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PubMed2001

Fast high-resolution magnetic resonance imaging demonstrates fractality of myocardial perfusion in microscopic dimensions.

The fractal nature of heterogeneity of myocardial blood flow and its implications for the healthy and diseased heart is not yet understood. The main hindrance for investigation of blood flow heterogeneity and its role in physiology and pathophysiology is that conventional methods for determination of myocardial perfusion have severe limitations concerning temporal and spatial resolution and invasiveness. In isolated rat hearts, we developed a nuclear magnetic resonance technique that does not depend on contrast agents and in which the apparent longitudinal relaxation time is made perfusion sensitive by selective preparation of the imaging slice. This perfusion-sensitive relaxation time is determined within 40 seconds as a map with a high spatial in-plane resolution of 140x140 microm(2) and a thickness of 1.5 mm. Perfusion imaging was validated with the established microsphere technique. Additionally, the congruence between perfusion-sensitive T:(1) maps and first-pass perfusion imaging was demonstrated. As an application of high-resolution perfusion imaging, fractal analysis of the spatial distribution of perfusion was performed. We were able to demonstrate that the fractality of this distribution exists even in microscopic dimensions. Vasodilation by nitroglycerin modulated the fractal pattern of perfusion, and the decrease of the fractal dimension indicated a shift toward homogeneity. This implies that parameters of the fractal distribution depend on the microvascular tone rather than on anatomic preformations; ie, fractality is a functional characteristic of perfusion.

Animals↗

PubMed2005

Physiological background of the loss of fractal heart rate dynamics.

BACKGROUND: Altered fractal heart rate (HR) dynamics occur during various disease states, but the physiological background of abnormal fractal HR behavior is not well known. We tested the hypothesis that the fractal organization of human HR dynamics is determined by the balance between sympathetic and vagal outflow. METHODS AND RESULTS: A short-term fractal scaling exponent (alpha1) of HR dynamics, analyzed by the detrended fluctuation analysis (DFA) method, and the high-frequency (HF) and low-frequency (LF) spectral components of R-R intervals (0.15 to 0.4 Hz; n=13), along with muscle sympathetic nervous activity (MSNA) from the peroneus nerve (n=11), were assessed at rest and during cold face and cold hand immersion in healthy subjects. During cold face immersion, HF power increased (from 6.9+/-1.3 to 7.6+/-1.2 ln ms2, P<0.01), as did MSNA (from 32+/-17 to 44+/-14 bursts/100 heartbeats, P<0.001), and LF/HF ratio decreased (P<0.01). Cold hand immersion resulted in a similar increase in MSNA (from 34+/-17 to 52+/-19 bursts/100 heartbeats, P<0.001) but a decrease in HF spectral power (from 7.0+/-1.3 to 6.5+/-1.1 ln ms2, P<0.05) and an increase in the LF/HF ratio (P<0.05). The fractal scaling index alpha1 decreased in all subjects (from 0.85+/-0.27 to 0.67+/-0.30, P<0.0001) during cold face immersion but increased during cold hand immersion (from 0.77+/-0.22 to 0.97+/-0.20, P<0.01). CONCLUSIONS: The fractal organization of human HR dynamics is determined by a delicate interplay between sympathetic and vagal outflow, with the breakdown of fractal HR behavior toward more random dynamics occurring during coactivation of sympathetic and vagal outflow.

Adult↗

PubMed2006

[Fractal features of soil particle size distribution in degraded sandy grassland during reclamation and enclosure].

Based on the analysis of 42 soil samples from the grassland reclaimed and enclosed different years, this paper studied the fractal features of soil particle size distribution and the relationships between fractal features and soil properties. The results showed that the fractal dimension of soil particle size distribution in 0 approximately 30 cm soil layer ranged from 2.378 in reclaimed land to 2.588 in non-grazed enclosure, while that in 0 approximately 15 cm soil layer decreased from 2.441 in grazed grassland to 2.387 after 20 years of reclamation, and increased to 2.588 after 11 years of enclosure. No significant changes were found in fractal dimension in 15 approximately 30 cm soil layer between reclamation and enclosure. Statistic analysis indicated that the fractal dimension could objectively reflect soil texture, mainly the content of clay, and then silt content. There existed a remarkable quantitative correlation between fractal dimension and soil physical, chemical and biological properties, which meant that the fractal dimension could be used as a comprehensive and quantitative index in evaluating soil properties and grassland degradation degree.

