Variance inflation in sequential calculations of body surface area, plasma volume, and prostate-specific antigen mass


Mahendra Bhandari, Vattikuti Urology Institute, Henry Ford Hospital, Detroit, MI, USA.



To assess the magnitude of variability among 11 formulae for human body surface area (BSA) and then among eight for plasma volume (PV), as used to represent physiological indices for body metabolism, drug dosages and body fluid management, and to evaluate the potential cumulative effect of variance inflation with prostate-specific antigen (PSA) mass as an endpoint.


In 3020 men undergoing robotic radical prostatectomy (RRP) at the Vattikuti Urology Institute between 2001 and 2008, the variation in BSA and PV formulae was calculated, as well as PSA mass, using analysis of variance (anova), Bland-Altman plots, linear regression, and correlation analyses.


For estimating BSA, anova indicated significant variance among the 11 formulae used (P < 0.001) with a between-groups variance of 5.45. Bland-Altman plots reported bias when the Dubois formula was compared to other BSA formulae. Furthermore the anova for PV, with BSA as a predictor, indicated significant variance among the eight formulae used (P < 0.001), with a mean between-group variance of 444.4 and a mean inflation factor of 81.5. Scatter plots between one PV formula (Boer) and others had a good linear fit. For PSA mass, anova indicated significant variance (P < 0.001) using PV as a predictor, with a mean between-group variance of 16 799.6 and a mean variance inflation factor of 37.8.


There is significant variation in the BSA calculated by commonly used formulae. This variation is carried over and further magnified in the sequential calculation of PV and PSA mass. Hence arbitrary selection of BSA and PV formulae is likely to affect inferences.


robotic radical prostatectomy


body surface area


plasma volume


body mass index




root mean-squared errors.


Anthropometric measurements of height and weight have been used to calculate body surface area (BSA) and plasma volume (PV), and further applied for calculating PSA mass [1], dose titration of chemotherapy drugs [2], and estimating fluid replacement needs in critically ill patients [3]. The direct measurement of BSA is possible by coating methods (alginate, paper, inelastic tape, aluminium foil) [4] and three-dimensional (3D) whole-body scanning [5]. However, the application of these methods in a clinical setting is cumbersome and impractical. In practice, BSA is routinely calculated from height and weight using various predictive bi-exponential formulae [5–27] constructed through population studies. The first such formula, having achieved an iconic status, is that of Du Bois and Du Bois [7], an approximation developed with nine subjects, including a child and a cretin.

PV can be directly measured with the indicator-dilution technique (Evans blue and indocyanine green) and single or dual isotope tracers such as 51Cr for tagging red blood cells or 131I/125I for tagging human serum albumin [28]. However, for retrospective data, PV is calculated with BSA as a predictor [29–40]. In previous reports there are both contradictions on the reliability of the Du Bois BSA approximation [6,21,23,41] and many formulae for estimating PV, with no accepted guideline; this presents a conundrum of choice. Based on this subjectivity, significant variation in the calculation of end products such as PSA mass [1] could result in far-reaching clinical implications.

We report the application of BSA, PV and PSA mass in a group of patients being treated for prostate cancer at our institute to: (i) calculate the extent and significance of variation in commonly used formulae for BSA and PV; and (ii) to determine whether this variation affects an analytical marker of prostate cancer, the PSA mass.


We analysed 3020 patients treated with robotic radical prostatectomy (RRP) between 2001 and 2008 at the authors’ institution. The data were obtained from the ROBOSURG database (approved by the institutional review board). Of 3033 patients, 13 had missing data on PSA and were excluded; the characteristics of the patients are shown in Table 1.

Table 1.  Demographic and clinical characteristics of the 3020 men treated with RRP
Mean (sd):
 Age at surgery, years  62.5 (7.5)
 Height, m   1.79 (0.07)
 Weight, kg  88.4 (13.8)
Median (interquartile range)
 Year of surgery2005 (2004–2007)
 PSA, ng/mL   5.1 (4.1–6.9)

BSA was estimated using formulae specified in the previous reports (Table 2) [5,7–27], all involving multiplication of height and weight raised to exponents. The estimation of PV usually involved multiplying BSA by a factor specific to the formula (Table 3) [29–40]. PSA mass (mg) represents the total circulating amount of PSA and was determined as a product of serum PSA concentration (in ng/mL) and PV [1].

