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PubMed2002

Effects of nicardipine and diltiazem on fractal features of short-term heart rate variability--application of coarse graining spectral analysis.

PURPOSE: This study was designed to investigate the effects of nicardipine and diltiazem on the fractal features of short-term heart rate variability (HRV), using coarse graining spectral analysis (CGSA). METHODS: Eighteen healthy volunteers participated in this study; they were divided into two groups according to the drug administered. Five-minute electrocardiogram and arterial pressure recordings were made during stepwise infusions of either nicardipine (0.4, 0.8, 1.6, and 3.2 microg.kg(-1).min(-1)) or diltiazem (2, 4, 8, and 16 microg.kg(-1).min(-1)) under rate-controlled breathing at 0.25 Hz. CGSA broke down the total power of the time series into harmonic (low frequency [0.0-0.15 Hz; LF] and high frequency [0.15-0.5 Hz; HF]) and nonharmonic (fractal) components. Cardiac sympathetic nervous system (SNS) and parasympathetic nervous system (PNS) activity indicators were evaluated as the ratios LF/HF and HF/TP (total spectral power), respectively. Fractal components were evaluated as %fractal and the spectral exponent beta of 1/f(beta). RESULTS: Compared with control measurements, the maximum dose of nicardipine infusion caused a significant decrease in systolic arterial pressure, a significant increase in the mean heart rate, and a significant increase in plasma norepinephrine level, findings that were associated with significant increases in %fractal and beta values (54.2 +/- 13.3 vs 75.6 +/- 9.8, and 0.86 +/- 0.22 vs 1.32 +/- 0.46, respectively; P < 0.05). PNS and SNS indicators showed decreased and increased values, respectively. Diltiazem caused a reduction in arterial pressure; however, no other parameters, including the nonharmonic components of HRV, were affected by this drug. CONCLUSIONS: These findings strongly suggest that nicardipine suppresses vagal cardiac neural outflow and activate the SNS, an action which, subsequently, causes changes in the fractal features of HRV. Although diltiazem reduces arterial pressure, it preserves the basic neural balance of the autonomic nervous system in regard to heart rate control.

Journal Article↗
PubMed2005

Fractal stochastic modeling of spiking activity in suprachiasmatic nucleus neurons.

Individual neurons in the suprachiasmatic nucleus (SCN), the master biological clock in mammals, autonomously produce highly complex patterns of spikes. We have shown that most (approximately 90%) SCN neurons exhibit truly stochastic interspike interval (ISI) patterns. The aim of this study was to understand the stochastic nature of the firing patterns in SCN neurons by analyzing the ISI sequences of 150 SCN neurons in hypothalamic slices. Fractal analysis, using the periodogram, Fano factor, and Allan factor, revealed the presence of a 1/f-type power-law (fractal) behavior in the ISI sequences. This fractal nature was persistent after the application of the GABAA receptor antagonist bicuculline, suggesting that the fractal stochastic activity is an intrinsic property of individual SCN neurons. Based on these physiological findings, we developed a computational model for the stochastic SCN neurons to find that their stochastic spiking activity was best described by a gamma point process whose mean firing rate was modulated by a fractal binomial noise. Taken together, we suggest that SCN neurons generate temporal spiking patterns using the fractal stochastic point process.

Action Potentials↗
PubMed1990

An application of fractal dimension to the detection of transients in the electroencephalogram.

It is sometimes desirable to identify a brief seizure, occasional spike, single evoked potential, or other transient in the EEG. Transient detection in an EEG can be a difficult task, often requiring prior knowledge of the characteristics of the transient. A fractal is a shape which retains structural detail despite magnification (scaling). The complexity of the structure of such a set, invariant under this scaling, can be characterized by a single number: the fractal dimension. Regarding the EEG as a fractal, we have shown that transient deterministic data in the EEG have a fractal dimension different from the quasirandom background. An extensive introduction to fractals is presented with the assumption that the reader is unfamiliar with the theory. In the preliminary results presented here, analysis by fractal dimension is shown to be a promising method of transient detection, requiring no prior knowledge of the characteristics of the transient. Possible applications of the technique to evoked potential technology and epilepsy surgery are discussed. Other applications to biology, neuroscience and medicine are reviewed.

