**Gas flow across a barrier is proportional to diffusing capacity and concentration gradient (Fick's law)**

Although, early on, physiologists debated whether the lung actively secretes O_{2} into the blood, we now know that the movements of both O_{2} and CO_{2} across the alveolar blood-gas barrier occur by simple **diffusion** (see __p. 108__). Random motion alone causes a net movement of molecules from areas of high concentration to areas of low concentration. Although diffusion per se involves no expenditure of energy, the body must do work—in the form of ventilation and circulation—to create the concentration gradients down which O_{2} and CO_{2} diffuse. Over short distances, diffusion can be highly effective.

Suppose that a barrier that is permeable to O_{2} separates two air-filled compartments (__Fig. 30-1__*A*). The partial pressures (see __p. 593__) of O_{2} on the two sides are P_{1} and P_{2}. The probability that an O_{2} molecule on side 1 will collide with the barrier and move to the opposite side is proportional to P_{1}:

**(30-1)**

The **unidirectional movement** of O_{2} in the opposite direction, from side 2 to side 1, is proportional to the partial pressure of O_{2} on side 2:

**(30-2)**

The **net movement** of O_{2} from side 1 to side 2 is the difference between the two unidirectional flows:

**(30-3)**

**FIGURE 30-1** Diffusion of a gas across a barrier.

Note that net flow is proportional to the difference in partial pressures, not the ratio. Thus, when P_{1} is 100 mm Hg (or torr) and P_{2} is 95 mm Hg (ratio of 1.05), the net flow is 5-fold greater than when P_{1} is 2 mm Hg and P_{2} is 1 mm Hg (ratio of 2).

The term **flow** describes the number of O_{2} molecules moving across the entire area of the barrier per unit time (*units:* moles/s). If we normalize flow for the area of the barrier, the result is a **flux** (*units:* moles/[cm^{2} ⋅ s]). Respiratory physiologists usually measure the flow of a gas such as O_{2} as the volume of gas (measured at standard temperature and pressure/dry; see __Box 26-3__) moving per unit time. V refers to the volume and is its time derivative (volume of gas moving per unit time), or flow.

The proportionality constant in __Equation 30-3__ is the **diffusing capacity** for the lung, D_{L} (*units:* mL/[min ⋅ mm Hg]). Thus, the flow of gas becomes

**(30-4)**

This equation is a simplified version of **Fick's law** (see __p. 108__), which states that net flow is proportional to the concentration gradient, expressed here as the partial-pressure gradient.

Applying Fick's law to the diffusion of gas across the alveolar wall requires that we extend our model somewhat. Rather than a simple barrier separating two compartments filled with dry gas, a wet barrier covered with a film of water on one side will separate a volume filled with moist air from a volume of blood plasma at 37°C (see __Fig. 30-1__*B*). Now we can examine how the physical characteristics of the gas and the barrier contribute to D_{L}.

Two *properties of the gas* contribute to D_{L}—molecular weight (MW) and solubility in water. First, the mobility of the gas should decrease as its molecular weight increases. Indeed, **Graham's law** states that diffusion is inversely proportional to the square root of molecular weight. Second, Fick's law states that the flow of the gas across the wet barrier is proportional to the *concentration* gradient of the gas dissolved in water. According to **Henry's law** (see __Box 26-2__), these concentrations are proportional to the respective partial pressures, and the proportionality constant is the solubility of the gas (*s*). Therefore, poorly soluble gases (e.g., N_{2}, He) diffuse poorly across the alveolar wall.

Two *properties of the barrier* contribute to D_{L}—area and thickness. First, the net flow of O_{2} is proportional to the **area (A)** of the barrier, describing the odds that an O_{2} molecule will collide with the barrier. Second, the net flow is inversely proportional to the **thickness (a)** of the barrier, including the water layer. The thicker the barrier, the smaller the O_{2} partial-pressure gradient () through the barrier (__Fig. 30-2__). An analogy is the slope of the trail that a skier takes from a mountain peak to the base. Whether the skier takes a steep “expert” trail or a shallow “beginner's” trail, the end points of the journey are the same. However, the trip is much faster along the steeper trail!

