The bidomain model is a mathematical model to define the electrical activity of the heart. 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.

The bidomain model was first proposed by Schmitt in 1969 before being formulated mathematically in the late 1970s.

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.

Many of the interesting properties of the bidomain model arise from the condition of unequal anisotropy ratios. The electrical conductivity in anisotropic tissues is not unique in all directions, but it is different in parallel and perpendicular direction with respect to the fiber one. Moreover, in tissues with unequal anisotropy ratios, the ratio of conductivities parallel and perpendicular to the fibers are different in the intracellular and extracellular spaces. For instance, in cardiac tissue, the anisotropy ratio in the intracellular space is about 10:1, while in the extracellular space it is about 5:2. Mathematically, unequal anisotropy ratios means that the effect of anisotropy cannot be removed by a change in the distance scale in one direction. Instead, the anisotropy has a more profound influence on the electrical behavior.

Three examples of the impact of unequal anisotropy ratios are

• the distribution of transmembrane potential during unipolar stimulation of a sheet of cardiac tissue,

• the magnetic field produced by an action potential wave front propagating through cardiac tissue,

• the effect of fiber curvature on the transmembrane potential distribution during an electric shock.

Formulation

Bidomain domain

The bidomain domain is principally represented by two main regions: the cardiac cells, called intracellular domain, and the space surrounding them, called extracellular domain. Moreover, usually another region is considered, called extramyocardial region. The intracellular and extracellular domains, which are separate by the cellular membrane). 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.

Unknowns and parameters

and the transmembrane potential.

Moreover, some important parameters need to be taken in account, especially the intracellular conductivity tensor matrix the extracellular conductivity tensor matrix need to be considered to derive the bidomain model formulation, which is done in the following section.

Standard formulation

The bidomain model is defined through two partial differential equations (PDE) the first of which is a reaction diffusion equation in terms of the transmembrane potential, while the second one computes the extracellular potential starting from a given transmembran potential distribution.

can be defined as applied external stimulus currents.

Ionic current equation

The ionic current is usually represented by an ionic model through a system of ordinary differential equations, the system reads

Different ionic models have been proposed:

• phenomenological models, which are the simplest ones and used to reproduce macroscopic behavior of the cell.

• physiological models, which take into account both macroscopic behaviour and cell physiology with a quite detailed description of the most important ionic current.

Model of an extramyocardial region

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.

Usually, this equation is a simple generalized Laplace equation of type is the corresponding conductivity tensor.

being the unit normal directed outside of the extramyocardial domain.

If the extramyocardial region is the human torso, this model gives rise to the forward problem of electrocardiology.

Derivation

The bidomain equations are derived from the Maxwell's equations of the electromagnetism, considering some simplifications.

The first assumption is that the intracellular current can flow only between the intracellular and extracellular regions, while the intracellular and extramyocardial regions can communicate between them, so that the current can flow into and from the extramyocardial regions but only in the extracellular space.

Using Ohm's law and a quasi-static assumption, the gradient of a scalar potential, which means that

represent the current density of the electric field, two equations can be obtained represent the intracellular and extracellular quantities respectively.

per unit area, which means that

By combining the previous assumptions, the conservation of current densities is obtained, namely

from which, summing the two equations

This equation states exactly that all currents exiting one domain must enter the other.

from both sides. In fact, Then, knowing the transmembral potential, one can recover the extracellular potential.

Then, the current that flows across the cell membrane can be modelled with the cable equation,

Combining equations () and () gives, one can get the first equation of the bidomain model which describes the evolution of the transmembrane potential in time.

The final formulation described in the standard formulation.

Boundary conditions

In order to solve the model, boundary conditions are needed. The more classical boundary conditions are the following ones, formulated by Tung.

First of all, as state before in the derive section, there ca not been any flow of current between the intracellular and extramyocardial domains. This can be mathematically described as, namely

For the extracellular potential, if the myocardial region is presented, a balance in the flow between the extracellular and the extramyocardial regions is considered Here the normal vectors from the perspective of both domains are considered, thus the negative sign are necessary. Moreover, a perfect transmission of the potential on the cardiac boundary is necessary, which gives

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

Reduction to monodomain model

, the model can be reduced to one single equation, called monodomain equation

Formulation with boundary conditions in an isolated domain

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

Numerical solution

There are various possible techniques to solve the bidomain equations. Between them, one can find finite difference schemes, finite element schemes and also finite volume schemes. Special considerations can be made for the numerical solution of these equations, due to the high time and space resolution needed for numerical convergence.

See also

• Monodomain model

• Forward problem of electrocardiology

References

External links

• Scholarpedia article about the bidomain model