Fractals in biology and medicine vol 4

Mathematics and Biosciences in Interaction

Managing Editor

Wolfgang Alt

Division of Theoretical Biology

Botanical Institute

University of Bonn

Kirschallee 1

D-53115 Bonn

e-mail: [email protected]

Editorial Board

Fred Adler (Dept. Mathematics, Salt Lake City)

Mark Chaplain (Dept. Math. & Computer Sciences, Dundee)

Andreas Deutsch (Div. Theoretical Biology, Bonn)

Andreas Dress (Center for Interdisciplinary Research for Structure Formation (CIRSF), Bielefeld)

David Krakauer (Dept. of Zoology, Oxford)

Robert T. Tranquillo (Dept. Chem. Engineering, Minneapolis)

Mathematics and Biosciences in Interaction is devoted to the publication of advanced textbooks, mono￾graphs, and multi-authored volumes on mathematical concepts in the biological sciences. It concentra￾tes 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 works￾hops. These can feature concepts and open problems as a result of such collaborative work, possibly

illustrated with computer software providing statistical analyses, simulations or visualizations.

The envisaged readership includes researchers and advanced students in applied mathematics –

numerical analysis as well as statistics, genetics, cell biology, neurobiology, bioinformatics, biophysics,

bio(medical) engineering, biotechnology, evolution and behavioral sciences, theoretical biology, system

theory.

Gabriele A. Losa

Danilo Merlini

Theo F. Nonnenmacher

Ewald R. Weibel

Editors

Birkhäuser

Basel • Boston • Berlin

FRACTALS in

BIOLOGYand

MEDICINE

Volume IV

Editors:

Prof. Dr. Gabriele A. Losa

Institute for Scientific Interdisciplinary Studies (ISSI)

via F. Rusca 1

CH-6600 Locarno

and

Faculty of Biology and Medicine

University of Lausanne

CH-1000 Lausanne

e-mail: [email protected]

Prof. Dr. Theo F. Nonnenmacher

Department of Mathematical Physics

University of Ulm

Albert-Einstein-Allee 11

D-89069 Ulm

A CIP catalogue record for this book is available form the Library of Congress, Washington D.C., USA

Bibliographic information published by Die Deutsche Bibliothek

Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;

detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.

The use of registered names, trademarks etc. in this publication, even if not identified as such, does not imply

that they are exempt from the relevant protective laws and regulations or free for general use.

ISBN 3-7643-7172-2 Birkhäuser Verlag, Basel - Boston - Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,

specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on

microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner

must be obtained.

© 2005 Birkhäuser Verlag, P.O. Box 133, CH-4001 Basel, Switzerland

Part of Springer Science+Business Media

Printed on acid-free paper produced from chlorine-free pulp. TFC '

Cover design: Armando Losa, SGF Graphic Designer, Locarno

Cover illustration: With the friendly permission of Gabriele A. Losa

Printed in Germany

ISBN-10: 3-7643-7172-2

ISBN-13: 978-3-7643-7172-2

9 8 7 6 5 4 3 2 1 www.birkhauser.ch

Prof. Dr. Danilo Merlini

Research Center for Physics and Mathematics

via F. Rusca 1

CH-6600 Locarno

Prof. Dr. Ewald R. Weibel

Institute of Anatomy

University of Berne

Baltzerstrasse 2

CH-3000 Bern 9

Contents

Foreword ...................................................................................................................... ix

Fractal Structures in Biological Systems ................................................................... 1

Mandelbrot’s Fractals and the Geometry of Life: A Tribute to Benoît Mandelbrot

on his 80th Birthday

E.R. Weibel ..................................................................................................................... 3

Gas Diffusion through the Fractal Landscape of the Lung: How Deep Does

Oxygen Enter the Alveolar System?

C. Hou, S. Gheorghiu, M.-O. Coppens, V.H. Huxley and P. Pfeifer ............................. 17

Is the Lung an Optimal Gas Exchanger?

