Fig 1.
Anatomic schematic of the airways of human lungs.
(A) Schematic illustration of conducting airways (blue box) and acinar airways (red box). The hierarchical airway network consists of dichotomous trees with 23 generations. The transferred air diffuses to capillaries enclosed in the alveoli, most of which are attached to the late generations of the airways. The airway lengths of the 16th-23rd generations are 1.33, 1.12, 0.93, 0.83, 0.7, 0.7, 0.7, and 0.7 mm in the order [12], and an average diameter of alveoli is 200 μm [34]. (B) Reduction in normalized airway diameter along with airway generation. The diameter reduction ratio is 0.79 in the conducting airways, whereas it shifts to 0.94 in the acinar airways. Note that Murray’s law for diffusion in insects, k = 0.71, cannot explain the acinar airways reduction ratio (red dashed line). Data were taken from Finlay [8] and Weibel [14].
Fig 2.
Schematic illustration of a trumpet model for acinar airways.
The acinar airways consist of eight generations of airways and alveoli. The acinar airways can be assumed as a bundle of rectangular channels with the identical cross-sectional area in the same generation. The sidewalls of the rectangular channels do not affect the vertical diffusion, so it can be assumed to be a single trumpet channel that expands like an exponential function involving the reduction ratio k and single channel length l.
Fig 3.
Cumulated alveolar surface area.
The circles denote the data obtained from [12], and the line is a spline interpolant Aa(x).
Fig 4.
Profiles of oxygen partial pressure and oxygen transfer rate.
(A) Temporal change in the distribution of the oxygen partial pressure for k = 0.9. (B) Profiles of the oxygen partial pressure for various diameter reduction ratios (k = 0.7, 0.9, and 1.1). (C) The dependence of the oxygen transfer to the blood capillaries on the airway depth. Note that the area below the curves is equal to unity by the definition of .
Fig 5.
Optimal diameter reduction ratio for maximizing the oxygen transfer rate per airway surface area.
(A) Oxygen transfer rate per surface area of acinar airways and alveoli versus diameter reduction ratio. The oxygen transfer rate per surface area peaks at a diameter reduction ratio of 0.94, for which the energy cost for transporting a given amount of oxygen is minimized, provided that the energy investment is proportional to the surface area of acinar airways. (B) The dependence of oxygen transfer rate and surface area of acinar airways on the diameter reduction ratio. The values are normalized by their maximum values, respectively.
Fig 6.
Dependence of optimal diameter reduction ratio on the alveolar membrane permeability.
(A) Oxygen transfer rate per surface area versus diameter reduction ratio for various permeability. (B) The optimal diameter reduction ratios for various permeability.