## Abstract

None of the equations frequently used to predict body surface area (BSA) has been validated for obese patients. We applied the principles of body size scaling to derive an improved equation predicting BSA solely from a patient's weight. Forty-five patients weighing from 51.3 to 248.6 kg had their height and weight measured on a calibrated scale and their BSA calculated by a geometric method. Data were combined with a large series of published BSA estimates. BSA prediction with the commonly used Du Bois equation underestimated BSA in obese patients by as much as 20%. The equation we derived to relate BSA to body weight was a power function: BSA (m^{2}) = 0.1173 × Wt (kg)^{0.6466}. Below 10 kg, this equation deviated significantly from the BSA vs. body weight curve, necessitating a different set of coefficients: BSA (m^{2}) = 0.1037 × Wt (kg)^{0.6724}. Covariance of height and weight for patients weighing <80 kg reduced the Du Bois BSA-predicting equation to a power function, explaining why it provides good BSA predictions for normal-size patients but fails with obesity.

- body height
- body weight
- anthropometry
- biological models

to be clinically useful, various physiological measurements must be corrected for different patient sizes. By the same token, there are highly toxic drugs with narrow therapeutic windows that must be precisely administered to avoid complications. In clinical practice, patient size differences are accounted for by dividing the physiological measurement or drug dose by the patient's body surface area (BSA). Because of the critical nature of these measurements, BSA predictions should be as accurate as possible. However, BSA is difficult to measure. Precise measurements require body casting with subsequent planometric measurements. Du Bois and Du Bois (6) correlated the casting technique to estimates derived from geometric calculations applied to individual body segment length and diameter measurements. The geometrically calculated BSA estimates closely approximated the values measured by casting (6). Geometric estimation of BSA requires numerous body segment measurements, making it impractical for BSA determination in clinical medicine. Du Bois derived a mathematical equation relating height and weight to BSA to facilitate BSA prediction from easily obtainable body height and weight measurements. Du Bois's equation was derived empirically and was based on measurements from only nine patients. Despite these limitations, it remains the most commonly used equation for BSA prediction in clinical medicine.

Larger series of BSA estimation by geometric methods revealed that the Du Bois equation predicted falsely low BSA values. BSA predictions were improved by modifying the Du Bois equation coefficients. The basic form of the Du Bois equation, relating height and weight to BSA, has not been challenged (7, 8). Most physiological measurements can be related to weight by a power function irrespective of body height (2, 12, 13). Because of the geometric relationship between three-dimensional body volume (with volume being proportional to weight) and two-dimensional BSA, BSA should be proportional to the body weight raised to the two-thirds (
) power (5). Thus a power function relating BSA to weight^{2/3} should predict BSA.

No BSA study has included a population of obese patients. With obesity, weight increases without a proportional increase in height. Consequently, it is possible that the Du Bois-type BSA-predicting equations, including height coefficients, could systematically miscalculate BSA for obese patients. Because many clinically important measurements are indexed to BSA, systematic errors in BSA estimation can adversely affect the clinical care of obese patients. Using geometric techniques, we estimated BSA in a population of obese patients. Combining our estimates with those from a large, published data set enabled modeling of the BSA-weight relationship to a power function by nonlinear regression. The resultant equation predicted BSA from weight and had an exponent close to 2/3, as predicted by the dimensional relationship between body volume and surface area.

## METHODS

#### Patients and measurements.

Consecutive patients seen in medicine and surgery clinics during a 2-mo period were examined. Patients were selected to exclude those with diagnoses that affect body composition, such as cancer. Patients presenting for morbid obesity surgery evaluation were included. Patients with body mass index (BMI) values >40 were alternated with those <40. All patients were weighed on a calibrated scale. Body segment lengths and circumferences were measured to the nearest millimeter, as was the total body height. Du Bois's method for converting these measurements to total BSA was used (6). With this geometric technique, surface area was estimated by multiplying the circumference of a body segment by its length and by a factor correcting for shape. The various surface areas were summed to determine the total BSA (Table 1). BMI was determined by dividing the weight by the square of the height in meters.

