This volume is number four in a series of proceedings volumes from the International Symposia on Fractals in Biology and Medicine in Ascona, Switzerland which have been inspired by the work of Benoît Mandelbrot seeking to extend the concepts towards the life sciences. It highlights the potential that fractal geometry offers for elucidating and explaining the complex make-up of cells, tissues and biological organisms either in normal or in pathological conditions, including the structural changes that occur in tumours. It helps develop the concepts, questions and methods required in research on fractal biology and natural phenomena and to evidence the pitfalls of a too simplistic application of these principles in investigating topical subjects of biology and medicine. It discusses present and future applications of fractal geometry, bringing together cellular and molecular biology, engineering, mathematics, physics, medicine and other disciplines and allowing an interdisciplinary vision. The book should be of interest to researchers and students from molecular and cell biology, biomedicine, biomathematics, analytical morphology, immunology and neurology who are interested in the combination of mathematics and life sciences.

Zusammenfassung
This book is a compilation of the presentations given at the Fourth International Symposium on Fractals in Biology and Medicine held in Ascona, Switzerland on - th 13 March 2004 and was dedicated to Professor Benoît Mandelbrot in honour of his 80 birthday. The Symposium was the fourth of a series that originated back in 1993, always in Ascona. The fourth volume consists of 29 contributions organized under four sections: Fractal structures in biological systems Fractal structures in neurosciences Fractal structures in tumours and diseases The fractal paradigm Mandelbrot's concepts such as scale invariance, self-similarity, irregularity and iterative processes as tackled by fractal geometry have prompted innovative ways to promote a real progress in biomedical sciences, namely by understanding and analytically describing complex hierarchical scaling processes, chaotic disordered systems, non-linear dynamic phenomena, standard and anomalous transport diffusion events through membrane surfaces, morphological structures and biological shapes either in physiological or in diseased states. While most of biologic processes could be described by models based on power law behaviour and quantified by a single characteristic parameter [the fractal dimension D], other models were devised for describing fractional time dynamics and fractional space behaviour or both (- fractional mechanisms), that allow to combine the interaction between spatial and functional effects by introducing two fractional parameters. Diverse aspects that were addressed by all bio-medical subjects discussed during the symposium.

Inhalt
Fractal Structures in Biological Systems.- Mandelbrot's Fractals and the Geometry of Life: A Tribute to Benoît Mandelbrot on his 80th Birthday.- Gas Diffusion through the Fractal Landscape of the Lung: How Deep Does Oxygen Enter the Alveolar System?.- Is the Lung an Optimal Gas Exchanger?.- 3D Hydrodynamics in the Upper Human Bronchial Tree: Interplay between Geometry and Flow Distribution.- Fractal Aspects of Three-Dimensional Vascular Constructive Optimization.- Cognition Network Technology: Object Orientation and Fractal Topology in Biomedical Image Analysis. Method and Applications.- The Use of Fractal Analysis for the Quantification of Oocyte Cytoplasm Morphology.- Fractal Structures in Neurosciences.- Fractal Analysis: Pitfalls and Revelations in Neuroscience.- Ongoing Hippocampal Neuronal Activity in Human: Is it Noise or Correlated Fractal Process?.- Do Mental and Social Processes have a Self-similar Structure? The Hypothesis of Fractal Affect-Logic.- Scaling Properties of Cerebral Hemodynamics.- A Multifractal Dynamical Model of Human Gait.- Dual Antagonistic Autonomic Control Necessary for 1/f Scaling in Heart Rate.- Fractal Structures in Tumours and Diseases.- Tissue Architecture and Cell Morphology of Squamous Cell Carcinomas Compared to Granular Cell Tumours' Pseudo-epitheliomatous Hyperplasia and to Normal Oral Mucosae.- Statistical Shape Analysis Applied to Automatic Recognition of Tumor Cells.- Fractal Analysis of Monolayer Cell Nuclei from Two Different Prognostic Classes of Early Ovarian Cancer.- Fractal Analysis of Vascular Network Pattern in Human Diseases.- Quantification of Local Architecture Changes Associated with Neoplastic Progression in Oral Epithelium using Graph Theory.- Fractal Analysis of Canine Trichoblastoma.- Fractal Dimension as aNovel Clinical Parameter in Evaluation of the Urodynamic Curves.- Nonlinear Dynamics in Uterine Contractions Analysis.- Computer-Aided Estimate and Modelling of the Geometrical Complexity of the Corneal Stroma.- The Fractal Paradigm.- Complex-Dynamical Extension of the Fractal Paradigm and its Applications in Life Sciences.- Fractal-like Features of Dinosaur Eggshells.- Evolution and Regulation of Metabolic Networks.- Cytoskeleton as a Fractal Percolation Cluster: Some Biological Remarks.- A Mystery of the Gompertz Function.- Fractional Calculus and Symbolic Solution of Fractional Differential Equations.- Fox-Function Representation of a Generalized Arrhenius Law and Applications.