2001 Fractal Modeling a Biomedical Engineering Perspective

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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 (*)
(*) Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted provided that: • Copies are not made or distributed for profit or commercial advantage. • Copies bear this notice and full citation on the first page. To copy otherwise, to republish, to post on services, or to redistribute to lists, require specific permission and/or fee.

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

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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.

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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 )

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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…

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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)

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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
INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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Ha – Hydroxiapatite comp 2 – composite 2 glass

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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)

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

4501

1

20

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

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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)

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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.

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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.

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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Foetal Heart Attractor
(Stage A; Delay= 8 samples ≈ 20 s)

C. Felgueiras, J.P. Marques de Sá (1998)

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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.

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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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.

INEB – Instituto de Engenharia Biomédica, Porto, Portugal

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