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rCBV
rCBV
PET Quantification
Quantitative analysis
The uptake and distribution of the tracer must quantitatively and accurately reflect the
concentration of available binding sites or the rate of some biochemical processes. The
extraction of quantitative values from dynamic PET imaging data requires the fitting of the
data to a mathematical model that describes the uptake and retention of the tracer in tissue.
The model should have several essential features, as given below, that help in deriving
valuable information.
1. The derived results must reflect, in some quantitative manner, the concentration of
available binding sites or the rate of biochemical activity.
2. The derived results must be accurate that is, the parameters derived from fitting the
model to the data must be well defined, with relatively small, well-understood
variance and covariance.
3. The derived results must be consistent across studies and across subjects—that is,
the test–retest reliability must be good.
4. Simplifications of the mathematical model can be made to improve the fit to the
data or to simplify the imaging and analysis protocol, but those simplifications must
be physically meaningful and must be validated against the complete model.
Three specific examples are described below demonstrating important issues involved in
tracer kinetic applications.
Cerebral blood volume estimation
If a tracer is administered that binds to red cells and remains within the vascular compartment,
the volume of distribution (or blood volume) can be calculated by,
Vd (ml) = [A/C] at equilibrium,
Where Vd is the distribution volume, A (mCi or any other unit) is the amount of tracer
injected and C (mCi/ml) is the tracer concentration. This is the basis for the dilution principle,
which provides a convenient method for determining the distribution volume of a closed
compartment.
A related, but different, quantity of interest is the amount of blood in a given volume or
mass of tissue. For example, in the brain, the cerebral blood volume per unit mass of tissue
consists of arterial, capillary and venous blood. Because of the large volume of the dural
venous sinuses, most of the cerebral blood volume (averaged over the entire brain) is venous,
although the relative distribution between arterial, capillary and venous changes in different
locations within the brain. If carbon monoxide labeled with 11C or 15O is inhaled, it will
bind tightly to hemoglobin and the resultant PET images will represent regional blood
volume
. The
(blood volume per unit mass of tissue) can be calculated as,
GSPant\Newbook\Final-2008\30-chp\462
PET Quantification
( / ) / *0.85*
rCBV ml gm C C d
t
C
b
C
1
K
2
k
f
C
463
t b
Where
concentration factor for the difference between peripheral and central hematocrits and d is
the density of brain tissue (1.04 gm/ml).
is the tissue activity concentration,
is the peripheral venous blood activity
Quantification of tissue metabolism using FDG PET
FDG-PET can provide useful information about the glucose metabolism. In fact, the
knowledge of tissue glucose metabolism has proven to be of great practical use in oncology,
neurology and cardiology. Its use is so widespread that we often forget that the quantification
of glucose metabolism using 18F-FDG PET is by no means straightforward. The estimation
of the regional Metabolic Rate of Glucose (MRG) from the FDG images has been the focus
of most quantitative methods. Hence we’ll first briefly discuss the models used in MRG
estimation and then some recently developed quantification methods.
Glucose is metabolized in the body by different biochemical pathways, the principal
pathway being the metabolism along citric acid cycle. Unlike glucose, FDG does not undergo
all the metabolic steps in glucose metabolism. After crossing the cell membranes via the
glucose transporters, Fluorodeoxyglucose (FDG) is converted to FDG-6-phosphate by
hexokinase. FDG-6-phosphate is not the substrate for the next downstream enzyme,
phosphoglucoisomerase and thus it stays in the cell as FDG-6-phosphate. In addition,
dephosphorylation of FDG-6-phosphate via glucose-6-phosphatase occurs only in liver and
kidney. Thus in most tissues, injected FDG gets trapped inside the cells as FDG-6-phosphate.
Using these properties of FDG, Sokoloff first developed an autoradiographic method to
quantify regional cerebral glucose metabolic rate (1). Briefly, he described the kinetics of
FDG by the following two tissue compartmental model.
The plasma FDG concentration is indicated by Cp and it represents the input function.
