Evaluation Of Compartmental And Spectral Analysis Models ... · 1987, where two-compartmental...

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 45, NO. 12, DECEMBER 1998 1429

Evaluation of Compartmental and SpectralAnalysis Models of [ F]FDG Kineticsfor Heart and Brain Studies with PET

Alessandra Bertoldo, Paolo Vicini, Gianmario Sambuceti, Adriaan Anthonius Lammertsma,Oberdan Parodi, and Claudio Cobelli,*Senior Member, IEEE

Abstract—Various models have been proposed to quantitatefrom [ 18F]-Fluoro-Deoxy-Glucose ([18F]FDG) positron emissiontomography (PET) data glucose regional metabolic rate. Weevaluate here four models, a three-rate constants (3K) model,a four-rate constants (4K) model, an heterogeneous model (TH)and a spectral analysis (SA) model. The data base consists of[18F]FDG dynamic data obtained in the myocardium and braingray and white matter. All models were identified by nonlinearweighted least squares with weights chosen optimally. We showthat: 1) 3K and 4K models are indistinguishable in terms ofparsimony criteria and choice should be made on parameterprecision and physiological plausibility; in the gray matter a morecomplex model than the 3K one is resolvable; 2) the TH model isresolvable in the gray but not in the white matter; 3) the classicSA approach has some unnecessary hypotheses built in and can bein principle misleading; we propose here a new SA model whichis more theoretically sound; 4) this new SA approach supportsthe use of a 3K model in the heart with a 60 min experimentalperiod; it also indicates that heterogeneity in the brain is modestin the white matter; 5) [18F]FDG fractional uptake estimates ofthe four models are very close in the heart, but not in the brain;6) a higher than 60 min experimental time is preferable for brainstudies.

Index Terms—Fluoro-deoxy-glucose (FDG), glucose, kineticsmetabolism, model identification, parameter estimation, physio-logical model, tracer.

I. INTRODUCTION

T HE use of the positron emitting glucose analogue [F]2-Fluoro-2-Deoxy-D-Glucose ([F]FDG) together with

positron emission tomography (PET) makes it possible toimage regional metabolism in the brain and in other organs,

Manuscript received December 17, 1996; revised May 7, 1998. This workwas supported in part by a grant from the Italian Ministero della Universitae della Ricerca Scientifica e Tecnologica (MURST 40%) on “Bioingegneriadei Sistemi Metabolici e Cellulari” and by the National Institutes of Health(NIH) under Grant RR02176.Asterisk indicates corresponding author.

A. Bertoldo is with the Department of Electronics and Informatics, Univer-sity of Padova, 35131 Padova, Italy

P. Vicini is with the Department of Electronics and Informatics, Universityof Padova, 35131 Padova, Italy; and the Department of Bioengineering,University of Washington, Seattle, WA 98195 USA.

G. Sambuceti and O. Parodi are with the CNR Institute of ClinicalPhysiology, 56125 Pisa, Italy.

A. A. Lammertsma was with MRC Cyclotron Unit, Hammersmith Hospital,London, U.K. He is now with the PET Centre, Free University Hospital,1007MB Amsterdam, the Netherlands.

*C. Cobelli is with the Department of Electronics and Informatics, Uni-versity of Padova, via Gradenigo 6/A, 35131 Padova, Italy (e-mail: [email protected]).

Publisher Item Identifier S 0018-9294(98)08845-4.

such as heart, muscle and liver. Dynamic PET data, analyzedwith mathematical models of [F]FDG kinetics, allow theestimation of tissue fractional uptake of [F]FDG and,thus, of the regional metabolic rate of glucose, by usingthe so-called lumped constant (LC), a scale factor betweenglucose and [ F]FDG metabolism. Normally only a singlemacroscopic parameter, i.e., [F]FDG fractional uptake,is calculated. However, most of the models also have thepotential to provide a much more intimate picture of thesystem, e.g., degree of tissue heterogeneity and rate constantsof blood-tissue exchange. Recently, the model by Sokoloffet al. [26] has been used to evaluate microscopic parametersin muscle [12], heart [9], [19], and liver [25] in variouspathophysiological conditions. A thorough comparison ofmodels of [ F]FDG kinetics is, however, lacking, the onlyavailable study being that of Lammertsmaet al. [13] from1987, where two-compartmental models, the Sokoloffet al.[26] and the Phelpset al. [20], and the Patlak graphicalmethod were investigated for brain data only. In recent years,new models have been proposed, notably the Schmidtet al.[21] model and the spectral analysis method [6].

In this study, PET dynamic data obtained in the brain andin the myocardium were used to evaluate and compare theperformance of four models of [F]FDG kinetics: 1) the three-compartment model of Sokoloffet al. [26], exhibiting threerate constants (3K); 2) the three-compartment model of Phelpset al. [20], exhibiting four rate constants (4K); 3) the model ofSchmidtet al. [21], which introduces tissue heterogeneity (TH)into the 3K model; 4) the recently proposed spectral analysis(SA) method of Cunningham and Jones [6] and Turkheimeret al. [28]. In the following, issues related to the LC willnot be discussed. Clearly, since the LC is a scale factor forpassing from [ F]FDG to a glucose metabolism picture, it isof utmost importance to correctly assess its value and ascertainits stability during different experimental conditions (e.g., highlevels of insulin, workload, etc.) [10], [12], [24], [25].

