Gas exchange: VO₂ calculation example

This example shows how to derive the equation for converting flow rate and the animal-induced change in oxygen concentration into the rate of oxygen consumption (VO₂, ml/min). Please assume that the V symbols represent rates (and therefore should have a dot over the V, if I knew how to do that in html).
The arrangement of the respirometry system is the LabAnalyst 'Mode 1' configuration, with the flowmeter upstream from the animal chamber and both CO₂ and water vapor absorbed before O₂ concentration is measured:

Obviously VO₂ is simply the difference between the amount of O₂ entering the metabolism chamber and the amount of O₂ leaving the chamber (all in ml/min):

(1) VO₂ = O_2in - O_2out
More specifically:
O_2in = FLOW_in * F_iO₂ and O_2out = FLOW_out * FeO₂

Where FLOW_in is the incurrent air flow into the chamber (ml/min), F_iO₂ is the concentration of O₂ in incurrent air (usually .2095), FLOW_out = the excurrent air flow out of the chamber (ml/min), and F_eO₂ is the O₂ concentration in excurrent air.

In the Mode 1 arrangement, FLOW_in is measured, but FLOW_out is not. But we can calculate FLOW_out as:

FLOW_out = FLOW_in + VCO₂ + VH₂O - VO₂

Where VCO₂ is CO₂ production by the animal and VH₂O is water loss from the animal (both in ml/min). Since the Mode 1 arrangement uses scrubbers to remove both water and CO₂ from excurrent air:

FLOW_out = FLOW_in - VO₂

Accordingly, we can re-write equation (1) as:

(2) VO₂ = FLOW_in * F_iO₂ - (FLOW_in - VO₂) * F_eO₂

rearranging:

(3) VO₂ = FLOW_in * F_iO₂ - FLOW_in * F_eO₂ + VO₂ * F_eO₂

and:

(4) VO₂ = FLOW_in * (F_iO₂ - FeO2) + VO₂ * F_eO₂

rearranging again:

(5) VO₂ - (VO₂ * F_eO₂) = FLOW_in * (F_iO₂ - F_eO₂)
(6) VO₂ * (1 - FeO2) = FLOW_in * (F_iO₂ - F_eO₂)

and finally:

(7) VO₂ = FLOW_in * (F_iO₂ - F_eO₂) / (1 - F_eO₂)

Reminder: all of the V symbols should have a dot over the letter V, indicating a rate instead of a volume.