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# (p. 3) Respiratory Mechanics

Respiratory Mechanics
Chapter:
Respiratory Mechanics
DOI:
10.1093/med/9780190670085.003.0001
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date: 11 July 2020

The respiratory system consists of the lungs and the chest wall. The chest wall includes the rib cage and all the tissues and muscles attached to it, including the diaphragm. The function of the respiratory system is to remove carbon dioxide (CO2) from, and add oxygen (O2) to, the mixed venous blood that is pumped through the pulmonary circulation by the right ventricle. To do this, two interrelated processes must occur:

• Ventilation—the repetitive bulk movement of gas into and out of the lungs

• Gas exchange—several processes that together allow the respiratory system to maintain a normal arterial partial pressure of O2 and CO2

Ventilation can occur only when the respiratory system expands above and then returns to its resting or equilibrium volume. This is just a fancy way of saying that ventilation depends on our ability to breathe. Although for most people, breathing requires very little effort and even less thought, it’s nevertheless a fairly complex process. In fact, ventilation can occur only when sufficient pressure is applied to overcome two forces that oppose the movement of the respiratory system. The interaction of these applied and opposing forces is referred to as the “mechanics of ventilation,” or respiratory mechanics.

## Opposing Forces

### Elastic Recoil

If you were to watch lung transplant surgery or an autopsy, you would see that the lungs deflate when they’re taken out of the thoracic cavity. If you looked closely, you would also notice that the chest wall increases in volume once the lungs are removed. This occurs because the isolated lungs and chest wall each have their own resting or equilibrium volumes. As you can see from Figure 1.1, any change from these volumes requires an increasing amount of applied pressure. So, if you think about it, the lungs and the chest wall act just like metal springs. The more they’re stretched or compressed, the greater the amount of pressure needed to overcome their inherent elastic recoil.

Figure 1.1 When separated from each other, the lungs recoil inward and the chest wall expands outward to reach their individual equilibrium volumes (double-sided arrows). Any change from these volumes requires an increasing amount of applied pressure to balance increasing inward or outward elastic recoil (arrows). In this way, the lungs and chest wall act just like metal springs.

The elastic recoil of the lungs and chest wall has two sources:

• Tissue forces result from the stretching of so-called elastic elements—elastin and collagen in the lungs, and cartilage, bone, and muscle in the chest wall.

• Surface forces are unique to the lungs and result from the surface tension generated by the layer of surfactant that coats the inside of each alveolus.

#### Pressure–Volume Relationships

The elastic recoil of the lungs, chest wall, and intact respiratory system is commonly depicted by graphs that show the pressure needed to maintain a specific volume. To help you understand these volume–pressure curves, I first want to spend some time looking at the properties of the lung spring and the chest wall spring shown in Figure 1.1. The relationship between the length of each “spring” and the pressure needed to balance its elastic recoil (also known as the elastic recoil pressure) is shown in Figure 1.2. As you can see, as the lung spring is stretched, more and more applied pressure (PL) is needed. Similarly, increasing outward or inward pressure (PCW) is needed to lengthen or shorten the chest wall spring. Notice that the resting or equilibrium length of each spring is the point at which it crosses the Y-axis and applied pressure (and elastic recoil) is zero.

Figure 1.2 Relationship between pressure (P) and length (L) of the lung spring (PL), the chest wall spring (PCW), and the “respiratory system” (PRS). Note that, at any length, PRS is the sum of PL and PCW. The resting or equilibrium length is the point at which each line crosses the Y-axis and P = 0. Increasing outward (+) or inward (–) pressure is required to balance elastic recoil as the chest wall spring and the “respiratory system” are stretched above or compressed below their equilibrium lengths. The “respiratory system” reaches its equilibrium (EQ) length when the inward recoil of the lung spring is exactly balanced by the outward recoil of the chest wall spring.

