A computational method for quantifying morphological variation in scleractinian corals

Morphological variation in marine sessile organisms is frequently related to environmental factors. Quantifying such variation is relevant in a range of ecological studies. For example, analyzing the growth form of fossil organisms may indicate the state of the physical environment in which the organism lived. A quantitative morphological comparison is important in studies where marine sessile organisms are transplanted from one environment to another. This study presents a method for the quantitative analysis of three-dimensional (3D) images of scleractinian corals obtained with X-ray Computed Tomography scanning techniques. The advantage of Computed Tomography scanning is that a full 3D image of a complex branching object, including internal structures, can be obtained with a very high precision. There are several complications in the analysis of this data set. In the analysis of a complex branching object, landmark-based methods usually do not work and different approaches are required where various artifacts (for example cavities, holes in the skeleton, scanning artifacts, etc.) in the data set have to be removed before the analysis. A method is presented, which is based on the construction of a medial axis and a combination of image-processing techniques for the analysis of a 3D image of a complex branching object where the complications mentioned above can be overcome. The method is tested on a range of 3D images of samples of the branching scleractinian coral Madracis mirabilis collected at different depths. It is demonstrated that the morphological variation of these samples can be quantified, and that biologically relevant morphological characteristics, like branch-spacing and surface/volume ratios, can be computed. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Morphological characterization of <Emphasis Type="Italic">Cupriavidus necator</Emphasis> DSM 545 flocs through image analysis

Flocculating agents are used as auxiliary to recover bacterial cells in downstream processes for polyhydroxyalkanoate production. However little is known about the Curpiavidus necator flocs. In this work a new procedure for floc characterization through digital image analysis is presented and validated using the batch settling test. Average diameter, particle size distribution and morphological characteristics of the microbial aggregates were obtained from the flocculation/sedimentation process of the Cupriavidus necator DSM 545 cells by the use of tannin as flocculating agent. The experimental results demonstrated that the proposed method is adequate to determine the average floc diameter with values around 150 μm in accordance with the value obtained from the batch settling test. Nevertheless a morphological characterization of Cupriavidus necator DSM 545 bioaggregates in terms of size distribution and regularity could only be performed by an image analysis procedure. The procedure allowed us to describe the regularity of bacterial flocs through the quantification of morphological parameters of Euclidean [convexity (Conv) and form factor (FF)] and fractal geometry [surface fractal dimension (D _BS)], which are important factors to be considered in the settling efficiency of aggregates.     

Branching tree model with fractal vascular resistance explains fractal perfusion heterogeneity

Perfusion heterogeneities in organs such as the heart obey a power law as a function of scale, a behavior termed "fractal." An explanation of why vascular systems produce such a specific perfusion pattern is still lacking. An intuitive branching tree model is presented that reveals how this behavior can be generated as a consequence of scale-independent branching asymmetry and fractal vessel resistance. Comparison of computer simulations to experimental data from the sheep heart shows that the values of the two free model parameters are realistic. Branching asymmetry within the model is defined by the relative tissue volume being fed by each branch. Vessel ordering for fractal analysis of morphology based on fed or drained tissue volumes is preferable to the commonly used Strahler system, which is shown to depend on branching asymmetry. Recently, noninvasive imaging techniques such as PET and MRI have been used to measure perfusion heterogeneity. The model allows a physiological interpretation of the measured fractal parameters, which could in turn be used to characterize vascular morphology and function.     

Simulation of growth of a filamentous fungus in 3 dimensions

The tridimensional growth of a filamentous fungus was simulated, based on a model for the evolution of the microscopic morphology of Trichoderma reesei. When supplemented with a spatial representation of growth, the model correctly simulates the evolution from a single spore to a pellet. Diffusion of oxygen is included in the model. The simulated tridimensional structures have a fractal nature; and the fractal dimension, determined by a box-counting method, increases during growth. The fractal dimension only depends on the mass of the pellet and is not affected by model parameters such as tip extension rate and branching frequency. Realistic pictures are obtained and the radius of the pellet increases at a constant rate. The influence of model parameters (tip extension rate, branching frequency, minimum porosity) on dissolved oxygen concentration profiles, biomass concentration profiles, rate at which the pellet diameter increases, and the evolution of the fractal dimension was determined. The dissolved oxygen profiles were found to be very different from the profiles, obtained by assuming a homogenous biomass distribution within the pellet. Finally, the formation of pellets from spore aggregates is calculated and the size of the spore aggregate is found to only influence the time needed before the appearance of a pellet and not its morphology. (c) 1997 John Wiley & Sons, Inc.     

