2008 IEEE Nuclear Science Symposium Conference Record M03-8

Spatially Variant Positron Range ModelingDerived from CT for PET Image Reconstruction

Adam Alessio, Member, IEEE , and Lawrence MacDonald, Member, IEEE

TABLE ICO~IP:\RISON OF ,\VERAGE POSITRO;-'; RA~GE ESTI~IATES (~I~I) IN WATER

where the energy of the positron, E, is in MeV and the maxi­mum range, Rmax is independent of material. The evaluationof (1) over the positron energy spectra for common isotopeslead to ID annihilation probability density functions presentedin figure 1. The mean of these range distributions are inkeeping with published data as presented in table I.

Abstract- The influence of a finite positron annihilation dis­tance represents a fundamental limit to the spatial resolutionof PET scanners. It is appreciated that this effect is a minorconcern in whole-body F18 imaging, but it does represent an issuewhen imaging with higher energy isotopes such as N13 or Rb82.This effect is especially relevant for imaging tasks along tissuegradients such as the cardiac/lung boundary and diaphragm/lungboundary. This work presents a method to determine the positronrange effect from a CT scan and to model this effect as shift­variant, anisotropic kernels. The positron annihilation distanceacross boundaries of tissues is represented with a simple model,which can be quickly derived from CT scans and applied in thereconstruction of PET images. The positron range compensationmap is applied in a modified OSEM algorithm to simulated andmeasured data.

IsotopeF18N13Rb82

Proposed0.611.566.57

Bai [6]0.511.31

Champion [7]0.661.737.49

Partridge [8]0.61.55.9

I. INTRODUCTION

0.01 ::; E ::; 2.5l\leV (1)2.5MeV < E < 20MeV

Fig. 1. Positron 10 annihilation density in water based on positron spectraand eq (1).

15

- - - F18- - C11-N13

Rb82

10distance (mm)

o "o

0.03 ,\, ,0,025'1 \

1 \1 \

0.02" \

§ :: " ~ \~ O,015~ I \\

o " \ \0.. ~I \ \

0.01 ~ ,,\

\ \0.005 \

The Katz and Penfold empirical equation is based on thetheory that the ability to stop beta particles depends primarilyon the density of electrons in the media (density of themedia in mg/cm3). Using (1) and positron energy spectra,we computed the average positron annihilation distance as afunction of absorber density as shown in figure 2.

To test the validity of the model, we compared the modelwith simulations using GEANT4 in the GATE simulationpackage [9]. GEANT4 uses a multiple scattering model basedon Lewis's theory [10), and we performed condensed simu­lations of the transport of positrons through matter. Figure 3presents the GEANT4 and the proposed model ID annihilationdistance in terms of the fraction of the number of positronsannihilated versus distance for different media. Comparison ofcurves shows agreement between the simulations and the Katz& Penfold empirical decay.

The previous discussion has focused on the 1D annihilationprobability: the probability that a positron will annihilate ata distance, d, from the site of decay. To form the positronrange blurring kernels, we determine the 3D annihilation

II. POSITRON RANGE IN UNIFORM MEDIA

Positron range degrades the spatial resolution in PET imag­ing. The majority of work modeling positron range for imagingtasks assumes that the medium is homogenous with the densityof water [4]. In homogenous tissue, the influence of positronrange can be modeled as a isotropic 3D kernel. We computedthe maximum positron range from a formula fit to empiricalbeta decay data by Katz and Penfold [5):

Rmax [mg/crn2] =

{

412El,26Ei-O,09Ei4lnE

530E - 106

Positron range is generally accepted as a minor concernin the resolution of Fl8 imaging with whole-body PET sys­tems [1]-[4]. For isotopes with higher positron energies, theinfluence of positron range can result in resolution degradationwhich is spatially variant dependent on the variations inabsorber media. This effect is especially relevant for imagingtasks along tissue gradients such as the cardiac/lung boundaryand diaphragm/lung boundary. This work adopts a simplemodel of the positron annihilation range for different densitytissues and relates these variant ranges to the Hounsfield unitsin the CT image. The model allows for the CT image froma PET/CT exam to guide the generation of a shift-variant,anisotropic positron range compensation map. The positronrange map is applied in a modified OSEM algorithm tosimulated and measured data.

Manuscript for 2008 IEEE NSS/MIC proceedings received November 7,2008. This work was supported by NIH grant HL086713.

A. Alessio and L. MacDonald are with the Dept. of Radiology, Universityof Washington, Seattle, WA 98195 USA. (telephone: (206)543-2419, email:aalessio@u.washington.edu)

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0.18

0.16 0.18

0.16

--------~

0.140.12

-------------------

0.04 0.06 0.08 0.1 0.12 0.14Il @ 511 keV (1Iem)

0.04 0.06 0.08 0.10.02

0.2·

0.02

0.8(1"I

0.61

0 0 .4

Fig. 5. Positron annihilation fitted parameters as a function of material asdefined by its linear attenuation coefficient of 51lkeV photons.

