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.     

Link between truncated fractals and coupled oscillators in biological systems

This article aims at providing a new theoretical insight into the fundamental question of the origin of truncated fractals in biological systems. It is well known that fractal geometry is one of the characteristics of living organisms. However, contrary to mathematical fractals which are self-similar at all scales, the biological fractals are truncated, i.e. their self-similarity extends at most over a few orders of magnitude of separation. We show that nonlinear coupled oscillators, modeling one of the basic features of biological systems, may generate truncated fractals: a truncated fractal pattern for basin boundaries appears in a simple mathematical model of two coupled nonlinear oscillators with weak dissipation. This fractal pattern can be considered as a particular hidden fractal property. At the level of sufficiently fine precision technique the truncated fractality acts as a simple structure, leading to predictability, but at a lower level of precision it is effectively fractal, limiting the predictability of the long-term behavior of biological systems. We point out to the generic nature of our result.     

Fern leaves and cauliflower curds are not fractals

The popular demonstration of drawing a mature fern leaf as expressed by Barnsley''s fractal method is mathematically and visually very attractive but anatomically and developmentally misleading, and thus has limited, if any, biological significance. The same is true for the fractal demonstration of the external features of cauliflower curds. Actual fern leaves and cauliflower curds have a very small number of anatomically variable and non-iterating bifurcations, which superficially look self-similar, but do not allow for scaling down of their structure as real fractals do. Moreover, fern leaves and cauliflower curds develop from the inside out through a process totally different from fractal drawing procedures. The above cases demonstrate a general problem of using mathematical tools to investigate or illustrate biological phenomena in an irrelevant manner. A realistic set of mathematical equations to describe fern leaf or cauliflower curd development is needed.     

利用IKONOS卫星数据和分形方法研究南麂岛土地覆盖状况

研究南麂岛的土地覆盖类型及其空间分布和结构特征。利用具有1米空间分辨率的IKONOS卫星遥感数据,提取南麂岛的植被覆盖和土地利用信息,获得草地、灌木林地、庄稼地和居民地等主要土地覆盖类型及其分布图。然后利用分形几何方法建立南麂岛土地覆盖类型特性分析模型,从斑块的面积效应、覆盖类型的分形分析、单个斑块的分形分析和覆盖类型分形特征差异显著性等方面进行分析讨论。研究结果表明,南麂岛的草地和灌木林地的分形维数较大,而庄稼地和居民地的分形维数较小,说明草地和灌木林地的斑块的结构特征和边界比庄稼地和居民地更为复杂。进一步研究表明,斑块的分形特性与其受人类活动的干扰程度密切相关。     

Power-law species richness accumulation as manifestation of the fractal community structure

Applications of the fractal to describing the species structure of communities are discussed. Fundamental notions of fractal geometry are explained in the first part. The problem of applying the concept of fractal to describe the spatial allocation of particular species and of community as a whole is reviewed in the second part. In the final part, the usage of the selfsimirity principle for analyzing community organization is substantiated, and evidence of the fractal structure of biocenoses is presented according to Whittaker's concept of alpha diversity. It is shown that community is characterized, as a fractal object, by scale invariance, by power function relationship between the number of structural elements of the community (individuals, populations, species) and the scale (sampling effort), and, finally, by fractional value of the power (fractal dimension). Power function is the formula the takes into account the share of rare species, or species represented by a single individual. providing for no saturation of the function f(x). This formula also does not contradict the A.P. Levich's "rule of ecological non-additivity" and, lastly, allows the application of fractal formalism to characterize the species structure of a community. It is concluded that the mathematical image of species richness is a monofractal, i.e., a set characterised by only one parameter, fractal dimension. Thus, the species structure of a community (as well as the pattern of its spatial allocation) displays self-similarity and is a fractal.     