Now let’s see what happens when we hook these two springs together in parallel (side by side). After all, the real lungs and chest wall are attached by a very thin layer of pleural fluid and function together as a single unit. Figure 1.2 shows that the elastic properties of this “respiratory system” are determined by the sum of its two individual pressure–length curves. In other words, at any length, the pressure needed to balance the elastic recoil of the “respiratory system” (PRS) is the sum of PL and PCW. Notice that the resting length of the “respiratory system” is the point at which the inward recoil of the lung spring is exactly balanced by the outward recoil of the chest wall spring and PRS is zero.

How can this possibly be relevant to pulmonary physiology? It turns out that the length–pressure curves of our springs are remarkably similar to the volume–pressure curves of the lungs, chest wall, and respiratory system. So if you understand the concepts shown in Figure 1.2, you’re well on your way to understanding everything you need to know about the elastic properties of the respiratory system.

Skeptical? Take a look at Figure 1.3, which shows the elastic recoil pressure of the respiratory system and its components at every volume between total lung capacity (TLC) and residual volume (RV). These curves are generated by having a subject relax his or her respiratory muscles at a number of different volumes while a shutter attached to a mouthpiece prevents exhalation.

Figure 1.3 Relationship between transmural pressure (PTM) and volume (V) of the lungs (plTM), chest wall (PcwTM), and respiratory system (PrsTM). Each curve shows how much outward (+) or inward () pressure is needed to balance elastic recoil at volumes between residual volume (RV) and total lung capacity (TLC). At any volume, PrsTM is the sum of PlTM and PcwTM. The resting or equilibrium volume is the point at which each line crosses the Y-axis and PTM = 0. The respiratory system reaches its equilibrium volume at functional residual capacity (FRC) when the inward recoil of the lungs is exactly balanced by the outward recoil of the chest wall.

At each volume, the pressure in the pleural space (PPL) and the airway (PAW) just proximal to the shutter are measured, and the transmural pressure (the internal or intramural pressure minus the external or extramural pressure) of the lungs (PlTM), chest wall (PcwTM), and respiratory system (PrsTM) are calculated (Figure 1.4).

Figure 1.4 The transmural pressure of the lungs (PlTM), chest wall (PcwTM), and respiratory system (PrsTM) is calculated by subtracting the pressure “outside” from the pressure “inside” each structure. Pressure at the body surface (PBS) is equal to atmospheric pressure and assigned a value of zero. Pleural pressure (PPL) is estimated by measuring the pressure in the lower esophagus (PES). When there is no air flow, alveolar pressure (PALV) and airway pressure (PAW) are equal.

Note that: (1) in the absence of air flow, PAW and alveolar pressure (PALV) are equal; (2) PPL is estimated by measuring the pressure in the esophagus (PES) with a thin, balloon-tipped catheter; and (3) pressure at the body surface (PBS) is normally atmospheric pressure (PATM), which is assigned a value of zero. It’s important to understand that these measurements must be performed under static (no-flow) conditions if they are to reflect only the pressure needed to overcome elastic recoil—but more about that later.

Look how much you already know about the elastic properties of the respiratory system. Just like in our spring model, the pressure needed to maintain the respiratory system at any volume is the sum of the elastic recoil pressures of the lungs and chest wall. The volume reached at the end of a relaxed or passive expiration (functional residual capacity; FRC) is the point at which the inward recoil of the lungs is exactly balanced by the outward recoil of the chest wall (PrsTM = 0). Until it reaches its equilibrium volume (PcwTM = 0), the outward recoil of the chest wall actually assists with lung inflation. At higher volumes, sufficient pressure must be applied to overcome the inward recoil of both the lungs and the chest wall. Below FRC, pressure must be applied to balance the increasing outward recoil of the chest wall.

### Viscous Forces

A spring is a great metaphor for elastic recoil, because it’s easy to understand that a certain amount of pressure is needed to keep it at a specific length. When we breathe, though, we have to do more than just overcome the elastic recoil of the respiratory system. We also have to drive gas into and out of the lungs through the tracheobronchial tree. This requires additional pressure to overcome both the friction generated by gas molecules as they move over the surface of the airways, and the cohesive forces between these molecules. Together, these are referred to as viscous forces.

