Download Latest Developments in Diffraction Enhanced Imaging

Interaction of X-rays with
Matter and Imaging
Gocha Khelashvili
Assistant Research Professor of Physics
Illinois Institute of Technology
Research Medical Physicist
EXELAR Medical Corporation
The Plan
•
•
•
•
•
•
X-ray Interactions with Matter Used at Imaging Energies
Photoelectric Effect
Coherent Scattering
Incoherent Scattering
Pair and Triplet Production
Refraction
Small- and Ultra-small Angle Scattering
Radiography
How does it work?
Imaging Parameters and Sources of X-ray contrast
Drawbacks of Radiography
Diffraction Enhanced Imaging (DEI)
How does it work?
Imaging Parameters and Sources of X-ray contrast
Drawbacks of DEI
Multiple Image Radiography (MIR-Planar Mode)
How does it work?
Sources of X-ray contrast
MIR parameters and images
MIR Model Based on Discrete Scatterers
Multiple scattering series approach and MIR transport equation
Solution of MIR transport equation
Imaging Parameters
Summary
Photoelectric Effect
M
L
Photoelectric Absorption
K
Fluorescent X-ray emission
e
A
K
Zn
 h 
M
L
2
m
(cm /atom)
M
L
K
 n  4 at h  0.1 MeV, gradually rising to 4.6 at 3 MeV.
for 
m  3 at hv  0.1 MeV, gradually decreasing to 1 at 5 MeV
for h  0.1 MeV, photoelectric efect most important
e
A
Z4
 h 
3
3

Z
(cm 2 /atom) 
A
(cm 2  g -1 )
3

 h 
K
K
Thompson (Classical) Scattering
d 1 2
 r0 1  cos 2 
d 2


ke2
15
where r0 

2.817

10
m Classical electron radius
2
me c
No energy loss by photon - No recoil by electron.
Thompson (Classical) Scattering
d 1 2
 r0 1  cos 2  2 sin 
d 2


8 2
 0   r0  66.525 1030 m2
3
Rayleigh Scattering (Coherent Scattering)
1 . Photons are scattered by bound electrons
2. Atoms are neither excited or ionized
3. Scattering from different parts of electron cloud - coherent scattering
Rayleigh Scattering (Coherent Scattering)
d 1 2
1

q
2
2
 r0 1  cos   F ( x, Z )  2 sin  where x  sin =
d 2

2 2

F ( x, Z )

F (q , Z )    (r ) exp  iq  r  d 3r - atomic form factor
where  ( r )   ( x, y, z ) - total electron density

sin qr 2
F (q, Z )  4   (r )
r dr - for spherical symmetry
qr
0
Z
F ( x, Z )
F ( q , Z )    0 | exp  iq  rn | 0 
n 1
(atomic scattering factor, atomic structure factor)
 0 - ground state WF calculated from Hartree-Fock theory
R 
Z2
 h 
2
(cm 2 / atom)
Compton Scattering (Incoherent Scattering)
1. Energy is transfered to electron
2. Electron recoils from collision
3. Electron considered at rest before collision (No bounding effects)
4. Electron deposits dose in the medium
Compton Scattering (Incoherent Scattering)
d 1 2  h   h h 

2
 r0 
 sin   - Klein-Nishina Cross Section
  
d  2  h  h
h

d 1 2
 r0 1  cos 2   FKN
d 2
2
2
2


 1  cos  

 
1
FKN  
 1 

2
1   (1  cos  )   1   1  cos    1  cos  
h
h (in MeV)


2
me c
0.511




Effects of Binding Energy in Compton (Incoherent)
Scattering
1. Electrons are in constant motion in atoms (binding effect)
2. Electrons recoil after collision
3. Energy is transfered to electrons
4. X-ray photon looses part of its energy
d inc d KN

 S  x, Z 
d
d
S (q , Z )   F (q , Z ) and S (q , Z )  
2
 0
S ( x, Z )
Z
 exp iq  r  0
Z
j 1
j
S ( q , Z ) - incoherent scattering function
Z
S  q , Z     0 | exp iq (rm  rn )  |  0   F (q , Z )
m 1 n 1
2
Effects of Binding Energy in Compton (Incoherent)
Scattering
 inc