肝脏超声图象局部分形指数研究

为了研究人体组织超声图象的局部特征,并为进行人体组织定征研究提供新的参数,提出了一个分析超声图象局部分形指数的新方法－局部分形指数小波分析法ＬＦＷＡＭ（ＬｏｃａｌＦｒａｃｔａｌＳｃａｌｅＷａｖｅｌｅｔＡｎａｌｙｓｉｓＭｅｔｈｏｄ）。应用此法研究了人体肝脏组织超声图象分形体的构造规则;进行了局部分形指数的分析。验证了ＬＦＷＡＭ法分析肝脏超声图象局部特性的有效性,得出了局部分形指数更能全面、细致地刻画肝脏组织超声图象分形特征的结论,为进而研究局部病变的识别与图象的分割提供了基础。     

Fractal and spectral dimension analysis of liver fibrosis in needle biopsy specimens

OBJECTIVE: To evaluate the usefulness of a reliable and reproducible mathematical scoring system based on fractal geometry for quantifying the irregular pattern in fibrosis commonly seen in liver biopsy specimens from chronic liver diseases. STUDY DESIGN: The study used 26 standard liver biopsy specimens obtained from patients with chronic hepatitis C virus-related liver disease. The degree of fibrosis in each specimen was estimated using a quantitative scoring system based on the computer-assisted evaluation of both the fractal and spectral dimensions of deposited collagen. The fractal dimension was then compared with the percent area of collagen measured using an image analysis system. RESULTS: The fractional dimension of its irregular shape defines fibrosis as a natural fractal structure. The complex distribution of its collagenous components (unmeasurable by means of the usual morphometric parameters) can be optimally quantified using a single numerical score that seems to be a better alternative to the semiquantitative methods adopted so far. The proposed method is reproducible, rapid and inexpensive; furthermore, supported by specific software, its mathematical approach excludes subjectivity and eliminates the external factors capable of influencing staging and classification. CONCLUSION: This study demonstrated that it is possible to quantify the irregularity of the structures of the liver in an objective manner and that the box-counting fractal dimension does not depend on the amount of collagen deposited on the slide. Furthermore, as has been found in other fields of investigation, study of the fractal properties of the liver is likely to reveal more about its structure and the pathogenesis of liver diseases.     

A differential geometry approach for biomedical image processing

We show in this paper how simple considerations about bio-arrays images lead to a peak segmentation allowing the genes activity analysis. Bio-arrays images have a particular structure and the aim of the paper is to present a mathematical method allowing their automatic processing. The differential geometry approach used here can be also employed for other types of images presenting grey level peaks corresponding to a functional activity or to a chemical concentration. The mathematical method is based on elementary techniques of differential geometry and dynamical systems theory and provides a simple efficient algorithm when the peaks to segment are isolated.     

Fractal dimension of K1735 mouse melanoma clones and spheroid invasion in vitro

An in vitro tumour-host confrontation method to investigate the invasion behaviour of cancer has been applied to K1735 mouse melanomas. Fluorescently labelled spheroids of cancer cells and host cells were confronted and the temporal course of cancer invasion into the host was investigated using confocal laser scanning microscopy. To improve the quantitative data of this method, the boundary images of the fluorescently labelled confrontation pairs were treated as fractals. The physical and mathematical framework for determination of the fractal capacity dimension is widely used in biology and medicine and has proved to be a very useful tool for describing the cancer invasion process. The fractal capacity dimension determination was carried out by dilation of the binary boundaries of the objects, which were treated as an estimate of the Minkowski-Bouligand dimension. The fractal dimension correlated well with the degree of invasion of the K1735-M2 clone. Control experiments, with host-host confrontations and various K1735 clones with reduced invasiveness, support these results.     

Using the concept of pseudo amino acid composition to predict resistance gene against Xanthomonas oryzae pv. oryzae in rice: an approach from chaos games representation

To evaluate the possibility of an unknown protein to be a resistant gene against Xanthomonas oryzae pv. oryzae, a different mode of pseudo amino acid composition (PseAAC) is proposed to formulate the protein samples by integrating the amino acid composition, as well as the Chaos games representation (CGR) method. Some numerical comparisons of triangle, quadrangle and 12-vertex polygon CGR are carried to evaluate the efficiency of using these fractal figures in classifiers. The numerical results show that among the three polygon methods, triangle method owns a good fractal visualization and performs the best in the classifier construction. By using triangle + 12-vertex polygon CGR as the mathematical feature, the classifier achieves 98.13% in Jackknife test and MCC achieves 0.8462.     

