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PubMed2005

Fractal analysis of acceleration signals from patients with CPPD, rheumatoid arthritis, and spondyloarthroparthy of the finger joint.

Arthritis is one of the leading causes of disability and affects a major segment of the population. Consequently, accurate diagnosis of arthritis is important. Arthritis due to calcium pyrophosphate deposition disease (CPPD), rheumatoid arthritis, and spondyloarthropathy, induce complex changes in the cartilage and the articular surface. The fractal dimension provides a measure of the complexity of a signal. Recently, we have developed non-invasive acceleration measurements to characterize the arthritic patients. The question remains if the fractal dimension of the acceleration signal is different for different arthritis conditions. The purpose of this study was to distinguish between different types of arthritis of the finger joint using the fractal dimension of the acceleration signal obtained from the finger joint of the arthritic patients. Acceleration signals were obtained from the finger joint of arthritis patients with rheumatoid arthritis, spondyloarthropathy, and calcium pyrophosphate deposition disease of the finger joint. ANOVA results showed that there were significant differences between the fractal dimension of acceleration signals from patients having calcium pyrophosphate deposition disease and rheumatoid arthritis and spondyloarthropathy. Fractal dimension of acceleration signals, in concert with other clinical symptoms, can be used to classify different types of arthritis.

Arthritis, Rheumatoid↗
PubMed2004

Fractal analysis of narwhal space use patterns.

Quantifying animal movement in response to a spatially and temporally heterogeneous environment is critical to understanding the structural and functional landscape influences on population viability. Generalities of landscape structure can easily be extended to the marine environment, as marine predators inhabit a patchy, dynamic system, which influences animal choice and behavior. An innovative use of the fractal measure of complexity, indexing the linearity of movement paths over replicate temporal scales, was applied to satellite tracking data collected from narwhals (Monodon monoceros) (n = 20) in West Greenland and the eastern Canadian high Arctic. Daily movements of individuals were obtained using polar orbiting satellites via the ARGOS data location and collection system. Geographic positions were filtered to obtain a daily good quality position for each whale. The length of total pathway was measured over seven different temporal length scales (step lengths), ranging from one day to one week, and a seasonal mean was calculated. Fractal dimension (D) was significantly different between seasons, highest during summer (D = 1.61, SE 0.04) and winter (D = 1.69, SE 0.06) when whales made convoluted movements in focal areas. Fractal dimension was lowest during fall (D = 1.34, SE 0.03) when whales were migrating south ahead of the forming sea ice. There were no significant effects of size category or sex on fractal dimension by season. The greater linearity of movement during the migration period suggests individuals do not intensively forage on patchy resources until they arrive at summer or winter sites. The highly convoluted movements observed during summer and winter suggest foraging or searching efforts in localized areas. Significant differences between the fractal dimensions on two separate wintering grounds in Baffin Bay suggest differential movement patterns in response to the dynamics of sea ice.

Journal Article↗
PubMed1987

Fractal analysis of a voltage-dependent potassium channel from cultured mouse hippocampal neurons.

The kinetics of ion channels have been widely modeled as a Markov process. In these models it is assumed that the channel protein has a small number of discrete conformational states and the kinetic rate constants connecting these states are constant. In the alternative fractal model the spontaneous fluctuations of the channel protein at many different time scales are represented by a kinetic rate constant k = At1-D, where A is the kinetic setpoint and D the fractal dimension. Single-channel currents were recorded at 146 mM external K+ from an inwardly rectifying, 120 pS, K+ selective, voltage-sensitive channel in cultured mouse hippocampal neurons. The kinetics of these channels were found to be statistically self-similar at different time scales as predicted by the fractal model. The fractal dimensions were approximately 2 for the closed times and approximately 1 for the open times and did not depend on voltage. For both the open and closed times the logarithm of the kinetic setpoint was found to be proportional to the applied voltage, which indicates that the gating of this channel involves the net inward movement of approximately one negative charge when this channel opens. Thus, the open and closed times and the voltage dependence of the gating of this channel are well described by the fractal model.

Animals↗
PubMed2003

Linear chains and chain-like fractals from electrostatic heteroaggregation.

