## Abstract

The alveolar gas equation, the focus of a classic paper by Fenn, Rahn, and Otis, provides a framework for understanding the mechanisms involved in pulmonary gas exchange as well as the limits of human performance. The classic 1946 paper by Fehn, Rahn, and Otis gives your students an opportunity to learn about the alveolar gas equation from the physiologists who pioneered it and demonstrates that mathematics and data graphics are fundamental tools with which to learn respiratory physiology. In this essay, I outline avenues of discovery by which your students can explore the alveolar gas equation. Meaningful learning stems from inspiration: to learn, you must be inspired to learn. If anyone can inspire learning in respiratory physiology, it is Wallace Fenn, Hermann Rahn, and Arthur Otis.

- altitude
- pulmonary gas exchange
- respiratory physiology
- teaching

in respiratory physiology, pulmonary gas exchange is the name of the game. The composition of gas in the alveoli–the partial pressures of O_{2} and CO_{2}–drives and reflects pulmonary gas exchange, and it provides a framework for understanding the theoretical mechanisms involved in gas exchange and for understanding the practical limits of human performance.

It was a quest for ways to extend the practical limits of human performance that moved Wallace Fenn, Hermann Rahn, and Arthur Otis (Fig. 1) to lay one of the cornerstones for understanding pulmonary gas exchange: the alveolar gas equation (2).^{1},^{2} The introduction to their classic paper was brief: In our studies of high altitude physiology we have found an oxygen-carbon dioxide diagram a very great aid to accurate thinking. It has been useful in so many different problems that we consider it worth while to describe it in this paper so that it can be made available to others.

This introduction is modest but perhaps presumptuous: the paper was Fenn's first publication in respiratory physiology. What the introduction left unsaid was that Fehn, Rahn, and Otis had been studying respiratory physiology for several years under the confidential auspices of the Office of Scientific Research and Development (4).

Why is this classic paper by Fehn, Rahn, and Otis a valuable educational resource? It is valuable because it demonstrates the application of critical-thinking skills to a new scientific problem: the composition of gas in the alveoli at altitude. It is valuable because it illustrates that mathematics and data graphics are fundamental tools with which to learn respiratory physiology. And last, it is valuable because it gives your students an opportunity to learn about the alveolar gas equation from the physiologists who pioneered it.

## Background

Before I plunge into a description of ways in which your students can explore the alveolar gas equation, have you ever wondered how Fehn, Rahn, and Otis came to study it? Your students might too. Who better than one of the players–Hermann Rahn (10)–to provide some background: The entrance of Wallace O. Fenn into the history of respiratory physiology can be precisely dated. It was within days after the US entry into World War II. At that time he was forty-eight years old and had established himself as the acknowledged leader in the physiology of muscle and electrolytes…

Wallace Fenn was drawn into respiratory physiology by his desire to contribute to the war effort. This was to be largely a war in the air, and from a military point of view, supremacy in altitude tolerance meant supremacy of air power. The airplanes of that day did not yet have pressurized cabins, but the possibility occurred that the human lung might be pressurized by application of positive pressure breathing. The question was whether [human] lungs could tolerate a sufficient amount of pressure to raise the partial pressure of oxygen to a significant degree, or would the lungs rupture, or would the circulation stop?

…Wallace Fenn had never worked in the field of human respiration. The equipment in his laboratory would be regarded as primitive by current standards. Among the more useful items were a few assorted spirometers, two or three Haldane machines, an equal number of Van Slykes, and several U-tube manometers…

In addition to this modest inventory of equipment, Fenn had three young instructors, all trained in biology departments. They knew all about such things as how fast the drosophila can beat its wings [Chadwick]

^{3}, how and why the rattlesnake changes color [Rahn], and how to activate or inhibit enzymes found in grasshopper eggs [Otis], but none of them had ever blown a vital capacity…[Chadwick, Otis, and Rahn], living with their wives on postdoctoral stipends which were only a fraction of what a graduate student receives today, were the most unlikely crew to have been assembled for the unknown job that lay ahead of them.Neither the equipment nor the staff was very impressive, and it seems doubtful that by present standards the project could have qualified for a National Institutes of Health grant.

Apparently it was not the planned endeavor we might have imagined.

Later in his career, Arthur Otis (4) wrote about the value of his inaugural publication in respiratory physiology (6): This paper was not a great contribution to the big world of science. It was important to me. I had discovered that I could play the game.

All of them–Wallace Fenn, Hermann Rahn, and Arthur Otis–could play the game. They helped create the game.

### Alveolar Gas Equation

In physiology textbooks (3, 14), the basic version of the alveolar gas equation is often written 1, where Pa_{O2} and Pa_{CO2} are the partial pressures of O_{2} and CO_{2} in the alveolar gas, Pi_{O2} is the partial pressure of O_{2} in the inspired gas, and R is the respiratory exchange ratio V̇co_{2}/V̇o_{2}, in which V̇co_{2} is the rate of CO_{2} elimination from the lungs and V̇o_{2} is the rate of O_{2} uptake from the alveolar gas.

