A simple anthropometric estimation formula for healthy female labourers’ leg volume

Abstract Background Leg volume (LV) is an important reference in nutrition, physiology in exercise, or clinical diagnosis. Therefore, how to evaluate LV easily and quickly with accuracy is important in these areas. Aim To develop a simple anthropometric estimation formula with ease of use and good accuracy for leg volume (LV) of female labourers. Subjects One hundred and thirty female labourers (110 subjects for formula regression procedure and 20 subjects for the comparison phase) were recruited as subjects with no reported leg surgery history, trauma, or deformity. Methods A set of 3 D scanners was used to measure the range data of each subject’s leg. Results The resultant LV estimation formula is LV = 0.215 × LL × CTH1.620 with R2 = 0.967, in which LL stands for leg length and CTH for circumference of thigh. Mean error of this LV estimation is 0.10% and much smaller than that of the previous study (25.11% with significant difference). Conclusion An anthropometric estimation formula for female labourers’ leg volume was developed in this study. Estimation mean error of this formula is much smaller than the one in the previous study. This formula is easy to use and shows good accuracy in estimating female labourers’ leg volume.

Direct measuring methods and anthropometric estimation methods are usually used in LV evaluation.Direct measuring methods usually include water volumetry (Brijker et al. 2000;Stanton et al. 2000;Rabe et al. 2010;Nishimura et al. 2020) and optoelectronic methods (Stanton et al. 2000;Tan et al. 2013) in LV evaluation studies.Water volumetry (or water displacement measurement) has been taken as the gold standard in measuring the volume of objects for a long time; an object is put into a tank full of water, and the volume of water flowing out is calculated as the estimated volume of the object.Researchers have used this method to evaluate LV of patients (Stanton et al. 2000;Friends et al. 2008;Brodovicz et al. 2009;Mosti and Partsch 2013;Mosti and Caggiati 2021).This method has low cost, but its procedure is messy and time-consuming and usually cannot be applied on subjects with ulcers or open wounds.Recently, researchers have adopted the optoelectronic method as new technology in LV measurement and evaluation (Mayrovitz et al. 2000;Cau et al. 2016;Sharkey et al. 2018;Rabe et al. 2018;Tischer et al. 2020).This method uses the principle of stereoscopic vision to measure the 3D range data of the target object by projecting laser or structured light onto the target object, and the 3D form as well as the related measurements, such as length, area or volume, can then be calculated with computer software.Stanton et al. (2000) and Tan et al. (2013) reported that the optoelectronic method, also called three-dimensional (3D) scanning method, is of advantage to measure and evaluate lower limb volume, and could be taken as the new gold standard in measuring LV.Sharkey et al. (2018) recommended that, for increased interobserver reliability, this method provides a more reliable standard of limb volume measurement.Tischer et al. (2020) reported that this method could be considered as a valuable measurement tool for use in various research and clinical applications.With the advent of 3D scanning technology, a 3D scanning system with good accuracy is more easily available nowadays with decreasing cost and is usually adopted in measuring 3D forms of the human body or segments, especially in studies with a large number of subjects (Yu and Tu 2009;Yu et al. 2010;Liu and Tu 2017;Tu et al. 2021), however, its procedure is still somewhat messy and inconvenient for patients or subjects.Therefore anthropometric estimation methods are usually considered the best option for use in clinical diagnosis with emergency and convenience.
Anthropometric estimation methods use one-dimensional (1D) measurements of the leg, such as circumferential measurements, to calculate LV with estimation equations.These methods, depending on their estimation equations, usually include the truncated conical frustum model, disc model, geometric algorithm model, and linear regression model.The truncated conical frustum model and disc model have both been used in clinical diagnosis (Kaulesar et al. 1993;Mayrovitz 2012;Williams and Whitaker 2015).In the truncated conical frustum model, the leg was taken as the summation of many truncated conical frustums with heights, and the summation of calculated volumes of these truncated conical frustums was LV.The volume equation of the truncated conical frustum was used in this model.Similarly, in the disc model, a disc or cylindrical form was used to replace the truncated conical frustum, and the summation of calculated volumes of these discs was LV.The volume equation of the disc was used in this model.These two models were usually referred to as tape measurement or circumferential measurement methods because the upper and lower circumferences of one truncated conical frustum or the circumference of one disc were taped or measured from some segment of the leg.In clinical diagnosis, these methods have good accuracy but they are time-consuming.The geometric algorithm model took the geometric form of the whole leg as leg and foot, and the summation of calculated volumes of them was LV (Jones and Pearson 1969;Mayrovitz 2012).Jones and Pearson (1969) used the truncated conical frustum model to calculate the volume of the leg and a wedge form as the foot in calculating foot volume; summation of their volumes was LV.Mayrovitz (2012) also used the truncated conical frustum model to calculate the volume of leg and a complex algorithm in calculating foot volume; summation of their volumes is LV.The geometric algorithm model is also time-consuming in clinical diagnosis.To solve the timeconsuming issue in estimating LV, Katch et al. (1973) proposed a linear regression model as the LV estimation formula in their study.This estimation formula required only body weight and calf circumference of the subject as variables with R 2 equal to 0.865 (calculated from R ¼ 0.93).They also reported in their study that if leg length and thigh circumference were added as variables in the regression, the resultant formula could have a higher R 2 equal to 0.903 (calculated from R ¼ 0.95) in estimating LV.However, thinking about the formula to "be the most practical equation for predicting leg volume," they proposed the regression model with lesser variables but lower R 2 as LV estimation formula in their study.The explaining power of this estimation equation (0.865), however, might not be good enough for application circumstances with the accuracy needed and will counterbalance its practical strength.With the principle of dimensional analysis/homogeneity in Åstrand and Rodahl (1986, table 9-1, page 406), a volume should consist of L 3 , where L represents a linear dimension of body size, and this could be used to "facilitate the correct formulation of biological problems."Therefore, the LV estimation formula with good accuracy and ease of use is hypothesised to have a nonlinear form consisting of 1-D foot or leg measurement and complying with dimensional homogeneity.To eliminate the gender factor in analysis between this study and Katch et al. (1973), this study recruited female labourers as subjects.
Considering there is still a lack of LV estimation formulae with good accuracy and ease of use, the purpose of this study, therefore, is to develop a simple estimation formula with ease of use and good accuracy for LV of female labourers.