A class of flow bifurcation models with lognormal distribution and fractal dispersion

We report a quantitative analysis of a simple dichotomous branching tree model for blood flow in vascular networks. Using the method of moment-generating function and geometric Brownian motion from stochastic mathematics, our analysis shows that a vascular network with asymmetric branching and random variation at each bifurcating point gives rise to an asymptotic lognormal flow distribution with a positive skewness. The model exhibits a fractal scaling in the dispersion of the regional flow in the branches. Experimentally measurable fractal dimension of the relative dispersion in regional flow is analytically calculated in terms of the asymmetry and the variance at local bifurcation; hence the model suggests a powerful method to obtain the physiological information on local flow bifurcation in terms of flow dispersion analysis. Both the fractal behavior and the lognormal distribution are intimately related to the fact that it is the logarithm of flow, rather than flow itself, which is the natural variable in the tree models. The kinetics of tracer washout is also discussed in terms of the lognormal distribution.     

Analysis of fractal dimension of O2A glial cells differentiating in vitro

Fractal dimension is a quantitative measure of morphological complexity. Glial cells of the oligodendrocyte-type 2 astrocyte (O2A) lineage exhibit increasing morphological complexity as they differentiate in vitro. Enriched populations of O2A progenitor cells isolated from neonatal rat cerebral hemispheres or optic nerves were allowed to differentiate in vitro, and their fractal dimensions were measured over time. The fractal dimensions of the maturing cells correlated with perceived complexity; cells with elaborate process branching had larger fractal dimensions than cells with a simpler morphology. An analysis of changes in fractal dimension revealed distinct rates of growth for both oligodendrocytes and type 2 astrocytes. The fractal dimension remained constant over a 10-fold range in optical magnification, demonstrating that cultured O2A glial cells exhibit self-similarity, a defining characteristic of fractal objects. These results illustrate that fractal dimension analysis of maturing cell populations is a useful method for quantitatively describing the process of cell differentiation.     

Spatial Structure of the Branching Streamer Channels in a Corona Discharge

The dendritic structure of streamer channels in a corona discharge is described by using fractal theory. It is found that, for a needle-plane discharge, the fractal dimension of the plasma structure is D = 2.16 ± 0.05. The computed spatial distributions of the branching ratios are compared with the available experimental data. The influence of the branching processes on the distribution of chemically active radicals in streamer corona discharges is studied.     

A temporal model of animal behavior based on a fractality in the feeding of Drosophila melanogaster

We present a new temporal model of animal behavior based on the ethological idea that the internal states of the individual essentially determine the behavior. The internal states, however, are conditioned by the external stimuli. This model, including environmental and internal parameters, predicts a fractal property of the behavior, that is, an inverse power law distribution of the duration. Being consistent with the model, we have found a fractal property of feeding in Drosophila melanogaster: The dwelling time of starved flies on food showed a clear inverse power law distribution. The dependence of the fractal dimension on the intensity of food stimuli has been observed, and the predicted change into an exponential distribution was proved.     

Arterial branching within the confines of fractal L-system formalism

Parametric Lindenmayer systems (L-systems) are formulated to generate branching tree structures that can incorporate the physiological laws of arterial branching. By construction, the generated trees are de facto fractal structures, and with appropriate choice of parameters, they can be made to exhibit some of the branching patterns of arterial trees, particularly those with a preponderant value of the asymmetry ratio. The question of whether arterial trees in general have these fractal characteristics is examined by comparison of pattern with vasculature from the cardiovascular system. The results suggest that parametric L-systems can be used to produce fractal tree structures but not with the variability in branching parameters observed in arterial trees. These parameters include the asymmetry ratio, the area ratio, branch diameters, and branching angles. The key issue is that the source of variability in these parameters is not known and, hence, it cannot be accurately reproduced in a model. L-systems with a random choice of parameters can be made to mimic some of the observed variability, but the legitimacy of that choice is not clear.     

