Wave propagation in ventricular muscle is rendered highly anisotropic by the intramural rotation of the fiber This rotational anisotropy is especially important because it can produce a twist of electrical vortices, which measures the rate of rotation (in degree/mm) of activation wavefronts in successive planes perpendicular to a line of phase singularity, or filament This twist can then significantly alter the dynamics of the filament This paper explores this dynamics via numerical simulation After a review of the literature, we present modeling tools that include: (i) a simplified ionic model with three membrane currents that approximates well the restitution properties and spiral wave behavior of more complex ionic models of cardiac action potential (Beeler-Reuter and others), and (ii) a semi-implicit algorithm for the fast solution of monodomain cable equations with rotational anisotropy We then discuss selected results of a simulation study of vortex dynamics in a parallelepipedal slab of ventricular muscle of varying wall thickness (S) and fiber rotation rate (theta(z)) The main finding is that rotational anisotropy generates a sufficiently large twist to destabilize a single transmural filament and cause a transition to a wave turbulent state characterized by a high density of chaotically moving filaments This instability is manifested by the propagation of localized disturbances along the filament and has no previously known analog in isotropic excitable media These disturbances correspond to highly twisted and distorted regions of filament, or "twistons," that create vortex rings when colliding with the natural boundaries of the ventricle Moreover, when sufficiently twisted, these rings expand and create additional filaments by further colliding with boundaries This instability mechanism is distinct from the commonly invoked patchy failure or wave breakup that is not observed here during the initial instability For modified Beeler-Reuter-like kinetics with stable reentry in two dimensions, decay into turbulence occurs in the left ventricle in about one second above a critical wall thickness in the range of 4-6 mm that matches experiment However this decay is suppressed by uniformly decreasing excitability Specific experiments to test these results, and a method to characterize the filament density during fibrillation are discussed Results are contrasted with other mechanisms of fibrillation and future prospects are summarized (c)1998 American Institute of Physics

Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation.

We modified the FitzHugh-Nagumo model of an excitable medium so that it describes adequately the dymanics of pulse propagation in the canine myocardium. The modified model is simple enough to be used for intensive 3-dimensional (3D) computations of the whole heart. It simulates the pulse shape and the restitution property of the canine myocardium with good precision.

A simple two-variable model of cardiac excitation

This article focuses on the regional heterogeneity of the mammalian sinoatrial (SA) node in terms of cell morphology, pacemaker activity, action potential configuration and conduction, densities of ionic currents (i(Na), i(Ca,L), i(to), i(K,r), i(K,s) and i(f)), expression of gap junction proteins (Cx40, Cx43 and Cx45), autonomic regulation, and ageing. Experimental studies on the single SA node cell to the whole animal are reviewed. The heterogeneity is considered in terms of the gradient model of the SA node, in which there is gradual change in the intrinsic properties of SA node cells from periphery to centre, and the alternative mosaic model, in which there is a variable mix of atrial and SA node cells from periphery to centre. The heterogeneity is important for the dependable functioning of the SA node as the pacemaker for the heart, because (i) via multiple mechanisms, it allows the SA node to drive the surrounding atrial muscle without being suppressed electrotonically; (ii) via an action potential duration gradient and a conduction block zone, it promotes antegrade propagation of excitation from the SA node to the right atrium and prevents reentry of excitation; and (iii) via pacemaker shift, it allows pacemaking to continue under diverse pathophysiological circumstances.

The sinoatrial node, a heterogeneous pacemaker structure.

The reduction in switchable polarization of ferroelectric thin films due to electrical stress (polarization fatigue) is a major problem in ferroelectric nonvolatile memories. There is a large body of available experimental data and a number of existing models which address this issue, however the origin of this phenomena is still not properly understood. This work synthesizes the current experimental data, models, and approaches in order to draw conclusions on the relative importance of different macro- and microscopic scenarios of fatigue. Special attention is paid to the role of oxygen vacancy migration and electron injection into the film and it is concluded that the latter plays the predominant role. Experiments and problems for theoretical investigations, which can contribute to the further elucidation of polarization fatigue mechanisms in ferroelectric thin films, are suggested.

