Yesterday Greg, Dave and I pulled into Port Angeles, WA en route to Victoria, BC. We decided to stay a day and do the Hurricane Ridge hike in the Olympic National Park. This hike is a 3 mile in and out with a 900 foot elevation gain starting at about 4350 feet and ending at just about a mile high. We’ll call it 5240 feet because that’s what it says on the map.
The hike felt higher. Like Machu Picccu higher. The last 1/2 mile we met with all the elevation gain and it was steep. Midway I started to breathe heavy. 3/4ths up I was gasping. Another 200 meters I lost the ability to speak. Another 100 meters and I decided I would die on that mountain and hopefully be remembered bravely in a John Krakauer novel.
At the top, I crawled to a bench and collapsed there while a noisy child screeched wildly and was chastised by what must have been his great grandfather who shook his cane at him in annoyance.
“Wow you’re out of shape,” said Greg.
“It’s… the… altitude!”
“That guy made it no problem,” he said gesturing to great grampa.
“He probably… has.. polycythemia,” I stretched.
“Yeah that must be it.”
“Radio my wife,” I said, “I’ll never make it off this mountain and I want to say goodbye.”
“You don’t have a wife. I’m your husband. And you are a drama queen.”
One of the fundamental equations studied in respiratory therapy school is The Alveolar Air Equation. I’ve seen a lot of students struggle with this. So I am not only going to explain it in a clear way, I am also going to vindicate myself and prove to both Greg and his elderly, cane-wielding drag friend Polly Cythemia that I am indeed NOT out of shape and that I was suffering from an acute and debilitating case altitude sickness.
Despite what you may have been told, the actual equation is this:
PAO2 = ( FiO2 * (PB- PH2O)) – (PaCO2 / RQ)
Yes I know the back half of that may be shocking, even disturbing to those of you who learned the PaCO2 * 1.25 method. You can unclench. It’s the same thing.
But let’s start at the very beginning. According to Rogers and Hammerstein, it’s a very good place to start. Working from left to right we have PAO2
PAO2 represents the partial pressure of oxygen in the alveoli. P standing for both Partial and Pressure because saying PP always turns grown men into giggling adolescents. A stands for Alveoli and O2 standing for Oxygen, because, as you should know, oxygen atoms are unstable alone and stabilize by forming a double covalent bond with another oxygen atom. So each molecule of oxygen in the air is actually made up of two oxygen atoms. Hence O2.
Dry air contains 78.09% nitrogen, 20.95% oxygen, 0.93% argon, 0.04% carbon dioxide. For the purposes of respiratory math we can just round the O2 percentage to 21% and and the CO2 down to 0% for inhaled air.
Air, like life, work, school, money and fat-shaming spouses at the tops of mountains, exerts pressure on you. At sea level, that pressure is one atmosphere or 760 torr which we use in this equation because using 1 would be too easy. Torr is a unit of pressure measured in mmHg (millimeters of mercury). The symbol for mercury is Hg because Me would be too easy. In respiratory therapy we often switch between Torr and cm H2O, yet another unit of pressure, because again, being consistent would be too easy.
Anyway, while the units of measure invented by scientists with competing egos may be a convoluted mess, physics is much nicer to you. The pressure of a specific gas in a mixture of gases is exactly proportional to the percentage of that gas in the mixture. Also ‘gases’ is the plural of ‘gas’. ‘Gasses’ is a verb in the third person present tense. To use both in a sentence: The gassy gascon gasses us with his gases. Also his miasma triggers my asthma. But that’s another story….
Back to partial pressure. For example, if a mixture of gas is exerting 1000 torr, and 10% of that gas is O2, then the partial pressure of that gas is 10% of the total pressure which is 100 torr. Easy as pi.
So real world example, the pressure of air at sea level is 760 Torr. O2 is 21% of that mixture. Therefore the partial pressure of oxygen in air, with 0% humidity, is 21% of 760 torr which is 160 torr. And I hope you can see where I am headed here by now.
Fraction of Inspired Oxygen
Moving to the other side of the equation the first variable we hit is FiO2, or Fraction of inspired oxygen. In normal air, as we discussed earlier, this is ~.21. As you know, .21 is the same as 21%. But when talking in Fraction of Inspired Oxygen, we notate it as the decimal rather than the percentage. This is because people are petty semantic absolutists who, insecure about their level of knowledge, feign indignant superiority when faced with something they actually know about that you might be doing wrong. Drama ensues. Careers are ruined. The popular RT’s won’t let you sit at their lunch table. I don’t give a personal crap whether you speak in percentages or decimals, just know that in this equation requires the decimal form and that kids are mean.
