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 Cell

APD versus DI

Restitution 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.

APD 、 DI and T(CL)

Restitution Curve A+D=T 1:1 2:2

Conduction 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:

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. Reference

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) Normal Beats (NB) NB to VT VT to VF

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

Spiral Waves ventricle cells 2 days in culture 8 day old embryo recorded temp: 36 deg C. each frame is approximately 1 cm square Reference ： Optical Mapping Image Database

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

Experimental Results for Multi-armed Spirals in Cardiac Tissue Reference: PNAS, vol. 101, p15530 (2004).

Aliev-Panfilov Model

Cable Theory

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 ~ 1 0 ~ Herbivores ~ 3 0 ~ Plants ~ 9 Rules of Cellular Automata

The next population in a cell. Time step = 1. = Value(now) + Changes Value(next) = Value(now) + Changes now next Update the Population

Plants dominated by season and herbivores. Roughly separating the season into two parts. Pla.(next) = Pla.(now) + Changes Changes Summer +1 –Her. Winter –Her. The Rules of Plants

The affection coming from neighboring cells. Define the local sum (L) of the population densities. Eight Neighbors L(i) = Value(i) + Value(j) j = Neibors j = Neibors The Neighbors of a Fixed Cell

Her.(next) = Her.(now) + Changes If Pla. GE. Her. Changes Car. = 0 ; H 0 ~L(H)~H 1 +1 Otherwise -1 The Rules of Herbivores

Her.(next) = Her.(now) + Changes If Pla. LT. Her. Changes -(Her. – Pla.) – Car. -(Her. – Pla.) – Car. The Rules of Herbivores

Her.(next) = Her.(now) + Changes Changes Her. > 0 ; C 0 ~L(C)~C 1 +1 Otherwise -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 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>0.5 Random noise: H=0.5 Anti-persistent noise: H<0.5 S(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