On fractal properties of arterial trees

The question of fractal properties of arterial trees is considered in light of data from the extensive tree structure of the right coronary artery of a human heart. Because of the highly non-uniform structure of this tree, the study focuses on the purely geometrical rather than statistical aspects of fractal properties. The large number of arterial bifurcations comprising the tree were found to have a mixed degree of asymmetry at all levels of the tree, including the depth of the tree where it has been generally supposed that they would be symmetrical. Cross-sectional area ratios of daughter to parent vessels were also found to be highly mixed at all levels, having values both above and below 1.0, rather than consistently above as has been generally supposed in the past. Calculated values of the power law index which describes the theoretical relation between the diameters of the three vessel segments at an arterial bifurcation were found to range far beyond the two values associated with the cube and square laws, and not clearly favoring one or the other. On the whole the tree structure was found to have what we have termed "pseudo-fractal" properties, in the sense that vessels of different calibers displayed the same branching pattern but with a range of values of the branching parameters. The results suggest that a higher degree of fractal character, one in which the branching parameters are constant throughout the tree structure, is unlikely to be attained in non-uniform vascular structures.     

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.     

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.     

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.     

Development of a mathematical method for classifying and comparing tree architecture using parameters from a topological model of a trifurcating botanical tree

This paper describes a model for the topological mapping of trifurcating botanical trees. The model was based on a system of modular units that represented the interconnectivity of shoot meristems (terminal segments) and internodes (internal segments) within whole plant canopies, organized with increasing centrifugal ordering. The model was capable of describing the dynamics of plant growth as expressed by changes in topological parameters over time. Preliminary calculations for experimental trees indicated that the model represents growth in a biologically sound manner. Methods are described for the calculation of the architecture parameters size, size-complexity, structural complexity, and tree asymmetry index (TAI). Parameter calculations were based on the mathematical principles developed for the classification of bifurcating dendrite trees, and were designed to both extract structural information, and to enable statistical comparison between trees of different size. Parameters were mathematically adjusted for trifurcation, and appeared to be able to represent quantitatively the architectural properties of tree structures. In addition to the calculation of the TAI for trifurcating trees, new methods were developed to enable comparisons to be made of the architectural complexity of trifurcating trees of differing size. These were based on the principle of the pair-wise comparison of the mean centrifugal order number (MCON) with respect to segments against highest order number. We argue and illustrate that this principle can be more informative than that of pair-wise comparison of the MCON against tree degree (topological size). Further improvements to this method were made by examining branching points (vertices) rather than segments (links) to calculate the MCON.     

Tree asymmetry—A sensitive and practical measure for binary topological trees

The topological structure of a binary tree is characterized by a measure called tree asymmetry, defined as the mean value of the asymmetry of its partitions. The statistical properties of this tree-asymmetry measure have been studied using a growth model for binary trees. The tree-asymmetry measure appears to be sensitive for topological differences and the tree-asymmetry expectation for the growth model that we used appears to be almost independent of the size of the trees. These properties and the simple definition make the measure suitable for practical use, for instance for characterizing, comparing and interpreting sets of branching patterns. Examples are given of the analysis of three sets of neuronal branching patterns. It is shown that the variance in tree-asymmetry values for these observed branching patterns corresponds perfectly with the variance predicted by the used growth model.     

Using L-systems for modeling source-sink interactions, architecture and physiology of growing trees: the L-PEACH model

Functional-structural plant models simulate the development of plant structure, taking into account plant physiology and environmental factors. The L-PEACH model is based on the development of peach trees. It demonstrates the usefulness of L-systems in constructing functional-structural models. L-PEACH uses L-systems both to simulate the development of tree structure and to solve differential equations for carbohydrate flow and allocation. New L-system-based algorithms are devised for simulating the behavior of dynamically changing structures made of hundreds of interacting, time-varying, nonlinear components. L-PEACH incorporates a carbon-allocation model driven by source-sink interactions between tree components. Storage and mobilization of carbohydrates during the annual life cycle of a tree are taken into account. Carbohydrate production in the leaves is simulated based on the availability of water and light. Apices, internodes, leaves and fruit grow according to the resulting local carbohydrate supply. L-PEACH outputs an animated three-dimensional visual representation of the growing tree and user-specified statistics that characterize selected stages of plant development. The model is applied to simulate a tree's response to fruit thinning and changes in water stress. L-PEACH may be used to assist in horticultural decision-making processes after being calibrated to specific trees.     

