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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5255_Библиотеки_им_академика_М_И_Перельмана
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432
( ) ( )
I H t d
( ) ( )
I H t d
( )
0
if t
( ).
d
0
if t
Deconvolution Analysis And Renal Transit Time Parameters
of radiopharmaceutical through the kidney. The configuration of the renogram not only
depends on the kidney function but also upon the manner in which the radiotracer arrives in
the kidney (input function). The input itself depends upon the site and speed of the bolus
injection and also the systemic blood re-circulation. Moreover, not all the counts recorded
over the kidney regions are related to the kidney function. The activity in the blood vessels
and the overlapping tissues (background) has a significant contribution to the configuration
of a renogram. Due to this complexity, the direct quantitation of renogram leads to generation
of the parameters, which may not have a physiological interpretation. They simply quantify
the renogram and not necessarily the kidney function. Though, they are still useful parameters
for the evaluation of relative kidney function but in cases where both the kidneys or a
solitary kidney have/has an abnormal renogram pattern, the test loses its objectivity. This is
particularly so in the event of injection failure where the renogram may show a false/
unnatural pattern.
To preserve the objectivity of renography and to calculate the physiologically meaningful
parameters two types of corrections (deconvolution and background subtraction) should be
done. Before describing the method of deconvolution some basic concepts need to be
understood.
Stimulation of a system with a known input and analyses of the response produced by
the system (output) is a general method to study the function of a system as a whole (blackbox). In a linear and stationary (shift invariant) system the relationship between input function
I(t), output function R(t) and system function H(t) is given by the convolution integral;
R(t) =
(1)
where is a dummy variable (time). Generally there is no similarity between system function
and output functions but in an ideal situation where the input is in the form of a -function
(Dirac function) they may be quite similar. Normally the -function is defined as a function
which has an undefined amplitude at the time of occurrence (t0) and zero elsewhere, with an
additional property that the area under the function is unity.
(2)
(3)
R(t) =
=
=
0
1
Delta function can be considered as an impulse of large magnitude and infinitely short
duration. Regarding the linearity property the product of -function and an ordinary continuous
function e.g. H() may be defined by;
Deconvolution Analysis And Renal Transit Time Parameters
( ) ( )
0
( )
H t
( ) ( )
H t d
0
( )
H t t
433
H
=
(4)
In equation-4 if the function H() is changed to H(t – ) or in other words shifted by
time t (shift invariant property) then the said equation may be written as;
=
(5)
Comparison of the equation-1 and equation-5 reveals that if the input function is in the
form of a -function then the output represents the system function. Due to this similarity
the system function is usually called impulse response function (IRF). Obviously, the function does not exist hence; the output is always the convolution of the input function and
system function. However, in real situation (where the input is not a -function) it can be
assumed that instead of one -function (at an specific time) there is a train of -functions
spreading from time zero to to. The amplitude of each is equal to the height of the
corresponding point on the input function. Each of these imaginary inputs produces its
corresponding output. Hence there will be a train of output functions with different amplitudes
spreading in time. The amplitudes are proportional to the amplitudes of corresponding
points on the input function. The linear sum of all these imaginary outputs is the convolution
of input function and system function (Figure 1).
Figure 1: Top portion shows a dirac function and its corresponding output function (system
response). Bottom portion shows that the input curve of a renogram can be assumed to be made
of a series of unit input functions for which there are corresponding output or system response
functions. The right hand side of lower part shows that renogram is a convolution of input and
system response function.
434
{ ( )}
L I t
I s I t e dt
Deconvolution Analysis And Renal Transit Time Parameters
In renography a small volume of activity is administered intravenously to the patient. At
the time of injection, the bolus may approximately represent a -function. However, due to
the mixing effect, the bolus is rapidly smeared and hardly remains a bolus by the time it
reaches the kidney. Moreover, not all the activity that enters the kidney is filtered (or
excreted), a fraction of it always remains in the blood circulation. Due to these two phenomena
(mixing and re-circulation) renogram can never represent the kidney retention function
(IRF) and is the convolution of retention function and input function of the kidney (1,2).
Deconvolution (the inverse operation of convolution) is a mathematical procedure for
derivation of either the input or system function from the two other functions. In renography
deconvolution is used to extract the impulse response function of the kidney from convoluted
renogram. The calculated IRF represents the form of renogram that would be obtained if an
injection could be given directly into the renal artery and if re-circulation could be prevented.
The major advantage of deconvolution over direct quantitation of renogram is that the
parameters derived from IRF have physiological meaning unlike those derived from the
renogram. Deconvolution provides quantitative information about the absolute functional
status of the kidney to help the physician for a better objective interpretation of the renogram.
