2001 Fractal Modeling a Biomedical Engineering Perspective
Transcript of 2001 Fractal Modeling a Biomedical Engineering Perspective
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Fractals in Physiology
A Biomedical Engineering Perspective
J.P. Marques de S [email protected]
INEB Instituto de Engenharia Biomdica
Faculdade de Engenharia da Universidade do Porto
1999-2001 J.P. Marques de S; all rights reserved (*)
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1. Motivation
2. Fractals in Physiology
3. Self-Similarity and Fractal Dimension
4. Statistical Self-Similarity
5. Self-Affinity and the Fractional
Brownian Model
6. Chaotic Systems and Fractals
7. Why Nature uses Fractals?
8. Fractals in BME
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1 - MOTIVATION
Many physiological signals exhibit almost random
behaviour
Foetal Heart Rate
C. Felgueiras, J.P. Marques de S (1998)
but with statistical self-similarity
Model: Temporal Fractal
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Many physiological structures exhibit a high degree of spatial
complexity
Human Retina Image
Retina angiogram. A.M. Mendona, A. Campilho (1997)
But some repeating law
may be present
Diffusion Limited Aggregation Process: a particle falls randomly in a circle of radiusR and diffuses
until anchoring to another particle or getting out of the circle. Nr of points available for growth in a
circle of radius r:DrrN )(
Model: Spatial Fractal
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Hydroxyapatite Growth on Bioactive Surfaces
Substrate: hydroxyapatite Substrate: composite
C.Felgueiras, J.P. Marques de S, 1998
Fractal modelling may help
to assess the growth conditions
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2 - FRACTALS IN PHYSIOLOGY
(Some Examples)
Temporal Fractals
Heart beat sequences
Electroencephalograms Time intervals between actionpotentials
Respiratory tidal volumes Ion channel kinetics in cell
membranes
Glycolysis metabolism DNA sequences mapping
Spatial Fractals
Intestine and placenta linings Airways in lungs Arterial system in kidneys Ducts in lever His-Purkinje fibres in the heart Blood vessels in circulatory system Neuronal growth patterns Retinal vasculature
Reference: Bassingthwaighte JB, Liebovitch, L. S., West B. J. (1994) FractalPhysiology. Oxford University Press.
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3 - SELF-SIMILARITY
AND
FRACTAL DIMENSION
Koch curve
Scale: r a.r 0 < a < 1
Self-similarity dimension:How many parts in an object are similar to the whole ?
Simple Objects
Dimension = D = 1
N = 1 N = 3 = (1/3) 1
r = 1 r = 1/3
Dimension = D = 2
N = 9 = (1/3) 2N = 1
Number of exact replicas N(r) when the resolution is r :
DD
rr
rN =
=
1)(
)/1log(
))(log(
r
rND =
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Fractal Objects(with exact self-similarity)
Fractal:
Any object with D having a fraction part?
Not always
Any object with D > topological dimension?
Not always
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Length of the object :
r
N(r)
D
rrNrrL
==
1
)(.)( ln(L(r)) = (D-1) . ln(1/r)
ln(1)=0 ln(3)=0.47
ln(4/3)
=0.12
slope = D - 1 = 0.26
ln(1/r)
ln(L(r))
Any object with a property L(r) satisfying a Power Law
Scaling:
rrL )( ,
is said to have a fractal property L(r).
