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PubMedDate not supplied

Fractal organization of feline oocyte cytoplasm.

The present study aimed at verifying whether immature cat oocytes with morphologic irregular cytoplasm display self-similar features which can be analytically described by fractal analysis. Original images of oocytes collected by ovariectomy were acquired at a final magnification of 400x with a CCD video camera connected to an optic microscope. After greyscale thresholding segmentation of cytoplasm, image profiles were submitted to fractal analysis using FANAL++, a program which provided an analytical standard procedure for determining the fractal dimension (FD). The presentation of the oocyte influenced the magnitude of the fractal dimension with the highest FD of 1.91 measured on grey-dark cytoplasm characterized by a highly connected network of lipid droplets and intracellular membranes. Fractal analysis provides an effective quantitative descriptor of the real cytoplasm morphology, which can influence the acquirement of in vitro developmental competence, without introducing any bias or shape approximation and thus contributes to an objective and reliable classification of feline oocytes.

Animals↗
PubMed1993

The concept of fractals in the pharmaceutical sciences.

An old problem is sometimes better understood when it is viewed from a new perspective than by investigating it repeatedly with classical methods. Such a new perspective is provided by the concept of fractals. Briefly, this term defines a spread class of geometric shapes, whose subunits replicate the structure of the larger unit in accordance with the formalization of the idea of self-similarity. This concept was introduced by Mandel-brot in respond to the need for a more sophisticated explanation of numerous phenomena commonly encountered in nature. In this review we will provide a short introduction into the concept of fractals and a concise overview of those studies where fractal geometry has been employed so far in pharmaceutical research. We believe that a review of the most recent findings in the application of fractal geometry to problems encountered in the pharmaceutical sciences might be helpful also to those people who are involved in the search for novel drug substances, as well as in the investigation of their performance in vitro and in vivo. A few suggestions will be made where this concept probably could be helpful in the future and instructions will be given on how the fractal dimension of rugged objects can be estimated practically.

Animals↗
PubMed1996

Self-similarity and fractal irregularity in pathologic tissues.

The irregularity and self-similarity under scale changes are the main attributes of the morphologic complexity of cells and tissues, either normal or pathologic. In other words, the shape of a self-similar object does not change when scales of measure change because any part of it might be similar to the original object. Size and geometric parameters of an irregular object, however, differ when inspected at increasing resolution, which reveals more details. Significant progress has been made over the past three decades in understanding how to analyze irregular shapes and structures in the physical and biologic sciences. Dominant influences have been the discovery by B.B. Mandelbrot of a new, practical geometry of nature, now called fractal geometry, and the continuous improvements in computational capabilities. The application of the principles of fractal geometry, unlike the conventional Euclidean geometry developed for describing regular and ideal geometric shapes practically unknown in nature, enables one to measure the fractal dimension, contour length, surface area, and other dimensional parameters of almost all irregular and complex biologic tissues. During the past decade, a large amount of experimental evidence has accumulated showing that even in biomedical sciences fractal patterns could be observed. Through several examples borrowed from the recent literature, we focus on the application of the fractal approach to measuring irregular and complex features of pathologic cells and tissues and also on its potential role in the understanding of tumor biology.

Fractals↗
PubMed1996

Quantification of nuclear pleomorphism using an asymptotic fractal model.

OBJECTIVE: Nuclear pleomorphism (nuclear membrane irregularity) was investigated using transmission electron microscope (TEM) micrographs of 1,419 nuclei (32 oral carcinomas and normal cells from surgical margins). STUDY DESIGN: Nuclear profiles (1,400 x) were digitized (1 pixel = 35 nm) and fractal dimension estimated using the "yardstick" method. RESULTS: Log-log plots of yardstick length vs. perimeter showed a significant effect on length measurement typical of fractals at low resolutions (large yardsticks), but this effect disappeared at higher resolution (small yardsticks); that is compatible with Rigaut's asymptotic fractal model. Analysis of the asymptotic fractal parameters c, L and Bm showed that c was higher in normal nuclei, but log(L) and Bm were higher in malignant nuclei. A linear discriminant analysis using c, log(L) and Bm reclassified correctly 78.8% of the nuclei (normal 88.0%, tumor 70.2%). CONCLUSION: Asymptotic fractal analysis of nuclear profiles appears to show great potential for quantitative discrimination of oral cancer cell features.

Carcinoma, Squamous Cell↗
PubMed1996

Using fractals and nonlinear dynamics to determine the physical properties of ion channel proteins.

