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Background: Body mass index (BMI) is a squared-height power function. Nevertheless, some studies show a significant exponential weight-height correlation. Objectives: To demonstrate that the weight-height relationship from 2 to 20 years of age is better expressed by an exponential function. Design: 5th, 50th and 85th percentile weight-height curves according CDC 2000 Growth Charts. A theoretical curve was created with the data on the 50th percentiles of weight and height for each age, equivalent to the 50th percentile of the weight-for-height curve. The statistical analysis was performed applying regression analysis of the curve estimation in the power and exponential models. Results: The exponential model correlation coefficient is higher than the power model. The exponential model variable (1.9 in boys, 2 in girls) was standardized to 2 to establish the body mass exponential index (BMEI): weight/exp(2*height). Weight-for-age and exponential height-for-age fiftieth percentile curves show a stable age-independent ratio near 2. These ratios are 1.5 and 2.5 for the 5th and 85th percentiles, respectively. The shape of the well-known curve BMI-for-age is due to the disparity between a exponential curve and a power curve. Conclusions: An exponential function expresses the weight-height relationship during growth better than a power function. A BMEI of 2 with limits of 1.5 and 2.5 is useful for screening nutritional status during growth, and the weight-for-height chart is an ideal substitute for the BMI-for-age chart. The BMI-for-age curve shape and the disproportional BMI in taller children are mathematical artifacts without biological meanings.

The body mass index (BMI), named by Keys in 1972 [^{2} [

The BMI has been criticized to the extreme of being considered an obsolete index that underestimates overweight [

The BMI is not a perfect measurement in children because it is height-dependent; however, it has been chosen due to its reproducibility and validity in easily measuring body fat [^{3}) [^{p}) [^{3.7} index [

In a mathematical sense, the BMI is a specific case of the general power function model used to explain allometric growth [

The purpose of this study is to demonstrate that the weight-height growth relationship between 2 and 20 years of age is best expressed with an exponential function, enabling the creation of a more accurate nutritional index than the BMI. The study is performed by analyzing the data from the CDC 2000 growth chart [

The CDC 2000 growth chart for children and teenagers from 2 to 20 years of age is available at the webpage cdc.gov/growthcharts and displays the weight-for-age, height-for-age, and BMI-for-age charts, as well as the weight-for-height chart between 77 and 121 cm, equivalent to the 2 to 5 years of age. The data were gathered from five national studies from 1963 to 1994 [

The final curves represent the 3rd, 5th, 10th, 25th, 50th, 75th, 90th, 95th and 97th percentiles, distributed according to age and gender, as well as the 85th percentile in the BMI-for-age and weight-for-height charts. The 85th percentile of the weight-for-age distribution is not shown on the percentile tables and was calculated by applying the LMS method, as noted on the CDC webpage. The values are available for each month of age between month 24 and month 240, although this study used 6-month intervals to simplify calculations.

The weight-for-height curves exclude growth stages by being interrupted at 121.5 cm. To include all growth stages, a theoretical curve was created with the data on the 50th percentiles of weight and height for each age, equivalent to the 50th percentile of the weight-for-height curve.

The statistical analysis was performed using the SPSS version 20 statistical package by applying regression analysis of the curve estimation in the power and exponential models.

The 50th percentile values of weight-for-age in relation to the 50th percentile of height-for-age formed an ascending curve, on which a regression analysis with a curve estimation of the two models, power and exponential, was performed. A significant weight-height relationship was obtained for these two models, but a stronger relationship was obtained for the exponential model (

Before plotting the charts, the equations were modified to match the power model with the BMI and the exponential model with Henneberg’s proposed equation [^{2} ratio was the BMI, and the W/exp(2H) ratio was termed the body mass exponential index (BMEI).

The subsequent step consisted of comparing the increases with age of W and the functions of H on the charts (^{2} curves the match was lower on the ratio 20:1.

