1

Direct Patlak Reconstruction from Dynamic PETData Using the Kernel Method with MRIInformation Based on Structural Similarity

Kuang Gong, Jinxiu Cheng-Liao, Guobao Wang, Kevin T. Chen, Ciprian Catana and Jinyi Qi*

Abstract—Positron Emission Tomography (PET) is a functionalimaging modality widely used in oncology, cardiology, andneuroscience. It is highly sensitive, but suffers from relativelypoor spatial resolution, as compared with anatomical imagingmodalities, such as magnetic resonance imaging (MRI). Withthe recent development of combined PET/MR systems, we canimprove the PET image quality by incorporating MR informationinto image reconstruction. Previously kernel learning has beensuccessfully embedded into static and dynamic PET imagereconstruction using either PET temporal or MRI information.Here we combine both PET temporal and MRI informationadaptively to improve the quality of direct Patlak reconstruction.We examined different approaches to combine the PET and MRIinformation in kernel learning to address the issue of potentialmismatches between MRI and PET signals. Computer simula-tions and hybrid real patient data acquired on a simultaneousPET/MR scanner were used to evaluate the proposed methods.Results show that the method that combines PET temporalinformation and MRI spatial information adaptively based onthe structure similarity index (SSIM) has the best performancein terms of noise reduction and resolution improvement.

Index Terms—Positron emission tomography, Patlak directreconstruction, Kernel method, MRI, structure similarity

I. INTRODUCTION

Positron Emission Tomography (PET) can produce threedimensional functional images of a living subject through theinjection of a radioactive tracer. It has wide applications inoncology [1], cardiology [2] and neurology [3]. Though PETis highly sensitive, in the picomolar range, the image qualityis still limited by various resolution degradation factors [4].Compared with PET, x-ray Computed Tomography (CT) andMagnetic Resonance Imaging (MRI) have better anatomicalimage resolution. With the wide availability of PET/CT andthe new arrival of PET/MR systems, anatomical informationfrom CT or MRI are readily available for use in the PETreconstruction. Research in this area has been going on fordecades and different methods have been developed based on

Copyright (c) 2017 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

K. Gong, J. Cheng-Liao and J. Qi* are with the Department of Biomed-ical Engineering, University of California, Davis, CA 95616 USA (e-mail:qi@ucdavis.edu).

G. Wang is with the Department of Radiology, University of California,Davis, CA 95616 USA.

K. T. Chen and C. Catana are with Martinos Center for BiomedicalImaging, Department of Radiology, Massachusetts General Hospital andHarvard Medical School, Charlestown, MA 02129 USA.

approaches such as mutual information or joint entropy [5]–[7], segmentation or boundary extraction [8]–[12], gradient in-formation from a prior image [13], [14], intensity similarity ina prior image [15]–[19], wavelet [20], dictionary learning [21],[22], image sparse representation [23], convolutional neuralnetwork [24] and the kernel method [25]–[28]. The kernelmethod is a well-studied framework in machine learning thatimplicitly transfers the raw data points to a higher dimensionalfeature space using a kernel function. It has recently beenapplied to dynamic image reconstruction [25] and successfullyextended to MRI aided PET reconstruction [26]–[28].

Here we combine the above two approaches to furtherimprove the quality of dynamic PET reconstruction. Comparedwith static PET, dynamic PET provides additional temporal in-formation of the tracer kinetics, which can be useful for tumordetection and treatment monitoring [29]–[31]. Conventionallydynamic PET images are reconstructed frame by frame, andtime activity curves (TAC) from regions of interest (ROI) or in-dividual pixels are fitted to a linear or nonlinear kinetic model.Direct reconstruction methods have been developed recently toestimate parametric images directly from the measured sino-gram by combining the PET imaging model and tracer kineticsin an integrated framework [32]–[36]. It has been shown thatdirect reconstruction methods can reduce noise amplificationcompared with traditional indirect methods [37]. Anatomicalinformation can be used to regularize the parametric images indirect reconstruction [28], [38], [39]. In this work, we focus onthe Patlak model [40]–[44], which is a widely used graphicalmodel. The slope of Patlak plot is a useful quantitative metricwhen the tracer uptake is irreversible. One challenge in usinganatomical information in PET reconstruction is that there areoften mismatches between MRI and PET signals in real data.For example, tumor regions showing up in a PET image maynot appear in a co-registered MRI image [45]. It is thus crucialto account for this mismatch by exploiting both anatomical andPET information to construct the kernel matrix. In a previouswork [27], spectral models were employed to model the PETtemporal information and the kernel matrix were constructedusing MRI information. The work here aims at combiningthe MRI based kernel matrix and dynamic PET based kernelmatrix adaptively using a local similarity measure.

