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.
MATERIALS AND METHODS
Specific pathogen-free adult male Sprague–Dawley rats, approximately 9–10 weeks of age and weighing 300 ± 30 g, were obtained (Charles River Laboratory, Raleigh, NC). On arrival, all animals were allowed free access to food and water and were housed in suspended plastic cages with chipped bedding in rooms maintained at 21°C ± 2°C and 50% ± 10% relative humidity with a 12-hr light/dark cycle. The animal facility is accredited by the American Association for Accreditation of Laboratory Animal Care (AAALAC).
The three 6-month-old male Rhesus monkeys used for this study were born and raised at the California National Primate Research Center (CNPRC) under the provisions of the University of California at Davis and CNPRC Institute of Laboratory Animal Resources, conforming to practices established by AAALAC. Body weight for subjects 1, 2, and 3 were 1.74 kg, 1.30 kg, and 1.79 kg, respectively. Subjects 1 and 3 were controls, while subject 2 was exposed to ozone as part of an ozone exposure study. The ozone monkey was exposed to 0.5 ppm ozone for 6 hr/day for 5 days, followed by 9 days of filtered air. This sequence of 5 ozone exposures followed by 9 exposures to filtered air was repeated for 11 cycles. Prior to preparation of lung casts, the animals were sedated with Telazol (8 mg/kg IM), and anesthetized with Diprivan [0.1–0.2 mg/(kg min), IV], with the dose adjusted as necessary by the attending veterinarian. They were then euthanized with an overdose of pentobarbital followed by exsanguination through the abdominal aorta.
All animal protocols were approved by the Institutional Animal Care and Use Committees at Pacific Northwest National Laboratory and the University of California at Davis, and studies were performed in accordance with the National Research Council (NRC) guidelines for the care and use of laboratory animals (NRC, 2011).
Lung Cast Preparation
Rat lung cast preparation
The procedure for creating rigid casts of rat lungs in situ is based on the work of Phalen et al. (1978). Rats were sacrificed with CO2 asphyxiation; then a 14 gauge catheter tube was surgically inserted just below the larynx. The skin covering the thorax was resected, and several slits were made in the intercostal muscles penetrating into the thoracic cavity. The lungs were then inflated with saline to a pressure of ∼ 25 cm H2O to open the airways and to remove residual gas. Approximately 5 mL of a combination of ∼ 90% silicone-based 5-min epoxy and ∼ 10% low viscosity silicone oil (Dow Corning 200® Fluid, 20 cst) was mixed thoroughly and degassed under vacuum. The mixture was drawn into a syringe that was then attached to a valve at the trachea tube. The epoxy mixture was then slowly pushed into the lung. The amount of casting material used was determined by measurements of the pulmonary dead space volume and not by pressure. From 3He MR images of airway trees, we calculated that a typical volume of casting material needed for a 330 g Sprague–Dawley rat is ≈1.2 cc. Once the casting material was injected, the valve was closed, and the mixture was allowed to cure in situ for ≈24 hr. The lungs were then carefully removed en bloc from the chest cavity and immersed in undiluted, household liquid bleach for 4–8 hr, until all the tissue was removed. Casts were gently rinsed with water and allowed to air-dry prior to imaging.
Monkey lung preparation
Postexsanguination, the lungs were immediately inflation-fixed via tracheal cannula for 4 hours at 30 cm fluid pressure with 1% glutaraldehyde/1% paraformaldehyde in cacodylate buffer (adjusted to pH 7.4, 330 mOsm). Lung volume was measured by fluid displacement after fixation and lungs were stored in fixative at 4°C until casting. Before casting, lungs were removed from fixative and rinsed in phosphate buffered saline. Airway casts were made using a modification of the negative-pressure injection technique of Perry et al. (2000). Briefly, Dow Corning® 734 Flowable Sealant (Dow Corning, Midland, MI) and Dow Corning 200® Fluid, 200cs (Dow Corning) were mixed in a heat-sealed bag until a homogenous, bubble-free mix was obtained. The lung was suspended by the tracheal cannula in a modified desiccator attached to a vacuum pump. The silicon mixture was pulled into the lungs by negative pressure (−100 mmHg) until the first sight of silicone at the pleural surface (10–20 min). The desiccator was then slowly brought back to atmospheric pressure. The silicone in the lung was allowed to polymerize overnight, and then excess tissue was removed by soaking in bleach. The silicone cast of airways was rinsed thoroughly in water and air dried.
Rat CT imaging
Rat lung casts underwent X-ray microcomputed tomography using a Skyscan 1076c in vivo microCT scanner. Casts were scanned at 35-μm resolution using a source setting of 50 kV without any filter. Individual images were sequentially captured using a stepped rotation of 0.6 degrees with no frame averaging. Raw scan data were converted into slice datasets using Nrecon reconstruction software (Skyscan, Belgium) using the following settings: 30% beam hardening, 8% ring artifact reduction and grayscale conversion.
Monkey MR imaging
Casts for Monkeys 1 and 2 were imaged using a 2 T Varian UnityPlus MRI spectrometer (Bruker Instruments, Fremont, CA). To facilitate cast visualization, each sample was mounted inside the magnet bore on a translation stage so it could be moved vertically between successive three-dimensional (3D) acquisitions. Generally, this was necessary as each cast was longer than the viewable imaging region. Each 3D dataset was collected on a 256 × 256 × 256 matrix in 3.5 hr using a standard spin-echo sequence with two averages, a 12-ms echo time, and a 100-ms repetition time. Typically, three separate 3D datasets were required to cover the entire cast. Afterward, each was Fourier reconstructed on a 512 × 512 × 256 matrix showing a 3.2 cm × 3.2 cm × 3.2 cm field of view. Images were then stored as 256 separate slices, and after all data were reconstructed, data from multiple image sets were concatenated to form a single stack of contiguous, two-dimensional slices.