1
  1  cos2   FKN  S ( x, Z )  2 sin  d
2
0
Pair Production
h
 2 p cos 
c
h
v
 2 p cos   h  2 pc cos   2mvc cos   2mc 2  cos  , where  
c
c
  1 and cos   1  h  2mc 2 (contradiction)
Pair production cannot conserve both energy and momentum unless some other object is
involved in the process to carry away part of initial photon momentum
Energy Conservation - h  2mc 2 ; Momentum Conservation:
Pair Production in Nuclear Coulomb Force Field
Pair Production in Nuclear Coulomb Force Field
Bethe - Heitler Cross Section - EGS4, EGSnrc, GEANT4
d ( k , Z , E  )

dE 

 r02 Z 2 
k
4
4
 2


2
2 
 E  E 1    ln Z  4 f c  Z , k    E E 2    ln Z  4 f c  Z , k   
3
3

 3



1
km
=
- fine structure constant, r0 - classical radius of electron,  =Z-1/3
137
 E E


f c  Z , k  - Coulomb correction, 1   and 2   functions acounting for screening effects
Threshold Photon Energy is: h  2me c 2

Average energy of e and e

hv  2me c 2
 E
2
Pair Production in the Electric Field (Triplet Production)
m
m0
1 
2
,
  
1 
- Doppler Effect
1 
h min  4me c 2
Radiography Setup
h  10 KeV are absorbed by primary collimators

Average eneregy of beam increases - "hard" x-rays

penetrate deeper
Radiography Setup and Imaging Principles
x
Radiology Setup
Object
Double Crystal
Monochromator
Si(333)
z
y
Incident X-ray beam
Area Detector
Attenuation Law
I A ( x, y , z )  I 0 e   ( x , y ) z
Image Contrast
Image
1 I A ( x, y , z )
 ( x, y, z )   ln
z
I0
I A ( x, y )
  x, y  
IA
Drawbacks of Radiography
Incoherently
Scattered Beam
x
x
Detector
Pixel
Object Pixel
Attenuated
Beam
y
I R  I A  I Scat
Image Contrast
(by absorption)
y
1
z
 ( x, y )   ln
I A  I Scat

I A  I Scat
I A  I Scat
I0
DEI Setup and Imaging Principles
x
z
DEI Setup
Area Detector
Object
Double Crystal
Monochromator
Si(333)
Analyzer Crystal
Si(333)
Incident X-ray beam
y
Formation of DE Images
x
Incoherently
Scattered Beam is
Blocked by Crystal
Detector
Pixel
x
B
Object Pixel
y
Enhanced
Attenuated Beam
y
Physics of DEI
Pisano, Johnston(UNC); Sayers(NCSU); Zhong (BNL);
Thomlinson (ESRF); Chapman(IIT)
Low Angle
Side
High Angle
Side
Relative Intensity I/Io
1.00
0.80
0.60
0.40
0.20
0.00
-10
-5
0
5
10
Analyzer Angle (radians)
Data from NSLS X27
Calculation of DEI Images
D
Low Angle
Side
High Angle
Side
1.00
Relative Intensity
I/Io
0.80
0.60
L  B 
D
0.40
H  B 
0.20
2
0.00
-10
I L  I R R  L   Z   I R [ R  L  
I R  x, y  
-5
0
5
Analyzer Angle (rad)
dR
 L   Z ]
d
 Z  x, y  
R( L )
( H )  R( H )
dR
d
I H  I R R  H   Z   I R [ R  H  
( L )
I H  x, y  R ( L )  I L  x, y  R ( H )
I L  x, y 
dR
d
( H )  I H  x, y 
2
10
I L  x, y  ddR ( H )  I H  x, y  ddR ( L )
dR
d
D
dR
d
( L )
- Absorption
- Refraction
dR
 H   Z ]
d
Comparison - Conventional and DEI
ACR - Phantom
610 - 054
Map
Conventional
DEI
ACR Phantom (Gammex RMI - Model 156) - tumor-like masses, microcalsifications,
cylindrical nylon fibrids  40-45 mm thick compressed breast.
Conventional Radiography - Synchrotron at 18 keV.
DEI image of ACR phantom
- smallest calcifications
Data from NSLS X27
Cancer in Breast Tissue
Pisano, Johnston(UNC); Sayers(NCSU); Zhong (BNL); Thomlinson (ESRF); Chapman(IIT)
Conventional
DEI - Absorption
BNL Sept 1997
DEI - Refraction
Drawbacks of DEI
Detector
Pixel
Object Pixel
x
x
y
y
Experimental Evidence of Problems in
DEI
18 keV x-ray beam
-9.6 to 8.8  rad (0.8  rad)
1256  444 pixels image
50 m  50 m each pixel
I ( , x, y ) 

