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X-ray Scatter-to-Primary Ratio versus Thickness Two analytic Models Evaluated Against Monte Carlo Calculations

J. Eric TkaczykGE Research, Niskayuna, NY

•Detectors at GE Research•Relevance of Scatter/Absorptive Problem To Detectors•Experimental Data•Two analytic models•Comparison to Monte Carlo Calculations•Conclusions

Outline

tkaczyk@crd.ge.com

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Niskayuna, NY – World Headquarters

Shanghai, ChinaBangalore, India

Global Research OrganizationGlobal Research Organization

Technical RepresentationTechnical RepresentationChemistryChemistry 18%18%MechanicalMechanical 17%17%PhysicsPhysics 9% 9%ElectricalElectrical 18%18%Computer Computer SciSci.. 17%17%All OtherAll Other 21%21%

9/12/02 3

g GE Research a-Si Flat Panel Digital X-ray Imager

Row & ColumnReadout

Electronics

Projected evolution for digital x-ray imaging•higher resolution pixels•greater patient coverage•faster scanning

Projected evolution for digital x-ray imaging•higher resolution pixels•greater patient coverage•faster scanning

CsI Scintillatoron Monolithic Array

ofa-Si Photodiodesand FET Switches

9/12/02 4

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Scintillator Matrix

Photodiode

ReadoutElectronics

Projected evolution isfrom multi-slice to volume

CT

•higher resolution pixels•much greater patientcoverage•faster scanning speed

Projected evolution isfrom multi-slice to volume

CT

•higher resolution pixels•much greater patientcoverage•faster scanning speed

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Combined Absorption and Scattering Processes

thin sheet

1) transmission(i.e. primary)

2) absorption 3) scattering

T

The radiation density profilebecomes a remarkably

difficult to calculate for non-infinitesimal layers

The radiation density profilebecomes a remarkably

difficult to calculate for non-infinitesimal layers

particle of interest could be •x-ray in patient•light photon in scintillator•e/hole in semiconductor•neutrons through absorber

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x-ray in patient

Combined absorption/scatter phenomena is ubiquitous for detectors and imaging.The albedo = W=mscatter/mtotal is useful to divide the problem space

Detector studies seek to maximize the gain efficiency and minimize crosstalk

Combined absorption/scatter phenomena is ubiquitous for detectors and imaging.The albedo = W=mscatter/mtotal is useful to divide the problem space

Detector studies seek to maximize the gain efficiency and minimize crosstalk

light photon in scintillator

e/hole transport in unbiased semiconductor

light incident to a planetaryatmosphere

Combined Absorption and Scattering Processes

S. Chandrasekhar, Radiative TransferDover Pub. 1960

R. Swank, Appl. Optics, 12, 1865 (1973).

120cm

0.150mm

0.300mm

100km

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xDetectorPbSheet0.3cmStandAcrylic Sheetsair gap20mm Al(optional)SID=137cmSensor Panel Height~0.5cmthicknessFocal Spot HeightFilter Turret44.8cm28.6cm108.4cmTop Detector Height

Experimental Study of Scatter

•Measurements with and without Pb Sheet Allows Scatter Measurement•Experimental Results were simulated with MC calculation using GEANT4•Measurements with and without Pb Sheet Allows Scatter Measurement•Experimental Results were simulated with MC calculation using GEANT4

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0.001

0.01

0.1

1

10

0

2

4

6

8

10

0 50 100 150 200 250

Response (Me/mAs)

Scatter-to-Primary

Acrylic Thickness (mm)

P=23.7e-0.0261 T

76kvp, noAl, noGrid

SPR=

0.01028T + 9.141x10-5 T2

Total

Primary

Scatter

Experimental Study of Scatter

Data shows exponentialattenuation of the primary

Scatter is non-monotonic ... it is linearly increasing at small thicknessand exponentially decreasing at large thickness

Scatter is non-monotonic ... it is linearly increasing at small thicknessand exponentially decreasing at large thickness

non-monotonicdependence

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r

r ‘

General Formal Solution to the Coupled Scatter/Absorption Problem is an Integral Equation

( )2

)',(

'2

)0,(

')','()',('')0(),(

rr

erIpddr

r

eIrI

rr

rr

V S

r

source-

WWWW+=W-

-

-

Ú Útt

Primary=source attenuated

by object Scatter from other points in the object

S. Chandrasekhar, Radiative Transfer, Dover Pub. 1960

Total Signal is Self-ConsistentSum of Primary from the sourceand Scatter from all other points

Total Signal is Self-ConsistentSum of Primary from the sourceand Scatter from all other points

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First Order Solution: Diffusion Equation

( )2

)',(

'2

)0,(

')','()',('')0(),(

rr

erIpddr

r

eIrI

rr

rr

V S

r

source-

WWWW+=W-

-

-

Ú Útt

)()()( 22 rSrr =+— fsf

General Solution

Analytic Model 1... Diffusion Equation

conditions of slow variation (i.e. not valid near surfaces and sources)

