Quantitative whole-body parametric PET imaging incorporating a generalized Patlak model

Nicolas A. Karakatsanis1, Member, IEEE, Yun Zhou1, Martin A. Lodge1, Michael E. Casey2, Richard L. Wahl1, Arman Rahmim1,3, Senior Member, IEEE

Abstract–Recently, we proposed a dynamic multi-bed PET imaging and analysis framework enabling clinically feasible whole-body parametric imaging. The standard Patlak linear graphical analysis allows for efficient modeling of whole-body tracer kinetics by directly estimating the uptake rate constant Ki and blood distribution volume V, based on a common two-compartment kinetic model. However, this model does not account for reversible uptake (e.g. dephosphorylation in FDG), thus underestimating Ki in this case, a finding observed in a number of published FDG or similar tracer studies. We propose a novel generalized PET parametric imaging framework enabling truly quantitative whole-body Patlak imaging including in regions exhibiting reversibility. For this purpose: a) an extended non-linear Patlak model has been utilized, enriched with the net efflux rate constant kloss, (b) a basis function method has been applied to linearize the estimation process through a computationally efficient algorithm, and (c) a hybrid Ki imaging technique is introduced based on the Patlak correlation-coefficient to enhance robustness to noise. Our evaluation included both simulated and real subject clinical studies. A set of published kinetic parameter values and the XCAT phantom were employed to generate realistic simulation data for 2 dynamic 7-bed acquisition protocols (0-45min and 30-90min post-injection). Quantitative analysis on the Ki images suggests superior quantitative performance of the generalized Patlak in comparison to the standard Patlak imaging in both acquisitions, even when kloss is comparable to Ki. In addition, validation on three dynamic whole-body patient datasets demonstrated clinical feasibility and increased focal uptake with potential for enhanced diagnosis and treatment response monitoring.

I. INTRODUCTION

YNAMIC PET enables quantitative parametric imaging (e.g. FDG uptake rate constant Ki), and has gained continued interest in the context of single-bed

acquisition. On the other hand, single-frame SUV PET imaging is routinely invoked in the context of multi-bed acquisition [1]. Recently, we have proposed [2],[3] a framework enabling clinically feasible whole-body parametric PET imaging, thus combining the benefits of multi-bed acquisition and estimation of quantitative tracer kinetic parameters from dynamic scans.

Manuscript received November 15, 2013. This work was supported in part

by Siemens Medical Solutions and the NIH grant 1S10RR023623. 1 N. A. Karakatsanis (e-mail: nkarakatsanis@jhmi.edu), Y. Zhou, M. A.

Lodge, R. L. Wahl and A. Rahmim are with the Department of Radiology, School of Medicine, Johns Hopkins University, Baltimore, MD, USA

2 M. E. Casey is with Siemens Medical Solutions, Knoxville, TN, USA 3 A. Rahmim is also with the Department of Electrical & Computer

Engineering, Johns Hopkins University, Baltimore, MD, USA.

The standard Patlak linear graphical analysis was chosen as a robust approach to model whole-body tracer kinetics . It directly estimates the tracer influx or uptake rate constant Ki and blood distribution volume V by assuming a two-compartment kinetic model with an irreversible compartment, a commonly invoked model for organs and tumors exhibiting FDG uptake in PET human studies [4]. However a considerable number of published studies report kinetic parameter data for certain regions which suggest a non-negligible degree of uptake reversibility (dephosphorylation) [9]-[11].

The linear Patlak model assumes no reversible uptake and it has been observed [12]-[14] to underestimate Ki to account for lack of reversibility modeling, thus compromising quantitative accuracy in those regions, an important property when the imaging task requires quantification, such as in the case of treatment response monitoring. Furthermore, when tumor regions exhibit net uptake reversibility, then contrast,, and thus tumor detectability, may also be compromised. Moreover, despite the fact that these regions are not common, their presence within the field of view (FOV) becomes more probable in the case of whole-body imaging. Consequently, the need to account for uptake reversibility becomes more evident in whole body PET parametric imaging.

In this study we propose a novel generalized PET parametric imaging framework to allow for truly quantitative whole body Patlak imaging including in regions exhibiting uptake reversibility (i.e. non-negligible FDG dephosphorylation rates; though this approach is also applicable to other tracers such as FLT). We focus on oncology applications when the imaging task involves tumor detection or treatment response assessment, but the kinetic model and the associated parametric imaging framework can be applied for other type of PET studies as well. For this purpose, a generalized non-linear Patlak graphical analysis method, originally proposed by Patlak and Blasberg [5], is utilized. The model is equipped with an additional net efflux rate constant kloss to properly account for reversibility in the trapping or metabolism of the tracer. This model has been used in the past by a limited number of animal and brain PET studies for ROI-based kinetic analysis [6]-[8], but not within a parametric imaging framework, which involves kinetic parameters estimation at the voxel level, and not for whole-body imaging and oncology applications.