**FIGURE 30-2** Effect of barrier thickness.

Finally, a combined property of both the barrier and the gas also contributes to D_{L}, a proportionality constant *k* that describes the interaction of the gas with the barrier.

Replacing D_{L} in __Equation 30-4__ with an area, solubility, thickness, molecular weight, and the proportionality constant yields

**(30-5)**

__Equations 30-4__ and __30-5__ are analogous to **Ohm's law** for electricity:

**(30-6)**

Electrical current (I) in Ohm's law corresponds to the net flow of gas (); the reciprocal of resistance (i.e., conductance) corresponds to diffusing capacity (D_{L}); and the voltage difference (ΔV) that drives electrical current corresponds to the pressure difference (P_{1} − P_{2}, or ΔP).

**The total flux of a gas between alveolar air and blood is the summation of multiple diffusion events along each pulmonary capillary during the respiratory cycle**

__Equation 30-5__ describes O_{2} diffusion between two compartments whose properties are uniform both spatially and temporally. Does this equation work for the lungs? If we assume that the alveolar air, blood-gas barrier, and pulmonary-capillary blood are uniform in space and time, then the net diffusion of O_{2} () from alveolar air to pulmonary-capillary blood is

**(30-7)**

is the diffusing capacity for O_{2}, is the O_{2} partial pressure in the alveolar air, and is the comparable parameter in pulmonary-capillary blood. Although __Equation 30-7__ may seem sophisticated enough, a closer examination reveals that , , and are each even more complicated than they at first appear.

Among the five terms that make up , two vary both temporally (during the respiratory cycle) and spatially (from one piece of alveolar wall to another). During inspiration, lung expansion causes the surface area (A) available for diffusion to increase and the thickness of the barrier (a) to decrease (__Fig. 30-3__*A*). Because of these *temporal* differences, should be maximal at the end of inspiration. However, even at one instant in time, barrier thickness and the surface area of alveolar wall differ among pieces of alveolar wall. These *spatial* differences exist both at rest and during the respiratory cycle. **N30-1**

**FIGURE 30-3** Complications of using Fick's law.

**N30-1**

**Spatial Differences in Alveolar Dimensions**

*Contributed by Emile Boulpaep, Walter Boron*

The total area of the lungs is not distributed evenly among all alveoli. First, all else being equal, some alveoli are “naturally” larger than others. Thus, some have a greater area for diffusion than do others, and some have a thinner wall than do others.

Second, the position of an alveolus in the lung can affect its size. As we saw in __Chapter 27__, when a person is positioned vertically, the effects of gravity cause the intrapleural pressure to be more negative near the apex of the lung than near the base (see __Fig. 27-2__). Thus, other things being equal, alveoli near the apex of the lung tend to be more inflated, so that they have a greater area and smaller thickness compared to alveoli near the base of the lung.

Third, during inspiration, alveoli undergo an increase in volume that causes their surface area to increase and their wall thickness to decrease. However, these changes are not uniform among alveoli. Again, the differences can be purely anatomical: all other things being equal, some alveoli “naturally” have a greater static compliance (see __p. 610__) than others. Thus, during inspiration, their area will increase more, and their wall thickness will decrease more. However, other things being equal, the compliance of an alveolus also depends on its position in the lung. We will see in __Chapter 31__ that the relatively overinflated alveoli near the apex of the lung (in an upright individual) have a relatively low compliance. In other words, during inspiration these apical alveoli have a smaller volume increase (see __Fig. 31-5__*D*). Thus, their area for diffusion undergoes a relatively smaller increase, and their wall thickness undergoes a relatively smaller decrease, than alveoli near the base of the lung.

In summary, for all of the reasons we have discussed, the area and thickness parameters vary widely among alveoli at the end of a quiet inspiration, and the relation among these differences changes dynamically during a respiratory cycle.

