Bidomain model
It consists in a continuum (volume-average) approach in which the cardiac microstructure is defined in terms of muscle fibers grouped in sheets, creating a complex three-dimensional structure with anisotropical properties.
Then, to define the electrical activity, two interpenetrating domains are considered, which are the intracellular and extracellular domains, representing respectively the space inside the cells and the region between them.
[1] The bidomain model was first proposed by Schmitt in 1969[2] before being formulated mathematically in the late 1970s.
[3][4][5][6][7][8][9][10] Since it is a continuum model, rather than describing each cell individually, it represents the average properties and behaviour of group of cells organized in complex structure.
Thus, the model results to be a complex one and can be seen as a generalization of the cable theory to higher dimensions and, going to define the so-called bidomain equations.
[11][12] Many of the interesting properties of the bidomain model arise from the condition of unequal anisotropy ratios.
The intracellular and extracellular domains, which are separate by the cellular membrane, are considered to be a unique physical space representing the heart (
), while the extramyocardial domain is a unique physical space adjacent of them (
The extramyocardial region can be considered as a fluid bath, especially when one wants to simulate experimental conditions, or as a human torso to simulate physiological conditions.
[12] The boundary of the two principal physical domains defined are important to solve the bidomain model.
[12] Moreover, some important parameters need to be taken in account, especially the intracellular conductivity tensor matrix
need to be considered to derive the bidomain model formulation, which is done in the following section.
Different ionic models have been proposed:[19] In some cases, an extramyocardial region is considered.
This implies the addition to the bidomain model of an equation describing the potential propagation inside the extramyocardial domain.
Moreover, an isolated domain assumption is considered, which means that the following boundary conditions are added
[12] If the extramyocardial region is the human torso, this model gives rise to the forward problem of electrocardiology.
[12] Using Ohm's law and a quasi-static assumption, the gradient of a scalar potential field
[12] The second assumption is that the heart is isolated so that the current that leaves one region need to flow into the other.
Then, the current density in each of the intracellular and extracellular domain must be equal in magnitude but opposite in sign, and can be defined as the product of the surface to volume ratio of the cell membrane and the transmembrane ionic current density
By combining the previous assumptions, the conservation of current densities is obtained, namely[12] from which, summing the two equations[12] This equation states exactly that all currents exiting one domain must enter the other.
[12] From here, it is easy to find the second equation of the bidomain model subtracting
The final formulation described in the standard formulation section is obtained through a generalization, considering possible external stimulus which can be given through the external applied currents
[12] In order to solve the model, boundary conditions are needed.
[6] First of all, as state before in the derive section, there ca not been any flow of current between the intracellular and extramyocardial domains.
is the vector that represents the outwardly unit normal to the myocardial surface of the heart.
Here the normal vectors from the perspective of both domains are considered, thus the negative sign are necessary.
Instead, if the heart is considered as isolated, which means that no myocardial region is presented, a possible boundary condition for the extracellular problem is[12]
By assuming equal anisotropy ratios for the intra- and extracellular domains, i.e.
[12] If the heart is considered as an isolated tissue, which means that no current can flow outside of it, the final formulation with boundary conditions reads[12]
Special considerations can be made for the numerical solution of these equations, due to the high time and space resolution needed for numerical convergence.
