# Bidomain model

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The bidomain model is a mathematical model of the electrical properties of cardiac muscle that takes into account the anisotropy of both the intracellular and extracellular spaces.

The bidomain model was developed in the late 1970s. [1] [2] [3] [4] [5] [6] [7] [8] It is a generalization of one-dimensional cable theory. The bidomain model is a continuum model, meaning that it represents the average properties of many cells, rather than describing each cell individually. [9]

Many of the interesting properties of the bidomain model arise from the condition of unequal anisotropy ratios. The electrical conductivity in anisotropic tissue is different parallel and perpendicular to the fiber direction. In a tissue with unequal anisotropy ratios, the ratio of conductivities parallel and perpendicular to the fibers is 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. [10] Mathematically, unequal anisotropy ratios means that the effect of anisotropy cannot be removed by a change in the distance scale in one direction. [11] Instead, the anisotropy has a more profound influence on the electrical behavior. [12]

Three examples of the impact of unequal anisotropy ratios are

• the distribution of transmembrane potential during unipolar stimulation of a sheet of cardiac tissue,[13]
• the magnetic field produced by an action potential wave front propagating through cardiac tissue,[14]
• the effect of fiber curvature on the transmembrane potential distribution during an electric shock.[15]

The bidomain model is now widely used to model defibrillation of the heart.

## Formulation

### Standard formulation

The bidomain model can be formulated as follows:

{\displaystyle {\begin{alignedat}{2}\nabla \cdot \left({\mathbf {\Sigma } }_{i}\nabla v\right)+\nabla \cdot \left({\mathbf {\Sigma } }_{i}\nabla v_{e}\right)&=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{ion}\right)\\\nabla \cdot \left({\mathbf {\Sigma } }_{i}\nabla v\right)+\nabla \cdot \left(\left({\mathbf {\Sigma } }_{i}+{\mathbf {\Sigma } }_{e}\right)\nabla v_{e}\right)&=0\end{alignedat}}}

### Formulation with boundary conditions and surrounding tissue

The surrounding tissue ${\displaystyle \mathbb {T} }$ can be included to give reasonable boundary conditions to make the system solvable:

{\displaystyle {\begin{alignedat}{4}\nabla \cdot \left({\mathbf {\Sigma } }_{i}\nabla v\right)+\nabla \cdot \left({\mathbf {\Sigma } }_{i}\nabla v_{e}\right)&=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{ion}\right)&\,\,\,\,\,\,\,&{\mathbf {x} }\in {\mathbb {H} }\\\nabla \cdot \left({\mathbf {\Sigma } }_{i}\nabla v\right)+\nabla \cdot \left(\left({\mathbf {\Sigma } }_{i}+{\mathbf {\Sigma } }_{e}\right)\nabla v_{e}\right)&=0&&{\mathbf {x} }\in {\mathbb {H} }\\\nabla \cdot \left({\mathbf {\Sigma } }_{0}\nabla v_{0}\right)&=0&&{\mathbf {x} }\in {\mathbb {T} }\\{\vec {n}}\cdot \left({\mathbf {\Sigma } }_{0}\nabla v_{0}\right)&=0&&{\mathbf {x} }\in \partial {\mathbb {T} }\\{\vec {n}}\cdot \left({\mathbf {\Sigma } }_{0}\nabla v_{0}\right)-{\vec {n}}\cdot \left({\mathbf {\Sigma } }_{e}\nabla v_{e}\right)&=0&&{\mathbf {x} }\in \partial {\mathbb {H} }\\{\vec {n}}\cdot \left({\mathbf {\Sigma } }_{i}\nabla v\right)+{\vec {n}}\cdot \left({\mathbf {\Sigma } }_{i}\nabla v_{e}\right)&=0&&{\mathbf {x} }\in \partial {\mathbb {H} }\end{alignedat}}}

## Derivation

Let ${\displaystyle \mathbb {H} }$ with boundary ${\displaystyle \partial \mathbb {H} }$ be the set of all points ${\displaystyle \mathbf {x} }$ in the heart. In each point in ${\displaystyle \mathbb {H} }$ there is an intra- and extracellular voltage and current, denoted by ${\displaystyle v_{i}}$, ${\displaystyle v_{e}}$, ${\displaystyle J_{i}}$ and ${\displaystyle J_{e}}$ respectively. Let ${\displaystyle \mathbf {\Sigma } _{i}}$ and ${\displaystyle {\mathbf {\Sigma } }_{e}}$ be the intra- end extracellular conductivity tensor matrices respectively.

