Kinetic modeling of [18F]FDG in skeletal muscle by
PET: a four-compartment five-rate-constant model
A.
Bertoldo1,
P.
Peltoniemi2,3,
V.
Oikonen2,
J.
Knuuti2,
P.
Nuutila2,3, and
C.
Cobelli1
1 Department of Electronics and Informatics, University of
Padova, Padua 35131, Italy; and 2 Turku PET Centre and
3 Department of Medicine, University of Turku, F-20520 Turku,
Finland
 |
ABSTRACT |
Various modeling
strategies have been developed to convert regional
[18F]fluorodeoxyglucose ([18F]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 [18F]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
[18F]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
 |
INTRODUCTION |
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-[3H]mannitol), a transportable but not
metabolizable (e.g.,
[methyl-3-O-14C]glucose), and a
metabolizable (e.g., D-[3H]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
[18F]fluorodeoxyglucose ([18F]FDG), a
glucose analog. By interpreting [18F]FDG plasma
concentration and tissue activity curves with compartmental models with
three (3K model) or four (4K model) rate constants, one arrives at
[18F]FDG transport and phosphorylation. In contrast to
the triple-tracer dilution technique, which provides transport and
phosphorylation fluxes of glucose, the PET [18F]FDG
technique can provide only rate constants of transport and phosphorylation of [18F]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 [18F]FDG individual unit
processes exhibit different affinity. However, [18F]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 [18F]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 [18F]FDG PET studies performed in normal humans
in basal and insulin-stimulated states, we will study which is the most
accurate model for describing [18F]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/m2). 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 [15O]H2O and [18F]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 68Ge.
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 [18F]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.
[18F]FDG Studies
For [18F]FDG studies, 0.16-0.28 GBq of
[18F]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 [18F]FDG exceeded 98%.
Arterial blood samples for measurement of plasma radioactivity were
withdrawn as previously described (21).
[15O]H2O Studies
For measurement of blood flow, 1.2-1.5 GBq of
[15O]H2O 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 [15O]H2O 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 [18F]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 [18F]FDG
kinetics in the skeletal muscle. Briefly, if the impulse response
[h(t)] of the system is written as
|
(1)
|
with
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 [Cp(t)] plus a term taking into
account the vascular component present in the ROI
|
(2)
|
where Vb is the fraction of the total volume
occupied by the blood pool and Cb(t) is the
arterial blood tracer concentration calculated as follows:
Cb(t) = Cp(
)(1
0.3H)
(8), where H is the subject's hematocrit. The method
estimates the number M of nonzero values of
j that (together with the corresponding
j) best describes the data providing, at the
same time, useful insight into the system structure. In fact, the
number of amplitudes
j corresponding to the
eigenvalues
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
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
corresponding to the zero eigenvalue (
= 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
1,
1,
2,
2, and Vb; then one tries
M = 3 and estimates
1,
1,
2,
2,
3,
3,
and Vb and so on. To select the best model, parsimony criteria are used (see Parameter Estimation).
Compartmental modeling of [18F]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 [18F]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 [18F]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 A)
|
(3)
|
|
(4)
|
where Cp is [18F]FDG plasma arterial
concentration, Ci is extracellular concentration of
[18F]FDG normalized to intracellular volume,
Ce is [18F]FDG intracellular concentration,
Cm is [18F]FDG 6-phosphate
([18F]FDG-6-P) intracellular concentration, C
is total 18F activity concentration in the ROI,
K1
(ml · ml
1 · min
1) and
k2 (min
1) are exchange parameters
between plasma and extracellular space, k3
(min
1) and k4 (min
1)
are transport parameters into and out of the cell, and
k5 (min
1) is the phosphorylation
parameter (Vb and Cb have the same meaning as
in Eq. 2).

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Fig. 1.
Four-compartment 5-rate-constant (5K) model for measuring
the metabolic rate of glucose with
[18F]fluorodeoxyglucose ([18F]FDG).
Cp, arterial plasma [18F]FDG concentration;
Ci, extracellular concentration of [18F]FDG
normalized to intracellular volume; Ce,
[18F]FDG intracellular concentration; Cm,
[18F]FDG 6-phosphate intracellular concentration;
K1
(ml · ml 1 · min 1) and
k2 (min 1), exchange parameters
between plasma and extracellular space; k3
(min 1) and k4
(min 1), transport parameters into and out of the cell;
k5 (min 1), phosphorylation
parameter.
|
|

