Pitfalls of single-sample determination of renal clearance

Sabine Zitta1, Willibald Estelberger3, Herwig Holzer1, Rainer W. Lipp2, Karl Oettl3 and Gilbert Reibnegger3,

1 Department of Internal Medicine, Division of Nephrology, 2 Department of Internal Medicine, Division of Endocrinology and Nuclear Medicine and 3 Institute of Medical Chemistry and Pregl Laboratory, Karl-Franzens-University of Graz, Graz, Austria

Abstract

Background. Single-sample techniques are widely used for determination of renal clearance by elimination kinetics of radiolabelled marker substances. Frequently, however, formulae for transforming single-time measurement values into estimates of kinetic function, such as renal clearance, are being established exclusively by data reduction methods and are devoid of any physiological meaning.

Methods. Using 11 subjects with normal or impaired renal function, we compared one such method using 99mTc-labelled mercaptoacetyltriglycine (99mTc-MAG3) for single-sample determination of tubular extraction rates with a more elaborate computer-based system identification technique. This latter method yields measures for glomerular filtration rate as well as effective renal plasma flow based on elimination kinetics of sinistrin and p-aminohippuric acid.

Results. When applying the single-sample technique, two of the 11 estimated values for tubular extraction rate were negative, indicating an erroneous analysis of kinetic behaviour. This single-sample method failure was not caused by the marker, but rather by the specific mathematical procedure used for the evaluation. Importantly, evaluation of the same experimental data with a conventional two-sample technique would eliminate the principal mathematical defect and produce physiologically reasonable results, without requiring additional effort.

Conclusions. Our study does not criticize the 99mTc-MAG3 technique per se. Rather, these findings indicate that usage of single-sample techniques for determination of inherently kinetic phenomena may produce incorrect results. Therefore, despite their obvious practical advantages, such simplified methods should be performed with great caution.

Keywords: kinetic models; regression models; renal clearance; system identification; tubular extraction rate

Introduction

Measurement of kidney function is important in the diagnosis and management of renal diseases. Various techniques are available for measuring different aspects of renal function. Generally, a marker substance is injected intravenously and its concentration in blood is measured as a function of time. From such a temporal concentration profile, one attempts to extract information concerning renal function by means of mathematical analysis.

The methods used in renal function assessment differ greatly according to the marker substances employed and therefore the mode of renal function investigated. This is also true for the effort required in establishing concentration vs time dependence.

99mTc-labelled mercaptoacetyltriglycine (99mTc-MAG3) is a renal tubular agent widely used for renal imaging that is also exploited for quantitative measurement of kidney function. This agent has been recommended by several authors [13] and others have suggested its use as a particularly simple method of choice using various single blood-sample procedures [47]. The procedure by Bubeck [5,6] is quite popular.

As in most single-sample techniques, Bubeck's model relates the dose of the radiolabelled marker substance to the serum concentration (activity) of the marker, determined at a pre-specified time point after its intravenous administration: the ratio dose/activity can be interpreted as a hypothetical volume of distribution. This ratio is then subjected to a simple logarithmic transformation to provide the requested tubular extraction rate (TER; for details see Subjects and methods).

Unfortunately, certain logarithm function arguments will at times yield negative values. In Bubeck's formula, when the ratio between dose and time-dependent activity falls below a distinct threshold (which is slightly time-dependent; see Subjects and methods), a negative result with mathematical certainty is obtained for TER.

To investigate whether this type of error in Bubeck's single-sample technique occurs in clinical practice, we compared TER from 99mTc-MAG3 using the single blood-sample technique suggested by Bubeck with measurements of glomerular filtration rate (GFR) and effective renal plasma flow (ERPF) in 11 subjects with either normal or impaired renal function. For GFR determination, an inulin-like polyfructosan, sinistrin, was employed, and for ERPF, we used p-aminohippuric acid (PAH). A computer-assisted system-identification procedure, which is based on repeated marker measurements from blood samples following intravenous bolus injections, was employed [810].

Subjects and methods

Subjects
Eleven subjects (six females and five males; age ranging from 40 to 76 years with a median of 51 years) with normal or impaired kidney function were enrolled into this comparative study. Table 1Go shows basic characteristics of the subjects.


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Table 1.  Basic characteristics of the study patients

 

Chemicals
Sinistrin, an inulin-like polyfructosan with improved clinical properties over inulin (Fresenius Pharma Austria, Linz, Austria), PAH (Merck & Co., West Point, PA, USA) and MAG3 (Mallinckrodt Radiopharma GmbH, Petten, The Netherlands) were used as marker substances.

