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3.4. Comprehensive evaluation of dynamic renal scintigraphy
Radioisotope methods of kidney examination are among the most frequent procedures in nuclear medicine. In comparison with already practically almost deserted renography easily by using a pair of collimated scintillation detectors using scintillation cameras provides the opportunity to visually assess Scintigraphic images of the kidneys and a detailed and comprehensive analysis of the kinetics of the administered radiopharmaceuticals le d trespasses, parts and urinary tract. Dynamic scintigraphy of the kidneys therefore serves to qualitatively and quantitatively evaluate the functional ability of the kidneys , their perfusion and the kinetics of the upper urinary tract . A comprehensive mathematical evaluation of this dynamic study includes the following main points :
Radio indicator application and data storage
Investigations carried out either sitting or lying down, the detector scintillation camera fitted with a corresponding collimator (by energy gamma radionuclide used) is attached to the back of the patient so that the kidneys were just under half of the field of view (in the upper part of the visual field is then sufficient margin to be displayed n in the blood pool). Prepare an appropriate radio indicator with an activity of approx. 100300 MBq for ^{99m}TcDTPA or MAG3 in the syringe , or about 1520 MBq for ^{131}Ihippuran. The position of the detector is specified by applying a syringe with prepared activity above the proc before application. xyphoides  on a monitor or persistent oscilloscope screen, the syringe should be displayed at the cranial border of 1/3 to 1/4 of the midfield. For precise positioning (especially for atypical placement of the kidneys), the method of d and preapplication of a small amount of radioindicator (23 MBq  does not significantly affect the dynamic study itself), which makes the kidneys visible on the camera monitor after 24 minutes , which we then set to the field of view without any problems. We then quickly apply the indicator (we recommend rinsing the syringe with saline  a threeway valve is advantageous) and start accumulation of dynamic scintigraphic study ^{*)} .
^{*)} If we want to determine the absolute
value of kidney function (clearance or glomerular filtration) by blood sampling , measure the syringe with a radio
indicator in a suitable standardized way (eg calibrated
scintillation detector  see note)   we obtain the value of
applied activity in relative units [number of pulses / unit of
time]. Measure the activity remaining in the syringe after
application (including the background) and subtract it from the
applied activity. During the examination, we will take one or two
blood samples in the twentieth to thirtieth minute. After
centrifugation, measure the activity of 1 ml. plasma by the above
detector in the same relative units (see note). After the end of the dynamic
study, we insert into the commentary to the study the measured
relative value of the applied activity, the remaining activity
after application, the measured activity of 1 ml. plasma together
with time data of the sampling time from the application and the
time interval between the measurement of the applied activity and
the 1 ml sample. plasma.
Note: To
measure the applied activity and the activity of a 1 ml sample.
plasma, we recommend using a well scintillation detector (eg NKG
314) equipped with a suitable lead collimation insert (hole
diameter approx. 4 mm) and a crossbar. The syringe with the
activity for application is measured on a crossbar at a height of
approx. 4O50 cm above the detector with the collimation insert
fitted. In the same arrangement, we measure the syringe with the
remaining activity after application. Sample 1 ml. plasma
measured in a test tube inserted é into the well of the
scintillation detector. Between measurements of activity in both
of these diametrically different geometric arrangements, a
conversion factor of high value (of the order of tens of
thousands) applies. This factor is best determined by dilution
measurement: Draw a suitable radionuclide activity into the
syringe (eg approx. 2050 MBq ^{99m }Tc) and measure the number of
pulses on the crossbar via the collimation insert. Then we mix
this activity perfectly in a larger volume (preferably in 10
liters) of water. Then take a 1 ml sample. and measure the number
of pulses in the well of the scintillation detector. The
conversion factor F is then determined from the relation F = VN _{o }/ N _{1 }, where V is the
volume of water in ml., N _{o }is the number of syringe pulses on the
crossbar and N _{1}is the number of sample pulses
of 1 ml. in the detector well. If we keep the geometry of the
measurement, this factor will apply in the long run  we
recommend recalibrating it about twice a year. However, it is
advisable to perform a simple check of the detection efficiency
of the measuring system before each series of measurements.
Recommended storage mode :
180 images after 10 sec. , 64 x 64 matrix , 16 bit.
If we need to quantify
perfusion, we store in two groups:
Group 1: 60 images for 1 sec.
Group 2: 174 frames after 10 sec.
The recommended storage time is 30 minutes. However, if we see a sufficiently fast passage of the radio indicator on the display during storage and we do not need an accurate determination of the global function, the examination can be terminated earlier, eg in the 20th minute. Conversely, when we observe the retention of the radioindicator, it is appropriate to apply a diuretic (furosemide) in the 15th20th minute (we will write down the time of diuretic application) and continue to save until 30.m i n. When evaluating the study, we then evaluate the response of retention to the diuretic. If necessary, we can also make static scintigrams of late phases, which can then be evaluated simultaneously with the dynamic study.