The internal structure of materials prepared by aggregation of oppositely charged polystyrene spheres (electrostatic heteroaggregation) is investigated by static light scattering, optical microscopy, and Brownian dynamics simulation. Light scattering indicates ultralow mass fractal dimensions, as low as 1.2. Such low fractal dimensions, approaching the theoretical limit of a linear object, imply a chaining mechanism. Optical micrographs reveal linear chains with the particle charge alternating down the chains. Brownian dynamics simulation gives additional support for a chaining mechanism. For the polystyrene system (120-nm primary particle diameters), the fractal dimension is found to increase from 1.2 to 1.7 as the background electrolyte is increased. In terms of electrostatic screening, the results match those reported recently for larger polystyrene spheres. The low fractal dimensions appear to represent a crossover from linear chains to a structure of diffusion-limited aggregates; however, experiments under density-neutral conditions imply that sedimentation plays an important role in the formation of ultralow fractal dimensions. The practical implication is that microcomposites with a locally uniform distribution of starting materials and almost any degree of branching can be prepared from oppositely charged particles.

Journal Article↗
PubMed1995

Carrier-mediated transport can obey fractal kinetics.

A model based on the fractal methodology is proposed for the kinetic study of carrier-mediated transport under heterogeneous conditions, i.e., when the drug-carrier interaction occurs at an interface with an effective dimensionality smaller than the embedding dimension of d = 2. A model equation is derived for the flux, based on a similar approach for an analogous equation for enzyme kinetics. It is shown that the total flux-solute concentration plots are curvilinear when the fractal dimension is smaller than unity while they become biexponential, with ascending and descending limbs, when the fractal dimension D is in the range 1 < D < 2. Nonlinear Lineweaver-Burk plots are obtained when this fractal kinetics approach is used. Good fittings are obtained when the fractal model is applied to literature data previously analysed with a combined transport mechanism, revealing experimental systems that display a D value in the range 1 < D < 2. It is suggested that transport studies should be carried out at a wider working solute concentration range and various agitation and incubation conditions in order to derive definite conclusions for the transport pathways.

Animals↗
PubMed1990

The fractal properties of retinal vessels: embryological and clinical implications.

The branching patterns of retinal arterial and venous systems have characteristics of a fractal, a geometrical pattern whose parts resemble the whole. Fluorescein angiogram collages were digitised and analysed, demonstrating that retinal arterial and venous patterns have fractal dimensions of 1.63 +/- 0.05 and 1.71 +/- 0.07, respectively, consistent with the 1.68 +/- 0.05 dimension of diffusion limited aggregation. This finding prompts speculation that factors controlling retinal angiogenesis may obey Laplace's equation, with fluctuations in the distribution of embryonic cell-free spaces providing the randomness needed for fractal behaviour and for the uniqueness of each individual's retinal vascular pattern. Since fractal dimensions characterise how completely vascular patterns span the retina, they can provide insight into the relationship between vascular patterns and retinal disease. Fractal geometry offers a more accurate description of ocular anatomy and pathology than classical geometry, and provides a new language for posing questions about the complex geometrical patterns that are seen in ophthalmic practice.

Computer Graphics↗
PubMed2004

Mass-fractal clustering and power-law decay of cluster size in 1-propanol aqueous solution.

Mesoscale structure of 1-propanol aqueous solutions with propanol mole fraction xp ranging from 0.1 to 0.33 has been studied by means of small angle neutron scattering (SANS) and large-scale reverse Monte Carlo (RMC) technique. Analysis of the SANS intensities in terms of a fractal model shows that the fractal dimension df of mesoscale structure of the solution is about 1.8-1.9 for water-rich solution and about 1.5 for propanol-rich solution. Percolation analysis on the RMC results reveals that the water molecules and the propanol molecules cluster, respectively, as a mass fractal, the dimension dM of which is about 2.3-2.5 for both clusters for water-rich solution. Furthermore, the distribution of the cluster size is expressed by a simple power law with an exponent tau of about 1.35-1.5 for the propanol clusters and 1.05-1.2 for the water clusters. These results imply that the current solution is characterized by polydisperse mass fractals. In fact, a theoretical relation for polydisperse system of mass fractals, df = dM(2-tau), holds well in the current solution. The characteristic change in df from 1.8-1.9 to 1.5 described above is attributed to the crossover between the water-rich regime and the propanol-rich regime. Most of the water molecules and the propanol molecules are located on the interface between clusters, and the water molecules form thin layers of about 10 A thick irrespective of 1-propanol content studied.

Journal Article↗
PubMed1990

Surface fractality of proteins from theory and NMR data.