The main contributions of this paper include (1) combiningthe PET information and the MRI information adaptivelybased on local similarity and (2) applying this hybrid kernelto the direct Patlak reconstruction. This paper is organized asfollows. Section 2 introduces the Patlak model and the kernel

2

method. Section 3 describes the computer simulations andhybrid real data used in the evaluation. Experimental resultsare shown in section 4, followed by discussions in section 5.Finally conclusions are drawn in Section 6.

II. METHOD

A. Dynamic PET model

Dynamic PET data in the kth time frame, yk ∈ RM×1, canbe modeled as a collection of independent Poisson randomvariables with the mean yk ∈ RM×1 related to the unknowntracer distribution in the kth time frame, xk ∈ RN×1, throughan affine transform

yk = NkPxk + sk + rk (1)

where Nk ∈ RM×M is the time dependent normalizationmatrix. P ∈ RM×N is the detection probability matrix withthe (i, j)th element pij being the probability of detecting aphoton pair produced in voxel j by detector pair i. sk ∈ RM×1and rk ∈ RM×1 denote the expectation of scattered andrandom coincidences in the kth time frame. M is the numberof lines of response (LOR) and N is the number of imagevoxels. When concatenating T time frames together, we canget

y = N(IT ⊗ P )x+ s+ r, (2)

where N = diag[N1,N2, ...NT ], ⊗ denotes the Kroneckerproduct, IT is a T × T identity matrix, y = [y′1, y

′2, ...y

′T ]′,

x = [x′1, x′2, ...x

′T ]′, r = [r′1, r

′2, ...r

′T ]′, s = [s′1, s

′2, ...sT

′]′,and “′” denotes matrix or vector transpose. Let y =[y′1,y

′2, ...y

′T ]′ denote the measured dynamic data. The log-

likelihood function of y can be written as

L(y|x) =

T∑k=1

M∑i=1

(yk)i log(yk)i − (yk)i − log(yk)i! . (3)

The maximum likelihood estimate of the unknown image xcan be found by

x = arg maxx≥0

L(y|x). (4)

B. Patlak model

The Patlak graphical method [40] has been widely usedfor compartment models that contain at least one irreversiblecompartment. Under this model, the tracer concentration attime t, c(t) ∈ RN×1, can be represented by a weighted sum ofthe blood input function Cp(t) and its integral after a sufficientlength of time t? as

c(t) = κ

∫ t

0

Cp(τ)dτ + bCp(t), t > t?, (5)

where κ ∈ RN×1 is the Patlak slope image and b ∈ RN×1is the Patlak intercept image. The Patlak slope image κ canreflect the influx rate of the PET tracer and is a very importantquantitative metric in practice. Considering the radioactivedecay, the image intensity at the kth frame, xk, can beexpressed as

xk =

∫ te,k

ts,k

c(τ)e−λτdτ = Sp(k)κ+ Cp(k)b (6)

where ts,k and te,k are the start time and end time of framek, λ is the decay constant of the PET tracer, and

Sp(k) =

∫ te,k

ts,k

∫ τ

0

Cp(τ1)dτ1e−λτdτ, (7)

Cp(k) =

∫ te,k

ts,k

Cp(τ)e−λτdτ. (8)

Putting all time frames together, we can have

x = (A⊗ IN )θ, (9)

where θ = [κ′, b′]′, IN is an N ×N identity matrix, and

A =

Sp(1) Cp(1)Sp(2) Cp(2)

......

Sp(T ) Cp(T )

. (10)

Conventionally, images are reconstructed frame-by-frame ac-cording to (4) and then the Patlak parameters are estimatedby fitting the time activity curves to the Patlak model (9). Theleast squares solution of the fitting is

θ = ((A⊗ IN )′(A⊗ IN ))−1

(A⊗ IN )′x. (11)

C. Direct reconstruction using the kernel method

Here we choose the direct reconstruction approach bysubstituting the Patlak graphical model (9) into (2) to get:

y = N(IT ⊗ P )(A⊗ IN )θ + s+ r. (12)

Previously efforts have been made to add regularization in theparametric space by using a penalty function, which aims toenforce smoothness inside the region and allow sharp transi-tion near the edges [38], [39]. Here we choose to regularize thedirect reconstruction through the use of a kernel matrix [25].The main idea is to represent the unknown parametric imagesby a linear combination of transformed features calculatedfrom prior information. Using the kernel trick, we can expressthe Patlak slope and intercept images as