Monkey CT imaging
The lung cast from Monkey 3 was imaged by X-ray computed tomography using a Metris XT H 255 CT scanner. Casts were scanned at 46-μm isotropic resolution using a source setting of 80 kV and 320 μA. In total, 3,142 projections were acquired with 360 degrees of rotation at 4 frames per projection for 1,000 ms of exposure.
Segmentation, Centerline Extraction, and Centerline Analysis
Our approach to segmentation (Carson et al., 2010), centerline extraction (Jiao et al., 2009, 2010) and centerline analysis (Einstein et al., 2008) can be found in the relevant references. Figure 1 illustrates the result of the rendering for Monkey 1 and Rat 1.
Partitioning Segments into Branches
All imaged airway segments were partitioned into branches. We define branch as an assemblage of segments, spawning possibly multiple offshoots, that runs from a bifurcation (or n-furcation in general) off of a parent branch down to the terminal bronchioles (Einstein, 2008). Branches were assembled from segments according to the following simple recursive definition: (1) the trachea is the first segment of the main branch; (2) at each junction, the daughter segment with the largest hydraulic diameter is added to the branch. If several daughter segments are tied for the largest hydraulic diameter, the longest daughter segment is chosen among the tied daughter segments. The remaining daughter segments at that junction are taken as the beginning of new, daughter branches. For example, the main branch is defined as an assemblage of consecutive segments of the lung tree starting at the trachea segment and continuing with the widest daughter segment at each junction (or the widest daughter segment if there is a tie). Figure 2 illustrates the main branch for one of the monkeys analyzed in this work. This definition extends to both binary and nonbinary trees. We would like to emphasize that the term branch used in this analysis has a different meaning than in generation average, where the term branch is used interchangeably with the term segment. Work by Einstein et al. (2008) showed that our definition is valuable in modeling some important morphometric relationships.
Nonlinear least squares regression, nonlinear mixed effects model, and truncated regression were used to fit various models to the data, such as the models for diameter narrowing. Nonlinear least squares regression models were fit with the Gauss–Newton algorithm (Lindstrom, 1990); nonlinear mixed effects models were fit using restricted maximum likelihood with the adaptive Gauss–Hermite approximation to the log-likelihood (Pinheiro, 1995); and truncated regression models (Greene, 2003; Davidson and MacKinnon, 1993) were estimated by the maximum likelihood methods (Newton–Raphson algorithm).
All computations were implemented in the R programming environment version 2.12.0 (Vienna, Austria), except for empirical Bayesian estimates of parameters of nonlinear mixed models, which were estimated with WINBUGS version 1.4.3 (Cambridge, UK), and for the truncated regression, which was carried out in STATA version 10.1 (College Station, TX). Package lme4 in the R programming environment was used for fitting nonlinear mixed models (Bates and Sarkar, 2006).
To avoid burdening the text with a full description of statistical methods, certain technical parts of the methodology are presented in more detail in Appendix.
The segmented data in the five 3D images consisted of 29,940 segments for Rat 1, 37,685 segments for Rat 2, 11,212 segments for Monkey 1, 17,803 segments for Monkey 2, and 24,172 segments for Monkey 3. All segments were partitioned into branches (by the recursive definition described in Partitioning Segments into Branches section), with Rat 1 containing 17,415 branches, Rat 2: 22,555 branches, Monkey 1: 6,213 branches, Monkey 2: 9,754 branches, and Monkey 3: 13,005 branches. The majority of branches were peripheral branches that consisted of a small number of segments (usually one or two segments).
To provide reasonably stable estimates of model parameters for branch tapering as a function of distance from the beginning of the branch [Eqs. (1)–(3) below] and to obtain convergence of parameter estimates in model fitting, only branches with 12 or more segments were used for fitting diameter-narrowing models. This restriction was not applied for the daughter branch diameter models [Eqs. (4) and (5)], for both (1) models for daughter segments off the main branch [Eq. (4)] or (2) models for daughter diameters where all parent–daughter pairs were included [Eq. (5)].
Models for Branch Diameter Narrowing
Irregularities at the beginning of the pulmonary tree
For the two rats, the dimensions of the first few segments of the main branch (including the trachea) and of the first offshoot of the main branch revealed an initial increase in airway diameter (see, e.g., Rat 1 in Appendix Fig. A1). These first few segments were excluded from our modeling and constitute a special case not addressed here. These irregularities in the diameters of first segments in the main branch and its first offshoot branch were absent in the three monkeys.
We initially fit a power curve model that was introduced by Karau et al. (2001) for pulmonary arterial morphometry and later applied to the pulmonary tree (Einstein et al., 2008).
A single branch
To describe a common geometry for the tapering of diameter for all branches, the model predicts the diameter D of a single branch as a power function of the distance x from the beginning of the branch:
where D0 is the branch diameter at x = 0; L is the total length of the branch; and c is the exponent of the power curve.
When c < 1, the curvature of the taper curve (x vs. diameter) is convex to the x-axis, that is, the diameter gets smaller at an increasing rate for each fixed increment of x. When c = 1, the curve of diameter versusx is linear. When c > 1, the curve is concave to the x-axis, that is, the rate of diameter narrowing decreases with x. The error term ε in Eq. (1) is the deviation of the observed diameter from a perfect fit to the model, due to either true biological variation or measurement error. The term ε is assumed to have a distribution with a mean of zero and a standard deviation of σε. In the nonlinear regression terminology, the standard deviation of the ε distribution is referred to as the standard error of the estimate (SEE) (σε) and describes the “noise” around the fitted power curve fit.
To extend the single branch model [Eq. (1)] to a model for all branches, Karau considered the following model:
where D0, L, c, and ε are the same as in Eq. (1) and are assumed to be parameters that are common to all branches; si is a shift in path length along the main branch to the location where the main branch expected diameter is equivalent to the initial expected diameter of the ith branch. Heuristically, it is as if the ith branch could be grafted onto the main branch at a distance si down the main branch, and the starting diameter of branch i would match the diameter of the main branch at that point. In effect, the model assumed that the subsequent tapering of branch i would match that of the main branch. The term self-consistency comes specifically from this behavior.
