## INTRODUCTION

Models for lung airway morphometry have played many important roles in pulmonary studies of respiratory disease and in understanding the impact of environmental exposures. Morphometry-based models provide numerical constraints for the geometry of the pulmonary tree and, thus, define the setting in which environmental and therapeutic exposures take place. The models find use in space-filling algorithms that enable extrapolation of the observed geometry to the complete lung. Pulmonary morphometric models also enable an efficient data reduction of extensive datasets of airway segment dimensions into a much smaller set of parameters. Model parameters estimated from a sample of animals in a population provide geometric information about a “normative” animal as well as means of describing differences and variability within and between different subpopulations, including diseased and genetically modified animals.

The morphometry of mammalian conducting airways has been researched in numerous studies. We will present only a brief overview of these studies, focusing on two different approaches to modeling. The reader can find an extensive description on these studies in our previous paper (Einstein, 2008).

Early models for pulmonary morphometry were built on the concept of the pulmonary branch and airway generation to model geometry (Weibel, 1963; Horsfield et al., 1971; Phalen et al., 1978; Haefeli-Bleuer and Weibel, 1988; Kitaoka and Suki, 1997; Mauroy et al., 2004; Majumdar et al., 2005). In this framework, the branch is defined as an airway segment between two consecutive bifurcations, and a generation is the inclusive number of branches between a given branch and the trachea. For each generation, the airway geometry is summarized by the average dimension (diameter, length, branch angle, and gravity angle). An advantage of the generation-average analysis is that it is simple to carry out and it results in easy-to-understand summaries of the observed data. However, this model fails to properly categorize airways (particularly in monopodial airways) averaging across various subpopulations of branches. Averaging over many airways in a generation may also obscure severe geometry changes that occur in a small fraction of a generation's airways (Lee, 2011). Finally, the generation-average model does not address measurement error. Thus, the model describes the observed lung and not the true lung in the absence of measurement error, an oversight that may result in incorrect inferences from the observed lung to the entire lung.

Our previous work (Einstein et al., 2008) introduced an alternative approach that was inspired by parallel work on arterial geometry (Karau, 2001; Molthen et al., 2004). In contrast to generation-average analysis, where the terms “branch” and “segment” are equivalent, we define the term branch as a sequence of consecutive segments that begins with a bifurcation off a parent branch, following the largest daughter segment of each bifurcation, down to the terminal bronchioles. This definition of branch necessitates a development of two equations to describe the geometry of diameters: (1) an equation for the narrowing of the pulmonary diameter down a single branch and (2) an equation for the initial diameter of a daughter branch. Einstein et al. (2008) showed that this branch-based approach provides a simple and highly predictive means to capture the airway geometry.

A weakness of our original branch-based approach (Einstein et al., 2008) was that the model heavily relied on the concept of “self-consistency” (Fredberg and Hoenig, 1978; Fredberg, 1989). The advantage of the self-consistency assumption was that the geometry of the entire pulmonary tree could be estimated from a few so-called principal pathways and could be summarized by a very small number of parameters. The teleological argument for self-consistency in the airway tree is that such structures have been shown to adhere to optimized laws of fluid mechanics (Murray and Cecil, 1926a, b). However, the underlying self-consistency assumptions, if not true, may lead to an incorrect representation of the airway geometry. Our original approach (Einstein et al., 2008) did not offer much to assess the validity of the self-consistency assumptions—the assumptions were either validated by a visual assessment of plots (for the diameter narrowing equation) or were not validated at all (for the daughter branch diameter equation).

Moreover, our original “branch-based” approach did not account for measurement error. Historically, the initial research on pulmonary morphometry relied on only one lung cast of a female Long Evans rat (Raabe et al., 1976). Recent progress in imaging has led to an easier and spatially more extensive collection of airway data. These extensive data sets cover larger portions of the lung and, thus, are more dominated by airway segments on the periphery of the lung. The peripheral airway segments tend to have more substantial measurement error. The potential causes of the measurement error include lung casting, imaging, and image processing with relative errors being greater in smaller structures. Measurement errors can be categorized into two groups: (1) errors due to the difference between the measured and true dimensions of diameters, lengths, angles, and so on, and (2) errors due to completely missed segments. Both of these errors may be random and/or systematic. In developing a model for the airway geometry, one needs to first determine if the measurement errors have an impact on the model estimates. If they do, one needs to take the errors into account.

In this work, we present a new model for airway diameters of the entire pulmonary tree. Similar to our original approach (Einstein et al., 2008), it has two parts: (1) a model for diameter narrowing within a single branch and (2) a model for the initial diameter of a daughter branch. The first model, the model for diameter narrowing, is a natural extension of the previous self-consistency model. The new approach enables a test of the self-consistency assumption and presents a more general model that can be used when departures from self-consistency are found. To estimate this more general model, we introduce statistical methods that separate the true between-branch differences in geometry from within-branch variation and measurement error within the lung. We also introduce a model for the initial diameter of daughter branches. This model extends the estimation of daughter branch diameters from the main branch only (the original, main branch approach) to daughter branch diameters in the entire lung (complete lung models). Statistical methods are introduced for this model to account for measurement error, specifically for the occurrence of missed segments. Computational issues involved in estimating both models are discussed. The models are illustrated on data from two rats and three (two normal and one ozone-exposed) monkeys.

In Materials and Methods section, from Animal Care to Segmentation, Centerline Extraction, and Centerline Analysis sections, we describe laboratory and imaging procedures. Partitioning Segments into Branches section defines partitioning of lung segments into branches. Statistical Methods section is a brief overview of statistical methods (more technical aspects of the statistical methods are presented in Appendix).

Cast Data section describes the cast data. Models for Branch Diameter Narrowing section presents models for branch diameter narrowing. Irregularities at the beginning of the pulmonary tree subsection describes irregularities at the beginning of the pulmonary tree. Self-consistency model subsection presents the self-consistency model as previously developed in the literature. Visual assessment of the self-consistency assumption and mathematical definition of self-consistency Subsection shows that self-consistency is a graphical artifact that is violated by individual branches and defines necessary and sufficient conditions for self-consistency. Variance model subsection introduces a more general model that we call the variance model. This model allows for potential departures from self-consistency and provides a test for self-consistency. The result of the test, the estimates of the magnitude of the departure from self-consistency, and other important differences between the self-consistency and variance models are presented in Test for self-consistency and magnitude of departures from self-consistency subsection. Models for the Daughter Branch Diameter section deals with models for daughter/parent diameter ratios. Main branch model subsection presents a model based on data from the main branch. Problems with the main branch model subsection discusses problems with the main branch model. Complete lung models—effect of missed daughter branches subsection discusses challenges in the complete lung models due to presence of missed daughter branches. Complete lung models—Truncation model with constant diameter ratio and Complete lung models—Truncation model with nonconstant diameter ratio subsections introduce two solutions to the missed branch problem using truncated regression with either constant or nonconstant mean daughter/parent diameter ratios. Test for the constancy of mean daughter/parent diameter ratio subsection uses the results of the truncated regression with nonconstant mean ratios to test for the constancy of the ratios.