Study evaluation
After invoking a scintigraphic study in the basic OSTNUCLINE system, we will launch a comprehensive program RENDYN  dynamic scintigraphy of the kidneys .
Visual evaluation of sequential images
First, a series of appropriately absorbed images (together with the values ??of the respective time intervals) is created on the screen, capturing the distribution and course of accumulation of the radioindicator in the kidneys and its gradual excretion into the bladder. This sequence of images to our modulation objectively reflect the dynamics of tracer concentration, we can set scaling individually characterized ch images to a common maximum (usually chooses the most out of the picture in 3 to 5 min.) According to these images, select the preverbal assessment , both implicit standard formulation of normal evaluation , e.g.
“After intravenous administration of
the radioindicator, the kidneys of the usual
shape, size
and placement
are displayed , without focal changes. The
nephrographic curves of both kidneys have a normal course, we do
not observe a slowdown of drainage or retention
of the radioindicator in the hollow kidney system.
Conclusion:
Visual evaluation with equivalent scintigrams as well
as quantitative analysis of
nephrographic curves indicate good function of both kidneys,
rapid
transit through the parenchyma and free drainage of the hollow
system.
Signature: MUDr. ” ,
thus inserting nonstandard text describing the relevant pathology. A series of images together with a verbal evaluation can be printed for documentation (Fig.3.4.0), but this is usually not necessary, as images of significant phases are included in the resulting protocol.
Designation of areas of interest and creation of curves
The following areas of interest are characterized for the quantitative analysis of dynamics :
Bloodstream .........
ROI 1
Left kidney ........... ROI 2
Right kidney .......... ROI 3
Optional (if
retention):
Left renal cortex ...... ROI 4
Right kidney cortex ..... ROI 5
Tissue background .......... RO I 6
For the area of interest of the blood pool, we will use the images immediately after the arrival of the activity. As ROI1, we mark the perfused structures above the kidneys here, it is desirable to include the heart area if it is in the field of view. The areas of interest of the kidneys are marked in the following images of the parenchymatous phase. If we are interested in the transit functions and times of the parenchyma and pelvis separately, we mark the ROI of the cortex of the right and left kidneys in the images in the excretion phase. These ROIs will be moonshaped, with the inner part avoiding the pancreas and the urinary tract, the outer part running along the ROI2 or ROI3 of the whole kidney. When (optional) marking the ROI6 of the tissue background, we must take care to avoid the kidneys, strongly perfused areas and urinary tract (we recommend mlunarshaped areas far enough from the outer lower side of the kidneys). The program creates curves of the time course of radioactivity from the marked areas of interest, which eventually corrects on a tissue background.
Before the actual mathematical processing of dynamic curves , the program asks what kind of sequential scintigraphy of the kidneys it is: whether it is an examination of glomerular filtration (using ^{99m}Tc  DTPA), an examination of tubular function (using ^{99m}Tc  MAG 3), an examination using ^{131}I  hippuran. Furthermore, whether it is a native study or after exposure to captopril. According to these answers, the execution of calculations is adapted, the respective Save Area is installed and the terminology of the calculated parameters is generated.
The mathematical processing of the curves follows . At all stages of processing below, the program first asks if we want to execute them, and they are executed only if the answer is yes. This saves evaluation time in cases where we are only interested in some parameters, or only visual and qualitative data.
If a fast group of images capturing the perfusion phase has been recorded, the renal perfusion can be (optionally) analyzed. Displays the initial (i.e. perfusion) phase curves of both the kidney and can set the display scale in the horizontal direction (compressionexpansion) for optimal P r ezentaci perfusion dynamics. On perfusion curves of both kidneys is then automatically (with manual correction) feature points of arrival radiotracer top of the first pass of the bolus and "valley" plata separating perfusion and parenchymal phase of nefrografic to curve. For the left and right kidneys, indices quantifying perfusion are calculated: Washida's perfusionflow index, which, based on the ratio between the descending and ascending arms of the peak of the first bolus pass, quantifies the part of the blood  borne radio indicator that flows through the kidney and continues through the bloodstream, in contrast to the second part that is filtered by the kidney (Fig.3.4.1a). Furthermore, the relative perfusion index is calculated , which, based on the proportion of integrals (areas) of the ascending parts of the perfusion curve of the left and right kidneys, calculates in percent the relative blood flow of the left and right kidneys (Fig.3.4.1b).
Fig.3.4.1. Quantification of renal
perfusion by analysis of the firstpass bolus of the
radiolabel of the radiolabel. a) Washida's perfusionflow index quantifies the ratio of filtration and perfusion of the kidney. b) Relative renal perfusion is quantified by the ratio of the areas under the perfusion onset curves. 
