Revisiting the effects of the reciprocal function between alveolar ventilation and CO₂ partial pressure ($\mathrm{\text{P}}{\mathrm{\text{A}}}_{\mathrm{\text{C}}{\mathrm{\text{O}}}_2}$) on $\mathrm{\text{P}}{\mathrm{\text{A}}}_{\mathrm{\text{C}}{\mathrm{\text{O}}}_2}$ homeostasis at rest and in exercise

This paper discusses some of the clinically relevant effects of the “reciprocal (1/x) nature” of the function linking alveolar ventilation (V̇A, the volume of gas mobilized in the lungs, per unit of time, contributing to the pulmonary gas exchanges) to the mean partial pressure of CO₂ in the alveolar gas ($\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$). This reciprocal function exerts fundamental constraints on the control of respiration, which are essential for alveolar gases homeostasis and stability, at rest and in exercise. This function also imposes major ventilatory challenges during heavy exercise, as hypocapnia develops.

THE PLANT GAIN OF THE RESPIRATORY SYSTEM IS A RECIPROCAL SQUARED FUNCTION OF ALVEOLAR VENTILATION: THE MAGNITUDE OF CHANGES IN $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ (FOR A GIVEN CHANGE IN ALVEOLAR VENTILATION) INCREASES AS V̇A DECREASES

If one neglects the insignificant amount of CO₂ in the air, $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ and thus ${\mathrm{P}}_{{\mathrm{C}\mathrm{O}}_2}$ in the arterial blood ($\mathrm{P}{\mathrm{a}}_{\mathrm{C}{\mathrm{O}}_2}$) are dictated, in steady-state conditions, by the ratio between the volume of CO₂ exchanged in the lungs per unit of time [carbon dioxide output (V̇co₂)] and V̇A (1):

$$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}=\text{ K}·{\dot{\mathrm{V}}\text{CO}}_{\text{2}}·{\dot{\mathrm{V}}\mathrm{A}}^{-1}.$$ (1)

If V̇co₂ is expressed in milliliter per minute in STPD conditions and V̇A in L/min in BTPS conditions, then K which has the dimension of a pressure is equal to 0.863 (1). This relationship is displayed in Fig. 1. In a resting subject (V̇co₂ ∼ 230 mL/min), V̇A of ∼5 L/min is required to maintain isocapnia ($\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ ∼ 40 mmHg). This figure shows that the magnitude of the change in $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ for 1 L/min change in alveolar ventilation, referred to as the respiratory plant gain for CO₂ (${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$), varies according to the level of alveolar ventilation in a nonlinear manner. Mathematically, the slope of the tangent of the V̇A/$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ relationship, i.e., ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$, is the first derivative of $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ over V̇A (d$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$/dV̇A) as illustrated in Fig. 1, A and B:

$$\mathrm{d}\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}/\mathrm{d}\dot{\mathrm{V}}\mathrm{A}=(-(\text{K}·{\dot{\mathrm{V}}\mathrm{\text{CO}}}_{\text{2}}·{\dot{\mathrm{V}}\mathrm{A}}^{-\text{1}})·{\dot{\mathrm{V}}\mathrm{A}}^{-\text{1}})=-(\text{K}·{\dot{\mathrm{V}}\text{CO}}_{\text{2}}·{\dot{\mathrm{V}}\mathrm{A}}^{-\text{2}}).$$ (2)

Figure 1.