= 1-D for lengths
= 2-D for areas = 3-D for volumes
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The scaling property can be written as:
L(ar) = k L(r) ,
with a < 1, k=a
and
k, iteration factor, independent of r
For the Koch curve:
r = 1 L(1) = 1
r = 1/3 L(1/3) = 4/3 = 4/3 . L(1)
r = 1/9 L(1/9) = 16/9 = 4/3 . L(1/3)
L(r/3) = 4/3 . L(r)
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Cardiac Pacing Electrode
Zr Interface impedance with roughness effect
Zi Interface impedance per unit of true interface area
Zr = a (Zi)
= 1/(D-1)
Roughening the surface D 3 0.5
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Estimating the Fractal Dimension
Length: DrrrNrL = 1)()(
Number of covering boxes or spheres: DrrN )(
Box-counting: number of squares containing a piece of the
object
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Statistical Self-Similarity
Generalisation:
)/1ln(/))(ln(lim0
rrNDr
cap
= , capacity dimension
(Grassberg and Procaccia)
Result of box-counting methodapplied to retina angiogram
0
2
4
6
8
10
12
14
-10 -8 -6 -4 -2 0
log2(1/r)
log2
(N(r))
C.Felgueiras, J.P. Marques de S, A.M. Mendona (1999)
D = 1.82
Number of branching sites (cluster diameter)D
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Generalised Fractal Dimensions
(In a coverage of N(r) boxes of size r)
Information dimension:
)ln(/lnlim)ln(/)(lim)(
100inf rpprrSD
rN
iii
rr=
==
S(r) Entropy; pi - probability for a point to lie in box i
Correlation dimension:
)ln(/lnlim)ln(/))(ln(lim)(
1
2
00rprrCD
rN
ii
rrcorr
===
C(r) : Number of pairs of points in box i
Generalised dimensions:
)/1ln(
)(lim
)/1ln(
ln
1
1lim
0
)(
1
0 r
rF
r
p
qD
q
r
rN
i
qi
rq
=
=
=
D0 = Dcap ; D1 = Dinf; D2 = Dcorr
Dq decreases with q, e.g.: corrcap DDD inf
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Hydroxyapatite Growth on Bioactive Surfaces
0
2
4
6
8
10
12
14
16
-8 -7 -6 -5 -4 -3 -2 -1 0
log2(1/ )
Fq(
)
0
1
2
3
4
5
6
1,55
1,6
1,65
1,7
1,75
1,8
0 1 2 3 4 5 6
q
Dq
comp3c
comp14
ha3a
ha14b
ha hydroxiapatite substrate
comp composite glass substrate
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Ha Hydroxiapatite
comp 2 composite 2 glass
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5 - SELF-AFFINITY
AND THE
FRACTIONAL BROWNIAN MODEL
The symmetric random walk
2
1)1()1(;)(
1
======
ii
n
ii xPxPxnx
n
x(n)
nnnnkkn
n
knxPn
,2,...,2,2
1
2
))(( +=
+==
[ ] [ ] nnxVnxE == )(0)(
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Probability of the random walk for n=16.
0
0.05
0.1
0.15
0.2
0.25
-1
6
-1
2 -8 -4 0 4 8
1
2
1
6
k
P
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Taking: r points per unit time
jumps
With: 22;0, rr
Symmetric Random Walk Brownian Motion Function
txe
txtB
222/
2
1),(
=
Properties:
B(t+t)-B(t): Gaussian increments
E[B(t+t)-B(t)] = 0
V[B(t+t)-B(t)] = 2
. t [B(t+t)-B(t)] t
scaling
[B(t+t)-B(t)] t
[B(t+rt)-B(t)] tr = r 21
. [B(t+t)-B(t)]
Self-Affinity
t r.t ; (B) r 21
. (B)
Iteration factor dependent on r (compare with page 9)
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-50
0
50
100
150
1
501
1001
1501
2001
2501
3001
3501
4001
4501
0
1 0
2 0
3 0
4 0
5 0
1 2 0 1 4 0 1 6 0 1 8 0 1
0
5
1 0
1 5
2 0
151
101
151
201
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Two processes:
Self-similar: )(2
1
2rL
rL =
Self-affine: )(22
rLrr
L =
0
0.1
0.2
0.3
0.4
0.5
0.60.7
0.8
0.9
1
1 2 4 8 16 32
Self-similar
Self-affine
1/r
L(r)
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Fractional Brownian Motion
Def.: Moving average of B(t,x) with increments weighted by2/1)( Hst
- random walk with memory
( )
t
HH xsdBstxtB ),()(,
2/1 ] [1,0H
Hurst coefficient
Self-affine process with :H
H tB )(
H = Brownian motion
0 < H < fBm with anti-correlated
samples
< H < 1 fBm with correlated samples
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BH(t)
H
H tB )( Kernel
H=0.2, D=1.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
149
97
145
193
241
289
337
385
433
481
-2000
-1000
0
1000
2000
1 101 201 301 401
H=0.5, D=1.5
0
0.2
0.4
0.6
0.8
1
1.2
149
97
145
193
241
289
337
385
433
481
-30
-25
-20
-15
-10
-5
0
5
1
H=0.8, D=1.2
0
1
2
3
4
5
6
7
152
1
03
1
54
2
05
2
56
3
07
3
58
4
09
4
60
-30000
-20000
-10000
0
10000
20000
30000
1 101 201 301
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Computing the fractal dimension of the fBm :
t = 1
x = 1
t = 1/N
x
o
o
ForH
H
Ntx
Nt
===
11
The region tx is covered by:
HN
Nsquares of size
NL
1=
The whole segment is covered by:
H
H
H LN
N
NNLN
==
=
2
2 1.)(
Therefore: D = 2 H
Higushi method: L(k)=sk1-D
=skH-1
1 100
L(k)
k
S o
oo
oo o o
o
o
0 10
ln(L(k))
ln(k)
slope=1-D
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6 - Chaotic Systems and Fractality
Chaotic system:
Deterministic dynamic system with high
sensitivity to initial conditions.
