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Use of running fractal dimension for the analysis of changing patterns in electroencephalograms.

Running fractal dimensions were measured on four channels of an electroencephalogram (EEG) recorded from a normal volunteer. The changes in the background activity due to eye closure were clearly differentiated by the fractal method. The compressed spectral array (CSA) and the running fractal dimensions of the EEG showed corresponding changes with respect to change in the background activity. The fractal method was also successful in detecting low amplitude spikes and the changes in the patterns in the EEG. The effects of different window lengths and shifts on the running fractal dimension have also been studied. The utility of fractal method for EEG data compression is highlighted.

Alpha Rhythm↗

Determination of fractal dimension of physiologically characterized neurons in two and three dimensions.

Although there is a growing interest in the application of fractal analysis in neurobiology, questions about the methodology have restricted its wider application. In this report we discuss some of the underlying principles for fractal analysis, we propose the cumulative-mass method as a standard method and we extend the applicability of fractal analysis to both 2 and 3 dimensions. We have examined the relationship between the method of log-log Sholl analysis and fractal analysis and have found that they correlate well. Measurements of physiologically characterized retinal ganglion cells indicate that different cell types can have significantly different fractal dimensions. Such differences may allow the correlation of the physiological type of a neuron with its morphological fractal dimension.

Animals↗

Fractal analysis of the electromyographic interference pattern.

Evaluation of motor unit recruitment is an important component of the clinical EMG exam. Typically this is assessed qualitatively using auditory features and estimates of the visual complexity of the EMG waveform. Recent advances in nonlinear dynamics have led to the development of the concept of fractals which can be used to quantify complexity and space filling features of various structures. This study was undertaken to determine if the normal EMG interference pattern (IP) has fractal characteristics that might be helpful in quantitative analysis. EMG activity was recorded from the 9 normal biceps muscles as force was varied from 10 to 90% of maximal. Using a box count algorithm, the fractal dimension was calculated. The EMG IP displays fractal characteristics with a dimension that is highly correlated with force and ranges from 1.1 to 1.4 as force increases from 10 to 90% MVC. The fractal dimension (FD)-force relationship is similar to that observed with other methods of IP analysis and suggests that the fractal dimension can be used to quantify and capture the essence of the 'complexity' of motor unit recruitment patterns.

Adult↗

Reliability of traditional and fractal dimension measures of quiet stance center of pressure in young, healthy people.

OBJECTIVES: To assess reliability of traditional and fractal dimension measures of quiet stance center of pressure (COP). DESIGN: Cross-sectional study. SETTING: University laboratory. PARTICIPANTS: Thirty young healthy men (n=20) and women (n=10) (mean age, 23 y). INTERVENTIONS: Not applicable. MAIN OUTCOME MEASURES: COP was recorded for 3 trials across 4 conditions: eyes open and eyes closed standing on firm and foam surfaces. Traditional COP variables--peak sway velocity and range of sway, both in the anteroposterior (AP) and mediolateral (ML) directions, and total excursion area, and fractal dimension of the COP in the AP and ML directions--were calculated. Reliability statistics were calculated. RESULTS: Range of sway (AP) was the most reliable traditional variable (intraclass correlation coefficient model 2,1 [ICC(2,1)] range -.28 to .72.). Peak sway velocity (AP) had poorest reliability (ICC(2,1) range, .05-.29). Only 1 of the traditional variables had excellent reliability; total excursion area (firm, eyes closed) (ICC(2,1)=.95). All bar 1 fractal dimension measures had excellent ICCs. Relative technical error of measurement ranged from 4% to 7% for the fractal dimension measures. Coefficients of variation were also very good, ranging from 1.8% to 6.7%. CONCLUSIONS: Fractal dimension measures were more reliable than traditional measures of COP. Although traditional measures are used extensively to assess COP, their reliability is questionable. Fractal dimension measures show promise to reliably quantify COP and warrant further investigation.

Adult↗

Binding and dissociation kinetics using fractals: an analysis of electrostatic effects and randomly coupled and oriented coupled receptors on biosensor surfaces.

