How to reliably estimate the tortuosity of an animal's path: straightness, sinuosity, or fractal dimension?

The tortuosity of an animal's path is a key parameter in orientation and searching behaviours. The tortuosity of an oriented path is inversely related to the efficiency of the orientation mechanism involved, the best mechanism being assumed to allow the animal to reach its goal along a straight line movement. The tortuosity of a random search path controls the local searching intensity, allowing the animal to adjust its search effort to the local profitability of the environment. This paper shows that (1) the efficiency of an oriented path can be reliably estimated by a straightness index computed as the ratio between the distance from the starting point to the goal and the path length travelled to reach the goal, but such a simple index, ranging between 0 and 1, cannot be applied to random search paths; (2) the tortuosity of a random search path, ranging between straight line movement and Brownian motion, can be reliably estimated by a sinuosity index which combines the mean cosine of changes of direction with the mean step length; and (3) in the current state of the art, the fractal analysis of animals' paths, which may appear as an alternative and promising way to measure the tortuosity of a random search path as a fractal dimension ranging between 1 (straight line movement) and 2 (Brownian motion), is only liable to generate artifactual results. This paper also provides some help for distinguishing between oriented and random search paths, and depicts a general, comprehensive framework for analysing individual animals' paths in a two-dimensional space.     

Effect of Colony, Patch Distance, And Trajectory Sense on Movement Complexity in Foraging Ants

Animals display foraging trajectories when exploiting food patches. These displacements have been studied, particularly in organisms that forage from a central place. The complexity of a movement path may be analyzed by means of the fractal dimension, an index that estimates the tortuosity of a trajectory. In field experiments we studied the effect of trajectory sense, distance to the resource patch and colony on the movement complexity in a common ant of central Chile. We found that these factors and their interactions significantly affected the complexity of movement paths. We discuss whether mortality risk determines more linear trajectories when the resource patch is distant from the nest, and whether the information acquired from the environment by workers determines less complex return trips.     

Fractals and the analysis of growth paths

A simple practical method exists for classifying and comparing planar curves composed of connected line segments. This method assigns, a single numberD, the fractal dimension, to each curve.D=log(n)/[log(n)+log(d/L)], where:n is the number of line segments,L is the total length of the line segments, andd is the planar diameter of the curve (the greatest distance between any two endpoints). At one end of the spectrum, for straight line curves,D=1; at the other end of the spectrum, for random walk curves,D→2. Standard statistics are done on the logarithms of the fractal dimension [log(D)]. With this measure, trails of biological movement, such as the growth paths of the cells and the paths of wandering organisms, can be analyzed to determine the likelihood that these trails are random walks and also to compare the straightness of the trails before and after experimental interventions.     

The Quasi-Fractal Structure of Fish Brain Neurons

The box-counting method for calculating the fractal dimension (D) with the ImageJ 1.20s software is used as a tool for quantitative analysis of the neuronal morphology in the fish brain. The fractal dimension was determined for several types of neurons in the brain of two teleost species, Pholidapus dybowskii and Oncorhynchus keta. These results were compared with those obtained for some neurons of the human brain. The fractal (fractional) dimension (D), as a quantitative index of filling of two-dimensional space by the black and white image of a cell, is shown to vary from 1.22 to 1.72 depending on the type of neuron. The fractal dimension reaches its maximum in less specialized neurons that carry out a number of different functions. On the other hand, highly specialized neurons display a relatively low fractal dimension. Thus, the fractal dimension serves as a numerical measure of the spatial complexity of the neuron and correlates with the morphofunctional organization of the cell.     