#### Pressure–Flow Relationships

The best way to understand viscous forces is to think about blowing (or sucking) air through a tube (Figure 1.5A). Air will flow through the tube only if there’s a difference in intramural pressure (∆PIM) between its two ends. Just how much of a pressure gradient is needed depends on several factors, which are shown in this simplification of Poiseuille’s equation:

Figure 1.5 (A) The pressure gradient between the ends of a tube (P1 – P2) is determined by the rate of gas flow ($V˙$) and the radius (r) and length (L) of the tube.

(B) During laminar flow, gas moves in concentric sheets, and velocity increases toward the center of the airway.

(C) Chaotic or turbulent flow requires a much higher pressure gradient.

$Display mathematics$
(1.1)

Here, $V˙$ is the flow rate, and L is the length and r the radius of the tube. Don’t worry about memorizing this equation, because, believe it or not, you already know what it says. Think about it. It simply says that you have to blow or suck harder if you want to generate a high flow rate or if the tube is either very long or very narrow. The only thing you really need to remember is that radius is the most important determinant of the pressure gradient. It’s a lot harder to blow through a coffee stirrer than through a drinking straw!

Of course, the tracheobronchial tree is much more complex than a simple tube. The good news is that flow into and out of the lungs is governed by exactly the same principles. It’s important to recognize, though, that Equation 1.1 is true only when flow is laminar—that is, when gas moves in orderly, concentric sheets (Figure 1.5B). If flow is chaotic or turbulent (Figure 1.5C), ∆PIM varies directly with $V˙2$ and inversely with the 5th power of the airway radius. High flow, high gas density, and branching of the airways predispose to turbulent flow.

### Compliance and Resistance

Elastic recoil and viscous forces play a very important role in the mechanics of ventilation, so it’s helpful to be able to quantify them. Elastic recoil is most often expressed in terms of compliance (C), which is the ratio of the volume change (∆V) produced by a change in transmural pressure (∆PTM).

$Display mathematics$
(1.2)

Notice that compliance and elastic recoil are inversely related. When elastic recoil is high, a given pressure change produces a relatively small change in volume, and compliance is low. When elastic recoil is low, the same pressure change produces a much greater change in volume, and compliance is high. By definition, compliance is a static measurement. In other words, it can only be calculated in the absence of flow. Since compliance is the ratio of volume and transmural pressure, it is equal to the slope of the volume–pressure curves in Figure 1.3. Notice that respiratory system compliance is highest in the tidal volume range and decreases at higher volumes.

Viscous forces are quantified by resistance (R), which is the ratio of the intramural pressure gradient (∆PIM) and the resulting flow ($V˙$).

$Display mathematics$
(1.3)

When resistance is low, a small pressure gradient is needed to generate flow. When resistance is high, a larger pressure gradient is needed. Note that resistance is a dynamic measurement because it can only be calculated in the presence of flow.

Although it’s probably clear, I want to emphasize that ∆P is not the same in Equations 1.2 and 1.3. When used to calculate compliance, ∆P is the change in transmural pressure needed to balance elastic recoil. When calculating resistance, ∆P is the intramural pressure gradient needed to overcome viscous forces. The methods used to calculate the compliance and resistance of the respiratory system are discussed in Chapter 9.

## Applied Forces

At any time during inspiration and expiration, sufficient pressure must be applied (PAPP) to overcome the viscous forces (PV) and elastic recoil (PER) of the lungs and chest wall.

$Display mathematics$
(1.4)

Based on our previous discussion, PER is the transmural pressure of the respiratory system in the absence of gas flow, while PV is the intramural pressure gradient driving flow. Equation 1.2 tells us that PER is equal to the change in volume (∆V) divided by respiratory system compliance (CRS), and from Equation 1.3, we know that PV equals the product of resistance (R) and flow ($V˙$). So, we can rewrite Equation 1.4 as:

$Display mathematics$
(1.5)

This is called the equation of motion of the respiratory system. It tells us that at any time during the respiratory cycle, the applied pressure must vary directly with resistance, flow rate, and volume and inversely with respiratory system compliance.

