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Analysis of the uterine contractility in non-pregnant women provides information about physiological changes during menstrual cycle. Spontaneous uterine activity was recorded directly by a micro-tip catheter (Millar Instruments, Inc. USA). The sensor produced an electrical signal, which varied in direct proportion to the magnitude of measured pressure. The study was approved by the regional ethics committee. We used the techniques of surrogate data analysis to testing for nonlinearity in the uterine contraction signals. Approximate entropy was the test statistic. For this analysis a healthy patient with normal contractions, a patient with dysmenorrhea and a patient with fibromyomas in the follicular phase were selected. The results showed that the spontaneous uterine contractions are considered to contain nonlinear features.

Despite the fact that all anatomical forms are characterised by non-polyhedral volumes, rough surfaces and irregular outlines, it has been suggested that sophisticated computer-aided analytical systems based on the Euclidean principles of regularity, smoothness and linearity can be used in human quantitative anatomy. However, the new fractal geometry is a more powerful means of quantifying the spatial complexity of real objects. The present study introduces the surface fractal dimension as a numerical index of the complex architecture of the corneal stroma, and investigates its behaviour during computer-simulated changes in keratocyte density and distribution, and in the heterogeneous composition of the extracellular matrix. We found that the surface fractal dimension depends on keratocyte density and distribution, as well as on the different concentrations of the constituents making up the extracellular matrix. Our results show that the surface fractal dimension could be widely used in ophthalmology not only because of its ability to quantify drug-correlated architectural changes, but also because it can stage corneal stroma alterations and predict disease evolution.

Complex-dynamical fractal is a hierarchy of permanently, chaotically changing versions of system structure, obtained as the unreduced, causally probabilistic general solution to an arbitrary interaction problem. Intrinsic creativity of this extension of usual fractality determines its exponentially high operation efficiency, which underlies many specific functions of living systems, such as autonomous adaptability, “purposeful” development, intelligence and consciousness (at higher complexity levels). We outline in more detail genetic applications of complex-dynamic fractality, demonstrate the dominating role of genome interactions, and show that further progressive development of genetic research, as well as other life-science applications, should be based on the dynamically fractal structure analysis of interaction processes involved. We finally summarise the obtained extension of mathematical concepts and approaches closely related to their biological applications.

We carried out measurements on serial sections through dinosaur eggshells found in the Hateg basin of Romania. The pore structure of these eggshells exhibits peculiar hierarchical selfsimilarity, from millimeter scale to nanoscale. At optical scale, the eggshell is built up by packing bundles of calcite carrots, well aligned with their axis perpendicular to the eggshell surface, with “gaps” (macro pores) in between. Each calcite carrot is about 0.5 mm in diameter and has the length equal to the eggshell thickness. Scanning electron microscopy (SEM) reveals that these carrots are in turn micro porous spongy calcite structure, with 1μm average pore diameter. The structure on an even smaller scale is studied by transmission electron microscopy (TEM) on thin sections prepared by ion milling, using methods from material science, revealing yet another layer of complexity. The observed features lead us to the conclusion that calcite crystallization leading to the carrot morphology is controlled from the nano- to micro-scale by the structure of the collagen net developed in the eggshell cells.

The analysis of metabolic processes, gene expression patterns, and protein-protein interactions in different organisms indicates that cellular metabolic networks have a scale-free and hierarchical topology described by power laws. The dynamics of these networks might be produced by a fractal organization of an autoregulatory loop, named metabolic hypercycle, between opposite redox processes of anabolic and catabolic types. This fractal architecture allows the formation of a long range correlated state of cellular networks which is globally regulated by a critical hub sensitive to the redox state. In prokaryotic cells this fundamental regulator is generally a two-component kinase system while in eukaryotic cells it is likely that casein kinase-2 and glycogen synthase kinase-3 play a central role in metabolism control. Both prokaryotes and eukaryotes share the same conserved sequence signatures, the PAS domain, in the main sensors of the changes in redox potential. Many experimental data support the hypothesis that the developmental pathways of cells and complex organisms are the results of conserved biological clocks based on metabolic hypercycles organized in fractal networks.

The possibility being discussed is that the cytoskeleton, the intricate polymeric meshwork which spans the cytoplasm, may be regarded as a percolation system and that at the edge of the percolation transition mechanotransduction may be enhanced. Since calcium ion can be considered the main factor controlling the state of the cytoskeletal network, it is hypothesized that the increase of free intracellular calcium which follows a mechanical stimulus may serve to “ loosen” the cytoskeletal network into a fractal percolation cluster, a partial sol state at which mechanotransduction is most efficient. It is also suggested that such a critical state represents an optimal condition for generation of mechanical forces.

The Gompertz function describes global dynamics of many natural processes including growth of normal and malignant tissues. On one hand, the Gompertz function defines a fractal. The fractal structure of time-space is a prerequisite condition for the coupling and Gompertzian growth. On the other hand, the Gompertz function is a probability function. Its derivative is a probability density function. Gompertzian dynamics emerges as a result of the co-existence of at least two antagonistic processes with the complex coupling of their probabilities. This dynamics implicates a coupling between time and space through a linear function of their logarithms. The spatial fractal dimension is a function of both scalar time and the temporal fractal dimension. The Gompertz function reflects the equilibrium between regular states with predictable dynamics and chaotic states with unpredictable dynamics; a fact important for cancer chemoprevention. We conclude that the fractal-stochastic dualism is a universal natural law of biological complexity.

The aim of this article is to present a practical introduction to fractional calculus. Fractional calculus is an old mathematical subject concerned with fractional derivatives. Fractional derivatives used in this paper are restricted to the Riemann-Liouville type. Based on the Riemann-Liouville calculus, we formulate fractional differential equations. Fractional differential equations are applied to models in relaxation and diffusion problems. Fractional calculus is used to formulate and to solve different physical models allowing a continuous transition from relaxation to oscillation phenomena. An application to an anomalous diffusion process demonstrates that the method used is also useful for more than one independent variable. Based on the theory of fractional derivatives and linear transformation theory, we demonstrate how symbolic calculations on a computer can be used to support practical calculations. The symbolic program FractionalCalculus based on Mathematica is used to demonstrate the solution of fractional differential equations step by step. The key method applied is linear transformation theory in connection with generalized functions.

In this contribution we will present a FOX H-FUNCTION formulation of a generalized exponential function (Arrhenius Law), which describes the central concept of anomalous particle transport including anomalous relaxation / diffusion processes, in disordered but scaling materials. We will develop a fractional concept for the mathematical description of anomalous relaxation processes based on linear fractional differential equations of type d ^α/ dt ^α where, 0 < α < 1, α is the order of fractional differentiation (α ≠ 1) : We also will present a transformation procedure for semi-fractional (α = 1/2, 3/2,...) linear differential equations to a system of integer number ordinary differential equations. This last formulation of the relaxation problem takes the term “fractals” out of the picture. As examples we compare our theoretical results on mechanical stress relaxation of a plastic material, and to the rebinding process of CO to myoglobin (Mb) after photodissociation for a test of the generalized Arrhenius Law.