The essential condition for validity of convolution integral is that the system should be
linear and stationary. Hence for application of deconvolution technique in renal analysis the
kidney must be assumed as a linear and stationary system. However, this assumption may
not be valid in some of the kidney conditions. The time activity curve recorded over the
cardiac region is normally used to represent the kidney input. The heart should, therefore, be
included in the field of view of gamma camera during dynamic image acquisition.
Methods of deconvolution
Except in some extraordinary situations, where the input and output functions have very
simple equation there is no analytical (exact) solution for the convolution integral. However,
it is possible to develop alternative methods, which are approximations over the exact
solution. For this purpose the convolution integral has to be reformulated so that it can be
solved by arithmetic operations. Different approaches have been proposed for deconvolution
as mentioned below.
Laplace transformation
The Laplace transform of a function I(t) is defined by :
=
( ) ( )
0
where I(s) is the Laplace transform of the I(t). One of the most important properties here is
that the Laplace transform of the convolution of two function is the product of the Laplace
transforms of the two functions (Lassen 1978). Hence the convolution integral in equation-1
may be changed into the simple product of two function as;
st
(6)
Deconvolution Analysis And Renal Transit Time Parameters
( )
R s=( ). ( )
I s H s
( )
H s= ( )/ ( )
R s I s
{ ( )}
F I t
I I t e dt
435
(7)
where I(s), R(s) and H(s) are the Laplace transform of the input, output and system
function respectively. If R(s) and I(s) are known, it is easy to obtain H(s) using the
equation;
(8)
Inverse transformation of H(s) will provide the desired impulse retention function of the
system. However, in renography this method has never been used in its original form due to
complexity in transformation of the input function and inverse transformation of the resultant
function. The approximation of the proposed method was found to have limited application
(3,4). Both the methods require an assumption that the input and IRF have a predefined
constant pattern.
Fourier transformation
The Fourier transform of a function I(t) is defined by
=
( ) ( )
0
where the I() is the Fourier transform of the I(t). This transformation has also the same
property that the product of individual Fourier transform of the input function I() and
system-response function H() is equal to the Fourier transformation of the output function
R(). Mathematically:
i t
(9)
R() = I() . H() (10)
Hence,
H() = R() / I() (11)
The inverse Fourier transformation of H() results in the generation of IRF [H(t)]. The
main advantage of this technique is that there are standard numerical algorithms available
for transformation of the functions and therefore, no approximation or simplification is
required. However, for successful transformation of the function some required promises are
difficult to be satisfied in routine renography (5).
Matrix method
If I0 is the first input, (H0) the corresponding IRF then the output function can be given in
expanded form as:
R0= I0. H
0
436
1
R
1 0 0 1
. .
I H I H
n
H
0 0 0 1. 1 0.
/ / /
R I I H I I H I
n
H
(1/ ).[ ]
o n n k k
I R I H
n
H
(1/ . ).[ ]
o n n k k
I t R I H
Deconvolution Analysis And Renal Transit Time Parameters
Similarly
=
----- ---------------------------------
----- ---------------------------------
HIHIHIR ...
nnnn
0110
Convolution integral in equation-1 is in the continuous form and can be expressed in
discrete form as:
n
HIR
k
knn
0
n
tHIR
..
k
knn
0
R , kI and knH
n
are the discrete values of
or
being the frame length in seconds and
t
,
)(tR
)(I
equation for
and
H yields:
n
k
k
respectively, in the nth, kth and (n-k)th intervals. Solving this
)(
tH
(12)
/ IRH
000
----- ---------------------------------
----- ---------------------------------
=
n n n
=
Incorporating the frame length we get
=
// IHIIRH
001011
n
.
1
k
n
k
.
1
(13)
Deconvolution Analysis And Renal Transit Time Parameters
{ ( . )}
Z I nT
( )
I z
( , )
n
I n T z
( 0, 0)
0
H
/ .
R I t
n
H
n n k k
I t R I H
( 0)
( 0)
0
H
n
H
n n k k
I t R I H
( 0
0)
0
H
/ .
R I t
n
H
n n k k
I t R I H
437
The accuracy of this method depends on three factors; sampling interval, noise in the
output curve and selection/accuracy of the first point of input curve (6).
Z-Transformation
The z-transformation of a discrete-time function I(n, T) is defined as ;
=
where I(z) is the z-transform of the I(n.T), z is complex variable and T is the time interval
(t) and n is an integer. Application of this transformation on the discrete form of convolution
integral given in equation-12 leads to three distinct algorithms for calculation of IRF (7).
