Clearance calculation based on a single sample

Summary

A formula has been derived for the calculation of renal clearance with the use of a single plasma sample. The formula is based on a one-compartment model. A small correction for non-immediate mixing and non-uniform distribution of the tracer was calculated from empirical data. The accuracy in the calculation method depends on how exactly the distribution volume is known and at what time the blood sample is taken. The expected standard deviation in the clearance value was calculated from data of mean value and spread for the distribution volume of $^{99}\text{Tc}^m\text{-DTPA}$. In an investigation of 39 subjects with $^{99}\text{Tc}^m\text{-DTPA}$, a standard deviation of 5 to 6 ml/min was obtained in comparison with a standard method for clearance calculation. This value is in good agreement with the expected one.

Introduction

It has previously been shown that the tracer concentration in a single plasma sample taken 3-5 h after the injection is correlated to the renal clearance (Tauxe et al., 1971; Fisher & Veall, 1975; Dakubu et al., 1980). Recently, Groth and Aasted (1981) have derived a nomogram and a formula based on a comparison between the tracer concentration in a single sample and the clearance value calculated according to the Bröchner-Mortensen method (1972).

The formula by Groth and Aasted is empirical and not based on any clearance model. In the present communication, a formula is derived from a simple one-compartment model. With this formula it is possible to take into account different distribution volumes correctly and to determine the optimum time for taking the plasma sample. Further, the expected accuracy in the one-sample method can be determined.

Method

The method is based on a one-compartment model. The distribution volume of the tracer is described by the compartment volume $V$. The disappearance of the tracer, caused by the renal excretion, is described by a disappearance rate constant $k$.

Firstly, it is assumed that the tracer is immediately and uniformly mixed in the distribution volume after the injection. After that, a correction for non-immediate mixing is applied and finally the effect of non-uniform distribution is taken into account.

Immediate and Uniform Mixing

According to the definition, the clearance value $Cl$ is given by:

$Cl = \frac{\dot{Q}(t)}{C(t)} \quad (1)$

where $C(t)$ is the tracer concentration in plasma and $\dot{Q}(t)$ is the excreted amount of tracer per unit time.

Assuming uniform and immediate mixing of the tracer in the volume, the following is valid:

(i) The tracer concentration at any time $t$ is given by:

$C(t) = \frac{Q(t)}{V} \quad (2)$

where $Q(t)$ is the amount of tracer.

(ii) The disappearance of the tracer is monoexponential from the injection time and the disappearance rate $k = \dot{Q}(t)/Q(t)$ can be written:

$k = \frac{1}{t} \ln \left[ \frac{C(0)}{C(t)} \right] \quad (3)$

Combining (1) and (2), the clearance value can be written:

$Cl = k \times V \quad (4)$

and with the use of (3):

$Cl = \frac{V}{t} \times \ln \left( \frac{Q(0)}{V \times C(t)} \right) \quad (5)$

$Q(0)$ is equivalent to the injected amount of tracer.

Correction for Non-Immediate Mixing

During the first minutes after the injection, before the tracer is completely mixed, the concentration in plasma is greater than during immediate mixing. The greater concentration gives an increased renal excretion which can be interpreted as a loss of tracer activity. The injected amount of tracer, $Q(0)$ in formula (5), therefore shall be reduced by a reduction factor $r$. The corrected clearance value then can be written:

$Cl_r = \frac{V}{t} \times \ln \left( \frac{r \times Q(0)}{V \times C(t)} \right) \quad (6)$

Correction for Non-Uniform Distribution

After equilibration, when mixing is complete, the tracer concentration is less in plasma than in the distribution volume as a whole (Chantler et al., 1969). The formula (2) therefore shall be modified to:

$C(t) = m \times \frac{Q(t)}{V} \quad (7)$

where $m$ is a factor less than 1. The factor depends on the clearance value, but is supposed to be constant with time after equilibration. The formulas (4) and (6) then will be altered to:

$Cl = k \times V' \quad (8)$

and

$Cl = \frac{V'}{t} \times \ln \left( \frac{r \times Q(0)}{V' \times C(t)} \right) \quad (9)$

where $V' = V/m$.