S. Gheorghiu, S. Kjelstrup, P. Pfeifer and M.-O. Coppens ........................................... 31

3D Hydrodynamics in the Upper Human Bronchial Tree: Interplay between

Geometry and Flow Distribution

B. Mauroy ...................................................................................................................... 43

Fractal Aspects of Three-Dimensional Vascular Constructive Optimization

H.K. Hahn, M. Georg and H.-O. Peitgen ...................................................................... 55

Cognition Network Technology: Object Orientation and Fractal Topology in

Biomedical Image Analysis. Method and Applications

M. Baatz, A. Schäpe, G. Schmidt, M. Athelogou and G. Binnig .................................... 67

The Use of Fractal Analysis for the Quantification of Oocyte Cytoplasm

Morphology

G.A. Losa, V. Peretti, F. Ciotola, N. Cocchia and G. De Vico....................................... 75

Fractal Structures in Neurosciences ........................................................................... 83

Fractal Analysis: Pitfalls and Revelations in Neuroscience

H.F. Jelinek, N. Elston and B. Zietsch ............................................................................ 85

Ongoing Hippocampal Neuronal Activity in Human: Is it Noise or Correlated

Fractal Process?

J. Bhattacharya, J. Edwards, A. Mamelak and E.M. Schuamn ...................................... 95

Do Mental and Social Processes have a Self-Similar Structure? The Hypothesis

of Fractal Affect-Logic

L. Ciompi and M. Baatz ................................................................................................ 107

vi Contents

Scaling Properties of Cerebral Hemodynamics

M. Latka, M. Turalska, D. Kolodziej, D. Latka, B. Goldstein and B.J. West .............. 121

A Multifractal Dynamical Model of Human Gait

B.J. West and N. Scafetta ............................................................................................ 131

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

Z.R. Struzi, J. Hayano, S. Sakata, S. Kwak, Y. Yamamoto ......................................... 141

Fractal Structures in Tumours and Diseases .......................................................... 153

Tissue Architecture and Cell Morphology of Squamous Cell Carcinomas

Compared to Granular Cell Tmours’ Pseudo-epitheliomatous Hyperplasia

and to Normal Oral Mucosae

R. Abu-Eid and G. Landini ........................................................................................... 155

Statistical Shape Analysis Applied to Automatic Recognition of

Tumor Cells

A. Micheletti .................................................................................................................. 165

Fractal Analysis of Monolayer Cell Nuclei from Two Different Prognostic

Classes of Early Ovarian Cancer

B. Nielsen, F. Albregtsen and H.E. Danielsen ..............................................................175

Fractal Analysis of Vascular Network Pattern in Human Diseases

G. Bianciardi, C. De Felice, R. Cattaneo, S. Parrini, A. Monaco and G. Latini ......... 187

Quantification of Local Architecture Changes Associated with Neoplastic

Progression in Oral Epithelium using Graph Theory

G. Landini and I.E. Othman ......................................................................................... 193

Fractal Analysis of Canine Trichoblastoma

G. de Vico, M. Cataldi, P. Maiolino, S. Beltraminelli and G.A. Losa ......................... 203

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

Curves

P. Waliszewski, U. Rebmann and J. Konarski ............................................................. 209

Nonlinear Dynamics in Uterine Contractions Analysis

E. Oczeretko, A. Kitlas, J. Swiatecka, M. Borowska and T. Laudanski ...................... 215

Computer-Aided Estimate and Modelling of the Geometrical Complexity

of the Corneal Stroma

F. Grizzi, C. Russo, I. Torres-Munoz, B. Franceschini, P. Vinciguerra

and N. Dioguardi ........................................................................................................ 223

Contents vii

The Fractal Paradigm ................................................................................................ 231

Complex-Dynamical Extension of the Fractal Paradigm and Its Applications

in Life Science

A.P. Kirilyuk ................................................................................................................. 233

Fractal-like Features of Diosaur Eggshells

M.V. Rusu and S.Gheorghiu ......................................................................................... 245

Evolution and Regulation of Metabolic Networks

G. Damiani ................................................................................................................... 257

