Fractal Scaling Properties Research Articles

The anomalous pharmacokinetics of amiodarone explained by nonexponential tissue trapping.

Conventional pharmacokinetic (PK) concepts fail to describe the long-term pharmacokinetics of the extremely cationic amphiphilic drug amiodarone. A nonclassical model based on the phenomenon of trapping at tissue binding sites with very long release times is presented, which implies that a volume of distribution and a steady-state level cannot be defined. In agreement with clinical PK data available in the literature, the model well describes not only single-dose disposition curves but also the persistently increasing plasma concentration-time curve during long-term treatment (up to 5 years) and the washout curve following cessation of therapy. The novel aspect is a long-tailed tissue residence time distribution which is incorporated into a recirculatory model leaving the initial distribution process and the clearance concept unchanged. The underlying theoretical approach, which is known as "strange or anomalous" kinetics in physical sciences, and the fractal scaling property of the model may enhance our understanding of the PK of extremely hydrophobic xenobiotics.

Nucleotide composition effects on the long-range correlations in human genes

We use the wavelet transform to investigate the fractal scaling properties of coding and noncoding human DNA sequences. We find that the strength of the long-range correlations observed in the introns increases with the guanine-cytosine (GC) content, while coding sequences show no such correlations at any GC content. However, we demonstrate that long-range correlations can be detected when the coding sequences are undersampled by retaining the third base of each codon only. This strongly suggests that the observed correlations are not likely to be due to insertion-deletion mechanisms. We comment about the origin of these correlations in terms of putative dynamical processes that could produce the isochore structure of the human genome.

Exact solution of linear transport equations in fractal media—III. Adsorption and chemical reaction

Absorption kinetics and first-order reaction models on fractal lattices are studied. Adsorption kinetics in macro/microporous fractal solids showing fractal scaling properties and a complex topological pore-network structure are developed by considering the coupling of two fractal graphs. For first-order reactions on fractal substrata, the exact determination of the behavior of the effectiveness factor vs the Thiele modulus is obtained. A model for analyzing the spatial concentration distribution in a fractal catalyst is proposed.

Heterogeneity of the pore and solid volume of soil: distinguishing a fractal space from its non-fractal complement

The scaling properties of fractals are formally defined only in the limit of infinite or infinitessimal length scales, while measurements are necessarily restricted to finite scales. The implications of this limitation are studied from the perspective of characterising the heterogeneity of a fractal and its complement. An examination of the scaling properties of the solid and pore space of soil aggregates is presented as a case study. A relationship is derived between the heterogeneity of the solid volume and that of the pore space, under the assumption that either the solid or pore can be approximated by a fractal. If the solid is fractal, then the pore volume (its space complement) is asymptotically homogeneous as the sample volume increases to infinity i.e. the porosity is effectively scale-independent for large sample sizes. For smaller sample volumes, however, the porosity is scale-dependent, and this dependency is a function of the fractal dimension of the solid. Where the solid has a fractal dimension close to the Euclidean value, the pore volume behaves like a fractal, and this presents a problem from a methodological point of view, in distinguishing between a fractal and its complement. It is shown how misinterpretation can lead to errors in the prediction of moisture distribution and transport processes. A method for making the distinction is presented here, and is tested using image analysis of soil thin sections.

Wavelet based fractal analysis of DNA sequences

Abstract The fractal scaling properties of DNA sequences are analyzed using the wavelet transform. Mapping nucleotide sequences onto a “DNA walk” produces fractal landscapes that can be studied quantitatively by applying the so-called wavelet transform modulus maxima method. This method provides a natural generalization of the classical box-counting techniques to fractal signals, the wavelets playing the role of “generalized oscillating boxes”. From the scaling behavior of partition functions that are defined from the wavelet transform modulus maxima, this method allows us to determine the singularity spectrum of the considered signal and thereby to achieve a complete multifractal analysis. Moreover, by considering analyzing wavelets that make the “wavelet transform microscope” blind to “patches” of different nucleotide composition that are observed in genomic sequences, we demonstrate and quantify the existence of long-range correlations in the noncoding regions. Although the fluctuations in the patchy landscape of the DNA walks reconstructed from both noncoding and (protein) coding regions are found homogeneous with Gaussian statistics, our wavelet-based analysis allows us to discriminate unambiguously between the fluctuations of the former which behave like fractional Brownian motions, from those of the latter which cannot be distinguished from uncorrelated random Brownian walks. We discuss the robustness of these results with respect to various legitimate codings of the DNA sequences. Finally, we comment about the possible understanding of the origin of the observed long-range correlations in noncoding DNA sequences in terms of the nonequilibrium dynamical processes that produce the “isochore structre of the genome”.