Our framework also includes application of the basis function method [15] to effectively linearize the parameter estimation method in a computationally efficient algorithm. Finally we introduce a hybrid parametric imaging method which efficiently utilizes correlation-coefficient-based binary

D

clustering to reduce the noise in the baretaining accuracy in tumor and high uptake r

II. METHODS AND MATERIAL

A. Linear and non-linear Patlak graphical

The linear Patlak model utilizes dynamic

and the time course of blood plasma trac(input function) to estimate through lineakinetic macro-parameters of tracer net influxand the total blood distribution volume Vemploying the following equation [4]:

*0i

( )( ) , > ,( ) ( )

kt

Pkk

P k P k

C dC t K V t tC t C t

τ τ= +∫

where C(t) is the measured time activity curvvoxel, CP(t) is the blood plasma TAC oestimated either from an image region-of-ifrom blood sampling, tk with k=1..m denopoints for the m dynamic PET frames/measuthe time after which relative kinetic equilibrblood and the reversible compartment is attainPatlak equation describes the linear relationthe ratio of the measured C(t) to the concentration CP(t) and (ii) the ratio of the ruthe plasma TAC to the plasma activity, the launits of time and sometimes denoted as Equation 1 is based on the following definitio

( ) ( )i 0 0 pK C C dτ τ∞

= ∞ ∫

where C0(∞) denotes the tracer concentration at infinite time, assuming uptake was irreradioactivity decay. Later, retaining the same(equation 2), Patlak and Blasberg [5] introduand more general graphical analysis model tpresence of mildly reversible kinetics. Indenote this model as the generalized Paadditional kinetic parameter was introduced,kloss, to describe the net rate constant for mloss to the blood plasma (net efflux ratassuming kloss<<Ki the following non-linearcan be obtained for the kinetic analysis of m d

( )0

i

( )( ) , >( ) ( )

k loss kt k t

Pkk

P k P k

e C dC t K V tC t C t

τ τ τ− −

= +∫

As the newly introduced kloss parameter iexponential term, the estimation problem nolinear. As it will be demonstrated later, the acPatlak models depends on the ratio of kloss/Kparameter with a key role. Sossi et al , in parstudied the effect of this ratio, which they tdopamine turnover" (EDT), in neurological Parkinson's disease evaluations [8].

ackground while regions.

LS

l analysis

c PET image data cer concentration ar regression the x rate constant Ki V at each voxel

, 1..k m= (1)

ve (TAC) at each or input function interest (ROI) or ote the mid-time urements and t* is rium between the ned. The standard nship between (i)

plasma activity unning integral of atter expressed in "stretched time".

on for Ki [5]:

(2)

left in the system eversible and no e definition for Ki uced an extended to account for the n this study, we atlak model. An , here denoted as

metabolized tracer te constant). By r Patlak equation dynamic frames:

*> , 1..t k m= (3)

s contained in an ow becomes non-ccuracy of the two Ki , an important rticular, have also erm as "effective PET imaging for

B. Simplified Patlak Kinetic Model

The standard Patlak model ass

kinetic model involving three rate with the second compartment cons(Fig. 1, top left). The measured trequations 1-3 is defined as the sumand the metabolized Cm(t) tracer coequation 1 effectively simplifies thinto a single compartment irreversibtop right) involving a single rate coafter reformulating equation 1 as fol

i 0

i

( ) ( ) (

( ) ( ),

kt

k P P k

P k P k

C t K C d VC t

K C t VC t

τ τ= +

= ⊗ +∫

where ⊗ denotes the mathematical In this context, the tracer uptake considered as the impulse response Patlak Kinetic Model (SPKM1) foessentially is a constant function of t

Fig. 1: (left) Two-compartment kinetic mreversibility for FDG tracer and (right) the eSPKM1 and (d) SPKM2 models under Patlak

On the other hand, the non-linea

two compartment kinetic model wconstant (Fig. 1, bottom left). Sequation 3:

( )

( )i 0

i

( ) ( )

( )

k loss k

loss k

t k tk P

k tP k P

C t K e C d

K e C t VC

τ τ τ− −

−

= +

= ⊗ +

∫

we are able to see that the non-lineareduces from a two compartmentsingle compartment model (Fig 1, bo

Thus, lossk tiK e− can be consider

function of the Simplified Patlak Kikinetics (SPKM2) and it is a non-lindecreasing with time.

