The flow of a liquid film over a rapidly rotating horizontal disk has many applications inmedical, industrial, and engineering fields. A specific example is the heat and mass transfer processes between expanded liquid and surrounded dense gas. Diferent wave regimes of a liquid film depend on a flow conditions such as the properties of a liquid, its initial speed,parameters of environment, etc. Therefore, experimental investigation of the film flow over a spinning disk is needed to both validate theoretical predictions and establish methods for fluid flow monitoring.This thesis presents novel video-based algorithms for detection and tracking wave structural data of the liquid film flowing over a spinning disk reactor. The algorithms are based on the spiral model of wave and the quasi-optimal method for estimation of a wave velocity as ill-posed problem. Their performance is compared with results predicted by the fluid dynamics based on the Navier-Stokes equations in the case of thin film.Using experimental video data, the developed models and algorithms allow investigators to estimate the characteristics of wave regimes such as wavelengths, inclination angles, and the radial and azimuthal velocity components of the fluid. The accuracy of estimated characteristics was analyzed. It was shown that average distance between consecutive two waves,their spiral shapes, and the radial velocities of waves confirm the theoretical results and predictions. In particular, computed wavelength is within 1% and a change of the inclination angles is within 2% of the predicted values.

Reliable signal transmission among cells is important for long-range coordination. While higher organisms have designated structures for signal transmission, such as axons, it remains unclear how simpler communities of cells are organized to relay signals. Furthermore, many biological systems exhibit spatial heterogeneity, which can interrupt signal propagation. In this thesis, we investigate this problem by modeling the spatial organization and dynamics of electrochemical signaling, and we compare our results to experiments from our collaborators on Bacillus subtilis bacterial biofilms. The experiments show that only a fraction of cells participates in signal propagation and that these cells are spatially clustered with a size distribution that follows a power-law decay. These observations suggest that the fraction of participating cells is just at the tipping point between a disconnected and a fully connected conduit for signal transmission. We utilize percolation theory and a minimal FitzHugh-Nagumo-type excitable dynamics model to test this hypothesis, and genetically modified biofilms with altered structure and dynamics to validate our modeling. Our results suggest that the biofilm is organized near the critical percolation point in order to negotiate the benefit and cost of long-range signal transmission. Then, more detailed experiments show that the participation probability is correlated from cell to cell and varies in space. We use these observations to develop an enhanced percolation model, and show using simulations and a renormalization argument that the main conclusions are unaffected by these features. Finally, we use our dynamic model to investigate the effects of heterogeneity beyond the radial wave regime and into the spiral wave regime. We find that spatial correlations in the heterogeneity promote or suppress spiraling depending on the parameters, a surprising feature that we explain by demonstrating that these spirals form by distinct mechanisms. We characterize the dependence of the spiral period on the heterogeneity using techniques from percolation theory. Taken together, our results reveal that the spatial structure of cell-to-cell heterogeneity can have important consequences for signal propagation in cellular communities.<br>

There is a growing consensus that life-threatening 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 spiral-wave 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 wave-dynamics. Chapter 2 contains our study of “Spiral-Wave 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 “Spiral-wave 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 “Spiral-wave 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 “Spiral-wave 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 state-of-the-art mathematical model of human ventricular tissue due to ten-Tusscher, 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 circular-wave dynamics in the TNNP04 model for cardiac tissue model. We present a detailed and systematic study of spiral-wave turbulence and spa- tiotemporal chaos in two mathematical models for human cardiac tissue due to (a) ten-Tusscher and Panfilov (the TP06 model) and (b) ten-Tusscher, 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 low-amplitude 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 life-threatening 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 spiral-wave 