Fractal Dynamics in Physiology Alterations with Disease and Aging

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Fractal Dynamics in PhysiologyAlterations with
Disease and Aging

• Presentation by Furkan KIRAÇ

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OUTLINE

• What is fractal?
• Its roots and motivation
• Fractal Dimension
• Mandelbrot Set
• Julia Set
• What is chaos? How is it related to real life?
• Logistic Equation
• Feigenbaums Constant
• How are fractals related to chaos and life?
• The paper Fractal Dynamics in Physiology

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How long is the coast of Britain?

• Suppose the coast of Britain is measured using a
  200 km ruler, specifying that both ends of the
  ruler must touch the coast. Now cut the ruler in
  half and repeat the measurement, then repeat
  again

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What is fractal?

• Mandelbrot invented the word fractal in 1967
• Latin adjective fractus.
• The corresponding Latin verb frangere means to
  create irregular fragments
• Fractal means a composition of irregular
  fragments.

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Fractal Dimension (FD)

• FD can be calculated by taking the limit of the
  quotient of the log change in object size and the
  log change in measurement scale, as the
  measurement scale approaches zero.

Consider a straight line.
Consider a square
blow up the line by a factor of two.
The line is now twice as long as before.
FD Log 2 / Log 2 1
FD Log 4 / Log 2 2
Consider a snowflake curve formed by repeatedly
replacing ___ with _/\_
FD Log 4 / Log 3 1.261
Since the dimension 1.261 is larger than the
dimension 1 of the lines making up the curve, the
snowflake curve is a fractal.
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Mandelbrot Set

• How is it actually computed? The basic algorithm
  is
• For each pixel c, start with z0.
• Repeat z z2 c up to N times, exiting if the
  magnitude of z gets large.
• If loop reaches to N, the point is probably
  inside the Mandelbrot set.
• If point exits the view, it can be colored
  according to how many iterations were completed.

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Why do you start with z0

• Zero is the critical point of z2c, that is, a
  point where d/dz (z2c) 0. If you replace z2c
  with a different function, the starting value
  will have to be modified.
• E.g. for z z2zc, the critical point is given
  by 2z10, so start with z-1/2.
• In some cases, there may be multiple critical
  values, so they all should be tested.

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What is the area of Mandelbrot set?

• Ewing and Schober computed an area estimate using
  240,000 terms of the Laurent series. The result
  is 1.7274... However, the Laurent series
  converges very slowly, so this is a poor
  estimate.
• A project to measure the area via counting pixels
  on a very dense grid shows an area around 1.5066.
• Hill and Fisher used distance estimation
  techniques to rigorously bound the area and found
  the area is between 1.503 and 1.5701.

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Julia Set

• The Mandelbrot set iterates z2c with z starting
  at 0 and varying c.
• The Julia set iterates z2c for fixed c and
  varying starting z values.
• Mandelbrot set is in parameter space (c-plane)
• while the Julia set is in variable space
  (z-plane).

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Julia Set Example
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What is Chaos Theory?

• A non-linear dynamical system can exhibit one or
  more of the following types of behaviour
• forever at rest
• forever expanding
• periodic motion
• quasi-periodic motion
• chaotic motion
• The type of behavior may depend on the initial
  state of the system and the values of its
  parameters, if any.

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Demonstration of chaos!

• Chaos is unpredictable behavior arising in a
  deterministic system because of great
  sensitivity to initial conditions.
• Chaos arises if two arbitrarily close starting
  points diverge exponentially, so that future
  behavior is unpredictable.
• Lets consider the following sequence xn1
  4.xn.(1-xn)

Iter Sequence 1 Sequence 2 difference 1 0.7
000000000 0.7000000001 0.0000000001 2 0.8400000
000 0.8399999998 0.0000000002 3 0.5376000000 0
.5376000004 0.0000000004
20 0.3793606672 0.3794161825 0.0000555153 21 0
.9417846056 0.9418381718 0.0000535662 22 0.2193
054491 0.2191161197 0.0001893294
80 0.3149359471 0.8782328851 0.5632969380 81 0
.8630051853 0.4277595387 0.4352456466 82 0.4729
089416 0.9791252630 0.5062163214 83 0.997064298
2 0.0817559295 0.9153083687
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A Well Known Chaotic System

• Weather is considered chaotic since arbitrarily
  small variations in initial conditions can
  result in radically different weather later. This
  may limit the possibilities of long-term weather
  forecasting.
• The canonical example is the possibility of a
  butterfly's sneeze affecting the weather enough
  to cause a hurricane weeks later.
• Lorenz Model was the model that proved this idea.