Algorithms↗
PubMed1994

Quantification of tight junction complexity by means of fractal analysis.

The concept of fractal geometry provides an elegant tool for the quantitative and objective structural description of various objects, the fractal analysis. Fractal analysis quantifies the structural complexity of objects by a characteristic singular value, the fractal dimension (FD). It can be estimated, e.g. by the box-counting method and provides a highly integrated measure in the range 1 < FD < 2 for curves extending within a plane. In this study, fractal analysis is used for the first time to evaluate the complexity of the tight junction network between adjoining cells. Bovine brain endothelial cells were cultured under various experimental conditions and the tight junctions were drawn to scale as visualized by the freeze fracture technique. These drawings were analyzed by fractal analysis, and by two other methods commonly used in this field, viz. the strand counting (SC) and complexity index (CI) methods. In contrast to the latter methods, the FD shows no directional preference and therefore no assumptions on the dynamic properties of the network's complexity are required. Thus, FD is demonstrated to provide the most sensitive, reliable and complete measure of tight junction complexity. In combination with SC and CI, additional information can be achieved concerning the directionality of the altered arrangement of tight junctional strands. Our analysis allows for the following conclusions. (1) Defined experimental influences can modify the complexity of tight junctions that are formed between endothelial cells in vitro, and (2) these structural modifications of the tight junctions are mainly due to an altered strand branching pattern.

Animals↗
PubMed2005

Fractality of dendritic arborization of spinal cord neurons.

Skeletonized images of Golgi impregnated neurons from the human, monkey, cat and rat dorsal horns were subjected to fractal analysis. These neurons have sparse branching of dendrite arbors. It is noticed that, in certain neuronal samples, some authors report that scaling range of experimentally declared fractals is extremely limited and spanned approximately between 0.5 and 2.0 decades. In order to retain our hypothesis that neurons with dendrites of uncomplicated shapes can be considered fractal over three decades of scale, we conducted four procedures: (i) we used the box-counting method, (ii) we scaled the box sizes as a power of 2, (iii) we chose the coefficient of correlation, measuring the "goodness of fit" of experimental data points to regression straight line, to be equal to or larger than 0.995, and (iv) we pointed out that all the neurons analyzed have a single fractal dimension measuring a global fractality showing no linear regions. As a control, we used some cerebellar Purkinje cells whose dendrite trees show much more complex structure and profuseness of branching. Since, generally, the neuronal structure is among the most complex of all cellular morphologies, we believe that supporting this hypothesis we advance the neuroscience and fractal theory.

Animals↗
PubMed2001

Fractal analysis and time- and frequency-domain measures of heart rate variability as predictors of mortality in patients with heart failure.

Time-domain measures of heart rate (HR) variability provide prognostic information among patients with congestive heart failure (CHF). The prognostic power of spectral and fractal analytic methods of HR variability has not been studied in the patients with chronic CHF. The aim of this study was to assess whether traditional and fractal analytic methods of HR variability predict mortality among a population of patients with CHF. The standard deviation of RR intervals, HR variability index, frequency-domain indexes, and the short-term fractal scaling exponent of RR intervals were studied from 24-hour Holter recordings in 499 patients with CHF and left ventricular ejection fraction < or =35%. During a mean follow-up of 665 +/- 374 days, 210 deaths (42%) occurred in this population. Conventional and fractal HR variability indexes predicted mortality by univariate analysis. For example, a short-term fractal scaling exponent <0.90 had a risk ratio (RR) of 1.9 (95% confidence interval [CI] 1.4 to 2.5) and the SD of all RR intervals <80 ms had an RR of 1.7 (95% CI 1.2 to 2.1). After adjusting for age, functional class, medication, and left ventricular ejection fraction in the multivariate proportional-hazards analysis, the reduced short-term fractal exponent remained the independent predictor of mortality, RR 1.4 (95% CI 1.0 to 1.9; p <0.05). All HR variability indexes were more significant univariate predictors of mortality in functional class II than in class III or IV. Among patients with moderate heart failure, HR variability measurements provide prognostic information, but all HR variability indexes fail to provide independent prognostic information in patients with the most severe functional impairment.

Aged↗
PubMed1988

Fractal models are inadequate for the kinetics of four different ion channels.

