Beyond the principle of similitude: renormalization in the bronchial tree

The average dimensions (diameter, length, and volume) of the airways in the mammalian bronchial tree, long thought to be exponential functions of the generation number, are shown to be power laws in generation number modulated by a harmonic variation. These data are satisfactorily described by means of a functional scaling relation--renormalization group property--between successive generations for the average variable of interest. This type of scaling may provide a mechanism for the morphogenesis of complex but highly stable structures.     

Pulmonary acinus: geometry and morphometry of the peripheral airway system in rat and rabbit

The geometry and morphometry of intraacinar airways in rat and rabbit lungs were studied from silicone rubber casts. Acini, defined as the complex of alveolated airways distal to the "terminal" bronchiole, were trimmed off the bronchial tree. In both species, the acinar volume followed a log-normal distribution over a range in size of one order of magnitude. At an inflation level of 60% total lung capacity, their mean volume was 1.86 mm3 in the rat and 3.46 mm3 in the rabbit. On a representative sample of acini of different volumes, the branching pattern was characterized as irregular dichotomy, and the segment length and inner and outer diameters were measured. The average acinus had a mean of six generations in the rat and seven in the rabbit. Both showed a decrease in segment length and inner diameter with each generation. The mean longitudinal pathway length--that is, the distance from the initial acinar segment to the terminal sacs--was found to depend on the cube root of the acinar volume in both species. It was calculated at 1.46 and 1.95 mm for rat and rabbit, respectively.     

Distribution of airway narrowing responses across generations and at branching points,assessed in vitro by anatomical optical coherence tomography

Background

Previous histological and imaging studies have shown the presence of variability in the degree of bronchoconstriction of airways sampled at different locations in the lung (i.e., heterogeneity). Heterogeneity can occur at different airway generations and at branching points in the bronchial tree. Whilst heterogeneity has been detected by previous experimental approaches, its spatial relationship either within or between airways is unknown.

Methods

In this study, distribution of airway narrowing responses across a portion of the porcine bronchial tree was determined in vitro. The portion comprised contiguous airways spanning bronchial generations (#3-11), including the associated side branches. We used a recent optical imaging technique, anatomical optical coherence tomography, to image the bronchial tree in three dimensions. Bronchoconstriction was produced by carbachol administered to either the adventitial or luminal surface of the airway. Luminal cross sectional area was measured before and at different time points after constriction to carbachol and airway narrowing calculated from the percent decrease in luminal cross sectional area.

Results

When administered to the adventitial surface, the degree of airway narrowing was progressively increased from proximal to distal generations (r = 0.80 to 0.98, P < 0.05 to 0.001). This ''serial heterogeneity'' was also apparent when carbachol was administered via the lumen, though it was less pronounced. In contrast, airway narrowing was not different at side branches, and was uniform both in the parent and daughter airways.

Conclusions

Our findings demonstrate that the bronchial tree expresses intrinsic serial heterogeneity, such that narrowing increases from proximal to distal airways, a relationship that is influenced by the route of drug administration but not by structural variations accompanying branching sites.     

Computational model of airflow in upper 17 generations of human respiratory tract

Computational fluid dynamics (CFD) studies of airflow in a digital reference model of the 17-generation airway (bronchial tree) were accomplished using the FLUENT computational code, based on the anatomical model by Schmidt et al. [2004. A digital reference model of the human bronchial tree. Computerized Medical Imaging and Graphics 28, 203-211]. The lung model consists of 6.744 x 10(6) unstructured tetrahedral computational cells. A steady-state airflow rate of 28.3L/min was used to simulate the transient turbulent flow regime using a large eddy simulation (LES) turbulence model. This CFD mesh represents the anatomically realistic asymmetrical branching pattern of the larger airways. It is demonstrated that the nature of the secondary vortical flows, which develop in such asymmetric airways, varies with the specific anatomical characteristics of the branching conduits.     

Tree roots as self-similar branching structures: axis differentiation and segment tapering in coarse roots of three boreal forest tree species

We applied a fractal root model to the 3D architecture of the coarse root systems of Betula pendula Roth, Picea abies (L.) H. Karst., and Pinus sylvestris L. in mixed boreal forests. Our dataset consisted of 60 root systems excavated in five different mixed forest stands. We analyzed the variability of the model parameters with respect to species, site type, and different root axes. According to our results, the cross-sectional area of root segments (i.e. second power of diameter) was a suitable variable for analyzing the values of parameters of the fractal model. The parameter values varied with generation and order of root segments; the roots thus did not follow the simple fractal branching. The variation of parameters along the root axes showed the existence of a zone of rapid tapering in all tree species. The model was, with parameter values analyzed from the data, moderately capable of accounting for the main coarse root characteristics. It was important for model predictions to take into account the tapering of root segments. We conclude that, in boreal forests, tree root systems are the output of the axis-specific morphogenetic branching rules and functional adaptation to spatial heterogeneity in the soil.     

