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442
Deconvolution Analysis And Renal Transit Time Parameters
Figure 3: Upper part shows the renogram showing right and left kidney curves, input (heart) and
background curves. Lower part of the figure shows the transit curves for left and right kidney,
kidney parenchyma and pelvis respectively with their values in table shown.
Deconvolution Analysis And Renal Transit Time Parameters
443
Figure 4: Shows a single (left) kidney, input and background curves (Top), the corresponding
transit time curves are shown at the bottom of the renogram. Values of transit time parameters
(abnormal) are also given.
between these two ranges were assumed to be the borderline values (overlapping values of
normal and abnormal transit times) and should be interpreted individually for each patient.
Very short transit times should not be included in the normal range as such results manifest
the systematic error rather than abnormality in the kidney function. The error may be due to
incorrect algorithm or method used in the software (frequent occurrence) or may simply be
due to the non linear behaviour of the kidney function (rare event).
Figure 3 shows a typical normal renogram with its IRF curves for whole kidney,
parenchyma and pelvis, which are presented separately. The corresponding transit time
values (normal) are also presented. Figure 4 is renogram of a patient with single functioning
kidney. Both the whole kidney and parenchymal transit times are exceeding their normal
range.
Suitability of a renogram for deconvolution
Though there is a sound basis for using the deconvolution analysis in renography, the
results produced have not been consistent enough to find a widespread acceptance. While
444
Deconvolution Analysis And Renal Transit Time Parameters
some reports claim the successful discrimination of kidney disease with the help of transit
times (12) others argue that standard renography is better than the deconvolution technique
(13,14). Some have gone a step further by raising a skeptical question about the suitability
of renogram for deconvolution (15). They feel that the kidney is neither a linear nor a
stationary system.
It is not irrelevant to mention that for all types of quantitation some degree of
approximation is inevitable otherwise the physiological phenomena are too complicated to
be quantified by any technique. Application of deconvolution technique assumes that the
kidney is a linear and stationary system. This is a necessary assumption for all forms of
kidney function measurements and is not exclusively for deconvolution analysis. In recalculation of Rutland’s formula we could prove that this assumption is as necessary for
intravascular background subtraction as for deconvolution (10). Unfortunately this simple
fact has always been overlooked that if the kidney cannot (approximately) be regarded as
linear and stationary system then the reproducibility of renography, accuracy of GFR
measurement and many other quantitations would be under question. With a good
approximation kidney can be regarded as a linear and stationary system. Though, there are
situations where this assumption does not remain valid. Fortunately these situations are
often recognisable from the unusual pattern of IRF curves hence the result may be rejected
to avoid misinterpretation.
Virtually most of the evidences, which have been accumulated against the basic
assumption, are irrelevant because deconvolution is a linear operation hence, impurities
(background) in the input or output curves do not interfere with linear and stationary behaviour
of the kidney. Background is simply a source of systematic error in deconvolution analysis
as well as in any other types of quantitation.
Variation in renal extraction of radiopharmaceutical or changes in urine flow rate or any
other physiological variation may invalidate the assumption only if they have sufficient
amplitude. It has already been proved that physiological variation (of any kind) up to a
certain limit do not have significant effect on calculation of transit time parameters (16).
We agree that the cardiac time activity curve do not exactly represent the real input of
the kidney but it can be used to represent the input with a good degree approximation. It
could be shown that the cardiac curves used as input function could perfectly reflect the
activity in the kidney vasculature and consequently the input of the kidney (10).
If one does not accept the kidney as a linear and stationary system then a logical
question arises, why Tmax with no physiological interpretation is accepted as a reliable
clinical parameter whereas the mean transit time, which has a physiological basis, is not
accepted as meaningful parameter as it should have been. We believe that this is entirely
due to inadequate application of deconvolution technique.
A thorough review of literature on the subject shows that the inconsistency about the
usefulness of the deconvolution in renography originates from the incorrect algorithms and/
Deconvolution Analysis And Renal Transit Time Parameters
445
or methods used in the procedure and not from the basic assumptions in the application of
deconvolution. Though the technique is rather simple in principle but practically very difficult
to implement as an accurate and stable procedure. There are certain sources of error in
calculation of transit times, which different workers use different method to deal with. Many
efforts have been made in clinical application of the technique while little has been done to
improve and standardise the technique.