Different approaches to study protein surface fractality are considered. An approach based on analysis of surface versus molecular weight dependence is shown to be an informative tool for investigation of protein surface behaviour. An evidence for protein surface fractality, obtained with the use of this analysis from the data of both NMR measurements in protein solutions and computer analysis of protein structures, is presented. Obtained value of fractal dimension of protein surface (ds congruent to 2.2) is in a good agreement with the results of conventional approach (with variation of yardstick length) to protein surface fractality. A conclusion is made that surface enlargement due to the rise of protein molecular weight is accompanied by the increase of maximum scale of irregularities on protein surface. Possible effect of surface fractality on hydrodynamic characteristics of protein molecules in solution is discussed.

Chemical Phenomena↗
PubMed1999

Characterization of Branch Complexity by Fractal Analyses.

The comparison between complexity in the sense of space occupancy (box-counting fractal dimension Dc and information dimension DI) and heterogeneity in the sense of space distribution (average evenness index J&d1; and evenness variation coefficient JCV) were investigated in mathematical fractal objects and natural branch structures. In general, increased fractal dimension was paired with low heterogeneity. Comparisons between branch architecture in Anthyllis cytisoides under different slope exposure and grazing impact revealed that branches were more complex and more homogeneously distributed for plants on northern exposures than southern, while grazing had no impact during a wet year. Developmental instability was also investigated by the statistical noise of the allometric relation between internode length and node order. In conclusion, our study demonstrated that fractal dimension of branch structure can be used to analyze the structural organization of plants, especially if we consider not only fractal dimension but also shoot distribution within the canopy (lacunarity). These indexes together with developmental instability analyses are good indicators of growth responses to the environment.

Journal Article↗
PubMed1992

Fractal properties of herpes simplex dendritic keratitis.

Fractal geometry can be used to analyze the complexity and self-similarity of many natural forms and structures. By using three techniques for fractal dimension estimation, a group of 11 herpes simplex virus dendritic corneal ulcers was analyzed and found to possess outlines with fractal properties. The mean surface fractal dimension approached 1.4 over 1.5 decades of scale. The demonstration and quantification of the fractal dimension of dendritic lesions is important in the understanding of their behaviour and, at a more fundamental level, the planar spread of viral infection.

Humans↗
PubMed1991

A method for determination of fractal dimensions of sialographic images.

A method for measuring the fractal dimension of radiographic images was devised by the modified pixel dilation (PD) method, and its accuracy was examined by two types of image analyzers: one controlled by a minicomputer and based on the hexagonal array of the pixels, and the other controlled by a personal computer and based on the square array of the pixels. It was shown that the fractal dimensions measured by using the image analyzers for three curves (straight line, Koch curve, and Sierpinski gasket) were deviated by less than 2.1% from the theoretical dimension. The time required for the measurements fell within 10 seconds by the minicomputer-controlled image analyzer, and within 12 minutes by the personal computer-controlled image analyzer, which contrasts to the 30 minutes or more required for measuring fractal dimensions by other available methods. Analysis of the ductal patterns in sialograms by this method indicated that the fractal dimensions were significantly different between normal parotids and parotids with Sjögren syndrome. The modified PD method can be used for measuring the fractal dimension of ductal patterns in sialograms. The measurements may serve as numerical gradings of the progression of disease.

Humans↗
PubMed1999

Transient fractal behavior in snapshot attractors of driven chaotic systems.

Snapshot attractors, i.e., attractors formed by a cloud of trajectories at the same instants of time, are usually employed to reveal the fractal structure of randomly or chaotically driven dynamical systems. A necessary condition for the underlying fractal structure to be observed is that the ensemble of particles utilized in the construction of the snapshot attractors are subject to identical perturbation at any instant of time. We examine the influence of small phase-space inhomogeneity in the chaotic perturbation on the observability of the snapshot fractal attractors. We find that, typically, fractal structure can be seen in only a transient period of time. The scaling of the transient time with the amount of inhomogeneity is investigated. Implication to experimental observation of fractal structure in physical systems is pointed out.