Computer simulation of the geometry of the human bronchial tree

Some morphological features of the human bronchial tree were simulated by computergenerated trees. The trees were ordered by the methods of Horsfield and Strahler. Delta, the difference between the Horsfield orders of the two branches at a bifurcation, was determined by pseudorandom numbers generated according to a distribution of probabilities defined on input. By trial and error a distribution was found which resulted in trees being generated with average Strahler order branching ratios of 2.82, similar to a real bronchial tree. Branching angles and length ratio could also be defined on input. By varying these input parameters it was found that the form of the tree was quite sensitive to them, and that by a suitable choice the intrasegmental part of the bronchial tree could be simulated. It is concluded that branching ratio, length ratio, mean branching angles and distribution of delta are controlled within tight limits in the bronchial tree, and this may support the concept of optimal design.     

A fractal analysis of the radial distribution of bronchial capillaries around large airways.

We analyzed published measurements of the bronchial circulation and airway wall (Anderson JC, Bernard SL, Luchtel DL, Babb AL, and Hlastala MP. Respir Physiol Neurobiol 132: 329-339, 2002) and determined that the radial distribution of bronchial capillary cross-sectional area was fractal. We limited our analysis to bronchial capillaries, diameter < or =10 mum, that resided between the epithelial basement membrane and adventitia-alveolar boundary, the airway wall tissue. Thirteen different radial distributions of capillary-to-tissue area were constructed simply by changing the number of annuli (i.e., the annular size) used to form each distribution. For the 13 distributions created, these annuli ranged in size from to of the size of the airway wall area. Radial distributions were excluded from the fractal analysis if the sectioning procedure resulted in an annulus with a radial thickness less than the diameter of a capillary. To determine the fractal dimension for a given airway, the coefficient of variation (CV) for each distribution was calculated, and ln(CV) was plotted against the logarithm of the relative piece area. For airways with diameter >2.4 mm, this relationship was linear, which indicated the radial distribution of bronchial capillary cross-sectional area was fractal with an average fractal dimension of 1.27. The radial distribution of bronchial capillary cross-sectional area was not fractal around airways with diameter <1.5 mm. We speculated on how the fractal nature of this circulation impacts the distribution of bronchial blood flow and the efficiency of mass transport during health and disease. A fractal analysis can be used as a tool to quantify and summarize investigations of the bronchial circulation.     

Size-dependent morphology of the conductive bronchial tree in four species of myomorph rodents

Size-dependent structural patterns in the conductive bronchial tree of four species of myomorph rodents of different body weight were determined by lung casts. The lungs of the harvest mouse, Micromys minutus, body weight 5–7 g, the house mouse, Mus musculus, body weight 35–45 g, the brown rat, Rattus norvegicus, body weight 200–400 g, and the African giant pouched rat, Cricetomys gambianus, body weight 1,200–1,800 g, were inflated to 20 cm H₂O, frozen, freeze-dried, hardened, and filled with silicone rubber. The casts were pruned, and branching pattern, diameter, and volume of the conductive bronchial tree were determined using a binocular magnifier. All four species have four lobes on the right lung and an undivided left lung, and the central bronchial tree on either side shows an identical monopodial branching pattern. Although the ramification of the central conductive bronchi is not size-dependent, the diameter and volume are. The diameter of the left main bronchus equals 1.24% of body length in Micromys and 0.6% in Cricetomys, and the conductive bronchial tree makes up 13% of the total lung volume in Micromys and 6% in Cricetomys. Relatively wider airways and a decline in airway resistance with declining body mass in small mammals compared to large ones result in a high ventilatory dead space, which is compensated for by a higher breathing frequency. © 1996 Wiley-Liss, Inc.     

Why are there so many threat displays?

The morphology of branching trees in general, and of the bronchial tree in particular, can be described in terms of three parameters, the diameter, length and branching ratios. These are the factors by which mean diameter and mean length increase in successive orders towards the trachea, and by which the number of branches increases in successive orders away from the trachea. Orders of branching are counted from the periphery towards the trachea, according to the method of Strahler. A model of from two to nine orders, and of constant total length and volume, was used to investigate the effect of varying the above parameters on the calculated pressure difference across the model during flow. In particular, the branching ratio was set at known values for dog and human lungs, and diameter and length ratios were independently varied. Known data from dog and human lungs were found to be close to the points predicted by the model where the lines of minimal resistance and minimal entropy production crossed. Other factors which may affect the values of these parameters are discussed.     

