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PubMed1993

Quantitation of the renal arterial tree by fractal analysis.

To determine whether the renal arterial system has a fractal structure, the fractal dimension of renal angiograms from 52 necropsy cases was measured using an implementation of the box-counting method on an image analysis system. The method was validated using objects with known fractal dimensions. The method was accurate with errors of less than 1.5 per cent and reproducible with initial values within 1.2 per cent of the mean of ten sets of measurements (reliability coefficient 0.968, 95 per cent confidence limits 0.911-0.984). In the 36 satisfactory angiograms the mean fractal dimension was 1.61 (SD 0.06), which was significantly greater than the topological dimension of 1 (P < 0.0001), indicating that the renal arterial tree has a fractal structure. There was no significant relationship between age (P = 0.494), sex (P = 0.136), or systolic (P = 0.069) or diastolic (P = 0.990) blood pressure, but two congenitally abnormal kidneys (hypoplastic dysplasia and renal artery stenosis) had fractal dimensions at the lower end of the normal range (third percentile). Since the renal arterial tree has a fractal structure, Euclidean geometric measurements, such as area and boundary length, are invalid outside precisely defined conditions of magnification and resolution.

Adult↗
PubMed1994

Fractal geometric analysis of material from molar and non-molar pregnancies.

Histological sections from 25 non-molar pregnancies, nine partial hydatidiform moles, and 16 complete hydatidiform moles were examined (diagnosis was taken as the consensus of seven experienced histopathologists) and the fractal dimension was measured using a box-counting method implemented on a microcomputer-based image analysis system. The fractal dimensions of the different diagnostic categories were normally distributed with a mean of 1.50 for non-molar pregnancies, 1.44 for partial moles, and 1.42 for complete moles. All the measured fractal dimensions were greater than the topological dimension (1), demonstrating that the specimens had a fractal element to their structure. There was a significant difference between the fractal dimensions of non-molar pregnancies and complete moles (P = 0.0005), but not between partial moles and non-molar pregnancies (P = 0.0823) or complete and partial moles (P = 0.4400). Using the fractal dimension to predict the histopathological diagnosis assigned 56 per cent of the cases to the correct category with a kappa statistic of 0.26, so the fractal dimension, used alone, is not a useful morphometric discriminant in the diagnosis of molar and non-molar pregnancy.

Female↗
PubMed1994

Fractal geometry in rat chimeras demonstrates that a repetitive cell division program may generate liver parenchyma.

In the development of mammalian organs, a rapid and robust expansion of the parenchymal compartment must occur following allocation of organ primordia. This expansion must be regulated so that sufficient tissue mass is generated for further organization into functional tissues. The discovery that mosaic patches in the liver of rat chimeras are fractal (a geometric form with characteristic complexity) suggests a possible information storage scheme for programs of parenchyma generation. Since fractal objects are produced by the repetitive application of specific rules, it is possible that such a mechanism is responsible for the generation of organ parenchyma. The model cell division program for the generation of organ parenchyma considered here is to choose a cell at random to divide and place the daughter cell in a randomly chosen adjacent position displacing other cells which might occupy the chosen position. The completion of the division creates a new population of cells representing the input conditions for the next division. When this is repeated over and over in a tissue comprising two genetically distinguishable populations of cells, analysis of the geometry of the mosaic pattern obtained should fulfill specific predictions. If cell division occurred in this manner, the complexity of patch boundaries (patches are contiguous aggregates of cells of the same marker lineage in tissue from a chimera) should be independent of the proportion of the two parental cell lineages which make up the chimera's tissue. However, the complexity of the entire patch pattern should be dependent on this proportion. The complexity of the spatial distribution of the patches within a chimera's tissue should also be dependent on the proportion of the two parental lineages. We have measured the complexity of patch boundaries (surface fractal dimension), the complexity of entire fields of patches (mass fractal dimension), and the complexity of the spatial distribution of patches (fractal fragmentation) in rat liver from chimeras. We have established that the surface fractal dimension does not change as the proportion of the two parental lineages in the chimera's tissue changes, that there is a simple relationship between the complexity of entire patches and this proportion, and that the patches are fractally fragmented. These results are consistent with the hypothesis that repetitive application of this simple cell division program accounts for the generation of liver parenchyma.

