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PubMed2000

The recording of monophasic action potentials with fractal-coated iridium electrodes in humans.

The present study was designed to evaluate the feasibility of the recording of monophasic action potentials (MAP) with fractal-coated iridium electrodes in a clinical setting. In 18 patients who underwent an electrophysiological study for various arrhythmias, we performed MAP recordings with both 1.3-mm2 and 6-mm2 tip surface area fractal-coated iridium and standard silver--silver chloride (Ag/AgCl) electrodes in the high right atrium and two ventricular positions. Amplitude and MAP duration at 90%, 50%, and 25% of repolarization were calculated during steady-state pacing at 600, 500, and 400 ms cycle lengths with extrastimuli application. Morphology comparisons of MAP signals recorded with both types of electrodes were performed by regression analysis using 5% of the repolarization segments of the MAP trajectory. Differences between MAP duration at 90%, 50%, and 25% of repolarization recorded with fractal-coated and Ag/AgCl electrodes were statistically insignificant. Amplitude values recorded with 6-mm2 tip electrodes were significantly smaller than those recorded with Ag/AgCl electrodes for all comparisons. During steady-state pacing, the correlation coefficients between Ag/AgCl and fractal-coated 1.3-mm2 and 6-mm2 tip electrodes were within the range of 0.93-0.999 and 0.87-0.999, respectively. The correlation of MAP amplitude and duration at 90%, 50%, and 25% of repolarization following the extrastimulus S2, recorded with both types of electrodes, was significantly weaker for right atrial recordings (r value range 0.78-0.92) as compared to ventricular recordings (r value range 0.92-0.99). The MAP sensing features of fractal-coated iridium and Ag/AgCl electrodes are comparable. The best results for recording of MAPs with fractal-coated electrodes can be achieved with small surface area tip electrodes.

Action Potentials↗

PubMed2002

Ship recognition via its radiated sound: the fractal based approaches.

Due to the complexity of its radiated sound, ship recognition is difficult. Fractal approaches are proposed in this study, including fractal Brownian motion based analysis, fractal dimension analysis, and wavelet analysis, to augment existing feature extraction methods that are based on spectrum analysis. Experimental results show that fractal approaches are effective. When used to augment two traditional features, line and average spectra, fractal approaches led to better classification results. This implies that fractal approaches can capture some information not detected by traditional approaches alone.

Journal Article↗

PubMed1991

Fractal geometry: a design principle for living organisms.

Fractal geometry allows structures to be quantitatively characterized in geometric terms even if their form is not even or regular, because fractal geometry deals with the geometry of hierarchies and random processes. The hypothesis is explored that fractal geometry serves as a design principle in biological organisms. The internal membrane surface of cells, or the inner lung surface, are difficult to describe in terms of classical geometry, but they are found to show properties describable by fractal geometry, at least sectionwise and within certain bounds set by deterministic design properties. Concepts of fractal geometry are most useful in characterizing the structure of branching trees, such as those found in pulmonary airways and in blood vessels. This explains how the large internal gas exchange surface of the lung can be homogeneously and efficiently ventilated and perfused at low energetic cost. It is concluded that to consider fractal geometry as a biological design principle is heuristically most productive and provides insights into possibilities of efficient genetic programming of biological form.

Animals↗

PubMed1990

Fractal properties of pulmonary blood flow: characterization of spatial heterogeneity.

The heterogeneity of pulmonary blood flow was examined using a fractal analytic procedure, and the results were compared with the traditional gravitational model of flow distribution. 99mTc-labeled macroaggregate was injected intravenously at functional residual capacity in six supine anesthetized dogs. The lungs were fixed in situ and sliced in transverse sections. The slices were imaged on a planar gamma camera, and a three-dimensional array of blood flow measurements was reconstructed for each lung. Fractal analysis was used to examine the spatial heterogeneity or RDs (relative dispersion = SD/mean) as a function of the number of pieces into which the flow array was subdivided. RDs was fractal and could be characterized by a fractal dimension (Ds) of 1.09 +/- 0.02, where a Ds of 1.0 reflects homogeneous flow and 1.5 indicates a random flow distribution. The data fit the fractal model exceptionally well with an average r = 0.98. RDs was examined in gravitational and isogravitational planes and as expected was greatest in the gravitational direction. However, the difference was small, suggesting that gravitation plays a secondary role to an underlying process producing heterogeneity. Within the limits of resolution attained by this study (piece volumes greater than 0.25 cm3), the heterogeneity of pulmonary blood flow is well characterized by a fractal model.