κ = Kακ, (13)b = Kαb, (14)

where K ∈ RN×N is the kernel matrix, ακ ∈ RN×1and αb ∈ RN×1 denote the kernel coefficient vectors of κand b, respectively. Construction of the kernel matrix K isdescribed in Section II.D. Here the same kernel matrix isused for the Patlak slope and intercept because we expect thatthey share the same regional boundaries, although they mayhave different intensities and contrasts. However, the proposedalgorithm is amendable to use different kernel matrices for thePatlak slope and intercept, if necessary. In the implementation,we require the kernel coefficients to be non-negative, whichguarantees non-negative Patlak slope and intercept. Definingα = [α′κ,α

′b]′, θ can be expressed as

θ = (I2 ⊗K)α. (15)

Substituting (15) into (12), we get

y = N [A⊗ (PK)]α+ s+ r, (16)

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Algorithm 1 Algorithm for direct Patlak reconstruction usingthe kernel methodInput: Maximum iteration number MaxIt, sub-iteration

number SubIt, starting time frame FraSta and endingtime frame FraEnd

1: Initialize α0,SubIt = 1N2: for n = 1 to MaxIt do3: xn = (A⊗K)αn−1,SubIt

4: αn,0 = αn−1,SubIt

5: for k = FraSta to FraEnd do6: xn+1

k =xn

k

(NkPK)′1M

[(NkPK)′ y

(NkPK)xnk+s+r

]7: end for8: for q = 1 to SubIt do9: αn,q = αn,q−1

[A⊗(PK)]′N1MT([A⊗ (PK)]′N1MT

xn+1

(A⊗IN )αn,q−1

), where 1MT

is a MT × 1 all-one vector.10: end for11: end for12: return θ = (I2 ⊗K)αMaxIt,SubIt

where we use the following property of Kronecker product

(IT ⊗ P )(A⊗ IN )(I2 ⊗K) = A⊗ (PK). (17)

To estimate α, we use the nested EM algorithm [46] for fasterconvergence. The resulting algorithm flowchart is shown inAlgorithm 1.

D. Kernel construction

Following the original paper [25], we use a radial Gaussiankernel. The (i, j)th element of the kernel matrix K is

K(fi,fj) = exp

(−||fi − fj ||

2

2Nfσ2

), (18)

where fi ∈ RNf×1 and fj ∈ RNf×1 represent the featurevectors of voxel i and voxel j, respectively, σ2 is the varianceof the image and Nf is the number of voxels in a featurevector. Two types of feature vectors have been proposedbefore. In [25], one-hour long dynamic PET data were re-binned into three twenty-minutes frames and the reconstructedimages xreb

m ,m = 1, 2, 3, were used to form the feature vectorsfPi ,

fPi = {(xreb1 )i, (x

reb2 )i, (x

reb3 )i}. (19)

Hereinafter, this method is referred to as KPET. The secondmethod extracts the feature vectors from a co-registered MRimage [26], [27],

fMi = {(xM )i}, (20)

where (xM )i is a local patch centered at voxel i from the MRimage. This method is referred to as KMRI. In this paper, a3×3×3 local patch was extracted for each voxel to form thefeature vectors. Before the feature extraction, the re-binnedPET images and MR image were normalized so that the imagestandard variation inside the brain region is 1 and we set σ = 1in (18).

As shown in previous publications [26], [27], KMRI canpreserve sharp boundaries in PET images. However, it can

also introduce bias when PET tracer uptake does not share thesame boundary with the MRI image. A better approach is tocombine the two types of kernels so that we can minimize therisk of smoothing out PET signals while exploiting anatomicalinformation from KMRI. An intuitive choice is to weight MRIand PET features equally, which we refer to as K-PET-MRI-0.5. This method would unfortunately reduce the image qualityin the regions where PET image and MRI image share sameboundaries compared with that using the KMRI alone, becausePET features are often noisier than MRI features. A betterapproach is to assign a spatially variant weight to PET andMRI features based on a similarity measure ρ. Ideally thesimilarity measure ρ should be high (close to 1) in regionswhere PET and MRI images share the same boundary andbe low (close to 0) in regions where their boundaries do notmatch.

