monte carlo simulation for polychromatic x-ray...

Research ArticleMonte Carlo Simulation for Polychromatic X-Ray FluorescenceComputed Tomography with Sheet-Beam Geometry

Shanghai Jiang1 Peng He12 Luzhen Deng13 Mianyi Chen1 and BiaoWei12

1Key Lab of Optoelectronic Technology and Systems Ministry of Education Chongqing University Chongqing 400044 China2Engineering Research Center of Industrial Computed Tomography Nondestructive Testing Ministry of EducationChongqing University Chongqing 400044 China3Department of Radiation Physics The University of Texas MD Anderson Cancer Center Houston TX 77030 USA

Correspondence should be addressed to Peng He penghecqueducn and Biao Wei weibaocqueducn

Received 4 March 2017 Accepted 16 April 2017 Published 8 May 2017

Academic Editor Jyh-Cheng Chen

Copyright copy 2017 Shanghai Jiang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

X-ray fluorescence computed tomography (XFCT) based on sheet beam can save a huge amount of time to obtain a wholeset of projections using synchrotron However it is clearly unpractical for most biomedical research laboratories In this paperpolychromatic X-ray fluorescence computed tomography with sheet-beam geometry is tested by Monte Carlo simulation Firsttwo phantoms (119860 and 119861) filled with PMMA are used to simulate imaging process through GEANT 4 Phantom 119860 contains severalGNP-loaded regions with the same size (10mm) in height and diameter but different Auweight concentration ranging from 03 to18 Phantom119861 contains twelve GNP-loaded regions with the sameAuweight concentration (16) but different diameter rangingfrom 1mm to 9mm Second discretized presentation of imaging model is established to reconstruct more accurate XFCT imagesThird XFCT images of phantoms 119860 and 119861 are reconstructed by filter back-projection (FBP) and maximum likelihood expectationmaximization (MLEM) with and without correction respectively Contrast-to-noise ratio (CNR) is calculated to evaluate all thereconstructed images Our results show that it is feasible for sheet-beam XFCT system based on polychromatic X-ray source andthe discretized imaging model can be used to reconstruct more accurate images

1 Introduction

As a promising imaging modality X-ray computed tomogra-phy combining X-ray analysis and tomographic reconstruc-tion algorithm has attracted wide concern in recent yearsIt can not only measure the distribution of elements butalso the content of elements within samples in a nonde-structive and noninvasivemanner [1ndash3] Conventional XFCTtechniques with synchrotron source scan samples using thetranslation-rotation method which is obviously unsuitablefor most biomedical research laboratories due to its huge andexpensive equipment

Some improvements were proposed by other researchersThe synchrotron source is replaced by X-ray tube and sim-ulated and experimental demonstration of polychromatic-source XFCT were implemented to reduce dose and scantime which makes benchtop system feasible [4ndash6] AlthoughXFCT based on sheet-beam geometry using synchrotron

were also developed fewer researches were done with poly-chromatic X-ray source [7]

As a contrast agent gold nanoparticles (GNPs) haveattracted wide concern due to their application in cancerdetection and therapy [8ndash10] Au within samples exposedby X-ray beam mainly emits K-shell X-rays fluorescencewhich can be detected to reconstruct XFCT images In thisstudy sheet-beam XFCT system based on polychromatic X-ray source was verified by Monte Carlo simulation First twophantoms which contained several GNP-loaded regions wereimaged through GEANT 4 The XFCT images were recon-structed by FBP and MLEM algorithms Second discretizedpresentation of sheet-beam geometry was established toreconstruct more accurate XFCT images Contrast-to-noiseratio (CNR) was used to evaluate images quality At lastCNR for the reconstructed images as functions of Au weightconcentration and size of GNP-loaded regions was discussed

HindawiInternational Journal of Biomedical ImagingVolume 2017 Article ID 7916260 10 pageshttpsdoiorg10115520177916260

2 International Journal of Biomedical Imaging

PC

Spectrometer

CollimatorsPolychromatic X-ray source

CCDRotationSheet beam

Array detectors

middot middot middot


Figure 1 Schematic diagram of XFCT imaging system using sheet-beam and linear detector arrays

2 Principles and Methods

Principles and simulations of polychromatic X-ray fluores-cence computed tomography with sheet-beam geometry arepresented in this paper The data analysis techniques andimage reconstruction algorithm are described in Sections21ndash24

21 Image System The schematic diagram of sheet-beam CTsystemproposed in our study is shown in Figure 1The systemincludes polychromatic sheet-beam X-ray source parallelcollimator array detectors spectrometer and computer

Polychromatic X-ray from X-ray tube is collimated intosheet beam and then impinges on the object to cover thewhole cross-section GNPs exposed by X-ray beam canisotropically emit characteristic X-ray photons Linear arrayphoton-counting detectors with energy resolution are posi-tioned perpendicular to the beam propagation direction forX-ray fluorescent spectra [11]

211 Monte Carlo Model The Monte Carlo simulation wascompleted by GEANT 4 software and ROOT software [12]Because simulations of projection at each angle are indepen-dent of each other we replace the X-ray source in Figure 1with a virtual source to reduce simulation time of GEANT4 The spectrum of sheet-beam X-ray source was calculatedby SpekCalc software which simulates X-ray spectra emittedfrom thick-target tungsten anode X-ray tubes [10 13] In thisstudy electron beam of 120 keV interacted with tungstenThen the emitted X-ray photons were filtered by Sn withthickness of 1mm The spectrum of X-ray source was shownin Figure 2 Here the width and thickness of sheet beamwereset to 64 cm and 1mm respectively

