2001 fractal modeling a biomedical engineering perspective

7/31/2019 2001 Fractal Modeling a Biomedical Engineering Perspective

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INEB Instituto de Engenharia Biomdica, Porto, Portugal

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.

2001 fractal modeling a biomedical engineering perspective

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