Modeling of Excitable Media

t φ

0 mV

-90 mV ^fastINa

+

slowI_Ca2+

I_ir,K+ blocked

IV,K⁺

reactivated

Figure 2.4: A schematic of the human cardiac action potential. Its shape emerge from the interaction of the cardiac ion channels. After the fast Na⁺ channels initiate the upstroke, the plateau phase is maintained by slower Ca²⁺channels and the blocking of inward rectifying K⁺ channels. At the end of the action potential, additional K⁺

channels open to guarantee quick repolarization.

Figure 2.5: Gap junctions (in violet) connect neighboring cardiomyocytes and thus allow direct communication of action potentials, mainly in fiber direction.

charges ares sufficient for significant voltage changes.

The action potential in the heart spreads to neighboring cells through the extracellular space, but also directly from cell to cell. Cardiomyocytes are elongated cells of 50 to 70 µm length, their cytoplasm is connected directly to their neighbors through gap junctions, mainly at the ends (figure 2.5). This anisotropy leads to significant differences in the excitation propagation velocities along and perpendicular to the muscle fiber orientations.

The described mechanisms, excitation amplification, depolarization and later repolar-ization and relaying this excitation to neighbors cause the spread of excitation as waves in cardiac tissue. In the section “Excitation Pathways in the Heart”, I will explain the excitation of the whole heart in more detail and section “Pathological Heart Activations”

will cover pathological activation patterns.

2.2 Modeling of Excitable Media

Considering the medical importance of a functioning heart, it is understandable that ex-tensive efforts have been made to describe the functional behavior of the cardiomyocytes

2 The Heart, an Excitable Medium 2.2. Modeling of Excitable Media in varying levels of detail and complexity. Mathematical models allow to understand the mechanisms underlying cardiac behavior, in particular pathologic activation patterns and methods to terminate them. Here I will focus on the electrical aspects, the equally important electro-mechanical coupling is beyond the scope of this work.

2.2.1 Bidomain and Monodomain Models

In a first approximation, it makes sense to consider mainly the electric potential on the inside and outside of cardiac muscle cells. On a macroscopic level, one can make the assumption that both the intracellular and extracellular domain pervade the whole geometric space, i.e. the state of each point consists of the intra- and extracellular state at that point.

This idea can be formulated in thebidomain model of cardiac tissue:

j_i and j_e are the current densities within the intra- and extracellular space, respec-tively. These current densities are governed by the potentials φ_. and conductivities σ_. (conductivity tensors in the case of anisotropic media):

j_i=−σ_i∇φ_i (2.6)

j_e=−σ_e∇φ_e (2.7)

Interaction between the two domains is mediated through the transmembrane current densityj_m, which scales with the surface to volume ratioχ:

∇σ_i∇φ_i=χj_m=−∇σ_e∇φ_e (2.8)

If the intracellular domain does not cover the same physical space as the extracellular domain, no-flux boundary conditions for the intracellular medium are assumed on the boundary (see figure 4.1, boundary effects are discussed in more detail in section 4.1

“Circular Heterogeneities Modeled”):

n_i· ∇φ_i ∂Ωi

= 0 (2.9)

The extracellular medium’s potential and current are assumed to be identical on both sides of this boundary.

Introducing the (trans)membrane potentialφ_m=φ_i−φ_eand the membrane capacitance per areaC_m,j_m can also be written as:

j_m=C_m∂φ_m

∂t +j_ion(φ_m) (2.10)

2.2. Modeling of Excitable Media

Here the ionic membrane currentj_ionincludes the linear and nonlinear effects that make up the cells’ behavior. In most models the overall transmembrane current is composed of currents for different types of ion channels and pumps, many of which depend not only resistively on the membrane voltage but also on the internal state of the cell.

Usually the previous equations are rewritten in a slightly different way which is closer to typical numerical approaches to solve them:

∂φ_m

The equations (2.11) and (2.12) constitute thebidomain equations. The most common way to solve such a bidomain system is for each time step to first integrate (2.11) and then solve (2.12) for φ_e.

Since especially the latter step is computationally expensive for large systems, it makes sense to consider the special case where σ_e and σ_i are not independent of each other, but are only scaled differently as in [25]:

σ_i=kσ_e (2.13)

Equation (2.16), the monodomain equation, depends only on the membrane potential φ_m which makes solving it numerically a lot simpler. Although the assumption (2.13) does not hold physiologically in many cases, the monodomain representation can still reproduce a broad range of physiological behavior qualitatively and quantitatively.

The mono- and bidomain equations are compatible with the formal requirements for a reaction-diffusion system (equation (2.2)) with just one or two diffusive variables. Addi-tional variables u are “hidden” in the ion current dynamics ofj_ion =j_ion(φm(x),u(x)) and evolve locally according to some ^∂u_∂t^(x) =f(φ(x),u(x)).

Together with boundary and initial conditions and a suitable f, the mono- and bido-main equations can describe the evolution of an excitable medium which is continuously

2 The Heart, an Excitable Medium 2.3. Excitation Pathways in the Heart

Figure 2.6: Numerical integration of the Barkley model on a square domain. The color, from blue to red, indicates the value of the activation variable. Left: A plane wave traveling in upward direction across the tissue Middle: A self-sustained spiral, in clockwise rotation. Right: Turbulent activation in the Bär-Eiswirth model[26], a modified Barkley model with an additional nonlinear term. The model systems were

integrated using the MediaSim software framework (main author: Philip Bittihn).

extended in space. Several methods exist to handle spatially extended systems numeri-cally, most of them discretize space so that the system only needs to be calculated at a finite number of sites. Typical examples of a particularly simple model (as proposed by Barkley [9]) system are shown in figure 2.6.