Two GNP-loaded PMMA phantoms are shown in Fig-ure 3 The PMMA phantom is 64 cm in both height anddiameter The phantom on the left side contained several

0

02

04

06

08

1

Nor

mal

ized

coun

ts

40 60 80 100 12020Energy (keV)

Figure 2 Spectrum of incident polychromatic X-ray source

GNP-loaded regions which has the same size (10mm) inheight and diameter but different gold concentration (mixedwith water) ranging from 03 to 18 The GNP-loadedregions in right phantom have the same concentration (15)but different diameter ranging from 1mm to 9mm

The emitted fluorescence photons were detected by aseries of energy-sensitive tallies (shown in Figure 1) Theywere positioned 1mm behind lead collimator with a seriesof pinhole openings of diameter 05mm The whole XFCTscanning procedurewas divided into independent simulationfor each projection angle The fluorescent detector arrayincludes 64 energy-sensitive detectors where each detectorhas the same sensitive area (05mm times 05mm) and energyresolution (05 keV) [14]When 10Mhistories (photons) werecalculated for each simulation the uncertainty was less than5 for relevant photon energies (50ndash75 keV)

International Journal of Biomedical Imaging 3

Oslash64

Oslash45

Oslash10

09 06

03

1815

12+ +

(a)

Oslash64Oslash45

Oslash9

Oslash8

Oslash7Oslash6

Oslash5

Oslash4

Oslash35

Oslash3

Oslash25

Oslash2Oslash15

Oslash1

++

+

+

+

++

+

+

+

+

+

+

GNPs 15

(b)

Figure 3 Phantoms contained GNP-loaded regions (a) GNP-loaded regions with same size in height and diameter but different Au weightconcentrations (b) GNP-loaded regions with same Au weight concentration but different diameters

212 Data Acquisition The X-ray photons arriving at thedetectors mainly come from both Compton scatter andcharacteristic X-ray photons Considering fluorescent fieldand the attenuation of low-energy photon in the phantomthe gold K120572 lines (670 and 688 keV) are the best candidatesto reconstruct XFCT images in our simulations To extractfluorescent signal count cubic polynomial was used to fit thepoints around the gold fluorescent peaks The fluorescencesignal counts measured of each projection during the simula-tion were the difference between the measured signal countsand the fitted counts [10 15] A sinogram of the gold fluo-rescence signal counts was reconstructed using the extractedgold fluorescence signal from each projected simulation

22 The Imaging Model of Sheet-Beam XFCT The imagingmodel of sheet-beam XFCT was established previously bysome researches [7 11] and its geometry is presented in Fig-ure 4While the xy-coordinate system is attached to an objectthe st-coordinate system is spun with the data acquisitionsystem and can be at any instant obtained by rotating the xy-coordinate system by an angle 120579 counterclockwise [16] Thatis their relationship can be expressed as follows

(119904119905) = ( cos 120579 sin 120579

minus sin 120579 cos 120579)(119909119910) (1)

According to the results of previous research [11 17ndash19] the total photons of fluorescent X-ray reaching the 119894thdetector are represented as follows

119868119894V (120579 119904)= int+infinminusinfin

119891 (120572 119904 119905) ∙ 119892 (120572 119904 119905) ∙ Ω (119904 119905) 120588 (119909 119910) 119889119905 (2)

where

119891 (120579 119904 119905) = 1198680 exp [minusint119905minusinfin

120583119868 (119909 119910) 119889119904] (3)

119892 (120579 119904 119905) = 120583119901ℎ120596int120574max

120574min

exp [minusintinfin0

120583119865 (119909 119910) 119889119887] 119889120574 (4)

s

t

O

x

y

QP

S

mth detector

I0

Ii

Δd

훾min훾max

휃

Figure 4 Schematic diagram of XFCT imaging geometry usingsheet beam

120596 is the yield of characteristicX-ray photonsΩ is the solidangle at which the point 119876 is viewed by the119898th fluorescencedetector120583119901ℎ is the photoelectric linear attenuation coefficientof Au The 120588(119909 119910) 120583119868(119909 119910) 120583119865(119909 119910) are the distribution ofAu weight concentration linear attenuation coefficient ofincident X-ray energy and linear coefficient of fluorescentX-ray Here in order to simplify reconstruction (4) can beexpressed approximately as follows

119868119894V (120579 119904) asymp 120583119901ℎ1205961198680Ωint+infinminusinfin

120588 (119909 119910) 119889119905 (5)

Thus the measured process by the XFCT based onsheet-beam geometry can be viewed approximately as Radontransform and FBP algorithm can be used to reconstruct


1 2 3

J

1 2 3

J

1 2 3

J

middot middot middot middot middot middot middot middot middot

휌j j j= 1 2 3 J 휇Ij = 1 2 3 J 휇F

j j = 1 2 3 J

Figure 5 The matrices defined for XFCT

XFCT images Here the reconstructed images are usuallyconsidered to be uncorrected

To acquire the corrected images it is necessary to obtaindiscretized representation of (2) During the process weassume that the phantom is two-dimensionalThree matricesshown in Figure 5 including 120588119895 120583119868119895 and 120583119865119895 corresponding to120588(119909 119910) 120583119868(119909 119910) and 120583119865(119909 119910) are used to describe the wholephantom where j (119895 = 1 2 3 119869) represents the number ofeach pixelWe assume sheet-beamX-ray incident phantom at119873 angles where 119899 (119899 = 1 2 3 119873) represents the numberof each angle The sheet-beam source is considered as 119875 X-rays at each incident direction shown in Figure 6(a) Here 1199011015840is the number of the 1199011015840th X-ray We consider the process ofthe 119901th incident X-ray interacted with phantomTherefore

