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PubMed1994

Quantification of occipital EEG changes in Alzheimer's disease utilizing a new metric: the fractal dimension.

In a retrospective preliminary investigation, the occipital electroencephalogram (EEG) changes associated with Alzheimer's disease (AD) were examined using a novel quantification metric: the fractal dimension. The mean occipital EEG fractal dimension was determined for each of three patient groups representing a spectrum of clinical and EEG pathology: controls, probable AD, and autopsy-confirmed AD. The fractal dimension was significantly reduced in each of the AD groups with respect to controls (p < 0.001); and within the AD groups, it was significantly reduced in autopsy-confirmed AD relative to probable AD (p < 0.01). The fractal dimension findings paralleled the manifest EEG abnormalities in a way that suggests it has potential clinical utility in metric studies on the EEG, especially when applied to the dementias. Additionally, the EEG pathology studied here was particularly well-described by the fractal dimension, providing further support for a nonlinear approach to the background activity of the EEG.

Alzheimer Disease↗
PubMed1994

The application of fractal geometric analysis to microscopic images.

Fractal geometry is a relatively new tool for the quantitative microscopist that is a more valid way of measuring dimensions of complex irregular objects than the integer-dimensional geometries (such as Euclidean geometry). This review discusses the theory of fractal geometry using the classic examples of the Von Koch curve, the Cantor set and the Sierpinski gasket. The problems of describing the dimensions of these objects are discussed and the concept of fractal dimensionality is introduced. Methods for measuring fractal dimensions are discussed, including their implementation on microcomputer-based image analysis systems . The advantages and problems of fractal geometric analysis are discussed and current applications in the field of microscopy are reviewed.

Animals↗
PubMed2004

Analyte-receptor binding on surface plasmon resonance biosensors: a fractal analysis of Cre-loxP interactions and the influence of Cl, O, and S on drug-liposome interactions.

A fractal analysis of the association and dissociation (whereever applicable) of Cre-loxP interactions and drug-liposome interactions on a sensor chip surface is presented. In both of these cases a dual-fractal analysis is required to adequately describe the association kinetics. The dissociation kinetics for Cre-loxP interactions is adequately described by a single-fractal analysis. The dual-fractal analysis used to describe the association kinetics of Cre-loxP interactions is consistent with the original two-step mechanism presented using a surface plasmon resonance biosensor. Our analysis includes both diffusion and surface effects by introducing the fractal dimension which makes quantitative the degree of heterogeneity on the sensor chip surface. Affinities are provided. Only the association kinetics were analysed for drug-liposome interactions since the initial sections of the dissociation curves were too steep to obtain reasonable drug-liposome complex concentration values on the sensor chip with time. Attempts made to relate the association rate coefficients with the molecular weight of the drug were unsuccessful. On using desipramine and imipramine as "arbitrarily selected standards" or "references" (only C, H, and N atoms present), it was noticed from the data analysed that the inclusion of the O and S atoms in the drug leads to a decrease in the association rate coefficients, ka1 (or k1) and ka2 (or k2) (compared with the arbitrarily selected standards or references). Similarly, the addition of the Cl atom in the drug leads to an increase in the association rate coefficient (compared with the arbitrarily selected standards or references). More data needs to be analysed to determine whether this is true for other drugs also.

Chlorine↗
PubMedDate not supplied

Classification of breast ultrasound images using fractal feature.

Fractal analyses have been applied successfully for the image compression, texture analysis, and texture image segmentation. The fractal dimension could be used to quantify the texture information. In this study, the differences of gray value of neighboring pixels are used to estimate the fractal dimension of an ultrasound image of breast lesion by using the fractal Brownian motion. Furthermore, a computer-aided diagnosis (CAD) system based on the fractal analysis is proposed to classify the breast lesions into two classes: benign and malignant. To improve the classification performances, the ultrasound images are preprocessed by using morphology operations and histogram equalization. Finally, the k-means classification method is used to classify benign tumors from malignant ones. The US breast image databases include only histologically confirmed cases: 110 malignant and 140 benign tumors, which were recorded. All the digital images were obtained prior to biopsy using by an ATL HDI 3000 system. The receiver operator characteristic (ROC) area index AZ is 0.9218, which represents the diagnostic performance.

Adolescent↗
PubMed2006

A fractal analysis approach to viscoelasticity of physically cross-linked barley beta-glucan gel networks.

