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。     

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.     

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.     

桂花叶片的分形特征

提供了一种叶片分形维数的测定方法,并用其测定了桂花(Osmanthus fragrans Lour)的叶片分形维数。通过分析发现：(1)桂花叶片具有典型的分形特征并且这种分形特征可以用分形理论的一般方法加以研究;(2)同一株桂花树的成熟叶片在统计意义上具有相同的分形维数;(3)对于桂花这个物种的成熟叶片在统计意义上具有相同的分形维数。由此推测同一种植物的成熟叶片(或者其他构件)具有相同分形维数。最后提出创建植物分形分类学的设想。     

桂花叶片的分形特征

提供了一种叶片分形维数的测定方法,并用其测定了桂花(Osmanthus fragrans Lour)的叶片分形维数。通过分析发现:(1)桂花叶片具有典型的分形特征并且这种分形特征可以用分形理论的一般方法加以研究;(2)同一株桂花树的成熟叶片在统计意义上具有相同的分形维数;(3)对于桂花这个物种的成熟叶片在统计意义上具有相同的分形维数。由此推测同一种植物的成熟叶片(或者其他构件)具有相同分形维数。最后提出创建植物分形分类学的设想。     

Fractal analysis of the uterine contractions.

The fractal dimension D may be calculated in many ways, since its strict definition, the Hausdorff definition is too complicated for practical estimation. In this paper we perform a comparative study often methods of fractal analysis of time series. In Benoit, a commercial program for fractal analysis, five methods of computing fractal dimension of time series (rescaled range analysis, power spectral analysis, roughness-length, variogram methods and wavelet method) are available. We have implemented some other algorithms for calculating D: Higuchi's fractal dimension, relative dispersion analysis, running fractal dimension, method based on mathematical morphology and method based on intensity differences. For biomedical signals results obtained by means of different algorithms are different, but consistent.     

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.     

Surface fractality as a guide for studying protein-protein interactions

The concept of fractal dimension is applied to protein surfaces. Satellite tobacco necrosis virus, prealhumin, retinol binding protein and lysozyme have been studied. A residue fractal index has been defined, which provides a suitable colour code when using computer graphics for visualizing surfaces. Some provisions are made that render the MS algorithm useful to calculate protein surface fractal dimensions. It has been found that a correlation exists between regions of high fractal dimension and those involved in protein-protein interactions. The usefulness of surface fractality in this context is demonstrated by a molecular docking experiment.     

遗传算法支持下土地利用空间分形特征尺度域的识别

针对土地利用空间分形特征只存在于有限尺度域的现象,采用无标度区内离散点拟合的离差平方和平均值最小作为目标函数,提出了一种基于遗传算法的土地利用空间分形特征尺度域的识别方法,用于准确计算分形维数的有效区间范围。以武汉市武昌区水域空间分形特征为例,利用Quickbird多光谱遥感影像提取土地利用空间信息,重点讨论了基于遗传算法识别土地利用空间分形特征尺度域范围的总体思路、适应度函数和遗传算子等环节;然后分别从测定系数、标准差和无标度区间3个角度,将其同人工判断法、相关系数法以及强化系数法进行对比讨论;并结合研究区域实际的水域空间分布格局,分析不同方法计算所得半径维数的合理性。结果表明,土地利用分形特征尺度域的范围对分形维数计算结果有较大影响;相对于传统计算方法来说,遗传算法在尺度无标度区间识别上具有更高的精度,可以为土地利用空间格局分形特征的研究提供客观指导意见。     

A holistic approach to determine tree structural complexity based on laser scanning data and fractal analysis

The three‐dimensional forest structure affects many ecosystem functions and services provided by forests. As forests are made of trees it seems reasonable to approach their structure by investigating individual tree structure. Based on three‐dimensional point clouds from laser scanning, a newly developed holistic approach is presented that enables to calculate the box dimension as a measure of structural complexity of individual trees using fractal analysis. It was found that the box dimension of trees was significantly different among the tested species, among trees belonging to the same species but exposed to different growing conditions (at gap vs. forest interior) or to different kinds of competition (intraspecific vs. interspecific). Furthermore, it was shown that the box dimension is positively related to the trees’ growth rate. The box dimension was identified as an easy to calculate measure that integrates the effect of several external drivers of tree structure, such as competition strength and type, while simultaneously providing information on structure‐related properties, like tree growth.     