Fractal dimension and histogram method: Algorithm and some preliminary results of noise-like time series analysis

In the present work a methodological background for the histogram method of time series analysis is developed. Connection between shapes of smoothed histograms constructed on the basis of short segments of time series of fluctuations and the fractal dimension of the segments is studied. It is shown that the fractal dimension possesses all main properties of the histogram method. Based on it a further development of fractal dimension determination algorithm is proposed. This algorithm allows more precision determination of the fractal dimension by using the “all possible combination” method. The application of the method to noise-like time series analysis leads to results, which could be obtained earlier only by means of the histogram method based on human expert comparisons of histograms shapes.     

Correcting for finite spatial scales of self-similarity when calculating fractal dimensions of real-world structures

Fractal geometry is a potentially valuable tool for quantitatively characterizing complex structures. The fractal dimension (D) can be used as a simple, single index for summarizing properties of real and abstract structures in space and time. Applications in the fields of biology and ecology range from neurobiology to plant architecture, landscape structure, taxonomy and species diversity. However, methods to estimate the D have often been applied in an uncritical manner, violating assumptions about the nature of fractal structures. The most common error involves ignoring the fact that ideal, i.e. infinitely nested, fractal structures exhibit self-similarity over any range of scales. Unlike ideal fractals, real-world structures exhibit self-similarity only over a finite range of scales.Here we present a new technique for quantitatively determining the scales over which real-world structures show statistical self-similarity. The new technique uses a combination of curve-fitting and tests of curvilinearity of residuals to identify the largest range of contiguous scales that exhibit statistical self-similarity. Consequently, we estimate D only over the statistically identified region of self-similarity and introduce the finite scale- corrected dimension (FSCD). We demonstrate the use of this method in two steps. First, using mathematical fractal curves with known but variable spatial scales of self-similarity (achieved by varying the iteration level used for creating the curves), we demonstrate that our method can reliably quantify the spatial scales of self-similarity. This technique therefore allows accurate empirical quantification of theoretical Ds. Secondly, we apply the technique to digital images of the rhizome systems of goldenrod (Solidago altissima). The technique significantly reduced variations in estimated fractal dimensions arising from variations in the method of preparing digital images. Overall, the revised method has the potential to significantly improve repeatability and reliability for deriving fractal dimensions of real-world branching structures.     

Applications of fractal analysis to physiology

This review describes approaches to the analysis of fractal properties of physiological observations. Fractals are useful to describe the natural irregularity of physiological systems because their irregularity is not truly random and can be demonstrated to have spatial or temporal correlation. The concepts of fractal analysis are introduced from intuitive, visual, and mathematical perspectives. The regional heterogeneities of pulmonary and myocardial flows are discussed as applications of spatial fractal analysis, and methods for estimating a fractal dimension from physiological data are presented. Although the methods used for fractal analyses of physiological data are still under development and will require additional validation, they appear to have great potential for the study of physiology at scales of resolution ranging from the microcirculation to the intact organism.     

瞬态诱发耳声发射分形特性研究

耳声发射的产生是耳蜗非线性动力学机制作用的结果 ,是外毛细胞的非线性生物机构放大过程的一种能量泄露。我们将分形理论用于对耳声发射信号波形形态的研究 ,并通过基于数学形态学的分形维数计算方法对数例瞬态诱发耳声发射信号进行了分析。实验结果表明该信号具有较好的分形特性 ,为耳声发射进一步的深入研究和理解提供了一定的帮助。     

Explicitly Accounting for Pixel Dimension in Calculating Classical and Fractal Landscape Shape Metrics

Different summarized shape indices, like mean shape index (MSI) and area weighted mean shape index (AWMSI) can change over multiple size scales. This variation is important to describe scale heterogeneity of landscapes, but the exact mathematical form of the dependence is rarely known. In this paper, the use of fractal geometry (by the perimeter and area Hausdorff dimensions) made us able to describe the scale dependence of these indices. Moreover, we showed how fractal dimensions can be deducted from existing MSI and AWMSI data. In this way, the equality of a multiscale tabulated MSI and AWMSI dataset and two scale-invariant fractal dimensions has been demonstrated.     