Journal Article↗
PubMed2000

Anomalous burning rates of flamelets induced by self-similar multiple scale (fractal and spiral) initial fields

In contrast to the classical problem of a single idealized flamelet (which is described by a nonlinear reaction-diffusion equation of motion) which propagates at a constant burning rate, self-similar multiple scale fields, whether fractal or nonfractal, induce anomalous rates of burning determined by the space-filling properties of the initial field. We compare the regimes induced by (line-cuts through) three specific geometries with distinct space-filling characteristics: (1) an algebraic spiral which has capacity (box-counting dimension) D(k)>0, and fractal dimension H=0; (2) an exponential spiral which has D(k)=0 and H=0, and geometric ratio R>1; (3) a fractal Cantor dust which has D(k)=H>0. The (nondimensional) burning rate U(B) induced by all three geometries takes the general form U(B) approximately F(tau(-zeta)), where F is a function whose form depends on the specific geometry, zeta is an exponent that contains the space-filling characteristic of the geometry, and tau is a nondimensional time. (1) For the algebraic spiral, F(x)=1(x), and zeta=D(k); F is continuous. (2) For the exponential spiral, F(x)=ln(x), and zeta=1/(R-1); F is continuous. (3) For the fractal Cantor dust, F(u)(x)=1(x), and zeta=H (for the envelope); F itself is a step-like discontinuous function. Thus, as D(k)-->0, or as H-->0, or as R-->infinity, then zeta-->0 and U(B)-->const; and as D(k)-->1, or as H-->1, (space filling) then zeta-->1; and as R-->1 (space filling) then zeta-->infinity. Two numerical methods, a fundamental (Eulerian) solution to the equation of motion and a Lagrangian model for flamelet propagation, confirm these theoretical predictions. The Lagrangian model is based on the idealized flamelet as a "point" with finite flame thickness Delta(L), (which is determined by the two-flamelet collision process), propagating with a given flame speed U(L). The Lagrangian model allows simulations in parameter ranges not easily accessible by the fundamental method (such as the case for the fractal Cantor dust). Interestingly, the linear regime of scalar diffusion in an algebraic spiral field displays the same dependence on D(k) as in the present reaction-diffusion case. The nonlinear regime of advection-diffusion (Burger turbulence) shows a different dependence on D(k).

Journal Article↗
PubMed2001

Catastrophic bifurcation from riddled to fractal basins.

Most existing works on riddling assume that the underlying dynamical system possesses an invariant subspace that usually results from a symmetry. In realistic applications of chaotic systems, however, there exists no perfect symmetry. The aim of this paper is to examine the consequences of symmetry-breaking on riddling. In particular, we consider smooth deterministic perturbations that destroy the existence of invariant subspace, and identify, as a symmetry-breaking parameter is increased from zero, two distinct bifurcations. In the first case, the chaotic attractor in the invariant subspace is transversely stable so that the basin is riddled. We find that a bifurcation from riddled to fractal basins can occur in the sense that an arbitrarily small amount of symmetry breaking can replace the riddled basin by fractal basins. We call this a catastrophe of riddling. In the second case, where the chaotic attractor in the invariant subspace is transversely unstable so that there is no riddling in the unperturbed system, the presence of a symmetry breaking, no matter how small, can immediately create fractal basins in the vicinity of the original invariant subspace. This is a smooth-fractal basin boundary metamorphosis. We analyze the dynamical mechanisms for both catastrophes of riddling and basin boundary metamorphoses, derive scaling laws to characterize the fractal basins induced by symmetry breaking, and provide numerical confirmations. The main implication of our results is that while riddling is robust against perturbations that preserve the system symmetry, riddled basins of chaotic attractors in the invariant subspace, on which most existing works are focused, are structurally unstable against symmetry-breaking perturbations.

Journal Article↗
PubMed2001

Fractal properties of the lattice Lotka-Volterra model.

The lattice Lotka-Volterra (LLV) model is studied using mean-field analysis and Monte Carlo simulations. While the mean-field phase portrait consists of a center surrounded by an infinity of closed trajectories, when the process is restricted to a two-dimensional (2D) square lattice, local inhomogeneities/fluctuations appear. Spontaneous local clustering is observed on lattice and homogeneous initial distributions turn into clustered structures. Reactions take place only at the interfaces between different species and the borders adopt locally fractal structure. Intercluster surface reactions are responsible for the formation of local fluctuations of the species concentrations. The box-counting fractal dimension of the LLV dynamics on a 2D support is found to depend on the reaction constants while the upper bound of fractality determines the size of the local oscillators. Lacunarity analysis is used to determine the degree of clustering of homologous species. Besides the spontaneous clustering that takes place on a regular 2D lattice, the effects of fractal supports on the dynamics of the LLV are studied. For supports of dimensionality D(s)<2 the lattice can, for certain domains of the reaction constants, adopt a poisoned state where only one of the species survives. By appropriately selecting the fractal dimension of the substrate, it is possible to direct the system into a poisoned or oscillatory steady state at will.

