Interaction of X-rays with Matter and Imaging Gocha Khelashvili Assistant Research Professor of Physics Illinois Institute of Technology Research Physicist.

Interaction of X-rays with Matter and Imaging

Gocha Khelashvili

Assistant Research Professor of Physics

Illinois Institute of Technology

Research Physicist

EXELAR Medical Corporation

The Plan• X-ray Interactions with Matter Used at Imaging Energies Photoelectric Effect Coherent Scattering Incoherent Scattering 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

• Laboratory DEI / MIR Machine• Summary

Photoelectric Effect

K

L

M

L

L

KK

MM

Kα Kβ

Photoelectric Absorption

Fluorescent X-ray emission

( )

42

3

0.1 MeV (cm /atom) Z

h Ah

ν τν

≤ → ∝

Thompson (Classical) Scattering

( )2

2 2 150 0 2

11 cos where 2.817 10 m Classical electron radius

2

No energy loss by photon - No recoil by electron.e

d ker r

d m c

σθ −= + = = ×

Ω

Thompson (Classical) Scattering

( )2 2 2 30 20 0 0

1 81 cos 2 sin 66.525 10

2 3

dr r m

d

σθ π θ σ π

θ−= + = = ×

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)

( )[ ]

( )

22 20

3

2

1 1 q1 cos ( , ) 2 sin where sin =

2 2 2

( , ) ( , ) ( ) exp - atomic form factor

where ( ) ( , , ) - total electron density

sin ( , ) 4 ( ) - for spheric

dr F x Z x

d

F x Z F q Z r iq r d r

r x y z

qrF q Z r r dr

qr

σ θθ π θ

θ λ

ρ

ρ ρ

π ρ

= + =

= ⋅

=

=

∫

h

r r r r r:

r

( )

0

0 01

0

al symmetry

( , ) ( , ) |exp |

(atomic scattering factor, atomic structure factor)

- ground state WF calculated from Hartree-Fock theory

Z

nn

F x Z F q Z iq r

∞

=

= ⟨Ψ ⋅ Ψ ⟩

Ψ

∫

∑r r r:

( )

22

2 (cm / )R

Zatom

hσ

ν∝

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)

( )( )

( ) ( )

2 20

2 20

2 22

2

2

1sin - Klein-Nishina Cross Section

2

11 cos

2

1 cos11

1 (1 cos ) 1 1 cos 1 cos

(in MeV)

0.511

KN

KN

e

d h h hr

d h h h

dr F

d

F

h h

m c

σ ν ν νϕ

ν ν ν

σθ

α θ

α θ α θ θ

ν να

′ ′⎛ ⎞⎛ ⎞= + −⎜ ⎟⎜ ⎟′Ω ⎝ ⎠⎝ ⎠

= + ×Ω

⎧ ⎫−⎧ ⎫ ⎪ ⎪= +⎨ ⎬ ⎨ ⎬

+ − ⎡ ⎤+ − +⎩ ⎭ ⎪ ⎪⎣ ⎦⎩ ⎭

= =

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

inc KNd d

d d

σ σθ θ

= ( )

( )

( ) [ ]

2

0 1

20 0

1 1

,

( , ) ( , ) and ( , ) exp 0

( , ) ( , ) - incoherent scattering function

, |exp ( ) | ( , )

Z

jj

Z Z

m nm n

S x Z

S q Z F q Z S q Z iq r

S x Z S q Z

S q Z iq r r F q Z

εε

ε> =

= =

×

= =⟨ ⋅ ⟩

= ⟨Ψ − Ψ ⟩−

∑ ∑

∑∑

r r r r r

r:

r r r r r

Effects of Binding Energy in Compton (Incoherent) Scattering

( )2

0

11 cos ( , ) 2 sin

2inc KNF S x Z dπ

σ θ π θ θ= + × × ×∫

Radiography Setup

10 KeV are absorbed by primary collimators

Average eneregy of beam increases - "hard" x-rays

penetrate deeper

hν ≤

⇓

⇓

Radiography Setup and Imaging Principles

0

( , , )1( , ) ln AI x y zx y

z Iμ =−

( , )0( , , ) x y z

AI x y z I e μ−=

Double Crystal Monochromator

Si(333)

Object

Radiology Setup

Area Detector

Incident X-ray beam

Attenuation Law Image

z

x

y

A

A

I

I

ΔΣ =Image Contrast

Drawbacks of Radiography

yΔ

xΔ

R A ScatI I I= +

Object Pixel

yΔ

xΔ

Detector Pixel

Attenuated Beam

(by absorption)

