Fractals in Physiology

Transcription

1 Fractals in Physiology A Biomedical Engineering Perspective J.P. Marques de Sá jmsa@fe.up.pt INEB Instituto de Engenharia Biomédica Faculdade de Engenharia da Universidade do Porto 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.. 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

2 2 - MOTIVATION Many physiological signals exhibit almost random behaviour Foetal Heart Rate C. Felgueiras, J.P. Marques de Sá (998) but with statistical self-similarity Model: Temporal Fractal

3 3 Many physiological structures exhibit a high degree of spatial complexity Human Retina Image Retina angiogram. A.M. Mendonça, A. Campilho (997) 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

4 4 Hydroxyapatite Growth on Bioactive Surfaces Substrate: hydroxyapatite Substrate: composite C.Felgueiras, J.P. Marques de Sá, 998 Fractal modelling may help to assess the growth conditions

5 5 2 - FRACTALS IN PHYSIOLOGY (Some Examples) Temporal Fractals Heart beat sequences Electroencephalograms Time intervals between action potentials 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. (994) Fractal Physiology. Oxford University Press.

6 6 3 - SELF-SIMILARITY AND FRACTAL DIMENSION Koch curve Scale: r a.r < a < Self-similarity dimension: How many parts in an object are similar to the whole? Simple Objects Dimension = D = r = r = /3 Dimension = D = 2 N = N = 3 = (/3) N = N = 9 = (/3) 2 Number of exact replicas N(r) when the resolution is r : D D ( ) = = r N r r log( N( r)) D = log(/ r)

7 7 (with exact self-similarity) Fractal Objects Fractal: Any object with D having a fraction part? Not always Any object with D > topological dimension? Not always

8 8 Length of the object : N(r) r D L( r) = r. N( r) = r ln(l(r)) = (D-). ln(/r) ln(l(r)) ln(4/3) =.2 ln()= slope = D - =.26 ln(3)=.47 ln(/r) Any object with a property L(r) satisfying a Power Law Scaling: α L ( r) r, is said to have a fractal property L(r). α = -D α = 2-D α = 3-D for lengths for areas for volumes

9 9 The scaling property can be written as: L(ar) = k L(r), with a <, k=a α and k, iteration factor, independent of r For the Koch curve: r = L() = r = /3 r = /9 L(/3) = 4/3 = 4/3. L() L(/9) = 6/9 = 4/3. L(/3) L(r/3) = 4/3. L(r)

10 Cardiac Pacing Electrode Zr Interface impedance with roughness effect Zi Interface impedance per unit of true interface area Zr = a (Zi)β β = /(D-) Roughening the surface D 3 β.5

11 Estimating the Fractal Dimension Length: L( r) = rn( r) r D Number of covering boxes or spheres: N( r) r D Box-counting: number of squares containing a piece of the object

12 2 Statistical Self-Similarity Generalisation: D cap = lim ln( N( r)) / ln(/ r), capacity dimension r (Grassberg and Procaccia) Result of box-counting method applied to retina angiogram 4 2 log2(n(r)) log2(/r) C.Felgueiras, J.P. Marques de Sá, A.M. Mendonça (999) D =.82 Number of branching sites (cluster diameter) D

13 3 Generalised Fractal Dimensions (In a coverage of N(r) boxes of size r) Information dimension: D inf = lim r S( r) / ln( r) = lim r N( r) i= p i ln p i / ln( r) S(r) Entropy; p i - probability for a point to lie in box i Correlation dimension: D corr = lim r ln( C( r)) / ln( r) = lim ln r N( r) i= p 2 i / ln( r) C(r) : Number of pairs of points in box i Generalised dimensions: Dq = N( r) q ln pi F ( r) lim i= q = lim r q ln(/ r) r ln(/ r) D = D cap ; D = D inf ; D 2 = D corr D q decreases with q, e.g.: Dcap D inf Dcorr

14 4 Hydroxyapatite Growth on Bioactive Surfaces 6 4 Fq( ) log2(/ε ),8,75 Dq,7,65 comp3c comp4 ha3a ha4b,6, q ha hydroxiapatite substrate comp composite glass substrate

