The arterial pattern and fractal dimension of the dog kidney.

A method has been developed by which it is possible to measure the fractal dimension of the arterial tree of the kidney. The objective of this work is to determine a method which permits us to discriminate between the architectures of specific organs by reference to a unique number, namely the fractal dimension of the arterial tree of that organ. This method opens the possibility of a new taxonomy for normal organs and for the pathological injiries related to the vascular morphology of those organs. The method that we have devised uses as its input the volume which is taken up by the arterial tree of the kidney. In order to calculate this volume we first obtained a plastic cast (the arteries were filled with Araldite CY233 plastic resin after which the organic tissues were corroded); thereafter we constructed a theoretical arterial tree having the same volume as the renal one. From this simplified tree, we were able to calculate its fractal dimension. The complete process of constructing the theoretical arterial tree and the subsequent calculation of its fractal dimension was carried out automatically by way of a computer programme to which we have given the name fractal program.     

Ontogenetic changes in the fractal geometry of the bronchial tree in Rattus norvegicus

Respiration and metabolism change dramatically over the course of the development of vertebrates. In mammals these changes may be ascribed to organogenesis and differentiation of structures involved in gas exchange and transport and the increase in size. Since young as well as mature individuals must be well-designed if the species is to survive, the physiological changes during the development should be matched with geometrical or structural adjustments of the respiratory system. The aim of this study was to evaluate changes in the fractal geometry of the bronchial tree during the postnatal development of the rat. The average fractal dimension of the bronchial tree of the rats was 1.587, but that of juveniles was larger than that of the adults. We found a significant negative correlation between age and fractal dimension. This correlation could be considered be misleading because of the difficulty of separating age/body size effects. Nevertheless, because fractal dimensions of the bronchial tree of rabbits and humans are known to be similar, 1.58 and 1.57 respectively, the body size effect may be nil. To our knowledge, this is the first report of ontogenetic changes in the fractal dimension of the bronchial tree in mammals.     

求解分形介质中分数阶扩散模型的相似方法(英文)

分形介质中输运现象的分数阶扩散方程是一个积分-偏微分方程,含有由分形Hausdorff维数d_f和反常扩散指数d_w确定的参数.对于这类方程的求解问题,给出了尺度变换群的不变子并且导出了关于尺度不变解的积分-常微分方程.最后利用Mellin变换和Fox函数得到尺度不变解.     

The Fractal Dimension as a Measure of the Quality of Habitats

Habitat fragmentation produces isolated patches characterized by increased edge effects from an originally continuous habitat. The shapes of these patches often show a high degree of irregularity: their shapes deviate significantly from regular geometrical shapes such as rectangular and elliptical ones. In fractal theory, the geometry of patches created by a common landscape transformation process should be statistically similar, i.e. their fractal dimensions and their form factors should be equal. In this paper, we analyze 49 woodlot fragments ( Pinus sylvestris L. ) in the Belgian Kempen region to study the direct relationship between a transformation process and the concomitant patch geometry. Although the fractal dimension of the woodlots is scattered (i.e. they are not statistically similar), the perimeter-area relation of the fragments is characterized by a single, 'dimension-like' exponent. This exponent suggests a certain shape homogeneity among the patches, which is confirmed by the absence of hierarchical levels associated with sharp increases of the fractal dimension at scale transitions. The interaction of different natural (soil factor, vegetation type) and anthropogenic (afforestation, urbanization) processes during patch development is assumed to have generated this feature. Comparison of the area and perimeter fractal dimension with an ecological index for habitat quality, the interior-to-edge ratio, shows that the fractal dimension is suitable for predicting interior habitat presence, which is more likely for patches with smooth perimeters and compact areas. The ratio of the area to the perimeter fractal dimension confirms this observation, with high values for high interior-to-edge ratios, characteristic for regularly shaped patches.     

恢复演替中草地斑块动态及尺度转换分析

研究了９ａ草地恢复油潜系列中斑块边界形状和斑块面积分布动态,并进行了凡度转换分析。獐茅斑块的边界分维数和斑块化指数最高,羊草斑块的较低,碱蓬斑块的斑块化指数略小于羊草斑块,边界分维数大于洋草斑块。共他类型斑块的两种指数基本上介于獐茅斑块之间,斑块化指数的年限变动滞后于斑块边界分维数,斑块边发维数在整个试验的尺度范围内符合同一自相似规律,斑块的面积分布格局在不同的尺度上有不同的自相似规律。     

Spatial leaf distribution and self-thinning exponent of<Emphasis Type="Italic"> Pinus banksiana</Emphasis> and<Emphasis Type="Italic"> Populus tremuloides</Emphasis>

