Diffusion-Reaction in Branched Structures: Theory and Application to the Lung Acinus D. S. Grebenkov, 1, * M. Filoche, 1,2 B. Sapoval, 1,2 and M. Felici 1 1 Laboratoire de Physique de la Matie `re Condense ´e, C.N.R.S. Ecole Polytechnique, 91128 Palaiseau, France 2 Centre de Mathe ´matiques et de leurs Applications, C.N.R.S. Ecole Normale Supe ´rieure, 94140 Cachan, France (Received 17 June 2004; published 11 February 2005) An exact ‘‘branch by branch’’ calculation of the diffusional flux is proposed for partially absorbed random walks on arbitrary tree structures. In the particular case of symmetric trees, an explicit analytical expression is found which is valid whatever the size of the tree. Its application to the respiratory phenomena in pulmonary acini gives an analytical description of the crossover regime governing the human lung efficiency. DOI: 10.1103/PhysRevLett.94.050602 PACS numbers: 05.40.Fb, 02.50.–r, 05.10.–a, 05.60.–k The supply of nutritive 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 respi- ratory systems, as well as river basins. A ramified geometry ensures capillary distribution to large exchange surfaces. In mammalian lungs, the bronchial tree is responsible for the convective transport of fresh air from the mouth to the gas exchange units, called pulmonary ‘‘acini’’. The acini com- prise the last generations of airways where oxygen is trans- ported by molecular diffusion in air and transferred to blood through the alveolar membrane [1]. The diffusion transport in branched structures has been addressed in several papers (see [2,3], and references therein). The important point here is that we consider the stationary diffusion described by the Laplace equation for oxygen concentration, with a finite absorption rate at the alveolar membrane. Recently, it has been shown that this partial differential equations problem (PDE) in a branched ge- ometry can be mapped into a discrete problem defined by random walks on a finite Cayley tree, obtained from the skeleton of the three-dimensional branched structure. Random walks figuring oxygen diffusion on this skeleton tree were used to compute the human acinus efficiency [4]. The very slow decrease of branch diameters into the acinus is known to be irrelevant for diffusional flow (the problem of extreme sensitivity of a branched system to hydrody- namic flow is discussed in [5]). In the present Letter, we describe an efficient ‘‘branch by branch’’ procedure providing an exact resolution of this discrete problem. The flux of particles diffusing on arbi- trary trees with partial absorption at the boundary is de- rived analytically. Its application to symmetric trees provides an exact explicit relation for this diffusional flux. The branch by branch approach can also be used for asymmetric trees and is applied to calculate the flux into the real pulmonary acinus described in [6]. The PDE problem for oxygen concentration is expressed by the Laplace equation c  0 with the mixed boundary condition at the partially absorbing surface @c @n  1  c; (1) where n is the normal to the surface. A fixed concentration c 0 is set at the source of diffusion. The parameter  is the ratio D=W of the oxygen diffusivity in air D and membrane permeability W [7]. In the healthy human lungs, the value of  is around 30 cm. It is of practical importance to know how the transport properties of the human acini depend on this physical (and physiological) parameter as, for ex- ample, pulmonary edema degrades the membrane perme- ability leading to a significant increase of . The oxygen flux through the membrane of total surface S is given by   W R cdS. The system behavior as a gas exchanger is well described by a quantity  called efficiency and de- fined as    Wc 0 S : (2) Therefore,  is a number between zero and one represent- ing the fraction of the surface which is active. It only depends on the physical parameter  and the morphology of the branched structure [7]. Let us introduce the discrete representation of this PDE problem. First, we note that the stationary diffusion with partial absorption at the boundary can be modeled by d-dimensional partially absorbed random walks on a lattice of parameter a [Fig. 1(a)]. In this frame, the mixed bound- ary condition (1) means that a particle hitting the boundary can be absorbed with probability , or reflected to its preceding position with probability 1  . The absorp- tion probability  is related to the parameter  of the equivalent PDE problem by the following relation:   1  =a 1 [8]. Dealing with thin channels of square profiles, there are 2d  2 directions to the boundary. Then, the total probability to be absorbed at one step is 2d  2=2d. So, d-dimensional random walks in a thin channel of ‘‘diameter’’ a can be considered as one- dimensional longitudinal walks with the following dy- namic ‘‘rules’’: being on an intermediate site k, the random particle can jump to the left (site k  1) with probability PRL 94, 050602 (2005) PHYSICAL REVIEW LETTERS week ending 11 FEBRUARY 2005 0031-9007= 05=94(5)=050602(4)$23.00 050602-1  2005 The American Physical Society