Perfusion curves together with the calculated perfusion parameters are displayed and can be printed graphically (however, we do not normally perform this printing, relative perfusion indices will be included in the final report).
Quantification of the excretion phase of nephrographic curves
Important parameters for the assessment of renal excretory activity are the time at which the maximum is reached and the value of the excretion halflife of the radioindicator from the kidney. On the nephrographic curves of the left and right kidneys (generated from ROI1 and ROI2), the maximum point and the start and end point of the section for quantification of kidney excretion are automatically defined (with the possibility of manual modification). Exponential functions are interpolated by this least squares method and gr and f fitings are plotted. The maximum time and halflife of the radiolabel excretion from the kidney are calculated. If a diuretic was administered during the study, the excretion halflives can be calculated separately for the descending sections of the curves before and after diuretic administration  by comparing these halflives, we can quantify renal response to diuretics .
Fig.Exkr0. From the interpolation of the exponential function by the excretory sections of the nephrographic curve before and after the application of the diuretic, a good reaction of the kidney can be seen here  the halflife of the excretion is from the slowed value of 12 min. shortened to 2 minutes. 
Note: Quantification of the maximum and excretion phase is optional  if one of the nephrographic curves (or both) is constantly rising, we will not perform this calculation and the resulting protocol will use Tmax instead of parameters. and T1 / 2 (or T1 / 2nat., T1 / 2diur) prints "Excretion not quantified".
Analytical method for calculating the separated function
The separated renal function, ie the determination of the relative share of the left and right kidneys in the total (global ) renal clearance, can be determined by analyzing the ascending (parenchymatous, secretory) sections of the nephrographic curves corrected for blood background. ^{*) }
The analytical method , developed within the framework of our research task 198590, is schematically outlined in Fig.3.4.2.
Fig.3.4.2. Analytical method for calculating the separated renal function. t  time, 
To calculate, we have nephrographic curves R(t) of both kidneys and a curve of the time course of radioactivity in the blood pool (blood pool) B(t). We start from the basic differential equation of radiolabel secretion in the kidney, according to which the instantaneous rate of increase of radioactivity in the kidney is directly proportional to the instantaneous concentration of radioactivity B(t) in the bloodstream with the GFR coefficient characterizing the kidney clearance :
d
R(t)
¾¾¾ =
GFR. B(t).
dt
By integrating this equation and dividing both sides by the concentrations of B (t) of the radio indicator in the blood pool, we get the transformed equation
R(t) ò B(t) dt
¾ ¾ ¾ = P. ¾ ¾ ¾ ¾ ¾ ,
B(t) B(t)
which after transformation [ ò B(t) dt ] / B(t) ® x, R(t) / B(t) ® y is the equation of the line y = P. x in the transformed variables x and y, whose P direction indicates the paired GFR function of the kidney. Thus, this transformed equation describes a linearized initial section of the nephrographic curve.
However, the measured nephrographic curve R¢(t) consists of the actual nephrographic curve R (t), on which the background p (t) is superimposed: R¢(t) = R(t) + p(t). We can make a fairly reasonable assumption that in the initial ascending phase of the nephrogram, the background in the kidney area consists mainly of the radioactivity of blood contained in the vascular system of the kidney ( (t), where Q is a constant. By integrating and modifying the equation of the measured nephrographic curve, we obtain a transformed equation
R¢(t) ò B(t) dt
¾ ¾ ¾ = P. ¾ ¾ ¾ ¾ ¾ + Q,
B(t) B(t)
in which the last term Q = p(t) / B(t) is constant due to the assumption of the vascular nature of the background p(t). After the transformation [ ò B(t) dt ] / B(t) ® x, R¢(t) / B(t) ® y, this equation is then the equation of the line y = Px + Q in the transformed variables x and y, whose direction P gives separated clearance of GFR of the given kidney and the parameter Q indicates the fraction of blood background in the kidney (details of derivation are given in Fig.3.4.2).
According to this
theoretical derivation, we determine the separated kidney
function as follows: We transform the nephrographic curve of each
kidney by dividing it by the blood pool curve and at the same
time transform the time coordinate ^{**)}
. We obtain the socalled linearized nephrographic curve ,
which always contains a significant and clearly visible linearly
ascending section, beginning shortly after the arrival of
activity in the kidney and ending at the moment when excretion
begins. The beginning and end of a linear section are quite well
defined both manually (or visually) and automatically.
**)The total time course of the concentration
of the radio indicator B(t) in the bloodstream is composed of
several exponential components, but during a relatively short
period of linear rise of the transformed nephrographic curve, B
(t) can be considered practically monoexponential: B(t) @ B. e ^{ }^{l
}^{.t} . Then [ ò B(t) dt ] / B(t) @ l .t is
linear and the time coordinate transformation does not need to be
performed.
