This PDE system consists in the coupling of a bidomain model in a cardiac tissue $\Omega_{\mathrm{t}}$ , to a quasi-static electric potential equation in a contiguous extra-cardiac medium $\Omega_{\mathrm{ext}}$ (e.g. blood, pericardium, torso).

Equations

$\left\{\begin{array}{ll} \partial_t v + \dfrac{1}{C_\mathrm{m}}\left(I_\mathrm{ion}(v,w) + I_\mathrm{app}\right) - \dfrac{1}{A_\mathrm{m}C_\mathrm{m}}\nabla\cdot(\sigma_\mathrm{i}\nabla (v+u_\mathrm{e})) = 0 & \text{on }\Omega,\\ - \dfrac{1}{A_\mathrm{m}C_\mathrm{m}}\left[\nabla\cdot(\sigma_\mathrm{i}\nabla v) + \nabla\cdot((\sigma_\mathrm{i}+\sigma_\mathrm{e})\nabla u_\mathrm{e})\right] = 0 & \text{on }\Omega,\\[+5pt] \nabla\cdot(\sigma_{\mathrm{e}}\nabla u_{\mathrm{e}}) = 0 &\text{on }\Omega_{\mathrm{ext}},\\ \partial_t w + g(v,w) = 0 &\text{on }\Omega_{\mathrm{t}},\\ \sigma_\mathrm{i}\partial_{\bm{n}} (v+u_\mathrm{e}) = 0&\text{on }\partial\Omega_{\mathrm{t}},\\ + \text{ boundary conditions for }u_\mathrm{e},\\ + \text{ initial conditions.} \end{array}\right.$

The stimulation can be passed as boundary condition using the anode/cathode parameters, as in the flux anode cathode problem.

Unknowns

CEPS ID	Name	Math symbol	Unit
0	`Transmembrane voltage`	$v(\bm{x},t)$	$\si{\milli\volt}$
1	`Extracellular potential`	$u_\mathrm{e}(\bm{x},t)$	$\si{\milli\volt}$
-	ionic state variables	$w(\bm{x},t)$	-

Tissue properties:

$A_\mathrm{m}$ is the surface to volume ratio of cell membranes, in $\si{\per\centi\metre}$ ,
$C_\mathrm{m}$ is the cell membrane surfacic capacitance, in $\si{\micro\farad\per\centi\metre\squared}$ ,
$\sigma_\mathrm{i}(\bm{x})$ the tensorial intracellular conductivity, in $\si{\milli\siemens\per\centi\metre}$ . Usually, the tensor is the same everywhere when expressed in the local frame defined by fiber orientation. However, alterations of the tissue conductivity can be introduced by using volume fraction.
$\sigma_\mathrm{e}(\bm{x})$ the tensorial extracellular conductivity, in $\si{\milli\siemens\per\centi\metre}$ . Similar to $\sigma_\mathrm{i}$ , extended in extracardiac conductivity.

Other functions

$I_\mathrm{ion}$ is the current due to ionic gates, in $\si{\micro\ampere\per\centi\metre\squared}$ ,
$I_\mathrm{app}(\bm{x},t)$ is the user-defined applied current used to stimulate a specific region of the domain,
is the evolution function of ionic variables.

Usage

In addition to the bidomain parameters, the following keys can be set to parametrize the domains in which the problem is extended with Poisson equation:

blood attributes : <integers>
torso attributes : <integers>
extracardiac attributes : <integers>

If none of these attribute keys is found in the parameter file, the extension is made on the whole computational domain.

For stimulations, use the following:

electrodes stimulation : <options of time function>

The options are those of a compact support function, without specifying the TIME keyword.

problem type : extended bidomain
 
#-----------------
# Output controls
 
output file name : ./results/extBidomain
output period : 0.5
 
activation time data : 0 -20 -40 0.75
 
#-----------------
# Simulation time
 
# in ms
PDE start time : 0.0
PDE end time   : 50
 
#----------
# Geometry
 
3d mesh : ./meshes/catheterBloodTissue.vtk
geometry scale : 0.1
 
#--------
# Tissue
 
Am : CONSTANT 200.0
 
tissue attributes : 7
extracellular conductivity : CPIECEWISE \
                             6  13. 0. 0. 0. 13 0. 0. 0. 13, \
                             -1 4. 0. 0. 0. 2. 0. 0. 0. 2.
intracellular conductivity : CONSTANT 3.5 0. 0. 0. 1. 0. 0. 0. 1.
 
#-------------
# Ionic model
 
ionic model : 0 MS
 
#-------------
# Stimulation
 
anode attributes : 3
cathode attributes : 2
electrodes stimulation : TIME PROFILE CONSTANT START 0.04 DURATION 2. AMPLITUDE -15000.
 
#----------------
# solver options
 
spatial discretization : FE 1
PDE solver             : SBDF 2
PDE time step          : 0.05
 
linear solver absolute tolerance : 1.E-12
linear solver relative tolerance : 1.E-12
linear solver type               : GMRES
 
ionic model solver : RL 2