A Discrete Model of the Dynamic Behavior of the Cardiac Muscle Emilia Entcheva and Frank J. Claydon Dept. of Biomedical Engineering, The University of Memphis, Memphis, TN, USA Abstract The objective of this study is to construct a discrete finite state model of the electrical activity of the heart, which is directly related to the FitzHugh-Nagumo equations, but offers both a faster and more stable solution. We set up a 2D two variable model with non-nearest neighbor interactions to include curvature dependence of the wave front. For better approximation of the electrotonic propagation, a hexagonal grid and distancedependent weighting coefficients are used. Despite its simplicity the model follows the local dynamics prescribed by the modified FitzHugh-Nagumo equations and its parameters are directly derived from the original ones. The model was validated by comparison to existing experimental data for the propagation velocity, curvaturevelocity relationship, rotation speed and core size. It was successfully used to study spiral wave interactions with nonexcitable obstacles of different shapes.
1. Introduction It has been shown recently that irregular wave propagation and the discreteness of cardiac tissue might be pertinent to the mechanism of cardiac arrhythmias [2, 6]. The onset of spiral waves in the heart and their breakdown have been demonstrated both experimentally [2, 3] and numerically [1, 6]. It is widely accepted that cardiac tissue can be perceived as generic excitable medium and cellular automata (CA) methods can be applied for modeling purposes. Despite wide acceptance, objections have been raised due to the fact that ad hoc rules are usually used and some important features, such as curvature dependence and dispersion are missing in the above caricature models. Several attempts have been made to better relate the CA models to the physiologically realistic membrane dynamics since Gerhardt’s paper [6], most of them related to the simple 2-variable FitzHugh representation. The goal of this study is to preserve the advantages of the CA method, while at the same time improves the representation of the local dynamics and the electrotonic propaga-
tion. The result yielding a relatively simple and fast tool for simulation of cardiac arrhythmia related phenomena.
2.
The Model
2.1.
Local Dynamics
. In order to approximate the local dynamics we use a simplified version of the FitzHugh equations [1]: ut = Lu - u(u-1)(u-Uth)/ ε vt = u - v
(1) (2)
Equations (1) and (2) represent the dynamics of the transmembrane potential (u) and recovery variable (v) [5]. Lu is the diffusion term and its approximation is discussed in section 2.2. As suggested by FitzHugh [5] the threshold phenomenon of the 2-parameter system is of the “quasi” type and intermediate values for the transmembrane potential can be observed only for very fine settings of the stimuli. This justifies the discontinuity in time of the u-variable between the two stable branches and the linear approximation for the threshold line: Uth = m + nv (the middle unstable branch of the u-nullcline, Figure 1). Geometrically we obtain the coefficients m = ∆ (the difference between the first two resting states for the transmembrane potential, Figure 1); n = (u1-∆)/Vmax , where u1 is the maximum value of the u-variable, Vmax is the maximum value of the recovery variable. Parameter ε represents the ratio of the time constants for the fast (u) and slow (v) variable. The CA rules are based on the estimated value of the u variable compared to the threshold Uth , depending on the stage of recovery. The excitation and recovery thresholds for the v-variable are also expressed in terms of the above parameters: Vexc = ∆, Vrec = Vmax - Vexc.
rel. refractory v
absolute refractory
active
v
# 1 d r w 43
u
Uth ∆ u regenerative period Figure 1. Local dynamics represented by the nullclines in the phase space for the transmembrane potential and the recovery variable according to the FitzHugh model [5].
2.2.
