) 
                                   
                                 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 - 
                                 
                                   
                                     R 
                                     12 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       j 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               RW 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 j 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ⅇ 
                               
                                 
                                   - 
                                   2 
                                 
                                 ⁢ 
                                 
                                   
                                     α 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     d 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 - 
                                 
                                   
                                     R 
                                     12 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   310 
                   ) 
                 
               
             
           
         
       
     
     If the parameters α 1 (e j )d 1 (e j ) produce a relatively small amount of attenuation, then relation (309) can be further simplified to be 
     
       
         
           
             
               
                 
                   
                     1 
                     + 
                     
                       
                         γ 
                         1 
                       
                       · 
                       
                         [ 
                         
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 e 
                                 i 
                                 2 
                               
                             
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     j 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 e 
                                 j 
                                 2 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         γ 
                         2 
                       
                       · 
                       
                         [ 
                         
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 e 
                                 i 
                               
                             
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     j 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 e 
                                 j 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         γ 
                         3 
                       
                       · 
                       
                         [ 
                         
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 d 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               
                                 d 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   j 
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           RW 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             i 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ⅇ 
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 α 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   j 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 d 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   j 
                                 
                                 ) 
                               
                             
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             - 
                             
                               
                                 R 
                                 12 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   j 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           RW 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             j 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ⅇ 
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 α 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 d 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             - 
                             
                               
                                 R 
                                 12 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   311 
                   ) 
                 
               
             
           
         
       
     
     When the influence of the wave attenuation in surrounding medium  44  can be neglected (e −2α     1     (e)d     1     (e) ≈1), relations (310) and (311) can be further simplified to be: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           γ 
                           1 
                         
                         · 
                         
                           [ 
                           
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   
                                     d 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                                 · 
                                 
                                   e 
                                   i 
                                   2 
                                 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 
                                   
                                     d 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       j 
                                     
                                     ) 
                                   
                                 
                                 · 
                                 
                                   e 
                                   j 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         
                           γ 
                           2 
                         
                         · 
                         
                           [ 
                           
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   
                                     d 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                                 · 
                                 
                                   e 
                                   i 
                                 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 
                                   
                                     d 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       j 
                                     
                                     ) 
                                   
                                 
                                 · 
                                 
                                   e 
                                   j 
                                 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         
                           γ 
                           3 
                         
                         · 
                         
                           [ 
                           
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     j 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                     ≈ 
                     
                       log 
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 RW 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 - 
                                 
                                   
                                     R 
                                     12 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       j 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             
                               
                                 RW 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   j 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 - 
                                 
                                   
                                     R 
                                     12 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       e 
                                       i 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   312 
                   ) 
                 
               
             
             
               
                 
                   
                     1 
                     + 
                     
                       
                         γ 
                         1 
                       
                       · 
                       
                         [ 
                         
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 e 
                                 i 
                                 2 
                               
                             
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     j 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 e 
                                 j 
                                 2 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         γ 
                         2 
                       
                       · 
                       
                         [ 
                         
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     i 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 e 
                                 i 
                               
                             
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     e 
                                     j 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 e 
                                 j 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         γ 
                         3 
                       
                       · 
                       
                         [ 
                         
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               
                                 d 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   i 
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               
                                 d 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   j 
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           RW 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             i 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           - 
                           
                             
                               R 
                               12 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 j 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     
                       
                         
                           RW 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             j 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           - 
                           
                             
                               R 
                               12 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 i 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   313 
                   ) 
                 
               
             
           
         
       
     
     The thickness of target  45  is treated as known parameter through this derivation. The target  44  thickness for ASG case can be given by eq. (14) based on a known target medium density and eq. (217) for SGI based on the condition of stress continuity. 
     (Implementation) 
     Step 1. Measure wave signals at multiple different unknown strain levels and evaluate reflection coefficients R 12  (e) and target medium thicknesses d 2  (e) at each strain level. 
     For attenuation evaluations based on eq. (305), three separate measurements are required. 
     Yet four sets of measurements are required for evaluation based on eq. (309). 
     In ASG, the target medium thickness should be evaluated by eq. (14). The thickness of target medium in SGI should be evaluated by eq. (217). 
     Step 2. Set up system of equations for attenuation constants (γ 1 , γ 2  and γ 3 ) evaluation. 
     For technique based on eq. (305), following three equations are required: 
     
       
         
           
             
               
                 
                   γ 
                   1 
                 
                 · 
                 
                   e 
                   A 
                   2 
                 
               
               + 
               
                 
                   γ 
                   2 
                 
                 · 
                 
                   e 
                   A 
                 
               
               + 
               
                 γ 
                 3 
               
             
             = 
             
               
                 1 
                 
                   
                     - 
                     2 
                   
                   ⁢ 
                   
                     
                       d 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         A 
                       
                       ) 
                     
                   
                 