 I  ( ) f (   , x, y)d   I  ( )  f ( , x, y)
0
0

I0    I0   R1   R2  
g  , x, y   I 0    f  , x, y   RA  
M. Wernick et al "Multiple - Image Radiography" Phys. Med. Biol. 48 (2003) 3875
Experimental Results
f ( ; x, y )
f ( ; x, y )
600
Rod, off-center
Background
600
400
400
200
200
-1
-1
-0.6
-0.2
0.2

0.6
-0.6
-0.2
0.2

0.6
1-5
x 10
1
-5
f ( ; x, y )
x 10
f ( ; x, y )
600
600
Thick Paper
Rod and Paper
400
400
200
200
-1
-1
-0.6
-0.2

0.2
0.6
1-5
x 10
-0.6
-0.2

0.2
M. Wernick et al "Multiple - Image Radiography" Phys. Med. Biol. 48 (2003) 3875
0.6
1-5
x 10
Refraction images
Profiles
no paper
1
MIR
0.8
MIR
0.6
0.4
DEI
DEI
thin paper
DEI
0.2
thick paper
0
0
50
100
150
200
Position (pixels)
M. Wernick et al "Multiple - Image Radiography" Phys. Med. Biol. 48 (2003) 3875
a( x, y )   ln
T  x, y 
I0
 1 

  ln    f  , x, y  d  - Attenuation Image
 I 0 

 f  , x, y   R   
  T ( x, y)  I0  d - Refraction Image

r ( x, y) 

w( x, y) 
   r ( x, y) 

2
f  , x, y 
T  x, y 
d - Ultra-Small Angle Scattering Image
M. Wernick et al "Multiple - Image Radiography" Phys. Med. Biol. 48 (2003) 3875
Generalization to CT Reconstruction
I A  I 0e  z
I0
I A  I 0 e  ( 1  2 
I0
1  2  3  4
z
z
N
z
z
I A ( x, y, z )  I 0e 1z e 2z e 3z
e 4z  I 0e

N
 n z
n1
 IA  N
p( x, y)   ln     n z    ( x, y, z)dz
 I 0  n1
L
r ( x, y)   gradr ln n( x, y, z ) dz
L
 grad   ( x, y, z) dz
r
L
w( x, y )? ? z
N ) z
A Physical Model of MIR
Khelashvili, Brankov (IIT), Chapman (U.Sask), Anastasio, Yang (IIT), Zang (BNL), Wernick (IIT)
Phys. Med. Biol. 51 (2006) 221-236
Object Voxel
- Scattering Centers -  a ,  s , nsp
- Nonscattering Medium  ( z ), grad n
Multiple Ultra-Small Angle Scattering
• Radiation Transport Theory Approach
I (r , sˆ )  I (r ,  ,  ) - Specific Intensity (radiance or brightness)
Radiation intensity of x-ray beam in at position r in the direction
sˆ  (sin  cos  ,sin  sin  , cos  )
ds
 ds - particles in ds volume
d
sˆ
d
p( s , s)- phase function - fraction of the radiation
sˆ
 

scattered from d  sˆ into d sˆ .
MIR Radiation Transfer Equation
sˆ  r I (r , sˆ )    (r ) I (r , sˆ )   n ( r ) a I ( r , sˆ )   n ( r ) s I ( r , sˆ ) 
 n ext
ˆ
ˆ
+b (r , s )  sˆ I ( r , s ) 
4

4
p( sˆ , sˆ)I ( r , sˆ) d  
Ultra-Small Angle Approximation

d
I ( z,  , s )  s t I ( z,  , s )     n ext    I ( z,  , s )  b ( z,  , s )
I ( z,  , s ) 
z
ds
n ext

4
 

p (s  s)I ( z ,  , s)d 2 s
 


r    zk  xi  yj  zk , t  i 
j , s  sin  cos  i  sin  sin  j
x
y
d 2 s  dsx ds y
b  r ln n( x, y)
n
n
n( x, y)  n0  x 
y  n0  nx x  n y y
x
y
General Solution


1


I ( z,  , s ) 
d

d
q
exp

(
i




is

q
)

exp

(
ibq

ib

z
)
z



2 



(2 ) 
2



1
2
2
 exp  (is   n t   ) z   F0 ( z,  , q ) K ( z ,  , q )
F0 ( , q )   I 0 (  , s ) exp(i    is  q )d 2  d 2 s
  n ext z