黑土表层土壤颗粒的分形特征

基于第二次全国土壤普查结果,应用土壤颗粒的质量分布计算了36个典型剖面表层土壤颗粒的分形维数值.结果表明:土壤颗粒分形维数值D在2.5831～2.8230,其变异性极弱,且分形维数值随质地变细而增大;土壤机械组成中,砂粒(2～0.02mm)含量、粉粒(0.02～0.002mm)含量与分形维数值均呈显著负相关(P<0.05);粘粒(<0.002mm)含量与土壤分形维数值呈极显著正相关(P<0.01);分形维数值D与土壤中的有机质、全N、全P、全K含量及pH值相关性均不显著.土壤分布的分形维数可以作为反映黑土退化程度的一个综合性定量指标.     

间作对桑树和谷子生长和光合日变化的影响

以桑树和谷子为研究材料,探讨了大田条件下,桑树-谷子间作对桑树和谷子的干物质生产、土地利用率和光合日变化的影响.结果表明：桑树-谷子间作条件下,间作桑树的株高、地茎、根长和枝条数分别比单作桑树增加了6.0%、13.7%、6.8%和14.8%,且间作桑树的产叶量比单作桑树增加了31.3%;间作谷子与单作比较,其株高和根长的变化不大.桑树-谷子间作增加了土地当量比,提高了土地利用率.单作、间作桑树和谷子叶片在12:00时均表现出明显的光合午休现象,且单作桑树的光合午休现象比间作桑树严重.桑树-谷子间作提高了中午时桑树叶片气孔导度和水分利用率,增加了桑树光合碳同化能力,抑制了桑树叶片实际光化学效率、电子传递速率和最大光化学效率的下降,从而减缓了桑树的“光合午休”现象.桑树 谷子间作能明显提高桑树叶片的光合生产能力.     

木本植物对喀斯特石质生境岩石形态结构的适应性

为弄清植物对喀斯特岩石形态结构的适应性,选择3种岩石类型及7种木本群落为研究对象,基于分形、空间句法理论研究岩石结构面孔裂隙形态结构及其与群落特征的关系。结果表明:石灰岩结构面孔裂隙间隙度大、白云岩居中、白云质砂岩小,分维数则反之;白云质砂岩结构面孔裂隙整合度、控制值、密度值大,白云岩居中,石灰岩小;白云岩生境下群落物种多样性及优势种碳、氮含量高、白云质砂岩次之、石灰岩低;分形与空间句法指数间呈显著相关关系,两者能较好的揭示岩石形态结构;总体上3种岩性生境下物种多样性及优势种碳、氮含量与岩石结构面孔裂隙分形及空间句法指数具较强相关关系;大致相同的环境下白云岩形态结构适宜植物生长、白云质砂岩次之、石灰岩差;研究岩石形态结构对喀斯特植被恢复具有重要意义。     

Fractal properties of forest spatial structure

The definition of fractal dimension of natural objects, which enables to deal with scale dependence of fractal dimension is discussed. Abrupt changes of fractal dimension of spatial structure of terrestrial ecosystems are considered in the context of hierarchical paradigm. On this ground the procedure is proposed for segmentation of a territory, which takes into account the scale dependence of spatial variability of ecological parameters. Using remotely sensed data — normalized difference vegetation index (NDVI) and thermal radiation in the infrared band — fractal dimensions and critical scales are evaluated for different forest types with the help of software, developed for this purpose. The results obtained corroborate the potentialities of fractal approach in ecology. These methods and results can be used for discrimination of remotely sensed data; but further investigations, including detailed comparison of fractal characteristics of remotely sensed forest images with results of on-site field studies are necessary to validate them.     

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.     

