Fractal geometry of root systems: Field observations of contrasting genotypes of common bean (Phaseolus vulgaris L.) grown under different phosphorus regimes

Root growth and architecture are important for phosphorus acquisition due to the relative immobility of P in the soil. Fractal geometry is a potential new approach to the analysis of root architecture. Substantial genetic variation in root growth and architecture has been observed in common bean. Common bean (Phaseolus vulgaris L.) genotypes with contrasting root architecture were grown under moderate and low P conditions in a field experiment. Linear and planar fractal dimension were measured by tracing root intercepts with vertical planes. Linear fractal dimension increased over time in efficient genotypes, but remained fairly constant over time in inefficient genotypes. Planar fractal dimension increased over time for all genotypes, but was higher in efficient than inefficient genotypes at the end of the experiment. Planar fractal dimension of medium P plants was found to correlate with shoot P content indicating fractal dimension to be a possible indicator for root P uptake. The increasing fractal dimension over time indicates that fractal analysis is a sensitive measure of root branching intensity. A less destructive method for acquisition of data that allows for continuous analysis of fractal geometry and thereby screening for more P efficient genotypes in the field is suggested. This method will allow the researcher to conduct fractal analysis and still complete field trials with final yield evaluation.     

Microbial growth patterns described by fractal geometry.

Fractal geometry has made important contributions to understanding the growth of inorganic systems in such processes as aggregation, cluster formation, and dendritic growth. In biology, fractal geometry was previously applied to describe, for instance, the branching system in the lung airways and the backbone structure of proteins as well as their surface irregularity. This investigation applies the fractal concept to the growth patterns of two microbial species, Streptomyces griseus and Ashbya gossypii. It is a first example showing fractal aggregates in biological systems, with a cell as the smallest aggregating unit and the colony as an aggregate. We find that the global structure of sufficiently branched mycelia can be described by a fractal dimension, D, which increases during growth up to 1.5. D is therefore a new growth parameter. Two different box-counting methods (one applied to the whole mass of the mycelium and the other applied to the surface of the system) enable us to evaluate fractal dimensions for the aggregates in this analysis in the region of D = 1.3 to 2. Comparison of both box-counting methods shows that the mycelial structure changes during growth from a mass fractal to a surface fractal.     

Modelling applicability of fractal analysis to efficiency of soil exploration by roots

BACKGROUND AND AIMS: Fractal analysis allows calculation of fractal dimension, fractal abundance and lacunarity. Fractal analysis of plant roots has revealed correlations of fractal dimension with age, topology or genotypic variation, while fractal abundance has been associated with root length. Lacunarity is associated with heterogeneity of distribution, and has yet to be utilized in analysis of roots. In this study, fractal analysis was applied to the study of root architecture and acquisition of diffusion-limited nutrients. The hypothesis that soil depletion and root competition are more closely correlated with a combination of fractal parameters than by any one alone was tested. MODEL: The geometric simulation model SimRoot was used to dynamically model roots of various architectures growing for up to 16 d in three soil types with contrasting nutrient mobility. Fractal parameters were calculated for whole roots, projections of roots and vertical slices of roots taken at 0, 2.5 and 5 cm from the root origin. Nutrient depletion volumes, competition volumes, and relative competition were regressed against fractal parameters and root length. KEY RESULTS: Root length was correlated with depletion volume, competition volume and relative competition at all times. In analysis of three-dimensional, projected roots and 0 cm slices, log(fractal abundance) was highly correlated with log(depletion volume) when times were pooled. Other than this, multiple regression yielded better correlations than regression with single fractal parameters. Correlations decreased with age of roots and distance of vertical slices from the root origin. Field data were also examined to see if fractal dimension, fractal abundance and lacunarity can be used to distinguish common bean genotypes in field situations. There were significant differences in fractal dimension and fractal abundance, but not in lacunarity. CONCLUSIONS: These results suggest that applying fractal analysis to research of soil exploration by root systems should include fractal abundance, and possibly lacunarity, along with fractal dimension.     