References
1. Stephenson JL. Theory of the measurement of blood flow by the dilution of an indicator. Bulletin of
Mathematical Biophysics 1948; 10: 117-121.
2. Zierler KL. Equation for measuring blood flow by external monitoring of radioisotopes. Circ Res
1965; 16: 309-321.
3. Van Stekelenburg LHM. Hippuran transit times : A new approch. Phys Med Biol 1978; 23 (2): 291-
301.
4. Fleming JS, Goddard BA. A technique for the deconvolution of the renogram. Phys Med Biol 1974;
19(4): 546-549.
5. Niemi, AJ. On discrete deconvolution. Med Biol Enginee 1976; 14: 582-584.
6. Valentinuzi ME, Montaldo Volachec E. Discrete deconvolution. Med Biol Enginee 1975; 13: 123-125.
7. Bererhi H. Z-Transform method for deconvolution as applied to the renogram. Nucl Med Commun
1995; 16: 161-167.
8. Rajabi H and Pant GS. Optimum filtration for time activity curves in nuclear medicine. Nucl Med
Commn 2000; 21: 823-828.
9. Sutton DG, Kempi V. Constrained least-squares restoration and renogram deconvolution: a comparison
by simulation. Phys Med Biol 1992; 37(1): 53-67.
10. Rajabi H. Deconvolution analysis in Renal Disorders, Ph.D. thesis, AIIMS, New Delhi (India) Jan,
1998.
11. Britton KE, Nimmon CC, Whitfield HN, Hendry WF. Obstructive nephropathy: Successful evaluation
with radionuclides. Lancet 1979; 28: 905-907.
12. Mizuiri S, Hayashi I, Takano M, Ban R, Ohara T, Sasaki Y, Hasegawa A. Fractional mean transit time
in transplanted kidneys studied by technetium-99m-DTPA: comparison of clinical and biospy findings.
J Nucl Med 1994; 34(1): 84-89.
13. Kempi V, Sutton DG. Estimating the diagnostic yields resulting from renography and deconvolution
parameters : A logistic regression analysis. J Nucl Med 1995; 36(1): 147-152.
14. Russell CD, Japanwalla M, Khan S, Scoot JW, Dubovsky EV. Thechniques for measuring renal
transit time. Eur J Nucl Med 1995; 22: 1372-1378.
15. Ham HR. Is renography suitable for deconvolution analysis? (Letter to the editor). J Nucl Med 1996;
37: 403-404.
16. Gullquist RR, Fleming JS. Error analysis by simulation studies in renography deconvolution. Phys
Med Biol 1987; 32(3): 383-395.
Radiotracer Kinetics: Applications
in Nuclear Medicine
J. Bharathi Dasan
The principles of tracer kinetics have many applications in nuclear medicine. A list of
nuclear medicine quantification techniques, which make use of these principles, would be
exhaustive and would encompass almost every quantitative procedure used in this field.
Therefore instead of making this chapter into a boring litany of such procedures let us try to
understand the ways in which the principles of tracer kinetics are applied in day to day
nuclear medicine procedures by way of a simple example in the form of GFR estimation. We
will look at how the process of GFR estimation has been simplified over the years and the
assumptions and mathematical models, which have enabled such simplification. At the end
of this chapter I hope the reader will have a fair idea on how these assumptions and
applications can be used in other areas of nuclear medicine.
The kidney clears the plasma of its impurities; therefore renal clearance is the ideal
method for quantitation of renal function. To know what clearance is, we must start from
Fick’s law, which implies that;
At steady state (a state when input into a system is equal to the output from the system)
amount of substance extracted by an organ from the blood (Q) per minute is given by:
Q= [Arterial concentration of substance (A) – venous concentration of substance (V)]
Blood flow (BF) (All with respect
If the organ in question is the kidney and if the substance is dissolved only in plasma then
t in the particular organ in question).
o
RPFVAQ
])()([
PP
])()[(/,
VAQRPFTherefore
PP
44 6
Radiotracer Kinetics: Applications in Nuclear Medicine
RPF
( ) ( )
p p
Cu Vu
A V
( )pCu Vu
''P
Cu Vu
447
Where
plasma concentration of the substance and
is Renal arterial plasma concentration of the substance,
pA)(
is the renal plasma flow.
is renal venous
pV)(
For a tracer excreted by the kidneys, amount of tracer removed from plasma per minute
‘Q’= Amount of tracer excreted into urine per minute = urine concentration of substance
x urine volume formed per minute
Therefore RPF =
.
