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PubMed2004

Fractal branching pattern of the monopodial canine airway.

Unlike the human lung, monopodial canine airway branching follows an irregular dichotomized pattern with fractal features. We studied three canine airway molds and found a self-similarity feature from macro- to microscopic scales, which formed a fractal set up to seven scales in the airways. At each fractal scale, lateral branches evenly lined up along an approximately straight main trunk to form three to four two-dimensional structures, and each lateral branch was the monopodial main trunk of the next fractal scale. We defined this pattern as the fractal main lateral-branching pattern, which exhibited similar structures from macro- to microscopic scales, including lobes, sublobes, sub-sublobes, etc. We speculate that it, rather than a mother-daughter pattern, could better describe the actual asymmetrical architecture of the monopodial canine airway.

Animals↗
PubMed2003

Fractal properties of sympathetic nerve discharge.

Fano factor analysis and dispersional analysis were used to characterize time series of single and multifiber spikes recorded from the preganglionic cervical sympathetic nerve and cardiac-related slow-wave activity of the whole postganglionic sympathetic vertebral nerve (VN) in anesthetized cats. Fluctuations in spike counts and interspike intervals for single preganglionic fibers proved to be fractal (i.e., time-scale invariant), as reflected by a power law relationship between indices of the variance of these properties and the window size used to make the measurements. Importantly, random shuffling of the data eliminated the power law relationships. Fluctuations in spike counts in preganglionic multifiber activity also were fractal, as were fluctuations in the height and of the area of cardiac-related slow waves recorded from the whole postganglionic VN. These fractal fluctuations were persistent (i.e., positively correlated), as reflected by a Hurst exponent significantly >0.5. Although fluctuations in the interval between cardiac-related VN slow waves were random, those in the interval between heart beats were fractal and persistent. These results demonstrate for the first time that apparently random fluctuations in sympathetic nerve discharge are, in fact, dictated by a complex deterministic process that imparts "long-term" memory to the system. Whether such time-scale invariant behavior plays a role in generating the fractal component of heart rate variability remains to be determined.

Action Potentials↗
PubMed1999

The use of fractal features from the periphery of cell nuclei as a classification tool.

A polygonization-based method is used to estimate the fractal dimension and several new scalar lacunarity features from digitized transmission electron micrographs (TEM) of mouse liver cell nuclei. The fractal features have been estimated in different segments of 1D curves obtained by scanning the 2D cell nuclei in a spiral-like fashion called "peel-off scanning". This is a venue to separate estimates of fractal features in the center and periphery of a cell nucleus. Our aim was to see if a small set of fractal features could discriminate between samples from normal liver, hyperplastic nodules and hepatocellular carcinomas. The Bhattacharyya distance was used to evaluate the features. Bayesian classification with pooled co-variance matrix and equal prior probabilities was used as the rule for classification. Several single fractal features estimated from the periphery of the cell nuclei discriminated samples from the hyperplastic nodules and hepatocellular carcinomas from normal ones. The outer 25-30% of the cell nuclei contained important texture information about the differences between the classes. The polygonization-based method was also used as an analysis tool to relate the differences between the classes to differences in the chromatin structure.

Animals↗
PubMedDate not supplied

Fractal correlation properties of R-R interval dynamics and mortality in patients with depressed left ventricular function after an acute myocardial infarction.

BACKGROUND: Preliminary data suggest that the analysis of R-R interval variability by fractal analysis methods may provide clinically useful information on patients with heart failure. The purpose of this study was to compare the prognostic power of new fractal and traditional measures of R-R interval variability as predictors of death after acute myocardial infarction. METHODS AND RESULTS: Time and frequency domain heart rate (HR) variability measures, along with short- and long-term correlation (fractal) properties of R-R intervals (exponents alpha(1) and alpha(2)) and power-law scaling of the power spectra (exponent beta), were assessed from 24-hour Holter recordings in 446 survivors of acute myocardial infarction with a depressed left ventricular function (ejection fraction </=35%). During a mean+/-SD follow-up period of 685+/-360 days, 114 patients died (25.6%), with 75 deaths classified as arrhythmic (17.0%) and 28 as nonarrhythmic (6.3%) cardiac deaths. Several traditional and fractal measures of R-R interval variability were significant univariate predictors of all-cause mortality. Reduced short-term scaling exponent alpha(1) was the most powerful R-R interval variability measure as a predictor of all-cause mortality (alpha(1) <0.75, relative risk 3.0, 95% confidence interval 2.5 to 4.2, P<0.001). It remained an independent predictor of death (P<0.001) after adjustment for other postinfarction risk markers, such as age, ejection fraction, NYHA class, and medication. Reduced alpha(1) predicted both arrhythmic death (P<0.001) and nonarrhythmic cardiac death (P<0.001). CONCLUSIONS: Analysis of the fractal characteristics of short-term R-R interval dynamics yields more powerful prognostic information than the traditional measures of HR variability among patients with depressed left ventricular function after an acute myocardial infarction.

