Numerical Simulations of Cardiac Dynamics. What Can We Learn from Simple and Complex Models

Modeling the electrical activity of the heart, and the complex signaling patterns that underlay dangerous arrhythmias such as tachycardia and fibrillation, requires a quantitative model of action potential propagation. At present, there exist detailed ioni

Numerical Simulations of Cardiac Dynamics. What Can We Learn from Simple

and Complex Models?

F H Fenton

Hofstra University, Hempstead NY, USA

Abstract

Modeling the electrical activity of the heart, and the complex signaling patterns that underlay dangerous arrhythmias such as tachycardia and fibrillation, requires a quantitative model of action potential propagation. At present, there exist detailed ionic models of the Hodgkin-Huxley form that accurately reproduce dynamical features of the action potential at a single cell level. However, such models are very time consuming in computer simulations. We show how simplified models can help on the study of cardiac arrhythmias. In particular, how breakup of spiral waves, believed as the dynamics underlying the transition from tachycardia to fibrillation, can occur as function of tissue size and shape. We also discuss some of the limitations in these models and some differences in dynamics between simple and complex models.

1.Introduction

The study of cardiac arrhythmias using computer models originated in the mid 1940’s when Wiener and Rosenblueth [1] used a very simple cellular automata model to describe the propagation of action potentials and investigate under which conditions flutter and fibrillation could arise. Eventhough their model considered homogenous cells with only three dynamical states (excited, refractory and unexcited) and produced a constant conduction velocity as well as a constant duration of action potential. It was enough to explain how electrical waves could re-circulate around obstacles and give a theoretical frame work for the circulating impulses in cardiac tissue observed experimentally by Mines [2] and Garrey [3] in 1914.

Since then, many numerical simulations of cardiac dynamics have helped in many ways to our understanding of arrhythmias. For example in 1964 Moe et al. [4] used a cellular automata similar to [1] but added two more rules, (i) the absolutely-refractory state lasted certain amount of time R which varied randomly from cell to cell, and (ii) four extra intermediate “relatively-refractory” states where considered. This produced a spatially inhomogeneous tissue with dispersion of recovery and variable conduction speed (CV). As the dispersion in values of the refractory times R increases, Moe’s model can produce circulating waves with out the need of any anatomical obstacle (spiral waves), further more at a higher dispersion, breakup of spiral waves occurs leading to a state of multiple wavelets. Moe’s model shows how tissue irregularities and variability in CV facilitates the induction of fibrillation. In the late 1960’s and 1970’s many numerical simulations of continuos generic excitable models showed the existence of spiral waves in isotropic tissue. And in the early 1990’s simulations with ionic models showed that steep restitution of action potential duration (APD) (slope > 1) can produce large wave oscillations which can then lead to conduction blocks and spiral wave breakup [5-6] (see Fig.1).

1.1.Ionic Models.

In 1952 Hodking and Huxley [7] introduced the first continuos mathematical model designed to reproduce cell membrane action potentials. Since then, there has been many complex models developed for cardiac cells following their approach. Most of these models can be separated into three classes. 1) “First generation” of ionic models, which reproduce basic ionic currents and concentrations like the Beeler-Reuter (BR) [8] and the Luo-Rudy-I (LR-I) [9] models. 2) ”Second generation”models, which are more robust since they not only include more currents but also pumps and exchangers for dynamic ionic concentrations like the DiFrancesco-Noble [10]. And 3) simplified models that only use the minimum set of phenomenological currents necessary to reproduce mesoescopic characteristics of cell dynamics such as AP and CV restitutions [11-12].

Simulations using first and second generation models are in general very computationally intensive, specially in 2 and 3 dimensions, therefore it is always desirable to isolate the minimum key features necessary to characterize specific phenomena by using simplified models . Nevertheless it is important to keep in mind that when reducing a complex model some intrinsic behavioral dynamics may be lost. The porpoise of this latter is to show some of the advantages and limitations when using simplified models of cardiac action potentials.

2. What can we learn from simplified

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