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452
Radiotracer Kinetics: Applications in Nuclear Medicine
.dtGCdLtt
This is the fraction of the substance cleared from
estimate the overall total excretion of the substance from
during the time dt. From this we can
V
1
during the time interval from
V
1
the start of the experiment to t (Lt) by integrating both sides of the equation with respect to
time, i. e.
t
0
in (4)
2
dtCGLt
t
we get
C
2
t
t
0
V
2
VCdtCGI
11
(6)
Substituting (6) in (5) and solving for
C
2
Substituting this value of
C
We get
dC
1
V
1
dt
C
I
11
---GC
V
V
t
dtCG
t
0
22
VaC
11
-
V
2
Differentiation of the above with respect to time will give us the following equation
2
Cd
V
1
1
2
dt
dC
)-(G
dt
GC
11
2
dCV
11
dtVV
2
Rearranging the above and dividing it with
2
Cd
1
2
dt
This equation is of the form
G
VV
21
22
V
dC
dt
will give
1
GC
VV
11
0
21
0//
cdtBdydtyAd
This is a linear differential equation of the second order and has the general
tt
solution
and
1
can be obtained if the plasma concentration is plotted against time in a semilog
2
21
CBeAe
where
t
C
is the concentration at time t. The constants A ,B,
t
paper ( Figure 2).
This model was validated by Sapirstien when he found that, in keeping with the above
prediction, the plasma concentration of injected creatinine declined in a biexponential manner.
Therefore following single injection of a tracer similar to creatinine, it disappears from
plasma in a bi-exponential manner. The components which give rise to this biexponential
clearance are:
GSPant\Newbook\Final-2008\29-chp\452
Radiotracer Kinetics: Applications in Nuclear Medicine
Figure 2: Change in plasma concentration with time (biexponential function showing two
cpmpartments)
453
i) An earlier faster rate of clearance during which, the passage of the tracer from the
plasma into the interstitial fluid predominates. This constitutes the so called fast
component of the biexponential curve. In case of tracers like OIH which are more
completely and rapidly extracted by the kidneys, there is a substantial contribution
to this component from renal clearance. On the other hand, with a tracer like DTPA,
the renal contribution to this component is very small. In any case this faster
component continues till a point when the plasma and the interstitial fluid together
start to behave like a single compartment.
ii) Subsequent to the initial faster component there is fall in the rate of decrease in
tracer concentration, during this second slower phase, the tracer is eliminated by the
kidneys. This phase constitutes the so-called slower component of the bi-exponential
curve. This fact is illustrated by the following graphs, (Figure 3) which illustrate the
behaviour of hippuran after intravenous injection when during the first phase, it
participates in the process of equilibration with the second compartment, and during
the second phase, it is predominantly cleared from the plasma by the kidneys.
During this second slow component of the curve, tracer concentration in plasma and
the second compartment decrease at the same rate. Hence both the plasma and the
second compartment behave as a single compartment with respect to tracer kinetics
at this phase.
GSPant\Newbook\Final-200\29-chp\453
454
Figure 3: Elimination of tracer concentration from kidney with fast (first compartment) and slow
component (second compartment)
Radiotracer Kinetics: Applications in Nuclear Medicine
Our ability to make the above predictions eliminates the continuous infusion of tracer
as a prerequisite for performing renal clearance studies with tracers that behave reasonably
similar to creatinine within the human body. The DTPA, OIH etc are such tracers.
Two compartment models such as the one used by Sapirstein require that multiple
plasma samples have to be taken during the first and the second parts of the biexponential
curve. Russel in 1993 used a two compartment model and calculated the variation of the
standard error in clearance estimation with varying sampling times and number of samples
with I
131
OIH,
99m
Tc DTPA (2). From their experiments, they found that measurement of the
slow component requires measurement long after injection and measuring the slow
compartment is important for both glomerular and tubular agents. Accuracy in measurement
of the fast component requires that data must be obtained soon after injection. The fast
component less important for agents such as DTPA but more important for tubular agents.
They further concluded that
(i) It is more important to cover a wide interval than have a large number of samples.