Calculation of $r$ and $m$

The factors $r$ and $m$ were calculated by means of two empirical formulas by Bröchner-Mortensen (1980):

$Cl = 0.9908 \times Cl_1 - 0.0012 \times Cl_1^2 \quad (10)$

and

$V_s = \frac{Cl_s}{b} \times \left[ \left( \frac{Cl_s}{Cl_1} \right)^2 - \frac{Cl_s}{Cl_1} + 1 \right] \quad (11)$

where $Cl_1$ is clearance calculated from a one pool-system and is given by:

$Cl_1 = b \times \frac{Q(0)}{C_0} \quad (12)$

where $b$ is the final slope of the plasma curve, $C_0$ is the extrapolated concentration at time 0, $V_s$ is the calculated distribution volume, and $Cl_s$ is clearance according to (10), '$Cl$', with a small correction for different plasma volumes. For a plasma volume of 3000 ml, $Cl_s$ is equal to '$Cl$'. In the further calculations, $Cl_s$ has been approximated to '$Cl$'.

The reduction factor can be applied on formula (12), which will give a corrected clearance value equal to:

$Cl = b \times \frac{r \times Q(0)}{C_0} \quad (13)$

The factor $r$ is therefore equal to $Cl/Cl_1$. By using (10), the factor is obtained as:

$r = 0.495 \times (1 + \sqrt{1 - 0.0049 \times Cl}) \quad (14)$

The value of $r$ is nearly one for low clearance values and decreases to about 0.8 at high values.

Combining (10) and (11), the distribution volume can be written as:

$V = \frac{Cl}{b} \left(0.991 - 0.00122 \times Cl \right) \quad (15)$

According to (8) and assuming $k = b$, the modifying factor is given by:

$m = 0.991 - 0.00122 \times Cl \quad (16)$

The value of $m$ is nearly one for small clearance values and decreases to about 0.85 at high values.

Clearance Computation

The formulas (14) and (16) cannot be used directly in the calculation of clearance, as $Cl$ is not known in advance. The following procedure therefore has been used for the clearance computation.

Rearranging (9), the clearance value can be written:

$Cl = \frac{1}{\frac{t}{V'} - \frac{\ln r}{Cl}} \times \ln \left( \frac{Q(0)}{V' \times C(t)} \right) \quad (17)$

The calculated value of the quotient $\ln r/Cl$ is shown as a function of $Cl$ in Fig. 1. The value is nearly constant for $Cl > 20$ ml/min and can, with good approximation, be set equal to $-0.0016$ (ml/min)$^{-1}$ for all clearance values.

The corrected clearance value then can be written:

$Cl = \frac{1}{t/V' + 0.0016} \ln \left( \frac{Q(0)}{V' \times C(t)} \right) \quad (18)$

The effect of the modifying factor is accounted for by the following procedure: (i) An approximate clearance value, $Cl_r$, is calculated according to (18) but with the use of the true distribution volume $V$; (ii) $m$ is calculated according to (16) with the use of $Cl_r$; (iii) The corrected clearance value is calculated according to (18) with the use of $V' = V/m$.

The error in $Cl$ caused by this procedure is in general less than 1%.

Dependence on Error in V

The distribution volume $V$ is dependent on the type of tracer. It has been shown that the volume corresponds to $12.4 \pm 2.4$ (SD) % of body weight ($m$) for inulin (Ladegaard-Pedersen, 1972). For $^{51}\text{Cr-EDTA}$, the following data has been found: $21.8 \pm 5.1$ (SD) % (Ladegaard-Pedersen, 1971), $20.3 \pm 2.0$ (SD) % for males and $18.8 \pm 2.4$ (SD) % for females (Bröchner-Mortensen, 1980). In this investigation (see clinical application), a value $24.6 \pm 6.9$ (SD) % was found for $^{99}\text{Tc}^m\text{-DTPA}$. The very large standard deviations found in these investigations implies that the estimation of $V$ for an individual is uncertain with up to 50% variation. However, the calculation of clearance according to (18) is fairly insensitive on the precise value of $V$, especially if the blood sample is taken at a proper time.

A calculation of clearance using different values of $V$ is shown in Fig. 2. For a clearance value of 100 ml/min and a sample time of 200 min, a 50% error in $V$ will give an error in clearance value of less than 15%.

The uncertainty in $Cl$ caused by the uncertainty in $V$ was calculated using the above mentioned mean value and standard deviation of $V$ for $^{99}\text{Tc}^m\text{-DTPA}$. Fig. 3 shows the results of such a calculation made for two different sample times. The calculated standard deviation, $\sigma_{Cl}$, is thus around 12 ml/min at $Cl = 100$ ml/min and around 4 ml/min at $Cl = 30$ ml/min for a sample time of 300 to 500 min after injection.

The Optimal Sample Time

The optimal time, $t_{opt}$, for taking the sample is when the clearance calculation is least dependent on $V$. This happens when the derivative $dCl/dV'$ of formula (18) is zero, i.e. when $t = V'/Cl$. Fig. 4 shows $t_{opt}$ as a function of $Cl$ for two distribution volumes $V$. For clearance values around 100 ml/min, $t_{opt}$ is about 3 h, and for clearance values around 30 ml/min $t_{opt}$ is about 10 h.