A fractal analysis is used to analyze the influence of: (a) electrostatic interactions on binding and dissociation rate coefficients for antibodies HH8, HH10, and HH26 in solution to hen egg-white lysozyme (HEL) immobilized on a sensor chip surface [Biophys. J. 83 (2002) 2946]; and (b) the binding and dissociation of recombinant Fab in solution to random NHS-coupled Cys-HEL and oriented thiol-coupled Cys-HEL immobilized on a sensor chip surface [Methods 20 (2000) 310]. Single- and dual-fractal models were employed to fit the data. Values of the binding and the dissociation rate coefficient(s) and the fractal dimensions were obtained from a regression analysis provided by Corel Quattro Pro 8.0 (Corel Corporation Limited, Ottawa, Canada. 1997). The binding rate coefficients are quite sensitive to the degree of heterogeneity on the sensor chip surface. It is of interest to compare the results obtained by the fractal analysis with that of the original analysis [Biophys. J. 83 (2002) 2946]. For example, as one goes from the binding of 21 nM HH10/HEL to the binding of 640 nM HH10/HEL(K97A), Sinha et al. [Biophys. J. 83 (2002) 29461 indicate that the enhancement of diffusional encounter rates may be due to 'electrostatic steering' (a long-range interaction). Our analysis indicates that there is an increase in the value of the fractal dimension, Df1 by a factor of 1.12 from a value of 2.133-2.385. This increase in the degree of heterogeneity on the surface leads to an increase in the binding rate coefficient, k1 by a factor of 1.59 from 12.92 to 20.57. The fractal analysis of binding and dissociation of recombinant Fab in solution to random NHS-coupled Cys-HEL and oriented thiol-coupled Cys-HEL immobilized on a sensor chip [Methods 20 (2000) 310] surface are consistent with the degree of heterogeneity present on the sensor chip surface for the random and the oriented case. As expected, the random case will exhibit a higher degree of heterogeneity than the oriented case, leading to subsequently a higher binding rate coefficient.

Antibodies↗

Involvement of fractal geometry on solute permeation through porous poly (2-hydroxyethyl methacrylate) membranes.

Fractal geometry was applied to quantify the complexity of an internal structure of porous membranes prepared with poly(2-hydroxyethyl methacrylate) (pHEMA). The porous pHEMA membranes were synthesized by means of free-radical solution polymerization. Boundary lines of the porous structures in the pHEMA membrane were taken by a scanning electron microscope as image data, and these images were fed into a computer to estimate the fractal dimension. The boundary images of porous pHEMA membranes were observed to be a typical fractal and their complexity was quantified as a non-integral fractal dimension. The permeation of fluorescein isothiocyanate-labeled dextran, molecular weight 4400 (FD-4) as a model penetrant through the porous pHEMA membrane was determined using water-jacket type two-chamber diffusion cells. A fairly good negative relationship between the permeability coefficient of FD-4 and the fractal dimension was observed, suggesting the usefulness of the fractal dimension as a novel means for evaluating solute permeation through the porous membranes.

Algorithms↗

A new image analysis method based on topological and fractal parameters to evaluate the angiostatic activity of docetaxel by using the Matrigel assay in vitro.

Angiogenesis is regulated both by positive and negative factors that modulate the migration, proliferation, proteolytic activity, and differentiation of endothelial cells. Various in vivo and in vitro models have been developed to test the effect of angiogenic agonists or antagonists. Until very recently, morphometric methods to evaluate the effects of angiogenic inducers and inhibitors were mostly descriptive. Here, we have described a new, automatic, image analysis method to evaluate the angiostatic activity of docetaxel by using the in vitro Matrigel assay. This method allows to establish several parameters, such as dimensional (area % covered by endothelial cells and the total length of the cellular network per field), topological (the number of meshes and the number of branching points per field), and fractal (fractal dimension, lacunarity) of the capillary-like network. Following docetaxel treatment, a reduction of about 25% in the size parameters of cell network, of about 45% in branching points per field, and more 50% in mesh number were observed. The resulting fractal analysis was consistent with these findings and fractal dimension significantly correlated with both the number of branching points and the number of meshes. These results indicate that both topological and fractal parameters allowed a characterization of the spatial texture generated by endothelial cells during in vitro angiogenesis, the former being easier to implement in a computer program and faster to calculate. Finally, topological parameters seem to exhibit a wider dynamic range than the fractal ones.

Angiogenesis Inhibitors↗

Fractal dimension in human cerebellum measured by magnetic resonance imaging.

