Mathematics in Biology and Medicine
(Biomechanics, Cardiovascular System, Lungs)
Monday, June 17th
Salle Du Manège
10h00
Modelling Calcium Dynamics in Cardiac Purkinje cells
Bungay, Sharene, Kazi Haq and Bruno Stuyvers
Memorial University of Newfoundland

Cardiac Purkinje cells are responsible for transmitting electrical signals (action potentials) from the pacemaker cells, down the centre of the heart to the apex, and out to the ventricular endocardium, where they trigger the contraction of myocardial cells.

In recent years, research has been focused on determining how Purkinje cells function to fulfill their role, with many questions yet unanswered. Notably, a more complete understanding of the mechanisms of calcium regulation in Purkinje cells is required to understand, and possibly prevent, ventricular arrhythmias that are thought to originate in abnormal Ca2+ handling of Purkinje cells. A mathematical model consisting of a system of PDEs that describe calcium dynamics in Purkinje cells will be presented. This model includes: calcium release (from the sarcoplasmic reticulum (SR) - the main calcium storage organelle); calcium diffusion; calcium uptake (into the SR); calcium leak into the cytosol; as well as the interaction of calcium with various buffers (both mobile and stationary) within the cell. Numerical results show that the model is able to reproduce both spontaneous and evoked calcium events that have been observed experimentally.

10h30
Rate of electrical activity in a spatially heterogeneous distribution of connected autonomous and non-autonomous cardiac cells: a mathematical modeling study with application to biopacemakers
Comtois, Philippe
Département de Physiologie, Institut de Génie Biomédical, Université de Montréal

Cell therapy and tissue engineering applied to cardiac tissue underlie the use of cultured cells. These cells show intrinsic variability in their contractile or electrical properties. An interesting innovative application of tissue engineering is the biopacemaker patch consisting of autonomous electrical cardiac cells that could ultimately serve as a replacement for electronic pacemakers. Experiments show that initial seeding of cells is composed of mainly two populations: electrically spontaneous cells and excitable but non-autonomous cells. Random initial deposition followed by an unknown state-dependent cell division process creates a spatially heterogeneous monolayer with intrinsic rate of electrical activity. A mathematical model of stochastic dispersion of autonomous cells was studied to investigate the effects of varying spatial patterns of cell and its influence on inter-sample rate. As a first step, cells are modeled with modified Fitzhugh-Nagumo models and deposited via a stochastic algorithm to form a completely full square monolayer. As expected, higher density of spontaneous cell increases the rate of activity of the monolayer and faster changes with increasing density are found for cases with increased growth of existing clusters. Interestingly, inter-sample variability is greater for distribution showing lower fractal dimension for identical density of spontaneous cell. These results confirm the need for a better understanding of cell characteristics and spatial patterning within the cell monolayer to optimize the biopacemaker function.

11h00
Automated initiation of multiple spiral waves in reaction-diffusion systems
Jacquemet, Vincent
Université de Montréal

Phase singularity analysis provides a quantitative description of spiral wave patterns observed in chemical or biological excitable media, for instance in a cardiac tissue during an arrhythmia. The configuration of phase singularities (locations and directions of rotation) is easily derived from phase maps in two-dimensional manifolds. The question arises whether one can construct a phase map with a given configuration of phase singularities. We will present a constructive mathematical approach to numerically solve this problem in geometries relevant to atrial anatomy. This tool can notably be used to create initial conditions with controllable spiral wave configuration for cardiac propagation models and thus help in the design of computer experiments in atrial electrophysiology.

11h30
Multistability and Automaticity during ischemia and adrenergic stimulation in ionic models of cardiac myocytes
Vinet, Alain, S. Bouchard and F. Grégoire-Lacoste
Université de Montréal

Cardiac arrhythmias are often triggered by the spontaneous firing of otherwise excitable cells. Such abnormal automaticity can occur in the ventricles during the early phase of ischemia or in the auricles following the activation of the cardiac sympathetic autonomic nervous system (CSAN). In ischemia, the concentration of potassium increases in the ischemic zone, resulting in a depolarization of the cells and the creation of an injury current flowing from the impaired to normal tissue. On the other hand, CSAN activation increases the conductance of K+ and Ca++ membrane currents. We show that these pairs of parameters can induce multi-stability and automaticity in an ionic model of the human ventricular myocyte and of canine atrial myocyte respectively, both describes by a high dimension differential algebraic system. The analysis of simplified models shows that, in both cases, the automatic regimes involve the interaction of a fast and a slow oscillator. The fast oscillator is driven by the K+ and Ca++ currents and their gate variables. The slow one implies either the spontaneous release of the Ca++ by the sarcoplasmic reticulum in the ventricular model, or the Na-Ca and Na-K exchanger in the atrial model. Each oscillator is associated to a pair of Hopf bifurcations whose relative locations vary as a function of the parameters, generating diverse complex scenarios of bifurcation. These can be understood by following the path follow by these bifurcations in the parameter plane.