Table 2.  BSA formulae applied to the patient sample; anova for selected formulae, P < 0.05
TypeSourceFormula for BSA (unit)Mean (sd), m2
  1. NS, not selected for inclusion in study; H, height; W, weight. Bold, highest mean; italic, lowest mean. *Formula specific for males. †Adult formula (>20 years old). Formulae and table format adapted from [4] and [6], respectively. The mean (sd) of all BSA formulae except [7] was 2.08 (0.19) m2. Bonferroni post-hoc analysis indicated that [16] was similar to [10], [17] and was similar to [10] and [16], [23] was similar to [10], [7] was similar to [24] and [21] was similar to [13].

1[8](cm2) 5.4 [W (g) × H (cm)]0.52.14 (0.19)
[9](m2) [H (cm) × W (kg)/3600]0.52.09 (0.18)
[5](cm2) 159.25[W (g) × H (cm)]0.52.00 (0.17)
2[7](cm2) 71.84 × H (cm)0.725 × W (kg)0.4252.07 (0.17)
[10](cm2) 0.0003207 × H (cm)0.3 × W (g) (0.7285 − (0.0188 × log w (g))2.11 (0.19)
[11](cm2) 240 × H (cm)0.4 + W (kg)0.53NS
[12](cm2) 74.66 × H (cm)0.725 × W (kg)0.425NS
[13]*(cm2) 59.02 × H (cm)0.776 × W (kg)0.4252.04 (0.16)
[14](cm2) 113.1 × H (cm)0.6468 × W (kg)0.4092NS
[15](cm2) 88.83 × H (cm)0.663 × W (kg)0.444NS
[16](cm2) 235 × H (cm)0.42246 × W (kg)0.514562.10 (0.18)
[17](cm2) 242.65 × H (cm)0.3964 × W (kg)0.53782.11 (0.19)
[18](cm2) 13.15 × H (cm)1.2139 × W (kg)0.2620NS
[19](cm2) 100.315 × H (cm)0.383 × W (kg)0.693NS
[20](cm2) 239 × H (cm)0.417 × W (kg)0.517NS
[21](cm2) 94.9 × H (cm)0.655 × W (kg)0.4412.04 (0.17)
[22]*(cm2) 128.1 × H (cm)0.6 × W (kg)0.44NS
3[23](m2) 0.1173 × W (kg)0.64662.12 (0.21)
4[24](m2) 0.0087 (Hcm + Wkg) – 0.262.06 (0.16)
[25](m2) 0.0097 (Hcm + Wkg) −0.545NS
[26](m2) (Hcm + Wkg-60)/100NS
5[27](cm2) − 2142 + 0.2453 × H (cm)2 + 617 × W (kg)2  + 0.6825 × head circumference (cm)2NS
Table 3.  Previously published formulae for PV; anova for selected formulae, P < 0.001
SourceFormula for PV (unit)
  1. H, height; W, weight; PCV, plasma cell volume. *Male-specific formula. †Used for analysis.

[29]*(L) 1.67 × BSA (m2)
[30]*(mL) 995e0.6085 × BSA (m2)
[31]*(mL) 1630 × BSA (m2)
[32]*(mL) 1580 × BSA (m2)-520
[33]*(mL) 2294 × BSA (m2)-928
[34]*(mL) 1846 × BSA (m2)-522
[35]*(mL) − [(366.9 × H (m) + 32.19 × W (kg) + 604) × 0.47 × 0.91]
[36]*(mL) 1487 × BSA (m2) + 68
[37](mL) 41 × W (kg)
[38](mL) 44 × W (kg)
[39](mL) 45 × W (kg)
[40](mL) [(41 × W (kg)) × (100-PCV)]/(100–47)

