**2.2 The formula of DuBois & DuBois (1916)**

Meeh's formula remained standard until 1916 when the DuBois brothers published several manuscripts exploring different formulas to measure the BSA. To determine the surface area

Is Body Surface Area the Appropriate Index for Glomerular Filtration Rate? 5

3a + b gives a value of 1.95, the formula of Gehan & George gives a value of 1.97 and Haycock's equation gives 2.01. In 1987, Mosteller also used dimension analysis to derive a simplified formula from that of Gehan & George, which is easier to remember. (Mosteller, 1987) In 2000, Shuter & Aslani presented the original results of the DuBois & DuBois formula on a more robust statistical footing (non-linear regression), yielding values for the model constants that would have been obtained if the brothers had had access to modern statistical methods and modern computers. (Shuter & Aslani, 2000) Since fitting BSA estimates directly by non-linear regression is more accurate, this was the technique Livingston & Lee preferred in 2001 to derive their formula which, like Meeh's formula, only relies on body weight. Based on the data of 47 patients, they developed an equation, which

In general, the correlation between all these formulas is high (Verbraecken et al., 2006), but there are no clear advantages of the others over the DuBois formula, which remains the best known and most used formula today. Although, it is unrealistic to expect that one single height-weight formula predicts BSA with the same accuracy in all people, since humans change shape as they grow and age. Errors in height-weight formulas will be exaggerated in very small people like children or in obese patients. It has been shown that for children weighing below 10 kg and in the obese population, the DuBois formula does not give the best results. For children with a weight below 10 kg the formulas of Haycock or Mosteller are preferred (van der Sijs & Guchelaar, 2002), for obese persons the formula of Livingston & Lee is advised. (Livingston & Lee, 2001) A simulation with virtual data showed us that the formulas of Boyd, Gehan & George, Haycock and Mosteller overestimate the BSA as compared to the BSA calculated with the DuBois & DuBois formula. In Figure 1 one may observe an elliptical shape of the data in the plot of the BSA calculated with the Gehan & George equation against the DuBois & DuBois calculated BSA. The graph also illustrates that a BSA of 1.73m² calculated by DuBois & DuBois can vary between 1.65m² and 2.15m² if calculated by the Gehan & George equation. This elliptical shape is also seen when plotting the formulas of Boyd, Gehan & George, Haycock et al. and Mosteller against the DuBois &

Fig. 1. BSA calculated with the formula of Gehan & George plotted against the DuBois & DuBois calculated BSA. The solid line is the identity line. The dotted lines indicate a body

is especially useful to predict BSA in obese patients. (Livingston & Lee, 2001)

DuBois formula (graphs are not shown).

surface area of 1.73m².

of the various parts of the body, the brothers tightly covered patients' bodies with manila paper molds. The area of the mold was determined by cutting it in pieces and placing the pieces flat on photographic film which was exposed to sunlight. The unexposed paper was then cut and weighed. The BSA was derived from the weight divided by the average density of the photographic paper. The brothers used 19 detailed measurements from 7 different body parts of 5 patients to derive a geometric formula to predict BSA that had an acceptably limited error compared with the true BSA. (DuBois & DuBois, 1915) One year later, DuBois & DuBois added 4 patients to their 5 initial patients and they derived, by iteration, a 'height-weight' formula from this dataset. Among the 9 patients, there was a child of 2 suffering from rickets, an obese adult female, a 36 year old adult with mental and physical development of an 8 year old and a diabetic patient of 18 with a very low body mass index. (DuBois & DuBois, 1916)

