A single-cell model to study changes in neuronal fractal dimension

Many methods have been developed to quantify neuronal morphology: measurement of neurite length, neurite number, etc. However, none of these approaches provides a comprehensive view of the complexity of neuronal morphology. In this work we have analyzed the evaluation of fractal dimension (D) as a tool to represent and quantify changes in complexity of the dendritic arbor, in in vitro cultures grown under low-density conditions. Neurons grown in isolation developed a bipolar morphology corresponding to a fractal dimension close to the unit. The analysis showed that neuronal complexity increased when cells were incubated with a depolarizing potassium concentration and there was a correlation with an increase in fractal dimension (D5 mM KCl = 1.08 +/- 0.01, D25 mM KCl =1.25 +/- 0.01). We conclude that fractal dimension is a suitable parameter to quantify changes in neuronal morphological complexity.     

Wavelet packet fractal analysis of neuronal morphology

An image analysis method called two-dimensional wavelet packet analysis (2D WPA) is introduced to quantify branching complexity of neurons. Both binary silhouettes and contour profiles of neurons were analyzed to determine accuracy and precision of the fractal dimension in cell classification tasks. Two-dimensional WPA plotted the slope of decay for a sorted list of discrete wavelet packet coefficients belonging to the adapted wavelet best basis to obtain the fractal dimension for test images and binary representations of neurons. Two-dimensional WPA was compared with box counting and mass-radius algorithms. The results for 2D WPA showed that it could differentiate between neural branching complexity in cells of different type in agreement with accepted methods. The importance of the 2D WPA method is that it performs multiresolution decomposition in the horizontal, vertical, and diagonal orientations.     

Nuclear patterns of human breast cancer cells during apoptosis: characterisation by fractal dimension and co-occurrence matrix statistics

An analytical strategy combining fractal geometry and grey-level co-occurrence matrix (GLCM) statistics was devised to investigate ultrastructural changes in oestrogen-insensitive SK-BR3 human breast cancer cells undergoing apoptosis in vitro. Apoptosis was induced by 1 μM calcimycin (A23187 Ca²⁺ ionophore) and assessed by measuring conventional cellular parameters during the culture period. SK-BR3 cells entered the early stage of apoptosis within 24 h of treatment with calcimycin, which induced detectable changes in nuclear components, as documented by increased values of most GLCM parameters and by the general reduction of the fractal dimensions. In these affected cells, morphonuclear traits were accompanied by the reduction of distinct gangliosides and loss of unidentifiable glycolipid molecules at the cell surface. All these changes were shown to be involved in apoptosis before the detection of conventional markers, which were only measurable during the active phases of apoptotic cell death. In overtly apoptotic cells treated with 1 μM calcimycin for 72 h, most nuclear components underwent dramatic ultrastructural changes, including marginalisation and condensation of chromatin, as reflected in a significant reduction of their fractal dimensions. Hence, both fractal and GLCM analyses confirm that the morphological reorganisation of nuclei, attributable to a loss of structural complexity, occurs early in apoptosis. This work was supported by grant 31-57626.99 from the Swiss National Science Foundation and grant FOR 450.94 from the Swiss Cancer League.     

Power scaling of ammonitic suture patterns from Cretaceous Ancyloceratina: constraints on septal/sutural complexity

