Vol. 275, Issue 2, H668-H679, August 1998
Estimating glucose metabolism using glucose analogs and two
tracer kinetic models in isolated rabbit heart
Robert C.
Marshall1,2,3,
Patricia
Powers-Risius1,
Ronald H.
Huesman1,
Bryan W.
Reutter1,
Scott E.
Taylor1,
Heidi E.
Maurer1,
Michelle K.
Huesman1, and
Thomas F.
Budinger1
1 Lawrence Berkeley National
Laboratory, University of California, Center for Functional Imaging,
Berkeley 94720; 2 Martinez
Veterans Affairs, Northern California Health Care System, Martinez
94553; and 3 University of
California at Davis, School of Medicine, Davis, California 95616
 |
ABSTRACT |
The purpose of this
investigation was to 1) evaluate the
relative accuracy of the Sokoloff and Patlak tracer kinetic models in
estimating glucose metabolic rate (GMR) in the presence and absence of
insulin; 2) evaluate the effect of
nutritional state on the lumped constant (LC); and
3) compare the kinetics of
2-fluoro-2-deoxy-D-[14C]glucose
(FDG) and
2-deoxy-D-[3H]glucose
(DG) membrane transport and phosphorylation. The experimental preparation was the isolated, red blood cell-albumin-perfused rabbit
heart. Our results showed that both tracer kinetic models provided GMR
estimates that correlated well with the Fick method (for FDG,
R = 0.84 and 0.91 for the Sokoloff and
Patlak models, respectively); nutritional state did not affect the LC;
and FDG and DG have different transport and/or phosphorylation
parameters. We also observed that 1)
the addition of a fourth compartment to the Sokoloff model reduced the
mean squared error between measured and modeled data by a factor of
7.4; 2) a longer time (21.8 min) was
required to obtain a linear phase of the Patlak plot than is allowed in
clinical studies; and 3) accurate
GMR estimates were obtained only by using different LCs reflecting
insulin's presence or absence. Our results indicate potential sources
of error in the use of FDG and positron emission tomography to quantify GMR in patients.
fluorodeoxyglucose; deoxyglucose; positron emission tomography
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INTRODUCTION |
ALTHOUGH POSITRON EMISSION tomography (PET) and
2-[18F]fluoro-2-deoxy-D-glucose
are in general clinical use (43), the question remains as to how well
this approach quantifies myocardial glucose metabolism. Accurate
quantification of glucose metabolism with PET and FDG requires an
appropriate tracer kinetic model. Evaluation of the accuracy of a
tracer kinetic model in patients is difficult because of limitations in
spatial and temporal resolution with PET (5), the need to be invasive
to independently measure glucose consumption with the Fick principle,
and the potential for uncontrolled factors affecting glucose
and/or FDG uptake and metabolism. In this investigation, we
evaluated, in a well-controlled in vitro environment, two tracer
kinetic models that have been used with PET and FDG in patients (13,
14, 19, 24) and compared the results with the Fick-determined glucose
metabolic rate (GMR).
The first of these tracer kinetic models was originally developed by
Sokoloff et al. (41) for analysis of glucose utilization in the brain
using
2-deoxy-D-[14C]glucose
and autoradiography. This approach employs a three-compartment model to
assess tracer transport from blood into the cell and phosphorylation in
the cytosol. Patlak and co-workers (3, 33, 34) developed a second
model, also for use in the brain, which uses an unrestricted number of
reversible compartments to represent tracer transport and one
irreversible phosphorylation compartment. In contrast to the Sokoloff
model in which it is necessary to estimate transport and
phosphorylation parameters from kinetic analysis of the entire tissue
FDG uptake curve, the Patlak technique estimates GMR from the
"steady-state" portion of FDG myocardial and blood curves using a
graphical approach. Although there are potential errors related to the
difficulty in attaining a true tracer steady state in vivo (23), the
Patlak graphical analysis has been used preferentially in recent
clinical studies evaluating glucose metabolism with FDG and PET (14,
19, 24). In the study by Gambhir et al. (13), both tracer kinetic
models were employed, but the results were not validated against the
Fick principle. The relative accuracy of these two tracer kinetic
models in the heart has not been evaluated and compared against an
independent measure of glucose consumption.
To quantify GMR, both the Sokoloff and Patlak models depend on a term
known as the lumped constant to correct for differences in the kinetics
of FDG and glucose membrane transport and phosphorylation. Two recent
reports indicate that the value of the lumped constant changes in the
presence vs. the absence of insulin (16, 31). However, neither of these
studies quantified glucose metabolism using a tracer kinetic model:
both investigations employed the crystalloid-perfused working rat
heart, which has a high coronary flow rate, making it difficult to
simultaneously perform kinetic analysis of FDG accumulation curves to
estimate glucose consumption and measure arteriovenous differences of
both tracer and glucose to assess the lumped constant. There have been
no reports demonstrating that the use of different, condition-specific
lumped constants results in improved accuracy of GMR estimates using
either tracer kinetic model.
The first goal of this investigation was to evaluate the relative
accuracy of the Sokoloff and Patlak methodologies in estimating myocardial GMR in the presence and absence of insulin using different measured lumped constant values. Second, feeding vs. fasting has been
reported to mask the effect of insulin on glucose metabolism in
isolated hearts (39, 40). We therefore compared results obtained in the
presence of insulin with those observed in its absence in hearts from
fed vs. fasted rabbits to determine if the nutritional state of the
animal independently affected lumped constant values. Third, because
the kinetics of FDG vs. DG membrane transport and phosphorylation have
not been compared in the heart, we assessed possible differences in
computed transport and phosphorylation parameters between these two
glucose analogs.