In 1946, Fehn, Rahn, and Otis would have written the basic alveolar gas equation (*Eq. 1*) 2, where *pO* and *pC* are the partial pressures of O_{2} and CO_{2} in the alveolar gas, p′O is the partial pressure of O_{2} in the inspired gas, and *Q* is the respiratory exchange ratio *y*/*x* in which *y* is the amount of CO_{2} eliminated from the lungs and *x* is the amount of O_{2} removed from the alveolar gas. Instead, they presented the entire alveolar gas equation 3, where *f*′O is the fraction of O_{2} in the inspired gas (see *Eq. 7* in Ref. 2).

Do you marvel at the symbols used by Fehn, Rahn, and Otis in their 1946 paper? I did. In the 1940s, there was no coherent nomenclature for symbols of respiratory physiology. As you might imagine, this created problems. In 1950, a committee chaired by John Pappenheimer^{4} provided the fix: standardized symbols (8). Table 1 lists current renditions of the symbols used by Fehn, Rahn, and Otis in their classic paper.

### Strategies for Students

Graduate students and undergraduate students that know some respiratory physiology can benefit from reading this paper by Fehn, Rahn, and Otis. I suggest your students read the entire paper using this strategy: read carefully pages 637–643 and skim the rest. This will provide your students with sufficient background for the questions that follow.

Your students can answer the questions individually or in small groups. Either way, your entire class will benefit by discussing the questions and answers. Bear in mind that the process of developing a reasonable answer to each question is more important than the answer itself. After all, that is what discovery learning is all about.

### Opportunities for Discovery Learning

Because this paper by Fehn, Rahn, and Otis provides learning opportunities that differ from those embedded in other classic papers, I have listed questions and answers in a manner that differs from the approach used in similar *Advances* essays (see Ref. 9). I have prepared a handout that includes a brief introduction, the questions that follow, and Fig. 2 from the Fehn, Rahn, and Otis paper (2). The handout is available at *Advances in Physiology* website.

*1. Discuss the symbols of respiratory physiology used by Fehn, Rahn, and Otis. In what ways were the symbols useful? In what ways could the symbols have been improved?*

In scientific writing, symbols clarify reasoning and simplify communication. Can you imagine writing even the basic alveolar gas equation (*Eq. 1*) without symbols? The difficulty with the symbols created by Fehn, Rahn, and Otis is that the symbols are abstract. How is a reader supposed to differentiate between *pO* and *p*′*O*? And what does the ′ represent? It has no obvious meaning in the symbol *p*′*O*.

The Atlantic City Committee (8) used a logical framework to standardize the symbols of respiratory physiology. If your students want to consider the creation of effective symbols, *How to Write Mathematics* (13) is a helpful resource.

*2. In his second teaching point, Raff (9) writes that control groups are mandatory for all physiological studies. Fehn, Rahn, and Otis did not have a control group. Why not?*

Fehn, Rahn, and Otis did not have a control group because theirs was a theoretical study of the composition of alveolar gas at altitude. There was no need to control for factors that could have unsuspectingly affected the variables of interest.

*3. Researchers are encouraged to report estimates of variability and uncertainty (1). Fehn, Rahn, and Otis reported estimates of neither. Why not?*

Fehn, Rahn, and Otis did not report estimates of variability or uncertainty for the same reason they did not use a control group: theirs was a theoretical study. As a result, there was no measurement error or random variation that would have created variability among sample observations or that would have mandated the need to estimate the uncertainty about the value of some population parameter.

*4. In words, describe the basic alveolar gas equation* . Your students could simply replace the symbols with words, but that result would be nonsensical. Your students can, however, express the basic alveolar gas equation in words that do make sense if they eliminate the respiratory exchange ratio R from the equation. These are the steps to do that.

*Step 1*. Into the basic alveolar gas equation S1substitute V̇ co_{2}/V̇o_{2} for the respiratory exchange ratio R: S2.

*Step 2*. But and where Pb is barometric pressure. Therefore, S3.

*Step 3*. Substitute *Eq. S3* into *Eq. S2* and simplify the result to get S4.

Now that the basic alveolar gas equation (*Eq. S1*) has been converted to *Eq. S4*, it can be translated: the Po_{2} of the alveolar gas is what remains after the Po_{2} associated with O_{2} uptake has been removed from the Po_{2} of the inspired gas.

*5. In words, describe the role of the term* *in the entire alveolar gas equation. How much can it affect the magnitude of Pa _{O2}*

To address this question, your students must take a different tack than the one they took in *problem 4*. One way to deduce the role of this term is to make it disappear: it disappears if R = 1.^{5} More formally, if R = 1, then . Now your students can describe the role of the term in the alveolar gas equation: the term accounts for the change in alveolar Po_{2} that occurs when the respiratory exchange ratio R differs from 1. For some level of O_{2} uptake, if R < 1, then less CO_{2} enters the alveoli and total alveolar gas volume decreases; as a result, Pa_{O2} increases. In contrast, if R > 1, then more CO_{2} enters the alveoli and total alveolar gas volume increases; as a result, Pa_{O2} decreases.

