Elasticity measuring method and elasticity measuring apparatus

There are provided an elasticity measuring method of measuring an elastic constant of elasticity of an elastic body, for example, an elastic rubber, an organism or a living body, etc., which is soft and rather incompressible, by means of measuring an internal strain using energy such as ultrasounds, an X-ray, and magnetism, where the elastic constant includes, for example, typically shear modulus, Young's modulus, etc.; and an elasticity measuring apparatus adapted to practice such an elasticity measuring method. According to such method and apparatus, the elastic constant distribution within the subject is detected only through measuring the strain distribution within the subject, without measuring the stress distribution. The elasticity measuring method is of detecting the ratio of elastic value-to-elastic value, which are representative of levels of elasticity involved in a reference point on a predetermined straight line extending inside of a subject and a predetermined observation point on the straight line, respectively, by means of detecting the ratio of strain-to-strain, which are involved in said referencepoint and said observation point, respectively, with respect to a direction toward which the straight line extends.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates to an elasticity measuring method of 
measuring a value representative of a level of elasticity of an elastic 
body, for example, an elastic rubber, an organism or a living body, etc., 
which is soft and rather incompressible, by means of measuring an internal 
strain using energy such as ultrasounds, an X-ray, and magnetism, where 
said value may be referred to as "elastic value" or "elastic constant" 
hereinafter, and includes, for example, typically shear modulus, Young's 
modulus, etc.; and an elasticity measuring apparatus, which is adapted to 
practice such an elasticity measuring method, and to be incorporated into, 
for example, a non-destructive inspection equipment, an image diagnostic 
system in which the elastic constant of the human body is measured, and 
upon being regarded as an organized body, an organized body diagnosis is 
implemented, and the like. 
2. Description of the Related Art 
Hitherto, the elastic constant of a uniform quality of elastic rubber or 
the like is determined in such a manner that a test material shaped as a 
board is set up to a tension tester to measure both stress and strain , so 
that the elastic constant is calculated. With regard to a living body, for 
example, in case of a checkup for breast cancer, a doctor examines 
stiffness of the object through the palpation. 
With respect to industrial products such as the elastic rubber, it is 
theoretically possible, after they are processed in a format of the 
product such as a tire or the like, to conduct the diagnosis as to whether 
the inside of the product is provided with a uniform elasticity. This will 
be implemented in such a way that the force is applied from the outside to 
the industrial product to induce the strain , and at that time the stress 
distribution of the product and the strain distribution of the inside 
thereof are all measured so that the distribution of the elastic constant 
can be detected and visibly displayed on a display unit. However, indeed, 
it is very difficult to measure the stress distribution of the surface of 
the products as well as the stress distribution of the inside of the 
products. Thus, it is actually almost impossible to implement the 
diagnosis as to whether the inside of the product is provided with a 
uniform elasticity. 
Further, the palpation practiced for a living body involves problems, 
because it depends on a doctor's sense or feeling and thus lacks a 
determinability. Furthermore, it is difficult to diagnose a lesion 
organization, such as the liver cancer, which hides far back in the 
internal organs. Still further, even if it is desired that the 
distribution of the elastic constant is detected in accordance with the 
stress distribution of a surface of the living body and the strain 
distribution of the inside thereof, it is more difficult, in comparison 
with that in case of the industrial products, to measure the stress 
distribution of a surface of the living body. For example, now, even if it 
is desired that the elastic constant of the liver tissue is detected in 
accordance with the strain of the liver due to the pulsation of the heart, 
it is impossible to measure the stress distribution of the liver. It may 
be considered that a low frequency of motion is generated by means of 
applying from the outside the vibration with a predetermined force and a 
predetermined velocity using a vibrator-like something, or a static 
deformation is provided by means of pushing using a spread board-like 
configuration of something. However, it is almost impossible to measure 
the stress distribution of the inside and it is even difficult to measure 
the stress distribution of a skin surface of the living body. 
SUMMARY OF THE INVENTION 
In view of the foregoing, it is therefore an object of the present 
invention to provide an elasticity measuring method capable of determining 
an elastic constant distribution of the inside of the subject through only 
a measurement of the strain distribution of the inside of the subject, 
without necessity of measurement of the stress distribution, and an 
elasticity measuring apparatus, which is adapted to practice such an 
elasticity measuring method. 
To achieve the above-mentioned objects, according to the present invention, 
there is provided the first elasticity measuring method of detecting the 
ratio of elastic value-to-elastic value, which are representative of 
levels of elasticity involved in a reference point on a predetermined 
straight line extending inside of a subject and a predetermined 
observation point on the straight line, respectively, by means of 
detecting the ratio of strain-to-strain , which are involved in said 
reference point and said observation point, respectively, with respect to 
a direction toward which the straight line extends. 
Here, it is noted that the above-referenced terminology "elastic value" may 
be representative of a level of elasticity, and includes in technical 
concept, for example, typically shear modulus G, Young's modulus E, and 
arithmetic operation results including those, for example, G.sup.2, 
E.sup.2, 1n G, 1n E, aG+b (a, b each constant), etc. Further, for example, 
as shown in the expression (23) which will be described later, there will 
be a case where the ratio of strains is directly equivalent to the ratio 
of elastic values. In such a case, according to the present invention, the 
aforementioned terminology "the ratio of elastic value-to-elastic value" 
includes in technical concept the ratio of strains also. This is the 
similar as to the matter of terminology such as "elastic value", "the 
ratio of elastic value-to-elastic value", etc., which will be described 
hereinafter. While the term "elastic value" implies rather variable which 
will take different values on the respective points inside of the subject, 
it happens here that such terminology is referred to as "elastic constant" 
which is more general term. 
Further, according to the present invention, there is provided the second 
elasticity measuring method wherein a reference member, of which elastic 
value representative of a level of elasticity is known, is disposed on a 
reference point on a predetermined straight line extending inside of a 
subject; and an elastic value on a predetermined observation point on the 
straight line within the subject is detected on the basis of the ratio of 
strain-to-strain, which are involved in said reference point and said 
observation point, respectively, with respect to a direction toward which 
the straight line extends, and the elastic value of said reference member. 
In the first and second elasticity measuring method as mentioned above, it 
is preferable that it is determined whether an absolute value of the 
strain on the observation point exceeds a predetermined threshold, and if 
the absolute value is less than the threshold, a predetermined value by 
which the ratio of strain-to-strain respectively involved in said 
reference point and said observation point is replaced is associated with 
said observation point, instead of said ratio of strain-to-strain. 
Furthermore, according to the present invention, there is provided the 
third elasticity measuring method comprising the steps of: 
detecting strain.epsilon..sub.xx (x) as to points on a predetermined 
straight line extending inside of a subject from a reference point A up to 
a predetermined observation point X, with respect to a direction toward 
which the straight line extends, and a differential coefficient 
.epsilon..sub.xx (x), .sub.x of the strain .epsilon..sub.xx (x) with 
respect to the direction toward which the straight line extends, and 
detecting an integral value of the ratio of the strain.epsilon..sub.xx (x) 
to the differential coefficient.epsilon..sub.xx (x),.sub.x along the 
straight line from the reference point A up to the observation point X, 
whereby the ratio of elastic value-to-elastic value, which are 
representative of levels of elasticity involved in the reference point A 
and the observation point X, respectively, is detected. 
Still further, according to the present invention, there is provided the 
fourth elasticity measuring method wherein a reference member, of which 
elastic value representative of a level of elasticity is known, is 
disposed on a reference point A on a predetermined straight line extending 
inside of a subject, said method comprising the steps of: 
detecting strain .epsilon..sub.xx (x) as to points on the straight line 
extending inside of the subject from the reference point A up to a 
predetermined observation point X, with respect to a direction toward 
which the straight line extends, and a differential coefficient 
.epsilon..sub.xx (x), .sub.x of the strain.epsilon..sub.xx (x) with 
respect to the direction toward which the straight line extends, 
detecting an integral value of the ratio of the strain.epsilon..sub.xx (x) 
to the differential coefficient.epsilon..sub.xx (x),.sub.x along the 
straight line from the reference point A up to the observation point X, 
and 
detecting an elastic value on the observation point X on the basis of both 
the integral value and the elastic value of said reference member. 
Also in the third and fourth elasticity measuring method as mentioned 
above, similar to the first and second elasticity measuring method, it is 
preferable that it is determined whether absolute values of the strain 
.epsilon..sub.xx (x) on the points x exceed a predetermined threshold, and 
if an absolute value of strain .epsilon..sub.xx (x.sub.0) as to a 
predetermined point x.sub.0 of any of the points x is less than the 
threshold, the ratio of the strain .epsilon..sub.xx (x.sub.0) as to the 
predetermined point x.sub.0 to the differential coefficient 
.epsilon..sub.xx (x.sub.0),.sub.x is replaced by a predetermined value to 
detect the integral value. 
Still further, according to the present invention, there is provided the 
fifth elasticity measuring method comprising the steps of: 
detecting strains, .epsilon..sub.xx (x,y), .epsilon..sub.xy (x,y), 
.epsilon..sub.yx (x,y) and .epsilon..sub.yy (x,y) as to points (x, y) on 
an arbitrary route C in a predetermined two-demensional plane spreading 
within a subject from a reference point (A, B) up to a predetermined 
observation point (X,Y), and their associated differential 
coefficient.epsilon..sub.xx (x,y), .sub.x , .epsilon..sub.xx (x,y), 
.sub.y, .epsilon..sub.xy (x,y), .sub.x, .epsilon..sub.xy (x,y), .sub.y, 
.epsilon..sub.yy (x,y), .sub.x, .epsilon..sub.yy (x,y), .sub.y ; and 
detecting a curvilinear integral value on the route C from the reference 
point (A, B) up to the observation point (X,Y), where the curvilinear 
integral value is given by the following expression 
##EQU1## 
whereby the ratio of elastic value-to-elastic value, which are 
representative of levels of elasticity involved in the reference point (A, 
B) and the observation point (X, Y), respectively, is detected. 