Adrenergic beta-Antagonists↗
PubMed1993

Potential of fractal analysis for lesion detection in echographic images.

The application of fractal analysis to parametric imaging of B-mode echograms and to differentiation of echographic speckle textures was investigated. Echograms were obtained from realistic simulations and from a clinical study on diffuse liver disease. The simulations comprised tissue models with randomly positioned scatterers in a 3-D volume in which the number density was varied over a range from 0.5 to 25 mm-3. The clinical echograms comprised both normals and patients with liver cirrhosis. Three methods of estimating the fractal dimension were investigated, two in the spatial image domain and one in the spatial frequency domain. The results of these methods are compared and the applicability and the limitations of texture differentiation using fractal analysis is discussed. The main conclusion is that fractal analysis offers no obvious advantage over statistical analysis of the texture of echographic images. Its use for parametric imaging is further limited by the need to use relatively large windows for local estimation of the fractal dimension.

Fractals↗
PubMed2000

A method to determine the fractal dimension of the cross-sectional jaggedness of the infarct scar edge.

This study describes the use of a shareware software package available from the National Institutes of Health for computing the fractal dimension. Specifically, when fractal analysis is used in its correct context it provides for a quantitative description of the space filling properties of two-dimensional objects. A rabbit model of post myocardial infarction is described where the cross-sectional infarct edge is reconstructed and its jaggedness determined by calculating its fractal dimension via the pixel dilation method. The fractal dimensions of the anterior and posterior lateral infarct edges were calculated to have a mean of 1.16 and 1.29, respectively. In conclusion, the fractal technique can be used to describe the complex jaggedness of the infarct edge. This case study also illustrates the fact that the complexity of an infarcted area is not uniform across the scar. For example, we found that the space filling properties of the anterior and posterior borders of a myocardial infarct can differ by more than 2-fold.

Animals↗
PubMed1995

[Measurement and analysis of monophasic action potentials using fractally coated electrodes--I].

The monophasic action potential (MAP) contains a wealth of information about the stat of the myocardium, which makes it very useful for numerous diagnostic and therapeutic applications in patients with heart disease. The silver-silver chloride electrodes which are currently used for the measurement of MAP have poor long-term stability in contact with biological tissue. This study was therefore undertaken with the aim of investigating the electrochemical behaviour of fractally coated leads in terms of their signal-detection performance. Experience gained with these leads in cardiac pacemakers has already demonstrated the long-term stability and biocompatibility of the fractally coated leads. Present results show that, due to their large electrochemically active surface area, fractally coated leads have a very low impedance over a wide frequency range. The negligible polarization artifact of these leads permits the measurement of cardiac potentials immediately after a stimulus. Fractally coated leads are thus highly suitable for the measurement of MAP, and have clear advantages over Ag/AgCl electrodes. The second part of this study reports on the results of MAP measurements using fractally coated leads.

Electrocardiography↗
PubMed2003

A fractal analysis of human cranial sutures.

OBJECTIVES: Many biological structures are products of repeated iteration functions. As such, they demonstrate characteristic, scale-invariant features. Fractal analysis of these features elucidates the mechanism of their formation. The objectives of this project were to determine whether human cranial sutures demonstrate self-similarity and measure their exponents of similarity (fractal dimensions). DESIGN: One hundred three documented human skulls from the Terry Collection of the Smithsonian Institution were used. Their sagittal sutures were digitized and the data converted to bitmap images for analysis using box-counting method of fractal software. RESULTS: The log-log plots of the number of boxes containing the sutural pattern, N(r), and the size of the boxes, r, were all linear, indicating that human sagittal sutures possess scale-invariant features and thus are fractals. The linear portion of these log-log plots has limits because of the finite resolution used for data acquisition. The mean box dimension, D(b), was 1.29289 +/- 0.078457 with a 95% confidence interval of 1.27634 to 1.30944. CONCLUSIONS: Human sagittal sutures are self-similar and have a fractal dimension of 1.29 by the box-counting method. The significance of these findings includes: sutural morphogenesis can be described as a repeated iteration function, and mathematical models can be constructed to produce self-similar curves with such D(b). This elucidates the mechanism of actual pattern formation. Whatever the mechanisms at the cellular and molecular levels, human sagittal suture follows the equation log N(r) = 1.29 log 1/r, where N(r) is the number of square boxes with sides r that are needed to contain the sutural pattern and r equals the length of the sides of the boxes.

