-0.4 -0.2 0 0.2 0.4 0.6 0.8
0 50 100 150 200 250 300 350 400
current (pA/pF)
time (ms) INaCa
IKr
IKs
INaK
IK1×0.5
(a)
-0.4 -0.2 0 0.2 0.4 0.6
0 50 100 150 200 250 300 350 400
current (pA/pF)
time (ms) ICa,L
ICa,T
If,K
If,Na
IKr+IKs
Ito+Isus
(b)
Fig. 5.11.Ionic curents of (a) the ten Tusscher et al. and (b) the central Zhang et al. model.
If= If,K + If,N a
If,K = gf,K,maxy (Vm− EK) , If,N a = gf,N a,maxy (Vm− EN a) αy = exp
−Vm+ 78.91 26.62
, βy = exp Vm+ 75.13 21.25
The T-type Ca2+ current was proven to have an important contribution in SAN cells (fig. 5.11 b) [127]. If this current is blocked, a negative chronotropic effect is present in the SAN. ICa,T is implemented in the Zhang et al. model in a similar way than the L-type Ca2+
currents in other models.
Some of the maximum conductances of the model were differing to describe the cells from central and peripheral regions. Mainly the maximum conductance of the channels depicting the currents ICa,L, Ito, Isus, IKr, IKs, and If are reduced for the central cell compared to the peripheral cell. IN a is not expressed in the central SAN myocytes and the membrane capacitance is reduced since central cells are smaller than peripheral.
5.6. Excitation Conduction 77
(a) (b) (c)
(d) (e) (f)
Fig. 5.12. Schematic functioning of a cellular automaton. The green dots are active elements, red dots represent the refractory state. After an initial stimulus (a) the excitation is conducted to neighboring cells (b). The elements are refractory during the plateau phase (c,d). After repolarization (e,f) the cells are again excitable. Fig. from [264].
Microscopic models allow the individual modeling of myocytes with several elements con-sidering different ionic mechanisms and gap junction densities in each node [140, 263]. As microscopic modeling is not in the scope of this work, it is not further detailed in this section.
5.6.1 Cellular Automata
The application of rule based models of the excitation conduction like cellular automata al-lows the efficient simulation of the cardiac electrophysiology [265, 266, 267, 268, 269, 270].
The geometrical information of the model usually considers the anisotropic and heteroge-neous anatomical features of the heart. Furthermore, tissue dependent electrophysiological parameters like refractory periods, action potential courses, excitation conduction velocities, and autorhythmic properties are considered, which are taken from literature.
A cellular automaton can be divided into two components [271, 272]: A regular, discrete, finite network representing the underlying spatial structure and a finite automaton working at each node of the network. The network consists e.g. of an anatomical model of the human heart.
The tissue specific excitation conduction velocity and the anisotropy of the myocardium can be considered based on myocyte orientation data. The finite automaton at each node point represents the physiological state by modeling the course of the transmembrane voltage depending on tissue type, stimulation frequency and refractory periods. The scheme of a cellular automaton is depicted in fig. 5.12.
5.6.2 Simplified Reactions-Diffusion Systems
Modeling of excitation conduction with simplified reaction-diffusion systems can reconstruct the physiological and pathological excitation on tissue and organ level [138, 273, 274]. The reaction-diffusion systems consists of two components: One represents the membrane activity (reaction part), and the other describes the electrical interaction of the tissue (diffusion part).
An early representative is the two state FitzHugh-Nagumo model [275]:
∂u
∂t = u − u33 − v
+ D∇2u, ∂v
∂t = (u + β − γv)
with the state variable u for the activation of transmembrane voltage and the state variable v for inactivation. The diffusion term is formulated for isotropic media with the scalar diffusion coefficient D. The parameters β, γ, and are the so-called membrane parameters and are influencing the course of the AP. The diffusion coefficient and the membrane parameters are time dependent.
Modifications of the FitzHugh-Nagumo equations allow a more realistic description of the conduction in the myocardium [276, 277]. This model considers the anisotropic features of the tissue by using a diffusion tensor. It includes more membrane parameters to adapt the AP course to measured data.
5.6.3 Models of the Electrical Flow
A common approach to reproduce the activity of cardiac tissue consists of a model describing the anisotropic excitation conduction in combination with an electrophysiologically accurate ionic model. The ionic models are explained in section 5.5. The excitation conduction model could be either based on resistor networks or can consider the electrical flow of ions be-tween adjacent cells using Poisson’s equation. The advantage of resistor networks to describe electrical coupling of cardiac cells is the lower computational cost compared to approaches based on Poisson’s equation. The most important disadvantage is that varying anisotropic conductivities in the tissue are not reproducible with the resistor network approaches.
A further distinction is made between models considering the extracellular space to be on constant potential and thus calculating the current flow through the intracellular space and through gap junctions (monodomain model) and approaches taking the current flow in intra-and extracellular space as well as through gap junctions into account (bidomain model). Both, monodomain and bidomain model can be represented by resistor network or Poisson’s equa-tion based approaches. As the implemented method in this work is based on the calculaequa-tion of Poisson’s equation, this approach is detailed further.
5.6.3.1 Bidomain Model
The discrete electrical interaction of the tissue can be approximated by the continuous ap-proach of the bidomain model [278, 279, 280, 281, 282]. The bidomain model considers the current flow in the intra- and the extracellular domain independently. The two domains are separated by the cell membrane (fig. 5.13) through which currents flow between the do-mains. Hence, the intercellular stimulus current, which drives the conduction of excitation is constructed with the influences of both domains.