Still further, according to the present invention, there is provided the 
sixth elasticity measuring method where in a reference member, of which 
elastic value representative of a level of elasticity is known, is 
disposed on a reference point (A, B) in a predetermined two-dimensional 
plane spreading within a subject, said method comprising the steps of: 
detecting strains, .epsilon..sub.xx (x,y), .epsilon..sub.xy (x,y), 
.epsilon..sub.yx (x,y) and .epsilon..sub.yy (x,y) as to points (x, y) on 
an arbitrary route C in the two-demensional plane from a reference point 
(A, B) up to a predetermined observation point (X,Y), and their associated 
differential coefficients .epsilon..sub.xx (x,y), .sub.x, .epsilon..sub.xx 
(x,y), .sub.y, .epsilon..sub.xy (x,y), .sub.x, .epsilon..sub.xy (x,y), 
.sub.y, .epsilon..sub.yy (x,y), .sub.x, .epsilon..sub.yy (x,y), .sub.y ; 
detecting a curvilinear integral value on the route C from the reference 
point (A, B) up to the observation point (X,Y), the curvilinear integral 
value being given by the following expression; and 
##EQU2## 
detecting an elastic value on the observation point (X, Y) on the basis of 
both the integral value and the elastic value of said reference member. 
Also in the fifth and sixth elasticity measuring method as mentioned above, 
similar to the first to fourth elasticity measuring method, it is 
preferable that it is determined whether absolute values of the detonator 
of the integral kernel of said curvilinear integral, 
det={2.epsilon..sub.xx (x,y)+.epsilon..sub.yy 
(x,y)}.multidot.{.epsilon..sub.xx (x,y)+2 .epsilon..sub.yy 
(x,y)}-.epsilon..sub.xy (x,y).multidot..epsilon..sub.yx (x,y), with 
respect to the points (x, y) exceed a predetermined threshold, and if the 
absolute value of a detonator det={2.epsilon..sub.xx 
(x.sub.0,y.sub.0)+.epsilon..sub.yy 
(x.sub.0,y.sub.o)}.multidot.{.epsilon..sub.xx (x.sub.0,y.sub.0)+2 
.epsilon..sub.yy ( x.sub.0, y.sub.0)}-.epsilon..sub.xy 
(x.sub.0,y.sub.0).multidot..epsilon..sub.yx (x.sub.0,y.sub.0) as to a 
predetermined point (x.sub.0,y.sub.0) of any of the points (x, y) is less 
than the threshold, the integral kernel as to the predetermined point 
(x.sub.0,y.sub.0) is replaced by a predetermined value to detect the 
integral value. 
Still further, according to the present invention, there is provided the 
first elasticity measuring apparatus comprising: 
(1) ultrasonic wave transmitting and receiving means for transmitting 
ultrasonic waves a plurality of number of times in the direction along a 
predetermined straight line extending inside of a subject and receiving 
the reflected ultrasonic waves to obtain an ultrasonic received signal; 
(2) strain arithmetic means for detecting based on the ultrasonic received 
signal strains.epsilon..sub.xx (X) and .epsilon..sub.xx (A) of the subject 
as to an observation point X and a reference point A on the predetermined 
straight line, respectively, with respect to a direction toward which the 
straight line extends; 
(3) ratio arithmetic means for detecting the ratio of the strains 
.epsilon..sub.xx (X) and .epsilon..sub.xx (A) to determine the ratio of 
elastic value-to-elastic value, which are representative of levels of 
elasticity involved in the observation point X and the reference point A, 
respectively; and 
(4) display means for displaying the ratio of elastic value-to-elastic 
value determined by said ratio arithmetic means. 
Still further, according to the present invention, there is provided the 
second elasticity measuring apparatus comprising: 
(1) strain arithmetic means for detecting, based on ultrasonic received 
signals each representative of a tomographic image measured at intervals 
of a specified time, which tomographic image is involved in a 
predetermined two-dimensional plane spreading within a subject, strains 
.epsilon..sub.xx (X) and .epsilon..sub.xx (A) of the subject as to an 
observation point X and a reference point A in the two-dimensionalplane, 
respectively, with respect to a direction toward which a straight line 
coupling the observation point X and the reference point A extends; 
(2) ratio arithmetic means for detecting the ratio of the 
strains.epsilon..sub.xx (X) and .epsilon..sub.xx (A) to determine the 
ratio of elastic value-to-elastic value, which are representative of 
levels of elasticity involved in the observation point X and the reference 
point A, respectively; and 
(3) display means for displaying the ratio of elastic value-to-elastic 
value determined by said ratio arithmetic means. 
In the first and second elasticity measuring apparatus as mentioned above, 
it is preferable to provide such an arrangement that the elasticity 
measuring apparatus further comprises determining means for determining 
whether an absolute value of the strain .epsilon..sub.xx (X) on the 
observation point X exceeds a predetermined threshold, and said ratio 
arithmetic means outputs, if the absolute value is less than the 
threshold, a predetermined value by which the ratio of strain-to-strain 
respectively involved in said observation point X and said reference point 
A is replaced, instead of said ratio of strain-to-strain. 
Still further, according to the present invention, there is provided the 
third elasticity measuring apparatus comprising: 
(1) ultrasonic wave transmitting and receiving means for transmitting 
ultrasonic waves a plurality of number of times in the direction along a 
predetermined straight line extending inside of a subject and receiving 
the reflected ultrasonic waves to obtain an ultrasonic received signal; 
(2) strain detecting means for detecting strain.epsilon..sub.xx (x) as to 
points on the straight line extending inside of the subject from the 
reference point A up to a predetermined observation point X, with respect 
to a direction toward which the straight line extends; 
(3) differential coefficient arithmetic means for differentiating the 
strain.epsilon..sub.xx (x) in the direction toward which the straight line 
extends to detect a differential coefficient .epsilon..sub.xx (x), .sub.x 
of the strain .epsilon..sub.xx (x) with respect to the direction toward 
which the straight line extends; 
(4) ratio arithmetic means for detecting an integral value of the ratio of 
the strain .epsilon..sub.xx (x) to the differential coefficient 
.epsilon..sub.xx (x), .sub.x along the straight line from the reference 
point A up to the observation point X, so that the ratio of elastic 
value-to-elastic value, which are representative of levels of elasticity 
involved in the reference point A and the observation point X, 
respectively, is detected; and 
(5) display means for displaying the ratio of elastic value-to-elastic 
value determined by said ratio arithmetic means. 
Still further, according to the present invention, there is provided the 
fourth elasticity measuring apparatus comprising: 
(1) strain detecting means for detecting, based on ultrasonic received 
signals each representative of a tomographic image measured at intervals 
of a specified time, which tomographic image is involved in a 
predetermined two-dimensional plane spreading within a subject, strain 
.epsilon..sub.xx (X) of the subject as to points on a straight line 
coupling a predetermined observation point X and a reference point A in 
the two-dimensional plane from the reference point A up to the observation 
point X, with respect to a direction toward which the straight line 
extends; 
(2) differential coefficient arithmetic means for differentiating the 
strain.epsilon..sub.xx (x) in the direction toward which the straight line 
extends to detect a differential coefficient .epsilon..sub.xx (x), 
.epsilon. of the strain .epsilon..sub.xx (x) with respect to the direction 
toward which the straight line extends; 
(3) ratio arithmetic means for detecting an integral value of the ratio of 
the strain .epsilon..sub.xx (x) to the differential 
coefficient.epsilon..sub.xx (x),.sub.x along the straight line from the 
reference point A up to the observation point X, so that the ratio of 
elastic value-to-elastic value, which are representative of levels of 
elasticity-involved in the reference point A and the observation point X, 
respectively, is detected; and 
(4) display means for displaying the ratio of elastic value-to-elastic 
value determined by said ratio arithmetic means. 
In the third and fourth elasticity measuring apparatus as mentioned above, 
it is preferable to provide such an arrangement that the elasticity 
measuring apparatus further comprises determining means for determining 
whether absolute values of the strain .epsilon..sub.xx (x) on the points x 
exceed a predetermined threshold, and if an absolute value of strain 
.epsilon..sub.xx (x.sub.0) as to a predetermined point x.sub.0 of any of 
the points x is less than the threshold, said ratio arithmetic means 
detects the integral value through replacing the ratio of the strain 
.epsilon..sub.xx (x.sub.0) as to the predetermined point x.sub.0 to the 
differential coefficient .epsilon..sub.xx (x.sub.0),.sub.x by a 
predetermined value. 
It is acceptable that the first to fourth elasticity measuring apparatus as 
mentioned above each further comprise: 
(5) preset means for presetting an elastic value involved in the reference 
point A; 
(6) elasticity arithmetic means for detecting an elastic value on the 
observation point X on the basis of both the ratio of elastic 
value-to-elastic value and the elastic value involved in the reference 
point A; and 
(7) additional display means for displaying the elastic value on the 
observation point X, instead of said display means. 
Still further, according to the present invention, there is provided the 
fifth elasticity measuring apparatus comprising: 
(1) ultrasonic wave transmitting and receiving means for transmitting 
ultrasonic waves a plurality of number of times in directions along a 
plurality of straight lines in a predetermined two-dimensional plane 
spreading within a subject and receiving the reflected ultrasonic waves to 
obtain ultrasonic received signals; 
(2) strain detecting means for detecting strains, .epsilon..sub.xx (x,y), 
.epsilon..sub.xy (x,y), .epsilon..sub.yx (x,y) and .epsilon..sub.yy (x,y) 
as to points (x, y) on an arbitrary route C in a predetermined 
two-dimensional plane spreading within a subject from a reference point 
(A, B) up to a predetermined observation point (X,Y); 
(3) differential coefficient arithmetic means for differentiating the 
strains, .epsilon..sub.xx (x,y), .epsilon..sub.xy (x,y), .epsilon..sub.yx 
(x,y) and .epsilon..sub.yy (x,y) to detect their associated differential 
coefficients .epsilon..sub.xx ( x,y), .sub.x, .epsilon..sub.xx 
(x,y),.sub.y, .epsilon..sub.xy (x,y), .sub.x, .epsilon..sub.xy (x,y), 
.sub.y, .epsilon..sub.yy (x,y), .sub.x, .epsilon..sub.xy (x,y), .sub.y ; 
(4) ratio arithmetic means for detecting a curvilinear integral value on 
the route C from the reference point (A,B) up to the observation point 
(X,Y), where the curvilinear integral value is given by the following 
expression 
##EQU3## 
whereby the ratio of elastic value-to-elastic value, which are 
representative of levels of elasticity involved in the reference point (A, 
B) and the observation point (X, Y), respectively, is detected; and 
(5) display means for displaying the ratio of elastic value-to-elastic 
value determined by said ratio arithmetic means. 