Journal Article↗
PubMed2004

Description of diffusive and propagative behavior on fractals.

The known properties of diffusion on fractals are reviewed in order to give a general outlook of these dynamic processes. After that, we propose a description developed in the context of the intrinsic metric of fractals, which leads us to a differential equation able to describe diffusion in real fractals in the asymptotic regime. We show that our approach has a stronger physical justification than previous works on this field. The most important result we present is the introduction of a dependence on time and space for the conductivity in fractals, which is deduced by scaling arguments and supported by computer simulations. Finally, the diffusion equation is used to introduce the possibility of reaction-diffusion processes on fractals and analyze their properties. Specifically, an analytic expression for the speed of the corresponding travelling fronts, which can be of great interest for application purposes, is derived.

Journal Article↗
PubMed2006

Spacing distributions for point processes on a regular fractal.

The homogeneous Poisson point process in Rd (denoted by Pd) is a basic model of stochastic geometry and modern statistical physics. Using ideas from fractal geometry, geometrical statistics, and random matrix theory, we introduce the model of random points on a self-similar fractal as a model of intermediate statistics, in the sense that the interpoint spacing statistics of the model are intermediate between those of P1 and P2 when the fractal dimension is in between 1 and 2, and intermediate between those of P2 and P3 when the fractal dimension is in between 2 and 3, and so on. We also introduce the idea of using a continuous family of such models to interpolate between P1 and P2 and thereby effectuate crossover transitions between P1 statistics and P2 statistics. We first derive the kth-nearest-neighbor spacing distribution for the general model, and then study the interpoint spacing statistics of several realizations of the model involving Sierpinski fractals in R2 and R3. We also study a realization of a continuous interpolation between P1 and P2, in particular a continuous interpolation between a point process on a line and a point process on a plane-filling curve, using the continuous family of self-similar Koch curves in R2. In the latter study, we specifically analyze the second-nearest-neighbor interpoint spacing statistics, which undergo a crossover transition between semi-Poisson and Ginibre statistics.

Journal Article↗
PubMed1989

Fractal character of the auditory neural spike train.

Long-counting-time pulse-number distributions (PND's) were measured from a broad variety of cat primary auditory fibers using different tone and noise stimuli, counting times T, and number of samples NT. Whereas short-counting-time PND's (T approximately 50 ms) manifest the presence of spike pairs (an enhancement of even over odd-count probabilities), the irregular shapes of the long-counting-time PND's (T approximately greater than 0.1 s) reveal that the underlying sequence of action potentials consists of spike clusters when viewed on a longer time scale. For all units measured, the count variance-to-mean ratio (Fano factor) F(T) varied little over some 90 dB change in the stimulus level. On the other hand, F(T) increased substantially as T and/or NT were increased, corresponding to the capture of larger and larger spike clusters in the counting time. A relationship is developed between the Fano-time function F(T) and the normalized coincidence rate function, g(tau) versus delay time tau. A plausible form for g(tau) leads to a Fano-time function in good accord with the data. The observed power-law growth of the Fano factor for large counting times [F(T) approximately T alpha where 0 less than alpha less than 1] is accompanied by a power-law decay of the coincidence rate for large delay times [g(tau) approximately tau alpha -1] and a power-law form for the power spectral density at low frequencies [S(f) approximately f -alpha]. The behavior of the PND's and the scale invariance implicit in these fractional-power-law relationships suggest that the neural events on all primary auditory fibers exhibit fractal behavior for sufficiently large times (sufficiently low frequencies). The spike pairs and spike clusters in the PND's are natural consequences of this behavior. The fractal dimension D identical to alpha is estimated to be in the range of 0.3 approximately less than D approximately less than 0.9 for counting times in the range 0.1-10 s. The fractal dimension provides a measure of the degree of event clustering, or irregularity of a sequence of events, that is preserved over different time scales. PND's from low-skew vestibular units, in contrast, do not exhibit fractal behavior. It is suggested that auditory neural-firing patterns may serve to efficiently sample natural fractal noises.

Action Potentials↗