With this line segment we interpolate the linear regression function by the least squares method, thus obtaining its parameters P and Q. The slope parameter of the line P represents a separate kidney function. The parameter Q, which is the intersection of the fitted line with the vertical axis at the moment of arrival of radioactivity indicates the fraction of blood background in the kidney, i.e., the coefficient which j e multiply the curve of blood pool in order to get the curve of the blood background in the kidney. By subtracting the blood pool curve thus multiplied from the measured nephrographic curve, we obtain a "pure" nephrographic curve corrected for blood background . Separate the valuetherefore, the clearance is determined without the need for correction on the vascular background. Blood background correction is a welcome "byproduct" of calculating separate clearance; All other calculations, especially deconvolution and construction of transit functions and determination trance and ther times, so they can run longer on a "clean" nefrografických curves.
We will now return to the implementation of this analytical method of calculating the separated function in our RENDYN program for complex mathematical evaluation of dynamic renal scintigraphy. The stage of quantification of the separated function begins with the simultaneous display of both nephrographic curves and the question of whether we want to calculate the separated function.
In the positive case, transformed (linearized) nephrographic curves created by dividing the original nephrographic curves by the blood pool activity curve and coordinate transformations are created and displayed for the left and right kidneys, respectively. Curve so formed possess the important property that they always include the well defined linear segment corresponding own secretion radioin d diacetylated monoglycerides in the kidney. The beginning of the linear section corresponds to the interface of the venous and secretory phase, when an equilibrium mixing of the radioindicator in the bloodstream has already taken place. The end of the linear portion corresponds to the situation where the secretory phase will be influenced beginning excretion from kidney and linear curve starts to bend.
The program automatically defines this linear section on the first run and interpolates the linear regression function with it using the least squares method. The fit graph, the equation of the linear function and the question of whether we agree with the interpolation are displayed. We answer in the affirmative when the line passes well through the linear section of the curve. In the case of not entirely accurate fitting, the interpolation can be repeated, while the program offers the possibility of manual modification of the start and end point for interpolation of the linear function.
In this
way, the curves from both kidneys are processed. Separated
clearance is calculated based on the guidelines of linear
functions interspersed with the respective linear sections of the
transformed curves of both kidneys. The analytical method used,
based on the differential equation c of the
kidney learance, in
addition to calculating the separated function, also performs an
exact correction of nephrographic curves on the venous
background, as described above.
The step of determining the separated function ends with the
display of both "pure" nephrographic curves (corrected
for background) together with the results of the determination of
the separated function (in% of the total function). The
calculation can be repeated as required.
*) Why a
mathematically more complex analytical method?
The hitherto standard method of determining the separated renal
function is outlined in Fig ..... The point of the interface of
the perfusion and secretion phase is first determined on the
nephrographic curves. It is assumed that the activity in the
kidney in this phase is given only by its blood circulation and
at this point the time course of the radioactivity in the
bloodstream is drawn. Furthermore, the second important point is
determined on the nephrographic curve (approximately in the 2nd
minute) before the onset of the peak, where there is already
sufficient secretion of r and dioindicator in the kidney,
but we assume that excretion from the kidney has not yet
occurred. The ratio of the areas enclosed by the nephrographic
curves and the interleaved blood background curves between the
two significant points then indicates the separate function of
the right and left kidneys .
However, this
method has two pitfallsrelated to the definition of
both significant points on the nephrographic curve. The first
significant point  the interface between the perfusion and
secretory phases  is often not clearly expressed on the
nephrographic curve, which makes it impossible to define it
precisely and can introduce a significant error in the
subtraction of the blood background. Defining the second required
point can also be problematic, as we do not have a clear control
on how early before the peak of the nephrographic curve the excretion of the radioindicator from the
kidney begins to be applied secretly . Both
of these difficulties are eliminated by the analytical method for
determining the separated kidney function implemented in our
program. It can therefore be considered more objective and exact.
Its further contribution stems from the basic text.
Kidney excretory fraction  output efficiency
For an objective assessment of renal drainage , the analysis of the nephrographic curves themselves can sometimes be misleading. The shape of the nephrographic curve is constantly influenced by the mutual "balance" of the function (accumulation) of the kidney and its drainage (excretion). Especially when there is impaired renal function and an overall flat nephrographic curve, it is difficult to assess the degree of obstruction of the ducts. To some extent, deconvolution transit analysis (described below) can help. However, the combined mathematical analysis of nephrographic curves with the curve from the bloodstream according to Fig.3.4.2 offers yet another possibility to quantify the excretion of the radioindicator from the kidneys. If we multiply the linear function y = Px + Q (fitted by a straight line of the transformed nephrographic curve) by the curve of the blood pool B(t), we get a pure accumulation curve Ac(t), expressing the accumulation of the radioindicator in the kidney under hypothetical situation, if there were no drainageexcretion: this is what the nephrographic curve of the given kidney would look like in case of total obstruction  see Fig.Excr1.