Branching tree model with fractal vascular resistance explains fractal perfusion heterogeneity

Perfusion heterogeneities in organs such as the heart obey a power law as a function of scale, a behavior termed "fractal." An explanation of why vascular systems produce such a specific perfusion pattern is still lacking. An intuitive branching tree model is presented that reveals how this behavior can be generated as a consequence of scale-independent branching asymmetry and fractal vessel resistance. Comparison of computer simulations to experimental data from the sheep heart shows that the values of the two free model parameters are realistic. Branching asymmetry within the model is defined by the relative tissue volume being fed by each branch. Vessel ordering for fractal analysis of morphology based on fed or drained tissue volumes is preferable to the commonly used Strahler system, which is shown to depend on branching asymmetry. Recently, noninvasive imaging techniques such as PET and MRI have been used to measure perfusion heterogeneity. The model allows a physiological interpretation of the measured fractal parameters, which could in turn be used to characterize vascular morphology and function.     

Fractal modeling of pulmonary blood flow heterogeneity

The heterogeneity of pulmonary blood flow is not adequately described by gravitational forces alone. We investigated the flow distributions predicted by two fractally branching vascular models to determine how well such networks could explain the observed heterogeneity. The distribution of flow was modeled with a dichotomously branching tree in which the fraction of blood flow from the parent to the daughter branches was gamma and 1-gamma repeatedly at each generation. In one model gamma was held constant throughout the network, and in the other model gamma varied about a mean of 0.5 with a standard deviation of sigma. Both gamma and sigma were optimized in each model for the best fit to pulmonary blood flow data from experimental animals. The predicted relative dispersion of flow from the two model fractal networks produced an excellent fit to the observed data. These fractally branching models relate structure and function of the pulmonary vascular tree and provide a mechanism to describe the spatially correlated distribution of flow and the gravity-independent heterogeneity of blood flow.     

Form of branching of pulmonary air pathways in the canine lung]

The degree of regularity in bronchial dichotomy has been studied in 5 corrosive preparations of the dog bronchial tree by the method of E. R. Weibel. It has been demonstrated that a high degree of regularity is specific for the dog lungs that makes 0.82 +/- 0.12 regarding small and large diameters. Approximation in distribution of bronchi having the diameter of 3--3.5 mm in 1--9 generations by means of binomial distribution gives a good result. The greatest bronchial rate with the diameter mentioned above has been registered in 5--7 generations. Theoretical data are presented to demonstrate the necessity in regularity of bronchial branching in order to maintain an optimal gaseous exchange in lungs.     

Measurement of axial diffusivities in a model of the bronchial airways.

Values for the effective axial diffusivity D for laminar flow of a gas species in the bronchial airways have been obtained as a function of the mean axial gas velocity u by experiment measurements of benzene vapor dispersion in a five generation glass tube model of the bronchial tree. For both inspiration and expiration D is seen to be approximately a linear function of u over the range of Reynolds' numbers 30-2,000 corresponding to peak flows in bronchial generations 0-13 under resting breathing conditions. The diffusivity for expiration is seen to be approximately one-third that for inspiration due presumably to increased radial mixing at bifurcations during expiration. The effective diffusivities relative to the molecular diffusivity can be expressed by the formulas D/Dmol = 1 + 1.08 NPe for inspiration and D/Dmol = 1 + .37 N-Pe for expiration. These velocity dependent diffusivities help to explain the short transit times of gas boluses from mouth to alveoli and will aid in the analysis of airway gas mixing by mathematical transport equations.     

Three-dimensional visualization and morphometry of small airways from microfocal X-ray computed tomography

Physiological morphometry is a critical factor in the flow dynamics in small airways. In this study, we visualized and analyzed the three-dimensional structure of the small airways without dehydration and fixation. We developed a two-step method to visualize small airways in detail by staining the lung tissue with a radiopaque solution and then visualizing the tissue with a cone-beam microfocal X-ray computed tomographic (CT) system. To verify the applicability of this staining and CT imaging (SCT) method, we used the method to visualize small airways in excised rat lungs. By using the SCT method to obtain continuous CT images, three-dimensional branching and merging bronchi ranging from 500 to 150 microm (the airway generation=8-16) were successfully reconstructed. The morphometry of the small airways (diameter, length, branching angle and gravity angle between the gravity direction and airway vector) was analyzed using the three-dimensional thinning algorithm. The diameter and length exponentially decreased with the airway generation. The asymmetry of the bifurcation decreased with generation and one branching angle decided the other pair branching angle. The SCT method is the first reported method that yields faithful high-resolution images of soft tissue geometry without fixation and the three-dimensional morphometry of small airways is useful for studying the biomechanical dynamics in small airways.     

