## Abstract

Various modeling strategies have been developed to convert regional [^{18}F]fluorodeoxyglucose ([^{18}F]FDG) concentration measured by positron emission tomography (PET) to a measurement of physiological parameters. However, all the proposed models have been developed and tested mostly for brain studies. The purpose of the present study is to select the most accurate model for describing [^{18}F]FDG kinetics in human skeletal muscle. The database consists of basal and hyperinsulinemic-euglycemic studies performed in normal subjects. PET data were first analyzed by an input-output modeling technique (often called spectral analysis). These results provided guidelines for developing a compartmental model. A new model with four compartments and five rate constants (5K model) emerged as the best. By accounting for plasma and extracellular and intracellular kinetics, this model allows, for the first time, PET assessment of the individual steps of [^{18}F]FDG kinetics in human skeletal muscle, from plasma to extracellular space to transmembrane transport into the cell to intracellular phosphorylation. Insulin is shown to affect transport and phosphorylation but not extracellular kinetics, with the transport step becoming the main site of control. The 5K model also allows definition of the domain of validity of the classic three-compartment three- or four-rate-constant models. These models are candidates for an investigative tool to quantitatively assess insulin control on individual metabolic steps in human muscle in normal and physiopathological states.

- positron emission tomography
- parameter estimation
- glucose
- compartmental model
- insulin

quantitative assessment of the individual steps of glucose metabolism in human skeletal muscle, i.e., transport from plasma to extracellular space and back, transport across the cell membrane, and intracellular phosphorylation, is crucial for understanding insulin action in normal and physiopathological states such as diabetes. Because direct measurement of these individual steps in vivo in humans is not possible, several indirect approaches have been proposed (for review see Refs. 5 and 29). Among the quantitative approaches, two model-based techniques have emerged: the triple-tracer dilution technique (4, 29) and a technique based on positron emission tomography (PET) (14, 23, 26, 30, 34). The first approach consists of injecting into the forearm of the subject three tracers: an extracellular (e.g.,d-[^{3}H]mannitol), a transportable but not metabolizable (e.g., [*methyl*-3-*O*-^{14}C]glucose), and a metabolizable (e.g., d-[^{3}H]glucose) tracer. The three dilution curves are then analyzed with a physiologically based multicompartmental model of regional blood-tissue exchange, which provides estimates of glucose transmembrane transport and intracellular phosphorylation fluxes. The second approach is based on PET noninvasive measurement in leg skeletal muscle of injected [^{18}F]fluorodeoxyglucose ([^{18}F]FDG), a glucose analog. By interpreting [^{18}F]FDG plasma concentration and tissue activity curves with compartmental models with three (3K model) or four (4K model) rate constants, one arrives at [^{18}F]FDG transport and phosphorylation. In contrast to the triple-tracer dilution technique, which provides transport and phosphorylation fluxes of glucose, the PET [^{18}F]FDG technique can provide only rate constants of transport and phosphorylation of [^{18}F]FDG, since the tracer is an analog. Muscle glucose uptake can then be assessed by using the so-called lumped constant. Recent reports indicate that the lumped constant is relatively insensitive to insulin (16, 23,33), but glucose and [^{18}F]FDG individual unit processes exhibit different affinity. However, [^{18}F]FDG has proven to be a sensitive probe for assessing the individual steps of metabolism in human skeletal muscle in various physiopathological states (14, 15, 19, 26, 30, 34).

In human skeletal muscle [^{18}F]FDG PET studies, the 3K and 4K models are used. These models are, in a way, a heritage of the brain studies where these two models, originally proposed by Sokoloff et al. (31) and Phelps et al. (24), respectively, have become a paradigm. To the best of our knowledge, whether the 3K and 4K models also are the models of choice in human skeletal muscle has not been thoroughly addressed. This is the purpose of the present study.

By using [^{18}F]FDG PET studies performed in normal humans in basal and insulin-stimulated states, we will study which is the most accurate model for describing [^{18}F]FDG kinetics in the human skeletal muscle. To do so, we first resort to a quasi-model-independent input-output technique [referred to in the PET literature as spectral analysis (SA)]. Then, we move on with compartmental modeling by following guidelines dictated by SA. A new model emerges that accounts for plasma and extracellular and intracellular kinetics and has five rate constants (5K model). The physiology underlying the 5K model, as well as its predictions in terms of insulin control, is discussed. Finally, the rich 5K model picture allows us to revisit the classic 3K and 4K models to better understand the physiological meaning of their parameters.

## MATERIALS AND METHODS

### Subjects

Eleven men volunteered for the study (age = 26 ± 1 yr, body mass index = 22.3 ± 1.0 kg/m^{2}). Five subjects were studied in the fasting state and six during a euglycemic-hyperinsulinemic clamp. The subjects were healthy as judged by history, physical examination, and routine laboratory tests and were not taking any medications. The nature, purpose, and potential risks of the study were explained to all subjects before they gave their written informed consent to participate. The study was approved by the Joint Commission of Ethics of the University of Turku and Turku University Central Hospital.