- ~ ~ f - - C-11 ,~

~ 10~ -N13 .. .. .-Js 8t - - -Rb82 :.-:-:-:---- I

~..... :I... __~=--.:.=~-~-,"-~~~~: =-c=. - -d~.:.-· - .- -1.: -. - =~- -- -.=-= -0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

In an effort to model the positron annihilation density fordifferent absorbers, we fit this dual exponential function tothe annihilation probability density derived from the Katz &Penfold empirical equation for a range of materials.

We define the absorbing material with the CT image ac­quired during a PET/CT exam. Considering that CT imagingis non-quantitative, there is no direct link between CT number(Hounsfield units) and tissue density (mg/cm 3). We assumethat the bilinear/trilinear scaling methods which convert theCT image to linear attenuation coefficients at 51lkeV (fJ@ 511 keV) for conventional PET attenuation correction areaccurate (This assumption generally holds for physiologictissues) [11]. The mass attenuation coefficients (fJ/density) at511 keV of tissues of interest are all close to 0.096 cm 2/g.Dividing the fJ @ 5IlkeV map (l/cm) by 0.096 cm 2/gprovides an approximate map of the density of the tissues.Figure 5 provides the values of the parameters (k 1 , k2 , C) thatfit the annihilation probability densities as a function of anabsorber's linear attenuation coefficient at 511keV. This figureprovides a lookup table of the anticipated annihilation densityfor a given PET attenuation value.

- - -F18- - C11-N13

Rb82

F18, Katz & Penfold

Rb82, Katz & Penfold

RlbBoneWaterLung

RlbBoneWater

··Lung10 i~~ -,(1

F18, GEANT4

Rb82, GEANT4

35

Fig. 3. Fraction of positrons annihilated within fixed distances derived fromGEANT4 simulations and proposed empirical model using Katz & Penfold.

probability: the probability that a positron will annihilate ata location (x, y, Z)T when it originated at the site (x, y, z)o.The 3D annihilation density is formed by normalizing the IDannihilation density by the surface area of the sphere at eachlocation. Figure 4 compares the 3D annihilation probabilitydensity of the GEANT4 simulations and the proposed use ofKatz & Penfold's eq( 1).

Fig. 2. Average positron range in media of different density based on eq(1).

Rb82, GEANT4 Rb82, Katz & PenfoldIII. POSITRON RANGE IN NON-UNIFORM MEDIA

10 15 203DanrllhllatlooastanCe(mml

Fig. 4. Comparison of 3D annihilation density functions of the GEANT4simulations and proposed empirical model for Rb82.

Derenzo [3J proposed that these 3D annihilation probabili­ties can be fitted by the sum of two exponentials,

(2)

In non-homogenous tissue, the positron range could bemodeled as anisotropic kernels which ideally would be de­rived from Monte Carlo simulations for each patient's tissuedistribution. The Monte Carlo approach is generally too com­putationally intensive for clinical PET imaging. A primarychallenge of quickly determining the range effect in non­homogenous media is how to model the range across bound­aries.

Bai et al. have explored modeling positron range innon-homogenous materials for a small animal system [6].They compensated for variant positron ranges by either A)anisotropically truncating an isotropic point probability densityfunction dependent on tissue type, or B) performing successive

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Fig. 7. Phantom used in GATE simulations with rectangular air, lung, boneand water regions. Uniform activity was simulated within each dotted regionin the central water region to mimic activity in the myocardium next towater, lung, and bone. The right activity region contained twice the activityconcentration as the left region. Colorbar presents density values (g/crfl3).

2

o

1.8

1.4

0.8

1.6

0.6

1.2

0.2

0.4

500400200 300(mm)

100

50

300

100

500

150

450

400

350

200

E250g

in a modified OSEM reconstruction such that the mean ofthe Poisson distributed PET measurements, y, are given byy == PgeoPx, where Pgeo is the geometric projection matrix,P is the proposed blurring kernels, and x is the image.

B. GATE Simulations

We applied the proposed positron range compensation tothe reconstruction of data from GATE simulations. A phantompresented in figure 7 with lung, bone, and water regions wassimulated to generate Rb82, 2D PET data for a cylindrical,clinical PET system (70cm transaxial FOY, 15cm axial FOY).

101

40,

30fI

convolution operations of tissue dependent range kernels todetermine range models across tissue boundaries.