BSA formulae published previously were grouped as: (1) ‘√(weight × height)’; (2) ‘weighta× heightb’; (3) ‘weighta’; (4) ‘weight + height’; and (5) three explanatory variables (Table 2). From these, 11 formulae were selected for testing, based on previous standardization, availability of error analysis data, and popularity. The formula of Du Bois and Du Bois [7], its recently revised version [21], and other modifications [9,10,16,17] have been used extensively. Other formulae for BSA [8,13,24] were selected as they had the least percentage error compared to the alginate method [4]. The formula from [5] qualified for inclusion on the basis of standardization with independent 3D whole-body scanning in a Chinese population. Despite our intentions to use the formula of Takai and Shimaguchi [27] from group 5, head circumference was not available in our database. The eight PV formulae were selected for analysis in a manner similar to that used to select BSA formula (Table 3).

anova was used on the 11 selected BSA formulae, with the Bonferroni post-hoc test for all anova analyses that returned significant variance at the α-level of 0.05. We assessed the agreement between the Du Bois formula [7] and other BSA formulae individually using Bland and Altman plots, as well as the Pitman test of difference in variance [42]. The Bland-Altman plot was also used to compare the Du Bois BSA formula to the mean of the other 10 BSA formulae. To examine how PV formulae correlated with each other, using the Du Bois formula as a BSA predictor, linear correlation analysis was used to compare the PV estimation of Boer [29] with the seven other formulae, and to the mean of these seven PV estimates. anova was used to assess all PV formulae and for PSA mass estimations calculated using PV as a predictor. To quantify the distribution in BSA, PV and PSA mass, we used four randomly selected patients as representative examples. We recorded their minimum and maximum values in each formula category. Variance inflation, a measure of the change in variance between BSA and PV, was assessed as ‘between-groups variance PV/between-groups variance BSA’. Variance inflation in calculating PSA mass from PV was similarly assessed.


The mean (sd) height and weight of the patients were 1.79 (0.07) m and 88.4 (13.8) kg, respectively; other salient clinical characteristics of the patients are listed in Table 1. The variance in BSA was significant between groups (P < 0.001), with rejection of equal variance according to the Bartlett’s test (P < 0.001). The post-hoc Bonferroni test indicated that some BSA formula groups were similar (Table 2). The mean squares between-groups variance for BSA formulae was 5.45.

To further examine the agreement between the widely accepted Du Bois [7] and other BSA formulae, Bland and Altman plots were used to examine how the mean between the Du Bois and other formulae changed relative to their difference (Fig. 1). The Du Bois formula produced higher bias at a high BSA (underestimation) than that from [10] (mean difference = −0.041, r = −0.460), [17] (mean difference = −0.036, r = −0.585), [16] (mean difference = −0.038, r = −0.495), [9] (mean difference = −0.022, r = −0.536), [8] (mean difference = −0.073, r = −0.645), [23] (mean difference = −0.055, r = −0.573) and other BSA formulae (mean difference =−0.014, r = −0.434; Fig. 1A–D,F,H,K, respectively). The Du Bois formula showed a higher bias at high BSA (overestimation) than that from [24] (mean difference = 0.004, r = 0.792, Fig. 1E) and [13] (mean difference = 0.026, r = 0.683, Fig. 1G). The best agreements, considering the scarcity of data points (bias) beyond the ±1.96 sd limits, were for [7] vs [5] (mean difference = 0.071, r = −0.238, Fig. 1I) and [21] (mean difference = 0.027, r = 0.215, Fig. 1J).

Figure 1.

Figure 1.

Bland and Altman plots of the difference in BSA as a function of the mean BSA based on the formula from [7] and [10] (A), [17] (B), [16] (C), [9] (D), [24] (E), [8] (F), [13] (G), [23] (H), [5] (I), [21] (J), and the mean BSA of previous formulae (K). Plots show mean (sd) bias (CIs). The negative regression trend line indicates underestimation bias and the positive regression trend line overestimation bias.

Figure 1.

Figure 1.