On the relationship between fractal geometry of space and time in which a system of interacting cells exists and dynamics of gene expression.

We report that both space and time, in which a system of interacting cells exists, possess fractal structure. Each single cell of the system can restore the hierarchical organization and dynamic features of the entire tumor. There is a relationship between dynamics of gene expression and connectivity (i.e., interconnectedness which denotes the existence of complex, dynamic relationships in a population of cells leading to the emergence of global features in the system that would never appear in a single cell existing out of the system). Fractal structure emerges owing to non-bijectivity of dynamic cellular network of genes and their regulatory elements. It disappears during tumor progression. This latter state is characterized by damped dynamics of gene expression, loss of connectivity, loss of collectivity (i.e., capability of the interconnected cells to interact in a common mode), and metastatic phenotype. Fractal structure of both space and time is necessary for a cellular system to self-organize. Our findings indicate that results of molecular studies on gene expression should be interpreted in terms of space-time geometry of the cellular system. In particular, the dynamics of gene expression in cancer cells existing in a malignant tumor is not identical with the dynamics of gene expression in the same cells cultured in the monolayer system.     

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.     

Fractal geometry in mosaic organs: a new interpretation of mosaic pattern

Fractal geometries have been widely observed in nature. The formulation of mathematical treatments of non-Euclidean geometry has generated models of highly complex natural phenomena. In the field of developmental biology, branching morphogenesis has been explained in terms of self-similar iterating branching rules that have done much toward explaining branch patterns observed in a range of real tissue. In solid viscera the problem is more complicated because there is no readily available marker of geometry in parenchymal tissue. Mosaic pattern provides such a marker. The patches observed in mosaic liver are shown to be fractal, indicating that the pattern may have arisen from a self-similar process (i.e., a process that creates an object in which small areas are representative of, although not necessarily identical to, the whole object). This observation offers a new analytical approach to the study of biologic structure in organogenesis.     

Dynamic contrast-enhanced MRI and fractal characteristics of percolation clusters in two-dimensional tumor blood perfusion.

Dynamic contrast-enhanced magnetic resonance imaging (DE-MRI) of the tumor blood pool is used to study tumor tissue perfusion. The results are then analyzed using percolation models. Percolation cluster geometry is depicted using the wash-in component of MRI contrast signal intensity. Fractal characteristics are determined for each two-dimensional cluster. The invasion percolation model is used to describe the evolution of the tumor perfusion front. Although tumor perfusion can be depicted rigorously only in three dimensions, two-dimensional cases are used to validate the methodology. It is concluded that the blood perfusion in a two-dimensional tumor vessel network has a fractal structure and that the evolution of the perfusion front can be characterized using invasion percolation. For all the cases studied, the front starts to grow from the periphery of the tumor (where the feeding vessel was assumed to lie) and continues to grow toward the center of the tumor, accounting for the well-documented perfused periphery and necrotic core of the tumor tissue.     

Wavelet-Based 3D Reconstruction of Microcalcification Clusters from Two Mammographic Views: New Evidence That Fractal Tumors Are Malignant and Euclidean Tumors Are Benign

