Evaluation of Compartmental and Spectral Analysis Models of [ F]FDG Kinetics for Heart and Brain Studies with PET

Transcription

1 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 45, NO. 12, DECEMBER Evaluation of Compartmental and Spectral Analysis Models of [ F]FDG Kinetics for 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 quantitate from [ 18 F]-Fluoro-Deoxy-Glucose ([ 18 F]FDG) positron emission tomography (PET) data glucose regional metabolic rate. We evaluate 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 [ 18 F]FDG dynamic data obtained in the myocardium and brain gray and white matter. All models were identified by nonlinear weighted least squares with weights chosen optimally. We show that: 1) 3K and 4K models are indistinguishable in terms of parsimony criteria and choice should be made on parameter precision and physiological plausibility; in the gray matter a more complex model than the 3K one is resolvable; 2) the TH model is resolvable in the gray but not in the white matter; 3) the classic SA approach has some unnecessary hypotheses built in and can be in principle misleading; we propose here a new SA model which is more theoretically sound; 4) this new SA approach supports the use of a 3K model in the heart with a 60 min experimental period; it also indicates that heterogeneity in the brain is modest in the white matter; 5) [ 18 F]FDG fractional uptake estimates of the four models are very close in the heart, but not in the brain; 6) a higher than 60 min experimental time is preferable for brain studies. Index Terms Fluoro-deoxy-glucose (FDG), glucose, kinetics metabolism, model identification, parameter estimation, physiological model, tracer. I. INTRODUCTION THE 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 to image regional metabolism in the brain and in other organs, Manuscript received December 17, 1996; revised May 7, This work was supported in part by a grant from the Italian Ministero della Università e della Ricerca Scientifica e Tecnologica (MURST 40%) on Bioingegneria dei Sistemi Metabolici e Cellulari and by the National Institutes of Health (NIH) under Grant RR Asterisk indicates corresponding author. A. Bertoldo is with the Department of Electronics and Informatics, University of Padova, Padova, Italy P. Vicini is with the Department of Electronics and Informatics, University of Padova, Padova, Italy; and the Department of Bioengineering, University of Washington, Seattle, WA USA. G. Sambuceti and O. Parodi are with the CNR Institute of Clinical Physiology, 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, University of Padova, via Gradenigo 6/A, Padova, Italy ( cobelli@dei.unipd.it). Publisher Item Identifier S (98) such as heart, muscle and liver. Dynamic PET data, analyzed with mathematical models of [ F]FDG kinetics, allow the estimation of tissue fractional uptake of [ F]FDG and, thus, of the regional metabolic rate of glucose, by using the so-called lumped constant (LC), a scale factor between glucose and [ F]FDG metabolism. Normally only a single macroscopic parameter, i.e., [ F]FDG fractional uptake, is calculated. However, most of the models also have the potential to provide a much more intimate picture of the system, e.g., degree of tissue heterogeneity and rate constants of blood-tissue exchange. Recently, the model by Sokoloff et al. [26] has been used to evaluate microscopic parameters in muscle [12], heart [9], [19], and liver [25] in various pathophysiological conditions. A thorough comparison of models of [ F]FDG kinetics is, however, lacking, the only available study being that of Lammertsma et al. [13] from 1987, where two-compartmental models, the Sokoloff et al. [26] and the Phelps et al. [20], and the Patlak graphical method were investigated for brain data only. In recent years, new models have been proposed, notably the Schmidt et al. [21] model and the spectral analysis method [6]. In this study, PET dynamic data obtained in the brain and in the myocardium were used to evaluate and compare the performance of four models of [ F]FDG kinetics: 1) the threecompartment model of Sokoloff et al. [26], exhibiting three rate constants (3K); 2) the three-compartment model of Phelps et al. [20], exhibiting four rate constants (4K); 3) the model of Schmidt et 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 Turkheimer et al. [28]. In the following, issues related to the LC will not be discussed. Clearly, since the LC is a scale factor for passing from [ F]FDG to a glucose metabolism picture, it is of utmost importance to correctly assess its value and ascertain its stability during different experimental conditions (e.g., high levels of insulin, workload, etc.) [10], [12], [24], [25]. II. THE DATA The heart data are shown in Fig. 1 and refer to six subjects (H1, H2, H3, H4, H5, and H6) with a myocardial infarction history. PET scans were performed with a ECAT III Positron Tomograph (CTI Inc., Knoxville, TN), and they were reconstructed using a Hanning filter with a cutoff of 0.5 (Nyquist frequency), thus, resulting in a transaxial spa /98$ IEEE

2 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 gr glucose 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 the tracer. The doses were: 8.5 mci for H1, 9.5 mci for H2, 8 mci 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).

