ASSIG 3

ILLUMINATIONS Curricular Integration of Physiology

Teaching an intuitive derivation of the clinical alveolar equations: mass balance as a fundamental physiological principle

Michael C. Wang, Thomas C. Corbridge, X Donald R. McCrimmon, and X James M. Walter Northwestern University, Feinberg School of Medicine, Chicago, Illinois

Submitted 16 May 2019; accepted in final form 2 January 2020

INTRODUCTION

Every student of pulmonary physiology comes across two essential equations of ventilation and gas exchange, the alve- olar ventilation equation (“the CO2 equation”) and the alveolar gas equation (“the O2 equation”). These equations are partic- ularly relevant for health professions students, given both their physiological significance and tremendous utility in clinical practice.

For equations of such importance, it may be surprising that, in educational resources, they are typically presented outright with minimal justification (2) or derived with abbreviated or simplified methods (3); seldom, even in medical textbooks, are complete and rigorous derivations presented.1 Sometimes der- ivations are omitted when teaching equations because they are either not intuitive or overly complex. However, in the case of the alveolar equations, there is a wealth of physiological reasoning behind their derivations, which involve only algebra, and are accessible across all levels of medical training.

Central to deriving and understanding the equations is the principle of mass balance, the idea that substances accumulate within and leave systems based on the relative rates with which they enter and exit these systems. Steady-state equilibrium occurs when the rates of entry and exit are equal. The concept of mass balance is relevant to the physiology of multiple organ systems.

The intuition and utility of mass balance is perhaps best illustrated with the pulmonary alveolar equations because of the multistep process needed for their derivations. While the seminal paper by Fenn et al. (5) on the alveolar gas equation presents fascinating graphs that have been cited as a learning opportunity (4), their derivation uses antiquated notation, is difficult to follow for the typical medical student, and does not include the alveolar ventilation equation. This paper presents a unified framework for deriving both alveolar equations (and their clinical simplifications) that pedagogically highlights mass balance through the novel concept of “ventilation iden- tities.” We coin this term to emphasize the fundamental im- portance of equations for nitrogen, oxygen, and carbon dioxide

that are the building blocks for the derivations and come directly from the application of mass balance.

Here, we demonstrate how we teach the derivations and highlight how we link applications of these equations to clin- ical practice. While the clinical examples provided often center on a critically ill patient, we believe the foundational principles discussed can and should be used to guide clinical decision making outside of the intensive care unit. This approach was presented as an optional learning supplement to first-year medical students at the Northwestern University Feinberg School of Medicine, which follows an organ system-based curriculum, as a resource for the pulmonary module in Febru- ary 2019.

Review of Foundational Concepts

We introduce terms (Table 1) and prerequisite physiological concepts by briefly reviewing three essential pulmonary topics: dead space, the respiratory quotient, and ventilation-perfusion mismatch. Consistent with the modern literature, we use New- ton’s dot notation with volumes to convey rates of volume change or flow.

Dead space. The ultimate goal of breathing is to maintain homeostasis by replenishing O2 and eliminating CO2 in pro- portion to the demands of ongoing aerobic respiration. The motivation for considering dead space is that gas exchange only occurs in the alveoli, whereas standing in between the environment and the alveoli are the conducting airways, air passages with no gas exchange potential. This means that the rate at which air is moved in and out of the lungs (the minute ventilation, V̇E) is not the rate at which fresh air is moved in and out of the alveoli (the alveolar ventilation, V̇A). This is because a portion of every breath can be thought of as “wasted,” or stuck in the conducting airways without any gas exchange potential, the so-called dead space ventilation (V̇D). Thus, we can write:

V̇E � V̇A � V̇D

V̇A � V̇E � V̇D

(1)

During normal breathing, the brain controls the respiratory rate (RR) and the volume of air inhaled/exhaled, termed the tidal volume (VT). Just as cardiac output can be broken down into heart rate times stroke volume, V̇E can be expressed as RR times VT, and V̇D as RR times dead space volume (VD). Thus, with RR, VT, and VD, we can rewrite Eq. 1 as:

Address for reprint requests and other correspondence: J. M. Walter, Northwestern University, Feinberg School of Medicine, Div. of Pulmonary and Critical Care Medicine, 240 East Huron St., Suite M-300, Chicago, IL, 60611 (e-mail: [email protected]).

1 Boron and Boulpaep, authors of one of the most comprehensive physiol- ogy texts (1), only include them in online-only web notes. The framework presented here draws inspiration from their derivation.

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V̇A � RR�VT � VD�

V̇A � RR · VT�1 � VD

VT � (2)

These forms of the dead space equation tell us three impor- tant things:

• If all else is constant, V̇A decreases with increasing VD. • If all else is constant, V̇A increases with increasing RR and VT. • Holding V̇E � RR·VT constant, V̇A decreases with increasing

RR (and corresponding decreasing VT).