119901 = 119875 (119899 minus 1) + 1199011015840 (6)

Step 1 Let119876119901119895 be the subset of119876119901 which consists of the lightblue pixels shown in Figure 6(b) 119871119901119895 is the intersected lengthof the119901thX-ray and the 119895th pixelThe incidentX-ray intensitybefore reaching the 119895th pixel is expressed as follows

119891119901119895 = 1198680 exp(minus sum119896isin119876119901119895

120583119868119896119871119868119901119896) (7)

Step 2 The total X-ray counts emitted from the 119895th pixelare proportional to the product of 120588119895 and 120596120583119901ℎ119891119901119895119871119868119901119895 Let120575119894119895 be the angle viewed by the detector corresponding to 119894thprojection at the jth pixel Here we assume that the X-raycounts emitted from the 119895th pixel can be recorded by the119898thdetector where number 119894 can be calculated by (6) The X-raycounts measured by the detector are written as follows

120583119901ℎ120596120575119894119895119891119901119895120588119895119871119868119901119895 (8)

Step 3 Not all X-ray fluorescence photons emitted from jthpixel can be detected by detectors Here we assume thatthe X-ray photos by the jth pixel can be detected within 120575119894119895when the line passing through the center of the jth pixel andparalleling to the lead hole can reach detector without blockAttenuation of fluorescent X-ray from jth to detector must bealso considered during the further process In Figure 6(c) the

fan-shaped X-ray can be divided intoK (K is positive integer)individual X-rays Let Δ120575 = 120575119901119895119870 and l (1 le 119897 le 119870) is thenumber of individual fluorescent X-rays The attenuation ofthe lth fluorescent X-ray can be expressed as follows

exp(minus sum119902isin119879119901119895119897

120583119865119902119871119865119901119895119902) (9)

where Tpjl is defined as the set consisting of the pixels (lightblue squares shown in Figure 6(d)) intersected with the lthfluorescent X-ray 119871119865119901119895119902 is described as the intersected lengthof the lth fluorescent X-ray with the 119902th pixel (119902 isin 119879119901119895119897)

119892119901119894119895 = 120583119901ℎ120596120575119901119895 119870sum119897=1


120583119865119902119871119865119901119895119902) (10)

Step 4 We consider the process of the pth incident X-rayinteracted with phantom Let us define that 119894 is the numberof 119894th projection ranging from 1 to 119868 The contribution ℎ119894119895 ofthe jth pixel to the 119894th projection at pth incident X-ray can beexpressed as follows

119894 = 119872 (119899 minus 1) + 119898ℎ119894119895 = 119891119901119895119892119901119894119895120575119894119895 (11)

Accordingly the discretized representation of (4) is writ-ten as follows

119868119894 = sum119895

ℎ119894119895120588119895 (119894 = 1 2 119868) (12)

The matrix representation of (9) is

I = H120588 (13)

where

H = (ℎ119894119895) (1 le 119894 le 119872 1 le 119895 le 119873)I = (119868119894) (1 le 119894 le 119872) 120588 = (120588119895) (1 le 119895 le 119873)

(14)


1 2 3

J

Qp


(a)

jth pixel

kth pixelLI

pk

(b)

1 2 3

J

jth pixel

pth incident X-rayO

x

yt

s

lth fluorescence X-ray

mth detector

훿pj

Ii = IM(nminus1)+m


(c)

lth fluorescent X-ray

mth detector

qth pixelLF

pjq

(d)Figure 6 The parameters defined for discretized presentation (a) A set 119876119901 defined for blue squares intersected by the pth incident X-ray(b) A set 119876119901119895 defined for light blue squares intersected by the pth incident X-ray (c) Definition of 120575119894119895 m (d) A set Tpjq defined for light bluesquares intersected by lth fluorescent X-ray

23 XFCT Image Reconstruction Here we assume that themaps of 120583119868 and 120583119865 are known in our study Two algorithmsincluding FBP and MELM were used to reconstruct XFCTimages with correction and without correction respectivelyFirst sinograms of two phantoms with GNP-loaded regionswere acquired by the describedmethod in Section 212ThenXFCT images with 64 times 64 pixels of 64 times 64mm2 werereconstructed by FBP and MLEM without correction Themore accurate projection matrix described in (6)ndash(15) wascalculated to correct the effect of attenuation for high qualityimages

24 XFCT Image Analysis The reconstructed XFCT imagesare evaluated as CNR by calculating the ratio of differencebetween the mean value of each GNP-loaded region andbackground (PMMA) and standard deviation of backgroundCNR is defined as follows [16 20]

CNR = ΨRegion minus Ψ119861119870119881119861119870 (15)

where ΨRegion and Ψ119861119870 are mean reconstructed values ofGNP-loaded region and background and 119881119861119870 is standarddeviation of background (PMMA) According to the Rosecriterion imaging sensitivity limit of the system proposedwas determined using CNR of 4 [21]

3 Results

31 XFCT Image Reconstruction Sinograms of two phantomswith GNP-loaded regions are shown in Figure 7 Figure 7(a)shows the sinogram of phantom 119860 and Figure 7(b) is thesinogram of phantom 119861 The reconstructed XFCT images ofphantom 119860 are shown in Figure 8 where Figures 8(a) and8(c) are the images reconstructed by FBP and MLEM (100iterations) without correction respectively CorrespondinglyFigures 9(b) and 9(d) are reconstructed by FBP and MLEM(100 iterations) with correction respectively Similar repre-sentation is also shown in Figure 9 Obviously grey valuesof GNP-loaded regions decrease with reduction of Au weightconcentration shown in Figure 8


0

500

1500

1000

(a)

10020030040050060070080090010001100

(b)