               
               ⁢ 
               
                 log 
                 ⁡ 
                 
                   [ 
                   
                     
                        
                       
                         
                           
                             RW 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               A 
                             
                             ) 
                           
                         
                         
                           
                             RW 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               A 
                             
                             ) 
                           
                         
                       
                        
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             R 
                             12 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               A 
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   γ 
                   1 
                 
                 · 
                 
                   e 
                   B 
                   2 
                 
               
               + 
               
                 
                   γ 
                   2 
                 
                 · 
                 
                   e 
                   B 
                 
               
               + 
               
                 γ 
                 3 
               
             
             = 
             
               
                 1 
                 
                   
                     - 
                     2 
                   
                   ⁢ 
                   
                     
                       d 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         B 
                       
                       ) 
                     
                   
                 
               
               ⁢ 
               
                 log 
                 ⁡ 
                 
                   [ 
                   
                     
                        
                       
                         
                           
                             RW 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               B 
                             
                             ) 
                           
                         
                         
                           
                             RW 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               B 
                             
                             ) 
                           
                         
                       
                        
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             R 
                             12 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               B 
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   γ 
                   1 
                 
                 · 
                 
                   e 
                   C 
                   2 
                 
               
               + 
               
                 
                   γ 
                   2 
                 
                 · 
                 
                   e 
                   C 
                 
               
               + 
               
                 γ 
                 3 
               
             
             = 
             
               
                 1 
                 
                   
                     - 
                     2 
                   
                   ⁢ 
                   
                     
                       d 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         e 
                         C 
                       
                       ) 
                     
                   
                 
               
               ⁢ 
               
                 log 
                 ⁡ 
                 
                   [ 
                   
                     
                        
                       
                         
                           
                             RW 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               C 
                             
                             ) 
                           
                         
                         
                           
                             RW 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               C 
                             
                             ) 
                           
                         
                       
                        
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             R 
                             12 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               C 
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                   ] 
                 
               
             
           
         
       
     
     For technique based on eq. (309), following three equations are utilized: 
     
       
         
           
             
               
                 
                   γ 
                   1 
                 
                 · 
                 
                   [ 
                   
                     
                       
                         - 
                         2 
                       
                       ⁢ 
                       
                         
                           
                             d 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               A 
                             
                             ) 
                           
                         
                         · 
                         
                           e 
                           A 
                           2 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           
                             d 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               B 
                             
                             ) 
                           
                         
                         · 
                         
                           e 
                           B 
                           2 
                         
                       
                     
                   
                   ] 
                 
               
               + 
               
                 
                   γ 
                   2 
                 
                 · 
                 
                   [ 
                   
                     
                       
                         - 
                         2 
                       
                       ⁢ 
                       
                         
                           
                             d 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               A 
                             
                             ) 
                           
                         
                         · 
                         
                           e 
                           A 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           
                             d 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               B 
                             
                             ) 
                           
                         
                         · 
                         
                           e 
                           B 
                         
                       
                     
                   
                   ] 
                 
               
               + 
               
                 
                   γ 
                   3 
                 
                 · 
                 
                   [ 
                   
                     
                       
                         - 
                         2 
                       
                       ⁢ 
                       
                         
                           d 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             A 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           d 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             B 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
               
             
             = 
             
               log 
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         RW 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           A 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         ⅇ 
                         
                           
                             - 
                             2 
                           
                           ⁢ 
                           
                             
                               α 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 B 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               d 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 B 
                               
                               ) 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             - 
                             
                               
                                 R 
                                 12 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   B 
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               R 
                               12 
                               3 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 B 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       
                         RW 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           B 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         ⅇ 
                         
                           
                             - 
                             2 
                           
                           ⁢ 
                           
                             
                               α 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 A 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               d 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 A 
                               
                               ) 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             - 
                             
                               
                                 R 
                                 12 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   A 
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               R 
                               12 
                               3 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 A 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
               
                 
                   γ 
                   1 
                 
                 · 
                 
                   [ 
                   
                     
                       
                         - 
                         2 
                       
                       ⁢ 
                       
                         
                           
                             d 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               A 
                             
                             ) 
                           
                         
                         · 
                         
                           e 
                           A 
                           2 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           
                             d 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               C 
                             
                             ) 
                           
                         
                         · 
                         
                           e 
                           C 
                           2 
                         
                       
                     
                   
                   ] 
                 
               
               + 
               
                 
                   γ 
                   2 
                 
                 · 
                 
                   [ 
                   
                     
                       
                         - 
                         2 
                       
                       ⁢ 
                       
                         
                           
                             d 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               A 
                             
                             ) 
                           
                         
                         · 
                         
                           e 
                           A 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           
                             d 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               C 
                             
                             ) 
                           
                         
                         · 
                         
                           e 
                           C 
                         
                       
                     
                   
                   ] 
                 