Animals↗
PubMed1999

Fractal analysis: an objective method for identifying atypical nuclei in dysplastic lesions of the cervix uteri.

OBJECTIVES: Fractal geometry is a tool used to characterize irregularly shaped and complex figures. It can be used not only to generate biological structures (e.g., the human renal artery tree), but also to derive parameters such as the fractal dimension in order to quantify the shapes of structures. As such, it allows user-independent evaluation and does not rely on the experience level of the examiner. METHODS: We applied a box-counting algorithm to determine the fractal dimension of atypical nuclei in dysplastic cervical epithelium. An automatic algorithm was used to determine the fractal dimension of nuclei in order to prevent errors from manual segmentation. Four groups of patients (CIN 1-3 and control) with 10 subjects each were examined. In total, the fractal dimensions of 1200 nuclei were calculated. RESULTS: We found that the fractal dimensions of the nuclei increased as the degree of dysplasia increased. There were significant differences between control and atypical nuclei found by an analysis of variance. Atypical nuclei associated with CIN 1, CIN 2, and CIN 3 also differed significantly among these groups. CONCLUSION: We conclude that the fractal dimension is a valuable tool for detecting irregularities in atypical nuclei of the cervix uteri and thus allows objective nuclear grading.

Cell Nucleus↗
PubMed1998

Evaluation of self-similar features in time series of serum growth hormone and prolactin levels by fractal analysis: effect of delayed sleep and complexity of diurnal variation.

We assayed the diurnal concentrations of growth hormone (GH) and prolactin (PRL) in 6 healthy male volunteers to evaluate the self-similar features in the time series of each hormone on the basis of fractal theory and to determine the fractal dimension as an index of the complexity of the diurnal variation. In addition, we assessed the effects of a 6-hour delay in the sleep period on the complexity of the diurnal variation of these hormones. There was a statistically significant fractal feature in the serum levels of GH both under the nocturnal-sleep and delayed-sleep conditions in all subjects. The time series of the serum PRL concentrations also showed a statistically significant fractal feature under the nocturnal-sleep and delayed-sleep conditions in all subjects. The fractal dimensions of the patterns of the GH or PRL levels were 1.879 and 1.929 or 1.754 and 1.785 under the nocturnal-sleep and delayed-sleep conditions, respectively. Two-way ANOVA revealed no significant difference in the fractal dimension between the two sleep conditions but did reveal a significant difference between the fractal dimensions of the GH and PRL levels. These results showed (1) that delayed sleep had no significant effect on the complexity of the diurnal pattern of these hormones, and (2) that the diurnal pattern of the GH levels was more complex than that of the PRL levels.

Adult↗
PubMed2005

Fractal characterisation of boundary irregularity in skin pigmented lesions.

A growing literature shows researcher's interest in fractal analysis, arising from its ability to describe and characterise quantitatively the complexity of several tumour profiles. The aim of the work was to investigate the fractal properties of skin pigmented lesion boundaries. Although melanoma is one of the most aggressive tumours, early detection and a high rate of diagnostic accuracy, followed by timely excision, can allow complete recovery in melanoma patients. A modified approach to fractal dimension estimation was performed that was able to consider, in a data fit procedure, the range in which lesions show fractal properties. Identification of this zone is the most important step towards a correct fractal analysis procedure. The method was checked against a known fractal dimension object (Koch's curve) with an error of 0.007. The fractal dimension was estimated in 110 skin pigmented lesions and showed a significantly increasing linear regression (p < 0.05), from common naevi to naevi with dysplasia to melanomas. This result is important for screening, as it can inform the decision to excise precociously malignant lesions or to avoid unnecessary removal of benign ones. The limitations of the method are discussed.

Diagnosis, Differential↗
PubMed1998

Fractal analysis of bone texture on os calcis radiographs compared with trabecular microarchitecture analyzed by histomorphometry.