Adolescent↗
PubMed2005

Neuro-immunity in stress-related oral ulcerations: a fractal analysis.

By testing archival paraffinized biopsy blocks obtained from the oral pathology library with immunohistochemistry, we tested the hypothesis that substantial alterations are demonstrable in the cross-talk between sympathetic (VMAT) and para-sympathetic innervation (VAchT), and resident CD3+ T cells in the mucosa from oral lichen planus (OLP), compared to recurrent aphthous stomatitis (RAS) and control biopsies. We quantified fractal dimension and Euclidean dimension of CD3+ cells between the two pathologies, and across the set of CD3+ cells proximal to the vesicles of monoamines transport (VMAT)+ or the vesicles of acetylcholine transport (VAchT)+ innervation, compared to cells relatively distal to the nerve endings. The data show exquisite organization of the punctuate sympathetic and para-sympathetic staining about the resident CD3+ T cells in the OLP lesions, but not in the aphthous lesions or in control mucosa. Fractal analysis reveals that aphthous lesions are characterized by CD3+ T cells of larger size (Euclidean dimensional map), compared to control mucosa. CD3+ T cells in OLP lesions are also found to be significantly larger than those found in control lesions, when they are not proximal to sympathetic or para-sympathetic vesicles. The membrane of CD3+ T cells is overall more complex (fractal dimension) in aphthous lesions, compared to control sections. A similar trend is apparent, albeit not statistically significant, in CD3+ T cells resident in OLP lesions, whether or not they are located proximal to nerve endings. An overall decrease in the ratio of fractal dimension-over-topological dimension was also observed across the pathological lesions, compared to control. Taken together, these data indicate that as CD3+ T cells grow larger in the pathological conditions, they, in effect, stretch their plasma membrane, and that the cells may be at different stage of the cell cycle, relative to their position vis a' vis nerve endings. Because fractal analysis is performed on individual cells, it has the potential of being developed in a novel diagnostic test, as well as a prognostic tool for monitoring the etiology and the course of treatment at the individual cellular level. Our findings also open new frontiers of fundamental, clinical and translational biosciences of OLP.

Cells, Cultured↗
PubMed1993

The fractal geometry of proliferative diabetic retinopathy: implications for the diagnosis and the process of retinal vasculogenesis.

10 retinal vessel patterns with neovascularisation at or near the optic disk (NVD) from eyes of patients with diabetic retinopathy were compared with vascular patterns from 14 normal eyes. The vascular patterns were taken from low angle fundus photographs. After digitizing, the fractal dimensions were calculated by means of the density-density correlation function method. The fractal dimension was found to be significantly higher for vessel patterns with NVD [D = 1.845 +/- 0.056 (m +/- sd)] as compared with the normal control group (D = 1.708 +/- 0.073) (p < 0.001). The fractal dimension of 1.8 appears to be a cutoff value. Higher values may indicate proliferative changes. Under these conditions the sensitivity of the method for the detection of NVD > or = Grade 3 in the Early Treatment Diabetic Retinopathy Study (ETDRS) grading system is 90%. The presence of such NVD in an eye is a "high risk characteristic" for severe visual loss, which requires panretinal laser treatment. Fractal analysis is therefore a possible new strategy for computer assisted "automated" detection and quantification of proliferative diabetic retinopathy. The fractal dimension of the new vessels suggests possible mechanisms involved in retinal vasculogenesis.

Diabetic Retinopathy↗
PubMedDate not supplied

Fractal geometric analysis of the renal arterial tree in infants and fetuses.