Subjects
One hundred and ten female labourers were recruited as subjects in this study.All subjects had no reported history of leg surgery, trauma, or deformity.Each subject was informed and signed the consent form before measurement.This study complied with the tenets of the Declaration of Helsinki and was approved by the Institutional Review Board at Jen-Ai Hospital, Taichung City, Taiwan (Code 105-17).Height and weight of subjects recruited in this study were compared for representativeness with those of female subjects in the 3D anthropometric Taiwanese labourer BodyBank, which is a nation-wide 3D anthropometric database established by Institute of Labour, Occupational Safety and Health, Ministry of Labour, Taiwan (Liu and Tu 2017).

Apparatus
A 3D body scanner (LT Body Scanner) and 3D foot scanner (LT-FS-L300) developed by Logistic Technology# company were used in this study to scan the subject's whole right leg.The 3D body scanner (Figure 1), consisting of eight cameras and a control and analysis software (Beauty3D#), was used to scan the 3D range data of the right leg.Its accuracy is reported as 1.0 mm (Logistic Technology Corporation 2020).Each scan could be finished in about 5 s.The 3D foot scanner (Figure 2) was used to scan the 3D range data of the foot.It is equipped with tempered glass as a standing plate for the subject.Four cameras surrounding the standing plate were fixed on a sliding fixture and could scan the subject's foot on the plate.Its accuracy is reported as 0.25 mm (Logistic Technology Corporation 2020).Each foot scanning time is about 6 s.Measurement error of the 3D scanners was ICC(3,1) ¼ 0.99 reported in the previous study (Tu 2014).

Measurement procedures
Each subject was asked to change her clothes and wear only underwear in preparation for this study.After the subject's clothes were changed, the subject's stature and body weight were measured and body mass index (BMI) was then calculated.Each subject was scanned by the 3D body scanner in a standing posture with both legs separated by about 30 cm.After 3D body scanning, subjects were asked to stand with their right leg on the 3D foot scanner, and their right foot was then scanned.

Measurement items
The 3D range data of each subject's whole right leg used for further analysis was automatically merged with the data of the right leg from the 3D body scanner and the data of the right foot from the 3D foot scanner by software (Beauty3D#).LV and eleven 1D leg measurements were extracted based on the definition as described in Katch et al. (1973) and Friends et al. (2008).LV was defined as the whole volume below the level of the crotch, as shown in Figure 3. Eleven 1D leg measurements, as shown in Figure 4, included leg length (LL), circumference of hip side (CHP), circumference of thigh (CTH), knee circumference upper (CKU), knee circumference middle (CKM), knee circumference lower (CKL), circumference of shank (CSH), ankle girth (AG), instep girth (IG), ball girth (BG), and foot length (FL).