The pressure required during inspiration is normally supplied by the diaphragm and the other inspiratory muscles. When they are unable to perform this function, pressure must be provided by a mechanical ventilator. Let’s look at how applied and opposing forces interact during both normal or spontaneous breathing and mechanical ventilation.

### Spontaneous Ventilation

#### Inspiration

Figure 1.6 shows how PPL, PALV, flow, and volume change throughout inspiration. Remember that the inspiratory muscles don’t inflate the lungs directly. Rather, they expand the chest wall, and lung volume increases because the visceral and parietal pleura are attached by a thin layer of pleural fluid. Pleural pressure is normally negative (sub-atmospheric) at end-expiration. That’s because the opposing elastic recoil of the lungs and chest wall pulls the visceral and parietal pleura in opposite directions, which slightly increases the volume of the pleural space and decreases its pressure. As the inspiratory muscles expand the chest wall, lung volume and lung elastic recoil increase. This causes a further drop in PPL, which reaches its lowest (most negative) value at the end of inspiration.

Figure 1.6 The change in pleural (PPL) and alveolar (PALV) pressure, flow, and volume during a spontaneous breath. Pressure at the mouth (PAW) remains zero (atmospheric pressure) during spontaneous ventilation.

At end-expiration, the respiratory system is normally at its equilibrium volume, and both PALV and PAW are zero (atmospheric pressure). As the inspiratory muscles expand the chest wall, the volume of the lungs increases faster than they can fill with air, and PALV falls. Since PAW remains zero, this produces a pressure gradient that overcomes viscous forces and drives air into the lungs. As the lungs fill with air, PALV rises until both it and air flow return to zero at the end of inspiration. Since flow is zero at end-expiration and end-inspiration, the tidal volume during a spontaneous breath (VT) depends only on the change in respiratory system transmural pressure and respiratory system compliance, as shown in this modification of Equation 1.2:

$Display mathematics$
(1.6)

Watch out! Make sure you don’t get confused by the differences in the pressures shown in Figures 1.3 and 1.6. Specifically, in Figure 1.3, PALV (PrsTM) and PPL (PcwTM) increase with lung volume, and PALV is always equal to PAW. In Figure 1.6, both PALV and PPL fall, and PALV and PAW are the same only at end-expiration and end-inspiration. These differences are due solely to the conditions under which the pressures are measured. Remember that the curves in Figure 1.3 are generated by having a subject relax his or her respiratory muscles at different lung volumes while a shutter prevents air from entering or leaving the lungs. The curves shown in Figure 1.6 represent “real-time” pressures during spontaneous breathing.

Since the respiratory muscles generate all the pressure (PMUS) required during inspiration, the equation of motion during spontaneous ventilation can be written as:

$Display mathematics$
(1.7)

#### Expiration

As gas leaves the lungs and the respiratory system returns toward its equilibrium volume, pressure is required only to overcome the viscous forces produced by air flow. In the absence of expiratory muscle activity, this pressure is provided solely by the stored elastic recoil of the respiratory system. Now lung volume falls faster than air can leave, and PALV rises above PAW (Figure 1.6). During such a passive exhalation, PALV and flow fall exponentially and reach zero only when the respiratory system has returned to its equilibrium position. As lung volume and elastic recoil fall throughout expiration, PPL also becomes less negative and gradually returns to its baseline value.