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) two-dimensional (2D) arrays of such units; our models build on earlier ones and allow for no, one-way, or two-way MF couplings. Our studies of MF units elucidate the dependence of the action-potential (AP) morphology on parameters such as Ef , the fibroblast resting membrane potential, the fibroblast conductance Gf , and the MF gap-junctional 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, one-way, and two-way connections of fibroblasts with myocytes. We show, in particular, that the plane-wave conduction velocity CV decreases as a function of Ggap, for no and one-way couplings; however, for two-sided 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 low-amplitude control scheme, which has been suggested for the control of spiral-wave 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 spiral-wave dynamics in the presence of periodic deformation (PD) in two state-of-the-art 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 spiral-turbulence (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 n-cycle temporal evolution (here n is a positive integer), (b) rotating-spiral 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 low-amplitude control scheme that has been suggested, hitherto only without PD, for the control of spiral-wave turbulence, via low-amplitude current pulses applied on a square mesh, in mathematical models for cardiac tissue. We also develop line-mesh and rectangular-mesh variants of this control scheme. We find that square- and line-mesh-based, low-amplitude control schemes suppress spiral-wave turbulence in both the TP06 and TNNP04 models in the absence of PD; however, we show that the line-based scheme works with PD only if the PD is applied along one spatial direction. We then demonstrate that a minor modification of our line-based control scheme can suppress spiral-wave turbulence: in particular, we introduce a rectangular-mesh-based control scheme, in which we add a few control lines perpendicular to the parallel lines of the line- based control scheme; this rectangular-mesh scheme is a significant improvement over the square-mesh 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 two-dimensional 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 EP-bilayer domains. We examine this dependence on myocyte parameters by using the three different parameter sets P1, P2, and P3; to elucidate the initial-condition dependence we vary the time at which we apply the S2 pulse in our S1-S2 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 gap-junctional conductance Dgap. We perform a set of simulations to understand the source-sink relation between the E and P cells in an EP composite; such a source-sink 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 EP-composite 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 stimulation-current 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 spiral-wave dynamics in a variety of state-of-the- art ionic models for ventricular tissue will stimulate more experimental studies that examine such dynamics.

No description available.

There is a growing consensus that life-threatening cardiac arrhythmias like ventricular 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 inhomogeneities in cardiac tissue can have dramatic effects on such spiral waves. In this thesis we try to understand these experimental results by carrying out detailed and systematic studies of the interaction of spiral waves with different types of inhomogeneities in mathematical models for cardiac tissue. In Chapter 1 we begin with a general introduction to cardiac arrhythmias, the cardiac conduction system, and the connection between electrical activation waves in cardiac tissue and cardiac arrhythmias. As we have noted above, VT and VF are believed to be associated with spiral waves of electrical activation on cardiac tissue; such spiral waves form because cardiac tissue is an excitable medium. Thus we give an overview of excitable media, in which sub-threshold perturbations decay but super-threshold perturbations lead to an action potential that consists of a rapid stage of depolarization of cardiac cells followed by a slow phase of repolarization. During this repolarization phase the cells are refractory. We then give an overview of earlier studies of the effects of inhomogeneities in cardiac tissue; and we end with a brief description of the principal problems we study here. Chapter 2 describes the models we use in our work. We start with a general introduction to the cable equation and then discuss the Hodgkin-Huxley-formalism for the transport of ions across a cell membrane through voltage-gated ion channels. We then describe in detail the three models that we use for cardiac tissue, which are, in order of increasing complexity, the Panfilov model, the Luo Rudy Phase I (LRI) model, and the reduced Priebe Beuckelmann (RPB)model. We then give the numerical schemes we use for solving these model equations and the initial conditions that lead to