Figure 7 The reconstructed sinograms of (a) phantom 119860 and (b) phantom 119861

0

001

002

003

004

005

(a)

0

5

10

15

times10minus3

(b)

0

002

004

006

008

(c)

0005

001

0015

002

0025

(d)

Figure 8 ReconstructedXFCT images of phantom119860 (a) Reconstructed by FBPwithout correction (b) reconstructed by FBPwith correction(c) reconstructed by MLEM without correction and (d) reconstructed by MLEM with correction

The reconstructed Au concentration calculated from themean value of each GNP-loaded region in phantom 119860 isplotted in Figure 10(a) (acquired by FBP) and Figure 10(b)(acquired by MLEM) Both figures show that uncorrectedAu weight concentration has larger error than correctedconcentration which may mean that our corrected modelcan provide more accurate results For the same Au weight

concentration the reconstructed values in Figure 10(d)acquired by MLEM algorithm have more stable values thanin Figure 10(c) acquired by FBP algorithm

32 XFCT Image Analysis CNR for reconstructed XFCTimages with FBP and MLEM as a function of Au weight


0

001

002

003

004

005

006

(a)0

001

002

003

004

(b)

0

0005

001

0015

(c)0

001

002

003

004

(d)


concentration are presented in Figures 11(a) and 11(b) respec-tively Both of bar charts show us that values of CNR in oursetup have higher than 4 when Au weight concentrationis greater than 06 To detect lower concentration thesetup needs to be modified including length and diameterof collimators spectrum of X-ray source and distance fromX-ray source to the center of phantom [22] Figures 11(a) and11(b) can also indicate that CNR of GNP-loaded region forsame size is linearly proportional to Au weight concentration(1198772 ge 09992) According to Rose criterion (CNR gt 4)not all GNP-loaded regions in Figure 8 were detectable anddetection limits from Figures 8(a) to 8(d) were 059 062060 and 056 respectively

CNR for reconstructed XFCT images with FBP andMLEM as a function of each GNP-loaded region size arealso presented in Figures 11(c) and 11(d) respectively Whenthe diameter of GNP-loaded region is ranging from 7mm to9mm the values of CNR increase with increase of diameterbut fluctuate largely from 1mm to 6mm The phenomenon

illustrates that size ofGNP-loaded region can influence valuesof CNR when incident X-ray width is invariant

Algorithms can influence values of CNR In Figures11(a) and 11(c) the corrected images have lower CNR thanuncorrected ones with FBP On the contrary the correctedimages with MLEM have greater than the uncorrected onesin Figures 11(b) and 11(d) However the corrected concentra-tions have more accurate than the uncorrected ones

4 Discussion

We have presented a benchtop system for polychromaticX-ray fluorescence computed tomography with sheet-beamgeometry through Monte Carlo simulation The discretizedmodel with sheet-beam XFCT is also described in our study

In the simulation we used polychromatic X-rays (X-raytubes) instead of synchrotron radiation in similar imagingsystem proposed previously by others [7] which make itpossible to reduce costs and size of apparatus Another


0

1

2

3

4

5

6Re

cons

truc

ted

Au co

ncen

trat

ion

()

02 04 06 08 10 1816 200 12 14True Au concentration ()

UncorrectedCorrected

(a)

0

1

2

3

4

5

6

7

8

Reco

nstr

ucte

d Au

conc

entr

atio

n (

)



(b)

05

10

15

20

25

30

35

40

45

Reco

nstr

ucte

d Au

conc

entr

atio

n (

)

9 8 36 010 5 2 147Diameter of GNP-loaded regions (mm)


(c)


0

05

10

15

20

25

30Re

cons

truc

ted

Au co

ncen

trat

ion


(d)

Figure 10 Reconstructed Auweight concentration (a) and (c) Acquired by FBP algorithmwith and without correction (b) and (d) Acquiredby MLEM algorithm with and without correction

advantage of XFCT imaging with sheet-beam geometry isa drastic reduction of overall scanning time compared totraditional XFCT [10] Although the proposed XFCT systemmay not be demonstrated by experimental study it mayprovide valuable method for optimization of XFCT system

However a technical challenge for the proposed system isimprovement of detection limit and CNR First they may beimproved further by additional modifications to the currentsetup such as quasi-monochromatization of incident X-rayspectrum and further optimization of detector collimation[7 19] Second the optimized algorithm may improve imagequality According to our results different algorithms influ-ence the values of CNR which is similar with conclusionof Di et al [23] Thirdly X-ray detector with higher energy

resolution is used during the process Another potentialapproach is to consider both 119870120572 peaks and 119870120573 peaks at thesame time

Our future work will consist of optimizing polychromaticXFCT with sheet-beam geometry such as spectrum of X-ray source and length of collimators We will also build animaging system based on our simulations to demonstrate thefeasibility by experiment

5 Conclusion

In this investigation the feasibility of polychromatic sheet-beamXFCT system proposed in this study was demonstratedby Monte Carlo method Two phantoms which contained


20


03 06 09 15 180 12Au weight concentration ()

0

2

4

6

8

10

12

14CN

R

(a)



0

2

4

6

8

10

12

14

CNR

(b)


0123456789

101112

CNR

9 8 7 6 5 4 3 2 1 010Diameter of GNP-loaded regions (mm)

(c)


0123456789

1011

CNR


(d)

Figure 11 CNR (a) and (c) Acquired by FBP algorithm with and without correction (b) and (d) Acquired by MLEM algorithm with andwithout correction

several GNP-loaded regions were imaged using GEANT 4Accurate images were reconstructed by FBP and MLEMwith and without correction respectively Our results mayprovide necessary justification for the design of benchtopXFCT imaging system for in vivo imaging