### (p. 13) Mechanical Ventilation

#### Inspiration

Mechanical ventilators apply positive (supra-atmospheric) pressure to the airway. In the absence of patient effort, the pressure supplied by the ventilator (PAW) during inspiration must at all times equal the sum of the pressures needed to balance elastic recoil and overcome viscous forces. During such a passive inflation, PER is equal to PALV, so the equation of motion becomes:

$Display mathematics$
(1.8)

Figure 1.7 shows plots of PAW, PALV, PPL, flow, and volume during a passive mechanical breath with constant inspiratory flow. An end-inspiratory pause is also shown, during which the delivered volume is held in the lungs for a short time before expiration begins. Since flow is constant, lung volume increases at a constant rate. If we assume that compliance doesn’t change during inspiration, there must be a linear rise in PALV (which equals Δ‎V/CRS). If we also assume that resistance doesn’t change, $PV(whichisR×V˙)$ will also be constant. Since it is the sum of PV and PALV, PAW must also rise at a constant rate. Pleural pressure increases throughout inspiration as the lungs are inflated and the visceral and pariental pleura are forced closer together. Pleural pressure becomes positive once the chest wall exceeds its equilibrium volume. Finally, as lung volume increases, there must be a progressive rise in lung transmural pressure (i.e., the gradient between PALV and PPL).

Figure 1.7 Schematic diagram of airway (PAW), alveolar (PALV), and pleural (PPL) pressure, flow, and volume versus time during a passive mechanical breath with constant inspiratory flow. Peak (PPEAK) and plateau (PPLAT) pressure and the pressure needed to balance elastic recoil (PER) and overcome viscous forces (PV) are shown.

Now let’s examine what happens during the end-inspiratory pause. When inspiratory flow stops and the inspired volume is held in the lungs, PAW rapidly falls from its maximum or peak pressure (PPEAK) to a so-called plateau pressure (PPLAT). This pressure drop occurs because there are no viscous forces in the absence of gas flow, and pressure is needed only to balance the elastic recoil of the respiratory system. In other words, PPLAT is simply PALV (and PER) at the end of inspiration. The difference between PPEAK and PPLAT must then be the pressure needed to overcome viscous forces (PV).

In Figure 1.7, I have put pressure on the Y-axis and time on the X-axis because that’s how pressure curves are shown on the ventilator interface, but it’s important to recognize that the same information can be displayed using volume–pressure curves like those shown in Figure 1.3. In Figure 1.8, I have removed the curve showing lung transmural pressure (PlTM) and added a curve showing the total pressure generated during a mechanical breath (PAW). Since the transmural pressure of the chest wall (PcwTM) and respiratory system (PrsTM) in Figure 1.3 are, in fact, PPL and PALV, they have been relabeled in Figure 1.8. Later in this chapter and in several subsequent chapters, this alternative view of the pressure changes during a mechanical breath will be used to examine the effect of inspiration and positive end-expiratory pressure on the change in PPL and PlTM.

Figure 1.8 Plots of lung volume (V) versus alveolar (PALV) and pleural (PPL) pressure between residual volume (RV) and total lung capacity (TLC). The total pressure supplied by the ventilator (airway pressure; PAW) during passive inflation and peak (PPEAK) and plateau (PPLAT) pressure are also shown. Note the similarities between Figures 1.3 and 1.7.

Figure 1.9 shows how PAW, PALV, PV, and PER are affected by changes in resistance, compliance, tidal volume, and flow rate. By now, it’s obvious that I like to use mechanical models, and I’m going to use another one to help you understand these pressure curves. This one consists of a balloon to represent the elastic elements of the respiratory system and a straw to simulate the airways of the lungs (Figure 1.10). Just like during mechanical ventilation, if you blow up a balloon through a straw, the pressure inside your mouth (PM) must always equal the sum of the pressures needed to overcome the elastic recoil of the balloon (PER) and the viscous forces of the straw (PV). In this model, PM is analogous to PAW in Equation 1.8.

Figure 1.9 The effect of changes in compliance, resistance, volume, and flow on the pressure required to balance elastic recoil (PER) and overcome viscous forces (PV) and on peak (PPEAK) and plateau (PPLAT) pressures. PER increases with a fall in compliance and an increase in tidal volume. PV increases with resistance and flow.

Figure 1.10 Balloon-and-straw model of the respiratory system. When inflating the balloon, the pressure in the mouth (PM) must always equal the sum of the pressures needed to overcome viscous forces (PV) and elastic recoil (PER).

