Elasticity Imaging:
Principles
Stanislav (Stas) Emelianov
emelian@mail.utexas.edu
Department of Biomedical Engineering
The University of Texas at Austin
Department of Imaging Physics
The University of Texas M.D. Anderson Cancer Center
Elasticity Imaging – Goal
Remote
non-invasive (or adjunct to invasive)
imaging (or sensing) of
mechanical properties of
tissue for
clinical applications
Elasticity
Changes in tissue elasticity are related to pathological changes
… Such swellings as are soft, free from pain, and yield to the finger, ... and
are less dangerous than the others.
... then, as are painful, hard, and large, indicate danger of speedy death;
but such as are soft, free of pain, and yield when pressed with the finger,
are more chronic than these.
THE BOOK OF PROGNOSTICS, Hippocrates, 400 B.C.
Hippocrates
It is the business of the physician to know, in the first place, things similar
and things dissimilar; … which are to be seen, touched, and heard; which
are to be perceived in the sight, and the touch, and the hearing, … which
are to be known by all the means we know other things.
ON THE SURGERY, Hippocrates, 400 B.C.
Hippocrates, 400 B.C.
1
Elasticity Imaging – Glance at History
Hippocrates, circa 460-377 B.C.
Tissue Elasticity
Ultrasound
MRI
Oestreicher, 1951
Mechanical Properties of Tissue
Mechanical Properties of Tissue
Elasticity (e.g., bulk and shear moduli)
Elasticity (e.g., bulk and shear moduli)
Viscosity (e.g., bulk and shear viscosities)
Viscosity (e.g., bulk and shear viscosities)
Nonlinearity (e.g., strain hardening)
Nonlinearity (e.g., strain hardening)
Other (e.g., anisotropy, pseudoelasticity)
Other (e.g., anisotropy, pseudoelasticity)
Other methods
Frank et al, 1948
Biomechanics
(muscle, skin, ...)
Sarvazyan et al, 1975
Frizzell et al, 1976
Fung, 1981
Thompson et al, 1981
Wilson and Robinson, 1982
Dickinson and Hill, 1982
Eisensher et al, 1983
Tristam et al, 1986
Sarvazyan et al, 1984
Krouskop, (Le)vinson, 1987
Lerner and Parker, 1988
Yamakoshi et al, 1988
Adler et al, 1989
Meunier and Bertrand, 1989
Duck, 1990
Parker et al, 1990
Ophir et al, 1991
Sarvazyan and Skovoroda, 1991
Avalanche of papers
Krouskop et al, 1998
Erkamp et al, 1998
Pereira et al, 1990
DeJong et al, 1990
Fowlkes et al, 1992
Sarvazyan et al, 1995
Fowlkes et al, 1995
Muthupillai et al, 1995
Plewes et al, 1995
Chenevert et al, 1998
Many papers
2
Which Elastic Moduli?
Elasticity
Human sense of touch – what do we feel?
R
Relations between various elastic constants
Constant
l
m
E
n
K
G
Common Pair
E, n
l, m
m
nE
(1 n )(1  2n )
E
2(1 n )
m (3l  2m )
(l  m )
E
l
l
2(l  m )
 0.5 
l  23 m
m
m
2l  m 
n
E
3(1  2n )
E
2(1 n )
K, G
K  23 G
•
G
9 KG
3K  G
3K  2G
6 K  2G
K
G
•
x2
=
=
Poisson’s ratio (n)
Bulk modulus (K)
Shear modulus (mG)
Young’s modulus (E)
F1
Rigid circular die
G

 cl 
K  32 G

G  K
G
0
K
x1
x3
F
F1  3m 0
Most soft tissues are incompressible, i.e.,
deformation produces no volume change
ct 
Incompressible material
W
n
In addition, due to tissue incompressibility,
Young’s modulus = 3 · shear modulus
1
2
E  3m
Semi-infinite elastic medium
K – bulk modulus
m – shear modulus
G


