34 Body Density from Hydrostatic Weighing

In the previous chapter we learned how to calculate the average density of an object using hydrostatic weighing. Now we are ready to understand how body fat percentage is determined from hydrostatic weighing. Here is the hydrostatic density formula that we arrived at previously:

    \begin{equation*} \rho = \frac{F_g}{F_g-F_a}\rho_w \end{equation*}

The above formula will provide the average density of an entire object, but in order to estimate body fat percentage we need to know the average density of the body tissues, so we need to not counting the volume residual gas in the lungs (RV) and digestive tract (RG). We need to correct our hydrostatic density formula by subtracting these volumes from volume of our object. Doing so gives us the following formula for body density. (The steps taken to make that correction are shown at the end of this chapter.)

\rho_b = \frac{F_g}{\frac{F_g-F_a}{\rho_w}-g(RV-RG)}

The volume of gas in the digestive track gas is typically assumed to be 0.1 L and the residual lung volume can be estimated using, once again, a set of empirical models[1]:

For females: RV (\bold{L}) = 0.009 \times age (years) + 0.08128 \times height (in) -3.9

For  males: RV (\bold{L}) = 0.017 \times age (years) + 0.06858 \times height (in) -3.447

Note that the empirical models shown above will estimated the residual gas volume in liters, so the corrected body density formula also uses liters as the volume unit and produces an output in units of kg/L. Finally, we can use the calculated body density to interpolate body fat percentage using an empirical formula such as the one introduced at the start of the Unit:

BF\% =495/\rho_b (\bold{kg/L})-450

The lab for this unit involve some of these formulas and if you are curious you can read more about those formulas, play with a simulation of hydrostatic weighing, check out a website that does the calculations for you, and see that different formulas have been developed for different population sets in an effort to increase accuracy.[2]. Determining body fat percentage from body density is not something that Jolene would do on the MED floor of a hospital, but athletic training facilities and clinics specializing in care associated with body composition might use this method.

We will not work out the correct the formula for average body tissue density that we  as to account for residual gas volumes follows. We will start out with the same first couple of steps that we took to find the original formula. We start with the definition of the density of an object.

    \begin{equation*} \rho = \frac{m}{V} \end{equation*}

Using the standard model for weight near Earth’s surface we know that mass is can be calculated as weight divided by g.

Inserting that result for mass into the density equation we have:

    \begin{equation*} \rho = \frac{F_g}{gV} \end{equation*}

For a completely submerged object the volume of water displaced is equal to the volume of the object, so we can replace V with V_d.

    \begin{equation*} \rho = \frac{F_g}{gV_d} \end{equation*}

To find the density of the body tissues, not including any residual gasses, we need to subtract the residual gas volume from the total displaced volume:

    \begin{equation*} \rho_b = \frac{F_g}{g(V_d-RV-RG)} \end{equation*}

Using the definition of density again, we can replace V_d with the displaced water mass (m_d) divided by water density (\rho_w):

    \begin{equation*} \rho_b = \frac{F_g}{g(\frac{m_d}{\rho_w}-RV-RG)} \end{equation*}

Next we multiplying the g through the parentheses:

    \begin{equation*} \frac{F_g}{\frac{m_dg}{\rho_w}-g(RV-RG)} \end{equation*}

Notice that we now have the mass of displaced water multiplied by g. That is exactly how we would calculate the weight of the displaced water (F_d), so we could make that substitution. However, we also know that Archimedes’ principle tells us that the weight of displaced water is equal to the buoyant force F_b, so we can actually replace the displaced water weight (m_dg) with F_b

    \begin{equation*} \rho_b = \frac{F_g}{\frac{F_b}{\rho_w}-g(RV-RG)} \end{equation*}

We also learned in the last chapter that the buoyant must be the difference in in the magnitudes of the weight and the apparent weight.

    \begin{equation*} F_b = F_g - F_a \end{equation*}

Making that replacement in our density equation we have:

    \begin{equation*} \rho_b = \frac{F_g}{\frac{F_g-F_a}{\rho_w}-g(RV-RG)} \end{equation*}

And there we have the hydrostatic body density equation. You may see it written differently in different sources because simply dividing all terms in the equation by g, and defining F_a/g as the “apparent mass” (m_a) the formula can be written in terms of mass instead of weight.

    \begin{equation*} \rho_b = \frac{m}{\frac{m-m_a}{\rho_w}-(RV-RG)} \end{equation*}


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