China↗

PubMed2006

[Application of fractal theory to spectral signal recognition].

The fractal theory is a discipline that studies a kind of irregular and chaotic object with similarity of its part and whole. The fractal dimension is a basic index mark of irregularity and self-similarity in the fractal theory. The present paper takes the spectral signal according with Lambert-beer' law as the object, introduces the basic theory of fractal geometry in brief, puts forward the method of fractal dimension as the feature of spectral signal recognition, makes use of reconstructing phase space to gain the fractal dimension of spectral signal, compares different values of the fractal dimension to recognize different spectral signal, and gives an example for explanation.

English Abstract↗

PubMed2025

Cardiac CT fractal analysis of LV noncompaction and common cardiomyopathies.

BACKGROUND: Left ventricular noncompaction (LVNC), or hypertrabeculation, is a myocardial condition that remains challenging to diagnose and differentiate from other cardiomyopathies. This study evaluated the ability of cardiac CT to differentiate between LVNC, hypertrophic cardiomyopathy (HCM), dilated cardiomyopathy (DCM), and controls using fractal analysis of LV trabeculae. METHODS: Subjects with LVNC, HCM, DCM, as well as controls, who underwent coronary CT angiography were included. LV trabecular structure was quantified using fractal analysis on a stack of 15 short-axis CT images. For each subject, maximum (FDmax) and average (FDglobal) fractal dimensions were reported. A subset of subjects also had clinically acquired cardiac MRI (CMR) exams for comparison. One-way ANOVA, Pearson correlation, and Bland-Altman analysis were used for statistical analysis. RESULTS: The study included 313 subjects (median age: 58.8 [48.1-68.0] years, 153 male) categorized into Control (89), LVNC (46), HCM (106), and DCM (72) cohorts. FDmax was significantly higher in LVNC (1.379 ± 0.047) than in Control (1.305 ± 0.033), HCM (1.321 ± 0.040), and DCM (1.344 ± 0.054) cohorts; all p < 0.001. Similarly, FDglobal was significantly higher in LVNC (1.279 ± 0.041) than in the other cohorts; all p < 0.05. In a subset of 132 subjects with both CT and CMR exams, fractal dimensions from the two modalities were strongly correlated (r = 0.63, p < 0.0001), with CT-derived values being higher (1.337 ± 0.049 vs. 1.262 ± 0.045, p < 0.0001). CONCLUSIONS: CT-derived fractal dimensions of LV trabecular structure were significantly higher in LVNC compared to control subjects, HCM, and DCM. CT-derived fractal dimensions strongly correlated with, but were higher than, those from cardiac MRI in the same subjects.

Humans↗

PubMed2001

Fractal analysis of lumbar vertebral cancellous bone architecture.

Osteoporosis is characterized by bone mineral density (BMD) decreasing and spongy bone rearrangement with consequent loss of elasticity and increased bone fragility. Quantitative computed tomography (QCT) quantifies bone mineral content but does not describe spongy architecture. Analysis of trabecular pattern may provide additional information to evaluate osteoporosis. The aim of this study was to determine whether the fractal analysis of the microradiography of lumbar vertebrae provides a reliable assessment of bone texture, which correlates with the BMD. The lumbar segment of the spine was removed from 22 cadavers with no history of back pain and examined with standard x-ray, traditional tomography, and quantitative computed tomography to measure BMD. The fractal dimension, which quantifies the image fractal complexity, was calculated on microradiographs of axial sections of the fourth lumbar vertebra to determine its characteristic spongy network. The relationship between the values of the BMD and those of the fractal dimension was evaluated by linear regression and a statistically significant correlation (R = 0.96) was found. These findings suggest that the application of fractal analysis to radiological analyses can provide valuable information on the trabecular pattern of vertebrae. Thus, fractal dimensions of trabecular bone structure should be considered as a supplement to BMD evaluation in the assessment of osteoporosis.