The 2D Wavelet-Transform Modulus Maxima (WTMM) method was used to detect microcalcifications (MC) in human breast tissue seen in mammograms and to characterize the fractal geometry of benign and malignant MC clusters. This was done in the context of a preliminary analysis of a small dataset, via a novel way to partition the wavelet-transform space-scale skeleton. For the first time, the estimated 3D fractal structure of a breast lesion was inferred by pairing the information from two separate 2D projected mammographic views of the same breast, i.e. the cranial-caudal (CC) and mediolateral-oblique (MLO) views. As a novelty, we define the “CC-MLO fractal dimension plot”, where a “fractal zone” and “Euclidean zones” (non-fractal) are defined. 118 images (59 cases, 25 malignant and 34 benign) obtained from a digital databank of mammograms with known radiologist diagnostics were analyzed to determine which cases would be plotted in the fractal zone and which cases would fall in the Euclidean zones. 92% of malignant breast lesions studied (23 out of 25 cases) were in the fractal zone while 88% of the benign lesions were in the Euclidean zones (30 out of 34 cases). Furthermore, a Bayesian statistical analysis shows that, with 95% credibility, the probability that fractal breast lesions are malignant is between 74% and 98%. Alternatively, with 95% credibility, the probability that Euclidean breast lesions are benign is between 76% and 96%. These results support the notion that the fractal structure of malignant tumors is more likely to be associated with an invasive behavior into the surrounding tissue compared to the less invasive, Euclidean structure of benign tumors. Finally, based on indirect 3D reconstructions from the 2D views, we conjecture that all breast tumors considered in this study, benign and malignant, fractal or Euclidean, restrict their growth to 2-dimensional manifolds within the breast tissue.     

Nuclear texture features for classifying benign vs. dysplastic or malignant squamous epithelium of the larynx.

OBJECTIVE: To search for nuclear features and feature combinations able to assess malignancy and premalignant changes on tissue sections of laryngeal squamous epithelium. STUDY DESIGN: A total of 139 lesions of benign changes (BC) (n = 44), epithelial dysplasias (ED) (n = 50) and invasive laryngeal cancer (LC) (n = 45) were retrieved from archival pathology specimens. The goal of this study was to identify the best features or feature combinations that discriminate BC from LC and also reflect the degree of ED. In order to verify the results on independent data, the groups were split into two separate subgroups, one for training and one for testing. RESULTS: On the test set of slides, the overall correct classification of BC vs. LC cases was 82% using only one feature, fractal2_area. This classification rate could be increased to 91% when a discriminant function based on 10 features was used. However, this gain was not significant. CONCLUSION: Fractal texture features can be used to assess malignancy on tissue sections as an alternative to DNA measurement. In this study feature combinations did not significantly improve classification rates.     

Fractal dimension in human cerebellum measured by magnetic resonance imaging

Fractal dimension has been used to quantify the structures of a wide range of objects in biology and medicine. We measured fractal dimension of human cerebellum (CB) in magnetic resonance images of 24 healthy young subjects (12 men and 12 women). CB images were resampled to a series of image sets with different 3D resolutions. At each resolution, the skeleton of the CB white matter was obtained and the number of pixels belonging to the skeleton was determined. Fractal dimension of the CB skeleton was calculated using the box-counting method. The results indicated that the CB skeleton is a highly fractal structure, with a fractal dimension of 2.57 +/- 0.01. No significant difference in the CB fractal dimension was observed between men and women.     

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.     

Importance of interphase nucleolar organizer regions in tumor pathology

The importance of the distribution of silverstained nucleolar organizer regions (Ag-NORs) in interphase nuclei for diagnostic and prognostic purposes in tumor pathology has been reviewed. The available data demonstrated that interphase Ag-NOR evaluation may be of help in distinguishing malignant from hyperplastic or normal cells. On the other hand, there is increasing evidence that a relationship exists between the quantity of interphase Ag-NORs and the prognosis of malignant tumors: the greater the number of interphase Ag-NORs, the worse is the prognosis. This can be explained by the observation that the interphase Ag-NOR quantity is strictly related to the cell proliferation rate. The procedures used for the measurement of the interphase Ag-NOR quantity are also critically discussed.     

Importance of interphase nucleolar organizer regions in tumor pathology.

The importance of the distribution of silver-stained nucleolar organizer regions (Ag-NORs) in interphase nuclei for diagnostic and prognostic purposes in tumor pathology has been reviewed. The available data demonstrated that interphase Ag-NOR evaluation may be of help in distinguishing malignant from hyperplastic or normal cells. On the other hand, there is increasing evidence that a relationship exists between the quantity of interphase Ag-NORs and the prognosis of malignant tumors: the greater the number of interphase Ag-NORs, the worse is the prognosis. This can be explained by the observation that the interphase Ag-NOR quantity is strictly related to the cell proliferation rate. The procedures used for the measurement of the interphase Ag-NOR quantity are also critically discussed.     

