Fractal dimension in aspiration cytology smears of breast and cervical lesions

OBJECTIVE: To standardize the automated measurement of fractal dimension on cytologic smears and compare the fractal dimension of benign and malignant breast cells and cervical lesions on cytologic material to evaluate its role in the discrimination of benign from malignant cells. STUDY DESIGN: We randomly selected fine needle aspiration cytology smears of 42 cases of infiltrating duct carcinoma and 38 cases of fibroadenoma of the breast. Similarly, 16 cervical carcinoma and 20 normal cervical smears were selected for study. Ten cells were selected randomly from each case. Box counting of fractal dimension of malignant and benign cells was achieved with an image cytometer (Leica, Cambridge, England) using Quantimet 600 software (Leica). Then a well-spaced grid with multiple small boxes of a particular pixel length was superimposed on the cell. The dimension of the box was selected as 4, 8 and 16 pixels. With the help of a logical "AND" operation, we counted the number of boxes touching the peripheral margin of the cell nuclei. For each cell, the log-log graph of 1 per box size was plotted against the number of boxes touching the peripheral rim of the cell. The slope of each graph was identified using the least-squares method of regression analysis. RESULTS: The mean fractal dimension of malignant cells was 0.8536 +/- 0.1120 as compared to 0.8403 +/- 0.1115 in benign cell groups. The Mann-Whitney U test showed a significant difference in fractal dimension in these 2 groups (P = .05). The mean fractal dimension of malignant cells from the cervix was 0.8656 +/- 0.1499 as compared to 0.8315 +/- 0.1312 in benign cells. The Mann-Whitney U test showed a significant difference in fractal dimension in these 2 groups (P < .02). CONCLUSION: Fractal dimension may be a helpful adjunctive technique to discriminate between benign and malignant cells.     

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.     

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.     

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。     

Microbial growth patterns described by fractal geometry.

Fractal geometry has made important contributions to understanding the growth of inorganic systems in such processes as aggregation, cluster formation, and dendritic growth. In biology, fractal geometry was previously applied to describe, for instance, the branching system in the lung airways and the backbone structure of proteins as well as their surface irregularity. This investigation applies the fractal concept to the growth patterns of two microbial species, Streptomyces griseus and Ashbya gossypii. It is a first example showing fractal aggregates in biological systems, with a cell as the smallest aggregating unit and the colony as an aggregate. We find that the global structure of sufficiently branched mycelia can be described by a fractal dimension, D, which increases during growth up to 1.5. D is therefore a new growth parameter. Two different box-counting methods (one applied to the whole mass of the mycelium and the other applied to the surface of the system) enable us to evaluate fractal dimensions for the aggregates in this analysis in the region of D = 1.3 to 2. Comparison of both box-counting methods shows that the mycelial structure changes during growth from a mass fractal to a surface fractal.     

A fractal analysis of the radial distribution of bronchial capillaries around large airways.

We analyzed published measurements of the bronchial circulation and airway wall (Anderson JC, Bernard SL, Luchtel DL, Babb AL, and Hlastala MP. Respir Physiol Neurobiol 132: 329-339, 2002) and determined that the radial distribution of bronchial capillary cross-sectional area was fractal. We limited our analysis to bronchial capillaries, diameter < or =10 mum, that resided between the epithelial basement membrane and adventitia-alveolar boundary, the airway wall tissue. Thirteen different radial distributions of capillary-to-tissue area were constructed simply by changing the number of annuli (i.e., the annular size) used to form each distribution. For the 13 distributions created, these annuli ranged in size from to of the size of the airway wall area. Radial distributions were excluded from the fractal analysis if the sectioning procedure resulted in an annulus with a radial thickness less than the diameter of a capillary. To determine the fractal dimension for a given airway, the coefficient of variation (CV) for each distribution was calculated, and ln(CV) was plotted against the logarithm of the relative piece area. For airways with diameter >2.4 mm, this relationship was linear, which indicated the radial distribution of bronchial capillary cross-sectional area was fractal with an average fractal dimension of 1.27. The radial distribution of bronchial capillary cross-sectional area was not fractal around airways with diameter <1.5 mm. We speculated on how the fractal nature of this circulation impacts the distribution of bronchial blood flow and the efficiency of mass transport during health and disease. A fractal analysis can be used as a tool to quantify and summarize investigations of the bronchial circulation.     