=
n
(14)
0
1. The case where the first point of input and output are zero
=
1 1
n
(1/ . )
= 1 1 1.
2. The case where the first point of input is non zero
0
k
I
0
I R
and output is zero
then:
= 0
1
n
(1/ . )
=
0
k
1
3. The case where the first point of input and output are nonzero
=
0 0
n
=
(1/ . )
0
k
1
0 0
I
0
,
R
then :
0
R
0
then:
(15)
(16)
(17)
Comparison of the equation-17 and equation-13 clearly shows that they are exactly the
same. Equations-15 and 16 can easily be obtained by shifting the values of the input and/or
output function.
Preparation of the data for deconvolution
It is mandatory for deconvolution that all the image frames are of the same duration. If some
initial frames are of shorter duration then they have to be binned so as to make the frames of
438
Deconvolution Analysis And Renal Transit Time Parameters
equal length. While processing the image data for deconvolution special care has to be taken
to contain the noise. Presence of even minor error in the raw data may result in large
perturbation in IRF and may even destroy the results. Matrix method is particularly sensitive
to the noise in input rather than the output curve. Noise is superimposed on the renogram
and deconvolution operation is severely affected with the presence of noise. The calculated
IRF may be considered as superimposition of two components, deconvoluted renogram (real
IRF) and the deconvoluted noise. Using the convolution theorem it can be shown that;
H() = Hr() + N() / I() (18)
where H() is the Fourier transform of calculated IRF, Hr() is the Fourier transform of
real IRF, I() is the Fourier transform of input function and N() is the Fourier transform of
noise in renogram. It is clear that the difference between real and calculated IRF is highly
dependent on relative amount of noise in renogram N(). Two approaches may be used,
which are described below to reduce the noise component and improve the signal to noise
ratio in IRF.
Filtering of noise in renogram
If the signal to noise ratio in input is improved then the noise component (N()/I()) of
IRF decreases. Digital filters are the mathematical tools that are used to eliminate these
unwanted components of the data. Unfortunately the complete removal of the noise is possible
only if the noise and real data have separate frequency components. Virtually the renogram
and incorporated noise have some common frequency components. Removal of the noise in
overlapping frequencies results in distortion of the curve due to elimination of the common
frequency components of the real data. Performing the optimum filtering in such situations
is a very difficult task particularly, if a filter with a short cut-off frequency and a slow rolloff is to be used. A new method of noise reduction has been suggested to perform the
optimum level of smoothing (8).
Constrained deconvolution
An alternative method is to find an estimation of H() which minimises some predetermined
criteria. Sutton and Kempi (9) used the Wiener filter to minimise the square mean error
between calculated and real IRF. For this purpose the renogram must be free of background
which is practically unachievable.
Background subtraction
Background subtraction is not directly related to deconvolution analysis but background has
a great influence on accuracy of the result. The effect of background can be analyzed in
simulated and real data. For the calculation of transit time parameters modified method of
Rutland may be used to subtract the intravascular background as has been described in the
Deconvolution Analysis And Renal Transit Time Parameters
439
chapter of “Background Subtraction” in this book. This really helps in establishing a stable
method of deconvolution for estimating renal transit time parameters (10).
Deconvolution software
A deconvolution software should be very stable and accurate under the existing circumstances
at a given centre. The inconsistency in selection of the methods and design of the algorithms
used, have caused a significant discrepancy in the results produced by different centres. We
analysed these procedures in great detail and developed a method with some modification.
We feel that the technique can be used anywhere with the same amount of accuracy and
reproducibility. While developing software for deconvolution the following points should be
kept in mind. The developed software should be validated using simulated data (10). The
technique should then be standardized in normal healthy donors and also for generating the
range of normal transit time parameters. For clinical validation the technique may be evaluated
in some of the proven cases of kidney disorders particularly in obstructive uropathy.
Data acquisition protocol
Patients are normally positioned in supine position in front of the camera in such a way that
heart is also included in the field of view of the camera. They should be hydrated prior to
image acquisition. A photopeak at 140 keV with 20% window is normally used for data
acquisition with LEAP/LEGP collimator. Intravenous administration of 200-300 MBq
DTPA (or 100 MBq of
99m
Tc MAG3) in a 0.4-0.5ml bolus may be adequate. Sequential
images should be recorded in a 6464 or 128128 matrix. The first few frames may be of
shorter duration followed by 120 or more frames with longer duration (10 sec). The total
duration of dynamic acquisition may vary from 20 – 30 minutes depending upon the protocol
used at a given centre.
