Cell-to-Cell Modeling of the Interface Between Atrial and Sinoatrial Anisotropic Heterogeneous Nets

Abstract

Abstract. The transition between Sinoatrial cells and Atrial cells in the Heart is not fully understood. Here we focus in cell-to-cell mathematical models involving typical Sinoatrial cells and Atrial cells connected with experimentally observed conductance values exclusively. We are interested mainly in the geometry of the microstructure of the conduction paths in the Sinoatrial Node. We show with some models that appropriate source-sink relationships between Atrial and Sinoatrial cells may occur according to certain geometric arrangements. Inter-phase between Atrial and Sinoatrial cellsIt has been observed in different species that going from the center of the sinoatrial node (SAN) in the heart toward the atrium, there is a transitional zone of cells having morphological and electrophysiological properties in-between to that of typical sinoatrial (SA) and atrial (A) cells [16]. The transitional cells have an aspect intermediate between that of typical nodal cells and that of the common atrial cells. Typical nodal cells have poor development of the contractile system and is assumed in general that they do not contract, moreover the existence of connexin43 is undetectable in the SA node center (see [3] and references therein), but posses automaticity in firing its action potential. On the contrary, atrial cells do contract themselves, but they require of an stimulus in order to contract, and they contain mainly connexin-43. This characteristics are included in the cells models used in this paper. A whole range of intermediate cells have been reported, but more important to the models in this paper, cells with one end connected to SA cells and the other end with A cells have been found [16]. The basic structure conforming the cytoarchitecture of this groups of cells consists of interdigitations of nodal and atrial bundles forming histological connections between nodal and atrial myocytes at regular distances [18].In [29] the authors, introduce a model of strands of atrial cells penetrating the SA node observed in the Pig. The model was constructed with 101 × 101 atrial and SA cells modeled with Oxsoft HEART V4.5. The lattice so constructed has a center of SA cells forming a circle of 30 cells of radius with twelve atrial interdigitations positioned at 30 deg intervals, where interdigitations are defined as sets of atrial cells at least ten cells distant from the node centre, and which are subtended by an angle of 15 deg. In that paper SA to SA conductance is g SA = 10 nS and the A to A conductance g A , varies from 10 nS to 250 nS.

Cell-to-cell Mathematical modeling of arrhythmia phenomena in excitable media

“…Although simplified models in one or two dimensions may reflect certain experimental data [12], it may be that such simplifications do not represent the actual behavior of big groups of cells, for instance, in the transmission of the action potential of the system formed by mixing pacemaker and atrial cells. For example, in [20] it is shown that for specific mathematical models of atrial and sinus cells, the activation of the complex depends on the number of cells involved, as well as on the geometric distribution of the cells in the network. Taking this into account, modeling diffusive media with cells including a big number of observables may not be a trivial task, since even the stability of the numerical methods involved may be in chalenge.…”

“…However, in the case of fibrillation, Wiener and Rosenblueth do not arrive at anything conclusive, since they affirm that solving the obtained equations "is a question for a more complex and difficult mathematical work" a work that, in the opinion of the author, it was never achieved. Nowadays there are articles that attempt to model more complex arrangements in cellular systems [20], but computational constraints strongly determine the possibility of more realistic models. This is one of the reasons that explain why cellular automaton are still in use.…”

Cell-to-cell Modeling of the interface between Atrial and Sinoatrial Anisotropic Heterogeneous Nets II: Mechanisms Underlying Nets Dynamics

“…The cell-to-cell approach requires: a) individual cell dynamics, modeled by Hodgking-Huxley type equations. For SA node cells, in [24] we used the model of Severi et al [36], described in that paper A model of idealized two-dimensional arrangements of cells using a similar structure can be found in [39]; b) In order to implement better models, an approximate number of cells in SA node is required. This number may be estimated to be in the order of million, and we give an estimation in the Conclusions section.…”

“…Considering the cytoarchitecture of SA node not only the number of cells to which each cell is connected is important (already cited), but also the geometric distribution of each connection. Here we should remark that due to the use of connections matrices for the models of [24], the inherent three dimensionality of the cytoarchitecture (see [7] where the necessity of the 3D approach in order to understand the human SA node structure is amply discussed) does not require a special treatment as is the case for PDE, in which a tensor is required to describe the complex geometrical distribution of cells in the heart [11], [12], [13], [33].…”

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In [24] we studied the transitional zone between Sinoatrial cells and Atrial cells in the Heart. The present paper study the mechanisms underlying the dynamics of the modeled nets in that paper.2000 Mathematics Subject Classification.

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