Arterial branching in man and monkey

Vessel diameters and branching angles are measured from a large number of arterial bifurcations in the retina of a normal human subject and in that of a rhesus monkey. The results are compared with each other and with theoretical results on this subject.     

Nonsymmetrical bifurcations in arterial branching

The results of optimality studies of the branching angles of arterial bifurcations are extended to nonsymmetrical bifurcations. Predicted nonsymmetrical bifurcations are found to be not unlike those observed in the cardiovascular system.     

植物分枝模型

从生物体总是最有效地利用物质的思想出发,对植物分枝形状建立了一个数学模型。该模型认为,当主干与侧枝的截面积之间存在类似平行四边形法则的关系时,分枝的体积取极小值。该模型揭示植物分枝形态不仅符合力学平衡的原则,在进化上也有显著生物学意义。根据实测数据提出了偏移度的概念,认为分枝形态建成与个体内部枝条相互作用有关,植物分枝取向受空间效应影响时仍满足体积最小的原则。由于此模型具有随机性,遵循此模型的分枝可呈现千变万化的形式。模型将分枝角度与分枝截面积有机结合起来,可作为计算机模拟植物的方法。     

Which diameter and angle rule provides optimal flow patterns in a coronary bifurcation?

The branching angle and diameter ratio in epicardial coronary artery bifurcations are two important determinants of atherogenesis. Murray's cubed diameter law and bifurcation angle have been assumed to yield optimal flows through a bifurcation. In contrast, we have recently shown a 7/3 diameter law (HK diameter model), based on minimum energy hypothesis in an entire tree structure. Here, we derive a bifurcation angle rule corresponding to the HK diameter model and critically evaluate the streamline flow through HK and Murray-type bifurcations. The bifurcations from coronary casts were found to obey the HK diameter model and angle rule much more than Murray's model. A finite element model was used to investigate flow patterns for coronary artery bifurcations of various types. The inlet velocity and pressure boundary conditions were measured by ComboWire. Y-bifurcation of Murray type decreased wall shear stress-WSS (10%-40%) and created an increased oscillatory shear index-OSI in atherosclerosis-prone regions as compared with HK-type bifurcations. The HK-type bifurcations were found to have more optimal flow patterns (i.e., higher WSS and lower OSI) than Murray-type bifurcations which have been traditionally believed to be optimized. This study has implications for changes in bifurcation angles and diameters in percutaneous coronary intervention.     

Arterial bifurcations in the cardiovascular system of a rat

Arterial bifurcations in the cardiovascular system of a rat were studied, using a resin cast of the entire arterial tree. At each bifurcation, measurements were made of the diameters of the three vessels involved, the two branching angles, and the angle delta, which the parent artery makes with the plane containing the two branches. The results were found to be consistent with those reported previously in man and monkey. In addition, measurements of delta in the present study indicate that arterial bifurcations are mostly two dimensional.     

Basal EDRF activity helps to keep the geometrical configuration of arterial bifurcations close to the Murray optimum

We have used X-ray microangiography to investigate the hypothesis that the potent endogenous vasodilator endothelium-derived relaxing factor (EDRF) contributes to the maintenance of "optimality" in vascular branching by modulating the diameters of the parent (D0) and daughter (D1 and D2) arteries at bifurcations. Five anatomically different types of bifurcation were studied in buffer-perfused rabbit ear preparations both under resting conditions and after pharmacological constriction by 5-hydroxytryptamine (5HT). A range of flow rates (1-5 ml min-1) was employed as release of EDRF from endothelial cells is stimulated by shear stress. Experimental data obtained in the presence and absence of EDRF activity were compared with theoretical predictions in three ways. (1) Junction exponents (x) were determined at each bifurcation from the equation Dx1 + Dx2 = Dx0, and their frequency distributions constructed. Murray (1926a, Proc. natn. Acad. Sci., U.S.A. 12, 207-214; 1926b, J. gen. Physiol. 9, 835-841.) proposed that x will be exactly 3 if power losses and intravascular volume are minimized simultaneously. In unconstricted preparations, either in the presence or absence of EDRF activity, and in preparations constricted by 0.1 microM 5HT in the presence of EDRF activity, the modes and medians of the frequency distributions of x were found to be close to 3 at all flow rates. In contrast, in 0.1 microM 5HT-constricted preparations in the absence of EDRF activity, no single mode common to all flow rates was apparent and medians were significantly larger at all flow rates. (2) Theoretically "optimal" branching angles were derived from experimental diameter measurements using four mathematical models which minimize respectively the total surface area, total volume, total drag (shear stress) and total power losses at bifurcations (Murray, 1926b). These calculated branching angles were then compared with actual branching angles. EDRF activity was found to be necessary for accurate prediction of branching angles by the minimum volume and power loss models in 5HT-constricted but not in resting preparations. (3) For each model or "minimization principle", there is an optimal mathematical relationship between the junction exponent, x, and the angle between daughter arteries, psi 12, at a bifurcation (Roy & Woldenberg, 1982, Bull. math. Biol. 44, 349-360.) Experimentally determined values of x and psi 12 agreed closely with those predicted both by the minimum volume and the minimum power loss principles, except again in 5HT-constricted preparations in the absence of EDRF activity.(ABSTRACT TRUNCATED AT 250 WORDS)     