F  8RWG 1  K
 1 G

3K

E  3m
1


 
G  8 RWm
0

K


Static deformation of
(nearly) incompressible material is
primarily determined by
shear or Young’s modulus
Sarvazyan et al, 1995
3
Breast Tissue Elasticity and Pathology
Breast Tissue Elasticity and Pathology
Skovoroda et al., 1995,
Wellman et al., 1999,
Biophysics, 40(6):1359-1364.
Breast Tissue
Type
Normal
gland
Infiltrative ductal
cancer with alveolar
tissue predominating.
Fibroadenomas of
glandular origin
Infiltrative ductal cancer
with fibrous tissue
predominating.
Ductal
fibroadenoma
Young’s
Modulus (kPa)
0.5-1.5
1.0-1.5
1.5-2.5
2.0-3.0
8.0-12.0
Elastic Properties of Prostate Gland
Aglyamov S.R. and Skovoroda A.R., 2000, Biophysics,
2000, 45(6).
Harvard BioRobotics Laboratory Technical Report.
Krouskop et al., 1998, Ultrasonic Imaging, 20:260-274.
Samani et al., 2003, Physics in Medicine and Biology, 48:2183-2198
.
4
Elastic Properties of Arteries
Sarvazyan A.P., 2001,” In: Handbook of Elastic Properties of Solids, Liquids and Gases.
Type of soft tissue
E, kPa
Comments
Contrast in Elasticity Imaging
Reference
Artery
human, in vitro
Thoracic aorta
Abdominal aorta
Iliac artery
Femoral artery
Liquids
300-940
980-1,420
1,100-3,500
1,230-5,500
For normal physiological conditions of longitudinal
tension and distending blood pressure, below 200 mm
Hg.
Ascending aorta
183-582
M-mode echocardiography in different age groups.
Alessandri et al., 1995
Coronary artery
1,060-4,110
Represents values for various ages (0-80 y.o.) and
atherosclerosis conditions in right and left arteries.
Ozolanta et al., 1998
Equilibrated at intraluminal pressure of 45 mmHg,
media thickness and lumen diameter were measured in
the applied 3 -140 mmHg intraluminal pressure range.
Intengan et al., 1998
rat, in vitro
mesenteric small arteries
aortic wall
porcine, in vitro
thoracic aorta
intima-medial layer
adventitia layer
ascending aorta
intima-medial layer
adventitia layer
descending aorta
intima-medial layer
adventitia layer
Contrast Mechanism in
Other Imaging Modalities
100-1,500
700-1,600
All Soft Tissues
Range includes measurements for normal vs
spontaneously hypertensive animals.
248
69
Bending experiments were used to impose various
strains on different layers of tested blood vessels.
spatial distribution of the absorption (density)
Bone
102 103 104 105 106 107 108 109 1010
Bulk Modulus (Pa)
MRI:
proton spin density and relaxation time constants
Marque et al., 1999
43
4.7
447
112
Computerized Tomography:
McDonald, 1974
Fung, 1993
Shear Modulus (Pa)
Blood
Vitreous Humor
Dermis
Connective Tissue
Contracted Muscle
Palpable Nodules
Glandular Tissue of Breast
Liver
Relaxed Muscle
Fat
Bone
Ultrasound Imaging:
variation in acoustical impedance
(bulk modulus and density)
Epidermis
Cartilage
Cornea
Optical Imaging:
refraction index, absorption/scattering
Sarvazyan et al, 1995
5
Strain
Theory of Elasticity
Stress
1-D deformation
L0
•
•
•
•
•
•
Strain
Stress
Constitutive relationships
Equations of equilibrium
Equations of motion (wave equation)
References
L=L0+DL
Strain Tensor
1-D example
  11  12  13 


  21  22  23 


 31  32  33 
L0
General:
Lagrangian strain:
1  u
1  L2  L20 

  
2  L20 
u
u u 
 ij   i  j  k k 
2  x j xi xi x j 
Symmetry:
Other strains:
L  L0 L  L0 1  L  L 

;
; 
L0
L
2  L2 
2
2
0
Small deformations:
DL
1  L2  L20  L  L0 1  L2  L20 


 1  
 
L0
2  L20 
L0
2  L2 
 ij   ji
Normal strain components:
11;  22;  33
Shear strain components:
12   21;  23   32;  31  13
F
Stress Tensor
F
  11  12  13 