The parameter
the reverse transport from the tissue to plasma. Once inside the cell, the free FDG
concentration is indicated by
GSPant\Newbook\Final 2008\30-chp\463
C
K
1
k
3
p
k
2
Figure 1: Two tissue compartmental model of FDG
corresponds to the FDG transport across the glucose transporter and
whereas the FDG-6-phosphate (product of hexokinase
C
f
C
b
464
3
k
1 3 2 3
* /( )
K complex K k k k
max[ ]
[ ]
V S
Km S
PET Quantification
reaction) concentration is denoted by Cb. The rate constant for hexokinase reaction is denoted
by
. C
represents the total activity as observed in the tissue time activity curves.
tot
By measuring the FDG time activity curves and arterial plasma FDG concentration the
model parameters K1-k3 can be estimated. It can then be shown that the unidirectional FDG
uptake rate constant (referred to as K-complex or Ki) is given by the following equation:
The regional metabolic rate of glucose (MRG) is then given by
MRG = K complex * plasma glucose /lumped constant
The lumped constant is necessary to account for the differences in the uptake and
phosphorylation between FDG and glucose. Even though the mechanism of FDG uptake and
phosphorylation is similar as for glucose, there are some differences in the corresponding
rates of uptake and phosphorylation between FDG and glucose. Hence, the lumped constant
is used to account for these differences.
It is important to recognize that the metabolic rate of glucose may not be the most stable
quantity of interest in FDG studies under all circumstances. Glucose metabolic rate as
determined from FDG PET is a composite measure reflecting the uptake rate and
phosphorylation rate. Both the glucose uptake and phosphorylation are known to follow
non-linear Michaelis-Menten kinetics.
Michaelis Menten equation is given by
V
(3)
Where V is the rate of reaction (e.g., phosphorylation rate)
Vmax is the maximum rate of reaction
Km is the equilibrium constant
[S] is the substrate concentration (e.g., glucose concentration)
Because the substrate concentration [S] term appears in the denominator, the equation is
non-linear. However, Michaelis Menten equation can be further analyzed by assuming
specific relationship between Km and [S]. When [S] << Km, one can see that [S] term in the
denominator can be neglected. This results in reaction rate being linearly dependent on
substrate concentration. Under normal physiologic conditions, the linearity assumption seems
to hold true for kinetics of glucose uptake. This is because the plasma glucose concentration
is significantly lower than Km of glucose transporter and hexokinase. With further increase
in plasma glucose levels there is progressively less increase in metabolic rate suggesting
that the binding sites in the transporter and enzyme are getting occupied. This is illustrated
GSPant\Newbook\Final-2008\30-chp\464
PET Quantification
Metabolic Rate of Glucose (MRG)
Ctot Cp K complex Cp dt Cp Vd
in figure 2. Once all the binding sites are filled, metabolic rate becomes independent of
plasma glucose levels. At this plasma glucose levels (as in diabetics), MRG may no longer
be the most useful measure from a clinical perspective. In this population, K-complex may
be a more important measure of glucose metabolism. This issue is particularly important if
one attempts to quantify change in glucose metabolism with respect to treatment.
Figure 2: The theoretical relationship between plasma glucose and Metabolic Rate of Glucose
(MRG). At low plasma glucose concentration, MRG is approximately linear with plasma glucose
465
Plasma Glucose
but it is saturated at higher plasma glucose concentration. Under normal physiologic conditions,
the Km of glucose transporter and hexokinase are approximately equal to plasma glucose, hence
the linear relationship between plasma glucose and MRG holds.
In addition to the compartmental analysis described above, Patlak et al described a
graphical method to estimate the glucose metabolic rate (2). Soon after the FDG injection,
the plasma FDG concentration (Cp) and free tissue FDG concentration (Cf) follow different
time courses. However, once these two compartments are in equilibrium, Patlak et al showed
that the unidirectional uptake rate constant (K-complex) can be directly derived from the
plasma and tissue time activity curves without the need to estimate the individual model
parameters. MRG can then be estimated from the K-complex as described in equation-1.
The Patlak equation is given by:
/ / )
where Ctot represents the total FDG uptake in the region of interest (obtained from the time
activity curves), Cp is the plasma FDG concentration (obtained by arterial blood sampling).
(4)
GSPant\Newbook\Final 2008\30-chp\465
466
Injected Dose Patient s weight
PET Quantification
Thus, the two parameters of Patlak equation, K-complex and Vd(volume of distribution)
can be directly derived by a simple straight line fit. K- complex derived from Patlak analysis
has been shown to have very good correlation with K complex derived from Compartmental
analysis.
Patlak analysis and compartmental analysis are generally considered gold-standard
methods for quantification of glucose metabolism but are cumbersome to use for clinical
purposes. Both these methods require long dynamic acquisition as well as the knowledge of
arterial plasma FDG concentration to estimate the glucose metabolic rate.