Adult↗

PubMed2001

Fractal dimension of trabecular bone: comparison of three histomorphometric computed techniques for measuring the architectural two-dimensional complexity.

Trabecular bone has been reported as having two-dimensional (2-D) fractal characteristics at the histological level, a finding correlated with biomechanical properties. However, several fractal dimensions (D) are known and computational ways to obtain them vary considerably. This study compared three algorithms on the same series of bone biopsies, to obtain the Kolmogorov, Minkowski-Bouligand, and mass-radius fractal dimensions. The relationships with histomorphometric descriptors of the 2-D trabecular architecture were investigated. Bone biopsies were obtained from 148 osteoporotic male patients. Bone volume (BV/TV), trabecular characteristics (Tb.N, Tb.Sp, Tb.Th), strut analysis, star volumes (marrow spaces and trabeculae), inter-connectivity index, and Euler-Poincaré number were computed. The box-counting method was used to obtain the Kolmogorov dimension (D(k)), the dilatation method for the Minkowski-Bouligand dimension (D(MB)), and the sandbox for the mass-radius dimension (D(MR)) and lacunarity (L). Logarithmic relationships were observed between BV/TV and the fractal dimensions. The best correlation was obtained with D(MR) and the lowest with D(MB). Lacunarity was correlated with descriptors of the marrow cavities (ICI, star volume, Tb.Sp). Linear relationships were observed among the three fractal techniques which appeared highly correlated. A cluster analysis of all histomorphometric parameters provided a tree with three groups of descriptors: for trabeculae (Tb.Th, strut); for marrow cavities (Euler, ICI, Tb.Sp, star volume, L); and for the complexity of the network (Tb.N and the three D's). A sole fractal dimension cannot be used instead of the classic 2-D descriptors of architecture; D rather reflects the complexity of branching trabeculae. Computation time is also an important determinant when choosing one of these methods.

Cluster Analysis↗

PubMed2006

Neuronal differentiation and synapse formation in the space-time with temporal fractal dimension.

An improvement of the Waliszewski and Konarski approach ([2002] Synapse 43:252-258) to determine the temporal fractal dimension b(t) and scaling factor a(t) for the process of neuronal differentiation and synapse formation in the fractal space-time is presented. In particular the analytical formulae describing the time-dependence of b(t)(t) and a(t)(t), which satisfy the appropriate boundary conditions for t-->0 and t-->infinity, are derived. They have been used to determine the temporal fractal dimension and scaling factor from the two-parametric Gompertz function fitted to experimental data obtained by Jones-Villeneuve et al. ([1982] J Cell Biol 94:253-262) for embryonal carcinoma P19 cells treated by retinoic acid. The results of the calculations differ from those obtained previously by making use of the three- and four-parametric Gompertz function as well as other S-shape functions (Chapman, Hill, Logistic, Sigmoid) evaluated by the fitting of the experimental curve. The temporal fractal dimension can be used as a numerical measure of the neuronal complexity emerging in the process of differentiation, which can be related to the morphofunctional cell organization. A hypothesis is formulated that neuronal differentiation and synapse formation have a lot in common with the process of tumorigenesis. They are qualitatively described by the same Gompertz function of growth and take place in the fractal space-time whose mean temporal fractal dimension is lost during progression.

Algorithms↗

PubMed2001

A kinetic study of analyte-receptor binding and dissociation, and dissociation alone, for biosensor applications: a fractal analysis.