Incoherently Scattered Beam

0

1( , ) ln A ScatI Ix y

z Iμ +

=−

A Scat

A Scat

I I

I I

Δ +ΔΣ =

+Image Contrast

DEI Setup and Imaging Principles

z

x

y

Area Detector

Object

Analyzer Crystal Si(333)

DEI Setup

Incident X-ray beam

Double Crystal Monochromator

Si(333)

Formation of DE Images

yΔ

xΔ

Object Pixel

yΔ

xΔ

Detector Pixel

Enhanced Attenuated Beam

Incoherently Scattered Beam is Blocked by Crystal

Bθ

1050-10 -5

Low AngleSide

High AngleSide

0.40

0.20

0.00

0.60

0.80

1.00

Analyzer Angle (μradians)

Physics of DEI

Pisano, Johnston(UNC); Sayers(NCSU); Zhong (BNL); Thomlinson (ESRF); Chapman(IIT)

Rel

ati

ve

Inte

nsi

ty

I/I

o

Data from NSLS X27

Calculation of DEI Images

( ) ( ) ( )[ ]L R L Z R L L Z

dRI I R I R

dθ θ θ θ θ

θ= + Δ = + Δ ( ) ( ) ( )[ ]H R H Z R H H Z

dRI I R I R

dθ θ θ θ θ

θ= + Δ = + Δ

( ) ( ) ( )

( ) ( ) ( )( ) ( )

, ( ) , ( ), - Absorption

( ) ( ) ( ) ( )

, ( ) , ( ), - Refraction

, ( ) , ( )

dR dRL H H Ld d

R dR dRL H H Ld d

H L L HZ dR dR

L H H Ld d

I x y I x yI x y

R R

I x y R I x y Rx y

I x y I x y

θ θ

θ θ

θ θ

θ θ

θ θ θ θ

θ θθ

θ θ

−=

−

−Δ =

−

1050-10 -5

Low AngleSide

High AngleSide

0.40

0.20

0.00

0.60

0.80

1.00

Analyzer Angle (μrad)

Re

lati

ve

In

ten

sit

y

I/

Io2D

L B

θθ θ= −2D

H B

θθ θ= +

Dθ

6 1 0 - 0 5 4

Map Conventional DEI

Comparison - Conventional and DEIACR - Phantom

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

Conventional DEI - Absorption DEI - Refraction

Cancer in Breast Tissue

BNL Sept 1997

Pisano, Johnston(UNC); Sayers(NCSU); Zhong (BNL); Thomlinson (ESRF); Chapman(IIT)

Drawbacks of DEI

yΔ

xΔ

Object Pixel

yΔ

xΔ

Detector Pixel

Experimental Evidence of Problems in DEI

0 0( , , ) ( ) ( , , ) ( ) ( , , )I x y I f x y d I f x yθ θ θ θ θ θ θ+∞

−∞

′ ′′ ′ ′ ′= − = ∗∫

( ) ( ) ( ) ( )0 0 1 2I I R Rθ θ θ θ′ =

1235678

Sheets of paper

Lucite substrate

Lucite rod

Alignment target

4

( ) ( ) ( ) ( )0, , , , Ag x y I f x y Rθ θ θ θ′= ∗ ∗

Experimental Results

-1 -0.6 -0.2 0.2 0.6 1x 10

-5

Rod, off-center

x 10-1 -0.6 -0.2 0.2 0.6 1

-5

Thick Paper

200

400

600

-1 -0.6 -0.2 0.2 0.6 1

x 10-5

Background

200

400

600

x 10-1 -0.6 -0.2 0.2 0.6 1

-5

Rod and Paper

200

400

600

( ; , )f x yθ

( ; , )f x yθ

( ; , )f x yθ( ; , )f x yθ

θ

θ

θ

θ

200

400

600

Refraction images Profiles

thick paper

thin paper

no paper

0 50 100 150 200

0

0.2

0.4

0.6

0.8

1

Position (pixels)

MIR

DEIDEI

MIR

DEI

( ) ( )0 0

, 1( , ) ln ln , , - Attenuation Image

T x ya x y f x y d

I Iθ θ

+∞

−∞

⎡ ⎤= − = − ×⎢ ⎥

⎣ ⎦∫

( ) ( )0

, ,( , ) - Refraction Image

( , )

f x y Rr x y d

T x y I

θ θ θ θθ

+∞

−∞

⎡ ⎤= −⎢ ⎥

⎣ ⎦∫

Generalization to CT Reconstruction

zΔ zΔzΔ zΔ

K0

zAI I e μ−=

0I 0I 1 2( )0

N zAI I e μ μ μ− + += K

1μ 2μ 3μ 4μ Nμ

3 11 2 40 0( , , )