II. THE DATA

The heart data are shown in Fig. 1 and refer to six sub-jects (H1, H2, H3, H4, H5, and H6) with a myocardialinfarction history. PET scans were performed with a ECATIII Positron Tomograph (CTI Inc., Knoxville, TN), and theywere reconstructed using a Hanning filter with a cutoff of0.5 (Nyquist frequency), thus, resulting in a transaxial spa-

0018–9294/98$10.00 1998 IEEE

1430 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 45, NO. 12, DECEMBER 1998

Fig. 1. The heart data. Blood () and time (�) data are expressed in (counts/second/pixel) while time is in minutes.

tial resolution of 9-mm full-width–half-maximum (FWHM).One hour before the study, subjects were given a 50 grglucose load. Dynamic PET images of a normally perfused(from a previous study with Nitrogen-13-Ammonia tracer)region of interest (ROI), the free wall of the myocardium,were measured (counts/second/pixel) following an injection(2 min) of [ F]FDG. The scanning proceeded according to

the following schedule: eight scans of 15 s, four scans of 30 s,one scan of 60 s, five scans of 120 s, and eight scans of 300 s.Scanning was completed within 55 min after injection of thetracer. The doses were: 8.5 mCi for H1, 9.5 mCi for H2, 8mCi for H3, 7 mCi for H4, 6 mCi for H5, and 7 mCi for H6.

Timed arterial blood concentrations (counts/second/pixel)were obtained by PET imaging of the left ventricle (LV).

BERTOLDO et al.: COMPARTMENTAL AND SPECTRAL ANALYSIS MODELS FOR STUDIES WITH PET 1431

Fig. 2. The brain data. Blood () and time (�) data are expressed in (nCi/ml) while time is in minutes.

A small ROI was drawn within the left ventricular cavityin order to minimize spillover from the wall and avoidunderestimation of the count density. Processing of the heartdata was performed as described in [2]. In particular, forthe partial volume correction, the wall activity was dividedby a recovery coefficient as a function of the thickness (themeasured thickness was in all cases greater then the 9 mm

FWHM). The recovery function was previously determinedrunning by phantom study.

The spillover radioactivity from LV to tissue [F]FDG wasaccounted for in the measurement equation (see Section III),while that from tissue to LV was assumed negligible.

The brain data are shown in Fig. 2 and refer to twonormal male subjects and and one patient with


epilepsy, each with two ROI’s, one gray matter (visual cortex)and one white matter (right hemisphere for and ; lefthemisphere for ). The two ROI’s were manually drawn onthe image from the final emission scan and then transferred tothe images from each of the other emission scans to obtain thetime activity curve. Even if the two ROI’s were drawn so asto have gray and white matter, the presence of heterogeneoustissue is expected above all in the gray region. Al the subjectswere fasted overnight. The data were published in [8] to whichwe refer for further details. Briefly, for each subject, a bolus ofFDG (10 mCi) was injected intravenously and arterial bloodsamples were taken concurrent with the injection of the tracer.The samples of blood were taken at 0.25, 0.5, 0.75, 1, 1.25, 1.5,1.75, 2, 2.5, 3, 4, 5, 6.5, 10, 15, 20, 30, 60, 90, and 120 min.The samples were immediately placed on ice and the plasmasubsequently separated for the determination of [F]FDGconcentration. In the three subjects scanning proceeded ac-cording to the following schedule: ten scans of 0.2 min,2 0.5 min, 2 1 min, 1 1.5 min, 1 3.5 min, 2 5 min,1 10 min and 3 30 min. Scanning was completed within120 min after injection of the tracer. The PET scanner used,a GE/Scanditronix PC4096-15WB was an eight-ring, 15-slicemachine, and the scans were reconstructed on a 128128matrix using a Hanning filter with a cutoff frequency of 0.5(Nyquist frequency), resulting in a spatial resolution in theimage plane of 6.5 mm FWHM. The number of pixels includedand averaged in the ROI’s ranged from 80–290 pixels. Thesize of a pixel in these studies was 2 mm2 mm. Tissue Fconcentrations (nCi/ml) and plasma concentrations (nCi/ml)were corrected for decay. The plasma data set was correctedalso for external delay.

III. T HE MODELS

All models assume that glucose metabolism is in steady-state and tracer theory predicts that [F]FDG kinetics aredescribed by linear, time-invariant differential equations [3].

A. The Three-Compartment—Three Rate Constants Model

The two-tissue-compartment model proposed by Sokoloffetal. in 1977 [26] is shown in Fig. 3. The model was originallydeveloped for autoradiographic studies in the brain with 2-[14C]Deoxyglucose as tracer, and subsequently used for PET[ F]FDG studies in the brain and other tissues/organs. The3K model assumes that [F]Fluorodeoxyglucose-6-phosphate([ F]FDG-6-P) is irreversibly trapped in tissue for the dura-tion of the experiment.

The model and the input–output experiment are described by

(1)

where and are rate constants, is the concen-tration of [ F]FDG in plasma, is the concentration of[ F]FDG in tissue, is the concentration of [F]FDG-6-P in tissue. The model is called 3K model, because it hasthree rate constants. Note that there is no equation for the

Fig. 3. The 3K model.

first compartment, because in PET studies is assumedto be known and used as the input for model identification.The output (tissue measurement) equation used for modelidentification was, for the brain [13]

(2)

and, for the heart

(3)

where is [ F]FDG in blood and (unitless) accountsin the brain data for the vascular volume present in the tissueROI, while in the heart data it is dominated by the effect ofspillover from blood to tissue (negligible in the brain) [14].

