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Fractals in Physiology
A Biomedical Engineering Perspective
J.P. Marques de Sá –
[email protected] INEB – Instituto de Engenharia Biomédica Faculdade de Engenharia da Universidade do Porto © 1999-2001 J.P. Marques de Sá; all rights reserved (*)
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1. 2. 3. 4. 5. 6. 7. 8.
Motivation Fractals in Physiology Self-Similarity and Fractal Dimension Statistical Self-Similarity Self-Affinity and the Fractional Brownian Model Chaotic Systems and Fractals Why Nature uses Fractals? 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 Sá (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. Mendonça, A. Campilho (1997)
But some “repeating” law may be present…
Diffusion Limited Aggregation Process: a particle falls randomly in a circle of radius R 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:
N (r ) ≈ r D
Model: Spatial Fractal
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Hydroxyapatite Growth on Bioactive Surfaces
Substrate: hydroxyapatite
Substrate: composite
C.Felgueiras, J.P. Marques de Sá, 1998
Fractal modelling may help to assess the growth conditions
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2 - FRACTALS IN PHYSIOLOGY
(Some Examples) Temporal Fractals • • • • • • • Spatial Fractals • • • • • • • •
•
Heart beat sequences Electroencephalograms Time intervals between action potentials Respiratory tidal volumes Ion channel kinetics in cell membranes Glycolysis metabolism DNA sequences mapping
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) Fractal Physiology. 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
r=1 N=1 r = 1/3 N = 3 = (1/3) 1
Dimension = D = 2
N=1
N = 9 = (1/3) 2
Number of exact replicas N(r) when the resolution is r :
1 N (r ) = = r − D r
D
⇒
D=
log(N (r )) log(1 / r )
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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 :
N(r) r
L(r ) = r . N(r ) = r1− D ln(L(r)) = (D-1) . ln(1/r)
ln(L(r))
ln(4/3) =0.12 ln(1)=0 slope = D - 1 = 0.26
ln(1/r)
ln(3)=0.47
Any object with a property L(r) satisfying a Power Law Scaling:
L(r ) ∝ r α ,
is said to have a fractal property L(r).
α = 1-D α = 2-D α = 3-D
for lengths for areas for volumes
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The scaling property can be written as: L(ar) = k L(r) ,
with a < 1, k=aα and
k, iteration factor, independent of r
For the Koch curve: r=1 ⇒ L(1) = 1 L(1/3) = 4/3 = 4/3 . L(1) L(1/9) = 16/9 = 4/3 . L(1/3)
r = 1/3 ⇒ r = 1/9 ⇒
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)β
β = 1/(D-1)
Roughening the surface → D ≈ 3 → β ≈ 0.5
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Estimating the Fractal Dimension
Length: L(r ) = rN(r ) ∝ r1− D
Number of covering boxes or spheres: N(r ) ∝ r − D
Box-counting: number of squares containing a piece of the object
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Statistical Self-Similarity Generalisation:
Dcap = lim ln(N(r )) / ln(1 / r )
r →0
,
capacity dimension
(Grassberg and Procaccia)
Result of box-counting method applied to retina angiogram
14 12 10 log2(N(r)) 8 6 4 2 0 -10
-8
-6 log2(1/r)
-4
-2
0
C.Felgueiras, J.P. Marques de Sá, A.M. Mendonça (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:
N (r ) i =1
Dinf = lim S(r ) / ln(r ) = lim −
r →0 r →0
∑ pi ln pi / ln(r )
S(r) – Entropy; pi - probability for a point to lie in box i
Correlation dimension:
Dcorr = lim ln(C(r )) / ln(r ) = lim ln
r →0 r →0
N (r ) i =1
∑
2 p i / ln(r )
C(r) : Number of pairs of points in box i
Generalised dimensions:
N (r )
Dq = lim
Fq (r ) 1 i =1 = lim r → 0 1 − q ln(1 / r ) r → 0 ln(1 / r )
ln
∑
pq i
D0 = Dcap ; D1 = Dinf ; D2 = Dcorr Dq decreases with q, e.g.:
Dcap ≥ Dinf ≥ Dcorr
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Hydroxyapatite Growth on Bioactive Surfaces
16 14 12 10 Fq( ) 8 6 4 2 0 -8 -7 -6 -5 -4 log2(1/ε ) -3 -2 -1 0 0 1 2 3 4 5 6
1,8
1,75
1,7 Dq
comp3c comp14 ha3a
1,65
ha14b
1,6
1,55 0 1 2 3 q 4 5 6
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
n
x(n) = ∑ x i
i =1
;
P(x i = 1) = P(x i = −1) =
1 2
x(n)
n
n 1 P ( x ( n) = k ) = n + k 2n 2
k = − n,− n + 2,..., n − 2, n
E[x(n)] = 0
V[x(n)] = n
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Probability of the random walk for n=16.