3 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 cavity in order to minimize spillover from the wall and avoid underestimation of the count density. Processing of the heart data was performed as described in [2]. In particular, for the partial volume correction, the wall activity was divided by a recovery coefficient as a function of the thickness (the measured thickness was in all cases greater then the 9 mm FWHM). The recovery function was previously determined running by phantom study. The spillover radioactivity from LV to tissue [ F]FDG was accounted 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 two normal male subjects and and one patient with

4 1432 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 45, NO. 12, DECEMBER 1998 epilepsy, each with two ROI s, one gray matter (visual cortex) and one white matter (right hemisphere for and ; left hemisphere for ). The two ROI s were manually drawn on the image from the final emission scan and then transferred to the images from each of the other emission scans to obtain the time activity curve. Even if the two ROI s were drawn so as to have gray and white matter, the presence of heterogeneous tissue is expected above all in the gray region. Al the subjects were fasted overnight. The data were published in [8] to which we refer for further details. Briefly, for each subject, a bolus of FDG (10 mci) was injected intravenously and arterial blood samples 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 plasma subsequently separated for the determination of [ F]FDG concentration. In the three subjects scanning proceeded according to the following schedule: ten scans of 0.2 min, min, 2 1 min, min, min, 2 5 min, 1 10 min and 3 30 min. Scanning was completed within 120 min after injection of the tracer. The PET scanner used, a GE/Scanditronix PC WB was an eight-ring, 15-slice machine, and the scans were reconstructed on a matrix using a Hanning filter with a cutoff frequency of 0.5 (Nyquist frequency), resulting in a spatial resolution in the image plane of 6.5 mm FWHM. The number of pixels included and averaged in the ROI s ranged from pixels. The size of a pixel in these studies was 2 mm 2 mm. Tissue F concentrations (nci/ml) and plasma concentrations (nci/ml) were corrected for decay. The plasma data set was corrected also for external delay. III. THE MODELS All models assume that glucose metabolism is in steadystate and tracer theory predicts that [ F]FDG kinetics are described by linear, time-invariant differential equations [3]. A. The Three-Compartment Three Rate Constants Model The two-tissue-compartment model proposed by Sokoloff et al. in 1977 [26] is shown in Fig. 3. The model was originally developed for autoradiographic studies in the brain with 2- [ 14 C]Deoxyglucose as tracer, and subsequently used for PET [ F]FDG studies in the brain and other tissues/organs. The 3K model assumes that [ F]Fluorodeoxyglucose-6-phosphate ([ F]FDG-6-P) is irreversibly trapped in tissue for the duration of the experiment. The model and the input output experiment are described by where and are rate constants, is the concentration 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 has three rate constants. Note that there is no equation for the (1) Fig. 3. The 3K model. first compartment, because in PET studies is assumed to be known and used as the input for model identification. The output (tissue measurement) equation used for model identification was, for the brain [13] and, for the heart where is [ F]FDG in blood and (unitless) accounts in the brain data for the vascular volume present in the tissue ROI, while in the heart data it is dominated by the effect of spillover from blood to tissue (negligible in the brain) [14]. All the four model parameters and are a priori uniquely identifiable [3]. The model allows to calculate the fractional uptake of [ F]FDG [24] Once is known, one can then calculate the local metabolic rate of glucose by assuming a value for the lumped constant LC and by using the steady-state glucose concentration in plasma. Glucose metabolism in the heart is in a quasi-steadystate and we refer to [1] and [19] for a discussion of this assumption. The model was identified from [ F]FDG brain data, using the measured (assumed error-free) plasma time activity curve as forcing input of the model. In the brain measurement equation (2), the whole-blood time activity was obtained as [5] where is the subject s measured hematocrit (the assumption is made that in the capillaries equals that of the large vessels; other investigators have used different approaches, e.g., Schmidt et al. [22] have used in rats a fixed capillary of 0.30, while Phelps et al. [20] have used in dogs, monkeys, and humans a capillary equal to 0.85 that of large vessels and assumed that [ F]FDG does not distribute into red blood cells). In the heart measurement equation (3), was directly provided by the left ventricle PET measurement, and it was used instead of in (1). Use of the left ventricle PET measurement permits the noninvasive determination of the input function but the definition of an error-free curve for the heart blood time activity is limited by the presence, above all in the last scan frames, (2) (3) (4) (5)