Based on previous Ki definitioshown [4],[5] that the macro-paraconstant Ki can also be expressindividual rate constants (micro-para

1 3

2 3i

K kKk k

=+

Furthermore, when we equateKi/kloss and (K1/k2)(1+k3/k4) of the ab

ls (SPKMs)

sumes a two compartment constants (K1, k2 and k3)

sidered irreversible (k4=0) racer concentration C(t) in

m of the extravascular Ce(t) ncentration. Linear Patlak e two compartment model ble model for t>t* (Fig. 1, onstant Ki ; this is evident llows:

*

)

> , 1..

k

kt t k m= (4)

operation of convolution. rate constant Ki, can be function of the Simplified

or irreversible kinetics and time.

models (a) without and (c) with quivalent single-compartment (b)

k graphical analysis assumptions

ar Patlak model assumes a with a non-negative k4 rate

imilarly, by restructuring

*

( )

( ), > , 1..

P k

k k

VC t

t t t k m= (5)

ar Patlak model effectively t model into a reversible ottom right).

red as the impulse response inetic Model for reversible near function exponentially

on (equation 2), it can be ameter of net influx rate sed with respect to the ameters) as follows:

(6)

the distribution volumes bove two reversible models

(figure 1(c) and (d)), we derive the relationship between net efflux rate constant kloss and the individual micro-parameters:

3 32 4

3 4 2 3 3 4

Patlakloss loss

k kk kk kk k k k k k

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟+ + +⎝ ⎠ ⎝ ⎠

(7)

where Patlaklossk is the definition for kloss as originally suggested

by Patlak and Balsberg [5]. Note also that kloss approaches Patlaklossk when k4<<k3. C. Basis Function Method to efficiently estimate Ki, kloss

and V parameters of generalized Patlak model

In the case of the linear Patlak model (equation 1 or 4), the two parameters of interest, Ki and V, can be estimated directly using linear regression methods such as Ordinary Least Squares (OLS). Let us consider the following standard Patlak regression model

( ) ( )

( )

( )[ ]

1

1

10

0

,

( )

,

( )m

Ts m

tP P

Ts i

tP P m

C t C t

C d C t

K V

C d C t

τ τ

τ τ

⎡ ⎤= + = ⎣ ⎦⎡ ⎤⎢ ⎥⎢ ⎥= =⎢ ⎥⎢ ⎥⎣ ⎦

∫

∫

Y X β ε Y

X β (8a)

This linear relationship is equivalent to equation 4:

( ) *

0( ) ( ) , > , 1..kt

k i P P k k kC t K C d VC t t t k mτ τ ε= + + =∫ (8b)

However, for the non-linear Patlak model (equation 2 or 5), we propose an efficient implementation of the Basis Function Method (BFM) [15]. Initially a global range of n (e.g. n=1000) discrete candidate kloss values (e.g. [1e-4,1e-1]) is determined from equation 7 using a collection of k-values, as reported in literature [9]-[11], a subset of which was later utilized to generate TACs and conduct simulations (Table I). Then a set of n basis functions, each representing one of the n candidate kloss values, are constructed to linearize the problem:

( ) , 1.. , 1..j

klossk tj k Pt C e j n k mω −= ⊗ = = (9)

After replacing the exponential term in equation 5 with each of the n basis functions (equation 9) the following set of n bilinear Patlak equations is constructed:

( ) ( )

( ) ( )

1 110

0

,

( )

( )

loss

m loss m

Tj j j j jg i

t k tP P

jg

t k tP P m

K V

e C d C t

e C d C t

τ

τ

τ τ

τ τ

− −

− −

⎡ ⎤= + = ⎣ ⎦⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

∫

∫

Y X β ε β

X (10a)

or equivalently

( ) *( ) ( ) , > , 1..j jk i j k P k k kC t K t V C t t t k mω ε= + + = (10b)

Thus the original non-linear parameter estimation problem (equation 5) has now been translated to a set of n linear estimation problems (equation 10). Then, OLS regression can be applied to each of the n linearized Patlak equations in order to estimate the j-th set of Ki and V parameters. Subsequently, the corresponding residual sum of squares RSSj is calculated for each of the n estimation problems.

( ) ( )ˆ ˆ ˆRSS , ,TT j j

j j j j iK V⎡ ⎤= − − = ⎢ ⎦⎣Y Xβ Y Xβ β (11)

From all sets of estimated parameters ˆjβ and j

lossk the parameter set yielding the minimum RSSj is finally selected:

{ } { }BFMˆ ˆ , , arg min RSSopt

opt

jj opt jloss

jk jβ= =β (12)

By repeating this algorithm for each voxel the Ki , kloss and V parametric images are estimated.

D. Hybrid generalized Patlak imaging Multi-bed dynamic PET acquisition involves short frames

(e.g. 45sec in this study) sparsely acquired over time at each bed position. In addition, the generalized Patlak model is non-linear and involves three parameters as opposed to two for standard Patlak. As a result the noise in the estimated parameters can be significant. Therefore, we propose a novel hybrid imaging method by selectively applying either the standard or the generalized Patlak model based on the Patlak weighted linear correlation-coefficient WR as defined next.

