Exploring Modelling Analogies
between Living and Non-Living
Systems
The paradigms of Cancer and Alzheimer
By:
I. Nikolaides
H. Ayfantes
A. Heliopoulos
Presentation structure
Part 1: What is a signal, anyway?
Part 2: Fractal and informational analysis
Part 3: Applications and neuro-degenerative
diseases
Part 1:
Signals and signal analysis
What is a signal, anyway?
Signal = Something that gives us information
Signal analysis = Extracting the information from
the signal
Signals can be surprisingly deep!
Both humans and computers can analyse signals,
and each is good at something else
Examples of signals
Essentially: How many independent and
dependent variables do we have?
1+1
2D
3+1
4D
2+1
3D
2+3
5D
Computerised signal analysis
Fourier analysis: “What notes does this sound
have?”
Stochastic analysis: “What is the average of this,
and how spread out is it?”
Informational analysis: “How predictable is this
signal?”
Fractal analysis: “How rough/smooth is this
signal?”
Examples of applied signal analysis
Economics
Seismology
Bio-imaging
Material Mechanics
Part 2:
Introduction to Fractal and
Informational analysis
Smooth shapes and their length
Q: How do we measure the length of a smooth
shape?
A: Measure it with ever-shrinking “yardsticks” and
see where the length converges to.
6r
6,21r
Convergence
limit: 2πr
Non-smooth shapes
But when we do the same for (say) a coastline...
2400 km
3400 km
2900 km
...the measured length increases without limit!
The concept of “dimension”
So, is there any benefit to those
measured lengths?
First, let's look at the way scaling
affects Euclidean shapes:
Each time the scaling decreases nD
fold, the measurements increase n fold.
And, since the measured length of
Great Britain increases 2.38-fold for
every time the yardstick is halved...
...we
1.25 can define D as 1.25, as
2 =2.38.
D shall be referred to as the shape's fractal dimension.
Living vs Non-living fractal systems
Living Systems
Non-living Systems
Romanesco
Brocolli
D ≃ 2.6
“Frozen
Lightning”
D ≃ 2.5
Brain
D ≃ 2.8
Coastline of
Norway
D ≃ 1.5
A little more detail
D
If n =S, then
Integer D leads to Euclidean shapes: Lines,
planes, spaces etc.
Non-integer D...
D≈1.26
D≈1.58
...leads to things in-between.
D≈1.93
Analytic computation of FD
A unit square is composed of 4 smaller squares
equal to it (S=4) each scaled by half (n=2).
so D=log (4)=2.
2
A Sierpinski triangle is composed of 3 smaller
triangles (S=3) each scaled by half (n=2)...
...so D=log2(3)=1.58.
N=3
Fractal dimension measurement
The most common method is box-counting. It consists of covering the shape with
boxes/cubes/hypercubes of ever-decreasing size...
... and measuring how many of them contain the shape inside them.
Box-counting in higher dimensions
Box-counting is extremely easy to extend to more
dimensions!
Lacunarity
What happens when we want to compare fractals
of similar fractal dimension?
D=2
D=2
For that, Mandelbrot proposed to count how
many pixels each box contains. Then,
Methods of fractal measurement
Many methods exist that can measure FD and
Lacunarity
Invariably, those methods are
slow
approximate
not applicable to higher-dimension signals
or, usually, a combination of those.
Our own methods have none of those constraints
The same method can be applied to any signal
we want!
Part 3:
→Possible
applications
→Current results
→Future research
Possible applications 1/5
Optical microscope images:
vs.
Which one has cancer?
Possible applications 2/5
AFM images:
3 measurements examined at once!
Possible applications 3/5
Magnetic Resonance Images:
Is there anything wrong with this brain?
Possible applications 4/5
Electroencephalograms and functional MRIs
Can we derive useful information?
Possible applications 5/5
Electroencephalograms!
Approach no. 1:
Approach no. 2: an
16 different dimensions entire EEG turned into a
3-D shape
What has been studied?
Two things: Optical microscope images, and
MRIs of brains.
The rest will be studied in due time.
Immunohistochemistry
Immunohistochemistry is a chemical method for
detecting specific tissues using antibodies.
( Ελληνιστί: Ανοσοϊστοχημεία)
First the antibodies attach to the tissues, then
they are dyed to make them visible.
Results about unknown images:
“Not very serious cancer”:
“Really serious cancer”:
Lacunarity≈15
Lacunarity≈5
“Nothing too worrying”: Lacunarity≈
8
Magnetic Resonance Imaging
Essentially, we magnetise the human body really
strongly and see how it reacts.
Really good results in neuroimaging, which is
why we focused on MRIs of brains.
We had 11 patients and 5 healthy people to
examine.
FD of brains,
depending on Alzheimer's
3.32 seems to be a limit between sufferers and
non-sufferers
9/11 sufferers had FD between 3.14 and 3.32
4/5 healthy had FD between 3.32 and 3.38
Very slight difference, but it gave an accuracy of
over 80%!
(There was sadly no time to examine the
lacunarities)
Conclusions:
Fractal dimension is a measure of how "rough" or
"smooth" a shape is
Lacunarity is a measure of how "homogeneous"
or "non-homogeneous" a shape is
FD can be approximated by counting boxes at
various scales
Lacunarity can be approximated by counting box
weights at various scales
Our algorithm is fast because it eliminates all
excessive logical steps
Fractal analysis has shown promising results with
regards to immunohistochemistry and MRIs of
Fin!
Thank you for your patience.