2008 IEEE Nuclear Science Symposium Conference Record M13·2

DIRECT 4D RECONSTRUCTION OFPARAMETRIC IMAGES INCORPORATINGANATO-FUNCTIONAL JOINT ENTROPY

ling Tang, Member, IEEE, Hiroto Kuwabara, Dean F. Wong, and Arman Rahmim, Member, IEEE

(1)t > t*

where

SP(t) = l CP(t), (2)

t* is the time for the tracer to reach steady state (determinedvisually), and Kj and bj are the Patlak slope and interceptparameters (as seen when dividing both sides of Eqn. 1 byCP(t)). In particular, the parameter of interest I'\,j indicates thefractional rate of uptake from the plasma to the tissue. Next,

II. METHODS

A. 4D Reconstruction Algorithm

Patlak and Blasberg presented a linear model for the timeactivity curves in a region of interest and the blood inputfunction, which applies to a tissue-compartment model thatcontains at least one irreversible compartment [11]. Based onthis Patlak model, the relation between the time-activity curvein a voxel j, Cj(t), and the blood input function, CP(t), canbe written as:

The Patlak linear model for irreversible tracers was recentlyincluded in a direct parametric estimation task, wherein theauthors expanded the objective function for the reconstructiontask to directly relate the Patlak parameters across the image tomeasured data, and used a preconditioned conjugate gradientalgorithm to find the optimum solution [5]. In this work, wehave alternatively extended the system matrix formulation andderived a direct 4D EM parametric reconstruction algorithm,with the advantages of being closed-form and accurate.

An area of growing interest has been that of developingreconstruction algorithms to incorporate anatomical informa­tion, yet taking into account the differences between PET andanatomical segmentation [6], [7]. In particular, Somayajula etal. [8] proposed to use the mutual information (MI) betweenfeatures (e.g. intensities or intensity gradients) of MRICT andPET images as priors. It was later suggested that the useof only the joint entropy (JE) component of the MI metricwas better conditioned to minimize bias in the reconstructedimages [9]. In [10], we designed and investigated a one-step­late (OSL) maximum a posterior (MAP) algorithm with theJE between features (such as intensity) of the anato-functionalimage pairs as the prior. In this work, we have implementeda new approach in which this methodology is directly appliedto the parametric PET images within the proposed 4D imagereconstruction algorithm.

I. INTRODUCTION

Dynamic PET imaging provides the capability to extractimportant physiological and biochemical parameters of interest[1]. This is most commonly achieved via compartmentalmodeling methods. There have been a number of differentformulations to directly estimate such kinetic parameters fromsinogram data (e.g. [2], [3]). The direct approach has theimportant advantages of: (i) not assuming the object beingstatic within a frame as it incorporates time-integrals ofthe tracer dynamics within each frame, and (ii) accuratelymodeling the Poisson noise distribution in PET data (whereasindirect methods face the very difficult task of estimatingnoise correlations in reconstructed images, often neglected forsimplicity).

The majority of direct parametric reconstruction methods inthe past utilized nonlinear kinetic models to estimate individ­ual kinetic parameters. Nonetheless, a number of graphicalmodeling methods have been developed that yield simplelinear/visual techniques for estimation/evaluation of kineticproperties of various PET tracers (e.g. see [4] as a review).

Manuscript received November 14, 2008.This work was supported by the Siemens Medical Solutions USA, INC. The

authors are with the Department of Radiology, The Johns Hopkins University,Baltimore, MD 21287 USA, e-mail: jtangI8@jhmi.edu.

Abstract- We developed a closed-form 40 algorithm to di­rectly reconstruct parametric images as obtained using the Patlakgraphical method for (nearly) irreversible tracers. Conventionalmethods consist of individually reconstructing 20/30 PET data,followed by graphical analysis on the sequence of reconstructedimages. The proposed approach maintains the simplicity andaccuracy of the EM algorithm by extending the system matrixto include the relation between the parametric images and themeasured data. The proposed technique achieves a closed-formsolution by utilizing a different hidden complete-data formulationwithin the EM framework. Additionally, the method is extendedto maximum a posterior (MAP) reconstruction via incorporatingMR image information, with the joint entropy between the MRand parametric PET features. A Parzen window method was usedto estimate the joint probability density of the MR and paramet­ric PET images. Using realistic simulated [llC]-Naitrindole PETand MR brain images/data, the quantitative performance of theproposed methods was investigated. Significant improvements interms of noise vs. bias performance have been achieved, whenperforming direct parametric reconstruction, and additionallywhen extending the algorithm to its Bayesian counter-part usingMR-PET join entropy.