A fractal analysis is presented for (a) analyte-receptor binding and dissociation kinetics and (b) dissociation kinetics alone for biosensor applications. Emphasis is placed on dissociation kinetics. 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. A single-fractal analysis is adequate to model the dissociation kinetics in the examples presented. Predictive relationships developed for the dissociation rate coefficient(s) as a function of the analyte concentration are of particular value since they provide a means by which the dissociation rate coefficients may be manipulated. Relationships are also presented for the binding and dissociation rate coefficients as a function of their corresponding fractal dimension, D(f), 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 D(f)) leads to an increase in the binding and in the dissociation rate coefficient.

Animals↗

PubMed2001

Fractal methods to analyze ion channel kinetics.

We describe the traditional nonfractal and the new fractal methods used to analyze the currents through ion channels in the cell membrane. We discuss the hidden assumptions used in these methods and how those assumptions lead to different interpretations of the same experimental data. The nonfractal methods assumed that channel proteins have a small number of discrete states separated by fixed energy barriers. The goal was to determine the parameters of the kinetic diagram, which are the number of states, the pathways between them, and the kinetic rate constants of those pathways. The discovery that these data have fractal characteristics suggested that fractal approaches might provide more appropriate tools to analyze and interpret these data. The fractal methods determine the characteristics of the data over a broad range of time scales and how those characteristics depend on the time scale at which they are measured. This is done by using a multiscale method to accurately determine the probability density function over many time scales and by determining how the effective kinetic rate constant, the probability of switching states, depends on the effective time scale at which it is measured. These fractal methods have led to new information about the physical properties of channel proteins in terms of the number of conformational substates, the distribution of energy barriers between those states, and how those energy barriers change with time. The new methods developed from the fractal paradigm shifted the analysis of channel data from determining the parameters of a kinetic diagram to determining the physical properties of channel proteins in terms of the distribution of energy barriers and/or their time dependence.

Animals↗

PubMed1993

Characterisation of the neovascularisation process in diabetic retinopathy by means of fractal geometry: diagnostic implications.

The neovascularisation formation and regression process of the peripheral retina in diabetic retinopathy was studied by means of fractal analysis. The fractal dimension of the local retinal vessel pattern was calculated to be significantly lower before formation of relevant neovascularisations than 2.5 years later, after formation of strong preretinal neovascularisations. Another year later the new vessels had regressed partially and the fractal dimension was again significantly reduced. This behaviour is almost independent of the representation of the vessel thickness during calculation. Since the retinal vasculature is a fractal, the fractal dimension appears as the "natural" measure of proliferative retinal vessel changes. It is demonstrated that the fractal dimension can be applied to characterise proliferative diabetic retinopathy. These features offer the possibility for computer-driven ("automated") quantitative characterisation of the treatment effect in proliferative diabetic retinopathy and possibly automated detection of proliferative diabetic retinopathy in the future. The limitations of the method are discussed.

Diabetic Retinopathy↗

PubMed1998

Classification of foetal heart rate sequences based on fractal features.

Visual inspection of foetal heart rate (FHR) sequences is an important means of foetal well-being evaluation. The application of fractal features for classifying physiologically relevant FHR sequence patterns is reported. The use of fractal features is motivated by the difficulties exhibited by traditional classification schemes to discriminate some classes of FHR sequence and by the recognition that this type of signal exhibits features on different scales of observation, just as fractal signals do. To characterise the signals by fractal features, two approaches are taken. The first modes the FHR sequences as temporal fractals. The second uses techniques from the chaos-theory field and aims to model the attractor based on FHR sequences. The fractal features determined by both approaches are used to design a Bayesian classification scheme. Classification results for three classes are presented; they are quite satisfactory and illustrate the importance of this type of methodology.

Bayes Theorem↗

PubMed1996

Antigen-antibody binding kinetics for biosensors. Changes in the fractal dimension (surface roughness) and in the binding rate coefficient.