Emergence of matched airway and vascular trees from fractal rules

The bronchial, arterial, and venous trees of the lung are complex interwoven structures. Their geometries are created during fetal development through common processes of branching morphogenesis. Insights from fractal geometry suggest that these extensively arborizing trees may be created through simple recursive rules. Mathematical models of Turing have demonstrated how only a few proteins could interact to direct this branching morphogenesis. Development of the airway and vascular trees could, therefore, be considered an example of emergent behavior as complex structures are created from the interaction of only a few processes. However, unlike inanimate emergent structures, the geometries of the airway and vascular trees are highly stereotyped. This review will integrate the concepts of emergence, fractals, and evolution to demonstrate how the complex branching geometries of the airway and vascular trees are ideally suited for gas exchange in the lung. The review will also speculate on how the heterogeneity of blood flow and ventilation created by the vascular and airway trees is overcome through their coordinated construction during fetal development.     

Subdivision in an ancestral species creates asymmetry in gene trees

We consider gene trees in three species for which the species tree is known. We show that population subdivision in ancestral species can lead to asymmetry in the frequencies of the two gene trees not concordant with the species tree and, if subdivision is extreme, cause the one of the nonconcordant gene trees to be more probable than the concordant gene tree. Although published data for the human-chimp-gorilla clade and for three species of Drosophila show asymmetry consistent with our model, sequencing error could also account for observed patterns. We show that substantial levels of persistent ancestral subdivision are needed to account for the observed levels of asymmetry found in these two studies.     

Evaluating a non-destructive method for calibrating tree biomass equations derived from tree branching architecture

Key message

Functional branch analysis (FBA) is a promising non-destructive method that can produce accurate tree biomass equations when applied to trees which exhibit fractal branching architecture.

Abstract

Functional branch analysis (FBA) is a promising non-destructive alternative to the standard destructive method of tree biomass equation development. In FBA, a theoretical model of tree branching architecture is calibrated with measurements of tree stems and branches to estimate the coefficients of the biomass equation. In this study, species-specific and mixed-species tree biomass equations were derived from destructive sampling of trees in Western Kenya and compared to tree biomass equations derived non-destructively from FBA. The results indicated that the non-destructive FBA method can produce biomass equations that are similar to, but less accurate than, those derived from standard methods. FBA biomass prediction bias was attributed to the fact that real trees diverged from fractal branching architecture due to highly variable length–diameter relationships of stems and branches and inaccurate scaling relationships for the lengths of tree crowns and trunks assumed under the FBA model.     

Quantifying the degree of self-nestedness of trees: application to the structural analysis of plants

In this paper, we are interested in the problem of approximating trees by trees with a particular self-nested structure. Self-nested trees are such that all their subtrees of a given height are isomorphic. We show that these trees present remarkable compression properties, with high compression rates. In order to measure how far a tree is from being a self-nested tree, we then study how to quantify the degree of self-nestedness of any tree. For this, we define a measure of the self-nestedness of a tree by constructing a self-nested tree that minimizes the distance of the original tree to the set of self-nested trees that embed the initial tree. We show that this measure can be computed in polynomial time and depict the corresponding algorithm. The distance to this nearest embedding self-nested tree (NEST) is then used to define compression coefficients that reflect the compressibility of a tree. To illustrate this approach, we then apply these notions to the analysis of plant branching structures. Based on a database of simulated theoretical plants in which different levels of noise have been introduced, we evaluate the method and show that the NESTs of such branching structures restore partly or completely the original, noiseless, branching structures. The whole approach is then applied to the analysis of a real plant (a rice panicle) whose topological structure was completely measured. We show that the NEST of this plant may be interpreted in biological terms and may be used to reveal important aspects of the plant growth.     

Airway branching morphology of mature and immature rabbit lungs.

The scheme of Horsfield et al. for describing the pulmonary airway tree (J Appl Physiol 52: 21-26, 1982) catalogs each airway according to its order and the difference in order of its two daughters (denoted Delta). Although this scheme captures the natural asymmetry in the airway tree, it is still deterministic, because it assumes that all airways of a given order are the same; yet such variability is extremely important in determining the overall behavior of the lungs. We therefore analyzed complete lung lobes from three mature and two immature rabbits and determined the Horsfield order and Delta of every airway down to the terminal bronchioles. We also measured the diameter of each airway. This allowed us to determine the average structure of the rabbit airway tree, the variation about this average, and also how the structures of mature and immature airway trees compare. We found some variation in branching asymmetry and airway diameter at a given order between animals but no evidence of systematic differences in structure between mature and immature lungs. We found evidence of a difference in the branching structure of the peripheral vs. the central part of the airway tree (the break point being around order 20). We also determined the nature of the variation in Delta and diameter as a function of order, which should be valuable for the development of computer models seeking to encapsulate the naturally occurring regional variation in airway geometry in the normal rabbit lung.     

