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PubMed1997

The effect of varying the region of interest on calculations of fractal index.

OBJECTIVES: To compare the effect of using regions of interest (ROIs) of different size and shape on the fractal index of alveolar bone. STUDY DESIGN: Two sets of clinical posterior bitewing radiographs were used to calculate the fractal index (S). Two comparisons were made. First, S was calculated from large interdental ROIs that included small amounts of root structure and compared with S from small ROIs that included no root structures. Then S was calculated from large interdental ROIs (similar to those used for the first set) and compared with S calculated from ROIs that included nearly all of the mandibular alveolar bone (and adjacent root) present on the bitewing. RESULTS: For the first comparison, paired t-tests showed that fractal indices calculated with the large ROIs were significantly different from the respective indices calculated from the small ROIs (P < 0.001). For the second comparison, the fractal indices calculated from the large quadrant ROIs were not significantly different from those calculated from the large ROIs (P = 0.120). CONCLUSION: ROI size and shape may affect the results of fractal analysis of alveolar bone.

Alveolar Process↗
PubMed1999

Fractal dimension calculated from two types of region of interest.

OBJECTIVES: To compare the effect of using two region of interest (ROI) types on calculating the fractal dimensions of trabecular bone during simulated osteoporosis. METHODS: Ten 5 mm thick cross-sections from the long bone of a cow were progressively decalcified in 0.1 N Hcl for 5, 10, 20, 30, 60 and 90 min intervals, and radiographed using 0 degrees projection angle in a specially designed device. Two types of ROI (the ruggedness of the boundary and bone profiles) were placed on each digital image. Fractal dimensions and variance in mean pixel intensity were computed from each ROI using the caliper method in ImageFractal (National Institutes of Health, Washington, DC, USA). Correlation analysis quantified the relationship between changes in variance and fractal dimensions of the two types of ROI. RESULTS: A Strong correlation (r=0.90 approximately 0.98, P</=0.0026) was found between variance in mean pixel intensity and decreasing fractal dimensions calculated from bone profiles. There was no consistency in the ruggedness of the boundary and increasing decalcification. CONCLUSIONS: The fractal dimensions used to characterize the ruggedness of a boundary of trabecular bone in a two-dimensional ROI are inadequate for the detection of osteoporosis, but those calculated from bone profiles may be a sensitive descriptor of trabecular bone structure.

Animals↗
PubMed2000

Fractal properties of bone.

OBJECTIVE: To introduce the applications of fractal analysis of bone in medical imaging in general and dental radiographs in particular. RESULTS: Various methods for measuring fractal dimension have been used to compare normal with osteoporotic bone with contradictory results. This disparity may be attributed to differences in the anatomical sites studied and differences in methods used to obtain the 2D images. However, differences in methods used to measure fractal dimension can also be responsible. CONCLUSION: When fractal dimension is used to study bone it must be remembered that all stages in the analytical chain have an impact on the results. Thus, to obtain more conclusive results, studies on the fractal dimension of bone should be carefully designed and individual methods thoroughly evaluated.

Algorithms↗
PubMed2005

The fractal structure of polycation-DNA complexes.

We used static light scattering to obtain new measurements on the internal structure of aggregated non-viral gene-delivery particles in colloidal suspension. The vector particles are prepared by charge neutralization of plasmid DNA either by poly-L-lysine or by a Lipofectin/integrin-targeting peptide. We use established theories of the stability of colloidal particles and fractal concepts to explain the aggregation processes and demonstrate the existence of a new property (fractal dimension) of the aggregated vector particles. Aggregation is shown to produce particles with fractal dimensions in the range between 1.8 and 2.4; the former suggests a loose three-dimensional structure and the latter characterizes an aggregation process that leads to the formation of particles with tightly packed structures. We show that the fractal dimension of the vector particles is sensitive to changes in physicochemical conditions (ionic strength) of the buffer solution and propose that fractal dimension may provide a useful means of monitoring the physical state of non-viral delivery-vector particles during preparation and storage.

Cations↗
PubMed2001

Effects of pharmacological adrenergic and vagal modulation on fractal heart rate dynamics.