Still further, according to the present invention, there is provided the 
sixth elasticity measuring apparatus comprising: 
(1) strain detecting means for detecting, based on ultrasonic received 
signals each representative of a tomographic image measured at intervals 
of a specified time, which tomographic image is involved in a 
predetermined two-dimensional plane spreading within a subject, strains, 
.epsilon..sub.xx (x,y), .epsilon..sub.xy (x,y), .epsilon..sub.yx (x,y), 
and .epsilon..sub.yy (x,y) as to points (x, y) on an arbitrary route C in 
a predetermined two-dimensional plane spreading within a subject from a 
reference point (A, B) up to a predetermined observation point (X,Y); 
(2) differential coefficient arithmetic means for differentiating the 
strains, .epsilon..sub.xx (x,y), .epsilon..sub.xy (x,y), .epsilon..sub.yx 
(x,y) and .epsilon..sub.yy (x,y) to detect their associated differential 
coefficients .epsilon..sub.xx (x,y), .sub.x, .epsilon..sub.xx 
(x,y),.sub.y, .epsilon..sub.xy (x,y), .sub.x, .epsilon..sub.xy (x,y), 
.sub.y, .epsilon..sub.yy (x,y),.sub.x, .epsilon..sub.yy (x,y), .sub.y ; 
(3) ratio arithmetic means for detecting a curvilinear integral value on 
the route C from the reference point (A,B) up to the observation point 
(X,Y), where the curvilinear integral value is given by the following 
expression 
##EQU4## 
whereby the ratio of elastic value-to-elastic value, which are 
representative of levels of elasticity involved in the reference point (A, 
B) and the observation point (X, Y), respectively, is detected; and 
(4) display means for displaying the ratio of elastic value-to-elastic 
value determined by said ratio arithmetic means. 
In the fifth and sixth elasticity measuring apparatus as mentioned above, 
it is preferable to provide such an arrangement that the elasticity 
measuring apparatus further comprises determining means for determining 
whether absolute values of the detonator of the integral kernel of said 
curvilinear integral, det={2 .epsilon..sub.xx (x,y)+.epsilon..sub.yy 
(x,y)}.multidot.{.epsilon..sub.xx (x,y)+2 .epsilon..sub.yy 
(x,y)}-.epsilon..sub.xy (x,y).multidot..epsilon..sub.yx (x,y), with 
respect to the points (x, y) exceed a predetermined threshold, and if the 
absolute value of a detonator det={2 .epsilon..sub.xx 
(x.sub.0,y.sub.0)+.epsilon..sub.yy 
(x.sub.0,y.sub.0)}.multidot.{.epsilon..sub.xx (x.sub.0,y.sub.0)+2 
.epsilon..sub.yy (x.sub.0,y.sub.0)}-.epsilon..sub.xy 
(x.sub.0,y.sub.0).multidot..epsilon..sub.yx (x.sub.0,y.sub.0) as to a 
predetermined point (x.sub.0,y.sub.0) of any of the points (x, y) is less 
than the threshold, said ratio arithmetic means detects the integral value 
through replacing the integral kernel as to the predetermined point 
(x.sub.0,y.sub.0) by a predetermined value. 
It is acceptable that the fifth and sixth elasticity measuring apparatus as 
mentioned above each further comprise: 
(6) preset means for presetting an elastic value involved in the reference 
point (A, B); 
(7) elasticity arithmetic means for detecting an elastic value on the 
observation point (X, Y) on the basis of both the ratio of elastic 
value-to-elastic value and the elastic value involved in the reference 
point (A, B); and 
(8) additional display means for displaying the elastic value on the 
observation point (X, Y), wherein said display means is replaced by said 
additional display means. 
According to the present invention as described above, it is possible to 
implement a quantitative measurement of elastic constants, which is 
hitherto considered that it is difficult to be attained without measuring 
a stress distribution, through only a measurement of distortion. 
Hereinafter, a principle of the measurement of the elastic constant 
involved in the present invention will be described citing an example in 
which an elastic constant distribution of the tissue of a living body is 
measured. 
Taking orthogonal coordinates axes as 1, 2 and 3, and assuming that a 
strain tensor is given by 
##EQU5## 
and a stress tensor is given by 
##EQU6## 
The strain tensor .epsilon..sub.ij (i, j=1-3) can be determined in 
accordance with the following equation (1), depending on the respective 
elements u.sub.i (i, j=1-3) of the measured displacement vector. 
EQU .epsilon..sub.ij =(u.sub.i,j +u.sub.j,i)/2 (1) 
where ".sub.,j " and ".sub.,i " imply the differential with respect to the 
j-direction and the differential with respect to the i-direction, 
respectively. 
The strain tensor .epsilon..sub.ij (i, j=1-3) is expressed with the bulk 
strain .epsilon..sub.aa =.epsilon..sub.11 +.epsilon..sub.22 
+.epsilon..sub.33 and the deviation strain e.sub.ij as follow: 
EQU .epsilon..sub.ij =.epsilon..sub.aa /3+e.sub.ij ( 2) 
and the stress tensor .sigma..sub.ij (i, j=1-3) is expressed with the means 
normal stress p=.sigma..sub.aa /3=(.sigma..sub.11 +.sigma..sub.22 
+.sigma..sub.33 /3 and the deviation stress b .sub.ij as follows: 
EQU .sigma..sub.ij =p.delta..sub.ij +b.sub.ij ( 3) 
where .delta..sub.ij is given by 
##EQU7## 
Thus, the strain and the stress have relation to each other through the 
bulk modulus K and the shear modulus G. 
Specifically, the bulk modulus K and the shear modulus G are defined as the 
ratio of the means normal stress p to the bulk strain .epsilon..sub.aa and 
the ratio of the deviation stress b .sub.ij to the deviation strain e 
.sub.ij, respectively. 
EQU p=K.epsilon..sub.aa ( 5) 
EQU b.sub.ij =2G e.sub.ij ( 6) 
Since the tissue of a living body is regarded as an incompressible one, 
when Poisson ratio.upsilon. is given with 0.5, 
EQU .epsilon..sub.aa =0 (7) 
substituting this for the related item in equation (2), 
EQU .epsilon..sub.ij =e.sub.ij ( 8) 
and thus, 
EQU K=2G (1+.upsilon.)/3 (1-2.upsilon.)=.infin. (9) 
finally, the mean normal stress P is indeterminate in view of equation (5). 
From equations (3), (6) and (8) 
EQU .sigma..sub.ij =p .delta..sub.ij 30 2G .epsilon..sub.ij ( 10) 
It may be considered that an area within the tissue of a living body, for 
which the elastic constant distribution is intended to be detected, 
involves no mechanical source, and is independent of inertia because of a 
low frequency of movement. Thus, the following equilibrium equations 
exist. 
EQU .sigma..sub.11,1 +.sigma..sub.12,2 +.sigma..sub.13,3 =0 
EQU .sigma..sub.21,1 +.sigma..sub.22,2 +.sigma..sub.23,3 =0 
EQU .sigma..sub.31,1 +.sigma..sub.32,2 +.sigma..sub.33,3 =0 
The expression of the above equilibrium equations according to the sum rule 
is given by the following formula. 
EQU .sigma..sub.ij,j =0 (11) 
Next, the explanation will be made separately on different two cases (A) 
and (B). 
(A) A case in which only a displacement on the primary axis (1-axis) with 
respect to the 1-axis direction can be measured, for example, such a case 
that by the use of a ultrasonic Doppler system, only a displacement on the 
points on ultrasound beams with respect to the ultrasound beam direction 
can be measured. 
In such a case, since only the 1-axis direction is considered, it may be 
considered that there is no variation in medium with respect to the 
secondary axis (2-axis) direction and the tertiary axis direction. Thus, 
EQU G.sub.,2 =G.sub.,3 =0 
Assuming that the expansion or compression as to the primary axis direction 
brings escape of the force in the secondary axis direction and the 
tertiary axis (3-axis) direction, 
EQU .sigma..sub.12 =.sigma..sub.13 =.sigma..sub.23 =.sigma..sub.22 
.sigma..sub.33 =0 (12) 
EQU .sigma..sub.21,1 =.sigma..sub.31,1 =.sigma..sub.ij,2 =.sigma..sub.ij,3 
=0(13) 
From the equation (10), the stress .sigma..sub.11 is given by 
EQU .sigma..sub.11 =p+2 G .epsilon..sub.11 ( 14) 
On the other hand, the mean normal stress p is expressed, on the basis of 
the condition of equation (12), as follows. 
EQU p=.sigma..sub.aa /3=(.sigma..sub.11 +.sigma..sub.22 
+.sigma..sub.33)/3=.sigma..sub.11 /3 (15) 
Thus, equation (14) is expressed as follow; 
EQU .sigma..sub.11 =.sigma..sub.11 /3+2G .epsilon..sub.11 ( 16) 
and then 
EQU .sigma..sub.11 =3G .epsilon..sub.11 ( 17) 
In the differential of formula (17) in the primary axis direction, 
EQU .sigma..sub.11,1 =3)G.sub.,1 .epsilon..sub.11 +G .epsilon..sub.11,1)(18) 
Formula (11) in the form of equilibrium equation means that the left side 
in the above-noted equation, .sigma..sub.11,1, equals zero, that is, 
.sigma..sub.11,1 =0 and thus, 
EQU G.sub.,1 .epsilon..sub.11 +G .epsilon..sub.11,1 =0 
hence, 
EQU G.sub.,1 /G=-.epsilon..sub.11,1 /.epsilon..sub.11 ( 19) 
Upon changing the notation of the coordinates axis from the primary axis, 
the secondary axis and the tertiary axis to the x-axis, the y-axis and the 
z-axis, respectively, now integrating both the sides of equation (19), 
##EQU8## 
In the execution of the integral of the left part of equation (20), 1n 
G(x)/G(A) is derived. The strain or strain .epsilon..sub.xx (x) is 
equivalent to one which is obtained through differentiating x-component 
u.sub.x (x) of the displacement vector with respect to x. 