The gating kinetics of single ion channels have been well described by models which assume that channels exist in a number of discrete kinetic states, with the rate constants for transitions among the states remaining constant in time. In contrast to such discrete Markov models, it has recently been considered whether gating might arise from transitions among a continuum of states, with the effective rate constants for leaving the collections of states given by a fractal scaling equation (Liebovitch, L.S., J. Fischbarg, J.P. Koniarek, I. Todorova, and M. Wang. 1987. Biochim. Biophys. Acta. 896:173-180; Liebovitch, L.S., and J.M. Sullivan. 1987. Biophys. J. 52:979-988). The present study compares discrete Markov with fractal continuum models to determine which best describes the gating kinetics of four different ion channels: GABA-activated Cl channels, ACh-activated end-plate channels, large conductance Ca-activated K (BK) channels, and fast Cl channels. Discrete Markov models always gave excellent descriptions of the distributions of open and shut times for all four channels. Fractal continuum models typically gave very poor descriptions of the shut times for all four channels, and also of the open times from end-plate and BK channels. The descriptions of the open times from GABA-activated and fast Cl channels by the fractal and Markov models were usually not significantly different. If the same model accounts for gating motions in proteins for both the open and shut states, then the Markov model ranked above the fractal model in 35 of 36 data sets of combined open and shut intervals, with the Markov model being tens to thousands of orders of magnitude more probable. We suggest that the examined fractal continuum model is unlikely to serve as a general mechanism for the gating of these four ion channels.

Animals↗
PubMed1989

Determination of the fractal dimension of membrane protein aggregates using fluorescence energy transfer.

It is demonstrated that fluorescence resonance energy transfer may be used to determine the fractal dimension of aggregates of membrane-bound proteins. Theoretical and experimental results are presented for two different experimental designs: energy transfer between proteins and energy transfer from lipids to proteins. For energy transfer between proteins the lattice spacing must be known independently for a fractal dimension to be uniquely determined, and this represents a disadvantage to this experimental design. Results are presented for the calcium ATPase and a fractal dimension of 1.9 is estimated for ATPase aggregates by assuming a lattice spacing of 50 A. Energy transfer from lipids to protein provides a means of estimating the length of the "coast-line" of the aggregate. In this case the fractal dimension is uniquely determined from a log-log plot. An analysis of data for bacteriohodopsin reconstituted in phospholipid vesicles gives a fractal dimension of 1.6. The structural basis of the value for the fractal dimension is discussed for these two systems. These techniques provide a means of assessing the nature of protein-protein interactions in membranous systems.

Calcium-Transporting ATPases↗
PubMed1998

Fractal dimension of the bone marrow in metastatic lesions.

We previously reported that the distribution of the cells in normal bone marrow is fractal and self-similar. The purpose of this study was to determine whether the same is true in metastatic tumors. Thirty-two bone marrow biopsy sections (3 to 5 microm thick) of 28 patients were used to measure the fractal dimensions of the metastatic tumor cells' distribution. Microscopic images were obtained and were used for the fractal measurements. In the two-dimensional images, the fractal dimensions were 1.98 +/- 0.02 (95% +/- 5% cellularity), suggesting a compact nonfractal structure. The dimensions, however, were 1.72 +/- 0.1 (56% +/- 11% cellularity) for the normal components, with a P-value of <.0001 that is in agreement with our previous study. These results suggest that loss of the fractal structure in the metastatic lesions may be attributable to loss or suppression of the regulatory mechanisms maintaining the fractal morphogenesis of the bone marrow. This report provides a novel objective approach in the study of pathophysiology of the bone marrow disorders.

Adolescent↗
PubMed2001

The fractal nature of Escherichia coli biological flocs.