Fractal and spectral dimension analysis of liver fibrosis in needle biopsy specimens

OBJECTIVE: To evaluate the usefulness of a reliable and reproducible mathematical scoring system based on fractal geometry for quantifying the irregular pattern in fibrosis commonly seen in liver biopsy specimens from chronic liver diseases. STUDY DESIGN: The study used 26 standard liver biopsy specimens obtained from patients with chronic hepatitis C virus-related liver disease. The degree of fibrosis in each specimen was estimated using a quantitative scoring system based on the computer-assisted evaluation of both the fractal and spectral dimensions of deposited collagen. The fractal dimension was then compared with the percent area of collagen measured using an image analysis system. RESULTS: The fractional dimension of its irregular shape defines fibrosis as a natural fractal structure. The complex distribution of its collagenous components (unmeasurable by means of the usual morphometric parameters) can be optimally quantified using a single numerical score that seems to be a better alternative to the semiquantitative methods adopted so far. The proposed method is reproducible, rapid and inexpensive; furthermore, supported by specific software, its mathematical approach excludes subjectivity and eliminates the external factors capable of influencing staging and classification. CONCLUSION: This study demonstrated that it is possible to quantify the irregularity of the structures of the liver in an objective manner and that the box-counting fractal dimension does not depend on the amount of collagen deposited on the slide. Furthermore, as has been found in other fields of investigation, study of the fractal properties of the liver is likely to reveal more about its structure and the pathogenesis of liver diseases.     

Fractal analysis of vascular networks: Insights from morphogenesis

Considering their extremely complicated and hierarchical structure, a long standing question in vascular physio-pathology is how to characterize blood vessels patterns, including which parameters to use. Another question is how to define a pertinent taxonomy, with applications to normal development and to diagnosis and/or staging of diseases.To address these issues, fractal analysis has been applied by previous investigators to a large variety of healthy or pathologic vascular networks whose fractal dimensions have been sought. A review of the results obtained on healthy vascular networks first shows that no consensus has emerged about whether normal networks must be considered as fractals or not.Based on a review of previous theoretical work on vascular morphogenesis, we argue that these divergences are the signature of a two-step morphogenesis process, where vascular networks form via progressive penetration of arterial and venous quasi-fractal arborescences into a pre-existing homogeneous capillary mesh. Adopting this perspective, we study the multi-scale behavior of generic patterns (model structures constructed as the superposition of homogeneous meshes and quasi-fractal trees) and of healthy intracortical networks in order to determine the artifactual and true components of their multi-scale behavior. We demonstrate that, at least in the brain, healthy vascular structures are a superposition of two components: at low scale, a mesh-like capillary component which becomes homogeneous and space-filling over a cut-off length of order of its characteristic length; at larger scale, quasi-fractal branched (tree-like) structures. Such complex structures are consistent with all previous studies on the multi-scale behavior of vascular structures at different scales, resolving the apparent contradiction about their fractal nature.Consequences regarding the way fractal analysis of vascular networks should be conducted to provide meaningful results are presented. Finally, consequences for vascular morphogenesis or hemodynamics are discussed, as well as implications in case of pathological conditions, such as cancer.     

Computer simulation of the geometry of the human bronchial tree

Some morphological features of the human bronchial tree were simulated by computergenerated trees. The trees were ordered by the methods of Horsfield and Strahler. Delta, the difference between the Horsfield orders of the two branches at a bifurcation, was determined by pseudorandom numbers generated according to a distribution of probabilities defined on input. By trial and error a distribution was found which resulted in trees being generated with average Strahler order branching ratios of 2.82, similar to a real bronchial tree. Branching angles and length ratio could also be defined on input. By varying these input parameters it was found that the form of the tree was quite sensitive to them, and that by a suitable choice the intrasegmental part of the bronchial tree could be simulated. It is concluded that branching ratio, length ratio, mean branching angles and distribution of delta are controlled within tight limits in the bronchial tree, and this may support the concept of optimal design.     

Modeling of Branching Patterns in Plants

A major determinant of plant architecture is the arrangement of branches around the stem, known as phyllotaxis. However, the specific form of branching conditions is not known. Here we discuss this question and suggest a branching model which seems to be in agreement with biological observations. Recently, a number of models connected with the genetic network or molecular biology regulation of the processes of pattern formation appeared. Most of these models consider the plant hormone, auxin, transport and distribution in the apical meristem as the main factors for pattern formation and phyllotaxis. However, all these models do not take into consideration the whole plant morphogenesis, concentrating on the events in the shoot or root apex. On the other hand, other approaches for modeling phyllotaxis, where the whole plant is considered, usually are mostly phenomenological, and due to it, do not describe the details of plant growth and branching mechanism. In this work, we develop a mathematical model and study pattern formation of the whole, though simplified, plant organism where the main physiological factors of plant growth and development are taken into consideration. We model a growing plant as a system of intervals, which we will consider as branches. We assume that the number and location of the branches are not given a priori, but appear and grow according to certain rules, elucidated by the application of mathematical modeling. Four variables are included in our model: concentrations of the plant hormones auxin and cytokinin, proliferation and growth factor, and nutrients—we observe a wide variety of plant forms and study more specifically the involvement of each variable in the branching process. Analysis of the numerical simulations shows that the process of pattern formation in plants depends on the interaction of all these variables. While concentrations of auxin and cytokinin determine the appearance of a new bud, its growth is determined by the concentrations of nutrients and proliferation factors. Possible mechanisms of apical domination in the frame of our model are discussed.     