Data analysis
A total of fourteen measurements, including the subject's own stature height, body weight, BMI, and eleven 1D leg  measurements, were used in the regression analyses of the LV estimation formula using the NLR module in IBM SPSS Statistics 22.The first step of the regression analyses is to regress each of these fourteen measurements into one single one on LV.If any single measurement was able to explain more than 95% of the variance of LV (R 2 > 0.95), the significant measurement with highest R 2 would be chosen as the estimator.The regression model would be in the nonlinear form of LV ¼ a Â Mrmt p , in which Mrmt was one of the fourteen measurements, and a and p were coefficients.If not, these fourteen measurements would be used as candidate estimators for subsequent analyses: (1) two measurements as a pair, and (3) three measurements as a group.The regression model for two measurements as a pair would be in the form of LV ¼ a Â Mrmt p i Â Mrmt q j , where i and j designated different measurements, and a, p, and q were coefficients.The regression model for three measurements as a group would be in the form of LV ¼ a Â Mrmt p i Â Mrmt q j Â Mrmt r k , where i, j, and k designated different measurements, and a, p, q, and r were coefficients.If there was more than one regression model which could explain more than 95% variance of LV, the final LV estimation formula would be chosen with the following criteria for ease of use: (1) the one with the least number of variables and coefficients, and (2) the one with better physical meaning.If there was no regression model with R 2 more than 0.95, the one with the highest R 2 among these regression models would be chosen as the final LV estimation formula in this study.After the final LV estimation formula was chosen, an additional twenty female labourers were recruited and their data extracted and used to calculate the LV estimation errors by this final estimation formula and by the one in Katch et al. (1973), with t-test for comparison and validation.

Demographic data of subjects
The demographic data of 110 female labourers are shown in Table 1.Their mean age is 39.02 years with standard deviation (S.D.) of 12.76 years.Their mean stature height is 159.40 cm with S.D. of 5.79 cm.Their mean body weight is 60.74 kg with S.D. of 16.24 kg.Their mean BMI is 23.87 kg/m 2 with S.D. of 6.13 kg/m 2 .The analysis of variance shown in Table 2 indicates that the sample population is not significantly different from the 3D anthropometric Taiwanese labourer BodyBank.It can be said that the sample population is a good representation of the Taiwanese female labourer population.

Data of LV and 1D leg measurement items
The results of LV and eleven 1D leg measurements extracted from the 3D range data of 110 female labourers are shown in Table 3. Mean LV is 7721.42cm 3 with S.D. as 1827.21cm 3 , and mean LL is 67.95 cm with S.D. as 3.93 cm, for example.

Regression models with one single measurement
Fourteen regression models with one single measurement each (C 14 1 ¼ 14) on LV are shown in Table 4. Regression models in Table 4 are ranked in descending order of R 2 , and the measurement error of coefficients a and p are shown in terms of standard error.Regression model 1 has the highest R 2 of 0.911, and the measurement used as estimator in this model is CTH.Since there is no regression model with R 2 more than 0.95 in Table 4, the regression models with two measurements as a pair and three measurements as a group are analysed further.

Regression models with two measurements as a pair
Ninety-one regression models with two measurements as a pair (C 14 2 ¼ 91) on LV were analysed, and 25 models with R 2 no less than 0.911 (the lowest R 2 in Table 5) are shown in Table 5. Regression models in Table 5 are ranked in descending order of R 2 , and measurement errors of coefficients a, p, and q are shown in terms of standard error.There are two regression models with R 2 greater than 0.95 in   Three hundred and sixty-four regression models with three measurements as a group (C 14 3 ¼364) on LV were analysed, and 16 models with R 2 no less than 0.967 (the lowest R 2 in Table 5) are shown in Table 6.Regression models in Table 6 are ranked in descending order of R 2 , and measurement errors of coefficients a, p, q, and r are shown in terms of standard error.In these models, regression model 1 has the highest R 2 of 0.984 with LL, CTH, and CSH as estimators.

Selection of LV estimation formula in this study
Eighteen regression models with R 2 more than 0.95 were candidates for the final LV estimation formula, including model 1 and model 2 in Table 5, and all models in Table 6.Following criterion (1) described in Section 2.5, model 1 and model 2 in Table 5 were chosen for their least numbers, and model 1 was then chosen as the final LV estimation formula in this study for its higher R 2 as Equation 1. Equation 1 would be used further in comparison with the formula in Katch et al.(1973).

Comparison with Katch et al.'s (1973) formula
The data from an additional twenty female labourers extracted and used in the comparison are shown in Table 7.
The results of the LV estimated errors by Equation 1and by the formula in Katch et al. (1973) with t-test are shown in Table 8.The estimated mean error by estimation formula in this study is 0.10%, which is smaller than that by the formula in Katch et al. (1973) which was 25.11% with a significant difference (p-value < 0.0001).