The predatory mite <Emphasis Type="Italic">Neoseiulus womersleyi</Emphasis> (Acari: Phytoseiidae) follows extracts of trails left by the two-spotted spider mite <Emphasis Type="Italic">Tetranychus urticae</Emphasis> (Acari: Tetranychidae)

As it walks, the two-spotted spider mite Tetranychus urticae Koch (Acari: Tetranychidae) spins a trail of silk threads, that is followed by the predatory mite, Neoseiulus womersleyi Schicha (Acari: Phytoseiidae). Starved adult female N. womersleyi followed T. urticae trails laid down by five T. urticae females but did not follow a trail of one T. urticae female, suggesting that the amount of spun threads and their chemical components should correlate positively with the number of T. urticae individuals. To examine whether chemical components of T. urticae trails are responsible for the predatory mite’s trail following, we collected separate T. urticae threads from the exuviae and eggs, and then washed the threads with methanol to separate chemical components from physical attributes of the threads. Female N. womersleyi did not follow T. urticae trails that had been washed with methanol but contained physical residues, but they did follow the direction to which the methanol extracts of the T. urticae trails was applied. These results suggest that the predatory mite follows chemical, not physical, attributes of T. urticae trails.     

What if it gets crowded? Density‐dependent tortuosity in individual movements of a Neotropical mammal

Effects of density dependence on animal movements have received much attention in ecology, but it is still debated to what extent dispersal and movements in general are density dependent, and their potential contribution to population regulation processes. Here, we determine the occurrence and nature of density dependence in the movements of a Neotropical marsupial, the black‐eared opossum Didelphis aurita Wied‐Neuwied 1826. Using spool‐and‐line tracking devices, we estimated the tortuosity of fine‐scale movements of 149 individuals by their fractal dimension D. We evaluated the relative importance of population size, reproductive or climatic seasons and reproductive maturity of individuals as determinants of movement tortuosity, using a model selection approach. Population size was the most important determinant of movement tortuosity, with season (climatic seasons for females, reproductive seasons for males) and reproductive maturity as secondary but also important variables. We detected a positive density‐dependent effect on movement tortuosity, resulting in more intensive use of areas by individuals during periods of high population size. This positive association between movement tortuosity and population size is more likely to result from intraspecific competition, which forces individuals to explore their environment more intensively during high‐density periods. Therefore, despite being density dependent, movements in D. aurita apparently do not contribute to population regulation mechanisms.     

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.     

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.     

退化沙质草地开垦和围封过程中的土壤颗粒分形特征

研究了科尔沁退化沙质草地开垦和围封过程中土壤颗粒分形维数的变化特征,以及分形维数与土壤性状的关系.结果表明,不同开垦和围封年限的土壤颗粒分形维数(0～30cm)介于2.387～2.588之间.随着开垦年限的增加,0～15 cm层土壤颗粒分形维数从2.441降至2.387;围封11年后,0～15 cm层土壤颗粒分形维数增加到2.588.15～30 cm层土壤颗粒分形维数无明显变化.土壤颗粒分形维数是反映土壤质地的一个较好指标,重点反映粘粒含量,其次是粉粒含量.分形维数的变化能够很好地表征退化沙质草地土壤的化学、物理和生物学性状的变化趋势,可以作为评价退化沙质草地土壤性状的一个综合指标.     

Fractal nature of chromatin organization in interphase chicken erythrocyte nuclei: DNA structure exhibits biphasic fractal properties

Arrangement of chromatin in intact chicken erythrocyte nuclei was investigated by small angle neutron scattering. The scattering spectra have revealed that on the scales between 15 nm and 1.5 microm the interior of the nucleus exhibited properties of a mass fractal. The fractal dimension of the protein component of cell nucleus held constant at approximately 2.5, while the DNA organization was biphasic, with the fractal dimension slightly higher than 2 on the scales smaller than 300 nm and approaching 3 on the larger scales.     

退化沙质草地开垦和围封过程中的土壤颗粒分形特征

研究了科尔沁退化沙质草地开垦和围封过程中土壤颗粒分形维数的变化特征,以及分形维数与土壤性状的关系.结果表明,不同开垦和围封年限的土壤颗粒分形维数(0～30cm)介于2.387～2.588之间.随着开垦年限的增加,0～15 cm层土壤颗粒分形维数从2.441降至2.387;围封11年后,0～15 cm层土壤颗粒分形维数增加到2.588.15～30 cm层土壤颗粒分形维数无明显变化.土壤颗粒分形维数是反映土壤质地的一个较好指标,重点反映粘粒含量,其次是粉粒含量.分形维数的变化能够很好地表征退化沙质草地土壤的化学、物理和生物学性状的变化趋势,可以作为评价退化沙质草地土壤性状的一个综合指标.     