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.     

元宝山南方红豆杉克隆种群分布格局的分形特征

南方红豆杉(Taxus chinensis var. mairei)是提取抗癌药物紫杉醇的珍贵资源植物。分布于广西元宝山自然保护区的南方红豆杉,自然状态下以克隆繁殖方式进行种群更新,研究其种群克隆生长的空间分布格局,可揭示种群克隆繁殖特性和生态适应机制,对南方红豆杉这一珍稀濒危物种的保护和管理具有重要意义。利用分形理论中的计盒维数和信息维数,对元宝山的南方红豆杉克隆种群空间分布格局的分形特征进行了分析,结果表明:种群空间分布格局存在分形特征,其计盒维数介于0.993 1~1.353 1之间,信息维数介于1.350 8~1.652 1。南方红豆杉克隆种群格局的计盒维数远离最大理论维数值2,种群总体对空间的占据能力较弱。克隆构型指数与种群计盒维数密切相关。分析结果显示,克隆构型趋于“密集型”的种群具有较强的空间占据能力,而克隆构型趋于“游击型”的种群其空间占据能力较弱。信息维数与种群的集聚强度密切相关,其差异反映了种群格局强度的变化,较大的信息维数反映种群个体的集聚强度较高,而较小的信息维数则反映出种群个体间的集聚强度较弱,其维数值大小与分株种群数量、分株种群斑块的空间配置密切相关。     

Teleonomical optimization of a fractal model of the pulmonary arterial bed

Modeling the pulmonary arterial tree (PAT) is considered here as an optimal synthesis problem. Firstly, a class of candidate models is specified: the three-dimensional symmetric dichotomous fractal trees of elastic tubes described by Womersley's equations. Secondly, the parameters are shown to be constrained by interactions of PAT with the rest of the body; these constraints are used to limit the volume of the parametric space to which attention will be directed in the synthesis step. Thirdly, a teleonomical hypothesis is proposed: a naturally selected PAT must have a minimal input impedance under conditions keeping total arterial volume and distensibility as small as possible. This hypothesis is translated in mathematical terms and the resulting cost-function minimized in the limited parametric volume. The optimal model has parameter values and an impedance spectrum corresponding satisfactorily with real data. Moreover this model gives a clear picture of the internal hemodynamic behavior of PAT as an impedance matching device.     

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.     

Fractal dimension of the middle meningeal vessels: variation and evolution in Homo erectus, Neanderthals, and modern humans

The middle meningeal vascular network leaves its traces on the endocranial surface because of the tight relationship between neurocranial development and brain growth. Analysing the endocast of fossil specimens, it is therefore possible to describe the morphology of these structures, leading inferences on the cerebral physiology and metabolism in extinct human groups. In this paper, general features of the meningeal vascular traces are described for specimens included in the Homo erectus, Homo neanderthalensis, and Homo sapiens hypodigms. The complexity of the arterial network is quantified by its fractal dimension, calculated through the box-counting method. Modern humans show significant differences from the other two taxa because of the anterior vascular dominance and the larger fractal dimension. Neither the fractal dimension nor the anterior development are merely associated with cranial size increase. Considering the differences between Neanderthals and modern humans, these results may be interpreted in terms of phylogeny, cerebral functions, or cranial structural network.     

Nonlinear fractal dynamics of human colonic pressure activity based upon the box-counting method

The computational fractal dimension of human colonic pressure activity acquired by a telemetric capsule robot under normal physiological conditions was studied using the box-counting method. The fractal dimension is a numeric value that quantifies to measure how rough the signal is from nonlinear dynamics, rather than its amplitude or other linear statistical features. The colonic pressure activities from the healthy subject during three typical periods were analysed. The results showed that the activity might be fractal with a non-integer fractal dimension after it being integrated over time using the cumsum method, which was never revealed before. Moreover, the activity (after it being integrated) acquired soon after wakening up was the roughest (also the most complex one) with the largest fractal dimension, closely followed by that acquired during sleep with that acquired long time after awakening up (in the daytime) ranking third with the smallest fractal dimension. Fractal estimation might provide a new method to learn the nonlinear dynamics of human gastrointestinal pressure recordings.     