By assuming the standard Patlak model equation 1 can be rewritten, for a given dynamic frame k (k=1...m), according to the following linear relationship:

( )( )

0( )

, , , 1..( )

ktPk

k i k k kP k P k

C dC ty K x V y x k m

C t C t

τ τ= + = = =∫ (13)

Then the weighted correlation coefficient WR can be calculated at each reconstructed voxel TAC, as follows [16]:

2 22 2

k k k k k k k kk k k k

k k k k k k k k k kk k k k k k

w w x y w x w yWR

w w x w x w w y w y

−=

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞− −⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑ ∑ ∑

(14)

where wk=(framek_duration)2/(framek_counts). Thus a WR-image is created from the available dynamic image set.

Subsequently, an initial range of possible WR values (0.7 - 0.95) is defined. Then, a WR-threshold is picked from that range to classify each time the voxels of the WR-image in two clusters, one of relatively high (hWR cluster) and one of low Patlak correlation (lWR cluster). Later, generalized Patlak modeling is assumed for the voxels belonging to hWR cluster, while standard Patlak is considered for the lWR cluster. Thus, non-linear BFM is employed for the estimation of the Ki, kloss and V parameters in the hWR cluster, while OLS is applied for the Ki and V parameter estimation in the lWR cluster. The kloss parameter is assumed to be zero for the latter case. Thus a set of hybrid Ki, kloss and V parametric images are created each time. Finally we repeat this process for a number of WR

thresholds within the initial range to generatehybrid parametric images and then corresponding to the best weighted average ratio (CNR) for the region of interests, witdefined as the size, in number of voxels, ointerest. A flow chart describing the principroposed hybrid parametric imaging methodfigure 2.

Fig. 2: The flow chart for the WR-based hybrid Patla We have systematically observed [3],[16]

correlation clusters are likely associated withusually corresponding to tumors or regiouptake in general, while low Patlak correlatiocharacterizes voxel TACs in regions of lowbackground regions with small activity concehigh noise levels. Due to the sensitivity to nnon-linear estimation, we propose BFM apprhigh Patlak correlation voxels to ensureestimation in tumors or regions of high uptaklevels), while simple OLS Ki regression fosmoothing is suggested for the remaining vorobustness to noise.

E. Dynamic Multi-bed Acquisition Time W

In this study two dynamic multi-bed acquwere employed to study the effect of the injection time window on the model (noise-estimated Ki values of the tumor regions standard and generalized Patlak models. In binitial single-bed dynamic scan over the heato acquire the initial phase (first 6-min postinput function. Subsequently, the followingschemes were employed:

a) m=6 whole-body passes over corresponding to a time window ofinjection, and

b) m=13 whole-body passes over corresponding to a time-window ofinjection

The previous acquisition protocols according to the system specifications of thescanner. Moreover the first protocol has beenon noise-bias performance of Ki parametricscanner [2]. However, it is expected that thprotocols will vary, to a certain extent, for dand system configurations. The time flow dynamic acquisition over each of the 7 beds fois illustrated in figure 4. Finally, realistic P

e multiple sets of select the set contrast to noise th weights being

of each region of ipal steps of the d is illustrated in

ak imaging method

] that high Patlak h low noise TACs ons of sufficient on cluster usually w uptake such as ntration, and thus

noise of the BFM roach only to the quantitative Ki ke (i.e. low noise ollowed by post-oxels to preserve

Window

uisition protocols acquisition post--free) bias of the when using the

both protocols, an art was conducted t-injection) of the g two acquisition

7 beds each, f 10-45min post-

7 beds each, f 30-90min post-

were designed e clinical PET/CT n optimized based c images for that he recommended different scanners of the multi-bed

for the 2 protocols oisson noise was

added to each simulated dynamic scanner specifications and each prot

Figure 4: A flow chart of the second phase acquisition protocols involving a total of m peach scanned for 45sec.

F. Simulations set-up An extensive literature review w

characteristic set of published FDG

TABLE I. K RATE PARegions K1 k2

Normal Liver 0.864 0.981 0Liver Tumor 0.243 0.78 Normal Lung 0.108 0.735 0Lung Tumor 0.044 0.231 1Myocardium 0.6 1.2

The kinetic data were used toget

4-parameter kinetic model (figure noise-free TACs (figure 3(a)) [9]-[1to the XCAT phantom including lunof 15mm, 10mm and 5mm diame(figure 3(b)).

Figure 3: (a) Noise-free time activity curves fusing a 2 compartment kinetic model and XCAT phantom where the TACs were assign

dataset, according to the tocol characteristics.

of the two whole-body dynamic

passes, each consisting of 7 beds,

was conducted to collect a k-values (Table I).