)(Vu
(1)
Therefore in order to estimate the renal plasma flow the renal artery, renal vein and
urine will have to be sampled. When the substance is almost completely extracted by the
kidney during a single transit through the kidney, the renal vein plasma concentration of the
substance Vp is negligible.
The RPF then can be expressed as:
RPF
A
(2)
)(Cu
If the tracer is excreted exclusively by kidney, the renal arterial concentration
A)(
is the
p
same as peripheral venous plasma concentration
Therefore RPF =
(3)
P
Since the extraction fraction [(arterial – venous concentration)/ arterial concentration] is
never 100% for any tracer, some of the plasma flow to the kidney perfuses into nonexcretory areas like renal pelvis, peri-renal fat and the renal capsule the value obtained by
equation- 3 is called effective renal plasma flow (ERPF). The ERPF can otherwise be called
the renal clearance value of a substance, which is almost completely extracted from the
plasma during a single transit of the plasma through the kidney. Here we have chanced upon
the term renal clearance. So let us define this term before proceeding further.
Renal Clearance, ERPF and GFR
The kidney can clear the tracer from the plasma exclusively by glomerular filtration
(e.g., inulin) or by a combination of glomerular filtration and tubular secretion (e.g.,
creatinine).
GSPant\Newbook\Final-200\29-chp\447
448
Vu Cu
.,.e
i
Radiotracer Kinetics: Applications in Nuclear Medicine
If the tracer is almost entirely cleared from the plasma during a single transit, then the
clearance gives the ERPF. Para amino hippuric acid (PAH) is one such tracer, and
measurements of ERPF obtained using PAH and continuous infusion techniques are taken to
constitute the gold standard in ERPF measurements. Among radiotracers,
131I
OIH (Ortho
iodo Hippurate) closely resembles PAH in its behavior.
If a tracer is freely filtered but neither secreted nor-reabsorbed, then it is cleared
exclusively from the fraction of plasma volume filtered across the glomerulus. The rate of
clearance of such a tracer therefore corresponds to the glomerular filtration rate (GFR).
Inulin is one such tracer, and continuous infusion estimates using inulin are taken to
constitute the gold standard in GFR measurements. Among various radiotracers, the
behavior of
99m
Tc-DTPA closely approximates that of inulin in various experimental
studies. In this discussion let us go back to equation-3 for ERPF.
ERPF =
P
This equation requires that a) we sample the urine, and b) the peripheral venous plasma
concentration of the substance be kept constant. Since urine sampling and continuous infusion
are cumbersome, other methods have been developed which seek to bypass this hurdle.
From urine sampling to plasma sampling
The rate at which the tracer is excreted into urine matches its rate of dissapearance from
plasma when the tracer is eliminated exclusively by the kidneys, a condition satisfied by
many tracers like DTPA and OIH. Hence plasma can be sampled instead of urine.
From continuous infusion to single injection
Now before we proceed further, let us define some useful frrequently used terms. These
definitions deal with the spatial and temporal variation in tracer distribution within a system
(biological or otherwise).
Some definitions
Equilibrium : Tracer distribution in a system is invariant in time and space
no input or output.
Steady state : There is spatial but not temporal variation in tracer distribution within a
system i.e. input equals output; for example during continuous infusion of a strictly
intravascular tracer, there is variation in tracer concentration between intravascular and
extravascular spaces but a constant concentration within each.
there is
Compartment : A system can be thought to be made up of a series of compartments,
each of which is characterized by uniform spatial concentration of tracer at any instance in
GSPant\Newbook\Final-2008\29-chp\448
Radiotracer Kinetics: Applications in Nuclear Medicine
449
time, although the tracer concentration may vary with time. When different tracers are used,
a varying number of compartments are assumed (e,g, intravascular vs extravascular or
intracellular vs extracellular), thereafter we mathematically model the behaviour of the
tracer within the body by calculating the rates of exrchanges of tracer in between these
compartments. For example after injection of tracer into the plasma, the plasma can be
assumed to be one compartment and the red cells to be another . The interstitial fluids can
be taken to be yet another compartment. Therefore if such a tracer were to be excreted into
the urine its behaviour inside the first compartment i.e., plasma will not only depend on its
excretion into the urine but will also depend upon the rates at which it can go to and forth
between this compartment and the others. However at this juncture a couple of points need
to be highlighted,
1. A Compartment is not always anatomical, it can be physiological or chemical.
2. Compartments are hypothetical, they do not truly exist in the body since distributions
of tracer in the body vary continuously in space rather than being uniform (eg.
capillaries in different parts of the body have different permeability – parts of Extra
Cellular Fluid near capillaries have different concentration than those a little further
away).
