Cardiac Dynamics with Computational Models

he aim of this thesis is to analyze the effects of time variations of the gap junctions (GJ) conductivity. In particular, we explore the dynamics situations that exist in unhealthy tissue. In Chapter 1, we present a brief introduction of the heart organ and its importance, from a biological point of view. We begin the study with a simple explanation on how the heart pumps the blood, due to a synchronized contraction triggered by a regular electrical propagation through the cardiac tissue. We also described the several connexin types found in the GJ, and other factors that alter the GJ channels, and finally modify the electric propagation in the cardiac tissue. The connexin type in the GJ, and other factors that alter the GJ channels, influence the effectiveness of electric propagation in the heart. We review Peñaranda et al. membrane model in Chapter 2. It is simulated in a one dimensional strand under healthy conditions to help to familiarize with the normal behavior of the cardiac electrical activity under healthy conditions. In Chapter 3, we introduce the GJ dynamics. Then, we couple the membrane model with the GJ dynamics. This allows the conductivity values to vary in time during the simulations. We present the results of the test for the system of membrane model + GJ dynamics for two formulations, the monodomain and bidomain formulations. We explain the main differences between the two formulations. We conclude that the monodomain formulation is adequate for the purpose of our study, while being less computationally demanding as the bidomain formulation. We simulate the membrane model + GJ dynamics under unhealthy conditions. These situations are known to favor the creation of arrhythmia. To do this, we modified three parameters to mimic unhealthy cardiac tissue. The first parameter is the one associated with an overall reduction of the conductance (like it appears in an ischemic tissue). The second modified parameter changes the time scale associated with the GJ dynamics. The third modified parameter produces a reduction of the upper plateau of the connexin characteristics (quantified here by the shrinking factor F S). These modifications are somewhat physiologically justified but are rarely present all at once in an experiment with animal or human tissue. A simulated strand of ventricular tissue is initially set with uniform values of conductances for all cells. In the diseased situation that we have explored, these values significantly vary after certain time of simulation and will eventually reach one or two steady limit cycle. In general, we observe two different values to which the conductances converge, i.e. 0.1 and 0.4. The convergence of the conductance in a cell to a certain value is influenced by their neighboring cells. Thus, we have put in evidence a mechanism for the dynamical dispersion of the conductance among neighboring cells in the cardiac tissue. This effect is characterized in both the asymmetrical Cx43 − 45 as well as in the symmetrical Cx43 − 43 GJ type of connexins. In addition, we present the results of a statistical study of the conductance values distributions at the end of very long simulations. This is performed in order to quantify the phenomenon and determine the range of FS at which we can observe stability to one value or the other, or bistability. In addition, we explore the influence of the initial noise added to the conductance values. Finally, in Chapter 4, we present a new simple model that we have developed in order to perform longer simulations at a lower computational cost. Although the simple model is not in perfect agreement with the full model, it is a good approximation that matches quantitatively in the small noise cases