This problem couples a bidomain equation, with an ODE model representing a simplification of the internal circuitry of a pacemaker. The coupling is made through boundary conditions which account for the polarization of the electrodes by considering the saline/electrode interface as a parallel R-C model.

Details on how the stimulation of a pacemker cycle consists in can be found in [1].

Equations

On ,

$\begin{array}{ll} \partial_t v + \dfrac{1}{C_\mathrm{m}}\left(I_\mathrm{ion}(v,w) + I_\mathrm{app}\right) - \dfrac{1}{A_\mathrm{m}C_\mathrm{m}}\nabla\cdot(\sigma_\mathrm{i}\nabla (v+u_\mathrm{e})) = 0 & \text{on }\Omega,\\ - \dfrac{1}{A_\mathrm{m}C_\mathrm{m}}\left[\nabla\cdot(\sigma_\mathrm{i}\nabla v) + \nabla\cdot((\sigma_\mathrm{i}+\sigma_\mathrm{e})\nabla u_\mathrm{e})\right] = 0 & \text{on }\Omega,\\[+5pt] -\sigma_\mathrm{e} \nabla u_\mathrm{e} \cdot n = \mu(x)u_\mathrm{e}&\text{on }\partial\Omega\setminus\{\Gamma^+\cup\Gamma^-\},\\ -\sigma_\mathrm{e} \nabla u_\mathrm{e} \cdot n = c^- \partial_t \left(u_\mathrm{e} - U^-\right) + g^+\left(u_\mathrm{e} - U^-\right) &\text{on }\Gamma^-,\\ -\sigma_\mathrm{e} \nabla u_\mathrm{e} \cdot n = c^+ \partial_t \left(u_\mathrm{e} - U^+\right) + g^+\left(u_\mathrm{e} - U^-\right) &\text{on }\Gamma^+,\\ C\mathrm{d}_t U^- + \frac{1}{R}U^+ = 0\\ + \text{ initial conditions}. \end{array}$

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	$w(\bm{x},t)$	-
2	`Anode potential`		$\si{\milli\volt}$
3	`Cathode potential`		$\si{\milli\volt}$
4	`CETS Voltage`	$V_{\mathrm{CETS}}$	$\si{\milli\volt}$
5	`CTS Voltage`	$V_{\mathrm{CTS}}$	$\si{\milli\volt}$

Model parameters

$A_\mathrm{m}$ 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 $\sigma_\mathrm{i}$ .
$c^\pm$ and $g^\pm$ are the equivalent capacitance and conductance of the electrode/bath contacts, seen as a R//C model.
and are the capacitance and resistance of the device, which change during each of the phases of the stimulation cycle.

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 bidomain outputs, the time series file for real values (suffixed by _probes.dat), contains the values of:

and ,
the voltage seen by the CETS and CTS capacitors,
the current going through the circuit (and into the tissue)
the average extracardiac potential on anode and cathode
the voltage measured by the device
an integer describing in each phase of the stimulation cycle the pacemaker is (0: offset, 1: pulse, 2: switch, 3: ocd, 4: waiting next pulse)

The electric potential distribution generated by the pacemaker as if the domain was complitely passive (ie linear equation), can be output as well, with the "compute electric field" option set to yes.

problem type : pacemaker bidomain
 
# Geometry
3D mesh            : ./meshes/catheterBloodTissue.vtk
geometry scale     : 0.1
tissue attributes  : 7
anode attributes   : 2 
cathode attributes : 3 
 
# Outputs
output file name       : ./results/pacemakerBidomain
output period          : 1
activation time data   : 0 -70 -42 0.75
probe points           : 0. 0. -0.1, 0. 0. -0.6, 0. 0. -1.19, -2.25 -2.25 -0.1, -2.25 -2.25 -0.6, -2.25 -2.25 -1.19
 
# Main problem parameters
pde start time              : 0.0
pde end time                : 50.0
stop at complete activation : yes
compute electric field      : yes 
 
# Pacemaker properties
 
# Stimulation
pacemaker stimulation amplitude : 5000.0
pacemaker stimulation duration  : 2.0
pacemaker stimulation nspikes   : 1.0
 
# Other phases of stim cycle durations
pacemaker offset duration : 1.0
pacemaker period duration : 666.0
pacemaker switch duration : 0.122
pacemaker ocd    duration : 12.2
 
# Device electronics (microFarad, kOhm)
pacemaker internal cets    : 9.3639
pacemaker internal cts     : 10.6248
pacemaker internal rgnd    : 0.018
pacemaker internal rpulse  : 0.007
pacemaker internal rwa     : 0.080
pacemaker internal rwc     : 0.20
pacemaker internal rbig    : 80.0
pacemaker internal rlittle : 0.005
 
# Contact properties
 
anode resistance: 2.0
anode capacitance: 18.74
# anode time constant: 37.48 # replace resistance or capacitance
 
cathode resistance: 0.03
cathode capacitance: 5.55
# cathode time constant: 0.1665 # replace resistance or capacitance
 
# Tissue
ionic model : 0 BR77
 
extracellular conductivity : CPIECEWISE \
     7 3.91 0. 0. 0. 1.97 0. 0. 0. 1.97,\
    -1 6.67 0. 0. 0. 6.67 0. 0. 0. 6.67
intracellular conductivity : CPIECEWISE \
     7 1.74 0. 0. 0. 1.19 0. 0. 0. 1.19,\
    -1 0.   0. 0. 0. 0.   0. 0. 0. 0.
 
Am : CONSTANT 200.0
 
# Solvers parameters
 
linear solver type               : BICGSTAB
linear solver relative tolerance : 1.E-12
 
ionic model solver : FBE
 
pde solver    :  FBE
pde time step : 0.01
# time step can be overriden by the pacemaker
pacemaker offset  time step : 0.01
pacemaker pulse   time step : 0.01
pacemaker switch  time step : 0.01
pacemaker ocd     time step : 0.01
pacemaker waiting time step : 0.01

References

[1] Pannetier, V. et al. (2023). Modeling Cardiac Stimulation by a Pacemaker, with Accurate Tissue-Electrode Interface. In: Bernard, O., Clarysse, P., Duchateau, N., Ohayon, J., Viallon, M. (eds) Functional Imaging and Modeling of the Heart. FIMH 2023. Lecture Notes in Computer Science, vol 13958. Springer, Cham. https://doi.org/10.1007/978-3-031-35302-4_20