  21  22  23 


 31  32  33 
S0
S
L=L0+DL
Symmetry:
 ij   ji
Cauchy stress:
F

S
Other stresses:
F F L0
;
S0 S0 L
Small deformations:
DL
F F F L0
 1  

L0
S S0 S0 L
Normal and Shear Stresses
Normal stress components:
 11; 22; 33
Shear stress components:
12   21; 23   32; 31  13
6
Constitutive Relations
General Concept
Uniaxial Deformation
L0
F
F
 ij 
S0

 ij
n 
n – Poisson’s ratio
This is not a definition of a Poisson’s ratio
Stress
 22
11
1
  0  Cijmn ij mn
2

81
constants
Cijmn  C jimn
Cijmn  Cijnm
Cijmn  Cmnij
Examples:
Rubber – Mooney-Rivlin function
Foam – Blatz-K(a)o potential
Tissue – ??? (Fung 1981)
l
2
1-D example
F
one plane (monoclinic) – 13
two orthogonal planes (orthotropic) – 9
one axis (transversely isotropic) – 5
(cubic) – 3
 11( x)  const 
 11
0
x1
Remarks:
Nonlinearity
  ( I1 , I 2 , I 3 )
Stress-strain relations:
Nonlinear
 x
i
 fi  0
j
fi – body (volumetric) forces, e.g., gravity
Constitutive (stress-strain) relations:
Two elastic constants (l and m, for example)
are needed to describe linear isotropic material
Strain energy:
Equilibrium (Newton’s 2nd law)
 ij
 ij  l ii ij  2m ij
21
constants
General
F
 ii2  m ij2
Stress-strain relations:
Symmetry:
Strain
Equations of Equilibrium
Hooke’s Law
Strain energy (linear, isotropic material):
  0 
Stress-strain relations:
 ij  Cijmn mn
L=L0+DL
 11  E11
E – Young’s modulus
Generalized Hooke’s Law
Strain energy (linear, anisotropic material):
 – strain energy function or potential
S
Constitutive Relations
• Generally, both elastic moduli are functions of
coordinate: l=l(x1,x2,x3) and m=m(x1,x2,x3)
• In forward problem formulation, there are
15 equations and 15 unknowns (ij, ij, ui)
• The existence and uniqueness of the solution
can be demonstrated
 ij 

e.g.,  ij  l ii ij  2m ij 
 ij
Strain-displacement relations:

1 u
 ij   i 
2  x j
u j
xi

uk uk 
xi x j 
Derive 1-D equation of equilibrium
for E=const
7
Equations of Equilibrium
Small deformations of linear, isotropic (i.e., Hookean) material

u u u

u

u u

u u
(l ( 1  2  3 )) 
(2m 1 ) 
( m ( 1  2 )) 
( m ( 1  3 ))  f1  0
x1
x1 x2 x3
x1
x1 x2
x2 x1
x3
x3 x1

u u u

u u

u

u u
(l ( 1  2  3 )) 
( m ( 2  1 )) 
(2m 2 ) 
( m ( 2  3 ))  f 2  0
x2
x1 x2 x3
x1
x1 x2
x2
x2
x3
x3 x2
u u
u u
u

u u u



(l ( 1  2  3 )) 
( m ( 3  1 )) 
( m ( 3  2 )) 
(2m 3 )  f 3  0
x3
x1 x2 x3
x1
x1 x3
x2
x2 x3
x3
x3
Boundary (force and/or displacement) conditions