Fractal dimension has been used to quantify the structures of a wide range of objects in biology and medicine. We measured fractal dimension of human cerebellum (CB) in magnetic resonance images of 24 healthy young subjects (12 men and 12 women). CB images were resampled to a series of image sets with different 3D resolutions. At each resolution, the skeleton of the CB white matter was obtained and the number of pixels belonging to the skeleton was determined. Fractal dimension of the CB skeleton was calculated using the box-counting method. The results indicated that the CB skeleton is a highly fractal structure, with a fractal dimension of 2.57 +/- 0.01. No significant difference in the CB fractal dimension was observed between men and women.

Adolescent↗

Permeability of fractal aggregates.

It is well known that the permeability and density of an aggregate decreases with its size, affecting its settling velocity and coagulation rate (rate of particle capture) with other particles. This change in aggregate density with size can be described by fractal scaling relationships. Two distinctly different fractal scaling approaches, however, have been used to describe aggregate permeability. In one approach (single-particle-fractal model), the permeability is calculated by assuming primary particles are uniformly distributed in the aggregate. In the other approach (cluster-fractal model), it is assumed that aggregates are composed of primary particles separated into individual clusters that are less permeable than the aggregate. The overall permeability of the aggregate is dependent on the number and sizes of these clusters. Using three different permeability correlations (Brinkman, Happel and Carmen-Kozeny), it is demonstrated through comparison with aggregate settling velocity data that the single-particle-fractal model does not provide realistic predictions of settling velocity as a function of aggregate size. In addition, it is shown that the Carmen-Kozeny permeability equation does not produce realistic settling velocity relationships. The transport settling velocity and capture rate of sinking aggregates in natural and engineered environments should therefore only be calculated using the Happel or Brinkman equations and a cluster-fractal model.

Chemical Precipitation↗

Changes in fractal dimension during aggregation.

Experiments were performed to evaluate temporal changes in the fractal dimension of aggregates formed during flocculation of an initially monodisperse suspension of latex microspheres. Particle size distributions and aggregate geometrical information at different mixing times were obtained using a non-intrusive optical sampling and digital image analysis technique, under variable conditions of mixing speed, coagulant (alum) dose and particle concentration. Pixel resolution required to determine aggregate size and geometric measures including the fractal dimension is discussed and a quantitative measure of accuracy is developed. The two-dimensional fractal dimension was found to range from 1.94 to 1.48, corresponding to aggregates that are either relatively compact or loosely structured, respectively. Changes in fractal dimension are explained using a conceptual model, which describes changes in fractal dimension associated with aggregate growth and changes in aggregate structure. For aggregation of an initially monodisperse suspension, the fractal dimension was found to decrease over time in the initial stages of floc formation.

Chemical Precipitation↗

Influence of diffusion to fractal surfaces on the binding kinetics for antibody-antigen, analyte-receptor, and analyte-receptorless (protein) systems.

The diffusion-limited binding kinetics of antigen-antibody, ligand-receptor, analyte-receptorless systems for biosensor applications is analyzed within a fractal framework. The analysis presented applies equally well to these types of systems. For example, for the binding of 2-(p-toluidiny)-naphthalene-6-sulfonic acid (2,6-TNS) to beta-cyclodextrin (ligand-receptor system) immobilized on a fiber-optic base inclusate biosensor, an increase in temperature from 4 to 30 degrees C leads to an increase in the fractal dimension, D1 and to a decrease in the binding rate coefficient, k1. For the binding of TRITC-labeled low-density proteins (LDL) directly to an optical fiber-based sensor (analyte-receptorless system), an increase in the LDL concentration from 5 to 50 micrograms ml-1 in solution leads to a decrease in the fractal dimension, D1 and to an increase in the binding rate coefficient, k1. Also, during the binding of human chorionic gonadotropin (hCG) to anti-hCG antibody immobilized on a HPLC column (antigen-antibody system), an increase in temperature from 4 to 50 degrees C leads to an increase in the fractal dimension, D1 and in the binding rate coefficient. k1. The different examples analyzed and presented together for the three different types of systems provide one means of a 'unified analysis,' and a method by which the forward binding rate coefficient, k1 may be controlled, that is, by changing the fractal dimension or 'disorder' on the surface. The analysis should assist in improving the stability, sensitivity, and response time of biosensors wherein different types of binding systems are utilized in the analysis method. More-or-less all of the treatment presented should be applicable to the above types of binding systems occurring in non-biosensor applications also. However, the single-fractal analysis is unable to describe the data over the full time course of some of the experiments.

Antibodies↗

Fractal dimension estimation of carotid atherosclerotic plaques from B-mode ultrasound: a pilot study.