ARAMETERS k3 k4 VB

0.005 0.016 - 0.1 0 -

0.016 0.013 0.017 1.149 0.259 - 0.1 0.001 -

ther with a 2-compartment 1(c)) to generate realistic 1], that were later assigned

ng and liver tumors of sizes eter in different locations

for different regions, as generated the k-values of Table I. (b) the

ned to produce the dynamic data.

III. RESULTS AND DISCUSSIO

Figure 4(a) presents four Patlak plots, in

from a lung tumor kinetic (k-values obtaineROI analysis on noise-free and reconstructeimages, employing either the standard or Patlak model. They are all plotted togethdiagram, as the definition for the stretched timx-axis is a generalized one for both models, i.the stretched time definition of the standardwe assume kloss=0.

By comparing the noise-free plot data ofwe can observe a certain degree of bending toplot of the standard model, while the plot fomodel is a straight line. That is expected, aassume kloss=0 (equation 1) while in reality kltrue kloss=0.035) i.e. the model does not accoumetabolized tracer activity concentration aresults to overestimation of the stretched timevariable) for a given ratio of tissue to concentration (vertical axis variable). On thegeneralized model effectively recovers therelationship in the Patlak plot by correctingmetabolized tracer after multiplying the strethe exponential term exp(-klosst) (equation 3)the plot to the right in the case of the standardirect effect of the overestimation of theprovides an intuitive graphical explanunderestimation of fitted Ki parameter in these

The same effect is observed for the plokinetic ROI analysis on the noisy reconstrucwhen ROI analysis is performed on noisy datthe standard Patlak plot is also efficientllinearity is achieved in the plot after the ingeneralized Patlak model.

Moreover, the comparison between the Ptrue and noisy reconstructed ROI data in suggests, equally for both models, aunderestimation of tumor ROI mean Ki estimto true Ki values, due to the partial voluminherent in the tomographic image reconsIndeed, it has been observed here that PVEsmall tumor regions (e.g. the 10mm diameteof this study) to a larger extent than plasmCP(t) measurements (sampled from large cardROIs), thus resulting in equivalent undereestimated values for both models, as aexpected by model equations 1 and 3.

The two noise-free lung tumor TACs co4b were generated i) by the complete 2parameter kinetic model (figure 1c) and iPatlak model (figure 1d, equations 3 and 5respective k-values of Table I. The two TACfor the whole duration of the acquisition, a that the generalized Patlak can provide simplification of the complete kinetic modother similar tracers. This property, when coPatlak support for clinically feasible muacquisitions, makes the proposed generalized

ON

n total, as derived ed from Table I) ed noisy dynamic

the generalized her in the same me variable in the e. it also includes

d Patlak model, if

f the two models, o the right for the or the generalized s the former plot loss>0 (lung tumor unt for the loss of and consequently e (horizontal axis

plasma activity e other hand, the

e expected linear g for the loss of etched time with ). The bending of rd linear model, a e stretched time nation for the e cases. ots derived from cted data. In fact ta, the bending of ly corrected and

ntroduction of the

Patlak analysis on figure 4(a) also

a non-negligible mates with respect me effect (PVE) struction process. E affected C(t) of r spherical tumor

ma input function diac left-ventricle estimation of Ki

also theoretically

ompared in figure 2-compartment-4-ii) the simplified ), employing the s are very similar strong indication a very accurate

del for FDG and ombined with the ulti-bed dynamic d model an ideal

graphical analysis method for thquantitative whole-body PET paramclinical context.

In figure 4c the true Ki imagrepresentative set of simulated recoas produced after employing OLSBFM/OLS parametric imaging meththresholds. Note the enhanced tumratio (TBR) especially for the lunghybrid Patlak method is applied witwhich was derived from the previously described in section II.Dlung tumor is visible in the hybrid Bassuming the generalized model, buand OLS estimation is employed.

Figure 4: (a) Patlak plots for the standarafter analyzing simulated lung tumor ROIs aand noisy dynamic images, the latter recalgorithm, (b) noise-free simulated lung tumoa 2 compartment kinetic model or the simpequation (equation 5) of generalized Patlak employed for both TACs). (c) True and the rimages of a cardiac bed employing differenThe tumor to background contrast ratio (TBtumor, whose Patlak plot ROI data and TACrespectively.

he performance of truly metric imaging even in the

ge is compared against a onstructed noisy Ki images, S or repeating the hybrid hod with two different WR

mor to background contrast g tumor regions when the th a WR threshold of 0.95, WR selection algorithm D. In fact, the designated

BFM/OLS imaging method ut not when standard model

rd and generalized Patlak model

as extracted from noise-free (true) constructed using STIR OSEM or TACs as generated using either plified single compartment model analysis (the same k-values were reconstructed simulated Patlak Ki

nt parametric estimation methods. R) is evaluated for a 10mm lung Cs were presented in (a) and (b)

The % error (bias) quantitative analysis dynamic data for the Ki and the kloss estimafigures 5 and 6, respectively, evidentlconsiderable reduction of the % bias for botfor a large range of kloss/Ki ratios when the greplaces the standard model.

