Anatomically based finite element models of the human pulmonary arterial and venous trees including supernumerary vessels.

Studies of the origin of pulmonary blood flow heterogeneity have highlighted the significant role of vessel branching structure on flow distribution. To enable more detailed investigation of structure-function relationships in the pulmonary circulation, an anatomically based finite element model of the arterial and venous networks has been developed to more accurately reflect the geometry found in vivo. Geometric models of the arterial and venous tree structures are created using a combination of multidetector row X-ray computed tomography imaging to define around 2,500 vessels from each tree, a volume-filling branching algorithm to generate the remaining accompanying conducting vessels, and an empirically based algorithm to generate the supernumerary vessel geometry. The explicit generation of supernumerary vessels is a unique feature of the computational model. Analysis of branching properties and geometric parameters demonstrates close correlation between the model geometry and anatomical measures of human pulmonary blood vessels. A total of 12 Strahler orders for the arterial system and 10 Strahler orders for the venous system are generated, down to the equivalent level of the terminal bronchioles in the bronchial tree. A simple Poiseuille flow solution, assuming rigid vessels, is obtained within the arterial geometry of the left lung, demonstrating a large amount of heterogeneity in the flow distribution, especially with inclusion of supernumerary vessels. This model has been constructed to accurately represent available morphometric data derived from the complex asymmetric branching structure of the human pulmonary vasculature in a form that will be suitable for application in functional simulations.     

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.     

Fractal dimensions and multifractility in vascular branching

A definition for the fractal dimension of a vascular tree is proposed based on the hemodynamic function of the tree and in terms of two key branching parameters: the asymmetry ratio of arterial bifurcations and the power law exponent governing the relation between vessel diameter and flow. Data from the cardiovascular system, which generally exhibit considerable scatter in the values of these two parameters, are found to produce the same degree of scatter in the value of the fractal dimension. When this scatter is explored for a multifractal pattern, however, it is found that the required collapse onto a single curve is achieved in terms of the coarse H?lder exponent. Thus, the presence of multifractility is confirmed, and the legitimacy of the defined dimension is affirmed in the sense of the theoretical Hausdorff limit in as much as this limit can be reached with experimental data.     

The multifractal structure of arterial trees

Fractal properties of arterial trees are analysed using the cascade model of turbulence theory. It is shown that the branching process leads to a non-uniform structure at the micro-level meaning that blood supply to the tissue varies in space. From the model it is concluded that, depending on the branching parameter, vessels of a specific size contribute dominantly to the blood supply of tissue. The corresponding tissue elements form a dense set in the tissue. Furthermore, if blood flow in vessels can get obstructed with some probability, the above set of tissue elements may not be dense anymore. Then there is the risk that, spread out over the tissue, nutrient and gas exchange fall short.     

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.     

A model for renal arterial branching based on graph theory

The kidney is one of the most complicated organs in terms of structure and physiology, in part because it is highly vascularized. The renal vascular development occurs through two mechanisms that sometimes overlap: vasculogenesis and angiogenesis. Here, we consider angiogenesis to model the renal arterial tree with the two processes of vascular angiogenesis: sprouting and splitting. We recognize the vessels are not tubes with ends that get glued but physiological factors are relevant into the vascular development. Our contribution integrates the graph theory and physiological information to derive a quantitative model for the vascular tree in the sense that the vertices and edges represent, respectively, a branching point and a vessel. From such a premise, development of the arterial vascular tree of the kidney is mathematically expressed, including physiological processes as the effect of the vascular endothelial growth factor (VEGF) on the vessel length. A definition of the graph is used to visualize the topology of vascular tree in kidney providing physiological information into the edges. Thus, renal arterial branching is modeled as a graph where edges are labeled and oriented.     

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.     

The branching angles in computer-generated optimized models of arterial trees

The structure of a complex arterial tree model is generated on the computer using the newly developed method of "constrained constructive optimization." The model tree is grown step by step, at each stage of development fulfilling invariant boundary conditions for pressures and flows. The development of structure is governed by adopting minimum volume inside the vessels as target function. The resulting model tree is analyzed regarding the relations between branching angles and segment radii. Results show good agreement with morphometric measurements on corrosion casts of human coronary arteries reported in the literature.     

