Methods: The method is based on renographies lasting 40 min in which regions of interest (ROIs) are manually created over selected parts of certain blood pools (e.g. heart, lungs, spleen, and liver). For each ROI the corresponding time-activity curve (TAC) was generated, decay corrected and exposed to a monoexponential fit in the time interval 10 to 40 min postinjection. The rate constant in min^-1 of the monoexponential fit was denoted BETA. Following an iterative procedure comprising usually 5-10 manually created ROIs, the monoexponential fit with the maximum rate constant (BETA_max) was used for estimation of GFR.

Results: In a patient material of 54 adult subjects in whom GFR was determined with multiple or one sample techniques with [⁵¹Cr]-ethylenediaminetetraacetic acid ([⁵¹Cr]-EDTA) the regression curve of standard GFR (GFR_std) (i.e. GFR adjusted to 1.73 m² body surface area) showed a close, non-linear relationship with BETA_max with a correlation coefficient of 95%. The standard errors of estimate (SEE) were 6.6, 10.6 and 16.8 for GFR_std equal to 30, 60, and 120 ml/(min·1.73 m²), respectively. The corresponding SEE values for almost the same patient material using Gates' method were 8.4, 11.9, and 16.8 ml/(min·1.73 m²).

Conclusions: The alternative rate constant method yields estimates of GFR_std with SEE values equal to or slightly smaller than in Gates' method. The two methods provide statistically uncorrelated estimates of GFR_std. Therefore, pooled estimates of GFR_std can be calculated with SEE values approximately 1.41 times smaller than those mentioned above. The reliabilities of the pooled estimate of GFR_std separately and of the multiple samples method are of the same magnitude. Therefore, [^99mTc]-DTPA renography could replace the multiple samples method for GFR determination. In addition, the renography requires fewer resources from patient and staff and offers more clinical results as regards renal uptake function and renal outflow.

Key Words: [^99mTc]-DTPA renography, glomerular filtration rate, rate constant method, Gates' method.

A gamma camera (Genesys, Cirrus or Vertex, Philips Medical Systems, the Netherlands) with a 40-cm field of view and mounted with a low-energy 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 [^99mTc]-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 [^99mTc] photon peak.

Processing the [^99mTc]-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 time-activity curve was subsequently corrected for decay of the [^99mTc] 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 10-40 min is fitted with a monoexponential curve (shown in yellow). The rate constant BETA is 0.01187 1/min. BETAmax 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 10-40 min is fitted with a monoexponential curve (shown in yellow). The rate constant BETA is 0.01177 1/min. BETAmax is 0.01187 1/min.

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:

or

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 time-activity 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:

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 [^99mTc]-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 extra-vascular volume.

Immediately following the radionuclide bolus injection a considerable part of [^99mTc]-DTPA enters the EVV owing to the large differences in concentrations of [^99mTc]-DTPA between PV and EVV. As in the GFR determination with [⁵¹Cr]-EDTA using the multiple samples technique, the concentration equilibrium of the radionuclide indicator between PV and EVV is attained after 3 hours. The extra-vascular distribution volume of [^99mTc]-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 [^99mTc]-DTPA between PV and PVDV from 10 min to 40 min postinjection is so small, that the decline in concentration of [^99mTc]-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 extra-vascularly distributed [^99mTc]-DTPA within any ROI over a blood pool further complicates things. This extra-vascularly distributed [^99mTc]-DTPA resides predominantly in the PVDV during the 40 min long renography. Let PVDV_roi denote the perivascularly distributed [^99mTc]-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:

Define the volume ratio (VR) as the ratio of PVDV_roi to PV_roi. Then Eq. 5 can be rewritten as

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

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 m² body surface area:

Using Eqs. 1 and 3 and insertion of Eqs. 8 and 9 into Eq. 7 yields

Solution of Eq. 10 with respect to GFR_std gives

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 m² (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 [^99mTc]-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 [^99mTc] 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:

and for the right kidney as

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 [^99mTc] 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

 and

Finally, the total cleared renal fraction (TCRF) is calculated as

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

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:

where

and

The standard deviation of the pooled variable (SD_pooled) is calculated from Eq. 18 as

  A computational example illustrates the advantages of using a pooled estimate:

and

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 m²). 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 17-74 years, mean age 53 years). The subjects were selected from adult patients referred to our department for routine renography for various nephro-urological disorders in whom the glomerular filtration rate was determined simultaneously using the multiple samples or the one sample techniques with [⁵¹Cr]-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.