5 BERTOLDO et al.: COMPARTMENTAL AND SPECTRAL ANALYSIS MODELS FOR STUDIES WITH PET 1433 of the spillover of tissue radioactivity into blood. This can affect the parameter estimates of each model (in particular the phosphorylation constant) but not the conclusions regarding the choice of the best model since the same input function is assumed for all models. The model parameters were estimated by weighted nonlinear least squares. Tissue activity curves are described by where is the midscan time, is the measurement error at time, and is the number of data. Thus, the cost function to be minimized is (6) WRSS (7) WRSS denotes weighted residual sum of squares, is the weight of the th datum, and the vector of unknown model parameters of dimension Measurement error was assumed to be additive, uncorrelated, Gaussian, zero mean, and with a variance described as proposed in [15] where is the length of the scanning interval relative to and is an unknown proportionality constant. Weights were chosen optimally [3] as (8) (9) and the scale factor of (8) was estimated a posteriori [3] as WRSS (10) where WRSS is the value of the cost function evaluated at the minimum, i.e., for equal to the vector of estimated model parameters WRSS (11) Fig. 4. (a) (a) Mean weighted residuals for the heart 3K, 4K, and SA models. WRSS has been minimized by using the Levenberg Marquardt algorithm as implemented in [11]. Precision of the parameter estimates was evaluated from the inverse of the Fisher information matrix by COV (12) The model was successfully identified in all subjects. 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. Parameter values for all cases are shown in Table I(a) for the heart and in Table I(b) for the brain. In the brain (two ROI) parameters were also estimated from two shorter time interval, 0 90 and 0 60 min, to evaluate the impact of the experiment duration on parameters estimates. Precision was poor (CV well above 100%) in and for in -right hemisphere (0 120 min), -visual cortex (0 90 min) and -visual cortex (0 60 min). In agreement with [22], and decreased for a longer observation interval in the gray ROI, while in the white ROI they were relatively constant. decreased with the observation interval in the gray ROI, while in the white ROI it was relatively constant. As expected, is higher in the heart than in the brain. The fractional FDG uptake with its precision (obtained by error propagation) is shown in Table VI(a) and (b) for the heart data and brain, respectively. B. The Three Compartment Four Rate Constants Model In 1979, Phelps et al. [20] proposed a modification of the model of [ F]FDG kinetics (Fig. 5) using the observation that, following a pulse of [ F]FDG, total tissue activity was observed to decline after 120 min, thus, indicating a loss of product. This model does not require that [ F]FDG-6-P is irreversibly trapped in tissue for the whole duration of the

6 1434 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 45, NO. 12, DECEMBER 1998 Fig. 4. (b) (Continued.) (b) Mean weighted residuals for the gray brain 3K, 4K, SA, and TH models. experiment, but that it can be dephosphorylated. The model and input output experiment equations are (13) where and are rate constants, is the concentration of [ F]FDG in plasma, is the concentration of [ F]FDG in tissue, and is the concentration of [ F]FDG-6-P in tissue. This model is called 4K model, because it has four rate constants. The PET tissue measurement equations were the same as those of the 3K model. All model parameters ( and ) are a priori uniquely identifiable [3]. Like for the 3K model, the fractional uptake of [ F]FDG is (14) The model was numerically identified like the 3K model. Mean weighted residuals are shown in Fig. 4; in particular,

7 BERTOLDO et al.: COMPARTMENTAL AND SPECTRAL ANALYSIS MODELS FOR STUDIES WITH PET 1435 Fig. 4. (c) (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, while parameter estimates are shown in Table II(a) and (b). In general, precision of parameter estimates degraded with respect to that of the 3K model. This is expected, since the 4K model has an additional parameter with respect the 3K model. The model was identified in the heart with a generally poor precision for since was rather small. In the brain, a better performance of the model was noted indicating that the data could allow one to resolve a more complex model than the 3K one. Some problems with were observed in the -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 two ROI s of the subject. As far as observation interval is considered, the parameter estimates exhibit a pattern similar to the 3K model for the gray matter: decreased (but less in magnitude) and for the results at 120 min or 90 min remained the same. The white matter showed a relatively constant value for all the parameters. The FDG fractional uptake with its precision (obtained by error propagation) is shown in Table VI(a) and (b) for the heart and brain, respectively.

