This volume is number four in a series of proceedings volumes from the International Symposia on Fractals in Biology and Medicine in Ascona, Switzerland. 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, abnormal and tumoral conditions. 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.

In March 2000 leading scientists gathered at the Centro Seminariale Monte Verita, Ascona, Switzerland, for the Third International Symposium on "Fractals 2000 in Biology and Medicine." This interdisciplinary conference was held over a four-day period and provided stimulating contributions from the very topical field Fractals in Biology and Medicine. This Volume III in the MBI series highlights the growing power and efficacy of the fractal geometry in understanding how to analyze living phenomena and complex shapes. Many biological objects, previously considered as hopelessly far from any quantitative description, are now being investigated by means of fractal methods. Researchers currently used fractals both as theoretical tools, to shed light on living systems self-organization and evolution, and as useful techniques, capable of quantitatively analyzing physiological and pathological cell states, shapes and ultrastructures. The book should be of interest to researchers and students from Molecular and C"

"Fractals in Biology and Medicine, Volume 2" explores the potential of the fractal geometry in understanding how to analyse natural shapes. The volume devotes special emphasis to the complex field of human tumours.

Stereologic techniques begin to play an increasing role in biologic morphology, particularly there where correlation of structure and function on a quantitative basis is sought. These powerful methods have been in use for many years - partly even for many decades - in geology, mineralogy and metallurgy, while attempts to introduce them into histology have remained rather rare until a few years ago. In order to stimulate discussion among anatomists about stereo logic methods the International Society for Stereology, an interdisciplinary society, organized a Sym posium on Quantitative Methods in Morphology which took place on August 10, 1965 in the framework of the Eighth International Congress of Anatomists in Wiesbaden, Germany. The papers presented at this symposium are published in this volume in slightly extended form. Some of the papers of this volume are of rather specialized nature and presume a basic knowledge of stereology. The first chapter on general stereological principles has therefore been considerably extended and short introductory review paragraphs have been added to a few subsequent chapters to help those who are not yet familiar with this new field in understanding the more specialized original articles. Long discussion periods formed an essential part of the symposium. However, they were conducted very informally and hence it would not have been profitable to reproduce them in extenso, particularly since the major results of discussion have been incorporated by the authors into the expanded manuscripts presented here.

The work presented in this monograph marks a new era, we believe, both in the development of quantitative anatomy of the lung, and in the correlation of anatomy with physiology. For many years, physiologists interested in the overall functioning of the lung have felt a need for better quantitative descriptions of pulmonary anatomy. As physiologists, we know a good deal about the forces operating to producepulmonary ventilation, and the quantities that define this function in rest and exercise ; and the same for effective distribution of air within the lung - "alveolar" ventilation-, and for the exchange of respiratory gases between air and blood. There have been no correspondingly precise quantitative measurements of the pulmonary structures that serve theese functions. The great advances in the study of pulmonary anatomy in the past decade have been chiefly in the realm of "fine structure". This has tended to bring together anatomy and biochemistry or physical chemistry, rather than anatomy and physiology. This conjunction has aided, for example, the conception of diffusion as a physicochemical process, but not that of diffusion as a metabolic bodily function. It was, therefore, a remarkably fortunate circumstance which brought together in our laboratory, about three years ago, Professor DoMINGO GoMEZ and Dr. EWALD R. WEIBEL: Professor GoMEZ a mathematician and biophysicist of dis tinction and long experience; Dr. WEIBEL a young anatomist trained under Pro fessor GIAN ToeNDURY in Zurich, and with additional research experience with Professor A VERILL LIEBOW at Y ale.

"Fractals in Biology and Medicine" explores the potential of fractal geometry for describing and understanding biological organisms, their development and growth as well as their structural design and functional properties. It extends these notions to assess changes associated with disease in the hope to contribute to the understanding of pathogenetic processes in medicine. The book is the first comprehensive presentation of the importance of the new concept of fractal geometry for biological and medical sciences. It collates in a logical sequence extended papers based on invited lectures and free communications presented at a symposium in Ascona, Switzerland, attended by leading scientists in this field, among them the originator of fractal geometry, Benoit Mandelbrot. "Fractals in Biology and Medicine" begins by asking how the theoretical construct of fractal geometry can be applied to biomedical sciences and then addresses the role of fractals in the design and morphogenesis of biological organisms as well as in molecular and cell biology. The consideration of fractal structure in understanding metabolic functions and pathological changes is a particularly promising avenue for future research.