Microarchitecture of trabecular bone is an important determinant of bone fragility; to date, its evaluation requires bone biopsy with histomorphometry analysis. Methods of noninvasive characterization of trabecular bone microarchitecture are in development and we have developed and validated a bone texture analysis applied to bone radiographs and based on fractal geometry. The aim of our study was to compare this fractal analysis of trabecular bone texture on radiographs to the trabecular microarchitecture analyzed by bone histomorphometry on os calcis biopsies. Thirty eight ossa calcis from 19 human cadavers were studied. Fractal analysis of the trabecular bone of os calcis radiographs was performed by the maximum likelihood estimator following the fractional brownian motion model. The ossa calcis were dissected, then transcortical biopsy cores focused on the fractal analysis region of interest were obtained. Structural and connectivity parameters were measured with both automatic and semiautomatic analyzers. We have found a significant relationship between the fractal Hmean parameter and structural histomorphometric indices; the best correlation was found with trabecular separation (r = -0.55; P = 0.0004). Based on a stepwise regression analysis, trabecular spacing and trabeculae number together would explain 38% of the variance of the fractal parameter. Although the relationship with connectivity indices was poor, our fractal analysis of os calcis trabecular bone texture on radiographs seemed to partially reflect the trabecular bone microarchitecture.

Aged↗
PubMed2007

Fractal analysis of contours of breast masses in mammograms.

Fractal analysis has been shown to be useful in image processing for characterizing shape and gray-scale complexity. Breast masses present shape and gray-scale characteristics that vary between benign masses and malignant tumors in mammograms. Limited studies have been conducted on the application of fractal analysis specifically for classifying breast masses based on shape. The fractal dimension of the contour of a mass may be computed either directly from the 2-dimensional (2D) contour or from a 1-dimensional (1D) signature derived from the contour. We present a study of four methods to compute the fractal dimension of the contours of breast masses, including the ruler method and the box counting method applied to 1D and 2D representations of the contours. The methods were applied to a data set of 111 contours of breast masses. Receiver operating characteristics (ROC) analysis was performed to assess and compare the performance of fractal dimension and four previously developed shape factors in the classification of breast masses as benign or malignant. Fractal dimension was observed to complement the other shape factors, in particular fractional concavity, in the representation of the complexity of the contours. The combination of fractal dimension with fractional concavity yielded the highest area (A ( z )) under the ROC curve of 0.93; the two measures, on their own, resulted in A ( z ) values of 0.89 and 0.88, respectively.

Breast Neoplasms↗
PubMed2005

Fractal analysis of the PDL-bone interface and implications for orthodontic tooth movement.

BACKGROUND: The periodontal ligament (PDL) is a soft tissue interposed between the tooth and the alveolar bone. It is responsible for transmission of forces in vivo; this promotes bone remodeling. The purpose our study was to use fractal analysis to quantify the complex morphology of the PDL-bone interface. METHODS: We used Scion Image (Scion Corp, Frederick, Md) and Benoit fractal analysis (Tru Soft International, St. Petersburg, Fla) programs to calculate the fractal dimension of the PDL-bone interface in rats via the box-counting method. Rats in the experimental groups received an initial force of 0.1N or 0.5N with customized springs for 6 hours. RESULTS: Our studies showed an increase in normal fractal dimension at the root apices of the rats' maxillary molars. We also found evidence that the fractal dimension varies along the entire root length from the apex to the cementoenamel junction. CONCLUSIONS: Mechanical loading might lead to an increase in fractal dimension at the PDL-bone interface apart from mechanisms of bone cell directed remodeling. These changes in fractal dimension are proportional to loading and could provide a new parameter for force determination in orthodontic tooth movement.

Alveolar Process↗
PubMed2004

The fractal dimension as a measure of the quality of habitats.

