## Fractals in Biology and Medicine, Volume 4Gabriele A. Losa, Theo F. Nonnenmacher, Ewald R. Weibel 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. |

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### Contents

Fractal Structures in Biological Systems | 1 |

A Tribute to Benoit Mandelbrot on his 80th Birthday | 3 |

How Deep Does Oxygen Enter the Alveolar System? | 17 |

Is the Lung an Optimal Gas Exchanger? | 31 |

Interplay between Geometry and Flow Distribution | 43 |

Fractal Aspects of ThreeDimensional Vascular Constructive Optimization | 55 |

Object Orientation and Fractal Topology in Biomedical Image Analysis Method and Applications | 67 |

The Use of Fractal Analysis for the Quantification of Oocyte Cytoplasm Morphology | 75 |

Fractal Analysis of Monolayer Cell Nuclei from Two Different Prognostic Classes of Early Ovarian Cancer | 175 |

Fractal Analysis of Vascular Network Pattern in Human Diseases | 187 |

Quantification of Local Architecture Changes Associated with Neoplastic Progression in Oral Epithelium using Graph Theory | 193 |

Fractal Analysis of Canine Trichoblastoma | 203 |

Fractal Dimension as a Novel Clinical Parameter in Evaluation of the Urodynamic Curves | 209 |

Nonlinear Dynamics in Uterine Contractions Analysis | 215 |

Computeraided Estimate and Modelling of the Geometrical Complexity of the Corneal Stroma | 223 |

The Fractal Paradigm | 231 |

Fractal Structures in Neurosciences | 83 |

Pitfalls and Revelations in Neuroscience | 85 |

Is it Noise or Correlated Fractal Process? | 95 |

Do Mental and Social Processes have a Selfsimilar Structure? The Hypothesis of Fractal AffectLogic | 107 |

Scaling Properties of Cerebral Hemodynamics | 121 |

A Multifractal Dynamical Model of Human Gait | 131 |

Dual Antagonistic Autonomic Control Necessary for 1f Scaling in Heart Rate | 141 |

Fractal Structures in Tumours and Diseases | 153 |

Tissue Architecture and Cell Morphology of Squamous Cell Carcinomas Compared to Granular Cell Tumours Pseudoepitheliomatous Hyperplasia an... | 155 |

Statistical Shape Analysis Applied to Automatic Recognition of Tumor Cells | 165 |

ComplexDynamical Extension of the Fractal Paradigm and its Applications in Life Sciences | 233 |

Fractallike Features of Dinosaur Eggshells | 245 |

Evolution and Regulation of Metabolic Networks | 257 |

Some Biological Remarks | 269 |

A Mystery of the Gompertz Function | 277 |

Fractional Calculus and Symbolic Solution of Fractional Differential Equations | 287 |

FoxFunction Representation of a Generalized Arrhenius Law and Applications | 299 |

309 | |

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acinus airways algorithm alveolar arterial biological blood branching bronchial tree calculated cancer cells cellular coefficient cognitive complexity components computed corneal stroma cytoplasm cytoskeleton described diameter diffusion distribution dynamics eggshell entropy production epithelial equation exponent Figure flow fractal analysis fractal dimension fractal geometry fractal structure fractional calculus fractional differential equations frequency gait gas exchange global Gompertz function heart rate human Hurst exponent hypercycle image object interaction ISI sequence length linear long-range correlations Losa lung Mandelbrot mathematical matrix mean mechanical mechanotransduction membrane metabolic method morphological multifractal neurons nodes Nonnenmacher normal nuclear nuclei obtained oocytes optimal oral ovarian cancer oxygen parameters patients pattern percolation phase Phys Physiol physiological pixels pore power law pressure processes profiles properties random scaling segments self-similar signals simulations solution space spatial statistical Strahler stride interval surface surrogates tissue tumor urodynamic values variability vascular network velocity Weibel

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Page ii - David Krakauer (Dept. of Zoology, Oxford) Robert T. Tranquillo (Dept. Chem. Engineering, Minneapolis) Mathematics and Biosciences in lnteraction is devoted to the publication of advanced textbooks, monographs, and multi-authored volumes on...

Page ii - ... concentrates on truly interdisciplinary research presenting currently important biological fields and relevant methods being developed and refined in close relation to problems and results relevant for experimental bioscientists. The series aims at publishing not only monographs by individual authors presenting their own results, but welcomes, in particular, volumes arising from collaborations, joint research programs or workshops. These can feature concepts and open problems as a result of such...