the formation of spiral waves. For all these models we give representative results from our simulations and compare the states with spiral turbulence. In Chapter 3 we investigate the effects of conduction inhomogeneities (obstacles) in the three models introduced in Chapter 2. We outline first the experimental results that have provided the motivation for our study. We then discuss how we introduce obstacles in our simulations of the Panffilov, LRI, and RPB models for cardiac tissue. Next we present the results of our numerical studies of the effects, on spiral-wave dynamics, of the sizes, shapes, and positions of the obstacles. Our Principal result is that spiral-wave dynamics in these models depends sensitively on the position of the obstacle. We find, in particular, that, merely by changing the position of a conduction inhomogeneity, we may convert spiral turbulence (the analogue in our models of VF) to a single rotating spiral (the analogue of VT) anchored to the obstacle or vice versa; even more exciting is the possibility that, at the boundary between these two types of behaviour, we find a quiescent state Q with no spiral waves. Thus our study obtains all the possible qualitative behaviours found in experiments, namely, (1) VF might persist even in the presence of an obstacle, (2) it might be suppressed partially and become VT, or (3) it might be eliminated completely. In Chapter 4 we extend our work on conduction inhomogeneities (Chapter 3) to ionic inhomogeneities. Unlike conduction inhomogeneities, ionic inhomogeneities allow the conduction of activation waves. We find, nevertheless, that they too can lead to the anchoring of spiral waves or even the complete elimination of spiral-wave turbulence. Since spiral waves can enter the region in which there is an ionic inhomogeneity, their behaviours in the presence of such an inhomogeneity are richer than those with conduction inhomogeneities. We find, in particular, that a single spiral wave anchored at an ionic inhomogeneity can show temporal evolution that may be periodic, quasiperiodic, or even chaotic. In the last case the spiral wave shows a chaotic pattern inside the ionic inhomogeneity and a regular one outside it. Defibrillation is the control of arrhythmias such as VF. Most often defibrillation is effected electrically by administering a shock, either externally or via an internally implanted defibrillator. The development of low-amplitude defibrillation schemes, which minimise the deleterious effects of the applied shock, is a major challenge in the treatment of cardiac arrhythmias. Numerical studies of models for cardiac tissue provide us with convenient means of studying the elimination of spiral-wave turbulence by the application of external electrical stimuli; this is the numerical analogue of defibrillation. Over the years some low-amplitude defibrillation schemes have been suggested on the basis of such numerical studies. In Chapter 5 we discuss two such schemes that have been shown to suppress spiral-wave turbulence in two-dimensional models for cardiac tissue and also scroll-wave turbulence in three-dimensional models. One of these schemes uses local electrical pacing, typically in the centre of the simulation domain; the other applies the external electrical stimuli over a mesh. We study the efficacy of these schemes in the presence of conduction inhomogeneities. We find, in particular, that the local-pacing scheme, though effective in a homogeneous simulation domain, fails to control spiral turbulence in the presence of an obstacle and, indeed, might even facilitate spiral-wave break up. By contrast, the second scheme, which uses a mesh, succeeds in eliminating spiral-wave turbulence even in the presence of an obstacle. We end with some concluding remarks about the possible experimental implications of our study in Chapter 6.

This thesis is devoted to the study of the dynamics of spiral and scroll waves in a mathematical model for cardiac tissue. We study the effects of the presence of heterogeneities on electrical-wave dynamics. The heterogeneities in the medium occur because of the variation in the electrophysiological properties of the constituent myocytes in the tissue, or because of the presence of cells like fibroblasts and pathological myocytes that can trigger early afterdepolarizations (EADs). We study how these heterogeneities can lead to the formation of spiral and scroll waves and how they can affect the stability of the spiral and scroll waves in cardiac tissue. We also investigate the role of abnormal cells, which can trigger pathological excitations like EADs, on the formation of spiral and scroll waves, and how such cells can trigger premature electrical pulses like premature-ventricular-complexes (PVCs) in cardiac tissue. Earlier studies have examined the role of ionic heterogeneities on spiral-wave initiation and their effects on spiral-wave stability. However, none of these studies has calculated, in a controlled way, the effects of individual