### Study Protocol

Studies were performed after a overnight fast. Alcohol and caffeine were prohibited 24 h before the study, and the subjects were instructed to avoid strenuous physical activity 1 day before the study. The subjects were lying supine during the study. Two catheters were inserted: one in an antecubital vein of the left hand for the infusion of glucose and insulin and injections of [^{15}O]H_{2}O and [^{18}F]FDG and one in the radial artery for blood sampling. The 120-min euglycemic-hyperinsulinemic (1 mU · kg^{−1} · min^{−1}) clamp technique (10) was used, with insulin infused at 1 mU · kg^{−1} · min^{−1} for 180 min starting 60 min before tracer injection and euglycemia maintained with a variable rate of infusion of 20% glucose.

### Imaging

An eight-ring tomograph (model ECAT 931/08, Siemens/CTI, Knoxville, TN) was used for image acquisition. The scanner has an axial resolution of 6.7 mm and an in-plane resolution of 6.5 mm (32). The subject was positioned in the tomograph with the femoral region within the gantry. Before the emission scanning, a transmission scan for correction of photon attenuation was performed for 20 min with a removable ring source containing ^{68}Ge. All data were corrected for dead time, decay, and measured photon attenuation. For image processing, a recently developed Bayesian iterative reconstruction algorithm using median root prior with 150 iterations and the Bayesian coefficient of 0.3 was applied (1). Regions of interest (ROIs) were drawn in the anteromedial muscle compartments of both femoral regions in four cross-sectional slices, with care taken to avoid large blood vessels (25). Localization of the muscle compartments was verified by comparing the flow images with the transmission image, which provides a topographical distribution of tissue density. The ROIs outlined in the flow images were copied to the [^{18}F]FDG images to obtain quantitative data from identical regions. The time-activity curve, obtained from the weighted mean of the six tissue ROIs by using the size of the ROIs as weights, was used.

### [^{18}F]FDG Studies

For [^{18}F]FDG studies, 0.16–0.28 GBq of [^{18}F]FDG was injected intravenously over 2 min, and dynamic scanning for 120 min was started (12 × 15, 4 × 30, 3 × 60, 1 × 120, and 22 × 300 s frames). The radiochemical purity of [^{18}F]FDG exceeded 98%. Arterial blood samples for measurement of plasma radioactivity were withdrawn as previously described (21).

### [^{15}O]H_{2}O Studies

For measurement of blood flow, 1.2–1.5 GBq of [^{15}O]H_{2}O were injected intravenously, and dynamic scanning was performed for 6 min (6 × 5, 6 × 15, and 8 × 30 s frames). To determine the input function, blood from the radial artery was continuously withdrawn using a pump at a speed of 6 ml/min. The radioactivity concentration was measured using a two-channel detector system (Scanditronix, Uppsala, Sweden) calibrated to the well counter (Wizard 1480, Wallac, Turku, Finland) and the PET scanner, as previously described (27).

### Modeling of [^{15}O]H_{2}O Kinetics

The arterial input curve was corrected for dispersion and delay, as previously described (22). The autoradiographic method and a 250-s integration time were applied to calculate blood flow pixel-by-pixel. This method has been previously validated against the steady-state method (27), and blood flow was measured using strain-gauge plethysmography (22).

### Modeling of [^{18}F]FDG Kinetics

#### Input-output modeling.

An input-output modeling approach, usually referred to in the PET literature as SA (8), was first used as proposed previously (3) to characterize the reversible and irreversible components of the system and to estimate the minimum number of compartments needed to describe the [^{18}F]FDG kinetics in the skeletal muscle. Briefly, if the impulse response [*h*(*t*)] of the system is written as
Equation 1with β_{j}≥ 0 for every *j*. The total activity in the ROI [C(*t*)] is the convolution of*h*(*t*) with the arterial plasma tracer concentration [C_{p}(*t*)] plus a term taking into account the vascular component present in the ROI
Equation 2where V_{b} is the fraction of the total volume occupied by the blood pool and C_{b}(*t*) is the arterial blood tracer concentration calculated as follows: C_{b}(*t*) = C_{p}(τ)(1 − 0.3H) (8), where H is the subject's hematocrit. The method estimates the number *M* of nonzero values of α_{j} that (together with the corresponding β_{j}) best describes the data providing, at the same time, useful insight into the system structure. In fact, the number of amplitudes α_{j} corresponding to the eigenvalues β_{j}≠ 0 gives the number of reversible compartments (i.e., the tracer can reach these pools directly or indirectly from plasma and return to plasma) that can be discriminated in the tissue. However, nothing can be said in terms of compartment connectivity; e.g., two amplitudes at the β_{j}≠ 0 frequencies do not establish whether the corresponding reversible tissue compartments are in parallel (heterogeneous tissue) or in cascade (homogeneous tissue), since these two structures are kinetically indistinguishable. Finally, the amplitude α corresponding to the zero eigenvalue (β = 0) reveals the presence of an irreversible process; i.e., the tracer cannot leave the tissue. The method starts by using *M* = 2 and estimates the values of α_{1}, β_{1}, α_{2}, β_{2}, and V_{b}; then one tries*M* = 3 and estimates α_{1}, β_{1}, α_{2}, β_{2}, α_{3}, β_{3}, and V_{b} and so on. To select the best model, parsimony criteria are used (see *Parameter Estimation*).

