**6.1 Comparison of indexing GFR for BSA, height and ECV**

In this section we compare indexing GFR for BSA and for the alternatives height and ECV. True mathematical evidence for normalizing a physical quantity by any index is wellknown. (Turner & Reilly, 1995) The uncorrected quantity should be a linear function of the indexator with zero intercept (Figure 3A). After indexation the relationship between the indexed quantity and the indexator then completely disappears (Figure 3B).

Fig. 3. (A) Linear regression of GFR versus the indexator; (B) Disappearance of the relationship indexed GFR-indexator.

Because of their rapidly increasing size and renal maturation, children may give insight into the properties of different normalization indexes. Publicly available data are used to test the mathematical requirements for the indexators BSA, height and ECV. The dataset contains data for healthy children (between 0 and 15 years) of absolute and BSA-corrected median GFR values (51Cr-EDTA), median heights and weights. (Pottel et al., 2010) ECV of the children was calculated with the ECV formula of Bird et al. (Bird, 2003)

### **6.1.1 Indexing GFR for BSA**

When considering median absolute GFR values versus BSA, one may observe a linear relationship (y = 59.96x with R² = 0.96) (Figure 4A). When GFR is indexed for BSA, the relation GFR-BSA disappears once the kidneys reach maturity (Figure 4B). So, the mathematical requirement for an indexator is fulfilled.

### **6.1.2 Indexing GFR for height**

The most evoked factor to index GFR in obese patients is height. (Anastasio, 2000; Schmieder, 1995) Again, we studied the fundamental prerequisite relationship GFR-height and the lack of relationship between GFR indexed for height and height. When the

Is Body Surface Area the Appropriate Index for Glomerular Filtration Rate? 13

Delanaye et al. studied the fundamental mathematical requirement between GFR and height in a limited obese population, but did not find a satisfying linear relationship. (Delanaye, 2005) A better relationship may be found when squared height is used instead of height. It is known that correcting GFR for BSA may cause problems when considering obese or anorectic people. Indexing for height may lead to the same errors, especially in populations

The earliest study of ECV normalisation is by Newman et al., who argued that since the function of the kidneys is to sustain the chemical composition of the extracellular fluid, ECV is more closely related to kidney function than BSA. (Newman et al., 1944) Also the research group of Peters et al. intensively contemplated a change to using ECV to index the GFR. (Bird, 2003; Peters, 1992, 1994b, 2000) Furthermore, several scientists showed a high correlation between BSA and ECV. (Abraham et al., 2011; Newman, 1944; Peters, 2004; White & Strydom, 1991) One could therefore argue that since this equivalence exists, either ECV or BSA can be used to index the GFR. Various studies suggested that ECV is at least a more appropriate index for GFR in children. (Bird, 2003; Friis-Hansen, 1961; Peters, 1994b, 2000) According to Peters et al. this arises from the fact that children have a higher body surface area than adults in relation to their weight, which leads to an overcorrection of the GFR when BSA is used as an indexator. It has also been shown that humans change shape as they grow, which undermines the validity of BSA as an indexation variable. (Peters, 2004) Another argument to prefer ECV above BSA is the fact that ECV is three-dimensional

Mathematical evidence for normalizing GFR for ECV based on our children's database can

Fig. 6. (A) Linear relationship between absolute GFR and ECV; (B) Disappearance of the

In 1961, Friis-Hansen developed a formula to estimate ECV in children. The formula is, like the DuBois & DuBois formula, of the form constant \* weighta \* heightb. (Friis-Hansen, 1961) In 2003, a similar height-weight ECV equation was published by Bird et al. (Bird, 2003) In 2011, Abraham et al. found that height provided most of the information in the estimation of ECV and that ECV could be simply estimated from ECV = height \* √weight.

with extreme height.

be found in Figure 6.

(Abraham, 2011)

**6.1.3 Indexing GFR for ECV** 

wheras BSA is two-dimensional. (Peters, 2000)

A. B.

relationship between GFR and ECV, when GFR is corrected for ECV.

Fig. 4. (A) Linear relationship between absolute GFR and BSA; (B) Disappearance of the relationship between GFR and BSA, when GFR is corrected for BSA.

absolute median GFR values of children were plotted versus height, the results were disappointing (y = 42.01x with R² = 0.77) (Figure 5A). But when the absolute median GFR values of the children are plotted against the squared height, the regression is better (y = 36.89x with R² = 0.99) (Figure 5C) than the regression of GFR against BSA (Figure 4A).