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Fig. 2.
Three-compartment 3-rate-constant (3K) model for
measuring the metabolic rate of glucose with
[18F]FDG. Cp, arterial plasma
[18F]FDG concentration; C'e,
[18F]FDG intracellular concentration;
C'm, [18F]FDG 6-phosphate
intracellular concentration; K'1
(ml · ml 1 · min 1) and
k'2 (min 1),
[18F]FDG transport from plasma to intracellular space and
back, respectively; k'3
(min 1), [18F]FDG phosphorylation.
|
|
All six model parameters, K1,
k2, k3,
k4, k5, and
Vb, are a priori uniquely identifiable (7)
(see APPENDIX A).
From the model, one can calculate the fractional uptake of
[18F]FDG, K
(ml · ml
1 · min
1) (see
APPENDIX B)
|
(5)
|
In 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
|
(6)
|
Of interest is the ratio
V
/V
, which can be shown to be the
ratio of "true" [18F]FDG mass in the intracellular
space to true [18F]FDG mass in the extracellular space.
In fact, one has
|
(7)
|
3K AND 4K MODELS.
The 3K (31) and 4K (24) models were
originally proposed for [18F]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
|
(8)
|
|
(9)
|
where Cp is [18F]FDG plasma arterial
concentration, C'e is [18F]FDG
intracellular concentration, C'm is
[18F]FDG-6-P intracellular concentration, C is
total 18F activity concentration in the ROI,
K'1
(ml · ml
1 · min
1) and
k'2 (min
1) describe
[18F]FDG transport from plasma to tissue and back,
respectively, k'3 (min
1) is
[18F]FDG phosphorylation, and Vb and
Cb have the same meaning as in Eq. 4.
All four model parameters, K'1,
k'2, k'3,
and Vb, are a priori uniquely identifiable
(7). The model allows calculation of the fractional uptake
of [18F]FDG as
|
(10)
|
The 4K model is an extension of the 3K model by allowing
dephosphorylation of [18F]FDG-6-P to be
present
|
(11)
|
where the meaning of variables and rate constants is the same as
for the 3K model, with k'4 denoting
[18F]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 (Cobs) was described
as
|
(12)
|
where e(tj) is the measurement error at
time k1, assumed to be independent, Gaussian,
zero mean and with a variance (
) given by (3)
|
(13)
|
where
tj is the length of the scanning
interval relative to Cobs(tj) and
is an unknown proportionality constant to be estimated a posteriori
(7) as
|
(14)
|
WRSS(
) is the weighted residual sum of squares
evaluated at the minimum, i.e., for p equal to the estimated
|
(15)
|
where wj is the weight of the
jth datum [wj =
tj/Cobs(tj)],
tj is the midscan time, 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 tj is defined as
|
(16)
|
Residuals 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
|
(17)
|
|
(18)
|
The 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 |
[18F]FDG Kinetics
Mean plasma and tissue activity concentrations of
[18F]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.

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Fig. 3.
Individual [18F]FDG plasma and tissue
time-activity (normalized to dose) curves in basal state (A,
subjects 1-5) and during insulin stimulation (B,
subjects 6-11).
|
|
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. Table 1 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 [18F]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 [18F]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
[18F]FDG exchanging reversibly with plasma plus a
compartment where [18F]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 [18F]FDG tissue
pool and with [18F]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 Table
3, together with their precision. Mean
weighted residuals are shown in Fig. 4
for 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 Vb
value in subject 5; in subjects 2 and
3, Vb was not resolvable. In the insulin
studies, parameter precision generally degraded with respect to the
basal value but was poor only for k4 in
subjects 7-9 and for k5 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, Vb was hardly resolvable. K was
estimated very precisely because of its macroscopic nature. The 5K
model results indicate that insulin does not affect
K1 and k2, but it
significantly increases k3 (P < 0.02). It also increases k5 (P < 0.02) and decreases k4 (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).

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Fig. 4.
Weighted residuals, i.e., difference between data and
model predictions divided by the measurement error standard deviation
averaged over the subjects, of 5K and 3K models in basal (A)
and insulin stimulation (B) studies.
|
|
3K Model
Parameter estimates are shown in Table
4. 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
K1: 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 [18F]FDG extracellular and
[18F]FDG and [18F]FDG-6-P
intracellular concentrations are shown in Fig.
5.

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Fig. 5.
5K mean model prediction of concentrations,
Ci, Ce, and Cm (normalized to
dose), in basal (A) and insulin stimulation (B)
studies.
|
|
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 [18F]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 [18F]FDG across the cell membrane is not
sufficiently fast compared with delivery into the extracellular space
to allow lumping of extracellular space into the [18F]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
[18F]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
[18F]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 [18F]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 [18F]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
[18F]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 [18F]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 (K1 and
k2) and transmembrane (k3
and k4) transport as well as intracellular phosphorylation (k5). 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 (Ci prediction, Fig. 5), at variance
with the 3K and 4K models, which require no (a negligible) gradient.
K1, which describes delivery of
[18F]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
[15O]H2O 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
K1 (or on blood flow) in the present
experimental situation. The parameter k3
represents transport of [18F]FDG into the cell and is
always resolvable from the data. Insulin significantly stimulates
k3 (0.049 ± 0.007 vs. 0.151 ± 0.011, P < 0.02). The parameter k5
represents intracellular phosphorylation; it is always possible to
estimate k5 reliably in the basal state, while
during insulin stimulation, it was reliably estimated in four of the
six subjects. When it is resolvable, k5 appears
to be significantly stimulated by insulin (0.032 ± 0.002 vs.
0.069 ± 0.007, P < 0.02). We speculate that
k5 was not resolvable in subjects 7 and 8, because k5 tended to become
very high and k4 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 [18F]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 [18F]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 [18F]FDG transport and
phosphorylation rate constants; e.g., one could apply the triple-tracer
technique originally proposed for glucose in the forearm
(29) to [18F]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 K1, 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 k3 and
k5 [0.049 and 0.032 min
1 (basal),
0.151 and 0.069 min
1 (insulin), respectively] compare
well with glucose kin and
kmet 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 [18F]FDG transport and phosphorylation
allows one to infer which of these processes is the rate-limiting step
of [18F]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
(CT) and phosphorylation
(CP) as
|
(19)
|
|
(20)
|
The values of the control coefficients are shown in Fig.
6. The results indicate that while, in
the basal state, transport ([18F]FDG transporters) and
phosphorylation (hexokinase) contribute almost equally to the control
of [18F]FDG kinetics in human skeletal muscle,
hyperinsulinemia shifts the main site of control to the transport step.