Given a good similarity measure, we can weight PET andMRI features based on the similarity. The hybrid kernel matrixis then formed as

K(fi,fj) = exp

(− Dij

2Nfσ2

). (21)

where

Dij =(1− ρi)(1− ρj)||fPi − fPj ||2+3ρiρj ||fMi − fMj ||2

(1− ρi)(1− ρj) + 3ρiρj. (22)

The factor 3 is included because there are three PET imagesin (19) and only one MRI image in (20). When multipleMRI images are available (e.g., [47]), the factor should beadjusted accordingly. Equation (22) is constructed so that Dij

is symmetric.A crucial task is to find an appropriate similarity measure

ρ. We studied two approaches to calculating ρ. The first onewas based on the correlation between the PET kernel and MRIkernel vectors [48] and the second method was based on thestructural similarity [49] between the reconstructed images ofthe last frame data using KPET and KMRI. We found that thekernel-correlation based result is not always satisfactory asthe correlation between matched flat regions is dominated byimage noise. Therefore, we focus on the structural similaritybased method in this paper. This method is referred to as K-PET-MRI-S.

The structural similarity (SSIM) index [49] is a widely usedimage quality index for comparing images. Before calculatingthe SSIM index map, the input images are scaled to have amean value of one inside the brain regions. Given two localpatches extracted from the input images, x and y, the generaldefinition of SSIM is

SSIM(x,y) = [l(x,y)]α · [c(x,y)]

β · [s(x,y)]γ, (23)

where l(x,y) =2µxµy+C1

µx2+µy

2+C1, c(x,y) =

2σxσy+C2

σx2+σy

2+C2and

s(x,y) =σxy+C3

σxσy+C3are the luminance, contrast and similarity

measures. Here µx, µy , σ2x, σ2

y are the means and variancesof x and y, and σxy is the covariance between x and y.To avoid block artifacts, neighboring voxels are weightedby an isotropic Gaussian function. The standard deviation ofthe Gaussian function was set to 1 pixel in our experiment,because we found that the default value (1.5 pixels) suggested

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by [49] resulted in a blurry SSIM map for our application. C1,C2 and C3 are small constants used to stabilize the results. Weused the MATLAB implementation [49] with C1 = (0.01L)2,C2 = (0.03L)2 and C3 = C2/2, where L is a constant value.Based on our observations, the most influencing componentof SSIM(x,y) for distinguishing an unmatched region frommatched regions is the similarity measure s(x,y). Hence wefixed α and β to 1 and adjusted γ and L. We evaluated themean SSIM values for a simulated unmatched tumor regionand matched normal region for a wide range of γ and L values.We found that setting γ = 12 and L = 50 is a reasonablechoice, which resulted in a low SSIM value for the unmatchedregions while keeping the SSIM value for the matched regionsto be greater than 0.95. More details of the calculation aregiven in the supplemental materials.

After obtaining SSIM index, we calculate ρ by scaling theSSIM index

ρi = max

(0,

SSIMi − C1− C

), (24)

where C was set to 0.2. The reason for scaling is that theminimum value of the SSIM value is around 0.2 while therange of ρi should be between 0 and 1.

For efficient computation, all kernel matrices were con-structed using a K-Nearest-Neighbor (KNN) search in a7×7×7 search window with K = 50, i.e.,

Kij =

{K(fi,fj), fj ∈ KNN of fi,0, otherwise.

(25)

Following [25], all kernel matrices were normalized by

K = diag−1[K1N ]K (26)

and K was then used in the image reconstruction.

III. EXPERIMENT EVALUATIONS

A. Computer simulation study

A 3D brain phantom from the Brainweb [50] was used to inthe simulation study. The scanner geometry models a SiemensmCT scanner [51]. The system matrix P was computed byusing the multi-ray tracing method [52]. The image matrixsize is 125 × 125 × 105 and the voxel size is 2 × 2 ×2 mm3. The time activity curves of the gray matter andwhite matter were generated mimicking an FDG scan usinga two-tissue-compartment model with the kinetic parametersshown in Table I and an analytic blood input function [53]in Fig. 1. To simulate mismatches between the MR and PETimages, twelve hot spheres of diameter 16 mm were insertedinto the PET image as tumor regions, which are not visiblein the MR image. The dynamic PET scan was divided into24 time frames: 4×20 s, 4×40s, 4×60 s, 4×180 s, and8×300 s. For the direct Patlak reconstruction, only the last5 frames for a total duration of 25 minutes were used. Threeorthogonal slices of the MRI prior image, PET attenuationmap and PET activity image at the 24th frame are shown inFig. 2. Noise-free sinogram data were generated by forward-projecting the ground-truth images using the system matrixand the attenuation map. Poisson noise was then introduced

0 10 20 30 40 50 60

Scan time (min)

0

20

40

60

80

100

120

140

Act

ivity

(a.

u.)