Discussion
Ease of use is important in developing the LV estimation formula for applications and it should be comprised of memorability and accessibility.An estimation formula that is easily memorised and accessed could be called simple for application.Memorability of estimation formula could be quantified as the number of variables and coefficients.The smaller the number is, the easier the formula would be to memorise.There were eighteen candidate regression models with R 2 more than 0.95, including two models with two measurements as regression variables (model 1 and model 2 in Table 5) and 16 models with three measurements as regression variables (all models in Table 6).The number of variables and coefficients in models with two measurements as regression variables was 5 (two variables and three coefficients), and the number in models with three measurements as regression variables was 7 (three variables and four coefficients).Based on selection criterion (1), model 1 and model 2 were selected, and on criterion (2), model 1 in Table 5 was of higher R 2 and then selected as the final estimation formula with LL and CTH as estimators (see also Equation 1).In fact, if the power coefficient of LL as 1.000 in Equation 1 could be omitted as in mathematical denotation, the number of this LV estimation formula could be counted as 4 equivalently, and it was then the one with the least number of variables and coefficients among all the 18 regression models.In addition to memorability, the accessibility of an estimation formula is related to the measurability of the variables used in the estimation formula.The estimators of this final formula, LL and CTH, are both easily measured in application.
With good memorability and accessibility, Equation 1 is a simple LV estimation formula with good accuracy.Dimensional analysis/homogeneity is very important in developing and facilitating LV estimation formula.The results of this study validated the hypothesis that the LV estimation formula with good accuracy and ease of use is of nonlinear form consisting of 1D foot or leg measurements and complying with dimensional homogeneity.For the resultant LV estimation formula (see also Equation 1), there were two 1D leg measurements on the right side of the formula, and summation of their power coefficients was 1 þ 1.62 ¼ 2.62, that is approximating to 3, which was the dimension of leg volume on the left side of the formula.For the regression model with highest R 2 in this study, that is Equation 4, its summation of the power coefficients is 2.78.Actually, in regression models with R 2 > 0.95 in Tables 5 and 6, the summations of the power coefficients of the variables on the right side were in the range of 2.612-2.93,which are all approximating to 3.
Compromising between memorability and explaining power (R 2 ) of the LV regression model might result in a different final estimation model.By applying selection criterion (1) to all the 18 candidate regression models with R 2 more than 0.95, two models in Table 5 were chosen for the same least number of variables and coefficients, and then model 1 in Table 5 was chosen (better memorability).This criterion, however, denied all the regression models in Table 6, even though they had R 2 as 0.984 which was higher than the two former models.Selection criterion (1) in this study followed the decision principle of Katch et al. (1973) as described in the introduction section and concerned memorability better than explaining the power of the estimation formula.If explaining power is considered more important than memorability, model 1 in Table 6 would be chosen as it has the highest R 2 at 0.984 (better-explaining power) in all regression models with the highest number of variance and coefficients as well, making it less memorable.
To compromise memorability and the explaining power of a formula, R 2 effectiveness is suggested to be adopted as a critical index in the regression model selection criteria.R 2 effectiveness is the difference of the R 2 s with respect to the difference of the numbers of variables and coefficients between two regression models and could be calculated as Equation 2, where NVC is the number of variables and coefficients of the regression model, and i and j designate different LV regression models.To calculate R 2 effectiveness of one regression model, say model 2 in Table 6, by Equation 2, model 2 in Table 6 is taken as regression model i, and model 1 in Table 4 is taken as regression model j in Equation 2 as a constant basis, as model 1 in Table 4 is the regression model with highest R 2 although it is less than 0.95.After calculation, R 2 effectiveness of all the 18 candidate estimation models ranges from 0.014 to 0.008 as shown in Figure 5, and their detailed values are shown in Table 9.In Table 9, model M1T5 stands for model 1 in Table 5 with its R 2 effectiveness value as 0.014, and M1T6 for model 1 in Table 6 with its R 2 effectiveness value as 0.010429, et al.It could be observed easily from Figure 5 that the model with maximum R 2 effectiveness is M1T5 (model 1 in Table 5).The result is equivalent to the results presented in this study.Therefore, R 2 effectiveness is suggested to be used in selection criteria as an index compromising memorability and explaining the power of formula in future studies.
Measurements with poor correlation coefficients to LV might still be good estimators in the LV estimation formula.Two measurements (LL, CTH) have been used as estimators in the LV estimation formula in this study.Pearson's correlation coefficients of LL and CTH can be seen in Table 10.In Table 10, CTH had a high correlation coefficient of 0.943, but LL had the lowest correlation coefficient of all at 0.051.However, LL was regressed and shown as one estimator in the final LV estimation formula in this study.LL was even shown as an estimator in five of the 25 regression models in Table 5 and all 16 regression models in Table 6.If the correlation coefficient of some certain measurement was used as the premier criterion to select the measurement as a step-in variable in the stepwise regression process as Katch et al. (1973) did, LL would be dropped in the very first step and not be included in the further regression analyses.In the case of this study, the final LV estimation formula (Equation 1) would not be in the feasible domain.Therefore, it is suggested that if finding the estimation formula of some body segment is the study purpose, in constraint of the reasonable number of variables, models with all combinations of the variables (measurements) concerned should be tested or regressed to find the optimal estimation formula instead of stepwise regression analyses.There are more regression models derived in this study that could be used as LV estimation formula under circumstances requiring different memorability or explaining power (accuracy).In circumstances requiring good memorability but enough accuracy, say 90% of explaining power on LV, model 1 in Table 3, as Equation 3, with only CTH as estimator and R 2 equal to 0.911 could be considered and applied.In circumstances requiring very high R 2 but less memorability, model 1 in Table 6, as Equation 4, with three variables (LL, CTH, CSH) and four coefficients with R 2 equal to 0.984 could be considered and applied.In circumstances requiring compromised memorability and accuracy, Equation 1 would be a good choice.In fact, in circumstances with different variable (measurement) accessibility, models with various combinations of variables in Table 5 and Table 6 could also be considered as needed.
LV ¼ 19:312 Â CST 1:548 (3) The effect of ethnicity on the LV estimation formula could not be validated in this study.With data of 20 additional females, results of the t-test (see also Table 8) showed that mean error of LV estimation by the formula in this study is 0.10%, which is much smaller than that by the formula used in Katch et al. (1973) which was 25.11%, and with a significant difference.However, subjects recruited in this study to drive the final LV estimation formula were Asian females in Taiwan, and the subjects in Katch et al. (1973) were Caucasian females, whose raw data could not be accessed in their original report.Cross validation by subjects with different ethnicity between this study and Katch et al.'s study could not be conducted, and the effect of ethnicity, therefore, could not be distinguished.To discriminate the effect of the ethnicity of LV estimation formula, further study is needed.Katch et al. (1973), variable representing leg volume (LV) in their estimation formula is V, and circumference of shank is C.These two variables, however, are denoted as LV and CSH respectively in this study.In this table, the variables of the estimation formula in Katch et al. (1973) are transformed correspondingly for comparison.A simple estimation formula for female labourers' leg volume with good accuracy was developed in this study using a 3D scanner.The results showed that the final LV estimation formula is LV ¼ 0.215 Â LL Â CTH 1.620 with R 2 equal to 0.967.Twenty extra female subjects were recruited to compare this final estimation formula with that of Katch et al. (1973).The results showed that the mean error of LV estimation by formula in this study is 0.10%, which is much smaller than that in Katch et al. (1973), which was 25.11% with a significant difference.This formula has ease of use and good accuracy to estimate female labourers' leg volume.Other regression models derived in this study could also be used as LV estimation formula under circumstances requiring different memorability or explaining power.

