SpiralWave Dynamics in Ionically Realistic Mathematical Models for Human Ventricular Tissue
Abstract
There is a growing consensus that lifethreatening cardiac arrhythmias like ven tricular tachycardia (VT) or ventricular ﬁbrillation (VF) arise because of the formation of spiral waves of electrical activation in cardiac tissue; unbroken spiral waves are associated with VT and broken ones with VF. Several experimental studies have shown that in homogeneities in cardiac tissue can have dramatic effects on such spiral waves.
In this thesis we focus on spiralwave dynamics in mathematical models of human ventricular tissue which contain (a) conduction in homogeneities, (b) ionic in homogeneities, (c) ﬁbroblasts, (d) Purkinje ﬁbers. We also study the effect of a periodic deformation of the simulation domain on spiral wavedynamics. Chapter 2 contains our study of “SpiralWave Dynamics and Its Control in the Presence of In homogeneities in Two Mathematical Models for Human Cardiac Tissue”; this chapter follows closely parts of a paper we have published [1]. Chapter 3 contains our study of “Spiralwave dynamics in a Mathematical Model of Human Ventricular Tissue with Myocytes and Fibroblasts”; this chapter follows closely a paper that we have submitted for publication. Chapter 4 contains our study of “Spiralwave Dynamics in Ionically Realistic Mathematical Models for Human Ventricular Tis sue: The Effects of Periodic Deformation”; this chapter follows closely a paper that we have submitted for publication. Chapter 5 contains our study of “Spiralwave dynamics in a Mathematical Model of Human Ventricular Tissue with Myocytes and Purkinje ﬁbers”; this chapter follows closely a paper that we will submit for publication soon.
In chapter 2, we study systematically the AP morphology in a stateoftheart mathematical model of human ventricular tissue due to tenTusscher, Noble, Noble, and Panﬁlov (the TNNP04 model); we also look at the contribution of individual ionic currents to the AP by partially or completely blocking ion channels associated with the ionic currents. We then carry out systematic studies of plane wave and circularwave dynamics in the TNNP04 model for cardiac tissue model. We present a detailed and systematic study of spiralwave turbulence and spa tiotemporal chaos in two mathematical models for human cardiac tissue due to (a) tenTusscher and Panﬁlov (the TP06 model) and (b) tenTusscher, Noble, Noble, and Panﬁlov (the TNNP04 model). In particular, we use extensive numerical simulations to elucidate the interaction of spiral waves in these models with conduction and ionic in homogeneities. Our central qualitative result is that, in all these models, the dynamics of such spiral waves depends very sensitively on such in homogeneities. A major goal here is to develop low amplitude deﬁbrillation schemes for the elimination of VT and VF, especially in the presence of in homogeneities that occur commonly in cardiac tissue. Therefore, we study a control scheme that has been suggested for the control of spiral turbulence, via lowamplitude current pulses, in such mathematical models for cardiac tissue; our investigations here are designed to examine the efficacy of such control scheme in the presence of in homogeneities in biophysical realistic models. We ﬁnd that a scheme that uses control pulses on a spatially extended mesh is more successful in the elimination of spiral turbulence than other control schemes. We discuss the theoretical and experimental implications of our study that have a direct bearing on deﬁbrillation, the control of lifethreatening cardiac arrhythmias such as ventricular ﬁbrillation.
In chapter 3, we study the role of cardiac ﬁbroblasts in ventricular tissue; we use the TNNP04 model for the myocyte cell, and the ﬁbroblasts are modelled as passive cells. Cardiac ﬁbroblasts, when coupled functionally with myocytes, can modulate their electrophysiological properties at both cellular and tissue levels. Therefore, it is important to study the effects of such ﬁbroblasts when they are coupled with myocytes. Chapter 3 contains our detailed and systematic study of spiralwave dynamics in the presence of ﬁbroblasts in both homogeneous and inhomogeneous domains of the TNNP04 model for cardiac tissue. We carry out extensive numerical studies of such modulation of electrophysiological properties in mathematical models for (a) single myocyte ﬁbroblast (MF) units and (b) twodimensional (2D) arrays of such units; our models build on earlier ones and allow for no, oneway, or twoway MF couplings. Our studies of MF units elucidate the dependence of the actionpotential (AP) morphology on parameters such as Ef , the ﬁbroblast resting membrane potential, the ﬁbroblast conductance Gf , and the MF gapjunctional coupling Ggap. Furthermore, we ﬁnd that our MF composite can show autorhythmic and oscillatory behaviors in addition to an excitable response. Our 2D studies use
(a) both homogeneous and inhomogeneous distributions of ﬁbroblasts, (b) various ranges for parameters such as Ggap, Gf , and Ef , and (c) intercellular couplings that can be no, oneway, and twoway connections of ﬁbroblasts with myocytes. We show, in particular, that the planewave conduction velocity CV decreases as a function of Ggap, for no and oneway couplings; however, for twosided coupling, CV decreases initially and then increases as a function of Ggap, and, eventually, we observe that conduction failure occurs for low values of Ggap. In our homogeneous studies, we ﬁnd that the rotation speed and stability of a spiral wave can be controlled either by controlling Ggap or Ef . Our studies with ﬁbroblast inhomogeneities show that a spiral wave can get anchored to a local ﬁbroblast inhomogeneity. We also study the efﬁcacy of a lowamplitude control scheme, which has been suggested for the control of spiralwave turbulence in mathematical models for cardiac tissue, in our MF model both with and without heterogeneities.