The diffusion-limited binding kinetics of antigen in solution to antibody immobilized on a biosensor surface is analyzed within a fractal framework. Changes in the fractal dimension, Df observed are in the same and in the reverse directions as the forward binding rate coefficient k. For example, an increase in the concentration of the isoenzyme human creatine kinase isoenzyme MB form (CK-MB) (antigen) solution from 0.1 to 50 ng/mL and bound to anti-CK-MB antibody immobilized on fused silica fiber rods leads to increases in the fractal dimension Df from 0.294 to 0.5080, and in the forward binding rate coefficient k from 0.1194 to 9.716, respectively. The error in the fractal dimension Df decreases with an increase in the CK-MB isoenzyme concentration in solution. An increase in the concentration of human chorionic gonadotrophin (hCG) in solution from 4000 to 6000 mIU/mL hCG and bound to anti-hCG antibody immobilized on a fluorescence capillary fill device leads to a decrease in the fractal dimension Df from 2.6806 to 2.6164, and to an increase in the forward binding rate coefficient k from 3.571 to 4.033, respectively. The different examples analyzed and presented together indicate one means by which the forward binding rate coefficient k may be controlled, that is by changing the fractal dimension or the "disorder' on the surface. The analysis should assist in helping to improve the stability, the sensitivity, and the response time of biosensors.

Antibodies↗

PubMed1998

Fractal analysis of trabecular bone texture on radiographs: discriminant value in postmenopausal osteoporosis.

Trabecular bone microarchitecture cannot be routinely evaluated. We have developed and validated a fractal analysis of trabecular bone texture on calcaneus radiographs. The aim of this work was to evaluate the ability of the fractal analysis to discriminate a group of 39 postmenopausal women with osteoporotic (OP) vertebral crush fractures (68.0 +/- 10.8 years) from an age-matched control group of 39 women (68.0 +/- 10.7 years). The value of the fractal analysis was compared with the value of the femoral neck bone mineral density (FNBMD) and trochanteric bone mineral density (TRBMD). The result is expressed by the parameter Hmean (Hmean = 2 - fractal dimension). Hmean value was 0.691 +/- 0.050 in the OP group versus 0.739 +/- 0.024 in the controls, while FNBMD was 0.598 +/- 0.113 g/cm2 versus 0.645 +/- 0.109 g/cm2 and TRBMD was 0.512 +/- 0.108 g/cm2 versus 0.594 +/- 0.106 g/cm2 respectively. The statistical significance of the Hmean test (p < 0.0001) was higher than for FNBMD (p < 0.05) and for TRBMD (p = 0.0004). We used a receiver operating characteristic (ROC) curve to check this superiority. The area under the ROC curve was 0.824 for Hmean, 0.633 for FNBMD and 0.727 for TRBMD. This superiority of the Hmean ROC curve was statistically significant versus FNBMD, but not versus TRBMD. In a second analysis, we studied the subgroups of OP patients and controls with overlapping FNBMD or TRBMD values to check whether the fractal dimension test could be discriminant in these subgroups. Significant statistical differences were found for Hmean between OP patients and controls in the overlapping subgroup for FNBMD or TRBMD (respectively p = 0.006 and p < 0.02). These data confirm that the fractal analysis of texture on calcaneus radiographs is able to discriminate OP patients with vertebral crush fracture from controls. This discrimination was stronger than that obtained by FNBMD or TRBMD alone. It was also present when we compared subgroups with overlapping values of FNBMD or TRBMD.

Absorptiometry, Photon↗

PubMed1999

Fractal analysis of proximal femur radiographs: correlation with biomechanical properties and bone mineral density.

Conventional radiography and fractal analysis were used to quantify trabecular texture patterns in human femur specimens and these measures were used in conjunction with bone mineral density (BMD) to predict bone strength. Radiographs were obtained from 51 human femur specimens (25 male, 26 female). The radiographs were analyzed using three different fractal geometry based techniques, namely semi-variance, surface area and Fourier analysis. Maximum compressive strength (MCS) and shear stress (MSS) were determined with a material testing machine: BMD was measured using quantitative computed tomography (QCT). MCS and MSS both correlated significantly with BMD (MCS: R = 0.49-0.54; MSS: R = 0.69-0.72). Fractal dimension also correlated significantly with both biomechanical properties (MCS: R = 0.49-0.56; MSS: R = 0.47-0.54). Using multivariate regression analysis, the fractal dimension in addition to BMD improved correlations versus biomechanical properties. Both BMD and fractal dimension showed statistically significant correlation with bone strength. The fractal dimension provided additional information beyond BMD in correlating with biomechanical properties.