(ii) Six or more samples must be obtained at sampling times forming a geometric progression
between 5 and 40 min for ERPF agents and between 10 min and 240 min for GFR
agents.
(iii) ERPF –sampling must start within 5 min after injection
(iv) GFR – sampling must be done till at least 3 hours.
When the importance of the fast exponential term in plasma clearance was calculated
the follwing results were obtained.
GSPant\Newbook\Final-2008\29-chp\454
Radiotracer Kinetics: Applications in Nuclear Medicine
-C C ( )
t
RENAL FUNCTION Anephric One third normal Normal
131
I
- OIH 0% 13% 37%
99m
Tc-DTPA 0% 3% 11%
99m
Tc MAG-3 0% 10% 31%
455
From the data
(i) Fast component is more important for tubular than for glomerular agents.
(ii) Fast compartment mesurement becomes progressively less important with decreasing
renal function.
99m
For
Tc DTPA the error introduced by neglecting the fast component is a maximum of
11% and decreased to almost zero as renal function dropped. Therefore we can approximate the
behaviour of DTPA to be monoexponential , i.e., the behaviour of a tracer which distributes into
only a single compartment. This is the basis of single compartment models.
Plasma sampling in single compartment models
Assumption : Clearance agent is distributed in a single compartment from which it is
eliminated exclusively by kidneys.
Figure 4: Change in plasma concentration with time in a single compartment model
dC t
( )
1
dt
GSPant\Newbook\Final-200\29-chp\455
1 2
456
0 979
Radiotracer Kinetics: Applications in Nuclear Medicine
C (t) concentration in compartment at tme t.
= clearance
C
1
This is of the form dy /dx = y with the general solution y = Ae
So C = Ae
-t
x
where A = intercept
= slope of lnC (Figure 4)
The line obtained when lnC is plotted against t is identical to the second, slower exponential
of two compartment model. A minimum of two samples are required since for plotting any line
two points must be defined.
When using the two sample method for single compartment model the plasma
concentration of tracer is related to the sampling time as represented below.
t
1
AeP
1
t
2
AeP
2
(7) where P1 is plasma concentration at first sampling time t
1
(8) where P2 is plasma concentration at second sampling time t
2
‘A’ in the above equation stands for the plasma tracer concentration at time t=0. This
value can be obtained by plotting the natural logarithm of the plasma concentration (lnP)
against time in a semi-log paper when the y intercept of the resulting straight line will give
the value of ‘A’ (a hypothetical value).
Russel et al. studied the two sample method for single compartment, compared it to the two
compartment model and suggested various correction factors for various sampling times (3). They
suggested that the second sample should be collected only after a minimum of 3hrs has elapsed
from the injection time and gave a correction factor of 0.979 if the sample were to be taken at 3
hours.The first sample is usually taken 60 mins post-injection
Therefore G.F.R. can be calculated by using two plasma samples from the formula (3)
.
GFR
P
1
ln
D
P
2
T T T T
2 1 2 1
ln ln
T P T P
1 2 2 1
exp
D = dose, counts/min
P1= plasma activity at time T1, counts/min-ml
P2= plasma activity at time T2, counts/min-ml
GFR is in ml/min
GSPant\Newbook\Final-2008\29-chp\456
Radiotracer Kinetics: Applications in Nuclear Medicine
(injected dose)
C (plasma conc.)
AVD where C/D is the specific plasma concentrat
ion
457
Now let us look at some non compartmental models, which are used to study tracer kinetics.
We shall continue to illustrate their utility by studying the application of these models to GFR
measurement. Again we shall start by defining some common terms. These models will enable
GFR estimation with a single sample.
Theoretical Volume of Distribution
It was shown in 1963 by Blaufox et al. that there existed an inverse relationship between the
specific concentration of OIH and the ERPF after a single injection of OIH (4).
The theoretical volume of distribution of a substance at a particular time t is the hypothetical
total volume in which the total administered amount of a substance would be distributed, if it
were to exist everywhere at the same concentration as its plasma concentration. It is otherwise
called the Apparent volume of distribution or AVD.