In chapter 4, we carry out a detailed, systematic study of spiralwave dynamics in the presence of periodic deformation (PD) in two stateoftheart mathematical models of human ventricular tissue, namely, the TNNP04 model and the TP06 model. To the best of our knowledge, our work is the ﬁrst, systematic study of the dynamics of spiral waves of electrical activation and their transitions, in the presence of PD, in such biophysically realistic mathematical models of cardiac tissue. In our studies, we use three types of initial conditions whose time evolutions lead to the following states in the absence of PD: (a) a single rotating spiral (RS),
(b) a spiralturbulence (ST) state, with a single meandering spiral, and (c) an ST state with multiple broken spirals for both these models. We then show that the imposition of PD in these three cases leads to a rich variety of spatiotemporal pat terns in the transmembrane potential including states with (a) an RS state with ncycle temporal evolution (here n is a positive integer), (b) rotatingspiral states with quasiperiodic (QP) temporal evolution, (c) a state with a single meandering spiral MS, which displays spatiotemporal chaos, (d) an ST state, with multiple bro ken spirals, and (e) a quiescent state in which all spirals are absorbed (SA). For all three initial conditions, precisely which one of the states is obtained depends on the amplitudes and the frequencies of the PD in the x and y directions. We also suggest speciﬁc experiments that can test the results of our simulations. We also study, in the presence of PD, the efﬁcacy of a lowamplitude control scheme that has been suggested, hitherto only without PD, for the control of spiralwave turbulence, via lowamplitude current pulses applied on a square mesh, in mathematical models for cardiac tissue. We also develop linemesh and rectangularmesh variants of this control scheme. We ﬁnd that square and linemeshbased, lowamplitude control schemes suppress spiralwave turbulence in both the TP06 and TNNP04 models in the absence of PD; however, we show that the linebased scheme works with PD only if the PD is applied along one spatial direction. We then demonstrate that a minor modiﬁcation of our linebased control scheme can suppress spiralwave turbulence: in particular, we introduce a rectangularmeshbased control scheme, in which we add a few control lines perpendicular to the parallel lines of the line based control scheme; this rectangularmesh scheme is a signiﬁcant improvement over the squaremesh scheme because it uses fewer control lines than the one based on a square mesh.
In chapter 5, we have carried out detailed numerical studies of (a) a single unit of an endocardial cell and Purkinje cell (EP) composite and (b) a twodimensional bilayer, which contains such EP composites at each site. We have considered bio physically realistic ionic models for human endocardial cells (Ecells) and Purkinje cells (Pcells) to model EP composites. Our study has been designed to elucidate the sensitive dependence, on parameters and initial conditions, of (a) the dynamics of EP composites and (b) the spatiotemporal evolution of spiral waves of electrical activation in EPbilayer domains. We examine this dependence on myocyte parameters by using the three different parameter sets P1, P2, and P3; to elucidate the initialcondition dependence we vary the time at which we apply the S2 pulse in our S1S2 protocol; we also investigate the dependence of the spatiotemporal dynamics of our system on the EP coupling Dgap, and on the number of Purkinje ventricular junctions (PVJs), which are measured here by the ratio R, the ratio of the total number of sites to the number of PVJs in our simulation domain.
Our studies on EP composites show that the frequency of autorhythmic activity of a P cell depends on the diffusive gapjunctional conductance Dgap. We perform a set of simulations to understand the sourcesink relation between the E and P cells in an EP composite; such a sourcesink relation is an important determinant of wave dynamics at the tissue level. Furthermore, we have studied the restitution properties of an isolated E cell and a composite EP unit to uncover this effect on wave dynamics in 2D, bilayers of EP composites.
Autorhythmicity is an important property of Purkinje cell; it helps to carry electrical signals rapidly from bundle of His to the endocardium. Our investigation of an EP composite shows that the cycle length (CL) of autorhythmic activity decreases, compared to that of an uncoupled Purkinje cell. Furthermore, we ﬁnd that the APD increases for an EP composite, compared to that of an uncoupled P cell. In our second set of simulations for an EPcomposite unit, we have obtained the AP behaviors and the amount of ﬂux that ﬂows from the E to the P cell during the course of the AP. The direction of ﬂow of this ﬂux is an important quantity that identiﬁes which one of these cells act as a source or a sink in this EP composite. We have found that the P cell in an EP composite acts as a stimulationcurrent source for the E cell in the depolarization phase of the AP, when the stimulus is applied to both cells or to the P cell only. However, the P cell behaves both as a source and a sink when the stimulus is applied to the E cell only. In our third set of simulations for an EP composite unit, we have calculated the restitution of the APD; this plays an important role in deciding the stability of spiral waves in mathematical models for cardiac tissue. Our simulation shows that, for the EP composite with high coupling (Dgap = Dmm~10), the APDR slope decreases, relative to its value for an isolated E cell, for parameter sets P1 and P2, and ﬁrst increases (for 50 ≤ DI ≤ 100 ms) and then decreases for the parameter set P3 ; however, for low coupling (Dgap = Dmm~100), the variation of the AP D as function of DI, for an EP composite, shows biphasic behavior for all these three parameter sets. We found that the above dynamics in EP cable type domains, with EP composites, depends sensitively on R.
We hope our in silico studies of spiralwave dynamics in a variety of stateofthe art ionic models for ventricular tissue will stimulate more experimental studies that examine such dynamics.
Collections
 Physics (PHY) [300]
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