In particular, the noise-free Ki % bias isrange of possible kloss/Ki ratios and wheacquisition is applied (figure 5a). The % biafor both models when true kloss=0, irrespectivexpected from equations 1 and 3. Howevincreases (assuming only k4 increases, withparameters constant) Ki % bias, as predictemodel, increases dramatically, while for the git does not exceed 5%, when true kloss is comparable to true Ki. In addition, a furthesuch that 0<kloss/Ki <2 , assuming K1 and k2 coadditional but relatively quite milder increasenot exceeding 10% for the generalized modelwould like to note that, based on the collectiok-values reported in literature, the range of kthat always 0<kloss/Ki<1.5. Thus, in this initialized our investigation with a represevalues from literature (Table I) and then, by kconstant, we carefully expanded our searcconstraint on k4 and k3 values, such that alway

Figure 5: Noise-free Ki estimation error (% bias) curveffectively increasing k4 for each curve) for (b) 10-4acquisition protocol. The different color curves in both (to a range of k3 published values such that true 0<kloss/Kare constant and correspond to the lung tumor case (Tabl

The previous Ki % bias analysis for the sa

was also repeated for the 30-90min post-inje

on the noise-free ates presented in ly suggests the th parameters and generalized Patlak

s evaluated for a en the 10-45min as is equally zero ve of k3 or Ki, as ver, as true kloss h the rest of the ed with standard

generalized model smaller or even

er decrease of k3, onstant, causes an e of the Ki % bias l. At this point we on of FDG tracer

kloss and Ki is such study we have

ntative set of k-keeping K1 and k2 ch space with a ys 0<kloss/Ki<2.

ves vs. true kloss/Ki (by 45m and (c) 30-90m (a) and (b) correspond Ki<2 . Also K1 and k2 le I).

ame tumor region ection acquisition

time window (figure 5b). The conanalysis were also confirmed here. Hstandard model, the Ki % bias obprotocol was relatively larger and tmodest kloss/Ki ratios. On the otherbias was observed for the generalizthe acquisition time window, indithat model for quantitative estimdynamic acquisition takes place injection times, as is inherently dynamic protocols.

The reason for the larger bias standard model, when dynamicallylies in equations 1 and 4 and is grPatlak plots of figure 4a. At later poof metabolized tracer activity concebetween the standard and the generawill become more significant, resuleffect on the standard Patlak curve tthe linear fitting, by assuming a linOLS method, on a bended-to-the-rigunderestimation of Ki slope, as the right. Effectively, the need for "kstandard stretched time by theequivalently, the need for the gbecomes more evident with increasilater time windows.

Figure 6: (a) Correlation between estimakloss estimation error (% bias) as a functacquisition protocol. The estimated data arcurves in both (a) and (b) correspond to a rthat true 0<kloss/Ki<2 . Also K1 and k2 are contumor case (Table I).

A similar % bias analysis for noi

repeated for the kloss parameter as wegood correlation between the estimatrue kloss value, as the latter increas

nclusions of the previous However, in the case of the served with the 30-90min thus considerable even for r hand, a similarly low % zed model, irrespective of cating the significance of

mation, particularly when at relatively later post-required in whole-body

in the Ki estimates of the y acquiring at later times, raphically illustrated at the ost-injection times, the loss entration and the mismatch alized model stretched time lting in a stronger bending to the right. Consequently, near model and employing ght curve will cause further time window slides to the

kinetic correction" of the e factor exp(-klosst), or generalized Patlak model ng t, i.e. when acquiring at

ated and true kloss and (b) plot of tion of true kloss/Ki for 30-90m e noise-free. The different color ange of k3 published values such nstant and correspond to the lung

ise-free simulated data was ell. Figure 6a shows a very ated noise-free kloss and the es, for a range of true kloss

values commonly found in literature (0<kloss<the accuracy of the two models depends not juratio of kloss/Ki. Therefore, the % bias in kloss kloss/Ki increases, assuming K1, k2 and k3 cerror plot and only decreasing k3 once betweein a similar way as for figures 5. The resfigure 6b show that the kloss % bias, though increasing kloss/Ki, it never exceeds 10% wheall combinations of k4 and k3 examined. Thsearch space as for the Ki bias analysis has be

The enhanced quantitative estimation oregions is also clinically demonstrated in figcharacteristic whole-body patient dynamic dathe Johns Hopkins PET center. In the first cobody correlation-coefficient (WR) images forpatients are shown. The WR images have beeWR threshold, as calculated from the WR described in section II.D. For illustration purcorresponding to WR less than the thresholdwhile the remaining voxels were set to inspecting the WR images, we can observe thhigh focal uptake, suspected for the presencWR thresholding clusters the tumor voxels effectively filters out the surrounding backgrthus the background noise.

