Annals of Biomedical Engineering, Vol. 28, pp. 755–771, 2000 0090-6964/2000/28͑7͒/755/17/$15.00 Printed in the USA. All rights reserved. Copyright © 2000 Biomedical Engineering Society Origins of Spiral Wave Meander and Breakup in a Two-Dimensional Cardiac Tissue Model ZHILIN QU,FAGEN XIE,ALAN GARFINKEL, and JAMES N. WEISS Cardiovascular Research Laboratory, Departments of Medicine ͑Cardiology͒, Physiology, and Physiological Science, University of California, Los Angeles, CA (Received 22 November 1999; accepted 23 June 2000) Abstract—We studied the stability of spiral waves in homo- trocardiographic recordings show a transition from order geneous two-dimensional cardiac tissue using phase I of the to disorder when VT degenerates to VF. This transition Luo–Rudy ventricular action potential model. By changing the is analogous to the transition from order to chaos in conductance and the relaxation time constants of the ion chan- nels, various spiral wave phenotypes, including stable, quasi- generic nonlinear dynamical systems. Chaos in cardiac periodically meandering, chaotically meandering, and breakup arrhythmias has been investigated by several were observed. Stable and quasiperiodically meandering spiral authors.5,15,23,28,30,34,37 Recent combined experimental and waves occurred when the slope of action potential duration theoretical evidence12,21,30 suggests that the transition ͑APD͒ restitution was Ͻ1 over all diastolic intervals visited from VT to VF may represent a quasiperiodic transition during reentry; chaotic meander and spiral wave breakup oc- curred when the slope of APD restitution exceeded 1. Curva- to spatiotemporal chaos. ture of the wave changes both conduction velocity and APD, Previous experimental and theoretical work has iden- and their restitution properties, thereby modulating local stabil- tified the restitution properties of action potential dura- ity in a spiral wave, resulting in distinct spiral wave pheno- tion ͑APD͒ and conduction velocity ͑CV͒ as key param- types. In the LR1 model, quasiperiodic meander is most sensi- eters influencing the stability of cardiac tive to the Naϩ current, whereas chaotic meander and breakup arrhythmias.7,8,16,23,28,29,34,37 APD restitution is generally are more dependent on the Ca2ϩ and Kϩ currents. © 2000 Biomedical Engineering Society. ͓S0090-6964͑00͒00807-9͔ defined as the curve relating the present APD to the previous diastolic interval ͑DI͒, the interval from the end of the previous action potential to the next excitation Keywords—Reentry, Arrhythmias, Restitution, Stability, Elec- trophysiology, Simulation APDnϩ1ϭ f ͑DIn͒ϭ f ͑CLnϪAPDn͒ϭF͑APDn͒, ͑1͒ INTRODUCTION where CL is cycle length. Similarly, CV restitution is Ventricular fibrillation ͑VF͒ is the single most com- defined as mon cause of sudden cardiac death, yet its mechanisms are poorly understood. Increasing evidence suggests that CV ϭg͑DI ͒. ͑2͒ spiral waves, a generic property of excitable media, are a nϩ1 n major form of reentry underlying common cardiac 19 arrhythmias.4,9,14 It has been conjectured11,14,25 that ͓In the literature of excitable media, CVnϩ1ϭg(CLn) monomorphic tachycardia may correspond to a stationary was generally called the ‘‘dispersion relation.’’ Here we anchored spiral wave, and polymorphic tachycardia to a prefer the term CV restitution because the term disper- meandering spiral wave. The clinical observation that sion is widely used by cardiologists in a different con- disordered VF is almost always preceded by ventricular text, such as ‘‘dispersion of refractoriness.’’͔ It has been tachycardia ͑VT͒38 raises the possibility that the transi- shown in paced cells and in one-dimensional ͑1D͒ rings tion from VT to VF may correspond to spiral wave that the equilibrium state loses its stability when the breakup, in which an initiated single spiral wave ͑the VT slope of APD restitution Ͼ1, leading to complex dynam- ics, such as alternans and chaos.8,23,28,34,37 In 2D cardiac phase͒ breaks up after several rotations into multiple 7,16,29 spiral waves ͑the VF phase͒. In addition, clinical elec- tissue models, numerical simulations showed that spiral wave breakup was caused by steep APD restitu- tion. However, how restitution properties relate to the Address correspondence to: Zhilin Qu, Cardiovascular Research Laboratory, MRL 3645, UCLA School of Medicine, 675 Charles E. various spiral wave behaviors is not well understood. Young Dr. South, Los Angeles, CA 90095-1760. Electronic mail: In this paper, we study spiral wave dynamics, the [email protected] transition to spatiotemporal chaos, and their relationship 755 756 QU et al. to cardiac electrical restitution properties in a homoge- Numerical Simulation neous 2D cardiac tissue, using phase I of the Luo–Rudy ͑LR1͒ ventricular action potential model.24 We carried out numerical simulations of the isolated cell, of 1D cable, and of 