Breakdown of short-term fractal-like behaviour of HR indicates an increased risk for adverse cardiovascular events and mortality, but the pathophysiological background for altered fractal HR dynamics is not known. Our aim was to study the effects of pharmacological modulation of autonomic function on fractal correlation properties of heart rate (HR) variability in healthy subjects. Short-term fractal scaling exponent (alpha1) along with spectral components of HR variability were analysed during the following pharmacological interventions in healthy subjects: (i) noradrenaline (NE) infusion (n=22), (ii) NE infusion after phentolamine (PHE) (n=8), (iii) combined NE + adrenaline (EPI) infusion (n=12), (iv) vagal blockade with high dose of atropine (n=10), (v) and vagal activation by low dose of atropine (n=10). Then alpha1 decreased progressively during the incremental doses of NE (from 0.85 +/- 0.250 to 0.55 +/- 0.23, P<0.0001). NE also decreased the average HR (P<0.001) and increased the high frequency spectral power (P<0.001). Vagal blockade with atropine increased the alpha1 value (from 0.82 +/- 0.22 to 1.24 +/- 0.41, P<0.05). Combined NE + EPI infusion and vagal activation with a low dose atropine did not result in any changes in alpha1, and alpha-adrenergic blockade by PHE did not completely reverse the effects of NE on alpha1. Increased levels of circulating NE result in reduction of short-term correlation properties of HR dynamics. The results suggest that coactivation of cardiac vagal outflow at the time of high levels of a circulating sympathetic transmitter explains the breakdown of fractal-like behaviour of human HR dynamics.

Adrenergic Fibers↗
PubMed2002

Fractal analysis of mandibular bony healing after orthognathic surgery.

OBJECTIVES: We evaluated the radiographic changes to the operational sites after orthognathic surgery by using fractal analysis. STUDY DESIGN: Panoramic radiographs from 35 patients who underwent orthognathic surgery on the mandible without any complications during the osseous healing process were selected. The radiographs taken before the operation (stage 0) and 1 or 2 days (stage 1), 1 month (stage 2), 6 months (stage 3), and 12 months (stage 4) after the operation were digitized at 600 dpi with 256 gray levels. The fractal dimension was calculated by means of a tile-counting method in the region of interest centered on the operational site and was statistically analyzed according to its stages. RESULTS: The fractal dimension was decreased immediately after the operation and increased gradually according to the time lapse (P <.05). The fractal dimension in the region of interest 12 months postoperatively was similar to that preoperatively. CONCLUSION: This result suggests that fractal dimension can be used to evaluate the bony healing process after orthognathic surgery.

Adult↗
PubMed2006

Deposition of fractal-like aerosol aggregates in a model of human nasal cavity.

Toxicity of diesel exhaust is related to the inhalation of nano-sized fractal-like aerosol aggregates. Their complex behavior (in comparison to spherical particles) should be taken into account in deposition modeling. The deposition of aerosol fractal-like aggregates in the model of a human nose was studied numerically for the flow rate corresponding to breathing conditions. The simplified geometry of the human nasal replica was implemented in the computational fluid dynamics (CFD) code (FLUENT) used for calculation of the three-dimensional airflow structure. A Brownian dynamic (BD) algorithm was applied for determination of the aggregates deposition in the nasal cavity during inhalation. These calculations were carried out for several populations of aggregates. The values of parameters used in the BD simulations for characterization of fractal-like aggregates, that is, fractal dimension (Df) and the radius of gyration (Rg), were in the range of 1.7-2.1 and 0.24-0.36 microm, respectively. These are the representative values for soot aggregates emitted from diesel engines. The results of computation show approximately 20% penetration of submicrometer aggregates through the nose and a weak dependence of deposition efficiency on Df and Rg values. The proposed methodology may lead to a more realistic description of deposition of nonspherical aerosol particles in the respiratory system. A more sophisticated approach for description of fractal-like aggregates dynamics is suggested for future studies.

Aerosols↗
PubMed2006

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

A fractal analysis is used to model the binding and dissociation kinetics of connective tissue interstitial glucose, adipose tissue interstitial glucose, insulin, and other related analytes on biosensor surfaces. The analysis provides insights into diffusion-limited analyte-receptor reactions occurring on heterogeneous biosensor surfaces. Numerical values obtained for the binding and the dissociation rate coefficients are linked to the degree of heterogeneity or roughness [fractal dimension (D(f))] present on the biosensor chip surface. The binding and dissociation rate coefficients are sensitive to the degree of heterogeneity on the surface. For example, for the binding of plasma insulin, as the fractal dimension value increases by a factor of 2.47 from D(f1)=0.6827 to D(f2)=1.6852, the binding rate coefficient increases by a factor of 4.92 from k(1)=1.0232 to k(2)=5.0388. An increase in the degree of heterogeneity on the probe surface leads to an increase in the binding rate coefficient. A dual-fractal analysis is required to fit the binding kinetics in most of the cases presented. A single fractal analysis is adequate to describe the dissociation kinetics. Affinity (ratio of the binding to the dissociation rate coefficient) values are also presented. Interferents for glucose, such as uric acid and ascorbic acid, were also detected by using glucose biosensors based on carbon nanotube (CNT) nanoelectrode ensembles (NEEs) (Lin Y, Lu F, Tu Y, Ren Z).