The experimental preparation was the isolated, isovolumic, retrograde
red blood cell (RBC) plus albumin perfused rabbit heart (26, 27), which
has more physiological perfusion rates than crystalloid-perfused
hearts. Lumped constants were determined with and without insulin in
the perfusate in hearts from fed and fasted rabbits using a
model-independent approach. The GMRs estimated using the Sokoloff and
Patlak tracer kinetic models and average lumped constant values
determined to be appropriate for each experimental condition were
compared with results obtained with the Fick principle. Our results
indicate that 1) the original
Sokoloff model needed to be modified to include a fourth compartment to
account for tracer delivery and distribution in the heart;
2) the modified, four-compartment
model derived from Sokoloff and the Patlak graphical approach yielded
comparably accurate estimates of myocardial GMR in the presence and
absence of insulin provided the appropriate lumped constant relevant to
each condition was used; 3) the
nutritional state of the animal had no effect on the lumped constant;
and 4) FDG and DG have different
lumped constant and combined rate constant values, reflecting the fact
that fluorine substitution in 2-deoxy-D-glucose has an
effect on membrane transport and/or phosphorylation.
| |
METHODS |
Experimental Preparation
Preparation of isovolumic, retrograde RBC-albumin-perfused rabbit
hearts was similar to that reported previously (25-27). All procedures were done in accordance with institutional guidelines for
animal research. Male New Zealand White rabbits (3.5-4.5 kg) were
given 4,000 U heparin sodium (Upjohn, Kalamazoo, MI) and 250 mg
pentobarbital sodium (Abbott, North Chicago, IL) via an ear vein. The
heart was immediately excised through a median sternotomy, arrested in
ice-cold buffer, and rapidly attached to a cannula to allow retrograde
perfusion. An apical drain was inserted into the left ventricle (LV).
The atrioventricular node was crushed to allow controlled stimulation,
and a fluid-filled latex balloon connected to a Gould-Statham P23ID
pressure transducer (Gould, Oxnard, CA) was inserted into the LV. The
balloon was inflated to maintain diastolic pressure at 8-10 mmHg.
A coronary venous sampling catheter and needle thermistor (Bailey
Instrument, Saddlebrook, NJ) were inserted into the right ventricle.
The venae cavae and pulmonary artery were ligated so all coronary
venous drainage flowed out of the sampling catheter. Stimulating
electrodes connected to a Grass SD44 stimulator were placed against the
left and right ventricles, and 4-V, 4-ms stimuli were delivered at a
rate of 180 min
1.
Temperature was maintained between 36 and 38°C with a
water-jacketed heating coil and heart chamber. Coronary flow was held
constant with a peristaltic pump (Rainin Instrument, Woburn, MA). The
RBC-albumin perfusate was not recirculated. The rate of coronary blood
flow was measured as the volume of blood discharged per minute from the
coronary venous sampling catheter. Control plasma flow rate was ~1.2
ml · min1 · g
LV wet wt1. The perfusion
line included an in-line 20-µm blood transfusion filter (Statcorp,
Columbia, TN) to remove RBC aggregates.
Hearts were perfused with a modified Tyrode solution containing
oxygenated bovine RBCs and 22 g/l bovine serum albumin (fraction V,
fatty-acid free; Sigma Chemical, St. Louis, MO). The bovine serum
albumin was dialyzed overnight at 4°C against buffer and filtered
through a 0.8-µm Millipore filter. Bovine RBCs were separated from
whole blood by centrifugation in 250 ml polyethylene bottles at 2,800 g for 20 min at 4°C and then were
washed, resuspended with oxygenated ice-cold buffer, and spun again;
this separation procedure was repeated four times. The specific
electrolyte concentrations of the buffer solution were (in mmol/l) 110 NaCl, 2.5 CaCl2, 6 KCl, 1 MgCl2, 0.435 NaH2PO4,
and 28 NaHCO3. The pH and oxygen tension were measured on the RBC-albumin perfusate using an IRMA Blood
Gas Analyzer (Diametrics Medical, St. Paul, MN). The mean ± SD pH
value was 7.38 ± 0.04, and the
PO2 value was 303 ± 72 mmHg. The
concentration of RBCs in the perfusate buffer was adjusted to a
hematocrit of 0.17-0.25. The flask containing the RBC-albumin
perfusate was gassed with a mixture of 98%
O2-2% CO2 during the experiment.
Hearts from rabbits that were fed ad libitum or fasted overnight (~18
h) were studied in the presence or absence of insulin in the perfusate.
There were three experimental groups:
1) hearts from fed rabbits perfused
with insulin (5 mU/ml); 2) hearts
from fed rabbits perfused without insulin; and
3) hearts from fasted rabbits
perfused without insulin. A high insulin concentration was used to
ensure a maximum insulin effect, since Taegtmeyer et al. (42) observed
that insulin binds to an unpredictable extent to glassware and tubing.
Plasma glucose concentration in hearts perfused with insulin was 5.3 ± 0.5 mmol/l (range, 4.5-6.5 mmol/l). Work by Ng et al. (31)
has shown that reducing perfusate glucose concentration in the presence
of insulin affects lumped constant values. To study the effect of
reducing perfusate glucose concentration in the absence of insulin, we
varied perfusate glucose concentration from 2.8 to 6.2 mmol/l (mean,
4.9 ± 1.3 mmol/l) in hearts from fed rabbits and 1.9 to 6.4 mmol/l
(mean, 3.1 ± 1.5 mmol/l) in hearts from fasted rabbits perfused
without insulin. To study the effect of flow on the lumped constant,
plasma flow was varied from ~0.5 to 1.5 ml · min1 · g
LV wet wt1 in all three
experimental groups.
Radiopharmaceuticals and Synthesis
of 131I-Labeled Albumin
Radioisotopes were purchased from the following sources:
2-fluoro-2-deoxy-D-[U-14C]glucose
(FDG) and
2-deoxy-D6-3H]glucose
(DG) from American Radiolabeled Chemicals (St. Louis, MO), and
131I from Du Pont-NEN Research
Products. Bovine serum albumin was labeled with
131I utilizing the IODO-GEN-based
protein iodination technique (29).