A: relationship between V̇A and $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ in a resting adult subject. The 1/x nature of the relationship between V̇A and $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ ($\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ = K·V̇co₂·V̇A⁻¹) accounts for the fact that when V̇A decreases, any given change in V̇A produces larger and larger increase in $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$: the respiratory plant gain for CO₂ (${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$), i.e., the slope of tangent of V̇A-$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ relationship (shown in blue dotted line) increases as V̇A decreases. B: values of $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ as well as the slopes of the tangent of the V̇A-$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ relationship (${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$) at three different levels of V̇A (5 L/min, 2.5 L/min, and 1.25 L/min) when a subject hypoventilates. C: effect of increasing inspired ${\mathrm{P}}_{{\mathrm{C}\mathrm{O}}_2}$ (${\text{P}\mathrm{I}}_{\text{CO}}{}_{{}_2}$) by 15 or 40 mmHg on the V̇A-$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ function at rest. A higher $\mathrm{P}{\mathrm{I}}_{\mathrm{C}{\mathrm{O}}_2}$ shifts up the V̇A-$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ relationship by an amount corresponding to $\mathrm{P}{\mathrm{I}}_{\mathrm{C}{\mathrm{O}}_2}$ but does not alter the slope of the tangent of V̇A-$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ function (at any given level of V̇A, V̇co₂/V̇A remains the same as during air breathing). Since an increase in ${\text{P}\mathrm{I}}_{\text{CO}}{}_{{}_2}$ rises mean $\mathrm{P}{\mathrm{a}}_{\mathrm{C}{\mathrm{O}}_2}$, any ventilatory stimulation resulting from this increase in $\mathrm{P}{\mathrm{a}}_{\mathrm{C}{\mathrm{O}}_2}$ allows the respiratory system to operate in the “hyperventilation range,” where ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$ is lower. The dotted lines represent the range of the controller gain values from 1 L/min/mmHg (low sensitivity) to 3 L/min/mmHg (high sensitivity). A B, and C are the ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$ values for V̇A ∼ 5, 8, and 35 L/min, respectively; note that ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$ is independent of the mechanism producing the increase in alveolar ventilation and thus of mean $\mathrm{P}\mathrm{A}{(\mathrm{a})}_{{\mathrm{C}\mathrm{O}}_2}$. D: example of the effects of an increase in metabolism, produced by a muscular exercise, on the relationship between V̇A and $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$. The relationship shifts up and to the right during a muscular exercise in keeping with the level of V̇co₂. However, 1) $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ is maintained constant (or even decreased at high intensity exercise) due to the proportional increase in V̇A and V̇co₂, a fundamental tenet of exercise hyperpnea, 2) due to higher metabolism, the slope of the tangent (blue dotted lines) of the V̇A-$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ relationship (${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$), at a CO₂ set point of 40 mmHg, decreases during exercise. E: tangent values of the V̇A-$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ relationship (${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$) at the two different levels of metabolism shown in C, illustrating how much change in V̇A would be needed to keep $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ within a normal range at rest and in exercise: fluctuations of V̇A by ∼ 10 L/min can lead to changes in $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ between 37 and 44 mmHg during exercise (V̇co₂ = 2.3 L/min), whereas similar changes in $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ would be produced by fluctuations of less than 1 L/min at rest (V̇co₂ = 0.23 L/min). F: values of the slope of the tangent of V̇A-$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ relationship (${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$) as a function of V̇co₂ at two different levels of $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ (40 and 30 mmHg). Note that the Y axis, −d$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$/dV̇A, is expressed with positive values for clarity. ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$ decreases as a function of V̇co₂. $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$, partial pressure of CO₂ in the alveolar gas; $\mathrm{P}{\mathrm{a}}_{\mathrm{C}{\mathrm{O}}_2}$, ${\mathrm{P}}_{{\mathrm{C}\mathrm{O}}_2}$ in the arterial blood; $\mathrm{P}{\mathrm{I}}_{\mathrm{C}{\mathrm{O}}_2}$, inspired ${\mathrm{P}}_{{\mathrm{C}\mathrm{O}}_2}$; ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$, respiratory plant gain for CO₂; V̇A, alveolar ventilation; V̇co₂, carbon dioxide output.

${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$ is therefore a function of V̇A⁻². In other words, ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$ is low when V̇A is high but dramatically increases when V̇A decreases, as illustrated in Fig. 1, A and B. For instance, a decrease in V̇A by half (from 5 to 2.5 L/min) doubles $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ (from 40 to 80 mmHg), leading to an average change in $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ of 16 mmHg for every liter per minute change in alveolar ventilation. An additional reduction in V̇A by half (from 2.5 to 1.25 L/min), will again double $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$, i.e., from 80 to 160 mmHg; as a result, $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ now has changed by 64 mmHg for every liter per minute change in alveolar ventilation. As shown in Fig. 1B, the absolute value of the slope of the tangent of V̇A-$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ function (${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$) at V̇A of, for instance, 5, 2.5, and 1.25 L/min ($\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ of 40, 80, and 160 mmHg) averages 8, 32, and 128 mmHg/L/min, respectively, increasing dramatically as V̇A decreases.