Visual assessment of the self-consistency assumption and mathematical definition of self-consistency
As has been illustrated by Karau et al. (2001) and by Einstein et al. (2008), the self-consistency assumption can be visually interpreted on the graph of xversusD: when the curve for the ith branch is shifted horizontally by si, it will (according to the model) coincide with the curve for the main branch. Figure 3A,C shows the graph of x + siversusD for Rat 1 and Monkey 1, respectively, along with the fitted self-consistency curve from Eq. (2). On initial examination, the fitted curves look very reasonable and suggest that there may be self-consistency among the branches. The problem with this type of display, however, is that the overlay of a cloud of points from various branches conceals details from the individual branches that are inconsistent with self-consistency.
When individual branches are plotted separately, the plot shows important differences among the shapes of the diameter curves for the individual branches. These differences are illustrated on curves for selected branches (Fig. 3B for Rat 1 and Fig. 3D for Monkey 1). The three branches in each plot display a very different curve of diameter tapering, including both concave and convex patterns. As we will show in Test for self-consistency and magnitude of departures from self-consistency subsection, these curves do not differ from each other by mere noise. The problem with visual assessment of the self-consistency assumption using displays such as Fig. 3A,C is that the graphs are driven mainly by the main branch and a few longer branches, giving the appearance that the points lie along the fitted self-consistency curve. However, when curves for other individual branches are isolated, they reveal quite different curve shapes (as shown in Fig. 3B,D). In Variance model subsection, we outline a formal approach to quantify the differences in the curvature of tapering among the diversity of branches. This approach relies on the following mathematical formulation of self-consistency.
Self-consistency of the diameter-tapering curve across different branches can be mathematically formulated by equality of two branch-specific parameters in Eq. (1). It can be shown (see Appendix) that a necessary and sufficient condition for self-consistency across different branches is that parameters c and l = Lc/D0 from Eq. (1) are constant across branches. The parameter c was described in Self-consistency model subsection. The parameter l captures the total length (L) of a given branch relative to the initial diameter (D0) of the branch, but it also incorporates the exponent c. Thus, there are two possible inconsistencies among branches: the inconsistency in c and the inconsistency in l. Both are illustrated in Fig. 4 that shows curves for three hypothetical branches. The inconsistency in l is illustrated on branches 1 and 2. Both branches have the same convex power exponent c (c1 = c2), but branch 2 has larger l than branch 1 (l2 > l1) reflecting that branch 2 is longer than branch 1. Inconsistency in c is illustrated by branch 3, which has a larger, concave power exponent c (c3 > 1) compared to branches 1 and 2. (Both branches 1 and 2 have an identical convex power exponent c; c1 = c2 < 1.)
The parameterization introduced in the previous paragraph leads to the following model, which incorporates potential nonconstancy in the parameters c and l across branches:
This model is derived from Eq. (1) and the fact that . The parameters D0i, li, and ci are branch-specific (the subscript i denoting the branch). Parameters D0i capture the initial diameter of each branch and are of interest in modeling the daughter–parent diameter relationship described later in this article. Under self-consistency, parameters li and ci are constant across branches. If, on the other hand, departures from self-consistency are present, they can be quantified by the amount of variation in li and ci. The result is a more general model that allows for a different diameter-narrowing curvature in different branches. The fit of Eq. (3) is performed by fitting a nonlinear mixed model that uses a log-value parameterization summarized in Appendix.
Test for self-consistency and magnitude of departures from self-consistency
Parameter estimates for the self-consistency model [Eq. (2)] and the variance model [Eq. (3)] are presented in Table 1. The estimates of σβ and σγ from fitting Eq. (3) to each segmented and partitioned lung cast dataset determine the level of self-consistency in log(l) and log(c) across branches. With true self-consistency σβ and σγ would be zero. To test for self-consistency, we calculated the 95% confidence intervals for σβ and σγ (see Appendix for details) and examined if they contained zero. For all five animals, the estimates of the σγ and σβ parameters in the variance model suggest very substantial variation in the model parameters c and l across branches. The 95% confidence intervals for σγ and σβ show that this magnitude of variation in c and l is not compatible with the self-consistency model (self-consistency model implies that σγ and σβ are equal to zero). The σγ value was around 0.30 for the two rats, whereas it was over 0.60 for the three monkeys (1.09 for the ozone-exposed Monkey 2 was the largest value). These values of σγ mean that for each of the two rats, it was common for the branch-specific parameter ci to differ by (e0.30 − 1) × 100% = 35% from the mean ci, and for the three monkeys, the “typical” departures in ci were even larger: 129% from the mean ci for Monkey 1, 197% for Monkey 2, and 84% for Monkey 3. Substantial between-branch heterogeneity in the parameter li was found too. Again, the between-branch heterogeneity was more substantial for the three monkeys than for the two rats (the largest between-branch heterogeneity in li being in Monkey 2).
Table 1. Estimated self-consistency and variance models
In the variance model mean log(c), mean log(l), SD of log(c), SD of log(l), and correlation of log(l) and log(c) correspond to μγ, μβ, σγ, σβ, and ρβγ in Eq. (3) parameterization.
