Figure 1.
Oxygen transport in the lung, as a function of branching generation z.
The transition from convection (heavy lines) to diffusion (light lines) occurs at generation zcd, with at rest, moderate exercise, heavy exercise, and maximum exercise, respectively (Table 2, see Methods section). The bronchial ducts (z = 0–14) do not carry alveoli; the acinar ducts (z = 15–23) are lined with alveoli (circles). In the ducts with z≥zcd, O2 diffuses through the airways (diffusion coefficient of O2 in air, Da), crosses the alveolar membrane (air-erythrocyte barrier including the alveolar tissue and plasma; hatched; permeability W), and binds to an erythrocyte in the capillary.
Table 1.
Transport parameters to compute O2 currents at temperature 310K.
Figure 2.
O2 current across a 1/8 acinus (gas exchanger at rest).
The O2 current as a function of permeability, W, illustrates decreasing screening from right to left. Far right: diffusion-limited current; far left: reaction-limited current; dashed line: physiological permeability, Wp = 0.007 cm/s (Table 1). Four insets, in which the square networks are 2-dimensional schematic representations of 1/8 acinus membrane surface, illustrate how accessible regions (from left to right) change as permeability increases. When permeability is small (the inset at far left), the O2 diffusion paths of length Λ (green) is long, the oxygen concentration c(x) = ca, and the membrane is unscreened (red). As permeability increases, the O2 diffusion paths of length are shortened and the red/unscreened areas decrease (the insets from left to right). Symbols n.s., w.p.s., s.p.s., and c.s. denote no screening, weak partial screening, strong partial screening and complete screening, respectively. The calculations from Eqs. 4 and 5 are carried out for a 3-dimension acinus, and the fractal dimension, Df, is taken to be 3. Except for W, which is treated as a variable, the values of all structural and transport parameters in Eqs. 4 and 5 are taken from Table 1 and 2 for a 1/8 acinus (gas exchanger at rest, second column in Table 2).
Table 2.
Structural and physiological data to compute O2 currents and compare computed and experimental values.
Table 3.
The diffusion coefficients and solubilities of O2, CO, and CO2 in water and air at 300 K.
Figure 3.
Computed and experimental oxygen currents and pulmonary efficiencies.
(A) Computed and experimental O2 currents for the whole lung and (B) pulmonary efficiencies, at different levels of exercise (levels of exercise are defined in the Methods section). The currents are computed from Eqs. 4 and 5. The computed pulmonary efficiency is obtained from Eqs. 5 and 7a, and the experimental value of the efficiency is obtained from Eq. 7c. In this figure, we compute the oxygen current with the physiological permeability, so instead of treating W as a variable, we take W = Wp (Table 1). Other transport parameters in the equations are listed in Table 1. The structural parameters at different levels of exercise are listed in Table 2 (column 2, 3, 4, and 5). The fractal dimension, Df, is taken to be 3.
Figure 4.
Computed oxygen currents for the lung.
Currents for the whole lung are computed for variable W and compared with experimental values (dots). Regions A, B, and C show the effects of operating the lung at the physiological value of W, Wp, and far way away from Wp (see text). At moderate, heavy, and maximum exercise, the O2 partial pressure difference across the membrane remains essentially constant,
(Table 2). Accordingly, the respective currents are computed replacing the three
values by their average,
, which makes the currents in the no-screening regime,
, coincide. The currents are computed from Eqs. 4 and 5. Permeability, W, is treated as a variable. The transport parameters in the equations are listed in Table 1, and the structural parameters at different levels of exercise are listed in Table 2 (column 2, 3, 4, and 5). The fractal dimension, Df, is taken to be 3.
Figure 5.
Computed oxygen currents for alternative architectures.
Currents as a function of W are computed from Eqs. 4 and 5 for different fractal dimensions, Df, at fixed and side length of the gas exchanger (side length of 1/64 acinus, column 5 of Table 2). Other transport parameters in the equations are from Table 1.
Figure 6.
O2 currents, at rest, across other gas-exchanger models.
(A) Random-walk computations for largest and smallest specimen of eight 1/8 human acini, modeled as network of acinar ducts [21], [22]. (B) Finite-element computations for Sierpinski's plane-filling curves [32], as planar models of a 1/8 and 1/128 acinus (Df = 2, [4]). Both panels also show the respective currents from the renormalization method. Inset in (B): concentration field, (c(x)−cbβa/β,)/(ca−cbβa/β,), in the 1/8 acinus (column 2 of Table 2).