Fig. Excr1. Calculation of the excretion fraction of the kidney (follows on from the Fig.3.4.2.). 
Kidney excretion can now be assessed by
comparing the calculated accumulation curve Ac(t) with the actual
nephrographic curve R(t); we will use the areas under both curves
for this. The excretion fraction EXF(T) is the
part of the total radioactivity accumulated by the kidney
at time T that is actually excreted from the
kidney. We calculate it so that the difference between
the areas under the accumulation curve Ac(t) and the
actual nephrographic curve R(t) is expressed as a percentage of
the area under the accumulation curve  Fig. Excr1. The value of
the excretion fraction quantifies the efficiency of kidney
excretion  output efficiency  of the kidney.
The reference time T for which the excretion
fraction is calculated is customary to take T = 30min.
For a kidney with wellfunctioning drainage, the excretion
fraction is above 80%. In Fig. Excr2 we see the results of the
excretion analysis in the case where the left kidney showed
normal drainage, while in the right kidney there was a serious
drainage disorder  obstruction of the excretory tract.
Fig. Excr2. Intermediate results of kidney drainage analysis. 
The kidney excretion curve can also be useful for assessing renal drainage  it arises by plotting the difference between the value of the accumulation curve Ac(t) and the value of the actual nephrographic curve R(t). In Fig. Excr3, the excretion curves of the pathological case from the previous figure are plotted.
Fig. Excr3. Excretion curves of boot kidneys . 
Bloodpool curve processing and determination of global function
By mathematical analysis of the rate of
decrease of the concentration of nephrotropic radioindicator in the bloodstream, it is
possible to determine the total clearance
of the kidneys,
ie GFR, TER resp. ERPF  depending on the radiopharmaceutical
used.
The time course curve of the radioindicator concentration in the
bloodstream (from ROI1) has a typical shape (Fig.3.4.2).
Immediately after the application, it reaches the maximum value
almost abruptly (when applying the radioindicator in the form of
a discrete bolus, oscillations corresponding to primocirculation
and recirculation in central hemodynamics may occur at this
maximum for our problem). Then the curve decreases relatively
quickly mainly due to the distribution of the
radioindicator into the extracellular space. The rate of decline
decreases until after about 20 minutes it reaches a steady
monoexponential course given by the clearance of the
radioindicator from the bloodstream due to the actual filtration
activity of the kidneys. And it is the rate of this
monoexponential decline that is proportional to the global
functional fitness of the kidneys  their clearance.
The course of the blood pool curve in the first approximation can be approximated by a biexponential  the sum of two exponential functions, the first of which has a high velocity coefficient in the exponent and corresponds to the initial sharp decline, the second slower exponential has a significantly lower velocity coefficient given by total kidney clearance. However, in a more detailed analysis, we find that the initial sharp decline s is not exactly monoexponential, because it involves more compartments of the distribution dynamics of the radiopharmaceutical. We have found that the shape function is well suited for accurate modeling of the leakage curve of a radiolabel from the bloodstream.
^{ }^{[R2 / (1+ tR3)] . t}^{ }  ^{ }^{R5 . t}^{ }  
B (t) =  R1 . e  + R4 . e 
B (t) = R1 . e ^{ }^{[R2 / (1+ tR3)] . t} + R4 . e ^{ }^{R5 . t} ,
which we call "multiexponential". It is actually a combined exponential function with a continuously decreasing rate coefficient R2 / (1 + t ^{R3} ) at the first exponential term, which asymptotically transitions into a monoexponential function (second term) with a constant rate coefficient R5, which is determined by the global renal clearance GFR:
R5 = GFR / V _{D} ,
where V_{D} is the distribution volume of the radio indicator used.
In our program, we process the curve of the time course of radioactivity in the bloodstream (from ROI1) as follows:
At the beginning of the curve, shortly after the arrival of the activity in the place where the possible chaotic course of the curve has already stabilized on the monotonic descent, the starting point of the fit is defined. The end point of the fit is defined at the very end of the curve (around 30 minutes). Automatically defined points can be modified manually.