K ( z ,  , q )  exp 
P(q   z )dz 

 4 0

 
P(q ) 

 
p( s ) exp(is  q )d 2 s
Phase Function
a
a
p( s , s) 
d
4 2 a 2

d
4 2  s 2


 s d
 a2 d 
a  103  106 
1 N 2
nsp  1    1 
re 
2 V
  105
2
2
2
2
2
2
,
s

s

s

sin



,
d
s  dsx ds y  sin  d d   d d ;
x
y
2
 s d
s
4 2
4 2
p( s , s)  2
 p( s)  4
 4W0
2
2
2
2
2
2
 a d
 ext  4  s 
 4  s 
Phase Function
 
2
p
2
2
|
s
|
p
(
s
)
d
s

2
p
(
s
)
d
s



p( s)  4 pW0 exp  p s ,
2
1
2 
 4 ln   1


2
2
2
|
s
|
p
(
s
)
d
s

 p(s)d
1
p  2 
p
2 
2
4 ln   1


2
s

1
p
Plane Wave Solution
I 0 (  , s )  I 0δ( s )
 

I
I ( z ,  , s )  0 2 exp( )   exp iq ( s  bz )
(2 )
 


( W0 ) k exp(kq 2 / 4 p )
k!
k 0
s
W0 
,
 ext
   (  n ext   ) z and    n ext z,
 
  exp  iq (s  bz )   exp(kq
 
2
/ 4 p )d q 
2
4 p
k
s  bz  ( s  bz ) 2x  ( s  bz ) 2y
2
 p
2
exp  
s  bz ,
 k

d 2q
Plane Wave Solution
I ( z,  , s ) 
I0


exp( )
k 0
( W0 )k  p
k k!

 p
exp  
s  bz
 k
2

if  W0   n s z  0 (   (  n ext   ) z and    n ext z )
 ext   a   s
I ( z,  , s )  I 0 exp(   z)δ( s  bz)



Imaging Parameters
IT      I ( z,  , s ) d 2 s  I 0 exp     n a    z 
2
 IT
AMIR (  )   ln 
 I0
 z
     n  z   a    z   dz - Absorption Image
 0
z
1
 x ( x, y, z ) 
IT
2
s
I
(
z
,

,
s
)
d
s   bx ( z )dz;
 x
1
 y ( x, y, z ) 
IT
 s I ( z,  , s )d
2
0
y
2
Refraction - Angle Image
z
2
s   by ( z )dz;
0
  n ( z ) s ( z ) 
2 s  bz I ( z,  , s )d s  0   p  d z


Ultra-Small Angle Scattering Image
1
w( x, y, z ) 
Ir


2
z
2
Experimental Conformation
Lucite container – wedge shaped.
Polymethylmethacrylate (PMMA) microspheres in glycerin.
Experimental Conformation
labDEI System
Morrison, Nesch, Torres, Khelashvili (IIT), Hasnah (U. Qatar) Chapman (U.Sask)
X-ray Source
Detector
Analyzer
Pre-mono &
Mono
Lab DEI System
tissue images
1cm
Morrison, Nesch, Torres, Khelashvili, Chapman (IIT)
Muehleman (Rush Medical College)
Human tissue image using prototype
cartilage
laboratory DEI system using Mo K
bone
(17.5keV) radiation. Image is of a
section of a knee joint immersed
in formalin showing cartilage.
Summary
• First reliable Theoretical Model of DEI – MIR has been developed that Model
can be used to simulate experiments starting from source, through crystals
(this was known), through object (was unknown), through analyzer crystal
(partially known – dynamical theory of diffraction – but crystal and beam
specific calculations need to be done).
• Monte – Carlo simulation of DEI/MIR imaging system (G.Khelashvili – U.
Saskatchewan / Nesch LLC)
• CT reconstructions – some steps are already taken in this direction – J.
Brankov at all “A computed tomography implementation of multiple-image
radiography” Med. Phys. 33 (2006) 278-289
• CSRRI (IIT) / Nesch LLC – are developing in-lab research DEI instrument
Acknowledgements
L.D. Chapman (Anatomy and Cell Biology, University of Saskatchewan, Canada)
J. Brankov, M. Wernick, Y. Yang, M. Anastasio (Biomed. Engineering, IIT)
T. Morrison (CSRRI, IIT) and I. Nesch (Nesch LLC)
C. Muehleman (Department of Anatomy and Cell Biology, Rush Medical College)