8 1436 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 45, NO. 12, DECEMBER 1998 TABLE I PARAMETER ESTIMATES AND THEIR PRECISION (EXPRESSED AS PERCENT CV). (a) HEART 3K MODEL AND (b) BRAIN 3K MODEL (a) (b) model is an extension of the 3K model to an heterogeneous tissue and is shown in Fig. 6(a). A number of weighted subregions, each described by a 3K model is assumed and the resulting model [Fig. 6(b)], obtained by making some assumptions on the [21], becomes time-variant Fig. 5. The 4K model. C. The Heterogeneous Tissue Compartmental Model Schmidt et al. proposed [21], [22] a model which takes into account the heterogeneous composure of the brain tissue. The (15) where and represent the weighted average of the tissue concentrations in every subregion; is the rate constant

9 BERTOLDO et al.: COMPARTMENTAL AND SPECTRAL ANALYSIS MODELS FOR STUDIES WITH PET 1437 TABLE II PARAMETER ESTIMATES AND THEIR PRECISION (EXPRESSED AS PERCENT CV). (a) HEART 4K MODEL AND (b) BRAIN 4K MODEL (a) (b) for transport of FDG into the mixed tissue (a mass-weighted average of the subregion transport rate constants), and The equation used for model identification was (17) are the time-varying rate parameters. (16) All model parameters, and, are a priori uniquely identifiable [3].

10 1438 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 45, NO. 12, DECEMBER 1998 TABLE III BRAIN TH MODEL: PARAMETER ESTIMATES AND THEIR PRECISION (EXPRESSED AS PERCENT CV) The FDG fractional uptake for the TH model can be calculated (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 normal myocardium does not appear to be an issue. 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, and results in Table III. A posteriori identifiability of this model was generally good with precision of parameter estimates of the gray ROI s exhibiting poor values only for in the 0 60 observation interval, and for also in the 0 90 and intervals. In the white ROI s, the situation was different and, generally, the parameter estimates exhibited a poor precision. The FDG fractional uptake parameter with its precision (obtained by error propagation) is shown in Table VI(b).

11 BERTOLDO et al.: COMPARTMENTAL AND SPECTRAL ANALYSIS MODELS FOR STUDIES WITH PET 1439 TABLE IV PARAMETER ESTIMATES AND THEIR PRECISION (EXPRESSED AS PERCENT CV); AND HAVE DIMENSION min 01 EXCEPT RELATIVE TO!1WHICH IS UNITLESS. (a) HEART SPECTRAL ANALYSIS (INPUT OUTPUT) MODEL Fig. 6. The TH model. D. The Spectral Analysis Model The so-called spectral analysis (SA) method, basically an input output model, was introduced by Cunningham and Jones in 1993 [6] and has been used to determine local metabolic rate of glucose in the brain [28]. This new technique aims to identify the kinetic components of the tissue tracer activity without specific model assumptions, e.g., presence or absence of FDG dephosphorylation or homogeneity in the tissue. Below, the fundamentals of the technique [6], [28] are first briefly described. 1) Fundamentals: Suppose that the impulse response can be described by a sum of distinct exponential terms (19) with for every Then, the measured time activity 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 eigenvalues and then to estimate their associated amplitudes i.e., the spectral content. Let us now derive an expression for the tissue (a) measurement based on these premises. To do so, let us start with simple cases. What happens if a certain eigenvalue of the impulse response is close to infinity (i.e., has a very large value)? Then, the corresponding term in (20) is proportional to via and can be viewed as a high-frequency component, i.e., accounting for the fast passage of tracer in the vascular space of the ROI. It is a vascular volume or spillover component and. Conversely, what happens if a certain eigenvalue of the impulse response is close to zero? Then, the corresponding term in (20) is proportional to via and can be viewed as a low-frequency component, i.e., accounting for a (quasi)trap for the tracer. Lastly, the components corresponding to the intermediate values ( intermediate frequency components) will reflect the uptake of tracer within the tissue with their number corresponding to the number of distinct tissue compartments