#### Compartmental modeling of [^{18}F]FDG kinetics.

Compartmental models are widely used to describe PET tracer data, since they provide insight into the system structure and function (6,12). SA revealed (see results) that a 5K model is resolvable from the data and is a good candidate model to describe [^{18}F]FDG kinetics in skeletal muscle. We also analyzed the classic 3K and 4K models (see below).

##### 5K MODEL.

The 5K model is shown in Fig. 1 and can be viewed as an extension of the classic 3K model by Sokoloff et al. (31) (Fig. 2), originally proposed in the brain. The novelty of the 5K model lies in its explicit accounting of an extracellular compartment; i.e., it assumes that, in skeletal muscle, it is possible to distinguish the kinetics steps of delivery of [^{18}F]FDG to the extracellular space, its transport from the extracellular to the intracellular space, and its intracellular phosphorylation. The 5K model is described by (see appendix
)
Equation 3
Equation 4where C_{p} is [^{18}F]FDG plasma arterial concentration, C_{i} is extracellular concentration of [^{18}F]FDG normalized to intracellular volume, C_{e} is [^{18}F]FDG intracellular concentration, C_{m} is [^{18}F]FDG 6-phosphate ([^{18}F]FDG-6-*P*) intracellular concentration, C is total ^{18}F activity concentration in the ROI,*K*
_{1}(ml · ml^{−1} · min^{−1}) and*k*
_{2} (min^{−1}) are exchange parameters between plasma and extracellular space, *k*
_{3}(min^{−1}) and *k*
_{4} (min^{−1}) are transport parameters into and out of the cell, and*k*
_{5} (min^{−1}) is the phosphorylation parameter (V_{b} and C_{b} have the same meaning as in *Eq. 2
*).

All six model parameters, *K*
_{1},*k*
_{2}, *k*
_{3},*k*
_{4}, *k*
_{5}, and V_{b}, are a priori uniquely identifiable (7) (see appendix
).

From the model, one can calculate the fractional uptake of [^{18}F]FDG, *K*(ml · ml^{−1} · min^{−1}) (seeappendix
)
Equation 5In PET studies, equilibrium operational volumes are normally calculated (12). The model allows the calculation of two operational distribution volumes, one related to the extracellular (V
) and one to the intracellular (V
) space; by solving *Eq. 3
* at steady state one has
Equation 6
Of interest is the ratio V
/V
, which can be shown to be the ratio of “true” [^{18}F]FDG mass in the intracellular space to true [^{18}F]FDG mass in the extracellular space. In fact, one has
Equation 7

##### 3K AND 4K MODELS.

The 3K (31) and 4K (24) models were originally proposed for [^{18}F]FDG quantification in the brain, but they are also widely used for quantification of glucose transport and phosphorylation in human skeletal muscle. We will use these two models as they are normally employed in the literature (14, 26, 30, 34) and will reconcile them with the 5K model description in the discussion. An apex symbol will be introduced to characterize the 3K and 4K model variables and parameters to make the comparison easier.

The 3K model (Fig. 2) is described by
Equation 8
Equation 9where C_{p} is [^{18}F]FDG plasma arterial concentration, C′_{e} is [^{18}F]FDG intracellular concentration, C′_{m} is [^{18}F]FDG-6-*P* intracellular concentration, C is total ^{18}F activity concentration in the ROI,*K*′_{1}(ml · ml^{−1} · min^{−1}) and*k*′_{2} (min^{−1}) describe [^{18}F]FDG transport from plasma to tissue and back, respectively, *k*′_{3} (min^{−1}) is [^{18}F]FDG phosphorylation, and V_{b} and C_{b} have the same meaning as in *Eq. 4
*.

All four model parameters, *K*′_{1},*k*′_{2}, *k*′_{3}, and V_{b}, are a priori uniquely identifiable (7). The model allows calculation of the fractional uptake of [^{18}F]FDG as
Equation 10The 4K model is an extension of the 3K model by allowing dephosphorylation of [^{18}F]FDG-6-*P* to be present
Equation 11where the meaning of variables and rate constants is the same as for the 3K model, with *k*′_{4} denoting [^{18}F]FDG-6-*P* dephosphorylation. The measurement equation and the expression of fractional tracer uptake are the same as for the 3K model, i.e., *Eqs. 9
* and *
10
*.