Experimental Protocol
After the heart was prepared, an equilibration period of at least 15 min preceded all experimental interventions. A heart was acceptable for
study if it developed at least 60 mmHg pressure (peak systolic 
diastolic). After equilibration, myocardial perfusion was gradually
changed to the experimental flow rate and subsequently maintained
constant throughout the remainder of the experiment. Constant infusion
of FDG and DG (~12 µCi of each isotope/l) was initiated 5-10
min after equilibration at the experimental flow rate and continued for
60 min. In some experiments, only FDG was infused. Isotope infusion was
initiated as a step function by employing two parallel perfusion
circuits (one with and one without isotope) and two, in-line, three-way
stopcocks placed just above the aortic cannula. Rapid venous sampling
(~7-15 s/sample, depending on the flow rate) from the right
ventricular cannula into microcentrifuge vials commenced with
radioisotope introduction and continued uninterrupted for 2-3 min.
The interval between samples was subsequently lengthened (from 15 to
240 s) until ~55-60 venous samples were taken in each experiment. Samples from the perfusate flask were taken at 4-min intervals ("arterial samples"). All samples were immediately
chilled and centrifuged in an Eppendorf microcentrifuge. Glucose and
lactate concentrations in the supernatant were assayed in duplicate
using a YSI Biochemistry Analyzer (model 2700; Yellow Springs
Instrument, Yellow Springs, OH).
At the end of an experiment, the LV was separated from the heart,
blotted dry, and weighed (wet weight). The LV was sliced into ~10
pieces, dried for 24 h at 90°C, and reweighed (dry weight).
Tissue FDG and DG Content
Tissue content of FDG or DG was computed as the summed product of
plasma flow, the arteriovenous concentration difference in plasma
14C or
3H activity, and sampling interval
at each sampling time
![A<SUB>T</SUB>(<IT>t<SUB>k</SUB></IT>) = <LIM><OP>∑</OP><LL><IT>j</IT>=1</LL><UL><IT>k</IT></UL></LIM> F[C<SUB>A</SUB> − C<SUB>V</SUB>(<IT>t</IT><SUB><IT>j</IT></SUB>)]&Dgr;<IT>t</IT><SUB><IT>j</IT></SUB>](/content/vol275/issue2/fulltext/H668/img001.gif)
|
(1)
|
where
AT(tk)
is the tissue FDG or DG content, F is plasma flow,
CA is the arterial FDG or DG
activity,
CV(tj)
is the venous FDG or DG activity, and
tj is the
sampling interval.
Lumped Constant
Lumped constants were computed as the steady-state extraction fraction
ratio of FDG to glucose or DG to glucose
![<FR><NU>{[C<SUB>A</SUB> − C<SUB>V</SUB>(<IT>t</IT>)]/C<SUB>A</SUB>}<SUB>analog</SUB></NU><DE>[(C<SUB>A</SUB> − C<SUB>V</SUB>)/C<SUB>A</SUB>]<SUB>glucose</SUB></DE></FR>](/content/vol275/issue2/fulltext/H668/img002.gif)
|
(2)
|
where
CV(t)
is venous tracer concentration as a function of time and
CV is constant venous glucose
concentration. In all experiments, the final steady-state extraction
fraction value for FDG and DG used to compute the lumped constant was
reached ~20-25 min after initiation of isotope infusion. This
method of determining the lumped constant is model independent and has
been used before by Sokoloff et al. (41) in the brain and by Ng et al.
(31) in the heart.
Patlak Graphical Analysis
As described by Patlak et al. (34), FDG-DG phosphorylation rate was
estimated by inspection of the asymptotic behavior of the plot of
tissue tracer content vs. integrated plasma tracer content after both
were normalized by arterial plasma activity, i.e.,
AT(t)/CA
vs.
, where
AT(t)
is tissue tracer content as a function of time. The linear portion of
the plot can be expressed by the equation
|
(3)
|
where
Ki
and
Vi
are the slope and ordinate intercept of the linear asymptote,
respectively.
Ki
is the influx constant and provides a measure of the rate of FDG-DG
phosphorylation.
Vi
is equal to or less than the reversible tissue FDG-DG distribution volume. To accurately estimate the FDG-DG phosphorylation rate from the
Patlak graph, tracer must have equilibrated between plasma and
reversible tissue regions before the linear phase of the graph used to
estimate the
Ki
(see DISCUSSION). Under the experimental conditions
employed here (i.e., constant isotope infusion), once tracer
equilibration has developed, it should continue through the end of the
experiment. Therefore,
Ki
and
Vi
were estimated as the slope and ordinate intercept of the straight line
fitted to the final "steady-state" portion of the graph starting
at a point when the curve first appeared linear (usually ~20 min
after isotope injection) through the end of the experiment (60 min). The lowest correlation coefficient was >0.99.
Four-Compartment Model
Figure 1 shows the four-compartment model
that was used to describe the kinetics of DG and FDG phosphorylation in
the isolated rabbit heart. The principles of this model are derived
from Sokoloff et al. (41). However, the form of the model and fitting
procedures are different from that originally described by Sokoloff.
The specific mathematical approach is outlined in the
APPENDIX. Briefly, the fractional utilization (FU) of
FDG-DG per unit flow was estimated as the asymptotic rate of isotope
accumulation in the fourth compartment. This number multiplied by flow
provides a combined rate constant, KFUR, that
estimates the fractional phosphorylation rate of FDG-DG.
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Fig. 1.
Schematic of four-compartment model.
Q2(t),
Q3(t),
and
Q4(t)
are the tissue compartment activity contents per unit flow as a
function of time. The fourth compartment represents the phosphorylated
state. Each k is a first-order rate
constant describing intercompartmental transport in the designated
direction. FDG-6-PO4,
fluoro-2-deoxy-D-[14C]glucose
6-phosphate; DG-6-PO4,
2-deoxy-D-[3H]glucose
6-phosphate;
CA(t),
arterial tracer concentration as a function of time.
|
|
GMR
Patlak graphical analysis and four-compartment
model.