Habitat fragmentation produces isolated patches characterized by increased edge effects from an originally continuous habitat. The shapes of these patches often show a high degree of irregularity: their shapes deviate significantly from regular geometrical shapes such as rectangular and elliptical ones. In fractal theory, the geometry of patches created by a common landscape transformation process should be statistically similar, i.e. their fractal dimensions and their form factors should be equal. In this paper, we analyze 49 woodlot fragments ( Pinus sylvestris L. ) in the Belgian Kempen region to study the direct relationship between a transformation process and the concomitant patch geometry. Although the fractal dimension of the woodlots is scattered (i.e. they are not statistically similar), the perimeter-area relation of the fragments is characterized by a single, 'dimension-like' exponent. This exponent suggests a certain shape homogeneity among the patches, which is confirmed by the absence of hierarchical levels associated with sharp increases of the fractal dimension at scale transitions. The interaction of different natural (soil factor, vegetation type) and anthropogenic (afforestation, urbanization) processes during patch development is assumed to have generated this feature. Comparison of the area and perimeter fractal dimension with an ecological index for habitat quality, the interior-to-edge ratio, shows that the fractal dimension is suitable for predicting interior habitat presence, which is more likely for patches with smooth perimeters and compact areas. The ratio of the area to the perimeter fractal dimension confirms this observation, with high values for high interior-to-edge ratios, characteristic for regularly shaped patches.

Environment↗
PubMed2001

A kinetic analysis using fractals of cellular analyte-receptor binding and dissociation.

A fractal analysis is presented for cellular analyte-receptor binding and dissociation kinetics using a biosensor. Data taken from the literature may be modelled, 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. The predictive relationship developed for the equilibrium constant, K (affinity which is equal to k(d)/k(1or2)), as a function of the analyte concentration is of particular value since it provides a means by which the affinity may be manipulated. This should be of assistance in cell-surface reactions, drug-candidate optimization and for the design of immunodiagnostic devices. 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, and the analyte concentration in solution. When analyte-receptor binding or dissociation is involved, an increase in the heterogeneity on the surface (increase in D(f) or D(fd) as the case may be) leads to an increase in the binding and the dissociation rate coefficients. It is suggested that an increase in the degree of heterogeneity on the surface leads to an increase in the turbulence on the surface owing to the irregularities on the surface. This turbulence promotes mixing, minimizes diffusional limitations and leads subsequently to an increase in the binding and the dissociation rate coefficients. The binding and dissociation rate coefficients are rather sensitive to the degree of heterogeneity, D(f) and D(fd), respectively, that exists on the biosensor surface. The heterogeneity on the surface in general affects the binding and dissociation rate coefficients differently. In general, the analyte concentration in solution has a mild affect on the fractal dimension for binding or the fractal dimension for dissociation. This is indicated by the low values of the exponent in the predictive relationships developed.

Fractals↗
PubMed2001

Alveolar process fractal dimension and postcranial bone density.

OBJECTIVES: Our goal was to determine whether the radiographic fractal dimension of the maxillary and mandibular alveolar processes is related to bone density of the alveolar processes, spine, hip, and radius in healthy women. STUDY DESIGN: Thirty-seven dentate healthy white women aged 20 to 78 years underwent assessment of systemic and alveolar process bone. After a periodontal examination, D-speed vertical bite-wing and periapical radiographs with aluminum step wedges and a density correction algorithm were used to make alveolar process fractal dimension and density calculations within regions of interest that avoided crestal bone and intrabony defects. Anteroposterior (L1-L4) and lateral (L2-L4) lumbar spine, total hip, and total wrist densities were determined by means of dual energy x-ray absorptiometry. RESULTS: Correlation analysis revealed significant relationships between maxillary alveolar process fractal dimension and maxillary alveolar process density (r = 0.47, P < or =.01), mandibular alveolar process density (r = 0.48, P < or =.01), and mandibular alveolar process fractal dimension (r = 0.44, P < or =.05); and between mandibular alveolar process fractal dimension and maxillary alveolar process density (r = 0.54, P < or =.01) and mandibular alveolar process density (r = 0.58, P < or =.001). No significant relationships were found between the maxillary alveolar process or the mandibular alveolar process fractal dimension and the density of any postcranial regions. CONCLUSIONS: In healthy women, the alveolar process radiographic fractal dimension is significantly related to the alveolar process density but is not related to the density of the spine, hip, or radius.