Passive Propagation
In order to avoid the artificial anisotropy characteristic for rectangular grids we use a hexagonal grid. In order to include curvature and dispersion (speed dependence on the extent of recovery) in the model we consider nonnearest interactions in a “vicinity” radius from 1 to 4 (Figure 2).The diffusion and electrotonic propagation in the cardiac muscle are modeled with an exponential neighborhood function. We assume circular diffusion and quasistatic approximation for the diffusion coefficients [9]. from ut = Lu: u = (1 / 2 πdt ) . exp(-r2/4dt) = c. exp(- r2/4dt) (3) 2
For t=1, we obtain diffusion coefficient D=c.exp(- r /4d). If the coefficient c is big enough (assume c=100), the diffusion coefficient D approaches d. In Table 1 are shown the normalized calculated weights for an isotropic medium. To represent the flux only, we take the difference between the weighted potential of the neighborhood and the potential of the center and use that value in the CA rules of excitation. In Figure 2 a piece of the hexagonal grid and the four different neighborhood radii used in this study are shown.
r1
r2 1
3 2
r4
r3
8
5 4
7
6 Figure 2. Hexagonal grid and neighborhood radii. Table 1. Neighborhood points location (distance d from the center) and normalized weights (w) for exponential weighting.
2 r√3 26
3 2r 20
4 r√7 10
5 3r 6
6 r2√3 3
7 r√13 2
8 4r 1
The CA model was constructed on hexagonal lattice with variable dimensions (up to 1000 by 1000) to test the sensitivity to boundary influences. All the simulations reported below were done on a lattice 200 by 100. Three types of boundary conditions were implemented: Dirichlet, Neumann (mirror) and periodic (torus).
2.3.
Model parameters
The parameter u was normalized, assuming values of 0 and 1 [1]; Vmax=0.3 was chosen as a typical value in [5], ∆ was assigned values in the range 0.01 to 0.1. All of the above were multiplied by a proportionality coefficient 200 in order to work with integer computations. The parameter ε was assigned values between 100-200 (200 was used in [1]). Exponential decay was modeled with a space constant of the order of the tissue space constant (1mm), with resulting diffusion coefficient D=1su2/1tu (see section 4 for the real value), su=space unit, tu=time unit.
3.
Simulations
The above described model was applied to simulate the onset and development of spiral waves and in particular to study their interaction with inexcitable obstacles of different shapes. The spiral waves were induced by standard cross-field stimulation.
3.1.
Interactions with obstacles
As suggested by Wiener [9], the shape of the obstacles is expected to influence wave propagation. We studied the interaction of an already existing rotating wave with three types of nonexcitable obstacles: convex, concave and with sharp edges. Case 1: Convex obstacles A convex obstacle has to have a big enough diameter in order to cause a stable rotating wave to change its behavior. Small convex obstacles (the size of the core) have little effect on the rotors, besides some curling on their back (with respect to the rotor) side. Obstacles of bigger circumference cause the existing spiral wave to drift along their border, being rigidly attached to it. Figure 3(a) shows the interaction of the rotor with convex obstacle above the critical size.
Case 2: Concave obstacles Small concave obstacles (of the magnitude of the core size) behave in a very similar way as the convex ones. Concave obstacles of bigger size, depending on their curvature and position with respect to the rotor can annihilate or anchor it (Figure 4). Case 3: Sharp edged obstacles Thin, sharp edged obstacles (they can represent sclerotic patches or fibrotic tissue) were found to induce vortex shedding after considerable sodium channel blockage in cardiac tissue [2]. We tested the effect of the sharp edged obstacles on an already established rotor. They have to be positioned at least partially laterally to the existing rotor in order to affect its stability. Diffraction effects are visible on Figure 5.
(a)
(b)
(a)
Figure 4. Interaction with concave obstacles: (a) Small concave obstacles have similar effect to the convex ones; (b) bigger concave obstacles can annihilate the rotor (periodic boundary conditions used above) or anchor it.
(a) (b) Figure 3. Interaction with convex obstacles: (a) Obstacles with big circumference anchor the rotor; (b) small obstacles have little effect on it.
(b) Figure 5. Interaction with thin sharp obstacles: (a) diffraction effects; (b) different phase on both sides of the obstacle.
4. Results and Discussion
related to the modified FitzHugh model; and 3). minimum possible number of parameters. Thus we believe our model includes all the important features necessary to effectively simulate the activity in a block of cardiac tissue.