### Parameter Estimation

Input-output and compartmental model parameters were estimated by weighted nonlinear least squares as implemented in SAAM II (2,28). The measured PET activity (C^{obs}) was described as
Equation 12where e(*t _{j}
*) is the measurement error at time

*k*

_{1}, assumed to be independent, Gaussian, zero mean and with a variance (ς) given by (3) Equation 13where Δ

*t*is the length of the scanning interval relative to C

_{j}^{obs}(

*t*) and γ is an unknown proportionality constant to be estimated a posteriori (7) as Equation 14WRSS(

_{j}**p̂**) is the weighted residual sum of squares evaluated at the minimum, i.e., for

**p**equal to the estimated

**p̂**Equation 15where w

_{j}is the weight of the

*j*th datum [w

_{j}= Δ

*t*/C

_{j}^{obs}(

*t*)],

_{j}*t*is the midscan time,

_{j}*N*is the number of scans, and

*P*is the number of parameters. Parameter precision was evaluated from the inverse of the Fisher information matrix (7).

To select the best model, in addition to parameter precision, weighted residuals inspection and parsimony criteria were used, in particular, the Akaike information criterion (AIC) and the Schwarz criterion (SC) (7, 18).

The weighted residual at time *t _{j}
* is defined as
Equation 16Residuals must reflect, if the model is correct, the assumptions on the measurement error, i.e., to be a zero mean and independent process.

AIC and SC were computed as Equation 17 Equation 18The model with the smallest AIC and SC is considered to be the most parsimonious, i.e., to best fit the data with the fewest parameters. In the following, for the sake of space, we report only the AIC values, since the SC resulted in similar conclusions.

### Statistics

Results are means ± SE. Significance of differences has been determined by using the Mann-Whitney or, when appropriate, the Wilcoxon rank test. The runs test (7) was used to verify the independence of residuals. *P* < 0.05 was considered to be significant.

## RESULTS

### [^{18}F]FDG Kinetics

Mean plasma and tissue activity concentrations of [^{18}F]FDG normalized for the dose are shown in Fig.3. Plasma glucose was 89 ± 4 mg/dl in the basal state and 99 ± 2 mg/dl during insulin stimulation. During insulin stimulation, plasma insulin concentration rose from 6.1 ± 1.3 to 42 ± 2 μU/ml with glucose infused at 30.8 ± 2.4 μmol · kg^{−1} · min^{−1}.

### Input-Output (SA)

The system was described by one-, two-, three-, and four-exponential models. The one- and four-exponential models were rejected, because they were either too simplistic or too complex. Table1 shows the AIC values of the two- and three-exponential models. In the basal studies the three-exponential model was always the best. In the insulin studies the three-exponential model was superior in *subjects 6, 10*, and *11*, while in *subjects 7, 9*, and *10* the two-exponential model was the most parsimonious.

The parameter values of the two- and three-exponential models are shown in Table 2. In the basal studies, three (*subjects 1, 3*, and *5*) of the five subjects show the presence of two reversible processes, an irreversible process, and a blood component. All of the insulin studies show the presence of an irreversible process and a blood component. In three (*subjects 6, 10*, and *11*) of the six subjects it is also possible to detect, as in the basal studies, two distinct reversible processes, while in the other three subjects (*subjects 7–9*) only one equilibrating component is detected.

### From SA to Compartmental Modeling

SA gives the minimum number of compartments necessary to describe the [^{18}F]FDG data, together with some information on the compartmental structure. Because it is not possible to analyze ROIs within femoral muscles with respect to the ability of different muscle fiber types to use [^{18}F]FDG, even by using the most advanced of the current generation of PET scanner, the skeletal muscle tissue within the ROIs was assumed to be homogeneous. With this assumption, SA results of basal studies of *subjects 1, 3*, and *5* can be represented by using the 5K model, i.e., two extravascular compartments in cascade describing the [^{18}F]FDG exchanging reversibly with plasma plus a compartment where [^{18}F]FDG-6-*P* is irreversibly trapped for the duration of the experiment. In the remaining two basal studies (*subjects 2* and *4*), SA detects the presence of three reversible processes, no irreversible process, and a blood component. However, in these two subjects, the slowest eigenvalue is ∼100-fold smaller with respect to the second value, and thus, in relative terms, it is practically equal to zero. Consequently, with the assumption of a negligible glucose-6-phosphatase activity, these results can also be interpreted within the 5K model framework. As far as the insulin studies are concerned, the 5K model can still be used to explain the SA results; however, results of *subjects 7* and*8* point toward a collapsed 3K version of the 5K model, i.e., a model with a single reversible free [^{18}F]FDG tissue pool and with [^{18}F]FDG-6-*P* irreversibly trapped in tissue. SA results are not compatible with a 4K model structure, i.e., a model with two reversible processes and no integral component. Therefore, 4K model results are not reported.

