Breathing is a rhythmic activity that has many aspects that can be described mathematically. At rest, most adults breath in and out at about 10 to 20 times each minute. By drawing an analogy to the rhythmic tidal motions of the sea, this breathing volume is appropriately called a "tidal volume". Each breath has a volume of about 400 to 600 cc or about half a liter. From this data, we can calculate what is called a minute volume, which is the amount of air breathed in and out over the course of one minute. From the previous data, this would be 400 x 10 to 600 x 20 or 4,000 to 12,000 cc per minute as a relatively normal range for the minute volume. For those so inclined, you may try calculating how much air you breath in and out in one day. An interesting fact is that we breath so many air molecules in our lives, that we are most probably breathing at least one molecule of air each day, that all of the famous people who have ever lived also breathed.
Since the air is not one hundred percent oxygen, we are also breathing in about 79% nitrogen along with 21% oxygen (O2). The oxygen combines with an iron atom within our hemoglobin molecules that are within our red blood cells (RBCs). Interestingly, astronomers have discovered that the only way that the element iron is formed in the universe is from the supernova explosions of dying stars. This suggests that we are truly made of star stuff!
Our RBCs carry the hemoglobin that binds with a vital oxygen molecule to our tissues where it is released into our cells. The oxygen is used by the cells' mitochondria to create energy for the cell. At the end of this process, half of the oxygen molecule (O2) ends up in a water molecule (H2O). Truly amazing! We breath in a gas (O2) and convert it into water (H2O) with every breath without even thinking about it!
There are many mathematical relationships that can be used to understand our breathing. One interesting discovery is that because dogs breath in and out rapidly or "pant", it was thought that mechanical respirators that help patients to breath could work that way too. Older machines delivered the relatively normal tidal volumes of 400 to 600 cc per breath at rates of up to 30 to 40 breaths per minute. The new machines were called "jet ventilators" because they could deliver over 100 breaths per minute similar to the panting of a dog, but the tidal volumes were much smaller - only about 50 cc per breath.
Initially it wasn't sure these new ventilators would work since they gave such relatively small tidal volumes of 50 cc compare to 500 cc from the conventional ventilators. But this is where the math comes in. If we calculate the minute volume for the jet ventilator, we see that it is 50 x 100 = 5,000 cc per minute, which is within our normal range of 4,000 to 12,000 cc per minute calculated above! We deliver the same minute volumes, but with smaller breaths. This is useful in many critical care situations. Some patients can tolerate only relatively small breaths. In the operating room, it is sometimes important to reduce the movement of the lungs during some surgical procedures. The jet ventilator is perfect in these situations as it delivers very small volumes to the lungs and therefore doesn't disturb the surgical field.
This is only a brief introduction to the mathematics of breathing, but it demonstrates the interesting ways that the physiological act of breathing and mathematical concepts relate to one another in quite fascinating ways.