Applications of fractal analysis to physiology

This review describes approaches to the analysis of fractal properties of physiological observations. Fractals are useful to describe the natural irregularity of physiological systems because their irregularity is not truly random and can be demonstrated to have spatial or temporal correlation. The concepts of fractal analysis are introduced from intuitive, visual, and mathematical perspectives. The regional heterogeneities of pulmonary and myocardial flows are discussed as applications of spatial fractal analysis, and methods for estimating a fractal dimension from physiological data are presented. Although the methods used for fractal analyses of physiological data are still under development and will require additional validation, they appear to have great potential for the study of physiology at scales of resolution ranging from the microcirculation to the intact organism.     

Lag time in microbe growth as an age-dependent branching process with two phase types

We study a two-type, age-dependent branching process in which the branching probabilities of one of the types may vary with time. Specifically this modification of the Bellman-Harris process starts with a Type I particle which may either die or change to a Type II particle depending upon a time varying probability. A Type II particle may either die or reproduce with fixed probabilities but may not return to a particle of Type I. In this way the process models the lag phenomenon observed in microbe growth subsequent to transfer to a new culture medium while the organism is adapting to its new environment. We show that if the mean reproduction rate of Type II particles exceeds 1, then the population size grows exponentially. Further the extinction probability for this process is related to that of the Bellman-Harris process. Finally the governing equations are solved for several choices of the growth parameters and the solutions are graphically displayed showing that a wide variety of behavior can be modeled by this process.     

北京东灵山地区辽东栎林种群空间分布分形分析

通过植物个体的坐标点与模拟冠幅的分形维数的分析,对辽东栎林内不同种群沿海拔梯度变化的空间分布格局和种群动态进行了比较和讨论。结果表明:模拟冠幅分形分析方法更适于分析具有大小不等、复杂多样的冠幅的植物个体空间分布格局。随着海拔的升高,不同种群的空间占据能力也随之变化。辽东栎种群的空间分布占绝对优势地位,分形维数可达1.9811;五角枫种群则逐步下降,分形维数最低为0.1170。低海拔山坡六道木种群空间占据能力较强,高海拔山坡照山白种群有较大的空间占据能力。在一定的环境条件下,不同的种群可能具有相近的空间占据能力,但对乔木层与灌木层来说,相同的分形维数的内涵是不同的。冠幅的分形维数作为表征植物种群空间占据能力的工具,是种群动态分析和种群分布格局研究的的重要指标之一。     

The theory of functional systems and probabilistic prognosis of behavior

The article is dedicated to possible probability prediction of behavior in system organization of behavioral acts. System mechanisms of anticipation of required results of behavioral activity by living organisms adapted to stable and changing conditions of life are discussed. The author proposes that in all forms of behavior, an organism strictly predicts parameters for satisfaction of dominating needs of results that constitute the goal of a given form of behavior. In author's opinion, the probabilistic prognosis concerns only methods, acts and attendant emotional states, as well as possible ways of attaining the results (i.e., the means rather than action parameters).     

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.     

民勤荒漠植物枝系构型的分类研究

荒漠植物的枝系构型因素包括各级分枝角度、各级分枝长度、枝径比、逐步分枝率和总体分枝率、分枝分维数和计盒维数等16个指标。采用组内欧式距离法进行聚类,把荒漠植物依构型指标分为4个类型：第一类型包括霸王（Zygophyllum xanthoxylum）、黄刺条（Caragana frutex）等14种荒漠植物;第二类型包括秦晋锦鸡儿（C．purdomii）、荒漠锦鸡儿（C．roborovskyi）等11种荒漠植物;第三类型包括网状沙拐枣（Calligonum cancellatum）、黄花草木樨（Melilotus suaveolens）等9种荒漠植物;第四类型包括扁果木蓼（Atraphaxis replicta）、洋白蜡（Frawinus amerirana）等14种荒漠植物。不同的枝系构型类型反映了不同荒漠植物对于空间资源与环境的长期适应对策。