##EQU9## 
In a case where a zero-point exists with respect to the item u.sub.x 
(x),.sub.x, it will be possible to avoid the problems by means of adopting 
such a scheme that the item [u.sub.x (x),.sub.x ],.sub.x /u.sub.x 
(x),.sub.x is given by zero, alternatively, such item is replaced by the 
value [u.sub.x (x+dx),.sub.x ],.sub.x /u.sub.x (x+dx) involved in the 
neighboring point (x+dx). 
Equation (21) may be rewritten as follows: 
##EQU10## 
Accordingly, it will be understood that the following expression is also 
applicable to the determination. 
##EQU11## 
Further, it is possible to detect an absolute shear modulus by means of 
putting on the point A the material of which the shear modulus is known. 
Assuming that the shear modulus of such a material is given with G.sub.0, 
it is possible to determine the logarithm of the shear modulus, or the 
shear modulus in the form of the following expressions: 
##EQU12## 
The above explanation concerns a principle of the measurement of the 
elastic constant involved in the first to fourth elasticity measuring 
method, and the first to fourth elasticity measuring apparatus according 
to the present invention. 
(B) A case in which two-way components of a displacement vector on the 
points in a two-dimensional plane can be measured, for example, such a 
case that the use of a two-dimensional cross-correlation for a 
two-dimensional small area permits the detection of a displacement with 
respect to the ultrasound beam direction and an additional displacement 
with respect to the direction which falls at right angles with the 
ultrasound beam direction. 
In such a case, the shear modulus G distributes in a plane (1, 2) when 
taking orthogonal coordinates axes as 1, 2 and 3, and there is no 
variation with respect to the 3-axis direction. Thus, 
EQU G.sub.,3 =0 
Effect from the exterior merely causes the expansion or compression to 
emanate within the plane (1, 2), and the force escapes in the 3-axis 
direction, that is, 
EQU .sigma..sub.13 =.sigma..sub.23 =.sigma..sub.33 =0 (26) 
EQU .sigma..sub.31,1 =.sigma..sub.32,2 =.sigma..sub.ij,3 =0 (27) 
From the equation (10), the stress .sigma..sub.11, .sigma..sub.22, 
.sigma..sub.12, .sigma..sub.21 are given by 
EQU .sigma..sub.11 =p+2G .epsilon..sub.11 ( 28) 
EQU .sigma..sub.22 =p+2G .epsilon..sub.22 ( 29) 
EQU .sigma..sub.12 =2G .epsilon..sub.12 ( 30) 
EQU .sigma..sub.21 =2G .epsilon..sub.21 ( 31) 
On the other hand, mean normal stress p is expressed, on the basis of the 
condition of equation (20), as follows. 
EQU p=.sigma..sub.aa /3=(.sigma..sub.11 +.sigma..sub.22 
+.sigma..sub.33)/3=(.sigma..sub.11 +.sigma..sub.22)/3 (32) 
Thus, substituting equations (28) and (29) for equation (32), 
EQU p=(p+2G.epsilon..sub.11 +p+2G.epsilon..sub.22)/3 
EQU 3p=2p+2G(.epsilon..sub.11 +.epsilon..sub.22) 
EQU .thrfore.p2G(.epsilon..sub.11 +.epsilon..sub.22) (33) 
substituting the above equation for equations (28) and (29), 
EQU .sigma..sub.11 =2G(2 .epsilon..sub.11 +.epsilon..sub.22) (34) 
EQU .sigma..sub.22 =2G(.epsilon..sub.11 +2 .epsilon..sub.22) (35) 
In the differential of formulas (34), (30) , (31) and (35) in the 1-axis 
direction or the 2-axis direction, 
EQU .sigma..sub.11,1 =2(G.sub.,1 (2.epsilon..sub.11 
+.epsilon..sub.22)+G(2.epsilon..sub.11 +.epsilon..sub.22).sub.,1)(36) 
EQU .sigma..sub.12,2 =2(G.sub.,2 .epsilon..sub.12 +G .epsilon..sub.12,2)(37) 
EQU .sigma..sub.21,1 =2)G.sub.,1 .epsilon..sub.21 +G .epsilon..sub.21,1)(38) 
EQU .sigma..sub.22,2 =2(G.sub.,2 (.epsilon..sub.11 +2 
.epsilon..sub.22)+G(.epsilon..sub.11 +2 .epsilon..sub.22).sub.,2)(39) 
From equilibrium equation (11) and (27), 
EQU .sigma..sub.11,1 +.sigma..sub.12,2 =0 
EQU .sigma..sub.21,1 +.sigma..sub.22,2 =0 
and thus, 
##EQU13## 
Formula (40) can be solved as to an area in which determinant, that is, the 
following det is not zero: 
##EQU14## 
and the gradient .gradient. (1 n G) is determined in accordance with the 
following formula: 
##EQU15## 
Upon changing the notation of the coordinates axis from the 1-axis, the 
2-axis and the 3-axis to the x-axis, the y-axis and the z-axis, 
respectively, when conducting the curvilinear integral with respect to an 
arbitrary route C, the relative shear modulus with respect to the 
reference point (A, B) is determined in accordance with the following 
formula: 
##EQU16## 
It is preferable that the route C of the integral is selected in such a 
manner that the determinant does not become zero. If it is difficult in 
structure of the system to avoid the point on which the determinant 
becomes zero, it will be possible to avoid the problems by means of 
adopting such a scheme that the item .gradient. (1 n G (x,y)) is given by 
zero, alternatively, such item is replaced by the value involved in the 
neighboring point. 
Further, it is also possible, on a similar basis as discussed above, to 
detect an absolute shear modulus by means of putting on the point (A, B) 
the material of which the shear modulus is known. When the curvilinear 
integral value in formula (42) is given with S, formula (42) is denoted as 
follows; 
EQU 1 n G (x, y)=S+1 n G (A, B) 
and when G (A, B) is given with the known shear modulus G.sub.0, the above 
equation is rewritten as follows: 
EQU 1 n G (x, y)=S+1 n G .sub.0 ( 43) 
The above explanation concerns a principle of the measurement of the 
elastic constant involved in the fifth and sixth elasticity measuring 
method, and the fifth and sixth elasticity measuring apparatus according 
to the present invention. 
While both the above-described cases (A) and (B) are each to detect the 
shear modulus G, Young's modulus E is expressed with the shear modulus G 
and Poisson ratio.upsilon. as follows: 
EQU E=2(1+.upsilon.) G (44) 
and Poisson ratio is regarded as 0.5. Hence, 
EQU E=3G (45) 
Accordingly, it is possible to present information in the form of Young's 
modulus E. Further, it is noted that the present invention does not always 
require to detect or measure the strictly defined elastic constant, such 
as the shear modulus G, Young's modulus E and the like, and it is 
acceptable to detect, for example, the logarithm in G of the shear 
modulus, and the like, as mentioned above.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
Hereinafter, there will be described embodiments of the present invention. 
FIG. 1 is a flowchart useful for understanding an elasticity measuring 
method according to the first embodiment concerning a method of the 
present invention. FIGS. 2 (A)-(D) are each a view showing the function in 
the flowchart of FIG. 1 by way of example. 
Now, prior to practicing the present embodiment shown in FIG. 1, there is 
determined displacement U.sub.x (x) on the respective points on a 
predetermined straight line (an x-direction) extending in the subject, for 
example, as shown in FIG. 2(A). The displacement U.sub.x (x) may be 
measured, irrespective of a way of determination, by the use of, for 
example, a so-called ultrasonic diagnostic system in which ultrasonic 
acoustic waves are transmitted into the subject, and upon receipt of the 
reflected ultrasonic acoustic waves, images involved in the inside of the 
subject are obtained; a X-ray CT (computed tomography) in which X-rays are 
irradiated to the subject to derive tomographic images of the inside of 
the subject; or an MRI (magnetic resonance imaging) in which tomographic 
images of the inside of the subject are derived utilizing the nuclear 
magnetic resonance. 
In step (1.sub.-- 1) shown in FIG. 1, displacement U.sub.x (x) is 
differentiated with respect to the x-direction to determine strain 
.epsilon..sub.xx (x) involved in the x-direction. The strain 
.epsilon..sub.xx (x) is shown, for example, in FIG. 2(B). In step 
(1.sub.-- 2), the absolute value .vertline..epsilon..sub.xx (x).vertline. 
of the determined strain .epsilon..sub.xx (x) is compared with a threshold 
.DELTA.. If the absolute value .vertline..epsilon..sub.xx 
(x).vertline..gtoreq..DELTA., the process goes to the step (1.sub.-- 3) in 
which the ratio of the shear modulus G(x) at the observation point x to 
the shear modulus G(A) at the reference point A are determined in 
accordance with the aforementioned equation (23) (refer to FIG. 2(C)). 
If the absolute value .vertline..epsilon..sub.xx (x).vertline.&lt;.DELTA., the 
process goes to the step (1.sub.-- 4) in which there is given G(x)/G(A)=0, 
since the performance of the arithmetic operation according to the step 
(1.sub.-- 3) will give too large value for G(x)/G(A) to expect accuracy. 
In the step (1.sub.-- 5), assuming that the shear modulus G(A) at the 
reference point A is given by the known value G.sub.0, the shear modulus 
G(x) at the observation point x is calculated (refer to FIG. 2(D)). 
Incidentally, for example, in a case where an elastic constant distribution 
of the tissue of a living body is detected to be available for the 
diagnosis, it is sufficient that a relative elastic constant such that to 
what extent a portion deemed be the lesion is hardened in comparison with 
its periphery is seen, and it often happens that there is no need to 
detect the elastic constant itself. In such a case, the step (1.sub.-- 5) 
shown in FIG. 1 is not needed. 
Further, according to the present embodiment, there is shown an example 
such that first, displacement U.sub.x (x) is determined, and the 
displacement U.sub.x (x) is differentiated with respect to the x-direction 
to determine the strain .epsilon..sub.xx (x) involved in the x-direction. 
However, the present invention is not restricted to such an example. For 
example, there is known a photoelasticity method such that a strain 
distribution of a transparent body such as glass, celluloid and the like 
is determined by utilizing of the phenomenon that when such a transparent 
body is strained, a polarization state of the light passing through the 
transparent body is varied. The present invention does not exclude such a 
modification that the strain .epsilon..sub.xx (x) is directly determined, 
without determining the displacement U.sub.x (x), by the use of the 
technique as mentioned above. Further, it is noted that as shown in 
another embodiment, which will be described later, there is a technique of 
directly determining the strain on the basis of a signal carrying internal 
information of the subject, such as an ultrasonic received signal and the 
like. 