### 5K Model

Parameter estimates are shown in Table3, together with their precision. Mean weighted residuals are shown in Fig. 4for basal and insulin-stimulated states. Residuals satisfy the hypothesis of independence (*P* = 0.5 and 0.8 for basal and insulin studies, respectively). Numerical identifiability was satisfactory in the basal studies, except for the small V_{b}value in *subject 5*; in *subjects 2* and*3*, V_{b} was not resolvable. In the insulin studies, parameter precision generally degraded with respect to the basal value but was poor only for *k*
_{4} in*subjects 7–9* and for *k*
_{5} in*subjects 7* and *8*. This is expected from SA results, which suggested a model simpler than the 5K model. Similar to the basal state, V_{b} was hardly resolvable. *K* was estimated very precisely because of its macroscopic nature. The 5K model results indicate that insulin does not affect*K*
_{1} and *k*
_{2}, but it significantly increases *k*
_{3} (*P* < 0.02). It also increases *k*
_{5} (*P*< 0.02) and decreases *k*
_{4} (albeit not significantly, *P* = 0.17) when they are resolvable. Insulin significantly increases *K* (*P* < 0.02). Also, insulin does not affect V
(0.074 ± 0.002 and 0.063 ± 0.013 ml/ml for basal and insulin studies, respectively, *P* = 0.46), while it significantly increases V
(0.049 ± 0.002 and 0.103 ± 0.011 ml/ml for basal and insulin studies, respectively,*P* < 0.02) as well as V
/V
(0.672 ± 0.035 and 1.742 ± 0.188 for basal and insulin studies, respectively,*P* < 0.02).

### 3K Model

Parameter estimates are shown in Table4. Mean weighted residuals are shown in Fig. 4 for basal and insulin-stimulated states. Residuals do not satisfy the hypothesis of independence. Precision was very good for all the parameters in both studies, although the insulin studies showed a degraded precision as in the 5K model.*K*′_{1} and *k*′_{2}do not change significantly in response to insulin, while*k*′_{3} and *K*′ significantly increase (*P* < 0.007 and *P* < 0.007).*K*′ is very similar to the corresponding 5K parameter*K*; this is expected from theory, since fractional uptake is largely a model-independent parameter. As far as the other rate parameters are concerned, it is difficult to compare their values against those of the 5K model, since they have a different meaning. Interestingly, *K*′_{1} is not different from*K*
_{1}: 0.026 ± 0.004 vs. 0.019 ± 0.002 for the basal state (*P* = 0.04) and 0.031 ± 0.008 vs. 0.021 ± 0.003 for the insulin state (*P* = 0.03).

### 5K vs. 3K Model

On the basis of the results of the runs test, the 5K model has to be preferred to the 3K model. AIC values of the 3K and 5K models are shown in Table 5 and indicate that the 5K model is better than the 3K model in the basal state, while in the insulin studies the 5K model is better in four (*subjects 6, 8, 10*, and *11*) of the six subjects. The 5K mean model predictions of [^{18}F]FDG extracellular and [^{18}F]FDG and [^{18}F]FDG-6-*P*intracellular concentrations are shown in Fig.5.

### Blood Flow

Flow values were 0.023 ± 0.006 and 0.031 ± 0.007 ml · ml^{−1} · min^{−1} in the basal and insulin studies, respectively, and were statistically not significantly different.

## DISCUSSION

The present study provides evidence that a four-compartment five-rate-constant model describing kinetic events occurring in plasma and extracellular and intracellular space is the most reliable description of [^{18}F]FDG kinetics in skeletal muscle in normal humans. The novelty of the model lies in its explicit accounting of the extracellular space. In particular, our results show that transport of [^{18}F]FDG across the cell membrane is not sufficiently fast compared with delivery into the extracellular space to allow lumping of extracellular space into the [^{18}F]FDG intracellular compartment, an assumption of the 3K and 4K models originally developed for brain studies (12). Explicitly accounting for the extracellular space permits one to follow [^{18}F]FDG when it is first delivered into the extracellular space and then crosses the cell membrane, thus allowing an accurate assessment of the effect of insulin on [^{18}F]FDG transport and phosphorylation. Finally, the 5K model, because of its physiologically sound basis, allows a better understanding of the physiological meaning of the values generated by the 3K and 4K model parameters.