Think about blowing up a balloon through a straw and look again at Equation 1.8. You would have to blow really hard (high PM) to overcome viscous forces (PV) if the straw were long and narrow (high R) or if you wanted to inflate the balloon very quickly (high $V˙$). If you put a lot of air in the balloon (high ∆V), or if the balloon were very stiff (low C), you would need a lot of pressure to overcome elastic recoil (PER).

Now let’s go back to Figure 1.9. Just like in our balloon and straw model, an increase in resistance or inspiratory flow increases PV and PPEAK without changing PER or PPLAT. When compliance falls or tidal volume rises, PPEAK increases with PER and PPLAT, but there is no change in PV. A decrease in resistance, flow, and volume, and an increase in compliance have just the opposite effects.

Figure 1.11 shows how a change in the profile of inspiratory flow affects PAW and PALV when all other parameters remain constant. Since resistance doesn’t change, PV varies only with the rate of gas flow. As we saw in Figures 1.7 and 1.9, if flow from the ventilator is constant (Figure 1.11A), there will be a progressive, linear rise in both PALV and PAW as lung volume and elastic recoil increase, but the difference between them (PV) won’t change. If flow falls but never stops during inspiration (Figure 1.11B), PV also falls, and PALV approaches but never equals PAW. When flow is initially very rapid and falls to zero (Figure 1.11C), PAW quickly reaches and then maintains its peak level. As flow decreases, PV falls to zero, and PALV increases in a curvilinear fashion until it equals PAW. Since tidal volume and compliance don’t change, PPLAT is the same in Figures 1.11A, 1.11B, and 1.11C, but notice that there’s a progressive fall in PPEAK. That’s because less and less pressure is needed as flow and PV fall. When end-inspiratory flow is zero, PPEAK equals PPLAT.

Figure 1.11 The pressure within the ventilator circuit (PAW) and alveoli (PALV), flow, and volume during positive pressure mechanical breaths with three different inspiratory flow profiles.

#### Expiration

Like spontaneous breathing, expiration is normally passive during mechanical ventilation, and gas flow is driven by the stored elastic recoil of the respiratory system. As shown in Figure 1.7, flow reaches zero, and PALV and PPL return to their baseline levels only when the entire tidal volume has been exhaled and the respiratory system has returned to its equilibrium position. Notice that PAW reaches zero well before PALV does. The reason for this, and its importance, will be discussed in Chapters 9 and 10.

#### Positive End-Expiratory Pressure

Recall that the respiratory system reaches its equilibrium volume when the elastic recoil of the lungs and the chest wall are equal and opposite (Figure 1.3). At that point, the respiratory system has no remaining elastic recoil, and PALV is zero (atmospheric pressure). Mechanical ventilators can, however, be set to increase the equilibrium volume by maintaining positive (supra-atmospheric) pressure throughout expiration. This is referred to as positive end-expiratory pressure (PEEP). Figure 1.12 shows that PEEP raises end-expiratory alveolar pressure, which increases PALV, PAW, and PPL throughout the entire mechanical breath. If we now switch to a volume–pressure curve (Figure 1.13), you can see how increasing end-expiratory alveolar pressure creates a new, higher equilibrium volume and increases end-inspiratory lung volume. PEEP is used to open or “recruit” atelectatic (collapsed) alveoli and is discussed many more times throughout this book.

Figure 1.12 Simultaneous plots of airway (PAW), alveolar (PALV), and pleural (PPL) pressure versus time during a passive mechanical breath with constant inspiratory flow before (A) and after (B) the addition of 5 cmH2O PEEP. Peak (PPEAK) and plateau (PPLAT) pressure and the pressure needed to balance elastic recoil (PER) and overcome viscous forces (PV) are shown.

Figure 1.13 Modification of Figure 1.8 showing that PEEP (arrow) creates a new equilibrium volume (EV) for the respiratory system and increases end-inspiratory volume (VEI).