FIG. 3 is a view showing a step by which the step (1.sub.-- 4) in the 
flowchart of FIG. 1 can be replaced. 
In the step (1.sub.-- 2), if .vertline..epsilon..sub.xx 
(x).vertline.&lt;.DELTA., it is acceptable to replace the equation 
G(x)/G(A)=0 by the equation G(x)/G(A)=G(x-dx)/G(A), where G(x-dx) is the 
shear modulus at the point (x-dx) near the observation point x. 
Incidentally, according to the present invention, the step (1.sub.-- 2) 
shown in FIG. 1 does not necessarily have to be provided. It is acceptable 
to arrange the system in such a way that an arithmetic operation according 
to the step (1.sub.-- 3) is executed omitting the step (1.sub.-- 2). For 
example, in a case where it is expected that the strain .epsilon..sub.xx 
(x) always exceeds the threshold .DELTA., that is, 
.vertline..epsilon..sub.xx (x).vertline..gtoreq..DELTA., the step 
(1.sub.-- 2) is not of course needed. On the other hand, even if there is 
expected the possibility of .vertline..epsilon..sub.xx 
(x).vertline.&lt;.DELTA., the extremely large value of G(x) detected may be 
neglected. But, with respect to .epsilon..sub.xx (x)=0 itself, it is 
preferable to avoid this term, since the execution of the division in the 
step (1.sub.-- 3) becomes impossible. 
FIG. 4 is a flowchart useful for understanding an elasticity measuring 
method according to the second embodiment concerning a method of the 
present invention; 
In step (4.sub.-- 1) shown in FIG. 4, displacement U.sub.x (x) of the 
subject on a predetermined straight line extending toward inside of the 
subject in the x-direction which the straight line extends in is 
differentiated with respect to the x-direction to determine strain 
.epsilon..sub.xx (x) involved in the x-direction. In step (4.sub.-- 2), 
the strain .epsilon..sub.xx (x) is differentiated with respect to the 
x-direction to determine a differential coefficient .epsilon..sub.xx (x), 
.sub.x of the strain .epsilon..sub.xx (x). 
In step (4.sub.-- 3), the absolute value .vertline..epsilon..sub.xx 
(x).vertline. of the determined strain .epsilon..sub.xx (x) is compared 
with a threshold .DELTA.. If the absolute value .vertline..epsilon..sub.xx 
(x).vertline..gtoreq..DELTA., the process goes to the step(4.sub.-- 4) in 
which the following formula is calculated. 
EQU R(x)=.epsilon..sub.xx (x), .sub.x /.epsilon..sub.xx (x) 
If the absolute value .vertline..epsilon..sub.xx (x).vertline.&lt;.DELTA., the 
process goes to the step (4.sub.-- 5) in which there is given 
EQU R(x)=0 
In step (4.sub.-- 6), the initialization is conducted so as to provide x=A, 
and S=0. In step (4.sub.-- 7), the following formula is calculated: 
EQU S=S-R(x).multidot..DELTA.x.multidot.dx 
where .DELTA.x is a sampling interval in the x-direction and dx=1 or 0 or 
-1. In step (4.sub.-- 8), the value x is increased by the infinitely small 
increment dx. In step (4.sub.-- 9), it is determined whether or not the 
value x has reached the maximum value MAX. The steps (4.sub.-- 7), 
(4.sub.-- 8) and (4.sub.-- 9) are repeated until the condition x&gt;MAX is 
satisfied. 
This is to determine logarithm of the ratio of the shear modulus G(x) at 
the point x to the shear modulus G(A) at the point A in accordance with 
the aforementioned equation (21), that is, 
EQU S=1n{G(x)/G(A)} 
FIG. 5 is a view showing a step by which the step (4.sub.-- 5) in the 
flowchart of FIG. 4 can be replaced. In this step, when 
.vertline..epsilon..sub.xx (x).vertline.&lt;.DELTA., R(x) is replaced by 
R(x-dx) which is involved in the value near the point x. 
FIG. 6 is a view showing a step by which the step (4.sub.-- 6) in the 
flowchart of FIG. 4 can be replaced, in a case where material having a 
known shear modulus G.sub.0 is put on a point A. In this step, when the 
initial value for S is given by S=1n G.sub.0, the logarithm of the shear 
modulus G(x) at the point x, that is, S=1n G (x) can be calculated. 
FIG. 7 is a flowchart useful for understand ing an elasticity measuring 
method according to the third embodiment concerning a method of the 
present invention. This embodiment shows an example in which displacement 
distributions U.sub.x (x, y) and U.sub.y (x, y) in a predetermined two 
dimensional plane spreading inside of the subject are determined. This is 
to determine logarithm of the ratio of the shear modulus G(x, y) at the 
point (x, y) to the shear modulus G(A, B) at the point (A, B) in 
accordance with the aforementioned equation (42), that is, 1n 
{(G(x,y)/G(A,B)}. 
In step (7.sub.-- 1), first, strain distributions are determined in 
accordance with the arithmetic expressions as set forth below: 
EQU .epsilon..sub.xx (x, y)=(.delta./.delta.x) u.sub.x (x, y) 
EQU .epsilon..sub.yy (x, y)=(.delta./.delta.y) u.sub.y (x, y) 
EQU .epsilon..sub.xy (x, y)=.epsilon..sub.yx (x, y)=1/2{(.delta./.delta.x) 
u.sub.y (x, y)+(.delta./.delta.y) u.sub.x (x, y)} 
In step (7.sub.-- 2), the strain distributions obtained in the step 
(7.sub.-- 1) are further differentiated so that differential coefficients 
of the strain are determined. The equations are given by 
EQU .epsilon..sub.xx (x,y).sub.,x =(.delta./.delta.x) .epsilon..sub.xx (x,y) 
EQU .epsilon..sub.xx (x,y).sub.,y =(.delta./.delta.y) .epsilon..sub.xx (x,y) 
EQU .epsilon..sub.yy (x,y).sub.,x =(.delta./.delta.x) .epsilon..sub.yy (x,y) 
EQU .epsilon..sub.yy (x,y).sub.,y =(.delta./.delta.y) .epsilon..sub.yy (x,y) 
EQU .epsilon..sub.xy (x,y).sub.,y =(.delta./.delta.y) .epsilon..sub.xy (x,y) 
EQU .epsilon..sub.yx (x,y).sub.,x =(.delta./.delta.x) .epsilon..sub.yx (x,y) 
In step (7.sub.-- 3), the initialization is effected to provide x=A, and 
y=B so that a reference point (A, B) is designated, and in addition to 
provide S=0. 
In step (7.sub.-- 4), the following arithmetic operation is carried out: 
EQU det={2.epsilon..sub.xx (x,y)+.epsilon..sub.yy 
(x,y)}.multidot.{.epsilon..sub.xx (x,y)+2 .epsilon..sub.yy 
(x,y)}-.epsilon..sub.xy (, x,y).multidot..epsilon..sub.yx (x,y) 
where "det" corresponds to a denominator of the integral kernel in the 
aforementioned equation (42). 
In step (7.sub.-- 5), the determination as to 
.vertline.det.vertline..gtoreq..DELTA. is conducted for the purpose of 
avoiding such a situation that the integral kernel in the equation (42) 
becomes extremely large. 
In steps (7.sub.-- 6) and (7.sub.-- 7), the following values are 
determined, respectively. 
##EQU17## 
In step (7.sub.-- 8), arithmetic operation according to the following 
formula is carried out. 
EQU S=S-(1/det)(dX dY).multidot.A.multidot.B 
wherein dX=.DELTA.x dx, dY=.DELTA.ydy, and .DELTA.x and .DELTA.y are 
sampling units. 
In step (7.sub.-- 9), infinitely small increments, dx and dy, are 
determined in such a manner that the infinitely small increment vector 
(dx, dy) is orientated toward the tangential direction of a predetermined 
route C. 
FIG. 8 is a typical illustration useful for understanding how infinitely 
small increments, dx and dy, are determined. 
In case of advancement along the route C on the two dimensional plane C, 
there are given dx=0, dy=-1 for the section (a), dx=1, dy=0 for the 
section (b), and dx=0, dy=1 for the section (c). 
Incidentally, if the route C is determined previously, it is possible to 
calculate beforehand also the infinitely small increments (dx, dy) along 
the route C. Thus, it is acceptable to store the previously calculated (dx 
dy) in a ROM or the like. 
Now, again referring to FIG. 7, when the infinitely small increments (dx, 
dy) is given in step (7.sub.-- 9), in step (7.sub.-- 10), x and y are 
increased by infinitely small increments (dx, dy), respectively. In step 
(7.sub.-- 9), it is determined as to whether or not the process advances 
up to the final course of the route C, and if not yet, the program returns 
to step (7.sub.-- 4). Thus, through the arithmetic operation, the equation 
(42) is executed so that the values on point (x, y) along the route C is 
obtained according to the following formula. 
EQU S=1n{G)x,y)/G(A,B)} 
In step (7.sub.-- 5), if it is determined as 
.vertline.det.vertline.&lt;.DELTA., the process goes to the step (7.sub.-- 
9), without calculating the new S in the step (7.sub.-- 8). This means 
that in case of .vertline.det.vertline.&lt;.DELTA., as the value S (x, y) of 
S of the point (x, y), the value S (x-dx, y-dy) of S of the just prior 
point (x-dx, y-dy) is used as it is, in other words, the integral kernel 
of the equation (42) is replaced by zero with respect to the point (x, y). 
FIG. 9 is a view showing steps by which the steps (7.sub.-- 5) to (7.sub.-- 
8) circled with a dashed line in the flowchart of FIG. 7 can be replaced. 
The difference of the replaced portion from the associated portion of FIG. 
7 resides in the point that there are provided two steps (7.sub.-- 
8.sub.-- 1) and (7.sub.-- 8.sub.-- 2) instead of the step (7.sub.-- 8) of 
FIG. 7, and in the step (7.sub.-- 5) when it is decided as 
.vertline.det.vertline.&lt;.DELTA., then, directly, the step (7.sub.-- 
8.sub.-- 2) is executed. 