The fractal structures of Escherichia coli biological flocs were characterised in terms of fractal dimension, which is a measurement of how the bacteria in the flocs occupy space. The dimensional analysis methods, based on power law correlations between floc perimeter, projected area and maximum length, were used to determine the one- and two-dimensional fractal dimensions (D(1) and D(2)) of E. coli flocs formed by flocculation in chitosan solution with a concentration of 10.0 mg chitosan per g dry cell weight (DCW), giving D(1)=1.07+/-0.06 and D(2)=1.70+/-0.08 (+/-S.D.). The three-dimensional fractal dimension (D(3)) of the E. coli flocs was determined by the two-slopes method, using cumulative size distributions of floc length and solid volume, to be 1.99+/-0.08 (+/-S.D.), which is close to the value of D(3)=2.14+/-0.04 (+/-S.D.) measured by the small angle light scattering method. The results demonstrate that E. coli flocs flocculated with chitosan have a fractal nature, as their fractal dimensions D(1), D(2) and D(3) differ from the values of 1, 2 and 3 expected for the spherical Euclidean object, respectively.

Journal Article↗
PubMed2005

Fe3O4 nanocrystals with novel fractal.

Fe3O4 novel fractal nanocrystals have been synthesized by a surfactant-assisted solvothermal process for the first time. X-ray diffraction (XRD), X-ray photoelectron spectra (XPS), Mössbauer spectroscopy (MS), scanning electron microscopy (SEM), and transmission electron microscopy (TEM) have been used to investigate the novel fractal nanocrystals. The lengths of the fractals are about 2-3 microm, and the trunks and branches of Fe3O4 fractals have almost the same diameters of ca. 30-50 nm. The roles of surfactant PEG-20000 and N2H4 have been discussed in detail. One key fact has been found that the ferrocene concentration has a vital effect on the morphologies of the products. The side-branching process and the oscillation of the concentration have been proposed to illustrate the formation mechanisms of the fractal nanocrystals. In addition, magnetic properties of Fe3O4 fractal nanocrystals have also been detected by a vibrating sample magnetometer, showing relatively high saturation magnetization (Ms) of ca. 78.75 emu/g.

Journal Article↗
PubMed2006

Fractal Analysis: revisiting Pollock's drip paintings.

We investigate the contentions that Jackson Pollock's drip paintings are fractals produced by the artist's Lévy distributed motion and that fractal analysis may be used to authenticate works of uncertain provenance. We find that the paintings exhibit fractal characteristics over too small a range to be usefully considered as fractal; their limited fractal characteristics are easily generated without Lévy motion, both by freehand drawing and gaussian random motion. Several problems must therefore be addressed before fractal analysis can be used to authenticate paintings.

Journal Article↗
PubMed2002

Cord-like mosaic patches in the adrenal cortex are fractal: implications for growth and development.

Organogenesis proceeds rapidly and faithfully during fetal development. The process includes generation of parenchyma, followed by organization into functional tissues. The method by which the growth of organ parenchyma is regulated is not known, but insight into this regulation has been obtained by studying mosaic tissues of experimental chimeras and transgenic mosaics. The patterns revealed by this procedure offer an indication of how the parenchyma was generated. In the liver, the pattern appears as islands of one cell type in a sea of the other cell type, while in the adrenal cortex the pattern is one of alternating cords of one cell type adjacent to the other cell type. We have established previously that mosaic patches in the liver are fractal. The fractal dimensions of patches in the liver are consistent with an iterative, recursive growth model with simple stereotypical division rules. Here we report that the patches in mosaic adrenal cortex of the mouse and rat are also fractal and that the fractal dimension of the surface of the patches is lower than that in the liver. Fetal development and fractal dimensions of adrenal cortical mosaic patches are consistent with an algorithmic cell division model in which parenchymal growth is constrained to edges of growth centers forcing cord structures to form. Fractal analysis of the geometry of mosaic patches in tissues of experimental chimeras is helpful in constructing hypotheses of organ growth.

Adrenal Cortex↗
PubMed1992

Fractal dimensions of ductal patterns in the parotid glands of normal subjects and patients with Sjögren syndrome.

RATIONALE AND OBJECTIVES: The sialographic ductal patterns of the parotid glands in patients with Sjögren syndrome were compared with those of normal patients by measuring the fractal dimensions. METHODS: Fractal dimensions were estimated using the modified pixel dilation method. RESULTS: The mean fractal dimension was 1.64 +/- 0.06 for the normal glands and 1.39 +/- 0.10 for the glands with Sjögren syndrome (P < .005). No correlation between the age or sex and fractal dimension was observed for both groups. In Sjögren syndrome, a significant difference in the fractal dimension was observed between the subgroup having punctuate fillings with a diameter less than 1 mm and the subgroup from 1 to 2 mm (P < .001). CONCLUSION: The fractal dimension is useful as a numeric grading of the complexity of the ductal pattern and the progression of parotid disease.