   ij n j  Fi  (ui  ui0 )  0


 j

 11n1   12n2   13n3  0
n=(n1,n2,n3) – unit normal vector at the surface
ui0 – surface displacement components
Free surface (i.e., Fi=0):   ij n j  0   21n1   22n2   23n3  0
j
 31n1   32n2   33n3  0
u1  u
Displaced surface:
0
1
0
2
0
3
 (ui  ui0 )  0  u2  u
Mixed boundary conditions:
 11n1   12n2   13n3  F1
u2  0
u3  0
To describe elastic properties of
linear isotropic material,
how many moduli of elasticity are required:
To describe elastic properties of linear isotropic
material, how many moduli of elasticity are required:
Generalized Hooke’s Law
Strain energy (linear, anisotropic material):
1
  0  Cijmn ij mn
2
21%
7%
29%
21%
21%
1.
2.
3.
4.
5.
One modulus
Two moduli
Three moduli
Four moduli
Five moduli
  0 
Stress-strain relations:
Cijmn  C jimn
Cijmn  Cijnm
Cijmn  Cmnij
l
2
 ii2  m ij2
Stress-strain relations:
 ij  l ii ij  2m ij
 ij  Cijmn mn
81
constants
Hooke’s Law
Strain energy (linear, isotropic material):
21
constants
Symmetry:
one plane (monoclinic) – 13
two orthogonal planes (orthotropic) – 9
one axis (transversely isotropic) – 5
(cubic) – 3
Two elastic constants (l and m, for example)
are needed to describe linear isotropic material
Nonlinearity
Strain energy:
  ( I1 , I 2 , I 3 )
Stress-strain relations:
Nonlinear
u3  u
8
Viscosity
Mechanical Properties of Tissue
Deformation
Creep
Time
Equilibrium (Newton’s 2nd law)
 ij
 x
i
j
 fi  
 2 ui
t 2
Loading Cycles
Constitutive (stress-strain) relations:
Stress
 ij  l ii ij  2m ij  
Stress Relaxation
Stress
Load
Deformation
Other (e.g., anisotropy, pseudoelasticity)
Creep
Constant Load
Hysteresis loops:
independent of the rate of loading
(most soft tissues)

 ii
 ij  2 ij
t
t
12
• Bulk and shear viscosities are introduced
Strain
Constant Deformation
8
1st cycle
Cornea
Stress Relaxation
Time
• Is there a characteristic time?
0
• Four (4) parameters are needed to
describe mechanical properties of the
viscoelastic material
5th
4
30
60
90
>10th
Load
Viscosity (e.g., bulk and shear viscosities)
•
 – shear viscosity
 – bulk viscosity
Force (N)
Elasticity (e.g., bulk and shear moduli)
Nonlinearity (e.g., strain hardening)
Viscosity  Equations of Motion
• Shear viscosity: shear waves
• Bulk viscosity: longitudinal ultrasound waves
• Viscoelastic models:
Maxwell, Voigt, Kelvin, KVFD, etc.
cycle
• There are various models to describe
constitutive relations, e.g., Voight model,
Maxwell model, Kelvin model, etc.
Time (min)
Elongation
9
To describe viscoelastic properties of
linear isotropic material,
how many parameters are required:
To describe viscoelastic properties of linear isotropic
material, how many parameters are required:
 ij
 x
i
0%
23%
18%
0%
18%
1.
2.
3.
4.
5.
One parameter
Two parameters
Three parameters
Four parameters
Five parameters
Mechanical Properties of Tissue
Equilibrium (Newton’s 2nd law)
j
 fi  
 2 ui
t 2
Elasticity (e.g., bulk and shear moduli)
Constitutive (stress-strain) relations:
 ij  l ii ij  2m ij  

 ii
 ij  2 ij
t
t
 – shear viscosity
 – bulk viscosity
Viscosity (e.g., bulk and shear viscosities)
• Bulk and shear viscosities are introduced
• Four (4) parameters are needed to
describe mechanical properties of the
viscoelastic material
• There are various models to describe
constitutive relations, e.g., Voight model,
Maxwell model, Kelvin model, etc.
Nonlinearity (e.g., strain hardening)
Other (e.g., anisotropy, pseudoelasticity)
10
Nonlinearity  Constitutive Relations
General Concept
Uniaxial Deformation
L0
F
F

 ij 
 ij
S0
 – strain energy function or potential
S
•
•
Most soft tissues exhibit strain hardening
Tissues of organs with primary “mechanical” functions (muscle, skin,
tendon, etc.)
Other tissues (kidney, brain, blood clot, etc.)
Strain hardening is more common although strain softening is possible
•
•
L=L0+DL
 22
11
n – Poisson’s ratio
This is not a definition of a Poisson’s ratio