#### Previously published measurements.

Boyd (4) published a series of tables containing BSA measurements and estimates performed by a number of investigators. This is the same data set utilized by Gehan and George (7). Height, weight, and measured BSA data were provided for 413 measurements from patients weighing from 1 g to 98 kg. These were entered into a database along with our data.

#### Equations.

We modified the basic scaling relationship
to
where BSA is the body surface area in meters squared,*a* is a dimensionless coefficient, Wt is the weight in kilograms, and *b* is a dimensionless scaling coefficient. This will be referred to as the scaling model.

The Du Bois form of the BSA equation was also fitted to the data
where *a*, *b,* and *c* are dimensionless coefficients, and Ht is the height in meters. We refer to this as the standard or Du Bois model.

#### Covariance of height and weight.

Plotting height and weight measurements against each other assessed the interdependence of weight on height. After application of a moving average filter, a first-derivative plot was generated. The height-weight curve flattens where height no longer covaries with weight. The weight at which this occurs is determined from the point where the derivative plot crosses zero. Regression analysis was performed separately for the weights above and below the derivative plot zero crossing.

#### Mathematical modeling.

Data were fitted by nonlinear regression (NCSS, Kaysville, UT). Geometric BSA estimates served as the dependent variable. The independent variables were weight (Wt) and height (Ht). The parameters*a*, *b*, and *c* were determined by the nonlinear regression computer program. Goodness of fit for the models was determined from the *R*
^{2} [1 − residual sum of the squares (RSS)/total sum of the squares (TSS)]. TSS was corrected for the mean of a data set to allow comparison of data with different ranges.

When mathematical equations are compared by nonlinear regression, the one with the smallest RSS best fits the data and therefore represents the best model. For equations with different numbers of variables, this may not be true; the equation with more variables might result in a better fit, because there are more degrees of freedom.*F*-testing is performed to determine whether the equation with more variables better fits the data because of the variable effect or because it represents a better model for the data (11).*F*-testing of the RSS was performed when the Du Bois and scaling equations were compared to determine which best described the geometrically derived BSA data.

#### Error analysis.

Performance of the BSA-estimating models was determined by measuring the percentage of difference between the models' predicted BSA and the BSA estimated by the geometric method Patients were segregated into four groups: 0–9 kg, 10–49 kg, 50–79 kg, and >80 kg. Means ± SE of the percentage differences were compared by ANOVA, with contrasts to determine statistical significance.

## RESULTS

#### Measurements.

Forty-seven patients (29 female and 18 male) were included in this study. Their weights ranged from 51.3 to 248.6 kg. Heights ranged from 152 to 182 cm, with a mean of 167 ± 1.4. The lowest BMI was 18.3 and the highest was 91.3. Eighteen patients had BMIs <29 (mean BMI = 24.9 ± 0.8). Six were obese, with BMIs ranging from 30 to 39 (mean BMI = 32.6 ± 2.0), and 23 were seriously obese, having BMIs exceeding 40 (mean BMI = 56.3 ± 2.6). Surface areas for the various body segments and their contribution to the total %BSA are presented in Table 2.

#### Regression analysis for BSA.

Our BSA estimates were added to 413 estimates previously published (4). Nonlinear regression of the data for the scaling model is presented in Fig. 1. The formula and its coefficients are
Equation 1Data were also fitted to the standard Du Bois-type height-weight model
Equation 2
*F*-test analysis revealed that both models fit the data equally well.

#### Height-weight covariance.

The first derivative of the filtered height vs. weight curve approached zero when the weight exceeded 80 kg. Regression analysis for the relationship between height and weight revealed correlation coefficients of 0.98 for weights ≤80 kg (*n* = 420) and 0.04 for weights >80 kg (*n* = 40). Because height and weight were highly correlated for weights <80 kg, we were able to derive an equation relating these two measurements
Equation 3Figure 2 demonstrates the height-weight relationship and the regression curve for the above equation.