BloodGray matterWhite matterTumor

Fig. 1: The simulated time activity curves.

to the noise-free data by setting the total count level to beequivalent to an 1-hour FDG scan with 5 mCi injection.Uniform random events were simulated and accounted for30 percent of the noise free data in all time frames. Scatterswere not simulated in this study. During image reconstruction,the blood input function and all the correction factors wereassumed to be known exactly. Apart from the tumor study,we also performed an neural activation simulation study bysimulating the activation regions according to [17]. Resultsare shown in the supplemental materials.

TABLE I: The simulated kinetic parameters of FDG

Tissue fv K1 k2 k3 k4Gray matter 0.03 0.071 0.086 0.055 0.001White matter 0.03 0.046 0.080 0.052 0.001

Tumor 0.05 0.082 0.055 0.085 0.002

To quantify the noise properties across different realiza-tions, multiple independent and identically distributed (i.i.d.)dynamic data sets were generated. The KPET kernel matrixand the hybrid kernels were calculated for each realization.The KMRI kernel matrix was kept the same in all realizations.When constructing the PET features for each realization, thefirst, second and last twenty-minutes frame were reconstructedusing the ordered-subsets EM (OSEM) algorithm (10 iterationsand 6 subsets), and smoothed by a Gaussian filter with aFWHM of 3 mm. We then used the kernels to reconstructthe last static frame as well as to perform the direct Patlakreconstruction using the last 5 frames. For the static recon-struction, 20 iterations with 6 subsets were run, while for thedirect Patlak reconstruction, 100 iterations of the nested EMalgorithm with Ns = 3 sub-iterations were run.

For quantitative comparison, contrast recovery coefficient(CRC) vs. the standard deviation (STD) curves were plotted.The CRC was computed between selected gray matter regionsand white matter regions as

CRC =1

R

R∑r=1

(arbr− 1

)/

(atrue

btrue − 1

). (27)

Here R is the number of realizations, which is equal to50 in both the static reconstruction and the direct Patlakreconstruction. ar = 1/Ka

∑Ka

k=1 ar,k is the average uptake

5

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1000

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0.005

0.01

0.015

(b) Attenuation, transverse20 40 60 80 100 120

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Fig. 2: Three orthogonal slices of the MR prior image (first column),attenuation map (second column) and the 24th frame of the PETground truth image (third column).

of the gray matter over Ka ROIs in realization r. The ROIswere drawn in both matched gray matter regions and theunmatched tumor regions. The reason to quantify both thegray matter and tumor regions is to compare the performanceof different methods under the situations with either matchedor unmatched MR information. For the case of matched graymatter, we had Ka = 10, and for the unmatched tumorregions, we had Ka = 12. When choosing the matched graymatter regions, only those pixels inside the predefined 20-mm-diameter spheres and containing 80% of gray matter wereincluded. br = 1/Kb

∑Kb

k=1 br,k is the average value of thebackground ROIs in realization r, and Kb = 37 is the totalnumber of background ROIs. The background ROIs were 12-mm spheres drawn in the white matter. The background STDwas computed as

STD =1

Kb

Kb∑k=1

√1

R−1∑Rr=1(br,k − bk)2

bk, (28)

where bk = 1/R∑Rr=1 br,k is the average of the kth back-

ground ROI over realizations, and Kb = 37 is the total numberof background ROIs.

B. Hybrid real data

A 70-minutes dynamic FDG PET scan of a human subjectwas acquired on a Siemens Brain MR-PET scanner after a5-mCi injection. The dynamic PET data were divided into25 frames: 4×20 s, 4×40 s, 4×60 s, 4×180 s, 8×300 sand 1×600 s. The data were reconstructed into an imagearray of 256×256×153 voxels with a voxel size of 1.25 ×1.25 × 1.25 mm3 using different kernel matrices as those

(a)Time (min)

0 10 20 30 40 50 60 70

Activity(a

.u.)