Three examples are given of how concepts from fractals and nonlinear dynamics have been used to analyze the voltages and currents recorded through ion channels in an attempt to determine the physical properties of ion channel proteins. (1) Early models had assumed that the switching of the ion channel protein from one conformational state to another can be represented by a Markov process that has no long-term correlations. However, one support for the existence of long-term correlations in channel function is that the currents recorded through individual ion channels have self-similar properties. These fractal properties can be characterized by a scaling function determined from the distribution of open and closed time intervals, which provides information on the distribution of activation energy barriers between the open and closed conformational substates of the ion channel protein and/or on how those energy barriers change in time. (2) Another support for such long-term correlations is that the whole-cell membrane voltage recorded across many channels at once may also have a fractal form. The Hurst rescaled range analysis of these fluctuations provides information on the type and degree of correlation in time of the functioning of ion channels. (3) The early models had also assumed that the switching from one state to another is an inherently random process driven by the energy from thermal fluctuations. More recently developed models have shown that deterministic dynamics may also produce the same distributions of open and closed times as those previously attributed to random events. This raises the possibility that the deterministic atomic and electrostatic forces play a role in switching the channel protein from one conformational shape to another. Debate exists about whether random, fractal, or deterministic models best represent the functioning of ion channels. However, fractal and deterministic dynamics provide a new approach to the study of ion channels that should be seriously considered by neuroscientists.

Animals↗
PubMed1997

[A study on fractals in the epidemics of measles].

Fractal is a new field of science to study the complexity and irregularity of events. The basic principles of fractal are introduced in this paper and used in the analyses of measles epidemics in Guangdong Province and Nanping district of Fujian Province, based on the monthly reported data on incidence, from 1951 to 1989. The results show that there are fractals in measles epidemics, the fractal dimensions are 1.588 and 1.626 before vaccination, and change to 1.716 and 1.758 after the vaccination respectively, indicating that the epidemics of mealses have become more complex and irregular than before. It is shown that fractal study is useful in expressing the complexity and irregularity of mealses epidemics.

China↗
PubMed1999

Analysis of analyte-receptor binding kinetics for biosensor applications: an overview of the influence of the fractal dimension on the surface on the binding rate coefficient.

An overview of fractal analysis is presented for analyte-receptor binding kinetics for different types of biosensor application. Data taken from the literature can be modelled by using (1) a single-fractal analysis, (2) a single- and a dual-fractal analysis, and (3) a dual-fractal analysis. Cases (2) and (3) represent a change in the binding mechanism as the reaction progresses on the surface. Predictive relationships developed for the binding rate coefficient as a function of the analyte concentration are of particular value because they provide a means by which the binding rate coefficients can be manipulated. Relationships are presented for the binding rate coefficients as a function of the fractal dimension, Df, or the degree of heterogeneity that exists on the surface. The binding rate coefficient is rather sensitive to the degree of heterogeneity, Df, that exists on the biosensor surface. For the examples analysed, the order of dependence of the binding rate coefficient on Df ranges from 1.4770 (k1), for the binding of intercalators and metabolites in solution to DNA immobilized at a positively charged surface, to 4.9434 for the binding of 5 nM nucleotide+GroEL in solution to GroES immobilized on a Ni2+-nitriloacetic acid sensor chip [Nieba, Nieba-Axmann, Persson, Hamalainen, Edebratt, Hansson, Lidholm, Magnusson, Karlsson and Pluckhun (1997) Anal. Biochem. 252, 217-228]. GroEl and GroES are two proteins (chaperones) which facilitate protein folding in the cell in an ATP-dependent manner [Hemmingson, Woolford, van der Vies, Tilly, Dennis, Georgopoulos, Henfrix and Ellis (1988) Nature (London) 333, 330-334]. The overview provides an overall analysis of the reaction parameters of importance observed and how they are influenced in antigen-antibody-binding kinetics for different biosensor applications. The predictive relationships presented provide further physical insights into the binding reactions on the surface, and should assist in enhancing biosensor performance. In general, the technique and the overview presented are applicable for the most part to other reactions occurring on different types of surface, for example cell-surface reactions.

Antigen-Antibody Reactions↗
PubMed1992

The fractal dimension may be a useful morphometric discriminant in histopathology.

The hypothesis is presented that fractal geometry, and the fractal dimension in particular, is likely to be a more useful method of image analysis in histopathology than the use of regular Euclidean geometry. The concept of the fractal dimension is described and it is suggested that many human tumours have a fractal structure over a wide range of scales. It is postulated that the fractal dimension will be useful in discriminating between different diagnostic categories and may also allow more precise prognostic stratification within single diagnostic groups.

Humans↗
PubMed1997

Binding Kinetics for Biosensor Applications Utilizing Fractals: A Categorization

Analyte-receptor binding reactions occurring on biosensor surfaces are categorized using a fractal analysis. Both a single- and a dual-fractal analysis are utilized to describe the analyte-receptor binding kinetics. The three categories presented to describe the binding kinetics include (a) case I-only a single-fractal analysis is utilized, (b) case II-a single- and a dual-fractal analysis are utilized (implying a change in the binding mechanism), and (c) case III-only a dual-fractal analysis is utilized. This is an initial attempt to categorize surface binding kinetics. The categorization provides one possible means of classifying analyte-receptor reactions occurring on biosensor surfaces. The analysis should help to improve biosensor performance parameters. The analysis is, in general, extendable to analyte-receptor reactions occurring on other surfaces.