Breathing faster or deeper can both enhance gas exchange. However, rapid shallow breathing tends to be less efficient at gas exchange than slow deep breathing. This is important in the Intensive Care Unit (ICU), where assessing a ventilated pa- tient’s status requires review of not only V̇E but also VT.

In addition to the anatomic dead space of the conducting airways, physiological VD includes the volume of alveolar air that does not participate in normal alveolar respiratory gas exchange. For example, destruction of alveolar walls can result in the coalescing of multiple alveoli, giving rise to enlarged air spaces that are relatively poorly perfused (e.g., emphysema). Conditions such as pulmonary embolism may entirely block perfusion to alveolar capillary units. Increased alveolar dead space explains how these diseases lead to decreased gas ex- change with an unchanged V̇E. Anatomic and alveolar dead space together are referred to as physiological dead space, which is the true value of VD as represented in the equations.

The respiratory quotient. The respiratory quotient (RQ) is the ratio of CO2 generated to O2 consumed in metabolism:

RQ � V̇CO2

V̇O2

(3)

Where V̇CO2 is the rate of CO2 generation, and V̇O2 is the rate of O2 consumption. The value of RQ is substrate dependent and changes with the diet. RQ is exactly 1 for all carbohy- drates, �0.8 for proteins, and �0.7 for fats. We demonstrate this concept by considering the balanced chemical equation for the complete oxidation of glucose:

C6H12O6 � 6O2 ¡ 6CO2 � 6H2O

The coefficients in this equation show that, for glucose, an equal amount of CO2 is generated as O2 is consumed (RQ � 1). By changing the starting material, one can derive the RQ for any substrate; for example, RQ � 0.7 for myristic acid. This explains why we adjust RQ for a hospitalized patient on intravenous sugar solution (RQ � 1) or a patient with starva- tion ketosis (RQ � 0.7). RQ � 0.8 is a clinically accepted value for a Western diet.

Ventilation-perfusion mismatch. Thus far, the discussion has treated the lungs as a single unit of gas exchange that ventilates at V̇A. In fact, gas exchange occurs within heterogeneous alveolar-capillary units that differ not only in their V̇A, but also their capillary perfusion Q̇ (largely due to gravity). As a result, ventilation relative to perfusion, or the V̇A/Q̇ ratio, varies within the lungs. If this ratio is high, then ventilation domi- nates, and alveolar gas more closely resembles atmospheric gas, with low CO2 and high O2 (the extreme case where Q̇ � 0 then V̇A/Q̇ ¡ � is dead space). If this ratio is low, then perfusion dominates, and alveolar gas more closely resembles venous gas, with high CO2 and low O2 (the extreme case of V̇A/Q̇ ¡ 0 is shunt). Put another way, the gas composition of each alveolar-capillary unit depends on its individual V̇A/Q̇ ratio, and all units together compose overall gas exchange.

This concept has major clinical significance because CO2 and O2 are transported differently in the blood. A significant fraction of blood CO2 is transported as bicarbonate, with smaller fractions transported as carbamino compounds (including carbaminohemo- globin) and dissolved CO2. The dissociation curve for CO2 trans- ported in this fashion is roughly linear in the physiological range. Consequently, high V̇A/Q̇ units compensate for low V̇A/Q̇ units in overall CO2 elimination. In marked contrast, blood O2 is mostly transported as oxyhemoglobin, with a miniscule dissolved com- ponent. The oxyhemoglobin dissociation curve is markedly non- linear in the physiological range, being quite flat at PO2 � 60 mmHg and quite steep at PO2 � 60 mmHg. Thus high V̇A/Q̇ units with a PO2 � 60 mmHg are on the flat portion of the curve, and there is little increase in O2 content as PO2 is increased further above 60 mmHg. In contrast, V̇A/Q̇ units with a PO2 � 60 mmHg are on the steep portion of the oxyhemoglobin dissociation curve, and small further decreases in PO2 correspond to comparatively large decreases in O2 content. Because of this asymmetry in the effects of changes in PO2 on O2 content between high and low V̇A/Q̇ units, increasing PO2 in high V̇A/Q̇ units cannot compensate

Table 1. Terms and definitions

Ventilations and Rates Other Terms

Term Definition Term Definition

V̇A Alveolar ventilation (L/min) VT Tidal volume (Liters)

V̇AI Inspired alveolar ventilation (L/min) VD Dead space volume (Liters)

V̇AE Expired alveolar ventilation (L/min) RR Respiratory rate (1/min)

