Deriving the Respirometry Equations

At an abstract level, nearly all respiratory systems can be approximated by a simple one compartment model for the air in the chamber. Only three fluxes exist in this model for each gas, Vi, the in-current flux of fresh air entering the chamber; the fluxes of respiratory gases in and out of the subject that we are most interested in, Vo2 and Vco2; and Ve, the excurrent flux of air leaving the chamber.

Assuming that temperature, volume and pressure of the chamber do not change, this means that all fluxes for each gas need to perfectly balance out at all times. Note the different flux directions, which determines the sign in the balance equation. Green fluxes into the chamber get a positive sign, while fluxes out of the chamber get a negative sign. This is important when looking at different gas fluxes to and from the subject. Rearranging the formula allows the calculation of subject flux from the difference of in-current and excurrent fluxes.

We cannot measure the fluxes directly. Commonly, a combination of flow rate measurement of an airstream and measuring the fraction of the gas of interest in the air is used. Thus, we need to express our fluxes in these terms. Luckily, the conversion is straightforward. The flux of the gas in question is simply the product of the overall air flow rate, FR, and the fraction of the gas, F.

Doing it for each relevant gas gives us the following two equations. Note that the signs appear reversed for CO2 due to the reverse direction of the CO2 flux. While O2 is consumed from the chamber, CO2 gets released into the chamber. Getting the fractional concentration of the gases requires that we add a gas analyzer for each gas to the system. Usually, it is placed downstream of the chamber. To get the in-current concentrations, it will occasionally be switched to a different sampling line in a process called baselining via a channel bypassing the animal chamber.

If we place a flow meter in the in-current tubing lines to the chamber, we get a measurement of the in-current flow rate. This configuration is called a push-mode respiratory system. In a pull-mode respiratory system, the flow meter is placed downstream from the animal chamber, measuring excurrent air flow rate pulled from the chamber. Likewise, if we place a flow meter into the excurrent air stream, we can directly measure the excurrent flow rate, FRe, a pull-mode system.

Now, can we assume that in-current and excurrent flow rates are equal? This is usually not the case. For instance, if the respiratory exchange ratio is not exactly 1, more oxygen gets consumed by the subject than is replaced by carbon dioxide. As a consequence, the air flow into the chamber would need to be larger than the excurrent flow from the chamber. However, there is still an elegant escape from having to measure both flow rates.

Haldane realized that our biological systems contain an inert gas fraction consisting mostly of nitrogen that is assumed to neither be consumed nor released by the subject. Expressed in mathematical terms, this means that the subject flux, Vs, for nitrogen is always zero, and thus the in-current and excurrent fluxes are equal. Measuring the nitrogen fraction is typically not feasible due to the very fact that it is an inert gas, but we can express it through the fractions of all other gases in our A-stream. It is what is left when removing all other gas fractions, mathematically one minus or 100% minus the fractions of each other gas. This gives us an equation that relates the in-current flow rate, FRi, and the excurrent flow rate, FRe, when all gas concentrations are known.

In a push system we are measuring FRi. The only unknown is FRe, which we can get by rearranging the Haldane equation. Likewise, in a pull system FRe is measured, and we can rearrange to get FRi from the Haldane equation. To reiterate, with one flow rate measured, the unknown can be derived with the Haldane assumption. As a little helper, observe how the index in the numerator always matches the index of the flow rate on the left side of the equation. The indices of the fractions in the denominator are always the opposite.

We need flux equations (one per gas), the Haldane assumption, fractional contents (two per gas, in-current and excurrent), and a flow rate. The procedure to follow, derive the equation for the unknown flow rate from the Haldane assumption, insert that into the flux equations, and simplify toward the final equation.

For pull mode, we assume a system where CO2 is chemically scrubbed from the in-current air stream, and all water vapor is scrubbed from both in-current and excurrent flows. We can now derive VO2 for this pull system as follows. Following our recipe, we would first write down the generic respiratory equation for our two gases. Most quantities in them will already be measured directly. Since we scrubbed CO2 in the in-current air stream, FiCO2 is zero, that’s the in-current fractional content of CO2. This leaves one quantity which is undetermined – the in-current air flow rate, FRi. Using the Haldane assumption for pull systems, we can derive an expression for FRi in terms of FRe, which we measure. Substituting this into the generic respirometry equation for VO2, we yield. Note the common flow rate factor. By multiplying the second term in the sum with an elaborate 1, we can then bring both terms into a single fraction. Watching closely, two terms cancel in the sum, and we arrive at a final form of the equation.

Note how the terms that make up the equation have specific meaning. The first two are what is often called the delta O2, the difference between in-current and excurrent oxygen concentration. Next, there is a correction for the fact that release of CO2 by the subject appears as a reduction of oxygen fraction, even though it is unrelated to the oxygen uptake. The denominator is known as the Haldane correction. As you saw, it simply follows naturally from the derivation. VCO2 is derived as follows. The final equation is quite a bit simpler due to the original CO2 scrubbing.

For a push mode example, we have a slightly more complex system. All water is scrubbed from the in-current and excurrent air streams, but CO2 is measured both in-current and excurrent. VO2 in a push system is derived as follows. Again, we first write down the generic respirometry equations for our two gases. Most quantities in them will already be measured directly, including CO2. This leaves one quantity which is undetermined – the in-current flow rate, FRe. Using the Haldane assumption for pull systems, we can derive an expression for FRe in terms of measured FRi. Substituting this into the generic respiratory equation for VO2, we yield. Note the common flow rate factor. Again, multiplying the second term in the sum with an elaborate 1, we can bring the terms into a single fraction. Here also two terms cancel in the sum, and we arrive at the final form of the equation.

In the final equation for VO2, we again have the elements of delta O2, fractional content differences, the correction for changes in CO2 and the Haldane correction. VCO2 being measured in-current and excurrent is derived similarly. In the final equation for VCO2 we again have the element of delta CO2, this is the fractional content differences, the correction for changes in O2 and the Haldane correction.

Beyond VO2 and VCO2: VO2 and VCO2 form the basis of several other respiratory and metabolic equations, respiratory exchange ratio and respiratory quotient. The energy expenditure calculated with the Weir equation and its various permutations. And net glucose and fat oxidation rates. We will just briefly show these equations here and save their derivations for follow-up videos. Thank you.