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

models?

The simplified model used in here is a three variable simple ionic model (3V-SIM) developed in [11] which can reproduce arbitrary monophasic APD and CV restitutions by varying some of its parameters. The total membrane current of the model is given by the sum of three phenomenological independent currents:=total I I_fi(U,v)+I_so(U)+I_si(U,w). Where U represents the membrane voltage and, v and w two gate variables.I_fi=-4*p*v*(U-.1)*(1-U) corresponds to a fast inward (Na) current , I_so=0.04*(U-0.03272/0.04)*exp(-2U) +0.03272, corresponds to a slow outward (K) current, and I_si=-w*(1+tanh(10*(U-0.859)))/44.15 a slow inward (Ca) current. The gates v and w govern the activation and inactivation of their corresponding currents and their kinetics are given by =∂v t (1-p)*(1-v)/((1-q)*18.5 +q*177)–p*v/3 and w t ∂=(1-p)*(1-w)/37-p*w/645.13.The membrane voltage follows the cable equation m total xj ij xi t C I U D U /)(−∂∂=∂. Where ij D is the diffusion tensor, m

C

=12/cm F µ, p and q are

Heaviside step functions defined by p=1 (p=0) for U ≥Uq (U<Uq) and q=1 (q=0) for U ≥ Up (U<Up),Uq=0.1, Up=0.033 and U needs to be scale as U*100-85when comparing with ionic models. The model as given above, reproduces the same APD and CV restitution as

the eight-variable Beeler-Reuter model[8] (see Fig. 1)

Fig. 1. APD restitution function indicates the duration of an AP as function of the time between activations,known as diastolic interval (DI) and CV restitution indicates the velocity of the wave front as function of the DI. In A) we show the APD restitution and in B) the CV restitution for the 3V-SIM and the BR-model. The curves were obtained by pacing at shorter and shorter DI’s as in [11]. Using sec /12cm D i =, dx=0.25 and dt= 0.13, 0.75 for the 3V-SIM and the BR-model respectively.

2.1. One-dimensional rings of cardiac tissue

In 1988 Frame and Simson [13] not only showed, as Mines [2] and Garrey [3], circulating impulses using

rings of canine heart muscle, but also oscillations on the impulse rotation cycle. These oscillations can be explained in terms of the APD and CV restitution [14].The oscillations are the result of a Hopf bifurcation present when the APD restitution has a slope >1. Figure 2 shows these oscillations using the BR-model and the 3V-model. We can see that the 3V-model reproduces in a good approximation the amplitude and modulations of the APD oscillations obtained with the BR-model. This oscillations increase as the ring diameter decreases, up to a size of 13.25 cm, where the ring is too small to support a propagating wave. It can be easily shown [5] that by decreasing the steepness of APD restitution, the oscillations can be suppressed. It is important to note that the time needed to simulate300 rotations using a 600MHz Alpha is about 6.5 min. for the 3V-model and 54min. for the BR-model (using ∞=m t m )(,which allows the use of a larger dt [11]), a ratio of about 8.3 times faster using the 3V-model. This ratio becomes very significant when attempting large-scale simulations in

three-dimensions.

Fig. 2. Oscillations of APD as function of cycle number (one beat to the next) for a simulated ring of cardiac tissue with L= 12.2 cm using the BR-model (A) and the 3V-model (B).

2.2. Two-dimensional sheets of cardiac tissue

Simulations of the BR-model in two-dimensional homogeneous sheets of tissue [15] have shown that spiral waves are unstable and readily breakup into multiples because of the AP oscillations produced by the steep APD restitution [5]. The 3V-model also produces unstable spiral waves that break much in the same way.Courtemanche and Winfree [15] showed that when the calcium current in the BR-model is speeded up by two (known as MBR-model), spiral waves become stable.Similarly with the 3V-model, one can vary some of its parameters [11-12] and closely reproduce the restitutions of the MBR-model. In this case the simplified model also leads to stable spiral waves with the same range of frequency rotation, shape and size as the MBR-model 11-12].

Since simulations using the 3V-model are much faster

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