Animals↗

PubMed1991

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.

Animals↗

PubMed1993

Fractal morphogenesis by a bacterial cell population.

Many species of bacteria have been found to form fractal colonies. Environmental (physicochemical) and biological factors for this fractal morphogenesis have been examined for their roles in the genesis of fractal and pattern diversity. Morphology of a bacterial colony on a solid agar medium depends on the nutrient diffusion field (two-dimensional). When concentrations of nutrients are low, point-inoculated bacteria (e.g., Bacillus subtilis) exert diffusion-limited growth. A self-similar fractal colony formed slowly under such a condition has the same morphology as one made by the diffusion-limited aggregation (DLA) model. The value of fractal dimensions (ca. 1.72) and the appearance of specific phenomena (screening and repulsion effects) are consistent with computer simulations of the DLA fractal model. On the other hand, a round colony recognized on an agar-rich medium was considered to be the product of reaction-limited growth and was simulated by the Eden model. When motile bacteria are point inoculated onto semi-solid agar media, bacterial spreading behavior also is morphogenic. Branching patterns with various morphologies (e.g., dense-branching morphology) have been recognized and examined for factors responsible for pattern changes. By microscopic inspection of the extending branch, multicellular behavior of bacteria has been observed in the structured cell distribution. Besides cell division and translocation activities, wetting agents produced by some species of bacteria (e.g., serrawettins produced by Serratia marcescens) are considered to be important microbial factors for efficient space occupation and specific cell transpositions in various surface environments.

Bacillus subtilis↗

PubMed2004

Discriminating between elderly and young using a fractal dimension analysis of centre of pressure.

The aim of this project was to evaluate the use of a new analysis technique, fractal dimension analysis, for quantification of quiet stance centre of pressure (COP). By using a fractal dimension analysis of COP, it might be possible to gain more information about control during quiet stance than traditional analyses have previously allowed. The current project considered a group of young healthy participants and a group of elderly healthy participants to compare traditional measures of COP against a fractal dimension analysis of COP. Results indicated that both types of analyses are able to distinguish between eyes open and eyes closed in the elderly group. However, the fractal dimension analysis more accurately detected differences between the participant groups when standing with their eyes closed. Based on these results it is suggested that fractal dimension analysis is more informative about posture control than traditional measures. It is suggested that a fractal dimension type of analysis can be incorporated into clinical testing to identify patients with pathologies.

Journal Article↗

PubMed2003

[Fractal characteristics of particle size distributions of mangroves soils in Yingluo Bay].

Based on fractal theory, the fractal characteristics of particle-size distributions of mangrove soils in Yingluo Bay (21 degrees 28'N, 109 degrees 43'E) were studied. The results show that the fractal dimensions of the soils ranged from 2.6837 to 2.8834, and decreased in the order of sand loam < light loam < medium loam < heavy loam < light clay. The fractal dimensions of the soils on exterior beach were lower than those on middle and inner beaches. There was a significant positive linear relationship between fractal dimension and soil salinity and organic matter content. The major factors that influenced the fractal dimensions of the soils were community type, soil texture, beach position, soil salinity, and soil organic matter content.

Algorithms↗

PubMed2005

[Fractal characteristics and scale dependence of mature granular sludge in anaerobic baffled reactor (ABR)].