Cytoskeleton as a Fractal Percolation Cluster: Some Biological Remarks

S. Traverso ................................................................................................................... 269

A Mistery of the Gompertz Function

P.Waliszewski and J. Konarski .................................................................................... 277

Fractional Calculus and Symbolic Solution of Fractional Differential Equations

G. Baumann .................................................................................................................. 287

Fox-Function Representation of a Generalized Arrhenius Law and Applications

T.F. Nonnenmacher ...................................................................................................... 299

Index ............................................................................................................................ 309

Foreword

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 10-

13 March 2004 and was dedicated to Professor Benoît Mandelbrot in honour of his 80th

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:

x Fractal structures in biological systems

x Fractal structures in neurosciences

x Fractal structures in tumours and diseases

x 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 (bi￾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.

We are especially grateful to Professor Benoît Mandelbrot for his public

presentation on < Fractales, Hasard et Aléas de la Bourse > held in the Palazzo

Corporazione Borghese at Locarno and for his active and critical participation during all

the Symposium as well.

We are particularly indebted to the following institutions for their support:

International Society for Stereology, Swiss National Science Foundation, Italian Society

for Microscopic Sciences, Institute for Scientific Interdisciplinary Studies, Research

Center for Mathematics and Physics, Rete Due of the Swiss Italian Broadcasting, and

Department of Education Culture and Sport of the Republic of Cantone Ticino, who

accepted to confer their scientific and cultural patronage and also to the sponsors,

Department of Education Culture and Sport of the Republic of Cantone Ticino, Swiss

National Science Foundation, the Majors of Ascona, Bellinzona and Locarno, Rete Due

of the Swiss Italian Broadcasting, UBS SA Locarno, and Cagi Cantina Giubiasco.

Our thanks are also due to Professor Mauro Martinoni, head of the Ufficio Studi

Universitari of the Cantone Ticino for his kind collaboration and precious support.

Monte Verità, Ascona 2004 The Editors

Fractal Structures in Biological Systems

Mandelbrot's Fractals and the Geometry of Life:

A Tribute to Benoît Mandelbrot on his 80th Birthday

Ewald R. Weibel

Department of Anatomy, University of Berne, Bühlstrasse 26, CH-3000 Bern 9, Switzerland

Summary. The concept of fractal geometry advanced by Mandelbrot since 1977 has brought new insight

into the design of biological structures. Two fundamental geometrical forms abound: interfaces between

different compartments with a very large surface within finite space, and branched trees that distribute

blood and air into the tissue space. These structures show a level of complexity that is best described by

fractal geometry. Thus, the surface area of cellular membranes as well as the gas exchange surface of the

lung have a fractal dimension which is larger than 2. The design of the airway tree is described in quanti￾tative terms and the functional consequences are discussed, both with respect to airflow in the bronchi and

gas exchange in the acini. Similar conditions are described with respect to the blood vascular network. It

is finally discussed whether fractal geometry plays a role in designing animals of greatly different body

size from 2 g in a shrew to 500 kg in horses and steers. The scaling exponent of 3/4 for metabolic rate has

been explained on a basis of two fractal models, but it is shown that this does not hold for maximal meta￾bolic rate which is directly proportional to the surface of inner mitochondrial membrane that in turn has

fractal properties. The concept of fractal geometry is valuable in understanding the design of biological

structures at all levels of organization.

1 Introduction

I first met Benoît Mandelbrot in 1977 in Paris at a symposium on "Geometric

probability and biological structures" organized to commemorate 200 years of the Buf￾fon needle problem, the first exercise in geometrical statistics [1]. Mandelbrot was one

of the keynote speakers (Fig. 1). It was the year in which he published his book "Frac￾tals: Form, Chance and Dimension" [2] which marked the beginning of a new way of

describing the structure of natural objects. It turned out that Mandelbrot's concept of

fractal geometry gained significant influence on the way we now describe the geometric

design of living systems, of what forms the geometry of life. But it also had a signifi￾cant impact on the further development of a quantitative approach to the study of inter￾nal life forms with the methods of stereology. The Buffon symposium was related to the

theoretical foundations of stereology, but it was somehow akin to the concept of frac￾tals. In 1777, the great naturalist Buffon asked the French Academy of Sciences: what is

the probability P that a randomly tossed needle of length l intersects a set of parallel

lines spaced at an equal distance d, e.g. the seam lines of a parquet floor? Buffon solved

the problem himself: considering the chances to have the needle at different orientations