A. Ishimaru, Wave Propagation and Scattering in RandomMedia, Academic Press, Inc. N.Y. 1978.

reciprocal diffusion length source function

Diffusion equation used by Swank to modellight propagation in scintillators

Diffusion equation used by Swank to modellight propagation in scintillators

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( )TTo

T

z

zTzCoF

pp

pp

eeW

I

eedzIS

mhmh

mhm

h

m

)1(

0

)(

1)1(2

2

---

=

---

-˙˚

˘ÍÎ

È

-=

= Ú

Analytic Model 2... after Smith and Kruger

T z

T z

T s

Process F1

Process F2

Process B

P

S P

S P

S

z

Multiple Scattering istreated by an effective

attenuation coefficient forscatter which is less

then attenuationcoefficient for primary

h<1

describes the observed non-monotonic dependence

T-z

0.001

0.01

0.1

1

10

0

2

4

6

8

10

0 50 100 150 200 250

Response (Me/mAs)

Scatter-to-Primary

Acrylic Thickness (mm)

P=23.7e-0.0261 T

76kvp, noAl, noGrid

SPR=

0.01028T + 9.141x10-5 T2

Total

Primary

Scatter

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T z

T z

T z

Process F1

Process F2

Process B

P

S P

S P

S

z

[1] S.W. Smith and R.A. Kruger, “A signal processing model of diagnostic x-ray scatter,” Med. Phys. 13, 831 (1986).

“1st Forward Scatter”included by Smith & Kruger

“1st Back-Scatter”added for more complete treatment

Augment the Method of Smith and Kruger

by including Back-Scattered Processes

Augment the Method of Smith and Kruger

by including Back-Scattered Processes

Analytic Model 2... after Smith and Kruger

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˙˙

˚

˘

ÍÍ

Î

È +--˜̃

¯

ˆÁÁË

Ê

+

-+

--

+

++-

¥˜̃¯

ˆÁÁË

Ê

++

+-

-=

Tpe

Tpe

Tpe

WIS oT

mhx

hhmh

xhhx

hxxhxmh

h

)1()1(

1

1

2

1)1(

)1(2

)1)(1(1

)1()1(

)1(4

)1(

2/2

R.K. Swank, "Calculation of Modulation Transfer Function of X-ray Fluorescent Screens,“ (equation 23 was used), Appl. Optics, 12, 1865 (1973)

Smith and Kruger model extended to include backscatter

Diffusion equation for slab geometry with perpendicular incidence.

˙˙

˚

˘

ÍÍ

Î

È +-

++

+˜̃¯

ˆÁÁË

Ê

+

-+

--

++

++-

¥˜̃¯

ˆÁÁË

Ê

+

++-

-=

Tpe

c

cTpe

c

c

cTpe

WIS oT

mh

hh

hhmh

hh

hhmh

hb

)1(

)1(1

1

1

1

2

)1(

)1(1

)1)(2/1(1

1

)1(1

)1(

What is striking about this comparison is the similar exponentialdependences obtained from such different approaches

The parameter h appears is both expressions and represents therelative attenuation of scatter compared to primary

What is striking about this comparison is the similar exponentialdependences obtained from such different approaches

The parameter h appears is both expressions and represents therelative attenuation of scatter compared to primary

Compare Two Analytic Models

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10

100

1000

0 1 2 3 4 5 6 7

monte carlo, 60kevmonte carlo, 50eVmonte carlo, 40keVmonte carlo, 30keVSK model-forwardSK model-backwardSK model-total

Scatter Counts at Detector

Thickness x Primary Attenuation Constant

Monte Carlo Calculations

Smith and KrugerModel Fit to MC Calculations

Smith and KrugerModel Fit to MC Calculations

Forward & Back ScatteredContributions

T s

T s

T s

Process F1

Process F2

Process B

P

S P

S P

S

z

Compare Monte Carlo to Analytic Model 2

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Monte Carlo Params 1D model Fits Diffusion Model Fitskev ?? photoele.?? compton?? total albedo ? c ? ? ?? expected??expected

30 0.1410 0.183 0.324 0.56 1.1 0 0.9 0.2 1.14 0.84722240 0.0543 0.184 0.238 0.77 0.9 0.0 0.85 0.4 0.83 1.15777350 0.0265 0.187 0.213 0.88 0.85 0.5 0.85 1.0 0.61 1.3133860 0.0151 0.181 0.196 0.92 0.8 1.0 0.85 3.0 0.48 1.384439

•Fits to the analytic model to the MC calculation relatesparameters in the analytic model to cross-section for

scatter and absorption•Models can be used in studies of scintillator and

photodiode performance

•Fits to the analytic model to the MC calculation relatesparameters in the analytic model to cross-section for

scatter and absorption•Models can be used in studies of scintillator and

photodiode performance

Conclusion0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

0.5 0.6 0.7 0.8 0.9 1

Attenuation Coefficent of Scatter

Albedo=scatter xSection/ absorption xSection

Smith & Kruger

Diffusion

Final fit parametersh

Albedo