异质景观条件下江汉平原土壤的空间分异

景观异质与土壤分异息息相关。以DEM数据为基础,以海拔50m、100m为界,将江汉平原划分为平原湖区景观、平缓岗地景观及起伏丘陵景观3种类型。在ArcG IS 10支持下,将江汉平原土壤图与景观类型图进行叠置分析,提取不同景观类型片区各土壤亚类斑块周长、面积等信息,计算了各景观类型片区各土壤亚类的分维数、平均斑块面积、稳定度等信息,定量分析了江汉平原各景观类型片区土壤空间分异特征,结论如下:(1)不同景观类型区各土壤亚类分布差异明显,起伏丘陵景观区主要以红壤和黄棕壤地带性土壤为主;平原湖区潮土和水稻土等耕作土非常发育;平缓岗地区地带性土壤和耕作土壤平分秋色。(2)连片性较好的土壤亚类呈现不同的景观选择性:耕作土集中分布于平原湖区景观片区;地带性土壤多集中分布于丘陵和岗地景观片区。(3)平原湖区面积很大,各类土壤都有发育的空间,土壤亚类之间分维数和稳定度差别比较大;平缓岗地景观区由于面积非常局限,土壤亚类发生发育受到空间的限制,边界破碎化,分维数平均都比较大,斑块镶嵌结构均比较复杂,稳定度差别较小。(4)主要土壤亚类的分维数和稳定性指数值一定程度地反映了各主要土壤亚类的最匹配的景观类型,即能够提供其发生发育的最佳条件。研究有利于深入认识土壤形成和演化规律,为土壤资源的合理利用及定向培育服务。     

Quantification of Tortuosity and Fractal Dimension of the Lung Vessels in Pulmonary Hypertension Patients

Pulmonary hypertension (PH) can result in vascular pruning and increased tortuosity of the blood vessels. In this study we examined whether automatic extraction of lung vessels from contrast-enhanced thoracic computed tomography (CT) scans and calculation of tortuosity as well as 3D fractal dimension of the segmented lung vessels results in measures associated with PH.In this pilot study, 24 patients (18 with and 6 without PH) were examined with thorax CT following their diagnostic or follow-up right-sided heart catheterisation (RHC). Images of the whole thorax were acquired with a 128-slice dual-energy CT scanner. After lung identification, a vessel enhancement filter was used to estimate the lung vessel centerlines. From these, the vascular trees were generated. For each vessel segment the tortuosity was calculated using distance metric. Fractal dimension was computed using 3D box counting. Hemodynamic data from RHC was used for correlation analysis.Distance metric, the readout of vessel tortuosity, correlated with mean pulmonary arterial pressure (Spearman correlation coefficient: ρ = 0.60) and other relevant parameters, like pulmonary vascular resistance (ρ = 0.59), arterio-venous difference in oxygen (ρ = 0.54), arterial (ρ = −0.54) and venous oxygen saturation (ρ = −0.68). Moreover, distance metric increased with increase of WHO functional class. In contrast, 3D fractal dimension was only significantly correlated with arterial oxygen saturation (ρ = 0.47).Automatic detection of the lung vascular tree can provide clinically relevant measures of blood vessel morphology. Non-invasive quantification of pulmonary vessel tortuosity may provide a tool to evaluate the severity of pulmonary hypertension.

Trial Registration

ClinicalTrials.gov {"type":"clinical-trial","attrs":{"text":"NCT01607489","term_id":"NCT01607489"}}NCT01607489     

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.