All the four model parameters and area prioriuniquely identifiable [3].

The model allows to calculate the fractional uptake of[ F]FDG [24]

(4)

Once is known, one can then calculate the local metabolicrate of glucose by assuming a value for the lumped constantLC and by using the steady-state glucose concentration inplasma. Glucose metabolism in the heart is in a quasi-steady-state and we refer to [1] and [19] for a discussion of thisassumption.

The model was identified from [F]FDG brain data, usingthe measured (assumed error-free) plasma time activity curveas forcing input of the model. In the brain measurementequation (2), the whole-blood time activity was obtained as[5]

(5)

where is the subject’s measured hematocrit (the assumptionis made that in the capillaries equals that of the largevessels; other investigators have used different approaches,e.g., Schmidtet al. [22] have used in rats a fixed capillaryof 0.30, while Phelpset al. [20] have used in dogs, monkeys,and humans a capillary equal to 0.85 that of large vesselsand assumed that [F]FDG does not distribute into red bloodcells). In the heart measurement equation (3), wasdirectly provided by the left ventricle PET measurement, andit was used instead of in (1).

Use of the left ventricle PET measurement permits the non-invasive determination of the input function but the definitionof an error-free curve for the heart blood time activity islimited by the presence, above all in the last scan frames,


of the spillover of tissue radioactivity into blood. This canaffect the parameter estimates of each model (in particular thephosphorylation constant) but not the conclusions regardingthe choice of the best model since the same input function isassumed for all models.

The model parameters were estimated by weighted nonlin-ear least squares. Tissue activity curves are described by

(6)

where is the midscan time, is the measurement error attime , and is the number of data. Thus, the cost functionto be minimized is

WRSS (7)

WRSS denotes weighted residual sum of squares,is theweight of the th datum, and the vector of unknown modelparameters of dimension

Measurement error was assumed to be additive, uncorre-lated, Gaussian, zero mean, and with a variance described asproposed in [15]

(8)

where is the length of the scanning interval relative toand is an unknown proportionality constant.

Weights were chosen optimally [3] as

(9)

and the scale factor of (8) was estimateda posteriori [3] as

WRSS(10)

where WRSS is the value of the cost function evaluatedat the minimum, i.e., for equal to the vector of estimatedmodel parameters

WRSS (11)

WRSS has been minimized by using the Levenberg–Marquardtalgorithm as implemented in [11].

Precision of the parameter estimates was evaluated from theinverse of the Fisher information matrix by

COV (12)

The model was successfully identified in all subjects. Meanweighted residuals are shown in Fig. 4, in particular, Fig. 4(a)for the heart and Fig. 4(b) and (c) for the brain. Parametervalues for all cases are shown in Table I(a) for the heart andin Table I(b) for the brain. In the brain (two ROI) parameterswere also estimated from two shorter time interval, 0–90 and0–60 min, to evaluate the impact of the experiment durationon parameters estimates. Precision was poor (CV well above100%) in and for in -right hemisphere (0–120 min),

-visual cortex (0–90 min) and -visual cortex (0–60 min).

(a)

Fig. 4. (a) Mean weighted residuals for the heart 3K, 4K, and SA models.

In agreement with [22], and decreased for a longerobservation interval in the gray ROI, while in the whiteROI they were relatively constant. decreased with theobservation interval in the gray ROI, while in the whiteROI it was relatively constant. As expected, is higher inthe heart than in the brain. The fractional FDG uptakewith its precision (obtained by error propagation) is shown inTable VI(a) and (b) for the heart data and brain, respectively.

B. The Three Compartment—Four Rate Constants Model

In 1979, Phelpset al. [20] proposed a modification of themodel of [ F]FDG kinetics (Fig. 5) using the observationthat, following a pulse of [ F]FDG, total tissue activity wasobserved to decline after 120 min, thus, indicating a loss ofproduct. This model does not require that [F]FDG-6-P isirreversibly trapped in tissue for the whole duration of the


(b)

Fig. 4. (Continued.) (b) Mean weighted residuals for the gray brain 3K, 4K, SA, and TH models.

experiment, but that it can be dephosphorylated. The modeland input–output experiment equations are

(13)

where and are rate constants, is theconcentration of [ F]FDG in plasma, is the concen-tration of [ F]FDG in tissue, and is the concentrationof [ F]FDG-6-P in tissue. This model is called 4K model,

because it has four rate constants. The PET tissue measurementequations were the same as those of the 3K model.

All model parameters ( and ) are a prioriuniquely identifiable [3]. Like for the 3K model, the fractionaluptake of [ F]FDG is

(14)

The model was numerically identified like the 3K model.Mean weighted residuals are shown in Fig. 4; in particular,


(c)

Fig. 4. (Continued.) (c) Mean weighted residuals for the white brain 3K, 4K, SA, and TH models.

Fig. 4(a) for the heart and Fig. 4(b) and (c) for the brain, whileparameter estimates are shown in Table II(a) and (b).