V̇E Minute ventilation (L/min) RQ Respiratory quotient

V̇D Dead space ventilation (L/min) FIX Inspired fraction of gas X

V̇N2 N2 ventilation (L/min) FAX Expired (alveolar) fraction of gas X

V̇O2 Rate of O2 consumption (L/min) PIX Inspired pressure of gas X (mmHg)

V̇CO2 Rate of CO2 production (L/min) PAX Alveolar pressure of gas X (mmHg)

Q̇ Perfusion or cardiac output (L/min) PaX Arterial pressure of gas X (mmHg)

PB Barometric (atmospheric) pressure (mmHg)

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for low V̇A/Q̇ units in overall blood oxygenation; the high V̇A/Q̇ units contribute far less additional O2 content relative to the decreased O2 content contributed by the low V̇A/Q̇ units. Thus any exacerbation of baseline V̇A/Q̇ heterogeneity in the lungs, termed V̇A/Q̇ mismatch, decreases overall blood oxygenation, even at a constant overall V̇A and Q̇. It is important to emphasize that mismatch occurs not only in pathological states, but also in normal, healthy lungs; understanding this fact is key to under- standing the clinical utility of the alveolar equations.

The Ventilation Identities

In this section, we split V̇A into its component gases by using the principle of mass balance to derive fundamental statements about the relationship between the gases in the atmosphere, the gases in the body, and V̇A. Together, the following three equations regarding N2, O2, and CO2 (Eqs. 4, 5, and 6, below) underpin the clinical equations to follow. To highlight their importance, we call them the “ventilation identities.”

We derive the “ventilation identities” with several assump- tions:

1. Gas transfer in the lung is at steady state. Steady-state equilibrium is safely assumed in most clinical scenarios. However, with acute changes, such as during the transi- tion between different activity levels or during interval training incorporating bursts of high-intensity exercise, steady state may not be achieved. Sometimes this even leads to the respiratory exchange ratio, the empiric mea- surement of CO2 production versus O2 consumption from inspired and expired gases, differing from RQ at the cellular level due to, e.g., lactate production or removal and subsequent bicarbonate buffering. The mathematics of non-steady-state solutions are complex and not ade- quately represented by the equations we present.

2. Argon and other gases present in trace amounts in the atmosphere are considered as part of nitrogen. This sim- plifying assumption groups the metabolically inactive gases together (defined in Nitrogen below).

3. There is no CO2 or water vapor in the atmosphere. For simplicity, we assume CO2 is only expired, not inspired, and the only gases contributing to atmospheric pressure are N2 and O2.

4. All gases are dry when measured. The airways humidify air, and the presence of water vapor would complicate the application of mass balance to the gases of interest. However, imagine that, for all measured quantities, the water vapor is removed, and the gas is then expanded to occupy its initial volume (which laboratories actually perform!). Because the saturated partial pressure of water vapor is constant at a given temperature, we can make this assumption and then account for water vapor in a simple way in the derivation of the alveolar gas equation.

The reasoning behind the “ventilation identities” can then be expressed as follows: Consider an arbitrary gas X that may be present in the environment or in the body. As a person inspires at inspired alveolar ventilation (V̇AI), some molar fraction of the inspired breath consists of X; call it FIX. Similarly, as a person expires at expired alveolar ventilation (V̇AE), some molar fraction of expired breath consists of X; call it FAX (capital A for “alveolar” to emphasize that the source for non-dead space expiration is alveolar air).

Then, we can express inspired and expired ventilation of gas X, respectively, as FIX · V̇AI and FAX · V̇AE. It is important to separate these terms because V̇AI is not necessarily equal to V̇AE. Ignoring water vapor (assumption 2), V̇AE is usually slightly less than V̇AI, since for RQ � 0.8, on average there are only about 8 molecules of CO2 expired for every 10 molecules of O2 inspired. The difference is frequently ignored in casual discussions of breathing, but is crucially important for the derivations that follow.

On balance, the body may consume X, generate X, or not interact with X at all, and each of the three major gases involved in human respiration interacts with the body in one of these three ways. One confusing point for learners is that, depending on the gas X, the quantity V̇X will not necessarily denote a ventilation, i.e., not necessarily a volume inhaled or exhaled like V̇A; it may represent a rate of generation or consumption, depending on the gas.

Nitrogen. N2 is the most abundant gas in the atmosphere, but, for the respiratory system, it is considered a metabolically inactive gas. It is neither produced nor consumed by the body, but rather freely diffuses down its partial pressure difference to equilibrium, such that, at steady state, the same amount of N2 is inspired as expired.