This simple observation helps us to understand why a patient who is already hypoventilating (already hypercapnic) “requires” smaller decrements in alveolar ventilation than if normocapnic, to increase $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$. Conversely, a smaller increment in alveolar ventilation will be needed to bring $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ from 100 to 60 mmHg than from 50 to 40 mmHg. In the “hyperventilation” range, ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$ continues to decrease as V̇A increases, dropping for instance from 8 to 2 mmHg/L/min as V̇A rises from 5 to 10 L/min.

Of note, increasing inspired ${\mathrm{P}}_{{\mathrm{C}\mathrm{O}}_2}$ ($\mathrm{P}{\mathrm{I}}_{\mathrm{C}{\mathrm{O}}_2}$) has an intriguing effect on ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$. As illustrated in Fig. 1C, a higher $\mathrm{P}{\mathrm{I}}_{\mathrm{C}{\mathrm{O}}_2}$ does not alter the slope of the tangent of the V̇A-$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ function, as at any given level of V̇A, the ratio V̇co₂/V̇A remains unchanged regardless of $\mathrm{P}{\mathrm{I}}_{\mathrm{C}{\mathrm{O}}_2}$. During inhalation of CO₂, the V̇A-$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ relationship is shifted up by a value corresponding to $\mathrm{P}{\mathrm{I}}_{\mathrm{C}{\mathrm{O}}_2}$:

$$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}\text{\hspace{0.17em}= }[\text{K}·{\dot{\mathrm{V}}\text{CO}}_{\text{2}}·{\dot{\mathrm{V}}\mathrm{A}}^{-\text{1}}]+{\text{P}\mathrm{I}}_{\text{CO}}{}_{{}_2}.$$ (3)

However, any additional stimulation of V̇A resulting from an increase in mean $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ will displace the new ${\mathrm{P}}_{{\mathrm{C}\mathrm{O}}_2}$ set point on the V̇A-$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ relationship toward the “high ventilation range” (with respect to V̇co₂) leading to a reduced ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$_. Therefore, any increased ventilation produced by an increase in mean $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$, resulting from CO₂ inhalation, decreases ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$. In other words, for a given V̇co₂, ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$ is dictated by the absolute level of ventilation reached in response to an increased mean $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$. For instance (Fig. 1C), assuming that inhaling a given % in CO₂ results in a new $\mathrm{P}{\mathrm{a}}_{\mathrm{C}{\mathrm{O}}_2}$ set point of 50 mmHg, which would have increased V̇A to ∼ 20 L/min, ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$ would drop by 20 times (0.4 mmHg/L/min versus 8 mmHg/L/min during air breathing with a $\mathrm{P}{\mathrm{a}}_{\mathrm{C}{\mathrm{O}}_2}$ set point ∼ 40 mmHg and V̇A ∼ 5 L/min). This effect should not be confused with conditions wherein $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ increases due to hypoventilation, which is always associated to an increase in ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$ (Fig. 1, A and B). The consequence of this effect on periodic breathing (PB) is discussed in the following paragraph.

THE PLANT GAIN OF THE RESPIRATORY SYSTEM IS A RECIPROCAL FUNCTION OF EXERCISE INTENSITY: THE CHALLENGE OF MAINTAINING ISOCAPNIA (OR PRODUCING HYPOCAPNIA) WHEN V̇co₂ INCREASES

