Abstract
The space-filling fractal network in the human lung creates a remarkable distribution system for gas exchange. Landmark studies have illuminated how the fractal network guarantees minimum energy dissipation, slows air down with minimum hardware, maximizes the gas- exchange surface area, and creates respiratory flexibility between rest and exercise. In this paper, we investigate how the fractal architecture affects oxygen transport and exchange under varying physiological conditions, with respect to performance metrics not previously studied.We present a renormalization treatment of the diffusion-reaction equation which describes how oxygen concentrations drop in the airways as oxygen crosses the alveolar membrane system. The treatment predicts oxygen currents across the lung at different levels of exercise which agree with measured values within a few percent. The results exhibit wide-ranging adaptation to changing process parameters, including maximum oxygen uptake rate at minimum alveolar membrane permeability, the ability to rapidly switch from a low oxygen uptake rate at rest to high rates at exercise, and the ability to maintain a constant oxygen uptake rate in the event of a change in permeability or surface area. We show that alternative, less than space-filling architectures perform sub-optimally and that optimal performance of the space-filling architecture results from a competition between underexploration and overexploration of the surface by oxygen molecules.
Highlights
On its way from the trachea to blood in the lung, oxygen (O2) travels through 14 generations of branching ducts forming the bronchial airways; 9 generations of ducts forming the acinar airways, which end in 300 million alveoli; and across the thin walls separating alveoli and blood capillaries (Fig. 1; [1,2])
The metrics will be in terms of diffusive transport, local oxygen currents, and total oxygen currents
The line W ~Wp~0:007 cm=s intersects all four plateaus and runs almost through the ‘‘knee point’’ at maximum exercise. This optimizes switching from rest to exercise in extraordinary ways: If the lung operated in region C, it would have maximum fault tolerance over a maximum interval of permeabilities at maximum exercise, but waste more than 99% of its surface area due to massive screening (Equations 5b, 5c, 7a)
Summary
On its way from the trachea to blood in the lung, oxygen (O2) travels through 14 generations of branching ducts forming the bronchial airways; 9 generations of ducts forming the acinar airways, which end in 300 million alveoli; and across the thin walls separating alveoli and blood capillaries (Fig. 1; [1,2]) This architecture is a space-filling, fractal network at two levels (Fig. 1; [1,2,3,4]), each creating a remarkable distribution system: The spacefilling bronchial tree, in which transport is by convection, guarantees minimum dissipation (pressure-driven flow [5,6,7]), including the 3/ 4 power law for metabolic rates [8], and slows air down with a minimum number of ducts [4].
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