A hypothesis that the pattern of spatial leaf distribution in forest canopies is numerically related to the exponent of the self-thinning relationship in even-aged monocultures was tested by evaluating the crown fractal dimension of Pinus banksiana (jack pine) and Populus tremuloides (quaking aspen) in Wood Buffalo National Park, Canada. Pure species stands that were considered the most dense for a given mean tree size were measured to establish the empirical self-thinning relationships. The value of the self-thinning exponent was estimated as –1.42 with 95% Confidence Interval (CI) (–1.47, –1.36) for Pinus banksiana, and –1.29 with 95% CI (–1.45, –1.14) for Populus tremuloides. For each species the box dimension of spatial leaf distribution was estimated from unit cylinders described by sequentially lowering in forest canopies, horizontal flaps of one of various diameters attached to the top of a height-measuring pole. The box dimension appeared as 1.95 (1.84, 2.06) for Pinus banksiana, and 2.24 (2.05, 2.43) for Populus tremuloides. By assuming that the box dimension is equivalent to the fractal dimension at the inter-population level, the self-thinning exponent was predicted to be –1.53 (–1.62, –1.45) for Pinus banksiana, and –1.33 (–1.45, –1.23) for Populus tremuloides. The empirical exponent was equivalent to that predicted from the box fractal dimension, as judged by the 95% CI of the dimensions. We conclude that spatial patterns of leaf distribution in forest canopies, as being characterized by the box fractal dimension, are closely related to the value of the self-thinning exponent in the dense monocultures of the species we examined.     

樟子松人工林树冠结构的分形分析

基于樟子松人工林7块固定标准地中的31株解析木的树冠体积和叶量,以幂函数关系(F=Av^(D/3))建立了预估树冠表面积的分形维数。同时根据生物量实测数据,建立预估叶量的生物模型Lw=0.180397D^3045903H^-1.67348。基于枝解析、树干解析数据,动态地预估了一年、二年、三年前的树冠体积,并结合树冠体积、叶量的这种幂函数关系可以动态地预估一年、二年、三年前树冠表面积的分形维数,从而反映出树冠结构的动态变化规律。为了了解不同分级样木的分维数变化情况,利用2003年调查的4块生物量标准地数据,根据单株树木各个枝条占据的空间体积与该枝条的带叶枝干重的关系,计算了各标准地不同分级样木树冠的分维数。为探讨单株样木树冠的分维数的计算提供了一种可行方法。树冠的分维数作为表征树冠的动态生长变化是一有用和可靠的指标。     

Changes of the morphometric indices and fractal dimension of the spinal cord neurons during ontogenesis of the cherry salmon Oncorhynchus masou

For the quantitative characteristics of the morphology of two groups of spinal cord neurons in the cherry salmon at the early stages of ontogenesis, the fractal dimension and some more traditional morphometric indices, such as the total length of neuronal branches, the number of terminal branches and branchpoints, and the cell area, are determined. The values of the morphometric indices and the fractal dimension of the bivariate neuronal pattern are shown to increase in the course of ontogenesis from the first to the second year of fish life. The correlation between the main morphometric value and the fractal dimension is found, and their correspondence to the morphometric modifications of the dendrite tree in the neurons investigated during ontogenesis is detected.     

Pair-preferences: a Quantitative Measure of Regularities in Protein Sequences

Abstract

We present here the results obtained by applying several different methods to quantitatively measure regularities in protein sequences based on pair-preferences. We have studied the distribution of amino acid residues, singly as well as in pairs in a large data base and have attempted this task. We confirmed the existence of well-defined pair-preferences in proteins which were shown to be remarkably absent in simulated random sequences of similar amino acid distribution. The analysis of the sequences from the SWISS-PROT data base using simple statistical tests, Fourier analysis, fractal analysis and statistical thermodynamical tests were used to derive parameters to define a natural sequence. As a consequence of the existence of pair-preferences, parameters like fractal dimension (D), spectral exponent (β), scaling parameter (H) and entropy (statistical) were found to be characteristic for natural sequences. For a reference state we chose a randomised state devoid of any pair-preference. The pair-preferences qualified well to be used as quantitative measures of regularities in protein sequences.     

A mathematical model for the distribution of hemodynamic parameters in the human retinal microvascular network

To quantitatively assess the arteriovenous distribution of hemodynamic parameters throughout the microvascular network of the human retina, we constructed a retinal microcirculatory model consisting of a dichotomous symmetric branching system. This system is characterized by a diameter exponent of 2.85, instead of 3 as dictated by Murray’s law, except for the capillary networks. The value of 2.85 was the sum of a fractal dimension (1.70) and a branch exponent (1.15) of the retinal vasculature. Following the feeding artery (central retinal artery), each bifurcation was recursively developed at a distance of an individual branch length [L(r) = 7.4r ^1.15] by a centrifugal scheme. The venular tree was formed in the same way. Using this model, we evaluated hemodynamic parameters, including blood pressure, blood flow, blood velocity, shear rate, and shear stress, within the retinal microcirculatory network as a function of vessel diameter. The arteriovenous distributions of blood pressure and velocity in the simulation were consistent with in vivo measurements in the human retina and other vascular beds of small animals. We therefore conclude that the current theoretical model was useful for quantifying hemodynamics as a function of vessel diameter within the retinal microvascular network.     