In general, precision of parameter estimates degraded withrespect to that of the 3K model. This is expected, since the4K model has an additional parameter with respect the 3Kmodel. The model was identified in the heart with a generallypoor precision for since was rather small. In the brain,a better performance of the model was noted indicating thatthe data could allow one to resolve a more complex modelthan the 3K one. Some problems with were observed inthe -right hemisphere (120 and 90 min), in the -left

hemisphere (90 and 60 min), and in the-right hemisphere(90 min). was estimated with a poor precision in the twoROI’s of the subject. As far as observation interval isconsidered, the parameter estimates exhibit a pattern similarto the 3K model for the gray matter: decreased(but less in magnitude) and for the results at 120 min or 90min remained the same. The white matter showed a relativelyconstant value for all the parameters. The FDG fractionaluptake with its precision (obtained by error propagation)is shown in Table VI(a) and (b) for the heart and brain,respectively.


TABLE IPARAMETER ESTIMATES AND THEIR PRECISION (EXPRESSED ASPERCENT CV). (a) HEART 3K MODEL AND (b) BRAIN 3K MODEL

(a)

(b)

Fig. 5. The 4K model.

C. The Heterogeneous Tissue Compartmental Model

Schmidtet al. proposed [21], [22] a model which takes intoaccount the heterogeneous composure of the brain tissue. The

model is an extension of the 3K model to an heterogeneoustissue and is shown in Fig. 6(a). A number of weightedsubregions, each described by a 3K model is assumed andthe resulting model [Fig. 6(b)], obtained by making someassumptions on the [21], becomes time-variant

(15)

where and represent the weighted average of thetissue concentrations in every subregion;is the rate constant


TABLE IIPARAMETER ESTIMATES AND THEIR PRECISION (EXPRESSED ASPERCENT CV). (a) HEART 4K MODEL AND (b) BRAIN 4K MODEL

(a)

(b)

for transport of FDG into the mixed tissue (a mass-weightedaverage of the subregion transport rate constants), and

(16)

are the time-varying rate parameters.

The equation used for model identification was

(17)

All model parameters, and, are a prioriuniquely identifiable [3].


TABLE IIIBRAIN TH MODEL: PARAMETER ESTIMATES AND THEIR PRECISION (EXPRESSED ASPERCENT CV)

The FDG fractional uptake for the TH model can becalculated (see the Appendix for details) as

(18)

The TH model has been numerically identified [see (6)–(8)]from the brain data since tissue heterogeneity in a nor-mal myocardium does not appear to be an issue. Meanweighted residuals are shown in Fig. 4, in particular Fig. 4(a)

for the heart and Fig. 4(b) and (c) for the brain, and re-sults in Table III. A posteriori identifiability of this modelwas generally good with precision of parameter estimatesof the gray ROI’s exhibiting poor values only for inthe 0–60 observation interval, and for also in the 0–90and 0–120 intervals. In the white ROI’s, the situation wasdifferent and, generally, the parameter estimates exhibiteda poor precision. The FDG fractional uptake parameterwith its precision (obtained by error propagation) is shownin Table VI(b).


Fig. 6. The TH model.

D. The Spectral Analysis Model

The so-called spectral analysis (SA) method, basically aninput–output model, was introduced by Cunningham and Jonesin 1993 [6] and has been used to determine local metabolicrate of glucose in the brain [28]. This new technique aimsto identify the kinetic components of the tissue tracer activitywithout specific model assumptions, e.g., presence or absenceof FDG dephosphorylation or homogeneity in the tissue.Below, the fundamentals of the technique [6], [28] are firstbriefly described.

1) Fundamentals:Suppose that the impulse responsecan be described by a sum of distinct exponential terms

(19)

with for every Then, the measured timeactivity curve can be described by the convolution of

and the plasma time activity curve

(20)

The idea is to find a fixed grid of “possible” eigenvaluesand then to estimate their associated amplitudesi.e., thespectral content. Let us now derive an expression for the tissue

TABLE IVPARAMETER ESTIMATES AND THEIR PRECISION (EXPRESSED ASPERCENT CV); �

AND � HAVE DIMENSION min�1 EXCEPT � RELATIVE TO � !1 WHICH IS

UNITLESS. (a) HEART SPECTRAL ANALYSIS (INPUT–OUTPUT) MODEL

(a)

measurement based on these premises. To do so, let us startwith simple cases.

What happens if a certain eigenvalue of the impulseresponse is close to infinity (i.e., has a very large value)?Then, the corresponding term in (20) is proportional tovia and can be viewed as a “high-frequency”component, i.e., accounting for the fast passage of tracer in thevascular space of the ROI. It is a vascular volume or spillovercomponent and .

Conversely, what happens if a certain eigenvalue of theimpulse response is close to zero? Then, the correspondingterm in (20) is proportional to via and can beviewed as a “low-frequency” component, i.e., accounting fora (quasi)trap for the tracer.

Lastly, the components corresponding to the intermediatevalues (“intermediate frequency” components) will reflectthe uptake of tracer within the tissue with their numbercorresponding to the number of distinct tissue compartments


TABLE IV ( Continued.)PARAMETER ESTIMATES AND THEIR PRECISION (EXPRESSED ASPERCENT CV); � AND � HAVE DIMENSION ml/ml/min

EXCEPT � RELATIVE TO � ! 1 WHICH IS UNITLESS. (b) BRAIN SPECTRAL ANALYSIS (INPUT–OUTPUT) MODEL

(b)

within the ROI exchanging with plasma. Therefore, they givean indication of tissue heterogeneity.

We can now write an expression for , by explicatingthe contribution of the terms corresponding to and to

as in [28]

(21)

As already noted, the amplitude relative to correspondsto the parameter which was defined as in the 3K, 4K, andTH models.