Like area and thickness, alveolar varies both temporally and spatially (see __Fig. 30-3__*B*). In any given alveolus, is greatest during inspiration (when O_{2}-rich air enters the lungs) and least just before the initiation of the next inspiration (after perfusion has maximally drained O_{2} from the alveoli), as discussed on __page 676__. These are *temporal* differences. We will see that when an individual is standing, is greatest near the lung apex and least near the base (see __pp. 681–682__). Moreover, mechanical variations in the resistance of conducting airways (see __pp. 681–682__) and the compliance of alveoli (see __p. 597__ or __pp. 608–610__) cause ventilation—and thus (see __p. 610__)—to vary among alveoli. These are *spatial* differences.

As discussed below, as the blood flows down the capillary, capillary rises to match (see __Fig. 30-3__*C*). Therefore, O_{2} diffusion is maximal at the beginning of the pulmonary capillary and gradually falls to zero farther along the capillary. Moreover, this profile varies during the respiratory cycle.

The complications that we have raised for O_{2} diffusion apply as well to CO_{2} diffusion. Of these complications, by far the most serious is the change in with distance along the pulmonary capillary. How, then, can we use Fick's law to understand the diffusion of O_{2} and CO_{2}? Clearly, we cannot insert a single set of fixed values for , , and into __Equation 30-7__ and hope to describe the overall flow of O_{2} between all alveoli and their pulmonary capillaries throughout the entire respiratory cycle. However, Fick's law does describe gas flow between air and blood for a *single piece of alveolar wall* (and its apposed capillary wall) at a *single time* during the respiratory cycle. For O_{2},

**(30-8)**

For one piece of alveolar wall and at one instant in time, A and a (and thus ) have well defined values, as do and . The total amount of O_{2} flowing from all alveoli to all pulmonary capillaries throughout the entire respiratory cycle is simply the sum of all individual diffusion events, added up over all pieces of alveolar wall (and their apposed pieces of capillary wall) and over all times in the respiratory cycle:

**(30-9)**

Here, , , and are the “microscopic” values for one piece of alveolar wall, at one instant in time.

Even though the version of Fick's law in __Equation 30-9__ does indeed describe O_{2} diffusion from alveolar air to pulmonary-capillary blood—and a comparable equation would do the same for CO_{2} diffusion in the opposite direction—it is not of much practical value for *predicting* O_{2} uptake. However, we can easily compute the uptake of O_{2} *that has already taken place* by use of the **Fick principle** (see __p. 423__). The rate of O_{2}uptake by the lungs is the difference between the rate at which O_{2} leaves the lungs via the pulmonary veins and the rate at which O_{2} enters the lungs via the pulmonary arteries. The rate of O_{2} *departure from* the lungs is the product of blood flow (i.e., cardiac output, ) and the O_{2} content of pulmonary venous blood, which is virtually the same as that of systemic arterial blood (). Remember that “content” (see __p. 650__) is the sum of dissolved O_{2} and O_{2} bound to hemoglobin (Hb). Similarly, the rate of O_{2} *delivery to* the lungs is the product of and the O_{2} content of pulmonary arterial blood, which is the same as that of the mixed-venous blood (). Thus, the difference between the rates of O_{2} departure and O_{2} delivery is

**(30-10)**

For a cardiac output of 5 L/min, a of 20 mL O_{2}/dL blood, and a of 15 mL O_{2}/dL blood, the rate of O_{2} uptake by the pulmonary-capillary blood is

**(30-11)**

Obviously, the amount of O_{2} that the lungs take up must be the same regardless of whether we *predict* it by repeated application of Fick's law of diffusion (see __Equation 30-9__) or *measure* it by use of the Fick principle (see __Equation 30-10__):

**(30-12)**

**The flow of O**_{2}**, CO, and CO**_{2}** between alveolar air and blood depends on the interaction of these gases with red blood cells**

We have been treating O_{2} transport as if it involved only the diffusion of the gas across a homogeneous barrier. In fact, the barrier is a three-ply structure **N30-2** comprising an alveolar epithelial cell, a capillary endothelial cell, and the intervening interstitial space containing extracellular matrix. The barrier is remarkable not only for its impressive surface area (50 to 100 m^{2}) and thinness (~0.6 µm) but also for its strength, which derives mainly from type IV collagen in the lamina densa of the basement membrane (often <50 nm) within the extracellular matrix.