Estimation of tree root lengths using fractal branching rules: a comparison with soil coring for Grevillea robusta

Previous theoretical research has suggested that lengths of tree roots can be estimated on the basis of their branching characteristics, if branching has a fractal pattern that is independent of root diameter. This theory and its underlying assumptions was tested for Grevillea robusta trees at a site in Kenya by comparing estimates of root length from conventional soil coring and the output of a fractal branching algorithm. The trees were in a 4-year-old stand established on a 3 × 4 m planting grid. Root lengths (L _r) in four units of the planting grid were estimated by soil coring. Branching characteristics determined by examination of 32 excavated roots from 16 trees were: The number of branches at each branching point; the length of links between branching points (L _l); the diameter of root tips; and parameters which describe the change in diameter at each branching point. Each was found to be independent of root size. These data were used to parameterise a branching algorithm, which was then used to estimate numbers of root links in the four grid units (n _l) from root diameters at the bases of the four trees at the corners of each unit. Root lengths, from L _r = n₁ L₁, severely underestimated L _r. This discrepancy probably resulted from inaccuracy in the parameterisation of the branching algorithm, as output from the algorithm was very sensitive to small changes in parameter values. Use of fractal branching rules alone to estimate roots length does not appear possible unless the algorithm is calibrated to adjust for errors in parameter estimation. Calibration can be achieved by calculation of an 'effective link length', L ^eff ₁, from L _r/n _l, where L _r is measured by a reference method such as soil coring.     

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 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.     

The subscapular arterial tree as a source of microvascular arterial grafts

The subscapular arterial tree may be used as a source of microvascular grafts to replace damaged or diseased portions of arteries, particularly in the hand and forearm. By studying cadaver dissections, it is possible to estimate the number of branches that may be found at different arterial segment lengths from the origin of the subscapular artery. Fifty-five preserved cadaver subscapular arterial trees were dissected, and the branching patterns were documented. Three major arterial branching patterns of the subscapular artery were observed with one, two, and three major branches to the serratus anterior in 60 percent, 29 percent, and 9 percent of the cases, respectively. The authors determined the number of 1-mm-diameter, 1-cm-long branches arising from each of six 3-cm regions of the arterial tree measured from the origin of the subscapular artery to the end of the longest terminal branch. The probability of finding at least one usable terminal branch that is at least 12.0 cm in length was found to be 98 percent. Typically, there are two to five useful branches at this distance. Such information may help surgeons fine tune their process of selecting an appropriate arterial donor site for a particular arterial defect and supports the use of the subscapular arterial tree as a donor site for microvascular arterial grafts.     

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.     

Local Diameter Fully Constrains Dendritic Size in Basal but not Apical Trees of CA1 Pyramidal Neurons

Computational modeling of dendritic morphology is a powerful tool for quantitatively describing complex geometrical relationships, uncovering principles of dendritic development, and synthesizing virtual neurons to systematically investigate cellular biophysics and network dynamics. A feature common to many morphological models is a dependence of the branching probability on local diameter. Previous models of this type have been able to recreate a wide variety of dendritic morphologies. However, these diameter-dependent models have so far failed to properly constrain branching when applied to hippocampal CA1 pyramidal cells, leading to explosive growth. Here we present a simple modification of this basic approach, in which all parameter sampling, not just bifurcation probability, depends on branch diameter. This added constraint prevents explosive growth in both apical and basal trees of simulated CA1 neurons, yielding arborizations with average numbers and patterns of bifurcations extremely close to those observed in real cells. However, simulated apical trees are much more varied in size than the corresponding real dendrites. We show that, in this model, the excessive variability of simulated trees is a direct consequence of the natural variability of diameter changes at and between bifurcations observed in apical, but not basal, dendrites. Conversely, some aspects of branch distribution were better matched by virtual apical trees than by virtual basal trees. Dendritic morphometrics related to spatial position, such as path distance from the soma or branch order, may be necessary to fully constrain CA1 apical tree size and basal branching pattern.     

The shape of evolution: systematic tree topology

Three hypotheses that predict probabilities associated with various tree shapes, or topologies, are compared with observed topology frequencies for a large number of 4, 5, 6 and 7-member trees. The united data on these n-member trees demonstrate that both the equiprobable and proportional-to-distinguishable-types hypotheses poorly predict tree topologies, while all observed topology frequencies are similar to predictions of a simple Markovian dichotomous branching hypothesis. Differences in topology frequencies between phenetic and non-phenetic trees are observed, but their statistical significance is uncertain. Relative frequencies of highly asymmetrical topologies are larger, and those of symmetrical topologies are smaller, in phenetic than in non-phenetic trees. The fact that a simple Markovian branching process, which assumes that each species has an equal probability of speciating in each time period, can predict tree topologies offers promise. Refinement of Markovian branching hypotheses to include the possibility of multiple furcations, differential speciation and extinction rates for different groups of organisms as well as for a single group through geological time, hybrid speciation, introgression, and lineage fusion will be necessary to produce realistic models of lineage diversification.