We assume Ohmic current-voltage relationship and get

{\displaystyle {\begin{alignedat}{2}J_{i}&=-{\mathbf {\Sigma } }_{i}\nabla v_{i}\\J_{e}&=-{\mathbf {\Sigma } }_{e}\nabla v_{e}.\end{alignedat}}}

We require that there is no accumulation of charge anywhere in ${\displaystyle \mathbb {H} }$, and therefore that

{\displaystyle {\begin{alignedat}{2}\nabla \cdot \left(J_{i}+J_{e}\right)&=0\\\nabla \cdot \left(-{\mathbf {\Sigma } }_{i}\nabla v_{i}-{\mathbf {\Sigma } }_{e}\nabla v_{e}\right)&=0\end{alignedat}}}

giving one of the model equations: Template:NumBlk This equation states that all current exiting one domain must enter the other.

The transmembrane current is given by Template:NumBlk

We model the membrane similarly to that of the cable equation, Template:NumBlk where ${\displaystyle \chi }$ is the surface to volume ratio of the membrane, ${\displaystyle C_{m}}$ is the electrical capacitance per unit area, ${\displaystyle v=v_{i}-v_{e}}$ and ${\displaystyle I_{ion}}$ is the ionic current over the membrane per unit area.

Combining equations (Template:EquationNote) and (Template:EquationNote) gives

${\displaystyle \nabla \cdot \left({\mathbf {\Sigma } }_{i}\nabla v_{i}\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{ion}\right),}$

which can be rearranged using ${\displaystyle v=v_{i}-v_{e}}$ to get another model equation: Template:NumBlk

### Boundary conditions

In order to solve the model, boundary conditions are needed. One way to define the boundary condition is to extend the model with a volume ${\displaystyle \mathbb {T} }$ with perimeter ${\displaystyle \partial \mathbb {T} }$ that surrounds the heart and represent the body tissue.

As was the case for ${\displaystyle \mathbb {H} }$, we assume no accumulation of charge in ${\displaystyle \mathbb {T} }$, i.e. Template:NumBlk where ${\displaystyle {\mathbf {\Sigma } }_{0}}$ is the conductance tensor of the body tissue and ${\displaystyle v_{0}}$ is the voltage in ${\displaystyle \mathbb {T} }$.

Assuming that the body is electrically surrounded from the environment, there can be no current component on the surface ${\displaystyle \partial \mathbb {T} }$ in the normal direction, hence: Template:NumBlk

On the surface of the heart, a common assumption is that there is a direct connection between the surrounding tissue and the extracellular domain. This means that the potentials ${\displaystyle v_{e}}$ and ${\displaystyle v_{0}}$ must be equal on the heart surface, i.e. Template:NumBlk

This direct connection also require that all ionic current exiting ${\displaystyle \mathbb {T} }$ on the heart surface, must enter the extracellular domain, and vica versa. This gives another boundary condition: Template:NumBlk

Finally, we assume that there is a complete isolation of the intracellular domain and the surrounding tissue. Similarly to equation (Template:EquationNote), we get

${\displaystyle {\vec {n}}\cdot \left(\mathbf {\Sigma } _{i}\nabla v_{i}\right)=0\,\,\,\,\,\,\,\mathbf {x} \in \partial \mathbb {H} }$

which can be rewritten using ${\displaystyle v=v_{i}-v_{e}}$ to Template:NumBlk

Extending the model to include equations (Template:EquationNote)-(Template:EquationNote) gives a solvable system of equations.

## Reduction to monodomain model

By assuming equal anisotropy ratios for the intra- and extracellular domains, i.e. ${\displaystyle \mathbf {\Sigma } _{i}=\alpha \mathbf {\Sigma } _{e}}$ for some scalar ${\displaystyle \alpha }$, the model can be reduced to the monodomain model.

## References

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