Absorptiometry, Photon↗
PubMed1995

Fractal and integer-dimensional geometric analysis of pigmented skin lesions.

Accurate in vivo diagnosis of pigmented skin lesions is required to identify and excise malignant melanomas but to avoid unnecessary excision of benign lesions; the published rates of clinical diagnostic accuracy are about 65%. This study investigates whether fractal geometric analysis of pigmented skin lesions can improve the rate of diagnostic accuracy. Forty-two pigmented skin lesions (15 malignant melanomas, 21 melanocytic naevi, and 6 basal cell papillomas) on patients attending a dermatology clinic were photographed, excised, and sent for histopathological examination. The fractal dimension of the boundary of the lesions was measured using a box-counting method implemented on a microcomputer-based image analysis system. Euclidean geometric parameters were also measured. The fractal dimension of all the lesions was greater than the topological dimension (one), indicating that there is a fractal element to their structure. Using all measured parameters together, multivariate linear discriminant analysis produced a confusion matrix in which 45% of the lesions were assigned to the correct diagnostic group with a kappa statistic of 0.33. There was no significant difference between the fractal dimension of melanocytic naevi and that of malignant melanomas (p = 0.18). Although pigmented skin lesions have a fractal element to their structure, the fractal dimension of their boundaries is not a useful morphometric discriminant between the diagnostic groups of malignant melanomas and benign melanocytic naevi.

Algorithms↗
PubMed2001

Fractal branching pattern in the pial vasculature in the cat.

Arborization pattern was studied in pial vascular networks by treating them as fractals. Rather than applying elaborate taxonomy assembled from measures from individual vessel segments and bifurcations arranged in their branching order, the authors' approach captured the structural details at once in high-resolution digital images processed for the skeleton of the networks. The pial networks appear random and at the same time having structural elements similar to each other when viewed at different scales--a property known as self-similarity revealed by the geometry of fractals. Fractal (capacity) dimension, Dcap, was calculated to evaluate the network's spatial complexity by the box counting method (BCM) and its variant, the extended counting method (XCM). Box counting method and XCM were subject to numerical testing on ideal fractals of known D. The authors found that precision of these fractal methods depends on the fractal character (branching, nonbranching) of the structure they evaluate. Dcaps (group mean +/- SD) for the arterial and venous pial networks in the cat (n = 6) are 1.37 +/- 0.04, 1.37 +/- 0.02 by XCM, and 1.30 +/- 0.04, 1.31 +/- 0.03 by BCM, respectively. The arterial and venous systems thus appear to be developed according to the same fractal generation rule in the cat.

Animals↗
PubMed2006

Fractal dimension of canine mammary gland epithelial tumors on cytologic smears.

BACKGROUND: Fractal geometry is a tool that can be used for describing, modeling, analyzing, and processing irregular and complex figures. Past investigations in medicine have revealed that fractal analysis could also be applied in tumor pathology to characterize irregular boundaries of the nuclei of tumor cells. OBJECTIVE: The aim of this study was to define whether the fractal dimension parameter could be used on cytologic specimens to differentiate benign from malignant canine mammary gland epithelial tumors. METHODS: The fractal dimension of nuclear surface was determined by computer-assisted morphometry on Hemacolor-stained cytologic smears obtained by fine needle aspiration of normal canine mammary gland epithelial cells, and cells in mammary adenomas, tubulopapillary carcinomas, solid carcinomas, and anaplastic carcinomas. Data were analyzed by Mann-Whitney U test. RESULTS: Significant differences (P <.001) were observed in mean fractal dimension among all tumor types and in comparison with normal canine mammary gland epithelial cells (except for the fractal dimension between solid carcinomas and anaplastic carcinomas). CONCLUSION: The morphometric parameter, fractal dimension, could help in the diagnostic discrimination between benign and malignant canine mammary gland epithelial tumors on cytologic specimens.