We discuss a 2D cellular automaton type of model, that is directly related to the modified FitzHugh dynamics equations and includes curvature and dispersion effects through non-nearest neighbor interactions. Electrotonic propagation is modeled by exponential weighting for the cells in the neighborhood. The model was validated by comparing its parameters to experimentally observed ones. As reference values to parametrize our model we use: 1) action potential duration of 180ms (refractory period 130ms) as reported in [3] and 2) propagation velocity 0.34mm/ms [4] or 0.5mm/ms [10]. Under these conditions one time unit in our model corresponds to 3ms, and one space unit to 0.5 or 0.9mm (for propagation velocity 0.34mm/ms and 0.5mm/ms resp.). Quite different values for the diffusion coefficient of the fast u-variable have been reported: 0.06mm2/ms [7], 0.25mm2/ms [8]. For a space unit equal to 0.9mm, we simulate a diffusion coefficient of 0.27mm2/ms. The curvature dependence in the model is approximately 1:2.2 (convex-to-concave waves), in the range of the values shown in [6]. The rotation period for a stabilized spiral wave is appr. 200ms for a typical setting of parameters (ε = 100, ∆ = 0.1), slightly above the reported in [2] and [3]. The core size is about 10-15mm. In absence of nonhomogeneities (variance of the refractory period of the cells or presence of inexcitable obstacles) stable spiral waves were observed in all cases. Davidenko et al. [3] reported predominantly rigid rotation in normal sheep myocardium. The model was applied to study the interaction of a rotor with convex, concave and sharp edged obstacles. Various big blood vessels can be represented by the above obstacles. We found that convex and concave obstacles above some critical size (approximately 3 times the core size) are likely to serve as anchors for existing rotors. In most cases the rotor was anchored to the obstacle, rotating rigidly for a long time, as shown experimentally [3]. For smaller sizes propagation was only slightly disturbed in the region behind the obstacle with respect to the spiral. Concave obstacles can annihilate rotors. Thin sharp obstacles exhibit diffraction effects.
[1] Barkley D. A model for fast computer simulation of waves in excitable media, Physica D, 1991, 49:61-70. [2] Cabo C., Pertsov A. M., Davidenko J. M., Baxter W., Gray R. and Jalife J. Vortex Shedding as a Precursor of Turbulent Electrical Activity in Cardiac Muscle. Biophysical J., 1996, 70:1105-1111. [3] Davidenko J. M., Pertsov A. V., Salomonsz R., Baxter W., and Jalife J. Stationary and drifting spiral waves of excitation in isolated cardiac muscle. Nature, 1992, 355:349-351. [4] Delgrado C, Steinhaus B, Delmar M., Chivalo D.R., and Jalife J., Directional differences in excitability and margin of safety for propagation in sheep ventricular epicardial muscle. Circ. Research, 1990, 67:97-110. [5] FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane. Biophysical J., 1961, 1:445466. [6] Gerhardt M., Schuster H., and Tyson J. J. A cellular automaton model of excitable media including curvature and dispersion. Science, 1990, 247:1563-1566. [7] Kawato M., Yamanaka A., Urishibara S., Nagata O., Irisawa H., and Suzuki R., Simulation analysis of excitation conduction in the heart: propagation of excitation in different tissues. J. Theor. Biol. 1986, 120:389-409. [8] Tyson J.J. and Keener J.P., Singular perturbation theory of traveling waves in excitable media, Physica D, 1988, 32:327-361. [9] Weimar J. R., Tyson J. J., and Watson L. T. Diffusion and wave propagation in cellular automaton models of excitable media, Physica D, 1992, 55:309-327. [10] Wiener N. and Rosenblueth A., The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle, Arch. Inst. Cardiol. Mexico, 1946, 14, no.3-4, 205265.
5. Conclusions
Address for correspondence:
When compared to the existing cellular automaton type models of the cardiac dynamics we claim that our model offers: 1) a better approximation of the spatial component of the excitation variable by means of the hexagonal grid, wider neighborhood function and with exponentially declining weights implemented; 2). a physiologically realistic local dynamics which is directly
Emilia Entcheva Dept. of Biomedical Engineering The University of Memphis Memphis, TN 38152, USA
[email protected]
Acknowledgments Thanks to Dr. R. Malkin for the valuable discussions and to Qiuying Huang for the help with the XWindows visualization.
References