Characterizing long-range correlations in DNA sequences from wavelet analysis.

The fractal scaling properties of DNA sequences are analyzed using the wavelet transform. Because the wavelet transform microscope can be made blind to the ``patchiness'' of genomic sequences, we demonstrate and quantify the existence of long-range correlations in genes containing introns and noncoding regions. Moreover, the fluctuations in the patchy landscapes of DNA walks are found to be homogeneous with Gaussian statistics.

Heterogeneity and implicated surface effects: statistical, fractal formulation and relevant analytical solution

The aim of this paper is to examine the implications of material heterogeneity on brittle material response and on relevant surface effects. Statistical and fractal concepts are used for this purpose. In the statistical formulation the displacement gradients of the micro-medium are considered to be random fields characterized by stationary exponential or Gaussian auto-covariance and by the relevant correlation length or scale of fluctuation. Through Taylor series expansion around the mean of the random field, an important analogy is found between the statistical formulation and the micro-structural theory, originally introduced by Mindlin, where higher order gradients of deformation appear in the constitutive equations. The analogy is valid only when fluctuations are small, so that some simplifications are allowed. It is found that the so-called internal length appearing in the micro-structural formulation is analogous to the correlation length in the statistical one. In the statistical approach there are no extra boundary conditions in the formulation, as is the case when higher order gradients are introduced. However, what is know as “conditioning” of the field at the boundaries effects its behavior near/on them. The statistical approach can provide further information in the form of higher order moments not captured by the gradient theory. Material/structure response is strongly dependent on the aforementioned scale. Its effect is most pronounced near the boundaries of a structure where its role on surface related phenomena is paramount. In order to study heterogeneity at a hierarchy of scales, i.e. absence of characteristic length, complex disorderly system, fractal concepts and relevant power decay laws are considered. The formulation introduces the fractal dimension of the heterogeneous displacement gradient of the micro-medium, a length describing the overall size of the structure, and the lower cutoff of the scaling law. The physical interpretation of the lower cutoff is the lower limit of applicability of the power scaling law. Mathematically it is important since in this case the fractal can be “followed” in the spatial domain. Similarly to the statistical case, an analogy between the fractal formulation and gradient theories is identified. No extra boundary conditions appear in the fractal formulation. However, there are still open questions with respect to the behavior of a fractal after conditioning, as is the case on boundaries. The analytical solution of a relevant surface instability problem for the gradient, statistical, and fractal formulation is presented. The solution was obtained through symbolic computations by computer because the analytical work is tedious and error prone. The analytical solution provides significant insight into the problem of heterogeneity and skin effects in brittle materials, internal length estimation, and the role of fractal scaling properties. Finally, the concepts introduced herein are discussed with respect to experimental information and numerical implementation.

Fractal Description of Cerebral Cortical Patterns in Frontal Lobe Epilepsy

The convoluted pattern of normal cerebral cortex resists description by Euclidean geometry. However, subtle abnormalities often unrecognised in vivo may underlie a wide range of neurological disorders. Abnormalities of cortical morphology frequently characterise these lesions, and we applied fractal geometry in an attempt to quantify these abnormalities. We have demonstrated typical fractal scaling properties for the cortical-white matter interface on axial and coronal MR images, with a fractal dimension (Df) of 1.45 +0.06 (mean +1 standard deviation) in normal subjects. We have also applied this to 16 patients with frontal lobe epilepsy, a condition associated with subtle disruptions of the cortical ribbon, all of whom had no obvious abnormality on visual inspection of the images. Nine of these 16 had a Df < 1.27 (mean -3 standard deviations). A control group of 10 patients with temporal lobe epilepsy with a focal foreign tissue lesion (vascular or tumoural) all had Df in the normal range. Analysis of shape using this method identifies subtle abnormalities of the cortical ribbon, and has potential application to images of the human brain in a wide range of clinical and pathophysiological settings.

New method of structural functions for analysing fractal scaling properties of natural processes

A method to reveal and estimate the fractal scaling properties of positive and negative increments in one-valued functions that describe natural processes is proposed. Structural functions, which were introduced by Kolmogorov (1941) for analysing the scaling properties of small-scale turbulence, provide the basis of the method. Examples are given to illustrate the application of the proposed method for analysing the trace of one of the coordinates of Brownian motion, simulated asymmetric wave forms, internal waves, sand waves in one-directional streams and river turbulence.

