Spiral Wave Meandering in Models of Ventricular Tissue as an Antiarrhythmic Target GP Kremmydas1, AV Holden2 1LUMC-KI, Argostoli, Greece 2University of Leeds, Leeds, UK Abstract version of the Luo-Rudy phase II biophysical equations are used as idealised virtual tissue tools. The effect We characterise the effects of ionic conductance and of altering model parameters on the spatial extent of concentration changes on the meandering patterns and spiral wave meander is used to assess the effectiveness dynamics of re-entrant excitation in models of mammalian of pharmacological manipulation of the corresponding ventricular tissue. An increase in the spatial extent of biophysical parameters in altering the likelihood of spiral wave meander might lead to an increased likelihood self-termination of re-entrant excitation. An increased of self-termination of the re-entrant excitation, since the extent of meander increases the probability of spiral probability of the spiral core meeting and annihilating at wave core meeting and annihilating at the in-excitable the in-excitable tissue boundaries is increased. Thus, the tissue boundaries. Therefore, spiral wave meandering spiral-wave meandering behaviour and dynamics provides behaviour and dynamics provides a potential target for a potential target for antiarrhythmic drugs. antiarrhythmic drugs. Such detailed investigation of antiarrhythmic action on models of cardiac excitation provides a useful virtual tissue engineering tool for the 1. Introduction initial evaluation of appropriate drug therapies. Ventricularre-entrant arrhythmias can lead to fibrillation and cardiac death. Experimental and theoretical evidence 2. Methods suggests that such arrhythmias can appear in the absence A two dimensional isotropic and homogenous excitable of anatomical obstacles and even in apparently healthy medium describing cardiac excitation can be modelled by myocardium. The underlying mechanism of re-entrant the reaction-diffusion system arrhythmias is believed to be the establishment of spiral waves of excitation in ventricular muscle [1]. The m 2 existence of spiral wave solutions is a common feature ∂tV = −1/C Iion,i + D∇ V in reaction-diffusion equations of excitable media and i=1 the ability to support spiral wave excitation has been ∂tuj = fj(u1,u2,...,uj,...,un−1,V) (1) experimentally verified for cardiac tissue. Many basic j =1, 2,...,n − 1 characteristics of re-entrant arrhythmias can be understood by using homogenous reaction-diffusion models of cardiac where V is the membrane potential, C is the specific tissue with reaction kinetics based on detailed biophysical membrane capacitance, Iion,i = gi(u1,u2,...,un−1,V) equations of cardiac cell excitation. This approach the set of transmembrane ionic currents, D the diffusion 2 can serve as the first step towards the computational coefficient, ∇ the Laplacian operator and uj the gating reconstruction of virtual tissue and organ models in which variables and ionic concentrations of the given model. further idiosyncratic features of cardiac tissue as an The exact form of the equations (1) for the Oxsoft excitable medium such as anisotropy and inhomogeneity Heart single guinea-pig ventricle cell (GPV) are are incorporated. provided in [2] while for the Luo-Rudy model (LRII) in In this study we characterise the effects of ionic [3, 4, 5]. The equations for LRII used here were extracted conductance and concentration changes on the meandering from the source code in C++ available from the web-site patterns and dynamics of re-entrant excitation in models of http://www.cwru.edu/med/CBRTC/LRdOnline. mammalian ventricular tissue. Two-dimensional reaction- For both models, spiral waves were initiated by a diffusion systems with kinetics based on the Oxsoft phase distribution method [2] in a 20mm×20mm medium Heart single guinea-pig ventricle cell and a recent discretised with a space step of 100 µm using a standard 0276−6547/03 $17.00 ©2003 IEEE 785 Computers in Cardiology 2003;30:785−788. very close to its critical value and excitation propagation slows down. 100 80 LRII + 60 [Na ]o 40 a b meand 20 2+ R [Ca ]o ∆ % 0 Figure 1. Tip trajectories for the GPV (a) and the LRII (b) model (size for both plots is 3mm×3mm) defined as -20 [K+] the intersection of the corresponding V isolines with the -40 o isoline of an appropriately selected gating variable (I Ca -60 gating variable f=0.5 for GPV and ICa,L gating variable -50 -25 0 25 50 ∆ f=0.4 for LRII). % [ion]o 100 80 GPV 5-point discretisation for the Laplacian operator (see fig. + [Na ]o 1) and non-flux boundary conditions. The equations 60 (1) where integrated using a semi-explicit Euler method 40 2+ (gating variables were calculated implicitly at every time [Ca ]o 2 meand 20 step) with ∆t =10µs and D=60mm /s for LRII and R ∆ ∆t =50µs and D=31.25mm2/s for GPV (the selected % 0 D values correspond to solitary plane wave propagation -20 + velocity of ≈ 0.4m/s). [K ]o -40 Spiral wave tip trajectories were defined as the intersection of V =-10mV and u =0.5 isolines (u -60 GPV GPV -50 -25 0 25 50 is a gating variable for ICa) for GPV and V =-40mV and ∆ % [ion]o uLRII=0.4 isolines (gating variable for the L-type calcium current ICa,L) for LRII. Figure 2. Effects of extracellular ion concentration Spiral-wave re-entrant activity in each model was changes on the spatial extent of spiral wave meandering simulated for 2s and for each simulation a transmembrane in LRII (top) and GPV (bottom) model expressed ionic conductance (gNa, gKr, gKs), an extracellular as % change in Rmeand versus % change in ion + 2+ + ionic concentration ([Na ]o, [Ca ]o, [K ]o) or the concentrations. Standard values (0% ∆[ion]o) are 2+ + 2+ + permeability of the cardiac cell membrane to Ca for GPV: [Na ]o=140mM, [Ca ]o=2mM, [K ]o=4mM + 2+ ions PCa was altered and the spiral tip trajectory and for LRII: [Na ]o=140mM, [Ca ]o=1.8mM, + (xtip(t),ytip(t)) was monitored on the (x, y)-plane. [K ]o=4.5mM. Among these parameters gKr and gKs are unique to LRII since GPV does not incorporate the corresponding ionic Using the two idealised virtual-tissue models a set currents (IKr and IKs). The spatial extent of spiral wave of in silico “experiments” were conducted to investigate meander was quantified as the minimum radius Rmeand of the effects of parameter changes on the dynamics a circle enclosing the tip trajectory over a whole period at of spiral wave re-entry. Figure 2 summarizes the the end of each simulation. results of experiments investigating the effects of altering extracellular ionic concentrations on the spatial extent of 3. Results spiral wave meandering. Per cent changes of meandering size ∆Rmeand are plotted versus per cent change in the Tip trajectories for spiral wave simulations with corresponding extracellular ionic concentrations. In both + 2+ “standard” parameters for GPV and LRII are shown in the GPV and LRII model increasing [Na ]o and [Ca ]o figure 1. A characteristic five-lobe meandering pattern is led to an increase of meandering extent while the increase + apparent in the GPV model with Rmeand=1.14mm (1.a) of [K ]o had an opposite effect. while in the LRII model (1.b) the spiral wave rigidly The sodium current INa activates during the rising rotates around a circular core (Rmeand=0.43mm). Near the phase of the action potential and its amplitude affects spiral wave tip, the curvature of the wavefront becomes the maximum depolarisation rate (maximum dV/dt)and 786 100 100 LRII 80 80 g 60 60 Ks PCa 40 40 meand 20 meand R R 20 ∆ g ∆ Na g % 0 % Kr 0 -20 -20 -40 -40 -60 -50 -25 0 25 50 -50 -25 0 25 50 ∆ ∆ ∆ ∆ % PCa , % gNa % gKr , % gKs 100 Figure 4. Effects of altering gKr and gKs on meander size GPV 80 PCa in LRII. 60 gNa 2+ 40 here is that PCa is implicated in the equation for the Ca - meand 20 mediated slow inward current ICa of the GPV model while R 2+ ∆ in the LRII model it affects the Ca -mediated component % 0 of the L-type calcium current ICa,L. The L-type current -20 together with the T-type (transient) calcium current I Ca,T -40 have in more recent models replaced the older slow inward I current. This difference could explain the higher -60 Ca -50 -25 0 25 50 extent of meandering in GPV compared to LRII model for ∆ ∆ % PCa , % gNa similar changes in PCa: in GPV the “bulk” ICa current is affected while in LRII only the calcium part of one of its Figure 3. Effect of changes in sodium conductance 2+ components (ICa,L) is affected (the total ICa,L current has g and Ca permeability P on spatial extent + Na Ca also K+ and Na mediated components controlled by the of meandering. Standard values are for LRII: −4 corresponding ion permeability). gNa=16mS/µF PCa=5.4×10 cm/s and GPV: gNa=2.5µS, PCa=0.25nA/mM LRII GPV thus the conduction velocity of activation wavefronts in cardiac tissue. Persistent inactivation of INa occurs inside the inexcitable spiral core around which spiral waves persistently rotate. Pathophysiological conditions or pharmacological interventions that alter the sodium conductance gNa would significantly affect the dynamics of spiral wave reentry in cardiac muscle. Figure 3 summarises the results of numerical experiments on the Figure 5. Meandering patterns for the LRII and GPV effects of altering the fast sodium conductance (g Na) and models. Tip trajectories (plotted in different spatial scale) 2+ membrane permeability to Ca ions (PCa). In LRII and correspond to 700ms of re-entrant activity in simulations GPV increasing gNa and PCa extends the meander size of in which PCa was increased by 50% with respect to the spiral waves.