Cost of departure from optimality in arterial branching

Measurements of branching angles in the arterial tree have in the past indicated a great deal of scatter away from what is expected to be optimum on theoretical grounds. In this study the cost penalty of nonoptimum branching angles is calculated for the first time to determine how far from optimum these angles are. The results lead to the remarkable conclusion that while the scatter of the measured branching angles is fairly large, they represent deviations from optimum angles which correspond to only 2% or so penalty in cost.     

Spatial and temporal variation of secondary flow during oscillatory flow in model human central airways

Axial and secondary velocity profiles were measured in a model human central airway to clarify the oscillatory flow structure during high-frequency oscillation. We used a rigid model of human airways consisting of asymmetrical bifurcations up to third generation. Velocities in each branch of the bifurcations were measured by two-color laser-Doppler velocimeter. The secondary velocity magnitudes and the deflection of axial velocity were dependent not only on the branching angle and curvature ratio of each bifurcation, but also strongly depended on the shape of the path generated by the cascade of branches. Secondary flow velocities were higher in the left bronchus than in the right bronchus. This spatial variation of secondary flow was well correlated with differing gas transport rates between the left and right main bronchus.     

Roots and calibers of the human coronary arteries

The anatomical structure of the coronary-aortic junctions in humans is studied by using corrosion casts of the coronary network. A model is proposed for the specification of these junctions in terms of vessel diameters and branching angles, and the model is used to produce morphological data on these junctions which hitherto have not been available. This anatomical model correlates poorly with the accepted theoretical model of arterial bifurcations in the cardiovascular system. The results suggest that the structure of the coronary-aortic junctions is very different from the structure of typical arterial bifurcations and, by implication, that the flow conditions under which they function are very different. A good understanding of these junctions is important in coronary bypass surgery, where the coronary-aortic junctions are emulated by creating a new anastomosis for the graft at the base of the ascending aorta, and in coronary artery disease, where atherosclerotic lesions occur not far from the coronary-aortic junctions.     

Branching pattern of pulmonary arterial tree in anesthetized dogs

The geometry of the pulmonary arterial tree of six adult dogs was measured by a high-speed, volume-scanning, X-ray tomographic technique. After the dogs were anesthetized a catheter was advanced to the right ventricular outflow tract and 2 mL/kg Renovist contrast agent injected rapidly. During the subsequent pulmonary arterial phase of the angiogram the dogs were scanned. Three-dimensional geometry of the pulmonary arterial tree was measured in terms of vessel segment cross-sectional area, branching angles and interbranch segment lengths along axial pathways. The effect of lung inflation and phase of the cardiac cycle on geometry was shown to be most marked on vessel cross-sectional area. The geometric branching patterns in all dogs were similar. The observed, in-vivo branching pattern behaved somewhat like the branching pattern predicted from optimized models proposed by Murray, Zamir, and Uylings.     

Sex differences in intracranial arterial bifurcations

Background: Subarachnoid hemorrhage (SAH) is a serious condition, occurring more frequently in females than in males. SAH is mainly caused by rupture of an intracranial aneurysm, which is formed by localized dilation of the intracranial arterial vessel wall, usually at the apex of the arterial bifurcation. The female preponderance is usually explained by systemic factors (hormonal influences and intrinsic wall weakness); however, the uneven sex distribution of intracranial aneurysms suggests a possible physiologic factor—a local sex difference in the intracranial arteries.Objective: The aim of this study was to explore sex variation in the bifurcation anatomy of the middle cerebral artery (MCA) and internal carotid artery (ICA), and the subsequent hemodynamic impact.Methods: Vessel radii and bifurcation angles were measured in patients with MCA and ICA bifurcations. Data from a previously published study of 55 patients undergoing diagnostic cerebral digital subtraction angiography at Dalcross Private Hospital in Sydney, Australia, between 2002 and 2003, were available for analysis. The measurements were used to create idealized, averaged bifurcations of the MCA and ICA for females and males. Computational fluid dynamics simulations were performed to calculate hemodynamic forces in the models.Results: The vessel radii and bifurcation angles of 47 MCA and 52 ICA bifurcations in 49 patients (32 females, 17 males; mean age, 53 years; age range, 14–86 years) were measured. Statistically significant sex differences were found in vessel diameter (males larger than females; P < 0.05), but not in bifurcation angle. Computational fluid dynamics simulations revealed higher wall shear stress in the female MCA (19%) and ICA (50%) bifurcations compared with the male bifurcations.Conclusions: This study of MCA and ICA bifurcations in female and male patients suggests that sex differences in vessel size and blood flow velocity result in higher hemodynamic forces acting on the vessel wall in females. This new hypothesis may partly explain why intracranial aneurysms and SAH are more likely to occur in females than in males.     

