Models of the Heart in Vulnerability and Defibrillation Studies

This section provides an overview of the remainder of the modeling tools used in bidomain studies of defibrillation.

Description of Myocardial Geometry and Fiber Architecture

The 3D organ-level simulations of defibrillation have used Vetter and McCulloch's37 3D model of anatomically based rabbit ventricular geometry and fiber orientation. The original geometry was defined in prolate spheroidal coordinates, which were then translated into Cartesian. MSC Patran (MSC-Software Corporation) was used to generate the computational grids. The surfaces and the "solids" of the mesh were created with Patran functions. Additional "solids" were generated to represent the ventricular cavities and the perfusing bath; the conductivity of these regions was assigned the value of blood. Using Patran, surfaces and solids were meshed to create unstructured triangular and tetrahedral meshes, respectively, with an average element edge length of 300-500 |im in the tissue and 1 mm in cavities and perfusing bath. Once the entire mesh was complete, files containing node coordinates, elemental connectivity, and the original finite element to which each tetrahedral element in the new mesh belongs, were generated. This information was then used to determine, with the combination of two multidimensional root-finding algorithms, fiber orientation at the centroid of each tetrahedron. Local material properties were assigned to each element using the fiber orientations. Fig. 2 illustrates the 3D geometry and fiber orientation in the rabbit ventricles as determined by this algorithm.

Figure 2: (a) Geometry (semitransparent rendering) and posterior and anterior views of epicardial fiber orientation (short white lines) in the rabbit ventricular model. (b) Configuration of the perfusing chamber (3.92 cm wide) and shock electrodes. The ventricles are paced at the apex, and the colors represent the distribution of transmembrane voltage during a paced beat

Figure 2: (a) Geometry (semitransparent rendering) and posterior and anterior views of epicardial fiber orientation (short white lines) in the rabbit ventricular model. (b) Configuration of the perfusing chamber (3.92 cm wide) and shock electrodes. The ventricles are paced at the apex, and the colors represent the distribution of transmembrane voltage during a paced beat

Representation of Ionic Currents and Membrane Electroporation

As already discussed above, for defibrillation studies, all membrane models need to be modified to ensure stability during the shock, when a dramatic change in transmembrane potential takes place.24 It is important to understand, however, that defibrillation shocks induce complex changes in transmembrane potential, some of which have been consistently observed in experiments, but never reproduced by membrane models. Some of these observed phenomena are listed below. First, strong shocks applied during action potential plateau in isolated guinea pig papillary muscle,38 cultured neonatal rat myocyte strands,39 and isolated guinea pig myocytes23 induce asymmetrical changes in transmembrane potential, Vm, with the negative transmembrane potential change, AVm, being larger than the positive (i.e., AVm > AVj+). Second, with increase in shock strength, AVm magnitude does not increase proportionally but instead saturates.38'39 Third, for large shock strengths AVm exhibits non-monotonic behavior with initial rapid increase and then a decrease.23'39 None of the available membrane models reproduce these responses to strong shocks, necessitating modifications in the available membrane models.

Using a recent version of the Luo-Rudy dynamic (LRd) model, Faber and Rudy40 found that the negative bias in AVm asymmetry could not be reproduced by the natural addition of electroporation (model LRd + EP), the latter documented to always take place following strong shocks.41 Only when the outward current activated upon strong shock-induced depolarization was incorporated, Ia, first suggested by Cheng et al.,23 that a match between simulation and experiment was achieved (Fig. 3), provided it was assumed that Ia was part of the K+ flow through the L-type Ca+-channel augmented LRd, or (aLRd model). With the use of the new aLRd model it was possible to reproduce the experimentally observed rectangularly-shaped positive AVm transient, negative-to-positive AVm ratio near 2,39,42 stronger electroporation at the anode,43 and dependence of the AVm magnitude on field strength (compare Fig. 3 to Fast et al.39 and Cheek et al.)42 To conduct simulations with the rabbit ventricular model, similar changes were incorporated in the Puglisi-Bers ventricular myocyte ionic membrane model.44 Being equipped with a membrane model that can accurately reproduce the membrane responses to shocks is essential to simulate tissue and organ behavior observed experimentally, and thus, to provide insight about behavior in the depth of the tissue that cannot be assessed by the current experimental techniques.

Shock Electrodes and Waveforms

In the rabbit model, shock electrodes are represented as 3D iso-current density or iso-voltage surfaces within the 3D computational grid (ventricles plus perfusing bath and blood in cavities); these are chosen to mimic geometry and location of electrodes in optical mapping experiments (far-field). The examples included in this chapter use external plate (at the boundaries of the perfusing chamber) electrodes; a right ventricle (RV) catheter and a return electrode in the posterior bath as well as cuff electrodes have also been implemented.45'46 The shock waveforms include square waves as well as truncated-exponential (62% tilt) mono- and biphasic shocks of different polarities and durations.47-52 All shocks were given

Figure 3: Responses to shocks of various strengths E delivered during the action potential plateau (coupling interval of 10 ms) in an 800 ^m-long fiber. (a) Superposition of shock-induced virtual electrode polarization (VEP) at the fiber ends. Membrane kinetics is represented by the LRd, LRd with electroporation (LRd + EP), and augmented LRd (aLRd) models. Shock duration is 10 ms and strengths are 8, 12, and 16V/cm-1 (th in, thicker, and thickest solid lines, respectively). APA, action potential amplitude. Vertical dotted line indicates time of AVm (change in transmembrane potential Vm) measurement, 3 ms after shock onset. (b-d) Shock-induced positive and negative AVm and AVm as a function of shock strength for the three models. Characters II and III denote types of nonlinear responses, as per Fast et al.39 (e) AV—/AV+ ratio as function of shock strength in the three models. (Figure modified from Ashihara and Trayanova)25

Figure 3: Responses to shocks of various strengths E delivered during the action potential plateau (coupling interval of 10 ms) in an 800 ^m-long fiber. (a) Superposition of shock-induced virtual electrode polarization (VEP) at the fiber ends. Membrane kinetics is represented by the LRd, LRd with electroporation (LRd + EP), and augmented LRd (aLRd) models. Shock duration is 10 ms and strengths are 8, 12, and 16V/cm-1 (th in, thicker, and thickest solid lines, respectively). APA, action potential amplitude. Vertical dotted line indicates time of AVm (change in transmembrane potential Vm) measurement, 3 ms after shock onset. (b-d) Shock-induced positive and negative AVm and AVm as a function of shock strength for the three models. Characters II and III denote types of nonlinear responses, as per Fast et al.39 (e) AV—/AV+ ratio as function of shock strength in the three models. (Figure modified from Ashihara and Trayanova)25

to ventricles paced eight to ten times at the apex of a basic cycle length of 300 ms. The shocks were administered at various coupling intervals with respect to the last paced beat.

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