2D tissue. Here we specify the numerical details for the simulations. To simulate an METHODS isolated cell, we integrated the following differential equation The Cardiac Model Cardiac cells are resistively connected by gap junc- ϭϪ ϩ tions between cells. Ignoring the detailed structure of the dV/dt ͑Iion Isti͒/Cm , ͑6͒ real tissue, we consider a homogeneous continuous con- duction model7,16,29 in which: where Isti is the external stimulus current pulse density we applied to the system. The duration of the pulse is 2 tϭϪI /C ϩDٌ2V, ͑3͒ץ/Vץ ion m ms and the strength is Ϫ40 ␮A/cm2 ͑about two times threshold stimulus strength͒.Weusedafourthorder where V is the transmembrane potential. C ϭ1␮F/cm2 m Runge–Kutta method to integrate Eq. ͑6͒ with a time is the membrane capacitance, and D is the diffusion step ⌬tϭ0.01 ms. constant determined by gap junction resistance, surface- To simulate the effects of curvature, we used the to-volume ratio, and membrane capacitance.7,29 We use 2 following 1D cable equation: Dϭ0.001 cm /ms. Iion is the total ionic current density of the membrane from the LR1 model, which is: IionϭINa ¯ 3 x2, ͑7͒ץ/2VץxϩDץ/VץtϭϪI /C ϩD␬ץ/Vץ ϩIsiϩIKϩIK1ϩIKpϩIb . INaϭGNam hj(VϪENa) is the ϩ ¯ ion m fast inward Na current; IsiϭGsidf(VϪEsi) is the slow inward current, assumed to be the L-type Ca2ϩ current; ¯ IKϭGKxx1(VϪEK) is the slow outward time-dependent where ␬ is the curvature. Equation ͑7͒ was adopted from ϩ ¯ 39 K current; IK1ϭGK1K1ϱ(VϪEK1) is the time- Zykov to study the effects of curvature on CV and ϩ 6 independent K current; IKpϭ0.0183 Kp(VϪEKp) is the APD, and was used recently by Comtois and Vinet. ϩ plateau K current; and Ibϭ0.03921(Vϩ59.87) is the Equation ͑7͒ was derived by Zykov for a stationary wave total background current. m, h, j, d, f, and x are gating with constant ␬.39 Equation ͑7͒ was integrated using the variables satisfying the following type of differential conventional Euler method with ⌬tϭ0.005 ms and ⌬x equation ϭ0.015 cm. For 2D simulation, the conventional Euler method dy/dtϭ͑y ϱϪy͒/␶y , ͑4͒ was computationally too tedious and costly to integrate Eq. ͑3͒. We integrated Eqs. ͑3͒–͑5͒ using operator split- where y represents the gating variable. The ionic concen- ting and adaptive time step methods. We use ⌬xϭ⌬y ϭ0.015 cm. The ordinary differential equations were in- trations are set as ͓Na͔iϭ18 mM, ͓Na͔oϭ140 mM, tegrated with a time step which varied from 0.005 to 0.1 ͓K͔iϭ145 mM, ͓K͔oϭ5.4 mM, while the intracelluar Ca2ϩ concentration obeys ms, and the partial differential equation was integrated using the alternating direction implicit method with a Ϫ Ϫ time step of 0.1 ms. Details and numerical accuracy were d͓Ca͔ /dtϭϪ10 4I ϩ0.07͑10 4Ϫ͓Ca͔ ͒. ͑5͒ i si i discussed previously.27 For simulation of an obstacle in the tissue, we elec- Details of the LR1 action potential model were presented trically disconnected a circular area in the center of the in Table I of Luo and Rudy’s paper.24 By setting ͓K͔ o tissue by setting the diffusion constant to zero ͑i.e., no- ϭ5.4 mM, the maximum conductance of I and I are K K1 flux boundary condition͒, and used the same numerical ¯ 2 ¯ 2 GKϭ0.282 mS/cm and GK1ϭ0.6047 mS/cm . In Luo method to integrate the system. 24 ¯ 2 ¯ and Rudy’s paper, GNaϭ23 mS/cm and Gsiϭ0.09 Tip trajectories of spiral waves were traced using the 2 ¯ ¯ ¯ mS/cm . In this paper, we change GNa , Gsi , and GK , intersection point of successive contour lines of voltage and the relaxation time constants ␶y (yϭm, h, j, d, f, x͒ corresponding to Ϫ35 mV, measured every 2 ms. The to create different spiral wave behaviors. Unless explic- intersection points of these successive contour lines form itly stated either in the text or in the figure captions, a tip trajectory. APD was defined as the duration during parameter values are the same as specified in the original which VϾϪ72 mV, and DI as the duration during which LR1 model. We will omit the unit of the channel con- VϽϪ72 mV. The resting potential of the LR1 model is ductance (mS/cm2) in the rest of this paper. around Ϫ84 mV. Nonlinear Dynamics of Spiral Waves 757 length of 1 s, and S2 was delivered at progressively shorter coupling intervals scanning the diastolic interval until refractoriness was reached. To calculate APD and CV in tissue, we initiated a unidirectional wave in the 1D ring using Eq. ͑7͒. APD and CV restitution were obtained by reducing the ring length until conduction failed. APD and CV were mea- sured after 5–10 cycles of transient for each ring length. When the slope of APD restitution was Ͻ1 everywhere, the reentrant wave in 1D was stable, and the APD res- titution curve was single valued. When the APD restitu- tion slope exceeded 1, however, the unidirectional wave oscillated in a modulated alternans,8,18,35 yielding a double-valued APD restitution curve ͑due to memory effects arising from the alternating APD͒.