3D Structural Model and Visualization of Blood Vessels Based on L-System

The insight in structures of the blood vessels is a basis for study of blood flows to help understanding the abnormalities of blood vessels that can cause vascular diseases. Basic concept used for constructing structures of blood vessels in organs is arterial branching, which is usually characterized by fractal similarity in the bifurcation pattern. In this work, the concept of Lindenmayer system (L-system) is modified for three-dimensional (3D) tree-like structures to model structures of blood vessels in organs, and then, applied to construct and visualize structural blood vessels via our software created based on openGL and Lazarus program. The structure of blood vessels is constructed based on the physiological law of arterial branching proposed Murray (Murray’s law) under additional assumptions and constraints such as the spreading of blood vessels to cover all directions, the angle condition and the non-overlapping vessels condition. The concept is applied to simulate structures of blood vessels in 3 study cases, including symmetric arterial branching, non-symmetric arterial branching and structure of blood vessel on different domains. The results of structures of blood vessels generated from all cases are measured based on the number of segments, the total blood volume and the fractal dimension. The results of modeling and simulation in this work are illustrated by comparing with other results appeared literature. Moreover, the constructed structures of the blood vessels based on this 3D L-system could be useful for future research such as blood flow, pressure and other properties involving in structures of blood vessels in different organs of human and animals. HIGHLIGHTS A new 3D L-system is developed based on directional vectors for construction of 3D tree-like structures such as structures of blood vessels The model of structures of blood vessels is constructed based on the physiological laws of arterial branching (Murray’s law) with additional assumptions on the spreading of blood vessels, the angle condition, and the non-overlapping of blood vessels Algorithm and software are developed based on L-system to simulate and visualize 3D structures of blood vessels GRAPHICAL ABSTRACT

Numerical study of a fractal-like tree node micromixer based on Murray’s law

This article focuses on the influence of fractal-like tree node (FTN) on the mixing efficiency and pressure drop of the micromixer. The mixing efficiency of FTN micromixers with different branch angle δ = 30°, 60° and 90° are compared at six kinds of Reynolds (Res). We can get that the micromixer with δ = 90° has higher mixing efficiency at any Re. The mixing results of the center FTN and the stagger FTN micromixer show that the center FTN has better mixing effect. The angle of FTN and the number of FTN are the key to improve the mixing efficiency. They are also the key to change the pressure drop in the microchannel. The FTN can slow down the pressure drop and maintain the stable pressure drop between two adjacent FTNs. The way to obtain a more stable pressure range is to increase the distance between two adjacent FTN. This provides a reliable reference for maintaining a stable pressure in the microchannel.

Lepidoptera demonstrate the relevance of Murray’s Law to circulatory systems with tidal flow

Background Murray’s Law, which describes the branching architecture of bifurcating tubes, predicts the morphology of vessels in many amniotes and plants. Here, we use insects to explore the universality of Murray’s Law and to evaluate its predictive power for the wing venation of Lepidoptera, one of the most diverse insect orders. Lepidoptera are particularly relevant to the universality of Murray’s Law because their wing veins have tidal, or oscillatory, flow of air and hemolymph. We examined over one thousand wings representing 667 species of Lepidoptera. Results We found that veins with a diameter above approximately 50 microns conform to Murray’s Law, with veins below 50 microns in diameter becoming less and less likely to conform to Murray’s Law as they narrow. The minute veins that are most likely to deviate from Murray’s Law are also the most likely to have atrophied, which prevents efficient fluid transport regardless of branching architecture. However, the veins of many taxa continue to branch distally to the areas where they atrophied, and these too conform to Murray’s Law at larger diameters (e.g., Sesiidae). Conclusions This finding suggests that conformity to Murray’s Law in larger taxa may reflect requirements for structural support as much as fluid transport, or may indicate that selective pressures for fluid transport are stronger during the pupal stage—during wing development prior to vein atrophy—than the adult stage. Our results increase the taxonomic scope of Murray’s Law and provide greater clarity about the relevance of body size.

Multiobjective Optimization of Pin-Type Flow Channels Using a Reinterpretation of Murray’s Law

Biomimetics has been used to improve performance in several fields of engineering. For flow fields, Murray’s Law has been used to explore branching of channels that carry reactants and products. The applicability of Murray’s Law to flow fields was examined here. The pin-type flow field was used to explore variations and conflicting performance objectives: pressure drop, manufacturability, standard deviation of flow velocity, and channel area. NSGA-II was used to solve a multiobjective optimization problem. Two designs, 3 × 3 and 11 × 11, were analyzed. Results that were similar to previous single-objective studies were obtained, confirming the efficacy of Murray’s Law. Computational fluid dynamics simulations were used to compare optimized and unoptimized designs. The maximum velocity for the 3 × 3 and 11 × 11 cases was lower when Murray’s Law was followed, indicating that it effectively slowed down the flow. Similarly, the flow was much more uniform: the standard deviation of flow velocity in the channels was 94% and 57% lower, respectively, for both cases, compared to the unoptimized designed. Finally, a method to select one optimal solution from a front of non-dominated solutions, the nearest point method, was demonstrated.

Hierarchy in materials for maximized efficiency

This perspective article gives the future research direction on the application of the generalized Murray's law for the design of porous hierarchy in materials and the establishment of a general materials design theory 'law of hierarchy' taking four types of hierarchy into account.