As illustrated in Fig. 1, D and E, increasing V̇co₂ during a muscular exercise shifts the V̇A-$\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ relationship up and to the right, in turn dramatically decreasing ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$ at any given CO₂ set point. Indeed, the fundamental characteristic of the ventilatory response to a muscular exercise, at least in humans, is to maintain $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ constant (i.e., V̇A increases in proportion to V̇co₂, Fig. 1D; 2–4), whereas hypocapnia develops during heavy exercise (5). When V̇co₂ increases in isocapnic condition, ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$ displays an inverse function of the gas exchange rate (Fig. 1, E and F). Any given change in V̇A, therefore, has a much lower impact on $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ during exercise than at rest. For example, at a $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ ∼ 40 mmHg, the first derivative of $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ over V̇A will reach ∼8 mmHg/L/min at rest versus 0.8 mmHg/L/min for a V̇co₂ of 2.5 L/min. Another way to look at this property is illustrated in Fig. 1E, which shows that for keeping $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ between 37 and 44 mmHg (within the normal range) V̇A must change by ∼ 1 L/min at rest versus ∼ 9 L/min during our chosen level of exercise. This also implies that very large changes in V̇A must be generated to produce hypocapnia during heavy exercise (5; Fig. 1F): it takes more than 30 L/min of increment in alveolar ventilation (and much more in terms of minute ventilation) to only decrease $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ from 40 to 30 mmHg during heavy exercise compared with ∼ 2 L/min at rest; as a result, $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ is more “immune” to fluctuations in ventilation during exercise than at rest. This property is illustrated in Fig. 1F, displaying the changes in RPG_CO2 as a function of exercise intensity; for a V̇co₂ = 3.4 L/min, ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$ would be ∼ 30 times less than at rest.

EFFECTS OF THE RECIPROCAL FUNCTION BETWEEN ALVEOLAR VENTILATION AND CO₂ PARTIAL PRESSURE ON PERIODIC BREATHING AT REST AND IN EXERCISE

One of the widely accepted mechanisms put forward to account for the generation of periodic breathing (PB) assumes that PB can be generated by self-sustained cyclic oscillations in arterial ${\mathrm{P}}_{{\mathrm{C}\mathrm{O}}_2}$ and in minute ventilation (6). As ventilation diminishes, $\mathrm{P}\mathrm{A}{(\mathrm{a})}_{{\mathrm{C}\mathrm{O}}_2}$ increases; this transient hypercapnia produces in turn a stimulation of breathing leading to a reversal of the changes in $\mathrm{P}\mathrm{A}{(\mathrm{a})}_{{\mathrm{C}\mathrm{O}}_2}$ depressing breathing again, and a new cycle of oscillations can be produced (6). In this model, the generation and maintenance of PB therefore depends on three fundamental components: 1) the sensitivity of the structures involved in the chemical control of breathing usually referred as “the controller gain,” which can be described in terms of changes ventilation produced by given changes in $\mathrm{P}{\mathrm{a}}_{\mathrm{C}{\mathrm{O}}_2}$ (7); 2) the gain of the controlled system (plant gain), i.e., how much $\mathrm{P}{\mathrm{a}}_{\mathrm{C}{\mathrm{O}}_2}$ changes for a given change in ventilation (${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$); and 3) the frequency response (delays and time constant) of the chemical regulation of breathing as well as of the systems eluting then storing CO₂ in the lungs, blood, and tissues_.

Although associated with a reduced ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$, hypocapnia is present in various types of PB including altitude-induced hypoxemia or in heart failure (8). Increasing $\mathrm{P}{\mathrm{I}}_{\mathrm{C}{\mathrm{O}}_2}$ blunts or even suppresses the production of PB (8–10). The mechanisms through which increasing $\mathrm{P}{\mathrm{I}}_{\mathrm{C}{\mathrm{O}}_2}$ could affect PB can be quite counterintuitive. Preventing/limiting the decrease in $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ when V̇A is at its peak could certainly reduce the magnitude of $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ oscillations. In addition, as discussed earlier, rising mean $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ has a stimulatory effect on ventilation that results in an additional reduction in ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$ (Fig. 1B), which, in and of itself, can blunt PB production (11). This “protective” effect of an increased ventilation (Fig. 1, A and C) has already been put forward as a mechanism of reduction in the propensity for central and cyclical sleep apnea following acetazolamide-induced hyperventilation for instance (11). Correcting hypocapnia by increasing $\mathrm{P}{\mathrm{I}}_{\mathrm{C}{\mathrm{O}}_2}$ could therefore produce an additional decrease in ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$ via the resulting additional elevation in ventilation, in turn impeding PB production (8–10).