Strain
Examples:
Rubber – Mooney-Rivlin function
Foam – Blatz-K(a)o potential
Tissue – ??? (Fung 1981)
Vessel Composition (Fung, 1988):
muscle (soft), elastin (soft), collagen (hard)
Strain hardening in kidney samples
(Erkamp et al, 1998)
80
40
Stress (kPa)
E – Young’s modulus
n 
Stress
 11  E11
Strain hardening
Strain hardening
30
Collagen
40
20
10
Elastin
0
1.0
1.1
1.2
Stretch ratio (L/L0)
1.3
0
5
10
15
Strain [%]
Wellman et al., 1999
11
Strain hardening
Mechanical Properties of Tissue
This and other graphs as well as other literature data suggest that tissue
strain hardening (or nonlinearity in stress-strain relations) can be used for
tissue analysis including composition, differentiation, etc.
Anisotropy  Constitutive Relations
Generalized Hooke’s Law
Strain energy (linear, anisotropic material):
Elasticity (e.g., bulk and shear moduli)
1
  0  Cijmn ij mn
2
  0 
Stress-strain relations:
Viscosity (e.g., bulk and shear viscosities)
Nonlinearity (e.g., strain hardening)
Other (e.g., anisotropy, pseudoelasticity)
Cijmn  C jimn
Cijmn  Cijnm
Cijmn  Cmnij
l
2
 ii2  m ij2
Stress-strain relations:
 ij  l ii ij  2m ij
 ij  Cijmn mn
81
constants
Hooke’s Law
Strain energy (linear, isotropic material):
21
constants
Symmetry:
one plane (monoclinic) – 13
two orthogonal planes (orthotropic) – 9
one axis (transversely isotropic) – 5
(cubic) – 3
Two elastic constants (l and m, for example)
are needed to describe linear isotropic material
Nonlinearity
Strain energy:
  ( I1 , I 2 , I 3 )
Stress-strain relations:
Nonlinear
Krouskop et al, 1998
Ophir et al., 2001
12
To describe elastic properties of
linear anisotropic material, how many
parameters are required:
Anisotropy
•
•
Arterial wall: orthotropic material – 9 constants
two orthogonal planes of symmetry
Muscle: transversely isotropic – 5 constants
axis of symmetry
Isotropic – 2 constants
16
Stress (kPa)
•
Along
fibers
12
Across
fibers
8
4
0
0
5
10
Strain (%)
15
20
To describe elastic properties of linear anisotropic
material, how many parameters are required
Generalized Hooke’s Law
Strain energy (linear, anisotropic material):
8%
4%
4%
8%
21%
1.
2.
3.
4.
5.
One parameter
Two parameters
Nine parameters
Twenty one parameters
Infinite set of parameters
1
  0  Cijmn ij mn
2
  0 
Stress-strain relations:
Cijmn  C jimn
Cijmn  Cijnm
Cijmn  Cmnij
l
2
 ii2  m ij2
Stress-strain relations:
 ij  l ii ij  2m ij
 ij  Cijmn mn
81
constants
Hooke’s Law
Strain energy (linear, isotropic material):
21
constants
Symmetry:
one plane (monoclinic) – 13
two orthogonal planes (orthotropic) – 9
one axis (transversely isotropic) – 5
(cubic) – 3
Two elastic constants (l and m, for example)
are needed to describe linear isotropic material
Nonlinearity
Strain energy:
  ( I1 , I 2 , I 3 )
Stress-strain relations:
Nonlinear
13
Mechanical Properties of Tissue:
References
Theory of Elasticity: References
• Landau LD and Lifshitz EM. Theory of elasticity. Pergamon
Press; New York, 1986
• Saada AS. Elasticity Theory and Applications. Krieger
Publishing Company; Malabar, Florida, 1993
• Nowazki W. Dynamics of elastic systems. Wiley;
New York, 1963
• Nowazki W. Thermoelasticity. Pergamon Press;
New York, 1983
•
Duck, F.A. Physical properties of tissues. Academic press; New York, 1990
•
Fung YC. Biomechanics – mechanical properties of living tissues. SpringerVerlag; New York, 1981
•
Krouskop TA, Wheeler TM, Kallel F, Garra BS, Hall TJ, “The elastic moduli
of breast and prostate tissues under compression,” Ultrasonic Imaging vol. 20
pp. 260-274, 1998
•
Aglyamov SR, Skovoroda AR, “Mechanical properties of soft biological
tissues,” Biophysics, Pergamon, 45(6):1103-1111, 2000.
•
Sarvazyan AP, “Elastic properties of soft tissues,” In: Handbook of Elastic
Properties of Solids, Liquids and Gases, Volume III, Chapter 5, 107-127, eds.
Levy, Bass and Stern, Academic Press, 2001
•
Other text books and archival publications
• There are many other textbooks and archival publications
Elasticity Imaging – Approaches
Static (strain-based, or reconstructive)
imaging internal motion under static deformation
Dynamic (wave-based)
imaging shear wave propagation
Mechanical (stress-based, also reconstructive)
measuring tissue response at the surface
14
Elasticity Imaging – Approaches
How … Static Elasticity Imaging?
How … Static Elasticity Imaging?
Displacement
Displacement
Strain Amplitude
Strain
Static (strain-based, or reconstructive)
Soft
imaging internal motion under static deformation
Dynamic (wave-based)
Hard
imaging shear wave propagation
Mechanical (stress-based, also reconstructive)
Range
Range
measuring tissue response at the surface
15
How … Static Elasticity Imaging?
Static or Reconstructive
Ultrasound Elasticity Imaging
Elasticity Reconstruction:
back to Equations of Equilibrium
Small deformations of linear, isotropic (i.e., Hookean) material
B-Scan
Displacements
Strains
Elasticity
• Capture data during deformation

