Bidomain model

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

The bidomain model was developed in the late 1970s.         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. 

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.  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.

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

Formulation

Standard formulation

The bidomain model can be formulated as follows:

{\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_{\mathrm {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}} where $\chi$ is the membrane surface area per unit volume (of tissue), $C_{m}$ is the electrical capacitance of the membrane per unit area, $v=v_{i}-v_{e}$ where $v_{i}$ is the interstitial voltage and $v_{e}$ is the extracellular voltage, and $I_{\mathrm {ion} }$ is the ionic current over the membrane per unit area.

Formulation with boundary conditions and surrounding tissue

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

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

We assume Ohmic current-voltage relationship and get

{\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 $\mathbb {H}$ , and therefore that

{\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:

$\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{i}\right)+\nabla \cdot \left(\mathbf {\Sigma } _{e}\nabla v_{e}\right)=0.$ (1)

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

The transmembrane current is given by

$J_{t}=\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{i}\right)=-\nabla \cdot \left(\mathbf {\Sigma } _{e}\nabla v_{e}\right).$ (2)

We model the membrane similarly to that of the cable equation,

$J_{t}=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\mathrm {ion} }\right),$ (3)

where $\chi$ is the membrane surface area per unit volume (of tissue), $C_{m}$ is the electrical capacitance of the membrane per unit area, $v=v_{i}-v_{e}$ and $I_{\mathrm {ion} }$ is the ionic current over the membrane per unit area.

Combining equations (2) and (3) gives

$\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{i}\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\mathrm {ion} }\right),$ which can be rearranged using $v=v_{i}-v_{e}$ to get another model equation:

$\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{e}\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\mathrm {ion} }\right).$ (4)

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 $\mathbb {T}$ with perimeter $\partial \mathbb {T}$ that surrounds the heart and represent the body tissue.

As was the case for $\mathbb {H}$ , we assume no accumulation of charge in $\mathbb {T}$ , i.e.

$\nabla \cdot \left(\mathbf {\Sigma } _{0}\nabla v_{0}\right)=0\,\,\,\,\,\,\,\mathbf {x} \in \mathbb {T} ,$ (5)

where $\mathbf {\Sigma } _{0}$ is the conductance tensor of the body tissue and $v_{0}$ is the voltage in $\mathbb {T}$ .

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

${\vec {n}}\cdot \left(\mathbf {\Sigma } _{0}\nabla v_{0}\right)=0\,\,\,\,\,\,\,\mathbf {x} \in \partial \mathbb {T} .$ (6)

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 $v_{e}$ and $v_{0}$ must be equal on the heart surface, i.e.

$v_{e}=v_{0}\,\,\,\,\,\,\,\mathbf {x} \in \partial \mathbb {H} .$ (7)

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

${\vec {n}}\cdot \left(\mathbf {\Sigma } _{0}\nabla v_{0}\right)={\vec {n}}\cdot \left(\mathbf {\Sigma } _{e}\nabla v_{e}\right)\,\,\,\,\,\,\,\mathbf {x} \in \partial \mathbb {H} .$ (8)

Finally, we assume that there is a complete isolation of the intracellular domain and the surrounding tissue. Similarly to equation (2), we get

${\vec {n}}\cdot \left(\mathbf {\Sigma } _{i}\nabla v_{i}\right)=0\,\,\,\,\,\,\,\mathbf {x} \in \partial \mathbb {H}$ which can be rewritten using $v=v_{i}-v_{e}$ to

${\vec {n}}\cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+{\vec {n}}\cdot \left(\mathbf {\Sigma } _{i}\nabla v_{e}\right)=0\,\,\,\,\,\,\,\mathbf {x} \in \partial \mathbb {H} .$ (9)

Extending the model to include equations (5)-(9) gives a solvable system of equations.

Reduction to monodomain model

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

Numerical solution

There are some special considerations for numerical solution of these equations, due to high time and space resolution needed for numerical convergence.