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 Mandelbrots 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.

Fractal Structures in Biological Systems.- Mandelbrots 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.