0

0.005

0.01

0.015

0.02

0.025

0.03

Blood

(b)

∫ t

0 Cp(τ)dτ/Cp(t)0 20 40 60 80 100 120

c(t)/C

p(t)

0

1

2

3

4

5

6

Putamen

Linear fitting

(c)

∫ t

0 Cp(τ)dτ/Cp(t)0 20 40 60 80 100 120

c(t)/C

p(t)

0

1

2

3

4

5

6

7

8

9

10

11

Inserted tumor

Linear fitting

(d)

Fig. 3: (a) The segmented MRI blood vessels; (b) The extractedblood input curves based on the MRI segmented vessels; (c) ThePatlak plot for the left putamen TAC; and (d) the Patlak plot for theinserted tumor TAC.

used in the simulation study. Correction factors for randoms,scatters were estimated using the standard software providedby the manufacturer and included in the forward modelduring reconstruction [54], [55]. The motion correction wasperformed in the LOR space based on the simultaneouslyacquired MR navigator signal [56]. Attenuation was derivedfrom a T1-weighted MR image using the SPM based atlasmethod [57]. The first 20-minutes, middle 20-minutes and last30-minutes PET data were reconstructed separately to formthe PET features. The MR features were extracted from theT1-weighted MR image, which has the same voxel size asthe PET images. To obtain the blood input function, bloodregions were segmented from the MR image by thresholdinginside a predefined mask as the blood vessel is bright in theT1-weighted MR image. The blood input function was thencomputed from the frame-by-frame OSEM reconstruction. Amaximum intensity projection of the segmented blood regionoverlaid with the MRI image is shown in Fig. 3(a) and theextracted blood input function without decay correction isshown in Fig. 3(b). As the ground truth of the regional uptakeis unknown, a hot sphere of diameter 12.5 mm, mimickinga tumor, was added to the PET data (invisible in the MRIimage). It simulates the case where MRI and PET informationdoes not match. The location of the tumor is shown in Fig. 7.The TAC of the hot sphere was set to the TAC of the graymatter, so the final TAC of the simulated tumor region is higherthan that of the gray matter because of the superposition. Thesimulated dynamic tumor images were forward-projected togenerate a set of noise-free sinograms, taking into account ofdetector normalization and patient attenuation. Randoms andscatters from the inserted tumor were not simulated becausethey would be negligible compared with the scattered and

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random events from the patient background. Poisson noisewas then introduced and finally the tumor sinograms wereadded to the original patient sinograms to generate the hybridreal data sets. Apart from the inserted tumor, the left putamenregion was also used in the quantification, considering only itsabsolute uptake as the ground truth is unknown. The Patlakplots of the left putamen and the inserted tumor along withthe fitted lines are shown in Fig. 3(c,d), respectively, whichindicate that the Patlak model is reasonable.

We evaluated both the static reconstruction and direct Patlakreconstruction. To generate multiple realizations for the staticreconstruction, the last 40 minutes PET data were pooledtogether and resampled with a 3/40 ratio to obtained 50 i.i.d.datasets that mimic 3-minutes frames. Specifically, for everylist-mode event, we generated a random number uniformlydistributed between 0 and 1 and the event was accepted onlyif the random number is less than 3/40 [58]. This process wasrepeated 50 times. The data were reconstructed using differentmethods to compare the bias versus standard deviation curvesof the tumor uptake.

For the direct Patlak reconstruction, the last six dynamicframes were used. For each frame, 50 independent realizationswere generated by randomly sampling the data with a 1/5ratio. The resampled data mimic a set of dynamic data with a1-mCi FDG injection. For tumor quantification, images withand without the inserted tumor were reconstructed and thedifference was taken to obtain the tumor only image andcompared with the ground truth. The tumor contrast recoverywas calculated as

Contrast Recovery =1

R

R∑r=1

lr/ltrue (29)

where lr is the mean tumor intensity inside the tumor ROI,ltrue is the ground truth of the tumor intensity, and R is thenumber of the realizations. For the background, 23 sphericalROIs with a diameter of 12.5 mm were drawn in the whitematter and the standard deviation was calculated according to(28).

IV. RESULTS

A. Simulation results

Fig. 4 shows the similarity map computed in K-PET-MRI-S. Most tumor regions are successfully identified by lowvalues in the similarity map. The region in the coronal slicethat shows low similarity but is outside the tumor region iscaused by the blurring of a low-similarity tumor region in aneighboring coronal slice. Fig. 5 shows three orthogonal slicesof the Patlak slope images reconstructed by different methodstogether with the ground truth images. Besides kernel basedreconstructions, we also included the post-smoothed direct EMreconstruction (denoted by ‘EM+filter’). Note that the directEM reconstruction was simply a special case of the kernelEM algorithm given in Algorithm 1 with K = I . A Gaussianfilter was used to reduce the variance of the direct EMreconstruction to a comparable range to the kernel methods. Byvisual inspection, the Patlak slope image reconstructed usingKMRI has a higher resolution than the KPET method. The fine

Fig. 4: The similarity maps used in K-PET-MRI-S for thesimulated data set. Circles indicate the locations of the insertedtumors.

details are also preserved by the hybrid kernels as pointed bythe arrows.