Considering inspired and expired N2 content as equal, we can express the volume of N2 inspired or expired in 1 min as (see Fig. 1A):

N2 ventilation � inspired N2 � expired N2

V̇N2 � FIN2

· V̇AI � FAN2 · V̇AE (4)

where, as above, F is mole fraction, A is alveolar, I is inspired, and E is expired. V̇N2

is defined, consistent with terms like V̇A

and V̇E, as a ventilation. In the derivation of the alveolar gas equation, the ventilation

identity for N2 balances the V̇AI and V̇AE values. Teaching this equation also allows students to think explicitly about the physiology of N2 in respiration, presenting an opening to discuss important clinical considerations for N2:

• Nitrogen’s inertness is thought to be a desirable property in reducing atelectasis; the presence of N2 helps maintain a sufficient gas tension within the alveoli to resist collapse (6).

• Nitrogen’s inertness also explains the use of O2 therapy in the management of pneumothorax: washing out the nitrogenous com- ponent of the ectopic gas facilitates absorption back into the bloodstream and a decrease in the pneumothorax volume (8).

Oxygen. O2 is inspired from the environment and flows to the alveoli (Fig. 1B), where some of it diffuses into the bloodstream, mostly binds to hemoglobin, and diffuses into metabolically active tissues to become the final electron accep- tor of the electron transport chain. Some O2 remains in the alveoli and forms part of the expired breath. Thus, like N2, O2 is present in both inspired and expired gas, but, unlike N2, O2 is consumed by the body, so less is expired than inspired.

We can express the volume of O2 consumed per minute as:

O2 consumption � inspired O2 � expired O2

V̇O2 � FIO2 · V̇AI � FAO2

· V̇AE (5)

where it is important to note that V̇O2 is not a ventilation: it denotes rate of O2 consumption, not inspiration or expiration.

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Carbon dioxide. CO2 is present in negligible amounts in the atmosphere, and we can safely assume it is not a significant part of inspired gas (assumption 3). Assuming a constant V̇A, it reaches a steady-state pressure in the alveoli, such that it is eliminated at the same rate that it is generated, and thus we can express both the volume of CO2 eliminated per minute and the volume of CO2 generated per minute as (see Fig. 1C):

CO2 production � CO2 expiration

V̇CO2 � FACO2 · V̇AE (6)

Note that there is no inspired term because we consider the atmosphere to have negligible CO2 content. Also note that the definition of V̇CO2 is again different, this time denoting rate of CO2 generation or expiration, not inspiration or consumption.

Accounting identities. The “ventilation identities” addressed mass balance for individual gases. Their independent contri- butions to V̇AI and V̇AE compose total ventilation (Dalton’s law). We can easily derive:

V̇AI � inspired O2 � inspired N2

V̇AI � FIO2 · V̇AI � FIN2

· V̇AI

1 � FIO2 � FIN2

(7)

V̇AE � expired O2 � expired N2 � expired CO2

V̇AE � FAO2 · V̇AE � FAN2

· V̇AE � FACO2 · V̇AE

1 � FAO2 � FAN2

� FACO2

(8)

obtaining the simple intuition that the mole fractions of con- stituent gases sum to 1.

The Clinical Alveolar Equations

We now have all of the ingredients necessary to derive the alveolar equations used in clinical practice and conceptually understand them; Table 2 summarizes the building blocks of these equations. The basic road map is as follows: the alveolar ventilation equation is a simple transformation of the ventila-

tion identity for CO2. Substituting the dead space Eq. 2 for V̇A

yields the clinically useful form. The alveolar gas equation results from the combination of all

of the “ventilation identities.” A simplifying assumption yields the clinically used form and understanding the nature of this assumption is crucial for grasping its limitations.

The alveolar ventilation equation. DERIVATION. Consider Eq. 6, the ventilation identity for CO2. By Dalton’s law of partial pressures, the partial pressure of a gas (PX) is directly propor- tional to its mole fraction (FX).

We then have:

V̇CO2 � kPACO2 · V̇AE

where P denotes partial pressure, and k is a proportionality constant. Rearranging terms, we have:

PACO2 �

kV̇CO2

V̇A

(9)

where it is understood that V̇A refers to V̇AE, and we redefine k as its reciprocal to follow convention. This is the basic alveolar ventilation equation.

Fig. 1. Mass balance diagrams for N2, CO2, and O2 in a representative alveolar-capillary unit. The principle of mass balance implies that, at steady state, the rate at which each gas enters the alveolus is equal to the rate at which each gas exits the alveolus. A: the rate N2 is inspired equals the rate it is expired, denoted by V̇N2

. There is no net movement of N2 between the alveolus (green outline) and the associated capillary (c). B: O2 enters the alveolus by inspiration and leaves by two routes: 1) net diffusion into the capillary, denoted by V̇O2, and 2) expiration. C: CO2 present in inspired gas is negligible. There is net diffusion of CO2

from the capillary into the alveolus, a quantity denoted by V̇CO2, which equals the rate of CO2 expiration. See Table 1 for definitions of terms used.