To implement the classic SA model the first step is to definea grid of ’s. The range of the was chosenas defined in [28]. The lower limit was , where

is the end time of the experiment. The upper limit waswhere is the duration of the first scan (15

s in the heart, 0.2 min in the brain data). The spacing of the

’s was fixed as in [28]

(22)with The unknown values of the variouskinetic components were estimated via nonnegative linearweighted squares as implemented in MATLAB (The Math-works, Sherborn, MA) [18]. The components for andfor were explicitly included. Precision of the

’s was obtained from the inverse of the Fisher informationmatrix [3]. Note that the error of the ’s with classical SA isunderestimated since the eigenvalues are constrained.

Mean weighted residuals are shown in Fig. 4, in particular,Fig. 4(a) for the heart and Fig. 4(b) and (c) for the brain.Results are reported for the heart in Table IV(a) and forthe brain in Table IV(b) for the three 120-, 90-, and 60-minobservation intervals. We first discuss the heart results.

a) Heart: A component at was always de-tectable. The fact that a component at is always


present, except for , indicates that [ F]FDG is irreversiblytrapped in the tissue. It can be noted that also for the subject

there are actually two eigenvalues very close to zero[Table IV(a)]. The number of distinct ’s is indicative oftissue heterogeneity: however, it can be seen that the’s,apart from those at and at , are poorly estimated.Their meaning is, therefore, uncertain.

If we inspect the results, we can observe that some couplesof ’s are actually next to each other on the eigenvaluegrid (e.g., in , there are two ’s for ’s 6 and 7).It is likely that the poor precision of the ’s stems fromthe fact that the approach cannot attribute to a certainthe right , but divides the ’s between two ’s next toeach other: we will call this phenomenon “line doubling.” Toavoid this phenomenon, the identification for each subject wasrepeated redefining the ’s grid to contain only the following:zero, infinity, ’s which were unequivocally estimated before[Table IV(a)] and ’s which were the arithmetic mean ofthe double lines from the results in Table IV(a) (e.g., for

the average of the ’s numbered with 6). With thisprocedure, an improved precision for the’s [Table V(a)]was obtained. However, the precision for the amplitude valuesis underestimated since the eigenvalues are fixed. Inand

there are altogether three distinct ’s, of which justone is, therefore, relative to tissue. For the other subjectsthere are four or five spectral lines. The presence of twoor three tissue components, in addition to the component at

and at is likely not to be attributable totissue heterogeneity (hardly, physiologically credible exceptfor a possible endo/epicardial gradient or for some pathologicalstates), but to the effects of noise in the scan data. Anotherpossibility could be that the true model is an homogeneousone but different from the 3K and 4K ones.

b) Brain: For brain data, results are quite different. Firstof all, the components at intermediate values ofare morefrequent than in the heart [Table IV(b)]. In particular, thenumber of the spectral lines increases (particularly in thegray matter) with the increasing of the observation interval(from 60–120 min), e.g., in -visual cortex the 60-min twocomponents became three for the 120 min. The componentat is resolvable in three over six cases for 120 min,in four for 90 min, and in two for 60 min. However, in theother cases we can find onecomponent at a very low value.This may indicate that the tracer is irreversibly trapped in thetissue according with 3K and TH hypothesis. Thecomponentrelative to was found for and gray matterfor all observation intervals, and for white matter. In theremaining cases we found some very highvalues. Precisionof ’s outside zero and infinity is, however, most often poor:if, however, the ’s grid is reduced in a similar manner as forthe heart, the [Table V(b)] results are poor only for whitematter for a 90-min observation interval.

2) Estimation of : The SA model provides an estimate ofonly if specific assumptions are made on the system, i.e.,

a 3K, a TH or a different model structure must be postulated.Turkheimeret al. [28] assumed a TH model and in this case,one can show that is given by the amplitude correspondingto . In case that no spectral line is detectable at ,

as a result of noise, Turkheimeret al. [28] have discussedthe use of three filtering techniques. First a cutoff value for

needs to be chosen, e.g., here the one suggested in[28] was used

(23)

Next, the following filtering method based on least squareswas used: the plasma component associated with frequenciesgreater than were subtracted from the tissue dataso that only the component relative to the integral ofplus, eventually, experimental noise, was left. Finally, usingweighted least squares, the function to be minimized was

(24)

where

with

(25)

The values are reported in Table IV(a) and (b) for the heartand brain, respectively. The variation in the heart values mightbe due to hibernating myocardium.

IV. EVALUATION OF MODELS

A. Estimation of Fractional Uptake

The FDG fractional uptake values are shown for the heartand brain in Table VI(a) and (b), respectively. The estimatesare always very precise, except for three cases (-visualcortex, TH model; -left hemisphere, 4K model; -righthemisphere, TH model) where the CV is Note that the

estimate obtained with SA looks better; however, one hasto remind that the SA filter forces a model structure on thedata and that parameter estimation is forced to be linear dueto the fixing of the eigenvalues.

The estimates from the different models are very similarin the heart: only in one case (H6) the SA model gives ahigher value. For the brain, the situation is different. The 4Kestimate is often greater than the 3K one, an effect possiblydue to compensation through parameter for the presence oftissue heterogeneity [13], [21], [23] or to the possibility of a“real” [20]. The SA value also tends to be lower than that ofthe 3K and 4K models, and similar to the TH value. However,there is a different trend between white and gray matter: inthe gray matter the value decreases with increasing of theobservation interval, while this trend is not so clearly presentin white matter, where the values also tend to be more similarbetween the 3K and 4K models.


