C e n t r e  d e  r e c h e r c h e s  m a t h é m a t i q u e s
en français

Workshop

Fractals and Modeling in Structural and Dynamical Analysis

November 11-14, 2000 
Montréal, Québec, Canada

 

Tiling of the plane by Levy curve


 
 
 
 
 
With development of  new  methods in imaging and elaborated materials in the applied sciences (drug delivery, porous mediaÖ),  some limits of the classical approaches have been reached. Fractal concepts and nonlinear models successfully gave a new impetus to these classical problems. This workshop will cover the latest theoretical developments, their contribution in the biomedical field and future direction of investigation.
We hope that this interdisciplinary meeting will attract contribution from participants of  various fields and milieu, including industry.
The format will consist of morning and  afternoon lectures and a poster presentation.  In this conference, researchers from either mathematical, physical or biological sciences will present  recent results essentially obtained with these techniques. 
  • Organizers
  • Invited Speakers
  • Titles/Abstracts of Some Lectures
  • Abstract Submission for a Poster
  • Registration and Hotel Information
  • Financial Support
  • Our Coordinates

  • Organizers


    Invited Speakers


    Titles/Abstracts of Some Lectures

    Lacunarity and Detrended Fluctuation Analysis of Chromatin Texture for Diagnosis in Breast Cytology

    Andrew J. EINSTEIN

    Identification of malignancy in fine-needle aspiration biopsy of the breast has profound therapeutic implications, but poses a diagnostic dilemma. In this talk, a variety of nonlinear approaches will be applied to the characterization of nuclear morphology, including fractal dimensions, lacunarity, and detrended fluctuation analysis. Using logistic regression to classify cases on the basis of these measures of nuclear morphology, as well as nuclear size, we will demonstrate that an accurate diagnosis of malignancy can be achieved in breast cytology.

    Long-range correlations in genomic DNA: a signature of the nucleosomal structure

    Alain Arneodo 

    The packaging of the eucaryotic genomic DNA involves the wrapping around the histone proteins followed by the successive foldings of higher order stuctured
    nucleoprotein complexes. The bending properties of DNA play an essential role in these compaction processes. This hierarchically organized pathway is likely to be reflected in the fractal behavior of DNA bending signals in eucaryotic genomes, but the challenge is to somehow extract this structural information by a clever reading of the DNA sequences. We show that when using an adapted mathematical tool, the "wavelet transform microscope", to explore the fluctuations of bending profiles, one reveals a characteristic scale of 100-200bp that separates two different regimes of power-law correlations (PLC) that are common to eucaryotic as well as eubacterial and archaeal genomes. The same analysis of the DNA text yields strikingly similar results to those obtained with the bending profiles, and this for the three kingdoms. In the small-scale regime, PLC are observed in eucaryotic genomes, in nuclear replicating DNA viruses and in archaeal
    genomes, which contrasts with their total absence in the genomes of eubacteria and their viruses, thus indicating that small-scale PLC are likely to be related to the mechanisms underlying the wrapping of DNA around histone proteins. Actually these small-scale PLC are shown to provide a very efficient diagnostic of the nucleosomal structure.
     

    Three-dimensional model of the human airway tree based on a fractal branching algorithm

    Hiroko Kitaoka 

    A three-dimensional (3D) model of the human airway tree is proposed using a deterministic algorithm that can generate a branching duct system in an organ providing a homogeneous and effective fluid delivery to the whole organ. The algorithm is based on a principle that the amount of fluid delivery through a branch is proportional to the volume of the region it supplies. This principle defines the basic process of branching: generation of the dimensions and directionality of two daughter branches is governed by the properties of the parent branch. 
    When the contour of an organ and the position of the trunk are specified, branches are successively generated by the algorithm. The condition of no more branching is given by the threshold flow rate through the branch. A tree structure generated with the algorithm has self-similar property, because the branching is 
    recursively performed with the same rules independent of its size. 
    Applied to the human lung, the algorithm generates an airway tree. It contains about 27,000 terminal branches correspond to the terminal bronchioles. The outlook of the airway tree model is quite similar to the real one and its morphometric characteristics are in good agreement with those reported in the literature. The 
    fractal dimension of the airway tree in this model is measured with a 3-D box-counting method. The obtained value of 1.73 is the same as those in the real adult 
    and fetal human airway tree. Although the algorithm is completely deterministic, it is possible to generate many airway trees having inter-individual differences 
    by adding a small amount of fluctuations in certain parameters. The algorithm proposed here is very useful to study structure and function of branching systems in living bodies. 