15 5 Ha Hydroxiapatite comp 2 composite 2 glass

16 6 5 - SELF-AFFINITY AND THE FRACTIONAL BROWNIAN MODEL The symmetric random walk n x ( n) = x i = i ; P( x i = ) = P( x i = ) = 2 x(n) n n P( x( n) = k) = k = n, n + 2,..., n 2, n n n + k 2 2 [ x( n) ] = V[ x( n) ] n E =

17 7 Probability of the random walk for n= P k 6

18 8 Taking: r points per unit time ±ε jumps 2 2 With: r, ε ; rε σ Symmetric Random Walk Brownian Motion Function Properties: B( t, x) = σ 2πt e x 2 / 2 σ 2 t B(t+ t)-b(t): Gaussian increments E[B(t+ t)-b(t)] = V[B(t+ t)-b(t)] = σ 2. t σ[b(t+ t)-b(t)] t scaling σ[b(t+ t)-b(t)] t σ[b(t+r t)-b(t)] r t = r 2. σ[b(t+ t)-b(t)] Self-Affinity t r. t ; σ(b) r 2. σ(b) Iteration factor dependent on r (compare with page 9)

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20 2 Two processes: r 2 2 Self-similar: L = L( r) r 2 r 2 Self-affine: L = L( r) L(r) /r Self-similar Self-affine

21 2 Fractional Brownian Motion Def.: Moving average of B(t,x) with increments weighted by / 2 ( t s) H - random walk with memory t ( t, x) B ( t s) db( s, x) H H / 2 H ], [ Hurst coefficient Self-affine process with : σ ( BH ) H t H = ½ Brownian motion < H < ½ fbm with anti-correlated samples ½ < H < fbm with correlated samples

22 22 B H (t) Kernel σ ( BH ) H t H=.2, D= H=.5, D= H=.8, D=

23 23 Computing the fractal dimension of the fbm : x x o= t = /N t o= For t = N x = t H = N H The region x t is covered by: N squares of size H N The whole segment is covered by: N N( L) = N. = N H N Therefore: D = 2 H L = N 2 H = 2 H L Higushi method: L(k)=sk -D =sk H- L(k) ln(l(k)) S o o o o o o o o o k slope=-d ln(k)

24 24 Chaotic system: 6 - Chaotic Systems and Fractality Deterministic dynamic system with high sensitivity to initial conditions. Example: Hénon process: x(n+) = y(n) + -.4x(n) 2 ; y(n+) =.3x(n)..5.5 x [n] n a) b) c) Figure 9. a) Hénon process signals with initial values differing in -2 ; b) Hénon 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.

25 25 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+.

26 26 7 Fractal Modelling of Foetal Heart Rate FHR (bpm) Class A Class FS Class B Class A NEM sleep 3 cases Class B REM sleep 5 cases Class FS - Neuro-cardiac depression 43 cases Time (min) FHR modelling by a bi-scale fbm fractal 2 5 Class A Class B Class FS L(k) k H Results of Higushi method ( L ( k) = Sk ) showing two scaling regions

27 27 FHR bi-scale behaviour ln(l(k)) ln(s2) ln(s) H- H2- ln(k) STV LTV H, H2,8,6,4,2 H-A H-B H-FS H2-A H2-B H2-FS S, S S-A S-B S-FS S2-A S2-B S2-FS

28 28 Foetal Heart Attractor (Stage A; Delay= 8 samples 2 s) C. Felgueiras, J.P. Marques de Sá (998)

29 29 FHR Classification Results 7 Features (3 temporal; 4 attractor) Actual No. of Predicted Class Class Cases A B FS A 3 26 (84%) B 5 2 (4%) 5 (6%) 48 (96%) (%) (%) FS 43 2 (5%) (%) 4 (95%) Training set error: 7.3% - Test set error: 5.4% Comparison with traditional clinical method: FHR No. of Pe % Class Cases Fractal STV p A <. B <. FS Total <. C. Felgueiras, J.P. Marques de Sá, J. Bernardes, S. Gama (998) Classification of Foetal Heart Rate Sequences Based on Fractal Features, Medical & Biological Eng. & Comp., 36(2): 97-2.

30 3 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.