Figure 1 .
Figure 1.3D body scanner used in this study consisted of eight cameras and control software.

Figure 2 .
Figure2.The 3D foot scanner used in this study.It is equipped with tempered glass as a standing plate for subject.Four cameras surrounding the standing plate were fixed on a sliding fixture and could scan the subject's foot as she stood on the plate.

Figure 3 .
Figure 3. Leg volume (LV) was defined as the whole volume below the level of the crotch as shown in pink colour.

Figure 5 .
Figure 5. R 2 effectiveness of all eighteen candidate LV estimation models.

Table 5 ,
including model 1 and model 2. Regression model 1 has the highest R 2 of 0.967 with LL and CTH as estimators, and regression model 2 has the highest R 2 of 0.957 with LL and CKU as estimators.

Table 1 .
Demographic data of 110 female subjects.

Table 2 .
Comparison of sample distribution of height and weight between this study and 3D anthropometric Taiwanese labourer BodyBank.

Table 4 .
Regression models with one single measurement each on LV.

Table 5 .
Regression models with two measurements as a pair on LV (R 2 м 0.911).

Table 6 .
Regression models with three measurements as a group on LV (R 2 м 0.967).

Table 7 .
Related data of extra 20 female labourers for formulae comparison.

Table 9 .
R 2 effectiveness values of all eighteen candidate LV estimation models.

Table 10 .
Pearson's correlation coefficients of leg measurements with LV.