A gamma camera (Genesys, Cirrus or Vertex, Philips Medical Systems, the Netherlands) with a 40cm field of view and mounted with a lowenergy high resolution parallel hole collimator was used. The patient was supine with the detector head opposite the kidney region from the dorsal side. Following a bolus injection in adults of about 150 MBq [^{99m}Tc]DTPA (TechneScan® DTPA, Mallinckrodt Medical, The Netherlands) into the medial cubital vein, digital images were recorded for 40 min with 10 sec/frame. Images were recorded as 64 x 64 matrices in word mode with a 20% energy window around the [^{99m}Tc] photon peak.
Processing the [^{99m}Tc]DTPA renography with respect to the rate constant method
A region of interest was created over a part of a blood pool (left ventricle, the whole heart, spleen, lungs, or liver). The corresponding timeactivity curve was subsequently corrected for decay of the [^{99m}Tc] radioisotope. A monoexponential fit of the TAC with time constant BETA in min^{1} was made in the time interval 10 min to 40 min postinjection. In the first iteration the parameter BETA_{max} was assigned the value of BETA (Figure 1).

Figure 1  First iterative step with an ROI over the spleen. The TAC from 1040 min is fitted with a monoexponential curve (shown in yellow). The rate constant BETA is 0.01187 1/min. BETA_{max} is 0.01187 1/min.

A second ROI over a vascular region was created. The above computational step was repeated for the new ROI yielding a new value of the time constant BETA. If BETA was greater than BETA_{max}, then BETA_{max} was assigned the value of BETA (Figure 2). This procedure was repeated about 5 to 10 times until it was no longer possible to find a ROI yielding a BETA greater than the current BETA_{max} (Figure 3).

Figure 2  Second iterative step with an ROI over the liver. The TAC from 1040 min is fitted with a monoexponential curve (shown in yellow). The rate constant BETA is 0.01177 1/min. BETA_{max} is 0.01187 1/min.


Figure 3  Sixth iterative step with an ROI over the left ventricle. The TAC from 1040 min is fitted with a monoexponential curve (shown in yellow). The rate constant BETA_{max} is 0.01354 1/min and this value was the maximum BETA value of 6 iterations. Therefore, BETA_{max} is 0.01354 1/min. This value yields a GFR_{std} equal to 78.9 ± 13.0 ml/(min·1.73 m^{2}) (Eqs. 2425).

The relationship between BETA_{max} and GFR
Define an optimum rate constant BETA_{opt} by the ratio
where GFR is the glomerular filtration rate in ml/min and PV denotes the plasma volume in ml. The formula for BETA_{opt} can be rearranged as follows:
BETA_{opt}= [GFR · (PV_{roi}/PV)] / [PV · (PV_{roi}/PV)] 
Eq. 2

or
BETA_{opt}= GFR_{roi} / PV_{roi} 
Eq. 3

where PV_{roi} represents the plasma volume within the ROI over the blood pool and GFR_{roi} represents a virtual GFR, i.e. the fraction of GFR which PV_{roi} represents of PV.
The timeactivity curve (TAC), which the gamma camera records within PV_{roi} from 10 to 40 min postinjection and after decay correction can be expressed as function of time t:
TAC(t)= ALFA · exp(BETA · t) 
Eq. 4

where ALFA is a factor with the dimension of a count rate and BETA a rate constant in min^{1.}
If there was no exchange of [^{99m}Tc]DTPA between the vascular and extravascular volumes, then BETA could be identified with BETA_{opt}, i.e. the ratio GFR/PV. Let EVV denote the large extravascular volume.
Immediately following the radionuclide bolus injection a considerable part of [^{99m}Tc]DTPA enters the EVV owing to the large differences in concentrations of [^{99m}Tc]DTPA between PV and EVV. As in the GFR determination with [^{51}Cr]EDTA using the multiple samples technique, the concentration equilibrium of the radionuclide indicator between PV and EVV is attained after 3 hours. The extravascular distribution volume of [^{99m}Tc]DTPA in the time interval from 10 min to 40 min postinjection will be called the perivascular distribution volume and denoted PVDV.
It is assumed that the net exchange of [^{99m}Tc]DTPA between PV and PVDV from 10 min to 40 min postinjection is so small, that the decline in concentration of [^{99m}Tc]DTPA in PV is approximately due to renal uptake alone. As a consequence of this, the rate constant BETA of a monoexponential fit to the TAC from 10 min to 40 min postinjection corresponds to the ratio for BETA_{opt} in Eq. 3 as regards the numerator.
However, the inevitable presence of extravascularly distributed [^{99m}Tc]DTPA within any ROI over a blood pool further complicates things. This extravascularly distributed [^{99m}Tc]DTPA resides predominantly in the PVDV during the 40 min long renography. Let PVDV_{roi} denote the perivascularly distributed [^{99m}Tc]DTPA within a ROI over a blood pool.
Taking the above view points into consideration the rate constant BETA in Eq. 4 can be expressed as follows:
BETA= GFR_{roi} / (PV_{roi} + PVDV_{roi}) 
Eq. 5

