Models of Local Cell Dynamics

Equations (2.34) - (2.36) and (2.45) - (2.46) are denoted as “reaction-diffusion equations”, since they contain both, a diffusive part, and a local reaction part. In order to solve Eq. (2.45) or Eq. (2.35), the ion channel dynamics is still needed (I_Ion), which results in additional differential equations. Here we discuss models which describe the local ion chan-nel dynamics on a cellular level. As described in section 2.1.1 on page 11 the membrane potential V_m of a cell is composed of ion concentrations of different types (e.g. sodium, calcium or potassium). These concentrations change due to diverse mechanisms, like leak channels, the Na⁺/K⁺-ATPase or voltage gated ion channels.

17The conductivities are also spatially dependent, thus they are actually local variables (σ=σ(r)).

2.4. Numerical Simulations The models discussed here aim to describe these underlying mechanisms on a more or less detailed and realistic level (depending on the actual model), in order to reproduce the characteristic behavior of a single cell. Various features of the cell dynamics that shall be reproduced by the models are the upstroke, duration, and refractory period of an action potential, for example.

Sets of coupled differential equations are used for this purpose. From a mathematical point of view it turns out, that for modeling basic features like the refractory behavior more, at least two dynamical variables and two equations are necessary. That means, additionally to the diffusive Eqs. (2.35) or Eq. (2.45), respectively, which describe the local dynamics and the diffusive behavior of the membrane potentialV_m, at least one additional equation is needed, describing the evolution of at least one additional variable. The need for additional variables can be explained by the refractive property of the tissue which demands some kind of memory of the past. Accordingly, the simplest models which describe excitable media are two-dimensional (the membrane potentialV_m and one additional variable).

Simple models which are commonly used to describe excitable media (also apart from cardiac tissue) are for example the Barkley model [71] (three parameters) or the Aliev-Panfilov model [72] (five parameters), which provide only basic features of excitable media, but are extremely fast to compute. However, more underlying equations and variables can be included into the models, in order to be more realistic and to reproduce more details of the dynamics of actual cardiac tissue. With this gain of realism and level of detail, also the computational cost increases. The class of “ionic models” comprises individual equations and variables for single ion channels and specific ion concentrations. The Luo-Rudy II model [73] (fifteen dynamical variables) is an example for such a rather detailed model. The choice of the proper cardiac cell model used for a numerical study or, more general, for the respective scientific objective is discussed in section 2.3 on page 40. In the following, the Aliev-Panfilov model and the Fenton-Karma model [26] are presented as examples for a rather simple model and a model with moderate level of complexity, respectively, in order to elucidate the general form of the underlying equations.

Aliev-Panfilov Model

Equations (2.47)-(2.49) depict the Aliev-Panfilov model, here used in the context of the monodomain approach:

∂V_m

∂t =∇ ·D∇V_m−kV_m(V_m−1)(V_m−a)−V_mv , (2.47)

∂v

∂t =(V_m, v)(−v−kV_m(V_m−a−1)). (2.48) Thus, Eq. (2.47) corresponds to Eq. (2.45), where the first term still describes the diffusion of the first membrane potential, but the local reaction part (second and third term) is now specifically given by the Aliev-Panfilov model. The evolution of the second variable v is furthermore described in Eq. (2.48), and the expression for (V_m, v) reads

0 30 60 90 120 t [a.u.]

0.0 0.2 0.4 0.6 0.8 1.0

Vm[a.u.]

Threshold

0.0 0.5 1.0 1.5 2.0

v[a.u.]

Figure 2.20: A representative action potential using the Aliev-Panfilov model, with the parameters a= 0.05, ₀ = 0.002, µ₁ = 0.2, µ₂ = 0.3 and k= 8. The membrane potential V_m (black) and the second dynamical variable v (green) are shown. The action potential was triggered by an external increase of the membrane potential from zero toV_m= 0.1 a.u. att= 20 a.u. exceeding the excitation threshold (a= 0.05).

(V_m, v) =₀+ µ₁v

V_m+µ₂. (2.49)

These simple models have also the advantage (beyond their fast computation time), that some of the parameters have a direct physical meaning. In this case, parameter a is the excitation threshold and parameter k determines the excitability (excitation threshold) of the cell (see section 2.1.1 on page 11) . However, the other parameters µ₁, µ₂ and ₀ are used to fit the shape of the cardiac action potential. In Fig. 2.20 a representative action potential is shown, with both dynamical variablesV_m and v.¹⁸

Fenton-Karma Model

The Fenton-Karma model (or 3V-model) [26] is a cardiac cell model with a moderate level of complexity (three dynamical variables and thirteen parameters) but still offers a reasonable computational demand. It describes the transmembrane currentI_Ion in Eq. (2.45) by three distinct current densities: the fast inward currentI_fi, the slow outward currentI_so and the slow inward currentI_si in Eq. (2.53):

18This action potential is already computed numerically. Details about the numerical algorithms used to solve the underlying differential equations will be given later.

2.4. Numerical Simulations

Figure 2.21: A representative action potential using the Fenton-Karma model, with the the parameter setFK1(see A.2 in the Appendix A on page 131). The membrane potential V_m(black) and the two other dynamical variablesv (green) andw(orange) are shown. The action potential was triggered by an external increase of the membrane potential from zero toV_m= 0.1 a.u.att= 100 a.u..

These currents aim to model the most prevalent dynamics of sodium, potassium and calcium (discussed in section 2.1.1 on page 11) and are explicitly given in Eqs. (2.50)-(2.52). The evolution of two additional dynamical variables (v, w) is shown in Eqs. (2.54)-(2.55):

∂V_m

In Fig. 2.21, a representative action potential is shown, with all three dynamical variables (Vm,v, and w).

All sets of parameters, and further cardiac cell models used in this thesis are presented in the Appendix A on page 131.

Impact of the Choice of Parameters

Every cardiac cell models has a specific set of parameters which needs to be chosen before a simulation. It is important to understand, that the choice of parameters determines the local dynamics of the cell, thus the shape and properties of an action potential, but beyond also has a huge impact onto the dynamical behavior of spiral or scroll waves. The break-up mechanism, propagation behavior and filament tension of scroll waves [26] are for example important features, which are mainly determined by the local parameters. Fenton et al., for example, performed for that reason a single study only to investigate this dependence by testing ten different parameter sets for the Fenton-Karma model [74].

With properties or rules which are defined only locally but determine the dynamics on a global scale, this feature on the one hand illustrates in an interesting way the nature of complex systems. However, the sensitive dependence of parameters stresses also their significance concerning the design of proper numerical simulations which aim to address a specific scientific objective. The Fenton-Karma model combines a reasonable computation time, with a moderate level of complexity (thirteen parameters), which allows to reproduce diverse dynamics of spiral/scroll waves. That is, why this model was used for many studies of this thesis. More details about the choice of the cardiac cell model and its parameters are given in section 2.3 on page 40.