Inspection of a plot of the data of GFR_std versus BETA_max revealed a distinct non-linear relationship between the two parameters. For that reason GFR_std was fitted to BETA_max with a monoexponential function including a constant term:

with a standard error of estimate of GFR_std around the monoexponential fit

Figure 4 shows the data of GFR_std  and  BETA_max for the patient material comprising 54 subjects including the regression curve and the 95% tolerance limits of GFR_std for a given value of BETA_max. A single data point is outside the tolerance limits.

Gates' method

Inspection of a plot of the data of GFR_std versus TCRF revealed a linear relationship between the two parameters. Linear regression of GFR_std versus TCRF yielded a correlation coefficient of 91% and the following regression line:

with a standard error of estimate of GFR_std around the linear fit

The intercept of the regression line in Eq. 26 is not significantly different from zero at the 5% significance level. Hence, TCRF represents an unbiased estimate of GFR_std.

Figure 5 shows the data of GFR_std and TCRF for the patient material comprising 52 subjects including the regression line and the 95% tolerance limits of GFR_std for a given value of TCRF. Three data points are outside, that is above, the tolerance limits.

Pooled estimates of GFR_std using the rate constant and Gates' methods

The patient material originally consisted of 52 subjects in whom both GFR_beta and GFR_tcrf were determined. Pooled estimates of GFR_std were calculated according to Eq. 18. Four patients had to be excluded since they showed significantly different values of GFR_beta and GFR_tcrf (Eq. 17). These four patients comprise the four patients with estimated GFR_std outside the 95% tolerance limits in Figs. 4 and 5. Having excluded these four patients, 48 patients remain in whom GFR_std and GFR_pooled have been calculated.

Figure 5 - Linear regression of the standard glomerular filtration rate (GFRstd) versus the Total Cleared Renal Fraction (TCRF) in Gates' method in 52 adult subjects (Eqs. 26-27). The dotted curves represent the 95% tolerance limits of GFRstd for a given TCRF.

Linear regression of GFR_std versus GFR_pooled yields a correlation coefficient as high as 97% and the regression line:

with a standard error of estimate of GFR_std around the linear fit

Figure 6 shows the data of GFR_std and GFR_pooled for the patient material comprising 48 subjects including the regression line and the 95% tolerance limits of GFR_std for a given value of GFR_pooled. No data points are outside the tolerance limits.

The slope and intercept of the regression line in Eq. 28 are not significantly different from unity and zero, respectively, at the 5% significance level. Hence, the regression line of GFR_std versus GFR_pooled is not significantly different from the line of identity.

Figure 6 - Linear regression of the standard glomerular filtration rate (GFRstd) versus the pooled glomerular filtration rate (GFRpooled) in 48 adult subjects (Eqs. 28-29). The dotted curves represent the 95% tolerance limits of GFRstd for a given GFRpooled.

The multiple samples method is known to have a reliability of about 10% for GFR of about 30 ml/min and about 7.7% for values exceeding 30 ml/min (10). For a considerably decreased, a moderately decreased and a normal GFR_std value of 30, 60 and 120 ml/(min · 1.73 m²), respectively, the SEE values of the rate constant method, Gates' method, and of GFR_pooled were calculated from Eqs. 24-29. These SEE values include the reliabilities of the multiple samples method. Therefore they are regarded as the total SEE values.