B. Model Structure

The various model structures must be compared by usingquantitative criteria such as the plot of the weighted residuals,the precision of estimates and the value of parsimony criteria.It is worth emphasizing the crucial importance of a correctchoice of the weighting scheme for arriving at statisticallysound parameter estimates and precision. This has also beenrecently strengthened in [4]. Briefly, if one has an exactknowledge of the variance of the measurement error (assumingthis is the dominant noise component) theory indicates that theweights should be chosen equal to the inverse of this variance.This is generally possible but primary data should be collectedaccordingly [4]. Often a “relative” (instead of an “absolute”)knowledge is available, i.e., the measurement error varianceis known apart for a scale factor , whereis unknown. In this case, theory indicates that weights shouldbe chosen as Here, we have chosen a “relative”weighting scheme and, in particular, the one proposed in [17]and [7] which appears to be a reasonable one for PET dynamicdata. Clearly, a different choice of weights, e.g., the frequentlyused which corresponds to a constant varianceassumption [3], would have produced different parameterestimates and precisions.

The comparison of the mean weighted residuals for the heartand for the white matter of the 3K and 4K models [Fig. 4(a)and (c)] is not conclusive, since the two profiles are virtuallyequivalent. Also, the SA model residuals show a very similarrandom pattern. In contrast, for the gray matter, the weightedresiduals of TH and SA show a pattern better than the 4K,which in turn is better than the 3K.

Parameter precision is better for the 3K model, which is thesimpler of the three models. To make a comparison amongthem in terms of model parsimony, the Akaike informationcriterion (AIC) [3] was used

AIC WRSS (26)

where WRSS is the weighted residual sum of squares of the(11), is the number of parameters ( for the 3K model,

for the 4K model and for the TH model) andis the number of the data points (frames). Note that the AICdifference ( AIC) between two models and , having

and ( ) parameters, respectively, has astandard deviation approximately equal to [3].

The results are shown in Table VII. In the heart the 3K andthe 4K models are indistinguishable and the mean differencebetween the 3K and 4K AIC values (AIC) is smallerthan its standard deviation. In the brain, generally, the 3Kperforms marginally better that the 4K model, which in turnis marginally better than the TH model; the meanAICbetween 3K and 4K is, also for the brain, smaller than itsstandard deviation while this is not true for the TH model asthe difference between 3K and TH (or/and 4K and TH) AICvalues is marginally larger than the own standard deviation. So,considering AIC and parameter precision, in heart and brainthe 3K is a good candidate model. However, it must be notedthat brain AIC results only (like for the heart study) are not

TABLE VPARAMETER ESTIMATES AND THEIR PRECISION; � AND �

HAVE DIMENSION min�1 EXCEPT � TO � ! 1 WHICH IS

UNITLESS. (a) HEART SPECTRAL ANALYSIS (INPUT-OUTPUT) MODEL

(a)

conclusive. If we also take into account the results of the SAmethod, we can say that the brain data are probably compatiblewith a more complex model than the 3K one, and particularlyso in the gray matter where SA results generally showed twospectral lines (in the white matter, the SA results showed apoor spectral content). So, for the gray matter of the brain itis possible that the TH model is more physiological than the3K one, although estimating six parameters with precision issometimes difficult.

If physiological plausibility is called, then the studies bySchmidt and coworkers [16], [21]–[23] should be kept in mind.They have shown that, when generating synthetic data withthe TH model (thus, accounting for tissue heterogeneity) andidentifying them with the 4K model, nonzero estimates ofare found even in the total absence of dephosphorylation. Thus,

is possibly a model artifact entirely due some undermod-eling of the system: in [21]–[23], the authors speculate onthe possible effect of neglecting tissue heterogeneity duringrelatively short PET experiments. Further claims that a nonzero

could be a model artifact come from the evidence of verylow activity of glucose-6-phosphatase in the brain [26], [27]both for glucose and deoxyglucose.


TABLE V (Continued.)PARAMETER ESTIMATES AND THEIR PRECISION; � AND � HAVE DIMENSION ml/ml/min EXCEPT

� TO � ! 1 WHICH IS UNITLESS. (b) BRAIN SPECTRAL ANALYSIS (INPUT-OUTPUT) MODEL

(b)

C. A New Spectral Analysis Model

SA results point out a number of problems. First the classicSA requires to be positive. Second, it give rise towhich show very poor precision and even if one can improvethe estimation by using new grids to avoid line doubling(compare Tables V with IV) some bias may occur since the“true” is not necessarily the arithmetic mean of the doubledline eigenvalues. Finally, the estimation ofwith the filteringtechnique [28] requires the assumption of a specific modelstructure. Below a new SA method is proposed by revisitingit as an exponential impulse response identification problem.

The restriction to positive in SA is not necessary and,in principle, can be misleading. From compartmental theory[3], it is known that, while the impulse response of a genericcompartmental model is always positive, its coefficients do nothave to be positive, unless input and output are in the samecompartment.