# Branches/# segments
0.31 (0.25, 0.36)
0.35 (0.29, 0.40)
0.68 (0.57, 0.78)
1.34 (1.10, 1.53)
0.51 (0.44, 0.57)
0.08 (−0.09, 0.28)
−0.06 (−0.20, 0.10)
−0.24 (−0.63, 0.18)
−0.89 (−1.22, −0.55)
−0.46 (−0.63, −0.27)
4.07 (3.75, 4.39)
4.38 (4.08, 4.68)
8.54 (7.40, 9.69)
20.27 (15.05, 25.49)
6.39 (5.84, 6.94)
3.30 (2.73, 4.14)
2.92 (2.55, 3.42)
4.56 (2.83, 7.97)
4.39 (2.59, 9.22)
2.74 (2.32, 3.30)
SD of log(c)
0.32 (0.22, 0.46)
0.28 (0.19, 0.40)
0.83 (0.52, 1.22)
1.09 (0.84, 1.38)
0.61 (0.48, 0.77)
SD of log(l)
1.16 (0.68, 2.10)
0.89 (0.50, 1.58)
5.82 (2.26, 15.93)
14.73 (4.88, 46.93)
2.06 (1.33, 3.15)
Correlation of log(l) and log(c)
0.71 (0.50, 0.84)
0.81 (0.64, 0.90)
0.29 (−0.00, 0.64)
0.51 (0.18, 0.96)
0.61 (0.42, 0.77)
In addition to the problem with a strong departure from self-consistency, the self-consistency model did not provide a good approximation to fitting an “average branch.” This lack of fit is shown in Fig. 5A (Rat 1) and in Fig. 5C (Monkey 3). The plots show the joint distributions of log(ci) and log(li) for individual branches (estimated by the variance model). The constant estimate of log(c) and log(l) from the self-consistency model is also placed on the graph, and the inconsistencies of individual branches with this single, constant estimate are obvious. The self-consistency estimate was also found to be higher than mean log(ci) (i.e., μγ) and mean log(li) (i.e., μβ), averaged across all branches that were fit. We found the same discrepancy in all five animals (see Table 1).
The better fit of the variance model compared to the self-consistency model is reflected in the 17%–37% smaller residual variation (σε) for the variance model relative to the self-consistency model (Table 1). This difference in fit is illustrated in Fig. 5B,D, which depicts the fit of both models for two selected branches of Rat 1 and Monkey 1, respectively. As can be seen from the plots, the self-consistency model fit (dashed line) performs quite poorly compared to the variance model fit (solid line). This, again reflects the fact that the self-consistency model attempts to fit all branches with the same curvature and is bound to underperform, as is the case here, when curvatures vary substantially across branches.
An interesting finding was a high positive correlation ρβγ between β (i.e., log(ci)) and γ (i.e., log(li)). The correlation ranged from 0.29 to 0.81 across the five animals. This result indicates that ci and li are not independent of each other but are rather closely related, which is not surprising because ci < 1 (accelerated diameter tapering) would tend to lead to shorter branches and ci > 1 (decelerated diameter tapering) to longer branches.
Models for the Daughter Branch Diameter
This section describes models for the daughter branch diameters. Equations (2) and (3) presented in the previous section described the tapering of diameter within a given branch. To describe the complete diameter geometry of the entire pulmonary tree, an additional model, a model for the initial diameter D0 of each branch, is needed.
Main branch model
Karau (2001) introduced a model for the initial diameter of the daughter branch in relation to the diameter of the parent branch. This model is estimated by simultaneously fitting Eq. (1) and the following Eq. (4) to the data from the main branch [Eq. (1)] and from the first segment of each daughter (offshoot) branch coming of the main branch [Eq. (4)].
In Eq. (4), Dbr(x) is the expected diameter of the offshoot branch at distance x along the main branch and Dbr0 is the estimate of the expected diameter of the offshoot if it had occurred at the beginning of the main branch. The estimated ratio Dbr0/D0 [D0 estimated from Eq. (1)] represents an estimate of the average ratio of the offshoot-to-parent branch diameters, because the ratio does not depend on x. This is true because E[Dbr0(x)]/E[D0(x)] = [Dbr0(1 − x/L)c]/[D0(1 − x/L)c] = Dbr0/D0. Figure 6A shows the fitted model for Rat 1 and Fig. 6C shows the fitted model for Monkey 1 [Eqs. (1) and (4) fitted simultaneously].
The estimates of Dbr0/D0 for all five animals are presented in Table 2. The estimated mean daughter-to-parent diameter ratio for the two rats was considerably smaller (0.41 and 0.47) than for the monkeys (0.61–0.67). Among the three monkeys, the ratio for the ozone-exposed Monkey 2 (0.67) was slightly higher than for the two normal monkeys (0.61 and 0.64).
Table 2. Estimates of mean daughter-to-parent diameter ratio parameters by the main branch and the truncation model with constant diameter ratio
Number of daughter-parent pairs used in the estimation.
Derived as from the estimated truncated regression coefficients ( and ), 95% confidence intervals based on the 95% confidence intervals for μ and on σ.
0.70 (0.69, 0.70) for truncation point = 20× voxel size (i.e., 0.4 mm) and 0.64 (0.57, 0.72) for truncation point = 50 × voxel size (i.e., 1.0 mm)
Bounds calculated across a range of truncation points: lower bound = minimum of lower 95% bounds for , upper bound = maximum of upper 95% bounds for ; minima and maxima of the bounds taken across truncation point varying from 0 to 5 × voxel size.
Bounds of (0.57, 0.74) for truncation points up to 50 × voxel size (i.e., up to 1.0 mm).
The main branch model [Eqs. (1) and (4)] has three shortcomings:
1The model assumes that the daughter–parent geometry of the entire lung can be captured by the daughter–parent geometry of the main branch, that is, the lung has a constant daughter–parent relationship for all branches.
2The model-fitting ignores the occurrence of smaller daughter branches that might have been missed during data acquisition due to resolution issues.
3The diameter-narrowing power-curve relationship (1 − x/L)c used in the model [Eqs. (1) and (4)] is an unnecessary additional assumption and can be avoided by placing no additional constraint on the parent and daughter diameters in relation to the distance x.
Our major concern was the first point above. The main branch model disregards diameter data from offshoots other than the offshoots of the main branch. The model estimates the mean daughter-to-parent diameter ratio from the main branch and its daughter segments and assumes that the same ratio will hold in the remainder of the lung. This assumption is also referred to as self-consistency, although it means something else than self-consistency in the context of diameter narrowing within a single branch described earlier. Given the lung's diversity of diameter-tapering patterns demonstrated in the previous section, use of daughter and parent diameters from the entire lung—rather than just the main branch—would provide a more robust and representative estimate of this key ratio and its variation across the lung.