Blood flow downstream of a two-dimensional bifurcation

Many cardiovascular lesions such as aneurysms, intimal cushions, and atherosclerotic plaques tend to occur near bifurcations. This suggests that hemodynamic factors may be involved. Since measuring devices (such as anemometers) are still too large to allow local measurements of flow disturbances, we have attempted to predict the nature of these factors mathematically. Biological variables include pulsatile flow of a nonNewtonian fluid in distensible branching vessels with different angles and flow rates. Our initial analysis considers the flow in a two-dimensional bifurcation with a symmetrical flow divider perfused with steady flow at variable Reynolds numbers. At all flows, high shear forces develop on either side of the flow divider (i.e. at the apex of the bifurcation). With high flows, regions of sluggish or reverse flow develop near the outer walls of the bifurcation. The analysis confirms that the flow at the apex is quite different from that at the outer angles and that the latter varies more with flow rate than the former.     

A two-phase growth strategy in cultured neuronal networks as reflected by the distribution of neurite branching angles

Neurite outgrowth and branching patterns are instrumental in dictating the wiring diagram of developing neuronal networks. We study the self-organization of single cultured neurons into complex networks focusing on factors governing the branching of a neurite into its daughter branches. Neurite branching angles of insect ganglion neurons in vitro were comparatively measured in two neuronal categories: neurons in dense cultures that bifurcated under the presence of extrinsic (cellular environment) cues versus neurons in practical isolation that developed their neurites following predominantly intrinsic cues. Our experimental results were complemented by theoretical modeling and computer simulations. A preferred regime of branching angles was found in isolated neurons. A model based on biophysical constraints predicted a preferred bifurcation angle that was consistent with this range shown by our real neurons. In order to examine the origin of the preferred regime of angles we constructed simulations of neurite outgrowth in a developing network and compared the simulated developing neurons with our experimental results. We tested cost functions for neuronal growth that would be optimized at a specific regime of angles. Our results suggest two phases in the process of neuronal development. In the first, reflected by our isolated neurons, neurons are tuned to make first contact with a target cell as soon as possible, to minimize the time of growth. After contact is made, that is, after neuronal interconnections are formed, a second branching strategy is adopted, favoring higher efficiency in neurite length and volume. The two-phase development theory is discussed in relation to previous results.     

A two‐phase growth strategy in cultured neuronal networks as reflected by the distribution of neurite branching angles

Neurite outgrowth and branching patterns are instrumental in dictating the wiring diagram of developing neuronal networks. We study the self‐organization of single cultured neurons into complex networks focusing on factors governing the branching of a neurite into its daughter branches. Neurite branching angles of insect ganglion neurons in vitro were comparatively measured in two neuronal categories: neurons in dense cultures that bifurcated under the presence of extrinsic (cellular environment) cues versus neurons in practical isolation that developed their neurites following predominantly intrinsic cues. Our experimental results were complemented by theoretical modeling and computer simulations. A preferred regime of branching angles was found in isolated neurons. A model based on biophysical constraints predicted a preferred bifurcation angle that was consistent with this range shown by our real neurons. In order to examine the origin of the preferred regime of angles we constructed simulations of neurite outgrowth in a developing network and compared the simulated developing neurons with our experimental results. We tested cost functions for neuronal growth that would be optimized at a specific regime of angles. Our results suggest two phases in the process of neuronal development. In the first, reflected by our isolated neurons, neurons are tuned to make first contact with a target cell as soon as possible, to minimize the time of growth. After contact is made, that is, after neuronal interconnections are formed, a second branching strategy is adopted, favoring higher efficiency in neurite length and volume. The two‐phase development theory is discussed in relation to previous results. © 2004 Wiley Periodicals, Inc. J Neurobiol, 2005     

Orientational preferences shown by microspikes of growing nerve cells in vitro

The shapes of microspikes on single neurones cultured in vitro have been analysed with respect to angles of bending and branching. Characteristic frequency distributions were found in all of the four categories of angles. Certain `preferred' orientations of bending and branching were seen to center about 60°, 90°, and 120°. The possible cellular basis for such behavior is discussed, as well as the similarity of bending and branching in the axonal-microspike system of the single nerve cell to the analogous branching in the vertebrate blood vascular system.     

