Problem 32 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

The Body Mass Index, $$\mathrm{BMI}=\frac{\text { Mass }}{\text { Height }^{2}}$$ was introduced by Adolphe Quetelet, a French mathematician and statistician in \(1869 .\) The Center for Disease Control and Prevention (CDC) notes that BMI is a helpful indicator of overweight and obesity in adults. From simple allometric considerations, BMI3 = Mass/Height \(^{3}\) should be approximately a constant, \(C\). If \(B M I 3=\) Mass/Height \(^{3}=C\) then \(B M I=\) Mass/Height \(^{2}=C\) Height. so that BMI should increase with height. CDC also states that "... women are more likely to have a higher percentage of body fat than men for the same BMI." If a male and a female both have \(\mathrm{BMI}=23\) and are of average height for their sex \((1.77\) meters for males and 1.63 meters for females), then BMI3 for the male \(=\frac{23}{1.77}=13.0\) and BMI3 for the female \(=\frac{23}{1.63}=14.1\) Thus BMI3 is larger for the female than for the male and may indicate a larger percentage of body fat for the female. Shown are four Age, and 50 th percentile Weight, Height data points for boys and for girls. Compute BMI and BMI3 for the four points and plot the sixteen points on a graph. Which of the two indices, BMI or BMI3, remains relatively constant with age? Data are from the Centers for Disease Control and Prevention, http://www.cdc.gov/growthcharts/data/set1clinical/cj41c021.pdf and \(\cdots\) cj41c022.pdf. $$ \begin{array}{|l|rrrr|} \hline \text { Age (Boys) } & 8 & 12 & 16 & 20 \\ \hline \text { Weight (kg) (50 percentile) } & 26 & 41 & 62 & 71 \\ \text { Height (cm) (50 percentile) } & 128 & 149 & 174 & 177 \\ \text { BMI kg/m }^{2} & & & & \\ \text { BMI3 kg/m }^{3} & & & & \\ \hline & & & 12 & 16 & 20 \\ \text { Age (Girls) } & 26 & 41 & 54 & 58 \\ \hline \text { Weight (kg) (50 percentile) } & 128 & 151 & 162 & 163 \\ \text { Height (cm) (50 percentile) } & & & & \\ \text { BMI kg/m }^{2} & & & & \\ \text { BMI3 kg/m }^{3} & & & & \\ & & & & \\ \hline \end{array} $$ We suggest that BMI3 might be more useful than BMI as an index of body fat. Other indices of body fat that have been suggested include \(M / H, M^{1 / 3} / H, H / M^{1 / 3},\) and \(c M^{1.2} / H^{3.3}\). The interested reader should visit the web site cdc.gov/ncedphp/dnpa/bmi and read the references there.

Step-by-Step Solution

Answer

BMI3 remains relatively constant with age for both boys and girls.

Step 1: Convert Heights to Meters

To begin with, we need to convert the given heights in centimeters to meters for both boys and girls so that we can calculate BMI and BMI3 in the required units. This conversion is done by dividing the height in centimeters by 100.

Step 2: Calculate BMI for Boys

Use the formula \( \mathrm{BMI} = \frac{\text{Mass}}{\text{Height}^2} \) to calculate BMI for each age point in boys. Substitute the weights in kilograms and converted heights in meters into the formula.

Step 3: Calculate BMI for Girls

Similar to boys, calculate the BMI for each age point for girls using \( \mathrm{BMI} = \frac{\text{Mass}}{\text{Height}^2} \). Use the given weights and converted heights for girls.

Step 4: Calculate BMI3 for Boys

Using the formula \( \mathrm{BMI3} = \frac{\text{Mass}}{\text{Height}^3} \), calculate BMI3 for each age point in boys. Use the weights in kilograms and the heights in meters.

Step 5: Calculate BMI3 for Girls

Calculate BMI3 for each age point in girls using \( \mathrm{BMI3} = \frac{\text{Mass}}{\text{Height}^3} \). Again, use the respective weights and heights for girls in meters.

Step 6: Plot the BMI and BMI3

Now, plot the BMI and BMI3 values for boys and girls on a graph. Separate lines for BMI and BMI3 are to be plotted to observe the trend as age progresses.

Step 7: Analyze the Indices

Compare the constancy of BMI and BMI3 with age for both boys and girls by observing the trend on the graph. The index that remains more constant across ages indicates a better measure.

Key Concepts

BMI calculation

The Body Mass Index, commonly abbreviated as BMI, is a numerical value that provides a quick way to estimate healthy body weight based on an individual's height. The formula for BMI is simple:

• Mass (in kilograms) divided by Height (in meters squared): \( \mathrm{BMI} = \frac{\text{Mass}}{\text{Height}^2} \).

To illustrate, if someone weighs 70 kilograms and is 1.75 meters tall, their BMI would be calculated as follows:

• 70 ÷ (1.75 × 1.75) = 22.86.

A BMI between 18.5 and 24.9 is generally considered normal. However, it is important to note that BMI does not directly measure body fat % or account for muscle mass. Thus, it should be considered a guideline rather than an absolute measure of health.

Allometric considerations

Allometry involves studying the relative growth of a part of an organism in relation to the entire organism. The concept suggests that body proportions change in a predictable way as size increases or decreases across individuals or species.

• In the context of BMI, allometric considerations show that the calculation might need adjustment for better accuracy.

Traditionally, BMI uses height squared in its formula, but allometry hints that height cubed \( \frac{\text{Mass}}{\text{Height}^3} \) might provide a more constant index across different heights and potentially reflect body composition more accurately.

• This idea suggests that BMI3 could hold value in better understanding body shape and composition changes not well captured by traditional BMI metrics.

Age and growth analysis

Understanding how BMI and other indices change with age provides insight into growth patterns and health status, especially during development years.

• Children and teens naturally undergo several changes in height and weight, making BMI variations typical as they grow.

Analyzing BMI across various ages allows us to monitor healthy growth trajectories and identify potential health concerns early.

• In this analysis, we compare BMI and BMI3 to see which more consistently reflects changes or remains stable across different age groups.
• Such analyses are crucial for framing age-appropriate health guidelines and understanding developmental health.

Body fat indices

Body fat indices provide estimates of body composition beyond simple weight and height metrics. They attempt to assess body fat mass relative to size, potentially offering a more comprehensive health evaluation.

• Different indices have been proposed, such as \( M / H \), \( M^{1 / 3} / H \), \( H / M^{1 / 3} \), and \( c M^{1.2} / H^{3.3} \).

Each index aims to improve upon the limitations of BMI by accounting for individual differences in body composition. For example:

• While BMI provides a quick scan, it can misrepresent body fat levels in muscular individuals or those with a higher percentage of body fat.
  More refined indices like BMI3 or the aforementioned measures consider density and distribution better, potentially offering insight into health beyond what BMI alone can.