The structure and gelation kinetics of mixed linkage barley beta-glucans of varying Mw have been investigated. The fractal concept has been applied to describe the structure development of barley beta-glucan gels using a scaling model and dynamic rheometry data. The model supports that the gel structure consists of fractal clusters that upon aggregation lead to a three-dimensional network. The analysis showed that with increasing Mw a denser (more packed) network is formed as indicated by the corresponding fractal dimension (df) values. The microelastic parameter of the model, alpha, showed that all gel structures were in the transition regime implying structural reordering upon ageing. The description of the microstructure as a fractal network seems to be able to explain syneresis and other observations from large deformation testing of such systems. The molecular treatment of the gelation kinetics suggests that the gelling behavior is governed by the probability of collision of chain fragments with consecutive cellotriosyl units. This is greater for small chains due to their higher diffusion rates, for chains having lower amounts of cellulose like fragments and finally for those showing smaller degree of intrachain interactions. As a result, the faster gelling systems exhibit lower fractal dimensionality (more disordered systems) something that is in accordance with current kinetic theories.

Algorithms↗
PubMed2004

Fractal dynamics and synchronization of rhythms in urodynamics of female Wistar rats.

Two phenomena are ubiquitous in physiological signals. First, they have temporal fractal structure. Secondly, they are rhythmic but with fluctuations. The fractal structure and rhythms can alter from normal to abnormal with disease or injury. Here, we demonstrate that a temporal "coherent" fractal structure and synchronization of rhythms are two essential features for concerted systemic functions in urodynamics of female Wistar rats. We report evidence on the dynamic interaction in the micturition reflex of lower urinary tract: the rhythms of detrusor of the bladder and external urethral sphincter are synchronized around 7 Hz and their fractal dimensions as functions of time are temporally "coherent" under 1.5 during voiding in order to facilitate the concerted process. On the other hand, in the spinal cord injured rats, either the temporal coherent structure of fractal or the underlying synchronization of rhythms is impaired, weakened, or absent. These results have potential implications in the diagnosis of the so-called "dynamical" diseases.

Animals↗
PubMed2004

Fractal characteristics of breath to breath timing in sleeping infants.

We examined interbreath interval (IBI) time series of 19 term infants during active and quiet sleep for fractal properties using Fano factor analysis. For each time series we calculated the fractal exponent (alpha), comparing alpha for the original time series with two forms of surrogate data, a temporally independent surrogate set and an autoregressive surrogate set. alpha values were normally distributed between 0.79 and -0.22, and did not differ with sleep state. The fractal characteristics of the original time series were not retained in the temporally independent surrogate time series indicating that the distribution of intervals alone was not fractal, but were retained using autoregressive surrogates with an order of 10, suggesting that the fractal properties of the IBI time series were related to correlations between successive breaths. These observations suggest that some of the respiratory variability that occurs during sleep in infants, which in the past has been regarded as stochastic noise, may be the product of deterministic processes.

Factor Analysis, Statistical↗
PubMed2005

Application of fractal flocculation and vertical transport model to aquatic sol-sediment systems.

In estuarine and coastal environments, flocculation occurs between particles of different fractal dimensions and of different densities. Questions remain concerning the level of detail required to model particle flocculation and settling in these heterogeneous systems. This paper compares the goodness of fit between two flocculation models, using measured time series particle size distribution data collected from clay, colloidal silica, emulsified crude oil, clay-crude oil, and silica-crude oil systems. The coalesced sphere (CS) flocculation model includes the effects of heterogeneous particle size and density; the modified coalesced fractal sphere (mCFS) model adds effects due to heterogeneous fractal dimension. Goodness of fit was quantified using values of a minimized objective function, the mean of the sum of the square of the relative residuals (MSSRR). For nearly all tested experimental conditions, MSSRR values varied less than 5% between the CS and mCFS flocculation models. Additionally, collision efficiency values for single-particle-type (alpha(HOMOO)) and dual-particle-type (alpha(HETT)) systems were obtained through parameter regression using the CS and mCFS models. Using the mCFS model, estimated fractal dimension (D) values obtained for clay and clay-oil systems were between 2.6 and 3.0, lower than that postulated by the CS model but higher than that estimated experimentally by the particle concentration technique. The Stokes settling velocity of a clay aggregate of a given mass is reduced with decreased fractal dimension. This results in clay-oil flocculation occurring faster than floc sedimentation in the studied hydrodynamic range. Thus, the mCFS model provides insights to the fate of spilled oil in inland and coastal waters.