The spatial scaling of 77 hemisutures from 65 species of Cretaceous heteromorphic ammonites was quantified with the fractal box‐counting method. Fractal dimensions within Baculites compressus did not significantly differ between adult hemisutures; however, the juvenile suture of this species did exhibit a significantly lower fractal dimension. This suggests that variation in sutural complexity between explicitly adult ontogenetic stages may not contribute to significant noise in comparisons between other species/morphotypes. High‐spired, three‐dimensionally coiled heteromorphs with a larger degree of septal asymmetry exhibit higher fractal dimensions in outer whorl hemisutures than inner whorl hemisutures due to their elongation and improved space occupation over a larger whorl surface. Three‐dimensionally coiled ammonites also have higher fractal dimensions on average (mean D_b = 1.45) with respect to their 2‐D coiled counterparts (mean D_b = 1.38). All ammonites in this study exhibit a positive trend between sutural complexity and shell size (proxied by whorl height). These relationships suggest that septal frilling is constrained by shell morphology and whorl section geometry during septal morphogenesis. This, in turn, influences the scaling, space‐filling properties and scaling limits of ammonitic suture patterns. Sutural/septal complexity is also found to positively influence the amount of liquid retained in marginal septal recesses. However, as these septa approach larger scales, less cameral liquid is retained per septal mass. This may further explain the positive relationship between sutural complexity and shell size.     

A fractal model for the characterization of mycelial morphology

A new technique based on a fractal model has been developed for the quantification of the macroscopic morophology of mycelia. The morphological structuring is treated as a fractal object, and the fractal dimension, determined by an ultrasonic scattering procedure developed for the purpose, serves as a quantitative morphological index. Experimental observations reported earlier and simulations of mycelial growth, carried out using a probabilistic-geometric growth model developed for the purpose, both validate the applicability of the fractal model. In experiments with three different species, the fractal dimensions of pelletous structures were found to be in the range 1.45-2.0 and those of filamentous structures were in the range 1.9-2.7, with values around 2.0 representing mixed morphologies. Fractal dimensions calculated from simulated mycelia are in rough agreement with these ranges. The fractal dimension is also found to be relatively insensitive to the biomass concentration, as seen by dilution of the original broths. The relation between morphology and filtration properties of the broths has also been studied. The fractal dimension shows a strong correlation with the index of cake compressibility and with the Kozeny constant, two filtration parameters that are known to be morphology dependent. This technique could thus be used to develop correlations between the morphology, represented by the fractal dimension, and important morphology-dependent process variables. (c) 1993 John Wiley & Sons, Inc.     

Fractal dimension and vessel complexity in patients with cerebral arteriovenous malformations

The fractal dimension (FD) can be used as a measure for morphological complexity in biological systems. The aim of this study was to test the usefulness of this quantitative parameter in the context of cerebral vascular complexity. Fractal analysis was applied on ten patients with cerebral arteriovenous malformations (AVM) and ten healthy controls. Maximum intensity projections from Time-of-Flight MRI scans were analyzed using different measurements of FD, the Box-counting dimension, the Minkowski dimension and generalized dimensions evaluated by means of multifractal analysis. The physiological significance of this parameter was investigated by comparing values of FD first, with the maximum slope of contrast media transit obtained from dynamic contrast-enhanced MRI data and second, with the nidus size obtained from X-ray angiography data. We found that for all methods, the Box-counting dimension, the Minkowski dimension and the generalized dimensions FD was significantly higher in the hemisphere with AVM compared to the hemisphere without AVM indicating that FD is a sensitive parameter to capture vascular complexity. Furthermore we found a high correlation between FD and the maximum slope of contrast media transit and between FD and the size of the central nidus pointing out the physiological relevance of FD. The proposed method may therefore serve as an additional objective parameter, which can be assessed automatically and might assist in the complex workup of AVMs.     

红树植物木榄幼树斑块形状的分形分析

应用分形理论分析了山口国家级红树林自然保护区木榄幼树斑块形状的分形特征。木榄幼树斑块在整体水平上的分形维数为 1 .1 2 ,而单个斑块的分形维数介于 1 .1 7～ 1 .3 7之间。分形维数的大小与斑块形状的复杂程度密切相关。频谱法和周长 -面积法都适用于分析幼树斑块形状的分形特征。     