Conductivity tensors for both domains are defined to consider the anisotropic electrical prop-erties of cardiac tissue and the surrounding medium e.g. blood, bath, and air. The
extracel-Fig. 5.13.The two domains of the bidomain model. The intracellular space is separated from the extra-cellular space by a membrane. Fig. from [270].
5.6. Excitation Conduction 79 Table 5.2.Intra- and extracellular conductivities(S/m)[281, 283].σil andσel are the longitudinal intra-and extracellular conductivities.σit andσet are the transversal intra- and extracellular conductivities.
Clerc (1976) Roberts and Scher (1982) Colli-Franzone (1993)
σil 0.174 0.344 0.3
σit 0.0193 0.0596 0.031525
σel 0.625 0.117 0.2
σet 0.236 0.0802 0.13514
lular anisotropy is determined by the cylindrical shape of the cells in combination with the fiber orientation of the cells and the sheet orientation of a myocardial layer. The intracellular anisotropic conductivity tensor represents density and distribution of gap junctions and also the oriented intracellular structures of the cell with respect to the orientation of the my-ocytes. As the microscopic intra- and extracellular conductivities vary between adjacent cells the bidomain model uses a mean conductivity tensor to describe the macroscopic anisotropy in both domains.
For each domain, Poisson’s equation is defined with index e for the extracellular domain and index i for the intracellular domain:
∇ (σe∇Φe) = −βIm− Ise (5.16)
∇ (σi∇Φi) = βIm − Isi (5.17)
with the intra- and extracellular potentials Φi and Φe, respectively, the corresponding con-ductivity tensors σi and σe, the transmembrane current density Im, the membrane surface to cell volume ratio β, and externally applied current sources Isi and Ise in the intra- and extracellular domain, respectively.
The conductivity tensors σi and σerepresent the anisotropic electrical properties of the tissue and depend on the tissue type and the orientation of the main axis of the myocytes. Table 5.2 depicts an overview of commonly used values for the conductivities.
Based on the assumption of orthotropic conductivity distribution in cardiac myocytes and thus the rotational symmetry, the conductivity tensors σi and σecan be constructed with only the longitudinal and transversal conductivities of the intracellular (σil and σit, respectively) and the extracellular space (σel and σet, respectively):
σi,e= R
σ(i,e)l 0 0 0 σ(i,e)t 0 0 0 σ(i,e)t
RT
whereas the rotation matrix R is constructed of the two components Rxy and Rxz with R = RxzRxy:
Rxy =
cos ϕ sin ϕ 0
− sin ϕ cos ϕ 0
0 0 1
, Rxz =
cos ϑ 0 sin ϑ
0 1 0
− sin ϑ 0 cos ϑ
ϕ and ϑ describe the orientation of the fiber’s main axis in the global space with ϕ being the angle in the (x,y)-plane and ϑ being the angle in the (x,z)-plane.
A mathematically useful reformulation of the two Poisson’s equations (eqs. 5.16 and 5.17) is performed to achieve the bidomain equations. The two domains are coupled via the trans-membrane voltage Vm, which is defined by:
Vm = Φi− Φe (5.18)
Furthermore, the transmembrane current density Im, which flows into one domain must come from the other. Thus, in the summation of the two Poisson’s equations (eqs. 5.16 and 5.17) Im vanishes:
∇ (σi∇Φi) + ∇ (σe∇Φe) = −Isi− Ise
Under the assumption that no externally applied current sources are present and with Vm = Φi− Φe, this can be transfered into:
∇ (σi∇Φi) = −∇ (σe∇Φe)
∇ (σi∇(Vm+ Φe)) = −∇ (σe∇Φe)
∇ ((σi+ σe) ∇Φe) = −∇ (σi∇Vm) (5.19) Equation 5.19 is the first part of the bidomain model describing the influence of the trans-membrane voltage on the extracellular potential. Vm is commonly calculated with the ionic models presented in section 5.5. The second part of the bidomain equations is constructed based on eq. 5.17 under consideration of eq. 5.18:
∇ (σi∇Φi) = ∇ (σi∇ (Vm+ Φe)) = ∇ (σi∇Vm) + ∇ (σi∇Φe)
= βIm− Isi = β
Cm
dVm
dt + Imem
− Isi
since the current Im describes the summed up flow through the membrane given by the electrophysiological models:
Im =
Cm
dVm
dt + Imem
Thus to conclude, the second part of the bidomain equations is given by:
∇ (σi∇Vm) + ∇ (σi∇Φe) = β
Cm
dVm
dt + Imem
− Isi (5.20)
It describes in which way the currents in intra and extracellular space are determining the intercellular stimulus current as an input for the electrophysiological models.
The tissue is normally surrounded by a bath medium. In the heart, this bath is equivalent to the blood at the endocardial boundary and the pericardium at the epicardial bound-ary. The bath and the extracellular space of the tissue build a continuous domain in the bidomain model but with different conductivity tensors. The conductivity in the isotropic bath is assumed to be a scalar value. The intracellular domain is only defined in the active tissue since the bath has no coupled intracellular space. To consider different media, bound-ary conditions need to be included. The first boundbound-ary condition is a grounded potential in the bathing medium to provide a reference potential. A further boundary condition is the continuity of the normal current at the tissue-bath interface in the extracellular domain:
nTσe∇Φe = nTσb∇Φe
with σb being the scalar conductivity of the bath and n is the unit vector normal to the boundary surface between the two media.
As the intracellular space is only defined in the tissue, the intracellular current density is zero at the tissue-bath interface:
nTσi∇Φi = 0
5.7. Pathoelectrophysiology 81