恢复演替中草地斑块动态及尺度转换分析

研究了９ａ草地恢复油潜系列中斑块边界形状和斑块面积分布动态,并进行了凡度转换分析。獐茅斑块的边界分维数和斑块化指数最高,羊草斑块的较低,碱蓬斑块的斑块化指数略小于羊草斑块,边界分维数大于洋草斑块。共他类型斑块的两种指数基本上介于獐茅斑块之间,斑块化指数的年限变动滞后于斑块边界分维数,斑块边发维数在整个试验的尺度范围内符合同一自相似规律,斑块的面积分布格局在不同的尺度上有不同的自相似规律。     

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.     

Fundamental unpredictability in multispecies competition

One of the central goals of ecology is to predict the distribution and abundance of organisms. Here, we show that, in ecosystems of high biodiversity, the outcome of multispecies competition can be fundamentally unpredictable. We consider a competition model widely applied in phytoplankton ecology and plant ecology in which multiple species compete for three resources. We show that this competition model may have several alternative outcomes, that the dynamics leading to these alternative outcomes may exhibit transient chaos, and that the basins of attraction of these alternative outcomes may have an intermingled fractal geometry. As a consequence of this fractal geometry, it is impossible to predict the winners of multispecies competition in advance.     

Fractal dimension in endometrial carcinoma

OBJECTIVE: To mathematically assess in a pilot study, endometrial glandular margin irregularity in simple hyperplasia, complex atypical hyperplasia and well-differentiated endometrial carcinoma with the help of box counting of fractal dimension and to discriminate these lesions on the basis of box counting of fractal dimension of the gland. STUDY DESIGN: Ten cases each of endometrial simple hyperplasia (without atypia), complex hyperplasia with atypia and endometrial carcinoma (well-differentiated, endometrioid) were assessed in the study. Five fields at 20 x magnification from each case were randomly selected, and the glands were outlined with the help of a pointer. Using the box counting method, the fractal dimension of each case was measured. RESULTS: Mean fractal dimension in simple hyperplasia, complex atypical hyperplasia and endometrial carcinoma was, 0.899 +/- 0.13, 0.932 +/- 0.042 and 0.939 +/- 0.02, respectively. Statistical analysis showed that the fractal dimension of glands of simple hyperplasia were significantly different from that of complex atypical hyperplasia and endometrial carcinoma (P = .041 and .013, respectively, ANOVA). However, there was no significant difference in fractal dimension between glands of complex hyperplasia and of endometrial carcinoma (P = .659, ANOVA). CONCLUSION: This study provides mathematical (objective) assessment of the measurement of glandular margin irregularities in simple hyperplasia, complex atypical hyperplasia and endometrial carcinoma. Fractal dimension of gland margin may have diagnostic potential in the future.     

Paleopathological study on malignant bone tumor in Japan. Differential diagnosis on osteolytic lesions in the skull

This report concerns a probable case of metastatic malignant bone tumor in the skull of a prehistoric skeleton from Honshu Island in Japan. In gross observation, a fragmental skull of an adult showed many osteolytic lesions without any healing processes which could be also ascertained by roentgenological studies. Besides this case, four cases with a diagnosis of malignant bone tumor have so far been reported among archeological skeletal remains in Japan. This case from the prehistoric "Jomon" period is certainly the oldest case showing such malignant tumorous change in the skeleton. In this report, the osteolytic changes in the "Jomon" skull are described in detail and compared with two other cases showing the same osteolytic changes. The morphology and distribution of the lesions as well as the sex and age of the individual are discussed to make an adequate differential diagnosis for malignant osteolytic lesions in the skeleton.