Table 2. Building blocks of the alveolar equations

Equation Description

V̇N2 � FIN2

·V̇AI � FAN2 ·V̇AE Ventilation identity for N2

V̇CO2 � FACO2 ·V̇AE Ventilation identity for CO2

V̇O2 � FIO2 ·V̇AI � FAO2

·V̇AE Ventilation identity for O2

1 � FIO2 � FIN2

Accounting identity for inspiration

1 � FAO2 � FAN2

� FACO2 Accounting identity for expiration

RQ � V̇CO2

V̇O2

Respiratory quotient definition

V̇A � RR·VT�1 � VD

VT � Dead space clinical equation

See Table 1 for definition of terms used.

148 INTUITIVE DERIVATIONS OF THE CLINICAL ALVEOLAR EQUATIONS

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At body temperature (TB) of 37°C (310K), k � 863 mmHg. This constant comes from the fact that, by historical conven- tion, the three quantities in Eq. 9 are expressed under different conditions of temperature, pressure, and humidity. The con- stant k accounts for physical conditions as well as for propor- tionality. It is possible to derive k � RT � (760/273) TB, where R is the ideal gas constant expressed in L BTPS·mmHg·L STPD

�1·K�1, 760 mmHg is standard pressure, and 273K is standard temperature (5, 9). Thus, while k is TB-dependent, the physiological range of TB, e.g., during a fever, is relatively small, making k � 863 mmHg an appropriate value for most physiological applications of the alveolar ventilation equation.

Two adjustments to Eq. 9 greatly increase its clinical utility. First, due to the approximately linear CO2 dissociation curve and the resultant compensation of high V̇A/Q̇ units for low V̇A/Q̇ units with respect to CO2 elimination, the arterial CO2

pressure (PaCO2 ) is a reasonable surrogate for alveolar CO2 pres-

sure (PACO2 ). Second, while V̇A is not a readily accessible value in

clinical practice, Eq. 2 expresses V̇A in terms of clinical parame- ters. Substituting PaCO2

for PACO2 and Eq. 2 for V̇A, we have:

PaCO2 �

kV̇CO2

RR · VT�1 � VD

VT � (10)

We call Eq. 10 the clinical alveolar ventilation equation be- cause of its utility in clinical practice. PaCO2

can be measured on a routine blood gas or approximated through waveform capnog-

raphy. Additionally, patients on ventilators have both known RR and VT values. Thus most of the values in the equation are known, especially for mechanically ventilated patients. As such, while Eq. 9 clearly illustrates the fundamental relationship between the generation of CO2, the elimination of CO2 through breathing, and the CO2 that remains in the body, Eq. 10 provides a more clinically oriented variant that we believe can help clinicians reason through the differential diagnosis of perturbations in PaCO2

. CLINICAL APPLICATIONS: THE DETERMINATION OF CARBON DIOX-

IDE STATUS. We present three short clinical applications to illustrate the utility of the clinical V̇A equation to students:

1. V̇CO2 can be measured with exhaled gas analysis, but is more commonly estimated in clinical practice. Perhaps even more powerful than knowing this parameter at discrete points in time is the fact that it depends directly on the rate of metabolism. This means that the more metabolically active a person is, the greater the rate at which he/she will generate CO2 (and require O2). That is why intensive care unit clinicians may deeply sedate or even paralyze severely hypoxemic patients; by lowering their metabolic rate, they are helping to keep their CO2 levels down (and O2 levels up) and reduce their ventilatory demand (Fig. 2A).

2. A clinical case example: An intern leaves a mechanically ventilated patient stable, sedated, and paralyzed at the end of a shift. The next morning, an arterial blood gas is performed, and the PaCO2

has risen substantially, despite no change in ventilator settings. What happened overnight? From Eq. 10, the right-hand side of the equation must have increased.

Fig. 2. Key relationships in the clinical alveolar ventilation (V̇A) equation. These stylized relationships assume all variables not in the figures are constant and do not provide physiological compensation. Default values are as follows: k � 863 mmHg; V̇CO2 � 0.2 L/min; RR � 12/min; VT � 0.5 Liters; VT � 0.15 Liters. A: at constant V̇A, the higher the rate of CO2 generation (V̇CO2), the higher the PaCO2

. Metabolic rate determines V̇CO2, such that agitation increases V̇CO2, and sedation decreases V̇CO2. B: PaCO2

increases as dead space (VD) increases, decreasing V̇A when V̇E is constant. Dead space diseases such as pulmonary embolism can cause hypercapnia through this mechanism. C: the same V̇E can result in markedly different PaCO2

, depending on the respiratory rate (RR) and VT. As RR increases at a constant V̇E, VT necessarily decreases, resulting in increased dead space ventilation (V̇D) and increased PaCO2

. Thus, in general, slow, deep breathing is more efficient for gas exchange than rapid, shallow breathing. See Table 1 for definitions of terms used.

