A Model of 3D Propagation in Discrete Cardiac Tissue JG Stinstra1, SF Roberts2, JB Pormann2, RS MacLeod1, CS Henriquez2 1University of Utah, UT, USA 2Duke University, NC, USAAbstract A model was developed of a bundle of cardiac fibers embedded in an extracellular space. In contrast to the classical bidomain approach, the model is constructed such that the intracellular and extracellular spaces are spatially distinct. The model was used to test the hypothesis that the distribution of the extracellular fluid in the tissue can affect the conduction velocity. The preliminary results suggest that under nominally normal conditions, the propagation speed depends on the fraction of extracellular space and less on the actual distribution of extracellular space within the bundle. 1. Introduction Computer models of the heart have become useful tools in analyzing mechanisms of arrhythmias and anti-arrhythmic therapies. One of the most commonly used model to describe the electrophysiology of cardiac tissue is the so-called bidomain model [1]. This model describes cardiac tissue as a set of two homogeneous isotropic or anisotropic spaces, that span the entire simulated tissue volume. Both spaces are connected at each point in space by a membrane model describing the electrophysiology of the ion channels in the membrane and using this approach one can simulate the propagation of depolarization waves in the myocardium. One of the limitations of the bidomain model is that the discrete cellular structure is not explicitly taken into account. Because of this limitation, the bidomain cannot account for variations in the distribution of gap junctions, cell sizes extracellular space around cells that can arise under certain arrhythmogenic conditions, such as ischemia [2]. In order to further investigate how the action potential propagates in cardiac tissue we have developed a new model in which the intracellular and extracellular spaces are spatially distinct and separated by a membrane [2,3]. In contrast to classical models, the model allows for realistically shaped myocytes in three dimensions surrounded by non-uniform extracellular space, more closely mimicking the actual tissue architecture (Figure 1). For example, the extracellular space varies in thickness around a myocyte, with a larger fraction of space surrounding structures like capillaries. This asymmetric distribution suggests that conduction may be preferentially faster along the more open spaces, as the resistance is effectively lower. In this paper we present a short overview of the methods used to create models at a cellular scale in three dimensions. The model is used to test the hypothesis that the distribution of extracellular space affects the speed of conduction of action potentials in tissue. 2. Methods To simulate the changes in the propagation of action potentials for different distributions of the interstitial space, we created a 3D model of a strand of myocytes. The model consisted of two discrete volumes: the Figure 1 Image showing the difference between the continuous bidomain approach and the discrete bidomain approach. ISSN 0276−6547 41 Computers in Cardiology 2006;33:41−44.cytoplasm of the myocytes and the interstitial space surrounding the myocytes separated by the cell membrane. The model was setup in such a way that some myocytes touch neighboring myocytes (see Figure 2), whereas other parts are surrounded by the extracellular space. In cardiac tissue the part of the cell membrane, which connects two adjoining myocytes, is infused with gap junctions that form a low impedance connection between cells. In the model we therefore subdivided the membrane into a part that connects two myocytes and into a part that interfaces to the extracellular space (Figure 2). The portion of the membrane that connects two myocytes is modeled by a surface resistor and capacitor, which represent the low impedance pathway through the gap junctions averaged out over a piece of membrane. The surface between the interstitial space and the cytoplasm contains the ionic channels, and hence is modeled using an ion channel model (current sources). Both surfaces were assumed to have the same amount of membrane capacitance. In this implementation, the volumes enclosed by these surfaces were assumed to be isotropic homogeneous ohmic volume conductors, characterized by a single conductivity value. As the intracellular space does not only contain cytoplasmic fluid, but also myofibrils and mitochondria, the conductive properties of the cytoplasmic fluid is averaged out over the total volume of the cell. In a similar way the extracellular space is homogenized as well. An overview of the equations used to describe the electrical potential in each compartment is given in Figure 2. In the model we assume the cell membranes behave according to the model specified by Luo and Rudy [4] for cardiac myocytes in Guinea Pigs. For the model presented in this paper we assumed a simple geometry in which the myocytes have a brick like shape and line up to form a strand of myocytes, see Figure 3. Every myocyte in the model had the same length of 100 µm and a cross section of 300 µm2, values which were derived from histology literature on cardiac tissue [3]. The model encompassed a strand of 3 by 3 by 25 myocytes with the myocytes connected at the ends and lateral adjoining boundaries by gap junctions (see Figure 3). We chose an average surface connectivity through the gap junction infused surfaces of 0.0015 kΩcm2, a value we used in an earlier modeling study to predict the impedance of cardiac tissue in a range that was consistent with literature values [3]. The conductivity of the extracellular space was assumed to be 2.0 S/m. This value was chosen assuming that the sodium and chloride ions are mainly responsible for the conductive properties of the extracellular fluid. Similarly a value of 0.3 S/m was chosen for the intracellular space based on the potassium