The AVD is given by :
AVD
D
1
C/D
As the clearance of the tracer falls, the plasma concentration rises and the AVD falls. Thus
from the AVD at a particular time after injection and by comparison with AVD in normal
subjects at the same time, the clearance can be estimated. This is the basis of the single sample
method.
The plasma concentration in an individual varies not only with clearance but also with
plasma volume. The plasma volume in turn varies with the surface area of the patient. Therefore
the AVD has to be normalized to the surface area of the individual. Then single sample method
can be used even in children. The same logic holds good for ERPF or GFR estimation with
single sample methods. Various algorithms have been given for estimation of GFR by the single
sample method by various authors.
eg. cl = + ln [1000 x 1 D/C1], where D is the dose and C1 is the plasma concentration.
= 2.866t – 1222.9 - 16820 / t
= -0.278t + 119.1 + 2405/t , where tis the time of sampling
As given by Russel et al. (3)
Pathological Factors Affecting AVD
In patients with reduced plasma proteins or with pathologically altered plasma protein composition
(eg.) proteinuria, ascites, edema) there is
GSPant\Newbook\Final-200\29-chp\457
458
× 100
Radiotracer Kinetics: Applications in Nuclear Medicine
(i) An enlarged ECV
(ii) Change in concentration of intravascular fraction (esp if protein bound substance is
used).
Both these will influence measured volume of distribution and thus the clearance irrespective
of renal function.
In suspected or known cases with poor renal function blood should be withdrawn later than
in patients with normal renal function.
So far we have decreased the number of plasma samples required for GFR estimation,
let us now see if we can eliminate plasma sampling completely, and learn about the principles,
which are behind this elimination.
The tracer leaves the plasma for the urine via the nephrons. Therefore by monitoring the
renal tracer uptake during the first few minutes after intravenous tracer renal clearance of the
tracer can be measured. The following method applies this principle.
Gates method
In the original study by Gates (5) the following protocol was followed.
One minute pre-injection count of the administered dose was performed by placing the
syringe 30cm from the centre of a parallel hole collimator of the gamma camera (128 x 128
matrix).
99m
1.
TcDTPA was rapidly injected i.v. into the patients who were positioned over the
gamma camera and the images were acquired in a 128 x 128 matrix at 4 frames per
minute. The study was carried out for 6 minutes a composite image of the entire
study was created. ROIs were drawn around each kidney with appropriate background.
2. A 1 min post-injection syringe count was obtained.
3. Net radionuclide counts within the kidneys were determined after background
subtraction (the background ROI areas were normalized to the respective renal areas).
4. This was done for various time intervals after arrival of the tracer in the kidneys 01 min, 1-2 min, 2-3 min, 3-4 min, 1-3 min and 1-4 min.
5. Tracer arrival is usually about 15 sec post-injection.
The percentage of injected activity which constitutes the total renal uptake can then be
calculated using the formula:
(R. K. counts-R. K.Bkg counts) + (L.. K.counts-L. K. Bkg counts)
Pre injection counts-post injection counts
where R.K. stands for right kidney and L.K. for left kidney
GSPant\Newbook\Final-2008\29-chp\458
Radiotracer Kinetics: Applications in Nuclear Medicine
Renal count-B kg count
( . . ) ( . . )
R K counts Bkg counts L K counts Bkg counts
100 9 75621 6 19483
459
(1) However, there is gamma ray attenuation by the soft tissues interposed between the
kidney centre and the scintillation camera.
Since the depth of the kidneys from the skin surface varies from patient to patient there
is variability in gamma ray attenuation. Various formulae have been suggested to estimate
the renal depth. The Tonnesen formula given below.
Tonnesen formula (for normally positioned kidneys).
R.K.Depth = 13.3 (weight / height) + 0.7 (weight in kg)
L.K. Depth = 13.2 (weight / height) + 0.7 (height in cm)
For transplanted kidneys, depth is calculated with ultrasonography.