In clinical imaging, static image acquisition is most often used and arterial blood samples
are not collected. Hence compartmental analysis and Patlak analysis are usually restricted to
research settings. Simpler quantification methods were proposed to take into account the
limitations in clinical PET acquisition. Of these measures, the most widely used quantitative
measure is the Standardized Uptake value (SUV). It is given by
SUV
Tissue FDGuptake
(5)
/ '
The tissue FDG uptake will increase with increase in the dose or decrease in the patient’s
weight. Thus, the SUV “standardizes” the FDG uptake by taking into account these two
factors. Even though SUV is the term normally used for the expression represented by
equation-2, alternative SUV schemes have also been suggested. Since there is minimal
uptake of FDG by the adipose tissue, it has been proposed that a more appropriate SUV
technique should correct for the lean body mass rather than total body weight (3). Anther
normalization scheme is to use body surface area instead of body weight (4). However,
because of its simplicity, SUV (body weight) is the most common quantitative measure used
for clinical purposes. The SUV in brain, heart and tumors has been found to have good, but
not perfect, correlation with metabolic rate by several groups. In a study of renal cell cancer
patients treated with anti-VEGF antibody, we found that in general SUV correlates well with
K-complex (5). Over all tumors, SUV and K-complex correlated well (r=0.97, P<0.0001).
However, change in SUV with treatment over all tumor scan pairs was much less well
correlated with the corresponding change in K-complex (r=0.73, P<0.0001). Thus, when
monitoring individual patient therapy serially, large differences in the percentage changes in
the two indices were occasionally found, sometimes sufficient to produce opposing
conclusions regarding the progression of disease.
This study demonstrated that two principal factors account for the difference between
SUV and K-complex. Firstly, SUV does not account for physiologic changes in the available
dose to the tumor (i.e., it does not correct for variations in the integral of the input function).
This may be referred as “available dose” problem. As described in the compartmental analysis
section, knowledge of plasma FDG concentration (input function) is required to accurately
estimate the glucose metabolic rate. In the SUV calculation, the available dose is
GSPant\Newbook\Final-2008\30-chp\466
PET Quantification
467
approximated by normalizing the injected dose by patient’s body weight (denominator of
equation-2). In other words, it is assumed that the injected dose gets uniformly distributed
throughout the patient’s body. However, if the uptake of glucose in different organs varies
across patients significantly, it would vary the blood FDG concentration (since the blood
FDG concentration is determined by the balance between tissue uptake and excretion). This
violates the uniformity assumption of SUV. As we found in our previous study described
earlier, even though SUV correlates well with K-complex across group of patients, the
correlation decreases significantly when individual patients are serially followed for
monitoring response to therapy. This limits the usefulness of SUV in monitoring response to
therapy.
Secondly, the numerator (FDG uptake) in SUV is the sum of the 18F-FDG that has been
metabolized and trapped by the tumor cells plus the free 18F-FDG, which has not been
metabolized and is in the intravascular, interstitial and unmetabolized intracellular
compartments. Thus, SUV represents not just the metabolized FDG required for MRG
estimation but also the free unmetabolized FDG. We refer to it as “unmetabolized FDG”
problem. If images could be acquired at very late times (e.g., many hours after injection), it
is possible that the second of these deficiencies could be made negligible (6), but late
acquisition is often impractical.
In order to overcome these limitations of SUV, different techniques collectively called
“simplified kinetic analysis (SKA)” have been proposed. For example, the method of Hunter
et al. (7), which we refer to as SKA-S method, is one such method. By using a standardized
functional form for the input function, based on a population average, and by taking a single
late venous blood sample, SKA-S attempts to compensate for potential changes in the
integral of the input function (the available dose). However, because SKA-S does not follow
tissue uptake dynamically, it does not correct for unmetabolized 18F-FDG. Thus SKA-S
corrects for the available dose but fails to take into account the unmetabolized FDG fraction.
The unmetabolized 18F-FDG fraction has been estimated to vary from 6% to 67% (5,6).
Hence, the unmetabolized free 18F-FDG fraction is often a considerable and variable fraction
of uptake, even after 50–60 min of injection.
In order to correct for the unmetabolized FDG and still keep the method simple, we
developed an alternative simplified kinetic analysis method, referred to as “SKA-M method”
(8). In the SKA-M method, we tried to combine some of the advantages of SKA-S method
with the accuracy of the Patlak method. SKA-M is a modified Patlak analysis in which the
input function from the individual patient no longer has to be measured and instead is
replaced with a single population-averaged input function, scaled with a late venous blood
sample. In addition, the SKA-M method uses a few dynamic images acquired late (e.g.,
beginning 25 min or later after injection) to monitor the rate of uptake of 18F-FDG by the
tumor. We hypothesized that by combining late dynamic imaging with a scaled population
input function, SKA-M could correct for both available dose and unmetabolized 18F-FDG
and so would agree better with Patlak analysis.