149INTUITIVE DERIVATIONS OF THE CLINICAL ALVEOLAR EQUATIONS

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Assuming the patient was sedated, still, afebrile, and had no change in nutrition, then V̇CO2 should not have changed. Additionally, the ventilator maintained a constant RR and VT. Therefore, the physiological dead space must have increased. A sudden rise in physiological dead space in a critically ill patient should raise concern for venous throm- boembolism (Fig. 2B).

3. An oft-cautioned fatal mistake in clinical medicine is the case of a patient who presents to the emergency room in distress and breathing rapidly. The trainee who sees him/her prescribes a benzodiazepine to help calm his/her “hyperven- tilation.” The patient subsequently becomes severely aci- demic. It turns out that the rapid breathing was not hyper- ventilation, but rather a physiological response to an ele- vated CO2. When respiratory drive was depressed by the drug, CO2 levels increased even further. For example:

RR � 30 breaths/min �tachypnea� VT � 0.3 Liters �shallow breathing� VD � 0.2 Liters �increased dead space�

V̇CO2 � 0.25 L/min �elevated metabolism; in distress�

PaCO2 �

kV̇CO2

RR · VT�1 � VD

VT �

� 863 · 0.25

30 · 0.3�1 � 0.2

0.3� � 72 mmHg (normative : 40 mmHg)

This case illustrates the fact that multiple variables impact CO2 status, and, specifically, a tachypneic patient is not necessarily hypocapnic (Fig. 2C). A disciplined analysis of Eq. 10 shows that RR is only part of the expression that determines PaCO2

, which means one cannot assume a carbon dioxide level by simply observing a patient’s RR. The only way to know PaCO2 is to draw a blood gas (or approximate a level through wave- form capnography), highlighting the value of blood-gas anal- ysis in clinical care.

The alveolar gas equation. RATIONALE AND METHOD. Unlike the case for CO2, a simple blood test is not a good proxy for the alveolar partial pressure of O2 (PAO2

); arterial partial pressure of O2 (PaCO2

) is normally less than PAO2 . This is largely due to

the relatively flat oxyhemoglobin dissociation curve at PO2 values greater than 60 mmHg. Recall from Ventilation-perfu- sion mismatch above that, for overall blood oxygenation, high V̇A/Q̇ units cannot compensate for low V̇A/Q̇ units. Therefore, when oxygenated blood from all the alveolar-capillary units mixes in the systemic arterial system, O2 content is less than that which corresponds to overall PAO2

(PaO2 � PAO2

). There is also a small contribution from natural shunts of venous blood from the bronchial and thebesian veins to arterial blood. These vascular shunts occur downstream from the pulmonary capil- laries, decreasing arterial O2 saturation from what would be expected based on pulmonary gas exchange.

The difference between PAO2 and PaCO2

is referred to as the (A-a)PO2 difference:

�A-a�PO2 difference � PAO2 � PaO2

(11)

PaO2 is determined with an arterial blood-gas measurement.

Getting at PAO2 is the goal of the alveolar gas equation. We will

exploit the fact that the RQ ties O2 and CO2 levels together and is a clinically obtainable value.

DERIVATION. Almost all of the equations in Table 2, the “ventilation identities” (Eqs. 4–6), the accounting identities (Eqs. 7 and 8), and the RQ formula (Eq. 3), compose this derivation. Specifically, substituting all of these equations into each other yields the alveolar gas equation.

Ultimately, we want to isolate FAO2 , which is directly pro-

portional to our quantity of interest, PAO2 . The easiest place to

start is by substituting Eqs. 5 and 6 into Eq. 3:

RQ � V̇CO2

V̇O2

� FACO2

· V̇AE

FIO2 · V̇AI � FAO2

· V̇AE

(12)

We know RQ (we are prepared to approximate it as 0.8), we know FIO2

(0.21 in ambient air, or set by an assisted breathing device), and we know FACO2

(determined by PACO2 ). What to

do about V̇AI and V̇AE? They are related by the fractional N2 concentration because, by Eq. 4, N2 content is equal in inspired and expired air (Fig. 1A), and by Eqs. 7 and 8, N2 is related to O2 and CO2 by their mole fractions summing to 1! By Eq. 4:

V̇AI � V̇N2

FIN2

V̇AE � V̇N2

FAN2

And substituting in Eqs. 7 and 8:

V̇AI � V̇N2

1 � FIO2

(13)

V̇AE � V̇N2

1 � FAO2 � FACO2

(14)

Substituting Eqs. 13 and 14 into Eq. 12, we get:

RQ �

FACO2 ·

V̇N2

1 � FAO2 � FACO2

FIO2 ·

V̇N2

1 � FIO2

� FAO2 ·

V̇N2

1 � FAO2 � FACO2

�

FACO2

1 � FAO2 � FACO2

FIO2

1 � FIO2

� FAO2

1 � FAO2 � FACO2

From here, careful algebraic manipulation isolates FAO2 , giv-

ing:

150 INTUITIVE DERIVATIONS OF THE CLINICAL ALVEOLAR EQUATIONS

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FAO2 � FIO2

� FACO2�FIO2 �

1 � FIO2

RQ � (15)

The final step is to transform mole fractions into pressures. What’s the proportionality constant? It may at first appear to be the barometric pressure PB. However, as stated in assumption 4, we have assumed that gases are dry, whereas the alveoli actually contain “wet” air; that is, air saturated with water vapor, which at TB is approximately 47 mmHg. That means, of PB, only (PB � 47) mmHg is available for other gases.

Multiply the fractional concentrations in Eq. 15 by (PB � 47) to obtain pressures:

PAO2 � PIO2

� PACO2�FIO2 �

1 � FIO2

RQ � (16)

where PIO2 � FIO2

(PB � 47). This is the full alveolar gas equation. With information

about inspired O2, RQ, and PACO2 , one can calculate PAO2

. As was true for the alveolar ventilation equation, a few modifica- tions will increase its clinical utility. First, as before, we can substitute PaCO2

for PACO2 . Next, focusing on the term in

parentheses:

FIO2 �

1 � FIO2

RQ �

FIO2 · RQ � �1 � FIO2�

RQ

� 1

RQ �

FIO2 · �1 � RQ�

RQ

As RQ approaches 1 and FIO2 approaches 0, the second term

approaches 0. For a person with a mixed diet and breathing room air, when RQ � 0.8 and FIO2

� 0.21, the second term is 0.21· (1 � 0.8)/0.8 � 0.05. For these values, the approximation is good enough to simplify the expression to 1/RQ.

Making the simplifying substitutions, we get:

PAO2 � PIO2

� PaCO2

RQ (17)

where PIO2 � FIO2

(PB � 47).

We call Eq. 17 the clinical alveolar gas equation because of its utility in quickly generating an estimate for PAO2

at the bedside. We use this along with PaO2

to compute the (A-a)PO2

difference. In brief, the utility of this number is that we can separate problems with oxygenation into problems with O2 supply to the alveoli (associated with a normal (A-a)PO2 difference) and problems with gas exchange between the alve- oli and pulmonary capillaries (associated with an elevated (A-a)PO2 difference). The former includes low PB, low FIO2 (Fig. 3A), and hypoventilation (Fig. 3B). None of these affect gas exchange, so PAO2

and PaO2 drop about equally, and the

(A-a)PO2 difference is unchanged. The latter includes shunt, V̇A/Q̇ mismatch, and diffusion limitation. With respect to O2, unaffected lung has little ability to compensate for affected lung, so PaO2

drops without major effects on PAO2 , widening

the (A-a)PO2 difference. Thus the (A-a)PO2 difference is a useful tool in narrowing the differential diagnosis of hypox- emia.

Equation 17 should be used with caution for patients on supplemental O2, for two reasons. First, as FIO2

increases, the assumption that simplifies Eq. 16 to Eq. 17 becomes increas- ingly inaccurate (although this inaccuracy is negligible when low levels of supplemental oxygen are used). Using Eq. 16 overcomes this limitation. Second, the (A-a)PO2 difference in healthy lungs increases as FIO2

increases due to amplified effects of physiological V̇A/Q̇ mismatch and physiological vascular shunts. (A-a)PO2 difference increases from roughly 10 mmHg at FIO2

to 40 mmHg or higher at FIO2 � 0.9 (7). Thus the

evaluation of whether a patient on supplemental oxygen has an abnormal (A-a)PO2 difference requires adjustment for FIO2

, and there is no universally accepted clinical standard for making this adjustment. Rather than revert to Eq. 16 and reference an FIO2

-dependent (A-a)PO2 difference, clinicians more com- monly use the P/F ratio, defined by P/F ratio � PaO2

/FIO2 . This

indicator has utility in diagnosing gas exchange pathology in patients on supplemental O2, but, as merely an estimate itself, also has important limitations (10).