This is followed by an iterative interpolation of the multiexponential function by this section of the bloodpool curve. First, the program uses a suitable algorithm to determine the point in time (ie the point of the curve) where the faster unbalance components have disappeared and starting from which the curve has a virtually monoexponential course (the faster unbalance phase is indicated by dots below the curve). The monoexponential function R4.exp (R5.t) is interpolated with this slower component, it is subtracted from the primary blood pool curve and the exponential function R1.exp (R2.t / (1 + t ^{R3} )) is interpolated by the resulting difference curve . with variable rate coefficient R2 / (1 + t ^{R3} ). The resulting interpolated function is given by the sum of both exponential functions. A fit graph is plotted and the clearance halflife value T_{1/2} = ln2 / R5 is displayed below, which basically indicates how long it takes the kidney to cleanse half the volume of distribution of a given substance.
Next, the program asks if we want to calculate the global function. According to the above relation, the global renal clearance is given by the product of the rate coefficient R5 of the asymptotic exponential and the distribution volume of the radio indicator V_{D }:
GFR = R5. V _{D} .
We have calculated the value of R5 fitace curve of bloodpool, we need more value distribution volume V_{D} . In the program we can choose two methods for determining the volume of distribution of a radiopharmaceutical: calculation based on the value of applied activity and measured activity of a blood sample, or determination (or estimation) of volume of distribution by empirical formula from patient height and weight .
In the first method, the program measures the required measured data: the value of the applied activity and the residue in the syringe after application, the calibration factor between the measurement of the applied activity and the activity of the blood sample (check the default value of the calibration factor), the activity value of 1 ml. plasma sample, time interval between application and collection and time interval between measurement of applied and collected activity (for correction for ^{99m}Tc decay ). Based on the interpolated multiexponential function of the decrease in radioactivity in the bloodstream, the activity of the sample taken was 1 ml. plasma at a defined time after application along this function extrapolates to the moment of entry of activity, there it is related to the value of applied activity, which calculates the distribution volume V_{D} and thus the value of global clearance in [ ml./sec. ] .
The abovedescribed sample method for determining global kidney function is correct, but not every workplace has the ability to accurately measure the activity of blood samples taken and applied activity. In addition, taking a blood sample is burdensome for the patient, and sampling methods in general are often too complicated for the current routine operation of nuclear medicine facilities. Therefore, most offices prefer somewhat less accurate but much easier way to quantify f esterification global renal function when volum distribution of radiopharmaceuticals is determined using an empirical formula of height and weight patient without the need for sampling and knowledge of the applied activity. The empirical formula, which is part of our program, contains parameters whose numerical values were obtained by correlation analysis of a number of examinations, in which the global function was determined at our workplace by both the sample method (taken as a reference) and the empirical method of determination V_{D} from height and weight. The empirical formula for the distribution volume V_{D} depends, of course, on the type of radiopharmaceutical used, in our program it is implemented for MAG3 and for DTPA.
If a separate function has been calculated, the absolute global function is calculated for the left and right kidneys in [ml./sec.] . The results of the global and separate functions are shown on the display, along with colorcoded nephrographic curves and a bloodstream curve (can be printed for documentation).
Calculation of transit functions and transit times
Qualitative evaluation of renal function is normally performed on the basis of assessing the shape of nephrographic curves. Also, most quantitative parameters of renal function are determined by mathematical analysis of nephrographic curves. Nefrografická curve, however, is determined not only the function of the le d guilt, but also depends on the way in which the tracer in the kidneys gets. The kidneys are "saturated" with radioactivity from the blood pool, as shown in the upper left part of Fig.3.4.3. Radioactivity in the bloodstream changes significantly over time (in which the kidneys also participate). The scanned nephrographic curve is then the result of a certain composition of the kidney's own response function  the socalled transit function and time course curves of radioactivity in the bloodstream. From a mathematical point of view, the nephrographic curve is created by the composition, the socalled convolution, of the kidney's own transit function and the course of radioactivity in the bloodstream. It is interesting and useful to extract the hidden selfresponse function of the kidney  the transit function  simulating the hypothetical situation shown in the picture in the middle left, when we would apply the bolus of the radioindicator directly to the renal artery and monitor its transit through the kidney. The typical shape of such a transit function is drawn next to the right: after the arrival of the bolus, the radioactivity increases abruptly, maintains a constant value during transport by the parenchyma, and after penetrating the hollow system leaves the kidneys.
Fig.3.4.3. By deconvolving the nephrographic curve with the curve of the concentration of the radioindicator in the bloodstream, we calculate the transit function, which models the passage of the bolus of the radioindicator in the hypothetic situation if we applied it directly to the renal artery. 
The lower part of Fig.3.4.3 briefly outlines the mathematical method of calculating the transit function. First, the convolution integral expressing the formation of the nephrographic curve R(t) by composing the kidney's own transit function F(t) and the timevarying radioactivity in the bloodstream B(t) as an input function is given. The extracted transit function can be used with an appropriate functional transformation m ation of all three stakeholders such functions for which it is valid so. Convolution theorem : convolution of two functions is equal to the product of paintings both functions. In our case, the Laplace transform is used. It follows from the convolution theorem that the desired "extraction" of the transit function F(t), ie the deconvolution of the nephrographic curve, is achieved by the following procedure: ^{*)}
*) We developed the method and program at our workplace in the years 198082 and included it in the program for a comprehensive analysis of dynamic scintigraphy of the kidneys on the GAMMA11 device. It later became part of the OSTNUCLINE system on a PC.