In order to construct their formula, the brothers concluded that the error in Meeh's formula could be reduced by also taking height into consideration, since adding height made the formula more applicable for patients of the same general shape but differing somewhat in relative dimensions. DuBois & DuBois assumed that area could be estimated from the formula BSA = CWaHb, with C a constant, W = weight in kg and H = height in cm. As the left side in this equation has the dimensions of an area (squared length L2), the right side needs to have the same dimensionality. As weight is considered proportional to a volume (L3) when mass density is considered constant, the following dimensionality condition is obtained: 2 = 3a + b. This constraint reduces the complexity of the problem, as only two parameters (the constant C and a or b) have to be obtained from the data. Logarithmic transformation of both sides of the equation reduces the problem even further to a simple linear regression fit. With the aid of a computer this is easily performed with modern statistical methodology, but back in 1916, DuBois & DuBois had to use repetitive combinations for a and b to arrive at their final and today's well known DuBois & DuBois formula: BSA = 0.007184 \* W 0.425 \* H 0.725 .

#### **2.3 The search for a better formula (1935-2010)**

In the following years, several authors have proposed other formulas using more sophisticated statistical techniques and studying larger populations. In 1935, Boyd listed 401 direct measurements of surface area obtained by direct coating, triangulation or surface integrator methods. (Boyd, 1935) Boyd recommended 2 formulas for calculating surface area, one based on height and weight and one based on weight only. The formula involving height and weight was superior to that based on weight alone. In 1970, Gehan & George used Boyd's database to refine the exponents in the equation proposed by DuBois & DuBois. (Gehan & George, 1970) Although Gehan & George did find that their equation failed for small children and obese subjects, no further attempt was made to assess other models relating height and weight to BSA. In 1978, Haycock et al. started calculating BSA by using a geometric method with schematic reduction of body segments to cylinders and a sphere. (Haycock et al., 1978) Validation in 81 persons, from premature infants to adults, was done by comparison with the DuBois & DuBois formula (DuBois & DuBois, 1916) for adults and with the Faber & Melcher formula (Faber & Melcher, 1921) for infants. The formulas of Boyd, Gehan & George and Haycock rely on linear regression of logarithmically transformed weight and height measurements. The authors published equations of the type of DuBois & DuBois, in which different values for the constants a, b and C were obtained. Although the constraint 3a + b = 2 was not used in the construction of these formulas, one may notice that the dimensionality condition is nearly always fulfilled. For Boyd's formula

of the various parts of the body, the brothers tightly covered patients' bodies with manila paper molds. The area of the mold was determined by cutting it in pieces and placing the pieces flat on photographic film which was exposed to sunlight. The unexposed paper was then cut and weighed. The BSA was derived from the weight divided by the average density of the photographic paper. The brothers used 19 detailed measurements from 7 different body parts of 5 patients to derive a geometric formula to predict BSA that had an acceptably limited error compared with the true BSA. (DuBois & DuBois, 1915) One year later, DuBois & DuBois added 4 patients to their 5 initial patients and they derived, by iteration, a 'height-weight' formula from this dataset. Among the 9 patients, there was a child of 2 suffering from rickets, an obese adult female, a 36 year old adult with mental and physical development of an 8 year old and a diabetic patient of 18 with a very low body

In order to construct their formula, the brothers concluded that the error in Meeh's formula could be reduced by also taking height into consideration, since adding height made the formula more applicable for patients of the same general shape but differing somewhat in relative dimensions. DuBois & DuBois assumed that area could be estimated from the formula BSA = CWaHb, with C a constant, W = weight in kg and H = height in cm. As the left side in this equation has the dimensions of an area (squared length L2), the right side needs to have the same dimensionality. As weight is considered proportional to a volume (L3) when mass density is considered constant, the following dimensionality condition is obtained: 2 = 3a + b. This constraint reduces the complexity of the problem, as only two parameters (the constant C and a or b) have to be obtained from the data. Logarithmic transformation of both sides of the equation reduces the problem even further to a simple linear regression fit. With the aid of a computer this is easily performed with modern statistical methodology, but back in 1916, DuBois & DuBois had to use repetitive combinations for a and b to arrive at their final and

today's well known DuBois & DuBois formula: BSA = 0.007184 \* W 0.425 \* H 0.725 .