99m
Tc-
Static images in 256256 matrix may also be acquired as per protocol though static
images are not required for deconvolution analysis. They are acquired to see the delayed
excretion of radioactivity from the kidneys. The first static image is usually taken immediately
after completion of the dynamic study, the second after 2 hr and the third after 24 hr, if
needed.
Data processing
Three regions of interest (ROI) are drawn for each kidney namely the whole kidney (WK),
kidney parenchyme and kidney pelvis. Whole kidney (WK) ROIs are normally drawn on
the 1-3 minute composite image. Whenever the edge of the kidney is not clearly visible
(poor functioning kidney) the software may be used to generate the mean time image.
Kidney pelvis ROIs (we used in our work) are drawn on a mean time image (11). Whenever
the mean time image could not delineate the pelvis from parenchyma perfectly the last
minute composite images can be used. Background ROIs are drawn on a sub-renal area
440
Deconvolution Analysis And Renal Transit Time Parameters
avoiding the ureteric coverage. The whole procedure is repeated for other kidney separately.
Cardiac ROI is drawn on 0-1 minute composite image. The corresponding time activity
curves (TAC) for each reigion are generated and the pelvis curves are subtracted from the
corresponding whole kidney curves to obtain the parenchymal time activity curve. The
kidney, background and cardiac curves are scaled to counts per second, and temporarily
filtered and displayed on the monitor for visual inspection and direct quantitation.
The original curves can then be transformed into isotime with frame duration of 10
second. A suitable filter may be applied to all the curves. Cardiac curve filtered from point
4 onward and renal curves from the minimum point after the vascular peak. This is quite
necessary due to the fact that the initial parts of the curves are composed of very high
frequency components. Low-pass filtering causes unacceptable distortion in this part of the
curves. The data may be rebounded to make the frame length as 20,30 or 40 second per
frame depending upon the need. After rebounding, the data can again be filtered using a
proper filter.
Deconvolution is performed for whole kidney, parenchyma and pelvis curves up to the
last point. For the whole kidney and parenchyma, the cardiac curve should be used as the
input. But for pelvis the parenchymal curve after proper background subtraction can be used
as the input. The quantitation is performed on the resultant IRF curves after filtering and
results are displayed on the monitor.
The height of plateau can be calculated taking the average of the three points between 1
to 2 minutes (10). Minimum transit time (Min TT) is defined by the first point on the
plateau such that its next adjacent point has a height 5% less than the plateau height.
Maximum transit time (Max TT) is defined as the first point on the curve with a height 97%
less than plateau height. Whenever such a point is not found (when the actual point is
beyond the study duration) the last but one point can be selected as MaxTT and the physician
may be informed that the actual values of the parameters are more than the calculated one.
Mean transit times (MTT) is calculated using standard arithmetic method. Transit time
index (TTI) can be calculated by dividing the area below the IRF curve between Min TT to
Max TT by the height of plateau as shown in figure 2.
Background subtraction
One may ignore to subtract extra vascular background but intra-vascular background
subtraction is important. The details of background subtraction (extra and intra-vascular)
have been described elsewhere (10) and also as separate chapter in this book.
Simulated study
Before applying the software to clinical studies simulated study may be performed. For this
purpose a software phantom may be developed to generate data of different types. The
method of generation of input, output, system response, background and noise is described
elsewhere (10).
Deconvolution Analysis And Renal Transit Time Parameters
Figure 2: Shows the graphical representation of IRF curve, where transit time parameters are
also defined.
441
Standardisation of normal values
Normal transit time parameters should be derived from studies on normal volunteers. We
used data from 36 donors using 72 normal kidney renograms after subtraction of intravascular
background. All the values had an approximate normal distribution around the corresponding
average values. There was a negligible difference between average values, median and
geometrical mean for all the transit times calculated. The skewness was used to evaluate the
symmetry of data around the mean. Almost all the parameters had a very small skewness
towards the right. Maximum skewness was observed for maximum transit time. The
observation was quite expected due to the presence of extravascular background. The mean
values with standard deviation for normal range of all parameters are presented in table-1.
Table 1 : Normal transit time parameters (10).
Transit times Whole kidney Kidney parenchyma
Min TT 139 ± 32 118 ± 20
TTI 117 ± 25 101 ± 23
MTT 237 ± 50 199 ± 35
Max TT 336 ± 80 315 ± 70
For all the parameters (approximately) the mean ± 2 s.d. was accepted as normal range
(95% confidence interval) and values beyond ±3 s.d. were treated as abnormal. The values
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