Measurement of sinistrin and PAH
Details of the concentration measurements have been previously described [8,9]. Briefly, sinistrin was detected by enzymatic methods employing hydrolysis of the polyfructosan, epimerization of fructose into glucose, and glucose oxidation according to Kuehnle et al. [11] and Gretz et al. [12]. PAH was measured according to the method of Bratton and Marshall [13] with modifications by Smith et al. [14]. The possibility that PAH was conjugated into its acetylated form was controlled by incubating samples in 150 g/l trichloroacetic acid for 1 h at 65°C, thus removing the acetyl residue by acid hydrolysis.

Calculation of TER—Bubeck's procedure
According to Bubeck [5,6], two estimates of TER are independently computed from each of the two blood concentrations of the radiopharmaceutical obtained at 20 and 40 min. These estimates are corrected for decay of the counts (half-life of 99mTc is 6.03 h). The calculation employs a single-sample approach based on a regression formula: the essential step in Bubeck's model is the following computation:Go

001
where TER is the tubular extraction rate, standardized to body surface; the dose is the injected activity of 99mTc-MAG3; the concentration is the activity of 99mTc measured in plasma 20 (or 40) min after injection, standardized to body surface; and A and B are functions of time t after injection (min):Go

002

At each time point (20 and 40 min after injection), a separate estimate for TER is obtained; the results are averaged to yield a final TER value.

Under conditions when TER=0, Bubeck's equation can be rewritten to yield a ‘critical’ ratio between dose and activity:Go

003

Thus, TER becomes negative whenever the ratio between dose and concentration falls below:Go

004

Figure 1Go demonstrates the dose/activity dependence of TER for 20 and 40 min. At 20 min, the critical ratio is 6.94, and at 40 min the critical ratio is 8.50. The more impaired the renal elimination, the smaller the ratio (because the plasma concentration remains high for a prolonged period of time). This is why Bubeck's model inevitably fails in subjects attaining certain levels of impaired renal function.



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Fig. 1.  General behaviour of Bubeck's regression formula. Dependent on the exact time point of collecting blood for measurement, at a dose/activity ratio below a time-dependent threshold the computed TER values become negative. Closed circles represent the behaviour for blood collected at 20 min; open circles denote the situation at 40 min.

 

Calculation of TER—conventional two-sample kinetic model
In addition to Bubeck's procedure for TER estimation from two independent single-sample determinations, we analysed TER using a simplified two-sample kinetic model. The 20 and 40 min TER estimations were considered representative for a simple mono-exponential decay curve to obtain an estimate for the clearance of 99mTc-MAG3 according to well-known procedures.

Determination of GFR and ERPF using sinistrin and PAH
GFR and ERPF were assessed from 10 measurements of plasma concentrations of sinistrin and PAH taken during a 2 h period after intravenous bolus injection of both marker substances (2500 mg sinistrin, and 500 to 1000 mg PAH, depending on body weight). Both sinistrin and PAH concentration profiles were subjected to a computer-assisted system identification process applying the basic two-compartment model of pharmacokinetics involving a central, rapidly perfused compartment corresponding to the intravascular space, coupled to a less rapidly perfused peripheral compartment. The physiological basis of such compartmental models has been summarized by Atkinson et al. [15]. More details have been published elsewhere [8,9,16,17]. Final values of GFR and ERPF were standardized according to body surface.

Results

Table 1Go summarizes the basic demographic and laboratory information of the subjects.

Comparison of TER (99mTc-MAG3), GFR (sinistrin) and ERPF (PAH)
Clearance measurements are summarized in Table 2Go. Problems associated with TER calculation using 99mTc-MAG3 are obvious. For example, in some patients, one or even both of the two single-sample estimations of TER yield negative results, which is physiologically impossible and unrealistic in view of the measured activities: for each patient the count rate in the 20 min blood sample exceeds the rate at 40 min. Thus, although the time course of blood activity behaves as expected, negative results for TER were obtained. In fact, it is the mathematical model that transforms the measured activities into TER values that produces the negative TER values.


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Table 2.  Results of the various clearance measurements (ml/min)

 
Furthermore, in nine of 11 cases, TER values estimated after 20 min did not match the 40 min values. In general, the TER20 values were much lower. Only two patients had reasonably matching TER20 and TER40 values. Therefore, the expression of final results simply as the arithmetic mean of both single-sample values as prescribed by Bubeck may be prone to error.

Improvement of TER estimation by using a two-sample model
Results from 99mTc-MAG3 TER estimations may be considerably improved by replacing the Bubeck model with a simple pharmacokinetic two-sample approach to the 20 and 40 min 99mTc-MAG3 activities.