An Experimental Investigation of the Resistance of Model Root Systems to Uprooting

The architecture of a tree root system may influence its abilityto withstand uprooting by wind loading. To determine how theroot branching pattern may alter the anchorage efficiency ofa tree, artificial model root systems with different topologiesand branching angles were built. The root systems were embeddedat various depths in wet sand and the pull-out resistance measured.A model to predict the uprooting resistance from the data collectedwas designed, allowing predictions of anchorage strength withregards to architecture. The dominant factors influencing pull-outresistance were the depth and length of roots in the soil. Themost efficient type of branching pattern predicted by the programwas one with an increased number of roots deep in the soil.The optimum branching angle most likely to resist pull-out isa vertical angle of 90° between a lateral and the main axis.The predicted mechanically optimal radial angle between a lateralbranch and its daughter is between 0 and 20°. Values ofbranching angle are compared with those measured in real woodyroot systems of European larch and Sitka spruce. Root architecture; root anchorage; pull-out resistance; windthrow; Picea sitchensis ; Larix decidua     

BRANCH GEOMETRY AND EFFECTIVE LEAF AREA: A STUDY OF TERMINAUA-BRANCHING PATTERN. 1. THEORETICAL TREES

Single lateral branches and branch tiers of Terminalia catappa L. are simulated and drawn by computer. Leaf clusters on the branches are approximated by discs, and the effective leaf areas are determined by use of Dirichlet domains. Theoretical optimal branching angles which produce the maximum effective leaf area are obtained from simulations. Symmetrical and asymmetrical branching angles are contrasted; the latter characterize real trees. Varying leaf disc radius and ratio of branch-unit lengths affects optimal branching angles, as does the symmetry of a tier of five branches. Leaf area indices for individual branches and branch tiers are given for all simulations. The number of branches in a tier has a major effect on leaf area index and effective leaf area. The theoretical optimal branching angles of many simulations are very close to the values observed in real trees of T. catappa. We conclude that the observed branching angles and number of branches in a tier of this species optimize light interception within constraints of a fixed pattern of branching, one that is widespread among tropical trees.     

Predicting root biomass from branching patterns of Douglas-fir root systems

There are many examples of branching networks in nature, such as tree crowns, river systems, arteries and lungs. These networks have often been described as being self-similar, or following scale-invariant branching rules, and this property has been used to derive several scaling laws. In this paper we model root systems of Douglas-fir ( Pseudotsuga menziesii var. glauca (Beissn.) Franco) as branching networks following several simple branching rules. Our objective is to establish a relationship between trunk diameter and root biomass. We explore the effect of the self-similar branching assumption on this relationship. Using data collected from a mature stand in British Columbia, we find that branching asymmetry and the rate of root taper change with root size, thereby violating the assumption of self-similarity. However, the data are in general agreement with Leonardo da Vinci's area-preserving branching hypothesis. We use the field data to parameterize two models, one assuming self-similar branching and a second incorporating the measured size dependencies of branching parameters. The two models differ by only a small amount (≈8%) in their predictions. For both models, the predicted relationship between trunk diameter and root biomass is in good concordance with previously published empirical data. We conclude that the assumption of self-similar branching, although violated by the data, nevertheless provides a useful tool for predicting the allometric relationship between trunk diameter and root biomass. Finally, we use our models to show that the geometric properties of individual bifurcations fundamentally change the root biomass cost of different root topologies.     

Cost analysis of arterial branching in the cardiovascular systems of man and animals

A new scheme is presented whereby data on arterial branching can be interpreted in terms of direct cost to the physiological system. The scheme makes it possible to assess, at a glance, the true degree of optimality of an arterial network. Departure from optimality is indicated in terms of cost, rather than in terms of the difference between theoretical and measured branching angles. The scheme is applied to several groups of biological data and new conclusions are reached with regard to their degrees of optimality.     

Lateral inhibition in the astogeny of conical multistiped graptolites

Conical multistiped graptolites (in particular Rhabdinopora) grow in patterns which can be understood using a lateral inhibition model based on diffusion of nutrients or pheromones around the colony. Many aspects of rhabdosome morphology are explained with this model. These include branching and the distribution of branching points, the morphology of bifurcations, the almost constant spacing of stipes, the high rate of branching in early stages, the variation in expansion rate, and regeneration patterns after damage. For the first time, a full three-dimensional computer simulation of graptolite growth has been accomplished, assuming biologically justified processes instead of formal growth rules. The simulations indicate that dissepiments were instrumental in controlling rhabdosome shape, but not sufficient to keep its horizontal section perfectly circular.