Complete lung models—effect of missed daughter branches
We introduce two models that extend the estimation of the average daughter-to-parent diameter ratio beyond the main branch, including all observed daughter–parent pairs. To aid the prediction of the initial diameter of a daughter branch, we define the parent segment diameter Dp and the daughter segment diameter Dd to correspond to the n-furcation where the parent and daughter segments intersect (see Appendix for details).
The ability to estimate the average daughter-to-parent ratio using all observed daughter–parent pairs does not come without challenges. The problem is the occurrence of missed daughter segments. Calculations that rely only on observed daughter–parent pairs and ignore unobserved (missed) daughter–parent pairs might lead to biased estimation of the daughter-to-parent diameter ratio for the population of all (observed and unobserved) daughter–parent pairs. The problem is illustrated in Fig. 6B (Rat 1) and Fig. 6D (Monkey 1). Here, the ratio Dd/Dp for observed daughter segments is plotted in relation to the parent diameter Dd. A smoothed mean line is plotted to highlight the trends in mean Dd/Dp as a function of Dd. For large parent diameters Dd, the mean ratio Dd/Dp seems fairly constant, but it increases rapidly for small values of Dd. For very small values of Dd, the mean ratio Dd/Dp even exceeds one, which is highly suspect. Larger Dd/Dp ratios for lower values of Dp among observed daughter–parent pairs would be expected due to unobserved daughter–parent pairs. The unobserved daughter–parent pairs likely tend to have smaller diameters Dd and smaller ratios Dd/Dp and, therefore their exclusion would lead to an overestimate of the mean ratio Dd/Dp. The need to adjust for this bias leads to a method of estimation of the ratio that takes account of unobserved daughter segments.
Complete lung models—truncation model with constant diameter ratio
To estimate the mean daughter-to-parent diameter ratio, we propose to fit the equation
with a truncation regression methodology that accommodates a truncated daughter diameter distribution arising from the missed daughter segments.
In Eq. (5), μ is the mean of log(Dd/Dp) and ε is assumed to have a normal distribution with a zero mean and standard deviation σ. The standard deviation includes both biological variation in the daughter-to-parent diameter ratio and measurement error. Equation 5 is formulated in terms of log-diameters to enable the use of the truncation regression methodology, which requires the truncated variable (in this case Dd) to be the left side of the equation.
A fit of a truncated regression requires a specification of a truncation point. We conducted a sensitivity analysis with regard to the choice of the truncation point and present the estimates (and 95% confidence intervals) of the mean daughter/parent diameter ratio for several choices of the truncation point and a range of plausible estimates (that we call the plausible bounds). The plausible bounds combine (append) the 95% confidence intervals for the mean daughter/parent diameter ratio across a wide range of choices for the truncation point. Further details on these calculations are provided in Appendix.
Results of the truncation regression are shown in Table 2, bottom panel. The table shows the estimates of the mean Dd/Dp for three selected truncation points. The three truncation points included: 0 (the analysis ignores truncation and is equivalent to standard linear regression), 2× imaging voxel size and 5× imaging voxel size. The estimated mean Dd/Dp for truncated regression with truncation point at 0 mm is higher (0.72–0.74) than for truncated regressions with non-zero truncation points. As we increased the truncation point to fivefold the voxel size (5×), the estimated mean ratio Dd/Dp decreased down to 0.67–0.73 for the five animals. Although the largest ratio at 5× truncation point was observed for Monkey 3, the same animal also had much smaller voxel sizes. When we increased the truncation point to 50× voxel size for Monkey 3, the estimated mean ratio Dd/Dp for Monkey 3 decreased to 0.64 (95% CI 0.57, 0.72)—a smaller ratio compared to the remaining animals.
The plausible bounds for mean Dd/Dp in Table 2 provide a range of plausible values for mean Dd/Dp and express the uncertainty in mean Dd/Dp due to (1) the estimation from a finite sample and (2) the unknown extent of missed daughter branches. We observed that the plausible bounds were quite wide. This means that, unless one imposed additional assumptions on the truncation process and/or on the domain of the lung, very different daughter–parent geometries would be consistent with the data. One possible reason for the wide plausible bounds is that the mean Dd/Dp may not be constant throughout the lung (a model accommodating this departure from constancy is introduced in the next subsection).
We also observed that an overwhelming portion of the intervals delineated by the plausible bounds extend below the truncation-ignoring estimate (i.e., truncation point = 0 mm), which suggests an evidence for missed daughter branches.
In contrast to the truncation model, the daughter/parent diameter ratio estimates from the main branch model tended to lie toward the lower plausible bound for mean Dd/Dp (monkeys) or even below it (rats). This difference suggests that the main branch model maybe underestimating mean Dd/Dp.
Complete lung models—truncation model with nonconstant diameter ratio
As observed earlier, the data suggest that the mean daughter/parent diameter ratio may not be constant throughout the lung. We therefore grouped daughter–parent pairs into 3–4 groups (scales 1 through 4) according to the size of the parent diameter Dp (Table 3). The size of Dp reflects how distal the pair is from the terminus—thus, the scale 1 group would correspond to bigger, less distal segments and scale 4 would correspond to smaller, more distal segments. We then proceeded to calculated mean ratio Dd/Dp separately for each group g by fitting the equation
with the truncation regression methodology. The parameters in Eq. (6) have the same meaning as in Eq. (5), except there are separate parameters for each group g (denoted by the subscripts).
Table 3. Estimates of mean daughter-to-parent diameter ratio parameters by the truncation model with nonconstant diameter ratio
Truncated regression with truncation point = 5 × voxel size. Nt = number of observations available above the truncation point. Estimate derived as from the estimated truncated regression coefficients ( and ), 95% confidence intervals based on the 95% confidence intervals for μ and on σ. See Table 2 for the voxel size for each animal.