It may be tempting to examine the inverse relationship between PACO2

and PAO2 in the alveolar gas equation and con-

Fig. 3. Key relationships in the clinical alveolar gas equation. These stylized relationships assume all variables not in the figures are constant and do not contribute to physiological compensation. Default values are as follows: PB � 760 mmHg; FIO2

� 0.21; PaCO2 � 40 mmHg; RQ � 0.8. A: PAO2

changes linearly as a function of the fraction of inspired oxygen (FIO2

). Thus, supplemental oxygen, e.g., nasal cannula, readily raises PAO2 , while a fire may introduce additional

gas into the local environment, lowering FIO2 and, therefore, PAO2

. At altitude, each breath contains less gas overall, resulting in a lower PAO2 at the same FIO2

. Airline cabins are pressurized, but not to sea level, which can be problematic for passengers with underlying lung disease. B: PAO2

decreases as PaCO2 increases;

here, PaCO2 is a proxy for ventilation status. For example, hypoventilation simultaneously increases PaCO2

and decreases PACO2 . Respiratory quotient can also

affect PAO2 , as it indicates the rate of CO2 elimination relative to O2 consumption. Thus, in fasting ketosis (RQ � 0.7), V̇CO2 is reduced relative to V̇O2, and,

if PaCO2 were to be maintained at 40 mmHg, the reduction in alveolar ventilation would reduce PAO2

. See Table 1 for definitions of terms used.

151INTUITIVE DERIVATIONS OF THE CLINICAL ALVEOLAR EQUATIONS

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clude it is causal. However, the concept of RQ implies that CO2 production and O2 consumption are tied together in metabolism by the contents of the diet and cannot vary inde- pendently. A common misconception identifies disease pro- cesses that cause both hypercapnia and hypoxemia, such as chronic obstructive pulmonary disease and global hypoventi- lation, as primary problems with excess carbon dioxide. This reasoning contains an error of confounding; it fails to take into account that alveolar carbon dioxide cannot exogenously increase (unless it is given by medical intervention), but rather is endogenously determined by factors in Eq. 10, such as V̇A and dead space, which simultaneously affect alveolar oxygen (Fig. 3B).

CLINICAL APPLICATION: TWO PATIENTS WITH HEROIN OVERDOSE.

We use the following simple application with example num- bers to show students how calculation of the (A-a)PO2 differ- ence can inform the care of an acutely ill patient.

Assume two patients present simultaneously to an Emer- gency Department with a heroin overdose. They are breathing ambient air but appear somnolent with slow, shallow breathing.

You perform an arterial blood gas on each patient:

• Patient A: PaCO2 � 60 mmHg; pH � 7.24; PaO2

� 65 mmHg • Patient B: PaCO2

� 60 mmHg; pH � 7.24; PaO2 � 45 mmHg

From the elevated PaCO2 , you conclude both patients are

hypoventilating; however, you wish to know if further evalu- ation is necessary. You now calculate the alveolar O2 tension (note that it is the same for both patients) assuming RQ � 0.8 and a PB near sea level:

PAO2 � PIO2

� PaCO2

RQ

� FIO2 �PB � 47� �

60

0.8

� (0.21)(760 � 47) � 75

�75 mmHg

Finally, you calculate the (A-a)PO2 difference for each patient:

• Patient A: (A-a)PO2 difference � 75 � 65 � 10 mmHg • Patient B: (A-a)PO2 difference � 75 � 45 � 30 mmHg

Patient A has a nonelevated (A-a)PO2 difference and most likely is simply hypoventilating due to the effects of heroin. Patient B, on the other hand, has an elevated (A-a)PO2 differ- ence, which should raise concern for pulmonary parenchymal pathology in addition to hypoventilation. Could he/she have aspirated? Does he/she have an underlying respiratory condi- tion? For patient B, these possibilities need to be explored.

DISCUSSION

In this paper, we present a framework for thinking about gas exchange based on the principle of mass balance, with the “ventilation identities,” and then use it to derive two key equations for CO2 and O2: the alveolar ventilation equation and the alveolar gas equation. This process, involving only algebra, allows educators to organically teach many important concepts

in respiratory physiology and link these concepts to the care of acutely ill patients.

Although asking students to independently derive these equations will never be a required part of a health professions curriculum (and, indeed, is not the goal of this approach), many find their understanding of pulmonary physiology enhanced by exposure to the derivations. For the pulmonary module of our organ system-based curriculum, we offered a document cov- ering the derivations and a review session covering the docu- ment as optional resources to students. Students who attended rated this session 4.08/5 and found it a helpful adjunct to the content provided in lectures. Narrative feedback focused on the utility of linking foundational physiology with clinical practice and the satisfaction of knowing the origin of these key equa- tions.

We encourage educators to consider incorporating the framework in this document when teaching pulmonary physi- ology. We hope the clinical applications that were mentioned, in particular, help curriculum leaders in the health professions think about integration of physiology with clinical medicine across the basic medical sciences.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

M.C.W. conceived and designed research; M.C.W. and D.R.M. prepared figures; M.C.W. drafted manuscript; M.C.W., T.C.C., D.R.M., and J.M.W. edited and revised manuscript; M.C.W., T.C.C., D.R.M., and J.M.W. approved final version of manuscript.

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