Example:
Hnon process:
x(n+1) = y(n) + 1 - 1.4x(n)2;
y(n+1) = 0.3x(n).
0 10 20 30 40 50 60 70 80 90 100-1.5
-1
-0.5
0
0.5
1
1.5
n
x[n]
a) b) c)
Figure 9. a) Hnon process signals with initial values differing in 10-12;
b) Hnon attractor; c) A detail of the attractor.
Some properties: High sensitivity to initial conditions. Values not predictable in the long run. System state describes intricate trajectories (attractor) in the
phase space.
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Analysis tool: Attractors in the phase-space
Takens Theorem:
The set {x[i],x[i+k],,x[i+dk]} is topologically equivalent to the attractor in an
embedding space of dimension d+1.
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7 Fractal Modelling of Foetal Heart Rate
0 5 10 15 20
133
148
154
Time (min)
FHR
(bpm
)
Class B
Class FS
Class A
Class A NEM sleep
31 cases
Class B REM sleep
50 cases
Class FS - Neuro-cardiac
depression
43 cases
FHR modelling by a bi-scale fBm fractal
0 20 40 60 80 1000
500
1000
1500
2000
k
L(k)
Class AClass BClass FS
Results of Higushi method ( 1)( = HSkkL )
showing two scaling regions
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FHR bi-scale behaviour
10
ln(L(k))
ln(k)
H1-1
H2-1
0
ln(S1)
ln(S2)
STV LTV
H1, H2
0
0,2
0,4
0,6
0,8
H1-A H1-B H1-FS H2-A H2-B H2-FS
S1, S2
5
6
7
8
9
10
S1-A S1-B S1-FS S2-A S2-B S2-FS
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Foetal Heart Attractor
(Stage A; Delay= 8 samples 20 s)
C. Felgueiras, J.P. Marques de S (1998)
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FHR Classification Results
7 Features (3 temporal; 4 attractor)
Actual No. of Predicted Class
Class Cases A B FS
A 31 26
(84%)
5
(16%)
0
(0%)B 50 2
(4%)
48
(96%)
0
(0%)
FS 43 2
(5%)
0
(0%)
41
(95%)
Training set error: 7.3% - Test set error: 15.4%
Comparison with traditional clinical method:
FHR No. of Pe %
Class Cases Fractal STV p
A 31 16.1 67.7 < 0.001
B 50 4.0 26.0 < 0.001
FS 43 4.7 18.6 0.021
Total 124 7.3 33.9 < 0.001
C. Felgueiras, J.P. Marques de S, J. Bernardes, S. Gama (1998) Classification of Foetal Heart RateSequences Based on Fractal Features, Medical & Biological Eng. & Comp., 36(2): 197-201.
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8 - FINAL COMMENT
Interest of Fractal Modelling in BME
Reveal the mechanisms that producephysiologic signals and structures.
Clarify the meaning of measurements. Clarify the existence of deterministic
causes in apparent random behaviour.
Derive model parameters to classifydata.