Optimal Branching Structure of Fluidic Networks with Permeable Walls

An assessment of branching asymmetry of the tracheobronchial tree

The tracheobronchial tree is commonly seen to have a systematic branching symmetry, despite being known to have an asymmetrical design. Branching asymmetry allows for uniform airflow and provides robustness against the morphogenesis-related size variability. Here, a constructal approach is used to tracheobronchial tree analysis, and a general model based on entropy generation during breathing process is provided, which holds with asymmetric characteristics of the tree, and the change for inhaling and exhaling air. In contrast to traditional models available in the literature, the entropy generation of inspiration and expiration processes is compared for symmetry and asymmetric designs. This approach unravels the fundamental consequences of asymmetric constraint in the process of breathing and provides justification for the tracheobronchial tree having the same number of bifurcation levels as optimized symmetrical trees.

Dermal Lymphatic Capillaries Do Not Obey Murray's Law

Lymphatic vessels serve as a major conduit for the transport of interstitial fluid, immune cells, lipids and drugs. Therefore, increased knowledge about their development and function is relevant to clinical issues ranging from chronic inflammation and edema, to cancer metastasis to targeted drug delivery. Murray's Law is a widely-applied branching rule upheld in diverse circulatory systems including leaf venation, sponge canals, and various human organs for optimal fluid transport. Considering the unique and diverse functions of lymphatic fluid transport, we specifically address the branching of developing lymphatic capillaries, and the flow of lymph through these vessels. Using an empirically-generated dataset from wild type and genetic lymphatic insufficiency mouse models we confirmed that branching blood capillaries consistently follow Murray's Law. However surprisingly, we found that the optimization law for lymphatic vessels follows a different pattern, namely a Murray's Law exponent of ~1.45. In this case, the daughter vessels are smaller relative to the parent than would be predicted by the hypothesized radius-cubed law for impermeable vessels. By implementing a computational fluid dynamics model, we further examined the extent to which the assumptions of Murray's Law were violated. We found that the flow profiles were predominantly parabolic and reasonably followed the assumptions of Murray's Law. These data suggest an alternate hypothesis for optimization of the branching structure of the lymphatic system, which may have bearing on the unique physiological functions of lymphatics compared to the blood vascular system. Thus, it may be the case that the lymphatic branching structure is optimized to enhance lymph mixing, particle exchange, or immune cell transport, which are particularly germane to the use of lymphatics as drug delivery routes.

Fractal analysis of concurrently prepared latex rubber casts of the bronchial and vascular systems of the human lung

Fractal geometry (FG) is a branch of mathematics that instructively characterizes structural complexity. Branched structures are ubiquitous in both the physical and the biological realms. Fractility has therefore been termed nature's design. The fractal properties of the bronchial (airway) system, the pulmonary artery and the pulmonary vein of the human lung generates large respiratory surface area that is crammed in the lung. Also, it permits the inhaled air to intimately approximate the pulmonary capillary blood across a very thin blood-gas barrier through which gas exchange to occur by diffusion. Here, the bronchial (airway) and vascular systems were simultaneously cast with latex rubber. After corrosion, the bronchial and vascular system casts were physically separated and cleared to expose the branches. The morphogenetic (Weibel's) ordering method was used to categorize the branches on which the diameters and the lengths, as well as the angles of bifurcation, were measured. The fractal dimensions (DF) were determined by plotting the total branch measurements against the mean branch diameters on double logarithmic coordinates (axes). The diameter-determined DF values were 2.714 for the bronchial system, 2.882 for the pulmonary artery and 2.334 for the pulmonary vein while the respective values from lengths were 3.098, 3.916 and 4.041. The diameters yielded DF values that were consistent with the properties of fractal structures (i.e. self-similarity and space-filling). The data obtained here compellingly suggest that the design of the bronchial system, the pulmonary artery and the pulmonary vein of the human lung functionally comply with the Hess-Murray law or 'the principle of minimum work'.

AVATREE: An open-source computational modelling framework modelling Anatomically Valid Airway TREE conformations

This paper presents AVATREE, a computational modelling framework that generates Anatomically Valid Airway tree conformations and provides capabilities for simulation of broncho-constriction apparent in obstructive pulmonary conditions. Such conformations are obtained from the personalized 3D geometry generated from computed tomography (CT) data through image segmentation. The patient-specific representation of the bronchial tree structure is extended beyond the visible airway generation depth using a knowledge-based technique built from morphometric studies. Additional functionalities of AVATREE include visualization of spatial probability maps for the airway generations projected on the CT imaging data, and visualization of the airway tree based on local structure properties. Furthermore, the proposed toolbox supports the simulation of broncho-constriction apparent in pulmonary diseases, such as chronic obstructive pulmonary disease (COPD) and asthma. AVATREE is provided as an open-source toolbox in C++ and is supported by a graphical user interface integrating the modelling functionalities. It can be exploited in studies of gas flow, gas mixing, ventilation patterns and particle deposition in the pulmonary system, with the aim to improve clinical decision making.