Bounds calculated across a range of truncation points: lower bound = minimum of lower 95% bounds for , upper bound = maximum of upper 95% bounds for ; minima and maxima of the bounds taken across truncation point varying from 0 to 5 × voxel size.
Scale 1: ≥0.35 mm
0.59 (0.58, 0.61)
Scale 2: (0.2 mm, 0.35 mm)
0.78 (0.77, 0.80)
Scale 3: <0.2 mm
0.60 (0.28, 1.27)
Scale 1: ≥0.35 mm
0.62 (0.60, 0.63)
Scale 2: (0.2 mm, 0.35 mm)
0.72 (0.70, 0.74)
Scale 3: <0.2 mm
0.60 (0.34, 1.08)
Scale 1: ≥1.0 mm
0.69 (0.66, 0.72)
Scale 2: (0.8 mm, 1.0 mm)
0.92 (0.62, 1.36)
Scale 3: (0.6 mm, 0.8 mm)
Scale 4: <0.6 mm
Scale 1: ≥1.0 mm
0.70 (0.66, 0.73)
Scale 2: (0.8 mm, 1.0 mm)
0.96 (0.74, 1.25)
Scale 3: (0.6 mm, 0.8 mm)
Scale 4: <0.6 mm
Scale 1: ≥1.0 mm
0.66 (0.63, 0.68)
Scale 2: (0.8 mm, 1.0 mm)
0.65 (0.62, 0.69)
Scale 3: (0.6 mm, 0.8 mm)
0.70 (0.69, 0.71)
Scale 4: <0.6 mm
0.74 (0.73, 0.74)
The results of fitting Eq. (6) are shown in Table 3. For brevity, we only present (1) estimates of mean ratio Dd/Dp at truncation point equal to 5× voxel size and (2) the plausible bounds. The results provide important information concerning the assumption of constancy of the mean daughter/parent diameter ratio throughout the lung.
Test for the constancy of mean daughter/parent diameter ratio
For both rats and monkeys, the results shown in Table 3 suggest departures from constancy of the mean daughter-to-parent diameter ratio. The mean ratio Dd/Dp (estimated at 5× truncation point) at scale 1 (least distant daughter–parent pairs) was lower than at scale 2 (second least distant pairs) among four of the five animals. For neither of the two rats did the plausible bounds for the two scales overlap. The ratio for most distant pairs (scales 3 and 4) was not very well determined or could not be determined at all due to a small or nonexistent sample size, except for the CT-imaged monkey (Monkey 3). The mean ratio Dd/Dp for Monkey 3 tended to increase with the scale. Overall, both rats and monkeys featured important differences in mean ratio Dd/Dp throughout the lung, a violation of the constant mean ratio assumption.
We introduced a new approach for modeling diameters of airways and applied it to lung casts of two rats and three monkeys imaged by either MRI or CT. The new approach addressed issues of self-consistency and measurement error and resulted in an improved representation of the geometry and important differences compared to the self-consistency model.
Our new models built on previous models by our group (Einstein, 2008). Both (old and new) models are based on a definition of a branch as a sequence of consecutive segments that begins with a n-furcation off a parent branch, following the largest daughter segment of each n-furcation, down to the terminal bronchioles. With this definition, the goal of describing the entire diameter geometry divides into the development of two models: a model for diameter narrowing within a single branch, and a model for a daughter branch diameter. An important feature of the branch definition is that it organizes segments in a manner that facilitates approximation of diameters within a branch by a simple power-curve function. Previous models using the branch definition (Einstein, 2008) relied on the assumptions of self-consistency. We found that, for the five animals analyzed in this work, the departures from self-consistency were real and needed to be included in the modeling. We developed new models that incorporate these departures.
Self-Consistency—Diameter Narrowing Within a Branch
Our new model for diameter narrowing within a single branch (variance model) showed that, contrary to the self-consistency assumption, the different airway branches had a substantial diversity in the shape of the diameter-narrowing curve. We found that the curves differed in how convex/concave they were (curvature power exponent c) and in how far they extended (parameter l). We also observed that the tapering of an average branch under the variance model was more convex and rapid than the tapering predicted by the self-consistency model.
The difference between the variance and self-consistency models seemed to be a result of a misrepresentation of the geometry by the self-consistency model. The misrepresentation was caused by forcing a nonconsistent phenomenon into the self-consistent model. This behavior is illustrated on the fit of the self-consistency model to Rat 1 in Fig. 3A. As is obvious from the plot, the shape of the curve estimated from the self-consistency fit was concave (log(c) = 0.31). However, when curves for individual branches were examined, the curves spanned a wide range of convexity (see y-axis of Fig. 5A for log(c) of the individual curves).
A limitation of the variance model is that it describes variation of the diameter-narrowing curvatures across different branches but is unable to explain the differences. If one was able to attribute these differences to some factors (e.g., location within the lung), these factors could be used in space-filling predictions.
A limitation of our implementation of the variance model was that we had to limit the data only to branches with at least 12 segments, as fitting the model with larger datasets was not computationally feasible. Although it is possible that branches with fewer segments may be different in their diameter geometry, we did not find any suggestion of this difference for the five animals in this study (data not shown).
We caution researchers against ignoring the variability of diameter narrowing. Two lungs may have quite a different distribution of diameter-narrowing across their branches but have the same mean narrowing. As a result, the prevalence of branches of a certain rate of narrowing may differ between the two lungs, which may, in turn, have an impact on the physiological properties of each lung. The morphometric variability can be readily implemented in applications of the models (e.g., space-filling or generation of phantom lungs) by stochastic generation of the diameters from variable, branch-specific parameters. To examine the impact of the uncertainty in the parameter estimates, it is also advisable to conduct a sensitivity analysis by varying the morphometric parameters throughout their 95% confidence intervals.