Results from this re-evaluation are also summarized in Table 2Go. This simple and well-known kinetic approach yields uniformly positive values for TER because the radioactivity in blood decreased over time in all subjects. Importantly, no extra measurements were necessary, demonstrating that this method simply makes better use of the experimental findings.

Discussion

For the proper understanding of dynamic physiological processes, such as renal elimination of marker substances, repeated measurements of critical system variables are required to describe the temporal evolution of a marker in the circulation. Several methods of markedly varying conceptual depth and complexity are available for mathematical analysis of time-dependent concentrations. The least sophisticated methods, such as interpolation of measured data, apply only data modelling. Such models describe observed marker concentration profiles in a purely phenomenological manner and permit, at best, the ‘drawing of a smooth curve through data points' by means of an analytic expression for an interpolation function having no physiological meaning. Nevertheless, these methods are easily performed in everyday clinical practice.

In contrast, dynamic input–output models, such as the two-compartment model used in this study for the measurement of GFR and ERPF, are designed to represent at least the most fundamental mechanistic features of the involved physiological system. This is generally accomplished by modelling physiological compartments exchanging the relevant substances. When these models are expressed with an appropriate equation system incorporating adequate model parameters, analysis of measured system-relevant data by system identification techniques is possible. Such a model describes the temporal sequence of the process in mathematical terms and reflects physiological relationships. From the measured data, it extracts meaningful quantitative parameters describing subject situations and is able to predict the behaviour of individual patients following various modes of external disturbances.

The 99mTc-MAG3 TER determination suggested by Bubeck represents data modelling since the 99mTc-activity measured at 20 min (or, independently, 40 min) after marker injection was subjected to a regression equation obtained by the author in a specific group of individuals. Importantly, this regression equation was chosen from a number of other mathematical equations because of its relative computational simplicity. Parameters A and B in the regression equation are devoid of physiological meaning.

A simple mathematical inspection of Bubeck's algorithm reveals that, during precise conditions, this procedure necessarily yields negative TER values. Our data from 11 patients demonstrated that this problem can indeed occur in clinical practice. To show that this failure was not restricted to patients with extremely weak renal function, we studied GFR and ERPF using more sophisticated techniques. In clinical practice, negative TER values typically go undetected unless calculations are done explicitly by hand. This is because computerized analyses routinely yield outputs of ‘TER<30’ when computed TER values fall below 30 ml/min.

Importantly, our criticism pertains exclusively to Bubeck's evaluation formula and does not extend to the 99mTc-MAG3 TER method per se. In support of this, the 40 min radioactivity was invariably below the activity at 20 min. Thus, the raw activity data never indicated a negative clearance and the failure of the technique was due exclusively to the mathematical evaluation. Indeed, as shown in the Subjects and methods section, an expression from Bubeck's formula can be derived which yields a time-dependent critical ratio between injected dose and measured plasma activity at which TER estimates become zero. Whenever the ratio between dose and activity falls below this critical value due to impaired renal elimination of the marker, TER estimates become negative.

Results from the 99mTc-MAG3 method were markedly improved by applying the conventional two-sample model fitting a mono-exponential decay process to the 20 and 40 min values of 99mTc-activity, and negative TER-estimates were thus excluded. Moreover, this two-sample technique did not increase the workload of the procedure; it simply made better use of the measurement data. Furthermore, this technique provided additional physiologically relevant information. For example, the apparent marker activity at time zero together with the applied marker dose allowed rough estimation of the distribution volume, and the decay constant (elimination rate constant) together with the initial distribution volume yielded an estimate for the clearance. Although it is well-known that simple mono-exponential models have problems and weaknesses, they certainly avoid the occurrence of negative results and give more realistic estimates of TER.

The present study showed that extreme caution is necessary when single-sample techniques are used to explore time-dependent phenomena such as renal elimination kinetics. These methods, by definition, require a form of population-based regression algorithm and are prone to fail in individual situations when users are not adequately warned. Two-sample approaches improve the data because at least some temporal marker behaviour information from the investigated individual can be included. However, elimination kinetics are often more complex than simple first-order processes, and systematic deviations of results from two-sample methods are practically unavoidable.

Dynamic physiological processes such as renal elimination require more than two temporarily spaced experimental samples that must be properly analysed. For ideal results, a physiologically meaningful dynamic model should be used.

Notes

Correspondence and offprint requests to: Dr Gilbert Reibnegger, Institut für Med. Chemie und Pregl-Laboratorium, Harrachgasse 21/II, A-8010 Graz, Austria. Email: gilbert.reibnegger{at}kfunigraz.ac.at Back

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Received for publication: 15. 8.01
Revision received 18. 6.02.



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