Mathematical characterization of three-dimensional gene expression patterns

MOTIVATION: The importance of a systematic methodology for the mathematical characterization of three-dimensional gene expression patterns in embryonic development. METHODS: By combining lacunarity and multiscale fractal dimension analyses with computer-based methods of three-dimensional reconstruction, it becomes possible to extract new information from in situ hybridization studies. Lacunarity and fractality are appropriate measures for the cloud-like gene activation signals in embryonic tissues. The newly introduced multiscale method provides a natural extension of the fractal dimension concept, being capable of characterizing the fractality of geometrical patterns in terms of spatial scale. This tool can be systematically applied to three-dimensional patterns of gene expression. RESULTS: Applications are illustrated using the three-dimensional expression patterns of the myogenic marker gene Myf5 in a series of differentiating somites of a mouse embryo.     

Differentiation of linearly correlated noise from chaos in a biologic system using surrogate data

Experimental time-series of human H-reflexes were analyzed for the presence of fractal structure or deterministic chaos. Surrogate data sets consisting of stochastic time-series with preservation of selected properties of the experimental time-series were used as mathematical controls. Artifacts generated during the analysis of the experimental data are identified, and shown to be due to linear correlation in the original time-series. The method is simple and generally applicable to the non-linear analysis of time-series from any experimental system.     

Use of fractal dimensions to quantify coral shape

A morphometrical method to quantify and characterize coral corallites using Richardson Plots and Kaye’s notion of fractal dimensions is presented. A Jurassic coral species (Aplosmilia spinosa) and five Recent coral species were compared using the Box-Counting Method. This method enables the characterization of their morphologies at calicular and septal levels by their fractal dimensions (structural and textural). Moreover, it is possible to determine differences between species of Montastraea and to tackle the high phenotypic plasticity of Montastraea annularis. The use of fractal dimensions versus conventional methods (e.g., measurements of linear dimensions with a calliper, landmarks, Fourier analyses) to explore a rugged boundary object is discussed. It appears that fractal methods have the potential to considerably simplify the morphometrical and statistical approaches, and be a valuable addition to methods based on Euclidian geometry.     

Fractal dimension in endometrial carcinoma

OBJECTIVE: To mathematically assess in a pilot study, endometrial glandular margin irregularity in simple hyperplasia, complex atypical hyperplasia and well-differentiated endometrial carcinoma with the help of box counting of fractal dimension and to discriminate these lesions on the basis of box counting of fractal dimension of the gland. STUDY DESIGN: Ten cases each of endometrial simple hyperplasia (without atypia), complex hyperplasia with atypia and endometrial carcinoma (well-differentiated, endometrioid) were assessed in the study. Five fields at 20 x magnification from each case were randomly selected, and the glands were outlined with the help of a pointer. Using the box counting method, the fractal dimension of each case was measured. RESULTS: Mean fractal dimension in simple hyperplasia, complex atypical hyperplasia and endometrial carcinoma was, 0.899 +/- 0.13, 0.932 +/- 0.042 and 0.939 +/- 0.02, respectively. Statistical analysis showed that the fractal dimension of glands of simple hyperplasia were significantly different from that of complex atypical hyperplasia and endometrial carcinoma (P = .041 and .013, respectively, ANOVA). However, there was no significant difference in fractal dimension between glands of complex hyperplasia and of endometrial carcinoma (P = .659, ANOVA). CONCLUSION: This study provides mathematical (objective) assessment of the measurement of glandular margin irregularities in simple hyperplasia, complex atypical hyperplasia and endometrial carcinoma. Fractal dimension of gland margin may have diagnostic potential in the future.     

The fractal structure of glycogen: A clever solution to optimize cell metabolism.

Fractal objects are complex structures built with a simple procedure involving very little information. This has an obvious interest for living beings, because they are splendid examples of optimization to achieve the most efficient structure for a number of goals by means of the most economic way. The lung alveolar structure, the capillary network, and the structure of several parts of higher plant organization, such as ears, spikes, umbels, etc., are supposed to be fractals, and, in fact, mathematical functions based on fractal geometry algorithms can be developed to simulate them. However, the statement that a given biological structure is fractal should imply that the iterative process of its construction has a real biological meaning, i.e., that its construction in nature is achieved by means of a single genetic, enzymatic, or biophysical mechanism successively repeated; thus, such an iterative process should not be just an abstract mathematical tool to reproduce that object. This property has not been proven at present for any biological structure, because the mechanisms that build the objects mentioned above are unknown in detail. In this work, we present results that show that the glycogen molecule could be the first known real biological fractal structure.