Aluminum Silicates↗
PubMed2002

Fractal analysis of heart rate variability and mortality after an acute myocardial infarction.

The recently developed fractal analysis of heart rate (HR) variability has been suggested to provide prognostic information about patients with heart failure. This prospective multicenter study was designed to assess the prognostic significance of fractal and traditional HR variability parameters in a large, consecutive series of survivors of an acute myocardial infarction (AMI). A consecutive series of 697 patients were recruited to participate 2 to 7 days after an AMI in 3 Nordic university hospitals. The conventional time-domain and spectral parameters and the newer fractal scaling indexes of HR variability were analyzed from 24-hour RR interval recordings. During the mean follow-up of 18.4 +/- 6.5 months, 49 patients (7.0%) died. Of all the risk variables, a reduced short-term fractal scaling exponent (alpha(1) <0.65), measured by detrended fluctuation analysis, was the most powerful predictor of mortality (univariate relative risk 5.05, 95% confidence intervals [CI] 2.87 to 8.89, p <0.001). A low scaling exponent alpha(1) predicted death in the patients with and without depressed left ventricular function (p <0.001 and p <0.01, respectively). Several other HR variability parameters also predicted mortality in univariate analyses, but in a multivariate analysis after adjustments for clinical variables and left ventricular ejection fraction, alpha(1) was the most significant independent HR variability index that predicted subsequent mortality (relative risk 3.90, 95% CI 2.03 to 7.49, p <0.001). Short-term fractal scaling analysis of HR variability is a powerful predictor of mortality among patients surviving an acute myocardial infarction.

Aged↗
PubMed1999

The fractal structure of glycogen: A clever solution to optimize cell metabolism.

Fractal objects are complex structures built with a simple procedure involving very little information. This has an obvious interest for living beings, because they are splendid examples of optimization to achieve the most efficient structure for a number of goals by means of the most economic way. The lung alveolar structure, the capillary network, and the structure of several parts of higher plant organization, such as ears, spikes, umbels, etc., are supposed to be fractals, and, in fact, mathematical functions based on fractal geometry algorithms can be developed to simulate them. However, the statement that a given biological structure is fractal should imply that the iterative process of its construction has a real biological meaning, i.e., that its construction in nature is achieved by means of a single genetic, enzymatic, or biophysical mechanism successively repeated; thus, such an iterative process should not be just an abstract mathematical tool to reproduce that object. This property has not been proven at present for any biological structure, because the mechanisms that build the objects mentioned above are unknown in detail. In this work, we present results that show that the glycogen molecule could be the first known real biological fractal structure.

Carbohydrate Conformation↗
PubMed1997

Fractal dimensions of short EEG time series in humans.

Fractal dimensions has been proposed as a useful measure for the characterisation of electrophysiological time series. But one of the problems of this approach, is the difficulty to record time series long enough of determine the 'real' fractal dimension. Nevertheless it is possible to calculate fractal dimensions for very short data-segments. Using time series of different length it is possible to show, that there is a monotoneous relation between fractal dimension and the number of data-points. This relation could be further interpreted with the help of an extrapolation scheme. In addition this effect is also seen with surrogate data, generated from that signal. We conclude that it is feasible to use fractal dimension as a tool to characterise the complexity for short electroencephalographic (EEG) time series, but it is not possible to decide whether the brain is a chaotic system or not.

Electroencephalography↗
PubMed2001

Application of fractal geometry to dissolution kinetic study of a sweetener excipient.

In the context of relationship study between dissolution kinetic and particle morphology using the fractal geometry tool, we use a commercially available quality of saccharin powder. The characterization of molecular feature and image analysis study allows us to conclude to the statistic self-similarity of particles of four sieved particles size fractions, permitting the fractal approach. Calculation of reactive fractal dimension is performed using two forms of mass transfer equation: -dQ/dt=kQ(D(R)/3)DeltaC and -dQ/dt=k'R(D(R)-3)DeltaC, with DeltaC=(C(f)/[lnC(s)/(C(s)-C(f))]). Based on comparison of the surface fractal dimension D(S) on the two values of reactive fractal dimension D(R), a dissolution mechanism can be drawn: the dissolution starts at the whole surface of particles and is further governed by digging into holes that involve inner mass of particles. S.E.M. observations confirm this hypothesis. The confrontation between the D(R) values provided by the two ways of determination is essential for a good prediction of the mechanism.