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is partially supported by the National NaturalScience Foundation of China (61401049) the Graduate Sci-entific Research and Innovation Foundation of Chongqing

China (Grant no CYB16044) the Fundamental ResearchFunds for the Central Universities (106112016CDJXY120003)Chongqing Strategic Industry KeyGeneric Technology Inno-vation project (no cstc2015zdcy-ztzxX0002)

References

[1] G R Pereira R T Lopes M J Anjos H S Rocha andC A Perez ldquoX-ray fluorescence microtomography analyzingreference samplerdquo Nuclear Instruments amp Methods in PhysicsResearch vol 579 no 1 pp 322ndash325 2007

[2] D H Mcnear P Edward E Jeff et al ldquoApplication of quan-titative fluorescence and absorption-edge computed microto-mography to image metal compartmentalization in Alyssummuralerdquo Environmental Science amp Technology vol 39 no 7 pp2210ndash2218 2005


[3] T Takeda Q Yu T Yashiro et al ldquoIodine imaging in thyroid byfluorescent X-ray CT with 005mm spatial resolutionrdquo NuclearInstruments amp Methods in Physics Research vol 467ndash468 no 7pp 1318ndash1321 2001

[4] K Ricketts C Guazzoni A Castoldi and G Royle ldquoA bench-top K X-ray fluorescence system for quantitative measurementof gold nanoparticles for biologicalrdquo Nuclear Instruments andMethods in Physics Research Section A Accelerators Spectrom-eters Detectors and Associated Equipment vol 816 pp 25ndash322016

[5] N Manohar and S H Cho ldquoQuality of micro-CT imagesacquired from simultaneous micro-CT and benchtop x-rayfluorescence computed tomography (XFCT) A preliminaryMonteCarlo studyrdquo inProceedings of the 2013 60th IEEENuclearScience Symposium andMedical Imaging Conference (NSSMICrsquo13) Korea November 2013

[6] Y Kuang G Pratx M Bazalova B Meng J Qian and L XingldquoFirst demonstration of multiplexed X-ray fluorescence com-puted tomography (XFCT) imagingpdfrdquo IEEE Transactions onMedical Imaging vol 32 no 2 pp 262ndash267 2013

[7] Q Huo T Yuasa T Akatsuka et al ldquoSheet-beam geometryfor in vivo fluorescent X-ray computed tomography proof-of-concept experiment in molecular imagingrdquo Optics Letters vol33 no 21 pp 2494ndash2496 2008

[8] M J Pushie I J Pickering M Korbas M J Hackett and G NGeorge ldquoElemental and chemically specific X-ray fluorescenceimaging of biological systemsrdquo Chemical Reviews vol 114 no17 pp 8499ndash8541 2014

[9] L E Cole R D Ross J M Tilley T Vargo-Gogola and R KRoeder ldquoGold nanoparticles as contrast agents inX-ray imagingand computed tomographyrdquo Nanomedicine vol 10 no 2 pp321ndash341 2015

[10] B L Jones and S H Cho ldquoThe feasibility of polychromaticcone-beam X-ray fluorescence computed tomography (XFCT)imaging of gold nanoparticle-loaded objects a Monte CarloStudyrdquo Physics in Medicine and Biology vol 56 no 12 pp 3719ndash3730 2011

[11] S Nakamura Q Huo and T Yuasa ldquoReconstruction techniqueof fluorescent X-ray computed tomography using sheet beamrdquoinProceedings of the 22ndEuropean Signal ProcessingConference(EUSIPCO rsquo14) pp 1975ndash1979

[12] H Von Busch G Harding G Martens J-P Schlomka andB Schweizer ldquoInvestigation of externally activated X-ray flu-orescence tomography for use in medical diagnosticsrdquo inMedical Imaging 2005mdashPhysics of Medical Imaging vol 5745 ofProceedings of SPIE pp 90ndash101 SanDiego Calif USA February2005

[13] G Poludniowski G Landry F Deblois P M Evans and FVerhaegen ldquoSpekCalc a program to calculate photon spectrafrom tungsten anode X-ray tubesrdquo Physics in Medicine andBiology vol 54 no 19 pp N433ndashN438 2009

[14] N Sunaguchi T Yuasa KHyodo andT Zeniya ldquoThe feasibilitystudy on 3-dimensional fluorescent X-ray computed tomogra-phy using the pinhole effect for biomedical applicationsrdquo inProceedings of the 2013 35th Annual International Conference ofthe IEEE Engineering in Medicine and Biology Society (EMBCrsquo13) pp 2348ndash2351 Japan July 2013

[15] S-K Cheong B L Jones A K Siddiqi F Liu N Manohar andS H Cho ldquoX-ray fluorescence computed tomography (XFCT)imaging of gold nanoparticle-loaded objects using 110 kVp X-raysrdquoPhysics inMedicine and Biology vol 55 no 3 pp 647ndash6622010

[16] P Feng W Cong B Wei and G Wang ldquoAnalytic comparisonbetween X-ray fluorescence CT and K-edge CTrdquo IEEE Trans-actions on Biomedical Engineering vol 61 no 3 pp 975ndash9852014

[17] Q Yang B Deng W Lv et al ldquoFast and accurate X-ray fluores-cence computed tomography imaging with the ordered-subsetsexpectation maximization algorithmrdquo Journal of SynchrotronRadiation vol 19 no 2 pp 210ndash215 2012

[18] P J Riviere P Vargas G Fu and L J Meng ldquoAccelerating X-ray fluorescence computed tomographyrdquo in Proceedings of theIEEE Annual International Conference of the IEEE Engineeringin Medicine and Biology Society pp 1000ndash1003 MinneapolisMinn USA September 2009