Although it is usually applied intentionally, positive end-expiratory pressure can also occur as an unintended consequence of mechanical ventilation. When the time available for expiration (expiratory time; TE) is insufficient to allow the respiratory system to return to its equilibrium volume (with or without PEEP), flow persists at end-expiration, and elastic recoil pressure (PALV) exceeds PEEP. This additional alveolar pressure is called intrinsic PEEP (PEEPI) to distinguish it from intentionally added extrinsic PEEP (PEEPE). The sum of intrinsic and extrinsic PEEP is referred to as total PEEP (PEEPT). As shown in Figure 1.14, PEEPI further increases end-inspiratory and end-expiratory lung volume, PAW, PALV, and PPL. Intrinsic PEEP results from a process called dynamic hyperinflation, which will be covered in Chapter 10.

Figure 1.14 Plots of airway (PAW), alveolar (PALV), and pleural (PPL) pressure, flow, and volume with PEEPE of 5 cmH2O and PEEPI of zero (A) and 5 cmH2O (B). PEEPI occurs when there is insufficient time for complete exhalation. This is indicated by persistent flow at end-expiration (arrow). PEEPI further increases PAW, PALV, PPL, and end-inspiratory and end-expiratory lung volume. Alveolar pressure at end-expiration is the sum of PEEPE and PEEPI. EV is the equilibrium volume produced by PEEPE.

Notice in Figures 1.12 and 1.14 that when PEEPE or PEEPI is present, PPLAT is no longer equal to PER, which is the pressure needed to overcome the elastic recoil produced by the delivered tidal volume. Instead PPLAT equals the sum of PER and PEEPT, which is the total elastic recoil pressure of the respiratory system. This leads to an important modification of the equation of motion.

$Display mathematics$
(1.9)

#### Patient Effort During Mechanical Ventilation

Consider what happens when a patient makes an inspiratory effort during a mechanical breath. Now, a portion of the pressure needed for inspiration is provided by contraction of the respiratory muscles (PMUS). This means that we can rewrite the equation of motion once again:

$Display mathematics$
(1.10)

$Display mathematics$
(1.11)

Equation 1.11 simply tells us that the ventilator must provide less and less positive pressure as patient effort increases. As you can probably imagine, this can have major effects on PAW, PALV, PPL, flow rate, and volume. The importance of patient–ventilator interactions will be discussed in several subsequent chapters.

## Time Constant

Before leaving the discussion of respiratory mechanics, I want to cover one more topic, and that’s the time constant of the respiratory system. Let’s go back to the balloon and straw model of the respiratory system, but now let’s imagine that the balloon is inflated and you’ve covered the end of the straw with your thumb. Now take your thumb away and let the air come out. It’s probably easy to recognize that there are two factors that determine how fast the balloon deflates—the compliance of the balloon and the resistance of the straw.

If the elastic recoil of the balloon is high (low compliance) or the straw has a large diameter (low resistance), flow will be rapid and the balloon will empty quickly. If the balloon has little elastic recoil (high compliance) or the straw has a very small lumen (high resistance), flow will be slow, and the balloon will empty slowly.

The same is true for the respiratory system during passive expiration. In fact, it turns out that the product of respiratory system compliance and resistance determines how quickly expiration occurs. This is referred to as the time constant (τ‎) and has units of time (seconds).

$Display mathematics$
(1.12)

During passive expiration, the volume of inspired gas remaining in the lungs (V) at any time (t) is determined by the inspired (tidal) volume (Vi) and the time constant.

$Display mathematics$
(1.13)

In this equation, e is the base of natural logarithms (approximately 2.72) and t is the time from the start of expiration (in seconds). Fortunately, you don’t have to use or even remember this equation. All you have to remember is that during a passive expiration, approximately 37%, 14%, and 5% of the tidal volume remains in the lungs after 1, 2, and 3 time constants. Because expiratory flow also decays exponentially, an identical relationship exists between the initial (maximum) flow $(V˙i)$ and the flow $(V˙)$ at any time (t) during expiration.

$Display mathematics$
(1.14)