From the transit functions we can subtract three significant time moments, characteristic for the dynamics of the passage of the radioindicator through the kidney:
The minimum transit time (beginning of the decrease in the transit function) indicates that the radio indicator has already passed through the kidney (or parenchyma) and is starting to leave. The mean transit time (at the point where the transit function is halved) indicates the time taken for half of the input amount of the radio indicator to pass through the kidney (or parenchyma). The maximum transit time (the point where the transit function drops to practically zero) indicates the time during which all the input radio indicator has already passed through the kidney or a given part of it.
Fig.3.4.4. Typical forms of transit functions of the whole kidney (top) and parenchyma (bottom) for different types of nephropathy. 
Transit function and times are useful to calculate for the whole kidney as well as for the parenchyma and pelvis of both kidneys. It allows to assess whether any extension of transit tracer kidney is already at the level of glomerular and tubular, or is caused by dilation Fri n lid or an obstruction of urinary tract. In Fig.3.4.4 we have in the upper part the transit function always from the whole kidney, at the bottom below the corresponding transit function only of the parenchyma. We can distinguish about four cases, always listed above the vertical pair of transit functions:
In this context, it should be noted that careful delineation of the ROI of the parenchyma so that no part of the hollow system is included should be considered, especially in distopic or rotated kidneys.
In our experience, transit functions and the resulting transit times are very sensitive parameters of kidney function. There is a big difference between normal and pathological values of transit times. Regional analysis for the whole kidney, parenchyma and pelvis can be a valuable aid in the differential diagnosis of the causes of nephropathy.
The calculation of transit functions and transit times of the passage of the radioindicator through the kidneys and their parts is optional in our program. Transit functions are constructed using the abovementioned Laplace deconvolution of nephrographic curves (corrected for tissue and intravascular background) with an interpolated exponential bloodstream curve, which is taken as the input function. The resulting transit function reconstructs the situation in which the tracer would be administered as a bolus injected Pøím of arterial renal artery. The points of minimum, mean and maximum transit time are automatically defined on the transit curves (with the possibility of manual modification). If renal parenchymal ROIs have been marked, transit curves and times are calculated for t yit curves, from the differences indirectly for the pans. We can thus assess whether the possible extension of transit occurs at the level of the parenchyma or the hollow system.
Fig.3.4.5. Intermediate results of transit functions and transit times in deconvolution analysis of dynamic renal scintigraphy. 
After processing all transit curves, these curves are displayed together in a clear graph (Fig.3.4.5) together with the values of transit times, from which we can infer a possible absolute and mutual extension of the radioindicator transit through the kidneys and their parts (for documentation we can print).
Evaluation of significant images and nephrographic curves
On the b Ada analysis of sections of curves in calculating the separated functions and halfexcretion is nasumují relevant images and thus create images of renal perfusion, secretory and early and late excretory phase, wherein the bottom of the screen offering to review and edit text and visual slope evaluation of these paintings. If still images, eg of later phases, have been stored and preselected before starting the RENDYN program, they will also be displayed and can be evaluated visually.
For the same purpose, nephrographic curves are displayed together with a query about or. application of diuretics. If the diuretic has been applied, enter the time of its application. On the display, the colordifferentiated nephrographic curves of both kidneys are enlarged in a common graph, and if the areas of interest of parenchymes (ROI3 and 4 ) have been marked , the curves of the time course of radioactivity in parenchyms are also plotted in dotted lines. A significant vertical line indicates the moment of diuretic application. In the text, visual assessment, which is offered for editing in the bottom of the screen, we can assess the shape nefrogra f ical waveform including eventually. diuretic responses.
Assessment of the renovascular origin of hypertension
In renal artery stenosis , the affected kidney increases production in the reninangiotensin system and thus increases the filtration pressure. In this way, the body tries to correct kidney function, but at the cost of systemic hypertension . With simple (native) functional renal scintigraphy, even with significant renal artery stenosis (due to the above correction mechanism), the result may be practically normal or only nonspecifically reduced. The situation is different if we administer an ACE inhibitor before dynamic functional scintigraphy. ACE inhibitorsthey inhibit the conversion of angiotensin and lower systemic blood pressure. There is a release of vas efferens and thus a reduction in filtration pressure at the glomerular level in the kidney affected by renal artery stenosis, while in the other kidney with a normal blood supply there are only slight changes. Thus, after administration of an ACE inhibitor, there is a change in the transport of the applied nephrotropic radioindicator through this kidney, which is reflected in a change in the shape of the nephrographic curve .