Ki
and KFUR are
combined rate constants that provide a measure of the rate of FDG-DG
phosphorylation. To convert these estimates of FDG-DG phosphorylation
rate to GMR, the following relationship was used
|
(4)
|
where
K is either
Ki
or KFUR,
CA is the arterial glucose
concentration, and LC is the lumped constant. Lumped constant values for each experimental group were computed by averaging the results from
the corresponding individual experiments.
Fick GMR. GMR was computed
using the Fick equation, arterial and venous chemical glucose
concentrations, and plasma flow
|
(5)
|
The
GMR is expressed as micromoles per minute per gram LV wet weight.
Statistical Methods
Data are expressed as means ± SD. Statistical significance was
analyzed using both parametric and nonparametric methods. Paired observation tests (paired t-test and
Wilcoxon signed rank) were used to compare FDG vs. DG within each
experimental group; differences between experimental groups were
analyzed using an unpaired t-test and
Mann-Whitney U-test. In all cases, the
results using parametric and nonparametric methods were in agreement.
Reported P values were obtained using
the nonparametric approach, since these were consistently higher than
those obtained with the parametric method. A
P value <0.05 was considered
significant.
| |
RESULTS |
RBC Metabolism and FDG-DG Uptake
Lactate was present in arterial plasma samples due to RBC metabolism
occurring during both storage and the experiment. Lactate concentration
(mmol/l) averaged 0.11 ± 0.06 in hearts perfused with insulin, 0.13 ± 0.05 in hearts from fed rabbits perfused without insulin, and
0.11 ± 0.04 in hearts from fasted rabbits without insulin in the
perfusate (no statistically significant difference between groups;
smallest P value >0.15). These
lactate concentrations are more than two orders of magnitude below that shown to alter lumped constant values (16).
We evaluated FDG-DG uptake in bovine RBCs at 37°C by
1) incubating RBCs (0.20 hematocrit)
with FDG and DG for 15 min and acquiring samples at 5-min intervals and
2) pumping RBC-albumin perfusate containing FDG and DG through the perfusion apparatus (no heart attached) for up to 70 min at speeds ranging from 2 to 20 ml/min with
"venous" (exit drain) samples acquired at 5- to 10-min intervals. Collected samples were immediately chilled, centrifuged, and counted. No significant FDG and DG uptake into RBCs was observed. Because FDG
uptake in RBCs has been observed in vivo during PET studies (35), the
absence of demonstrable FDG or DG uptake in our experiments presumably
relates to species differences and to the fact that our RBCs were
harvested and stored at 6°C before in vitro use.
GMR
The Fick GMR
(µmol · min1 · g
LV wet wt1) was higher in
hearts from fed rabbits perfused with insulin (0.50 ± 0.12;
n = 18) than in hearts from fed
rabbits perfused without insulin (0.29 ± 0.11;
n = 12;
P < 0.001) and in hearts from fasted
rabbits perfused without insulin (0.15 ± 0.05;
n = 15;
P < 0.001; see Table
1). Comparing the two groups perfused
without insulin, the GMR was statistically higher in hearts from fed
rabbits than from fasted rabbits (P < 0.002).
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Table 1.
Glucose metabolic rates as assessed using the Fick principle,
Patlak graphical analysis, and four-compartment model
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Lumped Constant
Lumped constant values were compared between the three groups to assess
the effect of feeding, fasting, and insulin on computed transport and
phosphorylation parameters for both glucose analogs. The results are
illustrated in Fig. 2. For both FDG and DG,
the lumped constant values in hearts from fed rabbits perfused with insulin (FDG, 0.45 ± 0.09; DG, 0.32 ± 0.08) were significantly less than in hearts from fed rabbits perfused without insulin (FDG,
1.05 ± 0.27, P < 0.001; and DG,
0.96 ± 0.32, P < 0.001) and in
hearts from fasted rabbits perfused without insulin (FDG, 0.92 ± 0.16, P < 0.001; and DG,
0.82 ± 0.17, P < 0.003). There was no significant difference between the fed and fasted groups perfused without insulin. These results indicate that the presence vs.
the absence of insulin has a significant effect on the value of the
lumped constant for both FDG and DG while the nutritional state of the
animal does not.
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Fig. 2.
Bar graph of mean lumped constants for the following three experimental
groups: fed with insulin and fasted and fed without insulin. Shaded
bars, data for DG; open bars, data for FDG. Error bars are SD;
n, no. of hearts/group.
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In each group, plasma flow ranged from ~0.5 to 1.5 ml · min1 · g
LV wet wt1. In the two
groups perfused without insulin, plasma glucose concentration ranged
from 1.9 to 6.4 mmol/l. Neither plasma flow rate nor perfusate glucose
concentration in the absence of insulin had a significant effect on
lumped constant values (data not shown).
Patlak Graphical Analysis
A graphical representation of a typical Patlak analysis of an
experiment from a fed rabbit heart perfused with insulin is shown in
Fig. 3. Visual inspection of these curves
indicates that a linear phase of the Patlak graph for both glucose
analogs begins ~1,200 s (20 min) after isotope introduction. For all
experiments, the time taken to develop a linear phase was identical for
both isotopes (average, 21.8 min; range, 15-26 min) and did not
vary between experimental groups. These results provide an estimate of
the time required to achieve plasma-tissue free FDG-DG equilibration under the constant isotope infusion conditions used here.
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Fig. 3.
Results of Patlak analysis.  , FDG data;  , DG data. Equations are
for the regression lines drawn through the data points used for Patlak
analysis. y-Axis, normalized tissue
FDG or DG content; x-axis,
experimental elapsed time (because a constant infusion was used, the
normalized, integrated FDG input activity was equivalent to
experimental elapsed time).
AT(t),
tissue tracer content as a function of time; CA, arterial
tissue concentration.