GSPant\Newbook\Final 2008\30-chp\467
468
PET Quantification
Figure 3: Correlation between SKA-S and SKA-M with unidirectional Uptake rate constant
As shown in figure 3, both SKA-M and SKA-S correlated well with K-complex (Ki) (r =
0.99, P <0.001 for SKA-M and r = 0.96, P < 0.001, for SKA-S). The scatter of the SKA-S
values about the regression line was somewhat higher than that found for SKA-M (12.9%
vs. 4–5%). In addition, the slope of the SKA-S regression line differed significantly from
unity (slope = 0.807), with a significant intercept (0.0049, or 23.3% of the mean value). In
contrast, the SKA-M method had slopes closer to unity (>0.930) and an intercept closer to
zero (<0.0007, or <3.7% of the mean). The positive intercept and nonunity slope resulted in
a bias of SKA-S compared with Ki of about 15% (range, -18% to 113%), versus a bias of
about 1% for SKA-M. Because of this bias, SKA-S differed from Ki by greater than 20% in
10 of 27 studies. The SKA-M method differed from Ki by greater than 20% in only 2 of 27
studies. We speculate that these differences occurred because SKA-S does not correct for
unmetabolized 18F-FDG, whereas SKA-M does.
When tumors have low metabolic activity, even a small amount of unmetabolized 18FFDG becomes a significant fraction of total 18F-FDG uptake in the tumor. For example, we
found that SKA-S differed, on average, from Ki by 45% + 7.3% when Ki was less than
0.015 mL/g/min but by only 1.7% + 2.5% when Ki was more than 0.015 mL/g/min. On the
other hand, the SKA-M method differed from Ki by 7% + 3.5% when Ki was less than 0.015
mL/g/min and by -4.7% +
0.7% when Ki was more than 0.015 mL/g/min. Hence, for tissues
with low uptake, neglecting the unmetabolized 18F-FDG concentration may cause large
percentage errors in SKA-S. The SKA-M may be particularly useful in this situation.
Conversely, if the patient population had contained fewer regions with low uptake (e.g.,
<0.015 mL/g/min), SKA-S may have had a lower percentage disagreement with Patlak
analysis. The values of SUV, and to a lesser extent SKA-S, depend on the time at which
imaging is performed. At the usual acquisition time of around 60 min after injection, many
tumors have not reached their uptake plateau. Reaching the plateau has been found to take
as long as 256–340 min after injection for many tumors (9,6).
In summary, both SKA-M and SKA-S correlated well with Patlak analysis. SKA-M had
less variability about the regression line, a regression slope closer to unity, and no significant
GSPant\Newbook\Final-2008\30-chp\468
PET Quantification
469
intercept. In addition, because SKA-M follows the rate of uptake, it can account for
unmetabolized 18F-FDG, which may explain the smaller bias (1% vs. 15%) compared with
SKA-S. By measuring rate of uptake, SKA-M, in principle, removes the dependence on
uptake time that SUV exhibits. The SKA-M method reduces imaging time by more than
40% and, equally important, avoids the necessity of measuring the input function.
The SKA methods fall between SUV and Compartmental/Patlak analysis in terms of
complexity of quantification and attempt to address weaknesses of SUV without significantly
increasing the complexity of PET acquisition. Further experience with the SKA methods in
various patient populations is required to define their role in clinical practice.
Quantification of cerebral protein synthesis using L-[1-11C]-Leucine PET
Positron emission tomography (PET) imaging methods suitable for identification of
abnormalities in brain protein synthesis were developed more than two decades ago (10-12).
Application of these methods was, however, hindered by three issues:
(1) Difficulty in the synthesis of L-[1-11C]-leucine (12).
(2) Identification of appropriate tracer kinetic model of protein synthesis. Unlike glucose,
which does not re-circulate into the precursor pool after metabolism, leucine enters
the precursor pool by proteolysis and dilutes the specific activity of the tracer in the
precursor pool. Until now, there is no reliable method to estimate the fraction of the
amino acid precursor pool for protein synthesis derived from tissue proteolysis
(referred to as ) (12,13).
(3) The influence of other large neutral amino acid (LNAA) levels on leucine transport
via LNAA transporter. However, recent developments in the field have overcome
the difficulties associated with chemical synthesis of L-[1-11C]-leucine (14), a method
for quantifying the endogenous brain amino acid precursor pool has been described
and validated (15,16) and the effect of plasma LNAA on accuracy of quantification
has also been clarified recently (17).