Based on the SEM-images of mature granular sludge in ABR, the research process on fractal characteristics of these granules was built. The treatment and fractal dimensions calculation of the images were carried on through Photoshop, Newscan and Fips programs. The results showed that the boxing-counting dimensions of the mature granular sludge were more than 1.85, and the granules formed in the third and fifth compartments were more compact. The boundaries of the same granular sludge were fractal with the fractal dimensions of approximate 1.10, demonstrating the irregularity of the boundaries or surfaces, furthermore the surface of these granules formed in the first and fifth compartments was more irregular. It seemed that the compass dimensions of the granules boundary were a proof of irregularity of their surface. Moreover, there existed two scale scopes in calculating the compass dimensions of each granule, especially some of them with a range of one order of magnitude. A little discrepancy was appeared between box-counting and compass ways to calculating fractal dimensions. It will be important to build relationship between fractal dimensions with different scales and composition, structure, physico-chemical characteristics of granular sludge.

Bacteria, Anaerobic↗

PubMed1991

Emerging fractal properties in gorgonian growth forms (Cnidaria: Octocorallia).

Gorgonian growth has been hitherto studied mostly by investigating the influence of different environmental conditions on shape and orientation. We have made an attempt at analyzing the intrinsic growth properties of these organisms, by using a method capable of pointing out a fractal arrangement. Colony outlines of the gorgonian species Eunicella singularis, Eunicella cavolinii, Paramuricea clavata, and Lophogorgia ceratophyta have been digitized and their length measured by traveling along them with a series of logarithmically increasing steps. When this procedure is used on fractal lines, logarithmic plots of estimated length vs. step show linear trends, whose slope yields the fractal dimension. In gorgonian outlines, due to their limited multi-scale organization, we have found curved trends typical of non-fractal lines. However, non-linear regression analysis has shown that the degree of bending is variable from curve to curve, with differences among species. also, in P. clavata and L. ceratophyta there is a positive correlation between the degree of development and the tendency to assume a fractal geometry. This suggests that gorgonian growth mechanisms retain a self-similar design, which becomes evident only in species combining a large size with a high branch density. Based on these data, possibilities of modeling biological patterns through fractal growth models are questioned.

Animals↗

PubMed1997

Fractals in pathology.

Many natural objects, including most objects studied in pathology, have complex structural characteristics and the complexity of their structures, for example the degree of branching of vessels or the irregularity of a tumour boundary, remains at a constant level over a wide range of magnifications. These structures also have patterns that repeat themselves at different magnifications, a property known as scaling self-similarity. This has important implications for measurement of parameters such as length and area, since Euclidean measurements of these may be invalid. The fractal system of geometry overcomes the limitations of the Euclidean geometry for such objects and measurement of the fractal dimension gives an index of their space-filling properties. The fractal dimension may be measured using image analysis systems and the box-counting, divider (perimeter-stepping) and pixel dilation methods have all been described in the published literature. Fractal analysis has found applications in the detection of coding of coding regions in DNA and measurement of the space-filling properties of tumours, blood vessels and neurones. Fractal concepts have also been usefully incorporated into models of biological processes, including epithelial cell growth, blood vessel growth, periodontal disease and viral infections.

Bone Diseases↗

PubMed2003

Fractal dimensions of breast lesions on cytology smears.

In the present study, box-counting fractal dimensions of benign and malignant cells of breast tumors on cytology material were measured and compared. We selected fine-needle aspiration cytology smears of 14 cases of histopathology-proven infiltrating duct carcinomas and 7 cases of fibroadenoma of the breast. Five cells were randomly selected from each case. Box-counting fractal dimensions of all cells were measured with the help of an image cytometer (Leica, Cambridge, UK), using Quantimet 600 software (Leica). In total, 70 malignant cells and 35 benign cells were studied. The mean fractal dimensions of malignant cells and benign cells were 0.9571 +/- 0.1265 and 0.8354 +/- 0.1367, respectively. There was significant difference in the fractal dimension of malignant and benign cells (P = 0.006, Mann-Whitney nonparametric test). The measurement of fractal dimension may be helpful in discriminating malignant from benign cells. This may be another discriminating feature of malignant cells, along with classic image morphometry based on Euclidean geometry.

Biopsy, Fine-Needle↗

PubMed1994

Fractal organization of trabecular bone images on calcaneus radiographs.