Excipients↗
PubMed1997

Predictive value for future arrhythmic events of fractal dimension, a measure of time clustering of ventricular premature complexes, after myocardial infarction. CAST Investigators. Cardia Arrhythmia Suppression Trial.

OBJECTIVES: Our objective was to test fractal dimension (D), a measure of clustering of ventricular premature complexes (VPCs), on entry Holter recording as a predictor of future arrhythmic death and other-cause mortality in postinfarction patients in the Cardiac Arrhythmic Suppression Trial (CAST). BACKGROUND: Nonlinear dynamic methods of signal processing are being applied in medicine to provide new insights into apparently "chaotic" biologic events, including cardiac arrhythmias. One such application is the derivation of a fractal D to describe the clustering of VPCs in time. METHODS: Baseline Holter recordings were analyzed in blinded manner for 484 patients: 237 died or had a resuscitated cardiac arrest during follow-up, and 247 matched patients had no events. Fractal D, measured in four ways, was assessed as a predictor using Cox regression. RESULTS: One measure of D (high resolution D) was a significant univariate (relative hazard ratio 0.79 per SD change, p = 0.011) and multivariate (hazard ratio 0.75, p = 0.046) predictor of arrhythmic death but not other death (univariate p = 0.95, relative hazard 0.95, p = 0.66). Fractal D was greater (VPCs less clustered) in those patients free of arrhythmic events. On subgroup analysis, the predictive value of D resided in the randomized patient group (i.e., those who showed VPC suppression during initial antiarrhythmic drug titration and were randomized to blinded therapy with active drug or placebo) (multivariate hazard ratio 0.57, p = 0.001). CONCLUSIONS: A high resolution fractal D was predictive of arrhythmic (but not nonarrhythmic) death in a large postinfarction cohort. Further study of this new signal processing approach to ambulatory electrocardiographic recording will be of interest.

Aged↗
PubMed2000

Analyte-receptor binding and dissociation kinetics for biosensor applications: a fractal analysis.

A fractal analysis of confirmative nature only is presented for analyte-receptor binding and dissociation kinetics for biosensor applications. Data taken from the literature may be modeled, in the case of binding using a single-fractal analysis or a dual-fractal analysis. The dual-fractal analysis represents a change in the binding mechanism as the reaction progresses on the surface. Relationships are presented for the binding and dissociation rate coefficients as a function of their corresponding fractal dimension, Df or the degree of heterogeneity that exists on the surface. When analyte-receptor binding or dissociation is involved, an increase in the heterogeneity on the surface (increase in Df) leads to an increase in the binding and in the dissociation rate coefficient. It is suggested that an increase in the degree of heterogeneity on the surface leads to an increase in the turbulence on the surface owing to the irregularities on the surface. This turbulence promotes mixing, minimizes diffusional limitations, and leads subsequently to an increase in the binding and in the dissociation rate coefficient (Martin S.J., Granstaff, V.E., Frye, G.C., Anal. Chem., 65, (1991) 2910). The binding and the dissociation rate coefficient are rather sensitive to the degree of heterogeneity, Df,bind and Df,diss respectively, that exists on the biosensor surface. For example, the order of dependence on Df,bind is 19.2 for the binding rate coefficient, kbind for the binding of 0.03-1.0 microM SH-2Ld in solution to 2C TCR immobilized on a surface plasmon resonance (SPR) biosensor (Corr, M., Salnetz, A.E., Boyd, L.F., Jelonek, M.T., Khilko, S., Al-Ramadi, B.K., Kim, Y.S., Maher, S.E., Bothwell, A.L.M., Margulies, D.H., Science, 265, (1994) 946). The order of dependence on Df,diss is -6.22 for the dissociation rate coefficient, kdiss for the dissociation of 250-1000 nM Sophora japonica agglutinin (SJA)-lactose complex from the SPR surface. In general, the technique is applicable to other reactions occurring on different types of surfaces, such as cell-surface reactions.

Biosensing Techniques↗
PubMed2003

Is the brain cortex a fractal?