### Model Structure

The use of compartmental models to describe [^{18}F]FDG kinetics in human skeletal muscle has recently seen important contributions (14, 15, 19, 26, 34). Common to all these reports is the tacit assumption that the 3K and 4K models, which were originally proposed to describe [^{18}F]FDG kinetics in the brain (24, 31), also provide the appropriate compartmental structure for studying human skeletal muscle. To test this assumption, we resorted to the input-output modeling technique (3, 9), usually referred to as SA, which provides, in a virtually model-independent way, precious information on the minimum number of compartments contained in a PET dynamic data set and some hints on their connectivity. With the assumption of tissue homogeneity, since it is difficult to analyze by PET ROIs within the femoral muscles with respect to the ability of different muscle fiber types to handle [^{18}F]FDG, SA shows that a cascade four-compartment five-rate-constant model is generally resolvable from PET data obtained in normal humans. This is unequivocally so in the basal state, while under an insulin stimulation of 1 mU · kg^{−1} · min^{−1}, we observed that in two very insulin-sensitive subjects it was possible to reliably resolve the 5K model only up to the transmembrane transport step; i.e., the phosphorylation parameter was numerically nonidentifiable. In other words, the 5K model collapses in these two cases into a 3K model, but it is worth anticipating (see below) that the parameters of this collapsed 5K model take on a physiological meaning different from that classically accepted for the 3K model. An interesting finding of SA is that the results tend not to support the other classic model of the literature, i.e., the 4K model. The fact that a 4K model has been numerically quantified in some reports (15) is not surprising, since in our hands an even richer compartmental structure is resolvable from the data. What is relevant here is that SA is not supporting the structure of the 4K model; i.e., the analysis predicts a negligible dephosphorylation rate constant in agreement with experimental reports indicating a negligible glucose-6-phosphatase activity in skeletal muscle (17).

### 5K Model

The 5K model of Fig. 1 allows a physiologically sound description of the major kinetic events of [^{18}F]FDG in human skeletal muscle. The novelty of the model lies in its ability to segregate plasma and extracellular and intracellular space, thus allowing a description of extracellular (*K*
_{1} and*k*
_{2}) and transmembrane (*k*
_{3}and *k*
_{4}) transport as well as intracellular phosphorylation (*k*
_{5}). The novel explicit description of the extracellular space of the 5K model relaxes the critical assumption of the 3K and 4K models that interstitial and intracellular concentrations are nearly in equilibrium at all times. This assumption is on firm ground in the brain, since capillary endothelium forms a tight blood-brain barrier (12), but is not tenable for skeletal muscle, where transcapillary transport occurs by diffusion. The 5K model, with its interstitial compartment, is also able to account for the large arteriovenous gradients present during insulin stimulation (C_{i} prediction, Fig. 5), at variance with the 3K and 4K models, which require no (a negligible) gradient.

*K*
_{1}, which describes delivery of [^{18}F]FDG from plasma to extracellular space, is related to tissue perfusion, and in fact its value is not significantly different from blood flow estimated from [^{15}O]H_{2}O in basal (0.026 ± 0.004 vs. 0.023 ± 0.006 ml · ml^{−1} · min^{−1}) and insulin-stimulated (0.031 ± 0.008 vs. 0.031 ± 0.007 ml · ml^{−1} · min^{−1}) states. Insulin does not appear to have an appreciable effect on*K*
_{1} (or on blood flow) in the present experimental situation. The parameter *k*
_{3}represents transport of [^{18}F]FDG into the cell and is always resolvable from the data. Insulin significantly stimulates*k*
_{3} (0.049 ± 0.007 vs. 0.151 ± 0.011,*P* < 0.02). The parameter *k*
_{5}represents intracellular phosphorylation; it is always possible to estimate *k*
_{5} reliably in the basal state, while during insulin stimulation, it was reliably estimated in four of the six subjects. When it is resolvable, *k*
_{5} appears to be significantly stimulated by insulin (0.032 ± 0.002 vs. 0.069 ± 0.007, *P* < 0.02). We speculate that*k*
_{5} was not resolvable in *subjects 7*and *8*, because *k*
_{5} tended to become very high and *k*
_{4} became very small, thus making it difficult to detect their reliably. This argument is supported by noting that *subjects 7* and *8* are among the most insulin-sensitive individuals and that a somewhat similar trend is also exhibited by the other most insulin-sensitive subject (*subject 9*; see Fig. 3 and Table 3, where their *K* is ≅0.01 ml · ml^{−1} · min^{−1}, which is almost twice the value of the other 3 subjects). It is worth remarking that the 5K model in *subjects 7* and *8* is still returning reliable kinetic information up to the transmembrane transport step. Finally, it was predicted that insulin has virtually no effect on the extracellular [^{18}F]FDG operational volume but almost doubles the intracellular operational volume. As a result, their ratio, which is the ratio of the intracellular to the extracellular [^{18}F]FDG masses, is increased by a factor of 2.6.