Having learnt what the term compartment stands for, let us see how compartmental
analysis can be used in GFR estimation using DTPA. The following discussion will highlight
some of the common place assumptions behind the use of compartmental analysis in various
quantification procedures in nuclear medicine.
GFR estimation by plasma sampling after single injection
So far in the measurement of clearance we have surmounted :
1. Renal vein sampling (by using a tracer with high extraction fraction)
2. Renal artery sampling (by using a tracer eliminated exclusively by kidneys and thus
substituting it with peripheral venous plasma)
3. Urine collection (by using a tracer eliminated exclusively by the kidneys). Now we
will supplant continuous infusion with single injection by studying the behaviour of
the tracer in the body after single injection.
Following single i.v. injection the rate of tracer elimination will be dependent on plasma
concentration and renal function alone, if and only if the tracer were to remain confined to
plasma alone during the study. This also requires that the tracer must be eliminated by the
kidney alone (i.e. no extra-renal clearance).
While the latter condition may be satisfied, the former is difficult because:
(i) Tracers leave the plasma to extra cellular fluid in capillary beds all over the body.
(ii) They may back diffuse from these tissue fluids once plasma concentration starts falling.
GSPant\Newbook\Final-200\29-chp\449
450
Radiotracer Kinetics: Applications in Nuclear Medicine
With this in mind Saperstein studied the distribution of tracers in the body in an
experiment, which substantiated certain crucial assumptions that are made in all clearance,
studies, used to estimate GFR or ERPF by single plasma injection of tracer (1).
Sapirstein’s model
Sapirstein (1) while studying the distribution and clearance of creatinine in dogs used a
two compartment model and made the following assumptions (which the study validated).
1. Upon introduction into the blood stream, the tracer is homogenously mixed through
a uniform compartment (first compartment).
2. The mixing throughout this compartment is extremely rapid in relation to removal
from this compartment.
3. It is assumed that the compartments penetrated by creatinine are arranged in series
rather than in parallel. That is to say; the penetration of the second compartment can
occur only by way of the first. Its implication is that the entire first volume of tracer
distribution becomes homogenously mixed before the second is significantly
penetrated.
4. It is assumed that the tracer leaves its first volume of distribution for the bladder at
a rate which is proportional to its concentration in the first compartment of
distribution.
5. It is further assumed that the tracer leaves this compartment for the second
compartment at a rate proportional to the difference in its concentrations between
the first and second compartment.
6. It is assumed that the tracer is not metabolized to any significant extent.
7. It is assumed that throughout the period of observation, the bladder and
intracompartmental clearance of the tracer are stable.
The following discussion is based on the model illustrated by Figure 1
I injected dose
V1 volume of Ist compartment
V2 volume of 2nd compartment
G Excretory clearance from the first compartment
C1 Concentration in Ist compartment
C2 Concentration in 2nd compartment
kinetics between V1 and V
2
GSPant\Newbook\Final-2008\29-chp\450
Radiotracer Kinetics: Applications in Nuclear Medicine
dt
dt
Figure 1: Model showing two compartments
451
The rate at which the tracer leaves the compartment one = volume
in concentration
.
)/(
dtdC
1
x rate of change
)(1V
This depends on G, , C1 and C2 as follows :
dC
V
1
1
The negative sign indicates that in
The amount injected I = Amount in
dC
1
V
V
is decreasing with time.
1
+ Amount in
1
][-
CCGC
211
+ excreted amount till time t Lt at
V
2
which the measurements are being made.
LtVCVCI
2211
If we assume that there exists an infinitesimally small period of timedt between the
time tand
constant, then the excretory clearance from
as
dL can be got by
t
during which the tracer concentration
dtt
during this small period of time dt represented
V
1
C in volume
t
can be taken to be
V
1
(4)
(5)
GSPant\Newbook\Final-200\29-chp\451
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