细胞核内特征值的改变可引起DNA倍体的变化

目的通过测定不同DNA倍体细胞,研究细胞核内特征值的改变。方法用宫颈刷刷出宫颈细胞,经固定后,用涂片离心机制成二张玻片,一张行巴氏染色作TBS诊断,另一张行Feulgen染色做DNA定量测定。通过对宫颈细胞核图像内像素的统计,计算出细胞核内多种特征值,比较不同DNA倍体细胞内特征值的不同。结果 161873例妇女行宫颈细胞学检查,常规细胞学检查发现2454例低级别鳞状上皮内病变(low-grade squamous intraepithelial lesion,LSIL)和523例高级别鳞状上皮内病变(high-grade squamous intraepithelial lesion,HSIL);而DNA倍体分析发现3412例有3个以上>5c细胞。84%以上的LSIL和HSIL病例均可见倍体异常细胞。与2c细胞相比,4c、5c、7c及9c细胞核面积及核半径明显增大;7c、9c细胞核内平均光学密度和紧实度均值也有明显改变,而光密度方差和灰度熵无变化。结论宫颈细胞DNA倍体改变往往伴有细胞形态和DNA核内分布等特征值的改变。     

Fractal morphometry of cell complexity.

Irregularity and self-similarity under scale changes are the main attributes of the morphological complexity of both normal and abnormal cells and tissues. In other words, the shape of a self-similar object does not change when the scale of measurement changes, because each part of it looks similar to the original object. However, the size and geometrical parameters of an irregular object do differ when it is examined at increasing resolution, which reveals more details. Significant progress has been made over the past three decades in understanding how irregular shapes and structures in the physical and biological sciences can be analysed. Dominant influences have been the discovery of a new practical geometry of Nature, now known as fractal geometry, and the continuous improvements in computation capabilities. Unlike conventional Euclidean geometry, which was developed to describe regular and ideal geometrical shapes which are practically unknown in nature, fractal geometry can be used to measure the fractal dimension, contour length, surface area and other dimension parameters of almost all irregular and complex biological tissues. We have used selected examples to illustrate the application of the fractal principle to measuring irregular and complex membrane ultrastructures of cells at specific functional and pathological stage.     

Visual discrimination of fractal borders.

The ideas of fractals and fractal dimension are here translated into the realm of visual psychophysics. Borders between two fields of different luminance were used. Because of the finite grain of the visual system, fractal dimension need be defined only within a certain size range. For a fractal dimension of 1.15, the just-detectable difference in fractal dimension was found to be about 0.0085, rising to about 0.015 for a fractal dimension of 1.25. Reducing exposure duration from 1 s to 0.33 s decreases sensitivity to differences in fractal dimension, but there was no gain in increasing the exposure duration. Good visual observers who are naive to the task require some training before reaching optimal performance. The ability to discriminate fractal dimension differs between fractal edges of the same fractal dimension that were generated with differing statistical programs. Even after considerable training, an observer makes 29% errors when asked to distinguish a fractal edge generated with a Gaussian random walk from one with a rectangular random walk. Gaussian random walk fractals can be more easily distinguished from Poissonian and Cauchy ones.     

Effects of macrophyte heterogeneity and food availability on structural parameters of the macroinvertebrate community in a Pampean stream

Environmental heterogeneity in natural ecosystems influences several parameters at the population and community levels. In freshwater ecosystems, habitat heterogeneity can be provided by macrophyte species with different structural shapes. Previous studies suggest that aquatic plants with more complex architectures will support higher number, biomass, and taxon richness of macroinvertebrates than plants with simpler shape. We investigated the influence of macrophyte structural heterogeneity (quantified by fractal dimension) and food availability (represented by epiphytic biomass) on several parameters (number of individuals, biomass, body size distribution, taxon richness, and diversity) of the macroinvertebrate community in a Pampean stream. Four submerged macrophyte species (Egeria densa, Elodea ernstae, Ceratophyllum demersum, and Stuckenia striata) and associated macroinvertebrates were sampled in late spring, summer, and autumn. Plants were photographed and fractal dimension was estimated from the images by the box-counting method. Fractal dimension was independent of plant surface area per unit of macrophyte biomass and differed significantly among species. Mean fractal dimension varied between 1.29 and 1.62, and increased following the sequence E. densa → S. striata → E. ernstae → C. demersum. Macrophyte species with higher fractal dimension supported a greater abundance of macroinvertebrates, especially those of small body size (500–1,000 μm); but fractal dimension was unrelated to macroinvertebrate biomass, richness, and diversity. However, overall animal biomass was significantly associated to the epiphytic abundance. Consequently, macrophyte heterogeneity influences macroinvertebrate density and body size distribution, while animal biomass depends on epiphytic food resources provided by plants.     