(angles) and different distances from the lines he derived P = (2/ʌ)·(l/d). The further

developments of this principle have led to stereological methods by which, for example,

the surface area of membranes is estimated by probing the tissue with needles [4]. The

Buffon needle problem thus was perhaps the first realization of "Form, Chance and Di￾mension".

4 E.R. Weibel

Figure 1. Benoît Mandelbrot and participants of the Buffon symposium of 1977 beneath the statue of

Georges-Louis Leclerc Comte de Buffon at the Jardin des Plantes in Paris.

At the Buffon symposium Mandelbrot talked about "the fractal geometry of

trees and other natural phenomena" [5] among which were the structure of natural

boundaries which are never simple, and the systematic structure of trees that abound in

nature, in animals for example in the form of blood vessels and airways. By presenting

these fundamental concepts Mandelbrot opened the eyes of biological morphologists for

unpredicted complexities in the structure of internal organs. I will, in the following give

but a few indications to how the concept of fractal geometry has changed our views of

biological design.

2 A Fractal Look at Biological Surfaces

Many biological processes such as the exchange of substances or chemical reac￾tions, take place at interfaces between different compartments of the cell or the body

and this is why cellular membranes are such a prevalent and important structural entity.

The quantitative description of structure-function relationships therefore often depends

on the measurement of the surface area of such membranes. This is for example the case

in the lung where oxygen is transferred from the air to the blood across a large surface

area, or in the liver cells where an extended membrane system hosts complex metabolic

reactions [6]. In 1977, morphometric studies on liver cells presented controversial re￾sults because the surface area of the endoplasmic reticulum membrane, the site of drug

metabolism and of protein synthesis, had been estimated at 6 m2

/cm3

by Loud [7]

whereas we had obtained a value of 11 m2

/cm3

[6]. And yet, the methods used were the

same, with one exception: we obtained our measurements at 90'000 X magnification of

the electron microscope whereas the other group had used a lower magnification.

When, at the Buffon symposium, Mandelbrot showed on the example of

Richardson's problem of the indeterminate length of the coast of Britain, that the length

Mandelbrot’s Fractals and the Geometry of Life

5

5

of a boundary depends on the yardstick used to obtain the measurement and that this

was related to its fractal dimension [2] this pointed the way on how to resolve the para￾doxical results on cell membranes. In a systematic study of liver cell structure by

stereology using different electron microscopic magnifications (Fig. 2) Paumgartner et

al. [8] found the estimates of the surface of cellular membranes to increase with increas￾ing magnification or decreasing resolution scale (Fig. 3) concluding that the fractal di￾mension of the endoplasmic reticulum of liver cells was about 2.7 which fully explained

the differences in surface measurement obtained at different microscopic resolutions,

i.e.

Figure 2. Membrane system of liver cells at two different magnifications with grids for measuring sur￾face area by intersection counts. From [8].

Figure 3. Measured surface density of endoplasmic reticulum and inner mitochondrial membranes in￾crease with increasing magnification. The slope of the log-log regressions are related to the fractal dimen￾sion [8].

with yardsticks of different length. Similar results were obtained for other membranes

such that the fractal dimension of inner mitochondrial membranes was estimated at 2.54

(Fig. 3).

A very similar problem could then also be solved the same way. When human

lungs were studied morphometrically by light microscopy we had measured the internal

6 E.R. Weibel

surface area of an adult human lung at about 60 - 80 m2

[9] whereas later, using the

electron microscope with its higher resolving power, this estimate increased to 130 m2

,

the value now taken as real [10]. This too is related to the fact that the lung's internal

surface is a space-filling fractal surface whose dimension is estimated at 2.2. We will

return to this later.

