Spiral- and scroll-wave dynamics in mathematical models for canine and human ventricular tissue with varying Potassium and Calcium currents

Abstract

We conduct a systematic, direct-numerical-simulation (DNS) study, in mathematical models for ventricular tissue, of the dependence of spiral- and scroll-wave dynamics on $G_{Kr}$, the maximal conductance of the delayed rectifier Potassium current ($I_{Kr}$) channel, and the parameter ${\gamma }_{Cao}$, which determines the magnitude and shape of the current $I_{CaL}$ for the L-type calcium-current channel, in both square and anatomically realistic, whole-ventricle simulation domains. We study canine and human models. In the former, we use a canine-ventricular geometry, with fiber-orientation details, obtained from diffusion-tensor-magnetic-resonance-imaging (DTMRI) data; and we employ the physiologically realistic Hund-Rudy-Dynamic (HRD) model for a canine ventricular myocyte. To focus on the dependence of spiral- and scroll-wave dynamics on $G_{Kr}$ and ${\gamma }_{Cao}$, we restrict ourselves to an HRD-model parameter regime, which does not produce spiral- and scroll-wave instabilities because of other, well-studied causes like a very sharp action-potential-duration-restitution (APDR) curve or early after depolarizations (EADs) at the single-cell level. We find that spiral- or scroll-wave dynamics are affected predominantly by a simultaneous change in $I_{CaL}$ and $I_{Kr}$ from their original values in the model, rather than by a change in any one of these currents; other currents do not have such a large effect on these wave dynamics in this parameter regime of the HRD model. In particular, we examine spiral-wave dynamics for ten different values of $G_{Kr}$ and ten different values of ${\gamma }_{Cao}$ in our 2D DNSs. For our 3D DNSs in an anatomically realistic domain, we chose 16 parameter sets. In the parameter regime we begin with, the system displays broken spiral or scroll states with S1–S2 initial conditions (see below). We show that, by simultaneously increasing $G_{Kr}$ and reducing ${\gamma }_{Cao}$, we can get to a parameter regime in which the system displays single, stable rotating spirals or scroll waves. We obtain stability diagrams (or phase diagrams) in the $G_{Kr}-{\gamma }_{Cao}$ plane; and we find that these diagrams are significantly different in our 2D and 3D studies. In the 3D case, the geometry of the domain itself supports the confinement of the scroll waves and makes them more stable compared to their spiral-wave counterparts in our flat, 2D simulation domain. Thus, a combination of functional and geometrical mechanisms produce different dynamics for 3D scroll waves and their 2D spiral-wave counterparts. In particular, the former do not break easily because, in an anatomically realistic ventricular geometry, they are not easily absorbed at boundaries, nor do they break near boundaries. We have also carried out a comparison of our HRD results with their counterparts for the human-ventricular TP06 model; and we have found important differences between wave dynamics in these two models. The region in parameter space, where we obtain broken spiral or scroll waves in the HRD model is the region of stable rotating waves in the TP06 model; the default parameter values produce broken waves in the HRD model, but stable scrolls in the TP06 model. In both these models, to make a transition, (most simply, from broken-wave to stable-scroll states) we must simultaneously increase $I_{Kr}$ and decrease $I_{CaL}$; a modification of only one of these currents is not enough to effect this transition. Furthermore, the converse, i.e., an increase in $I_{CaL}$ along with a decrease in $I_{Kr}$ does not yield any interesting dynamical transitions in the HRD model, for, in this range of currents, this model does not sustain spiral or scroll waves or broken waves.