Since the total variance of, for example, the rate constant method, is equal to the squared sum of the variances of the rate constant method separately and the multiple samples method separately, the reliability or SEE value of the rate constant method separately can be determined from the expression:

where "tot" and "sep" refer to the total and separate SEE values, respectively. From equations similar to Eq. 30, the separate SEE values for Gates' method and GFR_pooled were also calculated (Table 1).

Table 1 - The standard errors of estimate of estimated GFR_std in the multiple samples method with [⁵¹Cr]-EDTA (SEE_msm), the rate constant method (SEE_beta), Gates' method (SEE_tcrf), and of GFR_pooled (SEE_pooled) for GFR_std equal to 30, 60 and 120 ml/(min · 1.73 m²). The total SEE values include the reliabilities of the multiple samples method, whereas the separate SEE values refer only to the method mentioned.

The following conclusions can be drawn from the table: a) The rate constant method possesses a reliability equal to or slightly better than that of Gates' method; b) The pooled GFR has a reliability about 30% better than that of the rate constant or Gates' methods; and c) The pooled GFR has a reliability slightly worse than that of the multiple samles method for moderately and considerably decreased GFR_std values but slightly better at normal glomerular filtration rates.

The apparent size of the perivascular distribution volume as function of GFR_std

Direct proportionality between GFR_std and BETA_max could be expected in Eq. 11 if the minimum volume ratio (VR_min) could be assumed to be constant from patient to patient. Fig. 4 clearly revealed that this assumption did not hold.

Solution of Eq. 11 with respect to VR_min gives:

VR_min is calculated for the patient material using estimated normal values for PV_std based on sex, height and weight (8). A graph of VR_min versus GFR_std is shown in Figure 7. VR_min is near zero level for extremely decreased GFR_std values and increases to about 2.3 at high GFR_std values. In other words, the perivascular distribution volume 10-40 min postinjection in the [^99mTc]-DTPA renography apparently increases with GFR_std. However, the relationship between the magnitude of [^99mTc]-DTPA filtered into the lumen of the Bowman capsules and the diffusion (and possible filtration) of [^99mTc]-DTPA from other capillary systems into the perivascular space is not clear to us.

Figure 7 - Parabolic regression of the minimum volume ratio VRmin versus GFRstd in 54 adult subjects (Eq. 31). The regression curve is indicated in the figure. The correlation coefficient is 89%.

On the apparent drawbacks of the rate constant method for estimation of GFR_std

It may be considered a drawback that our method is applied to [^99mTc]-DTPA while this radioactive indicator in many places has been replaced by [^99mTc]-MAG3 which yields better renal images since it has a renal clearance about 3 times that of [^99mTc]-DTPA or GFR. However, since the most important parameter of renal function is the glomerular filtration rate, it can hardly be a drawback to use a radionuclide indicator for assessment of GFR which has a clearance almost identical to GFR itself.

The present version of the rate constant method requires that the renography lasts 40 min. This can be considered a drawback since the duration of a renography in most nuclear medicine departments is 20 min. In our department the renography lasts 20 min in the large patient group where renography is used for screening for renovascular hypertension. In the other large group with patients suspected of nephro-urological disorders the renography lasts 40 min. In our experience decreased renal function occurs more frequently in the latter patient group than in the hypertensive patient group. Hence, the rate constant method can be a valuable tool for alternative GFR estimation in routine renography in patients with nephro-urological disorders.

Using renographies lasting 40 min in nephro-urological patients there is a possibility for a more reliable quantification of renal outflow and, hence, for a more accurate distinction between a kidney with an obstructed and a non-obstructed dilated renal pelvis. Very decreased renal mean transit times cannot reliably be determined from renographies lasting 20 min (11-12).

Based on the above view points we recommend nuclear medicine departments to consider the advantages of using renographies lasting 40 minutes instead of the conventional 20 min, in particular in patients suspected of nephro-urological disorders.