ion-channel conductances on spiral- and scroll-wave properties, such as the frequency of these waves, and the effects of the spatial gradients, in each ion-channel conductance, on their stability; we present these results in Chapter 2. Although many studies in the past have studied the effects of fibroblast coupling on wave-dynamics in cardiac tissue, a detailed study of spiral-wave dynamics in a medium with a well-defined, heterogeneous distribution of fibroblasts (e.g., with a gradient in the fibroblast density (GFD)) has not been performed; therefore, in Chapter 3 we present the effects of such GFD on spiral- and scroll-wave dynamics. Then, in Chapter 4, we present a systematic study of how a clump of fibroblasts can lead to spiral waves via high-frequency pacing. Some studies in the past have studied the role of early afterdepolarizations (EADs) in the formation of arrhythmias in cardiac tissue; we build on such studies and present a detailed study of the effects of EADs on the formation of spiral waves and their dynamics, in Chapter 5. Finally, in Chapter 6 we provide the results of our detailed investigation of the factors that assist the triggering of abnormal electrical pulses like premature ventricular complexes by a cluster of EAD-capable cells. A brief summary of the chapters is provided below: Chapter 2: In this chapter we investigate the effects of spatial gradients in the ion-channel conductances of various ionic currents on spiral-and scroll-wave dynamics. Ionic heterogeneities in cardiac tissue arise from spatial variations in the electrophysiological properties of cells in the tissue. Such variations, which are known to be arrhythmogenic, can be induced by diseases like ischemia. It is important, therefore, to understand the effects of such ionic heterogeneities on electrical-wave dynamics in cardiac tissue. To investigate such effects systematically, of changing the ion-channel properties by modifying the conductances of each ionic currents, on the action-potential duration (APD) of a myocyte cell. We then study how these changes in the APD affect the spiral-wave frequency ω in two-dimensional tissue. We also show that changing the ion-channel conductance not only changes ω but also the meandering pattern of the spiral wave. We then study how spatial gradients in the ion-channel conductances affect the spiral-wave stability. We find that the presence of this ionic gradient induces a spatial variation of the local ω, which leads to an anisotropic reduction of the spiral wavelength in the low-ω region and, thereby, leads to a breakup of the spiral wave. We find that the degree of the spiral-wave stability depends on the magnitude of the spatial variation in ω, induced by the gradient in the ion-channel conductances. We observe that ω varies most drastically with the ion-channel conductance of rapid delayed rectifier K+ current GKr, and, hence, a spiral wave is most unstable in the presence of a gradient in GKr (as compared to other ion-channel conductances). By contrast, we find that ω varies least prominently with the conductances of the transient outward K+ current Gto and the fast inward Na+ current (GNa); hence, gradients in these conduc-tances are least likely to lead to spiral-wave breaks. We also investigate scroll-wave instability in an anatomically-realistic human-ventricular heart model with an ionic gradient along the apico-basal direction. Finally, we show that gradients in the ion-channel densities can also lead to spontaneous initiation of spiral waves when we pace the medium at high frequency. Chapter 3: In this chapter we study the effects of gradients in the density of fibroblasts on wave-dynamics in cardiac tissue. The existence of fibroblast-myocyte coupling can modulate electrical-wave dynamics in cardiac tissue. In diseased hearts, the distribution of fibroblasts is heterogeneous, so there can be gradients in the fibroblast density (henceforth we call this GFD) especially from highly injured regions, like infarcted or ischemic zones, to less-wounded regions of the tissue. Fibrotic hearts are known to be prone to arrhythmias, so it is important to understand the effects of GFD in the formation and sustenance of arrhythmic re-entrant waves, like spiral or scroll waves. Therefore, we investigate the effects of GFD on the stability of spiral and scroll waves of electrical activation in a state-of-the-art mathematical model for cardiac tissue in which we also include fibroblasts. By introducing GFD in controlled ways, we show that spiral and scroll waves can be unstable in the presence of GFDs because of regions with varying spiral or scroll-wave frequency ω, induced by the GFD. We examine the effects of the resting membrane potential of the fibroblast and the number of fibroblasts attached to the myocytes on the stability of these waves. Finally, we show that the presence of GFDs can lead to the formation of spiral waves at high-frequency pacing. Chapter 4: In this chapter we study the arrhythmogenic