In the following years, several authors have proposed other formulas using more sophisticated statistical techniques and studying larger populations. In 1935, Boyd listed 401 direct measurements of surface area obtained by direct coating, triangulation or surface integrator methods. (Boyd, 1935) Boyd recommended 2 formulas for calculating surface area, one based on height and weight and one based on weight only. The formula involving height and weight was superior to that based on weight alone. In 1970, Gehan & George used Boyd's database to refine the exponents in the equation proposed by DuBois & DuBois. (Gehan & George, 1970) Although Gehan & George did find that their equation failed for small children and obese subjects, no further attempt was made to assess other models relating height and weight to BSA. In 1978, Haycock et al. started calculating BSA by using a geometric method with schematic reduction of body segments to cylinders and a sphere. (Haycock et al., 1978) Validation in 81 persons, from premature infants to adults, was done by comparison with the DuBois & DuBois formula (DuBois & DuBois, 1916) for adults and with the Faber & Melcher formula (Faber & Melcher, 1921) for infants. The formulas of Boyd, Gehan & George and Haycock rely on linear regression of logarithmically transformed weight and height measurements. The authors published equations of the type of DuBois & DuBois, in which different values for the constants a, b and C were obtained. Although the constraint 3a + b = 2 was not used in the construction of these formulas, one may notice that the dimensionality condition is nearly always fulfilled. For Boyd's formula

**2.3 The search for a better formula (1935-2010)** 

mass index. (DuBois & DuBois, 1916)

3a + b gives a value of 1.95, the formula of Gehan & George gives a value of 1.97 and Haycock's equation gives 2.01. In 1987, Mosteller also used dimension analysis to derive a simplified formula from that of Gehan & George, which is easier to remember. (Mosteller, 1987) In 2000, Shuter & Aslani presented the original results of the DuBois & DuBois formula on a more robust statistical footing (non-linear regression), yielding values for the model constants that would have been obtained if the brothers had had access to modern statistical methods and modern computers. (Shuter & Aslani, 2000) Since fitting BSA estimates directly by non-linear regression is more accurate, this was the technique Livingston & Lee preferred in 2001 to derive their formula which, like Meeh's formula, only relies on body weight. Based on the data of 47 patients, they developed an equation, which is especially useful to predict BSA in obese patients. (Livingston & Lee, 2001)

In general, the correlation between all these formulas is high (Verbraecken et al., 2006), but there are no clear advantages of the others over the DuBois formula, which remains the best known and most used formula today. Although, it is unrealistic to expect that one single height-weight formula predicts BSA with the same accuracy in all people, since humans change shape as they grow and age. Errors in height-weight formulas will be exaggerated in very small people like children or in obese patients. It has been shown that for children weighing below 10 kg and in the obese population, the DuBois formula does not give the best results. For children with a weight below 10 kg the formulas of Haycock or Mosteller are preferred (van der Sijs & Guchelaar, 2002), for obese persons the formula of Livingston & Lee is advised. (Livingston & Lee, 2001) A simulation with virtual data showed us that the formulas of Boyd, Gehan & George, Haycock and Mosteller overestimate the BSA as compared to the BSA calculated with the DuBois & DuBois formula. In Figure 1 one may observe an elliptical shape of the data in the plot of the BSA calculated with the Gehan & George equation against the DuBois & DuBois calculated BSA. The graph also illustrates that a BSA of 1.73m² calculated by DuBois & DuBois can vary between 1.65m² and 2.15m² if calculated by the Gehan & George equation. This elliptical shape is also seen when plotting the formulas of Boyd, Gehan & George, Haycock et al. and Mosteller against the DuBois & DuBois formula (graphs are not shown).

Fig. 1. BSA calculated with the formula of Gehan & George plotted against the DuBois & DuBois calculated BSA. The solid line is the identity line. The dotted lines indicate a body surface area of 1.73m².

