Numerical simulation of electrocardiograms - Modeling of Physiological Flows - page 79

with appropriate boundary conditions

σ i ∇ V m · n + σ i ∇ u e · n = 0 n Σ × ( 0 , T ) ,

(4.5)

σ e ∇ n

u e · = 0 n Σ × ( 0 , T ) ,

V m and w w 0 .Here and initial conditions V m | t = 0 = | t = 0 = ( 0 , T ) is the time interval of

def
=

def
=

A m i ion , the vector n stands for the outward unit normal

interest, χ A m C m , I ion m

def
= ∂Ω

H and V m , w 0 are given initial data.
The boundary conditions (4.5) 1 , 2 state that the intra- and extracellular currents do
not propagate outside the heart. While (4.5) 1 is a widely accepted condition (see, e.g.,
[35, 53, 61, 68]), the enforcement of (4.5) 2 is only justified under an isolated heart
assumption (see [53, 61]). The coupled system of Eqs. (4.4)-(4.5) is often known in
the literature as isolated bidomain model (see [15, 17, 61]).

to Σ

Remark 1. The complexity of (4.4)-(4.5) can be reduced by using the so-called mon-
odomain approximation:

div σ∇ V m =

+ ( , ) − ,

χ ∂ t V m I ion V m w I app in Ω

m H

(4.6)

σ∇ V m · n = 0 n Σ ,

def
= σ i ( σ i + σ e ) − 1

where σ e is known as the bulk conductivity tensor. Note that

(4.6) decouples the computation of V m from that of u e . Under the isolating condition
(4.5) 2 , problem (4.6) can be interpreted as the approximation of (4.4) 2 and (4.5) 1
of order zero with respect to a parameter

σ

; the latter measures the gap be-
tween the anisotropy ratios of the intra- and extracellular domains (see [14, 17] for
details). Although several simulation analyses (see, e.g., [14, 51]) suggest that the
monodomain approximation may be adequate for some propagation studies in iso-
lated hearts, it cannot be applied in all situations since it neglects the extracellular
feedback into V m (see, e.g., [14, 22, 51]).

ε ∈ [ 0 , 1 ]

4.2.2 Coupling with the torso: ECG modelling
The myocardium is surrounded by a volume conductor, Ω T , that contains all the

extramyocardial regions (see Fig. 4.2). For convenience,

T will be called the torso .
It is commonly modelled as a passive conductor (generalized Laplace equation). A
perfect electric heart-torso coupling, across the interface

Ω

Σ , is generally assumed

(see, e.g., [35, 53, 61, 68]):

=

Σ , u e u T on

(4.7)

σ ∇ u e · n = − σ ∇ u T · n T on Σ ,

e T

where σ T stands for the conductivity tensor of the torso tissue and n T for the outward

unit normal to ∂Ω T .