Adipose Tissue↗
PubMed1993

Fractal analysis of foetal heart rate variability.

Current methods for analysing foetal heart rate (FHR) patterns have yet to meet their full potential in the recognition of hypoxia in the foetus. Following the recent suggestion that fractal analysis can be applied to FHR recordings, the current paper describes a method for distinguishing two simultaneous fractal dimensions in FHR variation. An irregular line was plotted from 2500 consecutive foetal heart beat to beat intervals derived from an ultrasound source. A window of 500 intervals was moved along the line in steps of 20 intervals. At each step the Richardson technique was used to make estimates of the length of the line within the window using 40 different ruler lengths. When the estimates were plotted against the ruler lengths on log-log axes the resulting curve exhibited two distinct linear regions, each demonstrating an inverse power relationship. From the two slopes the fractal dimensions were derived for unspecified low- and high-frequency FHR variation in the current window. The values of both fractal dimensions were plotted simultaneously with the irregular FHR line and were found to accord with perceived changes in FHR variation. The method described is simply a measure of the irregularity in a series of foetal heart beat to beat intervals: the existence of fractal properties in the irregular line does not of itself imply underlying deterministic dynamics (e.g. chaos). This new method of observing FHR variability requires no preprocessing of the measured data, which are all taken into account. Not only does it represent a method for studying normal foetal behaviour but also has potential as a sensitive indicator of impending foetal compromise.

Female↗
PubMed2004

Analysis of sleep-stage characteristics in full-term newborns by means of spectral and fractal parameters.

STUDY OBJECTIVES: In this work, we studied the behavior of the fractal dimension during each of the neonatal electroencephalogram (EEG) sleep phases and during the awake state, comparing the results with those of the classical spectral parameters and with zero crossing values. DESIGN: Fractal dimension, zero crossing, and spectral parameters of the EEG bands were determined for each 2-second frame of the EEG sleep-time series. Eight channels of each EEG recording were examined. PARTICIPANTS: Twenty healthy full-term newborns (10 boys and 10 girls) with normal psychomotor development evaluated at 24 and 36 months of age, were chosen to participate in this study. MEASUREMENTS AND RESULTS: Fractal analysis showed that where rhythmic and regular activity are present, as during quiet sleep, the fractal dimension is low and rises when bioelectric activity is more variable and complex, reaching its maximum value during wakefulness. The discriminative value of this parameter was similar to that of some spectral bands. CONCLUSIONS: This work was an initial attempt to apply techniques derived from the nonlinear deterministic studies used to evaluate system complexity, to the neonatal EEG, in order to acquire a normative database that can be used as a reference in neurological pathologies. Fractal dimension alone or together with zero crossing and theta and delta bands could be used for computerized discrimination of neonatal EEG sleep phases.

Cerebral Cortex↗
PubMed1996

Classification of lung tumors on chest radiographs by fractal texture analysis.

RATIONALE AND OBJECTIVES: The authors determine the usefulness of fractal texture analysis in classification of lung tumors on chest radiographs. METHODS: A method of fractal texture analysis was applied for classification of lung tumors on digitized chest radiographs. It was performed in 15 cases of benign lesions and in 26 cases of malignant tumors. All lesions were proved by histology. For classification, the fractal distances of the tumors were calculated and an energy measure related to the variances as a second characteristic was applied. The results were compared with those of a conventional classification method based on the co-occurrence matrix. RESULTS: Although the conventional method failed in classification of lung tumors, a clear separation between benign and malignant lesions was attained by fractal analysis. A differentiation between primary lung cancer and metastases was not possible. CONCLUSIONS: The results of our study demonstrate the usefulness of fractal texture analysis for classification of lung tumors on chest radiographs.

Fractals↗
PubMed2005

Fractals analysis of cardiac arrhythmias.