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Visualizing Chaotic Motion Attractors

• One way of visualizing chaotic motion, or indeed
  any type of motion, is to make a phase diagram.
• In such a diagram time is implicit and each axis
  represents one dimension of the state.
• A phase diagram for a given system may depend on
  the initial state of the system (as well as on a
  set of parameters),
• Often phase diagrams reveal that
• the system ends up doing the same motion for all
  initial states in a region around the motion
• almost as though the system is attracted to that
  motion
• Such attractive motion is fittingly called an
  attractor for the system

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An Attractor Example

• A simple harmonic oscillation system is shown.
• Assume no friction
• Then the red circle is the attractor of the
  system.

v
x
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Strange Attractors

• While most of the motion types mentioned give
  rise to very simple attractors, chaotic motion
  gives rise to what are known as strange
  attractors that can have great detail and
  complexity.
• Strange attractors have fractal structure.

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Strange Attractors and Fractals
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Strange Attractors in Action!
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How are chaos and fractals are related to real
life?

• Lets answer this question with 2 distinct
  examples
• Logistic Equation used for animal population
  modeling
• Iterated Function Systems for rendering plants

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What is Logistic Equation?

• It models animal populations.
• The equation is xn1 A.xn.(1-xn) where x is
  the population (between 0 and 1) and A is a
  growth constant

Iteration of this equation yields the period
doubling route to chaos.
For A between 1 and 3, the population will settle
to a fixed value.
At 3.00, the period doubles to 2 one year the
population is very high, causing a low population
the next year, causing a high population the
following year.
At 3.45, the period doubles to 4, population has
a four year cycle. The period keeps doubling,
faster and faster, at 3.54, 3.564, 3.569, ...
Until At 3.57, chaos occurs the population never
settles to a fixed period. For most A values
between 3.57 and 4, the population is chaotic,
but there are also periodic regions.
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Attractor of Logistic Equation
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Periodic Behavior of Logistic Map with A3.5 and
X00.7

• 89 0.382819683017324 90
  0.82694070659143891 0.500884210307217 92
  0.87499726360246493 0.38281968301732494
  0.82694070659143895 0.50088421030721796
  0.87499726360246497 0.38281968301732498
  0.82694070659143899 0.500884210307217100
  0.874997263602464

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Visualization of Logistic Equation
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Feigenbaums Constant

• In a period doubling chaotic sequence, such as
  the logistic equation, consider the parameter
  values where period-doubling events occur
• e.g. r13, r23.45, r33.54, r43.564
• Lets look at the ratio of distances between
  consecutive doubling parameter valueslet
  deltan (rn1 - rn) / (rn2 - rn1)
• Then the limit as n goes to infinity is
  Feigenbaum's (delta) constant.
• It has the value 4.669201609102990671853...
• The interpretation of the delta constant is that
  as you approach chaos, each periodic region is
  smaller than the previous by a factor approaching
  4.669...

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Iterated Function Systems

• The Fern
• T1 x' 0x 0y .16, y' 0x 0y 0 1
• T2 x' .85x .04y 0, y' -.04x .85y
  1.6 85
• T3 x' .2x - .26y 0, y' .23 x .22y
  1.6 7
• T4 x' -.15x .28y 0, y' .26x .24y
  .44 7

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How does the Fractal Fern look?
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Other IFS Examples 1
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Other IFS Examples 2
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Fractals are related to natural laws!

• We find chaotic behavior in every part of life.

We find fractals in every part of chaotic
behavior.
It seems reasonable that the modeling method the
nature uses is fractal geometry.
Recursively applying same set of gravitational
transformations to an initial condition can
create a beautiful fern for example.
So a fern only stores those 24 numbers in its
genetic code in a certain way the nature can make
use of them.
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Humans are also chaotic!

• It is not so much suprising that the human body
  also acts chaotic, and uses fractal geometry.
• In fact human body seems to possess a lot of
  fractal geometry structures
• Arterial venous trees
• Cradiac muscle bundles
• His-Purkinje Conduction System
• Blood veins
• Nervous System

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How to make use of this chaos!

• Scientists have shown that with aging and disease
  the fractal like (chaotic) behavior of the
  systems degrade.
• So we can use this degradation as a feature for
  our disease diagnostic systems.

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Heart Rate Dynamics in Health and DiseaseA Time
Series Test
We can find patterns it is very predictable,
not chaotic.
HEALTHY!
We can find patterns it is very predictable,
not chaotic.
It acts in a random fashion, still not chaotic
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Spatial vs Temporal Self Similarity
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Wavelet Analysis of Human Heart Beat
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Detrended Fluctuation Analysis (DFA)
Human Heart Beat Electrical Signals Captured
Integrated to convert Heart Beat Velocities to
Heart Displacements
Slope Corresponds to different kinds of behavior
can be used as a feature
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DFA in General
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Fractal Dimension Diagnosis
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Human Gait (Walking) Dynamics
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Conclusion

• Nature uses chaotic systems in every part of our
  lives.
• Weather
• Animal Population Growth
• Human body is also built up of chaotic elements
• Nervous System
• Blood veins
• We can make use of fractals and chaos theory for
  diagnosing certain diseases.
• Heart Failures Atrial fibrilation,
• Huntington Disease Gait Dynamics

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Thanks

• Any Questions ?