Fig. 5. (A) Linear relationship between absolute GFR and height; (B) No disappearance of the relationship between GFR and height, when GFR is indexed for height; (C) Linear relationship between absolute GFR and squared height; (D) Disappearance of the relationship between GFR and squared height, when GFR is corrected for squared height.

Although height is the most often considered index in the obese population, further research whether height or squared height may serve as a correction factor for GFR is necessary. Delanaye et al. studied the fundamental mathematical requirement between GFR and height in a limited obese population, but did not find a satisfying linear relationship. (Delanaye, 2005) A better relationship may be found when squared height is used instead of height. It is known that correcting GFR for BSA may cause problems when considering obese or anorectic people. Indexing for height may lead to the same errors, especially in populations with extreme height.

#### **6.1.3 Indexing GFR for ECV**

12 Basic Nephrology and Acute Kidney Injury

Fig. 4. (A) Linear relationship between absolute GFR and BSA; (B) Disappearance of the

absolute median GFR values of children were plotted versus height, the results were disappointing (y = 42.01x with R² = 0.77) (Figure 5A). But when the absolute median GFR values of the children are plotted against the squared height, the regression is better (y = 36.89x with R² = 0.99) (Figure 5C) than the regression of GFR against BSA (Figure 4A).

Fig. 5. (A) Linear relationship between absolute GFR and height; (B) No disappearance of the relationship between GFR and height, when GFR is indexed for height; (C) Linear relationship between absolute GFR and squared height; (D) Disappearance of the

relationship between GFR and squared height, when GFR is corrected for squared height. Although height is the most often considered index in the obese population, further research whether height or squared height may serve as a correction factor for GFR is necessary.

A. B.

relationship between GFR and BSA, when GFR is corrected for BSA.

A. B.

C. D.

The earliest study of ECV normalisation is by Newman et al., who argued that since the function of the kidneys is to sustain the chemical composition of the extracellular fluid, ECV is more closely related to kidney function than BSA. (Newman et al., 1944) Also the research group of Peters et al. intensively contemplated a change to using ECV to index the GFR. (Bird, 2003; Peters, 1992, 1994b, 2000) Furthermore, several scientists showed a high correlation between BSA and ECV. (Abraham et al., 2011; Newman, 1944; Peters, 2004; White & Strydom, 1991) One could therefore argue that since this equivalence exists, either ECV or BSA can be used to index the GFR. Various studies suggested that ECV is at least a more appropriate index for GFR in children. (Bird, 2003; Friis-Hansen, 1961; Peters, 1994b, 2000) According to Peters et al. this arises from the fact that children have a higher body surface area than adults in relation to their weight, which leads to an overcorrection of the GFR when BSA is used as an indexator. It has also been shown that humans change shape as they grow, which undermines the validity of BSA as an indexation variable. (Peters, 2004) Another argument to prefer ECV above BSA is the fact that ECV is three-dimensional wheras BSA is two-dimensional. (Peters, 2000)

Mathematical evidence for normalizing GFR for ECV based on our children's database can be found in Figure 6.

Fig. 6. (A) Linear relationship between absolute GFR and ECV; (B) Disappearance of the relationship between GFR and ECV, when GFR is corrected for ECV.

In 1961, Friis-Hansen developed a formula to estimate ECV in children. The formula is, like the DuBois & DuBois formula, of the form constant \* weighta \* heightb. (Friis-Hansen, 1961) In 2003, a similar height-weight ECV equation was published by Bird et al. (Bird, 2003) In 2011, Abraham et al. found that height provided most of the information in the estimation of ECV and that ECV could be simply estimated from ECV = height \* √weight. (Abraham, 2011)

Is Body Surface Area the Appropriate Index for Glomerular Filtration Rate? 15

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

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

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

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

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

. The fast first exponential decay represents mixing of the tracer in the

the concentration of the filtration marker to 37% of its original value.

indicator for kidney function, which is in fact the GFR indexed for ECV.

c(t) = Ae-α<sup>t</sup>

+ Be-β<sup>t</sup>

area under the bi-exponential curve.


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 to question the BSA correction.