Bland and Altman plots of the difference in BSA as a function of the mean BSA based on the formula from [7] and [10] (A), [17] (B), [16] (C), [9] (D), [24] (E), [8] (F), [13] (G), [23] (H), [5] (I), [21] (J), and the mean BSA of previous formulae (K). Plots show mean (sd) bias (CIs). The negative regression trend line indicates underestimation bias and the positive regression trend line overestimation bias.

As the most recently used PV formula was by Boer [29], using the formula from [7] as a BSA predictor, correlational analysis was used to assess the agreement between that PV formula and others (Fig. 2). The Boer PV was most consistently correlated with those from [31–34,36], with all correlations having an r2 of 1.00 and root mean-squared errors (RMSE) of <0.001 (Fig. 2A–G, respectively). Moreover, the Boer PV was generally well correlated with other PV formulae (r2 = 0.99, RMSE 0.024). However, the pattern of agreement was curvilinear when comparing the Boer PV with that from [30] (r2 = 0.99, RMSE 0.025, Fig. 2D) and showed wide variation when comparing it with that from [39] (r2 = 0.89, RMSE 0.09, Fig. 2F).

Figure 2.

Correlation of PV from the formula of [29] based on the BSA calculation from [7]; correlation analysis between PV [29] and PV based on different PV formulae; [36] (A), [32] (B), [33] (C), [30] (D), [34] (E), [39] (F), [31] (G), and the mean PV calculated from different formulae (H).

To test the variance among the eight PV formulae, anova was used to assess the results of PV derived from each of the 11 BSA formulae (data not shown). The variance between groups was 444 (21.9) with a range of 67.3 (P < 0.001) and unequal by Bartlett’s test (P < 0.001). Furthermore, Bonferroni analysis indicated that all groups were significantly different from each other.

To associate the statistical significance of variance inflation with clinical relevance, we followed the BSA, PV and PSA mass data of four randomly selected patients (Table 4). For all patients, the range in values increased for each category as BSA, PV and PSA mass were sequentially calculated. For example, the BSA range of patient no. 3 was 2.32–2.56 m2, and on calculating the endpoint PSA mass, this dramatically increased from 139.50 to 235.10 µg.

Table 4.  The range in BSA, PV and PSA mass of four randomly selected patients
PatientBSA, m2PV, L*PSA mass, µg*
  • *

    PV and PSA mass ranges were calculated using all BSA predictors.

13.15–3.714.60–9.41 24.84–50.81
21.55–1.711.92–3.01 18.62–29.20
41.99–2.132.62–3.96  0.26–0.40

Variance inflation was calculated to assess the change in variance when calculating PV from BSA; the mean factor was 81.5 with a range of 12.3 (Table 5). The formula from [8] produced the lowest and that from [5] produced the highest PV variance and variance inflation factors, respectively. anova of PSA mass approximations from PV derived from 11 BSA formulae showed a mean (sd) between-group variance of 16 799.6 (756.3) (Table 5), with a range of 2576.8 (P < 0.001), and with unequal variances according to Bartlett’s test (P < 0.001). The Bonferroni test showed similarity between some PSA mass formula groups (Table 4). Notably, PSA mass calculated from [5] produced the greatest variance, while that calculated from [8] produced the least variance. Variance inflation was greatest using the formula from [13] as a BSA predictor (38.2) while that from [10] produced the least variance inflation (34.2). Variance produced in BSA is carried over and inflated when calculating PV and PSA mass. Total variance inflation on PSA mass was maximal using [13] as a starting point and at minimal with [5], regardless of the PV formula used. anova to assess the between-group variation in PSA mass was significant (P < 0.001), which indicated that deviations in PSA mass as a result of formula variance inflation cannot be neglected.

Table 5.  Variance inflation factors dependent on BSA calculations
BSA formulaPV varianceVIFPSA mass varianceVIFTotal variance inflation*
  1. VIF, variance inflation factor. *Variance inflation in calculating PV from BSA × variance inflation in calculating PSA mass from PV. Bold, maximum in column; italic, minimum. Bonferroni post-hoc analysis on PSA mass calculations indicated derivation from the following sets were similar: [34] and [29], [36] and [31], [31] and [29], [31] and [30], [30] and [29], and [30] and [39].

