Veterans Affairs Medical Center, Greater Los Angeles Health Care
System, and the University of California Los Angeles Center for Human
Nutrition, Los Angeles, California 90073
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 (m2) = 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
(m2) = 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.
 |
INTRODUCTION |
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 weight2/3 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 R2 [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
|
(1)
|
Data were also fitted to the standard Du Bois-type height-weight
model
|
(2)
|
F-test analysis revealed that both models fit the data
equally well.

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Fig. 1.
Nonlinear regression of estimated body surface area (BSA)
and body weight. Data points, estimated values for BSA plotted against
the patient's body weight. Solid line, fitted regression line for the
scaling equation: BSA (m2) = 0.1173 × Wt
(kg)0.6466. R2 for the regression's
fit to the data is 0.9914.
|
|
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
|
(3)
|
Figure 2 demonstrates the
height-weight relationship and the regression curve for the above
equation.

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Fig. 2.
Plot of height (cm) vs. weight (kg). Data points,
measurement for individual patients. Solid line, plot of the regression
equation: Ht (cm) = 33.34 × Wt (kg)0.3922
(R2 = 0.9807). For weights <80 kg, height
and weight both increase proportionately, such that the scaling
equation Ht (cm) = 33.34 × Wt (kg)0.3922 closely
fits the data. Beyond 80 kg, weight increases without any significant
increase in height.
|
|
This relationship can be substituted into the Du Bois-type
equation (Eq. 2)
yielding
|
(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 (Table
3) 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%.

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Fig. 3.
Plot of BSA predicted from the various equations vs. BSA
estimated from body segment measurements. , Predictions
derived from the scaling equation; , from Du Bois's
equation; , from Haycock's equation; ,
from Gehan's equation. Solid line, slope of 1, where the predicted BSA
equals the BSA estimated from measurements. Above BSA of 2.0, all of
the equations derived from Du Bois's original model underestimate BSA.
This underestimation worsens with increasing BSA, is most significant
for Du Bois's coefficients, and is only slightly improved by
Haycock's or Gehan's modifications.
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|
 |
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 (m2) = 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 × Wt2/3 (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
+
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.
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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