Journal Article↗
PubMed1998

Flocculation of Hematite with Polyacrylic Acid: Fractal Structures in the Reaction- and Diffusion-Limited Aggregation Regimes.

The structure of hematite aggregates in the presence of fairly monodisperse polyacrylic acid (PAA) with two different molecular weights (Mw = 1.36 x 10(6), Mw/Mn = 1.53; Mw = 3.69 x 10(4), Mw/Mn = 1.60) was studied using static light scattering (SLS). The fractal dimensions were calculated from the scattering exponents, after taking into account the finite size of aggregates, using exponential and Gaussian cutoff functions. Three flocculation regimes, namely, pre-DLA, DLA (diffusion-limited aggregation), and post-DLA, were defined based on the polymer concentration. In the DLA regime, fractal dimension values, Df = 1.84 +/- 0.02 and 1.73 +/- 0.02, were obtained using exponential and Gaussian cutoff functions, respectively. A fractal dimension of approximately 2.0 was found, as expected, in the pre-DLA regime (at PAA concentrations lower than the optimal dosage for a DLA regime) where the flocculation rate was reaction limited. In contrast, in the post-DLA regime, the flocculation was slow but the structure of aggregates was as tenuous as in the DLA regime with a fractal dimension Df approximately 1.8. Moreover, for all three regimes, the Df values were independent of the molecular weights of PAA. The lower fractal dimension in post-DLA was probably due to the increased concentration of polymer chains between adjacent particles in aggregates. The steric hindrance favored tip-to-tip aggregation, leading to a more tenuous structure. Copyright 1998 Academic Press.

Journal Article↗
PubMed2000

Measurement of Fractal Aggregates of Polydisperse Particles Using Small-Angle Light Scattering.

The effect of primary particle polydispersity on the structure of fractal aggregates has been investigated through the salt-induced, diffusion-limited aggregation of mixtures of hematite. The fractal dimension was determined experimentally using three independent methods: q dependence of static light scattering, kinetic scaling, and correlation of aggregate mass and linear size both determined from Guinier scattering. The fractal dimensions D(f) obtained were 1.75+/-0.03, 1.76+/-0.03, and 1.70+/-0.05, respectively. The use of a previously derived fractal mean particle size was validated in allowing data collapse to master curves for the aggregation kinetics data. The fractal mean particle size is shown to have general utility by taking a number weighting to describe polydisperse aggregation kinetics and a mass weighting to describe small q scattering behavior. Copyright 2000 Academic Press.

Journal Article↗
PubMed2000

Influence of Calcination Procedure on Porosity and Surface Fractal Dimensions of Silica Xerogels Prepared under Different Evaporation Conditions.

Silica xerogels were prepared under three different solvent evaporation conditions before gelation and two calcination procedures after aging and drying, using a hydrochloric acid-catalyzed tetraethyl orthosilicate system with acetone as the solvent. The influence of calcination procedures on the percentage porosity and surface fractal dimensions of the resultant xerogels is investigated. The evaluation of percentage porosity and surface fractal dimensions based on field emission gun-scanning electron microscopy indicates that calcination generally causes an increase in percentage porosity and surface fractal dimensions. Calcination at 700 degrees C without an intermediate stay at 600 degrees C leads to larger extent of further condensation that hinders the increase of porosity and surface fractal dimensions, in comparison with calcination with an intermediate stay. The difference in porosity and surface fractal dimensions derived from calcination procedures is also influenced by the solvent evaporation conditions before gelation in preparation of the samples. When the solvent evaporation rate is slower, the difference is slight because further condensation in the resultant finer gel textures proceeds more efficiently. Copyright 2000 Academic Press.

Journal Article↗
PubMed1987

The fractal dimension of a test signal: implications for system identification procedures.

The experimental identification of a non-linear biologic transducer is often approached via consideration of its response to a stochastic test ensemble, such as Gaussian white noise (Marmarelis and Marmarelis 1978). In this approach, the input-output relationship a deterministic transducer is described by an orthogonal series of functionals. Laboratory implementation of such procedures requires the use of a particular test signal drawn from the idealized stochastic ensemble; the statistics of the particular test signal necessarily deviate from the statistics of the ensemble. The notion of a fractal dimension (specifically the capacity dimension) is a means to characterize a complex time series. It characterizes one aspect of the difference between a specific example of a test signal and the test ensemble from which it is drawn: the fractal dimension of ideal Gaussian white noise is infinite, while the fractal dimension of a particular test signal is finite. This paper shows that the fractal dimension of a test signal is a key descriptor of its departure from ideality: the fractal dimension of the test signal bounds the number of terms that can reliably be identified in the orthogonal functional series of an unknown transducer.