[19] Q Huo H Sato T Yuasa et al ldquoFirst experimental result withfluorescent X-ray CT based on sheet-beam geometryrdquo X-RaySpectrometry vol 38 no 5 pp 439ndash445 2009

[20] M Bazalova-Carter M Ahmad T Matsuura et al ldquoProton-induced x-ray fluorescence CT imagingrdquo Medical Physics vol42 no 2 pp 900ndash907 2015

[21] A Rose ldquoVision human and electronicrdquo Physics Today vol 28no 12 pp 49ndash50 2008

[22] M Bazalova Y Kuang G Pratx and L Xing ldquoInvestigation ofX-ray fluorescence computed tomography (XFCT) and K-edgeimagingrdquo IEEE Transactions on Medical Imaging vol 31 no 8pp 1620ndash1627 2012

[23] Z Di S Leyffer and SMWild ldquoOptimization-based approachfor joint X-ray fluorescence and transmission tomographicinversionrdquo SIAM Journal on Imaging Sciences vol 9 no 1 pp1ndash23 2016

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



PC

Spectrometer

CollimatorsPolychromatic X-ray source

CCDRotationSheet beam

Array detectors



Figure 1 Schematic diagram of XFCT imaging system using sheet-beam and linear detector arrays

2 Principles and Methods

Principles and simulations of polychromatic X-ray fluores-cence computed tomography with sheet-beam geometry arepresented in this paper The data analysis techniques andimage reconstruction algorithm are described in Sections21ndash24

21 Image System The schematic diagram of sheet-beam CTsystemproposed in our study is shown in Figure 1The systemincludes polychromatic sheet-beam X-ray source parallelcollimator array detectors spectrometer and computer

Polychromatic X-ray from X-ray tube is collimated intosheet beam and then impinges on the object to cover thewhole cross-section GNPs exposed by X-ray beam canisotropically emit characteristic X-ray photons Linear arrayphoton-counting detectors with energy resolution are posi-tioned perpendicular to the beam propagation direction forX-ray fluorescent spectra [11]

211 Monte Carlo Model The Monte Carlo simulation wascompleted by GEANT 4 software and ROOT software [12]Because simulations of projection at each angle are indepen-dent of each other we replace the X-ray source in Figure 1with a virtual source to reduce simulation time of GEANT4 The spectrum of sheet-beam X-ray source was calculatedby SpekCalc software which simulates X-ray spectra emittedfrom thick-target tungsten anode X-ray tubes [10 13] In thisstudy electron beam of 120 keV interacted with tungstenThen the emitted X-ray photons were filtered by Sn withthickness of 1mm The spectrum of X-ray source was shownin Figure 2 Here the width and thickness of sheet beamwereset to 64 cm and 1mm respectively

Two GNP-loaded PMMA phantoms are shown in Fig-ure 3 The PMMA phantom is 64 cm in both height anddiameter The phantom on the left side contained several

0

02

04

06

08

1

Nor

mal

ized

coun

ts

40 60 80 100 12020Energy (keV)

Figure 2 Spectrum of incident polychromatic X-ray source

GNP-loaded regions which has the same size (10mm) inheight and diameter but different gold concentration (mixedwith water) ranging from 03 to 18 The GNP-loadedregions in right phantom have the same concentration (15)but different diameter ranging from 1mm to 9mm

The emitted fluorescence photons were detected by aseries of energy-sensitive tallies (shown in Figure 1) Theywere positioned 1mm behind lead collimator with a seriesof pinhole openings of diameter 05mm The whole XFCTscanning procedurewas divided into independent simulationfor each projection angle The fluorescent detector arrayincludes 64 energy-sensitive detectors where each detectorhas the same sensitive area (05mm times 05mm) and energyresolution (05 keV) [14]When 10Mhistories (photons) werecalculated for each simulation the uncertainty was less than5 for relevant photon energies (50ndash75 keV)


Oslash64

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09 06

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1815

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(a)

Oslash64Oslash45

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++

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+

+

+

+

+

GNPs 15

(b)




(119904119905) = ( cos 120579 sin 120579

minus sin 120579 cos 120579)(119909119910) (1)



119891 (120572 119904 119905) ∙ 119892 (120572 119904 119905) ∙ Ω (119904 119905) 120588 (119909 119910) 119889119905 (2)

where


120583119868 (119909 119910) 119889119904] (3)

119892 (120579 119904 119905) = 120583119901ℎ120596int120574max

120574min

exp [minusintinfin0

120583119865 (119909 119910) 119889119887] 119889120574 (4)

s

t

O

x

y

QP

S

mth detector

I0

Ii

Δd

훾min훾max

휃




120588 (119909 119910) 119889119905 (5)



1 2 3

J

1 2 3

J

1 2 3

J


휌j j j= 1 2 3 J 휇Ij = 1 2 3 J 휇F

j j = 1 2 3 J




119901 = 119875 (119899 minus 1) + 1199011015840 (6)



120583119868119896119871119868119901119896) (7)


120583119901ℎ120596120575119894119895119891119901119895120588119895119871119868119901119895 (8)




120583119865119902119871119865119901119895119902) (9)


119892119901119894119895 = 120583119901ℎ120596120575119901119895 119870sum119897=1


120583119865119902119871119865119901119895119902) (10)


119894 = 119872 (119899 minus 1) + 119898ℎ119894119895 = 119891119901119895119892119901119894119895120575119894119895 (11)


119868119894 = sum119895

ℎ119894119895120588119895 (119894 = 1 2 119868) (12)


I = H120588 (13)

where


(14)


1 2 3

J

Qp


(a)

jth pixel

kth pixelLI

pk

(b)