Adult↗

PubMed2004

Fractal dimension in butterflies' wings: a novel approach to understanding wing patterns?

The geometrical complexity in the wings of several, taxonomically different butterflies, is analyzed in terms of their fractal dimension. Preliminary results provide some evidence on important questions about the (dis)similarity of the wing patterns in terms of their fractal dimension. The analysis is restricted to two groups which are widely used in the literature as typical examples of mimicry, and a small number of unrelated species, thus implying the consideration of only a fraction of the wing pattern diversity. The members of the first mimicry ring, composed by the species Danaus plexippus (better known as the monarch butterfly), and the two subspecies Basilarchia archippus obsoleta (or northern viceroy) and Basilarchia archippus hoffmanni (or tropical viceroy), are found to have a very similar value for the fractal dimension of their wing patterns, even though they do not look very similar at first sight. It is also found that the female of another species (Neophasia terlootii), which looks similar to the members of the previous group, does not share the same feature, while the Lycorea ilione albescens does share it. For the members of the second group of mimicry related butterflies, the Greta nero nero and the Hypoleria cassotis, it is shown that they also have very close values for the fractal dimension of their wing patterns. Finally, it is shown that other species, which apparently have very similar wing patterns, do not have the same fractal dimension. A possible, not completely tested hypothesis is then conjectured: the formation of groups by individuals whose wing patterns have an almost equal fractal dimension may be due to the fact that they do share the same developmental raw material, and that this common feature is posteriorly modified by natural selection, possibly through predation.

Animals↗

PubMed2002

Fractal and nonfractal analysis of cell images: comparison and application to neuronal dendritic arborization.

Neurons of the rat spinal cord were stained using the Golgi impregnation method. Successfully impregnated neurons from laminae II, III, and VI were subjected to fractal and nonfractal analyses. Fractal analysis was performed using length-related techniques. Since an application of fractal methods to the analysis of dendrite arbor structures requires caution, we adopted as appropriate a nonfractal method proposing a generalized power-law model with two main nonfractal parameters: (i) the anfractuosity, characterizing the degree of dendritic deviation from straight lines; and (ii) an estimate of the total length of arbor dendrites. The anfractuosity can distinguish between two sets of drawings where the fractal methods failed. We also redefine some basic concepts of fractal geometry, present the ruler-counting method, and propose a new definition of fractal dimension.

Animals↗

PubMed2006

Fractal michaelis-menten kinetics under steady state conditions: Application to mibefradil.

PURPOSE: To provide the first application of fractal kinetics under steady state conditions to pharmacokinetics as a model for the enzymatic elimination of a drug from the body. MATERIALS AND METHODS: A one-compartment model following fractal Michaelis-Menten kinetics under a steady state is developed and applied to concentration-time data for the cardiac drug mibefradil in dogs. The model predicts a fractal reaction order and a power law asymptotic time-dependence of the drug concentration, therefore a mathematical relationship between the fractal reaction order and the power law exponent is derived. The goodness-of-fit of the model is assessed and compared to that of four other models suggested in the literature. RESULTS: The proposed model provided the best fit to the data. In addition, it correctly predicted the power law shape of the tail of the concentration-time curve. CONCLUSION: A simple one-compartment model with steady state fractal Michaelis-Menten kinetics describing drug elimination from the body most accurately describes the pharmacokinetics of mibefradil in dogs. The new fractal reaction order can be explained in terms of the complex geometry of the liver, the organ responsible for eliminating the drug.

Algorithms↗