In this paper, a pilot study regarding carotid atherosclerotic plaque instability using B-mode ultrasound (US) images was carried out. The fractal dimension of plaques obtained from ten symptomatic subjects (i.e., subjects having experienced neurological symptoms) as well as from nine asymptomatic subjects, was estimated using a novel method, called the kth nearest neighbour (KNN) method. The results indicated a significant difference, as per the fractal dimension, between the two groups, providing a significantly lower value for the asymptomatic group. Moreover, the phase of the cardiac cycle (systole/diastole) during which the fractal dimension was estimated had no systematic effect on the calculations. The fractal dimension of the plaques was also estimated using a well-known method, namely the box-counting (BC) method. No significant differences between the two groups, as per the fractal dimension, were observed using the BC method. The presented pilot study suggests that the fractal dimension, estimated by the proposed method, could be used as a single determinant for the discrimination of symptomatic and asymptomatic subjects.

Aged↗

A fractal analysis of analyte-estrogen receptor binding and dissociation kinetics using biosensors: environmental and biomedical effects.

A fractal analysis is used to model the binding and dissociation kinetics between analytes in solution and estrogen receptors (ERs) immobilized on a sensor chip of a surface plasmon resonance (SPR) biosensor. The influence of different ligands is also analyzed. A better understanding of the kinetics provides physical insights into the interactions, and suggests means by which appropriate interactions (to promote correct signaling) and inappropriate interactions such as with xenoestrogens (to minimize inappropriate and deleterious to health signaling) may be better controlled. The fractal approach is applied to analyte-ER interaction data available in the literature. The units for the different parameters (rate coefficients and affinities) in fractal-type kinetics are different from those obtained in classical kinetics. Numerical values obtained for the binding and the dissociation rate coefficients are linked to the degree of roughness or heterogeneity (fractal dimension, D(f)) present on the biosensor chip surface. In general, the binding and the dissociation rate coefficients are very sensitive to the degree of heterogeneity on the surface. A single-fractal analysis is adequate in some cases. In others (that exhibit complexities in the binding or the dissociation curves) a dual-fractal analysis is required to obtain a better fit. This has biomedical and environmental implications in that the dissociation (and the binding) rate coefficient may be used to alleviate (deleterious effects) or enhance (beneficial effects) by selective modulation of the surface. The affinity values obtained in the analysis are consistent with the numbers required to (a). promote signaling between the correct analyte and the estrogen receptor, and (b). minimize the signaling between xenoestrogens and the estrogen receptor.

Biomedical Technology↗

The effect of image variations on fractal dimension calculations.

OBJECTIVES: We used digitized dental radiographs of alveolar bone to test the hypothesis that the fractal dimension, as calculated with the program "ImageFractal" was independent of variations in X-ray exposure, beam alignment, and region of interest placement. STUDY DESIGN: The radiographic data set consisted of 72 radiographs digitized with 200 microns pixels. Radiographs were obtained with the use of three time settings and two alignments. Rectangular regions of interest were placed on each digital image over the interdental bone between the mandibular first and second molars on six hemimandibles. Each of six hemimandibles had identical copies of a unique region of interest placed on every image in its series. New regions were made 3 months later. A fractal dimension was computed from each region of interest with the caliper method included in ImageFractal, a public domain program available through National Institutes of Health. The resulting fractal dimensions were evaluated with two repeated measures analysis of variance. RESULTS: No significant differences were found between the fractal dimensions calculated for baseline images and those from overexposed and underexposed images, from images with 4 to 6 degrees of alignment variations, or from repeat regions of interest. CONCLUSION: The results support the hypothesis that fractal dimensions derived from digitized dental radiographs are not affected by variations in exposure or small variations in alignment and imply an absolute region of interest placement may not be necessary. However, caution should be used with the use of the fractal dimension to discriminate among alveolar bone variations until further research is performed.

Absorptiometry, Photon↗

Fractal analysis of surface EMG signals from the biceps.

Nonlinear analysis techniques are necessary to understand the complexity of the EMG. The purpose of the present study was to determine the fractal dimension of surface EMG obtained from the biceps brachii of normal subjects during isokinetic flexion-extension of the arm. The measurements were obtained with different loading conditions on the arm and for various rates of flexion-extension. Fractal dimensions of the surface EMG signals were calculated for each of these conditions. ANOVA results showed statistically significant differences between the fractal dimensions calculated for different loading conditions and rates of flexion-extensin (P < or = 0.005). Linear regression analysis showed a correlation coefficient of 0.99 between the fractal dimension and the load, and a correlation coefficient of 0.98 between the fractal dimension and the rate of flexion-extension. The results of the study show that the fractal dimension can be used along with other parameters to characterize the EMG signal.