     

    Extended, Universal Fractality by the Dynamic Redundance Concept of Complexity and Its Applications in Medicine and Biology

    Andrei P. Kyrilyuk

    Most efficient and truly reliable applications of knowledge to living systems monitoring and change (biology, medicine, ecology) demand the causally complete, fundamental and rigorously substantiated understanding of their origin and detailed operation actually absent at the basically empirical modern level of biosciences. We show that the essence of the phenomenon of life as such, as well as the detailed living system dynamics, can be rigorously described in terms of the recently proposed universal, reality-based concept of dynamic complexity [1] based on the qualitatively new phenomenon of dynamic redundance (multivaluedness). The
    latter naturally emerges simply due to the unreduced, universally nonperturbative analysis of the driving system interactions that reveals the general mechanism of autonomous dynamic entanglement between the interacting real entities and shows that this entanglement always emerges in a redundant (excessive) number of system versions, or realisations, with respect to actually available places for the interaction products. Therefore the internally entangled, irregularly configured system realisations are forced, by the driving system interactions, to permanently replace each other in a causally random order, with equal and a priori determined probabilities. This universal way of interaction process development gives rise to the dynamic emergence of the observed system structure in the form of extended dynamical fractal which is characterised by the dynamically probabilistic, unpredictably moving and self-developing branches/realisations that naturally form hierarchical structure of any, even apparently 'non-fractal' ('smooth'), type of appearing objects. The dynamic complexity as such can be rigorously and universally defined as any growing function of the total number of system realisations (finest fractal branches') or the related rate of their (probabilistic) change, equal to
    zero for the unrealistically simplified case of only one system realisation. It is this, unrealistic limiting case that is actually used in the canonical, dynamically single-valued science for the effectively one-dimensional imitation of the dynamically multivalued reality, including artificial 'modelling' of the scholar 'science of complexity' (conventional concepts of 'chaos', 'self-organisation', 'adaptability', etc.). The 'emergent' and causally probabilistic (dynamically redundant) properties of the extended dynamical fractal determine its essential difference with respect to the conventional, dynamically single-valued fractality and fill the 'modelled', artificially fixed form of the latter with life (now rigorously defined). Indeed, the extended dynamical fractal shows inbred dynamic adaptability, 'individuality' (intrinsic unpredictability of the detailed structure/behaviour) combined with the auto-programmed general development, and general absence (or essential violation) of the 'scale symmetry' (self-similarity), the irreducible properties of living (and actually any real) systems which are not (correctly) reproduced by the
    usual fractals. The extended dynamic fractality, rigorously obtained as the unreduced result ('general solution') of any unrestricted interaction process, represents the observed structure of the universal dynamic complexity of the world which is none other than the exact dynamical structure of all really existing entities. Living organisms and related systems correspond to high enough levels of this unified, fractal hierarchy of unreduced dynamic complexity of the world. The qualitatively extended and totally adequate understanding of dynamic complexity of any real system leads to well-specified and practically efficient applications of the
    concept of extended dynamic fractality to living systems which include causally complete understanding of genetic mechanisms and intrinsically "creative evolution" of life, objective definition and reliable, conscious control of 'ill' and 'healthy' states of an organism, new types of objectively 'precise', complex-dynamical medicine and treatment for particular illnesses, exact artificial reconstitution of an organism and any its part, complete understanding of functioning of the brain (nervous
    system), causally complete and creative ecology, etc. In this way, the proposed concepts of extended dynamic fractality and universal dynamic complexity open a realistic way towards transformation of biology, medicine and ecology into the superior, fundamentally substantiated and integral, type of knowledge intrinsically unified within the universal science of complexity [1].

    [1] A.P. Kirilyuk, Universal Concept of Complexity by the Dynamic
    Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the
    Ultimate Unification of Knowledge (Naukova Dumka, Kiev, 1997), 550 p., in
    English. For a non-technical review see also: e-print physics/9806002
    (accessible through http://arXiv.org/abs/physics/9806002).