Measurement of subvisual changes in cervical squamous metaplastic cells for detecting abnormality.

Quantitative measures of visually normal squamous metaplastic cells exfoliated from the uterine cervix were obtained to test the hypothesis that these cells, like intermediate squamous and endocervical columnar cells, show subvisual evidence of atypia in cases of bonafide squamous intraepithelial neoplasia. The cells identified as squamous metaplastic were obtained from 14 abnormal (dysplastic) and 9 diagnostically negative cases. Although the cell populations so grouped showed no statistically significant differences in overall cell size, nuclear area or nuclear/cytoplasmic ratios, there were significant differences in nuclear and cytoplasmic densitometric features and in nuclear texture features. A combination of three features (nuclear density, texture and cytoplasmic density) permitted 76% of the cells to be categorized correctly as originating in normal versus abnormal slides. It is concluded that selected quantitative features of exfoliated metaplastic cell populations may contribute to improved diagnostic accuracy in automated screening for cervical abnormalities.     

Trends in soil fractal parameters caused by accumulation of soil organic matter as resulting from the analysis of water vapor adsorption isotherms

The experimental adsorption isotherms are used to evaluate the specific surface area and the surface fractal dimensions of several samples containing organic matter. The aim of the investigations has been to search for correlations between specific surface area and geometrical heterogeneity, as characterized by the surface fractal dimension and the content of organic matter. Besides natural organic soils (peats, forest humuses and brown coal from mines) we also investigate controlled systems obtained by adding humic acids to kaolin and quartz, organic compost to soils and by mixing sandy soil and peat at different ratios. The aim of the investigations of controlled systems was to discover some general trends in the dependencies of the specific surface area and the surface fractal dimension on the content of organic matter.     

长江源区高寒草原和高寒草甸土壤粒径分布特征

为探究长江源区主要下垫面土壤空间异质性与粒径分布(PSD)非均匀性,运用分形理论描述高寒草原和高寒草甸2种下垫面土壤粒径分布特征,分析了 2种下垫面土壤的分形维数特征差异及其与土壤颗粒组成的关系.结果表明:研究区土壤颗粒粒径主要分布于100-800 μm,高寒草原土壤单重分形维数(Dv)为2.429～2.508,高寒草...     

科尔沁沙地农田沙漠化演变中土壤颗粒分形特征

研究了科尔沁沙地农田沙漠化过程中土壤的粗粒化和养分的贫瘠化特征 ,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 :土壤沙粒含量越高 ,土壤分形维数越低 ,表征农田沙漠化程度越高 ;土壤颗粒分形维数与土壤有机 C、全 N、粘粉粒含量之间存在显著的线性关系。说明分形维数能很好地表征农田沙漠化演变中土壤结构和养分状况以及沙漠化的程度 ,可作为评价土壤沙漠化演变的一项综合性定量指标。     

Modeling brewers' yeast flocculation

Flocculation of yeast cells occurs during the fermentation of beer. Partway through the fermentation the cells become flocculent and start to form flocs. If the environmental conditions, such as medium composition and fluid velocities in the tank, are optimal, the flocs will grow in size large enough to settle. After settling of the main part of the yeast the green beer is left, containing only a small amount of yeast necessary for rest conversions during the next process step, the lagering. The physical process of flocculation is a dynamic equilibrium of floc formation and floc breakup resulting in a bimodal size distribution containing single cells and flocs. The floc size distribution and the single cell amount were measured under the different conditions that occur during full scale fermentation. Influences on flocculation such as floc strength, specific power input, and total number of yeast cells in suspension were studied. A flocculation model was developed, and the measured data used for validation. Yeast floc formation can be described with the collision theory assuming a constant collision efficiency. The breakup of flocs appears to occur mainly via two mechanisms, the splitting of flocs and the erosion of yeast cells from the floc surface. The splitting rate determines the average floc size and the erosion rate determines the number of single cells. Regarding the size of the flocs with respect to the scale of turbulence, only the viscous subrange needs to be considered. With the model, the floc size distribution and the number of single cells can be predicted at a certain point during the fermentation. For this, the bond strength between the cells, the fractal dimension of the yeast, the specific power input in the tank and the number of yeast cells that are in suspension in the tank have to be known. Copyright 1998 John Wiley & Sons, Inc.     

