|CARP provides several solution methods to solve the cardiac bidomain and monodomain equations. The default numerical scheme is based on the following linear transformation:|
are the extracellular potential and transmembrane voltage respectively;
the ionic current variables;
and are conductivity tensors of intracellular
and extracellular spaces respectively;
is the isotropic conductivity of the fluid in which the heart is immersed
(bath and cavities); is the capacitance
per unit area and ß is surface to volume ratio;
and g model ionic currents and specify the cell membrane model.
Numerically, the bidomain equations are decoupled by an operator splitting approach that leads to a three step scheme involving the solution of a parabolic PDE, an elliptic PDE and a non-linear system of ODEs at each time step. In the simplest case both the parabolic PDE and the non-linear ODE systems are solved via the explicit forward-Euler scheme
|where is the discretized
is the time step; ,
are the temporal discretizations of ,
respectively, for time equal to .
For meshes of finer spatial discretization the parabolic PDE is solved employing
a implicite Crank-Nicholson approach.
The system of equations is solved using the finite element method, employing linear elements and lumped mass matrices.