u u u

u

u u

u u
(l ( 1  2  3 )) 
(2m 1 ) 
( m ( 1  2 )) 
( m ( 1  3 ))  f1  0
x1
x1 x2 x3
x1
x1 x2
x2 x1
x3
x3 x1

u u u

u u

u

u u
(l ( 1  2  3 )) 
( m ( 2  1 )) 
(2m 2 ) 
( m ( 2  3 ))  f 2  0
x2
x1 x2 x3
x1
x1 x2
x2
x2
x3
x3 x2
• Estimate displacement vector (3-D)
u u
u u
u

u u u



(l ( 1  2  3 )) 
( m ( 3  1 )) 
( m ( 3  2 )) 
(2m 3 )  f 3  0
x3
x1 x2 x3
x1
x1 x3
x2
x2 x3
x3
x3
• Compute strain tensor (3-D)
Eliminating “non-measurable” parameters and assuming “2-D” case:
  
u1

u u    
u2 u1

u 
)
( m ( 1  2 )) 

)) 
(2m 2 )  0
 (2m
 (m (
x2  x1
x1 x2
x2 x1  x1  x1
x1 x2
x2
x2 
• Reconstruct mechanical properties
Deform and image
Track internal motion
Evaluate deformations
Reconstruct
Young’s or shear
elastic modulus
Second-order hyperbolic partial differential equation:
mechanical boundary conditions are required along two intersecting characteristics
no assumptions are made regarding either l(x1,x2) or Poisson’s ratio
16
L0
Elasticity Imaging – Approaches
F
F
S0
S
Theory of Elasticity
(Dynamic Approach)
Example: plane waves
• Infinite homogeneous (l,m=const) elastic medium (i.e., ignore bulk and shear
viscosities), no body forces (fi=0)
L=L0+DL
Static (strain-based, or reconstructive)
imaging internal motion under static deformation
Dynamic (wave-based)
imaging shear wave propagation
General (3-D) case
• Strain
11 
1  u
L  L0
L0
• Stress
 11 
u
F
S
 11  E11
E – Young’s modulus
 ij  l ii ij  2m ij  
measuring tissue response at the surface
• Equations of motion
 11
 2u
  21
x1
t