Define the volume ratio (VR) as the ratio of PVDV_{roi} to PV_{roi}. Then Eq. 5 can be rewritten as
BETA= (1/(1 + VR)) · GFR_{roi} / PV_{roi} 
Eq. 6

Equation 6 fully describes in mathematical terms the iterative nature of the procedure for determination of a value for BETA as close as possible to BETA_{opt} in Eq. 3. The smaller VR is, the closer BETA will be to BETA_{opt}. Hence, the purpose of the iterative method is not to find a ROI with a minimim size of PVDV_{roi} but rather a minimum value for VR. This value will be denoted VR_{min}. Of course, when the iterations are stopped since a minimum value for VR has been obtained, this value corresponds to the above maximum BETA value (BETA_{max}).
We have that
BETA_{max}= (1/(1 + VR_{min})) · GFR_{roi} / PV_{roi} 
Eq. 7

Let BSA denote the body surface area in square meters. Introduce the standard plasma volume (PV_{std}) as the plasma volume adjusted to 1.73 m^{2} body surface area:
PV_{std}= (1.73/BSA) · PV 
Eq. 8

The standard glomerular filtration rate (GFR_{std}) is the glomerular filtration rate adjusted to 1.73 m^{2} body surface area:
GFR_{std}= (1.73/BSA) · GFR 
Eq. 9

Using Eqs. 1 and 3 and insertion of Eqs. 8 and 9 into Eq. 7 yields
BETA_{max}= (1/(1 + VR_{min})) · GFR_{std} / PV_{std} 
Eq. 10

Solution of Eq. 10 with respect to GFR_{std} gives
GFR_{std}= (1 + VR_{min}) · PV_{std} · BETA_{max} 
Eq. 11

The standard plasma volume can be estimated based on sex, height and weight in normal adults (8). If our patient material mentioned below is regarded as normal with respect to the plasma volume, the value for estimated PV_{std} can be calculated as 2283 ± 82 ml/1.73 m^{2} (mean ± 1 SD), i.e. with a coefficient of variation SD/mean of only 4%. Hence, it is reasonable to expect a fairly constant value for PV_{std} in Eq. 11. Equation 11 apparently establishes a direct proportionality between GFR_{std} and BETA_{max}. However, this does not hold true since it will be shown in the discussion section that VR_{min} is an increasing function of GFR_{std}.
Processing the [^{99m}Tc]DTPA renography with respect to the Gates' method
Regions of interest were created around both kidneys and narrow perirenal background ROIs were drawn almost around the whole of the kidneys. The background ROIs were used for calculation of the net count rates in cps at 2 min postinjection for the left and right kidney (NCR_{l} and NCR_{r}). The count rates were corrected for the minimal decay of the [^{99m}Tc] radioisotope in the course of the 2 min. The kidney centre distances in dorsal projection of the left and right kidneys (KCD_{l} and KCD_{r}) were estimated based on patient's height and weight (9). Let S_{gc} denote the sensitivity in cps/MBq of the gamma camera at the surface of the collimator.
The kidney geometry factors for the measurements of the left and right kidneys (KGF_{l} and KGF_{r}) can be expressed as follows for the left kidney:
KGF_{l}= TF_{it} · S_{gc} · exp(0.117 · KCD_{l}) 
Eq. 12

and for the right kidney as
KGF_{r}= TF_{it} · S_{gc} · exp(0.117 · KCD_{r}) 
Eq. 13