Structural complexity in mussel beds: the fractal geometry of surface topography

Seafloor topographic complexity is ecologically important because it provides habitat structure and alters boundary-layer flow over the bottom. Despite its importance, there is little agreement on how to define and measure surface complexity. The purpose of this investigation was to utilize fractal geometry of vertical cross-section profiles to characterize the surface topography of the soft-bottom mussel bed (Mytilus edulis L.) at Bob's Cove, ME, USA. Mussels there have been shown previously to have spatially ordered fractal characteristics in the horizontal plane. Two hypotheses were tested. The first was that the bed surface is fractal over the spatial scale of 1.44-200 mm, with fractal dimension less than or equal to 1.26, the value for the Koch curve, our model for bed profiles. The second was that bed surface topography (i.e., in vertical profile) is less complex than the mussel bed spatial pattern (i.e., aerial view in the horizontal plane). Both hypotheses were supported. Cross-sections of plaster casts of the bed produced 88 surface profiles, all of which were fractal over the entire spatial scale of more two orders of magnitude employed in the analysis. Fractal dimension values (D) for individual profiles ranged from 1.031 to 1.310. Fractal dimensions of entire casts ranged up to mean (1.242+/-0.046) and median (1.251) values similar to 1.26, the theoretical value of the Koch curve. The bed surface was less complex than the bed spatial pattern because every profile had D<1.36, the smallest value previously obtained from aerial views of the bed. The investigation demonstrated for the first time that surface topography of a soft-bottom mussel bed was fractal at a spatial scale relevant to hydrodynamic processes and habitat structure important for benthic organisms. The technique of using cross-section profiles from casts of the bed surface avoided possible underestimates of fractal dimension that can result from other profiling methods reported in the literature. The results demonstrate that fractal dimension can be useful in the analysis of habitat space and water flow over any irregular seafloor surface because it incorporates the size, shape, and scale of roughness elements into a simple, numerical metric.     

Effects of macrophyte architecture and leaf shape complexity on structural parameters of the epiphytic algal community in a Pampean stream

Habitat heterogeneity is one of the main factors determining distribution of organisms, and vegetation is of primary importance in shaping the structural environment in aquatic systems. The effect of macrophyte complexity on macroinvertebrates has been well researched; however, much remains to be revealed about the influence of complexity on epiphytic algae. Here, we used fractal dimension to study the effect of complexity at two scales, macrophyte architecture and leaf shape, on several parameters of the epiphytic algal community (number of individuals, biomass, taxon richness and diversity) in a Pampean stream. Four submerged macrophyte species with different complexities and associated algae were sampled in late spring, summer and autumn. Important differences were found in fractal dimension of the whole plant and leaves among macrophyte species. The particulate organic matter and chlorophyll a associated positively to leaf fractal dimension, but not to plant fractal dimension, partially supporting the hypothesis of a positive effect of macrophyte complexity on periphyton biomass. No association was found in fractal dimension with algal abundance, taxon richness or diversity. Complementary, a mesocosm experiment was performed with plastic imitations of different plant fractal dimensions. After four weeks, there were differences in chlorophyll a and autotrophy index between treatments that suggested a positive effect of complexity on autotrophic periphyton biomass. These results indicate that the well-known positive effect of macrophyte complexity on macroinvertebrates might be partially explained by a positive effect of complexity on periphyton biomass.     

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 variaton 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.     