    Fractals in the Timing of Events that Disrupt the Rhythm of the Beating of the Heart

    Larry Liebovitch
     

    Smaller is better: the role of screening in the transfer of oxygen in the mammalian acinus

    Bernard Sapoval

    Issues Relating to Fractal Segmentation of Natural Phenomenal;including Synthetic Aperture Radar, Weather Patterns, Porous Silicon and Cell Boundaries.

    Martin Turner

    When Weierstrss (1815-1897) created one of the first continuous but nowhere differentiable functions his work was groundbreaking in the change of mathematical thinking, as it destroyed the property of differentiation as a constant. The uses of these functions at the time were not apparent and many mathematicians were vocally alarmed. Poincare (1854-1912) wrote in his collected works “Yesterday, if a new function was invented it was to serve some practical end; today they are specially invented only to show up the arguments of our fathers, and they will never have any other use.” Hilbert (1862-1943) and other mathematicians experimented with various functions to create mathematical mappings that are still used in practice today. It took the modelling of Brownian motion, by Levy and Wiener, as a non-differentiable walk, to show that nature can be modelled successfully with these new formulas. Twenty-five years ago Mandelbrot (1924-) published the major work “Les objets fractals: forme, hasard et dimension” (1975) that first defined the term based on the Latin adjective fractus derived from the verb frangere meaning to break. Since the invention of the graphical computer there has been an exponential revolution in the use of fractal functions. Now virtually every modelling and animation package has a fractal texture generator to create synthetic ‘natural like’ objects. The functions to create these textures can be extremely simple and designers need only modify a dial on a simple graphical interface and admire the results. Designers now do not need to understand the first part of the mathematics behind its creation, only the aesthetics. Many types of natural objects have been synthesised to create comparable models that compare with real data and are now acceptable in many fields. These range from crystalline and coral formation, to cell and tree growths and natural noise field. A large related field has been the inclusion of self-affine signals in art and music as well as science, which is often applied, at times with great effort, again without appreciating the fundamental formulaic background. During the same time, methods were created to take signal and image data and categorise them according to certain fractal characteristics; the most common being the fractal dimension. These aimed to apply the inverse of the signal creation techniques and therefore compete with Euclidean detection systems. Although fractal creation techniques are common the corresponding fractal segmentation operators have not found themselves in many image processing toolkits. It is this question of why fractal creation has become more popular than fractal detection that I hope to address. Over the last five years researchers at the Institute of Simulation Sciences have been active on various projects to both create synthetic fractal models and detect fractal characteristics within real data. A list of issues and problems in these techniques will be given with some examples, as well as a series of tools and tricks that have been used to help reduce errors.

    • Choice of dimension, and comparison of techniques
    • Data windowing and sampling methods
    • Instability of the log-linear regression process
    • Additive deterministic and noise structures
    • Discontinuity in samples
    • Use of pre-affine transformations
    • Goodchild and Shelberg et al dimension reduction
    • Use of higher order dimensions and multi-fractals.
    Both successes and failures will be shown to highlight these issues.



    Abstract Submission for a Poster

    Contributors are welcome to send an abstract for the poster session to  by e-mail to pelletl@crm.umontreal.ca as attached document or using the  on-line submission

    Registration and Hotel Information

    Basic registration information, including name, status, institutional affiliation, mailing address, daytime telephone number, fax number and email address, may be conveyed to the CRM by mail, fax (see Our Coordinates at the bottom of this page) or on-line registration

    Financial Support

    Partial support will be available for those graduate students who submit a letter of interest and an abstract of one page to be considered for a comminication,  by e-mail as attached document or using  on-line submission


    Our Coordinates

    Centre de recherches mathématiques (CRM)
    Université de Montréal
    C.P. 6128, Succ. centre-ville
    Montréal (Québec) CANADA H3C 3J7
    E-mail:                 ACTIVITES@CRM.UMontreal.CA
    
    World Wide Web:         http://www.CRM.UMontreal.CA/biomath
    
    Telephone:              (514) 343-2197
    
    Fax:                    (514) 343-2254

    [Page d'accueil du CRM][CRM Home Page]
    13 octobre 2000, webmester@CRM.UMontreal.CA