The parametric portrait provided by the 5K model is new, and its validation would require a comparative study with an independent technique for measuring in vivo [^{18}F]FDG transport and phosphorylation rate constants; e.g., one could apply the triple-tracer technique originally proposed for glucose in the forearm (29) to [^{18}F]FDG in leg skeletal muscle. However, the credibility of the numerical 5K model is on reasonably firm ground. The only independent knowledge available on the model parameters is for *K*
_{1}, which, as discussed above, well describes blood flow. As far as the other parameters are concerned, one can only speculate with caution, since we are measuring FDG rate constants (min^{−1}) in the leg muscle, while previous triple-tracer studies (29) were concerned with glucose rate constants (min^{−1}) in the forearm. With this premise, our FDG *k*
_{3} and*k*
_{5} [0.049 and 0.032 min^{−1} (basal), 0.151 and 0.069 min^{−1} (insulin), respectively] compare well with glucose *k*
_{in} and*k*
_{met} of another study (29) [0.063 and 0.024 min^{−1} (basal), 0.156 and 0.107 min^{−1} (insulin)]. Even more speculative is the comparison of the operational extracellular and intracellular volumes, since the triple-tracer volumes (29) are true volumes; however, the trend is similar, since insulin only increases the intracellular volume. Of interest is also the 5K model prediction that insulin increases the ratio of intracellular to extracellular FDG masses by a factor of ∼2.6. This result is different from that observed previously (4, 29), where a twofold increase was observed in the ratio of intracellular to extracellular volumes but not in the ratio of the masses. However, we are confident in the credibility of the model, although further studies are certainly desirable to enrich the single-tracer PET portrait.

### Rate-Limiting Steps

The ability of the 5K model to provide a clear segregation of transmembrane [^{18}F]FDG transport and phosphorylation allows one to infer which of these processes is the rate-limiting step of [^{18}F]FDG kinetics in human skeletal muscle. The picture is obviously articulated, and there is no single rate-limiting step; i.e., control is not concentrated on a single step but is, rather, distributed between the transport and phosphorylation steps. Following the metabolic network control theories developed independently elsewhere (11, 13), one can calculate from the model the control coefficients of transmembrane transport (*C*
^{T}) and phosphorylation (*C*
^{P}) as
Equation 19
Equation 20The values of the control coefficients are shown in Fig.6. The results indicate that while, in the basal state, transport ([^{18}F]FDG transporters) and phosphorylation (hexokinase) contribute almost equally to the control of [^{18}F]FDG kinetics in human skeletal muscle, hyperinsulinemia shifts the main site of control to the transport step.

### 3K Model Revisited

The 3K model of Fig. 2 is the most widely used description in PET studies of [^{18}F]FDG kinetics in the human skeletal muscle (14, 30, 34). This is the major reason why we have analyzed our data also with this model. It is important to use the richer kinetic picture provided by the 5K model to revisit the simpler 3K model structure to gain insight into the physiological meaning of its parameters; in addition, there is the need to reconcile the two 5K and 3K kinetic pictures in the two cases where the 5K structure collapsed into the 3K model. Before specifically addressing these issues, it is worth noting that our 3K model results (Table 4) are in agreement with those recently reported by Kelley and co-workers (14, 15), whose dose-response studies are the only ones with which we can compare our basal and 1 mU · kg^{−1} · min^{−1} euglycemic insulin-clamp studies. Comparing the 5K with the 3K model is not straightforward, but here one is helped by the striking similarity of the 3K model parameter *K*′_{1} to the*K*
_{1} of the 5K model and also to the [^{15}O]H_{2}O blood flow value in the basal {0.019 ± 0.002 (*K*′_{1}) vs. 0.026 ± 0.004 (*K*
_{1}) vs. 0.023 ± 0.006 ml · ml^{−1} · min^{−1}([^{15}O]H_{2}O)} and insulin-stimulated {0.021 ± 0.003 (*K*′_{1}) vs. 0.031 ± 0.008 (*K*
_{1}) vs. 0.031 ± 0.007 ml · ml^{−1} · min^{−1}([^{15}O]H_{2}O)} states. These values support a tissue perfusion meaning of the 3K model parameter*K*′_{1}, rather than its believed transmembrane transport meaning. For the remaining two rate constants,*k*′_{2} and*k*′_{3}, it is impossible to find a one-to-one correspondence with *k*
_{2},*k*
_{3}, *k*
_{4}, and*k*
_{5}, since their values in basal and insulin studies are different. One is tempted to speculate that the 3K model rate constant *k*′_{3} is an aggregated parameter of the two 5K model rate constants representing inward transmembrane transport (*k*
_{3}) and intracellular phosphorylation (*k*
_{5}) and that the 3K rate constant *k*′_{2} is an aggregated parameter of the two 5K model rate constants describing outward transmembrane transport (*k*
_{4}) and transport from extracellular space back to plasma (*k*
_{2}). This would question the currently accepted notion that the 3K model parameters*K*′_{1} and *k*′_{2}represent inward and outward transmembrane transport, respectively, and*k*′_{3} represents intracellular phosphorylation. The likely aggregated nature of*k*′_{2} and *k*′_{3}also bears a less clear-cut physiological picture: the only significant effect of insulin is the stimulation of*k*′_{3}, but if the above revisitation holds, we are left with the impossibility of segregating the individual effects on transport and phosphorylation.