1 2 3

J

jth pixel

pth incident X-rayO

x

yt

s


mth detector

훿pj

Ii = IM(nminus1)+m


(c)


mth detector

qth pixelLF

pjq






3 Results



0

500

1500

1000

(a)

10020030040050060070080090010001100

(b)


0

001

002

003

004

005

(a)

0

5

10

15

times10minus3

(b)

0

002

004

006

008

(c)

0005

001

0015

002

0025

(d)






0

001

002

003

004

005

006

(a)0

001

002

003

004

(b)

0

0005

001

0015

(c)0

001

002

003

004

(d)






4 Discussion




0

1

2

3

4

5

6Re

cons

truc

ted

Au co

ncen

trat

ion

()



(a)

0

1

2

3

4

5

6

7

8

Reco

nstr

ucte

d Au

conc

entr

atio

n (

)



(b)

05

10

15

20

25

30

35

40

45

Reco

nstr

ucte

d Au

conc

entr

atio

n (

)



(c)


0

05

10

15

20

25

30Re

cons

truc

ted

Au co

ncen

trat

ion


(d)






5 Conclusion



20



0

2

4

6

8

10

12

14CN

R

(a)



0

2

4

6

8

10

12

14

CNR

(b)


0123456789

101112

CNR


(c)


0123456789

1011

CNR


(d)





Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Oslash64

Oslash45

Oslash10

09 06

03

1815

12+ +

(a)

Oslash64Oslash45

Oslash9

Oslash8

Oslash7Oslash6

Oslash5

Oslash4

Oslash35

Oslash3

Oslash25

Oslash2Oslash15

Oslash1

++

+

+

+

++

+

+

+

+

+

+

GNPs 15

(b)




(119904119905) = ( cos 120579 sin 120579

minus sin 120579 cos 120579)(119909119910) (1)



119891 (120572 119904 119905) ∙ 119892 (120572 119904 119905) ∙ Ω (119904 119905) 120588 (119909 119910) 119889119905 (2)

where


120583119868 (119909 119910) 119889119904] (3)

119892 (120579 119904 119905) = 120583119901ℎ120596int120574max

120574min

exp [minusintinfin0

120583119865 (119909 119910) 119889119887] 119889120574 (4)

s

t

O

x

y

QP

S

mth detector

I0

Ii

Δd

훾min훾max

휃




120588 (119909 119910) 119889119905 (5)



1 2 3

J

1 2 3

J

1 2 3

J


휌j j j= 1 2 3 J 휇Ij = 1 2 3 J 휇F

j j = 1 2 3 J




119901 = 119875 (119899 minus 1) + 1199011015840 (6)



120583119868119896119871119868119901119896) (7)


120583119901ℎ120596120575119894119895119891119901119895120588119895119871119868119901119895 (8)




120583119865119902119871119865119901119895119902) (9)


119892119901119894119895 = 120583119901ℎ120596120575119901119895 119870sum119897=1


120583119865119902119871119865119901119895119902) (10)


119894 = 119872 (119899 minus 1) + 119898ℎ119894119895 = 119891119901119895119892119901119894119895120575119894119895 (11)


119868119894 = sum119895

ℎ119894119895120588119895 (119894 = 1 2 119868) (12)


I = H120588 (13)

where


(14)


1 2 3

J

Qp


(a)

jth pixel

kth pixelLI

pk

(b)

1 2 3

J

jth pixel

pth incident X-rayO

x

yt

s


mth detector

훿pj

Ii = IM(nminus1)+m


(c)


mth detector

qth pixelLF

pjq






3 Results



0

500

1500

1000

(a)

10020030040050060070080090010001100

(b)


0

001

002

003

004

005

(a)

0

5

10

15

times10minus3

(b)

0

002

004

006

008

(c)

0005

001

0015

002

0025

(d)






0

001

002

003

004

005

006

(a)0

001

002

003

004

(b)

0

0005

001

0015

(c)0

001

002

003

004

(d)






4 Discussion




0

1

2

3

4

5

6Re

cons

truc

ted

Au co

ncen

trat

ion

()



(a)

0

1

2

3

4

5

6

7

8

Reco

nstr

ucte

d Au

conc

entr

atio

n (

)



(b)

05

10

15

20

25

30

35

40

45

Reco

nstr

ucte

d Au

conc

entr

atio

n (

)



(c)


0

05

10

15

20

25

30Re

cons

truc

ted

Au co

ncen

trat

ion


(d)






5 Conclusion



20



0

2

4

6

8

10

12

14CN

R

(a)



0

2

4

6

8

10

12

14

CNR

(b)


0123456789

101112

CNR


(c)


0123456789

1011

CNR


(d)





Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2 3

J

1 2 3

J

1 2 3

J


휌j j j= 1 2 3 J 휇Ij = 1 2 3 J 휇F

j j = 1 2 3 J




119901 = 119875 (119899 minus 1) + 1199011015840 (6)



120583119868119896119871119868119901119896) (7)


120583119901ℎ120596120575119894119895119891119901119895120588119895119871119868119901119895 (8)




120583119865119902119871119865119901119895119902) (9)


119892119901119894119895 = 120583119901ℎ120596120575119901119895 119870sum119897=1


120583119865119902119871119865119901119895119902) (10)


119894 = 119872 (119899 minus 1) + 119898ℎ119894119895 = 119891119901119895119892119901119894119895120575119894119895 (11)


119868119894 = sum119895

ℎ119894119895120588119895 (119894 = 1 2 119868) (12)


I = H120588 (13)

where


(14)


1 2 3

J

Qp


(a)

jth pixel

kth pixelLI

pk

(b)