12 1440 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 45, NO. 12, DECEMBER 1998 TABLE IV (Continued.) PARAMETER ESTIMATES AND THEIR PRECISION (EXPRESSED AS PERCENT CV); AND HAVE DIMENSION ml/ml/min EXCEPT RELATIVE TO!1WHICH IS UNITLESS. (b) BRAIN SPECTRAL ANALYSIS (INPUT OUTPUT) MODEL (b) within the ROI exchanging with plasma. Therefore, they give an indication of tissue heterogeneity. We can now write an expression for, by explicating the contribution of the terms corresponding to and to as in [28] (21) As already noted, the amplitude relative to corresponds to the parameter which was defined as in the 3K, 4K, and TH models. To implement the classic SA model the first step is to define a grid of s. The range of the was chosen as defined in [28]. The lower limit was, where is the end time of the experiment. The upper limit was where 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 various kinetic components were estimated via nonnegative linear weighted squares as implemented in MATLAB (The Mathworks, Sherborn, MA) [18]. The components for and for were explicitly included. Precision of the s was obtained from the inverse of the Fisher information matrix [3]. Note that the error of the s with classical SA is underestimated 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 for the brain in Table IV(b) for the three 120-, 90-, and 60-min observation intervals. We first discuss the heart results. a) Heart: A component at was always detectable. The fact that a component at is always

13 BERTOLDO et al.: COMPARTMENTAL AND SPECTRAL ANALYSIS MODELS FOR STUDIES WITH PET 1441 present, except for, indicates that [ F]FDG is irreversibly trapped 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 of tissue 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 couples of s are actually next to each other on the eigenvalue grid (e.g., in, there are two s for s 6 and 7). It is likely that the poor precision of the s stems from the fact that the approach cannot attribute to a certain the right, but divides the s between two s next to each other: we will call this phenomenon line doubling. To avoid this phenomenon, the identification for each subject was repeated 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 of the double lines from the results in Table IV(a) (e.g., for the average of the s numbered with 6). With this procedure, an improved precision for the s [Table V(a)] was obtained. However, the precision for the amplitude values is underestimated since the eigenvalues are fixed. In and there are altogether three distinct s, of which just one is, therefore, relative to tissue. For the other subjects there are four or five spectral lines. The presence of two or three tissue components, in addition to the component at and at is likely not to be attributable to tissue heterogeneity (hardly, physiologically credible except for a possible endo/epicardial gradient or for some pathological states), but to the effects of noise in the scan data. Another possibility could be that the true model is an homogeneous one but different from the 3K and 4K ones. b) Brain: For brain data, results are quite different. First of all, the components at intermediate values of are more frequent than in the heart [Table IV(b)]. In particular, the number of the spectral lines increases (particularly in the gray matter) with the increasing of the observation interval (from min), e.g., in -visual cortex the 60-min two components became three for the 120 min. The component at is resolvable in three over six cases for 120 min, in four for 90 min, and in two for 60 min. However, in the other cases we can find one component at a very low value. This may indicate that the tracer is irreversibly trapped in the tissue according with 3K and TH hypothesis. The component relative to was found for and gray matter for all observation intervals, and for white matter. In the remaining cases we found some very high values. Precision of s outside zero and infinity is, however, most often poor: if, however, the s grid is reduced in a similar manner as for the heart, the [Table V(b)] results are poor only for white matter for a 90-min observation interval. 2) Estimation of : The SA model provides an estimate of only if specific assumptions are made on the system, i.e., a 3K, a TH or a different model structure must be postulated. Turkheimer et al. [28] assumed a TH model and in this case, one can show that is given by the amplitude corresponding to. In case that no spectral line is detectable at, as a result of noise, Turkheimer et al. [28] have discussed the 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 squares was used: the plasma component associated with frequencies greater than were subtracted from the tissue data so that only the component relative to the integral of plus, eventually, experimental noise, was left. Finally, using weighted least squares, the function to be minimized was where with (24) (25) The values are reported in Table IV(a) and (b) for the heart and brain, respectively. The variation in the heart values might be due to hibernating myocardium. IV. EVALUATION OF MODELS A. Estimation of Fractional Uptake The FDG fractional uptake values are shown for the heart and brain in Table VI(a) and (b), respectively. The estimates are always very precise, except for three cases ( -visual cortex, TH model; -left hemisphere, 4K model; -right hemisphere, TH model) where the CV is Note that the estimate obtained with SA looks better; however, one has to remind that the SA filter forces a model structure on the data and that parameter estimation is forced to be linear due to the fixing of the eigenvalues. The estimates from the different models are very similar in the heart: only in one case (H6) the SA model gives a higher value. For the brain, the situation is different. The 4K estimate is often greater than the 3K one, an effect possibly due to compensation through parameter for the presence of tissue heterogeneity [13], [21], [23] or to the possibility of a real [20]. The SA value also tends to be lower than that of the 3K and 4K models, and similar to the TH value. However, there is a different trend between white and gray matter: in the gray matter the value decreases with increasing of the observation interval, while this trend is not so clearly present in white matter, where the values also tend to be more similar between the 3K and 4K models.

