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The concept of fractal geometry advanced by Mandelbrot since 1977 has brought new insight into the design of biological structures. Two fundamental geometrical forms abound: interfaces between different compartments with a very large surface within finite space, and branched trees that distribute blood and air into the tissue space. These structures show a level of complexity that is best described by fractal geometry. Thus, the surface area of cellular membranes as well as the gas exchange surface of the lung have a fractal dimension which is larger than 2. The design of the airway tree is described in quantitative terms and the functional consequences are discussed, both with respect to airflow in the bronchi and gas exchange in the acini. Similar conditions are described with respect to the blood vascular network. It is finally discussed whether fractal geometry plays a role in designing animals of greatly different body size from 2 g in a shrew to 500 kg in horses and steers. The scaling exponent of 3/4 for metabolic rate has been explained on a basis of two fractal models, but it is shown that this does not hold for maximal metabolic rate which is directly proportional to the surface of inner mitochondrial membrane that in turn has fractal properties. The concept of fractal geometry is valuable in understanding the design of biological structures at all levels of organization.

We investigate oxygen transport to and across alveolar membranes in the human lung, the last step in the chain of events that takes oxygen through the bronchial airways to the peripheral, acinar airways. This step occurs by diffusion. We carry out analytic and numerical computations of the oxygen current for fractal, space-filling models of the acinus, based on morphological data of the acinus and appropriate values for the transport constants, without adjustable parameters. The computations address the question whether incoming oxygen reaches the entire available membrane surface (reaction-limited, unscreened oxygen current), a large part of the surface (mixed reaction/diffusion-limited, partly screened current), or only the surface near the entrance of the acinus (diffusion-limited, completely screened current). The analytic treatment identifies the three cases as sharply delineated screening regimes and finds that the lung operates in the partial-screening regime, close to the transition to no screening, for respiration at rest; and in the no-screening regime for respiration at exercise. The resulting currents agree well with experimental values. We test the analytic treatment by comparing it with numerical results for two-dimensional acinus models and find very good agreement. The results provide quantitative support for the conclusion, obtained in other work, that the space-filling fractal architecture of the lung is optimal with respect to active membrane surface area and minimum power dissipation. At the level of the bronchial tree, we show that the space-filling architecture provides optimal slowing down of the airflow from convection in the bronchial airways to diffusion in the acinar airways.

We investigate gas transport and exchange in a model of the mammalian lung, from the perspective of thermodynamic optimization (second law energy efficiency). This approach to modeling the structure-function relation of the lung exploits the analogy between the respiratory organs and a chemical membrane reactor, and reveals that the design of the lung may be optimal for its function. We use methods from irreversible thermodynamics to give approximate expressions for the entropy production rate in the lung, and a variational approach to minimize the rate under meaningful functional constraints. The large-scale bronchial tree and small-scale alveolar sponge are modeled separately, to account for the different nature of mass-transport at the two scales (pressure-driven flow and diffusion, respectively). We prove that maximum energy efficiency requires equipartition of thermodynamic forces: pressure drop must be uniformly distributed across all the branches of the bronchial tree, and oxygen concentration drop must be uniformly distributed over the lung membrane. We show that the fractal-like architecture of the lung, the particular size of the gas-exchange units, and the subtle interplay between the airway tree and its vascular network are highly compatible with these requirements of equipartition.

Uniform flow distribution in a symmetric volume can be realized through a symmetric branched tree. It is shown however, by 3D numerical simulation of the Navier-Stokes equations, that the flow partitioning can be highly sensitive to deviations from exact symmetry if inertial effects are present. The flow asymmetry is quantified and found to depend on the Reynolds number. Moreover, for a given Reynolds number, we show that the flow distribution depends on the aspect ratio of the branching elements as well as their angular arrangement. Our results indicate that physiological variability should be severely restricted in order to ensure adequate fluid distribution through a tree. Time-dependant simulations have also been performed and have shown that inspiration and expiration flows are both subject to inertial effects but with completely different velocities structures.

We study various properties of constructive optimization in 3D vascular systems. After some remarks on existing approaches to vascular modeling and on the theory of vascular optimality, we briefly describe an algorithm called Global Constructive Optimization (GCO). Twenty-one vascular systems are modeled in three different groups: planar, spherical, and liver shaped. Based on the Strahler ordering scheme, these models are characterized and compared to data from liver corrosion casts. A good correspondence could be observed between modeled and real portal venous systems. The branching characteristics of the hepatic vein still pose open questions. Finally, a concept for the modeling of vascular interdigitation based on optimality principles is suggested.