u u u

u

u u

u u
 2u
(l ( 1  2  3 )) 
(2m 1 ) 
( m ( 1  2 )) 
( m ( 1  3 ))   21
x1
x1 x2 x3
x1
x1 x2
x2 x1
x3
x3 x1
t
 ij
 x
i
j
 fi  

u u u

u u

u

u u
 2u
(l ( 1  2  3 )) 
( m ( 2  1 )) 
(2m 2 ) 
( m ( 2  3 ))   22
x2
x1 x2 x3
x1
x1 x2
x2
x2
x3
x3 x2
t
  11  12  13 


  21  22  23 


 31  32  33 
  11  12  13 


  21  22  23 


 31  32  33 
• Constitutive relationships
Mechanical (stress-based, also reconstructive)
• Assume that u1=u1(x1,t), u2=u2(x1,t), and u3=u3(x1,t)
u u 
 ij   i  j  k k 
2  x j xi xi x j 
 2ui
t 2

 ii
 ij  2 ij
t
t
u u
u u
u
 2u

u u u



(l ( 1  2  3 )) 
( m ( 3  1 )) 
( m ( 3  2 )) 
(2m 3 )   23
x3
x1 x2 x3
x1
x1 x3
x2
x2 x3
x3
x3
t
l  2m   u21    u21  0
m
2
2
x1
t
 2 u 2 ( 3)
x12

 2 u 2 ( 3)
t 2

 2u1 1  2u1
l  2m

 0 where cl 
x12 cl2 t 2

0 
 2 u 2 ( 3)
x12

Longitudinal wave
(ultrasound)
2
1  u 2 ( 3)
m
 0 where c t 
ct2 t 2

Shear waves !
17
Speed of shear waves is most closely related to:
Speed of shear waves is most closely related to
• Infinite homogeneous (l,m=const) elastic medium (i.e., ignore bulk and shear
viscosities), no body forces (fi=0)
Shear Wave
Imaging
• Assume that u1=u1(x1,t), u2=u2(x1,t), and u3=u3(x1,t)
18%
11%
7%
11%
11%
1.
2.
3.
4.
5.
Poisson’s ratio
Bulk modulus
Young’s modulus
Compressibility
Shear viscosity

u u u

u

u u

u u
 2u
(l ( 1  2  3 )) 
(2m 1 ) 
( m ( 1  2 )) 
( m ( 1  3 ))   21
x1
x1 x2 x3
x1
x1 x2
x2 x1
x3
x3 x1
t

u u u

u u

u

u u
 2u
(l ( 1  2  3 )) 
( m ( 2  1 )) 
(2m 2 ) 
( m ( 2  3 ))   22
x2
x1 x2 x3
x1
x1 x2
x2
x2
x3
x3 x2
t
u u
u u
u
 2u

u u u



(l ( 1  2  3 )) 
( m ( 3  1 )) 
( m ( 3  2 )) 
(2m 3 )   23
x3
x1 x2 x3
x1
x1 x3
x2
x2 x3
x3
x3
t
l  2m   u21    u21  0
m
2
2
x1
t
 2 u 2 ( 3)
x12

 2 u 2 ( 3)
t 2

 2u1 1  2u1
l  2m

 0 where cl 
x12 cl2 t 2

0 
 2 u 2 ( 3)
x12

Longitudinal wave
(ultrasound)
2
1  u 2 ( 3)
m
 0 where c t 
ct2 t 2

Shear waves !
E  3m
Bercoff et al. 2004
18
Supersonic
Shear Wave Imaging
Elasticity Imaging:
Principles
Stanislav (Stas) Emelianov
emelian@mail.utexas.edu
Department of Biomedical Engineering
The University of Texas at Austin
Bercoff et al. 2004
Department of Imaging Physics
The University of Texas M.D. Anderson Cancer Center
19