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The
Ki
and reversible distribution volumes
(Vi)
determined from Patlak graphical analysis of FDG-DG tissue accumulation
curves for each of the three groups are listed in Table
2. For both FDG and DG,
Ki
did not vary significantly between the fed with insulin, fed without
insulin, and fasted without insulin groups, indicating that the
presence or absence of insulin and the nutritional state of the animal
did not affect
Ki
values for either isotope. Similarly, flow did not affect
Ki
values for either isotope, and the perfusate glucose concentration did
not affect
Ki
values in the two groups perfused without insulin (data not shown).
Vi
provides a lower-limit estimate of the reversible distribution volume
of nonphosphorylated FDG and DG (3, 33, 34). Possibly because of
variable results, neither insulin in the perfusate nor the nutritional
state of the animal had a significant effect on
Vi
values (Table 2). Plasma flow rate and perfusate glucose concentration
also had no effect on
Vi
values (data not shown). Similarly, there were no significant
differences in the
Vi
values for FDG vs. DG in any of the groups. The reversible distribution volume of free FDG-DG, as estimated by
Vi,
exceeded the free water space in the isolated rabbit heart (0.79 ± 0.03 ml/g, computed as 1 dry wt/wet wt) in all three groups.
These results indicate that the reversible distribution volume of
nonphosphorylated FDG-DG is greater than the tissue water space and is
unaffected by insulin in the perfusate or the nutritional state of the
animal.
With the use of Eq. 4 and the averaged
lumped constant for each group, the mean GMRs for the three
experimental groups estimated from the Patlak graphical analysis agreed
well with those determined from the Fick equation (Table 1) for both
FDG and DG. As observed with the Fick method, there were significant
differences in mean GMRs estimated from the Patlak plot between the fed
with insulin, fed without insulin, and fasted without insulin groups
for both FDG and DG.
Four-Compartment Model
Figure 4 illustrates the results of fitting
a three-compartment and a four-compartment model to an experimentally
derived curve for DG. The four-compartment model and the measured data are concordant. In contrast, the three-compartment model underestimates the experimental curve between 200 and 500 s and overestimates the
entire terminal portion of the curve. Because the terminal phase of the
curve is crucial to estimating the asymptotic rate of filling of the
fourth compartment, overestimation of the later part of the curve
corresponds to an increased estimate of GMR. These results were
observed consistently for both FDG and DG in all experiments.
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Fig. 4.
Data points from one experimental curve for DG in a fed rabbit heart
plotted as the normalized arteriovenous (AV) difference vs. time
together with the best fits of the 3-compartment and the 4-compartment
models. Data are measured in triplicate, and  represents the mean
value at each time point. Error bars are SD.
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Quantitatively, using the four- vs. the three-compartment model reduced
the mean squared error between the measured and modeled normalized
arteriovenous difference by an average factor of 6.8 for FDG (range,
1.5-32) and 8.4 for DG (range, 1.4-39). Estimates of GMR
increased by an average of 43% for FDG (range, 12-240%) and 48%
for DG (range, 9.3-150%) when using the three-compartment model
rather than the four-compartment model.
Table 3 lists mean values for both the
individual and combined rate constants for FDG-DG estimated with the
four-compartment model for each of the three experimental groups. As
outlined in Fig. 1, the individual rate constants describe
unidirectional transport between individual compartments. The last term
in Table 3, KFUR,
is a combined constant that provides a measure of the fractional FDG-DG
phosphorylation rate and is similar to a term reported by Ratib et al.
(36). In our study, as well as in the one by Ratib et al., the combined
rate constant could be estimated with greater certainty than the
individual rate constants. Consistent with this observation is the fact
that widely divergent values for the first three individual rate
constants in hearts from fasted rabbits perfused without insulin were
not accompanied by comparable scatter in
KFUR values.
Similar to
Ki,
KFUR values did
not vary significantly between the three experimental groups. Similar
to the results with the Patlak graphical analysis, mean GMR values
estimated using Eq. 4 and average
lumped constants for each of the three experimental groups agreed well
with the results using the Fick equation (Table 1).
FDG vs. DG. Table
4 illustrates the comparison of lumped
constants and combined rate constants for FDG vs. DG in hearts perfused with both isotopes. In each group, the lumped constant values for FDG
were greater than those for DG. Similarly,
Ki
and KFUR values
for FDG were higher than those for DG in all three experimental groups.
Taken together, the results for the lumped constant and combined rate
constants for FDG vs. DG indicate that these two glucose analogs have
different transport and/or phosphorylation parameters and
suggest that these two glucose analogs should not be used
interchangeably.
Comparison of GMR Estimates Using Patlak Graphical and
Four-Compartment Analyses
The correlation between GMR estimated by the Patlak graphical approach,
the four-compartment model, and the Fick method for both FDG and DG is
illustrated in Fig. 5. The three
experimental groups are combined, and each data point represents an
individual experiment. GMR was computed using Eq. 4, the appropriate average lumped constant value
determined for each experimental group, and the individual
Ki
values or KFUR
values computed for each experiment. Because FDG and DG were shown to
have different lumped constants, a total of six lumped constant values
were used (3 each for FDG and DG). The equations for the linear
regression and correlation coefficient are shown in Fig. 5,
A-D. For both tracer kinetic models
and both isotopes, the slopes of the linear regression were not
significantly different from one (smallest P > 0.2), and the ordinate
intercepts were not significantly different from zero (smallest
P > 0.5). These results indicate
that both tracer kinetic models provided comparably accurate GMR
estimates using both glucose analogs.
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Fig. 5.
Correlation between glucose metabolic rate (GMR;
µmol · min1 · g
LV wet wt1) estimated by
the Patlak graphical approach vs. Fick-determined GMR
(A and
C) and the four-compartment model
vs. the Fick-determined GMR (B and
D). FDG data are represented in
A and
B, and DG data are represented in
C and
D.
|
|
In clinical studies evaluating myocardial glucose metabolism with PET
and FDG, a single lumped constant value of 0.67 has been used to
compute GMR (13, 14, 19, 24). To compare our results with those
obtained using an invariant lumped constant, we computed GMRs using
Eq. 4, the single lumped constant
value of 0.67, and the individual combined rate constants determined in
this study using both the Sokoloff and Patlak tracer kinetic models.