Dependence on Errors in $Q(0)$ and $C(t)$

The error $\Delta Cl$ which appears in $Cl$ for a certain error in $Q(0)$ or $C(t)$ can be written:

$\Delta Cl = \frac{1}{t/V' + 0.0016} \times \left( \frac{\Delta Q(0)}{Q(0)} - \frac{\Delta C(t)}{C(t)} \right)$

The error is nearly independent on the clearance value. For $V' = 15,000$ ml, a relative error of 10% in $Q(0)$ or $C(t)$ will give errors of 5 ml/min for $t = 300$ min and 3 ml/min for $t = 500$ min.

Clinical Application

The method was tested on 39 patients, examined with 150 MBq of $^{99}\text{Tc}^m\text{-DTPA}$. Four blood samples were drawn 240, 260, 280 and 300 min after the injection. One millilitre of plasma from each sample was measured in a gamma well counter for 1 min on the day after the examination. From the activity concentration in the four samples, clearance was calculated according to the Bröchner-Mortensen method (1972). The final slope $b$ of the plasma curve was obtained by the least squares method and the ‘one-pool clearance’, $Cl_1$, was calculated from formula (12). The corrected clearance value was calculated according to formula (10). Clearance, according to the one-sample method [formula (18)] was calculated for the 240 and 300 min samples. The distribution volume used in the calculation was 24.6% of the body mass. This value was obtained as the mean value of the calculated distribution volumes [formula (11)] for the 39 patients.

Fig. 5 shows the comparison between clearance values obtained with the Bröchner-Mortensen method and the one-sample method with the sample time $t = 300$ min. The mean difference between the two values was $2.3 \pm 1.0$ (SEM) ml/min and the standard deviation was 6.1 ml/min. The mean difference between the clearance values calculated from the Bröchner-Mortensen method and the 240 min sample was $2.4 \pm 0.8$ (SEM) ml/min and the standard deviation was 5.3 ml/min.

Discussion

The formula derived for clearance determination with one sample requires that the distribution volume, $V$, is known. The accuracy in the estimation of $V$ for an individual determines the accuracy in the method. In this investigation, a great spread was found in the quotient $V/m$ for $^{99}\text{Tc}^m\text{-DTPA}$ compared to an investigation with $^{51}\text{Cr-EDTA}$ (Bröchner-Mortensen, 1980). If the difference in spread is real, the accuracy in the one-sample method should be about three times better for $^{51}\text{Cr-EDTA}$.

The correction for non-immediate mixing and non-uniform distribution are derived from empirical formulas of Bröchner-Mortensen. These formulas were calculated for $^{51}\text{Cr-EDTA}$ and may not be exactly the same for $^{99}\text{Tc}^m\text{-DTPA}$. However, a rather great relative error in the corrections can be tolerated, as the corrections are small, typically less than 10%. The empirical formulas by Bröchner-Mortensen indicate that the values of $r$ and $m$ are dependent on the plasma volume. The variation with the volume, however, is small and therefore the effect has been neglected.

The best accuracy in the method is obtained when the plasma sample is taken at the optimal time $t_{opt}$. For expected clearance values below 30 ml/min, $t_{opt}$ is 10 h or more which may be an impracticably long time. However, even a moderate increase from the commonly used time 4-5 h to 8 h, will decrease the uncertainty in the clearance value by a factor of about two. For clearance values above 100 ml/min, $t_{opt}$ is less than 2 h. This may be a too short time for equilibration between the different fluid components (Chantler et al., 1969). If equilibration is not achieved, the clearance value will be underestimated.

The formula derived is fairly insensitive to the precision in the quotient $Q(0)/C(t)$. It is sufficient to keep the error in the quotient within $\pm 10\%$. This means that the commonly used measuring time in a well counter can be considerably reduced.

In the investigation of the 39 subjects with $^{99}\text{Tc}^m\text{-DTPA}$, the result obtained at high clearance values is somewhat better than the predicted one. At low clearance values the accuracy also seems to be at least as good as the predicted one, for the few subjects examined. However, further investigations are needed, with blood sampling at the later time of 8 h or more after the injection to control the reliability of the method at low clearance values.

In summary, the derived formula is useful in calculating clearance using a single sample drawn at any time after the equilibration. Accuracy in the calculation will be improved when a more exact distribution volume is known and with the sample being drawn nearer to the optimal time.

References

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