Persistent behavior in a phase-shift sequence of periodical biochemical oscillations

We present the analysis of a phase-shift sequence obtained from random transitions between periodic solutions of a biochemical dynamical model, formed by a system of three differential equations and which represent an instability-generating multienzymatic mechanism. The phase-shift series was studied in terms of Hurst’s rescaled range analysis. We found that the data were characterized by a Hurst exponent H = 0.69, which was clearly indicative of long-term trends. This result had a high significance level, as was confirmed through Monte Carlo simulations in which the data were scrambled in the series, destroying its original ordering. For these series we obtained a Hurst exponent which was consistent with the expectation of H = 0.5 for a random independent process. This clearly showed that, although the transitions between the periodic solutions were provoked randomly, the stochastic process obtained exhibited long-term persistence. The fractal dimension was also estimated and found to be consistent with the value of the Hurst exponent.     

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.     

A fractal model for the characterization of mycelial morphology

A new technique based on a fractal model has been developed for the quantification of the macroscopic morophology of mycelia. The morphological structuring is treated as a fractal object, and the fractal dimension, determined by an ultrasonic scattering procedure developed for the purpose, serves as a quantitative morphological index. Experimental observations reported earlier and simulations of mycelial growth, carried out using a probabilistic-geometric growth model developed for the purpose, both validate the applicability of the fractal model. In experiments with three different species, the fractal dimensions of pelletous structures were found to be in the range 1.45-2.0 and those of filamentous structures were in the range 1.9-2.7, with values around 2.0 representing mixed morphologies. Fractal dimensions calculated from simulated mycelia are in rough agreement with these ranges. The fractal dimension is also found to be relatively insensitive to the biomass concentration, as seen by dilution of the original broths. The relation between morphology and filtration properties of the broths has also been studied. The fractal dimension shows a strong correlation with the index of cake compressibility and with the Kozeny constant, two filtration parameters that are known to be morphology dependent. This technique could thus be used to develop correlations between the morphology, represented by the fractal dimension, and important morphology-dependent process variables. (c) 1993 John Wiley & Sons, Inc.     

Quantitative assessments of morphological and functional properties of biological trees based on their fractal nature.

The branching systems in our body (vascular and bronchial trees) and those in the environment (plant trees and river systems) are characterized by a fractal nature: the self-similarity in the bifurcation pattern. They increase their branch density toward terminals according to a power function with the exponent called fractal dimension (D). From a stochastic model based-on this feature, we formulated the fractal-based integrals to calculate such morphological parameters as aggregated branch length, surface area, and content volume for any given range of radius (r). It was followed by the derivation of branch number and cross-sectional area, by virtue of the logarithmic sectioning of the r axis and of the branch radius-length relation also given by a power function of r with an exponent (alpha). These derivatives allowed us to quantify various hydrodynamic parameters of vascular and bronchial trees as fluid conduit systems, including the individual branch flow rate, mean flow velocity, wall shear rate and stress, internal pressure, and circumferential tension. The validity of these expressions was verified by comparing the outcomes with actual data measured in vivo in the vascular beds. From additional analyses of the terminal branch number, we found a simple equation relating the exponent (m) of the empirical power law (Murray's so-called cube law) to the other exponents as (m=D+alpha). Finally, allometric studies of mammalian vascular trees revealed uniform and scale-independent distributions of terminal arterioles in organs, which afforded an infarct index, reflecting the severity of tissue damage following arterial infarction.     

Fractal Dimension in Butterflies’ Wings: a novel approach to understanding wing patterns ?