The notion of fractal has been largely used to describe geometrical properties of complex objects in biology and medicine. In the present study the question is addressed whether the human cerebral cortex is self-similar in a statistical sense, which is commonly referred to as being a fractal. A new calculational method is presented, which is volumetric and based on the fast Fourier transform (FFT) of segmented three-dimensional high-resolution magnetic resonance images. The analysis covers a wide range of spatial scales from the size of the whole cortex to the ultimate pixel size. Results obtained in six subjects confirm the fractal nature of the human cerebral cortex down to a spatial scale of 3 mm. The obtained fractal dimension is D = 2.80 +/- 0.05, which is in reasonable agreement with previously reported results. Deployment of FFT enables a simple interpretation of the results and yields a high performance, which is necessary to analyze the entire cortex. Thus the FFT-based analysis of segmented MR images offers a comprehensive approach to study neurodevelopmental and neurodegenerative changes in the fractal geometry of the cerebral cortex.

Algorithms↗
PubMed1997

Fractal texture analysis in computer-aided diagnosis of solitary pulmonary nodules.

RATIONALE AND OBJECTIVES: The authors investigated the use of fractal texture characterization to improve the accuracy of solitary pulmonary nodule computer-aided diagnosis (CAD) systems. METHODS: Thirty chest radiographs were acquired from patients who had no pulmonary nodules. Thirty regions were selected that were considered remotely suspicious-looking for nodules. Artificial nodules of multiple shapes, sizes, and orientations were added at subtle levels of contrast to 30 non-suspicious-looking regions of the radiographs. Fractal dimensions of the 60 "nodule candidates" were calculated to quantify the texture of each region. Four radiologists also interpreted the images. RESULTS: The fractal dimension of each possible nodule provided statistically significant (P < .05) differentiation between regions that contained an artificial nodule and those that did not. The area under the receiver operating characteristic curve for the fractal analysis was significantly better (P < .05) than that for the radiologists. CONCLUSION: Fractal texture characterization provides useful information for the classification of potential solitary pulmonary nodules with CAD algorithms.

Diagnosis, Computer-Assisted↗
PubMed1998

A comparison of porosity, fabric and fractal dimension as predictors of the Young's modulus of equine cancellous bone.

The purpose of this study was to compare the structural parameters of fabric and fractal dimension as predictors of the Young's modulus of equine cancellous bone. Eight 15 mm cubes of cancellous bone were obtained from three equine third metacarpal bones. Young's modulus was determined for the three orthogonal directions. The fabric and fractal dimension were calculated for each of the six exposed faces of each cube. Fractal dimension plus porosity provided a higher explanatory power for Young's modulus (R2 = 78.7%. P < 0.0001) than fabric plus porosity (R2 = 69.2%, P < 0.0001). Fractal dimension was also significantly correlated with fabric (R2 = 53.8%, P < 0.0001). Although this novel method for combining fractal dimension data into a pseudo-directionally dependent predictor of Young's modulus requires further validation over a greater range of porosities and differing cancellous bone tissues, its potential has been demonstrated.

Animals↗
PubMedDate not supplied

Detection of glucose and related analytes by biosensors: a fractal analysis.

A fractal analysis is used to model the binding and dissociation kinetics of connective tissue interstitial glucose, adipose tissue interstitial glucose, insulin, and other related analytes on biosensor surfaces. The analysis provides insights into diffusion-limited analyte-receptor reactions occurring on heterogeneous biosensor surfaces. Numerical values obtained for the binding and the dissociation rate coefficients are linked to the degree of heterogeneity or roughness (fractal dimension, Df) present on the biosensor chip surface. The binding and dissociation rate coefficients are sensitive to the degree of heterogeneity on the surface. For example, for the binding of plasma insulin, as the fractal dimension value increases by a factor of 2.47 from Df1 equal to 0.6827 to Df2 equal to 1.6852, the binding rate coefficient increases by a factor of 4.92 from k1 equal to 1.0232 to k2 equal to 5.0388. An increase in the degree of heterogeneity on the probe surface leads to an increase in the binding rate coefficient. A dual-fractal analysis is required to fit the binding kinetics in most of the cases presented. A single fractal analysis is adequate to describe the dissociation kinetics. Affinity (ratio of the binding to the dissociation rate coefficient) values are also presented. Interferents for glucose such as uric acid and ascorbic acid were also detected using glucose biosensors based on carbon nanotube (CNT) nanoelectrode ensembles (NEEs) (29) (Lin, Y.; Lu, F.; Tu, Y.; Ren, Z. Nano Lett. 2004, 4 (2), 191-195). Attempts are made to standardize biosensor properties in terms of diffusion characteristics on in vivo responsiveness.

Acetaminophen↗