Adult↗
PubMed2002

Pernicious effect of physical cutoffs in fractal analysis.

Fractal scaling appears ubiquitous, but the typical extension of the scaling range observed is just one to two decades. A recent study has shown that an apparent fractal scaling spanning a similar range can emerge from the randomness in dilute sets. We show that this occurs also in most kinds of nonfractal sets irrespective of defining the fractal dimension by box counting, minimal covering, the Minkowski sausage, Walker's ruler, or the correlation dimension. We trace this to the presence of physical cutoffs, which induce smooth changes in the scaling, and a bias over a couple of decades around some characteristic length. The latter affects also the practical measure of fractality of truly fractal objects. A defensive strategy against artifacts and bias consists in carefully identifying the cutoffs and a quick-and-dirty thumb rule requires to observe fractal scaling over at least three decades.

Journal Article↗
PubMed2003

Random walk through fractal environments.

We analyze random walk through fractal environments, embedded in three-dimensional, permeable space. Particles travel freely and are scattered off into random directions when they hit the fractal. The statistical distribution of the flight increments (i.e., of the displacements between two consecutive hittings) is analytically derived from a common, practical definition of fractal dimension, and it turns out to approximate quite well a power-law in the case where the dimension D(F) of the fractal is less than 2, there is though, always a finite rate of unaffected escape. Random walks through fractal sets with D(F)< or =2 can thus be considered as defective Levy walks. The distribution of jump increments for D(F)>2 is decaying exponentially. The diffusive behavior of the random walk is analyzed in the frame of continuous time random walk, which we generalize to include the case of defective distributions of walk increments. It is shown that the particles undergo anomalous, enhanced diffusion for D(F)<2, the diffusion is dominated by the finite escape rate. Diffusion for D(F)>2 is normal for large times, enhanced though for small and intermediate times. In particular, it follows that fractals generated by a particular class of self-organized criticality models give rise to enhanced diffusion. The analytical results are illustrated by Monte Carlo simulations.

Journal Article↗
PubMed2004

Fractal growth of a conventional calamitic liquid crystal.

We report observation of liquid crystal phase ordering via fractal growth aggregates for a calamitic, non-bent-core mesogen. Fractal growth of a conventional smectic- C (Sm-C) phase from the isotropic melt after a temperature quench was experimentally investigated with respect to time, cell gap, quench depth, and quench rate. The determined fractal dimensions relating to the area as well as the perimeter of the growing aggregates suggest a phase formation process via a percolation mechanism. Computer simulations of the phase ordering process give further evidence for percolation growth, qualitatively reproducing the observed textures and quantitatively leading to the same fractal dimensions. We propose a general model of fractal smectic liquid crystal growth, which accounts for all of the different systems observed so far, bent-core or "banana" phases as well as the observation of fractal phase ordering of a conventional Sm-C phase. The model is based on the "breaking" of the commonly observed growth anisotropy by strong in-layer molecular interactions. These are provided by hydrogen bonding in the Sm-C case discussed here and by steric interactions in the case of the bent-core phases discussed in previous publications.

Journal Article↗
PubMed1997

Fractal modeling and segmentation for the enhancement of microcalcifications in digital mammograms.

The objective of this research is to model the mammographic parenchymal, ductal patterns and enhance the microcalcifications using deterministic fractal approach. According to the theory of deterministic fractal geometry, images can be modeled by deterministic fractal objects which are attractors of sets of two-dimensional (2-D) affine transformations. The iterated functions systems and the collage theorem are the mathematical foundations of fractal image modeling. In this paper, a methodology based on fractal image modeling is developed to analyze and model breast background structures. We show that general mammographic parenchymal and ductal patterns can be well modeled by a set of parameters of affine transformations. Therefore, microcalcifications can be enhanced by taking the difference between the original image and the modeled image. Our results are compared with those of the partial wavelet reconstruction and morphological operation approaches. The results demonstrate that the fractal modeling method is an effective way to enhance microcalcifications. It may also be able to improve the detection and classification of microcalcifications in a computer-aided diagnosis system.

Algorithms↗