Fractal dimension in aspiration cytology smears of breast and cervical lesions

OBJECTIVE: To standardize the automated measurement of fractal dimension on cytologic smears and compare the fractal dimension of benign and malignant breast cells and cervical lesions on cytologic material to evaluate its role in the discrimination of benign from malignant cells. STUDY DESIGN: We randomly selected fine needle aspiration cytology smears of 42 cases of infiltrating duct carcinoma and 38 cases of fibroadenoma of the breast. Similarly, 16 cervical carcinoma and 20 normal cervical smears were selected for study. Ten cells were selected randomly from each case. Box counting of fractal dimension of malignant and benign cells was achieved with an image cytometer (Leica, Cambridge, England) using Quantimet 600 software (Leica). Then a well-spaced grid with multiple small boxes of a particular pixel length was superimposed on the cell. The dimension of the box was selected as 4, 8 and 16 pixels. With the help of a logical "AND" operation, we counted the number of boxes touching the peripheral margin of the cell nuclei. For each cell, the log-log graph of 1 per box size was plotted against the number of boxes touching the peripheral rim of the cell. The slope of each graph was identified using the least-squares method of regression analysis. RESULTS: The mean fractal dimension of malignant cells was 0.8536 +/- 0.1120 as compared to 0.8403 +/- 0.1115 in benign cell groups. The Mann-Whitney U test showed a significant difference in fractal dimension in these 2 groups (P = .05). The mean fractal dimension of malignant cells from the cervix was 0.8656 +/- 0.1499 as compared to 0.8315 +/- 0.1312 in benign cells. The Mann-Whitney U test showed a significant difference in fractal dimension in these 2 groups (P < .02). CONCLUSION: Fractal dimension may be a helpful adjunctive technique to discriminate between benign and malignant cells.     

Correcting for finite spatial scales of self-similarity when calculating fractal dimensions of real-world structures

Fractal geometry is a potentially valuable tool for quantitatively characterizing complex structures. The fractal dimension (D) can be used as a simple, single index for summarizing properties of real and abstract structures in space and time. Applications in the fields of biology and ecology range from neurobiology to plant architecture, landscape structure, taxonomy and species diversity. However, methods to estimate the D have often been applied in an uncritical manner, violating assumptions about the nature of fractal structures. The most common error involves ignoring the fact that ideal, i.e. infinitely nested, fractal structures exhibit self-similarity over any range of scales. Unlike ideal fractals, real-world structures exhibit self-similarity only over a finite range of scales.Here we present a new technique for quantitatively determining the scales over which real-world structures show statistical self-similarity. The new technique uses a combination of curve-fitting and tests of curvilinearity of residuals to identify the largest range of contiguous scales that exhibit statistical self-similarity. Consequently, we estimate D only over the statistically identified region of self-similarity and introduce the finite scale- corrected dimension (FSCD). We demonstrate the use of this method in two steps. First, using mathematical fractal curves with known but variable spatial scales of self-similarity (achieved by varying the iteration level used for creating the curves), we demonstrate that our method can reliably quantify the spatial scales of self-similarity. This technique therefore allows accurate empirical quantification of theoretical Ds. Secondly, we apply the technique to digital images of the rhizome systems of goldenrod (Solidago altissima). The technique significantly reduced variations in estimated fractal dimensions arising from variations in the method of preparing digital images. Overall, the revised method has the potential to significantly improve repeatability and reliability for deriving fractal dimensions of real-world branching structures.     

Fractal analysis of the Orce skull sutures

Methods of fractal geometry (Mandelbrot, 1983) are used here to analyse the relative complexity of the sagittal and lambdoid sutures visible in the skull fragment formed by parts of an occipital squame and parietals found in a sealed deposit at the early Lower Pleistocene site of Venta Micena (Orce, Granada, Spain), generally regarded as human bone but occasionally suggested as belonging to an equid. For comparison with the fossil, corresponding sutures of various primates (hominids, pongids and cercopithecids) and two other groups of mammals (equids and ruminants) were analysed using the computer program FRACTAL-D (Slice, 1989) in order to determine their fractal dimensions as a measure of differential sutural design complexity. The results show that the fractal dimension of the Venta Micena skull sutures lies within the range of variation for infant specimens of both modern and Plio-Pleistocene hominids. Sutural complexity in young pongids and cercopithecids overlaps the range of fractal dimensions found in hominids, whereas values obtained from equids and ruminants are significantly greater than those for all the primates analysed here. Therefore, in terms of fractal dimension measures of relative complexity, the sutures preserved in the Venta Micena fossil could not have belonged to an equid (pace Agusti & Moyà-Sola, 1987); rather, its fractal dimension is consistent with the attribution of the fossil to an infant of Homo sp.     