Comparisons of mixing in chaotic and turbulent flows

Results are presented from an experimental investigation comparing geometric scaling properties created by the mixing of dynamically passive tracers in chaotic flows with those resulting at the small scales of fully developed turbulent flows. The low Reynolds number, two-dimensional, time-periodic, closed flow between eccentric rotating cylinders is taken as the archetypal chaotic flow. The turbulent flow for comparison is the high Reynolds number, three-dimensional, unsteady, open flow in the self-similar far field of a steady axisymmetric jet. For each flow, the concentration field ζ( x, t) resulting from the mixing of a conserved scalar quantity is used to measure scaling properties of the support set on which the corresponding scalar energy dissipation rate field ( ReSc) -1∇ζ·∇ζ( x, t) is concentrated. The distributions of dissipation layer separations obtained for both flows are found to be identical. Contrary to central limit arguments for multiplicative quantities, the ensemble-averaged distributions in both flows have a —3 power law scaling for all but the smallest separations; classical log-normal scaling for multiplicative processes is found only in regions having undergone extensive stretching and folding. A statistical assessment of the fractal scaling properties based on one-dimensional intersections with the dissipation support set demonstrates that the chaotic flow at this stage of development approaches a global fractal dimension only in these same regions. Unlike previous studies of the fractal scaling of scalar isosurfaces in turbulent flows, the results for the turbulent flow presented here show no strong evidence for global fractal scaling in the dissipation support set.

Fractal properties of homologous series of structures

The concept of lattice-graph generation considered earlier (S. EI-Basil, J. Chem. Soc., Faraday Trans. 1993, 89, 909) has been studied further. It is demonstrated, using methods of symbolic dynamics and block-renaming, that the sequence which defines these graphs is a fractal, while the generation operation bears remarkable similarity to the various stages of the Cantor dust. Furthermore, Kekule counts of several homologous series of quasicrystal-like benzenoids form mathematical structures which possess the scaling properties of fractals. Self-similarity is extended to other homologous series of graphs (which represent chemical species) and it is demonstrated in all cases that the golden mean (τ= 1.618033989) is their characteristic scaling factor.

Dimensionality analysis of patterns: fractal measurements

Since the advent of Mandelbrot's fractal geometry in 1975, many geological and geophysical patterns have been characterized by their fractal scaling properties. The purpose of this paper is to present and compare standard and new methods used to estimate the fractal dimension of random (natural) fractals. Examples demonstrate the relative limitations of the techniques, and the modifications necessary to more accurately estimate the scaling properties of certain types of fractals. Discussion and examples include branching diffusion-limited aggregates, fracture surfaces, and fractional Brownian surfaces.

Testing fractal and multifractal concepts for seafloor topography.

Scattering and reverberation properties of the seafloor depend on the high-resolution spatial structure of submarine topography. Whether concepts of fractal geometry, scaling properties, and multifractal theory provide possibilities to describe the seafloor across several scales of resolution is investigated. In an attempt to predict the spatial structure of the seafloor beyond the scale resolved by survey systems, it has been proposed that the seafloor can be simulated using the self-similarity property of a fractal. (By definition, a fractal is an object of Hausdorff dimension strictly exceeding its topological dimension.) However, results from analyses of bathymetric data from the East Pacific Rise at 13° N/104° W and from the Juan de Fuca Ridge at 45° N/130° W show scale-dependent spatial structures, contradicting a simple self-similarity or self-affinity. Consequently, the seafloor probably has noninteger (fractal) dimension, but with different geologic processes dominating specific ranges of scale (rather than scale invariance). To capture the complexity of seafloor generating processes, the more versatile concept of multifractals is introduced and explored for seafloor topography description.

Optical wavelet transform and local scaling properties of fractals

The wavelet transform has recently received much attention in the fractals world due to its ability to characterize the scaling properties of multifractal aggregates. This paper is devoted to the optical version of the wavelet transform, with a special emphasis on instrumental aspects. The optical wavelet transform amounts to a series of distinct bandpass filterings of the optical Fourier spectrum of the system under study. One wavelet component is given by one filtered image of the system. The way in which the intensity at any point depends on the filter size directly gives the local scaling behavior around this point.