The above interpretative framework is further elucidated by examination of the insulin studies of *subjects 7* and*8*, where the 5K model collapses into a 3K structure. One expects from theory that when the 5K model parameters of outward transmembrane transport (*k*
_{4}) and phosphorylation (*k*
_{5}) are not resolvable, the 3K model parameters*k*′_{2} and *k*′_{3}would take on the values of 5K model rate constants*k*
_{2} and *k*
_{3}. This is exactly what is happening, since one can note in *subjects 7*and *8* a remarkable similarity between*K*′_{1} and *K*
_{1},*k*′_{2} and *k*
_{2},*k*′_{3} and *k*
_{3}.

In conclusion, a new model of [^{18}F]FDG kinetics in human skeletal muscle has been proposed with plasma and extracellular and intracellular space as compartments with five rate constants describing the individual steps of the kinetics, from plasma to extracellular space to transmembrane transport to intracellular phosphorylation. This model provides a rich kinetic picture, including a perfusion parameter reflecting essentially blood flow and parameters describing transport into and out of the cell membrane and intracellular phosphorylation. Use of the 5K model framework has clarified the physiological meaning of the classical 3K model parameters. The 5K model provides insight into insulin control of metabolism by predicting that transport and phosphorylation are stimulated by insulin, with control metabolic theory showing that hyperinsulinemia shifts the main site of control to the transport step. This first report is very encouraging, but further studies are necessary to define the domain of validity of the model and to ascertain its power also in studying various physiopathological states such as obesity and diabetes.

## Acknowledgments

This work was supported by European Community Project BMH4-97-2726 and by grants from the Yrjö Jahnsson Foundation (P. Peltoniemi), the Novo Nordisk Foundation (P. Nuutila), and the Academy of Finland (P. Nuutila).

## Appendix

To derive the 5K model in terms of concentration, we start by formulating the model in terms of mass. The mass representation is shown in Fig. 7. The mass balance equations are
Equation A1where q_{p}, q_{x}, and q_{e}represent the amount of [^{18}F]FDG in plasma and extracellular and intracellular space, respectively, and q_{m}is the amount of phosphorylated [^{18}F]FDG in the intracellular space (notation has been chosen to maintain the traditional meaning in the PET literature for q_{e} and q_{m}). Denoting with V_{p}, V_{x}, and V_{c} capillary plasma and extracellular and intracellular volume, respectively, one has
Equation A2Dividing by V_{c} and defining C_{i}(*t*) = (V_{x}/V_{c})C_{x}(*t*), one can write the mass balance equations in terms of concentration
Equation A3The total amount measured by PET, q(t), is the summation of the amounts present in the vascular, extracellular, and intracellular space of the ROI. By expressing also q(t) in terms of concentration, i.e., q(*t*) = VC(*t*), where V is the total volume and C the concentration, one has
Equation A4The total volume V can be written as
Equation A5where *a* = V_{x}/V_{c}. Thus
Equation A6Substituting *Eq. EA6
* into *Eq. EA4
*, one has
Equation A7and thus
Equation A8We show below that the model defined by *Eqs. EA3
* and *
EA8
* is a priori nonidentifiable. We use the transfer function method (7) to analyze whether the model is a priori uniquely identifiable. By taking Laplace transforms of *Eq.EA3
* and rearranging one has
Equation A9
Solving for C_{i}, C_{e}, and C_{m}
Equation A10
Thus the Laplace transform of C can be written as
Equation A11The exhaustive summary of the model is
Equation A12
where φ_{1},…,φ_{6} are the known observational parameters. The model is not identifiable, because it is not possible to solve for all nine unknown model parameters. However, if one defines
Equation A13the exhaustive summary becomes
Equation A14
It is easy to show that all six model parameters,*K*
_{1}, *k*
_{2},*k*
_{3}, *k*
_{4},*k*
_{5}, and V_{b}, are now a priori uniquely identifiable. In fact, the solution of the system of *Eq.EA14
* gives
Equation A15
The reparameterized uniquely identifiable model is thus
Equation A16
Equation A17This is the 5K model shown in Fig. 1 and described by *Eqs.3
* and *
4
*.

## Appendix

If a tracer steady state is assumed, the metabolic rate (MR) of [^{18}F]FDG is
Equation B1By solving the steady-state model equations (*Eq. 3
*), one has
Equation B2
and thus
Equation B3Substituting *Eq. EB2
* into *Eq. EB1
* one has
Equation B4and thus fractional [^{18}F]FDG uptake, which is given by
Equation B5

## Footnotes

Address for reprint requests and other correspondence: C. Cobelli, Dept. of Electronics and Informatics, University of Padova, Via Gradenigo 6/A, Padua 35131, Italy (E-mail:cobelli{at}dei.unipd.it).

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