14 1442 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 45, NO. 12, DECEMBER 1998 B. Model Structure The various model structures must be compared by using quantitative 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 correct choice of the weighting scheme for arriving at statistically sound parameter estimates and precision. This has also been recently strengthened in [4]. Briefly, if one has an exact knowledge of the variance of the measurement error (assuming this is the dominant noise component) theory indicates that the weights should be chosen equal to the inverse of this variance. This is generally possible but primary data should be collected accordingly [4]. Often a relative (instead of an absolute ) knowledge is available, i.e., the measurement error variance is known apart for a scale factor, where is unknown. In this case, theory indicates that weights should be 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 dynamic data. Clearly, a different choice of weights, e.g., the frequently used which corresponds to a constant variance assumption [3], would have produced different parameter estimates and precisions. The comparison of the mean weighted residuals for the heart and for the white matter of the 3K and 4K models [Fig. 4(a) and (c)] is not conclusive, since the two profiles are virtually equivalent. Also, the SA model residuals show a very similar random pattern. In contrast, for the gray matter, the weighted residuals 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 the simpler of the three models. To make a comparison among them in terms of model parsimony, the Akaike information criterion (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) and is the number of the data points (frames). Note that the AIC difference ( AIC) between two models and, having and ( ) parameters, respectively, has a standard deviation approximately equal to [3]. The results are shown in Table VII. In the heart the 3K and the 4K models are indistinguishable and the mean difference between the 3K and 4K AIC values ( AIC) is smaller than its standard deviation. In the brain, generally, the 3K performs marginally better that the 4K model, which in turn is marginally better than the TH model; the mean AIC between 3K and 4K is, also for the brain, smaller than its standard deviation while this is not true for the TH model as the difference between 3K and TH (or/and 4K and TH) AIC values is marginally larger than the own standard deviation. So, considering AIC and parameter precision, in heart and brain the 3K is a good candidate model. However, it must be noted that brain AIC results only (like for the heart study) are not TABLE V PARAMETER ESTIMATES AND THEIR PRECISION; AND HAVE DIMENSION min 01 EXCEPT TO!1WHICH IS UNITLESS. (a) HEART SPECTRAL ANALYSIS (INPUT-OUTPUT) MODEL (a) conclusive. If we also take into account the results of the SA method, we can say that the brain data are probably compatible with a more complex model than the 3K one, and particularly so in the gray matter where SA results generally showed two spectral lines (in the white matter, the SA results showed a poor spectral content). So, for the gray matter of the brain it is possible that the TH model is more physiological than the 3K one, although estimating six parameters with precision is sometimes difficult. If physiological plausibility is called, then the studies by Schmidt and coworkers [16], [21] [23] should be kept in mind. They have shown that, when generating synthetic data with the TH model (thus, accounting for tissue heterogeneity) and identifying them with the 4K model, nonzero estimates of are found even in the total absence of dephosphorylation. Thus, is possibly a model artifact entirely due some undermodeling of the system: in [21] [23], the authors speculate on the possible effect of neglecting tissue heterogeneity during relatively short PET experiments. Further claims that a nonzero could be a model artifact come from the evidence of very low activity of glucose-6-phosphatase in the brain [26], [27] both for glucose and deoxyglucose.