Bone density is not the unique factor conditioning bone strength. Trabecular bone microarchitecture also plays an important role. We have developed a fractal evaluation of trabecular bone microarchitecture on calcaneus radiographs. Fractal models may provide a single numeric evaluation (the fractal dimension) of such complex structures. Our evaluation results from an analysis of images with a varying range of gray levels, without binarization of the image. It is based on the fractional brownian motion model, or more precisely on the analysis of its increment, the fractional gaussian noise (FGN). The use of this model may be considered validated if two conditions are fulfilled: the gaussian repartition and the self-similarity of our data. The gaussian repartition of intermediate lines of these images was tested on a sample of 32,800 lines from 82 images. Following a chi-square goodness-of-fit test, it was checked in 86% of these lines for alpha = 0.01. The self-similarity was tested on 20 images by two estimators, the variance method of Pentland and the spectrum method of Fourier. Self-similarity is defined by lined-up points in a log-log plot of the FGN spectrum or of the variance as a function of the lag. We found two self-similarity areas between scales of analysis ranging from 105 to 420 microns, then above 900 microns, where linear regression produced high mean correlation coefficients (r > or = 0.97). Following this validation, we studied the reproducibility of this new technique. Intra- and interobserver reproducibility, influence of transferring the region of interest, and long-term reproducibility were assessed and given CV of 0.61 +/- 0.15, 0.68 +/- 0.47, 0.53 +/- 0.16, and 2.07 +/- 0.84%, respectively. These data have allowed us to validate the use of this fractal model by checking the fractal organization of our radiographic images analyzed by the model. The good reproducibility of successive x-rays in the same subject allows us to undertake population studies and to envisage longitudinal series.

Calcaneus↗

PubMed1993

Influence of fractal kinetics on molecular recognition.

Molecular recognition is a central issue for nearly every biological mechanism. The analysis of molecular recognition to date has been conducted within the framework of classical chemical kinetics, in which the kinetic orders of a reaction have positive integer values. However, recent theoretical and experimental advances have shown that the assumptions inherent in this classical framework are invalid under a variety of conditions in which the reaction environment may be considered nonideal. A good example is provided by reactions that are spatially constrained and diffusion limited. Bimolecular reactions confined within two-dimensional membranes, one-dimensional channels or fractal surfaces in general exhibit kinetic orders that are noninteger. An appropriate framework for the study of these nonideal phenomena is provided by the Power-Law formalism, which includes as special cases the Mass-Action formalism of chemical kinetics and the Michaelis-Menten formalism of enzyme kinetics. The Power-Law formalism is an appropriate representation not only for fractal kinetics per se, but also for other nonideal kinetic phenomena, provided the range of variation in concentration is not too large. After defining some elementary concepts of molecular recognition, and showing how these are manifested in classical kinetic terms, this paper contrasts the implications of classical and fractal kinetics in a few simple cases. The principal distinction lies in the ability of fractal kinetics to nonlinearly transform, rather than proportionally transmit, the input S/N ratio. As a consequence, fractal kinetics create a threshold for the input signal below which no recognition occurs and above which amplified recognition takes place.(ABSTRACT TRUNCATED AT 250 WORDS)

Enzymes↗

PubMed2001

Use of fractal theory in neuroscience: methods, advantages, and potential problems.

Fractal analysis has already found widespread application in the field of neuroscience and is being used in many other areas. Applications are many and include ion channel kinetics of biological membranes and classification of neurons according to their branching characteristics. In this article we review some practical methods that are now available to allow the determination of the complexity and scaling relationships of anatomical and physiological patterns. The problems of describing fractal dimensions are discussed and the concept of fractal dimensionality is introduced. Several related methodological considerations, such as preparation of the image and estimation of the fractal dimensions from the data points, as well as the advantages and problems of fractal geometric analysis, are discussed.

Algorithms↗

PubMed1996

Supraspinal effects on the fractal correlation in human H-reflex.