In March 2000 leading scientists gathered at the Centro Seminariale Monte Verita, Ascona, Switzerland, for the Third International Symposium on "Fractals 2000 in Biology and Medicine." This interdisciplinary conference was held over a four-day period and provided stimulating contributions from the very topical field Fractals in Biology and Medicine. This Volume III in the MBI series highlights the growing power and efficacy of the fractal geometry in understanding how to analyze living phenomena and complex shapes. Many biological objects, previously considered as hopelessly far from any quantitative description, are now being investigated by means of fractal methods. Researchers currently used fractals both as theoretical tools, to shed light on living systems self-organization and evolution, and as useful techniques, capable of quantitatively analyzing physiological and pathological cell states, shapes and ultrastructures. The book should be of interest to researchers and students from Molecular and C"

The book discusses the controversial issue of whether animals are designed according to the same rules that engineers use in building machines, namely that materials and energy are used economically while attempting to achieve a high level of performance. There is considerable scientific controversy surrounding this question because, although there is much evidence suggesting that animals are indeed well designed, evolutionary biology tells us that animals are not 'engineered' but result from evolution by natural selection. This book collates this evidence which is discussed by a group of eminent biologists from many different biological disciplines."

"Fractals in Biology and Medicine" explores the potential of fractal geometry for describing and understanding biological organisms, their development and growth as well as their structural design and functional properties. It extends these notions to assess changes associated with disease in the hope to contribute to the understanding of pathogenetic processes in medicine. The book is the first comprehensive presentation of the importance of the new concept of fractal geometry for biological and medical sciences. It collates in a logical sequence extended papers based on invited lectures and free communications presented at a symposium in Ascona, Switzerland, attended by leading scientists in this field, among them the originator of fractal geometry, Benoit Mandelbrot. "Fractals in Biology and Medicine" begins by asking how the theoretical construct of fractal geometry can be applied to biomedical sciences and then addresses the role of fractals in the design and morphogenesis of biological organisms as well as in molecular and cell biology. The consideration of fractal structure in understanding metabolic functions and pathological changes is a particularly promising avenue for future research.

This book discusses the controversial issue of whether animals are designed according to the same rules that engineers use in building machines, namely that materials and energy are used economically while attempting to achieve a high level of performance. There is considerable scientific controversy surrounding this question because, although there is much evidence suggesting that animals are indeed well designed, evolutionary biology tells us that animals are not "engineered" but result from evolution by natural selection. In this volume these highly controversial questions are debated by eminent experts on the basis of a wealth of evidence ranging from the molecular biology and biochemistry of enzyme systems through the study of bone and muscle to the design and function of integrated systems of energy supply and the nervous system. The authors have made a special effort to present the chapters in a form that is accessible to a broad readership of biologists interested in basic principles.

"Fractals in Biology and Medicine, Volume 2" explores the potential of the fractal geometry in understanding how to analyse natural shapes. The volume devotes special emphasis to the complex field of human tumours.

It is rare indeed for one book to be both a first-rate classroom text and a major contribution to scholarship. "The Pathway for Oxygen" is such a book, offering a new approach to respiratory physiology and morphology that quantitatively links the two. Professionalism in science has led to a compartmentalization of biology. Function is the domain of the physiologist, structure that of the morphologist, and they often operate with vastly disparate concepts and procedures. Yet the performance of the respiratory system depends both on structural and on functional properties that cannot be separated.

The first chapter of "The Pathway for Oxygen" engages the student with the design and function of the vertebrate respiratory organs from a comparative viewpoint. The second chapter adds to that foundation the link between cell energetics and oxygen needs of the whole animal. With Chapter 3 the excitement begins--new ideas, fresh attacks on old problems, and a fuller account of the power of the quantitative approach Dr. Weibel has pioneered.

"The Pathway for Oxygen" will be read eagerly by medical students, graduate students, advanced undergraduates in zoology--and by their professors.

This book addresses a simple question: Are animals designed economically? The pronghorn can run at speeds of up to 60 kilometers an hour and can maintain this speed for nearly a full hour. Clearly, the form of this elegant animal is beautifully matched to the function it needs to perform.

This is symmorphosis. The theory of symmorphosis predicts that the size of the parts in a system must be matched to the overall functional demand. Moreover, it predicts that animals must provide their complex systems with a functional capacity that can cope with the highest expected functional demands, possibly including some safety margin to prevent the system from failing when it is overloaded. In "Symmorphosis," Ewald Weibel tests these predictions by working out the quantitative relations between form and function.

Physiologists will value this book because Weibel shows them that morphological information can be as quantitative as physiological data. Anatomists will value the book for its demonstration that advanced integrative physiology crucially depends on adequate but rigorously quantitative and testable information on structural design. Finally, anyone interested in the origins of the diverse forms of animals will be fascinated by Weibel's demonstrations that show how animals as different as shrews, pronghorns, dogs, goats--even humans--all develop from essentially the same blueprint by variation of design. This is a hidden beauty of the animal kingdom, which can be uncovered by a rigorous investigation of the quantitative relations of form and function.