长江下游江心洲土壤颗粒特征及分形规律

研究了长江下游马鞍山段成长型江心洲洲头、洲中央和洲尾3个部位100cm深度范围内的土壤颗粒特征及分形规律。结果表明:江心洲土壤颗粒总体偏砂性,主要由粉粒、极细砂和细砂组成,颗粒最大粒径为350μm;在剖面上,3类颗粒的比例洲中央部位变化较小,洲头和洲尾变化较大;颗粒随深度的增加逐渐变粗;粉粒平均含量表现为洲中央>洲头>洲尾,分别为75.42%、43.58%和22.13%;土壤颗粒分形维数表现为洲中央>洲头>洲尾,分别为2.28、2.17和2.05,小于安徽省耕作表层土壤的分形维数平均值,与单位体积比表面积呈极显著正相关,与粘粒、粉粒含量极显著正相关,与极细砂、细砂极显著负相关。江心洲不同部位的土壤颗粒特征与江心洲的发育演变之间存在对应关系。     

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.     

Structural complexity in mussel beds: the fractal geometry of surface topography

Seafloor topographic complexity is ecologically important because it provides habitat structure and alters boundary-layer flow over the bottom. Despite its importance, there is little agreement on how to define and measure surface complexity. The purpose of this investigation was to utilize fractal geometry of vertical cross-section profiles to characterize the surface topography of the soft-bottom mussel bed (Mytilus edulis L.) at Bob's Cove, ME, USA. Mussels there have been shown previously to have spatially ordered fractal characteristics in the horizontal plane. Two hypotheses were tested. The first was that the bed surface is fractal over the spatial scale of 1.44-200 mm, with fractal dimension less than or equal to 1.26, the value for the Koch curve, our model for bed profiles. The second was that bed surface topography (i.e., in vertical profile) is less complex than the mussel bed spatial pattern (i.e., aerial view in the horizontal plane). Both hypotheses were supported. Cross-sections of plaster casts of the bed produced 88 surface profiles, all of which were fractal over the entire spatial scale of more two orders of magnitude employed in the analysis. Fractal dimension values (D) for individual profiles ranged from 1.031 to 1.310. Fractal dimensions of entire casts ranged up to mean (1.242+/-0.046) and median (1.251) values similar to 1.26, the theoretical value of the Koch curve. The bed surface was less complex than the bed spatial pattern because every profile had D<1.36, the smallest value previously obtained from aerial views of the bed. The investigation demonstrated for the first time that surface topography of a soft-bottom mussel bed was fractal at a spatial scale relevant to hydrodynamic processes and habitat structure important for benthic organisms. The technique of using cross-section profiles from casts of the bed surface avoided possible underestimates of fractal dimension that can result from other profiling methods reported in the literature. The results demonstrate that fractal dimension can be useful in the analysis of habitat space and water flow over any irregular seafloor surface because it incorporates the size, shape, and scale of roughness elements into a simple, numerical metric.     

Microstructural characterization of chitosan and alginate films by microscopy techniques and texture image analysis

The aim of this work is to characterize the microstructure of chitosan and alginate edible films by microscopy techniques and texture image analysis. Edible films were obtained by solution casting and solvent evaporation. The microscopy techniques used in this work were: light, environmental scanning electron and atomic force microscopy. Textural features and fractal dimension were extracted from the images. Entropy and fractal dimension were more useful to evaluate the complexity and roughness of films. The highest values of entropy and fractal dimension corresponded to alginate/chitosan, followed of alginate and chitosan films. An entropy/fractal dimension ratio, proposed here, was useful to characterize the degree of image complexity and roughness of edible films at different magnifications. It was possible to postulate that microscopy techniques combined with texture image analysis are efficient tools to quantitatively evaluate the surface morphology of edible films made of chitosan and alginate.