Fig. 6 shows the CRC vs. STD curves for the matched gray-matter regions and tumor regions in the last static frame andthe Patlak slope images. The error bars depict the standarddeviation of the CRC values computed from 50 independentrealizations (the gray-matter regions have larger error bars thanthe tumor regions because the tumor regions have more voxelsand also higher intensity). We can see that for the matchedgray-matter regions, KMRI method has the highest CRC value.The performance of K-PET-MRI-S is very close to that ofKMRI. Both have much lower noise than either K-PET-MRI-0.5 and KPET, which are better than the EM + filter. For thetumor regions that are not visible in the MRI prior image,KMRI method has the lowest CRC, while KPET and K-PET-MRI-0.5 have higher CRC but also higher noise. By usingthe spatially adaptive weighting, K-PET-MRI-S achieves aslightly higher CRC than both KPET and KMRI while keepingthe noise at the same level as that of KMRI. The noise inK-PET-MRI-0.5 method is very close to that of the KPETmethod because the MRI intensity in the white matter is almostuniform and does not have much influence in determiningthe kernel matrix when combined with PET features withequal weights. Overall, the K-PET-MRI-S method has the bestperformance.

B. Hybrid real data results

Fig. 7 shows the similarity map in K-PET-MRI-S for the hy-brid real data. The artificially inserted tumor was successfullyidentified by low similarity values. In other regions, we can seethat the similarity values are lower than those of the simulationdata. This could be caused by mismatch between the real MRIand PET data. Fig. 8 shows the reconstructed Patlak slopeimages of the patient data by different methods. Using theKMRI method, the cortices can be seen more clearly, but theshape of the tumor is distorted as the spherical tumor doesnot exist in the MRI image. The KPET method can preservethe spherical shape of the tumor, but the details of the cortexare lost. By using the hybrid kernels, the shapes of both thecortexand the tumor can be recovered.

Fig. 9 shows the tumor constrast recovery vs. STD curves,as well as the left putamen absolute uptake vs. STD curves,for the last frame static reconstruction and Patlak directreconstruction. We can see that KMRI method has the lowest

7

Ground Truth EM + filter KMRI KPET K-PET-MRI-0.5 K-PET-MRI-S

Ground Truth EM + filter KMRI KPET K-PET-MRI-0.5 K-PET-MRI-S

Ground Truth EM + filter KMRI KPET K-PET-MRI-0.5 K-PET-MRI-S

Fig. 5: Three orthogonal slices of the reconstructed Patlak slopes using different methods for the simulation dataset.

noise, but also the lowest uptake ratio. KPET has highercontrast, but also higher noise. The performance of K-PET-MRI-0.5 is slightly better than that of KPET, but the noise isstill higher than that of KMRI. Among all the methods, K-PET-MRI-S has the best performance with the high contrastclose to that of KPET and low noise close to that of KMRI.

Comparing the results in Fig. 6 and Fig. 9, we can seethat the noise reduction between the KMRI method and theKPET method is larger in the simulation than in the real data.We hypothesized that the reason for the difference was thatthe KPET kernel was recomputed for each realization in thesimulation study while it was computed once using the fulldynamic data in the real data experiment. This kernel vari-ability in the simulation study introduces extra fluctuations inthe reconstructed images of KPET. To test this hypothesis, wereprocessed the simulation data using the same procedure used

in the real data experiment: (1) in the static reconstruction, 5min data set were resampled from the last 40 min scan; (2)in the direct Patlak reconstruction, each of the last five frameswas resampled to one-fifth of the events. The KPET kernel wascomputed once using the 60-minute data. Fig. 10 shows thenoise comparison between KPET and KMRI methods usingthe resampling approach with a fixed KPET kernel. In staticreconstruction, the noise reduction between KMRI and KPETis 16% (down from 25% in Fig. 6), and in the Patlak directreconstruction, the noise reduction factor is 17% (down from37% in Fig. 6). These values are consistent with the noisereduction factors that we saw in the real data experiment(Fig. 9).

8

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Fig. 6: CRC-STD curves at the gray matter region (left column) andthe tumor region (right column) for the last frame reconstruction (toprow) and the Patlak direct reconstruction (bottom row). For the staticresults, markers are plotted every two iterations with the lowest pointcorresponding to the 6th iteration. For the Patlak results, markers areplotted every ten iterations with the lowest point corresponding tothe 30th iteration.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Sagittal

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Axial

Fig. 7: The similarity maps used in K-PET-MRI-S for thehybrid real patient data. The tumor region is marked by theblack circles and also indicated by the arrows.