This relationship can be substituted into the Du Bois-type equation (*Eq. 2
*)
yielding
Equation 4

#### Error analysis of the models.

Figure 3 compares estimated BSA to results derived from the BSA-predicting equations examined in this study. With increasing body size, the Du Bois and Du Bois (6), Haycock et al. (8), and Gehan and George (7) equations underestimate BSA. The scaling equation prediction overlies the identity line. Error analysis (Table3) demonstrated that the scaling model performed better than the standard Du Bois model for weights >80 kg. All three modifications of the Du Bois model significantly underestimated BSA compared with the scaling model (*P*< 0.001, ANOVA with contrasts). Between 10 and 80 kg, both models provided reasonable BSA estimates. For patients with weights <10 kg, the Haycock equation fit the BSA data best; the original Du Bois equation significantly underestimated BSA relative to the Haycock equation (*P* < 0.001, ANOVA with contrasts). The scaling and Gehan equations significantly overestimated BSA relative to the Haycock equation (*P* < 0.001, ANOVA with contrasts). Because the scaling model performed less well for weights <10 kg, a secondary curve-fitting procedure was performed excluding BSA estimates for patients weighing >10 kg to improve the BSA-estimating equation for patients weighing <10 kg
For weights <10 kg, this equation improved the %error to 2.23 ± 0.89%.

## DISCUSSION

We derived an equation predicting BSA over a wide range of weights and confirmed that BSA is related to the two-thirds power of weight as proposed by Meeh more than a century ago (10). Because BSA is a two-dimensional measurement, Meeh assumed that BSA would be proportional to the two-thirds power of the body weight. Early investigations yielded inconsistent estimates for *a* in Meeh's equation: SA = *a *× Wt^{
}. Obesity-related variations in total body-specific gravity and decreased surface area-to-volume ratio were hypothesized to cause variable*a* estimates (4). Very few obese subjects were included in these early investigations, resulting in abandonment of Meeh's equation for hypothetical reasons without the benefit of actual experimental evidence. Using BSA measurements from nine patients, Du Bois and Du Bois (6) empirically derived the equation that remains the most frequently used in clinical practice for predicting BSA: BSA (m^{2}) = 0.007184 × Wt (kg)^{0.425} × Ht(cm)^{0.725}. The Du Bois equation was not derived from any known physiological relationships but rather from an assumed necessary mathematical form. Du Bois hypothesized that BSA would be proportional to body weight and that changes in specific gravity and surface area-volume ratios occurring with increased size would be accounted for by the height term. Analyzing a large compilation of published BSA estimates (7), Gehan and George refined the exponents in the equation proposed by Du Bois by performing multiple regression on logarithmically transformed height and weight measurements. No attempt was made to assess other models relating height and weight to BSA. Gehan and George did find that the equation failed for small children and obese subjects. Underestimates in predicting BSA for children were subsequently improved by parameter revision of the Du Bois equation (8).

Previous derivations of BSA-predicting equations relied on linear regression of logarithmically transformed weight and height measurements (6-8). Linear regression of logarithmically transformed data is less optimal than direct curve fitting by nonlinear regression. When logarithmically transformed, data with small values are compressed, minimizing their influence on the final regression line's slope. Data with large values have their errors magnified by logarithmic transformation, such that a relatively small number of points have undue influence on the regression line's slope (9). Fitting BSA estimates directly by nonlinear regression without transforming their values eliminates these potential errors, and our study is the first to utilize these techniques to derive a BSA-predicting equation. Fitting a curve's shape with nonlinear regression is facilitated by having independent variables ranging over several orders of magnitude. With the five-order-of-magnitude difference in weights used in our study, the nonlinear regression analysis converged on an exponent for the scaling equation that was very close to the two-thirds anticipated from the geometric relationship between BSA and volume. Our data are also consistent with animal BSA-weight scaling studies. When the BSA-weight relationship was examined across species ranging in size from rats to cows, the equation was found to be BSA = 0.103 × Wt^{2/3} (5). The coefficient and exponent for this equation closely resemble those we derived for humans, which ranged over five orders of magnitude in size.