The correlation between each model-estimated and Fick-determined GMR
using this single lumped constant value for FDG is illustrated in Fig.
6. The format of this illustration is the
same as in Fig. 5, A and
B. In contrast to the results observed
with different lumped constants, the slopes of the linear regressions
are significantly different from one
(P < 0.001), and the ordinate
intercepts are significantly different from zero
(P < 0.001) for both tracer kinetic
models. Consequently, slower GMRs are overestimated, and faster GMRs
are underestimated. These results clearly demonstrate that accurate GMR
estimates can only be obtained using different lumped constant values
when evaluating glucose metabolism in the presence vs. the absence of
insulin.
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Fig. 6.
Fick GMR vs. Patlak (A) or
compartment (B) GMR. Compartment and
Patlak GMRs were calculated using a single lumped constant value of
0.67. Equation for the linear regression is included in
A and
B.
|
|
| |
DISCUSSION |
In this investigation, we have evaluated the accuracy of the Sokoloff
compartment model and the Patlak graphical analysis in estimating
myocardial GMR in the presence and absence of insulin in hearts from
fed and fasted rabbits using both FDG and DG as tracers. To fit
experimental heart curves, the original Sokoloff model had to be
altered to include a fourth compartment to account for tracer delivery
and distribution. We found that accurate GMR estimates were produced by
both tracer kinetic models only by using the appropriate lumped
constant for each experimental condition. In contrast to insulin, the
nutritional state of the animal did not appear to have any effect on
the lumped constant. FDG and DG were also observed to have different
transport and/or phosphorylation parameters.
Compartment Model
A three-compartment model has been used to quantify GMR with PET and
FDG (13, 17, 36). In our analysis of glucose metabolism in the isolated
heart, a fourth compartment was required to obtain satisfactory fits to
the data. The simplest explanation for the need to use an additional
compartment is that tissue FDG and DG accumulation curves were computed
using in vitro arteriovenous differences instead of in vivo residue
detection with PET. Compared with the present data, in vivo myocardial
and blood FDG curves acquired with PET are degraded due to limitations
in spatial and temporal resolution and statistical uncertainties. As a
result, parameter estimation is less accurate, reducing the number of compartments that can be fit to the data.
Two previous studies used the three-compartment model to quantify GMR
in an isolated heart preparation (20, 21). In contrast to our
investigation, these studies evaluated FDG accumulation kinetics from
externally detected tissue residue curves using coincidence detection.
Before myocardial FDG-6-phosphate accumulation, perfusate FDG content
represents a significant fraction of detected activity when measured by
external detection. In contrast, tissue FDG and DG accumulation
determined from arteriovenous differences excludes plasma tracer
activity from computed residue curves. Therefore, the early dynamics of
FDG and DG exchange between plasma and tissue are probably better
assessed from arterial and venous concentrations, which might have
contributed to the need to use an additional compartment in this study.
One of the fundamental assumptions of the original Sokoloff model is
that it is possible to estimate capillary FDG, DG, and glucose
concentrations from arterial measurements (41). This assumption is
based on the concept that brain uptake of these substances is
completely barrier limited and independent of the rate of blood flow.
However, because of the structural and functional dissimilarities of
the two organs, it is not clear if this assumption is valid in the
heart. One important difference is that glucose enters the brain by
facilitated diffusion through the blood-brain barrier, whereas entry
into the heart is via diffusion through pores in the myocardial
capillary wall. Because of this difference, FDG-DG might diffuse more
rapidly into the heart than into the brain. If diffusion into the heart
is rapid, then myocardial uptake of FDG-DG is not completely barrier
limited, invalidating Sokoloff's original assumption for use in the
heart.
To test the validity of this assumption in the heart, we assessed the
rapidity of FDG diffusion by measuring its peak instantaneous extraction relative to an intravascular reference tracer (1). [The peak instantaneous extractions of FDG and DG are expected to
be similar in the heart, differing only by their respective diffusion
coefficients (22).] Using
131I-labeled albumin as the
intravascular tracer, we measured the peak instantaneous extraction of
FDG in eight additional hearts using published techniques (1, 9, 25, 27; see Fig. 7). An average of 72% of the
FDG diffused out of the capillary during its initial transit through
the heart. This value is approximately two times that reported for
glucose in brain at comparable blood flow rates and plasma glucose
concentrations (9, 44). Because of the rapid extravascular diffusion in
the heart, initial distribution of FDG (and DG) is not completely
barrier limited, indicating that initial capillary FDG (and DG)
concentration cannot be estimated from arterial measurements. In the
present study, the addition of a fourth compartment and the inclusion
of flow in the model (see APPENDIX) allowed us to avoid
this assumption.
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Fig. 7.
Peak instantaneous extraction
[E(t)]
as a function of plasma flow. , Data from fasted without insulin
rabbits; , data from fed with
insulin rabbits. Arterial glucose concentration was 5.6 mmol/l.
|
|
Patlak Graphical Analysis
The Patlak graphical approach provides an alternative method to that of
Sokoloff for measuring GMR from FDG-DG tissue and blood curves (3, 33,
34). Compared with the Sokoloff three-compartment model, the Patlak
model is more general with an unrestricted number of reversible
compartments and one irreversible compartment. The advantages of the
Patlak approach are that it does not depend on an explicit compartment
model, and it is easier to use than the Sokoloff model. However, one of
the fundamental requirements for successful use of the Patlak graphical
analysis is that there is a relatively rapid equilibration of FDG
between plasma and reversible tissue regions so that a unidirectional
flux of tracer into the irreversible compartment dominates the curve
for a measurable time period during the experiment.