Heart rhythms are generated by complex self-regulating systems governed by the laws of chaos. Consequently, heart rhythms have fractal organization, characterized by self-similar dynamics with long-range order operating over multiple time scales. This allows for the self-organization and adaptability of heart rhythms under stress. Breakdown of this fractal organization into excessive order or uncorrelated randomness leads to a less-adaptable system, characteristic of aging and disease. With the tools of nonlinear dynamics, this fractal breakdown can be quantified with potential applications to diagnostic and prognostic clinical assessment. In this paper, I review the methodologies for fractal analysis of cardiac rhythms and the current literature on their applications in the clinical context. A brief overview of the basic mathematics of fractals is also included. Furthermore, I illustrate the usefulness of these powerful tools to clinical medicine by describing a novel noninvasive technique to monitor drug therapy in atrial fibrillation.

Adult↗
PubMed1994

Temporal heterogeneity of the blood flow velocity at the middle cerebral artery in the normal human characterized by fractal analysis.

The objective of this study is to characterize the temporal fluctuation of the axial blood flow velocity (BFV) at the middle cerebral artery (MCA). Biological observables such as BFV present complex oscillations. The irregularity of physiological systems may be assessed by fractal analysis by computing the fractal dimension (D gamma) and the corresponding temporal correlation (r gamma). The BFV at the MCA was registered with transcranial Doppler ultrasonography (TCD) in four adult volunteers. As fractal processes are assumed to have no absolute time scale, two time scales were compared. The digitized signal was averaged respectively at 1-s intervals and for each heart beat. D gamma and r gamma were determined using relative dispersion analysis. The results were D gamma = 1.24 +/- 0.09 and r gamma = 0.45 +/- 0.19 (mean +/- SD) for the 1-s based time scale and D(r) = 1.17 +/- 0.09 and r gamma = 0.57 +/- 0.20 for the heart-beat scale. We conclude that the temporal heterogeneity of the BFV at the MCA in the normal human has fractal properties. Fractal analysis of TCD data may become useful in clinical diagnosis because loss of complexity in physiological systems has been linked to senescence or disease conditions. Wide variations of the so called normal values of BFV measured by TCD have been reported. The physiological BFV fluctuations may explain, in part, the variability of values recorded during routine TCD diagnostic examinations. Our observations may also be of value for understanding the interaction of the vascular endothelium and the blood flow stream (shear stress).

Adult↗
PubMed2000

In vivo assessment of trabecular bone structure using fractal analysis of distal radius radiographs.

Our purpose in this study was (i) to measure trabecular bone structure using fractal analysis of distal radius radiographs in subjects with and without osteoporotic hip fractures, and (ii) to compare these measures with bone mineral density (BMD) as well as with measures of trabecular bone structure derived from high resolution magnetic resonance (MR) images. Distal radius radiographs were obtained using semi-industrial films (55 kVp, 400 mAs) in 30 postmenopausal patients, who had suffered osteoporotic hip fractures (74.8+/-8.2 years) in the last 24 months and 27 postmenopausal age-matched (74.6+/-6.6 yr) normal volunteers. Radiographs were digitized at 50 microm. A Fourier power spectrum-based fractal dimension (FD) characterizing the trabecular pattern was measured in a region of interest proximal to the joint line. The fractal dimension was calculated over two spatial frequency (f) ranges: FD1 was calculated over 0.5<log(f)<l.0, FD2 over the higher range 1.0<log(f)<1.5. Trabecular BMD in the radius was obtained using peripheral quantitative computed tomography (pQCT) (Stratec GmbH, Germany). In addition BMD of the proximal femur was determined using dual x-ray absorptiometry (DXA) (QDR 2000, Hologic, MA). In a subset of patients (16 controls and 18 with hip fractures), high resolution MR imaging of the distal radius (spatial resolution of 156 x 156 x 500 microm) was used to obtain measures analogous to bone histomorphometry. There were significant differences (p<0.05) between the fracture and nonfracture groups in the total femur BMD (13%), trabecular BMD in the distal radius (4%), and the fractal dimension in the radiographs (FD2) (3%). The correlations between FD2 and the total femur BMD as well as trabecular bone BMD in the distal radius were -0.48 (p<0.006) and -0.22 (p<0.33); respectively; FD1 increased with BMD and showed lower correlations. FD2 showed good correlations with App. Tb.N (-0.71) and App. Tb.Sp (0.69) (p<0.01), moderate correlation with App BV/TV (-0.53) (p<0.05), and no significant correlation with App. Tb.Th. The correlations between structural measures and FD1 showed the inverse trend and were typically lower. The odds ratios for a hip fracture were 2.44 for total femur BMD, 1.5 for trabecular BMD (radius), and 1.5 for FD2, respectively. In summary, the fractal measures derived from radiographs of the radius show differences between subjects with and without hip fractures, the predictive power of measures in the distal radius are comparable to radial trabecular BMD but lower than that of total hip BMD.