[10]476.2387.38216 275.1334.1752986
[16]427.9478.52116 342.7038.1892999
[17]427.3378.40916 318.7238.1882994
 [9]433.2579.49516 547.9838.1953036
[24]444.2781.51716 978.2638.2163115
[13]456.1083.68817 431.1438.2183198
 [5]484.0088.80718 490.8638.2043393
 [8]416.7476.46615 914.0138.1872920
[23]422.7677.57116 136.9938.1712961
 [7]442.7581.23916 917.5838.2103104
[21]456.4983.76017 442.5438.2103200


Crucial clinical inferences depend on indirect measurements of BSA and PV; while direct methods of calculating both BSA and PV are available, they are usually not applicable in retrospective and critical care situations [3,4]. Indirect estimates of BSA from height and weight have been coupled with estimates of PV using BSA as a predictor. Du Bois and Du Bois [7] reported the first indirect formula for estimating BSA. In a study of 3020 patients being treated for prostate cancer, we report for the first time that variance inflation is present in sequential calculations of BSA, PV and PSA mass.

Due to the multitude of equations available to estimate BSA, we examined 11 selected and representative equations for effects of variance and agreement. anova of the BSA formulae indicated a significant between-group variation, which was logically expected, given the nature of variation in the formula types and coefficients. Bland and Altman agreement plots showed that the BSA formula from [7] produces a significant bias compared to most other selected BSA formulae. As PV can be calculated from BSA using many formulae, it is logical to expect some form of variance transfer in this sequential calculation. anova of PV formulae showed significant between-group variance, which was greater than the variance in BSA. The PV formula of Boer [29] correlated well, in general, with other PV formulae, except those from [30] and [39], consistent with the fact that most of the PV formulae used BSA as a predictor. PSA mass was recently used as an analytical marker for prostate cancer [1]. As this marker depends on PV and PSA, both of which show significant variance, it is important to evaluate whether the cumulative variance in PV using BSA as a predictor affects the calculated PSA mass. Notably, the variance inflation apparent in calculating PV from BSA was retained and further increased when calculating PSA mass from PV.

Other previous studies have indicated shortcomings in the Du Bois BSA formula, e.g. underestimation in obese individuals [23], children [17], different nutritional status among individuals [21], ethnic differences [4], and statistical concerns [10,16]. Other investigators have questioned the estimation of BSA with just two explanatory variables, i.e. height and weight [27]. Sardhina et al.[43] used two- and five-compartment models to show that the Du Bois formula underestimated BSA in obese patients, which was corroborated by the comparative analysis of Verbraecken et al.[6]. Investigators have developed direct methods of measurement, such as 3D whole-body scanning [5] and the use of alginate [4], but such methods are cumbersome and impractical for retrospective studies.

The results of the Bland-Altman plots were consistent with the findings by Verbraecken et al.[6] that the BSA formula from [9] overestimated values compared to the Du Bois formula [7]. As the formula from [21] is a nonlinear revision of the Du Bois formula, agreement is expected on the whole. The error analysis of Lee et al.[4], using the alginate method for comparison, indicates that the BSA formula from [5] and [8] produced a 1.9% and 6.0% absolute error, respectively. While the present analysis indicates minimal variance inflation in PV using [5] as a BSA predictor, it is possible that the 3D body scanning technique of [5] underestimates BSA due to skin folds [4].

PV is required to adequately exchange plasma and clotting factors in critical-care situations, e.g. in the management of haemophilia [44]. With many PV formulae available [29–40], researchers have arbitrarily chosen to use certain PV formulae [1]. Previously, a report compiled by an expert panel on radionuclides of the International Council for Standardization in Hematology [45] mandated the correction of estimated PV by 23.9%, which is too broad for statistical analysis and clinical application. The PV formula from [30] was devised based on the measurement of PV in normal New Zealand males and females with a broad age of 15–>90 years. However, only 13% of males in their study population were aged >50 years. As the present patients comprised men being treated for prostate cancer and who were predominantly in the older group, it is possible that the PV formula from [30] might not be applicable in this case. Interestingly, Ockelford and Johns [44] reported a significant error in previous PV formulae that used explanatory variables present in predictive equations for BSA, and instead advocated using body weight alone, with a multiplicative factor to determine PV. Thus, it is logical that the formula of Kjellman [39] produced the lowest correlation with the other PV formulae.