Fractal perspectives in pulmonary physiology

Like other organs that exchange substantial quantities of material with blood, the lung accommodates a large two-dimensional surface in a component three-dimensional volume. The lung's structure shows a resemblance to certain one- and two-dimensional mathematical functions that possess plane- and space-filling properties. When viewed from a conventional geometric perspective, many of the familiar forms and functions of pulmonary tissue appear to possess unusual qualities that defy explanation. Mathematically, they behave as though they had a fractional geometric dimension. This property is shared by a class of functions known as fractals. Fractals are described, and practical techniques are presented to measure the properties of the edges and surfaces of the lung. The consequences of fractal structure are also considered for the bronchial tree, pulmonary vasculature, and microcirculation. Insights arising from viewing the lung in this new perspective are summarized.     

Fractal analysis on root systems of rice plants in response to drought stress

A fractal analytical method was used to examine the developmental responses of root systems in upland rice genotype CT9993-5-10-1-M (japonica) and lowland genotype IR62266-42-6-2 (indica) (abbreviated as CT9993 and IR62266, respectively) to soil water stress. The root systems were grown for one month in root boxes with 25 cm in length, 2 cm in width and 40 cm in depth, which were filled with soil. The root systems were sampled by following the needle-pinboard method, and then spread on the transparent plastic films with nets after carefully washing out the soils. The two-dimensional images of root systems were digitized by using a scanner. The digitized images were used for analysis based on fractal geometry with the box-counting method. The reductions in shoot dry weight, photosynthesis rate and transpiration rate of IR62266 by soil drought were greater than those of CT9993. The change of fractal parameters in response to soil moisture conditions differed between the two rice genotypes. The values of fractal abundance (FA) and fractal dimension (FD) in well-watered IR62266 plants were larger than in CT9993. The value of FA of IR62266 was decreased more by drought stress than that of CT9993, indicating that the volume of soils explored by the whole root systems of CT9993 was maintained or less decreased under drought stress in comparison to IR62266. Moreover, the values of FD tended to increase in CT9993 while it tended to decrease in IR62266 in response to drought. These root responses detected by the fractal analysis in CT9993 may be advantageous for its extracting more water from drying soils, which explains its better growth under drought-stressed condition.     

Fractal root system of Melica przewalskyi along different aspect in degraded grassland

Aims Fractal root system is phenotypic plasticity result of plant root architecture to respond to environmental heterogeneity, may reflect the growth strategy of plants to adapt to environmental conditions. Our objective was to explore the relationship between root fractal dimension and fractal abundance of fractal root system of Melica przewalskyi population in response to aspect variation in the northwest of China. Methods The study site was located in a degraded alpine grassland on the northern slope in Qilian Mountains, Gansu Province, China. Survey and sampling were carried out at 40 plots which were set up along four slope aspects transects with 20 m distance between adjacent plots. Handheld GPS was used to determine the elevation, longitude and latitude of each plot. ArcGIS was used to set up digital elevation model (DEM). Community traits were investigated and six individuals roots of M. przewalskyi were collected randomly at each plot. The samples were cleaned and divided into different organs, then scanning the root with the Win-RHIZO for measurements of fractal dimension and fractal abundance in laboratory, and their biomass were then measured after being dried at 80 °C in an oven. Important findings With the slope aspect turned from north to east, west, and south, the density, height and soil moisture content of the plant community displayed a pattern of initial decline, the height, density, root fractal abundance of M. przewalskyi increased and the root fractal dimension decreased. The root fractal dimension was negatively associated with the fractal abundance in all aspects, but the relationship varied along the slope aspects gradient; there was a highly significant negative correlation (p < 0.01) between the root fractal dimension and fractal abundance at north slope and south slope aspect, whereas the correlation only reached a significant level (p < 0.05) at the east slope aspect and west slope aspect; indicating that there is a trade-off between the root fractal dimension and fractal abundance. In addition, when the slope aspect changed from north to east, west and south, the standardized major axis (SMA) slope of the regression equation in the scaling relationships between root fractal dimension and fractal abundance increased (p < 0.05), indicating that the roots of M. przewalskyi at the droughty southern slope have less branch and more sparse in the same soil volume of root exploitation and utilization. Consequently, the resource allocation pattern on reasonable trade-off between root fractal dimension and fractal abundance in different slope aspect of M. przewalskyi, reflects the relationship between the income and the cost of construction of plant root architecture.