Animals↗
PubMed1994

On the fractal nature of trabecular structure.

Fractal analysis has recently been suggested [Med. Phys. 20, 1611-1619 (1993)] as a means to characterize the structure of cancellous bone by measuring the fractal dimension using a box counting algorithm. This work re-examines the possible fractal nature of such structures on nuclear magnetic resonance (NMR) images of cancellous bone by estimating the trabecular boundary length as a function of box size under various experimental conditions. On high-resolution images (pixel sizes on the order of 50 microns) and signal-to-noise ratios of 30, the trabecular boundary turns out to be a smooth surface relative to the achievable resolution and is thus nonfractal. The fractal dimension of the trabecular structure is undefined and can vary significantly as a function of image signal-to-noise ratio. The present work further indicates the "apparent" fractal dimension obtained by box counting to be a reflection of marrow pore size. In conclusion, the results indicate that, at the currently achievable resolution, the box counting algorithm is not suitable for fractal analysis on images of cancellous bone and that the fractal appearance of the trabecular network reported previously is artifactual.

Algorithms↗
PubMed2003

Fractal properties of human muscle sympathetic nerve activity.

Muscle sympathetic nerve activity (MSNA) in resting humans is characterized by cardiac-related bursts of variable amplitude that occur sporadically or in clusters. The present study was designed to characterize the fluctuations in the number of MSNA bursts, interburst interval, and burst amplitude recorded from the peroneal nerve of 15 awake, healthy human subjects. For this purpose, we used the Allan and Fano factor analysis and dispersional analysis to test whether the fluctuations were time-scale invariant (i.e., fractal) or random in occurrence. Specifically, we measured the slopes of the power laws in the Allan factor, Fano factor, and dispersional analysis curves. In addition, the Hurst exponent was calculated from the slope of the power law in the Allan factor curve. Whether the original time series contained fractal fluctuations was decided on the basis of a comparison of the values of these parameters with those for surrogate data blocks. The results can be summarized as follows. Fluctuations in the number of MSNA bursts and interburst interval were fractal in each of the subjects, and fluctuations in burst amplitude were fractal in four of the subjects. We also found that fluctuations in the number of heartbeats and heart period (R-R interval) were fractal in each of the subjects. These results demonstrate for the first time that apparently random fluctuations in human MSNA are, in fact, dictated by a time-scale-invariant process that imparts "long-term memory" to the sequence of cardiac-related bursts. Whether sympathetic outflow to the heart also is fractal and contributes to the fractal component of heart rate variability remains an open question.

Adult↗
PubMed1995

On the fractal nature of heart rate variability in humans: effects of vagal blockade.

The purpose of the present study was to investigate the effects of the vagal blocker atropine on the fractal nature of human heart rate variability (HRV) at rest. Approximately 10-min segments of beat-to-beat heartbeat intervals, i.e., HRV, of 10 normal subjects and 11 cardiac disease patients were measured before and after intravenous injection of 0.5-0.75 mg atropine sulfate. HRV data were analyzed by coarse graining spectral analysis (Y. Yamamoto and R. L. Hughson, Physica 68D: 250-264, 1993) to break down their total power into harmonic and nonharmonic (fractal) components. The harmonic component was used to calculate the contribution of high (> 0.15 Hz)-frequency components to total HRV power (%HF). From the fractal component, the contribution of the fractal component to total HRV power (%fractal), the spectral exponent beta, and Hurst scaling exponent (H) were calculated. For both normal subjects and cardiac patients, atropine resulted in significantly (P < 0.05) less mean HRV and significantly (P < 0.05) greater beta compared with control, whereas mean values for %fractal were as high as 70% and were not significantly (P > 0.05) different between atropine and control. The mean value of H with atropine was significantly (P < 0.05) greater than that for control. Directional changes in %HF and beta were consistent with only one exception for a patient who had the smallest change in log %HF by atropine. The normally irregular fractal pattern of resting HRV was decreased by atropine as shown by the decrease in %HF and the increase in beta.(ABSTRACT TRUNCATED AT 250 WORDS)

Adult↗