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Monodomain model

Model

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Source updated December 30, 2025

The monodomain model is a reduction of the bidomain model of the electrical propagation in myocardial tissue. The reduction comes from assuming that the intra- and extracellular domains have equal anisotropy ratios. Although not as physiologically accurate as the bidomain model, it is still adequate in some cases, and has reduced complexity.

01Formulation

Being $\mathbb {T}$ the spatial domain, and $T$ the final time, the monodomain model can be formulated as follows ${\frac {\lambda }{1+\lambda }}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\text{ion}}\right)\quad \quad {\text{in }}\mathbb {T} \times (0,T),$

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

Derivation

The monodomain model can be easily derived from the bidomain model. This last one can be written as ${\begin{aligned}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{e}\right)&=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\text{ion}}\right)\\\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+\nabla \cdot \left(\left(\mathbf {\Sigma } _{i}+\mathbf {\Sigma } _{e}\right)\nabla v_{e}\right)&=0\end{aligned}}$

Assuming equal anisotropy ratios, i.e. $\mathbf {\Sigma } _{e}=\lambda \mathbf {\Sigma } _{i}$ , the second equation can be written as $\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{e}\right)=-{\frac {\lambda }{1+\lambda }}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right).$

Then, inserting this into the first bidomain equation gives the unique equation of the monodomain model ${\frac {1}{1+\lambda }}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\text{ion}}\right).$

02Boundary conditions

Differently from the bidomain model, the monodomain model is usually equipped with an isolated boundary condition, which means that it is assumed that there is not current that can flow from or to the domain (usually the heart). Mathematically, this is done imposing a zero transmembrane potential flux (homogeneous Neumann boundary condition), i.e.:

$(\mathbf {\Sigma } _{i}\nabla v)\cdot \mathbf {n} =0\quad \quad {\text{on }}\partial \mathbb {T} \times (0,T)$

where $\mathbf {n}$ is the unit outward normal of the domain and $\partial \mathbb {T}$ is the domain boundary.

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Sources and credits

This article is adapted from the Wikipedia article “Monodomain model”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Monodomain model

01Formulation

Derivation

02Boundary conditions

Sources and credits

Related topics

Bidomain model

Forward problem of electrocardiology