The fact that the Patlak graphical analysis provided GMR estimates that
correlated well with results obtained with the Fick principle suggests
that the requirement for rapid tracer equilibration was fulfilled in
the isolated rabbit heart. These results were obtained during constant
isotope infusion. In contrast, isotopes are usually delivered as a
bolus in patients such that blood tracer activity constantly declines
after introduction. In the first clinical application of the Patlak
graphical analysis to the heart, Gambhir et al. (13) recommended that
scans acquired as early as 12 min after FDG injection be used to
quantify GMR. However, the accuracy of the GMR estimates obtained with
the Patlak technique in that study was not validated against results
obtained with the Fick principle. In the present study, >20 min were
required to achieve tracer equilibration during constant isotope
infusion (Fig. 3). Because of the differences in tracer introduction
and delivery, neither the time at which tracer equilibration occurs in
vivo nor the errors caused by tracer nonequilibration are known. These
considerations indicate that careful evaluation and validation of the
Patlak graphical analysis under appropriate experimental and clinical
conditions should be completed before final acceptance of this
approach.
The ordinate intercept of the linear portion of the Patlak plot is
equal to or less than the sum of the plasma volume and reversible
tissue distribution volume of free FDG-DG (3, 33, 34). In our studies,
this combined volume averaged >1.0 ml/g LV wet wt (Table 2). Because
total tissue water space in the RBC-albumin-perfused rabbit heart is
0.79 ml/g, the average value for the summed FDG-DG plasma and tissue
distribution volume exceeds tissue water space. Given that membrane
transport prefers FDG-DG to glucose while phosphorylation by hexokinase
favors glucose over FDG-DG (8), it is predictable that the
concentration of FDG-DG in the cytoplasm should be greater than that
for glucose. However, these relative affinities are insufficient to
account for the fact that the distribution volumes for free FDG and DG were greater than total tissue water space. For this latter observation to be valid, the myocyte must concentrate FDG-DG relative to the plasma. In order for the cell to concentrate FDG-DG, membrane transport
has to be asymmetric with the rate of inward transport exceeding
outward transport. If correct, this unexpectedly high distribution
volume for unphosphorylated FDG-DG could have an effect on the
successful use of the Sokoloff compartment model to estimate GMR.
Effect of Insulin on the Lumped Constant
The lumped constant is a combined factor that converts FDG-DG
phosphorylation rate into the net rate of glucose utilization (41). It
is composed of six terms in the following order:
Km/
Vm
, where is the ratio of distribution volumes for FDG-DG to glucose, Km/Vm
are the maximum velocities and Michaelis constants of hexokinase for
FDG-DG and glucose (symbols with * refer to FDG-DG; those without refer
to glucose, respectively), and is the fraction of phosphorylated
glucose that proceeds through glycolysis. Because the value for is
close to one (11) and is not known to be affected by insulin, the terms
most likely to be altered by insulin are and
Km/Vm. In the brain, the ratio of distribution volumes for FDG-DG to glucose
() has been shown to be a function of tissue glucose concentration
(10, 15). Because insulin alters tissue glucose concentration in the
heart, it is possible that insulin changes the value for the lumped
constant through its effect on .
In contrast to the brain, insulin is known to affect the activity of
hexokinase in the heart (2). Russell et al. (38) investigated the
possibility that insulin's effect on the lumped constant was due to a
change in the relative affinity of hexokinase for DG vs. glucose
(Km/).
These investigators reported that insulin increased the fraction of
hexokinase bound to mitochondria and that bound hexokinase had a
decreased affinity for DG relative to glucose. Based on this
observation, they proposed that the reduction in the lumped constant in
hearts perfused with insulin was due to an insulin-related increase in
of mitochondrial-bound
hexokinase for DG. At present, there is insufficient information to
elucidate the mechanism whereby insulin alters the value of the lumped
constant in the heart. Additional research is needed to assess the
effect of insulin on both and Km/Vm
before this mechanism can be more precisely understood.
Nutritional State
An increase in GMR in the presence of insulin has been observed in
isolated hearts from fasted but not fed animals (32, 39, 40). One
explanation for this finding is that hearts from fed animals maintain a
"metabolic memory" due to a residual insulin effect acquired
during in vivo feeding (39). Direct comparison of the GMR results
obtained in the present study with those previously published is
difficult because of differences in experimental design. However, the
observation that the nutritional state of the animal had no effect on
the lumped constant is inconsistent with the concept of a metabolic
memory, i.e., the presence of a residual insulin effect should have
lowered the lumped constant in hearts from fed compared with fasted
rabbits perfused without insulin. A possible explanation for this
discrepancy is that a different experimental preparation
(crystalloid-perfused working rat heart) was employed in the previous
studies than was employed here.
FDG vs. DG
The present data indicate that FDG and DG have different lumped
constant and combined rate constant values in the isolated RBC-albumin-perfused rabbit heart. In all cases, the values for FDG
were higher than those for DG, reflecting a preference of either
membrane transport or phosphorylation for FDG over DG. Although these
two glucose analogs have not been compared in the heart, Reivich et al.
(37) compared [18F]FDG
and [11C]DG in the
brain and did not observe any difference in lumped constant values. In
their study, lumped constants were measured in different experimental
groups (humans) with PET and then compared, whereas, in our study, FDG
and DG were infused simultaneously, and lumped constant values for the
two isotopes were determined from results obtained in a single heart.
One probable explanation for the discrepant results between heart and
brain is in the experimental design. Experimental errors and biological
variability might obscure differences in FDG vs. DG lumped constants
when different experimental subsets are compared; these may become more
apparent when paired observations made in a single experiment are
compared. It is also possible that transport and phosphorylation for
human brain and rabbit myocardium might have different relative
affinities for FDG vs. DG.
Clinical Implications
The results of the present study provide evidence that different lumped
constant values are required to obtain accurate estimates of GMR with
PET and FDG in the presence vs. the absence of insulin. It is also
possible that factors other than insulin could affect the value of the
lumped constant. High concentrations of lactate and 
-hydroxybutyrate
have been shown to affect the value of the lumped constant for FDG
(16). In two recent studies, free fatty acids were observed to affect
the value of the lumped constant (4, 12). Furthermore, the effects on
the lumped constant of myocardial pathology such as acute low-flow
ischemia and "chronic" stunning and/or
hibernation have not been reported. Unfortunately, it is not clear what
lumped constant value should be used under many of the conditions that
are routinely encountered in the clinical applications of PET and FDG
to study glucose metabolism. However, our value of 0.45 in the presence
of insulin, combined with a previously reported value of 0.33 (31),
suggests that more accurate quantification of GMR with FDG and PET in
normal myocardium during euglycemic hyperinsulinemic clamp might be
achieved using a lumped constant value of ~0.4.