The geometrical complexity in the wings of several, taxonomically different butterflies, is analyzed in terms of their fractal dimension. Preliminary results provide some evidence on important questions about the (dis)similarity of the wing patterns in terms of their fractal dimension. The analysis is restricted to two groups which are widely used in the literature as typical examples of mimicry, and a small number of unrelated species, thus implying the consideration of only a fraction of the wing pattern diversity. The members of the first mimicry ring, composed by the species Danaus plexippus (better known as the monarch butterfly), and the two subspecies Basilarchia archippus obsoleta (or northern viceroy) and Basilarchia archippus hoffmanni (or tropical viceroy), are found to have a very similar value for the fractal dimension of their wing patterns, even though they do not look very similar at first sight. It is also found that the female of another species (Neophasia terlootii), which looks similar to the members of the previous group, does not share the same feature, while the Lycorea ilione albescens does share it. For the members of the second group of mimicry related butterflies, the Greta nero nero and the Hypoleria cassotis, it is shown that they also have very close values for the fractal dimension of their wing patterns. Finally, it is shown that other species, which apparently have very similar wing patterns, do not have the same fractal dimension. A possible, not completely tested hypothesis is then conjectured: the formation of groups by individuals whose wing patterns have an almost equal fractal dimension may be due to the fact that they do share the same developmental raw material, and that this common feature is posteriorly modified by natural selection, possibly through predation.We sincerely acknowledge the invaluable help of Adolfo Ibarra Vázquez, senior curator of the Lepidopterous collection at the Instituto de Biología, Universidad Nacional Autónoma de México, and the comments made by two anonymous referees     

The nonlinear parameters of interference EMG of two day old human newborns

Temporal structure of interference electromyogram (iEMG) was studied in healthy two days old human newborns (n = 76) using the non-linear parameters (correlation dimension, fractal dimension, correlation entropy). It has been found that the non-linear parameters of iEMG were time-dependent because they were decreasing within the first two days of life. Also, these parameters were sensitive to muscle function, because correlation dimension, fractal dimension, and correlation entropy of iEMG in gastrocnemius muscle differed from the other muscles. The non-linear parameters were proven to be independent of the iEMG amplitude. That model of early ontogenesis may be of potential use for investigation of anti-gravitation activity.     

The nonlinear parameters of an interference electromyogram in two-day-old human newborns

The temporal structure of an interference electromyogram (iEMG) was studied in healthy one- and two-day-old human newborns (n = 76) using nonlinear parameters (correlation and fractal dimension, correlation entropy). It was found that the nonlinear parameters of iEMG were time-dependent because they decreased during the first two days of life. These parameters were also sensitive to the muscle function, because the correlation dimension, fractal dimension, and correlation entropy of iEMG of the gastrocnemius muscle differed from the same parameters of other muscles. Some nonlinear parameters were found to be independent of the iEMG amplitude. This model of early ontogenesis is expected to be useful for studying antigravitational activity.     

Fractal dimension of antibody‐PEG precipitate: Light microscopy for the reconstruction of 3D precipitate structures

Protein and in particular antibody precipitation by PEG is a cost‐effective alternative for the first capture step. The 3D structure of precipitates has a large impact on the process parameters for the recovery and dissolution, but current technologies for determination of precipitate structures are either very time consuming (cryo‐TEM) or only generate an average fractal dimension (light scattering). We developed a light microscopy based reconstruction of 3D structures of individual particles with a resolution of 0.1–0.2 µm and used this method to characterize particle populations generated by batch as well as continuous precipitation in different shear stress environments. The resulting precipitate structures show a broad distribution in terms of fractal dimension. While the average fractal dimension is significantly different for batch and continuous precipitation, the distribution is broad and samples overlap significantly. The precipitate flocs were monofractal from micro‐ to nanoscale showing a random but consistent nature of precipitate formation. We showed that the fractal dimension and 3D reconstruction is a valuable tool for characterization of protein precipitate processes. The current switch from batch to continuous manufacturing has to take the 3D structure and population of different protein precipitates into account in their design, engineering, and scale up.     

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.     

Spatial distribution of leaf form and the self-thinning exponent are affected by the sensitivity of the response to abscisic acid in anArabidopsis thaliana population

The spatial distribution of leaves is related to the exponent of the self-thinning relationship in plant populations. In this study, we evaluated the fractal dimension of rosette leaves of wild-type (WT)Arabidopsis thaliana and of an abscisic acid (ABA) -insensitive mutant (abi2-1) to test a model of the spatial distribution of leaf form in anArabidopsis population based on subdivision of a cube surrounding the leaf into uniform boxes and to investigate ABA’s affect on this model of the leaf. The values of the self-thinning exponent were -1.31 and -1.45 for WT andabi2-1. The mean dimensions of the box used to model the spatial distribution of leaf form, estimated using our model, were 2.08 and 2.03, respectively. By assuming that the box dimension equals the fractal dimension within the populations, the predicted self-thinning exponent equaled -1.40 for WT and -1.49 forabi2-1. When exogenous ABA was applied to both genotypes, the self-thinning exponent became -1.26 and -1.43 for WT andabi2-1, and the exponents predicted using the dimensions of the box were -1.37 and -1.46, respectively. The empirically predicted exponent equaled that predicted using the dimensions of the box (95% confidence interval). Empirical prediction of the spatial pattern using the two genotypes with and without ABA showed that ABA influenced the spatial form of the rosette leaves. Therefore, sensitivity to ABA can affect self-thinning through genetically determined changes in leaf form and its spatial distribution.