effects of lo-calized fibrobblast clumps. Localized heterogeneities, caused by the regional proliferation of fibroblasts, occur in mammalian hearts because of diseases like myocardial infarction. Such fibroblast clumps can become sources of pathological reentrant activities, e.g., spiral or scroll waves of electrical activation in cardiac tissue. The occurrence of reentry in cardiac tissue with heterogeneities, such as fibroblast clumps, can depend on the frequency at which the medium is paced. Therefore, it is important to study the reentry-initiating potential of such fibroblast clumps at different frequencies of pacing. We investigate the arrhythmogenic effects of fibroblast clumps at high- and low-frequency pacing. We find that reentrant waves are induced in the medium more prominently at high-frequency pacing than with low-frequency pacing. We also study the other factors that affect the potential of fibroblast clumps to induce reentry in cardiac tissue. In particular, we show that the ability of a fibroblast clump to induce reentry depends on the size of the clump, the distribution and percentage of fibroblasts in the clump, and the excitability of the medium. We study the process of reentry in two-dimensional and a three-dimensional mathematical models for cardiac tissue. Chapter 5: In this chapter we investigate the role of early afterdepolarizations (EADs) on the formation of spiral and scroll waves. Early after depolarizations, which are abnormal oscillations of the membrane poten-tial at the plateau phase of an action potential, are implicated in the de-velopment of cardiac arrhythmias like Torsade de Pointes. We carry out extensive numerical simulations of the TP06 and ORd mathematical models for human ventricular cells with EADs. We investigate the different regimes in both these models, namely, the parameter regimes where they exhibit (1) a normal action potential (AP) with no EADs, (2) an AP with EADs, and (3) an AP with EADs that does not go back to the resting potential. We also study the dependence of EADs on the rate of at which we pace a cell, with the specific goal of elucidating EADs that are induced by slow or fast rate pacing. In our simulations in two- and three-dimensional domains, in the presence of EADs, we find the following wave types: (A) waves driven by the fast sodium current and the L-type calcium current (Na-Ca-mediated waves); (B) waves driven only by the L-type calcium current (Ca-mediated waves); (C) phase waves, which are pseudo-travelling waves. Furthermore, we compare the wave patterns of the various wave-types (Na-Ca-mediated, Ca-mediated, and phase waves) in both these models. We find that the two models produce qualitatively similar results in terms of exhibiting Na-Ca- mediated wave patterns that are more chaotic than those for the Ca-mediated and phase waves. However, there are quantitative differences in the wave patterns of each wave type. The Na-Ca-mediated waves in the ORd model show short-lived spirals but the TP06 model does not. The TP06 model supports more Ca-mediated spirals than those in the ORd model, and the TP06 model exhibits more phase-wave patterns than does the ORd model. Chapter 6: In this chapter we study the role of EAD-capable cells, and fibroblasts on the trigerring of abnormal electrical pulses called premature ventricular complexes (PVCs). Premature ventricular complexes, which are abnormal impulse propagations in cardiac tissue, can develop because of various reasons including early afterdepolarizations (EADs). We show how a cluster of EAD-generating cells (EAD clump) can lead to PVCs in a model of cardiac tissue, and also investigate the factors that assist such clumps in triggering PVCs. In particular, we study, through computer simulations, the effects of the following factors on the PVC-triggering ability of an EAD clump: (1) the repolarization reserve (RR) of the EAD cells; (2) the size of the EAD clump; (3) the coupling strength between the EAD cells in the clump; and (4) the presence of fibroblasts in the EAD clump. We find that, although a low value of RR is necessary to generate EADs and hence PVCs, a very low value of RR leads to low-amplitude EAD oscillations that decay with time and do not lead to PVCs. We demonstrate that a certain threshold size of the EAD clump, or a reduction in the coupling strength between the EAD cells, in the clump, is required to trigger PVCs. We illustrate how randomly distributed inexcitable obstacles, which we use to model collagen deposits, affect PVC-triggering by an EAD clump. We show that the gap-junctional coupling of fibroblasts with myocytes can either assist or impede the PVC-triggering ability of an EAD clump, depending on the resting membrane potential of the fibroblasts and the coupling strength between the myocyte and fibroblasts. We also find that the triggering of PVCs by an EAD clump depends sensitively on factors like the pacing cycle length and the distribution pattern of the fibroblasts.

No description available.