3 The Lung's Airway Tree

One of the most influential fractal models has been the Koch tree model (Fig.

4a), a self-similar space-filling fractal based on dichotomous branching whereby the

size of the daughter-branches is reduced by the same factor from one generation to the

next. Even though the airway tree of the human lung shows considerable irregularity the

principle of a systematic reduction of airway size seems to apply (Fig. 4b). In introduc￾ing this model in 1977 Mandelbrot remarked that "the lung can be self-similar and it is".

On that basis it was later demonstrated by a systematic analysis that the airway tree in

different species shows a common fractal structure, in spite of some gross differences in

airway morphology [11].

The reduction of airway diameter and length by a constant factor is of func￾tional significance, both in blood vessels and in airways. It was proposed on theoretical

grounds by W.R. Hess [12] and C.D. Murray [13] that the dissipation of energy due to

the flow of blood or air in a branched tube system can be minimized if the diameter of

the two daughter-branches d1 and d2 are related to the diameter of the parent branch do

as do

3

= d1

3 + d2

3

. If we consider a simplified symmetric branching tree, where the two

daughter-branches have equal diameter and length, then the diameter of the daughter

branch is reduced with respect to do by a factor 2-1/3. In the context of fractal geometry

the reduction factor depends on the fractal dimension of the branching tree so

Figure 4. (a) Koch tree model of the airways (from [2]) compared to (b) cast of human airway tree.

that the correct formula is: d1 = do

.

2-1/D. In the case of the Hess-Murray law D = 3 be￾cause the tree is considered to be space-filling.

Does this law apply in the airways of the human lung? In 1962 Domingo Go￾mez and I had analyzed the dimensions of the human bronchial tree [14]. We found that

it branches over 23 dichotomous generations (note that the Koch tree in Fig. 4a has 12

generations). When the average diameters of the airways were plotted semi-

Mandelbrot’s Fractals and the Geometry of Life

7

7

logarithmically against the generations (Fig. 5) we observed that the data points lie

closely around a straight line down to generation 14, the last generation of so-called

conducting airways, and the slope of this line was 2-z/3. The average diameter of airways

in each generation can thus be predicted from the Hess-Murray law.

This then allowed us to conclude that the conducting airways of the human lung

are designed as a self-similar and space-filling fractal tree. This way the airways reach

into all corners of the lung's space with similar distances of all tips from the origin of

the airways in the trachea – the Koch tree of Fig. 4a is a reasonable scheme of airway

morphology. Furthermore, we found that the airways are designed for efficient ventila￾tion because they abide to the Hess-Murray law.

Figure 5. Semi-log plot of average airway diameter in human lungs against generations of branching.

After [14].

But precisely because of the optimisation conditions defined by the Hess￾Murray law it has been suggested that "an optimal bronchial tree may be dangerous"

[15]. The reason is that the reduction factor 2-1/3 = 0.79 is critical with respect to defin￾ing the airway resistance that varies inversely with the fourth power of the airway di￾ameter. Thus, a very small reduction of this factor would cause the airway resistance to

increase very dramatically in the smaller peripheral bronchioles. This could lead to

catastrophic situations, for example in asthma a pathological condition characterized by

progressive narrowing of small airways. Fig. 6 shows that a small reduction in this fac￾tor causes the airway resistance to increase drastically. It turns out, however, that the

bronchial tree is built with a certain safety factor in that respect. A closer analysis of the

data in Fig. 5 shows that the homothety factor corresponds about to the critical value of

0.79 in generation 6 but then slowly increases to about 0.9 in the 16th generation [15].

The average factor for small airways is therefore about 0.85 (Fig. 6) and this means that

(1) the flow resistance decreases in the small airways and (2) that a small reduction in

the homothety factor does not have a serious effect on lung function [15].

This larger than optimal factor of diameter reduction has as a consequence that

the fractal dimension of the conducting airway tree must be larger than 3 (see equation

above). This "unrealistic" proposition is only possible because the bronchial tree is trun￾cated at about generation 17 beyond which we find six generations of airways with a