If we perform dynamic sequential scintigraphy of the kidneys in a native pair  after administration of captopril (which is the most commonly used ACEinhibitor), then in the subsequent evaluation of both studies (but it is necessary to enter the study specifications correctly) it is possible to display a native nephrographic curve captopril always for the left and right kidneys, which allows you to compare the effect of this drug on kidney function. In the text of the visual evaluation at the bottom of the screen, we can talk about the event. renovascular origin of hypertension .
Note on the order
of the two studies :
The first dynamic scintigraphy
should be performed after administration of Captopril, as it is
no longer necessary to repeat the test under native conditions if
the result is normal. If the result of functional scintigraphy
after premedication with an ACEinhibitor is not completely
normal, then another native examination (ie without premedication
with an ACEinhibitor) will be performed at the appropriate time
interval (preferably at least 24 hours) under the same conditions
(especially hydration of the patient).
The resulting protocol
Finally, a summary image is displayed on the screen containing images of the kidneys in significant phases of the dynamic study (images of the perfusion and parenchymatous phases, early and late excretion phases), nephrographic curves and levin transit functions. Below them is an overview of the most important quantitative parameters calculated by the program. At the bottom of the screen is the text of the verbal evaluation, which can be modified and supplemented. The same applies to the text of the "Conclusion" if its standard wording has been generated ; otherwise we will insert the text of the final evaluation here, including the doctor's signature. Finally, a final report is printed (in the required number of copies) containing significant images, curves, calculated parameters and verbal evaluation, including the conclusion and signature of the doctor (Fig.3.4.6) :
Department
of Nuclear Medicine, University
Hospital Ostrava Date:
.............. Name of patient: ....................... . Birth
certificate number: .........................


Mathematical analysis and complex evaluation of dynamic functional scintigraphy of kidneys  MAG3  
Evaluation:
After intravenous administration of the radioindicator,
the kidneys of the usual shape, size and placement are
displayed, without focal changes. The nephrographic curve
of the left kidney has a normal course, on the curve of
the right kidney we observe a slowdown of drainage and
retention, disappearing after diuretics. Conclusion: Visual
evaluation of sequential images and quantitative analysis
of nephrographic curves indicate good function of
both kidneys, rapid transit through the parenchyma and
free drainage of the hollow system. In the right kidney,
a slight slowing of dilatation type
drainage . 
Department
of Nuclear Medicine, University
Hospital Ostrava
Date:
.............. Name of patient: ....................... . Birth
certificate number: .........................


Mathematical analysis and complex evaluation of dynamic functional scintigraphy of kidneys  MAG3  
Evaluation:
After intravenous administration of the radioindicator, a
wellaccumulating left kidney of the usual shape and size
was displayed, without focal changes. The right kidney
appears late as markedly hypofunctional
and inhomogeneous  only the narrow margin of the
functional parenchyma around the markedly dilated
excavated hollow system with significant retention
is preserved . The nephrographic curve of the left kidney
has a physiological course. The nephrogram of the right
kidney has a markedly flat shape with a low functional
segment, the curve has a permanently ascending course,
unresponsive to the application of a diuretic in the 17th
minute. Conclusion:
Visual evaluation of sequential images and quantitative
analysis of nephrographic curves indicate good left
kidney function, but severely hypofunctional
right kidney with marked renal parenchymal
atrophy. Left renal drainage
physiological, right obstructive drainage
disorder , no response to diuretic. Global
kidney function is almost normal due to age. 
Structure of the RENDYN
program
The RENDYN program consists of the following
parts:
RENDYN 1  display of a series of images,
verbal evaluation, ROI marking, creation of curves
RENDYN 2  mathematical processing of curves, quantification of
global and separated functions
RENDYN 3  Laplace deconvolution, calculation of transit
functions and times
RENDYN 4  summation of images of secretory and excretion phase,
verbal evaluation
RENDYN 5  comparison of native and after captopril
curves RENDYN 6  display of results, text editing, report
printing
At the same time, this structure shows how to proceed when the calculation is interrupted or when the program is restarted in order to repeat a certain part of the calculations. E.g. to change the texts of the verbal evaluation, just run RENDYN5, to repeat the calculation of the parameters of the global and separate functions, run RENDYN2 (after which RENDYN3 and 4 can be omitted and continue by running RENDYN5).
Occupancy of SAVE AREA after the end of the
RENDYN program :
SA 1,2,3  ROI, curves
SA 10  image of perfusion phase
SA 11  image of secretory phase
SA 12  image of early excretion phase
SA 11  image of late excretion phase