The factor TF_{it} represents the transmission factor of the imaging table. TF_{it} was typically 0.90 for the three gamma cameras used in the study. The number 0.117 is the linear attenuation coefficient of [^{99m}Tc] in the body taking into account scattered radiation (4). The letters "exp" denote the exponential function.
The cleared renal fractions of the left and right kidneys at 2 min postinjection (CRF_{l} and CRF_{r}) are then calculated as
CRF_{l}= NCR_{l} / (KGF_{I} · Q) 
Eq. 14

and
CFR_{r}= NCR_{r} / (KGF_{r} · Q) 
Eq. 15

where Q is the net injected dose in MBq.
Finally, the total cleared renal fraction (TCRF) is calculated as
TCRF= CRF_{l} + CFR_{r} 
Eq. 16

In Gates' method TCRF is used as an estimate of GFR_{std}.
Pooled value of GFR_{std} based on the rate constant and Gates' methods
Since the estimated values for GFR_{std} are statistically uncorrelated in the rate constant and Gates' method, a pooled estimate for GFR_{std} can be determined. Before a pooled estimated is calculated, a statistical test is performed with a view to deciding whether the two estimates are significantly different at the 5% significance level. If the two estimates are significantly different, no pooled estimate will be determined. In this situation it is concluded, that a least one of the two estimates is in error.
Let GFR_{beta} and GFR_{tcrf} denote the estimated values of GFR_{std} in the rate constant and Gates' methods, respectively. Further, the corresponding standard deviations are denoted SD_{beta} and SD_{tcrf}. The test variable SU measures the difference between the two estimates of GFR_{std} in standard units since SU is calculated as
SU= (GFR_{beta}  GFR_{tcrf}) / (SD_{beta}^{2} + SD_{tcrf}^{2})^{½} 
Eq. 17

If SU is above 2 or below 2 the conclusion is drawn that the two estimates are significantly different.
If they are not significantly different, the procedure for determination of a pooled estimate makes use of statistical weighting of the data, i.e. the weights of each of the two estimates is proportional to the inverse of the variance of each statistical variable. Let W_{beta} and W_{tcrf} denote the weights of GFR_{beta} and GFR_{tcrf} in the pooled estimate GFR_{pooled}:
GFR_{pooled}= W_{beta} · GFR_{beta} + W_{tcrf} · GFR_{tcrf} 
Eq. 18

where
W_{beta}= (1/SD_{beta}^{2}) / (1/SD_{beta}^{2} + 1/SD_{tcrf}^{2}) 
Eq. 19

and
W_{tcrf}= (1/SD_{tcrf}^{2}) / (1/SD_{beta}^{2} + 1/SD_{tcrf}^{2}) 
Eq. 20

The standard deviation of the pooled variable (SD_{pooled}) is calculated from Eq. 18 as
SD_{pooled}= (W_{beta}^{2} · SD_{beta}^{2} + W_{tcrf}^{2} . SD_{tcrf}^{2})^{½} 
Eq. 21

A computational example illustrates the advantages of using a pooled estimate:
GFR_{beta}= 81.7 ± 13.1 ml/(min · 1.73 m^{2}) 
Eq. 22

and
GFR_{tcrf}= 88.5 ± 14.6 ml/(min · 1.73 m^{2}) 
Eq. 23

Insertion of the variable values from Eqs. 22 and 23 into Eq. 17 yields SU equal to 0.35, i.e. there is no significant difference between GFR_{beta} and GFR_{tcrf}. Insertion of the variable values from Eqs. 22 and 23 into Eqs. 18 and 21 gives GFR_{pooled} equal to 84.7 ± 9.8 ml/(min · 1.73 m^{2}). In comparison with Eqs. 22 and 23, the standard deviation of GFR_{pooled} has been reduced with the factor 1.41 (i.e. the square root of 2) on the average. Hence, the use of a pooled estimate yields a more correct mean value for GFR_{std} with a standard deviation of the mean about 1.41 times smaller.
Patients
The patient material comprised 54 adult subjects (18 females, 36 males; age range 1774 years, mean age 53 years). The subjects were selected from adult patients referred to our department for routine renography for various nephrourological disorders in whom the glomerular filtration rate was determined simultaneously using the multiple samples or the one sample techniques with [^{51}Cr]EDTA. The majority of the GFR determinations employed the multiple samples technique, and all patients with an increased serum creatinine concentrations were examined using this method. In the multiple samples method 4 blood samples were drawn 3 hours postinjection with a time interval of 20 minutes. If the estimated endogenous creatinine clearance was below 30 ml/min, an additional blood sample was drawn 20 min after the 4th blood sample.
In two patients a part of the kidneys were outside the gamma camera field of view and, therefore, the patient material in Gates' method with determination of TCRF comprises only 52 patients.