PB also occurs during exercise in patients with chronic heart failure (CHF; 12, 13), producing large oscillations in ventilation with periods of ∼ 45–60 s and 10 L/min amplitude (14, 15). PB in cardiac patients is typically associated with a poor prognosis (14, 16). The dramatic reduction of ${\mathrm{R}\mathrm{P}\mathrm{G}}_{{\mathrm{C}\mathrm{O}}_2}$ as exercise intensity increases (Fig. 1F) makes the controller system more critically dependent on its response kinetics than at rest for generating self-sustained PB, since changes in $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ for a given change in ventilation are considerably decreased when compared with rest. Indeed, in unsteady conditions, the amplitudes of oscillations in ventilation are not only dictated by the amplitudes of the changes in $\mathrm{P}\mathrm{A}{(\mathrm{a})}_{{\mathrm{C}\mathrm{O}}_2}$ at the level of chemoreceptors, but also by the frequency response of V̇e response to $\mathrm{P}\mathrm{A}{(\mathrm{a})}_{{\mathrm{C}\mathrm{O}}_2}$ oscillations. As a first approximation, the time constant (τ) of the V̇e response to a change in $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ can be estimated in healthy adult subjects to be around 70 s; this figure was based on the studies of the V̇e response to $\mathrm{P}{\mathrm{a}}_{\mathrm{C}\mathrm{O}}{}_{{}_2}$ in response to sinusoidal, impulse, or step forcing [for review, see Ref (7)]. Considering that the ventilatory response to a given change in $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ can be described as a first order system (17), the amplitude (A) of ventilatory oscillations, resulting from oscillations in $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ can be written as

$$A=A_{\mathrm{S}\mathrm{S}}/{[1+2\text{π}·f·\text{τ})}^2{]}^{0.5}$$ (4)

where A_SS is the amplitude response in steady-state condition (17), f is the frequency of $\mathrm{P}{\mathrm{a}}_{\mathrm{C}{\mathrm{O}}_2}$ oscillations, and τ is the overall time constant of the V̇e response to $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ changes. Steady-state chemosensitive gains (A_SS) averaging 2.4 L/min/mmHg have been reported in patients with heart failure presenting PB versus 0.85 L/min/mmHg in patients with CHF with no PB (18). Even with such a high gain, the amplitude of V̇e oscillations would at best reach 0.40 L/min/mmHg in response to 45–60-s period blood gas oscillations (in keeping with Eq. 4). The combination of a very low plant gain during exercise and a low amplitude response of the chemical control system (in keeping with its frequency response and the periods of oscillations) certainly represents a major challenge to sustain PB during a muscle exercise, at least via the sole cyclic self-maintenance of oscillations in arterial ${\mathrm{P}}_{{\mathrm{C}\mathrm{O}}_2}$ and ventilation. The observation that primary large swings in V̇co₂ (and V̇o₂) out of phase with the changes in ventilation, are present during PB in exercise (19), changing the numerator of the alveolar gas equation (Eq. 1), could allow for a simple solution to this conundrum. Indeed, even if the mechanisms leading to the production of large swings in V̇co₂ and V̇o₂ during PB are not fully understood, out of phase oscillations in V̇o₂ and ventilation can certainly result in swings in $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ and thus could still fit with the general model of Cherniack and Longobardo (6). Alternatively, one can propose that the presence of large oscillations in V̇co₂ and V̇o₂ during PB (20), or one of their circulatory surrogates (19), could “force” ventilation to oscillate, regardless of any CO₂ error signal, akin to the still debated mechanisms of exercise hyperpnea (2–4). This could represent a mechanism of production of PB, revealed by exercise, whose role, if any, in resting conditions would need to be investigated.

In conclusion, the nature of the function linking V̇A and $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ affects the stability of the respiratory system: the capacity of ventilation to elute CO₂, for a given change in alveolar ventilation, is reduced in exercise, as well as during any stimulation of ventilation out of proportion of the pulmonary gas exchange rate (including stimulations produced by increased inspired CO₂), whereas it is enhanced by several folds during hypoventilation. This effect could be seen as a rudimentary mechanism contributing to $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ homeostasis during exercise-induced hyperpnea by rendering changes in $\mathrm{P}{\mathrm{A}}_{\mathrm{C}{\mathrm{O}}_2}$ more immune from fluctuations in ventilation than at rest, at the expense of a heavy ventilatory requirement, when intensity increases and hypocapnia develops.

DISCLOSURES

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

AUTHOR CONTRIBUTIONS

P.H. prepared figure; drafted manuscript; edited and revised manuscript; approved final version of manuscript.