Applications of Cellular Automata in Cardiac and Ecological Systems

國立東華大學物理系

蕭又新4/28/2006

Outline: Cardiac Systems

Heart rate variability Action potential and Cardiac cells Arrhythmias and spiral waves Spiral waves described by partial differential

equations Cellular automata approach

Cardiac activity and ECG

正常人的心率及 R-R分佈圖

食用搖頭丸的女性患者

Action Potential in a Ventricular Action Potential in a Ventricular Cell Cell

動作電位週期

動作電位週期

舒張區間

舒張區間

APD versus DIAPD versus DI

Restitution Curve by ExperimentRestitution Curve by Experiment

Restitution Curve in canine endocardial muscle

Koller, Marcus L. et al. Dynamic restitution of action potential duration during electrical alternans and ventricular fibrillation. Am. J. Physiol. 275(Heart Circ. Physiol. 44): H1635-H1642, 1998.

APDAPD 、、 DI and T(CL)DI and T(CL)

TDA

DRA

nn

nn

)(1

Restitution CurveRestitution Curve

Conduction BlockConduction Block

Spatially distributed action potential dynamics in a canine cardiac Purkinje fiber

Jeffrey J. Fox et al. Spatiotemporal Transition to Conduction Block in Canine Ventricle. Circ Res. 2002;90:289-296

Normal Rhythm and Arrhythmias

Normal sinus rhythm

60~100 beats per minute Ectopic rhythms

For examples ： Ventricular tachycardia(心室頻脈 )

Ventricular fibrillation(心室顫動 )

Ventricular Tachycardia (VT)

Ventricular tachycardia (VT) is a tachydysrhythmia originating from a ventricular ectopic focus, characterized by a rate typically greater than 120 beats per minute and wide QRS complexes.

VT may be monomorphic or polymorphic. Nonsustained VT is defined as a run of tachycardia of less than 30 seconds duration; a longer duration is considered sustained VT.

Referencehttp://www.emedicine.com/

Ventricular Fibrillation (VF)

What is ventricular fibrillation? The heart beats when electrical signals move through it. Ventricular fibrillation is a condition in which the heart's electrical activity becomes disordered. When this happens, the heart's lower chambers contract in a rapid, unsynchronized way. The heart pumps little or no blood.

VT and VF in Electrocardiogram

Reference: Chaos, Solitons and Fractals Vol.13 (2002) 1755.

Normal Rhythm

ventricle cells 2.5 days in culture 8 day old embryo 0.8 ml plating density recorded temp: 36 deg.

C each frame is

approximately 1 cm square

Reference ：Optical Mapping Image Database

http://www.cnd.mcgill.ca/bios/bub/imagebase.html

Spiral Waves

ventricle cells 2 days in culture 8 day old embryo recorded temp: 36 deg

C. each frame is approxim

ately 1 cm square

Reference ：Optical Mapping Image Database

http://www.cnd.mcgill.ca/bios/bub/imagebase.html

Spiral Waves Breakup

ventricle cells 2 days in culture 8 day old embryo 0.4 ml plating density alphaGA acid 50ul recorded temp: 36 deg C. each frame is approximately

1 cm square

Reference ：Optical Mapping Image Database

http://www.cnd.mcgill.ca/bios/bub/imagebase.html

Experimental Results for Multi-armed Spirals in Cardiac Tissue

Reference: PNAS, vol. 101, p15530 (2004).

Aliev-Panfilov Model

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Cable TheoryCable Theory

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Normal Rhythm and Conduction Block

Simulation results of normal rhythm and conduction block

Spiral Waves Formation and Breakup

Action Potential in Cardiac Muscle

Cellular Automata in Cardiac Tissue

Activation state (6 time units)

Refractory state (3 time units)

Rest state Nearest-neighbor

coupling

Target Waves

Spiral Waves Formation (I)

Spiral Waves Formation (II)

Wave Breaks by Considering Spatial-Modulation of the Refractory Period

Wave Breaks Occurring by Heterogeneity :Alain Karma, PNAS 97, 5687 (2000)

Simulated 3D Spirals Based on MRI Images

256X256 grids for each frame

Enjoy Music Coming from Your Heart

Outline: Ecological Systems

Complexity in laboratory insect populations Extinction in spatially structured populations Cellular automata approach in a modeling

ecology: grass, rabbit, and wolf Time-domain analysis: Hurst exponent Future works: computational epidemiology

Laboratory Insect Populations

Proc. R. Lond. B 264, 481 (1997)

Food Chain

Predator-Prey Mechanism

Species: grass, rabbit, and wolf

Season effect Nearest-neighbor

and next nearest-neighbor coupling: 8 cells

50x50 cells

The frame of CA (50 X 50). The components of the ecosystem.

