The supply of oxygen, nutrients or other substances from the source to the periphery of an extended biological system frequently involves branched structures. Typical examples are plant roots and branches, animal circulatory and respiratory systems, as well as river basins. A ramified geometry ensures rapid access to large exchange surfaces. In mammalian lungs, the acini comprise the last generations of airways where oxygen is transported by molecular diffusion in air and transferred to blood through the alveolar membrane.
In the first part of the talk, we consider stationary diffusion which can be described by the Laplace equation for the oxygen concentration inside the acinus, with a constant concentration at the entrance, and Robin boundary condition on the alveolar membrane. This partial differential equations (PDE) problem in the lung acinus can be mapped into a discrete problem on a finite skeleton tree of its three-dimensional branched structure. We describe an exact "branch by branch" calculation of the diffusional
(oxygen) flux on arbitrary tree structures. Its application to the respiratory processes gives an analytical description of the crossover regime governing the human lung efficiency.
In the second part of the talk, we discuss diffusion-weighted imaging of the lungs with hyperpolarized gases. In this case, one deals with nonstationary restricted diffusion in the presence of inhomogeneous magnetic fields. We present Monte Carlo simulations in a geometrical model of healthy and emphysematous acini.
Potential applications of these results to a reliable diagnosis of emphysema is discussed.
D. S. Grebenkov, M. Filoche, B. Sapoval, and M. Felici, Phys. Rev. Lett. 94, 050602 (2005).
D. S. Grebenkov, G. Guillot, and B. Sapoval, J. Magn. Reson. 184, 143-156 (2007).
(references and further information are available on