Numerical simulation of electrocardiograms - Modeling of Physiological Flows - page 80

Fig. 4.2. Two-dimensional geometrical description: heart domain Ω H , torso domain Ω T

(extramyocardial regions), heart-torso interface Σ and torso external boundary Γ ext

The resulting coupled system can be formulated in terms of V m , u e , w and the
torso potential u T as follows (see, e.g., [53, 61]):

⎨
⎩

∂ t w + g ( V m , w )= 0 , in Ω H × ( 0 , T ) ,
χ m ∂ t V m + I ion ( V m , w ) − div ( σ i ∇ V m ) − div ( σ i ∇ u e )= I app in Ω H × ( 0 , T ) ,
− div ( σ i + σ e ) ∇ u e − div ( σ i ∇ V m )= 0 nΩ H × ( 0 , T ) ,
− div ( σ T ∇ u T )= 0 nΩ T × ( 0 , T ) ,

(4.8)

σ T ∇ u T · n T = 0 n Γ ext × ( 0 , T ) ,

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

σ e ∇ u e · n = − σ T ∇ u T · n T

on Σ × ( 0 , T ) ,

V m , w w 0 . The boundary condition on

with the initial conditions V m | = | =

t = 0 t = 0

def
= ∂Ω

Γ \ Σ states that no current can flow from the external torso surface (see

ext T

Fig. 4.2).
The coupled system of Eqs. (4.8) is often known in the literature as the full bido-
main or coupled bidomain model (see, e.g., [14, 61]). It can be considered as the
“gold standard” in the modelling of the ECG (see, e.g., [38, 53, 61]).

Remark 2. A common approach to reduce the computational complexity of (4.8)
consists of uncoupling the computation of

from that of u T , by neglecting
the electrical torso-to-heart feedback (see, e.g., [14, 38, 50]). Thus, the coupling
conditions (4.7) are replaced by

( w , V m , u e )

u T = u e on Σ ,

(4.9)

σ e ∇ u e · n = 0 n Σ ,