Aged↗
PubMed2004

Fractal analysis of the vascular tree in the human retina.

The retinal circulation of the normal human retinal vasculature is statistically self-similar and fractal. Studies from several groups present strong evidence that the fractal dimension of the blood vessels in the normal human retina is approximately 1.7. This is the same fractal dimension that is found for a diffusion-limited growth process, and it may have implications for the embryological development of the retinal vascular system. The methods of determining the fractal dimension for branching trees are reviewed together with proposed models for the optimal formation (Murray Principle) of the branching vascular tree in the human retina and the branching pattern of the human bronchial tree. The limitations of fractal analysis of branching biological structures are evaluated. Understanding the design principles of branching vascular systems and the human bronchial tree may find applications in tissue and organ engineering, i.e., bioartificial organs for both liver and kidney.

Animals↗
PubMed2002

Evidence for fractal correlation properties in variations of peripheral arterial tone during REM sleep.

Previous studies utilizing detrended fluctuation analysis (DFA) of heart rate variability during sleep revealed a higher fractal exponent during rapid eye movement (REM) sleep than non-REM sleep. The aim of this study was to determine whether the same difference exists in the variations of peripheral arterial tone (PAT). Finger pulse wave measured by a novel plethysmographic technique was monitored during sleep in 12 chronic heart failure patients, 8 heavy snorers, and 12 healthy volunteers. For each subject, at least two 15-min time series were constructed from the interpulse intervals and from pulse wave amplitudes during REM and non-REM sleep. Fractal scaling exponents of both types of time series were significantly higher for REM than non-REM sleep in all groups. In each of the groups and in both sleep stages, the fractal scaling exponents based on pulse wave amplitude were significantly higher than those based on pulse rate variability. A repeat of the analysis for short-, intermediate-, and long-term intervals revealed that the fractal-like exponents were evident only in the short- and intermediate-term intervals. Because PAT is a surrogate of sympathetic activation, our results indicate that variations in sympathetic activation during REM sleep have a fractal-like behavior.

Adult↗
PubMed2001

Decreased fractal component of human heart rate variability during non-REM sleep.

The physiological significance of the fractal component of short-term, spontaneous heart rate variability (HRV) in humans remains unclear. The aim of the present study was to gain further information about the respective fractal components by extracting them from HRV, blood pressure variability (BPV), and instantaneous lung volume (ILV) time series via coarse graining spectral analysis in nine healthy subjects during waking and sleep states. The results show that the contribution made by the fractal component to the total variance in the beat-to-beat R-R interval declined significantly as the depth of non-rapid eye movement (non-REM) sleep increased, that the ILV time series was largely periodic (i.e., nonfractal), and that BPV was unaffected by sleep stage. Finally, the fractal component of HRV during REM sleep was found to be quite similar to that seen during waking. These results suggest that mechanisms involving electroencephalographic desynchronization and/or conscious states of the brain are reflected in the fractal component of HRV.

Adult↗
PubMed2000

Fractal nature of regional ventilation distribution.

High-resolution measurements of pulmonary perfusion reveal substantial spatial heterogeneity that is fractally distributed. This observation led to the hypothesis that the vascular tree is the principal determinant of regional blood flow. Recent studies using aerosol deposition show similar ventilation heterogeneity that is closely correlated with perfusion. We hypothesize that ventilation has fractal characteristics similar to blood flow. We measured regional ventilation and perfusion with aerosolized and injected fluorescent microspheres in six anesthetized, mechanically ventilated pigs in both prone and supine postures. Adjacent regions were clustered into progressively larger groups. Coefficients of variation were calculated for each cluster size to determine fractal dimensions. At the smallest size lung piece, local ventilation and perfusion are highly correlated, with no significant difference between ventilation and perfusion heterogeneity. On average, the fractal dimension of ventilation is 1.16 in the prone posture and 1. 09 in the supine posture. Ventilation has fractal properties similar to perfusion. Efficient gas exchange is preserved, despite ventilation and perfusion heterogeneity, through close correlation. One potential explanation is the similar geometry of bronchial and vascular structures.

Aerosols↗