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.
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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.
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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
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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.
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