When the renal depth is known, it can be used to correct the renal uptake as follows:
Corrected counts for each kidney
– x
e
where x = Renal depth
µ - linear attenuation coefficient for
99m
Tc gammas in soft tissues (= 0.153)
Hence the corrected % total uptake of DTPA =
100
1 2
e e
e injected counts Post injected counts
Pr
The percentage total renal uptake thus determined was plotted against 24 hours creatinine
clearance values and linear regression analysis of the variables of time, background
configuration and depth correction with 24 hours creatinine clearance values was performed.
From this Gates found that the optimal combination of variables was :
a) Depth correction
b) Analysis at 2-3 min post-injection
c) Semilunar Bkg.
The formula for calculation of GFR was derived from the results of the regression
analysis and was found to be
( . . ( . . )
R K counts Bkg counts L K counts Bkg counts
GFR
e e
Pr
153 13 3 7 153 13 2 7
. { .. ( / ) . } . { .. ( / ) . }
W H W H
e injected counts Post injected counts
. .
GSPant\Newbook\Final-200\29-chp\459
460
Radiotracer Kinetics: Applications in Nuclear Medicine
This brings us to the end of our discussion on GFR estimation. We have progressed from
multicompartmental analysis to unicompartmental analysis and carried on to noncompartmental methods. From this discussion on the methods for estimation of GFR we
hope the reader will have got a feel of the use of mathematical models, particularly
compartmental models in the field of nuclear medicine. Since the entire discipline of Nuclear
Medicine is built on the knowledgeable application of such tracer kinetic methodology, we
hope this discussion will have served as starting point for the reader to explore further and
further the unique, multidisciplinary and supremely elegant field of medicine called Nuclear
Medicine.
References
1. Sapirstein LA, Vidt DG, Mandel MJ, Hanusek G. Volumes of distribution and clearances of
intravenously injected creatinine in the dog. Am J Physiol 1955; 181: 330-336.
2. Russell CD. Optimum sample times for single –injection, multisample renal clearance methods. J
Nucl Med 1993; 34: 1761-1765.
3. Russell CD, Bischoff PG, Kontzen FN et al. Measurement of glomerular filtration rate: Single injection
plasma clearance method without urine collection. J Nucl Med 1985; 26: 1243-1247.
4. Blaufox MD, Frohmuller HGW, Campbell JC, Utz DC et al. A simplified method of estimating renal
function with iodohippurate
5. Gates GF. Glomerular filtration rate: estimation from fractional renal accumulation of 99m Tc-DTPA.
Am J Roentogenol 1982; 138(3): 565-570.
131
I. J Surg Res 1963; 3: 122-125.
GSPant\Newbook\Final-2008\29-chp\460
PET Quantification
S. Senthil Kumar and Ajay Kumar
Positron emission tomography (PET) is suitable for quantitative measurements of many
biochemical and physiological processes that are difficult to study in a non-invasive manner.
Perhaps the most frequent use of this tool is in the observation of physiological parameters
that are concerned with the transfer of fluid from one compartment of the body to another,
and in the biochemical observations of the formation of one compound from another.
Basic assumptions
As is true for most of the tracer kinetics, there are some basic assumptions in case of
PET quantification. The first basic assumption is that the tracer element follows its unlabeled
substance (“tracee”) faithfully in the process under investigation. Qualitatively there is
agreement on it; quantitatively, the rates of reaction are influenced by the isotopic composition
of the reacting molecule (known as isotope effect). Implicit in this primary assumption is
the condition that the injection of the tracer element shall not disturb, in any important
fashion, the normal metabolic behavior of the system. As a consequence of this assumption,
the tracer can be used to study the kinetics of tracee; the tracer provides the method of
measurement.
The second basic assumption is that in systems of constant volume the rate of flow of
tracee out of a compartment is proportional to the amount of tracee present in the
compartment. Thus the rate of flow may be assumed to be proportional to concentration.
A third basic assumption is that of uniform distribution throughout the compartments
(compartment is a volume or space within which the tracer rapidly becomes uniformly
distributed, i.e. it contains no significant concentration gradients). There are many
physiological conditions in which this assumption is invalid, such as, when the time of
mixing is long compared to the reaction rate. One of the challenges in developing PET
tracers is to ensure that these assumptions are not violated significantly.
46 1
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