In FIG. 9, it is noted that when .vertline.det.vertline.&lt;.DELTA., as the 
integral kernel of the point (x, y) on which the current arithmetic 
operation is performed, the integral kernel of the immediately prior point 
(x-dx, y-dy) is used as it is, so that S is determined. In this manner, 
according to the embodiment shown in FIG. 9, when 
.vertline.det.vertline.&lt;.DELTA., the integral kernel of the neighboring 
point (x-dx, y-dy) is used as it is. 
FIG. 10 is a view showing a step by which the step (7.sub.-- 3) in the 
flowchart of FIG. 7 can be replaced, in a case where material having a 
known shear modulus G.sub.0 is put on a point (A, B). When the initial 
value for S is given by S=1n G.sub.0, the logarithm of the shear modulus 
G(x, y) at the point (x, y), that is, S=1n G (x, y) can be calculated 
(refer to the equation (43)). 
FIGS. 11-14 are each a view showing exemplarily a route C starting from a 
point (A, B). As seen from these figures, it is possible to arbitrarily 
determine the route C. 
FIG. 15 is a view showing another example of a route C. In FIG. 15, the 
marks x indicate each a point involved in the fact that the denominator 
det of the integral kernel in the aforementioned equation (41) with 
respect to the associated point is zero. According to the present 
embodiment, the route C is adaptively determined, while avoiding a point, 
det=0. In this manner, in a case where the route C is determined so as to 
avoid the point, det=0, the step (7.sub.-- 5) in the embodiments shown in 
FIGS. 7 and 9 may be omitted. 
FIGS. 16 (A)-(B) are views useful for explaining a technique of determining 
a route C, while avoiding a point, det=0, by way of example. 
Now, there will be described, by way of example, as shown in FIG. 16 (A), 
such a case that the process advances in the x-direction and encounters 
the point, det=0. 
As shown in FIG. 16 (B), in step (16.sub.-- 1), there is given the 
increment vector (dx, dy)=(1, 0) with respect to the x-direction. In step 
(16.sub.-- 2), it is determined as to the increased point (x+dx, y+dy) 
whether or not .vertline.det.vertline.&lt;.DELTA.. If 
.vertline.det.vertline..gtoreq..DELTA., the process advances in the 
x-direction without change of the travelling direction. On the other hand, 
in a case where the process advances to the point of 
.vertline.det.vertline.&lt;.DELTA. such as the advancement from the point 
P.sub.0 to the point P.sub.1, the program proceeds to step (16.sub.-- 3) 
in which the process advances in the y-direction toward the point P.sub.2 
on the basis of the point P.sub.0 with the increment vector (dx, dy)=(0, 
1). It is determined as to the point P.sub.2 also whether or not 
.vertline.det.vertline.&lt;.DELTA.. If .vertline.det.vertline.&lt;.DELTA. also 
with respect to point P.sub.2, then the process advances in the minus 
y-direction toward the point P.sub.3 on the basis of the point P.sub.0 
with the increment vector (dx, dy)=(0, -1). 
In this manner, there will be formed a route C avoiding the point, 
.vertline.det.vertline.&lt;.DELTA., when an obstacle 
(.vertline.det.vertline.&lt;.DELTA.) exists in the travelling direction, 
through modifying the travelling direction. 
FIG. 17 is a block diagram showing the basic arrangement of constituents of 
an ultrasonic diagnostic system into which an elasticity measuring 
apparatus according to the first embodiment concerning a device of the 
present invention is incorporated. 
When an ultrasonic transmitted signal S1 is inputted to a pulser 11, a high 
voltage pulse is applied from the pulser 11 to a piezoelectric ultrasonic 
transducer (not illustrated) with which a probe 12 is equipped, so that an 
ultrasonic pulse U is radiated from the transducer toward the inside of 
the subject or the human body under examination. The ultrasonic pulse U 
advances within the subject in a predetermined direction (defined as the 
x-direction), while being reflected on tissues within the subject. The 
reflected ultrasounds are received by the transducer and converted into 
electric signals, which electric signals are passed via a receiving 
amplifier 13 to a displacement detecting means 14 and a detector circuit 
15. 
The displacement detecting means 14 detects displacement U.sub.x (x) on the 
respective points in the x-direction toward which the ultrasonic pulse 
advances within the subject while the ultrasonic pulse is transmitted and 
received a plurality of number of times in the same direction within the 
subject. As a way of detection of the displacement U.sub.x (x), there are 
known various techniques, for example, a Doppler effect, a 
cross-correlation, a phase-tracking, etc. The technique of detection of 
the displacement itself is not essential to the present invention. 
Accordingly, the detailed description of detection of the displacement 
will be omitted. 
The received signal, which is passed via the receiving amplifier 13, is 
also inputted to the detector circuit 15 which detects the entered 
received signal. In the detector circuit 15, tomographic images of the 
subject are issued through transmitting ultrasounds from the probe 12 in 
the various directions, for example, on a sheet of FIG. 17, within the 
subject, and receiving the reflected ultrasounds by the probe 12. 
The detector circuit 15, mechanism of transmission and reception of 
ultrasounds and an arrangement of an ultrasonic diagnostic system in its 
entirety are of a well known technology, and in addition they are not 
directly involved in the present invention. Hence, the more detailed 
description of those will be omitted. 
A signal representative of a tomographic image of the subject, which is 
issued by the detector circuit 15, is applied to a digital scan converter 
16 and converted into a signal for use in display. Thus, the tomographic 
image of the subject is displayed on a screen of a CRT display device 17. 
On the other hand, the displacement U.sub.x (x) along the straight line 
extending in the x-direction within the subject, which has been detected 
in the displacement detecting means 14, is inputted to a differentiator 21 
to perform a differential operation, so that strain .epsilon..sub.xx (x) 
involved in the x-direction is detected. 
EQU .epsilon..sub.xx (x)=(.delta./.delta.x)u.sub.x (x) 
As the differentiator 21, in case that an analog signal is treated, for 
example, an analog differential filter may be adopted, and in case that a 
digital signal is treated, for example, a digital filter may be adopted. 
Of the strain .epsilon..sub.xx (x) determined by the differentiator 21, 
strain .epsilon..sub.xx (A) involved in the reference point (A) is 
inputted to a sample hold circuit 22 and stored therein. As the sample 
hold circuit 22, in case that an analog signal is treated, for example, a 
sample hold circuit adapted for holding the analog signal may be adopted, 
and in case that a digital signal is treated, for example, a register 
adapted for storing the digital signal may be adopted. 
The strain .epsilon..sub.xx (A) involved in the point (A), which is stored 
in the sample hold circuit 22, and the distortion .epsilon..sub.xx (x) 
involved in the subsequent points (x) are inputted to a divider 23 to 
perform operation .epsilon..sub.xx (A)/.epsilon..sub.xx (x). This is 
equivalent to the ratio of the shear modulus G(x) at the point x to the 
shear modulus G(A) at the point A, or G(x)/G(A), which is operated on the 
basis of the equation (23). 
EQU G(x)/G(A)=.epsilon..sub.xx (A)/.epsilon..sub.xx (x) 
The strain .epsilon..sub.xx (x) outputted from the differentiator 21 is 
inputted to an absolute value circuit 24 to compute the absolute value 
.vertline..epsilon..sub.xx (x).vertline. of the strain .epsilon..sub.xx 
(x). The absolute value .vertline..epsilon..sub.xx (x).vertline. is 
supplied to a comparator 25 to compare it with a threshold.DELTA.. A 
switching circuit 26 is controlled in accordance with an output of the 
comparator 25. If .vertline.68 .sub.xx (x).vertline..gtoreq..DELTA., the 
output of the divider 23 is inputted to a digital scan converter 16, while 
if .vertline..epsilon..sub.xx (x).vertline.&lt;.DELTA., the switching circuit 
26 is switched to supply the value "0" instead of the output of the 
divider 23 to the digital scan converter 16. The absolute value circuit 
24, the comparator 25 and the switching circuit 26 are provided for the 
purpose of avoiding such a situation that the value of .epsilon..sub.xx 
(A)/.epsilon..sub.xx (x), which is computed by the divider 23, will 
overflow owing to the fact that the value of the strain .epsilon..sub.xx 
(x) is too small. 
The above-mentioned ratio G(x)/G(A), which is computed by the divider 23, 
is converted by the digital scan converter 16 into a signal for use in 
display. For example, such an information is superposed on the tomographic 
image of the subject, so that the hard portion within the subject will be 
color-displayed with the luminance according to the hardness of the 
associated portion. 
FIG. 18 is a block diagram showing a strain detecting circuit, by way of 
example, with which the displacement detecting means 14 and the 
differentiator 21 in the embodiment shown in FIG. 17 can be replaced. 
The strain detecting circuit 50 shown in FIG. 18 is based on the principle 
which will be described hereinafter. 
An arithmetic operation for complex auto-correlation of the received signal 
S (x) as to a predetermined direction (x-direction) is performed to 
determine an inphase component I(x) and a quadrature component Q (x). 
Through determination of the inverse tangent (arctan) of the ratio of 
these components, the displacement U.sub.x (x) involved in the x-direction 
at the point x is obtained in the form of the following equation: 
EQU U.sub.x (x)=arctan {I(x)/Q(x)} 
While, the strain .epsilon..sub.xx (x) involved in the x-direction at the 
point x is given by 
EQU .epsilon..sub.xx (x)=d U.sub.x (x)/dx 
The above expression can be rewritten as follows: 
##EQU18## 
The strain detecting circuit 50 shown in FIG. 18 performs an arithmetic 
operation on the basis of the modified formula noted above, and detects 
the strain .epsilon..sub.xx (x) without detecting the displacement U.sub.x 
(x). 
The received signal S (x) passing through the receiving amplifier 13 (refer 
to FIG. 17) is inputted to an auto-correlator 51 of the strain detecting 
circuit 50 shown in FIG. 18. The auto-correlator 51 performs an arithmetic 
operation for complex auto-correlation of the received signal S (x). The 
auto-correlator 51 itself is well known, and thus the illustration of the 
arrangement of constituents thereof and the associated description will be 
omitted. 
The auto-correlator 51 issues an inphase component I(x) and a quadrature 
component Q (x), which are passed via differentiators 52 and 53 to 
multipliers 54 and 55, respectively, and in addition directly to the 
multipliers 55 and 54, respectively. Further, these components I(x) and a 
quadrature component Q (x) are supplied to multipliers 56 and 57, 
respectively, so that {I(x)}.sup.2 +{Q(x)}.sup.2 is detected. 