Self-Consistency—Model for the Daughter Branch Diameter
Our previous model for the daughter branch diameter (the “main branch model”) assumed that the mean daughter/parent diameter ratio for the daughter branches of the main branch was representative of the entire lung. Contrary to this assumption, we found that the mean daughter/parent diameter ratio is likely not constant throughout the lung. The mean ratio tended to be smaller for the largest airways. In particular, the mean ratio estimated from the main branch resulted in smaller mean ratios for the entire lung, suggesting that the main branch features some of the smallest daughter-to-parent diameter ratios in the lung. These results emphasize the importance of critical assessment of the self-consistency (mean ratio constancy) assumption in modeling the daughter branch diameter, and the need to account for any departures from the self-consistency assumption when they occur.
In our last model for the daughter diameter, we allowed the mean daughter-to-parent diameter ratio to vary according to the parent segment diameter, providing a better-fitting model.
In our new approach to model the airways diameters, we also attempted to pay attention to measurement error and its impact on the estimated models. Measurement error arises from lung casting, imaging, and/or image processing and is particularly prevalent in peripheral airway segments. The peripheral airways, in turn, have a large impact on the model estimates as the data-rich MRI and CT datasets have a very high representation of peripheral airways.
We recognize two categories of measurement errors: an error due to the difference between the true and the observed dimensions of the airway and an error due to some of the airway segments being missed. We showed that the latter had a substantial effect on estimating the mean daughter-to-parent diameter ratio. Missed (unobserved) airways lead to left truncation of the observed distribution of the daughter segment diameters and, as a result, an overestimate of the true mean daughter-to-parent diameter ratio. Additional methodology is needed to address the impact of resolution on morphometry estimates.
The truncated regression methodology has its limits, especially when used for peripheral subsets of the lung that are dominated by very small airways. These regions have an extremely small number of observed segments below a certain dimension (a limitation related to the resolution limit of lung casting and imaging) and the estimate of the mean daughter-to-parent diameter ratio is either unreliable or unattainable.
There are measurement error issues that may be important that we have been unable to address in this analysis. For example, measurement error in the predictor variables in both equations for the diameter narrowing and for the daughter branch diameter may lead to some bias in the estimated model. To explore this effect in the future, it would be helpful if the methods of adjusting for measurement errors were tested on simulated data.
A limitation of this study is the inclusion of only five animals and the selection of these animals. The studies, which produced these casts, were not primarily designed to relate to the concept of self-consistency in the diameter geometry. The focus of this study, however, was on a new morphometric methodology and the five animals served as illustrations of benefits of this methodology. The fact that the assumption of self-consistency failed in all five animals in this study is an important finding and a warning about the self-consistency assumption. The animals that are included are not included to demonstrate more or less self-consistency for certain species, ages, or disease states. Rather, they should be considered an illustrative set of airways used to demonstrate methodological points that are discussed in this work.
A new branch-based approach to model airway diameters shows that there are substantial departures from self-consistency in two rodents and three monkeys. The new “variance” modeling fits data on branch diameters and daughter/parent diameter ratios better than the self-consistency model. Resolution limits in imaging lead to missed daughter branches and an overestimate of mean daughter/parent diameter ratios. Some correction of this bias can be achieved by use of truncated regression.
A portion of this research was performed using EMSL, a national scientific user facility sponsored by the Department of Energy's Office of Biological and Environmental Research and located at Pacific Northwest National Laboratory. The authors would like to gratefully acknowledge the help of Tamas Varga at EMSL.
To enforce the truncation at a given truncation point, the truncation regression ignores all data below the truncation point.
Necessary and Sufficient Condition for Self-consistency
In this section, we will mathematically formulate the self-consistency assumption for the diameter narrowing model. Let us consider two branches and their fits to Eq. (1): and . Consistency of the two curves can be summarized in terms of the model parameters by two equalities: c1 = c2
The first equality is obvious: the two equations cannot be consistent if the two power curves have different exponents. The second equality comes from the fact that diameters measured the same distance from the end of the branch in the two branches should be equal. If we set x equal to L1 − x′ and L2 − x′ in the two equations (x′ denotes the distance from the end of the branch), and use the first self-consistency equation (c1 = c2), we get that, canceling x′, yields the second equality. For a given branch, we will denote l = Lc/D0. Thus, self-consistency means that c and l are constant across branches.
The Variance Model
The variance model is a generalization of the self-consistency model that allows for variation in parameters l and c. The equation for the variance model is:
This model is derived from Eq. (1) and the fact that . The parameters D0i, li, and ci are branch-specific (the subscript i denoting the branch). Parameters D0i capture the initial diameter of each branch and are of interest in modeling the daughter–parent diameter relationship described later in this article. Under self-consistency, parameters li and ci are constant across branches. If, on the other hand, departures from self-consistency are present, they can be quantified by the amount of variation in li and ci. The result is a more general model that allows for a different diameter-narrowing curvature in different branches.
Because all three parameters (D0i, li, and ci) have to be positive and because we found their distributions to be right-skewed, we chose to estimate the distribution of their log values ( , , ). This is a particularly nice feature for the exponent parameter ci because γi = log(ci) is negative for convex curves, zero for linear curves and positive for concave curves. With the nonlinear mixed model, we estimate the distribution of the vector (αi, βi, and γi). The distribution is a multivariate normal distribution with a mean (μα, μβ, and μγ) and a variance matrix ∑. The variance matrix ∑ contains the variance of the parameters αi, βi, and γi (σα2, σβ2, and σγ2) and the covariances of these parameters. The estimates of σβ and σγ from fitting Eq. (3) determine the level of self-consistency in log(l) and log(c) across branches.
A nonlinear mixed model has to be used to fit Eq. (3). A common mistake is to fit the model by estimating the branch-specific parameters (D0i, li, and ci) from fitting separate fits of Eq. (1) to each branch and by calculating the joint distribution of log (D0i, li and ci) across the branch-specific estimate. The problem with the joint distribution of log (D0i, li, and ci) estimated in this fashion is that the variance of the distribution is biased upward because it is inflated by the variance of the estimates of the individual parameters (D0i, li, and ci) from fitting Eq. (1).