**N30-2**

**Three-Ply Structure of the Alveolar Barrier**

See the following review:

Maina JN, West JB: Thin and strong! The bioengineering dilemma in the structural and functional design of the blood-gas barrier. Physiol Rev 85:811–844, 2005.

One could imagine that, as O_{2} diffuses from the alveolar air to the Hb inside an erythrocyte (red blood cell, or RBC), the O_{2} must cross 12 discrete mini-barriers (__Fig. 30-4__). A mini-diffusing capacity (D_{1} to D_{12}) governs each of the 12 steps and contributes to a so-called **membrane diffusing capacity (D**_{M}**)** because it primarily describes how O_{2} diffuses through various membranes. How do these mini-diffusing capacities contribute to D_{M}? Returning to our electrical model (see __Equation 30-6__), we recognize that D is analogous to the reciprocal of resistance. Therefore, we can represent the 12 diffusive steps by 12 resistors in series. Because the total resistance is the sum of the individual resistances, the reciprocal of D_{M} is the sum of the reciprocals of the mini-diffusing capacities:

**(30-13)**

Of course, these parameters vary with location in the lung and position in the respiratory cycle.

**FIGURE 30-4** Transport of O_{2} from alveolar air to Hb. The 12 diffusion constants (D_{1} to D_{12}) govern 12 diffusive steps across a series of 12 barriers: (1) the interface between the alveolar air and water layer, (2) the water layer itself, (3–5) the two membranes and cytoplasm of the type I alveolar pneumocyte (i.e., epithelial cell), (6) the interstitial space containing the extracellular matrix, (7–9) the two membranes and cytoplasm of the capillary endothelial cell, (10) a thin layer of blood plasma (<0.2 µm in mammals), and (11, 12) the membrane and cytoplasm of the erythrocyte. θ ⋅ V_{c} describes how fast O_{2} binds to Hb.

For most of the O_{2} entering the blood, the final step is binding to Hb (see __p. 647__), which occurs at a finite rate:

**(30-14)**

θ is a rate constant that describes how many milliliters of O_{2} gas bind to the Hb in 1 mL of blood each minute, and for each millimeter of mercury (mm Hg) of partial pressure. V_{c} is the volume of blood in the pulmonary *capillaries.* The product θ ⋅ V_{c} has the same dimensions as D_{M} (*units:* mL/[min ⋅ mm Hg]), and both contribute to the overall diffusing capacity:

**(30-15)**

Because O_{2} binds to Hb so rapidly, its “Hb” term 1/(θ ⋅ V_{c}) is probably only ~5% as large as its “membrane” term 1/D_{M}.

For carbon monoxide (CO), which binds to Hb even more tightly than does O_{2} (see __pp. 654–655__)—but far more slowly—θ ⋅ V_{c} is quantitatively far more important. The overall uptake of CO, which pulmonary specialists use to compute D_{L} (see __p. 670__), depends about equally on the D_{M} and θ ⋅ V_{c} terms.

As far as the movement of CO_{2} is concerned, one might expect the D_{L} for CO_{2} to be substantially higher than that for O_{2}, inasmuch as the solubility of CO_{2} in water is ~23-fold higher than that of O_{2} (see __p. 593__). However, measurements show that is only 3- to 5-fold greater than . The likely explanation is that the interaction of CO_{2} with the RBC is more complicated than that of O_{2}, involving interactions with Hb, carbonic anhydrase, and the Cl-HCO_{3} exchanger (see __pp. 655–657__).

In summary, the movement of O_{2}, CO, and CO_{2} between the alveolus and the pulmonary capillary involves not only diffusion but also interactions with Hb. Although these interactions have only a minor effect on the diffusing capacity for O_{2}, they are extremely important for CO and CO_{2}. Although we will generally refer to “diffusing capacity” as if it represented only the diffusion across a homogeneous barrier, one must keep in mind its more complex nature.