The reason for this effect lies in the obseuptake voxels are usually associated with gooand therefore TACs of high WR which normthresholding and are assigned to the hWRbackground voxels of high noise and pooretend to get assigned to the lWR cluster. As a BFM/OLS method selectively applies the mothus quantitatively accurate, generalized Ptumor and high uptake regions avoiding lowwhose noise is highly susceptible. For thstandard Patlak and OLS estimation is preferrobust to noise.

Figure 7: Clinical demonstration of proposed wholeimaging method: Three characteristic clinical whole-bodslope Ki) images, employing an optimized 10-45m acqHopkins PET center. The first column presents, focorresponding correlation-coefficient (WR) whole-bodimages shown are filtered using the optimal threshold, aWR selection algorithm described in section II.D.

<0.04). However, ust on kloss but the was evaluated as

constant for each en different plots, ults presented in it increases with

n 0<kloss/Ki<1 for he same k-values en used here too. of Ki in tumor gure 7 with three

atasets acquired at olumn, the whole-r each of the three en filtered using a selection method rposes, all voxels d where set to 0,

1. By visually hat in regions with ce of tumors, the together while it round voxels and

ervation that high od count statistics mally survive the R cluster , while er count statistics result, the hybrid

ore complete, and Patlak model to

w count voxels to he latter voxels, rred, as it is more

e-body hybrid Patlak dy parametric (Patlak quisition at the Johns for each patient, the dy images. The WR as calculated with the

The next three columns presePatlak Ki images of the three patmethod, ii) hybrid BFM/OLS withand iii) hybrid BFM/OLS with the each patient. In all three patient cascontrast in the OLS images, were enhybrid BFM/OLS imaging was appoptimal WR threshold was applied. also been observed in optimal hywhich was not visible before. Note foci in the parametric images corresin the WR images for each patient.

TABLE II. PATIENT TUMOR TO BACKGR

foci ID OLS hybrid BFM/O

(WR=0.95)foci1 2.6 ± 0.6 5.4 ± 0.7 foci2a 2.4 ± 0.8 2.7 ± 0.7 foci3 1.7 ± 0.3 2.2 ± 0.4

In addition, the tumor to backgro

were evaluated (TBR mean ± bkgrdfoci, each for one of the three ppresented in Table II. In all casesimages was enhanced considerably,while the absolute noise (s.d.) resuggesting also a better CNR for ththe hybrid images with the optimcolumn in figure 7 and Table II) exh

The success of hybrid regressiextent, on the WR threshold level wclustering of the voxel TACs into thcorrelation respectively. A very lavalue very close to unity, may substthe background of the hybrid paramalso exclude certain tumor voxelestimated with the non-linear modetumor do not exhibit a very high uvolume effect. Besides, the calculatassumption of the linear (standard) tumor regions with non-negligible uwill never match unity even for higwe recommend that a WR threshold

On the other hand, a very low thmay ensure accurate modeling of thhigh or modest uptake regions, it manoise in the background induced byof kinetic parameters from very noi

Therefore, in order to efficiehybrid parametric imaging method tdata, we propose initially investthreshold values between 0.75 and 0one that yields hybrid parametricaverage tumor CNR. By reducingsearch WR space, we effectivequantification error on one side andthe background on the other side.

ent the final whole-body tients when using i) OLS h a random WR threshold

optimal WR threshold for ses, focal uptakes with low nhanced considerably when plied, especially when the In addition, a new foci has

ybrid images of patient B that all contrast-enhanced

spond to an equivalent foci

ROUND ROI CONTRAST RATIO

OLS

hybrid BFM/OLS (WR=0.98)

8.3 ± 0.9 3.1 ± 0.7 2.4 ± 0.5

ound contrast (TBR) levels d s.d.) for three designated patients of figure 7, and s, TBR of foci in hybrid , with respect to the OLS, emained at similar levels

he hybrid methods. Finally, mal WR clustering (third hibited the highest TBR. on depends, to a certain

which determines the binary hose of high and low Patlak arge threshold level, i.e. a antially reduce the noise in

metric images, while it may ls from being accurately el, especially if sections of uptake, e.g. due to partial tion of WR is based on the Patlak model and thus for

uptake reversibility the WR gh uptake rates. Therefore less or equal to 0.95.

hreshold value, although it he kinetics in all tumor and ay not sufficiently suppress y the non-linear estimation sy voxel TACs. ntly apply the presented to clinical patient dynamic tigating a range of WR 0.98 and later choosing the c images with the better g the extent of our initial ely limited the potential d the noise amplification in

Subsequently, and within

those safety margins, we selected the optimal WR threshold to maximize CNR performance in the regions of interest. Thus, the free nature of the kinetic correlation parameter allowed us to include into our proposed whole body PET parametric imaging framework a data-driven method to enhance robustness to noise, while retaining accuracy in the tumor or high uptake regions. By maximizing CNR through hybrid BFM/OLS regression, the performance in both the imaging tasks of tumor detectability for lesion detection at an early phase and tumor quantification for reliable treatment response assessment can be significantly enhanced.