15 BERTOLDO et al.: COMPARTMENTAL AND SPECTRAL ANALYSIS MODELS FOR STUDIES WITH PET 1443 TABLE V (Continued.) PARAMETER ESTIMATES AND THEIR PRECISION; AND HAVE DIMENSION ml/ml/min EXCEPT TO!1WHICH 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 classic SA requires to be positive. Second, it give rise to which show very poor precision and even if one can improve the 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 doubled line eigenvalues. Finally, the estimation of with the filtering technique [28] requires the assumption of a specific model structure. Below a new SA method is proposed by revisiting it 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 generic compartmental model is always positive, its coefficients do not have to be positive, unless input and output are in the same compartment. If the aim of SA is to find the number of exponential components of the system impulse response without having to define a structure beforehand, defining a grid of eigenvalues and estimating only the relative amplitudes is also not necessary. The right approach to follow is simply to estimate the number of exponentials necessary for (19) to give a good fit to the data by using models of increasing order. For instance, one can start first with a two-exponential model and estimate by nonlinear weighted least squares then try a three-exponential model (27) (28) and estimate and so on. Then, one can use standard model parsimony criteria techniques [3] to choose the best model. This way, we will have, as an additional bonus, not only the precisions of the s, but also of the s. In fact, if one fixes the s on a predetermined grid, it is not 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 sound model-independent information which can guide the selection of the most appropriate among the potential compartmental

16 1444 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 45, NO. 12, DECEMBER 1998 TABLE VI (a) HEART FRACTIONAL UPDATA K (min 01 )ESTIMATE AND ITS PRECISION OF THE 3K, 4K, AND SA MODELS. (b) BRAIN FRACTIONAL UPTAKE K (ml/ml/min) ESTIMATE AND ITS PRECISION OF THE 3K, 4K, TH, AND SA MODELS (a) (b) candidate structures. Let us consider first the 3K model. The model prediction for is relation between and the 3K model rate constants is (30) (29) assuming This means that if from SA one resolves three components, corresponding, respectively, to the eigenvalues an intermediate value, say and it is possible to interpret these results using the 3K model. The In case SA shows one component corresponding to two intermediate components, say and one corresponding to the TH model is a possible candidate. In fact, the TH model in presence of two homogeneous tissues gives (31)

17 BERTOLDO et al.: COMPARTMENTAL AND SPECTRAL ANALYSIS MODELS FOR STUDIES WITH PET 1445 where the tissues. are the relative mass weights of can be written as TABLE VII AIC OF THE 3K, 4K, AND TH MODELS IN THE HEART AND BRAIN (32) and it is now easy to write the SA versus TH model relationship (33) Finally, in the case where two intermediate components, say and one corresponding to are resolved by SA, the 4K model becomes a possible candidate. In this case, one has TABLE VIII ESTIMATION OF THE EXPONENTIAL IMPULSE RESPONSE IN (a) THE HEART (34) where are combinations of the 4K model parameters [20]. Thus, one has (35) It is important to emphasize that while SA is extremely helpful to discriminate among potential compartmental models, it does not give an unique answer, i.e., several structures are compatible with the same SA results. We have concentrated on the classic [ F]FDG kinetic models but other model structures are equally possible. For instance an SA result with one component corresponding to, two intermediate components, and one corresponding to is compatible with the TH model but also e.g., with a four-compartment catenary 5K model: both these compartmental realizations are indistinguishable in terms of SA results. (a) The parameters of (27) and (28) were estimated in all subjects and ROI s. Results of this new SA approach are reported for the heart in Table VIII(a), and for the brain in Table VIII(b). In the heart only a two exponential model was resolvable: one of the eigenvalue was very small and in most cases poorly estimated. It is of interest to note that a twoexponential model with a very small (ideally zero) becomes compatible with the 3K model, thus, confirming our 3K model identification results (see Table I).

