Gas Diffusion through the Fractal Landscape of the Lung: How Deep Does Oxygen Enter the Alveolar System?

We investigate oxygen transport to and across alveolar membranes in the human lung, the last step in the chain of events that takes oxygen through the bronchial airways to the peripheral, acinar airways. This step occurs by diffusion. We carry out analytic and numerical computations of the oxygen current for fractal, space-filling models of the acinus, based on morphological data of the acinus and appropriate values for the transport constants, without adjustable parameters. The computations address the question whether incoming oxygen reaches the entire available membrane surface (reaction-limited, unscreened oxygen current), a large part of the surface (mixed reaction/diffusion-limited, partly screened current), or only the surface near the entrance of the acinus (diffusion-limited, completely screened current). The analytic treatment identifies the three cases as sharply delineated screening regimes and finds that the lung operates in the partial-screening regime, close to the transition to no screening, for respiration at rest; and in the no-screening regime for respiration at exercise. The resulting currents agree well with experimental values. We test the analytic treatment by comparing it with numerical results for two-dimensional acinus models and find very good agreement. The results provide quantitative support for the conclusion, obtained in other work, that the space-filling fractal architecture of the lung is optimal with respect to active membrane surface area and minimum power dissipation. At the level of the bronchial tree, we show that the space-filling architecture provides optimal slowing down of the airflow from convection in the bronchial airways to diffusion in the acinar airways.

[1]  C D Murray,et al.  The Physiological Principle of Minimum Work: I. The Vascular System and the Cost of Blood Volume. , 1926, Proceedings of the National Academy of Sciences of the United States of America.

[2]  Alexander A. Maximow,et al.  A Textbook of Histology , 1935, The Indian Medical Gazette.

[3]  M. Delbruck,et al.  Structural Chemistry and Molecular Biology , 1968 .

[4]  U. Mark Kac,et al.  Probabilistic methods in some problems of scattering theory , 1974 .

[5]  H. Berg,et al.  Physics of chemoreception. , 1977, Biophysical journal.

[6]  B. Simon.  Functional integration and quantum physics , 1979 .

[7]  C. Sparrow.  The Fractal Geometry of Nature , 1984 .

[8]  N. G. Makarov,et al.  On the Distortion of Boundary Sets Under Conformal Mappings , 1985 .

[9]  P. Laszlo.  Preparative chemistry using supported reagents , 1987 .

[10]  P. Pfeifer.  2 – CHARACTERIZATION OF SURFACE IRREGULARITY , 1987 .

[11]  E R Weibel,et al.  Morphometry of the human pulmonary acinus , 1988, The Anatomical record.

[12]  Bruce J. West,et al.  FRACTAL PHYSIOLOGY AND CHAOS IN MEDICINE , 1990 .

[13]  S. Havlin,et al.  Fractals and Disordered Systems , 1991 .

[14]  E. Weibel.  Fractal geometry: a design principle for living organisms. , 1991, The American journal of physiology.

[15]  West,et al.  Complex fractal dimension of the bronchial tree. , 1991, Physical review letters.

[16]  Peter Pfeifer,et al.  Optimization of Diffusive Transport to Irregular Surfaces with Low Sticking Probability , 1994 .

[17]  Transfer to and across Irregular Membranes Modelled by Fractal Geometry , 1994 .

[18]  Sierpiński’s Space-Filling Curve , 1994 .

[19]  H. Sagan.  Space-filling curves , 1994 .

[20]  E. Weibel,et al.  Fractals in Biology and Medicine , 1994 .

[21]  R. Evershed,et al.  Mat Res Soc Symp Proc , 1995 .

[22]  Hans Frauenfelder,et al.  Complexity in proteins , 1995, Nature Structural Biology.

[23]  Catalyst design accounting for the fractal surface morphology , 1996 .

[24]  James H. Brown,et al.  A General Model for the Origin of Allometric Scaling Laws in Biology , 1997, Science.

[25]  B. Sapoval,et al.  A Simple Method to Compute the Response of Non-Homogeneous and Irregular Interfaces: Electrodes and Membranes , 1997 .

[26]  Paul Meakin,et al.  Fractals, scaling, and growth far from equilibrium , 1998 .

[27]  Can one hear the shape of an electrode? I. Numerical study of the active zone in Laplacian transfer , 1999 .

[28]  Nonstandard roughness of terraced surfaces , 2000, Physical review letters.

[29]  S. Gheorghiu,et al.  COUNTEREXAMPLES IN FRACTAL ROUGHNESS ANALYSIS AND THEIR PHYSICAL PROPERTIES , 2001 .

[30]  Bernard Sapoval,et al.  Smaller is better—but not too small: A physical scale for the design of the mammalian pulmonary acinus , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[31]  Albert Gjedde,et al.  The pathway for oxygen in brain. , 2003, APMIS. Supplementum.

[32]  B. Sapoval,et al.  Diffusional screening in the human pulmonary acinus. , 2003, Journal of applied physiology.

[33]  E. Weibel,et al.  An optimal bronchial tree may be dangerous , 2004, Nature.

[34]  Stefan Gheorghiu,et al.  Optimal bimodal pore networks for heterogeneous catalysis , 2004 .

[35]  M Filoche,et al.  Renormalized random walk study of oxygen absorption in the human lung. , 2004, Physical review letters.

[36]  S. Gheorghiu,et al.  Is the Lung an Optimal Gas Exchanger , 2005 .