This is the variable in this equations we as therapists modify to correct for hypoxia. In mechanical ventilation and with air entrainment devices we can set this with pretty high accuracy anywhere from 21% – 100%.
Atmospheric and Vapor Pressure
PB stands for Barometric Pressure. Or Pressure, Barometric more precisely. This is the total pressure of the gases inhaled. Whether on a vent or an entrainment device or a nasal cannula, at sea level this number is 760 torr and decreases as we get higher in altitude as discussed earlier.
PH2O represents the pressure of the water vapor suspended in the gas mixture. This is relevant to both humidity and temperature. At body temperature (37C) and 100% relative humidity, this is 47 torr. In the ICU we do our best to ensure this constant. In the dry, cold mountain air at 5200 feet, this may have have been zero for me. But in practice, just use 47 torr.
At constant body temperature and 100% humidity, how much this varies with altitude is a question I cannot seem to find the answer to. I assume it would be proportional to the the change in overall PB, but it is not a true gas, it is a liquid in suspension, and therefore assumptions should not be made. Anyone? Anyone? Bueller?
This is the partial pressure of carbon dioxide in arterial blood. This is the variable that must be obtained through an arterial line or through a needle stick usually to the radial artery of the wrist. In the ICUs I have been in most mechanically ventilated patients have an artline in place. These are great because constantly poking holes in people isn’t fun.
Now you may have learned this section of the equation as PaCO2/R with a given R as .8. This is the respiratory quotient. A true measurement of the RQ is the ratio of the volume of CO2 breathed out divided by the volume of O2 taken in. This is, mostly, a function of nutrition. Fats lower the number where carbohydrates raise it. A legitimate measurement requires indirect calorimetry but for a modern diet, the result will usually be around .8.
However, just using this as a constant instead of an accurate measurement makes an assumption about metabolism that adds a level of error to the calculation. If someone is compromised respiratorily and also has some metabolic derangement affecting the RQ, this could throw things way off. Metabolic derangements that could affect RQ on a cellular level as opposed to a ventilatory level: an interesting thing to look further into.
And to bring you back to familiarity, since the reciprocal of .8 is 1.25, you may often see the back half of this equation written as PaCO2 * 1.25 instead of PaCO2/.8. Same. Exact. thing.
How is this information helpful? Where I might just do this because I’m weird and find math fun, others may need some inspiration in the form of clinical significance. That inspiration comes in a measurement know as the A-a gradient. This is the partial pressure of alvolar oxygen minus the partial pressure of arterial oxygen or PAO2 – PaO2. This is a good assessment of the integrity of the alveoli. For a healthy young person, the difference should be between 5 – 10 Torr. After age 20, this gradient increases as the lungs gradually lose their diffusing capacity with age. A conservative estimate of normal A–a gradient is less than [age in years/4] + 4.
For example for a 40 year old, 40/4 + 4 = 14. the A-a gradient should be right in that ballpark if their lungs are not compromised.
In patients with compromised respiration, even though the alveolar air introduces a bit of uncertainty with its built in assumptions regarding RQ, we can trend this number to see if our therapies are working or not. If this gradient is shrinking, you are winning the fight.
At 5200 feet above sea level the air pressure drops to 630 Torr.
Cool Air pressure calculator here: https://www.omnicalculator.com/physics/air-pressure-at-altitude
At a temperature of 50F on top of that mountain the water vapor pressure at, lets assume my nose was functioning properly as the turbulent humidifier that it is, 100% humidity is 13.2
Cool water vapor pressure calculator here: http://www.endmemo.com/chem/vaporpressurewater.php
Assuming I am healthy we can use a PaCO2 of 40 torr and an RQ of .8. Our FiO2 stays constant at .21. Therefore:
PAO2 = .21(630 – 13.2) – 40/.8 = 80 torr.
Assuming at my age my A-a gradient is 15, then my PaO2 is 65 torr. Normal range for PaO2 is > 80. Mild hypoxia is 70 – 79 torr. Moderate hypoxia is 60 – 69 torr.
65 Torr translates to MODERATE HYPOXIA!
Vindication never felt this good. Or this bad. Kinda both. I guess I’m fine not being able to breathe as long as I am RIGHT.
But then again, why didn’t anyone else struggle?