Arterial branching within the confines of fractal L-system formalism

Parametric Lindenmayer systems (L-systems) are formulated to generate branching tree structures that can incorporate the physiological laws of arterial branching. By construction, the generated trees are de facto fractal structures, and with appropriate choice of parameters, they can be made to exhibit some of the branching patterns of arterial trees, particularly those with a preponderant value of the asymmetry ratio. The question of whether arterial trees in general have these fractal characteristics is examined by comparison of pattern with vasculature from the cardiovascular system. The results suggest that parametric L-systems can be used to produce fractal tree structures but not with the variability in branching parameters observed in arterial trees. These parameters include the asymmetry ratio, the area ratio, branch diameters, and branching angles. The key issue is that the source of variability in these parameters is not known and, hence, it cannot be accurately reproduced in a model. L-systems with a random choice of parameters can be made to mimic some of the observed variability, but the legitimacy of that choice is not clear.     

Sox2 is important for two crucial processes in lung development: branching morphogenesis and epithelial cell differentiation

The primary lung bud originates from the foregut and develops into the bronchial tree by repetitive branching and outgrowing of the airway. The Sry related HMG box protein Sox2 is expressed in a cyclic manner during initiation and branching morphogenesis of the lung. It is highly expressed in non-branching regions and absent from branching regions, suggesting that downregulation of Sox2 is mandatory for airway epithelium to respond to branch inducing signals. Therefore, we developed transgenic mice that express a doxycycline inducible Sox2 in the airway epithelium. Continuous expression of Sox2 hampers the branching process resulting in a severe reduction of the number of airways. In addition, the bronchioli transiently go over into enlarged, alveolar-like airspaces, a pathology described as bronchiolization of alveoli. Furthermore, a substantial increase was observed of cGRP positive neuroendocrine cells and ΔNp63 isoform expressing (pre-) basal cells, which are both committed precursor-like cells. Thus, Sox2 prevents airways from branching and prematurely drives cells into committed progenitors, apparently rendering these committed progenitors unresponsive to branch inducing signals. However, Sox2 overexpression does not lead to a complete abrogation of the epithelial differentiation program.     

A simplified method of casting the macroscopic airways of lungs

A technique for preparing casts of the macroscopic airways of mammalian lungs, which is both simplified and inexpensive in comparison with previous techniques, is described. The models are accurate, durable and flexible, and clearly demonstrate the orientation and branching pattern of the bronchial tree. The nature of the procedure also extends the availability of casts to laboratories or individuals with limited instrumentation and/or funding. Preliminary results using this technique to inject the lungs and certain air sacs of birds are also discussed.     

Distribution of end-points of a branching network with decaying branch length

The geometry of the human bronchial tree has been described as a network formed by successive dichotomous branching with constant branching angles and geometrically decaying branch lengths. Models having these properties and with randomly distributed branching planes are constructed. The distribution of the end points of the model networks are described by computing the variance of the distributions in the direction of the axis of the network and in the transverse directions. It is found that, for a given decay ratio, there is a branching angle for which the volume filled by the end points is a maximum. The advantages of the network with the decay ratio and branching angle of the human bronchial tree are discussed.     

A Simple Geometrical Pattern for the Branching Distribution of the Bronchial Tree, Useful to Estimate Optimality Departures

The design of the bronchial tree has largely been proposed as a model of optimal design from a physical-functional perspective. However, the distributive function of the airway may be more related to a geometrical than a physical problem. The bronchial tree must distribute a three dimensional volume of inspired air on a two dimensional alveolar surface, included in a limited volume. It is thus valid to ask whether an optimal bronchial tree from a physical perspective is also optimum from a geometrical point of view. In this paper we generate a simple geometric model for the branching pattern of the bronchial tree, deducing relationships that permit estimation of the departures from the geometrical optimum of each bifurcation. We also, for comparative purposes, estimate the departures from the physical optimum. From the geometrical assumptions: i) a symmetrical dichotomic fractal design, ii) with minimum volume and iii) maximum dispersion of the terminal points; and several simulations we suggest that the optimality is characterized by a bifurcation angle theta approximately 60 degrees and a length reduction scale gamma = (1/2)(1/3) = 0.7937. We propose distances from the physical and geometrical optimality defined as Euclidean distances from the expected optima. We show how the advanced relationships and the distances can be used to estimate departures from the optimality in bronchographs of four species. We found lower physical and geometrical departures in the distal zone than those of the proximal zones, as well as lower physical than geometrical departures from optimality.