Vessel tortuousity and reduced vascularization in the fetoplacental arterial tree after maternal exposure to polycyclic aromatic hydrocarbons

Polycyclic aromatic hydrocarbons (PAHs) are ubiquitous environmental pollutants and the main toxicants found in cigarettes. Women are often exposed to PAHs before pregnancy, typically via prepregnancy smoking. To determine how prepregnancy exposure affects the fetoplacental vasculature of the placenta, we exposed female mice to PAHs before conception, perfused the fetoplacental arterial trees with X-ray contrast agent, and imaged the vasculature ex vivo by microcomputed tomography (micro-CT) at embryonic day 15.5. Automated vascular segmentation and flow calculations revealed that in control trees, <40 chorionic plate vessels (diameter>180 μm) gave rise to ～1,300 intraplacental arteries (50-180 μm), predicting an arterial vascular resistance of 0.37±0.04 mmHg·s·μl(-1). PAH exposure increased vessel curvature of chorionic plate vessels and significantly increased the tortuousity ratio of the tree. Intraplacental arteries were reduced by 17%, primarily due to a 27% decrease in the number of arteriole-sized (50-100 μm) vessels. There were no changes in the number of chorionic vessels, the depth or span of the tree, the diameter scaling coefficient, or the segment length-to-diameter ratio. PAH exposure resulted in a tree with a similar size and dichotomous branching structure, but one that was comparatively sparse so that arterial vascular resistance was increased by 30%. Assuming the same pressure gradient, blood flow would be 19% lower. Low flow may contribute to the 23% reduction observed in fetal weight. New insights into the specific effects of PAH exposure on a developing arterial tree were achieved using micro-CT imaging and automated vascular segmentation analysis.     

A class of flow bifurcation models with lognormal distribution and fractal dispersion

We report a quantitative analysis of a simple dichotomous branching tree model for blood flow in vascular networks. Using the method of moment-generating function and geometric Brownian motion from stochastic mathematics, our analysis shows that a vascular network with asymmetric branching and random variation at each bifurcating point gives rise to an asymptotic lognormal flow distribution with a positive skewness. The model exhibits a fractal scaling in the dispersion of the regional flow in the branches. Experimentally measurable fractal dimension of the relative dispersion in regional flow is analytically calculated in terms of the asymmetry and the variance at local bifurcation; hence the model suggests a powerful method to obtain the physiological information on local flow bifurcation in terms of flow dispersion analysis. Both the fractal behavior and the lognormal distribution are intimately related to the fact that it is the logarithm of flow, rather than flow itself, which is the natural variable in the tree models. The kinetics of tracer washout is also discussed in terms of the lognormal distribution.     

兴安落叶松分枝格局的分形特征

对于树木分枝格局分形特征的定量描述,可以加深对树木生长过程的理解。本文采用分形几何学对兴安落叶松(Larix gmelini)的分枝格局进行研究,结果表明1)兴安落叶松分枝格局是一种分形结构,存在自相似性。2)兴安落叶松分枝格局的分形维数介于1.4~1.7之间,揭示了它的结构复杂性程度和占据生态空间、利用生态空间的能力。分形维数在树木光合作用及生长发育研究中是一个有用的参数。     

Modeling of intraorgan arterial vasculatures. II. Propagation of pressure waves

The dependences of the wave conductivity of self-similar dichotomous branching models of intraorgan arterial vasculatures on the model parameters have been calculated. It was found that, with different sets of parameters, it is possible to model the sucking effect connected with negative wave reflection at the arterial branching as well as the resonant properties of arterial beds. It was shown that the selection of an adequate model for a given intraorgan vasculature should be based on the agreement between the biophysical characteristics of the model and the vasculature that reflect the propagation and reflection of pulse waves.     

Relation between diameter and flow in major branches of the arch of the aorta.

In the analysis of arterial branching the classical "cube law' has provided a working model for the relation between the diameter of a blood vessel and the flow which the vessel carries on a long-term basis. The law has shown good agreement with biological data, but questions remain regarding its applicability to all levels of the arterial tree. The present study tests the hypothesis that the cube law may not be valid in the first few generations of the arterial tree, where vessel capacitance and gross anatomy may play important roles. Biological data have shown some support for this hypothesis in the past but the heterogeneity characteristic of past data has not allowed a conclusive test so far. We present new data which have been obtained from the same location on the arterial tree and in sufficient number to make this test possible for the first time. Also, while past tests have been based primarily on correlation of the measured data with an assumed power law, we show here that this can be misleading. The present data allow a simpler test which does not involve correlation and which leads to more direct conclusions. For the vessels surveyed, the results show unequivocally that the relation between diameter and flow is governed by a 'square law' rather than the classical cube law. Coupled with past findings this suggests that the square law may apply at the first few levels of the arterial tree, while the cube law continues from there to perhaps the precapillary levels.     