0.25 0.2 0.15 0.1 0.05 0
P
-8
-4
0
4
8
12
-16
-12
k
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Taking: With:
r ±ε
points per unit time jumps
r → ∞ , ε → 0 ; rε 2 → σ 2
Symmetric Random Walk → Brownian Motion Function
B (t , x )
Properties: B(t+∆t)-B(t): E[B(t+∆t)-B(t)] V[B(t+∆t)-B(t)]
∆
=
2 2 1 e − x / 2σ t σ 2πt
Gaussian increments =0 = σ2 . ∆t ⇒ σ[B(t+∆t)-B(t)] ∝ ∆t scaling
σ[B(t+∆t)-B(t)] ∝ σ[B(t+r∆t)-B(t)] ∝
∆t r∆t
=
1 r2
. σ[B(t+∆t)-B(t)]
Self-Affinity
∆t → r. ∆t ; σ(B) →
1 r2.
σ(B)
Iteration factor dependent on r (compare with page 9)
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150 100 50 0 501 1001 1501 2001 2501 3001 3501 4001
801
-50
50 40 30 20 10 0
1 201 401 601
20 15 10 5 0
101 151 201 1 51
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Two processes:
r 1 L = L ( r ) 2 2
Self-similar:
Self-affine:
r r L = L(r ) 2 2
L(r)
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 4 8 16 32 1/r
Self-similar Self-affine
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Fractional Brownian Motion
Def.: Moving average of B(t,x) with increments weighted by (t − s) H −1 / 2 - “random walk with memory”
B H (t, x ) ≈
−∞
∫ (t − s )
t
H −1 / 2
dB(s, x)
H ∈ ] 0,1 [
Hurst coefficient
Self-affine process with :
σ ( B H ) ∝ ∆t H
H=½ 0<H<½ ½<H<1
Brownian motion fBm with anti-correlated samples fBm with correlated samples
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BH(t)
Kernel
H=0.2, D=1.8
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 49 97 145 193 241 289 337 385 433 481
σ( B H )
2000 1000 0 -1000 -2000 1
∝
tH
101
201
301
401
H=0.5, D=1.5
1.2 1 0.8 0.6 0.4 0.2 0 1 49 97 145 193 241 289 337 385 433 481
5 0 -5 -10 -15 -20 -25 -30
H=0.8, D=1.2
7 6 5 4 3 2 1 0 1 52 103 154 205 256 307 358 409 460
30000 20000 10000 0 -10000 -20000 -30000 1 101 201 301
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Computing the fractal dimension of the fBm :
∆x
∆ xo= 1
∆ t = 1/N
∆ to = 1
For
1 ∆t = N
→
1 ∆x = ∆t = N
H
H
The region ∆x∆t is covered by:
N NH
squares of size L =
1 N
The whole segment is covered by: Therefore: D=2–H
1 N N(L) = N. H = N 2 − H = 2 − H L N
Higushi method: L(k)=sk1-D=skH-1
L(k) S o o o o o
1
ln(L(k)) slope=1-D o o o
100
o k
0 10
ln(k)
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6 - Chaotic Systems and Fractality
Chaotic system: Deterministic dynamic system with high sensitivity to initial conditions. Example: Hénon process: x(n+1) = y(n) + 1 - 1.4x(n)2; y(n+1) = 0.3x(n).
1.5
1
0.5
] n [ x
0
-0.5
-1
-1.5
0
10
20
30
40
50 n
60
70
80
90
100
a) b) Figure 9. a) Hénon process signals with initial values differing in 10-12; b) Hénon attractor; c) A detail of the attractor.
c)
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
Class A 154
Class A – NEM sleep 31 cases
Class FS
FHR (bpm)
148 Class B
Class B – REM sleep 50 cases Class FS - Neuro-cardiac depression 43 cases
133
0
5
10 Time (min)
15
20
FHR modelling by a bi-scale fBm fractal
2000 Class A Class B Class FS
1500
L(k)
1000
500
0
0
20
40 k
60
80
100
Results of Higushi method ( L(k ) = Sk H −1 ) showing two scaling regions
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FHR bi-scale behaviour
ln(L(k))
ln(S2) ln(S1)
H1-1 H2-1
ln(k)
0 10
STV
LTV H1, H2
0,8 0,6 0,4 0,2 0 H1-A H1-B H1-FS H2-A H2-B H2-FS
S1, S2
10 9 8 7 6 5 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 Sá (1998)
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FHR Classification Results
7 Features (3 temporal; 4 attractor)
Actual Class A B FS
No. of Cases 31 50 43 A
Predicted Class B 5 (16%) 48 (96%) 0 (0%) FS 0 (0%) 0 (0%) 41 (95%)
26 (84%) 2 (4%) 2 (5%)
Training set error: 7.3% - Test set error: 15.4%
Comparison with traditional clinical method: FHR Class A B FS Total No. of Cases 31 50 43 124 Pe % Fractal 16.1 4.0 4.7 7.3 STV 67.7 26.0 18.6 33.9 p < 0.001 < 0.001 0.021 < 0.001
C. Felgueiras, J.P. Marques de Sá, J. Bernardes, S. Gama (1998) Classification of Foetal Heart Rate Sequences 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 produce physiologic signals and structures. • Clarify the meaning of measurements. • Clarify the existence of deterministic causes in apparent random behaviour. • Derive model parameters to classify data.
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