The tracer L-[1-11C]-leucine satisfies most of the criteria required for the estimation of
protein synthesis rate in vivo (11,18). The tracer shows sufficient transport at the bloodbrain barrier, and in addition to protein incorporation, there is only a single pathway for its
metabolism for which the metabolic products are quickly removed from the brain. Moreover,
L-[1-14C]-leucine has been used in animal autoradiographic experiments in studies of protein
synthesis, and this tracer has been shown to provide reliable estimates of regional cerebral
protein synthesis (13,19,20).
Based on their previous work with autoradiography, Schmidt et al. (15) proposed a noninvasive compartmental model approach to estimate the fraction of leucine derived from
plasma and which is incorporated into proteins (l). Smith et al. (16) subsequently showed
the feasibility of this method in PET studies of rhesus monkeys. We recently estimated in
GSPant\Newbook\Final 2008\30-chp\469
470
PET Quantification
humans using the two-tissue compartment model introduced by Schmidt et al. and assessed
the effect of plasma leucine as well as total large neutral amino acid (LNAA) concentrations
on estimates of brain protein synthesis (17). In addition, we also assessed the accuracy of
the Patlak graphical analysis method in estimating the unidirectional uptake rate constant of
leucine, as implementation of this method would allow the use of a blood time activity
curve obtained from a left ventricular region of interest instead of arterial blood sampling,
and would yield a non-invasively obtained index of exogenous protein synthesis.
Kinetic model
For cerebral protein synthesis rate measurement, a two-tissue compartment model (Figure
4), similar to FDG model described above, characterized by the rate constants K1, k2 and k
adequately describes the obtained PET time-activity curves.
3
Figure 4: (Top) Simplified two-tissue compartment model for labeled leucine with k2 representing
the effective loss of the 11C tracer from tissue and k3 representing the effective incorporation of L[1-11C]-leucine into protein. Moreover, tracer concentration in a combined free and metabolic
compartment is denoted as C
(Bottom) Simplified two-tissue compartment model for unlabeled (cold) leucine with k
, whereas tracer concentration in the protein pool is denoted as Cb.
f+m
rec
representing recycled leucine as a product of brain protein breakdown. The models describing L[1-11C]-leucine and unlabeled (cold) leucine differ only with respect to k
11
C]-leucine.
which is zero for L-[1-
rec
Identifiability analysis of the parameter vector showed that the rate constants are well
defined and result in a computationally stable estimate of the Kcplx macroparameter (=
GSPant\Newbook\Final-2008\30-chp\470
PET Quantification
( )
( )
( ) ( )
f cold p cold
2 3
k
k k
( )
leucine M
471
K1k3/(k2+k3)), which represents the unidirectional uptake rate constant for leucine. The
incorporation of leucine into tissue can be subsequently calculated by multiplying the Kcplx
(or Kcplx’) with plasma leucine levels, however this expression only considers the contribution
of exogenous plasma leucine and does not account for the contribution of leucine originating
from endogenous tissue sources. To estimate the fraction of leucine in the precursor pool
that is derived from arterial plasma (), Smith et al. (13) proposed the following equation:
/
C C
lim
t
f f cold
/
C C
p p cold
(6)
Where C
f(cold)
and C
are the concentrations of unlabeled leucine in the free precursor
p(cold)
pool and in plasma, respectively, and Cf and Cp are the corresponding concentrations of the
labeled leucine. For the unlabeled (cold) leucine, a two-tissue compartment model can be
defined (Fig. 4 bottom) with k
pool. As unlabeled (cold) leucine in tissue is in a state of dynamic equilibrium (dC
dC
/dt = 0), the following relationship can be derived.
b(cold)
C C
representing the rate of leucine recycled from the protein
rec
f(cold)
K
1
k
2
/dt =
(7)
As Cf = [K1/(k2+k3)]Cp for labeled leucine, equation-3 yields an estimate of the fraction
of leucine in the precursor pool derived from exogenous sources termed :
2
(8)
Finally, under normal physiological conditions the total incorporation of leucine into
brain tissue equals the protein synthesis rate (PSR) which can be calculated as,
1
( /min) (min )
PSR M Kcplx
(9)
Effect of LNAA levels on kinetic parameters
One consequence of the high affinity of the cerebrovascular LNAA transport system is that
the LAT1 transporter is nearly saturated with LNAAs at normal plasma concentrations. The
percent saturation was previously estimated at around 96% (21). In order to correct for this
effect, Smith et al. (21) measured the KM for the LNAAs in rats and derived a formula for
the apparent KM (K
amino acids competing for the LAT1 transporter. Values for K
calculated as
GSPant\Newbook\Final 2008\30-chp\471
), that is, a KM value for a given amino acid in the presence of other
M(app)
for leucine can be
M(app)
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