18 1446 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 45, NO. 12, DECEMBER 1998 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 the results of the gray matter, we notice the presence of two different tissue line for the and 0 90 min observation intervals plus one line indicating the FDG trap in the tissue, except for the gray matter for 0 90 min. The fact that for 0 60 min the gray matter results do not show the trapping line may be due to a too short observation interval. So, these results show that the gray matter needs a more complex model than 3K one. For the white matter only a two-exponential model was resolvable from the data. However, often one of the two is close to zero. The white matter results are compatible more with a 3K model than with a more complex model (4K or TH). V. CONCLUSIONS In this work, the most commonly used models for the analysis of dynamic PET data of [ F]FDG kinetics were evaluated. In particular: 1) the two-tissue compartment of Sokoloff et al. [26], exhibiting three rate constants (3K); 2) the two-tissue compartment of Phelps et al. [20], exhibiting four rate constants (4K); 3) the model of Schmidt et al. [21] which introduces tissue heterogeneity into the 3K model (TH); and 4) the spectral analysis method by Cunningham and Jones [6] and Turkheimer et al. [28] (SA). In addition a new SA model has been proposed. All the models have been identified by weighted nonlinear least squares. Particular care has been devoted to reliably describe the measurement error and weights were chosen optimally. The results provide some useful guidelines for the analysis of dynamic PET data to arrive at a detailed picture of [ F]FDG kinetics and can be synthesized as follow: 1) a choice between the 3K, 4K, and TH models must be based on statistical criteria such as pattern of weighted residuals, precision of parameter estimates, and parsimony criteria, as well as physiological plausibility; in addition one should use the results of a model-independent approach such as the new SA 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 numerically identifiable in the gray but not the white matter of the brain; 3) the classic SA approach has some unnecessary hypotheses built in and could, in principle, be misleading; 4) we have suggested a new SA approach to determine the number of exponentials in the system response which is theoretically more sound. The results of this approach show that in the heart (60-min experiment) and in the white matter the 3K model is robust while in the gray matter there is room for a more complex model. Such a model is not necessarily the TH one and it may well be that a different model describing tissue heterogeneity would be a better representation; 5) estimates of FDG fractional uptake provided by the four models are very close in the heart but not in the brain. APPENDIX A If the heterogeneous tissue is an aggregation of homogeneous subregions, one has [21] smaller (A1)

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Agranoff and M. H. Aprison, Eds., vol. 4, New York: Plenum, 1982, pp [28] F. Turkheimer, R. M. Moresco, G. Lucignani, L. Sokoloff, F. Fazio, and K. Schmidt, The use of spectral analysis to determine regional cerebral glucose utilization with positron emission tomography and [ 18 F]fluorodeoxyglucose: Theory, implementation, and optimization procedures, J. Cereb. Blood Flow Metab., vol. 14, pp , Alessandra Bertoldo received the Doctoral degree (Laurea) in electronic engineering from the University of Padova, Padova, Italy, in Since She is currently working towards the Ph.D. degree in biomedical engineering at the same University. In 1997 she worked at the Laboratory of Cerebral Blood Flow and Metabolism at the National Institutes of Health (NIH), Bethesda, MD. Her research interests are mainly related to the quantification of PET images with emphasis on modeling [ 18 F]Fluorodeoxyglucose and tracer receptor kinetics. Paolo Vicini was born on April 15th, 1967 in Pordenone, Italy. On May 19, 1992, he received an advanced degree (Laurea) in electronics engineering from the University of Padova, Padova, Italy. While pursuing this degree, he attended one academic year ( ) at the University of California, Los Angeles, as an exchange student with the Education Abroad Program of the University of California. On September 24, 1996, he received the Ph.D. (Dottorato di Ricerca) degree in bioengineering from the Polytechnic of Milan, Milan, Italy. He is currently a Senior Fellow at the Department of Bioengineering at the University of Washington, Seattle, WA. His main research interests are in the area of mathematical modeling and identification of biological systems, with emphasis on metabolism and pharmacokinetics. Gianmario Sambuceti, photograph and biography not available at time of publication. Adriaan Anthonius Lammertsma was born in the Netherlands on November 28, He studied physics at the University of Groningen, Groningen, the Netherlands, and graduated in In 1984 he received the Ph.D. degree in medicine from the University of London, London, U.K. From 1978 until 1981, he was a Fellow of the Dutch Society for Cancer Research. From 1979 until 1996, he was at the MRC Cyclotron Unit, Hammersmith Hospital, London, London, U.K., initially as a Visiting Worker, later as Senior Scientist. During a sabbatical ( ) he was Visiting Worker, later as Senior Scientist. During a sabbatical ( ) he was Visiting Associate Professor at the Division 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 Medical Physics and Informatics. His main research interest is the development of new quantitative tracer kinetic methods and their application for research questions in clinical as well as experimental PET. Oberdan Parodi, photograph and biography not available at time of publication. Claudio Cobelli (S 67 M 70 SM 97), for a photograph and biography, see p. 47 of the January 1998 issue of this TRANSACTIONS.