Index Terms- 40 PET reconstruction, parametric image esti­mation, anato-functional joint entropy

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(9)

(10)

1 ~ 1· . 1

Pu,v(U,V) ==N L.)(21r)- (1IJhu 1lJ{,v)-2S jES

. exp(_~ (U ~ Uj)2 _ ~ (V ~ Vj)2 )].

2 1/luu 2 1/lvvUnlike other works in which the covariance matrix 1/Juu of

a PET image was assumed to have uniform diagonal entries,we chose the following model [10]:

B. Incorporation ofAnato-Metabolic Joint Entropy

We designed and investigated a one-step-Iate (OSL) MAPalgorithm with the JE between features (such as intensity)of the anato-functional image pairs as the prior [10]. In thepresent work, we seek to implement and study direct 40Bayesian parametric reconstruction incorporating JE betweenthe parametric image itself (u=", and b) and the MR anatom­ical image (v). The feature vectors u and v are consideredas realizations of the random feature variables U and V,respectively.

We used Parzen window (non-parametric) estimation of thejoint probability density (JPO) [14], [15] between the PETparametric and MR images, via a superposition of Gaussiandensities centered on the elements of a sample S drawn fromthe image pairs. Assuming that the convariance matrices (1/Juuand 1/Jvv) are diagonal, with the measured functional andanatomical intensity realizations u and v, the JPD Pu,v(U, V)is approximately evaluated as:

C. Simulated Tracer Study

Aside from the widely utilized tracer FOG, which is irre­versibly phosphorylated (k4=0), some other tracers of interestexhibit complete or near irreversibility (within scanning peri­ods). These tracers include [18F]-Spiperone, [18F]-CFf (WIN35428), 2-[18F]-FA, and e1C]-Naltrindole, the last of whichtargeting the delta opioid receptor was simulated in this work.

Le. each element of the covariance matrix was assumedproportional to squared image intensity (motivated by previousinvestigations into the EM algorithm [16]), where Q (theproportionality constant) was optimized.

With Pu,v(U, V) expressed as in Eqn. 9, the JE metric wasdefined as

Hu,v(U,V) == - L LPu,v(U,V)lnpu,v(U, V). (11)u v

The derivative of the JE metric was then incorporated withinthe OSL-MAP algorithm, as we first proposed in [10], but thistime for parametric image reconstruction; for instance, appliedto Eqn. 7 (u=",), this results in:

",old",new == (12)

s 2::=1 SP(n) + f3a~:.v IK=KOld

.~ §1'(n)pT [yn(,,;~;, bOld)] ,

where f3 is the weighting factor on the JE regularization.

(3)

(4)

(5)Y==PM

where

Y== (;~ )'M=(~)' and

( ~~~)PCP(l)P ).P== (6)

SP(N)P CP(N)P

where y(",old, bo1d)=p [Sp(n)",old + cP(n)b01d] and s is thestandard sensitivity image.

A different underlying framework: As can be seen above,we have derived a closed-form algorithm for direct parametricreconstruction from 4D data. It is noticed that applicationof the direct parametric EM scheme originally proposed byCarson and Lange [2] to this problem does not result in aclosed-form solution. The standard, physically-intuitive voxelcontributions to each bin i were used as the hidden completedata in Carson and Lange [2]. The present work employs thecontributions of each of the lij and bj elements to each bin ias the complete data, which is an unobservable complete dataspace. This implementation (upon going through the EM steps[12]) has produced the aforementioned closed-form algorithm.

It is worth noting that it was demonstrated by Fessler andHero [13] that it is possible and sometimes helpful to use alter­native complete-data formulations, to arrive at more effectivereconstruction algorithms. It can be seen that the present workproposes one such approach with potential benefits in the areaof direct parametric EM reconstruction.

where ts,n and te,n are the starting and ending times for framen (decay factors are not written here for simplicity, but needto be taken into account), and

note that the accumulated image xj in voxel j and frame ncan be written as:

Thus, using the standard system matrix P relating the imagevector x n at a particular frame n = (I. ..N) to the measureddata yn for that frame yn=pxn, we can arrive at the followingoverall 40 relation:

Thus, this approach effectively composes an extended systemmatrix P relating the combined parametric image M to theoverall 40 measured data Y. Using this extended formulationwithin the EM framework, after some simplification, onearrives at the following iterative algorithm:

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Fig. 1. Overall NSD (noise) vs. NMSE (bias) curves for the estimatedPatlak slope (generated by increasing iterations), from (i) 3D reconstructionfollowed by graphical modeling, and (ii) the proposed 40 direct parametricimage reconstruction.