Is Body Surface Area the Appropriate Index for Glomerular Filtration Rate? 7

and kidney weight and between BSA and urea excretion in humans. (MacKay, 1932). Based on these observations, the indexation of GFR for BSA became standard in the medical

McIntosh et al. also introduced the use of the reference surface area of 1.73m², which was the average calculated BSA of 25 year old Americans at that time. The value of 1.73m² has served the physiological community well for nearly 80 years, but is clearly no longer applicable to modern Western populations, as has been shown by Heaf et al. (Heaf, 2007) A value of 1.95m² would probably be more appropriate for the average BSA of today's 25 year old adults in America. Switching from 1.73m² to 1.95m² has severe repercussions for the current classification system for Chronic Kidney Disease, which is based on fixed limits of 15, 30, 60 and 90 ml/min/1.73m². The importance of 1.73m² or 1.95m² is not the value as such, but the fact that it serves as a reference point. Therefore, there is no need to change the

Recently, Delanaye et al. recalculated Taylor's correlation and noted that the correlation between BSA and kidney weight was not different from that between kidney weight and body weight. (Delanaye et al., 2009a) This indicates that the BSA-indexation theory of

In 1928, McIntosh et al. already noticed that indexing is not necessary for 'normally built' people. McIntosh stated: "*The nature of the standard clearance formula is such that correction for body size in persons between 62 and 71 inches in height does not exceed 5 per cent, and in tests of renal function may be neglected.*" (McIntosh et al., 1928) It follows that in longitudinal studies, the absolute GFR should be used for evaluating the kidney function, avoiding the use of BSA-indexed GFR which is affected by weight changes. On the other hand, indexation seems to be necessary to compare different patient values and to allow comparison with fixed reference values. Three cases will here be studied to illustrate these statements: (1) the GFR of a small and heavy person will be compared with each other, (2) the GFR evolution during childhood will be presented and (3) the GFR of two adult men, one with a stable

Imagine the body of a small and heavy person as a small and big pond with the kidneys as a pump and filter combination to clear the dirt out of the pond. The dirt is equally present in the pond and the pump sends a constant flow through the filter which is here assumed to have the same clearing efficiency, after which the cleared water is drained off in the pond again. Repeated cycles will diminish the concentration of dirt in the water. If the small and the big pond both have an equally working pump and filter combination of 60 ml/min, it will take much longer for the big pond to be cleared than it will take for the small pond. Or the pump of the big pond will have to work at a higher rate than the pump of the small one to clear all the dirt out of the water in the same time. This indicates that the function of the pumps must be corrected for a value that describes the size of the ponds in a certain way. If we normalize the absolute GFR of 60 ml/min of the small and heavy person for the BSA (Table 2), then we get a corrected cGFR of 73 ml/min/1.73m² for the small person as opposed to a much smaller cGFR of 46 ml/min/1.73m² for the heavy person. Once the GFR is BSA corrected, it becomes clear that the small person has a better kidney function than the

weight and one with a weight that increases with age, will be followed.

**4.1 Case 1: Comparison of the GFR of a small and heavy person** 

community.

reference value.

McIntosh et al. was based on false assumptions.

**4. Is it necessary to index GFR?** 

For the determination of the body surface area of Indians, Banerjee et al. updated the constant C of the formula of DuBois & DuBois. (Banerjee & Bhattacharya, 1961; Banerjee & Sen, 1955) Nwoye et al. computed new variables for height and weight formulas that accurately predict the surface area of Africans and Saudi males. (Nwoye, 1989; Nwoye & Al-Shehri, 2003) The formula of Fujimoto et al. was developed to calculate the BSA in the Japanese population (Fujimoto et al., 1968), while Stevenson developed a formula to estimate the BSA in Chinese people. (Stevenson, 1937) Interesting is that recently, a new 3Dscanning method for measuring BSA is introduced and used to propose new BSA formulas. (Tikuisis et al., 2001; Yu et al., 2010) An overview of a non exhaustive list of BSA formulas is given in Table 1.


Table 1. Overview of BSA formulas. BSA is expressed in m², weight in kg, height in cm.