N

nn

zzz z z

AI x y z I e e e e I eμ

μμ μ μ =

− Δ− Δ− Δ − Δ − Δ

∑= =K

10

( , ) ln ( , , )N

An

n L

Ip x y z x y z dz

Iμ μ

=

⎛ ⎞=− = Δ =⎜ ⎟

⎝ ⎠∑ ∫

[ ] [ ]( , ) ln ( , , ) ( , , )r r

L L

r x y grad n x y z dz grad x y z dzρ= ∫ ∫:

( , )? ? w x y z:

Discrete Scatterer Model

- Scattering Centers - , , a s spnσ σ

Object Voxel

- Nonscattering Medium , grad nμ

Khelashvili, Brankov (IIT), Chapman (U.Sask), Anastasio, Yang (IIT), Zang (BNL), Wernick (IIT)

Multiple Ultra-Small Angle Scattering

Radiation density of x-ray beam in at position in the direction

ˆ (sin cos ,sin sin ,cos )

r

s θ φ θ φ θ=

r

r

ˆ s′r

ˆ sr

• Radiation Transport Theory Approach

ˆ( , ) ( , , ) - Specific IntensityI r s I r θ φ=r r r

dω′

dω

d sr

- particles in volumeds dsρ

( ) ( )( , )- phase function - fraction of the radiation

ˆ ˆscattered from into .

p s s

d s d sω ω

′

′ ′

r r

r r

MIR Radiation Transfer Equation

( ) ˆ4

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( , ) ( , ) ( , )4n ext

r n ext ss I r s I r s b r s I r s p s s I r s d

π

ρ σρ σ μ ω

π′ ′ ′⋅∇ = − + + ⋅∇ + ∫r r

rr rr r r r r r r r r r r r r

ext a sσ σ σ= +

Ultra-Small Angle Approximation

( )

2

( , , ) ( , , ) ( , , ) ( , , ) ( , , )

( ) ( , , )4

t n ext

n ext

dI z s s I z s I z s b z s I z s

z d s

p s s I z s d s

ρ ρ ρ σ μ ρ ρ ρ

ρ σρ

π

+∞ +∞

−∞ −∞

∂+ ⋅∇ = − + + +

∂

′ ′ ′+ −∫ ∫

rr r r r rr r r r r rr

rr r r r

2

, , sin cos sin sin

t

x y

r zk xi yj zk i j s i jx y

d s ds ds

ρ θ φ θ φ∂ ∂= + = + + ∇ = + = +

∂ ∂

=

r rr r r r r rrr r

r

ln ( , )rb n x y=∇rr r

0 0( , ) x y

n nn x y n x y n n x n y

x y

∂ ∂= + + = + +

∂ ∂

General Solution

[ ]

[ ]

2 22

0

1 1( , , ) exp ( ) exp ( )

(2 ) 2

exp ( ) ( , , ) ( , , )n t

I z s d d q i is q ibq ib z z

is z F z q K z q

ρ κ κ ρ κπ

κ ρ σ μ κ κ

+∞ +∞

−∞ −∞

⎡ ⎤= − ⋅ + ⋅ ⋅ − − ×⎢ ⎥⎣ ⎦

× − − ⋅

∫ ∫r rr r r r rr r r r r

r r rr r r

2 20 0

0

2

( , ) ( , ) exp( )

( , , ) exp ( )4

( ) ( ) exp( )

zn ext

F q I s i is q d d s

K z q P q z dz

P q p s is q d s

κ ρ κ ρ ρ

ρ σκ κπ

+∞ +∞

−∞ −∞

= ⋅ + ⋅

⎡ ⎤′ ′= −⎢ ⎥

⎣ ⎦

= ⋅

∫∫

∫

∫∫

r r r r rr r r r r

r rr r

r r r r r

Phase Function

2( , ) s d

p s sa d

σ σπ

′ =Ω

r r

3 6

2

5

10 10

11 1

2

10

sp e

a

Nn r

V

λ

δ λπ

δ −

≈ −

= − ≈ −

≈

( )

( ) ( )

2 22 2 2 2 2 2

22 2

2 2

02 22 2 2 2 2

4, sin , sin ;

4

4 4 ( , ) ( ) 4 4

4 4

x y x y

s s

ext

d as s s d s ds ds d d d d

d s

dp s s p s W

a d s s

σ δθ θ θ θ φ θ θ φ

δ

σ σσ δ δ

π σ δ δ

= = + = ≈ = = ≈Ω +

′ = ⇒ = =Ω + +

r

r r

aa

Phase Function

2 2

2 2

2

| | ( ) 24 ln 1

( )p

s p s d s

p s d sθ δ

δ⎛ ⎞= = +⎜ ⎟⎝ ⎠

∫∫∫∫

r r

r

( )2 2

20 2

| | ( ) 1( ) 4 exp ,

( )p p

p

s p s d sp s W s

p s d sα α

α= − =∫∫

∫∫

r

r

22

1 12

4 ln 1p

p

αθ δ

δ

= =⎛ ⎞+⎜ ⎟⎝ ⎠

Khelashvili, Brankov (IIT), Chapman (U.Sask), Anastasio, Yang (IIT), Zang (BNL), Wernick (IIT)

Plane Wave Solution

0 0( , ) ( )I s I sρ = är r r

( )2

0 202

0

0

( ) exp( / 4 )( , , ) exp( ) exp ( )

(2 ) !