The overall NMSE and overall NSD were calculated byaveraging the individual regional values.

III. RESULTS

As depicted in Fig. 1, in terms of the NSD (noise) vs.NMSE (bias) tradeoff, the proposed 4D direct parametricimage reconstruction technique noticeably outperforms theconventional technique (3D reconstruction followed by mod­eling). We can see at the same time that it takes moreiterations for the proposed technique (e.g., 1"V5 iterations) toachieve the same NMSE as for the conventional technique (1"V2iterations). The point is that, with matching NMSE values, thenoise performance is considerably improved by the proposedtechnique. The observation of slower convergence rates is acommon one when more advanced reconstruction techniquesare employed [18], [19].

For a visual comparison, Fig. 2 depicts slices throughparametric images obtained by the above two methods. Havingcomparable bias, the parametric images estimated from theproposed 4D direct reconstruction technique are clearly seento be much less noisy than those from the conventional 3Dreconstruction followed by graphical modeling analysis. Fur­thermore, Fig. 3 depicts quantitative NSD (noise) vs. NMSE(bias) comparisons for the above two methods in 16 individualbrain regions.

At the end, application of the direct parametric MAPreconstruction (incorporating JE between the parametric PETand MR images) is seen to further improve the NSD andNMSE tradeoff in most of the brain regions, as shown inFig. 4.

IV. SUMMARY

We have developed a closed-form 4D algorithm to directlyreconstruct parametric images as obtained using the Patlakgraphical method for (nearly) irreversible tracers. The pro­posed approach was implemented through composing a systemmatrix relating the combined parametric images directly to the

where K-T == ~ L:~1 ""r and jiT ~ L:~~1 JLr, ""r and JLr

denote the ith estimated and reference true Patlak slope value,respectively; and n is the number of voxels in the ROI r. Foreach ROI, the NMSE value was plotted against the NSD value,as calculated using

D. Evaluation Metrics

To evaluate the estimated parametric Patlak slope imagesquantitatively, we used the tradeoff between normalized meansquared error (NMSE) and normalized standard deviation(NSD) for individual regions of the brain (with exact bound­aries) and the tradeoff between overall NMSE and overall NSDover the whole brain. The NMSET for each ROI r (rangingfrom 1 to 16) covering one region of the brain was calculatedusing

(iT - jiT)2

NMSET == ~ (13)

......................•0.2

0.4

1.4.----0;::::::==================:=;-),]

1

,,0 1 3D Recon + Modeling1.2 •• -40 Direct Parametric Estimation

\50,,4

0", 3

""°""""""0,2

"""""", Iteration 1"""'0

c 0.8tnz 0.6

Fig. 2. Transaxial/coronal/sagittal slices through the parametric image: (top)the simulated original brain phantom, as well as reconstructions using (middle)standard 3D reconstruction followed by modeling, and (bottom) the proposed40 direct parametric EM reconstruction. The latter two images were chosento have similar bias.

We used existing parametric brain images from a humanstudy (implemented on a mathematical brain phantom [17])as the starting truth in our simulations. The correspondingdynamic datasets were then generated including realistic noiserealizations. The simulated dedicated-brain PET scanner had323 projection angles and 315 radial bins, and covered transax­ial and axial fields-of-view (FOYs) of 35.0 cm and 15.3 cm,respectively.

Parametric images were obtained using (i) conventional dy­namic 3D image reconstruction, followed by graphic modelinganalysis, (ii) the proposed 4D direct parametric reconstructionalgorithm and (iii) the proposed 4D direct parametric MAPalgorithm incorporating JE between parametric PET and MAPimages. We used 21 ordered subsets in the reconstructions.

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Fig. 4. Regional NSD vs. NMSE curves obtained for the estimated Patlakslope images, from the 4D direct parametric reconstruction without and with(MAP) incorporating the MR image information (16 brain regions).