IV. CONCLUSIONS AND FUTURE PROSPECTS Quantitative parameter estimation (including at the

individual voxel level, i.e. parametric imaging) has significant potentials especially for prognostic and treatment response monitoring tasks. Standard Patlak graphical analysis, however, although a convenient method to analyze dynamic multi-bed data and linearly fit the kinetic parameters, does not account for reversible uptake, and thus, may compromise quantitative accuracy especially in whole body acquisitions where regions with reversible kinetics are more abundant.

In the current study we equip our recently proposed whole body parametric PET imaging framework [2], [3] with an extended generalized Patlak model to enhance tumor uptake quantification even in regions with non-negligible uptake reversibility for which the standard Patlak model underestimates Ki due to incomplete modeling of the reversible kinetics.

However, generalized Patlak involves a non-linear parameter estimation process, which is more susceptible to the high noise levels usually associated with multi-bed dynamic acquisition protocols. Therefore, to enhance robustness to noise, we propose a hybrid regression method that segments the voxel TACs of the dynamic image set, based on the Patlak correlation coefficient, in order later to selectively apply BFM for the highly correlated voxel TACs while OLS estimation is performed to the rest.

In comparison with our previously proposed whole body parametric PET imaging methods, where a standard linear Patlak model was employed, the results here suggest that the newly proposed generalized Patlak imaging framework is also clinically feasible and quantitative even for regions with non-negligible uptake reversibility. Furthermore, in terms of tumor CNR and therefore detectability, it is superior to SUV and standard Patlak imaging, not only in tumor regions surrounded by high background activity, such as the liver and the neck, but also in certain tumor regions previously appearing to exhibit low uptake when imaged and analyzed with conventional methods, due to the presence of uptake reversibility.

On the other hand, whole-body generalized Patlak imaging is less robust to the high noise levels inherently present in whole body dynamic data and more advanced regression methods are necessary. For this purpose, here we proposed a clinically adoptable hybrid regression method utilizing the

Patlak correlation information as extracted from the 4D acquired PET data to enhance noise robustness while retaining tumor uptake quantification. The results from ROI analysis on the derived noisy BFM parametric images as well as the TBR contrast analysis in tumor regions from both simulated and clinical whole body hybrid BFM/OLS parametric images demonstrated the enhanced quantitative performance of the proposed framework in tumor regions.

Currently, we are also investigating a novel direct 4D parametric reconstruction algorithm where the generalized Patlak model is efficiently incorporated within the maximum-likelihood expectation-maximization (ML-EM) scheme to enable reconstruction of truly quantitative Ki images, accounting for kloss, directly from the PET sinogram data [17]. Direct 4D parametric ML-EM reconstruction allows for more accurate modeling of the well-known Poisson noise distribution in the projection data and thus for more efficient reduction of the noise propagation in the reconstructed parametric images [18]-[24]. Recently, we confirmed the noise reduction in the direct 4D reconstruction scheme by applying it on whole-body dynamic PET data [25] and utilizing the standard linear Patlak model [18]-[23].

Furthermore, we aim not only at low noise levels but also at more accurate PET quantification and, therefore, our purpose is to evaluate the potential of directly reconstructing from sinograms not just standard but also generalized Patlak images, as an alternative to the correlation-based indirect hybrid parametric imaging from the dynamic images, which has been proposed here. It should be noted that both indirect and direct parameter estimation methodologies are designed such that they are compatible to single-bed as well as whole-body dynamic acquisitions and can support clinically feasible PET protocols.

However the optimization of the kinetics likelihood objective function can be very slow, due to the correlation between the estimated kinetic parameters in the Patlak model [22]-[24]. In addition, the non-linearity between the dynamic images and the kinetic parameters, as introduced by the generalized Patlak model, can further affect the convergence rate of the algorithm [22],[24]. Therefore, we are investigating utilization of the concept of optimization transfer and use of surrogate likelihood objective functions [17] to accelerate the convergence of the generalized Patlak 4D reconstruction through a nested ML-EM algorithm [22]-[24].

V. ACKNOWLEDGMENTS The authors would like to thank Dr Abdel Tahari for

assisting us in recruiting patients. This work was supported by Siemens Medical Solutions and the NIH grant 1S10RR023623.

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