Data analysis in general and image analysis in particular require multi-scale approaches when dealing with complex structures. Relational information between structures on different scales needs to be taken into account. In many application fields, automated image interpretation still is a significant bottleneck due to the lack of appropriate image analysis technology. A new approach, Cognition Network Technology, is presented that was developed to handle and analyze complex data. This contribution focuses on how it handles and analyzes image data based on an object oriented, hierarchical and networked data model. A specific programming language allows building a semantic knowledge base that is used to interpreting image data by creating and processing instances of this data model. In many operational analysis tasks the approach has proven to produce reliable results fully automatically. It especially extracts structures of interest even in challenging cases such as low signal to noise ratio images, heterogeneous or variable structures of interest or tasks which include a complex semantic.

The present study aimed at verifying whether immature cat oocytes with morphologic irregular cytoplasm display self-similar features to be analytically described by fractal analysis. Original images of oocytes collected by ovariectomy were acquired at a final magnification of 400 X with a CCD video camera connected to an optic microscope. After grey thresholding segmentation of cytoplasm, image profiles were submitted to fractal analysis by three different methods which yielded divergent fractal dimension (FD) values. The highest FD of 1.91 was measured on grey-dark cytoplasm characterized by highly connected network of lipid droplets and intracellular membranes. The fractal analysis provided an effective quantitative descriptor of the real cytoplasm morphology, without introducing any bias or shape approximation, which could contribute to an objective and reliable classification of feline oocytes.

Fractal analysis has become a popular method in all branches of scientific investigation including ecology, physics and medicine. The method is often used to determine effects such as impact of cattle grazing, the distribution of stars within a galaxy or whether tissue is pathological. However several aspects of fractal analysis are not often considered when interpreting results communicated in the literature. These include the concept that no presentation of any pattern on a computer, even for an ideal fractal, is truly fractal. Pre-processing that is also required, such as scanning of images and resizing play a role in the variation of the final fractal dimension. In addition D is also a function of the fractal analysis method used and how the final fractal dimension is determined. To obtain a better overview of the effects of the steps involved in fractal analysis and the utility of this method, this chapter describes, using biological material from neuroscience, a non fractal based method, Sholl analysis and continues by discussing various processing options and the results obtained using fractal analysis. The effect of different fractal analysis methods, different computer applications of the same method, scale and resolution as well as regression analysis, which is for most methods the final step in determining D are discussed. This provides a platform for a better understanding of fractal analysis in research fields other than physics and mathematics and a more meaningful interpretation of results.

The patterns of background or ongoing in vivo activity, even in the absence of any external stimulus, are quite irregular showing no clear structure or repetitiveness in the neuronal firing sequences. Consequently, the ongoing firing pattern of a neuron is mostly considered as a neuronal noise which is traditionally modeled as a stochastic Point process, i.e., renewal process which is devoid of any correlation between successive inter-spike-interval (ISI). But a recently emerging alternative view is that the ongoing activity may possess sptaio-temporally coherent patterns, a feature of fractal process with long-range correlation. Here, we investigated the nature of irregular fluctuations of ongoing neuronal firing pattern of neurons located in human hippocampus by the following methods: (i) detrended fluctuation analysis (DFA) , (ii) multiscale entropy (MSE) analysis, and (iii) convergence of the statistical moment analysis (CMA). Neuronal activity was recorded in the absence of any explicit cognitive task while the subjects were awake. Both the DFA and MSE analysis clearly show that the ongoing firing patterns are not well described by a renewal process, rather they show long-range power-law correlations, representing ongoing memory effects, which possibly arises from a fractal process. Further, these neurons showed slow convergence of statistical moments. Such long-range correlations are also corroborated by statistical control sequences. Neurons which exhibit long-range correlations also exhibit statistically nonsignificant correlations with other neighboring neurons. The presence of long-range correlations is a characteristic of fractal-like dynamics, representing memory or history in the firing patterns. We propose that this type of spatio-temporal correlations may be used to optimize information transfer and storage at hippocampal synapses. The presence of correlation in the ongoing pattern also suggests the influence of pre-stimulus sequence on shaping the post-stimulus responses. Further, these findings call for the modification of the existing neural modeling approaches.

The hypothesis is presented that psychosocial processes of any dimension have a common fractal structure, generated by self-similar interactions between emotion and cognition in both mental and social processes. According to the meta-theory of affect-logic, basic affects such as interest/curiosity, fear, rage, pleasure/joy and sadness represent specific patterns of energy dissipation, selected by evolution for their survival value. Omnipresent so-called operator effects of emotions on thought and behaviour appear as self-similar on any mental and social level, thus generating the postulated fractal structures. Empirical evidence from the following three domains of observation supporting this hypothesis is presented and discussed: Everyday mental and social phenomenology, preliminary results from a computer-simulation of elementary affect-like behaviour, and short-term and long-term evolution in schizophrenia.