Much research suggests that air pollutants lead to both acute and chronic health effects (Dockery and Pope, 1994; Anderson et al., 1997). Studies using models of intrapulmonary airways are an important research tool for predicting the fate of inhaled pollutants in human airways or for diagnostic purposes, especially when detailed flow patterns and particle deposition characteristics are of interest (Balashazy et al., 1996). Therefore, mathematical models of airway bifurcations are of great use to associated engineers and biologists. Bifurcation models are very useful for studies in which airway models with various shape are required to observe, for example, effect of bifurcation angle on flow dividing ratio, different flow patterns in successive airways depending on whether successive bifurcations are in the same plane or not, effect of carina shape on flow patterns, and particle deposition (Andrade et al., 1998; Heistracher and Hofmann, 1995; Lee et al., 2000).

Various types of model airways have been created for both experiments and numerical studies, and most are specifically constructed whenever different structures of model airways were necessary. Although these models often represent reasonable shapes, some have unrealistic structure such as symmetric branching, sharp carinas, or abrupt change of shape in the transition zone. Most of all, these models do not report information on detailed shape nor suggest a general mathematical description of airway structure (Isabey and Chang, 1982; Scherer and Haselton, 1982; Balashazy et al., 1996; Hideki et al., 2001; Lee and Lee, 2002; Zhang et al., 2002). A detailed review of previous bifurcation models was given by Hegedus et al. (2004).

Heistracher and Hofmann (1995) first suggested a general mathematical model for a physiologically realistic bifurcation geometry. In their model, the shape is automatically determined whether or not important geometric parameters are specified, taking into account the blunt shape of carina and smoothly transitioning from parent branch to daughter branches. However, unfortunately the inner surface coordinates in the transition zone are defined iteratively, not analytically. As discussed by Hegedus et al. (2004), this iterative process is a numerically sensitive process and hard to reproduce. To overcome this problem in the transition zone, Hegedus et al. (2004) suggested a mathematically rigorous way to round in the carinal region.

Smooth stitching of two daughter airways in the carinal region may be the most difficult part of the bifurcation model to express in a mathematically closed form and at the same time, it is open to various formulation possibilities. As Hegedus et al. (2004) point out, carinal rounding suggested in their study is not a perfect procedure, but instabilities can occur depending on chosen functions and input geometric parameters. Although the mathematical descriptions are quite different, both studies (Heistracher and Hofmann, 1995; Hegedus et al., 2004) used rounding circles to define the carinal ridge's shape. However, the carinal ridge's shape may not be well represented by this approach, such as when the minor daughter diameter is much smaller than the major one as is common in mammals with a monopodial branching structure (see the Results section for further discussion).

Observations of intrapulmonary airways show that the carinal ridge, dividing the transition zone into left and right parts, does not always start at the mid-position of the parent branch; it is shifted toward the smaller daughter branch. Previous models (Heistracher and Hofmann, 1995; Hegedus et al., 2004) set the carinal ridge starting point at the middle of parent branch, which produces unrealistic bifurcations when the bifurcation is highly asymmetric (see the Results section for further discussion). The degree of asymmetry varies substantially between species. For example, rat conducting airways are very monopodial with an average asymmetry factor (major to minor diameter ratio) as high as 2 (Raabe et al., 1976; Yeh et al., 1979; Phillips and Kaye, 1995), whereas in human, the branching is much more symmetric. Bifurcation models need to produce a reasonable structure not only for rather symmetric bifurcations but also for highly asymmetric ones. In addition, previous bifurcation models were not validated against real airways.

In this study, we suggest a new mathematical approach for general asymmetric bifurcations where shape of carinal ridge (inner transition zone) is defined at each airway cross-section and different formulas are introduced that handle highly asymmetric bifurcation. Furthermore, the carinal ridge starts at a position that realistically depends on the bifurcation asymmetry. Our model is improved so that it can produce reasonable airway structures ranging from symmetric to highly asymmetric. We also evaluated our bifurcation model by comparing it with airway image data. This technique can be used for computerized measurements of pulmonary airways, simulating airway structure for fluid flow or particle deposition simulations, as well as for evaluating other bifurcation models. Although we propose a new bifurcation model, the present study owes much to previous work (e.g., Heistracher and Hofmann, 1995; Hegedus et al., 2004) in which the bifurcation structure was very well developed. A glossary of terms is provided in Table 1.

h | y coordinate of boundary curve |

i | Equal to L for left branch, R for right branch |

x_{b} | x coordinate of the point where the boundary curve and carina circle meet |

x | x coordinates of the daughter toroid center in the main plane at angle ϕ_{I} |

ΔE_{i} | Error at the ith normal vector |

F | Objective function |

L_{i} | Distance between centerline of torus and boundary curve or distance between centerline of torus and carina circle |

L_{di,s} | Lengths of straight section of daughter airways |

L_{p,s} | Length of straight section of parent branch |

N_{p} | Number of normal vectors that were originated from parent branch |

N_{t} | Number of normal vectors that were originated from transition zone |

N_{L} | Number of normal vectors that were originated from left daughter branch |

N_{R} | Number of normal vectors that were originated from right daughter branch |

R_{di} | Radii of daughter branches |

R_{p} | Radius of parent airway |

R | Radii of curvature of the daughter airway toroids |

r_{c} | Radius of curvature of carina ridge in the main plane |

s_{i} | R − _{p}R_{di} |

β_{i} | Subtended angles of daughter airways |

ϕ | Angle at which two toroids divide into daughter branches |

α_{i} | Angle at which carina circle is in contact with two daughter branches |

α | Angle at which boundary curve meets carina circle |