V. DISCUSSION

From the simulation data sets, we found that the noisereduction by KMRI over KPET is greater in the Patlak recon-struction than in the static reconstruction. In static reconstruc-tion the noise reduction by KMRI over KPET is 25%, whilein Patlak direct reconstruction the noise reduction by KMRIover KPET is 37%. This means including MRI informationcan have a greater impact on the Patlak direct reconstructionthan on static reconstruction. One possible explanation for thisis that in Patlak reconstruction, temporal information has beenexploited through the Patlak model and the effect of temporalinformation derived from KPET is not as helpful as that instatic reconstruction. In comparison, the noise reduction byKMRI over the direct EM reconstruction is less affected bythe Patlak model than that of KPET because MRI informationis separate from PET data.

In the simulation study, all reconstruction methods used thesame system model that was used for the data generation,

so the inconsistency in the data was solely due to Poissonnoise. In the hybrid real data experiment, the reconstructionmodel was a ray-tracing model, since we do not know thetrue system model of the real scanner. Therefore, in additionto statistical noise, there also existed modeling error in the realdata experiment. Despite the difference between the computersimulation and real data experiments, the proposed methodworked reasonably well in both situations, demonstratingits robustness to the change of the system model used inreconstruction. While we did not use time-of-flight (TOF)information in our study, the proposed method can be directlyapplied to TOF reconstruction by adopting a TOF-basedsystem model. There would be no change to the kernel matrixcalculation because it is performed in the image space.

When combining MRI and PET information, assigning anequal weight to MRI and PET features (K-PET-MRI-0.5) canreduce the bias caused by missing information in the MRIprior image. However, for matched regions, K-PET-MRI-0.5cannot fully take advantage of the MRI information. This isthe reason that the K-PET-MRI-0.5 method does not improvethe noise performance very much compared with the KPETmethod. KPET-MRI-S method uses SSIM as an index tocompute the similarity between the last-frame reconstructionsof KPET and KMRI methods. The SSIM method works wellto identify the mismatched regions in both simulation andreal data sets. As a result, K-PET-MRI-S achieved the bestperformance among all methods. In the real data experiment,we did observe low similarity regions outside the artificiallyinserted tumor. Currently, we cannot verify whether they werecaused by real difference between MRI and PET information,or by the instrumentation resolution difference. More real dataevaluation is needed to answer this question.

Finally, we note that for dynamic PET studies, motioncorrection may be necessary due to involuntary subject move-ments during long dynamic scans. In our study, the real PETdata were corrected for motion using a simultaneously ac-quired MR navigator signal [56]. When we applied the kernelmethods to patient data without MR-guided motion correction,we observed MR-induced artifacts in KMRI reconstructedimages, possibly due to mismatches between MRI and PETdata.

VI. CONCLUSION

We have proposed a hybrid kernel approach to direct Patlakreconstruction using both PET and MRI information. Theresults from simulation and real data experiments show thatthe proposed K-PET-MRI-S method provides a robust wayto handle the potential mismatch between MRI and PETdata while using matched MRI information to improve imagequality. Future work will include more clinical evaluations.

ACKNOWLEDGEMENTS

This work is supported by the National Institutes of Healthunder grant number R01EB014894, R24 MH106057 andR01EB000194. KT Chen was supported by Department ofDefense (DoD) through the National Defense Science andEngineering Graduate Fellowship (NDSEG) Program.

9

EM + filter KMRI KPET K-PET-MRI-0.5 K-PET-MRI-S

EM + filter KMRI KPET K-PET-MRI-0.5 K-PET-MRI-S

EM + filter KMRI KPET K-PET-MRI-0.5 K-PET-MRI-S

Fig. 8: Three orthogonal slices of the reconstructed Patlak slopes for the hybrid real patient data. The first column is the MRprior image.

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Fig. 9: Quantification results of the hybrid real dataset for(a,c) last frame static reconstruction (markers are plotted forevery iteration with the lowest point corresponding to thethird iteration) and (b,d) Patlak direct reconstruction (markersare generated for every ten iterations with the lowest pointcorresponding to the 30th iteration).

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Fig. 10: Noise comparison between KPET and KMRI usingthe resampling method and a fixed KPET kernel. (a) Staticimage. (b) Patlak slope image. The red diamonds representthe noise reduction factor between KMRI and KPET.

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