Fractal geometry is a useful method of quantitating the space-filling properties of complex objects and has a particular advantage in pediatric pathology because it is independent of organ size. The fractal dimensions of angiographic images of 44 renal arterial trees from 23 consent pediatric autopsies were measured by the box-counting method. The mean fractal dimension was 1.64 and all values were greater than the topological dimension (one), indicating that the renal arterial tree in fetuses and infants has a fractal element to its structure. There was no significant association with size of the kidneys, confirming the size-independent nature of the fractal dimension. There was no significant association with age of the subject, and the mean value was not significantly different from values obtained in studies of adult kidneys, suggesting that the degree of branching, at a lobar and lobular level, does not increase after about the 21st week of gestation. The results are compatible with a diffusion-limited aggregation model of development.

Adult↗
PubMed2003

Effect of temperature and aggregation rate on the fractal dimension of renneted casein aggregates.

This study deals with the influence of chymosin/casein ratios and temperature on the fractal dimension of renneted casein aggregates in diluted milk. The angular dependence of static light scattering, from 50 to 110 degrees, was used to determine fractal dimension from double logarithmic plots of intensity versus scattering vector. No significant effect was observed when varying chymosin/casein ratios from 3.3 to 79.2 microg of chymosin/g of casein at 30 degrees C with a resultant fractal dimension value of 2.20. Temperature, on the other hand, influenced the aggregation mechanisms with fractal dimension values equaled to 2.32, 2.22, and 2.15 at 25, 30, and 35 degrees C, respectively, for a chymosin/casein ratio at 13.2 microg of chymosin/g of casein. The importance of hydrophobic interactions in fractal aggregation of renneted casein particles is discussed.

Animals↗
PubMed2000

Ontogenetic changes in the fractal geometry of the bronchial tree in Rattus norvegicus.

Respiration and metabolism change dramatically over the course of the development of vertebrates. In mammals these changes may be ascribed to organogenesis and differentiation of structures involved in gas exchange and transport and the increase in size. Since young as well as mature individuals must be well-designed if the species is to survive, the physiological changes during the development should be matched with geometrical or structural adjustments of the respiratory system. The aim of this study was to evaluate changes in the fractal geometry of the bronchial tree during the postnatal development of the rat. The average fractal dimension of the bronchial tree of the rats was 1.587, but that of juveniles was larger than that of the adults. We found a significant negative correlation between age and fractal dimension. This correlation could be considered be misleading because of the difficulty of separating age/body size effects. Nevertheless, because fractal dimensions of the bronchial tree of rabbits and humans are known to be similar, 1.58 and 1.57 respectively, the body size effect may be nil. To our knowledge, this is the first report of ontogenetic changes in the fractal dimension of the bronchial tree in mammals.

Animals↗
PubMedDate not supplied

Fractal-assisted EUS image-analysis in the evaluation of variceal eradication after elastic band ligation.

BACKGROUND/AIMS: Endoscopic ultrasonography (EUS) is a strongly operator-dependent method. Mathematical algorithms in image-analysis are likely to enhance diagnostic accuracy. Incomplete eradication of esophageal varices can be demonstrated with EUS. The presence of paraesophageal varices, visible only with EUS, has been recently identified as a predictor of recurrence. The object of this study was to evaluate the usefulness of fractal-geometry-algorithms for the interpretation of EUS images and their impact on treatment decisions. METHODOLOGY: EUS was performed in 5 consecutive patients in order to detect the presence of paraesophageal varices after complete eradication of esophageal varices with elastic banding. Static images were analyzed and the fractal dimension was calculated. Follow-up ranged from 8-12 months. RESULTS: Fractal dimension was related to a) the presence and size of paraesophageal varices b) Child-Pugh Score c) rebleeding rate. Patients with a fractal dimension > 1.5 rebled. CONCLUSIONS: The use of mathematical algorithms derived from fractal geometry enhances interpretation in endosonographic imaging, increases the diagnostic yield and may prove helpful in making individual-tailored treatment decisions.

Adult↗
PubMed2000

Fractal analysis of STM images of lignin polymer obtained by in vitro synthesis.

Lignin, the structural polymer of the plant cell walls, is produced by free radical polymerization of phenolic alcohols, catalyzed by different peroxidases. The mechanism and the structural organization of lignin in the cell have not been completely understood. In this study we applied fractal analysis to images of lignin polymer obtained using scanning tunneling microscope. The analysis showed the regularity of the polymer at different levels of organization. According to the results obtained, at the 95% confidence level, there is no significant difference in the fractal dimension between images representing different organizational levels of lignin. In other words, lignin produced in in vitro conditions has fractal structural organization and, consequently the polymer can be expected to be regular in in vivo conditions. The value of the fractal dimension 1.929 +/- 0.021 is in good agreement with the theoretically predicted value for polyaddition and polycondensation mechanism of polymerization. The mechanism of in vivo lignin synthesis is discussed in terms of various experimental and theoretical evidences. In this paper, we could show that fractal analysis of the lignin polymer is a useful complementary approach to the experimental data collection in structural and phenomenological studies.