In this work we adopt a crude, but fast, model to adjustfor variations across boundaries. We perform a average of thefitted parameters (from figure 5) of annihilation densities forthe originating voxel and the target voxel. This average definesa new dual-exponential function to describe the probabilitythat an positron annihilated in a target voxel. Specifically, todetermine the positron range blurring kernel for originatingvoxel, x o , we define its contributions to a target voxel Xt as

Pot == Cote-klotdot + (1- Cot)e-k2otdot (3)

where dot is the distance from voxel X o to voxel Xl, Cot isthe average of the C parameters from from voxels X 0 and Xt,

and k[1,2]ot is the average of the k parameters from voxelsX o and Xt. The 3D positron blurring kernel is evaluated foreach originating voxel over all potential target voxels. All ofthe kernels are normalize to have density equal to I leadingto P that provides an estimate of the positron range blurringfor each pixel. Figure 6 presents the blurring kernels Po* fora decay of Rb82 in water next to a bone boundary and a lungboundary.

Fig. 6. Kernel used for decay of Rb82 in a pixel in water (+) next to theboundary of lung (left half of plot), rib bone (upper right quadrant), and water(lower right quadrant). The FWTM contour is plotted as the 2nd contour fromcenter (green) and the FW at 200th maximum is the most extreme contourplotted.

10 15 20 25

mm30 35 40

C. Patient Data

We applied the proposed method to reconstruct patient datafrom a clinical Rb82 cardiac PETICT exam. We convertedthe patient CT image to the PET attenuation map for 51 IkeYphotons. The map guided the generation of unique blurringkernels, with a radius of 2cm, for each voxel location.

IY. METHODS

A. Application to PETICT Reconstruction

For a patient sPecific attenuation map, the positron blurringkernel can be calculated according to eq (3). This kernel is anobvious approximation based on the probability of positronannihilation in any given voxel being drawn from the averageprobability density functions of the source and target voxel.This approximation allows for a fast computation of theblurring kernel from an arbitrary 511keY attenuation map. Westore the entire blurring map in a sparse matrix and apply it

Y. RESULTS

The reconstructed images of the GATE simulated dataappear in figure 8. The use of the positron range comPensationyielded sharper boundary definition with little to no tails at theedge of the water lung boundary. On the negative side, in theabsence of post-reconstruction smoothing. the positron rangecompensation does lead to hyper-resolved boundaries.

The images from the patient data are presented in figure 9.The reconstruction parameters (number of iterations and post­recon smoothing) were set to have visually similar resolution(as shown in profile). With "matched" resolution, the positronrange method appears to have reduced noise.

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Fig. 8. Images reconstructed from GATE simulations of two hot sources inwater next to lungs and bone. Image A is reconstructed with conventionalOSEM; Image B is reconstructed with proposed positron range kernel.Horizontal profiles (left) and vertical profiles through lung-water-Iung regions(right) highlight hyper-resolved boundaries with positron range model (red).

VII. ACKNOWLEDGMENT

Authors are grateful to Rob deKemp of the Ottawa HeartInstitute for patient data sets.

[11 M. E. Phelps, E. J. Hoffman, S.-C. Huang, and M. M. Ter-Pogossian,"Effect of positron range on spatial resolution," J Nucl Med, vol. 16,no. 7, pp. 649-652, 1975.

[2] Z. H. Cho, J. K. Chan, L. Ericksson, M. Singh, S. Graham, N. S.MacDonald, and Y. Yano, "Positron ranges obtained from biomedicallyimportant positron-emitting radionuclides," J Nucl Med, vol. 16, no. 12,pp. 1174-1176, 1975.

[31 S. Derenzo, "Precision measurement of annihilation point spread distri­butions for medically important positron emitters," in Positron Annihi­lation, R. Fujiwara, Ed. Sendai, Japan: The Japan Institute of Metals,1979, p. 81923.

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[5] L. Katz and A. S. Penfold, "Range-energy relations for electrons and thedetermination of beta-ray end-point energies by absorption," Reviews ofModern Physics, vol. 24, no. 1, pp. 28 LP - 44, 1952.

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[8] M. Partridge, A. Spinelli, W. Ryder, and C. Hindorf, "The effect of[beta]+ energy on performance of a small animal PET camera," NuclInsfr Meth Phys Res A, vol. 568, no. 2, pp. 933-936, 2006.

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REFERENCES

VI. CONCLUSION

We presented a new method to link the CT image withan anticipated positron range effect. The method allows forthe rapid estimation of a range blurring comPensation mapwhich can be applied in a modified OSEM algorithm. Wepresented initial results of the positron range comPensationwith simulated and measured Rb82 cardiac PET/CT datawhich suggest improved resolution/noise properties with theproposed approach. A thorough image quality evaluation of theproposed method is needed to quantify the benefit for cardiacPET imaging.

B.A.

,00

Fig. 9. Transaxial slice through patient images of Rb82 with reconstructionby 2D OSEM (left) and 2D OSEM with proposed positron range kernel (right).Profile through right and left ventricle appears to have matched boundarydefinition with reduced noise in reconstruction with positron range (yellow).

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