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=Bidomain model of Myocardial Tissue=

Standard formulation
The monodomain model can be formulated as follows:

\begin{alignat}{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{alignat} $$

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{alignat}{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{alignat} $$

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- end extracellular conductivity tensor matrices respectively.

We assume Ohmic current-voltage relationship and get

\begin{alignat}{2} J_i & = -\mathbf\Sigma_i \nabla v_i \\ J_e & = -\mathbf\Sigma_e \nabla v_e. \end{alignat} $$

We require that there is no accumulation of charge anywhere in $$\mathbb H$$, and therfore that

\begin{alignat}{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{alignat} $$ giving one of the model equations:

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

The transmembrane current is given by

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

where $$\chi$$ is the surface to volume ratio of the membrane, $$C_m$$ is the electrical capacitance per unit area, $$v=v_i-v_e$$ and $$I_{ion}$$ is the ionic current over the membrane per unit area.

Combining equations ($$) and ($$) gives

\nabla \cdot \left( \mathbf\Sigma_i \nabla v_i \right) = \chi \left( C_m \frac{\partial v}{\partial t} + I_{ion} \right) ,$$ which can be rearranged using $$v=v_i-v_e$$ to get another model equation:

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 perimiter $$\partial \mathbb T$$ 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.

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:

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.

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

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

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

Reduction to monodomain model
By assuming equal anisotropy ratios for the intra- and extracellular domains, i.e. $$\mathbf\Sigma_i = \alpha\mathbf\Sigma_e$$ for some scalar $$\alpha$$, the model can be reduced to the monodomain model.