Because mathematical equations will yield results for almost any numbers entered into them, users of BSA-predicting equations must know the conditions for which the equation is valid. The Du Bois equation was empirically developed from only nine patients with a limited range of body sizes. Previous studies of children (8) and the current investigation of obesity revealed that the Du Bois equation seriously underestimated BSA. However, it did provide reasonable estimates for patients ranging in size similar to the original nine patients Du Bois studied. This illustrates how an equation can result in erroneous results because it was used beyond the boundaries for which it was tested.

Our scaling equation accurately predicted BSA for patients weighing between 10 and 250 kg. However, different parameters were necessary for children weighing <10 kg. All previous studies relating BSA to weight found that a unique set of coefficients was necessary to adequately describe the BSA-weight relationship for small children (3, 4,8). Boyd plotted the logarithms of BSA vs. age and found that the curve was bimodal, with an inflection point at 5 yr of age. For children >5, and all nonobese adults, the Du Bois equation adequately predicted BSA. The curve appeared parabolic for children younger than 5, leading Boyd to derive the equation: BSA = kW^{α+γ log W} for these small children. We also found that the equation that reliably estimates BSA from weight for patients weighing >10 kg performs less well for those <10 kg. In contrast to Boyd, we did not find the need for a different mathematical expression; rather, a different set of coefficients for the scaling equation resulted in excellent BSA predictions for these smaller patients.

For many years, the Du Bois equation and its modifications have been relied on to estimate BSA. For patients weighing <80 kg, these equations provided good estimates of BSA. However, our analysis demonstrated that the Du Bois-type equations underestimate BSA in obese people. The equation we derived was based on a body scaling principle: that most physiological measurements scale proportionally to body weight. The Du Bois equation was derived by empirically fitting BSA measurements to a formula without any known physiological relationship between BSA and the height and weight parameters in the equation. Underestimation of BSA has important clinical implications. For example, if a 300 lb (136 kg), 5'10” (178 cm) patient were to have a cardiac output of 6 l/min, the calculated cardiac index (CI = CO/BSA) would be 2.4 l · min^{−1} · m^{−2} if the Du Bois BSA prediction were used, and 2.1 l · min^{−1} · m^{−2}for our scaling equation. Vasopressor therapy is required for a cardiac index of 2.1 l · min^{−1} · m^{−2} but not for 2.4 l · min^{−1} · m^{−2}. In this case, underestimation of BSA by the Du Bois equation could potentially result in inadequate treatment for shock.

Covariance analysis of height and weight demonstrated that, for patients weighing <80 kg, the Du Bois equation is mathematically equivalent to the scaling equation we derived. Figure 2 and *Eq.3
* demonstrate that, up to 80 kg, patients' weight can be reliably predicted from their height. Substituting *Eq. 3
* into the Du Bois-type equation (*Eq. 2
*) results in *Eq. 4,*which differs from the scaling equation only slightly in its coefficients. Thus, because of the covariance of height and weight for patients weighing <80 kg, the scaling and Du Bois equations are mathematically equivalent. Bailey and Briars (1) derived the height-weight covariance relationship by linear regression of log-transformed height and weight measurements. They combined this with the previously published results of Du Bois and Du Bois (6), Gehan and George (7), and Haycock et al. (8) to derive an equation relating BSA to body weight. In this analysis, weight was raised to the 0.69 to 0.71 power, slightly higher than the expected 0.66 (1).

In conclusion, we have derived a new equation relating BSA to body weight. The equations currently used for BSA determination are unnecessarily complex and inaccurate for obese humans. The scaling equation is simpler to use and more accurately predicts BSA than those currently in use. Obesity is very common, and the equations currently used introduce significant errors in the BSA prediction for obese patients. This problem is overcome by the equation we propose.

## Footnotes

Address for reprint requests and other correspondence: E. H. Livingston, Dept. of Surgery, VA Greater Los Angeles Health Care System, 11301 Wilshire Blvd., Los Angeles, California 90073 (E-mail:elivingston{at}mednet.ucla.edu).

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- Copyright © 2001 the American Physiological Society