Mathematics↗
PubMed1990

Fractal character of pulmonary microvascular permeability.

The pulmonary microvasculature offers a heterogeneous barrier to the motion of large solutes as they pass between blood and lymph. While this barrier has been approximated by a few discrete pathways or by statistical ensembles of many pathways, these descriptions only partly capture the structural and functional properties of the pulmonary microcirculation. The concept that this barrier may be a fractal object is explored. Endothelial cleft geometry displays scaling in junctional path length and self-similarity in its spatial organization. It is shown that a fractal cleft produces heterogeneous spaces capable of transporting water and macromolecules. Cleft location, size, and depth are characterized, in part, by a fractal dimension of approximately 0.8. The consequences for transport through a fractal barrier are then determined. Predicted sieving of macromolecules by a fractal barrier is found to be consistent with lung microvascular transport data. Nonlinear transport phenomena are one consequence of a barrier having a fractional dimension.

Blood-Air Barrier↗
PubMed1992

Fractal dimension from radiographs of peridental alveolar bone. A possible diagnostic indicator of osteoporosis.

The purpose of this study was to investigate whether a radiographic estimate of osseous fractal dimension is useful in the characterization of structural changes in alveolar bone. Ten dry mandibular bone segments were radiographed from three controlled projection angles (-5, 0, +5 degrees), before and after acid-induced partial decalcification. Fractal dimension was estimated by regression analysis of power spectra computed by Fourier transform of selected regions of interest in digitized images of the radiographs. Repeated-measures ANOVA showed that fractal dimension so determined varied over anatomic locations (p less than .01), but increased after acid-induced demineralization (p less than .0005), irrespective of the radiographic projection angles (p greater than .99). In vivo fractal dimension was computed from randomly selected intraoral radiographs of six premenopausal (ages, 32.8 +/- 3.9) and six postmenopausal (ages, 62.5 +/- 4.1) women. A significantly (p less than .01) higher fractal dimension was observed in the older group.

Adult↗
PubMed1989

A fractal analysis of cell images.

Methods of digital image analysis have been adapted to measure the fractal dimension of cellular profiles. The fractal dimension is suggested as a useful measure of the complexity of a contour. Three methods produce similar results when applied to constructed, near-ideal fractal figures. Comparison of the measurements for a variety of image types indicates the measurement accuracy in each case and may help in interpreting the results when applied to real, non-ideal cell images of unknown fractal dimension. Two of the methods are currently adopted as appropriate for use on neuronal contours. A correlation exists between the complexity of these contours and the magnitude of the estimated fractal dimension.

Animals↗
PubMed1991

A fractal analysis of pyramidal neurons in mammalian motor cortex.

Pyramidal neurons in the mammalian cerebral cortex can be described by a fractal dimension (Mandelbrot, 1982), which is an objective, quantitative measure of the complexity of their soma/dendritic borders. In the cat, the fractal dimensions of lamina V cells, which include pyramidal tract neurons (PTN), indicate that these cells are more complex than other pyramidal neurons (PN) in the same region of motor cortex. The lamina V cells of the cat are also more complex than those in motor cortex of the monkey. Moreover, lamina III neurons in the monkey are more complex than monkey lamina V neurons. The fractal dimension of the intracortical axon collateral arborizations of the same pyramidal neurons indicated, in all cases, that the branching of these terminals is less complex than the branching of the dendrites of the same cells. In line with the observation that the fractal dimensions of some homologous cellular populations are different in different species, it is suggested that the fractal dimension and the degree of morphological complexity may relate to the requirement for the number of separable functions to be accommodated within one neuron. For example, as the size of the cortex and the number of neurons in a region increase, the opportunity exists within a given cortical zone, for individual functions to be segregated and for functional specialization to be accommodated with less morphological complexity of the individual neurons performing each of these functions.

Animals↗
PubMed2007

Quantitative study on the microstructure of colloidal fat crystal networks and fractal dimensions.

This paper highlights the most recent progress in the quantitative study on the microstructure and the rheological properties of colloidal fat crystal networks. Several physical models describing the structural hierarchy of the colloidal fat crystal networks are reviewed here, with particular emphasis on fractal model, which can be used to explain the scaling behavior of fat crystal networks. The concept of the fractal dimension has been extensively used in the quantitative study of the microstructure of fat crystal networks and other colloidal networks; however, the relationship between the fractal dimension value and microstructural characteristics remains somewhat nebulous. Recent computer simulation work from our laboratory will be presented relating simulated microstructural features to values of particular measure of fractality including the box-counting, particle counting and Fourier-transform fractal dimensions.

Journal Article↗