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|
Fig. 6.
Control coefficients of
transmembrane transport (A) and phosphorylation
(B) in basal and insulin stimulation studies.
|
|
3K Model Revisited
The 3K model of Fig. 2 is the most widely used
description in PET studies of [18F]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
K1 of the 5K model and also to the
[15O]H2O blood flow value in the basal
{0.019 ± 0.002 (K'1) vs.
0.026 ± 0.004 (K1) vs. 0.023 ± 0.006 ml · ml
1 · min
1
([15O]H2O)} and insulin-stimulated
{0.021 ± 0.003 (K'1) vs.
0.031 ± 0.008 (K1) vs. 0.031 ± 0.007 ml · ml
1 · min
1
([15O]H2O)} 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 k2,
k3, k4, and
k5, 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 (k3) and intracellular
phosphorylation (k5) and that the 3K rate
constant k'2 is an aggregated parameter
of the two 5K model rate constants describing outward transmembrane
transport (k4) and transport from extracellular
space back to plasma (k2). 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 (k4) and phosphorylation
(k5) are not resolvable, the 3K model parameters
k'2 and k'3
would take on the values of 5K model rate constants
k2 and k3. This is
exactly what is happening, since one can note in subjects 7 and 8 a remarkable similarity between K'1 and K1,
k'2 and k2,
k'3 and k3.
In conclusion, a new model of [18F]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.
 |
APPENDIX A |
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
|
(A1)
|
where qp, qx, and qe
represent the amount of [18F]FDG in plasma and
extracellular and intracellular space, respectively, and qm
is the amount of phosphorylated [18F]FDG in the
intracellular space (notation has been chosen to maintain the
traditional meaning in the PET literature for qe and
qm). Denoting with Vp, Vx, and
Vc capillary plasma and extracellular and intracellular
volume, respectively, one has
|
(A2)
|
Dividing by Vc and defining
Ci(t) = (Vx/Vc)Cx(t), one can
write the mass balance equations in terms of concentration
|
(A3)
|
The 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
|
(A4)
|
The total volume V can be written as
|
(A5)
|
where a = Vx/Vc. Thus
|
(A6)
|
Substituting Eq. A6 into Eq. A4, one has
|
(A7)
|
and thus
|
(A8)
|
We show below that the model defined by Eqs. A3 and A8 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. A3 and rearranging one has
|
(A9)
|
Solving for Ci, Ce, and
Cm
|
(A10)
|
Thus the Laplace transform of C can be written as
|
(A11)
|
The exhaustive summary of the model is
|
(A12)
|
where
1,...,
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
|
(A13)
|
the exhaustive summary becomes
|
(A14)
|
It is easy to show that all six model parameters,
K1, k2,
k3, k4,
k5, and Vb, are now a priori
uniquely identifiable. In fact, the solution of the system of Eq. A14 gives
|
(A15)
|
The reparameterized uniquely identifiable model is thus
|
(A16)
|
|
(A17)
|
This is the 5K model shown in Fig. 1 and described by Eqs.
3 and 4.

View larger version (15K):
[in this window]
[in a new window]
|
Fig. 7.
Top: 5K model in
terms of mass. qp, qec, and qe,
[18F]FDG masses in capillary (with capillary and arterial
concentrations assumed to be in equilibrium) and extracellular and
intracellular spaces; qm, phosphorylated amount of
[18F]FDG in the intracellular space;
k1, k2,
k3, k4, and
k5, rate constants (min 1) of
[18F]FDG transport into and out of the interstitium,
[18F]FDG transport into and out of the cell, and
[18F]FDG phosphorylation, respectively.
Bottom: representation of the model with graphics normally
used in positron emission tomography literature.
|
|
 |
APPENDIX B |
If a tracer steady state is assumed, the metabolic rate (MR) of
[18F]FDG is
|
(B1)
|
By solving the steady-state model equations (Eq. 3),
one has
|
(B2)
|
and thus
|
(B3)
|
Substituting Eq. B2 into Eq. B1 one has
|
(B4)
|
and thus fractional [18F]FDG uptake, which is
given by
|
(B5)
|
 |
ACKNOWLEDGEMENTS |
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).
 |
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).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 7 June 2000; accepted in final form 20 March 2001.
 |
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