Fig. 3. Regional NSD vs. NMSE curves obtained for the estimatedPatlak slope images, from standard 3D reconstruction followed by graphicalmodeling and from the proposed 4D direct parametric reconstruction (16 brainregions).

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[5J G. Wang, L. Fu, and 1. Qi, "Maximum a posteriori reconstruction of thePatlak parametric image from sinograms", Phys. Med. Bioi, pp. 593-604,2008.

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[7J 1. E. Bowsher et al., V. E. Johnson, T. G. Turkington, R. 1. Jaszczak,C. E. Floyd Jr., and R. E. Coleman, "Bayesian reconstruction and use ofanatomical a priori information for emission tomography", IEEE Trans.Med. Imag.,vol. 15, pp. 673-686, 1996.

[8J S. Somayajula, E. Asma, and R. M. Leahy, "PET image reconstructionusing anatomical information through mutual information based priors",IEEE NSSIMIC Conf Record, pp. 2722-26, 2005.

[9J 1. Nuyts, "The use of mutual information and joint entropy for anatomicalpriors in emission tomography", IEEE NSSIMIC Conf Record, Honolulu,2007.

[10J 1. Tang, B. M. W. Tsui, and A. Rahmim, "Bayesian PET imagereconstruction incorporating anato-functional joint entropy", IEEE Intern.Symp. Biomed. Imag. Proceeding, pp. 1043-46,2008.

[l1J C. S. Patlak and R. G. Blasberg RG, "Graphical evaluation of blood-to­brain transfer constants from multiple-time uptake data generalizations",J. Cereb. Blood Flow Metab., vol. 5, pp. 584-90, 1985.

1I2J T. Hebert and R. Leahy, "A generalized EM algorithm for 3-D Bayesianreconstruction from Poisson data using Gibbs priors", IEEE Trans. Med.Imag., vol. 8, pp. 194-202, 1989.

[13J J. A. Fessler and A. O. Hero, "Space-alternating generalized expectation­maximization algorithm", IEEE. Trans. sign. proc., vol. 42, pp. 2664-77,1994.

[14J R. O. Duda, P.E. Hart and D.G. Stork, Pattern Classification, SecondEdition, Wiley, 2001.

1I5J P. A. Viola, Alignment by maximation of mutual information, PhDthesis, Massachusetts Institute of Technology, 1995.

[16] H. H. Barrett, D. W. Wilson, and B. M. W. Tsui, "Noise properties ofthe EM algorithm: I. Theory", Phys. Med. Bioi., vol. 39, pp. 833-846,1994.

[17] A. Rahmim, K. Dinelle, 1. C. Cheng, M. A. Shilov, W. P. Segars, S. C.Lidstone, S. Blinder, O. G. Rousset, H. Vajihollahi, B. M. W. Tsui, D.F. Wong, and V. Sossi, "Accurate event-driven motion compensation inhigh-resolution PET incorporating scattered and random events", IEEETrans. Med. Imag., vol. 27, pp. 1018-1033,2008.

[18] A. Rahmim, 1. C. Cheng, and V. Sossi, "Improved noise propagationin statistical image reconstruction with resolution modeling", IEEE Nucl.Sci. Symp. Conf. Record, vol. 5, pp. 2576-2578, 2005.

[19J A. Rahmim, J. Tang, M. A. Lodge, S. Lashkari, M. R. Ay, R. Lautamaki,B. M. W. Tsui, and F. M. Bengel, "Analytic system matrix resolutionmodeling in PET: an application to Rb-82 cardiac imaging", Phys. Med.BioI., vol. 53, pp. 5947-5965, 2008.

achieved when the proposed MAP algorithm (incorporatingjoint entropy between parametric PET and MR information)was employed.

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measured 4D data. The closed-form solution was achieved bythe utilization of a different complete-data formulation withinthe EM method. Instead of using the standard, physically­intuitive voxel contributions to each bin i as the hiddencomplete data, the present framework effectively employs thecontributions of each of the Patlak elements ("'j and bj ) toeach bin i as the complete data. This has ultimately producedthe aforementioned closed-form algorithm.

Furthermore, the method was extended to maximum aposterior (MAP) reconstruction via incorporating MR imageinformation, making use of the joint entropy between theparametric PET and MR image features. The NSD (noise)vs. NMSE (bias) tradeoff of the directly estimated Patlakslope image was shown to be significantly better than thatestimated from graphical analysis on 3D reconstructed images.Additional improvement on the NSD vs. NMSE tradeoff was

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