1 2 3

J

jth pixel

pth incident X-rayO

x

yt

s


mth detector

훿pj

Ii = IM(nminus1)+m


(c)


mth detector

qth pixelLF

pjq






3 Results



0

500

1500

1000

(a)

10020030040050060070080090010001100

(b)


0

001

002

003

004

005

(a)

0

5

10

15

times10minus3

(b)

0

002

004

006

008

(c)

0005

001

0015

002

0025

(d)






0

001

002

003

004

005

006

(a)0

001

002

003

004

(b)

0

0005

001

0015

(c)0

001

002

003

004

(d)






4 Discussion




0

1

2

3

4

5

6Re

cons

truc

ted

Au co

ncen

trat

ion

()



(a)

0

1

2

3

4

5

6

7

8

Reco

nstr

ucte

d Au

conc

entr

atio

n (

)



(b)

05

10

15

20

25

30

35

40

45

Reco

nstr

ucte

d Au

conc

entr

atio

n (

)



(c)


0

05

10

15

20

25

30Re

cons

truc

ted

Au co

ncen

trat

ion


(d)






5 Conclusion



20



0

2

4

6

8

10

12

14CN

R

(a)



0

2

4

6

8

10

12

14

CNR

(b)


0123456789

101112

CNR


(c)


0123456789

1011

CNR


(d)





Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2 3

J

Qp


(a)

jth pixel

kth pixelLI

pk

(b)

1 2 3

J

jth pixel

pth incident X-rayO

x

yt

s


mth detector

훿pj

Ii = IM(nminus1)+m


(c)


mth detector

qth pixelLF

pjq






3 Results



0

500

1500

1000

(a)

10020030040050060070080090010001100

(b)


0

001

002

003

004

005

(a)

0

5

10

15

times10minus3

(b)

0

002

004

006

008

(c)

0005

001

0015

002

0025

(d)






0

001

002

003

004

005

006

(a)0

001

002

003

004

(b)

0

0005

001

0015

(c)0

001

002

003

004

(d)






4 Discussion




0

1

2

3

4

5

6Re

cons

truc

ted

Au co

ncen

trat

ion

()



(a)

0

1

2

3

4

5

6

7

8

Reco

nstr

ucte

d Au

conc

entr

atio

n (

)



(b)

05

10

15

20

25

30

35

40

45

Reco

nstr

ucte

d Au

conc

entr

atio

n (

)



(c)


0

05

10

15

20

25

30Re

cons

truc

ted

Au co

ncen

trat

ion


(d)






5 Conclusion



20



0

2

4

6

8

10

12

14CN

R

(a)



0

2

4

6

8

10

12

14

CNR

(b)


0123456789

101112

CNR


(c)


0123456789

1011

CNR


(d)





Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

500

1500

1000

(a)

10020030040050060070080090010001100

(b)


0

001

002

003

004

005

(a)

0

5

10

15

times10minus3

(b)

0

002

004

006

008

(c)

0005

001

0015

002

0025

(d)






0

001

002

003

004

005

006

(a)0

001

002

003

004

(b)

0

0005

001

0015

(c)0

001

002

003

004

(d)






4 Discussion




0

1

2

3

4

5

6Re

cons

truc

ted

Au co

ncen

trat

ion

()



(a)

0

1

2

3

4

5

6

7

8

Reco

nstr

ucte

d Au

conc

entr

atio

n (

)



(b)

05

10

15

20

25

30

35

40

45

Reco

nstr

ucte

d Au

conc

entr

atio

n (

)



(c)


0

05

10

15

20

25

30Re

cons

truc

ted

Au co

ncen

trat

ion


(d)






5 Conclusion



20



0

2

4

6

8

10

12

14CN

R

(a)



0

2

4

6

8

10

12

14

CNR

(b)


0123456789

101112

CNR


(c)


0123456789

1011

CNR


(d)





Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

001

002

003

004

005

006

(a)0

001

002

003

004

(b)

0

0005

001

0015

(c)0

001

002

003

004

(d)






4 Discussion




0

1

2

3

4

5

6Re

cons

truc

ted

Au co

ncen

trat

ion

()



(a)

0

1

2

3

4

5

6

7

8

Reco

nstr

ucte

d Au

conc

entr

atio

n (

)



(b)

05

10

15

20

25

30

35

40

45

Reco

nstr

ucte

d Au

conc

entr

atio

n (

)



(c)


0

05

10

15

20

25

30Re

cons

truc

ted

Au co

ncen

trat

ion


(d)






5 Conclusion



20



0

2

4

6

8

10

12

14CN

R

(a)



0

2

4

6

8

10

12

14

CNR

(b)


0123456789

101112

CNR


(c)


0123456789

1011

CNR


(d)





Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

1

2

3

4

5

6Re

cons

truc

ted

Au co

ncen

trat

ion

()



(a)

0

1

2

3

4

5

6

7

8

Reco

nstr

ucte

d Au

conc

entr

atio

n (

)



(b)

05

10

15

20

25

30

35

40

45

Reco

nstr

ucte

d Au

conc

entr

atio

n (

)



(c)


0

05

10

15

20

25

30Re

cons

truc

ted

Au co

ncen

trat

ion


(d)






5 Conclusion



20



0

2

4

6

8

10

12

14CN

R

(a)



0

2

4

6

8

10

12

14

CNR

(b)


0123456789

101112

CNR


(c)


0123456789

1011

CNR


(d)





Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










20



0

2

4

6

8

10

12

14CN

R

(a)



0

2

4

6

8

10

12

14

CNR

(b)


0123456789

101112

CNR


(c)


0123456789

1011

CNR


(d)





Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









monte carlo simulation for polychromatic x-ray...

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