To understand something about the transport of a gas across the capillary wall into the alveolar space, we begin with the simplest possible model. We suppose that a gas such as oxygen or carbon dioxide is dissolved in blood at some concentration U uniformly across the cross-section of the capillary. The blood is flowing along a capillary that is bounded by alveolar air space. The partial pressure of the gas in the alveolar space, Pg, is taken to be constant.

Consider a segment of the capillary, of length L, with constant cross-sectional area A and perimeter p. The total amount of the dissolved gas contained in the capillary at any time is A —(x, t)dx. Since mass is conserved, we have

A^ U(x,t) dx^ = v(0)AU(0,t) - v(L)A—(L,t) + p^ q(x,t) dt, (17.3)

d dt where v(x) is the velocity of the fluid in the capillary, and q is the flux (positive inward, with units of moles per time per unit area) of gas along the boundary of the capillary. This assumes that diffusion along the length of the capillary is negligible compared to diffusion across the capillary wall. Differentiating (17.3) with respect to L and replacing L by x gives the conservation law

Finally, if we assume that the flow velocity v is constant along the capillary, then using (17.2), we obtain

pDs(p - -A [Pg a where Dm = xDs/a, and x = p/A is the surface-to-volume ratio. Notice that Dm has units of (time)-1, so it is the inverse of a time constant, the membrane exchange rate.

In steady state (independent of time), the conservation law (17.5) reduces to the first-order, linear ordinary differential equation d—

Note that, as one would expect intuitively, the rate of change of — at the steady state is inversely proportional to the fluid velocity. Now we suppose that the concentration — at the inflow x = 0 is fixed at —0 (at partial pressure P0 = —0/a). In steady state, the concentration at each position x is given by the exponentially decaying function

If the exposed section of the capillary has length L, the total flux of gas across the wall is Q = p f^ qdx = vA[— (L) - —0], which is

Plotted in Fig. 17.3 is the nondimensional flux

Note that

Velocity (dimensionless)

Figure 17.3 Dimensionless transmural flux Q as a function of dimensionless flow velocity Drz from (17.9). m

in the limit DmL/v ^ to. Thus, an infinitely long capillary has only a finite total flux, as the dissolved gas concentration approaches the alveolar concentration along the length of the capillary.

Data on the diffusion of carbon dioxide from the pulmonary blood into the alveolus (Fig. 17.4) suggest that carbon dioxide is lost into the alveolus at an exponential rate, consistent with (17.7). Furthermore, because the solubility of carbon dioxide in water is quite high, the difference between the partial pressure for the entering blood and the alveolar air is small, about 5 mm Hg.

In contrast, the solubility of oxygen in blood is small (about 20 times smaller than carbon dioxide, see Table 16.1), and although the difference in partial pressures is larger, this is not adequate to account for the balance of oxygen inflow and carbon dioxide outflow. That is, if (17.10) is relevant, then a decrease in a by a factor of 20 requires a corresponding increase by a factor of 20 for the partial pressure differences to maintain similar transport. Thus, if this is the correct mechanism for carbon dioxide and oxygen transport, the difference P0 — Pg for oxygen should be about twenty times larger than for carbon dioxide. Since 104 — 40 ^ 20(45 — 40) (using typical numbers from Figs. 17.4 and 17.5), there is reason to doubt this model.

Second, the data in Fig. 17.5 suggest that the uptake of oxygen by the capillary blood is not exponential with distance, but nearly linear for the first third of the distance, where it becomes fully saturated. We consider a model of this below. First, however, we discuss the effects of blood chemistry on gas exchange, which was ignored in the above model.

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