,

( ) and ,

exp (

kp

k

s

ext

n ext n ext

W kqII z s iq s bz d q

k

W

z z

iq

τ αρ τ

π

σ

σ

τ ρ σ μ τ ρ σ

+∞ +∞ ∞

=−∞ −∞

−′= − − −

=

′ = + =

−

∑∫ ∫rr r r r r

r( ) 22 2

2 2 2

4) exp( /4 ) exp ,

( ) ( )

p pp

x y

s bz kq d q s bzk k

s bz s bz s bz

πα αα

+∞ +∞

−∞ −∞

⎛ ⎞− ⋅ − = − −⎜ ⎟

⎝ ⎠

− = − + −

∫∫rr r

Khelashvili, Brankov (IIT), Chapman (U.Sask), Anastasio, Yang (IIT), Zang (BNL), Wernick (IIT)

Plane Wave Solution

0

0

if 0 ( ( ) and )

( , , ) exp( ) ( )

n s n ext n ext

ext a s

W z z z

I z s I z s bz

τ ρ σ τ ρ σ μ τ ρ σ

σ σ σ

ρ μ

′= = = + =

= +

= − −ärr r r

Khelashvili, Brankov (IIT), Chapman (U.Sask), Anastasio, Yang (IIT), Zang (BNL), Wernick (IIT)

Imaging Parameters

( )( )

( )

2

20

0

( , , ) exp - Beer's Law

( , , ) ln ( , , ) ( , , ) - Absorption Image

T n a

Tn a

I I z s d s I z

IA x y z x y z x y x z

I

ρ ρ σ μ

ρ σ μ

= = − +

⎛ ⎞= − = +⎜ ⎟

⎝ ⎠

∫∫¡

r r r

2 2

2 21 1( , , ) ( , , ) ; ( , , ) ( , , )

Refraction Image

x x x y y yT T

x y z s I z s d s b z x y z s I z s d s b zI I

θ ρ θ ρΔ = = Δ = =∫∫ ∫∫¡ ¡

r rr r r r

( )2

22 ( , ) ( , )1

( , , ) ( , , )

Ultra-Small Angle Scattering Image

n s

r p

x y x yw x y z s bz I z s d s z

I

ρ σρ

α= − = ⋅∫∫

¡

r rr r r

Khelashvili, Brankov (IIT), Chapman (U.Sask), Anastasio, Yang (IIT), Zang (BNL), Wernick (IIT)

Experimental Conformation

Lucite container – wedge shaped.

Polymethylmethacrylate (PMMA) microspheres in glycerin.

Khelashvili, Brankov (IIT), Chapman (U.Sask), Anastasio, Yang (IIT), Zang (BNL), Wernick (IIT)

Experimental ConformationKhelashvili, Brankov (IIT), Chapman (U.Sask), Anastasio, Yang (IIT), Zang (BNL), Wernick (IIT)

labDEI System

Detector

Analyzer

Pre-mono & Mono

X-ray Source

Morrison, Nesch, Torres, Khelashvili (IIT), Hasnah (U. Qatar) Chapman (U.Sask)

1cm

cartilage

bone

Lab DEI System tissue images

Morrison, Nesch, Torres, Khelashvili, Chapman (IIT)Muehleman (Rush Medical College)

Human tissue image using prototype

laboratory DEI system using Mo K

(17.5keV) radiation. Image is of a

section of a knee joint immersed

in formalin showing cartilage.

α

Summary

• First reliable Theoretical Model of DEI – MIR has been developed.

• 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).

• CT reconstructions – some steps are already taken in this direction – Miles N. Wernick et al “Preliminary study of multiple-image computed tomography”

• CSRRI (IIT) / Nesch LLC – are developing in-lab research DEI instrument

Acknowledgements

Funded by NIH/NIAMS.

L.D. Chapman (Anatomy and Cell Biology, University of Saskatchewan, Canada)

J. Brankov, M. Wernick, Y. Yang, M. Anastasio (Biomed. Engineering, IIT)

T. Morrison and I. Nesch (CSRRI, IIT)

C. Muehleman (Department of Anatomy and Cell Biology, Rush Medical College)