Fractal properties of perfusion heterogeneity in optimized arterial trees: a model study

Regional blood flows in the heart muscle are remarkably heterogeneous. It is very likely that the most important factor for this heterogeneity is the metabolic need of the tissue rather than flow dispersion by the branching network of the coronary vasculature. To model the contribution of tissue needs to the observed flow heterogeneities we use arterial trees generated on the computer by constrained constructive optimization. This method allows to prescribe terminal flows as independent boundary conditions, rather than obtaining these flows by the dispersive effects of the tree structure. We study two specific cases: equal terminal flows (model 1) and terminal flows set proportional to the volumes of Voronoi polyhedra used as a model for blood supply regions of terminal segments (model 2). Model 1 predicts, depending on the number Nterm of end-points, fractal dimensions D of perfusion heterogeneities in the range 1.20 to 1.40 and positively correlated nearest-neighbor regional flows, in good agreement with experimental data of the normal heart. Although model 2 yields reasonable terminal flows well approximated by a lognormal distribution, it fails to predict D and nearest-neighbor correlation coefficients r1 of regional flows under normal physiologic conditions: model 2 gives D = 1.69 +/- 0.02 and r1 = -0.18 +/- 0.03 (n = 5), independent of Nterm and consistent with experimental data observed under coronary stenosis and under the reduction of coronary perfusion pressure. In conclusion, flow heterogeneity can be modeled by terminal positions compatible with an existing tree structure without resorting to the flow-dispersive effects of a specific branching tree model to assign terminal flows.     

Dominance of geometric overelastic factors in pulse transmission through arterial branching

The branching characteristic of the arterial system is such that blood pressure pulses propagate with minimum loss. This characteristic depends on the geometric and elastic properties of branching vessels. In the current investigation, mathematical relations of branching geometry and elastic properties are formulated and their relative contributions to pulse reflection at an arterial junction are analyzed. Results show that alteration of pulse transmission through the junction is more significantly affected by changes in branching vessel radii and wall thickness than by corresponding percentage changes in vessel wall elastic moduli.     

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.     

A non-uniform equivalent cable model of membrane voltage changes in a passive dendritic tree

A non-uniform equivalent cable model of membrane voltage changes in a passive dendritic tree extending Rall's equivalent cylinder model is presented. It is obtained from a combination of cable theory with the continuum approach. Replacing the fine structure of the branching dendrites by an equivalent, conductive medium characterized by averaged electrical parameters, the one-dimensional cable equations with spatially varying parameters are derived. While these equations can be solved in general only numerically, we were able to formulate a general branching condition (comprising Rall's 3/2 power relationship as a special case) under which analytical solutions can be deduced from those of the equivalent cylinder model. This model allows dendritic trees with a greater variety of branching patterns than before to be analytically treated.     

Diameters, generations, and orders of branches in the bronchial tree

Studies of bronchial tree data by West et al. (J. Appl. Physiol. 60: 1089-1097, 1986) have shown that plots of mean diameter against generation, using log-log scales, can be represented by a power function with harmonic modulations. Other studies have shown that the mean diameter of the airways is exponentially related to order of branching. This paper demonstrates that both observations are compatible with a fractal model of branching, and because airway branching is fractal, this may explain why both are also true of the bronchial tree. Furthermore, the exponential relationship of mean diameter with generation in the larger airways, demonstrated by Weibel, is shown to result from the exponential relation of diameter with order in the fractal model.     

Analysis of blood flow in the entire coronary arterial tree

A hemodynamic analysis of coronary blood flow must be based on the measured branching pattern and vascular geometry of the coronary vasculature. We recently developed a computer reconstruction of the entire coronary arterial tree of the porcine heart based on previously measured morphometric data. In the present study, we carried out an analysis of blood flow distribution through a network of millions of vessels that includes the entire coronary arterial tree down to the first capillary branch. The pressure and flow are computed throughout the coronary arterial tree based on conservation of mass and momentum and appropriate pressure boundary conditions. We found a power law relationship between the diameter and flow of each vessel branch. The exponent is approximately 2.2, which deviates from Murray's prediction of 3.0. Furthermore, we found the total arterial equivalent resistance to be 0.93, 0.77, and 1.28 mmHg.ml(-1).s(-1).g(-1) for the right coronary artery, left anterior descending coronary artery, and left circumflex artery, respectively. The significance of the present study is that it yields a predictive model that incorporates some of the factors controlling coronary blood flow. The model of normal hearts will serve as a physiological reference state. Pathological states can then be studied in relation to changes in model parameters that alter coronary perfusion.     

Network analysis of an arterial tree

The arterial tree of a Sprague-Dawley rat was casted and carefully mapped with the aim of comparing its network characteristics with those suggested by the classical model of an arterial tree. It is shown that if the tree is to be measured accurately, the concept of 'whole vessels' on which the classical model is based must be abandoned since such vessels do not actually exist in the network, nor can they be accurately defined. The concept of 'vessel segments' is proposed instead and its use is demonstrated. A total of 1313 vessel segments in the arterial tree of the rat are mapped and divided into well defined 'levels'. The length and diameter of each segment are measured and the distribution and averages of these at different levels are presented as indicators of the branching characteristics of the tree.