Strikingly, PV is highest in PV formulae that used [5] as a BSA predictor and lowest in those using [8] as a BSA predictor. This is consistent with reports of high and low mean BSA values in our initial analysis, and indicates that BSA might be the major driving force in the resulting variance in PV. There are other alternatives for normalizing PV to different groups of patients. Feldshuh and Enson [46] developed a technique of determining normal blood volume based on an estimate of body composition, using Metropolitan Life height and weight tables. More recently, a blood volume analyser was reported that rapidly calculates PV in individuals within 30 min of using the indicator-dilution technique and dual simultaneous radioisotope tagging [28]. In the absence of having such direct measurements, variance among PV formulae should be carefully considered before drawing inferences.

Variance inflation in calculating PSA mass from PV was highest using [13] as a BSA predictor and lowest using [10] a different inflation pattern from that observed in calculating PV from BSA. One possible explanation is that the Boyd formula [10] used a logarithmic transformation to calculate BSA, which could have residual effects on further calculations of PV. However, the total variance inflation was similar to the observed pattern of calculating PV from BSA, which confirms that the initial BSA calculation might be a strong driving force in overall variability.

Physiological variables such as PSA mass calculated using formulae with inherent variation should be critically assessed because of the clinical impact. Distressingly, in the present patients, variance inflation in calculating PSA mass from PV and BSA as predictors resulted in a progressively increasing range of values in all three categories. Bãnez et al.[1] recently reported, in a multicentre study, that haemodilution might explain the lower PSA levels observed in obese patients, as a result of an inverse relationship between body mass index (BMI) and PSA. That group also reported no association between BMI and PSA mass. However, the authors did not take into account the underestimation of BSA in obese individuals when using the original Du Bois formula. If variation in PSA mass affects statistical tests, then the effect of haemodilution in obese patients with prostate cancer will need to be re-examined.

There are other physiological variables that use only BSA as predictors and require accurate reporting. For example, the Cardiac Index is a widely used marker of cardiac function and is reported as a ratio of cardiac output to BSA. Livingston and Lee [23] report that underestimating the BSA produced by Du Bois formula, by 0.3 CI units, could result in inadequate vasopressor treatment of shock. Furthermore, the left ventricular mass index is reported as a ratio of left ventricular mass and BSA [47]; a difference in this index due to variations in BSA formula, in relation to a standard definition of left ventricular hypertrophy (>83 g/m2), has prognostic and therapeutic implications for predicting the risk of cardiovascular morbidity and mortality [23,48].

The present analysis might be limited by restricting the population to men who had RRP. Although estimation formulae have been calibrated for children and women, citing BSA differences in each [6], we expect similar variance inflation to occur because of the nature of differences in BSA and PV formulae. Furthermore, the present was not a validation study, as no actual BSA or PV measurements were made using a criterion method such as alginate or indicator dilution. We tried to overcome this by including BSA calculated from the formula of Yu et al.[5] by 3D body scans, by incorporating the mean of BSA and PV formulae, and by carefully selecting BSA and PV formulae. As we did not analyse all BSA and PV formulae, this could represent a potential source of selection bias. Head circumference and plasma cell volume measurements were not available from our retrospective database. Regardless, we expect no change in the overall significance of variation and thus, despite these limitations, we consider that our findings are valid.

In conclusion, BSA and PV formulae provide estimates that can affect medical decision-making, especially when further applied to calculate surrogate analytical markers. Based on our observations of significant variance inflation in the sequential calculation of BSA, PV and PSA mass, we advocate a rigorous statistical treatment of extremes in formulae that produce high and low variation, before choosing one set of formulae for retrospective analysis and clinical practice.


None declared.