The BMEI oscillates about a value of 2 (

Finally, a theoretical recreation was performed to demonstrate the cause of the shape of the BMI. On a chart, with the values between 0.7 and 2 on the X-axis representing the height limits in the growth stage, two curves H^{2} and 2・exp(2H) were traced (^{2}/2・exp(2H) formed a similar BMI-for-age curve, with H = 1 as a minimum.

Model | Sex | R^{2} | Constant k | b1 | Equation | Ratio W/H^{a} | Index^{b} |
---|---|---|---|---|---|---|---|

Potential | M | 0.985 | 15.1 | 2.5 | W = k・H^{2.5} | k = W/H^{2.5} | i = W/H^{2} |

Potential | F | 0.983 | 15.3 | 2.5 | W = k・H^{2.5} | k = W/H^{2.5} | i = W/H^{2} |

Exponential | M | 0.998 | 2.3 | 1.9 | W = k・exp(1.9H) | k = W/exp(1.9H) | i = W/exp(2H) |

Exponential | F | 0.996 | 2.1 | 2.0 | W = k・exp(2H) | k = W/exp(2H) | i = W/exp(2H) |

^{a}The equation was modified to isolate the constant and show the weight-height relationship. ^{b}The variables (b1) were standardized to 2 to consider the BMI in the power model and an index according to the Henneberg equation [

This study shows that the weight-height relationship from 2 to 20 years of age is better expressed by an exponential function than by a power function on which the BMI is based. The studied equation, W/exp(2H) is more accurate than any of the other equations, such as W/H^{2}, W/H^{3}, or W/H^{p}, in obtaining the nutritional index. The constant 2:1 proportionality between the weight and the exponential height (

This study shows that the peculiar shape of the BMI-for-age chart is a mathematical anomaly. The initial descent observed until 6 years of age and the later ascent is due to a mathematical artifact imitated when dividing an exponential curve by a power curve (

Since Quetelet proposed the W/H^{2} coefficient [^{2}/H^{5} [^{2.5}. This same value was obtained in this study (

The exponential weight-height relationship has been previously suggested [

The exponential function is peculiar, as it doubles its value at regular intervals and can be demonstrated without needing mathematical tools such as logarithmic calculation or curve estimation. By example, the curve published by Leung et al. in 1996 [

The supremacy of the BMI in nutritional assessment has cornered the weight-for-height charts. The studies comparing both methods show significant differences that endorse the use of the BMI-for-age as a better method [

The use of the weight-for-height chart makes the weight-for-age chart obsolete, even inconvenient. This can be explained through the following example. Perrin et al., in their survey on the use of the BMI [^{ }percentile) and 40 kg of weight (80th percentile), both within the normal percentiles, but overweight, as is only shown when observing that the BMI is over the 95th^{ }percentile. Overweight status is also shown directly on the weight-for-height chart (

The BMEI, suggested in this paper as an alternative for BMI, is more complicated to calculate, requiring a scientific calculator; but, this limitation may be considered attenuated, as it does not require a transfer to a chart.

A nutritional index that associates the weight and height with an exponential function, such as the BMEI, is more accurate from 2 to 20 years of age than the BMI or other power function indexs. BMEI of 2, with limits of 1.5 and 2.5, could be used as nutritional index without requiring an age chart.

The BMI-for-age curve shape and the disproportional BMI in taller children are mathematical artifacts without biological meanings.

Author thanks Pedro Saavedra-Santana, PhD (School of Mathematics, University of Las Palmas de Gran Canaria), Gloria González-Azpeitia, PhD and Luis Peña-Quintana, PhD (Maternal and Child Hospital, University of Las Palmas de Gran Canaria) for their suggestions during the preparation of the manuscript. No financial compensation was provided to any of these individuals.

Funds for translation, manuscript editing services and meetings with experts in the area were provided by Nestlé Spain (Barcelona) and Laboratorios Ordesa (Barcelona).

The translation into English was performed by American Journal Experts.

Independent peer review prior to the submission process was performed by Rubriq.

Manuel Cidrás, (2015) Body Mass Exponential Index: An Age-Independent Anthropometric Nutritional Assessment. Open Access Library Journal,02,1-8. doi: 10.4236/oalib.1101943