Algorithms↗

Degree of fractal behavior as a diagnostic aid in quantitative image analyses.

PURPOSE: Multiple strategies in diagnoses of different diseases from images can include their histogram analyses. Any fractal behavior in the latter is to be quantified as to extent here, with a view toward contributing to a diagnostic process. PROCEDURE: One tool in quantitative image analyses is the fractal dimension D of the pixel histogram, a measure of self-similarity over various scales in a fitted power-law behavior of pixel intensity cumulative probability distribution. Proposed and developed here as diagnostic markers are features of its determination process that indicate to what extent there is fractal behavior. One of these is the curvature c that exists in log-log plots used for extracting the fractal exponent D of power-law behavior. RESULTS: Specific implementations are given both for a general lognormal pixel intensity distribution and for lung images. Both Ds and cs are determined for: normals, pulmonary embolism, cystic fibrosis, as well as a theoretical lognormal distribution. It is shown that D and heterogeneity described by a standard deviation are reciprocally related and not typically independent markers. The added independent information from c has possibilities of assisting in discrimination of normal and pathologic conditions, such as in lung diseases. CONCLUSION: In addition to a histogram's fractal dimension itself, there are indications that measures of the degree of fractal behavior may also hold promise in image diagnoses.

Animals↗

Fractal dimension analysis of weight-bearing bones of rats during skeletal unloading.

Fractal analysis was used to quantify changes in trabecular bone induced through the use of a rat tail-suspension model to simulate microgravity-induced osteopenia. Fractal dimensions were estimated from digitized radiographs obtained from tail-suspended and ambulatory rats. Fifty 4-month-old male Sprague-Dawley rats were divided into groups of 24 ambulatory (control) and 26 suspended (test) animals. Rats of both groups were killed after periods of 1, 4, and 8 weeks. Femurs and tibiae were removed and radiographed with standard intraoral films and digitized using a flatbed scanner. Square regions of interest were cropped at proximal, middle, and distal areas of each bone. Fractal dimensions were estimated from slopes of regression lines fitted to circularly averaged plots of log power vs. log spatial frequency. The results showed that the computed fractal dimensions were significantly greater for images of trabecular bones from tail-suspended groups than for ambulatory groups (p < 0.01) at 1 week. Periods between 1 and 4 weeks likewise yielded significantly different estimates (p < 0.05), consistent with an increase in bone loss. In the tibiae, the proximal regions of the suspended group produced significantly greater fractal dimensions than other regions (p < 0.05), which suggests they were more susceptible to unloading. The data are consistent with other studies demonstrating osteopenia in microgravity environments and the regional response to skeletal unloading. Thus, fractal analysis could be a useful technique to evaluate the structural changes of bone.

Animals↗

Fractal volume of drug distribution: it scales proportionally to body mass.

PURPOSE: To develop the physiologically sound concept of fractal volume of drug distribution, vf, and evaluate its utility and applicability in interspecies pharmacokinetic scaling. METHODS: Estimates for vf of various drugs in different species were obtained from the relationship: vf = (v - Vpl)(Vap - Vpl)/V + Vpl where v is the total volume of the species (equivalent to its total mass assuming a uniform density Ig/mL), Vpl is the plasma volume of the species and Vap is the conventional volume of drug distribution. This equation was also used to calculate the fractal analogs of various volume terms of drug distribution (the volume of central compartment, Vc, the steady state volume of distribution, Vss, and the volume of distribution following pseudodistribution equilibrium, Vz). The calculated fractal volumes of drug distribution were correlated with body mass of different mammalian species and allometric exponents and coefficients were determined. RESULTS: The calculated values of vf for selected drugs in humans provided meaningful and physiologically sound estimates for the distribution of drugs in the human body. For all fractal volume terms utilized, the allometric exponents were found to be either one or close to unity. The estimates of the allometric coefficients were found to be in the interval (0,1). These decimal values correspond to a fixed fraction of the fractal volume term relative to body mass in each one of the species. CONCLUSIONS: Fractal volumes of drug distribution scale proportionally to mass. This confirms the theoretically expected relationship between volume and mass in mammalian species.

Animals↗