Polarization fatigue in ferroelectric films: Basic experimental findings, phenomenological scenarios, and microscopic features

Models of cardiac tissue electrophysiology are an important component of the Cardiac Physiome Project, which is an international effort to build biophysically based multi-scale mathematical models of the heart. Models of tissue electrophysiology can provide a bridge between electrophysiological cell models at smaller scales, and tissue mechanics, metabolism and blood flow at larger scales. This paper is a critical review of cardiac tissue electrophysiology models, focussing on the micro-structure of cardiac tissue, generic behaviours of action potential propagation, different models of cardiac tissue electrophysiology, the choice of parameter values and tissue geometry, emergent properties in tissue models, numerical techniques and computational issues. We propose a tentative list of information that could be included in published descriptions of tissue electrophysiology models, and used to support interpretation and evaluation of simulation results. We conclude with a discussion of challenges and open questions.

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Models of cardiac tissue electrophysiology: progress, challenges and open questions.

We developed a mathematical representation of ventricular geometry and muscle fiber organization using three-dimensional finite elements referred to a prolate spheroid coordinate system. Within elements, fields are approximated using basis functions with associated parameters defined at the element nodes. Four parameters per node are used to describe ventricular geometry. The radial coordinate is interpolated using cubic Hermite basis functions that preserve slope continuity, while the angular coordinates are interpolated linearly. Two further nodal parameters describe the orientation of myocardial fibers. The orientation of fibers within coordinate planes bounded by epicardial and endocardial surfaces is interpolated linearly, with transmural variation given by cubic Hermite basis functions. Left and right ventricular geometry and myocardial fiber orientations were characterized for a canine heart arrested in diastole and fixed at zero transmural pressure. The geometry was represented by a 24-element ensemble with 41 nodes. Nodal parameters fitted using least squares provided a realistic description of ventricular epicardial [root mean square (RMS) error less than 0.9 mm] and endocardial (RMS error less than 2.6 mm) surfaces. Measured fiber fields were also fitted (RMS error less than 17 degrees) with a 60-element, 99-node mesh obtained by subdividing the 24-element mesh. These methods provide a compact and accurate anatomic description of the ventricles suitable for use in finite element stress analysis, simulation of cardiac electrical activation, and other cardiac field modeling problems.

Mathematical model of geometry and fibrous structure of the heart.

Purpose To review the anatomical basis for anti-reflux operations and describe a project designed to develop a 3D micro-anatomical model of the gastro-oesophageal junction with integrated physiological events.



Methodology and Results  A realistic 3D anatomically based computer model of the gastro-oesophageal junction (GOJ) was constructed using cross-sectional digital images from the Visible Human Project. A finite element mesh was constructed and an iterative fitting process provided smooth surfaces. For accurate simulation of physiologic function it is necessary to augment this geometric model with detailed micro-structural information. A custom-built, semi-automated, extended-volume imaging system was used to obtain images of the GOJ. The device consists of a three-axis translation stage where a tissue sample embedded in wax can be sequentially imaged and milled. At each step the top surface was stained for fiber type and connective tissue and the image captured using a high resolution digital camera. Proof of principle was demonstrated with an en-bloc harvested sheep GOJ and then repeated with a fresh autopsy human GOJ. The 3D microstructural model was constructed using the images with a total volume of 35.1 × 106 voxels. Clinical manometric measurements were used to inform mathematical modeling to simulate physiological events, such as swallowing.



Conclusions  This model may provide opportunities to examine the relationship of anatomy and physiology in health and disease of the GOJ, enhance patient education, enable theoretical drug trials and determine the relative importance of anatomical elements of anti-reflux surgery.

I. J. Le Grice

Papers

Mathematical model of geometry and fibrous structure of the heart.

Hp12 the anatomical basis of the surgical treatment of gastro-oesophageal reflux: lessons from history and simulation