Another problem with fitting Eq. (3) had to do with the parameterization of the nonlinear mixed model. Although Eq. (3) provided key parameterization for determining the extent of self-consistency in parameters l and c, it was problematic to implement the fit in available nonlinear mixed effects software (packages nlme and lme4 in R and PROC NLMIXED in SAS). For the iterative process of the REML (restricted maximum likelihood), it is necessary that the expression in the brackets of Eq. (3) does not become negative. That is, we need that for all data points x during all iterations of the estimation algorithm. However, we were unable to enforce this constraint in the implementations of the estimation algorithm available to us and in all five animals analyzed here, our estimations would be terminated with an error message due to this issue.
To solve this problem, we developed a two-step process to arrive at the estimates of parameters in Eq. (3). For both steps, we use a reparameterization of Eq. (3):
where xmax,i is the largest x in branch i and ΔLi = Li − xmax,i (see Fig. A2 for a graphical definition of ΔLi and xmax,i). We define a new parameter β′i by , which guarantees that x < Li and avoids an error termination of the estimation algorithm.
In the first step, the model in Eq. (A1) can be fit by frequentist statistical methods for fitting nonlinear mixed models (we used the lme4 package in R). The mean (μα, μβ′, and μγ) and the variance matrix ∑′ for the multivariate normal distribution of the vector (αi, β′i, and γi) in Eq. (A1) can be estimated and used as mean of the prior distributions for the estimation in the second step. The second step consists of refitting Eq. (A1) as a hierarchical Bayesian model using WinBugs. For those interested to fit this model, the following Winbugs code shows some of the choices of prior distributions.
The model not only provides estimates of the mean and variation of α (the distribution of D0) and of γ (the distribution of c), as the frequentist methods do, but is also able to derive the mean and variation of β (the distribution of l) through an empirical estimate using a relationship between li, ΔLi, and xmax. For each branch li is back-calculated by the equation
Estimates of μβ, σ2β, and ρβγ are then calculated as the means of the empirical distributions of , , and , respectively.
Calculation of Parent and Daughter Diameters (Dp and Dd) for the Complete Lung Models
To aid the prediction of the initial diameter of a daughter branch, we define the parent segment diameter Dp and the daughter segment diameter Dd to correspond to the n-furcation, where the parent and daughter segments intersect. The location of diameters in this context is different compared to the models presented earlier, where the diameters are placed at the middle of each segment. Here, the diameters Dp and Dd are calculated by interpolation (Dp) or by extrapolation (Dd) of the middle-segment diameters on their respective branches and represent the parent and daughter branch diameters that would be expected at the n-furcation (a concept similar to D(x) and Dbr(x) in the main branch model).
The parent diameter (Dp) was calculated by a linear interpolation of the diameter of the middle of the parent segment and of the diameter of the middle of the immediately next segment on the same parent branch.
The daughter diameter (Dd) was calculated for each branch in the following fashion: First, we attempted to estimate Eq. (1) by fitting a nonlinear least squares regression to the daughter branch diameters. If the fit of Eq. (1) converged, the estimated D0 in Eq. (1) represented the expected diameter of the daughter branch at the n-furcation and was used to estimate Dd. If the fit did not converge but there were at least two segments on the daughter branch, we would estimate Dd by either linearly extrapolating from the diameters of the middle of the first two segments of the daughter branch or by the diameter of the middle of the first segment, whichever was larger. If the daughter branch consisted of only one segment, Dd was estimated by the diameter of the middle of this only segment.
The definition makes Dd correspond to the initial branch diameter D0i in Eq. (3) and connects the two aspects of the diameter model (branch narrowing and daughter branch diameter). The definition also accounts for the different lengths of the daughter and parent segments in different daughter–parent pairs that would have impacted the size of the diameters if they were placed at the middle of the daughter and parent segments.
Complete Lung Models—Truncation Model With Constant Diameter Ratio
The truncation model with constant diameter (daughter/parent) ratio fits the equation
with a truncation regression methodology. The methodology accommodates a truncated daughter diameter distribution arising from the missed daughter segments. The distribution of log(Dd/Dp) is assumed to be normally distributed with the mean μ and standard deviation σ.
In general, the truncated regression methodology (Davidson and MacKinnon, 1993; Greene, 2003) allows estimation of linear regression models when all data below a certain threshold (truncation point) are missing and all data above that threshold are observed. The situation for the daughter–parent diameter data is, however, slightly different. We expect that branches start to be missed at certain resolution and that the likelihood of missing branches increases with the decreasing size of the branch, that is, there is not a sharp bound below which all daughter segments are missed.
To accommodate the variable truncation of the data, we conducted a sensitivity analysis for the truncated regression model, varying the truncation point and estimating the mean Dd/Dp for each choice of the truncation point.1 The truncation points were expressed in multiples of the imaging voxel size to relate the truncation process to the resolution limits of the imaging process. Alternatively, one might express the truncation points in multiples of the resolution limits of the casting process (if available) or of some combination of imaging and casting processes.
We also calculated a somewhat broader summary of the sensitivity analysis, one that incorporates more choices of the truncation point than the three (0×, 2×, and 5×) presented so far. This summary, called plausible bounds for mean Dd/Dp, is presented in the last row of Table 2. To derive these bounds, we would vary the truncation points in small increments from 0× to 5× voxel size (also up to 50× voxel size for Monkey 3), refit the truncation model for each truncation point, and append the 95% confidence intervals for the mean Dd/Dp across the models. The plausible bounds for mean Dd/Dp estimate a range of plausible values of mean Dd/Dp across the models considered in the sensitivity analysis. The bounds reflect the uncertainty from both the finite dataset and the unknown truncation process.