If the aim of SA is to find the number of exponentialcomponents of the system impulse response without having todefine a structure beforehand, defining a grid of eigenvalues

and estimating only the relative amplitudes is also notnecessary. The right approach to follow is simply to estimatethe number of exponentials necessary for (19) to give a good fitto the data by using models of increasing order. For instance,

one can start first with a two-exponential model

(27)

and estimate by nonlinear weighted least squaresthen try a three-exponential model

(28)

and estimate and so on. Then, onecan use standard model parsimony criteria techniques [3] tochoose the best model. This way, we will have, as an additionalbonus, not only the precisions of the’s, but also of the ’s.In fact, if one fixes the ’s on a predetermined grid, it isnot possible to obtain a measure of their precision. Finally,estimation of the avoids the problem of line doubling.

This new SA model also provides a statistically soundmodel-independent information which can guide the selectionof the most appropriate among the potential compartmental


TABLE VI(a) HEART FRACTIONAL UPDATA K (min�1) ESTIMATE AND ITS PRECISION OF THE3K, 4K, AND SA MODELS. (b)BRAIN FRACTIONAL UPTAKE K (ml/ml/min) ESTIMATE AND ITS PRECISION OF THE3K, 4K, TH, AND SA MODELS

(a)

(b)

candidate structures. Let us consider first the 3K model. Themodel prediction for is

(29)

assuming This means that if from SA oneresolves three components, corresponding, respectively, to theeigenvalues an intermediate value, say andit is possible to interpret these results using the 3K model. The

relation between and the 3K model rate constants is

(30)

In case SA shows one component corresponding to twointermediate components, say and one correspondingto the TH model is a possible candidate. In fact, theTH model in presence of two homogeneous tissues gives

(31)


where are the relative mass weights ofthe tissues. can be written as

(32)

and it is now easy to write the SA versus TH model rela-tionship

(33)

Finally, in the case where two intermediate components, sayand one corresponding to are resolved by SA,

the 4K model becomes a possible candidate. In this case, onehas

(34)

where are combinations of the 4K model parameters[20].

Thus, one has

(35)

It is important to emphasize that while SA is extremely help-ful to discriminate among potential compartmental models, itdoes not give an unique answer, i.e., several structures arecompatible with the same SA results. We have concentratedon the classic [ F]FDG kinetic models but other modelstructures are equally possible. For instance an SA result withone component corresponding to , two intermediatecomponents, and one corresponding to is compatiblewith the TH model but also e.g., with a four-compartmentcatenary 5K model: both these compartmental realizations areindistinguishable in terms of SA results.

TABLE VIIAIC OF THE 3K, 4K, AND TH MODELS IN THE HEART AND BRAIN

TABLE VIIIESTIMATION OF THE EXPONENTIAL IMPULSE RESPONSE IN(a) THE HEART

(a)

The parameters of (27) and (28) were estimated in allsubjects and ROI’s. Results of this new SA approach arereported for the heart in Table VIII(a), and for the brain inTable VIII(b). In the heart only a two exponential model wasresolvable: one of the eigenvalue was very small and inmost cases poorly estimated. It is of interest to note that a two-exponential model with a very small (ideally zero) becomescompatible with the 3K model, thus, confirming our 3K modelidentification results (see Table I).


TABLE VIII ( Continued.)(b) ESTIMATION OF THE EXPONENTIAL IMPULSE RESPONSE IN(b) THE BRAIN

(b)

In the brain, the situation is different. If we look at theresults of the gray matter, we notice the presence of twodifferent tissue line for the 0–120 and 0–90 min observationintervals plus one line indicating the FDG trap in the tissue,except for the gray matter for 0–90 min. The fact that for0–60 min the gray matter results do not show the trapping linemay be due to a too short observation interval. So, these resultsshow that the gray matter needs a more complex model than3K one. For the white matter only a two-exponential modelwas resolvable from the data. However, often one of the twois close to zero. The white matter results are compatible morewith a 3K model than with a more complex model (4K or TH).

V. CONCLUSIONS

In this work, the most commonly used models for theanalysis of dynamic PET data of [F]FDG kinetics wereevaluated. In particular: 1) the two-tissue compartment ofSokoloff et al. [26], exhibiting three rate constants (3K); 2)the two-tissue compartment of Phelpset al. [20], exhibitingfour rate constants (4K); 3) the model of Schmidtet al. [21]which introduces tissue heterogeneity into the 3K model (TH);and 4) the spectral analysis method by Cunningham and Jones[6] and Turkheimeret al. [28] (SA). In addition a new SAmodel has been proposed. All the models have been identifiedby weighted nonlinear least squares. Particular care has beendevoted to reliably describe the measurement error and weightswere chosen optimally.

The results provide some useful guidelines for the analy-sis of dynamic PET data to arrive at a detailed picture of

[ F]FDG kinetics and can be synthesized as follow: 1) achoice between the 3K, 4K, and TH models must be basedon statistical criteria such as pattern of weighted residuals,precision of parameter estimates, and parsimony criteria, aswell as physiological plausibility; in addition one should usethe results of a model-independent approach such as the newSA model; in the heart it is difficult to resolve the 4K model,but this is not true in the brain; 2) the TH model is numericallyidentifiable in the gray but not the white matter of the brain;3) the classic SA approach has some unnecessary hypothesesbuilt in and could, in principle, be misleading; 4) we havesuggested a new SA approach to determine the number ofexponentials in the system response which is theoreticallymore sound. The results of this approach show that in theheart (60-min experiment) and in the white matter the 3Kmodel is robust while in the gray matter there is room for amore complex model. Such a model is not necessarily the THone and it may well be that a different model describing tissueheterogeneity would be a better representation; 5) estimates ofFDG fractional uptake provided by the four models are veryclose in the heart but not in the brain.

