**7. Using the slow rate constant or mean transit time as indicator for kidney function**

In 1980, Brochner-Mortenson already pointed out that it would be more rational to express GFR in relation to ECV than in relation to BSA. (Brochner-Mortensen, 1980) He showed that the ratio GFR/ECV could easily be determined when GFR is measured as the plasma clearance of a filtration marker. This is also clearly explained in a review written by Peters. (Peters, 2004)

Table 6. Height and weight based formulas to estimate ECV. ECV is expressed in L, weight in kg and height in cm, except in the formula of Abraham et al. where height is in meter.

When considering changing GFR indexing from BSA to ECV, the reference ECV value of an average man of 1.73m² yields 13.5L. The ECV corrected GFR would then be expressed in ml/min/13.5L. However, one may not forget that ECV is mainly studied as an alternative index in the pediatric population. Little proof exists for application of ECV to adult or obese populations. Nevertheless, the data show that ECV might be a promising index, fulfilling both mathematical requirements. However, one may question the calculation of ECV from a height-weight equation, especially in the obese, based on the same arguments that are used

When correcting for BSA, problems arise for obese or anorectic people. The data show that square height or ECV might be good alternatives to index the GFR (Figure 7). It seems more logical to index GFR with a measure of fluid volume since the purpose of GFR is to regulate body fluid composition. ECV might be a promising index, fulfilling the theoretical

Fig. 7. Comparison of GFR indexed for BSA, squared height and ECV.

**7. Using the slow rate constant or mean transit time as indicator for** 

In 1980, Brochner-Mortenson already pointed out that it would be more rational to express GFR in relation to ECV than in relation to BSA. (Brochner-Mortensen, 1980) He showed that the ratio GFR/ECV could easily be determined when GFR is measured as the plasma clearance of a filtration marker. This is also clearly explained in a review written by Peters.

**AUTHOR FORMULA**  Friis-Hansen (1961) ECV = 0.0682 \* weight 0.400 \* height 0.633 Bird et al. (2003) ECV = 0.0215 \* weight 0.647 \* height 0.742

Abraham et al. (2011) ECV = height \* √weight

to question the BSA correction.

requirements, even in the obese.

**6.2 Conclusion** 

**kidney function** 

(Peters, 2004)

The glomerular filtration rate of a person can be determined accurately by measuring the plasma clearance of a filtration marker such as iohexol or 51Cr-EDTA, which is generaly given by a single bolus injection. The clearance of the marker is followed from plasma sampling at multiple time points over several hours. In case the mixing is immediate, the concentration-time curve may be described as a mono-exponential decay: c(t) = Ae-αt. The distribution volume in this case is the plasma volume (PV) and the GFR equals the dose of the exogenous marker divided by the area under the concentration-time curve: GFR = Dose/(A/α). The concentration at time zero c(0) = A equals the dose divided by the distribution volume, and therefore GFR = α \* PV or GFR indexed by plasma volume (GFR/PV) equals the rate constant α. As α is expressed in min-1, the inverse T = 1/α expressed in minutes, is the mean transit time. T can be seen as the time needed to reduce the concentration of the filtration marker to 37% of its original value.

In case the mixing is not immediate (as is the case for iohexol and 51Cr-EDTA) and not limited to the plasma volume, the concentration-time curve should be described by a biexponentional decay. The concentration c of the tracer at the time t can then be written as c(t) = Ae-α<sup>t</sup> + Be-βt. The fast first exponential decay represents mixing of the tracer in the distribution volume of the body as well as filtration of the tracer by the kidneys. After the mixing is completed, the slow second exponential decay indicates that only the kidneys are responsible for the further decrease of the tracer in the plasma. The GFR is defined as the injected dose of the marker divided by the area under the plasma clearance curve. Since the area under the c(t) curve is now equal to A/α + B/β, the GFR can also be written as the injected dose divided by A/α + B/β. At time zero, the dose is equal to (A + B) \* PV, where PV is the plasma volume, so GFR = (A + B) \* PV/(A/α + B/β). The mean transit time T = (A/α² + B/β²)/(A/α + B/β), expressed in minutes, equals ECV/GFR, which reduces to PV/GFR in case of a mono-exponential decay (B = 0). In that case, ECV = PV mathematically. The mean transit time T or its inverse 1/T may be seen as an excellent indicator for kidney function, which is in fact the GFR indexed for ECV.

To determine T, a complex procedure with multiple blood samplings (up to 9 samples) is required to obtain the parameters A, α, B and β of the bi-exponential decay. Since it is known that the slow component has a larger contribution to the GFR, the rate constant β is by itself a close reflection of GFR/ECV = 1/T. Therefore, methods to reduce the blood sampling to only the slow exponential decay (2 or 3 samples) were proposed. This onecompartment clearance, also called the slope-intercept technique, always slightly overestimates GFR/ECV since the area under the one-compartment slope is lower than the area under the bi-exponential curve.

In 1972, Brochner-Mortensen published an equation to correct for this overestimation when a 51Cr-EDTA clearance is performed in adults (GFR = 0.990778 \* slowGFR - 0.001218 \* slow GFR²). (Brochner-Mortensen, 1972) Two years later, Brochner-Mortensen et al. performed the same study in children which resulted in the correction equation GFR = 1.01 \* slowGFR – 0.0017 \* slowGFR. (Brochner-Mortensen et al., 1974) In 1992, Peters proposed the regression equation GFR/ECV = -0.093 + 1.06β + 0.009β², relating β or the 'approximate' GFR/ECV to the 'true' GFR/ECV based on the bi-exponentional plasma clearance curve of the radio-active agent 99Tc-DTPA. (Peters, 1992) Over the years, improved equations for correcting the slope-intercept measurements in different clinical populations have been published. (Brochner-Mortensen & Jodal, 2009; Fleming, 2007; Fleming et al., 2004; Jodal & Brochner-Mortensen, 2009) Although the radio-active agent 51Cr-EDTA is an excellent filtration marker, it is not accepted in the USA. Lately, the constrast medium iohexol is seen

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as an excellent alternative for the radio-active markers. In 2010, Schwartz et al. showed how the GFR in children could be determined from the slow component of an iohexol clearance curve according to the equation GFR = 1.0019 \* slowGFR – 0.001258 \* slowGFR². (Schwartz et al., 2010) Since the literature often cautions that the equations are only valid in the populations similar to those in which they were developed, Derek et al. recently tried to develop, based on an iohexol study, an universal formula for use in adults as well as in children. (Derek et al., 2011) The equation of Derek et al. is already expressed in ml/min/1.73m² and is of the form: GFR = slowGFR/[1 + 0.12 (slowGFR/100)]. The rate constant β, or even T, may be a perfect indicator for kidney function, which immediately allows comparison of the kidney function of different persons with each other. Simulations showed us that the cut-off value of 60 ml/min/1.73m² agreed with a β value of 0.004 min-1 or a transit time of 4 hours, a cut-off value of 30 ml/min/1.73m² agreed with a β value of 0.002 min-1 or a transit time of 8 hours.