In our previous study, 1/f beta-type power spectrum with the spectral exponent beta significantly greater than zero was found in the variability of soleus H-reflex amplitudes. This result indicated that the H-reflex variability was time-correlated owing to fractal characteristics. Furthermore, it was also suggested that the fractal characteristics were generated at the spinal level. The purpose of the present study was to investigate whether the fractal nature of the H-reflex variability was influenced by the loss of supraspinal input. Six healthy normal subjects and seven patients with spinal cord injury participated in this study. Soleus H-reflexes were evoked every 1 s from both legs simultaneously (stimulation intensity: motor threshold) and 1050 successive amplitudes of the H-reflex were recorded. The H-reflex sequence evoked from each leg was analyzed by "coarse graining spectral analysis" to calculate the spectral exponent beta. The value of beta was used to evaluate the level of time-correlation (fractal correlation). Cross-spectral analysis was used to evaluate the degree of synchronization between the H-reflex sequences evoked from both legs. The beta values for normal subjects (0.84 +/- 0.33, left leg; 0.88 +/- 0.34. right leg) were significantly greater (P < 0.001) than those for patients (0.31 +/- 0.18, left leg; 0.32 +/- 0.14, right leg), suggesting that the H-reflex sequences for normal subjects were more time-correlated than for patients. In the frequency range less than 0.2 Hz, the coherence of both legs was high (0.41 +/- 0.14) for normal subjects as compared to 0.20 +/- 0.12 for patients (P < 0.005). In this frequency range, the phase was almost 0 rad for normal subjects, indicating that the H-reflex variabilities of both legs were synchronized. These results suggested that (1) the strong fractal correlation observed in the H-reflex sequences for normal subjects was associated with supraspinal input, and (2) such supraspinal input had equal influence on the reflex arcs of the soleus of both legs.

Adult↗

PubMed2002

Fractal analysis of trabecular bone texture on calcaneus radiographs: effects of age, time since menopause and hormone replacement therapy.

An analysis of trabecular bone texture based on fractal mathematics, when applied to trabecular bone images on plain radiographs, can be considered as a reflection of trabecular bone microarchitecture. It has been shown to be able to distinguish postmenopausal osteoporosis cases from controls. This cross-sectional study was carried out to investigate the influence of age, time since menopause and hormone replacement therapy (HRT) on the fractal dimension of trabecular bone texture at the calcaneus in a sample of 537 healthy women. Fractal analysis of texture was performed on calcaneus radiographs and the result expressed as the Hmean parameter (H = 2-fractal dimension). Total hip, femoral neck and lumbar spine bone mineral density (BMD) was measured by dual-energy X-ray absorptiometry. There was a statistically significant Hmean parameter decrease with age (p<0.0001) but the degree of correlation was low (r = -0.2) compared with the correlation between age and BMD (r = -0.36 to -0.61 according to the BMD site). We found a weak but statistically significant correlation between time since menopause and Hmean (r = -0.14, p = 0.03) in the 241 postmenopausal women included in the study. Hmean was significantly lower in a group of postmenopausal women without HRT (n = 110) compared with a group of age-matched postmenopausal women with HRT (n = 110): respectively 0.683 +/- 0.043 and 0.695 +/- 0.038 (p = 0.03). In conclusion, this study suggests that there is a menopause- and age-related decrease in the Hmean parameter and that HRT interferes with the results of the fractal analysis of trabecular bone texture on calcaneus radiographs.

Absorptiometry, Photon↗

PubMed1988

Fractal aggregates.

In recent years it has been shown that the structures of a wide variety of colloidal aggregates can be described in terms of the concepts of fractal geometry. The purpose of this review is to discuss some of the evidence for fractal geometry in experimental systems and indicate how fractal geometry can be used to develop a better understanding of their aggregation kinetics and physical properties. At the present time much of our understanding of the structure and properties of fractal aggregates has come from computer simulations. Consequently, a major part of this paper is concerned with the role played by simple computer models in understanding the origins of fractal aggregates as well as their physical and chemical properties. The main emphasis is on models for colloidal aggregation but a brief description of a few other models for non-equilibrium growth and aggregation processes which have been of particular interest during the past few years has been included.

Colloids↗