A model for estimating the anisotropy of the conduction velocity in cardiac tissue based on the tissue morphology

Abstract

IntroductionIn the quest to better understand the mechanisms underlying cardiac arrhythmias, the use of computer simulations is a common tool. Computer simulations have the advantage of being able to resolve potentials and currents with a finer level of detail than is possible with measurements. However, in order to be useful, simulations need to reproduce observations, in this case from electrophysiology, and hence they must produce results against which to compare with experimental findings.The most common approach to the simulation of the propagation of depolarization fronts inside the myocardium is the use of the "bidomain model" [1]. The bidomain model is based on the notion that the myocardium can be separated into intracellular and extracellular spaces, which are joined to each other by a membrane that acts both as a current source and a pathway allowing current to flow between both spaces. More specifically, the bidomain model assumes that the intracellular space, the extracellular space and the membrane all coexist at each point in space. Hence at each location the myocardium can be characterized by a conductivity tensor for the intracellular space, a conductivity tensor for the extracellular space, and an ionic model that describes the current flowing through the membrane.The two conductivity tensors describing the electrical properties of the tissue represent the homogenized conductivity of either the intracellular or the extracellular space. The most common way of estimating these parameters is to choose them so that simulated cardiac propagation speeds fit experimentally observed values. Although this is a valid way of setting these parameters, it fails to relate them to the underlying tissue structure and composition.In this paper, we describe a model that simulates cardiac propagation based on a more detailed description of the microscopic tissue morphology than that provided by the standard bidomain. One of the reasons for including more details into the model is to be able to simulate pathologies like ischemia based on their physiological origins rather than by assigning parameters to fit experiments. For instance, during an episode of]ischemia the amount of extracellular space changes with time [2] altering the way a depolarization front propagates along the fiber [3].In order to create such a model, we built on the models of Spach et al. [4], who created a 2D model of cardiac tissue that consisted of several hundred realistically shaped myocytes that were coupled by gap junctions. Their model was limited by the fact that it did not include an extracellular space. Two-dimensional models are intrinsically limited because the intracellular and extracellular current paths cannot cross each other, thereby limiting the amount of available extracellular and intracellular pathways.To avoid these limitations of a 2D model, we created a full 3D model of cardiac tissue that included realistically shaped myocytes coupled by gap junctions and embedded in an extracellular matrix. We present here pr...

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Numerical analysis of finite element methods for the cardiac extracellular-membrane-intracellular model: Steklov–Poincaré operator and spatial error estimates

“…, N Γ , N Γ + 1, • • • , N 𝑒ℎ } be the usual basis of P 𝑑 hat functions associated with the vertices {𝑥 𝑙 , 1 𝑙 N 𝑖ℎ } and {𝑥 𝑘 , 1 𝑘 N 𝑒ℎ }, respectively. Take V Γ ℎ as in (44). Let Φ Γ 𝑙 = Φ 𝑖,𝑙|Γ , be the restriction of the basis function Φ 𝑖,𝑙 on Γ, for 1 𝑙 N Γ .…”

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