To evaluate the impact of the term on alveolar Po_{2}, suppose Fi_{O2} = 0.21.^{6} Next, evaluate the term for different combinations of R and Pa_{CO2}. Fehn, Rahn, and Otis used R = 0.6, 0.8, …, 1.4 and Pa_{CO2} = 10, 20, …, 50 mmHg. If Pa_{CO2} = 50 mmHg, then the term can increase Pa_{O2} by 7 mmHg (R = 0.6) or decrease Pa_{O2} by 3 mmHg (R = 1.4) (see Fig. 2).

*6. Use the entire alveolar gas equation* *to reconstruct two lines from Fig. 2 in Fehn, Rahn, and Otis (2): R = 0.6 at 10,000 ft and R = 1.2 at 15,000 ft.*

At each altitude, the value for R is defined, but values for the other variables must be provided. For simplicity, assume Fi_{O2} = 0.21. At each altitude, . At 10,000 ft, Pb ≈ 23 mmHg, so Pi_{O2} = 100 mmHg. At 15,000 ft, Pb ≈ 428 mmHg, so Pi_{O2} = 80 mmHg. All that remains is to assign values for Pa_{CO2}: 10, 20, …, 50 mmHg are convenient.

Table 2 includes values for alveolar Pco_{2} and Po_{2} that will enable your students to reconstruct the two R lines from Fig. 2 in Fehn, Rahn, and Otis (Fig. 3).

*7. Suppose you measure Pa _{O2} = 54 mmHg and Pa_{CO2} = 30 mmHg in a subject. At what altitude is that subject?*

If the subject is breathing air and if you adopt the values of R used by Fehn, Rahn, and Otis, R = 0.6, 0.8, …, 1.4, then the subject could be at 10,000 ft with R = 0.6, or the subject could be at 15,000 ft with R = 1.2 (Table 2 and Fig. 4). Without estimates for V̇ co_{2} and V̇o_{2}, it is impossible to know which one.

If the subject is breathing pure O_{2}, then the subject is at roughly 41,540 ft (2, 12). At any altitude, if a subject breathes pure O_{2}, then (See Ref. 2.) This means that two altitudes are equivalent if the sum of Pa_{O2} and Pa_{CO2} at altitude *x* is equal to the sum of Pa_{O2} and Pa_{CO2} at altitude *y*:

### Summary

The alveolar gas equation, the topic of this classic paper by Fehn, Rahn, and Otis (2), is one cornerstone of pulmonary gas exchange, and it led eventually to the O_{2}-CO_{2} diagram (12), another classic. This is how Hermann Rahn (10) described the O_{2}-CO_{2} diagram: Fenn's second masterpiece, the O

How elegant! The same tribute applies to the alveolar gas equation and its graphic forms in this paper (2)._{2}-CO_{2} diagram, did for pulmonary gas exchange what the [pressure-volume] diagram [his first masterpiece (11)] did for respiratory mechanics. With it he could represent all parameters of the alveolar gas and ventilation equations. He never claimed to have originated these equations, but he derived them independently, made sure they were correct, and put them in graphic form. As somebody put it, “That's when he made them sing.” On the diagram he could show all possible compositions of alveolar gas and the arterial blood under any specified set of conditions. He could indicate normal ranges and limits of survival as well as the pathways followed during hyperventilation and asphyxia and during exposure to CO_{2}, altitude, or hyperbaric pressures. It could be used to demonstrate ranges of normal and impaired performance. It was indeed a theme that could be sung with many variations.

This classic paper by Fehn, Rahn, and Otis gives your students an opportunity to learn about the alveolar gas equation from the physiologists who pioneered it, but its educational riches run deeper than that. The paper demonstrates that mathematics and data graphics are fundamental tools with which to learn respiratory physiology. And the paper is a vehicle by which your students can read the memoirs of Wallace Fenn (10) and Hermann Rahn (7). Your students cannot read the memoirs of Arthur Otis: he is Professor Emeritus at the University of Florida College of Medicine and lives in Gainesville, FL.

I have always believed that learning stems from inspiration: to learn, you must be inspired to learn. If anyone can inspire learning in respiratory physiology, it is Wallace Fenn, Hermann Rahn, and Arthur Otis.

## Acknowledgments

I thank Arthur B. Otis (Department of Physiology and Functional Genomics, University of Florida College of Medicine) for graciously answering my e-mails and phone calls.

## Footnotes

↵1 John West (15) has written about this and three other cornerstones for understanding pulmonary gas exchange.

↵2 In their paper, Fehn, Rahn, and Otis referred to this equation as the alveolar air equation. Today, we refer to the equation as the alveolar gas equation.

↵3 One reviewer asked why Chadwick was not a coauthor on this classic paper. The answer may be simply that Chadwick was working on other projects and that Rahn and Otis worked extraordinarily well together (A. Otis, personal communication).

↵4 Hermann Rahn and Arthur Otis were members of the committee.

↵5 This term also disappears if Pa

_{CO2}= 0 or Fi_{O2}= 0, but these conditions are merely theoretical.↵6 If Fi

_{O2}< 0.21, then the impact of the term on alveolar Po_{2}will be smaller.

- © 2006 American Physiological Society