Fractal Organization of Cultivated in vitro Spiculogenous Cells and Larval Spicules of the Sea Urchin Strongylocentrotus nudus

The morphological patterns of the cultivated cells of primary mesenchyme and the spicules of the larval skeleton of the sea urchin Strongylocentrotus nudus were quantified, and the value of their fractal dimensions (D) was determined with ImageJ 1.20s software. It was shown that during cytodifferentiation, the values of D in the fractal (fractional) dimension, which reflects the complex spatial organization of the spiculogenous mesenchyme elements in two-dimensional space, increase to values close to 1.7. The invertible treatment with cytochalasin, which destroys the system of the actin filaments, suppresses the normal control of biomineralization and causes a complex form of spicules, the fractal dimension of which varies within 1.5–1.6. Thus, the determination of the fractal dimension value serves as evidence of the fractional essence of the patterns studied, quantifies the spatially complex organization of cells and their assemblies during morphogenesis, and allows us to estimate the variation in the spicule morphology after cytochalasin treatment.     

Fractal analysis of sagittal suture morphology

Quantifying shape is a broad problem in the morphological sciences. Most techniques for numerically describing shape abstract the shape into the most logical ideal Euclidean dimension. The fractional, or fractal, dimension is a simple computation that expresses shape in real, rather than ideal, space. The structured walk technique developed for the fractal analysis of rugged boundaries is applied here to the contour of the human sagittal suture in order to discriminate the separate morphological patterns of interfingering and interlocking. These attributes contribute differentially to the suture's “complexity,” a concept often used in biomechanical hypotheses. Previous techniques for estimating sutural complexity do not isolate small-scale from large-scale morphological patterns. Results indicate that despite the visual appearance of great variation, human sagittal sutures are remarkably consistent in the degree of complexity expressed separately by large-scale interfingering lateral excursions and small-scale interlocking ruggedness. There is no significant correlation between the absolute or bregma-lambda chord length of the human sagittal suture and its degree of complexity as determined by the structured walk technique.     

The Quasi-Fractal Structure of Fish Brain Neurons

The box-counting method for calculating the fractal dimension (D) with the ImageJ 1.20s software is used as a tool for quantitative analysis of the neuronal morphology in the fish brain. The fractal dimension was determined for several types of neurons in the brain of two teleost species, Pholidapus dybowskii and Oncorhynchus keta. These results were compared with those obtained for some neurons of the human brain. The fractal (fractional) dimension (D), as a quantitative index of filling of two-dimensional space by the black and white image of a cell, is shown to vary from 1.22 to 1.72 depending on the type of neuron. The fractal dimension reaches its maximum in less specialized neurons that carry out a number of different functions. On the other hand, highly specialized neurons display a relatively low fractal dimension. Thus, the fractal dimension serves as a numerical measure of the spatial complexity of the neuron and correlates with the morphofunctional organization of the cell.     

Ultrastructural complexity of nuclear components during early apoptotic phases in breast cancer cells.

Fractal morphometry was used to investigate the ultrastructural features of the plasma membrane, perinuclear membrane and nuclear chromatin in SK-BR-3 human breast cancer cells undergoing apoptosis. Cells were incubated with 1 microM calcimycin (A23187) for 24 h. Cells in the early stage of apoptosis had fractal dimension (FD) values indicating that their plasma membranes were less rough (lower FD) than those of control cells, while their perinuclear membranes were unaffected. Changes of the chromatin texture within the entire nucleus and in selected nuclear domains were more pronounced in treated cells. This confirms that the morphological reorganization imputable to a loss of structural complexity (reduced FD) occurs in the early stage of apoptosis, is accompanied by the inhibition of distinct enzymatic events and precedes the onset of conventional cellular markers, which can only be detected during the active phases of the apoptotic process.