               
               + 
               
                 
                   γ 
                   3 
                 
                 · 
                 
                   [ 
                   
                     
                       
                         - 
                         2 
                       
                       ⁢ 
                       
                         
                           d 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             A 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           d 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             C 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
               
             
             = 
             
               log 
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         RW 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           A 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         ⅇ 
                         
                           
                             - 
                             2 
                           
                           ⁢ 
                           
                             
                               α 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 C 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               d 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 C 
                               
                               ) 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             - 
                             
                               
                                 R 
                                 12 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   C 
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               R 
                               12 
                               3 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 C 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       
                         RW 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           C 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         ⅇ 
                         
                           
                             - 
                             2 
                           
                           ⁢ 
                           
                             
                               α 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 A 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               d 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 A 
                               
                               ) 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             - 
                             
                               
                                 R 
                                 12 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   A 
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               R 
                               12 
                               3 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 A 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
               
                 
                   γ 
                   1 
                 
                 · 
                 
                   [ 
                   
                     
                       
                         - 
                         2 
                       
                       ⁢ 
                       
                         
                           
                             d 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               A 
                             
                             ) 
                           
                         
                         · 
                         
                           e 
                           A 
                           2 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           
                             d 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               D 
                             
                             ) 
                           
                         
                         · 
                         
                           e 
                           D 
                           2 
                         
                       
                     
                   
                   ] 
                 
               
               + 
               
                 
                   γ 
                   2 
                 
                 · 
                 
                   [ 
                   
                     
                       
                         - 
                         2 
                       
                       ⁢ 
                       
                         
                           
                             d 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               A 
                             
                             ) 
                           
                         
                         · 
                         
                           e 
                           A 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           
                             d 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               e 
                               D 
                             
                             ) 
                           
                         
                         · 
                         
                           e 
                           D 
                         
                       
                     
                   
                   ] 
                 
               
               + 
               
                 
                   γ 
                   3 
                 
                 · 
                 
                   [ 
                   
                     
                       
                         - 
                         2 
                       
                       ⁢ 
                       
                         
                           d 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             A 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           d 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             e 
                             D 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
               
             
             = 
             
               log 
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         RW 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           A 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         ⅇ 
                         
                           
                             - 
                             2 
                           
                           ⁢ 
                           
                             
                               α 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 D 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               d 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 D 
                               
                               ) 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             - 
                             
                               
                                 R 
                                 12 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   D 
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               R 
                               12 
                               3 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 D 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       
                         RW 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           e 
                           D 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         ⅇ 
                         
                           
                             - 
                             2 
                           
                           ⁢ 
                           
                             
                               α 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 A 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               d 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 A 
                               
                               ) 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             - 
                             
                               
                                 R 
                                 12 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   A 
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               R 
                               12 
                               3 
                             
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 A 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Step 3. Solve the system of equations simultaneously to evaluate three constants γ 1 , γ 2  and γ 3 . 
     Referring to  FIG. 7 , each of these output values of stiffness, stiffness at no compression, density and wave attenuation, may be provided selectively or in combination by a selector means  89  implemented, for example within the program  72 , which is provided to a mapper program  90 , which maps the values of individual ones or combinations of strain  78  and material properties  74  including: stiffness gradient  82 , initial stiffness  84 , density  86  or wave attenuation  88  to a color or gray scale to produce a graphic image  92  on the display  36  of  FIG. 1 . The cursor control device  40  operating through a driver program  94  may allow for the placement of a region of interest cursor  96  on the image to direct the processor  33  to make a quantitative measurement display  98  of the tissue within the region of the cursor  96 . Location of the cursor  96  may, for example, be with respect to a standard B mode image obtained by the acoustoelastographic ultrasound system  10  or may be an image produced by any of the quantities described above. Selector means  89  may also include an input for standard B mode data. 
     Referring now to  FIG. 8 , experimental results have shown that the SGI technique is able to differentiate among tissue types that exhibit different stiffness gradients, that is, changes in stiffness as a function of strain that closely correspond to stiffness gradients actually tested ex vitro with a standard material testing machine. This ability to measure stiffness gradient offers the potential for better differentiation of tissue types. 
     Both ASG and SGI are examples of techniques that use acoustoelastic theory and information from both the reflection coefficient and wave propagation time (measured from a number of loaded states and an unloaded state) to compute strains and material properties. If the properties of the surrounding medium (or tissue) are unknown, this method could be used to compute the properties of the surrounding medium first (in comparison to a known coupling material). Then the target tissue can be serially analyzed and identified in comparison to the now known behavior of the surrounding medium. 
     These methods could be used for materials other than biological tissues. 
     It is specifically intended that the present invention not be limited to the embodiments and illustrations contained herein, but include modified forms of those embodiments including portions of the embodiments and combinations of elements of different embodiments as come within the scope of the following claims.