APPENDIX A

If the heterogeneous tissue is an aggregation ofsmallerhomogeneous subregions, one has [21]

(A1)


where are the relative massweights of the homogeneous subregions. Multiplying the lastterm by

(A2)

Equation (A1) becomes

(A3)

and, after simple manipulation, one has

(A4)

and

(A5)

and

(A6)

If we define

(A7)

and

(A8)

(A9)

the fractional extraction of the TH model is given by

(A10)

ACKNOWLEDGMENT

The authors would like to thank Dr. L.-c. Wu for makingavailable the brain data set originally published in [8].

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[11] Harwell Subroutine Library, Subroutine VB01A, Theoretical StudiesDepartment of AEA Industrial Technology, Oxfordshire, U.K., 1993.


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[24] F. Schuier, F. Orzi, S. Suda, G. Lucignani, C. Kennedy, and L. Sokoloff,“Influence of plasma glucose concentration on lumped constant of thedeoxyglucose method: Effects of hyperglycemia in the rat,”J. Cereb.Blood Flow Metab., vol. 10, pp. 765–773, 1990.

[25] O. Selberg, W. Burchert, J. Hoff, G. J. Meyer, H. Hundeshagen,E. Radoch, H.-J. Balks, and M. J. M¨uller, “Insulin resistance inliver cirrhosis. Positron-emission Tomography scan analysis of skeletalmuscle glucose metabolism,”J. Clin. Invest., vol. 91, pp. 1897–1902,1993.

[26] L. Sokoloff, M. Reivich, C. Kennedy, M. H. D. Rosiers, C. S. Patlak, K.D. Pettigrew, O. Sakurada, and M. Shinohara, “The [14C]deoxyglucosemethod for the measurement of local cerebral glucose utilization:Theory, procedure, and normal values in the conscious and anesthetizedalbino rat,” J. Neurochem., vol. 28, pp. 897–916, 1977.

[27] L. Sokoloff, “The radioactive deoxyglucose method. Theory, procedureand application for the measurement of local glucose utilization in thecentral nervous system,” inAdvances in Neurochemistry, B. W. Agranoffand M. H. Aprison, Eds., vol. 4, New York: Plenum, 1982, pp. 1–82.

[28] F. Turkheimer, R. M. Moresco, G. Lucignani, L. Sokoloff, F. Fazio,and K. Schmidt, “The use of spectral analysis to determine regionalcerebral glucose utilization with positron emission tomography and[18F]fluorodeoxyglucose: Theory, implementation, and optimizationprocedures,”J. Cereb. Blood Flow Metab., vol. 14, pp. 406–422, 1994.

Alessandra Bertoldo received the Doctoral degree(Laurea) in electronic engineering from the Univer-sity of Padova, Padova, Italy, in 1994. Since 1995.She is currently working towards the Ph.D. degreein biomedical engineering at the same University.

In 1997 she worked at the Laboratory of Cere-bral Blood Flow and Metabolism at the NationalInstitutes of Health (NIH), Bethesda, MD. Her re-search interests are mainly related to the quantifi-cation of PET images with emphasis on modeling[18F]Fluorodeoxyglucose and tracer receptor kinet-ics.

Paolo Vicini was born on April 15th, 1967 inPordenone, Italy. On May 19, 1992, he received anadvanced degree (Laurea) in electronics engineeringfrom the University of Padova, Padova, Italy. Whilepursuing this degree, he attended one academic year(1990–1991) at the University of California, LosAngeles, as an exchange student with the EducationAbroad Program of the University of California. OnSeptember 24, 1996, he received the Ph.D. (Dot-torato di Ricerca) degree in bioengineering from thePolytechnic of Milan, Milan, Italy.

He is currently a Senior Fellow at the Department of Bioengineering at theUniversity of Washington, Seattle, WA. His main research interests are in thearea of mathematical modeling and identification of biological systems, withemphasis on metabolism and pharmacokinetics.

Gianmario Sambuceti, photograph and biography not available at time ofpublication.

Adriaan Anthonius Lammertsma was born in theNetherlands on November 28, 1951. He studiedphysics at the University of Groningen, Groningen,the Netherlands, and graduated in 1977. In 1984he received the Ph.D. degree in medicine from theUniversity of London, London, U.K.

From 1978 until 1981, he was a Fellow of theDutch Society for Cancer Research. From 1979 until1996, he was at the MRC Cyclotron Unit, Hammer-smith Hospital, London, London, U.K., initially asa Visiting Worker, later as Senior Scientist. During

a sabbatical (1985–1986) he was Visiting Worker, later as Senior Scientist.During a sabbatical (1985–1986) he was Visiting Associate Professor at theDivision of Nuclear Medicine and Biophysics of UCLA School of Medicine,Los Angeles, CA. Since 1996 he is at the Free University in Amsterdam,where he is Head of Research of the PET Centre and Professor of MedicalPhysics and Informatics. His main research interest is the development of newquantitative tracer kinetic methods and their application for research questionsin clinical as well as experimental PET.

Oberdan Parodi, photograph and biography not available at time of publi-cation.

Claudio Cobelli (S’67–M’70–SM’97), for a photograph and biography, seep. 47 of the January 1998 issue of this TRANSACTIONS.

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