0 ~ Carnivores ~ 10 ~ Carnivores ~ 1

0 ~ Herbivores ~ 30 ~ Herbivores ~ 3

0 ~ Plants ~ 90 ~ Plants ~ 9

Rules of Cellular Automata

The next population in a cell. Time step = 1.

Value(next) = Value = Value(now)(now) + Changes + Changes

nownow nextnext

Update the Population

Plants dominated by season and herbivores. Roughly separating the season into two parts.

Pla.Pla.(next)(next) = Pla. = Pla.(now)(now) + Changes + Changes

{Changes Changes Summer +1 –Her.Summer +1 –Her.

Winter –Her. Winter –Her.

The Rules of Plants

The affection coming from neighboring cells. Define the local sum (L) of the population

densities.

Eight Eight NeighborsNeighbors

L(i)L(i) = Value(i) + = Value(i) + Value(j)Value(j)

j = Neiborsj = Neibors

The Neighbors of a Fixed Cell

Her.Her.(next)(next) = Her. = Her.(now)(now) + Changes + Changes

If Pla. GE. Her. If Pla. GE. Her.

{Changes Changes

Car. = 0 ; HCar. = 0 ; H00~L(H)~H~L(H)~H11 +1 +1

Otherwise -1Otherwise -1

The Rules of Herbivores

Her.Her.(next)(next) = Her. = Her.(now)(now) + Changes + Changes

If Pla. LT. Her. If Pla. LT. Her.

{Changes Changes -(Her. – Pla.) – Car.-(Her. – Pla.) – Car.

The Rules of Herbivores

Her.Her.(next)(next) = Her. = Her.(now)(now) + Changes + Changes

{Changes Changes Her. > 0 ; CHer. > 0 ; C00~L(C)~C~L(C)~C11 +1 +1

Otherwise -1Otherwise -1

The Rules of Carnivores

No wolf Summer period Complicated

fluctuations Anti-correlation in

between grass and rabbit

No extinction

Spatiotemporal Plot for Grass Evolution

Considering wolf Summer period Complicated

fluctuations Positive correlation in

between rabbit and wolf

No extinction

Spatiotemporal Plots for Grass Evolution

No wolf Considering winter

effect (W=1, S+W=10) Complicated

fluctuations No extinction Anti-correlation in

between grass and rabbit

Spatiotemporal Plots for Grass Evolution

Considering wolf Considering longer winter

(W=3, S+W=10) Complicated fluctuations Wolf extinction Anti-correlation in between

grass and rabbit Complicated correlation in

between wolf and rabbit

Spatiotemporal Evolution of Grass

No wolf Considering spatial

effect: uniformly distributed rabbit (R=1)

Summer period Complicated fluctuations In early stage rabbits

increase fast Rabbit extinction Anti-correlation in

between grass and rabbit

Spatiotemporal Plots for Grass Evolution

No wolf Considering spatial effect:

uniformly distributed rabbit (R=3)

Considering winter effect (W=1, S+W=10)

Complicated fluctuations Surprise! slow down rabbit

extinction Anti-correlation in between

grass and rabbit

Spatiotemporal Plots for Grass Evolution

It might be a good way to design tiles as well as carpets!

Spiral Waves in Ecology: SURPRISE!

Random Noise and Brownian Diffusion

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0 500 1000 1500 2000 2500 3000

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0 500 1000 1500 2000 2500 3000

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Gaussian random noise

Brownian trajectory

Hurst Exponent (I)

Hurst Exponent (II)

H=0.8

H=0.6

H=0.4

H=0.2

Persistent noise: H>0.5Random noise: H=0.5Anti-persistent noise: H<0.5S(f) ~ f-b, b = 2H – 1

Extinction Characterized by H: OK

Extinction Characterized by H: NOT OK

Computational Epidemiology

S: susceptible state (latent period)

I: Infectious state (infectious period)

R: recovery period

Measles and Vaccination