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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 動 作 電 位 週 期 舒 張 區 間 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) An 1 R( Dn ) An Dn T Restitution Curve 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;90:289-296 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. Reference http://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 approximately 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 e t r t 2e 2e d 2 2 ke(e a )( e 1) er y x 1 r r ke(e b 1) (2 e) Cable Theory 2V raxial I m 2 x I m I c I ion ; I c Cm dV dt 1 2V V Cm I ion 2 raxial x t V 2V I ion / Cm D 2 t x D 1 /(Cm raxial ) 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 SpatialModulation 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 Rules of Cellular Automata The frame of CA (50 X 50). The components of the ecosystem. 0 ~ Carnivores ~ 1 0 ~ Herbivores ~ 3 0~ Plants ~9 Update the Population The next population in a cell. Time step = 1. now next Value(next) = Value(now) + Changes The Rules of Plants Plants dominated by season and herbivores. Roughly separating the season into two parts. Pla.(next) = Pla.(now) + Changes { Changes Summer Winter +1 –Her. –Her. The Neighbors of a Fixed Cell The affection coming from neighboring cells. Define the local sum (L) of the population densities. L(i) = Value(i) + Value(j) j = Neibors Eight Neighbors The Rules of Herbivores Her.(next) = Her.(now) + Changes If Pla. GE. Her. Car. = 0 ; H0~L(H)~H1 +1 Otherwise -1 { Changes The Rules of Herbivores Her.(next) = Her.(now) + Changes If Pla. LT. Her. Changes { -(Her. – Pla.) – Car. The Rules of Carnivores Her.(next) = Her.(now) + Changes { Changes Her. > 0 ; C0~L(C)~C1 +1 Otherwise -1 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 60 50 40 4 30 3 20 10 2 0 1 -10 0 -20 0 -1 500 1000 1500 2000 2500 3000 Brownian trajectory -2 -3 -4 0 500 1000 1500 2000 2500 Gaussian random noise 3000 x x 2 1 p( x, t ) exp 2 2 t 2 t Hurst Exponent (I) Hurst Exponent (II) H=0.8 H=0.6 Persistent noise: H>0.5 Random noise: H=0.5 Anti-persistent noise: H<0.5 S(f) ~ f-b, b = 2H – 1 H=0.4 H=0.2 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