The signals {dI(x)/dx}.multidot.Q(x), and I(x).multidot.{d Q(x)/dx}, which 
are outputted from the multipliers 54 and 55, respectively, are subjected 
to the subtraction process by a subtracter 58, and then inputted to a 
divider 59. While the signals {I(x)}.sup.2, and {Q(x)}.sup.2, which are 
outputted from multipliers 56 and 57, respectively, are added to each 
other by an adder 59, and then inputted to the divider 59. Thus, divider 
59 may output the strain .epsilon..sub.xx (x) detected on the basis of the 
formula (46). 
In this manner, it is also possible to detect the strain without detecting 
the displacement U.sub.x (x). 
FIG. 19 is a block diagram showing the basic arrangement of constituents of 
an ultrasonic diagnostic system into which an elasticity measuring 
apparatus according to the second embodiment concerning a device of the 
present invention is incorporated. In the following figures, the same 
parts are denoted by the same reference numbers as those of FIG. 19. 
Specifically, to transmit the ultrasonic pulse U to the inside of the 
subject, material 30 having a known shear modulus G.sub.0 is sandwiched 
between the probe 12 and the subject and then the ultrasonic pulse U is 
radiated. Here, a specified point inside of the material 30 is selected as 
a reference point A. 
A multiplier 31 for multiplying by the shear modulus G.sub.0 of the 
material 30 is set up before the digital scan converter 16 with respect to 
the signal flow. In this manner, the shear modulus G(x) at the point x is 
determined in accordance with the formula (25), so that the shear modulus 
G(x) can be displayed on a screen of a CRT display 17. Specifically, the 
system may be so arranged that for example, when it is desired that a 
distribution of hardness of the tomographic image is seen in its entirety, 
such information is superposed on the tomographic image of the subject, so 
that the hard portion inside of the subject will be color-displayed with 
the luminance according to the hardness of the associated portion, whereas 
when it is desired that the shear modulus G(x) at the specified point x 
within the tomographic image is seen, the shear modulus G(x) involved in 
that point is displayed with the numeral through moving a cursor to that 
point. 
According to the embodiment shown in FIG. 19, there is provided a delay 
circuit 27 before the switching circuit 26 with respect to the signal 
flow. The reason why the delay circuit 26 is provided is that when the 
comparator determines .vertline..epsilon..sub.xx (x).vertline.&lt;.DELTA., 
the shear modulus G(x-dx) at a point (x-dx) near the point x is used as 
the shear modulus G(x) at the point x. 
FIG. 20 is a block diagram showing the basic arrangement of constituents of 
an image display apparatus into which an elasticity measuring apparatus 
according to the third embodiment concerning a device of the present 
invention is incorporated. 
The image display apparatus shown in FIG. 20 has a tomographic image memory 
40 adapted for temporarily storing an image signal representative of a 
tomographic image of the subject, which image signal will be transmitted 
to the image display apparatus. As means for generating such an image 
signal, there is no particular restriction. It is acceptable to adopt, as 
such an image signal, for example, the signal outputted from the detector 
circuit 15 of the ultrasonic diagnostic system as shown in FIGS. 17 and 
19, or signals which will be obtained by the use of other apparatuses such 
as an X-ray CT, an MRI and etc. 
The tomographic image memory 40 stores a plurality of image signals each 
representative of a tomographic image measured at intervals of time 
.DELTA. t. The displacement detecting means 34 detects the displacement of 
the tomographic image measured every time .DELTA. t in accordance with the 
image signals. A way of detection of the displacement is not restricted. 
It is possible, for example, to detect the displacement by means of 
performing an arithmetic operation for mutual-correlation between two 
tomographic images measured with an interval of time .DELTA.t. 
While the ultrasonic diagnostic system as shown in FIGS. 17 and 19 is 
provided with the detector circuit 15 for producing the image signal and 
the displacement detecting means 14 as well, in which the displacement is 
detected directly from the prior signal involved in producing the image 
signal, according to the embodiment shown in FIG. 20, the displacement 
detecting means 34 detects the displacement U.sub.x (x) on a predetermined 
straight line (x-direction) on the basis of the image signals each 
representative of the associated tomographic image, which image signals 
are stored in the tomographic image memory 40. 
FIG. 21 is a block diagram showing the basic arrangement of constituents of 
an ultrasonic diagnostic system into which an elasticity measuring 
apparatus according to the fourth embodiment concerning a device of the 
present invention is incorporated. 
The strain .epsilon..sub.xx (x) detected in the differentiator 21, which is 
involved in the point x on a straight line extending to the direction in 
which the ultrasonic pulse advances within the subject, is inputted to an 
additional differentiator 28 to detect a differential coefficient 
.epsilon..sub.xx (x), x of the strain .epsilon..sub.xx (x). The strain 
.epsilon..sub.xx (x) and the differential coefficients .epsilon..sub.xx 
(x), .sub.x are inputted to the divider 23 to perform the arithmetic 
operation for .epsilon..sub.xx (x), .sub.x /.epsilon..sub.xx (x). A result 
of the arithmetic operation is inputted to an integrator 29 to perform the 
integration from the reference point A to the point x along the straight 
line. In this manner, there is determined logarithm of the ratio of the 
shear modulus G(x) at the point x to the shear modulus G(A) at the point A 
in accordance with the aforementioned equation (21), that is, 1n 
{G(x)/G(A)}. 
FIG. 22A is a block diagram showing an integration circuit by way of 
example in a case where an analog signal is dealt with. FIG. 22B is a view 
useful for explanation of operation of the integration circuit shown in 
FIG. 22A. 
In a case where an analog signal is dealt with, it is possible to arrange 
an integrator, as shown in FIG. 22A, which comprises an integration 
circuit 291 using an operational amplifier, and a sign inverter 292 for 
inverting the sign of an output signal of the integration circuit 291. To 
provide a point A as a starting point of the integration, the integration 
circuit 291 is equipped with a switch 291a. As shown in FIG. 22B, the 
switch 291a is kept closed until the signal corresponding to the point A 
is inputted, and the switch 291a is opened at the time when the signal 
corresponding to the point A is inputted. When the ultrasonic pulse U is 
radiated to the inside of the subject, the deeper the the ultrasounds go 
into the subject, the later the reflected ultrasounds are received. Thus, 
the distance in the x-direction inside of the subject is in proportion to 
the receiving time x of the signal. Consequently, an arithmetic operation 
for the integration, taking the point A as the starting point, may be 
performed through opening the switch 291a at the time point corresponding 
to the point A. 
FIG. 23A is a block diagram showing an integrator by way of example in a 
case where a digital signal is dealt with. FIG. 23B is a view useful for 
explanation of operation of the integrator shown in FIG. 23A. 
Digital signals are inputted to an adder 293 as shown in FIG. 23A on a time 
sequential basis. An output of the adder 293 is stored in a register 294. 
The adder 293 serves to add the newly entered value to the old value 
stored in the register 294 and refresh the register 294 to store the 
additional result therein. Thus, the register 294 may store the 
accumulated value of the digital signals sequentially inputted, that is, 
the integral value. The register 294 has a clear terminal CLR to which as 
shown in FIG. 23B, a clear signal is being inputted up to the reference 
point A. At the time point that the signal corresponding to the point A is 
inputted, the clear is released so as to start the integral as to the 
input after that point. In this manner, the starting point of the integral 
is determined. 
FIG. 24 is a view useful for explaining a scheme in which a starting point 
of integral of the integrator adapted to deal with the digital signal is 
optionally determined. 
As a general rule, the integral equation of the function f(x) is given by 
the following expression: 
##EQU19## 
Hence, it is possible to determine the integral value in a case where the 
point A is selected as the starting point of the integral, even if the 
integral does not always start from the point A. An integrator 295 effects 
an integral taking the origin x=0 as the starting point, so that the 
integral values are sequentially stored in a delay circuit 296. With 
respect to the delay circuit 296, a delay time is set up in such a way 
that after completion of the integral up to the maximum point x=MAX, the 
delay circuit 296 sequentially outputs the integral values involved in the 
respective points from the origin x=0 up to the maximum point x=MAX. When 
the integrator 295 has completed the integral operation as to the 
respective points up to the point A, the obtained integral value is stored 
in a register 297. A subtracter 298 subtracts the integral value involved 
in the point A from the integral values involved in the respective points 
x which are sequentially outputted from the delay circuit 296. 
The integrator 29 shown in FIG. 21 may be constructed with the arrangement 
shown in FIG. 24. According to such an arrangement, it is possible to 
optionally set up the reference point A by means of selecting a timing in 
which the integral value is stored in a register 297. 
FIG. 25 is a block diagram showing the basic arrangement of constituents of 
an ultrasonic diagnostic system into which an elasticity measuring 
apparatus according to the fifth embodiment concerning a device of the 
present invention is incorporated. 
The material 30 having a known shear modulus G.sub.0 is sandwiched between 
the probe 12 and the subject. An adder 32 is connected between the 
switching circuit 26 and the digital scan converter 16. According to such 
an arrangement, it is possible to detect logarithm in G(x) of the shear 
modulus G(x) at the point x in accordance with the aforementioned equation 
(24). 
FIG. 26 is a block diagram showing the basic arrangement of constituents of 
an ultrasonic diagnostic system into which an elasticity measuring 
apparatus according to the sixth embodiment concerning a device of the 
present invention is incorporated. 
The present embodiment is equivalent to the combination of the embodiment 
shown in FIG. 20 and the embodiment in FIG. 25. Hence, the detailed 
description of the present embodiment will be omitted. 
FIG. 27 is a block diagram showing the basic arrangement of constituents of 
the first half part of a signal processing apparatus of an ultrasonic 
diagnostic system into which an elasticity measuring apparatus according 
to the seventh embodiment concerning a device of the present invention is 
incorporated. 
The ultrasonic pulse U is radiated from the probe 12 in the various 
directions on a predetermined two-dimensional plane spreading inside of 
the subject, so that the inside of the subject is scanned with ultrasonic 
pulse beams. 