The ability of both the Sokoloff and Patlak tracer kinetic models to
provide accurate GMR estimates suggests that the assumptions required
to obtain convergence of model-predicted and experimentally measured
curves were fulfilled under the experimental conditions evaluated in
the isolated RBC-albumin-perfused rabbit heart. Although the present in
vitro environment is dissimilar to that encountered in vivo (since
tissue residue curves were determined using arteriovenous sampling and
isotopes were delivered as a constant infusion), the current results
might be relevant to in vivo data acquired with PET. Because the
factors determining the kinetics of initial FDG delivery and
distribution are probably similar in vivo and in vitro, our need to use
an additional compartment to fit experimental heart curves could
indicate that more accurate GMR estimates would be obtained in vivo
using a four- rather than a three-compartment model and high time
resolution PET (6, 30). It is also possible that errors produced by
employing PET images acquired before FDG equilibration after bolus
introduction might account for the poor separation of viable vs.
nonviable dysfunctional myocardium noted in recent clinical studies
using the Patlak graphical analysis to quantify GMR (14, 19). However,
until further studies are completed comparing model-estimated and
Fick-determined GMR using high time resolution tomographs
and/or arteriovenous sampling during tracer nonsteady states,
it cannot be predicted that the present success in obtaining
accurate quantitative estimates of glucose metabolism with both the
Sokoloff and Patlak models will be reproduced in vivo with PET.
| |
APPENDIX |
Mathematical Description of the Multicompartment Model
In our experiments, the measured parameters were the arterial and
venous blood concentrations as a function of time, which are denoted as
CA(t)
and
CV(t),
respectively. The tissue activity content per unit flow as a function
of time, denoted as R(t), was
computed as
![R(<IT>t</IT>) = <LIM><OP>∫</OP><LL>0</LL><UL><IT>t</IT></UL></LIM> [C<SUB>A</SUB>(&tgr;) − C<SUB>V</SUB>(&tgr;)]d&tgr;](/content/vol275/issue2/fulltext/H668/img009.gif)
|
(A1)
|
Figure
1 shows the four-compartment model used to model the kinetics of FDG
and DG phosphorylation. The differential equations governing the time
courses of the compartment activity contents per unit flow as a
function of time, which are denoted as
Q2(t), Q3(t),
and
Q4(t),
are
|
(A2)
|
|
(A3)
|
|
(A4)
|
where
kij
is the transfer rate to compartment i
from compartment j. The modeled
tissue uptake 
(t) is
|
(A5)
|
It
can be shown that the solution to Eqs.
A2-A5 for
(t) can be written in the
form
|
(A6)
|
Thus
the stimulated tissue uptake
(t) is just the
convolution of the measured arterial blood input with a function
h(t) that depends on the set of
compartment model parameters
(k21, k12,
k32,
k23, and
k43). The
function h(t) is called the impulse response for the compartment model.
We used the software package RFIT (7, 18, 28) to estimate a set of
compartment model parameters
(k21,
k12,
k32,
k23, and
k43) that
locally minimizes the mean squared error between the time derivatives
of the measured tissue uptake R(t)
described by Eq. A1 and a simulated
tissue uptake (t)
described by Eq. A6, given the time
derivative of the measured blood input
CA(t). Taking the time derivative of both sides of Eqs.
A1 and A6, we obtain the equations
|
(A7)
|
|
(A8)
|
Thus
the compartment model impulse response
h(t), which relates the blood curve
CA(t)
to the simulated tissue curve
(t) via
Eq. A6, also relates the blood
derivative curve
dCA(t)/dt
to the simulated tissue derivative curve
d(t)/dt
via Eq. A8 because of the linearity of
convolution and differentiation.
Providing RFIT with the blood and tissue derivative curves
dCA(t)/dt
and
dR(t)/dt
is preferable in this case because our samples of the arteriovenous
difference (Eq. A7) have
uncorrelated errors, in keeping with the use of an unweighted least
squares curve fit. For a constant infusion
|
(A9)
|
where
CA0 is constant
arterial tracer concentration during constant infusion, and the blood
derivative curve is the Dirac delta function
|
(A10)
|
Having estimated a set of compartment model parameters
(k21,
k12,
k32,
k23, and
k43), we are
interested in using the model to calculate the fraction of FDG or DG
that is phosphorylated. This corresponds to the amount of isotope that
is trapped in compartment 4, given a
unit amount of isotope in the arterial blood input. It can be shown
that this fractional utilization, which we denote by FU, is given by
|
(A11)
|
As the equations have been developed,
k21 is
dimensionless and FU multiplied by flow provides an estimate of that
fraction of the total delivered FDG or DG that is taken up by the
heart. This term (FU × flow) has the dimensions of milliliters
per minute per gram LV wet weight and is denoted in this report as
KFUR.
| |
ACKNOWLEDGEMENTS |
This study was supported by National Heart, Lung, and Blood
Institute Grants PO1 HL-25840 and R01 HL-48068 and by the Director, Office of Energy Research, Office of Biological and Environmental Research (OBER), Medical Applications and Biophysical Research Division
of the United States Department of Energy under OBER contract
DE-AC03-76SF00098.
| |
FOOTNOTES |
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. §1734 solely to indicate this fact.
Address for reprint requests: R. C. Marshall, Lawrence Berkeley
National Laboratory, Bldg. 55-121, 1 Cyclotron Rd., Berkeley, CA
94720.
Received 9 February 1998; accepted in final form 30 April 1998.
| |
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Top
Abstract
Introduction
