Monodomain model

The monodomain model can be formulated as follows[1]

${\displaystyle {\frac {\lambda }{1+\lambda }}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\text{ion}}\right),}$

where ${\displaystyle \mathbf {\Sigma } _{i}}$ is the intracellular conductivity tensor, ${\displaystyle v}$ is the transmembrane potential, ${\displaystyle I_{\text{ion}}}$ is the transmembrane ionic current per unit area, ${\displaystyle C_{m}}$ is the membrane conductivity per unit area, ${\displaystyle \lambda }$ is the intra- to extracellular conductivity ratio, and ${\displaystyle \chi }$ is the membrane surface area per unit volume (of tissue).

The bidomain model can be written as

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

Assuming equal anisotropy ratios, i.e. ${\displaystyle \mathbf {\Sigma } _{e}=\lambda \mathbf {\Sigma } _{i}}$, the second equation can be written

${\displaystyle \nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{e}\right)=-{\frac {1}{1+\lambda }}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right).}$

Inserting this into the first bidomain equation gives

${\displaystyle {\frac {\lambda }{1+\lambda }}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\text{ion}}\right).}$