The received signal, which is passed via the receiving amplifier 13, is 
inputted to two-way displacement detecting means 141. The two-way 
displacement detecting means 141 detects an x-direction displacement 
U.sub.x (x, y), and a y-direction displacement U.sub.y (x, y) on the 
points in the two-dimensional plane. These displacements U.sub.x (x, y) 
and U.sub.y (x, y) can be determined by means of applying, for example, a 
two-dimensional cross-correlation scheme and the like. A way of detection 
of the displacements U.sub.x (x, y) and U.sub.y (x, y) are well known, and 
the technique of detection of the displacement itself is not essential to 
the present invention. Accordingly, the detailed description of detection 
of the displacement will be omitted. 
The displacements U.sub.x (x, y) and U.sub.y (x, y) on the points in the 
two-dimensional plane, which have been detected in the two-way 
displacement detecting means 141, are inputted to two pairs of 
differentiators 211, 212; and 213, 214, respectively, so that 
.epsilon..sub.xx (x, y ), u.sub.xy (x, y ), u.sub.yx (x, y) and 
.epsilon..sub.yy (x, y) are computed in accordance with the follo wing 
equations: 
EQU .epsilon..sub.xx (x,y)=(.delta./.delta.x)u.sub.x (x, y) 
EQU u.sub.xy (x,y)=(.delta./.delta.y)u.sub.x (x, y) 
EQU u.sub.yx (x,y)=(.delta./.delta.x)u.sub.y (x, y) 
EQU .epsilon..sub.yy (x,y)=(.delta./.delta.y)u.sub.y (x, y) 
The strain .epsilon..sub.xx (x, y), which is outputted from the 
differentiator 211, is inputted to differentiators 281 and 282, and 
arithmetic units 311 and 312 as well. The differentiators 281 and 282 
perform an arithmetic operation for the differential coefficients 
.epsilon..sub.xx (x,y), .sub.x and .epsilon..sub.xx (x,y), .sub.y of the 
strain .epsilon..sub.xx (x,y), respectively. 
EQU .epsilon..sub.xx (x,y), .sub.x =(.delta./.delta.x).epsilon..sub.xx (x,y) 
EQU .epsilon..sub.xx (x,y), .sub.y =(.delta./.delta.y).epsilon..sub.xx (x,y) 
The strains u.sub.x,y (x,y) and u.sub.yx (x,y), which are outputted from 
the differentiators 212 and 213, respectively, are added each other by an 
adder 301 and then multiplied by 1/2 with a multiplier 302, so that the 
strain .epsilon..sub.xy ={u.sub.xy (x,y)+u.sub.yx (x,y)}/2 can be 
determined. This result is inputted to two differentiators 283 and 284, 
and the arithmetic units 311 and 312 as well. The differentiators 283 and 
284 differentiate the strain .epsilon..sub.xy (x,y) in both the 
x-direction and the y-direction, and perform an arithmetic operation for 
the differential coefficients .epsilon..sub.xy (x,y), .sub.x and 
.epsilon..sub.xy (x,y), .sub.y, respectively. 
EQU .epsilon..sub.xy (x,y),.sub.x =.epsilon..sub.yx (x,y)..sub.x 
=(.delta./.delta.x){u.sub.xy (x, y)+u.sub.yx (x, y)}/2 
EQU .epsilon..sub.xy (x, y), .sub.y =.epsilon..sub.yx (x, y), .sub.y 
=(.delta./.delta.y){u.sub.xy (x,y)+u.sub.yx (x,y)}/2 
The strain .epsilon..sub.yy (x, y), which is outputted from the 
differentiator 214, is inputted to differentiators 285 and 286, and 
arithmetic units 311 and 312 as well. The differentiators 285 and 286 
perform an arithmetic operation for the differential coefficients 
.epsilon..sub.yy (x,y), .sub.x and .epsilon..sub.yy (x,y), .sub.y, 
respectively. 
EQU .epsilon..sub.yy (x,y), .sub.x =(.delta./.delta.x).epsilon..sub.yy (x,y) 
EQU .epsilon..sub.yy (x,y), .sub.y =(.delta./.delta.y).epsilon..sub.yy (x,y) 
The differential coefficients .epsilon..sub.xx (x,y), .sub.x, 
.epsilon..sub.xx (x,y), .sub.y, .epsilon..sub.xy (x,y), .sub.x, 
.epsilon..sub.xy (x,y), .sub.y, .epsilon..sub.yy (x,y), .sub.x, 
.epsilon..sub.yy (x,y), .sub.y, which are obtained by the differentiators 
281, 282; 283, 284; and 285, 286, are inputted to an arithmetic unit 313. 
The arithmetic unit 311 performs the arithmetic operation for the following 
formula on the basis of the entered strains, .epsilon..sub.xx (x,y), 
.epsilon..sub.xy (x,y)=.epsilon..sub.yx (x,y), and .epsilon..sub.yy (x,y). 
EQU det={2 .epsilon..sub.xx (x,y)+.epsilon..sub.yy 
(x,y)}.multidot.{.epsilon..sub.xx (x,y)+2 .epsilon..sub.yy 
(x,y)}-.epsilon..sub.xy (x,y).multidot..epsilon..sub.yx (x,y) 
where "det" corresponds to the denominator of the integral kernel of the 
equation (41). 
The arithmetic unit 312 performs the arithmetic operation for four elements 
constituting the following matrix on the basis of the entered distortions, 
.epsilon..sub.xx (x,y), .epsilon..sub.xy (x,y)=.epsilon..sub.yx (x,y) and 
.epsilon..sub.yy (x,y). 
##EQU20## 
The arithmetic unit 313 performs the arithmetic operation for two elements 
constituting the following matrix (vector) on the basis of the entered 
differential coefficients .epsilon..sub.xx (x,y), .sub.x, .epsilon..sub.xx 
(x,y), .sub.y, .epsilon..sub.xy (x,y), .sub.x, .epsilon..sub.xy (x,y), 
.sub.y, .epsilon..sub.yy (x,y), .sub.x, .epsilon..sub.yy (x,y), .sub.y,. 
##EQU21## 
FIG. 28 is a block diagram showing the basic arrangement of constituents of 
the latter half part of a signal processing apparatus of an ultrasonic 
diagnostic system into which an elasticity measuring apparatus according 
to the seventh embodiment concerning a device of the present invention is 
incorporated. 
The denominator det detected by the arithmetic unit 311 is inputted to the 
reciprocal arithmetic unit 321 to be converted into 1/det, and then passed 
to a multiplier 323 to which the infinitely small increment vector (dx, 
dy) stored in a ROM 322 is also inputted. An output of the multiplier 323 
is passed to the next stage of multiplier 324 to be multiplied by the 
matrix A. An output of the multiplier 324 is passed to the next stage of 
multiplier 325 to be multiplied by the matrix (vector) B. In this manner, 
the value of the integral kernel of the aforementioned equation (41) can 
be determined with respect to the point (x, y). The value of the integral 
kernel is integrated by the integrator 29, so that logarithm of the ratio 
of the shear modulus G(x, y) at the point (x, y) to the shear modulus G(A, 
B) at the reference point (A, B), that is, in {G(x, y)/G(A, B)} can be 
determined, as shown in the equation (42). 
The thus determined logarithm, 1n {G(x, y)/G(A, B)}, of the ratio of the 
shear modulus at the point (x, y) on a two-dimensional plane inside of the 
subject is passed via the digital scan converter 16 to the CRT display 17, 
so that the logarithmic information can be displayed on a screen of the 
CRT display 17. Specifically, the system may be so arranged that for 
example, similar to the arrangement shown in FIG. 17, there is provided 
the detector circuit 15, and when it is desired that a distribution of 
hardness of the tomographic image is seen in its entirety, such 
information is superposed on the tomographic image of the subject, so that 
the hard portion inside of the subject will be color-displayed with the 
luminance according to the hardness of the associated portion. 
FIG. 29 is a block diagram showing the basic arrangement of constituents of 
the latter half part of a signal processing apparatus of an ultrasonic 
diagnostic system into which an elasticity measuring apparatus according 
to the eighth embodiment concerning a device of the present invention is 
incorporated. Regarding the first half part of the signal processing 
apparatus, the equivalence of the arrangement shown in FIG. 27 may be 
used. Specifically, to transmit and receive ultrasonic acoustic waves, for 
example, as shown in FIG. 19, material having a known shear modulus 
G.sub.0 is sandwiched between the probe 12 and the subject. 
In the arrangement of constituents of the device shown in FIG. 29, the 
different point from that in FIG. 28 resides in the integrator 29. A 
register 299 stores beforehand logarithm in G.sub.0 of the shear modulus 
of the material sandwiched between the probe 12 and the subject. To 
conduct the integral operation, the value 1n G.sub.0 is transferred 
through a selector 300 to a register 294, and thereafter the selector 300 
is switched so as to receive an output of an adder 293. The adder 293 is 
set up with providing the value in G.sub.0 as an initial value and is 
operative to add to the initial value input values supplied through the 
switching circuit 26 on a cumulative basis. In this manner, as shown in 
the aforementioned equation (43), it is possible to determine logarithm in 
G (x, y) of the shear modulus involved in the point (x, y). 
While the embodiments shown in FIGS. 27-29 relate to an example in which 
the elasticity measuring apparatus according to the present invention is 
incorporated into the ultrasonic diagnostic system, it is of course 
acceptable to apply the two-dimensional signal processing as shown in 
FIGS. 27-29 to for example the image display apparatus as described 
referring to FIG. 20. 
Further, while all the above-described embodiments are involved in 
detecting the shear modulus G, its logarithm 1n G, or the ratio of such 
shear modulus to that on the reference point, it is of course acceptable 
to provide an arrangement in which Young's modulus E rather than the shear 
modulus is detected. Furthermore, the present invention does not 
necessarily have to detect the shear modulus, Young's modulus, logarithm 
of those and the like, and it is sufficient for the present invention to 
detect data representative of the level of elasticity on the points within 
the subject. 
As described in detail above, according to the present invention, it is 
possible to know the elastic level of the respective points within the 
subject only through measuring the distortion of those points, without 
measuring the stress distribution of those points, thereby contributing to 
the industrial inspection, the diagnosis of a living body and the like 
quite a lot. 
While the present invention has been described with reference to the 
particular illustrative embodiments, it is not to be restricted by those 
embodiments but only by the appended claims. It is to be appreciated that 
those skilled in the art can change or modify the embodiments without 
departing from the scope and spirit of the present invention.