Cell Wall↗
PubMed2001

Gating kinetics of potassium channel and effects of nerve growth factors in PC12 cells analyzed with fractal model.

AIM: To study the gating kinetics of voltage-dependent K+ channels i n clonal pheochromocytoma (PC12) cells and the effect of nerve growth factor (NGF) on it. METHODS: Outward currents of K+ channel were recorded in PC12 cells cultured with or without NGF using cell-attached patch-clamp technique. The kinetic features of K+ channel and the effect of NGF on them were analyzed based on t he fractal model. RESULTS: The fractal dimension D for the closed durations of K+ channel in PC12 cells cultured without NGF was proportional to the absolute value of the pipette potential (Vp). Whereas the fractal dimension D' was inversely proportional to it. Under pipette potentials of 0, -30, -50, and -70 mV, the values of D were 1.75, 1.88, 1.95, and 2; and of D' were 1.51, 1.40, 1.34, and 1.23. After the addition of NGF the changes in the fractal dimensions became more complex. At the same pipette potential, fractal dimensions for K+ channel in PC12 cells cultured with NGF for both the open and closed durations were not greater than those in PC12 cells cultured without NGF. The kinetic setpoints for both the close and open durations did not vary with the pipette potential with values as follows: A (without NGF) = 0.29 +\ -0.05, A(with NGF) = 0.71 +\- 0.06, A' without NGF) = 0.110 +\- 0.020, and A' (with NGF) = 0.38 +\- 0.08. CONCLUSION: The voltage dependence of D (without NGF) increases the probability of the channel for remaining closed for long durations as the patch is depolarized, and that of D' (without NGF) decreases the probability of the channel for remaining open for long durations as the patch is depolarized. The addition of NG F in the PC12 cell culture accelerates the dynamic process of the K+ channel occurring over long time scales.

Animals↗
PubMed2003

Health, 'small-worlds', fractals and complex networks: an emerging field.

The importance of 'small-worlds', fractals and complex networks to medicine are discussed. The interrelationship between the concepts is highlighted. 'Small-worlds'--where large populations are linked at the level of the individual have considerable importance for understanding disease transmission. Complex networks where linkages are based on the concept 'the rich get richer' are fundamental in the medical sciences--from enzymatic interactions at the subcellular level to social interactions such as sexual liaisons. Mathematically 'the rich get richer' can be modeled as a power law. Fractal architecture and time sequences can also be modeled by power laws and are ubiquitous in nature with many important examples in medicine. The potential of fractal life support--the return of physiological time sequences to devices such as mechanical ventilators and cardiopulmonary bypass pumps--is presented in the context of a failing complex network. Experimental work suggests that using fractal time sequences improves support of failing organs. Medicine, as a science has much to gain by embracing the interrelated concepts of 'small-worlds', fractals and complex networks. By so doing, medicine will move from the historical reductionist approach toward a more holistic one.

Fractals↗
PubMed2005

[Application of fractal dimension in heart rate variability analysis during head-down tilt].

OBJECTIVE: The paper applies the fractal dimension to heart rate variability (HRV), and discusses the changes of the fractal dimension of cardiovascular system during head down tilt (HDT). METHOD: ECG of 10 subjects were recorded during HDT, the experiment was repeated in each subject. Fractal dimension was used to analyze the HRV. RESULT: Compared to supine before HDT, the standard deviation of all normal to normal RR intervals (SDNN) at 15-20 min, 20-25 min, 25-30 min and restoration increased significantly; the root mean square of successive differences between adjacent normal cycles (rMSSD) at 20-25 min increased significantly; the fractal dimension at 0-5 min, 5-10 min, 10-15 min, 20-25 min, 25-30 min and restoration increased significantly. CONCLUSION: The result shows that HRV and the nonlinearity of cardiovascular system increase during HDT. It is feasible that the fractal dimension can be used for analyzing HRV.

Cardiovascular Physiological Phenomena↗