Invasion percolation on substrates with correlated disorder

Simulations have been carried out to investigate invasion percolation with trapping on two dimensional substrates with a multifractal distribution of bond threshold probabilities. Suprisingly, the spatial correlations in the bond thresholds appear not to influence or influence only slightly the fractal scaling properties. Fractal dimensionalities, D, of about 1.82 for the entire clusters, about 1.45–1.5 for the hull and about 1.4–1.45 for the outer hull were obtained for all bond invasion percolation models with and without spatial correlations. The exponent τ describing the power law distribution of trapped defender cluster sizes has a value of about 1.90 (( D + d)/ d where d is the Euclidean dimensionality). The value for D of 1.4–1.45 for the outer hull is regarded as being consistent with but not in good agreement with the theoretical value of 4 3 for ordinary percolation. The value of about 1.50 obtained for the total hull is much different from the value of 1.75 associated with ordinary percolation hulls. The difference between the value of 1.82 obtained from the entire cluster and the theoretical percolation value of vl is (1.895833) is also regarded as being significant. The simulation results also show that the fractal dimensionality for the total hull is much different for bond and site invasion percolation with trapping on substrates with uncorrelated disorder while the outer hulls have the same fractal dimensionality for both models. For the site invasion percolation model D hull ⋍ 7 4 while for the bond invasion percolation model D hull ⋍D outer hull .

Models for Material Failure and Deformation

Simple computer models have been used to investigate a variety of pattern formation processes associated with material failure and deformation. These models reproduce surprisingly well the characteristic morphologies observed in a wide range of real systems. They provide a sound basis for the development of more realistic models that can be used to develop a better understanding of the mechanical properties of real materials. The present algorithms are adequate for some purposes, but substantial improvements are needed if simulation results are to make a major contribution to our theoretical understanding of the asymptotic fractal scaling and universality properties of patterns generated by failure and deformation phenomena.

LOCAL SCALING PROPERTIES OF FRACTALS OBSERVED IN Ni-Zr ALLOY FILMS

Wavelet transform was performed based on the fractals observed in Ni-Zr alloy films during ion irradiation. The mass distribution measure and the Laplacian potential gradient measure were used to study the local scaling properties of the ion-induced fractals. The strength of singularities at each point was calculated according to the wavelet transform. The densities of the strength of singularities were also deduced and compared with the f-α spectra yielded by multifractal analysis. The results showed that the ion-induced fractals had a wide range of strength of singularities.

Sensing scaled scintillations

We review some of the ways in which the fractal concept has found application in wave-propagation contexts. The scaling properties of fractals in both geometrical and statistical situations are reviewed and the relation to inverse power laws discussed. The relationship among the self-similar scaling properties of fractals, Lévy distributions, and renormalized group theory is explored to provide a simple picture of wave propagation through multiscale media. Finally, the notion of using a wavelet transform in the processing of fractal time series is considered.

Fractal processes in soil water retention

Numerous empirical models exist for soil water retention and unsaturated hydraulic conductivity data. It has generally been recognized that the empirical fitting coefficients in these models are somehow related to soil texture. However, the fact that they are empirical means that elaborate laboratory experiments must be performed for each soil to obtain values for the parameters. Moreover, empirical models do not shed insight into the fundamental physical principles that govern the processes of unsaturated flow and drainage. We propose a physical conceptual model for soil texture and pore structure that is based on the concept of fractal geometry. The motivation for a fractal model of soil texture is that some particle size distributions in granular soils have already been shown to display self‐similar scaling that is typical of fractal objects. Hence it is reasonable to expect that pore size distributions may also display fractal scaling properties. The paradigm that we use for the soil pore size distribution is the Sierpinski carpet, which is a fractal that contains self similar “holes” (or pores) over a wide range of scales. We evaluate the water retention properties of regular and random Sierpinski carpets and relate these properties directly to the Brooks and Corey (or Campbell) empirical water retention model. We relate the water retention curves directly to the fractal dimension of the Sierpinski carpet and show that the fractal dimension strongly controls the water retention properties of the Sierpinski carpet “soil”. Higher fractal dimensions are shown to mimic clay‐type soils, with very slow dewatering characteristics and relatively low fractal dimensions are shown to mimic a sandy soil with relatively rapid dewatering characteristics. Our fractal model of soil water retention removes the empirical fitting parameters from the soil water retention models and provides parameters (fractal dimension) which are intrinsic to the nature of the fractal porous structure. The relative permeability functions of Burdine and Mualem are also shown to be fractal directly from fractal water retention results.

Wavelet transform of fractal aggregates

We apply the wavelet transform for characterizing the geometrical complexity of two-dimensional fractal aggregates. We illustrate the efficiency of this mathematical microscope to capture the local scaling properties of self-similar snowflake fractals. We report on the first unambiguous numerical evidence for the structural self-similarity of Witten and Sander diffusion-limited aggregates (DLA). We emphasize the wavelet transform to be readily applicable to experimental situations.